The mechanisms of fast reactions in solution

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The Mechanisms of Fast Reactions in Solution by

E.E Caldin MA, D .Phil. Late Professor of Physical Chemistry University of Kent, Canterbury, United Kingdom

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lOS Press

:~iwi Ohmsha

Amsterdam • Berlin • Oxford.

Tokyo. Washington, De

© 2001, H.P. Caldin All rights reserved. No part of this book may be reprodueed, stored in a retrieval system, or transmitted, in any form or by any means, without the prior written permission from the publisher. ISBN 1 58603 103 1 (lOS Press) ISBN 4 274 90409 1 C 3043 (Ohmsha) Library of Congress Catalog Card Number: 00-109666 Publisher lOS Press Nieuwe Hemweg 6B 1013 BG Amsterdam The Netherlands fax: +31206203419 e-mail: [email protected]

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v

Preface Professor Edward Caldin, Emeritus Professor of Chemistry at the University of Kent at Canterbury, died peacefully in December 1999. Ted, as he was universally known, was bom in August 1914. He won scholarships to St Paul's School, and to study as an undergraduate at The Queen's College, Oxford, where he gained a reputation for his boxing prowess. He was one of the first students of R.P. (Ronnie) Bell, the distinguished Oxford physical chemist. His thesis was concemed with the mechanism of decomposition of nitramide, and experimental proof of the concept of non-classical proton transfer, or 'proton tunnelling' , whereby the wave properties of the proton could enable it to move from an acid to a base with less than the classical activation energy. Such tunnelling was predicted to be more evident at low temperatures. His developing academic career was interrupted by the advent of World War 11. Ted spent the war in various military research establishments, particularly in South Wales, working on pyrotechnic compounds for flares and tracer bullets, which, with characteristic deprecating humour, he referred to as 'fireworks'. It was there that he met his future wife, Mary, with whom he had a long and very happy marriage. He also found the time to research material for a book on Chemical Thermodynamics, published by Oxford University Press in 1958. After the war, Ted was appointed to a Lectureship in Chemistry at the University of Leeds. There was asevere shortage of research materials - an entire physical chemistry undergraduate practical course had to be devised with sodium sulphate as the sole compound under investigation. Ted at this time enjoyed the challenge of doing research with minimal resources, and he was a master of the 'string and sealing wax' school, which he had leamed from his mentor Ronnie Bel!. One Leeds research student, Geoffrey (now Sir Geoffrey) Allen tells how he studied the thermodynamics of hydrogen-bonding through carboxylic acid dimerisation equipped only with a carefully calibrated thermometer and some benzene. In the 1950's, Ted resumed his experimental studies on proton tunnelling. He realised the need to study reactions over a wide temperature range in order to detect deviations from the Arrhenius equation. Accordingly, he devised equipment to study reactions at very low temperatures (~ - 100°C) in non-aqueous media. This need to study rates over a wide temperature range led him into the study of fast reactions, and he was a pioneer in the development of stopped-flow equipment in the 1950's to study diffusion-controlled reactions. At Leeds, he also developed a novel micro-wave temperature-jump apparatus for studying a range of reactions in non-aqueous solvents. He taught a postgraduate course on fast reaction techniques at Leeds and this gave rise to his classic text on "Fast Reactions in Solution", published in 1964. He spent his retirement years on a sequel to this book, "The Mechanisms of Fast Reactions in Solution", which was finished just before he died. In 1965, he moved to the newly founded University of Kent at Canterbury, first as Reader, and then as Professor of Physical Chemistry. The University appealed to Ted, as it was intended to be mn in the Oxbridge tradition, but this was not to be. The profound social changes of the late 1960's and early 1970's were not really to his taste. At Canterbury, Ted masterminded the activities of a group interested in the rates and mechanisms of fast reactions in solution. A number of distinguished visitors spent

Preface

vi

sabbaticals with the group, and Ted was the prime mover in setting up the very successful series of Royal Society of Chemistry meeting s on mechanisms of reactions in solution, the first of which took place in 1970, the last one being three years ago. He was also the First Chairman of the Fast Reactions in Solution (FRIS) Discussion Group, when it was founded in 1976, and he regularly attended its meeting s even when well into his eighties. In addition to his commitment to chemistry, Ted had a lifelong interest in the history of science and the link s between science and religion. He gave an unforgettable 'open' lecture on the former topic in the 1970's. Ted was a devout Catholic who insisted on the fundamentally rational nature of Christianity. Among his philosophical writings are to be found "The Power and Limits of Science" (1949) and "Science and the Christian Apologetic" (1953), as wel1 as numerous articles. Ted was a scholar and philosopher in the old academic tradition. His breadth of other interests, such as music, art, literature and history was vast, as evidenced by his library, and provided him with an apparently limitless source of reflection and humour; Milton and Samuel Johnson may have been among his favourites, but he was equal1y as likely to quote lines from Lewis Carrol1 and A.A. Milne. A research student who lodged in his house was amazed to find a set of the works of Thomas Aquinas, careful1y annotated in the margin in Ted's characteristic minute scrawling hand. Ted always took his pastoral duties very seriously, and was always concemed for the welfare of his students. He gave much time to university activities, especial1y in his later years, and he carried out these duties with conviction and enthusiasm. Mary Caldin died in 1997 after a long illness. They leave two sons, Hugh and Giles, to whom we extend our deepest sympathy.

Brian Robinson

John Crooks

vii

Foreword Ted Caldin read chemistry at The Queen's College, Oxford and started his research career under the supervision of R.p. (Ronnie) Bell. This was his introduction to fast reactions, for at that time Bell was preoccupied by the recently predicted isotope effect on proton exchange. Later, during World War 11,Ted was assigned to work on explosives in Ministry of Supply laboratories. In 1945 a lectureship at Leeds University brought him back to academic research and to teaching. The physical and inorganic department was led by the charismatic and forwardthinking M.G. Evans who was bent on creating a modem chemistry course. Ted's role was to follow M.G.'s all-pervading first year 'introductory' undergraduate course with a secondyear course on Chemical Thermodynamics - not an easy task. He was a shy man with a waspish sense of humour, his slight frame topped with a shock of black hair, who bustled fram laboratory to lecture theatre in a nervous frenzy. He taught thermodynamics with an infectious enthusiasm for the originalliterature and scathing criticism of the textbooks of the day. Above all, he maintained the momentum established by the suave head of the department and was equally popular with the undergraduates. Ted Caldin's interests ranged more widely than chemistry. Philosophy and especially its interface with the scientific method attracted him. He lectured in the Department of Philosophy and in 1949 published "The Power and Limits of Science". Extramural sessions at his home were par for the course. His research programme was really launched in 1949 with two projects. One was on the colligative praperties of non-aqueous solutions (carbox ylic acids in benzene) with pressure and temperature as variables. The other was the seed com for studies of fast reactions. Acid-base reactions involving a coloured reactant or product were followed by measuring the time-dependence of the UV spectrum of the mixture. In order to slow down the reaction, measurements were made at around 193 K. The reactors used were of 0.5 to 1.0 litre capacity with 10 cm vacuum-sealed double windows to prevent misting. So began Ted's magnum opus.

Sir Geoffrey Allen, Chancellor ofthe University of East Anglia

viii

Acknowledgements Grateful acknowledgements are due to the authors, editors, and publishers concemed for permission to reproduce certain figures and tables. References are given in the texto It is a pleasure too for Ted's family to thank those who have contributed so much to this book: Janet Pitcher, who patiently typed and retyped the manuscript; Peter Brown and his colleagues at lOS Press; Ted's colleagues who assisted with general research, in particular Ceri Gibson, and those who kindly read and commented on the text, in whole or in parto Grateful thanks go to friends such as Dom Cyprian Stockford OSB, Maurice and Joan Crosland and Derek and Christiana Crabtree, who provided so much support and encouragement throughout the writing process. Special thanks are due to Professor Brian Robinson who so generously and selflessly undertook almost the entire burden of producing the book after Ted's death. Finally, this book is published in memory of Ted and Mary themselves.

