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r
B Fig. 1.1. Electronic and vibrational energy levels (schematic, rotational are omitted for simplicity), So singlet ground state, Si excited singlet state (higher electronically excited states are omitted in this figure), Ti triplet electronically excited energy state. Straight lines symbolise radiative processes (absorption (Ab) as well as emission: F, fluorescence; P, phosphorescence). Wavy lines give the radiationless transitions, ic, internal conversion; isc, intersystem crossing; sd, radiationless deactivation; te, thermal equilibration.
Ch. 1
Photophysics
11
means the molecule marked as A* in the following and in the figure is 'hot' in comparison to its environment. For this reason it equilibrates its 'temperature' in a short time in the condensed phase. It deactivates without radiation to the lowest vibrational state of the actual electronic state (marked in Fig. 1.1 as te). By isoelectronic internal conversion (ic) it can pass over to a very high vibrational state of the next lower electronic energy state. These deactivations normally take place until the molecule has reached the vibrational ground state of the first excited electronic state Si. This overall deactivation process is called thermal relaxation (sd), which can be divided into the isoenergetic deactivation 'internal conversion' and the non-isoenergetic 'thermal equilibration'. Its time scale is 10"'^ to 10"^^ s. The process can be symbolised by A*->A'.
(1.12)
All these different deactivation pathways are given in Fig. 1.1. The molecule A' is in thermal equilibrium with its environment. Average lifetime lasts from 10"^ to 10"^ s. The excited state can lose its energy by fluorescence (F) according to A ' - > A + /iv'.
(1.13)
As a result the molecule reaches any one of the vibrational levels of the electronic ground state SQ. In a next step it adapts its temperature to that of the environment. Another deactivation pathway from the electronic excited to the ground state is a radiationless one (radiationless deactivation): A'-^A.
(1.14)
Besides the excited singlet states some excited tripled states T,, T2, etc. exist, which cannot be reached directly by absorption of radiation starting from the ground state SQ. This direct transition is forbidden [3]. In the case of radiationless transition this limitation is not valid. For this reason the molecule A' can radiationlessly deactivate to the first triplet state A'' (T,) (see Jablonski's diagram in Fig. 1.1) A'-->A".
(1.15)
12
Introduction
Ch. 1
This process is called intersystem crossing (isc). In general values between 10^ and 10^0 s~^ are discussed as rate constants for a monomolecular reaction as in eq. (1.15). The state A!' is meta-stable. For this reason molecules in this state have a longer lifetime than those in the state A'. In fluids the average lifetime is between 10"^ and 10^ s. It depends on the nature of the molecule as well as on the purity of the solvent. Molecules A" can also emit, and thereby return to the ground state. This process is also forbidden as for the absorptive transition [3]. Therefore its probability is very small. It is called ^- or low-temperature phosphorescence (Pin Fig. 1.1): M'-^A
+ hV.
(1.16)
Another deactivation process is radiationless: A"-^A.
(1.17)
If the energy difference between A!' and A' is less than 40 kJ/mol, the molecule A" can uptake thermal energy and cross over backward to the state A': A"->A'.
(1.18)
If in the next step fluorescence happens, it is called high-temperature or aphosphorescence (delayed fluorescence). All the processes discussed are included in the term diagram given in Fig. 1.1. If appropriate acceptor molecules B are present, the molecules A' and A" can transfer energy according to A' + B ^ B ' + A
(1.19)
A"+B->B" + A
(1.20)
and
respectively. The processes of eqs. (1.19) and (1.20) can be caused by collision. They have to happen within the lifetimes of A' and A", respectively. The maximal bimolecular rate constant k^ turns out to be diffusion controlled and can be approximated according to [3]
Ch. 1
Photochemistry and photokinetics
_ oJ\T *n,=-5— .
13
.^ ^.^ (1.21)
Tis the absolute temperature in Kelvin, R amounts to 8.31 J moh* K"' and rj is the viscosity in kg m"' s~' (Pa s). At room temperature and normal viscosity of the solvent, the rate constant becomes /:^« 10^^ P mol~^ s^K Energy transfer can be caused even by induced resonance without any impact up to distances of 5-10 nm, a process of quantum mechanical origin [8, 9]. Under these conditions the 'bimolecular rate constant' is independent of the viscosity of the solvent and approaches values up to 10^^ P mol"^ s"^ Strictly speaking this process cannot be treated as a normal bimolecular reaction. 1.4 PHOTOCHEMISTRY AND PHOTOKINETICS L4.1 Chemical primary processes Molecules which have been excited by absorption of light or energy transfer can also chemically react, in addition to the physical processes discussed in eqs. (1.12) to (1.20). This chemical reaction can even start during the thermal relaxation, i.e. during the process from A* to A'. That means the cross-over to the term system of the product B takes place starting at a higher vibrational level. Another possibility is the start of the chemical reaction from the states A' or A". In all these examples the term diagram given by reactant A is left and a reaction takes place to a new compound B characterised by the left side of the term diagram (B, see Fig. 1.1). Generalising these states by the symbol A*, the following chemical primary processes can be discussed: (1) the dissociation into radicals according to A"-^B"+C*,
(1.22)
(2) the dissociation into other instable molecules, e.g. carbenes according to R~CHN^ -^N2 +RCH, •• or into imines according to
(1.23)
14
Introduction
RN = N = N' -> N + RN".
Ch. 1
(1.24)
(3) Another pathway is the dissociation into ions according to A'-^B^+C",
(1.25)
(4) the dissociation into a positive ion and an electron according to A"-^A^+e".
(1.26)
(5) the formation of stable molecules A ' - > B + C,
(1.27)
(6) the photoisomerisation A'-^B,
(1.28)
(7) or a photoaddition or photodimerisation according to A"+B(A)-^C.
(1.29)
This bimolecular process requires the impact of the exited molecule during its lifetime with another molecule B(A). Assuming the lifetime of the molecule A' is 10'* s, a bimolecular reaction can only take place if the concentration of the partners of the reaction is lOr^ mol 1~* or larger. If the lifetime is remarkably longer, this bimolecular reaction becomes more probable. For a lifetime of KHs it can only react according to eq. (1.29), if the concentration of the partner amounts to KH mol 1"^ or more. In both cases it is assumed that the rate constant is diffusion controlled (/:„,« 10^^ 1 mol"^ s"^), if the viscosity of the solvent has a normal value. Another possibility for the reaction of the exited molecule is the abstraction of hydrogen to form radicals according to A ' + R H - > A H + R*.
(1.30)
It had been mentioned that the primary reactions (1.13) to (1.30) can also
Ch. 1
Photochemistry and photokinetics
15
start from the molecules B' or B", formed according to eqs. (1.19) and (1.20). If these reactions are followed by another chemical reaction, the photoreaction of B is called a (physically) sensitised reaction. On the other hand, if A^ shows a photoreaction, then the processes (1.19) and (1.20) are competitive reactions. Therefore the molecule B desensitises A^ and quenches the photoreaction starting at A. The process (1.20) is photochemically very important. It allows the transfer of a precise amount of energy to the molecule B. 1.4,2 Definition of the quantum yield In thermal kinetics the rate is proportional to concentration in the most simplest mechanism according to eq. (1.1). The proportionality constant is the rate constant k. In photokinetics, the equivalent proportionality constant is, according to eq. (1.2), the so-called photochemical quantum yield. In the literature, some different definitions of quantum yields are discussed but not always clearly distinguished. Therefore the problems with three different definitions are discussed here. Two others related to the partial reactions and independent of the time of the reaction are given in Section 2.1.2. 1.4,2.1 Apparent integral quantum yield The apparent integral quantum yield YQ is defined for any reaction partner Cby
rc=±^=±fi^.
0.3.)
ANc is the number of moles of the reaction partner C reacting during the time t. Nhv is the number of moles of light quanta (Einstein) absorbed in total by the system in the same time t. c{i) and CQ are the concentrations of C per unit volume at time t and before the irradiation starts (r = 0), respectively. Ij is the amount of light absorbed by the system (not only the interesting reactant) per second and unit of volume, measured in Einstein 1"^ s~^ In general, yields are defined as positive values. For this reason the plus sign is used if C is formed during the reaction and the minus sign is used if the concentration of C decreases. YC can be easily measured but has a complex dependency on the time of the reaction.
16
Introduction
Ch. 1
1.4.2.2 True integral quantum yield The true integral quantum yield is defined by
^^vA
JlUdt
y^ can be distinguished from the apparent integral quantum yield, because its turnover is related to the amount of light absorbed by reactant A. It is assumed that A starts the photochemical reaction. For this reason N/^^ and /^ are the numbers of moles of light quanta absorbed by reactant A per volume and time unit. The latter quantity is discussed in detail in Section 1.4.3. It depends on the experimental conditions. 1.4.2.3 Apparent differential quantum yield The apparent differential quantum yield
?>c=±-f
(1.33)
is calculated for infinitesimal small turnover, c is the change in concentration of C with time. The total amount of light absorbed (by all the reactants) per volume and time unit is taken. All the quantum yields defined so far cannot give a correct view of the photokinetics. C + D, eq. (2.2) has been given explicitly according to the reaction scheme B -1
D
-1
by
Aa =
Ab Ac = \-x
-1 =
1
U
30
General Approach in Formal Kinetics
Ch. 2
using the vector of stoichiometric coefficients related to the transpose "'*-:(-l V'
-1
1 1)
derived for the four reactants according to eq. (1.3). In this example, the vector x contains only one degree of advancement, that of the above assumed single step. Since the observation of just one component is sufficient to describe the progress of the reaction, the vector v" can be reduced. Selection of the reactant A results using eq. (1.9) Aa = (-l)x. By rearrangement one finds ;c = (-l)Aa. This can be inserted in the equation given above resulting in
Ar
(Aa^ f-V\ (-l)(Aa) ri\ (Aa) 1 -1 \Ab = pAa = = Ac = v°v-'Aa = 1 -1 [Ad, .1> .-1;
This can be achieved either by multiplication and following the steps discussed above or, on the other hand, using eq. (2.6) with the vectors Aa', v% and calculating v"* = - 1 . Furthermore Aa reduces to Aa. According to the definition of P, the new vector on the right-hand side in the above equation is obtained as
rn p=
1 -1
Thus a relationship between the changes of all the concentrations and the chosen Aa is given. Therefore using the above equation three linear
Ch. 2
Fundamentals
31
independent equations with mass balance are obtained besides the trivial correlation Aa = Aa: Afe = Aa,
Ac = -Aa
and
Arf = -Aa
2J.L2 Partial reactions In case where the number of reactions between the reactants exceeds one, eq. (1.3) has to be expanded to a system of r partial reactions between n reactants A,. For each of these, eq. (2.1) has to be valid. Therefore the selected kth equation of the reaction is given by ,^kA = 0
(* = l,2,...,r),
including another index for this partial reaction. The transposed stoichiometric matrix v** contains r rows (number of reactions k) and n columns (number of reactants i). In case the molecular weights are included within the column vector, even for complex systems the law of conservation of mass is satisfied according to eq. (2.1). The same is valid for eqs. (2.2) to (2.8), if r degrees of advancement x^ are included in the column vector x* (one for each reaction). Then v becomes a regular quadratic rxr matrix, which can be derived from the matrix v° and represents a matrix which cannot be reduced any more with respect to the reactant. The next example demonstrates the procedure to derive the equations with the mass balance mentioned above by use of eq. (2.8). Example 2.2: Derivation of change in concentration A system of two reactions is given:
X, X2
A -1 0
B 1 -1
C D E 1 0 0 ir, 0 1 1 X2
Thus the transposed matrix of the stoichiometric coefficients v°^ has to be formed for two reactions between the 5 compounds (the stoichiometric coefficients of the reactants form a row for each step):
32
General Approach in Formal Kinetics
Ch.2
- 1 1 1 0 0 V = 0 - 1 0 1 1 This demonstrates clearly that in general matrices v**^ and v% respectively, are singular. In consequence the changes in concentrations are according to eq. (2.4)
(Aa^ (-1 0^ 1 -1 A6 Ac = 1 0 0 1 \Ad ^Ae, . 0 ij In the chosen example the concentrations of two components a and b suffice to describe the progress of the reaction. The vector Aa" can be reduced and the equation above is written as Aa^
-1
0
Ab,
1
-1
V-'^2,
By inversion of the reduced stoichiometric matrix the relationship can be rearranged to an equation according to eq. (2.5): -1
0
Aa Ah
The latter equation can be inserted in the upper first one. The result is a matrix equation according to eq. (2.6):
(Aa" r-1 0^ - 1 0 YAa 1 -1 -1 -if^Ab A& Ac = 1 0 Arf 0 1 \Ae^ . 0 ij
Fundamentals
Ch. 2
33
Obviously the matrix is the inverse of the reduced stoichiometric matrix and the vector Aa is the reduced vector of changes in concentrations of relevant reactants. The more general approach uses eq. (2.6), the reduced Aa as well as v~'. Multiplication of r-1 I v°v-'=p = 1 0 0
0^ r-\ - 1 [-1 0 1
0 -1
(I 0
0^ 1
-1 -1
0 -1
l-l -ij
finally results in the following relationship according to eq. (2.8) fAa^ Ac Ad Ae
(\ 0 -1 -1 -1
OVAa> 1 0 -1 -I)
This is the same result as obtained by the use of eq. (2.5), however, applying matrix calculus with less expenditure. The only disadvantage is the matrix inversion, but modem software provides an easy way to do this (see software MAPLE used in Appendix 6.1). Assuming that the concentrations of the intermediate and of the final products equal zero at the beginning of the reaction (feo = co =rfo= ^Q = 0), the following three linear independent equations with massbalance result: Ac = c = - Aa = ao - a. Ad = d = - Aa — Ab = GQ - a- b, Ae = ^ = - Aa - Afo = aQ - a - fo. The example given above explains the use of eqs. (2.2)-(2.8) as well as the advantage of using reduced reaction schemes. The given procedure re-
General Approach in Formal Kinetics
34
Ch.2
quires one prerequisite. The rank of the stoichiometric matrix has to be equal to r, which is equivalent to the number of the reactions taking place in the system (for further details see Section 2.1.4). This means the matrices and vectors used have to be regular. The next example expands the procedure to three steps of the reaction and to demonstrate once more the approach to deduce the changes in concentration of the reactants by use of eq. (2.8). In addition, the possibility of formulating these changes dependent on the degrees of advancement is given. These allow another approach to obtain the changes in concentration. Example 2.3: Complex reaction scheme A further example with three steps of reaction is discussed in detail step by step in Appendix 6.1 jcp
2A-
B
X2: B 4 - C - » D + E .
x^: D + A - ^ F + G One obtains the reaction scheme
Xl Xy ^3
A -2 0 -1
B 1 -1 0
C 0 -1 0
D 0 1 -1
E F G 0 0 0 1 0 0 0 1 1
-^i X2
-^3
The calculation of the changes in the concentration is given in detail in Appendix 6.1. In principle these values can easily be obtained by a summation of the products of stoichiometric coefficients for the reactant under consideration with the degree of advancement in the column in the above reaction scheme according to eq. (2.4) A a = - 2 J C , - JC3,
At/ = JC2 - X'^y
Afe = jCj - ^ 2 , Ae^
X2,
A c = -jCj
Af = Ag = x^
Ch. 2
Fundamentals
35
Advice: The changes in concentration are simply obtained from the reaction scheme for each reactant by a summation in the respective column of the products of stoichiometric coefficient times the degree of advancement of each step of the reaction. Using the reduced matrix of the stoichiometric coefficients one finds according to eq. (2.8), demonstrated in Appendix 6.1: Ac = -JC2 = 1/3 ^a + 2/3 Afo -1/3 Arf Ae = ^2 = -1/3 ^a - 2/3 ^b + 1/3AJ = -Ac, A/ = ^g = -y3Aa
- 2/3 Afe - 2/3Arf
Thereby the calculation of the relationship between the concentration changes can be formally obtained by use of the reaction scheme and simple matrix algebra. 2.7.2 Quantum yields independent on reaction time In Section 1.4.2, in analogy to the thermal rate constants, photochemical quantum yields have been defined. It was mentioned that the amount of light absorbed varies according to Section 1.4.3 during the photoreaction. Accordingly the definitions given in Section 1.4.2 exhibit a dependence on time. It is assumed that a simple photoreaction stoichiometrically takes place according to the condition given in eq. (1.4). If the reactant Aj = A starts a photoreaction, then the degree of advancement is given according to eq. (1.8) and one finds the rate law as given in eq. (1.2) to
a,=±^f/,.
(2.9)
/^ has been defined by eq. (1.39) in Section 1.4.4. Thus for any reactant by use of eq. (1.10) a correlation with the degree of advancement is found and the quantum yield can be defined in a new way
\vA
vj
I.
36
General Approach in Formal Kinetics
Ch. 2
In eq. (2.10) the amounts of the stoichiometric coefficients have to be inserted, since the quantum yields are defined always as positive numbers. It makes more sense to relate the quantum yield to the degree of advancement X, Therefore by definition F
A Xx - 1 0 X'y
B -1 0
C 2
D 1
-1
0
E
F
0 -1
0 kiob 1 k^ce
Xk
Using the chemical reactions, the stoichiometric coefficients are put into the rows below the reactants according to (d). Finally concentra-
Ch. 2
Fundamentals
tions of components with negative coefficients are written as a product together with the rate constant (index: step of reaction) into the last row according to (e). The scheme is used to find the correlation between the change in concentration and the degree of advancement by use of eqs. (1.9), (2.2) or (2.4) and by stoichiometric summation of the ith column: Aa^a-QQ^
-jc,, a = GQ — jc,
Ab = b-bQ=
-JC,, fo = ^Q - JC|
Ac = c - 0 = 2JC, -X2,
C = 2JC,
- JCj
and accordingly (do = 0,/o = 0) rf = X,, ^ = ^o--^2'
/ = ^2-
Taking the concentration c as an example, in the first step of the reaction the stoichiometric coefficient amounts to +2 (therefore + 2JC, ) and in the 2nd step to -1 (therefore -X2) In analogy the rates are extracted by stoichiometric summation of the contents of last column to a = - i , = -k^aby b = -k^ab, c = Ik^ab - ^2^^* d = k^ab, e = -k2ce,
f = k2ce
The stoichiometric coefficient of the considered reactant A, is multiplied by the element of the last column, if this reactant takes part in the partial reaction. In particular the relationship for c explains the procedure (v,c= 2, V2c=-1). Example 2.5: Catalytic reaction
A ^ B +C A + B->2B + C
X,
Xo
A -1
B C Xk 1 1 fc,a -1 -1 1 k^ab +2
39
40
General Approach in Formal Kinetics
Ch. 2
The difference in the formal approach to these two reactions in comparison to the former examples is caused by the catalytic action of component B. One mole of this component is consumed in the second step first; later on two moles are formed. Therefore the two stoichiometric coefficients have to be included in the scheme for jCj to get the correct results for equations according to eqs. (1.9) or (2.2). On the other hand, in the stoichiometric matrix only the sum of these two factors ( - 1 + 2 = +!) is found and used for the formation of the rate equation according to (e). Thus a = —k^a - k2ab,
b = k^a + k2ab,
c = k^a + k2ab.
2.1.3.2 Bodenstein hypothesis The Bodenstein hypothesis [13] can also be called a hypothesis of the stationary or quasi-stationary state. It makes the following assumptions: (1) The concentrations z\ of the intermediates Z, are negligible within the limits of measurement in comparison to the concentrations of starting material and products, (2) changes of concentrations with time of the intermediates are negligible with respect to the changes in the concentrations of initial products and end-products with time. That means both the equations z,=0,
z,=0
(2.16)
are valid for all the intermediates which fulfil this hypothesis. 2.1.3.3 Simple photokinetic rate laws In contrast to thermal reactions, any photochemical reaction is accompanied by a number of photophysical processes which all have to be taken into account in the reaction scheme. Most of these processes are thermal reactions. The mechanism of the photoreaction and - as we will see in the examples - the photochemical quantum yield depends on these photophysical steps. One of the most simple photoreactions is a photo-isomerisation A—^il->B which can be assumed to consist of the following partial steps
Fundamentals
Ch. 2
A + hv
A* x\
A*
-^
A'
A'
->
A + hv'
A'
-» A
X\
A'
-^ B
x's
X'2 X',
41
A -1 0 1 1 0
A* 1 -1 0 0 0
A 0 1 -1 -1 -1
B 0 0 0 0 1
^\ JA
M* k^d k^d k^d
As in the case of thermal reactions, the reaction scheme introduced in Section 2.1.1.1 can be used to set up the differential equations. However, the degrees of advancement are primed, since the number of steps can be reduced as will be demonstrated by use of the Bodenstein hypothesis. In the last column of this scheme, the number of moles of light quanta are written for a photochemical step, which are absorbed by the reactant starting this photochemical step. According to this assumption and the different photophysical relaxation processes discussed in Section 1.3 the primary exited molecule A* completely deactivates into the lowest level of vibrational energy of the first exited singlet state. Three further steps are possible: (1) deactivation by fluorescence, (2) radiationless deactivation and (3) isomerisation to reactant B. Reactant A reacts by step 1 and will be re-formed by reactions 3 and 4. In consequence, the change of A in dependence on time is (summation of all the products stoichiometric coefficient in the first row times the element of the last row: reactions 1,3,4) a = -If^ +(^3
•¥k^)a\
(2.17)
During this process all the molecules of reactant A will react, since they absorb light. The amount of light absorbed is given according to the definition: If^ in mole 1"^ s-^ The rate laws for the photophysically exited molecules can be given accordingly a* = /^ - *2« *
a' = jk2a*-(Jfc3-h)k4+/:5)a' as well as the change in concentration of B
(2.18)
42
General Approach in Formal Kinetics
Ch. 2
The fluorescence quanta emitted (fluorescence intensity F) according to the radiative process (3) amount per second to
since each molecule A* reacting according to the third step emits a quantum of light. The lifetimes of the exited molecules A* and A' are extremely short in comparison to the time the reaction needs. Therefore both concentrations as well as their changes with time are very small as long as normal light sources are used for irradiation. For this reason, the Bodenstein hypothesis can be assumed, and one can set
In consequence, the stationary concentrations according to eq. (2.18) amount to ^*=_A_^
^ ^ _
Ky
Z K'i
I
KA
_ • Kc
A K-i
v
KA
' rCe
Therefore the change in concentration of compound A is (a' inserted in eq. (2.17)): a = -/A+(jfe2+*3)
= Ko "» f^A i" iCr
/A. K-y "T
KA
> Kr
andof reactantB: b=
'^ ACa "T*
KA
/A. • fCc
A comparison of the above equation with eq. (2.12) shows that the quantum yield for this simple photoreaction is given by
Ch. 2
Fundamentals
43
for the proposed mechanism. In this case - the quantum yield (p^ is smaller than 1, ~ it does not depend on the concentration of the reactants, and - it does not depend on 7^. It has been assumed that all primarily exited molecules reach the state A'. Therefore the quantum yield cannot depend on the wavelength of the exciting light. The approaches to use the reaction scheme introduced in thermal kinetics can be very helpful and allow the transformation of the mechanism into a differential equation. All the elementary processes have been included in the scheme. For this reason, constants of these elementary processes can be explicitly determined from the quantum yields by these means. However, many of these steps cancel out by using the Bodenstein hypothesis (see Section 2.1.3.2) and only effective steps of the reaction are found. Nevertheless, the total scheme explains dependencies of quantum yields on concentration as discussed in Chapter 3 and in the following example. Conclusion: Most of the photophysical steps are thermal reactions, for which the Bodenstein hypothesis is valid. The photochemical quantum yield depends on these elementary steps. The dependence of the photochemical quantum yield on the types of photophysical processes becomes obvious comparing the simple photoreaction
for two cases: the start of the photoreaction from the singlet and triplet state, respectively. The results for the latter are summarised in brief in Example 2.6 which is discussed in detail in Appendix 6.2.1 Example 2.6: Simple photoisomerisation Instead of the photoreaction of an isomerisation from the singlet state for
44
General Approach in Formal Kinetics
Ch. 2
a pathway via the triplet state is considered. According to the procedure explained in Section 2.1.1.1 the reaction scheme is set up. All the excited molecules are supposed to form A' first. Therefore the both steps A + Av
>A* and A *
>A'
can be combined to give the single step A-hAv
^A'.
Application of the Bodenstein hypothesis to the intermediates A' and A" results in y> =
K^K {k2-^k^+k^){ks+k^)
and the quantum yield depends on other photophysical processes than in the case of a pathway via the singlet state. 2.1.4 The principle of linear independent reactions Some multi-step reactions have been discussed above in the examples. In all cases the different partial steps were elementary reactions. None of the rate equations of these steps had any dependence on one another. It is obvious that this number of linearly independent steps of reaction (partial reactions) plays an essential part in the process to set up the rate equations. This number has to be known for the calculation of the yields as well as for a kinetic analysis of the reaction system. Linear dependencies exist between different rows of the stoichiometric matrix v*, if its rank is smaller than the number r of all the partial reactions possible. Under such circumstances the advancement of the reaction can be described by a reduced number 5" < r of degrees of advancement or concentrations {s defines the rank of the stoichiometric matrix). In the following, two examples are given to demonstrate the problems, which arise by this restriction. They are chosen to give information about various aspects of linear dependence.
Fundamentals
Ch. 2
45
2.1.4.1 Uniform reactions The number of linearly independent steps of a reaction is given by the number s of the independent degrees of advancement and therefore according to the statement above to the rank s of the reduced stoichiometric matrix. The following definitions are used, especially in the case of photokinetics: - A reaction of just one linear independent step (5 = 1) is call uniform. - A thermal reaction with just a single elementary step is call simple uniform', in the case of a photochemical reaction elementary reactions for which the intermediates can be omitted because of the Bodenstein hypothesis are taken also as a simple one. Examples are both the photoreactions given in Section 2.1.3.3. - When the reactions steps depend on each other linearly and by these means 5^ = 1 is obtained, such reactions are called complex uniform. Examples for such complex uniform reactions are given in the following sections. 2.1.4.2 Thermal reactions The simplest case of a complex uniform thermal reaction is the reversible reaction according to the following reaction scheme: A A^B,
4>
^1
Xy
B
-1 I 1 -1
• •
k^a k^b
The two rows of the reactants are proportional to each other (stoichiometric coefficients can be converted into each other by multiplication by the factor (-1)). jCj* and x*2 are linearly dependent. Therefore the rank of the matrix v"* reduces to 1. This is valid for all the reactants A, in the first two linear dependent steps 1 and 2: Vu = - V 2 / .
For this reason the relationship between concentrations and the degrees of advancement is reduced from (stoichiometric summation within one column) Aa, = v,,x
+ V2,JC2
46
General Approach in Formal Kinetics
Ch. 2
to Aa,. =v,,.(x,*-X2). Only the difference, taken from the degrees of advancement jc^* and jCj, arises in all the equations. The x* themselves can neither be measured nor their dependencies on time calculated independently of each other. For this reason it makes sense to define the following new, linear independent degrees of advancement (without the index *) •
*
Xt = X\ """ X2 •
In consequence the reaction scheme reduces to (the new element in the last column is obtained using both the equations above) B -1
k^a-k2b
As a result the new time dependence of the degree of advancement jc, becomes i, = x\ - x*2- k^a(t) - k2b{t) according to the definition. Advice: In practice one cancels the linear dependent line(s) in the scheme and reconsiders the expressions in the last column. In the following, three other examples are discussed to demonstrate this procedure of reduction of the reaction scheme. Details are given in Appendix 6.3.1. The first two cases are mechanisms with back-reactions. The last one treats the case of a parallel reaction.
Ch. 2
Fundamentals
Example 2.7: Consecutive reaction: linear dependency For the reaction A ^ B ~» C, jc,* and x^ are linearly dependent. Therefore the rank of the matrix v*** reduces to 2.
For this reason the relationship between concentrations and the degrees of advancement is reduced to
A new linear independent degree of advancement is defined X\ — X\
Xy )
X'^ — ^-1
and the reaction scheme is reduced to
X\ X2
A -1 0
B 1 -1
C 0 1
Xk kiO - ^ 2 ^ • kjb
As a result, the new time dependence of the degree of advancement jc, becomes ij = i* - i* = /cj . a{t) - k2b{t) according to the definition. Further details can be found in Appendix 6.3.1. Advice: In practice one cancels the linear dependent line(s) in the scheme and reconsiders the expressions in the last column. Example 2.8: Reaction cycle Even reactions with more steps and fully reversible forming a reaction cycle can be treated
47
General Approach in Formal Kinetics
48
Ch.2
A ;^ B ^r^ C 'Fi^ A.
The backward reactions cause a reduced reaction scheme. In addition, the third step of the reaction can be cancelled and one obtains two linear independent degrees of advancement Xi = Xi ~- x^
ano X2 ^^ Xy -- x-y
and the following representation
X\
A -1
B 1
X'j
0
-1
C Xk 0 kiO- /jj^ - k^c + k^a 1 k^b - k^c — k^c + k^a
is found. The reduced stoichiometric matrix is symbolised by v. In consequence eq. (2.8) with the definition of matrix p can be further used as (2.7)
p = v--v-\
only taking into account that stoichiometric matrices are reduced with respect to the number of linear independent steps of a reaction. Thus, with Ar=v'*.x*
and
Aa'=p.Aa
one finds (see Appendix 6.3.1) the following relationships: Aa = -jc,,
Afc = jc, - JC2,
Ac = jCj
and Ac = - Aa - Afe.
Conclusion: The stoichiometry fits, if the number of degrees of advancement is reduced to the rank of the stoichiometric matrix.
Fundamentals
Ch. 2
49
Such a simplification can be used in all mechanisms in which a backreaction takes place. However, further reduction might be necessary in kinetic analysis. This becomes obvious looking at a further example, which is discussed in detail in Appendix 6.3.2 for the following reaction scheme: Example 2.9: Parallel reaction The rank of the matrix V is equal to the number of the degrees of advancement in the following mechanism. Therefore one would expect that the conditions of equal number of r and rank of the matrix stated at the beginning of Section 2.1.4 is sufficient for the optimal reduced matrix.
A + B-^C A + B->D
JC, 2
A -1 -1
B -1 -1
C 1 0
D Xk 0 kfOb 1 k2ab
However, if the concentrations a and b are chosen to be independent, eq. (2.8) can be used to obtain the non-trivial balance-equations: Ab = Aa and
Ac = - Aa - Ad.
Both the degrees of advancement show a linear dependence for kinetic reasons. A comparison of the rates in the last column of the scheme yields
Integration results in
and Aa, = v,.x; + V2,X2 = (V,, + jV2,)x,
General Approach in Formal Kinetics
50
Ch.2
It turns out that this reaction system can be described by a single degree of advancement according to the scheme
A
c
B
D
Xk
-(!+;»:) -(\ + x) 1 X k^ab
Xl
Advice: The simplified scheme is obtained by multiplication of the second row by X ^^d addition of the result to the first row. In consequence the last row is cancelled according to the method explained above. This last column contains the rates and is not influenced by this procedure: Considering a as the independent concentration, the changes in the different concentrations are given by Afc = Aa,
Ac = —
Jti -^k^
Aa,
AJ = —
-Aa.
jfc, +Jt2
A and B have the same stoichiometric coefficients in the reduced scheme. For C and D the ratios have to be taken according to Ac ^a
k^-¥k2
and
Ad — =• Aa k^ -{- ^2
Even though some arbitrariness exists in the definition of linear independent degrees of advancement, the measured value is not influenced at all. For example, the above given degree of advancement x* can be substituted by
if all other stoichiometric coefficients are divided by (1 +x) in consequence. Therefore the brutto equations are given by
Fundamentals
Ch. 2
(^).
