Encyclopedia of Physical Science and Technology, 3e, Physical Chemistry

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Encyclopedia of Physical Science and Technology

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Chemical Kinetics, Experimentation Terence J. Kemp University of Warwick

I. II. III. IV. V. VI. VII. VIII. IX.

Introduction Mathematical Manipulation of Rate Data Computer Modeling of Reaction Systems General Methodology of Reaction Kinetics Flow Systems Pulse and Shock Methods Relaxation Methods Spectral Line-Broadening Methods Electrochemical Methods

GLOSSARY Chemiluminescence Emission of electromagnetic radiation in the ultraviolet, visible, or infrared regions from the products of a chemical reaction proceeding in the dark. Fluorescence Spontaneous emission of light by a molecule from an electronically excited state that has the same total electronic spin as the ground state. Laser Optical device achieving the stimulated emission of electromagnetic radiation from a molecule, ion, or chemical system following the generation of a nonBoltzmann “population inversion” between the ground and an excited state by means of some source of excitation (e.g., an electric discharge or a powerful flash of white light (optical pumping)).

Monochromator Optical device based on dispersion of white light by one or more prisms or diffraction gratings into its constituent wavelengths, which are utilized in turn. Phosphorescence Spontaneous emission of light by a molecule from an electronically excited state that has a different total electronic spin from the ground state. Photomultiplier tube Electronic device contained within an evacuated glass envelope, enabling the photoelectric conversion of incident light at a metallic surface into electric current via a number of stages of amplification. Reaction layer That layer of solution adjacent to an electrode within which a stationary distribution of electroactive species is established as the result of homogeneous reaction.

721

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Chemical Kinetics, Experimentation

Relaxation Process where by a molecule returns from a high-energy state to its normal state of lower energy. Relaxation time Time for the fraction 1/e of excited molecules to return by relaxation to their normal state.

etc., indicate the concentrations of the various reactants; α, β, etc. are the orders of reaction with respect to [A], [B], etc., respectively; α + β + · · · represents the total reaction order; and k is the rate constant. The units of k are dependent on the total reaction order; from Eq. (3) it can be seen that they are

EXPERIMENTATION IN CHEMICAL KINETICS is the methodology of measuring the rates of chemical reactions to yield rate constants, reaction orders, and activation parameters with the ultimate goal of establishing the mechanism of a particular reaction. It covers classical approaches based on conventional sampling and analytical methods, and the use of high pressures as well as the impressive array of fast reaction methods now available through developments in magnet technology, electronics, lasers, and small dedicated computers. These are grouped under flow systems, pulse and shock methods, relaxation methods, analysis of spectroscopic line-broadening, and electrochemical methods.

(mol dm−3 sec−1 )/(mol dm−3 )α+β+···

I. INTRODUCTION The task of the experimentalist in dealing with chemical systems is to determine first the rate of the reaction concerned and then its rate constant before proceeding to the ultimate goal of elucidating the reaction mechanism. Such a reaction rate will be determined under a set of closely specified conditions, particularly reactant concentrations, catalyst, temperature, solvent, gas pressure, ionic strength, nature of the reactor surface, radiant light intensity, etc., depending on the type of system under investigation. The accuracy and precision of the figure arrived at will also need to be specified. Many investigators will attempt to factorize the rate constant for the reaction into its components via the Arrhenius equation, k = Ae−E/RT

(1)

by measuring rates (and hence k) over as wide a temperature range as will give reasonable accuracy to the individual values for A and E, or, in the language of transitionstate theory, the enthalpy and entropy of activation, H = and S = , respectively: kB T S = /R −H = /RT e (2) e h It should be noted that in translating reaction rates into rate constants, the experimentalist needs the rate equation, which typically takes the form k=

v = k[A]α [B]β [C]γ · · ·

(3)

where v signifies the rate as measured by loss of a reactant, e.g., −d[A]/dt or formation of a product; [A], [B],

and for total reaction orders of zero, one, two, and three the units of k are mol dm−3 sec−1 , sec−1 , dm3 mol−1 sec−1 , and dm6 mol−2 sec−1 , respectively. It is important to note that while the rate of reaction depends on the concentrations of the reactants, the rate constant is independent of these and is the parameter commonly referred to in discussion of reaction kinetics. While for normal reaction systems the rate constant is naturally independent of time, for systems featuring an initially inhomogeneous distribution of reactants—as, for example, along the track of an α-particle or laser pulse immediately after discharge—the rate constant varies with time until homogeneity is achieved.

II. MATHEMATICAL MANIPULATION OF RATE DATA When a reaction is mechanistically simple, it is easy to determine its rate, reaction order, and rate constant. For example, if a molecule A is decomposing in an inert solvent in a unimolecular process, then the rate law is −d[A]/dt = k[A]

(4)

which can be rearranged to −d[A]/[A] = k dt

(5)

Equation (5) can be integrated: −ln[A] = kt + constant

(6)

The integration constant is found by noting that at time zero the value of [A] can be regarded as [A]0 ; thus the constant equals ( −ln[A]0 ) and Eq. (6) may be rewritten as Eq. (7): ln[A] − ln[A]0 = −kt

(7)

Thus a graphical plot of [A] versus time will give a straight line of slope, −k and an intercept at time zero of [A]0 . For a second-order reaction, the rate law could take the form −d[A]/dt = k[A]2

(8)


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or, where we are dealing with two different reactants, A and B, −d[A]/dt = −d[B]/dt = k[A][B]

(9)

The experimentalist will normally select concentrations such that [A] = [B], and then Eq. (9) becomes identical with Eq. (8). Rearranging Eq. (8), −d[A]/[A]2 = k dt

(10)

and then integrating, 1/[A] = kt + const

(11)

As before, the integration constant is determined by noting that at time zero the concentration of A can be written as [A]0 , and the constant therefore equals 1/[A]0 . Equation (11) can now be rewritten as 1/[A] − 1/[A]0 = kt

(12)

Hence a plot of 1/[A] versus time will give a straight line with a slope of k. Equations (7) and (12) not only provide the vehicle for extracting rate constants from experimental data, they also provide the test for whether the disappearance of a particular reactant follows first- or second-order kinetics. To apply this test with rigor, it is desirable to test the adherence of the data to a particular mathematical law for several half-lives, i.e., for 75–95% of total reaction. (One half-life is the time for the concentration of A to fall from [A]0 to [A]0 /2.) To cope with kinetically more complex reactions, the experimentalist needs to rely on one or more of several well-tested procedures for simplifying the analysis of data by manipulating the starting concentrations of the reactants. The best known of these devices is the “isolation method,” due originally to Ostwald. If a reaction involves several reactants or retardation by a product, then the resulting rate law may be too complex, mathematically, to test in the normal situation when all the reactant concentrations are changing simultaneously. However, if every reactant concentration is kept large except for one (say A), then during the reaction only [A] will change appreciably, the equivalent amounts of B, C, etc., being removed representing only a tiny fraction of [B]0 , [C]0 , etc., so that [B], [C], etc., can be regarded as remaining essentially unchanged throughout reaction. For example, if we have a rate law given by Eq. (13), v = k[A]2 [B][C]2

(13)

then if [B] ≈ [C] [A], v = k[A]2 [B]0 [C]20

(14)

or v = k [A]2

(15)

where k = k[B]0 [C]20 . Thus Eq. (15) takes the form of Eq. (8) and can be dealt with similarly. It is also possible to determine reaction orders by this approach; thus if [B]0 in Eq. (14) is doubled, then k is doubled, indicating a firstorder dependence on [B], while if [C]0 is doubled then k

is quadrupled, indicating a second-order dependence on [C]. In other examples, of course, the “simplified” rate equation might take the form v = k [A]

(16)

when the test given by Eq. (7) would apply. The other main device for dealing with highly complex reactions is the method of initial rates. When it is impossible to investigate a reaction by some simple mathematical test such as Eq. (7) or (12), which normally depends on following at least one of the reactants for up to 90% of its course of disappearance, then the method of initial rates is utilized. This depends on the idea that if a reaction is followed for only 5 or 10% of its total course, then a plot of the concentration of that reactant disappearing most quickly versus time will approximate to a straight line. (All reactant–time plots, except for zero-order reactions, are curved with −d[A]/dt decreasing as [A] decreases; however, the graph of first few percent of any such reaction approximates to the tangent to this curve at time zero.) Such a straight line yields an immediate value for k if [A], [B], and [C] are taken as [A]0 , [B]0 , [C]0 , . . . . β

γ

−d[A]/dt = k[A]α0 [B]0 [C]0 · · ·

(17)

The effect of variation of [B], [C], etc., separately on −d[A]/dt will then yield rate data enabling evaluation of β, γ , etc.

III. COMPUTER MODELING OF REACTION SYSTEMS This takes various forms, depending on the complexity of the reaction system. At its simplest, programs are available in the principal programming languages for any computer operating system to enable the testing of raw data in terms of the simplest types of kinetic rate law, Eqs. (7) and (12), and it is now universal for microcomputers interfaced with apparatus to be capable of recording, storing, and processing kinetic data in digital form or after analog-to-digital conversion. Programs giving complete solutions have also been written for more complex kinetic situations, of which the following are well-known examples.


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1. Series first-order reactions: k1

k2

A→B→C

(18)

2. Reversible first-order reactions: k1

A

B

(19)

k−1

(A notable example is conformational interchange by NMR line-broadening techniques; see Section VIII.) 3. Reversible reactions involving second-order steps in both directions. The following examples are taken from the many studies based on line-broadening analyses from nuclear magnetic resonance (NMR) and electron spin resonance (ESR) spectroscopy, respectively. (a) Proton transfer: k1

A + H3 O+

AH+ + H2 O

(20)

k−1

(b) Electron transfer: k1

A + B±

A± + B

(21)

k−1

4. Parallel second-order reactions occurring in two different zones of polymeric solids without mutual interference: k1

zone 1: R· + R· → RR k2

zone 2: R· + R· → RR

(22)

(23)

5. Relaxation from two excited states in equilibrium (both stages first-order): ∗A

k2

k1 k −1

A

∗B

k2 A

+ B

B

k1 k −1

1. Two parallel terms: k = A1 e−E1 /RT + A2 e−E2 /RT

(24)

∗AB

k3 A + B

(25)

7. Kinetics of photobleaching of a system of absorbance A under steady illumination (making allowance for increased transmission, and therefore reduced rate of photolysis, as photoreaction proceeds): −d A/dt = k(1 − 10−A )

(27)

This is often recognized from a plot of ln k against 1/T , which, instead of exhibiting a single line as for Eq. (1), will show two linear regions intersecting in a particular, narrow temperature range. 2. For solution reactions investigated over very wide temperature ranges, especially cryogenic temperatures when molecular diffusion can be “frozen” at a limiting temperature T0 : k = Ae−[B/(T −T0 )]

k3

6. Relaxation from two excited states in equilibrium (one stage second-order): ∗A

where dozens of processes are involved, some chemical and others, such as diffusion or turbulence, being physical. Here the task is not to express the situation exactly, as in Eqs. (18)–(26), but rather to approximate to the observed state by examining the effects of gradually introducing particular component terms. The computer is used to generate a numerical solution from a set of given rate equations, rate constants, and reaction concentrations for different reaction times, and a screen graphical presentation is often employed. This procedure is illustrated in Fig. 1 for the concentrations-time profiles for reactants and products on a photoirradiated mixture of trans-but-2ene, NO, NO2 , and air. The relevant equations are given in Fig. 2. Computer modeling is also helpful in unravelling the temperature dependence of complex systems. Fitting to the Arrhenius equation [Eq. (1)] is simple, but the number of parameters involved increases, and therefore the programs become lengthier, with the complexity of the system. Well-known variants of the simple Arrhenius equation to which data can be fitted by computer are as follows:

(28)

IV. GENERAL METHODOLOGY OF REACTION KINETICS In relatively simple systems, where reactants proceed directly to a small number of products without the intervention of intermediate species (and in most cases even when these do play some role), the main problem is that of determining the concentration of one or more reactants or products as a function of time. This becomes essentially an exercise in analytical chemistry, in that virtually all methods of chemical and physicochemical analysis have been applied at some time in determining reaction rates.

(26)

Computers prove invaluable in dealing with exceedingly complex kinetic situations such as are found in combustion or the chemistry of planetary atmospheres,

A. Conventional Methods The design of specialized kinetic apparatus is largely determined by the magnitude of the reaction rates concerned.


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FIGURE 1 Experimental (upper) and computer predictions (lower) for the photoreaction of a mixture of trans-but-2ene, NO, NO2 , and air. [From Kerr, J. A., Calvert, J. G., and Demerjian, K. L. (1972). Chem. Br. 8, 252.]

When reactions are “slow,” i.e., with half-lives of the order of minutes, hours, or longer, then conventional sampling methods are very appropriate; thus a thoroughly mixed reaction system is made up after thermostating the separate reactants to the desired temperature, and samples are taken at suitable, measured time intervals and submitted to analysis for a particular reactant or product. Because reaction continues within the sample while sampling is occurring, it is usually the case that one needs to “quench” the reaction either by cooling the sample rapidly to “freeze” reaction or by mixing the sample with another reagents, which halts reaction by consuming one reactant virtually instantaneously. This type of procedure is still widely used because of its simplicity, particularly when more sophisticated apparatus is unavailable. The analysis itself is typically by titration, ultraviolet (UV)–visible or infrared (IR) spectrophotometry, or gas-liquid chromatography. An important variant of this method is to use a nondestructive analytical method in which the composition

of the entire reaction mixture with respect to one or more components is monitored throughout the reaction, from time to time or even continuously, by some physical method such as spectrophotometry, conductivity, potentiometry, or NMR. Commercial spectrophotometers and magnetic resonance spectrometers are now widely used in which a kinetic mode is available for automatically determining optical or radiofrequency absorbances at preset band maxima as a function of time: the resulting data, sometimes running to thousands of points, can be stored and analyzed at leisure. Polymerization kinetics can be followed very simply using a dilatometer (a reaction vessel consisting of a bulb with a capillary side-arm) to measure total reaction volume, which decreases gradually but markedly as reaction proceeds. Many “slow” gas phase reactions have been studied simply by measuring the total pressure of a “static” reaction system at different times. Such an approach can be dangerously simplistic, however, and it is best


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FIGURE 2 Reaction schemes for the photoxidation of trans-but-2-ene in the presence of NO and NO2 . (Left) Initiation of O atoms or ozone. (Right) Initiation by hydroxyl radicals. (These are incorporated into the computer model employed in Fig. 1.) [From Kerr, J. A., Calvert, J. G., and Demerjian, K. L. (1972). Chem. Br. 8, 252.]

supplemented by an in situ analytical method such as spectroscopy or by sampling tiny quantities into a mass spectrometer, if necessary via a gas chromatograph. While the static system has the advantage that it is well-defined in terms of temperature, concentration, surface area, and material of the containing vessel, it suffers from the comparatively small amounts of material available for analysis. Flow systems for gases provide plenty of material but are less well-defined otherwise. An often useful compromise is found in the capacity-flow technique, in which reactants are conducted into the vessel, efficient stirring is provided, and the reacting mixture is withdrawn at such a rate as to set up a steady state.

Kinetic processes occuring within solids are inherently more difficult to monitor, but ESR spectroscopy provides a nondestructive, highly sensitive method for investigating free radical processes. The development in NMR spectroscopy of magic-angle spinning (MAS) enables solid-state transformations to be determined, conventional NMR having been limited previously to bulk physical processes such as the transition through the glass temperature of polymers. One nice application of MAS-NMR is the kinetics of the hardening of wet cement by following the 29 Si resonances. Optical methods are most easily applied to thin films of materials— for example, the kinetics of polymer degradation can be


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determined from the development of the IR peak of the carbonyl group frequency in degraded polyalkenes. B. Steady-State Methods Many complex reactions proceed at a relatively slow rate as measured by the gross disappearance of reactant or appearance of product, and yet the measured rate, characterized by an “experimental” rate constant k, can include components relating to very fast processes. This is the case especially for chain reactions, where the number of reaction chains existing at a given moment is very small, but the individual processes associated with each center occur extremely rapidly. To resolve such a composite rate constant into its individual components normally requires some fast-reactions technique that concentrates on the initiation or termination stages. Another such example is given by fluorescence quenching. If a molecule A yields a fluorescent excited state A∗ that is partially quenched by an added material Q, then a simple kinetic analysis of the quenching data will reveal the second-order quenching constant as follows. Consider the reaction scheme:

kF [A∗ ] (kF + kNR )[A∗ ]

(38)

The ratio of the fluorescence yields, and therefore measured fluorescence intensities, IF and IF0 , in the presence and absence, respectively, of the quencher molecule is given by ϕF0 I0 (kF + kNR + kQ [Q])[A∗ ] = F = ϕF IF (kF + kNR )[A∗ ]

(39)

which can be simplified to IF0 /IF = 1 +

(kQ [Q])[A∗ ] (kF + kNR )

(40)

Thus a plot of IF0 /IF versus [Q] will yield a straight line of slope kQ /(kF + kNR ). This is called the Stern–Volmer constant, and it can be measured using a conventional spectrofluorimeter. Stern–Volmer constants can be measured for the quenching of a single emitting species by a range of quenchers and then factorized into their components simply by measuring the lifetime τ0 of the emitting state A∗ in the absence of quencher by some fast-reactions technique (Section VI). The term τ0 is related to kF and kNR by τ0 = 1/(kF + kNR )

Rate Ia

(29)

kF

kF [A∗ ]

(30)

kNR

kNR [A∗ ]

(31)

kQ

kQ [A∗ ][Q]

(32)

C. High-Pressure Methods

(33)

The rate of a chemical reaction is sensitive not only to temperature [Eq. (1)] but also to pressure. Simple thermodynamic arguments lead to Eq. (42):

hν

A −→ A

∗

A∗ −→ A + hvF A∗ −→ A Q + A∗ −→ A + Q

The rate of formation of A∗ is given by d[A∗ ]/dt = Ia −d[A∗ ]/dt = (kF + kNR + kQ [Q])[A∗ ]

(34)

In the steady-state situation, these two rates are equal, i.e., Ia = (kF + kNR + kQ [Q])[A∗ ]

(35)

Now the fluorescence quantum yield is given by the ratio of the number of photons emitted per second to the number of photons absorbed per second: ϕF =

kF [A∗ ] Ia

(36)

Substituting Eq. (36) into Eq. (35) yields Eq. (37): kF [A∗ ] (kF + kNR + kQ [Q])[A∗ ]

(37)

In the absence of any quencher, the fluorescence yield, denoted ϕF0 , is given by Eq. (38):

(41)

Another type of steady-state method is that of linebroadening of magnetic resonance spectra, covered in detail in Section VIII.

ln(k/k0 ) = − pV = /RT

The rate of disappearance of A∗ is given by

ϕF =

ϕF0 =

(42)

Here k and k0 refer to the rate constants of the reaction at pressures of p and l atm, respectively, and V = denotes the volume of activation. Accuracy in values of V = requires the use of pressures of up to several thousand atmospheres, as the effect is generally rather small (a few cubic centimeters per mole), and the reaction chamber is usually of thick stainless steel. The term V = is considered in the case of solution reactions to be composite, i.e., the sum of (1) the change in volume of the reactants in forming the transition state and (2) the change in volume of the solvating molecules during this process; the latter is particularly important when ions are involved as the reactants. When a process of molecular dissociation is involved, V = will be positive, while when molecules associate to form the transition state, V = is negative; these effects have been used diagnostically in solution kinetic studies of inorganic complexes, e.g., by Henry Taube.


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Recent technical developments of the method involve the use of the NMR method (Section VIII) at high pressures, both in steady-state and stopped-flow applications. The sample is contained in an open glass sample tube situated within a metallic pressure chamber; the metal has to be nonmagnetic in this case, and beryllium–copper alloy or titanium are the favored materials.

V. FLOW SYSTEMS These have been used to great effect in both solution and gas-phase studies. A. Continuous Flow (Solutions) The first successful use of continuous-flow systems in solution was due to Hartridge and Roughton; the principle here is that if the reaction time is of the order of a few milliseconds to a few seconds, then it is impossible to mix the reactants manually and then record optical changes. Instead, the two reactants are fed by gravity or under pressure through narrow-bore tubes to a mixing chamber, after which observations are made at various points remote from the mixing point in order to ascertain the extent of reaction. Alternatively, the flow rate can be varied to display varying time domains of the reaction at a fixed observation point. The disadvantage of the continuous-flow configuration is that considerable quantities of reactants

are used and observations are needed at different points in the mixed stream. A useful variant of the continuous-flow method is that in which the stream of mixed reactants is conducted into a quenching medium, which terminates reaction by eliminating one reactant virtually instantaneously. An adaptation of this idea is to pass the reacting solution into a Dewar vessel containing liquid nitrogen at 77 K when the reaction is literally frozen, and the extent of reaction, or the presence of some intermediate, is ascertained by a solid-state technique such as ESR spectroscopy. B. Stopped-Flow Methods The disadvantages of the continuous-flow method are largely overcome in the stopped-flow technique. After the reactants have been forced from their “drive syringes” into the mixing chamber, adjacent to the optical cell where an analyzing light beam is situated, the emerging mixed solution runs into another vertically mounted “stop syringe,” which rises as it fills until it is arrested by a mechanical stop (Fig. 3). Flow then ceases, but the fast reaction occurring in the optical cell continues to completion, normally in a time of between 10 msec and 10 sec. In recent adaptations the analyzing light, instead of passing into a monochromator–photomultiplier tube–oscilloscope–microcomputer assembly, is passed to a diffraction granting, which disperses the light over the faces of a series of photomultiplier tubes or photodiodes

FIGURE 3 Schematic diagram of stopped-flow apparatus. A, reservoir syringes; D, drive syringes; R, hydraulic ram; 1–5, valves; S, stopping syringe; L, light source; MO, monochromator; C, quartz windows; M, mixing point; PM, photomultiplier tube; HT, power supply to PM; MS, microswitch; W, waste; MA, microammeter; CRO, cathode-ray oscilloscope; TR, transient recorder; CR, computer.


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FIGURE 4 Time-resolved spectra (0–2 sec after mixing, taken at 40 msec intervals) for the reaction of Zn2+ with 2,2 :6 ,2

–terpyridine (L) in methanol:water (80%:20% v/v) at 25◦ C (Zn2+ ] = 10−4 mol dm−3 , [L] = 10 − 5 mol dm−3 ). [From Priimov, G. U. Ph.D. thesis, University of Warwick, 1999.]

(in the so-called photodiode array), each of which is separately powered and from which the individual outputs feed into a multiplexer. The output from the letter is presented on an oscilloscope screen as a complete optical spectrum, each data point corresponding to the signal from one of the photodetectors. The use of electronic delays enables a complete time-resolution of the spectrum to be built up (Fig. 4): An important development in the stoppedflow technique has been the incorporation of Fouriertransform NMR spectroscopy as the detection system. Another development of stopped-flow spectrophotometry is the quenched-flow system. Here, two solutions A and B are mixed to produce a highly reactive intermediate, which persists until it encounters a third reactant C at a second mixer. The resulting reacting mixture is analyzed optically by the usual optical train (Fig. 5).

C. Gas-Flow Methods The possibility of flowing reactant gases at low pressures down long tubes has long been recognized as a means of controlling a fast reaction. An additional feature is the use of some excitation source, often a microwave discharge, at a suitable location in order to generate atoms or small molecules such as ·CN, ·OH, ·Cl, or ·H. Detection systems include optical spectroscopy (especially for chemiluminescent species), electron spin resonance spectroscopy, and atomic resonance fluorescence. Figure 6 illustrates the use of mass spectrometric detection in a gas-flow system. High-capacity pumps remove most of the gas flow,

with only a small fraction entering the very low pressure (∼10−6 torr) mass spectrometer. One example of the type of reaction that can be examined is that between ethyl radical ( C2 H5 ) and oxygen, for which the stages are x

(i) Generate chlorine atoms in microwave discharge: Cl2 −→ 2Cl

(43)

x

(ii) React Cl with ethane: x

Cl + C2 H6 −→ HCl + C2 H5 x

x

(44)

(iii) React the generated ·C2 H5 with oxygen: C2 H5 + O2 −→ C2 H5 O2

x

x

(45)

The C2 H5 radicals are ionized in the mass spectrometer to yield C2 H5 + ions which are detected at a mass/charge ratio of 29. The moveable injector controls the relative concentration of ethyl radicals. Resonance fluorescence detection is a highly sensitive method for observing atomic species. A trace of gas such as hydrogen is mixed with helium and passed into a microwave discharge in which some hydrogen is dissociated: x

H2 −→ 2H

x

(46)

Some of the H atoms are excited and emit their typical fluorescence ∗

H −→ H + hvF x

x

(47)

The so-called resonance emission enters the reaction zone (Fig. 7) where ground-state H atoms become excited


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distribution as possible by means of a velocity selector (a set of slotted rotating discs). Better velocity selection can be achieved by using a supersonic nozzle in which the gas expands from a high pressure through a small nozzle into a vacuum. A beam of molecular ions can replace one of the neutral molecular beams. The detection system relies on the ionization of the scattered fragments as they strike the surface. Alternatively, the scattered species can be submitted to electron bombardment followed by mass spectrometric examination of the resulting ions. The internal energy state of the product molecule is often determined by laserinduced fluorescence, although in some cases infrared chemiluminescence can be observed directly from vibrationally excited states of the products. The geometry of the experimental arrangement is illustrated in Fig. 8.

VI. PULSE AND SHOCK METHODS These depend on the virtually instantaneous generation of new chemical species, normally of high reactivity, in a system by delivery of a burst of energy sufficient to generate electronically excited states or to break chemical bonds to generate radical species. FIGURE 5 (Top) Side and (bottom) plan views of a quenchedflow apparatus. Key: RES, reservoirs (three); DS, drive syringes (three); MS, microswitch; R, rack; P, pinion; M, motor; TI, thermostating fluid inlet; TO, outlet; M1, first mixer; M2, second mixer; Q, quartz windows (two); W, waste outlet; BN, brass nut (three); PS, motor power supply; L, lamp; MO, monochromator; FL, focusing lens; PM, photomultiplier tube; MA, milliammeter; CB, control box; CRO, cathode-ray oscilloscope; DR, data recorder; CR, chart recorder; 1–6, taps; shaded area, PTFE blocks. [From Goodman, P. D., Kemp, T. J., and Pinot de Moira, P. (1981). J. Chem. Soc., Perkin Trans. 2, 1221.]

H + hvF −→ ∗ H

x

A. Flash Photolysis In the technique of flash photolysis, later developed to laser flash photolysis, an intense flash of light is absorbed by molecules in the system to give excited states: hν

M → M∗

These then lose energy either by light emission [Eq. (50)], by nonradiative decay [Eq. (51)], by dissociation to give radical species [Eq. (52)], or by attacking the solvent, denoted SH [Eq. (53)]:

(48)

The concentration of H atoms in the reaction cell can thus be monitored by their fluorescence in a repeat of Eq. (47), and their reactions with the various molecules in the reaction cell determined. D. Crossed Molecular Beams Extremely detailed kinetic information for gas-phase processes is gained by allowing two streams of molecules emanating from small ovens to intersect at right angles and by examining the scattering intensity and velocity distribution of the resulting fragments as a function of the scattering angle. The experiment is carried out at high vacuum (10 − 6 to 10 − 7 torr), and the molecules in one or both of the streams are placed in as narrow a velocity

(49)

M∗ → M + hν1

(or hνp )

∗

M → M.

(51)

M∗ → A· + B· ∗

(50)

·

M + SH → MH + S

(52) ·

(53)

By monitoring the optical or ESR absorption of the species A· , B· , MH· , or S· , it is possible to determine their kinetics for such processes as dimerization, as in Eqs. (54) and (55). S· + S· → S—S

(54)

MH· + MH· → HM—MH

(55)

or reaction with oxygen, S· + O2 → SO·2

(56)


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FIGURE 6 Schematic diagram showing combination of gas-flow tube with mass spectrometric detection. [Reproduced with permission from Pilling, M. J., and Seakins, P. W. (1995). “Reaction Kinetics,” Oxford University Press, Oxford.]

While many types of experimental layout have been used successfully, that shown in Fig. 9 is quite typical. A pulsed excimer laser generates light at a wavelength that can be controlled by using different halogens or rare gases. The uv pulse, of duration that can also be controlled, can be used either directly or to pump a dye laser, giving extensive control of the wavelength range. The emerging pulse is then fed by mirrors, prisms, or an optical fiber to the sample cell where it excites the molecule of interest (in fluid solution or in gaseous form). Species generated by Eq. (49)–(56) are monitored by means of an analyzing beam from a high-pressure pointsource lamp, the output of which passes through the sample cell and thence through a preset monochromator to a photomultiplier tube or spectography or diode array. The signal from the former, which is related to the intensity of the incident light, is taken to a cathode-ray oscilloscope where it is displayed and stored as a function of time immediately following the firing of the laser. The stored signal is then either photographed for manual analysis or taken via an analog-to-digital converter to a microcomputer, where it is stored and subsequently has largely been processed to yield rate constants. The other main instrumental development involving lasers has been the evolution of lasers pulsed to picosecond or even femtosecond time intervals by mode-locking. These yield rather weak pulses and are normally used in association with devices enabling data accumulation. The Nobel prize for chemistry for 1999 was awarded to Ahmed Zewail for his work in extending flash photolysis

into the femtosecond (fs) regime, enabling, for example, observation of a transition state in the breaking of the I-C bond in ICN. The concept underlying this work is schematized in Fig. 10. The laser system generates a pulse which is split into two components, a pump pulse which instigates chemical processes and a probe pulse which monitors them. The probe pulse is delayed behind the pump pulse by a few femtoseconds by lengthening its light path (Fig. 10). The probe pulse causes fragments generated by the pump pulse to emit light, the characteristics of which provide dynamical information. An example of the fundamental character of this type of investigation is illustrated in Fig. 11. Ion pairs Na+ I− are excited by the pump pulse to the excited form [Na I]∗ . The bond distance is very short at the moment of formation, and the excited molecule has a covalent character. As the molecule vibrates, charge moves toward the I atom, and at the furthest point of the ˚ apart, with the stretching vibration the nuclei are 10–15 A bonding now becoming fully ionic [Na+ · · ·I− ]. The atoms continue to vibrate between these two forms: [Na − l]∗

[Na+ . . . l− ]∗

(57)

˚ the energies of the exAt the critical distance of 6.9 A, cited and ground states are very close and there is a 20% chance that the excited state will dissociate to give separate Na and I atoms. The femtosecond technique detects a decaying signal from the excited state and an increasing signal as bursts of Na atoms appear from each vibration (Fig. 11).


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FIGURE 7 Schematic diagram illustrating the various processes occurring during reaction of H atoms with resonance fluorescence detection. [Reproduced with permission from Pilling, M. J., and Seakins, P. W. “Reaction Kinetics,” Oxford University Press, Oxford.]

Lasers are particularly well suited for the timeresolution of light emission [Eq. (50)], especially in the form of time-correlated single-photon counting (Fig. 9). In this technique, the weak output from a repetitively pulsed lamp (pulse width ∼1 nsec) or from a sub-nanosecond pulsed laser is divided such that part of the pulse is taken directly to one photodetector (PM 1 in Fig. 9) while the other part is taken to the sample cell. The signal detected by

PM 1 is conducted to a time-to-amplitude converter (TAC), which is triggered or “started” on its arrival. The photon arriving at the sample cell excites a potentially luminescent molecule [Eq. (49)], which, after some time, fluoresces [Eq. (50)] or phosphoresces [Eq. (50)]. The photons due to luminescence activate the second photodetector (PM 2 in Fig. 9), from which a pulse is fed to the TAC, which is then “stopped.” The time interval between “start” and


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by discharging a bank of capacitors through a quartz tube filled with Kr or Xe: this produces a broad-continuum flash of between a few microseconds and ∼100 µsec duration. The time-resolution of this apparatus is orders of magnitude inferior to that of laser-flash photolysis, but larger energies are available in view of the greater spectral width and timespan of the exciting source, i.e., hundreds or thousands of joules rather than fractions of a joule, and so the microsecond technique still enjoys considerable use for particular problems. B. Pulse Radiolysis

FIGURE 8 Schematic diagram illustrating laser flash photolysis. Laser pulses pass through the light guide to the sample cell. Optical monitoring of post-pulse events can be kinetic or spectrographic.

“stop” is stored in the multichannel pulse-height analyzer and the process is repeated for many thousands, if not hundreds of thousands, of times per second. The accumulated data yield the averaged luminescent response of the sample molecule and, after computer processing, can lead to lifetimes with very small standard deviations. In the more traditional form of flash photolysis, originated by R. W. Norrish and G. Porter, the flash is generated

The technique of pulse radiolysis is closely related to that of flash photolysis: the optical monitoring system is the same, but the source of light activation is replaced by one of high-energy radiation in the form of a short pulse of a few nanoseconds or microseconds of fast electrons (typically 3 MeV). The effect of such high-energy radiation is to excite and ionize the material in the sample cell, which may be in liquid, gaseous, or even (transparent) solid form. In the case of water, the initial act of radiolysis is given by Eqs. (58)–(60): H 2 O → H 2 O+ + e − x

+

·

H2 O + H2 O → OH + H3 O

+

− e− + H2 O → eaq

(58) (59) (60)

Optical studies reveal the presence of the solvated electron − eaq as a broad, intense absorption maximizing at ∼700 nm.

FIGURE 9 Block diagram of time-correlated single photon counting apparatus. Key: LS, lamp; LH, lamp housing; S, slits; HRM, high-radiance monochromator; C, sample cell; SH, sample housing; L, lenses; PM 1, PM 2, photomultiplier tubes; DISC 1, DISC 2, discriminators; TAC, time-to-amplitude converter; MCAPH, multichannel pulse-height analyzer.


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FIGURE 10 Schematic diagram illustrating layout of femtosecond flash photolysis technique. The pump and probe pulses are separated in time by adjusting the light path of the latter. Beams of molecules in the sample tube are excited or dissociated by the pump pulse, and the fragments monitored by the probe pulse on its path to the detector. Reproduced with permission from Scientific American (see Bibliography).

Addition of materials capable of reacting with electrons − , enabling kinetics of its fast rereduces the lifetime of eaq actions to be determined. The powerful oxidant ·OH (the hydroxyl radical) has only a weak absorption in the ultraviolet, and its reactivity is best measured by a competition method based on its very fast oxidation of thiocyanate ion − CNS− to yield the intensely absorbing (CNS)2 ion (λmax 472 nm). Addition of a second substrate X will provide competition for · OH, and the intensity of the absorption of − (CNS)2 will be systematically reduced as [X] is increased, enabling a rate constant to be derived. While optical methods remain the favored means of analysis in both flash photolysis and pulse radiolysis, other methods of detection have been used with great effectiveness from time to time, including conductivity and ESR spectroscopy. The latter technique, in association with flash photolysis in particular, has led to the observation of ESR signals with anomalous intensities, for example, appearing totally in emission, a phenomenon described as chemically induced dynamic electron polarization or CIDEP. x

x

Other developments include extension of the optical range into the near IR and the use of cryogenic equipment to examine radiolysis at temperatures as low as 4.2 K. C. Shock Tubes Shock tubes are applied to the study of gas phase reactions on the microsecond to millisecond timescale at temperatures of several thousand kelvins. The shock wave is generated by breaking a diaphragm that separates the “driver” gas, usually H2 or He at several atmospheres pressure, and the reactant gas diluted in argon at a few torr (1 torr = 10−6 mm Hg). The normal configuration for the apparatus is a hollow tube (Fig. 10), typically measuring 15 cm diameter and 6 m in length, which contains, in the reactant chamber, sensors for measuring the velocity of the shock wave, and an observation point. The shock front generated on rupture of the diaphragm travels at supersonic speed (several Mach) toward the reactant gas zone, compressing this and heating it to very high temperatures (which can be calculated), in much less than


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FIGURE 11 Femtosecond flash photolysis of gaseous sodium iodide. As the excited ion-pair vibrates, it gradually decays (lower curve) and the resulting free atoms are detected (upper curve).

1 msec. Detection of the species in the reaction zone is normally optical, especially as many of these emit light from their excited states, although absorption spectra can also be measured. Strong shocks in CH4 and CH4 − NH3 yield, respectively, the emission bands of C2 and CN, while Al2 O3 dispersed as a dust yields AlO. Alternatively, the reaction zone can be sampled by allowing materials to leak through a pinhole into a mass spectrometer.

VII. RELAXATION METHODS These were developed initially by Eigen and depend on the application of a small disturbance to a chemical system at equilibrium, normally by the dissipation of a pulse of energy in the temperature-jump method but also on occasion by a sudden change in pressure (pressure-jump) or electric field. The system then adapts (Fig. 11) to its new situation, normally a slightly higher temperature and hence a changed equilibrium constant, at a rate that can be measured either optically or conductimetrically. The

duration of the pulse should be much shorter than the relaxation time τ of the system and should approximate to a step-function as in Fig. 11. Interestingly, if the relative change in concentrations of the reactants is very small, then, whatever the rate law, the reaction curve will follow first-order kinetics described by the relaxation time τ , the time for A (the change in absorbance at time t) to fall to a value (l/e)(A0 ) (where A0 is the final total change in absorbance). The mathematics of rather complex systems undergoing relaxation have been established and are given by Hague. While the single-pulse methods described above are widely used, another approach is to apply the perturbation as a sine wave, in which the response will be in the form of another sine wave lagging consistently behind the perturbation to a degree related to the relaxation time of the system. The frequency of the applied perturbation is critical: if it is too slow, then the chemical response will “catch up” and no lag will be detected, but if it is too fast the chemical response will never attain a measurable amplitude before it must change direction. Only


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in the all-important intermediate situation will the amplitude of the chemical response be sufficient to enable determination of the phase lag. Perturbations can take the form of high-frequency oscillating electric fields, but the use of sound waves (to give the ultrasonic method) is that most widely adopted. The attenuation of the sound wave at a particular frequency is related to the chemical relaxation processes occurring in the solution. The excitation is via a quartz crystal generator, and detection is by measuring optical dispersion of light transmitted through the sample.

VIII. SPECTRAL LINE-BROADENING METHODS

N (υB) CH3

H3C

O

N

C H

O C

H3C

H (62)

and will resonate at different frequencies νA and νB giving two lines. Rotation about the N C bond is slow at room temperature, because of the partial double-bonding [Eq. (62)], but as the temperature is increased rotation becomes faster and finally so fast that the signals coalesce to a single line. Between these two extreme situations (known as slow and fast exchange, respectively) lies a continuum of intermediate situations with line shapes given by Eq. (63), g(ν) = [(1 + τ π )P + QR]/(4π 2P 2 + R 2 )

If the lifetime of a molecule is very short, say δt, then the uncertainty principle predicts a broadening of its absorption line δν (in hertz) given by Eq. (61). δν ≈ 1/(2π δt)

(υA ) CH3

(61)

If δt is ∼0.1 sec, then δν is ∼1 Hz, i.e., in the region of NMR spectroscopy. Accordingly, if a molecule is undergoing rapid (∼0.1 sec) interchange between two conformations, or is participating in a fast exchange reaction such as proton-transfer, then its NMR spectral linewidth provides a unique source of kinetic information. We cannot develop the theory of NMR spectroscopy here, but state simply that the nuclei of a sample in a strong magnetic field are excited by radiofrequency radiation of a particular frequency for each type of nucleus (called the Larmor frequency) into excited spin states, from which they return to lower spin states on losing spin energy by nuclear relaxation. Various factors contribute to such relaxation, especially the motion of surrounding solvent molecules designated (even in liquids) as the “lattice”; this process is called spin–lattice relaxation, and the time for l/e part of the excited nuclei to relax this way is denoted T1 . The additional process titled “spin–spin relaxation” is denoted T2 . Both of these are normally of several seconds duration in liquids, and, in accordance with Eq. (61). NMR lines are accordingly very narrow in liquid samples. Chemical pathways for reducing the lifetime reduce T2 and hence broaden the lines. Analysis of such broadening leads to values for the kinetics of the chemical pathway. Another factor influencing the appearance of NMR spectra is the exchange of nuclei (usually protons) between positions with different Larmor frequencies. In the case of dimethylformamide, the protons in the two methyl groups in the planar conformation are inequivalent.

(63)

where P = [0.252 − ν 2 + 0.25δν 2 )τ + /4π

(64)

Q = [−ν − 0.5( pA − PB )δν]τ

(65)

R = 0.5( pA − pB )δν − ν(1 + 2π τ )

(66)

The symbols are defined as follows: τ = τA τB /(τA + τB ), and τA and τB are the average lifetimes of the nuclei in positions A and B, respectively; pA and pB are the mole fractions of A and B, respectively; δν = νA − νB ; is the width (at half-height) of the signal in the absence of exchange; and ν is the variable frequency. The lineshape calculations are exemplified in Fig. 12. The ESR spectrum of a dilute solution of a radical anion such as naphthalene consists of many lines. Addition of further naphthalene results in electron exchange [Eq. (67)], which broadens the lines: k2

÷

C10 H÷ 8 + C10 H8 C10 H8 + C10 H8

(67)

k−2

This is a consequence of reducing the lifetime of a particular spin state: an analysis of the lineshape yields the rate constant k2 , which equals k−2 . Radicals able to undergo conformational change show a spectral phenomenon called the alternating linewidth effect. A temperaturedependence study of this effect will yield activation parameters [energy and entropy, Eq. (2)] for this first-order process.

IX. ELECTROCHEMICAL METHODS Electrochemical methods depend on a competition between an electrode reaction, e.g., Eq. (68),


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removed from the system. Hence the (measured) cell current is proportional to the rate at which species X arrives at the electrode surface. Motion of X is due to (1) diffusion, (2) convection, and (3) migration, and conditions are arranged such that diffusion dominates. Diffusion occurs because of the depletion in the concentration of X near the electrode, resulting in a concentration gradient. If X is participating in a dynamic equilibrium in the bulk electrolyte solution. Eq. (69) where Y is electrochemically inactive and with the equilibrium far to the right, i.e., k1 k−1 , k1

X

Y, K = k1 /k−1

(69)

k−1

FIGURE 12 Computed nuclear magnetic resonance spectra for the exchange process A

B as a function of the parameter τ . [From Gunther, ¨ H. (1980). “NMR Spectroscopy,” Wiley, Chichester.]

X + e− → R

(68)

and the diffusion of the electroactive species X to the electrode surface. The potential of the electrode is selected so that Eq. (68) is essentially rapid and irreversible, resulting in all species X encountering the electrode being

then the rate of arrival of X at the electrode depends on the rate of generation of X from Y, i.e., the situation is under kinetic control. The mean lifetime τ of X is given by 1/k1 . Species X will diffuse a distance µ = Dτ during its mean lifetime, where D is the diffusion coefficient. Thus, a fraction 1/e of species X within distance µ of the electrode will be removed, while those farther away will be transformed to Y before they can be removed. The layer of solution of thickness µ is termed the reaction layer (Fig. 13). The current at the electrode is then determined by the rate at which species X is generated in the reaction layer. A forced convection electrode is exemplified in the technique of polarography, the cathode consisting of a growing mercury drop at the tip of a fine capillary, which provides a constantly renewed, reproducible electrode surface. A steadily increasing potential is applied to the electrode and the resulting current measured, to give a polarographic

FIGURE 13 Steady-state concentration distribution (reaction layer) in the case of a chemical reaction preceding an electrode reaction. [Adapted from Koryta, J., and Dvorak, J. (1987), “Principles of Electrochemistry,” Wiley, Chichester, England.]


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wave. The relation between the diffusion current, i d and the reaction current i r provides the kinetic information via Eq. (70) 1/2

ir Ck1 t 1/2 = id − ir K 1/2

(70)

where C is a constant of approximately unity and t is the drop time.

SEE ALSO THE FOLLWING ARTICLES ELECTROCHEMISTRY • KINETICS (CHEMISTRY) • NUCLEAR MAGNETIC RESONANCE

BIBLIOGRAPHY Andrews, D. L. (1992). “Lasers in Chemistry,” 2nd ed., Springer–Verlag, Berlin. Ashfold, M. N. R., and Baggott, J. E. (1989). “Bimolecular Collisions,” Royal Society of Chemistry, London. Baxendale, J. H., and Busi, F. (1981). “The Study of Fast Processes and Transient Species by Electron Pulse Radiolysis,” Reidel, Dordrecht. Bensasson, R. V., Land, E. J., and Truscott, T. G. (1983). “Flash Photolysis and Pulse Radiolysis. Contributions to the Chemistry of Biology

and Medicine,” Pergamon, Oxford. Billing, G. D., and Mikkelsen, K. V. (1996). “Molecular Dynamics and Chemical Kinetics,” Wiley, New York. Demas, J. N. (1983). “Excited State Lifetime Measurements,” Academic Press, New York. Espenson, J. H. (1995). “Chemical Kinetics and Reaction Mechanisms,” 2nd ed; McGraw-Hill, New York. Greef, R., Peat, R., Peter, L. M., Pletcher, D., and Robinson, J. (1985). “Instrumental Methods in Electrochemistry,” Ellis Horwood, Chichester. Hammes, G. G., ed. (1974). “Techniques of Chemistry,” 3rd ed., Vol. VI. Part II, Wiley (Interscience), New York. House, J. E. (1997). “Principles of Chemical Kinetics,” W. C. Brown, Dubuque, Iowa. Lewis, E. S., ed. (1974). “Techniques of Chemistry,” 3rd ed., Vol. VI. Part I, Wiley (Interscience), New York. Ng, C.-Y., Baer, T, and Powis, I. (1994). “Unimolecular and Bimolecular Ion-Molecule Reaction Dynamics,” Wiley, Chichester. Pilling, M. J., and Seakins, P. W. (1995). “Reaction Kinetics,” Oxford University Press, Oxford. Sandström, J. (1982). “Dynamic NMR Spectroscopy,” Academic Press, New York. Steinfield, J. I., Francisco, J. S., and Hase, W. L. (1989). “Chemical Kinetics and Dynamics,” Prentice-Hall, Englewood Cliffs, New Jersey. Wilkins, R. G. (1991). “Kinetics and Mechanism of Reactions of Transition Metal Complexes,” VCH, Weinheim, Germany. Wright, M. R. (1999). “Fundamental Chemical Kinetics,” Horwood Publishing, Chichester. Zewail, A. (1990). “The Birth of Molecules,” Scientific American, December issue, 40–46.

P1: FYD Revised Pages

Qu: 00, 00, 00, 00


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Chemical Thermodynamics J. Barthel and R. Neueder University of Regensburg

I. Fundamentals II. Fundamental and Caloric Equations of Thermodynamics III. Partial Molar Quantities IV. Fugacities and Activities V. Thermodynamic Equilibrium

GLOSSARY Activity coefficient Ratio of the activity to the mole fraction of a component Yi in a mixture or solution; a measure of the departure from ideal behavior. Affinity Driving force of chemical reactions; equals zero at the equilibrium state. Chemical potential Content in Gibbs energy of 1 mol of a component Yi of a mixture or solution, that is, the change in the total Gibbs energy of the system at constant temperature and pressure when 1 mol of component Yi is added to an infinite amount of the system. Electrochemical system System consisting of electrically conducting phases. The electric potentials of the phases are called Galvani potentials. Excess property Difference between the actual property of a system and its hypothetical value calculated for an ideal mixture or solution at the same temperature, pressure, and mole fraction composition. Extensive property (variable) Property (variable) of a system that is proportional to mass. Heterogeneous system System consisting of phases that

are homogeneous systems and phase boundaries at which the intensive properties show discontinuities. Homogeneous system System of which all intensive properties are continuous functions of position throughout the system. Ideal mixture or solution Mixture or solution that shows no change in volume, enthalpy, or heat capacity when it is made up from its initially separated components. Intensive property (variable) Property (variable) of a system that is independent of mass. Partial molar quantity Increase in any extensive thermodynamic property of a system at constant pressure and temperature when 1 mol of component Yi is added to an infinite amount of the system. The chemical potential of component Yi is also the partial molar Gibbs energy of compound Yi . Phase See Heterogeneous system. Reversible process Process yielding the maximum of usable work. Thermodynamic equilibrium State of a system at which no measurable changes of its intensive properties occur and no measurable flow of matter or energy takes place during the period of observation.

767

P1: FYD Revised Pages Encyclopedia of Physical Science and Technology

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768 CHEMICAL THERMODYNAMICS is a phenomenological science. The object of investigation is the macroscopic properties of chemical systems in thermodynamic equilibrium. Today Gibbs formalism is generally used to yield the framework of thermodynamic compatibility throughout the phenomenological sciences and technology. A multitude of methods, partially based on extrathermodynamic assumptions, are applied to rationalize the observations of chemical systems and for the procurement of basic data for chemical thermodynamic calculations.

I. FUNDAMENTALS A. Thermodynamic Systems and Properties A chemical system is an accumulation of chemical materials limited by a boundary. The physical world outside this boundary constitutes the surroundings of the system. The system may (open system) or may not (isolated system) exchange mass and energy with its surroundings. A system that exchanges energy but not mass is called a closed system. A macroscopic property of a system may (for extensive properties) or may not (for intensive properties) depend on its mass. Examples of extensive properties are the volume of the system or its heat capacity; examples of intensive properties are permittivity, density, and specific heat. A system is considered homogeneous if all intensive properties are continuous functions of position throughout the system; otherwise it is considered heterogeneous. In a heterogeneous system the discontinuities of properties are situated on surfaces enclosing homogeneous regions, which are called phases of the heterogeneous system, with the surfaces being the phase boundaries. Phases are characterized throughout this article by Greek superscripts (e.g., V (α) , V (β) ), phase boundaries by symbols such as V (α/β) . A homogeneous system or a phase of a heterogeneous system can be a gaseous, liquid, or solid system. Systems showing electrical potentials (Galvani potentials) of their phases are called electrochemical systems. Galvani potentials are the result of movable charges (ions, electrons, etc.) within the phases. For electrochemical systems electrical potential differences are observed between electrically conducting phases (e.g., batteries). A system is in a state of thermodynamic equilibrium if during the period of observation (1) no measurable changes of its intensive properties occur and (2) no measurable flow of matter or energy takes place. The system is in a steady state if only the first condition is fulfilled. Electrochemical systems in equilibrium show constant Galvani potentials throughout each phase. Chemical thermodynamics uses observable quantities for the definition of the state of a system. The state of

Chemical Thermodynamics

a homogeneous system (phase) in equilibrium is defined unequivocally by a set of variables of state such as p (pressure), T (temperature), and n 1 , n 2 , . . . , n k (amount of substance). The mole number n i is a measure of the amount of substance of component Yi . The number of components of a system is the minimum number of independent chemical compounds Yi of which the system investigated can be made up. A system built by k components Y1 , Y2 , . . . , Yk is called a k-component mixture. Pure systems contain only a single component. The mole number n i relates the mass wi of component Yi to its molar mass Mi . n i = wi /Mi

(dimension of n i , moles).

(1)

According to the International Union of Pure and Applied Chemistry (IUPAC) Commission on Symbols, Terminology and Units, a mole is the amount of substance of a system that contains as many elementary entities as there are carbon atoms in 0.012 kg of 12 C. The elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles, as shown in the following tabulation: Elementary entity

Mass of 1 mol (g)

Hg HgCl Hg2+ 2 e− (electrons) Fe0.91 S Mixture of 78.09 mol % N2 , 20.95 mol % O2 , 0.93 mol % argon, and 0.03 mol % CO2

200.59 236.04 401.18 5.460 × 10−4 82.88 28.964

If the complete set of variables ( p, T, n 1 , . . . , n k ) of a k-component mixture is known, every property P of the system in its equilibrium state is unequivocally defined: P = P( p, T, n 1 , . . . , n k )

(2)

that is, an identical system can be prepared with the same properties by the specified set of variables. This statement is the basis of analytical chemistry. Infinitely small changes in the variables of state entail infinitely small changes in properties P: ∂P ∂P dP = dp + dT ∂ p T,ni ∂ T p,ni +

k ∂P i=1

∂n i

dn i

(3)

p,T,n j =n i

In Eq. (3) the partial derivatives indicate the changes in P with one of the variables of state, the others being unchanged; dP is an exact differential as a consequence of


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the unequivocally defined, continuous, and continuously differentiable function P( p, T, n 1 , . . . , n k ). Applying to Eq. (3) the cross-differentiation identities known as Schwarz relations yields the relations ∂ 2 P/∂ T ∂ p = ∂ 2 P/∂ p ∂ T

(4a)

∂ 2 P/∂ n i ∂ p = ∂ 2 P/∂ p ∂n i

(4b)

∂ P/∂ n i ∂ T = ∂ P/∂ T ∂n i

(4c)

2

2

tional composition variables, molality (m i , moles of solute Yi per kilogram of solvent), demality (also called molonity, m˜ i , moles of solute Yi per kilogram of solution), and molarity (ci , moles of solute Yi per cubic decimeter of solvent). If in a mixture component Y1 is considered to be the solvent, Y1 = S, and Y2 , . . . , Yk are the solutes, the composition variables of the solutes are defined by the relations m i = n i /n s Ms k m˜ i = n i n i Mi

where (i = 1, 2, . . . , k). As a consequence of Eqs. (3) and (4) every finite change in property P is independent of the process applied (path of integration) when passing from an arbitrary intial state I to an arbitrary final state II, II dP = PII − PI (5)

I

ci = n i /V

(6)

II

n s = n s1 + n s2 + · · · , Ms = xs1 Ms1 + xs2 Ms2 + · · · ,

In technological calculations the amounts of substance n i in the basic set defined in the preceding section are often replaced by other composition variables such as the weights of substance wi which are also extensive variables, or by intensive variables such as weight percent (wt. %), volume percent (vol %), or mole percent (mol %). A commonly used intensive composition variable in fundamental and applied research on mixed systems is the mole fraction xi : k k xi = n i ni ; xi = 1 (7) i=1

TABLE I Conversion of Concentration Scales in Binary Systemsa From To

x2

c2

m2

m˜ 2

x2 =

x2

M 1 c2 d + (M1 − M2 )c2

M1 m˜ 2 1 + (M1 − M2 )m˜ 2

c2 =

d x2 M1 + (M2 − M1 )x2 x2 M1 (1 − x2 ) x2 M1 + (M2 − M1 )x2

M1 m 2 1 + M1 m 2 dm 2 1 + M2 m 2

m˜ 2 =

(9)

where s1, s2 denote the components making up the mixed solvent. Electrochemistry uses mean composition variables based on the mole numbers of the ions in the solution produced by the electrolyte compounds that are the solutes. If the set of extensive composition variables (n 1 , . . . , n k ) is replaced by a set of intensive composition variables, for example, (x1 , . . . , xk ), where xi = 1, only the intensive properties of the system investigated are defined; extensive properties then are converted to appropriate intensive properties, which are the corresponding molar quantities (i.e., the amount of the extensive property per mole of substance).

Mixed systems distinguishing solvent and solute components are called solutions. Solution chemistry uses addi-

m2 =

(8c)

In Eq. (8c) the volume V of the solution is given in cubic decimeters. Conversion formulas for the frequently used scales of composition variables are given in Table I. If in a mixture the solvent itself is a mixture of components (mixed solvent), the quantities n s and Ms can be defined as “solvent quantities” when using the relations

B. Composition Variables

i=1

(8b)

i=1

I

and equals zero for a cyclic process, II I dP = dP + dP = 0

(8a)

c2 c2 d − M 2 c2 c2 d

d m˜ 2

m2

m˜ 2 1 − M2 m˜ 2

m2 1 + M2 m 2

m˜ 2

a Subscript 1 denotes solvent; subscript 2 denotes solute; d denotes density of the solution; M1 and M2 denote molar masses.


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C. Forms of Energy Energy can be transferred to and from a system and can be stored in a system in different forms: 1. 2. 3. 4. 5. 6. 7.

Mechanical energy, W mech Electrical energy, W el Electromagnetic energy, W elm Radiant energy, W rad Heat, W heat = Q Latent heat, W lat. heat = Q lat Chemical energy, W chem

tion of silver nitrate where silver deposition and zinc dissolution are observed. The chemical energy of the same process is used to a maximum as electrical energy by the help of an appropriate galvanic cell if the electrical current in the cell is kept infinitely small. This example illustrates the following features of reversible processes: 1. They are ideal limiting cases. 2. They are infinitely slow, passing from the initial to the final state through a series of equilibrium states. D. The Fundamental Laws of Thermodynamics

These can be converted totally or partially from one to another. By definition, any contribution that increases the energy of a system is counted positive. Any differential element of work, DW, is not an exact differential; that is, Schwarz relations are not applicable, and integration from the initial to the final state of the system depends on the path of integration (equation defining the particular process). If a system is subjected to a uniform, normal pressure p, the work executed on this system by an infinitesimal increase dV of its volume is DW mech = − p d V

(10)

If in a galvanic cell the infinitesimal increase of charge dq is subject to a Galvani potential , the electrical work is DW = dq el

(11)

The amount of heat passing to a system during a process is calculated from the change in temperature dT with the help of the heat capacity C of the system D Q = C dT.

(12)

Heat applied to a system for a change of phases (e.g., melting) without an increase in temperature is called latent heat. Energy produced in chemical reactions is chemical energy. Electromagnetic energy, radiant energy, as well as nuclear energy will not be discussed here. Since all forms of energy do not yield exact differentials, balances of energy forms require the specification of the path of the process studied—for example, isothermal process (T = const) or isobaric process ( p = const). A process in which the maximum of usable work is produced is called a reversible process, in contrast to irreversible processes, which waste usable energy. The usable chemical energy of the reaction Zn + 2AgNO3 (aq) ← → Zn(NO3 )2 (aq) + 2Ag

(13)

is wasted by transforming it to heat (unusuable energy) when a rod of solid zinc is inserted into an aqueous solu-

Thermodynamics is an empirical science. The quintessence of all practical knowledge is condensed in three fundamental laws (axioms), which are valid because so far they have not been found to be false. (Some textbooks quote an additional “zero law” concerning the definition of thermodynamic temperature.) 1. First Law In 1847 Helmholtz formulated his statement concerning the conservation of energy and the equivalence of work and heat: “Although energy may be converted from one form to another, it cannot be created or destroyed.” As a consequence, (dU )isolated system = 0 where dU =

DW i ;

i = mech, el, . . .

(14)

(15)

The differential dU of the total or internal energy U is an exact differential, in contrast to the differential elements DW i of the different forms of energy. The change in internal energy depends only on the initial and final states and not on the process path. 2. Second Law The second law of thermodynamics as stated by Clausius in 1854 concerns the spontaneous evolution of a system subject to processes. It defines the entropy of a system, which is related to the change in heat during the process and the temperature by the relation d S = D Q rev /T

(16)

Clausius’s formulation (d S)isolated system ≥ 0

(17)

is the most appropriate formulation of the second law for chemical thermodynamics.


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All the symbols in Table II have been explained in Section I, except µi (i = 1, . . . , k), which is the chemical potential of component Yi . In Table II electrical energy is separated from mechanical, heat, and chemical forms of energy, since its extensive variable q (charge) is not independent of mole numbers n i and its introduction as a variable of state would require the use of compatibility equations. To summarize, the set of variables of state ( p , T , n 1 , . . . , n k ) or any other set obtained by replacing one of the variables by its conjugate variable can be used to express the internal energy of the system as a function of its state. Only two functions are relevant in thermodynamics:

3. Third Law The third law is a summary of conclusions concerning the zero-temperature behavior of systems. Its formulation, that the zero-point entropy of perfect crystalline substances is zero, S0 = 0

(18)

as announced in 1923 by Lewis and Randall, indicates by simple application of thermodynamic equations that the heat capacity of such systems vanishes at the zero point of the thermodynamic temperature scale. Consequently, the zero point of temperature cannot be attained experimentally since the thermodynamic functions expressing maximum reversible work of a system also vanish with horizontal tangent. The internal energy and its Legendre transforms (see Section II) are not determined at absolute zero temperature.

U = U (S , V , n 1 , . . . , n k )

(20) U = U (T , V , n 1 , . . . , n k )

(caloric equation)

(21)

The other functions can be obtained by appropriate transformations of Eqs. (20) and (21).

II. FUNDAMENTAL AND CALORIC EQUATIONS OF THERMODYNAMICS

B. Fundamental Equation

Thermodynamics generally is based on two equations, the fundamental and the caloric equations.

1. Homogeneous Systems Equation (20), in which all variables of state are extensive variables, can be written using the notation of Table II,

A. Variables of State

U = U (ξi );

The basic set of variables of state can be extended by reflecting on the forms of energy, each of which is given by the product of an intensive (η) and an extensive (ξ ) variable [see Eqs. (10) and (11)]. Forms of energy yield differential elements DW i that are not exact differentials, but mathematics postulate that for such elements there exists at least one integrating factor that converts the nonexact differential form to an exact differential; for example, the integrating factor of heat is T −1 , yielding entropy as the exact differential d S, d S = T −1 D Q rev . Using this concept, every differential element DW i can be expressed with the help of the conjugate intensive and extensive variables of the corresponding form of energy (see Table II), where DW i = ηi d ξi .

(fundamental equation)

ξi = V, S, n i , . . . , n k

(22)

yielding the exact differential of the internal energy k+2 ∂U dU = dξi (23) ∂ξi ξ j =ξi i=1 which entails the relations (∂U/∂ V ) S,ni = − p

(24a)

(∂U/∂ S)V,ni = T

(24b)

(∂U/∂n i ) S,V,n j =ni = µi

(24c)

because the combination of Eqs. (15) and (19) yields the relation

(19)

TABLE II Conjugated Extensive and Intensive Variables of Forms of Energy Energy form Variable

Arbitrary form of energy W i

Mechanical energy

Heat energy

Chemical energy

Electrical energy

Extensive Intensive

ξi ηi

V

S

n1 , . . . , nk

q

–p

T

µ1 , . . . , µ k


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dU =

k +2

ηi d ξi

(25)

i =1

to which Eq. (23) can be compared. Because the internal energy U [Eq. (22)] is a homogeneous function of degree 1 of the variables ξi , Euler’s theorem of homogeneous functions yields k +2 k +2 ∂U U= ξi = ξi ηi , (26) ∂ξi ξ j =ξi i =1 i =1 which, as compared with Eq. (25), yields the Gibbs– Duhem–Margules equation: k ξi d ηi = −V dp + S dT + n i d µi = 0 (27) i =1

The replacement of an arbitrary number of extensive variables ξi (i = 1, . . . , m) by their conjugate intensive variables ηi yields new energy functions U (m) of the system U (m) = U (m) (η1 , . . . , ηm , ξm +1 , . . . , ξk +2 ).

(28)

These functions can be understood as the Legendre transforms of Eq. (25): U (m) = U (ξi ) −

m

ηi ξi

(29)

i =1

The exact differential of Eq. (28) is m ∂U (m) (m) dU = d ηi ∂ηi ξi ,η j =ηi i =1

k +2 ∂U (m) + d ξi ∂ξi η j ,ξ j =ξi i =m +1

and entails the Maxwell relations (m) ∂U = −ξi ∂ηi ξ j ,η j =ηi (m) ∂U = ηi ∂ξi η j ,ξ j =ξi

(∂U /∂ S)V ,ni = (∂ H /∂ S) p,ni = T (∂U /∂ V ) S ,ni = (∂ A /∂ V )T ,ni = − p (∂ T /∂ V ) S ,ni = −(∂ p /∂ S)V ,ni

(∂G /∂ T ) p,ni = (∂ A /∂ T )V ,ni = −S (∂G /∂ p)T ,ni = (∂ H /∂ p) S ,ni = V

(∂ S /∂ V )T ,ni = (∂ p /∂ T )V ,ni

(∂ S /∂ p)T ,ni = −(∂ V /∂ T ) p,ni

(∂ T /∂ p) S ,ni = (∂ V /∂ S) p,ni

µi = (∂U /∂n i ) S ,V ,n j =ni = (∂ H /∂n i ) S , p,n j =ni = (∂ A/∂n i )T,V,n j =ni = (∂G/∂n i )T, p,n j =ni

(35)

The exact differentials dU [Eq. (23)] and dU (m) [Eq. (30)] entail Schwarz relations. Maxwell relations and Schwarz relations are summarized in Table III. 2. Heterogeneous Systems The description of the state of a heterogeneous system requires a set of variables of the type given in Table II for every phase α. As a consequence, the internal energy of a heterogeneous system made up by ν phases is U = U (α) ξi(α) = U (ξ j ) (36) when numbering the variables ξ either per phase, ξi(α) (i = 1, . . . , k + 2; α = 1, . . . , ν) or throughout the system, ξ j [ j = 1, . . . , ν(k + 2)]. The exact differential

(30)

dU =

ν k+2

ηi(α) dξi(α) =

α=1 i=1

(31a)

(31b)

The Legendre transforms currently used in thermodynamics are enthalpy, Helmholtz energy, and Gibbs energy: H = U + pV ;

d H = V dp + T d S + µi dn i

A = U − T S;

d A = − p d V − S dT + µi dn i (33)

G = U + pV − T S;

TABLE III Partial Derivatives of Energy Functions and Maxwell Relations

(32)

dG = V dp − S dT + µi dn i (34)

The replacement of all extensive by intensive variables again yields the Gibbs–Duhem–Margules equation. The chemical potentials µi introduced as the conjugate variables of mole numbers n i are given by Eqs. (24c) and (31b):

ν(k+2)

η j dξ j

(37)

j=1

is of the type given by Eq. (25). Hence, all equations for homogeneous systems can be used for the phases of heterogeneous systems. Equations (36) and (37) neglect the contributions of the phase boundaries U (α/β) to the total energy U of the heterogeneous system. 3. Electrochemical Systems The internal energy of phase α in an electrochemical system depending on the charge q (α) of phase α and the Galvani potential (α) , (α) (α) U (α) = U (α) S (α) , V (α) , n (α) (38) 1 , . . . , nk , q yields the exact differential dU (α) = − p (α) d V (α) + T (α) d S (α) + µi(α) dn i(α) + (α) dq (α)

(39)

in which dq (α) is dependent on mole numbers n i(α) and charges F z i of the single species Yi . In Eq. (40) F is the


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Faraday number (charge of 1 mole of electrons) and z i is the electrical valence of particle Yi : dq (α) = F z i dn i(α)

(40)

intensive variable. The partial derivatives in Eq. (43) are material properties such as (∂U/∂ T )V = Cv

(heat capacity at constant volume) (44)

Combining Eqs. (39) and (40) gives dU (α) = − p (α) d V (α) + T (α) d S (α) +

k

µ ˜ i(α) dn i(α) (41)

or functions of material properties [see Eq. (46)], (∂U/∂ V )T = (α/β)T − p

i=1

where µ ˜ i(α)

=

µi(α)

+ z i F

(α)

(42)

is the electrochemical potential of species Yi in the phase α. C. Caloric Equation The function U (T, V, n 1 , . . . , n k ) [Eq. (21)] yields the exact differential ∂U ∂U dU = dT + dV ∂ T V,ni ∂ V V,ni +

k ∂U i=1

∂n i

dn i

in contrast to the partial derivatives of the fundamental equation, which are the positive or negative conjugate variables, respectively. Partial derivatives (∂ f /∂ xi )x j =xi can be transformed to (∂ f /∂ yi ) y j = yi with the help of functional determinants (Jacobi transformation) if the functions xi = xi (y j ) are known. For practical use all partial derivatives of energy functions U , H , A, and G and of entropy S are reduced to functions of the tabulated material properties α (thermal expansivity coefficient), β (isothermal compresibility coefficient), and C p (heat capacity at constant pressure):

(43)

α = (1/V )(∂ V /∂ T ) p

(46a)

β = −(1/V )(∂ V /∂ p)T

(46b)

C p = (∂ H/∂ T ) p

T,V,n j =n i

and the related Schwarz relations. The function U (T, V, n 1 , . . . , n k ) is not a homogeneous function since T is an

(46c)

The results of these calculations are compiled in Table IV for practical use.

TABLE IV Reduction of Frequently Used Partial Derivatives to Material Propertiesa (∂U/∂ T )V = C p − T V α 2 /β

(∂U/∂ T ) p = C p − V pα

(∂ H/∂ T )V = C p + V α/β − T V α 2 /β

(∂ H/∂ T ) p = C p

(∂ A/∂ T )V = −S

(∂ A/∂ T ) p = −S − V pα

(∂G/∂ T )V = −S + V α/β

(∂G/∂ T ) p = −S

(∂ S/∂ T )V = C p /T − V α 2 /β

(∂ S/∂ T ) p = C p /T

(∂ p/∂ T )V = α/β

(∂ p/∂ T )s = C p (T V α)

(∂ V /∂ T ) H = C p β + V α − T V α 2 (1 − T α)

(∂ V /∂ T )s = −C p β/(T α) + V α

(45)

(∂U/∂ p)V = C p β/α − T V α

(∂U/∂ p)T = pVβ − T V α

(∂ H/∂ p)V = C p β/α + V − T V α

(∂ H/∂ p)T = V − T V α

(∂ A/∂ p)V = −Sβ/α

(∂ A/∂ p)T = pVβ

(∂G/∂ p)V = V − Sβ/α

(∂G/∂ p)T = V

(∂ S/∂ p)V = C p β/(T α) − V α

(∂ S/∂ p)T = −V α

(∂ V /∂ p) S = T V 2 α 2 /C p − Vβ

(∂ V /∂ p) H = V 2 α(T α − 1)/C p − Vβ

(∂ T /∂ p) H = V (T α − 1)/C p (∂U/∂ V ) p = − p + C p /(V α)

(∂U/∂ V )T = − p + T α/β

(∂ H/∂ V ) p = C p /(V α)

(∂ H/∂ V )T = (T α − 1)/β

(∂ A/∂ V ) p = − p − S/(V α)

(∂ A/∂ V )T = − p

(∂G/∂ V ) p = −S/(V α)

(∂G/∂ V )T = −1/β

(∂ S/∂ V ) p = C p /(T V α)

(∂ S/∂ V )T = α/β

a The material properties here are heat capacity C , thermal expansion coefficient α, and p isothermal compressibility β.


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The caloric equation does not contain entropy as a variable of state and hence cannot be used for providing information either on thermodynamic equilibria or on chemical potentials, in contrast to the fundamental equation.

III. PARTIAL MOLAR QUANTITIES A. Definitions Extensive thermodynamic quantities Z ( p, T, n 1 , . . . , n k ) of the type given by Eq. (47), such as volume V , entropy S, enthalpy H , heat capacity C p , and Gibbs energy G, yield partial molar quantities: Z i = (∂ Z /∂n i ) p,T,n j =ni .

(47)

Partial molar quantities Z i are the contributions per mole of the components Y1 , . . . , Yk to the total quantity Z p,T (n 1 , . . . , n k ) of the phase at constant pressure and temperature: k Z p,T (n 1 , . . . , n k ) = ni Z i . (48) i=1

Equation (48) results from Euler’s theorem, since Z p,T is a homogeneous function of degree 1; it entails the relations d Z p,T =

k

Z i dn i

(49a)

FIGURE 1 Determination of partial molar quantities Z2 and apparent molar quantities z from measurements of extensive thermodynamic quantities Z p,T,n (n2 ) (also Z2 from Z; see Section III.B).

Figure 1 illustrates the notation; beginning with n 1 moles of pure component Y1 , the system is made up by the addition of arbitrary amounts n 2 of component Y2 . The measurable quantity Z of the phase is given by the relation Z = n1 Z 1 + n2 Z 2

From Fig. 1 it follows that Z 2 is given by the slope of the tangent at point n 2 , which in turn permits the determination of Z 1 via Eq. (52) or (49b). Solution chemistry also makes use of another type of molar quantity, the so-called apparent molar quantity Z of the solute Y, according to Z = n 1 Z 1∗ + n 2 Z

i=1 k

n i d Z i = 0.

(49b)

i=1

Equation (49a) is the exact differential of quantity Z [Eq. (3)] at constant pressure and temperature. Equation (49b) is a Gibbs–Duhem–Margules type of equation, indicating the mutual dependence of partial molar quantities. Using the set of variables (x1 , . . . , xk ) entails the relations Z¯ p,T (x1 , . . . , xk ) = xi Z i d Z¯ p,T = Z i d xi

Z l∗ = lim Z i ; xi →1

or Z Y∞ = lim Z Y . xs →1

Z 1 = Z 1∗ + n 2 (∂ Z /∂n 1 ) p,T,n 2 ; Z 2 = Z + n 2 (∂ Z /∂n 2 ) p,T,n 1

Z 2∞ = lim Z 2 xl →1

(51)

(54)

and the limiting value at infinite dilution Z 2∞ = lim Z 2 = lim Z = ∞ z xl →1

where Z¯ p,T is the mean value of quantity Z p,T per mole of phase, Z¯ p,T = Z p,T / n i . The following discussion is limited to two-component systems (binary systems) of components Y1 and Y2 , mixtures of Y1 and Y2 , or solutions of Y2 (solute Y) in Y1 (solvent S). The quantity Z per mole of pure component Yi will be called Z i∗ and that per mole of Y2 (or Y) in the infinitely diluted solution in solvent Y1 , (or S) will be called Z 2∞ (or Z Y∞ ):

(53)

Y1 being the solvent. The apparent molar quantity Z is also shown in Fig. 1. The advantage of apparent molar quantities is their direct access from experimental data,

Z = (Z − n 1 Z 1∗ )/n 2 Comparison of Eqs. (52) and (53) yields the relations

(50)

xi d Z i = 0

(52)

xs →1

(55)

Figure 2 illustrates the features of mean molar properties Z¯ p,T [Eq. (50)] for a system of completely miscible components Y1 and Y2 . At mole fraction x2 the quantity Z¯ p,T is given by the relation Z¯ p,T = x1 Z 1 + x2 Z 2

(56)

yielding Z 1 = Z¯ p,T − x2 (∂ Z¯ /∂ x2 ) p,T ; Z 2 = Z¯ p,T + x1 (∂ Z¯ /∂ x2 ) p,T

(57)

Equation (57) shows that the tangent at Z¯ p,T at point P(x2 ) yields intercepts with the axis at x2 = 0 and x2 = 1, which produce the partial molar quantities Z 1 and Z 2 .


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is the change in Z 2 accompanying the transfer of Y2 from the pure state to the infinitely dilute solution in Y1 ; Z 2∞ is determined by appropriate extrapolation methods. Combining Eqs. (60b), (61b), and (62) yields the relation ∞ ∞ mix Z = n 2 rel

rel (63) Z + Z 2 ; Z = Z − Z where − rel Z is the balance for the dilution process from concentration c2 of Y2 to infinite dilution FIGURE 2 Determination of the partial molar quantities Z1 and Z2 as a function of mole fraction x2 from mean molar quantities ¯ see Section III.B). Z¯ p,T (x2 ) (also Z1 and Z2 from Z:

Besides partial molar and apparent molar quantities, thermodynamics makes use of relative quantities such as (Z i − Z i∗ ), (Z i − Z i∞ ), or ( Z − ∞ Z ), resulting from mixing or solution processes. B. Mixing Processes Formally, property Z before and after mixing of the components Y1 and Y2 can be written as Z init = n 1 Z 1∗ + n 2 Z 2∗ Z fin = n 1 Z 1 + n 2 Z 2

(58)

or Z fin = n 1 Z 1∗ + n 2 Z

(59)

yielding the balances for the mixing process mix Z = n 1 Z 1 + n 2 Z 2 ; or

Z i = Z i − Z i∗ (60a)

mix Z = n 2 Z − Z 2∗

∞ ∗

Z − Z 2∗ = Z − ∞ Z + Z2 − Z2

(61b)

In Eqs. (61a) and (61b) the quantity Z 2∞ = Z 2∞ − Z 2∗

(62)

(64)

(e.g., heat of dilution). Dilution from concentration c2 , to c2 , c2 < c2 is then given by the relation mix Z (c2 )/n 2 − mix Z (c2 )/n 2 = (c2 ) − (c2 )

(65)

C. Thermodynamic Relations Using partial molar quantities Z i , the differential d Z of the thermodynamic quantity Z at arbitrary pressure and temperature is k ∂Z ∂Z dZ = dp + dT + Z i dn i (66) ∂ p T,ni ∂ T p,ni i=1 entailing Schwarz relations of the type given by Eqs. (4a)– (4c), where P ≡ Z . Hence, the relations existing for the extensive quantities Z can also be used for their partial molar quantities: for example, (∂G i /∂ T ) p = −Si (∂G i /∂ p)T = Vi (∂(G i /T )/∂ T ) p = − Hi /T 2 (∂ Hi /∂ T ) p = C pi

(60b)

Here, mix Z is the integral effect of mixing: Z i = Z i − Z i∗ are the differential effects. For enthalpies and Gibbs energies, Z = H or G, only the differential effects Z i can be determined by experiments and not the partial molar quantities Z i themselves, in contrast to volumes and heat capacities, where both quantities are available. Since the mixing process may be accompanied by a change of state of component Y2 (e.g., solid or gas state of pure Y2 may change to liquid state in the mixture), it is advantageous to separate the differential effect Z 2 into two steps: Z 2 = Z 2 − Z 2∞ + Z 2∞ − Z 2∗ (61a) and

∞ dil Z /n 2 = − rel Z (c2 ) = Z − Z (c2 )

(67a) (67b) (67c) (67d)

Equations (67a)–(67d) show the particular role of partial molar Gibbs energies G i . By their definition, quantities G i are both partial molar quantities and chemical potentials as defined by the fundamental equation of thermodynamics: G i ( p, T ) = (∂G/∂n i ) p,T,n j =ni = µi ( p, T )

(68)

D. Molar Quantities of Pure Compounds and Molar Quantities at Infinite Dilution Molar volumes of pure compounds Vi∗ and their temperature and pressure dependence are available from density measurements:

Vi∗ = Mi di∗ (69a) ∗ ∂ ln di∗ ∂ Vi = αi∗ Vi∗ = −Vi∗ (69b) ∂T p ∂T p ∗ ∂ Vi ∂ ln di∗ = −βi∗ Vi∗ = −Vi∗ (69c) ∂p T ∂p T


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Molar heat capacities C ∗pi are known from calorimetric measurements; their temperature and pressure dependence is tabulated for practical use. Molar entropies Si∗ are tabulated at the standard state ◦ ◦ as standard entropies S298 (Yi ), S298 (Yi ) = Si∗ (1 atm, 298.15 K). They are defined by the relation (◦ , standard) 298.15 ◦ S298 (Yi ) = c∗◦ (70) pi d(ln T ) 0

Molar enthalpies and Gibbs energies cannot be determined by experiments (third law of thermodynamics); only differences with respect to some reference state can be defined. The concept of formation reactions is frequently used to replace molar enthalpies Hi∗ and Gibbs energies G i∗ in thermodynamic balances by the standard enthalpies and standard Gibbs energies of formation, f H ◦ [Yi ] and f G ◦ [Yi ], respectively. The formation reaction at standard conditions (1 atm, 298.15 K) of a pure chemical compound Yi (e.g., CH3 OH) requires the formation of Yi from chemical elements in their most stable state at standard conditions; here, C(graphite) + 2H2 (g) + 0.5 O2 (g) = CH3 OH(1)

(71)

According to this definition, the formation of an element in its most stable state at standard conditions equals ◦ ◦ zero. Hence, f H298 [H2 (g)] = 0: f H298 [O2 (g)] = 0; ◦ ◦ f H298 [C(graphite)] = 0; but f H298 [C(diamond)] = 1.90 kJ mol−1 , this quantity being the heat of phase transition, C(graphite) → C(diamond), at standard conditions. Standard heats of formation and phase transition are tabu◦ lated as values f H298 (Yi ). They can be determined with high precision by combustion of compounds Yi in special calorimeters. Tabulated standard Gibbs energies of formation f G ◦298 (Yi ) are based on the same concept of a formation, reaction at standard conditions. The temperature and pressure dependence of Hi∗ and ∗ G i are given by the relations ∗ ∂ Hi ∂ T p = C ∗pi (72) ∗ ∂ Hi ∂ p T = Vi∗ 1 − αi∗ T

∂G i∗ ∂ T p = −Si∗ ∗ ∂G i ∂ p T = Vi∗

Z i∞

(73a) (73b)

at standard conditons as used in solution Quantities chemistry are also tubulated. Standard heats of formation ◦ of ions in aqueous solutions, f H298 [Yi (aq)], include the heat of formation of the pure compound Yi under standard conditions and the heat of transfer of pure compound Yi from its pure state to infinite dilution in solvent S, that is, the quantity Hi∞ − Hi∗ . The tables of single-ion quantities in aqueous solutions are based on the additional assumption that f H ◦ [H+ (aq)] = f G ◦ [H+ (aq)] = S◦

[H+ (aq)] = 0, thus avoiding the ambiguity resulting from electrolyte dissociation.

IV. FUGACITIES AND ACTIVITIES A. Chemical Potential of Pure Gases On the basis of Eq. (68), the chemical potential of an ideal pure gas is obtained by integration of Eq. (73b) after replacing Vi∗ with the help of the equation of state of an ideal gas, Vi∗id = RT / p: µi∗id ( p, T ) = µi∗id ( p + , T ) + RT ln( p/ p + ).

(74)

In Eq. (74) p + is an arbitrary reference pressure. The chemical potential of a real gas is obtained when using an appropriate equation of state of a real gas. Vi∗real , such as the van der Waals, the Redlich–Kwong, or the virial equation: p RT µi∗real ( p, T ) = µi∗id ( p, T ) + Vi∗real − d p (75) p p+ In Eq. (75) the reference pressure p + is chosen so low ( p + → 0) that the reference potentials µi∗ ( p + , T ) of ideal and real gases are equal. The integral expression in Eq. (75) is related to the so-called fugacity coefficient φi∗ by the help of the relation p RT RT ln φi∗ = Vi∗real − d p, (76) p 0 permitting Eq. (75) to be written in the form

µi∗real ( p, T ) = µi◦ (T ) + RT ln pφi∗ p ◦ .

(77)

Fugacity coefficients of numerous pure gases are tabulated from low (φi∗ = 1) to high pressures for practical use. In Eq. (77) p ◦ is the standard pressure, p ◦ = 1 atm; quantity pφi∗ is called the fugacity of the pure gas Yi . Comparison of Eqs. (74) and (77) shows that ideal and real gases can be treated as formally equal when pressures are replaced by fugacities. Using an appropriate equation of state of a real gas, for example, the virial equation, pVi∗ = RT + Bi∗ (T ) p + · · ·

(78)

the fugacity coefficient φi∗ can be approximated with the help of the relation

φi∗ = exp Bi∗ p RT (79) B. Chemical Potential in Gas Mixtures In a gas mixture of ideal gases Y1 , . . . , Yk , at pressure p and temperature T , component Yi is at partial pressure pi (Dalton’s law) and Eq. (74) can be appropriately used to


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yield the chemical potential µi ( p, T ) of Yi in this mixture, since µi ( p, T ) = µi∗ ( pi , T ): µiid ( p, T ) = µi∗id ( p, T ) + RT ln( pi / p) = µi◦ (T ) + RT ln pi p ◦

(80)

Equation (80) exhibits the dependence of µi on the phase composition ( pi = xi p, where xi is the mole fraction): µiid ( p, T )

=

µi∗id ( p, T )

+ RT ln xi

(81)

Equations of state of real gas mixtures, for example, the virial equation of state of a binary gas mixture, p V¯ = RT + B¯ p + · · · B¯ = x12 B1∗ + x22 B2∗ + 2x1 x2 B12

(82a) (82b)

permit the calculation of fugacities φi of components Yi in the real mixture p RT RT ln φi = Vireal − dp (83) p 0 with the help of Eq. (56), which yields the partial molar quantities needed for estimating the integral expression in Eq. (83). In Eq. (82b) B1∗ and B2∗ are the second virial coefficients of the pure gases Y1 and Y2 ; the cross term B12 , resulting from the interaction of unequal particles in the gas phase, can be obtained from statistical models. A rough approximation is due to Lewis: B12 = (B1∗ + B2∗ )/2. The chemical potential of component Yi in a real gas mixture is related to that in the ideal mixture by the relation µireal ( p, T ) = µiid ( p, T ) + RT ln φi

(84)

Combination of Eqs. (75), (76), (81), and (84) yields the appropriate expression of the chemical potential µi in real gas mixtures: µireal ( p, T ) = µi∗real ( p, T ) + RT ln xi + RT ln φi φi∗ (85) for the dependence of the chemical potential on the composition and molecular interactions. Equation (85) relates the chemical potential µireal ( p, T ) of component Yi in the gas mixture to that of its pure state, µi∗real ( p, T ); the quotient of the fugacities φi and φi∗ is called the activity coefficient f i ; the product of activity coefficient f i and mole fraction xi is the activity ai of component Yi in the mixture:

f i = φi φi∗ ; ai = xi f i (86) Combining Eqs. (85) and (86) yields the generally used expression µi ( p, T ) = µi∗ ( p, T ) + RT ln ai = µi∗ ( p, T ) + RT ln xi + RT ln f i (87)

The superscript “real” is omitted in Eq. (87) because this equation is also valid for ideal mixtures where f i = 1 (φi = φi∗ ). The activity coefficient of a pure phase (xi = 1, φi = φi∗ ) equals unity, in contrast to its fugacity coefficient [φi∗ ; see Eq. (76)]. C. Chemical Potentials in Condensed Phases The knowledge of equations of state for gas phases permits the calculation of activity coefficients via fugacity coefficients. Equations of state for general practical use such as the virial equation (and others) are not known for condensed phases (liquids and solids). However, as shown by Planck and Schottky, the passage from the gaseous to the liquid or solid state does not change the structure of Eq. (87) and leads to the general formulation for the chemical potentials, µi ( p, T ) = µi∗ ( p, T ) + RT ln xi + RT ln f i

(88a)

µi∗ ( p, T )

(88b)

= lim [µi ( p, T ) − RT ln xi ] xi →1

lim f i = 1

(88c)

xi →1

which can be used in gaseous, liquid, and solid phases. For solutions, the chemical potentials of the solutes (Y2 , . . . , Yk ), when referred to infinite dilution instead of pure phase, can be written µi ( p, T ) = µi∞ ( p, T ) + RT ln xi ∗ µ − µi∞ + RT ln f i exp i RT entailing the definition of an activity coefficient: f 0i = f i exp µi∗ − µi∞ RT .

(89)

(90)

The superscript (∗ ) in Eqs. (89) and (90) indicates that the solute Yi and the solution are considered in equal states of aggregation, which is not necessarily the stable state of the pure compound Yi with chemical potential µi∗ . The general definition of the chemical potential of a solute Yi in solution (Y1 is the solvent S) is based on the activity coefficient of the type given by Eq. (90). µi ( p, T ) = µi∞ ( p, T ) + RT ln xi + RT ln f 0i

(i = 2, . . . , k)

µi∞ ( p, T ) = lim [µi ( p, T ) − RT ln xi ] xs →1

lim f 0i = 1

xs →1

(91a) (91b) (91c)

In solutions, Eqs. (88a)–(88c) are used for the solvent and Eqs. (91a)–(91c) for the solutes. Identity within restricted


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concentration ranges for Eqs. (88a)–(88c) if f i = 1 and for Eqs. (91a)–(91c) if f 0i = 1: µiid ( p , T ) = µi ( p , T ) + RT ln xi .

Equation (96b) must be integrated from infinite dilution of component Y2 (x2 = 0, f 02 = 1) to the arbitrary composition x2 to yield x2 (ln f 02 )x =x2 = d ln f 02

(92)

According to IUPAC rules the superscript indicates an arbitrary reference state, here ∗ or ∞. Quantities µi ( p , T ) −

µiid ( p , T )

=

µiE ( p , T )

= RT ln f i

x2 =0

=−

(93a)

µi ( p , T ) − µiid ( p , T ) = µiE ( p , T ) = RT ln f 0i (93b)

(94)

The preceding considerations are based on the use of the mole fraction scale. Chemists and chemical engineers who use other scales for the composition of mixtures and solutions, for example, weight percent and mole percent for mixtures or molality, molonity, and molarity for solutions (see Section I.A) must convert chemical potentials and activity coefficients to these scales. Conversion is based on the fact that changes in composition scales do not change the chemical potential, for example, conversion from the mole fraction scale (µi∞ , f 0i ) to the molality scale (µi∞(m) , γi ): µi∞ + RT ln xi + RT ln f 0i

d µi∞ = 0,

and x1 d ln x1 + x2 d ln x2 = 0

(95)

the following equations are obtained for mixtures and solutions, respectively: x1 d ln f 2 = − d ln f 1 (96a) 1 − x1

= µi∞(m) +RT ln m i + RT ln γi

and

x1 d ln f 02 = − d ln f 1 (96b) 1 − x1 Integration of Eq. (96a) from the pure component Y2 (x2 = 1, f 2 = 1) to an arbitrary composition x2 yields the activity coefficient at this composition: x2 (ln f 2 )x =x2 = d ln f 2 =−

x1 =1−x2 x1 =0

x1 d ln f 1 1 − x1

µi∞(m) = µi∞ + RT ln Ms ;

γi = xs f 0i

(97)

From µ∞ 2

µ∞ 2 =

µ∞ 2

∞(c)

µ∞(c) 2 ∞(c)

µ2 M1 d1

µ∞ 2 + RT ln

∞(m)

=

µ∞ 2 + RT ln M1

µ2

˜ ∞(m)

=

µ∞ 2 + RT ln M1

µ2

µ2 µ2

µ∞(m) 2 d1 M1

∞(c)

=

µ2

+ RT ln

(100)

Yi (i = 2, . . .) being the solutes and S the solvent. For frequently needed conversion formulas of mole fractions x, molarities c, molalities m, and molonities m˜ and corresponding activity coefficients v, y, γ , and β, see Tables V and VI.

TABLE V Conversion of Reference Chemical Potentials of Binary Systemsa

To

(99)

Using Eq. (91c) and the conversion formula from xi to m i (Table I) entails

x2 =1

(98)

D. Conversion of Reference Potentials and Activity Coefficients

In Eq. (94) the chemical potentials can be expressed with the help of Eqs. (88a)–(88c) for mixtures and with Eqs. (91a)–(91c) solutes in solutions. Since d µi∗ = 0,

x1 =1

x1 d ln f 1 1 − x1

If the activity coefficient of component Y1 is known as a function of composition, the integrals in Eqs. (97) and (98) can be evaluated graphically or analytically to yield the activity coefficients f 2 and f 02 , respectively.

are referred to as excess chemical potentials. According to the Gibbs–Duhem equation the chemical potentials and hence the activity coefficients of a mixture or solution are not independent; for a binary system at constant pressure and temperature we have x 1 d µ1 + x 2 d µ2 = 0

x1 =1−x2

µ2

˜ µ2∞(m)

∞(m)

− RT ln M1

µ2

∞(m)

− RT ln d1

µ2

µ2

µ2

∞(c)

+ RT ln d1

µ2

∞(c)

+ RT ln d1

µ2

˜ ∞(m)

− RT ln M1

˜ ∞(m)

− RT ln d1

∞(m)

µ2

˜ ∞(m)

∞(m)

µ2

˜ ∞(m)

a Subscript 1 denotes solvent; subscript 2 denotes solute; d denotes density of pure solvent; M denotes molar 1 1 mass of solvent.


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Chemical Thermodynamics TABLE VI Conversion of Activity Coefficients of Different Concentration Scales (Binary Systems)a From To

f 02

y2

γ2

β2

f 02 =

f 02

d + (M1 − M2 )c2 y2 d1

(1 + M1 m 2 )γ2

[1 + (M1 − M2 )m˜ 2 ]β2

y2 =

d1 (M1 x1 + M2 x2 ) f 02 d M1

y2

(1 + M2 m 2 )d1 γ2 d

d1 β2 d

γ2 =

x1 f 02

γ2

(1 − M2 m˜ 2 )β2

β2 =

M1 x 1 + M2 x 2 f 02 m1

(1 + M2 m 2 )γ2

β2

d − M 2 c2 y2 d1 d y2 d1

a Subscript 1 denotes solvent; subscript 2 denotes solute; d denotes density of pure solvent; d denotes density of 1 solution; M1 and M2 denote molar masses.

E. Chemical Potentials of Electrolyte Compounds Electrochemistry uses chemical potentials of the type given by Eqs. (91a)–(91c) for single ions. The link to thermodynamics is established by the help of mean mole fractions x± and mean activity coefficients f ± . For a binary z+ z− electrolyte as the solute, Y2 = Cv+ Av− (z + , valent cation: z − , valent anion), for example, Na2 SO4 (where z + = +2, z − = −1, v+ = 2, v− = 1), the chemical potential µ2 ( p, T ) consists of the chemical potentials of cations and anions, µ+ ( p, T ) and µ− ( p, T )

In Eq. (103) the symbol δ indicates virtual processes. A virtual process is a process that is realizable, not depending on time, for which entropy is defined in every state of the system; δS is an infinitesimally small variation and in this respect equivalent to a differential of first order. B. Generalized Forces and Internal Variables

A. Equilibrium Conditions

Figure 3 illustrates the situation of a system in equilibrium; entropy S is represented as a function of an internal variable ζ . Internal variables ζ and their conjugate quantities, generalized forces , are produced by the system in nonequilibrium states. Generalized forces such as gradients of concentration, temperature, or pressure or affinities of chemical reactions are the driving forces to equilibrium. Generalized forces are functions of the intensive variables of state of the system and equal zero at equilibrium, = 0. Their conjugate internal variables ζ are functions of the extensive variables of state and of supplementary conditions characterizing the process studied. An example is given in Section V.F.1. Neither the generalized forces nor internal variables are variables of state; the system in its equilibrium state does not recognize them. The use of the notation of internal variables is exemplified in Fig. 3 by an equilibrium of the distribution of

According to Gibbs, thermodynamic equilibrium is defined by one of the following conditions: (δS)U,V,n ≤ 0 (103a) or (δU ) S,V,n ≥ 0 (103b) Equation (103a) refers to closed, Eq. (103b) to open, systems. Conditions equivalent to Eq. (103b) are obtained with the help of Legendre transformations of the fundamental equation (δ H ) S, p,n ≥ 0; (δ A)T,V,n ≥ 0; (103) (δG)T, p,n ≥ 0

FIGURE 3 Explanation of Gibbs’ equilibrium condition δS ≤ 0. For details see text.

µ2 ( p, T ) = v+ µ+ ( p, T ) + v− µ− ( p, T ) ∞ = v+ µ∞ + ( p, T ) + v− µ− ( p, T ) + v RT ln x± + v RT ln f =

(101)

In Eq. (101) the mean quantities are given by the relations 1/v 1 x± = x+v+ x−v− ; f ± = f +v+ f −v− ; (102) v = v+ + v−

V. THERMODYNAMIC EQUILIBRIUM


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a chemical compound Y in two immiscible solvents in (2) contact, S1 and S2 , n (1) Y and n Y moles of Y being the amounts of Y dissolved in S1 and S2 , respectively. Then the internal variable ζ is represented as the ratio of the (2) mole numbers, ζ = n (1) Y /n Y . If the state of equilibrium is attainable from both sides, ζ > ζ eq and ζ < ζ eq , the tangent at ζ eq is horizontal, (∂ S/∂ζ )eq = 0, and δS equals zero, δS = (∂ S/∂ζ )eq δζ = 0. If, however, compound Y is insoluble in one of the two solvents, the tangent at ζ eq is defined only from one side by the limit of (∂ S/∂ζ ), which is a negative quantity, and δS ≤ 0 (Gibbs’ equilibrium condition). Gibbs criteria of equilibrium, of course, are in agreement with the second law of thermodynamics, which gives evidence of the variation of entropy in spontaneous processes (entropy increase) but gives no explicit evidence on the state of equilibrium itself. C. Stability of Equilibrium Eqs. (103a) and (103b) do not provide information on the stability of equilibrium. Stability of equilibrium is recognized only when taking into account the variations of degrees greater than unity, δ 2 S, δ 3 S, . . . or δ 2 U , δ 3 U , . . . in the total variations, S = δS + δ S + δ S + · · ·

(104a)

U = δU + δ U + δ U + · · ·

(104b)

2

3

2

3

where δU =

∂U i

∂ξi

Mathematics provide a general criterion for stability. The quadratic form δ 2 U is positive for all considerable variations of the parameters ξi if, and only if, all the roots λ of the equation U12 ... U1,k+2 U11 − λ U21 U22 − λ . . . U2,k+2 . .. . = 0 (107) . . ∼ Uk+2,1 ... Uk+2,k+2 − λ are greater than zero (i.e., if the quadratic form is positive definite). Variations of order higher than 2 generally must not be considered. D. Equilibria at Phase Boundaries A heterogeneous system made up of two phases, α and β, and k components Yi is in equilibrium [Eq. (103b)] if k (α) (α) (α) (α) (α) (α) − p δV + T δS + µi δn i i=1

+ − p δV (β)

(β)

+T

(β)

δS

(β)

+

k

(β) (β) µi δn i

≥0

i=1

(108) All variations in Eq. (108) are under the constraints δS = 0;

δV = 0;

δn i = 0;

i = 1, . . . , k

(109)

yielding the relations dξi

(105a)

δS (α) = −δS (β) ;

δV (α) = −δV (β) ;

(β)

1 Ui j dξi dξ j 2 i j ∂ ∂U Ui j = ∂ξ j ∂ξi ξ j =ξi

δ2U =

δn i(α) = −δn i ; (105b)

(105c)

ξi =ξ j

(110)

i = 1, . . . , k

which permit us to write Eq. (108) in the form − p (α) − p (β) δV (α) + T (α) − T (β) δS (α) +

k

(β)

µi(α) − µi

δn i(α) ≥ 0

(111)

i=1

and so on. The equilibrium position is a position of stable equilibrium if δ2 S < 0

or

δ2U > 0

(106a)

or

δ2U = 0

(106b)

or of unstable equilibrium if δ2 S > 0

or

δS (α) = 0;

δn i(α) = 0

(112a)

B:

δV (α) = 0;

δn i(α)

=0

(112b)

C:

δS

δV

=0

(112c)

A:

of undetermined equilibrium if δ2 S = 0

Equation (111) is the basic equation for the discussion of equilibrium conditions at phase boundaries. For this purpose, three sets of variations are studied:

δ2U < 0

(106c)

Also metastable equilibria can be observed (e.g., supercooled liquids). Metastable equilibria are stable only with regard to infinitely neighboring states.