The Estate of Professor E.F. Caldin

ix

Contents Preface Foreword Acknowledgements Prologue

v vii viii xi

PartOne Chapter 1. Introduction: Origins, Methods, Mechanisms, Rate Constants 1.1. Introduction 1.2. The structure of this book 1.3. Theory of rate constants for diffusion-controlled reactions References

3 10 12 18

Chapter 2. The Rates of Diffusion-Controlled Reactions 2.1. Introduction 2.2. Application of the diffusion law to rates of encounter and of chemical reaction: the Smoluchowski equation 2.3. A molecular model for translational diffusion and diffusion-influenced rates of reaction: random-walk theory 2.4. Encounter in solution: the solvent 'cage' 2.5. The course of an encounter 2.6. Alternative theoretical approaches: refinement of theory References

31 33 36 45 48

Chapter 3. The Mathematical Theory of Diffusion and Diffusion-Controlled Reaction Rates 3.1. Introduction 3.2. Theory of rates of diffusion, in terms of concentration gradients 3.3. Calculation of diffusion-controlled rate constant 3.4. Diffusion coefficients: theory and experiment References

51 54 59 65 79

21 21

Chapter 4. Flash Photolysis Techniques 4.1. Introduction 4.2. Experimental flash techniques 4.3. Some applications of flash techniques References

81 85 98 115

Chapter 5. Initiation by High-Energy Radiation: Pulse Radiolysis 5.1. Introduction 5.2. Experimental techniques

119 122

x

Contents

5.3. Some applications of pulse radiolysis References

125 136

Part Two Chapter

6. Fluorescence Quenching and Energy- Transfer from Excited Molecules 6.1. General principIes 6.2. Theory of fluorescence quenching 6.3. Experimental investigation of fluorescence quenching 6.4. Kinetic and mechanistic applications of fluorescence measurements 6.5. Physical mechanisms for non-radiative energy transfer between molecules References

Chapter 7. DUrafast Processes: Sub·Picosecond and Femtosecond Techniques 7.1. Introduction 7.2. Femtosecond studies of the entire reaction path 7.3. Some systems studied by sub-picosecond and femtosecond techniques References

141 142 151 156 167 187

191 192 197 221

Part Three Chapter

8. Proton· Transfer and Group- Transfer Reactons in Solution: Marcus Theory (1) 8.1. Marcus theory. Introduction 8.2. Application of Marcus theory to proton-transfer reactions in solution 8.3. Application of Marcus theory to group transfer 8.4. Refinements and extensions of the Marcus treatment References

227 228 256 261 261

Chapter 9. Electron- Transfer Reactions: Marcus Theory (11) 9.1. Introduction I. Electron-transfer in outer-sphere reactions of metal ions 9.2 11.Electron-transfer reactions in organic systems 9.3. Tailpiece: a brief re-capitulation References

265 266 292 292 317 318

Epilogue

323

List of symbols used frequently Subject Index

in the Marcus theory of reactions

325 327

xi

Prologue Mechanisms of Fast Reactions in Solution This book is a sequel to the author's Fast Reactions in Solution (1964). It is not a revised edition of that text, but an entirely new book, with a different approach. Fast-reaction techniques are now well established and widely used, and it is timely to consider what they contribute to the study of reaction mechanisms, i.e., to our understanding of events on the molecular scale during reactions. They have in fact transformed our notion of what can be expected in mechanistic explanations. Experiment and theory alike have shown great progress. On the experimental side, techniques of monitoring and recording the course of chemical reactions have proliferated and become more sensitive and more accurate. Laser devices capable of producing very short light-pulses have led to remarkable advances. It has be come possible to detect and record the motions of atoms and electrons in reacting systems at intervals of les s than a million-millionth of a second, thus making possible direct monitoring ofbond-stretching and bond-formation. On the theoretical side, several interrelated developments call for remark. Moleculardynamics calculation (1inked to high-speed computers) have made possible the rapid testing of molecular models of the course of fast reactions; often it tums out that a simple classical model gives a reasonable approximation to the full quantum-mechanical calculation. Such models can be derived from 'Marcus theory', which is, like transition-state theory, a general framework relating reaction rates to energetics; it takes systematic account of the solvent reorganisation due to the changes of electron distribution required by reaction. Advances in spectroscopy have led to a much fuller knowledge of reaction energetics. These interlocking developments have greatly improved and facilitated the theoretical understanding of the kinetic data produced by experimentalists, who in tum have been enabled to devise new experiments likely to produce other important results. A formidable alliance of experimental and theoretical methods has thus developed. The course of events at molecular level for a considerable number of diverse types of reaction, can now be described in previously unheard-of detail. Our conception of chemical understanding has been enlarged. This book is intended to present these advances to the attention of a wider audience.

Part One

3

Chapter 1

Introduction: Origins, Methods, Mechanisms, Rate Constants 1.1. Introduction 1.1.1. Whatdoes

'fast' mean?

The term 'fast reaction' is not a precise one, but it is non e the less serviceable. It is commonly used to mean, broadly, a reaction that is more than half completed within a time comparable with that required to initiate a reaction and make a measurement of its progress. For reactions in solution, initiation is often done by mixing two reactant solutions, an operation which takes a few seconds unless special devices are used. If the course of reaction is monitored by withdrawing samples for analysis, or by taking readings of some kind, each such operation will add a few more seconds. Consequently, when these 'conventional' methods are used, accurate rate constants cannot be determined for reactions with half-times much less than 10 seconds. An individual reaction which is half completed in one second might, in a rough-and-ready way, be taken as marking the borderline between 'fast' and 'conventional' rates. But the observed rate will depend on the experimental conditions; a second-order reaction may have a high rate constant and yet take place comparatively slowly if the concentrations are small enough, or if the temperature is lowered. The term 'fast reaction' is therefore commonly used to inelude reactions which would be too fast for 'conventional' methods if conducted at 'ordinary' temperatures and concentrations. These are not precise definitions; they serve only to indicate, in a preliminary way, the range of rates for which the methods described in this book are required. Most of the fast-reaction techniques can measure half-times down to 10-7 s, several of them to 10-9 sor below, and a few to 10-12 sor below. The range of first-order rate constants amenable to these techniques is thus from about 1 s-1 to well above 109 s-I, so the accessible range has been extended by over ten powers of ten (see Figure 1.1).1 1.1.2. Origins The development of fast-reaction studies may be dated from investigations begun in the 1920's on bimolecular reactions using flow techniques. Reaction was initiated by rapidly 1 The time for half-change t1/2 of a first-order reaction is related to lhe first-order rate constant k by ktl/2 = In2 "" 0.7. For a second-order reaction, with each reagent at concentration a, lhe corresponding relation is ktl/2

= l/a.

4

Chapter 1

mixing reactant solutions by forcing them into a mixing chamber and thence down a glass tube of uniform bore at a velocity of several metres per second; thus distances along the tube corresponded to times from initiation, and measurements of (e.g.) optical absorbance at a series of points gave a reaction/time plot. Reactions with half-lives of the order of a millisecond could be studied; the limiting factor was the rate of mixing, determined by physical factors such as the viscosity of the solutions. The techniques were steadily developed and thoroughly tested, largely on biochemical problems [1], but they were not widely known to chemists until the 1950's. Apart from some investigations on fluorescence quenching and on photostationary states, and on the use of low temperatures to reduce rates to the conventionally-measurable range, no other special techniques had been developed for fast reactions in solution by 1939 when war broke out. By 1954, when the first international conference on fast reactions was held, by the Faraday Society [2], a range of radically new techniques had emerged, in which initiation is not dependent on mixing (see Section 1.1.4). Relaxation techniques, both singledisturbance and periodic-disturbance, had been developed, largely at Gottingen, and were unveiled in a paper by Eigen [2,a]. Porter and Norrish [2,b,c] at Cambridge were working on flash-photolysis techniques (see Chapters 4 and 7), and their apparatus was described. (Pulse radiolysis carne not long after; see Chapter 5.) Fluorescence-quenching techniques (see Chapter 6), although well established, were not represented. Electrochemical techniques had been developed, mainly by Czech workers, since the 1940's; the possibilities were here discussed. Nmr methods were coming to be applied to solutions; the kinetic uses of analysis of line-broadening measurements were illustrated by some measurements on proton-transfer in liquid ammonia. All the main strategies for fast-reaction research were exemplified at this meeting. In 1960 the publication of the papers presented at an international meeting at Hahnenklee in Germany [3] showed that these new methods were coming into wider use and had produced results of great interest. The more-developed methods (flow, relaxation, flash photolysis) were already being applied to quite complex systems (photosynthesis, enzyme reactions). Review volumes began to appear, making available to wider audiences the results published in primary research journals. The first volume of the Weissberger series [4] was published in 1963; it contained authoritative articles on various methods, with particularly full treatments of flow, relaxation and flash techniques. Porter's series Progress in Reaction Kinetics contained review articles and summary tables of data [5]. A singlevolume book was published in 1964 [6]. In 1967 the Nobel Foundation held a high-level symposium of leading scientists in the field of fast reactions [7], largely those in solution, and the Nobel Prize for chemistry was awarded jointly to Eigen, Norrish and Portero All the main techniques, whatever mode of initiation (small-perturbation or large-perturbation), had been developed and tested, and some had been widely used. Major themes that could be noticed [7] were the newly-realised possibility of studying directly the fast elementary steps in reactions (e.g., proton-transfer [7,e], electron transfer [7,b,f], the role of electronically-excited states [7,b,c,f], and the applications to reactions of biological interest (e.g., photosynthesis [7,b] enzyme reactions, [7,g] base-pairing of polynucleotides [7,e]). Reaction half-lives of a few nanoseconds had been measured, and some diffusion-controlled rate constants had been determined. This symposium is a landmark in the development and spread of fast-reaction techniques. The scientists concerned were mostly chemists interested in elucidating the mech-