: ^ L ± ^ ( A + B)->C + ^ D ,
(;c'):
A+B — ^ C
51
+ -^^—D. fC|1 "v ^ ^#C' ^2
fCj "T" fC^
There exist some other possibilities to reduce the number of linear independent reactions, degrees of advancement, or concentrations at conditions specifically obtained in kinetics. Further information is given in Section 2.3. Conclusion: If an extreme difference between the rate constants of different steps exists a simplification of the mechanism is possible.
2.1.4.3 Photochemical reactions The reduction of the reaction scheme because of linear dependencies between partial steps works for photoreactions in the same way as for thermal reactions. In addition it has to be considered that intermediates are negligible and allow the application of the Bodenstein hypothesis. Therefore the number of steps reduces to one per photoreactive step. Between these steps, linear relationships can happen. Photoreactions and true thermal reactions (not the photophysical steps) can exhibit linear dependencies. The simplest case is a photoisomerisation superimposed on a thermal back-reaction. Example 2.10: Photoisomerisation with superimposed thermal backreaction The combined photochemical and thermal reaction mechanism according to A ->B B-
->A
'zi »=»
s
o o 2j
M
O.
^ (/)
13 u
§
u •c
^
+
o -c 04
13
^ ?
l^
I II
-3
-iT II • ^ • fS 1 m 1 'H II I II 1 « «-« • ro ^ ^ ts 1 •H 'H II •>1< •K,^ •H II II II <s II 'H • Q •^
"^1
CQ
7
K1
«s»
C
.O
O
S
U-i
O.
•^ c c4 S HJ
% U1 H
Ch.2
Ch. 2
Fundamentals
55
Conclusion: (a) Photochemical steps of reaction include many thermal degradation processes, which do not need to be considered in the rate law, if the Bodenstein hypothesis is valid, (b) The mechanism has to be reduced as far as possible to avoid linear dependencies, (c) Thermal and photochemical reactions can be treated in principle by the same formalism. Rate constants times concentrations have to be substituted by the product of the partial photochemical quantum yield times the amount of light absorbed, which contains the concentration of the reactant which starts the photoreaction. 2.1.6 Use of symbols Different symbols have been used for the degree of advancement and the concentrations depending on the conditions. Furthermore vectors have been symbolised by bold letters with arrows, matrices by bold letters, variables as italics. Stoichiometric coefficients and the matrix p are written in Greek symbols. In addition, the concentrations of reactants to be determined and the number of degrees of advancement can be reduced because of the law of conservation of mass and linear dependencies between different steps of the reaction, respectively. Therefore some indices have been introduced to characterise the different variables. The different symbols are summarised in Table 2.2. 2.1.7 General rate laws for simple thermal reactions The differential equation of the overall thermal reaction can be expressed for an elementary reaction by i: = ika^^fc^«c^^...
(2.22)
Under these conditions the parameters r = 1 and /i, = |v,| have to be inserted into eq. (2.15). In general eq. (2.22) allows more complicated reaction schemes to be described, and //, need not be an integer (see Section 3.1.7). The exponents //, show the order with respect to the different reactants. //^ is the order of the reaction with respect to A, fi^ with respect to B and so on. A summation over all these orders results in the total order of the reaction
56
General Approach in Formal Kinetics
Ch. 2
TABLE 2.2 List of symbols used in the following chapters, the derivatives with respect to time of the degree of advancement (i:,) have to be handled in the same way Symbol
Meaning
a°, (Aa*)
Of all reactants taking part in the reaction scheme, no reduction with respect to conservation of mass (changes in the concentrations) Vector of concentrations of selected reactants taking part in the reaction scheme, reduction with respect to conservation of mass considered (changes in the concentrations) Vector of degrees of advancement of all the partial steps taking part in the reaction scheme, no reduction with respect to linear dependencies between some of these steps Degree of advancement of one partial step taking part in the reaction scheme, no reduction with respect to linear dependencies between some of these steps Degree of advancement of one photophysical step taking part in the reaction scheme, no reduction by approximation according to Bodenstein considered Vector of degrees of advancement of those partial steps taking part in the reaction scheme, which show no linear dependencies Degree of advancement of one partial step after reduction to avoid linear dependencies Stoichiometric coefficient of the kxh step of the reaction scheme for the ith reactant Vector of stoichiometric coefficients of all reactants in one reaction step Vector of stoichiometric coefficients of those reactants in one reaction step remaining after reduction according to the law of conservation of mass Matrix of stoichiometric coefficients of all reactants in all reaction steps, no reduction Matrix of stoichiometric coefficients of all reactants in those reaction steps, which do not linearly depend on each other Matrix of stoichiometric coefficients of reactants in all reaction steps, reduction with respect to conservation of mass Matrix of stoichiometric coefficients of reactants after reduction with respect to mass conservation and linear dependence between steps of reaction
a , (Aa) X* X* xj X xi Vi^i v° V o*
v° V* V
order of reaction = / B yields according to eq. (2.22) the differential equation -d = x = l? = ka.
General Approach in Formal Kinetics
58
Ch.2
In this case the rate constant k has the dimension [s~^]. By integration within the limits r = 0, a = ao,x = 0 and r, respectively, one obtains a = aQe'kt
jc = 6 = ^0(1-e *0»
(2.23)
First order reactions show no dependence of the half-life on the initial concentration. According to eq. (2.23), the half-life yields In 2
0.693
h/2 ~"
The slope at the beginning of the reaction is CIQ = -ka. The definition of the general coordinates a =
a
?=
P-
r -kt
yields the following relationship: ^ = /8 = 1 - exp(-r),
a = exp(-r).
Furthermore the initial slope of a with respect to r becomes da
= -1
at time approaching zero. 2,1.7.3 Second order reactions Two possible cases have to be distinguished: (1) Initially only a single compound is present: B 2A-»B
-2
ka'
(2) On the other hand a reaction with two different starting components,
Fundamentals
Ch. 2
59
A I B IC -1
A+ B^C
-1
1 kab'
takes place. The differential equation of the rate has to be solved by a separation of the variables and by formation of partial fractions. The derivations and calculations for both cases are given in detail in Appendix 6.4.1. The different parameters, the obtained rates, concentrations, and their values in generalised coordinates are given in Table 2.3. The rate constant has the unit 1 moh' s-^ The limits of integration are given by jc = 0, a = aQ, r = Oand t In the case of two different initial components, the dependence of a on r for different values of u = &o/«o is represented in Fig. 2.1. It can be seen that the turnover drastically increases with increasing u, the ratio of the initial concentrations. [^0 ~^] becomes constant in a first approximation, if b^ »aQ, The differential equation can be simplified under these conditions to i = - a = k{bQ - x){aQ - jc) = k^a^ - x). As can be seen, the reaction formally takes place according to first order kinetics. Eq. (2.25) cannot be used when GQ equals b^. This specific case simplifies the differential equation to (see Appendix 6.4.1) JC = c = —a = —fo = ka^. Therefore an analogous solution to eq. (2.24) (only one initial reactant) is found: GQ
kt
c = A: = 1 + GQkt'
(2.24)
a = -1 + a^kt
whereby the general coordinates are defined by
a=
a
^ b ^ =
Gn
C
y = —, ^0 Gn
r,
X
§ = —. ^0 Gn
X=
G^t
General Approach in Formal Kinetics
60
Ch.2
TABLE 2.3 Rates, half-lives, concentrations, and general coordinates of second order reactions, for details see Appendix 6.4.1 2A-^B
A+B
Rate 2 Concentration -time relationship
J
=^ k{aQ - x){bQ - x)
l_ _ ^kt
a I.
In
GQ
(^o--^K
V^^
b ^ X=
1 + la^kt (2.24) 1 + la^kt
. = — ^
^-floexp(-(^o-^)^0 ^-aoexp(-(^0-^)^0 ^o(^o - Q Q ) ^= ^0 - «o exp(-(^o - «o)^0
Half-life Concentration given in general coordinates
ha = larsk _
^1/2 =
^ QQ * x
ln(2-^)
«0 M=-
«0
dx
^0 flo
1 1 + T'
= -1
They depend on each other according to
l + r'
{aoC
a = -A:,a, i = Jt,a - kib, c^kib
« - = 0 and
2.15
A-^B B + C->D
a = -kiQ,
2.16
A + B-^C C -^ A + B C + D->E
b-kxa-
kibc, c = -d = -kibc
X\oo = flfo
and Ar2oo = Min(a^,co).
a = ^ = -it,a^ + V . ^ ^ f^^^ij ^ f^^c _ ^^^^^
Coo = Xx^ - Xioo - X-^^ = 0
-d = e = k'^cd.
floo =ao~^loo+JC2oo = 0
^o«=:v3oo = Min(ao,^o)
Ch. 2
General Approach to Linear Systems of Reactions
65
In particular Example 2.16 depends on the starting concentration d^ being higher or smaller than a^ and b^ which influences e^ and x^. In this table just one condition is summarised. Details are given in Appendix 6.4.2. Assuming that the initial concentration of D is smaller than that of A as well as of B, an equilibrium is obtained between the remaining amounts of reactants A, B, and C according to
c^ =
a^b^
(2.28)
This equation is the typical form of the law of conservation of mass. According to thermodynamics, eq. (2.28) is only valid in ideal or ideal diluted systems. The consequence is, that any fundamental assumptions in kinetics (especially eq. (2.15)) will only be valid in ideal systems. 2.2 GENERAL APPROACH TO LINEAR SYSTEMS OF REACTIONS In Sections 2.1.1.1, 2.1.3.1, and 2.1.3.3 simple relationships for sequences of elementary steps of reactions have been introduced and discussed. A generalised system has been given by eq. (2.15) and has been demonstrated using Examples 2.4-2.10. This approach can be used to consider linear reactions in general. It is of principal interest, since it can also be used in the treatment of quasi linear photoreactions [10,14-16]. 2.2.1 Differential equations as a general treatment in kinetics 2.2.1.1 Formalism In linear systems of reactions, the differentiations of the degrees of advancement with respect to time t depend linearly on the concentrations. According to the principles mentioned in Sections 2.1.3.1 and 2.1.4.2, the last column of the stoichiometric matrix representation (see Example 2.7) becomes for the ^th of the r different partial reactions (eq. (2.15)) K =*it^)k
(^ = l,2,...,r).
This linear system of equations can be written as a matrix equation:
66
General Approach in Formal Kinetics
i*=K;.a'.
Ch. 2
(2.29)
As defined in Table 2.2, in this equation x* symbolises the vector of all r degrees of advancement differentiated with respect to time, a is the vector of the concentrations of all the reactants. KQ represents the matrix of all the rate constants. This matrix will have n columns and r rows. It can be directly derived from the scheme discussed in Section 2.1.1.1. This differential equation is valid for all concentrations. By combination of eqs. (2.3) and (2.29) one obtains the differential equation v°*i* = d" = v'*Koa' = K'c^\
(2.30)
It has been mentioned that linear dependencies between partial reactions have to be avoided (see Section 2.1.4) and the conservation of mass between the concentrations has to be used. Under these conditions according to eq. (2.6) and the definition of P (defined by eq. (2.7) the following relationship is obtained: r=ao+PAa. The reduced matrix v has to be used, otherwise v"* cannot be formed, since a quadratic matrix is necessary for inversion! Therefore the change in concentration with time is given according to eq. (2.30) by ^°=KJ.(ao+pAa) = ^o + K^PAa
(2.31)
and the following symbols are defined as io=K^ao,
KJ,=vKo,
Aa = a - a o .
Furthermore the reaction scheme has to be written in the reduced form. Then the reduced stoichiometric coefficients, degrees of advancement, and matrix of rates have to be taken i = Koa
(2.32)
Because of matrix algebra, in the following equation, according to eq. (2.5),
Ch. 2
General Approach to Linear Systems of Reactions
67
the reduced stoichiometric matrix v for the reduced reaction scheme of the system of linear independent steps of reaction has to be chosen: Aa = vx. Accordingly eq. (2.30) has to be written as v i = ^ = vKoa = vKo(ao+pAa).
(2.33)
By defining a matrix in analogy to K^ as Kc=vKoP, the above equation can be written as ^= ^o + K^Aa.
(2.34)
with the definition ao=vKoao. This differential equation represents a set of linear independent concentrations. The matrix K^ becomes a regular sxs matrix. Eq. (2.33) allows a direct transformation to the differential equations for the linear independent degrees of advancement. Multiplying by the inverse of the reduced matrix of the stoichiometric coefficients v~\ one obtains v"**vx = v"*a = V^ao + v'^K^vx, i = v-^^o+v-^KcVx,
(2.35)
x = Xo +K-X. The matrix of the rate constants is defined by K = v-^KcV.
(2.36)
Both the matrices K and K^ have to be regular matrices with sxs rows and columns. In the example of a consecutive reaction, all the equations above
General Approach in Formal Kinetics
68
Ch. 2
are derived explicitly to demonstrate the principle of this approach. Some of these equations are summarised in Table 2.5 for different mechanisms. In future discussions such reduced vectors of concentrations and matrices in differential equations are assumed. Therefore from now on the symbols ^ and * are only used, if either linear dependencies exist between the partial reactions and/or the number of concentrations of reactants is not reduced to the necessary ones. Thus in general the following equation^ will be valid: XQ + K - X a = a^ + KQ ' Aa, Example 2.17: Matrix of rate constants for consecutive reactions For the reaction A ^ B •-> C, the reaction scheme can be written as or as a reduced case of Table 2.1: A * -1 ^1 •
X2
*
B 1 -1 -1
1 0
C 0 k^a 0 k^b 1 kjb
Xl
A -1
X2
0
B 1 -1
C ^k 0 kiO - k2b 1 k^b
containing two linear independent steps of the reaction (.; = 2). It cannot be reduced any more. The stoichiometric matrices are therefore according to the defmitions given in Section 2.1.1.1 M v°* = 1
1 -1
0^ -1
0
0
1
and
M v° = 1 0
0^ -1 1
The matrix of the thermal rate constants are given by (kx
K=
0
0 Qi] k2
[O k.
and
K„ =
,0
L
Ch. 2
General Approach to Linear Systems of Reactions
69
Next, the equations are given in matrix notation / ' • * \
X * = K ; a° = K^3J
fki 0 O'jj 0 Jfc^ 0 b \^ 0 kj 0 j
^—> i4,'^l i
>
Ai A-.i - ^ ^ ^
(2.68) - ^ A>i .
This scheme has the following meaning: reactant A^ is formed by just one or even many compounds A,_j,A,._j It can react to form A,^^,,A..^,,... Backward reactions are allowed in principle. However, they are not allowed to form Ai or to start from A,. Under these assumptions the differential equation (with respect to a,) is given by a,. + {k, +/:,.+... )a,.(0 = A:,._,a,_, (r) + ^,^^,a..,, ( 0 -
(2.69)
In the equation above, the aj{t) represent the interference functions (j = 1 - 1 , /' - 1 , . . . ) . They exhibit the dependence on time for those reactants which form A,. It is assumed that they are known. Therefore the result is the number of exponential functions
90
General Approaches in Formal Kinetics
Ch. 2
The exponential functions in the angle brackets represent the number of different exponential functions of those reactants produced immediately in the reaction sequence before the reactant Ai. This means that only the sum over all reactants formed has to be determined. In consequence the additional new exponential function takes the exponent
Therefore (1) The exponential function for any initial product which is not formed by any back reaction amounts to one, since the differential equation, eq. (2.69), is a homogeneous one under these conditions:
(2) Any final product A^ yields the exponential function
since under these circumstances the differential quotient a^ is the only element on the left-hand side of eq. (2.69). The solution of this equation is the integral of the right-hand side of the equation. (3) Therefore a simple non-branched consecutive reaction A,
>A^
>Aj^
>
>A^
yields ^. = /
(/ = 1,2,3,...,/!-!),
e^-n-\
(4) The following is always valid according to Section 2.2.2: \<ei<s. If any reactant A-^ is the starting material taking part in one or more back reactions, eq. (2.69) has to be replaced by a system of differential equations.
Ch. 2
Information about Reactions Supplied by Graphs
91
The following equation can be taken as an example, if the above model of the reaction system, eq. (2.68), is extended to a back reaction from Af to y4/_i: /
X
(2.71)
a, +^/-i«/-, +(/:, +/:Ja, = V,a,...,(0. k- symbolises the rate constant for the back reaction from A^ to Ai^i. One finds instead of eq. (2.70) ^/=(^i-2»^r-i--) + 2,
(2.72)
since the homogeneous system adds two further eigenvalues. ^, stays the same, because €^,2 ^^^ ^^ be inserted in eq. (2.72) instead of e,_,. In the case of a back reaction from A,^., to i4,, eq. (2.71) has to be modified to a,. + k.a^ - k^^^a,^^ = /:,.,«,_, (0 + k^._^a.._^ (0,
One finds instead of eq. (2.70) or (2.72) the following relationship ^ . = ( ^ i - P ^ r - P - - ) + 2.
(2.73)
A comparison of the results of eqs. (2.70), (2.72), and (2.73) makes obvious the fact that the exponential function e, is only enlarged, if the back reaction or an uninterrupted sequence of back reactions take place to the considered reactant At. 2.3,3 X' and K-diagrams Further information can be obtained, if instead of concentration-time diagrams, other types of graphical representations are used: ~ An X'diagram represents a plot of two degrees of advancement versus each other for a given system of reactions. - The graph of two concentrations versus each other is called a concentration-diagram [19], which is abbreviated in the following by the term Kdiagram.
92
General Approaches in Formal Kinetics
Ch. 2
In many cases both these representations characterise a system in a better way than concentration-time diagrams. A special advantage of these diagrams is their limited length in their axes. For example, it represents the total advancement of the reaction in contrast to the time axis of concentrationtime diagrams which in principle extends to infinity. Concentration diagrams as well as X-diagrams are given by two of the equations, eqs. (2.45) and (2.56), respectively. Thus the K-diagram of ^,(0 versus aj{t) is given by
t^
(2.74)
2.3.3.1 Initial and final values Initial and final points of a K-diagram are determined by the initial and final concentrations of reactants A, and A,. If the concentrations of two totally degrading intermediates are graphed versus each other, the results are closed curves starting and ending in the origin of the diagram. Under these conditions no additional information is obtained in comparison to the related a/(0 diagram. 2.3.3.2 The initial slope One can calculate the initial slope in a K-diagram by use of the system of differential equations according to da^ I _ a,0 daj
ajQ
In some cases, a division of zero by zero can happen. Under these circumstances the formalism of Bemoulli-L'Hospital has to be applied, which is discussed in detail in the next example. The following reaction mechanism is assumed: k!
' kUt
'^^ ku,
.
Thus the two differential equations can be derived:
(2.75)
Information about Reactions Supplied by Graphs
Ch. 2
93
The rate constants of the back reaction are symbolised by kj (y = 1,...,rj. For 7 = 1 (that means A, is the starting material), one obtains
and one finds La 10 da.
= -1
^1^10
as well as ^ ^ | = / - =0
(r = 3.4.5...).
Whereas the condition i = 2 gives the same result as above. 0 yda^
=0
(r = 3,4,5...).
ktU 1"10
the graph of the concentration of reactant A^ versus the third reaction product A3 (/ = 3) happens to cause a zero division: dttj.
da^
I
(r = 4,5,...).
According to Bernoulli and L'Hospital both the numerator and denominator are differentiated with respect to t. As soon as the time t approaches zero the numerator becomes zero. Then the denominator is given by ^30 •" ^ 1 ^ 2 ^ 1 0 -
94
General Approaches in Formal Kinetics
Ch. 2
Thus
K 1 =0
(r = 4,5....).
Accordingly, the initial slopes of a K-diagram can be calculated for any combination of reactants. 2. i. 3.3 The final slope In general the slope of a K-diagram is given by —- = =
d^j
—-.
(2.76)
2^^*P;*exp(r^0
Eq. (2.76) requires a summation from 1 to s. The final slope in eq. (2.76) becomes indefinite at the limit t->oo. Assuming ri is the smallest eigenvalue (by amount), one can divide the nominator as well as the denominator by exp(-r,0: da, ^ r,Pi, 4-^r^p,^exp[(r^ ~ri>]
In this case the sums have to be taken from 2 to s. Then all the exponents are negative. Consequently at the limit r —> «> all the exponential functions become zero; therefore the final slope is given by da, daj I-
=-^.
(2.77)
Pji
The slope depends on the value p,,. The following cases can be considered: (a) Pfi = 0, pji ^ 0: the final slope is zero. (b) p .J 9t 0 and py, ^ 0: the final slope is finite. (c) Pj\ = 0 and in consequence p,, ?& 0: the final slope becomes infinite. Examples for the values p,., and py, respectively are given in Table 2.6 and were used to calculate the concentration's dependence on time in Appendix 6.4.1.
Ch. 2
Information about Reactions Supplied by Graphs
95
Example 2.25 The determination of these final slopes is demonstrated using the mechanism of a multi-step consecutive reaction A—!-^B—^C-^-^D. The final slopes are given in Table 2.7 for a variety of cases. In the first column the concentrations are listed which are graphed versus each other to form the K-diagram [aXaj)]. The next three columns represent the three cases for which different rate constants are assumed to result in the smallest eigenvalue. The data used in Table 2.6 supply the values p,, and p^, which are taken to calculate the ratios of the different rate constants according to eq. (2.77). Thus the final slopes can be determined. The full calculation procedure is demonstrated in Appendix 6.4.2.
TABLE 2.7 Slopes at final values for 3 different rate constants taking the smallest value Smallest eigenvalue ks
*2-*l
c{d)
*1*2
{ki-k2){kf d(a)
-*3)
hh (*3-*l)(*I- -*2)
c(b)
h *3-*l
k2ki k^k^
d{b)
die)
^_^
k^
k^^
-1
96
General Approaches in Formal Kinetics
Ch. 2
2.3.3.4 The extrema According to eq. (2.74) different cases can be distinguished in a Kdiagram aXo,): - for a, = 0, the result is a horizontal tangent and - for fly = 0, a vertical tangent. For the compound Aj, which reacts according to the mechanism, eq. (2.75), the maximum concentration amounts to I
_ ^1-1 ^ M
•^^i-H^f-H
The maximum of the k-diagram is given by a: a:1-1
_
^M
jfc,-hit;
if a,^., is not the starting material of the backward reaction. This relationship can be easily visualised in a K-diagram ei,(a,_,): in the maximum of the Kdiagram the ratio of ordinate to abscissa equals the right-hand side of the relationship above. 2.3.3.5 Influence of dilution The vector of derivatives comparable to eq. (2.53) (s)
5,=Vp,, given by a, to the order s, is proportional to the initial concentrations. Since V contains no concentrations, the p, must also be proportional to the initial concentrations. After division of eq. (2.74) by the initial concentration GQ on both sides of the equation, values are found which are independent of concentration. For this reason a graph of aja^ versus ajja^ results in a reduced concentration diagram, which becomes independent of dilution. 2.3.3.6 The areas In the K-diagram some equations can be set up for the areas below the concentration functions. In the case of a simple consecutive reaction
Ch. 2
Information about Reactions Supplied by Graphs
97
A, — ^ ^2 — ^ A3 — ^
one obtains for the areas below the curves a^Ca,)
This equation can be verified by complete induction, since obviously the integral is given by
Jo 2
'
it,+Jk, 2
Since eq. (2.78) has been verified, one obtains for / + 1
Integration of this equation gives ~*i j^«i da^^i +*i£^i+i da, =*, J^«, da, -*,>i J^^w da,. Taking into account that the first integral on the left-hand side of the equation vanishes at the lower and upper integration limits and that the first integral on the right-hand side is given by eq. (2.78), a substitution of 1 by 1 + 1 gives eq. (2.78) back again after simple rearrangement. If the ith reactant is the final product, then the rate constant k^ becomes 0. In consequence the following equation is obtained:
Jo
Since n
i=2
da,=
2
3 * *'
i—1
2 '
98
General Approaches in Formal Kinetics
Ch. 2
is valid, for the simple consecutive reaction discussed above one obtains
In the case of the more complicated reactions the areas can be calculated accordingly. 2.3.3.7 Points of inflection In the a^{aj) diagram a point of inflection is found, if
These points of inflection are typical for the K-diagram. They are not found in concentration-time diagrams. It is difficult to define general rules. However, one is able to simulate Kdiagrams using a desk-top computer for assumed mechanisms and to analyse the curves for points of inflection. The principle approach was discussed in Section 2.3.2. Points of inflection are allowed in K- or X-diagrams only, if more than two linear independent steps of reaction (s > 2) take place. 2.3.3.8 Regions of existence in a K-diagram The K-diagram is usually limited to a triangular area below the straight hne «/=«o-^y
(2.79)
At this line the sum of the concentrations of A, and Aj equals the sum of the concentrations a^ at the beginning of the reaction. For this reason, in general, the diagram stays within this triangle given by the coordinates 0, 0; 0, aQ-, and UQ, 0. Exceptions to this theorem can be expected if two reactants are produced within one step of a reaction. The sum of the concentrations of the reactants can no longer stay constant, if the mechanism contains the following partial step:
Ch. 2
Information about Reactions Supplied by Graphs
>A,
>B + C
>
99
(2.80)
For this reason the above-mentioned straight line (eq. (2.79)) can no longer describe the above-mentioned limitation. The partial step, eq. (2.80), cannot be reversible since the system loses its linearity. 2,3.4 Transformation of linear systems K and K' are assumed to be regular Jacobi matrices of two different linear systems of equal rank s. Furthermore, all eigenvalues are assumed to be different. Under these conditions, similarity transformations S and S' are possible: D = SKS"'
and D' = S'K'S'"^
D and D' are diagonal matrices. If all eigenvalues of K and K' are equal, that means r^ = r/,
for A: = 1,..., 5,
then D = D' can be obtained. In this case
or by use of T = S-'S'
and
T'^S'-^S,
one finds K = TK'T-'
or
KT = TK'.
(2.81)
The transformation T is linear, if the primed system can be transformed into the non-primed system. In this case.
General Approaches in Formal Kinetics
100
x = Tx'
Ch.2
(2.82)
and x^=Txl,
x = Tx',
Xo=Txl..
(2.83)
If this transformation is applied to the fundamental eq. (2.35), then T i ' = i = T i i + TK'x' = *« + TK'T-*x = io + Kx.
(2.84)
The matrices K and K' are similar matrices; they have the same eigenvalues and principal minors [18]. Example 2,26 A system of two independent reactions A'
>B',
C
>D'
results in the simple rate law y, = a^(l - exp(-A:,0), x^ = Co(l ~ tx^{-k^t)). (2.85) One looks for the transformation T which transforms the system into the reaction ->B-
->C.
This reaction is represented by [14]
or
MO ^20
^11
^12 if "0 I "'22A^0/
•^21
•'22
•'21
(T^1
^12
T;1
^12
^21
^22, V^2^0>
V^21
^22
0
Information about Reactions Supplied by Graphs
Ch. 2
101
and
0 YT;, "^ijKTlX
T.AJT,, T,^\(-k{ ^22 7
V21
'22
0
0 -k'^
These equations are useful for the determination of the four elements r^, and for the correlation of the eigenvalues. They depend on each other because of eq. (2.38). The last equation can be written explicitly (a)
'^\^\\ —K^u
*2^12 ""^2•'22 —""^2^22
(^/
Equation (a) means that either Jt, =ik;
or
7;i=0.
In the first case 7,2 becomes 0 because of equation (b), since k^ ^^kj. Equation (d) gives A:2 = Acj. According to equation (c) one finds T -
'^ T =
k'
Eq. (2.85) results in ao = r,,a^
and
T^.k^^ + 722*2^0 = 0.
By this means the elements of T are found to be 0 x=
ki Ky
Kt
GQ
Ky
a^(l-exp(-A:,'0)^ c^(l-exp(-A:^0)
ao K\
CQ
j
The result fits into the formalism found in Section 2.2.2 and discussed in Example 2.22:
102
General Approaches in Fonnal Kinetics
Ch. 2
jc, =ao(l-exp(-A:,0) ^2 = T ^ V ( * 2 ( ^ " ^*P(-*i^)) - *i(l - exp(-*20))' Ky "~ /Cf If r,, becomes 0, then according to equation (b) ^12 ~
^22 •"
7
^22
77
and one finds the transformation T ^
0
^
T= ^1
^0
^2
^0
The same result as above is obtained. These transformations can be used to determine the rate equations of complex systems by use of the known differential equations of simple systems as has been demonstrated above. However, this procedure does not have any advantages in comparison to the other methods described previously. In the case of two linear independent partial reactions, one can set up the following theorems for linear transformations: (1) Straight lines are transformed into straight lines, therefore (a) tangents become tangents and (b) asymptotes become asymptotes. (2) Parallel straight lines are transformed into parallel straight lines. (3) The ratio of distances on straight lines stays constant. (4) The ratio of areas stays constant. (5) Conic sections are transformed into other conic sections. 2.4 APPLICATIONS TO SELECTED SYSTEMS In this section some of the essentials stated in the previous sections are reviewed and summarised. The equations derived are demonstrated for some
Applications to Selected Systems
Ch. 2
103
simple selected systems. As mentioned, the result does not depend in principle on whether thermal or photochemical reactions are considered and the quantum yield does not depend on the concentration. 2,4.1 Uniform reactions For 5 = 1 the differential equation, eq. (2.35), degenerates to the simple equation
Therefore eq. (2.38) is reduced to
whereby *
,
=
•
a,iO a- —alO
The simple reaction A —>B has been discussed in Section 2.1.7.2. Therefore only two further reactions have to be compared. 2.4.1.1 Parallel reaction The reaction A * -1 • -1 X2
>B >C
. •
B C 1 0 k^a 0 1 k2a
stays uniform, since ^2=X^\y
where% = -^.
In consequence one finds for the second variable by integration xl = x^x.
General Approaches in Formal Kinetics
104
Ch.2
Thus the reaction scheme can be reduced by analogy to Example 2.9 in Section 2.1.4.2: B
- a + l)
X k^a
For this reason the differential equation is given by X = k^{a^ - (x +1)^) = k^a^ - (/:, + k^)x (2.oo) and ^ = -(pc +1)^1^ = -(^1 + ^2 )^»
c = k^a. According to eq. (2.86) the Jacobi matrix is represented by K = /: = -(/:, + ^2) • By use of eq. (2.55) the following relationship is obtained:
^
k^^k^
The integral for the degree of advancement is ^Q/V,
A: = -
-(l~exp(-(/:,+A:2)0).
A:, + ^2
One obtains either by direct integration or by use of above equation, a = ^0 exp(-(/:i 4- k^ )0, /Cj ~r /C2
Applications to Selected Systems
Ch. 2
105
2.4.1.2 Back and forward reactions In Section 2.1.4.2 an equilibrium reaction has been mentioned, A ^ B with the reaction scheme
-1
B 1 k^a — k2b
consistent with the assumed mechanism. Therefore the degree of advancement depends on time according to X = k^a — k2b = k^(aQ — JC) —/c2-^ = ^i^o ~ ( ^ i
+^2)'^-
For this reason, the Jacobi matrix K is reduced to one element, k = -{k^ -••^i)-
The degree of advancement amounts in the photostationary steady state (t -^ oo) to
AC|
"T" K-J
By integration one obtains x = b = - ^ ^ ( 1 - exp(-(fc, + k^ )0) /Ci " r AC2
and a=
—{k2 + k^ exp(-(A:, + k2)t)).