(α)

= 0;

(α)

Set A entails the reduction of Eq. (111) to − p (α) − p (β) δV (α) ≥ 0

(113)

Equation (113) is fulfilled if the following conditions are met:


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1. Both variations, δV (α) > 0 and δV (α) < 0, are possible and p (α) = p (β) (deformable phase boundary). 2. Variation δV (α) is impossible, δV (α) = 0, and p (α) > p (β) , p (α) = p (β) , or p (α) < p (β) (undeformable or rigid phase boundary). 3. Variation δV (α) is possible only in one direction, δV (α) ≥ 0 and p (α) ≤ p (β) (semideformable phase boundary). Similar considerations concerning sets B and C yield the conditions for heat-conducting (T (α) = T (β) ), nonconducting (T (α) > T (β) , T (α) = T (β) , or T (α) < T (β) ), and semi(β) conducting (T (α) ≥ T (β) ) and for permeable (µi(α) = µi ), (β) (β) (β) (α) (α) (α) impermeable (µi > µi , µi = µi , or µi < µi ), and (β) semipermeable (µi(α) ≥ µi ) phase boundaries. E. Phase Equilibria 1. Gibbs Phase Rule Each of the v phases of a heterogeneous system of k components is subjected to a Gibbs–Duhem–Margules equation [see Eq. (27)]: k+2

ξi(α) dηi(α) = 0;

α = 1, . . . , v.

(114)

i=1

Hence the phase diagram of a heterogeneous system is defined by v equations of the type given by Eq. (114), showing in total v(k + 2) variables. If all phase boundaries are deformable, heat conducting, and permeable for all components Yi , the number of independent variables is reduced to f =k+2−v

(115)

FIGURE 4 Schematic phase diagram of a pure compound. The diagram exhibits a section around a triple point T where solid (s), liquid (l), and gaseous phases (g) are in equilibrium; C is the critical →g is the vapor pressure curve, s ← →g is the sublimation point; 1 ← →1 is the fusion pressure curve. pressure curve, and s ←

A one-component system k = 1, to begin with, exhibits three types of regions in its phase diagram depending on the variables of state p and T (Fig. 4): divariant (v = 1, f = 2), univariant (v = 2, f = 1), and nonvariant (v = 3, f = 0) regions. Divariant regions are the fields of solid, liquid, and gaseous states. One equation (v = 1) of type Eq. (116) shows that in these fields the chemical potential µ( p, T ) is a function of p and T , a well-known feature. Univariant regions, as given by the curves (evaporation, 1 ← → g; sublimation, s ← → g; fusion, s ← → 1) in Fig. 4 (for phase transitions s ← s, see Figs. 5 and 6), indicate → equilibrium of two phases. These curves are obtained by resolution of the appropriate system of two (v = 2) Gibbs– Duhem–Margules equations: −Vi(α) d p + Si(α) dT + dµi = 0 (β)

−Vi

(β)

d p + Si dT + dµi = 0

(117a) (117b)

Equations (117a) and (117b), can be transformed to yield the Clausius–Clapeyron differential equation,

Number f is the number of degrees of freedom: Eq. (115) is the Gibbs phase rule. If f = 0, 1, 2, the system is called nonvariant, univariant, or divariant, respectively. 2. Theory of Phase Diagrams The Gibbs–Duhem equations [Eq. (114)] for a heterogeneous system in which all phase boundaries are deformable, heat conducting, and permeable for all compounds can be written k

xi(α) −Vi(α) d p + Si(α) dT + dµi = 0;

i=1

α = 1, 2, . . . , v. (116) In Eq. (116) Vi(α) and Si(α) are the partial molar volume and entropy of component Yi in phase α. The set of v equations (116) is the basis for the theory of phase diagrams.

FIGURE 5 Stable (solid lines) and metastable (dashed lines) phase diagrams of sulfur. For explanation see text.


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diagrams, an arbitrary phase (β) is chosen to be the reference phase: its k + 1 independent variables, p, T , (β) (β) x1 , . . . , xk−1 are spanning the phase-diagram space. In order to transform the set of k + 2 variables of Eq. (116) to these reference variables, the chemical potential of component Yi in phase β is written (β) (β) ∂µi ∂µi (β) dµi = dp + dT ∂p ∂T T,x j

+

=

d xs(β)

(β)

s=1

FIGURE 6 Phase diagram of water. Ice modifications: I, II, III, V, VI, VII. At the scale of this figure, representation is not possible for both the vapor pressure curve, beginning at triple point T (273.16 K. 6.03 × 10−6 kbar) and ending at critical point C (647.2 K. 0.218 kbar), and the sublimation pressure curve.

p,x j

(β) k−1 ∂µi

(β) Vi

∂ xs

dp −

p,T,x j =xs

(β) Si

dT +

(β) k−1 ∂µi

d xs(β)

(β)

∂ xs

s=1

p,T,x j =xs

(119) (β)

and since dµi(α)

(α→β)

S − Si(α) Hi dp = i(β) = (α→β) (α) dT Vi − Vi T Vi

(118)

Nonvariant regions are points in the phase diagram (triple points T , v = 3) indicating equilibrium of three phases (g, 1, s or s, s, 1 or s, s, g, etc.). The vapor pressure curve (equilibrium: 1 ← → g) terminates in a critical point C. Above the critical temperature, liquid and gaseous states are indistinguishable (supercritical fluid state). Phase diagrams of pure compounds contain these elements in various combinations. Figure 5 (phase diagram of sulfur) and Fig. 6 (phase diagram of water) are examples. The phase diagram of sulfur shows the phenomenon called allotropy, occurring when the compound studied may exist in two different solid states (here monoclinic and rhombic). Starting at state A, very slowly increasing temperature yields the transition from rhombic to monoclinic sulfur at point B and the transition from monoclinic to liquid sulfur at point C, with every transition being (α→β) accompanied by a change in molar enthalpy (Hi ), (α→β) (α→β) ), entropy (Si ), and heat capacity volume (Vi (α→β) (c pi ). Transitions of this type are called transitions of first order, in contrast to transitions of second order, which show changes only in heat capacity (e.g., Curie point transition of iron, etc.). The phase diagram of stable equilibrium states (solid lines) of sulfur is superimposed by a phase diagram of metastable states (dashed lines) in which direct transition from rhombic to liquid sulfur can be formed by rapidly increasing the temperature from point A to C. This transition takes place at point D situated on the fusion pressure curve of the metastable rhombic sulfur, the corresponding metastable triple point being Tm (Fig. 5). Multicomponent heterogeneous systems require rather complex calculations. In a general treatment of their phase

(β) = dµi , Eq. (119) can be inserted into Eq.

(116) to yield k

k (β) (β) xi(α) Vi − Vi(α) d p − xi(α) Si − Si(α) dT

i=1

+

i=1 k−1

(β) G αβ s d x s = 0;

α = 1, . . . , v

(120a)

s=1

where G αβ s

=

k

xi(α)

i=1

(β)

∂µi

(120b)

(β)

∂ xs

p,T,x j =xs

Since Eq. (120a) reduces to identity if α = β, Eq. (120a) is a system of v − 1 differential equations in k + 1 vari(β) (β) ables ( p, T, x1 , . . . , xk−1 ). All p, T , x diagrams can be reproduced with the help of Eqs. (120a) and (120b). F. Chemical Reactions 1. Generalized Forces and Internal Variables in Chemical Reactions The internal variable of chemical reactions is the “extent of reaction” ζ , defined by the variables of state n i (amount of substance Yi ) and supplementary conditions, which are given by the stoichiometry of the chemical reaction investigated, ω1 Y1 + ω2 Y2 + · · · ← → ω n Yn + · · · + ω k Y k

(121)

which in a more condensed form can be written ωi Yi = 0

(122)

In Eq. (122) the stoichiometric coefficients ωi are positive for the final and negative for the initial products. The


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internal variable ζ of chemical reactions is given by the relation dζ = dn i /ωi

(123) 3. Heat Balance of Chemical Reactions

yielding the expression dU = − p d V + T d S − A dζ

(124)

for the internal energy, where A=−

k

ωi µi

(125a)

i=1 k v

ωi µi(α)

(125b)

α=1 i=1

is the generalized force, called affinity, of a homogeneous [Eq. (125a)] or a heterogeneous [Eq. (125b)] reaction. 2. Chemical Equilibrium The Gibbs equilibrium condition entails the relations A=−

k

ωi µi = 0

i=1

or A=−

v k

ωi µi(α) = 0

(126)

α=1 i=1

if equilibrium of the chemical reaction is attainable from both sides. Using the activity concept (Section IV), Eq. (126) is transformed to k

ωi µi ( p, T ) = −RT ln K a ;

i=1 k eq ωi Ka = ai

(127)

i=1 v k

At constant pressure and temperature the Gibbs energy of reaction R G equals the negative affinity [Eqs. (125a) and (125b)], yielding the relation R G =

k i=1

or A=−

be taken into account with the help of the corresponding transfer quantities, H (α→β) and V (α→β) , respectively.

ωi µi(α) ( p, T ) = −RT ln K a ;

α=1 i=1 v k (α)eq ωi Ka = ai

ωi µi =

k

ωi µi + RT ln xi f i

(129)

i=1

The enthalpy of reaction R H is obtained from the derivative of R G/T with respect to temperature: k k ∂ ln f i 2 R H = ωi Hi − RT ωi (130) ∂T p i=1 i=1 Replacing Hi at 1 atm by the enthalpy of formation (see Section III. D) yields the basic equation of thermochemistry, k k ∂ ln f i R H ◦ = ωi f H ◦ − RT 2 ωi (131) ∂T p i=1 i=1 where in most cases the effect resulting from the temperature dependence of the activity coefficients can be neglected when compared with the large enthalpies of formation. G. Equilibrium of Electrochemical Systems For electrochemical systems, electrochemical potentials µ ˜ i ( p, T ) (Section II) are used instead of chemical potentials. Under the action of driving forces, both chemical reactions (e.g., reaction in a galvanic cell) and charge transport (e.g., electron transport outside the cell from the anode to the cathode) may take place. The scheme Cu(1) | Zn(2) | ZnSO4 (aq)(3) CuSO4 (aq)(4) Cu(5) (132) shows an example of a Galvanic cell. The cell reaction

(128)

α=1 i=1

In Eqs. (127) and (128) K a is the equilibrium constants of eq (α)eq the chemical reaction and ai and ai are the activities at equilibrium concentrations of the reactants. Using the concept of standard reaction enthalpies, standard Gibbs reaction energies, and standard entropies (Section III), the quantities µi ( p, T ) can be calculated with the help of tabulated standard values (at 25◦ C and 1 atm) and c p or Vi functions. Phase transitions on the path of integration must

→ (5) + ZnSO(3) Zn(2) + CuSO(4) 4 ← Cu 4

(133)

is accompanied by the transport of two electrons from phase 1 to phase 5, which materially are identical. At cell equilibrium the driving force equals zero: v k

(1) ωi µi(α) + n µ ˜e −µ ˜ (v) =0 e

(134)

α=1 i=1

In Eq. (134) µ ˜ (α) e are the electrochemical potentials of the electrons, and n is the number of electrons transferred from the anode (phase 1) to the cathode (phase 5). Since


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the terminal phases are materially identical (here copper) they differ only in their Galvani potentials (α) . (1) µ ˜ (1) (135) ˜ (v) − (v) = −F E. e −µ e = F

used for the determination of activity coefficients from vapor-pressure measurements:

(140) f i(1) = pi xi(1) pi∗

In Eq. (135) E is the measured potential difference of the cell at zero current, the electromotive force of the cell. From Eqs. (134) and (135) follows the wellknown Nernst equation: v−1 k (α) ωi ◦ RT E=E − (136a) ln ai nF α=1 i=1

Starting with the equilibrium condition for a solution and its gaseous phase, ◦(g) d µi∞(1) + RT ln ai(1) = d µi + RT ln pi φi (141)

−n F E ◦ =

v−1 k

ωi vi(α)

(136b)

α=1 i=1

H. Use of Equilibria for the Determination of Activity Coefficients Fugacity coefficients and hence activity coefficients can be calculated with the help of appropriate equations of state (see Section IV). This is possible, however, only for the gas phase (van der Waals equation, Redlich–Kwong equation, virial equation); for condensed phases no useful general equations of state are available. Experimental determination of activity coefficients in condensed phases is based on the study of equilibria. There are numerous methods, but only typical examples will be given.

and equivalent approximations yield the relation

f 0i(1) = pi xi(1) K

(142)

where K is Henry’s constant. An ideal mixture ( f i = 1) or dilution ( f 0i = 1) reduces Eqs. (141) and (142) to pi = xi(1) pi∗

and

pi = xi(1) K

(143)

known as Raoult’s and Henry’s laws, respectively. Figure 7 illustrates the calculation of f i and f 0i according to Eqs. (140) and (142) as the deviations from ideality. 2. Liquid–Liquid Equilibria In osmotic pressure measurements a solution is separated from the pure solvent S by a nondeformable membrane permeable only to the solvent. The pressure of the pure

1. Vapor–Liquid Equilibria Equilibrium of a gaseous and a liquid phase in contact (permeable, deformable, and heat-conducting boundary) entails the conditions ◦(g) d µi∗(1) + RT ln ai(1) = d µi + RT ln pi φi (137) which at constant temperature can be transformed with the help of Eq. (67b), showing the pressure dependence of the chemical potentials, to yield the relation xi(1) f i(1) RT d ln = −Vi∗(1) d p (138) pi φi∗ Integration from xi(1) = 1 (where Pi = pi∗ ) to an arbitrary mole fraction xi(1) (partial pressure in the gas phase pi ) yields the relation ∗ pi Vi∗(1) pi φi (1) (1) xi f i = ∗ ∗ exp dp (139) pi φi RT pi The exponential term in Eq. (139), commonly called the Poynting correction, differs little from unity for temperatures not near the critical temperature. Assuming ideal behavior of the gas phase, fugacities can be replaced by pressures. Then Eq. (139) yields the relation commonly

FIGURE 7 Activity coefficients f 1 , f 2 (mixture), and f 02 (solution) from vapor-pressure measurements on binary systems of components Y1 (chloroform) and Y2 (acetone). p id , p1id , p2id : hypothetical total and partial vapor pressures of the ideal system ( piid , Raoult’s law); O1 E1 = p1∗ , O 2 E2 = p2∗ , vapor pressures of pure compounds Y1 (chloroform) and Y2 (acetone). p real , p1real , p2real : measured total and partial vapor pressures of the binary system chloroform–acetone. p2id.sol : hypothetical partial vapor pressure of the ideally dilute solution of Y2 (acetone, solute) in Y1 (chloroform, solvent) (Henry’s law); O 2 F = K (Henry’s constant). Activity coefficients [see Eqs. (140) and (142)]: f 1 = AC1 /AB 1 ; f 2 = AC 2 /AB 2 ; f 02 = AC 2 /AD.


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solvent is p, while in osmotic equilibrium the solution is subject to an additional pressure , the osmotic pressure, yielding the equilibrium condition µs ( p + , T ) = µ∗s ( p, T )

(144)

Using the pressure dependence of the chemical potential and assuming that the partial molar volume is independent of pressure yields the relation p+ µs ( p + , T ) = µs ( p, T ) + Vs d p p

= µs ( p, T ) + Vs

(145)

By combining Eqs. (144) and (145), one can determine the activity coefficient of the solvent from the osmotic pressure xs f s = exp(−Vs /RT ) (146) The solvent activity xs f s can be expressed with the help of the molal osmotic coefficient which is defined as ln xs f s

=− (147) m Ms where m is the molality of the solute and Ms is the molar mass of the solvent. Using Eq. (146) yields the following relationship between the osmotic coefficient and the osmotic pressure: Vs

= (148) m Ms RT The measurement of osmotic coefficients combined with the Gibbs–Duhem–Margules equation is a wellestablished method for the determination of the activity coefficients of solutes.

account, and the integral of Eq. (151) can be solved analytically to yield the activity coefficients f s from freezing point depressions. Another method using liquid–solid equilibria determines solute activity coefficients from temperaturedependent solubility data. The pure solute Yi is in equilibrium with the saturated solution. With reference to the state of the infinitely dilute solution [Eqs. (91a)–(91c)], the equilibrium condition is given by the relation µi∗(s) ( p, T ) = µi∞ ( p, T ) + RT ln(xi f 0i )sat

(152)

The differential of Eq. (152) can be expressed with the help of the temperature dependence of the chemical potentials, yielding at constant pressure Hi∞ − Hi∗(s) Hi∞ dT = dT (153) 2 RT RT 2 In Eq. (153) the quantity Hi∞ is the change in enthalpy for the transfer of component Yi from the pure solid state to the infinitely dilute solution. This quantity can be obtained from heat of solution experiments. If the temperature-dependent solubility measurements are started at a temperature T where the component Yi is very slightly soluble (xi ) and the activity coefficient f 0i approaches unity, Eq. (153) can be integrated to yield T Hi∞ xi ( f 0i )sat,T = exp dT (154) 2 xi sat T RT d ln(xi f 0i )sat =

4. Electrochemical System Equilibria 3. Liquid–Solid Equilibria In freezing point experiments an equilibrium between liquid solution and its pure solid solvent S is achieved. The equilibrium condition at constant pressure is ∗(1) µ∗(s) µ ( p, T ) ( p, T ) s s d =d + d(R ln xs f s ) T T (149) The temperature dependence of the chemical potential [Eq. (67c)] can be used in Eq. (149) to yield Hs∗(1) − Hs∗(s) fus Hs dT = dT (150) RT 2 RT 2 where fus Hs is the molar enthalpy of fusion of the pure solvent at temperature T . Equation (150) is integrated from xs = 1, where T = T ∗ (freezing temperature of pure solvent) to an arbitrary concentration xs with freezing temperature T to yield T fus Hs ln xs f s = dT (151) RT 2 T∗ d(ln xs f s )

Using the appropriate molar heat capacities, the temperature dependence of the enthalpy of fusion can be taken into

Electromotive force (emf) measurements are frequently used to determine activity coefficients of electrolyte solutions. Equation (136a) relates the emf to the activities of the reacting cell components. From concentrationdependent measurements the standard potential E ◦ of the cell reaction and the activity coefficients can be obtained. As an example, according to Eq. (136a), the emf of the Galvanic cell Pt(s) | K(Hg) | KCl (conc. in S) | AgCl(s) | Ag(s) | Pt(s) (155) can be written as 2RT (156a) ln(cy± ) F RT (156b) E ◦ = E ◦ + ln aK(Hg) F Here, y± is the mean activity coefficient of the electrolyte on the molar scale. In Eq. (155) K(Hg) is a potassium amalgam electrode connected to a solution of KCl at concentration c in solvent S; AgCl(s)/Ag(s) is the silver/silver chloride reference electrode. E = E ◦ −


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786 Data analysis of emf measurements according to Eq. (156a) yields the standard potential of the cell reaction and the activity coefficients at each concentration. Suffice it to mention that such measurements could also be used to determine the activity of potassium in the amalgam phase. A final remark: The IUPAC has changed the standard pressure from 1 atm to 105 Pa (105 Pa = 1 bar = 0.98692 atm). The numerical values of the tabulated standard data at 1 atm of condensed components generally are not affected, owing to their small pressure dependence in contrast to gases which undergo a slight change.

SEE ALSO THE FOLLOWING ARTICLES BATTERIES • CRITICAL DATA IN PHYSICS AND CHEMISTRY • ELECTROCHEMISTRY • ELECTROLYTE SOLUTIONS, THERMODYNAMICS • POTENTIAL ENERGY SURFACES


BIBLIOGRAPHY Denbigh, K. (1981). “The Principles of Chemical Equilibrium,” 4th ed., Cambridge Univ. Press, London. Gibbs, J. W. (1957). “Collected Works,” Vol. I, Yale Univ. Press, New Haven, CT. Guggenheim, E. A. (1986). “Thermodynamics,” 8th ed., North-Holland, Amsterdam. Haase, R. (1956). “Thermodynamik der Mischphasen,” Springer-Verlag, Berlin and New York. Honig, J. M. (1999). “Thermodynamics,” Academic Press, San Diego. Kirkwood, J. G., and Oppenheim, I. (1961). “Chemical Thermodynamics,” McGraw-Hill, New York. Klotz, I. M., and Rosenberg, R. M. (1994). “Chemical Thermodynamics—Basic Theory and Methods,” 5th ed., Benjamin, New York. Kondepudi, D., and Prigogine, I. (1998). “Modern Thermodynamics,” Wiley, New York. Lewis, G. N., and Randall, M. (1961). “Thermodynamics,” 2nd ed., McGraw-Hill, New York. Planck, M. (1964). “Vorlesungen u¨ ber Thermodynamik,” 11th ed., de Gruyter, Berlin. Pitzer, K. S. (1995). “Thermodynamics,” 3rd ed., McGraw-Hill, New York.

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Computational Chemistry Matthias Hofmann

Henry F. Schaefer III

Ruprecht-Karls-Universität

Center for Computational Quantum Chemistry

I. II. III. IV.

History of Computational Chemistry Methods Used in Computational Chemistry Applications of Computational Chemistry Outlook for Computational Chemistry

GLOSSARY ab initio Latin term meaning from the beginning. In the context of computational chemistry, computations from first principle without any empirical input except fundamental physical constants. Density functional theory (DFT) Theory based on the electron density as the crucial property rather than the wave function in traditional ab initio methods. Force field A set of equations describing the potential energy surface of a chemical system. Hamiltonian operator An operator that describes the kinetic and potential energy of a system treated by wave mechanics. Molecular mechanics Theoretical treatment of molecules by a force field based on classical mechanics and electrostatics. Molecular modeling Branch of computational chemistry concerned with computer-aided molecular design. Orbital Function to describe a single electron. Molecular orbitals (MOs) build the total wave function of a system and are expanded in terms of atomic orbitals, AOs (basis functions). Orbitals can be occupied or virtual. Quantum mechanics Mathematical treatment based on the wavelike nature of small particles.

Schrödinger equation A differential equation for the quantum-mechanical treatment of a system. Self-consistent field (SCF) method Method used to solve mathematical equations which depend on their own solution. Semiempirical Making use of experimental results to derive parameters for approximations made in quantummechanical methods.

COMPUTATIONAL CHEMISTRY is the scientific discipline of applying computers to gain chemical information. It is the link between theoretical and experimental chemistry. Theoretical chemistry is mainly concerned with the development of mathematical models which allow one to derive chemical properties from calculations and to interpret experimental observations. The mathematical models developed in theoretical chemistry are usually validated by comparison with experiment. Theoretical chemistry existed before the arrival of electronic computers. Computational chemistry, however, relies heavily on powerful microelectronics to cope with huge computational tasks. It focuses on the application of theoretical methods which require calculational treatments which are by far too large to be done without fast computers.

487

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488 Of course, the strict separation of theoretical, computational, and experimental chemistry is of an academic nature. In practice, theoreticians often not only develop a new method but also need to design more efficient algorithms to make the method applicable. Before computational results can be interpreted, computational chemists need to undertake benchmark studies to determine the limitations of a method. Without the knowledge about the accuracy of the applied mathematical model, any computational study is without scientific significance. Likewise, many experimentalists use computer programs to support or complement their experimental studies.

I. HISTORY OF COMPUTATIONAL CHEMISTRY As early as 1929, only 3 years after Schrödinger’s formulation of the fundamental equation that bears his name, Dirac stated correctly, The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.

Hence, the further development of quantum chemistry was aiming at approximate solutions of the Schrödinger equation by simplifying the required mathematical treatment. In the 1930s the basics for a wide range of computational methods based on quantum mechanics were laid by the development of the Hartree-Fock method. In 1951 Roothaan, for the first time, considered molecular orbitals as a linear combination of analytic atomic oneelectron functions, shifting the mathematical task from the numerical solution of coupled differential equations to the evaluation of integrals over basis functions. Introduction of approximations for the most difficult integrals through the use of suitable parameters led to the development of semiempirical methods beginning in the 1950s. The more rigorous ab initio methods benefited from the use of Gaussian- instead of Slater-type basis functions, as pointed out by Boys in 1950 but generally accepted only two decades later. Configuration interaction (CI) was the first theoretical level used to include electron correlation and was widely applied during the 1970s. In the late 1970s many-body perturbation theory (Møller-Plesset methods), and during the 1980s coupled cluster methods, became more popular because they are more economical and more rapidly convergent, respectively, than CI. The 1990s can be considered the decade of density functional theory, which by that time had become so-

Computational Chemistry

phisticated enough to be useful for applications in chemistry. Molecular mechanics emerged in the mid-1960s and has become more sophisticated and more useful with time. Due to the number of various approximations, early computations performed to try to reproduce experimental findings yielded varying degrees of success. Computational chemistry could become a recognized scientific discipline only after a real predictive power was established. Perhaps the first case in which theory proved to be accurate enough to challenge experiment was the structure determination of methylene (CH2 ). From a spectroscopic investigation the ground state of this molecule was first concluded to be linear. However, this interpretation had to be revised after reliable computations predicted a significantly bent structure in 1970. The development of different methods and their efficient implementation is only one reason for the success of computational chemistry. Another factor is the dramatic development of computer technology (i.e., computational speed as well as the amount of core memory and of disk storage). Today’s personal desktop computers provide many times the computer power of early “supercomputers” at a fraction of the price. The combined development of both software and hardware allowed computational chemistry to become for chemical research an indispensable tool which allows one to plan experiments more carefully and hence to optimize the use of laboratory resources. The importance of computational chemistry was honored when the 1998 Nobel Prize in Chemistry was awarded to two pioneers of the field, J. A. Pople and W. Kohn.

II. METHODS USED IN COMPUTATIONAL CHEMISTRY The methods used in computational chemistry can be classified according to the sophistication of the underlying model (Fig. 1). Molecular mechanics methods are based on classical mechanics and are computationally the fastest. Semiempirical methods are based on a wave function description, in which some integrals are approximated by means of parameters and many others are neglected to reduce the computational cost. Ab initio methods use only fundamental physical constants but no further experimental results; Hartree-Fock (H F) theory is the starting level, which can be improved upon by accounting for electron correlation in various ways. Density functional methods are also quantum mechanical but are based on the electron density to describe chemical systems. They are often considered “ab initio” although some empirical parameters enter the energy functionals.

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FIGURE 1 Classification of computational chemistry methods. AMBER, assisted model building with energy refinement; CHARMM, chemistry at Harvard molecular mechanics; GROMOS, Groningen molecular simulation; CNDO, complete neglect of differential overlap; INDO, intermediate neglect of differential overlap; NDDO, neglect of diatomic differential overlap; MNDO, modified neglect of differential overlap; AM1, Austin model 1; PM3, parametric method number 3; HF, Hartree-Fock; MP2, Møller-Plesset, second order; MP3, Møller-Plesset, third order; MP4, MøllerPlesset, fourth order; CISD, configuration interaction singly and doubly excited; CISD(T), configuration interaction singly, doubly, and triply (estimated) excited; CCSD, coupled cluster singly and doubly excited; CCSD(T), coupled cluster singly, doubly, and triply (estimated) excited; LDA, local density approximation; GGA, generalized gradient approximation; BLYP, Becke/Lee, Yang, and Parr; B3LYP, Becke three-parameter/Lee, Yang, and Parr.

A. Force Field Methods A force field (FF) is a set of equations describing the potential energy surface of a chemical system. A molecular mechanics (MM) method uses a force field based on a classical mechanical representation of molecular forces to calculate static properties of a molecule (e.g., structure and energy of an energy minimum structure). Molecular dynamics (MD) also implements a force field but generates dynamic properties (e.g., evolution of an structure in time) by calculating forces and velocities of atoms. In MM methods atoms are treated as “balls” of different masses and sizes, and bonds are “springs” connecting the balls without an explicit treatment of electrons. The main advantage of this simple classical approach is the small computational cost, which allows one to treat very large molecules. FFs are typically constructed to yield experimentally accurate structures and relative energies. Some FFs are generated to accurately compute other properties such as vibrational spectra. The observation that properties of chemical functional groups are normally transferable from one compound to another validates the MM approach. The most basic component in a FF is the atom type and one element usually contributes several atom types. Each bond is characterized by the atom types involved and has a “natural” bond length since the variation with the chemical environment is relatively small. Similarly, bond angles between atom types have typical values. The energy absorptions in in-

frared (IR) spectroscopy associated with a certain bond stretch or angle deformation also fall in narrow ranges, which demonstrates that the variation of force constants is also relatively small. The existence of an increment system for heats of formation, for example, shows that the energy behaves additively as well. Hence, in MM the energy is expressed classically as a function of geometric parameters. 1. Energy Terms Advanced force fields distinguish several atom types for each element (depending on hybridization and neighboring atoms) and introduce various energy contributions to the total force field energy, E FF : E FF = E str + E bend + E tors + E vdW + E elst + · · · , where E str and E bend are energy terms due to bond stretching and angle bending, respectively; E tors depends on torsional angles describing rotation about bonds; and E vdW and E elst describe (nonbonded) van der Waals and electrostatic interactions, respectively (Fig. 2). In addition to these basic terms common to all empirical force fields there may be extra terms to improve the performance for specific tasks. Each term is a function of the nuclear coordinates and a number of parameters. Once the parameters have been defined, the total energy, E FF , can be computed and subsequently minimized with respect to the coordinates.

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FIGURE 3 (Curve a) Harmonic potential, E(R) = k(R0 − R)2 ; (curve b) third-order polynomial anharmonic potential, E(R) = 2 3 k(R0 − R) √ + k (R0 − R) ; (curve c) Morse potential, E(R) = k/2D(R −R ) 2 0 ) . D(1 − e

with θ = θ0ABC − θ ABC . If higher accuracy is desired (e.g., for computing IR frequencies), a third-order term can be included with an anharmonicity constant set to be a fraction of k ABC . FIGURE 2 Most basic energy terms included in empirical force field (FF) methods.

a. Stretch energy. The harmonic approximation gives the stretch energy of a bond between atom types AB A and B, E str , as AB E str (R) = k AB R 2 ,

where k AB is the force constant and R = R0AB − R AB is the bond length deviation from the natural value, R0AB , for AB which E str is defined to be zero. Further improvement can be achieved by including higher anharmonic terms to the equation. While these expressions describe the potential well for R close to R0 , the energy goes to infinity for large distances (Fig. 3). In contrast, a morse potential allows the energy to approach the dissociation energy, D, as R increases: E Morse (R) = D(1 − e

c. Torsion energy. The torsional potential, due to the rotation of bonds A–B and C–D about bond B–C, is periodic in the torsional angle ω, which is defined as the angle between the projections of A–B and C–D onto a plane perpendicular to B–C. The torsional energy therefore is expressed as a Fourier series: ABCD E tors (ω) = Vn cos(nω), n

which allows the representation of potentials with various minima and maxima (Fig. 4). Three terms are enough to model the most common torsional potentials.

√ k/2DR 2

) ,

but it is much more expensive in terms of computational cost. b. Bending energy. The harmonic approximation ABC for the bending energy, E bend , due to the deformation of the angle between the A–B and B–C bonds, is sufficient for most purposes: ABC E bend (θ ) = k ABC θ 2

FIGURE 4 A three-minimum potential (bold line) represented as a three-term Fourier series: E(ω) = 3n=1Vn cos(n ω).

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d. van der Waals energy. The van der Waals term, E vdW , covers nonelectrostatic interactions between nonbonded atoms. The van der Waals energy is positive (repulsive) and very large at short distances, zero at large distances, but slightly negative (attractive) at moderate distances due to temporarily induced multipole attraction (dispersion force), the most important attractive contribution of which (dipole–dipole interaction) has an R −6 dependence. The Lennard-Jones potential for E vdW includes a repulsive term, which is set proportional to (R −6 )2 to grow faster than R −6 :  12 6  AB R0AB  AB R0 E LJ (R) = ε . −2 R R εAB determines the energy depth of the minimum. Alternatively, a Buckingham or Hill potential can be used that employs an exponential function for the repulsive term. For each atom type a van der Waals radius, R0 , and the atom softness, ε, have to be determined, from which the diatomic parameters are calculated according to √ R0AB = R0A + R0B and ε AB = ε A ε B . e. Electrostatic energy. The electrostatic energy, E elst , is due to the electrostatic interactions arising from polarized electron distributions based on electronegativity differences. It can be modeled by Coulomb interactions of point charges associated with individual atoms: QA QB , εR AB ε being a dielectric constant, which can be used to model the effect of the same or other molecules present (e.g., solvent). The atomic charges, Q, are commonly obtained by fitting to the electrostatic potential as calculated by an electronic structure method. An E elst description based on dipole–dipole interactions between polarized bonds can alternatively be employed. Hydrogen bonds are nonbonded interactions between a positively charged hydrogen atom and an electronegative atom with lone electron pairs (mostly oxygen or nitrogen) and can be adequately modeled by appropriately chosen atomic charges. Although a single hydrogen bond is a very weak interaction, the large number occurring in biomolecules (e.g., proteins) makes hydrogen bonding a very important factor. In the large size limit, the bonded interactions increase linearly with the system size, but the nonbonded interactions show a quadratic dependence and determine the computational cost. The van der Waals interactions quickly fall off with the distance (R −6 dependence) and may be neglected for large separations. The electrostatic interaction (proportional to R −1 ) is much more far reaching and E elst (R AB ) =

needs to be considered out to very long distances. Fast multipole methods (FMMs) can be applied to reduce the computational cost of evaluating E elst . f. Other energy contributions. So that the performance can be improved, force fields include further parameters to take care of special cases. For example, cross terms account for the interplay between different contributions (e.g., longer bonds for small angles). Correction terms may be introduced to describe substituent effects (e.g., anomeric effect). Additional terms may be introduced to adequately treat special cases like pyramidalization of sp2 hybridized atoms. Hydrogen bonding may be treated explicitly (in addition to the electrostatic interaction) with a special set of van der Waals interaction parameters. Pseudo atoms maybe introduced to model lone pairs. In addition, atoms in unusual bonding situations (three-membered rings, molecules with linearly conjugated π -systems, aromatic compounds, etc.), which are not described adequately by the normal parameters, can be defined as new atom types. The force field energy, E FF , corresponds to the energy relative to a molecule with noninteracting fragments. Therefore, only energies for molecular structures built from the same fragments (conformers) can be compared directly. So that energy between different molecules (isomers) can be compared, the energy scale is converted to heats of formation by adding bond increments (estimated from bond dissociation energies minus the heat of formations of the atoms involved) and possibly group increments (e.g., methyl group): Hf = E FF +

bonds

H AB +

groups

H G .

2. Parametrization Determining the parameters for a force field is a substantial task. In general, not all necessary data are available from (accurate) experiments. Modern electronic structure computations can provide unknown data relatively easily and with sufficient accuracy. Another problem is the large number of parameters: for a force field with N atom types, the number of different types of bonds, bond angles, and dihedral angles scales as N 2 , N 3 , and N 4 , respectively, each requiring several parameters. So that the number of parameters can be reduced, the atom dependency can be reduced (e.g., the torsional parameters may be treated as dependent on the B–C central bond only and not on the atom types A and D). The parametrization effort can be reduced further by defining “generic” parameters to be used for less common bond types or when no reference data are available. This, of course, reduces the quality of a calculation. By deriving the di-, tri-, and tetra-atomic parameters

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from atomic data (atom radii, electronegativities, etc.), universal force fields (UFFs) allow one to include basically all elements. The performance, however, is relatively poor. The kind of energy terms, their functional form, and how carefully (number, quality, and kind of reference data) the parameters were derived determine the quality of a force field. Accurate force fields exist for organic molecules (e.g., MM2, MM3), but more approximate force fields (e.g., with fixed bond distances) optimized for computational speed rather than accuracy [e.g., AMBER (assisted model building with energy refinement), CHARMM (chemistry at Harvard molecular mechanics), GROMOS (Groningen molecular simulation)] are the only practical choice for the treatment of large biomolecules. The type of molecular system to be studied determines the choice of the force field. One limitation of force field methods is that they can describe only well-known effects that have been observed for a large number of molecules (this is necessary for the parametrization). The predictive power of these methods is limited to extrapolation or interpolation of known effects. 3. Quantum-Mechanical and Molecular-Mechanical (QM/MM) Method Another limitation of MM is the inability to investigate reactions. While force field methods are capable of describing conformational changes, for which all bonds remain intact along the reaction coordinate, they are by construction not capable of treating reactions in which bonds are broken and/or formed. The classical model is not designed to describe the electronic rearrangement associated with bond breaking and bond formation. Such problems are better treated by electronic structure methods discussed below. For large systems, a combined quantummechanical and molecular-mechanical (QM/MM) method can be applied. In this approach the reactive part of the molecule to be studied is described by a quantummechanical (semiempirical, ab initio, or DFT) method while the rest of the system is treated by a force field. The problem with this approach is the “communication” between classical and quantum-mechanical potential (i.e., how the atoms close to the QM/MM border are treated). The total energy, Etot , may be computed as follows: Etot = EQM + EMM + EQM/MM , where the quantum-mechanical contribution, EQM , and the molecular-mechanical contribution, EMM , are defined by a QM method and a MM method, respectively. The coupling term, EQM/MM , includes parameters that can be fitted

to reproduce experimental results and are specific to the chosen combination of QM and MM methods. Alternatively, the total energy, Etot , may be extrapolated from QM and MM calculations on a small part and on the whole of a suitably partitioned system (IMOMMintegrated molecular orbital, molecular mechanics method) Etot = EQM (small) + EMM (whole) − EMM (small) B. Wave Function Quantum-Mechanical Methods The explicit treatment of electrons in atoms and molecules requires quantum mechanics, which invokes a wave function, , to describe the system of electrons and nuclei. The square of the wave function represents the probability of a particle’s being at a given position. The central goal becomes the solution of the (time-independent) Schrödinger equation, H = E , which relates the wave function, , to the energy, E, of the system. The Hamiltonian operator, H, consists of the kinetic (T) and the potential energy (V) operators: H = T + V. The fact that electrons instantly adjust to changes in nuclear positions due to the much greater masses of the nuclei allows the motions of electrons and nuclei to be separated (Born-Oppenheimer approximation). The electronic wave function depends on only the nuclear position, not on the nuclear momenta. The electronic Hamiltonian, He , in atomic units is given by He = Te + Vne + Vee + Vnn = − +

Nucl. Elec. a

+

i

Nucl. Nucl. a

b>a

1 Elec. ∇i2 2 i

Elec. Elec. Za 1 + |Ra − ri | |ri − r j | i j>i

Za Zb , |Ra − Rb |

where r and R represent the electronic and nuclear coordinates, respectively, and the Laplacian is defined as ∂2 ∂2 ∂2 2 ∇i = + 2+ 2 . ∂ xi2 ∂ yi ∂z i The nucleus–nucleus repulsion, Vnn , is constant for a given geometry, and the kinetic energy, Te , and the electron– nucleus attraction, Vne , are easy to evaluate. The electron– electron repulsion, Vee , however, depends on the distances between electrons and is the reason why the Schrödinger

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equation cannot be solved exactly for systems with more than one electron. The energy can be computed as the expectation value of the Hamiltonian operator:

∗

H dτ |H|

E= ∗ = , |

dτ where the common bra-ket notation is used. The variational principle states that any trial wave function will give an energy equal to or higher than the exact value because the real system will adopt the best possible wave function (which corresponds to the exact energy). Thus, a trial wave function constructed in terms of a number of parameters can be improved by minimizing the energy with respect to the parameters (MO coefficients). A meaningful trial wave function should approach zero as r goes to infinity; it should be normalized, that is,

For example, a Slater determinant for the ground state of the hydrogen molecule can be written as follow: 1 φg (1) φg (1) (1, 2) = √ 2 φg (2) φg (2) 1 = √ [φg (1)φg (2) − φg (2)φg (1)] = −(2, 1), 2 where φg represents the bonding molecular orbital (MO), the 1σg orbital. The electronic Hamiltonian can be written as sums of one-electron (hi ) and two-electron (gi j ) operator plus the constant nuclear–nuclear repulsion: H= hi + gi j + Vnn i

with

and gi j =

E=

N

α | β = β | α = 0

to give spin orbitals φα and φβ (or φ and φ¯ for short). This is appropriate for closed-shell species with paired electrons, but open-shell species with unpaired electrons cannot be expected to have identical α and β orbitals. The unrestricted Hartree-Fock (UHF) method allows a different spatial function for each electron. However, UHF wave functions can suffer from spin contamination (i.e., the spurious mixing of higher spin states into the desired one [more formally, the expectation value of the S2 operator is larger than the correct value of S(S + 1), S being the total spin]). Restricted open-shell Hartree-Fock (ROHF)– based methods avoid the problem of spin contamination but do not allow spin polarization.

φi |hi |φi +

i

N N 1 ( φi φ j |gi j |φi φ j 2 i=1 j=1

− φi φ j |gi j |φ j φi ) + Vnn =

N

hi

i

1. Hartree-Fock Method

α | α = β | β = 1;

1 . |ri − r j |

The Hartree-Fock energy expression becomes

+

Hartree-Fock theory employs a single Slater determinant. In the restricted Hartree-Fock (RHF) method, one spatial function φi is multiplied by an α (representing spin up, spin quantum number m s = + 12 ) or β (representing spin down, m s = − 12 ) spin function with the properties

j>1

Nucl. Za 1 hi = − ∇i2 − 2 |R − ri | a a

|

= 1 (meaning the probability that the system is located somewhere in space is one); and it should comply with the Pauli principle. The latter states that two electrons must differ in at least one quantum number. Furthermore, the wave function should be antisymmetric (i.e., it should change sign when two electrons are interchanged). This is a characteristic property of electrons. Antisymmetry can be ensured by using Slater determinants with one-electron functions (orbitals) φi in columns and electrons (1, 2, . . .) in rows.

i

N N 1 (Ji j − K i j ) + Vnn , 2 i=1 j=1

or E=

N i

φi |hi |φi +

N N 1 ( φ j |Ji |φ j 2 i=1 j=1

− φ j |Ki |φ j ) + Vnn . Ji j is called a Coulomb integral because it corresponds to the electronstatic repulsion of the charge distributions due to φi2 and φ 2j ; the exchange integral, K i j , has no classical equivalent. Ji and Ki are Coulomb and exchange operators, respectively. To find a minimum energy, one can vary the orbitals under the condition that they remain orthogonal by using the method of Lagrange multipliers. This leads to the definition of the Fock operator, Fi , which describes the kinetic energy, the nuclear attraction energy, and the electron repulsion energy of one electron in the field of the other electrons: N Fi = hi + (J j − K j ). j

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A set of Hartree-Fock equations is obtained, Fi φi =

N

λi j φ j ,

j

which can be simplified by a unitary transformation, which does not change the total wave function, to make the Lagrange multipliers diagonal: Fi φi = εi φi . These special MOs φi are called canonical MOs and the εi are the corresponding MO energies, the expectation values of the Fock operator in the MO basis: εi = φi |Fi |φi . According to Koopman’s theorem the orbital energy corresponds to the ionization energy for a particular electron (neglecting orbital relaxation). The Hartree-Fock equations can be solved only iteratively because the Fock operator depends on all the occupied MOs by means of the Coulomb and exchange operators. The molecular orbitals, φi , are constructed as a linear combination of atomic orbitals (LCAO), χα , which form the basis set (see below), φi =

M α

cαi χα .

This leads to the Roothaan-Hall equations, which correspond to the Fock equations in the AO basis: FC = SCε.

χα χγ |g|χβ χδ ≡ χα χγ | χβ χδ ≡ αγ | βδ 1 ≡ χα (1)χγ (2) χβ (1)χδ (2) dr1 dr2 . |r1 − r2 | The Roothaan-Hall equations give the orbital coefficients as eigenvectors of the Fock matrix. So that the Fock matrix can be constructed, however, the density matrix, D (i.e., the orbital coefficients), has to be known. To start the iterative procedure, one must make an initial guess (e.g., from another calculation, or D is just set to zero), from which a Fock matrix can be derived. Diagonalization of the Fock matrix gives new (improved) orbital coefficients which allow one to build a new density matrix and a new Fock matrix. The procedure must be continued until the change is less than a given threshold and a self-consistent field (SCF) is generated (Fig. 5). In the large basis set limit the Hartree-Fock method formally scales with the fourth power of the number of basis functions due to the two-electron integrals. In practice, computations have a more favorable scaling. Modern algorithms are close to linear scaling because the Coulomb part of the electron–electron interaction which contributes the most to the computational effort (the distances where exchange becomes negligible is relatively short) can be replaced by a multipole interaction for large distances (fast multipole method, FMM). In a conventional HF implementation the two-electron integrals are computed and stored on disk. In contrast, in the direct SCF method, the integrals are recomputed whenever they are needed

The elements of the overlap matrix, S, are defined as Sαβ = χα |χβ , and the Fock matrix elements are given by Fαβ = χα |h|χβ +

occ.MO

χα |J j − K j |χβ .

j

The energy in terms of integrals over basis functions is given by E=

M M α

+

Dαβ χα |h|χβ

β M M M M 1 Dαβ Dγ δ ( χα χγ |g|χβ χδ 2 α β γ δ

− χα χγ |g|χδ χβ ) + Vnn , which introduces the density matrix elements, Dγ δ , as Dγ δ =

occ.MO

c γ j cδ j .

j

The two-electron integrals are often written without the g operator, and for further simplification only the indices are given:

FIGURE 5 Schematic representation of the self-consistent field (SCF) procedure. MO, molecular orbital.

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to avoid the storage bottleneck and the slow input–output operations. It is also possible to effectively screen for integrals which contribute only negligibly and thus can be discarded. The use of symmetry, if present, also reduces the computational cost considerably. 2. Semiempirical Methods The HF method represents a point of departure in electronic structure theory. One direction involves improvement of the accuracy by including electron correlation (see Section II.B.3.). Semiempirical methods, however, try to provide moderate accuracy, but at much lower cost than that of ab initio methods. Therefore, only valence electrons are treated explicitly and core electrons are replaced by an effective core (covering nucleus plus core electrons) and a minimal basis of orthogonal Slater-type orbitals (usually only s and p types) is chosen to describe the valence electrons. The two-electron integrals require the main computational effort in a HF calculation and their number is significantly reduced in semiempirical methods by the zero differential overlap (ZDO) approximation. This basic semiempirical assumption sets products of functions for one electron but located at different atoms equal to zero (i.e. µA (1)νB (1) = 0, where µA and νB are two different orbitals located on centers A and B, respectively). The overlap matrix, S, is set equal to the unit matrix, Sµν = δµν , and the two-electron integrals µν | λσ are zero, unless µ = ν and λ = σ , that is, µν | λσ = δµν δλσ µµ | λλ , where δi j = 0 for i = j and δi j = 1 for i = j. All three-and four-center two-electron integrals vanish automatically. One-electron integrals involving three centers are also set to zero. The remaining integrals are handled as parameters which partly compensate the errors introduced by the ZDO approximation. The parameters are derived from experimental data on atoms or are fitted to reproduce experimental results for molecules. The various semiempirical methods introduce different approximations for the one- and two-electron parts of the Fock matrix elements, Fµν = µ|h|ν +

AO AO λ

σ

Dλσ ( µν | λσ − µλ | νσ ),

with the one-electron operator Z 1 1 a h = − ∇2 − = − ∇2 − Va , 2 2 a |Ra − r| a where Z a denotes the charge resulting from the nucleus plus the core electrons.

a. Complete neglect of differential overlap. The complete neglect of differential overlap (CNDO) approximation is the most rigorous: only the one- and two-center Coulomb terms among the two-electron integrals survive: µA νB | λC σD = δAC δBD δµλ δνσ µA νB | µA νB . µA νB | µA νB are independent of the orbital type (to guarantee rotational invariance) and there are only two parameters, µA νA | µA νA = γAA and µA νB | µA νB = γAB , for the two-electron integrals. The γAB depends only on the nature of the atoms A and B and the distance between them and can be interpreted as the average electrostatic repulsion of one electron at center A and one electron at center B. The integral γAA is the average repulsion of two electrons at one atom. The one-electron integrals are µA |h|νA = −δµν

Nucl.

µA |Va |µA .

a

The Pariser-Pople-Parr (PPP) method is a special case of CNDO, restricted to the treatment of π electrons. b. Intermediate neglect of differential overlap. In the intermediate neglect of differential overlap (INDO) approximation the two-electron integrals are limited to the Coulomb integrals. One-electron integrals involving different orbitals of one center and Va operator from another have to disappear to guarantee rotational invariance. The one-electron integrals are the same as in the CNDO approximation and the two-electron integrals are given by µA νB | λC σD = δµA λC δνB σD µA νB | µA νB and parametrized as γAB and γAA . INDO is comparable to CNDO in computational cost but has the advantage that electronic states of different multiplicities can be distinguished. MINDO/3 (modified intermediate neglect of differential overlap) was the first successful semiempirical method to give reasonable predictions of molecular properties. The main improvement over earlier methods was the use of molecular data rather than atomic data for the parametrization. However, the number of parameters to be determined in MINDO/3 increases with the square of the number of atoms included because one parameter depends on the type of bonded atoms. c. Neglect of diatomic differential overlap. Many of the shortcomings of MINDO/3 are corrected in the neglect of diatomic differential overlap (NDDO) approximation, which includes no further approximations beyond ZDO. Thus, all integrals involving any two orbitals on one

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center with any two orbitals on another center are kept, which increases the number of integrals dramatically. The one-electron integrals are

µA |h|νB = µA − 12 ∇ 2 − VA − VB νB and

µA |h|νA = δµν µA − 12 ∇ 2 − VA µA − µA |Va |νA . a=A

The two-electron integrals are given by µA νB | λC σD = δAC δBD µA νB | λA σB . d. Modified NDDO. The more successful semiempirical methods, MNDO (modified neglect of differential overlap), AM1 (Austin model 1), and PM3 (parametric method number 3), are all based on NDDO but differ in the treatment of core–core repulsion and how the parameters are assigned. There are only atomic parameters, no diatomic parameters as in MINDO/3. The “modified” NDDO methods calculate the overlap matrix, S, explicitly rather than using the unit matrix. The MNDO method tends to overestimate the repulsion between atoms separated by approximately the sum of their van der Waals radii. To correct for this deficiency, AM1 modifies the core–core term by Gaussian functions. PM3 is essentially equivalent to AM1 but uses (automated) full optimization of the parameter set against a much larger collection of experimental data while the AM1 parameters are tuned by hand. PM3 therefore on average gives results in somewhat better agreement with experiment. Extending the basis set to include d functions as in MNDO/d and PM3(tm) raises the number of integrals (i.e., the number of parameters) tremendously but allows a larger variety of applications, for example, those including transition metal compounds (albeit with variable accuracy) or hypervalent molecules. Semiempirical programs usually report heats of formation calculated from the electronic energies less the calculated energies for the atoms plus the experimental heat of formations for the atoms: atoms Hf = E calc. (molecule) − E calc. (atom) +

atoms

Hf (atom).

The semi ab initio model 1 (SAM1) is another modified NDDO method, but it does not replace integrals by parameters. The one- and two-center electron repulsion integrals are explicitly calculated from the basis functions [employing a standard STO-3G (Slater-type orbital from three Gaussian functions) Gaussian basis set] and scaled by a function which has to be parametrized. SAM1 calculations take about twice as long as AM1 or PM3 calculations do.

3. Electron-Correlated Methods In the Hartree-Fock approach the real electron–electron interaction is replaced by an interaction with an averaged field. This means HF suffers from an exaggeration of electron–electron repulsion. The difference between the energy obtained at the HF level and the exact (nonrelativistic) energy (for a given basis set) is defined as the correlation energy. The name reflects that this energy difference is connected to the correlated movement of electrons which is not considered in the HF method and which reduces the electron–electron repulsion. The HF description typically allows electrons to be unrealistically resident in the internuclear region. This leads to an underestimated nuclear–nuclear repulsion and to bond lengths that are too short. As a consequence, stretching force constants and harmonic stretching frequencies computed at the HF level are too large. Likewise, the polarity of bonds is overestimated (the more electronegative atom tolerates a higher electron density in the HF picture) and computed dipole moments are often too large. Dynamic electron correlation, which is connected to the correlated movement of electrons, can be distinguished from static (near-degeneracy) electron correlation, which deals with the insufficiency of the one-determinant approach. HF usually provides a suitable description of closed-shell molecules in their electronic ground state. However, the homolytic dissociation of such a molecule generates two electronic states which are very close in energy. This situation requires a description by more than one Slater determinant (i.e., at least a two-configuration method). The energy difference between the HF method and a multiconfigurational method is the static correlation energy. Accounting for electron correlation is essential for quantitative answers from electronic structure calculations. Different post-HF methods which attempt to recover all or part of the correlation energy are discussed in the following text. Within the closed-shell HF picture, molecular orbitals are occupied by either exactly two or exactly zero electrons represented by the variationally best one-determinant wave function. Correlated levels give a different electron density which cannot be represented by a single Slater determinant. A logical starting point to account for electron correlation is to expand a multideterminantal wave function with the HF wave function as a starting point:

= a0 HF + ai i , i=1

where a0 usually is close to 1. Because this is analogous to expanding one MO in terms of AOs, one speaks of the basis set as the one-electron basis (responsible for the one-electron functions, the MOs) while the number

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sity matrix) may be used, which promise faster convergence of the CI expansion. In general, several Slater determinants are contracted linearly to form eigenfunctions of the spin operators SZ and S2 and which are called spin adapted configurations, or configuration state functions (CSFs):

CI = ai i . i=0

FIGURE 6 Examples of singly, doubly, triply, and quadruply excited determinants derived from a Hartree-Fock (HF) reference.

of determinants included in a correlated wave function builds the many-electron basis. In closed-shell HF theory there is only one determinant which has the lowest MOs occupied (occupation number = 2). The remaining orbitals are empty or “virtual” (occupation number = 0). Additional determinants are generated by exciting one or more electrons from an occupied MO into an unoccupied (virtual) MO. According to the number of excited electrons, one speaks of singles, doubles, triples, quadruples, and so forth (S, D, T, Q, respectively; Fig. 6). The larger the basis set the more virtual MOs and the more excited Slater determinants can be generated. The quality of a calculation is determined by both the size of the basis set and the number of excited determinants that are considered. If all possible determinants together with an infinite basis set could be used, one would get the exact solution of the nonrelativistic Schrödinger equation within the Born-Oppenheimer approximation. Because a different chemical environment mostly affects the valence electrons, but does not influence the core electrons, the frozen core approximation includes only determinants with excited valence electrons. Also the highest virtual orbitals may be left unoccupied in all determinants (frozen virtuals). a. Configuration interaction. In the configuration interaction (CI) procedure the trial function is constructed as a linear combination of the ground (reference configuration) and excited Slater determinants. The MO coefficients remain fixed throughout the calculation and are usually taken from the HF orbitals. Alternatively, natural orbitals (which are defined as diagonalizing the one-electron den-

The expansion coefficients, ai , are then determined variationally to give the minimum energy. For a full CI (FCI) the number of determinants grows factorially with the size of the system. A full CI recovers all of the electron correlation energy (for a given basis set) but can be applied only to obtain benchmark results for very small molecules to assess the performance of more economical methods. For applications to larger molecules, the CI expansion has to be truncated to make the computation feasible. CIS, CISD, CISDT, and CISDTQ correspond to expansions through singly, doubly, triply, and quadruply excited CSFs, respectively. According to Brilluoin’s theorem, the CI matrix elements of a closed-shell restricted HF wave function with singly excited CSFs vanish. Hence, CIS does not improve the description of the ground state. Doubles are found to contribute most to the correlation energy and consequently CISD (including only singly and doubly excited determinants) is the most widely applied CI method because inclusion of triples and quadruples is typically computationally too demanding. FCI is size consistent, but truncated CI methods are not. This means the energy computed for two noninteracting molecules is not identical to the sum of the energies computed for the individual molecules. This unphysical behavior is a major drawback of any truncated CI. So that CISD can be made approximately size consistent, the Davidson correction can be applied in which the contribution of quadruples, E Q , is estimated from the correlation energy given at the CISD level, E CISD , and the coefficient of the reference configuration, a0 : E Q = (1 − a0 )E CISD . CISD was also extended to quadratic CISD (QCISD) by the inclusion of some higher-order terms to yield a sizeextensive method. b. Multiconfiguration self-consistent field. The HF method does not give a good first-order description when more than one nonequivalent resonance structure is important for the electronic structure of a molecule. A multiconfiguration self-consistent field (MCSCF) calculation may be used instead. Not only the coefficients for the determinants are optimized in MCSCF, but also the MO coefficients simultaneously. The selection of configurations

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is not trivial. One easy way to construct an MCSCF is the complete active space self-consistent field, or CASSCF (also called full optimized reaction space, or FORS). Instead of choosing configurations, one must select a set of “active” (occupied and unoccupied) orbitals and all possible (symmetry-adapted) configurations within this “active space” are automatically included in the MCSCF. The method is called restricted active space self-consistent field, or RASSCF, when subsets of the active orbitals are restricted to have a certain (minimum or maximum) number of electrons to reduce the computational cost. The MCSCF provides a good first-order description covering the static electron correlation due to degeneracy problems. Dynamic electron correlation should be addressed with the MCSCF wave function as a reference. The multireference configuration interaction, or MRCI, generates excited determinants from all (or selected) determinants included in the MCSCF. The complete active space perturbation theory, second order (CASPT2) is a more economical approach. Both methods can be applied to compute excited states. c. Many-body perturbation theory. Perturbation theory assumes that somehow an approximate solution to a problem can be found. The missing correction, which should be small, is then considered as a perturbation of the system. When the perturbation is to correct for the approximation of independent particles the method is called many-body perturbation theory, or MBPT. In electronic structure theory the Hamiltonian operator, H, is written as a combination of a reference Hamiltonian, H0 , which can be solved for, and a perturbation H : H = H0 + λH , λ being the perturbation parameter (0 ≤ λ ≤ 1) which determines the strength of the perturbation. The energy, E, and wave function, , are expanded as Taylor series in λ: E = E 0 + λE 1 + λ2 E 2 + λ3 E 3 + · · · and

= 0 + λ 1 + λ 2 + λ 3 + · · · . 2

3

The Schrödinger equation, H = E , gives 0 and E 0 as the solution in the absence of any perturbation (λ = 0). E 1 , E 2 , etc., and 1 , 2 , etc., are the first-, second-, etc., order corrections to the energy and wave function, respectively. For λ > 0 the Schrödinger equation becomes (H + λH ) 0 + λ 1 + λ2 2 + · · · = E 0 + λE 1 + λ2 E 2 + · · · 0 + λ 1 + λ2 2 + · · · .

Because this equation has to be true for all values of λ, the terms connected to the same power of λ can be separated: λ0: H0 0 = E 0 0 , λ1: H0 1 + H1 0 = E 0 1 + E 1 0 , λ2: H0 2 + H1 1 + H2 0 = E 0 2 + E 1 1 + E 2 0 , and so forth, which gives the zeroth-, first-, second-, etc., order perturbation equations. If one chooses the intermediate normalization condition, 0 | 0 = 1;

0 | i = 0,

i > 0,

simple energy expressions are obtained: E 0 = 0 |H0 | 0 , E 1 = 0 |H | 0 , E 2 = 0 |H | 1 , and so forth. Knowledge of wave function corrections up to order i allows calculation of the energy up to order (2i + 1). This relationship is known as the Wigner theorem. In Møller-Plesset (MP) perturbation theory the unperturbed Hamiltonian, H0 , is taken as the sum over n Fock operators (n = number of electrons) giving a total of twice the average electron–electron repulsion energy and the perturbation operator becomes the difference between the exact electron–electron repulsion and twice the average electron–electron repulsion. With this choice of H0 the zeroth-order energy is just the sum of MO energies and the first-order energy equals the Hartree-Fock energy. The second-order correction, E(MP2), is the first to contribute to the electron correlation energy and can be calculated from the two-electron integrals over MOs: E(MP2) =

virt occ [ φi φ j | φa φb − φi φ j | φb φa ]2 i< j a 4 are not used routinely because they are both very complex and expensive in terms of resources. MP theory is size extensive but not variational (i.e., there is no guarantee that the correct energy is lower than

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Computational Chemistry TABLE I Formal Scaling of Various Methods with the Size of the Molecular System, N, and Size-Extensive and Variational Properties Methoda

Scaling

Size extensive?

Variational?

HF MP2 MP3

N4 N5 N6

Yes Yes

Yes No

Yes

No

MP4SDQ

N6

MP4SDTQ MP5

N7 N8

Yes Yes

No No

MP6 CCSD

N9 N6

Yes Yes

No No

Yes

No

CCSD(T)

N7

CCSDT CISD

N8 N6

Yes Yes

No No

No

Yes

CISDT

N8

No

Yes

CISDTQ

N 10

No

Yes

a HF, Hartree-Fock; MP, Møller-Plesset (numbers 2– 6 refer to second through sixth order; S, singly excited; D, doubly excited; T, triply excited; and Q, quadruply excited; CC, coupled cluster (S, D, and T as for MP except T in parentheses is estimated); CI, configuration interaction (S, D, T, and Q as for MP).

an MPn energy). This is no problem because usually only relative energies are of interest. However, the MPn series does not necessarily converge. When the HF reference provides a poor description of the electronic structure, the MPn series may become divergent and produce even worse results than those of HF. d. Coupled cluster methods. Coupled cluster (CC) theory was originally formulated for nuclear physics and only later was applied to the electron correlation problem in quantum chemistry. Today it is the method of choice for highly accurate computations. CC theory uses an exponential expansion of a reference function 0 , usually the Hartree-Fock determinant (in contrast with the linear expansion of CI):

T = T1 + T2 + T3 + · · · T N . The excitation operators, Ti , generate all ith excited Slater determinants from the reference:

i

T2 0 =

tia ia ;

a

occ virt i< j a10 ps). If complex lifetimes exceed several rotational periods, product angular distributions from crossed beam reactions exhibit forward-backward symmetry in the center-of-mass frame of reference. Direct reaction An elementary bimolecular reaction proceeding via direct passage through the transition state region. The absence of long-lived intermediates in such reactions leads to anisotropic center-of-mass product angular distributions that often provide insight into the most favorable geometries for reaction. Free radical An atom or molecule possessing one or more unpaired electrons. Free radicals may either be

stable molecules (e.g., NO, O2 , or NO2 ), or highly reactive transitory chemical intermediates (e.g., H, Cl, CH3 ) that react on essentially every collision with stable molecules. Ionization The process by which one or more valence electrons are removed from an atom or molecule. Most often achieved by electron impact or absorption of one or more ultraviolet photons. Laser-induced fluorescence (LIF) A spectroscopic technique usually employing visible or UV laser light, in which the fluorescence emission from a gaseous, liquid, or solid sample is monitored. A fluorescence excitation spectrum is a plot of the total emitted fluorescence vs. excitation wavelength and provides information similar to an absorption spectrum. Molecular beam Collimated stream of gaseous molecules produced by expansion of a gas through an orifice into an evacuated chamber. A supersonic molecular 697

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Dynamics of Elementary Chemical Reactions

beam is characterized by a velocity distribution much narrower than a Boltzmann distribution. Potential energy surface Schematic two- or threedimensional representation of the total potential energy of a chemical system as a function of internuclear coordinates. Transition state Region of the PES corresponding to the critical geometry through which a reacting system must pass for reactants to become products.

AN ELEMENTARY CHEMICAL reaction is any process involving bond fission and/or bond formation following a single collision between two reactants. Chemical reactions occur in all three phases of matter (gas, liquid, and solid), and at their interfaces. Under the experimental conditions most commonly used to carry out reactions, the overall reaction usually consists of a sequence of two or more elementary reactions. For example, the reaction of gaseous hydrogen with chlorine forming hydrogen chloride is represented by the following balanced chemical equation: H2 + Cl2 → 2HCl.

(1)

This reaction proceeds by a chain mechanism involving a repetitive sequence of elementary reactions involving the three stable molecules listed in Eq. (1), as well as two short-lived free radical intermediates, i.e., chlorine atoms (Cl) and hydrogen atoms (H). The most important elementary steps in the overall reaction mechanism are: Initiation :

Cl2 → 2Cl.

Propagation : Cl + H2 → HCl + H

Termination :

presence of a third body (M), which may be a molecule or the wall of the container. Note that the two elementary propagating reactions may be added like mathematical equations, yielding the overall chemical reaction (1). For an overall reaction such as that in Eq. (1) involving a sequence of elementary steps, the overall rate of formation of products may be a complex function of reactant concentrations, because products are formed by several different elementary processes. In the previous example, the HCl products are formed by reactions (3) and (4), each of which has its own rate law and rate constant. Thus, for a complex multistep process such as reaction (1), the rate law can only be determined through experiment. For an elementary bimolecular reaction A + B → C + D, the reaction rate is proportional to the concentrations (denoted by [A], [B], etc.) of the reactants: d[D] d[A] d[B] d[C] = =− =− = k[A][B]. (5) dt dt dt dt The proportionality constant, k, is called the reaction rate constant. Since an elementary reaction involves a single bimolecular collision between A and B, the maximum possible rate constant is usually the frequency of collisions between reactants. A few simple atom-transfer reactions (e.g., F + H2 → HF + H) actually do occur on nearly every collision, and are said to proceed at or near the “gas kinetic limit.”

I. KINETICS AND COLLISION THEORY

(2)

(3a)

H + Cl2 → HCl + Cl.

(3b)

H + Cl + M → HCl + M

(4a)

Cl + Cl + M → Cl2 + M

(4b)

H + H + M → H2 + M.

(4c)

The chain reaction is initiated by dissociation of Cl2 , a stable molecule, to form two highly reactive chlorine atoms (Cl). Since chemical bond fission requires the input of energy, initiation may be achieved by heating the sample (see Lindemann mechanism) or by ultraviolet irradiation (photodissociation). Following initiation, the two elementary bimolecular propagating reactions will continue until either or both of the reactants (H2 and Cl2 ) are consumed, at which time the termination steps end the chain reaction. Termination typically involves termolecular recombination of two radicals to form a stable molecule in the

In order to estimate the frequency of collisions between gaseous A and B molecules, consider a beam of molecules of incident flux I A (molecules/cm2 · s) impinging on a static cell containing molecules at a concentration [B] (molecules/cm3 ). The particles interact in a volume element V . The collision rate per unit time, Z , is given by Z = σI A [B]V.

(6)

Here, σ is the collision cross section, which may be estimated using a simple hard sphere model for colliding particles (Fig. 1). Two particles collide with a relative velocity vector, g, the magnitude of which is denoted by g, and impact parameter b, also known as the “aiming error” of the collision. A hard sphere collision will occur provided 0 ≤ b ≤ (r A + r B ). The collision cross section is therefore the area of a circle of radius d AB = r A + r B , i.e., 2 . The incident flux, I A = [A]g A , may then be σh.s. = πd AB substituted into Eq. (6). If the rate of reaction between A and B is simply the collision rate, then −

d[A] = Z = σh.s. g A [A][B]. dt

(7)

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Whereas the hard sphere cross section depends only on the sum of the radii of colliding particles, a reaction cross section may depend strongly on the energy of the collision and therefore on the relative velocity of the colliding particles, the magnitude of which is given by g. As illustrated in Fig. 2, the relative velocity vector, g, may be decomposed into two perpendicular components. The first is a radial component, gr = r˙ , i.e., the time derivative of r , the distance between the particle centers. Perpendicular to the radial axis is a tangential component, g⊥ = r θ˙ , where θ˙ is the time derivative of θ , the angle between g and the radial axis. As a result, the total kinetic energy of collision, E kin = 1/2 µg 2 , can be thought of as a sum of a radial kinetic energy, Er = 1/2 µgr2

(11)

and an energy associated with the perpendicular velocity component, 2 E ⊥ = 1/2 µg⊥ = 1/2 µr 2 θ˙ 2 =

FIGURE 1 Depiction of hard sphere collision cross section. Note that collisions only occur for impact parameters b < dAB .

Comparison of Eqs. (5) and (7) indicates that the collision rate constant, k, is related to the collision cross section and relative velocity of colliding particles by k(g) = σ g .

(8)

In deriving Eq. (8), it is assumed that molecules A and B collide with a single relative velocity g. In a real gaseous sample containing both A and B molecules at thermal equilibrium, the distribution of relative velocities is described by the Maxwell–Boltzmann Distribution Law: 3/2 2 µ − µg f (g) = 4πg 2 e 2k B T , (9) 2πk B T

L2 , 2µr 2

(12)

where L = µr 2 θ˙ = µgb is the magnitude of the angular momentum associated with the colliding pair. Thus, for interaction potentials that depend solely on the distance between the colliding pair, V (r ), only gr is effective in surmounting potential energy barriers such as those associated with the energy required to break and form bonds during reaction; g⊥ is associated purely with rotational motion of the two particles. One way to model the energy dependence of σ is to assume that reaction can only occur if the component

where µ = m A m B /(m A + m B ) is the reduced mass of the colliding particles, k B is Boltzmann’s constant, and T is the temperature in Kelvin. The hard sphere collision rate constant, kh .s. , is thus temperature dependent, and may be evaluated explicitly by integrating over all possible relative velocities, g: ∞ 8k B T 1/2 2 kh .s. (T ) = σh .s. g f (g) dg = πd AB . πµ g=0 (10) Using typical molecular values of d AB ≈ 0.35 nm and µ = 14 amu for room temperature collisions between N2 molecules, one observes the magnitude of a hard sphere collision rate constant to be on the order of 2.6 × 10−10 cm3 /molecule · s.

FIGURE 2 Decomposition of relative velocity vector, g, into radial (gr ) and perpendicular (g⊥ ) components, with θ defined as the angle between g and the internuclear axis. At the moment of a hard sphere collision, sin θ = b/dAB .

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of kinetic energy along the axis of the collision (i.e., Er ), exceeds a critical energy, Vc . Note that during the course of a collision, θ varies from 0 to 180◦ as r changes from −∞ to +∞. However, if the particles are treated as hard spheres, the internuclear potential, V(r ), is zero for r > d AB , and there is no interaction between them until collision, at which time (sin θ) = b/d AB . Using the fact that gr = g cos θ , the radial kinetic energy at the moment of contact is b2 1 2 2 2 Er = /2 µg cos θ = E kin (1 − sin θ ) = E kin 1 − 2 . d AB (13) In order for the reaction to be successful, Er ≥ Vc . For a given collision energy, E kin , this implies that the impact parameter must be smaller than a critical impact parameter, bc , defined such that bc2 E kin 1 − 2 = Vc . (14) d AB Thus, the cross section for a reaction involving a critical energy, Vc , given by Vc 2 σ = π bc2 = πd AB 1− (15) E kin is expected to increase with energy, as illustrated in Fig. 3. Note that this model predicts that the threshold for reaction occurs at E kin = Vc , and that the cross section reaches half the hard sphere value at E kin = 2Vc , asymptotically approaching the hard sphere value as E kin → ∞. In many cases, reaction cross sections for real systems differ considerably from that shown in Fig. 3. For example, some processes, such as charge exchange (e.g., A+ + B− → A + B), proceed with reaction cross sections far exceeding the hard sphere limit. Here, the long-range Coulomb potential causes reactants to be attracted to one another at large distances, considerably increasing the reaction cross section. Even for neutral–neutral interactions, the interaction potential, V (r ), often differs substantially from that of hard spheres. Long-range induced dipole-

FIGURE 3 Cross section (σ ) dependence on kinetic energy, Ekin , for the hard sphere model requiring an energetic threshold, Vc .

FIGURE 4 Schematic internuclear potentials for different models of atomic and molecular interactions. The hard sphere model exhibits only a repulsive component at small r ; more realistic potentials exhibit attractive and repulsive components.

induced dipole interactions (van der Waals’ interactions) result in an attractive region of the potential surface at longer bond distances even in cases when formal bonds between the interacting pair cannot be formed. When strong bonds can be formed between colliding particles, as in the case of two halogen atoms like Cl, a strongly attractive component of the potential is present over a wide range of internuclear separations. In such cases, the hard sphere potential would only be useful in modeling the interaction potential at small distances where electron– electron repulsion becomes dominant, as demonstrated in Fig. 4. However, the magnitudes of most reaction cross sections are controlled predominantly by the form of the attractive component of the potential at longer internuclear separations. We now discuss a relatively simple model for reactions involving gaseous particles interacting through a potential V (r ) operating at long range. Recall from Eq. (11) and (12) that E kin can be written as a sum of radial energy, Er , and energy associated with rotational motion of the interacting particles. Thus, the total energy, E = E kin + V (r ), of the colliding partners is E = Er +

(µgb)2 + V (r ), 2µr 2

(16)

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making it convenient to conceptualize an effective potential, Veff (r ) =

(µgb)2 + V (r ), 2µr 2

(µgb)2 + V (rt. p. ). 2µrt.2 p.

(18)

This may be rearranged to solve for b = bmax , the maximum impact parameter for a given relative velocity, g, that allows the collision partners to reach a fixed critical distance for reaction, rc : 2V (rc ) 1/2 bmax (g) = rc 1 − , (19) µg 2 which holds for g ≥ gmin , where, in order to be physically meaningful, gmin = [2V (rc )/µ] /2 1

=0

if V (rc ) > 0 if V (rc ) ≤ 0.

(20)

The reaction rate constant may in both of these cases be determined analytically by integration over all relative velocities exceeding gmin : k(T ) =

∞

π {bmax (g)}2 g f (g) dg

gmin

=

πrc2

= πrc2

8k B T πµ 8k B T πµ

1/2 e

− Vk (rcT)

1/2

if V (rc ) > 0

B

V (rc ) 1− kB T

k = Ae−Ea /k B T ,

(17)

that governs the radial motion of the particles. At large internuclear distances, the total energy of the system, E = 1/2 µg 2 , is simply the radial kinetic energy, since lim V (r ) = 0. However, at smaller values of r , where r →∞ eff Veff (r ) > 0, some of the initial kinetic energy is converted into energy associated with the effective potential. The distance of closest approach, or turning point, rt. p. , is reached when the magnitude of the effective potential is equal to the initial radial kinetic energy, and the radial velocity becomes zero: E = 1/2 µg 2 =

to the empirical Arrhenius expression found to satisfactorily model a large number of chemical reactions:

(21) if V (rc ) ≤ 0. (22)

In cases where V (rc ) is positive, the critical distance rate constant expression [Eq. (21)], is similar to the hard sphere collision rate constant [Eq. (10)]; however, an additional exponential term is present. This term represents the fraction of molecules at temperature T having sufficient energy to react. This temperature dependence is thus similar

(23)

where A is the Arrhenius preexponential factor, and E a is the Arrhenius activation energy. These quantities are most readily determined by plotting ln k vs. 1/T , which should be linear with a slope −E a /k B and intercept ln A. Note that this purely empirical relationship often holds for elementary as well as multistep reactions. The obvious similarity between Eqs. (21) and (23) suggests that E a is at least loosely related to the height of the potential energy barrier for the rate-limiting step in the reaction. However, the Arrhenius parameters are only phenomenological quantities derived from the temperature dependence of reaction rate constants. In fact, Arrhenius plots are in many cases found to be markedly nonlinear, suggesting the occurrence of a multistep reaction mechanism or a mechanism that changes at different temperatures. The critical distance model can be used to derive an explicit formula for the temperature dependence of the reaction rate constant for charge transfer reactions of the form A+ + B− → A + B. Such interactions are subject to long-range Coulomb attractions of the form V (r ) = −q 2 /4π ε0r , where q is the charge of an electron. Taking the critical distance, rc , to be the ionic–covalent curve crossing radius (R), which corresponds to the distance at which the Coulomb attraction between ions balances the energy required for electron transfer, one obtains by substitution into Eq. (22) the following expression for the charge exchange rate constant: 8k B T 1/2 q2 k(T ) = πR 2 1+ . (24) πµ 4πεo Rk B T This reaction rate constant expression bears some similarity to a hard-sphere rate constant; however, an additional term (q 2 /4π εo Rk B T ) results from the long range attractive interaction, and is in general the dominant contribution to the reaction rate constant for reactions of this type. For attractive potentials of the form V (r ) = −a/r s , Veff is given by Veff (r ) =

L2 a (µgb)2 a − = − s. 2 s 2 2µr r 2µr r

(25)

Provided s ≥ 3, Veff has a local maximum for a given impact parameter, b, at a radial distance, rmax , determined by 2−s sa rmax = . (26) µg 2 b2 For close approach required for reaction, the two particles must overcome this maximum, Vmax = Veff (rmax ), as

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FIGURE 5 Relationship between Veff (r ) and V(r ) for a given impact parameter, b. Close approach necessary for reaction requires E >Vmax , where Vmax is the maximum in Veff (r ).

illustrated in Fig. 5. Classically, if the energy of the colliding reactants is exactly equal to Vmax , all of the initial kinetic energy is converted into Veff when r = rmax , and “orbiting” will occur; i.e., the interacting pair will rotate together for an infinite amount of time since both the radial kinetic energy and the centrifugal force acting on the particles (−d Veff /dr ) are zero. The orbiting impact parameter, bor b , defined as the impact parameter for a given initial collision energy, E = 1/2 µg 2 , at which orbiting occurs, can be determined by setting Vmax = E and rearranging to arrive at s a(s − 2) 2/s 2 bor b = . (27) µg 2 s −2 Since smaller impact parameter collisions will result in a smaller value of Vmax , only collisions occurring with impact parameters less than bor b will lead to a close collision 2 and reaction. Thus, the reaction cross section, σ = π bor b, which depends on the relative velocity g, may be integrated over all relative velocities to derive the rate constant temperature dependence: ∞ 2

k(T ) = π bor b g f (g) dg 0

=2

3s−4 2s

1/2 π s −2 s−4 2/s 2/s , (s − 2) a (k B T ) 2s µ s (28)

where the gamma function, , is available in mathematical tables. This equation predicts that reactions involving quenching of an electronically excited state, which can be modeled using s = 3, will show weak inverse temperature dependence, k(T ) ∝ T 1/6 . Ion–molecule reactions, having s = 4, are predicted to have rate constants independent of temperature. Reactions dominated by

van der Waals interactions (i.e., s = 6) are expected to show a small positive temperature dependence (k ∝ T 1/6 ). The profound effect of the exact form of the internuclear potential on the interaction between particles can be observed in elastic scattering experiments. These studies allow determination of the angle of deflection of a particle from its original direction upon interaction with the second particle. Conceptually, the scattering process can be understood by considering the effect of a particle of mass µ colliding with an infinitely massive particle fixed in space. The deflection angle, χ , defined as the angle between the initial and final relative velocity vectors of the colliding pair, will depend on the form of potential, V (r ), and, based on the impact parameter at which a given collision occurs, the region of V (r ) that is sampled by the colliding pair. Recall from Eq. (17) that the effective potential, Veff , governs the radial motion of the colliding particles, and therefore determines the radial turning point, rt. p. , for a given magnitude of initial collision energy, E. Figure 6 shows three effective potentials (for impact parameters b = 0, b1 , b2 , and b3 , where 0 < b1 < b2 < b3 ) for an internuclear potential, V (r ), with both long-range attractive and short-range repulsive components. If the two particles collide with a fixed collision energy (e.g., that denoted by E 1 ), the radial turning point, and therefore the regions of V (r ) accessed during the collision, will depend strongly on the magnitude of b. For very large values of b, (b ∼ b3 ), the turning point lies at very large r , and the particle experiences little deflection upon approach (χ ∼ 0). For smaller values of b (b ∼ b2 ), the turning point moves to smaller values of r , and the interacting particles therefore sample more of the long-range attractive part of V (r ), resulting in a deflection toward more negative angles. As b becomes even smaller, however, the effects of the repulsive part of the potential begin to play a role, and χ

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FIGURE 6 Dependence of Veff on impact parameter, b, where 0 < b 1 < b 2 < b 3 . Larger magnitudes of b result in larger values of the radial turning point for a given collision energy.

reaches a minimum value, called the “rainbow” angle, χr , by analogy to the optical rainbow resulting from scattering in small water droplets. The impact parameter that results in this most negative degree of deflection is referred to as the rainbow impact parameter, br . For impact parameters less than br (b ∼ b1 ), short-range repulsion begins to dominate, and the deflection angle becomes less negative, reaching zero at the impact parameter at which the attractive and repulsive forces during the collision exactly offset, the so-called “glory” impact parameter, bg . For impact parameters smaller than bg , repulsion dominates, and the particle is scattered to positive angles, reaching a max-

imum of χ = π for direct head-on collisions (b = 0). A pictorial depiction of the dependence of χ on the impact parameter, known as the deflection function, is shown in Fig. 7, where particle trajectories during the course of a collision are shown for a wide range of impact parameters. Note that this figure depicts the dependence of the deflection angle, χ , on b∗ = b/re , where re is the internuclear separation where V (r ) reaches a minimum. Although collision theory has provided considerable insight beyond simple hard spheres, it cannot properly address questions such as what fraction of collision geometries are likely to lead to reaction. Such issues cannot be

FIGURE 7 Deflection angle, χ , dependence on impact parameter. Trajectories depict the degree of deflection of impinging particle resulting from attractive and repulsive components of the interaction potential. (From Levine, R. D., and Bernstein, R. B. (1987). “Molecular Reaction Dynamics and Chemical Reactivity,” Oxford University Press, New York.)

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properly accounted for unless the details of the molecular structure of reactants are considered. Often, the rate constant calculated from collision theory must be reduced by inclusion of an ad hoc “steric factor” which represents the fraction of collisions that have proper geometry for reaction. This additional factor is an empirical factor used to bring experimental observation into line with collision theory, and accounts for those important factors not addressed by collision theory.

According to transition state theory, a fast equilibrium exists between reactants and molecules at the transition state, denoted by AHB‡ : k1

ω

A + HB AHB‡ → AH + B. k−1

(31)

The equilibrium concentration of molecules at the transition state is given by [AHB‡ ] = K ‡ [A][HB],

(32)

where K = k1 /k−1 . The overall rate of product formation depends on the rate constant, ω, with which the activated complexes cross over to products: d[AH] (33) = ω[AHB‡ ] = ωK ‡ [A][B]. dt A comparison of Eqs. (30) and (33) indicates that the rate constant k = ωK ‡ . The equilibrium constant for production of activated complexes K ‡ is related to molecular partition functions (Q), calculated using statistical mechanics: ‡

II. ACTIVATED COMPLEX THEORY Many models of chemical reactions are based on the concept of an “activated complex,” or “transition state,” which corresponds to the nuclear configuration with the highest potential energy of the system traversed during the course of the reaction. The transition state corresponds to a critical geometry of the reacting system marking the boundary between reactants and products. Consider the elementary reaction: A + HB → AH + B.

(29)

The overall rate for this elementary reaction is given by d[AH] = k[A][HB]. (30) dt The reaction coordinate for this H-atom transfer reaction may be considered to be translational motion of the H atom from B to A. A potential energy diagram for this process may be represented in 2D or in 3D, as shown in Fig. 8.

K‡ =

[AHB‡ ] Q AHB‡ −E/k B T e . = [A][HB] Q A Q HB

(34)

In the above equation, E = E AHB‡ − E A − E HB is the energy difference between the reactants and activated complex. Assuming that the reaction involves translational motion of the hydrogen atom from B to A, the rate of passage through the transition state is given by v k B T 1/2 1 ω= = , (35) δ 2πµ δ

FIGURE 8 Schematic potential energy surface of A + HB → AH + B reaction, shown as a two-dimensional contour plot (left) and a three-dimensional surface plot (right). Trajectory on contour plot corresponds to lowest energy pathway from reactants to products and traverses the region corresponding to the reaction transition state, AHB.‡

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where v is the velocity through the transition state region and δ is the “width” of transition state along the reaction coordinate. The contribution to the partition function of the activated complex corresponding to motion along the reaction coordinate, in this case translational motion, is factored out of the transition state partition function, Q AHB‡ , to yield: (2πµk B T ) /2 δ Q AHB‡ . = h (36) 1

Q AHB‡ =

qtrans,AHB‡ Q AHB‡

Note that the width parameter δ appears again in Eq. (36). This parameter ultimately factors out in the transition state theory rate constant, k TST , which is given by: k TST = ωK ‡ =

k B T Q AHB‡ −E/k B T e . h Q A Q HB

(37)

k2

A ∗ → P 1 + P2 .

(38)

Under steady-state conditions, the concentration of the collisionally activated molecule, A∗ is constant, i.e., the rate of its formation is exactly balanced by the rate of its destruction: d[A∗ ] = k1 [A][M] − k−1 [A∗ ][M] − k2 [A∗ ] = 0. dt

(39)

Rearranging: [A∗ ] =

k1 [A][M] . k−1 [M] + k2

k2

A + B AB∗ → P1 + P2 . k−1

(40)

(42)

Following formation of the AB∗ intermediate by bimolecular collision, the reaction dynamics are somewhat analogous to those in the Lindemann mechanism: an internally excited molecule may decay to products or reform reactants. In the absence of collisions, the dynamics are solely determined by the unimolecular dynamics of the complex. However, at high pressures, particularly if the lifetime of the AB complex is long, it may undergo collision with another body, possibly carrying away excess energy resulting in formation of stable AB: k3

Many early experiments showed that thermal decomposition of a molecule A, forming products P1 and P2 , often exhibits first order kinetics at high pressure, and second–order kinetics at low pressure. As first proposed by Lindemann, the mechanism involves collisional excitation of the reactant A: k−1

At high pressure, k−1 [M] k2 , and d[P1 ]/dt = (k1 k2 / k−1 )[A], yielding first-order kinetics. Under low-pressure conditions, k−1 [M] k2 and d[P1 ]/dt = k1 [A][M] resulting in second-order behavior. The important result is that reactions that appear to be unimolecular, exhibiting firstorder kinetics at high pressures, actually involve bimolecular processes. An important class of elementary bimolecular reactions are those that involve formation of persistent collision complexes, denoted AB∗ , that may ultimately form products C + D, or decay back to reactants:

AB∗ + M → AB + M.

III. UNIMOLECULAR VERSUS BIMOLECULAR REACTIONS

k1

k1 k2 [A][M] d[P2 ] d[A] d[P1 ] . = =− = k2 [A∗ ] = dt dt dt k−1 [M] + k2 (41)

k1

Although we have derived this equation assuming that the reaction coordinate for atom transfer corresponds to translational motion, the same expression is obtained if the reaction coordinate is assumed to be vibrational motion. According to Eq. (37), the reaction rate constant may be calculated using the relevant molecular partition functions, known from statistical mechanics, remembering that Q AHB‡ does not include the translational motion contribution to the transition state partition function.

A + M A∗ + M,

The overall rate of product formation is

(43)

IV. STATISTICAL THEORIES OF UNIMOLECULAR DECOMPOSITION The Lindemann mechanism as well as reactions occurring via formation of long-lived complexes involve participation of highly internally excited intermediate species that may ultimately dissociate by one or more chemical channels. For example, the intermediate complex AB∗ in reaction (42) may form new products P1 + P2 , or decay back to reactants, A + B. The total rate constant for decay of AB∗ is the sum of the two rate constants, k−1 + k2 , and the relative importance of these competing processes is defined as the product branching ratio k2 /k−1 . Of key importance in understanding these reactions are the reaction rate constants k−1 and k2 . A number of theories have been developed to quantitatively predict rate constants for unimolecular reactions. Rice, Ramsperger, and Kassel developed a simple theory, now known as RRK theory, which contains many fundamental elements underlying most modern theories of unimolecular reaction. According to RRK theory, the

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energized molecule produced by bimolecular collision may be considered to consist of a group of s identical harmonic oscillators, each of frequency ν. If any oscillator accumulates sufficient energy E o = mhν, where m is an integer and h is Planck’s constant, the energized molecule will dissociate. An underlying assumption of RRK (as well as other related theories) is that energy may flow freely between the oscillators in the molecule. The total energy of all of the oscillators is denoted by E = nhν. The number of ways that n quanta can be placed in a molecule consisting of s oscillators is: wn =

(n + s − 1)! . n!(s − 1)!

(44)

The number of ways that n quanta can be placed in a molecule such that at least m quanta are in one oscillator is given by wm =

(n − m + s − 1)! . (n − m)!(s − 1)!

(45)

The probability, P, of dissociation is the ratio of these quantities: P=

wm (n − m + s − 1)!n! = . wn (n − m)!(n + s − 1)!

(46)

If n and m are large, then Sterling’s approximation (x! ≈ x x /e x ) may be applied, and, if s is small relative to (n − m), then (n − m + s − 1) ≈ n − m. The probability of decomposition then reduces to n − m s−1 P= . (47) n The rate constant for unimolecular decomposition is the probability P multiplied by a frequency factor ν: n − m s−1 E − E o s−1 kRRK = ν =ν . (48) n E For a given value of s, the reaction rate constant increases with increasing energy E above threshold, E o . If E E o , the reaction rate constant approaches the frequency factor ν. On the other hand, for reactions involving two similar but different-sized molecules having the same E and E o , since the larger molecule has a greater number of oscillators, s, the reaction rate constant k is smaller (since (E − E o )/E < 1). In practice, to obtain agreement with experiment, it is necessary to use values of s in Eq. (48) which are approximately one half of the actual number of vibrational modes in the molecule. Of course, because RRK theory treats all oscillators as having the same vibrational frequency, the theory employs very simple equations that represent qualitatively but not quantitatively the behavior of real molecules.

With the development of computers, accurate calculations using theoretical models better able to represent the behavior of real molecules has become widespread. A very important extension of the original theory, due to Marcus, is known as RRKM theory. Here, the real vibrational frequencies are used to calculate the density of vibrational states of the activated molecule, N (E). The number of ways that the total energy can be distributed in the activated complex at the transition state is denoted W (E ). Note that the geometry of the transition state need not be known, but the vibrational frequencies must be estimated in order to calculate W (E ). In calculating the total number of available levels of the transition state, explicit consideration of the role of angular momentum is included. The RRKM reaction rate constant is given by: kRRKM =

W (E ) , h N (E)

(49)

where h is Planck’s constant.

V. REACTIONS IN SOLUTION The density of molecules is substantially higher in liquids than in the gas phase. However, for reactions carried out in solution under relatively dilute conditions, the concentrations of reactants are not appreciably different from in the gas phase. Since reactant molecules A and B must undergo collision in solution in order to react, many of the same principles developed for gas-phase reactions also apply in solution. However, the presence of solvent molecules leads to important differences between reactions in solution and in the gas phase. In solution, the rate of diffusion often limits the rate of approach of molecule A to within a sufficient distance to B for reaction to occur. Once an encounter pair AB is formed, however, the solvent may act as a “cage,” effectively holding them in close proximity, thereby increasing the probability of reaction. In solution, the overall reaction may again be broken down into a sequence of elementary steps: kd

A + M AB, k−d

kr

AB → P,

(50)

where, AB is an encounter pair, kd and k−d are rate constants for approach and separation of the reactant molecules by diffusion, and kr is the rate of conversion of encounter pairs to products. Applying the steady-state approximation to the concentration of encounter pairs, we obtain d[P] k d kr = [A][B] = k[A][B]. (51) dt k−d + kr If the activation energy for the reaction is large, kr kd and the reaction rate constant k ≈ kd kr /k−d . Alternatively,

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if kr kd , then k ≈ kd , and the reaction rate constant is the rate of diffusion of reactants to sufficiently close proximity to facilitate reaction. The rate of a diffusion controlled reaction is determined by the magnitude of the diffusion coefficients, D, for the reactants A and B: k = 4π(DA + DB )R.

(52)

The diffusion coefficients are related to the viscosity of the solvent, η, and hydrodynamic radius, r , of the diffusing species by the Stokes–Einstein Law: D=

kB T , πβηr

(53)

where β is a constant typically ranging from 4 to 6. Note that this derivation assumes that diffusing particles are equally likely to move in any direction. However, if the reactants are oppositely charged ions, Coulomb attraction substantially increases reaction rates relative to the rate of diffusion.

VI. EXPERIMENTAL TECHNIQUES It is well known from the empirical Arrhenius expression that reaction rate constants often increase with reactant temperature. One goal of molecular reaction dynamics is to understand the roles that different forms of reactant energy (translational, electronic, vibrational, and rotational) play in chemical reaction. Also, if a reaction is successful, how is the total available energy channeled into the various available degrees of freedom of the product molecules? Many of the microscopic details of a chemical reaction are related to the nature of the transition state. Most insight into the transition state region has been obtained through experiments focusing on the asymptotic limits of the reaction; i.e., reactants and products. Four general categories of experiments will be discussed here, all focusing on elementary gas-phase reactions. In the first type of experiment, the total cross sections for various chemical reaction channels are measured, usually as a function of collision energy. In the second type, the angular and velocity distributions of products from single reactive encounters are measured. In the third type of experiment, the quantum state distributions (vibrational and rotational) of products are measured using spectroscopic techniques. In this latter approach, velocity distributions may also be obtained using Doppler or velocity imaging methods. Finally, a relatively recent development is “transition state spectroscopy,” which focuses directly on the transition state itself, usually through spectroscopic measurements.

A. Cross Section Measurements The reaction cross section may be determined experimentally as a function of collision energy using a variety of methods. For the reaction H + D2 → HD + D, the reaction has been carried out in a flow cell containing mixtures of D2 and a stable H atom precursor such as HBr. The H atoms are produced by photolysis of HBr at various UV wavelengths. Reaction cross sections are determined by measuring the concentration of products relative to reactants directly. In the present case, this involves monitoring the relative concentrations of H and D atoms (i.e., reactants and products) as a function of time following photodissociation of reactant precursor. Both H and D atoms may be detected by laser induced fluorescence (LIF) excitation spectroscopy near 121 nm (Lyman-α). By choosing different photolysis wavelengths and H atom precursor molecules, reaction rate constants may be determined for different collision energies. In this case, reaction cross sections are determined directly from Eq. (8), since the relative velocity is well defined using photolytic reactants. In Fig. 9, experimental data from several different laboratories are shown as solid points surrounded by rectangles, and theoretical values are connected by a solid line. The reaction cross section is zero below 0.4 eV due to the presence of a potential energy barrier for reaction, as discussed in detail in Section VI.B. The reaction cross section increases with increasing energy above threshold, with behavior qualitatively similar to that predicted by Eq. (15).

FIGURE 9 Cross section for H + D2 → HD + D reaction vs. collision energy. Solid points surrounded by rectangles are experimental data and open points connected by solid line is theoretical calculation. (From Gerlach-Meyer, U., Kleinermanns, K., Linnebach, E., and Wolfrum, J. (1987). J. Chem. Phys. 86, 3047–3048.)