Introduction:

Origins, Methods, Mechanisms,

Rate Constants

5

anism of chemical reactions by showing them as combinations of simpler 'elementary' reactions, Le., as rearrangements of the atoms concerned. There were also physicists and chemical physicists concerned with the physical aspects of (for example) exchange of energy between molecules or spectroscopic determination of energy levels, and also theoreticians working on improvements in molecular dynamics or wave-mechanical calculations. Developments of two kinds ensued. There were many among these scientists who were able to devise new types of apparatus and experimental procedures, seeking to improve the speed, sensitivity, versatility, reliability and ease of operation of the techniques. There were many others, especial1y those already interested in particular classes of reactions, who saw how the new kinetic techniques would permit systematic investigation of wider ranges of reactions. Some of these were concerned with inorganic reactions [8], such as complexing between ligands and transition-metal ions in aqueous solution. Advances in the preceding two decades in studies of structure and thermodynamics were recognised as a 'renaissance' of inorganic chemistry. There had been thorough investigation of the complexation reactions of octahedral Co(III), whose kinetics could be studied by mixing the reactant solutions manual1y and making measurements on a spectrophotometer at intervals no shorter than a few seconds. The corresponding reactions of Ni(I1) were too fast for this technique, with half-lives typical1y in the millisecond region, but they could be studied with a stopped- flow apparatus. The reactions of Fe(I1I) were too fast for this, but could be studied with a temperature-jump apparatus; those of Cu(I1), faster still, by the ultrasonic-absorption method. Physical organic chemistry was similarly enlarged. For example, proton-transfer reactions of carbon acids and bases could be studied by conventional kinetic measurements, but those of oxygen and nitrogen acids and bases were inaccessible until relaxation techniques eould be applied (cf. Seetion 8.2.3.1). The historieal distinetions between inorganie, organie and physieal ehemistry had therefore beeome less applieable. One result of this reeognition of eommon ground between chemists segregated in traditional departments and institutes was that eonferences flourished. A eommon interest in keeping up with developments in teehniques, and a realisation of eommon purposes, brought together meehanistie ehemists with a wide range of preoeeupations, from the simplest systems to the most eomplieated. Natural1y these conferences were sometimes foeussed on a particular topie or teehnique [9], but wide-ranging conferenees have remained popular [10,11]. An excel1ent survey of the teehniques was published in 1992 [12]. 1.1.3. Reaction rates and reaction mechanisms Rate measurements are important because they enable us to test hypotheses about the molecular ehanges involved, i.e., the reaetion meehanism. They eontribute to an understanding of how bonds are made and broken, how the eorresponding shifts of distribution of electrons are brought about, and how energy is transferred from bond to bond within a reacting moleeule or between a reaetant molecule and the solvent. The maeroseopie observed rates are interpreted in terms of events at the mieroseopie moleeular leve!. The rate law wil1 involve one or more rate constants, and the ways in which these change with temperature and other adjustable parameters describing the initial state of the system will constitute the evidence against which a mechanistic hypothesis is to be tested. The ultimate goal is to calculate from the fundamental wave-mechanical properties of the reactants a predicted value for the rate constant in agreement with experimental results. For example, consider the

6

Chapter 1

recombination of iodine atoms after dissociation by a powerful flash of iodine dissolved in an inert solvent: h -+ 21. The recombination process (2 I -+ h) can be monitored by fast-detection techniques that enable us to record the value of the optical absorbance of a sample solution over short periods of time. Plots of these values against time give us the rates of the reaction in the chosen solvent at a series of iodine concentrations at a particular temperature. Plots of rate against concentration can then be made; they indicate that (minor deviations apart) the rate is proportional to the square of the iodine concentration; thus we can write the equation rate = k[I]2. This second-order rate law is readily interpreted by a molecular model in which reaction occurs only when two iodine atoms collide: 1+1 -+ h. This is an instance of a relation in chemical kinetics. The next step is to vary the conditions of the experiment, particularly the temperature, and interpret the results in terms of our knowledge of the forces involved. In the present case, for instance, we may expect that two iodine atoms will always recombine if they collide, so we can use our knowledge of diffusion to relate the rate constant for such a diffusion-controlled reaction (k3) to the viscosity of the solvent (see below, Section 1.3); for water and other common solvents, the predicted value is ~ 1010 dm3 mol-1 s-l, which is normally close to the experimental value. The ultimate goal is to calculate rate constants in agreement with experiment from the forces involved, both between the atoms and between the atoms and the solvent.2 We can now give precise definitions of some words and phrases that are often used loosely. A reaction rate is arate of change of the concentration of some chemical species at a particular moment; it is derived from a set of observations, in which the course of a reaction is monitored over a period of time. Such observations are the basic experimental data. By analysing the relation between these rates and the corresponding concentrations, one obtains arate law that fits both the data and (often) some standard form (e.g., first-order, in which the rate is proportional to the concentration of one of the reactants). These rate laws, along with known structural data, may be given some interpretation in terms of reaction kinetics; one would describe a scheme of molecular motions that would explain the rate law. Often there are several alternative schemes that would be consistent with a particular set of data; further experimentation would then be needed in order to choose between them. Such schemes in terms of chemical kinetics describe events on the microscopic scale, involving atoms and molecules, as distinct from rate laws which are expressed in terms of macromolecular quantities (time and concentration). The schemes may in turn be interpreted in terms of a reaction mechanism, which relates them to chemical dynamics, i.e., to theories of how molecules behave, in terms either of some particular model with limited scope (such as collison theory, or transition-state theory) or of the more fundamental body of theory based upon quantum mechanics. Few types of reaction are simple, however. Most require a combination of 'elementary' steps - reactions assumed to be single irreducible acts at the molecular level OCCUfring concurrently or in a sequence. The objective of an investigation of a reaction from a mechanistic point of view is to construct a scheme of elementary reactions that predicts correctly the observed behaviour. The closer the predictions are to the observations, the

2 Such a calculation is always a difficult task, and is even more difficult for reactions in solution than for gas reactions, but it has been achieved in the case of electron transfer between transition-metal ions in aqueous solution, a type ofreaction unique in its mechanism (see below, and Section 9.1.3.4).

Introduction:

Origins. Methods. Mechanisms,

Rate Constants

7

better supported is the mechanism. It remains, however, a theoretical construction, subject to correction or improvement.3,4 In parenthesis, we must note an ambiguity in the term 'fast'. A 'fast' reaction is, in common parlance, one whose observed macroscopic rate constant (k) approaches that calculated for a diffusion controlled reaction (kD). A 'fast' molecular process, however, is one which is soon over, i.e., requires little time (e.g., vibration within a molecule); it may usefully be called a 'short-time' or 'short-lived' process. Such a process does not necessarily lead to a fast reaction. For instance, the time required for a proton to migrate fram a carbon atom to an oxygen atom is around 10-13 s, but if it is coupled to slower concomitant changes (e.g., of configuration or solvation) the macroscopic rate constant may be far below the diffusion-controlled value, while approaching kD when such changes are minimal (see Chapter 8, Section 8.1.3). Conversely, however, an observed rate constant near kD requires that the molecular changes be short-lived, so that in all or most encounters between reactive molecules they are completed within the short lifetime of the encounter complex (cf. Section 2.5.1.1). Reaction rate measurements are essential to a full account of any reaction mechanism. One at least of the elementary reactions making up a given reaction must be at least as fast as the overall reaction. All reactions in solution, except rearrangements or configurational changes, necessarily involve diffusion: in bimolecular reactions the reactant molecules must meet each other in the course of random diffusional motions, and if there are two products (A + B -+ C + D) these willlikewise separate by diffusion; this will occur also in unimolecular reactions with more than one product (A -+ B + C, etc.). We therefore need as good a theory as possible of the process of diffusion and of the properties of any short-lived intermediate complexes. We must also take account of the fact that three of the main techniques - flash photolysis, pulse radiolysis, and fluorescence quenching - differ fram the rest in that they involve initiation of the reaction by absorption of energy from, for example, pulses of visible or ultraviolet light. This has two consequences: (i) since such pulses can be made very short-lived, 'ultrafast' phenomena can be studied, and (ii) initiation often produces excited states or free radicals, which may be highly reactive.5 These techniques are therefore particularly valuable in building up a body of knowledge about mechanisms. 1.1.4. The range of rate constants of fast reactions in solution Most of the fast-reaction techniques can determine half-times down to 10-7 s, several of them to 10-9 s or below, and a few to 10-12 s or below. The range of first-order rate 3 An excellent introductory account of these fundamentals is contained in the first four pages of D.N. Hague, Fast Reactions [6,b]. 4 An instance of such a revision occurred in connection with the reaction H2 + 12 --+ 2 HI in the gas phase. It was long thought to be a bimolecular, reaction between H2 and 12 molecules, but was shown in 1967 to occur mainly by a combination of two parallel steps each involving atoms, such as 12 ~ 21 and H2 + 21 --+ 2HI (cf. l.H. Sullivan, J. Chem. Phys. 46 (1967) 13). 5 The great majority of the second-order rate constants in solution recorded in Tables 01 Chemical Kinetics: Homogeneous Reactions (N.B.S. Circular 510; 1951) lie in the range 10-7 to 1.0 M-I s-l. The lowest directlydetermined value recorded in these standard tables is 5 x 10-10 M-1 s-l.