For this discussion it was assumed that at time r = 0 the concentration of the second reactant A2 becomes zero (&o = 0)-
106
General Approaches in Formal Kinetics
Ch. 2
2A.2 Two linear independent reactions 2A,2,1 Independent parallel reactions Two parallel steps demonstrate the simplest example A->B,
C^D
for two linear independent steps of a reaction. Their rate laws are A:J = Z? = ao(l - exp(-/:,r)),
a^a^ exp(-/:,0,
;c2 = t/ = CQ (1 - exp(~^2^))'
C^CQ exp(-/:2^)-
By use of the time equations, the X- and K-diagrams of the system are given in parametric form. By an elimination of time one obtains fln
Jt^ln^
M
=_ /:, In
Cn
X'y
or in general coordinates ^ • = - ^
S2 ~
*2
»
^2=l-(l-^,)^
(2.87)
In these equations, the symbols I represents the degree of advancement reduced with respect to the initial concentration. In consequence in the ^diagrams (reduced X-diagrams) the initial |o and the final |„ slopes are given according to eq. (2.87) as ^, ^.
0, X>1 =X
and
d^i
4,
«,
A: B -^ C yields the Jacobi matrix K=
-A:. 2
0
(2.88) 2
and the rate equations
a = k^a, c = ^2^.
108
General Approaches in Formal Kinetics
Ch. 2
By integration one obtains (see Example 2.19 in Section 2.2.1.3) Xt =ao(l-exp(-A:,0) JC2 = c = -r^{k2 K<j
(1 - exp(-*,0) - *, (l - cxp(-k2t)))
ACi
(2.89)
a = OQ exp(-/:,r) _
b=
QQ^I /Cj
(exp(-/:,r) - exp(-"^2^))
'Cj
Both concentrations can be graphed in a reduced form depending on the ratio of the rate constants of the individual steps. The functions are given in Figs. 2.5 and 2.6. Eq. (2.89) gives X- and K-diagrams in parametric form. The elimination of t using
^,=-^, an
^2=^ and
x-¥^1
results in the relationship ^ 2 = 1 . -
1.0
(i-i.r-(i-^.)
(2.90)
i-x -i—I—I—I—I—I—I—I—I—I—I—I—I—I
A
I
> B
Fig. 2.5.fi= b/ciQ. graphed as a function ofx according to eq. (2.89).
Ch. 2
Applications to Selected Systems
109
Fig. 2.6. y = c/flQ graphed as a function o{% according to eq. (2.89).
The eq. (2.90) above represents the solutions for the case xi^\. The case ;f = 1 yields dependencies discussed in Example 2.17 in Section 2.2.1.1. The corresponding ^-diagram is given in Fig. 2.7 using
Fig. 2.7. ^-diagram according to eq. (2.90). x >s the chosen parameter.
110
General Approaches in Formal Kinetics
Ch.2
^2=e,+(l-f.)ln(l-e.). The slopes in the ^-diagram yield %
d^. = 0 and d^x
x-i
d^.
X>1 X = iX- B —
• ^
1
1I
1 i^
1
117
1
1
1
1
^ u
1
A
1 1
0.5 -
B
c"
^'^^^ 0.0 0.0
1
—1
h— 1
1
h-•
1
— 1
1 — h—
¥— 1
1.0
1
1
.
2.0
1
'
3.0
Fig. 2.11. Dependence of the concentration on time for the consecutive reaction A —> B C -^ D. The parameters ki = 1, A:2 = 2, /:3 = 3 are assumed.
1.0
_i
A
• B
1
1
1
{-
>- C
. 1.0
Fig. 2.12. Concentration diagrams of the consecutive reaction A -> B -> C -> D. The symbols at the curves mean: 1. b(a)y 2. c(a), 3. d(a)y 4. c(b). The assumptions ki=^ 1, A:2 = 2/3, k^ = 1/3 are valid.
118
General Approach in Formal Kinetics
Ch. 2
2.5 METHODS OF APPROXIMATION Linear reaction systems allow the rate laws to be presented in a closed form even if the reaction procedure is complex. But non-linear systems cause extreme difficulties in the integration of even simple equations. Therefore quite a few methods are described in the literature to approximate the solution of the differential equation. Nowadays such iterations are no longer necessary, since the relationship between concentrations can be calculated in an easy way for given parameters. Nevertheless in kinetic analysis two questions are essential: (1) Up to what amount can one simplify the rate laws and the relationships between the concentrations, if the rate constants of the partial reactions differ? (2) At what conditions can one distinguish between a simple and a complex mechanism? 2.5.7 Negligible concentration of an intermediate For the consecutive reaction A -> B -> C, the concentration of the intermediate B is given by the difference between the degrees of advancement of the two steps
The maximum value of its concentration is reached at the time at which the change in concentration b becomes zero: b = k^a — k2b-0. Therefore the maximum concentration amounts to JIC2-
tn ~^\^i
Since a^ C. The reproducibility of the measurement is assumed to be limited to 1%. Therefore a discrimination between the consecutive and the simple reaction mechanism is only possible, if kjk2 is larger than 0.01. The ^-diagram has to be considered for the time functions cc = a(t) and y = y(r) accordingly. It degenerates for small values of kjk2 to a diagonal between the corners ^ , = ^ 2 = 0 and ^j = ^2 = 1* Then the following relationship between the concentrations is valid: a-{-c=^aQ.
Depending on the reproducibility, the two linear independent steps of reaction of the consecutive mechanism (existing in reality) appear as a uniform reaction. In the following, conditions are given for some examples of consecutive reactions, which can be treated in the same way. The conditions are chosen such that their apparent mechanisms cannot be distinguished from uniform overall reactions at the experimental limitation mentioned above (see mechanisms A-D):
120
General Approach in Formal Kinetics
Ch. 2
(A) 2A >B >C, (B) A + B >C >D, (C)A >B, 2B >D, (D)2A >B, 2B >C. The different equations for these four different consecutive reactions are summarised in Table 2.8 using the same nomenclature as above. All these reactions are examples for overall reactions proceeding stoichiometrically. Their rates can be expressed by equations derived in Section 2.1.7. In some examples the stoichiometry makes it obvious that the mechanism is not given by elementary reactions. The overall mechanism summarises many elementary reactions. The method explained has the advantage that it is free of any additional hypothesis. Its disadvantage is, that it cannot be applied in general. This fact will be discussed for the reaction A ^ B —> C. Reducing the number of linear independent steps of the reaction by X = X* - xl
and
X2 = ^ 3 ,
one obtains x^-X2^b
1
< «5
eg «
0^
1 §
4>
>
JUj)
il
I
11
o II ^ I
II
II
o
« I
^ j
"^1
VI I
^
t
w
II I
H
w U
t t
rj
I
r X 4J
VI
II
B
>C
>D,
134
General Approach in Formal Kinetics
The rate equations are given by a = —k^Cy
b = k^a - ^2^'
Therefore
^2
^3
^3
in consequence k^a
= -
k}a
The differential1 equations give
db
"
dc _ -k. db'
dc
7
dc
For this reason the set of equations is given by — • — = A:2 Afe,
^— = ^2 A& - ^3 Ac. ^^2
In consequence one finds k^
k^
k^
/Co
fCy
fC^/C-i
The relative error amounts to
*'•
Ch. 2
Ch. 2
I i cd
a
1 i 2
I §
1 U
J ^
- 3
H osi
\^
^
"-rh?
I
K
I II
I
^
ri
Methods of Approximation
«3
+
+
-iirn?
+
1CO
t + t
flto• ^^^ ^^is reason the chosen approximate solution of the system can no longer describe the progress of the reaction for large turnovers. For this reason the reaction does not come to an end in a finite time. 2.5,5 The approximation of the equilibrium If an equilibrium precedes or succeeds any reaction or intersects a sequence of reaction steps, the relationships can be simplified, since the rate constants of forward and backward reactions are larger than those of the other partial reaction steps. This fact is demonstrated for the reaction A ^ B —>C. In the case where A and B are always in equilibrium, a^Kb
with
K^^
is valid. Furthermore the mass balance is given by a +fo+ c = aQ+fcQ and the single kinetic equation
Ch. 2
Methods of Approximation
137
can describe the reaction. By combination of these equations one finds
or for the concentration of the component B
Introducing this result in the kinetic equation, one finds .-
^3
K-hl
-(«o+^o~c)»
which will yield after integration \A
r
c = K"+-^o) l - e x p — JJ
The equation with mass balance results a4-fe + (: = 0
or b = -
_
^3
-b.
The second equation gives, after integration,
and by use of the first equation a = Kb = aQ exp
(K + l)
In contrast to a treatment according to the Bodenstein hypothesis the changes b of concentration with time are finite. That means the factor
138
General Approach in Formal Kinetics
Ch. 2
in the exponent has to be substituted by the factor 1 3 Kt
r K-j
In the case where an equilibrium reaction intersects a consecutive reaction as in the system A —» B ^ C -^ D, one finds b = Kc,
b = Kc
by the equilibrium approximation discussed. The respective kinetic equations read
The equation with mass balance yields a + b + c-\-d = aQ or ti + (Ar + l)c +rf= aoi therefore a + (/i:-fl)c + d = 0. The kinetic equation for the first step is a = aQexp(-^,0
The differential equation of the Product C results in
Methods of Approximation
Ch. 2
139
Thus, by integration of this linear differential equation beginning at c = 0, r = 0, one finds
c=-
^1^0
ik4-it,(if + l)
exp(~^,0-exp -
k.
(^+1)))
and b=
Kkjap exp(-A:,0-exp k^-k^{K + l) I
(^ + 1) ))
as well as d = a, l - e x p ( - ^ , 0 -
K{K + \) exp(-A:,0-expl- ^ ^ k^-ktiK + i)
f )))
The examples demonstrate that the number of linear independent partial steps of a reaction is reduced by one, if the equilibrium approximation can be applied to isomeric equilibria. This statement is not valid for more complex equilibria. Treating the reaction 2A ^ B -» C according to the equilibrium approximation, one finds b^Ka^,
b = 2Kaa,
c = k^b
and
Therefore a + 2& + 2c = a(l + 4Ka) + 2k^b = 0 or a{l + 4Ka) = 2k^Ka\
140
General Approach in Formal Kinetics
Integration beginning at a = AQ, / = 0 yields - - + — + 4Kln— = -2k3Kt. a
OQ
OQ
Using reduced concentrations
the following relationship is obtained:
J_(l-l).2.na=. 2X The concentration of C is given by
or
The final concentration of C (c^) is reached, if a becomes 0. It is
c^=Y + %aoDefining c
the following relationship results:
Ch. 2
Methods of Approximation
Ch. 2
141
Fig. 2.15. y= cjc^ as function of r for the reaction 2A ^ B —> C (equilibrium approximation). X- KQQ is used as a parameter.
y=-
1
-(l-a)-f-
1 + 2;^^
'
2X ii-a^y \ + 2X
It is given in Fig. 2.15 as a function of r. 2,5.6 Reduction of the number of linear independent steps of a reaction If either the Bodenstein hypothesis can be applied to a reaction system or one of the steps is an isomeric equilibrium, the number of linear independent steps of reaction is reduced. The concentration of an unstable intermediate is according to the Bodenstein hypothesis with z = 0:
^=x^^^*=^k=\
This equation can be solved with respect to ^c^^, if Vj^ ^ 0:
(2.98)
142
General Approach in Formal Kinetics
Ch. 2
For any component A, the general relationship is valid: Aa, = ^uXi+-- • + Vri^r k-l
Aai
=
^^'1^1
X^:.
iXf
^/
^
^pA ^
^ = ^^A j
^
^A ^ ^4 '^6 "^ (^2+^3+^4)-(^5+^6)
> B via singlet state
A — 5 ^ B via triplet state
^
^5
3,1.1.1 Quenched photoisomerisations The two examples of a photoisomerisation given here are the simplest cases of photoreactions. As demonstrated above and in Section 2.1.3.3 the photophysical pathway can proceed via the singlet or the triplet state. During these isomerisation reactions the intermediates A' or A" are of great interest. Besides the given mechanism the stationary concentration of the excited photophysical intermediate is influenced in contact with a partner B. Energy transfer to an acceptor molecule B opens a new reaction channel, but does not form a new reaction product itself. Looking at the photoisomerisation, this energy transfer will reduce the concentration of the excited state. By this step the photoreaction is "desensitised" or quenched. This quenching can either happen in the singlet state A' or at the triplet state A". Both mechanisms are discussed in the following examples.
Example 3.1: Quenching of the photoisomerisation via the singlet state Using the mechanism A—!^^C the following photophysical steps and the reaction scheme can be derived:
Relationship Between Quantum Yield and Mechanism
Ch. 3
A + hv A'
>A'
A
A' + B
X2
>B' + A
B'
>B
A'
>C
^3
x'.
4
A -1 1
A' 1 -1
1 0 0
-1 0 -1
B 0 0 -1 1 0
147
B' C X', 0 0 /A 0 0 k2a' 1 0 k^a'b' - 1 0 k^b' 0 1 k^a'
These steps result in the following rates and quantum yields: - a = c = y) lyv.
A
^2
A'
>A"
X',
A"
>A
X',
A" + B
>B" + A
B"
>B
A"
>C
^5
< X'-,
A -1 1 0 1 1 0 0
A' 1 -1 -1 0 0 0 0
A" 0 0 1 -1 -1 0 -1
B B" 0 0 0 0 0 0 0 0 -1 1 1 -1 0 0
C xi 0 IA 0 kja' 0 kja' 0 k^a" 0 ksa"b 0 Kb" 1 kjo"
the following expressions are obtained: -a = c = (p / A .
y,A =
3 7
{k2 + k^){k^ + k^b-¥ k^)
(3.2)
with the ratio between the quantum yield of the normal isomerisation and the quenched one
Ch. 3
Relationship Between Quantum Yield and Mechanism
C
1: A + /iv
•A'
2: A '
>A
3: A '
^A"
4: A "
^A
5: A" + B 6: B " 7: A" + C
.
.
.
k,b
>B" + A >B •D
,
and for the photochemical quantum yield .A _
k^b ^2 + Ic^b
In this photoaddition the quantum yield depends on the concentration of the reactant B. Therefore this quantum yield can be used to estimate the minimal concentration which yields a reasonable turnover at optimal conditions. The dependence of the quantum yield on the concentration of compound B turns out to be
Ch. 3
Relationship Between Quantum Yield and Mechanism A B + D with k^, the quantum yield becomes
^
2k^^k^h
If the radicals deactivate according to step 5 as well as step 6, the relationship becomes more complex A_ 1__M__L M V>^= ^ +-
I
^
2/:3fc/A
k^ik^^-k^b)
k^a
2k^j
In this case the quantum yield depends on the concentrations a and b as well as on the amount of light absorbed by A. 3,1,1.4 Sensitised photoreactions In the following, four examples of different types of sensitised photoreactions are discussed first taking into account energy transfer. In all these cases lifetimes and rate constants of the intermediate steps, respectively, determine the concentrations necessary to obtain a noticeable turnover. The different parameters for these necessary concentrations are listed for these mechanisms in Table 3.3. Details are discussed in Appendix 6.6.1.2 for these four examples. A further example of a sensitised photoreaction is the case of an aromatic ketone (A), which acts as a chemical sensitiser in the photooxidation by molecular oxygen (C) of a hydrocarbon (B) to form a peroxide R-OOH (D)
Ch. 3
o -a
I
o x: ex
O
x: o
H CO
A
o
T CO o
O
B
m -a
n
-S 6 B o
1
I
^
O
1
^ 1 ^ »
-Ci
-^
+
X
+
•Ad
X
oo
•^ ^
+
U r•JX
5 A >E + F
F+C
>G
G+E
>A + D
A x[ - 1 A 1 X',
A A
0 0 1
A'
B
C
1 -1 -1 0 0
0 0 -1 0 0
0 0 0 -1 0
E
F
G
0 0 0 1 0 0 1 -1
0 0 1 -1 0
0 0
D 0 0
0 1 -1
K /A
^a' k^a'b kjc *5«^
Applying the Bodenstein hypothesis to A' and to the radical intermediates E, F, and G, one obtains Xt — Xy "^ X-r ^ U,
X-y *"" X^ - - U,
X-i ~~ Xc -— Uj
X^ ~ X^ ^ U,
or •^1 " X2 ^ X-^ = X^ =
X^,
Inserting the values for i^ the following relationship can be found:
Therefore
(i = JC5 = k^eg = k^a'b.
Ch. 3
Relationship Between Quantum Yield and Mechanism
157
Since a =-
/Co "t" K-iO
the result is A (p^ =
-^b^-c^d^ip^I^,
kob ^
(3.9)
^2 + kjb
In this case the quantum yield depends on the concentration of B and is independent of the concentration of C. Photoisomerisations can also be sensitised by radicals. A typical example is the photoisomerisation from cis- (B) to fmn^-stilbene (C) by iodine (A), forming as intermediates an atomic iodine and the radical J I P h — C — C — Ph. I I H H
Mechanism and details are given in the following example. Example 3.13: Sensitised photoisomerisation The photoisomerisation sensitised by radicals according to B
^ )C
gives the scheme
>2D
A + hv D + B-
-»E
E
>B + D
E
>C + D
2D-
->A
xl X2
x; x'. x's
A -1
B 0
0 0 0 1
-1 1 0 0
D 0 2 0 -1 0 1 1 1 0 -2
c
E 0 1 -1 -1 0
x'.
rU k^bd k^e k^e k,d'
Simple and Complex Photochemical Reactions
158
Ch. 3
The first line of the scheme notes if = r/^^. This term takes into account that the radicals D which are produced in the photochemical primary process stay in a solvent cage and can recombine to form A. For this reason the fraction of quanta causing photoreaction becomes r < 1. Applying the Bodenstein hypothesis to all intermediates the following relationships can be derived: e= therefore (3.10) In consequence the quantum yield is reciprocally proportional to the square root of the amount of light absorbed /A. 3A. 1.5 Radical photoreactions The products can be formed either by the reaction of a radical (produced within the photo-process) with the initial product or in a radical chain reaction. These two possible mechanisms are discussed in the following examples. Example S-M: Interaction with intermediate radicals The reaction 2A-
kv
•>B + C
can be represented by A A + /IV
>D + E
D+A
>B + E
X2
E+A
>C + D
3
D+E
>A
E
i*
-1
B C D 0 0 1 1 0 -1
1 1
riA kjod
-1 1
0 0
1
-1 -1
k^ae k^de
x{ - 1
A
A'
>B
A'
>C
A' B 1 0 -1 0 -1 1 -1 0
C K 0 h 0 k2a' 0 k^a' 1 k^a'
Taking into account the linear dependencies of photophysical processes, the overall mechanism can be represented by the gross reactions * B C 1 0 A
x:
A"
>B
xl
A"
>C
X2
A
-1 1 0 1 0 0
A' A" B c i ; 1 0 0 0 /A - 1 0 0 0 k^a' -1 1 0 0 k^a' 0 - 1 0 0 k^a" 0 - 1 1 0 k^a" 0 - 1 0 1 Ka"
As above one obtains two different quantum yields in two linear dependent steps of a reaction which form a uniform reaction with the following dependencies of concentrations on time: b=
'h'^5
^^6
/A=^./A. I
_ .«A
^A=^2^A.
(3.14)
and result in the relationship b = -^c. k. The third example is the formation of B via singlet A' and C via triplet A". This reaction is uniform also:
Relationship Between Quantum Yield and Mechanism
Ch.3
»A'
A + hvA'
>A
A'
>A"
A"-
-^A
A"-
^B
A"-
-^C
A -1 < 1 Xy x; 0 1 X', x's 0 0
A
A
1
-1
0
0
0
0
kjo'
A'
>A"
X',
0
-1
1
0
0
0
kja'
A'
>B
A A
0
-1
0
1
0
0
k^a'
1
0
-1
0
0
0
ksa"
x'e
0
0
-1
0
-1
1
k,a"c
A" A" + C
>A >D
This mechanism results in
Ch. 3
Relationship Between Quantum Yield and Mechanism
-C = d =
^3^6 ^A
167
Aj
^
(3.18)
^ = -K+^2)/A
and the following differential equation after reduction to two linear independent steps
dx,
k^ {k^ 4- k^c)
k^ (*5 + k^ (co - ^2))'
which describes the relationship between b and d as well as ;c, and Xj respectively. Integration starting at the limit X, = ^2 = 0
and use of the definitions &
t
4
AC3
5
^6^0
results in l,=X,?2-Z.%2ln[l-^]
(3.19)
which is represented in Fig. 3.2. In the next example both steps of the photochemical reaction include a partner, both being photoaddition reactions. In the case where in both of the reactions the same initial product starts the photoreaction and reacts with the same partner via the same type of excited state, the reactions become uniform, i.e. just one linear independent step is observed. This mechanism can be compared with a thermal reaction discussed in Section 2.4.1.1.
Simple and Complex Photochemical Reactions
168 ^ ^
I
1
1
1
1
I
I
1
Ch. 3 h-
•1 via singlet:
A
^ B hv
via triplet: A + C
•
D
0.5 4
Fig. 3.2. ^-diagram for parallel photoreactions according to eq. (3.19) for^i = 1, w = 1.5. X2 is the chosen parameter.
Example 3-18: Two parallel photoaddition reactions using the same partner via different excited states The photoreaction
A + B-
->C
A -1
A + B-
->D
* -1
B -1
C D 1 0
-1
0
1
.* Xk A
Xj
A'
>A
Xj
A' + BA" A'' + D
->C
JC4
x'f
>A >E
4
A -1 1 0
A' 1 -1 -1
A" 0 0 1
B 0 0 0
c
0 1 0
-1 0 0
0 -1 -1
-I 0 0
1 0 0 0 0 -1
0 0 0
D 0 0 0
E xi 0 JK 0 ^2^' 0 k^a' 0 k^a'b 0 k^a" 1 k^a"d
172
Simple and Complex Photochemical Reactions
Ch.3
The following rate laws are obtained: -b = c = •
——
k2+lc^+
k^b
= (*o+'^o)-
Furthermore using ^~i=-?—r-^2~ bo
Ofl
the initial slope is
d^2
=
X\+X2-
at
ao AQ 4- d^ 0,
Xi>^
d^2 can be derived. If ;|j, < 1, a point of inflection is found at
_ S2w ~ *
(4x'^-X^) :
•
This relationship is represented in Fig. 3.4 for X\ = 0.1 and for different values of Xi ^ 0.1.
1.0 I
1 —•— — 4 —
1
^, 4
—•— —•— ^
1
4
f ^ 1 %\
^
T
via singlet: hv
j/X 4
ly 0.5 4
y/^ 4
\y/^
A +B
•
C
•
E
via triplet: hv A +D
y 4
0.5^ 0.1
0.0 MTt^ 0.0
—4— — 4 —
1
1
0.5
— h - —•— - 4 —
—•—1
1.0
Fig. 3.4. ^-diagram according to eq. (3.23). ;f i = 0.1. ;if2 is the chosen parameter, f j is the ordinate. The points of inflection are marked.
Simple and Complex Photochemical Reactions
174
Ch. 3
TABLE 3.4 Comparison of simple non-uniform photoreactions Reduced degree of advancement
Mechanism A + /IV
>A'
A'
>A
A'
>B
A'-hC
>D
A^M^
,A'
A'-
->A"
A'-
->B
A"
>A
A" + C
>D
A + hv
^A'
A'
>A
A' + B-
->C
A'-
>»A"
A"
-^A
A" + B-
•^D
A + ;rv
>A'
A'
->A
A'
-»A"
A' + B A" A" + D
>C
Slopes
fi =«0-exp(-xf2))
^,^x^S2'X^X2l4l''^]
fl =
J \X\^ w>l loo Ui+0-");i^2. «
Xi
x(l-exp(-X2?2))-^2
?i=l-(l-?2)^'exp(-;t2?2)
^ = ;ci+;^2 ^^2 and for OQ > ( ^ + ^o)• 0. ; f i > i ^2
>A >E
These mechanisms given in this section are again compared in Table 3.4 with respect to mechanism, resulting in degrees of advancement and final slopes.
Relationship Between Quantum Yield and Mechanism
Ch. 3
175
3.L2.2 Physically sensitised photoreactions Both the photo-parallel reaction and the consecutive photoreaction, being physically sensitised, form two linear independent steps of the reaction. Their mechanisms and the reduced reaction scheme are listed in Table 3.5. The overall reaction schemes are derived in Appendix 6.6.2. Example 3-20: Physically sensitised parallel photoreaction The parallel photoreaction according to the details given in Table 3.5
TABLE 3.5 Mechanisms and reaction schemes of parallel and consecutive photoreactions, physically sensitised Mechanism
Reduced reaction scheme and dependencies of the degrees of advancement
B~^^C ^1
D ~ ^ E
^2
B -1 0
D 1 0 0 -1
c
E 0 1
^k
B' + A
B'
>B
B'
>C
A' + D
B--^C A
^1
^D
^2
B -1 0
c 1 -1
D Xk 0 D
D'
>E
A + /IV
A'
>A'
>A
A' + B
>B' + A
B'
>B
B'
>C
A' + C C
>C' + A >C
a—>D
176
Simple and Complex Photochemical Reactions
Ch. 3
and derived in Appendix 6.6.2.1 gives the following dependencies of the concentrations of the excited intermediates: /l'U
^A
—
'
,
_ k^a'd
h'-
U -
^2 + k^b + k^d
k^ +k^'
Kn
i" /Co
Therefore ^ _ ^ _
^3^5^^A
-(D^F
{k^ + k^b + k(^d){k^ ^M""^' ^ ^d = e = .
^6^8^^
- ^ = ¥>2U
(3.24)
is valid. The following differential equations give the relationship between jc, and.X j : dX2 _D
^C
A x^ -1 0 X'j
B -1 -1
C D 1 0 0 1
^k
vtU 9\h
in two linear independent steps, represented by the overall mechanism:
Simple and Complex Photochemical Reactions
180
A + Ziv
A -I 1
>A' ^;
A'
>A
A'
^A"
A"
>A
4
|„A,(,-,„),J°««'oO-exp(-^^», feo
p,3,
d
can be derived according to eq. (3.45). Taking reduced coordinates ^^A,
;f = i : ,
^_1000/o(l-exp(^£;))^
the final relationship can be given as r = -xlni8-()8-l)
(3.49)
The dependence on concentration derived above corresponds to the relationship of the Michaelis-Menten reaction, which was given in Section 2.5.1.2. Example 3.30: Physically sensitisedphotoaddition A physically sensitised photoaddition (see Example 3.10) gives according to eq. (3.7) 1 _ ^2 "*• *3^ K "^ *5^
(p^
k^b
k^c
Ch. 3
The Integration of the Differential Equation
195
By definition of X2 =
K
A = Co - ^0
one finds using c =fe+ A that ^ _ 2 I Xi , X2 I X\X2
b
B—^C
the differential equations 1000/c>f/o(ao --Xt) F{E'„)
(3.92)
and X'y ~~ rCI At "" ^0
)
are valid. The relationship between the degrees of advancement can no longer be calculated in such a system in a direct way, since the function F(E'^) is part of the differential equation. Using general coordinates the system of both differential equations
'
nK)
(3.93)
Ch. 3
The Integration of the Differential Equation
213
with the independent variables ^, and ^2 "^ust be numerically integrated, whereby eqs. (2.7) and (1.41) have to be taken into consideration. Once more the method according to Runge and Kutta is a useful procedure. The mechanism A—^B—5^C can be treated in a similar way. In general coordinates the differential equations amount to
,
.X(g,-g.)
(3.94)
with the definition ;f = 1000/c^^2%In this case the first equation can be solved directly. But this advantage has no consequence. For the reaction
one finds
nE„)
with
(3.95)
214
Simple and Complex Photochemical Reactions
Ch. 3
3,3.10 Reactions with a difference in the dependence on intensity In the assumed photochemical consecutive mechanism A—!iL->B—~^C the first partial reaction is assumed to be a simple photoisomerisation with constant quantum yield. The second partial reaction is taken to be a radically sensitised reaction as described in Example 3.9 in Section 3.1.1.4. One assumes that the E-radicals are produced by a photolysis of a reactant D according to D—^E. In this case the differential equations .
1000/c;y)f/o(l-g,) ^ "^
^2
(3.96)
F{E'J2)
can be integrated according to Runge-Kutta as discussed in the previous section. 3.3.11 An alternative: the transformation of time Up to now the function F(£^) was considered to be primarily a function of concentration. For this reason it was integrated together with the concentrations themselves. If no slow dark reactions have to be considered, and if all partial photochemical reactions depend in the same way on the intensity of irradiation, the fundamental differential equation, eq. (3.72), can be rearranged as ^
^ ^ ] = lOOOxK\Io-^ ^^'F{E:)
l-^[
>» o w
I ^&
-a
ii
X
II
II
if
CO
•§1
^ E
a'g
CA
-I -a
^^
+
, ^
Simple and Complex Photochemical Reactions
o 'a
J3
B CO *5
1
II
-* -ft 11
oo
^
11
Ch. 3
I
24> Tt »-H tS en O
o
1
H i2
r4
Ch. 3
\^\
CO
m
II
1-^
o
1 o
II
'^
o
oo
O
o
S 2
pSi
X I
I
^
f^
-^^
00
II
II
3
>
si >>
lit
I
II
I
I
br
II
» o
^^i^
? ^ --
1 ^w "TT
s
^
I
i
I
o X
+ -
II 9.
ts
15 CO
t
Ch. 3
Ch. 3
U
s
I
Ul
I
o
II
I
II
N5
I
^hy^ -I II N5
t*4
9^
5'
+
o ^ H I a II •iij)
ttj
en
I, I
X
PPrJ
I Vf
9-
l - a = N-^-e'-E
-^
a = N-'-8'-E
(4.10)
270
Experimental Techniques in Pliotokinetic Analysis
Ch. 4
Consequently a measurement of the absorbances at some wavelengths is necessary. Knowledge of the molar decadic absoq)tion coefficients at these wavelengths enables the vector of the concentrations of the reactants to be obtained at the chosen irradiation time. However, the inversion of the matrix causes some numerical problems. Therefore in the literature quite a few algorithms are described which try to successfully invert matrices even in the case of badly conditioned matrices of the absorption coefficients [39,41]. The calculation of the a -vector according to the procedure given above should result in values which are mainly influenced by numerical problems with respect to the inversion process. However, neither the measured absorbances in the course of kinetic analysis nor the absorption coefficients obtained by calibration procedures are faultless. On the contrary these errors have to be taken into consideration in eq. (4.4) by an additional vector e which takes into account noise in measurement and errors in calibration. It is a residual vector. In consequence, by use of the measured absorbances, in the following the concentration vector obtained is used to recalculate absorbances. The difference between calculated and measured absorbances correlates to the residual vector e=E-8a.
(4.11)
a illustrates that the calculated concentrations just result in approximate values. Optimal concentrations are obtained as soon as the variance is minimised. For that the residual vector is multiplied by its transpose: |e|^=(E-e-Sy(E-e.a).
(4.12)
This procedure is called multi-linear regression and uses least squares approximations. The calculation can be improved with respect to numerical problems by applying pivoting in the course of matrix inversion. Weighting of the wavelengths used dependending on the relative error takes into account that the standard deviation in the calibration matrix varies with wavelength. The variance becomes constant [39]. The result depends on the condition of the calibration matrix given, e.g. by Hadamard's condition value or by the specifity of the calibration matrix obtained by the determinant of the quadratic calibration matrix [65-67].
Ch. 4
Absorption Measurement
271
Another possibility to select optimal wavelengths is the variancecovariance matrix [68]. The fundamental problem of this approach is that the algorithm tries to minimise the residual vector. If an additional unknown concentration is hidden in the spectrum measured, interactions between the reactants exist which were not taken into account in the calibration. Therefore the algorithm has to result in an error. This means the solution I has to be erroneous. In this case the minimisation of the least squares sums, the variance, has to give an erroneous result in principle. This means that for some of the concentrations, a wrong amount is added to compensate for the additional compound or the interaction. To avoid these problems an inverse approach is sometimes used [69]. During the calibration procedure no pure reactants are used but various mixtures of them [39]. For this reason the calibration returns a matrix of absorbances for the different mixtures at many wavelengths. In this case the parameter vector does not represent the concentrations but its portion of the mixtures. According to the least squares approximation the variance is also minimised. However, in this case the variance represents the difference between the estimated and the real portion. Since mixtures are used for calibration, in principle - a priori non-linearities are not excluded, - unknown components are automatically included, and - the inversion of the covariance matrix (e' e) gives a relatively bad solution because of possible linear dependencies between the mixtures. In consequence the condition number can turn out to be high. In general two approaches are used in the literature to find optimal parameters [39,42,70]: - The principle component regression (PCR), which uses an eigenvectoreigenvalue decomposition or a single value decomposition or a singular value decomposition of the matrix E [38,71,72]. Both require a lot of calculation time. - In contrast, partial least squares (PLS) [73] reduces the calculation time by a different type of reduction of the E-matrix by use of the Lanczos bidiagonalisation. Both approaches adjust a different method of calculation to determine the estimated parameter value. In both cases a systematic error (bias) is superimposed to reduce the high variances, caused by the inversion of badly conditioned matrices. In Fig. 4.19 the different algorithms are compared schematically [38].
Experimental Techniques in Photokinetic Analysis
272
fMLR as BLUE
6VCR
Multivariate Linear Regression MLR
Pricipal Components Regression PCR
Ch.4 S^LS
Partial Least-Squares Regression PLS
D
n
Matrix inversion
Singular Value decomposition
Lanczosbidiagonalisation
a
0
D
Classic approach c = Xb
^
n Inverse approach c = Xb
a Multivariate data (e.g. spectroscopic data), Bouguer-Lambcrt-Bcer
Fig. 4.19. Schematic comparison of methods of multivariate data analysis using the classical and inverse approach. BLUE, best linear unbiased estimator; PRESS, predictive residual error sum of squares.
In principle both the classical and the inverse approach use a multivariate data set. But in the classical approach the variance is minimised, whereas in the inverse approach one tries to find an equilibrium between bias and variance. Therefore the bias is reduced and by the procedure of predictive receivable error sum of squares either via a singular value decomposition or the bidiagonalisation method estimated values, either according to principle component regression or partial least squares, are found. The multilinear regression on the other hand will find the best linear unbiased estimation as an approach to a "true" concentration. Besides applications in absorption spectroscopy, fluorescence spectra can also be evaluated [74]. Another approach to multicomponent analysis is the use of Kalman filters [75,76], applying neural networks [77,78], or even fuzzy logic [79]. The quality of these approaches depends on the number of reaction components, the condition of the calibration matrix, the errors in determination, the absorption coefficients, and the measurement of the absorbances during kinetics. In any case the use of any of these algorithms can cause problems depending on the chosen algorithm and the specific behaviour of the reaction
Ch. 4
Absorption Measurement
273
spectrum. Therefore in practice such muhicomponent analyses are restricted to well behaving matrices with absorption coefficients given by significant changes in absorbance at the wavelengths of measurement during the overall reaction. In general the more reactants are to be considered the greater the problems will be. Applications of various types of muhicomponent analysis in kinetics are published for principle component analysis combined with target transformation [80] and using neural networks [81]. It is obvious that during kinetics one or the other reactant will turn up in relatively small concentrations. This fact is an additional problem compared to muhicomponent analysis in stationary systems as in the normal analysis of sample mixtures in an analytical laboratory. The dynamic process - first hinders the experimentalist from taking as much time as necessary to increase the integration time for the measured signals and - second causes the relative change in concentrations, therefore one or the other selected wavelength will become better or worse for observation. For this reason one takes as many wavelengths as possible out of the reaction spectrum, uses this over-determined system of equations to calculate the best condition of the matrix, and tries for an optimisation of the covariance matrices. The use of pattern recognition principles [82] is intended to classify and discover the main features of the experiment. In this cases elements of factor analysis [38], [72] must also be used. It becomes obvious that the more complex a reaction is the more difficult evaluation will be. In any case one finds that the less the spectra of the various reactants differ, the fewer the number of steps of reactions and included components can be. The above-mentioned approaches take for granted that the spectra of all the reactants are known. This is not at all the case in kinetic analysis, although usually at least the intermediates are unknown. As is demonstrated in Chapter 5, even reactions can be evaluated under these circumstances. But usually the constants obtained from the over-determined system of differential equations are only proportional to the interesting kinetic constants or a combination of them. For this reason absorption spectroscopy is likely to be combined with other methods. An example of combination with fluorescence and with chromatographic principles is given in Chapter 5. 4.2 J Derivative spectroscopy In analytical chemistry derivative spectroscopy is frequently used if the
Experimental Techniques in Photokinetic Analysis
274
Ch.4
absorption bands show poor characteristics. A typical application is the examination of biological material in turbid solution. Differentiation of the signal visually enhances small but sharp peaks compared to broad bands with little curvature. In the literature the different principles of differentiation are discussed as optical, electronic, and mathematical methods [42,83,84]. In contrast to integration each differentiation increases the noise of the signal. For this reason opinions in the literature [85], that the amount of information increases with the order of differentiation, are infrequent. In general mathematical differentiation is superior to the differentiation of the analogue signal, because in the latter case the original data are lost and not available for later further manipulation. Furthermore differentiation happens with respect to time instead of wavelength in the case of used analogue signals. Time constants and any fluctuations of the electronic parts of the spectrometer influence the result. However, these aspects are discussed controversially in the literature [85,86].
I 0.04
-0.12 200
220
240
260
280
300
320
340
360
Wavelength [nm] Fig. 4.20. Derivation reaction spectrum of the photoreaction rra/w-stilbene to phenanthrene via cw-stilbene in n-hexane by irradiation at 313 nm.
Ch. 4
Fluorescence Measurements
275
Even derivative spectra do not produce an increase in information compared to zero order absorption spectra. Differentiation can visualise the wavelength regions in which spectral changes are exceptional. For this reason derivative spectroscopy has been applied to kinetic analysis [87]. Fig. 4.20 demonstrates an example of a derivative reaction spectrum given for the above mentioned photoreaction of stilbene. The second derivatives are shown. The advantage in selection of characteristic wavelengths improves the kinetic evaluation as discussed in Chapter 5. However, in each application the optimum between increased noise and extracted information has to be found.
4.3 FLUORESCENCE MEASUREMENTS 4.3.1 Correlation between fluorescence and structure As mentioned in Section 1.3.2 the chemical constitution of an organic molecule determines whether it fluoresces or not [88,89,90]. In the vapour phase and in solvents, intra-molecular quenching effects determine whether emission will take place in addition to radiationless transitions. Furthermore in solvents the interaction with non-excited molecules (mostly solvent molecules) can cause an intermolecular deactivation process, which also reduces the probability of fluorescence. In principle, high fluorescence yield can be expected, if - The absorption coefficient of the long wavelength absorption band is high. This correlates with a high oscillator strength (given by the integration of the absorption coefficients over wave number) and a high rate of fluorescence decay. - The transition at the lowest energy is at wavelengths as long as possible to reduce the probability of photo-dissociation. - The emission starts from a relatively weak bonded orbital. - The molecule lacks such functional groups which increase radiationless transition. These requirements can be summarised in the following rules: - Fluorescence overcomes other processes the better, the smaller the excitation energy is in comparison to the energy of bonding, the more delocalised the jr-orbital is, and if n ~> jr transitions are at shorter wavelength than JT —> jr transitions.
276
-
Experimental Techniques in Photokinetic analysis
Ch. 4
Furthermore low temperatures and solvents at high viscosity will increase the yield of fluorescence. - On the other hand, radiationless transitions will reduce fluorescence, if the molecule has substituents which are more weakly bonded than hydrogen; e.g. substituents like -I, -NOj, -NH3*. They cause predissociation, a process which also reduces fluorescence. One can distinguish a phenomenological and a physical theory. According to the first one gets a more qualitative impression of the correlation between fluorescence yield and chemical constitution. The term fluorophore is introduced in combination with auxochromic groups, whereby any possibility of exhibition of torsion or hindered rotation will reduce fluorescence drastically. In some cases, such as anthracene or perylene, substitution hinders fluorescence. But in erythrosin containing four iodine substituents one finds very good fluorescence. Any additional bonding between two rings via a second bridge, especially including heteroatoms, results in fluorescence. According to the physical theory, fluorescence is treated as a concurrent process to photochemistry, intersystem crossing to the triplet state, and radiationless transition. Therefore, any intramolecular processes will reduce the emission. Interaction and energy transfer either in the molecular skeleton itself or to other molecules can be considered reasons for reduced fluorescence. Some typical examples are given by molecules in which electron excitation energy is transformed into vibrational energy. The reason can be either energy levels of the atomic vibration at close distance or weak bonded substituents of large mass. Furthermore fluorescence yield is reduced, if vibrational equilibrium distances differ greatly for the ground and excited electronic states, since any electronic excitation is combined with highly excited vibrational states. In principle the potential hypersurfaces as well as coupling between the different states have to be considered. In general high energy of excitation in combination with high bond strength will increase the fluorescence. Furthermore good resonance can stabilise the system of bondings. This means normally any substitution reduces the bond strength. On the other hand, good selection of auxochromes can even increase the stability of the jtsystem. The following effects can cause a reduction of the fluorescence yield by intermolecular deactivation: - The solvent will drastically affect the fluorescence intensity of the solute. This can be either caused by its polarity which influences the ground
Ch. 4
-
-
Fluorescence Measurements
277
and the excited states differently, whereby a drastic wavelength shift happens. Solvents containing heavy atoms increase the intersystem crossing rates drastically. The pK values are different for the ground and the excited state, sometimes varying many orders of magnitude. Therefore any change in the pH will effect fluorescence yields. Another reduction of fluorescence can be caused by hydrogen bridging. Since the concurrent processes such as radiationless deactivation are dependent on temperature, a variation of the temperature will also influence the fluorescence. Paramagnetic O2, metal ions, or high concentration will reduce the fluorescence intensity either by electron energy charge transfer or by impact deactivation.
4.3.2 Fluorescence spectrometer In gases and liquids, molecules are randomly distributed. For this reason those organic molecules which exhibit fluorescence emit the absorbed electronic energy into the space with radiant distribution. Therefore an observation in the optical pathway of the beam of excitation adds both the fluorescence intensity and the amount of light penetrating the sample. In principle the wavelengths of excitation and fluorescence differ but there are many cases where the fluorescence spectrum overlaps with the excitation wavelength. For this reason the arrangement of fluorescence measurement is somewhat changed in comparison to absorption measurements. The sample is usually observed at a W angle with respect to the optical excitation light beam. The principle set-up is given in Fig. 4.21. Good instrumentation uses a monochromator in the emission pathway, since normally some amount of scattering is superimposed on the fluorescence. In contrast to absorption spectroscopy the use of a reference is rather difficult. It requires a sample for which absorption and fluorescence spectra both exhibit the same spectral distribution as the probe. This is quite impossible. For this reason in some instrumentation one tries to eliminate fluctuations in the intensity of the light source. The excitation beam is split and a small amount of the intensity of excitation is focused on a photodiode. Nevertheless neither the changes in the optics of the instrumentation nor the variance of the transmission curve with wavelength can be overcome. However there is a way to calibrate the spectral distribution of a fluorimeter in-
278
Experimental Techniques in Photokinetic analysis
?
fluorescent light
F or Ml
Selection of excitation wavelength \
Ch.4
^o» ••• ^ n ForM2
1
Selection of emission wavelength 7^
v±yPM Fig. 4.21. General set-up of a fluorimeter system containing filters or monochromators in both the excitation and emission pathway. Q, light source emitting polychromatic light in the wavelength range A], A^,..., A„; F, filters; or Ml, M2, monochromators; P, photodiode for control of excitation intensity fluctuations; A^, A^, emission and observation wavelength; fluorescence light in the wavelength range A^,..., AQ; P M , photomultiplier.
strumentation using standard fluorophores with known fluorescence quantum yield [91]. This task is very tiresome and expensive. Fortunately in reaction kinetics the absolute value of fluorescence intensity does not have to be known. Since fluorescence intensity is proportional to the concentration of the fluorophore in dilute solutions, changes in time dependence of fluorescence can be correlated with a change in concentration. If even one of the components fluoresces, photokinetic evaluation becomes very simple. Nevertheless the measurement of fluorescence intensity to obtain concentration changes in reaction kinetics has been considered rather inexact. In absorption measurements small spatial fluctuations of the light sources do not cause extreme problems, since the diameter of the measurement beam is small in comparison to the width of the cell and the overall dimension of the active surface of the photomultiplier. On the other hand, small fluctuations of the excitation light beam drastically influence the geometry in the case of fluorescence measurements. Therefore small changes in overall intensity of fluorescence caused by the fluctuation of the plasma light sources become large with respect to the different volume elements in the cell. These volume elements are monitored by the entrance slit of the observation monochro-
Ch.4
Fluorescence Measurements
279
mator. The small slit causes an extreme dependence on the radiant geometry. For this reason very dilute solutions are chosen to reduce an intensity distribution in the cell [92]. But good stabilisation of the light source, parallel measurement of the intensity of the excitation light source, and exact geometry of the optical pathway allow quantitative evaluation of fluorescence signals. In principle the wavelength of irradiation and excitation are the same. The excitation light source is used to cause the photoreaction. By these means monochromatic irradiation is used automatically. A typical cell compartment is given in Fig. 4.22. The sample is stirred and temperature controlled. Three positions are mounted tilted at 45'' relative to the optical axis of the excitation light beam. One position is for the cell with the solvent, one for the fluorescence standard, and the third for the sample itself. These three positions absorbance measurement thermostat
photomuiti plier monochromator
amplifier 1
photodiode
YT recorder ADC unit
TM 990 labdatastation
St 46 stable power supply
Fig. 4.22. Combined absorption,fluorescencemeasurement, and irradiation compartment, /Q is controlled by a split-beam and a photodiode. The 45° tilted bar contains three positions: (1) for afluorescencestandard (S), (2) for the sample (R), (3) for a reference blank (v). The sample can be stirred. The cell holder is temperature controlled. The irradiation source is either a ST 46 or ST 75 (Quarzlampengesellschaft Hanau).
Experimental Techniques in Photokinetic analysis
280
Ch.4
St 41
M i
1
I E i
1
VI
1
S 1
I)MR 2 1
Fig. 4.23. Combined absorption andfluorescencespectrometer device in a schematic representation. An automatic dual beam DMR 21-spectrometer and a single beam PMQIIphotometer are used. The cell holder ZFM4 combines both instruments.
allow the control of the relative intensity of the light source for different photoreactions by use of the fluorescence standard. Therefore comparable results for different reactions are possible [93]. This specific 45"* arrangement also allows transmission measurements. Another arrangement for combined measurement of absorption and fluorescence information is given in Fig. 4.23, using a DMR 21-spectrometer, a PMQII-photometer and a fluorescence cell holder (ZFM4) [94]. The principle fluorescence arrangement given in Fig. 4.21 allows the measurement of fluorescence and excitation spectra. In the case of fluorescence measurements, the wavelength of excitation is kept constant. Only the emission wavelength is varied to take a fluorescence spectrum. In the case of an excitation spectrum, the wavelength of observation is set to the maximum of the fluorescence spectrum and kept constant. During the measurement of the excitation spectrum, the wavelength of excitation is varied. This proce-
Ch. 4
Reflectance Measurements
281
dure allows spectra to be obtained which are caused by the variation of the absorption coefficient with wavelength. Accordingly the measured spectra show a resemblance to absorption spectra. This method can be used in multicomponent solutions in which just one of the unknowns fluoresces. In this case the principle appearance of its absorption spectra can be obtained.
4.4 REFLECTANCE MEASUREMENTS When the samples are transparent, reactions at the interfaces have to be monitored by reflection spectroscopy. In Fig. 4.11 a device is shown which, using fibre optics, allows the direct measurement of reflectance at the interface with a commercially available photodiode array. Some conventional recording spectrometers are also equipped with fibre optics. Such devices are used to measure colours at the surfaces of metals or clothes directly [95]. In principle in Section 4.2.5 it was discussed that reflectance at an interface usually disturbs the absorption measurement. In the present applications, however, this reflectance allows reactions at the surfaces to be monitored. Since nowadays solid samples are exhibited to photo-induced processes, such measurements are a convenient application. In principle two types of reflectance take place [42,96]: (1) The regular or specular reflectance at a mirror surface for which the geometric laws are valid which results in the same angle of exit as for the incidence. Examples of this type of reflectance are metals, glasses or crystals. (2) Diffuse reflectance at interfaces which are structured and are microscopically inhomogeneous such that a large number of micro-reflecting surfaces distribute the reflected light statistically into space. There are quite a few cases where these two types of reflectance appear in combination. 4,4,1 Principles of reflectometry 4,4.1.1 Diffuse reflectance The quantitative description of diffuse reflectance measurements is known from thin-layer chromatography. Taking a powdery sample with such a large thickness (rf = ©o) that no reflection at the support material can be observed, the reflectivity R^ of the sample is connected to the stray coefficients as well as with the absorption coefficient k . Kubelka and Munk [97]
Experimental Techniques in Photolcinetic analysis
282
Ch.4
have developed a correlation between these absorption and stray coefficients and the dependence of the reflectivity R^ according to
s
(4.13)
2R^
The theory requires the following prerequisites [96]: The Lambert cosine-law has to be valid (isotropic distribution of stray light), that means the regular reflectance can be neglected. - The scattering particles of the layer are distributed statistically. Their dimensions are smaller than the thickness of the layer examined. - This layer is irradiated by diffuse monochromatic light or by parallel light at 60** incidence. If the layers are too thin, the stray coefficients become so small that no isotropic distribution of stray light is formed. The equation given above allows a quantitative determination of the concentrations of the particles in such a layer. It can be used to follow a photoreaction in powdery medium. This type of layer is schematically represented in Fig. 4.24. The light path for the incident and diffuse reflected radiation are included in the figure. Problems with quantification are well known from the application of thin-
k
R
•I t'- '1 r i T Fig. 4.24. Schematic representation of the propagation of incident and reflected radiation in a diffuse scattering layer. I, light path in the incident direction; J, light path in the opposite direction; X, local element for calculation; I^, incident intensity; R, intensity of reflected radiation; R,, reflected radiation at the support layer; T, radiation penetrating the layer.
Ch. 4
Reflectance Measurements
283
layer chromatography. A principle problem of the direct observation of photoreactions on a silica or alumina thin-layer as used in thin-layer chromatography is that both types of substrate activate organic molecules, especially during irradiation. Thus, such a type of kinetic analysis will vary the kinetic mechanism. However, this fact has to be taken into account in any thin-layer chromatographic analysis of solutions done to separate the reaction solution. 4A. 1.2 Regular reflectance Diffuse reflectance spectrometers offer a tool to examine on-line photoprocesses on diffuse reflecting substances or to control the photostability of coloured materials as pigment or car lacquer. Either the set-up using fibre optics (see Fig. 4.11) or modified thin-layer equipment (Fig. 4.25) can be used.
photomultipiier helping mirror slit filter
uv-achromate
X y carrier with sample 3
filter
mirror quartz lens Fig. 4.25. Schematic pathway of radiation in a typical thin-layer chromatography scanner. This set-up allows the measurement in transmission, reflection and fluorescence. The optical arrangement and the light source are varied according to the problem.
284
Experimental Techniques in Photokinetic analysis
Ch. 4
At mirror surfaces, incident radiation can be reflected regularly. This reflectivity (the ratio of the reflected to the incident radiation power) depends on the refractive indices of the two layers forming the interface [96,98]. According to the Fresnel relationships, the reflectivity is given by R - ^ i ^
(4.14)
The equation above gives only the simplest case where the irradiation is incident perpendicularly and the medium is not absorbing. However, in photokinetics normally conditions where the sample shows absorption are of interest. For this reason another equation has to be used, given by /? = U
iZ-
L_
(4.15)
where K is the index of absorption [43,98]. This type of measurement requires a lot of expenditure to evaluate the quantitative changes in concentration. In principle, a transmission and a reflection measurement have to be combined [99]. However, in some cases the reflection spectra can be used to obtain the concentrations by measurement of regular reflection, if the absorption coefficients of the components are known. This type of evaluation has been applied to reaction dynamics and thin films [100]. It is rather difficult. Nevertheless, it offers the only chance to examine photoprocesses in photoresists on circuit boards, which are not transparent [101]. In contrast to this rather difficult determination of changes in the concentrations of reactant, in a few applications reflection spectroscopy is simply used for the measurement of absorbance by an indirect method using a mirror behind the cell. The light beam of measurement passes twice through the cell. It is either coUimated by lenses, or fibre optics are used. A typical application of such a type is reflectometry in chemical sensors [102]. Another prospective application of reflectance measurements in the case of strongly absorbing solutions takes advantage of the principle of attenuated total reflection (ATR) [98].
Ch.4
Reflectance Measurements
285
4,4.2 Set-ups in Reflectance Measurements In Fig. 4.25 a typical arrangement of the measurement of diffuse reflectance is given. Such a set-up is commercially available and any thin-layer spectrometer can be used which normally scans the thin-layer plates. The coUimated optics of the set-up demonstrated can be substituted by fibre optics. This type of equipment is shown in Fig. 4.11 using a bifurcated fibre and a diode array. Under these conditions white light has to be used to produce a reflection spectrum monitored by the diode array. However, white light causes undefined photochemical conditions. For this reason a third fibre arm is used, which guides monochromatic radiation to the sample. If the white radiation is blocked by a shutter, it may not disturb the monochromatic photoirradiation. Since modem diode arrays take spectra within a few milliseconds the time of measurement stays short compared to the total photochemical process time. This type of equipment can be used to monitor the photoprocess during irradiation of photoresists [101]. The spectroscopic observation of rather concentrated solutions is of broad interest in photometry. Because of the necessary transmission of solutions, the concentrations in normal direct spectroscopy measurements in photokinetics are restricted to approximately 10"^ molar solutions. In many experiments - especially in the case of preparative photochemistry (where often
Fig. 4.26. ATR-rod (attenuated total reflection) monitoring changes in the sample via multiple total reflection and evanescent fields in the reaction solution. Fibers are hooked to a highly optically polished glass, in which the radiation is guided in contact with the solution in the surroundings.
286
Experimental Techniques in Photokinetic analysis
Ch. 4
the optical turnover is intended to be controlled) - this concentration is too low. Either cells with extremely small optical thicknesses have to be used, which cause problems with leakage and flow-through effects, or the principle of attenuated total reflectance measurements has to be applied. If irradiation passes the interface from an optically denser medium to one of smaller refractive index, at angles larger than a certain limiting one (with respect to the perpendicular optical axis) the radiation will no longer leave the medium. It is totally reflected at the interface. Electrodynamics explain the fact that the reflected radiation couples to an evanescent field, which exponentially decreases into the medium of lower optical density. Thus the reflected beam gathers information about the optical properties of this medium across the interface. Such a device is demonstrated in Fig. 4.26 for a highly absorbent solution. In a fibre the radiation is guided by multiple total reflection. This analytical "finger" can measure changes in the solution [98,103]. 4.5 CHROMATOGRAPHIC METHODS 4.5.1 General considerations and comparison between different chromatographic methods In complex photoreactions many products can be produced simultaneously during the photoprocess. For this reason separation techniques are desirable in principle. However, these techniques must allow an automatic observation during the reaction. The first applications of gas chromatography [104] demonstrated some problems. Liquid chromatography offers better results, since photoreactions are usually examined in liquid phases. Two principles are presented in the literature in detail. Both high performance liquid chromatography (HPLC) and thin-layer chromatography (TLC) are frequently used in analytical applications [105,106]. However, a few essential points with respect to their applicability to photokinetics have to be discussed: - first, a possible automation has to be compared; - second, the correlation between the signal (peak area or peak high and concentration has to be examined; - furthermore, any separation technique will interact with the reaction solution. Therefore, neither TLC nor HPLC techniques can be used in such a way
Ch.4
Chromatographic Methods
287
that after detection the volume element containing the reactants is fed back into a photoprocess cycle. Recently, spectra identification by use of spectra libraries [107] and the use of multicomponent analysis using chromatographic data have increased applicability of chromatography in combination with UVA'^is spectrometry. 4,5.2 Thin-layer chromatography Commercially available thin-layer chromatography equipment is shown in Fig. 4.25. It allows the determination of absorbent and/or fluorescent products after a small part of the photosolution has been extracted and applied onto a thin-layer plate which is developed by a suitable eluent. This is an interactive process comparable with early techniques in kinetics where small portions of the reaction solution were frozen and conventionally analysed afterwards off-line. Since the determination of peak areas, peak detection, and baseline correction are well established methods, the determination of concentration is a smaller problem and results in values with small standard deviations. Nevertheless, TLC is not suitable for automation in its application to photokinetics since the photoactivation of the sample by the chromatographic material is not negligible [108]. The light paths for reflection and fluorescence measurements are given in principle in Fig. 4.27. The result of an examination using a TLC device with spectral detection using a photodiode array is given in Fig. 4.27. The advantage is the combined observation of absorption and fluorescence spectra using a xenon light fluorescence F
reflection
filter
Fig. 4.27. Lightpath for observing a thin-layer plate in reflection orfluorescence.In the case offluorescencemeasurements, the directly reflected intensity is cut off by a filter.
Experimental Techniques in Photokinetic analysis
288
Ch.4
Os
corr
I.O
•
absorption
^••^*mm,
-
luK.
0.8
SOOs fluorescence
500 sjM
0.6 0.4 0.2
\ I y os 1
,
,
,
1
1
1
1
1 —
700 wavelength [nm] Fig. 4.28. DEMC placed as a spot on a silica TLC plate irradiated by a xenon source to obtain spectra. The photodiode array allows the combined measurement of absorbance (reduced reflectance) and fluorescence (increased reflectance). Icon is the reflectivity corrected with respect to a spot without sample. Many spectra are taken during 500 s of irradiation. 350
400
500
600
source for the measurement. The diode array supplies both types of spectra taken from a spot containing a laser dye. The intense polychromatic measurement light causes a photoreaction catalysed by a silica sorbent material. Laser dyes are assumed to be relatively photostable. However, as Fig. 4.28 shows, the spectra change within a few seconds. Under these conditions 7,7'-diethylamino-4-methylcoumarine (DEMC) undergoes photodegradation. The different photodegradation products demonstrate a change in absorption and fluorescence maxima. In addition, their fluorescence quantum yields differ. This specific technique allows the determination of the different reactants and metabolites for different reaction conditions. However, in many cases the rates of photoreaction processes are not comparable to those in solution due to the photocatalytic behaviour of the sorbent material. Detectivity and limit of determination can be optimised by pixel bunching of the diode array. It measures concentrations as low as in the case of monochromatic detection using commercial equipment [108].
Chromatographic Methods
Ch.4
289
4,5.3 High performance liquid chromatography HPLC causes the same problem with a time delay between analysis and taking a specimen out of the photoreaction solution. However, in HPLC automation is easier to realise [109]. In Fig. 4.29 a combined irradiation and HPLC apparatus is given as a block diagram. It is controlled by a microprocessor and contains a cycle for the photoreaction including a mercury arc, a photoshutter with an interference filter, and a flow pump at normal pressure. The reaction solution circulates through a flow cell with 1 cm optical thickness, in which it is irradiated. The irradiation time is controlled by the microprocessor via the shutter. The volume of the circulating solution amounts to approximately 8 ml of solution, of which 3 ml are irradiated at a time in the cell. At chosen reaction times lOjulof the reaction solution are extracted using a manifold valve (see Fig. 4.29). This volume is injected into a high pressure
lamp
photoshutter
0
gradient ^ former
reactor
Uh
solvent
C
column
—
UV/Vis detector
inject injection valve
iwaste H
LXu X-t- recorder
selenoid valve
air-
I/O interface
TM 990 CPU
A/D converter
dialog terminal floppy controller plotter
floppy-disk
graphic interface
Mjigraphic monitor
Fig. 4.29. Block diagram of a combined HPLC apparatus and an irradiation set-up, which is controlled by a microprocessor.
290
Experimental Techniques in Photolcinetic analysis
Ch.4
A
50 inj ection Fig. 4.30. Dilution by injection of 20/41 of reaction solution and re-injection of the same volume of eluent into the photocycle. The straight line is reproducible.
flow cycle of a HPLC apparatus. In addition it can make use of a gradient former. Under these conditions, the lOjul volume taken out of the photocycle cannot be substituted, since the eluent would vary the solvent properties, most probably influencing the photoreaction. However, when the HPLC analysis uses just a defined eluent which is the same as the solvent of the photoreaction, the re-injection into the photocycle just causes a dilution, but keeps the volume irradiated constant. This dilution can be accounted for. As Fig. 4.30 demonstrates, a graph of the integrated peak area of a compound measured under such conditions (mercury arc stays switched off) linearly depends on the number of injections [110]. The relationship turns out to be linear. Since the calibration is reproducible, the dilution can take account of each calculation of concentration (see Section 5.2.1). Therefore the concentration of all the components is possible. The concentration-time curves ^ e directly obtained. The evaluation becomes simpler, in principle. However, the noise is larger than in pure photometric determinations using UVA^is absorption spectroscopy. Some experiments prove that the reproducibility of such a processcontrolled device fulfils the requirements for a photokinetic evaluation.