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708 In some cases, reaction may lead to more than one type of chemical product. For example, the reaction of chromium cations, Cr+ , with methane, CH4 , may lead + to production of CrCH+ 2 + H2 or CrH + CH3 . In order to study ion-neutral reactions such as these, a beam of mass-selected reactant ions, e.g., Cr+ , is accelerated to a well-defined laboratory kinetic energy. The ion beam encounters target molecules held in a gas cell, where bimolecular reaction occurs. The product ions are then extracted from the reaction volume, mass selected, and counted. A wide variety of reactions have been studied using such techniques. In Fig. 10, the cross sections for reactions of ground state Cr+ are shown using solid symbols. Due to the endoergicity of reaction, the formation of + CrCH+ 2 only occurs at energies above ∼2.3 eV, and CrH formation has a threshold just under 3.0 eV. Reactions of an electronically excited state of Cr+ with CH4 , on the other hand, have a large cross section for production of CrCH+ 2 even down to zero collision energy, as indicated by open symbols. Experiments such as this provide insight into the role of electronic state on chemical reactivity. Furthermore, by measuring energetic thresholds for reaction, thermodynamic quantities such as bond dissociation energies may be determined directly. B. Angular and Velocity Resolved Studies Crossed molecular beam reactive scattering facilitates experimental determination of the angular and velocity distributions of chemical products from elementary bimolecular reactions. The technique involves production of two molecular beams containing the reactants, initially moving at right angles relative to one another, in an evacuated

FIGURE 10 Experimental cross sections for Cr+ + CH4 reaction. Solid points denote reaction of ground electronic state Cr+ and open points denote reactions of electronically excited Cr+ . (From Armentrout, P. B. (1991). Science 251, 175–179.)


FIGURE 11 Schematic diagram of crossed molecular beam apparatus. Beams cross at right angles; products are detected by a detector that may be rotated with respect to the reactant beams.

( ε > 20.5 (25◦ C)] solutions at various temperatures. Broken line: limiting law at 25◦ C.

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Electrolyte Solutions, Transport Properties

D. Theory of Transference Numbers of Dilute Solutions The definition of transference numbers, Eqs. (52), entails for 1, 1 electrolytes the relationships λi Ii = , I λi∞ = ∞,

ti = ti∞

(66a)

ti − 0.5 ∞ . = ti∞ − 0.5 ∞ + el

(66b)

and = λ+ + λ− ,

The first is the contribution to the electrophoretic part of conductance, and the second results from the negligibly small space-dependent part of the interionic two-particle force, λel(2) ∼ = 0 (Chen effect). Combining Eqs. (66)–(68) yields the transferencenumber equation in the form

(66c)

where λi and are the single-ion and the electrolyte conductivities and λi∞ and ∞ the corresponding limiting values. Single-ion conductivities contain relaxation (λirel = λi∞ × E/E) and electrophoretic (λiel ) contributions, just as electrolyte conductivity does; E is the change in the electric field caused by the ion charges in the solution:

E λi = λel + λi∞ 1 + . (67) E The electrophoretic effects on anions and cations are equal. The electrophoretic contribution consists of two parts:

E el el(1) λ =λ 1+ E

E 2λel el(2) +λ 1+ (68) + ∞ . E

(69)

Equation (69) indicates that the relaxation effect does not influence transference numbers of symmetric electrolytes. Using Eq. (60b) for el yields the relationships √

ti − 0.5 S2 αc −1 ∞ ∞ − + Bαc, (70a) = ti∞ − 0.5 1 + κR KA =

1−α 1 , α 2 c y±2

and

(70b)

κq . (70c) 1 + κR In Eq. (70a) the coefficient B is not completely calculable; B results from the terms of improved theories that are neglected in Eq. (70a) and might include the consequence of unknown corrections needed in the experiments; S2 and κ R are given in Table I. Figure 5 shows the features of transference numbers. The symmetry of Eq. (70a) is obvious. If t+∞ > 0.5, the transference numbers increase with increasing concentration and decreasing temperature, and vice versa if t+∞ < 0.5. Transference-number measurement yield the − ln y± =

FIGURE 5 Temperature dependence and concentration dependence of cationic transference numbers of methanol solutions of Me4 NSCN ( ❤) and KSCN (♦) at various temperatures. The full lines are the computer plots according to Eqs. (70). [From Barthel, J., Stroder, ¨ U., Iberl, L., and Hammer, H. (1982). Ber. Bunsenges. Phys. Chem. 86, 636–645.]

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quantities ti∞ by extrapolation as the only characteristic quantities of the electrolyte solution; the small concentration dependence of the transference numbers for symmetric electrolytes does not permit the determination of association constants, in contrast with conductance measurements. The most important application of transference-number measurements for symmetric electrolytes is the determination of precise single-ion conductances. Transference numbers were also successfully applied in investigations of ion aggregates, which contribute to ion mobility by their charges, such as ion pairs of nonsymmetric electrolytes and triple ions of symmetric electrolytes. E. Triple-Ion and Higher-Ion Aggregate Formation For low-permittivity solutions, the highest concentration for which pairwise additivity of the potential functions is reasonable in MM-level Hamiltonian models is found at very low concentrations, for example, 10−4 M in Fig. 6. Figure 6 shows the dependence on concentration and temperature of the molar conductivity of 1,2dimethoxyethane solutions of LiBF4 from infinite dilution to saturation. The plots of versus c1/2 show a minimum at moderate concentrations and a maximum at high concentrations. Although the minimum is only weakly dependent on temperature, the maximum exhibits a strong displacement. The minimum is a general feature of bilateral triple-ion formation: −→ [C+ A− C+ ]+ ; K + [C+ A− ]0 + C+ ←− (71a) T

and −→ [A− C+ A− ]+ ; A− + [C+ A− ]0 ←−

K T− ,

(71b)

where commonly the two formation constants K T+ and K T− are supposed to be equal. The maximum results from the competition of ion aggregates of various types. The conductivity equation of Fuoss and Kraus, y± c1/2 1 − S(∞ )−3/2 [c(1 − /∞ )]1/2 =

∞ 1/2 KA

+ λ∞ T

KT 1/2 KA

(1 − /∞ )c,

(72)

is the appropriate equation for reproducing the conductivity curve up to concentrations near the conductivity minimum (cmin = 2.14 × 10−2 mol dm−3 at 25◦ C in Fig. 6). In Eq. (72), y± is the mean activity coefficient of the free ions; S the limiting slope (see Table I); λ∞ T the limiting value of the triple ions C+ A− C+ and A− C+ A− ; and K A and K T = K T+ = K T− are the equilibrium constants of ionpair, Eq. (63), and triple-ion, Eqs. (71), formation. At concentrations far below the conductivity minimum (c 10−3 mol dm−3 in Fig. 6), triple-ion formation can be neglected. Data analysis is possible with the help of Eqs. (65), in agreement with pairwise additive potential functions, and yields the values of ∞ in Table III. The ion-pair association constants K A(1) of these plots agree well with the K A(2) determined independently at higher concentrations with the help of Eq. (72), which takes into account both ion-pair and triple-ion formation. No method is known for the determination of the values of λ∞ T ; data analysis yields only the product λ∞ K , in which the quanT T ∞ tity λ∞ is commonly estimated to be 2 /3. Both ion-pair T and triple-ion formation decrease with decreasing temperature in accordance with increasing solvent permittivity. The conductivity equation for electrolytes undergoing unilateral triple-ion formation, Eq. (71a) or Eq. (71b), is given by the relationship 2 (y±2 )2 c(1 − /∞ ) 1 − S(∞ )2 [c (1 − /∞ )]1/2 =

(∞ )2 KT c ∞ ∞ + 2λT − (∞ )2 (1 − /∞ ). KA KA (73)

TABLE III Limiting Conductivities and Ion-Pair and TripleIon Formation Constants of LiBF4 Solutions in 1,2Dimethoxyethane at Various Temperatures from Conductivity Measurements

FIGURE 6 Ion-pair, triple-ion, and higher-ion aggregate formation exemplified by measurements of the equivalent conductivity of LiBF4 in 1,2-dimethoxyethane solutions at 25◦ C and −45◦ C. [From Barthel, J., Gerber, R., and Gores, H.-J. (1984). Ber. Bunsenges. Phys. Chem. 88, 616–622.]

Temperatures [◦ C] ∞ /(S cm2 mol−1 )

−45 46.5

−25

−5

68.8

94.1

15

25

123

139

10−6 K A /(mol−1 dm3 )

0.57

1.7

5.1

14.4

10−6 K A /(mol−1 dm3 ) K T /(mol−1 dm3 )

0.53

1.7

5.0

13.9

23

28

30

(1) (2)

15

19

23

24

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Triple-ion formation is commonly restricted to lowpermittivity solvents (ε < 15), but it is also known in highpermittivity solvents as a consequence of noncoulombic interactions.

IV. VISCOSITY The concentration dependence of the viscosity η of electrolyte solutions up to moderate electrolyte concentrations is commonly represented by the equation √ η(c) − 1 : A c + Bc. ηs

(74)

In Eq. (74), ηs is the viscosity of the solvent, η(c) is the viscosity of electrolyte concentration c, and A and B are constants. Coefficient A is available from the theoretical z+ z− limiting law for Cν+ Aν− electrolytes:

√ 1 p¯ 2 1 − p ∗ e02 NA η(c) − η = , κ p¯ + 4 √ 2 2 p¯ 1 + p ∗ 480π ν+ z + ν− z − (75) where 3 3 z+ z− p¯ = ν+ ∞ − ν− ∞ ; λ+ λ−

p¯ = ν+ 2

2 z+ ∞ λ+

2

+ ν−

2 z− ∞ λ−

2

(76a,b) and p∗ =

∞ ν+ z + λ∞ 1 + − ν − z − λ− . 2 2 ∞ ν+ z + + ν− z − λ∞ + /z + − λ− /z −

(76c)

κ is given by Eqs. (22). A rough estimation based on the approximation ∞ z + λ∞ + = |z − |λ− shows that the relative viscosity increases proportionally to the ratio of the radii of ion and ion cloud. Coefficient B is an empirical quantity which reflects the effects of ionic size, solvent structure, and ion–solvent interactions. Commonly it is split into ionic contributions B = ν+ B+ + ν− B− ,

(77)

which are related to crystallographic radii and structure parameters of ion–solvent interactions known from molar volumes.

V. TRANSPORT EQUATIONS FOR CONCENTRATED ELECTROLYTE SOLUTIONS Today, empirical transport equations of concentrated electrolyte solutions are available, as well as equations which

are rigorous statistical mechanical approaches. Only a few of those that have attracted the interest of applied research and engineering science are treated here. Three classes of transport equations can be found in the literature: molten salt approaches, empirical extensions of the equations for dilute solutions, and empirical equations just for fitting measured data. Molten salt approaches such as the Vogel-FulcherTamman (VFT) equation have been used repeatedly for analyzing the temperature dependence of transport properties W (T ) such as diffusion, conductance, and fluidity, or of relaxation processes: B . (78) W (T ) = A exp − R(T − T 0 ) Equation (78) can be deduced from the equilibrium distribution of an isothermal, isobaric ensemble of cooperatively rearranging domains in the liquid, which can undergo a transition to a new configuration without configurational change at and outside its boundary. At the glass transition temperature T 0 of the system the configurational part of entropy vanishes. It is assumed that the transition of the supercooled melt to the glass is a type of second-order transition to obtain Eq. (78), where B is a temperature-independent energy term of transport, R is the gas constant, and A is a temperature-independent quantity, depending on the composition of the solution. Equations based on empirical extensions of the equations for dilute solutions use the fact that the viscosity of the system is the most important effect on the transport properties and introduce appropriate viscosity corrections. A. Empirical Equations The representation of physical properties of electrolyte solutions by the use of fitting equations is commonly executed with polynomials of concentration, temperature, pressure, and so forth, or with mathematical functions known for the appropriate representation of the shape of the experimentally determined curves. One of the most useful expressions of this type is given by Amis and Casteel for the specific conductivity of concentrated solutions: κ m m−µ 2 = b(m − µ) − a . (79) κmax µ µ It makes use of four parameters (κmax , µ, a, b) for the representation of the measured data over wide concentration ranges and reproduces well the maximum of specific conductivity κmax and its position µ (see Figs. 7 and 9); a and b have no physical meaning. The specific conductivity of concentrated electrolyte solutions and its temperature dependence are of crucial

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For data analysis of specific conductivities based on the VFT equation, Eq. (78), the glass transition temperature of the electrolyte solution at molality m, T 0 (m), is assumed to be T 0 (m) = T 0 (0) + am + bm 2 , T 0 (0) = lim T 0 (m). m→0

(81)

In Eq. (81), T 0 (0) is the glass transition temperature of the pure solvent. The temperature dependence of specific conductivity at molality m, κm (T ), is then given by the relationship Bm(κ) (κ) κm (T ) = Am exp − . (82) R[T − T 0 (m)]

FIGURE 7 Specific conductivity of LiBF4 in propylene carbonate at various temperatures from dilute solutions to saturation. Points: experimental data; lines; Eq. (79). [From Barthel, J. (1985). Pure and Appl. Chem. 57, 355–367.]

A similar equation is obtained for viscosities and other transport properties. The glass transition temperature of the pure solvent T 0 (0), obtained by extrapolation (m → 0), is found to be independent of the solutes in a given solvent and equal to that from viscosity measurements, which shows that the glass transition temperature is the appropriate reference temperature for transport processes in the liquid state. Using this result in Eq. (82) yields the further important

interest in technology. Figure 7 shows the features of specific conductivity. The maximum of specific conductivity κmax is a feature of every electrolyte solution permitting sufficiently high solubility of the solute. It follows from the competition between the increase dc of the electrolyte concentration and the decrease d of the ion mobility when the electrolyte concentration increases. The variation dκ of the specific conductivity is given by the relationship dκ = dc + cd.

(80)

Equation (80), following from the definition of molar conductivity ∝ κ/c [Eq. (55)], shows that maximum specific conductivity κmax is attained at a concentration µ, at which dκ equals zero. The maximum of specific conductivity κmax and the concentration µ at which it is attained are correlated. Figure 8 indicates linear correlations as observed for propylene carbonate solutions at various temperatures. This correlation is due to the existence of an energy barrier depending on the temperature and on the solvent parameters, particularly viscosity. At concentration µ, the electrolyte shows an activation energy for the transport process equivalent to that of the barrier. The concentration µ at maximum specific conductivity decreases with decreasing temperature, cf. Fig. 8, which proves that viscosity is the most important conductancedetermining factor.

FIGURE 8 Linear correlation of maximum specific conductivity κmax and corresponding values of µ exemplified by propylene carbonate solutions of various 1,1 electrolytes at 25◦ C, −5◦ C, and −35◦ C. (1) LiBF4 , (2) LiClO4 , (3) Bu4 NPF6 , (4) KPF6 , (5) Pr4 NPF6 , and (6) Et4 NPF6 . [From Barthel, J., and Gores, H.-J. (1985). Pure and Appl. Chem. 57, 1071–1082.]

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λi =

λi∞

δµel 1 + ∞i µi

δE 1+ E

(83)

after the second-order effects are neglected. The first-order approximation to the electrophoretic effect can be inferred from the electrophoretic velocity correction δµiel . The MSA pair distribution functions permit an easy extension of Henry’s law of electrophoretic mobility, cf. Eq. (60b), δµiel kT , ∞ =− ∞ µi 3π ηDi 1 + σ

FIGURE 9 Specific conductivity κ versus molarity m at 25◦ C for Bu4 NClO4 in mixed solvents propylene carbonate–acetonitrile (mole fraction of acetonitrile is indicated on the right side of each curve). Points: measured data; full lines: Eq (79); dashed lines: MSA equation.

feature that the temperature coefficients of conductivity for all electrolyte solutions at infinite dilution in a given solvent, and of viscosity, are equal at every temperature; that is, infinitely dilute solutions are corresponding states in terms of transport energies. Suffice it to note that the values extrapolated toward zero concentration of transport energies from highly concentrated solutions based on the VFT equation equal those obtained from the conductivity equations based on MM-level Hamiltonian models. B. Statistical Mechanical Approaches

(84)

where µi∞ is the electrophoretic velocity, Di∞ is the ionic diffusion coefficient at infinite dilution [deducible from λi∞ ; see Eq. (25)], and is Blum’s screening parameter of the MSA: κ 2 = . (85) (1 + σ ) In Eq. (84) σ is an average ionic diameter calculated from the ionic diameters σi and the ion concentrations ci : 2 i z i σi ci σ = . (86) z i2 ci κ is Debye’s parameter [Eqs. (22)]. The first-order relaxation effect is obtained by the solution of the continuity equation at this level, which yields β 2 e02 |z i z j | δE 1 =− E 4π ε0 ε 6kT σ (1 + σ )2 ×

β2

1 − exp(−2βσ ) , + 2β + 2 2 [1 − exp(−βσ )]

(87)

where 2 ∞ 2 ∞ e02 NA ci z i Di + c j z j D j . ε0 εkT Di∞ + D ∞ j

Extended laws are available for the variation with concentration of the transport coefficients of strong and associated electrolyte solutions at moderate to high concentrations. Like the CM calculations, this work is based on the Fuoss-Onsager transport theory. The use of MSA pair distribution functions leads to analytical expressions. Ion association can be introduced with the help of the chemical method. A simplified version of the equations, by taking average ionic diameters, reduces the complexity of the original formulas without really reducing the accuracy of the description and is therefore recommendable for practical use for up to 1-M solutions.

Association can be included in the extended theory with the help of the CM, which yields the association constant given by Eq. (65b). For the calculation of the portion (1 − α) of ion pairs, Eq. (65b) is subjected to an iteration process beginning with yIP = y± = 1:

C. Conductivity Equations

(89) The single-ion and the ion-pair activity coefficients of MSA are made up by an electrostatic part and a hardsphere contribution:

For a completely dissociated electrolyte, the appropriate equivalent conductivity expression, = λi , follows from the ionic conductivities

β2 =

(88)

1/2 2 yIP 1 4yIP yIP 1−α =1+ − + . 2 K A2 c2 y±4 2K A cy±2 4K A cy±2

ln yi = ln yiel + ln yihs ;

i = +, −

or

IP.

(90)

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The electrostatic part yiel is given by the expression ln yiel = −

1 e02 z i2 , 4πε0 ε kT 1 + σ

(91)

el that is, yIP = 0 for a symmetric electrolyte. For the hardsphere contribution, only the ratio hs 2 y± 1 − 0.5ξ πσ 3 NA cj = ; ξ = (92) hs (1 − ξ )3 6 yIP

is needed for the calculation of the association constant. The data analysis with the help of Eqs. (83), (84), (87), and (88) where association constants are used with MSA activity coefficients [Eqs. (90), (91), (92)] yields a good reproduction of experimental data up to molar concentrations. As an example, Fig. 9 shows the conductivity of Bu4 NClO4 in the mixed solvent system acetonitrile– propylene carbonate. Comparison is made of measured data with MSA and the Amis-Casteel equation, which both exactly reproduce the conductivity maximum at every solvent composition. D. Self-Diffusion The self-diffusion coefficient is computed from the interdiffusion coefficient of two isotropically different species or from tracer diffusion, where a labeled ion in tracer amount diffuses in a large excess of another electrolyte of the same ion. The diffusion coefficient Di of the species Xi is given by the expression

δki , (93) Di = Di∞ 1 + ki where ki is the diffusive force acting on an ion Xi and δki is the relaxation force. In the framework of MSA, the first-order contribution to the relaxation effect for the ionic species Xi reads 2 z i2 e02 κ 2 − κdif δki 1 , =− 2 ki 4π ε0 ε 3k B T (1 + σ )2 κdif + 2(1 + σ )

(94) where 2 κdif =

e02 NA cn z n2 Dn∞ . ε0 εkT n Di∞ + Dn∞

is given by Eq. (85) and κ by Eqs. (22); σ is the average diameter, Eq. (86). E. Other Approaches Altenberger and Friedman have given a conductance equation based on an HNC approach which is also valid up to concentrations of about 1 mol dm−3 .

Ebeling and Kraeft developed a statistical theory for ion–dipole solutions (physical model) with the aim of taking into account ion–solvent interactions. Computer simulations such as molecular dynamics (MD) and Brownian dynamics (BD) permit the study of transport properties. Self-diffusion coefficients can easily be obtained by differentiation of mean-square displacements or by integration of the velocity self-correlation functions of the ion. In contrast, the evaluation of conductivity by means of cross-correlation functions is cumbersome and computer-time-consuming and can only scarcely be executed. The advantage of computer simulations is the possibility of obtaining transport data that cannot or can only barely be measured. It is possible in this way to simulate diffusion coefficients of solvent molecules in the ionic solvation shells and to compare them with those of the bulk solvent molecules and with those of the ions, or to study transport coefficients at different time scales.

SEE ALSO THE FOLLOWING ARTICLES CHEMICAL THERMODYNAMICS • ELECTROCHEMISTRY • ELECTROLYTE SOLUTIONS, THERMODYNAMICS • ELECTROPHORESIS • STATISTICAL MECHANICS

BIBLIOGRAPHY Barthel, J., Krienke, H., and Kunz, W. (1998). “Physical Chemistry of Electrolyte Solutions—Modern Aspects,” Steinkopf, Darmstadt/ Springer-Verlag, New York. Bernard, O., et al. (1992). J. Phys. Chem. 96, 398–403, 3833–3840. Bernard, O., Turq, P., and Blum, L. (1991). J. Phys. Chem. 95, 9508– 9513. Bockris, J. O., and Reddy, A. K. N. (1998). “Modern Electrochemistry,” Vol. 1, 2nd ed., Plenum, New York, London. Covington, A. K., and Dickinson, T., eds. (1973). “Physical Chemistry of Organic Solvent Systems,” Plenum, New York. Falkenhagen, H. (1971). “Theorie der Elektrolyte,” 2nd ed., Hirzel, Leipzig (Engl. ed., 1952). Friedman, H. L. (1985). “A Course in Statistical Mechanics,” Prentice Hall, Englewood Cliffs, N.J. Hansen, J. P., and McDonald, I. R. (1976). “Theory of Simple Liquids,” Academic Press, London. Justice, J. C. (1983). Conductance of electrolyte solutions. In “Comprehensive Treatise of Electrochemistry,” Vol. 5 (B. E. Conway, J. O. Bockris, and E. Yeager, eds.), pp. 233–337, Plenum, New York. Resibois, P. M. V. (1968). “Electrolyte Theory: An Elementary Introduction to a Microscopic Approach,” Harper & Row, New York. Robinson, R. A., and Stokes, R. H. (1970). “Electrolyte Solutions,” 2nd rev. ed., Butterworth, London. Spiro, M. (1984). Conductance and transference determinations. In “Physical Methods of Chemistry,” 5th ed. (B. W. Rossiter, J. F. Hamilton, eds.), Wiley (Interscience), New York. Turq, P., Barthel, J., and Chemla, M. (1987), “Transport, Relaxation and Kinetic Processes in Electrolyte Solutions,” Lecture Notes in Chemistry, Springer-Verlag, Heidelberg.

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Energy Transfer, Intramolecular Paul W. Brumer University of Toronto

I. II. III. IV. V. VI. VII.

Introduction Qualitative Dynamics Dynamics: Theory Statistical Approximations and Dynamics Experimental Studies Control of Dynamics Summary

GLOSSARY Adiabatic Process during which there is no change in the electronic configuration of the molecule. Constant of the motion Property of a system, expressible as a smooth function of the coordinates and momenta, whose numerical value remains unchanged during the course of the system dynamics. Ergodic Classical mechanical system in which a trajectory uniformly covers a specific surface in phase space. The physics literature utilizes this term to imply uniform coverage of the surface in phase space defined by fixed total energy. Integrable Also termed regular or quasiperiodic. Classical mechanical system characterized by the existence of a set of independent constants of the motion equal in number to the number of degrees of freedom of the system. Specific attributes of such systems are discussed in the text. Mixed state State of a system in which there is some information missing relative to a complete specification of the state.

Mixing Classical mechanical system that is ergodic and possesses additional properties associated with relaxation. Pure state State of a system in which all information about the state is known and specified. Relaxation Tendency of a system to evolve from a specific time-dependent initial state to a time-invariant final state. Statistical Qualitative term signifying models or dynamics with characteristics of ergodic and mixing behavior. Unimolecular decay Process in which an energized molecule breaks up into smaller molecular constituents. The molecular analog of nuclear fission.

INTRAMOLECULAR dynamics is the time evolution of rotational, vibrational, and electronic degrees of freedom of isolated individual molecules, that is, molecules in a collision-free environment. As in any mechanical system, one can consider the time evolution of any of a variety of system properties. Intramolecular energy transfer focuses

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478 on the dynamical flow of energy among subcomponents of the molecule as the primary property of interest.

I. INTRODUCTION Naturally occurring systems in the gaseous or liquid phase at typical temperatures are composed of vast numbers of atoms and molecules in constant motion. The well-defined system temperature is a reflection of the stored energy content, both in the form of intermolecular features such as molecule–molecule interactions and translational motion and in the form of motion internal to individual molecules. The latter, the motion of isolated molecules, is termed intramolecular dynamics. Included within this definition are both dynamics at energies below the dissociation energy of the molecule, in which case it remains perpetually bound, and dynamics at energies above dissociation, in which case the molecule can break up into different chemical products. Intramolecular energy transfer is the subset of intramolecular dynamics in which the focus of attention is on the flow of energy within the isolated molecule. As the simplest of models, one may imagine a linear molecule A B C as being three mass points coupled by springs. Initiating an oscillation of the model by stretching the A B spring and subsequently observing the time-dependent alteration of lengths of the A B and B C springs correspond to a simple model of an experiment on vibrational energy transfer. Indeed, modern experimental techniques allow for the creation of collision-free environments in which such isolated intramolecular dynamics may be studied and possibly externally influenced. The isolated molecule, comprised of a set of N atoms bound by interatomic forces, is a complex physical system. It possesses 3N degrees of freedom related to nuclear motion, three being associated with center-of-mass translation, three with rotation (or two if the molecule is linear), and the vast remainder with vibration. The molecule also has electronic degrees of freedom associated with the configuration of its electrons. Additional degrees of freedom, related to internal nuclear composition, are not readily altered at the energies of interest in molecular chemistry and physics and can therefore be neglected. Thus, even the simplest description of the dynamics of a typical small molecule such as benzene (C6 H6 ), which regards it as being composed of 12 atoms, involves the complex motion of 33 interacting degrees of freedom. It is convenient to visualize intramolecular dynamics in terms of three steps, not necessarily independent. First, the molecule is prepared in a time-dependent state by any of a variety of means (e.g., collisions, chemical reaction, laser excitation). Second, the molecule evolves in time in accor-

Energy Transfer, Intramolecular

dance with quantum mechanics. Third, the time-evolved state of the system is analyzed, that is, measured. Explicit emphasis in a measurement (and sometimes preparation) on the flow of energy among subcomponents of the molecule (e.g., chemical bonds or rotational degrees of freedom) constitutes the study of intramolecular energy transfer. It is advantageous to distinguish two qualitatively different types of intramolecular energy flow, which we shall term reversible and irreversible. In the former, energy flows from one part of the molecule to another, but then subsequently returns, reforming the initial state. That is, the system shows no long-term trend toward a final redistribution of energy among subcomponents of the molecule. In the latter, there is a transfer of energy within the molecule, with a general trend toward a stationary final state. The latter behavior is that associated with relaxation, or statistical, dynamics and has historically been assumed to occur in highly excited molecules. Conditions under which molecular systems appear to display reversible vs irreversible energy transfer are discussed in detail later. Intramolecular dynamics and intramolecular energy transfer have been, and continue to be, areas of intense scientific interest. Such interest falls into two categories, loosely termed “practical” and “fundamental.” From the practical viewpoint, note that the outcome of a molecular process is heavily linked to the flow of energy in the molecular participants. That is, an understanding of intramolecular energy transfer proves central to the interpretation of chemical processes and their dependence on system conditions. For example, unimolecular decay, in which an isolated energized molecule dissociates into chemical products (e.g., ABC → A + BC, where ABC, A and BC are arbitrary molecules), occurs via the concentration of sufficient energy in the A BC bond. Further practical interest arises from important developments in laser technology that permit the introduction of energy into molecules in a variety of controlled ways. This opens the possibility of externally influencing the outcome of a molecular process by varying the initial mechanism of preparation (e.g., producing AB + C, rather than A + BC, from ABC by judicious preparation of the initial state). Indeed, theoretical developments over the past three years have led to several proposals for controlling chemical reactions in this manner (see Section VI). From a fundamental viewpoint, intramolecular energy transfer links to three basic scientific issues: (1) reversible vs relaxation phenomena, (2) quantum vs classical chaos, and (3) quantum/classical correspondence. These are briefly introduced in Section II, where it becomes clear that intramolecular energy-transfer studies provide a useful laboratory for the study of fundamental questions in these areas.


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Experimental and theoretical studies of intramolecular energy transfer and intramolecular dynamics have a long history. This article, designed as an introduction rather than as a survey, focuses on recent directions in this field of research. Of specific interest are insights gained into general rules for understanding and measuring intramolecular dynamics. For this reason we cite only a few sample computations to illustrate relevant general features and provide only a brief qualitative discussion of experimental methods. References to more historical interests in the field are provided at the end of this article. The organization of this article is as follows. Section II is designed to provide qualitative insight into intramolecular energy transfer via two subsections. The first, Section II.A, discusses selected computational results on two molecular models, and the second, Section II.B, qualitatively introduces the fundamental problems alluded to above. The reader who is interested in a qualitative picture is urged to first focus on these sections. Section III contains a description of isolated molecule dynamics from both the quantum and the classical viewpoints. Emphasis here is on several general features of classical and quantum intramolecular dynamics. Finally, two brief sections, Sections IV and V, discuss statistical approximations to intramolecular energy transfer and the nature of modern experiments designed to probe molecular motion. Space limitations, coupled with the author’s intention to provide a useful introductory treatment, have led to restrictions on the material that can be covered. Thus, we focus throughout this article on adiabatic processes, that is, dynamics that take place without change in the electronic configuration of the molecule. When this is not the case, a remark to this effect is made. In addition, we assume throughout that the radiation field, be it associated with radiative absorption or emission, is sufficiently weak to be treatable as a perturbation. Further, the field of intramolecular dynamics is replete with model approaches that, albeit reasonable, have not been justified either theoretically or experimentally—the latter due principally to technological limitations. The modern focus on accurate dynamics is emphasized in this article, with the consequence that such simple models are necessarily slighted.

separate steps. In the first step, the forces between the atoms in the molecule are determined or modeled, while in the second step one considers the dynamics determined by these forces. This dynamics is done either quantum mechanically, which is correct but difficult, or via classical mechanics, which is often a good approximation to the quantum result. In either case, the forces describing the dynamics are sufficiently complex to necessitate numerical computer solutions. As an introduction to the nature of intramolecular dynamics and to issues of interest in this area, we consider two examples. A. Two Model Calculations As a first example, consider nuclear motion in a fouratom system, A B C D. Two qualitatively different energy ranges are possible. In the first, the system is provided with sufficient energy to induce vibrational and rotational motion but insufficient energy to break any of the bonds. This is bound-state intramolecular dynamics. In the second regime, there is sufficient energy to allow molecular dissociation to one or more of the molecular products (e.g., A B + C D). Typically, each of the interatomic bonds will have a different dissociation energy, the energy required to break the bond. Thus, several dissociation “channels” are possible, such as A B C D → A B+C D

E1

A B C D → A+B C D

E2

A B C D → A+B+C+D

E3

where the lowest energy required for each of the particular processes is arbitrarily labeled E 1 , E 2 , etc. Consider the specific case of NaBrKCl for which theoretical, classical dynamics studies are available at energies where two dissociative channels are energetically accessible. (Questions as to the validity of the classical picture are relegated to later sections.) This system possesses attractive forces between the atoms such that the bound NaBrKCl species lies at an energy of approximately 40 kcal/mol below NaBr + KCl or NaCl + KBr. Specifically, consider the case where energized NaBrKCl is formed by the collision NaBr + KCl → NaBrKCl

II. QUALITATIVE DYNAMICS Information regarding intramolecular dynamics is available from three sources: experimental studies on specific molecular systems, theoretical computations on specific systems or models, and formal studies of typical (“generic”) systems. At present, the latter two provide considerably greater detail than the first and involve two

with sufficient energy to dissociate as NaBrKCl → NaCl + KBr NaBrKCl → NaBr + KCl. The initial collision between NaBr and KCl is here regarded as the preparatory step to the subsequent intramolecular dynamics and decay of the intermediate


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FIGURE 1 Time dependence of each of the six interatomic distances during a trajectory describing the collision of NaBr with KCl. System energy is E = 0.0912 a.u. [From Brumer, P. (1972). Ph.D. dissertation, Harvard University.]

NaBrKCl species. An analogy to the collision, bound molecule dynamics, and subsequent decay is a system of two balls, attached by a spring, colliding with two other such balls on a billiard table containing a deep hole in the center. The springs are breakable and endowed with the ability to exchange between pairs of particles. The basic dynamics in classical mechanics is embodied in trajectories, that is, the dynamics following precise specification of the system initial conditions. Comparison with experiment then involves averaging results over a set of trajectories consistent with the specific experimental conditions. Two such trajectories are considered in Figs. 1 and 2, where we show the time dependence of the distances between all the atoms during the collision. In one case the initial conditions lead to a long-lived intermediate, and in the other case they lead to a short-lived species. Consider the shorter trajectory (Fig. 1) first. A careful examination of the figure indicates the initial oscillations of the bound NaBr and KCl, with the other atomic distances shrinking in time as the two diatomics approach one another. The collision between them occurs at approximately 12,000 atomic time units (denoted atu, where an atomic time unit is 2.4 × 10−17 sec) and is promptly followed by decay to NaCl + KBr. Thus, for these particular initial conditions, the intermediate species NaClKBr is only a fleeting phase in the collision. Results in Fig. 2 are in sharp contrast, showing a long-lived intermediate that displays complex dynamics in a bound energized molecule.


Here, the system forms a bound four-body molecule at approximately 38,000 atu that persists until t = 240,000 atu. During this time the various bond distances go through a variety of values between 4 and 11 Bohr radii, a sort of vibrant dance of four bound particles. Decay follows thereafter to the product diatomics. In a classical picture, knowledge of the distances and momenta of the atoms as a function of time permits complete knowledge of all properties. Figure 3 shows, for example, the calculated energies in each of the alkali–halide bonds during the course of the dynamics for the same collision as seen in Fig. 2. Figure 4 contains an analogous picture of the rotational energy of the four-atom system. The dynamics is clearly marked by extensive energy exchange between the bonds and among rotation and vibration. These examples provide a picture of the complexity of individual trajectories in bound intramolecular dynamics. Such a trajectory emerges from a precise specification of initial particle momenta and coordinates. A comparison with real phenomena requires, however, averaging over all initial conditions not precisely specified in the given experiment. For example, the experiment may only have initially fixed the translational and internal energies of the colliding partners. Figure 5 provides a typical example of the result of averaging over a set of trajectories where the trajectory in Fig. 2 is one participant. Here, we show a typical measurable in the decay of NaClKBr, that is, the probability of independently finding the products NaCl and KBr with particular vibrational energies. Note that the results are simpler than the complex underlying trajectories. This is a result of both the comparatively simpler question being asked and the averaging implicit in the computation. The details of the dynamics ongoing during the course of the collision also often simplify as a result of averaging over a set of trajectories. Consider, for example, Fig. 6, which shows the average vibrational energy in each of the four alkali–halide bonds during the collision; in this case the calculation only shows trajectories prior to dissociation. The results are to be compared to Fig. 3, although the overall energies in the two figures are somewhat different. One sees that each of the bond energies changes rapidly over the initial time period but then tends to level out. This is an example of apparent intramolecular energy relaxation during the course of the dynamics. That is, the system seems to reach, at least with respect to these variables, a relatively invariant state reminiscent of the behavior of macroscopic systems relaxing to equilibrium. The observation of relaxation behavior is, of course, not ubiquitous and is dependent on gross system conditions (e.g., total energy, total angular momentum, etc.). Understanding conditions under which intramolecular dynamics may be approximated by a statistical model is indeed one of the


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FIGURE 2 Similar to Fig. 1 but at a lower energy, E = 0.0227 a.u. [From Brumer, P. (1972). Ph.D. dissertation, Harvard University.]

main themes in intramolecular energy transfer. Such statistical models have both advantages and disadvantages. They are advantageous in that they considerably simplify the description of the dynamics. They are disadvantageous in that they imply that the final state of the system is relatively insensitive to the initial state. That is, the outcome of a chemical event is not readily influenced by altering initial conditions. The example discussed here displays a number of relevant features. Clearly noticeable is the complexity of individual trajectories modeling long-lived dynamics, as well as the possibility of simplifications that result if a statistical description that includes relaxation applies. The participation of rotations as well as vibrations in the dynamics is also evident. In addition, it makes clear the important

role of the relative rates of intramolecular energy transfer and other competitive processes. That is, the degree of intramolecular energy flow within the molecule depends on the length of time the system exists as a bound entity, as well as on some (as yet undefined) “rate of intramolecular energy flow.” In cases where competitive processes such as dissociation (or the ever-present radiative emission) are possible, effective intramolecular energy flow requires a larger rate of energy transfer than of competitive processes. (Note, however, that in the case of competitive dissociative or other nuclear rearrangement processes, both the energy transfer and the competitive process are governed by the same set of underlying dynamical equations for the nuclear motion. This leads to a natural difficulty associated with attempting to partition


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FIGURE 3 Vibrational energy in each of the four alkali–halide bonds, as a function of time, for the trajectory shown in Fig. 2. [From Brumer, P. (1972). Ph.D. dissertation, Harvard University.]

the dynamics into distinct parts labeled intramolecular dynamics and dissociation.) The computation described above is completely classical: the nuclear motion is assumed to be well described by Newton’s equations. The extent to which classical mechanics provides a useful description of intramolecular energy flow is another focus of current research in this area. As one example of the validity of classical mechanics, consider the bound-state dynamics of a three-atom system confined to a line, that is, A B C. Computations on the case where the A B and B C bonds are anharmonic have been performed using both classical and quantum

mechanics, where the initial state is a mixed state (see Section III.C). One useful measure of system dynamics is the probability of the system returning to the state from which it started. Figure 7 shows the time dependence of this probability for one case, where the quantum and classical results are seen to be in excellent agreement. In sharp contrast is the comparison shown in Fig. 8, which corresponds to the probability of return to another, higher energy initial mixed state where classical–quantum disagreement is substantial. The effect has also been seen for the higher energy decay of A B C to AB + C as shown in Figs. 9 and 10. Specifically, the probability of the

FIGURE 4 Total rotational energy associated with the NaClKBr system during the trajectory shown in Fig. 2. [From Brumer, P. (1972). Ph.D. dissertation, Harvard University.]


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FIGURE 5 Probability of producing a specific vibrational state of NaCl or of KBr from collisions of NaBr with KCl at E = 0.0411 a.u. [From Wardlaw, D. (1982). Ph.D. dissertation, University of Toronto.]

system remaining in its initial state is shown in Figs. 9 and 10 for dynamics initiated in two different mixed states. Once again, agreement is achieved between classical and quantum mechanics in one case, but there is substantial

disagreement in the other, with the classical being more statistical. The origin of this difference lies, in this case, in the existence of so-called quantum trapping states, which lead to classical results that behave more statistically than the quantum. Such states occur when there is a very asymmetric distribution of energy among the bonds in the molecule. Although here classical mechanics is more statistical than quantum mechanics, this is not always the case. The computations described above are useful in that they provide some qualitative insight into the nature of intramolecular dynamics and energy flow in particular systems. They are, in essence, theoretical experiments on given systems in that they provide only hints of the general rules that govern rates, nature, and degree of intramolecular energy interchange. In the sections that follow, we describe the current state of understanding on the general principles that underlie these processes. Prior to doing so, we emphasize, in the next section, some of the fundamental issues related to intramolecular energy transfer that have been alluded to above. B. Fundamental Issues: Qualitative Overview The goal of science is to provide a qualitative and quantitative description of natural phenomena. Such a description

FIGURE 6 Average energy in each of the four alkali–halide bonds as a function of time. Here, the average values are obtained from a set of NaBr + KCI trajectories. “Error bars” show the range of values associated with the set of trajectories incorporated in the calculation. [From Brumer, P. (1972). Ph.D. dissertation, Harvard University.]


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FIGURE 7 Classical (dashed) and quantum (solid) probability, as a function of time, of a coupled Morse oscillator system remaining in its original state. [From Kay, K. (1980). J. Chem. Phys. 72, 5955.]

is most useful if it is as simple as possible. For example, there is little reason to invoke relativistic quantum mechanics to describe planetary motion; classical mechanics will normally suffice. Furthermore, such a scientific description is most useful if it is computationally tractable. Thus, for example, thermodynamics often provides a more useful route to disallowing certain processes in a bulk system than does a full dynamics calculation involving the underlying Avogadro number of molecules. It is in this spirit that the utility of two approximate descriptions—classical mechanics in lieu of quantum mechanics, and statistical approaches in lieu of full dynamics calculations—is a central theme in contemporary intramolecular dynamics. Indeed, they intertwine in an interesting fashion. Consider first the essential difference between a dynamics description and a statistical description of a system. By

FIGURE 8 Same as Fig. 7, but for a different initial state. [From Kay, K. (1980). J. Chem. Phys. 72, 5955.]


the former we mean a description based on Newton’s equations in classical mechanics or Schrodinger’s equation in quantum mechanics. By the latter we mean descriptions characteristic of nonequilibrium statistical mechanics, such as the Boltzmann or Fokker–Planck equation. Dynamics (either quantum or classical) is based on a set of basic equations that are time-reversal invariant. This property means that the final state of a process may be time reversed to recover the initial state. It implies that the final state of dynamical evolution contains all the information associated with the initial state. As a consequence, relaxation to a final equilibrium state, independent of the fine details of the initial state, does not occur, and hence, the final state may be a sensitive function of the initial state. Relaxation to equilibrium is, however, a familiar feature in macroscopic systems, and the equations of statistical mechanics are designed to provide a nonmicroscopic description that encompasses the relaxation process. The link between the underlying time-reversible equations of motion and the macroscopic irreversible equations is not well established and has been the subject of extensive, long-standing discussions on the basic equations of nature. Typical questions include the following. Are time-reversible equations more fundamental than statistical relaxation equations, or do they have equal, but independent, roles as models of nature? Is observed relaxation a consequence of coarse graining associated with macroscopic measurements on intrinsically time-reversible systems? Isolated-molecule dynamics is expected to be a sufficiently elementary process to permit observation of microscopic reversibility in the dynamics and, hence, to display a dependence of the outcome of dynamics on initial conditions. This dependence is desirable since the ability to retain information about initial conditions is necessary in order to achieve the technologically desirable goal of externally influencing chemical reactions. However, a great many experiments, perhaps with insufficiently well-characterized preparation and measurement, have indicated that time-irreversible relaxation is a useful model for many intramolecular processes. Thus, isolatedmolecule intramolecular dynamics serves as a laboratory for the study of the inter-relationship between irreversible relaxation behavior in systems that are fundamentally describable by time-reversible equations of motion. It also presents an experimental challenge to prepare sufficiently well-characterized states to observe time reversibility and sensitivity to initial conditions. Two further issues of fundamental importance, that of quantum–classical correspondence and that of “quantum chaos,” are intimately linked to studies in intramolecular dynamics. Extensive theoretical studies, beginning in the early 1960s, showed that a great many phenomena involving the dynamics of atoms and molecules are well


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FIGURE 9 Classical (dashed) and quantum (solid) probability of a coupled Morse–harmonic oscillator system remaining in its initial state. The energy is sufficient to allow dissociation. [From Kay, K. (1984). J. Chem. Phys. 80, 4973.]

described by classical mechanics. That is, atomic and molecular dynamics are at the borderline between classical and quantum mechanics, in that dynamic phenomena in molecules are often well approximated by classical dynamics. This is not always the case, as shown in the example in Section II.A, and so studies in intramolecular dynamics provide insight into the utility of classical mechanics as an approximation to quantum mechanics. Phrased in this manner, this appears to be a question of relative accuracy, with the expectation that the process is qualitatively similar in both mechanics. Recently, however, major qualitative distinctions between classical and quantum mechanics have been noted. Specifically, it is now known that even small classical mechanical systems (e.g., two

degrees of freedom) can display highly statistical behavior, termed chaos. One can show formally, however (as will be described further below), that such chaotic behavior is not possible in bound-state quantum mechanics. Further, typical semiclassical schemes that base quantization on the classical system motion (e.g., Einstein–Brillouin– Keller quantization) do not hold in this classically chaotic regime. Computational studies have indicated that chaotic behavior is expected in classical mechanical descriptions of the motion of highly excited molecules. As a consequence, intramolecular dynamics relates directly to the fundamental issues of quantum vs classical chaos and semiclassical quantization. Practical implications are also clear: if classical mechanics is a useful description of intramolecular dynamics, it suggests that isolated-molecule dynamics is sufficiently complex to allow a statistical-type description in the chaotic regime, with associated relaxation to equilibrium, and a concomitant loss of controlled reaction selectivity.

III. DYNAMICS: THEORY This section provides an introduction to the theory of classical and quantum intramolecular dynamics, with emphasis on general principles. A. The Hamiltonian A complete description of the dynamics of any molecular system is contained in the Hamiltonian H , which is the energy operator in quantum mechanics or the energy function in classical mechanics. In general, the Hamiltonian is a function of the electronic and nuclear degrees of freedom, as is the description of the system dynamics. This complex problem simplifies through the adoption of the Born– Oppenheimer approximation, which is the assumption that nuclear and electronic motion are independent due to their substantially different time scales and masses. This assumption allows one to first solve for the dynamics of the electrons and then obtain the forces experienced by the nuclei as determined by this fixed-electron configuration. Within the Born–Oppenheimer approximation, the nuclear Hamiltonian may be written in the form H (q, p) = T (p) + V (q),

FIGURE 10 As in Fig. 9, but for a different initial state. [From Kay, K. (1984). J. Chem. Phys. 80, 4973.]

(1)

where T (p) is the nuclear kinetic energy and V (q) is the nuclear potential energy. Classically, p is the momentum of the nuclei, whereas in quantum mechanics it is the momentum operator. In general, q has 3N components for an N -atom system. However, it is always possible to eliminate three coordinates corresponding to the center of mass


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of the system, and we shall assume this reduction to the internal molecular coordinate system has been carried out; thus, q denotes 3(N − 1) coordinates. Central to the nature of dynamics is the notion of coupled vs uncoupled degrees of freedom. Consider, for example, a system whose Hamiltonian is the sum of two terms, such that H (q, p) = h 1 + h 2 ,

(2)

where h 1 depends on a different set of coordinates and momenta than h 2 . Under these circumstances, the dynamics of the degrees of freedom in h 1 are independent of those in h 2 . That is, the total system is composed of two independent subsystems. Introduction of a coupling term, to give H = h 1 + h 2 + V (1, 2),

(3)

where V (1, 2) is a term dependent on the coordinates and momenta of both h 1 and h 2 , induces energy exchange and interrelated dynamics among the entire system. Major qualitative changes in the dynamics can result from small perturbations or couplings. It is important to note that although Eq. (3) is conceptually pleasing, there is no unique division into subsystems and into intersubsystem coupling for a given physical system. Rather, such a division must be motivated by an experimental or theoretical interest in a particular property, such as the energy flow between subsystems h 1 and h 2 . Central to the nature of intramolecular energy transfer is an understanding of the effect of the coupling V (1, 2) on the energy flow among the subsystems represented by h 1 and h 2 .

p(t) = A = const 1 (5)

B. Classical Mechanics

q(t) = q(t, A).

There are a number of formulations of classical mechanics, each providing different insights into its nature. For example, Hamilton’s method, used here, describes dynamics in terms of trajectories in generalized coordinates and momenta. Consider an M degrees of freedom system with system Hamiltonian H (q, p), where (q, p) is a complete set of M conjugate generalized coordinates and momenta. The time evolution of the system is given by Hamilton’s equations, dqi /dt = ∂ H/∂ Pi

phase space, with (q, p) as coordinates. A trajectory is a curve in this space parametrized by the index t. Modeling a realistic system necessitates producing a set of trajectories with varying initial conditions and looking at averages of system properties over this set. It is this trajectory technique, where V (q) is a model or realistic potential and where Hamilton’s equations are solved numerically on a computer, that has been extensively used to study intramolecular dynamics in small molecules. Results typical of those obtained were shown in Section II.A. The flexibility of Hamilton’s approach lies in the appearance of generalized canonical coordinates (p, q). As a consequence of their generality, one may seek out the set of coordinates and momenta within which the dynamics is most easily performed and understood. For example, a conservative Hamiltonian system may have, along any trajectory, a constant value of total angular momentum J. Such a quantity is said to be a constant of the motion and can prove useful as a momentum, since the equation of motion for J is particularly simple, dJ/dt = 0. Indeed, the idea of seeking constants of motion for use as coordinates or momenta is the central goal of the Hamilton–Jacobi approach to classical dynamics. The essential approach is simple. One seeks a set of M constants of the motion A of the system via a systematic procedure. Once found, these momenta are known to be constant along any trajectory, and the time dependence of the conjugate coordinates is generally simple. With the constants of the motion known, along with the procedure for generating them, it is a relatively straightforward algebraic problem to express the desired q(t), p(t) in terms of these constants and their conjugate coordinates. That is, we have

d pi /dt = −∂ H/∂qi

(4)

with i = 1, . . . , M. Specifying the state of the system at t = 0 via the initial conditions q(t = 0) = q0 , p(t = 0) = p0 then leads to a solution, a trajectory p(t), q(t), from which the time dependence of all system properties along that trajectory can be computed. The motion may be best visualized as taking place in a 2M-dimensional space, termed

Although we shall not deal with Hamilton–Jacobi theory here, the concept of a set of constants of the motion is vital to an understanding of the issue of intramolecular energy flow and statistical vs nonstatistical behavior. In essence, the number of global constants of the motion provides a method for grouping systems into general categories. Consider an M-degrees-of-freedom system. How many constants of the motion can we identify for a given trajectory? The answer is clearly 2M, with the simplest set consisting of the initial values of the coordinates and momenta that define the trajectory and that surely constitute a set of 2M equations for q(t), p(t) in terms of constants. That is, q(t) = q(q0 , p0 ; t)

p(t) = p(q0 , p0 ; t).

These constants of the motion, however, are of little interest. Although knowing them does identify the trajectory


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uniquely, it provides no reduction of the complexity of solving the full problem for q(t), p(t). Clearly, what we seek is a set of global constraints that define smooth surfaces in phase space on which the trajectories lie. Members of such a set of functions are called global constants of the motion. The number of global constants of motion that a system possesses provides a means of generally classifying the system behavior. 1. Integrable Systems Consider first the case in which an M-degrees-of-freedom system possesses M independent global constants of the motion, denoted K. Selecting these as the M generalized momenta K, with conjugate momenta Q, allows us to write the Hamiltonian as H (K). Hamilton’s equations then become d K i /dt = −∂ H/∂ Q i = 0 (6) d Q i /dt = ∂ H/∂ K i = ωi (K), where the last equation defines the frequencies ωi (K). Systems allowing this description are termed integrable, or regular, and possess a number of important properties: 1. Trajectories lie on M-dimensional surfaces in phase space whose topology is that of a torus. The tori are labeled by the values of K. For example, in a two-degreesof-freedom system, the motion lies on the surface of a doughnut (see Fig. 11). 2. The frequencies of motion about the independent directions on the torus are given by ω(K). 3. The time dependence of the coordinates and momenta for a given trajectory are given by the Fourier series (where n is a vector of M integers) q(t) = qn (K)exp[in · ω(K)t] n

p(t) =

(7) pn (K)exp[in · ω(K)t].

n

Note then that trajectories in such systems come arbitrarily close, during the course of their dynamics, to their original starting position. For this reason, such dynamics is termed “quasiperiodic.” The vast majority of “textbook” problems dealt with in elementary and advanced analytical mechanics treatises are of this type. Examples include the hydrogen atom or the small-vibrations Hamiltonian, although these systems tend to be, in addition, separable, that is, of the form H = H (K 1 ) + H (K 2 ) + · · · . Trajectories in integrable systems are stable with respect to small changes in initial conditions. In particular, consider a trajectory [q(t), p(t)] emanating from initial conditions q(0) = q0 , p(0) = p0 and an initially close trajectory [q (t), p (t)] originating from q (0) = q0 + δ Q, p (0) = p0 + δ P with δ Q and δ P very small. Then define d(t) as the time-dependent “distance in phase space” between these two trajectories, d(t) = [ p1 (t) − p1 (t)]2 + [ p2 (t) − p2 (t)]2 + · · · + [q1 (t) − q1 (t)]2 + · · ·]1/2 .

This quantity measures the rate at which two nearby trajectories separate as a function of time. 4. Then, for regular systems, d(t) grows linearly with t, a relatively slow rate of separation characteristic of stability. All these features are, in a qualitative sense, indicative of essentially predictable motion in integrable systems. behavior is repetitious, or at least describable by a simple set of frequencies. The linear growth of d(t) suggests that knowledge of the behavior of a single trajectory allows prediction, over a fair length of time, of the behavior of its initially nearby neighbors. As a simple example of regular dynamics, to be embellished later, consider a system of two uncoupled oscillators, H = H1 ( p1 , q1 ) + H2 ( p2 , q2 ),

(9)

where the individual oscillators, with Hamiltonians Hi (qi , pi ) = T ( pi ) + V (qi ), have potentials terms of the Morse form, V (qi ) = Di [exp(−qi /ai ) − 1]2 .

FIGURE 11 Sample quasiperiodic trajectory in a two-degreesof-freedom system as it moves on the surface of a torus in phase space. The trajectory shown is actually periodic; in general, the trajectory will fill the entire torus surface.

(8)

(10)

Here, Di is the oscillator dissociation energy and ai is a system parameter. Equation (10) might, for example, model three atoms on a line where the coupling potential between the oscillators has been eliminated. The individual Hamiltonians H1 and H2 are then the conserved integrals, and their numerical values remain constant at their initial values throughout the dynamics. Since this is a two-degrees-of-freedom system and since two constants of the motion exist, the system is integrable.


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488 Consider a measurement on the dynamics of this system. If an experiment were able to begin with an initial state consisting of a single trajectory, then the time evolution would be on a torus in phase space and there would be no energy transfer between the subsystems H1 and H2 . Sufficiently long-time observations on the system would reveal repetitive quasiperiodic motion. Two remarks regarding such measurements are, however, important. First, if one is interested in the dynamics of a subcomponent other than H1 and H2—say one interrogates the time dependence of the harmonic oscillator p12 /2m 1 + q12 —then the measurement would reveal energy flow into and out of this subsystem. Quasiperiodicity would still be evident, however, after a suitable time. Second, if the time scale of the experiment is short compared to the relevant system frequencies, then this quasiperiodicity will not be manifest. The essential point then is that the nature of the measurement of interest determines whether the regular system behavior is fundamental to, or observable in, the particular experimental study. There is another feature of integrable systems that is important. Specifically, consider the concept of statistical behavior in dynamics. The fundamental features of such behavior are that a trajectory at energy E fills the entire volume of phase space associated with that energy E, that a set of trajectories relaxes to a long-time limit that no longer varies with time, and that the final state is dependent solely on the energy of the system. It is clear that the first of these properties is not satisfied by a regular system, since a trajectory lies on the surface of a torus of dimensionality M, whereas the constraint to constant energy would confine dynamics to a larger surface of dimension 2M−1. It would also appear, from the list of properties above, that a regular system does not relax. This is, in fact, not the case. That is, properties (1)–(4) constitute features of the trajectories of a regular system. As already remarked, however, typical comparisons with physical systems require information on the average behavior of the time development of a collection, or ensemble, of trajectories. It is therefore important to note that despite the quasiperiodic behavior of integrable systems, an ensemble of trajectories in a regular system can relax to a long-time, stationary distribution. The final relaxed state of the system is, however, intimately related to the initial conditions of the dynamics. This is clear from the simplest of considerations. That is, each of the trajectories in the set of trajectories retains its original values of the conserved quantities. Thus, the final state of the system will depend on more than just the total overall energy of the system. Each of properties (1)–(4) gives rise to useful computational tools for the theoretical identification of integrable behavior in models of molecular motion. Relationships


to actual experimental techniques and measurements are, however, not well formulated. The regular system constitutes one major category of observable dynamical behavior. Systems that are integrable are well known and have been experimentally observed, as discussed later. A second major category of Hamiltonian systems emerges from formal ergodic theory, which defines a set of increasingly idealized statistical systems. Such systems (in terms of increasing statistical characteristics) are termed ergodic, mixing, K-systems, and Bernoulli systems. Each category imposes additional conditions, leading to requirements difficult to verify for realistic systems. Thus, they are to be regarded as idealized models of statistical motion. 2. Ergodic Consider first the integrable system where each trajectory lies on the surface of a torus. Two conditions are possible. In the first, the trajectory wraps about the torus and closes on itself without covering the torus completely. An example is shown in Fig. 11, where it is clear that this property arises if the frequencies of motion about the torus are related to one another by the relation n 1 ω1 + n 2 ω2 . Such a set of frequencies is said to be rationally related and results in the trajectory returning exactly to its original position. On the other hand, the frequencies on the torus may not be rationally related, in which case the trajectory fills the entire surface of the torus. Under such conditions the dynamics is said to be ergodic on the torus. This formal terminology does not correspond to the historical use of the term ergodic as found in the physics literature. There, ergodic tends to mean ergodic on the energy hypersurface, that is, on the (2M−1)-dimensional surface in 2M-dimensional phase space that results from constraining the system to constant energy. For clarity, we shall term this E-ergodic. Thus, the characteristic of an E-ergodic system is the existence of a single trajectory at each energy E that comes arbitrarily close to all points on the energy hypersurface. It is important to note, however, that this property does not ensure that the system displays irreversible relaxation during the course of the dynamics. A pictorial analog of possible motion of an ergodic system is provided by imagining a speck of carbon in a continuously stirred fluid. The carbon speck, representative of the system in phase space, moves throughout the fluid without constraint, but does not settle down to some long-time stationary state. 3. Mixing A system that is ergodic but has the rudimentary properties associated with statistical irreversible behavior is the mixing system. Such a system displays the following properties.


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Consider the system at energy E. Denote the average over the energy surface by the expression f , where f (q, p) is any dynamical property. Then, 1. lim f [q(t), p(t)] = f t → ∞. 2. The correlation between any two dynamical properties, that is,

g[q(t), p(t)], f [q(0), p(0)] − g[q(t), p(t)] f [q(0), p(0)], goes to zero as t → ∞. 3. Subdivide the total phase space into regular regions of particular volume. Then the probability of going from region i to region j in the long-time limit depends only on the size of the phase space regions i and j.

tween nearby trajectories that grows exponentially, that is, d(t) = d(0)exp(kt), indicative of trajectory instability. This set of properties gives rise to useful theoretical indicators of irregular motion, but connections with actual experimental observables are not well established. From the viewpoint of measurement, if one were able to prepare a single trajectory as the initial state of an irregular system, then the subsequent measurement of any property, other than energy, would show continual variation with time. The trajectory would, in addition, display no tendency to return to the original state over any finite time. If one prepared an ensemble of trajectories, it would approach a long-time stationary distribution dependent solely on energy.

4. Typical Molecular Systems Thus, a mixing system satisfies a number of simple properties that are in qualitative agreement with statistical relaxation dynamics. A particle of soluble colored material stirred into water provides a pictorial analog of mixing dynamics. Once again, the fluid models the phase space. The system evolves over time to reach a final macroscopically invariant distribution of uniformly colored fluid throughout the container. It is unfortunate that the formal definitions of ergodic, mixing, etc. systems involve the infinite time limit. As a consequence, a system may, for example, still be mixing even if relaxation is not observed in the finite time associated with a realistic measurement. This limitation significantly reduces the practical utility of formal concepts such as mixing behavior. A host of other formal systems with additional, and hence stricter, requirements have been defined. Here, we only mention the C-system, which is ergodic and mixing and which possesses the important characteristic that the distance d(t) between any two initially close trajectories in phase space grows exponentially in time. This trajectory instability leads to the rapid parting of trajectories from one another and, hence, the inability to predict the dynamics of trajectories, even for a relatively short time period, from knowledge of the dynamics of their neighbors. A system that displays characteristics of mixing as well as exponential divergence of adjacent trajectories is termed irregular or chaotic. In contrast with the characteristic properties of a regular system, an irregular system displays (1) trajectories that lie upon the (2M−1)dimensional energy hypersurface in phase space (additional simple constants of the motion such as angular momentum may also be incorporated), (2) and (3) trajectory dynamics that cannot be written in terms of a Fourier series involving a simple set of discrete frequencies and their overtones and combinations, and (4) a distance d(t) be-

Both regular and irregular motions are extremes of behavior, and their relation to the dynamics of realistic systems has principally been established through numerical computer studies. These studies indicate that many, but certainly not all, molecular systems display behavior characterizable as regular at low energies and irregular at higher energies. The example of carbonyl sulfide, OCS, is shown in Fig. 12, where the percentage of phase space not showing exponential divergence is shown. The system is seen to display a transition to chaotic motion at an energy of approximately 14,000 cm−1 . By 20,000 cm−1 , close to dissociation, almost all of the phase space is irregular. To appreciate the origin of the regular behavior at low energies, we note two common approximations

FIGURE 12 Percent regular trajectories as a function of energy for model OCS. The symbol D denotes the molecular dissociation energy. [From Carter, D., and Brumer, P. (1982). J. Chem. Phys. 77, 4208.]


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in low-energy molecular motion. The first, rotation– vibration decoupling, assumes that the rotational and vibrational motions are essentially uncoupled at low energies, that is, that the Hamiltonian is the sum of vibrational and rotational parts: H = Hvib + Hrot .

(11)

Second, we recall the standard normal-mode procedure for small-amplitude vibrational motion wherein, at sufficiently small energy, the vibrational Hamiltonian is of the form Hvib = H0 1 2 Pi + λi Q i2 + V (P, Q), = 2 i

(12)

with V (P, Q) sufficiently small to be negligible. The Q, P are called normal coordinates and momenta. Thus, lowenergy vibration is well approximated by a sum of M harmonic-oscillator Hamiltonians. In some instances an alternative separable Hamiltonian, composed of the sum of bond Hamiltonians, provides a superior separable representation. In either case, the low-energy vibrational motion is regular and separable. The situation changes dramatically with increasing energy as V (P, Q) becomes larger and the system begins to exchange energy between the decoupled harmonic oscillators. The subsequent dynamics, as observed in the measurement of the energy in a normal mode, depends intimately on the nature of the coupling, which is typically expandable in the form V (P, Q) = Vn (P, Q), (13) n

where Vn (P, Q) denotes polynomial terms of the form Q ik P jm with k + m = n. As a simple example of the effect of coupling, consider a two-degrees-of-freedom system with V (P, Q) = V2 (P, Q) = AQ 1 Q 2 .

(14)

It is convenient to first identify the constants of the motion in the harmonic Hamiltonian and use them as the new momenta. Consider then the momenta Ii = (4λi )−1/2 Pi2 + λi Q i2 (15) and conjugate coordinates 1/2 θi = cot−1 −λi Q i Pi .

(16)

In these coordinates, H0 assumes the form H0 = ω1 I1 + ω2 I2 , 1/2

(17)

where ωi = λi . We shall assume ω1 and ω2 to be unequal. These specific types of momenta and coordinates I are termed action-angle variables.

The relationships in Eqs. (15) and (16) allow us to rewrite V2 (P, Q) as V2 = A(I1 I2 /ω1 ω2 )1/2 [cos(θ1 −θ2 )−cos(θ1 +θ2 )], (18) where A is a constant. Hamilton’s equations of motion [Eq. (4)] then provide expressions for d Ii /dt that are nonzero due to the coupling V2 . In the event that the coupling is small, one may approximate the solution for the time dependence of the angles as that of the time dependence in the absence of the perturbation. This approach, a classical perturbation theory, gives the following result for the time dependence of Ii (t):

I1 I2 1/2 I1 (t) = I1 (0) − A ω1 ω2 × cos (ω1 − ω2 )t + θ10 − θ20 (ω1 − ω2 ) − cos (ω1 + ω2 )t + θ10 + θ20 (ω1 + ω2 ), (19) where θ10 , θ20 are the initial values of the angles. The quantity I2 (t) is similar, but out of phase. Thus, the action variables oscillate about their unperturbed values with frequencies (ω1 − ω2 ) and (ω1 + ω2 ). Since ω1 − ω2 is assumed large, the total variation of I1 and I2 as a function of time is small. The result is quite different if the system is resonant, that is, ω1 = ω2 . In this case, the effect of the perturbation is more drastic, and energy can be exchanged completely, albeit periodically, between the two harmonic oscillators. There are several reasons why the example treated above is a gross oversimplification of the situation in molecules. First, the unperturbed system is assumed harmonic, that is, linear in I. Second, the perturbation has been assumed to be composed of a single term. Third, only one type of perturbation has been included. We now qualitatively examine the important effects associated with the breakdown of these simplifying assumptions. a. Anharmonicity of H0 . In general, H0 is not harmonic, but is rather of the general anharmonic form H (I). As a result, the zero-order system frequencies ωi (I) = ∂ H0 /∂ Ii are no longer independent of the actions I; that is, the frequencies depend on the energy content of the oscillators. As a consequence, anharmonic systems will display regions of the I1 , I2 space where ω1 (I) and ω2 (I) are resonant as well as other regions where they are not. Thus, in the course of the dynamics, the zero-order system can go into and out of resonance as the energy of the oscillator varies. Regions of I space where the system is resonant are called resonance zones. Note that despite the coupling, the dynamics within this resonance region is regular.


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A well-known example provides a picture of the resonance zone and the “trapping effect” associated with a nonlinear resonance. Consider a child’s swing being pushed at a fixed frequency. The nonlinear swing, well approximated by a pendulum, will gain energy from the “pusher” until the system is well out of resonance. At this stage the swing loses energy until it once again comes into resonance with the driving frequency. The system is therefore effectively trapped in a range of swing energies determined by the resonance zone associated with this driven pendulum. A similar effect is associated with the single resonance region associated with ω1 (I) = ω2 (I) in the example above, the system being essentially trapped in the region about the resonance center if the dynmics is initiated in that region. b. Other coupling contributions. The above discussion emphasizes the ω1 = ω2 resonance, which results from the assumed form of the coupling in Eq. (14). In general, the coupling is more complicated, but is still expected to be expandable in the form V (I, θ) = Vm,n (I1 , I2 ) exp(inθ1 + imθ2 ). (20) mn

The V2 coupling term in Eq. (14) is an example of |n| = 1, |m| = 1 contributions to this expansion and leads to the ω1 = ω2 condition for resonance. Similarly, the n, m term in this series leads to an “n, m resonance” at action variables satisfying nω1 (I) = mω2 (I). Once again, within the neighborhood of this single resonance, the system displays regular energy transfer between the zero-order oscillators. We note that the size of the resonance zones tends to decrease with increasing n, m. c. Overlapping resonances. When a few terms in Eq. (20) contribute to the coupling, there is little reason to expect that specific regions of I space are influenced by solely one resonance. Under rather general conditions, the resonance regions in phase space arising from different terms in the coupling expansion [Eq. (20)] may overlap. Numerical studies have shown that energy flow between the zero-order oscillators assumes chaotic characteristics in regions of overlapping resonances. As an example, consider the resonance and resonanceoverlap structure associated with a collinear molecule A B C where H0 is the sum of two Morse oscillators [Eq. (10)] corresponding to the bond potentials and the coupling term is the form Ap1 p2 . Here, p j is the momentum associated with the jth bond. With E i defined as the energy of the ith bond and E the total energy of the system, quantitative application of resonance theory allows for the explicit determination of E i , E regions dominated by either a single resonance or by overlapping resonances. Sample results are shown in Fig. 13, where the solid shad-

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FIGURE 13 Resonance structure of a model system A B C where each atom has a mass equal to that of carbon. The A B bond has frequency 1000 cm−1 and dissociation energy Dc , whereas the B C bond has corresponding parameters 1300 cm−1 , 1.5Dc . Black areas denote single-resonance regions, and cross-hatched areas denote regions of overlapping resonances. [From Oxtoby, D., and Rice, S. A. (1976). J. Chem. Phys. 65, 1676.]

ing indicates regions dominated by a single resonance and the cross-hatched areas are those dominated by overlapping resonances. In the case shown, there is a general trend toward overlapping resonances as the energy increases, consistent with the observation of increasing chaotic behavior with increasing energy. For the particular parameters shown, however, the system, even at energies near dissociation (E = 1.5Dc ), displays regions of regular behavior dominated by a single resonance. Alternate system parameters can result in larger or smaller contributions from overlapping resonances. In summary, the picture that emerges with respect to energy transfer between specified zero-order oscillators is qualitatively straightforward. The coupling between the specified oscillators induces nonresonant energy transfer between the oscillators if the system is initiated, and remains, within a nonresonant region of I values. Resonant energy transfer results if the system begins within a resonance zone, or enters the resonance zone, during the dynamics. In both cases, energy transfer has welldefined pathways: the energy transfer is well described in terms of the time-dependent energy content of the zero-order oscillators. Finally, chaotic energy transfer between the zero-order oscillators is expected in the I-space regime dominated by overlapping resonances. Computational results on bound molecules indicate that the volume of resonance regions increases with increasing system energy. One important aspect of this discussion is worthy of emphasis. Specifically, the subdivision of the system into zero-order oscillators and coupling terms, and the


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492 subsequent expansion of the coupling term, must be motivated by the experimentally measured quantities. Specifically, one may see apparent chaotic motion between particular zero-order oscillators even if the system is regular. This would be the case if the time scale of measurement is short and the observed oscillators are not those directly related to the conserved integrals of motion.This feature also makes clear that overlapping resonances do not necessarily ensure true irregular motion. Detailed studies on the dynamics of realistic molecular systems are just becoming available. As a consequence, it is unclear whether the vast majority of highly excited molecules are weakly coupled with few overlapping resonances or are strongly chaotic. As a specific example of resonant coupling with weak coupling characteristics, and hence a specific energy-transfer pathway, we discuss below the study of overtones in the benzene molecule. As an example of chaotic energy transfer, we call attention to the NaBrKCl example discussed in Section II. Recent experimental studies on benzene have shown that the absorption spectrum contains local mode features, that is, evidence of local isolated bond motions. In the benzene case, the C H bonds, if they contain sufficient energy, appear directly in the spectrum, as if they were decoupled from the remainder of the molecular framework. In particular, one sees evidence of excitation to the overtones of C H stretch, that is, five, six, seven, etc. quanta of energy in the bond. The experimental results further indicate that if energy is deposited in these bonds, it would transfer to the remainder of the benzene nuclear framework within about 10−13 sec. Although apparently rapid, this rate of energy transfer is substantially slower than that expected from a C H bond democratically linked to all the degrees of freedom in the ring. Detailed quantum and classical studies of the dynamics of benzene indicate the following picture. Consider first the immediate local environment of a C H bond attached to the ring (Fig. 14). Two C C bond distances are


FIGURE 15 A schematic of the coupling scheme linking the C H stretch to the various modes of the benzene ring. Schemes (a) and (b) represent the same coupling schemes described in two different zero-order mode languages. [From Sibert, E. L., Hynes, J. T., and Reinhardt, W. P. (1984). J. Chem. Phys. 81, 1135.]

labeled s1 and s6 , with the C H bond distance being labeled s. Also shown is the angle β associated with the wag motion of the C H relative to the ring. The picture that emerges from numerical studies is that with increasing excitation of the C H bond, the C H bond frequency comes into resonance with the wag, where the resonance is characterized as n = 2, m = 1. Energy transfer from the C H bond first occurs as resonant energy transfer to the wag. Energy is subsequently transferred from the CCH wag to the remainder of the modes of the molecule. This is pictorially shown in Fig. 15. Trajectory calculations of the time dependence of the flow of energy out of the excited C H mode, for various degrees of excitation, are shown in Fig. 16. A complementary picture of the growth of energy in the ring modes of the benzene framework and into the lower lying states of C H on the benzene ring is shown in Figs. 17 and 18. On the time scale shown, the energy flow out of the C H bond is irreversible. Agreement with experiment is good, providing evidence that energy flow in this case occurs through a well-defined pathway of resonances. The comparative quantum calculations are discussed later. Further experimental and theoretical efforts are underway to establish the extent to which energy-transfer mechanisms in molecules are either chaotic or rather specific in their nature. C. Quantum Dynamics

FIGURE 14 Coordinates defining the C H bond distance (s). C C bond distances s6 , s1 ; and wag angle β in benzene. [From Sibert, E. L., Hynes, J. T., and Reinhardt, W. P. (1984). J. Chem. Phys. 81, 1135.]

Molecules are, of course, properly described by quantum mechanics, and classical mechanics is recognized as a particular approximation. Nonetheless, an introductory description of intramolecular energy transfer via classical mechanics has proven useful since it contains few concepts that are truly unfamiliar to the macroscopic world.


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FIGURE 16 Time dependence of the average energy in the C H oscillator for three cases: excitation in the ninth C H vibrational level (v = 9), v = 6, and v = 5. [From Sibert, E. L., Hynes, J. T., and Reinhardt, W. P. (1984). J. Chem. Phys. 81, 1135.]

tions prevent a discussion of these quantum phenomena, and the interested reader is referred to the bibliography for details. We focus rather on the dynamical consequences of energy quantization in quantum mechanics. This property means that a system can only exist at specific energy values, a property shared by other observables as well. Energy is, however, intimately linked to dynamics, since the Hamiltonian determines system time propagation, as discussed later. One important remark is in order. That is, although quantum phenomena have been observed in molecular systems, we possess only the very qualitative “traditional” rules regarding conditions under which quantum effects predominate. Specifically, if the initial state involves large classical actions and the initial state is one that is allowed classically, then quantum effects tend to be small. Considerably more work is necessary, however, before more quantitative, predictive statements can be made and before our understanding of classical/quantum correspondence in bound molecular systems is complete. Considerations of the quantum dynamics of bound molecules shows that, in the absence of the emission of radiation from energized molecules, all dynamics is quasiperiodic and regular. That is, quantum mechanics does not admit the possibility of long-time relaxation to a time-independent stationary state, a property that characterizes a classical mixing system. This property creates

Although it is possible to cast both quantum and classical mechanics in a similar formal language (i.e., distributions in phase space and a Liouville propagator), standard quantum mechanics is based on a mathematical structure that is substantially different from that of classical Hamiltonian mechanics. We first provide a brief qualitative summary of some results of quantum investigations, and then we present details that can be best appreciated by the reader who is well versed in quantum mechanics. First and foremost, we note that classical mechanics does not allow a number of phenomena that occur in nature. A familiar example is tunneling, in which a system has finite probability of being in a region of phase space where it is not permitted classically. The simplest example of tunneling occurs in a system consisting of a particle moving in a potential that has two minima with a potential barrier between them. Classically, a particle initiated in one of the wells with energy below the barrier height is confined to that well forever. In the quantum case, however, the system flows between the two wells: it “tunnels” through the potential barrier. Tunneling effects are most certainly important in the intramolecular dynamics of systems at energies below such potential barriers. Less well known are symmetry effects, resonances, etc. that can play important roles in intramolecular dynamics. Space limita-

FIGURE 17 Growth of energy content of ring modes in benzene associated with the v = 6 case in Fig. 16. Note that only a few ring modes, labeled by their frequency, are shown. [From Sibert, E. L., Hynes, J. T., and Reinhardt, W. P. (1984). J. Chem. Phys. 81, 1135.]


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Quantum mechanics describes system behavior in terms of operators that represent measurable quantities, their eigenfunctions, which describe possible states of the system, and their eigenvalues, which correspond to allowable values of the measurable. If the system is presumed best described in terms of a specification of the system energy, then one looks for states |ψ that are eigenfunctions of the molecular Hamiltonian operator H. That is, one solves the problem H|ψ(t) = −i h ∂|ψ(t)/∂t, ✟

(21)

where t is the time. The nature of this equation is such as to admit the solutions |ψ j (t) = |ψ j exp(−i E j t/ h ), ✟

(22)

where |ψ j is the solution to the eigenvalue problem H|ψ j = E j |ψ j .

FIGURE 18 Probability of finding i quanta in the C H oscillator as a function of time for the case of initial v = 6. [From Sibert, E. L., Hynes, J. T., and Reinhardt, W. P. (1984). J. Chem. Phys. 81, 1135.]

a number of difficulties in understanding the formal relationship between classical and quantum mechanics, particularly for energized molecules that display classically chaotic behavior. Practically, however, one finds that if the system is close to the classical limit, then quantum and classical dynamics agree over a significant time scale. This time scale is expected, in the vast majority of typical chemical experiments, to be in excess of the time of interest for the process. Under these circumstances, the formal discrepancy between classical and quantum mechanics is irrelevant to the specific chemical problem. Nevertheless, since quantum mechanics does not admit anything other than quasiperiodic behavior, attention has recently been focused on other quantities that might provide the qualitative distinction between quantum systems that display, more or less, statistical behavior.

(23)

Here, the Hilbert space vector |ψ j has a coordinate space representation ψ j (q) = q|ψ j , and |ψ j (q)|2 is the probability of observing a given value of q when the system is in a state defined by energy E j . In general, the system may be degenerate, in which case several E j may have the same numerical value. This treatment and the one that follows provide an idealized picture in which the molecule is entirely isolated from external influences. Such an ideal picture cannot, in fact, apply. Specifically, although one may be able to experimentally isolate the molecule from interactions with other molecules (e.g., via high-vacuum techniques), the molecule will always interact with the background radiation field to radiate energy. In this discussion we regard this emission as a small perturbation that can be introduced as part of the measurement process. Consider time dependence in quantum mechanics, with the experimentally prepared initial state assumed completely specified as the Hilbert space vector |φ(0). That is, the system is initially in a pure state and may be expanded in a linear combination of energy eigenstates |ψ j as |φ(t = 0) = c j |ψ j c j = ψ j |φ(0). (24) j

Then, from Eq. (22), the subsequent time evolution is given by |φ(t) = c j |ψ j exp(−i E j t/ h ). (25) ✟

In the alternative case, the so-called mixed state, one does not have a complete specification of the initial state; that is, the initial state cannot be described as a single Hilbert space vector, nor can it be written in terms of a linear combination of |ψ j . It can, however, be written in terms of a density matrix,


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ρ(0) =

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wn |χn χn |,

(26)

n

where the states |χn can be written in terms of a linear combination of |ψ j and where n wn = 1, with wn equal to or greater than 0. The essential feature of such states is that ρ(0) is lacking information on the relative phases of the participating eigenstates |ψ j . We focus on the pure state, although it is the exception, rather than the rule, in experimentally prepared systems. In accordance with quantum mechanics, a measurement of a particular property during the course of the system time evolution consists of evaluating the average value of the corresponding operator. For example, if one measures the property described by the quantity F, then the average value of F as a function of time is given by

F(t) = φ(t)|F|φ(t) = di, j exp(iωi, j )

(27)

i, j

di, j = ψ j |F|ψi c∗j ci , where ωi, j = (E i − E j )/ h. Thus, F(t) may be written as a linear combination of terms involving a discrete set of frequencies ωi, j . By analogy with the discussion of classical systems, this sum is seen to be quasiperiodic. The number of terms contributing to the sum in Eq. (27) and the relationship between the frequencies ωi, j determine the kind of qualitative behavior observed. In the case where only a few terms contribute, the dynamics is almost periodic. Such behavior is observable as, for example, a periodic modulation of the fluorescence emitted from a molecule prepared in a linear combination of a few states, a phenomenon known as quantum beats. An example of beats in SO2 is shown in Fig. 19, where the fluorescence reflects the interference between two contributing levels.

FIGURE 19 The intensity of fluorescence, as a function of time, from SO2 created in a superposition state composed of two levels. [From Ivanco, M., Hager, J., Sharfin, W., and Wallace, S. C. (1983). J. Chem. Phys. 78, 6531.]

FIGURE 20 Density of states (i.e., number of states per unit energy interval) as a function of energy for a number of molecules. The abcissa is in number of photons, rather than energy, where the type of laser photon used depends on the particular molecule. For example, the state density for CF3 CH2 OH is plotted vs photons, from an HF laser, associated with the P1 (7) line. The energy of each of these photons is 0.01660 a.u. Other photons used are the P1 (6) line with a photon energy of 0.01683 a.u. and the CO2 10P(20) line with energy of 0.00430 a.u. per photon. [From McAlpine, R. D., Evans, D. K., and McClusky, F. K. (1980). J. Chem. Phys. 73, 1153.]

Such simple behavior emerges only when the initially created state is composed of a few levels. This is seldom the case, as seen from Fig. 20, which shows the density of states (i.e., the number of states per unit energy interval) D(E) for some typical molecules. The quantity D(E) is seen to be an increasing function of the size of the molecule and, for even small molecules such as SF6 , can reach very large values (e.g., 103 cm−1 ). As a result, chemical experiments, whose energy resolution is often not sharp, will typically involve an initial state composed of many many levels. The subsequent dynamics emerges through the simultaneous interference of a multitude of terms, and the resultant behavior is difficult to qualitatively extract from


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a formal sum [Eq. (27)] of contributions from individual levels. For example, the system can display short-time behavior reminiscent of relaxation. Consider the case where the coefficients di, j are distributed in a smooth fashion about a particular frequency value. Then one can show that for short times compared to the density of frequencies, F(t) decays smoothly as a function of time, with a time scale governed by the inverse of the frequency width of the coefficients di, j . This behavior is termed dephasing, to distinguish it from irreversible relaxation of the initial state. That is, despite this decay, the system will eventually reassemble to form the initial state, although this time scale may be exceedingly long. Thus, an experiment measuring

F(t) over a time scale short compared to the recurrence time will show apparent relaxation of F(t). Nonetheless, formally, the system is quasiperiodic. The facts that quantum bound-state dynamics is quasiperiodic, classical mechanics can be mixing, and the latter is expected to approximate the former, make up the essence of the important unresolved problem “what is quantum chaos?” The quantum picture of bound-state dynamics calls attention to an important aspect of intramolecular dynamics. Specifically, the state |φ(t) is composed of a linear combination of states |ψ j . The probability of observing the system in an exact eigenstate |ψ j at time t is given by |c j exp(−E j t/ h )|2 = |c j |2 . Thus, the population of each exact eigenstate does not change as a function of time. If, in fact, one were solely interested in the population of these exact levels, then there is no such thing as time dependence in the dynamics of bound molecules (other than radiative emission)! Clearly, the focus of intramolecular dynamics and energy transfer is on attributes other than exact eigenstates populations. To appreciate the desired description of energy flow in chemistry, recall the historical origin of the interest in intramolecular energy flow. The most prominant case is that of unimolecular decay, in which a molecule, sufficiently energized, breaks into a variety of products (e.g., ABC → A + BC). In this case the focus of attention, and therefore of the measurement, is on the energy content of the A B bond. This is typical of chemical descriptions in which the analysis is in terms of subunits of the molecule that are not, in themselves, naturally distinct subcomponents of the molecule. Such a description results when a zero-order basis set is used. Specifically, consider the Hamiltonian for a two-degrees-of-freedom system written, as in the classical case, in the form ✟

H = H0 + V

H0 = H1 + H2 ,

(28)

where H1 and H2 describe two distinct subcomponents of interest in a particular experiment and the eigenfunctions of Hi are denoted by |χ ij , where

Hi χ ij = εij χ ij

i = 1, 2.

(29)

The perturbation V couples the zero-order states so that exact-energy eigenstates |ψk are linear combinations of these zero-order states or vice versa. That is, 1 2 χ χ , = bk ψk . (30) i

j

i, j

k

Using this expression and Eq. (22) gives the following form for the time evolution of these zero-order basis-set states; 1 2 χ χ (t) = bk ψk exp(−i E k t/ h ). (31) ✟

i

j

i, j

k

If the initial state consists of a linear combination of the zero-order states, then the populations of the zero-order states are seen to be time dependent. The degree to which the zero-order states enter into the exact eigenstates [i.e., the nature of the sum in Eq. (30)] is a measure of the strength of the coupling and provides an insight into the nature of the exact eigenstates from the view-point of this particular zero-order basis. It essentially provides the time-independent picture of the possible zero-order states that can be coupled during the dynamical evolution of the system. Equation (28), treated quantum mechanically, admits the same kind of perturbation treatment as in the classical case, with a similar emphasis on isolated resonances and overlapping resonances emerging. As in the classical case, the question of the nature of intramolecular energy flow—whether it is statistical or whether it displays a specific pathway—is of interest. Unfortunately, few quantum calculations on realistic molecular systems have been performed. The example of the dynamics of benzene, initially prepared in an excited state of the C H bond, has, however, been treated quantum mechanically. Here, the relevant zero-order Hamiltonian is of the form H = HL + HN + HLN ,

(32)

where HL is the Hamiltonian for the local C H vibrational motion, HN is the Hamiltonian for the remainder of the molecule, and HLN is the coupling between them. Computations have been carried out, including the coupling between the C H vibration and the CCH wag motion, via terms in HLH . In the adopted model, HN is essentially harmonic so that the zero-order states are of the form |vL , kN = |vL |kN , denoting v quanta in the C H stretch and k quanta in the ring modes. As in the classical treatment in Section II, interest is in the dynamics of benzene in the energy range where the C H bond is prepared with considerable energy. Figure 21 shows some of the many zero-order energy eigenstates in the energy regime associated with the zero-order state |6L , 0N . Not shown are states with less than four quanta in HL , which, although coupled to the |6L , 0N state, were too


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FIGURE 21 (a) Some of the zero-order states of benzene in the energy neighborhood near that of excitation to the sixth vibrational level of C H. States are organized, for clarity, in ladders associated with the number of quanta in the C H mode. (b) Levels for the analogous monodeuterated benzene case. [From Sibert, E. L., Reinhardt, W. P., and Hynes, J. T. (1984). J. Chem. Phys. 81, 1115.]

numerous to be included in the computation. For convenience, the levels are stacked in ladders, or “tiers,” with the ladder labeled by the quanta of energy in the C H oscillator. Calculations show that the exact system eigenstates are a strongly coupled mixture of the zero-order states. Figure 22 shows the result of a calculation on the quantum dynamics of benzene initially excited to the sixth vibrational level of C H. The population of the sixth level (denoted |cCH (t)|2 ) is seen to decay rapidly in time, with concomitant growth of population in the various tiers. The essential features of this quantum calculation are in accord with the classical results by the same authors and describe the dynamics of benzene, initially energized in the C H vibration, as decay via a specific pathway of resonances between zero-order modes. Although specific examples of this kind are of considerable importance, general rules regarding quantum intramolecular dynamics would be far more useful. Indeed, this was the motivation for investigating ergodic, mixing, etc. systems classically. Similarly, it is the reason for the general interest in the question of “what is quantum chaos?” We comment briefly (and only qualitatively) on recent developments in this area, noting first that an appropriate definition of chaos can only involve properties of the central participants in the time evolution, that is, the energy eigenstates and eigenvalues.

As a starting point, note that chaotic classical systems possess a number of prominent features. First, the system relaxes to a long-time stationary distribution. Second, the long-time limit is sensitive to only a few simple constants of the motion (e.g., the total energy). Third, as a direct consequence, the computation of system properties can be done by replacing the actual dynamics by simplified statistical models. The first property is clearly not satisfied by a quantum bound system, which has been formally shown to be quasiperiodic. Nonetheless, one possibility is that the system approach a long-time value about which fluctuations are small and that this long-time value be sensitive to only a few simple properties. Indeed, if all wave functions in an energy interval are basically the same (i.e., have similar properties), then the system will evolve in a fashion that is relatively insensitive to the nature of the preparation. Further, if there are no integrals of motion other than the total energy, then one might expect the energy eigenvalues to display rather simple properties reflecting this characteristic. These remarks motivate one current view on the nature of quantum chaos. Specifically, in such a system the eigenfunctions are proposed to be similar to one another in character as the energy varies, and the probability of observing a particular level spacing is expected to be of a specific form (the Wigner adjacentlevel distribution). The former requirement, coupled with the condition that all eigenfunctions be orthogonal, that is,

ψi |ψ j = 0,

(33)

hints at the nature of these wave functions; that is, they display erratic nodal patterns. This is clearly not the case

FIGURE 22 The time dependence of various quantities associated with the quantum dynamics of benzene initially excited to the sixth level of C H bond excitation. Labels are defined in the text, other than Pi (t), which denotes the probability of being in the i th tier of energy levels. Only three tiers were included in this calculation. [From Sibert, E. L., Reinhardt, W. P., and Hynes, J. T. (1984). J. Chem. Phys. 81, 1115.]


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FIGURE 23 Contour plot of ψ(q) associated with a typical state of the particle in a stadium potential. Only positive contours are shown. Note the highly erratic pattern of contour lines proposed to be a characteristic of wave functions associated with chaotic classical systems. [From Brumer, P., and Taylor, R. D. (1983). Faraday Discuss Chem. Soc. 75, 171.]

for the vast majority of typical systems studied in elementary quantum mechanics (e.g., the H atom, the particle in a box), which are, in fact, integrable and separable in classical mechanics as well. Note that the condition of erratic nodal patterns makes sense with respect to the view resulting from an analysis in terms of a zero-order basis set. In particular, expanding such exact wave functions in any arbitrary basis is expected to yield populations spread over all zero-order wave functions in the energy neighborhood. Thus, there is a form of statistical coupling between all zero-order basis functions. Numerical computations on model systems have, in fact, revealed erratic wave functions for some systems that are classically mixing. An example is shown in Fig. 23, where the system is a particle confined by infinite potential walls to a region that is the shape of a racetrack (the so-called stadium system). The nodal patterns are clearly highly disordered and in marked contrast with the simple nodal lines associated with, for example, the separable particle in a rectangle case. Unfortunately, there is no one-to-one correspondence between systems that display chaotic classical behavior and the observation of erratic wave function nodal patterns.

IV. STATISTICAL APPROXIMATIONS AND DYNAMICS Accurate dynamical studies, such as those discussed above, are limited to small molecular systems and relatively short times (e.g., typically 100 vibrational periods). These restrictions stem from a number of sources. First, reliable computations of forces between atoms in large molecular systems are extremely difficult, and few quantitatively accurate models of interatomic forces exist. Sec-


ond, even if such potentials are available, large-molecule motion treated quantum mechanically involves huge numbers of participating energy eigenstates. Techniques for efficiently computing large numbers of eigenstates in systems with significant numbers of degrees of freedom are only now being developed. Alternative techniques, which rely on the direct, numerical, temporal propagation of initial states [i.e., via Eq. (21)], do not utilize eigenstates but suffer inaccuracies in long-time applications. An analogous difficulty exists in attempting numerical studies of dynamics using classical mechanics; in this case, the exponential growth of distances in phase space (see Section III) translates into the rapid growth, with time, of numerical errors. Since the vast majority of interesting molecules have many degrees of freedom, the need for models that simplify the dynamics is evident. The most popular of such models relies on a statistical assumption of relaxation during the course of the dynamics. That is, one assumes that after preparation of the energized molecule, the system relaxes to a well-defined state that is dependent on only gross features (e.g., the total energy) of the preparation. Assumptions of this kind predate detailed dynamical studies of molecular dynamics. To appreciate the simplifications resulting from such models, consider the paradigm case of unimolecular decomposition (A → B + C, where A, B, and C are molecules), where this assumption leads to statistical theories of the rate of unimolecular decay. Here, A, sufficiently energized to dissociate, is prepared by any of a variety of means (e.g., laser excitation, collisions, or as the product of a chemical reaction). A detailed computation of the dynamics of this process for a realistic molecule, an intractable computational feat, would entail the following steps. One first specifies the exact nature of the molecular potential, the nature of the process that prepares the excited molecule, and the state of the molecule prior to preparation. Second, the exact dynamics of the evolution of the molecule, from preparation to decay, is computed. Such a computation must be repeated for each and every type of initial state of the molecule and each and every type of state preparation. In contrast, the formulation of a typical statistical model proceeds as follows. First, define the region of phase space in which the molecule A is to be regarded as being bound and a complementary region in which it is characterizable as B + C. Then assume that the rate of dissociation of the excited molecule depends primarily on the magnitude of simple known constants of the motion, that is, energy E and total angular momentum J. Assume further that any initially prepared state rapidly relaxes to a uniform distribution over the surface in phase space characterized by fixed E, J and within the region where A is regarded as bound. The computation of the


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rate of decay of such a system then entails an analysis of the rate at which the relaxed uniform distribution crosses over into the dissociation region. Such a computation is straightforward in both classical and quantum mechanics, if a number of approximations on the structure of the molecule are made, and leads to statistical theories, such as RRKM theory, discussed in the bibliography. The essence of such theories is, then, that the system undergoes memory erosion during the dynamics so that information on initial conditions is lost; only knowledge of E and J remains. Similar theories have been proposed for a wide variety of processes that involve the participation of long-lived molecular intermediates. These include chemical reactions that proceed via collision complexes, photodissociation where molecular preparation is through controlled laser excitation, molecules adhering to surfaces where detachment is induced via a variety of means, etc. Such theories have the advantage of yielding rather general results, which are amenable to both theoretical and experimental analysis. For example, in the case of unimolecular decay, the rate constant for dissociation is found to increase with increasing energy and decrease with the number of participating degrees of freedom in the system. Experimental studies on the validity of such theories have been ongoing for many years, as discussed briefly in the next section. The theoretical examination of the validity of such approaches is more recent and links directly to the issues discussed in Section III. At present, classical mechanical studies have shown the possibility of both statistical and nonstatistical decays, depending on the degree and extent of exponential divergence of trajectories in phase space. The greater the degree and extent of exponential separation, the closer the agreement with statistical approaches. Similarly, quantum-mechanical studies have shown that model systems can display unimolecular rate constants whose energy dependence is inconsistent with that predicted by simple statistical theories. It is fair to say, however, that a clear understanding of the interrelationship between molecular properties and the validity of statistical theories is in its early stages of development.

V. EXPERIMENTAL STUDIES The ideal experiment on intramolecular energy transfer, as yet unachieved, entails a number of simple features. Specifically, the molecule is prepared in a well-defined and well-characterized state and evolves for a known time interval, after which the state of the system is precisely determined via a high-finesse experimental probe. The importance of each of these components to the resolution of even the most qualitative of questions (e.g., is the energy transfer statistical or not) should be clear. If, for exam-

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499 ple, as is the case in some of the available techniques, the initial state of the molecule is in itself highly randomized energetically, then a subsequent measurement that shows that the energy is statistically distributed among the system modes is of little consequence. That is, this particular observation would be a consequence of the preparation, as distinct from the evolution, of the molecule. The dynamics of molecules with low energy content is usually well described by a Hamiltonian that is the sum of independent degrees of freedom, and absorption spectroscopy has provided considerable information on the nature of such systems. The situation with respect to energized molecules is far more complex. A number of experimental tools have been utilized over the past 30 years to examine the nature of energy transfer in such systems, as discussed next. Many suffer from the inaccurate knowledge of the initial system state. In our brief description of several experimental methods it will become clear that experimental tools have been rapidly developing over the past few years and that an explosion of highly informative experimental data is now underway. First and foremost, note that interest is in the nature of intramolecular dynamics of molecules in isolation. That is, observations must be made over a time scale where the molecule does not collide with others in the reaction vessel. Modern techniques allow very low pressures under which such measurements can be made. Most desirable among these methods are beam techniques in which molecules are studied in a low-density beam produced, for example, by vaporizing molecules in an oven. Experiments prior to this “beam age” (circa 1960) often inferred information about intramolecular dynamics from bulk data, which contained effects due to collisions, with resultant loss in accuracy. Measurement techniques in typical experiments can be subdivided into two categories. The first and most modern probes the bound molecular dynamics directly, for example, by observing radiation emitted or absorbed during the course of the dynamics. The second infers information about the nature of bound molecular dynamics by indirect means, typically by analyzing the outcome of a process that involves the molecule of interest as a longlived intermediate. These latter types of measurement are readily clarified by considering the NaCl + KBr example discussed in Section II. Specifically, this particular reaction proceeds via the bound NaC1KBr intermediate to yield two different sets of products, either NaCl + KBr or NaBr + KCl. The probability of observing these two product “channels” depends on the nature of the NaCIKBr dynamics. That is, the product ratio will provide insight into whether intramolecular energy transfer in NaCIKBr is rapid or not. More extensive measurements will entail analysis of the internal states of the product molecules


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500 (see Fig. 5), as well as of the relative velocities of the products. Thus, these kinds of measurements allow one to infer information on the characteristics of the dynamics of the intermediate without directly observing it. Other examples of this kind include measurements on the products of unimolecular decay, photodissociation, etc. Similarly, most experiments on intramolecular energy transfer fall into one of two categories with respect to preparation of the molecule. In the crudest, the system is prepared by a “coarse” technique where little detail about the initial molecular state is available. Such, experiments include preparation via collision that is, where the molecule of interest, A, collides with, and absorbs energy from, another molecule B and preparation via reaction, where the molecule A is the product of a “precursor” reaction. Although the bulk of early work on intramolecular dynamics was carried out with these techniques, far greater insight emerges from modern experiments in which the molecule A is prepared by the absorption of radiation. The optimum experiment would therefore proceed by preparing the molecules in a precise state using beam methods and laser excitation, followed by measurement of the radiative emission as well as other properties of the bound molecule. Such experiments are, in fact, underway on a variety of molecules in several laboratories around the world. In addition, information on the bound-state dynamics of molecules has emerged from pump-probe techniques in which two lasers are utilized, one to prepare the molecule in the desired state and the second to interrogate the dynamics. Along with these experimental developments, we note a need for reliable theories to understand the interrelationship between the observed features and the nature of the dynamics. Such developments are in progress.