8

Chapter 1

Fast-reaction methods

Conventional methods

.1

I~ I

1

I

1 1

1

1.0 1

I

First-order rate constant (S-I) (logarithmic scale)

Conventional methods

Fast-reaction methods

-1--

14 I

_1 1

I

I

I

I

10-6 I

10-4 !

10-2 I

1.0 !

102 I

Second-order rate constant (M-I

4 10 I

S-I)

6 10 I

108 I

10 10 I

I

(logarithmic scale)

Figure 1.1 Ranges of rate constants accessible by conventional and by fast -reaction techniques.

constants amenable to these techniques is thus from about 1 s-I to above 109 s-I, so the accessible range has been extended by over ten powers of ten.6 The range of rate constants accessible only by means of special techniques is greater than the whole of the 'conventional' range. The longest half-time that is commonly convenient to measure is of the order of one day, or 105 s; by measuring initial rates it is possible to extend thus by a factor of perhaps 102. The range of first-order rate constants that are accessible by ordinary methods is therefore about 10-7 to 10-1 S-1; for second-order rate constants it is about 10-7 to 1 M-I s-l. These ranges are shorter than the ranges where measurement requires fast-reaction techniques (Figure 1.1). Since there is no evidence that reaction rates are grouped in any way, we may expect very many reactions to have rates in the 'fas!' range. 1.1.5. Strategies and methods for determining the rates of fast reactions in solution Methods for the determination of the rates of fast reactions exemplify four possible strategies or guiding principIes; these are as follows. 6 The corresponding range of second-order rate constants depends also on the rate measurements can be made, and therefore on the sensitivity of the detection 1011 M-I s-I have been measured by several techniques, and the range extends range, for which the maximum rate constant may be taken somewhat arbitrarily sponding to a half-time of 10 s at concentration 0.1 M. or 100 s at 0.01 M; thus range is again in the region of 1010.

lowest concentration at which system. Values up to 1010 or down to meet the conventional as around I M-I s-I, correthe extension of the accessible

Introduction:

Origins. Methods, Mechanisms,

Rate Constants

9

(i) The principIe on which some of the simplest methods are based is to bring the rate down into the 'conventional' range, where the course of reaction can be monitored without the use of special techniques. Examples of methods based on this strategy are the use of low concentrations or of low temperatures. (ii) Another principIe that leads to some relatively simple methods is to reduce the mixing time so that it becomes small compared with the half-time of the reaction. This is commonly done by flow techniques. The reaction may be monitored either (a) by some means that avoids the use of fast detection techniques, such as the continuous-flow technique, or (b) by some fast measuring device, as in the stopped-flow method. (iii) External initiation of reaction without mixing. The shortest time that can be measured by any of the preceding methods is determined by the least time of mixing of the solutions, which is not easily reduced much below a millisecond. There are, however, ways in which mixing can be avoided altogether, by initiating fram outside the reaction solution. Two different principIes are available, as follows. (a) In relaxation methods, the principIe is to disturb a system which is in chemical equilibrium, and monitor the resulting concentration changes as the system 'relaxes' towards the new equilibrium. In the temperature-jump method, for example, an equilibrium is perturbed by imposing a sudden change of temperature. In response to this, reaction must take place, in one direction or the other, until the new equilibrium concentrations have been attained, and the course of the concentration changes is monitored with the aid of a fast oscillographic or digital method. Pressure-jump and electric-field-jump methods are also well established. Altematively, a periodic (sinusoidal) variation of temperature and pressure can be effected by means of ultrasonic vibrations; when the half-time of the reaction is comparable with the period of the disturbance, there is a sharp increase in the power absorbed, and fram the power/frequency spectrum the rate constant can be deduced. An oscillating electric field may be similarly used. In all these techniques the displacements of equilibrium are kept small, since this greatly simplifies the mathematical treatment. (b) In photochemical and related methods the initiation of chemical change is achieved by irradiation. The absorption of light, or of high-energy radiation, or electrans, by a solution can lead to drastic changes, by praducing atoms or free radicals or excited states of molecules, which then set off further changes. In the jlash-photolysis and pulseradiolysis techniques, a single powerful pulse of energy is used, and the subsequent reactions may be followed by fast recording techniques. (iv) A fourth strategy is to make the fast chemical reaction compete with some other fast process whose rate can be separately determined. There are three methods which, while otherwise differing widely, depend on this principIe. (a) Fluorescence-quenching methods depend upon competition between reaction and fluorescence emission. When a substance fluoresces in solution, the excited molecules have a certain mean lifetime before they emit light; this can be determined, and is commonly of the order of 10-8 s. If a solute is added which reacts rapidly with these excited molecules, so that an appreciable number of them are destroyed before emission can take place, the average lifetime and the fluorescence intensity are reduced. Fram their variation with concentration, the rate constant of the reaction can be found.

10

Chapter 1

(b) Electrochemical processes can be arranged in which (for example) a current which would normally be controlled by the rate of diffusion of some species is affected also by the rate at which that species is produced by a reaction in the solution. The first technique to be so used was polarography; others are the rotating-disc, potentiostatic and galvanostatic techniques. (c) Nuclear magnetic resonance and electron-spin resonance (paramagnetic resonance) can be adapted to rate measurements. The width of a line in (for instance) a proton nrnr spectrum is related to the lifetime of the spinning pro ton in a particular spin state. If this lifetime is cut short by a reaction involving that proton, the corresponding line in the nmr spectrum is broadened, to an extent depending on the rate of the reaction. The same principIe can be applied to determine rates of electron-transfer reactions from epr spectra. 1.1.6. Reaction rates accessible by the various methods The reaction half-times and rate constants accessible by the various techniques are summarised in Table 1.1. The smallest half-times, from about 10-9 s down to 10-13 or even 10-14 s, have been determined by the flash-photolysis, fluorescence-quenching, ultrasonicabsorption, and esr methods; next come the temperature-jump and electric-impulse techniques (approximately 10-6 s). In terms of rate constants, for first-order reactions the upper limit for a given technique is approximately the reciprocal of the least half-time that can be measured (k ~ O.7t1/~); for second-orderrate constants, however, the upper limit depends also on the concentration at which the reaction can be observed, and hence on the sensitivity of the technique; for most of the techniques the maximum observed value is of the order of 109 M-I s-I or above, even for stopped-flow for which the smallest accessible half-time is little less than 10-3 s. The usefulness of a technique is not, of course, related solely to the maximum time resolution or observable rate constant; other factors such as versatility, precision, convenience of operation and availability can all be important. The stopped-flow method, for instance, which is the most widely used of all, owes its popularity to its adaptability, speed and convenience, robustness, and wide availability; it is suited not to the fastest reactions but to those with rate constants less than about 105 M-1 s-l. Conversely, fluorescencequenching methods are applicable only to very fast reactions, for which however they are very powerful; and flash techniques, uniquely, can follow changes down to 10-14 s. The availability of commercial equipment, and its cost, may also be important, especially where sophisticated electronic instrumentation is required.

1.2. The structure of this book These considerations, along with those outlined in the foreword, are the main influences in the design of this book, which may be summarised as follows. Diffusion, control of reaction rates by diffusion, and the properties of encounter complexes are considered in Chapters 2 and 3; Chapter 2 is largely descriptive, while the more mathematical aspects are in Chapter 3, which may be judiciously skimmed by readers allergic to mathematical equations. It is advantageous to have in mind a preliminary pictorial sketch, and this is presented in the present chapter. Next follow Chapters 4,5,6 on the three strong-perturbation

Introduction:

Origins, Methods, Mechanisms,

Rate Constants

Table 1.1 Ranges of reaetion half-time (11/2 in s) aeeessible by various teehniques. Fulllines apparatus; dotted extensions refer to speeial equipment. All values are orders of magnitude Method

Ref.