Ch. 4
Special Methods
291
4.6 SPECIAL METHODS
Many further analytical methods exist for monitoring a photochemical reaction. Some even offer some selectivity with respect to the reactants in the solution. However, in most cases on-line combinations of instrumental analysis and photochemical examination cause problems. Thus, infrared spectroscopy would be a preferable method with respect to the selective determination of reactants, especially after development of Fourier instrumentation, which allows quantification with similar standard deviations as UVA^is-photometry. However, solvents used in IR spectroscopy absorb at longer wavelengths in the UV, absorbing the irradiation of the photolamp and causing unwanted concurrent reactions (e.g., CI' from CHCI3 or CCI4). On the other hand, solvents suitable for UV and the photoreaction will absorb interesting regions of the IR spectrum. Furthermore, IR measurements in the liquid phase cause some problems with the cells depending on the solvent. In the examination of optically active compounds, optical rotational dispersion (ORD) and circular dichroism (CD) have been applied with success [111]. Analysis took place in a conventional way extracting samples out of the reaction solution with off-line analysis in CD equipment. NMR equipment has also been used in kinetic examinations. The recent development of pulse Fourier spectroscopy allows the application of NMR even to solids and in flow systems. Thus, *^C-NMR spectroscopy can offer the advantage of monitoring chemical pathways. Even quantitative results can be achieved to determine rates of reaction [112]. Another possibility of the application of NMR spectroscopy is via chemically induced dynamic nuclei polarisation (CIDNP), if a radical pair is formed during the photoreaction. The intermediate formation of a radical pair or a biradical in a magnetic field causes intermediate emission and enhanced absorption signals, respectively, which show up in the nuclear magnetic spectra of diamagnetic reactants or products. A nuclear spin selection takes place for the electronic transition. A direct relationship is found between the type of NMR signal, and the sign of the hyperfine coupling constant [113]. In *^C-NMR apparatus, a UV fibre optic is focused on the sample tube inside the magnetic field [114]. By this means information on the structure of photochemical primary products and their dynamics can be obtained.
292
Experimental Techniques in Photolcinetic analysis
Ch. 4
4.7 PROCESS CONTROL IN KINETICS
Quite a few different set-ups have been demonstrated, in which an irradiation apparatus was combined with different methods of spectroscopy. In all these combinations, process control either by microprocessors or by personal computers, was achieved. Modem instruments contain a microprocessor for process control of the spectrometer and data transfer to a dedicated personal computer. Quite a few years ago data stations were integrated in the spectrometers. However, their flexibility is poor and their software is restricted to the instrumentation making adaptations to user specified problems very difficult. In general nowadays most instruments contain a certain "intelligence" without a display. Process control and data acquisition is done by a personal computer interfaced to the instrument via an analog to digital converter board (ADU), which contains some input-output (lO) ports with TTL levels for control handshake. This board is plugged into the computer bus. The control menu is usually handled under Windows to make a convenient user interface available. Very few of the software packages allow subroutines to be added by the users to handle flow pumps, photoshutters, valves or other switches. For this reason, special applications need self-written programs in the personal computer software (if possible) or an additional microprocessor (programmed in assembler) for the special task. In Fig. 4.31 an example is given controlling a dye laser by one microprocessor and a UVA^is-spectrometer by another. These two microprocessors are interconnected via data and control lines. Their programs are synchronised. The first microprocessor controls the spectrometer, the second an optical multichannel analyser, which rapidly takes emission spectra during the time the laser pumps the dye solution. Both have to be synchronised with the spectrometer which takes the absorbance spectra. The laser has to be triggered by the optical multichannel analyser. Thus, the circulating dye solution flashed by the laser can be examined by a vidicon (during the flash) with respect to its laser activity. Using a flow cell with 0.1 mm optical path length because of high absorbance of the dye laser solution, in the spectrometer the photodegradation can be monitored at the same time. Fig. 4.7 gives a view of the cell holder. In the case of diode arrays used in applications given in Figs. 4.4, 4.11, 4.12, and 4.28, the overall process control and data handling was programmed in C running on personal computers. In this case the control of
Ch.4
Determination of Intensity of Irradiation
293
I r S 3 . SL—polychromator h T A I m-rrr-—uJlli control
Fig. 4.31. Block diagram of a dye laser using an irradiation source and UVA^is spectrometer for time-resolved examination of photo-degradations of the dye solution at laser conditions. Two synchronised microprocessors are used for control. The assembler program amounts to 32 KB of code.
additional parameters (pump, shutters) was handled by macros, which are user definable [115]. 4.8 DETERMINATION OF INTENSITY OF IRRADIATION In Chapters 2 and 3 partial photochemical quantum yields have been introduced as parameters which give information on photochemical reactions. According to eqs. (2.12) and (2.14) they depend on the change in concentration of the reactant which undergoes the photoreaction and on the light absorbed by this reactant. For this reason an essential of any photochemical examination is the determination of this amount of light absorbed. Two principles are known: - either time-resolving methods can be used controlling the intensity of the irradiation source by photodiodes, or
294
Experimental Techniques in Photokinetic analysis
Ch. 4
-
time integration methods have to be applied which determine the intensity falling onto the surface of the photoreaction cell. These two principles can be correlated to either physical or chemical methods. Both offer advantages and disadvantages which are discussed below [116].
4.8,1 Physical methods Totally absorbing material is heated by incident irradiation (bolometer). A relationship between intensity and temperature effects exists. Since temperature measurement are very sensitive, this effect is used to determine intensities. Semiconductor detectors either use the internal or external photoeffect [117,118]. In a photodiode, an incident photon causes a photocurrent by charge separation. It can be amplified and depends linearly on the number of incident photons. In photocathodes or photomultipliers the incident photons force electrons to leave the material. This external photoeffect can be calibrated and amplified by acceleration and multiplication of the electrons in multi-electrode arrangements (dynodes). These devices have very short response times and can be successfully used to control the stability of a light source. Therefore such devices are frequently included in commercially available set-ups. An example is given in Fig. 4.32 combining an irradiation source with a measurement set-up. This is commercially available [119] and allows simple control of a photoreaction. However, due to geometry and inhomogeneity of the sensitive layer of the photodiode, non-homogeneous irradiation can cause errors. These thermoelements, photoelements, and other physical measurement devices can cause some problems for the photochemist: - Sensitivity of the physical system depends on wavelength. - Sensitivity drastically decreases in the ultraviolet (UV). - Correction factors as well as the compensation plots in operation manuals only supply limited precision. - Calibration of such devices has to be repeated frequently since the material of the detector head exhibits photodegradation. - This photodegradation is extreme, if the device is exposed to highly intense sources which can damage the detector. This problem has to be taken into account when measuring laser sources.
Ch. 4
Determination of Intensity of Irradiation .14
295
13
Fig. 4.32. Irradiation set-up in which the photoreaction is controlled by spectroscopy. The intensity of the radiation source is monitored by a physical device measuring photocurrent (supplied by AMKO. Tomesch) [119].
-
The linear dynamic range is limited; it can be enlarged by using a preamplifier. - Any absolute determination of the radiant flux density or radiant flux needs frequent and troublesome re-calibration due to changing sensitivity of the detector surface. - Detector areas show spatial distribution of sensitivity. This amounts to 40% and more. This fact will cause problems especially if the light is focused only onto part of the target. - Multiple reflections in the solution or in thin layers influence the effective intensity in the sample. Physical methods cannot take account of this. These facts makes it obvious that physical methods are perfect to control the intensity of an irradiation source as a function of time. That means, they
296
Experimental Techniques in Photokinetic analysis
Ch. 4
can be conveniently used for a relative determination of radiant flux permanently. 4,8.2 Chemical actinometers Since the absolute intensity of an irradiation source is difficult to determine by physical methods, photochemists prefer the use of a simple photochemical reaction for the measurement of intensity. Any photoreaction for which the mechanism is known to be simple can be used as a chemical actinometer. It requires that the photochemical quantum yield is determined and independent of intensity or other effects. For many years chemists have looked for such types of reactions and have proposed a large variety of actinometric systems. The first were based on a colorimetric principle. A product is formed by the photoreaction which results in a coloured complex by addition of a chelating agent. The absorbance of this complex is linearly dependent on the intensity of irradiation [7,120]. Another possibility is the formation of a gas which can be measured volumetrically. A further possibility is the use of photochromic systems in which the thermal backward reaction is suppressed or very slow. These reactions can be monitored by applications of UVA^is-spectroscopy quantifying the change in absorbance at the chromophore's maximum. The advantages of such chemical actinometers are the following: - Absolute intensity of the irradiation source can be determined, since the number of photons incident onto the sample are measured. - Most of the chemicals necessary are very cheap, thus offering to replace the "detector" in case of damage. - There exist a lot of substances, which are utilisable many times. They exhibit photochromic behaviour and photoisomerise. - Chemical actinometry is especially suitable for photochemists, since the actinometric solution can be substituted by the sample of interest without changing the geometry and experimental conditions. This is of special value in the case of liquid solutions. By these means the problems with multiple reflection in solution or at thin layers as well as with inhomogeneous distribution of the irradiation or special geometries of the photoreactor can be overcome. Two typical reactions which are used in chemical actinometry, are the photoreactions of iron oxalate (Parker's solution) [7,121] and of potassium Reinecke's salt [122]. In both cases, photochemistry and analysis are sepa-
Ch. 4
Determination of Intensity of Irradiation
297
rated. In the literature some errors are reported which can be made [123] if optimal conditions are not achieved. Whereas Reinecke's salt is hardly used any more, Parker's solution has become the classical actinometer for photochemists. Iron(III) ions which are produced by irradiation have to be chelated by 1,10-phenanthroline in an acetate buffer after the photoreaction. The coloured complex must be photometrically determined at a wavelength of 510 nm after a complete conversion has taken place. The time delay has to be at least 1 h, overnight is preferable. The actinometric solution is restricted to wavelength areas below 450 nm. Some skill of the experimenter is required otherwise the errors amount to more than 10% [123,124]. Another example is azobenzene which is even commercially available [125]; its photoreaction has been examined in detail [126,127]. The same is valid for other systems such as heterocoerdianthrone [128] and mesodiphenylhelianthrene [129]. These are also commercially available [125] as are the aberchrome 540 and similar compounds [130] applying a simple evaluation using totally absorbent solutions. The methods demonstrated in this chapter are used in the applications discussed in the next chapter to obtain the necessary data for the following photokinetic evaluations.
Chapter 5
Applications of Kinetic Analysis to Photoreactions
Different approaches are given in Chapters 2 and 3 for the calculation of the relationship between the concentrations of the reactants and the reaction time, assuming mechanism and kinetic constants are known. Such rate equations can be formed for any complex mechanism, at least in principle, since there is a well defined route from mechanism to concentration and time laws. The aim of any kinetic analysis is to determine the mechanism using concentration-time measurements. Therefore a variety of different methods have been described in Chapter 4 to obtain these data. A kinetic analysis has turned out to be successful if the coefficient scheme according to Example 2.1 and explained in Section 2.1.1.1 together with the numerical values of the rate constants k,^ are determined for the reaction under examination. In general many mechanisms can result in the same functional relationship between concentration and time. For this reason the mechanism determined and in consequence the rate constants obtained are not all unambiguous. Furthermore accuracy and reproducibility of the measurements are limited. They vary with the method used for the determination of the concentrations with time. Thus the limitation in the measured signals and the variety of rate laws increase the many mechanisms fitting to the signals measured. For this reason the only positive result of any kinetic analysis can be in principle: The values measured do not contradict an assumed mechanism
5.1 EVALUATION OF MEASUREMENTS OF CONCENTRATIONS 5. LI Determination of the number of linear independent steps of reactions Any kinetic analysis starts with a hypothesis. One assumes a certain mechanism, determines the concentration and time laws, and checks whether
300
Applications of Kinetic Analysis to Photoreactions
Ch. 5
the measured values fulfil the calculated functional relationship. If the result is positive, the kinetic constants can be determined. In the case of a negative result, the hypothesis has to be discarded. First, the number of linear independent reactions has to be known before the examination of proposed rate laws can be achieved. The result depends on the existing information and is limited by the quality of the measurement. In the present section, it is assumed that the concentrations of all the reactants as functions of time are known. 5.7.7.7 Matrix rank analysis According to Sections 2.1.1, 2.1.4, and 2.5.6 the relationship between the concentrations and the linear independent degrees of advancement can be given by (see Section 2.2.1.1) Aa = vx.
(5.1)
Thus the vector of the differences of concentrations is a product of a matrix for the general stoichiometric coefficients and the vector of the linear independent degrees of advancements. This relationship is valid for each time ti. For various times given as f/ (/ = 0,l,2,...,m) these measured differences in concentration can be arranged in a matrix
'Aa„ Aa =
Aa2,
Aa,2 ...
n
and
m>s
Under these conditions the rank of the matrix in eq. (5.2) equals s according to definition. That means s equals the number of linear independent concentrations and the number of linear independent partial steps of reaction according to eq. (5.1), respectively. The numerical rank analysis is explained by an example [15]. Example 5.1: Example of a matrix rank analysis The following mechanism is assumed:
Only two of the four reactions are linear independent according to Section 2.1.4. The reaction can even be described by a single degree of advancement, if for example B is a short lived intermediate, which can be treated by use of the Bodenstein hypothesis. The experimentally determined concentration-difference matrix contains the following elements (each column correlates to a different time of reaction): 1 2 3 A 7 -0.21 -0.43 -0.86 0 Afe 0 0 0.14 0.28 Ac 0.06
4 5 -1.21 -1.61 0 0 0.41 0.53
Ad
0.80
0.15
0.29
0.56
1.07
Under these conditions, the number of measured signals is too small for a kinetic analysis. However, it is sufficient for the demonstration of
302
Applications of Kinetic Analysis to Photoreactions
Ch. 5
the principle. One realises that the reactant B occurs in analytically negligible quantities. This means the Bodenstein hypothesis has to be used, if the assumed mechanism really takes place. Under these conditions the rank of the matrix has to be 1. It will be determined as follows: (1) Rows and columns are rearranged that the absolutely largest element, the so-called Pivot element (-1.61), is placed in the left upper comer (Ab is omitted): 5 1 2 3 A 7 -1.61 -0.21 -0.43 -0.86 0.14 0.28 Ac 0.53 0.06 0.15 0.56 Ad 1.07 0.29
4 -1.21 0.41 0.80
(2) The new first row obtained is divided by this Pivot element and new values of the elements in the first row are given as 1 0.130 0.267
0.534
0.752
(3) The /th element (/ = 5,1,...,4) of the first row thus obtained is multiplied by the first element of the second row (0.53). Next this product is subtracted from the Zth element of the second row:
0.53-1 0.53-* 0.06-0.130 0.53
1 0.130 0.267 0.543 0 -0.009 0.002 0.003
-^
0.752 0.012
T
(4) The third and any other row are treated accordingly. 1
0.130
0 0
-0.009 0.010
0.267
0.534
0.752
0.002 0.003 0.012 0.004 0.012 0.004
(5) The sub-matrix which is printed in italics (a) containing the 2nd and 3rd row from column 2 to 5 is treated next according the preceding paragraphs 1 to 4.
(a)
Ch. 5
Evaluation of Measurements of Concentrations
303
One obtains: 1 0 0
0.130 0.267 -0.782 1 1
0.010
0.534 0.752 0.133 0.265
(b)
0.004 0.072
and by another repetition of the procedure: 1 0 0
0.130 0.267 -0.782 1 1 0
0.534 0.752 0.133 0.265 0.900 0.366
(c)
This treatment has to be redone until either the values 1 are found on the diagonal / = 1 (see (c)) or all elements starting at a certain row become 0. Then the rank of the matrix equals the number of rows which cannot be neglected (all elements: zeros). In the example given, the rank of the used measured values is 3. Nevertheless the mechanism cannot be discarded. This fact becomes obvious if
Fig. 5.1. Concentration-difference diagram for the measured values used in the Example 5.1: {\)Ad, (2) Ac plotted versus Aa.
304
Applications of Kinetic Analysis to PhotoreacUons
Ch. 5
one plots the concentration differences Aa, versus each other in a diagram. Fig. 5.1 makes it obvious that the differences in concentrations depend linearly on each other within a reproducibility of the measured signals of ±0.01. For this reason the reaction is uniform, and the rank of the matrix of the signals becomes 1. The errors in measurement are the reason that in (a) the elements of the second and third row do not become 0. Therefore, after each 4th step of calculation one has to check whether the remaining elements of the succeeding rows can be taken to 0 in the limit of error of measurement or not. Computer programs have been developed for the rank analysis and the error estimation. 5.7.7.2 Graphical methods The formalism of a numerical rank analysis can be substituted by graphical methods. Even though this graphical approach is formally equivalent to the numerical rank analysis, diagrams frequently are more descriptive and result in more information. Concentration diagrams [10] For a uniform reaction {s = 1) the concentrations of each reactant A^ form a linear function of the concentration of each reactant A, according to eq. (1.8): (5.3) This relationship can be graphically checked. One plots the concentration of one component versus the concentrations of all the other components in pairs in a rectangular coordinate system. If linear concentration diagrams (K-diagrams) result, the reaction is uniform. Additionally one obtains informations on the stoichiometric coefficients, the equilibrium constants, and constants of competitive reactions (if the reaction contains competitive steps). Example 5-2: K-diagrams for a single reaction step For the reaction 2A + B
>C
Evaluation of Measurements of Concentrations
Ch.5
305
Fig. 5.2. K-diagrams for the reactions 2A + B -> C. The initial concentrations are a^ = 2, ^^ = 1.5, c„ = 0 (in relative units).
the relationships
fo = &Q - c are valid according to eq. (5.3). Figs. 5.2a,b demonstrate the resulting K-diagrams. Since B exists at the start of the reaction in excess, at the end of the reaction the vessel also contains B. Example 5-3: K-diagramfor consecutive reaction For the reaction A^F^B-^C,
C-^D
one finds
if the equilibrium approximation is applied. Or
Applications of Kinetic Analysis to Photoreactions
306
Ch.5
a^Kb. Since the relationship exists,
In consequence the concentrations of C and D depend according to
and one finds
The use of the K-diagrams (see Fig. 5.3) for the dependence of a{b) allows the determination of K (a is the ordinate, b is the abscissa). Furthermore, the competitive constant ;f can be found for the K-graph d versus c. The latter relationship can be used for control. In this figure the concentrations are added in relative units at the axes.
Fig. 5.3. K-diagram for the reaction A ^ B ~ > C , B->Cat equilibrium approximation: a^ 0.6,^„=10,/i:=:0.6,;t = 0.2.
Ch. 5
Evaluation of Measurements of Concentrations
307
Even if 5 > 1, linear K-diagrams may result for the concentrations of some components. This can happen if the columns of these components depend on each other in the scheme in Section 2.1.1.1. Thus for the reaction A-^B+C,
B->D
with a + c = ^0 b+d =c a linear diagram a{c) will be obtained, whereas the diagrams b{c) and b{d) are curved. Non-linear K-diagrams are discussed in Section 2.3.3 in detail. The relationships given in this section for reactions of first order are valid without limitation also for quasi-linear photoreactions. Additional information in the case of photoreactions can result in an examination of the dependence of K-diagrams on intensity and wavelength of the irradiation source. The dependence on intensity K-diagrams do not depend on the intensity of the irradiation source, if the quantum yields of both the partial reactions depend in the same way on the amount of light absorbed. However, if there is a difference in the dependences of the quantum yields of the two partial reactions on the light intensity, or if linear independent photochemical and thermal reactions compete with each other, then the K-diagrams will depend on /Q. The dependence on wavelength Since /Q varies with the wavelength of the irradiation source, the Kdiagrams depend on the wavelength in those cases in which they depend on the intensity. Furthermore the K-diagram changes if more than one component absorbs the light and starts the photoreaction by this means. Example 5.4: Simple consecutive reaction For the reaction A—^!^B—5^C
308
Applications of Kinetic Analysis to Photoreactions
Ch. 5
which follows the mechanism treated in Section 3.2.3 in Example 3.25, the same initial slopes are obtained as for the analogous thermal reactions:
da 0
= 0, d,a
^
=0. to
The partial quantum yields do not depend on irradiation intensity according to eq. (3.29). For this reason the same situation is valid for Kdiagrams. The form of the K-diagrams is determined by eq. (3.40). The related constantx contains the ratio K'Q I ic\, which varies with wavelength. For this reason the K-diagrams have to depend on the wavelength of the irradiation source. Concentration difference quotient diagrams K-diagrams lose linearity, if 5 > 1. If j = 2 (just 2 linear independent steps of the reaction form the mechanism), the concentration of each component A, is a linear function of two suitable chosen components A., and A,-. For this reason a^^ p-\' a^a^ + ajfl/-,
(5.4)
whereby p, a, and a2 are constants which are determined by the mechanism for each combination of components. For example one obtains for the reaction
the relationship b = bQ-c d = dQ-aQ+a-{-c
if a and c are chosen to be independent concentrations. The diagram c{b) is linear, if all the concentrations fulfil eq. (5.3). Equations for d and e are examples for eq. (5.3). They result at r = 0 in
Ch. 5
Evaluation of Measurements of Concentrations
309
«,o = P + ^i^,o+B^C hv
one finds using eq. (2.30) the matrix of the reaction constants ^-it,
0
0^
Kj
—k'^ J
K^ = 0
Taking into account the factors m, one obtains
0 K^ =
0 ntf.
0
334
Applications of Kinetic Analysis to Pliotoreactions
Ch. 5
and the following linear integral equations
JF'{E)dM^=-kiJM^dt, \F\E)dMr, = - ^ ^ [M.dt-k^
[Mr,dt-¥^^^ [MrdU ntQ
jr{E)dMc =^^^jbdt'-k,JMcdt. This system of equations can be evaluated by linear regression. The rateconstants A:,, ^2 ^"d ^3 can be determined directly. If the proportionality constant of the initial component m^ is known, in addition the constants m^ and niQ can be determined. 5,2,6 Missing discrimination between two reactions If measured values of a certain type are used for kinetic analysis, the general question arises, up to what amount a discrimination between different reaction pathways is possible for these measured values used in evaluation. In general no answer can be given to this question. However, for the quasilinear photoreactions examined above a criterion can be stated. The properties of the transformation according to eq. (5.40) allow the conclusion that in the A/-system two reactions R and R' cannot be distinguished, if a diagonal matrix T exists which allows the two Jacobi matrices to be transformes into each other according to the following equation: (5.41)
K „ = T K ; T - '
To do this the following conditions are necessary according to eq. (5.33): (1) The diagonal element in the ^-matrices must be equal. (2) For both the reactions the zero-elements must be placed at the same positions in the matrices. Example 5.16: Comparison of parallel and consecutive mechanisms Two different mechanisms are assumed. The first /?':
A - ^ B + D,
B->C
Ch. S
Evaluation of Measurements of Concentrations
335
is given by the following matrix equation:
^d'\ f-k; b' K
0 —"fCn
0
c'
^2
0
\d'j
0 oY-'^ a 0 0 0 0 0 0
b' c'
whereas the second reaction
R: A—^B—^C,
A-^D
can be described by
(-{k\+h) b c ydj
0
0 oVa
-kj^
0
0
b
0
ik2
0
0
c
jfc,
0
0
0 d)
The necessary conditions stated above kU ^12 = 0
and
*, 0 1 r
y.^0
and therefore
crn: :i:]=e:)Thus, the inverse of the matrix containing the columns of the linear dependent concentrations becomes L2-> =
-1
1
1 or
and eq. (5.50) can be written explicitly
^^-1
lY«o)_f-l
[l
OXbJ [l
lYlMa)Jbo-aoYflMa
OAOJ
t a, J [-\)
Therefore eq. (5.53) can be written as
^a\
{
0 ^
rn 1
\^J
l-ij
Thus one finds using eq. (4.4) at A = 1
5.3,2.5 Absorbance and degree of advancement The change in absorbances during a chemical reaction can be calculated via the balanced equations as has been demonstrated in the proceeding ex-
Ch. 5
Evaluation of Linear Dependent Measured Values
343
ample. More convenient is to treat the absorbances as a function of linear independent degrees of advancement. The correlation between the concentrations and the linear independent degrees of advancement yields according to the treatment in Section 2.1.1.1 and eq. (2.5) a =ao+vx,
(5.57)
where linear dependencies between different steps of the reaction have also been omitted (see considerations in Section 2.2.1.1). By multiplication of both sides from the left by the absorption coefficients matrix e one obtains Je-a =E=
rfea0+rfe-vx
or E = Eo + Qx,
(5.58)
with the definitions Eosrfeao
and
Qsdev.
(5.59)
The vector x consists of exactly s elements. The vector E is represented by h elements, h can be any number. Under these conditions the matrix Q consists of h rows and s columns. The special case s = h is very essential. If the wavelengths A had been chosen such that Q is regular, a reciprocal matrix exists and using eq. (5.58) one obtains (5.60)
X= Q-*(E-EO).
Under these conditions one can determine the linear independent degrees of advancement by use of absorbance measurements if the elements of Q are known. Example 5.19: Transformation consecutive photoreaction For the reaction
concentration
to absorbance for
a
Applications of Kinetic Analysis to Photoreactions
344
Ch.5
one finds the following relationships according to the derivations given in Section 2.2.1.4 and including a reversible isomerisation for the first step, using the equations discussed above, v is given by
(-1 0^ v = 1 -1
u
0
whereby s equals 2. The measurement at two wavelengths results £ =
^lA
« IB
^2A
^ 2B
^IX
For Q =rf• e • V one obtains
Q=
(""^aA "^ ^2B j
( ^2B "^ ^2C j
According to the definition, eq. (5.59), the elements of Q are represented by n
(5.61) They are the stoichiometric sums of the elements C;^. and can be directly taken out of the general scheme explained in Section 2.1.1.1. For this reason one multiplies the stoichiometric coefficients in the *th row with the absorption coefficients of those reactants in the related columns. 5,3,3 Matrix rank analysis If h absorbencies E;^{t) (A = 1,2,..., A) have been measured at m different times tf (/ = 1,2,..., m), the matrix AE of the measured values
^;u^Ex{u)-E.M
Ch. 5
Evaluation of Linear Dependent Measured Values
345
can be rearranged according to eq. (4.4) to the form AE = rf e Aa.
(5.62)
If the rank of e is not smaller than the rank of Aa, then AE and Aa will take the same rank. For this reason one is able to determine the number s of linear independent steps of a reaction by measurement of absorbances without knowledge of the absorption coefficients. 5.5.5.1 Application of graphical methods using absorbances Isosbestic points [12] During thermal- and photoreaction periods a registration of spectra is performed. The obtained reaction spectra (see Section 4.2.6.1) can at one or some wavelength cross in just one point. These points are called isosbestic points. An example is given by the reaction spectrum of azobenzene (see Fig. 5.4). Such a point is found at a certain wavelength A if at all times A£,(r) = 0. o:u^^
E 0.7 0.6 0.5
/////^^
0.4 0.3 0.2 0.1
//y^
z^:;^^^! ^ ^ ^ 236
f//y^
7*"^^^ 268 ^ ^
^ ^ ^ ^ " " " ^ ' " ^ "^^^ 340
320
300
280
260
240
A[nm]
Fig. 5.4. Reaction spectrum of the photoisomerisation of azobenzene in methanol, irradiated at 313 nm.
346
Applications of Kinetic Analysis to Photoreactions
Ch. 5
Therefore according to eq. (5.58) A £ , ( 0 = eAi^i +GA2^2 -^'"^QXS^S = 0
(5.63)
has to be valid. Since according to the definition the linear independent degrees of advancement jc^ cannot be zero during the total reaction eq. (5.63) is only fulfilled if at the wavelength A GAit=^Z^/it^Ai=0
for /: = l , 2 , . . . , ^
(5.64)
is given. In the case 5 = 1, i.e. the reaction is uniform, eq. (5.64) can be fulfilled very easily. One finds an isosbestic point at all wavelengths at which the spectra of the starting compounds cross with the spectra of the final products. If 5> 1 and all reactants absorb then it is rather improbable that eq. (5.64) is fulfilled at any wavelength. For the consecutive reaction
for example
has to be fulfilled at the same time for one wavelength. If just a few reactants absorb in the certain spectral region one can find an isosbestic point even though ^ > 1. An example is the reaction system A+B—^D +E A+C—^D +F under the condition that, in the spectral range considered, only the reactants A and B absorb: A£:A(0 = 4 ^ A D - ^ A A ) ( ^ I + ^ 2 ) -
If the condition s^^ = E;^^ is fulfilled at just one single wavelength, one
Evaluation of Linear Dependent Measured Values
Ch. 5
220
240
260
280
347
300 wavelength [nm]
Fig. 5.5. Reaction spectrum of the photo-induced hydratisation of l-/?-D-arabinofuranosylcytosine, irradiation at 254 nm with a mercury arc (/Q = 2.78 x 10~^ Einstein cm~^ s~^).
finds an isosbestic point at that wavelength. In another spectral region in which further compounds absorb, the spectra will cross without forming an isosbestic point. In Fig. 5.5 an example is given for a reaction spectrum in which the isosbestic point is valid during the total reaction just in part of the spectrum. During the photo-induced addition of water to l-)8-D-arabinofuranosylcytosine and subsequent thermal consecutive reactions an isosbestic point is only valid for wavelengths longer 280 nm [142]. Two others below vanish during the reaction procedure. In general the existence of an isosbestic point or its absence can be used to draw the following conclusions: (1) If spectra cross in a single isosbestic point a possibility exists for reaction to be uniform. (2) The crossing of spectra in more than one isosbestic point increases the probability of the reaction to be uniform. (3) The crossing of spectra in certain wavelength regions without an isosbestic point requires that the reaction is not uniform even though in other spectral ranges isosbestic points show up. (4) The absence of any crossing of spectra hinders the possibility of uniformity of a reaction.
348
Applications of Kinetic Analysis to Photoreactions
Ch. 5
Sometimes one can obtain further information from reaction spectra. For the reaction
according to eqs. (5.58) and (5.61) ^Ej^ (t) = d{e^^ - e^f^ )x, -h d[e^^ - e^^^ )x^ is valid. If at a certain wavelength e^^^ equals e^^, one finds
At this wavelength the concentration of the intermediate is directly proportional to the change in absorbance. One can recognise such a wavelength by the fact that the initial spectrum for r = 0 will cross with the final spectrum at r —> 00, since at this wavelength the condition is given by ^ A ( O ) = ^AA^O = ^ A ( ~ ) = ^^AC^O-
Absorbance difference diagrams [134] As demonstrated above isosbestic points only give limited information about the uniformity of a chemical reaction. More reliable expressions can be made using absorbance difference diagrams (ED-diagrams). For two different wavelengths 1 and 2 according to eq. (5.63) the change in absorbance with time is given by (J.OJ)
AE^it) = 021^1 + Q22X2 + • • • + Qls^s
This equation states the following. A graph in a rectangular coordinate system of AE^ and AE2 results an affine distorted image for {s>2 degenerate) of the relationship between the linear independent degrees of advancement Xf^ in two-dimensional space. Whereas only the ratio of the axis is valid in the case of a graph taking the diagonal matrices, in this type of diagram the
Evaluation of Linear Dependent Measured Values
Ch. 5
349
axes are even rotated and stretched. The most essential properties of affine or linear transformations had been summarised in Section 2.3.4. Linear absorbance difference diagram If the reaction is uniform, the number of linear independent reactions becomes s = \. Therefore using eq. (5.65) one obtains ^E, = ^E^
(5.66)
The ED-diagram results in a straight line passing the zero point with the slope QuIQix (see Fig. 5.6). By these means any reaction can be tested in a simple way for uniformity using diagrams with statistically dependence of the wavelength on g^*' • It has been demonstrated that ED-diagrams always result in zero point straight lines if the rank of Q A[nml ^ 320 y / \ 310
ii
1.0 . 0.8 .
M 300
^
0.6 . ^ ^
0.4 -
290
0.2 • 0^
1»0
AE ^313
-0.2 4
^^^""^^"^^ 250 1
, 200
, , 500 800
1
Ms]
•
Fig. 5.6. Linear absorbance difference diagram for the photoisomerisation of azobenzene, irradiated in methanol at 313 nm.
350
Applications of Kinetic Analysis to Photoreactions
Ch. 5
is smaller than 2. This is always valid for 5^ = 1. For jr = 2 a single wavelength combination can result by chance |Q| = Gi,Q22-612621=0
(5.67)
Thereby a certain analogy to the discussion of the isosbestic point in the reaction spectrum can be found. Condition eq. (5.67) can be easily fulfilled, if in a certain wavelength range just a few compounds absorb. It is sufficient if in this spectral range all other elements QA* except two become zero in one row. This will happen if independent parallel reactions take place whose reactants do not absorb in the spectral range examined. However, the rank of Q can become 1 if the J2A* become equal in two columns. This can happen for the parallel reaction treated in Section 2.1.4.2, if in some spectral regions only compounds A and D absorb. Under these conditions
is valid. Therefore the rank of Q becomes 1. Even though if in a certain spectral region just one compound absorbs, the Qj^j^ in two columns become equal. This fact proves that one has to examine the ED-diagrams at as many wavelength combinations as possible (see Fig. 5.7). Absorbance difference diagrams for s ^2 By the transformation of eq. (5.65) the correlation between the degrees of advancement jCj and Xj are rotated and distorted. Thereby the characteristic properties of the X- and ^-diagrams are lost. Fig. 5.7 gives the standard example for a photoreaction with two linear independent steps (^ = 2) [60]. Fig. 5.8 is an example of an ED-diagram of a consecutive photoreaction which proceeds according to eq. (3.39) and fulfils eq. (3.40) with ;f l
and dx2 dxi
>oo for % 1 the slope depends on Q , in a very complicated matter. In the case of x 5 1 one finds
which means that the final slope in an ED-diagram follows the direction of the x,-axis. In such a reaction normally the direction of the x,axis is known since the initial slope is zero. Therefore one can take the relative values 5-, = x, /a, and 5, = x,/a, out of the ED-diagram . Reduced ED-diagrams If one varies the initial concentration for different series of measurements, but keeps constant the ratio of initial concentrations for many initial compounds, then one finds for the graph AE, I a,, versus AE, / a,, a type of reduced ED-diagram. These ED-diagrams have similar properties to the reduced K-diagrams. For a thermal reaction, reduced ED-diagrams are invariant against dilution if both partial reactions proceed with the same order of reaction. The diagrams vary if the two partial reactions exhibit different orders of reaction. In the case of photoreactions one has to examine whether the differential equations depend explicitly on the initial concentrations for the relationship in the 6 -diagrams. An example explains this in the following. Example 5.21: Reduced X-diagram for a simple consecutive photoreaction The above-mentioned consecutive photoreaction results in
Or using 5, = x, / a,,
5, = x,
1 a,
356
Applications of Kinetic Analysis to Photoreactions
Ch.5
^„.' i 8 ^395
6 -
4 -
^484
2 •
f\
0
"
>^^^ ^
4
2
.
L^ 10-
-2 -j
^313
^
Fig. 5.9. Reduced ED-diagram of the consecutive photoreaction on 2-methyl-anthraquinone in alkaline methanolic solution, NaOH (0.1 mol l'); wavelength of irradiation is 313 nm, the intensity of the light source is /^ = 1.24 x 10"* Einstein s"' cm"\ cone. (•), a^, = 5.81 x 10"* mol/1. (O): a„= 1.16 X 10"^ mol/l.
is given. The consecutive photoreaction of 2-methyl-anthraquinone in alkaline methanolic solution is a typical example. It is demonstrated in Fig. 5.9. If the quantum yields are independent of concentration - as has been presumed - then the reduced ED-diagram has to be invariant against dilution since no initial concentrations show up in the differential equations. This can be seen in Fig. 5.9 for two different concentrations. Dependencies on intensity and wavelength For the dependencies on intensity and wavelength in the ED-diagrams the same rules are valid as for K-diagrams. For comparison see Section 5.1.1.2. In Fig. 5.8 the dependence on intensity has been marked, demonstrating that the ED-diagrams do not differ.
Ch. 5
Evaluation of Linear Dependent Measured Values
357
Combined photo and thermal reactions If the photochemical reactions are superimposed by thermal reactions, in some cases additional information on the nature of the partial reactions can be obtained. This can be demonstrated in the case of the photoisomerisation of rmn^-azoxybenzene (A) to *s"s
0 Eo-E
a linear dependence is also obtained in the absorbance system. It represents a straight line with the slope -Kk^/k^ and the intercept k^/Qk^. If one knows Q, one finds K and k^/k^ as in the case of the evaluation by concentrations. This is the case if E^ and E^ are known. 5.4,2 Difference equations [15] Taking the solution, eq. (2.45), one finds the difference equations for the degrees of advancement in analogy to eq (5.13)
372
Applications of Kinetic Analysis to Photoreactions
A x s x ( 0 + A ) - x ( 0 ) = K^(x~x^),
Ch. 5
(5.87)
whereby K^=p(exp(rA)-l)p-V The matrices p are given by eq. (2.45). If one multiplies the difference equations (eq. (5.14)) from the left by Q and from the right by Q"* one can transform the above equation into the absorbance system. One finds Q.Ax = Q . K ^ . ( x - x ^ ) or AE = Q . K ^ . Q - * ( E - E O ) = Z ^ . ( E - E ^ ) .
(5.88)
All the elements of the K^ matrix are varied by this transformation in such a way that the calculation is only possible if all the elements of the matrix Q are known. Since this procedure is a similarity transformation, the matrices K^ and Z^ have the same eigenvalues. This means that the consecutive photoreaction treated in Section 5.1.2.1 (Example 5.9) results in exp(-/:, A) - 1
and
exp(-A:2^) " 1
as eigenvalues of Z as well as of K^. The absorbance measurements can also be evaluated by recursion equations. Rearrangement of eq. (5.88) in analogy to eq. (5.13) results in correspondence to eq. (5.14) x ( 0 + A) = K' • x(0) + u with K'sp(exp(rA))p->. By defining
(5.89)
Ch. 5
Evaluation of Absorbance-Time Measurements
373
one finds the recursion equation E(0 + A) = Z' • E(0) + u'
(5.90)
with u' = E ^ - Z ' E^. In this equation the elements of K"^ are also varied strongly by transformation. However, the eigenvalues of Z'^and K*^ are the same once more. 5.4.3 Difference equations of higher order [15] If one forms the difference equation of higher order for the degree of advancement one finds by analogy to eq, (5.17)
^'x,-st^'-'x,^'-^{-•\Ys^{x, ^x,jf
(5.91)
whereby the elements 5f have the same meaning as in eq. (5.17). Multiplication of eq. (5.91) by QAJ^ yields the difference equation of higher order for the absorbances A^£,-5,^A^-^£,+... + ( - l ) ' 5 , ^ ( £ , - £ , ^ ) .
(5.92)
Using this equation the absorbance measurements can be evaluated by linear regression. By this means one obtains the s eigenvalues r^ using the quantities 5f. The same methods can be used to transform the recursion equation, eq. (5.18), into another one for the absorbances. In the course of the evaluation of concentrations according to eq. (5.17) or of measured values proportional to concentration according to eq. (5.35), one obtains the number e^ which gives the number of the exponential functions necessary to describe the function a^{t). Furthermore the related eigenvalues are received. However, using these equations just the number / and all the eigenvalues can be determined. Concentrations can no longer be obtained. This is a disadvantage of this calculation.
374
Applications of Kinetic Analysis to Photoreactions
Ch. 5
5A,4 Formal integration 5,4,4,1 Simple uniform photoreactions Photoisomerisation In the case of the photoisomerisation A—!!^B with a quantum yield constant according to eq. (3.35), one finds, using Napierian units for the derivation, . _ 1000/c>^/oa
In this and the following example K\ and i?'describe the absorption coefficients and the absorbances at the wavelength of irradiation as derived in Chapter 3 (Napierian units). For any wavelength of measurement A one finds according to eqs. (3.56) and (3.62)(since a^ = 0)
Qx Thus one obtains £,
1000/c>^/o(£,,-£,)F(£0-'
Qx
Qx
However, it is preferable to use decadic units; then the above equation becomes £, Qx
1000g>Vo(£,,-£,)F(0 Qx
using the different definition for the photokinetic factor according to eq. (1.42). The absorbance-time differential equations result in the form
Ch. 5
Evaluation of Absorbance-Tlme Measurements
E,=z,,F{t)
+ z,^E,F{t)
375
(5.93)
with z,,^\Q00e\ip^hE,^
and
z,, = - 1 0 0 0 E : , ^ ^ / O .
The constants Zx\ and Zx2 can be determined by formal integration for the different wavelengths of measurement. Zxi does not depend on the wavelength of measurement. Furthermore one finds for the final absorbance
If £\ and /Q are known, the quantum yield tp^ can be determined using z^iThe photoreduction of substituted 9,10-anthraquinones in neutral methanol has been examined by application to this method [143]. The quantum yields for the photochemical reactions of some of the derivatives are listed in Table 5.1. In Fig. 5.16 the absorbance difference diagram is given as an example for a substituted anthraquinone. Fig. 5.17 demonstrates the absorbance-time curves. All wavelengths have a linear relationship. In Table 5.1 the result of an evaluation according to eq. (5.93) is given for the same compounds. The 1-chloro-substituted compounds demonstrated a comparable small differential yield ip^ in neutral methanol.
TABLE 5.1 Quantum yields of various anthraquinone derivatives -Anthraquinone
Quantum yield ^ j ^
1-Chloro 2-Chloro 1,5-Dichloro 2-Methyl 2,3-Dimethyl 2-Carbonic acid
0.22 0.72 0.28 0.66 0.53 0.50
Applications of Kinetic Analysis to Photoreactions
376
I I
I
0 10" 30"
I
I
I
r
V
4'
Ch. 5
I
w
8'
/
Fig. 5.16. ED-diagram of the photoreaction of 2-chloro-anthraquinone in neutral methanol,
'380
.^313
20
> 30 \ [min]
Fig. 5.17. Absorbance-time diagram of the photoreaction of 2-chloro-anthraquinone in neutral methanolic solution, irradiated at 313 nm with /o= 1.26 x 10"^ Einstein cm~^ 8"^ initial product concentration GQ = 8.24 x 10"^ mol T^
Ch. 5
Evaluation of Absorbance-Time Measurements
377
Physically sensitised photoisomerisation In the physically sensitised photoisomerisation treated in Section 3.1.1.4 the quantum yield is given according to eq. (3.6) by A
Kb k^ + kj^b
X contains the part of the quantum yield independent of concentration. Thus according to Example 3.8 one obtains, using decadic units, ^^
\m)e\xIoaok,b ^^^^ kj + k^b
Using eq. (3.63) and assuming b^^O one finds
or
E
-hilInEx.Fit)
with
^3
Rearrangement of eq. (5.94) into the form
(5.94)
378
Applications of Kinetic Analysis to Photoreactions ["' dE, = z„ r F{t)dt + z,,f Jti
Jt,
E,Fit)dt - zJ'' Jt,
Ch. 5
E,dE,
Jti
allows the determination of the constants Zxi^o z^^ by formal integration. The following relationships are valid: ~
"" ^Aoo»
ii2. = iooOe;/„aoxG„ ^A3 GA^O =^AOO -^AO-
If the quantities e'^,lQ,aQ,bQ and £;IQ are known, one can determine xFurthermore one finds according to
the ratio of the constants /:2/*3- Th© level of agreement of the result for the evaluation of different series of measurements for different wavelengths of observation gives an idea of the accuracy of the determined constants. Furthermore it proves whether the mechanism has been determined correctly. Photoaddition For the photoaddition A+ B—^C, the quantum yield is given according to eq. (3.3) by
k2 + k^b One finds using eqs. (3.35) and (1.42) that
Ch. 5
Evaluation of Absorbance-Time Measurements
a=
379
I000€'lr,xk.ba ^, , dLM^—/r(^). k2 H- k^b
If GQ < b^ (that means a^ = 0 and 6^ = feg - aj,) one finds according to eq.
(3.63) a = —^ Qx
and
b = —^=^ Qx
with
In consequence one obtains ^A =
1 0 0 0 e ; / o X ^ 3 ( £ , , - £ , ) ( £ , ^ - E,) . ^ • , /^ ;r^^ ^(0
or ^Ji "^ ^ J 9 ^ J "*" ^ J i ^ i ^^^^M-^A2^A-^A3-A^(^)
(5.95)
1 + ZA4^A
With ^ 1000e;/oXfe3£,^£;t, ^"
"
^ ^1000£;/oX^3(£^ + g , , )
KQx^K^x^
k^Q,+k,E,^
^''
'*
l^iQx+k^E,^
k^Q,+k,E,„
the constants Z},^ to Z;^4 can be determined by formal integration. The constants Zx\, Zx2 snd Zxj can be used to calculate the values of EJ^„ and E•^„. x is obtained using the relationship - ^ = 1000e;/oZ. ZA4
380
Applications of Kinetic Analysis to Photoreactions
Ch. 5
The ratio ^2/^3 can be determined by
The differential equation of all these reactions results in the general form 1
M
Ex=-^^ j^p+a
^(0-
(5.96)
This equation describes a large number of uniform photoreactions with quantum yields, which do not depend on the amount of light absorbed. Complex uniform photoreactions In Section 3.1.3 and Example 2.11 the photoisomerisation
is discussed. Its partial quantum yields are constant according to eq. (2.21). Using eq. (3.81) one finds the differential equation
Taking eq. (3.63) one obtains E, = 1000/o(^f £; + ^f«B)(^A- -
=
{zxx+Zx2E,)F{t)
defining z„=1000/o(^fe;+^%)£,..
E,)Fit)
Ch. 5
Evaluation of Absorbance-Time Measurements
381
z,2=-iooo/o(^^;-fy>%). This equation can be evaluated by formal integration. Zx2 has to be independent of the wavelength of measurement. £^^ can be calculated according to
^A2
Using z^r^ one finds
lOOO/o
^^ ^
^^ ^
The final absorbance amounts to ^Aco = ^«AA«o + GA^OO =rfe;iBao" CA^OO. Thus one obtains according to eq. (5.103) (5.98) If «o' ^AA» ^AB» ^A ^"d ^B ^^^ known the ratio of the quantum yields can be determined in this case. Fig. 5.18 represents the graphical evaluation of the absorbance measured for the trans'cis isomerisation of azobenzene in methanol according to eq. (5.97) [127,147]. The differential equation was evaluated directly. The differential quotients have been calculated by regression parabola which were constructed for any seven points of measurement next to each other. In Figs. 5.19 and 5.20 the method of formal integration had been applied. In Fig. 5.19 the integration took place between any two sequential points of measurement. In this case the points vary greatly since the errors have not been equalised. In Fig. 5.20 the integration took place between / = 0, /' = 1,2,..., m and / = m, /' = m, m ~ 1, m - 2,..., 1,0 (this procedure is called forward- and backward-integration, respectively).
382
Applications of Kinetic Analysis to Photoreactions
Ch. 5
Y
1.0
0.5
Fig. 5.18. Evaluation according to the differential equation, eq. (5.97) [127,147]. The axis is given by dE' 1 K= 7-[10~^5"M. dt 1-10"^'
The numerical evaluation according to the method of formal integration at eight wavelengths of observation and variance of evaluation prove that the measured values fulfil eq. (5.97) with good results. The constants determined according to the two integration principles mentioned above agree with each other to a large extent. Zx2 is independent of the wavelength of observation within the error of measurement. A further good argument for the simple differential equation, eq. (5.97), is that the final absorbance of the total concentration and thus the absorbance at the isosbestic are proportional (see Fig. 5.21). Furthermore the relationship, eq. (5.98), is fulfilled. Using the exactly determined absorbance coefficients of the cis and the trans form, the ratio of the quantum yields has been calculated according to eq. (5.98) taking the final absorbances at various wavelengths.
Evaluation of Absorbance-Time Measurements
Ch. 5
383
A£'
1(1-10"') J/ (io''r'i i.o
^***^^j
0.5 1.0 1.5 X Fig. 5.19. Evaluation according to eq. (5.97) by formal integration between two consecutive points of measurement.
5-20
Fig. 5.20. Evaluation according to eq. (5.97) by formal integration between different times (see text). Axes are the same as in Fig. 5.19.
384
Applications of Kinetic Analysis to Photoreactions
0
0.5
Ch.5
1.0 £266
Fig. 5.21. Final absorbances as a function of the absorbance in the isosbestic point [127,147].
Table 5.2 demonstrates that the ratio of the quantum yields (px^/(p\ is independent of the wavelength of observation. For further proof of the significance of the method, other differential equations for alternative mechanisms have been used. Taking into account bimolecular quenching, the differential equation, eq. (5.97), has to be enlarged to the form of eq. (5.96). The six differential equations, eq. (5.96), were evaluated according to different methods with the parameters p = 2, a = 0;
/? = 2, or = l;
/7 = 3, a = l;
p = 3, a = 2;
/7 = 4, a = 2;
p = 4, a = 3.
The result was surprising. The differential equation with an odd number of constants was not fulfilled by the measured values. The "mean error" of the constants amounted to more than 100%. The values of the constants determined according to different methods fluctuated drastically. In some cases even the sign was changed. The result is that the mechanisms used to propose the differential equations have to be discarded. In contrary all the dif-
Ch. 5
Evaluation of Absorbance-Time Measurements
385
TABLE 5.2 Ratio of quantum yields of photoisomerisation [127,147].
A[nm]
^iV^?
A[nm]
B< >C >Bf >B, A >C
6. 7. 8. 9. 10
AB< A< >B
C, >C, ->C,
AAAA-
Applications of Kinetic Analysis to Photoreactions
394
Ch.5 6JL
A
5" r ^
\
/
^^T*
/
A/y / ~*i>^^ trans will not disturb photokinetics at normal conditions because of a half-life of approximately 1 week. Therefore a concentrated solution of fran^-azobenzene in methanol at 6.4 x 10-^ mol l-^ which totally absorbs radiation between 345 and 240 nm, can be used taking eq. (5.107) and the approximation given by eq. (5.109). In a first approximation the change in absorbance with time at a wavelength of observation is proportional to the intensity of radiation. This proportionality includes the photochemical quantum yields of the trans —> cis isomerisation step, the factor 1000 and the absorption coefficients at the observation wavelengths of the trans and cis isomers. This is a relatively crude approximation. However, a graph of the change in absorbance versus time gives a linear slope for the beginning of the reaction under these conditions with a deviation less than 1 % in comparison to the correct evaluation. These factors can be calibrated by other actinometers as for example the Parker solution [121]. The values given in Table 5.4 can be used to determine some of the lines of a mercury arc. The dashed line in the table indicates that for the measurement of the 254 nm line the other band of the azobenzene solution has to be taken. A trans solution must be preirradiated at 313 nm to bring the solution into the photostationary state.
400
Applications of Kinetic Analysis to Photoreactions
Ch. 5
TABLE 5.4 Calibration factors for some mercury lines X [nm]
Factor x 10*^ [Einstein cm"^]
280 313 334 254
4.6 5.2 3.6 2.3
Afterwards irradiation at 254 nm will cause cis —> trans photoreaction. The necessary solutions are commercially available. An approximation using the photokinetic factor given by eq. (3.69) supplies an even better evaluation using azobenzene as an actinometric system. Using a linear interpolation between two points of measurement at time t\ and ^2^ the photokinetic factor according to eq. (5.108) can be approximated within this time domain as F=F(l)-h ^ ^ ^ ^ " " ^ i ^ \ r - r ( l ) ) ,
(5.113)
E\2)-E\\y where the indices 1 and 2 correlate to the two times at which the photokinetic factor F is calculated using the absorbances at the wavelength of irradiation E\ Using the abbreviation /? = 1000(/?,-f/?2) = 1 0 0 0 ( c > f + £ ' B ^ ^ )
(5.114)
and insertion into equations like eq. (5.20), (5.93), or (5.97) results in the intensity measured between times r, and t^
R\t^ -t2) whereby the factor P is given by ^ _ F(l) [£(2) - Ejs)] - F(2) [£(1) - Ejs)] ^^ £(1) - Ejs) £(2)-£(l) E{2)-E(sy
Ch. 5
Evaluation of Absorbance-Time Measurements
401
In the case where the change in absorbance at the wavelength of irradiation is relatively small, this wavelength and that for the observation have to be chosen differently. The method is successfully used in practice using factors P given in literature [116], [124]. Photoreaction of isopropyl-azobenzene If the photoproduct does not absorb at the wavelength of irradiation and that of observation (Cg = 0, £" = Ej), eq. (5.107) can be simplified to EXt) = ~ 1000/o^f £^(1 -10"^ ^'^)
(5.117)
Using a cell path length of 1 cm and integration between the time limits t and r = 0 gives in decadic units ln(10^'^'^ - 1 ) = ln(10^'^^^ - 1 ) - 2.303X 1000/o^f e^r
(5.118)
These assumptions are valid for 2,2',4,4'-tetra-isopropylazobenzene which shows a uniform photoreaction. The thermal backward reaction is negligible at room temperature. Thus a graph of ln(10^('> - 1) versus the reaction time initially exhibits a straight line. This derivative can be evaluated as a simple actinometric system for radiation in the range between 350 and 390 nm [149]. In Fig. 5.26 the reaction spectrum demonstrates the validity of the assumptions above. At high absorbance (3.5 x 10-^ molar solutions; total absorption at 365 nm) evaluation becomes even easier. Taking the approximation of eq. (5.109), for e;i3 = 0 at the beginning of the photoreaction, eq. (5.110) gives a straight line in a graph of absorbance at an observation wavelength nonequal to the irradiation wavelength in the range 350-390 nm versus time. This fact is demonstrated in Fig. 5.27 for 450 nm. Heterocoerdianthrone The photochromic system of heterocoerdianthrone (HCD; dibenzo[q/]perylen-l,16-dione) in toluene exhibits a photooxidation to an endo-peroxide during irradiation at long wavelengths. This reaction has been extensively investigated in the literature [150]. Even though this photoreaction is complex, under certain conditions (aerated solution, total absorption at wavelength of irradiation, photoproducts do not absorb at this wavelength) it can
Applications of Kinetic Analysis to Photoreactions
402
450
Ch.5
500
Fig. 5.26. Reaction spectrum of the photoreaction of 2,2^4,4'-tetra-isopropylazobenzene in n-heptane (5 x 10"^ mol"* 1"^).
' 450
Fig. 5.27. Plot of absorbance at 450 nm versus irradiation time (A' = 365 nm) for the photoreaction of 2,2',4,4'-tetra-isopropylazobenzene in n-heptane (3.5 x 10"^ mol"* l"^).
Evaluation of Absorbance-Time Measurements
Ch. 5
403
be handled by eq. (5.117). Since under these conditions the peroxide does not absorb at any irradiation wavelength between 400 and 580 nm, eq. (5.118) can be used for the evaluation. The error in calculation of the irradiation /Q stays less than 1% if just 20% of the overall reaction time is evaluated. The reaction spectrum of this photoreaction is given in Fig. 5.28. Mesodiphenylhelianthrene Mesodiphenylhelianthrene (MDH) exhibits upon irradiation with wavelengths in the range between 475 and 610 nm a sensitised photooxidation to an endo-peroxide (MDHPO) comparable to the heterocoerdianthrone mentioned above. However, according to a detailed kinetic examination the photochemical quantum yield at high concentration of mesodiphenylhelianthrene depends on the concentration of oxygen alone which can be taken as constant under normal conditions because of its large excess in toluene [128]. Mesodiphenylhelianthrene is soluble to a large enough extent, thus eq. (5.110) can be written as
600 X [nm] Fig. 5.28. Reaction spectnim of the photoreaction heterocoerdianthrone to its endo-peroxide [151,152]. 400
Applications of Kinetic Analysis to Photoreactions
404
Ch.5
1.0
0.8
0.6
0.2
0.0 240
320
400
480
560
640 X[nm]
Fig. 5.29. Reaction spectrum of the photooxidation of mesodiphenylhelianthrene of its endoperoxidc [128,152].
The wavelength 429 nm is chosen for the observation. The reaction spectrum of the well examined sensitised photooxidation is given in Fig. 5.29. The absorbance-time diagram is given in Fig. 5.30. It proves linearity of a graph absorbance versus time for a concentrated solution. Since the photochemical quantum yield is constant within the wavelength range mentioned above the reaction can be used as a quantum counter in the visible range [128]. 5,4 J Evaluation of photoreactions superimposed by thermal reactions The evaluation of kinetic data is more complicated if an additional thermal reaction is superimposed on the photoreaction. In addition to the EDdiagrams given in Section 5.3.3.1 some further examples of kinetic analysis can be given. Typical examples are the reactions of photochromic substances either belonging to the classes of dihydroindolizine derivatives, fulgides, or even azobenzene-derivatives. This thermal step either can be a backward
Ch. 5
Evaluation of Absorbance-Time Measurements
0
30
60
90
120
405
150
180
i/s
Fig. 5.30. Absorbance-time diagram of the photooxidation of MDH in toluene solution.
reaction in competition to the photoisomerisation or a degradation step. Then mechanisms comparable to those given in eq. (3.95) have to be applied. In the first case the photokinetic equation similar to eq. (5.106) has to be corrected by a thermal term: •E'(t)
a(0 = -1000/o^fe'Aa(0^
^^
E\t)
•f k^b
(5.119)
or taking a second photoreaction into account.
a(o = -iooo/o[^N'A«(0+^2«BMo]- E\t)
•¥k2b.
(5.120)
Conservation of mass a(0) = a{t) + b{t)
(5.121)
and application of the Bodenstein hypothesis (see Section 2.1.3.2) leads to an equation for the overall mechanism: a{t) = -RI^FiOait) + R2loF(t)a{0) + k^a^O) - k^ait).
(5.122)
406
Applications of Kinetic Analysis to Photoreactions
Ch. 5
where R is given by eq. (5.114), I^ represents the incident radiation in Einstein cm"2 s"^ and ki is the thermal reaction rate constant. According to eq. (5.25) a linear relationship between concentration and absorbance signal exists assuming the Lambert-Beer law is valid. Similar to eqs. (5.93) and (5.107) the relationship E,(t) = --IoRE,{t)F(t) + loPxE^m + k^E.iO) - k^^E.it)
(5.123)
is found, whereby Px is defined by Px = ^
/?,4-/?2.
(5.124)
K^XAJ
This equation [153] includes the thermal step compared to the pure photochemical approach given by E,{t)^-'I^RE,{t)F{t)
+ I^P,E,(0)F{t).
(5.125)
This is equivalent to eq. (5.97) using the photostationary state {s) and the abbreviations z,. (see eq. (5.131)), if Pj,E^{0) = REj,{s)
(5.126)
is considered to be valid. Since all these equations cannot be solved in a closed form, formal integration is applied. It turns out that dependent on the relative values of the thermal rate constant, the photochemical quantum yields, and the absorption coefficients, the quality of the result depends on the evaluation chosen. Rearrangement and formal integration supplies the following equations [153,154]: \dEx^-z,\Ex{t)F{t)dt'^Z,E^{s)\F{t)dt,
(5.127)
\dE, = - z,JE,it)F{t)dt + Z2JF{t)dt - z,JE,{t)dt + z,E,iO)jdt, (5.128)
Ch. 5
Evaluation of Absorbance-Time Measurements
407
J j £ , = ~ z J £ , ( 0 F ( 0 t / r + zJF(0rfr4-^3J[£,(0)-£,(0]t/^
(5.129)
jdE;^^-x^JE;^{t)dt-\-X2E;,(0)jdt.
(5.130)
In these equations the following coefficients are defined: ^
/? = 1000(/?,+/?2) = 1 0 0 0 ( £ > f 4-6^y?f),
P;i= p€^ / ? , + / ? , V^AA
Z, =(/?,+/?2)/0,
Z2=PxIoE;,(OX
Z,=k2, (5.131)
F is the averaged photokinetic factor within the time interval of measurement. In eq. (5.130) the thermal rate constant is determined independently. 5.4.7.1 Photochromic systems of dihydroindolizines Various derivatives of dihydroindolizines with its reversible photoreaction to betaine and the superimposed thermal reaction have been used to test the algorithms. Program system SIDYS A program system called SIDYS using simulation techniques and parameter identification was applied to the absorption data. It includes random search techniques, Rosenbrock strategy, quasi-Newton methods, and lattice search. It runs on a mainframe computer and provides good agreement between simulated and experimental data. On the other hand, it proves that the data of this type of mechanism are not well conditioned and the quality of the evaluation drastically depends on the parameter set chosen [155]. Evaluation by personal computer Because of small calculation capacity, any evaluation will need more CPU time. Therefore parameter estimation with highly sophisticated algorithms will consume too much time on small computers. The different equations, eqs. (5.127)-(5.130) applying formal integration provide a good tool
Applications of Kinetic Analysis to Photoreactions
408
Ch. 5
TABLE 5.5 Quality of the various differential equations given by eqs. (5.127)-(5.130) ki
Pure photochemical eq. (5.127)
A£:'>0 0 5%
++ + --
A£:' = 0 0 5%
++ + —
Incl. thermal 3 constant eq. (5.129)
Incl. thermal 4 constant eq. (5.128)
^2 independent determination, eq. (5.130)
to obtain results even for this complex mechanism. In Table 5.5 the benefits of the different equations are summarised for different ratios of z-values. Since the differential equation, eq. (5.127), is more stable, but does not fit a mechanism which includes thermal reactions, it supplies only sufficient
error z (%)
• pure photochem
3-const
Z3/Z1
(%)
Fig. 5.31. Deviation between the calculated parameter Z| and true one for different ratios of thermal to photochemical reaction given by Z3 and z\ using eqs. (5.127) and (5.129).
Evaluation of Absorbance-Time Measurements
Ch. 5
40H
409
A
1
errors [% ]
30H
20 '"•A.
10^3
o OfV
Zl
fid-O-Dti
0.0
0.2
0.4
0.6
z-ifzx
0.8
Fig. 5.32. Errors in all the parameters for both eqs. (5.127) and (5.129) for different ratios of thermal to photochemical reaction given by Z3 and z\.
quality when thermal rate constants are negligible. However, Fig. 5.31 proves that up to a ratio of zjz^ < 0.05 even in the case of a superimposed slow thermal reaction, the "wrong" evaluation by the pure photokinetic equation, eq. (5.127), gives better results for the photochemical parameters. The determination of z^ which refers to the thermal rate constant exhibits large errors (see Fig. 5.32). Therefore in principle an independent determination of this thermal rate constant is advisable [153,155]. Combined determination of reaction constants and absorption coefficients In many cases the absorption coefficients of the photo-product cannot be determined, since B cannot be isolated in a pure form. For these conditions quite a few methods have been proposed in the literature [156-160]. If the conditions required for these approximations do not fit, the thermal reverse reaction is used to shift the photostationary state by variation of temperature. The possibilities of such an approach are demonstrated by the photoreaction of dihydroindolizines [49,161]. Using eq. (5.119) and the approximation at the photostationary state
410
Applications of Kinetic Analysis to Photoreactions
Ch. S
a(s) ko b(s) (plOOOIoC'^Fis) where the Arrhenius equation is used for the dependence on temperature, a linear relationship is found ^-
= i / o - ^ e x p | - ^ l + nP,.
E(s)-EiO) \°Fis)
^^IrJ
(5.132)
P
The parameters are defined as follows: Po=
—r.
^1=-—.
^2=
r^
r - (^ = lcm)
with E^ the activation energy andfc,the frequency factor of the Arrhenius equation. On the other hand, the photochemical quantum yield can also be determined by formal integration of the equation jdE,-k,j[E,(0)-E,(t)]dt
=
ip 1000 /oe;{6,B a(0)JFit)dt -'JE,(t)F(t)dt] where the thermal rate constant is determined independently [49]. As well as the thermodynamic constants for the activation, the photochemical quantum yields, the absorption coefficients, and the thermal rate constants can be obtained. In Fig. 5.33 experimental and simulated data are compared for a reciprocal plot according to eq. (5.132). 5.4,7.2 Photoisomerisation of (Z,E,E)'4,4''distyrylazobenzene 4,4'-Distyrylazobenzenes exhibit a photokinetic equilibrium and a superimposed thermal backward reaction. The absorption coefficient of one of the partners in the equilibrium is principally unknown. Nevertheless, using the dependence of the photostationary state on the intensity of the irradiation, the photochemical quantum yields (p^ and (P2 have been determined, as well as the thermal rate constant k, depending on the wavelength of irradiation and wavelength of observation [50]. Fischer's method [156] to determine absorption coefficients requires in-
Evaluation of Absorbance-Time Measurements
411
1/T[K]
^•^
1/TlKI
Fig. 5.33. Graphical representation of an evaluation according to eq. (5.132) for the superimposed photochemical and thermal reaction of rH-2',3'-dimethoxycabonyl-spiro-[fluorene9,1 '-pyrrolo[ 1,2-6]pyridazin] [161].
dependence of the photochemical quantum yield on the wavelength and can even be used if a thermal backward reaction influences the photostationary state. In this case, the thermal reaction is frozen out by working at low temperatures. Then, it has to be assumed that e^ is independent of temperature. This thermochromism is valid for the Z,E,E-, but not for the Z,Z,£'-isomer of distyrylazobenzene. Even though e^ and e^ could be determined by temperature dependent measurements, Fischer's method can not be used, since first experiments proved a dependence of the photochemical quantum yield oooo o o
2.5
500
0
o
o
o
1000
o
t/s
1500
Fig. 5.42. Diagram of fluorescence intensity versus time for some laser dyes based upon substituted stilbene-1 derivatives [93].
known eg. In any case, however, the two slopes have to differ by at least one order of magnitude to reduce errors. For this reason a bi-exponential fit is preferable. Typically non-straight lines in the logarithmic diagram intensity versus time are given in Fig. 5.42. The consecutive mechanism gives the following rate equations d{t) = I[-R,a{t) + R^b{t)] m
= i[RM0(R2
+ R^Mt)]
c{t) = l[R,b(t)] assuming very dilute solution and using the definitions: /?,=1000£'A^f,
/?3=1000e>3.
426
Applications of Kinetic Analysis to Pliotoreactions
Ch. 5
This set of differential equations can be solved at the boundary conditions a(0) and b(0) = c(0) = 0 [177,178] with Wi2 = O-Sl"/?, + /?2/?3 ± ^ ( ^ , +/?2+/?3f-4/?,/? 1
«(0 = J^^l^
[iW^ - R,)txp(-IW,t) + (R, - W2)cxp(-IW,t)]
Ht) = .^^^[exp(-IW,t)
- cxp(-IW^t)]
c(0 = J ' ^ " ^ [W -W,)-W,
(5.147)
txpi-IW,t) + W^ exp(-/W;0]
With eq. (5.137) / a ' ' ( 0 - ^ ( 0 - ( / ? , ~ W,)exp(-/lV,0 + (W^, -/?,)exp(-/VV20 by measurement of the fluorescence intensity at a wavelength a the change in A'^ concentration with time can be correlated to a bi-exponential function, which contains four parameters pi. If an offset has to be taken into account, a further parameter p^ has to be added ^a ( 0 «= Pi exp(-/720 + P3 exp(-/74r) -h ^5. The defined parameters correlate with the partial photochemical quantum yields according to
and
R2 = W^+W2-R^- R^
(5.148)
Ch. 5
Fluorescence
427
A non-linear regression allows calculation of the parameters p^ to p^ for fluorescence-time curves. These equations correlate with eq. (5.141) in the case of the simple mechanism A''^ B'^. Under such conditions eq. (5.148) reduces to R^^
El^
(5.149)
assuming that in this mechanism p4 and W2 become zero (mono-exponential). Thus eq. (5.149) yields /;(0) = p,+P3+P3
with
/ ; ( 0 ) - / ; ( 5 ) = /7,
and
-^=\y,=^.
The same relationship is obtained, as was derived for eq. (5.145) ^
m/;(0)-/>)
'
/
/;(0)
Intensity diagrams for consecutive reactions For the consecutive reactions intensity diagrams are curved if besides A'" another reactant also fluoresces. Examples are given for laser dyes (see Fig. 5.43) [92,93] and for short wavelength irradiation of anthrone [180]. In the case of CPTC at irradiation by 254 nm even an intensity quotient diagram can be constructed (see Fig. 5.44). These diagrams correlate with those of extinction difference quotient diagrams (see Section 5.3.3.1). They are derived by combination of intensity differences at different reaction times taken at three wavelengths according to the following equations:
762? (0 _= «
Applications of Kinetic Analysis to Photoreactions
428 120
1
Uso [%I
'
' 1• " • •
-
; 550nm«530nm
•
I • •
80 h
,•
• H
'n
• •
" «
5 • • "•
M
• • , •
40 h
•
• •
;
•
A
•
•
•
• •
• '"
.••
4
.•!
«
5
A A A '
:
5 i
'
1
A A A
J
A^75nni A A
•
B •
•••
•
•
: ^
• • :500nm
.MSOnm
•
:
60
1
•
m m
100 L
'
r
Ch.5
A ^
-
J 1
A A A A A
1 1
A
_J
A A
• •
^2^'-" ft* m
1\
*•
.*
»* ft
A'=365nm
20 h
^/
_.
0
1
.
1
20
1
40
L
,
60
1
80
,
H 1
1
1 100
Fig. 5.43. Intensity-diagram of the photoreaction of 7,7'-N-diethylamino-4-methylcoumarin (DEMC), irradiated at 365 nm [181].
5,53 Errors due to E > 0.02 When the assumption F(0** = 2.303 is no longer valid, instead of eq. (5.137) the relationship / ; ( 0 = /^^1000/oa(OF(0
(5.150)
has to be used with (5.151) One can assume that Kf stays constant during the progress of the reaction as a first approximation. Thus the differentiation of this equation yields 1-10"^'
/ ; ( 0 = /^^1000/oa(0—-— b,
-KplOOOIoait)-i^ '^ °
^ ^-^—P. ^ - i (l-exp(-£:'))
(5.152)
Ch. 5
Fluorescence
L
120
429
500
100 80 60
>m
a
,•1*'
• 430nm •475 nm
40 20 0
1
•
^1
Ail!
LA:
1
1
1
.^_^ 450
-20
** \
-40 -60 20
-10
10
10 "*•
20
30
40
50
60
Fig. 5.44. Intensity quotient diagram of the photoreaction of CPTC, irradiated at 254 nm in methanol [182].
with
E
(5.153)
For this derivation the Napierian absorbance units are taken for convenience (the molar absorption coefficient e' is decadic). Using eqs. (4.2), (5.152), (5.153) and a rearrangement of eq. (5.150) a(0 =
/;(0
l-exp(-£0 1000/oJ?^F E'
one finds (5.154) The function S(£') is defined by
430
Applications of Kinetic Analysis to Photoreactions
S(E') = exp(-E') +
E'{s)\
E'
i^^f^-exp(-£-)].
Ch.5
(5.155)
using decadic units, one obtains l - 1 0 " ^•IO\.-r 5(0 = 10-^'+-^^ E' 2.303£'
CS. 156)
Using the mean value theorem within the time limits tx and ^2, eq. (5.154) can be integrated to
ln/;(r2) = ln/;(r,)-/?,/o5(F)(r2-r,)
(5.157)
whereby r, B'' the influences are as drastic as demonstrated. 5.5.4 Applications 5.5.4.1 Anthraquinones In photochemical reaction pathways, normally the number of reactants which fluoresce is smaller than those absorbing. For this reason intensity diagrams were introduced in Section 5.5.1.2 to supply additional mechanistic information. The photoreaction of the anthrone to tautomeric anthranol gives a linear intensity diagram because anthranol is the only fluorescent product. However during this photoreaction one realises that the oxygen of the solution will be used to photooxidise the reactant to the non-fluorescent anthraquinone. This reactant is stable as long as oxygen is in the solution. As soon as it is consumed, a blue-green fluorescent anthrahydroquinone is formed. The maximum of the wavelength of fluorescence is shifted to the ultraviolet in this consecutive photoreaction. At least at one certain wavelength of observation a third fluorescent compound can be monitored as is demonstrated by the intensity diagram in Fig. 5.46. In the case of the photoreactions of anthrone and anthraquinone only the intensity diagrams were used to support mechanistic considerations obtained by formal kinetic examinations using absorbance. However, in the process of the examination of the photostability of laser dyes, quantum yields have been determined applying the equations given in Sections 5.5.1 and 5.5.1.1. 5.5.4.2 Photostability of laser dyes The structures of two strongly fluorescent dyes are given as
432
C
o
o
o
I c o 'a o o
SI
o
d d d d
n o — ^ o o o o o o o o o o ^ d d ci e'^b{xj)]cbX +/o/?2&U,Oexp|-J'[2.303£A«(A:,0 -f 23Q3e'^b{x,t)dx]^
(5.172) with
Ch. 5
Photoreactions in Viscous Material
dt
465
dt
In these equations, the dependence of /g^s on the volume elements is taken into account. Since in the perpendicular arrangement a scanner allows one to determine absorbance as a function of the local element, application of good local resolution supplies the necessary data for the calculation. It requires many measurements at each irradiation time. A prerequisite for the evaluation mentioned is knowledge about the reaction mechanism. Linear absorbance diagrams proved the photoisomerisation taking place as in solutions. However, the siloxane matrix has to be fresh. Different types of siloxanes were tested, some photochemically polymerised, others fabricated by a catalyst induced process. In the latter case the Ptcatalyst must not overcome a concentration limit otherwise it influences the azobenzene photoreaction. Approximate evaluations at low absorption (assuming a irradiation intensity independent of the volume element) do not offer appropriate results because of measurement problems. Therefore a transformation of the "time" scale has been used, discussed in Section 5.7.3. 5.7.2.2 Dihydroindolizines in siloxanes Dihydroindolizines can be embbeded in siloxanes out of an ether solution during the process of polymerisation. Absorbance diagrams allow a qualitative comparison with those obtained in solution. In principle the quantum yield of the photoreaction is reduced for a large number of derivatives. The 1,2-di-cyano-4-methyl-aceto derivative no longer exhibits a backward thermal reaction from the betaine because of steric hindrance. Even though irradiation was chosen co-linear with the measurement direction and thus diffusion is not involved, the combinations of differential equations according to Section 3.4 cause numerical problems at evaluation. Therefore only approximate calculations can be done simplifying the mechanism to estimate reaction constants [153]. 5.7.3 Photokinetics in viscous media 5.7. i. 1 Azobenzene As derived in Chapter 3.4 the photokinetic factor depends on the volume
466
Application of Kinetic Analysis to Photoreactions
Ch. 5
element as the concentrations of the reactants depend on time and space, given as a general solution for a single step uniform photoreaction by eq. (3.102). With the boundary conditions, these equations can be solved calculating the functions H{t) and G{z) stepwise (see Section 3.4.1). In Section 3.4.3.1 possibilities of an evaluation have been given for an isomerisation caused by irradiation into an isosbestic point, thus allowing a relatively simple relationship to be set up which becomes more complex if integrals cannot be solved in a closed form as demonstrated in Section 3.4.3.2 by eqs. (3.116)~(3.123). Using those considerations for eq. (5.97) and using the transformation to a pseudo-time 0 once more, one finds [210] 0(jc,O = /olOOoJ'expj-j'[2.303e;^(;c,0 + 230'ie'j^b{x,t)]dx[dt (5.173) The concentrations dependent on time and location in the polymer are given by partial differential equations and using nomencalture introduced in eq. (5.131):
The solutions are a ( e ) = - ^ ( / J j + /?, exp(-/?e)) biG) = ^^R0.
- exp(--R0))
which are comparable to thermal reactions. As derived in the literature by consecutive differentiation, one obtains with respect to time first [212-214] — = /o lOOOexpf-J'cj + C2 exp(-;?0))d[xl and then with respect to volume element 5^0 30 — - = -(c, +C2exp(-^0))— dtdz dt
(5.175)
Ch. 5
Photoreactions in Viscous Material
467
with c, = - ^ ( 2 . 3 0 3 e ; / ? 2 +2.303£^/?,),
K
One can integrate eq. (5.175) with respect to time t and can determine the integration constant/(z) at initial conditions for time r = 0, where no gradient of concentration is given: boundary conditions:
'— = 0
and
©(z,0) = 0
dz thus:
— = - c , 0 + ^ e x p ( - / ? e ) + f(z). dz R
Integration within the limits 0 and z gives a time-dependent constant g(t), which can be determined at initial condition z = 0:
f Jo
de • = -z + git)
c,0 + ^(l-exp(/?0)) R 'olOOOf
^ 0
c,0 + ^(l-exp(/?e)) R
de
-I c,0 + ^ ( l - e x p ( / ? e ) ) + z = 0 R
This integral cannot be solved explicitly. A Newton-Raphson iteration is used first to determine 6 values, which are used to calculate the concentrations of a{6) and b{6) according to eq. (5.174). In Fig. 5.70 the result of such an examination is given. One finds a systematic deviation, which can be explained by a diffusion process actually happening for cis- and transazobenzene molecules.
Application of Kinetic Analysis to Photoreactions
468 l.I JO
1.0
Xi
0.9
0.8
1
'
1
'
"
1
'
1
1
Ch.5 1
experimental data with diffusion without diffusion
'•{
0.7 0.6
0.5 1
1
11
50
1
1
100
1
'
1
150
200 time [s]
Fig. 5.70. Azobenzene in polysiloxane: experimental data and simulated curves versus time and the residuals. The simulation was calculated with and without diffusion.
As soon as diffusion has to be taken into account, the equations derived above can no longer be used. Diffusion causes a time-dependent gradient and the second Pick's law has to be included in eq. (5.172) by an additional term:
A
dz'
and
A
dz'
Diffusion and photoprocesses can be separated at irradiation intensities large enough and irradiation sections short enough to be able to neglect diffusion during irradiation times. This requires numerical integration over all local elements within two dimensions (diffusion perpendicular to the incident irradiation isotropic). Furthermore all the reaction times have to be considered for the photodifferential equation [210,213,215]. Essential is the optimisation of the step width by which the local elements are taken for calculation. Too small elements cause numerical unsteadiness for the solution of the dif-
Photoreactions in Viscous Material
Ch. 5
469
600 wavelength [nm] Fig. 5.71. Reaction spectrum of the photoreaction of the Z-isomer in the polymer liquid crystal at 100°C. Irradiation at 365 nm [196]. 0.4
0.3 V
V
0.2 H
O.I
D
•
V VV
^ O O O O
0.0
-0.1
500
1000
1500
2000 2500 irradiation time [s]
Fig. 5.72. Concentration-time diagram for the photoreaction of the Z-isomer in the polymer liquid crystal at 100°C; irradiation at 365 nm [196].
470
Application of Kinetic Analysis to Photoreactions
Ch. 5
fusion transport equation. On the other hand, too large steps average the concentrations too much. The improvement in the simulation by taking diffusion processes into account is given in Fig. 5.70 [215,216]. 5.7,3.2 Photokinetics in liquid crystals If the fulgide given in the reaction scheme in Section 5.6.3 is connected to an oligomere backbone and mesomeric groups, liquid crystals are formed which exhibit photoreactivity. Application of photokinetic equations and methods given in Sections 5.6.3, 5.7.2.1 as well as 5.7.3.1 allows an evaluation even above the glass temperature. According to eq. (5.172) the system of differential equations
- s'Ee,iPEcf,F(ED + e'cC,iPcEliF{E^ + ^x T T OX
^
at
= -e',e,A
^4
A"
>B
B
Xs ^6
>A
-1 1
A' 1 -1
0 1 0 1
-1 0 0 0
A" B 0 0 0 0 1 0 -1 0 1 -I 0 -1
-^* /A Atjfl'
k^a' k,a" k,a"
Kb
As in Section 2.1.3.3 explained by use of the Bodenstein hypothesis a* = a' = 0 can be assumed, therefore one finds /.
a =• k^ + ^5
These equations can be used to calculate a and b as explained in Example 2.6 and introduced in Example 2.4 a = ~/A + k2a' ^-k^d '-\-k^b b--k^d'-k^b and ~a =fe=
^3^5 ^A
-hb^^xlK-Kb
This reaction turns out to be uniform since during the reaction
480
Appendix
Ch.6
a+b-ttn is permanently valid. Example 2.11 In case of a reversible photoisomerisation without a thermal backreaction one can define A to be the trans and B the cis form A Xx - 1 1 Xl
A—!21-^B B—!^A
B 1 -1
Xk
^f/A
^\h
Because of the photochemical back-reaction, a photostationary state is reached. If A " = B" = T is the common intermediate the scheme will be
A + Ziv
>A'
A'
^A
A'
>T
B + hv
>B'
B'
>B
B'
>T
T
>A
T
>B
A x{ - 1 1 x\ 0 X', A 0 A 0 A 0 x'n 1 A 0
A' 1 -1 -1 0 0 0 0 0
T 0 0 1 0 0 1 -1 -1
B
B'
0 0 0 -1 1 0 0 1
0 0 0 1 -1 -1 0 0
A /A
k^a' k^a'
h k^V
Kb' k-jt
kgt
Therefore with the usual assumptions, the concentrations of the intermediates can be caculated
a'=-A. k2 +K3
b'=-
/ p
k,a' + kj}' .^'^y
ks+K
and the change in the concentration of A and B is given by
Simple Photochemical Reactions
Ch. 6
d = ~ / ^ 4- k2a' + A:^^ = (*2+*3)(*7+^8) —
481
(*5+*6)(*7+*8)
s,^^
= -K/A+^2/B
and
In consequence the partial photochemical quantum yields are represented by
^r=-(A:2+*3)(*7+*8)' 3 8
^6^7
^ B = .
"'
(*5+*6)(*7+*8)
because of the law of conservation of mass fl + Z? = AQ .
During the total reaction this photoreaction will be uniform. 623 Complex non-uniform photoreactions A typical example is a photoreaction with a superimposed thermal reaction (see Section 2.1.4.4), in which the reaction scheme is given as A—^B
A -1 * 1 ^2 * 0 •
B—5^A B—^C
B 1 -1 -1
C 0 MinCa^.^^^, the result is ^oo — -^loo "" - ^ a o o " " -^Soo"~ ^
and either
doo^dQ
or
-aQ>0
is obtained. In consequence the final concentration of compound E is given by ^oo =A:3^=Min(ao,feo). Taking the other assumption d^>Min(a^,b^\ one finds the final concentration of D to be zero:
Ch. 6
Consecutive Reactions with Superimposed Back-Reactions
495
In consequence the final concentration of E is determined by the initial concentration of D: •^300 =
^00 =
" 0 •
Therefore an equilibrium is obtained between the remaining amounts of reactants A, B and C according to c^ =
koodoo-
6.5 CONSECUTIVE REACTIONS WITH SUPERIMPOSED BACK-REACTIONS Using the definitions given in Table 2.2 the following equations derived in Section 2.2.1.1 are taken for the calculations in the next sections: Degrees of advancement for all steps of a reaction In matrix notation
^t = kj^aj. {k = 1,2,..., r)
Differential equations with absorption coefficients Non-reduced concentrations
v°*i* = ^^ = v°*Kna* s Kca*" a° = aj + pAa
And its time dependence
^« = KtCaJ + pAa) = ao + K^pAa
Including the defmitions
i j = K^aJ
^* = Kja"
K'C = VKS
Aa = a ^ = K^a
SQ
Reduced time dependence of degree of advancement Changes of selected concentrations Reduced with respect to mass balance
Aa = vx v i = ^ = v Kni = v Kft(a° + BAa)
And reactions steps
K^ = V K Q P
Changes in concentrations
^ = ^0 + K^^Aa ao = v KoSo
Appendix
496
Ch.6
x = v"^ao+v"'^KcVx x = xo + K-x Jacobi matrix
K = v"*KcV
Example 2.17 In addition to the derivation given in Section 22.1 A aspects of Section 2.2.1.2 are included here for the reaction scheme for the reaction AB->C A -1 * 1 X2 *
0
B 1 -I -1
• 4>
C 0 kiO 0 k^b 1 k^b
Using eq. (2.40)
we obtain K=
(-{k^+k^) k,
k
and its inverse K-' =
•^/'' "IT
to find a.
A Xx - 1 0 X2
B 1 -1
C Xk 0 k^a — k^b 1 k^b
Consecutive Reactions with Superimposed Back-Reactions
Ch. 6
497
Example 2.19 For the most simple consecutive reaction A -^ B -> C one finds for the steps given in detail in Example 2.17 the reaction scheme
JC,
A -1
B 1
X2
0
-1
C ^k 0 kfO 1 ^2^
r-1 o\ V
1 0
=
-1 V = 1 0
-1 Ij
X = X* = K„a° =
(-ki a = A:,
n.
fc.
0
V-*2
0
Ky
0 -k^
0
0^ -1 1;
OVa^ 0 b
0^ -1
" 1 V .A -»C
A' + B
A
-1 1
X',
0
A"
A"
>A
A" + B-
•4C
A x[ - 1 x\ 1 A 0 A 1 A 0
A' A" B 0 1 0 -1 0 0 -1 1 0 0 -1 0 0 -1 -1
c
A
0 /A 0 k^a' 0 k^a' 0 k^a" 1 k^a"b
With desensitisation by an additional partner via singlet
A + hv A'
>A' >A
A' + B B' A' + C
»B' + A ^B >D
A' 1
B
B'
x{
A -1
0
0
A
1
-1
X^
1
-1
0 -1
0 1
A A
0 0
0 -1
1
-1
0
0
C 0
D
A
0
IA
0
0
k^a'
0 0 -1
0 0
k^a'b k^b'
1
k^a'c
Ch. 6
Relationship between Quantum Yield and PhotoMechanism
511
via triplet
A + hv
>A'
x;
A 1 A' 1 A"1 B -1 1 0 0 1 -1 0 0 0 -1 1 0
B" 0 0
D
0 0 0
0 0 0 0
k^a' k,a"
0
k^a"b
A'
>A
4
A'
>A"
X-i
A"
>A
B" + A >B
X',
A" + C
6.6.1.2
>D
Sensitisedphotoreactions
Example 3.8: Physically sensitisedphotoreaction The physically sensitised photoreaction B-
kv
^C
with
^A'
A + /iv A'
>A ->B' + A
A' + B B'
>B
B'
>C
A x[ - 1 A 1 A 1 A 0 A 0
A' B 1 0 -1 0 - 1 -1 0 1 0 0
B' 0 0 1 -1 -1
c
A
0 IK 0 k-jO.' 0 k^a'b 0 k^b' 1 ksb'
yields a = 0,
-^b-c-ip^Ip^,
kjk^b A'
A'
>A
A'
>A"
A"
>A
A'' + B
>B''-f A
B''
>B
B"
>C
A x{ - 1 1 ^2 0 ^3 1 X\ x's 1 0 < 0 X'-,
A' A" B 1 0 0 -1 0 0 -1 1 0 0 -1 0 0 -1 -1 0 0 1 0 0 0
B" 0 0 0 0 1 -1 -1
C 0 0 0 0
K /A
k^a' k^a' k^a" k,a"b
0 0 1
Kb" k^b"
with the rate laws a = 0,
^br:zc^ip^l^,
Jkjfc
(p^ ^ Ky
I K'i KA
t ^s
k-, 6
7
ithin the quantum yield the factor k^b k,-¥k^b
b
ii+* *,
determines the minimal necessary concentration of B to allow energy transfer. Concentration b must reach the order of magnitude of k^/k^. If ^4= 10^ s"* and k^- 10*^ 1 mol"* s~*, the concentration of component B must become approximately 10~^ mol 1"*.
Relationship between Quantum Yield and PliotoMechanism
Ch. 6
513
Example 3.10: Physically sensitised photoreaction with another partner Another possible scheme for the physically sensitised reaction B + C-
hv
^D
takes into account a bimolecular step of reaction and is given by the following partial steps of reaction:
-»A'
A + kv A'
A
X-i
^B' + A
A' + B B'
X<j
x;
>B
B' + C-
4
->D
A -1 1 1 0 0
A' B B' C 1 0 0 0 -1 0 0 0 -1 -1 1 0 0 0 -1 0 0 0 -1 -1
D K 0 /A 0 k^a' 0 k^a'b 0 k^b' 1 k^b'c
One finds
phenanthrene a
0.1
o
1 B
Fig. 5-59
NOg-
-wlt-*''*''"'^'^ -
2000
4000
r-2
6000
8000
/[si
HPLC-data
Ch. 6
Example of a Photokinetic Evaluation
527
6.7.4.4 Photokinetic equations The differential equations using concentrations are given by a = {-ip^E\a{t)+oooor-'QONm _ , . ON _ --H i - i T f t ^ o o g s Q O N t ^ T t o « n Q f n v o o o O N O O N O N O O « oONc nC<Jo N vo oo»-o«nio»r>»nvovovovovor^r^r--
S S S ? 8 S CS 0 0 § 8 S CS 0 0 sor^t^ooososOO'^cNfSenen
o o d o o o o o o o o o o o o o o o d o o c > o ^ ' - J
voooeninenvo*-^ t^»o»or*-'-dvo^* 0'-Jo6oNen^enr^Ti^'^r^c4d 0\rtQ»o—tr^Ttdt^enO TJ•«o^\Ot^t^OOO^O^O'-< ,-H,-^,—icsesr^enen
CS 0 0
i8S
CM 0 0 ON
o o cs 00 v O O O ^ ^ c S c S e n e n O N '-< '-^ •-< '
?8S
8S
^t^ON'^vO"^r-cS'-'OovovooocsoNvo^oOTt»-Hen^rf»OTj-cs OooTt»-Ht--rnON*r>ON'^ooenr-'—iTfoO'—"ent^Qen«nr*0\'-^en ^^raenenTtrtv->w-)VOvor-r-ooooooONONONpOOOp^'-«
§8S
^^esenenTj-Ti-invovor^r-
s o r«i 0 0
Appendix
530
Ch.6
0,0016 Y 0.0012
0,0008
0,0004
t
Y =-0,00094 X +0,00138 ^
+ J
calculated data
1
1
1
0,2
0.4
0,6
0.8
Since Zxi = kE^^^ and 1^2 - ~^ > ^^^ ^^'^^ constant amounts to 9.4 x 1(H s-' and £;,„= 1.468. In the case of a photoreaction the photokinetic factor has to be included. Various evaluations are possible according to Figs. 5.18-5.20, the simplest one being the use of formal integration like (5.93) f,.l-10-^^'"^ dt + 2,, I E, it„) ^=^J: " e(0
dt
or \"EM„)^—^
AE I •''1
•=
dt
Zi,+Zi
dt
I
E (t )
J/,
E(t„) -
dt
with Zxi=lOOOe\(p''IoE^
and
/ ,^Ai Z;^^ =-1000e>'^/o
Ch. 6
Example of a Photokinetic Evaluation
531
or by analogy with Fig. 5.20
In both cases the Macro can be used to calculate the integrals in a more complex way. 'NumlntVl.O 'Numerical Integration using the trapeziod method 'Author: Felix Heise 'The function requires a field with two rows (or two columns) as input parameter. 'The first row (column) should contain the time-values, the second (e.g.) the absorbance-values within the integral. The algorithm of the Macro requires at least three pairs of data Function Numlnt(ParField) Field = ParField If UBound(Field; 1) = 2 And UBound(Field; 2) > 2 Then For Count = 1 To (UBound(Field; 2) - 1) Numint = Numint + (Field(l; Count + 1) - Field(l; Count)) * (Field(2; Count + 1) + Field(2; Count)) / 2 If Field(l; Count) > Field(l; Count + 1) Then Numint = CVErr(xlErrValue) Exit For End If Next Count End If If UBound(Field; 2) = 2 And UBound(Field; 1) > 2 Then For Count = 1 To (UBound(Field; 1) - 1)
532
Appendix
Ch. 6
Numint = Numint + (Field(Count + 1; 1) - Field(Count; 1)) * (Field(Count + 1; 2) + Field(Count; 2)) / 2 If Field(Count; 1) > Field(Count + 1; 1) Then Numint = CVErr(xlErrValue) Exit For End If Next Count End If If UBound(Field; 1) > 2 And UBound(Field; 2) > 2 Then Numint = CVErr(xlErrValue) End Function This macro was used to calculate the results according to the figure on p. 530 using columns 3 and 4 of the table on p. 529.
Chapter 7
References
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G. P. Misra, D. Lavabre, J. C. Micheau, Mechanistic investigations and spectrokinetic parameter determination during thermoreversible photochromism with degradation, J. Photochem. Photobiol. A: Chem. 80 (1994) 251. [171] S. Gliesing, M. Reichenbacher, H.-D. Ilge, D. FaBler, Welleniangeneffekte auf die Quantenausbeuten der Photoisomerisierung von PrSvitamin D3, Z. Chem. Leipzig 29 (1989)21. [ 172] D, Rottger, PhD thesis, Hohenheim, 1995. [173] H.-D. Ilge, Bestimmung der UVA^IS-Extinktionskoeffizienten und der partiellen Quantenausbeuten eines Vierkomponentensystems, Z. Phys. Chem. Leipzig 262 (1981)385. [174] Th. Flohr, R. Helbig, H. J. Hoffmann, Rate equation for photochromic glasses considering both thermal and optical regeneration, J. Mater. Sci. 22 (1987) 2058. [175] E. Friz, G. Gauglitz, T. Klink, A. Lorch, Measurements of fast changes in absorption and fluorescence spectra using an OMA-system, Comput. Enhanced Spectrosc. 1 (1983)49. [176] G. Gauglitz, R. Goes, Photokinetic evaluation by microprocessor controlled fluorescence data acquisition, Comput. Enhanced Spectrosc. 2 (1984) 159. [177] M. Guther, Software for parameter estimation of consecutive reactions, Tubingen, 1987. [178] E. Uhlmann, Stilben-1: LdsungsmittelabhSngigkeit der partiellen photochemischen Quantenausbeuten, Masters thesis, Tiibingen, 1990. [179] G. Gauglitz, Absorptions- und fluoreszenzspektroskopische Methoden fOr komplizierte Photoreaktionen, PhD-thesis, Tubingen, 1972. [180] G. Gauglitz, Determination of photochemical quantum yields using fluorescence data, demonstrated by laser dyes. Z. Phys. Chem. (Wiesbaden) N. F. 113 (1978) 217. [181] G. Gauglitz, Spektroskopische Untersuchungen zur PhotostabilitSt, dargestellt an Laserfarbstoffen, Habilitationsschrift, Tubingen, 1979. [182] T. Suratwala, Z. Gardlund, K. Davidson, D. R. Uhlmann, J. Watson, S. Bonilla, N. Peyghambarian, Silylated coumarin dyes in sol-gel hosts. 2. Photostability and sol-gel processing, Chem. Mat. 10 (1998) 199-209 [183] R. Raue, Laser dyes, in: Ullmann's Encyclopedia of Industrial Chemistry, Vol. A 15, VCH, Weinheim, 1990, pp. 151ff. [184] K. Franke, Konzeption und Darstellung neuer Laserfarbstoffe auf Indenbasis und ihre photochemische Untersuchung, PhD thesis, TUbingen, 1989. [185] J. Riedt, Die Abhangigkeit der partiellen photochemischen Quantenausbeuten vom Medium dargestellt an einem Laserfarbstoff, Masters thesis, Tubingen, 1985. [186] G. Gauglitz and W. Schmid, HPLC in photokinetics: determination of reaction mechanism and photochemical quantum yields of (E)-l-lphenylpropene, Chromatographia23(1987)395. [187] G. Gauglitz, T. Klink, W. Schmid, Untersuchung der Photochemie von Polyenen mit Hilfe von mikroprozessor-gesteuerter HPLC, Fresenius* Z. Anal. Chem. 318 (1984) 298. [188] W. Schmid, Erweiterung photokinetischer Untersuchungsmethoden durch mikroprozessorgesteuerte HPLC mit spektraler Detektion, PhD thesis, Tubingen, 1987.