VI. CONTROL OF DYNAMICS We noted, in Section I, that the study of intramolecular energy transfer is linked to the practical goal of controlling the dynamics of molecules. Since the 1980s theoreticians have made giant strides which make control over mollecular dynamics feasible. Space restrictions prevent anything other than a brief comment; the reader is referred to the Accounts of Chemical Research reference for further details. Equation (27) makes clear that the dynamics of a molecular process is intimately linked to the phases of the system preparation, contained in the di, j coefficients. It is clear, then, that if one were able to control the phases and magnitude of these terms, the subsequent system dynamics would also be controlled. Recently, a number of theoreticians have noted that coherent laser sources transfer phase


information to the molecule upon which they impinge. As a consequence, by controlling the phases of these laser sources, one can affect the nature of the molecular dynamics. More careful examination indicates that the experimentalist must control the relative phases of two laser sources, rather than the absolute phase of a single source, a far more feasible prospect. These proposals for the control molecular dynamics and chemical reactions rely heavily on quantum interference phenomena similar to that seen in the famous “doubleslit” experiment. As a consequence they herald a new age in molecular reaction dynamics, one in which quantum aspects of molecular motion are utilized to alter molecular dynamics.

VII. SUMMARY Understanding the nature of intramolecular energy flow in isolated molecules is of great practical and fundamental interest. Early developments, both theoretical and experimental, were hampered by a number of technological problems now being overcome. As a consequence, general features that determine the rate and extent of intramolecular energy transfer are slowly emerging. These include generic features of classical Hamiltonian systems and the way in which coupling terms influence the nature of the dynamics, the dependence of observed energy transfer on the zero-order system, the interaction between state preparation and state measurement on the qualitative interpretation of the dynamics, and the differences and similarities between the quantum and classical views of dynamics. Nonetheless, general rules regarding the rates and extent of intramolecular energy flow have yet to be established. Similarly, a number of fundamental issues arising in the study of intramolecular energy flow have yet to be resolved. Rapid technological developments in computational and experimental tools hold great promise for substantial developments over the next decade.

ACKNOWLEDGMENT We are grateful to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of the research on which this overview is based.

SEE ALSO THE FOLLOWING ARTICLES ATOMIC AND MOLECULAR COLLISIONS • CHAOS • CHEMICAL KINETICS, EXPERIMENTATION • CHEMICAL PHYSICS • COLLISION-INDUCED SPECTROSCOPY • DYNAMICS OF


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ELEMENTARY CHEMICAL REACTIONS • MECHANICS, CLASSICAL • QUANTUM MECHANICS • STATISTICAL MECHANICS

BIBLIOGRAPHY Brumer, P. (1981). Adv. Chem. Phys. 47, 201. Brumer, P., and Shapiro, M. (1988). Adv. Chem. Phys. 70, 365. Brumer, P., and Shapiro, M. (1989). Accts. Chem. Res. 22, 407. Faraday Discuss. Chem. Soc. (1983). 75. Felker, P. M., and Zewail, A. M. (1988). Adv. Chem. Phys. 70, 265. Forst, W. (1973). “Theory of Unimolecular Reactions,” Academic Press, New York.

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501 Gruebele, M., and Bigwood, R. (1997). Int. Rev. Phys. Chem. 17, 91. Gruebele, M. (2000). Adv. Chem. Phys. 114, 193. Lehmann, K. K., Scoles, G., and Pate, B. H. (1994). Annu. Rev. Phys. Chem. 45, 241. Levine, R. D. (1969). “Quantum Mechanics of Molecular Rate Processes,” Oxford Univ. Press, London. Noid, D. W., Koszykowski, M. L., and Marcus, R. A. (1981). Annu. Rev. Phys. Chem. 32, 267. Rice, S. A. (1975). “Excited States” (E. C. Lim, ed.), Vol. 2, Academic Press, New York. Rice, S. A. (1981). Adv. Chem. Phys. 47, 117. Stechel, E. B., and Heller, E. J. (1984). Annu. Rev. Phys. Chem. 34, 563. Wyatt, R. E., Iung, C., and Leforestier, C. (1995). Accts. Chem. Res. 28, 423.

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Hydrogen Bond Krzysztof Szalewicz University of Delaware, Newark, Delaware

I. Historical Perspective II. Quantum Mechanical Description of Hydrogen Bond III. Analytic Representations of Potential Surfaces IV. Nature of Hydrogen Bond V. Properties of Hydrogen Bonds VI. Types of Hydrogen Bonds VII. Hydrogen Bonds in Clusters VIII. Hydrogen Bonds in Solids IX. Hydrogen Bonds in Liquids X. Proton Transfer XI. Hydrogen Bonds in Biological Structures

GLOSSARY Basis functions In the context of solutions of the electronic Schrödinger’s equation for hydrogen-bonded system, this term refers to Gaussian functions of the form exp[−αr2 ], where r is the position vector, multiplied by powers of the coordinates x, y, and z. The basis functions are usually located at the nuclear positions and near the midpoint of the hydrogen bond (midbond functions). Linear combinations of such basis functions form molecular orbitals. Coulomb interaction The interaction between two charged particles. According to Coulomb’s law of electrostatics, the energy of such interaction is q1 q2 /R, where qi is the charge of the ith particle and R is the particles’ separation.

Dimer A complex formed by two molecules that do not react chemically with each other. The two constituent molecules of the dimer, called monomers, are disturbed upon the formation of the dimer but preserve their identity. Electron density The probability density of finding any electron of a molecule at a given point in space. Often visualized as the “electron cloud.” Electron density can be obtained by integrating the square of the modulus of an electronic wave function over the coordinates of all electrons but one. Euler angles A set of three angles that uniquely determines the orientation of a solid body in space. Hamiltonian The operator appearing in the Schrödinger equation of quantum mechanics. Operators act on wave functions, transforming them into other wave

505

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506 functions. For an isolated molecule the Hamiltonian contains second partial derivatives with respect to coordinates of all constituent particles—which describe the kinetic energy—and the sum of Coulomb interactions between all the particles—which describes the potential energy. Hartree-Fock method A method of solving the Schrödinger equation that assumes that particles’ motions are independent of each other and a given particle interacts only with the averaged charge distribution of other particles. The electronic wave function can then be approximated by an antisymmetrized product of one-electron functions (orbitals). The effects neglected by the Hartree-Fock method are called correlation effects. Molecular simulations Computer modeling of the motion of an assembly of atoms or molecules. In molecular simulations only the motions of nuclei are considered, (i.e., one assumes that the electronic Schrödinger equation has been solved providing intermolecular interaction potentials.) In practice empirical interaction potentials are utilized in most cases. Two main approaches are used: the Monte Carlo (MC) method and molecular dynamics (MD). The former relies on statistical sampling of the configuration space of the systems, whereas the latter solves classical mechanics (Newton) equations to find trajectories of molecules. Perturbation expansion A method of solving the Schrödinger equation by dividing the Hamiltonian H into an unperturbed part H0 and the perturbation V . H0 is chosen such that an accurate solution of the “unperturbed” Schrödinger equation H0 0 = E 0 0 is possible. The wave function and the energy E which solve H = E are expanded as power series in V starting from 0 and E 0 , respectively, which leads to a hierarchical set of equations for the coefficients in these power series called the wave function and energy corrections, respectively. The simplest implementation of the perturbation method is called the RayleighSchrödinger perturbation theory. Quantum mechanics Theory describing behavior of matter and radiation on the atomic and subatomic scale. Applied to chemical problems, quantum mechanics accounts for the properties of atoms and molecules in terms of the interactions between the constituent particles: electrons and nuclei. The motion of these particles is described in the nonrelativistic quantum mechanics by the Schrödinger equation. Schrödinger equation Partial differential equation of quantum mechanics used to calculate wave functions and energies of atoms and molecules. In chemistry it is often sufficient to consider only the time-independent version of this equation which can be written as

Hydrogen Bond

H K = E K K , where H is the Hamiltonian, K is the wave function, and E K is the energy of the system in its K th state. The ground state of the system is denoted by K = 0. Supermolecular approach The method of calculating the interaction energies by subtracting the monomers’ total energies from the dimer total energy. Symmetry-adapted perturbation theory (SAPT) The method describing the intermolecular forces including the hydrogen bonds. Wave function A solution of the Schrödinger equation. The square of the absolute value of a one-particle wave function is the probability density of finding the particle at a given point in space. The wave functions replace the trajectories of classical mechanics.

IN SIMPLEST TERMS the hydrogen bond is a type of intermolecular interaction characterized by the equilibrium configuration involving a hydrogen atom located close to the line connecting the two nearest nonhydrogen atoms of the interacting molecules. Intermolecular interactions are interactions between molecules that do not lead to formation of chemical bonds. Such interactions are also called noncovalent interactions or intermolecular forces. To discuss properties of hydrogen bonds, it will be convenient to denote one of the monomers as R1 -X-H and another one as Y-R2 . R1 and R2 are arbitrarily large molecular fragments, X and Y are sufficiently electronegative atoms— in most cases oxygen, nitrogen, or fluorine—and H is a hydrogen atom. Usually there is a lone electron pair on atom Y. The hydrogen-bonded dimer can be then denoted as R1 -X-H· · ·Y-R2 , where R1 -X-H is called the hydrogen donor and Y-R2 is called the hydrogen acceptor. In a typical hydrogen-bonded complex, the atoms X-H· · ·Y form approximately a straight line. The name hydrogen bond— which comes obviously from the central position of the hydrogen atom in the dimer—is misleading because this atom as such has no particular bonding properties (sometimes the term hydrogen bridge is used, which perhaps would be more appropriate). Instead, the bond results from a balance of four fundamental physical interactions taking place between the whole monomers. The definition of the hydrogen bond will be discussed further in sections VI and VII. Examples of hydrogen-bonded systems are presented in Fig. 1. Two monomers can be connected by more than one hydrogen bond, as in the case of the formic acid dimer shown in Fig. 1(b). Sometimes also some interactions within a single molecule resemble hydrogen bonds, as in the case of salicylic acid shown in Fig. 1(d). While hydrogen bonds involving N, O, and F atoms (both as donors and acceptors) are most common, also C and P can


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Hydrogen Bond

FIGURE 1 Examples of hydrogen-bonded systems. The hydrogen bonds are indicated by broken lines. (a) Ammonia· · ·hydrogen chloride; (b) formic acid dimer; (c) formamide dimer; (d) intramonomer bond in salicylic acid.

play the role of donors, and S, Cl, and Br can appear in both roles. The hydrogen bond is a special case of intermolecular or interatomic interactions. Consider two monomers: each of them can be an atom or molecule, separated by a distance R. To be precise, let R denote the distance between the centers of masses of each monomer. When R is sufficiently large, the monomers will typically attract each other (exceptions include large-R interactions dominated by dipole–dipole terms that can be of either sign, depending on mutual orientation). When R is shortened, up to some point the attraction will keep increasing. Upon a further shortening of the intermonomer separation there are two possibilities. The monomers may undergo a chemical reaction, forming a new molecule and releasing energy on the order of 100 kcal/mol. This new molecule is bound by chemical forces. In most cases, however, the monomers just begin to repel each other, with the repulsion increasing very quickly with the decrease of R. The forces acting between two such monomers are called intermolecular forces, and a typical shape of the potential energy surface

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507 in this case is shown in Fig. 2. The value of R where the attraction changes to repulsion is sometimes called the σ point, Rσ , of the potential (it is, of course, dependent on the mutual orientation of the two molecules). This distance can also be called “the distance of closest approach” of the two monomers. Although monomers with enough energy can get a little closer, the repulsive wall is so steep that the differences compared to Rσ are minor. The two monomers interacting via intermolecular forces can be bound or not. In a dilute gas, most monomers will have a positive energy of relative motion (i.e., these monomers will be in a scattering state, getting momentarily close to each other and then separating into remote regions of space.) A smaller number of monomers will be in bound states, staying at a finite distance from each other that is close to the equilibrium distance at the minimum of the potential, Re . The energetic location of such a bound state and the vibrational wave function are indicated in Fig. 2. The intermolecular forces are much weaker than the chemical forces because the interaction energy at the point where the attraction is strongest is typically smaller than 20 kcal/mol. For the weakest intermolecular bonds, the depth of the interaction potential can be as small as 0.02 kcal/mol, as is the case for the helium dimer. Intermolecular forces are the most prevalent forces of nature around us. The structure of all liquids and of most solid matter (an exception are metals) is determined by intermolecular forces. These forces also play a major role in shaping the properties and functions of biological systems. The intermolecular forces are often called van der Waals forces, although some authors reserve this name only for the interactions not stronger than a couple of kcal/mol. The boundary between chemical forces and intermolecular forces is, of course, flexible. Systems exhibiting the strongest intermolecular interactions may also be classified as being bound by chemical forces. An additional, although still subjective, criterion in borderline cases can be the extent of the deformation of a monomer’s geometries and electron densities upon the formation of the complex. The physical processes leading to intermolecular forces are well understood. The total interaction energy is composed of four components: electrostatic, induction, dispersion, and exchange energies. The electrostatic component is due to Coulomb interactions of the unperturbed charge distributions of the monomers. The charge distributions do get perturbed during the interaction: the charge distribution on monomer A produces an electric field on monomer B which in turn induces a charge deformation (i.e., polarizes monomer B and vice versa). This process gives rise to the induction energy. The two discussed components are completely defined in the framework of classical electrostatics. The third component, the dispersion


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FIGURE 2 A generic one-dimensional interaction potential. The equilibrium separation Re and the separation Rσ , where the potential changes from repulsive to attractive, are marked. De denotes the potential depth at the equilibrium. The position of the ground vibrational level and the ground vibrational wave function are indicated. D0 is the dissociation energy.

energy, is of quantum mechanical origin. It results from correlations of the fluctuations of the electronic charges on monomers A and B. The exchange energy is also of quantum origin. This component is the consequence of the electrons tunneling through the potential barrier between the monomers. The name follows from the fact that the electrons are exchanged between monomers during this process. Sometimes the exchange energy is interpreted in terms of the “repulsion of the electron clouds.” This picture—which derives from the fact that the exchange energy is proportional to the overlap of electronic charge distributions—does not reflect the physics of the interaction as well as the tunneling picture. Hydrogen bonds result from the same physical forces as all other intermolecular interactions. Thus, from this point of view there would be no need to define hydrogen bonding as a distinct process. Therefore, the phenomenon of hydrogen bonding is related to the structural characteristics of hydrogen-bonded dimers rather than to the physical nature of the interaction. As we will discuss below, if two monomers can be brought together to form a hydrogenbonded structure, this structure will likely be close to the

minimum on the potential energy surface. Thus, the usefulness of the hydrogen bonding concept may simply result from our ability to predict structures of dimers and larger clusters. A further reason for recognizing hydrogen bonding as a phenomenon worth a separate treatment are spectroscopic properties of the X-H stretching motion when the hydrogen atom participates in a hydrogen bond. The fundamental frequency of this vibration is significantly lowered (red shifted) in the dimer compared to that in the isolated monomer. The shift is very sensitive to the molecular environment and therefore provides a major tool to investigate the structure of hydrogen-bonded clusters, liquids, and solids. The hydrogen bond has been the subject of numerous review papers and monographs. It is always treated in works devoted to intermolecular interactions [see, e.g., Hobza and Zahradnik (1988), Stone (1996), Jeziorski and Szalewicz (1998), Müller-Dethlefs and Hobza (2000), as well as a special issue of Chemical Reviews (1994)]. Monographs restricted to the hydrogen bond include Schuster (1984), Jeffrey (1997), Scheiner (1997), and Desiraju and Steiner (1999).


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I. HISTORICAL PERSPECTIVE The subject of hydrogen bond (or bonding) is certainly a popular one. The ISI Science Citation database lists more than 26,000 papers that use this phrase in the title, abstract, or as a keyword. In the year 1999 alone there were about 3,000 such papers published. The importance of the hydrogen bond stems mainly from the fact that for certain types of molecules formation of such bonds determines the structure of the dimers or of larger molecular clusters. Thus, hydrogen bonds are characteristic features of many clusters, biological aggregates, and of condensed phases. Chemical structures containing what we now call hydrogen bonds were considered already in the early 1900s. The concept itself appeared around 1920 in works of Huggins and of Latimer and Rodebush. The term hydrogen bond was used for the first time by Pauling in the early 1930s. In the same period it became clear that anomalous properties of bulk water are due to the formation of hydrogen bonds. An important paper on this issue was published in 1933 by Bernal and Fowler (despite their not using the term hydrogen bond). The idea was then extended to other “associated” fluids. The hydrogen bonding concept was popularized by Pauling’s 1939 book Nature of Chemical Bond. While initially the experimental evidence of hydrogen bonding was coming from thermodynamic measurements of anomalous properties and from X-ray measurements of crystal structure, in the 1930s it was realized that the formation of a hydrogen bond has a profound effect on the frequency of the X-H stretch. This started the infrared investigations of hydrogen bonds, which became the most sensitive and the most widely applied experimental method of studying this phenomenon in clusters and in the liquid and solid phases. In the condensed phase the vibrational spectra are often determined with the Raman techniques, which detect lines corresponding to vibrational transitions in scattered radiation of visible frequencies. Since the early 1950s the nuclear magnetic resonance (NMR) spectroscopy has been applied to hydrogen-bonded systems. This technique is, however, utilized less often than the infrared spectroscopy due to complexity of the spectra. In the 1960s the microwave spectra in gas phase for some fairly strongly bound clusters were measured and gave precise information about structures of the clusters. Starting in the 1970s, molecular beam techniques provided a major tool for investigating small clusters. Spectroscopic measurements in molecular beams produced—via the rotational spectra—a wealth of information on geometries of clusters, including even very weakly bound ones. Also in the 1970s, neutron scattering techniques were applied to the condensed phase containing hydrogen bonds (although the earliest such studies

date back to the 1950s). The method of neutron diffraction has an advantage over the X-ray diffraction—it gives information about the positions of hydrogen atoms. In the 1980s high-precision near-infrared spectra of clusters in molecular beams gave reliable information on the frequencies of intermonomer vibrations in hydrogen-bonded clusters. In the 1990s it became possible to measure the same frequencies directly using the techniques of far-infrared spectroscopy. Very recently these methods have enabled a rather complete spectroscopic characterization of small clusters such as the water dimer.

II. QUANTUM MECHANICAL DESCRIPTION OF HYDROGEN BOND The hydrogen bond can be completely described from first principles by solving the Schrödinger equation for a set of molecules. In practice such solutions involve several approximations. Despite these approximations, with the current computer capabilities the solutions predict properties of small clusters with accuracy approaching, in some cases, experimental accuracies and provide extremely useful information on hydrogen-bonded systems, including information on systems and properties that cannot be measured. In addition to numerical information, quantum mechanics provides the framework necessary to understand the hydrogen-bond phenomenon. The theoretical approach most useful for an analysis of hydrogen bonding is symmetry-adapted perturbation theory (SAPT). For a more detailed presentation of SAPT and references to the original papers, see Jeziorski and Szalewicz (1998). This approach serves four main purposes: (a) it provides the basic conceptual framework within which intermolecular interactions—including hydrogen bonds—are discussed; in particular it provides the standard division of the intermolecular interaction into four fundamental components: electrostatic, induction, dispersion, and exchange; (b) it is the source of models used in the construction of empirical potentials (empirical force fields); (c) it provides asymptotic constraints on any potential energy surface derived either from experiment or from theory; and, finally, (d) it can accurately predict the complete intermolecular potential energy surfaces for hydrogen-bonded molecular complexes. The last goal can also be obtained using the so-called supermolecular approach [Chalasinski and Szczesniak (1994), van Duijneveldt et al. (1994)].

A. Interaction Energy To study hydrogen bonding it is sufficient to use the time-independent, nonrelativistic Schrödinger equation.


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As usual for most of chemistry, the Born-Oppenheimer approximation is assumed. This approximation relies on the observation that the electronic motion is orders of magnitude faster than the nuclear motion. Therefore, the motion of electrons is computed by solving the electronic Schrödinger equation with nuclei clamped in space. The solution of this equation provides the potential energy surface governing motion of the nuclei. For investigations of intermolecular interactions, this surface is conveniently divided into the energies of monomers and the interaction energy E int . In the dimer case E AB (R, ω A , ω B ; Ξ A , Ξ B ) = E A (Ξ A ) + E B (Ξ B ) + E int (R, ω A , ω B ; Ξ A , Ξ B ),

(1)

where E AB is the total energy of the dimer, E X is the total energy of monomer X, ω X denotes the Euler angles defining the orientation of monomer X in a dimer-embedded coordinate system, and Ξ X is the set of internal coordinates of each monomer. The energies entering Eq. (1) are solutions to the electronic Schrödinger equation. It should be noted that in definition (1) the energies of the dimer and of the monomers correspond to the same values of the internal coordinates Ξ X . In most cases of hydrogen-bonded systems, the monomers are nearly rigid compared to the dimer (i.e., the amplitudes of intramonomer motions are much smaller than those of the intermonomer ones). This is reflected in the monomer vibrational frequencies being much higher than the intermolecular ones. Thus, one can expect that the dependence of the interaction energy on the intramonomer coordinates can be neglected and that a large body of spectroscopic, scattering, and bulk phase experiments can be interpreted in terms of effective potentials depending on intermolecular coordinates only. This approximation leads to a dramatic simplification in studies of hydrogen-bonded complexes. For general complexes consisting of N atoms, there are 3N nuclear degrees of freedom, (i.e., 3N coordinates are needed to describe the position of the system in space). However, different locations of the center of mass of the complex in space as well as rotations of the whole system do not change interaction potentials. Because of these translational and rotational invariances, the energy surface of a system containing N atoms depends on 3N − 6 coordinates (3N − 5, i.e., one coordinate for a diatomic dimer like Ar2 ). Even for relatively small systems like the water dimer 3N − 6 = 12 is a large number of degrees of freedom to treat. The “frozen” monomer approximation reduces the number of degrees of freedom to only 6 for any dimer consisting of two general molecules. The simplest and apparently most natural way of obtaining rigid-monomer potentials is to perform interaction energy calculations assuming equilibrium monomer coor-

dinates. The set of equilibrium coordinates, which will be denoted by re , describe the geometry of the monomer at its potential energy minimum. Another reasonable choice of intramonomer coordinates is the geometry averaged over monomer vibrations, r 0 . The two geometries will be different since the monomer vibrations are always to some extent anharmonic. However, intuitively one can expect that the best effective potential can be obtained by averaging of the complete, monomer geometry-dependent potential over an appropriate vibrational wave function of the monomer. Computation of such an averaged potential, E int 0 , although as expensive as the computation of the complete potential energy surface, represents a useful task because its availability simplifies dramatically the spectroscopic, scattering, and bulk phase computations. For atomdiatom complexes the E int 0 potentials predict spectral quantities that are only about 0.1% different from those obtained from full three-dimensional nuclear dynamics calculations. Recently it has been shown [Jeziorska et al. (2000)] that the r 0 geometry is the optimal choice if only a single monomer geometry can be considered. The spectra of atom-diatom complexes computed using r 0 potentials exhibit deviations from the spectra computed with three-dimensional potential, which are about four times smaller than the analogous deviations produced by the re potentials. The rigid-monomer approximation will not work well if the monomers are too floppy and for complexes involving charged monomers. In the latter case the reason is that the monomers may be fairly significantly distorted upon the formation of the complex. For biopolymers, certain intramolecular coordinates do vary significantly, and these coordinates have to be included in the description of hydrogen bonds in such systems. In the supermolecular method the interaction energy is computed by subtraction of the individual energies E AB , E A , and E B . These energies are in practice calculated using finite basis sets. As a result, the monomer part of the dimer energy is improved by utilizing the basis functions of the interacting partner. This leads to a spurious lowering of the interaction energy, referred to as the basis set superposition error (BSSE). This error can be removed using the counterpoise (CP) technique introduced by Boys and Bernardi. Denoting by E Sσ (G) the electronic energy of system S at geometry G computed with basis set σ , the definition of the CP correction takes the following form β

α∪β

α∪β

δ E CP = E αA (AB) + E B (AB) − E A (AB) − E B (AB), (2) where α and β stand for the basis sets used for monomers A and B, respectively. The use of AB geometry for a monomer means that the nuclei of this monomer are at the same relative positions as in the dimer AB, and the basis


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functions originating from the interacting partner (often called the “ghost” functions) are located at the same spatial positions as in the dimer. The SAPT approach by definition does not include BSSE, since it calculates the interaction energy directly rather than by subtraction of monomer energies from the energy of the dimer. This feature made it possible to prove the correctness of the CP method by comparisons of supermolecular and SAPT calculations [van Duijneveldt et al. (1994)]. B. Perturbation Theory of Hydrogen Bond In the SAPT method, the total Hamiltonian H of the dimer is partitioned as H = H A + H B + V , where H X , X = A or B, is the total Hamiltonian for monomer X , and V is the intermolecular interaction operator. The operator V collects Coulomb interactions of all the particles of monomer A with those of monomer B. This partition means that the unperturbed operator is chosen as H0 = H A + H B and V is the perturbation. The interaction energy is then obtained directly in the form of a perturbation series in V , (1) (1) (2) (2) E int = E pol + E exch + E pol + E exch + ···,

(3)

with each term of the sum having a well-defined physical meaning. The polarization components, denoted by the subscript “pol,” are determined by the standard Rayleigh(1) Schrödinger perturbation expansion. The correction E pol is the classical electrostatic (Coulomb) interaction energy of two unperturbed charge distributions and will be written (1) as E elst . The remaining components, denoted by the subscript “exch,” are the exchange corrections accounting for the effect of resonance tunneling of electrons between the interacting systems. These contributions originate from the antisymmetrization of the polarization corrections to the wave function. The second-order corrections can be further divided into induction and dispersion components: (2) (2) (2) E pol = E ind + E disp ,

(2) (2) (2) E exch = E exch−ind + E exch−disp .

(4) The polarization energies through second order have a simple physical interpretation and can be rigorously expressed through monomer properties. 1. Electrostatic Interaction The electrostatic energy, the lowest order polarization component, is defined as (1) E elst (5) = 0A 0B V 0A 0B , where 0X is the unperturbed wave function of monomer X and f | g = f ∗ gdτ with the integration extending over all electron coordinates. The electrostatic energy can be expressed in terms of the total charge distributions ρ A (r )

and ρ B (r ) of the monomers, showing clearly its relation to the Coulomb law 1 (1) E elst = ρ A (r 1 ) (6) ρ B (r 2 ) d 3r 1 d 3r 2 . |r 1 − r 2 | The total electric charge distribution ρ X (r ) for monomer X is the sum of the electronic contribution—which can be obtained from the wave function 0X —and the nuclear contribution. The electrostatic interaction plays a major role in determining the structure of dimers consisting of polar molecules, in particular hydrogen-bonded systems. The evaluation of the electrostatic interaction energy for such systems is often performed by approximating the electrostatic potential of a molecule by that resulting from a set of point charges or from a single-center or multicenter distribution of multipole moments. However, one should emphasize that the electrostatic energy contains also important short-range terms due to the mutual penetration (charge overlap) of monomers’ electron clouds. This short-range part of the electrostatic energy, neglected both in the monocentered and distributed multipole expansions, makes significant contributions to the stabilization energy of hydrogen-bonded systems. The electrostatic interaction can be either attractive or repulsive. This property is best illustrated by large-R interactions of two neutral molecules possessing dipole moments. The electrostatic dipole– dipole term dominates then the interaction energy. This term can be either attractive or repulsive, depending on the mutual orientation, with the maximum and minimum of equal magnitudes. Often the electrostatic interaction energy alone, or rather its asymptotic form, is used to determine the approximate equilibrium orientation (but not the equilibrium separation) of hydrogen-bonded clusters. Due to the importance of this interaction, such predictions are frequently correct, and a few examples will be given in section V. The electrostatic predictions are not correct when other components of the interaction energy have anisotropies significantly different from the anisotropy of the electrostatic component (cf. section V). In fact, those anisotropies are always to a smaller or larger extent different so that the exact minimum structure of a hydrogen-bonded cluster can be found only by taking into account all of them. 2. Induction Interaction (2) The second-order polarization energy E pol is given by the expression

(2) E pol

=

0A 0B V KA LB 2 KL

E 0A + E 0B − E KA − E LB

,

(7)


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512 where K and L label excited states of the monomer A and B, respectively, and the prime over the summation sign indicates that the term with K = 0 and L = 0 is excluded. X X Eigenfunctions M and eigenenergies E M are solutions of the Schrödinger equation, with the Hamiltonian H X for monomer X describing the ground and excited states of this system. The induction energy is obtained when the sum over states in Eq. (7) is restricted to functions excited on only one monomer (i.e., to the functions of the (2) form 0A LB and KA 0B ). The corresponding part of E pol , (2) (2) (2) (2) denoted by E ind , is given by E ind = E ind (A) + E ind (B), where (2) E ind (A) = 0A B (1) (8) ind (A) (2) and a similar expression holds for E ind (B). The induction (1) function ind (A) is defined by KA B 0A (1) ind (A) = KA . (9) A A E − E K 0 K =0

The operator B = 0B |V 0B is the electrostatic potential generated by the unperturbed monomer B. Equation (8) has the form of the second-order energy correction arising when monomer A is perturbed by the static electric field generated by the (unperturbed) monomer B. Notice that Eqs. (8) and (9) include only the coordinates of electrons belonging to system A, all the effects of system B entering via the potential B . The second-order induction energy results, thus, from the polarization of the monomers by the static electric fields of unperturbed partners (in older literature the induction contribution is sometimes referred to as the polarization energy). Asymptotically, at large intermolecular distances R, this effect is fully determined by the permanent multipole moments and static multipole polarizabilities of the monomers. At finite R, additional monomer information is needed to account for the short-range, penetration part (2) of E ind . Because monomers that form hydrogen bonds almost always are very polar, the induction energy for hydrogenbonded clusters makes a very significant contribution to the total interaction energy. It is also quite anisotropic although less than the electrostatic energy. Some authors proposed to identify within the induction energy a term called the charge transfer energy. This term was assumed to describe the interactions due to the transfer of a part of the electronic charge from monomer A to monomer B or vice versa. Although such a transfer certainly takes place to some extent, a quantitative determination of this component proved to be not possible thus far. Methods proposed by various authors lead to dramatically different estimates of the size of this effect. Therefore, we will not discuss the hypothetical charge-transfer energy com-

Hydrogen Bond

ponent any further. In contrast, the difference between the dimer electron density and the sum of the densities of the two monomers is perfectly well defined and can be computed to interpret experimental observations related to the charge density, like, for example, NMR spectra. 3. Dispersion Interaction (2) is defined as The second-order dispersion energy E disp the difference between the second-order polarization and (2) (2) (2) induction energies, E disp = E pol − E ind . Therefore it can be written as 0A 0B V KA LB 2 (2) E disp = E A + E 0B − E KA − E LB K =0 L =0 0 (10) = 0A 0B V (1) disp ,

where (1) disp

=

K =0 L =0

KA LB V 0A 0B

E 0A + E 0B − E KA − E LB

KA LB ,

(11)

is the “dispersion function” representing the leading intermolecular correlation contribution to the dimer wave function. This function is a sum of products of wave functions that are describing the electronic excitations on both the monomer A and B; therefore the dispersion interaction can be viewed as the stabilizing energetic effect of the correlation of instantaneous multipole moments of the monomers. Since the dispersion energy is a correlation effect, it cannot be reproduced at the Hartree–Fock level of theory. The dispersion force results from the dependence of electrons of system B on the position of an electron in molecule A; therefore it goes beyond the averaged charge distributions characteristic of the Hartree–Fock method. The dispersion energy is usually the most isotropic component of the interaction in hydrogen-bonded clusters. Although it does not have a large effect on the cluster’s equilibrium orientation, it contributes significantly to the energy of hydrogen bonds, being of similar size to the induction energy for medium-size systems but playing an increasingly larger role as the system size grows. 4. Exchange Interaction The sum of low-order polarization corrections, even when evaluated exactly without the use of the multipole expansion, is not able to predict the existence of the van der Waals minimum or the repulsive wall at shorter intermolecular separations. The repulsive contributions are due to the electron exchange (i.e., to the physical process of the [resonance] tunneling of electrons between interacting systems). The unperturbed function 0 = 0A 0B as well


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Hydrogen Bond

as higher-order polarization functions such as (1) ind (X ) and describe the situation when the electrons stay within (1) disp their monomers, but the permanent and instantaneous polarization is allowed (therefore the name “polarization expansion”). The true wave function contains also components corresponding to tunneling of one, two, or more electron pairs between interacting units. In the case of two hydrogen atoms, both electrons can simultaneously tunnel in opposite directions between the two potential energy wells in the six-dimensional configuration space (one well with electron 1 at nucleus A and electron 2 at nucleus B and the other with the electrons exchanged). Because the two wells are equivalent by symmetry, the tunneling leads to the resonance splitting of the unperturbed energy level into the energies of the singlet and the triplet states. Asymptotically at large R the exact wave function becomes a linear combination of two equally weighted “resonance strucA B A B tures” 0 = φ1s (r 1 )φ1s (r 2 ) and P12 0 ≡ φ1s (r 2 )φ1s (r 1 ), the second of which cannot be recovered by a low-order X polarization theory (φ1s is the 1s orbital of hydrogen centered on atom X). It is clear that A0 ≡ √12 (0 ± P12 0 ) rather than 0 should be used as the unperturbed wave function in the perturbation theory of intermolecular interactions. Unfortunately, the use of A0 in the conventional Rayleigh-Schrödinger perturbation procedure (employing the sum of the monomer Hamiltonians as the unperturbed operator) is impossible since A0 is not an eigenfunction of the zeroth-order operator H0 . Therefore, a modification of the perturbation procedure is required such that the function A0 can be utilized in the perturbation development. Such a modification, usually referred to as symmetry adaptation, was first attempted in 1930 by Eisenschitz and London, and there has been continued activity in this field since the late 1960s. The currently most popular variant of symmetry adaptation, called the Symmetrized Rayleigh– Schrödinger method, was proposed in 1978 by Jeziorski, Chalasinski, and the present author. Since the exchange effects decay exponentially with distance, they are negligible for large R. In the region of

cases the exchange contribution at the potential minimum is the component that is largest in magnitude. It is also significantly anisotropic. 5. Multipole Expansion As the distance between monomers grows larger, the importance of various components of the interaction energy changes. The exchange effects, dominating at short separations, become negligible due to their fast, exponential decay with R. The polarization components decay much slower. This is easy to understand for the electrostatic energy which—in the case of neutral polar monomers—has to reduce for very large distances to a dipole–dipole interaction proportional to 1/R 3 . One can show in general that at large R the interaction energy E int has the following asymptotic expansion in powers of 1/R: E int (R, ω A , ω B , Ξ A , Ξ B) ∼

∞

Cn(ω A , ω B , Ξ A , Ξ B)R −n.

n=1

(12) Although it is only asymptotically convergent in Eq. (12) can approximate the exact interaction energy arbitrarily closely when R is sufficiently large. Therefore, the knowledge of the Cn coefficients is very useful in estimating the interaction energy at large distances, and is necessary to guarantee the correct large-R asymptotic behavior of empirical or theoretically derived potential energy surfaces. The angular dependence of the coefficients Cn (ω A , ω B , Ξ A , Ξ B ) for fixed internal geometries Ξ A , Ξ B of the monomers can be expressed in a closed form. The coefficients Cn can be computed from properties of monomers such as multipole moments and polarizabilities. The relevant formulas are obtained from the polarization series truncated at some finite order by replacing the potential V by its asymptotic expansion in powers of 1/R. For the Coulomb potential 1/|r 1 −r 2 |, such expansion has the form

l< ∞ 1 (−1)l B (l A + l B )! = √ |r 1 − r 2 | l A ,l B =0 m=−l< (2l A + 1)(2l B + 1)(l A + m)!(l B + m)!(l A − m)!(l B − m)!

× r1l A r2l B YlmA (θ1 , φ1 )Yl−m (θ2 , φ2 )R −l A −l B −1 , B the potential minimum these effects are, however, always very important. In the case of hydrogen-bonded dimers near the potential minimum, the electrostatic, induction, and dispersion effects are all negative (the latter two are in fact always smaller than zero). The positive exchange energy cancels a large part of the attractive effect. In many

(13)

where ri , θi , φi are the polar coordinates of ith particle and l< denotes the smaller of l A and l B . The coordinates of particle 1 are measured in system A, while those of particle 2 are measured in system B. The two coordinate systems have their z axes along the same line, and the other axes of one system are parallel to the


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514 respective axes of the other system. The functions Ylm (θ, φ) are the standard spherical harmonics. The expansion is valid only when R > r1 + r2 . Combining the equivalents of Eq. (13) for other Coulombic terms in the intermolecular interaction operator V , one gets the asymptotic expansion of V . Individual terms in this expansion are interpreted as arising from the interaction of a 2l A pole moment on monomer A (i.e., an instantaneous dipole, quadrupole, etc., moment formed by an electron) with a 2l B -pole moment on monomer B. Therefore the expansion in Eq. (13) is often called the multipole expansion of the potential. Equation (13) shows that after the multipole expansion of the potential is performed, the coordinates of electrons of molecule A are separated from those of molecule B. Therefore, if this expansion is substituted in Eqs. (5), (8), (10), and (11), the asymptotic interaction energy is expressible in terms of integrals each involving only coordinates of one monomer. Thus, this energy is expressible in terms of monomer properties only. This fact makes the calculations in the asymptotic region much easier than for the finite distances, where this approximation cannot be applied. It is important to realize that the multipole expansion has built-in dependence on the mutual orientation of two monomers. Thus, asymptotically the anisotropy of the polarization part of the interaction energy is precisely predicted by this expansion. Although for finite separations this anisotropy is modified by the penetration effects discussed below, the asymptotic prediction remains useful at finite R. The lowest power of 1/R appearing in the asymptotic expansion determines the behavior of the energy for very large separations. Monomers forming hydrogen bonds almost always have nonzero dipole moments. The electrostatic dipole–dipole interaction becomes then the dominating long-range term at very large R, because it decays as 1/R 3 . The consecutive electrostatic terms, dipole– quadrupole and quadrupole–quadrupole, decay as 1/R 4 and 1/R 5 , respectively. For neutral monomers with nonvanishing dipole moments, the induction energy decays as 1/R 6 , whereas the dispersion energy always decays as 1/R 6 (i.e., both components decay faster than the electrostatic interaction of polar monomers). Thus, the electrostatics strongly dominate the long-range interaction energy for hydrogen-bonded systems. For charged systems the role of electrostatics is even more important: the charge–charge, charge–dipole, and charge-induced dipole terms decay as 1/R, 1/R 2 , and 1/R 4 , respectively. Notice that for polar monomers at some angular orientations the interaction energy will be positive rather than negative for large enough R due to the positive dipole– dipole contribution. When the intermolecular distance gets sufficiently short, the multipole expansion fails to provide a good ap-

Hydrogen Bond

proximation to the interaction energy. The distance where this starts to happen is significantly larger than the equilibrium separation. The quantitative measure of the deviation can be obtained by comparing the results of SAPT calculations and of the multipole expansion for a given configuration. At large enough distances SAPT agrees with the asymptotic expansion arbitrarily well. In addition, SAPT electrostatic, induction, and dispersion energy components can be directly compared with analogous components of the asymptotic expansion, and the agreement is reached for each component separately. As the distance decreases, the exchange effects are becoming important so that the total interaction energies predicted by SAPT and by the asymptotic expansion begin to differ. The individual components begin to differ as well, although these differences are smaller than the exchange effects. The discrepancies in the components appear because the terms neglected when assuming the multipole expansion of the interaction potential are not negligible anymore. These terms are related to the overlap of electron charge distributions and therefore are called the overlap or penetration effects. When analytic forms of the polarization components are developed using the multipole expansion, the penetration effects have to be taken into account. Most often this is done by multiplying the 1/R terms by the so-called damping functions, decreasing the magnitude of these components for smaller R. In addition to damped 1/R terms, the polarization energies contain also purely exponential components that have to be included for an accurate modeling of these energies. The type of analysis of the interaction energy presented here is very useful for obtaining analytic fits to potential energy surfaces.

C. Many-Body Effects The interaction energy of a system consisting of N molecules can be defined similarly to the dimer energy of Eq. (1) as E i (Ξi ), (14) E int (ξ 1 , . . . , ξ N ) = E tot (ξ 1 , . . . , ξ N ) − i

where E tot is the total energy of the N -mer, E i is the energy of the ith monomer, and ξi = (Ri , ω i , Ξi ) stands for the set of all coordinates needed to specify the spatial position Ri , orientation ω i , and the internal geometry Ξi of the ith monomer. The N -mer interaction energy can be expressed as a sum over 2, 3, . . . , N -body interactions: E int = E int [2, N ] + E int [3, N ] + · · · + E int [N , N ], (15) where K -body contributions to the N -mer energy, E int [K , N ], can be written as the following sums:


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E int [2, N ] =

E int (ξi , ξ j )[2, 2],

(16)

i< j

E int [3, N ] =

20:6

E int (ξi , ξ j , ξ k )[3, 3],

(17)

i< j 1, the relationship k − /k + = cmc may be used to extract both forward and reverse rate constants. In Table II, values of k + and k − are given for a series of sodium alkyl sulfates, along with values of σ, n, and the cmc. It shows a quite dramatic dependence of exitrate constants on chain length. On the other hand, k + values are slower than would be predicted by simple diffusion considerations, and it has been suggested that this is due to long-term repulsions by the surface potential field. The measurements of such processes may be made by various techniques, (e.g., p-jump and ultrasonic absorption). After the initial process (relaxation to the quasiequilibrium state) just described, a change in the number of micelles occurs, representing relaxation to the true

cmc (M)

nb σ c

k− (s−1 )

k+ (M−1 s−1 )

NaC6 SO4

0.42

17

6

1.32 × 109

3.2 × 109

NaC7 SO4

0.22

22 10

7.3 × 108

3.3 × 109

NaC8 SO4

0.13

27

—

1.0 × 108

7.7 × 108

NaC9 SO4 NaC10 SO4 d

6.10−2 3.3 × 10−2

33 41

— —

1.4 × 108 9 × 107

2.3 × 109 2.7 × 109

NaC11 SO4

1.6 × 10−2

52

—

4 × 107

2.6 × 109

NaC12 SO4

8.2 × 10−3

64 13

1.0 × 107

1.2 × 109

NaC14 SO4 NaC16 SO4 e

2.05 × 10−3 4.5 × 10−4

80 16.5 100 11

9.6 × 105 6 × 104

4.7 × 108 1.3 × 108

a Data from Aniansson, E. A. G. et al. (1976). J. Phys. Chem. 80, 905–917, as compiled by Lindman, B., and Wennerström, H. (1980). Top. Curr. Chem. 87, 1–83. b Mean aggregation number. c Standard deviation for the micelle size distribution. d 40◦ C. e 30◦ C.

equilibrium. The relaxation time τ2 is heavily dependent on the sum of dissociation rates kn− × Sn for aggregation numbers (n) between monomer and proper micelles. Characteristically, τ2 values range from 10−3 to l s−1 , increasing with temperature and decreasing with added salt. The effects of both relaxation steps on micelle size distribution are illustrated in Fig. 10.

V. SOLUBILIZATION In terms of utility, the most important feature of the micelle is its ability to take up or solubilize nonsurfactant materials. A large number of substances not readily soluble in water may be dissolved in surfactant solutions (e.g., hydrocarbon substances). A principal application of this characteristic is to household detergent, by which waterinsoluble “dirt and grease” are suspended in the micellar pseudo-phase and separated from the material to be cleaned. One technological variation of this use involves the recovery of petroleum, adherent to underground rock surfaces, by pumping detergent solutions into oil wells. Under appropriate economic conditions, this could bring into production once more, sites that have been abandoned when conventional techniques ceased to give adequate yield. It may be seen that in surfactant solutions, the solubility of various hydrophobic substances can increase rapidly only above the cmc and that association does not occur significantly with individual surfactant monomers. Figure 11 illustrates changes in decanol solubility as a function of surfactant concentration in several systems. Solubility of

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FIGURE 11 Solubility of decanol in solutions of (a) sodium octanoate (20◦ C), (b) sodium decanoate (40◦ C), (c) sodium dodecanoate (40◦ C), and (d) sodium tetradecanoate (40◦ C). [From Ekwall, P. et al. (1969). Mol. Cryst. Liq. Cryst. 8, 157–213.]

such substances in aqueous solution has been used as a measure of the cmc. Because the extent of solubilization is dependent upon the quantity of micellar surfactant, such behavior has been interpreted in terms of the phase separation model where the surfactant is considered to be a separate or pseudo-phase. Indeed, various spectroscopic observations indicate that many solubilizates move in a liquidlike environment, an observation that is consistent with the idea of a separate phase. One consequence of this approach has been to show the validity of a relationship analogous to Henry’s law: cmc(Xa) = cmc(Xa = 0) − k · Xa

(4)

for micellar systems, where Xa is the mole fraction of the material to be solubilized. As described previously, the micelle exhibits a surface region with significant polar character, while the interior is essentially hydrocarbon in nature. It is to be expected that materials associating with micelles would distribute themselves between these two regions depending on polarity or polarizability. Indeed, those materials such as aliphatic hydrocarbons that have no surface activity are to be found in the micellar core while the more polarizable aromatic hydrocarbons have been found to associate with the surface region. Many such species give evidence of lying in an alcohol-like environment. There are further energetic considerations related to the effects of surfactant packing on the micellar structure itself that influence the distribution

of a solubilizate. As an illustration, solute near the surface can produce a gain in energy by decreasing the ratio of gauche to trans configurations in hydrocarbon chains in that region. Rigid molecules such as planar aromatic hydrocarbons may perturb packing in the core, which may contribute to their unfavorable location in that domain. While various techniques, such as stopped flow, have been used to follow substrate kinetics, many kinetic measurements have involved the photophysical properties of solubilized probes. Because of the luminescent properties of their excited states, the aromatic hydrocarbons provide opportunities for monitoring movement of such probes across the micelle boundary. For example, long-lived phosphorescence of aromatic hydrocarbons has been monitored in micellar solutions containing ionic quenchers that themselves are repelled by the surfactant head groups. Since quenching must take place in the aqueous phase, phosphorescence lifetimes may be interpreted to provide rate constants for exit of the probe from the micelle. Some typical values obtained by this technique are given in Table III. Fluorescence data have also been used to obtain such information.

VI. MICELLAR CATALYSIS It has been shown that a number of chemical processes can be kinetically altered in the presence of micelles. Either

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671

Micelles TABLE III Solubilizate Entry and Exit Rate Constants in Micellar SDSa Solubilizate Oxygen

k+ (M−1 s−1 ) 1 × 1010

k− (s−1 ) 5 × 107

Benzene

4.4 × 106

Naphthalene

2.5 × 105

Anthracene Pyrene

1.7 × 104 4.1 × 103

1-Bromonaphthalene

3.3 × 104

Perylene

4.1 × 102

a

Compilation taken from Fendler, J. H. (1982). “Membrane Mimetic Chemistry,” Wiley, New York.

rates of reaction or the distribution of products can be significantly altered when either reactants or products associate with surfactant micelles. This recognition has led to increased investigation into such behavior, and some simple examples are presented here. Several aspects of micelle–reactant interaction are suggested as responsible for the phenomenon called micellan catalysis. It is suggested that the micellar medium can affect transition states of reactions, thereby altering their rates; and some cases are cited in the literature. Seemingly, however, the most important effect in micellar catalysis relates to the control of local concentrations of components in bimolecular reactions. If a reactant exhibits some hydrophobic character, it may preferentially solubilize in the micellar pseudo-phase where its concentration can be controlled. For charged micelles, ionic reaction components can be localized at or repelled from the micelle surface. One may note, for example, that experiments with pH-sensitive dyes have shown that pH at the micelle surface can be lowered by a couple of units for negatively charged micelles such as SDS and raised by two units in positively charged systems such as CTAB. One of the most widely discussed processes involving micellar catalysis deals with hydrolysis reactions of the form O O || || −

R C O R + OH R C O− + R OH

(5)

Of course, for esters that solubilize, it is expected that reaction rates will depend on concentrations of OH− local to the micelle. One would expect that such concentrations will be enhanced in the presence of positively charged micelles and inhibited by negatively charged ones. Figure 12 gives examples of systems that follow those expectations. It may be seen that the response to concentrations of surfactant depend on the length of the hydrocarbon chain in the ester, suggesting some variation in extent to which the esters are solubilized.

FIGURE 12 Plots of kobs for the hydrolysis of p-nitrophenyl acetate (curves A), mono- p-nitrophenyl dodecanedioate (curves B), and p-nitrophenyl octanoate (curves C) versus (a) concentration of sodium dodecanoate at pH 9.59 = 0.1 and 50◦ and versus (b) concentration of n-dodecyltrimethyl ammonium bromide (LTAB) at pH 10.49 = 0.2 and 50◦ . Values of kobs for the reaction of A with sodium laurate have been divided by 2.0 to bring the curve on scale. [From Menger, F. M., and Portnoy, C. E. (1967). J. Am. Chem. Soc. 89, 4698–4703.]

The data presented by such reactions were analyzed in terms of the simple kinetic scheme

Sn E kw

P

SnE km ,

(6)

P

where Sn is the micelle, E is the substrate, and P is the product. It may be shown that the rate constant for the reaction in the absence of micelles kw and the observed rate of reaction kobs may be related to the concentration of micelles by the expression

1 1 1 k w kobs k w km (k w km)K[Sn] ,

(7)

which, it may be noted, bears resemblance to treatments of enzyme kinetics. Under the right conditions, plots of (kw + kobs )−1 versus [Sn ]−1 may be seen to yield rate constants for the micelle catalyzed reaction km and the binding constant, K s , for the equation. However, the simple conditions defined in Eq. (6) are not met under conditions described by Fig. 12b where both OH− and substrate can bind to the micelle. One should note that overall parallels drawn between micellar catalysis and enzyme behavior must be treated with care because enzymes exhibit a degree of site selectivity not seen in micellar systems. It may be noted that such a simple unimolecular treatment holds when we consider partitioning one reaction component. In the late 1980s, much effort has been expended to develop sophisticated models for catalysis that deal with bimolecular reactions and address kinetic dependencies on surface charge, pH, buffers, chain length, and specific salt effects. The literature in this area has become rather extensive.

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In systems where more than one reaction pathway is possible, localization of reactant by micelles may govern the kinetics and determine product yield. An example of this may be nicely seen in the photolysis of ketone:

O CH2

C

CH2

CH3

For convenience, one may use the notation

A

CH2

and

B

CH2

CH3

and write the reaction

ACB

h

AA AB BB

(8)

O The distribution of products in homogeneous solution is governed by simple statistics as shown in Fig. 13. However, with the addition of surfactant up to a concentration that localizes each reactant molecule in a micellar “cage,” the yield of AB is totally dominant. A large number of studies have been carried out with fast-reaction kinetics techniques to characterize micellar

effects on reaction rates of excited states and radiolytically generated radicals. While these might appear to be highly specialized measurements, the time resolution that may be achieved provides insight into the catalysis of many micelle-related events on a very short time scale. For example, the movement of charged radicals, such as hydrated electrons, to reaction sites at positively charged micellar surfaces has been shown to increase up to two orders of magnitude over diffusion control. Disproportionation of Br− 2 radicals has been shown to occur much more rapidly on a CTAB micelle surface than in solution. Initial steps in the radical processes governing lipid peroxidation, a mechanism of particular biological interest, have been studied in micelles to determine the effects of a membrane-mimic environment on radical behavior. Micellar influence on kinetics of excited-state quenching by adsorbed anions, photoionization of sequestered chromophores, and many other photoprocesses have been investigated.

VII. MICELLES AT INTERFACES For all the structural and behavioral complexity that micelles present, they are the simplest assembly of amphiphilic molecules with which to deal experimentally. In most cases one merely dissolves the surfactant of interest in water, and the gods of thermodynamics do the rest. While only a limited number of biological amphiphiles actually aggregate in this way (bilesalts, fatty acid salts, and lysolecithin), micelles do present accessible model systems with which to approach phenomena governing more complex, extended assemblies such as biological membranes. Although the limitations of micelle–membrane comparisons must be kept in focus, a wide range of information characterizing hydrophobic interaction and water– lipid interfacial phenomena, which are highly relevant to biological systems, has been built up from the collective study of micelles.

VIII. SURFACTANT AGGREGATES AT SOLID-LIQUID INTERFACES

FIGURE 13 Dependence of product distribution from photolysis of dissymmetrical dibenzylketones on CTACI concentration. [From Turro, N. S., and Cherry, W. P. (1978). J. Am. Chem. Soc. 100, 7431–7432.]

Surfactants are generally attracted to solid-liquid and liquid-air interfaces, and this interfacial enrichment is vital to a large number of industrial applications. (Even the term surfactant—short for surface-active agent—betrays the central importance of interfaces.) One such application is foam flotation in the mining industry, in which surfactant adsorption to ore microparticles causes them to flocculate at the surfaces of air bubbles and rise to the foam layer, where they are skimmed from the remaining matrix.

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(hydrophilic surfaces)

Adsorption density

(hydrophobic surfaces)

CMC Surfactant concentration FIGURE 14 Schematic of “two-step” absorption isotherms for surfactants on hydrophobic surfaces (dashed line, right vertical scale) and on hydrophilic surfaces (solid line, left vertical scale). The absorption models are inferred from the measured amounts of surfactant absorbed per unit surface area.

Another is particulate detergency, in which surfactant infiltration between a soil particle and the substrate eventually desorbs the soil and prevents its readsorption. A final example is tertiary oil recovery, in which oil is dislodged from the microchannels of porous rocks by competitive surfactant adsorption at the pore surfaces and by emulsification of the oil. Despite the importance of these applications, knowledge of surfactant behavior at interfaces has historically lagged far behind that in bulk solutions. Which morphologies surfactant aggregates assumed at interfaces, or even whether well-defined aggregates analogous to bulk micelles existed at interfaces, remained open questions until fairly recently. The first evidence for hydrophobic association at interfaces came from adsorption isotherms—i.e., measurements of surface adsorption density as a function of surfactant concentration in solution. In a landmark 1955 paper, A. M. Gaudin and D. N. Fuerstenau noted that the surface density of SDS on alumina increased sharply as the concentration approached the cmc, and they interpreted this as evidence for hydrophobic association into interfacial aggregates termed hemimicelles. Confirming evidence for interfacial aggregation has since come from hundreds of adsorption studies on a vari-

ety of hydrophobic and hydrophilic surfaces. At the risk of overgeneralizing, adsorption isotherms typically follow a two-step pattern (Fig. 14) for both hydrophobic and hydrophilic surfaces. In both cases, a low-density plateau at low concentrations gives way to a high-density plateau as the concentration approaches the cmc; this final adsorption density remains approximately constant up to very high surfactant concentrations. The surfactant organization at each plateau is inferred from the measured surface density and from known interaction sites. Hydrophobic surfaces, which interact with tailgroups via hydrophobic interactions, display a low-density plateau consistent with a horizontal monolayer and a high-density plateau consistent with a vertical monolayer, with tailgroups facing the surface. This is in contrast with charged hydrophilic surfaces, which interact electrostatically with oppositely charged headgroups. Here the low-density plateau approximately corresponds to a vertical monolayer (headgroups facing the surface), whereas the high-density plateau is consistent with a vertical bilayer above the cmc. However, while these flat morphologies served as the standard models of interfacial aggregation, uncertainties in surface area determination (which can approach 30%) could not definitively exclude curved interfacial

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aggregates resembling bulk micelles. (Indeed, many workers found in adsorption isotherms consistent evidence for defective or incomplete layers or even for adsorbed micelles.) While hydrophobic association was agreed to be the mechanism for interfacial adsorption near the cmc, the corresponding surfactant structure was unknown until recently. A. Imaging Interfacial Aggregates by Atomic Force Microscopy In 1995, the high-density structures above the cmc were imaged directly by atomic force microscopy (AFM) for both hydrophilic and hydrophobic surfaces. These results showed for the first time that interfacial micelles existed in well-defined shapes and sizes, that they generally possessed spherical or cylindrical curvature (in contrast to the standard models), and that this curvature was a compromise between the spontaneous curvature of bulk micelles and the constraints imposed by the flat surface. Briefly, AFM is a technique that maps the topography of a surface by plotting (on a color scale) the measured force between the surface and a small tip attached to a sensitive cantilever spring. In most applications, the tip is in direct contact with the surface, and the AFM performs as a sensitive contact profilometer. For imaging interfacial surfactant structures, however, contact forces disrupt the liquid crystalline aggregates. Therefore the repulsive colloidal stabilization forces between the surfactant layers adsorbed to the tip and sample are used as the contrast mechanism during imaging. A simplified schematic of the imaging mechanism is shown in Fig. 15 for ionic surfactants. The tip and sample are immersed in surfactant solution above the cmc (where the high-density plateau occurs in the adsorption isotherm). Surfactant adsorption on the tip and sample charges both with the same sign, resulting in a longranged, screened electrostatic repulsion between the two (Fig. 15a). By fixing the imaging force (using a feedback loop) in this noncontact regime, the AFM “flies” the imaging probe above the aggregate layer while obtaining a “surface map” of the colloidal stabilization force. This map of the tip-sample repulsion reveals the surfactant aggregate structure at the interface (Fig. 15b). Comparing AFM images with other data (e.g., adsorption isotherms) usually fixes the interfacial micelle structure uniquely. AFM imaging has been used to identify interfacial aggregate structure above the cmc for a variety of ionic, nonionic, and zwitterionic surfactants on both hydrophobic and hydrophilic surfaces. (Structures far below the cmc, corresponding to the low-density adsorption plateau, cannot be imaged readily because the tip-sample force is strongly hydrophobic and attractive in this regime.) The

B FIGURE 15 AFM imaging mechanism for interfacial surfactant aggregates. (a) Surfactant adsorption on the tip and sample creates a long-range repulsion between the tip and sample, down to separations (around 5 nm) where the opposing surfactant layers touch and fuse together. Imaging in the noncontact regime allows the AFM tip to obtain a map of the surfactant aggregate structure. (b) A sample AFM image (200 × 200 nm) of spherical micelles in a hexagonal pattern at the mica-solution interface. The surfactant is a divalent cationic surfactant with a C18 tail.

most popular substrates have been layered crystals, owing to the ease of surface preparation (cleaving by adhesive tape) and the variety of available surface properties. A summary of observed aggregate structures follows. B. Aggregates at Hydrophobic Surfaces AFM results (Fig. 16a) show that the morphology of surfactant aggregates at hydrophobic surfaces depends

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A

B FIGURE 16 AFM images and schematics of alkyltrimethylammonium surfactant aggregates on (a) the hydrophobic graphite surface and (b) the hydrophilic mica surface. In (a), the bottom row of tails are aligned parallel to a surface symmetry axis, orienting the half-cylinder perpendicular to this axis. In (b), the smaller contact area between the aggregate and surface allows the cylindrical aggregate to meander over the surface.

heavily on the crystalline anisotropy of the substrate, with the surfactant geometry itself playing a comparatively minor role. Almost all surfactants—ionic, nonionic, and zwitterionic, with tail lengths ranging from 12 to 18 carbon atoms—aggregate in the form of half-cylindrical aggregates on cleaved crystals of graphite and MoS2 . This is evidenced by AFM images in the form of rigid, parallel stripes, separated by a little over twice the surfactant length, with stripe axes running perpendicular to the underlying lattice symmetry axes. Parallel half-cylinders are consistent with the observed stripe spacing and with the known adsorption density (roughly equivalent to a vertical monolayer) from isotherms. The cylindrical curvature is, however, initially surprising considering that these surfactants form spherical

micelles in bulk. The orientation of the aggregates with respect to the surface lattice suggests that the crystalline anisotropy of the substrate plays a central role in determining this curvature. This has been further confirmed by control experiments on amorphous hydrophobic surfaces; these show half-spherical micelles above the cmc, in agreement with the spontaneous curvature in bulk solution. The current understanding of the adsorption and aggregation process is as follows. At very low concentrations, hydrophobic attraction causes the tailgroup to adsorb horizontally on the surface, and an anisotropic interaction with the surface lattice causes the tailgroup to orient itself parallel to an underlying symmetry axis. Because the tail-surface interaction for this configuration is typically

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676 stronger than that between two tails, the molecule does not gradually desorb to a vertical orientation (as in Fig. 14) as the surfactant concentration is increased. Instead, each strip of horizontal molecules, arranged head-to-head and tail-to-tail along a symmetry axis, serves as a foundation for a half-cylindrical aggregate above the cmc (see the schematic of Fig. 16a). Apparently, the original alignment of tails parallel to a symmetry axis not only determines the direction of the final half-cylindrical aggregate, but also the fact that half-cylinders are the favored arrangement. Where the lattice alignment is absent—i.e., for isotropic substrates such as hydrophobized silica—the same surfactants form half-spherical aggregates. Both graphite and MoS2 —despite marked differences in lattice symmetry, spacing, and surface groups—give rise to similar half-cylindrical aggregates at the same perpendicular orientation relative to the underlying lattice. This suggests that strong tailgroup alignment, leading to half-cylindrical aggregates, may be a feature common to all crystalline hydrophobic adsorbents. Since hydrophobic surfaces interact with the entire length of the tailgroup, this large interaction area evidently leads to a high degree of surface control of the aggregate structure. C. Aggregates at Hydrophilic Surfaces Hydrophilic surfaces, on the other hand, interact with the much smaller surfactant headgroup. It is therefore natural to expect “full” aggregate structures, whose curvature is controlled as much by intermolecular interactions as by the surface. This is exactly what is observed by AFM. The vast majority of experiments in this category have investigated ionic surfactants on oppositely charged surfaces. On the anionic surface of silica, single-chain cationic surfactants self-assemble into spherical aggregates (resembling bulk micelles) above the cmc. These are thought to originate from the electrostatic binding between charge sites on the surface and individual surfactant molecules; the latter then serve as nucleation sites for micellar aggregation above the cmc. Similar results have been observed with anionic surfactants on the cationic surface of alumina. In both cases, the spherical interfacial aggregates are consistent with the curvature found in bulk solution. Micelle curvature is expected to be relatively unperturbed as long as the substrate charge density (i.e., adsorption site density) falls short of the charge density on the outer surface of the micelle. Most surfaces satisfy this requirement. A notable exception is the anionic cleavage plane of mica, where exchangeable surface ions give rise to a far higher adsorption density than on silica. Alkyltrimethylammonium surfactants on mica self-assemble into parallel cylindrical aggregates (see Fig. 16b)—a higher-density

Micelles

structure than close-packed spheres and a flatter curvature than is found for bulk micelles. The mica surface has been likened to a highly charged “planar counterion,” which induces a sphere-to-rod transition at the interface, in a similar way that multivalent counterions induce sphere-to-rod transitions in bulk micelles. In cases where the surfactant headgroups are highly repulsive or bulky, even the mica surface is unable to bring headgroups close enough to effect a sphere-to-cylinder transition, and interfacial micelles remain spherical. This is the case for divalent surfactants, as shown in Fig. 15b. In summary, the geometry of surfactant aggregates at charged surfaces is highly sensitive to “charge density matching” between the surface and free (unperturbed) micelles. The interfacial aggregate can have a flatter curvature than free micelles in cases where the adsorption density becomes comparable to the micelle charge density.

IX. CONCLUSION In the technological realm, an application that has generated much excitement is the synthesis of mesoscopic materials using surfactant micelles as templates, as first reported in 1992 by Beck et al. This process relies on inorganic polymerization at the interfacial region between a surfactant aggregate and a solution in which the inorganic precursors (usually silicate ions) are initially dispersed. By restricting the polymerization reaction to the micelle–solution interface, a complex nanocomposite is formed consisting of surfactant micelles embedded in an ordered array within a continuous inorganic (e.g., silica) “scaffold.” Pyrolyzing the surfactant finally results in a mesoporous material, with pore sizes of order 5 nm, which can serve as, for example, a molecular filter, catalytic support, or laser waveguide. However, aside from such applications and, of course, the industrial interest in detergent action, one should note that of the wide range of experimentalists and theoreticians that have been drawn to investigation of micelles, most have been attracted by the unique intellectual challenge such systems offer. The literature provides ample evidence that they have not been disappointed.

SEE ALSO THE FOLLOWING ARTICLES CHEMICAL KINETICS, EXPERIMENTATION • CHEMICAL THERMODYNAMICS • ELECTROPHORESIS • HYDROGEN BOND • KINETICS (CHEMISTRY) • MACROMOLECULES, STRUCTURE • PRECIPITATION REACTIONS • SILICONE

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(SILOXANE) SURFACTANTS • SURFACTANTS, INDUSTRIAL APPLICATIONS • SURFACE CHEMISTRY

BIBLIOGRAPHY Beck, J. S., et al. (1992). J. Am. Chem. Soc. 114, 10834–10843. Fendler, J. H. (1982). “Membrane Mimetic Chemistry,” Wiley, New York. Fendler, J. H., and Fendler, E. J. (1975). “Catalysis in Micellar and Macromolecular Systems,” Academic Press, New York. Gaudin, A. M., and Fuerstenau, D. W. (1955). Trans. AIME 202, 958– 962.

677 Klafter, J., and Drake, J. M. (1989). “Molecular Dynamics in Restricted Geometries,” Wiley, New York and Chichester. Lindman, B., and Wennerström, H. (1980). Top. Curr. Chem. 87, 1–83. Manne, S., and Gaub, H. E. (1995). Science 270, 1480–1482. Manne, S., and Warr, G. G. (1999). In “Supramolecular Structure in Confined Geometries” (S. Manne and G. G. Warr, eds.), pp. 2–23. American Chemical Society, Washington DC. Menger, F. M. (1977). In “Bioorganic Chemistry III. Macro- and Multimolecular Systems” (E. E. van Tamelen, ed.), pp. 137–152. Academic Press, New York. Tanford, C. (1980). “The Hydrophobic Effect: Formation of Micelles and Biological Membranes,” Wiley, New York. Wennerström, H., and Lindman, B. (1979). Phys. Rep. 52, 1–86.

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Physical Chemistry Douglas J. Henderson

Charles T. Rettner

Brigham Young University

IBM Almaden Research Center

I. II. III. IV.

Classical Mechanics Quantum Mechanics Statistical Thermodynamics Kinetics and Dynamics

GLOSSARY Activated complex Short-lived transition state that occurs at the point of maximum energy along a reaction path when the molecules in a chemical reaction can no longer be considered as reactants or products. Adsorption Adhesion of a gas or liquid at a surface resulting in an increased concentration of the gas in the vicinity of the surface; to be distinguished from absorption, which occurs throughout the solid or liquid. Critical point Point where two phases become identical and form one phase. Degrees of freedom Variables which must be determined to specify the state of a system. Elementary reaction Reaction concerning a single chemical step, such as dissociation or recombination, as distinct from complex reactions which occur through a series of separate elementary reactions. Equation of state Relation between the thermodynamic properties of a system. Equilibrium State of an isolated system which is specified by quantities which are independent of time. Isotherm Curve joining states for which the temperature is constant.

Kinetics Study of how chemical systems change, concerning the rate at which change occurs and the factors on which this rate depends. Also used to refer to the sequence of reactions by which a complex reaction occurs. Molecular beam Stream of molecules all traveling in the same direction in vacuum, used in studies of isolated molecules and to examine the dynamics of single molecular collisions. Normal mode One of a set of coordinates of a system that can be excited while the others remain at rest. Order of a transition Transition from one thermodynamic phase to another is of order n if the first discontinuous derivative of the free energy with respect to the thermodynamic variables is of order n. Phase, thermodynamic Region of the space specified by the thermodynamic degrees of freedom of system separated from the remainder by a clearly defined surface and within which the thermodynamic properties differ from those of the remainder. Rate constant Constant that gives a measure of the rate of a chemical reaction; the proportionality constant between the rate of product formation and the product of the reagent concentrations. If the rate expression involves N molecules of the same reagent, the concentration must be raised to the power of N .

59

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160 Reversible process A process in which a system changes from one thermodynamic state to another is reversible if the thermodynamic variables in the inverse process pass through the same values but in the inverse order and in which all exchanges of heat, work, etc., with the surroundings occur with reverse sign and in inverse order. Spectroscopy Analytical technique concerned with the measurement of the interaction of energy and matter, the development of instruments for such measurements, and the interpretation of such information for analysis of the structure or constituents of a system. Techniques, such as mass spectrometry, which do not involve energy are often called spectroscopic because they also yield output scans in the form of spectra. Spectrum Intensity of a signal due to a process such as optical absorption or emission displayed as a function of some varying characteristic such as wavelength, energy, or mass. Also used in quantum mechanics and applied mathematics to specify the pattern of eigenvalues of a linear operator and in electrodynamics to specify the range of frequencies of electromagnetic radiation. State function In thermodynamics a variable is a state function if, when all the thermodynamic variables are specified, it has a unique value. As a result, the change in any state function in a reversible cyclic process must be zero. Thermodynamics Study of the changes in the properties of a system, usually as a result of changes in temperature or pressure.

PHYSICAL CHEMISTRY is the branch of chemistry in which experimental and theoretical techniques of physics are used to investigate and interpret chemical phenomena. Physical chemistry has its origins in the late nineteenth century, where it was largely concerned with the application of classical thermodynamics to chemistry. Modern physical chemistry is based more on quantum and statistical mechanics, which were developed only during the twentieth century. The branch of physical chemistry that employs twentieth century physical techniques is sometimes called chemical physics, with physical chemistry being regarded as concerned only with classical techniques. However, the distinction is artificial. Physical chemistry and chemical physics are really the same field and are considered as such here. Experimental physical chemistry has been revolutionized by relatively recent advances in electronic instrumentation, vacuum technology, and by the introduction of lasers. Equally, advances in computer power have had a great impact on theoretical studies, with an increasing emphasis on computer simulations and the detailed modeling of chemical systems.

Physical Chemistry

I. CLASSICAL MECHANICS The dynamics (i.e., motion and energetics) of molecules and atoms and, at a more fundamental level, electrons, are the origin of chemical phenomena. Prior to the twentieth century it was believed that all of the dynamics of a system, whether astronomical or molecular, were described by Newton’s equation of motion. dv , (1) dt where F is the force, v the velocity, t the time, and the proportionality factor, m, the mass of the particle or object. The force and velocity are vectors, whose direction and magnitude are both of importance. In complex problems it is often preferable to reformulate classical mechanics in terms of a scalar, such as the energy, which is characterized only by its magnitude. This gives rise to the Lagrangian and Hamiltonian equations of motion. The latter equations are of most interest here and are ∂qi ∂ = , ∂t ∂ pi (2) ∂ pi ∂ , =− ∂t ∂qi F=m

where pi and qi are generalized momenta and positions, respectively, and , the Hamiltonian, is the total energy of the system using momenta and position as variables. The space spanned by the pi and qi is called phase space. The dynamics of a system are described by a path in phase space. If the system is periodic, as is the case for electrons in an atom or molecule, then the path is a closed orbit in phase space. The advantage of the Hamiltonian formulation in physical chemistry is the fact that all variables are treated on an equal footing. However, the Hamiltonian and Newtonian formulations of classical mechanics are completely equivalent.

II. QUANTUM MECHANICS A. Duality of Matter and Energy; Uncertainty Principle During the nineteenth century it was established that matter consists of atoms and chemically bound aggregates of atoms called molecules. At first, it was thought that atoms were structureless. However, by about the turn of the century, it was shown that atoms were miniature solar systems consisting of a positively charged nucleus, whose structure is irrelevant for chemical phenomena, which contains nearly all the atomic mass, and negatively charged “planetary” electrons which orbit the nucleus.

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Because of this, nuclei are slow moving, virtually motionless, on the timescale of electronic motions. For many chemical phenomena, the nuclei can be regarded as fixed in space. Only the electronic dynamics need be considered. This simplification is called the Born–Oppenheimer approximation. At first it was thought that the electronic motions could be described by classical mechanics. However, it is impossible to describe the microscopic world by classical mechanics. It became apparent, from for example the photoelectric effect, that electromagnetic radiation was not always wavelike, but, in some circumstances, consisted of discrete, particle-like, units of magnitude called quanta, E = hν,

(3)

where ν is the frequency of the radiation and h = 6.626 ×10−34 J sec is Planck’s constant. Conversely, it became apparent, through for example the diffraction of electrons, that matter was not always particle-like but, in some circumstances consisted of waves of wavelength λ = h/ p.

(4)

Such waves can be used to probe the structure of crystal surfaces, through low-energy electron diffraction (LEED) or atomic beam diffraction. The latter is usually confined to He atoms, but even Ar atom diffraction can be discerned in favorable cases. In other words, there is a duality of matter and energy. Whether the particle-like or the wavelike character of matter/energy is dominant depends on the experiment. In fact, the experiment itself interacts with the matter/energy and defines some aspect of the system at the cost of indefiniteness of some other aspect. This uncertainty principle was made precise by Heisenberg who showed that even under the most ideal circumstances pi qi = h/4π.

(5)

If the experiment defines the particle character of the system, the uncertainty of the positions, qi , is small and the uncertainty in momenta, pi , or frequency is large. However, if the experiment defines the wave character of the system, the reverse is true. The momenta pi and positions qi are called conjugate variables. Energy and time are also conjugate variables so that Et = h/4π.

(6)

Classical mechanics, where there is no uncertainty, is a limiting case in which the magnitudes of the variables are large compared to h. As a result, classical mechanics is appropriate for large macroscopic bodies.

B. Wave Equation In the earliest formulation of quantum mechanics, classical mechanics was assumed valid with the exception that some periodic variables were quantized (i.e., had discrete values). Their values could be determined by integrating the pi over their orbits in phase space, according to pi dqi = n i h, (7) where n i is an integer called a quantum number. The integral in Eq. (7) over a closed path is called a phase integral. However, as the implications of the duality of matter and energy and the uncertainty principle were accepted, it became apparent that one could refer only to the probability of finding the system in some configuration. Just as the wave nature of radiation meant that there was a wave equation for radiation, the wave nature of matter implied the existence of a new wave equation. This wave equation, called the Schrödinger equation, is formulated as an eigenvalue equation (eigen ≡ characteristic or proper) where the Hamiltonian operator “operates” on the probability function or wave function or eigenfunction, ψ, to give the energy eigenvalue, E, times ψ. Thus, the wave equation is ψ = Eψ

(8)

The wave function has the property that |ψ|2 gives the probability of the system having the eigenstate whose energy is E. The Hamiltonian operator is formed by replacing p j in the classical Hamiltonian by √ the operator −(h/i)(∂/∂q j ), where h = h/2π and i = − 1. The q j remain unchanged. Interestingly, the earlier phase integral formulation [Eq. (7)] becomes the approximate Wentzel–Kramers–Brillouin (WKB) method of solution of Schrödinger’s equation and remains useful in many problems in the sense that differences between quantized values of a phase integral are integral multiples of h except that there may be a zero point value of the phase integral given by a fractional value of h. C. Hydrogen-Like Atom; Electronic Transitions One of the first systems to which quantum mechanics was applied was the hydrogen-like atom consisting of a single electron orbiting a nucleus of charge Z e0 . The energy eigenvalues or levels are obtained by solving Schrödinger’s equation and are given by 2π 2 m 0 Z 2 e04 hc R Z 2 = − , (9) n2h2 n2 where e0 is the charge of an electron, n is an integer, c is the velocity of light, and R is called Rydberg’s constant. Strictly speaking we should not use the electronic mass m 0 E =−

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FIGURE 1 Energy-level diagram for the hydrogen atom showing allowed transitions. Insert displays the Balmer series spectrum that can be observed in the visible.

in this formula but the reduced electronic mass. However, the effect is small. This means that electronic transitions between states characterized by n 1 and n 2 emit or adsorb energy or radiation whose wavelength is given by 1 1 1 2 (10) =Z R 2− 2 . λ n1 n2 An energy-level diagram for the hydrogen atom is shown in Fig. 1, which also displays some of the allowed transitions. The energy levels of the hydrogen-like atom are degenerate because more than one state corresponds to a specific value of n (the principal quantum number). These degenerate states are characterized by the quantum numbers l and m, which characterize the spherical harmonics of the wave function. For each value of n there are n values of l(l = 0, . . . , n − 1) and for each value of l there are 2l + 1 values of m(m = − l, −l + 1, . . . , 0, . . . , l − 1, l) giving n 2 values of l and n for each value of n. D. Many-Electron Atoms; Pauli Principle; Electron Spin To understand more complex atoms containing many electrons, we must solve the many-electron Schrödinger equation. Even in classical mechanics, many-body problems are difficult, so it is not surprising that many-electron quantum mechanics, usually called quantum chemistry,

Physical Chemistry

is an active research field today. However, an understanding of the electronic structure of atoms can be understood in terms of the aufbau (building up) principle whereby electrons are added one at a time to the atom. However, two additional facts should be mentioned. First, the quantum numbers n, l, and m are not sufficient to specify the state of an electron. The spin of an electron must also be specified. Electrons can have one of two spins (say, up or down). This is specified by the spin quantum number, s = ± 12 (so that there are 2|s| + 1 = 2 spin states). Thus, the state of an electron is specified by n, l, m, and s. For historical reasons the values l = 0, 1, 2, 3, 4, . . . are specified by the spectroscopic notation s, p, d, e, f, . . . . Thus, an electron might be said to be in a 1s(n = 1, l = 0) or a 2 p(n = 2, l = 1), state. Similarly, states for the whole atom are termed, S, P, D, E, . . . corresponding to L = 0, 1, 2, 3, . . . , where L is the total orbital angular momentum for the atom, which is arrived at by combining the orbital angular momenta of the individual electrons. Second, electrons obey the Pauli exclusion principle. This means that only one electron can occupy a quantum state. Thus, as the aufbau principle is employed, the electrons are added one at a time to the state of lowest energy, each state being filled by one electron. From these principles a simple understanding of the periodic table is gained. Each electronic shell is specified by n and contains 2n 2 states. Thus, the first row of the periodic table corresponds to n = 1 and contains 2 elements (H and He). The electronic configurations of these elements are denoted 1s (H) and 1s 2 (He). The superscript indicates the number of electrons with the given value of l. The second row corresponds to n = 2 and contains 8 elements with configurations 1s 2 2s, 1s 2 2s 2 , 1s 2 2s 2 2 p, . . . , 1s 2 2s 2 2 p 6 . The subsequent rows contain 8 columns even though 2n 2 exceeds 8 because the energy of the electronic states is ordered (approximately) 1s/2s 2p/3s 3p/4s 3d4p/5s 4d 5p/6s 4f 5d 6p/7s . . . so that the third row is still filled with 8 elements. With potassium (Z = 19) the nineteenth electron goes into a 4s rather than a 3d level. The transition elements are regarded as occupying one position in the table since the outer shell configuration does not change as the d electrons are added and, as a result, they have similar chemical properties. As n increases, not only must the d electrons be accommodated in single positions, but the f electrons must also be accommodated in a single position, so that the table becomes more complex. However, the underlying principles are simple. E. Molecular Systems: Chemical Bond Many-electron molecular systems are even more complex than atomic systems. The theory of such systems

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is an active field of research. The potential energy terms in the wave equation for the molecule involve not only the Coulomb repulsions between the electrons and the Coulomb attractions of the electrons and nuclei but also the Coulomb repulsion between the nuclei. Given the repulsion between the nuclei, we are inclined to ask how atoms form a chemical bond in a molecule. A simple answer can be obtained by considering the one electron hydrogen molecule ion H2+ . The wave functions of each H atom separately are ψls (A) and ψls (B). An approximation to the H2+ wave function can be obtained by forming a molecular orbital from a linear combination of these two wave functions ψb = ψls (A) + ψls (B)

and they are termed doublet states. The former three have zero spin, a multiplicity of 1, and are termed singlet states, while states with a multiplicity of 3 and 4 are termed triplet and quartet states, respectively. Thus, the ground electronic state of NO is written as 2 1/2,3/2 , where the 2 refers to the doublet nature of the state, the to = 1, and the 12 and 32 refer to values of which arise from the two possible ways that the spin and orbital angular momentum can add. In addition to electronic energy states, molecules posses both rotational and vibrational energy levels. Assuming a fixed distance between two atoms (rigid rotor approximation), the Schrödinger equation yields for the allowed rotational energy levels of a diatomic molecule h2 J (J + 1), (12) 8π 2 I where I is the moment of inertia of the molecule, J the rotational quantum number and the quantity (h 2 /8π 2 I ) is termed the rotational constant for that particular electronic state of the molecule (usually given the symbol B). Vibrational energy levels can be estimated by inserting the Hooke’s law potential energy, U (r ) = 0.5k(r − re )2 , in the Schrödinger equation (harmonic oscillator approximation). This yields eigenvalues, E v , for the permissible energy levels, of E v = (h/2π ) k/µ v + 12 = hν0 v + 12 , (13) Er ≈

or ψa = ψls (A) − ψls (B).

(11)

In the first (bonding) orbital, the electron is concentrated between the nuclei, and is simultaneously attracted by both nuclei resulting in a lower electronic energy which more than offsets the repulsion of the nuclei. In other words when the electron is between the nuclei it acts as a cement holding them together. There is zero probability of finding the electron between the nuclei in the second (antibonding) orbital. As a result, a chemical bond is not formed by this orbital. F. Molecular Systems; Energy Levels When atoms combine to form molecules, the individual atomic energy levels give rise to discrete electronic energy levels or states of the molecule. The number of these molecular electronic states far exceeds those of the individual atoms because of the many different ways in which the atomic states can be combined. Electronic states of molecules are classified in terms of several molecular quantum numbers in a manner analogous to atomic electronic states. For a diatomic molecule these include the electronic orbital angular momentum, l, its component along the internuclear axis, λ, and the corresponding quantities for the molecule as a whole, L and . Just as l = 0, 1, 2, . . . gives rise to s, p, d, . . . electron states and L = 0, 1, 2 . . . gives S, P, D, . . . atomic states, so λ = 0, 1, 2 . . . yield σ, π, δ . . . electron states and = 0, 1, 2 . . . correspond to , , . . . molecular states. The component of the total electronic angular momentum along the internuclear axis, , is also of importance. The ground electronic states of H2 , O2 , and N2 are states, while those of OH and NO are states. These latter two molecules have open valence shells with net spin of 12 , so that the multiplicity, S, of these states is 2 (S = 2S + 1),

where µ is the reduced mass of the system [µ = m 1 m 2 /(m 1 + m 2 )] and ν0 is known as the fundamental vibrational frequency. The smallest amount of vibrational energy a molecule can possess is thus ν0 /2, termed the zero-point energy. Rather than the simple Hooke’s law potential we may consider more realistic molecular potential energy curves such as a Morse potential given by U (r ) = De {1 − exp[−β(r − re )]}2 ,

(14)

where De is the dissociation energy of the molecule and β is related to De and ν0 . This leads to a similar expression for E v , but with an additional quadratic term in (v + 12 ), which is negligible for low vibrational energies. Similarly, an accurate treatment of molecular rotation leads to additional terms in higher powers of the quantity {J (J + 1)}. G. Spectroscopy Atoms and molecules can adsorb and emit radiation to change their internal energy states. The electronic transitions of the hydrogen-like atoms have already been mentioned. The quantization of the energy levels restricts the possible wavelengths of the radiation to discrete spectral lines. Only certain transitions are allowed and these are given by separate selection rules for electronic,

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164 vibrational, and rotational transitions. For example, in atomic transitions l can only take values of ±1. In addition, each allowed transition has an associated transition linestrength. In electronic spectroscopy involving transitions between electronic states, these are termed transition moments, Franck–Condon factors and Hönl– London factors for electronic, vibrational, and rotational transitions, respectively. The spectrum of absorption lines associated with a given species can be used as a fingerprint for identification and quantification in the process of absorption spectroscopy. An absorption spectrum is usually recorded with an absorption spectrometer or, more recently, with a laser. Similarly, emission spectroscopy is concerned with the radiation that is emitted by an excited atom or molecule. Here radiation is generally spectrally resolved using a spectrograph. In the various spectral regions, these techniques may be referred to as vacuum ultraviolet (VUV), ultraviolet (UV), infrared (IR), and microwave spectroscopies. Microwave photons have energies of less than 1 meV and are associated with rotational transitions, while infrared photons have energies of up to 1 eV and are involved with vibrational transitions. More energetic radiation from the visible to VUV is usually associated with electronic transitions. In addition to identifying species, these and related spectroscopic techniques are frequently employed to obtain detailed structural information, such as bond lengths and bond angles, and to study the flow of energy in chemical reactions. Other spectroscopic techniques include: 1. Raman spectroscopy, which concerns the spectral analysis of radiation scattered by an atom or molecule. Recent developments are coherent anti-Stokes Raman spectroscopy (CARS) and surface enhanced Raman spectroscopy (SERS), both of which are sensitive laser-based techniques, and examples of laser spectroscopy. 2. Photoelectron spectroscopy, which is based on kinetic energy analysis of electrons ejected from an atom or molecule by an enegetic photon and provides information on the binding energies or ionization potentials of the ejected electrons. Recent developments include X-ray photoelectron spectroscopy (XPS) of surfaces, and the use of lasers as radiation sources. 3. Auger electron spectroscopy (AES), which involves the electron impact ionization of an atom to give an excited electronic state that decays by emission of a second electron whose energy is characteristic of the atom. This is most commonly used in surface analysis. 4. Spin resonance spectroscopy, which concerns the application of a magnetic field to split energy states associated with electron or nuclear spin orientations. Electron spin resonance (ESR) involves absorption of microwave

Physical Chemistry

radiation, while nuclear magnetic resonance (NMR) is a radio-frequency technique. 5. Mössbauer spectroscopy, which involves the resonant absorption of a γ -ray photon by a nucleus. The resonant condition is achieved via the Doppler effect, by sweeping the velocity of a sample relative to the source. The chemical environment of the nucleus causes characteristic frequency shifts.

III. STATISTICAL THERMODYNAMICS A. First and Second Laws of Thermodynamics; Entropy The first law of thermodynamics states the conservation of energy, δ Q = dU + δW,

(15)

where δ Q is the heat absorbed by the system, dU is the change in internal energy of the system, and δW is the work done by the system. The second law of thermodynamics states that heat cannot pass from a cold reservoir to a hot reservoir without the application of work. The change in entropy, d S, is just δ Q/T , where T is the temperature. The factor 1/T is an integrating factor that transforms δ Q into an exact differential just as 1/v 2 transforms vdu − udv into the exact differential d(u/v). Because the change in entropy, d S = δ Q/T , is an exact differential, the change in entropy in a reversible cyclic process is zero. The entropy of a thermodynamic state is a well-defined single-valued function and the entropy is said to be a state function. An equivalent statement of the second law of thermodynamics is S ≥ 0,

(16)

where the change in entropy is zero for a reversible cyclic process. The entropy increases in an irreversible process. B. Free Energy; Experimental Measurements The first and second laws of thermodynamics can be combined to give T d S = d E + δW.

(17)

If the only work done by the system results from an expansion d V or a change in the amount d Ni of the constituents, then Eq. (14) becomes T d S = dU + pd V −

m

µi d Ni ,

(18)

i=1

where p is the pressure, V the volume, µi the chemical potential of constituent i, and Ni the amount or concentration

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of constituent i. The chemical potential of a gas is the value of the Gibbs function for one mole. In a purely mechanical system, equilibrium is achieved when the energy is a minimum. In a thermodynamic system, entropy changes as well as energy changes must be considered. At constant temperature, volume, and concentration, the Helmholtz free energy,

the so-called gas constant (R) divided by Avogadro’s number, N A = 6.022 × 1023 mol−1 , the number of molecules in a mole. The thermodynamic properties are related to the energy levels E i through the partition function Z defined by Z = e−β A = exp(−β E i ), (23)

A = U − T S,

where the sum is over all the energy levels of the system. Thus, to obtain the thermodynamics of a system, all that is required is that Schrödinger’s equation for the system be solved and the partition function summed. For most systems this is a difficult task, often impossible without some approximations. There is also a relation between the entropy and the microscopic configurations of the system. The entropy is proportional to the logarithm of the number of accessible states, , of the system. Thus,

(19)

is a minimum at equilibrium, whereas at constant pressure and temperature the Gibbs free energy G = A + pV =

m

µi Ni

(20)

i=1

is a minimum at equilibrium. For a dilute gas, where the perfect gas law ( pV = n RT ) applies, the value of µ per mole at a pressure p is µ( p) = µ◦ + RT ln( p/atm),

(21)

where µ◦ is the value of µ at 1 atmosphere, which is the pressure at which the standard state is established. The value of µ◦ is often taken as zero for the elements. In Eqs. (20) and (21) R( = 8.3144 J mol−1 K−1 ) is the so-called gas constant. A mole of any substances is the amount with a mass in grams equal to its molecular weight in atomic mass units, so that a mole of molecular hydrogen has a mass of 0.002 kg. The value of G or µ can be determined for some pressure p by measuring the volume of the gas as a function of pressure up to p and integrating. Thus, 1 p µ( p) = µ◦ + V ( p)d p, (22) n p0 where p0 is the pressure of the standard state (usually 1 atm). The free energy of a condensed phase can be related to that of a dilute gas through the vapor pressure, the pressure of the gas in equilibrium with the condensed phase. Once the free energy of the condensed phase has been established, values for other states can be obtained by measuring pressure or energy through a sequence of states leading to the desired state. C. Statistical Mechanics; Partition Function The thermodynamic properties of a system result from the dynamics of its molecules. Since even a three-body system is difficult, statistical methods must be employed to treat the large number of molecules in a thermodynamic system. The fundamental result in statistical mechanics is the fact that the probability of a system occupying the energy level E i is proportional to the Boltzmann factor, exp(−β E i ), where β = 1/kT and k = 1.3804 ×10−23 J K−1 is the Boltzmann constant. The Boltzmann constant is

i

S = k ln .

(24)

Equation (24) is called Boltzmann’s relation. At absolute zero, the system is in its ground state, and the number of accessible states is unity. Thus, the entropy of a system tends to zero as the temperature goes to zero. This is called the third law of thermodynamics. The Boltzmann relation provides a statistical interpretation of the entropy. The greater the number of accessible states, the less our knowledge of the system and the more randomness or disorder in the system. This entropy is a measure of disorder. The tendency of the entropy to increase reflects the tendency of thermodynamic systems to increase in disorder just as an initially ordered deck of cards increases in disorder during a game of cards. If the system is classical, the energy levels merge into a continuum and an important simplification results. The sum in the partition function becomes an integral. Moreover, if the kinetic energy degrees of freedom (i.e., the momenta) are independent of the potential energy or internal degrees of freedom (i.e., the generalized positions) then the momenta can be integrated immediately. For the particular case in which there is only translational motion. λ− 3N Z= exp(−β) dr1 · · · dr N , (25) N! where λ = h/(2π mkT )1/2 , N is the number of molecules in the system, and = (r1 , . . . , r N ) is the potential energy. The factor N ! is required because states that differ only by an interchange of molecules are not distinguishable. From Eq. (20), it follows that the average kinetic energy of the system is KE = 32 N kT

(26)

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In other words, there is a statistical relation between the temperature and the average motion of the molecules. The greater the temperature, the more rapidly the molecules move and the greater their kinetic energy. The problem of predicting the thermodynamic properties of such a classical system becomes the problem of evaluation of the configuration integral, the integral over exp(−β). This is still a difficult task. In general, it can be done only through computer simulations (Monte Carlo and molecular dynamics methods). However, there are a few simple approximations which are helpful. D. Perfect and Imperfect Gases The simplest system is the perfect gas in which the molecules do not interact, i.e., = 0. Thus, the configuration integral is just the volume raised to the power N . Using Stirling’s approximation, N ! = (N /e) N , Z = λ− 3N (eV /N ) N

(27)

and pV = N kT.

(28)

If the molecules interact, then the problem is more complex. The gas is called imperfect because there are deviations from the perfect gas result. These deviations can be written as a power series in the density, ρ = N /V , called a virial series. For example, if the molecules are hard spheres such that the molecules collide elastically but exert no attractive forces on each other, then βp /ρ = 1 + ρb + 58 (ρb)2 + · · · .

(29)

For hard spheres, the coefficients of ρ , called virial coefficients, are independent of the temperature. For more complex gases the virial coefficients are temperature dependent. The virial coefficients can be related to the forces between the molecules. However, both the relation itself and the evaluation of the resultant integrals rapidly become complex as the power n of ρ n increases. In general, it is difficult to go beyond n = 4. The pressure of the hard-sphere gas exceeds that of the perfect gas at the same temperature and density. To a first approximation, this can be thought to be a result of a reduction in the volume available to the molecules because of the volume occupied by the molecules themselves. The hard spheres can be said to have less free volume than the perfect gas. The hard-sphere gas cannot be liquified. Liquification requires attractive forces. Attractive forces can also cause the pressure to be less than the perfect gas result. Interestingly, attractive forces are not required for the existence of a solid phase. If the hard sphere gas is compressed,

computer simulations show that it will freeze and exist as a close-packed solid. E. Liquids; van der Waals Theory; Critical Point; Renormalization Group In contrast to a gas, a liquid need not fill space but can exist in equilibrium with its vapor with a surface separating the liquid and vapor. The pressure at which the equilibrium occurs is called the vapor pressure. Below the vapor pressure, liquid will evaporate until equilibrium is reached. For pressures greater than the vapor pressure, there is no interface between liquid and vapor. The liquid fills the container and there is no clear distinction between liquid and gas. The liquid under pressure can be heated at constant volume to a temperature greater than the critical temperature (the highest temperature at which liquid–vapor coexistence can occur), then allowed to expand and cool to the original temperature and pressure without any transition from liquid to gas being observed. A continuity of states between liquid and gas is said to exist. This is illustrated in Fig. 2. The liquid–gas phase can be referred to by the single term fluid. Thus, a theory of the liquid state is of necessity also a theory of an imperfect gas. The earliest theory of the liquid state is that of van der Waals. Although more than a century old, with slight modifications it is viable today. The idea of van der Waals was that a liquid behaved as a hard sphere gas except that the pressure must include the internal pressure due to the attractive forces of the molecules in the liquid. It is reasonable to assume that the contribution of the internal pressure to the free energy is proportional to the density. Thus,

n

FIGURE 2 Phase diagram of a typical simple liquid. The shaded region is not thermodynamically stable.

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p = p0 − ρ 2 a ,

(30)

where p0 is the pressure of the hard-sphere gas and a is a constant depending on the nature of the attractive forces. In its original formulation, the van der Waals theory was only qualitatively successful because van der Waals approximated p0 by the perfect gas expression with a reduced free volume, i.e., βp0 1 = . (31) ρ 1 − ρb This expression gives only the second hardsphere virial coefficient correctly and seriously overestimates p0 . Much more satisfactory results can be obtained from the approximation βp0 1 = (32) ρ (1 − η)4 where η = 14 ρb. The van der Waals theory predicts that the equation of state of a liquid can be expressed in a universal form if the following reduced variables, T ∗ = bkT /a, p ∗ = b2 p/a, and ρ ∗ = ρb, are used. This is called the law of corresponding states. As is illustrated in Fig. 3, the theory also predicts that below the critical temperature there is a first-order phase transition between the liquid and vapor accompanied by a discontinuous change in the density ρ. At the critical temperature the transition becomes second order since the liquid and vapor have become identical. For temperatures above the critical temperature, there is no phase transition. In the van der Waals theory, the critical point occurs when 2 ∂p ∂ p = = 0, (33) ∂ρ T ∂ρ 2 T i.e., the pressure isotherms have a point of inflection at the critical point. Modern theories show that the van der Waals theory is a first approximation to a systematic approach, called perturbation theory, in which the pressure is obtained as a power series in 1/T . In the van der Waals theory, the first two derivatives of p at constant T with respect to the density vanish at the critical point. This is not just a prediction of the van der Waals theory. Any theory in which the equation of state is analytic at the critical point will yield this result. By analytic, it is meant that the pressure can be expanded as a power series about the critical point. Experimentally, the equation of state is not analytic at the critical point. The exponents in an expansion near the critical point are generally not integers. At least one, and possibly two, more derivatives of p with respect to the density at constant T vanish near the critical point. There has been a great deal of work on the fascinating properties of the equation of state in the vicinity of the critical point. The most far

FIGURE 3 Typical pressure isotherms as a function of the volume V in the van der Waals theory. The shaded region is not thermodynamically stable. Here Tc is the critical temperature.

reaching is the renormalization group approach in which a group of successive transformations is applied to the liquid, yielding ultimately a renormalized system in which only the long-range correlations typical of the critical point remain. In this system the critical point properties can be examined. F. Mixtures Mixtures of two gases or liquids can be treated by the same techniques as liquids. The analog of a perfect gas is the ideal mixture, where molecules of the components are very similar so that the partition function can be written (for the two-component case) as Z=

N! Z1 Z2 N1 !N2 !

where N = N1 + N2 and Ni is the number of molecules of species i. From this, it can be deduced that the partial pressures of the components are proportional to their concentrations. This result is known as Raoult’s law. The factor N !/N1 !N2 ! gives rise to the entropy of mixing S = −N k[x1 ln x1 + x2 ln x2 ], where xi = Ni /N .