Continuous flow Stopped flow Temperature-jump Pressure-jump Eleetric-field jump Ultrasonie relaxation Dieleetric relaxation Flash* Pulse radiolysis** Fluorescenee quenehing Eleetroehemieal E.p.r. N.m.r. (proton)

[a] lb] [e] [d] [e] [f] [g] [h] [j] [k] [1] [m] [n]

11 refer to standard

10-6 ..........

10-10

............

* Flash teehniques can be used for reaetions with tl/2 down to 2 x 10-13 s (see Seetions 4.3.6.2 and 7.3.4.3). ** Pulse radiolysis teehniques can be used for reaetions with tl/2 down to 3 x 10-11 s (see Seetion 5.2.2). [a] H. Strehlow, Ref. [12], Chapter 3; 1.F. Holzwarth, in Ref. [I0,a]. lb] B. Chanee, in Ref. [I,d], Chapter I1; B.H. Robinson, in Ref. [l,e], Chapter 1. [e] H. Strehlow, Ref. [12J, Chapter 4, p. 60 seq.; 0.0. Hammes, in Ref. [I,e], Chapter IV; D.H. Tumer, Ref. [I,e], Chapter III. [d] H. Strehlow, Ref. [12], Chapter 4, p. 64 seq.; W. Knoehe, in Ref. [I,e], Chapter V, and in Ref. [l,e], Chapter IV. [e] H. Strehlow, Ref. [12], Chapter 4, pp. 67-68; E.M. Eyring and P. Hemmes, in Ref. [I,e], Chapter V. [f] H. Strehlow, Ref. [12], Chapter 4, pp. 70-77; 1.E. Stuehr, in Ref. [l,e], Chapter VI. [g] H. Strehlow, Ref. [12], Chapter 4, pp. 77-80; 1. Everaert and A. Persoons, 1. Phys. Chem. 85 (1981) 3930; L. De Maeyer, R. Wolsehann and L. Hellemans, in Ref. [lO,a], p. 50\. [h] This book, Chapter 4 (esp. Tables 4.1 and 4.2) and Chapter 7; N. Hirota and H. Olya-Nishiguchi, in Ref. [\.e], Chapter Xl, Tables 6 and 7; and see footnote*. [j] This book, Chapter 5 (esp. Table 5.1). [k] This book, Chapter 6 (esp. Tables 6.1, 6.2, 6.3, 6.4). [1] H. Strehlow, Ref. [12], Chapter 8; H. Strehlow in Ref. [I,d], Chapter VIII; C.P. Andrieux and 1.M. Savéant in Ref. [1,eJ, Chapter VII. [m] N. Hirota and H. Olya-Nishiguru, in Ref. [I,e], Chapter XI, Tables 4 and 5. [n] K.R. lennings and R.B. Cundall, Prog. React. Kinet. 9 (1977) 2. Table 1.2 The time-spectrum

of moleeular proeesses

Rate scale (s-l)

Time seale (s)

t

t

1015 femto

Proeesses

10-15 (fs)

eleetronie motion eleetron orbital jumps

1012 pico

10-12 (ps)

vibrational motion bond cleavage (weak bonds) eleetron transfer proton transfer

109 nano

10-9 (ns)

rotational and translational motion (small molecules) bond cleavage (strong bonds) spin-orbit eoupling

106 miero

10-6 (¡.¡.s)

rotational and translational hyperfine eoupling

motion (large moleeules)

12

Chapter 1

techniques. Chapter 7 outlines the remarkable developments in studies of 'ultrafast' reactions due to the use of very fast detection devices and the combination of these with mass-spectrographic techniques. (We note the triple alliance of kinetic experiments with spectroscopy and molecular dynamics.) Chapters 8 and 9, which should be considered together, have a dual function. In the first place, they deal, respectively, with proton-transfer and electron-transfer reactions, in recognition of their important roles as elementary processes in numerous mechanisms. In the second place, they both deal with 'Marcus theory', which has been steadily developed from the 1950's onwards and is the leading general theory of reaction rates. (The Nobe1 prize was awarded in 1992 to R.A. Marcus, a leading participant in their development.) Marcus theory builds on transition-state theory, which relates rate constants and their temperature-dependence to quasi-thermodynamic activation parameters (activation free energy, entropy and enthalpy) in a formula to which experimental results on most reactions can be fitted,7 but it assumes a more specific model, in which effects of solvent reorganisation are made explicit as well as those of changes of electron distribution and of bond breaking and making.1t has extensive support on the experimental side from fast-reaction studies (cf. Tables 8.1 to 8.4 and 9.1 to 9.6), and on the theoretical side from quantum-mechanical calculations on potential-energy barriers. It is successful both with proton-transfer, where covalent bonds are made and broken, and with electron transfer, where (uniquely) they are noto Its greatest achievement is a successful calculation of rate constants for outer-sphere e1ectron-transfer reactions between transition-metal ions in aqueous solution, in terms of the fundamental wave-mechanical properties of reactant molecules and an e1ectrostatic model of the solvento The results are the closest approach yet made to the ultimate goal of a successful ab-initio calculation for a reaction in solution; the agreement between theory and experiment is within a factor of ~ 30, compared with a range of 15 powers of ten for the observed rate constant (cf. Section 9.1.3.4). The possibility of progress towards a universally-applicable theory can be envisaged. Our treatment begins (unconventionally) by outlining (in Chapter 8) the fundamental s of the theory as applied to proton transfer (and methyl-group transfer), and goes on to apply them to electron transfer (in Chapter 9).8

1.3. Theory of rate constants for diffusion-controlled

reactions

1.3.1. The energetics ofvery fast reactions There is evidently an upper limit to the rate at which a bimolecular reaction in solution can proceed, set by the rate at which reactant molecules encounter each other. Some reactions have rates close to this limit, but there are many which have much lower rates and yet are fast in the sense that special means are required to measure their rates. The temperaturevariation of the rates of these latter reactions is represented empirically by the Arrhenius 7 Transition-state theory was developed in lhe 1930's along with the wave-mechanical theory of covalent bonding (for an early survey, see: S. Glasstone, H. Eyring and K.J. Laidler, Theory 01 Rate Processes, McGraw-Hill, New York, 1941). 8 This order is lhe reverse of that of the historical development, which began with the problem of electron transfer; lhe lheory was later adapted to proton transfer and subsequent1y to group transfer (see Sections 8.1 seq., 8.3).

Introduction:

Origins, Methods, Mechanisms,

Table 1.3 Second-order rate constants at 298 K, calculated various values of EA, to nearest order of magnitude

50

60

40

Rate Constants from k = 1011 exp( -EA/

20

exp(-EA/RT)

2 x 10-11

3 x 10-4

k (M-1 s-l)

2

3 x 107

13 RT) for

o

equation k = A exp( - EA/ RT), where EA is taken to represent a critical energy without which a collision will not result in reaction, and exp( - EA/ RT) is the fraction of effective collisions. To illustrate the relationship between k and EA, Table 1.3 shows the secondorder rate constants at 298 K calculated from the Arrhenius equation for various values of EA, with A = 1011 M-1 s-l (a representative value for many reactions not involving two ions). A second-order rate constant higher than 102 M-1 s-l, for instance, may be attributable to an activation energy less than 50 kJ mol-l. For the fastest reactions, however, this interpretation of the Arrhenius equation breaks down. Many reactions are known with rate constants in the region of 1010 M-1 s-l (for some examples, see Table 2.1, in Chapter 2, Section 2.1). The rates of such reactions approximate to the rate of molecular encounter, which is dependent on the speeds at which the reactant molecules move about in the solution. A reaction taking place at practically every encounter is not subject to any appreciable energy barrier once the molecules have reached adjacent positions. The diffusion process itself, however, will require some activation energy, and indeed the rate is found to increase with temperature, varying inversely with the viscosity and giving an apparent activation energy of the order of 10 kJ mol-l. From the molecular point of view, diffusion is the random migration of molecules or small particles, arising from the motions which they continually undergo by virtue of their thermal energy. In the gas phase these motions lead to occasional collisions, one at a time; in solution, after a collision the molecules tend to stay close to each other and recollide several times before separating (see Sections 2.4, 2.5.2.2). If the concentration of the solution is not uniform, the molecules migrate along the concentration gradient at any point; this is the macroscopic change that can be directly observed experimentally. For example, if a drop of a coloured material is put into apure solvent, the colour spreads through the liquid as the molecules of the material, moving at random, gradually occupy the whole available volume uniformly. 1.3.2. Concentration-gradient

treatment

This treatment of the rate of diffusion-controlled encounters in solution was developed originally by Smoluchowski [13,a] for the rates of coagulation of colloidal solutions, and was later applied to reactions between molecules. It as sumes that the diffusive motions of molecules can be treated like those of macroscopic particles in a continuous viscous fluido A simplified version is as follows. Consider a solution containing two kinds of solute molecule, A and B (Figure 1.2), assumed to be spherical. Suppose that these can be treated as hard spheres in a continuous medium, and that intermolecular forces can be neglected. We wish to find the rate at which

Chapter 1

14

(B)r

o (a)

f

(b)

Figure 1.2 Diffusion-controlled encounter. (a) Schematic diagram showing solute B mo1ecu1es approaching an A mo1ecu1e; reaction occurs on contact, when the centre-to-centre A-B distance r becomes equal to rA + r¡¡ (= rAB). (b) Schematic p10t of (B)r, the local concentration of B mo1ecules, against r; at large distance (B)r will be the bulk concentration (B)o, but if r falls to rAB then reaction occurs and (B)r becomes zero. (After P.w. Atkins, Physical Chemistry, 4th edn. p. 847.)