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K. HeiB, Kombinierte spektroskpische und chromatographischc Verfahren zur Untersuchung photochemischer Reaktionen, PhD thesis, Tiibingen, 1989. G. Gauglitz, R. Goes, W. Schmid, Photochemical degradation mechnaism of stilbene1 examined by spectroscopic and chromatographic methods, J. Photochem. 34 (1986) 339. S. WeiB, Faktorenanalyse in der Kinetik, PhD thesis, Tiibingen, 1991. J. R. Hurley, R. B. Cattel, The procrustes program: producing direct rotation to test a hypothesized factor structure, Comp. Behav. Sci. 7 (1962) 258. G. Gauglitz, H.-P. Neubauer, Moglichkeiten in der Reaktionskinetik durch prozeBgesteuerte HPLC, GIT Fachz. Lab. 34 (1990) 786. H.-P. Neubauer, PhD thesis, Tubingen, in preparation. Y. Yokoyama, T. Goto, T. Inoue, M. Yokoyama, Y. Kurita, Fulgides as efficient photochromic compounds. Role of the substituent on furylalkylidene moiety of furylfulgides in the photoreaction, Chem. Lett. (1988) 1049. E. Uhlmann, Photokinetik von Aberchrome 540 in isotroper und anisotroper Umgebung, PhD thesis, Ttibingen, 1995. E. Uhlmann, G. Gauglitz, New aspects in the photokinetics of Aberchrome 540, J. Photochem. Photobiol. A: Chem 98 (1996) 45. H. D. Roth, Chemically induced dynamic nuclear polarization as a tool in mechanistic photochemistry, Mol. Photochem. 5 (1973) 91. G. Gauglitz, G. Kraus, A reflectometric sensor for ammonia and hydrocarbons, Fresenius' J. Anal. Chem. 346 (1993) 572. G. Kraus, G. Gauglitz, Application and comparison of algorithms for the evaluation of interferograms, Fresenius' Z. Anal. Chem. 344 (1992) 153. J. Nagel, Interferenzspektroskopische Methode zur kinetischen Untersuchung von Photoresisten und photochromen Systemen, PhD thesis, Tiibingen, 1995. H.-J. Timpe, A. G. Rajendran, Light-induced polymer and polymerization reactions, Eur. Polym.J.27(1991),77. C. Carre, D. J. Lougnot, J. P. Fouassier, Holography as a tool for mechanistic and kinetic studies of photopolymerization reactions: A theoretical and experimental approach, Macromol. 22 (1989) 791. D. B. Yang, Kinetic studies of photopolymerization using real time FT-IR spectroscopy, J. Polymer Sci. A 31 (1993) 199. H. C. Kessler, The kinetics of light absorption in photobleaching media, J. Phys. Chem. 77 (1967), 2736. J. R. Sheats, J. J. Diamond, J. M. Smith, Photochemistry in strongly absorbing media, J. Phys. Chem. 92 (1988), 4922. W. Nahm, WeiBlichtinterferenz an diinnen Schichten, PhD thesis, Tiibingen. St. Kohlhage, Charakterisierung diinner, geordneter und ungeordneter Siloxanfilme, PhD thesis, Tiibingen. V. Hoffmann, D. Frfthlich, U. Hellstem, S. Kohlhage, W. Nahm, K. NuBbaumer, D. Oelkrug, Spectroscopical examination of photopolymerization, structure and permeability of thin polymer films, J. Mol. Struct. 293 (1993) 253-256.
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H. Mauser, Zur formalen Photokinetik: IV. Der Ablauf einfacher Photoreaktionen in zMhen Medien, Z. Naturforschg. 22b (1967) 569. H. Lege, Charakterisierung und Photochemie azaaromatischer Schichten, PhD thesis, Tubingen, 1991. D. Fr5hlich, Photokinetische Untersuchungen an Azobenzol in fester und fltissiger Matrix bei Laserbestrahlung, Masters thesis, Tiibingen, 1986. M. Guther, Software for Photokinetics in Viscous Media, Tiibingen, 1987. G. Gauglitz, Photophysical, photochemical and photokinetic properties of photochromic systems, in: H. Dtirr, H. Bouas-Laurent (eds.), Photochromism, Elsevier, Amsterdam, 1990. D. Fr6hlich, PhD thesis, Tiibingen, in preparation. D. Fr5hlich, G. Gauglitz, Untersuchung des Reaktionsverhaltens von Azobenzol bei Laserbestrahlung in Festkorpern, Proceed. 12. GDCh-Fachgruppentagung Photochemie, Aachen, 1991.
Subject index
Absorbance, 257, 267, 337 - at wavelength of irradiation, 192 -
decadic, 267
- degree of advancement relationship, 343 - interpolated, 251 - Napierian units, 184, 197, 259 absorbance diagram, see E-diagram - higher order, 266 absorbance-difference diagram, see EDdiagram absorbance-concentration relationship, 197, 267, 340, 524 absorbance-time diagram, 251, 261, 265, 374,391,517,520 - synchronisation, 251, 256, 265 absorption, 244 - index of, 284 absorption coefficient, 19, 267, 275, 281, 337, 344, 388, 392, 396,438,471, 517 - decadic, 19, 184, 259, 268, 270, 374, 397, 522 - Napierian, 19, 74, 182, 198, 259 absorption spectroscopy, 244, 261 absorption spectrum, 281 actinometry - 2,2',4,4'-tetra-isopropylazobenzene, 401 - azobenzene, 399 - chemical, 296 ~ dibenzo[aj]perylen-1,16-dione (HCD), 401 - mesodiphenylhelianthrene (MDH), 404 - Parker's solution, 399 - physical methods, 294 advancement variable, 7 amount of light absorbed, 3, 16, 181,192, 237,293,416 approximation, 118
- in practice, 397 - photostationary state, 410 - photokinetic factor, 200 ATR (attenuated total reflection), 285 automation, 289 auxochromic, 276 Bemoulli-L'Hospital, 92, 311 Bodenstein hypothesis, 4, 40, 122, 126, 129, 142, 153, 156, 158, 183, 302,477 bolometer, 294 brutto equation, 50 Calibration, 290 cell compartment - absorption measurement, 249, 253, 255, 280 - fluorescence measurement, 279, 280 chemically induced dynamic nuclear polarisation (CIDNP), 291,456 chemometrics, 453 chromatographic reaction spectrum, 444 chromatography, 286 - high performance liquid (HPLC), 286, 289 - thin-layer, 281, 283, 286, 287 chromophore, 296 circular dichroism (CD), 291 coefficients - stoichiometric, 26,473 - stoichiometric, matrix of reduced, 66, 485 - stoichiometric, matrix representation, 28,475 - stoichiometric, rank of matrix, 44 - stoichiometric, reduction of, 32, 143, 475,483,488
546
Subject index
combination - absorbance and chromatography, 443 - absorbance and fluorescence measurements, 439 - absorbance-NMR, 456 - absorbance-reflectance, 457 combination of thermal and photochemical reactions, 357 concentration - at end of reaction, 63, 67 - final. 67, 141 - linear independent, 67 - reduced, consecutive reaction, 141 - reduced, Michaelis-Menten, 130 - reduced, photoaddition, chemically sensitised, 194 - reduced, photoaddition, physically sensitised, 195 - reduced, radical chain reaction, 127 concentration diagram, see K-diagram concentration measure, 338 concentration measurement, 327 concentration-difference diagram, 303 concentration-difference-quotient diagram. see KDQ-diagrams concentration-time curve, 290,451 - general coordinates, 62 concentration-time diagram, 87,442, 454 - extrema, 87 ~ consecutive reaction, 112, 525 - points of inflection, 87 concentration-time measurements, 322 consecutive photoreaction, 350 conservation of mass, 25 correction factor - fluorescence intensity, 430 curve-fitting techniques, 261,454 Definition of symbols, 495 degree of advancement, 6,7,27,66,73,75, 78, 85,198,203,215, 300,343,483,487 - absorbance relationship, 343 - consecutive photoreaction, 186, 523 - consecutive reaction, 122
-
consecutive, photochemical and thermal, 212 - parallel reaction, 106 - photoisomerisation, reversible, 208 - photoisomerisation, superimposed thermal backward, 211 - radical chain reaction, 127 - reduced, 120.203 - reduced parallel photoadditions, 172 - reduced, consecutive photoreaction. 187.234.240 - reduced, consecutive photoreactions. viscous medium, 234,239 ~ reduced, parallel mixed photoreactions. 189 - reduced, parallel photoadditions, 169, 172 - reduced, parallel photoreactions, different initial products, 188 - reduced, parallel photoreactions, including photoaddition. 165 - reduced, photoisomerisation, 239 - reduced, photoisomerisation physically sensitised, 206 - reduced, photoreactions. survey on mechanisms. 216 - time-dependence, 85 derivative spectra, 275 derivative spectroscopy, 273 detector, 295 determination of intensity of irradiation, - chemical method, 296 - physical method, 294 device - combined irradiation and measurement. 250 difference equations. 316.330. 371 - higher order. 319,331.373 differential equation. 89.118.140. 242. 522 - Bernoulli. 219.462 - complex, consecutive, 134 - complex, numerical integration. 132 - consecutive, fluorescence, 426
Subject index -
consecutive, in part sensitised by radicals, 214 - consecutive, photochemical and thermal, 212 - consecutive, viscous medium, 234 - eigenvalues, 89 - integration of. see integrals - partial, viscous medium, 219 - photoaddiation, general coordinates, 205 - photoaddition, 380 - photoaddition, chemically sensitised, 227 - photoisomerisation, 374 - photoisomerisation, physically sensitised, 315 - photoisomerisation, superimposed thermal backward, 211 - photoisomerisation, viscous medium, 224 - photoreactions, 187 ~ photoreactions, radical chains, 230 - photoreactions, survey on mechanisms, 216,240 - transformation of time, 214 - uniform photoreactions, 181 differentiation, 274 diffusion, 468 diffusion controlled, 12, 14, 149 dilution, 290 - influence of, 96, 450, 519 discrimination of mechanism, 393 E-diagram, 265, 266, 364, 395, 416,457, 517,520 - consecutive photoreaction, 368 ~ final points, 365 -- Gibbs-triangle, 369 - higher order, 265 - uniform reaction, 365 ED-diagram, 348, 352, 375,415, 521 - consecutive photoreaction, 354, 362 - dependence on wavelength and intensity, 356
547
- final slopes, 354 ~ initial slope, 352, 358 - photoisomerisation, 358, 376 - points of inclination, 353 ~ reduced, 355 EDQ(s)-diagram - of higher order, 363 EDQ-diagram, 357, 359, 360, 521 - azobenzene, 391 ~ consecutive photoreaction, 362 eigenvalues, 89, 124, 322, 392, 508 - complex, 83 ~ of differential equation, 89, 435 - equal, 81 Einstein, 8 energy transfer, 146, 512 equation - brutto, 50 - differential, 37, 41, 55, 65, 74 - differential, set-up of, 38 - differential, solution of, 75 - matrix, 440 - normal, 326 - photokinetic-thermal, comparison, 397 equilibrium - approximation of, 137 error ~ approximation, 125, 133 - in absorbance measurement, 261 ESR spectroscopy, 243 evanescent fields, 285 excitation spectra, 280 exponential integral equation, 226 extinction, 20, 259, 267, 337 extrema in concentration-time diagrams, 87 Factor analysis, 273,451 factor of 1000, 182 Pick's law, 468 field - evanescent, 285 flash photolysis, 263
548
Subject index
fluorescence, 11. 244, 257, 275, 277, 287, 416,440 - photokinetic equations, 418 fluorescence excitation spectra, 280 fluorescence quantum yield, 276,278, 288 fluorescence-concentration relationship, 416 fluorescence-time curves, 427 fluorophore, 276,418 formal integration, 4, 324, 332, 374, 381, 382, 386, 451, 455, 517, 523. see integrals - azobenzene, 383 - fluorescence measurements, 436 - photoreaction, consecutive, 326 formal kinetics - photoisomerisation, thermal backward, 407 Fourier analysis, 291,451 Fresnel, 284 fuzzy logic, 272 General coordinates, 57, 60, 61, 119, 369 - concentration-time curves, 62 - consecutive photoreactions, viscous medium, 234 - uniform photoreactions, 204 geometric effects, 417 Gibbs diagram, 369 Hadamard, 270 half-life, 61, 399 - first order reaction, 58 - second order reaction, 59 - zero order reaction, 57 heterocoerdianthrone, 401 high performance liquid chromatography (HPLC), 327 Index of absorption, 284 influence of dilution, 519 integrals, 315 - complex non-uniform photoreactions, 210
-
complex uniform photoreactions, 208 consecutive photoreaction, 210 consecutive photoreaction, physically sensitised, 209 - consecutive photoreaction, viscous medium, 235,240 - fluorescence measurement, 422 - formal, 242 - including photokinetic factor, 199 - numerical, 242 - numerical, not in closed form, 231 - parallel photoreaction including photoaddition, 209 - photoaddition, chemically sensitised, 228 - photoaddition, physically sesitised, 195 - photoisomerisation, 196, 324 - photoisomerisation, reversible, 208 - photoisomerisation, sensitised by radicals, 187, 193,215 -- photoisomerisation, superimposed thermal backward, 211 - photoisomerisation, viscous medium, 232, 239,240 - photoreaction, chemically sensitised, 194 - photoreactions in viscous medium, 221 - photoreactions, radical chain, 192, 207, 229 - photoreactions, simple uniform, 191 - simple non-uniform photoreaction, 208 - simple uniform photoreaction, 192, 204 - transformation of time, 214 integration - foxmdX.see integrals - numerical, see integrals - Runge-Kutta, 211 - using Simpson* method, 205,231 intensity diagrams, 423,427,436 intensity of irradiation, 293 intensity quotient diagram, 429 intensity-time diagrams, 418 interference spectrum, 459
Subject index intermediates, 118, 123, 126, 132, 142, 176,301 - of radical chain reaction, 143 ~ photoaddition, radicals, 153 intermediate radicals, 158 internal conversion, 11 intersystem crossing, 12, 276 IR spectroscopy, 243, 291 irradiance, 17 irradiation, 250, 293 - process-controlled, 254 - requirements, 245 - set-up combined with measurement device, 250 irradiation measurement - chemical method, 296 ~ physical method, 294 isochronic, 255 isoenergetic, 9 isosbestic point, 196, 200, 264, 266, 345, 384, 466 - requirements, 347 Jablonski-term diagram, 9 Jacobi matrix, 5, 72, 75, 84, 85, 99, 145, 241,334,388,392,396,517 - complex photoreaction, 116 - consecutive reaction, 111 - eigenvalues, 75, 78 - eigenvectors, 78 - isomerisation, 105 - main minors, 77 - parallel reaction, 104 - photochemical reactions, 74, 186 - thermal reactions, 74 Kalman filter, 272,454 K-diagram, 78, 91, 304, 329, 517 - areas, 96 - consecutive, 108,305 - dependence on intensity, 307 - dependence on wavelength, 307 - extrema, 87, 96 - final point, 92
549
- final slope, 94, 308, 312 - influence of dilution, 96 - initial point, 92 - initial slope, 92, 308, 311 - isomerisation, 305 - parallel reaction, 106 - points of inflection, 87, 98 - reduced, 96 KDQ(3)-diagram,314 KDQ-diagram, 308, 329 - bimolecular, parallel, 309 - consecutive photoreaction, 311, 362 - consecutive, multi-step, 313 kinetic analysis, 118, 243, 324, 334, 393, 523 Kramer-Kronigs relationship, 460 Kronecker and Capelli theorem, 360 Kubelka-Munk,281 Lambert cosine-law, 282 Lambert-Beer law, 19,197, 258, 337,464 - deviations, 259 - restrictions, 259 Lanczos-bidiagonalisation, 271 laser dye, 288,422 least squares approximation, 270 lifetime - excited state, 14 linear equations - inhomogeneous, 447 linear regression, 322 linear systems ~ transformation, 99 Mac Laurin series, 199 MAPLE - use of, 475 matrix - concentration-differences, 301 - elements for formal integration, 388 - inversion, 269 - normal, 269 - reduced, 68 - regular, 67
550
Subject index
- variance-covariancc, 271 matrix rank analysis, 300, 344 - graphical, 328 M-diagram, 328 MDQ-diagram, 329 measurement - asynchronous, 251 - fluorescence, 275 - synchronous, 251, 256, 265 measurement of intensity of irradiation - chemical method, 296 -- physical method, 294 mechanism, 299, 392, 393, 517, 522 micelles, 262 Michaelis-Menten, 128, 194 microprocessors, 292 microwave spectroscopy, 243 monochromator, 245, 247 - dual arrangement, 281 muUicomponent analysis, 243, 267 multi-linear regression, 270 Napierian units, 259, 374,429 neural networks, 272 NMR spectroscopy, 243, 291 normal equations, 338 numerical integration, 517 Optical rotational dispersion (ORD), 291 order of reaction, 55 oxygen - in air, 245 Parker's solution, 296 partial least squares (PLS), 271 partial reaction, 31, 65, 102 pattern recognition principles, 273 personal computers, 292 phosphorescence, 12 photo diode, 294 photo diode array, 248,256, 257, 285,445 photochemical reaction, 51,280, 291 - 1-hydroxy-benzotriazole, 457 - 2,5-distyrylpyrazine (DSP), 462
-
-
4,4'-distyrylazobenzenes, 411 addition, 149, 168, 324, 328, 341, 378 addition, chemically sensitised, 156, 194 addition, chemically sensitised, viscous medium, 227 addition, physically sensitised, 194, 206,513 addition, radical chain, 159 addition, radical intermediate, 153 addition, via singlet, 150, 510 addition, via triplet, 150, 205, 510 anthrone, 422 azobenzene, 464 Carbol,435 chemically sensitised, 156 cinnamic alcohol, 446 comparison stirred solution, viscous medium, 241 complex, comparison parallel, consecutive, 191 complex, non-uniform, 178 complex, uniform, 177 consecutive, 180, 186, 267, 307, 311, 318, 323, 326, 330, 333, 344, 352, 354, 369,388,517 consecutive and parallel, 301, 329, 334, 361 consecutive, fluorescence, 424 consecutive, in part sensitised by radicals, 214 consecutive, multi step, 313 consecutive, photochemical and thermal, 212 consecutive, physically sensitised, 176, 209,516 consecutive, reduced X-diagram, 355, 369 consecutive, viscous medium, 234 coumarin dyes, 446 coumarin-derivatives, 423 C-S cyclophane, 415 dihydroindolizine, 405,408 diphenyloctatetraene, 446
Subject index diphenylpyrazines, 453 fast, 251 fulgides,405,454,455,471 isomerisation, 14, 239, 374 isomerisation via singlet state, 146 isomerisation via triplet state, 146, 178, 477 isomerisation, oxindigo, 385 isomeriation, physically sensitised, 205, 329, 377 isomerisation, reversible, 208, 380, 480 isomerisation, reversible via triplet, 178 isomerisation, reversible, thermal consecutive, 178 isomerisation, sensitised by radicals, 193, 157,207 isomerisation, superimposed thermal backward, 210,405,478,481 isomerisation, thermal backward, 178 isomerisation, viscous medium, 219, 224,231,238 liquid crystals, 471 mechanisms, distiction of, 334 mechanisms, distinction of, 266, 392, 393 non-uniform, 163 parallel, 160, 163, 178, 267, 334, 353 parallel mixed mechanisms, 179 parallel photoadditions, 171 parallel via singlet and triplet, 162 parallel with different partners, 178 parallel, additional partner, 164 parallel, both photoaddiations, 168 parallel, different initial products, 188 parallel, including photoaddition, 165, 209 parallel, physically sensitised, 175, 515 parallel, via singlet, 161 parallel, via triplet, 162 p-distyryl-benzene, 453 physically sensitised, 154, 194, 511 polymerisation, 462 quasi linear, 215, 316, 330, 332, 386, 392
551
- quenched, 146 - radical, 158, 184 - radical chain, 159, 207 - radical chain, viscous medium, 119 - reduction, 152 - reduction, radical intermediates, 153 - sensitised, 154 - stilbene,275,445,517 - stilbene-1,422,441 - superimposed thermal, 255, 357 - survey on mechanisms, 216 - trans- to cw-stilbene, 275,445, 517 - umbelliferone, 437 - uniform, 145, 181, 197 - viscous media, 218,462,466 photochemical reactor - assumptions, 183 photochromism, 414 photo-degradation, 288 photodimerisation, 14. see photochemical reaction photo-effect - external, 294 photoisomerisation, 14,40, 51 - azobenzene, 349, 381, 389, 399 ~ quenched, 146 - quenched, via singlet, 146 - quenched, via triplet, 146 - reversible, azobenzene, 345 - trans- to cis stilbene, 263, 344, 350, 360 photokinetic factor, 19, 20, 192, 241, 374, 408,417 - approximation, 200, 398,428 - approximation at total absorption, 201 - approximation by interpolation, 203 ~ approximation using error functions, 202 - approximation using Taylor series, 202 - interpolation, 400 photokinetics, 249, 257, 516 - general approach, 3 - requirements, 242 photokinetic-thermal equations, 397 photometry, 243, 247, 290
552
Subject index
photophysical steps - photoisomerisation via singlet state, 146 - -photoisomerisation via triplet state, quenched, 148 photophysics, 7,417 photopolymerisation, 462 photoreaction. see , photochemical reaction photoresist, 284, 285,460 photostability, 283 ~ of laser dyes, 431,440 photostationary state, 410 pivoting, 270 pK values, 277 points - isosbestic, 266 - quasi-isosbestic, 264 points of inflection, 98 polychromator, 248 post steady state, 132 potential hyper surfaces, 276 predissociation, 276 principle component analysis, 241, 273 principle component regression (PCR), 271 process - chemical primary, 13 ~ diffusion controlled, 12 - photophysical, 40, 275 process control, 292 product - final, 90 - initial, 90 proteins, 262 Quantum yield, 15, 35, 293, 296, 331 - apparent differential, 16 - apparent integral, 15 - chemcially sensitised, 157 - consecutive photoreactions, 181, 187, 312,323,353,391 - consecutive photoreactions, viscous medium, 235, 239 - consecutive photoreactions, physically sensitised, 177 - dependent on irradiation, 196
~ -
dependent on amount of light absorbed, 184,223 depending on viscosity of solvent, 441 independent of amount of light absorbed, 182,222 independent of irradiation, 196 laser dyes, 436 liquid crystals, 471 parallel photoadditions, 169,173 parallel photoreaction via singlet, 161, 162 parallel photoreaction via singlet and triplet, 163 parallel photoreactions with mixed mechanisms, 180 parallel photoreactions, both photoadditions, 172 parallel photoreactions, including photoaddition, 165 parallel photoreactions, physically sensitised, 176 partial, 236,461, 527 partial photochemical, 4, 36, 165,421, 481 partial reactions, 438 photoaddition, 168 photoaddition via singlet, 150, 378 photoaddition via singlet, quenched, 151 photoaddition via triplet, 151, 324 photoaddition via triplet, quenched, 152 photoaddition, chemically sensitised, 157, 194 photoaddition, chemically sensitised, viscous medium, 227 photoaddition, radical chain, 160 photoisomerisation, 231,374, 377,385 photoisomerisation via singlet, 43, 238, 380 photoisomerisation via singlet, quenched, 147 photoisomerisation via triplet, 43, 238 photoisomerisation via triplet, quenched, 148
Subject index -
photoisomerisation, physically sensitised, 194, 206 - photoisomerisation, reversible via triplet, 178,^81 - photoisomerisation, sensitised by radicals, 158, 193 - photoisomerisation, thermal backward, 178 ~ photoisomerisation, viscous medium, 218,224,231 - photoreaction, radical, 185 - photoreduction, radical intermediates, 153 - radical chain, viscous medium, 229 - stilbene-1,443 - trans- to ci5-stilbene, 451 - true differential, 36, 323 ~ true integral, 16 - using chromatographic data, 447 quasi-isosbestic points, 264 quasi-stationary variables, 130 Rate - chemically sensitised photochemical reaction, 156 - consecutive phororeactions, 181, 186, 187, 240 - consecutive photoreactions, physically sensitised, 177 - equation, 102 - non-uniform photoreaction, 186 - parallel mixed photoreactions, 189 -- parallel photoreaction via singlet, 161 ~ parallel photoreaction via singlet and triplet, 163 - parallel photoreaction via triplet, 162 - parallel photoreactions with different partners, 179 - parallel photoreactions with mixed mechanisms, 180 - parallel photoreactions, both photoadditions, 169, 172 - parallel photoreactions, different intial products, 188
-
553
parallel photoreactions, including photoaddition, 165 - parallel photoreactions, physically sensitised, 176 - photoaddition, radical chain, 160 - photoaddition, radical intermediates, 153 ~ photoaddition, via singlet, 150 - photoaddition, via singlet, quenched, 152 - photoaddition, via triplet, 151 - photoaddition, via triplet, quenched, 152 - photoisomerisation via singlet, 240, 241 - photoisomerisation via triplet, 241 - photoisomerisation, radical intermediate, 159 - photoisomerisation, reversible via triplet, 178 - photoisomerisation, sensitised by radicals, 158 - photoisomerisation, thermal backward, 178 - thermal reaction, 67 - total, 6 rate constant - bimolecular, 13 - consecutive reaction, 68 - matrix of, 66 rate equations - consecutive, 425, 523 rate laws, 118,517 - concentrations, 397 - general, 55 - number of exponential functions, 89 - photokinetic, 40, 75, 523 - quasi-stationary state, 132 - thermal, 37, 45 rate of reaction, 7, 60 reactants - non-reduced set, 28 - reduction of, 26 reaction - addition, 341
554 -
Subject index
back- and forward, 105 bimolecular, 13, 14 catalytic, 39 complex, thermal, 139 consecutive and parallel, 301 consecutive, 47,68, 84,95,107, 118, 119, 307, 323, 482,492,496,497, 499, 501,508 - consecutive, multi step, 313 - cyclic mechanism, 47,483, 504 - diffusion controlled, 12,14 - first order, 57 - isomerisation, 105 - multi-step, 27 - order, 55 - parallel, 49, 103,486 - partial, 31, 65, 102,313 - photochemical and thermal superimposed, 251, 256, 357 - photochemical consecutive, 74 - photochemical, 3,51, 244. See photochemical reaction - photoisomerisation and superimposed thermal, 51 - photoisomerisation, reversible, 52 - photoisomeristaion, reversible, and consecutive thermal, 52 - radical chain, 126, 143 - reversible and consecutive, thermal, 137 - reversible, thermal and consecutive, 120 - reversible, thermal, 45 - second order, 58, 60,488 - thermal, 3, 251 - uniform, 45, 103, 304 - zero order, 57 reaction chromatogram, 444,445,450 reaction rate, 7 reaction scheme, 27, 38,41,143, 241,473, 483, 522 - reduced, 67, 143 - reduced, complex reaction, 115 reaction spectrum, 263, 264, 345,438, 456, 516,518
- derivative, 275 - fluorescence, 434 reaction steps - linear independent, number of, 4,44, 142,241,266 - spectroscopically linear independent, number of, 4 reaction time, 393 recursion equation, 317, 373 - higher order, 321,331 reduction - of reactants, 32 - of reaction steps, 46 reflectance, 244,257,281 - diffuse, 281 - regular, 283 - specular, 281 reflection, 287 reflection spectroscopy, 281, 284 reflections - multiple, 262 reflectometric interference spectroscopy (RlfS), 458 reflectometry, 282, 284 refractive indices, 261 regression - multi-linear. 270,440 Reinecke's salt, 296 residuals, 261 Runge-Kutta algorithm, 213,435,, 451, 517 Scattering, 263, 277 sensors - chemical, 284 singlet, 10 solutions - turbid. 262 spectrometer - double beam, 246, 254 - fluorescence, 277 - photo diode array, 249,256, 257 - reflectance, 283 - reflection, 285
Subject index - requirements, 245 - simultaneous, 257 ~ single beam, 246 spectroscopic data, 78 spectroscopically linear independent steps of reaction, 4 spectroscopically uniform, 265 spectroscopy, 291 - absorption, 277 - multiplex, 247 - reflection, 281 - sequential, 245, 249 steps of reaction. See reaction steps stoichiometry, 25 stray coefficients, 281 stray light, 260 symbols - definition of, 495 Target transformation, 273 Taylor series, 202 - expansion, 132 temperature control, 252 transformation, 350, 387, 396 transformation of linear systems, 99 transformation of time, 4, 214, 229,238, 315,371,439,466,517 transition - radiationless, 11, 276 - thermal relaxation, 11 triplet, 11,276 turnover, 7 Uniform, 265, 347 - complex, 45 - photoreactions, 204, 380 - simple, 45 uniform reaction, 45, 103, 119, 364, 374 - parallel, 103
555
UV spectroscopy, 243 UVA^is-spectrometer, 292 UV/Vis spectroscopy, 244, 250, 267, 290, 292, 296 Vandermond - determinant, 80, 81 - matrix, 79, 86 vector - reduced for concentrations, 68 - stoichiometric, 26 - transposed, 26 Vieta theorem, 77 Vis spectroscopy, 243 viscosity, 14 - of solvent, 441 viscous medium - general solution of differential equations, 222 visible spectroscopy, 243 X-diagram, 91 - consecutive, 108 - consecutive photoreactions, physically sensitised, 177 - consecutive reaction, 110 - consecutive reaction, slopes, 110 - parallel reaction, 106 - reduced, 106,355,369 ~ reduced, consecutive photoreactions, 188,355 - reduced, consecutive reaction, 113 - reduced, final slopes, 174 - reduced, mixed parallel photoreactions, 190 - reduced, parallel photoreaction, 168, 170, 173 - reduced, parallel photoreactions, physically sensitised, 176