(34)

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For a nonideal mixture, the original approach of van der Waals is still useful. In this approach N! Z N1 !N2 !

Z=

(35)

where Z is obtained from the van der Waals equation of state of a liquid, preferably with the more satisfactory expression for p0 , with a and b replaced by the concentration-dependent quantities a = xi x j ai j (36) ij

and b =

xi x j bi j

(37)

ij

√ with ai j aii a j j taken as a parameter. Assuming that the spherical cores of the molecules do not overlap, then bi j =

1/3 3

1/3

bii + b j j 2

.

(38)

This approach is satisfactory for mixtures of nonelectrolytes. The situation is a little more complex for mixtures of electrolytes because of the long range of the Coulomb potential. However, each ion in the mixture tends to be surrounded by ions of opposite charge, which causes the potential to decay exponentially with a decay factor, κ, the Debye parameter, which is proportional to the square root of the product of the density and T −1 . As a result, an appropriate expansion parameter for electrolytes is κ, or T −1/2 , which is different from nonelectrolytes, where T −1 is the expansion parameter. These ideas become quantitative in the Debye–Hückel theory. G. Solids In contrast to the disorder of gases and liquids, there is translational order in crystals. Disordered or amorphous solids (i.e., glasses) exist which lack this order. However, they are really highly viscous liquids. This translational order is such that the entire structure, or lattice, can be generated by repeated replication of a small regular figure, termed the unit cell. The planes of any crystalline structure can be specified using Miller indices, which also serve to identify single crystal faces. Miller indices are obtained by determining the intercepts of the plane with the unit cell axes in terms of the length of the cell in that direction, taking the reciprocal, and normalizing so the indices are all integers. The ordered structure, or lattice, of a solid can be determined by X-ray or neutron diffraction studies, in which a beam of X-rays of neutrons is scattered from the sample to

produce a diffraction pattern, which can be analyzed to reveal the crystal structure of the sample. All crystal lattices can be classified into 14 Bravais lattices belonging to seven systems. For example, the simple cubic, face-centered cubic and body-centered cubic lattices are the 3 lattices of the cubic system. Cubic and hexagonal close-packed structures have the structure of tightly packed spheres where each sphere touches 12 neighbors, 6 in the same plane and 3 above and 3 below. These two close-packed structures differ in the placement of successive planes or layers. For the cubic case, a third layer is laid down to reproduce the first layer, so that the structure could be represented by ABABAB. . . . For hexagonal close packing, the third layer is again displaced, corresponding to ABCABC. . . . No theory of freezing exists. That is, there is no partition function that encompasses both the solid and fluid phases. However, separate theories of solids and fluids can be developed and their solid–fluid coexistence examined. To that extent theories of melting or freezing exist. Since freezing can occur in the hard-sphere system, no critical point is expected for freezing. This transition is expected to be first order at all temperatures, as illustrated in Fig. 1. If a solid were classical, the heat capacity would be 3N k. This is indeed the case at high temperatures and is called the law of Dulong and Petit. However, the experimental heat capacity goes to zero at low temperatures. This can be explained by regarding the solid as a collection of quantized oscillators. The only difficulty is to determine the spectrum of frequencies of the oscillators. For many purposes, the solid can be regarded as an elastic continuum. The result is the Debye theory. If something more sophisticated is needed one must solve for the normal modes of the crystal, i.e., the method of lattice dynamics. The conduction of electricity in a metal is due to the presence of free or quasi-free electrons in the metal. Classically, free electrons would contribute 3nk/2 to the heat capacity, n being the number of free electrons. However, experimental evidence indicates that the electrons do not contribute significantly to the heat capacity of a metal. The reason for this is the exclusion principle. Although the electronic gas is in its ground state, because of the exclusion principle the electrons can each occupy one energy level. The electrons occupy the levels up to a maximum energy, called the Fermi energy, εF . Only the small number of electrons with energies near εF can be thermally excited and, as a result, the electronic heat capacity is small. If the exclusion principle is taken into account, treating the conduction electrons as free describes many of the electronic properties of a metal. To treat metals in a more sophisticated manner and to account for semiconductors, the structure of the solid must be included. If this is done, the electrons are not free but are restricted to bands of energy.

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an electrolyte, for example, electrons may leave it to reduce cations in the solution, giving it a net positive charge and making the solution slightly negative. These charges arrange themselves at the interface in two layers, known as the double layer. The most important property of this double layer is the variation of potential in its vicinity. The potential governs the rate at which ions can be transported through the interface, and so controls the rate of electrochemical processes.

FIGURE 4 Image of a single atom of xenon sitting on the surface of a platinum single crystal at 4 K, obtained with a scanning tunneling microscope. Xenon has an atomic radius of ∼1.2 × 10−10 m. Picture provided by Don Eigler of the Almaden Research Center.

H. Interfaces The study of interfaces is becoming an increasingly important area of physical chemistry. Of particular interest are gas–solid and gas–liquid interfaces. Both of these can now be imaged directly with scanning tunneling microscopy (STM). Here the contours of a surface may be determined by sensing the small current that tunnels across a vacuum gap to an atomically sharp tip as it is scanned across it. Figure 4 illustrates the remarkable resolution of this instrument. It shows a single atom of xenon sitting on the surface of a platinum crystal at 4 K. For the case of the liquid–solid interface, STM can be used to study biological samples or the electrodes of an electrochemical cell. Other important interfaces are those between a liquid and its vapor and between two imiscible liquids. Consider, for example, the physical adsorption of a gas by a solid. If the solid is regarded as a giant sphere, the adsorption of the gas can be regarded as the interaction of a gas with a single infinitely large molecule dissolved in that gas. If the simplest form of the van der Waals theory of mixtures is applied to that system, then the adsorption isotherm is just = ρβ

a12 , β(∂ p /∂ρ)T

(39)

is a constant. At low densities β(∂ p /∂ρ)T = 1 where a12 and the adsorption is proportional to the density (Henry’s law). However, at higher densities β∂ p /∂ρ is a function of the density, and deviations from Henry’s law are observed. Especially interesting is the region near the critical point of the gas where (∂ p /∂ρ)T → 0 and singularities in the adsorption are observed. Interfaces between dissimilar materials may also become electrically polarized, with a separation of charge occurring at the interface. When a metal is placed into

IV. KINETICS AND DYNAMICS The previous sections have dealt only with the equilibrium properties of a system of molecules. Such properties tell us nothing of the time required for equilibrium to be reached or about the dynamical properties of these systems. The rate at which change occurs is the province of kinetics. The detailed manner in which chemical forces act to bring about atomic and molecular motion is the province of chemical dynamics. This section deals with the motions of atoms and molecules and the processes associated with chemical change. A. Kinetic Theory of Gases The kinetic theory of gases assumes that molecules have negligible size compared to their separation, are in continuous random motion, and interact only via elastic scattering. These postulates permit the calculation of molecular speed and velocity distributions. The probability that a molecule has a speed between v and v + dv is found to be d F(v) = 4π (m/2π kT )3/2 v 2 exp(−mv 2 /2kT )dv,

(40)

where T is the gas temperature and m the molecular mass. This is the Maxwell distribution of molecular speeds. Figure 5 displays this distribution for nitrogen gas at 25 and 500◦ C. Notice that the velocities are in the range of hundreds to thousands of meters per second, which are typical of those for small molecules at ambient temperatures. Recent experiments using light pressure from lasers to slow down atoms have resulted in atoms moving with velocities comparable to walking speed (1 m/s) and below. Such slow species are ideal for spectroscopic studies, since their adsorption spectra are not blurred by the Doppler effect due to their motion. The Maxwell distribution of molecular speeds permits the evaluation of such important quantities as the pressure p exerted by a dilute gas and the collision frequency Z in the gas under given conditions. The pressure is then given by 2 p = 13 ρmvrms ,

(41)

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metals, obtaining the result that the ratio of the thermal and electrical conductivities is a universal constant times the temperature (the Wiedemann–Franz law). B. Transport Properties

FIGURE 5 Maxwell distributions of speeds for molecular nitrogen at 25◦ C (298 K) and 500◦ C (773 K). Arrows indicate v¯ and vrms for each case. The most probable velocity has been arbitrarily scaled to unity in each case.

while the collision frequency for a one-component gas is given by √ Z = 2σ v¯ ρ, (42) where ρ is the number of molecules per unit volume and σ is the collision cross section. The quantity vrms is the average square speed and is related to the average kinetic energy and thus to the temperature, so that 3kT vrms = v 2 = , (43) m where m is the molecular mass. This quantity differs from the average speed, 8kT v¯ = v = . πm Equation (43) is equivalent to Eq. (26). Knowing the collision frequency and the molecular speed, it is possible to estimate the mean free path between collisions, λ = v/Z , so that √ λ = 1/ 2ρσ, (44) which shows the expected behavior that λ must decrease as the diameter of the molecule increases or as the density of the gas increases. Helium and nitrogen have estimated self-collision cross sections of 0.13 and 0.31 nm2 and at a pressure of 1 Torr ( = 133.3 N m−2 = 1.32 × 10−3 atm) there are about 3 × 1022 molecules m−3 , giving mean free paths of ∼1.8 × 10−4 and 7.6 × 10−5 m for helium and nitrogen, respectively. At 25◦ C, these species have respective velocities of v¯ = 1254 and 474 m/sec, giving collision frequencies of 6.9 and 6.2 × 106 sec−1 per molecule. The kinetic theory of gases can also be applied to the free-electron gas to describe the transport properties of

The kinetic motion of molecules may cause them to change their spatial distribution through successive random movements. This is the process of diffusion, which is a transport property. Other transport properties include viscosity, electrical conductivity, and thermal conductivity. While diffusion is concerned with the transport of matter, these are associated with the transport of momentum, electrical charge, and heat energy, respectively. Transport is driven in each case by a gradient in the respective property. Thus, the diffusion rate of species A is given by Fick’s law, Jz (A) = −D[dρ(A)/dz]

(45)

where Jz (A) is the net flux of A molecules crossing unit area in the z direction and D is the diffusion coefficient; simply kinetic theory leads to D ≈ 13 v¯ λ

(46)

Derivation of other transport properties follow from similar relationships. The viscosity coefficient or viscosity of a gas is given by η ≈ 13 v¯ ρmλ,

(47)

while the thermal conductivity coefficient κ is given by κ ≈ 13 v¯ ρλCv = ηCv /m,

(48)

where Cv is the molar heat capacity of the gas at constant volume. Notice that since λ is inversely proportional to ρ, both η and κ are independent of the gas density. This will be true so long as λ is small compared to the dimensions of the apparatus. In solution, D is given by the Stokes–Einstein relation which relates D to the viscosity coefficient of the solution, η, and the effective hydrodynamic radius a, where D = kT /6π ηa

(49)

and by the Einstein–Smoluchowski relation: D = d 2 /2τ,

(50)

where τ is the characteristic time between jumps of distance d. More elaborate theories of transport phenomena make use of the Boltzmann transport equation or computer simulations.

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C. Chemical Kinetics; Activated Complex Theory The quantitative study of chemical reaction rates and the factors on which these rates depend constitutes the field of chemical kinetics. Chemical reaction rates can be expressed in terms of the rate of production of any reaction product or as the rate of decrease in the concentration of any reactant. The individual steps in a chemical reaction sequence are termed elementary reactions. A number of consecutive elementary reactions may be responsible for a given chemical change. If the observed rate is found to be proportional to the concentration of a given reactant raised to some power, α then α is said to be the order of the reaction with respect to that reagent. The sum of the orders over all reagents gives the overall order of the reaction. A complex reaction, involving a number of elementary steps, may have a noninteger order. Thus the order should not be confused with the molecularity of an elementary reaction, which is the number of reagent molecules involved in a single reaction step. The sequence of elementary reactions by which a reaction proceeds is termed the reaction mechanism, a term also used to describe the detailed process of bond breaking and formation in a single reactive collision. By way of illustration, consider the formation of nitrogen dioxide from nitric oxide and oxygen. This reaction is found to be third order, corresponding to 2NO + O2 = 2NO2

(51)

with a third-order rate law corresponding to d[NO2 ] (52) = keff [NO]2 [O2 ]. dt Here the constant keff is the reaction rate constant and square brackets indicate concentrations. If concentrations are given in moles per liter, the rate constants will have units of (mol/L)1−n sec−1 , where n is the order of the reaction. A likely mechanism for this process can be written in terms of the elementary steps: k1

NO + NO → N2 O2 k−1

N2 O2 → NO + NO k2

N2 O2 + O2 → 2NO2 ,

(53a) (53b) (53c)

which leads to the observed rate law if the first two steps are assumed to come to equilibrium prior to the third reaction, or if the steady-state approximation, which assumes that the rate of change of all concentrations is zero, is invoked. Reaction (53a) has a molecularity of 2 and is a bimolecular reaction, while reaction (53b) is an example of a unimolecular reaction, involving a single species. An important class of reaction mechanisms are those in which a reaction product from one step is a reagent

in a prior step. The species concerned is often a highly reactive molecule with a vacancy in its outermost shell of electrons, termed a free radical. Such processes are termed chain reactions. Chain reactions are very important in polymerization reactions, where a radical may add to another reactant to form another (larger) radical. In cases where more than one reagent species is formed as a product of a later step, the chain is said to be branched, and such branching chain reactions often lead to explosions. In other cases, explosions may occur as a result of a fast exothermic reaction which yields a net excess of energy in the form of heat and in a time too short for the energy to be dissipated. The increase in temperature then causes an increase in rate, and the cycle ends in a thermal explosion. In some mechanisms a species may be consumed in one step of a reaction only to be regenerated in a subsequent step. In cases where the presence of this species increases the overall reaction rate, it is termed a catalyst, which is defined as a species that increases the rate of a reaction without being consumed or changing the reaction products. A catalyst must increase the rate of both forward and backward reactions in any system at equilibrium and can be thought of as lowering E 0 (see below). An expression for bimolecular rate constants can be obtained by observing that along a reaction coordinate the energy surface consists of two wells, representing the reactants and products, separated by a saddle point, representing the maximum energy required to pass along the minimum energy path between reactants and products. If the height of this maximum, relative to the reactant well, is E 0 then only collisions where the energy exceeds E 0 can lead to reaction. Integrating the Boltzmann distribution of energies over all energies exceeding E 0 , shows that probability of a collision with energy in excess of E 0 is proportional to exp(−βE 0 ). This is consistent with the rate law of Arrhenius: k = A exp(−E 0 /kB T ),

(54)

where A is known as the pre-exponential factor, and the Boltzmann constant is written as kB here to avoid confusion; A can readily be estimated from collision theory, using the expression for the collision frequency for one reagent with another, Z 12 Z 12 = ρ1 ρ2 σ12 [8kB T /µπ ]1/2

(55)

which leads to A = Pσ12 N A [8kB T /µπ ]1/2 ,

(56)

where NA is Avogadro’s number, which converts ρ to molar units, and P is the so-called steric factor, which accounts for the fact that not all collisions lead to reaction. Alternatively, we can replace the product Pσ12 with σreac , where σreact is termed the reactive cross section.

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172 A more general treatment of detailed reaction rates is available in the activated complex theory of Eyring, which assumes that there is an intermediate state between the reactants and the products, called the activated complex or transition state which can be regarded as at least somewhat stable and which is in thermodynamic equilibrium with the reactants, thus permitting thermodynamics to be applied. Instead of an energy, we must use the free energy G (because the pressure is constant) in the exponential. This treatment yields kT G ‡0 k=κ exp − h RT kT S0‡ H0‡ =κ exp − exp − , (57) h R RT where R is the gas constant, κ is the transmission coefficient and G ‡0 S0‡ , and H0‡ refer to differences between the activated complex and the reactants. Extensions of this statistical thermodynamical approach to estimating reaction rates include the RRK and RRK M theories of unimolecular decay rates, and the information theoretic formulation of reaction dynamics. These theories are remarkably successful, although generally more successful at interpreting experimental data and correlating results than at deriving results a priori. D. Reaction Dynamics; Inelastic Collisions Kinetic measurements and knowledge of reaction products and yields can provide only rather limited insight into the molecular dynamics of chemical reactions. To understand the detailed manner in which atoms and molecules move together and come apart in the process of a chemical reaction, it is necessary to study the isolated elementary reactions in as much detail as possible. Such isolation is most often provided by a dilute gas environment. Ultimately, the hope is to understand reaction dynamics in terms of electronic structure and to be able to calculate this for a chosen system. The electronic structure or potential energy surface is the meeting ground between theory and experiment. Currently most studies seek to probe those factors, or states, which influence the rate of chemical reactions, such as vibration and translational energy, and to examine the manner in which energy and angular momentum are disposed among the product states for various processes. This is the area known as state-to-state chemistry. Molecular photodissociation is an ideal process for such studies and has been examined in considerable detail. This unimolecular event is sometimes considered as a “half collision,” where the absorption of a photon excites the system to a repulsive state that flies apart. A number of radiation sources have been employed for such photoly-

Physical Chemistry

sis experiments, including discharge lamps, flash lamps, and synchrotrons. However, most recent studies have concerned laser photolysis. The photofragments are detected, for example, by emission or laser spectroscopy, which provides information on the velocity and quantum-state distribution of the fragments, with respect to rotational, vibrational, and electronic states. It is even possible with the aid of femtosecond lasers to follow the photofragmentation process in real time. Such measurements can provide information on the shape of the excited state potential energy surface. In the last decade, researchers have taken these ideas one step further to use the coherent nature of laser light to control the outcome of a photochemical reaction such as photodissociation. In this work, one or more pulse of laser energy is used to drive a reaction to a desired outcome, opening up exciting possibilities for new methods of chemical synthesis. Bimolecular reactions are often studies by firing collimated streams of reagents at each other in the form of crossed molecular beams. The scattered reagents and products can be detected by a rotatable mass spectrometer in order to measure angular distributions. Such experiments have shown that many reactions occur in essentially a single encounter in a direct mechanism, while others proceed through a long-lived complex mechanism. In other experiments, spontaneous light emission from the unrelaxed, or nascent, products, termed chemiluminescence, has been analyzed to yield quantum-state distributions. Lasers are often used to probe internal states of products, for example, by inducing emission as in laser-induced fluorescence (LIF) detection, and to prepare molecules in specific states and with chosen orientations. Vibrational energy is often found to be more efficacious in promoting reaction than is translational or rotational energy, since it is more strongly coupled to the reaction coordinate, or path in phase space along which reaction takes place. Product distributions are frequently observed to be far from equilibrium. For example, in direct reactions, high vibrational levels are often found to be more populated than low ones. This so-called population inversion forms the basis of the chemical laser. Reaction rate constants cannot be used to describe such detailed processes. Instead the differential reaction cross section, σreact n 1 , n 2 , n 3 , . . . n 1 , n 2 , n 3 , is employed, where n i are various quantum numbers and the primed quantities refer to reaction products. Such cross sections represent the effective collision area for reagents with given n 1 , n 2 , . . . , to give specific products. Rate constants represent the effective average of the product of the cross section with the approach velocity taken over the calculated distribution of reagent quantum states. Cross sections can be predicted from semiclassical trajectory calculations, in which equations of motion are

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solved by numerical integration, or they can be obtained from quantum calculations via the time-dependent Schrödinger equation. Both approaches require a previously calculated potential energy surface. However, accurate potentials are currently available only for the H + H2 reaction and its isotopic analogues, for which precise quantum calculations can be made. For other systems approximate surfaces can be obtained either semiempirically (e.g., using the LEPS or BEBO functions), or from approximate a priori calculations. Statistical theories are also employed. These are based on the assumption that different reaction channels are populated in proportion to the volume of phase space with which they are associated, which is consistent with conservation of energy and linear and angular momentum. Dynamical factors may cause deviations from such statistical behavior, providing information on the reaction mechanism. This is recognized in the information theoretic approach where the fully statistical outcome, or prior distribution, is compared with observations in so-called surprisal plots which indicate the degree to which the data deviate from statistical behavior. This approach has proven very valuable in the correlation and extension of a wide body of data. Since reaction rates can depend not only on reagent energy, but also on the form in which it is available, a full understanding of chemical behavior requires knowledge of the manner in which energy in various forms is redistributed by collisions. This information must be obtained by studies of energy transfer associated with inelastic collisions. Experimental studies vary from kinetic measurements of decay rates, to full state-to-state studies. It is found that rotational energy is readily transferred from one molecule to another, occurring on almost every collision. Transfer from rotation to translation can be 102 times slower, while transfer to vibration may be 104 times slower. Transfer between translation and vibration occurs only about once in a million collisions at room temperature. In general, the rate of energy transfer decreases rapidly as the amount of energy transferred increases, following an approximate exponential gap rule. E. Reactions in Solution In principle, reactions in solution occur in a similar manner to those in the gas phase and in some favorable cases the observed rate constant is the same in both phases. For example, the unimolecular decomposition of N2 O5 yields similar A and E 0 values in the gas phase and in a large range of solvents. However, there are many important differences. Reactions of ionic species and of large molecules such as proteins and polymers are rare in gas-phase studies but are common in solution. Reactions in solution are

often catalyzed, for example, by protons in acid catalysis and by enzymes in many biological systems. Moreover, interactions with solvent molecules may grossly alter the potential energy surface on which reaction occurs, compared to the isolated gas-phase system. Such interactions are strongest for polar reagents and solvents. Reactions in solution are often diffusion controlled, where the limiting step is the rate at which reagents can find each other. In the absence of strong interactions such as those between ions, the rate constant may be estimated from Fick’s law [Eq. (45)] together with the Stokes– Einstein relation [Eq. (49)] giving: k = 8RT /3η,

(58)

where R is the gas constant. Since reactants are also slow to drift apart, the time-averaged collision frequency per molecule may be close to the gas-phase value, so that reactions with rate constants much smaller than given by Eq. (58) may be relatively insensitive to diffusion effects (in practice this means E 0 ≥ 40 kJ/mol). Reaction products may also be slow to move apart, thus in liquid-phase photodissociation, where the adsorption of light causes a molecule to fall apart, the surrounding solvent cage may hold the products together long enough for recombination to occur. If two reactions are in equilibrium with an equilibrium constant K , and the back reaction is held constant at the diffusion limit, kd , then the forward rate constant will be equal to K kd . More generally, it is often found that for a given reaction involving a series of similar reagents, k ∝ K α, log k = b + α log K .

(59) (60)

This is the Brønstead equation, and is an example of a linear free-energy relationship, since log K ∝ G 0 and log k ∝ G ‡0 , then Eq. (60) could be written as G ‡0 = b + αG 0

(61)

Related to the Brønstead equation is the Hammett equation, which expresses the rate constant k of one of a series of related reactions in terms of a specific reference reaction with rate k0 , giving log(k/k0 ) = ρσ,

(62)

where ρ is a characteristic of the type of reaction and σ is a characteristic of the specific system. Expressions such as the Brønstead and Hammett equations are particularly useful since the complex nature of the environment makes absolute rate theories such as the activated complex theory difficult or impossible to apply in solution. The rate constant for a bimolecular reaction in solution can be expressed in terms of the activity coefficients of

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the reagents, γ (A) and γ (B), and of the transition state, γ (AB)‡ , by k = k0 [γ (A)γ (B)/γ (AB)‡ ],

(63)

where an activity coefficient is defined as the ratio of the effective concentration to the actual concentration, as required to obey Raoult’s law. The Debye–Hückel limiting law gives γ in terms of the ionic charges z of the species: √ log γ (i) = −sz i2 I , (64) where I is the ionic strength: I =

=∞ 1 i xi z i2 , 2 i =0

in which xi indicates molar concentration, giving: √ log k = log k0 + 2sz A z B I .

(65)

(66)

This indicates that the rate constant for reactions in solution depends on the ionic strength of the solution, behavior referred to as the kinetic salt effect. Thus, addition of inert ions will increase the rate of reaction between ions of like charge and vice versa. Recent advances in theory √ have shown that log k ceases to be a linear function of I at large ionic strength. The reaction rate continues to increase with increasing ionic strength, but less rapidly than that predicted by the above relation. A careful examination of the experimental data confirms this prediction. F. Reactions at Surfaces; Heterogeneous Catalysis; Corrosion Atoms and molecules frequently adsorb on surfaces, where they may decompose and/or react with other adsorbed species. Modern technology is increasingly dependent on surface chemistry which underlies many industrially important processes as well as destructive processes such as corrosion. It is useful to distinguish two types of adsorption: physical adsorption, or physisorption, and chemical adsorption, or chemisorption. Physisorption is similar in nature to condensation and involves little chemical interaction with the surface, being associated with van der Waals forces. Chemisorption involves a true chemical interaction with the surface, with the formation of a chemical bond. Thus, the enthalpy of physisorption is usually of the order of 20 kJ mol−1 , while for chemisorption values are in the region of 200 kJ mol−1 . A chemisorbed molecule may either remain intact in molecular chemisorption, or fall apart in dissociative chemisorption. In an important recent development, it is now possible to identify individual molecular bonds of adsorbed molecules using STM

(see above). In this method, the STM current shows features associated with the vibration frequencies of chemical bonds. Figure 6(A) shows an STM image of the molecule HC2 D on a copper surface. Figure 6(B) shows an image of the CD bond of this molecule. The fraction of a surface covered by a gas, , is given in terms of monolayers (ML), where 1.0 ML represents complete coverage by a single layer. The pressure dependence of at a given temperature is termed an adsorption isotherm. If the rate of adsorption is ka p(1 − ), and the rate of desorption is kd , we obtain the Langmuir isotherm: Kp = (67) 1 + Kp where K = ka /kd . Thus, when K p is small, is simply proportional to the pressure. An adsorption dependence on 1 − arises in the ideal case in which each molecule adsorbs at and occupies a single surface site. If two adjacent sites are required for adsorption, a (1 − )2 dependence might hold. The probability of adsorption for a single collision is termed the sticking probability, which usually implies chemisorption and may range from unity, for say oxygen on a clean metal surface, to close to zero for an inert system. At low temperatures, even inert gases may trap into a physisorption state, in proportion to their trapping probability. With a sticking or trapping probability of unity, a monolayer will form in about 1 sec at 10−6 Torr, which means that studies of clean surfaces must be carried out under conditions of ultra high vacuum (UHV), or below ∼10−9 Torr. The ability of surfaces to promote chemical reactions stems largely from their ability to cause dissociation. Consider the decomposition of ammonia to nitrogen and hydrogen: 2NH3 → N2 + 3H2

(68)

In the gas phase, this process has an activation energy of ∼330 kJ mol−1 , whereas in the presence of on a tungsten surface this falls to ∼160 kJ mol−1 . By providing an alternative low-energy path for reaction, the surface causes a large increase in the reaction rate. This is an example of heterogeneous catalysis. (See Section IV.C for a definition of catalysis.) For such a reaction to occur at a surface, the ammonia must first diffuse to the surface, it must dissociatively chemisorb, the hydrogen and nitrogen atoms must then recombine, and they must desorb and diffuse away from the surface. The recombination and desorption may actually occur as one step, as the reverse of dissociative adsorption. Either the adsorption or desorption steps are rate-limiting in gas–surface reactions, although fast liquid–surface reactions may be diffusion

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FIGURE 6 Scanning tunneling microscope image of a singly deuterated acetylene molecule (HC2 D) molecule on a Cu(100) surface. (A) An image of the C–D bond obtained by setting the second differential of the tunneling current to 269 meV. [See Stipe, B. C., Rezaei, M. A., and Ho, W. (1999). Phys. Rev. Lett. 82, 1724; courtesy B. C. Stipe, with permission.]

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176 limited. This is consistent with the fact that dissociative chemisorption and recombinative desorption are often activated processes. Chemical reactions at the gas–surface interface can be followed by monitoring gas-phase products with, for example, a mass spectrometer, or by directly analyzing the surface with a spectroscopic technique such as Auger electron spectroscopy (AES), photoelectron spectroscopy (PES), or electron energy loss spectroscopy (EELS), all of which involve energy analysis of electrons, or by secondary ionization mass spectrometry (SIMS), which examines the masses of ions ejected by ion bombardment. Another widely used surface probe is low-energy electron diffraction (LEED), which can provide structural information via electron diffraction patterns. At the gas–liquid interface, optical reflection ellipsometry and optical spectroscopies are employed, such as Fourier transform infrared (FTIR) and laser Raman spectroscopies. Elastic and inelastic collisions of atoms and molecules at surfaces are also of importance. The scattering of hydrogen and helium from surfaces leads to diffraction patterns in the same manner as with LEED, but since the atoms penetrate the surface far less deeply than even low-energy electrons, the structures obtained reflect the very surface of the sample. The inelastic surface scattering of molecules can be examined in detail using laser and mass spectrometric detection for the scattered molecules. Such measurements can be used to model the form of the gas–surface interaction potential, knowledge of which is a prerequisite for any detailed picture of gas–surface reaction dynamics. Not all surface chemistry is catalytic. In many cases the surface itself may be consumed, in processes such as etching and corrosion. Etching is employed to fabricate devices where it provides for the controlled removal of material. In the semiconductor industry, for example, discharges containing fluorine are used to etch silicon by volatilization as silicon fluorides. Corrosion is generally an unwanted process whereby items are destroyed through dissolution and/or oxidation. Metals may corrode through many different (usually electrochemical) processes. For example, in the presence of oxygen, a metal may displace protons as water or reduce oxygen to OH− , in acid and alkaline environments, respectively. In principle, this process requires the additional presence of a second metal, with a lower electrochemical potential. However, all samples have regions of high and low strain, which will have slightly different potentials. A given metal can be protected by contact with another (sacrificial) metal with a more negative potential, which will be preferentially corroded. This is applied in the galvanizing of iron by zinc.

Physical Chemistry

SEE ALSO THE FOLLOWING ARTICLES ADSORPTION • ATOMIC SPECTROMETRY • CHEMICAL THERMODYNAMICS • CRYSTALLOGRAPHY • KINETICS (CHEMISTRY) • LIQUIDS, STRUCTURE AND DYNAMICS • MECHANICS, CLASSICAL • PERIODIC TABLE (CHEMISTRY) • QUANTUM CHEMISTRY • QUANTUM MECHANICS • STATISTICAL MECHANICS • SURFACE CHEMISTRY

BIBLIOGRAPHY Adamson, A. W. (1980). “Physical Chemistry of Surfaces,” Wiley, New York. Adamson, A. W. (1979). “A Textbook of Physical Chemistry,” 2nd ed., Academic Press, New York. Atkins, P. W. (1982). “Physical Chemistry,” 2nd ed., Freeman, San Francisco. Bernstein, R. B. (1982). “Chemical Dynamics via Molecular Beam and Laser Techniques,” Oxford Univ. Press, New York. Berry, R. S., Rice, S. A., and Ross, J. (1980). “Physical Chemistry,” Wiley, New York. Eyring, H. (1944). “Quantum Chemistry,” Wiley, New York. Eyring, H., Henderson, D., and Jost, W. (1967). “Physical Chemistry—An Advanced Treatise,” 15 vols., Academic Press, New York. Eyring, H., Henderson, D., Stover, B. J., and Eyring, E. M. (1982). “Statistical Mechanics and Dynamics,” 2nd ed., Wiley, New York. Herzberg, G. (1950). “Molecular Spectra and Molecular Structure: I. Spectra of Diatomic Molecules,” 2nd ed., Van Nostrand-Reinhold, New York. Kauzmann, W. (1957). “Quantum Chemistry,” Academic Press, New York. Laidler, K. J. (1965). “Chemical Kinetics,” 2nd ed., McGraw-Hill, New York. Levine, R. D., and Bernstein, R. B. (1974). “Molecular Reaction Dynamics,” Oxford Univ. Press, New York. Levine, R. D., and Bernstein, R. B. (1987). “Molecular Reaction Dynamics and Chemical Reactivity,” Oxford Univ. Press, New York. Linnett, J. W. (1960). “Wave Mechanics and Valency,” Methuen, London. McQuarrie, D. A. (1976). “Statistical Mechanics,” Harper and Row, New York. Moore, W. J. (1983). “Basic Physical Chemistry,” Prentice Hall, New York. Partington, J. R. (1954). “An Advanced Treatise on Physical Chemistry,” 5 vols., Wiley, New York. Rowlinson, J. S. (1982). “Liquids and Liquid Mixtures,” 3rd ed., Butterworths, London. Smith, I. W. M. (1980). “Kinetics and Dynamics of Elementary Gas Reactions,” Butterworths, London. Smith, R. A. (1961). “Wave Mechanics of Crystalline Solids,” Wiley, New York. Steinfeld, J. I. (1974). “Molecules and Radiation: An Introduction to Modern Molecular Spectroscopy,” MIT Press, Cambridge, Massachusetts.

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Potential Energy Surfaces Donald G. Truhlar University of Minnesota

I. Introduction II. Quantum Mechanical Basis for Adiabatic Potential Energy Surfaces III. Topology of Adiabatic Potential Energy Surfaces IV. Breakdown of the Adiabatic Approximation V. Shapes of Potential Energy Surfaces

GLOSSARY Adiabatic representation Representation in which the electronic wave functions are calculated for fixed (i.e., nonmoving) nuclei. Avoided intersection Case in which two potential energy surfaces come together but do not intersect. Conical intersection Case in which two potential energy surfaces intersect such that their separation decreases to zero linearly in the relevant nuclear coordinates. Diabatic representation Representation in which the electronic wave function is not adiabatic. Dunham expansion Taylor series expansion of a potential energy curve in the vicinity of its minimum. Electron affinity Binding energy of an electron to a neutral atom or molecule. Equilibrium configuration Geometry of a molecule’s nuclear framework corresponding to the minimum adiabatic energy. Force field The gradient of the potential energy surface.

Glancing intersection Case in which two potential energy surfaces intersect such that their separation decreases to zero quadratically in the relevant nuclear coordinates. Ionization energy Energy required to remove an electron from an atom or molecule.

A POTENTIAL ENERGY SURFACE is an effective potential function for molecular vibrational motion or atomic and molecular collisions as a function of internuclear coordinates. The concept of a potential energy surface is basic to the quantum mechanical and semiclassical description of molecular energy states and dynamical processes. It arises from the great mass disparity between nuclei and electrons (a factor of 1838 or more) and may be understood by considering electronic motions to be much faster than nuclear motions. (When we say nuclear motions and nuclear degrees of freedom in this article, we refer to motions of the nuclei considered as wholes, i.e., to atomic motions.) This difference in timescales leads to the so-called electronic adiabatic approximation and to

9

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10 electronic adiabatic potential energy surfaces. Under some circumstances, however, it is convenient to use more general definitions; this results in potential energy surfaces variously known as diabatic or nonadiabatic, the latter adjective being a useful double negative (it is convenient to use this term because, except for a few small terms, adiabatic surfaces may be defined uniquely but in general diabatic surfaces are not; nonadiabatic has the useful connotation then of “anything except the adiabatic”). The concept of potential energy surfaces may be generalized to systems in liquids, in which case one obtains potentials of mean force.

I. INTRODUCTION The separation of electronic and nuclear motions according to time scales and the consequent introduction of an effective potential energy surface for nuclear motion was first considered by Born and Oppenheimer. Although their method is seldom used in modern work, the modern equivalents are still commonly called Born–Oppenheimer approximations and Born–Oppenheimer potential energy surfaces. The modern form of the derivation, which is summarized below, dates to a later paper by Born and to work by Born and Huang. Occasionally the phrases Born–Oppenheimer and Born–Huang are used to specify whether certain small terms are included in the potential energy surfaces, although, as mentioned above, it is more common to refer to any adiabatic surfaces as Born– Oppenheimer surfaces. A potential energy surface is an effective potential energy function for the relevant nuclear degrees of freedom. The latter are usually defined as all nuclear degrees of freedom minus three overall translations and two or three rotations of the nuclear subsystem. If an atom has N nuclei, it is common to consider the potential energy as a function of 3N − 5 coordinates for N = 2 (since two nuclei always lie on a line and therefore, when considered as point masses, have only two rotational degrees of freedom) and 3N − 6 coordinates otherwise. Thus for N = 2 we actually have a potential energy curve (i.e., a function of one variable), whereas for N ≥ 3 we have a potential energy hypersurface (i.e., a function of three or more variables). A potential energy surface would strictly speaking denote the potential energy as a function of two coordinates in a two-dimensional cut through the (3N − 6)-dimensional internal-coordinate space. In this article, however, we follow the very common language by which any potential energy hypersurface or potential energy function is referred to as a surface.

Potential Energy Surfaces

II. QUANTUM MECHANICAL BASIS FOR ADIABATIC POTENTIAL ENERGY SURFACES The Schrödinger equation for a system of N nuclei and n electrons is (H − E)(r, R) = 0,

(1)

where H is the Hamiltonian or energy operator of the molecule: h2 2 h2 2 H =− ∇ − ∇ + V (r, R) + Hrel (r, R). (2) 2M R 2m r In these equations, h is Planck’s constant divided by 2π, and R denotes a 3N -dimensional vector of scaled nuclear coordinates: M1 1/2 R1 = X 1, (3a) M M1 1/2 R2 = Y1 , (3b) M .. . R3N =

MN M

1/2 ZN,

(3c)

where M j , X j , Y j , and Z j are the mass and Cartesian coordinates of the nucleus j, M is any of the nuclear masses or a convenient reduced nuclear mass, m is the electronic mass, r is a 3n-dimensional vector of electronic Cartesian coordinates {xk , yk , z k }nk=1 : r1 = x 1 ,

(4a)

r2 = y1 ,

(4b)

.. . r3n = z n ,

(4c)

V (r, R) is the sum of all coulomb forces between the particles, Hrel (r, R) is the energy operator for relativistic effects including mass-velocity and spin–orbit coupling, E is the total energy, and ψ(r, R) is the wave function. Notice that the first two terms in Eq. (1) represent the nonrelativistic kinetic energy of the nuclei and the electrons. Usually the change in ψ(r, R) is about the same order of magnitude when we move a nucleus a small amount as when we move an electron the same small amount. When this is the case, ∇R2 ψ(r, R)/M is smaller than ∇r2 ψ(r, R)/m by a factor of order m/M, which is less than about 10−3 ; thus the first term in Eq. (2) may be neglected to a first approximation. The physical interpretation of this is that because of their larger masses, the nuclei move so slowly compared

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11


with the electrons that the dependence of the wave function on electronic coordinates is essentially the same as if the nuclei were completely stationary (i.e., as if their mass were infinite so that their kinetic energy was zero). Then Eq. (1) becomes h2 2 − ∇r + V (r, R) + Hrel (r, R) − Uα (R) 2m × ψα (r; R) = 0,

(5)

where Uα (R) and ψα (r; R) denote the eigenvalue and eigenfunction, respectively, with index α. Notice that both of these depend parametrically on the nuclear positions R and also that the ψα (r, R) form a complete set of functions of r for any particular value of R. In general the spectrum {Uα (R)} contains a continuous part as well as a discrete part, but the discrete part is the important part for the concept of potential energy surfaces. We shall therefore use only a discrete index α. This is adequate for almost all practical work since the continuous part is usually neglected in calculations. For a more rigorous derivation, however, all sums over discrete α in the equations below must be replaced by a sum and an integral. To obtain the complete system wave function we choose a trial function ψ(r, R) = ψα (r; R)χα (R). (6) α

Since the {ψα (r; R)} are complete, this trial function yields the exact ψ(r, R) if we retain the complete set and solve for {χα (R)} by the variation method. The best {χα (R)} by this method satisfy the equation dr ψβ∗ (r; R)(H − E) ψα (r; R)χα (R) = 0, α

β = 1, 2, . . . , ∞.

(7)

If we carry out the indicated operations, using Eq. (5) and the orthogonality of the {ψα (r; R)} in r at fixed R, Eqs. (7) become h2 − ∇R2 χα (R) + 2 Fαβ (R) · ∇R χβ (R) 2M β + G αβ (R)χαβ (R) + [Uα (R) − E]χα (R) = 0, β

α = 1, 2, . . . , ∞, (8) where

Fαβ (R) =

and

Gαβ (R) =

dr ψα∗ (r; R)∇R ψβ (r; R)

(9)

dr ψα∗ (r; R)∇R2 ψβ (r; R).

(10)

By the same argument given above for the variation of ψ(r, R) with respect to r and R, we expect that the terms containing Fαβ (R) and G αβ (R) are usually much smaller than the term containing Uα (R). When this is so, we may neglect the small terms, and the set of coupled Eqs. (8) simplifies to a separate uncoupled equation for each χα (R), namely, h2 2 − (11) ∇ + Uα (R) − E χα (R) = 0. 2M R This has the form (Hnuc − E)χα (R) = 0,

(12)

where Hnuc is an effective Hamiltonian for nuclear motion given by h2 2 ∇ + Uα (R). (13) 2M R Since a Hamiltonian is usually the sum of a kinetic energy operator and a potential energy operator, we may interpret Uα (R) as an effective potential for nuclear motion. In fact, Uα (R) is the potential energy surface that we sought to derive. Recalling the origin of Uα (R), we see that it represents the total energy of the electrons, both kinetic and potential, plus all the rest of the potential energy, when the electrons are in state α. Alternatively, it represents the entire (coulombic plus relativistic) potential energy of all particles plus the electrons’ kinetic energy. When the equations for the χα (R) decouple, as in Eq. (12), the electronic state is preserved during the nuclear motion. The resulting quantized energy requirement of the electrons plus the rest of the potential energy (given in the absence of relativistic effects by the nuclear–nuclear coulombic interaction energy) together constitute an effective potential for nuclear motion. When nuclear motion can be approximated by classical mechanics (which is often reasonable, especially for atoms heavier than hydrogen), Eqs. (12) and (13) are replaced by Hnuc = −

¨ = −∇R Uα (R), MR

(14)

¨ is the nuclear acceleration, and the right-hand where R side is the force on the nuclei. Since Uα (R) generates the force function, it is sometimes called the force field. Inclusion of the spin–orbit and other relativistic terms in Eq. (5), as we have done, is, strictly speaking, the most correct approach. This yields, as we have seen, a set of nuclear wave functions χα (R) whose uncoupled motion is governed by the potentials Uα (R) and which are coupled only by the nuclear-derivative terms Fαβ (R) and G αβ (R). In practice, though, Hrel (R) is difficult to treat on an equal footing with the coulombic terms in the Hamiltonian. Therefore one sometimes works with

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12 nonrelativistic adiabatic potential surfaces, although the adjective nonrelativistic is seldom stated explicitly. In this approximation one temporarily neglects Hrel (r, R) to solve Eq. (5). This yields a new set of nuclear wave functions χα (R) and Uα (r, R) that are easier to work with, but the effect of Hrel (r, R) must be included later. The matrix elements Hrel,αβ (R) = dr ψα∗ (r; R)Hrel (r, R)ψβ (r; R) (15) provide both a perturbative correction to the nonrelativistic potential energy surfaces, for α = β, and an additional coupling mechanism that must be considered along with Fαβ (R) and Uαβ (R) for α = β. In the rest of this article we restrict our attention to the nonrelativistic approximation, and we assume that Hrel (r, R) has been neglected in solving Eq. (5).

III. TOPOLOGY OF ADIABATIC POTENTIAL ENERGY SURFACES As shown in Eq. (5), an adiabatic potential energy surface is an eigenvalue of a Hermitean operator, that is, one that has only real eigenvalues. In most cases these eigenvalues are nondegenerate. At some geometries R, however, two or more eigenvalues may be equal, which is called a degeneracy. Such degeneracies may be mandated by symmetry or may be accidental. Points where two or more eigenvalues are equal are particularly interesting, and we may categorize some features of the potential energy surfaces in the neighborhoods of these points on the basis of symmetry. The nuclear configuration R, which appears as a parameter in the eigenvalue Eq. (5), may be classified by a symmetry point group, for example, D∞h for a homonuclear diatomic molecule or another symmetric linear molecule, C∞v for a heteronuclear diatomic molecule or other nonsymmetric linear molecule, Td for a tetrahedral molecule, C2v for a symmetric nonlinear triatomic molecule, and Cs for a planar molecule. Since the operations of such a group commute with −(h 2 /2M)∇r2 + V (r, R), the eigenfunctions ψα (r, R) of this operator can be taken to transform as irreducible representations of the group. We may thus classify both the eigenfunctions ψα (r, R) and eigenvalues Uα (R) by these irreducible representations, e.g.,

+ g , u , or g for D∞h or A1 or B2 for C 2v . First consider the case of two nuclei. As already mentioned, the potential energy surfaces in this case are really curves; they depend on only one scalar variable, the internuclear distance, which we may call R. One can show, on general grounds, that for a system with an even number of electrons, two Uα ( R) may be accidentally equal


FIGURE 1 An avoided crossing for a diatomic molecule. Ua is the potential energy for electronic state a , R the internuclear distance, and R ∗ the distance corresponding to an avoided crossing.

at isolated values of R if they correspond to different symmetry but not if they correspond to the same symmetry. When two Uα ( R) are equal, that is called a curve crossing. Sometimes two Uα ( R) approach very closely as if they are about to cross but then avoid crossing. This is called an avoided crossing. An example is shown in Fig. 1. When the system has an odd number of electrons and exists in a magnetic-free region, the possibilities are the same except that all Uα ( R) occur in degenerate pairs. Imposition of a magnetic field removes the degeneracy. Now consider the case of N ≥ 3 for which the potential energy surfaces depend on three or more variables. Here we also find potential surface intersections of states belonging to different symmetries and avoided intersections of states belonging to the same symmetry, but there is also a third possibility, namely, intersections even of potential energy surfaces belonging to the same symmetry. Such intersections in general occur in subspaces of dimension 3N − 8 or lower. If we consider a subset of two degrees of freedom in which the intersection occurs at a single point, the shape of the surfaces in the vicinity of the intersection is as shown in Fig. 2, that is, the two surfaces form a double cone. Such intersections are called conical intersections. Another shape of intersection may occur at linear geometries of polyatomic molecules. In this case, the two surfaces may have zero slope at the point of intersection. Such intersections are called glancing rather than conical.

FIGURE 2 Portions of two potential energy surfaces exhibiting a conical intersection. Ua is the potential energy for electronic state a .

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In general, for systems of three or more nuclei, some potential energy surfaces that intersect when one neglects spin–orbit coupling avoid intersecting when one includes it.

IV. BREAKDOWN OF THE ADIABATIC APPROXIMATION We are now in a position to understand the limitations of the adiabatic potential energy surface concept. First, however, we should understand the physical origin of avoided crossings and avoided intersections. We begin by considering the diatomic molecule NaCl, and we let R denote the distance between the nuclei. At R = ∞, the energy of two neutral atoms is lower than the energy of a Na+ ion and a Cl− ion by the difference E of the ionization potential of Na and the electron affinity of Cl− . As R is decreased, however, the energy of the ionic state decreases rapidly because of the long-range coulomb attraction, which may be represented by −e2 / R where e is the electronic charge, while the energy of the neutral state stays approximately constant until much shorter distances where the covalent interaction becomes appreciable. Thus at some distance R ∗ given approximately by e2 E ∼ (16) = R ∗ the hypothetical purely ionic state and the hypothetical purely covalent state would have the same energy. Actually though, at this R the corresponding eigenfunctions of Eq. (5) have mixed character, partly covalent and partly ionic, with about

50% partial ionic character. Since both states have 1 + g character, their eigenvalues are different. We call the energies of the hypothetical states with pure valence characteristics U1d ( R) and U2d ( R), where d denotes diabatic (or nonadiabatic). Although U1d ( R) and U2d ( R) cross, the adiabatic curves U1 ( R) and U2 ( r ) avoid crossing, having the shapes shown in Fig. 1. For this case α = 1 corresponds to a covalent state to the right of R ∗ but to an ionic state to the left of R ∗ and vice versa for α = 2. Now recall the argument given above for neglecting F12 (R) and G 12 (R). At R = R ∗ , the wave function is changing character very rapidly as a function of R, and the action of ∇R on ψ(r, R) is unusually large; this means that F12 ( R) and G 12 ( R) need not be negligible. In such a case, the dynamics governed by the nuclearmotion wave functions χ1 (R) and χ2 (R) do not decouple into independent motions governed by effective potentials (i.e., the adiabatic potential energy surface concept breaks down). Although we have given the argument for a particular diatomic molecule, the effect is general, and the

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13 adiabatic approximation is expected to break down in the vicinity of any avoided crossing or avoided intersection, and also at actual intersections for which the symmetry of the two states is the same at the intersection. There are other circumstances under which the adiabatic approximation may break down. We have considered the case where ψ(r, R) varies rapidly with R because of the factor ψα (r; R). A second case where the term containing Fαβ (R) in Eq. (8) may be significant occurs when χα (R) varies unusually rapidly, for instance, in highenergy collisions of Na+ with Cl− . When the nuclear speed is large, χα (R) must vary significantly on the scale of a very small de Broglie wavelength. We may summarize the two cases in a simple but approximate way as follows: When the nuclear kinetic energy is much smaller than the spacings between the adiabatic electronic energy surfaces Uα (R), these surfaces serve as potential energy surfaces for nuclear motion. When the nuclear kinetic energy is comparable to or larger than the spacings between the Uα (R), the adiabatic approximation may, and often does, break down. The adiabatic potential energy surfaces need not be abandoned completely when the adiabatic approximation breaks down, especially if the region of breakdown is fairly localized, as it often is when the breakdown is due to an avoided or conical intersection. If the nonadiabatic behavior is localized to a small region, we often employ the model of surface hopping. In this model the nuclear motion is assumed to be governed by an adiabatic potential energy surface until a nonadiabatic region is reached. In such a region there is a nonzero quantum mechanical probability that the system “hops” to another surface. Based on this probability one portion of the quantum mechanical probability density exits the nonadiabatic region in one of the adiabatic electronic states, and the other portion exits in the other one or more coupled adiabatic electronic states. After this the nuclear motions again proceed independently as governed by single potential energy surfaces until another nonadiabatic region is reached. Although this model neglects certain coherency effects that may be important for quantitative work, it is often useful for qualitative discussions and semiquantitative calculations. Another concept often invoked for qualitative discussions and for calculations when the adiabatic approximation breaks down is that of diabatic potential energy surfaces. There are several nonequivalent ways of defining such surfaces, each of which may be useful under some circumstances. The simplest way is that already illustrated above in conjunction with the NaCl example: namely, a diabatic state is the effective potential energy function for nuclear motion when the electronic state is artificially constrained to a state of prespecified pure valency.

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A second way to define diabatic electronic states and potential energy surfaces is more mathematical. Notice that the valency-character prescription leads to states coupled by the operator h2 2 (17) ∇ + V (r, R) + Hrel (r, R) 2m r and also by −(h 2 /2M)∇R2 . From the point of view of the nuclear motion, the former is sometimes called potential coupling (since it involves only multiplicative operators in the nuclear coordinates), and the latter is called derivative coupling. Since electronically adiabatic states have derivative coupling but no electronic coupling, a natural question is whether useful diabatic states can be defined to have potential coupling but not derivative coupling. Unfortunately, this leads to states that are completely independent of R and are not useful. It is possible, however, to make one component (in one or another coordinate system) of the vector coupling operator Fαβ (R) vanish for all α, β. Furthermore, if Fαβ (R) is approximated as the gradient of a scalar (which can be a good approximation when nonadiabatic effects are dominated by a narrowly avoided intersection), then it is possible to make all components of Fαβ (R) vanish for all α, β. Both of these prescriptions are sometimes employed to obtain diabatic states. Consider, for example, the case where Fαβ (R) is the gradient of a scalar for all R; then it has zero curl. We define diabatic electronic states by φαd (r; R) = φβ (r; R)Tβα (R). (18) Hel = −

β

The states {φαd (r; R)χαd (R)} will be uncoupled by nuclear derivative operators if we choose Tβα (R) for all β and α such that ∇R Tαβ (R) = Fαγ (R)Tγβ (R), (19) γ

−

h2 2 ∇r + V (r, R) − Uαn (R) ψαn (r, R) = 0, (22) 2m

where the superscript n denotes nonrelativistic. The true adiabatic states are coupled only by −(h 2 /2M)∇R2 , but these nonrelativistic adiabatic states are coupled by both this operator and Hrel (r, R). Because of the latter coupling, the nonrelativistic adiabatic electronic states ψαn (r; R) and their associated potential energy curves Uαn (R), which are the most widely employed states and potential energy surfaces in quantum chemistry, are actually diabatic. They are nevertheless usually called adiabatic although nonrelativistic is adiabatic is technically more appropriate.

V. SHAPES OF POTENTIAL ENERGY SURFACES A. Diatomics A schematic illustration of some typically shaped adiabatic potential energy curves for a diatomic molecule is shown in Fig. 3. All five curves shown become large and positive at small internuclear distance R. This represents a repulsive force between the nuclei and is due to internuclear repulsion and the unfavorability of overlapping the atomic charge clouds of the two different centers. All five curves tend to constants at large R. This is because the atomic interaction energy eventually decreases to zero as the distance between the atoms is increased. The constant spacings between the curves at large R are equal to the atomic excitation energies. Curves 1 and 2 are effective potentials for the interaction of ground-state atoms, and curves 3–5 represent effective potentials for the case where at least one of the atoms is excited. The figure shows an avoided crossing between curves 2 and 3 and

and this set of coupled partial differential equations does have a solution if Fαβ (R) has zero curl. Furthermore, if the state expansion α, β, . . . is truncated to a finite number of computationally important states, then the diabatic electronic basis is not independent of R. In this way, one can define a diabatic basis by a transformation from an adiabatic one, and it spans the same space. The diabatic potential surfaces are given by d Uαα (R) = |Tγ α (R)|2 Uγ (R), (20) γ

and the potential couplings are given by Uαβ (R) = Tγ∗α (R)Uγ (R)Tγβ (R).

(21)

γ

As mentioned in Section II, the usual treatments of potential energy surfaces neglect Hrel (r, R) in Eq. (5). Thus one solves

FIGURE 3 Typical potential energy curves for a diatomic molecule ordinarily thought of as bound. Ua and R are as in Fig. 1; De is the equilibrium bond energy of the ground state. Curves 1–5 are discussed in the text.

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true crossings of curves 3 and 4 by curve 5. Curves with deep enough minima, such as 1 and 3, may possess bound states of nuclear motion. Classically, a pair of nuclei whose motion is governed by one of these curves could show simple, almost-harmonic motion in the vicinity of the minimum of the curve. Quantally, there could be bound vibrational states localized in these regions of R. Curve 1 has the typical shape for the ground electronic state of a stable molecule such as H2 , N2 , or HCl. Such a curve is often represented in the vicinity of the minimum by a quadratic potential: 1 Uα ( R) ∼ (23) = k( R − Re )2 , 2 where k is the force constant (actually the quadratic force constant) and Re the equilibrium internuclear distance. An analytic representation valid over a wider range of R is given by 1 1 Uα ( R) ∼ = k11 ( R − Re )2 + k111 ( R − Re )3 2 2 1 + k1111 ( R − Re )4 + · · · , (24) 2 where k has been renamed k11 , and k111 and k1111 are anharmonic (cubic and quartic) force constants. Sometimes the constant coefficient of ( R − Re ) j is written 1/j! instead of 12 so care must be exercised when using anharmonic force constants. Equation (24) is called a Dunham expansion. An approximate representation of a potential curve like curve 1 in Fig. 3 that gives its approximate shape over the full range of R is Uα ( R) = De {1 − exp[−α( R − Re )]}2 .

(25)

This is called a Morse curve, De the equilibrium dissociation energy, and α the Morse range parameter. More complicated analytic forms with more parameters are also used. Information about the various parameters (Re , k11 , k111 , . . . , De , and α) comes primarily from spectroscopy, scattering or kinetics experiments, and quantum machanical electronic structure calculations. These are also the sources for information about potential energy surfaces of systems with three or more atoms. Figure 4 shows, to about the same scale as Fig. 3, some typically shaped potential curves for a diatomic system usually thought of as unbound (e.g., He2 , HeNe, or ArH). Notice that the lowest potential energy curve has only a very small minimum at large R. When this minimum is important for the problem at hand, such a potential energy curve is often represented by a Lennard–Jones 12–6 potential: σ 12 σ 6 Uα ( R) = 4ε , (26) − R R

FIGURE 4 Typical potential energy curves for a pair of atoms ordinarily thought of as unbound. Ua and R are as in Fig. 1. See text for discussion of the curves.

where ε is the well depth and σ the collision diameter. Notice that the minimum of Uα (R) occurs at R = Rm where Rm = 21/6 σ

(27)

Uα ( R = Rm ) = −ε.

(28)

and that

When the minimum of a predominantly repulsive potential curve is considered negligible, it may be represented by a so-called anti-Morse curve: Uα ( R) = D AM {exp[−2β( R − R AM )] + 2 exp[−β( R − R AM )]} + C AM ,

(29)

where D AM , β, R AM , and C AM are constants, or even by the simpler β Uα ( R) = exp(−α R). (30) R Equation (29) or (30) could be applied to curve 4 of Fig. 3, to curve 1, 3, or 4, of Fig. 4, or even to curve 2 of Fig. 3, for which it might be useful in the region to the right of the avoided crossing and to the left of the shallow, large- R minimum. Notice that the zero of energy is arbitrary for potential energy surfaces as long as it is chosen consistently throughout a given calculation. In Fig. 3 we placed the zero of energy at the bottom of the lowest potential curve. In Fig. 4 we placed it at the energy of two separated groundstate atoms. B. Larger Molecules Potential energy surfaces for systems with three or more atoms are harder to illustrate because they depend on three or more internal coordinates. Analytic representations are also more complicated than for diatomics.

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Potential energy surfaces may be roughly classified into those with deep enough minima to support one or more strongly bound vibrational states and those without such minima. We shall call the former attractive surfaces because of the role played by the attractive forces between the atoms in creating the minimum. The latter will be called repulsive surfaces. Because spacings between potential energy curves are usually much larger than thermal energies, most molecular motions under ordinary conditions are usually governed by the lowest energy potential surface. Thus molecules that usually exist as bound, stable entities (e.g., H2 O, CO2 , or CH4 ) have attractive surfaces in their ground (i.e., lowest energy) electronic states, whereas molecules usually observed only as transient species during collisions (e.g., FH2 which exists during F + H2 or H + HF collisions or NeH2 O, which exists during Ne + H2 O collisions) have repulsive surfaces. Systems like Ne2 or NeH2 O may exist as stable but very weakly bound (and thus easily and usually dissociated) species because of shallow potential wells in predominantly repulsive potential energy surfaces. Such weakly bound species are called van der Waals molecules. Systems with attractive surfaces in their ground electronic states may have repulsive surfaces in excited (i.e., higherenergy) electronic states, and vice versa. Examples of van der Waals wells and repulsive excited states for the easily illustrated special case of diatomic molecules may be seen in Figs. 3 and 4. The most well understood region of attractive potential energy surfaces is usually the region near the minimum. One usually describes the potential energy surface in such a vicinity by a Taylor’s series about the minimum: Uα (R) = Ue +

1 k i j qi q j 2 i j

+

1 ki jk qi q j qk 2 i j k

+

1 ki jkl qi q j qk ql + · · · , 2 i j k l (31)

where Ue , ki j , kik j , . . . are constants, and the {q j } are suitable internal coordinates defined to vanish at the location of the minimum. As written, Eq. (31) contains no terms linear in the {q j }, but if these are not related to Cartesian coordinates by a linear transformation, it may be necessary to include linear terms. Equation (31) is called an anharmonic force field. If the coordinates are linear combinations of Cartesians and terms beyond the quadratic are neglected, it becomes a harmonic force field. In the harmonic approximation it is always possible to define the {q j } in such a way that the cross terms vanish (i.e., ki j = 0

FIGURE 5 Perspective view of potential energy surface for collinear H + HCl → H2 + Cl. The vertical axis is potential energy, and axes in the horizontal plane are nearest-neighbor distances.

if i = j). If this is done and cross terms vanish in the kinetic energy operator as well, Eq. (31) is called a normal-mode expansion. In the vicinity of the minimum of the surface, a twodimensional cut through an attractive potential energy surface has the shape of a distorted paraboloid of revolution. Figure 5 shows a perspective view of a cut through a potential energy surface for a chemical reaction; in particular it is based on an approximate surface for the reaction H + HCl → H2 + Cl. To represent the potential as a function of two internal coordinates, the three atoms are restricted for this figure to lie on a straight line. If the hydrogens are labeled Ha and Hb , the left–right axis is the Hb -to-Cl distance with large values at the left, and the front–back axis is the Ha -to-Hb distance with large values in the foreground; the third interpair distance is the sum of these two. The vertical axis is potential energy. The figure clearly shows the existence of a minimum-energy reaction path from reactants in the foreground to products at the back left. The highest energy point along the minimumenergy path is a saddle point. This point is sometimes called the transition state, and it primarily determines the threshold energy for reaction to occur. The shape of the reaction path is important for determining the reaction probability as a function of the vibrational and relative translational energy of the reactants. Figure 6 shows the same information as in Fig. 5 but in the form of a contour map (i.e., a set of isopotential contours). The horizontal axis is the distance from Cl to

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the center of mass of H2 , and the vertical axis is [(MCl + 2MH /4MCl ]1/2 times the Ha -to-Hb distance. Because of the mass scaling factor, the nuclear-motion Hamiltonian, Eq. (13), in this coordinate system becomes h2 ∂ 2 ∂2 Hnuc = − + Uα (x , y), + (32) 2µ ∂ x 2 ∂ y2 where µ=

2MCl MH . (MCl + 2MH )

(33)

Since the coefficient of both derivative operators is the same, this Hamiltonian is the same as that for a single particle in two dimensions under the influence of a potential function Uα (x , y). This analogy is very helpful in mentally visualizing the motion of a polyatomic system whose dynamics are governed by a multidimensional potential surface. In Fig. 6 the H + HCl reaction is initiated at the lower right, and products are formed when the system, having passed through or near the saddle point (denoted + in the figure), reaches the top. For solutes in the liquid phase (e.g., an organic molecule in aqueous solution), one can obtain an effective potential

FIGURE 6 Contour map of a potential energy surface for collinear H + HCl → H2 + Cl. x is the distance from H to the center of mass of HCl and y the mass-scaled distance from Cl to its nearest H. Both axes are given in units of a 0, where 1a 0 = 1 bohr = 0.5292 × 10−10 m.

function of the solute coordinates by adding the free energy of solution to the gas-phase potential surface. The resulting potential function may be used in Eq. (14), and it is called a potential of mean force.

SEE ALSO THE FOLLOWING ARTICLES ATOMIC AND MOLECULAR COLLISIONS • ORGANIC CHEMICAL SYSTEMS, THEORY • QUANTUM MECHANICS • SURFACE CHEMISTRY

BIBLIOGRAPHY Gao, J., and Thompson, M. A. (1998). “Combined Quantum Mechanical and Molecular Mechanical Models,” American Chemical Society, Washington, DC. Herzberg, G. (1966). “Molecular Spectra and Molecular Structure. III. Electronic Spectra and Electronic Structure of Polyatomic Molecules,” Van Nostrand Reinhold, New York. Kondratiev, V. N., and Nikitin, E. E. (1981). “Gas-Phase Reactions: Kinetics and Mechanisms,” Springer-Verlag, Berlin. Löwdin, P.-O., and Pullman, B., eds. (1983). “New Horizons of Quantum Chemistry,” D. Reidel, Dordrecht, Holland. Maitland, A., Rigby, M., Smith, E. B., and Wakeham, W. A. (1981). “Intermolecular Forces,” Clarendon, Oxford. Michl, J., and Bonaˇcić-Koutecký. (1990). “Electronic Aspects of Organic Photo Chemistry,” John Wiley & Sons, New York. Murrell, J. N., Carter, S., Farantos, S. C., Huxley, P., and Varandas, A. J. C. (1984). “Molecular Potential Energy Functions,” Wiley, Chichester. Salem, L. (1982). “Electrons in Chemical Reactions: First Principles,” Wiley-Interscience, New York. Simons, J. (1983). “Energetic Principles of Chemical Reactions,” Jones and Bartlett, Boston. Smith, I. W. M. (1980). “Kinetics and Dynamics of Elementary Gas Reactions,” Butterworths, London. Truhlar, D. G., ed. (1981). “Potential Energy Surfaces and Dynamics Calculations: For Chemical Reactions and Molecular Energy Transfer,” Plenum, New York. Truhlar, D. G., and Morokuma, K., eds. (1999). “Transition State Modeling for Catalysis,” American Chemical Society, Washington, DC.

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Solid-State Chemistry Bahne C. Cornilsen Michigan Technological University

I. The Structure–Property Relationship II. Solid-State Structure and Structural Characterization III. Preparation Techniques and Property Variation

GLOSSARY Condensed matter sciences Include solid-state physics, ceramic science and engineering, metallurgical engineering, polymers, and materials science, as well as solid-state chemistry. Sintering General process by which powders react and densify to form polycrystalline compacts or ceramics. Such compacts densify by grain growth and porosity reduction. Sintering can be reactive, i.e., involve the reaction of two or more solid components to form a product or products. Structure Used herein to refer to the molecular-level, crystallographic, three-dimensional arrangement of atoms, as controlled by the chemical bonding. Both long-range and short-range order/disorder, including crystal imperfections and defects, must be considered.

SOLID-STATE CHEMISTRY concerns the preparation, structure, and properties of solid materials, often focusing on the relationship between structure and properties. Unique solid-state properties are taken advantage of for numerous practical, technological applications. Many properties are structure sensitive, i.e., they are controlled by the chemical bonding and molecular-level structure.

The syntheses and fabrication procedures themselves often play a key structure-controlling role in the preparation of materials with optimum properties. Detailed characterization of the structure and bonding is of major concern to the solid-state chemist. Control of structure and bonding during preparation and processing allows control of the critical, technologically important properties. Such control is necessary to optimize material performance.

I. THE STRUCTURE–PROPERTY RELATIONSHIP Solid-state chemists characterize materials with respect to their chemical and physical properties, structure, and bonding, as well as define how these properties are controlled by the chemical bonding and microscopic structure. They Vplay an increasingly important role as part of a team of condensed matter specialists whose common goal is to design materials with optimum properties for critical applications. Improved economic performance in a globally competitive economy is strongly dependent upon materials development. Materials problems limit the development of a variety of technologies, including high-temperature superconductors, high-energy density batteries, structural and insulating materials able to

295

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296 withstand extremely high temperatures, improved heterogeneous catalysts, improved electronic materials, optics compatible with very high laser intensities, stabile nuclear waste containment materials, and longer lasting electrodes for fuel cells and magnetohydrodynamic generators. To solve such problems and to optimize performance, it is necessary to control the molecular-level structure and bonding and to understand the structure–property relationship. The syntheses and fabrication processes themselves are therefore critical for preparation of solid-state materials which display the desired properties. Properties of interest include the chemical reactivity as well as the electronic, magnetic, optical, and mechanical properties.

II. SOLID-STATE STRUCTURE AND STRUCTURAL CHARACTERIZATION Characterization of the detailed chemical structure and bonding of a solid is a prerequisite to the understanding and control of the chemical and physical properties. In general the properties of a solid are controlled by the macroscopic and the microscopic structures. The macroscopic characteristics (e.g., dislocations) predominantly influence the mechanical properties, such as strength for metals and ceramics. The microscopic structure (interatomic or molecular-level structure) is controlled by the chemical bonding. Solids are classified according to their chemical bonding as metals, semiconductors, or insulators. Complete structural characterization of a material involves not only the elemental composition for major components and a study of the crystal structure, but also the impurity content (impurities in solid solution and/or additional phases) and stoichiometry. Noncrystalline materials can display unique behavior, and noncrystalline second phases can alter properties. Both the long-range order and crystal imperfection or defects must be defined. For example, the structural details which influence properties of oxides include the impurity and dopant content, nonstoichiometry, and the oxidation states of cations and anions. These variables also influence the point-defect structure, which in turn influences chemical reactivity, and electrical, magnetic, catalytic, and optical properties. Point defects are imperfections in the actual crystalline architecture as compared to the ideal lattice in which each atom site is filled with the appropriate element. They can influence properties at extremely low levels (ppb or ppm). Typical point defects include crystal sites with missing atoms (vacancies), atoms positioned in sites that are not filled by the crystal structure in question (interstitials), crystal sites containing impurity atoms (dopants), and cations with different oxidation states. Because mass,

Solid-State Chemistry

charge, and the number of lattice sites must be conserved, unusual oxidation states can be introduced (in dopants or predominant cations) and nonstoichiometric compositions stabilized. For example, Ni(III) can be formed in NiO, which nominally contains Ni(II) ions, and nickel vacancies (VNi ) are formed according to Eq. 1. 3+ Ni2+ Ni(1−3x) NiNi(2x) VNi(x)

(1)

Point-defect ordering (e.g., vacancy-dopant pairs) leads to interesting complications. Preparation conditions themselves (e.g., oxygen partial pressure and temperature in oxides) thermodynamically define and control this defect content and structure. It is important to realize that point defects are thermodynamically allowed and defined; they are not anomalous in the least. Therefore, undoped, highpurity compounds may exhibit sizable nonstoichiometry due to intrinsic point defects. Doping (intentional addition of an impurity) allows one to precisely control the point-defect content and nonstoichiometry and, thereby, the properties. Transport properties are influenced by the point defects. Electrical conduction (hole or electron transport) and solid state diffusion of atoms generally vary with the quantity and type of point defects. The determination of how nonstoichiometry is accommodated (i.e., by what type and amount of defect) is an active research area. Nonstoichiometry can also be accommodated by subtle changes in structure known as extended defects or crystallographic shear. Crystallinity, impurity levels, point-defect structure, and nonstoichiometry are each controlled by or influenced by the preparation method; therefore, it is discussed further in Section III. Surface properties can differ from the bulk structurally, both as clean surfaces or because of products formed on reactive surfaces (physisorbed or chemisorbed). The former can experience relaxation, that is, surface reconstruction due to the distortion in bonding for surface atoms which are lacking bonds. Impurity segregation at a surface can further alter properties, as can second phases formed on a surface. The activity of heterogeneous catalysts and corrosion is controlled by such surface properties and by the bulk and surface point-defect structures. Phases formed on semiconductor surfaces can change the electrical properties in an uncontrolled, deleterious fashion. Oxide passivation layers on compound semiconductors (e.g., mercury cadmium telluride IR detectors or gallium arsenide solar cells) can be grown to impart protection to the surfaces and to stabilize electrical properties by preventing uncontrolled reactions. Interfaces between two bulk phases, between the bulk and a surface, or at grain boundaries can further complicate the chemistry. Grain boundaries in polycrystalline materials can contain second phases (crystalline or noncrystalline) and have significant width. This is termed an


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interphase. On the other extreme, highly ordered (coherent) interfaces can occur between two microcrystallites (or grains). A second phase in a boundary can impart unique electrical properties, significantly influencing the characteristics of a capacitor, for example. Furthermore, dopants may segregate to a grain boundary phase, affecting boundary and bulk phase properties. Positive temperature coefficient (PTC) resistors and boundary layer capacitor operation are based on such effects. Characterization of the preceding structural variables is complex and challenges an extensive array of modern analytical instrumentation. Diffraction techniques (X-ray, neutron, and electron) are basic to the study of crystal structures. Improved data analysis techniques make these methods even more powerful for the study of powders. Nuclear magnetic resonance (NMR) spectroscopy has become a powerful tool for the study of solids with the advent of magic angle spinning techniques. Neutron inelastic scattering, Raman scattering, and IR vibrational spectroscopic analyses have been traditionally used to study lattice dynamics and solid-state phase transformations. They can also provide information about dopants and point-defect structures through studies of local modes as well as the extensive crystal structure information. High-resolution electron microscopy lattice imaging has proven to be a powerful tool for the study of crystal structures and extended defects. Electron spin resonance remains an effective tool for the study of paramagnetic solids, including impurities or low-level dopant structures. A variety of X-ray and electron spectroscopic techniques have been developed. These are particularly useful for providing information about elemental composition, surface structures, and cation oxidation states.

III. PREPARATION TECHNIQUES AND PROPERTY VARIATION The crystal structure of a solid can influence the properties of a material, for example, the structure must be noncentric for a material to demonstrate antiferromagnetic, ferromagnetic, ferroelectric, or piezoelectric behavior. Rapid cooling of a sample from high temperature and/or high pressure can quench in a structure that is not stable at room temperature or atmospheric pressure. High-pressure oxide polymorphs, which are more dense, have been studied to model the earth’s interior. Furthermore, unique crystal structure characteristics of a material can allow structure– property variation, for example, insertion compound formation in layered materials. Solids can be prepared as single crystals, glasses, thin films, powders, or sintered powder compacts (ceramics). Powders may be noncrystalline or polycrystalline, exhibit-

15:22

297 ing varying degrees of order. Powders may be used as catalysts or as precursors for crystal growing, glass formation, or ceramic formation. Numerous technologically useful components are produced by sintering powders, taking advantage of the unique electrical, magnetic, optical, or mechanical properties. Particle size is also an important powder variable. Production of powders with a controlled particle size distribution allows production of ceramics with uniform microstructures, improving mechanical properties. Strength of zirconia and zirconia–alumina refractories can be improved another way. Phase transitions which involve large atomic displacements lead to microcracking during heating and cooling, which implement fracture. Smallparticle-sized zirconia stabilizes a high-temperature phase at lower temperatures. This behavior, called “phase transformation toughening,” has been explained on the basis of higher surface free energies for these systems. Since a phase transition from this high-temperature phase is eliminated, the structural properties are improved. Research has shown that novel chemical preparation methods allow the production of unique materials, demonstrating properties unattainable through more traditional methods. Traditional solid-state synthesis techniques require high temperatures to increase the kinetics and allow reaction in reasonably short times. It is common to react solid powders after mixing by grinding or ball-milling. To ensure complete reaction of two powdered reactants it may be necessary to carry out repeated grinding and heating cycles. The grinding is necessary to reduce diffusion distances and increase product homogeneity. Repeated grinding and high-temperature treatments introduce undesirable impurities. Low-temperature solid preparation methods (meaning from ∼900◦ C to cryogenic temperatures) can produce powders having fewer impurities, high surface areas, and other unique characteristics which have useful applications, such as reactive surfaces for sintering or for catalysis. Higher treatment temperatures can actually reduce such activity by changing the bulk and/or surface structures. Low-temperature syntheses can sometimes allow unique surface phases to be stable. Tetragonal barium titanate, prepared at ∼700◦ C, has hexagonal barium titanate on the surface, which is stabilized by a higher surface free energy. Normally this hexagonal phase is not formed below 1460◦ C. This hexagonal surface was also found to reversibly adsorb CO2 as a surface carbonate. A variety of solution (water or organic solvent) techniques have been devised to control composition (e.g., the Ca:Mn ratio in an oxide such as CaMnO3 ). Control of this ratio is important in terms of the compositionsensitive properties. This is especially true for transition


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298 metal–containing systems because several oxidation states are stable and available for most of them. Unusual oxidation states introduced by variation in cation ratio and oxygen nonstoichiometry can influence catalytic and electronic properties. The “solid-state precursor” method involves precipitation of a compound or solid solution (e.g., oxalate or carbonate) with the desired stoichiometry which is then thermally decomposed to form the desired product oxide. Atomic scale mixing of the components precludes long calcining times at high temperatures. Powders formed are extremely homogeneous and have high surface areas. The composition of the products, however, does depend on the solution solubilities, and the precision of these precursor methods is not always high. The so-called “liquid mix technique” does allow better precision as well as homogeneity because no precipitation occurs; rather, the solution is heated to a viscous, glasslike form which is then decomposed to the product oxide. No solution– solid partitioning occurs, giving a composition equal to that of the starting mixture. The latter can be weighed to high accuracy (hundreds of ppm or better for major components), providing precise control of product cation ratios. The sol–gel technique is a low temperature, solutionpreparation method which has been applied extensively to produce glasses, fibers, coatings (protective or dielectric), abrasive particles, and controlled-pore-size catalyst– substrates. This method is used to prepare glasses, for example, that cannot be obtained upon quenching from the melt. Either colloidal or polymeric gels are formed by gelation of a precursor solution, involving hydrolysis and condensation of colloidal sols of metal salt or hydroxide solutions or of metal alkoxides. Drying, solvent removal, and firing conditions are then chosen to provide the desired microstructures and properties. Another important solution technique which should be mentioned is the “homogeneous precipitation technique.” It favors formation of a more ordered, crystalline product when two solutions are mixed to form an insoluble compound. The principle is slow precipitation, avoiding instantaneous formation of a disordered product. The benefits of all of the low-temperature solution techniques include homogeneity (atomic scale mixing) and minimal introduction of impurities. Other methods of preparation include chemical vapor deposition (CVD) and electrochemical methods. The latter are used to form thin films and protective coatings as well as battery electrodes. Since this is generally a lowtemperature method, the structure can differ from that of the same material prepared at a higher temperature. It can be disordered or amorphous. Chemical vapor deposition involves vapor phase transport of volatile organometallics


or other metal-containing species to the reaction site, and is used in the production and development of semiconductor devices. In some instances materials with potentially useful properties have not been exploited until prepared as pure crystals and films. Examples of this include doped polythiazyl, (SN)x , and polyacetylene, (CH)x , which have metallic properties, including electrical conductivity. The use of polymer precursors for ceramics (e.g., silicon carbide) is another interesting solid-state preparation technique. An exciting example, demonstrative of every aspect of solid-state chemistry, is the development of hightemperature superconducting oxides, which has followed the 1986 discovery of superconducting YBa2 Cu3 O7−x . This oxide will conduct at temperatures much higher than previous superconducting metal alloys, thereby reducing cooling expense. The synthesis, purification, characterization, extension to other metal-oxide systems, and the eventual application of these oxides in devices is certain to become a classic example of solid-state chemical science and technology. Control of the structure during preparation and processing allows one to control the properties and to optimize material performance for particular applications. Thorough structural characterization is a prerequisite. Based on the knowledge of how synthesis and processing influence structure and of how structure controls properties, the structure can be tailored and materials can be designed for optimum performance.

SEE ALSO THE FOLLOWING ARTICLES ANALYTICAL CHEMISTRY • BONDING AND STRUCTURE IN SOLIDS • CRYSTALLOGRAPHY • LASERS, SOLID-STATE • MICROSCOPY (CHEMISTRY) • PHASE TRANSFORMATIONS, CRYSTALLOGRAPHIC ASPECT • PRECIPITATION REACTIONS • SOLID-STATE IMAGING DEVICES • SUPERCONDUCTORS, HIGH TEMPERATURE • SURFACE CHEMISTRY

BIBLIOGRAPHY Brinker, C. J., and Scherer, G. W. (1990). “Sol–Gel Science,” Academic Press, Boston. Corbett, J. K., ed. (1985). Symposium on metal–metal bonding in solidstate clusters and extended arrays. J. Solid State Chem. 57(1), 1. Etourneau, J. (1999). Novel synthesis methods for new materials in solidstate chemistry. Bull. Mater. Sci. 22(3), 165–174. Fischer, J. E. (1997). Fulleride solid-state chemistry: Gospel, heresies and mysteries. J. Phys. Chem. Solids 58(11), 1939–1947, Grasselli, R. K., and Brazdil, J. F., eds. (1985). “Solid State Chemistry in Catalysis,” Series 279, American Chemical Society, Washington, D.C.


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Solid-State Chemistry Holt, S. L., Milstein, J. B., and Robbins, M., eds. (1980). “Solid State Chemistry: A Contemporary Overview,” Advances in Chemistry Series No. 186, American Chemical Society, Washington, D.C. Honig, J. M., and Rao, C. N. R., eds. (1982). “Preparation and Characterization of Materials,” Academic Press, New York. Nelson, D. L., and George, T. F. (1988). “Chemistry of High-Temperature Superconductors II,” American Chemical Society, Washington, D.C. Pimentel, G. C., and Coonrod, J. A. (1987). “Opportunities in Chemistry, Today and Tomorrow,” National Academy Press, Washington, D.C.

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299 Snyder, R. L., Condrate, R. A., and Johnson, P. F., eds. (1985). “Advances in Materials Characterization II,” Vol. 19, Materials Science Research, Plenum, New York. Sorensen, O. T., ed. (1981). “Nonstoichiometric Oxides,” Academic Press, New York. State of the art symposium: Solid state. (1980). J. Chem. Educ. 57, 531– 590. West, A. R. (1984). “Solid State Chemistry and its Applications,” Wiley, New York. Zelinski, B. J., and Uhlmann, D. R. (1984). Gel technology in ceramics. J. Phys. Chem. Solids 45, 1069.

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Solid-State Electrochemistry Gunter ¨ Holzapfel ¨ Dortmund University

I. Introduction II. Disorder in Solids III. Transport Processes—Diffusion, Mobility, and Partial Conductivity IV. Solid Electrolytes, Solid Ionic Conductors, and Solid–Solution Electrodes V. Galvanic Cells with Solid Electrolytes VI. Technical Applications of Solid Electrolytes

GLOSSARY Disorder Deviations from the regular crystal structure in a crystalline compound. Galvanic cell Arrangement for the conversion of chemical energy to electrical energy using electrodes in electrolytes. Interstitial ion Ion in excess compared to the ideal lattice of a solid. Ionic couductivity Electrical conductivity caused by ions. Solid electrolyte Solid compound in which the electrical current is carried out by ions, associated with mass transfer. Solid solution electrode Solid compounds exhibiting mixed (ionic and electronic) conductivity, where it is possible to dissolve or remove additional ions. Structural disorder Special kind of disorder in a crystal in which practically all ions of one kind are mobile and statistically distributed among their available lattice sites.

Superionic conductor Solid electrolyte with very high ionic conductivity, comparable to liquid electrolytes. Vacancy Missing ion in comparison to the ideal lattice of a solid.

THE FIELD of solid-state electrochemistry deals with research on physical, chemical or electrochemical problems using solid electrolytes. Solid electrolytes, also called solid ionic conductors, are solid, generally crystalline compounds in which the electrical current is carried by ions. Therefore, the passage of current is associated with mass transfer. Solid electrolytes enable us to build galvanic cells similar to those in the electrochemistry of liquids. Such galvanic cells with solid electrolytes play a most important role for scientific investigations in the field of thermodynamics and kinetics as well as for technical applications like batteries, sensors, fuel cells, and chemotronic components.

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Solid-State Electrochemistry

I. INTRODUCTION The field of solid-state electrochemistry—the electrochemistry of solids—has developed rapidly since about the middle of the twentieth century. This was caused by the discovery of new solid electrolytes with high ionic conductivity and some fundamental publications which pointed out the importance of solid ionic conductors for the thermodynamic investigations. Between the electrochemistry of solids and the electrochemistry in liquids there exist large analogies in theoretical point of view as well as in experimental methods. This is shown in Table I. The theoretical treatment of the disorder in solids shows extensive analogies to the theory of electrolytic dissociation. In the case of liquid electrolytes, the dissociated ions enable the conduction of the electric current. In solid electrolytes the conductivity is caused by the thermodynamic disorder of the crystals. This means that there are vacancies or interstitial ions in the crystal lattice which enable electrical conductivity. A further analogy exists regarding galvanic cells. From a thermodyamic point of view the treatment of galvanic cells with solid electrolytes corresponds to that in the liqud phase. Furthermore, the applied measuring methods largely correspond to those known in the electrochemistry of liquids. In spite of these analogies the electrochemistry of solids is more complex than the electrochemistry in aqueous solutions. So it must be noted that apart from ionic conduction, solids often show an electronic conductivity, caused by electrons or electron defects, which may be predominant in many cases over the ionic conduction. In good solid electrolytes the conduction of the electrical current is caused exclusively by the ions—in most cases practically by only one kind of ion present in a crystal. To explain the ionic conductivity of the solid the components that influence it must be examined. For the ionic conductivity σi of an ionic species i to be valid: σi = z i Fu i ci .

(1)

Here F is the Faraday constant, z i is the valence, ci is the concentration, and u i is the electrical mobility of the ions. From this equation it can be seen that two important quan-

tities influence the partial conductivity σi . These quantities are the concentration ci of the particle i and the electrical mobility u i . The concentration ci corresponds to the disorder of the solid and the mobility u i to transport processes.

II. DISORDER IN SOLIDS Under real conditions all ordered compounds exhibit deviations from a regular crystal structure. These are of great importance particularly for the understanding of the thermodynamic and kinetic properties of crystalline compounds. Such deviations may be: 1. interstitial ions, which are ions in excess as compared to the ideal lattice; 2. charged vacancies, which are missing ions compared to the ideal lattice; 3. foreign ions, which may be on either interstitial or regular lattice sites; and 4. electron disorder, which means the presence of quasi-free electrons and electron defects. These so-called point defects can be described by structure elements according to Kröger or by building units according to Schottky. The symbols used for these descriptions are summarized in Tables II and III. Structure elements are defined relative to the empty space, and building units are defined relative to the ideal lattice. One building unit according to Schottky corresponds to a combination of structure elements (generally two) according to Kröger. This is shown in the third column of Table III. Possible defects are illustrated schematically in Fig. 1 for an AB crystal using Kröger symbols. The two-dimensional section through the AB lattice shows A and B particles mainly in their normal positions. These particles have not been assigned any electrical charge because it is only meaningful to express charge relative to the unperturbed lattice, and A and B ions in their normal sites are electrically neutral relative to the unperturbed lattice. The figure also shows two A ions at interstitial sites; these ions bear an TABLE II The Kroger ¨ Symbols for Neutral Structure Elements in an AB Latticea

TABLE I Analogies between the Electrochemistry of Liquids and Solids Liquids

Solids

Electrolytic dissociation Galvanic cells with liquid electrolytes Electrochemical kinetics, particularly electrode kinetics

Disorder in solid compounds Galvanic cells with solid electrolytes Reactions in and on solids

Particle A

Particle B

Vacancy V

Foreign particle C

A site B site

AA AB

BA BB

VA VB

CA CB

Interstitial

Ai

Bi

Vi

Ci

Sites

a The examples chosen are A and B particles, foreign particles C, and vacancies at different sites.

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Solid-State Electrochemistry TABLE III Notation of the Lattice Molecule and Typical Defects in an AB Lattice in Terms of Building Units Using the Old and New Schottky Symbols Compared to the Corresponding Structure Elements According to Kroger ¨ and Vink Schottky old

Schottky new

Kröger/Vink

AB lattice molecule Neutral A particle on interstitial site Neutral A vacancy Neutral B particle on A-site

AB AO

AB A

Ai − Vi

A B•(A)

|A| B|A|

V A − AA BA − AA

Neutral C particle on A-site Quasi-free electron Electron hole

C•(A) ⊕

C|A| e |e· |

CA − AA e h

excess of positive charge denoted with. Two unoccupied sites can be seen; these vacancies are negatively charged relative to the unperturbed lattice and are denoted with . Structure elements with the corresponding symbols proposed by Kröger and Vink are easy to memorize; this notation is now also widely used in the literature. However, it is very important to know that the numbers of the various structure elements in a crystalline compound are not independent of one another. This is due to the fixed ratio of the number of A and B sites in an AB lattice. It is therefore generally impossible to change the concentration of only one type of structure element in a crystal. Thus, to create a vacancy on an A site we must either remove an A particle or simultaneously add a B particle to a new B site, so that the ratio of A to B sites remains unchanged. The generation of an interstitial particle leads to the destruction of a vacancy in the interstitial lattice. Thus, if we increase the size of a crystal or change the number of defects contained in it, one must either add or remove combinations of structure elements, this means that in general building units have to be used. Similarly, the use of structure elements to describe reactions of defects generally involves the use of building units.

FIGURE 1 Possible defects in an AB crystal using Kroger ¨ symbols.

The concentrations of charged atomic defects—point defects—follow the law of mass action. The considerations of thermodynamic equilibria can be applied to disorder equilibria in solid crystalline compounds, the socalled ordered mixtures. Point defects can be regarded as quasi-chemical species with which chemical reactions can be formulated. This has led to the so-called imperfection chemistry. As an example, the disorder equilibrium between vacancies and interstitial particles—the so-called Frenkel equilibrium—will be regarded. In this case a particle A moves from an A lattice site to an interstitial site whereby, for example, with respect to the unperturbed lattice a single positively charged interstitial . particle Ai is formed, a vacancy Vi in the interstitial lattice is destroyed, and a negatively charged vacancy VA on an A site is left. This exchange process can be written in the form of a chemical reaction, a so-called disorder reaction . A + |A| = 0 (2) using the new Schottky notation, we have . AA + Vi = Ai + VA .

(3)

Here the reaction is formulated in terms of structure elements, which for a thermodynamical treatment must be combined with building units as follows: . (Ai − Vi ) + (VA − AA ) = 0. (4) The concentration of the interstitial particles or vacancies as building units is identical with the concentration as structure elements. Symbolizing the concentrations by square brackets the law of mass action corresponding to Eq. (4) can thus be formulated as follows: . [Ai ][VA ] = const. (5) For all crystals there exists at equilibrium a constant product of the concentrations of interstitial particles and the corresponding vacancies, the so-called Frenkel equilibrium which depends on temperature and pressure. For higher concentrations of defects the equation must be written in terms of activities instead of concentrations. Frenkel disorder occurs in the silver halides, for example, in AgCl. Another kind of disorder equilibrium exists between A vacancies and B vacancies in a binary AB crystal—the so-called Schottky equilibrium. In this case the exchange of particles between A and B vacancies and the crystal is considered. This means that either A or B particles are transferred to the surface of the crystal, which is thus enlarged, while A or B vacancies are generated, or vice versa. For example, the A vacancies may be singly negative and the B vacancies singly positive. When the particles are brought to the surface, two of the particles there will be

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converted to volume particles, that is, the exchange effectively leads to the generation of a new lattice molecule and two vacancies. This can be written in terms of the new Schottky notation as . O = |A| + |B| + AB, (6) or using structure elements: . AA + BB = VA + VB + AB.

(7)

Here AB denotes the lattice molecule. This equation can be written in terms of building units: . O = (VA − AA ) + (VB − BB ) + AB. (8) Since the concentration of AB molecules can be regarded as constant at low defect concentration it can be included into the constant of the law of mass action, and we obtain: . [VA ][VB ] = const, (9) which means the product of A and B vacancies in an AB crystal is constant in this case. At higher concentrations of the defects activities must be used again. Schottky disorder is to be found in alkali metal halides such as NaCl and KCl. Aside from the described kinds of disorder there exist various other types. As an example in which electrons and the environment are playing a role in disorder, let us consider ZnO. At a temperature of 900◦ C the most important disorder centers in this compound are single positively charged zinc ions on interstitial sites and free electrons. The following equation representing the incorporation of oxygen, for example, from the air, into ZnO may therefore be formulated: . 1 O (g) + Zni + e = Vi + ZnO, (10) 2 2 or using building units: 1 O (g) 2 2

. + (Zni − Vi ) + e = ZnO.

(10a)

Because of electrical neutrality it follows that in ZnO . [Zni ] ∼ (11) = [e], that is, the concentration of zinc ions on interstitial sites is virtually equal to that of free electrons. The law of mass action corresponding to Eq. (10) is given by . 1/2 pO2 [e][Zni ] = const, (12) which may be simplified using Eq. (11) to yield 1/2

pO2 [e]2 = const

(13)

or −1/4

[e] ∼ pO2 .

(14)

According to Eq. (14) the concentration of free electrons is proportional to the inverse fourth root of the oxygen partial pressure; this is shown in Fig. 2. The conductivity

FIGURE 2 Dependence of the concentration of the electrons of ZnO at 900◦ C on the oxygen partial pressure.

of ZnO, which is mainly due to the partial conductivity of the free electrons, decreases with increasing oxygen partial pressures according to Eq. (14). The concentration of imperfection centers increases with rising temperature. The limiting case is crystals in which the concentrations of vacancies and interstitial particles become comparable; there then occurs a statistical distribution of particles in normal lattice positions and in interstitial positions. In this case it is no longer reasonable to distinguish between regular and interstitial lattice positions. The total number of positions a single type of particle can occupy may be several times higher than the number of such particles in the crystal. There exist crystals which have much more equivalent lattice sites available for one type of ion than ions are present in the lattice. At sufficiently high temperatures all ions are mobile and may be statistically distributed among these lattice sites. In this case we say a partial lattice of the crystal is in a quasi-molten state, the crystal has now a structural disorder. If such a disorder is present, the natural limit of the concentration of the mobile species is reached because all ions of one kind are now mobile. The best electrolytes known have such a disorder. Examples for such types are among others RbAg4 I5 at room temperature and AgI above 149◦ C. At this temperature AgI makes a transition from the β to the α phase; the partial lattice of Ag becomes quasi-molten, and there exist regions throughout which the silver ions virtually perform random motions as shown in Fig. 3. The idea that the silver ions in α-AgI are already in a quasi-molten state is supported by thermodynamic values and the diffusion coefficients, which are of the order of 10−5 cm2 /s similar to a liquid. The entropy for the transition from β-AgI to α-AgI amounts to 14.5 J/K mol, the entropy of fusion is 11.3 J/K mol. It is assumed that silver iodide melts in two stages. In the β–α transition the silver partial lattice passes into

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j = −D dc/d x,

(15)

where D is called Fick’s diffusion coefficient. If there is a concentration coefficient in any direction in space then we must use the general form of Fick’s first law: dc dc dc , , . (16) j = −D grad c = −D d x dy dz

FIGURE 3 Lattice of silver iodide with regions in which the silver ions can move.

the quasi-molten state (structural disorder), whereas the iodide partial lattice does not become liquid below the actual melting point. Comparable crystals that do not show the special behavior of structural disorder before they melt have an entropy of fusion about as large as the sum of the transition entropy and entropy of fusion in AgI, that is, about twice as large as the residual entropy of fusion in AgI. For completeness let us mention that besides the point defects discussed in this chapter there exist still other defects in crystals. These include 1. one-dimensional defects such as edge of screw dislocations, 2. two-dimensional defects such as surfaces or grain boundaries, and 3. three-dimensional defects such as cavities. These defects are not discussed here because they are not properties of the thermodynamic equilibrium.