A and B molecules will encounter each other by diffusion. The simplest assumption about the rate of diffusion of any species in solution is that the rate is proportional to the concentration gradient (Fick's law of diffusion). A local concentration gradient is set up when a pair of molecules A and B meet each other and react; the disappearance of the B molecule depletes the concentration of B in the immediate neighbourhood of the A molecule, thus producing a non-equilibrium spatial distribution of B molecules, Le., a concentration gradient, and similarly for the A molecules. This will result in a net flux of reactant molecules into the depleted region, tending to restore the equilibrium distribution. A steady state is reached when the average depletion is just enough to provide the concentration gradient needed to maintain a flux of molecules equal to the rate at which the molecules are lost by reaction. The mathematical theory of diffusion shows that the number of encounters between A and B molecules per second per unit volume is then (Section 2.2): (1.1)

Here nA and nB are the numbers of A and B molecules per unit volume, fA and fB are the radii of the molecules, DA and DB are their diffusion coefficients, and (DA + DB) represents approximately the relative diffusion coefficient. If reaction occurs at every encounter between A and B, the rate of encounter given by Equation (1.1) is also the rate of reaction. In terms of arate constant kD, this rate is given by the following expression (in which NA is the Avogadro number): 0.2)

Introduction:

Origins, Methods, Mechanisms,

Rate Constants

15

Equating the expressions (1.1) and (1.2), we find that the rate constant for a diffusioncontrolled reaction is given by the following, often called the Smoluchowski equation [13,a]: (1.3)

If we write r AB for the centre-to-centre encounter distance (r A + rB) and D AB for (DA + DB), we obtain the following compact expression for kD in terms of molecules per unit volume per unit time: (1.3a) or in terms of moles per unit volume per unit time: (1.3b) It must be remembered that in the derivation of these equations it was assumed that reaction occurs at every encounter. The calculated value of kD is thus an upper limit for reactions where reactant molecules meet by diffusion. Most reactions have smaller rate constants, often because they require activation energy, or because the reactant molecules have limited reactive sites which impose a need for correct orientation on collision. Such effects are considered in Chapter 2. 1.3.3. Random-walk treatment of dijfusion It is useful to complement this rather abstract treatment with one based on a more easily visualised model, which treats diffusion as due to random movements of the solute molecules, related to Brownian motion.9 The path of an individual molecule will be a three-dimensional zigzag, with abrupt changes of direction, similar to the two-dimensional zigzag shown in Figure 1.3 [14].1f the changes of direction are entirely uncorrelated, i.e., if each is completely independent of its predecessors, one can apply the mathematical theory of 'random walks'. A particular solute molecule may find itself, after a given time interval, anywhere within a wide range of distances from its initial position; but the average distance can be calculated. The result is that, in the simplest case, the mean-square displacement (r2) of the centre of mas s is related to the time (t) through the translational diffusion coefficient (D), by an equation derived by Einstein in 1905:10 r2

= 6Dt,

whencer~(r2)

-

1/2

~(6Dt)1/2.

(l.4)

9 Brownian molion is lhe name given lo lhe small random molions of micron-sized particles in suspension in liquids, which can be observed by high-power microscopes. It is altribuled lo lhe random molions of!he molecules of lhe surrounding liquid, whose impacts on !he (much larger) colloidal particle do nol always cancel out. 10 For a derivalion of lhis relalion, based on a simplified one-dimensional model, see P.W. Atkins, Physical Chemistry, 5th edn., Oxford Universily Press, 1994, p. A39. Essentially one calculates the probability !hal a molecule will be found al a given dislance r from lhe origin at a time t. For references see Chapler 2, Seclion 2.3.

16

Chapter 1

Figure 1.3 A plOl of a compuler-simulaled lwo-dimensional random walk of n = 18.050 sleps. The walk slarts al lhe upper lefl-hand comer of lhe track and works its way lo lhe righl-hand edge. (Some regions are complelely black; this is due lo repeated traversals.) The straight-line distance, 'as lhe crow flies', is only 196 slep lenglhs. This is in agreement wilh lhe expected root-mean-square displacemenl which is (2n) I/2 = 190 slep lenglhs. (Diagram from H.C. Berg, Ref. [13J.)

This relation enables us to calculate the average volume swept out, or 'searched through', by a reactant molecule in unit time, and hence the average number of encounters per unit time with molecules of the other reactant. Assuming that reaction occurs at every encounter, we can then calculate the diffusion-controlled rate constant. On carrying through the algebra (see Chapter 2, Section 2.3), we obtain an equation of exactly the same form as the one derived from the concentration-gradient approach (Equation (1.3) or (1.3a)) and with much the same numerical coefficient (which in any event is only approximate). This is not in fact surprising, because Fick's law of diffusion can itself be deduced from the random-walk model, which indeed provides the simpJest molecular interpretation of that law. 1.3.4. Rate constant and viscosity Application of the Smoluchowski equation (1.3) requires a knowledge of the sizes and diffusion coefficients of molecules. An estimate of the effective values of r can usualIy be made from molecular volumes. Diffusion coefficients present more of a problem, since not very many have been experimentally determined over a range of temperature. They may, however, be eliminated from Equation (1.3) by using the Stokes-Einstein relation, which is theoreticalIy derived from the same molecular model and is often in fairly good accord with experimental data; it expresses D in terms of the viscosity of the solvent (r¡) and the

Introduction:

molecular radii:

Origins. Methods, Mechanisms.

Rate Constants

17

11

kT

DA=--, 4Jrr¡TA

kT DB=--.

(1.5)

4Jr1)fB

Gn substituting for DA and DB in Equation (1.3), one obtains for the rate constant calculated according to the Stokes-Einstein model: (1.6)

(1.6a)

This equation shows that ko at a given temperature depends primarily on the viscosity 1) of the solvento Variations in the sizes of the molecules do not affect the calculated rate unless the radius ratio is altered; even then the term in the second bracket is not much affected. The reason for this insensitivity of the rate to molecular size is that, on the model assumed, a larger molecule will move about more slowly than a smaller one in a solvent of given viscosity, but will present a bigger target for encounter with another solute molecule; these two effects of molecular size on the encounter rate willlargely compensate. If the molecular radii of A and B are assumed equal, we obtain from Equation (l.6a) simple expression for ko: 12 4RT ko=--.

(1.7)

1)

Since ko is not sensitive to molecular size, this is a useful though approximate relation. It predicts that the rate constant of such a diffusion-controlled reaction will be inversely proportional to the viscosity of the solvent; its temperature-coefficient will be comparable with that of the viscosity, and therefore small compared with that of most reactions. The numerical value calculated for ko in water at 25 °e is 1.1 x 1010 M-1 s-I ; the values in many common organic solvents are of the same order. All these conclusions agree semiquantitatively with the experimental results on very fast reactions, and in some cases the agreement is close (cf. Table 2.2)0 The expression for ko in Equation (106) or (lo7) cannot be more than an approximation, from the nature of its assumptions, which ignore the molecular structure of the liquid and the consequences of solute-solvent and solute-solute interactions, and take no account of the fact that reactant molecules are seldom spherical and commonly have localised reaction siteso None the less, it provides a general guide to the effects of various factorso An observed rate constant comparable with the calculated value of ko is a useful indication of diffusion control. II The factor 4 is related to the assumption that the diffusing molecule 'slips' rather than 'sticks', when passing a solvent molecule. Ifone as sumes that it 'sticks', the factor 6 is more appropriate. See Section 3.4.1 12 If the factor 6 is used instead of 4 in Equation (l.4), the factor 4 in Equation (1.7) must evidently be rep1aced by 8/3. This appears in various texts.