III. TRANSPORT PROCESSES—DIFFUSION, MOBILITY, AND PARTIAL CONDUCTIVITY In this section the transport of ions in an electrical field and their diffusion in a concentration or activity gradient will be treated. The expressions derived are valid for the fluxes of each type of ion or electron separately. From these expressions equations for an interconnected transport of different types of particles can be derived. In the following a phenomenological treatment and an outlook on the statistical treatment will be given. A. Transport by Diffusion Fick stated an empirical relationship for the diffusion flux j in a concentration gradient dc/d x in the x direction. This is known as Fick’s first law:

In what follows only gradients in one direction (the x direction) will be regarded. The time dependence of the concentration is given by Fick’s second law: ∂c ∂ 2c (17) = D 2. ∂t ∂x These laws are valid in the case of ideal behavior that is, when the chemical potential µ of the particles holds: c µ = µ0 + RT ln 0 , (18) c where µ0 is the chemical potential in the standard state, c is the concentration of the particles, c0 is their concentration in the standard state, R is the general gas constant, and T is the absolute temperature. Using this equation Fick’s first law [Eqs. (15)] can be written as Dc dµ . (19) RT d x In the case of nonideal behavior, that is, for higher concentrations of the mobile particles, the more general expression jx = −

µ = µ0 + RT ln a

(20)

must be used for the chemical potential of the diffusing species, where a is the activity. A component diffusion coefficient D K is then defined in such a manner that an expression analogous to Eq. (19) is obtained: D K c dµ . (21) RT d x A relationship between D and D K can be reached in the following way. Since dc is equal to c d ln c, Eq. (15) can be expressed as j =−

d ln c . (22) dx From Eqs. (21) and (20) an expression for the particle flux which contains the component diffusion coefficient D K is obtained: d ln a j = −D K c . (23) dx A comparison of Eqs. (22) and (23) gives the relationship between D K and Fick’s diffusion coefficient D: d ln a D = DK . (24) d ln c j = −Dc

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As can be seen from Eq. (24), D = D K if the activity a is proportional to the concentration c. The factor d ln a/d ln c is called the thermodynamic factor. Another diffusion coefficient—the tracer—or self-diffusion coefficient DTr is defined, which can be measured at electrochemical equilibrium by using a radioactive isotope. Under certain conditions, for example, for vacancy diffusion, the tracer diffusion coefficient DTr is related to the component diffusion coefficient by: DTr = f D K ,

B. Transport in an Electrical Field If an electrical field only is present in the crystal and no gradient in the concentration or activity of the particles, the electrical current density i can be related to the electrical field strength E by Ohm’s law i = σE

(26)

i = −σ (dϕ/d x),

(27)

or where ϕ denotes the electrical potential and σ the electrical conductivity. The electrical current density i is connected with the current density j of the particles (28)

Here F denotes the Faraday constant, and z is the charge of the particles. From Eqs. (27) and (28) it follows that j =−

σ dϕ . zF dx

(29)

σi dϕ D K ,i ci dµi − . RT d x zi F d x

Eq. (30) may be reduced to give ci D K ,i dµi z i F dϕ ji = − + . RT dx dx With the definition of the electrochemical potential

(34)

or ji =

σi 2 2 zi F

dηi dx

(34a)

or using the electrical mobility according to Eq. (1): ji =

u i ci dηi zi F d x

(35)

From Eqs. (34) and (35) it can be seen that the discussion of mobility of the particles can be reduced to the discussion of the diffusion coefficient D K ,i . D. Chemical Diffusion Under certain conditions the fluxes of ions and electrons are related to each other by the conditions of electrical neutrality. This holds especially in the case when local differences in stoichiometry equilibrate. Here metal ions and electrons or nonmetal ions and electrons diffuse simultaneously. These transient phenomena are described ˜ In the by the so-called chemical diffusion coefficient D. ˜ following the result for the relationship between D and the component diffusion coefficient will be given for a compound whose partial conductivity σX− of the nonmetal ions is negligible in comparison with that of metal ions σMe+ , and at the same time their electron partial conductivity is much larger than that of metal ions: (36)

d ln aMe D˜ = D K ,Me , d ln cMe

(37)

where aMe and cMe are the activity of the metal and the concentration, respectively. Examples are the compounds FeO and Ag2 S.

(30) E. Atomistic Interpretation of the Transport of Ions in Solids

Using Eq. (1) and the Nernst–Einstein equation D K ,i = (u i /z i F)RT.

ci D K ,i dηi RT d x

In this case we obtain:

If besides a gradient of the electrical potential, ϕ, a gradient of the chemical potential µ or activity a of the particles is also present in the compound, we must write the more general equation ji = −

ji = −

σi σMe+ σX− .

C. Transport in an Electrical Field and in a Concentration- or Activity-Gradient

(33)

the expression in parentheses in Eq. (32) can be summarized to dηi /d x and we get:

(25)

where f is the so-called correlation factor. The correlation factor f is of the order of one for most cases.

j = i/z F.

ηi = µi + z i Fϕ

(31)

(32)

To understand the high ionic conductivity of some solid compounds an atomistic interpretation of the transport of ions in solids should be regarded. From the atomistic point of view the movement of ions in solids can be regarded as successive jumps between lattice sites or interstitial sites. For random motion of a

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particle in an isotropic crystal the diffusion coefficient can be expressed by D K ,i = 16 νa 2

(38)

if there is no correlation between the jumps; ν denotes the jump frequency and a denotes the distance for each jump. Using Eqs. (38), (31), and (1) we obtain for the partial conductivity σi = ci νa 2 z i2 F 2 6RT (39) and for the electrical mobility:

u i = z i F νa 2 6RT .

(40)

From Eqs. (39) and (40) we can see that the important quantity for the conductivity and the mobility is the product of the jump frequency of an ion and the square of the jump distance. The jump distance is of the order of the lattice parameter. It is possible to estimate an upper limit of the jump frequency. The maximum jump frequency νmax , that is, the highest frequency of change of a lattice site, results if the particles move with thermal speed v between the lattice sites without performing oscillations on such a site. Thus νmax = v/a .

(41)

For a given temperature this natural limit cannot be exceeded. The maximum conductivity denoted as σi (max), is then given according to Eqs. (40) and (41) as ci vaz i2 F 2 . (42) 6RT Herewith we can calculate a maximum possible partial conductivity for a substance, for example, silver iodide. Assuming that the silver ions migrate with thermal velocity v from one lattice site to another without oscillating at each lattice site, we get a jump frequency ν = v/a = 3.4 × 1012 s−1 at a temperature of 300◦ C, a diffusion coefficient of D K ,i = 5.6 × 10−5 cm2 /s, and the maximum conductivity is σ(max) = 2.8 −1 cm−1 for a jump distance ˚ The measured conductivity is σ = 1.97 −1 cm−1 , of 1 A. which is not much less than the calculated value. Many efforts have been made to improve this very simple but already good model. Besides jump and lattice gas models continuous models have been made. These models rely on the fact that the diffusion of an ion is not represented by instantaneous jumps from an equilibrium site to another one but by a continuous motion in between. From these considerations it can be seen that there is a natural upper limit for the value of the ionic conductivity of solid compounds. This upper limit is between 1 and 10 −1 cm −1 corresponding to a component diffusion coefficient of about 2 to 20 × 10−5 cm2 s. These values correspond to those in liquid electrolytes. Good σi (max) =

solid electrolytes reach these values or come near them. They are sometimes called super ionic conductors.

IV. SOLID ELECTROLYTES, SOLID IONIC CONDUCTORS, AND SOLID–SOLUTION ELECTRODES Having discussed in Sections II and III concentration and mobility, which influence the conductivity of solid electrolytes, a compilation of solid ionic conductors will be given in this section. This compilation does not presume to be complete because new solid electrolytes are discovered and developed continuously. In Fig. 4, the conductivities of some of the most important ones are shown as a function of temperaure and reciprocal temperature. The conductivity of liquid sulfuric acid is included for comparison. In the following, several important solid electrolytes will be treated according to the type of mobile ions that cause the ionic conductivity. A. Silver Ion Conductors One of the first solid electrolytes exhibiting a very high ionic conductivity, found in 1914, is α-Agl. This conducting α-phase is stable above 149◦ C and its high conductivity is caused by structural disorder. A similar disorder exists in RbAg4 I5 . This solid electrolyte exhibits the highest silver ion conductivity at room temperature at present. Therefore, it is of great technical interest. A

FIGURE 4 Conductivity of some very common solid electrolytes; H2 SO4 included for comparison.

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Solid-State Electrochemistry TABLE IV Conductivity of Several Silver Ion Conductors Compound

Temperature (◦ C)

σ ×102 /Ω cm) (×

150–500 −15–270

130–260 14–100

Agl RbAg4 I5 Ag3 SI Ag3 SBr Ag2 Hgl4 C5 H5 NHAg5 I6

TABLE VI Conductivity of Several Sodium Ion Conductors Compound

250–440

84–100

β-NaAl11 O17 Na5 GdSi4 O12 Na3 Zr2 Si2 PO12

20–300

0.1–8.2

NaSbO3 · 16 NaF

50–90 −20–480

Temperature (◦ C)

σ ×102 /Ω cm) (×

20–640 40–210

1.5–55 0.3–10

40–220

0.3–7.5

40–220

0.001–0.7

0.001–0.4 0.1–440

number of other silver ion conductors have been developed. Some of them and their conductivities for the given temperature ranges are listed in Table IV.

to obtain three-dimensional conductivity. This electrolyte has become of great interest for building sodium sulfur batteries, as will be discussed in Section VI.B. The conductivities of β-Al2 O3 and some other sodium ion conductors are shown in Table VI.

B. Oxygen Ion Conductors

D. Copper Ion Conductors

Doped zirconia dioxide and thorium dioxide are important solid electrolytes that owe their conductivity to transport of oxygen ions. They can be used between 600 and 1600◦ C. They are also an interesting example of how high ionic conductivity can arise by processes other than structural disorder. The disorder centers responsible for the ionic conductivity of zirconium dioxide are charged oxygen ion vacancies. These are produced by dissolution of CaO, MgO, or Y2 O3 in the zirconium dioxide. The calcium is incorporated at zirconium positions, since, however, only one oxygen ion is introduced with each calcium ion, one oxygen position remains unoccupied for each calcium atom introduced. The amount of doping is of the order of 10 mol%; in this way it can be seen why doping produces a very large number of vacancies in the zirconium dioxide. In Table V the conductivities of several oxygen conductors in given temperature ranges are listed.

The first solid electrolytes with high copper ion conductivity at room temperature were discovered in 1973. An example is 7CuBrC6 H12 N4 CH3 Br, whose conductivity at room temperature is 0.017 −1 cm−1 . Several other copper ion conductors have since been described. One of these conductors represented by the formula Rb4 Cu16 I7 Cl13 has a conductivity of 0.34 −1 cm−1 at 25◦ C. This is the solid electrolyte with the highest conductivity at room temperature known at present. In Table VII the conductivity of some copper ion conductors are listed.

C. Sodium Ion Conductors The most important Na+ ion conductor is Na2 O 11Al2 O3 , the so-called β-Al2 O3 . The mobile sodium ions are incorporated into planes of the lattice and can therefore move only in two dimensions; polycrystalline material is used TABLE V Conductivity of Several Oxygen Ion Conductors Compound

Temperature σ ×102 /Ω cm) (× (◦ C)

E. Proton Conductors Solid-state proton conductors with high ionic conductivity are eagerly sought because they could have important practical applications, for example, in fuel cells, water electrolyzers, and sensors. Substances with an appreciable proton conductivity known today are hydrogen uranyl phosphate tetrahydrate (HUP) HUO2 PO4 ·4H2 O— usable above 1◦ C—with an ionic conductivity of σ = 4 × 10−3 −1 cm−1 at 20◦ C and hydrogen uranyl arsenate tetrahydrate HUO2 AsO4 ·4H2 O—usable above 29◦ C— with an ionic conductivity of σ = 6 × 10−3 −1 40◦ C. Another kind of solid proton conductor is the protonic β-alumina. It can be produced by exchanging the TABLE VII Conductivity of Several Copper Ion Conductors Compound

ZrO2 (10 mol% Sc2 O3 ) ZrO2 (10 mol% Y2 O3 ) ZrO2 (13 mol% CaO) ThO2 (7.5 mol% Y2 O3 )

600–1400 600–1400

2–100 0.3–50

640–1400

0.2–30

1000–1500

Bi2 O3 (20 mol% Er2 O3 )

270–730

1.3–12 0.001–45

Temperature σ ×102 /Ω cm) (◦ C) (×

Rb4 Cu16 I7 Cl13 7CuBr·C6 H12 N4 CH3 Br

10–110

28–62

10–130

1.5–14

7CuCl·C6 H12 N4 HCl 17Cul·3C6 H12 N4 CH3 I

20–110

0.4–5

20–140

0.1–2.2

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whole sodium content of the solid sodium ion conductor Na–β-A12 O3 (see Section IV.C) for H+ , H+ (H2 O)n , or NH+ 4 ions. The ionic conductivity of these compounds is in the region of 10−4 –10−5 −1 cm−1 at room temperature. Various possible mechanisms of the conduction process in the different proton conductors are being discussed. The longtime stability of these compounds, which is especially important for possible technical applications, has not been clearly established. There exist still other solid proton conductors but as their conductivity in general is not as high as in the compounds mentioned earlier they will not be treated here. In addition to the solid electrolytes already described there exist solids in which other ions (e.g., lithium, fluoride or chloride ions) are mobile. These solid ionic conductors will not be treated here. F. Solid–Solution Electrodes Another class of important solids is mixed conducting solids exhibiting both fast ion transport and electronic conductivity. If, in addition to these properties a sufficiently large range of stoichiometry is present, these mixed conducting solids can be used as electrodes, for example, for batteries, the so-called solid–solution electrodes. One of the most promising solid–solution electrodes is based on titanium disulfide, in which it is possible to dissolve relatively large amounts of lithium metal in the TiS2 phase. There is a continuous range of nonstoichiometry from TiS2 to LiTiS2 . The structure of TiS2 and other similar chalcogenides of transition metals can be described as a sequence of layers held together by van der Waals forces only. The lithium is dissolved between the layers whereby the distance between the layers is slightly increased. Other changes in the crystal structure do not occur. Compounds of this kind are sometimes called insertion or intercalation compounds. Many other solid–solution electrodes have been investigated. Examples of other layer compounds besides TiS2 that are able to dissolve certain alkali metals and in some cases Cu+ or Ag+ ions include TiSe2 , MoS2 , WS2 , TaS2 , ZrS2 , NbS2 , VSe2 , MoSe2 , WSe2 , and CrS2 . In this regard we should also mention a related series of certain transition metal oxides that do not exhibit layered structures but are able to insert alkali metals or copper or silver, for example, TiO2 , MnO2 , MoO3 , WO3−y , V2 O5 , and Ta2 O5 .

V. GALVANIC CELLS WITH SOLID ELECTROLYTES The existence of solid ionic conductors has made possible the development of the electrochemistry of solids. In the

electrochemistry of solids galvanic cells with solid electrolytes play a very important role. In analogy to galvanic cells with liquid electrolytes, those with solid electrolytes consist of at least two electrodes separated by an electrolyte, which in this case is a solid ionic conductor. The important properties of such cells will be discussed. For this purpose the following galvanic cell will be considered as an example: pO , Pt / ZrO2 (+Y2 O3 ) / Pt, pO 2 2

←−−− ←−−−−−−− ←−−−. 4e−

2O2−

4e−

I It consists of doped zirconium dioxide as solid electrolyte with practically pure ionic conduction for oxygen ions. On the two sides there are porous, electronically conducting electrodes (e.g., consisting of porous platinum), surrounded by gaseous oxygen at different partial pressures. The zirconium dioxide must separate the electrode spaces from one another in gas-tight manner. For example, a tube of zirconium dioxide can be used that carries one electrode on the inner side and the second on the outer side, the outside and the inside being surrounded by gaseous oxygen at different partial pressures. If in this galvanic cell a positive electrical current flows from the left to the right electrode, 1 mol of O2 is transported from the right to the left electrode space by the passage of 4 faradays (4f). The following cell reaction occurs: O2 (right electrode) → O2 (left electrode).

(43)

Considering this cell reaction, we can obtain two properties of cell I; the first property is a thermodynamic one and the second a kinetic one. The thermodynamic property is the following. The electrical work that in the reversible case of the galvanic cell can be performed with the passage of 4 faradays amounts to 4EF, where E is the emf of the galvanic cell, defined as the electrical potential of the right electrode minus that of the left electrode. This work is equal to the negative Gibbs energy G of cell reaction (43): G(cel1 reaction) = −4EF.

(44)

Here G can be related to the chemical potentials µO2 of oxygen, and we then have, for Eq. (44), the following expression: µO2 (left electrode) − µO2 (right electrode) = −4EF. (45) The chemical potential µO2 is connected with the oxygen partial pressure pO2 by the following equation: µO2 = µ0O2 + RT ln pO2 pO0 2 . (46) Here µ0O2 denotes the standard chemical potential of oxygen, corresponding to a partial pressure pO0 2 of 1 atm; R is

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the gas constant and T is the absolute temperature. Substitution of Eq. (46) into Eq. (45) gives: E=

pO RT pO (right electrode) RT ln 2 = ln 2 . (47) 4F pO2 (left electrode) 4F p O2

This shows a relation existing between E, the emf of the galvanic cell, and the ratio of oxygen partial pressures at the two electrodes. If one oxygen partial pressure is known the other one can be determined. According to Eqs. (45)–(47) the emf of a galvanic cell contains different thermodynamic information. The emf makes it possible to determine the Gibbs energy of the cell reaction and the chemical potentials of the electrode components or the partial pressures of gases. It shall be mentioned here that reaction enthalpies and reaction entropies can also be deduced from the temperature dependence of the emf. More details on thermodynamic investigations will be given in Section V.A. The kinetic properties of galvanic cell I with doped zirconia as solid electrolyte arise from the fact that the flux of current through a cell such as cell I is a measure of the reaction rate by which oxygen is passed from one side of the cell to the other. Only oxygen ions can flow through the electrolyte when the electrical circuit is closed. For the rate of transport of the O2− particles in moles per unit time J through the electrolyte we can write: J = I /z F,

(48)

where I is the electrical current; in the case of oxygen the valence z is −2. The transport of mass in the form of ions through the electrolyte can often be attributed to a chemical reaction or a transport process at an electrode. In this way reaction rates can be measured electrically. It is often possible to analyze reaction mechanisms in detail by a combination of rate measurements by means of the electrical current with measurements of thermodynamic quantities—in particular, chemical potentials—by means of the emf of the galvanic cell. More details on kinetic investigations using galvanic cells will be given in Section V.B. A. Thermodynamic Investigations As a typical example for thermodynamic investigations using solid electrolytes the determination of the Gibbs energy G 0NiO at temperatures of 800–1000◦ C will be considered. The following cell with doped ZrO2 as solid electrolyte for oxygen ions can be used: Pt, Ni, NiO / ZrO2 (+Y2 O3 ) / Pt, pO2 = 1atm

2f: ←−−−−− ←−−−−−−− ←−−−−−−−. 2e−

O2−

II

2e−

One way to obtain a relation between the Gibbs energy and the emf of the cell is to regard the so-called virtual cell reaction. We assume that a certain amount of charge is passed through the cell as a current. In this example, the cell reaction is the formation of 1 mole NiO from solid nickel and oxygen by passing a flow of electricity of 2 f through the cell: Ni + 12 O2 = NiO · · · G NiO . This reaction does not take place under open-circuit conditions. It would take place under current flow, but then, in general, polarization effects will occur. The maximum possible electrical energy that we could obtain from the cell for the virtual reaction is the measured emf under open-circuit conditions multiplied by 2 f in this example. This electrical energy is related to the Gibbs energy G of the cell reaction; in this case to the Gibbs energy of formation of NiO from solid nickel and oxygen by: G 0NiO = −2EF.

(49)

Here G 0NiO is written because NiO as well as Ni and O2 are in their standard states. Similar investigations have been carried out on many other systems, for example, Cu2 O, FeO, PbO, In2 O3 , WO2 , ZnO, SiO2 , MoO2 , NiCl2 O4 , FeCr2 O4 , NiAl2 O4 , MgF2 , ThF4 , UF3 , and AlF3 . Furthermore, enthalpies and entropies of reaction as well as partial molar enthalpies and entropies and partial molar volumina can be measured using similar cells. B. Kinetic Investigations According to the electrochemistry of liquids kinetic investigations using solid electrolytes can be carried out in different ways. 1. Measurements at Zero Current In the case of zero-current measurements the electrical potential difference between the two end phases of the cell is measured under open circuit conditions. Information about thermodynamic quantities of reaction systems, for example, about chemical potentials, activities, or partial pressures, is obtained from such measurements. This was already described. 2. Measurements under Steady-State Conditions In this case there is no time dependence of currents and potentials. Therefore, it is experimentally unimportant whether the potentials or the currents are controlled. Here the steady-state current that represents a reaction rate may be measured as a function of the potential or vice versa.

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3. Measurements under Controlled Potential In this case a constant potential difference is applied to the cell and the resulting current is measured as a function of time. These investigations are called potentiostatic. 4. Measurements under Controlled Current In this case a constant current is passed through the cell and the resulting potential difference is measured as a function of time. These investigations are called galvanostatic. If galvanostatic or potentiostatic measurements or investigations under steady-state conditions are carried out it is often preferable to use separate cells of the same type under zero current to measure potential differences, so that the reference electrode is not polarized by a current. In the following, a few examples of kinetic investigations using solid electrolytes will be discussed. C. Electrochemical Measurements of Oxygen Diffusion in Metals The principle of the electrochemical measurement of oxygen diffusion in a metal consists in bringing the metal from a well-defined state into another well-defined state and following the diffusion-controlled relaxation process electrochemically. For example, the metal sample is placed on one side of the solid electrolyte ZrO2 and functions as one electrode of a galvanic cell. On the other side of the electrolyte there is a practically unpolarizable electrode such as porous platinum in contact with air, or an Fe/FeO electrode, which has a fixed oxygen partial pressure of about 10−19 atm at 800◦ C. The following cell may be used: Fe, FeO / ZrO2 (+Y2 O3 ) / Me + O(dissolved)

←−−− ←−−−−−−− ←−−−−−−−−− 2e−

O

O2−

←−−−−−−−−−. 2e−

III In cell III there is an Fe/FeO electrode on one side and a metal containing dissolved oxygen on the other side. The emf of the cell before beginning the experiment is a measure of the initial activity or concentration of the dissolved oxygen. At a certain time an emf is applied to the cell to make the oxygen activity at the metal/electrolyte interface very small. Then oxygen diffuses out of the metal and is carried as an electrical current through the electrolyte to the other side of the cell. In this way the diffusion current is transformed into an electrical current and can be measured. From the time dependence of the current the diffusion coefficient can be calculated using suitable diffusion equa-

TABLE VIII Determination of the Diffusion Coefficient of Oxygen in Various Solid and Liquid Metals

Metal

Solid or liquid

Temperature range (◦ C)

Diffusion coefficient (cm2 /s)

Ag Ag

s l

760–900 970–1200

1.5 × 10−5 –2.9 × 10−5 8.2 × 10−5 –1.7 × 10−4

Cu

l

990–1220

1.4 × 10−4 –2.2 × 10−4

Cu

s

800–1030

9.3 × 10−6 –3.5 × 10−5

Sn Ni

l s

730–930 1393

4.5 × 10−5 –7.4 × 10−5 1.3 × 10−6

Pb

l

800–1100

1.0 × 10−5 –1.7 × 10−5

Fe

l

1620

1.5 × 10−4

Sb Bi

l l

750–950 750–950

1.4 × 10−6 –2.9 × 10−6 8.6 × 10−6 –1.4 × 10−5

tions. Several systems have been investigated in this way. Some results are shown in Table VIII. D. Measurements of Chemical Diffusion Coefficients The process of attaining a uniform composition, which occurs in compounds where an existing gradient of stoichiometry is allowed to equalize, can be described by ˜ discussed in Secthe chemical diffusion coefficient D tion III.D. For such equilibration processes it is necessary, on the grounds of electrical neutrality, that both ions and electrons or electron defects must migrate simultaneously whereby the fluxes of ions and electrons are related to one another. The electrochemical method for the determination of chemical diffusion coefficients D˜ will be shown here as an example for the mixed conductor wüstite FeO. The basic element for the investigation of wüstite Fe1−δ O (δ = deviation from ideal stoichiometry) is the solid-state galvanic cell pO 2 , Pt/ZrO2 (+Y2 O3 )/Fe1−δ O)/Pt IV with doped ZrO2 as solid electrolyte, an electrode consisting of porous platinum in contact with air at one side and the wüstite being investigated as electrode at the other side. The experimental setup is shown in Fig. 5. The principle of the measurement is that, starting from a suitable initial state, the potential difference E of cell IV or the current I , respectively, are varied systematically. The other variable I or E is measured as a function of time. From the obtained results the chemical diffusion coefficient D˜ of wüstite can be calculated. In a potentiostatic experiment a definite value of the chemical potential of oxygen, corresponding to a certain deviation δ from the ideal stoichiometry,

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The chemical diffusion coefficients D˜ of Ag2+δ S measured at 200 and 300◦ C as a function of the deviation from the ideal stoichiometry are shown in Fig. 6. Another method for determining chemical diffusion coefficients is to use the frequency dependence of the cell impedance, which is obtained by ac measurements. This will not be treated here. E. Electrochemical Investigations of Formation of Surface Layers on Metals FIGURE 5 Experimental setup for the measurement of the chemical-diffusion coefficient of wustite: ¨ W, working electrode; R, reference electrode; C, counter electrode.

is set up in the wüstite before starting the measurement. Then a sudden change of the potential of the cell stipulated potentiostatically causes a current flow that is measured as a function of time. This current is primarily a measure of the addition or removal of oxygen at the phase boundary ZrO2 /wüstite. As a consequence of this, iron diffuses within the wüstite to or from the phase boundary. Thus, the current is equivalent to the diffusion current of iron, During this process the stoichiometry in the compound changes with time until a new δ value is attained. From the solution of the diffusion equations for this problem the chemical diffusion coefficient D˜ can be calculated. The result obtained for wüstite at 1000◦ C and a deviation δ = 0.106 from ideal stoichiometry is D˜ = 3.2 × 10−6 cm2 /s. Similar investigations with an improved experimental setup allowed the determination of the chemical diffusion coefficient D˜ of Ag2+δ S over the total range of stoichiometric composition of this compound.

The formation of nickel sulfide on nickel will be discussed as an example. The experimental arrangement is shown in Fig. 7. Silver iodide was used as the solid electrolyte, being a pure Ag+ ionic conductor under the experimental conditions. The negative pole of a power source was connected to the left-hand side of the arrangement and the positive pole to the right-hand side. The electrolytic cell itself consisted of tablets pressed together in a glass tube furnace flushed with nitrogen. An electrical current passing through the cell is a measure of the rate at which silver is removed from the silver sulfide, since silver ions migrate through the AgI and electrons through the external circuit. In this case, however, the rate of loss of the silver corresponds to the rate of the formation of nickel sulfide on nickel, since in this reaction nickel displaces the silver from the Ag2 S. The experiments were carrid out using the galvanostatic and potentiostatic methods. The important quantities are the current, which is a measure of the reaction rate (here of the formation of nickel sulfide), and the cell emf, which is not only a measure of the chemical potential of the silver in silver sulfide, but,

FIGURE 6 Chemical-diffusion coefficients of Ag2+δ S as a function of the deviation δ from stoichiometry at 200 and 300◦ C.

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FIGURE 7 Galvanic cell for the electrochemical investigation of the formation of NiS on nickel.

voltage U . The voltage is proportional to the ratio of mass to electrical charge of the silver ions and to the length of the rod. To verify Eq. (52) we should know the acceleration a at every time. In the experiments, however, the acceleration itself was not measured, rather it was the velocity before striking the ground that was measured. The accleration a and the velocity v are related by a = d v/dt ,

in view of the Gibbs–Duhem equation, also is a measure of the chemical potential of the sulfur in this compound. Thus, because of the high mobility of the silver in Ag2 S, it is also a measure of the chemical potential of the sulfur at the Ag2 S/NiS phase boundary. In this way the parabolic constant k for the rate of formation of NiS on nickel at 400◦ C could be measured as a function of the chemical potential of the sulfur on the outside of NiS. The results provided information about the disorder in NiS. F. Investigations on the Forces of Inertia of the Mobile Ions in Solid Ionic Conductors As an example, a rod of RbAg4 I5 with silver electrodes at both ends is exposed to an acceleration in the longitudinal direction. This acceleration may be produced, for example, by the braking of the rod dropped from different heights to a plastic material on the ground. Because of the high mobility of the silver ions it may be assumed that in the moment of striking the silver ions will be shifted only a little against the iodide lattice. Thus, an electrical field having the field strength E is built up with the result that the silver ions will be decelerated in the same way as the rigid iodide lattice. The sum of all forces acting on the silver ions must be zero. In this case the essential forces are the electrical force and the force of inertia. Quantitatively the electrical charge e multiplied by the electrical fields strength E is equal to the mass m of the silver ions multiplied by the acceleration a: eE = ma.

(53)

where t is the time. By integration of Eq. (53) over the whole time of striking we obtain the negative velocity −v: after strike a dt = −v (54) before strike

because the rod RbAg4 I5 has velocity v = 0 after the decleration. Using this we obtain by integration of Eq. (52) after strike ml v. (55) U dt = e before strike This equation was used for the interpretation of the experiments. Figure 8 shows a voltage–time curve measured at 25◦ C with a rod of RbAg4 I5 , which has a length of 7.4 cm. Thevelocity before striking was 1.44 m/s. The voltage pulse U dt is given by the area below the curve and has a value of 1.22 × 10−7 V s. According to Eq. (55) the area below the curve depends only on the velocity before striking for a given length of the rod. Figure 9 shows the measured voltage pulses from dropping experiments at 25◦ C using a RbAg4 I5 rod of 7.4-cm length as a function of the velocity before striking. The full line holds for the limiting case of inelastic striking, calculated with the help of Eq. (55) where m is the mass of

(50)

Since it can be assumed that the electrical field strength is constant along the rod of length l, an integration of E over l delivers the negative volgate −U , which can be measured between the ends of the rod: l E dl = El = −U . (51) 0

Inserting Eq. (51) in Eq. (50) we obtain: ml a. (52) e Equation (52) shows the relationship between the acceleration acting on the rod of RbAg4 I5 and the electrical −U =

FIGURE 8 Example of a measured voltage-time curve of a dropping experiment (T = 25◦ C; length of the rod = 7.4 cm; velocity before striking = 1.44 m/s).

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FIGURE 10 Zirconium dioxide sensor for the measurement of partial pressures of oxygen. The housing and packing are electrically conducting.

FIGURE 9 Voltage pulse as a function of the v before striking at 25◦ C. (Length of the rod = 7.4 cm; ——, calculated; ×, measured).

a silver atom. It is seen that the results of the measurements and the calculated voltage pulses are in agreement. Furthermore, it should be noticed that neither in Eq. (55) nor in Eq. (52) is the termperature included as a variable. That means the values of the voltage pulses should be independent of the temperature. This could be confirmed by measurements at 25 and 120◦ C and may be of interest for practical applications, as will be seen. In addition to the investigations described, other kinetic experiments have been carried out with the help of solid-electrolyte galvanic cells. The investigations include phase-boundary reactions at the solid–gas phase boundary (including measurements of evaportion and condensation rates) and phase-boundary reactions at the solid–solid phase boundary. These investigations will not be discussed here.

VI. TECHNICAL APPLICATIONS OF SOLID ELECTROLYTES Solid electrolytes are also of great technological importance. Some examples of applications will be described in the following sections. A. Sensors Galvanic cells with solid electrolytes can be used for direct measurement of partial pressures in gases and concentrations in liquids and melts. An important example is cell I, which contains doped zirconium dioxide as solid electrolyte. By using cells of this type a wide range of oxygen partial pressures in gases (down to 10−16 atm) can be determined. The zirconium dioxide probe for such work is used at temperatures between about 500 and 1000◦ C.

The wide range of oxygen partial pressures makes such cells an excellent analytical instrument for measurements of gases. It should be mentioned that the measured value is obtained almost instantaneously. In such measurements a reference atmosphere, for example, air or a metal–metal oxide mixture, is located on one side of the zirconium dioxide, so that the oxygen partial pressure is already known at one electrode; the oxygen partial pressure at the other electrode is then measured by the emf in accordance with Eq. (47). In the measurement the gas is passed to the sensor. Interesting industrial applications lie in the analysis of exhaust gases from furnaces or combustion engines. It is advisable to arrange the zirconium dioxide probe in the exhaust gas stream near the reaction space. Figure 10 shows such a sensor that can be used for the control and regulation of the combustion process in automobile engines. The possibility of very exact control of gasoline– air mixtures is of special interest in connection with the control of air pollution. A similar probe can be used to measure the concentration or thermodynamic activity of oxygen in liquid metals. For example, if the probe is dipped in the melt during steel production, the oxygen activity can be measured directly, which is of appreciable advantage. A further possibility to measure partial pressures of gases is given by the use of galvanic cells, in which another kind of ions than the corresponding species in the gas is mobile in the solid electrolyte. As an example, the following galvanic cell for measuring chlorine partial pressures will be regarded Ag/RbAg4 I5 /AgCl, Cl2 pCl2 , V where RbAg4 I5 is used as a silver ion conducting solid electrolyte. Cell V is in principle a concentration (activity) cell for silver ions analogous to cell I delivering the emf

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E=

RT aAg (AgCl) ln , F aAg (Ag)

(56)

where aAg (AgCl) denotes the activity of silver in the righthand AgCl-electrode, and aAg (Ag) denotes the activity of silver in the left-hand silver electrode, respectively, which has a value of one in this case because of the pure Ag metal. The activity of silver in AgCl, which is in equilibrium with the gas-atmosphere, having a certain chlorine partial pressure pCl2 , is given by −1/2 aAg (AgCl) = pCl2 /po exp G oAgCl RT , (57) where p o = 1 atm and G oAgCl is the Standard-Gibbs energy of formation of AgCl. Inserting Eq. (57) into Eq. (56), we obtain a relation between the chlorine partial pressure to be determined and the measured emf E RT E= (58) ln pCl2 /po − G oAgCl F . 2F Also, the partial pressures of other gases, for example, NO2 , O2 , or sulfur, can be measured by such sensors. Another method of determining the partial pressure of a gas is given by measuring the current flowing through a suitable galvanic cell with a solid electrolyte. This principle will be discussed in the following exemplified at an oxygen sensor. In principle, cell I discussed in Section V can be used, where pO 2 is the oxygen partial pressure to be measured. In this case, a current is passed through the cell so that oxygen is transported from the right-hand side of the cell to the left-hand side. For this an outer electrical voltage has to be applied to the cell. To get a defined correlation between pO 2 and the electrical current flowing through the cell, a so-called “diffusion-barrier” has to be arranged in front of the electrode at the right-hand side. This may, for example, consist of a porous ceramic material. The external electrical voltage is chosen in such a way that each oxygen molecule reaching the surface of the right-hand electrode immediately reacts electrochemically to an oxygen ion, which is then transported through the solid electrolyte. Under these conditions, the flowing current is proportional to the partial pressure pO 2 and so can be used to measure pO 2 . Another type of sensor—an accelerometer—can be constructed by applying the principles used in the case of the investigations on the forces of inertia of the mobile ions in solid ion conductors, described in Section V.F. As shown in Eq. (52) the voltage U measured at both ends of a rod of a solid electrolyte is a direct measure of the acceleration acting on this rod at every time; this means that such an arrangement, consisting of a rod of RbAg4 I5 as solid electrolyte with silver wires at both ends as electrodes for measuring the voltage U , can be used as an accelerometer. This is of interest for practical applications. The principle of such an accelerometer is shown in Fig. 11.

FIGURE 11 Schematic diagram of a RbAg4 I5 accelerometer.

B. Solid Electrolyte Batteries: The Sodium–Sulfur Cell The principle of a sodium–sulfur cell is shown in Fig. 12. The solid electrolyte is a Na+ ion conductor, consisting of β-Al2 O3 . It is generally used as a tube closed at one end and filled with liquid sodium as the anode. An iron sponge, which absorbs the liquid sodium, serves to improve the wetting of the electrolyte and to improve safety. A metal wire leads out of the anode to carry the current. The cathode consists of liquid sodium polysulfide and sulfur inserted in porous graphite. The working temperature of the sodium–sulfur cell is around 300◦ C. In the cell reaction sodium ions pass through the electrolyte and electrons through the external circuit, so that sodium is dissolved in sodium polysulfide. In this way electrical energy can be liberated. The energy density of the sodium–sulfur cell is many times greater than that

FIGURE 12 Sodium–sulfur cell.

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of the customary lead batteries, and the materials needed for the electrolytes and electrodes are available in large quantities. The cell can be recharged by changing the direction of the current. The sodium–sulfur cell is of great interest for large-scale energy storage and for electrotraction for electric vehicles. Prototypes have already been built. In addition to the sodium–sulfur cell other cell systems have been developed using other solid electrolytes. C. Fuel Cells and Electrolyzers Figure 13 shows schematically a high-temperature fuel cell incorporating zirconium dioxide as the solid electrolyte. The zirconium dioxide in a tube or disk form separates two electrode compartments, one containing air or pure oxygen and the other a fuel gas (e.g., hydrogen). The zirconium oxide carries two porous electrodes: nickel can be used on the fuel side, and on the oxygen side lanthanum–nickel oxide or some other electronconducting oxide. In a high-temperature fuel cell the oxygen takes up electrons at one electrode, then passes through the electrolyte as ions and combines at the other electrode with H2 to give H2 O, whereby the electrons are given up and flow again to the other electrode in the external current circuit. In this way energy is made available to the user. The advantages of the high-temperature fuel cell are that little or no polarization occurs at the electrodes and high current densities can be achieved. By reversing the direction of the current flow in a hightemperature fuel cell, that is, by supplying the cell with electrical energy, steam can be decomposed and the cell can thus be used as an electrolyzer. The hydrogen produced can be stored or conducted by pipelines to remote sites where it can serve for the production of energy in a

high-temperature fuel cell if required. This principle is being discussed in connection with large-scale energy storage and transport of energy. D. Chemotronic Components Galvanic cells containing solid electrolytes, which find use in electrical circuits, are often called chemotronic components. Next we describe as an example coulometers and time switches. However, there exist more chemotronic building units containing solid electrolytes such as analog memories and capacitors. They will not be described here. The galvanic cell Ag/RbAg4 I5 /Au VI can be used as a coulometer or time switch. Here the electrolyte RbAg4 I5 is a good Ag+ ion conductor even at room temperature. By passing a current through this cell with the negative pole at the silver side, silver is deposited on the gold electrode. The time switch is then in the loaded state. The silver can be transported back to the original silver electrode by a current flowing in the reverse direction. In this stripping process the cell potential is determined mainly by ohmic losses in the solid electrolyte that lie in the millivolt range. When all the silver has been stripped from the gold electrode the cell shows a sudden rise in potential, which can be used as a signal. Such electrochemical switches are suitable for times in the region of seconds to months. Cell VI can also be used as a coulometer; the amount of a current flowing through the cell during charging is then determined by the discharge process.

SEE ALSO THE FOLLOWING ARTICLES CRYSTALLOGRAPHY • ELECTROCHEMISTRY • ELECTROLYTE SOLUTIONS, TRANSPORT PROPERTIES • LASERS, SOLID-STATE • SOLID-STATE CHEMISTRY • SOLID-STATE IMAGING DEVICES

BIBLIOGRAPHY

FIGURE 13 High-temperature fuel cell.

Bard, A. J., and Faulkner, L. R. (2001). “Electrochemical Methods: Fundamentals and Applications,” Wiley, New York. Bruce, P. G. (1995). “Solid-State Electrochemistry,” Cambridge University Press, Cambridge. Gellings, P. J., and Boumeester, H. J. M. (eds.) (1997). “The CRC Handbook of Solid-State Electrochemistry,” CRC Press, Boca Raton, Florida. Stimming, U., Singhal, S. C., Tagawa, H., and Lehnert, W. (eds.) (1997). “Proceedings of the Fifth International symposium on Solid Oxide Fuel Cells (SOFC-V),” Electrochemical Society, Pennington, New Jersey.

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Superacids George A. Olah G. K. Surya Prakash University of Southern California

I. Acid Strength and Acidity Scale II. Superacid Systems III. Application of Superacids

GLOSSARY Brönsted superacids Proton donor acids stronger than 100% sulfuric acid. Carbenium ions Compounds containing a trivalent, tricoordinate carbon bearing a positive charge. Also called “classical cations.” Carbocations Compounds containing carbon bearing a positive charge which encompass both carbenium and carbonium ions. Carbonium ions Compounds containing high coordinate carbon bearing a positive charge with multicenter bonding. Also called “nonclassical” cations. Conjugate Brönsted–Lewis superacids Superacidic proton donor acids comprised of a combination of Brönsted and Lewis acids. Hammet’s acidity constant, H 0 A logarithmic thermodynamic scale used to relate acidity of proton donor acids. Immobilized superacids Superacids (both Brönsted and Lewis types) bound to inert supports such as graphite, fluorinated graphite, etc. Lewis superacids Electron acceptor acids stronger than aluminum trichloride. Solid superacids Solid materials possessing superacid

sites. May be of the Brönsted or the Lewis superacid type. Superacids Acid systems that encompass both Brönsted and Lewis superacids as well as their conjugate combinations. Superelectrophiles Electrophiles that are further activated by Brönsted or Lewis superacid complexation.

CHEMISTS long considered mineral acids such as sulfuric and nitric acids to be the strongest protic acids to exist. More recently this view has changed considerably with the discovery of extremely strong acid systems that are hundreds of millions, even billions, of times stronger than 100% sulfuric acid. Such acid systems are termed “superacids.” The term “superacids” was first suggested by Conant and Hull in 1927 to describe acids such as perchloric acid in glacial acetic acid that were capable of protonating certain weak bases such as aldehydes and ketones. Superacids encompass both Brönsted (proton donor) and Lewis (electron acceptor) acids as well as their conjugate pairs. The concept of acidity and acid strength can be defined only in relation to a reference base. According to an arbitrary but widely accepted suggestion

175

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176 by Gillespie, all Brönsted (protic) acids stronger than 100% sulfuric acid are classified as superacids. Various methods are available to measure protic superacid strengths (vide infra). Lewis acids also cover a wide range of acidities extending beyond the strength of the most frequently used systems such as AlCl3 and BF3 . Olah et al. (1985) suggested the use of anhydrous aluminum trichloride, the most widely used Friedel–Crafts catalyst, as the arbitrary unit to define Lewis superacids. Lewis acids stronger than anhydrous aluminum trichloride are considered Lewis superacids. There remain, however, many difficulties in measuring the strength of Lewis acid (vide infra). The high acidity and the extremely low nucleophilicity of the counterions of superacidic systems are especially useful for the preparation of stable, electron-deficient cations, including carbocations. Many of these cations, which were formerly suggested only as fleeting metastable intermediates and were detectable only in the gas phase in mass spectrometric studies, can be conveniently studied in superacid solutions. New chemical transformations and syntheses that are not possible using conventional acids can also be achieved with superacids. These include transformations and syntheses of many industrially important hydrocarbons. The unique ability of superacids to bring about hydrocarbon transformations, even to activate methane (the principal component of natural gas) for electrophilic reactions, has opened up a fascinating new field in chemistry.

I. ACID STRENGTH AND ACIDITY SCALE The chemical species that plays the key role in Brönsted acids is the hydrogen ion, that is, the proton: H+ . Since the proton is the hydrogen nucleus with no electron in its 1s orbital, it is not prone to electronic repulsion. The proton consequently exercises a powerful polarizing effect. Due to its extreme electron affinity, proton cannot be found as a free “naked” species in the condensed state. It is always associated with one or more molecules of acid or the solvent (or any other nucleophile present). The strength of protic acid thus depends on the degree of association of the proton in the condensed state. Free protons can exist only in the gas phase and represent the ultimate acidity. Due to the very small size of a proton (105 times smaller then any other cation) and the fact that only 1s orbital is used in bonding by hydrogen, proton transfer is a very facile reaction, reaching diffusion-controlled rates, and does not necessitate important reorganization of the electronic valence shells. Understanding the nature of the proton is important when generalizing quantitative relationships in acidity measurements.

Superacids

A number of methods are available for estimating acidity of protic acids in solution. The best known is the direct measurement of the hydrogen ion activity used in defining pH [Eq. (1)]. pH = log aH+ .

(1)

This can be achieved by measuring the potential of a hydrogen electrode in equilibrium with a dilute acid solution. In highly concentrated acid solutions, however, the pH concept is no longer applicable, and the acidity must be related very closely to the degree of transformation of a base with its conjugate acid, keeping in mind that this will depend on the base itself and on medium effects. The advantage of this method was shown in the 1930s by Hammett and Deyrup, who investigated the proton donor ability of the H2 O–H2 SO4 system over the whole concentration range by measuring the extent to which a series of nitroanilines were protonated. This was the first application of the very useful Hammett acidity function [Eq. (2)]. BH+ . (2) B The pKBH+ is the dissociation constant of the conjugate acid (BH+ ) and BH+ /B is the ionization ratio, which is generally measured by spectroscopic means [ultraviolet, nuclear magnetic resonance (NMR), and dynamic NMR]. Hammett’s “H0 ” scale is a logarithmic scale on which 100% sulfuric acid has an H0 value of −12.0. Various other techniques are also available for acidity measurements of protic acids. These include electrochemical methods, kinetic rate measurements, and heats of protonation of weak bases. Even with all these techniques it is still difficult to measure the acidity of extremely acidic superacids, because of the unavailability of suitable weak reference bases. In contrast to protic (Brönsted) acids, a common quantitative method to determine the strength of Lewis acids does not exist. Whereas the Brönsted acid–base interaction always involves a common denominator—the proton (H+ ) transfer, which allows direct comparison—no such common relationship exists in the Lewis acid–base interaction. The result is that the definition of “strength” has no real meaning with Lewis acids. The “strength” or “coordinating power” of different Lewis acids can vary widely against different Lewis bases. Despite the apparent difficulties, a number of qualitative relationships have been developed to characterize Lewis acids. Schwarzenbach and Chatt classified Lewis acids into two types: class a and class b. Class a Lewis acids form their most stable complexes with the donors in the first row of the periodic table—N, O, and F. Class b acids, on the other hand, complex best with donors in the second or subsequent row—Cl, Br, I, P, S, etc. Guttmann has H0 = pKBH+ − log

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introduced a series of donor numbers (DN) and acceptor numbers (AN) for various solvents in an attempt to quantify complexing tendencies of Lewis acids. Based on a similar premise, Drago came up with parameter E, which measures the covalent bonding potential of each series of Lewis acids as well as bases. Pearson has proposed a qualitative scheme in which a Lewis acid and base are characterized by two parameters, one of which is referred to as strength and the other as softness. Thus, the equilibrium constant for a simple Lewis acid–base reaction would be a function of four parameters, two for each partner. Subsequently, Pearson introduced the hard and soft acids and bases (HSAB) principle to rationalize behavior and reactivity in a qualitative way. Hard acids correspond roughly in their behavior to Schwarzenbach and Chatt’s class a acids. They are characterized by small acceptor atoms that have outer electrons that are not easily excited and that bear a considerable positive charge. Soft acids, which correspond to class b acids, have acceptor atoms of a lower positive charge and a large size, with easily excited outer electrons. Hard and soft bases are defined accordingly. Pearson’s HSAB principle states that hard acids prefer to bind to hard bases and soft acids prefer to bind to soft bases. The principle has proved useful in rationalizing and classifying a large number of chemical reactions involving acid–base interactions in a qualitative manner, but it gives no basis for quantitative treatment. Many attempts have been made in the literature to rate qualitatively the activity of Lewis acid catalysts in Friedel–Crafts-type reactions. However, such ratings depend largely on the nature of the reaction for which the Lewis acid catalyst is employed. Thus, the classification of Lewis superacids as those stronger than anhydrous aluminum trichloride is only arbitrary. Just as in the case of Gillespie’s classification of Brönsted superacids, it is important to recognize that acids stronger than conventional Lewis acid halides exit, with increasingly unique properties. Another area of difficulty is measuring the acid strength of solid superacids. Since solid superacid catalysts are used extensively in the chemical industry, particularly in the petroleum field, a reliable method for measuring the acidity of solids would be extremely useful. The main difficulty to start with is that the activity coefficients for solid species are unknown and thus no thermodynamic acidity function can be properly defined. On the other hand, because the solid by definition is heterogeneous, acidic and basic sites can coexist with variable strength. The surface area available for colorimetric determinations may have acidic properties widely different from those of the bulk material; this is especially true for well-structured solids such as zeolites.

The complete description of the acidic properties of a solid requires the determination of the acid strengths as well as the number of acid sites. The methods that have been used to answer these questions are basically the same as those used for the liquid acids. Three methods are generally quoted: (1) rate measurement to relate the catalytic activity to the acidity, (2) the spectrophotometric method to estimate the acidity from the color change of adequate indicators, and (3) titration by a strong enough base for the measurement of the amount of acid. The above experimental techniques vary somewhat, but all the results obtained should be interpreted with caution because of the complexity of the solid acid catalysts. The presence of various sites of different activity on the same solid acid, the change in activity with temperature, and the difficulty of knowing the precise structure of the catalyst are some of the major handicaps in the determination of the strength of solid superacids.

II. SUPERACID SYSTEMS Following Conant’s early work, the field of superacids, which had been dormant till the late 1950s, started to undergo rapid development in the early 1960s, involving the discovery of new systems and an understanding of their nature as well as their chemistry. As mentioned, superacids encompass both Brönsted and Lewis types and their conjugate combinations. A. Bronsted ¨ Superacids Using Gillespie’s arbitrary definition, Brönsted superacids are those with an acidity exceeding that of 100% sulfuric acid (H0 , −12). These include perchloric acid (HClO4 ), fluorosulfuric acid (FSO3 H), trifluoromethanesulfonic acid (CF3 SO3 H), and higher perfluoroalkanesulfonic acid (Cn Fn +2 SO3 H). Physical properties of some of the most commonly used superacids are listed in Table I. Studies by Gillespie have shown that truly anhydrous hydrogen fluoride (HF), which is extremely difficult to obtain in the pure form, has a Hammett acidity constant (H0 ) of −15.1 rather than the −11.0 found for the usual anhydrous acid. However, traces of water impurity drop the acidity to the generally observed value. Thus for practical purposes, hydrogen fluoride, which always contains some water impurity, is not discussed here, as its acidity of H0 = −11.0 is lower than that of H2 SO4 . Teflic acid (TeF5 OH) has been suggested to have an acidity comparable to that of fluorosulfuric acid. However, no concrete acidity measurements are available to support such a claim. A number of carbocationic salts bearing carborane anions [CB11 H6 Cl− 6 , etc.] have been studied. However, their parent Bronsted acids,

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Superacids TABLE I Physical Properties of Bronsted ¨ Superacids Property Melting point (◦ C) Boiling point (◦ C) Density (25◦ C), g/cm3 Viscosity (25◦ C), cP Dielectric constant Specific conductance (20◦ C), −1 · cm−1 −H0 (neat) a b

HClO4

ClSO3 H

HSO3 F

CF3 SO3 H

−112

−81

−89

−34

110 (Explosive)

151–152 (Decomposing)

162.7

162

1.767a — — —

1.753 3.0b 60 ± 10 0.2–0.3 × 10−3

1.726 1.56 120 1.1 × 10−4

1.698 2.87 2.0 × 10−4

≈13.0

13.8

15.1

14.1

At 20◦ C. At 15◦ C.

which can be considered as potential superacids, are still unknown. 1. Perchloric Acid (HClO4 ) Commercially, perchloric acid is manufactured by either reaction of alkali perchlorates with hydrochloric acid or direct electrolytic oxidation of 0.5 N hydrochloric acid. Another commercially attractive method is the direct electrolysis of chlorine gas (Cl2 ) dissolved in cold, dilute perchloric acid. Perchloric acid is commercially available in a concentration of 70% (by weight) in water, although 90% perchloric acid also had limited availability (due to its explosive hazard, it is no longer provided at this strength); for 70–72% HClO4 , an azeotrope of 28.4% H2 O, 71.6% HClO4 , boiling at 203◦ C is safe for usual applications. It is a strong oxidizing agent, however, and must be handled with care. Anhydrous acid (100% HClO4 ) is prepared by vacuum distillation of the concentrated acid solution with a dehydrating agent such as Mg(ClO4 )2 . It is stable only at low temperatures for a few days, decomposing to give HClO4 · H2 O (84.6% acid) and ClO2 . Perchloric acid is extremely hygroscopic and a very powerful oxidizer. Contact of organic materials with anhydrous or concentrated perchloric acid can lead to violent explosions. For this reason, the application of perchloric acid has serious limitations. The acid strength, although not reported, can be estimated to be around H0 = −13 for the anhydrous acid. Although various cation salts can be prepared with perchlorate gegen ions, the ionic salts tend to be unstable (explosive) due to their equilibria with covalent perchlorates. The main use of perchloric acid is in the preparation of − its salts, such as NH+ 4 ClO4 , a powerful oxidant in rocket fuels and pyrotechniques. 2. Chlorosulfuric Acid (ClSO3 H) Chlorosulfuric acid, the monochloride of sulfuric acid, is a strong acid containing a relatively weak sulfur–chlorine

bond. It is prepared by the direct combination of sulfur trioxide and dry hydrogen chloride gas. The reaction is very exothermic and reversible, making it difficult to obtain chlorosulfuric acid free of SO3 and HCl. On distillation, even in a good vacuum, some dissociation is inevitable. The acid is a powerful sulfating and sulfonating agent as well as a strong dehydrating agent and a specialized chlorinating agent. Because of these properties, chlorosulfuric acid is rarely used for its protonating superacid properties. Gillespie and co-workers have measured systematically the acid strength of the H2 SO4 –ClSO3 H system using aromatic nitro compounds as indicators. They found an H0 value of −13.8 for 100% ClSO3 H. 3. Fluorosulfuric Acid (HSO3 F) Fluorosulfuric acid, HSO3 F, is a mobile colorless liquid that fumes in moist air and has a sharp odor. It may be regarded as a mixed anhydride of sulfuric and hydrofluoric acid. It has been known since 1892 and is prepared commercially from SO3 and HF in a stream of HSO3 F. It is readily purified by distillation, although the last traces of SO3 are difficult to remove. When water is excluded, it may be handled and stored in glass containers, but for safety reasons the container should always be cooled before opening because gas pressure may have developed from hydrolysis. HSO3 F + H2 O H2 SO4 + HF Fluorosulfuric acid generally also contains hydrogen fluoride as an impurity, but according to Gillespie the hydrogen fluoride can be removed by repeated distillation under anhydrous conditions. The equilibrium HSO3 F SO3 + HF always produces traces of SO3 and HF in stored HSO3 F samples. When kept in glass for a long time, SiF4 and H2 SiF6 are also formed (secondary reactions due to HF). Fluorosulfuric acid is employed as a catalyst and chemical reagent in various chemical processes including

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alkylation, acylation, polymerization, sulfonation, isomerization, and production of organic fluorosulfates. It is insoluble in carbon disulfide, carbon tetrachloride, chloroform, and tetrachloroethane, but it dissolves most organic compounds that are potential proton acceptors. The acid can be dehydrated to give S2 O5 F2 . Electrolysis of fluorosulfuric acid gives S2 O6 F2 or SO2 F2 + F2 O, depending on the conditions employed. HSO3 F has a wide liquid range (mp = −89◦ C, bp = +162.7◦ C), making it advantageous as a superacid solvent for the protonation of a large variety of weak bases.

TABLE II Characteristics of Perfluoroalkanesulfonic Acids bp (◦ C) (760 mm Hg)

Compound CF3 SO3 H C2 F5 SO3 H C4 F9 SO3 H C5 F11 SO3 H C6 F13 SO3 H C8 F17 SO3 H

Cl2

Cl2

CF3 SSCF3 −→ CF3 SCl −→ CF3 SO2 Cl H2 O

aq. KOH

−−−→ CF3 SO3 H CF3 SO3 H is a stable, hygroscopic liquid that fumes in moist air and readily forms the stable monohydrate (hydronium triflate), which is a solid at room temperature (mp, 34◦ C; bp, 96◦ C/1 mm Hg). The acidity of the neat acid as measured by UV spectroscopy with a Hammett indicator indeed shows an H0 value of −14.1. It is miscible with water in all proportions and soluble in many polar organic compounds, such as dimethylformamide, dimethylsulfoxide, and acetonitrile. It is generally a very good solvent for organic compounds that are capable of acting as proton acceptors in the medium. The exceptional leaving-group properties of the triflate anion, CF3 SO− 3, make triflate esters excellent alkylating agents. The acid and its conjugate base do not provide a source of fluoride ion even in the presence of strong nucleophiles. Furthermore, as it lacks the sulfonating properties of oleums an HSO3 F, it has gained a wide range of application as a catalyst in Friedel–Crafts alkylation, polymerization, and organometallic chemistry. 5. Higher Homologous Perfluoroalkanesulfonic Acids Higher homologous perfluoroalkanesulfonic acids (see Table II) are hygroscopic oily liquids or waxy solids. They are prepared by the distillation of their salts from H2 SO4 , giving stable hydrates that are difficult to dehydrate. The acids show the same polar solvent solubilities as trifluoromethanesulfonic acid but are quite insoluble in benzene,

H0 (22◦ C)

161

1.70

−14.1

170 198

1.75 1.82

−14.0 −13.2

212 222

−12.3

249 241

4. Trifluoromethanesulfonic Acid (CF3 SO3 H) Trifluoromethanesulfonic acid (CF3 SO3 H, triflic acid), the first member in the perfluoroalkanesulfonic acid series, has been studied extensively. Besides its preparation by electrochemical fluorination of methanesulfonyl halides, triflic acid may also be prepared from trifluoromethanesulfenyl chloride.

Density (25◦ C)

257

heptane, carbon tetrachloride, and perfluorinated liquids. Many of the perfluoroalkanesulfonic acids have been prepared by the electrochemical fluorination reaction of the corresponding alkanesulfonic acids (or conversion of the corresponding perfluoroalkane iodides to their sulfonyl halides). α,ω-Perfluoroalkanedisulfonic acids have been prepared by aqueous alkaline permanganate oxidation of the compounds, Rf SO2 (CF2 CF2 )n –SO2 F. C8 F17 SO3 H and higher perfluoroalkanesulfonic acids are surface-active agents and form the basis for a number of commercial fluorochemical surfactants. B. Lewis Superacids Lewis superacids are arbitrarily defined as those stronger than anhydrous aluminum trichloride, the most commonly used Friedel–Crafts catalyst. Some of the physical properties of the commonly used Lewis superacids are given in Table III. 1. Antimony Pentafluoride (SbF5 ) Antimony pentafluoride is a colorless, highly viscous liquid at room temperature. Its viscosity is 460 cP at 20◦ C, which is close to that of glycerol. The pure liquid can be handled and distilled in glass if moisture is excluded. TABLE III Physical Properties of Some Lewis Superacids Property mp (◦ C) bp (◦ C) Specific gravity at 15◦ C (g/cc) a

At the bp.

SbF5

AsF5

TaF5

NbF5

B(OSO2 CF3 )3

7.0

−79.8

97

72–73

43–45

142.7 3.145

−52.8 2.33a

229 3.9

236 2.7

68–83 (0.5 Torr) —

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180 The polymeric structure of the liquid SbF5 has been established by 19 F NMR spectroscopy and is shown to have the following frameworks: a cis-fluorine bridged structure is found in which each antimony atom is surrounded by six fluorine atoms in an octahedral arrangement.

Superacids

ciation, but it is a monomeric covalent compound with a high degree of coordinating ability. It is prepared by reacting fluorine with arsenic metal or arsenic trifluoride. As a strong Lewis acid fluoride, it is used in the preparation of ionic complexes and, in conjunction with Brönsted acids, forms conjugate superacids. It also forms, with graphite, stable intercalation compounds that show an electrical conductivity comparable to that of silver. Great care should be exercised in handling any arsenic compound because of its potential high toxicity. 3. Tantalum and Niobium Pentafluoride

Antimony pentafluoride is a powerful oxidizing and a moderate fluorinating agent. It readily forms stable intercalation compounds with graphite, and it spontaneously inflames phosphorus and sodium. It reacts with water to form SbF5 · 2H2 O, an unusually stable solid hydrate (probably a hydronium salt, H3 O+ SbF5 OH) that reacts violently with excess water to form a clear solution. Slow hydrolysis can be achieved in the presence of dilute NaOH and forms Sb(OH)− 6 . Sulfur dioxide and nitrogen dioxide form 1:1 adducts, SbF5 :SO2 and SbF5 :NO2 , as do practically all nonbonded electron-pair donor compounds. The exceptional ability of SbF5 to complex and subsequently ionize nonbonded electron-pair donors (such as halides, alcohols, ethers, sulfides, and amines) to carbocations, first recognized by Olah in the early 1960s, has made in one of the most widely used Lewis halides in the study of cationic intermediates and catalytic reactions. Vapor density measurements suggest a molecular association corresponding to (SbF5 )3 at 150◦ C and (SbF5 )2 at 250◦ C. On cooling, SbF5 gives a nonionic solid composed of trigonal bipyramidal molecules. Antimony pentafluoride is prepared by the direct fluorination of antimony metal or antimony trifluoride (SbF3 ). It can also be prepared by the reaction of SbCl5 with anhydrous HF, but the exchange of the fifth chloride is difficult, and the product is generally SbF4 Cl. As shown by conductometric, cryoscopic, and related acidity measurements, it appears that antimony pentafluoride is by far one of the strongest Lewis acids known. Antimony pentafluoride is also a strong oxidizing agent, allowing, for example, preparation of arene dications. At the same time, its easy reducibility to antimony trifluoride represents a limitation in many applications, although it can be easily refluorinated. 2. Arsenic Pentafluoride (AsF5 ) Arsenic pentafluoride (AsF5 ) is a colorless gas at room temperature, condensing to a yellow liquid at −53◦ C. Vapor density measurements indicate some degree of asso-

The close similarity of the atomic and ionic radii of niobium and tantalum are reflected by the similar properties of tantalum and niobium pentafluorides. They are thermally stable white solids that may be prepared either by the direct fluorination of the corresponding metals or by reacting the metal pentachlorides with HF. Surprisingly, even reacting metals with HF gives the corresponding pentafluorides.They both are strong Lewis acids, complexing a wide variety of donors such ethers, sulfides, amines, and halides. They both coordinate with fluoride ions to form anions of the type (MF6 )− . TaF5 is a somewhat stronger acid than NbF5 , as shown by acidity measurements in HF. The solubility of TaF5 and NbF5 in HF and HSO3 F is much more limited than that of SbF5 or other Lewis acid fluorides, restricting their use to some extent. At the same time, their high redox potentials and more limited volatility make them catalysts of choice in certain hydrocarbon conversions, particularly in combination with solid catalysts. 4. Boron tris(Trifluoromethanesulfonate) [B(OSO2 CF3 )3 ] Boron tris (trifluoromethanesulfonate) was first prepared by Engelbrecht and Tschager in trifluoromethanesulfonic acid solution (vide infra) as a conjugate acid system. Olah and co-workers have isolated B(OSO2 CF3 )3 in pure form by treating boron trihalides (chlorides, bromides) with 3 equiv of triflic acid in Freon 113 or SO2 ClF solution. BX3 + 3CF3 SO3 H → B(OSO2 CF3 )3 + 3HX Boron tris(trifluoromethanesulfonate) is a colorless low-melting compound [mp, 43–45◦ C; bp, 68–73◦ C (0.5 Torr)] which decomposes on heating above 100◦ C at atmospheric pressure. It is extremely hygroscopic and is readily soluble in methylene chloride, 1,1,2trifluorotrichloroethane (Freon 113), SO2 , and SO2 ClF. Boron tris(trifluoromethanesulfonate) is a strong nonoxidizing Lewis acid and an efficient Friedel–Crafts catalyst. Apart from the discussed Lewis acids, other highly acidic systems such as Au(OSO2 F)3 , Ta(OSO2 F)5 , Pt(OSO2 F)4 , and Nb(OSO2 F)5 have been reported as

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their conjugate acids in HSO3 F solution. All the above conjugate superacids were found to be highly conducting and strongly ionizing over the entire conecentration range. C. Conjugate Bronsted–Lewis ¨ Superacids 1. Oleums: Polysulfuric Acids SO3 -containing sulfuric acid (oleum) has long been considered the strongest mineral acid and one of the earliest superacid systems to be recognized. The concentration of SO3 in sulfuric acid can be determined by weight or by electrical conductivity measurement. The most accurate H0 values for oleums so far have been published by Gillespie and co-workers (Table IV). The increase in acidity on the addition of SO3 to sulfuric acid is substantial, and an H0 value of −14.5 is reached with 50 mol% SO3 . The main component up to this SO3 concentration is pyrosulfuric (or disulfuric) acid H2 S2 O7 . On heating or in the presence of water, it decomposes and behaves like a mixture of sulfuric acid and sulfur trioxide. In sulfuric acid, it ionizes as a stronger acid: + − H2 S2 O7 + H2 SO4 H3 SO4 + HS2 O7

(K = 1.4 × 10−2 ) At higher SO3 concentrations, a series of higher polysulfuric acids such as H2 S3 O10 and H2 S4 O13 is formed and a corresponding increase in acidity occurs. However, as can be seen from Table IV, the acidity increase is very small after reaching 50 mol% of SO3 , and no data are available beyond 75%. Despite its high acidity, oleum has found little application as a superacid catalyst, mainly because of its strong oxidizing power. Also, its high melting point and viscosity have considerably hampered its use for spectroscopic study of ionic intermediates and in synthesis, except as an oxidizing or sulfonating agent. 2. Tetra(hydrogensulfato)Boric Acid–Sulfuric Acid HB(HSO4 )4 prepared by treating boric acid [B(OH)3 ] with sulfuric acid ionizes in sulfuric acid as shown by acidity measurements.

+ − HB(HSO4 )4 + H2 SO4 H3 SO4 + B(HSO4 )4

The increase in acidity is, however, limited to H0 = −13.6 as a result of insoluble complexes that precipitate when the concentration of the boric acid approaches 30 mol%. 3. Fluorosulfuric Acid–Antimony Pentafluoride (Magic Acid) Of all superacids, “Magic Acid,” a mixture of fluorosulfuric acid and antimony pentafluoride, is probably the most widely used medium for the spectroscopic observation of stable carbocations. The fluorosulfuric acid–antimony pentafluoride system was developed in the early 1960s by Olah for the study of stable carbocations and was studied by Gillespie for electron-deficient inorganic cations. The name Magic Acid originated in Olah’s laboratory at Case Western Reserve University in the winter of 1966. The HSO3 F:SbF5 mixture was used extensively by his group to generate stable carbocations. J. Lukas, a German postdoctoral fellow, put a small piece of Christmas candle left over from a lab party into the acid system and found that it dissolved readily. He then ran a 1 H NMR spectrum of the solution. To everybody’s amazement, he obtained a sharp spectrum of the t-butyl cation. The long-chain paraffin, of which the candle was made, had obviously undergone extensive cleavage and isomerization to the more stable tertiary ion. It impressed Lukas and others in the laboratory so much that they started to nickname the acid system Magic Acid. The name stuck, and soon others started to use it too. It is now a registered trade name and has found its way into the chemical literature. The acidity of the Magic Acid system as a function of the SbF5 content has been measured successively by Gillespie, Sommer, Gold, and their co-workers. The increase in acidity is very sharp at a low SbF5 concentration (≈10%) and continues up to the estimated value of H0 = −26.5 for a 90% SbF5 content. The initial ionization of HSO3 F:SbF5 is as follows. 2HSO3 F + SbF5 H2 SO3 F+ + SbF5 (SO3 F)− At higher concentrations of SbF5 , complex polyantimony fluorosulfate ions are formed. SbF5 + SbF5 (SO3 F)− Sb2 F10 (SO3 F)−

TABLE IV H0 Values for the H2 SO4 –SO3 System Mol% SO3

H0

Mol% SO3

H0

Mol% SO3

H0

1.00 2.00 5.00 10.00 15.00 20.00

−12.24 −12.42 −12.73 −13.03 −13.23 −13.41

25.00 30.00 35.00 40.00 45.00 50.00

−13.58 −13.76 −13.94 −14.11 −14.28 −14.44

55.00 60.00 65.00 70.00 75.00

−14.50 −14.74 −14.84 −14.92 −14.90

Due to these equilibria, the composition of the HSO3 F: SbF5 system is very complex and depends on the SbF5 content. Aubke and co-workers have investigated the structures of complex anions in the Magic Acid system by modern 19 F NMR studies. The major reason for the wide application of the Magic Acid system compared with others (besides its very high acidity) is probably the large temperature range in which

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182 it can be used. In the liquid state, NMR spectra have been recorded from temperatures as low as −160◦ C (acid diluted with SO2 F2 and SO2 ClF) and up to +80◦ C (neat acid in a sealed NMR glass tube). Glass is attacked by the acid very slowly when moisture is excluded. The Magic Acid system can also be an oxidizing agent that results in reduction to antimony trifluoride and sulfur dioxide. On occasion this represents a limitation. 4. Fluorosulfuric Acid–Sulfur Trioxide Freezing-point and conductivity measurements have shown that SO3 behaves as a nonelectrolyte in HSO3 F. Acidity measurements show a small increase in acidity that is attributed to the formation of polysulfuric acids HS2 O6 F and HS3 O9 F up to HS7 O21 F. The acidity of these solutions reaches a maximum of −15.5 on the H0 scale for 4 mol% SO3 and does not increase any further. 5. Fluorosulfuric Acid–Arsenic Pentafluoride AsF5 ionizes in FSO3 H, and the AsF5 FSO− 3 anion has an octahedral structure. The H0 acidity function increases up to 5 mol% AsF5 , with a value of −16.6. 6. Hydrogen Fluoride–Antimony Pentafluoride (Fluoroantimonic Acid) The HF:SbF5 (fluoroantimonic acid) system is considered the strongest liquid superacid and also the one that has the widest acidity range. Due to the excellent solvent properties of hydrogen fluoride, HF:SbF5 is used advantageously for a variety of catalytic and synthetic applications. Anhydrous hydrogen fluoride is an excellent solvent for organic compounds with a wide liquid range. The acidity of HF, initially estimated as H0 ≈ −11, has now been revised to an H0 of −15.1 for highly purified anhydrous HF. A dramatic increase in acidity (H0 ≈ −20.5) is observed when 1 mol% SbF5 is added to anhydrous HF. The initial sharp increase in acidity is apparently due to the removal of residual moisture impurity. For more concentrated solutions, only kinetic data are available, mainly from the work of Brouwer and co-workers, who estimated the relative acidity ratio of 1:1 HF:SbF5 and 5:1 HSO3 F:SbF5 to be 5 × 108 :1. This means an H0 value in excess of −30 on the Hammett scale for the 1:1 composition. The acidity may increase still further for higher SbF5 concentrations. It has been shown by infrared measurements that an 80% SbF5 solution has the maximum concentration of H2 F+ . In any case, even for the composition range of 1–50% SbF5 , this is the largest range of acidity known. The same infrared study has also shown that the predominant cationic species (i.e., solvated proton) in 0–40 mol% SbF5

Superacids + is the H3 F+ 2 ions. The H2 F ion is observed only in highly concentrated solutions (40–100 mol% SbF5 ), contrary to the widespread belief that it is the only proton-solvated species in HF:SbF5 solutions. Ionization in dilute HF solutions of SbF5 (1–20% SbF5 ) is thus − + SbF5 + 3HF SbF6 + H3 F2

The structure of the hexafluoroantimonate and of its − higher homologous anions Sb2 F− 11 and Sb3 F16 , which are formed when the SbF5 content is increased, have been determined by 19 F NMR studies.

7. HSO3 F:HF:SbF5 When Magic Acid is prepared from fluorosulfuric acid not carefully purified (which always contains HF), on addition of SbF5 the ternary superacid system HSO3 F:HF:SbF5 is formed. Because HF is a weaker Brönsted acid, it ionizes fluorosulfuric acid, which, on addition of SbF5 , results in a high-acidity superacid system at low SbF5 concentrations. 19 F NMR studies on the system have indicated the − presence of SbF− 6 and Sb2 F11 anions, although these can result from the disproportionality of SbF5 (FSO3 )− and Sb2 F10 (FSO3 )− anions. 8. HSO3 F:SbF5 :SO3 When sulfur trioxide is added to a solution of SbF5 in HSO3 F, there is a marked increase in conductivity that continues until approximately 3 mol of SO3 has been added per mol of SbF5 . This increase in conductivity has been attributed to an increase in H2 SO3 F+ concentration arising from the formation of a much stronger acid than Magic Acid. Acidity measurements have confirmed the increase in acidity with SO3 :SbF5 in the HSO3 F system. This has been attributed to the presence of a series of acids of the type H[SbF4 (SO3 F)2 ], H[SbF3 (SO3 F)3 ], H[SbF2 (SO3 F)4 ] of increasing acidity. Of all the fluorosulfuric acid-based superacid systems, sulfur trioxide-containing acid mixtures are, however, difficult to handle and cause extensive oxidative side reactions on contact with organic compounds.

9. HSO3 F–Nb(SO3 F)5 and HSO3 H–Ta(SO3 F)5 The in situ oxidation of niobium and tantalum metals in HSO3 F by bis(fluorosulfuryl)peroxide, S2 O6 F2 , gives the solvated Lewis acids M(SO3 F)5 , M = Nb or Ta. These acid systems have been shown to be highly acidic by conductivity studies.

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10. HSO3 F–Au(SO3 F)3 and HSO3 F–Pt(SO3 F)4 These superacids based on gold and platinum have been developed. They show a high acidity and good thermal stability. However, the high cost of the metals involved precludes their widespread use.

tion studies indicate that the HF:TaF5 system is a weaker superacid than HF:SbF5 . 13. Hydrogen Fluoride–Boron Trifluoride (Tetrafluoroboric Acid) Boron trifluoride ionizes anhydrous HF as follows:

11. Perfluoroalkanesulfonic Acid-Based Systems a. CF3 SO3 H:SbF5 . CF3 SO3 H:SbF5 (n = 1) was introduced by Olah as an effective superacid catalyst for isomerizations and alkylations. The composition and acidity of systems where n = 1, 2, 4 have been studied by Commeyras and co-workers. The change in composition of the triflic acid–antimony pentafluoride system depending on the SbF5 content has been studied. For the 1:1 composition, the main counteranion is [CF3 SO3 SbF5 ]− , and for the 1:2 composition [CF3 SO3 (Sb2 F11 )]− is predominant. With increasing SbF5 concentration, the anionic species grow larger and anions containing up to 5 SbF5 units have been found. In no circumstances could free SbF5 be detected. b. CF3 SO3 H:B(SO3 CF3 )3 . The acidity of triflic acid can also be substantially increased by the addition of boron triflate B(OSO2 CF3 )3 as indicated by Engelbrecht and Tschager. The increase in acidity is explained by the ionization equilibrium: B(OSO2 CF3 )3 + 2HSO3 CF3 + 2HSO3 CF3 + − B(SO3 CF3 )4

The measurements were limited due to the lack of a suitable indicator base, and even 1,3,5-trinitrobenzene the weakest base used, was fully protonated (H0 ≈ −18.5) in a 22 mol% solution of boron triflate. The acid system has found many synthetic applications, due mainly to the efforts of Olah and co-workers. 12. Hydrogen Fluoride–Tantalum Pentafluoride HF:TaF5 is a catalyst for various hydrocarbon conversions of practical importance. In contrast to antimony pentafluoride, tantalum pentafluoride is stable in a reducing environment. The HF:TaF5 superacid system has attracted attention mainly through the studies concerning alkane alkylation and aromatic protonation. Generally, heterogeneous mixtures such as 10:1 and 30:1 HF:TaF5 have been used because of the low solubility of TaF5 in HF (0.9% at 19◦ C and 0.6% at 0◦ C). For this reason, acidity measurements have been limited to very dilute solutions, and an H0 value of −18.85 has been found for the 0.6% solution. Both electrochemical studies and aromatic protona-

− BF3 + 2HF BF4 + H2 F+

The stoichiometric compound exists only in an excess of HF or in the presence of suitable proton acceptors. The HF:BF3 (fluoroboric acid)-catalyzed reactions cover many of the Friedel–Crafts type reactions. One of the main advantages of this system is the high stability of HF and BF3 and their nonoxidizing nature. Both are gases at room temperature and are easily recovered from the reaction mixtures. Acidity measurements of the HF:BF3 system have been limited to electrochemical determinations, and a 7 mol% BF3 solution was found to have an acidity of H0 = −16.6. This indicates that BF3 is a much weaker Lewis acid compared with either SbF5 or TaF5 . Nevertheless, the HF:BF3 system is strong enough to protonate many weak bases and is an efficient and widely used catalyst. 14. Conjugate Friedel–Crafts Acids (HBr:AIBr3 , HCl:AICl3 , Etc.) The most widely used Friedel–Crafts catalyst systems are HCl:AlCl3 and HBr:AlBr3 . These systems are indeed superacids by Gillespie’s definition. However, experiments directed toward preparation from aluminium halides and hydrogen halides of the composition HAlX4 were unsuccessful in providing evidence that such conjugate acids are formed in the absence of proton acceptor bases. D. Solid Superacids The acidic sites of solid acids may be of either the Brönsted (proton donor, often OH group) or the Lewis type (electron acceptor). Both types have been identified by IR studies of solid surfaces absorbed with pyridine. Various solids displaying acidic properties, whose acidities can be enhanced to the superacidity range, are listed in Table V. 1. Immobilized Superacids (Bound to Inert Supports) Ways have been found to immobilize and/or to bind superacidic catalysts to an otherwise inert solid support. These include graphite intercalated superacids. Graphite possessing a layered structure can form intercalation compounds with Lewis acids such as AsF5 and SbF5 . These

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TABLE V Solid Acids

III. APPLICATION OF SUPERACIDS

1. Natural clay minerals: kaolinite, bentonite, attapulgite, montmorillonite, clarit, Fuller’s earth, zeolites, synthetic clays or zeolites 2. Metal oxides and sulfides: ZnO, CdO, Al2 O3 , CeO2 , ThO2 , TiO2 , ZrO2 , SnO2 , PbO, As2 O3 , Bi2 O3 , Sb2 O5 , V2 O5 , Cr2 O3 , MoO3 , WO3 , CdS, ZnS 3. Metal salts: MgSO4 , CaSO4 , SrSO4 , BaSO4 , CuSO4 , ZnSO4 , CdSO4 , Al2 (SO4 )3 , FeSO4 , Fe2 (SO4 )3 , CoSO4 , NiSO4 , Cr2 (SO4 )3 , KHSO4 , (NH4 )2 SO4 , Zn(NO3 )2 , Ca(NO3 )2 , K2 SO4 , Bi(NO3 )3 , Fe(NO3 )3 , CaCO3 , BPO4 , AlPO4 , CrPO4 , FePO4 , Cu3 (PO4 )2 , Zn3 (PO4 )2 , Mg3 (PO4 )2 , Ti3 (PO4 )4 , Zr3 (PO4 )4 , Ni3 (PO4 )2 , AgCl, CuCl, CaCl2 , AlCl3 ,TiCl3 , SnCl2 , CaF2 , BaF2 , AgClO4 , Mg(ClO4 )2 4. Mixed oxides: SiO2 :Al2 O3 , SiO2 :TiO2 , SiO2 :SnO2 , SiO2 :ZrO2 , SiO2 :BeO, SiO2 :MgO, SiO2 :CaO, SiO2 :SrO, SiO2 :ZnO, SiO2 :Ga2 O3 , SiO2 :Y2 O3 , SiO2 :La2 O3 , SiO2 :MoO3 , SiO2 :WO3 , SiO2 :V2 O5 , SiO2 :ThO2 , Al2 O3 :MgO, Al2 O3 :ZnO, Al2 O3 :CdO, Al2 O3 :B2 O3 , Al2 O3 :ThO2 , Al2 O3 :TiO2 , Al2 O3 :ZrO2 , Al2 O3 :V2 O5 , Al2 O3 :MoO3 , Al2 O3 :WO3 , Al2 O3 :Cr2 O3 , Al2 O3 :Mn2 O3 , Al2 O3 :Fe2 O3 , Al2 O3 :Co3 O4 , Al2 O3 :NiO, TiO2 :CuO, TiO2 :MgO, TiO2 :ZnO, TiO2 :CdO, TiO2 :ZrO2 , TiO2 :SnO2 , TiO2 :Bi2 O3 , TiO2 :Sb2 O5 , TiO2 :V2 O5 , TiO2 :Cr2 O3 , TiO2 :MoO3 , TiO2 :WO3 , TiO2 :Mn2 O3 , TiO2 :Fe2 O3 , TiO2 :Co3 O4 , TiO2 :NiO, ZrO2 :CdO, ZnO:MgO, ZnO:Fe2 O3 , MoO3 :CoO:Al2 O3 , MoO3 :NiO:Al2 O3 , TiO2 :SiO2 :MgO, MoO3 :Al2 O3 :MgO 5. Cation-exchange resins, polymeric perfluorinated resinsulfonic acids 6. Heteropolyacids (Keggin type) 7. Bis(perfluorosulfonyl)imides, bis- and tris(trifluoromethylsulfonyl) methanes

intercalates are not very stable, however, as the Lewis acid tends to leach out. Similar intercalates have been obtained with other Lewis acids such as AlCl3 , AlBr3 , NbF5 , and TaF5 and conjugate acid systems such as HF:SbF5 . Flourine-complexed acids such as SbF5 -fluorinated graphite and SbF− 5 -fluorinated alumina have been used for hydrocarbon isomerizations.

RH2

ArH2

RHX

A. Preparation of Stable Trivalent Carbocations Superacids such as Magic Acid and fluoroantimonic acid have made it possible to prepare stable, long-lived carbocations, which are too reactive to exist as stable species in more basic solvents. Stable superacidic solutions of a large variety of carbocations, including trivalent cations (also called carbenium ions) such as t-butyl cation 1 (trimethylcarbenium ion) and isopropyl cation 2 (dimethylcarbenium ion), have been obtained. Some of the carbocations, as well as related acyl cations and acidic carboxonium ions and other heteroatom stabilized carbocations, that have been prepared in superacidic solutions or even isolated from them as stable salts are shown in Fig. 1.

CH3

H3C

OH

RCH

OH

CH3

RX

B. Aromatic and Homoaromatic Cations and Carbodications According to Hückel’s (4n + 2) electron rule, if a carbocation has an aromatic character, it is stabilized by resonance.

ROH2

R2OH

ROH

R2O

RCHCH2

HSO3F-SbF5 or HF-SbF5

R2CX2

RSH2

R2S

X

RCONR2 RCOOR′

OH

RSH

RCHO

CH3 2

Spectroscopic techniques such as 1 H and 13 C NMR, infrared, ultraviolet, and X-ray photoelectron spectroscopy have been employed to characterize carbocations. In many cases cation salts can be isolated with the superacid gegen ion, and some of them are structurally characterized by X-ray crystallography.

R2CO

H3C

1

(RO)2CO

(RO)2C

C

RCHCH3 RH

R2C

C

ArH

H

RCOOH

RCOOH2 X∆

R

RCONR2 RCOOR′ R2CX R H H

R2SH

R

X

RCOH2O

R FIGURE 1 Some ways of generating carbocations generated in superacids.

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FIGURE 2 Aromatically stabilized cations and dications and some bridgehead dications.

Some aromatically stabilized Hückeloid systems generated in superacid media along with some carbodications are shown in Fig. 2. C. Static-Bridged or Equilibrating Carbocations Some carbocations tend to undergo fast degenerated rearrangements through intramolecular hydrogen or alkyl shifts to the related identical (degenerate) structures. The question arises whether these processes involve equilibrations between limiting “classical” ion intermediates (trivalent carbenium ions), whose structures can be adequately described by using only Lewis-type two-electron, twocenter bonds separated by low-energy transition states, or whether intermediate “nonclassical” hydrogen- or alkylbridged carbonium ions (higher coordinate carbonium ions) are involved, which also require the presence of twoelectron bonds between three or more centers for their description. It is difficult to answer this question by NMR spectroscopy because of its slow time scale; however, NMR has been used to delineate structures where degenerate rearrangements lead to averaged shifts and coupling constants. Solid-state 13 C NMR (using cross-polarization magicangle spinning techniques), isotopic substitution, and faster methods such as infrared, Raman, and, especially, X-ray photoelectron spectroscopy (ESCA) are particularly useful in investigating these systems. Some typical examples are depicted in Fig. 3. D. Hydrocarbon Transformations The astonishing acidity of Magic Acid and related superacids allows protonation of exceedingly weak bases. Not only all conceivable π -electron donors (such as olefins, acetylenes, and aromatics) and n-donors (such

FIGURE 3 Degenerate classical (carbenbium) and nonclassical (carbonium) carbocations.

as ethers, amines, and sulfides) but also weak σ -electron donors such as saturated hydrocarbons including the parent alkane and methane are protonated. The ability of superacids to protonated saturated hydrocarbons (alkanes) rests on the ability of the two-electron, two-center covalent bond to share its bonded electron pair with empty orbitals ( p or s) of a strongly electron-deficient reagent such as a protic acid:

R–H H

R

H H

Superacids are suitable reagents for chemical transformation, particularly of hydrocarbons. E. Isomerization The isomerization of hydrocarbons is of practical importance. Isomeric dialkylbenzenes, such as xylenes, are starting materials for plastics and other products. Generally, the need is for only one of the possible isomers, and thus there is a potential for intraconversion (isomerization). Straightchain alkanes with five to eight carbon atoms have considerably lower octane numbers than their branched isomers, and hence there is a need for higher-octane branched isomers. Isomerizations are generally carried out under thermodynamically controlled conditions and lead to equilibria. The ionic equilibria in superacid systems generally favor increasing amounts of the higher-octane branched isomers at lower temperatures. Lewis-acid-catalyzed isomerization of alkanes can be effected with various systems. Superacid-catalyzed reactions can be carried out at much lower temperatures, even at or below room temperature, and thus provide more of the branched isomers. This is of particular

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importance in preparing lead-free gasoline. Increasing the octane number by this means is preferable to the addition of higher-octane aromatics or olefins, which may pose environmental or health-hazard problems. Of the many important superacid-catalyzed isomerizations, the isomerization tricyclo [5.2.1.02,6 ] decane to adamantane is unique, a reaction discovered by Schleyer.

Isomerization of alkylaromatics can also be effectively carried out with superacids. F. Alkylation Alkylation of aromatics is carried out industrially on a large scale; an example is the reaction of ethylene with benzene to produce ethylbenzene, which is then dehydrogenated to styrene, the monomer used in producing polystyrene. Traditionally, these alkylations have been carried out in solution with a Friedel–Crafts acid catalyst such as AlCl3 . However, these processes are quite energy consuming and form a complex mixture of products requiring large amounts of catalyst, most of which is tied up as complexes and can be difficult or impossible to recover. The use of solid superacidic catalyst permits clean, efficient heterogeneous alkylations with no concomitant complex formation.

CH2

CH2

AlCl3–HCl

CH2CH3

Aliphatic alkylation is widely used to produce highoctane gasolines and other hydrocarbon products. Conventional paraffin (alkane)–olefin (alkene) alkylation is an acid-catalyzed reaction; it involves the addition of a tertiary alkyl cation, generated from an isoalkane (via hydride abstraction) to an olefin. An example of such a reaction is the isobutane–ethylene alkylation, yielding 2,3-dimethylbutane. The great interest in strong-acid chemistry is further exemplified by the discovery that lower alkanes such as methane and ethane can be polycondensed in Magic Acid at 50◦ C, yielding mainly C4 to C10 hydrocarbons of the gasoline range. The proposed mechanism (Fig. 4) necessitates the intermediacy of protonated alkanes (pentacoordinate carbonium ions), at least as high-lying intermediates or transition states. Hydrogen must be oxidatively removed (by either the excess superacid or added oxidants) to make the condensation of methane thermodynamically feasible.

FIGURE 4 Mechanism of oxidative methane oligocondensation.

Because of the high reactivity of primary and secondary ions under these conditions, the alkylation reaction is complicated by hydride transfer and related competing reactions. However, in this mechanism it is implicit that an energetic primary cation will react directly with methane or ethane. This opens the door to new chemistry through activation of these traditionally passive molecules. A convenient way to prepare an energetic primary cation is to react ethylene with superacid. This has been used with HF–TaF5 catalyst to achieve ethylation of methane in a flow system at 50◦ C. With a methane–ethylene mixture (85:14), propane is the major product. G. Polymerization The key initiation step in cationic polymerization of alkenes is the formation of a carbocationic intermediate, which can then interact with excess monomer to start propagation. The mechanism of the initiation of cationic polymerization and polycondensation has been extensively studied. Trivalent carbenium ions play the key role, not only in acid-catalyzed polymerization of alkenes, but also in polycondensation of arenes (π -bonded monomers), as well as in cationic polymerization of ethers, sulfides, and nitrogen compounds (nonbonded electron-pair donor monomers). Pentacoordinated carbonium ions, on the other hand, play the key role in the electrophilic reactions of σ -bonds (single bonds), including the oligocondensation of alkanes and the cocondensation of alkanes and alkenes. Alkylation and oligocondensation reactions of alkanes giving higher molecular weight alkanes have been achieved under superacid conditions. H. Superacids in Organic Syntheses and Superelectrophilic Activation Since the discovery of stable carbocations, they were known to be readily quenched by various nucleophiles. These reactions, which were first used to confirm the structure of the ions, proved to be very useful in organic synthesis. The selectivity of the reactions is based on the fact that generally only thermodynamically more stable ions are formed under the reaction conditions, resulting in

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a high selectivity. The new functional group created in a superacid medium will itself undergo protonation and, thus, be protected against any further electrophilic attack. In this way, a number of new selective reactions were achieved in a high yield, as shown by the examples below. Furthermore, electrophiles that contain nonbonded electron pairs, π -bonds, or even σ -bonds can be further activated by protonation or Lewis acid complexation leading to superelectrophiles. Such activated species react with many deactivated aromatic as well as aliphatic compounds. 1. Phenol–Dienone Rearrangement This isomerization is of substantial importance in natural product syntheses, usually catalyzed by a strong base. The reaction occurs with good yields in polycyclic systems under superacidic conditions, as shown by Gesson and Jacquesay. CH3 O

CH3 O HF-SbF5

H

H

H O

HO

2. Reduction Hydride ion transfer to carbocations is a well-known reaction in hydrocarbon chemistry. This reaction has been used successfully in superacid to reduce α,β-unsaturated ketones with methylcyclopentane as the hydride donor. Superacid-catalyzed reduction of aromatics, as shown by Wristers, requires both a hydride donor and hydrogen.

H HF-SbF5

O

O

H

H HF-SbF5

COOCH3

CO/CH3OH

O

O

H

Olah et al. have developed direct carbonylation of isoalkanes that lead to ketones in high conversion and high selectivity under HF:BF3 catalysis. The chemistry is unlike the Koch reaction and involves activated formyl cation inserting directly into the C–H σ -bond of isoalkanes, followed by strong acid-catalyzed rearrangement.

O H3C H3C H3C

H

HF:BF3 CO

H3C CH3 CH3

4. Oxidation Novel oxidations of hydrocarbons in superacids with ozone or hydrogen peroxide have been investigated. Proto+ nated ozone (O+ 3 H) or hydrogen peroxide (H3 O2 ) attacks the single σ -bond, resulting in oxygen insertion. These can be followed by protolytic transformation, such as the conversion of isobutane into acetone and methyl alcohol. (CH3)3C

(CH3)3CH H2O2 Magic acid

(CH3)2C

O CH3OH

H

O3 Magic acid

H2O

(CH3)2C

[(CH3)3C

H (CH3)3C

H2O2

O3H

O]

H2O

OOH

1,2CH3 shift

O CH3

By similar procedures aromatics are also hydroxylated in high yields at low temperatures. H2O2 HSO3F/SO2ClF

HF-TaF5 isopentane H2

CH 5. Superelectrophilic Activation

3. Carbonylation The reaction between carbocations and carbon monoxide affording oxocarbenium ions (acyl cations) is a key step in the well-known Koch–Haaf reaction for preparing carboxylic acids from alkenes. This reaction has been extensively studied under superacidic conditions. An example is indicate below.

+ + Electrophiles such as NO+ 2 , CH3 CO , and H3 O can be further activated in strong protic acids to their respective dications: NO2 H2+ , CH3 COH2+ , and H4 O2+ . Such superelectrophiles are responsible for the high electrophilic reactivity in superacids. For example, acetyl cation is a poor acetylating agent for chlorobenzene in trifluoroacetic acid. However, in superacidic trifluoromethanesulfonic acid medium, acetylation takes place with ease.

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Cl

Cl CH3COSbF6

COCH3

Olah and co-workers have shown hydrogen–deuterium exchange of molecular H2 and D2 , respectively, with 1:1 HF:SbF5 and HSO3 F:SbF5 at room temperature. The facile formation of HD does indicate that protonation or deuteration occurs involving 3 at least as a transition state in the kinetic exchange process. The H3 ion, 3, is the simplest two-electron, three-center bonded entity.

in CF3COOH at 60°C, 24 hr, 1% in CF3SO3H at 60°C, 30 min, 81%

H H

Similar activations have been proposed for Lewis acid complexations. I. Miscellaneous Reactions Many acid-catalyzed reactions can be advantageously carried out using solid superacids instead of conventional acid systems. The reactions can be carried out in either the gaseous or the liquid phase. Using the example Nafion-H (a perfluoroalkane resin sulfonic acid, developed by DuPont) solid acid, several simple procedures were reported to carry out alkylation, transbromination, nitration, acetalization, hydration, and so on. J. Superacids in Inorganic Chemistry 1. Halogen Cations It has often been postulated that the monoatomic ions I+ , Br+ , and Cl+ are the reactive intermediates in halogenation reactions of aromatics and alkenes. The search for the existence of such species has led to the discovery of I+ 2 and other related halogen cations, which are stable in superacids. The I+ 2 cation may be generated by the oxidation of I2 with S2 O6 F2 in HSO3 F solution, − 2I2 + S2 O6 F2 → 2I+ 2 + 2SO3 F

and a stable blue solution of this cation can also be obtained by oxidizing iodine with 65% oleum. In a less acidic medium, the I+ 2 cation disproportionates to more stable oxidation states. The electrophilic Br+ 2 cation is obtainable only in the very strong superacid Magic Acid or fluoroantimonic acid, and it disproportionates in HSO3 F. The Cl+ 2 cation, which is much more electrophilic, has not yet been observed in solution. Monoatomic halogen cations seem to be too unstable for direct observation. 2. The Trihydrogen Cation, H+ 3 The H3 ion, 3, was first discovered by Thompson in 1912 in hydrogen discharge studies. Actually, it was the first observed gaseous ion–molecule reaction product.

H 3

3. Cations of Other Nonmetallic Elements Elemental sulfur, selenium, and tellurium give colored solutions when dissolved in a number of strongly acidic me2+ 2+ 2+ 2+ dia. It has been shown that S2+ 16 , S8 , S4 , Se8 , Te4 , 2+ and Te6 are present in such solutions. These cations are formed by the oxidation of elements by H2 S2 O7 or S2 O6 F2 ; for example, − 4S + 6H2 S2 O7 → S2+ 4 + 2HS3 O10 + 5H2 SO4 + SO2

Like the halogen cations, the sulfur, selenium, and tellurium cations are highly electrophilic and undergo disproportionation in media with any appreciable basic properties, although, as would be anticipated, the ease of disproportionation increases in series tellurium < selenium < sulfur. 4. Noble Gas Cations Noble gas cationic salts of xenon and krypton have also been isolated from superacid medium. The examples + + + include XeF+ , Xe2 F+ 3 , HCNXeF , XeOF3 , KrF , and + Kr2 F3 .

SEE ALSO THE FOLLOWING ARTICLES NOBLE-GAS CHEMISTRY • ORGANOMETALLIC CHEMISTRY • PHYSICAL ORGANIC CHEMISTRY

BIBLIOGRAPHY Gillespie, R. J., and Peel, T. E. (1971). Adv. Phys. Org. Chem. 9, 1. Jost, R., and Sommer, J. (1988). Rev. Chem. Int. 9, 171. Olah, G. A. (1993). Angew. Chem. Int. Ed. Engl. 32, 767. Olah, G. A., Prakash, G. K. S., and Sommer, J. (1985). “Superacids,” Wiley, New York. Tanabe, K. (1970). “Solid Acids and Bases; Their Catalytic Properties,” Academic Press, New York. Vogel, P. (1985). “Carbocation Chemistry,” Elsevier, Amsterdam.

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Surface Chemistry Simon R. Bare G. A. Somorjai University of California, Berkeley, and Lawrence Berkeley Laboratory

I. Surface Structure of Clean Surfaces II. Surface Structure of Adsorbates on Solid Surfaces III. Thermodynamics of Surfaces IV. Electrical Properties of Surfaces V. Surface Dynamics

GLOSSARY Adsorbate Adsorbed atom or molecule. Adsorption Process by which molecules are taken up on the surface by chemical or physical action. Chemisorption Binding of molecules to surfaces by strong chemical forces. Desorption Process by which molecules are removed from the surface. Heat of adsorption Binding energy of the adsorbed species. Physisorption Binding of molecules to surfaces by weak chemical forces. Sticking probability Ratio of the rate of adsorption to the rate of collision of the gaseous molecule with the surface. Surface free energy Energy necessary to create a unit area of surface. Surface reconstruction Equilibration of surface atoms to new positions that changes the bond angles and rotational symmetry of the surface atoms.

.

Surface relaxation Equilibration of surface atoms to new positions that changes the interlayer distance between the first and second layers of atoms. Surface state Electronic state localized at the surface. Surface unit cell Two-dimensional repeating unit that fully describes the surface structure. Work function Minimum energy required to remove an electron from the surface into the vacuum outside the solid.

SURFACES constitute the boundaries of condensed matter, solids, and liquids. Surface chemistry explores the structure and composition of surfaces and the bonding and reactions of atoms and molecules on them. There are many macroscopic physical phenomena that occur on surfaces or are controlled by the electronic and physical properties of surfaces. These include heterogeneous catalysis, corrosion, crystal growth, evaporation, lubrication, adhesion, and integrated circuitry. Surface chemistry examines the science of these phenomena as well.

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I. SURFACE STRUCTURE OF CLEAN SURFACES A. Introduction To the naked eye the surface of a single crystal of a metal looks perfectly planar with no imperfections. If this crystal is now examined with an optical microscope features of the surfaces down to the wavelength of visible light ˚ can be resolved. The surface will look gran(∼5000 A) ular with distinct regions of crystallinity separated from each other by boundaries or dislocations. These dislocations indicate areas on the surface where there is a mismatch of the crystalline lattice, and they can take several forms, for example, edge dislocations or screw dislocations. The presence of these dislocations or defects can dominate certain physical properties of the material. Dislocation densities of the order of 106 –108 cm−2 are commonly found on metal single crystals, whereas the number is lower (104 –106 cm−2 ) on semiconductor surfaces due to the different nature of the bonding. These defect densities must be compared with the total concentration of surface atoms (about 1015 cm−2 ). On further magnification, for example, using a scanning electron microscope, ˚ and our features can be resolved down to about 1000 A, view of the surface changes further. The surface will look pitted, with distinct planar areas (terraces) bounded by walls many atomic layers in height. Thus, on the microscopic and submicroscopic scale the surface morphology appears to be heterogeneous, with many different surface sites that differ by the number of neighboring atoms surrounding them. What about the nature of the surface on an atomic scale? In order to be able to discuss and understand the structure of surfaces it is necessary to understand the techniques that are capable of viewing the surface on an atomic scale. We briefly describe such techniques, illustrating their capabilities with pertinent examples. The techniques more commonly used are field-ion microscopy (FIM), lowenergy electron diffraction (LEED), helium atom diffraction, and high-energy ion scattering. In addition, the relatively new technique of scanning tunneling microscopy (STM) is proving to be a very promising tool. Only brief descriptions are given here and the reader is referred to some of the excellent books on the subject given in the bibliography.

Surface Chemistry

vented by Müller in 1936. The basic microscope can be very simple. In an ultrahigh vacuum cell, a potential of about 10,000 V is applied between a hemispherical tip of refractory metal of radius ∼10−4 cm and a fluorescent screen. The tip is charged positively, and a gas (usually helium) is allowed to impinge on the surface. Under the influence of the very strong electric field helium atoms that are incident on the tip are ionized. The positive ions thus created are repelled radially from the surface and accelerated onto the fluorescent screen, where a greatly magnified image of the crystal tip is displayed. The ionization probability depends strongly on the local field variations induced by the atomic structure of the surface—protruding atoms generate stronger ionization than atoms embedded in close-packed planes, thereby producing individual bright spots on the screen. The small radius of the tip is needed to produce the large fields necessary for ionization, but it also permits the immense magnification of the microscope. The tip surface is directly imaged with magnification of about 107 . Figure 1 depicts the image of a tungsten field-ion tip. Well-defined atomic planes of the crystal tip can be readily identified, indicating that there is order of the atomic scale, i.e., most of the surface atoms in any crystal face are situated in ordered rows separated by well-defined interatomic distances. The technique is limited to the refractory metals (W, Ta, Ir, and Re) which can withstand the strong electric field at the tip without desorption or evaporation from the surface. However, its great advantage is that individual atoms can be imaged on the screen, which also allows studies of surface diffusion.

B. Techniques Sensitive to Surface Structure 1. Field-Ion Microscopy Field-ion microscopy is one of the oldest techniques used for surface structure determination, having been in-

FIGURE 1 Field-ion micrograph of a tungsten tip. Various crystal planes are labeled. (Courtesy of Lawrence Berkeley Laboratory.)

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2. Low-Energy Electron Diffraction Another method which has demonstrated that crystal surfaces are ordered on an atomic scale is low-energy electron diffraction (LEED). This method is the most frequently used technique, such that virtually all modern surface science laboratories now rely on it for surface structural information. In order to obtain diffraction from surfaces, the incident wave must satisfy the condition λ ≤ d, where λ is the wavelength of the incident beam and d the interatomic distance. In addition, the incident beam should not penetrate much below the surface plane but should backdiffract predominantly from the surface so that the scattered beam reflects the properties of the surface atoms and not those of the bulk. The deBroglie wavelength λ of electrons is given √ ˚ = 150/E, where E is in electron volts. Thus, by λ (in A) in the energy range 10–500 eV, the wavelength varies from ˚ which is smaller or equal to most interatomic 3.9 to 0.64 A, distances, and the escape depth of the backscattered elec˚ thereby providing trons in this energy range is 5–10 A, surface sensitivity. These elastically scattered low-energy electrons yield surface structural information. The technique of LEED is depicted schematically in Fig. 2, while a schematic of the LEED apparatus is shown in Fig. 3. A collimated primary beam of electrons with a diameter of 0.1–1 mm at energies of 15–350 eV is impinged on a surface and the elastically backscattered electrons, after traveling through a field-free region, are spatially analyzed. This is achieved most commonly (see Fig. 3) by passing the scattered electrons through four hemispherical grids. The first grid is at the crystal potential while the second and third are at a retarding voltage to eliminate inelastic electrons, and the fourth is at ground. After passing through these grids the diffracted beams are accelerated onto a hemispherical phosphor screen.

If the crystal surface is well-ordered, a diffraction pattern consisting of bright, well-defined spots will be displayed on the screen. The sharpness and overall intensity of the spots are related to the degree of order of the surface. When the surface is less ordered, the diffraction beams broaden and become less intense, while some diffuse intensity appears between the beams. A typical set of diffraction patterns from a well-ordered surface is shown in Fig. 4. The presence of the sharp diffraction spots clearly indicates that the surface is ordered on an atomic scale. Similar LEED patterns have been obtained from solid single-crystal surfaces of many types including metals, semiconductors, alloys, oxides, and intermetallics. Due to the importance of LEED in surface chemistry we briefly discuss other aspects of the technique which make it one of the most powerful surface sensitive tools. It is convenient to subdivide the technique into two-dimensional LEED and three-dimensional LEED. In two-dimensional LEED we observe only the symmetry of the diffraction pattern on the fluorescent screen. The bright spots which correspond to the two-dimensional reciprocal lattice belonging to the repetitive crystalline surface structure yield immediate information about the size and orientation of the surface unit cell, i.e., the geometry of the surface layer. This is important information since reconstructioninduced and adsorbate-induced new periodicities are immediately visible. The diffuse background intensity also contains information about the nature of any disorder present on the surface. In three-dimensional LEED the information gained from the two-dimensional pattern is supplemented by the intensities of the diffraction spots which are measured as a function of incident electron energy. By comparing these intensity-versus-voltage curves [I (V ) curves] with those simulated numerically with the help of a suitable theory, the precise location of atoms or molecules in the surface with respect to their neighbors is determined. Thus, the bond length and bond angles in the surface layer are calculated. It should be mentioned, however, that the analysis of the LEED beam intensities requires a theory of the diffraction process which is a nontrivial point due to multiple scattering of LEED electrons by the surface, and this is not simple to represent in a theory. 3. Atomic-Beam Diffraction Another technique that utilizes the principle of diffraction is atomic- or molecular-beam diffraction. The deBroglie wavelength λ associated with helium atoms is given by the following:

FIGURE 2 Scheme of the low-energy electron diffraction experiment. (Courtesy of Lawrence Berkeley Laboratory.)

˚ = λ(A)

h 0.14 = , 1/2 (2ME) E(eV)1/2

(1)

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FIGURE 3 Scheme of the low-energy electron diffraction apparatus employing the postacceleration technique.

FIGURE 4 LEED pattern from a Pt(111) crystal surface at (a) 51 eV, (b) 63.5 eV, (c) 160 eV, and (d) 181 eV incident electron energy. (Courtesy of Lawrence Berkeley Laboratory.)

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where h is Planck’s constant and M and E are the mass and energy, respectively, of the helium atom. Atoms with ˚ and can reada thermal energy of ∼20 meV have λ = 1 A ily diffract from surfaces. The information obtained from atomic-beam diffraction is similar to that from LEED, but there are differences between the two techniques. In LEED the relatively high-energy (20–200 eV) electrons used penetrate the crystal, multiple scattering events are important, and the LEED electrons are scattered primarily from the ion cores of the crystal lattice. In atom diffraction, there is virtually no penetration of the low-energy (10–200 meV) atomic beam, making it much more surface sensitive than the electron beam. The atomic beam is primarily scattered from the valence electrons of the surface atoms. In fact, their scattering is usually simulated by a “hard wall” around the atoms in the top layer of the surface so that diffraction is from a “corrugated hard wall” with the periodicity of the surface mesh. As in LEED the location of the diffracted beams indicates the surface periodicity. Their intensities are related to the structure of the scattering potential within a unit mesh, in this case to the relative amplitude and positions of corrugations around the surface atoms. The essential elements of the apparatus necessary to perform atomic-beam diffraction are an atomic beam of gas and a detector. The atomic beam is usually generated from a nozzle source incorporating several skimmers. The energy (wavelength) of the beam is varied by either heating or cooling the nozzle. The detector usually employed is a mass spectrometer, mounted on a rotatable device to enable it to be movable over a large range of scattering angles. The atomic beam is chopped with a variable-frequency chopper before it impinges on the surface. In this way, an alternating intensity of the beam is generated at the mass spectrometer detector, which is readily separated from the noise due to helium atoms in the background. To illustrate the type of data that is obtained, Fig. 5 shows the He diffraction traces from a Au(110)-(1 × 2) surface at two different wavelengths. Helium diffraction is especially sensitive to surface order on an atomic scale. On scattering from a well-ordered single crystal surface nearly 15% of the scattered helium atoms appear in the specular helium beam whereas this fraction can drop to 1% when the surface is disordered. Measurements of the fraction of specularly scattered helium can therefore provide information on the degree of atomic disorder in the solid surface. 4. Scanning Tunneling Microscopy The relatively new technique of scanning tunneling microscopy also clearly demonstrates order on an atomic

FIGURE 5 Helium diffraction traces for Au(110)-(1 × 2) at a surface temperature of 100 K with incident angle i = 48◦ . The wave˚ length λHe is (a) 1.09 A˚ and (b) 0.57 A.

scale on single-crystal surfaces. It images surface topogra˚ and phies in real space with a lateral resolution of ∼2 A ˚ vertical resolution of ∼0.05 A. The technique utilizes the tunnel effect. Due to the wave nature of electrons, they are not strictly confined to the interior bounded by the surface atoms. Therefore, the electron density does not drop to zero at the surface but decays exponentially on the outside with a decay length of a few angstroms. If two metals are approached to within a few angstroms, the overlap of their surrounding electron clouds becomes substantial, and a measurable current can be induced by applying a small voltage between them. This tunnel current is a measure of the wave-function overlap, and depends very strongly on the distance between the two metals. This is the physical basis of the scanning tunneling microscope. Experimentally one of the electrodes is sharpened to a pointed tip which is scanned over the surface to be investigated (the other electrode) at constant tunnel current. The tip thus traces contours of constant wave-function overlap, and in the case of constant decay length, the trace is an almost true image of the surface atomic positions, i.e., the surface topography. An example is shown in Fig. 6.

5. Ion Scattering Ion scattering from surfaces is usually subdivided into two scattering regions: low-energy ion scattering (LEIS), energies typically ∼1 keV, and high-energy scattering (HEIS), energies 0.1–1 MeV. High-energy ion scattering is a probe that tests the local position of surface atoms relative to their bulklike sites. In HEIS the velocity of the ion is such that it is moving fast compared to the thermal motions of the

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FIGURE 6 Scanning tunneling microscope picture of a clean (1 × 5) reconstructed Au(100) surface with monatomic ˚ with approximately 1.5 A˚ from scan to scan. The inset shows the LEED steps. Divisions on the crystal axes are 5 A, pattern of the predominant (1 × 5) corrugation. (Courtesy of Lawrence Berkeley Laboratory.)

atoms in the solid, thus the beam senses a frozen lattice. If the target is amorphous each atom would sense a uniform distribution of impact parameters of the ions and diffuse scattering results. However, the scattering spectrum from a single crystal aligned with a major symmetry axis parallel to the beam is drastically modified from that of the amorphous target. The impact parameter distribution is also uniform at the first monolayer, but the first atom shadows the second from the beam, and small angle scattering events determine the impact parameter distribution at the second atom. This results in a unique (nonuniform) flux distribution at the second atom. Figure 7 illustrates the effect of small-angle scattering in a two-atom model. Ions incident at the smallest impact parameter undergo largeangle scattering, those at large impact parameter suffer small deflections which determine the flux distribution of ions near the second atom. The closest approach of the ion to the second atom, R, can be approximated assuming Coulomb scattering as follows: 1/2 R = 2 Z 1 Z 2 e2 d E ,

proximation, can be written analytically which leads to an estimate of the two-atom surface peak intensity I :

R2 I =1+ 1+ 2 2ρ

−R exp , 2ρ 2

(3)

where ρ is the two-dimensional root mean square thermal vibrational amplitude. The first term represents the unit contribution from the first atom in the string, the second term represents the variable contribution from the second

(2)

where Z 1 and Z 2 are the masses of the incident and target atoms, respectively; d is the atomic spacing; and E is the incident ion energy. This gives rise to a shadow cone beneath the surface atom as illustrated in Fig. 7. The flux distribution at the second atom, within the Coulomb ap-

FIGURE 7 Schematic showing the interactions at the surface of an aligned single crystal and the formation of the shadow cone. The energy spectra for the aligned and nonaligned case are also shown.

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atom. While the two-atom Coulomb approximation is not adequate enough to compare to experiment, it illustrates that the surface peak intensity is a function of one parameter, ρ/R. The intensity of the surface peak is thus sensitive to the atomic arrangement on the surface, i.e., the positions of the surface atoms with respect to their bulklike positions. The effect on the surface peak of different surface structures is depicted in Fig. 8. The nature of reconstructed, relaxed, and adsorbate-covered surfaces are discussed in the sections below. High-energy ion scattering is also a sensitive tool for answering other important questions in surface chemistry, namely what type of atoms are present on the surface and how many are present. All of the techniques discussed so far indicate that the solid surface is ordered on an atomic scale. Most of the surface atoms occupy equilibrium atomic positions that are located in well-defined rows separated by equal interatomic positions. This atomic order is predominant despite the fact that there are large numbers of atomic positions on the surface where atoms have different numbers of neighbors. A pictorial representation of the topology of a monatomic crystal on an atomic scale is shown in

FIGURE 9 Model of a heterogeneous solid surface, depicting different surface sites. These sites are distinguishable by their number of nearest neighbors.

Fig. 9. The surface may have atoms in any of the positions shown in the figure. There are atoms in the surface at kink positions and in ledge positions, and there are adatoms adsorbed on the surface at various sites. Atomic movement from one position to another proceeds by surface diffusion. To the first approximation, the binding energy of the surface atoms is proportional to the number of nearest and next-nearest neighbors. Therefore, for example, atoms at a ledge are bound more strongly than are adatoms. In equilibrium there is a certain concentration of all these surface species, with those species predominating whose binding energies are greatest. Thus, the adatom concentration on clean well-equilibrated surfaces should be very small indeed. However, while these surfaces are ordered on an atomic scale their structure is not always one of simple termination of the bulk unit cell, relaxation or reconstruction being common. C. Surface Relaxation

FIGURE 8 Schematic of the dependence of the intensity of the surface peak (SP) on different crystal surface structures. (a) The ideal crystal SP from “bulklike” surface, (b) enhanced SP observed in normal incidence for a reconstructed surface, (c) enhanced SP observed in nonnormal incidence for a relaxed surface, and (d) reduced SP observed in normal incidence for a registered overlayer.

Generally, the surface unit cells of clean metal surfaces have been found to be those expected from the projection of the bulk X-ray unit cell onto the surface, referred to as a (1 × 1) structure (in Miller index notation), and the uppermost layer z spacing (spacing in the direction normal to the surface plane) is equal to the bulk value within about 5%. Such surfaces include the (111) crystal faces of face centered cubic aluminum, platinum, nickel, and rhodium, and the (0001) crystal faces of hexagonal close packed cadmium and beryllium. This information has almost exclusively been determined by a detailed intensity analysis of the diffraction beams in LEED as a function of incident electron energy, and the interatomic positions ˚ The in the surface layer are calculated to within 0.1 A. Al (110) surface shows a 5–15% contraction, the Mo(100) surface a 11–12% contraction, and the W(100) surface a 6% contraction of the top-layer z spacing with respect to the bulk, while retaining the (1 × 1) surface unit cell. Generally, crystal planes whose atoms are less densely packed [for example, bcc (100) and fcc (110) planes] will be more likely to show relaxation than the more densely

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380 packed planes. In forming a surface of the less densely packed planes it is necessary to remove a larger number of nearest-neighbor atoms. Thus, to minimize the surface free energy a relocation of the surface atoms from their bulk positions is quite likely. There are several explanations as to why surface relaxation is prevalent on the more open surfaces. First, it can be imagined that the electron cloud attempts to smooth its surface, thereby producing electrostatic forces that draw the surface atoms toward the substrate, the effect being stronger the less closely packed the surface. Second, with fewer neighbors the two-body repulsion energy is smaller, allowing greater atomic overlap and therefore more favorable bonding at shorter bond lengths. Third, for surface atoms the bonding electrons are partly redistributed from the broken bonds to the remaining unbroken bonds, thereby the charge content of the latter is increased, reducing the bond length. D. Surface Reconstruction Surface atoms in any crystal are in an anisotropic environment which is very different from that around bulk atoms. The crystal symmetry that is experienced by each surface atom is markedly lower than when the atom is in the bulk. This symmetry change and lack of neighbors in the direction perpendicular to the surface allows displacement of surface atoms in ways which are not allowed in the bulk. Surface relaxation is one consequence of this, the other major consequence being surface reconstruction. Here the two-dimensional surface unit cell is different from that given by the termination of the bulk structure on the plane of interest. Surface reconstruction can give rise to a multitude of different structures depending on the electronic structure of a given substance. The phenomenon is more frequent on semiconductor surfaces than on metal surfaces. While the geometry is readily observed in the LEED pattern, the actual elucidation of the real-space reconstructed surface structure is often extremely difficult, and in some cases even after years of study and many proposals of the structure, the true structure is still not known. Such a system is the Si(111) surface. Upon cleaving in UHV at room temperature, the surface exhibits a (2 × 1) surface structure. On heating to about 700 K the surface structure changes to one with (7 × 7) periodicity. This (7 × 7) structure is then the stable structure of the (111) face. While this surface has been studied by a multitude of techniques, including LEED, STM, and He atom diffraction there is still no generally accepted structure of either the (2 × 1) or (7 × 7) reconstructions. One of the best known examples of reconstruction of metallic surfaces is that for the (100) faces of three 5d transition metals that are neighbors on the periodic table:

Surface Chemistry

gold, platinum, and iridium. The ideal unreconstructed surfaces have a square net of atoms. Surface reconstruction produces a superlattice that is basically five times larger in one direction than for the ideal surface. For Ir(100) the superlattice is denoted (5 × 1), for Pt(100) by the ma1 trix notation ( −15 14 ), and for Au(100) by the superlattice (5 × 20). From LEED I (V ) analyses, evidence indicates that the nature of the reconstruction is similar on all three metals and consists of a close-packed hexagonal top layer that is positioned in slightly different ways on the square net substrate. These reconstructions are consistent with the knowledge that the (111) face of fcc metals is energetically the most favorable. It is worth noting here that the adsorption of gases such as oxygen, carbon monoxide, or hydrogen, or the presence of impurities can inhibit these surface reconstructions. On the other hand, the presence of such adsorbates can also induce different surface reconstructions. The nature and cause of these surface phase transformations are not well established at present. The case of structural change from metastable to stable on adsorption or removal of adsorbates indicates the likelihood of electronic transitions that accompany reconstruction. At the surface there are fewer nearest neighbors as compared to atoms in the bulk. The electronic structure that is the most stable in this reduced-symmetry environment may be substantially different from that of the bulk metal. Since the surface atoms are surrounded by atoms only on one side and there is vacuum on the other, they may change their coordination number by slight relocation with simultaneous changes in the electronic structure. It is indeed surprising that more surfaces do not show reconstruction. E. Stepped and Kinked Surfaces The close-packed faces of solids (low-Miller-index faces) have the lowest surface free energy, and therefore they are the most stable with respect to rearrangement on disordering up to or near the melting point. However, stepped and/or kinked surfaces (high-Miller-index faces), although of higher surface free energy, are very important. They are known to play important roles during evaporation, condensation, and melting. Steps and kinks are sites where atoms break away as an initial process leading to desorption, or where atoms migrate during condensation to be incorporated into the crystal lattice. Theories of crystal growth, evaporation, and the kinetics of melting have identified the significance of these lower coordination-number sites in controlling the rate processes associated with phase changes. In addition, studies of chemisorption and catalysis using single-crystal surfaces have revealed different binding energies and enhanced chemical activity at steps and kinks on high-Miller-index

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FIGURE 10 LEED patterns (left) and surface structures (right) of (a) flat, (b) stepped, and (c) kinked platinum surfaces. (Courtesy of Lawrence Berkeley Laboratory.)

transition metal surfaces as compared to low-Millerindex surfaces. Adsorption of diatomic or polyatomic molecules frequently leads to dissociation with greater probability of steps and kinks than on flat atomic terraces. The presence of steps on a single crystal surface is readily discernible by LEED. The LEED patterns differ from those expected from crystals with low-index faces in that the diffraction beams are split into doublets. This splitting (Fig. 10) is a function of ordered steps on the surface. The distance between the split beams is inversely related to the distance between the steps, i.e., the terrace width. From the variation of the intensity maximum of the doublet spots with electron energy the step height can be determined. Many stepped surfaces exhibit high thermal stability. In particular the one-atom-height step periodic terrace configuration appears to be the stable surface structure of many high-Miller-index surfaces. While most of the stepped surfaces are stable when clean in their one-atomheight step ordered terrace configuration, in the presence of a monolayer of carbon or oxygen many stepped surfaces undergo structural rearrangement. The step height and terrace width may double, or faceting may occur. Faceting occurs when the step orientation becomes as prominent as that of the terrace and new diffraction features become rec-

ognizable in LEED. Upon removing the impurities from the surface, the original one-atom-height step ordered terrace surface structure is usually regenerated.

II. SURFACE STRUCTURE OF ADSORBATES ON SOLID SURFACES A. Introduction While the knowledge of the structure of clean solid surfaces is important in its own right for determining various properties of those surfaces, many phenomena are associated with the presence of adsorbates on the surfaces. In fact, in the natural environment of our planet, surfaces are never truly free of adsorbates. On approaching a surface each atom or molecule encounters a net attractive potential. This results in a finite probability that it will be trapped on the surface. This trapping, adsorption, is always an exothermic process. At the low pressure of 1 × 10−6 torr, approximately 1 × 1015 gas molecules collide with each square centimeter of surface per second. Since the surface concentration of atoms is about 1015 cm−2 , at this pressure the surface may be covered with a monolayer of gas within seconds; this is the major reason why surface studies are performed under ultrahigh vacuum conditions (P < 1 × 10−8 torr). The very low

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382 pressure is needed to maintain clean surface conditions for a time long enough to perform experimental measurements. At atmospheric pressure the surface will be covered within a fraction of a second. The constant presence of the adsorbate layer influences the chemical, mechanical, and surface electronic properties. Adhesion, lubrication, and resistance to mechanical or chemical attack or photoconductivity are just a few of the many macroscopic surface processes that are controlled by various properties of monolayers. Two macroscopic experimentally determinable parameters characterize the adsorbed monolayer: the coverage and the heat of adsorption. The coverage is defined as the ratio of the number of adsorbed atoms or molecules to the total number of adsorption sites (usually taken as the number of atoms in the surface plane). The heat of adsorption Hads , is implicitly linked to the strength of the adsorbate–substrate bond. Knowledge of both parameters often reveals the nature of bonding in the adsorbed layer. Atoms or molecules that impinge on the solid surface from the gas phase will have a residence time τ on the surface. If the impinging molecules achieve thermal equilibrium with the surface atoms τ = τ0 exp Hads /RT , where τ0 is related to the average vibrational frequency associated with the adsorbate. The heat of adsorption is always positive and is defined as the binding energy of the adsorbed species. A larger Hads and lower temperature increase the residence time. For a given incident flux, larger Hads and lower temperature yield higher coverages. It is conventional to divide adsorption into two categories: physisorption and chemisorption. Physisorption (or physical adsorption) systems are characterized by weak interactions ( Hads < 15 kcal mol−1 , accompanied by short residence times) and require adsorption studies to be performed at low temperature and relatively high pressure (high flux). Adsorbates that are characterized by stronger chemical interactions ( Hads ≥ 15 kcal mol−1 ), where near-monolayer adsorption commences even at room temperature and at low pressures (≤10−6 torr), are called chemisorption systems. While the two names imply two distinct types of adsorption, there is a gradual change from the physisorption to the chemisorption regime. One of the most fascinating facts about the structure of these physisorbed and chemisorbed overlayers in the submonolayer to few monolayer regime is the preponderance of the formation of long-range ordered structures. Well over 1000 two-dimensional unit cells have been documented in the literature. While only the shape, size, and orientation of the cells are known for most of them, the adsorption site and bond lengths have been determined for about 500 of them.

Surface Chemistry

B. The Ordering of Adsorbed Monolayers The ordering process in the adlayer is due to an interplay of the bonding with the substrate and the bonding between the adatoms or admolecules. Once a molecule adsorbs it may diffuse on the surface or remain bound at a specific site during most of its residence time. Thermal equilibration among the adsorbate and between the adsorbate and substrate atoms (i.e., adsorption) is assured if Hads and ∗ , the activation energy for bulk diffusion, are high E D(bulk) enough as compared to kT (≥10kT ). However, ordering primarily depends on the depth of the potential energy barrier that keeps an atom or molecule from hopping to a neighboring site along the surface. The activation en∗ ergy for surface diffusion, E D(surface) , is an experimental parameter that is of the magnitude of this potential energy barrier. E D∗ can be experimentally determined on well-characterized surfaces by field-ion microscopy, for example, and for Ar and W adatoms and O atoms on tungsten surfaces has the value 2, 15, and 10 kcal mol−1 , ∗ respectively. For small values of E D(surface) ordering is restricted to low temperatures, since as the temperature is increased the adsorbate becomes very mobile. As the value ∗ of E D(surface) increases, ordering cannot commence at low temperature since the adsorbate atoms need to have a considerable mean free path along the surface to find their equilibrium position once they land on the surface at a different location. Naturally, if the temperature is too high, the adsorbate will desorb or vaporize. The binding forces of adsorbates on substrates have components perpendicular and parallel to the surface. The perpendicular component is primarily responsible for the binding energy ( Hads ), while the parallel component often determines the binding site on the surface. The binding site may also be affected by adsorbate–adsorbate interactions, which produce ordering within an overlayer. These interactions may be subdivided into direct adsorbate– adsorbate interactions (not involving the substrate at all) and substrate mediated interactions. The latter are complicated many-atom interactions, for example dipole–dipole interactions. The adsorbate–adsorbate interactions may be repulsive; they always are repulsive at sufficiently small adsorbate– adsorbate separations. At larger separations they may be attractive, giving rise to the possibility of island formation. They may also be oscillatory, moving back and forth between attractive and repulsive as a function of adsorbate– adsorbate separation, with a period of several angstroms giving rise, for example, to non-close-packed islands. Except for the strong repulsion at close separations, the adsorbate–adsorbate interactions are usually weak compared to the adsorbate–substrate interactions, even when we consider only the components of the forces parallel

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to the surface. In the case of chemisorption, where the adsorbate–substrate interaction dominates, the adsorbates usually choose an adsorption site that is independent of the coverage and of the overlayer arrangement (the positions the other adsorbates choose). This adsorption site is usually that location which provides the largest number of nearest substrate neighbors, which is independent of the position of other adsorbates. Adsorbates with these properties normally do not accept close packing; the substrate controls the overlayer geometry and imposes a unique adsorption site. Close packing of an adsorbate layer is also observed. In this case the overlayer chooses its own lattice (normally a hexagonal close-packed arrangement) with its own lattice constant independent of the substrate lattice and results in the formation of incommensurate lattices. In this case no unique adsorption site exists: each adsorbate is differently situated with respect to the substrate. This situation is especially common in the physisorption of rare gases. Their relatively weak adsorbate–substrate interactions allow the adsorbate–adsorbate interactions to play the dominant role in determining the overlayer geometry. The chemisorption case is exemplified by oxygen and sulfur on metals; the physisorption case by krypton and xenon on metals and graphite. Intermediate cases exist. Although undissociated CO on metals is not physisorbed but chemisorbed, it sometimes produces incommensurate close-packed hexagonal overlayers. 1. The Effect of Temperature on Ordering Temperature has a major effect on the ordering of adsorbed monolayers: all of the important ordering parameters (the rates of desorption and surface and bulk diffusion) are exponential functions of the temperature. The influence of temperature on the ordering of C3 –C8 saturated hydrocarbons on the Pt(111) crystal face is shown in Fig. 11. At the highest temperatures, adsorption may not take place, since under the exposure conditions the rate of desorption is greater than the rate of condensation of the vapor molecules. As the temperature is decreased, the surface coverage increases and ordering becomes possible. First, one-dimensional lines of molecules form; then at lower temperatures ordered two-dimensional surface structures form. Not surprisingly, the temperatures at which these ordering transitions occur depend on the molecular weights of the hydrocarbons, which also control their vapor pressure, heats of adsorption, and activation energies for surface diffusion. As the temperature is further decreased, multilayer adsorption may occur and epitaxial growth of crystalline thin films of hydrocarbon commences. Figure 11 clearly demonstrates the controlling effect of temperature on the ordering and the nature of ordering of

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FIGURE 11 Monolayer and multilayer phases of the n-paraffins C3 to C8 on Pt(111) and the temperatures at which they are observed at 10−7 torr.

the adsorbed monolayer. Although changing the pressure at a given temperature may be used to vary the coverage by small amounts and thereby change the surface structures in some cases, the variation of temperature has a much more drastic effect on ordering. Temperature also markedly influences chemical bonding to surfaces. There are adsorption states that can only be populated if the molecule overcomes a small potential energy barrier. The various bond-breaking processes are similarly activated. The adsorption of most reactive molecules on chemically active solid surfaces takes place without bond breaking at sufficiently low temperatures. As the temperature is increased, bond breaking occurs sequentially until the molecule is atomized. Thus, the chemical nature of the molecular fragments will be different at various temperatures. There is almost always a temperature range for the ordering of intact molecules in chemically reactive adsorbate–substrate systems. It appears that for these systems ordering is restricted to low temperatures below 150 K, and consideration of surface mobility becomes, perhaps, secondary. 2. The Effect of Surface Irregularities on Ordering A solid surface exhibits a large degree of roughness on a macroscopic scale. Therefore, it is to be expected that if nucleation is an important part of the ordering process, surface roughness is likely to play an important role in preparing ordered surface structures. The transformation

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temperature or pressure at which one adsorbate surface structure converts into another can also be affected by the presence of uncontrolled surface irregularities (surface defects). Other causes that could influence ordering are the presence of small amounts of surface impurities that block nucleation sites or interfere with the kinetics of ordering, or impurities below the surface that are pulled to the surface during adsorption and ordering. The effect of surface irregularities on ordering can be investigated in a more controlled way using stepped crystal surfaces. In general, the smaller the ordered terrace between the steps, the stronger the effect of steps on ordering. For instance, the ordering of small molecular adsorbates on a high-Miller-index Rh(S)-[6(111) × (100)] is largely unaffected by the presence of steps whereas on the Rh(S)[3(111) × (111)] the ordering is influenced by the higher step density. Steps can also affect the nucleation of ordered domains. It is frequently observed on W and Pt stepped surfaces that when two or three equivalent ordered domains may form in the absence of steps, only one of the ordered domains grows in the presence of steps. 3. The Effect of Coadsorbates on Ordering It has been found that although certain molecules may not order when present on their own on a surface they can be induced to order by coadsorption of either carbon monoxide or nitric oxide. For example Table I summarizes the ordered structures that have been observed by the coadsorption of alkylidynes, acetylene, aromatics, and alkalis with CO on Rh(111). At low temperature both Na and ethylidyne form (2 × 2) overlayers on Rh(111) but with increasing temperature begin to disorder. If CO is coadsorbed then the adsorbates can be reordered into a c(4 × 2) unit cell. It is thought that the nature of this type of ordering process is due to adsorbate–adsorbate interactions: A molecule that might not otherwise order due to weak adsorbate–adsorbate interactions is ordered by coadsorbTABLE I Ordered Structures Induced by CO on Rh(111) Type of molecule

LEED pattern

System

Alkylidynes

c(4 × 2) √ √ (2 3 × 2 3)R30◦ c(4 × 2)

3CCH2 CH3 + CO C2 H2 + CO

Acetylene Aromatics

Alkalis

(3 × 3) (3 × 3) √ c(2 3 × 4)Rect √ c(2 3 × 4)Rect √ ( 3 × 7)Rect c(4 × 2)

CCH3 + CO

C6 H5 F + 2CO C6 H6 + 2CO C6 H6 F + CO C6 H6 + CO Na + 7CO Na + CO

ing a molecule such as CO, which has interactions strong enough to induce ordering in the overlayer. Similar phenomena have been observed on Pt(111), and it is thought that this coadsorbate-induced ordering may prove to be a very general phenomenon. C. Ordered Adsorbate Structures As mentioned in the introduction to this section, well over 1000 ordered adsorbate structures have been observed with LEED. A full listing and discussion of these structures is outside the scope of this article. Instead, one example of each of the three following categories of adsorption are presented to give an illustrative indication of the types of structures found: (i) an ordered monolayer of atoms, (ii) an ordered organic monolayer, and (iii) an ordered molecular monolayer. First, a few generalities of the ordered adsorbate structures are discussed, based on the large number of LEED observations: the so-called “rules of ordering”: 1. The rule of close-packing. Adsorbed atoms or molecules tend to form surface structures characterized by the smallest unit cell permitted by the molecular dimensions and adsorbate–adsorbate and adsorbate–substrate interactions. They prefer close-packing arrangements. Large reciprocal unit meshes are uncommon and the most frequently observed meshes are the same size as the substrate mesh, i.e., (1 × 1) or are approximately √ √ twice as large, e.g., (2 × 2), c(2 × 2), (2 × 1), ( 3 × 3). 2. The rule of rotational symmetry. Adsorbed atoms or molecules form ordered structures that have the same rotational symmetry as the substrate surface. If the surface unit mesh has a lower symmetry than the substrate, then domains of the various possible mesh orientations are to be expected on different areas of the surface with a resulting increase in symmetry. 3. The rule of similar unit cell vectors. Adsorbed atoms as molecules in monolayer thickness tend to form ordered surface structures characterized by unit cell vectors closely related to the substrate unit cell vectors. The surface structure bears a closer resemblance to the substrate structure than to the structure of the bulk condensate. These are not hard-and-fast rules but rather are generalizations of a great many systems. 1. Ordered Atomic Monolayers Some important conclusions can be drawn from the known structures of atomic adsorbates on single-crystal surfaces. First, the adsorbed atoms tend to occupy sites where they are surrounded by the largest number of substrate atoms

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(largest coordination number). This site is usually the one that the bulk atoms would occupy in order to continue the bulk lattice into the overlayer. The tendency toward occupying the site with the largest coordination number during adsorption on metals holds independently of the crystallographic face of a given metal, the metal for a given crystallographic face, and the adsorbate for a given substrate. Second, the adsorbed atom–substrate atom bond lengths are similar to the bond lengths in organometallic compounds that contain the atom pairs under consideration. The most common adsorption geometries are displayed in Fig. 12. The threefold hollow sites on the fcc(111) and hcp(0001) and bcc(110) are shown both in top and side views. Similarly, the fourfold hollow sites on the fcc(100) and bcc(100) crystal faces are shown. Finally, the center, long-bridge, and short-bridge sites on the fcc(100) crystal face and the location of atoms in an underlayer in the hcp(0001) crystal face are also displayed. In addition to the situations discussed, there exist some unique atomic adsorbate geometries. For example, small atoms such as nitrogen and hydrogen often prefer to sit below the surface, as in the case of titanium single-crystal surfaces. Also in the presence of strong chemical inter-

FIGURE 13 Structure of the p (2 × 2) and c (2 × 2) sulphur overlayers on Ni(100).

actions there may be a rearrangement of the substrate layer (an adsorbate-induced reconstruction). One example is oxygen on the Fe(100) crystal face. As an example of ordered atomic adsorption Fig. 13 portrays the two structures of sulfur on Ni(100). At a coverage of one-quarter of a monolayer of S, a p(2 × 2) overlayer is formed, and at one-half of a monolayer a c(2 × 2) structure is observed. In both cases the S sits in the fourfold hollow site (highest coordination). A LEED intensity analysis has been performed for both structures, and within experimental error the bond lengths are the same for both structures.

FIGURE 12 Top and side views (in top and bottom sketches of each panel) of adsorption geometries on various metal surfaces. Adsorbates are drawn shaded. Dotted lines represent clean surface atomic positions; arrows show atomic displacements due to adsorption.

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2. Ordered Molecular Monolayers Molecules adsorbed on surfaces may retain their basic molecular identity, bonding as a whole to the substrate. They may dissociate into their constituent atoms, which bond individually to the substrate. Alternatively, molecules may break up into smaller fragments which become largely independent or recombine into other configurations. There are also cases of intermediate character where relatively strong bonding distorts the molecule. One example of ordered molecular monolayers is CO on Pd(111). The (111) surface of fcc metals is the closepacked plane and shows similar ordering for adsorbed √ √CO for a variety of transition metals. That is the ( 3 × 3)R30◦ structure, formed at a coverage of one-third of a monolayer on the (111) faces of Pd, Ni, Pt, Ir, Cu, and Rh. This similarity in ordering is probably due to their surfaces being rather smooth with respect to variations in the CO adsorption energy. Smaller diffusion barriers between different adsorption sites are to be expected, and for large coverages repulsive interactions will be mainly responsible for the arrangement of the adlayer. Figure 14 √ √ shows a schematic representation of this ( 3 × 3)-R30◦ CO structure on Pd(111) and the corresponding observed

LEED pattern. A LEED structural analysis has not been performed for this structure, but supporting evidence using infrared spectroscopy indicates that the CO molecules sit in the threefold hollow sites. As the coverage is increased to one-half of a monolayer the LEED pattern transforms to a c(4 × 2). In this structure the CO molecules all sit in twofold bridge sites. If the adsorption takes place at low temperature (90 K), incrcasing the CO coverage beyond = 0.5 leads to the appearance of a series of LEED patterns arising from hexagonal superstructures, which by a continuous compression and rotation of the c(4 × 2) unit cell lead to a (2 × 2) coincidence pattern at a coverage of = 0.75. These transformations are also shown schematically in Fig. 14.

3. Ordered Organic Monolayers The adsorption characteristics of organic molecules on solid surfaces are important in several areas of surface science. The nature of the chemical bonds between the substrate and the adsorbate and the ordering and orientation of the adsorbed organic molecules play important roles in adhesion, lubrication, and hydrocarbon catalysis.

FIGURE 14 Schematic representation of the CO on Pd(111) system. Structure models and observed LEED structures for the various CO coverages are shown.

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There are many examples in the literature of ordered structures observed by LEED, but only a few of these structures have been calculated from the diffraction beam intensities. However, the ordering characteristics and size and orientation of the unit cells have been determined from the geometry of the LEED patterns. By studying the systematic variation of their shape and bonding characteristics correlations can be made between these properties and their interactions with the metal surfaces. Examples of ordered organic monolayers are normal paraffins on platinum and silver (111) surfaces. If straightchain saturated hydrocarbon molecules from propane (C3 H8 ) to octane (C8 H18 ) are deposited from the vapor phase onto Pt or Ag (111) between 100 and 200 K ordered monolayers are produced. As the temperature is decreased a thick crystalline film can condense. The paraffins adsorb with their chain axis parallel to the platinum substrate, and their surface unit cell increases smoothly with increasing chain length as shown in Fig. 15. Multilayers condensed on top of the ordered monolayers maintain the same orientation and packing found in the monolayers. The monolayer structure determines the growth orientation and the surface structure of the growing organic crystal. This phenomenon is called pseudomorphism, and as a result, the surface structures of the growing organic crystals do not correspond to planes in their reported bulk crystal structures.

FIGURE 15 Observed surface unit cells for n-paraffins on Pt(111).

D. Vibrational Spectroscopy Vibrational frequencies have been used for many years by chemists to identify bonding arrangements in molecules. Each bond has its own frequency, so the vibrational spectrum yields information on the molecular structure. This same information can now be obtained when molecules are adsorbed on single-crystal surfaces and, when combined with another surface-structure-sensitive technique (e.g., LEED), gives a very powerful combination of surface-structure determination. Vibrational spectroscopy also provides significant information on the identity of the surface species; its geometric orientation; the adsorption site; the adsorption symmetry; the nature of the bonding involved; and, in some cases, bond lengths, bond angles, and bond energies. For example, if CO is adsorbed and we observe the C O stretching mode the adsorption is molecular, whereas if the individual modes of metal-C and metal-O are observed then dissociation has taken place. In addition, each of these vibrational modes (C, O, and CO) has a different frequency for each bonding site. The intensities also relate to the concentration of each species on the surface. Electrons scattering off surfaces can lose energy in various ways. One of these ways involves excitation of the vibrational modes of atoms and molecules on the surface. The technique to detect vibrational excitation from surfaces by incident electrons is called high-resolution electron energy loss spectroscopy (EELS). This is the most common type of vibrational spectroscopy used for studying surface–absorbate complexes on single-crystal surfaces. Experimentally, a highly monoenergetic beam of electrons is directed toward the surface, and the energy spectrum and angular distribution of electrons backscattered from the surface is measured. In a typical experiment the kinetic energy of the incident electron beam is in the range of 1–10 eV. Under these conditions the electrons penetrate only the outermost few layers of the crystal, and the backscattered electrons contain only surface information. The incident electrons, monochromatized typically between 3 and 10 meV (∼25–80 cm−1 , 1 meV = 8.065 cm−1 ) and with energy E i , can lose energy hω upon exciting a quantized vibrational mode. These backscattered electrons of energy E i − hω produce the vibrational spectrum. There are several designs of electron monochromator and electron energy analyzers for performing EELS, and one of the most common designs, that of a single-pass 127◦ cylindrical electrostatic deflector, is shown in Fig. 16. A typical EELS spectrum, that of CO on Rh(111) is shown in Fig. 17. The sensitivity of EELS in detecting submonolayer quantities of adsorbates on the sample depends on the particular parameters of the spectrometer, the sample, and the

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FIGURE 16 Schematic diagram of an EELS spectrometer of the single-pass 127◦ cylindrical electrostatic deflector type.

Surface Chemistry

adsorbate. However, typical sensitivity is quite high due in part to the high inelastic electron cross section. A detection limit of ∼10−4 monolayers can be achieved for a strong dipole scatterer such as CO. In addition, unlike many other surface spectroscopies, EELS is also capable of detecting adsorbed hydrogen, although at a lower sensitivity (typically 10−1 –10−2 monolayers). It is a nondestructive technique and can be used to explore the vibrational modes of weakly adsorbed species and those susceptible to beam damage, such as hydrocarbon overlayers. The spectral range accessible with high-resolution EELS is quite large. Typical experiments examine between 200 and 4000 cm−1 , but much larger regions can be analyzed. Vibrational modes as far out as 16,000 cm−1 have been examined. Besides fundamentals, energy losses due to overtones, combination bands, and multiple losses are distinguishable. A distinct advantage of EELS is that electrons can excite the vibrational modes of the surface by three different mechanisms: dipole scattering, impact scattering, and resonance scattering. By analyzing the angular dependence of the inelastically scattered electrons a complete symmetry assignment of the surface–adsorbate complex can be made. The restrictions on the adsorption system are minimal: ordered or disordered overlayers can be examined, as can either well-structured single crystal samples or optically rough surfaces. Hence, chemisorption on evaporated films can be studied, as can the nature of metal overlayer– semiconductor interactions. In addition, coadsorbed atoms and molecules can be studied without difficulty. The major disadvantage of EELS, especially compared to optical techniques, is the relatively poor instrumental resolution, which usually varies between 3 and 10 meV (25–80 cm−1 ). The spectral resolution hinders assignment of vibrations due to individual modes, although peak assignments can be made to within 10 cm−1 . The high sensitivity of EELS coupled with the advantages discussed above has encouraged rapid development and use of this technique, despite resolution limitations, such that it has now been used to study hundreds of adsorptions systems. As an example of the type of surface chemistry that can be followed using EELS, Fig. 18 shows a series of EELS spectra of the adsorption and thermal decomposition of ethylene on Rh(111).

III. THERMODYNAMICS OF SURFACES FIGURE 17 Electron energy loss spectrum of CO adsorbed on Rh(111). The loss peak at 468 cm−1 is due to the Rh–CO symmetric stretch, and that at 2036 cm−1 is due to the C–O symmetric stretch. The spectrum was recorded at a resolution of 30 cm−1 .

A. Introduction The environment of atoms in a surface is substantially different to that of atoms in the bulk of the solid. Surface

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Therefore, E S is the excess of total energy that the solid has over E 0 , which is the energy that the system would have if the surface were in the same thermodynamic state as the interior. The other surface thermodynamic functions are defined similarly, for example, the specific surface free energy G S is given by the following: G S = H S − T SS, (5) S S where H and S are the specific surface enthalpy and entropy, respectively. B. Surface Tension in a One-Component System Creating a surface involves breaking chemical bonds and removing neighboring atoms, and this requires work. Under conditions of constant temperature and pressure at equilibrium, the surface work δW S is given by the following: S δWT,P = d(G S A), (6) where A is the increase in the surface area. If G S is independent of the surface area, surface work is as follows: S δWT,P = G S d A.

FIGURE 18 EELS spectra of the adsorption and decomposition of ethylene (C2 H4 ) on Rh(111) at (a) 77 K, (b) 220 K, and (c) 450 K.

atoms are surrounded by fewer nearest neighbors than bulk atoms, and these neighbors are not distributed evenly around the surface atoms. An atom in the interior experiences no net forces, but these forces become unbalanced at the surface. Consequently the thermodynamic parameters used to describe surfaces are defined separately from those that characterize the bulk phase. The specific surface energy E S , the energy per surface area, is related to the total energy E by the following equation: E = NE 0 + AE S ,

(4)

where A is the surface area of a solid composed of N atoms, and E 0 is the energy of the bulk phase per atom.

(7)

In a one-component system the specific surface free energy, G S , is frequently called the surface tension or surface pressure and is denoted by γ . Here γ may be viewed as a pressure along the surface opposing the creation of new surface. It has dimensions of force per unit length (dynes per centimeter, ergs per square centimeter, or newtons per meter). The surface tension γ for an unstrained phase is also equal to the increase of the total free energy of the system per unit increase of the surface area as follows: ∂G S γ =G = . (8) ∂ A T,P The free energy of formation of a surface is always positive, since work is required in creating a new surface, which increases the total free energy of the system. In order to minimize their free energy solids or liquids assume shapes in equilibrium with the minimum exposed surface area as possible. For example, liquids tend to form a spherical shape and crystal faces which exhibit the closest packing of atoms tend to be the surfaces of lowest free energy of formation and thus the most stable. Surface tension is one of the most important thermodynamic parameters characterizing the condensed phase. Table II lists selected experimentally determined values of surface tensions of liquids and solids that were measured in equilibrium with their vapor. Comparing the surface tension values of metals and oxides in Table II it can be seen that oxides have in general

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Surface Chemistry TABLE II Selected Values of Surface Tension of Solids and Liquids Material He (1) N2 (1) Ethanol (1) Water Benzene n-Octane Carbon tetrachloride Bromine W (s) Nb (s) Au (s) Ag (s) Ag (l) Fe (s) Fe (l) Pt (s) Cu (s) Ni (s) Hg (l) NaCl (s) KCl (s) CaF2 (s) MgO (s) SiO2 (s) Al2 O3 (s) Polytetrafluoroethylene Polyethylene Polystyrene Poly(vinyl chloride)

γ (ergs cm−2 )

T (◦ C)

0.308

−270.5

9.71 22.75 72.75 28.88 21.80 26.95 41.5 2900 2100 1410 1140 879 2150 1880 2340 1670 1850 487 227 110

−195 20 20 20 20 20 20 1727 2250 1027 907 1100 1400 1535 1311 1047 1250 16.5 25 25

450 1200

−195 25

307

1300

690 18.5 31 33 39

2323 20 20 20 20

a low surface tension. Therefore, a reduction in the total free energy of the system can be achieved by oxidation of the surface and a uniform oxide layer covering the surface is expected under conditions near thermodynamic equilibrium. Similarly, deposition and growth of a metal film on a metallic substrate of higher surface tension should yield a uniform layer that is evenly spread to completely cover the substrate surface. Likewise, a very poor spreading of the film is expected on deposition of a metal of high surface tension on a low-surface-tension substrate. This latter condition results in “island growth” and the deposited highsurface-tension metal will grow as whiskers to expose as much of the low-surface-tension substrate during the growth as possible. These, of course, are surface thermodynamic predictions and may be overidden by the presence of impurities at the surface or difficulties of nucleation. Since atomic bonds must be broken to create surfaces, it is expected that the specific surface free energy will be

related to the heat of vaporization, which is related to the energy input necessary to break all the bonds of atoms in the condensed phase. In fact, it has been found experimentally that the molar surface free energy of a liquid metal can be estimated by the following: γlm = 0.15 Hvap ,

(9)

where Hvap is the heat of vaporization of the liquid, and the molar surface free energy of a solid metal is given by the following: γsm = 0.16 Hsub ,

(10)

where Hsub is the heat of sublimation of the solid. For other materials, oxides, or organic molecules, such a simple relationship does not work due to the complexity of bonding and the rearrangement or relaxation of surface atoms at the freshly created surfaces. C. Surface Tension of Multicomponent Systems In many important surface phenomena, such as heterogeneous catalysis or passivation of the surface by suitable protective coatings, the chemical composition of the topmost layer controls the surface properties and not the composition in the bulk. Thus, investigations of the physical– chemical parameters that control the surface composition are of great importance. One of the major driving forces for the surface segregation of impurities from the bulk and for the change of composition of alloys and other multicomponent systems is the need to minimize the surface free energy of the condensed phase system. The change of the total free energy of a multicomponent system can be expressed with the inclusion of the surface term as follows: dG = S dT + V d P + γ d A + µi dn i , (11) i

where µi is the chemical potential of the ith component and dn i is the change in the number of moles of the ith component. At constant temperature and pressure, Eq. (11) can be rewritten as follows: dG T,P = γ d A − µi dn i , (12) i

where the minus sign indicates the decrease of the bulk concentration of the ith component. Comparing this equation with Eq. (8), the surface tension γ is no longer equal to the specific surface free energy per unit area for a multicomponent system. Using simple arguments in which number of moles of the condensed phase are transferred to the freshly created surface, the Gibbs equation can be derived as follows: dγ = −S S dT = i dµi , (13)

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where i is the excess number of moles of compound i at the surface. Just like the free energy relations for bulk phases, the Gibbs equation predicts changes in surface tension as a function of experimental variables such as temperature and surface concentration of various components. As a result of the Gibbs equation, the surface composition in equilibrium with the bulk for a multicomponent system can be very different from the bulk composition. As an example we discuss the surface composition of an ideal binary solution. For such a solution at a constant temperature the Gibbs equation can be expressed as follows:

(l + m) X 2b (γ1 − γ2 )a X 2S exp = exp RT RT X 1S X 1b ×

X 1b

2

2 l S 2 S 2 , − X 2b + X2 − X1 RT (17)

where is the regular solution parameter and is directly related to the heat of mixing Hm by the following: =

Hm . X 1b l − X 1b

where γ1 is the surface tension of the pure component and a the surface area occupied by one mole of component 1. Perfect behavior is assumed, i.e., the surface areas occupied by the molecules in the two different components are the same (a1 = a2 = a); X 1S and X 1b are the atom fractions of component 1 in the surface and in the bulk, respectively. It is also assumed that the surface consists of only the topmost atomic layer. For a two-component system, Eq. (15) can be rewritten in the following form: X 1S X 1b (γ2 − γ1 )a = b exp , (16) RT X 2S X2

Here l is the fraction of nearest neighbors to an atom in the plane and m is the fraction of nearest neighbors below the plane containing the atom. In this approximation the surface composition becomes a fairly strong function of the heat of mixing, its sign, and its magnitude in addition to the surface tension difference and temperature. Auger electron spectroscopy (AES) and ion scattering spectroscopy (ISS) are two experimental techniques which are most frequently used for quantitative determination of the surface composition. Figure 19 shows the surface atom fraction of gold, determined by AES and ISS, plotted as a function of the bulk atom fraction for the Ag–Au system. The solid line gives the calculated surface composition using the regular solution model and the dashed line indicates the curve that would be obtained in the absence of surface enrichment. The regular solution model appears to overestimate somewhat the surface segregation in this case, although the surface is clearly enriched in silver. Table III lists several binary alloy systems that have been investigated experimentally by AES or ISS and

where X 1S , X 2S , X 1b , X 2b have their meaning defined above; γ1 and γ2 are the surface tensions of the pure components; and the other symbols have their usual meaning. From Eq. (16), it can be seen that the component that has the smaller surface tension will accumulate on the surface. Equation (16) also predicts that the surface composition of ideal solutions should be an exponential function of temperature. While the bulk composition of a multicomponent system is little affected by temperature, the surface concentration of the constituents may change markedly. The surface segregation of one of the constituents becomes more pronounced the larger the difference in surface tensions between the components that make up the solution. Surface segregation is expected to be prevalent for metal solutions, since metals have the highest surface tensions. In reality, however, metallic alloys are not ideal solutions since they have some finite heat of mixing. In such a case the surface composition can be approximated in the regular solution monolayer approximation

FIGURE 19 Surface phase diagram of Au–Ag alloy.

dγT = −1 dµ1 − 2 dµ2

(14)

and it has been shown that the surface tension of component 1 in an idea dilute solution is given by the following: γ = γ1 + (RT /a) ln X 1S X 1b , (15)

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Surface Chemistry TABLE III Surface Composition of Alloys: Experimental Results and Predictions of the Regular Solution Model Segregating constituent Alloy system

Predicted regular solution

Ag–Pd Ag–Au Au–Pd Ni–Pd Fe–Cr Au–Cu

Ag Ag Au Pd Cr Cu

Cu–Ni Au–Ni Au–Pt Pb–In Au–In Al–Cu Pt–Sn Fe–Sn Au–Sn

Cu Au Au Pb In Al Sn Sn Sn

Experimental Ag Ag Au Pd Cr Au, none, or Cu depending on composition Cu Au Au Pb In Al Sn Sn Sn

the segregating components that were experimentally observed and also predicted by the regular solution model. The agreement is certainly satisfactory. It appears that for binary metal alloy systems that exhibit regular solution behavior there are reliable methods to predict surface composition. So far we have discussed the surface composition of multicomponent systems that are in equilibrium with their vapor or in which clean surface–bulk equilibrium is obtained in ultrahigh vacuum. In most circumstances, however, the surface is covered with a monolayer of adsorbates that frequently form strong chemical bonds with the surface atoms. This solid–gas interaction can markedly change the surface composition in some cases. For example, carbon monoxide, when adsorbed on the surface of a Ag–Pd alloy, forms much stronger bonds with Pd. While the clean surface is enriched with Ag, in the presence of CO, Pd is attracted to the surface to form strong carbonyl bonds. When the adsorbed CO is removed, the composition returns to its original Ag-enriched state. Nonvolatile adsorbates, such as carbon or sulfur, may have a similar influence on the surface composition as long as their bonding to the various constituents of the multicomponent system is different. Adsorbates should therefore be viewed as an additional component of the multicomponent system. A strongly interacting adsorbate converts a binary system to a ternary

system. As a result, the surface composition may markedly change with changing ambient conditions. The mechanical properties of solids, embrittlement, and crack propagation, among others, depend markedly on the surface composition. These studies indicate that the surface composition and the mechanical properties of structural steels may change drastically when the ambient conditions are changed from reducing to oxidizing environments. D. Equilibrium Shape of a Crystal or a Liquid Droplet In equilibrium the crystal will take up a shape that corresponds to a minimum value of the total surface free energy. In order to have the equilibrium shape, the integral γ d A over all surfaces of the crystal must be a minimum. Crystal faces that have high atomic density have the lowest surface free energy and are therefore most stable. The plot of the surface free energy as a function of crystal orientation is called the γ plot. Solids and liquids will always tend to minimize their surface area in order to decrease the excess surface free energy. For liquids, therefore, the equilibrium surface becomes curved where the radius of curvature will depend on the pressure difference on the two sides of the interface and on the surface tension as follows: (Pin − Pext ) = 2γ /r,

(18)

where Pin and Pext are the internal and external pressures, respectively, and r is the radius of curvature. In equilibrium a pressure difference can be maintained across a curved surface. The pressure inside the liquid drop or gas bubble is higher than the external pressure, because of the surface tension. The smaller the droplet or larger the surface tension, the larger is the pressure difference that can be maintained. For a flat surface r = ∞, and the pressure difference normal to the interface vanishes. Let us now consider how the vapor pressure of a droplet depends on its radius of curvature r . We obtain the following: ln (P/P0 ) = 2γ Vin /RT r,

(19)

where Vin is the internal volume. This is the well-known Kelvin equation for describing the dependence of the vapor pressure of any spherical particle on its size. Small particles have higher vapor pressures than larger ones. Similarly, very small particles of solids have greater solubility than large particles. If we have a distribution of particles of different sizes, we will find that the larger particles will grow at the expense of the smaller ones. Nature’s way to avoid the sintering of small particles that would occur according to the Kelvin equation is to produce a system

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with particles of equal size. This is the world of colloids where particles are of equal size and therefore stabilized and are usually charged either all negative or all positive or to repel each other by long-range electrostatic forces. Milk and our blood are only two examples of systems that contain colloids. E. Adhesion and the Contact Angle Let us turn our attention to the interfacial tension, that is, the surface tension that exists at the interface of two condensed phases. Let us place a liquid droplet on a solid surface. The droplet either retains its shape and forms a curved surface or it is spread evenly over the solid. These two conditions indicate the lack of wetting or wetting of the solid by the liquid phase, respectively. The contact angle between the solid and the liquid, to a large extent, permits us to determine the interfacial tension between the solid and the liquid. The contact angle is defined by Fig. 20. If the contact angle is large ( approaching 90◦ ), the liquid does not readily wet the solid surface. If approaches zero, complete wetting of the solid surface takes place. For larger than 90◦ the liquid tends to form spherical droplets on the solid surface that may easily run off, i.e., the liquid does not wet the solid surface at all. Remembering that the surface tension always exerts a pressure tangentially along a surface, the surface free energy balance between the surface forces acting in opposite directions at the point where the three phases solid, liquid, and gas meet is given by the following: cos = (γsg − γsl )/γlg .

(20)

Here γlg is the interfacial tension at the liquid–gas interface and γsg and γsl are the interfacial tensions between the solid–gas and the solid–liquid interfaces, respectively. Knowing γlg and the contact angle in equilibrium at the solid–liquid–gas interface, we can determine the difference γsg − γsl but not their absolute values. Since the wet-

FIGURE 20 Definition of the contact angle between a liquid and solid and the balance of surface forces at the contact point among the three phases (solid, vapor, and liquid).

ting ability of the liquid at the solid interface is so important in practical problems of adhesion or lubrication, there is a great deal of work being carried out to determine the interfacial tensions for different combinations of interfaces. The usefulness of a lubricant is determined by the extent to which it wets the solid surface and maintains complete coverage of the surface under various conditions of use. The strength of an adhesive is determined by the extent to which it lowers the surface free energy by adsorption on the surface. The work of adhesion is defined as follows: WAs = γ1,0 + γs,0 − γsl ,

(21)

where γ1,0 and γs,0 are the surface tensions in vacuum of the liquid and solid, respectively. In general, solids and liquids that have large surface tension form strong adhesive bonds, i.e., have large work of adhesion. The work of adhesion is in the range of 40–150 ergs/cm2 for solid–liquid pairs of various types. Organic polymers often make excellent adhesives because of the large surface area covered by each organic molecule. The adhesive energy per mole is much larger than that for adhesion between two metal surfaces or between a liquid and a solid metal because of the many chemical bonds that may be formed between the substrate and the adsorbed organic molecule. F. Nucleation Another important phenomenon that owes its existence to positive surface free energy is nucleation. In the absence of a condensed phase, it is very difficult to nucleate one from vapor atoms because the small particles that would form have a very high surface area and dispersion and, as a result, a very large surface free energy. The total energy of a small spherical particle has two major components: its positive surface free energy, which is proportional to πr 2 γ , where r is the radius of the particle, and its negative free energy of formation of the particle with volume V . The volumetric energy term is proportional to −r 3 ln (P/Peq ), where P is the pressure over the system and Peq is the equilibrium vapor pressure: 4πr 3 P G(total) = − RT ln + 4πr 2 γ , (22) 3Vm Peq where Vm is the molar volume of the forming particle. Initially, the atomic aggregate is very small and the surface free energy term is the larger of the two terms. In this circumstance a condensate particle cannot form from the vapor even at relatively high saturation (P > Peq ). Similarly, a liquid may be cooled below its freezing point without solidification occurring. Above a critical size of the spherical particles the volumetric term becomes larger and dominates since it

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FIGURE 22 One-dimensional potential energy of an adatom in a physisorbed state on a planar surface as a function of its distance z from the surface.

FIGURE 21 Free energy of homogeneous nucleation as a function of particle size.

decreases as ∼r 3 , while the surface free energy term increases only as ∼r 2 . Therefore, a particle that is larger than this critical size grows spontaneously at P > Peq . This is shown in Fig. 21. Because of the difficulty of obtaining this critical size, which involves as many a 30–100 atoms or molecules, homogeneous nucleation is very difficult indeed. To avoid this problem we add to the system particles of larger than critical size that “seed” the condensation or solidification. The use of small particles to precipitate water vapor in clouds to start rain and the use of small crystallites as seeds in crystal growth are two examples of the application of heterogeneous nucleation. G. Physical and Chemical Adsorption The concepts of physical adsorption (physisorption) and chemical adsorption (chemisorption) were introduced above. The nature of the two classifications is linked to the heat of adsorption, Hads , which is defined as the binding energy of the adsorbed species. Physical adsorption is caused by secondary attractive forces (van der Waals) such as dipole–dipole interaction and induced dipoles and is similar in character to the condensation of vapor molecules onto a liquid of the same composition. The interaction can be described by the onedimensional potential energy diagram shown in Fig. 22.

An incoming molecule with kinetic energy E k must lose at least this amount of energy in order to stay on the surface. It loses energy by exciting lattice phonons in the substrate, for example, and the molecule comes to equilibrium in a state of oscillation in the potential well of depth equal to the binding energy or adsorption energy E a = Hads . In order to leave the surface (desorb) the molecule must acquire enough energy to surmount the potential-energy barrier E a . The desorption energy is equal to the adsorption energy. The binding energies of physisorbed molecules are typically ≤15 kcal mol−1 . Chemisorption involves chemical bonding; it is similar to a chemical reaction and involves transfer of electronic charge between adsorbent and adsorbate. The most extreme form of chemisorption occurs when integral numbers of electrons are transferred, forming a pure ionic bond. More usually there is an admixture of the wave functions of the valence electrons of the molecule with the valence electrons of the substrate into a new wave function. The electrons responsible for the bonding can then be thought of as moving in orbitals between substrate and adatoms and a covalent bond has been formed. Two examples of the potential energy diagrams for chemisorption are shown in Fig. 23. Some of the impinging molecules are accommodated by the surface and become weakly bound in a physisorbed state (also called a precursor state) with binding energy E p . During their stay time in this state, electronic or vibrational processes can occur which allow them to surmount the energy barrier, and electron exchange occurs between the adsorbate and substrate. The molecule, or adatom in the case of dissociative chemisorption, now finds itself in a much deeper well; it is chemisorbed. Figure 23a shows the case in which the energy barrier for chemisorption is less than E p , so there is no overall activation energy to chemisorption. Figure 23b illustrates

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the case of adsorption isotherms for physical adsorption, to determine the surface area of the adsorbing solid. Consider a uniform surface with a number n 0 of equivalent adsorption sites. The ratio of the number of adsorbed atoms or molecules n to n 0 is defined as the coverage, = n/n 0 . Atoms or molecules impinge on the surface from the gas phase, where they establish a surface concentration [n a ]s (molecules per square centimeter). Assuming that only one type of species of concentration [n a ]g (molecules per cubic centimeter) exists in the gas phase the adsorption process can be written as follows: FIGURE 23 One-dimensional potential energy curves for dissociative adsorption through a precursor or physisorbed state: (a) adsorption into the stable state with no activation energy and (b) adsorption into the chemisorption well with activation energy E ∗ .

the case in which there is an overall activation energy E ∗ to chemisorption. In the former case the activation energy for desorption E d is equal to the heat of adsorption, while in the latter case the heat of adsorption is given by the difference between the heat of desorption and the activation energy. The occurrence of an activation energy to chemisorption is by far the exception rather than the rule. From these considerations it is expected that to a first approximation physisorption will be nonspecific, any gas will adsorb on any solid under suitable circumstances. However, chemisorption will show a high degree of specificity. Not only will there be variations from metal surface to metal surface, as would be expected from the differences in chemistries between the metals, but also different surface planes of the same metal may show considerable differences in reactivity toward a particular gas. H. Adsorption Isotherms An adsorption isotherm is the relationship at constant temperature between the partial pressure of the adsorbate and the amount adsorbed at equilibrium. Similarly an adsorption isobar expresses the functional relationship between the amount adsorbed and the temperature at constant pressure, and an adsorption isostere relates the equilibrium pressure of the gaseous adsorbate to the temperature of the system for a constant amount of adsorbed phase. Usually it is easiest from an experimental viewpoint to determine isotherms. The coordinates of pressure at the different temperatures for a fixed amount adsorbed can then be interpolated to construct a set of isosteres, and similarly to obtain an isobaric series. Adsorption isotherms can be used to determine thermodynamic parameters that characterize the adsorbed layer (heats of adsorption and the entropy and heat capacity changes associated with the adsorption process), and in

k Agas Asurface k

and the net rate of adsorption as F (molecules cm−2 sec−1 ) = k[n a ]g − k [n a ]s ,

(23)

where k and k are the rate constants for adsorption and desorption, respectively. Starting with a nearly clean surface far from equilibrium, the rate of desorption may be taken as zero and Eq. (23) becomes the following: F (molecules cm−2 sec−1 ) = k[n a ]g ,

(24)

where k, derived from the kinetic theory of gases, equals α(RT /2π M)1/2 cm sec−1 , α is the adsorption coefficient, and M the molecular weight of the impinging molecules. The surface concentration [n a ]s under these conditions is the product of the incident flux F and the surface residence time τ : [n a ]s = Fτ.

(25)

Here τ is the surface residence time, given by: τ = τ0 exp( Hads /RT ).

(26)

Replacing [n a ]g by the pressure using the ideal gas law, Eq. (25) can be rewritten as follows: αPN A Hads [n a ]s = . (27) τ0 exp (2π MRT)1/2 RT The simplest adsorption isotherm is obtained from Eq. (27), which can be rewritten as = k P where k =

αNA 1 Hads . τ exp 0 n 0 (2π MRT )1/2 RT

(28)

(29)

The coverage is proportional to the first power of the pressure at a given temperature provided that there are an unlimited number of adsorption sites available and Hads does not change with coverage. The isotherm of Eq. (28) is unlikely to be suitable to describe the overall adsorption process, but the Langmuir isotherm is a simple modification which represents a more

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real situation. The Langmuir isotherm assumes that adsorption is terminated on completion of one molecular adsorbed gas layer (monolayer) by asserting that any gas molecule that strikes an adsorbed atom must reflect from the surface. All the other assumptions used to derive Eq. (28) are maintained (i.e., homogeneous surface and noninteracting adsorbed species). If [n 0 ] is the surface concentration of a completely covered surface, the number of surface sites available for adsorption, after adsorbing [n a ]s molecules is [n 0 ] − [n a ]s . Of the total flux incident on the surface, a fraction ([n a ]s /[n 0 ])F will strike molecules already adsorbed and, therefore, be reflected. Thus, a fraction (1 − [n a ]s /[n 0 ])F of the total incident flux will be available for adsorption. Equation (25) should then be modified as follows: [n a ]s [n a ]s = 1 − Fτ, (30) [n 0 ] which can be rearranged to give [n a ]s =

[n 0 ]Fτ [n 0 ]k P = [n 0 ] + Fτ [n 0 ] + k P

(31)

from which =

k P , 1 + k P

(32)

where k = k/[n 0 ]. Equation (32) is the Langmuir isotherm. The adsorption of CO on Pd(111) obeys the Langmuir isotherm, and typical isotherms from this system are shown in Fig. 24. It can readily be shown that in the case of dissociative adsorption the Langmuir isotherm becomes

2 1 P= k 1− or =

(k P)1/2 . 1 + (k P)1/2

(33)

A clear weakness of the Langmuir model is the assumption that the heat of adsorption is independent of coverage. Several other isotherms have been developed which are all modifications of the Langmuir model. For example, the Temkin isotherm can be derived if a linearly declining heat of adsorption is assumed, i.e., H = H0 (1−β), where H0 is the initial enthalpy of adsorption. The isotherm is =

RT ln A P, β H0

(34)

where A is a constant related to the enthalpy of adsorption. The possibility of multilayer adsorption is envisaged in the Brunauer–Emmett–Teller (BET) isotherm. The assumption is made that the first layer is adsorbed with a heat of adsorption H1 and the second and subsequent layers are all characterized by heats of adsorption equal to the latent heat of evaporation, HL . By considering the dynamic equilibrium between each layer and the gas phase the BET isotherm is obtained, p 1 c−1 p . = + V ( p0 − p) Vm c Vm c p0

(35)

In this equation V is the volume of gas adsorbed, p the pressure of gas, p0 the saturated vapor pressure of the liquid at the temperature of the experiment, and Vm the

FIGURE 24 Adsorption isotherms for CO on Pt(111) single-crystal surfaces.

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volume equivalent to an adsorbed monolayer. The BET constant c is given by the following: c = exp(H1 − HL )/RT.

(36)

The BET equation owes its importance to its wide use in measuring surface areas, especially of films and powders. The method followed is to record the uptake of an inert gas (Kr) or nitrogen at liquid nitrogen temperature (−195.8◦ C). A plot of p/V ( p0 − p) versus p/ p0 , usually for p/ p0 up to about 0.3, yields Vm , the monolayer uptake. This value is expressed as an area by assuming that ˚ and 25.6 A ˚ for the area per molecule for nitrogen is 16.2 A krypton. In general, the BET isotherm is most useful for describing physisorption for which H1 and HL are of the same order of magnitude while the preceding isotherms are more useful for chemisorption. It is worth noting that the BET isotherm reduces to the Langmuir isotherm when H1 HL . I. Heat of Adsorption An important physical–chemical property that characterizes the interaction of solid surfaces with gases is the bond energy of the adsorbed species. The determination of bond energy is usually made indirectly by measuring the heat of adsorption (or heat of desorption) of the gas. The heat of adsorption can be determined readily in equilibrium by measuring several adsorption isotherms. The Clausius– Clapeyron equation ∂(ln P) Hads = (37) ∂T RT 2 can be integrated to give P1 − Hads 1 1 ln . = − P2 R T1 T2 Measuring the adsorption isotherm at two different temperatures, provided that proper equilibrium is established between the adsorbed and gas phase, yields the heat of adsorption. The heat of adsorption can also be obtained by direct calorimetry. The method most commonly used consists of measuring the temperature rise caused by the addition of a known amount of gas to a film of the metal prepared by evaporation in vacuo. This measurement will yield the differential heat of adsorption qd at the particular value of . The differential heat of adsorption is related to the isosteric heat of adsorption by the following: q = qd + RT ;

(38)

the difference is only RT, which is within experimental error.

The last, and most common, method of determining the heat of adsorption is a kinetic method called temperature programmed desorption (TPD). The method is as follows. The sample is cleaned in ultrahigh vacuum and a gas is allowed to adsorb on the surface at known pressures while the surface is kept at a fixed temperature. The sample is then heated at a controlled rate, and the pressure changes during the desorption of the molecules are recorded as a function of time and temperature. The pressure–temperature profile is usually referred to as the desorption spectrum. The desorption rate F(t) is commonly expressed as follows: E des , (39) F(t) = v f (σ ) exp − RT where v is the preexponential factor and f (σ ) an adsorbate concentration-dependent function. The various procedures for determining these parameters are well described in the literature. Assuming that v and E des are independent of the adsorbate concentration σ and t, E des can be obtained for zeroth-, first-, and second-order desorption, respectively, as follows: E0 ν0 E (40) = exp − R σa RTp E1 v1 E1 (41) = exp − RTp2 α RTp E2 E2 v2 σ exp − , = RTp2 α RTp

(42)

where Tp is the temperature at which a desorption peak is at a maximum and σ is the initial adsorbate concentration. The subscripts 0, 1, and 2 denote the zeroth-, first-, or second-order desorption processes; α is a constant of proportionality for the temperature rise with time, usually T = T0 + αt; that is, the temperature of the sample is raised linearly with time. As seen from the equations, Tp is independent of σ for the first-order desorption process. Alternatively, Tp is increased or is decreased with σ0 for the zeroth- and second-order process, respectively. Equations (40)–(42) allow us to determine the activation energy and the preexponential factor and also to distinguish between zeroth-, first-, and second-order desorption processes from the measurements of the dependence of the peak temperatures on initial adsorbate concentrations and heating rate α. A typical TPD spectrum is shown in Fig. 25. The bond energy Hbond is readily extracted from the heat of adsorption. In the case of the chemisorption of a diatomic molecule X2 onto a site on a uniform solid surface M the molecule may adsorb without dissociation

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FIGURE 26 Isoteric heat of adsorption for CO on Pd(111) crystal face as a function of coverage.

FIGURE 25 Typical thermal desorption spectra of CO from a Pt(553) stepped crystal face as a function of coverage. The two peaks are indicative of CO bonding at step and terrace sites. The higher temperature peak corresponds to CO bound at step sites.

to form MX2 . In this case, the heat of adsorption, Hads is defined as the energy needed to break the M X2 bond Hads

MX2(ads) −−→ M + X2(gas) If the molecule adsorbs dissociatively, the heat of adsorption is defined as follows: Hads

2MX(ads) −−→ 2M + X2(gas) The energy of the surface chemical bond is then given by Hbond (MX2 ) = Hads for associative adsorption or Hads + DX2 Hbond (M X ) = 2 for dissociative adsorption, where DX2 is the dissociation energy of the X2 gas molecule. The heat of adsorption is not a constant, quantity for a particular adsorbate–substrate system; there are several factors which affect the value of Hads . First, the heat of adsorption can change markedly with the coverage of the adsorbed pahse. An example of this is shown in Fig. 26 for CO on a Pd(111) surface. Decreasing values of Hads

with increasing adsorbate coverage are commonly observed due to repulsive adsorbate–adsorbate interactions. Second, the surface is heterogeneous by nature. There are many sites where the adsorbed species have different binding energies. Perhaps the most striking effect is that for adsorption on stepped and kinked platinum and nickel single crystal surfaces where molecules dissociate in the presence of these surface irregularities while they remain intact on the smooth low-Miller-index surfaces. If a polycrystalline surface is utilized for chemisorption studies instead of a structurally well-characterized single-crystal surface the measured Hads will be an average of adsorption at the various binding sites. In fact, even on the same crystal surface molecules may occupy several different adsorption sites with different coordination numbers and rotational symmetries, and each site may exhibit a different binding energy and therefore a different heat of chemisorption. For example, on the (111) face of fcc metals the adsorbates may occupy a three-fold site, a twofold bridge site, or an on-top site. Figure 27 shows the measured heats of adsorption of CO or single-crystal surfaces for many different transition metals while Fig. 28 shows the heats of adsorption of CO on polycrystalline transition metal surfaces. The heats of chemisorption on single-crystal planes indicate the presence of binding sites on a given surface which differ by ∼20 kcal mol−1 . It is not possible to identify one value of the heat of chemisorption of an adsorbate on a given transition metal unless the binding state is specified or it is certain that only one binding state exists. A polycrystalline surface however exhibits all the adsorption sites of the faces from which it is composed. Since these sites are present simultaneously heats of chemisorption for these surfaces represent an average of the binding energies of the different surface sites. As a result the measured heats of adsorption of Fig. 28 do not show the large structural variations that can be seen in Fig. 27. The adsorbate may also change bonding as a function of temperature as well as the adsorbate concentration. For example, oxygen may be molecularly adsorbed at low temperatures while it dissociates at higher temperatures.

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FIGURE 27 Heats of adsorption of CO on single-crystal surfaces of transition metals.

IV. ELECTRICAL PROPERTIES OF SURFACES A. Introduction Many of the physical and chemical properties of solid surfaces are directly influenced by the concentration of mobile charge carriers (electrons and diffusing ions). The concentration of these free charge carrier varies widely

FIGURE 28 Heats of adsorption of CO on polycrystal-line transition-metal surfaces.

for materials of different types. Metals, which are good conductors of electricity with resistivities in the 10−4 m range, have large free electron concentrations; almost every atom contributes one electron to the lattice as a whole. For insulators, with a resistivity of 109 m, and semiconductors with intermediate values, often less than 1 of every 106 atoms may contribute a free electron. The temperature dependence of the carrier concentration and the conductivity may be different for different materials depending on the mechanism of excitation by which the mobile charge carriers are created. Under incident radiation or bombardment by an electron beam surfaces emit photons, electrons, or both. The emission properties of solid surfaces differ widely, just as their mechanisms or relaxation after excitation by highenergy radiation differ. Many surface-sensitive experimental techniques providing information related to the electronic properties of surfaces are based on these processes, for example, Auger electron spectroscopy (AES), X-ray photoelectron spectroscopy (XPS), and ultraviolet photoelectron spectroscopy (UPS). These are discussed below. The underlying reason for the differences of the conductivity mechanisms and emission properties on the surfaces of the different materials lies in the differences in their electronic band structure. The band structure model of solids has been successful in explaining many solidstate properties, and we may apply it with confidence in studies of solid surfaces. There are many excellent textbooks on the subject of solid-state physics giving detailed descriptions of the band theory of solids, and a description is not presented here. In the following section a basic understanding of electron bands is assumed. B. The Energy Level Diagram For many purposes, in analyzing the electrical properties of metals or semiconductors, we are not concerned with the detailed shape of the electronic bands. We may conveniently represent schematically the electronic bands by straight lines where the potential energy of the electron near the top of the valence band and at the bottom of the conduction band is plotted against distance x through the crystal starting from the surface (x = 0). The energy gap represents the minimum potential energy difference between the two bands. In this type of diagram the electron energy increases upward and the energy of the positive hole increases downward, as indicated in Fig. 29. For a homogeneous crystal the bands may be horizontal, as shown in this figure. At the surface the bands may vary in energy with respect to their value in the bulk of the solid since the free carrier concentrations at the surface may be different from those in the bulk of the crystal.

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FIGURE 29 Energy-level diagram as a function of distance x from the surface (x = 0).

C. Surface Dipole and Surface Space Charge The anisotropic environment of surface atoms not only gives rise to such processes as surface relaxation and surface reconstruction but also to a redistribution of charge density. For a metal this redistribution can be explained as follows. In the bulk of a metal each electron lowers its energy by “pushing” the other electrons aside to form an “exchange correlation hole.” This attractive interaction Vexch is lost when the electron leaves the solid, so there is a sharp potential barrier Vs at the surface. In a quantum mechanical description, the electrons are not totally trapped at the surface and there is a finite probability for them to spread out into the vacuum. This is depicted in Fig. 30. This charge redistribution induces a surface dipole Vdip that modifies the barrier potential. The work function φ (which will be discussed in detail below) is the minimum energy necessary to remove an electron at the Fermi energy E F from the metal into the vacuum. The magnitude of this induced surface dipole is different at various sites on the heterogeneous metal surface. For example, a step site on a tungsten surface has dipole of 0.37 Debye (D) per edge atom as measured by work function studies. At a tungsten adatom on the surface there is a dipole moment of 1 D. At semiconductor and insulator surfaces the separation of negative and positive charges leads to the formation of a space-charge region. This space-charge region near the surface is formed by the accumulation or depletion

FIGURE 30 Charge density oscillation and redistribution at a metal–vacuum interface.

Surface Chemistry

of charge carriers in the surface with respect to the bulk carrier concentration. Such a space charge may also be induced by the application of an external electric field or by the presence of a charged layer on the surface such as adsorbed ions or electronic surface states which act as a source or sink of electrons. The height of the surface potential barrier Vs and its distance of penetration into the bulk, d, depend on the concentration of mobile charge carriers in the surface region. It can be shown that 2εε0 Vs 1/2 d≈ , (43) en e (bulk) where ε is the dielectric constant in the solid, ε0 the permittivity of free space, and n e (bulk) the bulk carrier concentration. The higher the free carrier concentration in the material, the smaller is the penetration depth of the applied field into the medium. Using a typical value of ε = 16, for electron concentrations of 1017 cm−3 or larger, the space charge is restricted to distances on the order of one atomic layer or less. This is due to the large free carrier density screening the solid from the penetration of the electrostatic field caused by the charge imbalance. In most metals almost every atom contributes one free valence electron and since the typical atomic density is of the order of 1022 cm−3 the free carrier concentration in metals is in the range of 1020 –1022 cm−3 . Thus, Vs and d are so small that they can usually be neglected. For semiconductors, or insulators on the other hand, typical free carrier concentrations at room temperature are in the range of 1010 –1016 cm−3 . Therefore, at the surfaces of these materials, there is a space-charge barrier of appreciable height (several electron volts) and penetration depth that could extend over ˚ into the bulk. This thousands of atomic layers (≈104 A) is the reason for the sensitivity of semiconductor devices to ambient changes that affect the space-charge barrier height. There is an induced electric field at the surface under most experimental conditions due to the adsorption of gases or because of the presence of electronic surface states. The electronic and many other physicochemical properties of semiconductor and insulator surfaces depend very strongly on the properties of the space charge. For example, the conduction of free carriers across the solid or along its surface could become space-charge-limited. The rate of charge transfer from the solid to the adsorbed gas, which results in chemisorption or chemical reaction, can become limited by the transfer rate of electrons over the space-charge barrier. When the energy level diagram was introduced, it was assumed that the electron energy levels remained unchanged right to the surface (x = 0). However, the presence of the space charge (and also surface states) leads to a bending of the bands. If the surface region becomes

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FIGURE 31 Energy-level diagram (a) in the absence of any space charge and (b) with a surface space charge due to depletion of electrons in the surface region.

FIGURE 32 Potential energy diagram illustrating the work function. EF is the Fermi energy, φ is the work function, and W is the potential well bonding the conduction band electrons into the solid.

depleted of electrons it would require more energy to transfer an electron to the conduction band from, for example, the reference state E F , due to the space charge potential barrier. This is depicted schematically in Fig. 31. Conversely, it is now easier to transfer a hole to the surface since the difference between E F and E V becomes smaller. It is very likely that there is accumulation or depletion of charges at semiconductor or insulator surfaces under all ambient conditions. For surfaces under atmospheric conditions, adsorbed gases or liquid layers at the interface provide trapping of charges or become the source of free carriers. For clean surfaces in ultrahigh vacuum, there are electronic surface states that act as traps or sources of electrons and produce a space-charge layer of appreciable height. Thus, the mobile carriers from the surface layer are swept into the interior or are trapped at the surface as the space-charge layer consists dominantly of static charges, the one most frequently encountered in experimental situations. We have so far considered the space-charge layer properties only in the insulating solid, assuming that the surface layer that acts as a donor or the electron trap is of monolayer thickness. However, considering the properties of solid–liquid interfaces or semiconductor–insulator contacts, it should be recognized that the space-charge layer may extend to effective Debye lengths on both sides of the interface. This is a most important consideration when we investigate the surface properties of colloid systems or of semiconductor–electrolyte interfaces.

insulators it can be regarded as the difference in energy between an electron at rest in the vacuum just outside the solid (i.e., at the level of zero kinetic energy) and the most loosely bound electrons in the solid. Thus, the work function is evidently an important parameter in situations where electrons are removed from the solid. A schematic energy level diagram assuming the freeelectron model of a metal showing the work function is depicted in Fig. 32. From the figure it can be seen that the value of φ depends on W , the depth of the potential well bonding the conduction electrons into the solid. Here W is a bulk property determined by the attraction for its electrons of the lattice of positive ions as a whole; it has an energy of the order of a few electron volts. The origin of the work function itself can be considered as being due to the image potential of the escaping electron. Electrostatic theory shows that a charge −e outside a conductor is attracted by an image charge +e placed at the position of the optical image of −e in the conducting plane. If −e is a distance x from the plane the image force is e2 /16π ε0 x 2 . This force is experienced by the electron escaping into the vacuum and is negligible beyond 10−6 – 10−5 cm away from the surface. The image potential is a specific surface contribution to W , and a second surface contribution is the existence of a surface double layer or dipole layer. Surface atoms are in an unbalanced environment, they have other atoms on one side of them but not on the other; thus, the electron distribution around them will be unsymmetrical with respect to the positive ion cores. This leads to the formation of a double layer. Two important effects emanate from this; the work function is sensitive to both the crystallographic plane exposed and to the presence of adsorbates. The orientation of the exposed crystal face affects φ because the strength of the electric double layer depends on the density of positive ion cores which in turn will vary

D. Work Function and Contact Potential The work function of a solid is a fundamental physical property of the solid which is related to its electronic structure. It is defined as the potential that an electron at the Fermi level must overcome to reach the level of zero kinetic energy in the vacuum. In semiconductors and

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Surface Chemistry TABLE IV Work Functions Measured from Different Crystal Faces of Tungsten and Molybdenum Work function (eV) Crystal face

Tungsten

Molybdenum

(110) (112) (111) (001) (116)

4.68 4.69 4.39 4.56 4.39

5.00 4.55 4.10 4.40 —

from one face to another. The work function of various crystal planes of tungsten and molybdenum are listed in Table IV. It can be seen that there is more than 0.3 eV difference in work function values. This variation of work function from one crystal face to another can clearly be demonstrated using a field emission microscope (FEM). This microscope is identical in construction to the FIM described earlier. However, instead of having helium or another imaging gas in the vacuum, no gas is admitted. The potential on the sample tip is reversed so that electrons are accelerated out of it by a very high local electric field (∼ 4 × 107 V cm−1 ). The current emitted from the tip surface where the work function is φ is approximately proportional to exp(−Aφ 3/2 ) and is a very fast function of φ. The brightness observed on the fluorescent screen is a function of the value of φ at that place on the tip, and the FEM image will consist of darker and brighter areas, the brightness depending on the work function of each crystal face exposed. An image is shown in Fig. 33 which is the FEM image of

FIGURE 33 Field emission pattern of a tungsten tip. The (011) plane is in the center. (Courtesy of Lawrence Berkeley Laboratory.)

a tungsten tip. The changes in φ produced by adsorbed atoms or molecules can be followed in the FEM. The work function of a solid is also sensitive to the presence of adsorbates. In fact, in virtually all cases of adsorption the work function of the substrate either increases or decreases; the change being due to a modification of the surface dipole layer. The formation of a chemisorption bond is associated with a partial electron transfer between substrate and adsorbate and the work function will change. Two extreme cases are (i) the adsorbate may only be polarized by the attractive interaction with the surface giving rise to the build up of a dipole layer, as in the physisorption of rare gases on metal surfaces; and (ii) the adsorbate may be ionized by the substrate, as in the case of alkali metal adsorption on transition metal surfaces. If the adsorbate is polarized with the negative pole toward the vacuum the consequent electric fields will cause an increase in work function. Conversely, if the positive pole is toward the vacuum then the work function of the substrate will decrease. The work function is a rather complicated (and not fully understood) function of the surface composition and geometry. Nevertheless, general systematic observations of φ are quite helpful. For example, the sign of φ for atomic adsorption is mostly that implied by the magnitude of the ionization potential, electron affinity, or dipole moment of the adsorbates as one would expect. The most common usage of work function changes in surface chemistry is in the monitoring of the various stages of adsorption as a function of coverage. Often the work function change will go through a maximum or minimum at particular coverages corresponding to the completion of an ordered atomic arrangement. Experimentally, the most accurate way of measuring changes in work function is by the Kelvin method, which uses a vibrating probe as a variable capacitor. A contact potential difference is set up between two conductors connected externally and the sample and a reference electrode form a parallel plate condenser. The distance between the two is periodically varied, thus generating an alternating current in the connecting wire. If a voltage source is placed in the connecting circuit just balancing out the contact potential difference, no current will flow. Once this situation has been achieved for the clean surface, a change in work function due to adsorption is simply the additional voltage which needs to be applied to compensate the change and keep the current zero. Accuracies of φ to within ±1 meV are obtainable. Intimately linked to the concept of work function is the process of thermionic emission. Thermionic emission is, as the name suggests, the phenomenon whereby electrons are ejected from a metal when it is heated in vacuum. The electrons that require the least amount of thermal energy to overcome their binding energy in the solid and

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escape are those in the high-energy tail of their equilibrium distribution in the metal. Thermionic emission is the most frequently used method to produce electron beams, for instance, in oscilloscope tubes and electron microscopes. Refractory metals (e.g., W) have traditionally been used as filaments in electron guns, mainly due to the fact that they can be heated to high temperatures and thus produce a relatively intense thermionic current. Since the work function of W is relatively high W filaments are often coated with a metal for lower work function, for example, Th (φ = 2.7 eV) to enable them to be operated at lower temperature for the same current thereby extending their lifetime. E. Surface States In a bulk solid the infinite array of ion cores in crystallographic sites leads to a potential that varies in a three-dimensionally periodic manner. The solutions to Schrödinger’s equation for such a potential lead to allowed energy bands, which are occupied by the electrons in the solid, and to particular values of the wave vector k of the electron where no traveling-wave solutions exist. The absence of eigenstates for these values of k leads to the band gaps in the electronic structure of the solid. The solid, however, is not infinite but is bounded by surfaces. In turn, surface atoms have fewer nearest neighbors and are in an asymmetric environment. The introduction of such a discontinuity at the surface perturbs the periodic potential and gives rise to solutions of the wave equation that would not have existed for the infinite crystal. These are derived by using appropriate boundary conditions to terminate the crystal and are called surface-state wave functions. These special solutions are waves which can travel parallel to the surface but not into the solid. They are localized at the surface and can have energies within the band gap of the bulk band structure. These states can trap electrons or release them into the conduction band. The concentration of electronic surface states in clean surfaces can be equal to the concentration of surface atoms (∼1015 cm−2 ). Impurities or adsorbed gases can reduce the surface state density. One important consequence of the presence of electronic surface states is that the electron bands are modified at the surface even in the absence of a space charge or electron acceptor or donor species (such as adsorbed gases). The shape of the conduction band at the surface of an intrinsic semiconductor in the presence of electrondonor and electron-acceptor surface states is shown in the energy level diagrams in Figs. 34a and 34b. Surface states can be associated not only with the termination of a three-dimensional potential at a perfect clean bulk exposed plane but also with changes in the potential

FIGURE 34 Energy-level diagrams for an intrinsic semiconductor in the presence of (a) electron–donor or (b) electron–acceptor surface states.

due to relaxation, reconstruction, structural imperfections, or adsorbed impurities. If the charge associated with any of these surface states is different from the bulk charge distribution then band bending will occur. Surface states can be observed, for example, using ultraviolet photoelectron spectroscopy, which is discussed below. F. Electron Emission from Surfaces The most important methods of analyzing the surface electronic and chemical composition involve energy analysis of electrons emitted from a surface during its bombardment with electrons, ultraviolet photons, or X-ray photons. For example, part of the experimental verification of the band theory of metals comes from the measured intensity and energy distribution of electrons emitted under excitation by photons. It should be remembered that we have already mentioned two ways in which electrons can be emitted from surfaces; (i) by applying a very high electric field (∼107 V cm−1 ) which pulls electrons from the surface, as used in FEM, and (ii) by heating the solid as in thermionic emissions. Before discussing the two major electron emission techniques from surfaces, photoelectron spectroscopy and Auger electron spectroscopy (AES), it is pertinent to briefly discuss the surface sensitivity of the interaction of electrons with solids. Figure 35 shows the mean free path of electrons in metallic solids as a function of the electron energy. This curve is often called the “universal curve,” and shows a broad minimum in the energy range between 10 and 500 eV with the corresponding mean free ˚ Electron emission from solids path on the order of 4–20 A. with energy in this range must originate from the top few atomic layers. Therefore, all experimental techniques involving the incidence and/or convergence from surfaces of electrons having energy between 10 and 500 eV are surface sensitive. For incident electrons of higher energy (1–5 kV) the surface sensitivity can be enhanced by having the electron beam impinging on the surface at grazing

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FIGURE 35 Universal curve for the electron mean free path as a function of electron kinetic energy. Dots indicate individual measurements.

incidence. Photons have a much larger penetration depth into the solid due to the much smaller scattering cross section. However, electrons created by excitation below a few atomic layers from the surface cannot escape due to inelastic scattering within the solid. If a monoenergetic beam of electrons of energy E p strikes a metal surface then a typical plot of the number of scattered electrons N (E) as a function of their kinetic energy E is shown in Fig. 36. The curve is dominated by a strong peak at low energies due to secondary electrons created as a result of inelastic collisions between the incident electrons and electrons bound to the solid. Other features in the spectrum include (i) the elastic peak at E p that is utilized in LEED, (ii) inelastic peaks at loss energies of 10–500 meV which provide information about the vibrational structure of the surface–adsorbate complex utilized in EELS, (iii) inelastic peaks at greater loss energies (plasmon losses) which provide information about the electronic structure of surface atoms, and (iv) small peaks on the large secondary electron peak due to Auger electrons which provide information on the chemical composition of the surface.

5000 eV) strikes the atoms of a material, electrons that have binding energies less than the incident beam energy may be ejected from the inner atomic level. By this process a singly ionized, excited atom is created. The electron

Auger Electron Spectroscopy Auger electron spectroscopy is the most common technique for determining the composition of solid and liquid surfaces. Its sensitivity is about 1% of a monolayer, and it is a relatively simple technique to perform experimentally. Auger electron emission occurs in the following manner. When an energetic beam of electrons or X-rays (1000–

FIGURE 36 Experimental number of scattered electrons N(E) of energy E versus electron energy E curve. (Courtesy of Lawrence Berkeley Laboratory.)

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FIGURE 37 Scheme of the Auger electron emission process.

vacancy formed is filled by deexcitation of electrons from other electron energy states. The energy released in the resulting electronic transition can, by electrostatic interaction be transferred to still another electron in the same atom or in a different atom. If this electron has a binding energy that is less than the energy transferred to it from the deexcitation of the previous process that involves the filling of the deep-lying electron vacancy, it will be ejected into vacuum, leaving behind a doubly ionized atom. The electron that is ejected as a result of the deexcitation process is called an Auger electron, and its energy is primarily a function of the energy-level separations in the atom. These processes are schematically displayed in Fig. 37. To a first approximation the energy of the Auger electron depicted in Fig. 37 is given by E Auger = E K − E LI − E LIII

(44)

and is independent of the energy of the incident beam. This is an important difference between AES and photoelectron spectroscopy and means that it is not necessary to monochromatize the electron beam which adds to the experimental convenience. There are two major experimental designs for AES. One is using the retarding grid analyzer which uses the same electron optics as LEED, thus LEED and AES can be performed using the same apparatus. The second is the cylindrical mirror analyzer (CMA) which has an inherently better signal-to-noise ratio. Scanning Auger microprobes are now in widespread use in the microelectronics industry for spatial chemical analysis of surfaces. With the exception of hydrogen and helium, all other elements are detectable by Auger electron spectroscopy. The Auger spectrum is usually presented as the second derivative of intensity, d 2 I /d V 2 , as a function of electron energy (eV). This way the Auger peaks are readily separated from the background, due to other electron loss processes that occur simultaneously. A typical Auger spectrum of molybdenum is shown in Fig. 38. By suitable analysis of the experimental data, as well as by the use of suitable reference surfaces, the Auger electron spectroscopy study can provide quantitative chemi-

FIGURE 38 Typical Auger spectra from (a) a clean Mo(100) single-crystal and (b) a Mo(100) surface contaminated with sulfur.

cal analysis in addition to elemental compositional analysis of the surface. It is possible to separate the surface composition from the composition of layers below the surface by appropriate analysis of the Auger spectral intensities. In this way the surface composition as well as the composition in the near-surface region can be obtained. When chemical analysis is desired in the near-surface region, AES may be combined with ion sputtering to obtain a depth-profile analysis of the composition. Using high-energy ions, the surface is sputtered away layer by layer while, simultaneously, AES analysis detects the ˚ composition in depth. Sputtering rates of 100 A/min are usually possible and the depth resolution of the compo˚ which is mainly determined by the sition is about 10 A, statistical nature of the sputtering process. A different aspect of AES concerns shifts in the observed peak energies that are due to chemical shifts of atomic core levels (in a way analogous to X-ray photoelectron spectroscopy). For example, studies of different oxidation states of oxygen at metal surfaces have shown chemical shifts that grow with the formation of higher oxidation states. G. Photoelectron Spectroscopy Photoelectron spectroscopy is a technique whereby electrons directly ejected from the surface region of a solid by incident photons are energy analyzed and the spectrum is then related to the electron energy levels of the system. The field is usually arbitrarily divided into two classes: ultraviolet photoelectron spectroscopy (UPS) and X-ray photoelectron spectroscopy (XPS). The names derive from the energies of the photons used in the

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particular spectroscopy. Ultraviolet photoelectron spectroscopy studies the properties of valence electrons that are in the outermost shell of the atom and utilizes photons in the vacuum ultraviolet region of the electromagnetic spectrum [He I (21.22 eV), He II (40.8 eV), and Ne I (16.85 eV) resonance lamps are the most commonly used photon sources]. X-ray photoelectron spectroscopy investigates the properties in the inside shells of atoms and uses photons in the X-ray region [Mg Kα (1253.6 eV) and Al Kα (1486.6 eV) being the most common]. With the advent of synchrotron radiation, a polarized, tunable light source covering the entire useful energy range, the division is now somewhat redundant. In both types of spectroscopy, if the incident photon has enough energy hν it is able to ionize an electronic shell and an electron which was bound to the solid with energy E B is ejected into vacuum with kinetic energy E k . By conservation of energy: E k = hν − E B .

(45)

If the incident radiation is monochromatic and of known energy, and if E k can be measured using a high-resolution energy analyzer (such as either a concentric hemispherical or cylindrical mirror analyzer), then the binding energy E B can be deduced. Equation (45) gives a highly simplified relationship between the kinetic energy, E k , of the emitted photoelectrons and their binding energy; E k is modified by the work function of the energy analyzer and by several atomic parameters that are associated with the electron emission process. The ejection of one electron leaves behind an excited molecular ion. The electrons in the outermost and in other orbitals experience a change in the effective nuclear charge due to an alteration of screening by other electrons. This gives rise to satellite peaks near the main photoelectron peaks. Several other effects, including spin-orbit splitting, Jahn–Teller effect, and resonant absorption of the incident photon by the atom, influence the detected photoelectron spectra. One of the most important applications of XPS is the determination of the oxidation state of elements at the surface. The electronic binding energies for inner-shell electrons shift as a result of changes in the chemical environment. An example of these shifts can be seen in nitrogen, indicating the photoelectron energy for various chemical environments (Fig. 39). These energy shifts are closely related to charge transfer in the outer electronic level. The charge redistribution of valence electrons induces changes in the binding energy of the core electrons, so that information on the valence state of the element is readily obtainable. A loss of negative charge (oxidation)

FIGURE 39 Is electronic binding energy shifts in nitrogen, indicating the different photoelectron energies observed in various chemical environments.

is in general accompanied by an increase in the binding energy E B of the core electrons. Relative surface coverages can also be obtained with XPS by monitoring the intensities of the core level peaks. Absolute coverages can be obtained from the core level intensities, but it is usual to calibrate against another technique. As mentioned above UPS probes the valence electrons of the solid. It is these electrons which form a chemisorption bond and a knowledge of electronic density of states at a surface is of vital importance in attempts to understand the formation of chemical bonds between solid surfaces and adsorbed atoms or molecules; UPS can provide even more information about the system if the emitted electrons are both energy and spatially analyzed. This is known as angle-resolved UPS (ARUPS). Using ARUPS the band structures of clean and adsorbate-covered surfaces have been determined, mapping out the dispersion of electronic states; ARUPS also reveals directional effects due to the spatial distribution of electronic orbitals of atoms and molecules at the surface. By changing the angle of incidence and the angle of detection, the electronic orbitals from which the photoelectrons are ejected can be identified. In addition, ARUPS provides detailed information about the surface chemical bond including the direction of the bonding orbitals and the orientation of the molecular orbitals of adsorbed species on the surface.

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V. SURFACE DYNAMICS A. Atomic Vibrations Until now it has been convenient to discuss both the properties and methods in terms of rigid lattices of atoms or molecules. In reality, the atoms are in motion and this motion should be included in a complete treatment of any properties it may affect. In X-ray diffraction experiments it is well known that the intensity of the scattered rays decreases as the temperature is increased. Simultaneously, the intensity of the diffuse background of the diffraction pattern increases. The simplest explanation for this observation is that the atoms are not rigid, but are vibrating about their equilibrium positions, and as a result, the exact Bragg condition is not met. Scattered waves from the rigid lattice that were adding up in phase now have a phase difference fluctuating with time due to the atomic motion. The effect of this motion on the intensity of the elastically diffracted beams is described in most good solid-state physics texts. Briefly, if I0 is the intensity elastically diffracted by a rigid lattice then the intensity I due to scattering by the vibrating lattice in the direction determined by Bragg scattering due to a reciprocal-lattice vector g¯ is given by the following: I = I0 exp(−αu 2 |g|),

(46)

assuming that the atoms are in simple harmonic motion. u 2 is the mean-square amplitude of vibration in the direction g¯ and α is a constant related to the number of dimensions in which the atoms are allowed to vibrate. In one dimension α = 1; in three dimensions α = 13 . The exponential factor in Eq. (46) is called the Debye–Waller factor and is often denoted as exp(−2M). The same kind of effect is observed in LEED only because LEED intensities arise from the just few atomic layer of a crystal the value of u 2 is that for the surface atoms. Because of the absence of nearest neighbors on the vacuum side we expect that u 2 at the surface will be greater than in the bulk. By using the Debye model of the solid it is possible to relate the observed intensity of the elastically scattered electrons in LEED to measurable quantities. We obtain the following: 12h 2 cos φ 2 T I00 (T ) = I00 (0) exp − , (47) mk λ 2D where I00 (T ) is the temperature-dependent intensity of the (0, 0) beam resulting from a beam of electrons of wavelength λ incident on the surface at an angle relative to the surface normal. I00 (0) is the specularly reflected intensity from a rigid lattice, h is Planck’s constant, m is the atomic

mass, k is Boltzmann’s constant, T is the temperature, and D is the Debye temperature. (The Debye temperature is associated with the energy ωmax of the highest frequency phonon mode possible in the Debye model of vibrations in the solid, hωmax = kD .) Equation (47) implies that a plot of the logarithm of the intensity at a given energy (wavelength) as a function of temperature is a straight line, the slope of which yields D , a measure of the surface vibrational amplitude perpendicular to the surface. In reality, the electron-beam penetration varies as a function of energy, so that Eq. (47) provides, at any given energy, an effective Debye temperature, which is some average of the surface and bulk layers. In empirical fashion, however, we may arrive at a surface Debye temperature from the low-energy limit of this effective Debye temperature. Adsorbates should have a marked influence on surfaceatom vibrations, since they change the bonding environment with respect to that on the clean surface. The adsorption of oxygen on tungsten increases the surface Debye temperature with respect to the bulk value due to the stronger W O bond as compared to the W W bond. Studies of surface-atom vibrations in the presence of adsorbates provide information on the nature of the surface bond. B. Surface Diffusion As discussed above, at any finite temperature the atoms at the surface of a crystal are vibrating at some frequency ν0 . Thus, ν0 times every second each atom strikes the potential-energy barrier separating it from its nearest neighbors (Fig. 40). The thermal energy causing the atoms to oscillate with increasing amplitude as the temperature is increased is not sufficient to dislodge most of them from their equilibrium positions. The thermal energy (3RT ≈ 1.8 kcal mol−1 at 300 K) tied up in lattice vibrations is only a small fraction of the total energy necessary

FIGURE 40 One-dimensional potential energy diagram parallel to the surface plane.

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to break an atom from its neighbors and to move along the surface. This bond breaking energy is of the order 15– 50 kcal mol−1 for many metal surfaces. As the temperature of the surface is increased, more and more surface atoms may acquire enough activation energy to break bonds with their neighbors and move along the surface. Such surface diffusion plays an important role in many surface phenomena involving atomic transport, e.g., crystal growth, vaporization, and adsorption. The migration of atoms or molecules along the surface is one of the most important steps in surface reactions and has proved to be the ratelimiting step for many reactions that have been studied at low pressures. A surface contains many defects on an atomic scale. Atoms in different surface sites have different binding energies. Surface diffusion can be considered as a multistep process whereby atoms break away from their lattice position (e.g., a kink site at a ledge) and migrate along the surface until they find a new equilibrium site. The frequency f with which an atom will escape from a site will depend upon the height, E D∗ , of the potential energy barrier it has to climb in order to escape as follows: E D∗ f = zν0 exp − , (48) kB T where z is the number of equivalent neighboring sites. For a (111) face of an fcc metal, z = 6, the vibration frequency is of the order of 1012 sec−1 . Assuming that E D is 20 kcal mol−1 , at 300 K the atom makes one jump in every 50 sec, while at 1000 K one in 10−8 sec. Thus, the rate of surface diffusion varies rapidly with temperature. This is the case for a single jump to a neighboring equilibrium surface site. What is of great importance is the long-distance motion of a surface atom. The result is derived from considering a mathematical treatment of an atom executing a random walk for a time t over a mean-square distance X 2 . For a sixfold symmetrical surface, we obtain the following: X 2 = f td 2 /3.

(49)

The value of f d 2 is a property of the material that characterizes its atomic transport. Its value provides information about the mechanism of atomic transport, and it is customary to define the diffusion coefficient D as follows: D = f d 2 /2b,

(50)

where b is the number of coordinate directions in which diffusion jumps may occur with equal probability. Equation (50) can therefore be rewritten as D = D0 exp − E D∗ kB T , (51) where D0 = (v0 d 2 /6), (v0 d 2 /4), for sixfold or fourfold symmetry, respectively; D is usually given in units of

square centimeters per second. If D is determined experimentally as a function of temperature, then a plot of ln D versus 1/T will yield us the activation energy of the diffusion process, provided that the diffusion occurs by a single mechanism. The rms distance X 2 1/2 can be expressed in terms of the diffusion coefficient by substitution of Eq. (50) into Eq. (49) to give for b = 6 as follows: X 2 1/2 = (4Dt)1/2 .

(52)

From measurements of the mean travel distance of diffusing atoms the diffusion coefficient can be evaluated. Conversely, knowledge of the diffusion coefficient allows us to estimate the rms distance or the time necessary to carry out the diffusion. For example, the diffusion coefficients of silver ions on the surface of silver bromide can be estimated to be 10−19 and 10−13 cm2 /sec at 300 and 100 K, respectively. Assuming that a rms distance of 10−4 cm is required for silver particle aggregation (printout) to commence, of what duration are the light-exposure times required? Using Eq. (52) we have t = 5 sec and t = 5 × 104 sec at 300 and 100 K, respectively. The exponential temperature dependence of D is, of course, the reason that silver bromide photography cannot be carried out at low temperatures (much below 300 K) but is easily utilized at about room temperature. We can also see that at slightly elevated temperature (∼450 K) the thermal diffusion of silver particles should be rapid enough (D ≈ 3 × 10−7 cm2 sec−1 ) so that their aggregation will take place rapidly even in the dark (t ≈ 10−2 sec) in the absence of any photoreaction. Surface diffusion has so far been discussed in terms of a single surface atom. However, on a real surface many atoms diffuse simultaneously and in most diffusion experiments the measured diffusion distance after a given diffusion time is an average of the diffusion lengths of a large, statistical number of surface atoms. A thermodynamic treatment in terms of macroscopic parameters can be followed to yield the following: D = D0 exp(−Q/RT ),

(53)

where Q is the total activation energy for the overall diffusion process and only one diffusion mechanism is involved. Experimentally, the diffusion coefficient D is obtained by using a relationship between the diffusion rate and coverage gradient, namely Fick’s second law of diffusion in one dimension: ∂c/∂t = D(∂ 2 c/∂ x 2 ),

(54)

where c is the concentration of adatoms, t the time, and x the distance along the surface. In most surface diffusion studies the surface concentration of diffusing atoms, c, is

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measured as a function of distance x along the surface, and Eq. (54) is solved by the use of boundary conditions that approximate the experimental geometry. These experiments are by no means trivial, and many novel experimental techniques have been applied to study surface diffusion on single crystals. A technique which has been used to measure surface diffusion rates is scanning Auger electron spectroscopy, which can follow adsorbate diffusion. A particular Auger transition of the adsorbate under investigation is used as a monitor of relative concentration versus distance scanned across the surface. Profiles are recorded after heating periods to observe the change in concentration profile as a function of time and temperature. While this technique monitors mass transport, and values of D and Q are averaged values, field ion microscopy can be used to follow the diffusion of individual atoms across a surface. To study diffusion, the metal is vapor deposited onto the tip. The tip is then heated to remove evaporated adatoms until only one or two remain on the surface plane of interest. The diffusion is then examined by photographically recording the position of the adatom at low temperatures, removing the applied field, and heating to the desired temperature for a given time. The tip is then cooled, the field reapplied, and the field ion image examined to see if the atom has moved to a neighboring site. This process is then repeated many times to obtain useful values of diffusion rates, and by examining the diffusion over a temperature range, the activation barrier to surface diffusion can be determined. Figure 41 shows a series of field ion images of a Rh atom on the W(112)

plane at 327 K. The field ion images are taken at 1-min intervals and the Rh atom can clearly be seen to have diffused across the surface. Unfortunately because of the high field strengths employed, adsorbates such as O or N tend to be stripped from the surface as ions, so their microscopic diffusion cannot be studied by this method. An interesting result from FIM studies of metal adatoms on metals is the recognition of clusters as important contributions to material transport. It has been found that rhenium dimers diffuse more rapidly than single Rh atoms on the W(112) plane, and Rh trimers diffuse at roughly the same rate as dimers. This is not a general trend, however, as iridium dimers move much more slowly than singles. While the single adatom diffusion technique gives us detailed microscopic information, the mass transport techniques are of use as they help to give understanding of the technologically important processes such as sintering and creep. C. Surface Reactions Heterogeneous catalysis, corrosion, photosynthesis, and adhesion are examples of chemical processes that are partially or fully controlled by reactions at surfaces. For the case of gas–solid reactions the surface reactions can be divided into two major categories: (i) stoichiometric surface reactions where the solid surface participates directly in the reaction by compound formation and (ii) catalytic surface reactions where the reaction occurs at the solid surface but the surface does not undergo any net chemical change. In both cases gaseous molecules impinge on the surface, adsorb, react, and form various intermediates of varying stability, and then the products desorb into the gas phase if they are volatile. All surface reactions involve a sequence of elementary steps that begins with the collision of the incident atoms or molecules with the surface. As the gas species approaches the surface it experiences an attractive potential whose range depends upon the electronic and atomic structures of the gas and surface atoms. A certain fraction of the incident gas molecules is trapped in this attractive potential well with a sticking probability given by the following: S(, T ) = S0 (1 − ) exp(−E a /RT ),

FIGURE 41 Diffusion of rhenium atoms on W(211) at 327 K. Field ion images are taken after 60 sec diffusion intervals. (Courtesy of Lawrence Berkeley Laboratory.)

(55)

where S0 is the initial (zero coverage) sticking coefficient, the surface coverage (0 < < 1), and E a the activation energy for adsorption. If this force attraction is due to a van der Waals interaction, the trapping is due to physical adsorption. If the attraction is much stronger, having the character of chemical bonding then we have chemisorption and the process is known as sticking. The boundary between the two types of bonding is usually

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410 set at a binding energy of 15 kcal mol−1 . Sticking by chemisorption is often preceded by trapping into a physisorbed state, in which case the physisorbed state is known as a precursor state for chemisorption. The presence of a precursor state is indicated by a sticking coefficient that remains almost constant as surface coverage increases until a saturation coverage is reached, when it rapidly falls to zero. This behavior arises because molecules in the relatively mobile precursor state diffuse to parts of the surface which are not covered by chemisorbed molecules. In direct chemisorption the sticking coefficient varies strongely with coverage and with ordering of the chemisorbed layer. The adsorbed species may also desorb from the surface if its energy overcomes the attractive surface forces. When a surface reaction occurs a certain proportion of the adsorbed species either decomposes (unimolecular reaction) or reacts with a second adsorbed species (bimolecular reaction) before the product desorbs. During the initial interaction of the gas molecule with the surface as the incoming molecule falls into a potential well the kinetic energy normal to the surface increases. Unless this energy is transferred to some other degree of freedom the molecule will simply bounce off; there will be no trapping or sticking. In the case of physisorption energy transfer via phonons is usually most important while for chemisorption electronic excitation via electron–hole pairs is thought to be important. The exchange of translational energy T with the phonons Vs is called T − Vs energy exchange. The gas molecule may also exchange internal energy, rotation R or vibration V with the vibrating surface atoms. In this case there are also R − Vs and V − Vs energy transfer processes. In order to understand the dynamics of gas–surface interaction, it is necessary to determine how much energy is exchanged between the gas and surface atoms through the various energy-transfer channels. In addition the kinetic parameters (rate constants, activation energies, and preexponential factors) for each elementary surface step of adsorption, diffusion, and desorption are required in order to obtain a complete description of the gas–surface energy transfer process. Most surface reactions take place at high pressures (1– 100 atm) either because of the chemical environment of our planet or to establish optimum reaction rates in chemical processing. Under these conditions, surfaces are usually covered by at least one monolayer of adsorbed species. Since activation energies for adsorption and surface diffusion are generally small (a few kT ), equilibrium among the different surface species, reactants, reaction intermediates, and products, is readily established. In the simplest (but not general and important) case of localized, associative adsorption into a single state, the surface coverage by

Surface Chemistry

adsorbed species is given in terms of the gas pressure P by the Langmuir isotherms: = KP/(1 + KP),

(56)

where K is an equilibrium constant. Catalyzed surface reactions usually take place between two or more coadsorbed species which compete for adsorption sites on the surface. When j gases adsorb competitively and associatively, the surface coverage by species i is given by the following: 1+ (57) K j Pj . = K i Pi j

Many catalyzed surface reactions can be treated as a two-step process with an adsorption equilibrium followed by one rate-determining step (diffusion, surface reaction, or desorption). The surface reaction kinetics are usually discussed in terms of two limiting mechanisms, the Langmuir–Hinshelwood (LH) and Eley–Rideal (ER) mechanisms. In the LH mechanism, reaction takes place directly between species which are chemically bonded (chemisorbed) on the surface. For a bimolecular LH surface reaction. Aads + Bads → products, with competitive chemisorption of the reactants, the rate of reaction is given by the following expression: Rate = kR A B = kR K A K B PA PB /(1 + K A PA + K B PB )2 .

(58)

The reaction rate is proportional to the surface coverages A and B and to the reaction rate constant kR . For noncompetitive adsorption, the rate expression becomes the following: kR K A K B PA PB Rate = kR A B = . (59) (1 + K A PA )(1 + K B PB ) General rate expressions of the form given in equations and have been experimentally verified for many types of LH reactions. Similar but more complicated rate expressions are easily derived assuming different (non-Langmuir) isotherms, higher-order reaction steps, or dissociative chemisorption of the reactants. In the ER mechanism, surface reaction takes place between a chemisorbed species and a nonchemisorbed species, e.g., Aads + Bg → products. The nonchemisorbed species may be physisorbed or weakly held in a molecular precursor state. In this case, the rate expression for the surface reaction becomes Rate = kR A PB = kR K A PA PB /(1 + K A PA ).

(60)

Presently no proven examples exist in which surface reaction occurs by the ER mechanism. Surface reaction kinetics determined experimentally are often expressed in the form of a power rate law as follows:

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Rate = kR

Piαi ,

(61)

i

where kR is the apparent rate constant and αi is the experimental order of the reaction (positive, negative, integer, or fraction) with respect to the reactants and products. The apparent rate constant in Eq. (61) is not that of an elementary reaction step (it contains adsorption equilibrium constants), but it can usually be represented by an Arrhenius equation as follows: kR = A exp(−E R /RT ),

(62)

where A is an apparent preexponential factor and E R is the apparent activation energy for the surface reaction. The magnitude of A and E A can provide important information about the rate-determining step of a surface reaction, and very frequently kR and A display a compensation effect. A related quantity is the reaction probability, λi = (2π m K T )1/2 νR /Pi = rate/flux; that is, the probability that an incident reactant molecule will undergo reaction. The simplified isotherms and rate expressions developed in this section are extremely useful despite the implicit assumption that a single state exists for the adsorbed species. Real surfaces are heterogeneous on an atomic scale with a variety of distinguishable adsorption sites. Gas molecules adsorbed at each type of site may display a wide distribution of excited rotational, vibrational, and electronic states. Experimentally, we can measure meaningful rate and adsorption equilibrium constants provided that adsorption and desorption are fast compared with surface reactions so that an adsorption equilibrium exists. In this circumstance the kinetic parameters are an ensemble average over all surface sites and states of the system.

molecules, together with their angular and velocity distributions provide detailed information about the T –Vs energy transfer processes that occur during the gas–surface interaction. A complete dynamical description for this interaction (T –Vs plus R–Vs and V –Vs ) can be determined if the distribution of internal energy states for the product molecules is determined simultaneously with their velocity distributions. This type of detection is known as state selective detection. The angular distribution of scattered molecules is usually displayed by plotting the intensity of detected molecules per unit solid angle versus the angle of scattering r that is measured with respect to the surface normal. Angular distributions in the two limiting cases of gas–surface interaction, cosine and specular scattering, are shown in Fig. 42. The scattered intensity for the cosine distribution decreases as cos with respect to the surface normal. Cosine scattering is expected when the adsorbed species have long residence times or are strongly coupled to the vibrational states of the surface atoms. It is a necessary criterion for complete thermal accommodation, a situation in which the molecules desorb with a kinetic temperature or velocity distribution that is the same as the temperature of the solid surface. Specular scattering occurs when the scattered intensity is sharply peaked at the angle of incidence (specular angle). In this case the interaction is elastic or quasielastic and little or no energy transfer takes place between the incident gas molecules and the surface. Sharply peaked angular distributions for surface reaction products (I () ∼ cosm , m > 1) indicate that a repulsive barrier exists in the exit channel. Measurements of velocity distributions provide more

1. Molecular-Beam Scattering The most powerful experimental technique for investigating the dynamics of the gas–solid interaction is molecularbeam surface scattering (MBS). The experimental arrangement is similar to that already described for helium atom diffraction. Instead of using an atomic beam of a light molecular weight gas and observing diffraction effects, a well-collimated beam of molecules strikes the oriented, preferably single-crystal, surface, and the species that are scattered at a specific solid angle are detected by a mass spectrometer. The angular distribution of the scattered molecules can be obtained by rotation of the mass spectrometer about the sample. The velocity distribution of the molecules after scattering is deduced by chopping the scattered molecules and thereby measuring their time of flight to the detector. The surface residence times of the

FIGURE 42 Rectilinear plot displaying the (a) specular scattering and (b) cosine angular distribution of scattered beams. The arrow indicates the angle of incidence.

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412 direct information on inelastic scattering than angular distributions alone. Although considerable information can be gained from such studies it has been impossible to get state specific information and indeed it is often unclear whether internal states present more efficient energy transfer channels than phonons or vice versa. The difficulties in studying internal energy transfer in molecular collisions with surfaces can be resolved by the application of state-specific detection techniques. Laser-induced fluorescence, multiphonon ionization, IR excitation with bolometric detection, and IR emission techniques have all been used to obtain state-resolved measurements of the internal energy distributions of molecules scattering from surfaces. It has also been possible to separate experimentally direct inelastic and trapping-desorption scattering. In the direct inelastic scattering of diatomics, coupling to rotational energy has been found to be very important and to exhibit several interesting phenomena: rotational rainbows and the production of rotationally aligned molecules in scattering.

Surface Chemistry

Exchange of hydrogen and deuterium to form HD is one of the simplest reactions that can be catalyzed on clean metal surfaces at temperatures as low as 100 K. The same reaction is immeasurably slow in the gas phase due to the very high dissociation energies of the reacting molecules (103 kcal mol−1 ). the H2 –D2 exchange reaction has been studied over flat (111) and stepped (332) single crystal surfaces of platinum. The Pt(332) surface contains high concentrations of periodic surface irregularities (steps) that are one atom in height. Reaction probabilities averaged over the cosine HD angular distributions were 0.07 on the (111) surface and 0.35 on the (332) surface under identical experimental conditions (Ts = 1100 K, Tg = 300 K). The reaction probability on the stepped surface varied markedly with the angle of incidence of the mixed H2 –D2 molecular beam. This is shown in Fig. 43. The reaction probability was highest when the beam was incident on the open edge of the step and lowest when the bottom of the step was shadowed

2. Molecular-Beam Reactive Scattering While molecular-beam scattering has made great advances in our understanding of the energy exchange processes during the gas–surface collision, molecular-beam techniques have also made important contributions to the understanding of the mechanisms of chemical reactions occurring at surfaces in the form of molecular-beam reactive scattering (MBRS). The use of time-of-flight techniques permits measurement of product velocity distributions and the detailed time resolution of fast transient reactions. Also of great value is the use of state-specific detection methods to determine product vibrational and rotational states. Although MBRS can only be utilized at low pressures (≤10−4 torr) its pressure range permits wide variations of surface coverages. The reaction probabilities on a single scattering can be determined together with the surface residence times of adsorbates. The surface kinetic information is obtained by measurements of the intensity and the phase shift of the product molecules with respect to the reactant flux. Residence times in the range 10−6 –1 sec can be monitored with relative ease, and activation energy is determined from the temperature dependences of the intensities and the phase shifts. The phase shift of the product molecules is usually measured at different chopping frequencies of the incident beam. At a given chopping frequency, only those product molecules are detected that are formed in the surface process and desorbed in less time than the chopping period. As an example of an investigation of the dynamics of a catalyzed surface reaction studied by MBRS we will consider the isotope exchange reaction, H2 –D2 .

FIGURE 43 HD production as a function of angle of incidence of the molecular beam, normalized to the incident D2 intensity. (a) Pt(332) surface with the step edges perpendicular to the incident beam (φ = 90◦ ); (b) Pt(332) where the projection of the beam on the surface is parallel to the step edges (φ = 0◦ ); and (c) Pt(111).

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(curve a). When the H2 –D2 beam was incident parallel to the steps, the rate of HD production was independent of the angle of incident at all angles of crystal rotation (curve b). These results indicate that the atomic step sites are about seven times more active than the (111) terrace sites for the dissociative chemisorption of hydrogen and deuterium molecules. Detailed analysis of the scattering data revealed a barrier height of 4–8 kJ mol−1 for dissociative H2 chemisorption on the (111) surface. On the other hand, this barrier did not exist (E a = 0) on the stepped surface. This difference in activation energy alone accounts for the different reaction probabilities of the step and terrace sites. While the dissociation probability of hydrogen molecules was higher on the stepped surface than on Pt(111), the kinetics and mechanism of HD recombination appear to be identical over both surfaces once dissociation takes place. On both surfaces, HD formation follows a parallel LH mechanism with one of the reaction branches operative over the entire temperature range of 300–1075 K. This branch has an activation energy and pseudo-first-order preexponential factor of E a = 54 kJ mol−1 and A1 = 8 × 104 sec−1 for the stepped surface and E a = 65 kJ mol−1 and A1 = 3 × 105 sec−1 for the Pt(111) surface. A second branch is observed for temperatures above 575 K, but the kinetic parameters for this pathway could not be accurately determined. D. Stoichiometric Surface Reactions Stoichiometric surface reactions are those in which the surface participates directly in the reaction by compound formation. Oxidation and corrosion are the two most important classes of such reactions. Surface oxidation of metals encompasses a series of at least three reaction steps that include (1) dissociative chemisorption of oxygen on the metal surface, (2) rearrangement of the surface atoms with dissolution of oxygen into the near surface region, and (3) nucleation of oxide islands which grow laterally and eventually condense to produce continuous oxide films. The oxide islands appear to precipitate suddenly once a critical oxygen concentration is reached in the near surface region. Nucleation takes place most readily at surface irregularities such as atomic steps, dislocations, and stacking faults. At room temperature, noble metals such as Rh, Ir, Pd, and Pt display little tendency for oxygen incorporation or surface rearrangement. Initial heats of oxygen chemisorption on these metals are much greater than the heats of formation of the corresponding bulk oxides. Other metals such as Cr, Nb, Ta, Mo, W, Re, Ru, Co, and Ni, dissolve surface oxygen by a place exchange mechanism in which oxygen atoms interchange positions with underlying metal atoms. These

metals display heats of adsorption for oxygen that are comparable to the heats of formation of the stable metal oxides. Metals such as Ti, Zr, Mn, Al, Cu, and Fe dissolve oxygen more readily and form stable oxide films even at room temperature. At low oxygen pressures these films often assume a crystalline structure, whereas at higher pressures (>10−3 atm) the films tend to be amorphous. At higher temperatures (400–1000 K), oxide formation occurs readily on the surfaces of nearly all metals. Growth of surface oxide films takes place only if cations, anions, and electrons can diffuse through the oxide layer. The growth kinetics of very thin films ˚ often follow the Mott or Cabrera–Mott (∼10–50 A) mechanisms in which electrons tunnel through the film and associate with oxygen atoms to produce oxide ions at the surface. A large local electric field (106 –107 V/cm) results at the surface which facilitates cation diffusion from the metal–oxide interface to an interstitial site of the oxide. The film thickness Z at time t is given by Z = α1 ln (α2 t + 1)

(63)

or inverse logarithmic 1/Z = α3 − α4 ln t

(64)

law of growth depending on whether electron tunneling or cation diffusion is rate limiting. The constants α1 –α4 are determined by the material, its structure, and the reaction conditions. The electron field strength and rate of growth decrease exponentially as the film thickens, resulting in an effective limiting thickness for the surface oxide layer. In addition to surface oxides, a vast array of surface compounds can be produced from the reactions of halogens, chalcogenides, and carbon-containing molecules with metal surfaces. Chemisorption of chlorine near 300 K on Cu, Ti, W, Mo, Ta, Ni, Pd, and Au, for example, results in the formation of stable surface compounds which often evaporate as molecular chlorides upon heating at elevated temperatures. Chemisorption of chlorine at 300 K on Ag(100) and Ag(111) produces chemisorbed chlorine overlayers which react irreversibly at about 425 K to produce AgCl with an activation energy of 56 kJ mol−1 . Upon heating, AgCl desorbs at about 830 K with a desorption activation energy of 192 kJ mol−1 . MBRS has been used to investigate the dynamics of several surface corrosion reactions at low reactant pressures. Systems studied include the oxidation Si, Ge, Mo, and graphite, and the halogenation of Si, Ge, Ta, and Ni. With the exception of silicon and germanium oxidation, where dissociative chemisorption of oxygen is apparently rate limiting, the kinetics of these surface reactions generally appear to be controlled by surface or bulk diffusion of the reacting species.

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E. Catalytic Surface Reactions A major goal of basic surface chemistry is in trying to understand heterogeneous catalysis on an atomic scale. Virtually all chemical technologies and many technologies in other fields use catalysis as an essential part of the process. The most important catalytic processes are summarized in Table V. These processes are listed together with the pertinent chemical reactions, widely used catalysts, and typical reaction conditions. There are several definitions of a catalyst, one general definition being that it is a substance that accelerates a chemical reaction without visibly undergoing chemical change. Indeed, a major role of a catalyst is in accelerating the rate of approach to chemical equilibrium. However, a catalyst cannot change the ultimate equilibrium determined by thermodynamics. Another major function of a catalyst is to provide reaction selectivity. Under the conditions in which the reaction is to be carried out, there may be many reaction channels, each thermodynamically feasible, that lead to the formation of different products. The selective catalyst will accelerate the rate of only one of these reactions so that only the desired product molecules form with near-theoretical or 100% efficiency. One example is the dehydrocyclization of n-heptane to toluene: CH3 CH3

(CH2)5

CH3

4H2

This is a highly desirable reaction that converts aliphatic molecules to aromatic compounds. The larger concentration aromatic component in gasoline, for example, greatly improves its octane number. However, n-heptane may participate in several competing simpler reactions. These include hydrogenolysis, which involves C C bond scission to form smaller molecular weight fragments (methane, ethane, and propane); partial dehydrogenation, which produces various olefins; and isomerization, which yields branched chains. All of these reactions are thermodynamically feasible, and since they appear to be less complex than dehydrocyclization, they compete effectively. A properly prepared platinum catalyst surface catalyzes the selective conversion of n-heptane to toluene without permitting the formation of other products. The catalyst selectivity is equally important for the reactions of small molecules (such as the hydrogenation of CO to produce a desired hydrocarbon) or very large molecules of biological importance, where enzyme catalysts provide the desired selectivity. Catalysis is a kinetic phenomenon; we would like to carry out the same reaction at an optimum rate over and over again using the same catalyst. In most cases such a steady-state operation is desirable and aimed for. In

the sequence of elementary reactions that include adsorption, surface migration, chemical rearrangements, and reactions in the adsorbed state, and desorption of the products, the rate of each step must be of steady state. The rate of the overall catalytic reaction per unit area catalyst surface can be expressed as (moles of product/catalyst area × time). Another expression for catalytic rate is the turnover number or turnover frequency. This is the (number of molecules of product/number of catalyst sites × time). For most heterogeneous catalyzed small molecule reactions the turnover number varies between 10−2 and 102 sec−1 . The calculation of the turnover number is limited by the difficulty of determining the true number of active sites. The reaction probability reveals the overall efficiency of a catalyst. It is defined as follows: reaction probability rate of formation of product molecules rate of incidence of reactant molecules. The determination of the rates of the net catalytic reactions and how the rates change with temperature and pressure is of great practical importance. Although there are many excellent catalysts that permit the achievement of chemical equilibria (for example, Pt for oxidation of CO and hydrocarbons to CO2 and H2 O), most catalyzed reactions are still controlled by the kinetics of one of the surface processes. From the knowledge of the activation energy and the pressure dependencies of the overall reaction, the catalytic process can be modeled and the optimum reaction conditions can be calculated. Such kinetic analysis, based on the macroscopic rate parameters, is vital for developing chemical technologies based on catalytic reactions. The rates of reactions are extremely sensitive to small changes of chemical bonding of the surface species that participate in the surface reaction. Since the energy necessary to form or break the surface bonds appears in the exponent of the Arrhenius expression for the rate constant for the overall reaction, it can increase or decrease the rate exponentially. For example, a change of 3 kcal in the activation energy alters the reaction rate by over an order of magnitude at 500 K. Small variations of chemical bonding at different surface irregularities, steps, and kinks, as compared to atomic terraces, can give rise to a very strong structure sensitivity of the reaction rates and the product distribution. Rate measurements exponentially magnify the energetic alterations that occur on the surface and could provide a very sensitive probe of structural and electronic changes at the surface and changes of surface bonding on the molecular scale. One of the most important considerations in catalysis is the need to provide a large contact area between the =

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Surface Chemistry TABLE V Chemical Processes Based on Heterogeneous Catalysis Processes Ammonia synthesis

Typical reactions

Catalyst

N2 + 3H2 → 2NH3

Triply promoted iron (Fe K2 O Al2 O3 CaO);

Dehydrogenation

Fe2 O3 Cr2 O3 K2 O mixed metal oxides

H2

Reaction conditions

720–800 K 40–100 atm 800–900 K 10–50 atm

Epoxidation Fischer–Tropsch synthesis of hydrocarbons Fischer–Tropsch synthesis of oxygenates

C2 H4 + 12 O2 → C2 H4 O CO + H2 → alkanes olefins aromatics

AgCl K2 O/Al2 O3 Fe3 O4 K2 O/Al2 O3 supported Co, Ru, Ni, Rh

CO + H2 → aldehydes acids alcohols

Rh2 O3 ·H2 O K2 O LaRhO4 supported Pd, Pt

520–600 K 500–700 K 10–50 atm 500–700 K 10–50 atm

Hydrotreating (desulfurization and denitrification)

R S R + H2 → 2RH + H2 S R N—R + 32 H2 → 2RHH + NH3

Co Mo, Ni Mo,

570–770 K

Ni Co Mo/Al2 O3 Ni W/Al2 O3 , MoS2 , WS2

30–200 atm

Olefins

Solid acids, zeolites Group VIII metals

270–470 K 1–5 atm

Xylenes

ZSM-5-zeolites

480–580 K 2–5 atm

Alkanes

Zeolites, Pt/Al2 O3

570–770 K 5–50 atm

Isomerization

Methanol synthesis

CO + 2H2 → CH3 OH

ZnCrO3 ZnO Cu2 O Cr2 O3 ZnO Cu2 O Al2 O3

570–670 K

Methanol to gasoline

CH3 OH → aromatics olefins, H2 O

ZSM-5-zeolites

480–540 K 2–15 atm

NOx Reduction

NO + 52 H2 → NH3 + H2 O 2NO + 2H2 → N2 + 2H2 O

Ru, Rh, Pd, Pt/SiO2

370–520 K 1–10 atm 450–650 K 1–10 atm

Ru, Rh, metal oxides

2CO + 2NO → 2CO2 + N2 Oxidation

Olefins + Alkanes 2NH3 + CO +

+ O2 → CO2 + H2 O

5 2 O2

1 2 O2

100–600 atm

Group VIII metals

370–670 K

Ag, Fe2 (MoO4 )3

550–570 K

V2 O5

1–10 atm

→ NO + 3H2 O

→ CO2

Partial oxidations Alcohols

CH3 OH + 12 O2 → H2 CO + H2 O

O

o-Xylene

3O2

O 3H2O O

Olefins

C2 H4 + 12 O2 → CH3 CHO

Reforming Dehydrogenation

1 2

O2

V2 O5 SnO2 ·MoO3

O

O2 H2O

O

R

R

Bi2 O3 ·MoO3

Pt, Pt–Re, Pt–Ge

700–800 K

3H2 continues

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TABLE V (Continued ) Processes

Typical reactions

Catalyst

Dehydrocyclization

Pt Au, Pt Re Cu

Reaction conditions 5–50 atm

3H2 Dehydroisomerization

Ir Au/Al2 O3

3H2 Isomerization

H2

Hydrogenolysis

2C3H8

Hydrogenation

H2 Selective Hydrogenation Olefins Alkynes

H2

NiS

R C CH + H2 → RHC CH2

Pt/Al2 O3

Steam reforming

CH4 + H2 O → CO + 3H2

Ni K2 O/Al2 O3

Water gas

CO + H2 O → CO2 + H2

Fe2 O3 ·Cr2 O3 ZnO Cu2 O

reactants and the surface. The total rate (moles of product per time) is proportional to the surface area. As a consequence, a lot of effort is expended to prepare large surfaces area catalysts and to measure the surface area accurately. One example of high-surface-area catalysts is the group of catalysts known as zeolites, which are aluminosilicates used for the cracking of hydrocarbons. They have crystal ˚ in size. The structure of one structures full of pores 8–20 A of the many zeolites used for catalysis, faujasite, is shown in Fig. 44. Since the catalytic reactions occur inside the pores, an enormous inner surface area, of several hundred

FIGURE 44 Line drawing of the structure of the zeolite faujasite.

420–500 K 1–10 atm 220–250 K 1–10 atm 850–1100 K 30–100 atm 650–800 J 20–50 atm

square meters per gram of catalyst, is available in these catalyst systems. Transition-metal catalysts are generally ˚ employed in a small, 10- to 100-A-diameter particle form dispersed on large-surface-area supports. The support can be a specially prepared alumina or silica framework (or a zeolite) that can be produced with surface areas in the 102 -m2 /g range. These supported metal catalysts are often available with near-unity dispersion (dispersion is defined as the number of surface atoms per total number of atoms in the particle) of the metal particles and are usually very stable in this configuration during the catalytic reaction. The metal is frequently deposited from solution as a salt and then reduced under controlled conditions. Alloy catalysts and other multicomponent catalyst systems can also be prepared in such a way that small alloy clusters are formed on the large-surface-area oxide supports. Most catalytic reactions take place via the formation of intermediate compounds between the reactants or products and the surface. The surface atoms of the catalyst form strong chemical bonds with the incident molecules, and it is this strong chemical surface–adsorbate interaction which provides the driving force for breaking highbinding-energy chemical bonds (C C, C H, H H, N N, and C O bonds), which are often an important part of the catalytic reaction. A good catalyst will also permit rapid bond breaking between the adsorbed intermediates and the surface and

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the speedy release or desorption of the products. If the surface bonds are too strong, the reaction intermediates block the adsorption of new reactant molecules, and the reaction stops. For too-weak adsorbate-catalyst bonds, the necessary bond-scission processes may be absent. Hence, the catalytic reaction will not occur. A good catalyst is thought to be able to form chemical bonds of intermediate strength. These bonds should be strong enough to induce bond scission in the reactant molecules. However, the bond should not be too strong to ensure only short residence times for the surface intermediates and rapid desorption of the product molecules, so that the reaction can proceed with a large turnover number. Of course, activity is only one of many parameters that are important in catalysis. The selectivity of the catalyst, its thermal and chemical stability, and dispersion, are among the other factors that govern our choices. While macroscopic chemical-bonding arguments can explain catalytic activity in some cases, atomic-scale scrutiny of the surface intermediates, catalyst structure, and composition, and an understanding of the elementary rate processes are necessary to develop the optimum selective catalyst for any chemical reaction. One of the important directions of research in catalysis is the identification of the reaction intermediates. The surface residence times of many of these species are longer than 10−5 sec under most catalytic reaction conditions (as inferred from the turnover frequency). They may be detected by suitable spectroscopic techniques either during the steady-state reaction or when isolated by interrupting the catalytic process. The concept of active sites is an important one in catalysis. A surface generally possesses active sites in numbers that are smaller than the total number of surface atoms. The presence of unique atomic sites of low coordination and different valency that are very active in chemical reactions has been clearly demonstrated by atomic-scale studies of metal and oxide surfaces. A catalytic reaction is defined to be structure sensitive if the rate changes markedly as the particle size of the catalyst is changed. Conversely, the reaction is structure insensitive on a given catalyst if its rate is not influenced appreciably by changing the dispersion of the particles under the usual experimental conditions. In Table VI we list several reactions that belong to these two classes. Clearly, variations of particle size give rise to changes of atomic surface structure. The relative concentrations of atoms in steps, kinks, and terraces are altered. Nevertheless, no clear correlation has been made to date between variations of macroscopic particle size and the atomic surface structure. Most surface reactions and the formation of surface intermediates involve charge transfer, either an electron transfer or a proton transfer. These processes are often

TABLE VI Structure-Sensitive and StructureInsensitive Catalytic Reactions Structure sensitive

Structure insensitive

Hydrogenolysis Ethane: Ni Methylcyclopentane: Pt Hydrogenation Benzene: Ni Isomerization Isobutane: Pt Hexane: Pt Cyclization Hexane: Pt Heptane: Pt

Ring opening Cyclopropane: Pt Hydrogenation Benzene: Pt Dehydrogenation Cyclohexane: Pt

viewed as modified acid–base reactions. It is common to refer to an oxide catalyst as acidic or basic according to its ability to donate or accept electrons or protons. The electron transfer capability of a catalyst is expressed according to the Lewis definition. A Lewis acid is a surface site capable of receiving a pair of electrons from the adsorbate. A Lewis base is a site having a free pair of electrons that can be transferred to the adsorbate. The proton-transfer capability of a catalyst is expressed according to the Brønsted definition. A Brønsted acid is a surface site capable of losing a proton to the adsorbate while a Brønsted base is a site that can accept a proton from the adsorbed species. Perhaps the most widely used catalysts, the zeolites, best represent the group of oxides that exhibit acid–base catalysis. Zeolites are alumina silicates, some of which are among the more common minerals in nature. Modern synthesis techniques permit the preparation of families of zeolite compounds with different Si/Al ratios. Since the Al3+ ions lack one positive charge in the tetrahedrally coordinated silica, Si4+ , framework, they are sites of proton or alkali–metal affinity. Variation of the Si/Al ratio gives rise to a series of substances of controlled but different acidity. By using various organic molecules during the preparation of these compunds that build into the structure, subsequent decomposition leaves an open pore structure, where the pore size is controlled by the skeletal structure of the organic deposit. Very high internal surface area catalysts (102 m2 /g) can be obtained this way ˚ and controlled acidity with controlled pore sizes of 8–20 A [(Si/Al) ratio]. These catalysts are utilized in the cracking and isomerization of hydrocarbons that occur in a shape selective manner as a result of the uniform pore structure and are the largest volume catalysts in petroleum refining. They are also the first of the high-technology catalysts in which the chemical activity is tailored by atomic-scale

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418 study and control of the internal surface structure and composition. A catalyst used in industry is very rarely a pure element or compound. Most catalysts contain a complex mixture of chemical additives or modifiers that are essential ingredients for high activity and selectivity. Promoters are beneficial additives that increase activity, selectivity, or useful catalyst lifetime (stability). Structural promoters inhibit sintering of the active catalyst phase or present compound formation between the active component and the support. The most frequently used chemical promoters are electron donors such as the alkali metals or electron acceptors such as oxygen and chlorine. For example, in the petroleum industry, chlorine and oxygen are often added to commercial platinum catalysts used for reforming reactions by which aliphatic straight-chain hydrocarbons are converted to aromatic molecules (dehydrocyclization) and branched isomers (isomerization). These additives accomplish several tasks during the reaction. By changing the chemical bonding of some of the surface intermediates, the steady-state concentration of these intermediates may be altered, and thus a somewhat higher concentration of the catalytically active species is obtained. In this way the rate of the reaction is increased and the selectivity may be improved. Often multicomponent catalyst systems are utilized to carry out reactions consisting of two or more active metal components or both oxide and metal constituents. For example, a Pt–Rh catalyst facilitates the removal of pollutants from car exhausts. Platinum is very effective for oxidizing unburned hydrocarbons and CO to H2 O and CO2 , and rhodium is very efficient in reducing NO to N2 , even in the same oxidizing environment. Dual functional or multifunctional catalysts are frequently used to carry out complex chemical reactions. In this circumstance the various catalyst components should not be thought of as additives, since they are independently responsible for different catalytic activity. Often there are synergistic effects, however, whereby the various components beneficially influence each other’s catalytic activity to provide a combined additive and multifunctional catalytic effects. It should be clear from this discussion that the working, active, and selective catalyst is a complex, multicomponent chemical system. This system is finely tuned and buffered to carry out desirable chemical reactions with high turnover frequency and to block the reaction paths for other thermodynamically equally feasible but unwanted reactions. Thus, an iron catalyst or a platinum catalyst is composed not only of iron or platinum but of several other constituents as well to ensure the necessary surface structure and oxidation state of surface atoms for optimum catalytic behavior. Additives are often used to block sites,

Surface Chemistry

prevent side reactions, and alter the reaction paths in a variety of ways. While industrial catalytic systems are complex and are not readily suited to basic science studies to understand how they work on an atomic scale, one approach to their understanding is the synthetic approach. In this approach we begin with a very simple system then synthesize complexity from this. The catalyst particle is viewed as composed of single crystal surfaces, as shown in Fig. 45. Each surface has different reactivity and the product distribution reflects the chemistry of the different surface sites. We may start with the simplest single crystal surface [for example, the (111) crystal face of platinum] and examine its reactivity. It is expected that much of the chemistry of the dispersed catalyst system would be absent on such a homogeneous crystal surface. Then high-Miller-index crystal faces are prepared to expose surface irregularities, steps, and kinks of known structure and concentration, and their catalytic behavior is tested and compared with the activity of the dispersed supported catalyst under identical experimental conditions. If there are still differences, the surface composition is changed systematically or other variables are introduced until the chemistries of the model system and the working catalyst become identical. This approach is described by the following sequence:

FIGURE 45 Catalyst particle viewed as a crystallite, composed of well-defined atomic planes. (Courtesy of Lawrence Berkeley Laboratory.)

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structure of crystal surfaces and adsorbed gases ↓↑ surface reactions on crystals at low pressures (≤10−4 torr) ↓↑ surface reactions on crystals at high pressures (10+3 −10+5 torr) ↓↑ reactions on dispersed catalysts Investigations in the first step define the surface structure and composition on the atomic scale and the chemical bonding of adsorbates. Studies in the second step, which are carried out at low pressures, reveal many of the elementary surface reaction steps and the dynamics of surface reactions. Studies in the third and fourth steps establish the similarities and differences between the model system and the dispersed catalyst under practical reaction conditions. The advantage of using small-area catalyst samples is that their surface structure and composition can be prepared with uniformity and can be characterized by the many available surface diagnostic techniques. In this approach to catalytic reaction studies the surface composition and structure are determined in the same chamber where the reactions are performed, without exposing the crystal surface to the ambient atmosphere. This necessitates the combined use of an ultrahigh vacuum enclosure, where the surface characterization is carried out, and a high-pressure isolation cell, where the catalytic studies are performed. Such an apparatus is shown in Fig. 46. The small-surface-area (approximately 1-cm2 ) catalyst is placed in the middle of the chamber, which can be evacuated to 10−9 torr. The surface is characterized by LEED and AES and by other surface diagnostic techniques. The

lower part of the high-pressure isolation cell is then lifted to enclose the sample in a 30-cm3 volume. The isolation chamber can be pressurized to 100 atm if desired and is connected to a gas chromatograph that detects the product distribution as a function of time and surface temperature. The sample may be heated resistively both at high pressure or in ultrahigh vacuum. After the reaction study the isolation chamber is evacuated, opened, and the catalytic surface is again analyzed by the various surface-diagnostic techniques. Ion bombardment cleaning of the surface or means to introduce controlled amounts of surface additives by vaporization are also available. The reaction at high pressures may be studied in the batch or the flow mode. Typical catalytic reactions that have been investigated, in some detail, using this approach include hydrocarbon conversion on platinum and modified platinum surfaces (isomerization, hydrogenolysis, hydrogenation, dehydrogenation and cyclization), dehydrosulfurization on molybdenum, ammonia synthesis on iron, and carbon monoxide hydrogenation on iron. F. Photochemical Surface Reactions Photochemical surface reactions form their own class due to the fact that a thermodynamically uphill reaction ( G > 0) may be carried out with the aid of an external source of energy, light. In fact, one of the most important chemical reactions of our planet, photosynthesis, requires the input of 720 kcal/mol of energy to convert carbon dioxide and water to one mole of sugar: light

6CO2 → 6H2 O−−−−−−→6H12 O6 + 6O2 chlorophyll

FIGURE 46 Schematic representation of the experimental apparatus to carry out catalytic reaction-rate studies on single-crystal surfaces of low surface area at low and high pressures in the range 10−7 to 104 torr.

(65)

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It is useful to consider light as one of the reactants in photosynthesis. By adding the light energy to Eq. (65), the reaction becomes athermic or even exothermic if excess light energy is utilized, hv + H2 O + CO2 = CH2 + 32 O2 We may consider photon-assisted or photochemical reactions of many types that lead to the formation of lower molecular weight hydrocarbons and of other products. One of the simplest of these important new classes of reactions leads to the dissociation of water: hv + H2 O = H2 + 12 O2

(66)

Another leads to the formation of methane: hv + CO2 + 2H2 O = CH4 + 32 O2

(67)

or to the fixation of nitrogen: hv + 3H2 O + N2 = 2NH3 + 32 O2

(68)

Light as a reactant may be employed in two ways. The adsorbed molecules can be excited directly by photons of suitable energy to a higher vibrational or electronic states. The excited species then may undergo chemical rearrangements or interactions that are different from those in the ground vibrational or electronic states. Alternatively, the solid can be excited by light in the near-surface region. Photons of band-gap or greater energy may excite electron–hole pairs at the surface. As long as these charge carriers have a relatively long lifetime (i.e., they are trapped at the surface, so that their recombination is not an efficient process), there is a high probability of their capture by the adsorbed reactants. These, in turn, can undergo reduction or oxidation processes using the photogenerated electrons and holes, respectively. The photographic process is one example of this type of surface photochemical reaction. However, we would like the photogenerated electrons and holes to be captured by the adsorbed molecules in order to carry out photochemical surface reactions of the adsorbates instead of the photodecomposition of the solid at the surface. The cross sections for adsorption of band-gap or higher-than-band-gap energy photons are so large that the photogeneration of electron–hole pairs is a most efficient process. At present, this cannot be readily matched by the efficiency of direct photoexcitation of vibrational or electronic energy states of the adsorbed molecules. Many solid surfaces efficiently convert light to longlived electron–hole pairs that can induce the chemical changes leading to the reactions in Eqs. (66)–(68). In fact, inorganic photoreaction is one of the exciting new fields of surface science and heterogeneous catalysis. It is important to distinguish between thermodynamically uphill photochemical reactions and thermodynam-

ically allowed photon-assisted reactions. The latter reactions are thermodynamically feasible without any external energy input, but light is used to obtain certain product selectively. Excitation of selected vibrations, rotations, or electronic states of the incident or adsorbed molecules by light permits us to change the reaction path or increase the reaction rate. For example, the hydrogenation of acetylene or the oxidation of ammonia can be photon-assisted, leading to different reaction rates than in the absence of light. As an example of a photocatalyzed surface reaction we discuss the photoelectrochemical dissociation of water. It was shown in 1972 that upon illumination of reduced titanium oxide (TiO2 ), which served as the anode in basic electrolyte solution, oxygen evolution was detectable at the anode, and hydrogen evolved at a metal (platinum) cathode. This reaction requires an energy of 1.23 V/electron (a two-electron process per dissociated water molecule). In the presence of light of energy equal to or greater than the band-gap energy of titanium oxide (3.1 eV), an external voltage as low as 0.2 V was sufficient to dissociate water. The process stopped as soon as the light was turned off, and started again upon reillumination. Shortly after, several other systems showed the ability to carry out photon-assisted dissociation of water. When p-type gallium phosphide, GaP, was used as a cathode instead of platinum upon illumination of the TiO2 anode, O2 and H2 could be generated at the semiconductor anode and cathode, respectively, without the need of applying any external potential. When strontium titanate, SrTiO3 , was substituted for TiO2 as the anode, H2 O photodissociation was found to take place without external potential even when a platinum cathode was employed.

FIGURE 47 Energy conditions needed to reduce B+ to B and oxidize A− to A at a semiconductor surface. Electrons that are excited by photons into the conduction band ECB must be able to reduce B+ , and electron vacancies (holes) in the valence band E VB must be able to oxidize A− .

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Figure 47 shows a schematic energy diagram to indicate the conditions necessary to carry out photoelectrochemical reactions efficiently. If the band-gap energy is greater than the free energies for the reduction and oxidation reactions, the photoelectron that is excited into the conduction band by light could reduce B+ to B by electron transfer from the surface to the molecule. The photogenerated electron vacancies (holes) could also oxidize the A− anions to A by capturing the electron. For the photodissociation of water, the conduction band must be above the H+ /H2 potential and the valence band below the O2 /OH− potential to be able to carry out the photoreaction without an external potential. The band gap must be greater than 1.23 V and the flat-band potential of the conduction and valence bands energetically well placed with respect to the (H+ /H2 ) and O2 /OH− couples. The flat-band potentials can be obtained by capacitance measurements as a function of external potential. There is, of course, considerable band bending of the conduction and valence bands of any semiconductor at the surface. This is due to the presence of localized electronic surface states and to charge transfer between the adsorbates and semiconductor. Potential-energy diagrams that show the band positions schematically at an n-type or ptype semiconductor liquid interface are shown in Fig. 48. The band bending provides an efficient means of separating electron–hole pairs, since the potential gradient as shown for the n-type semiconductor drives the electrons away from the semiconductor surface while it attracts the holes in the valence band toward the semiconductor electrolyte interface. As a result, the oxidation reaction takes place at the oxide anode while the reduction reaction takes place at the cathode to which the photoelectron migrates along the external circuit. The magnitude of the band bending at the surface depends primarily on the carrier concentration in the semiconductor and on the electron-donating or -accepting abilities of the adsorbates at the surface. Semiconductors that are not likely to carry out the photodissociation of water, according to the location of their

FIGURE 48 Band bending at the n-type and p-type semiconductor interfaces.

flat-band potential, may become photochemically active as a result of strong band bending at the surface. Often the oxidation or reduction photoreactions lead to the decomposition of the semiconductor electrode material. Instead of the photoreactions of adsorbate ions or molecules, a solid-state photoreaction occurs. This is particularly noticeable at the surfaces of illuminated CdS, Si, and GaP. Much of the research is therefore directed toward stabilizing these photoelectrode materials by suitable adsorbates that could prevent the occurrence of photodecomposition by providing an alternative chemical route for the photoreduction or photooxidation.

ACKNOWLEDGMENT This work was supported by the Assistant Secretary for Energy Research, Office of Basic Energy Sciences, Materials Sciences Division of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.

SEE ALSO THE FOLLOWING ARTICLES ADHESION AND ADHESIVES • ADSORPTION • AUGER ELECTRON SPECTROSCOPY • BONDING AND STRUCTURE IN SOLIDS • CATALYSIS, INDUSTRIAL • CATALYST CHARACTERIZATION • CHEMICAL THERMODYNAMICS • CRYSTALLOGRAPHY • PHOTOCHEMISTRY, MOLECULAR • PHOTOELECTRON SPECTROSCOPY • SOLID-STATE ELECTROCHEMISTRY • TRIBOLOGY

BIBLIOGRAPHY Adamson, A. W. (1982). “Physical Chemistry of Surfaces,” 4th ed., Wiley, New York. Anderson, J. R., and Boudart, M. (1981). “Catalysis Science and Technology,” Vols. 1–7, Springer-Verlag, Berlin/New York. Ertl, G., and Gomer, R., eds. (1983). “Springer Series in Surface Sciences,” Vols. 1–4, Springer-Verlag, Berlin/New York. Ertl, G., and Kuppers, J. (1979). “Low Energy Electrons and Surface Chemistry,” Verlag Chemie, Weinheim. Feuerbacher, B., Fitton, B., and Willis, R. F. (1979). “Photoemission and the Electronic Properties of Surfaces,” Wiley, New York. King, D. A., and Woodruff, W. P., eds. (1983). “The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis,” Vols. 1–4, Elsevier, New York. Morrison, S. R. (1977). “The Chemical Physics of Surfaces,” Plenum, New York. Roberts, M. W., and McKee, C. S. (1978). “Chemistry of the Metal-Gas Interface,” Oxford Univ. Press, London. Somorjai, G. A. (1981). “Chemistry in Two Dimensions: Surfaces,” Cornell Univ. Press, Ithaca, NY. Tompkins, F. C. (1978). “Chemisorption of Gases on Metals,” Academic Press, NY. Vanselow, R., and Howe, R., eds. (1979). “Chemistry and Physics of Solid Surfaces,” Vols. 1–6, Springer-Verlag, Berlin/New York.

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Ligand Field Concept Gunter Gliemann

Yu Wang

University of Regensburg

National Taiwan University

I. II. III. IV. V.

Basic Experimental Findings Model and Theory Optical Properties of Complexes Magnetic Properties of Complexes Stabilization of Complexes

GLOSSARY Charge transfer transition Change of the electronic state of a complex ion by the transfer of an electron from mainly the central ion state to mainly the ligand system state or vice versa. Coordination number Number of ligands bound at the central ion. Electron spin Intrinsic angular momentum vector of the electron. It is a quantum phenomenon, which has no analog in classical mechanics. Jahn-Teller effect For a nonlinear molecule in an electronically degenerate ground state, distortion must occur to lower the symmetry and to lower the energy to a more stable nondegenerate ground state. Ligand Atomic ion, molecular ion, or molecule coordinated at the central ion of a complex. Magnetic moment A property associated with a magnetic domain. It is an experimental measure of the magnetism of a compound, generally measured in units of magnetons. Orbital angular momentum Mechanical vector quantity perpendicular to the orbit of a particle. Its magni-

tude depends on the orbit diameter and the mass and velocity of the particle. Spin crossover complex Complex undergoes a spin transition induced by certain external factor such as temperature, pressure, light, etc. Tanabe-Sugano diagram Term splitting as a function of ligand field strength for 3d-transition metal ions in octahedral field, originated by Tanabe and Sugano. Term Entity of states of equal energy. Transition metals Elements with incompletely filled d orbitals: scandium, titanium, through copper (3d series); yttrium, zirconium through silver (4d series); lanthanum, hafnium through gold (5d series).

THE LIGAND FIELD CONCEPT is the basis of a quantum theoretical model developed in the 1950s for describing the electron systems of transition metal complexes. A transition metal complex is composed of a transition metal ion (central ion) surrounded by a system of ligands (atomic ions, molecular ions, or molecules). The ligands produce the electrical field (the ligand field) acting on the electron system of the central ion. As the ligand field theory shows,

523

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the optical, magnetic, and stability properties of transition metal complexes strongly depend on the symmetry and strength of the ligand field.

I. BASIC EXPERIMENTAL FINDINGS Crystallized salts of metal complexes have an amazing variety of brilliant colors, which are expressed by such prefixes as violeo-, praseo-, luteo-, purpureo-, and roseo(Table I). Investigations of the underlying chemical structures reveal a close relationship between color and composition, an immediate challenge for the spectroscopist. Therefore, after the foundation of coordination theory by A. Werner (1907) extended studies on the absorption spectra of complexes were investigated. The aim of these studies was to derive relations between the number and the position of absorption bands and the nature of the central ion and the ligands. Because the techniques and apparatus for measuring absorption spectra were still rather undeveloped, endeavors to interpret the spectra according to the theory of electrons remained unsatisfactory until the 1940s. But it was soon recognized that the existence of d electrons is significant for the color of transition metal ion complexes. The outstanding work of M. Linhard and coworkers, starting in 1944 with the investigation of Co3+ and Cr3+ complexes, set the standard for the precision of absorption spectroscopy of dissolved complexes. One of the main results of Linhard’s work is indicated schematically in Fig. 1. The absorption spectra of transition metal complexes can be divided into two spectral regions. In the longwavelength region (λ ≥ 350–400 nm) one finds one or more weak bands (extinction coefficient ε = 1–102 liters mol−1 cm−1 ). These bands do not appear when the central ion is not a transition metal ion (e.g., Al3+ instead of Cr3+ ; see Fig. 2). Therefore, these weak bands were assigned to transitions involving d electrons of the central ion (central ion bands, d-d bands). In the short-wavelength region (λ ≤ 350–400 nm) strong absorption bands (ε ∼ 103 – 106 liters mol−1 cm−1 ) are observed. These strong bands (ligand bands) are usually charge transfer bands, due to electron transfer between the central ion and the ligand system, or intraligand bands, caused by excitation of the electron system of the ligands.

FIGURE 1 A schematic absorption spectrum of a transition metal complex ion in the UV-VIS region. The central ion bands correspond to d →d transitions; the ligand bands correspond to charge transfer transitions and/or to intraligand transitions.

Stimulated by Linhard’s work, in 1946 F. E. Ilse and H. Hartmann formulated the ligand field concept, based on the classical ionic model of transition metal complexes (W. Kossel and A. Magnus) and on appropriate group theoretical methods (H. Bethe).

TABLE I Nomenclature Describing the Colors of Some Transition Metal Complexes TM complexes

Color

Prefix

cis-[Co(NH3 )4 Cl2 ]+ trans-[Co(NH3 )4 Cl2 ]+

Violet

Violeo-

Green

Praseo-

[Cr(NH3 )6 ]3+

Yellow

Luteo-

[Co(NH3 )5 H2 O]3+ [Co(NH3 )5 Cl]2+

Rose Purple red

RoseoPurpureo-

FIGURE 2 Absorption spectra of [Cr(C2 O4 )3 ]3− (full line) and ¨ H. L. (1957). Z. Phys. [Al(C2 O4 )3 ]3− (dotted line). [From Schlafer, Chem. 11, 65.]

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FIGURE 3 L-edge absorption spectra of Fe2+ in Fe(phen)2 (NCS)2 at 298 K (solid line) and at 15 K (dotted line). [From Lee, J. J., Sheu, H. S., Lee, C. R., Chen, J. M., Liu, R. G., Lee, J. F., Wang, C. C., Huang, C. H., and Wang, Y. (2000). J. Am. Chem. Soc. 122, 5742.]

In addition to the absorption spectra at UV-VIS range, the absorption in the much higher energy range has also been observed in recent years. Here the electron transition is between the inner core orbitals and the valence orbitals of the central ions. An example of Fe L-edge absorption is given in Fig. 3 to display the electron transition between 2 p and 3d orbitals. The magnetic properties of the transition metal complexes are known to exhibit quite a variety even with the same metal ion, for example, diamagnetic, and paramagnetic are found in Fe2+ , Co3+ complexes of various ligands (low-spin, high-spin complexes). The uneven distribution of electron density around the metal ion in a complex is demonstrated by the deformation density at the metal center shown in Fig. 4, where the spherical electron density is subtracted from the observed molecular electron density. Since the spherical density means that even populations are among five degenerated d-orbitals, the deformation density may give the direct observation on the difference among d-orbital population. All these recent experimental findings can be rationalized by the ligand field concept.

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525 tron pairs. Neither the ionic model nor the covalent model provided a satisfactory interpretation of several optical properties, since these models were concerned primarily with the electronic ground state of the complex ions. For a large number of complexes certain spectral regions of the absorption spectra can be assigned alternatively to the different components of the complex (central ion, ligand system). On this account a model that starts with the assumption of separated electron systems for the central ion and for the ligands will be appropriate for a theoretical treatment. If one is interested primarily in the electronic states of the central ion bound in the complex, one has to consider the electric field generated by the ligand system, socalled the ligand field. The effect of this ligand field on the electronic states of the central ion is then taken into consideration. The theory treating this concept of the ligand field is the ligand field theory, where the electronic structure of the ligand system is taken explicitly into account. In other words, the complete molecular orbital treatment of the complex is undertaken. To a first approximation the charge distribution of the ligands is represented by point charges and/or point dipoles in their centers, the ligand field treatment can be reduced to an atomic orbital treatment of the central ion, this extended ionic model is designated as crystal field theory. The starting point of the crystal field theory is the description of the electronic states of the free, isolated transition metal ion (the central ion) in a complex. Information on the electronic ground state and the excited states

II. MODEL AND THEORY In the 1920s and 1930s two apparently contrary models, the ionic model and the covalent model, were developed to explain the binding between the central ion and the ligands, which were both represented by point charges and point dipoles. In the ionic model of Kossel and Magnus, the binding between the central ion and the ligands is due to the electrostatic forces between the components. In the covalent model of Sidgwick and Pauling, the binding between the central ion and the ligands is accomplished by elec-

FIGURE 4 Deformation density of Ni(disn)2 plane around Ni, solid line positive, dotted line negative, contour interval is 0.1 eA˚ −3 . [From Lee, C. S., Hwang, T. S., Wang, Y., Peng, S. M., and Hwang, C. S. (1996). J. Phys. Chem. 100, 2934–2941.]

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of the free ions is available from the analysis of the corresponding absorption or emission spectra and/or from quantum mechanical calculations. A. Electronic States of a Free Ion The electronic states of a free ion can be characterized by their energy and by their angular momentum, which includes the orbital angular momenta and the spins of the electrons involved. For atoms with atomic number Z ≤ 30, the Russell-Saunders coupling is a good approximation. In this approximation the orbital angular momenta of electrons are added vectorically to the total orbital angular momentum L and the spins of the electrons are coupled to the total spin S: Ij = L; sj = S j

j

where lj and sj are the orbital and the spin angular momentum of the jth electron, respectively. As shown by quantum mechanics the absolute values of the vectors L and S are restricted to certain discrete amounts |L| = h L(L + 1), L = 0, 1, 2, 3, . . . |S| = h S(S + 1), 0, 1, 2, 3, . . . for even number of electrons S= 1/2, 3/2, 5/2, 7/2, . . . for odd number of electrons where h is h/2π (h is Planck’s constant); L and S is the quantum number of the total orbital angular momentum and the total spin of the system of electrons, respectively. For the numerical values L = 0, 1, 2, 3, . . . it is conventional to use the letters S, P, D, F, . . . By quantum mechanical rules spin S with quantum number S can take 2S + 1 to different special orientations with respect to a given direction shown in Fig. 5. The 2S + 1 is denoted as multiplicity M. Correspondingly, an orbital angular momentum L with quantum number L can assume 2L + 1 different spatial orientations. Therefore, a given set of quantum numbers L and S can assume a total of (2L + 1)(2S + 1) states with different orientations of orbital and/or spin angular momentum. These states form a Russell-Saunders term, symbolized by 2S+1 L. All (2L + 1)(2S + 1) states of the same term have equal energies, if the coupling between L and S is ignored. They are energetically “degenerate.” Usually, different terms have different energies. By Hund’s rule the term with the highest multiplicity M and the highest L value is the ground-state term (energetically most stable term). The energy difference between the terms is expressed in terms of B, an energy parameter of electronelectron repulsion. For example, a d 2 ion (two d electrons) will result in various states with multiplicity 1 (singlet; spin configu-

FIGURE 5 Spatial orientations of a spin vector S with S = 32 . There are 2S + 12 = 4 different allowed values of projection on the z axis.

ration with the two electron spins antiparallel, ↑↓) and 3 (triplet, ↑↑). There are, in total, two triplet terms and three singlet terms: 3 P, 3 F, 1 S, 1 D, and 1 G. The 3 F [composed of (2 × 3 + 1)(2 × 1 + 1) = 21 states] is the groundstate term. The five Russell-Saunders term of a d 2 ion are shown in the energy-level diagram of Fig. 6. The complete sets of Russell-Saunders terms for the d N ions with N = 1, 2, . . . , 9 are given in Table II. TABLE II Russell-Saunders Terms for Free dN Ionsa Occupation of the d shell d1, d9 d2,

d8

Russell-saunders term 2D 3 F, 3 P 1 G, 1 D, 1 S

d3, d7

4 F, 4 P 2 H, 2 G, 2 F, a2 D, 2 P

d4, d6

5D 3 H, 3 G, a3 F,

b3 F, 3 D, a3 P, b3 P

1 I, a1 G, b1 G, 1 F, a1 D, b1 D, a1 S, b1 S

d5

6S 4 G, 4 F, 4 D, 4 P 2 I, 2 H, a2 G, b2 G, a2 F, b2 F, a2 D, b2 D, c2 D, 2 P, 2 S

a If for a d N ion (N = 1, . . . , 9) several terms with the same L value and the same S value exist, they are distinguished by prefixes a, b, c. Terms with underlined L symbols are the ground terms for the first d N configuration.

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FIGURE 6 Russell-Saunders terms of a d 2 free ion with and without spin-orbital coupling.

If the coupling between the total orbital angular momentum, L, and the total spin angular momentum, S, has to be taken into account, the total angular momentum J should be used. Where J is the vector sum of L and S. The quantum number of the vector J is again restricted to certain discrete amounts. J = L + S, J = |L + S|, |L + S − 1|, . . . . . . . |L − S| |J| = h J (J + 1) The energy term is symbolized as 2S+1 L J , for example, the ground state of d 2 (3 F) is split into 3 F4 , 3 F3 , and 3 F2 , where the order of energy is such that 3 F4 > 3 F3 > 3 F2 as shown in Fig. 6. Each term is in 2J + 1 degeneracy and can be separated by applying magnetic field, i.e., the Zeemann effect. However, the splitting due to the L − S coupling is much smaller than the splitting due to the electron-electron repulsion designated the energy difference between the Russell-Saunders terms.

B. Crystal-Field Theory In the course of forming a complex, the central ion is bound to be affected by the ligand field (electrical field of the ligands), and thereby the motion modes of the electrons of the central ion will be perturbed. Accordingly, the term system of the central ion will be changed. Some terms of the free ion are energetically merely shifted, while others are split into progeny terms 2S+1 i , i = 1, 2, . . . with different energies (intracomplex Stark effect). The symbol i describes the orbital state of the ith progeny term. The number of the progeny terms 2S+1 i can be exactly determined by the methods of group theory. It depends on the L value of the (parent) term 2S+1 L and on the symmetry of the ligand system: Number of progeny terms 2S+1 i = f (L, symmetry) 1. Term Splitting Under Oh Symmetry The general group theoretical result can be illustrated by a simple model system. First we consider a system of six

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FIGURE 7 Schematic probability distributions and energy states of a p electron (L = 1) in an octahedral and square-planar ligand field induced by negative point charges.

ligands (represented by electrical negative point charges) at the corners of a regular octahedron. That is, the central ion is subjected to an octahedral environment, Oh symmetry. The central ion within this system is to be varied with regard to its orbital angular momentum (quantum number L). To simplify the problem the central ion may contain only one electron, either a p electron (L = 1) or a d electron (L = 2). The corresponding probability distributions of the electron charge known from the theory of atoms are shown in Fig. 7 and Fig. 8. The p electron can occupy three different orbital states (2L + 1 = 3). All three distributions are in equivalent positions with regard to the ligands (Fig. 7). Therefore, the energies of the three


states will be shifted by the same amount when the ligand field is acting on the central ion. No splitting will be observed. For a d electron (2L + 1 = 5), however, there are distributions that are obviously not equivalent with regard to the ligands (Fig. 8). The maxima of the distributions dx y , dx z , and d yz are directed equivalently into the angular bisectors between the bonds. Therefore, these three states will have the same energy. The distribution dz 2 and dx 2 −y 2 have maxima along the bonds to the ligands located on the x, y, and z axis. It follows that the dx 2 −y 2 and dz 2 states will have higher energy than the dx y , dx z , and d yz states because of the stronger repulsion between the electron of central ion and the point charges of the ligands. Quantitative calculations show that the dz 2 state is energetically degenerate with the dx 2 −y 2 state. Therefore, the fivefold D term of a d 1 ion is split into a threefold state t2 and a twofold state e (Fig. 8). From these two examples we see that for the same symmetry of the ligand field the number of progeny terms depends on the quantum number L of the orbital angular momentum of the electron system. Under the influence of an octahedral symmetry of the ligand field, P terms (L = 1) are merely shifted, whereas D terms (L = 2) are split into a twofold (e) and a threefold ( t2 ) degenerate term. Table III summarizes the resulting term-splitting under the octahedral field for L values up to 4. The numbers in parentheses give the orbital degeneracy of the terms. A, B, E, and T symbolize different orbital symmetries of the terms, i.e., the progeny term 2S+1 i mentioned above, where A and B represent nondegenerate, E represents twofold degenerate, and T represents threefold degenerate terms. The term-splitting of the ground states of d N complexes with N = 1 to 5 in an octahedral ligand field are shown schematically in Fig. 9. The d N and d 10−N ions exhibit equivalent splitting diagrams, for example, the ground state of d 8 (3 F) exhibit the same splitting diagram as that of d 2 (T1 + T2 + A1 ) but the order of energy is inverted. Thereby the order of energy is T1 < T2 < A1 for d 2 , but is A2 < T2 < T1 for d 8 . This is a consequence of the so-called electron-hole correlation. 2. Term-Splitting Under D4h Symmetry

FIGURE 8 Schematic probability distributions and energy states of a d electron (L = 2) in an octahedral and square-planar ligand field induced by negative point charges.

For the same quantum number L of the orbital angular momentum, the number of progeny terms depends on the symmetry of the ligand system. When the octahedral symmetry of the ligand system is reduced to a square-planar (D4h ) symmetry by canceling the ligands on the z axis, the pz distribution is subjected to a weaker field than the px and p y distributions. Thus lowering of the symmetry from Oh to D4h yields a term-splitting of T1 to A1 + E. In the same way, the D terms (L = 2) are split further into A1 + B1 + B2 + E terms (Fig. 8). Such splitting is demonstrated by the uneven population of B1 (dx 2 −y 2 ) and B2

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Ligand Field Concept TABLE III Splitting of Orbital States with L = 0, 1, 2, 3, 4 in an Octahedral (Oh ) and a Square-Planar (D4h ) Ligand Field Orbital state of the free ion Quantum number L

State symbola

0 1 2

S(1) P(3) D(5)

→A1 (1) →T1 (3) →E(2) + T2 (3)

→A1 (1) →A2 (1) + E(2)

3

F(7)

4

G(9)

→A2 (1) + T1 (3) + T2 (3) →A1 (1) + E(2) + T1 (3) + T2 (3)

→A2 (1) + B1 (1) + B2 (1) + 2E(2) →2A1 (1) + A2 (1) + B1 (1) + B2 (1) + 2E(2)

a

Orbital states in an Oh ligand field

Orbital states in a D4h ligand field

→A1 (1) + B1 (1) + B2 (1) + E(2)

The numbers in parentheses give the orbital degeneracy of the states.

(dx y ) terms of Ni2+ in roughly D4h symmetry as shown in Fig. 4. As for the splitting of the other terms, they are listed in Table III. It is worth noticing that the lower the symmetry is, the more splitting of the terms occurs. In D4h , the highest degeneracy is E, whereas in Oh , it is T. 3. Weak- and Strong-Field Methods, Term Diagrams The amount of the energetic splitting or shifting can be (approximately) determined by the methods of quantum mechanical perturbation theory. Since the perturbation comes from the electrical interaction between the electrons of the central ion and the charge distribution within the ligand system, the magnitude of energetic splitting or shifting will depend on the central ion-ligand distance R and on the charges q and electrical dipole moments µ of the ligands: Magnitude of energetic splitting and/or shifting = F(R, q, µ) Since the charge distributions within the ligands are not known exactly, an absolute calculation of these energetic effects is very tedious and time-consuming. In practice these magnitudes (as functions of R, q, and µ) are taken

as parameters in the calculation and are fit to experimental data. Normally a value in Dq is represented. Since both the electron-electron interaction and the influence of the ligand field have to be treated, there are two methods for finding the term system of a transition metal complex when the central ion contains two or more d electrons. Both methods start with the free d N ion of which the electron-electron repulsion is not yet taken into account. They differ only in the order of which one is treated first. According to the expected relative amounts of these energetic quantities, the weak-field method is employed when the effect of electron-electron interaction dominates, whereas the strong-field method is appropriate when the influence of the ligand field is dominant. Complete treatments of a complex ion by both methods will ultimately yield the same results. The weak-field method is described by the following scheme: d N −−−−−−→ 2S+1 L −−−−−→ 2S+1 electr on−electr on interaction

ligand f ield

The first step considers the electron-electron interaction. This step gives result in the Russell-Saunders terms 2S+1 L of the free ion following the procedure described in section II.A. In the second step the splitting of these terms by the ligand field is determined, following the procedure given in Section II.B.1 or B.2. The two steps of the strong-field method have the opposite order: d N −−−−−→ t2n e N −n −−−−−→ 2S+1 ligand f ield(Oh )

FIGURE 9 Term-splitting of the ground states of d 1 to d 5 and d 8 in an octahedral ligand field. Term-splitting of the ground states of d N , N > 5 is the same as that in d 10−N , but with the energy order inverted, see d 8 versus d 2 .

electr on interaction

In the first step the splitting of one-electron states of the d shell in the ligand field is considered. Under the octahedral ligand field, the five orbitals of the d shell are split into a family of two degenerate states e(dx 2 −y 2 , dz 2 ) and a family of three degenerate states t2 (dx y , dx z , d yz ) shown in Fig. 8. By quantum mechanical perturbation theory the energy difference between e and t2 is calculated as ≡ 10 Dq. The N d electrons can occupy the levels e and t2 following Pauli’s principle, such as t2n e N −n . In the second step of the

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increasing Dq value the terms are either shifted (L ≤ 1) or split (L > 1). This Tanabe-Sugano diagram will provide nice interpretation to the optical properties of Cr3+ in ruby (Section III). However, when central ion is d N with N = 4 ∼ 7, not only the shifting and splitting of the terms occurred as indicated in d 3 ion. There also appears to be an apparent change at certain Dq/B value (VT ). An example of Fe2+ complex in an Oh environment, a Tanabe-Sugano diagram of d 6 , is shown in Fig. 12. The ground state of the strongfield (Dq/B > VT ) is 1 A1 , a low-spin state, but that of weak-field (Dq/B < VT ) is 5 T2 , a high-spin state. When the Dq/B value is very close to VT , the spin crossover phenomenon occurs, where the spin state of the central ion can be fine tuned by varying temperature or pressure. The light-induced excitation of the spin state was also observed.

FIGURE 10 Term correlation diagram of d 2 in weak- and strong ligand fields.

strong-field method, the electron-electron interaction is taken into account. Take d 2 as an example. The weak- and strong-field methods are illustrated on the left and right side of Fig. 10, respectively. The solid and dotted lines in the middle indicate the correlation between two methods. This means the two methods will ultimately yield exactly the same result in term splitting. The influence of the ligand field can be conveniently presented in the form of diagrams showing the term energies as functions of the strength of the ligand field. For systems with octahedral symmetry the energies of the terms 2S +1 depend on the single parameter 10 Dq defined above. An example of the term diagram for Cr3+ (d 3 ) in an octahedral environment is shown in Fig. 11, the Tanabe-Sugano diagram of d 3 . When Dq = 0, the termsplitting of the free Cr3+ ion are as in Table III. With

FIGURE 11 Term diagram of a Cr 3+ ion in an octahedral ligand field. Term energies as functions of the ligand field strength Dq, Tanabe-Sugano diagram of d 3 . [From Tanabe, Y., and Sugano, S. (1954). J. Phys. Soc. Jpn. 9, 753.]

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tor is a characteristic of the central ion. The influences of the central ion and of the ligands on Dq value are as follows: 1. Influence of the central ion on Dq. For the same ligand system, (a) The Dq values are very similar for transition metals of the same series and the same charge. For example, the Dq value of the complexes [Mn(H2 O)6 ]2+ , [Fe(H2 O)6 ]2+ , and [Ni(H2 O)6 ]2+ is 8500, 10,400 and 8500 cm−1 , respectively. This indicates that divalent 3d transition metal ions have roughly the same Dq value. (b) The Dq values increase when going from the first to the second and to third series of transition metal ions, i.e., for the analogous complexes, the trend being 3d < 4d < 5d. As an example, for [M(NH3 )6 ]3+ complexes, the Dq value is 2287, 3400 and 4120 cm−1 for M = Co3+ , Rh3+ , and Ir3+ , respectively. TABLE IV Dq Values of Six-Coordinate Transition Metal Complexes

FIGURE 12 Tanabe-Sugano diagram of d 6 . [From Tanabe, Y., and Sugano, S. (1954). J. Phys. Soc. Jpn. 9, 753.]

4. Dq Values and Spectrochemical Series For a given central ion the field strength parameter Dq depends on the central ion-ligand distance (R) and on the charge distribution within the ligands (q, µ). The value of Dq is determined from experimental data. By comparison of the optical absorption data of octahedral complex ions with the Tanabe-Sugano term diagram, the 10 Dq (or ) values for several transition metal complexes are given in Table IV. Jørgensen developed a means of estimating the value of 10 Dq for an octahedral complex by treating it as the product of two independent factors. 10 Dq ≈ f (ligand) × g(central ion) Where the factor f describes the field strength of ligand relative to water, which is assigned to 1.0. The g fac-

f

g (cm−1 )a

(cm−1 )b

[CrF6 ]3− [Cr(H2 O)6 ]3+ [Cr(en)3 ]3+ [Cr(CN)6 ]3−

0.9 1

17,400 17,400

15,060 17,400

1.28

17,400

22,300

1.7

17,400

26,600

[Mo(H2 O)6 ]3+ [MnF6 ]2−

1 0.9

24,600 23,000

26,000 21,800

[TcF6 ]2−

0.9

30,000

28,400

[Fe(H2 O)6 ]3+

1

14,000

14,000

[Fe(ox)3 ]3−

0.99

14,000

14,140

[Fe(CN)6 ]3−

1.7

14,000

35,000

[Ru(H2 O)6 ]2+

1

20,000

19,800

[Ru(CN)6 ]4−

1.7

20,000

33,800

[CoF6 ]3−

0.9

18,200

13,100

[Co(H2 O)6 ]3+

1

18,200

20,760

[Co(NH3 )6 ]3+

1.25

18,200

22,870

[Co(en)3 ]3+

1.28

18,200

23,160

[Co(H2 O)6 ]2+

1

9,000

92,00

[Co(NH3 )6 ]2+

1.25

9,000

10,200

[Rh(H2 O)6 ]3+

1

27,000

27,200

[Rh(NH3 )6 ]3+

1.25

27,000

34,100

[Ir(NH3 )6 ]3+

1.25

32,000

41,200

Note: ox = oxalate = C2 O2− 4 ; en = ethylenedia-mine = HN2 CH2 CH2 NH2 . a Jørgensen, C. K. (1971). “Modern Aspects of Ligand Field Theory,” Chap. 26, Elsevier, New York. b Experimentaln data from Lever, A. B. P. (1986). “Inorganic Electronic Spectroscopy,” 2nd ed., Chaps. 6 and 9, Elservier, New York.

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(c) The Dq values increase with increasing ionic charge on the central ion, for example, the Dq value of [CrF6 ]3− and [CrF6 ]2− is 1506 and 2200 cm−1 , respectively. 2. Influence of the ligands on Dq. For the same central ion, (a) The Dq values increase with the number of the ligands, for example, the Dq value of [Co(NH3 )4 ]2+ and [Co(NH3 )6 ]2+ is 590 and 1020 cm−1 , respectively. (b) The Dq value increases in the order of the spectrochemical series: − I− < Br− < C1− ∼ SCN− ∼ N− 3 < (C2 H5 O)2 PS2 − − − < F < (C2 H5 )2 NCS2 < (NH2 )2 CO < OH − − < (COO)2− 2 ∼ H2 O < NCS < NH2 CH2 COO + < NCSHg ∼ NH3 ∼ C5 H5 N < NH2 CH2 CH2 NH2 − − − − ∼ SO2− 3 < NH2 OH < NO2 < H ∼ CH3 < CN

An underlined atomic symbol indicates that the ligand is coordinated with that atom.

C. Ligand-Field theory In the crystal-field model, it is assumed that the electrons of central ion are perturbed by the crystal field in the form of point charges located at the coordinated atoms of the ligand. Basically the crystal-field approach is still limited at the atomic orbital level, nevertheless, it did interpret successfully on many properties of the transition metal complex (see the following sections). However, purely based on the point charge model, it is hard to rationalize why in the spectrochemical series CN− is such a strong field, but F− is such a weak field. Apparently there is a need of improving the crystal-field model in order to reason the order of spectrochemical series. To improve the model of the crystal-field theory, the electrons are allowed to move over the whole complex ion in molecular orbitals. In other words, we have to consider the complex at a molecular orbital level. In a common approximation, the molecular orbitals are described by suitable linear combination of atomic orbitals (LCAO approximation). In this approach, in addition to d, p, and s orbitals of the central ion, the orbitals of the ligand system will also be included. In recent years, due to the great improvement of both hardware and software in computation, an ab initio quantum calculation of such complexity becomes feasible. However, in this content we will keep it as conceptual as possible. Namely, the group orbitals of the ligand will be included only in the form of σ and π bond. As shown by quantum mechanics and group theory only certain linear combinations yield energetic effects:

FIGURE 13 Combinations of a central ion d : (a) dx2 −y2 (b) dxy orbital, and (c) the σ group orbitals of the ligand σx + σ−x − σ y − σ−y .

1. The combination orbitals must have the same symmetry. For example, in Fig. 13 the dx 2 −y 2 orbital (Fig. 13a) of the central ion has the same symmetry as the σ group orbitals of the ligand system shown in Fig. 13c, therefore a combination between these two can be made. But the dx y orbital (Fig. 13b) has different symmetry from the σ group orbitals; thus no combination can be formed between the dx y orbital and the σ group orbitals. 2. The energies of the combination orbitals must be of comparable magnitude for significant interaction to occur. 3. The combining orbitals must overlap. Every pair of orbitals suitable of forming linear combination yields a stabilized (bonding) and destabilized (anti-bonding) states shown in Fig. 14. The anti-bonding state is usually labeled by an asterisk. When the energy difference between the combining orbitals is small, the energy splitting becomes large and the interaction between two orbitals is strong (Fig. 14a). The energy-level diagram of the molecular orbitals of an octahedral complex is presented in Fig. 15a, where only σ bonds between the central ion and the ligand are considered. According to the group theoretical treatment, the group orbitals of the ligand in the form of σ bond consist of a1 + e + t1 orbitals. The corresponding orbitals of the central ion

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FIGURE 14 Two atomic orbitals form a bonding and an antibonding state: (a) with small difference in energy of A and B and (b) with large difference in energy of A and B.

include a1 (s), t1(u) ( px , p y , pz ) and e(dx 2 −y 2 , dz 2 ), i.e., an sp 3 d 2 hybrid. These orbitals combine with the group orbitals of ligand and yield six σ bonding and six σ ∗ anti-bonding states, each contains a1 , e, t1 , and a1∗ , e∗ , t1∗ , respectively. The central ion orbitals t2(g) (dx y , d yz , dzx ) are “nonbonding” since there are no σ orbitals of the ligands with suitable symmetry as indicated in Fig. 13. Each coordinated ligand atom contributes two σ -electrons (σ -donor) to fill up the σ -bonding orbitals a1 , t1 , e. Therefore, the d electrons of the central ion will occupy the t2(g) and e∗ orbitals. The energy gap between t2(g) and e∗ corresponds to the crystal field parameter 10 Dq. If a π -bond interaction between the central ion and the ligand is taken into account, the group orbitals of the ligand in the form of π , π ∗ consist of t1(g) , t2(g) , t1(u) , t2(u) orbitals. The t2(g) orbitals of the central ion (dx y , d yz , dzx ) are no

FIGURE 15 Molecular orbital diagram of an octahedral complex considering. (a) σ -bonding only. (b) σ - and π-bonding between the central ion and the ligand.

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FIGURE 16 Comparison of the ligand field strength, value of ligands with σ -donor, π -donor, and π-acceptor character.

longer nonbonding orbitals, they can combine with πL or πL∗ group orbitals of the ligand shown in Fig. 15b. This π -interaction can either increase or decrease the 10 Dq value (of which only σ -bond is considered) depending on whether the ligand is a π -acceptor or a π -donor. When the ligand is served as a π -acceptor, for example CN− , where a low-lying empty πL∗ is available, the t2(g) orbitals of the central ion are stabilized by the π -interaction, therefore increasing the 10 Dq value. On the other hand, when the ligand is served as a π -donor, for example, fluoride F− ion, the t2(g) orbitals of the central ion are destabilized and thus decrease the 10 Dq value. The effect of this πinteraction on the 10 Dq value is illustrated in Fig. 16. With this π -interaction in mind, it is not too hard to understand the order of ligand field strength in the spectrochemical series given above. The energetic order within the system of progeny terms 2S+1 i (A1 , T1 , E, etc.) determines the important properties of the complex ions: 1. Optical properties. The energetic differences between the ground state and the few lowest excited states, mainly d-d transition, correspond to the absorption bands in the spectral region with λ ≥ 300–700 nm, with extinction coefficient of 100 –102 . There are also bands with much higher extinction coefficient (102 –106 ) which normally correspond to charge transfer band or intraligand transition. These bands are responsible for the color of the compound (Section III). 2. Magnetism. The spin multiplicity M = 2S + 1 of the ground term determines roughly the magnetic behavior of the complex ion (Section IV). 3. Stability. The stabilization of the ground term by the ligand field stabilization energy represents an

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increase in the binding energy of the complex ion, in addition to that obtained by the ionic model of Kossel and Magnus (Section V).

III. OPTICAL PROPERTIES OF COMPLEXES The main features of the optical absorption and emission spectra of transition metal complexes can be interpreted on the basis of crystal-field or ligand-field theory. Generally, the energies of the absorption and the emission bands correspond to energetic differences between electronic states. Therefore, an interpretation of the optical spectra will start with a comparison between the experimental spectra and the term diagram of the complex ion, according to the crystal-field theory. This procedure will be demonstrated by two informative examples. The absorption spectrum of the [Ti(H2 O)6 ]3+ ion consisting of one d-d band with its maximum at 492 nm is shown in Fig. 17. The ground state of this Ti3+ (d 1 ) free ion is 2 D (Table III). In an octahedral ligand field the 2 D term is split into the low-lying 2 T2 ground state and a 2 E state at higher energies (Fig. 9). By the absorption of a photon with energy of E(2 E−2 T2 ), the complex ion will be excited from its ground state 2 T2 into the state 2 E. The excitation energy in an octahedral d 1 ion is by definition equal to 10 Dq. From the wavelength of the absorption maximum at 492 nm, it follows that Dq has a value of ∼ 2030 cm−1 . In the molecular orbital theory the 492-nm band corresponds to the electron transition between the nonbonding t2 state and the anti-bonding e∗ state (see Fig. 15). The

FIGURE 17 Absorption spectrum of [Ti(H2 O)6 ]3+ . [From Hartman, H., Schlafer, H. L., and Hansen, K. H. (1957). Z. Anorg. Chem. 40, 289.]

FIGURE 18 Absorption (a) and emission (b) spectra of Cr3+ ion in ruby.

strong increase in absorption below ∼ 350 nm belongs to the charge transfer band, possibly from t2 to empty ligand excited states of suitable symmetry and spin states. The second example is the absorption spectrum of Cr3+ in ruby shown in Fig. 18. Two relatively strong bands I and II and three very weak absorption J1 , J2 , and J3 can be seen. At wavelengths below 300 nm a very strong increase in the absorption due to charge transfer transitions is observed (not shown in Fig. 18). Ruby is an Al2 O3 crystal wherein some Al3+ ions are substituted by Cr3+ ions. The absorption spectrum (Fig. 18) is due to the Cr3+ ions (d 3 ions) since Al3+ does not have any d electrons. This can be confirmed by the absorption spectra of Cr(ox)3− 3 and 3+ Al(ox)3− shown in Fig. 2. Every Cr ion is surrounded 3 by six oxygen ions forming an octahedron. The three lowest terms of the free Cr3+ ions are 4 F (ground state), 4 P, and 2 G (Table II). In the presence of the Oh ligand field, the terms 4 F and 2 G are split into three (4 A2 , 4 T2 , 4 T1 ) and four (2 E1 , 2 T1 , 2 T2 , 2 A1 ) progeny terms, respectively. A comparison between this term diagram (Fig. 11) of Cr3+ in ruby and the absorption spectrum yields the following assignment. The relatively intense bands I and II belong to the spin-allowed transitions 4 A2 → 4 T2 and 4 A2 → 4 T1 , respectively, whereas J1 , J2 , and J3 are due to quartet-doublet spin-forbidden transitions (4 A2 → 2 E, 4 A2 → 2 T1 , 4 A2 →2 A1 ). On the basis of this assignment and the corresponding Tanabe-Sugano diagram (Fig. 11), one can determine the Dq and B values of the complex. The significant intensity difference between the highenergy charge transfer bands (λ < 300 nm), the bands I and II, and the weak bands J1 , J2 , and J3 originates in the different nature of the corresponding transitions. As shown by quantum theory, the absorption of photons by

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molecules is regulated by some general rules or the socalled spectra-selection rules: 1. The total spin of the electron system remains constant (spin-allowed). 2. The direct product of the symmetries corresponding to ground state, excited state, and the optical transition moment should be totally symmetric (symmetry allowed or LaPorte allowed) These two criteria are fulfilled by most of charge transfer bands. The d → d transitions, however, are distinctly weaker than the charge transfer transitions, since they are LaPorte forbidden. Among the d → d transitions the spinallowed transitions show a significantly stronger absorption (bands I and II) than the spin-forbidden transitions (bands J1 , J2 , and J3 ). Ruby was the first crystalline compound to exhibit optical laser properties. The ruby laser works as follows. The excited quartet terms 4 T2 and 4 T1 (ref. to Fig. 11) are populated by irradiation (optical pumping) using the broad absorption bands at I and II from the ground term 4 A2 . Then radiationless transitions into the 2 E state take place within a period of

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