18

Chapter 1 Table 1.4 Diffusion-eontrolled rate eonstant for reaetion between ions, eompared . ions uncharged . wJth that for uneharged moleeules. Values of kD / kD = 8/(e ¡; - 1), for lons of eharges ZA and ZB, with distanee of closest approaeh a (in Á), for water at 25°C kions / k uncharged D D

lA lB

a (Á)

+2 +1 -1 -2

2.0

5.0

7.5

10.0

0.005 0.10 3.7 7.1

0.17 0045 1.9

0.34 0.60 1.6

0045

3.0

2.2

0.69 1.4 1.9

1.3.5. Reactions between ions The strongest molecular interactions are those due to charges on ions. If A and B are both ions, the Coulombic interaction energy is considerable and it is necessary to modify expressions for the rate constant such as that given in Equation (1.7). A simple electrostatic calculation (which will be outlined in Section 2.5.3 below) shows that they should be multiplied by a factor involving the charges on the ions, the distance of closest approach, and the relative permittivity of the solvent [l3,b]. Table 2.3 gives some values of this factor for ionic compounds in water. Unless the ions are exceptionally small, the value of kD is changed by les s than a power of ten. The brief outline given in this section will serve as an introduction to the next chapter, which deals with the phenomena of diffusion-controlled reaction rates.

References [1.11

[1.2] [1.3] [lA]

(a) H. Hartridge and F.J.W. Roughton, Proe. R. Soe. A 104 (1923) 376; (b) H. Hartridge and F.J.W. Roughton, Proe. R. Soe. B 94 (1923) 336; for reviews, see, (e) F.J.W. Roughton and B. Chanee, in: S.L. Friess, E.S. Lewis and A.Weissberger (Eds.), Investigation of Rates and Mechanisms of Reactions, Part /l, 2nd edn., Interseienee, New York, 1963, pp. 703-792; (d) B. Chanee, in: G.G. Hammes (Ed.), Investigation of Rates and Mechanisms of Reactions, Part 11, 3rd edn., Wiley-Interseienee, New York, 1974, pp. 5-62; (e) B.H. Robinson, in: C.F. Bemaseoni (Ed.), Investigation of Rates and Mechanisms of Reactions, Part /l, 4th edn., Wiley-Interseienee, New York, 1986, pp. 9-26. Diseuss. Faraday Soe. 17 (1954) 114-234; (a) M. Eigen, p. 194; (b) R.G.W. Norrish and G. Porter, p. 40; (e) G. Porter and M.W. Windsor, p. 178. Z. Elektroehem. 64 (1960) 1-204. This series is frequently mentioned, and for brevity's sake we shall designate it simply as Weissberger. Eaeh book in the series forrns Part 11of a two-volume set entitled lnvestigation of Rates and Mechanisms of Reaction, Wiley Interseienee, New York. Three editions have been published, with different editors: S.L. Friess, E.S. Lewis and A. Weissberger (1963), G.G. Hammes (1974) and c.F. Bemaseoni (1986). We shall distinguish these volumes by their dates, sinee (eonfusingly) the first of them belongs to the seeond edition of the complete work, the seeond volume belongs to the third edition, and the third belongs to the fourth edition. Thus we designate the three fast-reaetion volumes as Weissberger (1963), Weissberger (1974), and Weissberger (1986), respectively.

Introduction: [1.5] [1.6]

[1.7]

[1.8]

[1.9]

[1.10]

[1.11] [1.12] [1.13] [1.14)

Origins, Methods, Mechanisms,

Rate Constants

19

G. Porter (Ed.), Progress in Reaction Kinetics, Pergamon Press, Oxford, Vol. 1, 1961, Vol. 2, 1964, Vol. 3,1965, Vol. 4,1967, ete. (a) E.E Caldin, Fast Reactions in Solution, Blaekwell Seientifie Publieations, Oxford 1964; (b) Another, eovering reaetions in the gas phase as well as in solution, was published in 1971: D.N. Hague, Fast Reactions, Wiley-Interseienee, 1971. S. Claesson (Ed.), Fast Reactions and Primary Processes in Chemical Kinetics, Nobel Symposium 5, Almqvist and Wiksell, Stoekholm, Interseienee Publishers, New York, 1967; (a) R.G.W. Norrish, p. 33; (b) H.T. Witt, pp. 81, 261; (e) G. Porter, pp. 141,469; (d) ES. Dainton, pp. 185,485; (e) M. Eigen, pp. 245, 333, 477; (f) A. Weller, p. 413; (g) B. Chanee et al., p. 437. (a) E Basolo and R.G. Pearson, Mechanisms of Inorganic Reactions, Wiley, New York, 1958, Chapter 3. This was a pioneering monograph. In their prefaee, the authors refer to this 'renaissanee' of inorganie ehemistry in the preeeding two deeades, with sueeessive periods of advanee in studies of strueture (d. L. Pauling, Nature of the Chemical Bond), of thermodynamies, and last1y of kinetie and meehanistie studies. For instanee, (a) T.A.M. Doust and M.A. West (Eds.), Picosecond Chemistry and Biology, Seienee Reviews Ltd, 1983; (b) Symposium on Flash Photolysis and its Applieations, E 82 (1986) 2065-2451; (e) M. Ebert, J.P. Keene, A.J. Swallow and J.H. Baxendale (Eds.), Pulse Radiolysis, Aeademie Press, 1965; (d) J.H. Baxendale and E Busi (Eds.), The Study of Fast Processes and Transient Species by Electron Pulse Radiolysis, D. Reidel, Dordreeht, 1982; (e) P. Laszlo (Ed.), Protons and Ions in Fast Dynamic Phenomena, Elsevier, Amsterdam, 1978, mainly on ionie reaetions. For instanee, eonferenees whose proeeedings were published as (a) W.J. Gettins and E. Wyn-Jones (Eds.), Techniques and Applications of Fast Reactions in Solution, D. Reidel, Dordreeht, 1978; (b) A eonferenee open to all fast-reaetion teehniques has been held every year sinee 1977 by the Fast Reaetions in Solution group sponsored by the Royal Soeiety of Chemistry. A eonferenee on 'ultrafast' reaetions held at Berlin was published in J. Phys. Chem. 97 (1993) 1242312645, and an exeellent survey by A.H. Zewail appeared in J. Phys. Chem. 100 (1996) 12701. H. Strehlow, Rapid Reactions in Solution, VCH, Weinheim, 1992. (a) M. von Smoluehowski, Z. Phys. Chem. 92 (1917) 129; (b) P. Debye, Trans. Eleetroehem. Soe. 82 (1942) 265; (e) M. Eigen, Z. Phys. Chem. N.E 1 (1954) 176. H.C. Berg, Random Walks in Biology, Prineeton University Press, 1983, p. 12.

21

Chapter 2

The Rates of Diffusion-Controlled Reactions 2.1. Introduction The foregoing brief discussion on diffusion-controlled reactions serves to call attention to the importance of random translational diffusion as a means whereby reactant molecules can come into contact, and to the infiuence of intermolecular forces. In the present chapter, we continue to explore, with the aid of simple models, the motions of solute and solvent molecules that govern the rate at which reactive collisions occur. Throughout this book we shall come across rate constants in the region of 1010 M-1 s-l, often inversely related to the viscosity of the solvento There is compelling evidence that these are controlled by the rates at which the reactant molecules meet by random diffusive motions. Some examples of simple bimolecular reactions are collected in Table 2.1.1 A preliminary account of the theory of such very fast reactions was given in Chapter 1 (Section 1.3); in the present chapter we consider them in more detail, with the help of simple models. The basic assumptions are: (a) that reactant molecules meet in the course of their diffusive motions; (b) that they stay close together long enough for a number of collisions (an 'encounter') to occur; and (e) that during an encounter they may react, with or without an energy-barrier. This model may be treated by various methods. In this chapter we shall consider two approaches, one macroscopic and the other molecular, based, respectively, on the classical theory of diffusion and on the theory of random motions. We shall postpone much of the mathematics of diffusion to Chapter 3, in the interests of developing a working picture of diffusion-controlled reactions.

2.2. Application of the diffusion law to rates of encounter and of chemical reaction: the Smoluchowski equation 2.2.1. 1ntroduction In the 'classical' approach to the rate of encounter of particles in liquid solutions, initiated by Smoluchowski [2], the random diffusive motions of particles are treated in terms of Fick's law of diffusion, which (summarising experimental evidence) states that 'rates of diffusion are proportional to concentration gradients'. Consider a bimolecular reaction between reactants A and B. When a pair of A and B molecules meet and react, the local 1

Reactions between ions have been excluded fram this table; they are briefly treated in Section 2.5.3.

22

Chapter 2

Table 2.1 Some reaetions with very high rale eonslanls. Values of k (M-1 s-l) al or near 298 K, eompared with ko == 4RT Ir¡. Values of kl ko are rounded. For referenees [l,a], [l,b] ete. for the rate eonstants, see Ref. [1]. These reaetions do not inelude any between two ions; sueh reaetions are treated separately in Seetion 2.5.3 Seetion

Solvent

10-9 k klko

Ref.

H+ + NMe3 ---*HNMej OH- + PhOH ---*PhO- + HZO e- + PhCHZCI---* PhCHi + CI-

8.1.2.1 8.1.2.1 5.3.1.1

water water ethanol

25 e.14 5.1

[1, al [1, b] [1, e]

Os(bipy)~+ + Fe(bipy) (CN)z 2PhCOzH ---*(PhCOZHlz 3-hydroxypyrenea + pyridine 2CMej ---*Me3CCMe3

9.1.1

water

6.2.4 4.6.2

CCl4 ete. benzene oetane ete.

3.2 5 9 3

Hm+CO---* HmCO IZ + (MeZNlzCS ---*(MeZNlzCS ... Iz Aeetylcholinesteraseb + HNMe-aer+

2.2.2 2.5.2.2 2.5.2.4

Reaetion type Proton transfer: Electron transfer: Hydrogen bonding: Radical recombn: Molecular combination: Haemoglobin Charge-transfer Enzyme

glyeerol n-BuCI water

0.002 10 1.2

2 1 0.6 0.3 0.5 0.5 0.7

[1, [1, [1, [1,

d] e] f] g]

0.3 0.3 0.1

[1, h] [l,j] [1,1]

a Exeited moleeules. b Miehaelis-Menten

kineties assumed; see, Ref. [31,a,b].

concentrations of both A and B are depleted, giving rise to concentration gradients, under which a net migration of A and B molecules to the region occurs by diffusion; a steady state is reached when the average rate of reaction is equal to that of migration. The reactant molecules are treated as hard spheres in a continuous medium. Fick's law is applied to derive from this model a differential equation which when sol ved leads to an expression for the encounter rate constant kD in terms of the translational diffusion coefficients (DA, DB) and the radii (rA, TB) ofthe molecules. This expression is mentioned in Chapter 1 (Section 1.2.1) and derived in Chapter 3 (Section 3.2.1). If we denote the centre-to-centre distance of closest approach of the reactants by r AB = r A + rB, and as sume that the relative diffusion coefficient DAB can be approximated as DAB = DA + DB the expression in terms of molecular concentrations is: (2.1)

or in terms of molar concentrations: (2.2)

If chemical reaction occurs at every encounter, kD is al so the rate constant for reaction. Equation (2.1) is often known as the Smoluchowski equation. Since experimental values of diffusion coefficients are known in relatively few instances, it is often convenient to express them in terms of the viscosity (r¡) of the medium, which is known for many solvents and is easy to measure. Application of classical hydrodynamic theory to the hard-sphere modelleads, as was shown in Section 1.3.2, to the Stokes-Einstein equation (1.3) for the relative diffusion coefficient DAB, which may be

The Rates of Diffusion-Controlled

Reactions

23

written DAB = kT /4lf17(rA + 113). On substituting this expression into Equation (2.1), we obtain (cf. Equation (1.6a» for the rate constant of a diffusion-controlled reaction: (2.3) For equal-sized molecules A and B, this takes the simple form: 4RT kD=--. r¡

(2.3a)

This Equation (2.3a) is also a good approximation when r A and 113 are different but comparable; for instance, even if r A is doubled, the expression for the rate constant will be identical in form to Equation (2.3a) and the numerical factor will change only from 4 to 4.5. These are theoretical equations, derived from a simplified model. They provide a useful starting-point; we shall examine, in the present chapter and the next, how far they agree with observation. At present we note that Equation (2.3a) not only predicts rate constants of the right order of magnitude for the fastest reactions in water and ordinary organic solvents; it predicts also that these high rate constants will ron parallel with the viscosity of the solvent, and will exhibit relatively low temperature-coefficients determined by that of the viscosity.2 Quantitatively the theory predicts that if the viscosity is varied, by changing the solvent (or the pressure), then the observed rate constant k will vary as l/r¡, while if the temperature is varied with a given solvent then k will vary as T Ir¡. There are a considerable number of reactions where these predictions are at least qualitatively fulfilled (see Table 2.1). Although many diffusion-controlled reactions are extremely fast, some are not; they may have rate constants down to 107 M-1 s-I in common solvents, yet show the viscositydependence characteristic of diffusion control (Section 2.5.2 below). This is because orientational constraints can in principIe decrease the rate constant by several orders of magnitude, by reducing the chance of reaction at a particular collision. Such orientational effects are important in the general theory of diffusion control, and are especially prominent for reactions of macromolecules (Section 2.5.2.4). The fundamental rate expression to be considered is the Smoluchowski relation k = 4rr N DABrAB (Equation (2.1». The derived expression 4RT Ir¡ (Equation (2.3a», is a useful approximation, but deviations from it are observed, because the Stokes-Einstein equation which is involved is derived by hydrodynamic theory for spherical particles moving in a continuous ftuid, and does not accurately represent the measured values of translational diffusion coefficients in real systems. Although the proportionality D ex l/TI is indeed a reasonable approximation for many solutes in common solvents, the numeral coefficient 1/4 is subject to uncertainty. In the first place, this theoretical value derives from the assumption that in translational motion there is no friction between a solute molecule and the first layer of solvent molecules surrounding it, i.e., that 'slip' conditions hold. If, however, one assumes instead that there is no slipping ('stick' conditions), so that momentum is 2 Many common organic solvents have viscosities in the region of 10-3 kgm-1 s-1 (l centipoise) at room temperature and 1 atm pressure. Their temperature-variation is often represented fairly well by an equation of the form r¡ = A exp( B / R T), where A and B are constants; B commonly lies in the range 4 to 12 kJ mol-1 .

24

Chapter 2

transferred between solute and solvent molecules, the coefficient becomes 1/6. Secondly, the experimental values of diffusion coefficients for many reactant molecules in ordinary organic solvents are 10-30% higher than the Stokes-Einstein 'slip' values, implying that the coefficient could be around 1/3, so that in place of Equation (2.3a) we should find kD ~ 5RT Ir¡. It is often convenient to write the numerical coefficient simply as n, so that: kT

(2.4)

DAB=---

nnr¡rAB

and kD =

(~6)(

Rr¡T).

(2.5)

Further, even the proportionality D ex 1/ r¡ breaks down when the solute molecules are small compared with the solvent molecules. These matters are considered further in Chapter 3 (Section 3.4.3). Measurements of diffusion coefficients, for which there is now a range of methods (Section 3.4.2), will permit the wider use of the Smoluchowski equation (2.1) in place of (2.3a), thus avoiding the assumption of Stokes-Einstein behaviour. Historical note The original treatment by Smoluchowski [2] was designed to account for the rates of coagulation of colloids, and was applied by him to experimental data on copper and gold soIs. An early application of the theory to homogeneous chemical processes was to the quenching of the fiuorescence of dye solutions, by Sveshnikoff in 1935 [3]. Debye [4] in 1942 replaced diffusion coefficients by the solvent viscosity, and introducted a correction term for interionic forces. The mathematical theory was improved by Collins and others (1949 seq.), who also considered random-walk theory. In the 1950's, the rapid development of experimental methods for the study of very fast reactions [5] was accompanied by theoretical advances; the position in 1961 was summed up in a seminal review by Noyes [6]. Since then there has been much interest in developing theoretical approaches [7] to the many problems thrown up by the increasing mass of experimental results, including those on macromolecular systems such as enzymes. The following account of the concentrationgradient approach draws largely on the treatment ofNoyes {6]. 2.2.2. Extension of diffusion theory to include activation requirements The assumption made in the simple version of diffusion theory outlined above, that every encounter results in reaction, refers to a limiting case; in general, there will be encounters which do not result in reaction before the molecules separate. This will be the case if either activation energy or a particular geometrical orientation is required for reaction. Moreover, Equation (2.2) leads to the anomalous result that as the viscosity approaches zero the calculated rate constant tends to infinity, whereas its maximum value must in fact be the collision number corresponding to absence of solvent, Le., the gas-phase value. An improved calcu-

The Rates of Dijfusion-Controlled

Reactions

25

lation removes this anomaly and allows us to refine the model so as to include activation requirements [8].3 Suppose we consider any bimolecular reaction in solution, whatever its rate, and apply Fick's laws to the general case where activation energy and orientational factors are important as well as diffusion. Carrying through the calculation (see below, Chapter 3, Section 3.3.2), one finds that in the steady state the observed second-order rate constant k will be given by (in place of Equation (2.1)) the following equation:

(2.6)

where the newly-introduced quantity ko is the value of the rate constant that would be observed if there were no diffusion effects and the rate were limited only by a collision number equal to that in the gas phase, along with any chemical interactions between molecules in collision, including both activation energy and orientational requirements. (In that imaginary situation, the equilibrium spatial distribution of molecules would be maintained; however fast the reaction, it would not lead to local depletion of reactant molecules.) The Equation (2.6) may be rewritten as: (2.7) With the express ion for kD in Equation (2.1), this becomes: (2.8)

The significance of this useful relation may be seen if we replace each rate constant by a reciprocal relaxation time «). Then Equation (2.8) becomes: