Reaction Kinetics, Vol. 1: A Review of the Recent Literature Published up to December 1973 (A Specialist Periodical Report) (v. 1)

A Specialist Periodical Report

Reactio n Ki netics Volume 1

A Review of the Recent Literature Published up to December 1973 Senior Reporter

P. G. Ashmore, Department of Chemistry, University of Manchester lnstit ute of Science and Technology Reporters

S. W. Benson, Stanford Research Institute, Menlo Park, California, U.S.A. 1. M. 7 . Davidson, University of Leicester

R. J. Donovan, University of Edinburgh H. M. Gillespie, University of Edinburgh B. F. Gray, University of Leeds A. Jones, Shell Research Limited, Thornton-le-Moors, Chester H. 0. Pritchard, York University, Downsview, Ontario, Canada P.J. Robinson, University of Manchester Institute of Science and Technology R. W. Walker, University of Hull

0 Copyright 1975

The Chemical Society Burlington House, London, W I V OBN

ISBN :0 85186 756 1

Printed in Great Britain by Billing & Sons Limited Guildford and London

Foreword There have been many changes of plan during the evolution of this first volume of Specialist Periodical Reports on Reaction Kinetics. As emphasised vividly by Professor Benson in the first chapter, chemical kinetics has ceased to be a small, self-contained section of physical chemistry, and is now of great influence in chemical engineering, in all branches of chemistry, and in biochemistry, as well as providing a basic tool for exploring the mechanisms of chemical reactions. It is not surprising, therefore, that preliminary searches and enquiries revealed that a good deal of current research on the rates of reactions is (or will be) reviewed in published or planned volumes of other Specialist Periodical Reports titles. Many aspects of recent work on reactions in solution, on photochemical reactions, and on surface reactions are covered in this way. The logical reason for another SPR title, specifically on reaction kinetics, is to give systematic reviews of progress in theoretical and experimental work on the rates of elementary reactions, especially in the gas phase, and to report on advances in modelling combinations of elementary processes to give the best description of macroscopic rates in various phases or at their interfaces. Unfortunately, in all these investigations we are far beyond the emergence of the grand basic ideas that altered the whole approach to the subject - ideas linked inescapably with the work of, infer alia, Guldberg and Waage and Arrhenius on rate expressions; of McC. Lewis and Lindemann on collision and unimolecular theories; of Eyring and Polanyi on transitionstate theory; of Bodenstein and Semenov on chain reactions; and of Langmuir and Taylor on surface reactions. Our efforts to understand and control the factors that affect rates of reaction have diverged, and in both theory and experiment we have (to quote Benson) ‘ . . . . . specialists . . . . and-subspecialists . . . . and mission-oriented kineticists . . . . It was clear from the start that there could not be a single theme, even for one volume. However, it was necessary to decide whether each article should be addressed only to its own practitioners, or be attractive and intelligible to those working in other kinetic studies. I have chosen the second plan, and I hope that the resulting mixture will appeal to a fairly wide audience. After the introductory chapter, the articles fall into two groups. Chapters 2-5 consist of critical reviews of rate data for particular kinds of reaction, with emphasis on the current methods used to determine ‘rate constants, and their Arrhenius parameters, mainly for reactions in the gas phase. The second group, chapters 6-8, contains three essays on recent treatments of sets of elementary reactions, two being oriented towards explaining particular rate phenomenon and the other dealing with modellingin general. Unfortunately, but inevitably, some important aspects of kinetic studies have not been covered in this volume.

iv

Foreword

There is no systematic review of work on rates in solution, but this is fortunately compensated by an article in the Annual Reports for 1973 (Section A), and another article there deals with recent extensive work on catalysis by oxides; other articles on the kinetics of surface reactions have appeared in the Specialist Periodical Reports on Surface and Defect Properties of Solids. However, it is hoped to cover certain aspects of researches on heterogeneous catalysis and on solution kinetics in the future volumes, and also to cover in more detail some theoretical treatments of elementary processes. Serious proposals for contributions on these or other topics for future volumes, to be published at approximately two-yearly intervals, will be very welcome. I am very grateful to Dr. P. J. Robinson for many helpful discussions during the planning of this volume. July 1974

P. G . ASHMORE

Contents Chapter 1 Chemical Kinetics-Retrospect By S. W . Benson

and Prospects

1

1 Introduction

1

2 Gas-phase Kinetics - Neutral Species

3

3 Gas-phase Kinetics - Charged Species

8

4 Symmetry Conservation

9

5 Condensed Phases

11

6 The Crystal Ball

12

Chapter 2 Reactions of Atoms in Ground and Electronically Excited States By R. J. Donovan and H. M.Gillespie

14

1 Introduction

14

2 Experimental Techniques Time-resolved Atomic Absorption Spectrophotometry Time-resolved Atomic Resonance Fluorescence Chemical Lasers Time-resolved Spontaneous Emission Molecular Beams Flow Systems Other Techniques

16 16 18 19 21 22

3 Hydrogen Atoms Ground State H( 1 %'+) Electronically Excited H(2T.J and H(22S+)

24 24 31

4 Alkali Atoms Ground state (n2S+)Atoms Electronically Excited (napJ)Atoms

32 32 34

5 Alkaline Earth Atoms

35

22

23

Contents

Vi

6 CarbonAtoms

39

7 Nitrogen, Phosphorus, and Arsenic Atoms

42

8 Oxygen Atoms Ground State O(z3PJ) Electronically Excited 0(2lD Electronically Excited O(2lS0)

49 49 56 59

9 Sulphur, Selenium, and Tellurium Atoms Ground State (n3Pz)Atoms Spin-orbit Excited States (n3P1,0) First Excited Singlet State (n1D2) Second Excited Singlet State (nlSo)

60 60 64 66 69

10 Halogen Atoms Fluorine Chlorine Bromine Iodine Halogen Atom Recombination

70 70 76 77 79 80

11 Noble Gas Atoms

82

12 Heavy Metal Atoms Mercury Tin Lead

85 85 90 92

Chapter 3 Unimolecular Reactions By P. J. Robinson

93

1 Introduction

93

2 General Theoretical Aspects

93

3 Intramolecular Energy Randomization Thermal Energization Chemical Activation Studies Hot-atom Substitution Products Photoactivation Crossed-beam Studies Unimolecular Reactions of Ions Laser Studies Trajectory Calculations Conclusions

96 97 97 1Q3 104 104 110 112 112 113

Contents

vii

4 Intermolecular (Collisional) Energy Transfer

115

5 Other Chemical Activation Studies Input Data for Calculation All Available Studies to Determine the Critical Energy Eo Studies to Determine A,

120 121 122 124

6 Thermal Unimolecular Reactions in the Fall-off Region

126

7 Thermal Unimolecular Reactions in the High-pressure Region Cyclopropane and iIs Derivatives Cyclobutane Derivatives Cyclobutene Derivatives Polycyclic Systems Olefins Heterocyclic Compounds Dehydrohalogenations and other Four-centre Eliminations Six-centre Eliminations Isocyanides Bond-fission Reactions Carbene-forming Reactions Reactions of Free Radicals

128 132 135 138 139 146 146

Appendix

150 151 155 155 159 159 160

Chapter 4 A Critical Survey of Rate Constants for Reactions in Gas-phase Hydrocarbon Oxidation 161 By R. W. Walker 1 Introduction

161

2 Outline of Experimental Methods I Direct Studies of Hydrocarbon Oxidation, Inside or Outside Cool-flame Limits I1 Addition of Traces of Hydrocarbons to Slowly Reacting Mixtures of Hydrogen and Oxygen in Aged Boric-acid-coated Vessels at 733-773 K I11 Direct Oxidation of Aldehydes IV Kinetic Spectroscopy V Chemical Shock Tubes VI Photochemical Oxidation VII Hydrocarbon Flames VIII Very Low Pressure Pyrolysis (VLPP) IX Free-radical Buffer System

164 164

165 165 166 166 167 167 167 168

viii

Contents

3 Thermochemical Aspects of Hydrocarbon Oxidation

169

4 The Initiation Process

172

5 Radical-Radical Reactions 177 Recombination of Alkyl Radicals 177 Radical-Radical Reactions not involving Recombination 181 6 Radical Decompositions and Isomerizations Radical Decompositions Isomerization Reactions of Alkyl Radicals

184 184 186

7 Radical Attack on Alkanes and Related Compounds

187

8 Oxidation Reactions of Alkyl Radicals General Discussion Oxidation Reactions of the Methyl Radical Oxidation Reactions of Higher Alkyl Radicals Overall Rate Constants Formation of Conjugate Olefins Reactions of RO;: and QOOH Radicals

196 196 197 201 201 202 204

Chapter 5 Kinetic Studies in Silicon Chemistry By 1. M. T. Davidson

212

1 Introduction

21 2

2 Short-lived Molecules withp,-p, Bonds to Silicon

212

3 Reactions of Silyl and Alkyl Radicals

Reactions of Alkyl Radicals with Silicon Compounds Reactions of Silyl Radicals

214

214 218

4 Silylenes and Carbenes Inorganic Silylenes: Non-kinetic Work Inorganic Silylenes: Kinetic Work Organic Silylenes Carbenes Reactions of Carbon and Silicon Vapour with Silanes Insertion of Methylene into Silanes: Chemical Activation

221 221 221 225 225 226

5 Pyrolysis of Silicon Compounds

228

6 Thermochemistry

234

227

ix

Contents 7 High-energy Chemistry Ion-Molecule Reactions

235 236

8 Rearrangements

237

9 Molecular Elimination Reactions

239 24 1

Elimination Reactions of ‘Hot’ Molecules

10 Conclusions:

Chapter 6 Network Effects in the Dissociation and Recombination of a Diatomic Gas By H. 0.Pritchard

241

243

1 Preface

243

2 Introduction

243

3 A Variety of Theoretical Approaches Equilibrium Theories Non-equilibrium Theories

247 247 25 3

4 The Philosophy of the Present Approach

255

5 The Diatomic Dissociation-Recombin Problem

256

6 Transition Probabilities Bound-Bound Transitions Bound-Unbound Transitions

260 260 264

7 Solving the Master Equation

266

8 How DisequiIibrium Affects the Arrhenius Temperature 272 Coefficient 9 The Emergence of Network Effects 10 Conclusions The Nature of the Dissociation Process Stumbling Blocks to Progress Network Effects in Chemical Physics

275 28 1 282 286 289

Contents

X

Chapter 7 Recent Advances in the Analysis of Kinetic Data 291 By A. Jones 1 Introduction The Model-building Process Computing Predictions from Rate Equations Fitting Models to Data Sensitivity Analysis Confidence Limits Model Discrimination

29 1 29 1 293 294 295 295 296

2 Computing Predictions for Models Represented by Ordinary Differential Equations 296 Integration Methods 296 Integration Step Size for Fast Processes 298 Implicit Integration Methods 299 The Phenomenon of Stiffness 300 A Multi-step Implicit Integration Method 301 3 Least-squares Fitting of Non-linear Models The Taylor Series Method The Steepest Descent Method Marquardt’s Algorithm Generalized Inverses

Chapter 8 Kinetics of Oscillating Reactions By B. F. Gray

303 305 306 306 307

309

1 Introduction

309

2 Experimental Facts The Decomposition of Hydrogen Peroxide The Belousov-Zhabotinskii Reaction The Sodium Dithionite Decomposition The Oxidation of Carbon Monoxide The Non-isothermal Oxidation of Organic Compounds Biochemical Systems, Closed and Open Oscillations in Open Reactions

310 310 312 318 318 321 330 332

3 Theory of Oscillating Reactions The Basic Equations Conservation of Mass Detailed Balancing Chemical Feasibility

333 333 335 336 338

xi

Contents Mathematical Techniques Two-variable Systems The First Method of Liapounov Limit Cycles Behaviour in the Large Parasitic Oscillations Piecewise Linear Systems N-Variable Systems Hurwitz and Similar Criteria The Second Method of Liapounov Closed Systems: General Results Stability of Equilibrium Damped Oscillations and Quasistationary States Open Systems: General Results

339 339 341 342 343 344 345 345 348 349 349 349 353 358

4 Specific Examples of Oscillating Reactions The Be1ousov-Z habotinskii Reaction The Catalytic Decomposition of Hydrogen Peroxide The Decomposition of Sodium Dithionite The Lotka Mechanism The Carbon Monoxide Oscillator End-product Inhibition Glycolytic Oscillations The Thermokinetic Oscillator of Salnikov Thermokinetic Oscillations in a CSTR Thermokinetic Models of Organic Oxidation Oscillations ‘The Brusselator’ Miscellaneous Oscillators

359 359 363 365 366 368 369 370 372 373 374 3 80 383

5 Conclusions

384

Author Index

387

1

Chemical Kinetics - Retrospect and Prospects BY S. W. BENSON

1 Introduction

From time to time my now 16-year-old son, repeating a time-honoured but half-forgotten ritual, will ask me what I do for a living and as I absentmindedly chant in reply the litany - ‘. . . chemistry . . . physical chemistry . . . chemical kinetics.. .speed of chemical reactions. .. I can see the expression of curiosity changing slowly and familiarly to one of resigned bafflement. The situation is quickly recovered with another set of words, ‘. . . rocket engines . . . fires . . . explosions . . . atomic bombs . . . digestion . . . but neither of us has yet had the courage to explore the gulf between these languages. Chemical kinetics as a formal science can be today reckoned to be about a hundred years old, but, despite its pervasive involvement with nearly every branch of science and technology, it has never stirred many sparks in the public imagination. Probably, it is too far behind the front lines. This is, I believe, an unfortunate situation since the demands of our growing, complex, industrial-technological society will place an increasing burden of responsibility on chemical kinetics to provide answers to problems which seem to grow exponentially in their molecular complexity. In the present Report I would like to review some of the changes and involvements which have occurred in chemical kinetics, and then, hopefully, to look into the crystal ball and try to make some educated guesses about what the future is likely to hold. Any special field of knowledge starts in the observer mode with an assembly of described experiences. These may become categorized if common variables can be discerned, and the final stage sees the emergence of quantitative relations which completely describe the observations and permit their expression in mathematical form. The variables in a kinetic system are basically chemical composition (reactants, products, catalysts), pressure, and temperature, ignoring for the moment physical state and external fields. Chemical kinetics emerged as a quantitative science with the statement of the law of mass action by Guldberg and Waage and attained some form of adolescence with Arrhenius’ expression for the temperature dependence of the rate of chemical reactions. If one delves into the history of this early period one cannot help but be struck by the incredibly small data base which served to inspire both of these fundamental generalizations. In the next 50 years (roughly 1870-1920) there followed a period of very slow growth. Very few scientists were attracted to study kinetic phenol

2

Reaction Kinetics

mena. On the other hand the same could probably be said of most fields of chemistry during this period. Chemistry had not yet made a great social impact. As a hobby, however, chemistry was still relatively inexpensive and any enthusiast with a stopwatch and the patience to perform many repetitive chemical analyses could engage in the action in kinetics. Adequate tools had not yet been developed to explore many of the most interesting phenomena such as combustion and explosion. However, the real flowering of physical chemistry during this period gave an early inspiration to the application of physico-chemical methods of analysis to the study of kinetic processes. Optical and electrical properties were used to follow the course of reactions in solutions while gas reactions could frequently be followed simply by observing pressure changes. World War I gave a large impetus to the acceleration of scientific research and witnessed the extensive involvement of both industry and government in the support of science. The period between World War I and World War I1 also witnessed the development of the conceptual basis for chemical kinetics. The methods of describing molecular behaviour, using the tools of statistical mechanics, were outlined in useful detail for gas reactions by the important contributions of Lindemann, Rice, Ramsperger, Kassel, Evans, Polanyi, and Eyring, leading to the most recent expressions for unimolecular reactions in the RRKM (Rice-Ramsperger-Kassel-Marcus) formulation of Transition State Theory. Transition State Theory itself has blossomed into a primary underpinning of all ‘equilibrium’ kinetic theory for both gases and liquids. World War I1 gave a further enormous impetus to the development of both science and technology. The status of science in various countries is measured today in terms of the fraction of the total national effort* (GNP) expended on science (more properly, technology). The very concept of such a measure would have been the occasion of great humour in scientific circles prior to 1930. Perhaps an even greater impact introduced-by World War I1 was the development of electronic tools for measurement. The unbelievable rate of growth and sophistication of electronic devices, particularly solidstate devices, since 1940 is probably the single biggest common feature in the research activity of the past three decades. In a very profound sense we, meaning all scientists, may be said to be in the ‘Electronic Age’. What this has done primarily has been to give us the ability to explore the details of chemical interactions on a molecular level and to answer questions which would have been considered moot or meaningless just a short time ago. It has not changed our conceptual understanding of chemical kinetics (which was basically complete with the Dirac Equation) but it has made it possible to use quantum theory to explore the rich and complex phenomena of manybody intcractioiis both in space and in time. One eloquent testimony to this rich development in chemical kinetics has been the growth of specialities. We have today specialists in gas kinetics,

* In so-called ‘advanced’ countries, it is common to find from 2 to 4?$ of the gross national product spent on research and development.

Chemical Kinetics - Retrospect and Prospects

3

solution kinetics, and catalysis, and in these areas we have sub-specialities of ion-molecule reactions, atom-electron reactions, free-radical reactions, and unimolecular reactions. There are kineticists today who have devoted almost their entire professional lives to the kinetics of energy-transfer processes. In the areas of solution kinetics, scientists who study ionic reactions or ‘redox’ reactions rarely talk to kineticists who deal with enzyme kinetics or ‘concerted’ reactions. Finally, there are ‘mission-oriented’kineticists whose field of application may cut diagonally across many of these specialities. Lasers, space travel, rocket engines, automobile exhausts, photochemical smog, electrical discharges, and most recently the spectroscopy of interstellar dust have all generated communities of scientists with much applied but little basic interest in the results of chemical kinetics. Let us look at some of these specialities in some greater detail. 2 Gas-phase Kinetics - Neutral Species Studies of gas-phase reactions have always been technically the most difficult and expensive of kinetic researches. Their results, however, have also been the simplest to interpret at the molecular level and so they have been, and will probably continue to be, at the forefront of our ‘basic researches’ in chemical kinetics. They have been the traditional testing ground of molecular theories of kinetic processes. All chemical reactions in dilute (i.e. <1 atmosphere pressure) gases can be looked upon as sequences of elementary steps involving either one or two molecules at a time. Thus an overall termolecular event such as atom recombination can be considered to be a sequence of two bimolecular collision events in the first of which a short-livedcollision complex is formed. If an isolated molecule has a sufficient amount of internal energy, it may localize this in such a way as to rearrange its atomic structure (i.e. isomerize) or break a bond, and this becomes a unimolecular process. Unimolecular reactions are perhaps one of the best understood processes in chemical kinetics. We have a quantitative theory, the transition state theory, which permits us, with some physically reasonable, empirical assumptions, to evaluate the Arrhenius A-factor for the unimolecular reaction of a Maxwell-Boltzmann, thermalized population of reactant molecules. The accuracy with which we can do this is probably in most cases as good as or better than that with which the A-factors can be measured. In simple bond-breaking reactions (e.g. C2Hs --t 2CH3) and a limited number of 1,2-elimination reactions (e.g. CH,CH2C1 -+C2H4 HCl) we can also predict the activation energies, and so predict quantitatively how the rate constants vary with temperature. For the numerous other categories of unimolecular reactions, e.g. complex rearrangements (cyclopropane -+ propylene), we have no theoretical model which allows us to predict activation energies. The best we can do in such cases is to predict the effects of substituents on changes in the activation

+

React ion Kinetics

4

energy and hence to predict, empirically, activation energies in homologou s series, knowing one of the members of the series. The extension of such methods to highly branched compounds and ‘unusually’ strained compounds still remains an incomplete task and one which will await a much more basic understanding of both non-bonded interactions and ‘poly-centre’ valency. Thus we have no reliable method for predicting the activation energy of a 1,2 atom or group transfer which can in principle occur in ions, in free radicals, and in molecules :

CH36HCHzCH3

CH,=CHCD,Br

CH$HCH,

e

BrCH,---CH=CD,

Electronic techniques when applied to the measurement of physical properties have given us an extraordinary facility in exploring events occurring in ever shorter intervals of time. Mass spectrometric methods of analysis have given us the possibility of sampling gas systems in times of the order of lom4s. Molecular spectroscopy together with ultrasensitive, rapid-response s and has been thus used to follow reactions detectors can resolve times of behind shock waves. Recent developments in nano- and pico-second pulse laser spectroscopy give promise of following chemical events down to times s. Such techniques have made it possible to identify, of the order of study, and measure the production and subsequent reactions of unstable intermediates produced in chemival reactions. Free radicals and vibrationally and electronically excited species have by these means been ‘seen’ in rapid chemical reactions and we today have a good deal of data on their reactivity. One of the very exciting recent developments in this area has been the mol use of resonance absorption to measure, in situ, very small ( dmm3)concentrations of atoms and small molecules and radicals. The companion technique of resonance fluorescence is simpler to use and even more sensitive. The development of tuneable lasers for the wavelength region 200-400 nm, which seems imminent, would trigger an avalanche of interesting kinetic investigations in gas-phase reactions probably comparable to that which followed the introduction of flash photolysis techniques. With so much of the basic creative urge in kinetic circles being devoted to exploring ever smaller intervals on the time scale it is rarely appreciated that a quite considerable scientific market exists for understanding kinetic N

Chemical Kinetics

- Retrospect and Prospects

5

phenomena over very long time periods. A reliable knowledge of very, very slow rate phenomena is of considerable interest to geologists trying to understand the evolution of the earth’s crust over the past loQf1 years and also to geochemists anxious to retrace the evolution of our planetary system from the presumed Laplacian dust clouds of 10rofl years ago, Although these problems do not seem to pose challenges of appreciable urgency now, the future (i.e. >106*1years) of life in this solar system may well rest on the success of distant generations in deciphering the cosmic record. More contemporary interest in such slow reactions is to be found among bridge builders trying to make more enduring structures, among communications engineers trying to plant more permanent telephone poles, and among biologists trying to estimate the cumulative effects of our now rapidly changing chemical environment on various physiological functions and structures. The measurement of very small rate constants in a ‘short’ time is a real challenge. The RRKM theory of unimolecular reactions gives us a theoretical handle for exploring unimolecular reactions of non-Maxwellian populations of energized molecules. This had been a popular subject for many years but had been restricted to either reactions of small molecules at moderate C1) or else large molecules at very low pressures (e.g. N0,CI + NO, pressure (10-1-10-2torr). By working at much higher temperatures and much lower pressures (10-3-10-4torr) the Reporter and his colleagues have been able to extend such studies to larger molecules with as many as 20-40 atoms. The technique is called ‘Very Low-pressure Pyrolysis’(V.L.P.P.) and it has been very valuable in demonstrating the validity of applying RRKM theory to such extreme conditions. In the same period another technique of preparing populations of very excited molecules in narrow energy ranges (chemical activation) has been exploited by Rabinovitch and his co-workers in testing various details of unimolecular rate theory such as energy transfer between molecules in gas-phase collisions and the rates at which energy is redistributed among the internal degrees of freedom of a large molecule. Infrared spectroscopy together with laser techniques have both inspired and made possible very extensive investigations of the exchanges of rotational and/or vibrational energy in collisions between small molecules. In the past, such processes had been studied indirectly via measurement of sound dispersion or shock propagation through gases, The laser techniques have made it possible to pinpoint the histories of individual vibronic states in diatomic and linear triatomic molecules. No one appears to have attempted to extend such techniques to larger molecules. The theory of such energytransfer processes seems to be reasonably well understood and when extended to include the appreciable dipole forces which exist in the collisions of polar molecules, such as HF or H,O, seems to be able to account quantitatively for the rates observed. A comparable theory of energy transfer in larger molecules neither exists nor currently seems capable of quantitative testing. Picosecond laser technology may soon change this.

+

Reaction Kinetics

6

Bimolecular events in the gas phase fall in the category either of an energytransfer process such as we have discussed or of an association reaction (complex formation) or finally, and perhaps the sole chemical event, a metathesis reaction in which an atom or group is transferred from one species to the other:

===

H+F,

M e + EtBr

H F + F MeBr + Et

These metathesis reactions are among the oldest categories of bimolecular reaction studied by gas-phase kineticists and are in part reasonably well described by transition state theory. Their A-factor corresponds to what are called ‘tight’ transition states* and can be estimated to within a factor of three, which is as good as most experimental uncertainties. The activation energies for these processes, however, continue to be elusive. A number of empirical techniques exist for either estimating or calculating these activation energies to within ca. +8 kJ mol-1 uncertainty. Although this seems quite good it must be reckoned against the observation that ‘intrinsic’ activation energies (i.e. in the exothermic direction) for metathesis fall in a very narrow range of ca. 30 10 kJ mol-1 for most examples and 0-65 kJ mol-1 for all known examples. This is a theoretical vacuum which up to now has attracted relatively few prophets. Recent progress in our ability to predict heats of formation of ions, molecules, and free radicals suggests that this situation is overdue for remedy. One of the most fascinating areas on the bimolecular scene has been the four-centre double metathesis reactions : AB + CD

* AC + BD

Considered once the prototype par excellence of the collision theory of chemical reaction rates, it is today more aptly dubbed the ‘abominable snowman’ of chemical kinetics, with no extant authentic examples for covalently bound molecules. When Sullivan demonstrated that the reversible reaction H, + I,

2HI

could be completely accounted for by the te.rmolecularmechanism I + H , + I

IH+HI

* A tight transition state may be described as one in which near-neighbour distances are within ca. 0.6 8, (0.06 nm) of normal covalent distances.

Chemical Kinetics - Retrospect and Prospects

7

he made illegitimate the last member of the royal line which had, in fact, inaugurated the modern study of gas reactions. A number of suggestions have been made to account for the fact that such reactions have very high activation energies. In the absence of valid examples it is difficult to decide which of these properly accounts for the remarkable coincidence that, in all of these reactions, the free-radical pathway is always at least 10-100-fold faster than the possible bimolecular pathway. Also relevant to the double metathesis is the recent series of inquiries into the different kinds of activation energy effective in chemical reactions. It has been known for some time that simple, exothermic metathesis reactions are often capable of producing product molecules with exceptionally large amounts of vibrational energy. Thus the reactions H

+

Ch

+ C1 + 188kJmol-' HF + F + 405kJmol-'

4 HCl

H + F, +

put ca. 60% or more of their excess energy into vibrational energy of the product hydrogen halides. A random partitioning of this energy would place ca. 3 3 % into vibration while a properly weighted statistical model would put less than 10% into vibration. Polanyi has shown, using various geometries for the transition state, that the short-range nature of the forces and the short life of the transition state make the dynamics of the collision decisive for the energy distribution. In the above examples an additional effect of the light mass of hydrogen favours the appearance of vibrational energy in the products HX. By detailed balancing we would decide that in the reverse reactions vibrational energy in the product .molecules would be more effective than translational or rotational energy, and such observations have touched off a number of investigations on the role of various partitionings of activation energy in effecting bimolecular chemical reactions. The recently burgeoning field of 'molecular beam' kinetics has become an appropriate tool for exploring such questions and a number of such effects have already been reported. A very impressive use of ultrasonic expansion techniques to obtain high densities of molecules with high translational energies has been used by Anderson and Jaffe. They found that HI + HI collisions failed to produce H2+ 21 even with centre-of-mass translational energies of up to 800 kJ mol-'. The conclusion is that in this particular case the HI vibrational energy is at least 10 times more effective than translational energy in producing reaction. Such considerations have given rise to the possibility of producing very specific, unique, chemical reactions at room temperatures by selective vibrational excitation of one of the reactant molecules. Chemical lasers would be a natural tool for such purposes. A number of reports have appeared in the literature in which laser excitation appears to have initiated such specific reactions. However, the reactions have been either complex chemically or marginal in yield and fully convincing examples have yet to be demonstrated.

8

Reaction Kine tics 3 Gas-phase Kinetics - Charged Species

Mass spectroscopists have for a long time been studying bimolecular, electron-molecule reactions and the subsequent unimolecular decay of the excited ions so produced. RRK and RRKM theories have both been successfully adapted to describing the absolute and relative reaction rates thus observed but the complexity of the overall systems has probably limited interest in the field. In the past few years related studies on ion-molecule reactions have yielded results of great interest to chemists. Two techniques have sparked these efforts. One has been the ion-cyclotron resonance method in which positive or negative ions produced in one region can be 'trapped' in another reaction region for periods of up to 0.1 s. The sensitivity of the method, which permits observation of ions at concentrations as low as 5 ions ~ r n -has ~ made possible the observation of bimolecular metathesis reactions where activation energies are less than ca. 16 kJ mol-l. AX'+B

A+BX'

AH + B-

A- + BH

In the exothermic directions these reactions have activation energies and usually zero. This result Can be rationalized relative to radical-molecule metathesis reactions (which are structurally similar) where intrinsic activation energies are generally low (ix. 0 < E < 64 kJ mol-l). The ion-molecule reaction has a collision energy of the order of 80 kJ mol-l, due solely to attractive polarization forces (ion-induced dipole),

< 16 kJ mol-l

Epol= c2a/2r4 and this energy is usually enough to overcome any small barrier. The collision cross-sections are in fact close to the so-called Langevin cross-sections calculated from the polarization attraction. What has made these reactions of great interest to chemists has been the recent discovery that many of these reactions can be studied in both directions, either directly if the exothermicity is small (< 16 kJ mol-l), indirectly via cycles, or by employing auxiliary data. This makes it possible to obtain equilibrium constants for gas-phase reactions of ionic species and from them the heats of formation of ionic species in the gas phase. Electron affinities, proton affinities, and heats of solvation of ions by one or two solvent molecules have been forthcoming. Comparison of these data with comparable data for ionic reactions in solution have for the first time made it possible to obtain directly and quantitatively the effect of the solvent on ionic equilibria and on ionic reactions. This is bound to provide an enormous impetus to the theoretical understanding of ionic solvation and ionic reactions which should unfold rapidly in the near future.

Chemical Kinetics - Retrospect and Prospects

9

What has already emerged is that some of the ‘classical’ interpretations of acidity and reactivity offered in terms of reactant structure were in error and the observed reactivities had nothing to do with reactant structure except in so far as it modified solvation energy. Thus ButOH is a much stronger acid in the gas phase than MeOH because the electron affinityof the ButO. radical is much greater than that of the MeOm radical. The more effective solvation of MeO- in solution reverses this result so that MeOH is a stronger acid in solution. Similar data are appearing on the relative proton affinities (i.e. base strengths) of the substituted amines. In the gas phase successive methyl substitution increases the proton affinity of NH, quite appreciably. This alone would suggest that Me,N should have a pK, in water solution some 10 units greater than that of NH,. In fact the two are very close in base strength, which can now be attributed to the counterbalancing effect of much stronger solvation of NH4+compared with Me,NH+.

4 Symmetry Conservation Kinetic data had been very slowly accumulating to indicate that some gross physical properties of molecules or collision complexes, such as electronic spin, were conserved during chemical reactions. It remained for the Woodward-Hoffman rules to indicate that certain more general features of electronic orbital symmetry were important in many large classes of reaction in regulating the overall course of the reaction. Reactions such as the 1,2 molecular addition of H, to olefins are now looked on as ‘forbidden’ and in fact it is known that despite their exothermicity (ca. 80-120 kJ mol-l) their activation energies exceed 180 kJ mol-l. Simple molecular orbital analysis of the reaction indicates that the orbital symmetry properties of the ground state and the transition state correlate with an excited state, not the ground state of the product molecule. Unfortunately such rules have been to date ‘heuristic’ and not quantitative. It is not known how much ‘forbiddenness’ adds to the activation energy of such reactions. Very ingenious experiments have been designed to force reactions to follow ‘non-allowed’ paths by imbedding a reactive function in a stiff ring system. A few such experiments have indicated that forbidden and non-forbidden paths in electrocyclic reactions may differ in activation energy by perhaps 55-60 kJ mol-l, but the rules have still to be given quantitative character either theoretically or by broad experimental example. The reactions of cyclic molecules have provided one very active field of study for such symmetry effects and the results at present can only be described as highly controversial. A simple example is afforded by the competing reactions of geometrical and structural isomerization in 1,2-dideuteriocyclopropane :

Reaction Kinetics

10

D D

D

(dideuterio) The cis-truns reaction is ca. 12 times faster at 500" C than the olefin formation but otherwise very similar in Arrhenius parameters. The suggestion of a common, metastable, trimethylene 1,3-biradical intermediate in the reaction requires either the postulation of a 40 kJ mol-1 activation energy for ring closure of this biradical to cyclopropane or else that the energy of breaking the terminal C-H bond in the n-propyl radical is some 40 kJ mol-l higher than that for the first bond.

PH2\ CH, CH3 /cH2\ CH3 qH,

-

'

CH

CH,

''(?I, + H

AH" = 410 kJmol-'

(observed)

CH2 4*CH, /'

'CH,

+ H

AH" = 450kJ mol-'

Molecular orbital calculations favour this second possibility and suggest that every competing path may proceed by an independent transition state with no common intermediates. Some stereochemical evidence favours such an interpretation of many such electrocyclic reactions, The data are, however, equally well interpreted by having the reaction proceed through a common biradical intermediate with conformational changes (i.e. rotation) in competition with the rate-determining final step. I n some cases thermochemical data indicate that a concerted reaction may indeed occur via an intimate ion pair. The isomerization of 1,l-dichlorocyclopropane, which goes quantitatively to a single unexpected product 2,3-dichloropropane, may be an example :

It is likely that this field will remain active for some time and also that better knowledge of bond strengths and the thermochemistry of ring systems will be required to resolve it.

Chemical Kinetics - Retrospect and Prospects

11

5 Condensed Phases

The studies of condensed-phase reactions have long outnumbered those of gas-phase reactions for many reasons - simplicity of study, economy of effort, and relevance to other phenomena being but a few of them. It is likely that this situation will continue in the future if only because of the growing importance of biochemical rate processes. In most cases the reactions observed in the gas phase rarely occur in solution and vice versa. A prominent example until recently has been ionic reactions seen usually in solution. Free-radical processes and electrocyclic reactions have been widely observed in both gas and condensed phases and often the rate constants are similar with closely corresponding Arrhenius parameters. In ionic reactions, where solvation effects are so large, the rate parameters are understandably very different. For non-polar, bimo€ecularreactions in non-polar solvents, gas and solution rates are expected to differ very little and this is a consequence of the fact that the solvation energy of the transition state is expected to be not very different from the sums of the solvation energies of the two reactants. However, the entropies of solvation are not expected to show such compensation and so it is surprising when A-factors for bimolecular reactions do appear the same in gas and in solution. In truth there has not been much study of the effects of phase on rate constants and much of what has been done lacks sufficient thermochemical data to make a complete interpretation possible. In recent years studies of radical recombination in the gas phase and in solution have yielded some major apparent differences in the rates of recombination of ethyl, isopropyl, and t-butyl radicals. In the solution studies the rates have been very close to the diffusion-controlled limits whereas in the gas phase the current data suggest values from one to three powers of 10 slower. The uncertainties in the gas-phase data are sufficient to make the divergence not quite so large but the distinct possibility remains that rhere may well be basic differences between gas and condensed phase. One very large difference between the two phases lies in the nature of a collision. A non-reactive gas-phase collision between two species usually lasts about 10-13-10-12s, much shorter than a rotation time. In solution diffusion is very slow and molecules which do encounter each other stay together for times of the order of 10-9-10-10 s, much longer than rotation times. In addition, once molecules find themselves in near-neighbour proximity in solution they are favoured to make re-encounters even after separation, a phenomenon which gives rise to the well-known ‘cage-effect’. It may very well be that radical recombination reactions which have very small or zero activation energies but some appreciable orientation requirements may be indeed favoured in solution by the long encounter time as compared with the short gas-phase collision. Perhaps nowhere has electronics played such an important role as it has

12

Reaction Kinetics

in the explosive development of the study of rapid reactions. In the 1920s, a primitive form of stopped-flow kinetics made possible the study of kinetic events in times of milliseconds. Since World War 11, the use of the so-called ‘relaxation’ techniques, such as temperature jump, have pushed this to microseconds. The revolutionary discoveries of electron and nuclear magnetic resonance spectroscopy have made commonplace the study of free radicals in solutions and very rapid uni- and bi-molecular reactions. The 1i.m.r. technique of line broadening makes possible studies of such rapid unimolecular reactions as rotation about single bonds. Associated techniques such as ‘spin-echo’ have given new life to the studies of translational and rotational diffusion of molecules in different solvents. Dramatic as these new experimental developments have been, the pace of their extension by coupling them to sophisticated electronic methods for signal/noise enhancements and Fourier transform spectroscopy shows no sign of slackening. About the only thing we can say with certainty now is that we are in the midst of the most profound revolution in the experimental art of studying chemical changes and the end is nowhere in sight. 6 The Crystal Ball During the past decade there have been published some 4000-5000 articles per year devoted to the subject of kinetics and mechanism. The recent drought in scientific funding has scarcely affected this output and I can see little reason to expect a significant decrease in the near future. On the contrary, in fact, each advance in the experimental art has seen its counterpart in the development of a new kinetic specialization. Kinetics is the study of chemical changes in time and it is probably one of the most important aspects of chemical reactions. Because of this the need for kinetic study and the technological applications of kinetics can only grow. The real problem for kineticists is how to increase their contributions to meet those future needs without simply growing arithmetically in numbers. The needs will become more and more acute in the relatively new fields of biochemical kinetics where analytical techniques have only recently reached the stage where molecular interpretations of rate processes have become possible. Every science matures when it achieves understanding of its basic phenomena. At this point it loses major interest for scientists and becomes primarily an applied or engineering art. It is difficult to say how close kinetics is to this maturity. The basic phenomena of kinetics are today well recognized. Many of them, particularly the concepts, are well understood. A number can be given quantitative expression. However, at the level of many-body interaction, with all its paradoxical phenomena, which is where much of physical chemistry stands today, we are still very far from the kind of maturity which can make kinetics a ready and available tool for the biologist, the geochemist, the combustion engineer, the chemical meteorologist, the astrophysicist, or the ecologist. We have had too many surprises

Chemical Kinetics - Retrospect and Prospects

13

in the past decade, too many upsets of cherished notions to look forward with any smugness towards an early completion of our understanding of chemical kinetics.*

* Similar reservations apply to our experimental techniques. In a famous kinetics conference held at Minneapolis in 1950, F. 0. Rice suggested that a good, heatable, greaseless stopcock and either a wall-less reactor or an understanding of wall reactions might represent an enormous advance for gas-phase kineticists. The same can still be said in 1974.

2 Reactions of Atoms in Ground and Electronically Excited States BY

R. J. DONOVAN AND H. M. GlLLESPlE

Don’t believe what your eyes are telling you. All they show is limitation. Look with your understanding. . . . Richard Bach

Jonathan Livingston Seagull 1 Introduction It has long been recognized that the study of atomic reactions in the gas phase offers the most promising means of obtaining a detailed understanding of reaction dynamics. Recent technical developments have brought that goal within sight and we are probably entering one of the most interesting and challenging periods in the growth of reaction kinetics. In this chapter we shall attempt to summarize the present position, and show how techniques as diverse as molecular beams, chemical lasers, and flash photolysis are together providing the information we require. We shall also attempt to give a reasonably up-to-date account of the reactions of the following groups of atoms: H ; Li-Cs; Ca-Ba; C-Pb; N-Bi; 0-Te; F-I; He-Xe; Hg. This task has been considerably eased by the appearance of several other reviews 1-3 which cover the literature up to the beginning of 1973. We will not re-cover this ground in a comprehensive manner, but select areas which we feel are of particular interest, and extend the literature survey up to January 1974. Two important experimental developments are worthy of special note in this section. Molecular beam studies have now emerged from the ‘alkali age’ into the ‘organic age’, and the Faraday Discussion on Molecular Beam Scattering4 served as a timely means of emphasizing this point. With reactions such as F + C6H6 being studied, the esoteric image of molecular beams has now been dispelled. The second important development involves the emergence of resonance fluorescence techniques. Atomic resonance R. J. Donovan, D. Husain, and L. J. Kirsch, Ann. Reports ( A ) , 1972, 69, 19. K. J. Donovan and D. Husain, Ann. Repurts ( A ) , 1971, 68, 124. K. J. Donovan and D. Husain, Chem. Rev., 1970, 70,489. Faraday D i x u s ~ i o n sof the Chemical Society, No. 5 5 , ‘Molecular Beam Scattering’, 1973.

14

Reactions of Atoms in Ground and Electronically Excited States

15

fluorescence has been used in both pulsed (flash photolysis) and flow experiments; it allows the detection of atomic concentrations which are at least one order of magnitude less than those for other spectroscopic techniques (see experimental section below). Furthermore, the use of high-intensity tunable dye lasers has permitted the detection of fluorescence from molecular fragments in specific vibrational levels even under molecular beam condition^.^ Emphasis has recently been placed on the use of correlation diagrams for the consideration of reactions involving atoms and small molecules. This has led to an understanding of why some highly excited atomic states, e.g. 0(21S0) and N(22PJ),are relatively unreactive despite the fact that product formation is thermodynamically f a ~ o u r a b l e .The ~ importance of considering the MO configuration of the collision complex has been stressed.6 Thus reactions involving N(z4S,) O,(alA,), N(Z4S,) i- O#A,) and O+(24S,) N,(XIC,f) are all highly exothermic and reactants apparently correlate with products via quartet surfaces (several other examples are discussed later). The application of Walsh’s rules shows, however, that there are no low-lying quartet states of the collision complex and reaction will only proceed if a large barrier is surmounted, or non-adiabatic transitions take place. Further considerations of this type lead to the general conclusion that the number of low-lying potential energy surfaces will frequently be less than the number given by correlation rules alone. Thus it is important to consider the MO configuration of the collision complex in constructing correlation diagrams. Herschbach and Grice have also shown that simple MO considerations can be used to explain the scattering distributions observed in crossed-beam studies of reactions involving H, 0, halogen atoms, and CH3 with halogen molecules. The occurrence of non-adiabatic processes, sometimes with high probability, weakens the predictive strength of the above considerations. However, sufficient data now exist to allow analogies to be drawn between atoms of the same group, and by making due allowance for changes in spin-orbit coupling, the (approximate) rate and electronic state of the products for many reactions can be predicted with reasonable confidence. To provide firmer predictions it will be essential to have a more detailed and quantitative knowledge of the terms in the appropriate Hamiltonian which couple nuclear and electronic motions. Information of this type may be derived from spectroscopic and molecular beam scattering data, as well as from kinetic studies, and the future must surely see an even greater degree of co-operation between these fields. An interesting short review and an evaluation of rate data for several important reactions l o have also been published recently.

+

lo

+

H. W. Cruse, P. J. Dagdigian, and R. N. Zare, ref. 4, p. 277. B. A. Thrush, Faraday Discuss. Chem. SOC.,1972, No. 53, p. 235. D. R. Herschbach, ref. 4, p. 233. R. Grice, C. F. Carter, and M. Levy, ref. 4, pp. 357, 381. A. A, Westenberg, Ann. Rev. Phys. Chem., 1973, 24, 77. K. Schofield, J , Phys. Chem. Ref. Data, 1973, 2, 2 5 .

16

Reaction Kinetics

2 Experimental Techniques Time-resolved Atomic Absorption Spec6rophotometry.-This is perhaps the simplest and most generally applicable technique. The essential features are illustrated in Figure 1, which has been taken from a recent article describing an arrangement suitable for use in undergraduate teaching laboratories,ll and which serves to emphasize both the simplicity and inexpensive nature of such equipment. The atomic species of interest is produced by conventional flash photolysis techniques and the rate of reaction monitored by observing the attenuation of a suitable line in the atomic resonance radiation which is Vacuum Iine

It# I I

M

i!

co

p Ji3 i CU Charging Unit

cu

HS AL RV C M PM CO

High Voltage switch Atomic Emission Lamp Reaction vessel Collimator Monochromator Photomultiplier Cathoderay Oscilloscope

Figure L Simple experimentdl arrangement for time-resolved atomic absorption spectrophotometry (Reproduced by permission from J . Chem. Educ., 1974, 51, 51)

passed through the reaction vessel. The atomic resonance radiation is produced by a microwave or r.f. discharge in one of the inert gases which contains a small proportion of the element or a suitable compound of the eJement.l2 A monochromator of low resolving power but high luminosity is generally used to disperse the radiation which is then presented to a photomultiplier. The use of an interference filter in place of the monochromator would allow further simplification and even higher signal-to-noise ratios to be achieved. In most conventional arrangements, which employ a photomultiplier as light detector, the highest signal-to-noise ratio is obtained by maximizing the photocathode current; however, this may lead to excessive anode currents when the photomultiplier is used in its linear voltage region and to avoid such conditions it is necessary to modify the circuit. This is generally achieved M. G . Stock, R. J . Donovan, and D. J. Little,J. Chem. Educ., 1974, 51, 5 1 D. D. Davis and W. Braun, A p p l . Optics, 1968, 7, 2071.

Reactions of Atoms in Ground and Electronically Excited States

17

by combining the last three or four dynodes with the anode, thereby reducing the gain of the multiplier, while remaining in the linear voltage region and retaining the maximum photocathode current. Further details of this type of circuit may be found in a recent review.13 Under the conditions used for most experiments the attenuation of atomic radiation is not a linear function of the atomic concentration (typically lo1’ atoms cm-3) and it is usual to employ a modified form of the BeerLambert relationship, viz: O.D.

=

In lo/l= a(c.l)Y

where y is a coefficient which is determined experimentally. This procedure for converting the observed optical densities into relative atomic concentrations has recently been criticized by Clyne.14 Whilst this criticism is valid in principle, other experimental errors have tended to dominate over the limited range of atomic concentrations which have generally been employed. However, as rate measurements become more precise it will become more important to determine the form of the ‘curve of growth’. A more sophisticated arrangement, which has been employed l5 to monitor the reactions of 0(2lO,), is illustrated in Figure 2. The essential

Figure 2 Block diagram of the apparatus for the kinetic study of O(2lD2),generated by repetitively pulsed photolytic initiation, and monitored in absorption bv timeresolved attenuation of atomic resonance radiation in the vacuum ultraviolet. (a) Lithium fluoride windows; (b) coaxial lamp and vessel assembly; (c) microwave cavity (Reproduced b b r permission from Chem. Phys. Letters, 1972, 16, 530) lS

l5

G. Porter and M. A. West, in ‘Investigation of Rates and Mechanisms of Reactions’, ed. G. G. Hammes, Wiley-Interscience, New York and London, 1973,2nd edn., part 11. M. A. A. Clyne and P. Bemand, J. C. S. Faraday I t , 1973, 69, 1643. D. Husain, J. Wiesenfeld, and R. Heidner, Chem. Phys. Letters, 1972, 16, 530.

18

Reaction Kinetics

differences between this and the previously described arrangement, lie in the use of signal-averaging to improve the signal-to-noise ratio (which is in general low for fast reactions), and in the use of a solar blind photomultiplier which reduces the effect of scattered light from the flash and thus minimizes the effective ‘dead time’ of the multiplier (the ‘dead time’ may exceed the length of the flash by an order of magnitude under some conditions). Another method for reducing the effects of scattered light is to use a gating circuit to reduce the multiplier gain during the flash.ls Time-resolved Atomic Resonance Fluorescence.-This technique is similar in many respects to that for absorption spectrophotometry, with atoms generally being produced by pulsed photochemical methods. Instead of monitoring the attenuation of resonance radiation which is incident on the cell, however, the fluorescence produced by the atoms which have absorbed resonance radiation is observed. The experimental arrangement thus consists of a flash lamp (dissipating ca. 10 J), an atomic emission lamp, and a photomultiplier, each pointing along three mutually perpendicular axes towards the centre of the reaction cell. The technique is inherently more sensitive than atomic absorption methods and concentrations of ca. 1O1O atoms have been monitored for a number of atomic species. In principle, atomic concentrations as low as lo4 atoms ~ m may - ~be observed, but in practice this has only been achieved with the use of dye lasers (owing to the requirement of extremely high intensities fur the exciting beam) and has been limited to the study of alkali-metal atoms. Frequency doubling techniques should allow the future use of pulsed dye lasers down to ca. 200 nm, and thus the possibility of studying a much wider range of atoms. The first application of this technique to the kinetic study of atomic reactions was reported by Braun and Lenzi.” Further developments, due mainly to Braun and Davis,18have followed rapidly and a recent experimental arrangement is shown in Figure 3. A particularly simple arrangement has been described by Tellinghuisen and Brewer l 9 for monitoring ground-state iodine atoms, and extremely good signal-to-noise ratios were achieved on single-shot recordings using an oscilloscope (i.e. without using multiscalers as generally employed by Davis et all8). The technique has been applied mainly to the study of ground-state atoms owing to difficulties associated with monochromating the input atomic resonance radiation, while retaining sufficient intensity to produce reasonable levels of fluorescence. Thus if excited-state and ground-state atoms are produced, and no attempt is made to monochromate the exciting beam, all states may fluoresce simultaneously.

l6 l7

l9

A. U. Acuna, D. Husain, and J. R. Wiesenfeld, J . Chem. Phys., 1973, 58, 494. W. Braun and M. Lenzi, Discuss. Faraday Soc., 1967, N o . 44, 252. See, e.g., D. D. Davis, R. E. Huie, J. T. Herron, M. J. Kurylo, and W. Braun,J. Chern. Phys., 1972, 56, 4868, and refs therein. J . B. Tellinghuisen and L. Brewer, J. Chern. Phys., 1971, 54, 5 1 3 3 .

Reactions of Atonts in Ground and Electronically Excited States

19

MG

DG

Figure 3 Schematic drawing of the flash photolysis-resonance fluorescence apparatus. RL, resonance lamp; MG, microwave generators; PS, power supply; FL, fldsh lamp; RXC, reaction cell; VH, vacuum housing; PD, photon detector; PC, photocell; DG, delay generator; PA, pulse amplifier; DI, voltdge discriminator; MCA, multichannel dnalyser ; TTY, teletype; and OLC, on-line 1108 Univac computer for data processing (Reproduced by permission from J . Chem. Phys., 1972, 56,4868)

This problem has been circumvented in one study 2 o which effectively employed the atmosphere as a cut-off filter. The radiation emitted by a microwave-powered atomic iodine lamp is essentially limited to a single line at 206.2 nm, shorter wavelength transitions being both weaker and absorbed by O2 in the atmosphere. The 206.2 nm line is associated with the excited 2P+state of the iodine atom, and has thus been used to monitor this state in fluorescence. The apparatus employed for this study was extremely simple and again demonstrates that direct studies can be achieved with inexpensive equipment .20 The main advantages of using resonance fluorescence techniques are associated with the low concentrations which may be observed. Thus radicalradical type reactions can be minimized, and it is generally found that the fluorescence intensity is a linear function of the atomic concentration (the curve of growth is essentially linear for low atomic concentrations). Furthermore, very fast atom-molecule reactions can be followed under pseudo-firstorder conditions and over a long enough time interval to allow very precise measurements. Chemical Lasers.-As a technique for the study of reaction kinetics, the chemical laser is essentially a logical extension of the more conventional chemiluminescence techniques, and employs the excited products of a 2o

R. H. Strain, J. McLean, and R. J. Donovan, Chem. Phvs. Letters, 1973, 20, 504.

20

Reaction Kinetics

reacting system as a means for amplifying the radiative output. Thus it is possible to arrange for information on the distribution of excited states to be extracted from a system before relaxation occurs significantly. It is a potentially powerful technique but has not been as widely exploited as it deserves, particularly with regard to deriving quantiiative kinetic data. The reasons for this presumably lie in a general lack of familiarity with the technique and the theory involved, although the expense and technical expertise required are often less than with the corresponding experiment using spontaneous emission. Light amplification from a chemical reaction may be achieved if an inverted population distribution of products is produced sufficiently rapidly for laser gain to rise above threshold and thus allow stimulated emission to compete successfully with relaxation processes. Even when the total population of the upper level, in a vibration-rotation laser, is not greater than that of the lower level, light amplification may still be achieved on P-branch transitions (owing to the more favourable degeneracy factor) provided that Tvi, > Trot.This condition is generally referred to as a 'partial inversion'. FOCUS I NG M I R R O R S

P-SPHERICAL

IN ADJUST1 n' ' MOUNT

h-

-0 GAS ..-_...,.. -VACUUM WAN DLI NG I

1 sysrm J

lSYSTEM

I

I

GLASS JOIP1.IT1 ~

Figure 4 Chemicdl laser system. Broken line represents the path of the laser radiation. (Optical bench and protective shielding have been omitted) (Reproduced by permission from J. Chem. Educ., 1971, 48, 659)

A simple experimental arrangement 21 for a chemical laser is shown in Figure 4. This type of arrangement yields only limited information on the distribution of excited states in the reaction products; for example, the most highly excited level to show laser action gives a lower bound to the maximum excitation produced in a given degree of freedom. In principle it is possible to derive level populations, and reaction and relaxation rates by measuring the gain on each transition as a function of time. In practice, however, the experimental accuracy achieved to date has not been sufficiently high to allow reliable quantitative data to be derived from such a simple arrangement. 2i

W. W. Rice and R. J. Jensen, J . Chem. Educ., 1971, 48, 659.

Reactions of Atoms in Ground and Electronically Excited States

21

To circumvent these problems a number of ingenious methods hhve been devised, and have been discussed by Berry 22 in a recent model paper on the F H2 chemical laser. Berry 22 outlines the equal-gain t e m p e r a t ~ r e24, ~ ~ ~ zero-gain f e m p e r a t ~ r e2K, ~and ~ ~ tandem chemical laser methods,26pointing to disadvantages which are inherent in the first of these. An entirely different and rather more direct approach was used iwBerry’s own work;22an intracavity grating was used to spoil selectively the cavity at all wavelengths other than that of the desired transition. Thus when individual vibration-rotation transitions of HF lase, their gains are not influenced by cascade and ‘cannibalization’ effects.22 The necessary condition that a population inversion, or partial inversion, be produced, clearly limits the chemical laser technique compared with many others. To overcome this limitation chemical lasers have also been used to probe other systems 28 in which a population inversion is not achieved. However, this essentially represents an extension of techniques such as time-resolved fluorescence, or time-resolved absorption, discussed above. For further general discussion of chemical lasers the reader is referred to reviews by Pimente1,29K~mpa,~O and

+

27t

Time-resolved Spontaneous Emission.-This technique has been used to monitor a number of metastable excited atomic states, including O(21S0),32 S(31S0),33 and I(52P3).34It suffers from the disadvantage that unless relatively high atomic concentrations are employed, the signal-to-noise ratio is likely to be low. Signal-averagingtechniques have been employed to overcome this problem;33 however, we would recommend that, for radiative lifetimes greater than one second, other techniques be tried in preference to this. An interesting extension of this technique allows ground state atoms to be monitored, and is based on the well-known chemiluminescence produced following some recombination reactions. Thus for example the addition of O(23PJ)to ethylene has been monitored 35 by observing the chemiluminescence produced (from NO:) by the presence of trace quantities of nitric oxide. There has, however, been some controversy over the precision of these results and the possible involvement of other chemiluminescent reactions 22

23 24 25

26

O7 28

29

30 31 32 33

35

M. J. Berry, J . Chem. Phys., 1973, 59, 6229. J. €3. Parker and G . C. Pimentel, J . Chem. Phys., 1969, 51, 91. R. D. Coombe and G . C. Pimentel, J. Chem. Phys., 1973,59, 251, 1535. M. J. Molina and G. C. Pimentel, I.E.E.E. J. Quantum Electron, 1973, QE-9, 64. M. J. Molina and G . C. Pimentel, J . Chem. Phys., 1972, 56, 3988. H. L. Chen and C. B. Moore, J. Chem. Phys. 1970,54,4072. C . B. Moore, Adv. Chem. Phys., 1973, 23, 41. G . C. Pimentel, Sci. Amer., 1966, April, p. 32. K. L. Kompa, Angew. Chem. Znternat. Edn., 1970,9, 773. C. B. Moore, Ann. Rev. Phys. Chem., 1971, 22, 387. R. Atkinson and K. H. Welge, J. Chem. Phys., 1972,57,3689. 0 . J. Dunn, S. V. Filseth, and R. A. Young, J . Chem. Phys., 1973,59, 2892. D. Husain and J. R. Wiesenfeld, Trans. Faraday SOC.,1967, 63, 1349. F. Stuhl and H. Niki, J . Chem. Phys., 1971, 55, 3943.

22

Reaction Kinetic

has been suggested36 and refuted.37 A more systematic name for thi technique might perhaps be ‘time-resolved chemiluminescence’.

Molecular Beqms.-The ‘supermachine’ with its ‘universal’ detector (mas spectrometer) has been described elsewhere in the 39 and th advantages of using molecular beams to study the detailed dynamics a atomic reactions have frequently been e m p h a s i ~ e d .The ~ ~ ability to velocity and state-select the reactants, and even orient the molecules in some cases provides a means of probing the molecular dynamics in a way that is ultimate1 impossible using bulk techniques. In the past, beam techniques have beel mainly limited to the study of reactions with large cross-sections, but thi restriction has been considerably relaxed by recent improvements in detectors The main problems are expense and the large investment in time which i required. Thus it is still advisable to try to extract the maximum informatioi from bulk experiments before turning to molecular beam studies. Interesting developments in the technique include the application o chemical lasers to pump specific vibration-rotation states of a reactan beam,41 and the previously mentioned use of laser-induced fluorescence tc observe the vibrationai energy distribution in the reactively scattered product from a number of atomic reaction^.^ The technique of laser-induced fluorescence should be generally applicablc to the study of high-lying metastable atomic states (e.g. metastable noblc gas, and mercury atoms), and could also be applied to the study of certair metastable molecular species (it is necessary that the upper state involved ir the optical transition give rise,to an appreciable amount of fluorescence, anc should not be heavily predissociated). The recent rapid growth in the study of non-alkali atom reactions is particularly gratifying and we may expect molecular beam studies to make increasingly wider contributions to our understanding of kinetic processes. Flow Systems.-Flow tubes essentially represent a means of transforming a time co-ordinate into a distance co-ordinate, and may therefore be used in conjunction with, any of the usual detectors or spectroscopic techniques employed for the study of atomic reactions (see reviews by Thrush and Campbell 42 and Clyne 43). Techniques which have been employed include ”

D. D. Davis, R. E. Huie, J. T. Herron, M. J. Kurylo, and W. Braun, J . Chem. Phys

”

F. Stuhl and H. Niki, J . Chem. Phys., 1972, 57, 5403. Y. T. Lee, J. D. McDonald, P. R. LeBreton, and D. R. Herschbach, Rev. Sci. Instr..

1972,56,4868.

”

1969, 40, 1402.

40

*‘ 43

D. L. McFadden, E. A. McCullough, F. Kalos, and J. Ross, J. Chem. Phys., 1973, 59, 121. See, e.g., ‘Chemical Applications of Molecular Beam Scattering’, by M. A . D. Fluendy and K. P. Lawley, Chapman and Hall, London, 1973. T. J. Odiorne, P. R. Brooks, and J. V. V. Kasper, J. Chem. Phys., 1971, 55, 1980. I. M. Campbell and B. A. Thrush, Ann. Reports, 1965, 62, 17. M. A. A. Clyne, in ‘Physical Chemistry of Fast Reactions’, ed. B. P. Levitt, Plenum, London, 1973, Vol. 1.

Reactions of A t o m in Ground and Electronically Excited States

23

atomic absorption spectros~opy,~4v 45 atomic resonance fl~orescence,~~ spontaneous emission,47 chemilumine~cence,~~ e.s.r. spectros~opy,~~ and mass spectrometric sampling.50The main problems associated with flow tubes arise from surface rccombination of atoms (although a numbzr of standard treatments are known which minimize these effects) and efficient surface quenching of excited states. The latter point makes flow systems well suited for the study of ground-state atoms; however, metastable excited states have also been studied in fast flow systems 44* 51 (generally of special design). As the reporters have little practical experience of this technique, they refer the reader to the work of K a ~ f m a n , ~ ~ C l ~ n e , ~46 ~and ~ Setser 51 for further details. 459

Other Techniques.-An increasing number of reports on the use of modulation techniques are appearing 52-58 and this potentially universal method should find wide use in the future. It may be used to monitor both the decay of atoms 55 and the formation of molecular products 54. The sensitivity is high and absorptions of one part in lo6 may be rzadily detected. In emission the sensitivity can be even higher. Rapid-scan spectrometers have been employed to study iodine atom combination59 but have not found general usage. It would appear that their use is more profitably applied to the study of transient molecular spectra in the i.r.,60but even here the molecular modulation technique offers a formidable challenge. Shock tubes 61 have been used in conjunction with time-resolved atomic absorption spectrophotometry and chemiluminescence detection techniques. Their application to the study of atomic recombination/moleci~lardissocia52p

44 45 46

47 48

4B 60

61 52

63 54

55

58

57 58

69

80

61

539

F. Kaufman and C.-L. Lin, J. Chem. Phys., 1971, 55, 3760. M. A. A. Clyne, H. W. Cruse, and R. T. Watson, J. C. S. Faraday II, 1972,68, 153. M. A. A. Clyne and H. W. Cruse, J. C . S. Faraday II, 1972, 68, 1377. J. F. Noxon, J. Chem. Phys., 1970, 52, 1852. B. A. Thrush, Ann. Rev. Phys. Chem., 1968, 19, 371. A. A. Westenberg, Progr. Reaction Kinetics, 1972, 7, 23. H. Gg. Wagner, Angew. Chem. Znternat. Edn., 1971, 10, 604, and references therein. D. W. Setser, W. C. Richardson, G. W. Taylor, and L. G. Piper, Faraday Discuss. Chem. SOC.,1972, No. 53, 100. L. F. Phillips, Progr. Reaction Kinetics, 1973, 7, 83. H. S. Johnston, E, D. Morris, and J. Van den Bogaerde, J. Amer. Chem. Soc., 1969, 91, 7712. T. T. Paukert and H. S . Johnston, J. Chem. Phys., 1972,56,2824. L. F. Phillips, C. G. Freeman, M. J. McEwan, and R. F. C. Claridge, Trans. Faraday SOC.,1971, 67, 2004. R. Atkinson and R. J. Cvetanovii, J. Chem. Phys., 1971, 55, 659. Y. Fushiki and S . Tsuchiya, Chem. Phys. Letters, 1973, 22, 47. D. A. Parkes, D. M. Paul, C. P. Quinn, and R. C. Robson, Chem. Phys. Letters, 1973, 23, 425. E. W. Abrahamson, D. Husain, and J. R. Wiesenfeld, Trans. Faraday SOC.,1968, 64, 833. G. C. Pimentel, Appl. Optics, 1968,7,2155; A. S . Lefohn and G. C. Pimentel, J. Chem. Phys., 1971, 55, 1213. D. Bradley, Ann. Reports, 1965, 62, 63.

24

Reaction Kinetics

tion has provided an interesting controversy over the physical significance of activation energies.62 An arrangement for achieving monochromatic flash photolysis has been described.63 High intensities are produced and give rise to a sufficient concentration of transient species to allow their detection using flash spectroscopy (photographic recording). Further developments in this technique and its combination with the more sensitive absorption or fluorescence methods would be valuable. The use of time-resolved mass spectrometry has been described by a number of w ~ r k e r s , ~ * but - ~the ~ problems associated with electrical pick-up from the flash-lamp circuit appear to be more severe than for most other techniques and its use has been rather limited. Similar remarks apply to the use of time-resolved e.s.r. spectroscopy. Steady-state fluorescence methods are now standard and require no further comment (the work of Krause et aZ.67may be consulted for information on the more sophisticated experimental arrangements). However, one particularly elegant experiment employs a triple Fabry-Perot interferometer to examine the extreme wing line-broadening in Cs-inert gas mixtures.68 Finally we should acknowledge the continuing and important contributions being made by workers with classical end-product-analysis techniques. One such contribution is the determination of branching ratios 6 g (the relative rates into different reaction channels); most direct techniques yield either the total rate of removal for a given atomic state, or the rate into only one channel. Furthermore, the precision of the relative rate data obtained is often greater than from comparisons of absolute data, and thus provides a helpful check on reliability. 3 Hydrogen Atoms Ground State H(12S+).-The class of reaction H XY (where X and Y are halogen atoms) has been particularly important in the development of our understanding of reaction dynamics, and a great deal of information on the partitioning of excess energy into vibrational and rotational excitation in some of these reactions has been provided by Polanyi and co-workers 70-72

+

62

63 84 65

66 O7 88

89

70

71

72

H. Johnston and J. Birks, Accounts Chem. Res., 1972, 5, 327. A. B. Callear and J. C. McGurk, J. C . S. Faraday 11, 1972, 68, 289. D. Price and J. M. Lippiatt, Dynamic Mass Spectrometry, 1971, 2, 135. W. J. R. Tyerman, W. B. O’Callaghan, P. Kebarle, 0. P. Strausz, and H. E. Gunning, J . Amer. Chem. SOC.,1966, 88, 4277. R. T. Meyer, internat. J. Chem. Kinet., 1974, 6, 297. L. Krause and I. N. Siara, Canad. J. Phys., 1973, 51, 257 and references therein. R. E. M. Hedges, D . L. Drummond, and A. Gallagher, Phys. Rev. ( A ) , 1972, 6 , 1519. R. J . Cvetanovik and P. Michaud, J. Phys. Chem., 1972, 76, 1375. K. G. Anlauf, P. J. Kuntz, D . H. Maylotte, P. D. Pacey, and J. C. Polanyi, Discuss. Faraday SOC.,1967, 44, 183. K. G. Anlauf, P. E. Charters, D . S. Horne, R. G. McDonald, D. H. Maylotte, J. C. Polanyi, W. J. Skrlac, D. C. Tardy, and K. B. Woodall, J. Chem. Phys., 1970,53,4091. A. M. G. Ding, L. J. Kirsch, D . S. Perry, J. C. Polanyi, and J. L. Schreiber, ref. 4, p. 252.

Reactions of Atoms in Ground and Electronically Excited States

25

using the i.r. chemiluminescence technique. More recently, molecular beam studies employing mass spectrometric detection have begun to provide new information on these systems. Grosser and Haberland 73 have reported product angular distributions for reactions of H and D atoms with Cl, and Br,, finding that most of the hydrogen halide products recoil backwards with respect to the hydrogen atom beam, with no change in angular distribution (within experimental error) on changing from H to D ; these results are consistent with largely repulsive energy release. McDonald et al.'* have determined angular and velocity distributions for reactions of D atoms with Cl,, Br,, I,, ICl, and IBr. These results have also been discussed recently by Herschbach and a number of interesting points emerged. In the reaction of hydrogen atoms with Cl,, where about 40% of the available energy is released as translation with the remainder appearing as vibrational and rotational excitation of HCl, the recoil angular and velocity distribution is remarkably similar to that for photodissociation of C1,. A rationale for this has been provided in terms of the MO's of HCI,; the lowest unoccupied (20) orbital in C1, (to which an electron is excited in photodissociation) has a node between the atoms, while the highest occupied orbital of HCl, corresponds mainly to overlap of this antibonding orbital with the (Hls) orbital and also has a node between the C1 atoms. Herschbach has also used MO considerations for collision complexes of the type HX2 to explain the observed trend from backwards to sidewards peaking as X2 is changed from Cl, to Br, to I,, and the striking observation that, in reaction of D atoms with ICl, production of DI is favoured over formation of DCI by a factor of 3 or 4, despite the fact that formation of the latter is considerably more exothermic. (This effect has been noted also in reactions of Br, 0, and CH, with ICl.) The bonding l o and In orbitals of ICI have mainly C1 atomic orbital character, while the antibonding 20. and 272 orbitals have I atomic orbital character so that reaction at the iodine atom end is favoured. It should be noted, however, that Horgan et a1.75have reported that the reaction H

+ ICN +HCN + I

(1 1

occurs to the virtual exclusion of formation of HI. The problem of inverting experimental data to obtain a good representation of the potential energy hypersurface is relatively tractable for reactions of H atoms with halogen molecules because dynamical effects, which become important for other combinations of masses, are small. It is clear that, for the reaction H + Cl,, collinear approach is favoured with substantially repulsive energy release. Experimental and theoretical work by Polanyi and co-workers 7 2 indicates that both translational and vibrational energy in 73

'*

75

J. Grosser and H. Haberland, Chem. Phys. Letters, 1970,7,442. 3. D. McDonald, P. R. LeBreton, Y. T. Lee, and D. R. Herschbach, J. Chem. Phys., 1972, 56, 769. G. P. Horgan, M. R. Dunn, C. G. Freeman, M. J. McEwan, and L. F. Phillips, J . Phys. Chem., 1972,76, 1392.

26

Reaction Kinetics

excess of the threshold increase the reaction rate constant, with the former being more effective. Both experiment and theory show that excess translational energy appears mainly as product translation and only partly as rotation. Experiments further show that excess reactant vibrational energy is channelled about equally into product translation and vibration. The results of Herschbach and co-workers 7 4 suggest that as C12 is replaced by Br, and I,, the collinear H C1, surface should be replaced by a surface favouring a more sideways approach; the change to bent HX2 geometry is consistent with simple MO considerations. The effect of reagent translational energy on the reaction of hydrogen atoms with ClF, emerges from comparison of the molecular beam scattering results of Cross,76who found an angular distribution of D F product from the reaction D + CIF3 j .DF products (2)

+

+

+

similar to that obtained by McDonald et aL7* for D C1, at the same collision energy (42 kJ mol-l), with the observation by Rabideau 7 7 that no HF was produced at room temperature in reaction of H with ClF, (HCl was a product, but could not have been formed in a simple abstraction reaction). McDonald e f aZ.74found no detectable DCI yield at low collision energies (5 kJ mol-l) for reaction of D with C1,; reaction (2) is thus probably analogous in being promoted by excess translational energy. The complete in-plane angular distribution of T atoms formed in the reaction H+T2+HT+T (3) has been The deduced backward scattering of product HT molecules in the centre of mass system is in agreement with theoretical calculations 7 9 for H H, and recent experimental studies 8 o for the reaction D H, + D H + H (4)

+

+

This reaction has also been studied 81 using a flow system with e.s.r. detection, as earlier employed by Westenberg and de Haas,*, over the temperature range 167-346 K. The results, when compared with the higher temperature 83 give a smooth, gently curved Arrhenius plot from 167-745 K adequately described by the empirical expression k = (2.72 f 0.17) x lo-” ( T / K ) 2exp( -22.4 f.0.2 kJ rnol-I/RT) cm3 molecule-l s-l. Sevsral other recent studies have employed flow systems with e.s.r. 76

77 78

70

82

83

J. B. Cross, J. Chem. Phys., 1973,59,966. S . W. Rabideau, J . Chem. Phys., 1973, 59, 1533. G. H. Kwei, V. W. S. Lo, and E. A. Entemann, J. Chem. Phys., 1973, 59, 3421. K. T. Tang and M. Karplus, Phys. Rev. ( A ) , 1971, 4, 1844. J . Geddes, H. F. Krause, and W. L. Fite, J . Chem. Phys., 1972, 56, 3298. D. N. Mitchell and D. J. LeRoy, J . Chem. Phys., 1973, 58, 3449. A. A. Westenberg and N. de Haas, J . Chem. Phys., 1967, 47, 1393. B. A. Ridley, Ph.D. Thesis, University of Toronto, 1968.

Reactions of Atoms in Ground and Electronically Excited States

27

detection of H atoms. Takacs and Glass 84 monitored the reactions of hydrogen atoms and hydroxyl radicals with hydrogen bromide at 295K; fcr the reaction H HBr + H2 Br (5)

+

+

a rate constant of 3.4 f 0.8 x cm3molecule-l s-l was obtained, (a fluorinated halogenocarbon wall coating was used to suppress Br(32P+) recombination). No Br(32P4) was detected, but its formation was not precluded since rapid quenching by atomic hydrogen or bromine, or at the wall, may have occurred. Indeed, Br(32P3) has previously been detected in this reaction,s5and Davies et aLa6have reported that its formation rate is 8 % of that for Br(32P+). The reaction of H and D atoms with benzene and [2H,]benzene has been st~died,~ using ' mass spectrometry to monitor the benzene concentration in addition to e.s.r. detection of atoms. The activation energies for D and H atom reactions are equal within experimental error, 11.3 f 0.8 kJ mol-l. It was concluded that the rate-determining step is addition of a hydrogen atom to the benzene ring. Pre-exponential factors for reaction of H and D with C6H, were found to be 10.0 f 1.7 x cm3molecule-ls-l and 1.8 f 0.3 x cm3molecule-l s-l, respectively, the higher value in the latter case reflecting the occurrence of isotopic exchange reactions. Atkinson and CvetanoviC 8 8 have used a modulation technique to obtain the sum of rate constants for the reactions H

+ NO + M +HNO* + M ( M = H,) +HNO

+M

(6a) (6b)

generating hydrogen atoms by mercury photosensitization of hydrogen and measuring the phase shift between the weak red HNO emission and the cm6 incident 253.7 nm radiation. The value k, = 5.93 f 0.36 x molecule-2s-1 at 298 K was obtained, with an activation energy of -2.3 & 0.6 kJ mo1-1 in the temperature range 286-390 K. The value of the rate constant at room temperature is close to that obtained by Hartley and but rather higher than the recent value of Hikida et aLgO The activation energy for the process

+

+

+

H CO M +HCO M (7) has been determined 91 as 8.4 f 1.7 kJ mol-1 (298-373 K) with M = H2, using the sensitive Lyman-a absorption spectrophotometry technique to

88

*O

91

G . A. Takacs and G . P. Glass, J. Phys. Chem., 1973,77, 1060. J . R.Airey, P. D. Pacey, and J. C. Polanyi, Eleventh Combustion Symposium, 1967,85. P. B. Davies, B. A. Thrush, A. J. Stone, and F. D. Wayne, Chem. Phys. Letters, to be published (ref. cited in ref. 84). P. Kim, J. H. Lee, R. J. Bonanno, and R. B. Timmons, J. Chem. Phys., 1973,59,4593. R. Atkinson and R. J. CvetanoviC, Canad. J . Chem., 1973,51, 370. D. B. Hartley and B. A. Thrush, Proc. Roy. SOC.,1967, A297, 520. T.Hikida, J,. A. Eyre, and L. M. Dorfman, J. Chem. Phys., 1971,54,3422. H.Y.Wang, J. A. Eyre, and L. M. Dorfman, J . Chem. Phys., 1973,59,5199.

28

Reaction Kinetics

I

HCO

H + CO

Figure 5 Correlation diagram for H - CO, showing the predissociatiort level in the 2A” state of HCO, and the potential barrier formed by the avoided intersection of the 2A‘ states (after refs. 91 and 92)

monitor hydrogen atoms generated by pulse radiolysis. The presence of a barrier in the surface for recombination may be understood by considering the correlation diagram given in Figure 5 . Thus, for near-linear configurations, the ground-state reactants H(12S,) + CO(XIC+) correlate with an excited 2C+ state of HCO, while the ground state of HCO correlates with H( 1”+) + CO(a311). For non-linear configurations the intersection of the 2Z+ and one component of the 211 state (symmetry 2A’) is avoided. The resulting barrier (8.4 kJ) is located well below the previously known 92 heterogeneous predissociation in the 2A” (211n/2X-) state (at 148 kJ). The reported low pre-exponential factor (ca. 3 x cma s-l) for this reaction is interesting and suggests that regions of the potential surface close to the avoided intersection lead predominantly to reflection of the reactants back into the entrance channel. This would imply that the geometry of HCO in its ground state is significantly different from that in the region of the avoided intersection (the C-0 bond length in HCO is in fact known to be longer than in carbon monoxide). The potential energy surface for H + CO recombination has recently been determined, 93 using a double-zeta basis set augmented byp-functions on hydrogen to take account of polariza92

93

G . Herzberg, ‘Molecular Spectra and Molecular Structure. 111: Electronic Spectra and Electronic Structure of Polyatomic Molecules’, Van Nostrand, New York, 1966, pp. 469, 495. S. D. Peyerimhoff and R. J. Buenker, Ber. Bunsengesellschaft phys. Chem., 1974, 78, 119.

Reactions of Atoms in Ground and Electronically Excited States

29

tion effects. These calculations clearly show the potential barrier for HCO formation and further demonstrate that the HOC configuration is very unfavourable (the minimum for HOC lies above the energy for the separated fragments, H + CO). Further calculations with this surface may reveal more clearly why the pre-exponential factor is low. Lyman-a spectrophotometry has also been employed g4 to study the H + C,H, system. Computer simulation was used to make allowance for depletion of hydrogen atom concentration by reactions subsequent to the initial step H C2H4 + C2H5 (8)

+

A high-pressure limit for k, of 16.1 f 3.2 x

cm3molecule-l s-l was obtained from experiments in the presence of high pressures ( < 80 kN m-,) of He. Using this value, the relative collisional deactivation (stabilization) efficiencies of various third-body molecules were obtained. The pressure dependence of k, has been used 95 to assist the assignment of activated complex parameters in RRKM calculations on ethyl radical decomposition. Despite the many studies which have been undertaken on hydrogen atom recombination there is still considerable uncertainty over much of the rate data and there have been surprisingly few systematic studies of the temperature dependence of the reaction or the relative efficiencies of various third bodies. Experimental results up to mid-1971 both from studies near room temperature and from flame and shock-tube work have been reviewed critically.gs Of the low-temperature data obtained in flow systems for M = H2, the most reliable are those of Larkin and T h r ~ s h , ~and ~ - Ham, ~ ~ Trainor, and Kaufman,loOwho used a calorimetric probe to monitor hydrogen atom concentrations (the result that Bennet and Blackmore lo2obtained from e.s.r. measurements, is a factor of 2 lower). Baulch et aLg6recommend the value 8.3 x cm6 s-l, with suggested error limits of f 50%, at 300 K. It appears that H atoms are important as a third body at very high temperatures (T > 3000 K) but have a minor role at lower temperatures. The recent work of Trainor, Ham, and Kaufman lo3is of particular interest : rate constants for H + H + M with M = H,, He, or Ar, and for the corresponding process for deuterium atoms with M = D, were measured over the range 77-298 K; atom concentrationswere measured using a catalytic probe. Some of the rate constants obtained are shown in Table 1. These lo19

J. V. Michael, D. T. Osborne, and G . N. Suess, J . Chern. Phys., 1973, 58, 2800. J. V. Michael and G . N. Suess, J . Chem. Phys., 1973, 58, 2807. O6 D. L. Baulch, D. D. Drysdale, D. G . Horne, and A. C. Lloyd, ‘Evaluated Kinetic Data for High Temperature Reactions’, Butterworths, London, 1973, Vol. 1. s7 F. S. Larkin and B. A. Thrush, Discuss. Faraday SOC.,1964,37, 112. O* F. S. Larkin and B. A, Thrush, Tenth Combustion Symposium, 1965, 397. s9 F. S. Larkin, Canad. J . Chem., 1968, 46, 1005. loo D. 0. Ham, D. W. Trainor, and F. Kaufman, J. Chem. Phvs., 1970, 53, 4395. lol J. E. Bennet and D. R. Blackmore, Proc. Roy. SOC.,1968, A305, 553. lo2 J. E. Bennet and D. R. Blackmore, J . Chem. Phys., 1970,53, 4400. lo3 D. W. Trainor, D. 0. Ham, and F. Kaufman, J . Chern. Phys., 1973,58, 4599. s4

95

30

Reaction Kinetics

Table 1 Rate h t a for the recombination of H or D atoms Io3 M H+H+H,

H+H+He H+H+Ar D+D+D,

1033k/~m6 8.1 & 0.4 18.5 & 2.2 7.0 i 0.4 12.0 & 1.5 9.2 & 0.6 27.4 i 4.6 6.1 A 0.3 15.1 i 1.0

s-l

T/K 298 77 298 77 298 77 298 77

l‘

1

Other values of k within this range are also given.

results provide a useful basis for comparison of current theoretical models of hydrogen atom recombination. While good agreement with the absolute magnitude of the rate constant for M = H, has been obtained in several theoretical investigations, these have been much less successful in describing the detailed nature of the dependence of the rate on temperature and the third body. Recent approaches include the application of classical variational phase space theory lo3 (this has also been used lo5 for the case where H atom is the third body), which gives moderately good agreement with experiment over a wide temperature range and reproduces the shape of the temperature dependence curve of Trainor et aZ.103 The predicted values of rate constants in the low-temperature range are too small by 20-30% despite the inclusion of a minimum in the H-H, interaction (i.e. a radical complex contribution) which is probably too strongly attractive. Another line of approach has been the quantum mechanical ‘orbiting-resonance’ theory,los involving an energy-transfer mechanism proceeding via relatively long-lived ‘orbiting-resonances’ (quasibound states formed by tunnelling through the rotational barrier), and its modification lo’ to improve the model for rotational relaxation, Using a repulsive potential, a maximum in the recombination rate, between 65 and 1 0 0 K, is predicted which is inconsistent with the experimental results. Inclusion of a radical-complex contribution l o 8gives a better representation of the shape of the experimental curve although the absolute agreement is not especially good. It would appear that there must be some radicalcomplex contribution for M = H, (and for M = Ar) at low temperatures, but some doubt remains, and the well depths in the H-H2 potential which have been used to reproduce the experimental behaviour may be unrealistic. (a)V. H. Shui, J. P. Appleton, and J. C. Keck, Thirteenth Combustion Symposium, The Combustion Institute, 1971, 21; (6) V. H . Shui and J . P. Appleton, J . Chem. Phys., 1971, 55, 3126. ln5 V. H. Shui, J . Chem. Phys., 1973, 58, 4868. lo6 R. E. Roberts, R. B. Bernstein, and C. F. C. Curtiss, J. Chem. Phys., 1969, 50, 516. lo7 P. A. Whitlock, J. T. Muckerman, and R. E. Roberts, Chem. Phys. Letters, 1972, 16, In4

460. log

R. T. Pack, R. L. Snow, and W. D. Smith, J . Chern. Phys., 1972, 56, 926.

Reactions of Atoms in Ground and Electronically Excited States

31

Further refinement of the quantum theoretical model and extensions of experimental measurements are clearly desirable. The master equation approach to dissociation-recombination,loO~ 110 while restricted at present to consideration of the energy transfer mechanism at infinite dilution, highlights certain features of the energy transfer process. It further suggests some fruitful areas for experiment, and emphasizes possible shortcomings of the above approaches. In particular it appears (although the evidence is circumstantial) that for recombination there will always be some bottleneck in the deactivation 'ladder' so that the rate of formation of an orbiting resonance, or of its loss of one or two quanta of rotational or vibrational energy, cannot determine the rate of formation of ground-state H2. The phase-space theory (while it may be of use in many cases) is criticized on the grounds that it will not be applicable to hydrogen atom recombination below 500 K (or perhaps higher) because the implicit assumption that the upper vibration-rotation levels of H2 can be treated classically is quite unrealistic. Although any contribution from radical complex processes implies a negative activation energy for recombination, the calculations of Pritchard and co-workers suggest that, where there is a large energy transfer component, the temperature dependence is related to temperature variations in rotational disequilibrium. The reader is referred to the most recent papers in this series 110 for a more complete discussion, and to Chapter 6 in this volume. Electronically Excited H(22P,) and H(22S+).-Four isotopic variants of the reactions of excited H(22P,22S)atoms with H2(H*,D* H2, and D2) have been investigated ll1 in a flow system by monitoring the quenching of H and D Lyman-a fluorescence. An optically thin syste m was used to avoid radiationtrapping problems, and both H(22P) and D(22P) Lyman-a emissions were measured simultaneously using two detectors, one with a filter to remove almost all the H Lyman-a radiation. Although measurements were made on the 22P state, collisional interconversion of the 22P and 22S states of H and D is rapid so that the results relate to pairs of processes involving these states. For the reaction

+

H*fDz+HD+D*

(9)

a cross-section of 0.0028 nm2 (with an uncertainty of 25%) was obtained. Total cross-sections for removal varied from 0.84 nm2 (H* f H2and D,) to 0.91 nm2 (D* H2), with uncertainties of 5%. Experimental cross-sections for associative ionization processes such as

+

H* + D , + H D ; lo9 ll0

111

f e

(10)

T. Ashton, D. L. S. McElwain, and H. 0. Pritchard, Canad. J . Chern., 1973, 51, 237. E. Kamaratos and H. 0. Pritchard, Canad. J . Chem., 1973,51, 1923. G. V a n Volkenburgh, T. Carrington, and R. A. Young, J. Chem. Phys., 1973, 59, 6035.

32

Reaction Kinetics

were also measured by ion collection techniques (valuesrange from 8.2 x to 1.10 x 10-2nm2). The various possible reaction channels are discussed in terms of intersections of qualitative potential energy surfaces: most of the processes discussed cannot occur on a single adiabatic surface, and transitions between surfaces are most probable in the vicinity of intersections. It is suggested ll1 that most of the quenching leads to production of three groundstate atoms ; long-range interaction between the three 2A’surfaces correlating with H* H2, and the surface H H2(b3&+)which is attractive at long range, is discussed. While this and other channels can be discussed in a useful way, the isotope effects for total quenching, which are in any case fairly small, are more difficult to interpret because of the number of surfaces and different types of trajectories involved. A number of recent studies of reactions of translationally hot hydrogen atoms merit attention. Berry112 has used chemical laser methods to study the reaction H SF, at relative translational energies up to 427 kJ mol-l; hot H atoms were produced by photodissociation of HI. It was concluded that collision-induced dissociation.

+

+

+

H(K.E.)

+ SF,

-+

H

+ F t SF,

(1 1)

is the dominant reaction channel at high translational energies. Abstraction of F, which is observed in thermal reactions, was not detected, suggesting that relative translational energy is not efficient in promoting this (endoergic) reaction, and that internal energy of SF, may be required as is found in other endoergic bimolecular reactions (compare later comments on the reaction Br HCI). Results from the reactions of photolytically produced hot hydrogen atoms in HCl and HBr have been reported, and features of the cross-sections The cross-section for reaction of H with HCI appears to increase with energy over most of the range 0.1-1.1 eV. Reactions of recoil tritium atoms with various hydrocarbons have been studied, and mechanisms for formation of CH3T discussed.l14

+

4 Alkali Atoms

Interest in alkali atom reactions divides into two main areas; firstly, reactive scattering studies involving ground state atoms, and secondly, energy transfer studies involving electronically excited states. Ground-state (na5’+) atoms.-We shall not attempt to review the extensive literature on molecular beam studies involving alkali-metal atoms. Instead 112 ’13

M. J. Berry, Chern. Phys. Letters, 1973, 22, 507. D. K. Jardine, N. M. Ballash, and D. A. Armstrong, Cunnd. J . Chem., 1973, 51, 656. J. L. Williams, S. H. Daniel, and Y.-N. Tang, J . Phys. Clrem., 1973, 77, 2464.

Reactions of Atoms in Ground and Electronically Excited States

33

we discuss briefly a few of the most recent studies and rqfer the reader to reviews 115 and papers which act as key reference sources.116 Recent molecular ‘beam .studies 117 of the reactions of potassium atoms (supersonic beam with kinetic energy ca. 25 kJ mol- l) with mercuric halide molecules demonstrate that the reaction dynamics are similar to the wellknown dynamics for K X2 (where X2 is a halogen molecule). Electron transfer from K to HgX, takes place at long range in the entrance valley of the potential surface, resulting in total reaction cross-sections of ca. 1.SO nm2. The dissociation of HgX; in the field of K+ gives rise to stripping dynamics peaked mainly in the forward direction (a major fraction of the exoergicity appearing in KX vibration). Backward peaking of KX becomes more pronounced for the heavier halogens (HgBr, and Hg12), particularly at low energies (E N 8 kJ mol-l), indicating that a complex with lifetime approaching one rotational period is involved. This provides further support for the proposal that HgX; is formed in a bound state, ca. 130 kJ mol-l below the dissociation asymptote 117(unlike Br; which is formed close to the asymptote). Thus although the forces involved in the reactions K HgX2 are different from those for K + Xz,the effect on the reaction dynamics is only really apparent at low energies.ll Studies of the alkali dimer K2 with HgX, molecules show that two KX molecules are formed during one encounter,ll* viz.

+

+

K2

+ HgX2

+

Kl

+ HgX,

+ HgX + HgX-

K2X K2X+

-+

--+

+2KX

+ Hg

(12)

Bernstein and Rulis 119 have summarized the present status of experimental and theoretical work on the K CHJ reaction and point out that, despite the appearance of over thirty reports on this reaction, many aspects still require further investigation. Brooks has reported 120 an elegant study of oriented CFJ K, and has shown that the product KI recoils backwards in the centre of mass system when the I end of the molecule is closest to the incoming K atom, as expected. However, KI formation is also observed when the CF, end is closest to the incoming K atom, but is now forward scattered. Furthermore, the cross-sections for reaction of both orientations of CF31 are comparable. This is in contrast to the oriented CHJ Rb reaction for which no forward scattering is observed and the cross-sections are significantly different for the two orientations. Brooks has discussed 120 two possible reasons for these observed differences; the first of these involves the precessional motion of CXJ about the total angular momentum vector

+

+

+

J. L. Kinsey, in ‘Chemical Kinetics’, ed. J. C. Polanyi, M.T.P. International Review of Science, Butterworths, Oxford, 1972. Physical Chemistry Series 1, Vol. 9, p. 213. u6 Ref. 4 contains several useful review articles and papers which may be used as key reference sources. 117 R. Grice, D. R. Hardin, and K. B. Woodall, Mul. Phys., 1973, 26, 1073. 118 D. R. Hardin, K. B. Woodall, and R. Grice, Mol. Phys., 1973, 26, 1057. R. B. Bernstein and A. M. Rulis, ref. 4, p. 293. I2O P. R. Brooks, ref. 4, p. 299.

*I5

34

Reaction Kinetics

which will cause the CH3 group to sweep out a wider cone than the CF, group, Thus the iodine atom is not as well shielded in CF31 as in CH31. The second point relates to the range at which electron transfer can take place, and it was suggested 1 2 0 that the larger electron affinity expected for CF31 would allow this to occur at longer range than for CH31. Since the electron enters an antibonding orbital on CF31the molecule should decompose into CF, and I-, with KI being scattered along the axis of the originally orientated CF,I. Clearly, if the electron jump takes place at a sufficiently large distance the reaction probability will be the same at both ‘ends’ of the molecule.

-

-

Electronically excited (n2PJ) Atoms.-The quenching (n2PPJ n2S,) and spin-orbit relaxation (n2P+ n2P3) of electronically excited alkali atoms have been studied extensively using resonance fluorescence techniques, particularly by Krause and co-workers. Several important points emerge from these studies : (i) Cross-sections for spin-orbit relaxation by the noble gases 121are large when the spin-orbit splitting is small compared with the mean thermal collision energy (a 2: 0.5 nm2 for A E E 10-60 cm-l at ca. 400 K; e.g. Na and K), but fall rapidly as the splitting becomes comparable with kT (c _” 10-6-10-7 nm2 for A E N 250-600 cm-l; e,g. Rb and Cs).121 (ii) For large spin-orbit splittings (AE 21 250-600 cm-l) the crosssections for spin-orbit relaxation by the noble gases increase strongly with increasing translational energy,122while for small splittings the cross-sections show little ~ a r i a t i 0 n . l ~ ~ (iii) Spin-orbit relaxation by molecular species is significantly more efficient than for the noble gases when the spin-orbit splitting is 1 a ~ g e .125 l~~~ (iv) Resonance energy transfer processes involving the excitation of nuclear motion (rotation) in the quenching molecule appear to be important for large spin-orbit energies. Isotope effects observed for Cs Ha, HD, and D2 support these proposals.126 (v) Cross-sections for quenching to the ground electronic state are generally smaller than for spin-orbit In addition to the work summarized above, a number of other studies which provide information on the importance of different energy transfer channels deserve mention. Quenching of Na(32PJ) by N2 has received particular attention, both experimentally and theoretically. A model involving curve-crossing with an ionic state (Na+N;) appears to be satisfactory and

+

L. Krause, Appl. Optics, 1966, 5, 1375. A. Gallagher, Phys. Rev., 1968, 172, 88. l Z 5 P. L. Lijnse, Thesis, Rijksuniversiteit, Utrecht, Netherlands, 1973; P. L. Lijnse, J . Quant. Spec. Rad. Trans., in the press. lZ4 L. Krause, in ‘Physics of Electronic and Atomic Collisions, VH’, ICPEAC, 1971, North Holland, 1972. 125 D. A. McGillis and L. Krause, Canad. J . Phys., 1968, 46, 1051. 126 I. N. Siara and L. Krause, Canad. J. Phys., 1973, 51, 257. lZ1

122

Reactions of Atoms in Ground and Electronically Excited States

35

reveals that non-resonant energy transfer to vibration in N, takes place.l2'9 128 Crossed-beam studies in which excitation of Na(32PJ) by internally excited N,, H,, and D2is observed, give further support to this m0de1.l~~ Excitation cross-sections as a function of w and Aw were given for N2.129 The emission from alkali atoms excited by the Lewis-Rayleigh afterglow has been attributed 130 to N2(A3Ct) and not vibrationally excited N, as suggested earlier. This agrees with the findings of Krause et aZ.,129who observed that large kinetic energies in addition to vibrational excitation are required to excite sodium atoms. The variation in cross-section for quenching of Na(32PJ) by N, as a function of initial relative kinetic energy has also been investigated.lgl The velocity of Na(32PJ)was varied over the range 0.08-0.35 eV by photolysing NaI at different wavelengths (this technique has also-been applied to studies involving a number of other quenching gases).132 The cross-sections were found to fall with increasing translational energy, giving a linear correlation with the reciprocal relative translational energy.131 A comparison with previous shock-tube and flame data shows that the vibrational and rotational distributions in the quenching molecule have a pronounced effect on the cross-section. A model which places emphasis on the internal energy distribution of N2 has been given by Lijnse et al.133 This treatment was later revised,la8but a reappraisal of the earlier in the light of Barker and Weston's results,131would now appear to be worthwhile. Vibrationally excited H2 formed by quenching of electronically excited Na, Rb, and Cs, has been observed directly using time-resolved absorption spectrophotometry in the vacuum u.v., but the initial vibrational distribution could not be determined.134 The interaction potentials for Cs(6,PJ) and C S ( ~ ~ Swith + ) noble gas atoms have been determined by an elegant method involving observations on the extreme wings of the resonance line (up to 100 nm from the line centre).68

5 Alkaline Earth Atoms A number of exoergic reactions involving alkaline earth atoms are known to

give rise to intense. chemiluminescence and have attracted considerable recent interest. The large cross-sections associated with these reactions make them well suited for molecular beam studies and a number of detailed investigations have now been reported. Of particular importance is the E. Bauer, E. R. Fisher, and F. R. Gilmore, J. Chem. Phys., 1969, 51, 4173. P. L. Lijnse, Chem. Phys. Letters, 1973, 18, 73. 12$ H. F. Krause, J. Fricke, and W. L. Fite, J. Chem. Phys., 1972, 56, 4593. 130 C. J. Duthler and H. P. Broida, J. Chem. Phys., 1973, 59, 167. 131 J. R. Barker and R. E. Weston, Chem. Phys. Letters, 1973, 19, 235. 13* B. L. Earl, R. R. Herm, S.-M. Lin, and C. A. Mims, J. Chem. Phys., 1972, 56, 867. lS3 P. L. Lijnse and R. J. Elsenaar, J . Quant. Spectrosc. Rad. Trans., 1972, 12, 1115. lS4 P. H. Lee, H. P. Broida, W. Braun, and J. T. Herron, J , Phutuchem., 1973, 2, 165. 12'

lt8

36

Reaction Kinetics

demonstration that laser-induced resonance fluorescence may be employed to study product energy distributions under molecular beam conditions. The reaction of ground-state barium atoms with chlorine has been the subject of several recent investigations. There is evidence for three processes : Ba

+ C1, -+ BaCl(X2X) + C1

(13)

+=

BaC1(C21T) + C1

(14)

-+

BaCI*,

Jonah and Zare 135 have observed chemiluminescence in crossed molecular beam experiments; both a broad featureless emission, assigned to an excited state of BaCI,, and BaCl(C211-+ X2X) emission bands were reported. Their conclusion, that radiative two-body recombination is the main source of chemiluminescence, is of interest since such processes are regarded as unusual, requiring emission to occur within the collision lifetime. The main process was deduced to be formation of vibrationally excited ground-state BaCl, and the overall cross-section was estimated to be 0.60nm2. The predominance of this channel has been confirmed by crossed-beam experim e n t ~ which , ~ ~ ~reveaIed recoil energy distributions of MX product in the reactions of Ba, Sr, Ca, and Mg with Clzand Br,; in all cases forward scattering with fairly low recoil energies (l0-20% of the reaction exoergicity) was observed. The results indicate that MX, can only be formed in less than 5 % of all collisions. In contrast to the direct formation of ground-state BaCl, where a high degree of product internal excitation occurs, Menzinger and Wren 13' have analysed the BaCl(C2113+ X 2 Z + )emission in a flow apparatus, finding an almost statistical vibrational distribution, which suggests that formation of BaCI(C211) occurs via a fairly long-lived complex. It does not appear to be formed from the emitting electronic state of BaClz, since Wren and Menzinger found that BaCl: was quenched by He or N, preferentially to BaCl*; it was also estimated that the radiative lifetime of BaClg is long (zR > 100,us) and, if correct, this would strongly suggest that BaCI,*emission does not result from two-body combination. A more detailed knowledge of the relevant potential energy surfaces and intersections may help to clarify this point. A qualitative discussion of the surface for the perpendicular approach of Ba to CI, has been given by Yarkony et aZ.,139who have also performed ab initio calculations for the model system Ca F,. Mims, Lin and Herm140 have also obtained primitive product angular distributions for the reactions M ICI (M = Ba, Sr, Ca, or Mg) and Ba f BrCN. With the exception of the reaction of Mg with ICI, where no MgI was detected, the ratio MC1:MI was found to be ca. 2, while the product

+

+

la5 13g

13'

15*

lS9 140

C. D. Jonah and R. N. Zare, Chem. Phys. Letters, 1971, 9, 65. S.-M. Lin, C. A. Mims, and R. R. Herm, J. Chem. Phys., 1973, 58, 327. M. Menzinger and D. J. Wren, Chem. Phys. Letters, 1973, 18, 431. D. J. Wren and M. Menzinger, Chem. Phys. Letters, 1973, 20, 471. D. R. Yarkony, W. J. Hunt, and H. F. Schaefer, tert., Mol. Phys., 1973, 26, 941. C. A. Mims, S.-M. Lin, and R. R. Herm, J. Chem. Phys., 1973,58, 1983.

Reactions of Atoms in Ground and Electronically Excited States

37

ratio BaCN:BaBr was ca. 1O:l. Forward MX scattering was observed in varying degree for all these reactions; possible mechanisms were discussed and the need for more detailed experiments indicated. These workers have also performed exploratory molecular beam studies 141 on the reactions of various alkaline earth metals with CH,T, CFJ, CC14, SFs, PCl,, NO2, Me,CHN02, and CC13N02. There appear to be some analogies with the reactions of alkali atoms, for example in the reactions of Li and alkaline earth atoms with NO2, but quite different behaviour is observed in many other cases. As noted above, laser-induced resonance fluorescence to determine internal state distributions in the products of molecular beam reactions is of particular interest. The reaction Ba O2 BaO 0 (1 6 )

+

--f

+

was studied using this technique lp2 and, more recently, measurements of BaX internal excitations following the reaction of Ba atoms with hydrogen halides, in both crossed-beam and a scattering chamber apparatus, have been reported.6 The reaction zone was probed with a tuneable dye laser source: as this is scanned in wavelength, absorption by BaX(X2C+)occurs and the subsequent fluorescent emission is detected. A typical spectrum is shown in Figure 6. It was found that little of the reaction exoergicity appeared as product vibrational and rotational excitation, and that the fraction of exoergicity appearing as translational energy showed a positive correlation with cross-section. Pronounced non-statistical vibrational distributions were (5,5)

I

522

I

I

I

521

520

519

---

I

I

I

I

515

514

513

512

laser wavelengh / nm

Figure 6 Laser-induced fluorescence from BaCl (C211, + X2C , AV = 0 and -1 sequences). The wavelength scale refers to the input from a tunable dye laser (after ref. 5 ) +

141 14a

(a) S.-M. Lin, C. A. Mims, and R. R. Herm, J. Phys. Chem., 1973,77, 569; (b) R. R. Herm, S.-M. Lin and C. A. Mims, ibid., p. 2931. A. Schultz, H. W. Cruse, and R. N. Zare, J. Chem. Phys., 1972, 57, 1354.

38

Reaction Kinetics

obtained, with definite population inversion in the cases BaCI, BaBr, and Bar. These reactions are thought to proceed on highly repulsive surfaces with little mixed energy release. Use of a low-pressure diffusion flame reactor with photon counting has determined that, in the reaction Ca

+ ONCl

-+

NO

+ CaCl

(17)

CaCl was produced in ground ( X 2 C ) and excited (A211, B2C) states in the order X > A > B, i.e. there appear to be larger energy barriers for formation of products in higher However, for Ba

+ ONCl + N O

BaCl

(18)

emission from BaC1(C211) but not from BaCl(A211) was detected (the B2C state of BaCl is not known), so that product yields were in the order X > C $ A 21 0. These workers have also studied 144 the processes Ba

+ N,O

+ N2 NO, + BaO(AIC, X l C ) + NO --f

BaO(AIC, XIC)

(19) (20)

at low pressures, obtaining photon yields for AIC -+ XIC emission of 3.9 x and 3.3 x for N,O and NO2, respectively. Jones and Broida 145 have determined photon emission rates for these processes over a much wider pressure range, observing that the yield with Ba t N,O rises to a maximum with increasing pressure (about 0.19 at 10 Torr with argon) beyond which quenching reduces the efficiency. Jones and Broida suggest that BaO is formed in transitory excited states (from which they observed an underlying, many-lined emission at lower pressures) correlating with ground-state Ba and N20, which undergo collisional conversion into BaO(AIC) as the pressure is increased. (Broida 146 also mentions indirect evidence that the reaction of Ba + N 2 0 produces BaO in the a311 state.) The (AlC + X l C ) photon yields for Ba + NO, were about 0.1 of those for Ba NzO. With other oxidants (02,03,and NO) yields were considerably lower. The formation of various diatomic oxides and halides of Ba, Ca, and other metals in reactions of the metal atoms has been observed, using the technique of Microwave Optical Double Resonance.147 Microwave radiation was passed through the reaction zone, which was probed by visible light from a continuous wave laser (argon-ion or pumped dye laser) which pumps groundstate molecules to a specific V-R level of an excited electronic state. Changes in wavelength, intensity, or polarization of the detected photoluminuescence

+

R. H. Obenauf, C. J. HSU,and H. B. Palmer, J . Chem. Phys., 1973,58,4693. R. H. Obenauf, C. J. Hsu and H. B. Palmer, J. Chem. Phys., 1972, 57, 5607; 1973, 58, 2674. lP5 C. R. Jones and H. P. Broida, J. Chem. Phys., 1973, 59, 6677. 14& H. P. Broida, Ber. Bunsengesellschaft phys. Chem., 1974, 78, 152. l Q 7 D. 0. Harris, R. W. Field, and H. P. Broida, Ber. Bunsengesellschuft phys. Chem., 1974, 78, 146. lQS

Reactions of Atoms in Ground and Electronically Excited States

39

relative to the incident laser radiation can be related to transitions from rotational levels adjacent to the depopulated ground-state level or the pumped excited-state level. In this way, several microwave rotational transitions in the XIC(v = 0, 1) and AIC(v = 0-5) states of BaO, formed in the reaction Ba 02,were observed.

+

6 Carbon Atoms The reactions of carbon atoms are of considerable general interest, but the clean production of a given electronic state (23PJ,2 l 4 , or 21S0) is extremely difficult to arrange. Flash photolysis of carbon suboxide gives rise to all three states simultaneously.148 Fortunately, however, the kinetic behaviour and relative populations of the three states are sufficiently different to allow kinetic studies without undue complications from cascade processes, and

CH,(38,

)

1

"'"'

Figure 7 Correlation didgramfor the reactions of carbon atoms, in various electronic states, with H2. (The surfaces correlating with C1(D ) are represented by dashed lines for clarity) 148

W. Braun, A. M. Bass, D. D. Davis, and J. D. Simmons, Proc. Roy. SOC.,1969, A312, 417.

40

Reaction Kinetics

this source has been used for a number of kinetic The most highly excited lS0state is apparently produced via the secondary photolysis of C,O, and is thus only likely to be of importance under flash photolysis conditions.154 The earliest direct observations on carbon atom reactions employed kinetic absorption spectroscopy 14* and demonstrated that the more highly excited lS0 state is less reactive with Ha, than the lD, state, despite the greater exothermicity and 'spin allowed' nature of reaction. A consideration of the relevant correlation diagram (Figure 7) reveals the reasons for this, and a number of other predictions based on these diagrams have been made.3 The predictions concerning the unreactive nature of the IS,,state have recently been confirmed by experimental observations using timeresolved atomic absorption ~pectrophotometry.~~~ Rate data from direct studies of carbon atom reactions are given in Table 2. The reactions of ground-state carbon atoms, C(23PJ), have also been studied in flow tubes. Kley et al. 156 have demonstrated that active nitrogen reacts with a number of carbon-containing compounds [HCN, (CN),, C2H2,CH4, C2H6, CzH4,C,H,] to yield C(PP,) and that atomic absorption spectroscopy (165.7 nm and 156.0 nm) may be employed to monitor the atoms formed. It was proposed that the reaction. N(2*S,) + CN(X2C+)+- C(Z3P,) + N,(XIC,f)

(21)

provides the main route for carbon atom production. Addition of small amounts of cyanogen to active nitrogen gives rise to almost stoicheiometric amounts of carbon atom, and Thrush et aI.l5' have recently employed this technique to study the further reaction of C(23P,) with 0,. 1.r. emission from vibrationally excited CO was observed up to, but not above, w" = 17, corresponding to the full exothermicity of channel c. The absence of higher

x3qO(2'SO) (22b)

0(21D2)(22c) %O(XlX+) 149

15('

lS1 152 lS3

lZ4 lS5 lj6

-

O(2,PJ) (22d)

D. Husain and L. J. Kirsch, Chem. Phys. Letters, 1971, 8, 543. D. Husajn and L. J. Kirsch, Chem. Phys. Letters, 1971, 9, 412. D. Husain and L. J. Kirsch, Trans. Faraday SOC.,1971, 67, 2025. D. Husain and L. J. Kirsch, Trans. Faraday Soc., 1971, 67, 2886. D. Husain and L. J. Kirsch, Trans. Faraday Soc., 1971, 67, 3166. D. Husain and L. J. Kirsch, J. Photochern., 1974, 2, 297. G. M. Meaburn and D. Perner, Nature, 1966, 212, 1042. D. Kley, N. Washida, K. H. Becker, and W. Groth, Chem. Phys. Letters, 1972,15,45. E. A. Ogryzlo, J. P. Reilly, and B. A. Thrush, Chem. Phys. Letters, 1973, 23, 37.

Table 2 Rate constants k/cm3rnolec~le-~ s-l at 300 K for the reactions of carbon atoms C(23PJ,21D,,21S,) with various molecules (powers of 10 in parentheses) Molecule He Ne

Ref.

W3PJ)

Ar Kr Xe NO

co

7.3 f 2.2 (-11) 1.1(-10) 6.3 4.2.7 (-32) (M

=

He)"

149, 151 148 151

C(2lDZ) <3.0(-16) 1.1 f 0.4 (-15) < 1.0 (-15) 9.4 f 1.6 (-13) 1.1 f 0.3 (-10) 4.7 f 1.3 (-11)

Ref. 152 152 152 152 152 153

C(2lSO) < 2.0 (- 15)

Ref. 154

1.6 f 0.6 (-11)

153

G3.5 (-16)* G6.0 (-14) ca. 3.0 (-14)* < 1.0 (-11) Gl.0 (-16)*

155 154 155 154 155

ca. 5.0 (-lo)*

5.0 (-14)*

155 155

94.0 (- 14) G3.0 (-15)

154 154

CH4

~ 2 . (-15) 5

151

2.1 f 0.5 (-10)

153

COZ

c 1.0 (-14) 2.5 f 1.6(-11)

151 151

3.7 & 1.7 (-11) 1.4 f 0.5 (-10) ca. 3.7 (- 10)

153 153 153

3.3 f 1.5(-11) 3.3 (-11) ~ 3 . (-13) 6 7.1 f 2.5 (-32) (M 3.1 & 1.5 (-33) (M

151 148 151 151 151

CU.

2.6 (-11)

153

CU.

1.7 (-11) 2.6 f 0.3 (-10) 4.2 f 1.2 (-12)

153 150 150

NZO C3H6 0 2

Ha0 H Z

NZ

Third order, k/cm6 bined with half-life data.

a

s-l;

= =

He)" Ar)"

* Data for C(2lSJ

CU.

=P

8

from ref. 155 presented here by assuming first-order kinetics in atom concentration com-

P,

42

Reaction Kinetics

vibrational levels in CO strongly suggests that channel d is closed at thermal collision energies.15 This may be understood by applying MO considerations to the collision intermediate; the lowest energy state is expected to be lA’(lC+) with the triplet surfaces lying considerably higher.157 It would seem reasonable therefore that an appreciable energy barrier (activation energy) exists for reaction on the triplet surfaces. Furthermore, these results indicate that the collision complex for channel c has the configuration COO, rather than OCO; the latter configuration would allow the reaction to proceed through the regions of phase space where the efficient non-adiabatic process o(210,)

+ co

+ 0(23~,)

+ co

(23)

could occur, and thus allow the population of vibrational levels above v” = 17. Emission from CO(a311) was not observed in this work 157 and no evidence for the thermoneutral channel a was reported. This is in accord with the above discussion on triplet surfaces. A full correlation diagram for C + 0, is given in ref. 3. The reaction of CQ3P,) with NO yields predominantly CN radical^,^ despite the fact that this channel is 3.4 eV less exothermic than the channel correlating with CO.

+

+

~ ( 2 3 ~ NO(X~III) ~ ) + CN(X~C+) 0 ( 2 3 ~ , )

(24a)

This may again be understood in terms of the MO states for the collision complex; the channel leading to CO production must proceed through a quartet configuration which will lie at high energies, while the channel leading to CN may proceed through the low-lying doublet electronic configuration.

7 Nitrogen, Phosphorus and Arsenic Atoms Most of the rate data for reactions involving the 2P, ,0, and 4S states of nitrogen, phosphorus, and arsenic, together with data on highly excited states of antimony and bismuth atoms, have been reviewed el~ewhere.l-~ The present discussion will thus be restricted to the most recent data combined with comments on their general significance. Several processes involving ground-state nitrogen atoms, N(z4S,), have been studied in discharge flow systems. The reaction

+

~ ( 2 4 ~ ,+) CN(X~C.+) + ~ ( 2 3 ~ N,(x~c;) ~ )

(25)

originally proposed 15*as a chain-propagating step in the reaction of (CN), with ‘active nitrogen’, has recently been included in a modified and extended mechanism for this system by Berger and K i s t i a k o ~ s k y .Evidence ~~~ for its lS8 159

D. R. Safrany and W. Jaster, J . Phys. Chem., 1968,72,3305. M. Berger and G. B. Kistiakowsky, J. Phys. Chem., 1973,77, 1725.

Reactions of Atoms in Ground and Electronically Excited States

43

occurrence comes from the direct observation by Kley et al.15&of C atoms by absorption of carbon resonance radiation when trace amounts of various carbon-containing compounds, including (CN), itself, are added to a flowing nitrogen afterglow, and the observation of C(z3PP,and 2l0,) atoms using resonance fluorescence by Johnson and Fontijn 160 on adding C2F4 to discharged nitrogen. Roscoe and Roscoe 161have studied the rate of removal of N atoms in the reaction of ‘active nitrogen’ with various alcohols. The overall mechanism is not well established, and the rate constants in Arrhenius form which are presented (normalized for the overall stoicheiometry) cannot be simply related to the postulated initial step:

+

~ ( 2 4 s ~ )ROH

+R

+ HNO

(26)

Black et ~ 1 .have l ~ measured ~ the yield of N2(d’ = 1) in the reaction N(24Ss)

+ NO + N2(w”) + O(23PJ)

using Raman scattering in the Q branch of the anti-Stokes (d’= 1 + d’ = 0) line to determine the proportion of the exothermicity appearing as vibrational excitation of N,. Their result, that the fraction F = 0.25 f 0.03, is close to the earlier result of Schiff and co-workers;le3the experimental procedures employed rule out the possibility of vibrationally excited N2 being formed in collisions with translationally hot 0 atoms. Felder and Youngls4 have shown that reaction (27) is at least a factor of 10 faster than the corresponding reaction leading to formation of O(2l0,) which would require a crossing from a triplet to a singlet surface; the potential surfaces involved have been discussed by these authors. Campbell and Gray 165 have remeasured the rate constant for the recombination reaction to take account of atomic quenching effects on the intensity of NO(B211 + X211) emission accompanying reaction (28), obtaining (for M = N,) k,,

= 9.21

1.21

f 0.99 x 0.10 x

+

cm6 cm6

s-l at 298 K s - l at 196 K.

Brown lB6has determined upper and lower limits for the rate constant of the recombination N H M +NH + M (29)

+ +

S. E. Johnson and A. Fontijn, Chem. Phyv. Letters, 1973, 23, 252. J. M. Roscoe and S. G. Roscoe, Canad. J. Chem., 1973,51, 3671. Ie2 G . Black, R. Sharpless, and T. G. Slanger, J. Chem. Phys., 1973, 58, 4792. (a) J. E. Morgan, L. F. Phillips, and H. I. Schiff, Di~ctrss.Faraduy Soc., 1962,33,118; (6) L. F. Phillips and H. I. Schiff, J. Chem. Phys., 1962, 36, 3283; (c) J. E. Morgan and H. I. Schiff, Canad. J . Chem., 1963, 41, 903. 18* W. Felder and R. A. Young, J . Chem. Phys., 1972, 57, 572. Io5 I. M. Campbell and C. N. Gray, Chem. Phys. Letters, 1973, 18, 607. lo6 R. L. Brown, Znternat. J. Chem. Kinetics, 1973, 5, 663. le0

Table 3 Rate constants klcm3 molecule-l s-l at 300 K (unless otherwise stated) for and diatomic molecules (powers of 10 in parentheses) kfolecule N(22DJ) He <1.6(-16) <2 (-16) Ar 1.0 f 0.6 (-16) <2 (-16) Kr Xe

Ref. 174 175 174 175

2.1 f 0.3 (-12) 1.7 0.5 (-12) 5 (-12) 5 (-12)

176 177 174 175

W2PJ)

1.9 f 0.2(-15) 3.0 1.1 (-15)

P(32DJ)

Ref.

176 177

< 5 (16)

Ref. 171

< 5 (-16)

172

2.6 f 1.1 (-15) 1.7 & 0.3 (-11)

172 172

4.0 f 0.7 (-12)

171

D:

OQ CO

No

1.5 & 0.1 (-14) 2.3 f 1.1 (-14) 1.6 f 0.7 (-14) S 6 (-15) r5.2 f 0.4(-12) 1 9.3 -J 2.2I(-12) 4 6 f2(--12) 1 5 f 1(--12) (7 (-12) 6(-12) 2.1 f 0.2 (-12) 6.1 f 3.7(-11) 7 f 2.5 (-11) 1.8 (-10) k5.9 -J= 0.4 (- 11)

i

<3 (-16) -6 (-14) 176 177 174 178 175 175 176 177 174 175 176

176 177 <5(-16) 174 179

172

4.6 f 2.5 (-12) 2.6 f 0.2(-12)

177 176

1.4 & 0.2 (-11)

171

9.0 f 0.4 (-13)

176

1.5 f 0.4(-11)

172

3.4 f 1.1 (-11) 3.2 f 0.1 (-11)

177 176

5.5 f 0.6 (-11)

172

2.4 f 0.2 (-11) 1.8 f 0.2 (-11)

173 173

HCl C1,

Table 4 Rate constants k/cm3molecule-' s-l at 300 K for the removal of electronically in parentheses) Molecule

W2DJ) (1.8 f 0.2 (-13) 5 f 2(-13) 6 (-13) 4.8 f 0.9 (-12)

N,O 1.6 f 0.1 (-12)

C2H6 CzH4 CH,CH=CH, C2H2

SF, AsMe, AsCI,

Ref. N(22PJ) 176 174 S l . 1 (-15) 175 177 3.4 f 1.5(-12) 174 5 & 2(-14) 175 176

3 (-12)

175

1.2 (-10)

175

P(32DJ)

Ref.

Ref.

176 3.3 4 l.O(-12)

172

177 1.2 41 0,2(-11) 176

172

9.7 7.3 3.9 1.5 1.1

& 0.9 (-11) f 0.8 (-15) f 0.7 (-13) f 0.1 (-11) & 0.1 (-11)

171 173 173 173 172

6.0 1.5 1.3 8.7 1.5

f 0.7 (--11) & 0.1 (-10) f 0.1 (-10) f 0.7 (-11) f 0.3 (-15)

173 172 173 173 173

the removal of electronically excited nitrogen, phosphorus, and arsenic atoms by atoms p(3ap*) < 5 (-16)

Ref. As(~~D~) 171 g3.6 f 0.4 (-15)

< 5 (-16)

172

< 5 (-16) 2.0 f 0.3 (-13)

172 172

3.1 f 0.8 (-13)

171

< 5 (-16)

172

1.1 f 0.2 (-15) (296 K)

< 1 (-15) (296 K)

Ref.

As(~~P+) 170 <2(-16)

Ref. 170

168

1.7 &0.3(-12)(296K) 1.2 f 0.2 (-12)

168 168 <5(-15) 170

2.8 f 0.3 (-11) (296 K) 5.1 f 0.8 (-12)

168 169

2.8 f 0.5 (-12)

169

1.2 f 0.2 (-11) (296 K) 9.3 f 1.2 (-12)

169 170

1.9 f 0.2 (-12)

170

170

4.0 f 0.6 (-12) (296 K) 168 1.3 f 0.2 (-11) (403 K) 168 < 5 (-15) 5.3 f 0.8 (-12) 170

170

2.6 f 0.2 (-11)

172

2.2 f 0.4 (- 12)

170

4.4 f 0.3 (-11)

170

7 f 6(-16)

172

4.7 f 0.6 (-11) (296 K)

168

5.1 f 0.8 (-12)

169

3.0 & 0.5 (-11)

172

5.2 f 0.6 (-11)

170

5.5 f 0,7(-11)

170

6.0 f 0.3 (-12) 2.9 f 0.4 (-11)

173 173

excited nitrogen, phosphorus and arsenic atoms by polyatomic molecules (powers of 10 p(32p3)

Ref.

7.3 f 1.9(-14)

172

3.1 f 0.6 (-13)

172

1.1 f 0.2 (-10) 5 (-15) 1.9 f 0.2 (-12) 2.0 f 0.3 (-12) 2.8 f 0.5 (-12)

172 173 173 173

2.7 4.2 1.4 3.6 2.4

& 0.5 (-11)

f 1.2 (-11) f 0.2 (-10) f 0.4 (-11) f 0.5 (- 14)

As(~~D+) 7.8 f 1.2(-13)(296K) 3.3 f 0.5 (-13) 2.2 f 0.3 (-11)

1.9 f 0.4 (-12) (296 K) 1.4 f 0.4 (-12) 172 173 172 3.6 f 0.4(-11) 173 173 173 < 1 (-15) (296 K) 2.4 f 0.1 (-11)

{

Ref.

As(4'PS)

Ref.

1.8 f 0.2(-13)

170

7.4 f 0.7 (-13)

170

168 170

7.1 f 0.8 (-12)

170

170

4.9 f 0.5 (-11)

170

1.6 f 0.2 (-10)

170

168 170 170

168 170

Reaction Kinetics

46

for M = 50% N,, 50% H,, in a discharge-flow system using e.s.r. spectroscopy to determine absolute N and H atom concentrations at one point in the flow tube, and measuring relative N atom concentrations from the intensity of the nitrogen afterglow; absolute N atom concentrations were also deter< k2&m6 mined by titration with NO. The result, 3.1 f 1.0 x molecule-2s-1 < 6.4 f 1.5 x is appreciably higher than the value obtained by Mavroyannis and Winkler.lB7 This lower limit is larger than the rate constants for N or H atom self-recombination by a factor of about 3 ; for N N + M the collision frequency is lower by almost this amount. Brown 166 has suggested that H + H M is slower than the other recombinations (for the same third body), despite a larger collision frequency, because the only channel for stabilization of newly formed H, is by collision-induced relaxation in the ground state involving an exceptionally large vibrational spacing, whereas for NH (and N,) a possible additional channel is collisioninduced crossing to bound excited states whose potential curves dip below the dissociation limit of the ground state. There is no information on the reactions of ground-state (n4St) P and As atoms, but a body of data is being accumulated on the low-lying excited states arising from the np3 electronic configuration, n2D, and n2PJ,which may be usefully compared with available information on collisional deactivation of excited N atoms. The work of Callear and Oldman 168 on As(4,D4) atoms has now been supplemented by Bevan and Husain,ls4~170 who have monitored the reactions of A s ( ~ ~ D 42PJ) ,, with H, and CO by attenuation of atomic resonance radiation; it was shown that the effective first-order rate constants for removal of the individual spin-orbit levels of either state were equal within experimental error for given conditions, implying that equilibrium between the J levels is maintained over the period of decay. The results of Husain and co-workers 171-173 on quenching of electronically excited P(32DJ, 32PJ)atoms by a wide variety of gases have already been reviewed.’ Some of these results are presented in Tables 3 and 4. Correlation diagrams for some reactions of these species have been di~cussed,~ and differences in rates of quenching for different atomic states

+

16’ lG8 169

170 lil

li2 173 Ii*

l7>

176

177

178

+

C. Mavroyannis and C. A. Winkler, Canad. J. Chem., 1962, 40, 240. A. B. Callear and R. J. Oldman, Trans. Faraday SOC.,1968, 64,840. M. J . Bevan and D. Husain, J. C . S. Faraday ZI, 1974, in the press. M. J. Bevan and D. Husain, J. Photochem., 1974, 3, 1 . A. U. Acuna, D. Husain, and J. R. Wiesenfeld, J. Chem. Phys., 1973, 58, 494. A. U. Acuna, D. Husain, and J. R. Wiesenfeld, J. Chem. Phys., 1973, 58, 5272. A. U. Acuna and D. Husain, J. C. S . Faraday ZZ, 1973,69,585. C.-L. Lin and F. Knufmari, J. Chem. Phys., 1971, 55, 3760. G. Black, T. G. Slanger, G. A. St. John, and R. A. Young, J. Chem. Phys., 1969, 51, 116. D. Husain and A. Young, J. C. S. Faraday II, in the press. D. Husain, L. J. Kirsch, and J. R. Wiesenfeld, Faraday Discuss. Chem. SOC.,1972, No. 53, 201. F. Kaufman, ‘Atmospheric Reactions Involving Natural Species-An Evaluation’, Amer. Geophys. Union Meeting. San Francisco, 1968. J. F. Noxon, J. Chem. Phys., 1962, 36, 926.

Reactions of Atoms in Ground and Electronically Excited States

47

by atoms and di- and tri-atomics may be readily explained. The requirement for adiabatic correlation is strongest for nitrogen, and the effect of increasing spin-orbit coupling for phosphorus and arsenic may be seen in some of the data in Table 3 . Thus for the reaction of N(22PJ)with H2(X1Z;), there is no correlation with low-lying states of products, and reaction is about lo3 slower than that of N(22DJ)with H2. For P(32PJ),however, reaction with H2 is only a factor of 10 slower than that of P(32DJ),and for arsenic both excited states react at roughly comparable rates. The slight discrepancy between the two sets of results 168v lB9for reaction of A s ( ~ ~ D , ) H2, probably arises from the involvement of atom-radical reactions under the conditions employed for plate photometry.lB8 Quenching by He, Ar, and Kr is clearly inefficient in all cases. The relevant potential curves for N + He may be compared with those for the isoelectronic system O+ He, for which calculations have been performed by Augustin et al.lS00+(22DJ)+ He correlate with a weakly bound 211 state which has a crossing in low-order approximation with the repulsive *C- ground state correlating with O+(24S3)+ He; collisional deactivation of P(32D,) and A s ( ~ ~ Datoms , ) by xenon is favoured by mixing between quartet and doublet states by spin-orbit coupling and, presumably, by a deeper well in the attractive 211 curve giving crossing with the ground state at a lower energy. This mechanism for the transfer of large amounts of electronic energy into translational energy of the receding atoms is similar to that proposed lS1 for the quenching of O(2lD2) by xenon. The rate of quenching of P(32PJ) by xenon is moderately fast, about three orders of magnitude greater than for quenching by krypton; this must indicate that there is a potential curve crossing at an accessible energy involving some state correlating with P(32PJ)+ Xe. 0+(22PJ) He correlate with two repulsive states, but the potential energy diagram must be appreciably modified for P Xe; the magnitude of the rate constant might reflect either a crossingof repulsive states with a small activation energy, or transfer with relatively low probability involving a low-energy crossing (due to an attractive upper surface). A study of the temperature dependence would help to resolve this point. It may be noted that, in some cases where reaction of excited N, P, or As atoms with diatomics to give diatomic + atom is endothermic, quenching may nevertheless be quite fast owing to the formation of a metastable intermediate, for example the established species N3 and NCO in the reactions N(22D,) $- N2 and N(22DJ) CO. Thus the doublet surfaces correlating with these intermediates will cross the quartet surface correlating with N(24S3) and non-adiabatic transitions may take place (analogous to relaxation of O(2lD2) by N2or CO). Donovan and Husain have suggested that an intermediate AsCO, analogous to NCO(X211), may be involved in the quenching of A s ( ~ ~ Dby , ) C0.168vlB9It has been proposed that a similar

+

+

+

+

+

180

lal

S. D. Augustin, W. H. Miller, P. K. Pearson, and H. F. Schaefer, tert, J. Cltem. Phys., 1973, 58, 2845. R. J. Donovan, D. Husain, and L. J. Kirsch, Trans. Faraday SOC.,1970,66, 774.

48

Reaction Kinetics

mechanism may be involved in the fairly rapid quenching of P(32D,, 32P,) and A s ( ~ ~ Pby, )CO. For deactivation of P(32D,) and P(32P,) by polyatomic molecules, for which correlation diagrams cannot be readily constructed and where spin conservation and orbital symmetry requirements are less restrictive, it has been found that rates of deactivation of P(32D,) and P(32P,) atoms by a given molecule are comparable, in marked contrast to quenching by diatomics. In a few cases, e.g. reaction with CH4 and C2H4, rate constants for the two excited states differ by an order of magnitude. It appears that the local symmetry of the bond in CH4 reacting with the phosphorus atom is sufficient to impose a restriction in orbital symmetry analogous, in the limit, to the reaction P* + H2 where both 2PJand 2DJ species might give round state products exothermically but for which there is adiabatic correlation only for P(32D,) + H2. The interesting result with C2H4is difficult to discuss in the absence of knowledge on the geometry of the collision complex.

- 9.0

-

- I 1.0

+

," -13.0

d2

Q) L

-?i 0

E -15.0

-s

"E0 U

2

I

- 9.0

1

I

I

I

r

-11.0

0

U

- 13.0 - 15.0

9

I1 13 15 17 ionisat ion potential/eV

19

Figure 8 Correlation between the second order rate constants (k,) for the removal of (a) P(32DJ)and (b) P(32P,) and the ionization potential of the collision partner. 1 , CH,CH=CH,; 2, PCI3; 3, C2H4, 4, C2H2; 5, C2Ht3; 6, C1,; 7, HCI; 8, CHI; 9, CF3CI; 10, CF,H; 11, CF,; 12, SF, (Reproduced from J . C. S. Faraday ZI, 1973, 69, 585)

Reactions of Atoms in Ground and Electronically Excited States

49

The logarithms of the rate constants for removal of both P(3,DJ) and P(3,PJ) by the polyatomic species studied to date (Table 4) show a good linear correlation with the ionization potential of the quenching species (Figure 8). 8 Oxygen Atoms Ground State O(23PJ).-It was demonstrated some time ago182that, in the reaction 0 cs, -+ so cs (30)

+

+

ca. 8 and l8%, respectively, of the available exothermicity appears as vibrational excitation of CS and SO. This reaction has now been examined lS3 using crossed molecular beams (thermal energy); the dynamics appear to be those of a stripping reaction, with strong forward scattering of SO and < 5 kJ mol-l, from a total exothermicity of ca. 90 kJ mol-l, entering product translation. (With the apparatus employed, there may have been a backward scattered SO peak beyond the angular range of the detector, but the total cross-section obtained from the forward-scattered product was compatible with bulk kinetic measurements.) These results imply that a large proportion of the available energy appears as product rotation; this is an unusual observation, but some of the dynamic constraints imposed by conservation of angular momentum, in cases where there are both an atomic reactant and an atomic product, do not apply in this instance. The subsequent reaction

0

+ CS

-+

CO

+ S + 356 kJ mol-l

(31)

is now known to be the source of laser action in the intense CO fundamental band of the CW CS2-O2 chemical 1 a ~ e r . lCO* ~ ~ laser emission, following reaction (31), has also been studied recently,ls5as has the process 0

+ CSe

CO

+ Se + -494 kJ mo1-l.

(32) Crossed molecular beam studies have also been performed on the reactions of 0 atoms with halogen molecules. These appear to proceed via a long-lived complex, as indicated by the complete angle-velocity product distributions obtained 186* 18' for the reaction

O(23PJ)

-+

+ Br2(lX;)

-+

+ Br(4,PJ)

BrO(X211)

(33)

The reactions with ICI, IBr, Clz, and I2 have also been studied. More limited data for some of these processes have also been reported by Grice and co-workers,188who have suggested, from MO consideration for OXY, I. W. M. Smith, Discuss. Faraday SOC.,1967, 44, 194. P. L. Moore, P. N. Clough, and J. Geddes, Chem. Phys. Letters, 1972, 17, 608. lE4 F. G. Sadie, P. A. Buger, and 0. G. Malan, 2. Naturforsch, 1972, 27a, 1260. lE6 S. Rosenwaks and I. W. M. Smith, J. C. S. Faraday ZZ, 1973, 69, 1416. la* D. D. Parrish and D. R. Herschbach, J. Amer. Chem. SOC.,1973, 95, 6133. D. A. Dixon, D. D. Parrish, and D. R. Herschbach, ref. 4, p. 385. Iaa C. F. Carter, M. R. Levy, K. B. Woodall, and R. Grice, ref. 4, p. 381. laJ

50

Reaction Kinetics

that reaction on the lowest available triplet surface might give direct dynamics and that the 0 Br, and 0 I, reactions might involve a transition from the initial triplet surface to a lower singlet surface (ClOCl and BrOBr have stable singlet states). Both sets of workers have found, however, that the reaction 0 ICl gives mainly I 0 which precludes such a mechanism in this case. Dixon et aZ.ls7argue that, in reaction (33), the surface corresponding to symmetric BrOBr is unlikely to govern the reaction since orbital correlations predict a large energy barrier for insertion into Br,, and 'electronegativity rule' considerations suggest that end-on attack of 0 + Br, is favoured; this is also consistent with the observation that I 0 is the main product of the 0 ICl and 0 IBr reactions. It is suggested that the ground state of 0-Br-Br might well be a triplet, even if appreciably bent. CNDO-UHF calculations on triplet linear 0-Cl-Cl lend some support to this suggestion. These results emphasize that, increasingly, attention is being focused on gaining some knowledge of the geometries and energy levels of intermediate configurations in reactions. E.s.r. detection in a discharge flow system has been used l S 9to determine Arrhenius parameters for the reaction

+

+

+

+

+

0

+ C12 -+c10 + c1

(34)

of k,, = 1.02 ~f: 0.46 x 10-l' exp( -15.0 f 1.1 kJ mol-l/RT)cm3 molecule-l s-l, in good agreement with earlier results.1so For the process 0

+ ONCl

-+

ClO i- NO

(35)

k 3 , = 7.42 3 2.59 x 10-l2 exp( -12.6 f 0.7 kJ mol-l/RT)cm3 molecule-l s-l

was obtained. The estimated high collision efficiency of the subsequent reaction with C10 0 -t C10 + 0 2 -k CI (36) has been confirmed,lglwith k,, = (5.3 + 0.8) x 10-l' cm3molecule-ls-l at 298 K using atomic resonance fluorescenceand mass spectrometry in separate systems; this corresponds to a collision efficiency of about 0.1. These methods were used to study a number of other bimolecular processes, including 0 -1- OClO

-+

c10 + 0

2

(37)

for which the mean value k 3 7 = (5:;) x cm3 molecule-l s-' was reported. This result implies that the much higher value (5.0 x 10-11cm3 molecule-l s-l) obtained by Basco and Dogra l g 2following flash photolysis of CIOz is based on an erroneous interpretation of the mechanism; the observation of vibrationally excited O z ( X 3 C - ,'1 -= 14) must now be assigned 19" 19' 19"

J. N. Bradley, D. A. Whytock, and T. A. Zaleski, J. C. S. Faraday I, 1973, 69, 1251. M. A. A. Clyne and J. A. Coxon, Trans. Faraday SOC.,1966,62, 2175. P. P. Bemand, N.A. A. Clyne, and R. T. Watson, J. C. S. Faraday I, 1973, 69, 1356. N. Basco and S. K. Dogra, Proc. Roy. Snc.,1971, A323, 29, 417.

51

Reactions of Atoms in Ground and Electronically Excited States

+ OClO. The + 0 + c10

to reaction (36) rather than to 0 mechanism is thus OClO hv 0

+ c10

C10, flash photolysis

+

(38)

c1 + o2

(39)

+

c1 t OClO + 2 ClO

(40)

Mack and Thrush lg5have employed a fast flow system to obtain a value for the rate constant of the initial step in the reaction of oxygen atoms with formaldehyde 0 HzCO + HCO -t OH (41)

+

of k41 = 1.5 f 0.2 x cm3molecule-l s-l at 300 K, in good agreement with earlier mass spectrometric studies. Relative rate constants for the reactions 0 HCO + O H CO (42)

+ + 0 + HCO + H + CO, H + HCO + H , + CO

(43)

(44)

were also derived, k 4 2 : k 4 3 : k = 4 40.54:0.46:4.0. Following a crossed molecular beam study lS4it is now established that, in the reaction of ground-state oxygen atoms with ethylene, the initially formed C,H,O* fragments to give CH, HCO rather than CH, H,CO (a recent liquid-phase study lS5provides good evidence for a triplet biradical intermediate). As a corollary it seems fairly certain that the formaldehyde detected must be produced by

+

+

0

+ CH3

+-

H,CO

+H

(45)

Washida and Bayes lg6 have obtained a rate constant for this reaction of (1.23 &- 0.25) x 10-10cm3molecule-ls-l at total pressures of up to about 500 Nnr2. Methyl radical concentrations were determined by photoionization mass spectrometry. This rate constant corresponds to one half of all gas kinetic collisions, and these authors point out that simple considerations of spin statistics suggest that two-thirds of the collisions occur on a quartet surface which cannot correlate with ground-state products, so that virtually all collisions on the lower doublet surface must lead to reaction, and crossing from quartet to doublet surfaces may occur readily. Rate constants for the reactions of O(Z3PJ)atoms with various monoand di-substituted halogenoethylenes have been obtained l g 7 using a flow reactor with mass spectrometric detection; it is possible that, unlike the ethylene reaction, these involve the initial formation of aldehyde plus carbene, la3 lS4

la5

ln6

G. P. R. Mack and B. A. Thrush, J. C . S . Faraduy I, 1973, 69, 208. J. R. Kanofsky and D. Gutman, Chem. Phys. Letters, 1972, 15,236. S.-I.Hirokami and R. J. CvetanoviC, Cunud. J. Chem., 1973, 51, 373. N. Washida and K. D. Bayes, Chem. Phys. Letters, 1973, 23, 373. R. E. Huie, J. T. Herron, and D. D. Davis, Internat. J. Chem. Kinetics, 1972, 4, 521.

52

Reaction Kinetics

but the evidence is not conclusive. (The formation of CF2 in the reaction 0 C2F, has been observed lg8following flash photolysis of C,F,-NO, mixtures at wavelengths above 300 nm). These workers have also studied the reactions of oxygen atoms with but-1-ene, and cis-but-Zene and 2,3dimethyl-but-2-ene, using the flash photolysis-resonance fluorescence technique. They obtained a markedly curved Arrhenius plot for the reaction with but-1-ene over the temperature range 19-91 K, which was attributed to competition between addition and abstraction reactions, with respective rate parameters (3.7 f 1.8) x exp( -0.21 0.88*/RT) cm3molecule-l s-l and (1.6 f 0.9) x 10-l1 exp( -8.24 & 1.80/RT)cm3molecule-ls-l. A negative temperature-dependence was observed in the reactions with cisbut-2-ene and 2,3-dimethyl-but-2-ene, expressible in Arrhenius form by k = (9.69 & 0.96) x 10-l2 exp(1.33 i 0.26/RT)cm3molecule-ls-l and k = (5.58 i 1.07) x exp(6.57 & 0.50/RT)cm3molecule-l s-l, respectively. It seems firmly established by the wide range of conditions employed that these rate constants do indeed refer to bimolecular reactions; and while abstraction may compete with addition at higher temperatures, particularly for cis-but-2-ene, this does not affect the conclusion that there is a negative temperature-dependence for the addition reaction. Davis et all9 discussed possible explanations for this observation. Similar behaviour, noted in other reactions of Group VI atoms with alkenes, is discussed in the section on sulphur atom reactions. The reaction of OQ3PJ) atoms with acetylene has been investigated l g 9 using a discharge-flow reactor. The CO 4th Positive (All3 XIC) emission observed in this and other hydrocarbon oxidations, in atomic oxygen flames, has commonly been attributed to the reaction

+

--f

c20

+ 0 -+ CO(A1fl)+ CO(X1C)

but this is not consistent with the pressure dependence of the 4th Positive emission observed by these workers. They propose the modification 0 -t C,O CO*

+co*+ co

(47)

+ M -+CO(AlII) + M

(48)

where CO* is formed in one or more of the states having potential energy curves which overlap and perturb the A l l 3 state (d3A, e3C, PA) followed by collision-induced intersystem crossing via the mixed levels. This could explain the observed anomalies in the intensity distribution in the 4th Positive emission. The third-order rate coefficient for the reaction CO

+ 0 + M(=C02)

-+CO,

+M

(49)

* Activation energies in kJ mol-l. lBB ls9

(a) W. J. R. Tyerman, Trans. Furuduy SOC.,1969, 65, 163; (b) R. L. Mitchell and J. P. Simons, J . Chem. SOC.,1968, 1005.

A. Fontijn and S. E. Johnson, J. Chem. Phys., 1973, 59, 6193.

Reactions of Atoms in Ground and Electronically Excited States

53

has been determined 2oo as 5.4 f 0.7 x cms molecule-2s-l at 296 K by recording the chemiluminescent recombination emission following flash photolysis of CO,, a technique similar to that employed by Stuhl and NikiOS5 1.r. chemiluminescence following the process NO

+ 0 + M +NO2 + M

(50)

has bzen studied.201 Absolute values of 1; were determined which should enable the use of this reaction as a secondary emission standard in the wavelength region 1.3-3.3 pm. Basco and Morse ,02 have determined absolute concentrations of vibrationally excited 0, formed in flash photolysis of NO2, by the reaction O(z3PJ)

+ NO,

+

NO

+ 02*(v” < 11)

concluding that about 20 % of the exothermicity is partitioned into vibrational excitation of 0,.It was suggested that more highly excited 02(v” < 15) may be produced, by reaction of O(z3PJ)with N02(A2B1),when photolysis radiation ca. 400 nm is not filtered out. The rate constant for the reaction of OQ3P,) atoms with NO2 has been determined relative to the rate of thirdbody stabilized recombination of NO and 0, by Harker and in a molecular modulation study of NO, photolysis in a long-path i.r. cell. cm* s-l for the recombination rate Using a value of 6.9 x constant, k51 = 9.2 x cm3molecule-l s-l was obtained. This may be cm3mojecule-l s-l, determined using compared with the value 7.0 x e.s.r. detection.204Laser flash photolysis of ozone between 590 and 620 nm has been employed 205 as a clean source of O(FP,) atoms, and their decay due to the reaction 0 0 3+02 0 2 (52)

+

+

has been monitored using resonance fluorescence. The rate constant k,, = (2.02 f0.19) x 10-l’ exp( -18.92 f 0.88*/RT)cm3molecule-l s-l was obtained (220-353 K). Secondary processes, including the possible formation of O(23P,) by reaction of excited oxygen molecules with ozone, and radical-radical reactions, appear to have been negligible. Reaction (52) has now been studied using many techniques and the rate constant can be regarded as established within 15% over a temperature range of 300 K. Rate constants for the initial steps in the reactions of oxygen atoms with methane and chloromethanes have been determined 206 using a discharge flow reactor with mass spectrometric detection, over the temperature range 298-500 K. The values obtained are listed in Table 5.

* Activation 2oo

201 202

203 204

205 206

energies in kJ mol-’. E. C. Y. Inn, J. Chem. Phys., 1973,59, 5431. M. F. Golde, A. E. Roche, and F. Kaufman, J. Chem. Phys., 1973,59, 3953. N. Basco and R. D . Morse, Proc. Roy. SOC., 1973, A334, 553. A. B. Harker and H. S. Johnston, J . Phys. Chem., 1973,77, 1153. A. A. Westenberg and N. DeHass, J. Chem. Phys., 1969, 50, 707. D . D. Davis, W. Wong, and J. Lephardt, Chem. Phys. Letters, 1973, 22, 273. J. Barassin and J. Combourieu, Bull. SOC.chim. France, 1973, 2173; 1974, 1.

54

Reaction Kinetics

Table 5 Arrhenius parameters for the reactions of O(z3PJ)atoms with methane and chloromethanes 206 (powers of 10 in parentheses) A/cm3molecule-' s-l 5.8 (-11) 2.2 (-11) 9.9 (-12) 4.8 (- 12) 2.7 (-13)

Substrate

CH, CH3Cl CH,Cl, CHCl, CCll

E,,/kJ mol-l 37.3 28.9 23.8 20.9 18.0

E.s.r. spectroscopy has been employed 2 0 7 to study the reaction 0

+ HBr

+ OH

+ Br

(53)

and the subsequent processes. O(Z3P2),Br(42Pt), OH(X2T13) and H(12S+) were all monitored. This is the first direct study of reaction (53) and a rate constant k,, = 4.4 f 1.0 x cm3molecule-1 s-l at 298 K was obtained. OH(w = 0 and w = 1) were detected, and measurements of the relative populations gave a lower limit of 0.3 for the ratio of rate constants for formation of OH(w = 1) and OH(w = 0). Arnoldi and Wolfrum 208 have studied the reactions of 0 (and H) atoms with vibrationally excited HCl(w = l), in a discharge flow system. A pulsed chemical laser (C1 HI) was used to excite HCl, and the relative importance of the two pathways (and the corresponding H atom reactions)

+

0

+ HCl (W

=

1) + O H

0

+ HCl (w

=

1) -+0

+ Cl

+ HCl (V

(54) =

0)

(55)

was determined by observing the decay of HCl i.r. emission and by adding a small quantity of NO to obtain the 0 atom concentration. It was found that abstraction rather than collisional deactivation was the dominant removal process for 0 HCl*, so that the measured rate constant k,, k5, = 3.7 f 1.2 x 10-l2cm3molecule-l s-l is approximately equal to k54. (In contrast, vibrational deactivation of HCl was the main process in collisions with H atoms.) The rate constant k56 = 9.3 f 3.2 x 10-16cm3 molecule-ls-l at 298 K, with an activation energy of 18.8 kJ mol-l, has been reported 209 for the reaction

+

0

+

+ HCl (V = 0)

-+ OH

+ Cl

(56)

so that vibrational excitation of HC1 (the first vibrational quantum corresponds to 34.3 kJ mol-l) is extremely efficient in overcoming the energy barrier. The effect of reactant vibrational excitation on the rate of the reaction of oxygen atoms with CN(X2X+,w" = 0-7) has been studied 210 by flash *07 *08

2os

*lo

G. A. Takacs and G . P. Glass, J. Phys. Chem., 1973, 77, 1182. D. Arnoldi and J. Wolfrum, Chem. Phys. Letters, 1974, 24, 234. V. P. Balakhuin, V. I. Egorov, and E. 1. Intezarova, Kinetics and Catalysis (U.S.S.R.), 1971, 12, 258. H. Schacke, K. J. Schmatjko, and J. Wolfrum, Ber. Bunsengesehchaft phys. Chem., 1973, 77, 248.

Reactions of Atoms in Ground and Electronically Excited States

55

photolysis and kinetic absorption spectroscopy. For the exothermic reaction (AH: = -305 kJ mol-l) 0

+ CN + CO + N

(57)

no measurable change in the rate constants k(v”) = 2.09 x 10-l1cm3 molecule-l s-l for v” = 0,1, . . . 6 at 298 K could be detected. A significantly higher rate constant for the w” = 7 level was attributed to the effective use of this vibrational energy in the endothermic reaction (AH: = 138 kJ mol-l) O+CN+NO+C

(58)

[The interesting observation was also made that, in the reaction of 0 2 ( X 3 X ; ) with CN(X2X+,w’’ = 0-7), rate constants decreased monotonically with increasing CN * vibrational excitation, contrary to the earlier findings of Bullock and Cooper.211] Cramp and Lambert 212 have examined the vibrational relaxation at 298 K of C02(v3)by O(2,PJ) using laser-induced fluorescence. The efficiency of O(23PJ) was found to be an order of magnitude greater than for selfcollisions of C02, and two orders of magnitude greater than for C02 Ne collisions,where approximately the same reduced mass and collision frequency are involved. It has also been from i.r. emission measurements (2000-4000 K), that 0 atoms are an order of magnitude more efficient than Ar in relaxing the bending mode of C02. These observations are not readily interpreted in terms of conventional theories of vibrational relaxation and do not show any correlation with reduced mass. Cramp and Lambert suggest that either very strong intermolecular attractive forces (as proposed 214 for abnormally short relaxation times in NH,, H,O, CH,CN, and SO, selfcollisions) are involved, or vibronic transitions (as proposed by Nikitin 215) occur, the vibrational energy transfer being accompanied by a change in the electronic state of the system. This is allowed for species possessing resultant orbital angular momentum, such as NO, 0, and Fe. High vibrational relaxation efficiencies have also been found 216v 217 for collisions of 0 atoms with 02,CO, and, to a smaller extent, N2. It has been suggested 218 that the higher efficiency for CO + 0 and O2 + 0, compared with N2 + 0 relaxation, may be due to atom exchange reactions. Eckstrom has measured relaxation times for N2 0 from 120-3000 K in good agreement with high-temperature shock-tube results 216 and with low-temperature measurement^.^^ The observed weak temperature dependence is in marked

+

+

216 *I7

G . E. Bullock and R. Cooper, J. C. S. Furuday I, 1972, 68, 2175. J. H. W. Cramp and J. D. Lambert, Chem. Phys. Letters, 1973, 22, 146. R. E. Center, J. Chem. Phys., 1973, 59, 3523. J. D. Lambert, J. C . S. Furuday ZZ, 1972, 68, 364. (a) E. E. Nikitin, Mol. Phys., 1963, 7 , 389; (b) E. E. Nikitin and S. Ya. Umanski, Comments At. Mol. Phys., 1972, 3, 195. W. D. Breshears and P. F. Bird, J . Chem. Phys., 1968, 48, 4768. R. J. McNeal, M. E. Whitson, jun., and G. R. Cook, Chem. Phys. Letters, 1972, 16,

218

D. J. Eckstrom, J. Chem. Phys., 1973, 59, 2787.

211 212

213 21p

*15

507.

56

Reaction Kinetics

contrast to the predictions of the curve-crossing model of Fisher and Bauer.21g Electronically Excited 0(21D2).-Absolute rate data for a number of important reactions involving 0(21D2) have recently been determined,ls1220+ 221 using signal-averaging techniques together with time-resolved atomic absorption spectrophotometry at 115.2 nm (3lD; t 21D2). These data (Table 6) should

Table 6 Rate dirta for reaction and quenching of O(2lD2) by various gases studied by attenuation of atomic resonance radiation at A = 115.2 nm(O(3lD:) t 0(21D2)} and at 300 K (powers of 10 in parentheses) Molecule He Ne Ar Kr Xe 0 2

Nz

co CO,

klcm, molecule-' s-' G1.5 (-15) 1.1 i O . l (-14) 7.1 0.06 (-13) 1.5 f O . l (-11) 1.0 f 0.1 (-10) 7.0 f 0.5 (-11) 6.9 f 0.6 (- 11) 7.3 f 0.7 (-11) 2.1 f 0.2 (-10)

+

Ref. 15 221 221 221 221 220 220 220 222

Molecule H20 0, H, D2

NO N2O NO2 CH, (2302

k/cm3 molecule- s3.0 & 0.3 (- 10) 2.7 f 0.2 (-10) 2.7 -& 0.3 (-10) 1.8 f 0.2(-10) 9.4 f 0.8 (-11) 2.2 -& 0.2 (-10) 2.3 & 0.2 (-10) 3.1 & 0.4(-10) 4.0 I_t 0.4 (- 10)

Ref. 220 220 221 221 221 221 221 221 221

be regarded as the most reliable currently available. Investigations by other methods complement these results by providing information on competing reaction channels. In collisions with H20, the dominant process appears to be the rapid reaction to form two OH(X211) radicals; Simonaitis and Heicklen 222 have found that < 4 % of collisions at 373 K result in quenching to O(3P),and that the intermediate product (H202)in the reaction O(2lD2)

+ H20

+ H20:

(59)

can be stabilized at high pressures (20 f 10%with 650 Torr H 2 0 at 473 K). The relative rate of O(2lD2) reaction with H 2 0 and N2, kHSO:kN, = 3.5 f 1.5,223is in reasonable agreement with the ratio of 4.3 from the direct measurements of Heidner et al. Snelling 224 has also found the ratio kN2:ko, 0.7 f 0.25 (cf. Heidner et al., 0.99), and reports that 75% of O(2lD2) + O2(X3C;)collisions lead to formation of 02(b1Ct). Giachardi and Wayne 225 have reported this proportion as 50-60%. This reaction is of importance -7

219 220

221

222

223 224

225

E. R. Fisher and E. Bauer, J. Chem. Phys., 1972, 57, 1966. R. F. Heidner, tert., D. Husain, and J. R. Wiesenfeld, J . C. S. Faraday II, 1973, 69, 927. R. F. Heidner, tert. and D. Husain, Internat. J. Chem. Kinetics, 1973, 5 , 819; 1974, 6, 77. (a) R. Simonaitis and J. Heicklen, Internat. J. Chem. Kinetics, 1972, 4, 529; (b) R. Simonaitis and J. Heicklen, J. Phys. Chem., 1973, 77, 1096. C. J. Fortin, D. R. Snelling, and A. Tardif, Canad. J . Chem., 1972, 50, 2747. D. R. Snelling, Canad. J. Chem., 1974, 52, 257. D. J. Giachardi and R. P. Wayne, Proc. Roy. SOC.,1972, A330, 1 3 1 .

Reactions of Atoms in Ground and Electronically Excited States

57

in the upper atmosphere, being the main source of 02(b1C,+). Isotopic labelling experiments 226 have demonstrated that quenching of O(2’0,) by O2 proceeds mainly through an atom exchange process; flash photolysis of 1603-1802-Hemixtures showed that 85 % of the 1 8 0 2 had been converted into 1601*0. For the reaction O(2’0,)

+ 02(X3E;)

+=

O(23PJ)

+ O 2 ( P C ; ,v”

= n)

(60)

(note that this is spin allowed) formation of O z in the levels v” = 13, 14, and 15 was shown to occur with an efficiency in the range 0.3 < a, < 0.5. The excitation energy of O(2l0,) is only sufficient to produce O$ in levels up to v” = 10 and it was proposed that the additional energy comes from recoil of the O(2lD2)atoms produced in photolysis of 0,.The correlation diagram given in ref. 226 is unfortunately incorrect and assigns a higher degeneracy to many states of 0,than is actually the case. The correct correlation diagram for C8symmetry reveals that the surfaces for O(2lD2) 0 2 ( X 3 C - )cordate with very high-lying (unknown) states of 03,and do not lead to O(PP,) 0 2 ( P C ; ) via the 3B2states of O3 as proposed. We would suggest that it is more appropriate to consider vibronic surfaces in this case [i.e. surfaces correlating with vibrationally excited 0 2 ( X 3 C g )and O(z3PJ) should be considered]. Reaction of O(2lD2) with N20 proceeds by two channels with roughly equal cross-sections 22

+

+

O(2lD2)

+ N20(X1C+)

+=

--f

2NO(X211,v”

+

=

2 and 1)

02(a1Ag) N2(X1C,‘)

(61) (62)

The most recent value of the rate constant ratio for these two channels is 0.70 & 0.02 determined by Ghormley et a1.;228other reported ratios are 0.83 f 0.06 229 and 1.01 f 0.06.230Ghormley et a1.228found the relative rate of removal of O(2l0,) by O2 and N20 to be 0.31 i-0.01; the absolute rate constants of Heidner et a1.,220* 221 ko, = 7.0 f 0.5 x and kNa0= 2.2 ic 0.2 x 10-locm3molecule-ls-l, give the ratio 0.31, in excellent agreement. The data of Heidner et a1.220* 221 confirm that N20 and C 0 2 have comparable overall efficienciesfor removal of 0(21D2),as found by Loucks and Cvetano~i6,~~’ who obtained results suggesting that the only important channel with C02is 226

227

228

229

230

231

D. W. McCullough and W. D. McGrath, J. Photochem., 1973, 1, 241. J. P. Simons, C. Boxall and P. W. Tasker, Faraday Discuss. Chem. SOC., 1972, No. 53, 182. J. A. Ghormley, R. L. Ellsworth, and C. J. Hochanadel, J. Phys. Chem., 1973, 77, 1341. R. Simonaitis, R. I. Greenberg, and J. Heicklen, Internat. J. Chem. Kinetics, 1972, 4, 497. P. M. Scott, K. F. Preston, R. J. Anderson, and L. M. Quick, Canad. J . Chem., 1971, 49, 1808. L. F. Loucks and R. J. CvetanoviC, J. Chem. Phys., 1972,57, 1682.

Reaction Kinetics

58

o(210,)

+ co,

--f

0(23~,)

+ co,

(63)

The insertion reactions of 0(2ID2) with alkanes have received considerable attention in the past.69 More recent work 232 has revealed the presence of two other channels leading to elimination of H, and abstraction of a hydrogen atom to form OH. Lin and De More 232 have studied the reaction of O(2lD2) with CM, and have determined the relative rates into all three channels :

+ CH4

-+

CH,OH*

k,:k,:k,

=

1:1:0.25

O(2lD2)

(Ma)

It was concluded that direct abstraction (64b) can proceed independently from any fragmentation of the excited CH30H* intermediate.232 We would suggest that channel (64b) probably involves large impact parameter collisions, and should thus occur with a11 alkanes with approximately constant probability. The large rate constant 221 for overall removal of O(2l0,) by CH, [k = (3.1 f.0.4) x 10-10cm3molecule-1 s-l] implies a large cross-section for reaction and gives support to this proposal. Furthermore, suppression of this channel (64b) in the liquid phase would be expected, owing to the cage effect, and is in fact For smaller impact parameter collisions the force-field of CH4 will be drastically modified by the approaching O(2l0,) atom. The position of minimum potential energy will correspond to the CH,OH configuration, but the initially formed collision complex contains sufficient energy to fragment and, unless stabilized by collisions, will do so. The relatively large energies involved and smaller number of degrees of freedom available for CH,OH will lead to rapid fragmentation. Thus little stabilization is observed even at pressures greater than one atmosphere 232 (for the higher alkanes, the increasing number of degrees of freedom allows stabilization at progressively lower pressures.) Fragmentation may occur via three channels, CH30H* + CH, +CH, --f

+ O(2lD2)

C65a)

+ OH

C H 2 0 t- H2

(65c)

Channels (6%) and (65c) will be favoured relative to (65a), owing to the greater exoergicity. The experimental difficulties involved in separating the the direct reaction (64b and 64c) products from the indirect fragmentation (65b and 65c) products appear to have caused much confusion in the past. However, the work of Lin and De More 232 does much to clarify this situation. They have shown that the direct reaction (64c) accounts for ca. 9% of the 23'?

C.-L. Lin and W. B. DeMore, J . Phys. Chem., 1973, 77, 863 (and references therein).

Reactions of Atoms in Ground arid Electronically Excited States

59

reaction products and that this channel contributes equally in the gas and liquid phases. We may attempt to rationalize this by suggesting that, for a small number of low impact parameter collisions, the energy released by the approaching O(2l0,) atom is channelled directly into C-H motion and leads to the concerted elimination of H2 (the fact that vibrationally excited H2 can be formed allows access to quite large regions of phase space). This is consistent with the proposed short lifetime of the insertion product CH,OH* (cu. 8 x 10-13s). The persistence of this channel in the liquid phase is also expected, as the products are not free radicals and will not recombine in the solvent cage. Electronically Excited 0(21So).-The fairly substantial body of data on the rates of quenching of 0(2lSO)atoms by various gases (including the temperature dependence for some simple molecules) has been summarized in previous 0(21S0) atoms have generally been produced by photolysis of either C 0 2or N20. Recently, relative quantum yields have been determined233 for the production of O(lS0) in photolysis of C 0 2 ,02,03,and N 2 0 by 121.6 nm Lyman-a radiation. To date, 0(21S0) atoms have been monitored in kinetic experiments by the forbidden 21S0 -+ 2l& emission at 557.7 nm. Collisional enhancement of this emission by the noble gases and N2 and H2 is well d o ~ ~ m e n t e234 d . The ~ ~ ~effect ~ is proportional to pressure of added gas and has been shown, for Ar and N2, to be temperature independent in the range 200-380 K. It appears that, in the interaction with the inert gas, the angular momentum of the species 0(21So)--M is space quantized with component s1 = 0 along the internuclear axis, so that the transition is now allowed by the selection rule for Hund's coupling case (c), AQ = 0, +1. The quenching of 0(2lSO)by NO has been investigated 235 at 291 and 201 K ; the respective rate constants, (5.7 & 0.6) x 10-lo cm3 molecule-Ls-l and (5.0 & 0.6) x 10-lo cm3 molecule-' s-l, are in good agreement with previous determinations. The low flash energies employed (ca. 1 J) ensured that radical-radical reactions did not interfere. The quenching channels involved have not been established. A calculation of the quenching rate of O(2lS0) by O(23PJ)has been performed by Olson ;23e coupling between the product and reactant channels

is induced by spin-orbit interaction between 3rIgand C ' : states of 02.The Landau-Zener formula was used to calculate the transition probability at the crossing; ub initiu O2potential energy curves were employed. The predicted room temperature rate constant of 2.9 x lo-', cm3 molecule-l s-l is lower 233 2s4

2s5

B. A. Ridley, R. Atkinson, and K. H. Welge, J. Chem. Phys., 1973, 58, 3878. R. F. Hampson, jun., and H. Okabe, J. Chem. Phys., 1970, 52, 4; and ref. 32 and references therein. R. Atkinson and K. H. Welge, J. Photochem., 1973, 1, 341. R. E. Olson, Chem. Phys. Letters, 1973, 19, 137.

60

Reaction Kine tics

than the experimental value (7.5 x 10-l2cm3molecule-l s-l) obtained by Felder and Young,237although the model is no doubt essentially correct. A likely explanation for the discrepancy is that the activation energy obtained from the ab initio curves is too large. 9 Sulphur, Selenium and Tellurium Atoms Ground State (n3P2)Atoms.-The early pioneering work on the reactions of ground-state sulphur atoms, which yielded relative rate 2 3 9 has been largely superseded by work employing direct spectroscopic techniques, and absolute rate data are now available for the reactions of S(33P,) with 02,240-244 acetylene,245and a range of 01efins.~~~~ 247 The reaction of S(PP,) with 0, has been studied using flash time-resolved atomic absorption ~pectrophotometry,~~~ atomic resonance fluorescence,242and flow discharge All four techniques yield results which are essentially in agreement, and demonstrate that the earlier work by Homann et aZ.,244which was performed at high temperatures (675-1090 K), cannot be extrapolated to yield data for the region of 300 K. The data for 300 K and over the range 2 5 2 - 4 2 3 K are presented in Table 7.

Table 7 Rate data for the reaction S paren theses) k/cm3molecule-l s-l 1.7 & 0.4 (-12) 2.0 f 0.5 (-12) 2.8 0.3 (-12) J A = 2.24 f 0.27 (-12) = 0.0 f 1.7 kJ rnol-’)

+ 0 , + S O + 0 (powers TW 295 298 298

Ref. 241 243 240

252-423

242

of 10 in

The most precise data are those of Davis et aZ.,242 derived using the resonance fluorescence technique. Both of the transient products of this reaction

+ 02(x3c,-) -+ 0 ( 2 3 ~ , )+ so(x3c-)

~ ( 3 3 ~ ~ )

(67)

were observed, and the rate of formation of O(Z3PJ)was shown242to be 237

238

240

241 2p2

2d3 244

245 246

247

W. Felder and R. A. Young, J. Chem. Phys., 1972, 56, 6028. 0. P. Strausz and H. E. Gunning, Adv. Photochem., 1966, 4, 143. 0. P. Strausz, ‘Organosulphur Chemistry’, ed. M. J. Janssen, Interscience, New York, 1967, and references therein. R. W. Fair, A. van Roodselaar, and 0. P. Strausz, Cunud. J . Chem., 1971, 49, 1659. R. J. Donovan and D. J. Little, Chem. Phys. Letters, 1972, 13, 488. D. D. Davis, R. B. Klemm, and M. J. Pilling, Internut. J. Chem. Kinetics, 1972,4,367. R. W. Fair and B. A. Thrush, Trans. Faraduy SOC.,1969, 65, 1557. K. H. Homann, G. Krome, and H. G . Wagner, Ber. Bunsengesellschaft phys. Chem., 1968, 72, 998. D. J. Little and R. J. Donovan, J. Photochem., 1972, 1, 371. D. D. Davis, R. B. Klemm, W. Braun, and M. J. Pilling, Internat. J . Chem. Kinetics, 1972, 4, 383. R. B. Klemm and D. D. Davis, Innternat. J. Chern. Kinetics, 1973, 5, 375, 841.

Reactions of Atoms in Ground and Electronically Excited States

61

equal to the rate of decay of S(33P,). A detailed study of the dynamics of this reaction does not appear to have been made and, although vibrationally excited SO(X3Z:-)may be formed, the fraction of the total exothermicity which appears as vibration has not been established. The addition reactions of ground-state sulphur, selenium, and tellurium atoms, with olefins, are probably the most extensively studied reactions involving these a t o r n ~ . ~ ~ The ~-~~ most O detailed and precise studies are those of Davis et al.,24ss247 who used the resonance fluorescence technique to monitor the addition of S(33PJ)to a range of olefins over the temperature range 219-500K. The Arrhenius parameters derived from this work, -10.3,

- 10.5 -10.7 -1 0.9

d

/

-

-

-11.7

-

-11.9

=I

\

\

al

3 -12.1 -* 2 -12.3 !i

m

-12.5b

1

Figure 9 Arrhenius plot showing temperature dependence of reactions of groundstdte atomic sulphur,S(33P,), with olefins ethylene,propylene, but- l-ene, cis-but-Zene and tetramethylethylene (after ref. 247) 249 249a

A. B. Callear and W. J. R. Tyerman, Trans. Faraday Sac., 1966, 62, 371,2760. J. Connor, G. Greig, and 0. P. Strausz, J . Amer. Chem. SOC.,1969, 91, 5695. J. Connor, A. Van Roodselaar, W. Fair, and 0. P. Strausz, J . Amer. Chem. SOC., 1971, 93, 560. R. J. Donovan, D. Husain, R. W. Fair, 0. P. Strausz, and H. E. Gunning, Trans.

Faraday SOC.,1970,66, 1635.

-

Table 8 Arrheizius parameters for the reaction of s(33PJ), Se(43P,), aiid Te(Si3P,)with olefns Olefin

s(33PJ)(I

Se(43Pz)

Refs.

Refs.

I

1054/cm3 molecule-l s-'

kJ mol-1

Ethylene

7.13 f 0.74

6.6 f 0.3

246

1.8 f 0.6

11.8 f 0.8

Propylene

6.03 f 0.72

1.6 f 0.4

247

2.2 f 0.7

9.8 f 1.0

But-I-ene

7.41 f 1.15

1.5 f 0.4

247

5.2 rfr 1.7

cis-But-2-ene

4.68 f 0.70

-0.96 f 0.4 247

EactI

Refs.

1012A/cm3 molecule-ls-l

kJ mo1-1

248

ca. 1.4

ca. 10.5

249a

,, ,,

ca. 0.5

ca. 2.5

249a

9.4 f 1.0

3.3 & 1.4

5.1 f 1.1

,,

trans-But-2-ene

2.9 f 1.3

2.5 f 1.1

Isobutene

4.0 f 1.7

4.2 f 1.0

,, ,,

Buta-l,3-diene

8.8 rfr 3.8

3.7 f 1.0

,,

9.2 f 1.8 10.2 f 1.2

,, ,,

3.0 f 3.3

,,

Pent-1-ene

ca. 4

Vinyl chloride

1.3 i 0.7

Acrylonitrile Tetramethylethylene

1011~/cm3 molecule-l s'-l

Te(55P2)b 7

ca. 0.2

4.68 f 1.70

-5.4 i 1.0

247

Eactl

kJ mo1-'

EWtI

!a ca. 0.4

-6.7 f 5.8 249a

Spin-orbit relaxation of S ( 3 3 P ~is) rapid under the conditions employed for these experiments and thus all of the J states may take part in the reaction; Calculated from the data given in reference 249a.

a

8. P

2-

Reactions of Atoms in Ground and Electronically Excited States

63

together with those for Se(43P2)and Te(YP,) are given in Table 8. Previous kinetic studies of the addition reactions of S(33PJ)at 298 K, which employed flash spectroscopy in the vacuum U.V.,~~O are in reasonable agreement with the data of Davis et aLzas The Arrhenius plots from which the data247in Table 8 are derived, are illustrated in Figure 9, and demonstrate in a striking manner the negative activation energies which are found in some cases when the rate data are presented in Arrhenius form. Similar negative activation energies have also been found for addition of Te(YP,) to tetramethylethylene 24p and'0(2SPJ) to cis-but-Zene and tetramethylethylene.261This type of behaviour indicates that addition proceeds with zero threshold energy and that the crosssection falls rapidly with increasing relative kinetic energy, as for atomic recombination. A remarkably good linear correlation between the activation energy for S(33PJ)addition to a given olefin, and the ionization potential of the olefin, has been found 247 and is illustrated in Figure 10. These data should thus allow a precise estimate of the activation energy for olefins which have not been studied, by linear interpolation. Davis et aLza7also reemphasized previous comments on the increase in activation energy with decreasing electronegativityof the atom, and demonstrated that there is a linear correla-

6.04 .OF

Id

E 2.0-

7 Y \

U IJJ

0-

-2.0-

-4.0I

I

8.4

I

8.6

I

I

8.8 9.0

I

I

I

I

I

I

1

I

9.2 9.4 9.6 9.8 10.0 10.2 10.4 10.6 Ionization potential I eV

Figure 10 Plot showing the correlation of E, for reaction of S(3'PJ) with several olefins as a function of the ionization potential of each olefin (after ref, 247) 251

D. D. Davis, J. T. Herron, and R. E. Huie, J. Chem. Phys., 1973,59, 628.

64

Reaction Kinetics

tion between the activation energies for S(33PJ)and O(z3PJ).The reaction of S(3sPJ)with ethylene episulphide has been studied by two independent techniques 252 and shown to be extremely rapid [k = (4.47 f 0.26) x cm3 molecule-l s-l; the reaction was studied 252 over the range 298-355 K and the activation energy shown to be zero]. The addition of Se(43P2)to a wide range of olefins has been studied in some detail, using flash spectroscopy to monitor both the decay of the selenium atom and the formation of the metastable episelenides (Table 8). The latter were observed via intense diffuse bands in the U.V. and were found to persist for several minutes in some cases. A correlation between the ionization potential of the olefin and the activation energy for the addition reactions of Se(43P2)was found; however, in this case the correlation does not appear to be linear. This departure from linearity has been The data for Te(53P2)addition to ethylene, propylene, and tetramethylethylene 249 are consistent with the correlation for S(33PJ),and Arrhenius parameters are listed in Table 8. 2509

Spin-Orbit Excited States (n3Pl,,).-Observations on the spin-orbit multiplets of O(23PJ)show that Boltzmann equilibrium is maintained between these levels under the conditions of most e x p e r i r n e n f ~ . ~Furthermore, ~~ rapid relaxation between the spin-orbit multiplets of S(33PJ)has been and again indicates that any departure from an equilibrium distribution will be negligible under most conditions. The spin-orbit splittings for the 43PJ state of selenium are considerably larger than for the lighter elements of this group and a non-Boltzmann distribution has been observed 255 following the flash photolysis of CSe,. Spin-orbit quenching of Se(43P0)was studied using flash spectroscopy 255, 256 and rate data for the quenching gases employed are given in Table 9. Specific quenching channels were proposed, based on the assumption that resonant Table 9 Spin-orbit relaxation of Se(43Po)248 (powers of 10 in parentheses) Second-order quenching

Quenching species Ar se(43Po) NzO Se(43P0) co2 Se(43P0) Hz Se(4’Po) CO Se(43P0) Se(43P~) Nz Se(43P,) 0,

+ Ar

Proposed quenching process + Se(43P,)

+ Ar

+ N20(vz 0) Se(43P,)+ N,O(v, 1) Cody2 0) Se(43P,) + C 0 2 ( v 2 1) ++ H,(J) Se(43P,) + H,(J’) + CO(u” 0) Se(43P,) + CO(u” 1) + N2(u” 0) Se(43P2) t- N2(u” 1 ) + 02(u”

252

253 254

255

= =

= =

4

+

-+

=

-+

= -+ = 0) -+ Se(43P,)

==

+ O2(u”

=

=

0)

constant/ cm3 molecule- s2.4 I-t 0.3 ( - 14) 1.2 & 0.15 (-10) 1.4 5 0.1 (-10) 3.5 & 0.7 (-10) 1.1 & 0.3 (- 12) 3.0 i 0.3 (-- 12) 1.5 L 0.3 (-12)

D . D . Davis and R. B. Klemm, internat. J . Chem. Kinetics, 1973, 5 , 149. M. J. Kurylo, N. C. Peterson, and W. Braun, J. Chem. Phys., 1971, 54, 943. R. J. Donovan, Trans. Faraday. SOC.,1969, 65, 1419. A. B. Callear and W. J. R. Tyerman, Nature, 1964, 202, 1326.

Reactions of Atoms in Ground and Electronically Excited States

65

processes would be dominant,26obut the resulting distribution of energy in the quenching molecule was not observed, nor was the final state of the quenched atom (this could be 3P1or 3P2) and many interesting questions remain to be answered. Recent work on the spin-orbit relaxation of tellurium atoms has served to emphasize the large isotope effects which may occur in the quenching of excited Table 10 lists the cross-sections and ratios of cross-sections which have been reported for the quenching of various excited atoms by Table 10 Cross-sections for quenching at 300 K of I(YP+),Te(S3P1,,), and T1(62Ps) by H,, D, and HD (powers of 10 in parentheses) I(S2P+) -+ I(52P$ Te(S3Pl,,,)-+ Te(s3p2) T1(62P+) -+ T1(62P+)

AE/cm-l crH,/nm2 o,,/nm2 crHD/nm2 crH2/oD8Ref. 7603 7.4 ( - 5 ) 8.0 (-7) 2.1 (-4) 92 258,259 4707 5.8 (-3) 6.9 (-6) 840 257 7793 2.8 (-4) 3.9 (-5) 7 260

H,, D2, and HD. The data for relaxation of excited tellurium atoms were originally discussed in terms of a curve-crossing model, but more recently all the data presented in Table 10 have been discussed in a unified way in terms of resonant energy transfer processes.258 This discussion focused attention on specific resonant energy transfer channels of the type Te(53P~)+ H,(d’

= 0,J” =

1) Te(S3P2) H2(d = 1, J’

+

-j

=

3) - 6 cm-l

(68)

and has been successful in accounting for the data in Table 10 and also for the temperature dependences of the cross-sections where these are known.258 The excited spin-orbit states of tellurium, Te(53P1) and Te(S3PO),lie close in energy (AE = 44 cm-l) but are separated from the ground state Te(53P2), by ca. 4700 cm-l. Thus, while the 3P0 and 3P1states are in Boltzmann Table 11 Rate constants and quenching cross-sections 267 at 295 K for Te(53P1,,) (powers of 10 in parentheses) Quenching gas H2 D2 0 2

He Ar Xe 258

m7 25*

259 260

Rate constant/cms molecule- s1.03 f 0.15 (-11) 8.8 f 6.5 (-15) 1.28 f 0.55 (-14) 03.0 (-15) 1.38 f 0.28 (-15) c2.7 (-15)

Cross-section/nm2 5.8 (-3) 6.9 (-6) 2.6 (-5) G2.4 (-6) 3.0 (-6) G8.6 (-6)

A. B. Callear and W. J. R. Tyerman, Trans. Faraday Soc., 1965, 61, 2395; 1966, 62, 2313. R. J. Donovan and D. J. Little, J. C . S. Furaday 11, 1973, 69,952. R. J. Donovan, R. Butcher, and R. H. Strain, J. C.S. Furaduy ZI, 1974,in the press. J. J. Deakin and D. Husain, J. C. S. Faraday IZ, 1972, 68,41. J. R. Weisenfeld, Chern. Phys. Letters, 1973, 21, 517.

Reaction Kinetics

66

equilibrium on the time scale of a conventional flash-photolysis experiment, their relaxation to the ground state has been found to be slow. Studies using time-resolved atomic absorption spectrophstometry 25 have been made and yield the rate data for removal of Te(53P1,o) given in Table 11. Relaxation by the noble gases is less efficient than for Se(43P,), as expected from a consideration of the energy which must be converted into translation. However, relaxation of Te(53P1,o) by the noble gases is more efficient than for the halogens where [for C1(32P3)] less energy is converted into tran~lation.~~ This presumably indicates that curve crossing regions are accessible at 300 K for Group VI atoms, but not for the halogens. First Excited Singlet State, (n1D2)--The early work of Strausz and Gunning on the insertion reactions of S(31D2)into carbon-hydrogen bonds has been reviewed in detail el~ewhere.~~8~ 239 The numerous other studies concerned with quenching of S(31D2) by the noble gases and reactions with other molecules, which have been reported more recently, will be summarized here. Despite considerable effort in a number of laboratories, direct observations on S(31D2)have not yet been reported. This is due mainly to the fact that OCS [the cleanest photochemical source of S(31D2)]is a weak absorber in the u.v., thus requiring relatively high partial pressures for the production of measurable concentrations of S(31D2),and that the reaction S(3ID,)

+ OCS

-+

+

S2(a1Ag) CO

(69)

is extremely rapid. However, the formation of S2(a1Ag)has been observed spectroscopically 261 and the kinetics of its formation employed to obtain a lower limit for the rate of reaction (69) ( k > 6 x cm3molecule-l s-1).282 The decay of S2(a1Ag)was observed 262 to be extremely rapid ( T , 21 50 ,us) in sharp contrast to the slow decay generally observed for 0z(a1Ag).

The reaction between S(31D2) and N20 has been shown to proceed via three channels 263 S(31D2) N,O -+ N, SO(alA,) (70a)

+

+

+NStNO --f

+ N,O.

S(33PJ)

(7W (70c)

The first two channels were shown to have similar cross-sections 263 but jointly account for only 20% of the total cross-section, the channel leading to S(33PJ)being dominant. This contrasts with the reaction of O(2l0,) with N,O, where relaxation to O(23PJ)is negligible.264Three factors probably 0. P. Strausz, R. J. Donovan. and M. de Sorgo, Ber. Bitnsengesellschaft phys. Chem., 1968, 72, 253.

'Ifi2

2G3

R. J. Donovan, L. J. Kirsch, and D. Husain, Nature, 1969, 222, 1164. R. J. Donovan and W. H. Breckenridge, Chem. Phys. Letters, 1971, 11, 520. J. P. Simons, C. Boxall, and P. W. Tasker, Faraday Discuss. Chem. Soc., 1972, No. 53, p. 182.

Reactions of Atoms in Ground and Electronically Excited States

67

account for this marked difference: firstly the binding energy of the SNNO intermediate is expected to be greater than that for ONNO [the binding energy should lie between that for (NS), and that for (NO)& Secondly the exothermicities of the first two channels are less for S(31D2)than for O(2lD2); these two factors will lengthen the lifetime of the SNNO intermediate relative to that for ONNO. Thirdly, the spin-orbit coupling in SNNO will be greater than for ONNO and will thus facilitate crossing from the singlet potential surface to the triplet surface correlating with S(33P,). The formation of NS in the above reaction and the presence of an intense absorption system for this radical in the U.V. (A N 230 nm; see Figure 1l), has enabled relative rate data to be obtained 285 for a number of reactions (Table 12) by employing NS as a 'spectroscopic marker'. The efficiencies for Table 12 Rate constants (lower limits unless otherwise stated) for reactions of S(31D2)at 300 K (powers of 10 in parentheses)" 39265

Molecule

ocs

k/cm3molecule-l s-' 6.7 (- 11) (reaction) 1.3 ( - 11) (quenching) 4.0 (-11) 7.0 (- 12) 2.4 (-13) < 4 (-14) 5.5 (-11) (reaction quenching) 1.5 (-11) 4.7 (-11) 1.7 (-10) 6.0 (-11) 2.2 (- 11) (reaction quenching) 1.9 (-11) (reaction quenching) 4.2 (- 1 1 ) (reaction quenching) 1.5 (- 10) (reaction quenching) 2.5 (- 10) (reaction quenching) 5 (-13)

+

+ + + + +

+

Rate constants have calculated using the upper limit262for the reaction S(31D2) OCS, the relative rate data given in ref. 265, and the ratio ~ c ~ H ~ taken / ~ ~ from c B rcf. 238. a

quenching by the noble gases bear out the main predictions made ear lie^,^ based on the potential curves for 0 Xe. The other data are discussed in detail in ref. 265. The first excited singlet state of selenium Se(41D2) has recently been observed directly using time-resolved atomic absorption spectrophotometry266 and rate data determined with this technique are given in Table 13. It can be seen that quecching by the noble gases follows the same overall pattern to that observed for O(2lD2) and S(31D2), and that quenching by Xe is

+

*65

268

1972, D. J. Little, A. Dalgleish, and R. J. Donovan, Faraday Discuss. Chem. SOC., No. 53, p. 211. R. J. Donovan and D. J. Little, to be published.

Reaction Kinetics

68 2 2 2 nm

2 4 0 nm

230 nm

Before

4 ILS 10

29 49 104 460

1.28 ms

3.1 9 4.6 8

Figure 11 Formation and decay of the NS radical following the reaction of S(31D2) with N,O (Reproduced by permission from Chem. Phys. Letters, 1971,11, 520)

Table 13 Rate data 266 for the removal of Se(41D2) by various atoms and molecules at 300 K (powers of 10 in parentheses) Molecule He Ne Ar Kr Xe H2

k/cma molecule-' s-l 3.0 1.8 (-14) G3.6 (-13) 7.0 f.4.9 (-12) 2.2 f 1.2 (-11) 6.7 f 3.0 (-11) 2.1 f 0.8 (-10)

Molecule D 2

N2 0 2

co OCSe SF,

k/cm3 molecule-' s7.6 & 4.2 (-11) 1.7 f 0.8 (-10) 1.2 0.5 (-10) 1.4 f 1.1 (-10) ca.1 (-10) ca.3 (-12)

comparable in efficiency for all three atoms in the lD2 state. This latter point is perhaps a little surprising as the increase in spin-orbit coupling on going from oxygen to selenium would be expected to have a fairly marked effect on the probability for non-adiabatic transitions. Presumably changes in the slopes of the potential surfaces, and the range at which crossings between the singlet and triplet surfaces take place, off-set the increased spinorbit coupling term. A similar explanation may account for the low efficiency for quenching of Te(S1D2) by argon.267 Quenching of Se(41D2) by CO and N2 is seen to be extremely efficient (0 = 0.3 nm2) and it is clear that collisions for which the impact parameter is less than the crossing distance have a near unit probability for leading to relaxation.266 For OCSe the crossing of singlet and triplet surfaces will 267

R. J. Donovan, D. J. Little, and J. Konstantatos, J. Photochem., 1972, 1, 86.

Reactions of Atoms in Ground and Electronically Excited States

69

occur at relatively large internuclear separation owing to the substantial binding energy for the ground state of this molecule. It may thus be inferred that N,Se has a substantial binding energy, (but is a kinetically unstable species with respect to further reaction). The closely similar cross-sections observed 266 for Hzand Dzsuggest that reaction and not relaxation is the dominant channel for removal of Se(41D2) by these molecules. The reaction with 0, is fast;266however, the adiabatic channel leading to SeO(alA) is probably endothermic, and removal of Se(41D2) may involve the non-adiabatic channels leading to SeO(X3C-) or Se(43PJ);the latter channel may also yield 02(a1Ag). Reaction with OCSe is expected to yield Se, in the alA or blE states. Indirect evidence for this comes from observations on the rate of formation of Se,(X2C;); two maxima are observed under some. conditions and, by analogy with S2(a1Ag),the first maximum is attributed to the rapid relaxation of Se,(alA,), while the second maximum is consistent with the formation of Se,(X3C;) from atomic re~ombination.~ The reactions of Se(41D2)with propane, cyclobutane, ethane, and isobutane have been studied using flash photolysis with time-resolved mass spectrometry, and the formation of selenomercaptans was observed.268 Second Excited Singlet State (nlSo).-The earliest direct observations on S(3lSO)demonstrated in a striking manner that electronic excitation does Table 14 Rate constants k/cm3 molecule-l s-l at 298 K for the removal of S(3lSO)and O(T1S,) (powers of 10 in parentheses) s(31so) <3.5 1.5 (-17) < 5 (-15) 7.7 f 1.5 (-16) 6.0 & 0.6 (-13) 3.5 f 0.7 (-16) 3.2 & 0.4 (-10) 6.1 & 0.6 (-10) < 3 (-15) < 6 (-17) 4 -j= 2 (-13) 8.1 f 0.8 (-10) 1.0 f 0.2 (-10) 4.9 f 0.5 (-10) 1.5 f 0.2 (-15) 4.4 f 0.5 (-14) 1.3 f 0.2 (-13) 1.6 f 0.2 (-13) zBB

28B

Ref. 33 254 33 33 33 33 33 33

O(2lSO) 5.2 & 0.2 (-18)

Ref. 269u

33

2.8 (-16) 3.6 (-13) 1.0 (-14) 8.0 (-10) 5.0 (- 10) 1.1 (-11) 3.6 (-13)

2696 2696 2696 2696 2696 2696 2696

33 33 33 33 33 33 33 33

4.7 (- 14) 1.0 (-12) 9.6 (- 10) CU. 6 (-11)

2696 2696 2696 269c

W. J. R. Tyerman, W. B. O’Callaghan, P. Kebarle, 0. P. Strausz, and H. E. Gunning, J. Amer. Chem. SOC.,1966, 88, 4277. (a) A. Corney and 0. M. Williams, J. Phys. (B), 1972, 5, 686; ( 6 ) S. V. Filseth, F. Stuhl and K. H. Welge, J. Chem. Phys., 1970, 52, 239; 1972, 57, 4062; (c) R. A. Young, G. Black, and T. G. Slanger, J. Chem. Phys., 1969, 50, 309.

70

Reaction Kinetics

not necessarily lead to enhanced reactivity. Indeed, the lS0state was shown to be several orders of magnitude less reactive than the lD2state in many This work further served to emphasize the need to consider orbital symmetry in understanding the differences in reactivity between electronic states, and stimulated the use of correlation diagrams to assist in this. A more recent study of S(3lSO), which employed photon-counting techniques to observe the slow spontaneous emission from this has enabled many of the upper limits for rate constants given previously to be revised. [Note that collision-stimulated emission probably occurs with some gases; see comments on 0(2lS,,)]. The resulting data are given in Table 14. Although the lS0state of selenium has been observed using flash spectroscopy in the vacuum u.v.,~kinetic data have not yet been presented. 10 Halogen Atoms Fluorine.-The reaction F H, and its isotopic variants have been the subject of intensive experimental and theoretical investigation. Experimental techniques employed to study the microscopic reaction dynamics have included i.r. chemiluminescence, molecular beam reactive scattering, and chemical laser vibration-rotation emission. Chemical lasers can yield precise information on product vibrational state populations (given rigorous consideration of all possible population and depopulation channels), and are well suited to studies of the temperature dependence of reaction dynamics. The most detailed results obtained from a chemical laser system are those of Berry,22who has determined the complete product vibrational state ratios at room temperature for the reactions

+

F f H , +FH*+H

(71 a)

F+D2 +FD*+D

(71b)

F

+ HD +FH: + D

F+DH+FD*+H

(7 1c)

(7W

his most important conclusion being that there is complete parallelism of energv partitioning for all four reactions, with mass effects being secondary. The vibrational state ratios obtained, together with the results of other studies, are presented in Table 15. The possible effect of deactivation by radicals and molecules is discussed by Berry in some detail; he concludes that the nascent HF/DF population is not perturbed by vibrational relaxation processes under the conditions used for quantitative measurements. Suitably scaled plots of N, vs. f , (the fraction of energy partitioned into vibration) are very similar in shape for all four reactions with pronounced maxima at u = 2 (HF) and u = 3 (DF), implying that the energy partitioning is primarily governed by the potential energy surface (which is isotopically

Table 15 Prodirct vibrational-state ratios for the reactions F React ion

F

F

NQINs

+ Hz -+ H F + H

+ Dz -* DF + D

0.66 k 0.13 0.4 rrt 0.2

+ HD -+ HF + D F + H D + DF + H F

0.3 rrt 0.2

+ H2,D2,HD(300 K) NIINO

Technique"

Ref.

i .r. i.r.

27ob 271 272 213 22

NdN,

NZINL

0.58 f 0.12 0.53 f 0.10 0.56 f 0.06 0.48 f 0.01 0.63 f 0.04

3.6 f 0.2 3.4 f 0.7 3.4 f 0.3 (3.3)" 3.40 f 0.10

(1.5 f 0.3)d 1.5 f.0.1 1.80 f 0.10

(2.3)" 2.35 f 0.10

2.3 f 0.2

C.1.

271 273 22

0.15 f 0.05

3.1 f 0.2

5.2 & 0.4

C.1.

22

1.60 f 0.10

2.25 f 0.10

2.2 f 0.2

c.1.

22

i.r. c.1.

5.2 f 0.4

C.I.

i.r. c.1.

Values from refs. 270-272 are corrected for new Einstein transition probabilities. c.l., chemical laser; i.r., infrared cherniluminescence. Interpolated to room temperature. Lower limit Extrapolated value from higher N , / N , - , ratios. f Estimated from a crude population analysis (see ref. 22). a

@

270 271 272 273

N. Jonathan, C. M. Melliar-Smith, S. Okuda, D. H. Slater, and D. Timlin, Mul. Phys., 1971, 22, 561. J. C. Polanyi and K. B. Woodall, J . Chem. Phys., 1972, 57, 1574. H. W. Chang and D. W. Setser, J. Chem. Phys., 1973, 58, 2298. R. D. Coombe and G. C. Pimentel, J. Chem. Phys., 1973, 59,251.

5

9

tl

.9=

Reaction Kinetics

72

invariant) and that mass effects are secondary. This was clearly demonstrated by temperature parametrization using surprisal plots; very similar vibrational temperature parameters were obtained for all four reactions, with mass effects evident in small differences between the two HF-producing and the two DF-producing reactions. Berry compares these results with various published quasiclassical trajectory calculations; these do not predict the small difference between the two pairs of reactions, which may therefore be due to quanta1 rather than classical mass effects, For the intramolecular kinetic isotope effect, kFTHD/ki:DH, Berry has calculated 1.4 f: 1.0, in good agreement with the bulk kinetic result 1.45 i 0.03 obtained by Persky 2 7 4 a using a flow system with mass spectrometric detection. These values are to be preferred to the value 2.5 estimated by Kompa, Parker, and Pimente1;276UF,, the F atom source used in this early work, may have given rise to relaxation effects. Persky 2 7 4 b has also measured the kinetic isotope effect for F -t H2-D,, obtaining

kF+ --H, kF

I

=

(1.04 & 0.02) exp(1.55 i-

D,

Or4kJ

mol-1

),163-417 K

These results may be compared with the intramolecular kinetic isotope effect 2 7 6 for C1 H D (see below) of 1.75 at 297 K. Coombe and Pimentel 2 7 3 have used equal-gain and zero-gain techniques to study the temperature dependences of vibrational state ratios in the F 4-H, and F D, chemical lasers. While only a few transitions were accessible, and some of their results differ from the other values given in Table 15, particularly for HF* (N3/N2),the observed temperature variations were well outside the uncertainty of the experimental results. A decrease in vibrational population ratios was observed with increasing temperature H, which could be fitted by Arrhenius-type expressions for reaction of F

+

+

+

k3/kz

=

0.39 exp(0.490 kJ mol-l/RT)

k 2 / k , = 2.14 exp( 1.063 kJ mol-'/RT) while for F + Dz there was a pronounced increase in k y / k zwith increasing temperature. Coombe and Pimentel 2 7 7 have followed this with an elegant demonstration of the effect of rotational energy of a reactant (H2) upon the vibrational energy distribution in the product (HF) for reaction (71a), by studying the temperature variation of N3/N2and Nz/Nl using both normal hydrogen (25% para) and para-enriched hydrogen (85% para). N3/N2 and N2/Nl were consistently higher with para-enriched hydrogen and showed somewhat different temperature dependences. Coombe and Pimentel have suggested that the opposite temperature dependence for N J N , in the 27p

275 276

Persky, J . Chem. Phys., 1973, 5 9 , 5578; ( 6 ) A. Persky, ibid., p. 3612. K. L. Kompa, J. H. Parker, and G. C. Pimente1,J. Chem. Phys., 1968, 49, 4257 Y. Bar Yakov, A. Persky and F. S. Klein, J . Chern. Phys., 1 9 7 3 , 5 9 , 2415. R. D. Coombe and G. C. Pimentel, J . Chern. Phys., 1973, 59, 1535.

( u ) A.

Reactions of Atoms in Ground and Electronically Excited States

+

73

F D2reaction may be attributable to a combination of opposed rotational and translational effects: the smaller rotational spacing for D2 means that more J states are involved so that rotational population effects might be less pronounced. These chemical laser results emphasize a point made by P ~ l a n y i , ~that '~ the specificity of the product vibrational state populations for these exoergic reactions implies, via microscopic reversibility, pronounced selectivity of energy requirements for the reverse processes. In some cases there may be a 'stereochemical effect' with the requirement for extended bonds (vibrational excitation) for endothermic reactions, as was suggested by Berry and Pimentel 2 7 9 for the addition of HCI to chloroacetylene. This suggestion was made following a study of chemical laser emission from HCI produced by photodissociation of isomeric dichloroethylenes. A number of other exothermic H atom abstraction reactions have been studied using i.r. chemiluminescence measurements. The HF* vibrational energy distributions following reactions of F atoms with various primary C-H bonds and with PH,, SiH4, H20, H202,H2S, NH,, and N2H4 have been 0bserved.~'~7 2 8 0 In almost all cases a large fraction of the excess energy was partitioned into vibrational levels extending close to the thermochemical limit, with population inversion in several cases. For the primary C-H bonds there was a strong similarity in energy partitioning and no tendency for a smaller amount of HF* vibrational energy to be observed as the mass of the other fragment increased. Detailed differences due to thermochemistry, release of radical distortion energy, changes in potential surface, and mass effects are discussed. Although the rotational distribution was partially relaxed, it was clear that for reaction with (CH3)20,(CH3)$3,PH,, and SiH4 an appreciable amount of energy was released as rotation. Kim and Setser 281 have also observed i.r. chemiluminescence from HF* and DF* formed in the reaction of F with secondary C-H/C-D bonds in C,-Clo cycloalkanes and CD,CH2CD3; a lower percentage of available energy was released as vibration compared to reactions with primary C-H bonds. It appears that, to a first approximation, the reaction dynamics can be represented as mixed energy release on a repulsive three-body surface. This is confirmed by trajectory calculations performed 282 within a three-body (LEPS surface) formulation, and supports the suggestion that not all the available energy is accessible to HF* on the very short time scale required for transfer of H in the primary encounter, and that mass effects are minor. The reader is referred to this paper for detailed discussion. There is a marked absence of information on collisional deactivation of F(2?P3)and possible differences in reactivity compared with the ground state 278 27a 280

281 282

J. C. Polanyi, Accounts Chem. Res., 1972, 5 , 161. M. J. Berry and G . C. Pimentel, J. Chem. Phys., 1970, 53, 3453. W. H. Duewer and D. W. Setser, J. Chem. Phys., 1973, 58, 2310. K. C. Kim and D. W. Setser, J . Phys. Chem., 1973, 77, 2493. R. L. Johnson, K. C. Kim, and D. W. Setser, J. Phys. Chern., 1973, 77, 2499.

74

Reaction Kinetics

(22P+), from which it is separated by 404cm-l. The potential of e.p.r. spectrometry in this respect, as demonstrated by the work of Carrington et al.,283does not appear to have been exploited. A significant advance, therefore, is the detection of F(2,Pq,+) atoms using atomic resonance spectrometry in the far vacuum U.V. (95-98 nm).e84 The problem of differentially pumping a windowless spectrometer attached to a flow system, while maintaining an adequate optical aperture, was overcome by the use of commercially available collimated hole structures of low gas conductance and high light transmission. Using a fluorine resonance lamp with helium carrier gas, three resonance absorptions of F(2Pq) and one of F(2P+)were observed. Low intensity resonance fluorescence from F atoms could also be detected. Clyne et ul.255have measured the rate of the fast reaction F

+ C1,

-+

FCl t C1

(72)

in a fast flow system with mass spectrometric detection. Their result, k,, = (1.1 3 0.3) x 1O-locm3molecule-l s-l at 300 K, is fairly close to the value 8.6 x cm3 molecule-'s-l derived from the Arrhenius parameters for this reaction.286 Clyne et al. suggest that this may be a useful titration reaction and show that its rejection for this purpose by Wagner et ul. is based on erroneous data. Clyne et al. also measured the rate of the reaction F

+ CHF,

--jr

HF

+ CF,

(73)

in the range 301-667 K, and made less detailed measurements on reaction of F atoms with H2, CH4, CH,Cl, CH,C12, and CHCI,. Some of these results are included in Table 16. They are probably more precise than earlier data.

Other recent measurements from flow systems are included. Kaufman and co-workers 2 9 6 have monitored transient intermediates using 'molecular beam analysis', in which a variety of molecular beam techniques are used to separate neutral parents before they enter a high efficiency mass spectrometer ionizer; in particular, magnetic deflection with a hexapolar magnet allows only paramagnetic species to enter. The negative tempcrature dependence for reaction of F with CF,Br studied in this manner has been rationalized by Bozzelli et aLZg4using a mechanism involving the initial formation of a stable pseudotrihalogen radical CF,BrF analogous to FBrF : this is therefore similar to the initial reaction in the radical-complex recombination of halogen atoms in the presence of halogen molecules. (In support of this, CF,Br was found to be > 12 times more efficient than CF4 as a third body for recombination of F atoms.) For reaction with CC1,Br and CFJ, Bozzelli and Kaufman 2 9 2 were able to observe CX, radicals but 2H3

2*4

2H5 2x6

A. Carrington, D. H . Levy, and T. A. Miller, J. Chem. Phys., 1966, 45, 4093. P. P. Bemand and M. A . A. Clyne, Chem. Phys. Letters, 1973,21, 555. M. A. A. Clyne, D. J. McKenney and R. F. Walker, Canad. J. Chern., 1973,51,3596. J. Warnatz, H . Gg. Wagner, and C. Zetzsch, Ber. Bunsengesellschaft phys. Chem., 1971, 75, 119.

Reactions of Atoms in Ground and Electronically Excited States

75

Table 16 Rate constants at 298 K for reaction of F atoms with methane and halogenomethanes (powers of 10 in parentheses) Reactant CHI

CHCI, CHF,

CHClFp CFBI CC1,Br CF,Br

CCI,

k/cm3 molecule-l s-l

Ref.

4.9 (-13) A = 3.3 (-12) E,, = 4.81 kJ mol-l 4.3 (-13) 0.68 (-13) > 6 (-13) 5.25 (- 12) 1.87 (-13) A = 1.06 f 0.24 (-11) E,, = 10.0 kJ mol-l 2.3 (-12) 8.3 (-13) 1.2 & 0.6 (-10) 1.7 (-10) 9.3 & 4.7 (-11) 4.2 (- 16) (373 K) 1.3 (-14) (188 K) 6.6 (- 14) 4.0 (-16) 1.1 (-18)

287 288 289 290 285 285 290 291 292 293 292 294 294 295 296 297

no displaced halogen atoms, showing that these reactions proceed by abstraction rather than displacement. Pollock and Jones 2 9 0 determined rates relative to the chemiluminescent reaction F t NO -t- Ar +FNO* + Ar (74) Relative rates of abstraction of H and D atoms by thermal 18Fatoms have been reported 2 9 8 for the substrates H2, D,, CH4, CD4, C2H6,CH3CF3,H,S, and HI. Intermolecular kinetic isotope effects were found to be between 1.5 and 2.0. Substitutions, as opposed to H-abstraction reactions, have been observed 290 using thermal 18F atoms with CF3X, Me,_,H,,CX, and CH,X substrates. With F and Me substituents, yields were essentially zero but substitution 287

2so 281

2s2

294 295

296

2s7 2s8

2*s

H. Gg. Wagner, J. Warnatz, and C. Zetsch, Anales Asoc. quim. Argentina, 1971, 59, 169. K. L. Kompa and J. Wanner, Chem. Phys. Letters, 1972, 12, 560. R. Foon and G. P. Reid, Trans. Faraday SOC.,1971, 67, 3513. T. L. Pollock and W. E. Jones, Canad. J. Cltem., 1973, 51, 2041. R. Foon and N. A. McAskill, Trans. Furaday SOC.,1969, 65, 3005. J. W. Bozzelli and M. Kaufman, J. Phys. Chem., 1973, 77, 1748. I. 0.Leipunskii, I. 1. Morosov, and V. L. Tal’Roze, Doklady Akad. Nauk S.S.S.R., 1971, 198, 547. J. W. Bozzelli, C. E. Kolb, and M. Kaufmnn, J. G e m . Phys., 1973, 59, 3669. C. Zetsch, Dissertation, University of Gottingen, 1971. C . E. Kolb and M. Kaufman, J. Phys. Chem., 1972,76, 947. R. Foon and K. B. Tait, J. C. S. Faraday I , 1972, 68, 1121. R. L. Williams and F. S. Rowland, J . Phys. Chem., 1973, 77, 301. R. S. Iyer and F. S . Rowland, Chem. Phys. Letters, 1973, 21, 346.

76

Reaction Kinetics

occurred much more readily with X =H. These experiments do not provide direct evidence concerning possible inversion or retention of configuration at carbon; nevertheless these authors argue that the most plausible explanation of their observations is that substitution occurs by inversion through a trigonal-bipyramidal geometry, and that reaction is facilitated by the rapid response of light H atoms to changing force fields. Heavier substituents simply cannot relax to the new configuration fast enough, so that the reaction yield is close to zero despite the large exothermicity. In a series of papers,3ooresults are presented for a number of reactions between F atoms and polyatomic molecules studied using crossed molecular beams in which a long-lived intermediate complex appears to be formed. The results are compared with predictions of statistical (e.g. RRKM) theories for dissociation of the complex. An important point which emerges is that randomization of energy is not necessarily faster than decomposition of the complex. Ch1orhe.-Clyne and Walker301 have measured Arrhenius parameters for reactions of chlorine atoms with CH4, CH,Cl, CH,Cl,, and CHCl, in a fast flow system with mass spectrometric detection. In addition, a rate constant for the reaction C1 + CBrCl, + CC1, ClBr (75)

+

of k7&= (2.3 i-0.4) x 10-l3cm3molecule-ls-l at 652K was obtained (estimated room temperature value k ca. 2 x cm3 molecule-l s-l), and kinetic isotope (H-D) rate constant ratios for reactions with CH,-CD, and CHC13-CDC13 were measured over a wide temperature range. The magnitudes of the effects are in good agreement with predictions of BEBO calculations, although Arrhenius plots for the ratios showed no statistically significant curvature and could be expressed as kCH,/kCD, = 0.70 f 0.04 exp(3.2 f 0.1 kJ mol-l/RT)

kCHCI,IkCDCI, = 0.97 & 0.05 exp(l.4 f 0.1 kJ mol-l/RT) For the reaction with methane Cl

+ CH4

+

CH,

+ HCl

Clyne and Walker obtained k , , = (5.1 f 0.5) x 10-llexp(6.47 f 0.13 kJ at 300K), in very satisfactory r n ~ l - ~ / R T ) cmolecule-l~-~(1.3 m~ f 0.1 x agreement with the single temperature measurement O2 using the resonance cm3 molecule-l s-l at 298 K, fluorescencetechnique, k , , = 1.5 f 0.1 x 300

301

302

M. Parson, K. Shobatake, Y . T. Lee, and S. A. Rice, J. Chem. Phys., 1973, 59, 1402; (h) K. Shobatake, J. M. Parson, Y. T. Lee, and S. A. Rice, ibid., p. 1416; ( c ) K. Shobatake, J . M. Parson, Y .T. Lee, and S. A. Rice, ibid., p. 1427; ( d )K. Shobatake, Y . T. Lee, and S. A. Rice, ibid., p. 1435. M. A. A. Clyne and R. F. Walker, J . C . S. Faraday I , 1973, 69, 1547. D. D. Davis, W. Braun, and A. M. Bass, Internat. J. Chem. Kinetics, 1970, 2, 101. (a) J.

Reactions of Atoms in Ground and Electronically Excited States

77

and a recent also in a flow ystem with mass spectrometric detection, givingk,, = (1.84 f 0.14) x 10-11exp(ll.7 f 0.8 kJmol-l/RT)cm3 molecule-l s-'(1.5 x at 298 K). These results are important in view of the dearth of absolute rate measurements for chlorine atom abstraction reactions. Reaction (76) has been widely used as a secondary standard, the primary standard being the process C1

+ H,

--f

HCl

+H

(77)

Braun et al. obtained a rate constant for reaction (77) in good agreement with previous values, but combining these direct measurements for the reactions of chlorine atoms with CH, and H, gives a rate constant ratio which is a factor of two higher than previous relative measurements. Further direct substantiation of the value of k,, would be useful. Bar Yaakov et aZ.,', have measured the intramolecular kinetic isotope effectfor reaction of C1 with HD in the range 297-443 K, obtaining

NOz was used to scavenge the hydrogen atoms, preventing chain propagation. Bromine.-Rate constants for the electronic to vibrational energy transfer processes (X = C1 or Br)

+ = 0) -+ Br(42P.&+ HX (v = 1) Br(4,P3) + HX (v = 0) Br(4,P2) + HX (v = 0) Br(4,P3) + Br, Br(4,P*) + Br, Br(42P+) HX (v

--f

(78a) (78b)

-+

(79)

atoms were have been measured by Leone and W ~ d a r c z y k . Br(4,P+) ~~~ produced by pulsed-laser irradiation of Br, and time-resolved infrared emission from Br(4,P,) and HX(v = 1) was detected. At least 50% of the quenching by HC1 and HBr was by electronic to vibrational energy transfer. The rate of quenching by bromine is appreciably slower than that reported by Donovan and H u ~ a i n it; ~is~possible ~ that in the earlier work atom-atom reactions and/or quenching by impurities were important. Direct evidence has been obtained 308-for a large effect of reagent vibrational excitation on the rate of the reaction Br

+ HCl (v'

=

1-4)

--f

C1

+ HBr

(80)

by recording an i.r. chemiluminescence depZetion spectrum. The levels d = 3 and, especially v' = 4, which lie above the endothermicity of the reaction were substantially depleted, showing that reaction is rapid at thermal collision energies for molecules excited to these vibrational levels; it was also 303 ao4

305 306

G. Poulet, G. Le Bras, and J. Combourieu, J. Chim. Phys., 1974, 71, 101. S. R. Leone and F. J. Wodarczyk, J. Chem. Phys., 1974, 60, 314. R. J. Donovan and D. Husain, Trans. Faraday SOC.,1966, 62, 2643. D. J. Douglas, J. C. Polanyi, and J. J. Sloan, J. Chem. Phys., 1973, 59, 6679.

78

Reaction Kinetics

Table 17 Recommended rate constants for the collisional deactivation of electronically excited iodine atoms I(S2P3)at 300 K (powers of 10 in parentheses) Quenching species He AlXe I(5’P+) N,

co

co2

SF, CF, N2O CF,I CH31 CzHJ n-C,H i-C ,H n-C4H,I t-C,HJ HI DI HCI D2

H2

MD

k/cm3 Ref. molecule-l s-’ a <5(-18) 308 <2(-18) 309, 3lOa < 1.6 (-18) 3106 < 1.6 (-14) 310~ 6.5(-17) 311 1.2 (-15) 3106 1.3 (-16) 3106 1.7 (-16) 3096 2.4 (- 17) 3096 4.6 (-16) 3096 1.3(-15) 312 < 3.5 (- 16) 3096, 313a 2.8 (-13) 3136 1.9(-13) 314 3.2(-13) 315 2.0(--13) 314 2.0 (-- 13) 314 2.9 (-13) 314 3.8(--13) 314 1.3 (-13) 308 1.5(-13) 316 1.2(-13) 308 1.4(-14) 311 1.0(-15) 311 1.3(-13) 311 3.2(-13) 317

Quenching species

CH, CD, CF3H C2H6 C3H 8 n-C4H10 (CH3)3CH CH,= CH-CHBCI CH,= CH-CH2Br CH2=CH-CH21 CH, =CH, CF2=CFH CF2= CF2 CH,-CH=CH2 But-l-ene trans-Bu t-2-ene cis-But-2-ene Isobutene Tetramet hylethy lene C2H2

D2O H2O ICN 0 2

NO

Error range is typically &loyoof given values. factor of two, see ref. 311.

307

308 309

31n

311 312

318

314

316

316 31i

316 319

kl~rr.~ molecule-l s-’ 1.1 (-13) G2.2 (-15) 4.6 (- 14) 1.0 (-13) 1.7 (-13) 1.9 (-13) 2.7 (-13) 4.2 (-13) 1.3 (-13) 2.4 (-13) 4.2 (-13) 2.1 (--13) 5.3 (- 14)’ 3.7 (-15)’ 3.4(-13) 3.6 (-13) 2.1 (-13)’ 2.2 (-13)b 1.1(-12) 6.3 (-13)’ 3.1 (-14) 6.2 (- 14) 7.2 (-13) 6.0 (- 14) 2.6 (-11) 1.6 (-12)

Ref. 311 20 3106 31 1 31 1 3096 31 1 31 1 312 312 312 311 318 318 31 1 311 318 318 311 318 31 1 3106 311 312 319 311

These values are probably low by a

P. Bemand and M. A. A. Clyne, J . C . S. Faraday IZ, 1972, 68, 1758. R. J. Donovan and D. Husain, Trans. Faraday Soc., 1966, 62, 1050. (aj D. Husain and J. R. Wiesenfeld, Nature, 1967, 213, 1227; (b) D. Husain and J. R. Wiesenfeld, Trans. Faraday SOC.,1967, 63, 1349. ( a ) R. J. Donovan and D. Husain, Trans. Faraday Soc., 1966,62, I 1 ; (b) R. J. Donovan and D. Husain, ihid., p. 2023; (c) D. Husain and R. J. Donovan, Adv. Photochem., 1971, 8, 1. J. J. Deakin and D. Husain, J. C. S . Faraday ZI, 1972, 68, 41. F. G . M. Hathorn and D. Husain, Trans. Faradqv Soc., 1969, 65, 2678. ( a ) R. J. Donovan and R. H. Strain, to be published; (6) M. C. Stock, D. J. Little, and R. J. Donovan, J . Chem. Ecluc., 1974, 51, 51. R. J. Donovan, F. G . M. Hathorn, and D. Husain, Trans. Faraday SOC.,1968, 64, 3 192. R. J . Donovan and D. Husain, Nature, 1966, 209, 609. P. Cadman, J. C. Polanyi and I. W. M. Smith, J . Chim. p h y s . , 1967, 64, 1 1 1 . R. J. Butcher, R. J. Donovan, and R. H. Strain, in the press. R. J . Donovan, D. Husain, and C . D. Stevenson, Trans. Faraday SOC.,1969, 65,2941. J . J. Deakin, D. Husain, and J. R. Wiesenfeld, Chem. Phyr. Letters, 1971, 10, 146.

Reactions of Atoms in Ground and Electronically Excited States

79

shown that, within these levels, J’ = 1 and 2 were the most reactive rotational levels. Rates of quenching of Br(54P3)atoms by several gases have been obtained 3 0 7 using fluorescence techniques; the order of collisional efficiencies reported is He, Ar < H, < N2 < CO < SF6 < Br,. At the pressures employed (<3 k N r r 2 ) , quenching of Br(54P3) was not detected since this level has a shorter radiative lifetime than the moderately metastable Br(S4P+,+)states. More quenching channels must be energetically accessible for these highly excited atoms than for metastable Br(4T3) and, although overall rates of quenching are similar with some collision partners, quenching by N2and CO is considerably faster for Br(54P3)than for Br(42P+). 10dine.-1(5~P+) atoms have been monitored using time-resolved resonance fluorescence;20rate constants for quenching by C3Hsand CD, were obtained. cm3molecule-l s-l is in good The value for C3Hs, k = (1.6 f 0.4) x agreement with previous data (Table 17), while the rate constant for quenching by CD4, ic = (2.2 f 0.4)x cm3molecule-l s-l, is 50 times less than the value for CH, and provides another example of the very large isotope effects which may be observed in the quenching of electronicallyexcited atoms and small molecules. Although data for quenching of I(52P4.) by a wide range of molecules have been presented in previous there appears to be a need for a more critical compilation, as data obtained using different techniques do not always agree. In most cases the reasons for this lack of agreement almost certainly lie in the involvement of radical-radical effects with some techniques, and thus the data obtained using time-resolved atomic absorption spectrophotometry or resonance fluorescence are to be preferred (the atomic and radical concentrations are lowest using these techniques). The recommended rate data are given in Table 17. The Arrhenius parameters for quenching by a number of these gases have also been reported; while most hydrides show a small negative or zero ‘activation energy’, molecular hydrogen shows a relatively large positive activation energy. This has been discussed in terms of the changes in rotational state population with temperature, and it has been proposed 2 5 8 that resonant energy transfer channels dominate the overall removal processes [see above discussion of Te(53P1,o ) relaxation]. A quantum yield of 0.31 f 0.04 has been determined for formation of methyl iodide in the photolysis of I,-CH4 mixtures at 147.0nm.320The authors argue that, since quenching of I(52P-))is very much faster than its rate of abstraction from methane, methyl iodide is probably formed via reactions of I(64P3) atoms produced in the energetically feasible process 12(X1X:)

+ hv(147.0 nm)

--f

I(52P-)) + I(64P3)

The very large cross-section, 1.2 f 0.45 nm2, observed for quenching of s20

P. John, G. J. Kennedy, and B. G . Gowenlock, J.C.S. Chem. Comm., 1973, 683.

80

Reaction Kinetics

12(B31Tou+, v = 25) by I(2P,) has been attributed to the formation of long-

lived I3 collision complexes, giving a high probability for p r e d i s s ~ c i a t i o n . ~ ~ ~ Quenching is the only reaction channel observed, no vibrational relaxation of I t occurring on collision with I. Halogen Atom Recombination.-This topic continues to attract much theoretical at tention, but space precludes a lengthy discussion here. Chlorine atom recombination in the presence of a variety of gases has been studied 323 using conventional flash photolysis and monitoring C1, absorption for the first time. Cross-combination rate constants for I C1 in the presence of N, and COz, were found to be appreciably higher 324 than those for selfrecombination of iodine or chlorine atoms with the same third bodies; it is argued that this is consistent with the major contribution in these reactions being a radical-complex mechanism. Recent experimental results on the recombination of ground-state bromine atoms have been 326 There have been several further trajectory 328 In the most recent of these it studies on bromine atom is proposed that a more realistic division of contributions to the overall rate than the traditional energy transfer and radical-molecule complex mechanisms, is into triple-collision and bound-complex components. Wong and Burns 329 have also performed trajectory calculations for iodine atom recombination in the presence of inert gases. Of particular interest is the work on the recombination of iodine atoms in highly compressed gases and in liquids using conventional and laser flash photolysis;330very wide pressure ranges were studied, and in all cases changes in order of reaction with increasing concentration of the gas (M = He, Ne, Ar, Kr, Xe, H,, N,, O,, COZY CHI, C2H6, C3H8)were observed. There was no discontinuity between high-pressure gas- and liquid-phase data. The results are discussed in terms of contributions from radical-complex and energytransfer mechanisms, and estimated equilibrium constants for complex formation I T M+TM (82) and rate constants for I + I M -+I2 f M (83) 3229

+

are presented (see Table 18). Results for M a21

322 523 324 325

326 327

328 s29

330

=

Ar are shown in Figure 12.

A. N. Schweid and J. I. Steinfeld, J . Chem. Phys., 1973, 58, 844. R. P. Widman and B. A. DeGraff, J . Phys. Chem., 1973, 77, 1325. H. Hippler and J. Troe, Chem. Phys. Letters, 1973, 19, 607. H. N. Maier and F. W. Lampe, J . Phys. Chem., 1973, 77, 430. M. A. A. Clyne and A. R. Woon-Fat, J . C . S. Faradav / I , 1973, 59, 412. H. W.Chang and G. Burns, Canad. J. Chem., 1973,51, 3394. (a) A. G . Clarke and G . Burns, J . Chem. Phys., 1971, 55, 4177; (6) A. G. Clarke and G. Burns, ibid., 1972, 56, 4636; ( c ) A . G. Clarke and G. Burns, ibid., 1973, 58, 1908. W. H. Wong and G . Burns, J . Chem. Phys., 1973, 59, 2974. W. H. Wong and G. Burns, J . Chem. Phys., 1973, 58, 4459. H. Hippler, K. Luther, and J. Troe, Chem. Phys. Letters, 1972, 16, 174.

Reactions of Atoms in Ground and Electronically Excited States

81

Table 18 Equilibrium constants Keq = [IM]/[I][M] and rate constants 331 for reactions (83) at 3 1 4 K 1OZ3K,/cm3

M He Ne

molecule< 3.3 <5.8 22 f 5 23 f 12 33 f 12 25 f 10

Ar Kr Xe Ha

10l1k/cm3 molecule- s >3.3 >2.5 2.5 f 0.8 2.3 f 0.8 2.3 f 0.8 3.5 f 1.7 1

M Nz 0, CO, CHo CZH, C3H8

loz3K&m3 molecule40 f 12 28 f 8 58 f 8 53 12 51 f 17 <130

P /atm 10

*

100

lo1' k/cm3 molecule- s2.7 f 0.8 3.8 f 1.3 5.1 f 0.7 3.3 f 1.0 5.8 f 1.7 >3.3 1GOO

-

2 in'* v)

n

E, \

k

10 "

10 lo 10-5

10-4 IO-~ [Arl /ma1 ~ m - ~

10- I

Figure 12 Second-order rate constants for iodine atom recombination in Argon vs. pressure of Argon at 314 K ; a Laser flash photolysis; 0 Conventional flash photolysis ; A,A, v , other work (Reproduced by permission from Ber. Bunsengesellschaft Phys. Chem., 1973,77,1107)

It has been suggested that, in this case, the change from third order to an intermediate order, and then back to third order at about 1 atm pressure, is due to the energy transfer contribution reaching its limiting value while the radical complex component is still increasing with pressure. At even higher pressures, the radical complex rate constant reaches its limiting value so that a change to second order is observed. The situation is complicated by the possibility of collisions involvingmore than three bodies at very high pressures. H. Hippler, K. Luther, and J. Troe, Ber. Bunsengesellschaft phys. Chem., 1973, 77, 1104.

Reaction Kinetics

82

11 Noble Gas Atoms The chemistry of electronically excited metastable noble gas atoms is in general complex, with many reaction channels being accessible. Three broad categories may be distinguished;

A*

+ B,

+

+ B,* (energy transfer) +B (reaction) A + B2+ + e- (ionization) A

-+ AB +

The ionization channel (termed Penning ionization) has long been recognized and has been exploited, by mass spectroscopists, as a means of pumping metal vapour lasers, and in one type of detector used for gas chromatography. as have more detailed An introduction to the field has been ~ c c o u ~ ~ The s . review ~ ~ ~ by - ~Setser ~ ~ and Stedman 334 is concerned mainly with reactive processes and in particular with dissociative excitation, e.g.

A* -t B2 - + A + B"

i- B

Rundel and S t e b b i n g ~ ,and ~ ~ ~Muschlitz 335 concern themselves with the wider physical aspects including measurement of lifetimes of metastable states and cross-sections for individual channels. Two other 337 of particular relevance to gas kinetics, deal with chemi-ionization, but place emphasis on reactions involving metastable noble gas atoms, one 338 providing a useful summary of the theory. The simplest metastable noble gas atom reaction, involving He(z3S) and a ground-state hydrogen atom, has been discussed in detail by Miller et al.3387339 The total cross-sections for Penning ionization He(l~,2s,~S,) 4 H(ls2S,)

-+

He(ls2 ' S o )

+

H+ 4-e-

(84)

and for associative ionization He(ls,2s3S,) + H(ls2S+)+ HeH+

+ e-

(85)

have been calculated and the variations with collision energy derived. The cross-sections for both channels decrease with increasing energy, but that for associative ionization decreases most rapidly 339 above 0.05 eV. This is reasonable in physical terms as the HeH+ initially formed is less likely to survive in a bound state at high collision energies. The agreement between 332

333

194

335

33G

33i

33x

33y

E. E. Muschlitz, Science, 1968, 159, 599. R. D. Rundel and R. F. Stebbings, in 'Case Studies in Atomic Collision Physics', ed. E. W. McDaniel and M. R. C. McDowell, North Holland, Amsterdam 1972, vol. 2. D. W. Setser and D. H. Stedman, Progr. Reaction Kinetics, 1971, 6, 193. E. E. Muschlitz, A h . Chern. Phys., 1966, 10, 171. R. S. Berry, in 'Molecular Beams and Reaction Kinetics', ed. C. Ychlier, Academic Press, 1970, p. 193. A . Fontijn, Progr. Reaction Kinetics, 1971, 6, 75. W. W . Miller, J . Chern. Phyy., 1970, 52, 3563. W . H . Miller, C. A. Slocomb, and iH. F. Schaeffer, J . Chem. Phvs., 1972, 56, 1347.

Reactions of Atoms in Ground and Electronically Excited States

83

the calculated cross-sections and earlier experiments is good. More recently, Hotop et ul.340have presented experimental data for Penning ionization of H atoms by and He(21S0) The ratio of the cross-sections is in satisfactory agreement with the theory, as are most other aspects of the data. The radiative lifetime of the He(z3S,) state has been determined 3p1 as 4 x 103 s. Penning ionization of noble gas and mercury atoms has been extensively investigated by Hotop and N i e h a ~ s .The ~ ~ angular ~ distribution of electrons emitted by Ar on collision with a thermal beam of He(21S,23S) metastables was found to be asymmetric, and it was suggested that electron exchange takes place. Thus it was proposed that the electron from the outer shell of the target particle (in this case Ar), ‘tunnels’ to the hole in the He metastable (1s)-core with simultaneous ejection of the 2s electron. The observed enhancement of electron scattering in the He atom direction was then attributed 342 to the distortion of the outgoing electron wave by the Ar+ which essentially shields part of the solid angle. Cross-sections for Penning ionization of Xe and Ar by helium and neon metastables appear to rise to a maximum and then fall for kinetic energies in the range 1 0 - 1 0 0 eV.343 Investigations at high kinetic energy 344 (0.2-10 keV) have shown that metastable negative ion formation occurs for a number of processes of the type, He

+ He(K.E.)

--f

He+(ls) f He-*(ls2s2)

(86)

together with autoionization of the transient diatomic molecular system formed during the collision.344 The formation and decay of bound metastable noble gas molecules is of particular current interest in view of their possible involvement in the highpressure xenon laser (A = 173 nm).345-347Gerardo and Johnson 3*7 have shown that the cross-section for stimulated transitions from the lowest bound states of Xe,* to the repulsive ground state is larger than the photoionization cross-section, and that net gain can thus be achieved at ca. 173 nm. A detailed investigation of processes leading to emission from the 60 nm bands of He,* has been A detailed comparison of the Penning-electron and photoelectron spectra of H,, N2, and CO, has shown that, while the relative populations of the different electronic states of the molecular ions differ substantially for the two processes, the populations of the vibrational levels within a given 340

341 342

3p8

s44 346 346

347 348

H. Hotop, E. Illenberger, H. Morgner, and A. Niehaus, Chem. Phys. Letters, 1971, 10, 493. H. W. Moos and J. R. Woodworth, Phys. Rev. Letters, 1973, 30, 775. H. Hotop and A. Niehaus, Chem. Phys. Letters, 1971, 8, 497 (and refs. therein). M. L. Coleman, R. Hammond, and J. W. Dubrin, Chem. Phys. Letters, 1973, 19, 271. G. Gerber, R. Morgenstern and A. Niehaus, J. Phys. ( B ) , 1973, 6 , 4 9 3 . A. Gedanken, J. Jortner, B. Raz, and A. Szoke, J. Chem. Phys., 1972, 57, 3456. A. W. Johnson and J. B. Gerardo, J. Chem. Phys., 1973, 59, 1738. J. B. Gerardo and A. W. Johnson, I.E.E.E. J. Quant Electron., 1973, QE9, 748. W. B. Peatman and D. T. Wu, Chem. Phys., 1973, 2, 335.

Reaction Kinetics

84

electronic state are broadly similar.349The differences in electronic distribution are expected, as the matrix elements for transitions involving photoionization will be vanishingly small unless the symmetry species of the initial and final states are correct for an electric dipole transition (consider, for example, the spin restriction for light molecules; if the initial state of the molecule is a singlet, then only doublet states of the ion will be formed, states of other spin multiplicity having a very much lower probability). On the other hand, the electronic symmetry requirements for Penning ionization are much less restrictive owing to the presence of the excited noble gas atom, and a number of ion states of different multiplicity can be populated. The similar distribution of vibrational levels within a given electronic state observed both from Penning ionization and from photo-ionization is more surprising. Hotop and Niehaus 349 argue that, if the interaction leading to Penning ionization occurs at long range, then the presence of the excited atom will have little perturbing effect on the force field of the molecule as it ionizes, and a vertical (Franck-Condon) transition from the ground state will take place. While this may be true for a few special Penning reactions it is now abundantly clear that ‘strong interactions’ are involved in a large number of other reaction^.^^^-^^^ Thus broad distributions of vibrational and rotational states of products have been observed and, in some cases, large amounts of energy are partitioned into relative kinetic energy of the products. Another interesting finding is that dissociative excitation competes favourable with Penning ionization in some cases.334 It might be argued that Frarrck-Condon type Penning ionization is important for large impact parameter collisions, but the reactions with the largest known cross-sections [He(21So)+ HCl and HBr] give rise to marked deviations from Franck-Condon behaviour 353 These deviations persist 364 even for kinetic energies in the range 10-100 eV. In addition to the usual Penning ionization of O2by He(21S0)and He(z3S1),

+

He(21So,23Sl) O2 -+He

+ 0; + e-

(87)

+ O+ + 0-

(88)

ion-pair production

+

He(21So,23Ss,) O2

--f

He

but has a cross-section which is ahproximately one has also been order of magnitude less than for the Penning channel. 34s 550

351

352 353

354

355

H. Hotop and A. Niehaus, Znternat. J . Mass Spectrometry Ion Phys., 1970, 5 , 415. L. G. Piper, W. C. Richardson, G. W. Taylor, and D. W. Setser, Faraday Discuss. Chem. SOC.,1972, 53, 100 (and references therein). J. A. Coxon, D. W. Setser, and W. M. Duewer, J. Chem. Phys., 1973, 58, 2244. D. H. Stedrnan and D. W. Setser, J . Chem. Phys., 1970,52, 3957. Y . A. Bush, M. McFarland, D. L. Albritton, and A. L. S. Schmelrekopf, J . Chem. Pltys., 1973, 58, 4020. M. J. Haugh, J. Chem. Phys., 1973, 59, 37. Y . A. Bush, D. L. Albritton, F. C. Fehsenfeld, and A. L. Schmeltekopf, J . Chem. Phys., 1972, 57, 4501.

Reactions of Atoms in Ground and Electronically Excited States

85

The relative magnitudes of the cross-sections for singlet and triplet metastable states have been determined using both beam and flowing afterglow techniques, but the results do not agree.356 In addition to the optically metastable states with low principal quantum number, higher lying Rydberg states are also expected to be metastable. It is well known from the quantum-mechanical treatment of the hydrogen atom that the lifetime of Rydberg states increases as n4n5, so that lifetimes s are expected for n = 10. Hotop and Niehaus 3s7 have observed of ca. Penning ionization following quenching of such states and have shown that the cross-sections are immense (ca. 1 0 0 nm2).

12 Heavy Metal Atoms Mercury.-Mercury (63P1) was one of the first excited atoms to be studied and the use of 'mercury photosensitization' to produce other excited species has a long history. Understanding of the sometimes complex mechanisms involved was rendered difficuk, however, by the presence of three close-lying levels Hg(63P2,1, o ) and it is only recently that reliable data on the individual rates and mechanisms of reaction of the 63P1and 63P0levels (and to a lesser extent 63P2)have been obtained. This renewed interest has involved the application of such techniques as monochromatic flash photolysis, modulated excitation, and crossed-beam studies. It is now clear that, in many reactions of Hg(63PJ)atoms, complexes are formed (detectable in emission or absorptldn in some instances), and that competition between such processes as quenching to the ground state via chemical reaction, and spin-orbit relaxation in the upper levels, must be considered and will be governed in particular cases by the available potential energy surfaces and by energetic considerations. Several examples have been discussed in a recent paper.368 The formation of HgH following reaction of the 63P0and 63P1levels with H,, alkanes, and alkenes has now been firmly established. Results on formation 359 of HgH and HgD in reactions of Hg(63P,, o ) with H2, D,, and HD suggest a complex, formed by sideways approach of Hg to the hydrogen molecule, which fragments too quickly for equipartitioning of energy. The experimentally determined quantum yields were significantly different from those calculated using statistical theories : thus for reaction with HD, product ratios were HgD:HgH = 0.70 & 0.09:0.13 i 0.02 for Hg(63P1) atoms, and HgD:HgH = 0.82 i 0.08:0.175 & 0.02 for Hg(63Po) atoms. The authors point out that two-thirds of the total kinetic energy released will be imparted to the H atom and will thus strongly favour HgD formation, as observed. This effect could be of general significance in exothermic reactions involving excited atoms and HD. Quenching of Hg(6'Pl) is a factor of 9 s37 358

a59

J. P. Riola, J. S. Howard, R. D. Rundel, and R. F. Stebbings, J. Phys. (B), 1974,7,376. H. Hotop and A. Niehaus, J. Chem.Phys., 1967, 47,2506. A. B. Callear and J. C. McGurk, J. C. S. Faraday ZI, 1973, 69,97. A. B. Callear and J. C. McGurk, J. C.S. Faraday ZZ, 1972, 68, 289.

86

Reaction Kinetics

faster than quenching of Hg(6,Po) by H2; possible mechanisms have been discussed. The competing process Hg(6,P1)

+ H,

-+ Hg(61So)

+H +H

(89)

has been shown360to occur at about half the rate of HgH formation for Hg(63P1). Hong and Mains 361 have recently employed measurements of Lyman-a absorption by the H atoms produced in the quenching of Hg(63P1) by H2 and HD, obtaining quenching cross-sections of 0.100 f 0.004 and 0.109 f 0.012 nm2, respectively. The spin-orbit relaxation of Hg(63P1)to Hg(tj3Po)by various gases, relative to total quenching, has been i n v e ~ t i g a t e d362 . ~ ~Agreement ~~ between the two sets of results is generally very satisfactory. The reverse process, collisional excitation of Hg(fj3Po) to Hg(63P1), has been studied in collisions with N2.363The ratio of measured crossLsections for this process and its reverse was in good agreement with the value obtained from the principle of detailed balancing. The mechanism of reaction of Hg(63P1,o ) atoms with the alkanes is complex, but all reaction channels probably proceed via Hg*-H-R complexes with lifetimes of at least a few vibrational periods. Emission from these exciplexes has been observed in some cases.364 The observations of emission by excited HgH * in Hg(63Po)-photosensitizeddecomposition of H2 and some of the lower alkanes has been ascribed to secondary processes:365

+ HgH(X2C+)-+

Hg(63P0)

Hg(6lSo) + (HgH(A2u+,A2n+,B2C+) (90)

following formation of ground-state HgH in the primary process. The long-lived complexes (t ca. 1.8 ,us) of Hg(@Po)with NH3 and ND3have been well-~haracterized;~~~~ broad emission from the bound exciplex to the repulsive ground state is observed. Rates of formation, lifetimes, and dissociation energies of similar complexes formed in essentially chargetransfer interactions with water, alcohols, and amines were also and the spectra of clusters of NH, attached to Hg(6, Po)were also characterized (see Figure 13).368 In the quenching of Hg(6,Pl) by CH2F2,vibrationally excited HF(v” < 4) 360 381

362 363 364

365 366

367

A. B. Callear and P. M. Wood, J. C . S. Faraday II, 1972, 68, 302. J.-H. Hong and G . J. Mains, J. Photochem., 1973, 1, 463. A. C. Vikis, G. Torrie, and D. J. LeRoy, Canad. J. Chem., 1972, 50, 176. J. Pitre, K. Hammond, and L. Krause, Phys. Rev. (A), 1972, 6, 2101. S. Penzes, 0. P. Strausz, and H. E. Gunning, J. Chem. Phys., 1966, 45, 2322. A. C. Vikis and D. J. LeRoy, Canad. J. Chem., 1973, 51, 1207. ( a ) J. Koskikallio, A. B. Callear, and J. H. Connor, Chem. Phys. Letters, 1971, 8, 467; (b) A. B. Callear and J. H. Connor, ibid., 1972, 13, 245. (a) C. G. Freeman, M. J. McEwan, R. F. C. Claridge, and L. F. Philips, Chem. Phys. Letters, 1970, 5 , 5 5 5 ; (b) R. H. Newman, C. G . Freeman, M. J. McEwan, R. F. C. Claridge, and L. F. Phillips, Trans. Faraday SOC.,1970, 66, 2827; (c) C. G. Freeman, M. J. McEwan, R. F. C. Claridge, and L. F. Phillips, Trans. Furaduy SOC., 1971, 67, 67, 2004,2567, 3247.

368

P. N. Clough, J. C. Polanyi, and R. T. Taguchi, Cunud. J. Chem., 1970,48, 2919.

Reactions of A t o m s in Ground and Electronically Excited States

87

L

aJ

Q,

2-

160

.-g- u 40 QI Energy/crn-l x l o 3 Figure 13 Emission spectra of Hg(NH3),* complexes. (a) n = 1, unstabilized (b) n = 1, stabilized (c) n = 2 (d) n = 3 (e) n = 4 (f) n > 4 (Reproduced by permission from Chem. Phys. Letters, 1972, 13, 246)

The reaction is thought is formed with considerable rotational to proceed via vibrationally excited triplet CH2CFg

+

+

Hg(63P1) CH,F, Hg(61So) CH2CFX which then eliminates HF”. Cross-sections for formation of CO(w”) in the energy-transfer process

+

--f

+

Hg(63Po) CO(w” = 0) + Hg(61So) CO(w” < 9) (92) have been obtained using modulated excitation of the Hg atoms.369Intramultiplet quenching of Hg(63P1)to Hg(63Po)by a large excess of N2ensured that collisions of Hg(63P1) need not be considered. These results supplant earlier data 3 7 0 which were complicated by vibrational relaxation of CO. Although the impulsive ‘half-collision’ model used by Levine and Bernstein 371 (derived to fit the earlier experimental results) shows reasonable agreement with the more recently-obtained vibrational population distribution, the model of Simons and Tasker 372 is undoubtedly more realistic, and of wider applicability to situations where a system is prepared suddenly on a repulsive potential (e.g. photodissociation as well as the type of quenching presently under consideration.) In the general case it was assumed that, at the beginning of the recoil which transfers vibrational energy into the molecular fragment, this fragment is populated in a number of vibrational levels (governed by the appropriate Franck-Condon factors) by either absorption of a photon or radiationless transition. Change in equilibrium bond length of the molecular fragment is crucial in determining the shape of the final vibrational distribution. In the present example, the system is formed on a repulsive surface by intersystem crossing in the intermediate 8B9 370

371

Y.Fushiki and S. Tsuchiya, Chem. Phys. Letters, 1973, 22, 47. G. Karl, P. Kruus, and J. C . Polanyi, J . Chem. Phys., 1967, 46, 224. R. D. Levine and R. B. Bernstein, Chem. Phys. Letters, 1972, 15, 1. J. P. Simons and P. W. Tasker, Mol. Phys., 1973, 26, 1267.

88

Reaction Kinetics

Hg-CO complex (exciplex); there is direct evidence for the formation of such complexes in collisions of Hg(63Po)atoms with other species, as noted earlier. As it happens, however, the data of Fushiki and Tsuchiya are reproduced by the model with Ar, the change in CO bond length in the intermediate complex, equal to zero, which corresponds to the limiting situation considered by Levine and Bernstein. The overall yield of vibrational energy is comparatively small (10-15 %), consistent with CO bond distance in the complex similar to that in the CO molecule. The energy transfer process Ifg(63P~)t CN(X2C+)--+ Hg(6'So)

+ CN(B'C+)

(93)

has been Vibrational analysis of the CN(B2C++- X 2 C + )emission showed that the excited state was populated in a non-Franck-Condon manner. Again, it seems likely that a complex is formed; Vikis and LeRoy postulated a strong interaction such as Hg+CN-(2C+). Electronic energy transfer from Hg(3P1) to Na atoms Hg(63P,)

+ Na(32S+)

+=

Hg(61So)

+ Na* + A E

(94)

has been monitored using the subsequent fluorescent emission from Na atoms in S, P,and D Absolute cross-sections for the formation of 21 states were obtained, ranging from 0.0002 to 0.385 nm2 and showing a strong dependence on energy mismatch (Figure 14). These results do not support the intermediacy of Hg-Na quasi-molecules postulated by K r a ~ l i n y a . Krause ~ ~ ~ and co-workers 3 7 6 have reported results on the depolarization of mercury resonance radiation in collisions of Hg(63P1)atoms with the inert gases and nitrogen. Detailed work on the higher metastable component of the 63P state, Hg(63P2),has been restricted to molecular beam studies. Cross-sections for spin-orbit relaxation to Hg(63P,) as a function of initial collision energy with Ha, D,, N2,NO, and CH, have been obtained.377 In accord with the observed fairly weak dependence on energy, relative cross-sections derived from this velocity-selected work are in good agreement with the relative measurements of Van Itallie et ale3 In general, long-range resonant energy transfer does not appear to be important in these processes, and cannot explain the observed energy dependence of the quenching cross-sections. The model originally proposed by Bykhovskii and N i l ~ i t i n where , ~ ~ ~ relaxation occurs via potential energy surface crossings in the species Hg-M, is 373 374 375

376 377

378 s79

A. C. Vikis and D. J. LeRoy, Chem. Phys. Letters, 1973, 22, 587. M. Czajkowski, G. Skardis, and L. Krause, Canad. J. Phys., 1973, 51, 334. E. K. Kraulinya, 'Sensitised Fluorescence of Metal Vapour Mixtures 11', Latvian State University, Riga, 1969. Referred to in ref. 374. R. A. Phaneuf, J. Pitre, K. Hammond, and L. Krause, Canad. J. Phys., 1973,51, 724. H. F. Krause, S. Datz, and S. G. Johnson, J. Chem. Phys., 1973, 58, 367. F. J. Van Itallie, L. J. Doemeny, and R. M. Martin, J. Chem. Phys., 1972, 56, 3689. V. K. Bykhovskii and E. E. Nikitin, Optics and Spectroscopy, 1964,16, 111.

Reactions of Atoms in Ground and Electsonically Excited States

89

1 .o

10"

-4

E

J

lo-;

\

I 10-4

-1.2

Hg (63P, 1

- 0.8

-0.4

0

AZIeV

Figure 14 Cross-sectionsfor electronic energy transfer from Hg(6sP1)to S , P, and D states of sodium, plotted against the energy which must be converted into translation ( A E ) (after ref. 374)

in better agreement with the experimental data over much of the energy range investigated, although some features such as weak structure observed in the H2and Dzcross-sections remain unexplained. A simple, universal model is not really to be expected, however; specific interactions for each case must be considered, and a description of the potential energy surfaces involved for all channels available to a particular quencher, with all three spin-orbit levels should be sought. These molecular beam experiments do not provide information on competing channels for the fi3PZ level such as chemical reaction. Differential cross-sections for elastic scattering of Hg(6'Pz) atoms from Na, K, and Rb atoms in crossed molecular beams at thermal energies have been obtained.380In the attractive region of the interatomic potential probed at these energies, quenching by W transitions or electronic energy 980

T. A. Davidson, M. A. D. Fluendy, and K. P. Lawley, ref. 4, p. 158.

90

Reaction Kinetics

transfer does not occur and only elastic scattering, accompanied by possible Am, transitions, is observed. The results indicate that the potentials are very similar for all the alkali atoms; they also lead to an estimate that the total quenching cross-section cannot exceed gas kinetic values. This is not, however, a very severe restriction and it would appear likely that Penning ionization provides a channel for removal of Hg(63P,) by all the alkali atoms. Two fluorescence bands associated with transitions from excited states of Hg, to the repulsive ground state in irradiated Hg-N2 mixtures have been The formation of the 31, state was attributed to the reaction,

while the "0,- state was formed in subsequent molecular collisions with N,. Hg(6lPi) photosensitization of CO and N, has been investigated and Although the reaction detailed mechanisms are Hg(6lP1) f CO

-+

Hg(6'So) f CO(a311)

(96)

appears to occur to some extent, the principal primary process with CO is almost certainly formation of Hg(63P1). For the three reactions with N,

-+

Hg(63PJ)

+ N,

(97b)

it was concluded that reaction (97a) accounts for 28% and reaction (97b) > 50% of the total quenching. Formation of Hg(63P1), observed in emission, from Hg(6lPI) in collisions with a variety of monatomic, diatomic, and polyatomic gases has been observed.3a3 With the noble gases at least, only a one-step mechanism involving intersystem crossing in a collision complex (exciplex) is possible, and this may be of quite general occurrence. Tin.-Kinetic data on the reactions of Sn atoms in specific electronic states have hitherto been lacking. Husain and Brown 384 have now obtained rate constants for quenching of Sn atoms in the 5lDZ state, 1.068 eV above the ground state, by attenuation of atomic resonance radiation. The results are listed in Table 19. Quenching of O(2lD2) by Xe is extremely efficient, and this transfer of a substantial amount of energy into translation is adequately described by a curve-crossing me~hanism.~ Quenching of Sn(SID,) and Pb(6lDJ (see below) by Xe is inefficient, however, despite considerably larger spin-orbit coupling. This must imply that the appropriate states of XeSn and XePb correlating with Xe + M(nlD,) do not cross states correlating with ground-state atoms at an accessible energy, and are repulsive or only weakly bound with minima 381 38z

sBs 3y4

R. A. Phaneuf, J. Skonieczny, and L. Krause, Phys. Rev. (A). 1973, 8, 2980. R. Sirnonaitis and J. Heicklen, J. Photochem, 1973, 1, 181. V. Madhavan, N. N. Kichtin, and M. Z. Hoffman, J . Phys. Chern., 1973,77, 875. D. Husain and A. Brown, J . C.S. Faraday ZZ, in the press.

Table 20 Rate constants k/cm3 molecule-l s- at 300 K for the collisional removal of excited Pb atoms by various molecules (M) (powers of 10 in parentheses)

;P

85. x

Pb(63P1)

Ref.

Pb(Ci3P2)

Ref:

1.1 (-16) 0 f l.O(-16)

386 386

~ 1 . (-17) 3 2.0 f 0.5 (-15)

386 386

386 386 386 386 386 386 386 386 386 386 386 386 386 386

1 f l(-12) c 1 (-12) 8.0 f 8.0 (-15) 4.0 f l.O(-11) 4.7 f 0.3 (-13) 4.3 f 0.9 (-11) 2.2 f 0.4 (-11) 4 f l(-13) 1.6 f 0.3 (-11) 2.3 f 0.3 (-11) 1.7 f 0.2 (-11) 3 f 2(-11) 1.6 f 0.5 (-13) 5.8 f 0.6(-11)

386 386 386 386 386 386 386 386 386 386 386 386 386 386

2.9 f 0.4 (- 15) < 6 (-16) 2.0 f 2.0 (-15) 7.0 f 5.0 (-12) 2.3 f 0.7 (-12) 9.1 f l.O(-12) < 1 (-14) 2.8 f l.O(-14) 0 f 2 (-15) 8.2 f 0.2 (-13) 3.5 f 0.3 (-12) 0 f 4 (-15) 2.3 -f 0.3 (-15) 1.4 f 0.1 (-11) 385

986

Pb(6l D2)

Ref.

Pb(61So)

Ref.

< 2 (-16)

385a

< I (-15) < 1 (-14)

385a 385a

< 2 (-15) < 1 (--14)

385c 385c

<1(-15) 1.1 f 0.3 (-10) 9.3 f 0.2 (-12) 9.2 f 0.8 (-11) < 1 (-14) 4.5 f 0.8 (-12) < l(-14) 2.6 f 0.4(- 11) 2.0 & 0.6(-11) < 1 (-14) < 1 (-15) 5.6 f 1.2 (-11)

3856 385a 385b 3856 3856 3856 3856 3856 385b 3856 3856 385a

1.6 f 1.6(-15) 1.2 & 0.3 (-10) 6.3 f 0.4 (-12) 2.1 f 0.2 (-10) 3.5 f 0.5 (-15) 4.2 f 0.3 (-12) <1(-14) 2.3 & 0.3 (-12) 2.3 f 0.3 (-13) <5(-16) 1.7 f 0.2 (-15) 3.7 f 0.4(-11)

385d 385c 385d 385d 385d 385d 385d 385d 385d 385d 385d 38%

2 s

(a) D. Husain and J. G. F. Littler, Chem. Phys. Letters, 1972, 16, 145; (b) D. Husain and J. G . F. Littler, J. C. S. Furaday ZZ, 1972, 68, 2110; (c) D. Husain and J. G . F. Littler, J. Photochem., 1973,1, 327; ( d ) D. Husain and J. G . F. Littler, J. C. S. Faraday ZI, 1973, 69, 842. D. Husain and J. G. F. Littler, J . Photochern., in the press.

S c1

$ %

5

$. 3 a 2.

&

2 5

\D c

92

Reaction Kinetics

at large internuclear separation. The low rate of quenching by Nzlikewise suggests that there is no significant chemical interaction with Sn or Pb in the lDZstate.

Table 19 Rate constants klcm3 molecule-l s-l at 300 K for collisional removal of Sn(SID,) atoms by various molecules384(powers of 10 in parentheses) M

He Xe H2. 0 2

Nz NO

co

k

k

M

<2(-15) <1.7(-15) 5.5 & 0.1 (-11) 9.0 & 0.9 (-11) 8.5 ;t 1.1 (-14) 3.7 f 0.3 (-11) 1.9 i 0.1 (-11)

co2

NZO CH4 C2H4 C2Hz

SnMe,

8.4 f 0.8 (-12) 5.0 f 0.8 (-12) 8.7 & 1.2(-11) 5.3 f 0.3 (-11) 1.1 f 0.1 (-10) 2.0 k 0.2 (-10)

Lead.-A considerable body of data on the quenching oft he optically metastable states Pb(61D2) and Pb(GISo), respectively 2.66 and 3.65 eV above the 63P0ground state, has already been The same workers38s have now obtained quenching rate constants for the upper spin-orbit levels of the 63PJ configuration, Pb(63P2) and Pb(63P1), which have energies of 0.969 and 1.32 eV relative to the 63P0level. All of these results are presented in Table 20 for comparison. In addition, Husain and Littler 3 8 7 have monitored the removal of lead atoms in the lowest spin-orbit level (63P0) by time-resolved attenuation of atomic resonance radiation. The chemical processes involved are slow, and measurable third-order kinetics were observed in some cases. These results are given in Table 21. Table 21 Rate data 3 8 7 fur cullisional removal of Pb(63Po)atoms at 300 K (powers of 10 in parentheses)

(a) Third-order rate constants Collision partners NO He NO C02 0, co,

+ +

+

(b)

1.3 & 0.2 (-31) 2.1 & 0.3 (-30) 3.6 rf 1.8 (-32)

Second-order rate constants Collision partner He

co

co2

887

k/cm6molecule-2s - I

k!cm3 molecule-' s-l < 1 (-16) 5 (-16) 0 & 2 (-16) 1.8 (-15) 1(-16) 1.4 0.2 (-16) < 1 (-16)

N2O CH4 CZH, C2H2 D. Husain and J. G.F. Littler, J . Photochem., 1973, 2, 247.

3 Unimolecular Reactions BY P. J. ROBINSON

1 Introduction This Report covers the literature from 1971 to the end of 1973, and may thus be regarded as a supplement to the book by Robinson and Holbrook (a few errata to which are noted in the Appendix).l In this period there have been several other reviews touching on unimolecular reactions, and in particular the book by Forst must be standard reading for kineticists interested in the theoretical aspects. The dissociation of diatomic molecules is discussed elsewhere in this volume, while Troe and Wagner have reviewed the dissociation of small molecules (up to 4 atom^),^ and included the literature on unimolecular reactions to late 1971 in their general gas-kinetics review.* Setser has paid particular attention to unimolecular reactions in systems involving non-Boltzmann distributions of reactants.5 The present Report is intended to be fairly comprehensive (it is certainly long and contains a large number of references), and in particular it deals with both the ‘physical’and the ‘physical-organic’ aspects. Not all references are given when those cited provide keys to earlier papers, and inevitably some arbitrary selection has been necessary for work which is less central or less definitive, The section headings should reveal the layout of the review; Sections 2-4 discuss theoretical aspects and experimental work providing insights into the theory, while the more utilitarian and mechanistic aspects appear in the remainder of the Report. This is why some topics such as chemical activation appear in more than one place. My sincere thanks are offered to the many colleagues who assisted the work by providing reprints or advance manuscripts of their papers. 2 General Theoretical Aspects

The Rice-Ramsperger-Kassel-Marcus (RRKM)theory is now well established as a working predictive and correlative theory, and it is a noticeable feature of the recent literature that RRKM theory is increasingly being used as a routine calculational technique rather than an esoteric theory P. J. Robinson and K. A. Holbrook, ‘Unimolecular Reactions’, Wiley, London, 1972. W. Forst, ‘Theory of Unimolecular Reactions’, Academic, New York, 1973. J. Troe and H. Gg. Wagner, Phys. Chem. Fast Reactions, 1973, 1, 1. J. Troe and H. Gg. Wagner, Ann. Rev. Phys. Chem., 1972,23, 31 1. D. W. Setser, M.T.P. Internat. Rev. Sci.,Phys. Chem. Ser. 1, 1972, 9, 1.

93

94

React ion Kinetics

for the experts. Many examples of such uses will appear in the following pages, and while other more complex theories of unimolecular reactions have continued to be discussed,s they have not yet achieved the practical status of RRKM. Application of RRKM theory has been assisted by the comprehensive FORTRAN program of Bunker and Hase, which provides many different computational options including various methods of statecounting (exact count, classical, or semiclassical, with or without anharmonicity corrections), user specification of the activated complex or its selection by the criterion of minimum state density,8 and treatment of the overall rotations as inactive or adiabatic, with a special treatment for hot-atom substitution products (see Section 3). Various aspects of state-counting have been discussed,lo the most significant advance in this respect being the program of Stein and Rabinovitch l1which carries out virtually exact counting by an extremely efficient algorithm due to Beyer and Swinehart,12 and which is applicable to any system the energy levels of which can be explicitly stated (e.g. Morse oscillators or hindered rotors). For a simple approximate prediction of fall-off behaviour under ordinary thermal reaction conditions, Benson et al. have pointed out 13thatthe classical Kassel theory (referred to as ‘RRK theory’) gives a close approximation to the RRKM fall-off curve when the s parameter is suitably chosen, and that a good empirical rule is seff= C,,/R, where CVibis the vibrational molar heat capacity of the reactant molecule at the temperature in question. There is some theoretical justification for this suggestion, since the main failure of the classical RRK theories lies in the use of classical (rather than quantum) statistical mechanics, and the weakness of this assumption can be measured by the extent to which CVIb falls short of its classical value sR (where s is the total number of oscillators in the molecule). The application of this approach has been greatly aided by the publication of extensive tables and graphs of the Kassel Integral I[seff,Ea/RT, log,,(A oo/Zp)],14 and the fall-off curves predicted are demonstrably l3 of approximately the right shape and in approximately the correct pressure region for a wide range of thermal reaction types provided k , is greater than about lo--*s-l. Skinner and Rabinovitch have emphasized l5 that the success of this approach does not

* lo

l1

l2 l3 l4

l5

0. K. Rice, J . Chem. Phys., 1971,55,439; S . H. Lin and H. Eyring,Proc.Nut. Acad. Sci., 1972,69,3192;J. C. Light, Adv. Cltem. Phys., 1971,19, 1 ; R. G. Gilbert and I. G . Rees, Austral. J . Chem., 1971, 24, 1541. W. L. Hase and D. L. Bunker, Quantum Chemistry Program Exchange Cat. No. QCPE-2 34. D. L. Bunker and M. Pattengill, J. Chem. Phys., 1968,48, 772. For reviews of state-counting procedures, see ref. 1, Ch. 5 and W. Forst, Chem. Rev., 1971, 71, 399. W. Forst and Z. Prasil, J . Chem. Phys., 1970, 53, 3065; L. K. Huy, W. Forst, and Z . Prasil, Chem. Phys. Letters, 1971,9,476; S. E. Stein and B. S. Rabinovitch,J . Chem. Phys., 1974, 60, 908. S. E. Stein and B. S. Rabinovitch, J. Chem. Phys., 1973, 58, 2438. T. Beyer and D. F. Swinehart, Comm. A C M , 1973, 16, 379. D. M. Golden, R. K. Solly, and S. W. Benson, J. Phys. Chem., 1971, 75, 1333. G. Emmanuel, Internat. J. Chem. Kinetics, 1972, 4, 591. G . B. Skinner and B. S. Rabinovitch, J. Phys. Chem., 1972, 76, 2418.

Unimolecular Reactions

95

extend to the prediction of specific rate constants k(E) [k,(E) in ref. 11, nor to the energy distributions of reacting molecules, and have suggested a criterion C J R > E,/RTfor the conditions under which an approximateIy correct fall-off curve can be obtained. Solc has considered an inconsistency of Kassel theory which might be significant under some conditions.le Another approximate method of calculating fall-off curves has been proposed by Forst,17who showed that if k m is given by the strict Arrhenius form A, exp( -E,/RT), with A, and E m independent of temperature, then k(E) is given by equation (l), in which N ( x ) is the state density la

(E > E m ) (E < Em)

k(E) = A,N(E - E,)/N(E)

=o

(1)

of the reactant molecule at energy x. This expression, like the RRK results,1a involves only properties of the reactant molecules and the Arrhenius parameters of the reaction; indeed equation (1) reduces exactly to the classical Kassel result k ( E ) = A,[(E - E,)/E]'-l if the state density of classical harmonic oscillators, equation (2),20 is used. Thus Forst's expression can be

n a

N ( E ) = E"'/(s

-

l)!

hv,

1-1

regarded as a generalized quantum-mechanical form of the RRK results; like them it equates the critical energy Eo [below which k(E) = 01 to E,, and like them is only accurate if A, and E, are independent of temperature, which strictly they are not.21 In practice, the k(E) calculated from equation (1) using realistic N ( E ) expressions (e.g. those obtained by the steepest descent method), when integrated over the appropriate energy distribution, give a very good approximation to the fall-off curves calculated by more detailed methods such as RRKM. Forst has illustrated several such comparisons l 7and Frey et al. have discussed some practical computational Like the approximate RRK treatment of Benson discussed above, equation (1) is not applicable to the calculation of k(E) as a function of E per se, but only to the averaged k(E) relevant in thermal energization. In addition it will be inaccurate in the lower fall-off region, where molecules with energies between Eo and E, become more important. Nevertheless the method promises to be useful in correlating data in the upper fall-off regions, and has already been applied in a variety of investigations (see e.g. Section 7, p. 159) and notwithstanding the above comments, to chemical activation experiments.23 16 17 18

19 20

21 It

23

M. Solc, Coil. Czech. Chem. Comm., 1971, 36, 2327. W. Forst, J . Phys. Chem., 1972, 76, 342. Ref. 1, Ch.4. Ref. 1, Ch. 3. Ref. 1, Ch. 5. Ref. 1, Ch. 6. H. M. Frey, R. G. Hopkins, and I. C. Vinall, J. C. S. Faraday I , 1972,68, 1874. K. J. Mintz and R. J. Cvetanovic, Canad. J. Chem., 1973, 51, 3386; K. J. Mintz and D. J. LeRoy, ibid., p. 3534.

Reaction Kinetics

96

Orbital symmetry considerations 24 have become widely used in the consideration of reaction mechanisms, and a number of examples will be mentioned later. Group additivity methods for the estimation of thermo26 heats of chemical and kinetic parameters are very well formation and stabilization energies 2 7 of radicals and ions have been reviewed,28and the estimation of activation energies from bond properties has been further The detailed significance of activation energy and its variation with pressure has been while the symmetry properties of transition states 31 and general aspects of activated complex theory 32 still receive attention. Theoreticians continue to calculate increasingly reliable potential energy surfaces for complex reactions, and the state of the computational art is illustrated by the appearance 33 of the first trajectory calculations using CNDO to determine the force-constants and hence the mechanical behaviour of the system (H, CH, CH4) at each point in time. It appears from this work that mechanistic considerations based only on consideration of minimum-energy paths (‘reaction co-ordinates’) may be considerably misleading, dynamic considerations having a major influence on the course of any particular reaction event. Cross-connections with the work of Polanyi and others on vibrational excitation in bimolecular reactions,34and with laser-stimulation of unimolecular reactions,35may be expected to appear.

+

--f

3 Intramolecular Energy Randomization The RRKM and other statistical theories of unimolecular reactions are based on the assumption that internal energy is statistically redistributed among all the degrees of freedom of the energized molecules on a time scale which is short compared with the lifetime to unimolecular decomposition or isomerization. A number of different techniques are now being exploited 24

25 28

27 28

2B

30

31 s2

33 34

35

R. H. Wollenberg and R. Belloli, Chem. in Britain, 1974, 10, 95; T. F. George and J. Ross, J . Chem. Phys., 1971, 55, 3851; R. G. Pearson, J. Amer. Chem SOC.,1972, 94, 8287; Accounts Chem. Res., 1971, 4, 152. S. W. Benson, ‘Thermochemical Kinetics’, Wiley, New York, 1968. H. E. O’Neal and S . W. Benson, Free Radicals, 1973, 2, 275. A. S. Rodgers, M. C. R. Wu, and L. Kuitu, J. Phys. Chem., 1972, 76, 918. K. W. Egger and A. T. Cocks, Helv. Chim. Acta, 1973,56, 1516, 1537. V. I. Babayan and A. D. Stepukhovitch, Zhur. $2. Khim., 1973,47,2923; Z . G . Szabo and T. I. Konkoly, 2.phys. Chem. (Frankfurt), 1973,84,62; M. B. Pahari and R. Basu, J. Chim. phys., 1971, 68, 753. D. C. Tardy, Chem. Phys. Letters, 1972,17,431; R . G . Gilbert and I. G. Ross, J. Cltem. Phys., 1972, 57, 2299. J. W. McIver and R. E. Stanton, J. Amer. Chem. Soc., 1972, 94, 8618; J. N. Murrell, J. C. S. Chem. Comm., 1972, 1044. B. Widom, J. Chem. Phys., 1971, 55, 44; J. B. Anderson, ibid., 1973, 58, 4684; J. M. Perez, J. M. Figuera, and V. Menendez, Anales de Fis., 1973, 69, 233; K. Fukui, J. Phys. Chem., 1970, 74, 4161. I. S. Y.Wang and M. Karplus, J. Amer. Chem. SOC.,1973, 93, 8160. T. Carrington and J. C. Polanyi, M.T.P. Internat. Rev. Sci.,Phys. Chem. Ser. 1, 1972, 9, 135. M. F. Goodman, J. Stone, and E. Thiele, J. Chem. Phys., 1973, 59, 2919.

Unimoleculur Reactions

97

to obtain information on the validity of this assumption, and more specifically on the actual rates of intramolecular energy relaxation. The various techniques and the results obtained are discussed in varying degrees of detail in the following sections, and a summary of numerical conclusions is presented in the conclusion (Table 2, p. 114). Thermal Energization.-Satisfactory comparisons of unimolecular fall-off with RRKM theory provided indirect support for the assumptions of the theory, but will not be discussed in detail (see Section 6 for recent work in this area). More directly, non-random energization and subsequent nonRRKM behaviour could be detected by thermal experiments at sufficiently high pressures, where the initially formed non-randomized molecules might react with a rate-constant different from that found after randomization. Bunker and co-workers have examined the pyrolyses of N02C1and N20s by spectrophotometry at pressures up to 308 atm and 274 atm, respectively, of added nitrogen, giving collision frequencies up to 3 x 10l2s-l. For N02C1,no change in rate-constant was observed, confirming the applicability of the RRKM assumptions for the rate-determining process N02Cl --f NO2 Cl. In the case of N205the slight observed variation of rate-constant with pressure was thought to be probably significant, but the experiments refer less clearly to the ratedetermining step (4), since the reactants for this step are produced in the pre-equilibrium (3). Further work of this type is desirable. N20, NO2 NO, (3)

+

+

NO2

NO

+

+ NO3 NO2 + + 0 + NO, +2N02 -+

0 2

(4) (5)

Chemical Activation Studies.-The present section is restricted in the main to chemical activation experiments which provide direct tests of intramolecular energy randomization. A large numbzr of results for chemically activated molecules have been rationalized in terms of RRKM theory, and thus provide indirect evidence for the assumptions involved, but these will not be discussed here (see instead Section 5). In addition, the study of hot-atom substitution products, and the special techniques associated with chemical activation in crossed beams, are dealt with separately in Section 3 pp. 103 and 104. Recent chemical activation studies have provided the most direct measurements of rates of intramolecular energy transfer, one of the most valuable features of such experiments being the ability to vary and to calculate with good accuracy the lifetime available for energy relaxation to take place. In very favourable cases the time interval between collisions can be decreased to as low as 10-13s, and times in the range 10-lf-lO-Ss are accessible in a6

M.L. Dutton, D. L. Bunker,and H. H. Harris, J. Phys. Chem., 1972,76,2614.

Reaction Kinetics

98

many systems. The most definitive experimentsare those in which a symmetrical molecule is asymmetrically energized and labelled so that the competing unimolecular reaction paths can be distinguished. The most successful example of this technique to date is the work (now almost ‘classical’) of Rynbrandt and Rabinovitch 37 on Scheme 1 and its isotopic inverse (with H

+ CF,

CH,=CF-CF

/r,

+ CF,

\

CD,

Scheme 1

and D interchanged); other examples will be discussed later. In this work [2H,]hexafluorobicyclopropylwas energized in one ring and the ratio of the competing eliminations of CF2 from the two rings was measured by mass spectrometry of the products. The rate-ratio was found to deviate from the randomized value at collision intervals below about 10A9s,and the data were fitted to a model in which the excess energy in the initially excited ring decayed exponentially with a time constant of 0.9 x 10-l2 s; thus randomization was 99% complete in about 4 x 10-l2 s. Very recently, Rabinovitch and co-workers3* have given a very clear demonstration of energy relaxation in chemically activated compounds containing a reactive and initially energized group linked to a non-reactive ‘energy sink‘ of variable size. The systems, shown in Scheme 2, are related

RsF:CF,

+ CD,

__+I

RSF

,r*/

RFCF: CD, + CF,

\

Scheme 2 s7

J. D. Rynbrandt and B. S. Rabinovitch, J. Phys. Chem.. 1971, 75. 2164. B. S. Rabinovitch, J. F. Meagher, K. -J. Chao, and J. R.-Barker, Chem. Phys., 1974, 60,2932.

99

Unimolecular Reactions

to those in Scheme 1, but in this case randomization of the energy cannot be detected by observing reaction from the initially non-energized part of the molecule, since this is non-reactive. The primary experimental measurement is thus the decomposition/stabilizationratio D / S as a function of pressure or collision frequency w ; the apparent rate-constants wD/S are shown as the points in Figure 1, while the solid lines represent the prediction of the theoretical model used. The convergent value at high pressure corresponds to the rate-constant 3.5 x 10l1s-l calculated for decomposition of the initially formed ring before any energy has relaxed into the side-chain; it is the same for all the molecules since the energetics are virtually identical, and the same as for the initially energized molecules in Scheme 1. The asymptotic values at the low-pressure end are close to the rate constants calculated for the completely randomized molecules, and become progressively smaller as the size of RFis increased since the available energy is then distributed among many more degrees of freedom. The curves are calculated on the assumption that the energy Eo originally deposited in the ring decreases exponentiallyaccording to the equation Et = Eo exp( - A t ) , with the constant A equal to 1.1 x 10l2s-l, the value used to fit the results for Scheme 1. Thus randomization of the energy is again complete within about 4 x s. The continual change in w D / S at times many orders of magnitude greater than this is due not to continued energy relaxation, but to the long lifetimes to decomposition of the randomized molecules. Consequently, very low collision frequencies are required if all the randomized molecules are to be allowed to decompose before collisional 'stabilization. Since this last point is of considerable importance in the design and interpretation of other experiments concerning non-randomization, it will be pursued further by consideration of a very simple system in which all the molecules show a constant high rate-constant k* up to time t = t,, at which

'k

6

7

8

9

10

11

12

13

log,&J/s-'l

Figure 1 Apparent rate-constants wDIS for compounds shown in Scheme LS8

100

Reaction Kinetics

point all remaining molecules become instantly randomized and the rateconstant falls to the lower value k (Figure 2). In the absence of collisions, the fraction of the energized molecules remaining at time t, is exp( -k*t,), the fraction having decomposed non-randomly being N D = 1 - exp( -k*t,); N D will be small in most cases and can be approximated by k*t, if so.

Figure 2 Variation of rate-constant with time for hypothetical energized molecules undergoing energy randomization at time tr.

At time t = t , the fraction of (randomized) molecules remaining is x1 = exp( -k*t,) x exp[ -k(t - tr)], the total fraction then having decomposed being N D + R D = 1 - x t , where R D is the fraction of random decomposition. At time t, therefore, the observable ratio of non-random to total decomposition is given by equation (6). The behaviour of this function is more clearly seen for the case where the total fraction decomposed 1 - exp( -k*tJ

(1

> tr)

(6)

is not large (e.g. up to 0.5), in which case both exponential terms exp( - y ) can be reasonably approximated by 1 - y , giving equation (7). It now 1

(7)

becomes clear that the fraction of non-random decomposition can decrease with time long after t,, and that the extent to which this occurs will depend on the ratio of the randomized to non-randomized rate constants k/k*. For example, for the C5Fl1case in Figure 1, k/k* w 3.8 x lo5s-'/3.5 X 10" s-l = Hence N D / ( N D + R D ) will approach its limiting value only for t / t , of the order of lo6. Since t, is of the order of l/h = s, t must s and o about 106s-l. Similarly the apparent rate-constant be about w D / S should level out at around o = lo6s-l, and this is in good accord with the more sophisticated calculations as shown in Figure 1. For the CF, case, the ratio k/k* is about 0.02, and on the same basis one would

101

Unirnolecular Reactions

expect o D / S to become constant at o c 10IOs-l, again as observed. This model is of course very crude, but it does reproduce the correct orders of magnitude and shows clearly how the effects of initial non-randomization can still be seen at times long after randomization is complete. The importance of this discussion is in its relevance to the interpretation of less extensive data giving some information on intramolecular energy relaxation. For example, Avery and Doering 3 B have examined the systems shown in Scheme 3 and its isotopic inverse, which are somewhat similar to those in Scheme 1 . The experiments revealed evidence of only slight nonrandom isomerization at the shortest collision times used, about 3 x s.

F / + CD,

Scheme 3

In view of the previous discussion, this result implies complete energy randomization not only in 3 x 10-6s, but probably also in very much shorter time intervals, since the rate-constant for the isomerization of randomized molecules was estimated to be about lo5 times smaller than that for nonrandom isomerization on the initially formed molecules. Similarly, Kim and Setser 40 prepared energized CH,CF,CD, by combination of CHSCF2* and *CD3radicals and measured the ratio of HF to DF elimination by mass spectrometry of the olefins formed. The ratio of HF to DF elimination was independent of pressure, indicating randomization at the shortest collision interval, ca. s, and possibly much shorter times. The rate-constants o D / S for decomposition of this and other molecules were also in good agreement with RRKM calculations. Doering and co-workers 41 formed energized [2,2-2H,]spiropentanefrom CD2 plus methylenecyclopropane and found statistical scrambling of H and D in the methylenecyclobutane produced. This implies equal rates of ring opening of the >CCHzCHz and > CCHzCD2rings in the rate-determining step, with the corollary of energy randomization within the collision interval of about 3 x 10-lo s, and again possibly at much shorter times.

-

39 40

4l

-

N. L. Avery, Ph.D. Thesis, Harvard, 1972. K. C . Kim and D. W.Setser, J. Phys. Chem., 1973,77, 2021. W. von E. Doering, J. C. Gilbert, and P. A. Leermakers, Terrahedrun, 1968,24, 6863.

102

Reaction Kinetics

The work of Rabinovitch and co-workers on chemically activated 2-n-alkyl radicals supports randomization indirectly, by agreement of experiment with RRKM calculations, but the results are so extensive as to warrant mention in this section. Previous work on s-butyl radicals, formed by a large variety of H/D olefin reactions, had shown no sign of non-randomization effects even at collision times of 2 x s.42 The observations have now been extended 43 to a whole series of secondary radicals up to c.16H33; the relevant scheme is Scheme 4, D being measured by propene production and the total D + S by an internal standard technique. The

+

H*

+ RCH,CH:CH,

<

f R*

3

+ CH2:CHCH3

(D)

RCH2cHCH3*

ML RCH2cHCH3

(S)

Scheme 4

observed rate-constant toD/S falls from 2 x lo7s-I for C,H, to 3 x lo2 s-l for c&33 because of the increasing size of the energy sink attached to the reactive part of the molecule (there is also a smaller opposing effect due to the greater thermal energy of the larger reactant molecules). Vibration frequencies for the butyl radical and its activated complex were assigned so as to fit the experimental results, and for larger radicals the same models were modified only by the addition of more degrees of freedom for the nonreactive alkyl chain. The RRKM calculations were in excellent agreement with the experimental results for the whole series, with the strong implication of complete randomization at the shortest collision times, which were about s for the smaller radicals and low6s for the larger radicals. Experimental difficulties unfortunately precluded work at shorter collision times. Finally, the work of Moehlmann and McDonald4, on i.r. chemiluminescence from chemically energized cyclo-octanone [produced from O ( 3 P ) + cyclo-octene] represents an interesting technique in which randomization of energy was inferred from agreement of the relative emission intensities of different vibrational modes (C-H stretch, C = O stretch, and CH2 bend) with values calculated assuming statistical distribution of the internal energy. The calculation does of course require a postulation about the emission probabilities of excited states, but the agreement is nevertheless impressive. Unfortunately the time-scale available for randomization was not well defined in this work, but an upper limit of 10-3s is set by the residence time of the energized molecules in the system, and the lifetime to emission is probably very much smaller.

'* 43

44

I. Oref, D. Schuetzle, and B. S. Rabinovitch, J . Chem. Phys., 1971, 54, 575. E. A. Hardwidge, B. S. Rabinovitch, and R. C. Ireton, J. Chem. Phys., 1971, 58, 340. J. G. Moehlmann and J. D. McDonald, J. Chem. Phys., 1973,59, 6683.

Unimoleculur Reactions

103

Hot-atom Substitution Products.-A number of relevant studies have been made on molecules energized by mbstitution of a translationally hot atom produced by nuclear recoil.45For example, tritium produced by the SHe(n,p)T or 6Li(n,a)T reactions, or 18F from the l9F(n,2n)lsF reaction, undergo direct substitution reactions such as (8) to produce highly energized sub-

T- + CH,NC

-+

CH2TNC*

+ H*

(8)

strates. A number of instances of apparent non-RRKM behaviour have been recorded in such systems, but it has now been shown that the apparent failure to randomize may be due to abnormally large rotational effects in moleculeswhich actually have randomized energies. In the case of CH2TNC*, for example, the gas-phase product was entirely CH,TCN, even with 5 atm argon present, and this required a high isomerization rate ( >lOIOs-l) of the energized species.46 Using a simple RRKM treatment, the excited molecules must therefore have a high internal energy, well over 360kJmol-l on average (cf. the critical energy of 157 kJ mol-l). But at this high energy, the quantum-state densities of MeNC and MeCN are such that a significant proportion of the molecules should exist as MeNC, and collisional quenching should give an observable yield of CH2TNC (observed in the liquid but not the gas phase). Bunker 4 7 has subsequently rationalized these results in terms of rotational phenomena, which are unusually emphasized by the large angular momentum likely to be found in hot-atom substitution products. For CH2TNC the major effect is the large relative change in the I, moment of inertia when the energized molecule is converted into an activated complex (Z:/I: = 13.6/5.0). Since this rotation is adiabatic, the rotational momentum stays constant and the rotational energy therefore falls dramatically, the released energy being distributed among the other internal degrees of freedom. A semiquantitative treatment of these effects is consistent with the observed results. In the case of CH3CF218Fand CH218FCF3,the product yields were at variance with standard RRKM predictions; for example, the former would be expected to react predominantly by C-C fission rather than HF elimination at the energies involved, whereas HF elimination was experimentally the sole observed reaction.48 Again, however, consideration of the effects of the overall rotations (in this case inactive rather than adiabatic) permits rationalization of the results in terms of a reasonable model involving randomization of energy. To quote Bunker,47‘it is mostly a matter of taste whether or not the present treatment is viewed as an extended RRKM theory’.

47

48

For a review of hot-atom techniques see F. S. Rowland, M.T.P. Internat. Rev. Sci., Phys. Chem. Ser. 1, 1972, 9, 109. C. T. Ting and F. S. Rowland, J. Phys. Chem., 1970, 74,4080. D. L. Bunker, J. Chem. Phys., 1972, 57, 332. K. A. Krohn, N. J. Pukes, and J. W. Root, J. Chem. Phys., 1971,55,5785, and earlier papers cited therein.

104

Reaction Kinetics

In view of these problems, the apparent non-randomization 4 9 of r3H,]spiropentane at times below 2 x 10-los may not be truly symptomatic of non-RRKM behaviour, and evefi if it is, the discussion of the previous section indicates that randomization of the energy may be complete in a much shorter time. As an upper limit only, the figure is supported by the chemical activation experiments already referred Dissociation of recoil-tritium-labelled [3Hl]neopentane gave methyl radicals (trapped as methyl bromide) enriched in tritium, implying predominant dissociation of the C-C bond nearest the labelling site and thus non-randomization of energy in the available time; the half life to dissociation was estimated at ca. S.~O Photoactivation.-Conventional photoactivation experiments involve light absorption to give an electronically excited species which undergoes rapid internal conversion to a vibrationally excited ground-state molecule of known total energy. If this species then undergoes a unimolecular reaction, data on the rate of reaction (relative to the rate of collisional quenching) can provide a test of rate theory if strong collisions are assumed (or a test of collisional deactivation efficiency if rate theory is assumed, as in Section 4). Data of this type for cyclobutanone decomposition 51, 52 give k ( E ) values fitting reasonably to RRKM calculations for collision intervals down to 10-lo s. Apparently contradictory conclusions can be drawn from the nonradiative lifetimes of benzene, perdeuteriobenzene, and fluorobenzene, measured by fluorescence decay.53 Excitation of these molecules to specific vibronic states gave decay lifetimes which were characteristic of the individual states, rather than being dependent only on the total energy of the molecule, and this has been taken as evidence of non-randomization within the available time of 10-7-10-8 s.54 The nature of the non-radiative processes is not known. Crossed-beam Studies.-A new type of technique which is now providing information on intramolecular energy relaxation in very short time intervals is the study of ‘long-lived’ collision complexes in crossed molecular beam experiments (note that the ‘complex’ in this sense is an energized species, and not an activated complex or transition state). In suitable systems a reaction between beams of A and BC to give products AB + C may proceed by way of an intermediate complex ABC* (the * indicating energization) 40

51 52

53

54

Y.-N. Tsang and Y . Y . Su, J. Chem. Phys., 1972, 57,4048; J . Phys. Chem., 1972, 76, 2187. G. K. Winter and D. S. Urch, J. C. S. Chem. Comm., 1973, 474. F. H. Dorer, J . Phys. Chern., 1973,77, 954. N. E. Lee and E. K. C. Lee, J . Chem. Phys., 1969, 50, 2094; J. C. Hemminger and E. K. C. Lee, ibid., 1972, 56, 5284. K. G. Spears and S. A. Rice, J . Chern. Phys., 1971, 55, 5561; B. K. Selinger and W. Ware, ibid., 1970,53, 3160; A. S. Abramson, K. G . Spears, and S. A. Rice, ibid., 1972, 56, 2291. K. Shobatake, J. M. Parson, Y. T. Lee, and S. A. Rice, J. Chem. Phys., 1973,59, 1427.

105

Unimolecular Reactions

with a lifetime long enough for it to undergo several rotations (typical period 5 x s) and many vibrations (typical period s). Such behaviour is readily apparent from the angular distribution of product species, since by the time the complex decomposes it will have ‘forgotten’ the directions from which the reactant species arrived; the product fragments will therefore tend to depart in random directions when viewed from the centre of mass of the system. In fact, owing to the need for conservation of angular momentum, the product tends to ‘peak‘ in the forward and backward direction^,^^ but the peaking is symmetrical with respect to the 4~90”plane. There are many well-documented examples of this type of behaviour; see e.g. p. 108, Table 1. If, by contrast, the collision complex has an extremely short life, the product will peak in the forward direction, the limiting case being that of a ‘stripping reaction’. For some reactant pairs a transition between the two types of behaviour can be observed when the energy of the collision is varied; for example the reaction C + + D, -+ (CD;) + CD+ D shows a symmetrical product distribution at a collision energy of 3.5 eV but strong forward peaking at > 7 eV where the lifetime of the complex is much It is interesting to note that only in the present decade has intermediate complex formation become a clearly recognized phenomenon in ion-molecule reactions (the review by Friedman and Reuben 5 7 gives a useful account of work published before 1970). For reactions which proceed by way of a long-lived complex, information on energy randomization can be obtained by studying in various ways the properties of the dissociation fragments. The time scale is very short, though unfortunately not well defined, and the variation of the behaviour with energy can be explored by the use of velocity-selected beams. One possible approach is the comparison of competing reaction paths, as has been done in conventional chemical activation experiments, and some examples of this will be found below. At present, however, the method most used involves measurement of the translational energy distribution of the product fragments by time-of-flight analysis. One simple diagnostic feature is that the kinetic energy of the products will be relatively low if energy randomization has been able to redistribute the collision energy among the internal degrees of freedom of the complex. For example, reaction (9) is exothermic by about 167 M mol-1, yet only about 6 kJ mol-1 appears as relative translational energy

+

CS f SF6 -+ (CSSF,)*

+ CSF

+ SF5

(9)

of the product fragments; similar remarks apply to the K and Rb analogues.5s The translational energy distribution is also independent of the product b5

66

57

s8

W. B. Miller, S. A. Safron, and D. R. Herschbach, Discuss. Faraday SOC.,1967, No. 44, p. 108. C. R. Iden, R. Liatdon, and W. S. Koski, J. Chem. Phys., 1972,56,851; see also B. H. Mahan and T. M. Sloane, ibid., 1973, 59, 5661. L. Friedman and B. G. Reuben, Adv. Chern. Phys., 1971, 19, 33. S. J. Riley and D. R. Herschbach, J . Chem. Phys., 1973, 58, 27.

106

Reaction Kinetics

scattering angle because of the lack of influence of the history of the complex on the fragmentation process. For the reaction between Cs and SF6,a detailed survey has also been made of the vibrational and rotational states of the CsF The energy in these modes is also low, and is in good agreement with calculations based on randomization of internal energy between all the degrees of freedom of the complex. For contrast, the reaction of Li with SF6 produces an unsymmetrical distribution of products,6oand the vibrational energy in the LiF product is only about half that calculated on the same basis? there is no long-lived complex in this system. On a more quantitative basis, it is possible to use various theories of the fragmentation process to predict product kinetic energies, and comparison of the predictions with experimental data then provides information about the validity of the theory and its parameters. The RRKM theory does not in its basic form make predictions about any events happening after the system has passed through the critical configuration. It is relatively easy to extract information about the internal energies and relative kinetic energy of the fragments at this point 62 (although rotational effects can be complicated), but if there is a significant energy barrier to the reverse reaction, the motion of the separating but still-coupled fragments down the potentialenergy surface introduces a further energy partitioning whose characteristics are not well understood and which must be modelled in order to provide a theoretical kinetic-energy distribution for comparison with e ~ p e r i m e n t . ~ ~ An alternative approach is the ‘phase space theory’,64in which the formation of products in the various possible energy states is determined simply by the numbers of such states available, subject to the conservation of total energy and momentum. In this type of theory there is no reference to the properties of a critical configuration and no calculation of the lifetime of the complex. To date, crossed-beam experiments have not provided extremely definitive results on intramolecular energy randomization. This is partly due to the great experimental difficulties, partly because of the theoretical problems discussed above, and partly because of the inherent lack of control of the time scale available for randomization and the lack of precise knowledge of this time scale. The results so far obtained can be divided for present purposes into three groups :

(i) No Collision Complex Observed. This is not in general very helpful from the present viewpoint, but a few cases have provided evidence for energy 69

S. M. Freund, G . A. Fisk, D. R. Herschbach, and W. Klemperer, J. Chem. Phys., 1971, 54,2510.

6o

62

83

D. D. Parrish and R. R. Herm, J. Chem. Phys., 1971, 54, 2519. R. P. Marielle, D. R. Herschbach, and W. Klemperer, J. Chem. Phys., 1973,58, 3785. K. H. Lau and S. H. Lin, J. Phys. Chem., 1971, 75,2458 and references therein; C. E. Klots, Z . Naturforsch., 1972, 27a, 553; J. Chem. Phys., 1973, 58, 5364. S. A. Safron, N. D. Weinstein, D. R. Herschbach, and J. C . Tully, Chem. Phys. Letters, 1972, 12, 564.

R. A. White and J. C. Light, J . Chem. Phys., 1971,55,379; J. C . Light, Discuss.Faraday SOC.,1967, No. 44, p. 14; P. Pechukas, J. C. Light, and C. Rankin, J. Chem. Phys., 1966, 44, 794; F. H. Mies, ibid., 1969, 51, 787.

Unimolecular Reactions

107

exchange even during the very short times involved. Thus reaction (10)

showed no evidence of complex formation yet the product NH+ contained substantially more internal excitation than would be expected for a pure stripping process.6s Reaction (ll), at energies from 1-10 eV, showed no

complex formation 66* 6 7 but independent studies by quadrupole mass spectrometry and ion cyclotron resonance showed that the correspondingreactions of CD,’ with CH, or of CHZ with CD, proceed with complete isotopic scrambling. If the atoms themselves are randomized this must surely imply at least very extensive energy randomization as well. Reaction (12) showed

similar characteristics to reaction (1l), except that there was evidence for complex formation at the lowest collision 69 The time scale for such processes is not well defined, but must be at most about 10-l2s, and possibly much lower. (ii) Collision Complex Formed; Results Consistent with Randomization. In this type of situation the time can only be given a lower limit, usually quoted as ‘at least a few rotational periods’, i.e. >4 x s. It would certainly be a major advance if the products could be examined as a function of the time allowed to the collision complex, in a manner analogous to conventional chemical activation experiments carried out over a range of pressures. There is also the point that results ‘consistent with randomization’ are not necessarily inconsistent with non-randomization, especially in view of the highly convoluted form in which the reaction characteristics appear in the experimental measurements, and the theoretical modelling which is necessary to obtain the desired comparison. Some studies providing this type of support for energy randomization are listed in Table 1. (iii) Collision Complex Formed; Evidence Found for Nan-randomization. Many examples of this type have appeared recently, due particularly to the work of Parson, Lee, Shobatake, and S.A. Rice on crossed beams of fluorine atoms with a wide range of organic compounds including ethylene and C2D4;O 13~ 1313

13’ 13*

‘O

E. A. Gislason, B. H. Mahan, and A. S. Werner, J. Chem. Phys., 1971, 54, 3897. Z. Herman, P. Hierl, A. Lee, and R. Wolfgang, J. Chem. Phys., 1969, 51, 454. S.-L. Chong and J. L. Franklin, J . Chem. Phys., 1971,55, 641. W. T. Huntress, J. Chem. Phys., 1972, 56, 51 11. Z . Herman, A. Lee,and R. Wolfgang, J . Chem. Phys., 1969,51,452. J. M. Parson and Y.T. Lee,J . Chem. Phys., 1972,56,4658.

108

Reaction Kinetics

Table 1 Crossed-beam studies with long-lived complexes and results consistent with randomization Beams Products OBr Br 0 Br, CSF SF5 CS -k SF6 M SF6 (M = CS, Rb or K) MF SF5 M SnCl, (M = Cs, Rb or K) ? Li 3- M X ( M = K or Cs; LiX M X = C1, Br or I) CsCl KX (X = C1 or I) CsX KCl X+ N2(X = He, Ne, Ar, X N,+ or Kr) C1 CHF:CHCl F CHC1:CHCl (cis and trans) F CH,:CCI, I C l CH,:CHF 1 I H CHF:CCl, J B; R’CH:CClR C1 RCH:CBrR (R’, R) = (H, H), (H, Me) or (Me, H)

+

+ +

+ + + + +

+ + + + +

+

+ + + +

Method of study angle K.E. b angle K.E. angle K.E. angle

+ + +

angle

Ref. a 59

58 58 C

+ K.E.

d

e

e

angle

74

angle

74

angle

+ K.E.

76

D. D. Parrish and D. R. Herschbach, J. Amer. Chem. Soc., 1973,95, 6133; Ir Studied by electric resonance spectroscopy of CsF; A. B. Lees and G . H. Kwei, J. Chem. Phys., 1973, 58, 1710; W. B. Miller, S. A. Safron, and D. R. Herschbach, J. Chem. Phys., 1972, 56, 3581; Scattering cross-sections measured by D. C. Fullerton and D. F. Moran, J. Chem. Phys., 1971,54, 5221 (and other work cited therein), and vibrationd-rotational states of N,+ by T. F. Moran and D. C. Fullerton, ibid., 1972, 56, 21 (and references therein). a

butenes,71 other olefins and d i e n e ~C6D6,64 , ~ ~ various aromatic and heterocyclic compounds,73 and dichloroethylenes.74 In all these systems the product angular distributions were indicative of long-lived complex formation, and the angular distributions and translational energy distributions have been compared with the predictions of an RRKM-based theory and a phase-space theory. In all cases tested, the RRKM-based theory failed to reproduce the experimental results with adequate realism ; the phase-space theory was generally able to fit the experimental curves, although it was sometimes necessary to reduce the number of active degrees of freedom to obtain agreement, the details of the models being somewhat arbitrary. The general conclusion has been drawn 74 that when a light atom or group (e.g. H, CH3) is emitted from the collision complex, only a fraction of the degrees of freedom are active, whereas if the leaving group is massive (e.g. Cl), most degrees of freedom are involved in energy transfer. The question as to whether or not this also implies intramolecular energy randomization has not yet been Another type of study was spectroscopic measure73p

71 72

73 74

J. M. Parson, K. Shobatake, Y . T. Lee, and s. A. Rice, J . Chem. Phys., 1973,59, 1402. K. Shobatake, J. M. Parson, Y. T. Lee, and S. A. Rice, J. Chem. Phys.. 1973,59, 1416. K. Shobatake, Y . T. Lee, and S. A. Rice, J . Chem. Phys., 1973,59, 1435. K. Shobatake, Y . T. Lee, and S. A. Rice, J . Chem. Phys., 1973,59, 6104.

Unimolecular Reactions

109

ment of the vibrational states of NZ formed in reaction (13);76 this indicated Ne+

+ N,

+=

NZ

+ Ne

(13)

increasing non-randomization at low collision energies. It must be stressed that the RRKM-based theories referred to above do have additional assumptions added to the basic RRKM theory used in the interpretation of thermal and conventional chemical activation experiments. There are in fact a number of aspects of these crossed-beam results for which RRKM theory does give a reasonable prediction of events up to passage through the critical configuration. Thus the relative rates of the competing reactions in Schemes 5 71 and 6 71 are in reasonable agreement with RRKM F*

+ CH,CH:CHCH,

<

f

+ CH,CHF&-€CH,*

(cis or trans)

CH,CF:CHCH,

h CHFzCHCH,

+ H*

+ CH3

Scheme 5

F*

+ RC6H4X+ (complex)*/ \

f RCsH,(X)F

+ H-

+

4 RCBH4F X-

Scheme 6

+

predictions. On the other hand, the analogue of Scheme 5 for F' tetramethylethylene showed more than an order of magnitude error in the predicted rate ratio if all degrees of freedom were assumed In Scheme 7, RRKM with all modes active was in agreement with the experimental limit of 1/540 for H loss/Cl Herschbach and co-workers 76 used their own F*+ CHC1:CHCl

-+ CHClFcHCl*

,

/

f CHF:CHCl

X '

CC1F:CHCl

+ C1+ H-

Scheme 7

RRKM-based theory 63 and found reasonable agreement with experimental angle and kinetic energy distributions for reaction of chlorine atoms with bromo-olefins (see Table 1). They also applied their theory to reaction (14), .in which a long-lived collision complex is not formed, and concluded that randomization had not For the complex formed from C,H: and C,H4, the kinetic energy distributions at several collision energies were consistent with RRKM calculations only on the basis of a reduced number 75

G. H. Saban and T. F. Moran, J. Chem. Phys., 1972,57, 895, 5622; R. P. Lowe, ibid.,

76

J. T. Cheung, J. D. McDonald, and D. R. Herschbach, J . Amer. Chem. Soc., 1973,95,

I972,57, 5621.

7889.

I10

Reaction Kinetics

of effective oscillators, the decay times being estimated at 0.4-1.6 x CI. t CH,:CHCH,Br

+ CH,:CHCH,Cl

-tBr.

s." (14)

In conclusion it is clear that crossed-beam studies can provide a wealth of information of relevance to unimolecular rate theory. The information available to data has provided serious doubts about the efficiency of intramolecular relaxation in the 'long-lived' collision complexes studied, i.e. in times of the order of 10-12-10-11 s. It is to be hoped that experimental and theoretical developments will be forthcoming to separate the roles of energy randomization before and after passage through the critical configuration, to define more closely the rates of randomization, and to provide utilitarian links with the more usual type of thermal and chemical activation studies.

Unimolecular Reactions of Ions.-Apart from the study of ionic intermediates produced in crossed beams (see above), statistical theories of unimolecular decomposition have been widely applied to the interpretation of ion fragmentation patterns observed mass-spectrometrically (the theory then being generally known as Quasi-Equilibrium Theory or QET). In recent examples, the isotope effect in the loss of H or D from various deuteriated toluene ions was found to agree well with QET calculations, the degree of H / D scrambling varying qualitatively as expected with the energy and hence the lifetime of the ions.7s The fragmentation of several alkyl ions energized by charge exchange with Hg+ varied markedly with temperature owing to the effectiveness in the fragmentation process of the internal energy of the alkane prior to ionization, and the results were in good agreement with QET calculations previously applied to photoionization data. Chargeexchange ionization of benzonitrile and benzene by a range of ions of different electron affinities has provided well-defined k ( E ) vs. E curves for the ions involved 8 o and such experiments should provide useful tests of theory in the future. Lin and Rabinovitch rationalized the variation in relative abundances of the ion C,H,O+ and the metastable from its fragmentation into C2H, H,O+ in terms of a statistical theory involving randomization of energy in the original parent RCH,CH,OH+ ions.81 The technique of field-ionization mass spectrometry promises to push the available time resolution down to as low as s. Thus Derrick and co-workers 82 observed fragmentation products of [3,3,6,6-2H,]cyclohexeneions at 1O-l1-s, and found that the deuterium labelling was completely scrambled during this period. This presumably indicates extensive energy randomization as well. +

7i

?@

R1 82

A. Lee, R. L. LeRoy. 2.Herman, and R. Wolfgang, Chem. Phys. Letters, 1972,12,569. I. Howe and F. W. McLafferty, J. Amer. Chem. Soc., 1971, 93, 99. W. A. Chupka, J . Chem. Phys., 1971,54, 1936. B. Andlauer and Ch. Ottinger, J . Chem. Phys., 1971, 55, 1471. Y.N. Lin and B. S. Rabinovitch, J. Phys. Chem., 1970, 74, 1769. P. J. Derrick, A. M. Falick, and A. L. Burlingame, J . Amer. Chem. Soc., 1972,94,6794.

111

Unimolecular Reactions

Comparisons of the above type have not in general been a sensitive test of unimolecular rate theory because the data obtained are the convoluted representation of k(E) averaged over the poorly known energy distribution of the reacting ions. The new technique of photoelectron-photo-ion coincidence mass spectrometry now promises to yield very specific information about ions of closely selected energy, somewhat analogous to molecules excited by photoactivation. In this t e c h n i q ~ emolecules ,~~ are photoionized by monochromatic radiation under conditions such that essentially single ionization events are observed. The photoelectrons are separated by a conventional electron-energy analyser, while the positive ions and their fragmentation products are simultaneously examined, for example by time-of-flight mass spectrometry. The energy of the photo-ion is accurately calculable as the difference between the quantum energy of the ionizing radiation and the kinetic energy of the photoelectron emitted. The coincidence feature of the technique consists of counting only those ions which are detected at the appropriate time after the arrival of a photoelectron with energy in a specified narrow band. Thus over a suitable period one measures the production of fragments only from parent ions of specific and calculable energy, and by varying the energy band specified the appropriate measurement can be made as a function of energy. Alternatively, the photoionizing radiation can be varied in wavelength and the ions counted which correspond to threshold electrons (those produced with kinetic energy close to zero). This technique has not yet provided detailed tests of unimolecular rate theories but seems likely to do so. One recent example of its application is the work of Stockbauer on light and heavy methane and ethane; QET was found qualitatively to describe the fragmentation behaviour. Danby and Eland have illustrated the investigation of fragment kinetic energy via the distribution of flight times for ions from methylene chloride and oxygen. Simm et al. have shown 86 that the ion CzFG+formed with sufficient photolytic energy dissociates from an excited electronic state without prior internal conversion to the ground state. The authors considered that this constituted a violation of QET, since it is usually assumed in QET calculations that all the reacting ions decompose from the electronic ground state. The point seems to be one of terminology, since the separate electronic states could each have randomized vibrational-rotational energies, and be treatable by a statistical theory known as QET, without requiring in addition equilibration of the electronic energy. The measurement of fragment kinetic energy (cf. the previous section, where theoretical aspects are discussed) is much simpler for ions than for neutral molecules, and such measurements should be able to provide 8369

83

84

(a) J. H. D. Eland, Znternat. J . Mass Spectrometry Ion Phys., 1972, 8, 143; (b) C. J. Danby and J. H. D. Eland, ibid., p. 153.

R. Stockbauer, J. Chem. Phys., 1973,58, 3800. I. G. Simm, C. J. Danby, and J. H. D. Eland, J . C. S. Chem. Comm., 1973, 832.

Reaction Kinetics

112

useful tests of unimolecular rate theory (as well as information on mechanistic and thermochemical aspects 86). In this context, a procedure for evaluating product kinetic-energy distributions is built into Bunker and Hase’s RRKM program. LeRoy has interpreted data on various ion decompositions as indicating incomplete energy randomization in ‘very short-lived (activated) complexes’. The lifetime of an activated complex was measured by t/6 = (2EGanJp)-+ (cf. ref. 18), which is in fact the reciprocal of the speed with which a system passes through the critical configuration. The lifetime of an activated complex is infinitesimally small (since 6 0 and therefore t -+ 0), and it is meaningless to discuss energy randomization in a transient configuration with no life; in RRKM theory it is the energized molecules, not the activated complexes, which undergo energy randomization. LeRoy concluded that ,:s < s+ (the number of oscillators in the complex) for some of the systems studied, since a reduced value was needed to make the calculation reproduce the experimental kinetic energy E:,,,. The extent to which this occurred correlated with the ‘lifetime’ z/6 values, but since these are so closely related to the I?;,,, values taken, it rnay be that the correlation is an artefact of uncertainties in the experimental data or the calculational parameters. A correlation with the genuine unimolecular lifetime would at least need to be established before the conclusion of non-randomization could be accepted. --f

Laser Studies.-The study of chemical laser systems 8 8 forms a rapidly developing technique for the convenient measurement of the relative populations of different energy states of the products of chemical reactions, and such experiments are now being increasingly applied to unimolecular chemical laser^.^^^ It is to be expected therefore that information on intramolecular energy relaxation will be forthcoming, although few quantitative data in this respect have so far been reported. Padrick and Pimentel found that the HF eliminated from chemically energized CH3NF2 contained 20-25 kJ mol-l of vibrational energy, which corresponded to equal partitioning of the available 326 kJ mol-l between the 15 internal degrees of freedom of the molecule. Although the unimolecular lifetime of the energized molecules in this system is not known with certainty, and more refined interpretations will obviously be required, the implication is complete energy randomization at least within the collision interval of ca. s. Trajectory Calculations.-Although not experimental in the usual sense of the 86

87

89

E. G. Jones, J. H. Beynon, and R. G. Cooks, J . Chem. Phys., 1972, 57, 2652; R. G. Cooks,M. Bertrand, J. H. Beynon, M. E. Rennekamp, and D. W. Setser, J . Amer. Chem. SOC.,1973, 95, 1732. R. L. LeRoy, J. Chem. Phys., 1970,553, 846; 1971,55, 1476; see also R. L. LeRoy and D. C. Conway, J. Chem. Phys., 1972,56, 5199. C . B. Moore, Ann. Rev. Phys. Chem., 1971, 22, 387. T. D. Padrick and G. C. Pimentel, J . Phys. Chem., 1972, 76, 3125. M. J. MoIina and G . C. Pimentel, J. Chem. Phys., 1972, 56, 3988.

Unimolecular Reactions

113

word, the trajectory calculations of Bunker and co-workers 92 for the methyl isocyanide molecule must certainly be taken as evidence relevant to the question of intramolecular energy randomization. Earlier trajectory calculations for some model triatomic systems had shown near-RRKM behaviour,* but such calculations had not been made for a real molecule whose unimolecular reaction has been subjected to thorough experimental study. The method consists of setting up a potential-energy surface for the six-atom system (several variations were used, all with considerable anharmonic character to encourage energy randomization), then choosing initial energy distributions and geometries in a random (Monte Carlo) fashion (three variations were used), and following the mechanical behaviour of the molecule by integrating the equations of motion. The lifetime to isomerization is recorded for each trajectory, and when enough have been accumulated they provide a lifetime distribution for the given total energy, from which the specific rate constant k(E) could be obtained. At relatively low energies no isomerizations occurred in the time allowed (ca. 4 x s), and only an upper limit could be established. The results were interpreted as showing serious deviations from RRKM behaviour. For example, at 840 kJ mo1-1 the three initialization methods gave results which differed markedly from each other [k(E) = 3 x lolo, 1.5 X loll, and 4 x loll s-l] and from the RRKM prediction (3 X 10l2s-l). The rates of isomerization found were all lower than the RRKM predictions, and also lower than a theoretically estimated rate of energy relaxation. At 290 kJ mol-l, 5050 trajectories gave no isomerizations whereas 10 were expected from the RRKM calculation. At present there is no rationalization for the differing pictures presented by the trajectory calculations and the laboratory experiments for this molecule. Methyl isocyanide is not an ideal system for such comparisons in view of the unusual rotational effects in~olved,~' and the lack of chemical activation data for comparison with RRKM predictions at energies well above the threshold. The tritium-recoil work is unhelpful, again because of rotational effects(see Section 3, p. 103). Thus the trajectory results have to be compared with RRKM predictions at high energies, and the RRKM predictions with thermal data at low energies. Trajectory calculations for a system where they can be compared with experiment as well as with RRKM theory will be of very great interest. 919

Conclusions.-In Table 2 an attempt is made to summarize the numerical conclusions which can be drawn from the bewildering array of experimentation which has been discussed. The emphasis is very much on direct tests of energy randomization, and the most definitive of these from the chemist's point of view are the chemical activation experiments, especially those with H. H. Harris and D. L. Bunker, Chem. Phys. Letters, 1971,11,433.

@* D. L. Bunker and W. L. Hase, J. Chem. Phys.,

1973, 59,4621.

114

Reaction Kinetics

Table 2 Studies of intramolecular energy randomization u o r details see t e x t ) Reactant Thermolysis NO2Cl N205

Non-randomization Time for complete randomizationls Ref. observed ? no possibly

< 2 x 10-13 10-12?

36 36

Chemical activation 7-

CHaCFzCFCFCFzCDz

RCH&HCH ( R = n-CH3 to C13H27) Cyclo-octanone

Yes

ca.4 x

37

probably" noa

< 3 x 10-8 < 10-9

39 40

no"

< 3 x 10-10

41

nop

< <

43

noa

.g10-3

(small R) (large R )

44

Hot-atom substitution products CH2TNC CH 3CFz"F, CH218FCF3

nob

paD2

apparently"

<2 x

49

Me3CCH2T

apparently

>

50

Photoactivation CsH6, CsDs, C6H5F

apparently

> 10-7-10-8

46,47 48,47

nob

53, 54

Crossed beams These give an indication of relaxation occurring in times of the order of 10-13-10-11s. Some indicate randomization, some do not - see text. Reactions of ions

:.J=(*D CZFi

'yes' (electronic)

Lasers CH3NFz

no

Trajectory calculations CH,NC

Yes

" The times quoted may be very much upper limits; see text;

?

< 10-8 ?

85

89 91,92

Apparent non-randomization observed, but reinterpreted in terms of randomization; see text.

Unimolecular Reactions

115

the cyclopropane systems where observable chemical effects lead very directly to a measure of the actual rate of energy randomization. Hot-atom experiments have been shown to need very careful theoretical analysis, and the conclusions as to non-randomization in some systems can only be regarded as tentative. The large numbers of crossed-beam results (only summarized in Table 2) give conflicting results on randomization, and it is difficult to assess the validity and relevance of the various conclusions in view of the convoluted nature of the experimental results and uncertainties in the theoretical modelling used in their interpretation. The only studies indicating randomization times in the high region of loA8s are the fluorescence decay experiments and perhaps the trajectory calculations, and it is tempting to underweight these and the beam experiments in their present state of development compared with the more direct chemical activation results. Clearly, more work, both theoretical and experimental, will be needed to rationalize the apparently conflicting results from the different types of experiment. In the meantime one might draw the tentative and hopeful conclusion that while energy randomization may require times of 10-l’ s in some or many systems, it will usually be rapid enough for randomization to be assumable in calculations for thermal reactions and for many chemically activated reactions. On the other hand, the assumption may not be acceptable for the rapid reactions of very energetic species, and it should again be noted that in some systems non-randomization effects can be observed long after randomization is complete. 4 Intermolecular (Collisional) Energy Transfer Table 3 lists the various studies reported during the review period on collisiona1 energy transfer involving molecules which are energized in the unimolecular rate-theory sense. A good deal of work has also appeared on energy transfer to and from low-lying vibrational levels, but this will not be reviewed here. The most extensive and informative studies are those based on chemical activation or photoactivation, where in suitable cases the experimental data are able to distinguish between different models of the collisional deactivation process, giving a value for the average energy removed per collision, { A E ) , and possibly information about the distribution of stepsizes, usually characterized as step-ladder or exponential distributions.93 The previous indications that weak colliders tend to show the exponential type of distribution while strong colliders exhibit step-ladder behaviour have been convincingly confirmed in experiments with the 2,4dimethylpent2-yl radical. 94 Other studies provide relative values for collision efficiences, @, or A, defined as the apparent activation or deactivation efficiency relative to a reference bath gas; can be regarded loosely as the number of collisions required to de-energize the energized molecule. For thermal unimolecOg

O4

Ref. 1, Ch. 10. J. H. Georgakakos and B. S. Rabinovitch, J. Chem. Phys., 1972,56, 5921.

Table 3 Studies of collisional energy transfer Energized species Chemical Activation

Source

CH,ClCH,Cl, CD2ClCDzCl

2CHzC1or 2CD,CI

CH3CF3

CH3

CH,CHFCD, CH3CHFCH,CH3 S-cl6H33

2 ,CDimethylpent -2-y l

Methylcyclobutane Methylcyclopropane Methylcyclopropane

CL

(E*)/kJ mol-1

Eo/kJ mol-l

368

251

+ CF3 + +

CH,CHF CD3 CH3CHF CH3CHZ f C1GH32 H

+ Me,C:CMe,

H

+ olefin

+ +

CH2* C-C4H8 CH2* propene c - C ~ H ~ CH,

+

368 368 232

188

ca. 230 222 126

113

502,581 ca.250 389 255 372 255

Bath gas c-C~F~ SF6 CH,CI, CD3Cl CF4, Xe N2, co Ar, Kr Ne He

I

lH2 C, and C, ketones C, and Cs ketones H2

fCF4

4 D,

lH2 c-C4H, propene He

m

(AE)/kJ mol-l or B c ca. 50 29 25 21 17 17 12 6 42

Ref.

b

105

29 17 15 17-25 > 25-33 pc = 0.2-0.3 (CF4 = 1) p, = 1 pc = 0.2 (Me2C:CMe, = 1)

> 19 9 6 3, = 0 . 1 4 . 2 d 6 29-54 (Pc = 0.02-0.03)

C

40 40 43 43

E?} EX

Photoactivation

Cycloheptatriene(CHT)

460

II"""'

389

{

Pc = 0.5 pc = 0.1 Pc = 0.06

{ZH4

pc = 0.62

420

214

CHT n-CbHig, PhMe neo-C,H,, CO, Xe, Ar, Kr Ne, H2, Dz He Hz, Nz, 0 2 , co CBH14, C&le

Cycloheptatrienek(CHT)

Ar,Kr, Xe, 0,

Cyclobutanone

218

H,, Dz, Ne He

17 11.6 10.2 5.0 3.3-3.5 2.4-2.6 1.8 2-3'

Bc = 1.3-1.7 B, = 0.73-0.86 Pc = 0.38-0.53 pc = 0.12-0.22 fir. = 0.08-0.26 (CHT = 1)

(propene = 1)

2,3-Dimethylcyclobutanone (DMCB)

385

218

Ar

BC= 0.26 pc = 0.22

(DMCB = 1) Thermal energization at low pressures CH3NC

C2H6NC

i

Cz-C, alkanes and alkenes MeCN, PrCN Ng, HCN, Cog H,, He-Xe

= 0.34 (MeNC = 1) = 1.00 f 0.05

0.93, 1.1 0.45-0.70 0.26-0.46 (EtNC = 1)

Table 3 (cont.) Energized species

Source

E,/kJ mo1-1

Bath gas

He

1,2-Dideuteriocyclopropane

3,3,4,4-Tetrafluorocyclobutene (TFCB)

.

(AE)/kJ mol-l or p c p, = 0.037 (CIHB= 1) 8, = 0.72 pc = 0.20 (TFCB = 1) p, = 0.51

Is,

CF,CF,NN (TFDA)

Cyclobutane

(E*)/kJ mol-l

(VLPP)

= 0.13 (TFDA = 1) p, = 0 . 1 - 0 . 2 (wall = 1)

Model"

I

Ref. 133 22

m 100

SC = strong collision, EX = exponential, SL = stepladder; parentheses imply assumption rather than demonstration of the model used; D. W. 0.8-4.9 ofcollisions elastic, see text; Setser and E. E. Siefert,J. Chem. Phys., 1972,57,3613,3623; Ref. 5 , p. 12; E. Jakubowski, H. S. Sandhu, and 0. P. Strausz, J. Amer. Chem. Soc., 1971, 93,2610; S. W. Orchard and B. A. Thrush, Proc. Roy. SOC.,1972, A329,233; V Estimated from correlation of is with AE> ;05 S. H. Luu and J. Troe, Ber. Bunsengesellschaft phys. Chem., 1973,77, 325; data quoted are examples from 44 bath gases studied; N. E. Lee and E. K. C. Lee, J. Chem. Phys., 1969,50,2094; j J. Metcalfe, H. A. Carless, and E. K. C. Lee, J. Amer. Chem. SOC.,1972, 94,7235; see also J. M. Carless and E. K. C. Lee, ibid., p. 7221; F.-M. Wang, T. Fujimoto, and B. S. Rabinovitch, J. Phys. Chem., 1972,76, 1935; for a compilation of values for other bath gases see S. C. Chan, B. S. Rabinovitch, J. T. Bryant, L. D. Spicer, T. Fujimoto, Y.N. Lin, and S. P. Pavlou, S. P. Pavlou and B. S . Rabinovitch, J. Phrs. Chem., 1971,75, 3037; J. J. Cosa, H. E. Gsponer, E. H.Staricco, J. Phys. Chem., 1970,74, 3160; and C. A. Vallana, J. C. S. Faraday I , 1973, 69, 1817. a

<

Unimolecular Reactions

119

ular reactions at low pressures, Troe has developed an analytical correlation between b, and { A E } for a flexible dependence on E and E’ of the transition probability P(E’, E) between two states of energy E and E’, and this is expected to lead to easier correlations between thermal and chemical activation data. Also on the theoretical side, some progress has been made in the treatment of collisional energy transfer in terms of the statistical distribution of internal energy in the phase-space of the collision ~ o m p l e x g7. ~ For ~ ~ example, the high de-energization efficiency of propene relative to argon in deactivation of /3-naphthylamine(ca. 9 : 1 measured by phase-shift fluorimetry) is attributable almost entirely to the greatly increased phase-space available for energy redistribution.gs A survey of the available experimental data (Table 3) reveals a general agreement between the majority of { A E } values obtained in different studies, these ranging from 2 to 6 kJmol-l for the weakest collider (helium) to >25 kJ mol-1 for the largest molecules (the upper limit not being well defined). Two notable exceptions occur, however, for methylcyclobutane (studied by Frey et U Z . ~ ~ and ) methylcyclopropane (studied by Topor and Carr 9g). In both systems the thermal reaction is well characterized, so that RRKM calculations could be reliably based on models fitting the experimental A , and fall-off behaviour. For the overall reaction (15), the known 959

CH,CO

+ hv (325 nm) + c-C4Hs -+ CO + c-C4H,CH3*

(15)

thermochemistry permits calculation of the energy E* of the methylcyclobutane as 427-464l~Jmol-~,the limits corresponding to none or all of the 37 kJ mo1-l excess photolytic energy being carried into the energized product. By contrast, the E* required to reproduce the experimental decomposition rate on a strong-collision basis was about 502 kJ mol-l, some 38 kJ mol-1 greater than the maximum possible from the thermochemistry.sg In the similar system involving CH, plus p r ~ p e n ethe , ~ ~uncertainty concerning the excess photolytic energy was avoided in one set of results using 355 nm radiation, which is only just energetic enough to dissociate keten. On a virtually-strong-collision basis ( { A E ) = 42 kJ mol-l), the energy E* required to produce the experimental D / S data would again be impossibly high (427 kJ mol-l, which is 34 kJ mol-l higher than the estimated E* and would require an impossible heat of formation of lCH2). Another way of expressing these discrepancies is to note that the apparent (strong-collision) rateconstant o D / S is considerably less than the RRKM rate-constant calculated with apparent reliability from the thermochemistry. Since the experimental *6

p6 97

DS

J. Troe, Ber. Bunsengeseffschaftphys. Chem., 1973, 77, 665. H. von Weyssenhoff and E. W. Schlag, J. Chem. Phys., 1973,59, 729. Y. N. Lin and B. S. Rabinovitch, J. Phys. Chem., 1970,74,3151; K. Shobatake, S. A. Rice, and Y . T. Lee, J. Chem. Phys., 1973,59, 2483. H. M. Frey, G. R. Jackson, M. Thompson, and R. Walsh, J. C. S. Faraday I , 1973,69, 2054.

gg

M. G . Topor and R. W. Carr, J . Chem. Phys., 1973,58,757.

120

Reaction Kinetics

D / S ratios are also believed to be reliable, it appears that the collision frequency w must be an overestimate of the de-energization rate. Topor and Carr proceeded by fitting to their data a stepladder model, varying (AE) to obtain the best fit to the data with ( A E ) M 6 kJ mol-l, an abnormally low value by previously accepted standards. It is noteworthy that the theoretical plots of D / S against 1/ p are markedly curved, corresponding to the ‘turnup’ of the plots of o D / S against S / D (or w) at low The experimental data, on the other hand, would fit at least as well to a straight line. The data of Frey et al. are less scattered and define a line which is definitely not as curved as would be required for a stepladder model with ( A E ) low enough to fit the observed decomposition rate. The results therefore suggest a low collision efficiency arising from the transfer of relatively large amounts of energy on a relatively small fraction of the collisions, for example 1 = 0 . 1 4 . 2 . Similar collision efficiencies seem to be required to fit cyclobutane energization at very low pressures, measured by VLPP loo relative to a wall energization efficiency of unity, but this interpretation is not at present reconcilable with the higher pressure measurements. In contrast to the above conclusions there is a substantial body of evidence (Table 3 and ref. 93) indicating essentially strong collision behaviour in bath gases of the complexity of C3 and C, molecules, and Rabinovitch et al. have shown lo**g4 that in a number of weak-collider systems the data over a wide range of pressures are consistent with a small for all collisions rather than a large AE for occasional strong collisions. The methylcyclopropane and methylcyclobutane systems are sufficiently well defined that the discrepancies will need to be resolved before a reliable overall picture can be accepted; in particular, an extension of Frey’s work to lower pressures might be helpful. In the meantime, results extracted from such systems using the strong-collision assumption must be treated with caution (e.g. the methylcyclopropane data mentioned in Section 5). 5 Other Chemical Activation Studies In these studies, energized molecules are produced in a relatively narrow range of energies and their rate of reaction measured relative to collisional stabilization (or some other reference rate process). Most of the examples involve energization by an exothermic chemical reaction, but a number of cases involve both chemical and photolytic energy. The narrow energy distribution makes possible much more searching comparisons with theoretical models of the reaction than are possible from the broadly averaged rates measured in thermal energization systems. Various aspects of chemical activation have been reviewed in detail by Setser,6and in the present review loo

P. C. Beadle, D. M. Golden, K. D. King, and S.W. Benson, J. Amer. Chem. SOC.,1972,

lol

94, 2943. B. S. Rabinovitch, H. F. Carroll, J. D. Rynbrandt, J. H. Georgakakos, B. A. Thrush, and R. Atkinson, J. Phys. Chem., 1971, 75, 3376.

Unimolecular Reactions

121

we have already discussed the specially important aspects concerned with intramolecular energy relaxation and collisional energy transfer. In this section we deal relatively briefly with applications in which RRKM theory is taken as an accurate method for the calculation of k(E), and unit collisional de-energization efficiency is generally assumed. This last assumption may be less valid than is often thought (see Section 4), and to this extent some conclusions of the present section may be suspect. In reviewing this work, attention is concentrated on the broad types of use to which chemical activation studies can be put, rather than on the detailed findings for particular systems. The input data for calculation of an observable chemical-activation rate-constant
Input Data for Calculation All Avahble.-The ideal sources of information are reliable thermochemistry for the forming reaction, and good highpressure and fall-off data for the thermal reaction, leading to a reliable value for the critical energy and an activated-complex model consistent with the fall-off data (precise details of vibrational assignments being relatively unimportant provided AS* is correctly predicted 21). In the absence of experimental data, analogy with other similar systems (often by group-additivity estimates 26 of enthalpies and entropies) can often provide reliable values, the most difficult parameter being in general the critical energy. Often a series of chemically activated reactions can be ‘calibrated’ by fitting a model to one member of the series then keeping the arbitrary part constant for the rest of the series; good examples are the secondary alkyl radical series p8 and isotopically substituted fl~oroa1kanes.l~~~ lo4 The output of such calculations is a rate-constant for comparison with experiment, and satisfactory agreement is taken to provide support both for the parameters used and for the theoretical basis of the calculations. Other examples are provided by lo2 lo* lo4

Ref. 1, Ch. 8. B. D. Neely and H. Carmichael, J . Phys. Chem., 1973,77,307. K.C.Kim, D. W. Setser, and B. E. Holmes, J. Phys. Chem., 1973,77,725.

122

Reaction Kinetics

systems involving chemically activated fluoroalkane~,~~9 lo5* l O 6 polychloroethyl radicals,1o7 1,l-dichloro~yclopropane,~~* and hexyl radicals,lo0 the prediction of quenching cross-sections of Hg 6(3P1) and (3P0)by alkanes (by RRKM treatment of the Hg-H-R* complex)11o and the prediction (in agreement with experiment) that Si,H,* formed from 2SiH, would virtually all decompose to SiH, + SiH, at pressures below 1 atmosphere.lll In the case where the chemical energization occurs in a themal unimobcular reaction, the advantage of a narrow energy distribution is largely lost, but calculations are still able to give a useful prediction of the observed variations of product yields with pressure. For example, the reaction shown in Scheme 8 yields cyclohexa-1,4-diene with sufficient energy to undergo

Scheme 8

dehydrogenation unless stabilized by collision. The observed product varies from 88% cyclohexadiene at 8.4 Torr to 85% benzene at 0.3 Torr, and RRKM calculations based on minimal experimental information predicted the pressure variation satisfactorily apart from a shift of 0.45 units on the log,,(p/Torr) axis.112, Similarly, the production of both 2-methyland l-methyl-cyclopentadiene in the isomerization of l-methylbicyclo[2,1,0]pent-2-ene can be satisfactorily interpreted in terms of a scheme involving the chemically activated 1-methyl derivative as the sole primary product,l14and similar calculations have been carried out for spiropentane 115 and methy1spiropentane.ll6

Studies to Determine the Critical Energy Eo.-In some cases the energy E* of the chemically activated species can be accurately estimated from known 10s

106 107

108

109 110

111

112 11s 114

115

116

H. W. Chang, N. L. Craig, and D. W. Setser, J. Phys. Chem., 1972,76, 954. A. S. Rodgers and W. G. F. Ford, Znternat. J . Chem. Kinetics, 1973, 5, 965. G. B. Skinner and B. S. Rabinovitch, Bull. SOC.chim. belges, 1973, 82, 305. D. Seibt and H. Heydtmann, 2. phys. Chem. (Frankfurt), 1973,83,256. K. W. Watkins, J. Phys. Chem., 1973, 77, 2938. A. C. Vikis and H. C. Moser, J. Chem. Phys., 1970,53,2333. T . L. Pollock, H. S. Sandhu, A. Jodhau, and 0. P. Strausz, J. Amer. Chem. SOC.,1973, 95, 1017. M. C. Flowers, H. M. Frey, and H. Hopf, J . C. S. Chem. Comm., 1972, 1283; see also ref. 113. J. E. Baldwin and J. Ollershaw, Tetrahedron Letters, 1972, 3757. M. C. Flowers and H. M. Frey, J . Amer. Chem. SOC.,1972,94, 8636. M. C. Flowers and A. R. Gibbons, J. C . S. Perkin. ZZ, 1972, 548. M. C. Flowers and A. R. Gibbons, J. C. S. Perkin. ZZ, 1972, 555.

123

Unimolecular Reactions

thermochemistry; this is often true for example in the production of energized alkyl radicals by addition of H to olefins, or of energized alkanes by radical recombination. If in addition it is possible to construct reliable vibrationalrotational models of the reactant and activated complex, the critical energy Eo can be adjusted to obtain agreement of the calculated (k(E)> with the experimental CUDIS.This approach was much in evidence in earlier work on fluoroalkanes,lo2and has been applied to the isomerizations of n-pentyl 117 and 3-methylbut-lenyl 11* radicals and the elimination of H F from CH3NF2.11@ In other recent studies Eo was determined for the a,a- (carbeneforming) eliminations which are concurrent with the better-documented a,& (4-centre) eliminations in CH2C1CDC12120 and CD3CHF2,104 e.g. reactions (16)--(20). The a,a-elimination in each case had Eo higher by about 8 kJ mo1-1 than the a,&process.

CH,Cl + CDCl,

-

CH,ClCDC1, w/7

CH,ClCDCl,*

' -CH,:CCI, + DCl

(16)

(17)

CHC1:CDCl + HCl

(18)

CH,ClCCl + DCl

(19)

CHC1:CHCl

(20)

Studies to Determine the Energization Level E*.-When E,, is known and vibrational-rotational assignments can be made reliably, RRKM theory can be used to determine the energy level of the chemically activated molecules which fits the experimental CUDIS.This approach is at its best when A , and E m are known from good thermal studies, the molecule and complex models being selected to fit the experimental A S', and Eo following to be consistent with the observed E m . 2 1 Such studies therefore give information about the thermochemistry of the energizing reaction. An important example concerns chemical activation by reaction of photochemically generated species. Even though the basic thermochemistry may be known, part of the excess energy of the photolysis step appears as internal energy of the photolysis fragments, and the proportion is not calculable a priori. A common example concerns methylene generated by photolysis of diazomethane or keten (cyclopropane and propane have also been used 121). Insertion into alkanes (or addition to olefins) gives chemically activated molecules, and for their reactions Eo and activated-complex models can be constructed with reasonable reliability; e.g. reaction (21), where accurate thermal kinetics are available for methylcyclopropane. In recent studies, with oxygen present to trap 3CH2,the D / S ratio and hence ( k ( E ) ) was measured at different photolysing wavelengths and the variation used to determine the excess energy carried K. W. Watkins and D. R. Lawson, J. Phys. Chem., 1971,75, 1632. K. W. Watkins and L. A. O'Deen, J . Phys. Chem., 1971,75, 2665. D. S. Ross and R. Shaw, J. Phys. Chem., 1971,75, 1170. l P o K. C. Kim and D. W. Setser, J. Phys. Chem., 1972,76, 283. A. K. Dinghra, J. H. Vorachek, and R. D. Koob, Chem. Phys. Letters, 1971,9, 17. 11'

118

124

Reaction Kinetics

bu tenes

(22)

by the methylene into the energized The dependence of such results on the accuracy of the strong-collision assumption should be noted, however (see the discussion in Section 4). Other examples are excited propene from diazopropane p h o t ~ l y s i s ,cyclopropanes ~~~ from pyrazoline photol y ~ i sand , ~ ~ethylene ~ from photolysis of C,-C, paraffins.126 An important aspect of such studies is that they can be used to determine the excess energy carried by CH, under particular conditions, this information then being available to calculate E* for other reactions under similar conditions; this is the ‘E* Monitor’ method referred to in the next section. Similar applications can be made to determine the energy level of hot-atom substitution products 126 but the complications referred to in Section 3, p. 103 need to be noted.

Studies to Determine &,.-In this final case, knowledge is assumed of the energetics of formation of the chemically activated species and the critical energy for its decomposition. The latter information is most likely to be available for bond-fission processes where E,, can be calculated from the heats of formation of the radical products, and the approach has been extensively applied to the decomposition of chemically activated alkanes and silanes. In the case of energization by methylene insertion, allowance must be made for the excess photolytic energy carried by the CH2, and can be determined as noted above by comparison with a well-documented reference reaction. A representative example is a study of chemically activated ethane formed as in Scheme 9.12’ Since measurement of the decomposition to

c

CH,N, + hv

+

y czH6*

2CH,

CH,* i

G

a CH4

4

i-C,H,,*

A i-C,H,,

Scheme 9 lZ2 12J

lB4

lZ6 lZ6

12’

G. B. Kistiakowsky and B. B. Saunders, J. Phys. Chem., 1973, 77, 427. A. U. Acuna, J. M. Figuera, and V. Menendez, Anales de Quim., 1973, 69, 181. E. B. Klunder and R. W. Carr, J. Amer. Chem. SOC.,1973,95, 7386. H. Akimoto, K. Obi, and I. Tanaka, Bull. Chem. SOC.Japan. 1973,46,2267. See for example R. W. Weeks and J. K. Garland, J. Amer. Chem. SOC.,1971,93,2380; R. Kushner and F. S. Rowland, J . Phys. Chem., 1971,75,3771; G . Izawa, E.K.C. Lee and F. S. Rowland, ibid., 1973, 77, 1210. F. B. Growcock, W. L. Hase, and J. W. Simons, Znternat. J. Chem. Kinetics, 1973,5, 77; for other recent data on ethane see G. 0. Pritchard, P. C. Kobrinsky, and S. Toby, J . Phys. Chem., 1971,75, 2225, and F. R. Cala and S. Toby, ibid., p. 837.

125

Unimolecular Reactions

2CH3 would be impracticable, the required D / S data were obtained by comparing the rate of formation of C2H6 (S)with an internal reference reaction, the insertion of CH2 into isobutane to give isopentane, which is all stabilized under the conditions of the experiment. The rate-constant ( k ( E ) was found to be 4.6 x loQs-l. The energy (E*> of the C&* was calculated to be 481 kJ mol-1 from equation (23), in which & is a calculated correction

>

€or thermal internal energy of the CH, (7.5 kJ mol-l), and E*(lCH2) is the excess photolysis energy carried by the CH, into the chemically activated molecule. A value for the combined term [AH,",(lCH,) + E*(lCH2)] was obtained from a parallel study of an 'E* monitor system', the isomerizations of 1,2-dimethylcyclopropane in the CH,N2-but-2-ene system; note that a separate knowledge of AH;b(CH,) is not required. The critical energy was calculated from the CH3-CH3 bond-dissociation energy to be 359 kJ mo1-1 and activated-complex models giving the correct ( k ( E ) ) were found to correspond to log,,(A,/s-l) between 16.46 and 16.69, in reasonable agreement with a number of direct thermal measurements. Similar A, factors have been derived for alkenes 12*and for a whole series of alkanes.12QThese studies have confirmed the previous conclusion that the data for alkane dissociation and radical recombination can scarcely be reconciled with the conventional type of activated complex hitherto postulated. However, a more sophisticated treatment 130 is successful for ethane (see Section 7, p. 155), and may be fruitful for other alkane systems. Studies on chemically activated silanes l3l,132 formed by methylene insertion have led to A, values around 1015s-l for Si-C bond fission, which is significantly less than the values for C--C fission. In recent studies of chemically activated alcohols [from O(lD2) alkanes] and the isomerization of n-alkyl and alkenyl radicals, Forst's equation [equation (l)] has been used to determine A , ; this is particularly convenient since the trial-and-error aspect of fitting activated-complexmodels is eliminated. However, as noted in Section 2, the equation is not strictly applicable for calculating k ( E ) at a specific energy, only the thermal average.

+

G. W. Taylor and J. W. Simons, Internat. J. Chem. Kinetics, 1971,3, 453. W. L. Hase, R. L. Johnson, and J. W. Simons, Internat. J . Chem. Kinetics, 1972,4, 1 ; W. L. Hase and J. W. Simons, J. Chem. Phys., 1971,54,1277; F. B. Growcock, W. L. Hase, and J. W. Simons, J. Phys. Chem., 1972,76,607; J. A. Bell, ibid., 1971,75,1537. 13@ W . L. Hase, J . Chem. Phys., 1972, 57, 730. lal W. L. Hase, C. J. Mazac, and J. W. Simons, J . Amer. Chem. SOC.,1973, 95, 3454; W. L. Hase, W. G. Brieland, P. W. McGrath, and J. W. Simons, J. Phys, Chem., 1972, 76, 459. 132 W. L. Hase and J. W. Simons, J. Organometallic Chem., 1971, 32, 47.

lZ8 lZ9

Reaction Kinetics

126

6 Thermal Unimolecular Reactions in the Fall-off Region

During the review period there have been relatively few reports of studies giving accurate information about unimolecular fall-off behaviour (see Table 4). Data on small molecules ( < 4 atoms) are omitted, having been recently reviewed by Troe and W a g n e ~ .The ~ work of Waage and Rabinovitch on trans-l,2-dideuteriocyclopropane133 explores the potential usefulness of measuring-thefall-off of two competing reaction channels relative to each other (cf. multichannel chemical activation experiments). The rates of both geometrical and structural isomerization were measured in the lower fall-off region with and without helium present, and the relative rates can be correlated with details of the collisional energy-transfer process without using any independent measurement of collision cross-sections. Frey et aL2* found, in agreement with previous conclusions, that fall-off curves predicted for the isomerizations of 3,3,4,4-tetrafluorocyclobutenewere insensitive to even gross changes in the activated-complex model provided A S * had the correct value. The ‘Forst procedure’ (see Section 2) gave results in close agreement with full RRKM calculations and the calculated curve (strongcollision) deviated from the experimental results by at most 0.2 units on the log,,(p/Torr) axis (collision efficiency A 2 0.6). Shock-tube studies of unimolecular reactions are frequently complicated by fall-off, and corrections to obtain high-pressure rate-constants usually have to be made on a theoretical basis (see, for example, Section 7, pp. 132 and 155). The studies of Glanzer and Troe on nitro-alkanes (particularly nitromethane 134) are unusual in providing a substantial part of the actual fall-off curves under conditions very different from those studied in conventional systems. Further data of this sort would provide a very useful extension to the type of result available for tests of the predictive ability of current theories. The technique of VLPP (very low-pressure pyrolysis) has many interesting applications to unimolecular reactions, and a recent review by Benson and co-workers describes the method and its applications and provides a key to most of the published work. Although the reactants are usually energized solely by collision with the walls, it is believed that there is only rarely any chemical reaction with the walls. Thus the situation is analogous to that in a gas undergoing unimolecular reaction in the second-order region, and the measured product compositions give the low-pressure second-order rateconstant for the reaction. High-pressure Arrhenius parameters cannot be deduced unambiguously, but if A, is known or can be reliably estimated, E , can be varied to obtain a fit of the experimental rate-constant-temperature relationship. In all cases but one this method has given results in good agreelaa

lS6

E. V. Waage and B. s. Rabinovitch, J. Phys. Chem., 1972,76, 1695. K. Glanzer and J. Troe, Helv. Chim. A d a , 1972, 55, 2884. D. M. Golden, G. N. Spokes, and S. W. Benson, Angew. Chem. Znternat. Edn., 1973, 12, 534.

s

Table 4 Experimental studies of unimolecular reactions in their fall-of regions Reactant

Product ( s )

Jcis~compound propenes 3,3,4,4-tetrafluorocyclobutene l,lY4,4-tetrafluorobuta-l,3-diene methylenecyclobutane*1 ' spiropentane (allene ethylene?) J methylspiropentane (similar) trans-[2H2]-cyclopropane

1

+

CF2CF2N:N

+

CH,CH F2 CH&O

CH,:CHF HF CH, CO

Shock-tube studies: CHsNO2

CH,

+

+ NO2

Temperature"1K 753 607

621-660

0.001-lo00 (663 K) 0.005-6

393434

5-1 50

643-703

798 507

1000-1400 1loo 1650-1950

CH3F N2Fl

+

CH, HF 2NF2

{

Pressure rangelTorr 0.0004-0.003 0 . 0 2 4 in He 0.005-70

1820-2220 3514

0.2-9 41-720

2-001

P*lTorr (obs.)

}

Theoretical treatment

Ref.

-

RRKM

133

0.03

RRKM, Forst

22

(A = 0.25) RRKM (A = 0.15)

116

0.7 (663 K) 0.01 (642 K) 4 (414 K) 6 (425 K) 2.5 480"

x 104

690-6900 418-2645 2400-3 x 1 0 4 780-7300'

ca. 6 x lo4(&) (lo00 K) ca. 7000 (Ar) pi ca. 2200 Ar (1790 K) 1800(Ar, 1900K) ca. 2200 (Ar)

-

C

RRK,RRKM Forst

d 193

-

134

-

f

RRK

g

(Slater) RRKM

j

8

2

g @

8

h

+

a Temperatures at which useful fall-off data reported; Allene ethylene formed from chemically activated methylenecyclobutane and possibly also directly from spiropentane; J. J. Cosa, H. E. Gsponer, E. H. Staricco, and C. A. Vallana, J. C. S. Furuduy I , 1973, 69, 1817; B. Noble, H. Carmichael,and C. L. Bumgardner,J. Phys. Chem., 1972,76,1688; Equivalent pressure of strong collider (some weak-collision corrections made); f K. Glanzer and J. Troe, Helv. Chim. Acta, 1973,56,577; J. F. Bott, J. Chem. Phys., 1971,54, 181; K. P. Schug and H. Gg. Wagner, Z. phys. Chem. (Frankfurt),1973, 86, 59; Corrected to 351 K from data at various temperatures; f E. Tschuikow-Roux, K. 0.MacFadden, K. H. Jung, and D. A. Armstrong, J. Phys. Chem., 1973, 77, 734.

c.

tQ

4

128

Reaction Kinetics

ment with conventional higher-pressure measurements. Of particular interest is the work with cyc1obutane,lo0where gas-gas collisions were allowed to become significant and an estimate was obtained of their energization efficiency, relative to wall collisions. Again the measurement of such properties under ‘unusual’ conditions promises to be fruitful, although the data in this particular case are unfortunately inconsistent with higher-pressure studies. Other very recent papers on VLPP involve the pyrolysis of cyclohexa-1,3-diene,136giving results consistent with the mechanistic conclusions of higher-pressure studies (see Table 8), and the 1,4-elimination of hydrogen from cis-but-2-ene,ls7giving an estimated E , of 272 kJ mol-l. 7 Thermal Unbnolecular Reactions in the High-pressure Region This section deals with kinetic and to some extent mechanistic studies of reputedly unimolecular reactions in their first-order rtgimes. It is always difficult to draw the line between well-authenticated unimolecular reactions and others for which a unimolecular mechanism is plausible but less well established. The most convincing evidence is provided by gas-phase studies over a wide range of pressures and temperatures, with and without additives present, and in vessels of differing surface-to-volume ratio; such studies are likely to provide meaningful Arrhenius parameters for the elementary reactions involved. Studies in solution can also be reasonably thorough, particularly if the concentration is varied over an adequate range and changes of solvent are found to have little effect on the rate. The weakest rate data are provided by studies involving a small number of measurements, often on neat liquids, over a limited temperature range, with first-order behaviour demonstrated only for the course of individual runs. The decision as to which data to include in a compilation is necessarily subjective; the reactions listed here include those which the reviewer feels to be reasonably well authenticated as clean unimolecular reactions, and in addition some which are less well established but illustrate mechanistic points. The classification into reaction types is also difficult and sometimes rather arbitrary, and for ease of comparison with the earlier literature the presentation follows reasonably closely the layout used by Robinson and H01brook.l~~ It should be noted that the data given are only those reported during the review period, and not necessarily the only or the best data for a particular reaction. Where necessary the tabulated parameters have been calculated from the authors’ data using equations (24) and (25) 21 (the statistical factor L* being taken as l), or occasionally from reported rate-constants. E, log,,(A,/s-l) lS6

lS8

=

AH* -I- RT

=

10.753

+ log,,(TyK) + A.S*/19.144 J K-l

(24) mol-1

(25)

2. B. Alfassi, S. W. Benson, and D. M. Golden, J. Amer. Chem. Soc., 1973,95,4784. Z. B. Alfzsi, D. M. Golden, and S. W. Benson, Internat. J. Chem. Kinetics, 1973, 5 , 991. Ref. 1, Ch. 7.

Unimolecular Reactions

129

Cyclopropane and its Derivatives (Table %.-Several new investigations on the isomerization of cyclopropane have been published, including four shock-tube studies which are of particular interest since they involve much higher temperatures (860-1870K) than the earlier work in static systems (690-810K). Comparisons are complicated by the fact that at the high temperatures obtained in shock tubes the isomerization is significantly in the fall-off region even at pressures of several atmospheres. Corrections are therefore necessary to obtain k, values, and these have generally been somewhat empirical, e.g. based on RRK theory with s = C,,,/R. Jeffers et al. have recently summarized most of the available data 139 (see Figure 3) and concluded that up to 1300 K the shock-tube data are in satisfactory agree\

\

4

2

-

0

I

\

c

\

\ VI

cP

\

\

cn

2 -2

\

-4

-6 I

1

I

I

8

10

12

14

lo4 KIT

Figure 3 Arrhenius plot for cyclopropane i s o m e r i ~ a t i o n . ~The ~ ~ high-temperature results are, from top to bottom, the two sets of results of Jefers et al.,laSand those of Dorko et a1.148and Bradley and Frend;14=Barnard and Seebohm's results are not shown. The low-temperature results are those of earlier workers; the highpressure flow results of Johnson et al.las(not shown) lie in the middle of this band. laS

P. Jeffers, D. Lewis, and M.Sam, J. Phys. Chem., 1973, 77, 3037.

Table 5 Recent high-pressureArrhenius parametersfor reactions of cyclopropane derivatives

CI

ba

ReJand Reactant Biradicalring-opening

Product(s)

Cyclopropane

propene

Cyanocyclopropane

7 major >14 minor products 2,4-dimethyl-pen t-2-ene ally1 cyanide cis-crotononitrile trans-crotononitrile overall 3 pentenonesd [CH,CH:CHNH,] -+ obs. products

+

Acetylcyclopropane Cyclopropylamine [N-2H2]Cyclopropylamine Concerted ring-opening Pentachlorocyclopropane

1,1,3,3,3-pentachloropropene

trans-1,l-Dichloro-2,3-dimethylcyclopropane

3-chloropenta-l,3-diene + HCP

cis-1-Chloro-cis-2,3-dimethylcyclopropane 1-Chloro-trans-2,3-dimethylcyclopropane

penta-l,3-diene HC1' penta-l,3-diene+ HCl'

Ring-openingof cyclopropenes ( f)-1,3-Diethylcyclopropene

3-Methylcyclopropene

trans-3,4-dichloropent-2-ene

+

1trans-4-chloropent-2-ene

+

hept-3-yne hepta-2,4-dienes" but-1-yne(+ buta-1,3-diene)

loglo(Arn/s-l) E,/kJ mol-1 comments

14.2 253 11.9 230 14.5 272 14.8 268 14.5 265 15.3 280 15.5 277 14.5 226 14.8 260 14.6 14.0 14.1 14.6 244 242 14.4 242 15.1 kH/kD= 1.06at 649-678 K

=I

14.2 14.1 (14.1)J 13.9 13.8 14.6

168

200 1 (207)J j 200 1901 2001

144 145 143 139 139 141 142 a b C

e

f g

h k

147 147

kl

8

2

g3

10.4 13.5

135 158

n 0

g

2'

3-methylbut-1-yne 2-methylbuta-l,3-diene) 2-methyl penta-1,3-dienes (+ 2,3-dimethylbuta-l,3-diene) 2,3-dimethylpenta1,3-dienes (+ 3,4-dimethylpenta-l,3-diene)

13.0

154

13.4

164

12.5

168

trans-compound

12.4

138

P

144 138}

4

11.5

137

n

12.5 13.4 13.4

162 172 187 108 197" 213

(

1,3,3-Trimethylcyclopropene

+

Geometricalisomerization

12.1

Et (-)\

or (+)-compound'

Et

Ring-expansionreactions trans-1-Methoxy-2-vinylcyclopropene trans-1-Phenyl-2-vinylcyclopropane 1-Methoxy-1-vinylcyclopropane 1-Methylene-2-vinylcyclopropane 1-Ethylidene-2-vinylcyclopropane (syn and anti) 1,l-Divinylcyclopropane

4-methoxycyclopentene 4-phenylcyclopentene 1-methoxycyclopentene 3-methylenecyclopentene

J 3-ethylidenecyclopentene

1+ 4-methyl-3-methylenecyclopentene 1-vinylcyclopentene

11.5

}

11.1" 11.7 12.6

166

t L

V

w

CI

w

Table 5 (cont.) Reactant Cyclopropyl methyl ketone N-Propylidenecyclopropylamine

w

Product (s) 2,3-dihydro-5-methylfuran* 5-ethyl-1-pyrroline

logl,(A,/s-l)

E,/kJ rno1-l

13.9 14.1

231

ReJand comments e

200

W

12.8

128

X

11.8

104

Y

h,

Miscellaneous

D D2

4 CHZ ( S e e also Table 8 for Cope-type rearrangements of cyclopropane derivatives) (I

H. E. O'Neal and D. Henfling, Internat. J. Chem. Kinetics, 1972,4, 117;

Shock-tube work agreeing with earlier static studies; W. Tsang, Internat.

J. Chem. Kinetics, 1973,5 , 651; D. A. Luckraft and P. J. Robinson, Internat. J. Chem. Kinetics, 1973,5, 137; See also Ring-expansion section . of this table; A. T. Cocks and K. W. Egger, J. C. S. Perkin I I , 1973, 197, 199. f K.A. W. Parry and P. J. Robinson, Internat. J. Chem. Kinetics, 1973,5,27; D. A. Luckraft and P. J. Robinson, Internat. J. Chem. Kinetics, 1973,5,329; J. C . Ferrero, J. J. Cosa, and E. H. Staricco, J. C. S. Perkin I I , 1972,2382; Via ci5 -3,4-dichloropent-2-ene?;j Evaluated from reported product distribution and Arrhenius parameters for HCl elimination; R. P. Clifford and K. A. Holbrook, J . C. S. Perkin II, 1972, 1972; Via cis-4-chloropent-2-ene?; See also under Geometrical isomeriE. J. York, W. Dittrnar, R. J. Stevenson, and R. G. Bergman, J. Amer. Chem. SOC.,1973, 95, 5680; R. Srinivasan, Chem. 8 zation in this table; Cornm., 1971,1041; p Neat liquid; T. Sasaki, S. Eguchi, and M. Ohno, J. Org. Chem., 1972,37,466; M. Arai and R. J. Crawford, Canad. J. Chem., 1972,50,2158; ' See also under Ring-opening reactions in this table; J. M. Simpson and G. H. Richey, Tetrahedron Letters, 1973, 2545; Also 3 in solution; W.E. Billups, K. H. Leavell, E. S. Lewis, and S. Vanderpool, J. Amer. Chem. SOC.,1973,95,8096; Parameters for syn- and anti-isomers, Benzene solution, isotope effect also studied; W. R. Dolbier and J. H. Alonso, J. Amer. Chem. SOC.,1972, 94, 2544; but assignment not certain; A. T. Cocks and K. W. Egger, Internat. J. Chem. Kinetics, 1972,4,169; In decalin; 1. H. Sadler and J. A. G. Stewart, J. C. S. Perkin I I , 1973,278; ii' Y J. C. Gilbert and D. P. Higley, Tetrahedron Letters, 1973, 2075. h

2.

'

133

Unirnolecular Reactions

ment with the low-temperature work. Their own work in particular comprised two independent studies by the competitive shock-tube method, using different standard reactions and rather different conditions. The competitivc technique avoids much of the uncertainty associated with the temperature estimation,140and the results are in strikingly good agreement. Another briefer study using a third reference reaction was also in reasonable agreement,141and measurements at 0 . 6 1 3 7 atm in a flow system also agreed with previous The shock-tube measurements of Dorko et and of Barnard and Seebohm 144 are also very similar in this temperature range, although those of Bradley and Frend 145 are substantially lower. Dorko et studied i.r. emission and absorption from the v 5 + vl0 band of cyclopropane, and emission from the C = C stretch of propylene. Since absorption decreased continually from t = 0, while emission built up to a maximum then decayed, they concluded that vibrationally excited intermediates were involved, and fitted their data to Scheme 10. The rate parameters deduced

v v* v*+&

+Ar

+Ar

-

V**

V**+Ar

A

Scheme 10

are open to criticism, however.130 At temperatures above 1300 K both Bradley and Frend 145 and Barnard and Seebohm 144 have observed a sharp falling off of the Arrhenius plot; the former workers gave the experimental Arrhenius equation loglO(k/s-') = 4.75 - 48.5 kJ rnol-l/2.303 RT for the range 1350-1800 K. The results below 1300 K also fall consistently below the extrapolation from the static work, which may possibly indicate the onset of the same phenomenon. There are also similar effects at unduly high reactant concentrations, and one must conclude that there are still unappreciated factors involved in this reaction at high temperatures. Studies of substituent effects on the basic cyclopropane isomerization have been widely interpreted in terms of the biradical mechanism and have provided data on the stabilization of such radicals by substituent groups. W. Tsang, J. Chem Phys., 1965,43, 352; 1966, 44, 4283; 1967, 46, 2187; etc. P. Jeffers, C. Dasch, and S. H. Bauer, Internat. J . Chem. Kinetics, 1973, 5 , 545. l b 2 D. W. Johnson, 0. A. Pipkin, and C. M. Sliepcevitch, Ind. and Eng. Chem. (Fundamentals), 1972, 11, 244; 1973, 12, 262; D. M. Ruthven, ibid., p. 262. E. A. Dorko, D. B. McGhee, C. E. Painter, A. J. Caponecchi, and R. W. Crossley, J. Phys. Chem., 1971, 75, 2526. 144 J. A. Barnard and R. P. Seebohm, Symposium on Gas Kinetics, Szeged, Hungary, 1969. 145 J. N. Bradley and M. A. Frend, Trans. Faraday. SOC.,1971, 67, 72; note correction in ref. 139. lQ6 E. A. Dorko, R. W. Crossley, V. W. G r i m , G. W. Mueller, and K. Scheller, J . Phys. Chem., 1973,77, 143. 140 141

134

Reaction Kinetics

The isomerizations of chlorocyclopropanes are concerted processes, however, and further work (Table 5 ) has provided strong evidence for a semiionic reaction in which opening of the incipient cyclopropyl cation to an ally1 cation is controlled by Woodward-Hoffmann considerations, the allowed reaction being disrotatory with inward rotation of groups cis to the leaving halide. The relative rates of pyrolysis of various isomers correlate well 14' with the rates of ionic reactions such as the acetolysis of t~sylates.~** Isomerization in some cases produces chloro-olefins which can decompose to the observed products by rapid six-centre HCI elimination, e.g. reaction (26), particularly if chemically activated.

+He1

Racemization and isomerization of suitable cyclopropane derivatives (Table 5 and refs. 149, 150) have generally provided results compatible with biradical mechanisms and have continued to stimulate discussion of the precise nature of the biradical intermediates; the details are too involved to be discussed here. Useful reviews of theoretical aspects have appeared relS2 Substituent effects in the ring expansion of vinylcyclopropanes cently.lS1~ have been further studied (Table 5), and an interesting development is the discovery of similar unimolecular reactions for cyclopropane derivatives with double bonds involving heteroatoms, e.g. reaction (27). NzCH-Et

b

Et

-u /

(27)

Among the reactions listed as 'miscellaneous' in Table 5 , the methylenecyclopropane rearrangements have attracted much attention. Various P. J. Robinson and M. J. Waller, to be published. U. Schollkopf, Angew. Chem. Internat. Edn., 1968, 7 , 588. lQO J. M. Simpson and G . H. Richey, Tetrahedron Letters, 1973, 2545; N. E. Hare, E. W. Yankee, and D. J. Cram, J . Amer. Chem. SOC.,1973, 95, 4230; A. B. Chmurny and D. J. Cram, ibici.,p. 4237; T. Sasaki, S. Eguchi, and M. Ohno, J. Org. Chem., 1972, 37,466; M . Arai and R. J. Crawford, Canad. J. Chem., 1972,50,2158. lb0 C. Ullenius, P. W. Ford, and J. E. Baldwin, J. Amer. Chem. SOC.,1972, 94, 5910. 151 R. G. Bergman, Free Radicals, 1973, 1, 191. 152 L. Salem, Pure Appl. Chem., 1973,33, 317; L. Salem and C. Rowland, Angew. Chem. Internat. Edn., 1972, 11, 92; see also H. Kollmar, J . Amer. Chem. SOC.,1973,95,966. 14'

148

Unimolecular Reactions

135

theoretical papers have appeared,153and many non-kinetic stereochemical experiments have been reported ; for e x a m ~ l e , intricate l~~ experiments with r * the four geometrical isomers of CHMeCH(CN)C:CHMe. The detailed level of such considerations is suitably indicated by an extract from the authors' summary : 'it has been concluded that concerted birotational mechanisms involving transition states such as the 45" parallel twisted allylic (Woodward-Hoffmann, Mobius controlled) or the 45 perpendicularly twisted allylic (Benson-Salem subjacent orbital controlled) can be excluded from the presently acceptable set of structurally compatible, chiralityretaining, configuration-inverting mechanisms.' O

Cyclobutane Derivatives (Table @.-Much work has been carried out to reinforce and define more closely the biradical mechanism for decomposition of simple cyclobutane derivatives. Molecular orbital calculations 155 confirmed a very high energy barrier for the Woodward-Hoffmann-forbidden concerted process, while various ingenious stereochemical investigations have also been reported. Following the pattern of previous work,138reaction (28) was found lS6to give CHD:CHD that was 50% cis/50% trans, and _+

(2 CH,CH: CHD) or (but-2ene

+ CHD:CHD)

(28)

D but-2-ene with a similar cisltrans ratio, as expected for a biradical mechanism. The racemizations and isomerizations of optically active trans-l,2divinyl- and trans-l,2-dipropenyl-cyclobutaneshave been interpreted in terms of the detailed configurations of biradical intermediates,15 and full details on the isomerization of 1,2-bis(dideuteriomethylene)cyclobutane have appeared.lSB In reaction (29) the optically active reactant was found 158n

to undergo deuterium scrambling at a faster rate than racemization, implying that at least part of the scrambling must occur by a 1,3-sigmatropic process rather than via a biradical. Detailed investigations with a series of related compounds 1596 lead to the conclusion that this 1,3-shift is predominantly antarafacial with respect to the participating allylic component. lS8 W. W. Schoeller, Tetrahedron Letters, 1973, 2043; M.J. S. Dewar and J. S. Wasson, J . Amer. Chem. SOC.,1971,93,3081; A. S. Kende and E. E. Riecke, ibid., 1972,94,1397. W. von E. Doering and L. Birladeanu, Tetrahedron, 1973,29, 499. J. S. Wright and L. Salem, J. Amer. Chem. SOC.,1972, 94, 322; R. Hoffmann, S. Swaminathan, B. G. Odell, and R. Gleiter, ibid., 1970,92, 7091. 156 R. Srinivasan and J. N. C. Hsu, J. C . S. Chem. Comm., 1972, 1213. 15' J. A. Berson and P. B. Derran, J. Amer. Chem. SOC.,1973, 95, 267, 269. lS8 J. J. Gajewski and C. N. Shih, J. Amer. Chem. SOC.,1972, 94, 1675. lJg (a) J. E. Baldwin and R. H. Fleming, J. Amer. Chem. SOC.,1972, 94, 2141; (b) J. E. Baldwin and R. H. Fleming, ibid., 1973, 95, 5249, 5256, 5261. 15'

150

Table 6 Recent high-pressure Arrhenius parameters for reactions of C, ring systems Reactant Cyclobutanes Cyclobutane

1,l ,2-Trimethylcycl obutane Cyanocyclobutane trans-1,2-Dicyanocyclobutane 3-Cyano-methylenecyclobutane Cyclobutanones

Cyclobutanone

2,2,4,4-Tetra.methylcyclobutanone

2-Chlorocyclobutanone

Ref. and Ioglo(Am/s-l) E,/kJ mol-l comments

Product(s)

ethylene isobutene propene 2-methylbut-2-ene C2H4 vinyl cyanide C2H4 vinyl cyanide (no geom. isom.) vinyl cyanide allene

+

+

+

+

+

Jketen CIH4 1cyclopropane CO jMe,C:CH, Me,C:CO 11,1,2,2-tetramethylcyclopropane CO I(.) CH2C0 Me,C:CHOEt 1 ( b ) Me2C:C0 CH,:CHOEt J CHSCO C,H,Cl CHClCO CzH4

+

+ +

+ +

+

+ 1

(16.5)" 16.0 15.8 15.5 12.4 12.7

(274)" 267 252 J 237 172 205

100

14.6 14.2 14.9 14.9

2171 2401 234 249 J

d

13.6

169

ca. 15.2 ca. 14.2

1

1

ca. 2321 ca. 2121

b C C

C

e

160 f

23

2

z

5. a 2. b

cis-2,3-Dim thylcyclobu e tanone trans-2,3-Dimethylcyclobutanone cis- or trans-2,4-Dimethylcyclobutanone

+ +

MeCH:CO CsHs {CH,CO MeCH:CHMe MeCH:CH, MeCH:CO

+

}

(14.6)” (14.6)g (14.6)8

(192)q (205)‘ (214Y

(see Table 7 for polycyclic cyclobutanone derivatives) Cyclobutenes

1-Cyanocyclobutene 3,3,4,4-Tetrafluorocyclobutene

13.4 14.1

141 201

c

22h

a Estimated from VLPP - see Section 6; A. T. Cocks and H. M. Frey, J. Phys. Chem., 1971, 75, 1437; S. F. Sarner, D. M. Gale, H. K. Hall, and A. B. Richmond, J. Phys. Chem., 1972,76,2817; T. H. McGee and A. Schleifer, J. Phys. Chem., 1972,76,963; 1973,77, 1317; A.T. Blades and H. S. Sandhu, ibid., p. 1316; H. M. Frey and H. Hopf, J . C. S. Perkin, ZI, 1973, 2016; f J. Metcalfe and E. K. C. Lee,J. Amer. Chem. SOC.,1973, 95,43 16; fl Parameters obtained from measured rate-constant and estimated A factor; J. Metcalfe, H. A. J. Carless, and E. K. C. Lee, J. Amer. Chem. See also Table 4. SOC.,1972, 94, 7235;

138

Reaction Kinetics

A number of papers have appeared on the decomposition of cyclobutanone derivatives (Tables 6 and 7). The interesting feature here is the very clear evidence for a semi-ionic (zwitterionic) rather than biradical transition state in most of these reactions. For example, the great rate enhancement by ethoxy substitution is very characteristic of such a process [reaction (30) Me

+ CH,CO

i-J

EtO

(30)

EtO

Etd

shows the major reaction for one compound 160]. The last cyclobutanone in Table 7 IS an exception, the nature of the products being more consistent with a biradical mechanism.

Cyclobutene Derivatives(Table @.-Apart from these kinetic studies, theoretical aspects of the cyclobutene-butadiene system have been investigated by means of potential-energy-surface calculations 161 and by mechanistically oriented experiments. Isomerization of cis-3,4-dimethylcyclobutene was found 162 to give 0.005 % of the Woodward-Hoffmann-forbidden trans, trans-hexa-2,4-diene [reactions (31) and (32)] ; the forbidden disrotatory (31)

(cis, trans-)

r \ L

HRH f disrotatory forbidden

kCH,

*

(trans, trans-)

(32)

CH,

mechanism appears to be possible although penalized by an additional energy requirement of about 60 kJ mo1-1 relative to the allowed conrotatory isomerization. Reaction (33) gave slightly different amounts of the products

Hw

Me,SiO

OSiMe,

Aconrotation

D

160

161

16*

D

Me,SiO

OSiMe,

H &H

Me,SiO +

D D

D

x

r

3

H H

(33)

K. W. Egger, J. Amer. Cliem. SOC.,1973, 95, 1745. K. Hsu, R. J. Buenker, and S. D. Peyerimhoff, J. Amer. Chem. SOC.,1971, 93, 2117; M. J. S. Dewar and S. Kirschner, ibid., pp. 4290,4291; J. W. McIver and A. Komornicki, ibid., 1972,94,2625; A. Rastelli, A. S. Pouoli, and G. Del Re, J. C. S. Perkin II, 1972, 1571. J. I. Brauman and W. C. Archie, J. Amer. Chem. SOC.,1972,94,4262.

139

Unimolecular Reactions

from the two conrotatory routes, indicating a small kinetic selectivity due to the isotopic substitution.ls3 Isomerization by reversible ring-opening in cyclobutenes has been observed, e.g. reaction (34),le4this being in a sense the mechanistic inverse of the isomerization of dienes via cyclobutene formation [reaction (35) and Section 7, p. 1461.

Polycyclic Systems (Table 7).-Some polycyclic systems react essentially as simple ring systems with a geometrical restraint, e.g. the bicyclo[2,2,0] hexanes, study of which has provided information on the stabilization of Table 7 Recent high-pressure Arrhenius parameters for reactions of polycyclic compounds

Reactant Cyclopropanering reactions

Ref. and E,/W cornlog,,(A,/s-l) mol-l rnents

Pvoduct(s)

@h

(no cis-)

15.0

270

15.1

27 6

14.8

272

12.6

136

R. E. K. Winter and M. L. Honig, J. Amer. Chem. SOC.,1971, 93, 4616. J. E. Baldwin and M. S. Kaplan, J. Amer. Chem. SOC.,1972,94, 668.

lSJ

18*

H. M. Frey, A. M. Lamont, and R. Walsh, J. Chem. Soc.(A), 1971,2642.

Reaction Kinetics

140

Table 7 (cont.) Reactant

log,,(A,/s-l)

Product ( s )

Ph + 2 transisomers methylenecyclobutenes acyclic products

+

Cyclobutane ring reactions

[ Lo

C,H, +

0

I3

other products

I

Ref. and E,/kJ cornmol-1 ments

13.1

154

b

14.9

225

116

13.3

118

C

14.8

204

d

13.3e 13.2e 13 .9e 13.6' 13.6' 14.1e

139" 144" 154e 151' 1 26i 156"

f

13.7

154

113, (1 12)

--+

X

x = Cl Br Me Et C0,Me CH,OMe

Q'

(-+

C,H, + H,)

g

h

h

i k

141

Unimolecular Reactions Table 7 (cont.) Reactant

Product (s)

,O

Ref. a?Zd Em/W cornloglo(Am/s-l) mol-' ments

0

+ CH2C0

wo

0I

dfo \

\

+ Me,C:CO

14.2

203

I

13.2

157

m

12.9

158

n

114.5

184

0

15.4

180

P

12.6

161

P

13.9

164

P

13.9

168

4

12.7

103

r

>=c=o

Valence-bondbenzene isomers

Miscellaneous

eCN %" Q 0w +

(See also Tables 8 and 9)

\

a M. C. Flowers and D. C. Penny, Znternat. J . Chem. Kinetics, 1973,5, 469; Probably liquid phase; J . S. Swenton and A. Wexler, J . Amer. Chem. SOC.,1971, 93, 3066; In CsD6;A. De Meijere, D. Kaufmann, and 0. Schallner, Tetrahedron Letters, 1973, 553;

142

Reaction Kinetics

biradicals by various substituents. Labelling experiments can provide solid mechanistic information ; e.g. the product in reaction (36) demonstrates

clearly that the reaction proceeds by C-1-C-5 fission.ls6 The ring-opening reactions of the bicyclopentanes (A)-(C) to give the various isomers of hepta-2,5-diene proceed with considerable specificity, however (conrotatory/ disrotatory w 10 in each case), indicating a situation intermediate between the cyclobutane and cyclobutene p y r ~ l y s e s .The ~ ~ ~isomerization of benzene isomers is an interesting field in which little new work has appeared.16* Preliminary work in the reviewer’s laboratory indicates 169 that reaction (37)

-

is a reversible unimolecular reaction with E,, 188 kJ mol-1 and E-37 rn 153 kJ mol-l at 457-526 K. Thermal isomerization of Dewar-benzene and its l-chloro- and 1,4-dichloro-derivativeshas been found to yield at least a small proportion of an electronically excited (triplet) state of the benzene product A. T. Cocks and H. M. Frey, J. Chem. SOC.( A ) , 1971, 2564; Gas-phase results; also studied in solution; f E. N. Cain and R. K. Solly, J. Amer. Chem. SOC.,1972, 94, 3830; E. N. Cain and R. K. Solly, Znternat. J. Chem. Kinetics, 1972, 4, 159; E. N. Cain and R. K. Solly, J. Amer. Chem. SOC.,1973, 95, 7884; In two solvents; j E. N. Cain and R. K. Solly, J. Amer. Chem. SOC.,1973,95,4791; E. N. Cain and R. K. Solly, Austral. J. Chem., 1972, 25, 1443; A. T. Cocks and K. W. Egger, J. C . S. Perkin. 11, 1972, 2014; K. W. Egger and A. T. Cocks, J. C. S. Perkin IZ, 1972, 211; K. W. Egger, Internat. J. Chem. Kinetics, 1973, 5 , 285; O A. T. Cocks and K. W. Egger, J. C. S . Perkin If, 1973, 835; P By differential scanning calorimetry; D. M. Lemal and L. H. Dunlap, J. Amer. Chem. SOC.,1972, 94, 6562; S. F. Sarner, D. M. Gale, H. K. Hall, In hexane; G. D. Andrews, and ,4.B. Richmond, J. Phys. Chem., 1972, 76, 2817; M. Davalt, and J. E. Baldwin, J. Amer. Chem. SOC.,1973,95,5044;see also J. I. Brauman, W. E. Farneth, and M. B. D’Amore, ibid., 1973,95, 5043 for the same and related com7*

pounds. lG6

16’ lBH

lS9

170

J. E. Baldwin and G. D. Andrews, J. Org. Chem., 1973, 38, 1063. J. A. Berson, W. Bauer, and M. M. Campbell, J. Amer. Chem. Soc., 1970, 92, 7515. Further publications on previously reported reactions : E. Ratajczak, Roczniki Chem., 1971, 45, 257; Bull. Acad. polon. Sci., S6r. Sci. chim., 1973, 21, 691. A. M. Dabbagh, W. T. Flowers, R. N. Haszeldine, and P. J. Robinson, unpublished work. P. Lechtken, R. Breslow, A. H. Schmidt, and N. J. Turro, J. Amer. Chem. SOC.,1973, 95, 3025.

Unimolecular Reactions

143

Table 8 Recent high-pressure Arrhenius parameters for reactions of olefins

Product(s)

Reactant cis-trans Isomerizations cis-CHC1: CHCI

trans

trans-CHCI: CHCI cis-CF,CF: CFCFB

cis trans

trans-CF,CF: CFCF, tram-Hexa- 1,3,5- triene

cis cis-triene, cyclohexadiene trad cisd trad

cis-Penta-l,3-diene tram-Penta-l,3-diene cis-2,3-DimethylpentaI ,3-diene(A) trans-A

cisd

Cyclizations 2-alkyl and 5-alkyl 1-Alkyl- and 3-alkylcyclohexa-l,3-dienes hexa-1,3,5-trienes (alkyl = l-Me,-Et,-Bu', or 2-Me,-Et)

Ref. and E,/W cornloglo(Aoo/s-l) mol- ments 12.4 (13.2)" 12.3 13.2 (13.2)4 13.0 12.7

223 (240)' 220 229 (230)O 230 181

13.5 13.3 12.3

218 220 184

165 165 165

12.4

189

165

9.9-11.9

112124

b b b b c

e

1,5-Hydrogen transfer

F reverse reaction

Cope and related rearrangements

0

a

11.1

142

165

11.0

139

165

12.5

224

18961

11.6

105

g

10.3

132

g

Reaction Kinetics

144 Table 8 (cont.) Reactant

Product(s)

reverse reaction

0 Me,Si-

L

Miscellaneous Dicyclopentadiene

pentadiene

cis-Bu t-2-ene

buta-1,3-diene

+ H2

Ref. and E,/kJ comloglo(A,/s-l) mol-1 ments

14.3'

143

i

15.2'

152

i

12.1

84

j

9.9

164

173l

13.3" 12.7-14.7"

144" 138-} o 156"

(13.0)q

(272)q 137

Unimolecular Reactions

145

Table 8 (cunt.) Reactant

Product (s)

Ref. and EJkJ comments ~ o g l o ( ~ ~ / s -mol-l ') 13.1

192

r

12.5

183

r

13.0

203

r

258 (247)*

136

242

t

252

t

12.2

103

u

11.8 13.lW* 11.8"s 12.3" 12.2" 12.3" 12.0"

305 136w12 150"~ 200" 197" 200" 178"

0 Q Dz(5D2 D

D(jD D D

+

c

DJ

D

CH2:C:CH2 PhSCD,CH:CH, PhSCH,CH:CHMe CH,:CHCHMeSiMe, CH,:CHCHMeSiMe,Ph CH,:CMeCHMeSiMe,Ph CH, :CHCHPhSiMe

CH,C-CH PhSCH,CH:CD, PhSCHMeCH:CH2 MeCH:CHCH,SiMe, MeCH:CHCH,SiMe,Ph MeCH:CMeCH,SiMe,Ph PhCH:CHCH,SiMe,

v x x

y y y y

Parameters in parentheses are from shock-tube and earlier static data combined; S. W. Orchard and Shock-tube data; P. M. Jeffers, J. Phys. Chem., 1972, 76, 2829; B. A. Thrush, J. C. S. Chem. Comm., 1973, 14; Parameters for overall reactions proceeding by reversible cyclobuteneformation (Section 7, pp. 139 and 146); Neat liquids; C. W. Spangler, T. P. Jondahl, and B. Spangler, J. Org. Chem., 1973,38,2478; f ShockStudied by racemization of optically active tube; see also Table 12 for radical split; trans-reactant; P. S. Wharton and D. W. Johnson, J. Org. Chem., 1973, 38,4117; In CDCl,; L. A. Paquette and M. J. Epstein, J. Amer. Chem. SOC.,1973, 95, 6717; In a

Reaction Kinetics

146

Olefins (Table 8).-Shock-tube studies on the cis-trans isomerization of 1,2-dichloroethyIene and peifluorobut-Zene 171 have given results in fair agreement with the earlier low-temperature work, and the Arrhenius parameters obtained by combination of the various results can be regarded as well established, with A = 1013a2s-l in each case. There are now several examples of geometrical isomerization of buta-l,3-dienes by intermediate conrotatory formation of cyclobutenes. One example not shown in Table 8 is the case of 1,4-dideuteriobuta-1,3-diene,in which the ciqcis-isomer (for example) is found to give predominantly the trans,trans-isomer at first, the cis,transisomer appearing much more slowly [reaction (38)].172

Cope rearrangements and similar reactions provide much scope for kinetic and mechanistic work, interesting features being the appearance of heteroatoms and the inference of a double-bonded silicon compound as an intermediate formed by a Cope-type rearrangement in the cis-trans isomerization shown in reaction (39).173 Complex stereochemical experiments involving labelled and optically active compounds have been r e p ~ r t e d 174 .~~~~

Heterocyclic Compounds (Table 9).-The reactions in this list illustrate a wide assortment of reaction types and will not be discussed in any detail. decalin; Cope rearrangements in such systems are expected to have higher A factors than the more usual type; I. R. Bellobono, P. Beltrame, M. G. Cattania, and M. Simonetta, Tetrahedron, 1970,26,4407; j In CFCla; J. M. Brown, B. T. Golding, and J. J. Stofko, J. C . S. Chem. Comm., 1973, 319; Neat liquid; T. Sasaki, S. Eguchi, and M. Ohno, J. Org. Chem., 1972, 37, 466; By (k, k - for cis-trans isomerization, see text; g.c. in poly(pheny1 ether) stationary phase; In a variety of other stationary phases; S. H. Langer and J. E. Patton, J. Phys. Chem., 1972, 76, 2159; * A. T. Cocks and H. M. Frey, J . Chem. SOC.(A), 1971, 1661; q Estimated to fit VLPP data; A, T. Cocks, H. M. Frey, and R. G. Hopkins, J. C . S . Furuduy I , 1972, 68, 1287; Mixed pyrolysis gives mainly unscrambled H , Dz; D. A. Knecht, J. Amer. Chem. SOC., 1973, 95, 7933; ( k , k - , ) in hexane; G. D. Andrews, M. Davalt, and J. E. Baldwin, J. Amer. Chem. SOC.,1973,95, 5044; Shock-tube data; J. N. Bradley and K. 0. West, k W l ; In nitrobenzene; personal communication; parameters may refer to k l H. Kwart and N. Johnson, J. Amer. Chem. SOC.,1971, 92, 6064; Y H. Kwart and J. Slutsky, J. Amer. Chem. SOC.,1972, 94, 2515.

'

+

+

+

+

171

P. M. Jeffers, J. Phys. Chem., 1972, 76, 2829. L. M. Stephenson, R. V. Gemmer, and J. I. Brauman, J. Amer. Chem. SOC.,1972,94, 8620.

173 174

J. Slutsky and H. Kwart, J. Org. Chem., 1973, 38, 3658. J. M. Goldstein and M. S. Benson, J. Amer. Chem. SOC.,1972,94, 7147; M. Arai and R. J. Crawford, Cunad. J. Chem., 1972, 50, 2158.

Unimolecular Reactions

147

Table 9 Recent high-pressure Arrhenius parameters .for reactions of heterocyclic compounds Ref. and Reactant Three-membered rings

"\C/i

'Y

Product ( s )

Em/kJ COIIZloglo(Am/s-l) mol-1 ments

trans-e poxi de butan-2-one but-3-en-2-01 ethyl vinyl ether isobutyraldehyde

14.6 13.6 12.2 12.9 13.0

cis-epoxide butan-2-one but-3-en-2-0 1 ethyl vinyl ether

14.7 14.2 12.7 14.2

isobutyraldehyde isopropenyl methyl ether 2-methyl-prop-2-en-1-01

13.3 13.6 11.5

cyclohexanone cyclohexen-3-01

14.6 13.1

:CXY

23 1 234

a

263

;!}

b

208

+ Nz

\N

X,Y = F , F c1, cc13 Br, Ph CI, Ph CI, p-MeC6H4 C1, p-CICeH4 CI, p-OaNC6H4 C1, p-MeOC6H,

Q H

-M

N C1

reverse reaction

136 122 115 117 117 116 117 111

13.7

107

i

9.9

63

j

9.7

63

j

wC1 N

d e

13.4 13.8 13.8 13.9 13.9 13.8 13.8 13.4

f,g f,g

f,h f,h f, h f,h

148

Reaction Kinetics

Table 9 (cont.) Reactant Four-membered rings

Product (s)

g;

+

but-2-enes CH20 propene MeCHO

+

+

loglo(A,/s-l)

Ref. and E,/kJ commol-l ments

15.2 15.7

J

but-2-enes CHaO propene MeCHO

15.5 15.9

266 27 1

+ CH2S

13.0

202

I

13.0

107

m

16.4

163

n

ca. 12.5

ca. 232

0

14.9

241

P

15.6

269

Y

13.6

115

r

11.6

108

r

+

C2H4

BCF,-N

I

CF2-N

ll

C2F4

+ N2

Five-membered rings

-+ isomers W

O

(see Table 5 for reverse reaction)

0 CH2O

+ HCO2H

0-0

+ HC02H (CH20) + MeCOzH J MeCHO

0-0

Unimolecular Reactions

149

Table 9 (cont.) Product (s)

Reactant

Me,CO

log,,(Am/S-l)

+ HC02H

MeCHO

Ref. and E m / k J commol-l ments

9.6

100

r

+ MeC02H

8.9

98

r

+

14.4

203

s

15.2 12.2 11.9 10.3

153 98 82 74

t t t t

15.7

173

v

Six-membered rings

0

\0

MeOCH:CH, CH2:CHCHO

n=l n = 2" n = 3" bridge absent"

0 (86%)

M. C. Flowers and R. M. Parker, J. Chem. SOC.(B), 1971, 1980; M. C. Flowers and R. M. Parker, Znternut. J. Chem. Kinetics, 1971, 3, 443; M. C. Flowers, D. E. Penny, and J-C. Pommelet, Znternut. J. Chem. Kinetics, 1973, 5, 353; E. W. Neuvar and R. A. Mitsch, J. Phys. Chem,. 1967, 71, 1229; e I n two solvents; M. T. H. Liu and K. Toriyama, Internat. J. Chem. Kinetics, 1972, 4, 229; f In cyclohexene; other solvents also studied; g M. T. H. Liu and K. Toriyama, J. Phys. Chem., 1972,76, 797; A M. T. H. Liu and K. Toriyama, Cunud. J. Chem., 1972, 50, 3009; In four solvents; M. T. H. Liu and K. Toriyama, Cunud. J. Chem., 1973, 51, 2393; Neat liquids?; J. Ciabattoni and M.Cabell, J. Amer. Chem. SOC.,1971,93, 1482; cis-trans-Isomerization of reactant negligibly slow; K. A. Holbrook and D. A. Scott, J. C. S. Furuduy Z, 1974, 70, 43; Shock-tube; P. Jeffers, C. Dasch, and S. H. Bauer, Znternut. J. Chem. Kinetics, 1973, 5 , 545; In heptane; other solvents also studied; L. E. Friedrich and G. B. Schuster, J. Amer. Chem. Soc., 1971, 93, 4602; J. J. Cosa, H. E. Gsponer, E. M. Staricco, and C. A, Vallana, J. C. S. Furuduy Z, 1973, 69, 1817; A. T. Cocks and K. W. Egger, Helv. Chim. Actu, 1972, 55, 680; p A. T. Cocks and K. W. Egger, J. C. S. Perkin ZI, 1973, 197, 199; N. J. Daly and F. J. Ziolkowski, Austral. J. Chem., 1971, 24, 771; L. A. Hull, I. C. Hisatsune, and J. Heicklen, J. Phys. Chem., 1972, 76,2659; * H . M. Frey, R. G. Hopkins, and N. S. Isaacs, J. C. S . Perkin IZ, 1972,2082; E. L. Allred and A. L. Johnson, J. Amer. Chem. SOC.,1971, 93, 1300, and references cited therein; " In CDCl,; Phase uncertain; E. L. Allred and K. J. Voorhees, J. Amer. Chem. SOC.,1973, 95, 620. 1p

150

Reaction Kinetics

Dehydrohalogenations and Other Four-centre Reactions (Table lo).-Several shock-tube studies on fluoroalkanes have been reported, and Tsang has rationalized previous disparities by pointing out 175 that the data of Cadman et 0 1 . l ~imply ~ residence times which were impossibly low. Recalculation using more realistic parameters gave temperatures 5 0 0 - 6 0 0 K higher than originally reported and rate-constants (relative to the standard reaction) in reasonably good agreement with the results of Tschuikow-Roux et 178 for the decomposition of 1,I-difluoroethane and 1,l,l-trifluoroethane. Rodgers and Ford log have examined the various thermal and chemical activation experiments with 1,1,1-trifluoroethaneand correlated these satisfactorily in terms of a four-centre transition-state corresponding to log,,(k Js-l) = 14.6 - 303.8 kJ mol-l/2.303 RT. al.1777

Table 10 Recent high-pressure Arrhenius parameters for hydrogen halide and other four-centre eliminations Reactant H F Elimination CHSCHF2

Product (s)

CH,FCH,F CHF2CHF2 CF3CH2F CF3CHF2 CF3CH2Cl

+ HF CH,:CHF + HF CF,:CHF + HFc CF,:CHF + HF' CF,:CF, + HFc CF,:CHCl + HFP

HCI Elimination CH,CH,CI MeCH2CHC12 EtCH,CHCl, c~cZO-C,H~ ,C1 cyc10-C*HiSC1 CH,CHClF

C2H4 HCl 1-chloropropenes HCl 1-chlorobutenes HCI cycloheptene HCl cyclo-octene HCl CH,:CHF HCl

CH,:CHF

+

+ +

+ +

3-Chlorocyclohexene 4-Chloroc yclohexene

+ CH,:CF, + HCl cyclohexadiene + HCl cyclohexadiene + HCl

Miscellaneous EtCN Pr'CN

C2H4 HCN propene HCN

CH,CClF,

176

177

+

+

log,,(A,/s-l)

EJkJ mol-'

Ref. and comments

13.6

265

13.4 13.3 13.4 13.6 12.7

263 290 296 300 283

13.6 14.3 13.7 12.0 11.9 13.9

177 i i

11.2 13.2

234 237 225 183 177 239 252 223 155 202

13.1 12.2

291 268

n n

{ E::

a, (175) b d e

f h

i i k k 1 m m

W. Tsang, Internat. J. Chem. Kinetics, 1973, 5, 643. P. Cadman, M. Day, and A. F. Trotman-Dickenson,J. Chem. SOC.(A), 1970, 1356. E. Tschuikow-Roux and W. J. Quiring, J. Phys. Chern., 1971, 75, 295. E. Tschuikow-Roux, W. J. Quiring, and J. M. S i d e , J. Phys. Chem., 1970,74,2449.

Unimolecular Reactions

151

Table 10 (cont.)

Ref.

and

E,/kJ Reactant

Product (s)

ButCN

isobutene

CH,: CHCH(0H)CH

,

+ HCN

~

~

+

MeCONH, isobutene CH2C0 t-butylamine CHzCO CH,CO,H (CH3CO)2O Bui3Gaq [Bu*,GaH] isobutene C2H4 L-tBu',GaEt Bu',Bq [Bu!,BH] isobutene [Bu'EtBH] isobutene Bu',EtB BuiEt2B [Et,BH] isobutene L+Et,B MeAlBu,q (similar) Me,AlBu (similar)" Et,Al" NH, CH3NHNH2 JCH,:NH \CH,N:NH H, MeNHNHMe MeN:NMe H, MeCONHBut

+ +

+ + + +

+

+

+ +

logl,(A,/s-l) 12.2 12.9 ~

C O ~ -

mol-l ments n 266 233 190 ~

12.4 ca. 14.6

c a . Z }

0

11.3 11.6

135 127

p r

12.3 12.5 11.4

121 E}

s

10.9' 10.gu (13.2)" (13.5)" (13.5)"

116' 126"

t

(238)"

v

u

B. Noble, H. Carmichaei, and C. C. Bumgardner, J. Phys. Chem., 1972, 76, 1688; J. A. Kerr and D. M. Timlin, Internat J. Chem. Kinetics, 1971, 3, 427; For C-C fission see Table 12; Shock-tube; G. E. Millward, R. Hartig, and E. Tschuikow-Row, J. Phys. Chem., 1971,75,3195; Shock-tube; G. E. Millward and E. Tschuikow-Row, J. Phys. Chem., 1972, 76, 292; f Shock-tube; E. Tschuikow-Roux, G. E. Millward, and W. J. Quiring, J. Phys. Chem., 1971,75, 3493; V For HCl elimination see Table 13; Shock-tube; G. E. Millward and E. Tschuikow-Row, Internat. J. Chem. Kinetics, 1972, 4, 559; K. A. Holbrook and K. A. W. Parry, J. Chem. SOC.(B), 1971, 1762; J M. Dakabu and J. L. Holmes, J. Chem. SOC.(B), 1970, 1040; G. J. Martens, M. Godfroid, R. Decelle, and J. Verbeyst, Internat. J. Chem. Kinetics, 1972,4, 6 4 5 ; Y . K. Panshin and N. G. Panshina, Russ. J. Phys. Chem., 1970, 44, 783; J. L. Holmes and M. Dakabu, J. C. S. Perkin IZ, 1972,2110; P. N. Dastoor and E. U. Emovon, Cunad. J. Chem., 1973, 51, 366; O A. Maccoll and S. N. Nagra, J. C . S. Faraday I, 1973, 69, 1108; p P. G. Blake and A. Speis,J. Chem. SOC.(B), 1971,1877; In excess of ethylene; K. W. Egger, J. Chem. SOC.(A), 1971, 3603; A. T. Cocks and K. W. Egger, J. Chem. SOC.(A), 1971, 3606; ' k for loss of A1-Bu groups; K. W. Egger and A. T. Cocks, Trans. Faraday SOC.,1971,67,2629; In excess of propene; k for loss of A1-Et groups; A. T. Cocks and K. W. Egger, J. C . S. Faraday I , 1972, 68, 423; " Estimated to fit VLPP data; D. M. Golden, R. K. Solly, N. A. Gac, and S. W. Benson, Internat. J. Chem. Kinetics, 1972, 4, 433. a

t(

Six-centre Eliminations (Table 1l).-The six-centre mechanism is regarded as well established for most of the reactions listed in Table 11, and much of the work reported has been directed at the effects of substituents on the rate parameters. The decompositions of the compounds RNCO, RNCS, and RSCN all give HNCX rather than HXCN as the elimination product, but in the first two cases this is thought to result from secondary rearrangement of the tautomer originally produced.

Table 11 Recent high-pressure Arrhenius parameters for ester decomposition and other six-centre reactions Reactant A cetates EtOAc Pr' CH(OAc)Me ButCH(OAc)Me ButCH(OAc)Et ButCH(OAc)Pri ButCH(OAc)CH,CH:CH2 PhCH(0Ac)Me (BP)CH(OAc)Mef N N- Dimethylcarbamates Me,NCO,Et Me,NCO,Pr' Me,NCO,But

Thiolacetates Bu"SAc Bu'SAc Bu8SAc ButSAc Carbonates R10C02Ra

R1= R2= Et, Pr, BunyBu', or hexyl R1= RZ= Pr', [2Hs]Pri,Bun,or s-hexyl R1 = R2 = But or t-pentyl R1 = Me, RZ = hexyl, s-hexyl, or t-pentyl

Product(s)

+ C2H4 + Pr'CH:CH," + ButCH:CH,b + ButCH:CHMeb + ButCH:CMe,b + ButCH:CHCH:CH," + PhCH:CH,] + (BP)CH:CH,]

HOAC HOAc HOAc HOAc HOAc HOAc [HOAc [HOAc

+ CO, + C2H4 + CO, + C3Hs + CO, + i-C4Hs (Me,ND + CO, + i-CaDs) Me,NH Me,NH Me,NH

CH4

+ COS + butenes

alcohol

~OglO(AmlS-l)

w

VI ~.

EJkJ rn0l-l

(12.6) 13.1 12.5 13.1 13.1 14.1 12.8 12.6

(201)" 191 184 188 195 194 1831 1771

12.1 13.0

{::3 13.4

186 181 158 163 169

{I!;:

174

11.6

+ alkene + CO, 12.1-1 3.8' 12.8-14.4 12.1-1 3.4 12.6-13.3

182-204' 169-195 142-1 59 145-1 9 1

ReJand comments

Ally1formates ally1 formate a-methylallyl formate

CO, CO,

+ propene + isobutene

10.1 9.8

180 176

+

11.9

171

10.1 10.5 11.0

162

11.4 f 0.39 11.9 12.08 13.5

175 rt 39 171 176# 228

NH + propene

11.4

176

+ propene + isobutene + butenes + buta-lY3-diene + buta-lY3-diene + CSH4 + CH,:CD, (95%) + CBD4

'12.7 13.6 12.5 12.4 11.8 11.8 12.0 12.3

223 219 173 188 157 169 173 182

p- Hydroxy-olefns (CF&C(OH)CHsCH:CH2 CHI(OH)CH2C(CSH4X):CH2 X = H or rn-Me X = p-F, p-Me or rn-OMe X = p-C1 or rn-Br Unsaturated ethers and amines R1RZCHOCH2CH:CH, e.g., (RIRBCH)= Me, Et, Pr',t2H,]Pri, Bz or PhCD, PhCH20CH&-CH PhCD,OCH& CH Me,COCH,CH:CH,

Cyanates, thiocyanates, and isothiocyanates Pr'NCO ButNCO BuWCS CH*:CHCH,CH,NCS CH,:CHCHMeNCS CBHbSCN CHaCD2SCN GDBSCN

(CF3),C0 propene CH20 CH2:CMeC,H4X

+

+ propene PhCHO + allene (PhCDO + [2H,]allene) Me,CO + but-1-ene R1R2C0

HNCO HNCO HNCS HNCS HNCS HNCS HNCS HNCS

t

X X

c. v1

w

Table 11 (cont.)

L

UI

Reactant

Pr'SCN ButSCN Bu'SCN CHa:CHCH2CH2SCN

HNCS HNCS HNCS HNCS

+ + + +

Product (s) propene isobutene butenes4 buta-1,3-diene

logl,(A,/s-l) 12.3 12.3 12.3 12.3

Ref. and EJkJ mol-l comments 165 Y 156 Y 162 z 175 Y

Estimated to fit VLPP data; P. C. Beadle, D. M. Golden, and S . W. Benson, Internat. J. Chem. Kinetics, 1972, 4, 265; Some other olefins also produced; G. Chuchani, I. Martin, and A. Maccoll, J. C. S. Perkin I I , 1973,663; G. Chuchani, G. Martin, N. Barroeta, and A. Maccoll, J. C. S. Perkin ZZ, 1972, 2239; G. Chuchani, S. P. de Chang, and L. Lombana, J. C . S . Perkin II, 1973, 1961; f(BP) = 1-biphenylen-1-yl; H / D isotope effect also studied; R.Taylor, M. P. David, and J. F. W. McOmie, J. C. S. Perkin ZI, 1972,162; R.Taylor, ibid., p. 165; # N. J. Daly and F. Ziolkowkski, J. C. S. Chem. Comm., 1972,911; H. Kwart and J. Slutsky, J. C . S. Chem. Comm., 1972, 1182; Flow pyrolysis; D. B. Bigley and R. E. Gabbott, J. C. S. Perkin ZI, 1973, 1293; Parameters quoted are the ranges for groups of compounds studied in sealed-tube and flow experiments; D. B. Bigley and C:M. Wien, J. C . S. Perkin ZI, 1972,926; D. B. Bigley and C. M. Wien,J. C. S. Perkin I I , 1972, 1744; D. B. Bigley and C. M. Wien, J. C. S. Perkin ZI, 1972, 2359; * K. H. Leave11 and E. S. Lewis, Tetrahedron, 1972, 28, 1167; A. S. Gordon, Internat. J. Chem. Kinetics, 1972, 4, 541; p K. J. Voorhees and G. G. Smith, J. Org. Chem., 1971, 36, 1755; Rates very similar for these and five other compounds studied; H. Kwart, S. F. Sarner, and J. Slutsky, J. Amer. Chem. SOC.,1973, 95, 5234; Maximum theoretical isotope effect observed; H. Kwart, S. F. Sarner, and J. Slutsky, J. Amer. Chem. Soc., 1973, 95, 5242; K. W. Egger, J. C . S. Perkin II, 1973, 2007; N. Barroeta and A. Miralles, J. Org. Chem., 1972, 37, 2255; N. Barroeta, A. Maccoll, M. Cavazza, L. Congin, and A. Fava, J. Chem. SOC.( B ) , 1971, 1267; N. Barroeta and R. Mazzali, J. C. S. Perkin II, 1973, 839; N. Barroeta and A. Maccoll, J. Amer. Chem. SOC.,1971, 93, 5787; N. Barroeta, V. de Santis, and R. Mazzali, J. C. S. Perkin IZ, 1972, 769; No racemization of (-)-reactant; N. Barroeta, A. Maccoll, M. Cavazza, L. Congin, and A. Fava, J. Chem. Soc. ( B ) , 1971, 1264.

P

Unimolecular Reactions

155

1socyanides.-A review on the isomerization of isocyanides appeared in 1971,17s but there is little new information on high-pressure parameters in this important area. The most significant conclusion is probably the denial by Rabinovitch and co-workers 180 of any significant free-radical contribution to methyl isocyanide isomerization, but the discrepancy between these findings and those of Yip and Pritchard 181remains unexplained. Thermal explosions have recently been observed in isomerizing methyl isocyanide, and the reaction is thought to have ‘considerable promise of being the best one against which to make precise tests of thermal explosion theory’.182 Molecular orbital treatments of HNC 183 and MeNC 184 have appeared, and other theoretical considerations are discussed in Section 3, p. 113. The isomerizations of Ph3CNC and PhMeEtCNC have been investigated lB5 but do not provide unimolecular kinetic data. Bond-fission Reactions (Table 12).-The study of these radical-forming reactions is of particular importance since their activation energies are closely related to the corresponding bond dissociation energies and therefore provide an invaluable source of thermochemical data for free radicals. The pyrolysis of ethane itself at temperatures up to about 1400 K seems to be well understood now. Burcat et af.leshave made further shock-tube measurements and correlated most of the available data with an activated complex giving Ioglo(km/s-l) = 16.9 - 374kJm01-~/2.303RT; it is again to be noted that fall-off corrections are quite significant. As with cyclopropane pyrolysis, the Arrhenius plot for ethane pyrolysis curves over at high temperatures, with the ‘activation energy’, -R dln(k/s-l)/d(l/T), becoming nearly zero at around 1600 K.lS7This behaviour is presumed to result from the intrusion of further steps into the free-radical mechanism under these conditions. Hase la0has reconciled data for ethane dissociation and methyl recombination with a critical configuration which is consistent with Bunker’s minimum state-densitycriterion8 The model corresponds to loglo(k,/s-l) = 16.6 - 369 W mol-l/2.303 RT, corresponding well with Burcat’s preferred experimental parameters mentioned above. The energy of the critical configuration is 355 kJ mol-1 above the ground state, while the energy of two ground-state methyl radicals is a further 10 kJ mol-1 above the critical configuration. The recombination reaction has a critical energy which is negative ( -10 kJ mol-I), the critical configuration being that with the minimum number of K. M. Maloney and B. S. Rabinovitch, Zsonifrile Chem., 1971, 41. T. Fujimoto, F. M. Wang, and B. S. Rabinovitch, Canad. J. Chem., 1972,50, 3251. 181 C. K. Yip and H. 0. Pritchard, Canad, J. Chem., 1970,48, 2942. lg2 H. 0. Pritchard and B. J. Tyler, Canad. J . Chem., 1973, 51, 4001. D. Booth and J. N. Murrell, Mol. Phys., 1972, 24, 11 17. la* D. H. Liskow, C. F. Bender, and H. F. Schaefer, J. Amer. Chem. SOC.,1972,94,5178. la5 S. I. Yamada, M. Shibasaki, and S . Terashima, Chem. Comm., 1971, 1008; T. Austad and J. Songstad, Acfa Chem. Scand., 1972, 26, 3141. IB6 A. Burcat, G. B. Skinner, R. W. Crossley, and K. Scheller, Internat. J . Chem. Kinetics, 1973, 5, 345. 187 J. N. Bradley and M. A. Frend, J. Phys. Chem., 1971, 75, 1492. 17s

I8O

.

156

Reaction Kinetics

Table 12 Recent high-pressure Arrhenius parameters for bond-fission to form radicals Reactant Carbon-Carbon bonds CzH6 c2c16

ButBun Bu8Bu8 C-C6H11BUt neopentane Me,CHCH:CMe, Bu'CH,CHMe: CH2 CH,CHCICH:CH, CH,CH(OH)CH:CH, CF,CHO CHF2CHFp CF3CHzF CFsCHF2 MeCOCOMe CH,: C:0 Other C-X bonds PhCH,

Product ( s )

2CH3 2CC13 But Bu' 2Bum C-c~H11 But CH3 But CH, MeCHCHCMe,

+

E,/kJ mo1-'

16.9 17.7 16.3 16.3 16.3 17.7

374 286 152 159 156 356

16.0

372

14.5 16.3" 15.9" 15.5" 15.5" 15.4 15.3 15.1 14.7

179 245" 238" 230" 226" 191 224 152 207

a

b c c

c + d + +_________________---- ca. 16.0 ca.291 e 224 f 12.5 Bum+ CH,CHMeCH,f ____________-----_ 16.7 299 g CH3 + CHClCHCHz _________________-_--. 16.3 CH3 + CH(0H)CHCHS 290 190 ca. 17.0 ca. 347 h CF, + CHO 382 j 2CHFd 17.4 16.9 386 k CF, + CH2Fi 391 I 16.6 CF, + CHF,' 283 m 16.5 2MeCO 297" o 14.5" CHI + CO

+

PhCH2 H I + cH2-cHcH2

CH2:CHCHJ CHSN02 EtNO, Pr*NO, Pr'NO, HgEt2 SbMe, TIMe, PbMe,

CH3 NO2 Et NO2 Pr" NO, Pr' 4-NO, Et HgEt Me SbMe, Me TlMe, Me PbMe,

Azo-compounds R1N:NR2

R1

R1,R2= Et

log,,(A,/s-l)

Ref. and comments

+ + + + + .+ +

p

q

134 r s s

t,u t, v

t,w t,x

+ N2 + R2

(16.4) Pr' (16.6) (16.4) But 14.4 CH2:CHCMe215.0 CHI CCMe217.8 PhC(CF3)Z14.4 Me, CH2:CHCH2k H / k D = 1.28 Me, CH2:CHCD214.8 Pr', CH2:CHCH212.7 But, CH%:CHCH2(See also Table 9 for cyclic azo-compounds, and footnote dd for a azo-compounds)

(208)y

z z (179)@ z

(200)V

113 116 137 149 149 125

aa aa

bb cc cc cc cc

series of other

157

Unimolecular Reactions

Table 12 (cont.) Reactant Others ButOOBut

2But0

t-Amy100-t-Amy1 CF300CF3 Me3SiSiMeS CF3COON0 PhCH,NH, PhCH,NHMe PhCH,NMe, Me2NNH2 Me2NNMe, SF* N2F4 SO2F2

2-t-Amy10 2CF30 2SiMe3 CF3C02 NO“ PhCH, NH, PhCH, NHMe PhCH, NMe, Me,N NH, 2Me,N SF3 F 2NF2 S0,F F

+

Product (s)

+

+ + + +

+ +

Ref. and E,/H comloglo(A,/s-l) mol-l ments 15.8 15.6 (15.8) 15.2 17.2 14.3 (15.2) (15.1) (15.2) (17.0) (17.4) (13.6) 15.4” (13)

158 157 (152)’ 193 337 141 (301)Y (287)” (255)u (264)” (266)’ (331)” 83 (340)”

ee ee

fl gg

hh ii jj jj jj

kk

kk

ZZ mm nn

Shock-tube other data; ref. 186; Pyrolysis in presence of chlorine; M. L. White and R. R. Kuntz, Internat. J. Chem. Kinetics, 1973,5, 187; Shock-tube data; ref. 189; Flow pyrolysis; P. D. Pacey, Canad. J. Chem., 1973, 51, 2415; In shock-tube decomposition of tetramethylcyclopropane; w. Tsang, Znternat. J. Chem. Kinefics, 1973,5, 651; f Shock-tube data; see also Table 8 for molecular split; ref 189b; A. B. Trenwith, Internat. J. Chem. Kinetics, 1973, 5, 67; Ir M. T. H. Liu, L. F. Loucks, and R. C. Michaelson, Canad. J. Chem., 1973, 51, 2292; 4 Shock-tube data; see also Table 10 for H F elimination; 9 G. E. Millward, R. Hartig, and E. Tschuikow-Roux, J. Phys. Chem., 1971, 75, 3195; k G .E. Milward and E. Tschuikow-Roux, J. Phys. Chem., 1972, 76, 292; E. Tschuikow-Roux, G. E. Millward, and W. J. Quiring, J. Phys. Chem., 1971, 75, 3493; H. Knoll, K. Scherszer, and G. Geiseler, Znternat. J. Chem. Kinetics, 1973, 5, 271; Estimated to fit shock-tube data in fall-off region (close to koo for the nitropropanes); O H. Gg. Wagner and F. Zabel, Ber. Bunsengesellschaftphys. Chem., 1971, 75, 114; p C. T. Brooks, C. P. R. Cummins, and S. J. Peacock, Trans. Faraduy SOC., 1971, 67, 3265; * K. M. Maloney, H. B. Palmer, and D. J. Seery, Internat. J. Chem. Kinetics, 1972, 4, 87; ‘K. Glanzer and J. Troe, Helv. Chim. A m , 1973, 56, 577; * K. Glanzer and J. Troe, Helv. Chim. Acta, 1973, 56, 1691; Toluene carrier technique; S. J. W. Price and A. C. Lalonde and S. J. W. Price, Cnizad. J. Chem., 1971,49, 3367; J. P. Richard, Canad. J. Chem., 1972,34966; S . J. W. Price, J. P. Richard, and R. C. Rumfeldt, Canad. J. Chem., 1973, 51, 1397; K. M. Gilroy, S. J. Price, and N. J. Webster, Canad. J. Chem., 1972, 50, 2639; Estimated to fit VLPP data; M. J. Perona, P. C. Beadle, and D. M. Golden, Znternat. J. Chem. Kinetics, 1973,5,495; aa In xylene; P. S. Engel and D. J. Bishop, J. Amer. Chem. Soc., 1972,94,2148; bb In toluene; J. B. Levy and E. J. Lehmann, J. Amer. Chem. SOC.,1971, 93, 5790; cc R. J. Crawford and K. Takagi, J. Amer. Chem. SOC.,1972, 94, 7406; M. Prochazka, 0. Ryba, and D. Lim, Coll. Czech. Chem. Comm., 1971, 36, 3650, 2640; ee D. H. Shaw and H. 0. Pritchard, Canad. J . Chem., 1968, 46, 2722; C. K. Yip and H. 0. Pritchard, ibid., 1971, 49, 2290; see also C. K. Yip and H. 0. Pritchard, ibid., 1969, 47, 4708; 1972, 50, 1531; ff M. J. Perona and D. M. Golden, Internat. J. Chem. Kinetics, 1973, 5 , 5 5 ; gg R. C. Kennedy and J. B. Levy, J. Phys. Chem., 1972,76,3480; I h I. M. T.Davidson and A. B. Howard, J. C . S. Chem. Comm., 1973, 323; ii Rate-determining step in non-chain pyrolysis; R. Gibbs, R. N. Haszeldine, and R. F. Simmons, J. C. S. Perkin ZZ, 1972, 773; see also ibid., p. 1340; j j D. M. Golden, R. K. Solly, N. A. Gac, and S. W. Benson, J. Amer. Chem. SOC.,1972,94,363; kk D. M. Golden, R. K. Solly, N. A. Gac, and S. W. Benson, Internat. J. Chem. Kinetics, 1972, 4, 433; 11 J. F. Bott, J. Chem. Phys., 1971, 54,181 ; mm E. Tschuikow-Roux, K. 0. MacFadden, K. H. Jung, and D. A. Armstrong, J. Phys. Chem., 1973,77,734; n’a K. L. Wray and E. V . Feldman, J . Chem. Phys., 1971, 54, 3445.

158

Reaction Kinetics

states per unit energy range,I8 i.e. a sort of bottleneck in phase-space through which the molecule has to find its way. This model is consistent with a wide range of thermal and chemical activation data, and represents a significant advance over theories lS8requiring Eo > 0 for the recombination. The shock-tube studies of Tsang on alkanes and alkenes are noteworthy, leading as they do to heats of formation of alkyl and alkenyl radicals, and hence data on radical stabilization energies, and predicted Arrhenius parameters for a whole range of bond-fission processes as yet ~nstudied.'~~" Trenwith has concluded lS0from various pyrolysis studies that the resonance energy of the allyl and various substituted allyl radicals are all 53 k (6-8) kJ mol-l, and Tsang's recent figure 18sb of 50 kJ mol-1 for isobutenyl is also consistent with this generalization. Table 13 Recent high-pressure At-rhetzius parameters for 1,2-elimination to form carbenes Reactant Halogenoalkanes CHClF2 CHClzF CHBrF, CH,F CFaCH2CI CF,CHCI, Silicon Compounds Si2H6 CCl,SiCl, CCI,SiF, CHF2CF2SiF3 CHF,CF,SiMe, CCl,SiCl, CCI,Si(OEt), CCl,SiF,

Ref. and Product (s)

+ + + +

CF, HCI CClF HCI CF, HBr CH, HF CF,CH: HCI CF,CCl: HCI

+ + +

+ +

SiH, SiH4 CCI, Sic&* SiCIF, CCI, CHF,CF: SiF4 CHF,CF: SiFMe, CCl, Sic], CCI, SiCI(OEt), CC1, SiClF,

+ +

+

+ +

logl,(A,/s-')

E,/k3 C O ~ mol-' ments

12.6" 13.2 14.3 14 13.3 13.4

221" 219 233 356 274 264

14.5 10.0 12.8 13.1 13.9 15.2l 12.6'

206 124 126 138 198 164' 156l

a b c

d e

f g

108 i j

k

m m

10.8" 110" m a Probably not at high-pressure limit; G. R. Barnes, R. A. Cox, and R. F. Simmons, J. Chem. SOC.(B), 1971, 1176, and earlier work cited therein; I. D. Kushina, A. L. Bel'ferman, and V. U. Shevchuck, Kinetika i Kataliz, 1972, 13, 843; In presence of Estimated propene; R. A. Cox and R. F. Simmons, J. Chem. SOC.(B), 1971, 1625; to fit shock-tube data in fall-off region; K. P. Schug and H. Gg. Wagner, Z. phys. Chem. (Frankfurt), 1973, 86, 59; Shock-tube data (see also Table 10); G. E. Millward and E. Tschuikow-Roux, Internat. J. Chem. Kinetics, 1972, 4, 559; f Shock-tube data: M. V. C. Sekhar, G. E. Millward, and E. Tschuikow-Roux, Internat. J. Chem. Kinetics, 1973,5, 363; g M. Bowrey and J. H. Purnell, Proc. Roy. SOC.,1971, A321, 341; Also 10% split into CC19 + SiCl,; F. W. Anderson, J. M. Birchall, R. N. Haszeldine, and B. J. Tyler, to be published; R. N. Haszeldine, P. J. Robinson, and W. J. Williams, J. C. S. Perkin ZZ, 1973, 1013; R. N. Haszeldine, C. Parkinson, and P. J. Robinson, J. C. S. Perki 7 II, 1973, 1018; E. Lee and D. W. Roberts, In excess of liquid olefin; J. C. S . Perkiri I l , 1973, 437; 'Mainly vapour'. For a review see E. V. Waage and B. S . Rabinovitch, Internat. J. Chem. Kinetics, 1971, 3, 105. l t O (a) W. Tsang, J. Phys. Chem., 1972,76, 143; ( 6 ) W. Tsang, Internat. J. Chem. Kinetics, 1973, 5, 929. loo A. B. Trenwith, J. C. S. Faraday I. 1973, 69, 1737.

Unimolecular Reactions

159

Carbene-forming Reactions (Table 13).-There are now a number of wellauthenticated examples of these processes having A factors around loB s-l, which is a reasonable value for the three-centre transition-state usually proposed.lol The much lower A factor reported from gas-kinetic work108 on CC13SiC13 should probably be regarded with caution in view of the considerableexperimental difficultiesassociated with the handling and analysis of chlorosilanes. Reactions of Free Radicals lo2(Table 14).-A feature of Table 14 is the ‘normal’ A factors now found for the decomposition of the acetyl lo3and t-butoxyl lo4radicals in place of earlier lower values. Pritchard has commented, however, on the possible misinterpretation of heterogeneous effects as unimolecular fall-off in such Frey and Vinall lg6and Szirovicza and Walsh lo3have used a procedure based on Forst’s equation (1) for fitting high-pressure Arrhenius parameters to their fall-off data for the acetyl radical. It is not clear to what extent the results will be affected by the approximations involved, as a result of which equation (1) leads to an erroneous unimolecular rate constant in the low-pressure region (see Section 2); there is perhaps room for investigation of this point. RRKM theory has been used to correlate data on the thermal decomposition of ethyl radicals with measurements on the chemicallyactivated species produced from H C,H4.1s7

+

Table 14 Recent high-pressure Arrhenius parameters for isomerizution and decomposition of free radicals Ref. Reactanta Hex-1-yl

Product(s) hex-2-y1

+ +

__..------______-

CH2CMeCH, ButO c-CeH11CHClCHCl c-C~H~,CHCICCI~ CH,CO EtCO PhCO

and EJkJ comlog,,(A,/s-l) mol-’ ments 9.4 47 b ca. 13.6 ca.213 c 13.4 70 d 15.3 78 e 15.3 77 e 13.3 91 f 12.8 6 o g

allene CH, Me2C0 CHs c-CeH11CH:CHCl+ C1 C-C,H~~CH:CCI~ Cl CO CHs CO Et CO Ph

+ + +

+

14.6

+

h

123

Et,N2 C2H4; ref. 109; DecomposiMode of formation given with reference; But202 NF,; ref. 194; Radiotion of Bu’CH,CMe:CH,, seeTable 12; ref. 189b; lysis of C,H,, chloro-olefins; A. Horowitz and L. A. Rajbenbach, J. Amer. Chem. CH,CO; ref. 193. See also ref. 196; pEtBNp CO; Soc., 1973, 95, 6308; ’HI K. W. Watkins and W. W. Thompson, Internat. J . Chem. Kinetics, 1973, 5, 791; PhCHO 12; R. K. Solly and S. W. Benson, J . Amer. Chem. SOC.,1971,93,2127. a

+

+

+

+

+

S. W. Benson and H. E. O’Neal, ‘Kinetic Data on Gas-Phase Unimolecular Reactions’, NSRDS-NBS21, 1970, p. 12. lg2 For a review see J. A. Kerr, Free Radicals, 1973, 1, 1. lDs L. Szirovicza and R. Walsh, J. C . S. Faraday I , 1974,70, 33. lQ4P. Cadman, A. F. Trotman-Dickenson, and A. J. White, J. Chem. SOC.(A), 1971,2296. lQ6 C. K. Yip and H. 0. Pritchard, Canad. J. Chem., 1973,51,2138. lS6 H. M. Frey and I. C. Vinall, Internat. J. Chem. Kinetics, 1973, 5, 523. J. V. Michael and G. N . Suess, J. Chem. Phys., 1973,58,2807. lQ1

160

Reaction Kinetics

Appendix: Errata to P. J. Robinson and K. A. Holbrook, ‘Unimolecular Reactions’l p. 139 Equation (5.46): the top line should be exp( -QhvJ) p. 145 Equation (5.60): insert - sign before &hvd(first term in brackets) p. 145 Equation (5.63): delete subscript from E, on the right-hand side p. 236 Table 7.32: the reference for t-butylamine is 74,not d; footnote d should be deleted. p. 243 Should also refer to A. D. Kennedy and H. 0. Pritchard, J. Phys. Chem., 67, 161,(1963) p. 267 The page number for ref. 140 should be 4049. p. 318 Table 10.1: the reference for 1,2-dimethylcyclopropane should be 17, i p. 342 Insert the missing line: 4.80 0.096 0.094 0.092 0.091 0.089 0.088 0.086 0.085 0.0840.082

4 A Critical Survey of Rate Constants for Reactions in Gas-phase Hydrocarbon Oxidation BY R. W. WALKER

1 Introduction This Report attempts to cover the years 1967 to late 1973. A comprehensive review of all relevant rate data obtained over this period is clearly impossible. Further, the need to discuss some of the data critically has naturally placed limitations on the scope of the article, so that a personal selection of material became necessary. In particular, data concerned with liquid-phase oxidation have been excluded. In recent years, with the growth of modelling and computing techniques, a knowledge of accurate rate data has become of paramount importance. Groups such as CODATA,l NSRDS,2 and OSTI have assumed responsibility for the difficult task of the co-ordination, compilation, and assessment of rate constants. With the development of techniques such as electron spin resonance4 (e.s.r.) and kinetic spectro~copy,~ the absolute gas-phase concentration of many atoms and free radicals can now be measured. Unfortunately, because of the complexity of the spectra, e.s.r. in the gas phase is limited to atoms and diatomic radicals, and extension even to such simple radicals as CH3* and HOa*is not in prospect.* Similar restrictions apply to kinetic spectroscopy, although it has been used recently to measure the concentration of CH3*radicals.7-g Photochemical studies, shock tubes, flames, and discharge tubes combined with mass spectrometricand spectroscopic detection systems have all been a rich source of rate data for atom and radical reactions with hydrocarbonsand related compounds.10 Unfortunately, many of the reactions important in hydrocarbon oxidation (see Table 1) fall outside the scope of 1

I

lo

Committee on Data for Science and Technology of the International Council of Scientific Unions. Task Group on Data for Chemical Kinetics (Chairman, S . W. Benson, Stanford Research Institute, Menlo Park, Calif. 94025). National Standard Reference Data Series, U.S.Government Printing Office, Washington, D.C. Office for Scientific and Technical Information, High Temperature Rate Data, University of Leeds. A. A. Westenberg, Progr. Reaction Kinetics, 1973, 7, 23. N. R. Greiner, J. Chem. Phys., 1967, 46, 2795, 3389. A. A. Westenberg and N. de Haas, J. Phys. Chem., 1972,76, 1586. N. Basco, D. G . L. James, and R. D. Suart, Internut. J. Chem. Kinetics, 1970, 2, 215. N. Basco, D. G. L. James, and F. C. James, Internut. J. Chem. Kinetics, 1972,4, 129. H, E. van den Bergh and A. B. Callear, Trans. Furaduy Suc., 1971,67, 2017. Annual Reports on the Progress of Chemistry, The Chemical Society, London, 1967-1972.

161

162

Reaction Kinetics

these techniques. In consequence, there is a singular lack of data, and even those available are frequently unconfirmed and may be very inaccurate. Often the suggested rate constants for elementary steps in hydrocarbon oxidation have been derived indirectly from very complex mechanisms, which were frequently based on inadequate experimental data, such as the use of a single mixture composition only.ll This situation caused Steacie l2 to comment in 1954, ‘There is considerable evidence that atoms and radicals play a decisive role in many explosions, and in oxidation reactions. The systems are so complex, and our knowledge of them is so slight, however, that they cannot be used in practice as reliable sources of atoms and radicals.’ Sadly, this view is still largely true in 1974.

Table 1 Basic mechanism for hydrocarbon oxidation RH

+

+ +

Re H02* +XH+R* + R’. A B ( + M ) R*(+M) H. AB (+M) Re 0 2 - - -+AB HO,. - - -+ ABO + OH. R. 0 2 Re 0 2 - - + OR OH. R* 0 2 + AB,ABO, OR R* 0 2 (+M) + RO2. ( + M ) ROB. AB, ABO, OR RO2. RH + RO2H R. RO,. -+ QOOH QOOH + AB, ABO, OR QOOH 0 2 + .02QOOH R02. R02. +- chain termination ROOH RO OH. H202 M 20H. M H02- H 0 s H202 O2 0 2

X.+ RH R.( + M )

+ + + + +

+-

+-

+-

+

+

+ +

+

+

+

+

+

++-

+-

+ + +

(1) (2) (3) (4) (5-4) (5B) (5C)

(6A, B, C ) (7)

( 8 4 B, C ) (8D) (9)

(10A, B, C )

(10E) (11) (12) (1 3)

(14)

AB = olefin; ABO = carbonyl compound; OR = O-ring compound; R’, AB’ contain a smaller number of carbon atoms than R. Reactions for specific compounds are indicated by attaching an appropriate letter. Broken arrows in reactions (5A), (5B), and (5C) indicate overall processes.

The rate-constant vacuum in the oxidation area may be contrasted with the wealth of data available for reactions of atoms and free radicals with hydrocarbons. The sentiment expressed by Westenberg and de Haas,I3‘Confession being good for the soul and accuracy being good for science, we wish to report that we have repeated our earlier measurements of the rate coefficients of -0+ H, and -0+ CH,‘, can be extended to many similar reactions. Between 1967 and 1972 the rate constant for the addition of H* atoms to ethylene at room temperature was determined in noless than nineteen separate studies (and maybe more). The very important reaction between OH. radicals and carbon monoxide has also received considerable attention.l4 l1 l2

l4

A lengthy list! E. W. R. Steacie, ‘Atomic and Free Radical Reactions’, Reinhold, New York, 1954, 2nd edn., Vol. 1, p. 29. A. A. Westenberg and N. de Haas, .I. Chem. Phys., 1969, 50, 2512. W. E. Wilson, J. Phys. Chern. Ref. Data, 1972, 1, 535.

Rate Constants for Gas-phase Hydrocarbon Oxidation

163

Excellent reviews cover the reactions of atoms with hydrocarbons and related compounds in the gas phase.lo Baulch and co-workers have reviewed the elementary reactions involved in the oxidation of hydrogen l5 and of carbon monoxide;16 a complete understanding of hydrocarbon oxidation requires a precise knowledge of these reactions. Comprehensive tables of rate constants for bimolecular radical-molecule systems are available,179l a together with recent compilations for He -0. atoms,20 and OH~adica1s.l~Several reviews deal with alkyl radical abstraction reactions 21 and radical additions to double bonds have been surveyed.22 Radical decompositions have been extensively s t ~ d i e d , ~ although ~ - ~ ~ the rate constants are not always consistent with those of the reverse processes. Annual Reports of the Chemical Society and Chemicat Reviews of the period are excellent sources of recently determined rate constants. In selecting rate constants from review articles, it is necessary to distinguish between mere compilations l8 and data which have been subjected to critical scrutiny, such as the rate constants for He + CH4,26OH* C0,14 and for the 2CH3. CzH6 eq~ilibrium.~’ Table 1 sets out a basic mechanism for hydrocarbon oxidation; this Report is mainly concerned with the reactions therein. Several recent reviews on the mechanism of hydrocarbon oxidation are a ~ a i l a b l e . ~ ~ - ~ ~ 1 7 9

l5

lo

l7

l9 *O

21

22 28

2r 25 26

27 2s

8s

81

s2

+

+

D. L. Baulch, D. D. Drysdale, D. G. Horne, and A. C. Lloyd, ‘Evaluated Rate Data for High Temperature Reactions’, Vol. 1, Butterworths, London, 1972. D. L. Baulch, D. D. Drysdale, and A. C. Lloyd, O.S.T.I. Report No. 1, University of Leeds, 1968. V. N. Kondratiev, ‘VelocityConstants of Gas-Phase Reactions’, Nauka, Moscow, 1970. A. F. Trotman-Dickenson and G. S. Milne, ‘Tables of Birnolecular Gas Reactions’, National Standard Reference Data Series, NSRDS-NBS 9, U.S. Government Printing Office, Washington, D.C., 20402, 1967; A. F. Trotman-Dickenson and E. Ratajczak, ‘Supplementary Tables of Bimolecular Gas Reactions’, U.W.I.S.T., Cardiff, 1969; J. A. Kerr and E. Ratajczak, ‘Second Supplementary Tables of Bimolecular Gas Reactions’, University of Birmingham, Birmingham, 1972. B. A. Thrush, Progr. Reaction Kinetics, 1965,3,63; W. E. Jones, S. D. MacKnight, and L. Teng, Chem. Rev., 1973,73,407. F. Kaufman, Progr. Reaction Kinetics, 1961, 1, 1. A. F. Trotman-Dickinson, Adv. Free-Radical Chem., 1965,1, 1 ; P. Gray, A. A. Herod, and A. Jones, Chem. Rev., 1971, 71, 247. J. A. Kerr and M. J. Parsonage, ‘Evaluated Kinetic Data on Gas-Phase Addition Reactions’, Butterworths, London, 1972. S. W. Benson and H. E. O’Neal, ‘Kinetic Data on Gas Phase Unimolecular Reactions’, NSRDS-NBS 21, U.S. Government Printing Office, Washington, D.C., 1970 J. A. Kerr and A. C. Lloyd, Quart. Rev., 1968,22, 549. H. M. Frey and R. Walsh, Chem. Rev., 1969,69, 103. R. W. Walker, J. Chem. Soc. ( A ) , 1968, 2391. E. V. Waage and B. S. Rabinovitch, Internat. J. Chem. Kinetics, 1971, 3, 105. J. H. Knox, Combustion and Flame, 1965, 9, 297. A. Fish, ‘Organic Peroxides’, ed. D. Swern, Wiley, New York, 1970, vol. 1, p. 141. G. J. Minkoff and C. F. H. Tipper, ‘Chemistry of Combustion Reactions’, Butterworths, London, 1962. R. R. Baldwin and R. W. Walker, 14th Int. Combustion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1973, p. 241. ‘Oxidation of Organic Compounds’, ed. R. F. Gould, Advances in Chemistry Series, 1968, vol. 76.

Reaction Kinetics

164

2 Outline of Experimental Methods Apart from surface effects, and the corresponding lack of reproducibility, the most serious problem in direct investigations of hydrocarbon oxidation is the high reactivity of the products which causes increasing complexity as the reaction proceeds. This is particularly true with cool flames, where not only is the chemistry complex but temperature rises of up to 150-200 K occur even at sub-atmospheric pressures.33 Even slow combustion well outside the cool-flame alid ignition regions is rapidly dominated by products.28 For example, in the oxidation of propane at 708 K, the measured relative yield of propylene drops by about 30% over the first 10% of reaction.34 Similarly, autocatalysis due to secondary initiation by a product is a common feature.30 A number of attempts to study hydrocarbon oxidation at less than 1 % conversion have been made 35-37 using sensitive g.1.c. for analysis, but few rate constants have been obtained. Even during the induction period, up to 20 or 30 products may be formed, the majority of which have more than one route for their formation. Because of these difficulties, until recently, virtually no measured rate constants were available for reactions of alkyl radicals with oxygen, other than for the CH,. and C,H5-radicals.3s The main methods used for the determination of rate constants of interest in hydrocarbon oxidation will now be discussed. Although not of direct interest in hydrocarbon oxidation, methods VIII and IX are introduced here because they are concerned with rate constants for radical recombination reactions. Almost all absolute values for the rate constants of reactions of interest in hydrocarbon oxidation are deduced essentially from measurements relative to radical-radical processes. A firm knowledge of the rate constants for the latter is of paramount importance.

Method I: Direct Studies of Hydrocarbon Oxidation, Inside or Outside Cool-flame Limits.-Detailed product analyses have been made for a large 39-42 and related c o m p o u n d ~ . ~ 44 ~ ~Relative number of hydrocarbons rate constants are determined on the basis of an assumed mechanism. Frequently, analysis of products is only attempted after considerable reactant consumption, and with cool flames particularly, the precise temperature is 309

349

439

R. Hughes and R. F. Simmons, Combustion and Flame, 1970,14, 103. J. W. Falconer and J. H. Knox, Proc. Roy. Soc., 1959, A250, 493. 85 J . H. Knox and C. H. J. Wells, Trans. Faraday SOC.,1963, 59, 2786, 2801. 36 A. P. Zeelenberg and A. F. Bickel, J. Chem. SOC.,1961, 4014; A. P. Zeelenberg, Rec. Trav. chim., 1962, 81, 720. 37 J. E. Taylor and D. M. Kulich, Znternat. J . Chem. Kinetics, 1973, 5, 455. ag G. R. McMillan and J. G. Calvert, Oxidation Combustion Rev., 1965, 1, 84. ss A. Fish, Proc. Roy. SOC.,1967, A298, 204. O0 A. Fish, W. W. Haskell, and I. A. Read, Proc. Roy. SOC.,1969, A313, 261. T. Berry, C. F. Cullis, M. Saeed, and D. L. Trimm, iR ref. 32, p. 86. 42 J. A. Barnard and B. A. Harwood, Combustion and Flame, 1973, 21, 345. 4 3 M. Akbar and J. A. Barnard, Trans, Faraday Soc., 1968, 64, 3035; D. Anderson and D. E. Hoare, Combustion and Flame, 1969, 13, 51 1. O4 D. E. Hoare and M. KamiI, Combustion and Flame, 1970, 15, 61. 33 34

Rate Constantsfw Gas-phase Hydrocarbon Oxidation

165

not k ~ o w n Often . ~ ~ only one mixture composition is used, so that no kinetic evidence is provided for the suggested mechanism. Consequently, despite the vast number of published papers in this area, relatively few reliable rate constants have been obtained. Semi-quantitative rate constants for the steps in the reaction sequence: R. O2$ R02. + QOOH + oxidation products

+

have been estimated by Fish 45 and Knox 46 from direct studies of hydrocarbon oxidation; the values will be discussed in Section 8. 291

Method II: Addition of Traces of Hydrocarbons to Slowly Reacting Mixtures of Hydrogen and Oxygen in Aged Boric-acid-coated Vessels at 733-773 K.Information is given in two areas : (a) Measurements of the relative rate of consumption of additive and hydrogen are used to give rate constants for the reaction of H- and OHradicals with the additi~e.~’-~l (b) From a detailed analysis of the products over a wide range of mixture composition, reactions of the organic radical, formed from the additive, 50-53 in a hydrogen -k oxygen environment can be The mechanism of the hydrogen oxygen reaction has been quantitatively established under the conditions and this method provides a controlled, reproducible, and virtually constant environment for the oxidation of the additive. Normally, primary, secondary, and tertiary products are easily distinguished.

+

Method HI: Direct Oxidation of Aldehydes.-The basic mechanism for the oxidation of simple aldehydes in aged boric-acid-coated vessels over the temperature range 670-820K is now well known55 (see Table 1 and its footnotes). The oxidation of propionaldehyde has been used as a source of C2H5. and H02*radicals; the basic mechanism is given 5 7 [For reaction (5), the broken arrow indicates an overall reaction.] 45 46 47

Is

so

s2 63 64

s5

56

s7

A. Fish, in ref. 32, p. 69. J. H. Knox, in ref. 32, p. 1. R. R. Baldwin, D. E. Hopkins, A. C. Norris, and R. W. Walker, Combustion and FZame, 1970,15, 33. R. R. Baldwin, D. E. Hopkins, and R. W. Walker, Trans. Faraday SOC.,1970,66, 189. R. R. Baker, R. R. Baldwin, and R. W. Walker, Trans. Faraday Soc., 1970,66,2812. R. R. Baker, R. R. Baldwin, and R. W. Walker, 13th Int. Combustion Symp. The Combustion Institute, Pittsburgh, Pennsylvania, 1971, p. 291. R. R. Baldwin, C. J. Everett, D. E. Hopkins, and R. W. Walker, in ref. 32, p. 124. R. R. Baker, R. R. Baldwin, and R. W. Walker, Trans. Faraday Soc., 1970,66, 3016. R. R. Baker, R. R. Baldwin, and R. W. Walker, Combustion and FZame, 1970, 14, 31. R. R. Baldwin, D. Jackson, R. W. Walker, and S.J. Webster, Trans. Faraday SOC.,1967, 63, 1665, 1676. R. R. Baldwin, D. H. Langford, M. J. Matchan, R. W. Walker, and D. A. Yorke, 13th Int. Combustion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1971, p. 251. R. R. Baldwin, D. H. Langford, and R. W. Walker, Trans. Faraday SOC.,1969, 65, 792, 806. R. R. Baldwin, D. H. Langford, and R. W. Walker, Trans.Faraday SOC.,1969,65,2116.

166 C2H5CHO -t 0 2 C2H5CO. M CZH5. + 0 2 - H02- C2H5CH0 H02*+ HO,. H402 M’ OH. C2H5CH0

+

+

+

+

The oxidations of a~etaldehyde,~~ n-b~tyraldehyde,~~ and i-butyraldehyde,sO which proceed by similar mechanisms, provide convenient sources of CH3., n-C,H,*, and i-C3H,- radicals, respectively. Reactions of H02*radicals can be studied by using the oxidation of formaldehyde in potassium chloridecoated vessels between 620 and 820 K as the radical s o ~ r c e61s. ~62~ ~ Below 470 K, RCO radicals add oxygen in prefirence to decomposition by reaction (3), and aldehyde oxidation can thus be used as a source of RC03* radicals.83-65 RCO. + 0, RC03RC03* R’H -+ RC0,H + R’-

+

--f

Methods I1 and I11 have been used mainly with hydrocarbons and aldehydes containing four or fewer carbon atoms. Extension to compounds of higher molecular weight is highly desirable. Method IV: Kinetic Spectroscopy.-Atoms and radicals are produced by flash photolysis and estimated directly by spectroscopic methods. Although, in principle, this is a very general method which can be used over a wide temperature range,5*66 the only rate constants determined for alkyl radical reactions have been for CH3-.7-9 Method V : Chemicaf Shock Tubes.-From measurements of ignition delays in hydrocarbon oxygen mixtures,87* 68 and of i.r. emission from carbon monoxide and carbon d i o ~ i d e ‘O, ~and ~ ~ by the use of time-of-flight mass

+

b8

R. R. Baldwin, M. J. Matchan, and R. W. Walker, Combustion and Flame, 1970, 15, 109.

59 6o

62

63 64

65 66

67

70

R. R. Baldwin, R. W. Walker, and D. A Yorke, J.C.S. Faraday I, 1973,69, 826. R. R. Baldwin, C. J. Cleugh, and R. W. Walker, unpublished work. R. R. Baldwin, A. R. Fuller, D. Longthorn, and R. W. Walker, ‘Combustion Institute European Symposium’, Academic Press, London, 1973, p. 70. R. R. Baldwin, A. R. Fuller, D. Longthorn, and R. W. Walker, J.C.S. Faraday I, 1974, in the press. J. F. Griffiths and G. Skirrow, Oxidation Combitstion Rev., 1968, 3, 47. J. F. Griffiths, G. Skirrow, and C. F. H. Tipper, Combustion and Flame, 1968,12, 36q. D. J. M. Ray, R. Ruiz Diaz, and D. J. Waddington, 14th Int. Combustion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1973, p. 259. N. R. Greiner, J. Chem. Phys., 1970, 53, 1070. R. M. R. Higgin and A. Williams, 12th Int. Combustion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1969, p. 579. G. B. Skinner, A. Lifshitz, K. Scheller, and A. Burcat, J. Chem. Phys., 1972, 56, 3853. T. P. J. Izod, G. B. Kistiakowsky, and S . Matsuda, J. Chem. Phys., 1971, 55, 4425. A. M. Dean and G. B. Kistiakowsky, J. Chem. Phys., 1971, 54, 1718.

Rate Constants for Gas-phase Hydrocarbon OxicEation

167

~pectrometry,~~ information on a number of elementary reactions has been obtained over the temperature range 1100-2500 K. Rate constants are obtained by computer matching of experimental and calculated data. As the mechanisms are complex, the accuracy of the method is very dependent on the reliability of the mechanism and of the rate constants assumed to be known precisely. Extrapolation of shock-tube data to low temperatures must be considered very carefully. Again, rate constants for alkyl radicals are limited to those for CH3*reactions. Method VI: Photochemical Oxidation.-Photochemical initiation has been used extensively to study elementary oxidation reactions in the temperature range 270-420 K. Complex mechanisms operate, and rate constants are only available for reactions of CH,. and C2H5* and related radicals such as CH,6, C2H56,C H 3 c 0 , and C2H5&).389 73 729

Method VII: Hydrocarbon Flames.-Very few rate constants of interest in hydrocarbon oxidation have been obtained from flame studies since 1967 (for earlier work, see ref. 74). Method W: Very Low Pressure Pyrolysis (VLPP).-The very interesting VLPP method developed recently by Golden, Spokes, and Benson has been described as ‘the dream system’.75A reactive gas at very low pressure flows into a Knudsen cell, where unimolecular decompositions may occur. Reactants and products flow molecularly through an aperture into the ionization chamber of a mass spectrometer. At sufficiently low pressures, there are no collisions between the products from the initial unimolecular step and other species, and thus no secondary reactions. and the method has been very The scope of VLPP has been successfully applied to the study of unimolecular decompositions,76 radicalmolecule reactions, and radical-radical processes.75 For radical-radical processes, the following mechanism can be considered for the production of steady-state conditions: X -+ 2R* 769

R-+ R*+ R, R-+ R-+ disproportionation R

e

-+aperture

R2-+ aperture X aperture R- + surface --f

78

7‘

7s

T. C. Clark, T. P. J. Izod, and S. Matsuda, J. Chem. Phys., 1971,55, 4644. K. W. Watkins and W. W. Thompson, Internat. J. Chem. Kinetics, 1973,5, 791. D. E. Hoare and G . S. Pearson, Adv. Photochemistry, 1964,3, 83. R. M. Fristrom and A. A. Westenberg, ‘Flame Structure’, McGraw-Hill, New York, 1965; G. Dixon-Lewis and A. Williams, Quart. Rev., 1963, 17, 243. D. M. Golden, G. N. Spokes, and S. W. Benson, Angew. Chem. Internat. Edn., 1973, 12, 534.

76

77

P. C. Beadle, M. J. Perona, and D. M. Golden, Internat. J. Chem. Kinetics, 1973,5,495. N. A. Gac, D. M. Golden, and S. W. Benson, J. Amer. Chem. SOC.,1969,91, 3091.

Reaction Kinetics

168

X is the compound decomposed unimolecularly and R. is a free radical. If F i s the flow rate of X into the cell, F

= kX[W

+ k,X[Xl

(1 5 )

If conditions are chosen so that the steady-state assumption is valid, %,[XI

2(kr

+ kd)[R’I2-k (k,

4- kJR.1 As k,[R.I2 = ka,,[R2], then [Re] = (karJR2]/k,)*,so that =

kx[Xl/karp[R21 = (kr

f

kd)/kr

f (kar f

k8)/2kfk!rl[R21’

(16)

(17)

The left-hand side of equation (17) is obtained from the flow rate Fand the mass spectrometric analyses for X and Rz, so that a plot of k,[X]/k,l[R2] against [R,]- 4 gives a straight line of slope (k, k,)/2(krk,p)*. The values of k , and karPcan be calculated theoretically and k,, the first-order surface destruction constant, can be obtained by using different sizes of aperture. However, because of the uncertainty in k,, accurate values of k, can only be obtained if k, is relatively small. Suitable surfaces are difficult to find, and herein lies the main limitation of the method.

+

Method IX : Free-radical Buffer System.-The phase equilibrium: K(RR ) R’* + RI Re

rapidly established vapour-

+ + R’I

RR’

where RI and R I are alkyl iodides, provides a buffer system for the R - and R’.radicals, from which ratios of rate constants for radical recombinations can be ~btained.’~-*O 2R* -+ R, (r) 2R’* + Ri (r’) The relative rate of formation of R, and Ri is given by d[Rz]/d[RhJ = k,[R*]2/kr,[R’.]2 As reaction (RR’) is equilibrated, [R*]/[R’*]= KRRl[RI]/[R’I]

so that KRR’(kr’/kr)

=

(d[R@/d[R&*([RII/[R’II)

(1 8)

KRR, can be calculated from thermochemical data, and the relative rate of formation of R2 and Ri can be determined gas chromatographically. The absolute values of the rate constants estimated in this way ultimately depend on the rate constant for CH3 recombination, which is now accurately known (see Table 6). Owing to uncertainties in the calculated values of KRR,, the recombination rate constants are probably accurate to a factor of 10* l. 78

7D

R. Hiatt and S. W. Benson, J. Amer. Chem. SOC.,1972, 94, 25. R. Hiatt and S. W. Benson, Internat. J . Chem. Kinetics, 1972, 4, 151. R. Hiatt and S. W. Benson, Internat. J . Chem. Kinetics, 1973, 5, 385.

Rate Constants for Gas-phase Hydrocarbon Oxidation

169

3 Thermochemical Aspects of Hydrocarbon Oxidation The development by Benson and co-workers of group additivity rules for the estimation of heats of formation, entropies, and heat capacities of compounds has been one of the most outstanding contributions to kinetics over the past fifteen y e a r ~ . ~ lRecently, -~~ the rules have been extended to cover free radicals 86 so that heats of formation, entropies, and heat capacities, can be estimated over a wide temperature range for alkyl, alkylperoxy, and alkyl hydroperoxide radicals. As a consequence, absolute values of the equilibrium constants for the important reactions (7) and (9) can be estimated, and are considered reliable to within a factor of about 3 for normal temperatures. In the absence of experimental rate constants for many of the elementary reactions involved in the oxidation of alkyl radicals, the estimates obtained from thermochemical data are of paramount importance in unravelling the complications of hydrocarbon o ~ i d a t i o n 88 . ~ ~Equilibrium ~ constants for reaction (7) are given in Table 2 for a number of alkyl radicals. The position of equilibrium changes rather sharply with temperature, and an appreciation of this is central to an understanding of hydrocarbon oxidation. The values of K , are very similar for all alkyl radicals, but decrease markedly for ally1 radicals owing to their high resonance energy of stabikation of about 60 kJ m01-~.~@ 86r

Table 2 Estimated value of K,. for the equilibrium Re

+

O2$ RO,. 86 log1O(K7/mHgR. AH,o/kJmol-l 300K 500K 700K 753 K 111 9.65 1.90 -1.41 -1.99 CHs. C2HC 115 9.9 1.86 -1.58 -2.19 n-C,H,. 113 9.85 1.96 -1.42 -2.01 i-C3H,. 117 10.5 2.25 -1.25 -1.87 n-C4HB. 113 9.85 1.96 -1.42 -2.01 117 10.5 2.25 -1.25 -1.87 S-C~H,. i-C4H9114 9.6 1.66 -1.73 -2.33 t-C,H@. 117 10.1 1.72 -1.91 -2.53 (CH3)3CcH2 111 8.9 1.15 -2.16 -2.74 CH2=CHCHz 62.5 2.12 -2.13 -3.95 -4.18 a Extrapolation of K , to higher temperatures mu? allow for changes of AH,” and AS,” with temperature;

a1

a2

86

a6

861

n-C,H,* = CH3CH2CH2CH2, s-C,Hg. = CHSCHSCHCH,, i-C4Hg*= (CH,) &HCHz, t-C,Hg* = (CH,),C. S. W. Benson, ‘Thermochemical Kinetics’, Wiley, New York, 1968. S. W. Benson and J. H. Buss, J . Chem. Phys., 1958,29, 546. S . W. Benson, F. R. Cruikshank, D. M. Golden, G. R. Haugen, H. E. O’Neal, A. S. Rodgers, R. Shaw, and R. Walsh, Chern. Rev., 1969,69,279. H. K. Eigenmann, D. M. Golden, and S. W. Benson, J. Phys. Chern., 1973, 77, 1687. H. E. O’Neal and S. W. Benson, Internat. J . Chem. Kinetics, 1969,1, 221. S. W. Benson and R. Shaw, ‘Organic Peroxides’,ed. D. Swern, Wiley, New York, 1970, vol. 1, p. 105. S. W. Benson, ‘The Mechanisms of Pyrolysis, Oxidation and Burning of Organic Compounds’, N.B.S., U.S. Government Printing Office, Washington, D.C., 1970, p. 118. S. W. Benson, J, Amer. Chem. SOC.,1965,87,972. A. B. Trenwith, Trans. Faraday Soc., 1970,66,2805.

170

Reaction Kinetics

Entropies of activation can also be estimated for the calculation of A factors for unimolecular and bimolecular rate constants of both radical and molecule 81 A particularly important new method for the determination of heats of formation of alkyl radicals and related species, and the associated bond dissociation energies, has also been developed by Benson and co-workers. The method is based on the equilibrium: RH

+ I2 ft RI + HI

for which the following mechanism operates :

+ M + 21. + M + 1. + R. + Is R- + H I e R H + 1I2 RI

(20)

are considered to be 0 f 4 and 4 f 4 kJ mol-l, respectively. and Measurement of El, or L z O gives the enthalpy change in the reaction, and from the heat of formation of either RI or RH, the heat of formation of Rcan be calculated. The overall uncertainty in the values is 4-8 kJ mo1-I. Other workers have used the Evans-Polanyi relation to calculate bonddissociation energies from differencesin activation energies in homologous series of reactions 91* 92 (see Section 7). Table 3 lists some recently obtained values for C-H bond-dissociation energies, Dgg,(X-H), in hydrocarbons and related compounds. Where appropriate, recommended values based on all available data are also given.g3 There is every indication that the values of Dig8(X-H) for X = ethyl, isopropyl, and t-butyl are characteristic of primary, secondary, and tertiary C-H bonds, respectively, in alkanes to an accuracy of about 5 kJ mol-1. Further, it appears that the values can be extended to C-H bonds in alcohols and ketones (and probably therefore to the C-H bonds in the alkyl groups in aldehydes), although D&(X-H) for methanol, which is about 10-1 5 kJ mol-1 lower than in ethane, indicates that caution is necessary. D&,(RCO-H) is 364 k 4 kJmol-l for all aldehydes listed in Table 3, including benzaldehyde and acrolein, which suggests that the stability of RCOis independent of R and solely determined by the conjugation of the unpaired electron on the carbon atom with the lone-pair electrons on the oxygen atom.94 D,"g8(CH2=CHCH2--H) = 360 kJ mo1-I is particularly low, owing to resonance stabilization of the ally1 radical,8gbut as the propylene is further substituted with CH3 groups the primary C-H bond-dissociation energies are considered to increase.95

92

93 94

95

D. M. Golden and S. W. Benson, Chem. Rev., 1969, 69, 125. K. C. Ferguson and E. Whittle, Trans. Faraday Soc., 1971, 67, 2618. S. H. Jones and E. Whittle, Internat. J. Chem. Kinetics, 1970, 2, 479. J. A. Kerr, Chem. Rev., 1966, 66, 465. R. K. Solly and S. W. Benson, J . Amer. Chem. Soc., 1971, 93, 1592. H. P. Schuchmann and K. J. Laidler, Internat. J. Chem. Kinetics, 1972, 4, 49.

Rate Constantsfor Gas-phase Hydrocarbon Oxidation

171

Table 3 Bod-dissociation energies at 298 K, Do298 (X-H)"

X exptl.

CH3 CZH5 n-C3H7

435 f 5 438 f 7 410 f 4 412 4 414 f 4 398 f 6 398 f 6 413 f ? 395 f 6 383 f 5 412 f 6 412 f 6 421 f 5 405 f 5 400f5 397 5 394 f 5 397 f 5 393 f 5 411 f 7 410 f 5 386 f 6 399 f 8 389 f 6 378 f 5 390 & 5 366 f 6 360f4 365 rf 4 364 f 4 364 f 4 > 450 370 f 5 196 f 7 376 f 7

{ {

+

{ c-C~H~ c-C13H11 C-C7H13 CH3COCH2 CHSCOCHCHB CH20H CH3CHOH (CH3)2COH CH3OCH2 HCO CH3CO C2H5CO CH2=CHCO CeH5CO CH2=CH CH,=CHCH, 0 2

HO2

{ { {

+

recommended

435 410

} 123 }

410

148 h

410 395 380

L3

i

145 j 92 k 92

l-

11 I

92 92 92

:}

395

410

395 395 395 410

P

4 r S

t U V

W

X

90 Y

102 102

364 364 364 364 364 > 450 360 196 f 7 376 f 7

a Standard state, 1 atm; f5 kJ mol-', except where stated; D. M. Golden, R. Walsh, and S. W. Benson, J. Amer. Chem. SOC.,1965,87,4053; C. A. Goy and H. 0. Pritchud, J. Phys. Chem., 1965, 69, 3040; A. F. Trotman-Dickenson, Chem. and Ind. 1965, 379; fJ. H. Knox and R. G. Musgrave, Trans. Faraday SOC.,1967, 63, 2201; 9 P. S. Nangia and S. W. Benson, J. Amer. Chem. SOC.,1964,86, 2773; M. C. Lin and K. J. Laidler, Canad. J. Chem., 1967, 45, 1315; H. Teranishi and S. W. Benson, J. Amer. Chem. SOC., 1963, 85, 2887; 5 C . W. Larson, E. A. Hardwidge, and B. S . Rabinovitch, J. Chem. Phys., 1969, 50, 2769; D. F. McMillen, D. M. Golden, and S. W . Benson, Internat. J. Chem. Kinetics, 1972, 4,487; ' S . Furuyama, D. M. Golden, and S. W.'Benson, Internat. J. Chem. Kinetics, 1970,2,83; K. D. King, D. M. Golden, and S. W. Benson, J. Amer. Chem. SOC.,1970,92, 5541; R. K. Solly, D. M. Golden, R. K. Solly, D. M. Golden, and S. W. Benson, Internat. J. Chem. Kinetics, 1970,2, 11; and S. W. Benson, Internat. J. Chem. Kinetics, 1970, 2, 381 ; P F. R. Cruikshank and S. W. Benson, J. Phys. Chem., 1969,73, 733; * Z. B. Alfassi and S. W. Benson, J . Phys.

172

Reaction Kinetics

4 The Initiation Process Owing to the relatively high reactivity of the products, secondary initiation (degenerate branching) rapidly masks the primary initiation process in thermally induced hydrocarbon oxidation. At temperatures below 1200 K, the highly endothermic reaction (1) is generally considered to be the primary initiation process because of the observed sensitizing effect of oxygen on the pyrolysis of hydrocarbons.96 RH 0, -+ R* HO,* (1)

+

+

No experimentally determined values of kl for hydrocarbons are available. For aldehydes, the reaction is less endothermic (167 compared with 184-238 kJ mol-1 for alkanes), so that reaction (1) competes more successfully with secondary initiation.' At 753 K, the oxidation of propionaldehyde in aged boric-acid-coated vessels is autocatalytic, and the kinetic features can be quantitatively interpreted 5 7 using the mechanism given in Method 111 (Section 2). The autocatalysis is due to the decomposition reaction (13) of hydrogen peroxide, and the maximum rate is independent of k,, (reasonable values), being solely determined by k,, and k2p/k:4. As k,, is known a ~ c u r a t e l ykz,/kt4 , ~ ~ is obtained by computer treatment, and from the initial rate of oxidation given by expression (21), klp = 0.076 f 0.020 dm3mol-1 s-l. 56p

-d[C2HSCHO] dt C2H5CH0

-

k2pk$[C2H,CHO]*[02]

+ 0,

HO,.

*

k:4 -+

C,H5c0

+ HO,.

+ C2H,CH0 -+ H202+ CzH5C0

(21)

(1P> (2P)

In the temperature range 670-815 K, the oxidation of formaldehyde in potassium-chloride-coated vessels is extremely reproducible and shows no autocatalysis.61. 62 Using the mixture 2, 30, and 28 mmHg of formaldehyde, oxygen, and nitrogen, respectively, the chain length is only about 2, and is effectively reduced to zero when pressures down to 0.1 mmHg of formaldehyde are used. In consequence, the rate constant for the initiation process (If) can be accurately determined. Arrhenius parameters of A,, = 2.04 x 1O1O dm3mol-1 s-l and El,= 163 f 6 kJ mol-l are obtained;62the absence of a diameter effect confirms that the parameters refer to the homogeneous process. O2 + H e 0 H02* HCHO

+

+

~~

Chem., 1972, 76, 3314; R. Walsh and S. W. Benson, J. Amer. Chem. SOC.,1966, 88, 3480; * F. R. Cruikshank and S. W. Benson, Internat. J. Chem. Kinetics, 1969, 1, 381; J. A. Devore and R. Walsh and S. W. Benson, J. Amer. Chem. Sac., 1966, 88,4570; H. E. O'Neal, J. Phys. Chem., 1969,73,2644; K. W. Watkins and W. W. Thompson, Internat. J. Chem. Kinetics, 1973, 5 , 791 ; Z. B. Alfassi and S. W. Ben'son, J. Amer. Chem. SOC.,1973, 95, 319; R. K. Solly and S. W. Benson, J. Amer. Chem. SOC.,1971, 93, 1592; D. M. Golden, A. S. Rodgers, and S. W. Benson, J. Amer. Chem. SOC.,1966, 88, 3196. Q6

W. G. Appleby, W. H. Avery, W. K. Meerbott, and A. F. Sartor, J. Amer. Chem. SOC., 1953,75, 1809.

rrhenius parameters for initiation reactions

React ion

+ O2 + 0, + + + + O2 + 0, 0

0

0

H c 0 . f H02* + C2H5CO HO2. 2 HO2, He 2 2HO2. 2 + CzH5. HO,. -+ i-C3H,HO,. t-C,H,. HO,, +

-+

-+

:

+ + 0, + 0,

0 2 +

+ Nz

-+

-+

+

R* HO2. 20H.

coz + -0.

+ 20H.

+ -+ + Ar H2

OH

+ + + + +

-+

-+

-+

+ + +

N2 20H- HZ 20H. Ar t-CdHBO OH.

+

,4/dm3 mo1-1 s-' E/kJ mo1-1 2.0 x 1OO ' 163 f 6 k = 0.076 =t0.020 3.1 x 1Olo 239 3.2 x 1O1O 180 4 x 1010 213 4 x 1010 199 4 x 10'0 184 4 x 10'0 1.7 x 1Olo 2.5 x 109 3.1 x lo8 1.6 x 1O1O 2.5 x 109 3.5 x 109 3.2 x 1014 7.4 x 1014 3.2 x 1013 4 x 1015~

AH 201 163 159 171 201 209 196.5 196.5 180 176

Temp. rangelIC 670-81 5 713 300-800 300-1 250 670-815 670-8 15 670-8 15

Metho I11

670-8 15 1200-1 800 1400-2600 1400-25Oo 1200-1900 1200-1 900 1700-2600 650-900 650-900 950-1450 300-900

-

I11 -

V V V V V

V g g

V g

se reaction (see p. 174); Estimated (see p. 174); Units of s-l; Rate constants accurate to flOOyo in temperatur ants accurate to i -200-300~0 in temperature range quoted; f Rate constants accurate to f30% in temperature C. J. Jachimowski and W. Houghton, Combustion and Flame, 1971,17,25; D. L. Ripley and W. C. Gardiner, J. Ch W. C. Gardiner, M. McFarland, K. Moringa, T. Takeyama, and B. F. Walker, J. Phys. Chem., 1971, 75, 1504; R. ion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1967, p. 1063; A. M. Dean and G. B. Kistiakowsky, 0; E. Meyer, H. A. Olschewski, J. Troe, and H. Gg. Wagner, 12th Int. Combustion Symp., The Combustion Insti , 1969, p. 345; '' S. W. Benson and G. N. Spokes, J. Phys. Chem., 1968, 72, 1185.

1 74

Reaction Kinetics

As AH,(;= 167 jI 8 kJ mol-l, then Elf w AH:,, which is to be expected since El, m 0. Assuming A,, = A,,, then from k,, = 0.076 dm3 mo1-1 s - , at

713 K, Elp = 156 6 kJ mol-l, in good agreement with E,, = 163 i 6, as expected because of the almost identical aldehydic C-H bond-dissociation energies in the two aldehydes (Table 3). Based on the present parameters, older estimates 9 7 of k , for aldehydes at lower temperatures are clearly too high by factors in excess of los, suggesting either that the measured initial rates were affected by secondary initiation or that surface initiation was important. Recent low-temperature studies by Dixon, Skirrow, and Tipper 98 have confirmed this view. The measured value of kl, increases from 7.4 x to 2.9 x at 345 K and from 6.3 x to 5.1 x dm3 mo1-1 s-l at 393 K when the S/ Vratio increases from 0.6 to 6.1 cm-l. These values are 109-1010 times too high for homogeneous initiation by (la) and correspond to activation energies of only 45-65 kJ mol-l. CH,CHO + 0, + CH$O + HOP(la) Parameters for the reactions:

H2 + 0, -+ H* t H 0 2 -

and

H,O,

$.

0, -+2 H 0 2 .

( m (-14)

can be estimated using the calculated equilibrium constants and the rate constants of the reverse reactions. A value of k-lh = 1.7 x 1O1Odm3 mol-l s-l at 773 K is obtained 9 9 from a detailed interpretation of the second-limit explosion pressures of hydrogen oxygen mixtures in aged boric-acidcoated vessels. From AlberslOO results, k-lh e 1.7 x 1Olo at 300 K, so that EIh= 0 and El,,= AH,", = 239 kJ mol-l. From (standard state, 1 atm) = 5.0 J mo1-l deg-l lol, A l h l ' A - l h = 1.8, so that A,, = 3.1 x 1O1O dm3 mol-l s-l. Similarly, using lo2-lo4k 14 = A14 = 2.0 x lo9 (see Section 5, p. 181), then k-,, = 3.2 x 1Oloexp( -180 kJ mol-l/RT) dm3 mol-l s-l. The close agreement in the A factors for the three similar initiation reactions (If), (lh), and ( -14) suggests that the calculated values are reliable and further that the rate constants for the reverse reactions k-,,, and k , , are reasonably accurate. From the above data, the best Arrhenius parameters for the general reaction (1) are El = AH: and A , = 4 x 1 O 1 O dm3 mol-l s-l, the value A l ,

+

A. Combe, M. Niclause, and M. Letort, Rev. Inst. France Petrole, 1955, 10, 786, 929; J. B. Farmer and C. A McDowell, Trans. Furaduy Soc., 1952, 48, 624. g8 D. J. Dixon, G. Skirrow, and C. F. H. Tipper, 'Combustion Institute European Symposium', Academic Press, London, 1973, p. 94. 9B R. R. Baldwin, M. Fuller, J. S. Hillman, D. Jackson, and R. W. Walker, J.C.S. Fczraduy I, 1974, 70, 635. loo E. A. Albers, Dissertation, University of Gottingen, 1969. In1 J.A.N.A.F. Thermochemical Tables, Dow Chem. Co., Midland, Michigan, 1960. lo2 S. N. Foner and R. L. Hudson, Advances in Chemistry Series, 1962, vol. 36, p. 34. lU3 T. T. Paukert and H. S. Johnston, J. Chem. Phys., 1972,56,2824. lo4 J. Troe, Ber. Bunsengesellschaft Phys. Chem., 1969, 73, 946. 97

83 9

Table 5 Relative rates of initiation A1s-l or Reaction dma mol-l s-l E/kJ mol-l 196.5 7.4 x 1014 H202 M + 20H. M' 163 2 x 1O1O RCHO 0,+ R e 0 H02* 176 4 x 1015 BdOOH -+ But6 OH368 CH4 M + CH3. H* M 1.0 x 1014 4 x 10" 238 CH4 0 2 + CH3. HO2. 370 5.0 x l0l6 C2H6 + 2CH3. 163 213 4 x 1010 CzHG 0 2 + Et. HOz. 343 6.3 x 10" (CH3)dC -+CH3' But* 286 2.0 x 10'6 (CH3)3CC(CH3)3 + 2But. 342 1.9 x 1017 n-C4Hlo+ 2Et. 23 199 4 X 10" n-C4Hio 0 2 + Bu'* HO2. 345 6.3 x 1017 i-C4Hlo CH3- Pr'. 184 4 x 1Olo i-C4HI0 O2 -+ But. HOB. 359 1.3 x 10'' C3Hs + CH3. C2H3.r 163 4 x 1Olo C3H6 O2 allyl. HO,. 299 2.0 X 1016 1-C4H8 + allyl. CH3. 8 9 182 2C3H6 + C3H5. 4-Pr". lo6 2.5 x 1O1O

+

+ + ' + ' + + + ' + ' +

+ + + + + + +

--f

-+

+ + +

'

+ +

+

'

1

(ml,w 2 1 , or Ws-l"

is

A

r

573 K 5.0 x 6.0 x 2.6 x 10-l 1.6 x 10-22 1.2 x 10-13 9.1 x 10-l8 4.0 x 10-l2 3.4 x 10-15 1.5 x 10-10 1.2 x 8.6 x 10-l1 2.2 x 10-14 1.9 x 10-s 2.4 x 10-17 1.5 x 10-7 1.1 x 10-11 1.8 x 10-9

773 K 1.6 x 10-1 4.0 x 4.7 x lo3 5.4 x 10-14 2.4 x 4.9 x 10-s 3.2 x lo-' 4.0 x 10-7 8.5 x 10-4 1.4 x 3.2 x 3.0 x 10-6 3.0 x 6.9 x 10-0 7.9 x 10-4 1.2 x 10-4 1.9 x 10-4

1273 K 1.6 x lo4 5.1 2.3 x lo8 2.0 x 10-4 1.8 x 3.2 x lo1 9.0 x 5.2 x 102 3.5 x 104 1.7 x 103 3.6 x 10-1 4.4 x 109 1.4 2.2 x 101 1.0 x 101 1.1 x 104 1.1

1773 K 2.2 x lo6 2.8 x lo2 2.5 x 1O1O 2.5 6.0 6.2 x lo5 1.9 x lo1 4.9 x 106 7.2 x 107 1.5 x lo7 4.4 x lo1 4.3 x 107 1.2 x 102 3.3 x 105 5.4 x 102 3.1 x 107 1.0 x 102

2273 K ' $ 3.1 x lo7 2.5 X lo3 c] 3.5 x lo1' 8 4.9 x 102 4 7.3 x 101 1.6 x lo8 % 3.6 x lo2 '& 8.3 x 108 3 5.1 x 10s 8 2.6 x los 8.0 x 10' 7.4 x 109 1.7 x lo3 7.1 x 107 5.0 x 103 2.7 x 109 1.2 x 103

" [MI= 200, [O,] = 100 mmHg; See Table 4; R. Hartig, J. Troe, and H. Gg. Wagner, 13th Int. Combustion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1971, p. 147; F. Baronnet, M.Dzierzynski, G. M. Come, R. Martin, and M. Niclause, Internut. J. Chem. Kinetics, 1971, 3, 197; R. S . Konar, R. M. Marshall, and J. H. Purnell, Trans. Furuday SOC.,1968,64,405; f G. A. Chappell and H. Shaw, J. Phys. Chem., 1968, 72, 4672.

3

8 5 9

176

Reaction Kinetics

would take if El, = AH,, = 167 kJ mol-l (from Table 3) rather than the experimentally determined value of 163. No account is taken of the number of C-H bonds in RH in selecting the value of A,. Rate constants calculated from these parameters should be accurate to a factor of about 2 4 in the temperature range 650-900 K. As primary initiation is rapidly masked by secondary initiation, a precise value of kl will only be needed in the very early stages of reaction in computer modelling. Table 4 summarizes the rate data for reaction (l), together with rate constants obtained recently for other initiation reactions. Initiation by decomposition of hydrocarbons is considered separately in Table 5 . It is quite clear that the reactions H, + 0, and H,O, + 0, producing H 0 2 - radicals are much slower than the alternative initiation reactions producing OH- radicals. Although the Arrhenius parameters for CO O2 show considerable variation, they all predict the same rate constant at 2000 K within a factor of 2, with the exception of the parameters obtained by Brabbs, Belles, and Brokaw,lo5whose results are probably the least reliable. Similar consistency applies to the values of the rate constant for the reaction H2 -+ 0, = 20H- at about 1300 K. In the oxidation of hydrocarbons at high temperatures, primary initiation by decomposition of the hydrocarbon must be considered likely, particularly at low oxygen concentrations. Table 5 gives rate data over the temperature range 570-2270 K for possible homogeneous initiation reactions in the oxidation of various typical hydrocarbons. For direct comparison of rates, values of k[M] (unimolecular, second order), k [ 0 , ] (bimolecular), and k (unimolecular, first order) are given for M = 200, Po, = 100mmHg. The rate constants used for the pyrolysis reactions are considered reliable, but they have not been subjected to critical assessment. Several points may be made. (i) At low temperatures, initiation by organic peroxides completely dominates all other homogeneous processes, and any attempt to measure k, will be futile. However, as indicated by Baldwin and Walker,3f formation of ROOH becomes unimportant as the temperature increases to 650 K, where reaction (8D) is very slow conipared with alternative reactions of RO,. (see Section 8, p. 194)

+

R 0 2 *+ RH

+ ROOH t

R.

(8D)

(ii) Above 650 K, homogeneous dissociation of hydrogen peroxide becomes important, and at 770K the figures show that primary and secondary initiation rates become equal when the concentration of hydrogen peroxide reaches of the methane concentration and lo-,% of the isobutane concentration. Even in the special case of propylene, where reaction (1) is relatively fast, secondary initiation becomes dominant after 1 % conversion into hydrogen peroxide. Higher conversions into aldehyde, 0.03 % lo5

T. A. Brabbs, F. E. Belles, and R. S. Brokaw, 13th Int. Combustion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1971, p. 129.

Rate Constantsfor Gas-phase Hydrocarbon Oxidation

177

(methane)-10 % (isobutane), are necessary before secondary initiation by RCHO + O2becomes more important than reaction (1) at 750 K. (iii) As the temperature is raised above 750K, primary initiation occurs increasingly by decomposition reactions involving homolysis of the C--C bond. At 1250 K, under the conditions used for the calculations, reaction (1) can be completely ignored except for the special cases of methane and propylene. This is particularly true with hexamethylethane, where the rate of reaction (1) (k similar to that for CzHs 0,) is negligible even at 750 K, and with but-1-ene, where the pyrolysis rate is fast because of the resonance energy of stabilization of the ally1 radical. 8 9 The bimolecular initiation reaction involving two propylene molecules logcould be important at low oxygen concentrations in the middle temperature region. In summary, reaction (1) is of limited importance in the oxidation of hydrocarbons. Below 650 K, homogeneous primary initiation is normally swamped by surface processes or by secondary initiation; between 650 and 850 K, dissociation of hydrogen peroxide is likely to be dominant even in the early stages of reaction; and above 850K, pyrolysis reactions of hydrocarbons normally become faster than reaction (1). Taylor and Kulich 37 have demonstrated this latter point experimentally at 870-900 K using highly diluted ethane oxygen mixtures. Bradley and Durden l o 7consider that the only role of oxygen in the shock-tube oxidation of propane at 12001600 K is to promote the initiation reaction and to react with the products of pyrolysis. However, from the figures in Table 5 , it is unlikely that oxygen exerts any influence in the primary initiation step. General exceptions are the oxidation of methane; where reaction (1) is important between 1200 and 2200K,lo8 and the oxidation of aldehydes, where reaction (1) is less endothermic. In particular, if hydrogen peroxide is efficiently destroyed at the surface, then in the absence of other surface effects, reaction (1) may be the sole initiation process at about 750 K.61v 62

+

+

5 Radical-Radical Reactions

Recombination of Alkyl Radicals.-Relative

rate constants for disproportionation and recombination reactions of alkyl radicals and related species are accurately k n o ~ n . l110 ~ ~However, * almost all absolute rate constants obtained for radical-molecule reactions involving alkyl radicals are essentially based on measurements relative to radical-radical recombination processes. Unfortunately, radical-radical reactions have been extremely difficult to study on an absolute basis. Until recently, the only technique available was the rotating-sector method, with all its attendant difficulties.lll Erratic M. Simon and M. H. Back, Canad. J. Chem., 1973,51,2934. N. Bradley and D. A. Durden, Combustion and Flame, 1972, 19, 452. D. J. Seery and C. T. Bowman, Combustion and Flame, 1970,14, 37. lo9 J. A. Kerr and A. F. Trotman-Dickenson, Progr. Reaction Kinetics, 1961, 1, 105. J. A. Kerr, Ann. Reports (A)., 1968, 65, 189. ll1 S. W. Benson, 'Foundations of Chemical Kinetics', McGraw-Hill, New York, 1960, lo@

lo' J. lo8

p, 112.

178

Reaction Kinetics

results, often in error by factors of lo2-lo4, are frequently obtained, and only for CH3-radicals has the sector method produced undoubtedly reliable results.27 In consequence, it is usually assumed that all alkyl-radical recombinations have a temperature-independent rate constant of 1010-3 dm3 mol-l s-l, the accepted value for methyl recombination at about 300-450K.27 However, with the development of kinetic spectroscopy (Method IV), very low pressure pyrolysis (Method VIII), and the radical buffer technique (Method IX), there is little doubt that this assumption may be in error by several orders of magnitude.

Table 6 Secmd-order rate constants for the recombination of alkyl radicals" Radical CH,.

log,, (kr/dmsmol-'s-l) 10.39 f 0.05 10.38 j~0.04 10.42 f 0.06 10.38 f 0.04 10.62 f 0.06 9.92 f 0.15 (117 nmHg) 110.05 f 0.15 (234 mmH)} 8.6 f 1.1 9.6 f 0.9 10 7.5 f 1.0

I CzH5.

N

i-C,H7-

t-C,H,.

N

H.

+ C2H5.

{

1250 354 415 850 298

8.6 f 1.1 9.5 & 0.2 6.3 f 0.7

41 5 680-808 298

5.4 f 1.2 5.6 f 1.0 8.5 7.6 f 1

373 455 620 1100

10.56 f 0.18 (50 mmHg He) CH3. t-C,H,* 8.1 f 0.5 c H ~ = c H - ~ H ~ 9-87 f 0.20 9.70 f 0.20 9.9 0.3 CH i 11.1 f 0.3 Si(CH,), 2.5 f 2.0

+

TIK 293 295 298 313 298

298 770

1;:) 298 298 850

Method K.S. (IV) K.S. (IV) K.S. (IV) K.S. (IV)

K.S. (IV) Shock tube

Ref. b C

7 d e 112

Buffer (IX) f Buffer (ZX) 78 75 VLPP (VIII) Crossg combination Buffer (IX) 79 VLPP (vrIr) h Crossg combination Buffer (IX) 80 HCl pyrolysis 116 VLPP (VIII) h Reverse 117 reaction Discharge 122 Pyrolysis

118

VLPP

120

K.S. (IV) K.S. (IV) Reverse reaction

121 123 124

a -&-= dCR.l k, [R*I2; H. E. van den Berg, A. B. Callear, and R. J. Norstrom, Chem. dt Phys. Letters, 1968, 4, 101; F. Bayrakceken, J. H. Brophy, R. D. Fink and J. E., Nicholas, J.C.S. Faraday I, 1973, 69, 228; F. K. Truby and J. K. Rice, Internat. J. Chem. Kinetics, 1973,5, 721 ; W. Braun, A. M. Bass, and M. Pilling, J. Chem. Phys., 1970, 52, 5131; fR.Hiatt and S . W. Benson, J. Amer. Chem. Sac., 1972, 94, 6886; g D. C. Montague, Internat. J. Chem. Kinetics, 1973, 5, 513; D. M. Golden, personal communication.

Rate Constants for Gas-phase Hydrocarbon Oxidation

179

Table 6 summarizes the second-order rate constants obtained recently for the recombination of alkyl radicals. The rate constant for the general reaction : R- R- R2 (r) is defined by

+

--f

The room-temperature data for the CH,. radical are in excellent agreement, the recommended value being k,(2CH3*)= 1010-400*04 dm3mol-l s-l. Earlier values obtained by the rotating-sector method agree closely with this value.27 The collision frequency for CH,. radicals is about 101l.l at 298 K, so that CH,. radicals recombine in about 1 in 5 collisions, as predicted ~tatistically.~~ Any activation energy must undoubtedly be very low, as suggested by the shock-tube values of k,(2CH3.),112which although they show a slight pressure dependence must be near their high-pressure limit. Teng and Jones llSgive loglo(k,/dm3mol-l s-l) = 10.42 - (1.8 kJ rnol-l/2.303RT) for CH, radicals between 300 and 600 K from a discharge-tube study of the reaction between H- atoms and ethylene. The results are obtained from a computer fit of a very complex mechanism, and although the activation energy is reasonable (corresponding to a T* dependence), the value of k, at 298 K differs by a factor of 2 from the value recommended above. ll4* 115 The effect of pressure on k,(2CH3.) has been extensively but the results are inconclusive, although there is general agreement that P+ increases markedly with temperature, as predicted theoretically. Also, the experimental values of the A factor for ethane decomposition are 300-500 % lower than those calculated thermodynamically using the measured values of ~ , ( ~ C H , W As) k, . ~and ~ k - , are usually measured under completely different conditions, the magnitude of the discrepancy is perhaps not too surprising. As Table 6 shows, the values of k, for C2H5*,i-C3H7*,and t-C4H9-radicals are most uncertain; data for higher alkyl radicals are not available. However, it now seems likely that k,(2 i-C,H7*) and k,(2 t-C,H9*) are considerably below 1010-3by factors of 101-103 and 103-105, respectively, in the temperature range 300-1000 K. The error range in the radical buffer method is due to the uncertainty in the thermodynamic quantities necessary for the calculation of k,, and the main error in the VLPP method arises from wall effects. The rate constants are plotted in Arrhenius form in Figure 1. The pre-exponential A factors are unlikely to exceed 1010-5 dm3mol-l s-l and for C,H,* and i-C3H7. radicals this value is assumed, leading to activation energies of 7.4 and 14.9 kJ mol-l, respectively. The points for t-C4H,. are

'

112

T. C. Clark, T. P. J. Izod, M. A. di Valentin, and J. E. Dove, J. Chem. Phys., 1970,53, 2982.

llS 11*

115

L. Teng and W. E. Jones, Trans. Faraday Sac., 1972,68, 1267. H. Shaw and S. Toby, J . Phys. Chem., 1968,72,2337. P . C. Kobrinsky, G. 0. Pritchard, and S. Toby, J. Phys. Chem., 1971,75, 2225.

180

Reaction Kinetics

particularly scattered, and the line drawn in Figure 1 gives log,,(k,/dm3 mols-l) = 9.5 -(21.2 kJ mol-l/2.303 RT). With this expression, k,(2 t-C,H,*) should be reliable to an order of magnitude at all reasonable temperatures. Fortunately for most competitive studies, it is k! that appears in the rate molecule expression, so that the uncertainty in the rate constant for t-C,H,reactions is effectively reduced to a factor of about 3.

+

I

I

0

2

1

3

1

103 K I T

Figure 1 Arrhenius plots for the recombination of C2H5, of i-C3H7,and of t-C,H,

radicals.

-4 d[R1 -- kr[R.I2. Data given in Table 6. The vertical lines indicate error at

limits.

Thermochemicalcalculations 11* via the rate constant for the decomposition of hexamethylethane 117 suggest that Er(2t-CqHg*)= 20-25 W mol-l, in good agreement with the value above. This activation energy appears quite reasonable for the recombination of two bulky groups such as the t-C,H,radicals, although it implies a ‘tight’ transition state and a consequently lower A factor than for the recombination of CH3*radicals, where a ‘loose’ transition state is necessary. A similar argument may apply to the recombination of i-C3H7. radicals. Of related interest is the value of k, = 108.1dm3 mol-ls-l at 770K for the combination of CH3- and t-C,H,. radicals, obtained by Purnell et al.llRfrom a study of the pyrolysis of neopentane. Using the normally accepted l o g *110 geometric rule for the combination of unlike radicals, krra= 2(k,k,)* (22) 116 11’ 118

D. F. McMillen, D. M. Golden, and S. W. Benson, J. Amer. Chem. SOC.,1972,94,4403. W. Tsang, J. Chem. Phys., 1966, 44,4283. M. P. Halstead, R. S. Konar, D. A. Leathard, R. M. Marshall, and J. H. Purnell, Proc. Roy. SOC.,1969, A310, 525.

Rate Constants for Gas-phase Hydrocarbon Oxidation

181

then with kr(2CH3*)= 1010.4,kr(2 t-C,H,*) = 105s2dm3 mol-l s-l at 770 K, in general agreement with the observed ‘low’ values. Use of the geometric rule may, however, be inappropriate here. If the low value for kr(2t-C4Hg.) arises mainly from the steric repulsion between two t-C,H,* radicals, then since there will be little repulsion between CH3- and t-C4Hg*radicals, the rate constant for this cross-combination should be closer to k,(2CH3.) than to kr(2 t-C4Hg*). The geometric rule undoubtedly holds for the crosscombination of CH,. and i-C3H,. radi~a1s.l~~ It appears that k,(2CH2=CHcH,) 1 2 0 * lZ1is almost as high as k,(2CH3*), which may be significant because there should be little steric repulsion between two CH2=CHcH2 radicals on recombination. Thus the value of k,(2CH2=CHcH2) may be typical for the recombination of unbranched alkyl radicals such as n-C,H,* and n-C4H,*, and, in the absence of further information, a value of 10IO.Odm3mol-1 s-l is recommended for the recombination of these radicals in the temperature range 700-1000 K. The value lZ2of k,(H- C2H5.) = 1010-56 0.15 at 298 K and 50 mmHg must be close to its high-pressure limit, and from the cross-combination rule kr(2C2H5-)is at least dm3mol-1 s-’ at 298 K. The value for k,(2CHi) 123 is probably slightly high, as it is approximately equal to the collision frequency, but it provides support for the high values for kr(2CH3*).The very 125 for kr(2SiMe3)is only reliable to about lo2, but is consistent low value with the low values for k,(2 t-C4H,*).

+

*

lZ49

Radical-Radical Reactions not involving Recombination.-Table 7 gives the rate constants of some radical-radical reactions of importance in hydrocarbon oxidation. Radical-radical reactions in the hydrogen oxygen system have been reviewed recently by Baulch et al.l6 The situation with respect to the mutual reaction of H02*radicals is relatively satisfactory now that Foner and Hudson’s lo2original value for k,, has been confirmed. The discrepancy between the values obtained by Paukert and Johnston lo3and by Hochandel, Ghormley, and Ogren 126 is difficult to explain because spectroscopic detection of HO,. was used in both studies. It is significant, however, that Hochandel et al. give a value for k(OH. HO,.) that is certainly at least an order of magnitude too high, based not only on the other experimental values in Table 7, but also on the fact that on thermochemical grounds the A factor for the reverse reaction would be at least 10l2dm3mol-l s-l, which is at least a factor of 10 too high. At present, the Hochandel value for k14 must be considered of doubtful accuracy. Troe’s lo4 value of k14 = loge3* dm3mol-l s-l at 1200 K implies that E14 = 0.

+

+

Oe3

11* lZo

lZ1 lZ2 123 lZ4

lz6

J. Grotewold, E. A. Lissi, and M. G. Neumann, J. Chem. SOC.(A), 1968, 375. D. M. Golden and S. W. Benson, J, Amer. Chem. SOC.,1969,91,2136. H. E. Van den Bergh and A. B. Callear, Trans. Faruday SOC.,1970,66, 2681. M. J. Kurylo, N. C. Peterson, and W. Braun, J. Chem. Phys., 1970,53, 2776. W. Braun, J. R. McNesby, and A. M. Bass, J. Chem. Phys., 1967,46, 2071. J. C. J. Thynne, J. Organometallic Chem., 1969, 17, 155. M. C. Flowers, Ann. Reports (A), 1969, 66, 166. C. J. Hochandel, J. A. Ghormley, and P. J. Ogren, J. Chem. Phys., 1972, 56, 4426.

Reaction Kinetics

182

Table 7 Rate constants for. radical-radical reactions Reaction

2HO2. + H202

OH.

+

02"

+ HOz*+ H 2 0 + 0,

+

+ +

*On HO2. + OH. 0 2 CH,. -0. -+He HCHO 2C2H3. + C2H4 C2H2

He

+ HCO

+

-+

+ H2 + CO

log10 (kldm3 mol-ls -')

TIK

9.34 i 0.03 9.25 & ? 9.76 f 0.05 9.3 f 0.3 11.08 f ? 9.6 f 0.3 10.3 & 0.3 10.5 j=0.3 3 10.25 9.50 f 0.04

298 298 298 1 200 298 1050 1400 1050 300 298

10.3

+ 0.3

1000-1700

Method K.S. (IV)

Mass. Spec. K.S. (IV)

Spectroscopic K.S.

Flames Spectroscopic Flames Mass. Spec. Flash/Mass. Spec. Flames

Ref. 103 102 126 104 126 127 b 127 130 131 C

a - - d'Ho g 2 = k[HO2*IS; J. Troe, 14th Int. Combustion Symp., The Combustion dt Institute, Pittsburgh, Pennsylvania, 1973, p. 145; W. G. Browne, R. P. Porter, J. D. Verlin, and A. H. Clark, 12th Int. Combustion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1969, p. 1035.

+

Neglecting Hochandel's value, there is satisfactory agreement on k(OH. HO,.), but D i x o n - L e ~ i s " values ~~ of Q.0. H02.) = 10IOJ at 1050 K is probably a factor of 2-3 too high on thermochemical grounds. Considerable interest 128* 129 has been shown in the products of the reaction between CH,. atoms, which is frequently the main path for the removal of radicals and -0. CH,. radicals in hydrocarbon flames and oxygen-atom discharge tubes. Recently, Morris and Niki 130 have demonstrated by mass spectrometry that the major primary products in the CH,. .Om reaction are formaldehyde and the H atom: CH,. -0- HCHO + H*

+

+

+

--f

The high value 131 (109+5at 298 K) for the disproportionation of two vinyl radicals lends further support to a value of about 1O1O dm3 mol-' s-l for the recombination of C,HS. radicals. There are no direct experimental values of the absolute rate constants for radical-alkoxy (ROO)reactions or for radical-alkylperoxy (RO,.) reactions. Estimates of the recombination rate constants for RO. radicals have been made from the rate constants of the reverse reactions and thermochemical data. Gray, Shaw, and Thynne 132 suggest kr(2CH3t)) w kr(2n-C,H,t)) w

lZ8

12* 130

131

M. J. Day, K. Thompson, and G. Dixon-Lewis, 14th Int. Combustion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1973, p. 47. H. Niki, E. E. Daby, and B. Weinstock, J. Chem. Phys., 1968, 48, 5729. H. Niki, E. E. Daby, and B. Weinstock, 12th Int. Combustion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1969, p. 277. H. D. Morris and H. Niki, Internat. J. Chem. Kinetics, 1973, 5, 47. K. 0. McFadden and C. L. Curry, J. Chem. Phys., 1973, 58, 1213. P. Gray, R. Shaw, and J. C. J. Thynne, Progr. Reaction Kinetics, 1967, 4, 63.

Rate Constantsfor Gas-phase Hydrocarbon Oxidation

183

kr(2 t-C4HQo)w 109v3, and Heicklen 133 gives kr(2CH36) = 1 O 8 s 8 . More recently, Leggett and Thynne 134 have suggested values of 10Q.2(CH30), 10Q.8(C2H50),10g.'(i-C3H70), and 10Q.4(t-C4HQ0), and Phillips and COworkers have given 108'7(n-C3H76) 135 and 108.g(s-C4Hgb) 136 dm3mol-1 s-1, all in the temperature range 350-500 K. The values are reasonably consistent, and there is no indication that the rate constants decrease markedly as the alkyl group becomes branched. Although possibly of importance in competitive studies of Rb-molecule reactions, radical-ROO processes 133 will not in general be important at the higher temperatures in hydrocarbon oxidation because of the high reactivity of RO- radi~a1s.l~~ The lack of data on mutual reactions of RO,. radicals is particularly unfortunate, since it prevents the use of direct competitive studies for the determination of the absolute rate constants of R0,--molecule reactions. Heicklen 133 has suggested k24 = k25= logs5* at 300 K. RO2. RO2. + 2RO. 0, (24) ROz* HO2. + ROOH 0 2 (25) O a 5

+

+

+

+

The values are derived from photo-oxidation studies involving very complex mechanisms and are based on his values for the mutual reaction of RO* radicals. Knox46 has suggested that k25 = k14 = 10Q-5is acceptable, but considers that k24 may be much lower. Baldwin and Walker 137 have quantitatively re-interpreted results obtained by Euker and Leinroth lS8on the oxidation of n-butane at about 600 K, on the basis that radical-radical reactions involving RO,. are major propagating steps. It is an essential part of the interpretation that at least one of the rate constants kz4or k25 for the s-C4HQ02* radical should be close to lo9dm3mol-l s-l and it is entirely reasonable that k25 for s-C4Hg02should be at least loQsince reaction (25) is a simple, very exothermic, abstraction reaction. Recent studies 139 of the photo-oxidation of azomethane at 298 K have given the ratio k,6/(kz7k24m)* = 0.31. CH30,- CH30 -+CH300H HCHO (26) 2CH30 -+CH30H HCHO (27) 2CH302*-+~ C H ~+O O2 (24m)

+

+ +

Reactions (26) and (27) are very exothermic abstraction reactions, and assuming the reasonable values of k,, = lO8o5 and k2, = 10Q.O, then k24, = l O 9 . O dm3mol-l s-l, giving support to high values for the general reaction (24). Extensive reliable data exist for mutual reactions of R02- in the liquid J. Heicklen, in ref. 32, p. 23. C. Leggett and J. C. J. Thynne, Trans. Furuduy Soc., 1967, 63, 2504. lJS R. L. East and L. Phillips, J. Chem. Sac., ( A ) , 1970, 331. lS6 R. F. Walker and L. Phillips, J. Chem. SOC.( A ) , 1968, 2103. la7 R. R. Baldwin and R. W. Walker, Combustion and-Flame, 1973, 21, 5 5 . lS8 C. A. Euker and J. P. Leinroth, Combustion and Flume, 1970, 15, 275. 13* R. Shortridge and J. Heicklen, Cunud. J. Chem., 1973, 51, 2251. lJJ

lS4

184

Reaction Kinetics

pha~e.l*~y 141 Solvent effects preclude their use with gas-phase studies. Nevertheless, it is interesting to note that activation energies of about 34, 9, and 0 kJ mol-l have been suggested for the mutual reactions of tertiary R02. , secondary RO,. , and primary ROz*radicals in the liquid 6 Radical Decompositions and Isomerhations Radical Decompositions.-Radical decomposition by C-C homolysis competes with oxidation reactions of alkyl radicals at temperatures above about 650K.50-53 Homolysis of the C-H bonds is considerably slower at temperatures below 1500 K.23-25 From measurements of the relative yields of decomposition products and oxidation products, absolute rate constants can be obtained for the oxidation processes if the rate constants for the decomposition reactions are known. Radical-decomposition reactions have been reviewed r e c e r ~ t l y . ~There ~-~~ is considerable inconsistency in the rate data, and A factors, for example, vary by up to lo4for a particular radical decornpo~ition.~~ The major reasons for the lack of good data are fairly obvious. Clean, reliable sources of alkyl radicals are difficult to find, the temperature range of the study is often limited to about 330 to 450K, pressure dependences are uncertain, and frequently, the results are based on incomplete product analyses and doubtful mechanisms. Fortunately, some check on the reliability of the rate constants is usually available from thermochemical data and the Arrhenius parameters of the reverse r e a ~ t i o n . 2 ~ -Direct ~~ photolysis of azo-compounds and mercury-photosensitized decomposition of alkanes have so far provided the most reliable rate Table 8 Alkyl radical decompositions and isomerizations" Reaction A1s-l ElkJmol-'

+ + + +

n-C,H,. -+ C,H4 CH,. i-C3H,*-+ C2H4 + CH,. n-C4H,- -+ C2H4 C2H5. s-C4H,. -+ C3H64- CH,. i-C4H9- C,H, CH,. (CH,),CcH2 -+ i-C4H, CH,. l-n-C,H,,2-n-C5Hll. l-n-C6H,,* + 2-n-C6H,,--f

1.6 x 1014

126 &- 2 1 x 1014 > 188 2.5 x lo1, 120 f 4 7.3 X loi4 144 & 5 2.8 x 10l2 137 8 k = 3.53 x lo3 121 k 3 2.5 x 1013 1.0 x 101' 85 f 8 8.5 x lo* 47 f 8 (1.0 X loll) (61 & 8)

T/K 523-623 750-850 430-520 533-613 543-598 762 503-608 350-500 300-380 300-380

Ref. 6 , 143 23 b, h c, i c,j

d, 144 e , 145 f, 150 f,k g,k

First-order rate constants; ' Using k, = 101o.odm3 mol-'s-' (with similar units for other k, values); Using k, = 101o.s exp(--14.9 kJ mol-'/RT); Recalculated from original data, see text, p. 185; Using k,(CH,Me,CCH,) = 1010*34;f u s i n g k,(2C,H5-) = k,(CH,- +l-n-C,Hll-) = k,(C,H,* l-n-C,H,,.) = lolo; VSee text,p. 187; W. E. Morganroth and J. G. Calvert, J . Amer. Chem. Soc., 1966, 88, 5387; M. C. Lin and K. J. Laidler, Canad. J . Chem., 1967,45, 1315; f D. A. Slater, S. S. Collier, and J. G. Calvert, J. Amer. Chem. SOC.,1968, 90, 268; k K. W. Watkins, J. Phys. Chem.,1973,77,2938. 140 D. F. Bowman, T. Gillan, and K. U. Ingold, J . Amer. Chem. SOC., 1971,93, 6555. 141 J. A. Howerd, Adv. Free Radical Chem., 1971, 4, 49. 142 J. E. Bennett, D. M. Brown, and B. Mile, Trans. Faraday SOC.,1970, 66, 386.

+

+

Rate Constants for Gas-phase Hydrocarbon Oxidation

185

Table 8 gives the Arrhenius parameters for alkyl radical decompositions considered by the author to be the most reliable; these parameters are used later in the Report. The Arrhenius parameters for the n-C,H,* and n-C4H,decompositions have been recalculated from the original results using the recombination rate constants derived in the last section. It is assumed that k,(2 s-C4HQ-)= k,(2 i-C,H,-) = k,(2 i-C,H,-) = 1010.6exp(- 14.9 kJ mol-l/RT) dm3mol-1 s-l. The values of the decomposition rate constants so calculated are in good agreement with those derived from thermochemical data and the Arrhenius parameters of the reverse reactions.26 The Arrhenius parameters 23 given in Table 8 for reaction (28) almost certainly represent the maximum values for the rate constant, and it is now accepted that this type of decomposition is much slower than simple C - C h o m o l y ~ i s . ~ ~ ~ i-C3H7*-+ C2H4 CH,. (28)

+

Two values are given for the decomposition of the neopentyl radical, a single value of 3.53 x 103 at 762 K obtained from Benson and Anderson’s 144 study of the pyrolysis of neopentane in the presence and absence of hydrogen chloride, and ksn = 2.5 x 1013exp( -121 f 3 kJ mol-l/RT) s-’ over the range 500-610 K from a study of the mercury-photosensitized decomposition of neopentane by Furimsky and Laidler.146 Benson and Anderson’s value has been recalculated from their original ratio k,,/k3, = 1.95 x lo2 dm3mol-l, using more recent values of k2n.118* 147 CH3* (CH3)& + CH4 (CHd3CCH2 (2n) (CH3),CeH2 (CH3),C=CH2 CH3* (3n) Furimsky and Laidler’s Arrhenius parameters for reaction (3n) are obtained relative to k = 1010.34dm3mol-1 s-l for the cross-combination of CH3*and (CH3),CCH2 radicals, which may be too high by a factor of at least 10 (Section 5). As the combination rate constant enters their kinetic expression fully, and not as the half-power, then the values of k3, may also be at least an order of magnitude too high. Extrapolation of the Furimsky-Laidler values to 762 K and reduction by a factor of 10 gives k3n = 1.2 x lo4s-l, in tolerable agreement with the Benson-Anderson value. The latter value is preferred here, not only because of the uncertainty in the combination rate constant, but also because it has been determined in the temperature range where reaction (3n) is important in neopentane oxidation. No reliable, experimentally determined rate constants are available for the decomposition of higher alkyl radicals, but reasonable estimates can be obtained from thermochemical data and the rate constants of the reverse 1469

+

+

--f

+

148 143

145

140

ld7 148

M. M. Papic and K. J. Laidler, Canad. J. Chem., 1971, 49, 535, 549. K. H. Anderson and S. W. Benson, J . Chem. Phys., 1964,43, 3747. E. Furimsky and K. J. Laidler, Canad. J. Chem., 1972, 50, 1115, 1123. J. A. Kerr and D. Timlin, J . Chem. SOC.(A), 1969, 1241. R. R. Baker, R. R. Baldwin, C. J. Everett, and R. W. Walker, submitted for publication. K. W. Watkins and L. A. O’Deen, J. Phys. Chem., 1969,73, 4094.

186

Reaction Kinetics

Despite a number of careful investigation^,^^^ 149 high-pressure rate constants for the decomposition of alkoxyl radicals show serious disagreement.23 Commonly, the A factors are 3-4 orders of magnitude lower than expecred, a firm indication that many of the studies were carried out in pressure-dependent regions. Nevertheless, it is quite clear that alkoxyl radicals decompose relatively rapidly and, on thermochemical grounds, a rate constant of *') exp( -70 f 10 kJ mol-l/RT) s-' may be suggested for the firstorder decompositions: C2H5O -+ CH3. + HCHO (CH3),CH0 CH3* CH3CH0 (CH,),CO -+ CH3* CH3COCH3 CH3CH2CH26 C2H5* HCHO CH3CHCH2CH3 + C2H5* CH3CHO. -+

-+

I

+ + + +

0. At pressures greater than 100 mmHg and at temperatures in excess of 550 K, decomposition is the most likely reaction of alkoxyl radicals (except CH36) in oxidation environments. Isomerhation Reactions of Alkyl Radicals.-For the quantitative interpretation of hydrocarbon oxidation, it is necessary to know whether alkyl radical isomerization can compete with decomposition and oxidation processes. Unfortunately, few reliable rate constants are available. Early studies 1503 151 of the isomerizations (29) and (30) suggested abnormally low A factors of about lo7s-l, compared with estimates of 1011-1012 s-l. More recent studies,lS2and corrections to earlier data,lS0including the use of kr(C2H5. 1-n-C6Hls-)= lo1* instead of 1010*64 dm3mol-ls-l, have given the values shown in Table 8. 1-n-C5Hll*-+2-n-C5Hll* (29) 1-n-C6H13' + 2-n-C6HI3* (30) At 770 K, k29 = 2 x 105s-l and k30 = 5.5 x lo5 s-l, compared with k,, = 2 x lo5s-l, CH3CH2CHzCHz + CzH5. + C2H4

+

so that isomerizations (29) and (30) are likely to compete favourably with decomposition and oxidation reactions of these radicals at this temperature. The entropy change in both isomerizations is approximately zero, so that the equilibrium constants are determined solely by the enthalpy changes, which are both about 14.5 kJmol-l. At 770K, K2, = K30 = 0.10, so that k-29 = 2 x lo4 and k-30 = 5.5 x lo4s - l compared with k3b,= 1.5 x lo5 s149

15" 151

15*

l.

CH3cHCH,CH3 -+ CH3CH=CH2

+ CH3*

(3b')

M. J. Yee Quee and J. C. J. Thynne, J. Phys. Chem., 1968, 72, 2824; C. Leggett and J. C. J. Thynne, J . Chem. SOC.( A ) , 1970, 1188. K. W. Watkins, Canad. J . Chem., 1972, 50, 3738. L. Endrenyi and D. J. Le Roy, J. Phys. Chem., 1966, 70, 4081. K. W. Watkins, J . Phys. Chem., 1973, 77, 2938.

Rate Constantsfor Gas-phase Hydrocarbon Oxidation

187

As the temperature falls below 770 K, isomerizations will become slow relative to oxidation reactions of alkyl radicals, and at higher temperatures decomposition reactions will increasingly dominate because of their higher activation energy. The radical isomerizations of 1-n-C,HI1* and 1-n-C6H13*involve 1,4 and 1,5 intramolecular migration of a hydrogen atom: H

/H

/

CH,-C:-CH,

I

.---H

CH,-CH,

1-n€, H,,. 1,4 H transfer

l-n€, H,, 1,5 H transfer

Isomerization of CqHg-and C3H7-radicals would require 1,3 and 1,2 H-atom transfers with four- and three-membered-ring transition states, respectively, which involve very high strain energies leading to high activation energies. In consequence, isomerization of C3H7-and C4H9*radicals is slow relative to competing decomposition and oxidation reactions, as suggested by the results of McNesby and c o - w o r k e r ~ . ~ ~ ~ The strain energy involved in the formation of the ring transition states in isomerizations (29) and (30) can be calculated if it is assumed that the activation energy for these processes is the sum of the strain energy and the activation energy, EA, for an intermolecular hydrogen abstraction from a secondary C-H position by a primary aIkyl radical. If EA is taken as

+

+

n-C3H7* C3H8 + i-C3H7* C3H8

(31)

45 l d mol-l, the value for reaction (31),f54the strain energies are (2 4~8) and (40 f 8) kJ mol-l for the 1,5 and 1,4 H atom transfers respectively. The difference of 38 kJ mol-l is too high and probably arises because the A

factor for 1-n-C6H13* isomerization is too low. Taking the more reasonable value of A30 = loll s-l, the strain energy for 1,5 transfer becomes (16 f 8) kJ mol-l. These strain energies are of considerable interest to the discussion in Section 8. 7 Radical Attack on Alkanes and Related Compounds

Extensive information exists in the recent literature concerning the Arrhenius parameters for hydrogen abstraction from hydrocarbons knd related compounds by atoms and radicals.lO9 17--21 With the development of quantitative R. McNesby, C. M. Drew, and A. S. Gordon, J . Chem. Phys., 1956, 24, 1260; W. M. Jackson and J. R. McNesby, J . Amer. Chem. SOC.,1961, 83, 4891; J . Chem. Phys., 1962, 36, 2272; W. M. Jackson, J. R. McNesby, and B. de B. Derwent, ibid.,

us J.

1962,37, 2256.

R. E. Berkley, G. N. C . Woodall, 0. P. Strausz, and H. E. Gunning, Cunad. J . Chem., 1969,47, 3305.

188

Reaction Kinetics

gas chromatography and mass spectrometry, increased interest has been shown in the interpretation of the product distributiori in hydrocarbon oxidation. Consequently, much attention has been focussed recently on the determination of the parameters for attack at specific positions, so that the proportion of primary, secondary, and tertiary radicals can be estimated, and their role in the oxidation assessed. Studies with alkanes suggest that Arrhenius parameters for specific attack by H W , and * ~ OH* 66 can be obtained from an additivity rule by assuming that the contribution per C-H bond to the total rate constant is the same for all primary, all secondary, and all tertiary C-H bonds in each hydrocarbon. The overall rate constant is then given by expression (32), where n is the number of bonds of a specific type, A is the Arrhenius factor per C-H bond, and E is the corresponding activation energy. 499

k

=

npApexp( -E,/RT)

+ neAsexp( -E,/RT) + n,A, exp( -E,/RT)

(32)

The subscripts p, s, t, refer to attack at primary, secondary, and tertiary C-H bonds, respectively. Support for the additivity rule comes from studies of the reactions of C1- 156 atoms and CF,. 15' radicals with alkanes, where the specific parameters can be obtained directly from product analysis. Although no great precision is claimed, because of slight variations in entropy factors and in the dissociation energies of C-H bonds of the same type in different molecules, expression (32) represents the only method available for the calculation of the overall rate constant and the relative proportions of primary, secondary, and tertiary radicals produced at any temperature. Table 9 gives the specific parameters for a number of radicals of importance in hydrocarbon oxidation. Greiner's 66 parameters for the OH. radical are used in preference to those suggested by Baker, Baldwin, and Walker,49 whose rate constants are slightly high owing to self-heatingeffects in hydrogen + oxygen mixtures (Method 11). Allowance. for the self-heating gives values l5* very similar to those of Greiner. It i5 abundantly clear from Table 9 that radical selectivities for attack at different C-H sites are not solely determined by differences of activation energy. Two simple semi-empirical methods have been developed for the estimation of activation energies in radical-molecule reactions. First, equation (33) predicts the activation energy to within 4 kJ mo1-l of the experimental values, for attack on alkanes by radical X- = H . , CH3* C2H5*,1. , Cia , Bra, CCl,. , OH., or CH,O. .159 E 155 156

15' 158

159

= 51

- 30.10:,..

+ 2.30AH" + (AH0)2/(167 + 2.21AH"j)

(33)

J. T. Herron and R. E. Huie, J. Phys. Chem., 1969, 73, 3327. J. H. Knox and R. L. Nelson, Trans. Furaduy SOC.,1959, 55, 937; M. H. J. Wijnen, C. C. Kelly, and W. H. S. Yu, Canad. J. Chem., 1970,48, 603. M. Haluk Arican, E. Potter, and D. A. Whytock, J.C.S. Faraduy I, 1973, 69, 1811. R. R. Baldwin, M. Fuller, and R. W. Walker, unpublished work. R. R. Baldwin and R. W. Walker, J.C.S. Perkin 11, 1973, 361.

h

\

8

Table 9 Values of A (per C--H bond) and E for radical attack at primary, secondary, and tertiary bonds Primary Secondary Tertiary ,4/dm3mol-l s - l E/kJmo1-1 A/dm3 mol-’ s-l E/kJmol-’ A/dm3 rno1-I s-l E/kJmol-1 Reaction

+

H. RH *O*i RH OH. RH HO,. RH ROSS RH CH,. RH CH,O* RH

+ + + +

+

2.2 x 10’0 5.0 x 109 6.15 x lo8 1 x 109 1 x 109 4.9 x 108 5.3 x 107

40.5

24.2 6.9 81 .o 81 .o 49.0 29.5

5.0 x 1Olo 1.3 x 1Olo 1.4 x 109 1 x 109 1 x 109 3.3 x 108 3.6 x 1 0 7

35.0 18.8 3.6 71.3 71.3 42.3 18.7

See pp. 193 and 194; P. Gray, A. A. Herod, and A. Jones, Chem. Rev., 1971,71,247; Bimolecular Gas Reactions’ (see ref. 18).

8.7 x 1O1O 1.6 x 1O1O 1.25 x 109 1 x 109 1 x 109 2.4 x loE 1.9 x 107

29.3 13.8 -0.8 60.3 60.3 33.6 11.6

B s8

ct; Ref.

a

49 155 66

2

a

&

a

’

b

g

c

g.F

A. F. Trotman-Dickenson and G. S. Milne, ‘Tables of

5

190

Reaction Kinetics

Units of kJ mol-l are used throughout, and is the Taft parameter le0 for the XCH2-group and is a measure of the electronegativity of that radical relative to that of the CH,. radical. The term 30.1a~c,2. gives the lowering of the activation energy due to polar effects relative to E when X = H, so that the plot of (E,,, 30.lazCHz.)against AH" should give a common curve for all X* RH reactions, where Eexp is the experimental activation energy. This conclusion has been confirmed 169 with the above radical + alkane systems over the range AH" = + 125 to -125 kJ mol-l. It is clear that the EvansPolanyi type of relation, E = a + bAH", can only hold accurately over very narrow ranges of AH". Secondly, Alfassi and Benson la have obtained good agreement between observed and calculated activation energies for about 30 reactions of the type:

+

+

A.

f-

BC

-+

AB

+ C.

where B is a singly bonded atom, by the use of electron affinities of the groups A and C to allow for polar effects in the reaction. Recent evidence for significant deviation from linearity of conventional Arrhenius plots (non-Arrhenius effects) in the reactions of He atoms le2and simple alkyl radicals 1e2-1e5 with alkanes, and in the reaction of CH,* radicals with hydrogen,lse requires further confirmation. The case for significant non-Arrhenius effects in hydrogen-atom reactions le2 is very slender, and it is more likely that experimental inaccuracies and incorrectly assumed stoicheiometries at low temperatures are responsible for the curvature. If the effect is spurious in the case of H- CH,, then on thermochemical grounds there should be very little curvature below 2000 K for CH3*t H2. The reports of non-Arrhenius effects in alkyl radical alkane reactions mostly come from pyrolysis studies in the temperature range 900-1 100 K ; unfortunately, the mechanisms are complex and any conclusions drawn must be considered premature. Nevertheless, the question of non-Arrhenius effects for CH,. + H2 and CH,. alkane systems is particularly important in hydrocarbon oxidation, because CH3*radical concentrations are frequently monitored by measurement of the methane formed in such reactions. Owing to the difficulty of finding suitable radical sources, there is very limited information on the gas-phase reactions of HO,. , RO,. , and RC03with hydrocarbons and related compounds, despite their great importance in the oxidation of organic compounds. Unfortunately, c&~,. values for these radicals are not available, so that equation (33) cannot be used. Table 10 summarizes the rate constants available for HOz*reactions. The rate

+

+

1e391e5

+

160

162

163 le4 166 166

J. E. Leffler and E. Grunwald, 'Rates and Equilibria of Organic Reactions', Wiley, New York, 1963, p. 219. Z. B. Alfassi and S. W. Benson, Internut. J . Cltem. Kinetics, 1973, 5 , 879. T. C. Clark and J. E. Dove, Canad. J. Chem., 1973, 51, 2147. P. D. Pacey and J. H. Purnell, J.C.S. Furuduy I , 1972,68, 1462. P. D. Pacey and J. H. Purnell, Internat. J . Chem. Kinetics, 1972, 4, 657. P. D. Pacey, Canad. J. Chem., 1973, 51, 2415. T. C. Clark and J. E. Dove, Canad. J. Chem., 1973,51,2155.

Table 10 Rate constants for reactions of H02-radicals Reaction

+ C0-t CO, + OH*

HO,.

+ + +

+ 202 + OH. + SO, +He0

HOB. 0 9 OH. HO2. NO -+ NO, HOZ. SO, + OH. HO,. HCHO + Ha02

+

+

-+

HOB. C2H2CH0-+ H20, HO,. CSH, + H802 HO2. CSHB -t Ha02 HO,. i-C4Hlo-+ H20, HO2. H2 -+ H202

+ + +

+

+ C2H2c0 + C2H5. + CsH,, + C4HD. + H.

k/dms mol-l s-l < 3 x 10s > 1 x 108 < 7 x 105 <6 5.5 x 109 2.1 x 104 4.5 x 10s

-1.8 x lo8 5.2 x 105 1.01 x 106 1.9 x lo6 1.4 x 104 3.6 x 104 8.5 x 104 1.5 x 10s

Where appropriate, data corrected using kr(2H0,. $= Volman and R. A. Gorse, J. Phys. Chem., 1972,76,3301; J. Heicklen, J. Phys. Chem., 1973,77, 1932. @

T/K 373-473 298 298 300

:::}

E/kJ mol-'

-

1.0 x 10"

100 i 15

-

798

255-298 300 300 77G970

2.0 x 107

773 713 753 753 753 773

loo.*dm8mol-'s-l;

-

-

8.5 -

1.1 x 10'0

43.5

2.0 x 109 1.0 x 109 6.0 x 109

48.8b 37.1b 81.P

-

-

-

Method Photolysis Discharge Photolysis Photolysis H202/CO I1 Thermal, CHd02 Photolysis Photolysis Photolysis Thermal, HCHO/Op 111 I11 I11 I11 I11 111

Rex 171 167 C

d 168 168 169 e 170 170 172

173 56 55 55 55 54

Assuming A (per C-H bond) = 109*0 dm3mol-I s-1; D. H. D. D. Davis, W. A. Payne, and L. J. Stief, Science, 1973,179,280; R. Simonaitisand

192

Reaction Kinetics

parameters for the important reaction (34) are still very uncertain, although it is now clear that Westenberg and de Haas' 167 value at 298 K is many orders of magnitude too high. HO2* CO -+ CO, OH* (34)

+

+

Baldwin, Walker, and Webster's 168 estimate of k,, = 5.5 x lo3 at 713 K is almost certainly the most reliable value available, as it is determined directly (relative to k;.J from the yields of CO, in H,O,-CO-N, mixtures, and its magnitude has been confirmed by studies of the addition of carbon monoxide to slowly reacting mixtures of hydrogen and oxygen at 773 K.60 Hoare and Milne logquote k21fk34= 280 at 798 K from studies of the oxidation of = 48.8 kJ mol-l, (the recommended methane. Using Azt = 2.0 x log, parameters in Table 10) gives k,, = 4.5 x 103 dm3mo1-l s-l at 798 K, which is consistent with the BWW value 168 only if E3* w 0. However, the mechanism for methane oxidation is extremely complex, and little reliance can be placed on Hoare and Milne's value.

+ HCHO

HOz*

-+

H202

+He0

(2f)

Very tentative Arrhenius parameters of A , , = 1 x loll dm3mol-l s-l and E34 = 100 f 15 W mo1-1 may be suggested from the values of k34 at 713 and 773 K. Reaction (34) must thus be very slow at 300 K, and cannot be considered an important route for the conversion of CO into CO, in either the upper or the lower atmospheres. Reaction$ of HO,* radicals with nitric oxide l 7 O , 171and sulphur dioxide 1 7 0 are considerably faster than reaction (34), and almost certainly are important in the conversion of nitric oxide into nitrogen dioxide and of sulphur dioxide into sulphur trioxide in urban atmospheres through the cycle:

+

OH. CO -+ CO, H. + 0,+ M -+ HO,. H02* SO, +SO,

+ H+M + OH.

+ H02*+ NO -+NO, + OH-

(35) (36)

(37) (38)

The rate constant for reaction (35) is now well known,l* and at low temperatures is accurately given by k35 = 3.1 x lo8exp( -2.8 kJ mol-l/RT) dms mol-1 s-l. Reaction (34) is, therefore, very fast under atmospheric conditions because of the low activation energy. In contrast, above 700 K, because of the low A factor, OH. radicals react with CO much more slowly than with hydrocarbons, so that high concentrations of CO in the exhaust gases from combustion engines are unavoidable under rich conditions. A. A. Westenberg and N. de Haas, J. Chem. Phys., 1972,76, 1586. R. R. Baldwin, R. W. Walker, and S. J. Webster, Combustion and Flame, 1970,15, 167. la* D. E. Hoare and G. S. Milne, Trans. Furuduy Soc., 1967, 63, 101. 170 W. A. Payne, L. J. Stief, and D. D. Davis, J. Amer. Chem. SOC.,1973, 95, 7614. 171 R. Simonaitis and J. Heicklen, J. Phys. Chem., 1973, 77, 1096. le7

lB8

Rate Constantsfor Gas-phase Hydrocarbon Oxidation

193

Vardanyan, Sachyan, and Nalbandyan’s value of 1.1 x 1Olo exp( -43.5 W mol-l/RT) dm3mol-1 s-l for kzrwas obtained from a detailed study of the oxidation of formaldehyde between 773 and 973 K. Their parameters give a value of k,, at 773 K about 13 times higher than that obtained by (Table 10) from the effect of formaldehyde on the induction Baldwin et period of slowly reacting mixtures of hydrogen + oxygen (related to Method 11). The kinetics and mechanism of the slow reaction between hydrogen and oxygen under the conditions used are well establi~hed,~~ and it is vary unlikely that Baldwin’s figure is seriously in error. Unfortunately, therefore, Vardanyan, Sachyan, and Nalbandyan’s expression fork,, must be considered suspect, although their activation energy seems reasonable; complexities in the mechanism are the probable cause of the discrepancy. Assuming an A factor of 1.0 x loBper C-H bond, then Baldwin’s value of k,, = 1.01 x lo6 at 773 K gives A,, = 2.0 x loBdm3mol-1 s-l and E,, = 48.8 kJ mol-l. The value of k,, = 1.9 x 106 dm3mol-1 s-l at 713 K was obtained, relative to kid,from direct studies of the oxidation of propionaldehyde in aged boricacid-coated vessels su (Method 111), and the rate constant should be accurate to about 10%. H 0 2 * C2H6CH0-+ H202 C,H&O (2P)

+

+

+

For the HO,. alkane reactions, a competitive technique was used, in which the alkane was added to slowly reacting mixtures of formaldehyde and oxygen in potassium-chloride-coated vessel^.^^^ O2 In the case of ethane, the reactions of interest are as follows: 61v

+

H02* HCHO -+ H202 H e 0 0,--t HO,. HO2* CzHU --t HZO, C2H.5. 0 2 - - + C2H4

+

+

+

+He0

+ CO + C2H,*

+ HOz*

(20 (39)

(24 (5Ae)

The hydrogen peroxide is efficiently destroyed at the vessel surface before it can decompose by reaction (13) to give OH*radicals, so that the system is a ‘clean’ source of HO,. radicals. Under the conditions used,, reactions (39) and (5Ae) are effectively the sole reactions of H e 0 and C2H6*radicals, respectively, so that in the early stages of reaction, drC,H,l/drCOl

= ~2erC,HUI/~,fr~CHOl

(4)

Computer treatment allows for the carbon monoxide formed in the initiation process (If), and for the small amount of homogeneous decomposition of hydrogen peroxide which occurs at the temperatures used above 770 K.

+ +

HCHO 0, H e 0 H202 M + 2 0 H * 172

17s

+ HO2+M

(If) (13)

1. A. Vardanyan, G. A. Sachyan, and A. B. Nalbandyan, Combustion and Flame, 1971, 17, 315. R. R. Baldwin, A. R. Fuller, D. Longthorn, and R. W. Walker, J.C.S. Faraday I, 1972,68, 1362.

194

Reaction Kinetics

With propane and isobutane, allowance is necessary for the small amount of radical decomposition that occurs, and the ratios k,,,/k,, and k,Jk,, can then be accurately measured.

+

HO,* C3H8 + H202 HO,. i-C4Hlo + H,O,

+

+ C3H7. + C4HQ.

(2Pd (2i)

+

The results for k(HOz* alkane) must be regarded as preliminary, and further experiments to determine activation energies are in progress. The absolute values of k2e, k,,,, and kzi have been calculated using k,, = 2 x loQ exp( -48.8 kJ mol-l/RT) dm3mol-1 s-l, as recommended earlier. Assuming the additivity principle, then the specific rate constants, per C-H bond, for HOB*radical attack at the primary, secondary, and tertiary positions in alkanes are 2.3 x 105, 1.1 x lo4, and 6.4 x lo4dm3mol-l s-l, respectively, at 753 K. With A (per C-H bond) = 10Q.0dm3mol-1s-1,the specific activation energies are given in Table 9. The variation in the values of the activation energy from 49 for HO,. + HCHO to 81 kJmol-l for H02* C2H6 correlates well with the enthalpy changes in the reactions, although the absolute level may not be quite correct, For instance, the value of 60.3 kJ mo1-1 for the almost thermoneutral reaction HO,. isobutane appears a little high unless Do(HO2-H) is 10-20kJmol-1 lower than the recommended value of 376 kJ mol-1 (Table 3). However, relative rate constants calculated from the parameters for the HO,. reactions should be accurate, and even absolute values should be reliable over the limited temperature range 650-900K because errors in A and E will largely compensate. Knox 174 has pointed out that the rate constants given in Table 10 for H02* alkane are in remarkable agreement with relative rate constants obtained from studies of the initial stages (1 %) of competitive oxidations involving ethane, propane, and isobutane at 570-670 K, on the assumption that the hydrocarbons are removed solely by HO,. attack. No reliable experimental rate constants are available for RO,. abstraction reactions in the gas phase. However, if D"(R0,-H) = D0(HO2-H) as suggested by Benson 86 and by Golden,175then E ( R 0 , . + RH) M E(H02* RH), although the A factor for R 0 2 *reactions may be slightly lower than for HOz*reactions. Knox 46 has estimated kSDi= 20 dm3mol-l s-l at 373 K, from the results obtained by Allera et ~ 2 1 . lon ~ ~ the low-temperature oxidation of isobutane. At 373 K, attack at the primary C-H position is negligible, and the value may be compared with k,,, = 35 dm3mol-1 s-l, calculated from the specific HO,. parameters in Table 9.

+

+

+

+

+ +

+

t-C4HQ0,. i-C4Hlo + t-C,H,O,H t-C,HQ* H 0 2 * i-C,Hl,, + H202 t-C4H9* 174

175

+

(8Di) (2it)

J. H. Knox, 13th Int. Combustion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1971, p. 259. D. M. Golden, 14th Int. Combustion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1973, p. 121. D. L. Allera, T. Mill, G. Hendry, and F. R. Mayo, in ref. 32, p. 40.

Rate Constantsfor Gas-phase Hydrocarbon Oxidation

195

+

On the basis that the general reaction R029 RH is a perfectly normal radical-molecule metathesis reaction, Golden 175 has suggested that k,,, = 108s5exp( -59 kJ mol-l/RT) dm3mo1-I s-l, but reports a personal communication from Mill who suggests that there is evidence that k 8 D P may be more like exp( -71 kJ mol-l/RT). i-C3H70z* + C3H, -+ i-C3H700H

+ i-C3H,-

(8DP)

Allowing for the two secondary C-H bonds in propane, Mill’s expression for k8,, is in excellent agreement with the specific parameters given in Table 9 for the attack of HOz. radicals at a secondary C-H bond. In the absence of further information, it is recommended that the specific parameters for HOz* alkane reactions should also be used for R 0 2 * alkane reactions. Griffiths and Skirrow 63 give kql = 1.9 x 10gexp(-3OkJmol-l/RT) dm3mo1-1 s-l between 350 and 550 K, and Dixon, Skirrow, and Tipper g8 suggest k42/k41 = 2.4 and k4z/k,3 = 1.0 between 390 and 450 K, which suggests that reactions (41) to (43) have very similar Arrhenius parameters. The activation energy is somewhat below that suggested for HOz*attack on formaldehyde, but the difference is probably within experimental error, particularly in view of the assumed A factor for the HOz*reactions and the complexities involved in the low-temperature oxidation of 98 Further, RC03*radicals are more electronegative than H02. radicals, and a slightly smaller ( 6 8 kJ mol- l) activation energy for RC03* abstraction reactions is expected.lsg

+

+

84p

CH3C03*+ CH3CH0 + CH3C03H CH3C03*+ HCHO --+ CH3C03H C2H5C03- HCHO -+ C,H,C03H

+

+ CH3c0 +He0 + He0

(41) (42) (431

Only one or two rate constants are available for the addition of peroxy radicals to olefins, despite the importance of these reactions in hydrocarbon oxidation.28 From studies of the addition of ethylene to slowly reacting mixtures of formaldehyde and oxygen in potassium chloride-coated vessels at 773 K (Method 111), Baldwin, Fuller, Longthorn, and Walker 61 give 1.4 x lo4dm3mol-1 s-l for the overall rate constant of reaction (44). HOz.

+ C2H4 -+ CzH40+ OH’

Waddington and Ray 177 report a value of k45= 2 x lo6dm3mo1-1 s-I at 457 K, based on Griffiths and Skirrow’s value for kql (see above), from a study of the co-oxidation of acetaldehyde and cis-but-2-ene. CH,CO,* + cis€H,CH=CHCH,

17’

--+

CH,CH-CHCHJ

‘d

+ CH,CO,-

D. J. Waddington and D. J. M. Ray, J. Phys. Chem., 1972, 76, 3319.

(45)

196

Reaction Kinetics

Both rate constants represent overall processes, since the formation of oxirans must result from a sequence at least as complex as (46).

8 Oxidation Reactions of Alkyl Radicals General Discussion-Very few firm rate constants are available for oxidation reactions of alkyl radicals; this is not surprising as the mechanism of oxidation is still far from clear. It is necessary, as a consequence, to define clearly the rate constants that will be used. The formation of products is usually written in a formal way such as: R*

+ 0,- -

4

AB

+ HO2.

(5A)

where AB is the conjugate olefin. Each product could, however, be formed by the direct bimolecular reaction (6), via reaction (8) of RO,. , or via reaction (10) of the alkyl hydroperoxide (QOOH) radicals formed by the isomerization of R 0 2 *by intramolecular hydrogen-atom transfer : products

products

"/'

R* + 0,

k7

F====* k- 7

k8/'

R02.

---kg

products klO/f

QOOH

k-9

The overall rate constant, such as k,, for the formation of product A, is then given by equation (47),31

r

where xk8[X]and Ekl0[X]are the summations of all processes removing R 0 2 *,QOOH, by reactions of the type ROz- + X , QOOH + X , respectively. It should be noted that k,, will only be entirely independent of mixture composition if all reactions of type (8) and (10) are first-order, unimolecular processes. If, as is generally assumed, k8 is zero, and k - , is negligible, equation (47) simplifies to equation (48).

Rate Constantsfor Gas-phase Hydrocarbon Oxidation

197

Further, assuming k-7 B kQ,so that reaction (7) is effectivelyequilibrated, equation (48) can be approximated further to equation (49), where F is the

k,A

=

k 6 ~-I- K7kQF

(49)

fraction of QOOH radicals which forms the product in question. If this product is very dominant, then F approaches unity. Expressions considerably more complex than equations (47)449) operate if various different QOOH radicals are formed by the isomerization reaction (9), as is usually the case. Fish, Haskell, and Read have developed essentially similar equations to account for the products formed in the oxidation of 2-methyl~entane.~~ Oxidation Reactions of the Methyl Radicals.-Although the products from the oxidation of methyl radicals are relatively limited, the mechanism is still the subject of extensive 5 8 * 178 Two reactions are normally considered. CH3* 0 2 -k M -+ CH302* M (7m) CH,* 0,-+ HCHO OH* (6Bm) 479

+

503

+

+ +

Rate data obtained prior to 1965 have been reviewed by McMillen and Calvert 38 and by Hoare and P e a r ~ o n .Following ~~ a recent study by Basco et al. using kinetic spectroscopy,8reaction (7m) is established as the dominant reaction of CH3*radicals with oxygen at low temperatures. At 298 K, the order of the reaction lies between 2 and 3 over the pressure range 25-380 mmHg (M = nitrogen or neopentane) and the results are entirely consistent with the sequence: CH3* O2 CH30z3* CH3O;* M -+ CH302* M

+ + +

+

As the CH,. radical concentration is measured directly, in contrast to earlier methods, the rate constants from this study are recommended. The limiting low-pressure rate constants are (3.6 f 0.3) x 10" dms molF2s-l (M = neopentane) and (0.94 f 0.03) x loll dms mol-2 s-l (M = nitrogen) at 298 K. The limiting high-pressure rate constant obtained was (3.1 k 0.3) x lo8 dm3mo1-1 s-l. The limiting low-pressure rate constants are considerably 3 8 , 73 who used higher (factors of about 3-10) than those of earlier less reliable photolytic methods, but van den Bergh and Callear,Qwho also used kinetic spectroscopy, report a preliminary low-pressure value of approximately 2.2 x 10l1dm6mo1-2 s-l [M = propane] at 298 K. The agreement between the two studies is excellent. Basco et aL8 also report k6Bm < 2 x lo5dm3 mo1-1 s-l at 298 K, using direct observations of the OH. spectrum. The low rate constant for reaction (6Bm) is confirmed by VLPP studies 75 (Method VIII), over the temperature range 500-1400 K. No measurable formaldehyde was detected in over lo4 dm3 collisions between CH,. radicals and oxygen. Taking AGBm= 10Q,6 mol-ls-l, then EGBm > 100 kJ mol-l. Barnard and Cohen 178 have suggested 178

J. A. Barnard and A. Cohen, Trans. Faraday SOC.,1968,64, 396.

Reaction Kinetics

198

similar Arrhenius parameters to explain their results from the photooxidation of acetone between 470 and 670 K. The relative rate of loss of methane and hydrogen, when small amounts of the former are added to slowly reacting mixtures of hydrogen and oxygen at 500 mmHg and 773 K, can only be explained if the reaction between CH,. radicals and oxygen is predominantly a termolecular The marked temperature dependence of the relative yield of methane and oxygenated products in the oxidation of acetaldehyde between 713 and 813 K 6 8 can only be explained if the process:

CH,.

+ O2 -+ oxygenated products

(0x1

has an activation energy of about - 8 5 kJ mol-l. Reaction (6Bm) is clearly excluded by a negative activation energy of this magnitude, and the simplest explanation involves the sequence :

+

CH3* 0,+ M CH302*+ X

+ CH302*+ M -+

oxygenated products

(7m) (50)

Reaction (7m) is effectively equilibrated, so that k,, is given by:

and the large negative activation energy thus arises from the high value of E- 7 m w 111 kJ mol-l. The kinetics and products of oxidation 6 8 suggest that X is almost certainly a radical. Ethane, formed from the recombination of CH,. radicals, is produced in surprisingly high yields of about loo/,, so that X is most probably CH,. or CH302.radical. The high concentration of hydrogen peroxide in the oxidation products suggests that reaction (52) may

CH,.

+ H02*-+

CH,O

+ OH.

(52)

also be important. As CH,O radicals will also be formed in reaction (50) if X is CH,. or CH302.,then it is probable that methanol and formaldehyde, the main oxidation products of the CH,. radical at 700-800 K, actually arise from reactions of CH,O. radicals :

C H 3 0 + RH +- CH30H + R. CH& 0, HCHO H 0 2 CH,O + M -+HCHO + Ha M

+

-+

+

+

If radical-radical reactions are occurring to a high degree, considerable caution will be necessary in discussing the mechanism of CH,. oxidation, since radical concentrations may differ by orders of magnitude, and conclusions reached in one study may be completely inappropriate elsewhere. Despite the frustrating problem concerning the nature of reaction (50), it may be concluded quite firmly that (7m) is by far the dominant reaction between CH3*and oxygen up to 800 K. At higher temperatures, the mechanism becomes less obvious.

Rate Constantsfor Gas-phase Hydrocarbon Oxidation

199

Recently, high-temperature shock-tube studies have provided further information on the oxidation of CH,. radicals, which is considered to be adequately represented by the direct bimoleculax reaction (6Bm). The rate constants are summarized in Table 11. Dean and Kistiakowsky 7 0 (DK) shocked carbon monoxide-oxygen-argon mixtures containing traces of methane, and Izod, Kistiakowsky, and Matsuda8g (IKM) used the same basic mixture but with traces of ethane or azomethane. In both studies, the reaction was followed by measuring the i.r. emissions from carbon monoxide and carbon dioxide. The observed [COJ-time profiles were compared with those obtained by computer calculation. Clark, Izod, and Matsuda 71 (CIM) shocked azomethane-oxygen-inert gas mixtures, and monitored the products with a time-of-flight mass spectrometer. Formaldehyde was detected immediately after the induction time, and its presence was used as evidence for reaction (6Bm). Relatively complex mechanisms are involved, and difficulties are experienced in locating the most important rate-determining steps. The rate constants are probably accurate to a factor of 2 or 3, and the CIM value is the most reliable. Higgin and Williams 6 7 (HW) shocked methane-oxygen mixtures and compared measured and calculated ignition delays. Again a complex mechanism is involved, and because of the limited scope of ignition-delay measurements, their rate constant is probably less reliable than the other shock-tube values.

0

2

4

6

8

10' K I T

Figure 2 Arrheniusplot of the rate constantsfor the process CH,. OH. obtained from shock-tube studies. 0, DK;'O0, IKM;69 0, CIM;" A,HW;s7 (see p. 199).

+ 0,+ HCHO +

200

Reaction Kinetics

As shown in Figure 2 and Table 11, the rate constants are hardly in good agreement, although IKM and CIM predict the same value of k 6 B m within a factor of 2 at 1500 K. The line drawn in Figure 2 gives k6Bm = exp( -46 kJ mol-l/RT) dm3 mol-l s-I. Although apparently in serious disagreement with Benson's VLLP these parameters give k 6 B m = 7.8 x lo5 dm3 mol-1 s-l at 773 K, which is only slightly higher than the maximum value of 1.4 x lo5 given by Baldwin, Hopkins, Norris, and Walker 4 7 from the methane addition studies. Table 11 High-temperature" rate constantsfor CH,.

3.0 x 109 1.2 x 1 0 9

42 52

1750-2575 1400-2200 1350 1875

k = 2.0 x lo7 k = 8 X 10'

+ 02--+HCHO + OH-&

1.0 x lo8 1.8 x lo7 3.0 x 5.0 x losd

6.7 1.2 2.0 3.4

x x x x

lo* lo8 lo8 lo7

70 69 71 67

See pp. 199-201 for discussion of the nature of the reaction; Shock-tube studies; Using K , , = 0.15 Accuracy discussed on pp. 199, 200; Using E = 46 kJ mol-'; dm3mol-' (see p, 200).

Although the shock-tube results appear to support reaction (6Bm), it is more probable that the oxidation proceeds by intramolecular hydrogen atom transfer to give the cH,OOH radical, which then decomposes to give formaldehyde. CH,. 0, + M CH,O,. M CH302.-+ c H 2 0 0 H (9m) (10Bm) &I,OOH HCHO OH*

+

-+

--f

+

+

In the absence of other reactions of CH,O2- and c H 2 0 0 H radicals, and with k.-7m k9,, the observed rate constant (now written-as k5Bm) is given by: k5Bm =

LlkoIn

(53)

The values of kSBmare given in Table 11 for the typical temperature of 1500 K ; an activation energy of 46 kJ mol-1 was used for the CIM and HW values. From thermochemical data, K7, = 0.15 dm3 mol-1 at 1500 K, so that k,, can be calculated from the experimental values of ksBmusing equation (53). Independent values of ko, are not available, but a reasonable estimate can be made. As AH;, w 100 kJ mo1-l over the range 15002000 K, then from ESRm= 46 kJ mol-l (Figure 2), Egmw 146 kJ mol-'. Taking the reasonable value (see later) of A9, = 1012s5s-l, then kgmw 3 x lo7s-l at 1500 K compared (Table 11) with most reliable values, derived from k5Bm,of 1.2 x lo8 (IKM) and 2.0 x lo8 (CIM). The agreement in the case of the HW value is excellent, although possibly fortuitous. In view of the uncertainries in the experimental data (200--300%) and in the thermochemical calculations (300-500 %), formation of formaldehyde through the sequence (7m), (9m), and (10Bm) is entirely plausible. Benson's VLPP 75

201

Rate Constants for Gas-phase Hydrocarbon Oxidation

results would be explained if reaction (9m) is pressure-dependent at the very low pressures used. In short, there is no reliable evidence at all for the direct bimolecular reaction (6Bm).

Oxidation Reactions of Higher Akyl Radical s.-Overall Rate Constants. Table 12 summarizes the available rate con stants for the overall reaction (5A)forming conjugate olefin AB. R* 0 2 - - --t A B HO,* (5A)

+

+

Table 12 Rate constants for the fo rmation of conjugate olefin Method k/dms mol-l s-l Ref. TIK 753 I11 5.5 x 107 a, 56 180 896 1.0 x lo8 CZH6/02 I11 723 a, 59 CsH, 1.8 x 1 0 7 I1 a, 52 753 2.2 x 107 I1 a, 184 C4Hs-l 753 1.6 x lo8 I1 a, 184 s-C4H,. 0, trans-C4Hs-2 7.8 x 107 753 ~i~-C4Hs-2 I1 a, 184 4.3 x 107 753 I1 a, 184 CdHS-1 7 53 5.1 x 107 2.3 x 1 0 7 I1 i-C4H9* 0, i-C4H8 7 53 a, c 1.7 x 104 Photo-oxid. 6,181 3 13 a Recalculated from original data using the rate constants in Tables 6 and 8; a Recalculated from original data using k,(2 i-C4HB)= 101n.5exp(--14.9 kJ rnol-l/RT)dms rnol-'~-~; cPreliminary value, R. R. Baker, R. R. Baldwin, and R. W. Walker, unpublished work.

Reaction CZHS. 0,

+ n-C,H,. + O8 n-C,H,. + 0,

Product CzH4

+

+

The values have been recalculated from the original data using rate constants for radical combinations and decompositions given in Tables 6 and 8, respectively. Measurements based on the competition:

+

CzHs* 0 2 - - + CzH4 C,Hs* C2HsCH0 + CzHs

+

+ HO2. + C,H&O

644 (54)

in the oxidation of propionaldehyde at 713 K (Method 111) have been used 66 to determine kSAerelative to the known value of kS4.l7, The value of ksAe= 5.5 x lo7 is consistent with an earlier estimate of 1 x lo8 dm3 mol-l s-l obtained by Sampson lSo from a direct study of ethane oxidation 0,) = 1.8 x lo7 in a fast-flow system at 896 K. Similarly, k(n-C,H,. The other at 723 K from a study of the oxidation of n-b~tyraldehyde.~~ high-temperature rate constants in Table 12 have been obtained using Method 11. Relative yields of lower olefin and conjugate olefin are measured as a function of oxygen concentration, the results showing a competition between reactions (5A) and (3). R*-+ cracking products (3) l7# J. A. Kerr and A. F. Trotman-Dickenson, J . Chem. SOC.,1960, 1611.

+

R. J. Sampson, J . Chem. SOC.,1963, 5095.

lRn

202

Reaction Kinetics

A specific radical gives a specific lower olefin, for instance, n-C,H,*

+

+

C2H5. + C2H4, and s-C,H,- -+CH,. C3Hs only, and when allowance is made for the fact that both radicals (from propane and butane) give the conjugate olefin, accurate values (i10%) for k5A/k3can be obtained. The accuracy of the absolute values of k,, is limited by the uncertainty in the currently available values of k , (Table 8), which depend on radical-radical recombination rates (Table 6) as discussed earlier. A possible error generally of 200-300% in the values of k , , must be accepted, althwgh kSAeis probably firm to about f 50%. The agreement between the two values of k(n-C,H,. + 0,) is encouraging, although both are ultimately dependent on kr(2 n-C,H,-). The relatively high value of k(n-C4HQ* + 0,) offers some evidence that kr(2n-C4H,.) may be lower than the assumed value of 1O1O dm3mol-1 s-l (see Section 5, p. 181). From the only other available value of k,, = 1.7 x lo4 (recalculated) for i-C4H,. radicals at 313 K,lS1E s A = 32 kJ mol-l and A,, = 3.5 x lo9 dm3 mo1-1 s-l. These are reasonable Arrhenius parameters if conjugate olefin is formed in the direct bimolecular reaction (6A), but are not inconsistent with the formation via R 0 2 *46 (8A) or QOOH.31 Overall rate constants are also available for the formation of oxygenated products. Values are summarized in Table 13, and were obtained from competitive studies using Methods I1 and 111. In general, the rate constants are 10-100 times lower than for the conjugate-olefin-forming reactions, and reflect the very high yields of conjugate olefin in the products of the oxidation of ethane, propane, and butane. Table 13a*a Reaction C2H.j.

+

Overall rate constants for the formation of oxygenated products

0 2

+ 0,

n-C,H,-

+ 0, + 0,

n-C4H,s-C~H,.

See equation (47); and 10. a

Product C2H40 CH,CHO C,H,O

CZH,CHO tetrahydrofuran C2H,COCH ,

T/K 773 773 753 753

753 753

k/(dm3 mo1-1 s-l) 1.7 x 106 2.3 x 1.8 x 6.5 x 1.5 x 9.4 x

105

lo6 104 107 105

Ref. 56 56 52 52

184 184

All obtained using Method I1 and rate constants given in Tables 8

Formation of Conjugate Olefins. The nature of the precise mechanism for the

formation of conjugate olefin has received considerable attention. Knox 46 considers that formation via R 0 2 * is most consistent with the available experimental data at low temperatures. Fish,,, without any real evidence, suggests that the direct bimolecular reaction is important above 725 K, but that formation from QOOH is dominant at about 570 K. Berry, Cullis, and Trimm lS2 favour the direct bimolecular reaction even at 570 K, because it is lg2

D. H . Slater and J. G . Calvert, in ref. 32, p. 58 T. Berry, C. F. Cullis, and D. L. Trimm, Proc. Roy. SOC.,1970, A316, 377.

Rate Constantsfor Gas-phase Hydrocarbon Oxidation

203

difficult to explain the relatively high yields of butenes in the oxidation of n-butane if they are formed via the energetically unfavourable 1,4 H-transfer (see later) to give QOOH. Pollard et aZ.lS3have suggested that conjugate olefin is predominantly formed at the surface in the oxidations of ethane, propane, and butane at about 600 K. Recently, from a study of the addition of n-butane and of cis-but-2-ene to slowly reacting mixtures of hydrogen and oxygen at 753 K (Method 11), firm evidence has been obtained lS4to show that QOOH radicals are not a major source of conjugate olefin. Using a mixture containing 140, 70, 285, and 5 mmHg of hydrogen, oxygen, nitrogen, and hydrocarbon, respectively, with cis-but-2-ene the initial value of the product rztio [2,3-dimethyloxiran]/ Itrans-but-2-eneI is 1.35, in contrast to the value of 0.18 when n-butane is used. It is generally accepted that oxirans are formed via QOOH radicals, and Scheme 1 is required to interpret the results. trans-But-2-ene and 2,3dimethyloxiran cannot be formed from sequences involving n-C,H,* radicals, and addition of H atoms to cis-but-2-ene only gives s-C4H,* radicals. If trans-but-2-ene is formed solely from the s-~.,H,OOHradical, then the same value of [2,3-dimethyloxiran]/[trans-but-2-ene]should be obtained for both n-butane and cis-but-2-ene, and the high value for the latter can only be explained if trans-but-2-ene is formed predominantly in the bimolecular reaction or via s-C4H,02*radical. It is important to realize also that keg,, must be considerably lower than klOCb,otherwise reaction (9b) would be effectively equilibrated and the same ratio of products would be obtained even if no trans-but-2-ene is formed via QOOH. trans-c, Ha-2

H*/OH* + C4H,, + s-C,&*

t--

s-k4H800H +(10Cby

CH,CH-CHCH,

'0'

183

184

\

&C,H8-2

+ Ha

cis-C4Ha-2 + H02*

trans-C4Ha-2 Scheme 1

J. G . Atherton, A. J. Brown, G. A. Luckett, and R. T. Pollard, 14th Int. Combustion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1973, p. 5 13. R. R. Baker, R. R. Baldwin, A. R. Fuller, and R. W. Walker, submitted for publication.

.

Reaction Kinetics

204

The two possible routes for the formation of conjugate olefin, (6A) and @A), are virtually indistinguishable kinetically. However, Knox 46 has shown that the A factor required for RO,. decomposition to conjugate olefin in the case of t-C,H,O,* is lo2-lo3 times lower than expected from calculations by Benson and O’Neal.lsa Baldwin and Walker 31 have argued that a change in the mechanism of conjugate olefin formation between 570 and 870 K is most unlikely, and thus in the absence of further evidence, the overall reaction (5A) is best considered a direct bimoIecular process (6A) at all reasonable temperatures. It is possible to explain what Knox4s describes as ‘the puzzling feature of the oxidation of isobutane’. In brief, there is a remarkable transition from a reaction producing 1 % of isobutene at 428 K to one producing 80 % isobutene at 573 K. The relevant reactions are

+

(CH,),C. 0,+ (CH3),C=CH2 t H02. (CH3)3C. 0,-+ (CH3)3CO,. (CH3),CO,* RH + (CH3),C02H R. (CH&CO,* - - other products

+

+

+

--f

(70 (55)

At 428 K, t-butyl hydroperoxide is the dominant product 176 so that k,,,[RH] % k55,and as reaction (7t) will be equilibrated, then d[isobutene]/d[t-C,H,OOH]

=

k6At/K,tk8Dt[RH]

Using K7t = 2.7 x lo8 at 428 K and 8.7 x lo4dm3mol-l at 573 K (Table 2), k,,, = 1 x 10’exp( -60.3 kJ mol-I/RT) (Table 9), and k 6 A t = 3.5 x 10’ exp( -32 kJ mol-l/RT) dm3mol-l s-l (assumed, Section 8, p. 202), when [RH] = 100 mmHg, the relative rate of formation of isobutene and t-butyl hydroperoxide is 0.010 at 428 K and 5.4 at 573 K. Although other reactions of (CH,),CO,* will play some part at 573 K, it is quite clear that the dramatic change in the yield of isobutene is adequately explained. Reactions of RO,. and QOOH Radicals. Major yields of 0-heterocycles are found in the oxidation products of hydrocarbons of carbon number 5 or greater in the temperature region 570-700 K, n-butane (8 %), n-pentane (43 %), n-hexane (1 5 %), n-heptane (55 %), 2-methylpentane (30 %), 3ethylpentane (20-40 %), 3-methylpentane (1 5-25 %).lse Jt is generally accepted that 0-heterocycles are produced by the homogeneous formation 45 and subsequent decomposition of the QOOH radical (PRID R.

+ 0, + RO,.

RO,. QOOH la5 lB6

-+ -+

QOOH 0-heterocycle $- OH.

(7) (9)

(1OC)

S. W. Benson and H. E. O’Neal, J. Phys. Chem., 1967,71, 2903. P. Barat, C. F. Cullis, and R. T. Pollard, 13th Int. Combustion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1971, p. 179.

Rate Constantsfor Gas-phase Hydrocarbon Oxidation

205

For example, 1 3 hydrogen-atom transfer in the pentylperoxyl radical s-C5H1101* will lead to 2,4dimethyloxetan: 3

4

CH3-CH-CH,-CH,-CH3

I

5

6

CH,-

2 0

CH,

CH-C%-~H-

I 0

\0-H

‘0.1

CH,-CHCH,

I

CH,CH-0

I

+ OH-

The letters p, s, and t are used to signify transfer of primary, secondary, and tertiary H atoms, respectively. Oxirans, oxetans, tetrahydrofurdns, and tetrahydropyrans are formed from (1,4), (1,5), (1 $9, and (1,7) transfers, respectively. The relative rates of formation of the various QOOH radicals are determined mainly by three factors: (i) the relative ease of attack at the various C--H bonds in the hydrocarbon (Table 9); (ii) the strength of the C-H bond from which internal hydrogen-atom abstraction occurs (Table 3) ; (iii) the ring strain energy in the formation of the transition state. Activation energies for the isomerizations to QOOH can be calculated from the sum of the relevant strain energy and the activation energy of the corresponding intermolecular reactions ROz* + RH. Fish 40r 45 has assumed strain energies for the transition state identical with those in saturated cycloalkane systzms as follows: eight-membered(1,7 transfer), 42 kJ mol-l; seven-membered, 27 ;six-membered, 2.5 ;five-membered, 27 ; four-membered, 109. Fish assumed A, = lolls1, for all isomerizations, and activation energies of 32, 44, and 58.5 kJ mol-l for RO,. radicals attacking tertiary, secondary, and primary C-H bonds, respectively. His estimates of k, are summarized in Table 14. Fish has attempted to validate his choice of rate constants by a satisfactory semiquantitative prediction of the relative yields of 0-heterocycles in the oxidation of 2-methylpentane,29f 3s Considering reactions (7) and (9) to be equilibrated, Fish used equation (57) to calculate the relative rate of formation of the 0-heterocycles. 2sv

399

d[O-heterocycle]/dt = K7KPk10C[C6H13][021

(57)

However, in the use of the equation, two serious errors were made. First, with the high values used for some of the rate constants of the competing reactions, reactions (7) and (9) are no longer equilibrated, and equation (57) no longer applies. Secondly, the concentrations of the isomeric radicals, lEt

S. W. Benson, J . Chem. Pliys., 1961, 34, 521.

Reaction Kinetics

206

formed from the hydrocarbon, are incorrectly assumed to be proportional to their relative rates of formation. Although Fish later40 used a more appropriate equation to predict the relative yields of products in the highpressure oxidation of 2-methylpentane at 798 K, the mechanism used is almost certainly in error. Lower olefins are produced in high yield (about 20%) and are considered to come from C-C scission reactions of the OOOH radicals. However, studies of the addition of propane and butane to slowly reacting mixtures of hydrogen and oxygen at 753 K show quite definitely that the lower olefins are formed in alkyl-radical decompositions 53 under Fish's conditions. Indeed, a much closer prediction of the relative yields of the individual olefins is possible if they are considered to come from the decomposition of alkyl radicals. The rate constants in Table 14 appear semiquantitatively consistent with the relative yields of 0-heterocycles in the oxidations of n-pentane,lss 3-methylpentane,189and 3-ethylpentane at 570-670 K. However, because of the complexities of the mechanism, the lateness of the sampling, and the large temperature rises frequently observed, it is impossible to determine rate constants of the elementary steps quantitatively. Fish's rate constants extrapolated to 753 K (shown in Table 16, and discussed later) predict virtually 100% yields of 0-heterocycles in the oxidation of n-butane at 753 K. However, yields of less than 5 % are obtained 5 2 and it may be concluded that the values in Table 14 are in general at least lo2 times too high. 509

529

509

Table 14 Kinetic parameters"for the forniation of QOOH radicals at 550 K given by Fish 2 9 * 45 Nature of C-H bond broken

primary

I

secondary

tertiary

a

A

lB8

lBD loo

=

loll s-l

Type of H-atom transfer

E/kJ mol-I

167 88 63 1,5P 88 1,6P 100 1,7P 1,3s 155 71 1,4s 13s 46 71 1,6s 88 1,7s 142 59 33 59 75 1,7t assumed for all Hatom transfers. 1,3P 1,4P

g

log,o(k/s- l) -5.0 2.9 5.2 2.9 1.5 -3.6 4.3 6.6 4.3 2.9 -2.5 5.4 7.7 5.4 4.0

log10K9

-3.2 -3.2 -3.2 -3.2 -3.2 -1.8 -1.8 -1.8 -1.8 - 1.8 -0.7 -0.7 -0.7 -0.7 -0.7

C. F. Cullis, M. Saeed, and D. L. T r i m , Proc. Roy. Soc., 1967, A m , 455. P. Barat, C. F. Cullis, and R. T. Pollard, Proc. Roy. Soc., 1972, A329, 433. P. Barat, C. F. Cullis, and R. T. Pollard, Proc. Roy. Soc., 1971, A325,469.

Rate Constantsfor Gas-phase Hydrocarbon Oxidation

207

For the determination of the rate constants for reactions (9) and (lo), very simple systems are required. The oxidation of neopentane is almost ideal in this respect. Only one species of alkyl radical is formed, and as conjugate olefin formation is not possible, the yield of oxygenated products is magnified. Further, at high temperatures, the neopentyl radical decomposes readily, and from measurements of the relative amounts of decomposition and oxygenated products, rate constants for the oxidation reactions can be obtained using the known rate constant for the decomposition reaction. When traces of neopentane 147 are added to slowly reacting mixtures of hydrogen and oxygen at 753 K, the products methane, formaldehyde, 3,3-dimethyloxetan (DMO), isobutene, and acetone account for over 95 % of the neopentane lost in the early stages of reaction. Further, over a ten-fold range of oxygen concentration, 639

R1 = ([DMO] + [acetone])l[isobutenel = [021 and R,

= [acetone]/[DMO]

= [OJ

These observations are consistent with the mechanism:

689

147

+

C5Hll*+ CH3. (CHs),C=CH, C5H11* 0 2 += C6H,,O,*

+

C5H1102*+ C5HlOOOH C 6 ~ , , , 0 -+0 ~DMO + OH= C&IloOOH O2 + C5Hlo(OOH)02* C5Hlo(OOH)02* + CH3COCH3 2HCHO CH3* 0 2 - - + HCHO OH. CH3*+ H2 + CH4 + H*

+

+

+

+

(3n) (7n) (9n) (1 OCn) (1 OEn)

+ OH.

(58) (5Bm)

Using stationary-state treatment, it can be shown 147 that reaction (9n) is effectivelyirreversible, otherwise the kinetic expression for RLtends towards a square dependence on [OJ at high [OJ, in direct contrast to experimental observation that the dependence on [O,] falls slightly at the higher oxygen concentrations. Unless ksn is very high ( > 5 x lo8s-l), reaction (7n) is equilibrated and R, is given by equation (59). R1 =

K7nkdOJ/k3n

(59)

Using Benson’s value for k3n144(Table 8), with E3,, = 134 kJ mol-l, and K7,, = 86 dm3mol-1 from thermochemical data, kgn= 1.8 x 104 s-l at 753 K. Benson 8 7 suggests that Agn = 1012.1 s-l, so that Egn = 113 kJ mol-l. Thermochemical data give k-gn = 1.2 x lo5s-l, and to predict correctly the yield of acetone, assuming that klOEn= logdm3mol-1 s-l, kIo, = 1 x loos-l at 753 K.147 The rate constants obtained from the neopentane addition studies are summarized in Table 15.

Reaction Kinetics

208

Table 15 Rate constunts at 753 K from the neopentane studies 147

a

Reaction

Type of H-atom transfer

9n -9n lOEn lOCn

1,5P 5P,l -

k1s-l 1.8 x 104 1.2 x 105 1.0 x 109 1.0 x 106

Q

A1s-l a 1012.1 1010.6 1on.o b 1011.8

ElkJ mol-l 113 79 0 83.5

dm3 mol-'s-' units.

All A factors assumed, see ref. 147;

Use of Laidler's value 145 for ksn (Table 8) gives ken = 5 x lo2s-l, which, although considerably lower than Fish's extrapolated value for a 1,5p transition (Table 16), would lead to impossibly high yields of 2-methyloxetan when n-butane is added to slowly reacting mixtures of hydrogen and oxygen at 753 K.ls4 The validity of Laidler's Arrhenius parameters for reaction (3n) has been discussed in Section 6, p. 185. Table 16 Rate constantsfor the formation of QOOH radicals at 753 K Nature of C-H bond broken

Type of H-atom transfer

primary

r

secondary

-(

I r

tertiary Using A

=

193s

: :1 196s 1 3

Recommended kSls-' (1.3 x -lor) 2.2 x 10s 1.8 x 104 6.0 x lo4 (1.4 x lo2) 2.4 x 104 2.0 x 105 (6.5 x 105) 1.5 x 103 (2.6 x 105) (2.2 x 106) (7.1 x lo6)

s-' (see p. 207);

I,

Recommended E/kJ mo1-l a 159 125 113 105 144. 111 98 90 130 96 84

75

14'*la4

kSls-lb using Fish's values 2 x 10-1 1.5 x 105 6.5 x los 1 x 105 2.5 1.5 x lo6 6 x lo' 1.5 x lo6 1.5 x 10' 8 x lo6 4 x 108 8 x lo6

Table 14.

Using n-butane and cis- and trans-but-2-ene as additives in the hydrogenoxygen system (Method 11), values of k s for (1,4p), (1,4s) (oxirans) and (1,6p) (tetrahydrofuran) transitions have been obtained at 753 K.ls4 The rate constants for the (1,5p) and (1,5s) (2-methyloxetan) transitions cannot be isolated, but assuming that methyl ethyl ketone is farmed via a 1,3t transition from s-C,H,O,. radicals, then the rate constant for this highly strained transition can be determined. Using the values of k , determined directly from the butane and neopentane addition studies, it is possible to calculate the rate canstants of the other radical, k,(l ,4s)/k,( 1,4p) possible transitions as follows. For the s-C4H9OP* =- 11 at 753 K. Assuming equal A factors for the two transitions, &,(1,4p)

Rate Constants for Gas-phase Hydrocarbon Oxidation

209

-E,(1,4s) = 15 kJ mol-l, which corresponds to the difference in the bonddissociation energies of primary and secondary C-H bonds. As a tertiary C-H bond is weaker than the secondary bond by the same amount, it may be assumed that EB(1,4s)- EB(1,4t)= 15 kJ mol-1 (equal A factors), so that kl,&/kl,ds= 11 at 753 K. Combining k,(l,4p), kB(1,6p)and kB(1,5p) (knnfrom the neopentane studies, Table 15) with the assumption of equal A factors, then the difference in strain energies between five- and six-memberedring and between six- and seven-membered-ring transition states can be calculated. Table 16 gives the recommended rate constants for reaction (9) at 753 K. The activation energies are based on EB(1,5p)= 113 kJ mol-l, obtained from the neopentane work, and on the assumption that all A factors are 1012.1s-1. The rate constants obtained indirectly are shown in brackets. Assuming no error in the recombination rates of the relevant alkyl radicals (Table 6), it is probable that the relative values of k , determined directly are accurate to about *loo%, and the absolute values to about &3300%. It is very encouraging that the value of kg(1,3p) = 10l2,lexp( -159 kJ mol-l/RT) s-l is very close to the value of kg, = 1012s5exp( -146 kJ mol-l/RT) s-l needed to explain the high-temperature results on the oxidation of CH3*radicals (Section 8, p. 200). CH300.

--f

cH200H

(9m)

The rate constants and activation energies are completely different from those in Table 14. The rate constants for 1,4 and 1,5 transitions are about lo2times lower than Fish’s extrapolated values at 753 K and lo4times lower at 550 K. However, at 753 K, the rate constants for the 1,6 transitions are in tolerable agreement, and for the 1,3 transitions are higher than Fish’s values. The present results are consistent with strain energies of about 20, 30, 40, and 75 kJ mol-1 for (1,6), (1,5), (1,4), and (1,3) transitions, respectively. In view of the uncertainties in the values of the strain energies, arising from the necessary calculation from differences of large quantities, the values of 30 and 40 kJ mol-l are in acceptable agreement with values of 16 f 8 and 40 f 8 kJ mol-1 for the ring transition-state strain energies involved in the isomerization of l-hexyl and l-pentyl radicals which involve 1,5 and 1,4 transitions, respectively (Section 6.2). The rate constants in Table 16, when combined with the appropriate values of k6*,will predict to within a factor of 2 the observed yields of ethylene oxide and acetaldehyde from ethane,48 propylene oxide and oxetan from propane,52and of 2-methyloxetan from n-butane,laawhen the hydrocarbons are added to slowly reacting mixtures of hydrogen and oxygen at 753 K. Further, the high activation energies are consistent with Knox’s observation that the oxygenated products are formed mainly by reactions of R02-at the surface in the slow combustion of n-pentane below 570 K;46vlg1 Fish’s rate constants are far too high for surface processes to be important. lgl

J. H. Knox and C. G. Kinnear, 13th Int. Combustion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1971, p. 217.

210

Reaction Kinetics

Although it would be desirable to use the rate constants in Table 16 to predict the product distributions in cool-flame and slow-combustion studies at about 570K for a number of hydrocarbons, the mechanisms are so complex, the role of secondary products so uncertain, and the conditions so poorly defined that quantitative verification is not possible. Certainly, the new values of the rate constants give a semi-quantitative prediction of the relative yields of 0-heterocycles in the low-temperature studies, and in particular, they predict that tetrahydrofurans are always formed in greater amounts than oxetans and oxirans in the oxidation of pentanes l a 8and substituted pentanes.42* l a BlB0 $ Recent studies ls2 of the oxidation of 2,4-dimethylpentane at about 370 K in the liquid phase suggests k0(1,5t) = 1010.5expt-63 kJ rnol-l/RT) s-l, which at 753 K gives k9(1,5t) = 1.3 x 106 s-l, in excellent agreement with the value of 2.2 x lo6 s-l in Table 16, although the Arrhenius parameters given are somewhat different. Although the rate constants for a particular reaction in the liquid phase and the gas phase frequently differ by many orders of magnitude owing to solvent and cage effects in the liquid phase, Mayo lB3has shown that unimolecular reactions usually have similar rate constants in the two phases. Consequently, the agreement between the two values for k,(1,5t) lends support to the recommended rate constants given in Table 16. It is, however, quite possible that the accompanying Arrhenius parameters are slightly too high. At present, further discussion is premature. Mill lo2 also gives k,,, = 1011.6exp(-59 f 8 kJ mol-l/RT) s-l for the decomposition of the QOOH radical to give 2,2,4,4- tetramethyloxetan: 469

*

*

At 753 K, k,,, = 107m4 s-l, compared with a value of 1 O 6 e 0 s-l obtained 147 for the gas-phase decomposition of (CH,),C(CH,OOH)CH,* radicals into 3,3-dimethyloxetan. Both values are reasonably consistent with Fish's theoretical estimate of lo1' exp(-54 kJ mol-l/RT) s-' for decomposition of QOOH radicals into oxetans. Possibly, therefore, Fish's estimates of A = loll s-l and El,, = 59, 12.5, and 0 kJ mol-l for the decomposition of QOOH radicals into oxirans, tetrahydrofurans, and tetrahydropyrans, respectively, may give rate constants reliable to an order of magnitude (probably too high) over the temperature range 550-750 K. Scission reactions of QOOH radicals and transitions involving alkyl-group lg2 lo3

O e 6

O s 5

T. Mill, 13th Int. Combustion Symp., The Combustion Institute, Pittsburgh, Pennsylvania, 1971, p. 237. F. R. Mayo, J. Amer. Chem. SOC.,1967, 89, 2654.

Rate Constantsfor Gas-phase Hydrocarbon Oxiabtion

21 1

transfer have been discussed fully by Fish in relation to the product distributions in hydrocarbon oxidation, but only theoretical estimates of rate constants are available, and readers are referred to his review^.^^^ 46 Finally, now that the rate data have been assembled, it is worth considering the suggestion, made by Pollard lS3et al., that in the oxidation of isobutane at W K , formation of isobutene may occur predominantly by a surface reaction of t-C,H902*radicals. The relevant homogeneous and heterogeneous reactions are given below. (CH,),C* (CH3)3C* (CH3)3C02*

+ +

0 2 -+

0 2 +

+ HOz*

(CH,)&=CH2 (CH3)3C02*

(CH3),C02-+ (CH,),C(CH,)OOH + (CH3)3COaH (CH,),C.

+ (CH3)BCH

+

wall

(CH3)&02* -+ (CH&C=CH2

+ HO2*

(6At) (70 (9t) (8Dt)

(w)

Using a mixture containing 50 and 100 mmHg of isobutane and oxygen, respectively, the maximum estimate of k, is the diffusion-controlled value of 0.5 s-l. Using k,,, = lO@-O exp(-60.3 kJ mol-l/RT) dm3mol-1 s-l (Section 7) and k,, = 10laJexp(-125 kJ mol-l/RT) s-l [k9(l,4p) in Table 161, the relative rates of reactions (w), (8Dt), and (9t) are 0.5,7.5, and 13 s-l at 600 K. The relative rate of reactions (6At) and (w) is given by ksAt/K7,kW, as (7t) is certainly equilibrated. Using k6~t= 3.5 x lo9 exp( -32 kJ mol-l/RT) dm3 mol-1 s-l (Section 8, p. 202) and K,t = 2 x lo4 dm3 mol-1 (Table 2), the relative rate of (6At) m d (w) is about 570. Even allowing for uncertainties in the rate constants, it is quite clear that surface destruction of (CH3)3C02radicals, even at the maximum diffusion-controlledrate, is out of the question under the above conditions. A similar conclusion applies to the surface destruction of i-C4HgOa* radicals, which will also be formed in the oxidation of isobutane.

5

Kinetic Studies in Silicon Chemistry BY 1. M. T. DAVIDSON

1 Introduction Kinetic studies of the reactions of silicon compounds are of interest in their own right and in relation to the analogous reactions of carbon compounds. There has been considerable activity in this field since it was last reviewed and the coverage of the literature in this chapter is selective. The theme is the nature and reactions of isolated ( i x . non-solvated) intermediates in silicon chemistry; consequently most of the kinetic work described is in the gas phase, while some non-kinetic work is discussed if it provides information on the existence and reactions of short-lived species, viz. silylenes, silyl radicals, and molecules with pn-px bonds to silicon. The chemistry of these intermediates is first reviewed, followed by the reactions of their carbon analogues, alkyl radicals and carbenes, with silicon compounds, 1v

2 Short-lived Molecules with px-p, Bonds to Silicon Good evidence for the existence of Me,Si=CH, came originally from kinetic and trapping experiments ;3 confirmation has come from further pyrolysis and trapping studies of silacyclobutanes, in which distinctive adducts were formed by the addition of Me2Si=CH2to olefins and dienes,* e.g. generation of Me,Si===CH, in the presence of butadiene gave a silacyclohexene, probably by direct 1,4-~ycloaddition.~ Direct spectroscopic evidence for the existence of Me,Si=CH, in the pyrolysis of 1,l-dimethyl-1-silacyclobutane at 923 K has been obtained by low-temperature trapping.s More facile sources of Me,Si=CH, are now available, yielding the intermediate by pyrolysis at T w 700 K or by U.V.phot~lysis.~ There is some evidence for R. A. Jackson, in ‘Essays on Free-radical Chemistry’, Chemical Society Special Publ. No. 24, The Chemical Society, London, 1970, 299. I. M. T. Davidson, Quart. Rev., 1971,25, 111. (a) M. C. Flowers and L. E. Gusel’nikov, J. Chem. SOC. ( B ) . , 1968, 419; (6) W. J. Bailey and M. S. Kaufman, A.C.S. Meeting, 1969. N. S. Nametkin, R. L. Ushakova, L. E. Gusel’nikov, E. D. Babich, and V. M. Vdovin, Bull. Acad. Sci. U.S.S.R., Div. Chem. Sci., 1970, 19, 1589; N. S. Nametkin, L. E. Gusel’nikov, R. L. Ushakova, and V. M. Vdovin, ibid., 1971, 20, 1740. N. S. Nametkin, L. E. Gusel’nikov, R. L. Ushakova, and V. M. Vdovin, Doklady Chem., 1971, 201, 1056. T. J. Barton and C. L. McIntosh, J.C.S. Chem. Comm., 1972, 861. T. J. Barton, J. L. Witiak, and C. L. McIntosh, J. Amer. Chem. SOC., 1972,91, 6229; T. J. Barton and E. Kline, J. Organometallic Chem., 1972, 42, C21. 212

213

Kinetic Studies in Silicon Chemistry

the involvement of Me,Si=CH, in reactions in solution * and possibly in the rearrangement of silylmethyl-iron complexes. Sommer and co-workers have photolysed 1,l -diphenyl-14lacyclobutane at 328 K in cyclohexane-MeOD, obtaining a high yield of Ph,Si=CH2, trapped as Ph2Si(OMe)CH2D.10They have used the same technique to demonstrate the formation of Ph,Si=CH2 in the photolysis of MePh,Si,.ll Both Me,Si=CH, and Ph,Si=CH2 have been found to react with aldehydes and ketones, forming an olefin and a cyclic siloxane, probably via reaction (1).l2 If the ketone is enolizable, reaction route (2) is followed.

-

R2 R:Si=CH,

+

\

F=O R3

R2

R: Si-CH,

I

I IR3

0-C-R2

---+

\ C=CH, /

R3

+ RiSi-0

1 (RiSiO), (1)

R2

\

R3CH,/c=o

R2

\

/H

HO/c=c\R3

R:Si=CH, low temp.

R2

R3

Me,Si-=O has been suggested as an intermediate in the pyrolysis of the cyclic siloxanes, (Me,SiO), to (Me,SiO)7. (Me,SiO), was the most stable to pyrolysis in a static system ca. 820K, while the higher siloxanes were believed to break down by transannular reactions in which Me,Si=O was extruded.13 This reaction could be reversed by insertion of Me,Si=O into silicon-oxygen bonds. An overall first-order rate constant for this type of breakdown between 743 and 823 K was given by: 1oglo(k/s-l) = (12) - (255) kJ mol-l/2.303 RT Several cyclocarbosiloxanes have been pyrolysed in static and flow No kinetic data were reported, but there was evidence from product analysis that both Me,Si=CH, and Me,Si=O were present.

lo

l1 l2

Is 114

A. W. P. Jarvie and R. J. Rowley, J. Organometallic Chem., 1972, 34, C7. K. H. Pannell, J. Organometallic Chem., 1970, 21, P17. P. Boudjouk and L. H. Sommer, J.C.S. Chem. Comm., 1973, 54. P. Boudjouk, J. R. Roberts, C. M. Golino, and L. H. Sommer, J. Amer. Chem. SOC., 1972,94, 7926. D. N. Roark and L. H. Sommer, J.C.S. Chem. Comm., 1973, 167. L. E. Gusel'nikov, N. S. Nametkin, T. Kh. Islamov, A. A. Sobtsov, and V. M. Vdovin, Bull. Acad. Sci. U.S.S.R., Div. Chem. Sci., 1971, 20, 71. N. S. Niimetkin, T. Kh. Islamov, L. E. Gusel'nikov, A. A. Sobtsov, and V. M. Vdovin, Bull. Acad. Sci. U.S.S.R.,Div. Chem. Sci., 1971, 20, 76.

Reaction Kinetics

214

A detailed kinetic study of cyclic (Me,SiO), has been undertaken in a static system l6 between 766 and 842 K. Up to 4 % decomposition, the products (Me,SiO), and (Me,SiO), were formed in comparable amounts in a first order reaction with log,,(k/s-l)

=

(14.85 f 0.39) - (300.9 f 6.1) kJ mol-l/2.303 RT

Beyond 4% the rate of formation of (Me2Si0)3dropped below that expected from the first-order law, while the rate of formation of (Me,SiO), was reduced even more. With excess of added ethylene or propene the formation of (Me,SiO), was suppressed, while formation of (Me,SiO), now obeyed the first-order law up to at least 15% decomposition. These findings were taken as evidence for the following mechanism :

+ Me,Si=O + (Me,SiO), + (Me,SiO),

(Me2Si0)4 Me,SiO

+ (Me,SiO),

(3) ( -3) (4)

( -4)

reactions ( -3), (4), and of course ( -4) being inhibited by the trapping agent. Attempts to isolate adducts of Me,Si=O with olefins have not yet been consistently successful,le possibly because of the tendency, noted above,12 for cyclic silyl ethers to break up, regenerating Me,Si==O. From the activation energy for the pyrolysis of (Me,SiO),, the n-bond energy in Me,Si=O was estimated to be at least 158 kJ mol-l. The n-bond energy in Me2Si=CH, has been estimated l 7 as 138 f 22 kJ mol-l from an analysis of the kinetics of the gas-phase decomposition of 1,l-dimethyl-1silacyclobutane and trimethylsilane.18 However, the mechanism of the latter decomposition now has to be re-interpreted (see Section 5), and analysis of the former pyrolysis alone would lead to 156 f 37 kJ mol-1 for the n-bond energy in Me,Si=O. 3 Reactions of Silyl and Alkyl Radicals

Reactions of Alkyl Radicals with Silicon Compounds.-Several papers on the kinetics of hydrogen abstraction from silicon compounds by alkyl radicals in the gas phase have appeared since this topic was last reviewed.2 Kinetic results are collected in Table 1, along with some earlier relevant results for comparison. In the reactions of CF3 radicals with Me& Me,SiF, Me,SiF,, and MeSiF,, the steady increase in activation energy along the series was attributed largely to a polar effect;l9as methyl is replaced by fluorine there l5 lo

l7

l9

I. M. T. Davidson and J. F. Thompson, Chem. Comm., 1971, 251. I. M. T. Davidson and M. E. Delf, unpublished work. R. Walsh, J . Organornetallic Chem., 1972, 38, 245. I. M. T. Davidson and C. A. Lambert, J . Chem. SOC.( A ) . , 1971, 882. T. N. Bell and U. F. Zucker, J . Phys. Chem., 1970,74,979.

Kinetic Studies in Silicon Chemistry Arrheniusparameters for SH SiH4 SiH, MeSiH MeSiD MeSiD, Me2SiH2 Me,SiD2 Me,SiH Me,SiD Me4% Me,Si Me4% HSiCl, MeSiF, Me,SIF, Me,SiF HSiCl, Me4Si HSiCl, Me,Si SiH4 SiD4 HSiCl, MeSiF, MezSiFe Me,SiF MeSiCl, Me2SiCI2 Me,SiCl Me4Si SiH4 SiD4 HSiC1, Me4Si Si2Hs SizDB SiH,

, ,

SiDI

Me,COH

SiH, SiD, SiH4

R. + SH -+ RH

logl,(A/dm3 rno1-I s-l) 9.26 f 0.17 8.24 9.28 f 0.24 9.25 f 0.09 9.49 f 0.34 9.07 f 0.18 8.92 f 0.24 8.69 f 0.27 7.98 f 0.53 8.34 f 0.23 8.55 f 0.18 9.00 f 0.1 7.83 9.10 f 0.12 9.00 f 0.05 9.00 f 0.12 7.79 f 0.08 8.68 f 0.12 8.32 f 0.05 8.65 f 0.15 8.90 f 0.17 8.97 f 0.15 8.77 8.98 f 0.04 9.27 f 0.06 9.43 f 0.05 8.31 f 0.13 8.79 f 0.10 9.25 f 0.06 8.91 f 0.09 8.73 f 0.23 8.87 f 0.35 8.54 8.88 f 0.22 8.75 f 0.28 9.03 f 0.16 8.52 f 0.21 8.52 f 0.31 8.84 f 0.21 8.84 f 0.41 8.77

+ S. E/kJ mol-' 31.25 f 1.21 25.82 f 1.55 34.02 f 1.63 28.58 f 0.63 38.49 f 2.30 34.73 f 1.30 36.90 f 1.63 34.77 f 1.97 37.95 f 3.85 40.42 f 1.72 42.80 f 1.51 45.15 f 0.92 17.99 52.55 f 1.05 50.67 f 0.50 48.24 f 0.88 25.36 f 0.63 50.21 f 1.26 28.53 f 0.38 42.59 f 1.38 20.63 f 1.17 25.27 f 0.96 25.02 48.99 f 0.38 44.06 f 0.50 39.66 f 0.38 39.12 f 0.88 38.62 f 0.75 38.20 f 0.38 30.50 f 0.59 30.35 f 1.59 34.94 f 2.80 22.26 47.70 f 2.01 23.64 f 1.88 28.62 f 1.13 28.91 f 1.42 33.05 f 2.13 31.97 f 1.38 36.11 f 3.01 37.66 f 4.18

(a) Ref. 26; (b) ref. 87; (c) ref. 25; (d) ref. 24; (e) ref. 23; cf) J. A. Kerr, A. Stephens, and J. Young, Internat. J. Chem. Kinetics, 1969, 1, 371; (g) ref. 22; (h) ref. 20; ( i ) E. Jakubowski, H. S. Sandhu, H. E. Gunning, and 0. P. Strausz, J. Chem. Phys., 1970,52,4242; ( j ) ref. 19; (k) ref. 21 : ( I ) D-abstraction; (m)assuming loglokcomb = 10.34 and &,mb = 0;

(n) D-abstraction, calculated from isotope effect.

is an increasing inductive withdrawal of electrons, reducing the electron density on the hydrogen to be abstracted, and thus reducing the polar attraction between that hydrogen and the electrophilic CF3 radical. A similar

216

Reaction Kinetics

explanation was suggested for the trend in activation energy for hydrogen abstraction from Me4Si by the series of radicals CH,, CH,F, CHF,, and CF,, in which increasing electrophilic character in the radical may be correlated with decreasing activation energy.2o In hydrogen abstraction by CF, radicals from Me,Si, Me,SiCl, Me2SiC1,, and MeSiCI,, chlorine substitution raised the activation energy, but the increases were very This reduced effect compared with the methyl fluorosilanes was attributed to p,-d, interactions counteracting to some extent the inductive effect of chlorine. Small variations in activation energy were also observed 22 in the reactions of the fluoromethyl radicals, CH2F, CHF,, and CF, with HSiC13, possibly confirming the existence of two partially cancelling opposing factors in the siliconchlorine bond, but the activation energies are higher than for the reaction of CH, with HSiC1,. This was tentatively explained as being due to a polar repulsion, the hydrogen in HSiCl, being (d +) because of substantial inductive withdrawal of electrons. Support for this explanation comes from work of Bell and Platt 23 who studied hydrogen abstraction by CD, radicals from Me4Si, Me,SiF, Me,SiF,, and MeSiF,. Assuming no polar effect and applying the Polanyi equation, they calculated carbon-hydrogen bond dissociation energies as follows: in SiMe,, 403 kJ mol-l; in FSiMe,, 409 kJ mol-l; in F,SiMe,, 414 kJ mol-l; and in F,SiMe, 418 kJ mo1-l. The first of these values agrees well with an earlier estimate.,, From these values, the enthalpy effect in the reaction of CF, radicals with the same series of molecules lS could be estimated, and hence the polar contribution to the activation energy assessed. This contribution was found to reduce the activation energy below the value expected on enthalpy grounds alone in the case of Me,Si and Me,SiF, indicating a polar attraction; the polar effect with Me,SiF, was small, while with MeSiF, the activation energy was higher than the value based on enthalpy considerations, indicating a polar repulsion, as was found in the reactions of fluoromethyl radicals with HSiCl,.,, A striking feature of the results in Table 1 is the uniformity of logl,(A/dm3 mol-l s-l), almost all of the values lying within the range 8.8 5 0.5. In the abstractions by CH3 and CD, radicals from deuteriated and undeuteriated methylsilanes a regular trend in A factors was discerned, loglo(A/dm3mo1-1 s-l) decreasing by ca. 0.2 to 0.3 as the number of silicon-hydrogen bonds in the molecule decreased; the trend was followed both in deuterium abstraction and in hydrogen a b ~ t r a c t i o n .The ~ ~ absolute values for the rate constants for abstraction by higher alkyl radicals in Table 1 are in some doubt, because of the uncertainty surrounding the recombination rate parameters for these 2o

21 22 28

2p 25

J. A. Kerr and D. M. Timlin, Internut. J. Chem. Kinetics, 1971, 3, 69. T. N. Bell and U. F. Zucker, Cunud. J. Chern., 1970, 48, 1209. J. A. Kerr and D. M. Timlin, Znternut. J . Chem. Kinetics, 1971, 3, 1 . T. N. Bell and A. E. Platt, J . Phys. Chem., 1971, 75, 603. J. A. Kerr, A. Stephens, and J. C. Young, Internat. J. Chem. Kinetics, 1969, 1, 339. R. E. Berkley, I. Safarik, H. E. Gunning, and 0. P. Strausz, J . Phys. Chem., 1973, 77, 1734.

Kinetic Studies in Silicon Chemistry

217

radicals.26However, it is clear that the A factors are in the expected region, and that there is little variation in activation energy for different alkyl radicals attacking the same silicon-hydrogen bond. The kinetic isotope effect in these experiments was CQ. 3,26p26 considerably smaller than in hydrocarbon systems. Abstraction by hydrogen atoms from SiH, has been found to be extremely fast, being able to compete with addition to ethylene, even at 143 K; an addition4imination process, involving the SiH5 radical, has been suggesHydrogen abstraction from Me,SiH by deuterium atoms has been studied in a flow system associated with an e.s.r. spectrometer.28Abstraction was almost exclusively from silicon-hydrogen bonds, with a rate constant given by: log,,(k/dm3 mol-1 s-l)

=

(7.95)

-

(9.62 f 0.21) kJ mol-l/2.303 RT

Deuterium atoms, from the photolysis of C2D4,have also been allowed to react with SiH,, Si2H6, MeSiH,, Me2SiH2,and Me,SiH. Rate constants for hydrogen abstraction from each silane were measured at room temperature in competition with the addition of deuterium atoms to ethylene.29 The reactivities of the silanes and ethylene in their respective reactions with deuterium atoms were comparable, and the silane reactivities per siliconhydrogen bond were in the order: Si2H6> Me,SiH > Me,SiH, > MeSiH, > SiH,. In some gas-phase experiments on the reactions of methyl radicals with HSiCl,, HMeSiCl,, Me,SiCl,, and MeSiCl,, the expected abstraction and combination reactions occurred, abstraction from silicon-hydrogen bonds being of course more facile than from carbon-hydrogen. However, some other interesting .reactions were observed, resulting from the use of Me,Hg as the radical source.3o Methane and an organomercury compound were formed in two cases, MeSiCl, giving (Cl,SiCH,),Hg, and Me,SiCl, giving (MeC12SiCH2)2Hg.Chlorination reactions also occurred, e.g. MeSiHCl, + MeSiC1, and Ha. These reactions were greatly accelerated by Me,Hg, catalysis by mercury vapour being suggested as a possible reason. However, chlorine abstraction by the silyl radicals formed might also be important (see below). In solution, the relative rates of hydrogen abstraction by the CCl, radical from arylhydrosilanes have been measured at 353 K.31The results were consistent with electrophilic attack by CCl, at the silicon-hydrogen bond, with little charge separation in the transition state. These conclusions have 26

*' ao 31

R. E. Berkley, I. Safarik, 0.P. Strausz, and H. E. Gunning, J . Phys. Chem., 1973,77, 1741.

L. C. Glasgow, G. Olbrich, and P. Potzinger, Chem. Phys. Letters, 1972, 14, 466. M. A. Contineanu, D. Mihelcic, R. H. Schindler, and P. Potzinger, Ber. Bunsengesellschaft phys. Chem., 1971,75, 426. K. Obi, H. S. Sandhu, H. E, Gunning, and 0. P. Strausz,J . Phys. Chem., 1972,76,3911. J. Joklik and V. Bazant, Coll. Czech. Chem. Comm., 1972, 37, 3772. Y. Nagai, H. Matsumoto, M. Hayashi, E. Tajima, M. Ohtsuki, and N. Sekikawa, J. Organometallic Chem., 1971, 29, 209.

21 8

Reaction Kinetics

been confirmed in an extensive investigation of the reactions of CCl, radicals in refluxing carbon tetrachloride with 26 judiciously selected silicon compound~.,~ From experiments with pairs of these compounds competing for CCI, radicals a table of relative rate constants was compiled, revealing several interesting features. The relative rate constants for a group of seven substituted phenylsilanes showed a polar effect, with electron-withdrawing groups retarding the rate of abstraction as was found in the gas-phase studies discussed above. There was a steady increase in the relative rate constant along the series PhMe,SiH, Ph,MeSiH, and Ph,SiH, attributed to the electronreleasing resonance effect of the phenyl group counteracting the electronwithdrawing polar effect. In keeping with this explanation, a low relative rate constant was found for (PhCH2),SiH, where only the polar effect can operate. Progressive chlorine-substitution at silicon reduced the relative rate constant and the retarding effect of one fluorine atom was greater than that of one chlorine atom; these findings are also in accord with those in the gas phase. Comparison of the reactivity of a-NpPhMeSiH and a-NpPhMeSiD showed a primary isotope effect of 3.4, implying considerable siliconhydrogen bond stretching in the transition state. (Very similar isotope effects were observed in the gas 26 and the relatively high A factors in Table 1 are noteworthy in relation to the above comment on transition state geometry.) The molecule most reactive to hydrogen abstraction by CCl, was Ph,SiSiPhMeH, possibly because of stabilization of a partial positive charge on the silicon at reaction centre by the adjacent Ph,Si group. Reactions of Silyl Radicals.-Homolytic aromatic substitution by silyl radicals has been observed in as have some S,2 reactions at silicon.,* Sommer and Ulland generated various silyl radicals from a silicon hydride and a radical initiator and treated them with CCI,Br.22 The selectivity of the silyl radicals could then be measured from the ratio of silyl bromide to silyl chloride formed. The most striking variations were in a series of radicals with increasing numbers of chlorine atoms attached to silicon, where the selectivity in favour of bromine increased sharply until SiCI, radicals gave BrSiC1, only, no SiC14 being observed. The SiCl, radical was thus the most selective or ‘stable’, of those studied, a finding which corroborates earlier work.2 The rate constant for the combination of Me,% radicals has been measured in solution 36 and in the gas phase 3 7 by the rotating sector method, giving 36 5.5 x lo9 and 3 7 1.8 x lo1’ dm3mol-ls-l, respectively. These 3s9

32 33

34

35

3* 37

L. H. Sommer and L. A. Ulland, J. Amer. Chem. SOC.,1972, 94, 3803. S. W. Bennett, C. Eaborn, R. A. Jackson, and R. Pearce, J , Organometallic Chem., 1971,28, 59; H. Sakurai and A. Hosomi, J. Amer. Chem. SOC.,1971,93, 1709. A. Hosomi and H. Sakurai,J. Amer. Chem. SOC.,1972,94, 1384; D. J. Edge and J. K. Kochi, J.C.S. Perkin 11, 1973, 182. G. B. Watts and K. U. Ingold, J . Amer. Chem. SOC.,1972, 94, 491. P. P. Gaspar, A. D. Haizlip, and K. Y. Choo, J. Amer. Chem. SOC.,1972,94, 9032. P. Cadman, G. M. Tilsley, and A. F. Trotman-Dickenson, J.C.S. Furuduy I , 1972,68, 1849.

Kinetic Studies in Silicon Chemistry

219

results clearly show that combination of Me,Si radicals is as rapid a process as combination of methyl radicals, and that earlier indications of an abnormally low rate constant for this process were misleading.2 Also, in solution, SiH, radicals have been shown to combine more rapidly than Me,Si radical^.,^ These conclusions are important in relation to the interpretation of the results of pyrolysis of silanes, and are discussed further in Section 5. The gas-phase recombination rate constant 3 7 has been used to obtain absolute rate constants for chlorine abstraction from alkyl chlorides by the Me,Si radical,,* with the results given in Table 2. As befits a strongly

Table 2 Arrhenius parameters for Me,Si. RCl MeCl EtCl Pr"C1 Pr'CI BuTl

+ RCI

logl,(A/dm3 mol-1 s-l) 8.02 7.99 7.90 8.30 8.15

~f

Me3SiCl

+ R-

E/kJ mo1-I 16.98 15.93 14.47 15.79 12.37

exothermic reaction, the activation energies are small, while the A factors are well within the 'normal' range, although somewhat smaller than those in Table 1. There is the expected correlation between ease of chlorineabstraction and the carbon+hlorine bond dissociation energy, the results fitting the Polanyi equation. Comparison with earlier results for abstraction by SiCl, radicals confirmed that Me,Si is less selective, or more reactive, than SiCl,. The results permit some speculation about silicon+hlorine bond strengths, and approximate values of 400 kJ mol-1 for DH"(C1,Si-CI) and 440 kJ mol-1 for DH" (Me,Si-C1) were suggested. The well-established technique of generating silyl radicals by photolysing di-t-butyl peroxide with a silicon hydride in the cavity of $n e.s.r. spectrometer has been used to study some reacti addition reactions of Et,Si radicals to the esters was observed, but with halogeno-keto abstraction by Et,Si radicals occurred preferential in view of the Arrhenius parameters in Table sodium atoms with some chloroalkylsilanesand have been measured in the gas phase.40 The rate tion by sodium atoms from ClCH,SiMe, was 15 times greater than from ClCH,CMe,, whereas the corresponding ratio of rate constants for hydrogen abstraction from Me,Si and Me4C by various radicals does not exceed 4. Arguments were presented against this difference being due to differences in s8

P. Cadman, G . M. Tilsley, and A. F. Trotman-Dickenson, J.C.S. Faraday I, 1973,69,

914. so A. J. Bowles, A. Hudson, and R. A. Jackson, J . Chem. SOC.(B), 1971, 1947. 40 D. S. Boak and B. G. Gowenlock, J. Organometallic Chem., 1971,29, 385.

Reaction Kinetics

220

bond dissociation energies, a negative group effect in the silicon chloride being the favoured explanation. Whilst the reasons for rapid halogen abstraction by silyl radicals from carbon-halogen bonds are clear, it is slightly surprising to find that silyl radicals can abstract chlorine quite rapidly from silicon-chlorine bonds. Evidence for this comes from a preliminary gas-phase study ca. 500K in which Me,SiCl and MeSiC1, radicals were found to abstract chlorine in preference to hydrogen from R,SiHC1.41 Kinetic data are now being acquired 4 2 for the gas-phase reaction, Me,Si* t Me,SiCl,

+ Me3SiC1

+ Me,SiCl

Preliminary indications are that log,,(A/dm3 mol-1 s-l) is ca. 8 and E ca. 30 kJ mol-l. As well as the addition reactions of Et,Si radicals mentioned above, some addition reactions of SiH, and Me,Si radicals have been observed in the gas phase. Mercury-photosensitized reaction of SiH4 and nitric oxide has been studied at 298 K.43 Mercury photosensitization produced SiH, radicals, which reacted with SiH4in the absence of nitric oxide to give Si,H6. Addition of nitric oxide caused an induction period for the formation of SizH6,during which time di-, tri-, and tetra-silanes were shown to be present. Good evidence was presented for the following reactions : NO

+ SiH, + NOSH3

$JOSH,

+ NO -+ONNOSiH,

ONNOSiH,

--f

N,O

$-

*OSiH,

(5)

(6) (7)

Similar reactions occurred in the MeSiH,-nitric oxide system. The rate constant for reaction (7) was 4.7 x 1O-O s-l at 298 K (if logl,A,/s-l is ca. 13, E , would be ca. 90 kJ mol-l). The photosensitized reaction between Me,SiH and CzF4 has been studied in the gas phase at 300K.44 Direct photolysis was negligible, reaction resulting from the mercury-photosensitized dissociation of Me,SiH to Me3Si radicals and hydrogen atoms. The following addition and abstraction reactions were then observed :

+ C,F4 Me,SiC,F, Me,SiC,F, + C2F4-+ Me,SiC,F, Me3Si(C2F4); -k Me,SiH -+Me&- + Me,Si(C,F,),H Me,Si-

4*

43 44

--f

(8) (9) (10)

D. Atton, S. A. Bone, and I. M. T. Davidson, J . Organometallic Chem., 1972,39, C47. I. M. T. Davidson and J. I. Matthews, unpublished work. E. Kamaratos and F. W. Lampe, J . Phys. Chem., 1970, 74, 2267. R. N. Haszeldine, S. Lythgoe, and P. J. Robinson, J . Chem. SOC.( B ) , 1970, 1634.

Kinetic Studies in Silicon Chemistry

221

4 Silylenes and Carbenes

Inorganic Silylenes: Non-kinetic Work.-Several preparative studies have provided information on the reactions of silylenes which are important intermediates in silicon chemistry. Chernyshev er al.46obtained evidence for the formation of SiCl, in the gas-phase pyrolysis of HSiC13 and PhCH2SiCl,Hat T > 873 K. From co-pyrolysisof one of these silylene sources with substituted naphthalenes or with chlorobenzene, they obtained products which indicated that SiCl, could insert into alkyl carbon-hydrogen, aryl carbon-hydrogen, aryl carbon-chlorine, aryl carbonarbon, and siliconhydrogen bonds. Pyrolysis of Si2C1, also generates Sic&, which can be trapped by, added a ~ e t y l e n e . ~ ~ Margrave and his co-workers have continued their studies of the chemistry of SiF,; generation of SiF, in the presence of various methylsilanes showed no new product with Me,Si and only halogen exchange with Me,SiCl, indicating that SiF, did not insert into silicon-methyl or silicon-chlorine bonds. With Me,SiOMe several products were observed, consistent with insertion of SiF, into the oxygen-methyl bond, followed by secondary reaction~.~'With butadiene, two SiF, units add 1,4 to give a six-membered heterocyclic ring. * Some reactions of gaseous SiO, formed by heating solid silicon monoxide to ca. 1500 K, have been r e p ~ r t e d . ~Complex ~ product mixtures were obtained, but there was some evidence for addition of SiO across multiple carbon-carbon bonds and insertion of SiO into carbon-hydrogen, carbonhalogen, oxygen-hydrogen, and nitrogen-hydrogen bonds. SiO is more reactive and much less selective than the silylenes previously studied. Inorganic Silylenes: Kinetic Work.-There is now general agreement that pyrolysis of Si,H, proceeds by a silylene mechanism, forming SiH, and SiH4 in the primary decomposition step, and not SiH3 radical^;^^^ 61 Si2H 8 is indeed a convenient thermal source of SiH,. Ring and co-workers have pyrolysed a number of disilanes, alone, and in the presence of deuteriated and undeuteriated organosilanes.62 They obtained convincing evidence that the primary decomposition route of all of the disilanes chosen was to a silylene and a monosilane, and they measured relative rates in those cases 'r, 48

E. A. Chernyshev, N. G. Komalenkova, T. A. Klochkova, S. A. Shchepinov, and A. M. Mosin, J . Gen. Chem. U.S.S.R.,1971, 41, 117. E. A. Chernyshev, N. G . Kamalenkov, and S. A. Bashkerova, J. Gen. Chem. U.S.S.R., 1971,41, 1177.

47

48

61 ba

J. L. Margrave, D. L. Williams, and P. W. Wilson, Inorg. Nuclear Chem. Letters, 1971, 7 , 103. J. C. Thomson and J. L. Margrave, Inorg. Chem., 1972, 11, 913. E. T. Schaschel, D. N. Gray, and P. L. Timms, J . Organometallic Chem., 1972,35, 69. M . Bowrey and J. H. Purnell, Proc. Roy. SOC.,1971, A321, 341 and refs. therein. P. Estacio, M. D. Sefcik, E. K. Chan, and M. A. Ring, Inorg. Chem., 1970, 9, 1068. R. L. Jenkins, A. J. Vanderwielen, S. P. Ruis, S. R. Gird, and M. A. Ring, Inorg. Chern., 1973, 12, 2968.

Reaction Kinetics

222

where there were two possible silylene-forming routes. The observed primary decompositions were :

+ SiF, HF,SiSiH3 -+ SiH, + F2SiH, FSi,H, -+ SiH, + FSiH3 FSi,H5 + FSiH + SiH, HCI,SiSiH, + SiH, + CI,SiH, HCI2SiSiH3 SiCI, + SiH4 CISi2H, + SiH, + CISiH, ClSi,H, ClSiH + SiH, HClMeSiSiMeClH + ClSiMe + ClSiH,Me HClMeSiSiMeClH + HSiMe + C1,SiHMe Si2F6+ SiF,

(1 1)

(12) (13)

kl3/kli

=

5 f1

(14)

(15 ) kl5lkl6

--f

=

1

(16) (17)

k,,/k18

--f

=

0.8

zt

0.1 (18) (19)

kls/k,,

= 4.4

f 0.4 (20)

Excluding Si2Fs,all of the above reactions are 1,Zhydrogen shifts except for reaction (20), which is a 1,2-chlorine shift. The ratio kls/k,, shows that the chlorine shift is not facile, and in confirmation of this no products resulting from a 1,Zhalogen shift were found in the other halogenodisilanes. Experiments in which the above silylenes inserted into a silicon-hydrogen bond in ethylsilane led to the following reactivity sequence: SiH, > ClSiH > FSiH

> CI2Si,F,Si.

Relative rates of insertion of SiHz into various trapping agents have been measured in co-pyrolysis studies in a flow system at 620 K under conditions where the decomposition of Si2Hs, the source of SiH2, did not exceed 0.2%.53p 64 Results are in Table 3(i), along with one result for ,lSiHp generated by fast neutron irradiation of p h ~ s p h i n e .The ~ ~ reactivity sequence in Table 3(i) correlates with the order of hydridic character, Me3SiH > Me,SiH, > Si,H, > MeSiH, > SiH, > CISiH,, and with the decrease in hydridic character on replacing silicon with germanium and p h o s p h o r ~ s . ~ ~ Si3H8has been pyrolysed 53 at 603 K, alone and in the presence of MeSiD, and Me3SiD. SiH, appeared to be a primary product, and since no SiH,D was formed in experiments with MeSiD,, a silyl radical mechanism was discounted, the proposed primary steps being: Si3H8+ HSiSiH3

+ SiH,

Si3H8+ SiHz 4- Si,H6 63 64

65

(21) (22)

M. D. Sefcik and M. A. Ring,J . Amer. Chem. Sac., 1973,95, 5168. M. D. Sefcik and M. A. Ring, J . Orgunometallic Chem., 1973,59, 167. (a) P. P. Gaspar and P. Markusch, Chem. Comm., 1970, 1331; (6) P. P. Gaspar, P. Markusch, J. D. Holton, and J. J. Frost, J . Phys. Chem., 1972,76, 1352.

Kinetic Studies in Silicon Chemistry

223

From product analysis, k,,/k,, is ca. 2 ; supporting evidence for silylsilyene, HSiSiH,, was the formation of Me,Si,H,D in the co-pyrolysis of Si3Hs and Me,SiD. Arguments were presented in favour of the formulation HSiSiH, rather than H2Si=SiH2 or H2$i$iH2, and it was estimated that ca. 35% of the HSiSiH, underwent insertion reactions, the rest polymerizing. This contrast with the behaviour of methylcarbene, which isomerizes almost entirely to ethylene, is not unexpected. There were some indications in this work (from product analysis) that SiH, and HSiSiH3can insert into siliconsilicon bonds as well as into silicon-hydrogen bonds, but alternative explanations are possible (see below) and silylene insertion into silicon-silicon bonds cannot yet be accepted as an established reaction. The 1,Zhydrogen shifts forming silylenes in the pyrolysis of Si2Hs and Si3Hs were rationalized in terms of the hydridic character of the migrating hydrogen and the Lewis acidity of the acceptor site.

Table 3 Insertion reactions of SiH, at T ca. 600 K

*

none observed Me,GeSiH, H Me,GeSiH none observed

(ii) Si2H6 MeSiH, Si3Hs SiSHs

Si3H8 MeSi2H5 n-Si4Hlo i-Si4Hl0

1 .oo 1.4 2.5 1.5

d d d d

Si ,H Si2H6 SiHl

1.oo

0.33 2.4 x 10-3

e e e

(iii)

Si2H6 SiH4 H2

Insertion product Si3Hs Me3SiSiH3 HMe2SiSiH3 MeSi2H5

Relative rate (per M-H band)

Trapping agent (i) Si2H6 Me,SiH Me2SiH2 MeSiH, SiH4 ClSiH, Me,GeH Me2GeH2 MePH,

E/kJ mol-l

1 .oo 1.6 f 0.2 0.7 f 0.1 0.4 f 0.05 0.3 f 0.1 < 0.01 4 f 2 6.4 f 0.5 < 0.01

9.85

9.7 9.1

1.615

5.4 23.0

(a)Ref. 53; (b) ref. 55; (c) ref. 54; ( d ) ref. 50; (e) from absolute rate constants, ref. 56.

log,,(A/dm3 mol-' s-l).

Two thorough investigations of the pyrolysis of S&H6have been reported by Purnell and co-workers. In the first of these, a preliminary account of which has been reviewed previously,2 Si2& was pyrolysed between 556 and 613 K in a static system with and without added methylsilane~.~~ Pyrolysis of Si2H6alone gave H,, SiH4, Si3He, n-Si4HIo,and i-Si4Hlo. Kinetic data were obtained for the formation of the principal products H2, SiH4, and

224

Reaction Kinetics

Si,H,. All were formed in first-order reactions, rate constants for the formation of SiH4 being given by: log,,(k/s-l)

= (14.52

& 0.36)

- (206.0 f 4.6) kJ mol-l/2.303 RT

Arrhenius parameters for the formation of Si3HBwere the same within experimental error. Formation of hydrogen was shown to be a secondary process, viz. breakdown of a silicon hydride polymer on the walls of the reaction vessel. The kinetic data were shown to be consistent with primary decomposition to SiHz and SiH4 and not to SiH, radicals, and this conclusion was confirmed by co-pyrolysis of Si,D, and MeSiH,. All of the monosilane formed was SiD4, which not only confirmed the silylene mechanism, but also showed that SiH, did not participate in abstraction reactions. Copyrolysis of Si,H, with Me,Si gave no new products, indicating that SiH, did not insert into carbon-hydrogen or silicon-carbon bonds under these conditions, while co-pyrolysis with other methylsilanes gave insertion products from which the relative rates of insertion shown in Table 3(ii) were obtained. These authors, unlike Ring and co-worker~,~~ considered that variations in the ease of insertion into silicon-hydrogen bonds could explain the figures in Table 3(ii) without the need to invoke insertion into silicon-silicon bonds. This question is complicated by the discrepancy between the figures in Table 3(i) and 3(ii) for insertion into MeSiH,, which may be due to the substantially different experimental ~ o n d i t i o n s . ~ ~ Recently, John and Purnell s8 have measured the small temperature dependence of the relative rate of insertion of SiHz into SiH, and Si,H, between 578 and 607 K. Combining this result with their kinetic data for the pyrolysis of SiZH6,So established thermodynamic data for SiH4 and Si2H6,and a value for the entropy of SiH, from a statistical thermodynamic c a l c ~ l a t i o n , ~ ~ they obtained a value of 242.4 kJ mo1-1 for the enthalpy of formation of SiH,, and hence the Arrhenius parameters shown in Table 3(iii) for insertion by SiH,. The figures for insertion into H2 are based on a silylene mechanism for the pyrolysis of SiH4 (see Section 5). Comparison of the parameters for insertion into SiH4and Si2H6shows that the ratio of A factors is close to the statistical ratio expected on the basis of the number of silicon-hydrogen bonds, while the activation energies differ by ca. 4 kJ mol-l. It would seem perfectly reasonable for the silicon-hydrogen bond strengths in Si,H, and SiH4to differ by this amount, so that there seems to be no necessity to postulate insertion into silicon-silicon bonds. From the kinetic behaviour described above, and from spectroscopic data, silylene appears to be a bent molecule in the singlet state, undergoing insertion reactions but not abstraction 67 Triplet silylene may have been formed in some nuclear recoil experiments; fast neutron irradiation of phosphine gives high energy T i atoms, which pick up hydrogen to become ,lSiH,. In the presence of buta-1,3-diene, rapid formation of 589

66

57

P. John and J. H. Purnell, J.C.S. Faraday I, 1973,69, 1455. P. John and J. H. Purnell, J. Organometallic Chem., 1971, 29, 233.

Kinetic Studies in Silicon Chemistry

225

l-silacyclopent-3-ene was observed. The yield of this product was reduced to 20% of the original on addition of nitric oxide. If nitric oxide is assumed to be specific for species with unpaired electrons, the 31SiH, must then have been 80 % triplet and 20 % singlet, but the authors noted that this interpretation depends on the validity of that assumption.68 Organic Sily1enes.-Dimethylsilylene, Me,Si, is formed in the photolysis of cyclic 6 B and linear 8 o permethylated polysilanes. U.V. photolysis of cyclic (Me,Si), at 318 K gave the cyclic products (Me,Si), and (Me,Si),, with extrusion of Me2Si. The presence of Me,Si was established by adding suitable monosilanes, whereupon insertion of Me2Si into silicon-hydrogen or silicon-oxygen bonds was observed. When Me,SiCl, was added there was some evidence for direct reaction between photochemically excited polysilanes and Me,SiCl,. Photolysis of the linear polysilanes Me(Me,Si),,Me (n = 4-8) proceeded by chain contraction with loss of Me2% and incidentally by silicon-silicon bond scission. The latter process becomes more important with increasing chain length, while the yield of Me,Si is almost independent of chain length in the polysilanes. No evidence for insertion of Me,Si into silicon-silicon bonds was observed. A ‘silylenoid’ species has been reported in the disproportionation of M%SiSiMe,H catalysed by trans-[PtCl,(Et,P),], where the products were Me(Me,Si),H (n = 1-6). Formation of the products was explained in terms of an a-elimination of Me,Si, followed by Me2Si insertion into siliconhydrogen bonds. Although Me,% was trapped from the system by diphenylacetylene, the authors visualized the Me,Si to have been complexed with platinum and not free, hence the term ‘silylenoid’.

Carbenes.-The insertion of methylene into Me,%, in solution has been investigated.62EtMe,SiSiMe, was the only product, indicating that insertion into the silicon-silicon bond did not occur. Similarly, generation of methylene in Me,Si gave only EtSiMe,, showing that one silicon atom was enough to activate the a-carbon atom. Generation of CD, in Me&, gave HD2CCH2Si2Hs,showing that insertion was into the carbon-hydrogen bond and not into the silicon-carbon bond. The relative rates of insertion of tri tiated monochlorocarbene, generated by hot atom excitation, into silicon-hydrogen bonds have been reported. Per-bond reactivities, with insertion into ethylene as unity, were : Me,SiH, 6.2 f 1.9; Me,SiH,, 1.0 f 0.2; and SiH4 0.1. The apparent low reactivity 68

be 6o

62

G. P. Gennaro, Y.-Y.Su, 0. F. Zeck, S . H. Daniel, and Y.-N. Tang, J.C.S. Chem Comm., 1973,631. M. Ishikawa and M. Kumada, J. Organometallic Chem., 1972, 42, 325. M. Ishikawa, T. Takaoka, and M. Kumada, J. Organometallic Chem., 1972, 42, 333. K. Jamamoto, H. Okinoshima, and M. Kumada, J . Organometallic Chem., 1971, 27, C31. R. T. Conlin, P. P. Gaspar, R. H. Levin, and M. Jones, J . Amer. Chem. SOC.,1972, 94, 7165.

226

Reaction Kinetics

of SiHI may have been due to strong silicon-hydrogen bonds, or to secondary reaction^.^,

Relative rates have been reported for the addition of CCl, to methylvinylchlor~silanes,~~ H,C=CHSiMe,-,tCl, and to methylvinylflu~rosilanes,~~ H,C===CHSiMe,_,F, (n = 0-3). The results were discussed in terms of n-interactions between the vinyl group and the d-orbitals of silicon. These interactions are also important in the addition of CCl, to allylmethylchlorosilanes, as is the electrophilic character of CC1,.66 The reactions of CCI, with silacyclobutanes have been studied. If the silacyclobutane has a methyl group attached to the ring two concurrent insertion reactions are observed: into a silicon4arbon bond in the ring to form a silacyclopentane, and into a carbon-hydrogen bond B to the ring. It is likely that both processes occur with retention of configuration, but evidence on this point is not unarnbig~ous.~~ Photolysis of benzoyltrimethylsilane with various hydrosilanes in solution gives products of the type =SiC(HPh)OSiMe,. From relative rate measurements on a number of alkyl and aryl hydrosilanes it was concluded that a strongly nucleophilic siloxycarbene, PhCOSiMe,, was formed, the product then resulting from insertion of this carbene into the silicon-hydrogen bond+ in the hydrosilane.6s The zwitterion carbene, RCH=C=C: + RCHC-C [R = (CH,)Me or Me] has been found to react at ca. 350 K with Et,SiH by insertion into the silicon-hydrogen bond.69 Reactions of Carbon and Silicon Vapour with Manes.-Carbon vapour produced thermally or by e1ectri.c arc was co-deposited with Me,SiH and the products identified.'O Carbon vapour produced thermally consists mainly of C , with smaller quantities of Cz and C3, all in their electronic ground states. In carbon vapour produced from an arc there are in addition some electronic excited states. Two interesting products of the reaction of arcproduced singlet C1with Me,SiH were (Me3Si),CH2and HMe,SiCH=CH,. The former was believed to result from the insertion of C , into the siliconhydrogen bond, forming the carbene Me,SiCH which then inserted into another molecule of Me,SiH to give the product. The vinylsilane, on-the other hand, resulted from isomerization of another carbene, HMe,SiCH,CH, formed by C, insertion into a carbon-hydrogen bond. (Me,Si),CH, was the minor product, probably because carbene insertion into the second molecule 83 64

65

Y. -N. Tang, S. H. Daniel, and N. -B. Wong, J. Phys. Chem., 1970,74, 3148. P. Svoboda and V. Chvalovsky, Coll. Czech. Chem. Comm., 1972,37, 2253. P. Svoboda, V. Valsarova, and V. Chvalovsky, Coll. Czech. Chem. Comm., 1972, 37, 2258.

66

87

Es 'O

J. Koutkova and V. Chvalovsky, Coll. Czech. Chem. Comm., 1972, 37, 2100. D. Seyferth, H.-M. Shih, J . Dubac, P. Mazerolles, and B. Serres, J . Organometallic Chem., 1973,50, 39. H. Watanabe, T. Kogure, and Y. Nagai, J. Organometallic Chem., 1972, 43, 285. J. C. Craig and C. D. Beard, Chem. Comm., 1971, 692. P. S. Skell and P. W. Owen, J. Amer. Chem. SOC.,1972, 94, 1578.

Kinetic Studies in Silicon Chemistry

227

of Me3SiH was slow at the low temperature of the experiments. Triplet ground-state C, also inserted into silicon-hydrogen and carbon-hydrogen bonds in Me,SiH to give the two carbenes; final products were HMe,SiCH=CH, (as in the reaction with singlet C,) and Me,Si, the result of hydrogen abstraction by Me,SiCH. From the product yields, C, insertion into silicon-hydrogen bonds was ca. 9 times faster than into carbon-hydrogen. The products from C, and C3 arose almost entirely from insertion of these molecules into the silicon-hydrogen bond in Me,SiH. 7 0 Similar insertion reactions of small carbon molecules with SiCI4and SiH, have been observed.71 The same technique of co-deposition, followed by reaction in the condensed phase at or above 77 K, has been applied 72 to the reactions of atomic silicon with each of the following silicon hydrides: Me,SiH, Me,SiH,, MeSiH,, and Si,H,. All of the results were consistent with the initial formation of a silylsilylene, R3SiSiH (R = H, Me, or SiH,) by insertion of the silicon atom into a silicon-hydrogen bond, followed by insertion of the silylsilylene into the silicon-hydrogen bond of a second molecule of the substrate giving a trisilane, or by polymerization of the silylsilylene. The proportion of silylsilylene insertion increased with increasing methyl substitution of the substrate, indicating that silylsilylene, like SiH2 itself,53has electrophilic character. Silicon atoms have been generated from SiH4in a helium plasma, sustained by microwaves. Their reactions with SiH4, ultimately yielding SizHs, have been observed. 73 Insertion of Methylene into Silanes: Chemical Activation.-Simons and co-workers have continued their studies of the reactions of chemically activated silanes, produced by insertion of singlet methylene (from photolysis of diazomethane) into silicon-hydrogen or carbon-hydrogen bonds. 74-7 Added oxygen scavenged triplet methylene and radicals, simplifying the mechanism and enabling molecular and radical processes to be distinguished from each other. Chemically activated Me4Si was made by lCH2 insertion into Me,SiH, and the A factor for silicon-methyl bond rupture in the activated Me,Si estimated by a RRKM calculation. 74 Similar experiments and calculations were performed on activated EtSiMes (from Me,Si %Ha) both for silicon-thy1 and silicon-methyl bond rupture. 75 lCH2 insertion into Me2SiH, gave 76 activated Me,SiH and activated EtMeSiH,. With activated Me,SiH, silicon-carbon bond rupture was observed, as was a molecular decomposition route forming CH4 and Me,%. With activated EtMeSiH,

+

71 72

73 74 75

'I7

J. Binenboym and R. Schaeffer, Inorg. Chem., 1970,9, 1578. P. S. Skell and P. W. Owen, J . Amer. Chem. SOC., 1972, 94, 5434. P. P. Gaspar, K. Y. Choo, E. Y. Y. Lam, and A. P. Wolf, Chem. Comm., 1971, 1012. W. L. Hase and J. W. Simons, J. Chem. Phys., 1970,52,4004. W. L. Hase and J. W. Simons, J . Organometallic Chem., 1971,32, 47. W. L. Hase, W. G. Breiland, P. W. McGrath, and J. W. Simons, J . Phys. Chern., 1972, 76, 459. W. L. Hase, C. J. Mazac, and J. W. Simons, J. Amer. Chem. SOC.,1973, 95, 3454.

228

Reaction Kinetics

the main routes were loss of an ethyl radical and direct formation of molecular hydrogen with EtSiMe. RRKM calculations again yielded A factors for the silicon-carbon bond ruptures. Activated Me,SiH, and EtSiH, were formed by lCH, insertion into MeSiH,.77 The main mode of decomposition for the former was elimination of molecular hydrogen, while two lesser routes of comparable importance with each other were silicon<arbon bond rupture and direct formation of CHI with MeSiH. With activated EtSiH,, the main route was assumed from product balances to be the molecular &elimination to C2H4 and SiH,. Although some siliconarbon bond rupture was also observed, more important minor pathways were molecular formation of Ha and C2Hs with their corresponding silylenes. RRKM calculations were again performed on the individual bond-rupture processes. In all of the foregoing RRKM calculations limiting energies were estimated from the values of siliconarbon bond dissociation energies then current.2 These energies now require upward revision (see Section 5), and the RRKM calculations will no doubt be repeated, giving higher A factors.

5 Pyrolysis of Silicon Compounds

Detailed accounts have appeared of the pyrolysis of Me,SiH 7 8 and Me,Si 7 n in flow systems. As discussed previously,2 the pyrolysis of Me,SiH was interpreted as a radical non-chain process in which the formation of hydrogen and methane was rate-determined by the initial rupture of silicon-hydrogen and silicon-methyl bonds, respectively.7 8 On the other hand, Arrhenius parameters for the formation of methane in the pyrolysis of Me,Si were given by: log,,(k/s-l)

=

(14.3 f 0.23) - (282.8 f 3.3) kJ mol-l/2.303 RT

the activation energy being undoubtedly lower than the silicon-methyl bond dissociation energy. The product composition depended to some extent on the nature of the reaction vessel, but in the most inert vessels 10 products were detected; methane was the main product, while others formed in high yield were Me,SiH and 1,1,3,3-tetramethyl 1,3-disilacyclobutane (TMDS). 7 v Formation of TMDS implies the presence of Me,Si=CH,, which was also present, along with HMeSi=CH,, in the pyrolysis of Me,SiH.7s The Arrhenius parameters clearly indicate that there was at least a contribution from a chain mechanism in the pyrolysis of Me,Si, the following outline scheme being reasonable: Me,Si Me*

7g

+ Me,Si

+ Me,Si--f

CH,

-t Me-

+ Me,SicH2

(23) (24)

I. M. T. Davidson and C. A. Lambert, J . Chem. SOC.,( A ) , 1971, 882. R. P. Clifford, B. G. Gowenlock, C. A. F. Johnson, and J. Stevenson, J. Orgunometallic Cliem., 1972, 34, 53.

Kinetic Studies in SiIicon Chemistry

Me,Si-

+ Me,Si

Me,SicH,

229

+- Me,SiH -+

+ Me,SicH,

Me,Si=CH,

+ Me.

(26)

followed by dimerization of Me,Si=CH, to TMDS and by radical combination reactions to terminate the chain. The strong evidence for the occurrence of reaction (26) casts doubt on the interpretation of the mechanism of pyrolysis of Me,SiH 78 and of Me,Si,,80 in both of which this type of reaction was argued not to occur. In the latter pyrolysis the rate of formation of a product believed to be formed in a non-chain sequence rate-determined by * silicon-silicon bond rupture was measured, Qiving an A factor of s-l. This low A factor for silicon-silicon bond dissociation in Mes%, has not been corroborated by the more recent measurements of the rate of combination of Me,% radicals,35*3 7 which showed that the A factor for dissociation of Me6Si2 should be comparable with A factors for C-C dissociation, i.e. ca. lo4 times higher. For these reasons a re-investigation of the pyrolysis of Me&, was necessary; this has now been completed, the pyrolysis having been studied in a stirred-flow system between 770 and 872K with low partial pressures of Me,Si, in a nitrogen carrier-gas stream above atmospheric pressure. The results have as yet only been published in preliminary form,81 and in view of the complexity of the mechanism and the large quantity of new information obtained, only a brief account will be given here in advance of publication of the full details.82 The main products were Me,SiH, Me,Si, Me,SiCH,SiMe,H (which is an isomer of Me&,), and TMDS. Kinetic data were obtained for the formation of all of these products, but particularly for Me,SiH; Me,Si was the only product to be formed in a first-order reaction, the order for the formation of Me,SiH and TMDS being ca. 1.3 and for the isomerization reaction forming Me,SiCH,SiMe,H ca. 1.8. When excess rn-xylene was added the rate of formation of TMDS fell to zero while the rate of formation of the other products, except Me& was reduced, with a simplification in order to 1.0 for the formation of Me,SiH and 1.5 for the isomerization. In the absence of rn-xylene a short chain sequence is now believed to operate,81*82 as follows:

Me,Si.

Me,SiSiMe,

+

2 Me,Si-

+ Me,SiSiMe,

+-

Me,SiH

-+

Me,SiCH,SiMe,

-+

Me,Si*

-+

Me,SiCH,SiMe,H

Me,Si,CH, Me,SiCH,&Me, Me,SiCH,$iMe,

+ Me,Si,

(27)

+ Me5Si2cH2

(28) (29)

+ Me,Si=CH, + Me5Si,cH,

(30) (31)

followed by termination reactions, and by dimerization of Me,Si=CH, 8o

81 82

I. M. T. Davidson and I. L. Stephenson, J. Chem. SUC.( A ) , 1968,282. I. M. T. Davidson and A. V. Howard, J.C.S. Chem. Comm., 1973, 323. I. M. T. Davidson and A. V. Howard, J.C.S. Faraday IZ, (in the press).

to

230

Reaction Kinetics

form TMDS. Reaction (30) is in competition with reaction (31) and their relative rates obviously depend upon the concentration of Me,Si,. At high concentration (31) occurs to the exclusion of (30) forming the isomer of Me6Si2 in high yield with no TMDS and little Me3SiH, as was found experimentally in a detailed kinetic study of this isomerization reaction in a static system.83 At very low concentration, isomer formation was negligible (under these conditions TMDS was not observed either, probably because the Me,Si=CH, diffused to the walls, there being no added inert gas in these experiments). All of the main products except Me& were formed in one or other of the two chain cycles in the above sequence, (27) to (31), the non-integral orders arising because more than one termination reaction was important. When excess rn-xylene was added, reaction (32) could compete successfully with both (30) and (31): Me,SiCH,SiMe,

+ CsHlo

Me,SiCH,SiMe,H

+ CsHB-

(32) The suppression of reaction (30) not only stops the formation of Me,Si==CH, and hence TMDS, but also makes the formation of Me3SiH a non-chain process rate-determined by reaction (27) (in rn-xylene, Me,SiH resulted from hydrogen abstraction by Me,% radicals from the excess rn-xylene rather than from Me&,). Hence the Arrhenius parameters for reaction (27) may be deduced from the rate of formation of Me3SiHwith added rn-xylene, giving: --f

loglo(k27/~-l)= (17.2 f 0.3) - (336.6 i-4.0) kJ mol-l/2.303 RT This A factor is now in accord with those for the recombination of Me,Si radical^,^^^ s 7 while the activation energy can be identified with DH" (Me,%-SiMe,), thus representing a substantial increase on the earlier accepted value,,. 8o but much closer to DHO(H3Si-SiH3),84 which seems to be reasonable. This new value of DH"(Me,Si-SiMe,) together with recent appearance potential measurements 85 gave 368 kJ mol-l for DH"(Me,Si-H). Preliminary experiments on the rate of iodination of Me3SiH have been reported; although results were only obtained at one temperature, 546 K, necessitating some rather extensive assumptions in arriving at DH"(Me,Si-H), the figure obtained was ca. 376 kJ mol-l, in good agreement with the above result.86 As with the disilanes, the revised value for Me3SiH is quite close to that for SiHp.84$ Starting from these values, Arrhenius parameters have been estimated for all of the kinetically significant steps in the pyrolysis of Me,Si, and these estimates have been shown to be consistent with the overall kinetic results for the pyrolysis of 83

85 86

87

I. M. T. Davidson, C. Eaborn, and J. M. Simmie, J.C.S. Furuduy I, 1974,70,249, and refs. therein. W. C. Steele and F. G. A. Stone, J. Amer. Chem. SOC.,1962, 84, 3599; F. E. Saalfeld, and H. J. Svec, J. Phys. Chem., 1966,70, 1753. I. M. T. Davidson, M. Jones, and H. F. Tibbals, unpublished work. R. Walsh and J. M. Wells, J.C.S. Chem. Comm., 1973, 513. H, E. O'Neal, S. Pavlou, T. Lubin, M. A. Ring, and L. Batt, J . Phys. Chem., 1971,75, 3945.

23 1

Kinetic Studies in Silicon Chemistry

Me,Si, at high 83 and low 82 concentration. There is evidence that the Me,% is formed in the pyrolysis of Me,Si, by a unimolecular silylene elimination: Me,SiSiMe, -+Me,Si Me,Si (33)

+

which would be followed by insertion reactions of Me,Si. The product rate-measured in the earlier work on the pyrolysis of Me& could have resulted from insertion of Me,Si into Me~Si2,and indeed the Arrhenius parameters then obtained probably did relate to reaction (33) and not reaction (27), since they were identical with those found in the latest work for the formation of Me,Si, with and without added r n - ~ y l e n e . ~ ~ The increase in the value of DH"(Me,Si-H), and an equivalent increase in DH"(HMe,Si-Me), confirm the need to re-interpret the pyrolysis of Me,SiH. The revised values can be reconciled with the experimental results by invoking a chain-like sequence, in which methyl radicals and hydrogen atoms are regenerated by dissociation of HMe,SicH,, the other products of these dissociations being HMeSi=CH, and Me,Si=CH, which dimerize to form disilacyclobutanes.s2 It is no longer necessary to postulate that HMeSi=CH, and Me,Si=CH, result from radical disproportionation, as was originally suggested.'* It is to be expected that disilanes with at least one hydrogen, oxygen, nitrogen, or halogen attached to silicon will form silylenes on pyrolysis in preference to silyl radicals.88 This expectation was seen in Section 4 to have been fulfilled for the pyrolysis of Si2H6;50951 Si2C16;46 Si2Fs, HF,SiSiH,, FSi2H5,HC12SiSiH3,C1Si2H5,and HC1MeSiSiMeC1H.62 Silylene formation also occurs 89 with MeSi,H, and H,MeSiSiMeH,, as shown by pyrolysis studies between 600 and 700 K with and without added EtSiD,, thus: H,MeSiSiMeH, MeSi,H, MeSi,H,

--f

MeSiH

+ MeSiH,-

+ MeSiH, + MeSiH + SiH, --f

SiH,

(34) (35)

(36)

k,,/k,, was given as 1.7, but this figure has now been revised O 0 to 6.9. Possibly, the dependence of silylene insertion rate on hydridic character 53 is an important factor here, facilitating reaction (-35) relative to ( -36) and thus lowering E35 relative to E,,. Formation of silylene and analogous intermediates has also been observed in the pyrolysis of H3SiPH, and H,GeSiH,: H3SiPH, -+ SiHz PH, (37) H3SiPH24 H,SiGeH, 88

so

+ PH + SiH4

GeH,

+ SiH,

I. M.T. Davidson, J. Organometallic Chem., 1970, 24, 97. R. B. Baird, M. D. Sefcik, and M. A. Ring, Znorg. Chem., 1971, 10, 883. M. A. Ring, personal communication.

(38) (39)

Reaction Kinetics

232

H,SiGeH,

--t

SiH,

+ GeH,

(40)

k,,/k,, was 2.6, but k,* was much greater than k4,,, possibly for thermodynamic reasons. 91 Problems persist with the mechanism of pyrolysis of SiH4. Alternative explanations for the experimental results are a silylene mechanism (41)-(43), analogous to that agreed to occur in the pyrolysis of Si2H6,and a silyl radical mechanism (44)-(47). SiH4 $ SiH,

+ Hz

(411

+ SiH4+ SiH, + Si&, f Si,H, SiH,

(42) (43)

Ha

+ SiH4 -+ Hs + *SiH,

(45)

SiH,

+ SiH, + SizH6+ €3.

(46)

2 $iH3 --f SiZH6

(47)

Purnell and co-workers favour the silylene mechani~rn,~~9 while Ring et al. favour the silyl radical rnechani~m.~~ Basic thermodynamic data are not now in dispute; enthalpies of formation of SiH, and SizH6 are well established, and DH"(H,Si--H) and DHo(H3Si--SiH3),although measured some years ago by electron impact,84are probably quite reliably known. Hence A&(SiH,), may be calculated to be 211 kJ mol-l. John and Purnell have calculated A&(SiH& as 242 kJ mo1-l from experimental data for insertion reactions of SiHSgenerated by pyrolysis of SizH6,where the mechanistic interpretation is well established; this leads to a value of 211 kJmol-l for AH42. If the silylene mechanism is correct, AH41 and the experimental activation energy for the pyrolysis of SiH4 give an activation energy of 23 kJ mo1-1 for reaction ( -41), which is consistent with experimental observation^.^^ However, a more substantial point is that the above enthalpies, with calculated entropies, enable the equilibrium constants for reactions (48) and (49) to be obtained: 569

+ SiHs + SiH, + SiH, &HI + Ha + SiH, + Hz

SiH,

(48) (49)

At 600 K, K4* = lolla3,confirming the importance of SiH, relative to SiH,

which is strong evidence in the pyrolysis of Si,H,. Likewise, K49= in favour of the silylene mechanism for the pyrolysis of SiH4.57 The only disturbing feature in the above arguments concerns DH"(H,Si-H), which is calculated to be 249 kJ mol-l, much less than DH"(H,Si-H), which is 91

92

L. E. Elliott, P. Estacio,.and M. A. Ring, Znurg. Chem., 1973,12,2193. M. A. Ring, M. J. Puentes, and H. E. O'Neal, J . Arner. Chem. SOC.,1970,92,4845.

233

Kinetic Studies in Silicon Chemistry

398 kJmol-l. However, the 'bivalent state' in silicon is more stable than in carbon, and as sharp a drop as this from the first to the second bond dissociation energy in SiH, may be perfectly possible. Certainly, in all other respects the kinetic and thermodynamic arguments presented by Purnell and his co-workers offer a coherent and consistent explanation for the pyrolysis of Si2Hsand SiH4. Ring and co-workers have considered the kinetic features of the radical mechanism.92Reaction (46) is quite p l a ~ s i b l e34, ~but ~ ~there is some difficulty in estimating the Arrhenius parameters. Reasonable estimates lead to an activation energy close to the experimental value, but the experimental A factor cannot be reconciled with the mechanism unless Aq7 is ca. lo6dm3 m o l - l ~ - ~It. has recently been shown that the A factor for combination of Me3Si radicals is ca. 1O1O dm3mo1-l s-*, and there is evidence (in solution at least) that Ap7is higher than that.36 Hence kinetic arguments now weigh heavily in favour of the silylene mechanism. Arguments for the radical mechanism rest largely on the product composition in co-pyrolysis of SiH, and SiD,. Pyrolysis of an equimolar mixture in flow and static systems gave 92 some mixed silanes, SiH,-,D, (x = 1-3), smaller quantities of disilanes from SizDs to Si2H, and relatively large quantities of D, and HD, after 2 4 % decomposition. Appreciable quantities of Da and HD were also formed in the pyrolysis of SiD, in H2. However, the silylene mechanism could lead to extensive scrambling very soon after the start of the reaction, since reactions (-41) and (42) are fast, and it may be significant that the only disilane not observed in the co-pyrolysis of SiH, and SiDd after the 2% decomposition was H,SiSiD,, which would have been expected to be formed in good yield by reaction (47) if the radical mechanism was operating. On the other hand, the production of relatively large quantities of HD is much more consistent with the radical mechanism than with the silylene mechanism, and further experiments of this type would be desirable. Some experiments on the pyrolysis of EtSiC1, and of EtSiMe, at 823 K have been reported.03 In each case there was evidence for dehydrosilylation by a radical chain sequence: C2H5SiX3+ C 2 H 6+ Sixs X3Si

+ C2H5SiX3

--f

c2H4SiX33

+ C2H4SiX, X3Si + C2H4SiX3 X3!%

--f

--f

+ c2H4SiX3 CzH4+ s i x 3 X3SiH

X3SiC2H,SiX3 X3SiH

+ CH2=CHSiX3

(50) (51)

(52)

(53)

(54)

The reaction between Me3SiHand MeCl in the temperature range 702-767 K OS

C . Eaborn, J. M.Simmie, and I. M.T . Davidson,J . Organometallic Chem., 1973,47,45.

234

Reaction Kinetics

These were formed gave CH, and Me,SiCl as the predominant in a chain sequence propagated by reactions ( 5 5 ) and (56):

+ Me,SiH -+CH, + Me,Si. Me,Si* + MeCl -+ Me,SiCl + MeMe.

(55)

(56)

but the initiation and termination steps were not clearly established, although there was evidence that they were heterogeneous. In the gas-phase reaction of Me,SiH with NO2, a measurable rate was observed around 320 K, with ignition at T > 513 K. The main products were water, nitric oxide, and (Me,Si),O, with no carbon dioxide. An outline mechanism was suggested, featuring Me,Si and Me,SiO radicals, and the unstable adduct Me,SiON. Similar features were observed in the reaction of Et3SiH with NO,.95

6 Thermochemistry Acquisition of reliable thermochemical data for silicon compounds is inherently difficult, but there has been progress in obtaining enthalpies of formation, particularly by rotating bomb calorimetry or from heats of hydrolysis. 96 Several bond energy schemes for silicon compounds have been put forward. 9 7 Mass spectra and ionization potentials of the series of compounds Me3M1M2Me3,and appearance potential sof the fragments (Me3M1)+ and (Me3M2)+,have been measured (where M1 and M2were C, Si, Ge, Sn, and Pb). In conjunction with known or estimated enthalpies of formation of molecules and alkyl radicals the data enabled an overdetermined set of simultaneous equations to be set up, from which the optimum values for the enthalpies of formation of Me3M1h?L2Me3, *Me4M1,Me3M2,and (Me3M1)+could be obtained. Bond dissociation energies and average bond energies were then deduced. g g A similar method was used to compile the ‘CATCH’ Tables for silicon compounds, an excellent and very comprehensive collection of data, optimized by computer methods.99 It now requires some revision to take account 94

95

96

97

98

99

C. Eaborn, J. M. Simrnie, and I. M. T. Davidson, J. Organometallic Chem., 1972, 44, 273. G. L. Gagneja, B. G. Gowenlock, and C. A. F. Johnson, J . Organometallic Chem., 1973, 55, 249. S. H. Hajiev and M. J. Agarunov, J. Organometallic Cliem., 1970,22, 305; B. S. Iseard, J. B. Pedley, and J. A. Treverton, J . Chem. SOC.( A ) , 1971, 3095; J . C. Baldwin, M. F. Lappert, J. B. Pedley, and J. S. Poland, J.C.S. Dalton, 1972, 1943; L. P. Hunt and E. Sirtc, J. Electrochem. SOC.,1972, 119, 1741 ; B. S. Iseard, D.Phil. Thesis, University of Sussex, 1973. M. J. VanDalen and P. J. Van Den gergh, J. Organometallic Chem., 1970, 24, 277; D. Quane, J . Phys. Chem., 1971,75,2480; J. M. Gaidis, P. R. Briggs, andT. W. Shannon, ibid., p. 974. M. F. Lappert, J. B. Pedley, J. Simpson, and T. R. Spalding, J. Organornetallic Chem., 1971, 29, 195. J. B. Pedley and B. S. Iseard, ‘CATCH’tables for silicon compounds, University of Sussex, 1972.

Kinetic Studies in Silicon Chemistry

235

of the higher values of DH"(Me,Si-SiMe,), DH"(Me,Si-H), and related dissociation energies (see Section 5), but once this has been done, the 'CATCH' Tables will be the best available source of thermochemical data in silicon chemistry.

7 High-energy Chemistry Recoil tritium reactions with Me,SiF and with Me,Si, have been reported, With Me,SiF, abstraction and substitution at carbon-hydrogen bonds predominate over reactions at silicon-carbon and silicon-fluorine bonds. Reactivity is influenced by the silicon-fluorine inductive effect and by bond strength effects.loOThe latter also appear to be important in the Me,&, system, where hydrogen abstraction and substitution, heavy group displacement, and reaction at the silicon-silicon bond all occurred. Competitive experiments with neopentane showed that carbon-hydrogen bond reactivities in Me,%, and neopentane were about the same, carbon-silicon bond reactivity was greater than carbon-carbon bond reactivity, and silicon-silicon reactivity was greatest of all, probably due to a low threshold energy and to a high cross-section for reaction.lol The reactions of recoiling silicon atoms (generated by neutron irradiation of PH,) with SiH,, Si2H6,and Si3Hs have been studied.66There were some experiments with added nitric oxide as a scavenger, and some with the system PMe, and Me,SiH. From a thorough analysis of the composition of products from these experiments it was concluded that the reactions with the silanes Si,H2;+a involved only one type of reactive intermediate, and that there was good evidence for this being ,lSiH2 and not ,'SiH3. Si,H, was found to be relatively more reactive towards SiHz than was SiH,, but it was not possible to determine whether this higher reactivity was due to differences in the silicon-hydrogen bond reactivities in these two molecules or to a contribution from insertion into the silicon-silicon bond (cf. earlier refs. 53 and 56). The paper contains an excellent review of the chemistry of recoiling silicon atoms.65b The X-ray radiolysis of Me,Si has been investigated in the gas phase lo2 and in ~01ution.l~~ In the gas-phase experiments at 298 K equimolar mixtures of (CH,),Si and (CD,)$i were irradiated, with 'oxygen and nitric oxide as scavengers to distinguish radical and molecular product-forming mechanisms. The product composition was consistent with the formation of Me, Me,Si, and Me,SiCH2 radicals, with som; products due ta Me,SiCH,, probably in its biradical form. The major single source of methane was direct molecular elimination, a process which was not found to be important ',at lower energies in the decomposition of chemically activated Me,% loo lol

S. H. Daniel and Y.-N. Tang, J. Phys. Chem., 1971, 75, 301. S. H. Daniel, G. P. Gennaro, K. M. Ramik, and Y.-N. Tang, J . Phys. Chem., 1972, 76, 1249.

lo* lo*

G. J. Mains and J. Dedinas, J . Phys. Chem., 1970, 74, 3476. G. J. Mains and J. Dedinas, J . Phys. Chem., 1970,14, 3483.

236

Reaction Kinetics

Ion-Molecule Reactions.-Ion-molecule reactions, initiated by electron impact, have been studied in Me,SiH2, Me,SiH, and Me4Si.Io4 Many reactions were identified and their rate constants measured. With Me2SiH2, the three primary ions were Me2Si+,MeSiH+, and MeSiH;. Me,SiH+ was the main product ion, formed in a series of hydride transfer reactions: Me2Si+ t Me2SiH2 Me2SiH+ -+

MeSiH+

+ Me2SiH2

MeSiHl

+ Me2SiH2

+

--f

+ Me2SiH

+ MeSiH, Me,SiH+ + MeSiH, Me,SiH+

(57) (58) (59)

An insertion-elimination reaction was also observed :

+ Me2SiH2-+Me2Si2Me,++ H2

Me2Si+

Hydride transfer was also important with Me,SiH, converting the primary ions Me,SiH+ and Me2Si+into the secondary ion Me,Si+ :

+ Me,SiH + Me,Si+ i- Me2SiH Me,SiH+ + Me,SiH Me,Si+ + Me,SiH2 Me2Si+

-+

(60) (61)

No hydride ion transfers were observed with Me,Si, where the main reaction involved the molecule-ion, probably in a collisional dissociation :

Me&+

+ Me4Si -+ Me,Si+

-1- Me3Si.

+ C2H6

(62)

Ion-molecule reactions in SiH, have been studied by high-pressure mass spectrometry and tandem mass and by ion cyclotron resonance.lo6 Rate constants and cross-sections were measured. At low pressure (<6 x mm Hg) the primary ions SiH+ and Si+ react with SiH,:

+ SiHJ Si,Hz + H2 SiH+ + SiH4 +Si2H: + H2 Si+

SiH+ + SiH,

-+

+ Si2H+

+ 2H2

(63) (64)

(65)

Hydride transfer reactions also occur between SiH, and SiH+, SiH;, and SiH:, e.g.: SiH; -1- SiH, -+ SiH; + SiHs (66) At higher pressures,1o5SiHi was converted into Si2H; in a third-order reaction : SiH; + 2SiH, --+ Si2H: + SiH, (67) l'I5

P. Potzinger and F. W. Lampe, J . Phys. Chem., 1971, 75, 13. T.-Y. Yu, T. M. H. Cheng, V. Kempter, and F. W. Lampe, J. Phys. Chem., 1972, 76,

I"'

J. M. S. Henis, G. W. Stewart, M. K. Tripodi, and P. P. Gaspar, J . Chem. Phys., 1972,

3321.

57, 389.

Kinetic Studies in Silicon Chemistry

237

Similar reactions produced Si3Hi, Si4Hl+l,and Si,H&. Protonated molecules thus constitute an important group of products, whereas they are very minor products in the ion-molecule reactions of methane. Their facile formation in the SiH4 system has been suggested as the reason for the good yield of Si,H, in the radiolysis of SiH4.106Similarly, the high rate constant for the formation of products with two silicon atoms by the reaction of Si' with SiH, may account for the formation of higher silanes in discharges and nuclear recoil experiments.lo6 An ion cyclotron resonance study of the ion-molecule reactions in SiH4CH, mixtures enabled numerous reactions leading to mixed products (i.e. products containing carbon and silicon) to be identified, and their rate constants determined.lo7 In addition to the hydride transfer reactions described above, similar reactions of SiH, with CH;, CHf, and CHZ were identified. There were no product ions containing silicon-carbon bonds resulting from the reactions of Si+ or CH:, and thermodynamic arguments were adduced to show that such reactions would be too endothermic. In the formation of the mixed products which were observed, SiHl and CH: were the most reactive ions. Whilst hydride transfer appears to be a well-established process in the ion-molecule reactions of silicon compounds, and is plausible in relation to the electronegativities of carbon, silicon, and hydrogen, the products in the silane system might have resulted from proton transfer, e.g. : CH;

+ SiH, + SiH:*

-+

SiH:

+ Hz

or, in the case of reaction (66), by H-atom transfer. In order to distinguish between these possibilities ion-molecule reactions in a mixture of SiH, and CD, were studied; proton and H-atom transfer were shown to be unimportant, hydride transfer being indeed the main Notwithstanding the importance of hydride transfer, there is some evidence for the occurrence of protonation, as indicated by the formation of protonated disilanes and higher silanes. Protonation of SiH4 has now been observed as well, SiH: being formed in the reactions of SiH, with various ions.log From the observed relationship between kinetic energy and crosssection, formation of SiH; was found to be exothermic in reactions of SiH, with NHZ, C,H:, and CZH; ; on the other hand, the reaction with C,Hi was endothermic. Hence the proton affinity of SiH, was calculated to be between 628 and 653 kJ mol-l. Alternatively and equivalently, AHfO(SiHClg is between 908 and 975 kJ mol-l. 8 Rearrangements 1,ZRearrangements in silicon-containing molecules or radicals have been lo'

G. W. Stewart, J. M. S. Henis, and P. P. Gaspar, J . Chem. Phys., 1972,57, 1990.

lo8

G. W. Stewart, J. M. S. Henis, and P. P. Gaspar, J. Chem. Phys., 1972,57, 2241.

lo9

T. M. H. Cheng and F. W. Lampe, Chem. Phys. Letters, 1973,19, 532.

238

Reaction Kinetics

known for some time. Formation of silylenes by 1,Zshifts of hydrogen or halogen in disilanes are discussed above,60-62and an example in a siliconcontaining radical occurs in the free-radical isomerization of Me&, where a Me3Si group migrates :83 Me3SiSi(Me2)cH2 Me$iCH,SiMe, --f

(68)

A similar rearrangement has been suggested *l to account for the formation of Me,SiCl as the only product in the gas-phase reaction of methyl radicals with ClCH,SiMe2H:

CICH,SiMe,

--f

cH2Si(Me2)C1

(69)

In certain silicon-containing molecules interchange of groups between silicon and a carbon atom in the a-position can occur, via a cyclic transition state :llo RICXY.SiRt

R1CXR2.SiRiY (R2 = Halogen or Ph, Y

--f

=

F, C1, OAc, or OTs)

1,ZMigration of Me,Si was observed in the thermal isomerization of trimethylsilylindene in diphenyl ether between 419 and 474 K.ll1 The reaction was first order and intramolecular, with AG* around 150 kJ mol-l. A more complex example, akin to a 1,2-rearrangement, occurs in the thermolysis of tris(organosily1)hydroxylamines at 473 K : (R3Si),NOSiR3 -+ R3SiN(R)Si(R2)OSiR3 This is believed to occur intramolecularly by insertion of SiR3 into the nitrogen-oxygen bond and transfer of R from silicon to the a-nitrogen atom.l12 As well as the 1,Zmigrations in molecules and radicals noted above, there are examples of similar processes in anionic and solvolytic rearrangement~.~~~ A 1,3-migration of silicon has been observed in the thermal rearrangement of allyl silanes in the gaseous or liquid phase. Gas-phase experiments at 773 K gave evidence for a concerted unimolecular process, with an activation energy of ca. 200 kJ mol-l. The rearrangement, which was reversible, caused isomerization of substituted allyl silanes.lf4 An interesting redistribution reaction occurs when unsymmetrical disilanes are subjected to prolonged thermolysis at relatively low temperature :115

111

112 113

11*

115

W. I. Bevan, R. N. Haszeldine, J. Middleton, and A. E. Tipping, J. Organometallic Chem., 1970,23, C17, and refs. therein. P. E. Rakita and G. A. Taylor, Znorg. Chem., 1972, 11, 2136. P. Boudjouk and R. West, J . Amer. Chem. Soc., 1971, 93, 5901. M. A. Cook, C. Eaborn, and D. R. M. Walton, J. Organometallic Chem., 1970, 24, 301; M. S. Biernbaum and H. S. Mosher, J. Amer. Chem. SOC.,1971,93,6221; A. G. Brook and J. D. Pascoe, ibid., p-6224; R. West and B. Bichlmeir, ibid., 1972,94, 1649; J . J. Eisch and M. R. Tsai, ibid., 1973, 95, 4065. H. Kwart and J. Slutsky, J. Amer. Chem. SOC.,1972, 94, 2515. H. Sakurai and A. Hosomi, J. Organometallic Chem., 1972, 36, C15.

239

Kinetic Studies in Silicon Chemistry

2 RMe,SiSiMe,

+ (Me,Si), + (RMe,Si), (R

= Pr",

Pr", Ph, or PhCH,)

From product composition and from experiments in chlorobenzene which produced no chlorosilanes, it was established that the reaction occurred by a bimolecular four-centre process. The rate was rather low, 13% reaction being achieved after 73 days at 473 K, but the process is nevertheless much faster than dissociation of the silicon-silicon bond, 81 which has a half-life of 2 x lo1, years at 473 K. 9 MoIecular EIimination Reactions Unimolecular gas-phase @-eliminationreactions of 2-halogenoethylsilanes are well known :

+ R3SiX (X = F or C1, R = alkyl or X)

XCH2CH2SiR3+ C2H4

In one case, that of ClCH,CH,SiCl,, dehydrosilylation was also observed as a minor reaction, and was believed likewise to be unimolecular:

CICH,CH2SiC13

--f

HSiC13 + C2H3Cl

However, this has recently been shown to be a radical process and an explanation has been suggested for the non-occurrence of this type of reaction in other 2-chloroethylsilanes.93 Pyrolysis studies of CHF,CH,Si(OMe), between 515 and 601 K showed that it underwent homogeneous unimolecular @-eliminationto form CHF=CH2 and FSi(OMe),. First-order rate constants were given by: log,,(k/s-l)

=

(11.04 f 0.11) - (152.0 f 1.2) kJ mol-l/2.303 RT

Recent work on tetrafluoroethylsilanes has revealed another mode of thermal decomposition of this class of compound. Pyrolysis of CHF2CF2SiF3 between 413 and 473 K in static system with reactant pressures of 20-150 mm Hg gave SiF, and CHF=CF,, the expected &elimination products, in a homogeneous first-order reaction.l17 However, fluorinated cyclopropanes were observed as secondary products, while in experiments with added Me,SiH, Me,SiCHFCHF, was formed instead of CHF=CF, and the cyclopropanes. Experiments with added CHF=CF2 or propene gave analogous results to the experiments with Me,SiH. Although the product composition was drastically affected by these added compounds, the rate of decomposition of CHF2CF2SiF3was not. These results provide convincingevidence for an a-elimination, forming a carbene: CHF2CF2SiF, ll& 11'

--f

SiF4 + CHF,CF:

(70)

D. Graham, R. N. Haszeldine, and P. J. Robinson, J. Chem. SOC.(B), 1971, 611. R. N. Haszeldine, P. J. Robinson, and W. J. Williams, J.C.S. Perkin II, 1973, 1013.

240

Reaction Kinetics

In the absence of additives the carbene rearranged to CHF:CF, ; in the later stages of the reaction carbene insertion into CHF: CF2gave the observed cyclopropanes. With the additives, carbene insertion occurred instead of rearrangement, giving a characteristicnew product. First order rate constants for this unimolecular a-elimination were given by: log,,(k/s-l)

= (13.11

k 0.10) - (137.8 f 0.9) kJ mol-l/2.303 RT

The A factor is reasonable for this type of reaction and the low activation energy was attributed to the strength of the silicon-fluorine bond formed. The Arrhenius parameters are very similar to those for &elimination in similar compounds, but the experiments with additives showed that 8elimination occurred to ~ 5 of % the total reaction in this case. Similar experiments with CHF2CF2SiMe3between 573 and 643 K showed that it too underwent a-elimination by a unimolecular homogeneous reaction :llS CHF2CF2SiMe3-+ CHF,CF: Me,SiF (71)

+

First order rate constants were given by: log,,(k/s-l)

=

(13.93 f 0.22)

-

(197.7 A 2.5) kJ rnol-'/2.303 RT

The higher activation energy was attributed to a less electropositive silicon atom causing the forming silicon-fluorine bond to release less energy in the transition state. In this work, one of the additives was cis-but-2-ene to which the carbene added stereospecifically, indicating that it was formed in a singlet state. This conclusion was confirmed by experiments with added oxygen, which had no effect; oxygen would be expected to scavenge the carbene in its triplet form, which was, therefore, presumably absent. CI,CSiCl, has long been known llS to undergo a-elimination on pyrolysis, forming CCI, and SiCl,. A kinetic study of this reaction has now been reported. Between 423 and 463 K in the gas phase, with pressures from 1 to 120 mm Hg, the carbene elimination accounted for ca. 90% of the total reaction, first-order rate constants being given by : l ~ g , , ( k / s - ~=) (10.0 f 0.8) - (124.3 f 6.7) kJ mol-l/2.303 RT

The pyrolysis was performed in the presence of ethylene, which trapped the carbene to form a cyclopropane.120 Although (Me,Si),Hg gives Me3Si radicals on photolysis, its thermal reactions are predominantly mo1ecular.l In solution around room temperature a bimolecular four-centre reaction occurs between (Me,Si),Hg and 1,2-dibrornides, bringing about a stereospecific cis-elimination of bromine from the dibromide to form an alkene, Me,SiBr, and mercury.121 Similarly, R. N. Haszeldine, C. Parkinson, and P. J. Robinson, J.C.S. Perkin II, 1973, 1018. W. A. Bevan, R. N. Haszeldine, and J. C. Young, Chem. Ind., 1961, 789. lZo D. Seibt and H. Heydtmann, 2.phys. Chem. (Frankfurt), 1973, 83, 256. lZ1 S. W. Bennett, C. Eaborn, R. A. Jackson, and R. W. Walsingham, J . Organometallic Chem., 1971,27, 195.

118

11*

Kinetic Studies in Silicon Chemistry

24 1

in solvents such as toluene, anisole, and cyclohexane, (Me,Si),Hg undergoes self-reaction and reaction with the solvent by bimolecular four-centre mechanisms.122In cyclohexane at low concentration of (Me,Si),Hg a firstorder reaction due to dissociation of (Me,Si),Hg was also observed, from which DH”(Me,Si-HgSiMe,) was estimated to be > 200 kJ mol-l.

Elimination Reactions of ‘Hot’ Molecules.-Molecular elimination is an important route in the decomposition of some chemically activated silanes, generated by insertion of singlet methylene into silicon-hydrogen bonds. The most striking example is EtSiH3*which is believed to undergo a 8elimination as its main decomposition route: 7 7 EtSiH3*

--f

+ SiH,

C2H4

(72)

There are other routes forming molecular products directly along with a silylene in the decomposition of chemically activated silanes. Thus, H, and C2H6 are formed from EtSiH,* ;CH4and H, from Me,SiH,* ;H, and, to a small extent, CHI and C2H6from EtMeSiH,*; and CH, from Me,SiH*. Elimination reactions of ‘hot’ CF,CH,SiX,* (X = F or C1 and/or Me) forming CF,=CH, and FSiX, have been ~ b s e r v e d 21 , ~but ~ ~ it is not known whether these products are formed in a direct &elimination or in an aelimination followed by carbene rearrangement. 76p

10 Conclusions This chapter has given rather more attention to non-kinetic or semi-quantitative kinetic work than might have been expected in a Report on Chemical Kinetics. However, it is important for the kineticist working with silicon compounds to know about the chemistry of the different reactive intermediates containing silicon, and to appreciate how they may both resemble and differ from their carbon counterparts. It is to be hoped that the above account of the present state of knowledge will have gone some way towards providing this information, and may have helped kineticists to identify areas of the subject worthy of detailed quantitative study, so that the next volume of this Report can contain a larger quantity of detailed kinetic information from which firmer conclusions may be drawn. Progress in kinetic studies of silicon compounds should also be of more general interest. The history of this topic is relatively short, with the result that the same degree of development has not been reached as in some other areas of chemistry. For example, our understanding of gas-phase pyrolysis mechanisms of silicon compounds is roughly equivalent to that obtaining in hydrocarbon pyrolysis a decade or two ago. This displaced time-scale is not wholly beneficial; the availability of a fund of knowledge and experience from carbon chemistry can speed progress in comparable areas of silicon 12*

C. Eaborn, R. A. Jackson, and R. W. Walsingham, J.C.S. Perkin ZZ, 1973, 366.

242

Reaction Kinetics

chemistry, but it can also mislead if unjustifiably close analogies are drawn between the behaviour of silicon molecules and radicals and their carbon counterparts. The interplay between these factors makes an interesting and, in some respects, cautionary story, from which chemists opening up other areas of the Periodic Table might well benefit.

6 Network Effects in the Dissociation and Recombination of a Diatomic Gas BY H. 0. PRITCHARD

1 Preface The examination of the dissociation and recombination of a diatomic gas in terms of the master-equation approximation has revealed a number of unsuspected network effects which may have important consequences for the interpretation of other molecular relaxation processes. This chapter summarizes the application of the master equation to the description of diatomic dissociation-recombination processes, and its limitations and advantages are discussed: the ensuing network effects are described, and important areas where reinterpretation of non-equilibrium relaxation data may be required are suggested. Whilst covering developmentsup until the last quarter of 1973, it is not intended to be an exhaustive documentation - almost an impossibility in‘such a field - but rather an essay in which one particular aspect of the problem is highlighted and its ramifications examined : it is hoped therefore that its appeal will be to a wider audience than just those whose professional interest is in the problem of diatomic dissociation reactions.

2 Introduction We begin by marking out an area of chemistry and physics where rate processes can be described in terms of kinetic equations like the master equation, and hence in which network effects are likely to be important. The discussion will centre itself about the description of the dissociation and recombination of a diatomic gas, where these effects first showed up. Many measurements of the rates of dissociation of diatomic gases have been made using shock-tube techniques 1-6 and it is now firmly established that in the presence of an inert diluent, the Arrhenius temperature coefficients of such rates are always less than the known spectroscopic dissociation energy of the diatomic H. 0. Pritchard, Quart. Rev., 1960, 14,46. J. N. Bradley, ‘Shock Waves in Chemistry and Physics’, Methuen, London, 1962. A. G. Gaydon and I. R. Hurle, ‘The Shock Tube in High Temperature Chemistry and Physics’, Chapman and Hall, London, 1963. E. F. Greene and J. P. Toennies, ‘Chemical Reactions in Shock Waves’, Academic Press, New York, 1964. S. H. Bauer, Ann. Rev. Phys. Chem., 1965, 16,245.

243

244

Reaction Kinetics

mo1ecule.6-8 Since we shall continually refer to hydrogen as a model for a diatomic molecule, it will be convenient to bear in mind that whereas the spectroscopic dissociation energy is 36 117 cm-l, i.e. about 103.2kcal mol-l (or 431.9 kJ mol-l),$ one particular set of experiments lo yields an Arrhenius temperature coefficient for the rate of dissociation of hydrogen diluted in argon : Ar Hz + A r H H (1)

+

+ +

of about 92 kcal mol-l (385 kJ mol-l) over the temperature range 28005000K;nor would it be unfair to say that this is typical of most of the shockwave data on diatomic dissociation reactions. At about the same time as shock-tube techniques were being developed, measurements on the reverse reaction, recombination, were also being made and many examples have been studied.I1-l3 Some of the earliest work on the recombination of iodine,14-16

M

+ I + I + M + Iz

(2)

still stands, and the observation that near room temperature the recombination rate constants have small negative temperature coefficients of 1-2 kcal mol-l (5-10 kJ mol-l) is universally accepted. Then, for the best part of a decade, attempts to reconcile the dissociation and recombination data were clouded by much confusion : the shock-wave dissociation rate constants could be used to calculate recombination rate constants (kr,M)using the Rate Quotient law, i.e.

where Kc is the equilibrium constant for the reacti0n.l' However, because at that time the shock-wave data were of limited accuracy, the hightemperature recombination rate constants would not extrapolate satisfactorily towards the low-temperature data. A further consequence of the Rate Quotient Law is that the difference between Arrhenius temperature coefficients for dissociation and recombination must be (from the van't Hoff relationship) the Av for the dissociation reaction - thus, taking Hz as an example, the recombination rate constants would lie on a very curved

lo

l2

l3 l4

l6 l8 l7

J. Troe and H. G. Wagner, Ber. Bunsengesellschaft phys. Chem., 1967, 71, 930. A. C.Lloyd, Ititernat. J. Chem. Kinetics, 1971, 3, 39. H. S. Johnston and J. Birks, Accounts Chem. Res., 1972, 5, 327. G . Herzberg, J . Mol. Spectroscopy, 1970, 33, 147. A. L. Myerson and W. S. Watt, J. Chern. Phys., 1968, 49, 425. D . L. Baulch, D . D . Drysdale, and A. C. Lloyd, 'High Temperature Reaction Rate Data', Leeds University Reports, 1968 onwards. A. D. Stepukhovich and V. M. Umanskii, Wspekhi Khim., 1969, 38, 590. F. Kaufman, Ann. Rev. Phys. Chem., 1969, 20, 45. N. R. Davidson, R. Marshall, A. E. Larsh, and T. Carrington, J. Chem. Phys., 1951, 19, 1311. M. 1. Christie, R. G. W. Norrish, and G. Porter, Proc. Roy. Suc., 1953, A216, 152. K. E. Russell and J. Simons, Proc. Roy. SOC.,1953, A217, 271. 0.K. Rice, J . Phys. Chem., 1961,65, 1972; 1963, 67, 1733.

Network Eflects in Dissociation and Recombination of a Diatomic Gas

245

Arrhenius plot with an apparent activation energy of about -2 kcal mol-l ( - 10 kJ mol-l) at room temperature and about - 11 kcal mo1-1 ( -45 kJ mol-l) near 4000K;changes of activation energy of this magnitude were regarded with considerable unease. Not only were the experimental data regarded with suspicion, but the validity of the Rate Quotient Law was questioned by several authors, including the present one,ls*l Dalthough it was stoutly defended by Rice.17 With time, the situation has resolved itself satisfactorily: the experimental data can be accommodated on a single monotonically curved Arrhenius plot,l and, in a long and difficult series of experiments, Burns and co-workers have shown 20-22 that .recombination rate constants for bromine and iodine measured at very high temperatures can be consideredconsistent with shock-wavedata on the dissociation reaction ; furthermore, detailed calculations mentioned later in this report have shown that although there may be very small but real departures from the Rate Quotient Law in these systems, it is unlikely that they will ever be detected experimentally. Observations of dissociation and recombination are of course not confined to simple shock-tube and flash-photolysis experiments: dissociation occurs when a gas flows at hypersonic speed over an aerodynamic object, and recombination occurs when hot gases expand from a n ~ z z l e24. ~At ~ ~the same time, the ionization of inert gas atoms in strong shock waves in the temperature region 7000-1 2 OOO K also exhibits Arrhenius temperature coefficientsmarkedly less than the ionization potential^,^^ and the recombination of electronswith inert-gas ions to give neutral atoms shows overpopulation and bottleneck effects 2B very similar indeed to those calculated for diatomic recombination reactions. The common factor in all these experiments is that there is a fragmentation of a composite particle having internal degrees of freedom or a re-formation of such a particle from two structureless particles, and the rates of the redistribution of energy among the internal states and l8 l9 2o

22

2a 24 25 26

27 28 20

82

3s s4

s6 s6

H. 0. Pritchard, J. Phys. Chem., 1961, 65, 504. H. 0. Pritchard, J. Phys. Chem., 1962, 66, 2111. G. Burns and D. F. Hornig, Canad. J. Chem., 1960,38, 1702. J. K. K. Ip and G. Burns, J. Chem. Phys., 1969,51, 3414, 3425. H. W. Chang and G. Burns, Canad. J . Chem., 1973,51, 3394. J. G.Hall and C. E. Treanor, ‘Nonequilibrium Effects in Supersonic Nozzle Flows’, Cornell Aero. Lab. Report No. 163, 1968. J. W. Rich and C. E. Treanor, Ann. Rev. Fluid Mechanics, 1970,2, 355. T. I. McLaren and R. M. Hobson, Phys. Fluids, 1968,11, 2162. D. R. Bates, A. E. Kingston, and R. P. McWhirter, Proc. Roy. SOC.,1962, A267,297. W. G. Valance and E. W. Schlag, J. Chem. Phys., 1966,45,216. B. Widom, J. Chem. Phys., 1971,55,44. 0. K. Rice, J. Chem. Phys., 1971, 55, 439. R. G. Gilbert and I. G . Ross, J. Chem. Phys., 1972,57,2299. W. M. Gelbart, S. A. Rice, and K. F. Freed, J. Chem. Phys., 1972,57, 4699. H. Gebelin and J. Jortner, Theor. Chim. Acta, 1972, 25, 143. R. N. Kortzeborn and F. F. Abraham, J. Chem. Phys., 1973,58, 1529. J. B. Homer and A. Prothero, J.C.S. Faraday I , 1973,69, 673. J. D. Anderson, Amer. Inst. Aeronautics Astronautics J., 1970,8, 545. D. J. Spencer, T. A. Jacobs, H. Mirels, and R. W. F. Gross, Internat. J. Chem. Kinetics, 1969, 1, 493.

Reaction Kinetics

the fragmentation or re-formation are not independent of each other. Thus, if network effects are important in one case they are likely to be so in others. It is then only a short step in one direction to the description of the unimolecular dissociation of a polyatomic molecule: many recent discussions of such processes have a great deal in common with the treatment described below for diatomic p r o c e ~ s e s , ~and ~-~ so~we may expect that network effects will be important there also; the same may be true of the kinetics of nucleation and growth of droplets33 or smokes.34 Then, as will be shown later, these network effects are in fact a property of the internal relaxations of assemblies of molecules - and they are not confined to dissociation and recombination processes - hence the whole area of lasers, both gasdynamic 38 and the more conventional gas lasers,37and the treatment of ultrasonic absorption by gases 3 8 will not be complete (and could in fact be quite incorrect) without a proper consideration of these co-operative phenomena associated with large networks. On the other hand, although dissociation and recombination processes are important, say, in flames and explosions, disequilibrium of the internal degrees of freedom of the molecules is only of secondary importance, and network effects probably need not be invoked in the foreseeable future. The use of the term ‘network’ traditionally brings to mind the concept of an electrical network with resistive, inductive, and capacitative elements. However, in recent years, a thermodynamic or statistical mechanical approach has been developed in the treatment of certain other network phenomena. For example, in the ‘thermodynamic’ treatment of a telephone network, the idle customers’ needs are regarded as being the heat bath, the energy of the network i s defined as the number of calls in progress, and there are logical analogues to such thermodynamic concepts as the canonical distribution, the partition function, the temperature, detailed balancing, entropy, and so on :3 9 however, whilst these analogies are important and instructive, the molecular dissociation problem is considerably more sophisticated than the present state of the art in telephone networks, which is about equivalent to treating the molecule as a harmonic oscillator with only single-quantum jumps. Likewise, the concept of entropy is being used in discussing traffic flows in and in all three cases, be it a molecular, a telephone, or a city-street network, the concepts of maximum entropy and equilibrium are fundamental. The principle of maximum entropy stems essentially from the fact that all three kinds of process are stochastic and Markovian in character - a Markov process being one in which the next change in the properties of the system depends only upon the present state of the system and upon a set of transition rate coefficients that is characteristic of the heat-bath temperature; 357

37

39

40

R. E. Center and G . E. Caledonia, Appl. Optics, 1971, 10, 1795. H. J. Bauer, F. D. Shields, and H. E. Bass, J. Chem. Phys., 1972, 57, 4624. V. E. Benes, ‘The Mathematical Theory of Connecting Networks and Telephone Traffic’, Academic Press, New York, 1965. J. A. Tomlin and S. G . Tomlin, Nature, 1968, 220, 974.

Network Efects in Dissociation and Recombination of a Diatomic Gas

247

the system has no memory of how it arrived at its present state, and the past history is irrele~ant.~l9 4p It would be incorrect, however, to convey the impression that network ideas have not been applied to chemical problems on previous occasions: for example, Boudart discusses the isomerization of hydrocarbons in terms of network ideas,43but the concepts developed in this paper are rather different from those encountered in the isomerization problem. 3 A Variety of Theoretical Approaches Discussions of dissociation and recombination of diatomic molecules fall broadly into one of two classes, equilibrium or non-equilibrium: until recently, very little progress has been made in the latter. However, many gas-dynamic situations exist in which it is known that the internal degrees of freedom of the molecule are in equilibrium neither with the translational temperature nor with the degree of dissociation of the system 24 and so it has been common practice in the past, in order to make progress with various aerodynamic to use simple equilibrium theories of dissociation and recombination and to try to simulate the non-equilibrium properties by empirical approximation, using such concepts as the ‘sudden freezing’ of certain degrees of freedom. 23p

Equilibrium Theories.-In a dissociation reaction, it would seem logical to identify the critical energy required to dissociate the molecule with the dissociation energy ( D o ) of that molecule, and it would then follow that the rate constant for dissociation would behave as the product of a collision number (Z)and a Boltzmann factor (exp[ -Do/RT])representing the fraction of those collisions containing more than the required energy along the line of centres ; an Arrhenius temperature coefficient virtually indistinguishable from D owould therefore follow. Just about the same conclusion is reached if the vibrational energy contained in the molecule is allowed to contribute to the critical energy, correspondingly reducing the amount that has to come from the relative kinetic energy of the collision. At this level of approximation, it is however possible to contrive rationalizations for the reduced Arrhenius temperature coefficients so Cpmmonly observed, or the complementary negative temperature coefficients for recombination. For example, although three-body collisions decrease in frequency only very slowly as the temperature 46 one only needs, in 41

42 43

44 43

I. Oppenheim, K. E. Shuler, and G. H. Weiss, Adv. MoI. Relaxation Processes, 1967,1, 13. M. R. Hoare, Nature, 1970, 226, 599. M. Boudart, ‘Kinetics of Chemical Processes’, Prentice Hall, Englewood Cliffs, N.J., 1968. M. J. Lighthill, J. Fluid Mechanics, 1957, 2, 1 . L. S. Kassel, ‘Kinetics of Homogeneous Gas Reactions’, The Chemical Catalog Co., New York, 1932.

248

Reaction Kinetics

order to generate quite sizeable negative temperature coefficients, to add to this the very plausible suggestion that processes such as:

M

+ X + X + M + Xz

(4)

*' the difficult question become harder the more energy one has to is how much harder ? Another idea which enjoyed popularity for a while was that, naturally, all of the internal energy of the molecule could contribute towards the critical energy, i.e. rotation as well as vibration; the attractiveness of this idea was strengthened by the intuitive belief that obviously there must be some truth in this statement, and, as we shall see in the next paragraph, all that was wrong was that the inclusion of rotational effects was formulated incorrectly. Summarizing briefly, the standard statistical treatment of the distribution of the required critical energy E* over s squared terms in the molecule leads 4 8 to a rate in which the pre-exponential factor becomes multiplied by (&*/kT)"-l, which has the general effect of decreasing the Arrhenius temperature coefficient of the reaction 5 0 to somewhat below the critical energy - simply because some of the temperature dependence is now shifted into the pre-exponential factor. This led to the fashion for reporting experimental dissociation data in the form of AT" exp[ - D , / R T ] rather than in Arrhenius form, but unfortunately the required values of n needed to represent the data are often uncomfortably large * in comparison with the number of internal degrees of freedom that a diatomic molecule possesses, and this practice should be discouraged!51 The necessary reformulation of rotational effects showing that they do in fact lead to low Arrhenius temperature coefficients for dissociation, and negative temperature coefficients for recombination, is as follows. Because of the fact that each rotational state of a diatomic molecule is described by a different effective internuclear potential curve 52a [cf. Figure l(a)], each rotational state has its own effective dissociation energy D*(J) which is the difference between Umax(J),the effective rotational barrier, and the u = 0 level of that J state: Figure l(b) illustrates the situation for the J = 21 state of Hz,with 0*(21) = 0,(21) + Umax(21).These centrifugal barriers U,,,(J) increase in height with increasing J, and eventually so dominate the internuclear interaction that neither bound nor quasi-bound states can exist when J is very high;52athese properties have been thoroughly investigated 459

499

p6 p7

D. Husain and H. 0. PritchardiJ. Chem. Phys., 1959,30, 1101. W. H. Wong and G. Burns, Canad. J . Chem., 1973,51, 1 1 1 . R. H. Fowler and E. A. Guggenheim, 'Statistical Thermodynamics', Cambridge

University Press, Cambridge, l939. H. 0. Pritchard, Rec. Trav. chi.m., 1955, 74, 779. H. 0. Pritchard, J . Chem. Phys., 1956, 25, 267. S. H. Bauer, Proceedings AGAARD Conference No. 12, Oslo, May 1966, p. 147. G. Herzberg, 'Spectra of Diatomic Molecules', D . van Nostrand and Co., New York, 1950, (a)pp. 4 2 5 4 3 0 ; (b) pp. 99-101.

lU

51

52

Network Efects in Dissociation and Recombination of a Diatomic Gas

249

I

I

/M€RNUCL€AR SffAhM77ON /N A T i C Wl7S

Figure 1 (a) Efective potential energy curves for HZfor four values of J: 1 a.u. of energy = 219 474.72 cm-l; 1 a.u. of distance = 0.529167 X cm

theoretically for He and D2.63* 64 Thus, if we assume that all the rotationvibration levels of the molecule have their proper Boltzmann populations, then simple extension of the standard collision theory result shows that the rate of dissociation now behaves as :

where 2 is a collision number (roughly independent of J ) , and n(J) is the 6s g4

T. G. Waech and R. B. Bernstein, J. Chem. Phys., 1967, 46,4905. R. J. LeRoy, ‘Eigenvalues and Certain Expectation Values for all Bound and Quasibound Levels of the Ground State of HE, HD, and De), WIS-TCI-387, Theoretical Chemistry Institute, University of Wisconsin, Madison, Wisconsin, 1971.

250

Reaction Kinetics

/50001

v

-20000~ '

8

1

1

I

1

I

1

i

I

4 6 8 /O /NT€RNUCL€AR XPARAT/ON /N ATOMX' UN.7-S

2

I

/2

Vibrational energy-level diagram for the J = 21 state of H, showing two quasi-bound states, u = 1 and u = 8; the lifetimes of these states with respect to predissociation (tunnelling)are about 5 x and s, respectively

Figure 1 (b)

Network Eflects in Dissociation and Recombination of a Diatomic Gas

251

number of diatomic molecules in any particular rotational state J ; after a little manipulation 55j this gives a rate constant in the form:

where the denominator is the total rotation-vibration partition function (P.f.rot,vib) for the molecule. This form of the rate constant has an Arrhenius temperature coefficient very close to D o at low temperatures, but the higher the temperature, the more the temperature coefficient falls below D o : at the same time, recombination rate constants calculated from equation (6) and the Rate Quotient Law [equation (3)] decrease monotonically as the temperature increases, the Arrhenius slope becoming progressively more negative the higher the temperature; thus the behaviour of equation (6) with temperature strongly resembles the experimentally observed behaviour.' What is happening is that if we insist that the molecules have a Boltzmann rotational distribution, then the higher the temperature, the more molecules there are in high J states : however, the higher the value of J, the larger is the centrifugal barrier U,,,(J) that the molecule has to overcome in order to dissociate; thus, as the temperature is increased, the rate of dissociation does not increase quite as rapidly as it would if the critical energy were constant and equal to D o . This can be seen very simply by differentiating the expression k = const. exp[ - D * ( T ) / R T ] whence dk/dT is reduced if db*(T)/dT is positive [cf. also equation (15) later]. The complementary argument for recombination becoming slower as the temperature rises is as follows: an increase in the approach velocity of the two recombining atoms means that, except for a head-on collision, the encounter has more angular momentum: encounters of low J face virtually no centrifugal barrier, but as J increases the encounters face higher and higher (centrifugal) activation energy barriers U,,,(J) to recombination ;54 therefore, as the temperature rises, and gas-kinetic velocities increase, more and more encounters possess high angular momenta and so are forced to take these high-energy recombination paths, and the rate of recombination actually falls.* As we shall see later, this is not the whole story, but the heavier

* It was suggested recently that there is an effective change with temperature in the thermally averaged multiplicity of bound and unbound states of the molecule which correlate with the separated atoms when electronic multiplicity is present in the atoms :56 this effect is automatically included in the above description. (a) V. A. LoDato, D. L. S. McElwain, and H. 0. Pritchard, J . Amer. Chem. SOC.,1969, 91, 7688; (b) D. L. S. McElwain and H . 0. Pritchard, ibid., p. 7693; ( c ) D. L. S. McElwain and H. 0. Pritchard, 13th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, Pennsylvania, 1971, p. 37; ,( d )E. Kamaratos and H. 0. Pritchard, Cunad. J . Chem., 1971,49, 2617; ( e ) D. L. S . McElwain and H. 0. Pritchard, ibid., p. 3915; (f)D. L. S. McEIwain and H . 0. Pritchard, ibid., 1972, 50, 897; ( g ) N. I. Labib, D. L. S. McElwain, and H. 0. Pritchard, i b i d , p. 3832; (h) T. Ashton, D. L. S. McElwain, and H. 0. Pritchard, ibid., 1973, 51, 237; ( i ) E. Kamaratos and H . 0. Pritchard, ibid., p. 1923; ( j ) H . 0. Pritchard, ibid.,p. 3152; ( k ) N. I. Labib and H. 0. Pritchard, ibid., 1974, 52, 739. B. Kivel, J . Chem. Phys., 1973, 59, 3495.

252

React ion Kine tics

the molecule the more nearly will the non-equilibrium rate approach the ideal equilibrium rate [equation (6)] because of the network effects which we are going to describe, and therefore the more important will be this rotational mechanism in determining the temperature coefficients of dissociation and recombination rates. Incidentally, equation (6) is not new,57 but these properties have only recently been r e ~ o g n i z e d . ~ ~ j Finally, there is another equilibrium explanation for the anomalous temperature coefficients observed in dissociation and recombination reactions which has been known for more than a decade. If one approximates the manifold of processes represented by the two equations [V] (see p. 257) by

M -t X * MX

(7)

and assumes that the manifold of processes [VI], approximated as

X

+ MX+

M

+ Xr

(8)

is fast, then the rate of formation of X, depends upon the concentration of MX, which of course must decrease as the temperature rises, with a temperature coefficient equivalent to the binding energy of MX; this is the wellknown chaperon or radical-molecule-complex theory.58 However, if this were to be the only explanation, the binding energies for MX molecules would have to be much too large - for example in the case of H r noted earlier in this paper, the binding energy of the ArH complex would need to be a some 1-2 kcal mol-1 (5-10 kJ mol-l) at room temperature, but something like 1 1 kcal mol-1 (45 kJ mol-l) at high temperature, which is quite untenable! Thus, the chaperon theory by itself, and in its simple equilibrium form, is less successful than the simple equilibrium collision approach to the energytransfer theory, as described above ; recent comparisons of the relative importance of the energy-transfer and radical-molecule-complex mechanisms 59-61 probably therefore overestimate the contribution of the latter because of the hitherto inadequate formulation of the energy-transfer component. However, recent versions of the chaperon theory yield quite realistic Arrheni us temperature coefficients for recombination, becoming increasingly negative with increasing temperature : these versions have included an allowance for non-equilibrium effects, which are relatively unimportant, and a proper sampling of the rotational phase space in the Monte Carlo calculation of the reaction between MX and X, which is obviously most important. In fact (looking now at the dissociation process) it is intuitively obvious that when a proper account is taken of rotational motion, the equilibrium forms of the energy-transfer (i.e. simple collision) 57

68

59

6o

61

S. Glasstone, K. J. Laidler, and H. Eyring, ‘The Theory of Rate Processes’, McGrawHill, New York, 1941, p. 131. G. Porter, Discuss. Faraday Soc., 1962, No. 33, p. 198. R. T. Pack, R. L. Snow, and W. D. Smith, J . Chern. Phys., 1972, 56, 926. A. G. Clarke and G. Burns, J . Chem. Phys., 1973,58, 1908. W. H. Wong and G. Burns, J. Chem. Phys., 1973, 58, 4459; 1973, 59, 2974.

Network Eflects in Dissociation and Recombination of a Diatomic Gas

253

and chaperon theories taken together should have a temperature coefficient which does not differ markedly from that of the energy-transfer form alone, but the rate may be increased by a factor which arises from the difference in collision cross-section for collisions of M with Xzand with MX. Recombination processes are important in many areas of physics, and have therefore attracted much more attention than dissociation processes, although of course the two are complementary. In much of this work, the aim has been to calculate the rate at which encounters between pairs of X atoms lead to the formation of a molecular ~ t a t e . ~Such ~ - ~approaches ~ are incomplete, since it has been suspected for many years,66-68and convincingly established by recent theoretical that a very large fraction of newly formed molecules actually redissociate: to adapt an old proverb, one deactivation does not make a molecule! Nevertheless, rather as in a battle, the more molecules which actually cross the boundary, the more are likely to remain there permanently, and so these treatments must be considered among the building blocks of a complete theory of dissociation and recombination. 55h9

Non-equilibrium Theories.-It has been recognized for a long time that nonequilibrium effects could be important in dissociation and recombination processes, and much of the earlier work, and its flavour, has been summarized by Nikitin 7 0 and by In the past ten years, there have been two sustained attempts to evolve non-equilibrium theories of dissociation and recombination which are, on the face of it, very different indeed. One is the classical phase-space theory of Keck and Carrier,69in which it is argued, quite correctly, that the bound states of most diatomic molecules are so close together near the dissociation limit that the discrete states themselves can be regarded as a continuum. In its equilibrium form,72one has to solve for the trajectory of a representative mass point moving on a multidimensional surface representing the relative potential energies of the three particles M X X: thus recombination or dissociation by both the energytransfer mechanism [i.e. (4) and its reverse] and by the chaperon mechanism [i.e. (7) and (S)] are automatically included. The possibility that a nonequilibrium distribution of bound states exists among the upper bound levels of the molecule is then superimposed on the solution by assuming that motion

+ +

62

63 64

66 Oi

88

’O

71 72

J . Yates and G. Sandri, Phys. Rev., 1969, 188, 161. R. E. Roberts, R. B. Bernstein, and C. F. Curtiss, J . Chem. Phys., 1969, 50, 5163. R. L. Pope and L. J. Tassie, Proc. Roy. SOC.,1970, A320, 487. A. Jones and J. L. J. Rosenfeld, Proc. Roy. SOC.,1973, A333, 419. J. C. Polanyi, J . Chem. Phys., 1959, 31, 1338. E. E. Nikitin and N. D. Sokolov, J. Chem. Phys., 1959,31, 1371. A. 1. Osipov and E. V. Stupochenko, Soviet Phys. Uspekhi, 1963,6,47. J. Keck and G. F. Carrier, J . Chem. Phys., 1965, 43, 2284. E. E. Nikitin, ‘Theory of Thermally Induced Gas Phase Reactions’, Indiana University Press, Bloomington, Indiana, 1966. D. L. Bunker, ‘Theory of Elementary Gas Reaction Rates’, Pergamon, London, 1966. J. C. Keck, J . Chem. Phys., 1958, 29, 410.

254

Reaction Kinetics

among the discrete states, approximated in the theory as a continuum, is governed by a diffusion equation in which kT/2 of energy is exchanged between M and the vibrationally excited X, in each collision; suitable choice of the adjustable parameters in the theory has led to a satisfactory reproduction of the dissociation rates for a wide range of diatomic molecules. 72-76 The other theory explicitly maintains the discrete nature of every individual level, and in effect is nothing more or less than an attempt to solve the normal chemical kinetic differential equations, regarding each individual rotational and/or vibrational state as a distinct species: the equations have the form of a ‘master equation’ and such an approach is often identified by that label; conditions for the validity of the master-equation approach for these processes have been discussed e l ~ e w h e r e . ~ ~Retention ~~ of the discreteness of bound states (and their designation as being vibrationally excited, or rotationally excited, or both) is an idea that is conceptually attractive to chemists, but the cost is high: there is one differential equation to be included for each bound state considered, and most diatomic molecules have several thousand bound (and quasi-bound) states (H, has about 350, Dz about 700, and Iz about 55 000); thus one is faced with the problem of the solution of enormous sets of simultaneous differential equations. To compound the problem, these systems of equations suffer from a mathematical malady called stiffness (or alternatively are said to have parasitic eigenvalues), so that their solution numerically for temperatures much below the middle of the shocktube temperature range is particularly 65c, 55g, 5 5 h , 78, 7 9 In addition, one needs to know, at least approximately, transition rates between every pair of possible eigenstates, which requires the knowledge of many millions of transition probabilities in the general case. Nevertheless, the master-equation approach has some very definite advantages, other than as a vehicle for the advancement of numerical analytical techniques. In principle it is exact, and although achievement may be some way off, one can see that an a pviori calculation of the rate of dissociation (or recombination) of Hz or D2in a large excess of helium will eventually be feasible with considerably less investment in computer time than has been devoted to a priori quantum mechanical calculations of the structure of atoms and molecules in recent years ; it is, however, a many-faceted problem, and development or refinement of techniques in several of these areas should precede an attack on the complete problem. The master-equation approach leads to a better appreciation of the way the internal relaxations occur and the way in which they are coupled to the dissociation mode than is possible through the classical phase73

74 7J

7G

77 78

79

V. H. Shui, J. P. Appleton, and J. C. Keck, J . Chem. Phys., 1970,53, 2547. V. H. Shui and J. P. Appleton, J. Chem. Phys., 1971, 55, 3126. V. H. Shui, J. P. Appleton and J. C. Keck, 13th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, Pennsylvania, 1971, p. 21. V. H. Shui, J. Chem. Phys., 1973, 58, 4868. P. K. Davies and I. Oppenheim, J. Cliern. Phys., 1972, 56, 86. J. E. Dove and D. G. Jones, J. Chem. Phys., 1971,55, 1531. H. E. Bailey, Phys. Fluids, 1969, 12, 2292.

Network Eflects in Dissociation and Recombination of a Diatomic Gas

255

space theory: further, this approach makes it obvious that the level spacings in Hz itself are so far apart that they cannot be considered as a continuum below 500K, nor possibly even below 5000K,65h and in consequence the confidence expressed 76 in recent phase-space calculations of Hzrecombination is over-optimistic. Finally, it is only through the master-equation approach that the kind of network effects which are described in this chapter can be revealed: the consequences of these effects are sometimes surprising (for example the H2/D2isotope effect), and the insight gained through this approach may easily be taken over into the description of related problems, such as the theory of laser relaxations. 75p

4 The Philosophy of the Present Approach The philosophy of the present approach to the problem was stated by H. W. Emmons in his plenary Lecture to the 13th Combustion Symposium in Salt Lake City.80 In enumerating the problems of describing the gas-dynamic equations of a combusting gas mixture, he went on to say: ‘It is natural at this point to ask “Which of the many processes present should we retain in our equations?” It is no great effort to put in everything which might conceivably be of importance. While this approach to an answer to our question produces the most general results, it also makes nearly all problems unsolvable. The correct approach to the answer: “Put nothing into the equations which the problem at hand does not force you to include.” If the problem is solved inadequately because some essential effectis omitted, we can always try again to get an improved result.’ The techniques used in our work have evolved from the earlier elegant treatment of Montroll and Shuler:81in that early work the diatomic molecule was treated as a truncated simple harmonic oscillator, and dissociation was assumed to occur when molecules found themselves in the topmost level of the system, where they were trapped and could not return. Such an approach is consistent with Emmons’ philosophy, but would seem to lack, above all, one very important ingredient of chemical intuition - the fact that the vibrational energy levels become closer together as the dissociation limit is approached. Thus, our aim initially was to re-examine the problem, making due allowance for the anharmonicity of the vibrational motion of the molecule, and modifying and extending the Montroll-Shuler treatment where appropriate to achieve this end. At the same time, recognizing the enormous complexity of the problem and the difficulties of interpreting the complementary experimental data, no attempt was made to force the calculated results to match any experimental data: Dove and Jones likewise did not force their results 7 8 - 8 2 to match experiments, but Johnston and Birks fell into the

82

H. W. Emmons, 13th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, Pennsylvania, 1971, p. 1. E. W. Montroll and K. E. Shuler, Adv. Chem. Phys., 1958, 1, 361. J. E. Dove and D. G. Jones, Chem. Phys. Letters, 1972,17, 134.

256

Reaction Kinetics

trap, and, as we shall see later, were forced to make quite unrealistic assumptions in order to attain this matching because of the strength of some of these network effects. 5 The Diatomic Dissociation-Recombination Problem As Emmons has said, it is a fairly simple matter to write down all the processes which might be important in the dissociation of a homonuclear diatomic molecule X2in the presence of an inert gas M. If the molecule has internal (rotation-vibration) states denoted by i, the initial distribution in a shockwave dissociation experiment will be characterized by population of only a few states having low values of i (i.e. i = 0, 1, 2, 3 . . . up to some fairly low integer): in fact, the initial distribution is unimportant as far as reaction is concerned because after a period of the order of 10-100 vibrational relaxation times (either in a dissociative or a recombinative experiment) the system appears to get on to virtually the same track in its approach to equilibrium;55b the same phenomenon hasbeen described in other non-equilibrium situation^.^^ A corollary of this observation is that one cannot integrate the kinetic equations backwards in time - since the system could have arrived at the present distribution from almost any starting point: and of course, this inability to integrate the equations backwards is intimately connected with entropy production. When the shock front arrives, causing an almost instantaneous rise in temperature, Ihe Xzmolecules begin to adjust their population distribution towards the new required equilibrium distribution by the processes

M M

+ X2(0) + X(i)

M 4-X2(i) M

+ X2(j)

which, in effect, convert translational energy into internal energy. If the shock is a fairly strong one, and the temperature rise is such that some dissociation will result, then [I11

M

+ X,(i)

M

$-

X

+X

follows: often, in practice, the temperature rise is not ideally instantaneous,1-6 and instead of it taking some 5-10 collisions to achieve the new translational temperature, the shock front actually takes of the order of 1 ,us to pass, by which time processes of type [I] are essentially complete. We then enter a regime where molecules are dissociated by process [IT] and the states from which dissociation actually occurred are replenished by processes [I]. Soon, as the atom concentration builds up, another kind of process becomes important, i.e. X X 2 ( i ) X X&) [I’I

+ + + X + Xz(i) X + X + X

83

E. Paul and T. L. Hill, Proc. Nat. Acad. Sci. U.S.A., 1972, 69, 2246.

Network Eflects in Dissociation and Recombination of a Diatomic Gas

251

Of course, the gas is not infinitely dilute in X 2 , so that we must also consider the processes : EI”1

+ X 2 ( i )* X2(h) + X 2 ( j )

X4h)

Thus, by the time a steady rate of dissociation has been established, one needs three rate constants with which to define that rate, i.e. kd,Y, kd,x,, and kd,=; the last can be eliminated if measurements are made at early times, and both the later two if the measurements are made at early times in a very dilute mixture of M and X2. The equations [I”] and [II’? are, however, only special cases of a more general type:

EIVI

X2(h)

+ X2(i)*

X2(k)

+X +X

in which internal energy is exchanged, and any deficit is either absorbed from or transferred to the relative translational motion of the initial or resulting particles. If such processes were to be very fast, they can (as we shall see below) exert a subtle influence of the kinetics of the overall process. This is not all, however, because in general the atom X will have an affinity for the inert molecules M: thus:

[vl

M

+ MX(j)

M -I- MX(j) d M

+ MX(i)

M

+X +M

can occur, where in fact M in each equation can be considered to be either M or X z . Thus, as time goes on, and the atom concentration builds up, not only does re-formation of molecules become important through the reverse of processes [11], [11’], [II’7, and [Iv], but the reaction [VI] (as opposed to an EVII

X

+ M X ( j )S M + X2(i)

energy-transfer process) may become important also : this is the chaperon or radical-molecule-complex mechanism of recombination to which we have already referred. On top of all this, if X is an atom like N o r Br, then X atoms can also recombine into excited electronic states which may be stabilized either by collision or by radiating, and the complexity of the problem is increased accordingly.84 Were we to consider the dissociation of a heteronuclear diatomic molecule X Y , the analogue of process [11] is: M 84

+ XY(i)F?. M + X + Y

R. K. Boyd, G. Burns, T. R. Lawrence, and J. H. Lippiatt, J . Chem. Phys., 1968,49, 3804, 3822.

258

Reaction Kine tics

except that now there are three, not one, possible reverse reactions, since X and Y can also disappear via

M

+X +X

S M

+ X2(i)

+Y

M

+ Y2(i)

M -1- Y

and

as well: thus, whenever X z occurs in the homonuclear set of equations, one must now consider three parallel sets 55f with X 2 , X Y , and Y z ;likewise, X can be replaced by either X or Y . However, some of these perturbations are more substantive than this, and change the whole character of the problem: in particular, of the possible set of analogues of [1’], there is a subset:

WIII

X -1- Y z ( i )F1: X Y ( j )

+X

which forms a chain reaction through which equilibrium of X Y with X 2 and Y can be established, by-passing recombination processes. Another new reaction enters the picture, also, because in addition to all the X , Y permutations of [111] and [IV], we have to consider the possibility of

[VIII] CIXI

+ XY(i)+ X 4 j ) + Yz(k) X Y ( h ) + X Y ( i ) @ Xz(j) + Y + Y or Y 2 ( j )+ X + X XY(h)

Finally, since X Y will have a dipole moment, coupling of some of the rotationvibration states of X Y by the radiation field could be important under some conditions.* Thus the problem of describing in detail the dissociation of a heteronuclear diatomic molecule is very much more complicated than it is for a homonuclear one, and will not be discussed further, except to note in passing that depending upon the assumed relative importance of various types of process, time-dependent ‘rate constants’ are not ~ n l i k e l y . ~85~ f > Throughout this description of the diatomic dissociation, I have placed no restriction on the values that the indices i, j , etc. could take in any process. In many treatments of relaxation problems it is common, often as a matter of convenience, but also in reverence to the persuasive elegance of the early Landau-Teller treatments, to restrict transitions of type [ I ] to nearest-neighbour states, or to assume that the number of vibrational quanta is conserved in transitions of type [111] (or even [ V I I I ] ! ! ) ;similarly, transitions like [11] are often only considered when i denotes the topmost bound state (e.g. truncated harmonic oscillator) and the multitude of possible resonant transitions 86 of types [IV] and [ I X ] are almost universally ignored. As it

* At some stage, it would probably be worth simulating this effect by carrying out a calculation on Hz, artificially treating the levels as though they could be coupled by the radiation field: this would at least reveal the magnitudes of fluxes and level-population changes that one could expect in the real heteronuclear case. *j

R. D. Kerr and G . G. Nika, J. Phys. Chem., 1971,75, 171, 1615, 2541. S. McElwain and H. 0. Pritchard, J . Amer. Cliem. Soc., 1970, 92, 5027.

** D. L.

Network Eflects in Dissociation and Recombination of a Diatomic Gas

259

happens, the nearest-neighbour-only restriction has little effect on the rate of although it had long been feared that it ~ o u l d . ~8 8~ ~ Nevertheless, it is inviting difficulties to build in these restrictions, for not only is the problem oversimplified, but the network effects which are present become distorted or disguised; these network effects are particularly subtle and the resulting explanation of any phenomenon (particularly relaxation as opposed to reaction) may, as we shall see, become strained and possibly inconsistent. Perhaps it is worth while to pause and to elaborate further on the subject of persuasive elegance, because it appears to have some adverse effects on the development of this field, among others: it gives rise to the tendency to regard molecules as if they were constrained to behave like those equations which we can solve! Stemming from the early work of Montroll and Shuler using the harmonic oscillator model, several very elegant results were found : one in particular, that an assembly of molecules relaxes from one Boltzmann distribution to another through a succession of intermediate Boltzmann d i s t r i b ~ t i o n s , ~ has ~ ~ been shown to be false for anharmonic oscillators by numerical 55g1 55k However, the simplicity and elegance of this concept has led experimenters to try to interpret their data in terms of a vibrational temperature, when in some cases the actual population distribution may not even approximate to a Boltzmann distribution for any temperature. Similarly, retention of the concept of a vibrational temperature has caused Kiefer 91 to postulate the breakdown of detailed balancing, when in fact what has broken down is the concept of a vibrational temperature! In my view, progress will be faster if we use the concept of vibrational temperature with much more care; at the same time, we must discard the idea of conservation of vibrational quanta 93 - what in fact is likely to be partially conserved is vibrational energy, or maybe internal energy (i.e. including rotational energy, and in some cases perhaps even electronic energy 94). Thus, the complete kinetic description of even the dissociation of a homonuclear diatomic gas is revealed as a monumental problem. If we consider ortho-hydrogen (and assume that it cannot be converted into para-hydrogen by a collision of type [I]) there are 172 rotation-vibration states, and there are about 15 OOO forward processes of types [I] and [11] for each species of collision partner (M, H t , H), for which we need to know the transition probabilities. Then there are of the order of lo8 forward processes of types [HI] and [IV], not to mention an unknown number of types [V] and [VI]; 55c9

92y

88

91 O2

O4

R. Herman and K. E. Shuler, J. Chem. Phys., 1958, 29, 366. W. G . Valance, E. W. Schlag, and J. P. Elwood, J . Chem. Phys., 1967, 47, 3284. E. W. Montroll and K. E. Shuler, J . Chem. Phys., 1957,26,454. A. 1. Osipov, Doklady Akad. Nauk S.S.S.R., 1960, 130, 523. J. H. Kiefer, J . Chem. Phys., 1972,57, 1938. Y. Sat0 and S. Tsuchiya, J . Phys. SOC.Japan, 1971, 30, 1467. J. D. Kelley, J. Chem. Phys., 1972, 56, 6108. A. B. Callear and W. J. R. Tyerman, Trans. Faraday SOC.,1966, 62, 2313.

260

Reaction Kinetics

even then, we would be ignoring the virtual certainty that most of the processes [1’1, i.e. H + Hz(i) H He(j) (9)

+ +

could result in a nuclear spin conversion, i.e. that really we cannot treat the dissociation of ortho-hydrogen in isolation from that of para-hydrogen : thus even the dissociation of hydrogen has to be treated in many respects like the dissociation of a heteronuclear molecule, with the full range of processes [I] t o [IX]; however, this complication does not arise in many homonuclear diatomics, or else the distinction between ortho- and para- states cannot be made, and therefore it seems that a comprehensive treatment of any of the species ortho-Hi,, para-H2, ortho-D2,or para-D2in isolation would serve as a good enough model for the general homonuclear diatomic dissociation reaction. 6 Transition Probabilities

To pursue this approach, one needs to know, as we have said, some millions of transition probabilities, even for Hz,which is the simplest possible diatomic gas. However, the approach is feasible nevertheless, because as was pointed out by Carrington, it is an ill-conditioned problem to try to derive information about transition probabilities from observed relaxation behaviour. 95 The corollary is that the predicted relaxation behaviour will not be very sensitive to the assumed transition probabilities, and this turns out to be particularly true for the relationship between dissociation rates and transition probabilities, so much so in fact that it has been possible to deduce most of the important features of dissociation-recombination reactions with a very incomplete knowledge of the transition probabilities; we will return to the reasons for this state of affairs later. Thus, in the future, to perform a very accurate calculation of the rate of dissociation, one will only need to know relatively few key transition probabilities, and the remainder can be filled in approximately, according to the same pattern, or even, as we shall see below, ignored altogether. The theoretical calculation of transition probabilities is a field to which enormous effort is being devoted, and we can do no more than summarize the kinds of transition probability required for our particular problem, and indicate a few of the properties of those probabilities. Bound-Bound Transitions.-The conversion of the relative kinetic energy of approach of an inert atom and a diatomic molecule into vibrational excitation of the molecule is the area to which the most effort has been devoted in the past, and about which the most is known.96*9 7 This arises because it was g5 96

T. Carrington, J. Chern. Phys., 1961, 35, 807. K. F. Herzfeld and T. A. Litovitz, ‘Absorption and Dispersion of Ultrasonic Waves’, Academic Press, New York, 1959. T. L. Cottrell and J. C. McCoubrey, ‘Molecular Energy Transfer in Gases’, Butter-

worths, London, 1961.

Network Eflects in Dissociation and Recontbinaiion of a Diatomic Gas

261

shown by Jackson and Mott 9 8 many years ago that if the interaction between the atom and the diatom was a pure exponential repulsion, the probability of an exchange between translational and Vibrational energy of a harmonic oscillator could be calculated easily for collinear collisions in the distortedwave approximation; more recently, these techniques have been extended to molecules which are Morse oscillators rather than harmonic oscillators, thereby introducing more realism into the ~ a 1 ~ ~ l a t i o loo n ~ The . ~ ~basic ~ steps in the calculation are as follows.18 Consider two vibrational states i and j separated by an energy Aij ; then for each value of the relative approach velocity mi, the recoil velocity wj is fixed if the transition occurs, and the probability of such a transition occurring, pU(wi,wj), can be calculated by the perturbation methods already mentioned. The quantity p i j ( o i , oj)is symmetrical in the indices i and j , which means that for the same total energy (i.e. vibrational plus translational) the probability for deactivation is the same as the probability for activation. Detailed balancing arises because these probability functions have to be integrated over the Maxwell-Boltzmann distribution of relative approach velocities, and for activation the limits of integration are A, < E(w,) < 00 whereas for deactivation the limits are 0 < E(w,) < 0 0 ; thus, provided a translational temperature can be defined, transition probabilities must always obey detailed balancing, and this will be assumed to be the case throughout this discussion, (It may happen in an approximate calculation such as I have just outlined, since it is common to apply some kind of cos20averaging factor to the calculated head-on collision probabilities, that detailed balancing may only be approximate: however, a proper formulation of the three-dimensional collision problem, dispensing with the concept of a collision diameter, which is the cause of much confusion,lol must lead to transition rate coefficients which conform exactly to detailed balancing, and it has been proposed that such tests be applied to calculated transition rates.lo2) There is no evidence to suspect that dissociation and recombination reactions are so fast that the translational distribution will be d i ~ t u r b e d , except ~ ~ ~ -at~ the ~ ~ very highest temperatures: a reasonable criterion 7 7 for the maintenance of the translational distribution, and therefore the translational temperature and so detailed balancing, is that transition probabilities should not exceed the order of 0.1 per collision; in most of the model calculations to date 7 8 this is more or less the case. For a diatomic molecule which is a Morse oscillator, the basic properties of the transition probabilities are as follows. There are no selection rules - all transitions are possible, and there are no simple relationships between 999

553

J. M. Jackson and N. F. Mott, Proc. Roy. SOC.,1932, A137, 703. D.Rapp and T. Kassal, Chem. Rev., 1969, 69, 61. loo A. P. Clark and A. S. Dickinson, J . Phys. ( B ) , 1973, 6, 164. lol T. Carrington, J . Chem. Phys., 1972, 57, 2032. lo2 H.Tai and E. Gerjouy, J . Phys. (B), 1973, 6, 1426. lo3 R. D. Present, J . Chem. Phys., 1969, 51, 4862. lo* A. I. Osipov and E. V. Stupochenko, Teor. i eskp. Khim., 1970, 6, 753. lo5 B. Shizgal and M. Karplus, J. Chem. Phys., 1971, 54, 4345, 4357. g8

99

262

Reaction Kinetics

successive transition probabilities (unlike the harmonic oscillator case lo6). The two principal factors which determine a given T-V transition probability appear 55e to be the amount of energy which has to be exchanged between translation and vibration, and the temperature: thus, for deactivation, transition probabilities increase slowly with decreasing vibrational separation A, between the two states, and with increasing temperature; through detailed balancing, activation probabilities contain a Boltzmann factor exp[ -A,/kT], and therefore large energy changes are strongly discriminated against, in activation, at low temperatures. For the time being, for reasons which will become clearer as this chapter develops, such transition probabilities are adequate as a basis for the study of relaxation and dissociation problems, at least as far as processes of type [I] are concerned. Unfortunately, encounters between an already dissociated atom and the diatom feature strongly in many practical dissociation p r o c e ~ s e s lo* , ~ ~e.g. ~ ~H H2, 0 02,and, of course, in recombination processes where the atom concentration is necessarily high at early times,loSe.g. I I,; because such interactions are attractive or else complex, and do not approximate even roughly to an exponential repulsion, they are not amenable to these perturbation loo However, with the rapid developments in molecular dynamics and trajectory calculations, these semi-reactive-type collisions are beginning to receive attention : for example, some Monte Carlo trajectory calculations have been attempted for Br + Brz and for H H2,659 111 and obviously more will be forthcoming. It seems clear from these calculations, and from much other evidence als0,112-115 that the interconversion of translational and internal energy is much more efficiently catalysed in such collisions than when the colliding particle is inert - although experimental evidence on this point still remains a little less 116 than unanimous! Somewhat less well developed are the topics of translation-rotation (T-R) and vi bration-vibration (V-V) interchanges, and transition probabilities for such processes are perhaps best assigned at the moment from a mixture of theoretical considerations with considerable guidance from experiment. Theoretical work on low-lying transitions of rotating molecules is reasonably well a d v a n ~ e d , l ' ~ -and ~ ' ~ it is clear that for such transitions, probabilities are

+

+

+

+

H.K.Shin, J. Phys. Chem., 1973,77, 1394, 2657. I. R. Hurle, A. Jones, and J. L. J. Rosenfeld, Proc. Roy. SOC.,1969, A310, 253. Io8 J. H. Kiefer and R. W. Lutz, 11th Symposium (International) o n Combustion, The Combustion Institute, Pittsburgh, Pennsylvania, 1967, p. 67. D. L. Bunker and N. R. Davidson, J . Amer. Chem. SOC.,1958, 80, 5085, 5090. 11" A. G. Clarke and G . Burns, J. Chem. Phys., 1971, 55, 4717. I. W. M. Smith and P. M. Wood, Mol. Phys., 1973, 25, 441. 112 3. H. W.Cramp and J. D. Lambert, Chem. Phys. Letters, 1973, 22, 146. 113 A. N. Schweid and J. I. Steinfeld, J. Chem. Phys., 1973, 58, 844. llP R. E. Center, J. Chetn. Phys., 1973, 58, 1530; 1973, 59, 3523. ?15 D. J. Eckstrom, J . Chem. Phys., 1973, 59, 2787. 116 J. E. Breen, R. B. Quy, and G . P. Glass, J. Chem. Phys., 1973, 59, 556. 11' K . Takayanagi, Commetits Atomic Mol. Phys., 1973, 4, 59. W. Eastes, J. Chem. Phys., 1973, 59, 3534. 119 D. Secrest, Ann. Rev. Pliys. Chem., 1973, 24, 379. lo6

Io7

Network Efects in Dissociation and Recombination of a Diatomic Gas

263

much higher than for low-lying vibrational transitions : like the vibrational transitions, the probabilities for downward transitions increase slowly with temperature, and again, the larger the energy gap between the levels the less likely is there to be an exchange of energy with the translational mode. However, the spacing of the rotational levels increases with increasing excitation, in contrast with vibrational levels, which converge, and there is as yet no firm comparison which would show whether, for the same energy spacing, rotational or vibrational energy transfer is the more rapid, or whether they are about equally likely. This is a juncture at which considerable reorientation of our thinking is required: through a preoccupation with low-lying states, we have become accustomed to regarding rotational relaxation as being fast - which it is at low temperatures. However, because of the progressively increasing spacing between levels, the higher the temperature, the more molecules which have to be raised to high rotational levels by large steps: this causes the rotational relaxation time to increase with tempzrature beyond a certain temperature (which of course must be an individual characteristic of the particular molecule); we shall come back to this later. (An important consequence of large spacings between high rotational states will arise in chemical activation studies where a highly excited molecule might be formed with a large amount of angular momentum: it would be natural in the RRKM treatment of such a process to imagine that rotational excitation is readily dissipated, but this assumption will fail because dissipation of this rotational energy can only occur in very large, and therefore very slow, steps.) Likewise, there is intense activity to calculate probabilities of vibrationvibration interchanges between m 0 1 e c u l e s , ~ ~largely ~ - ~ ~ ~confined to low or moderate temperatures, low levels of excitation, and with a few exceptions 93 limited to transitions in which there is conservation of vibrational quanta; this is because most laser experiments are concerned with this kind of regime, and we can obviously look forward to a thorough matching of theory and experiment for such transitions sometime in the future. What is clear, as usual, is that the more of the total vibrational energy that has to be converted into translational motion, the less likely is the transition to take place;124-125 however, we are still on very shaky ground when it comes to making predictions for extremely high temperatures, such as occur in shock-wave dissociations, and for general changes in either the total internal energy or the total number of vibrational quanta. Finally, we come to T-VR and VR-VR transitions, i.e. ones in which the rotational and vibrational quantum numbers change simultaneously. Such 929

120 121

122 12$ 124 125

G. C. Berend and S. W. Benson, J . Chem. Phys., 1969,51, 1480. R. I. Morse, J . Chem. Phys., 1972, 56, 2329. R. D. Sharma, Phys. Rev., 1969, 177, 102;Phys. Rev. ( A ) , 1970,2, 173. T. A. Dillon and J. C. Stephenson, J. Chem. Phys., 1973, 58, 2056. A. B. Callear and G . J. Williams, Trans. Faraday Soc., 1966,62, 2030. J. C. Stephenson, J. Finzi, and C. B. Moore, J. Chem. Phys., 1972, 56, 5214.

264

Reaction Kinetics

transitions have received rather little theoretical attention,126$ l Z 7although evidence is beginning to accumulate from laser experiments that both intermolecular and intramolecular interconversions of vibrational and rotational energy are i m p ~ r t a n t . l ~ ~Obviously -l~~ such transitions must be important, so much so that it has even been suggested 13* that in future the principal effort in the theoretical treatment of transition probabilities should be devoted to the general problem, whence the T-V, T-R, V-V cases which we have already discussed, and also the V-R 136 just become isolated examples of the general problem. A specific challenge to theoretical prediction at the moment is to know whether (say) the transition : Hz(v

= 0, J =

27)

+MS H~(u

9, J

=

+

3) M ] A E [ = 37cm-l

(10) has a high transition probability because it is almost a resonant process, or a very low transition probability because there is a very large change in the angular momentum of the molecule (IAJI = 24): it has already been shown 55i that whether or not such transitions occur will have a discernible effect on both the dissociation-recombination and the relaxation behaviour of Hz- particularly the latter. In fact, it could well be that a proper quantummechanical formulation of the T-VR problem 134 might give a ‘yes’ or ‘no’ answer to this question without having to solve the actual Schrodinger equation for a specific collision process such as (10). 55hp

Bound-Unbound Transitions.-Wavefunctions for unbound states of molecules are quite difficult to handle, especially numerically, and in consequence there has been relatively little progress in the calculation of transition probabilities of type [IJ] processes, let alone the more complicated types which are also possible. Hence, it has usually been assumed that the probability of a collisional dissociation is the same as would be a transition to a hypothetical This is quite vibrational state situated at the dissociation limit.18+ arbitrary, but becomes reasonable once it is realized, as we discuss below, that the actual rate of dissociation is relatively insensitive to the actual values assigned to these probabilities. It can also be rationalized by arguing that if some kind of box normalization is applied to the manifold of translational states just above the dissociation limit, and if the box is made small enough, these continuum states will coalesce into a few vibration-like states whose 559

lZ6 lZ7

lZ8 129 150

132 133 134

KG 136

78v

H. K. Shin, J . Phys. Chem., 1972,76,2006. A. W. Young and H. K. Shin, Chem. Phys. Letters, 1973, 21, 267. R. B. Kurzel, J. I. Steinfeld, D. A. Hatzenbuhler, and G. E. Leroi, J. Chem. Phys., 1971,55, 4822. E. H. Fink, D. L. Akins, and C. B. Moore, J , Chem. Phys., 1972, 56, 900. J. C. Stephenson and C. B. Moore, J . Chem. Phys., 1972,56, 1295. M. Y. D. Chen and H. L. Chen, J . Chem. Phys., 1972,56, 3315. R. R. Stephens and T. A. Cool, J . Chem. Phys., 1972,56, 5863. B. M. Hopkins and H. L. Chen, J. Chem. Phys., 1973,58, 1277. E. E. Nikitin, Comments Atomic Mol. Phys., 1970,2, 59. J. D. Kelley, J . Chem. Phys., 1970, 53, 3864. H. K . Shin, J . Phys. Chem., 1973, 77, 346.

Network Efects in Dissociation and Recombination of a Diatomic Gas

265

amplitude density will be similar to the original amplitude density: thus, to treat these transitions into the continuum as vibration-like seems reasonable. The common practice of regarding these pseudo-vibrational continuum states as just a single level at the limit is subject to two criticisms. First, the higher the temperature, the more such levels are easily accessible, and perhaps the multiplicity of the assumed pseudo-state should be increased with temperature. Secondly, the placing of that single state at the limit is arbitrary, and it should more properly be placed kT above the limit. In fact, these two refinements would work in opposing directions - hence the common practice is probably as good as any, in the absence of better data. Very recently, theoretical calculation 13' has confirmed that the probabilities of such transitions are in fact like those commonly a ~ s u m e d 7,8~and ~ ~ are nowhere near as colossal as the values assumed by Johnston and Birks in an attempt to reconcile the vibrational model of dissociation with the experimental data. Before leaving this topic, it is perhaps useful to say something about the recombination process, for which probabilities must be assigned if the kinetic equations representing the dissociation are to bc integrated : in fact, this presents no problem, for if it is assumed that, on average, newly dissociated atoms recoil from each other according to a Maxwell-Boltzmann distribution of velocities, recombination rate coefficients follow from the assumed dissociation rate coefficients through detailed balancing; nevertheless, the pictorial visualization of the actual three-body recombination process is important. Dissociation by process [11] leaves us with the inert third body receding from the centre of mass of the system, and the pair of newly dissociated atoms in some condition where they must eventually separate: they may separate immediately, or may orbit each other for a time, depending upon how much angular momentum resides in the pair. The concept of pairs is fundamental to the description of recombination by the reverse of process [II]. Because of the strong attraction between the two atoms, the dissociated gas will contain, on average, a certain number of pairs of atoms X: orbiting around each other, and this number can be calculated by standard statistical mechanical mean^.^^^ 13@ In addition, when H or D atoms are involved, there will be a small fraction 65 of pairs already in quasi-bound states of the molecule, having arrived there by tunnelling through the effective potential barrier.63 It is the collisional deactivation of these pairs by M, removing either vibrational or rotational energy, or both, that leads to recombination.* The same 1389

* The existence of two types of pair leads to a formal ambiguity in defining the rate constant for recombination. If recombination is defined as the H, molecule having reached any of the negative energy states, 0r.a subsetof these then the rate is third-order; however, if as is the natural definition arising from eigenvalue formulations of the relaxation, H, molecules are defined to include all quasi-bound states as well, there is a small unimportant second-order component to the recombination rate.55h 138

130

J. L. Lin, Physica, 1972, 62, 369. D. G. Rush and H. 0. Pritchard, 11th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, Pennsylvania, 1967, p. 13. I. R. Hurle, P. Mackey, and J. L. J. Rosenfeld, Ber. Bunsengesellschaft phys. Chem., 1968, 72, 991.

266

Reaction Kinetics

considerations apply to the formation of MX; however, the equilibrium constant for the formation of MX* pairs is very much less than that for the formation of X t because of the weak interaction between M and X, but of course the concentration of M is very much higher than that of X in most experiments. At a further level of sophistication, one would have to consider a small component of the flux arising from long-lived states MXZ which could then break up by a unimolecular process either into M and Xe, MX + X, or M, X, and X.1407 141

7 Solving the Master Equation Consistent with the Emmons philosophy,80 we look for something relatively simple to solve - having already said that the harmonic-oscillator model was too far removed from reality. Thus, examination of the dissociation of Xe, infinitely diluted in M (to eliminate X2-X2 collisions), using a model which places the vibrational levels of the Xz molecule in their correct positions and assumes that the behaviour of a rotationless (i.e. J = 0 ) state would be a good guide to the true behaviour, seemed a reasonable choice. The arguments for assuming that the behaviour of the J 0 state would lead to an adequate description were : (i) that rotational relaxation was much faster than vibrational relaxation 9 7 and therefore there were in effect a series of parallel activation paths 142 and (ii) that Morse wavefunctions for J # 0 states were not known in analytic form, but for low J they must be qualitatively similar to those for the J = 0 state,18 and therefore transition probabilities for J J transitions would not be sensitive to J ( J J’ transitions being hidden by the assumption of rotational pre-equilibration). Thus, for He, which has n = 15 bound vibrational levels (conforming very closely j Z b to a Morse oscillator), we have to solve (n f 1) simultaneous differential equations of the kind:

-

-

i dn7l dt

-

=

0,1,2 ... (n

-

1)

n -- 1

=

[MI Z

C ( P , t j ~ j P,,t12) -

(12)

j = G

for the dissociation or recombination of H, at high dilution in M: briefly, the ni represent bound-state populations, n, represents the number of dissociated molecules, Z is the collision number for H, with M (assumed to be independent of i), Pi,. are probabilities for bound-bound transitions, and 140

142

J. Blauer, V. S. Engleman, and W. C . Solomon, 13th Symposium (International) o n Combustion, The Combustion Institute, Pittsburgh, Pennsylvania, 1971, p. 109. D. A. Micha, Accounts Chem. Res., 1973, 6, 138. H. 0. Pritchard, in ‘Transfer and Storage of Energy in Molecules’, ed. G. M. Burnett and A. M. North, John Wiley, London, 1969, vol. 2 , p. 368.

267

Network Eflects in Dissociation and Recombination of a Diatomic Gas

P are probabilities for bound-unbound collisions; the reader is referred to the original references for a more detailed discussion of these equation^.^^ It is by now well known that such sets of simultaneous differential equations are very difficult to solve by numerical i n t e g r a t i ~ n , ~ ~7?9 * 138 and we have shown that, using asymptotic expansion techniques, the same solutions can be achieved very much more quickly: the method is described in detail elsewhere 55c [apart from a misprint in equation (10) of that paper 55g] and it has since been confirmed by extensive numerical experiment that it is permissible to truncate the asymptotic expansion after the first term, even up to quite high temperature^.^^^ Fundamental to the method is the location of eigenvalues di of a matrix whose off-diagonal elements A, are the actual transition probabilities Pij for transitions between the bound states i and j (or some closely related quantities, e.g. Z[M]P,, the transition rates); the diagonal elements Aii are not in fact the probability that no change will occur, but are related to the need to conserve the total mass (or number) of H atoms during the evolution of the population A selection of eigenvalues for the model calculation on Hz is as follows (Z[M] 1.15 x 109T4s-' = - do): 789

55c9

65h9

i=

T/K 5000 2000 700

-do

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

-dn-2/s-1

1.4 x 1011...............3.6 x lo8 7.5 x 1010 ............... 4.1 x 107 3.5 x 1Olo ............... 1.6 x lo6

-dn

- 11s-

1.7 x 105 4.4 x 7.5 x 10-23

The range of eigenvalues dn-3 to d,, evenly spaced between dn-2and do, are not reproduced; also omitted is an implied zero eigenvalue which arises from the conservation condition and ensures that at infinite time we end up with the same number of (+H2 + H) as we started with. The significance of these eigenvalues is that they represent characteristic relaxation times for 55h the system, and as we have demonstrated

in conformity with equation (3), and in the rotationally equilibrated case:

We will comment on the extent of the failure of the latter equality later. The other eigenvalues relate to characteristic equilibration times for other 'normal modes' of relaxation 55r in the system - for example 4;'represents roughly the characteristic time for re-establishment of equilibrium between the atom population and the topmost bound levels of the molecule - and at the present time are of little observational significance. However, the quantity { d i l l is of extreme practical importance because many numerical integration techniques are only stable if the integration step length is less than the order of 0.21d;'!:79 hence, referring to the above eigenvalues, the characteristic time for 143

D. L. S. McElwain, unpublished results.

268

Reaction Kinetics

dissociation at 2000 K is about 20 s, but the step length has to be of the order of 2 i; lo-“ s, and the integration becomes impossibly time-consuming; the situation is even more catastrophic at 700 K, but on the other hand, direct integrations are quite feasible at 5000 Nevertheless, the location of these very small eigenvalues at 2000K and below is not an easy matter - the stiffness of the set of equations has re-emerged as an ill-conditioning of the set of eigenvalue equations, but it has been possible to localize the illconditioning and so deal with it : perhaps there is a ‘law’ of numerical analysis here ! The population distributions arising from these calculations have been tabulated extensively in the original publication, both for dissociation and for In summary, what appears to happen in dissociation at (say) 2000 K is as follows. Starting with all the H2 molecules in the ground vibrational state, there is a growth in the populations of all levels immediately the integration is commenced; successively, levels u 1, then u = 2 and so on up to about u = 8 or 9 come into equilibrium with the ground-state population, and in an induction period a little over T , , ~ or , ILI,;-’~I, a pseudosteady population distribution is reached in which the bottom-most 8 or 9 vibrational levels are very very closely in equilibrium with each other at the heat bath (translational) temperature; the next 2 or 3 levels are roughly in equilibrium with the heat bath, and the topmost 3 levels are progressively underpopulated by a factor of 2 or 3. This has all happened in less than the microsecond which it takes for the shock front to pass, and is therefore consistent with the experimental observation that vibrational relaxation precedes dissociation lP5,Ig4 - for the total departure from a true Boltzmann distribution at this temperature (i.e. the population deficits for =- 11-14) cannot amount to much more than about of the total H2 molecules s), when the extent of the reaction is present. At about this time (say about 2 x low6,the system has entered this pseudo-steady regime, and dissociation proceeds linearly with time up to s (2% reaction) and essentially linearly up to 10-1 s (20% reaction). In the linear regime, the vibrational population distribution remains almost unchanged, which can be interpreted simply to mean that just as quickly as molecules are supplied to the ‘interesting’ states (i.e. u = 11-14) from below, they dissociate. At later times, the repopulation of these upper states by recombination of already dissociated atoms becomes important and the kinetics then transform to those for a pair of opposed reactions with (for [MI = constant) orders of one and two, and all populations then approach their final equilibrium values monotonically. As far as the rate constant is concerned, although one can be defined theoretically at all times, it only has the required properties of constancy and of obeying the Rate Quotient Law 145 after the induction period has passed and the pseudo-steady regime is in progress. 14*

J. P. Appleton, J . Chem. Phys., 1967, 47, 3231. R. K. Boyd, J . Chem. Phys., 1974,60, 1214.

Network Egects in Dissociation and Recombination of a Diatomic Gas

269

The converse is true of recombination: starting from only atoms, the topmost vibrational levels come into equilibrium with them and the translational temperature very quickly indeed (- Id;'/), and there then follows a rapid complex relaxation lasting of the order of zvib or Id;?zl until a pseudosteady population regime is achieved.* Thereafter, the increase in population of Hz(u = 0) proceeds linearly until there has been a sufficient depopulation of atoms by recombination to modify the pseudo-linear rate law: again, no physically meaningful rate constant emerges until after the initial transients have settled down (t > ld;?21). These properties have been checked repeatedly in our calculations, and there is now no doubt that, to all intents and purposes, the rate constants for dissociation and for recombination remain constant, and obey the Rate Quotient 55A even under the most extreme flow conditions;55gl55k thus, these numerical calculations, invoking as they do only the requirement of detailed balancing of the individual energy-transfer processes (and therefore the existence of a translational temperature) strongly reinforce the conclusions of analytical reasoning,l79 148-151 and in fact the negligibly small discrepancies that do exist between our calculated rate constants and the Rate Quotient Law are consistent with the analysis outlined by Brau.150 As far as dissociation is concerned, this vibrational model has two distinct faults, and here our calculations 55c and those of Dove and Jones 78 are in complete accord. One is that although the vibrational relaxation time is essentially correct, related as it is to Plo and Pol [equation (14) above], the rate of dissociation is rather more than an order of magnitude too low. The other is that although the Arrhenius temperature coefficient for dissociation is less than the dissociation energy, it is only marginally less, i.e. 102.5 kcal mol-1 (429 kJ mol-I); likewise, the recombination rate has a negative temperature coefficient, but there is much too weak a variation with temperature. Thus, in terms of Emmons' philosophy there must still be something important lacking in the model, and we will proceed to pinpoint that deficiency before discussing the reasons for the temperature coefficients obtained in the above calculations. At this stage in the development of these investigations, it was not clear whether the next most important omission was the failure to consider vibration-vibration coupling or the neglect of rotational effects; 'conventional knowledge' insisted that vibration-vibration effects must be very important, and so attention was turned first to this problem, both by us 86 and by Kiefer.Q1 55c9

55f9

1459

* This observation tends to refute the suggestion that vibrational relaxation is very much faster in deactivation than in a ~ t i v a t i o n ,a~ conclusion "~ strongly confirmed in recent calculations on deactivation in a nozzle flow 55k and in recent experiment~.'"~ I. R. Hurle and A. L. Russo, J . Chem. Phys., 1965, 43,4434. T. I. McLaren and J. P. Appleton, J . Chem. Phys., 1970,53, 2850. 14* B. Widom, Science, 1965, 148, 1555. l*@N. S. Snider, J . Chem. Phys., 1966,45, 3299. 150 C. A. Brau, J . Chem. Phys., 1967, 47, 1153. 151 H. J. Kolker, Z . Naturforsch., 1968, 23a, 1102. 146 147

270

Reaction Kinetics

The work of Kiefer was aimed at achieving a match between theoretical calculation and experiment through parameter adjustment : it is therefore not an a priori treatment in the sense that we require here, and we now know that to some extent the unexpectedness of some of his conclusions results from his attempt to force an inadequate model to fit the experimental data. Our own approach, not knowing any relevant transition probabilities for exchange of vibrational energy in processes of types [111] and [IV], was to try to bracket the expected behaviour, choosing rather high and rather low probabilities in two parallel sets of calculations : further, since the expected temperature coefficients for these assumed transition probabilities were very uncertain, the calculations were only performed at one temperature, namely 2000 K, which is a little below the lowest experimental temperature for dissociation measurement,] O but one for which extensive calculations had been done in developing the computational techniques. The master equations [equation (1 1) and (12)] now become augmented with additional terms,ss and at the present time are only amenable to brute-force numerical integration; however, it is clear from a comparison of the population distributions obtained with and without V-V coupling 86 that these equations must be amenable to solution by perturbation techniques using the infinite-dilution (T-V) population distribution as a zeroth-order approximation. The results of these model calculations were as follows. If the redistribution of vibrational energy in a collision between two Hz molecules is relatively inefficient, then the system will be well-behaved in the sense that (at early times, before kd,H has also to be considered) the rate of dissociation can be described by two rate constants kd,M and kd,H,. If, however, the redistribution processes [I111 and [IV] are very efficient, then no simple rate law exists, and the rate of dissociation can only be cast in the form of const. x [MIa[H2l8, where a .c 1, 13 r 1, and ( a p) < 2: this is because for each M:H, ratio, the pseudo-steady vibrational population distribution is significantly different, and so the rate of dissociation does not change directly in proportion to changes in the M:H, ratio.ss The most recent experimental evidence 152 suggests that near-resonant V-V exchange for HCI is not super-efficient, and this conclusion is supported 153 by other work on HBr. For H2-Hz collisions, the polarity effects are much less important, and we may safely argue that, at most. V-V exchange for near-resonant collisions between H, molecules can only be moderately efficient: this is consistent with the fact that experimentally observed shock-wave dissociation reactions appear to obey normallooking rate laws, and we can therefore conclude that V-V coupling is not a dominating factor in these reactions. Nevertheless, no one has ever investigated how closely the orders in the reactants have integral values, and the possibility still remains that some part of the anomalous temperature coefficients could stem from a gradual and (so far) imperceptible change in

+

152

S. R. Leone and C. B. Moore, Chem. Pliys. Letters, 1973, 19, 340. Y . Noter, I. Burak, and A. Szoke, J . ghem. Phys., 1973, 59, 970.

Network Eflects in Dissociation and Rccombination of a Diatomic Gas

271

both individual and total orders of the reaction with temperature, caused by relative changes in the importance of T-V and V-V transitions. Subsequently, attention was turned to the inclusion of rotational effects in the dissociation of hydrogen, both by us (H, and D,) 55h and by Dove and Jones (Hzonly).sz The key feature of both calculations was the inclusion of the quasi-bound levels (although the overall contribution to the rate from predissociation or tunnelling from these levels is negligible 55h) and the development of methods able to solve very large sets of stiff simultaneous differential equations (about 170 for either ortho- or para-Hz, about 350 for either species of D,). In both calculations, performed independently and without consultation, the same two important factors emerged,K5h* 82 namely that, using the same techniques to assign transition probabilities as had been used in the earlier T-V calculations, not only was the rate constant now of the correct order of magnitude, but also the Arrhenius temperature coefficient was well below 100 kcal mol-l (420 kJ mol-l) in the experimental temperature range; additionally, we found that manipulation of only rotational probabilitieschanged the temperature coefficient. Full details are given in the original publication^,^^^^ 55h but Table 1 presents a summary of 55c9

Table 1 Comparison of five diflerent non-equilibrium models for the rate constant for the dissociation of hydrogen with the rotationally averaged equilibrium rate constant: values/cm3molecule- sT/K

Model 00 Model I Model II 5.5 x io-ia 2.4 x 10-15 4.7 x 10-15 3000 6.8 x 10-ls 3.3 x lo-'* 5.7 x 10-l8 2000 1.3 x 7.3 x 1.2 x 1500 2.1 x 10-23 1.4 x 1 0 - 2 5 2.5 x 10-25 5000

Model III 1.9 x 10-14 4.9 x 10-17 1.5 x 3.4 x 10-24

Model IV 2.4 x 10-14 5.8 x 10-17 1.6 x 3.7 x 10-24

Model V 3.6 x 10-14 6.8 x 1O-l' 1.9 x 4.4 x 10-24

Model 00 : rotationally equilibrated collision approximation [equation ( 6 ) ] . Model I : vibrational levels only; nearest-neighbour transitions only. Model I1 : vibrational levels only; all transitions allowed. Model 111: rotational and vibrational levels; transitions restricted to Av or AJ only. Model IV: rotational and vibrational levels; simultaneous changes in v and J allowed, but AJ restricted to be small. Model V: as model IV, but restriction on large AJ removed.

the rate constants obtained with five different models: comparison of the rates for Model I1 with those for Models IV and V shows the marked increase in predicted rate caused by including rotational levels specifically, and comparison between Models IV and V shows the effectof changing only some rotational transitional probabilities; comparison of the rates for 3000 and 5000 K shows also that the temperature coefficient of the rate is different for Models IV and V, and that both these temperature coefficients are less than for Model 11. Thus, both sets of authors concluded, rather tentatively at the time, that rotational effects were in some way a likely cause of the observed

272

Reaction Kine tics

anomalous Arrhenius 82 It is now my view, based on an examination of the parallel equilibrium rate,55jthat rotation, through equation (6), is the principal cause of low Arrhenius temperature coeficients, but as we shall see in the next two sections, the precise values of these coefficients are also determined by non-equilibrium or network effects. In passing we note that, with such large sets of simultaneous equations, it has not been possible to present detailed rotation-vibration population distributions, but it is clear that during a dissociation the pseudo-steady state is one in which:55h ( a ) all levels of low u and low J are in equilibrium with the heat bath; ( b ) levels of high u and low J are slightly depopulated (as in the simple T-V case) ; ( c ) levels of high J, for any u, are more severely depopulated (since, crudely, they have to be populated by ever increasing steps as J increases).

8 How Disequilibrium Meets the Arrhenius Temperature Coefficient If the levels from which dissociation is taking place are depopulated, the rate of dissociation will be less than the equilibrium rate: then, if the extent of this depopulation varies with temperature, the temperature coefficient of the reaction rate will differ from the equilibrium temperature coefficient. The two questions are: what is the extent of this depopulation? and, does it get better or worse as the temperature goes up? The answer to the second question appears unequivocally from the calculations done by Dove and Jones 7 8 , 82 and ourselves 55h that the disequilibrium must get worse as the temperature goes up, since the calculated Arrhenius temperature coefficients fall more and more below the equilibrium value with increasing temperature. The answer to the first question is that the extent of the nonequilibrium diminution in rate is a function of the network properties of the system of levels and the probabilities connecting them: the effect is largest for H2, where the non-equilibrium rate is only about 6 % of the equilibrium rate at 5000 K, 10%at 3000 K, 15% at 2000 K, and 30% at loo0 K, and for Dz one can approximately double these percentages; 5 5 jit is important to contrast this with the phase-space theory,7s which predicts the same nonequiiibrium diminution of the rate for both H2 and D2.In this section, we will try to answer in words the question as to why the diminution of the rate gets worse as the temperature rises, and leave until the next section the reasons why the extent of the diminution is different for H2 and for D2. Let us confine our introductory discussion to a consideration only of vibrational levels. Then a molecule finding itself in any state other than the ground state, upon collision with a third body M, stands a greater chance of losing rather than of gaining vibrational energy (but despite this, an assembly of molecules possessing too little vibrational energy still manages to accumulate more vibrational energy, approaching equilibrium in an exponential 55c9

Network Eflects in Dissociation and Recombination of a Diatomic Gas

273

manner). When we come to the ‘interesting’ levels, i.e. those from which dissociation is occurring, our picture is that molecules are dissociating from these states just about as quickly as they can be supplied from lower down the ladder:66bequilibrium is, and can only be achieved, when there are enough molecules coming down from above to balance those g3ing up. The transition probabilities are such that, for a molecule in an ‘interesting’ level, the ratio of its chance of going down to its chance of dissociating decreases as the temperature increases, i.e. the upward transition probabilities increase faster than do the downward transition probabilities as the temperature increases. Since saturation of downward transitions is essential to equilibrium in the dissociating gas, the result of this weighting towards upward transitions is that it is harder and harder to maintain the equilibrium distribution at the higher temperatures. We will now apply the same argument to recombination (as it is often extremely difficult to visualize the converse of a particular effect in this dissociation-recombination problem 15*). Because of the decreasing disparity in the probabilities of upward and downward jumps, the higher the temperature, the more nearly are the atoms and the upper vibrational levels in equilibrium with each other (cf. examples for 300, 1O00, and 3000 K in Table 36 of reference 5 5 4 : ignoring the T* dependence from the collision number, there is a constant specific rate of formation of molecules from atoms, but, as the temperature decreases, the poorer equilibration of atoms with molecules means that proportionately fewer escape again, and so the net rate of stabilization increases with decreasing temperature; this is essentially the picture proposed by Polanyi some time ago.ss Now what is the magnitude of this effect ? We have already noted that in calculations for Hz, the depression iil ‘activation energy’ was far too However, the effect on the temperature coefficient of the rate appears to be a function of the number of vibrational levels in the molecule, and we have performed some calculations for a model molecule having a fixed dissociation energy, but a variable number of vibrational levels,155with the apparent result that, when the potential well contains many ( > 100) vibrational levels, the Arrhenius temperature coefficient of the rate tends to a value of ( D o - RT). This limit appears to be the simple Tolman activation energy for the equilibrium case, i.e. the difference between the average energy of those reacting (identically Do in the model calculation) and the average energy of all those present, viz. RT, the vibrational energy; it would therefore be interesting to analyse the non-equilibrium analogue of the Tolman definition 156 and compare the results with the calculated temperature coefficients for the vibrational model of H,.* Thus, we can summarize the effect of vibrational disequilibrium a5j9

* We tried 55c to examine the disagreement between these temperature coefficients and the Tolman equilibrium definition, but did so incorrectly, taking the average energy of those molecules reacting to be the energy before the dissociative collision, instead of the total energy during the collision, i.e. Do. 154 J. L. J. Rosenfeld, unpublished comments. E. Kamaratos and H. 0. Pritchard, unpublished results. 158 Ref. 45, pp. 21-24.

274

Reaction Kinetics

on the Arrhenius temperature coefficient roughly in the following terms -in the limit of very few vibrational levels the coefficient will be D oand in the limit of many vibrational levels it will be ( D o - RT), with intermediate behaviour for an intermediate number of vibrational levels: the reason for this behaviour becomes understandable in the light of the properties of the vibrational activation network, as described in the next section. The situation is generally the same when rotational levels are included as well, although, as has been pointed out, there are some conditions under which a molecule stands a greater chance of being activated rather than deactivated by a collision with an inert body.15’ The most important example relates to the lowest J states of molecules, where it is clear that in order to have a maximum number of molecules at some non-zero J value, it is most probable that for (say) the J = 2 state of Hz, more molecules per second are making the J = 2 4 transition than are making the J = 2 + 0 transition according to Kondratiev, about twice as many.158 Since in general we will rarely be concerned with disequilibrium among states of low J, we may logically extend the conclusion of the preceding paragraph to the case of a molecule having both rotational and vibrational degrees of freedom: thus we would expect that in the limit of very large numbers of energy levels, the observed Arrhenius temperature coefficient would be depressed by 2RT, i.e. ths total internal (Vibrational rotational) energy, to give: --f

+

+ Rd[b*(T)]/d ( 1 / T ) ) - 2RT = E*(T) - 2RT

‘E’ = ( D * ( T )

(15)

where b*( T ) is the rotationally averaged equilibrium critical energy of equation ( 6 ) ;it should be possible to establish the equivalence of this equation to one given by Hay and Belford 159 some time ago. It would seem prudent to add that this reduction would only approach this limit under conditions where the dissociation reaction is virtualJy completely decoupled from the internal relaxations - which is reasonably the case in these reactions except at very high temperatures. Under the conditions which we have described above for HB,where the upper rotational levels appear to be severely depopulated but the vibrational levels are reasonably well equilibrated, we will not achieve this full contribution of the rotational energy to the Tolman s } R T ] , where activation enxgy, and would find only [B*(T) - (1 0 < 4 < 1. Finally, we must add just one more twist. If we had formulated our calculations more carefully, we would have realized, and would have built in to the model, the fact that all molecules which dissociate do not just ‘crawl’ out of the potential well with only enough energy to dissociate, and for example, wherever we have written Do,we should really write bo = ( D o R T ) - one can easily deduce this by considering what is the average

+

+

15’

lj9

T. W. Broadbent and A. B. Callear, J.C.S. Faraday ZZ, 1972, 68, 1367. V. N. Kondratiev, ‘Chemical Kinetics of Gas Reactions’, Pergamon, London, 1964. A. J. Hay and R. L. Belford, J. Chem. Fhys., 1967, 47, 3944.

Network Eflects in Dissociation and Recombination of a Diatomic Gas

275

energy of those atoms which recombine, assuming there is no marked selec47 tion of the kind we mentioned We conclude, therefore, that non-equilibrium effects can alter the Arrhenius temperature coefficients of reactions, but in the diatomic dissociation case, because the diatomic molecule has so few internal degrees of freedom, the expected range is within f R T of the effective critical energy, the precise value being determined by the actual form of the pseudo-steady population distribution. 9 The Emergence of Network Effects The behaviour we have just described is really a consequence of the network effects to which we have alluded frequently: the first clear indication of these effects in fact came from just the kind of calculation described in the preceding section. We had repeated our T-V calculation using D2, which has 22 vibrational levels instead of the 15 levels for H2, and found that the calculated rate of dissociation for D2 was higher than for H2, despite the fact that the dissociation energy Do was significantly greater.55e Exploring further, we 155 found that for a fixed value of Do,the more levels the molecule was assumed to have, the greater was the rate of dissociation, although understandably we did not continue our numerical experiments to such great numbers of levels that the limiting equilibrium rate was reached. There is obviously a variational theorem here, that if the number of connected levels in a potential well is increased, maintaining Do constant, the rate of dissociation will not decrease: the germ of the proof of this theorem already exists in the theorem concerning eigenvalues of the sum of two positive definite matrices,lso but at the present time all the necessary conditions for a complete proof have not been e~tab1ished.l~~ Physically, what this means is that apparently the onset of non-equilibrium depopulation of the upper levels is pushed to higher energies when the steps are smaller: we have just argued that the maintenance of equilibrium is favoured when the upward and downward transitions from a particular level have probabilities which are not too dissimilar, and this can only be so when the levels are close together. This faster rate for the dissociation of D2 persists in our T-VR calculations 55h at high temperatures where the Boltzmann factor exp[(Dp - D p ) / R T ]is not dominant, and it appears that the prediction of a faster rate for the dissociation of para-Hia than of ortho-H2 is another example of this network phenomenon.55h A second and not altogether unrelated network property is the gross insensitivity of the dissociation rate to large perturbations in the assumed transition probabilities - a corollary of Carrington’s conclusion 95 that it is an ill-conditioned problem to try to determine transition probabilities from relaxation times in such systems. Several examples of this insensitivity 160

R. Bellman, ‘introduction to Matrix Analysis’, McGraw-Hill, New York, 1970, p. 117.

276

Reaction Kinetics

have been published 55e and we have found many others. Suppose that we revert to the vibrational model for simplicity, and we concentrate on the levels u = 9, 10, 11, and 12: then we arbitrarily reduce the transition probability for u = 10 ++ u = 11 to zero (in both directions) and the rate of dissociation falls, but does not fall to zero because processes u = 9 u = 11 and u = 10 ++u = 12 are still allowed: trivial? However, if on the other hand we arbitrarily give the u = 10 u = 1 1 transition an almost infinitely large transition probability, the rate constant hardly increases at all, since it is determined co-operatively by all the transition probabilities in the network: not quite so trivial! We have shown that the eigenvalue equation for determining the rate constant is formally the same as that for determining the current through a resistance network if current = dissociation rate and conductance (i.e. resistance- I) 3 transition rate.55d Clearly, if we have an n-point resistance network with resistances connecting every pair of terminals, and if we either short-circuit or alternatively remove any particular resistance, usually we will not cause much of an effect on the total resistance of the network. Obviously this analogy could be extended further by associating with each point in the network a capacitance (which would determine the charge held at equilibrium = the equilibrium population distribution for each level) and used to illustrate the transient processes preceding the dissociation. One can also sifnulate these effects qualitatively in another way. Imagine a network of city streets having a North-American block structure, and consider the flow of traffic out towards a suburb at evening rush hour. The widest street will carry the heaviest Traffic, with lesser traffic density on the narrower ones: this is a fairly straightforward analogue of the molecular situation, where the greater part of the reaction flux is by nearest-neighbour transitions, with lesser fluxes reacting by multiple-quantum jumps. It is fairly obvious that (analogous to our u = 9, 10, 11, 12 example above) if one of the traffic lights on the main street gets stuck permanently on red, there will be chaos in the immediate vicinity of that junction, but after a short time (analogous to the vibrational relaxation time) traffic further back will begin to turn off the main street and by-pass the blockage - and although the rate at which vehicles reach the suburbs is now less, the traffic still flows. Even more obvious is that if the traffic light were to stick on green, the rate at which vehicles would reach the suburbs would hardly be affected at all, the rate then being determined co-operatively by the flow patterns at other traffic lights in the network. A variant on this theme is the result of manipulating the actual dissociation probabilities themselves: if we make them all very small, the rate goes to zero as it must, but if we make them all very large, the rate increases only very weakly;55call the traffic lights at the city perimeter sticking either at red or at green would mimic this behaviour in a crude qualitative fashion. Another example of this insensitivity to transition probabilities is shown by a nearest-neighbour-only T-V calculation, to which we have already alluded. Skeleton results are given in Table 1, where it is seen that even at 55r9

55dp

-

-

Network EHects in Dissociation and Recombination of a Diatomic Gas

277

4

I I

I I

I I

0

I I

I I

/72 I I

I

I I 1 1

\

\

Figure 2 Schematic representation of a sparse relaxation matrix. In this matrix, the energy levels are considered to be ordered with increasing energy, the transitions between low-lying levels being represented by elements in the top left-hand corner, and transitions between high rotation-vibration states being represented by elements at the bottom, towards the right. Thus, thejirst element to the right of the diagonal on the top row is the transition probability for the transition between the ground state and the next lowest-lying state: the last element on the top row is the transition probabilityfor a direct transitionfrom the ground state to the topmost (quasi-bound) state of the molecule; the probabilities for direct dissociation from each state are contained in the respective diagonal elements.65hNote that i f a = b = c, the matrix is a simple band matrix

worst, Models I and I1 do not differ by more than a factor of two (and in fact if a selection of non-nearest-neighbour transitions is allowed near the dissociation limit, even this factor of two disappears s6B): loosely, this can be taken as confirmation of the idea that most molecules which find themselves dissociating will have arrived there by single-step transitions, at least as long as the vibrational states are far apart. The vibrational model is, however, inadequate, as we now h o w , so further discussion and elaboration on these results would only be of academic interest, and for practical purposes would redundant in the light of the following parallel results obtained with the rotation-vibration Models IV and V.lS1 Figure 2 represents an approximation to the full 172"d-order relaxation matrix representing the dissociation la

G. McCart, T. Ashton, and H. 0. Pritchard, unpublished results.

Reaction Kinetics

278 Table 2

(a) Ability of banded relaxation matrices to reproduce key eigenvaules at 5000 K Matrix a 60 60 50 50 50 50 50

dimensions b C 90 60 75 60 50 90 75 50 40 75 75 30 25 75

Percentage of correct eigenvalue dn-1 dn-2 100 100 99.9 100 99.7 100 98.8 99.7 96.1 99.2 94.1 98.4 83.8 96.4

(b) Examples of catastrophic failures at 2000 K Percentage of correct eigenvalue Mutr ix dimensions a c 4 - 1 dn--g b 98.2 100 (i) 50 25 75 98.1 100 50 20 75 60.1 100 50 15 75 8.5 5 75 40.7 50 60.1 100 15 75 (ii) 50 100 15 50 24.6 15 15 5 50 1.3 x 10-4 4.7 x 10-4

(c) Minimum bandwidths needed to give acceptable dissociation and relaxation rates TemperaturelK

5000 2000 1500 700

Matrix a 50 50 50 50

dimensions b C 50 75 25 75 20 75 15 75

Percentage of correct eigenvalue dn-1 dn-2 98.8 99.7 98.2 100 99.2 100 100 100

of ortho-H2, in which the transitions are ordered according to the energy requirement for the transition (see legend to Figure 2 for further clarification). Since many of the transitions requiring large energies have very small transi55c they can safely be assumed to have zero probability tion probabilities,656, without affecting either the rate constant or the observable relaxation times. Table 2(a) shows that at 5000 K one only needs to consider a band of the 60 smallest-energy transitions, together with all the transitions between the topmost 90 levels, to reproduce both dnVL and dn-2to better than 0.1%: reducing either the width of the band to 50 or the number of freely interacting upper states to 75 reduces the rate constant without affecting the relaxation time; on the other hand, maintaining both a and c constant, reduction of the band width for the transitions connecting the lower to the upper states causes both the dissociation rate and the relaxation time to fall away. Table 2(b) shows that if the band of transitions in the middle of the network is made too narrow, there is a sudden and catastrophic fall in both eigenvalues, demonstrating convincingly that there is real and subtle coupling between the

Network Eflects in Dissociation and Recombination of a Diatomic Gas

279

dissociation and the internal relaxation despite the apparent and gross insensitivity of the rate of one process to probabilities assigned to components of the other p r ~ c e s s . ~ 55e ~ ~ Table ~ 2(c) shows that as the temperature is reduced, one can accept successively narrower bandwidths for the transition network connecting the lower and upper states. Two points are of interest: first the computational point that any ab initio calculation of the rate of dissociation requires the calculation of only a relatively small subset of all the transition probabilities; second, that the physical picture of the T-VR dissociation process is the logical extension of the nearest-neighbour concept, that most of the molecules which find their way to dissociation do so by accumulating rotational and vibrational energy in relatively small increments. One can also discern quite marked network qualities in the inter-relationships between transition probabilities and the internal relaxation times. The simplest example, and one that has been appreciated for some time,3* s5k is the slight lengthening of the vibrational relaxation time at high temperatures in comparison with the simple harmonic oscillator approximation, equation (14): this effect is shown in columcs two, three, and four of Table 3. Between lo00 and 1500 K, the relaxation times Galculated from vibrational S5d9

55c9

Table 3 The eigenvalue Id;221 for six diflerent relaxation models (valuesls) TIK Model 0 5000 1.6 x 10-9 3000 7.1 x 2000 2.3 X lo-' 1500 5.6 X lo-' 1000 2 . 0 , ~1 0 - 7 Model 0 : Models I-V:

Model I Model II 3.0 x 1 0 - 9 2.8 x 10-9 8.5 x 8.4 x 2.44 X lo-' 2.44 X lo-' 5.71X lo-' 5.71 X lo-' 2 . 0 ~x 10-7 2 . 0 ~x 1 0 ' 7

Model III 5.9 x 1 0 - 8 2.5 x lo-' 5.0 X 7.7 X 1.4 x 10-6

Model ZV 4.3 x 10-8 8.3 x lo-' 1.1 X 1.4 X lo-' 2.3 x 10-7

Model V 4.5 x 10-9 1.3 x lo-' 2.9 X lo-' 4.5 X lo-' 5.9 x 10-8

harmonic oscillator approximation [equation (14)l. see footnote to Table 1.

Models I and I1 begin to lengthen slightly with respect to the simple harmonic oscillator approximation (cf. also Figure 1 of ref. 55k), an effect which has been put down to an increasing complexity of the relaxation:3' 55c a similar effect occurs in rotational relaxation also, but the lengthening is more dramatic because of the progressively increasing spacing of the rotational energy level^.^^^^ 163 The small differences between Models I and I1 themselves at high temperatures also seem sensible, since the same relaxation must surely take longer if it has to take place by single-quantum jumps. Models 111, IV, and V differ from the others in that the rotational sublevels are explicitly included. Thus the essential characteristics of the relaxation are that now the molecules can accumulate internal energy in many more smaller steps (roughly 170 as opposed to 15): on the other hand, the amount of energy to be collected to effect completion of the relaxation is 162p

L. M. Raff and T. G. Winter, J. Chem. Phys., 1968,48, 3992. C . A. Boitnott and R. C. Warder, Phys. Fluids, 1971,14,2312.

lB2

280

Reaction Kinetics

(for H,) considerably more than twice as much; in Models I and I1 the molecules were ‘given’ their rotational energy RT before the vibrational relaxation commenced, but now they have to accumulate both rotational and vibrational energy, and in the special case of Hzthe latter is rather smzll, especially if the temperature is low. Model V makes little distinction between rotational and vibrational energy in the sense that the transition probabilities depend principally upon the energy separation between the two states in a manner which is characteristic of vibrational energy levels: the only difference is that to rectify a mismatch between the computed and observed probabilities for low rotational transitions, a scaling factor of e-5 w 1/150is applied to any transition in which J changes; thus, for the same energy separation between levels the computed probability is independent of the magnitude of the change in J. At 10oO K it seems to take about three or four times as long for the system to relax in large vibrational steps (Model I) as in the smaller rotational/vibrational mixture of steps (Model V), despite the fact that much less energy has to be accumulated, which does not surprise us in view of our experience with dissociation rates, However, above 2000 K the data cannot be rationalized in this way: it seems to take Model V about 1+ times as long to accumulate the necessary internal energy and one is forced to think (since the ratio of the rotational and vibrational heat capacities changes markedly) that, other things being equal, it must take longer to equilibrate a 170-point network than a 15-point network: there are more pigeon-holes to be filled and the ripples take longer to settle down? Perhaps it would be true that if all the telephone exchanges in a large city were ‘off the air’ and were switched on simultaneously, it would take longer to reach a steady state than if all the local exchanges, each already in its own local steady state (rotational equilibrium), simultaneously had their inter-exchange facilities restored ? Model IV, in which large changes in J are strongly forbidden, understandably always takes longer to relax than Model V, especially so at high temperatures where population of high J states is important. Model 111, in which either only u or only J c a n change in any one encounter, is even slower to relax than Model IV. At 5000 K the difference between Model 111 and Model IV is small, both much slower than Model V, indicating the marked contribution of transitions having smallish energy changes but very large AJ’s in the latter model: the small difference between Models I11 and IV suggests that otherwise the population of high J states takes quite a long time and that it does not matter if a small amount of extra vibrational energy can be taken up along with this rotational energy. At low temperatures, however, there are strong differences between Models 111, IV, and V, with Models 111 and IV actually being slower to relax than the simple vibrational models: it seems therefore that the restriction of these two models to virtually 65a only A J = f 2 effectively separates the rotational and vibrational relaxations. Perhaps we could use our city-street network analogy again - with say a network of restricted-access highways (vibrationally equilibrated?) superimposed on the ordinary network of city streets (rotationally equilibrated ?) :

Network Eflects in Dissociation and Recombination of a Diatomic Gas

281

a blockage in either network reflects itself in a redistribution of traffic within that network in a relaxation time characteristic of that network, but, much more slowly, there is an overall redistribution of traffic between the two networks to try to overcome the effects of the blockage. A word of caution is necessary here, concerning any attempt to identify relaxation times with specific microscopic molecular processes : we would not think of doing it with our street network! With each characteristic time in the molecular system, there is associated a normal mode of relaxation,55c and it so happens that below lo00 K for Models I and 11, one normal mode coincides (to all intents and purposes) with the v = 0 v = 1 relaxation; at higher temperatures, however, as we have already noted, the relaxation cannot be described by a single process. Likewise for smaller energy spacings, i.e. Models 111, IV, and V, virtually every characteristic time is associated with a very complicated normal mode of relaxation, and we have examined the ramifications of this behaviour in moderate detail e l ~ e w h e r e . One ~~~ important feature which emerges is that the longest non-reactive relaxation time contains both rotational and vibrational character, and recent experimental observations of ‘vibrational’ relaxation have been tentatively explained in such terms:133it appears that the leading term in the normal mode of the longest relaxation time is associated with the largest rotational gap in the v = 0 ++ v = 1 transition (which for ortho-H2 is v = 0, J = 5 v =0, J = 7); this is in fact the largest and hardest individual transition in the whole network of transitions if one does not arbitrarily restrict A J to small values (i.e. Model V). Another is that if one tries to describe the relaxation of a set of levels far from the dissociation limit in terms of a truncated set of energy levels, improper truncation of the set can introduce spurious and unwanted relaxations slower than any of the real relaxations in the Nevertheless, it is clear that a full understanding of the trends appearing in Table 3 still requires further work. Moreover, the Johnston-Birks strategy of using the observed relaxation time to define Pol and Plo via the harmonicoscillator approximation, equation (7), and forcing kd,Mto be the experimental value by manipulation of the P%iof equations (1 1) and (12) is revealed as a transparently nalve model of the dissociation process - particularly so when it is realized that relaxation times measured for the same temperature but by different techniques need not necessarily coincide:55ia fairly obvious result of increasing the dissociation probabilities p,tiis to strengthen the vibration-dissociation coupling, which begins to lengthen still further the already lengthened relaxation time over the value consistent with equation (7).16* +t

-

10 Conclusions In an essay dealing with a subject of such complexity, it is not possible to draw a single set of self-contained conclusions: rather, I will try to present lo*

N. I. Labib and H. 0. Pritchard, unpublished results.

282

Reaction Kiizefics

three groups of concluding remarks, the first setting out what I see to be the most realistic description of the diatomic dissociation process consistent with the present state of knowledge, the second collecting together the major stumbling blocks to further understanding and characterization of these processes, and the third suggesting some of the kinds of difficulties that might arise from network effects in other branches of chemical physics. The Nature of the Dissociation Process.-The fundamental property of the diatomic dissociation-recombination reaction, that the dissociation proceeds with an Arrhenius temperature coefficient which is a lot less than the spectroscopic dissociation energy, or that the recombination proceeds progressively more slowly as the temperature increases, can be explained essentially in terms of a purely equilibrium theory, as I have already described: other concepts, such as disequilibrium, tunnelling and quasi-bound resonances, chaperon effects, etc., which have been central in recent attempts to understand these observations, are in fact only peripheral in their importance and need only be invoked to account for small differences in behaviour for a given diatomic molecule as the third body M varies, or to account for differences in degree from diatom to diatom. The essential cause of low Arrhenius temperature coefficients in diatomic dissociation is the fact that if the molecules are forced to be rotationally equilibrated, those molecules in high J states must take paths of higher energy if they are to dissociate: the proportion of molecules with high J increases with temperature, which means that, in effect, the rotationally averaged critical energy for dissociation increases with temperature; thus the rate of dissociation does not increase with temperature as rapidly as it would if the critical energy were constant, and therefore the apparent ‘activation energy’ comes out to be ‘low’. The complementary explanation of the negative temperature coefficient for recombination is also readily formulated : simply, the higher the temperature the faster will the pairs of atoms approach each other and therefore, except for head-on collisions, the higher will be the proportion of two-atom encounters having high angular momentum; but higher angular momentum carries with it the requirement of overcoming a higher centrifugal energy barrier for the pair to enter the region of phase space from which it can be deactivated to form a molecule-proper, and therefore the rate of recombination actually falls as the temperature increa~?s.~~j In passing, one might note that the explanation of low Arrhenius temperature coefficients for ionization of inert-gas atoms 26 can probably be accommodated within a similar framework, although the required spectrum of auto-ionizing energy levels is perhaps insufficiently well known at the present time;165certainly, angular momentum restrictions on the auto-ionization of OD- ions having much more than the required threshold energy have been found,L66and it therefore seems prudent to suggest that all apparent negative temperature 165

166

R. F. Stebbings and F. B. Dunning, Phys. Rev. ( A ) , 1973, 8, 665. L. D. Doverspike, R. L. Champion, and S. K. Lam, J . Chem. Phys., 1973, 58, 1248.

Network Eflects in Dissociation and Recombination of a Diatomic Gas

203

coefficients 16' should be re-examined for angular momentum restrictions. Now we must consider disequilibrium, and will divide the discussion of its effects into two parts, first its effect on the rate, and then its effect on the temperature coefficient of that rate. We have seen that failure to maintain a Boltzmann distribution among the rotation-vibration states of the diatomic molecule causes the non-equilibrium rate of dissociation to fall below the rotationally averaged equilibrium rate, equation (6). The magnitude of this rate depression is large at high temperatures, but becomes less important at low temperature~.~~j Also, the rate depression is most severe for Hz (being, as we noted before, about a factor of 15 at 5000 K but only a factor of 3 at lo00 K) but is not nearly so bad for D2because Dz has many more levels in the network than does H, (cf. Figure 1 of ref. 5 3 ) : thus, we would expect that for a heavy molecule like Iz or Br,, having rotation-vibration spectra numbering many thousands of levels instead of a few hundreds, the rate of dissociation in the presence of a structureles third body M would approach very closely to the equilibrium rate, particularly so at low temperatures. In this light, it also becomes possible to comment on the behaviour of the temperature coefficients of these dissociation rates under non-equilibrium conditions. We have already discussed these temperature coefficients in terms of an extended form of the Tolman definition of 'activation energy', and have seen that in the vibrational case the Arrhenius temperature coefficient could fall somewhere between b, and ( D o - RT) depending upon (among other things) the number of vibrational energy levels in the molecule: we then conjectured, through equation (13, that in the rotation-vibration case there could be a variation of up to 2RT in the temperature coefficient; however, this is far too nai've an assumption. Before analysing this problem in detail, let us dispense with tunnelling and quasi-bound resonances : they do contribute to the dissociation process for Hz, but the net effect of these additional channels on the overall rate is rather Other hydrogen-containing molecules are heavier and have longer bond lengths; tunnelling is less important for heavier particles, and as is shown by comparison of H: lifetimes 16* with H, lifetimes,54tunnelling is less important when < R,> is larger; thus these effectscan be discounted for virtually every other molecule. It has always been taken for granted in the past that it was vibrational energy-transfer efficiency that was the key attribute causing third bodies M to vary in their ability to cause dissociation of X2or recombination to two X atoms. However, vibrational relaxation is completed long before dissociation has become significant ( ] ~ / , ~ - ~ 1 > ldn-ll),and clearly the ability of one M to effect an equilibration between the u = 0 and u = 1 states of Xzmore quickly than another M is largely irrelevant. What is decisive in fact is the ability of M to remove rotational energy from Xz, and not in small amounts of 16i

168

D. D. Davies, R. E. Huie, and J . T. Herron,J. Chem. Phys., 1973, 59, 628. G. Hunter, A. Y. Yau, and H. 0.Pritchard, Atomic Data and Nuclear Data Tables, 1974, 14, 11.

284

Reaction Kinetics

rotational energy but i n rather large amounts: for example, if we are talking about H,, it is not the ability to convert J = 2 into J = 0 effectively that is important, but the ability to cause interconversions between states of high J , where the energies involved are much larger. To be a little more specific, suppose that a molecule Hsis in the J = 34 state; it cannot dissociate (now ignoring tunnelling) unless it accumulates a total energy of ( D o 5352) cm-l: however, were it to suffer an encounter with M, M now having such internal structure that (maybe because of a favourable resonance) it could remove (say) 10 units of angular momentum - possibly also changing the vibrational energy up or down as well - then the molecule would only need to accumulate a total of ( D o i - 1626) cm-’ to dissociate; even efficient removal of rotational angular momentum in units of 2 would achieve the same end, although a little more slowly. Thus, a third body M having an efficient rotational energy transfer ability will provide molecules which are in high J states with an easier route to dissociation, and will tend to depopulate even further those high J states: the net consequence of this will be that as some molecules are now relieved of the necessity of dissociating via very high energy paths, the mean critical energy for dissociation averaged over the nonequilibrium rotational population distribution will be less than it would have to be for an equilibrium rotational distribution; hence the Arrhenius temperature coefficientwill be higher. Clearly, if u and Jchange simultaneously, so that AE remains small [e.g. reaction (lo)], the effect will be the most beneficial. In the limit of infinite rotational energy-transfer efficiency, virtually all the molecules could escape by the J = 0 route, and the critical energy would then be just Do, independent of temperature! It is possible that this effect has already been seen for H2for which it has been reported recently that with M = argon, the temperature coefficient is 88.9 kcal mol-l (327 kJmol-l) whereas with M = hydrogen, the temperature coefficient is very close to the spectroscopic dissociation energy,lgSand is therefore the nearest thing we have yet seen to a true activation energy! Finally, one need not always expect impurities to lower the ‘activation energy’ as was observed with hydrogen l o - they could also raise it,lio if the impurities are efficient rotational relaxants for the molecule concerned. This picture now has to be modified somewhat to allow for the possibility that M can react with X, to give M X X . Let us consider first the purely equilibrium situation with respect to X, and take the rotationally averaged equilibrium rate as our standard rate of dissociation. Then X, molecules in low Jstates will derive little benefit from an alternative pathway to dissociation via M X + X. However, those Xzmolecules in high J states, being forced to dissociate over a rather large rotational barrier, can now escape by a much easier route, even if most of the angular momentum contained in X, were to end up in the newly formed MX: this is because MX, being a much more

+

A

l7O

W. D. Breshears and P. F. Bird, 14th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, Pennsylvania, 1973, p. 21 1. A. M. Dean, J . Chem. Phys., 1973, 58, 5202.

Network Eflects in Dissociation and Recombination of a Diatomic Gas

285

weakly bound molecule than Xz, will therefore have a much lower series of rotational potential maxima than does Xz. We may distinguish two extreme types of case. First, if X2 is in a quasi-bound state and all of its angular momentum ends up in MX, and the total energy also remains about the same, then the new MX molecule will probably be above the rotational maximum for that amount of rotational angular momentum. If on the other hand Xg is in a bound state, but just below the spectroscopic dissociation limit, transferral to the MX ladder with about the same total energy and angular momentum will certainly mean that it now needs less energy to dissociate over the MX centrifugal barrier than it would have required over the Xz centrifugal barrier. Of course, neither will the molecular binding energy nor the molecular rotational angular momentum remain the same in such a reactive encounter, and the results of these changes will be very much indeed like those discussed in the preceding paragraphs for the energy-transfer mechanism: nor of course, will any of the ‘interesting’ levels remain in Boltzmann equilibrium with the heat-bath.temperature, but again the diminution of the rate from this 61 will be rather small for most X2. Thus, it is clear that the chaperon mechanism, by providing additional low-energy dissociative channels for Xz molecules having high values of J, must cause an acceleration in the dissociation rate, and since this acceleration will vary with temperature, the Arrhenius temperature coefficient will also be altered: in what way, it is difficult to say, since the effects will depend very much upon how the series of individual processes : M

+ X,(V,J ) + X + MX(V’,J’)

(1 4)

depend on the parameters u, J, u’, J’, and the temperature. In this section of the discussion, I have allocated most of the blame for the anomalous Arrhenius temperature coefficients of diatomic dissociation reactions to rotation and the way it causes the mean effective critical energy to increase with temperature: then both rotational energy transfer and the chaperon effect play rather similar roles in mitigating somewhat the consequences of these centrifugal barriers to dissociation. I do not regard this description of the dissociation process as being in any way in contradiction with the numerical treatments of Keck and of Burns and their respective Both seemingly different approaches are quite co-workers.21v60$ successful in reproducing both the overall rates and the temperature coefficients of these rates for a variety of molecules: the former approach, however, as we have mentioned earlier, is weak in its description of non-equilibrium effects,and has perhaps been over-enthusiastically applied to Hz, where such non-equilibrium effects are the most important and the assumption of classical behaviour is the most doubtful; likewise, the latter approach has tended to underestimate the importance of the energy-transfer mechanism in determining the temperature coefficients. Thus, assuming that the adjustable parameters which have to be chosen in these two theories are transferrable in the sense that in the course of time they will be found to be equally accept61p

72-759

286

Reaction Kinetics

able in describing transport or other properties in these same systems - then it is clear that not only are these two approaches roughly equivalent, but that they both contain a sufficiently proper treatment of rotational motion to be roughly equivalent also to the description I have just given: Wong and Burns have already noted the equivalence of seemingly different descriptions of their dissociation-recombination models ; however, it is unlikely that, after the fact, it would be possible to reanalyse the results of these two alternative approaches and recast them in terms of the concepts expounded here. Stumbling Blocks to Progress.-Let us define ‘progress’ in terms of the rather limited objective - that of achieving a complete a priori calculation of the rate of the dissociation or recombination of H2by an inert gas: this is the molecule for which the non-equilibrium effects are the most severe and for which both the chaperon mechanism and the phase-space theory can have only very limited applicability. A perennial complaint in the past has been concerned with our inability to define precisely what was occurring in a shock-tube dissociation experiment : temperatures were suspect because of boundary-layer 171 results were not very reproducible, possibly because of impurity effects,l0*170 and so on. However, with the recent theoretical progress on three fronts, as outlined in the preceding paragraphs, these problems assume a lesser importance. In the absence, until very r e ~ e n t l y , ~of~ j a comprehensive explanation of low Arrhenius temperature coefficients, one always had to keep in mind the possibility that they mighl be due to some experimental artefact, and this was the reason for the philosophy adopted by my group in approaching this problem -there was no point in trying to bend the theoretical res-Ats to match the experimental ones in case the latter might eventually be proved to be wrong! Thus, it seems much less urgent now than it did even a couple of years ago for someone to define the limits of the reactant orders in a dissociation, although such experiments are still well worth doing. Likewise, our ignorance as to whether war-resonant processes accompanied by large changes in angular momentum [e.g. reaction (lo)] are allowed or not is now less important to this particular probZem, although still of fundamental intrinsic interest : we have already commented on one theoretical possibility for solving this problem; another possibility would involve further numerical experimentation, trying to see which of Models IV or V in Table 3 matches the experimental ultrasonic relaxation behaviour the better. Of course, experimental evidence is probably not very far away from some processes (which for our purposes incidentally must refer to the ground electronic state and not to excited electronic states of the diatomic molecule 128* 120 - the nature of the interaction with M is very different for saturated and unsaturated electronic states); and perhaps with the ability to measure these transition probabilities will also come the ability to mezsure the populations of molecules in ‘interesting’ states during dissociation or recombination ! H. Mirels, Phys. Fluids, 1963, 6, 1201.

Network Efects in Dissociation and Recombination of a Diatomic Gas

287

The calculation of transition probabilities remains an area where progress is needed, but our results demonstrated in Figure 2 and Table 2 show that we really require far fewer probabilities than one might at one time have feared. Furthermore, the gross insensitivity of rates of dissociation to small errors in individual transition probabilities means that even of those probabilities required, one need only calculate a key selection, and assign the others by interpolation between the key values: the key values must of course include some probabilities for dissociation steps. Given such probabilities, it is clear that a simple repetition of the exploratory work we have undertaken 55 should yield the desired rate of dissociation or recombination of Ha by the inert gas - at least in the temperature range from about 700 K upwards. To complete the description of the process, one would like also to be able to calculate the rate of dissociation or recombination of H2 by H2 as a third body, and one would therefore need to graft on to this solution the perturbation effect of VR-VR coupling and of the radical-molecule-complex mechanism: contributions of the latter type would also need to be considered in extending the temperature range down to the experimental limits 172 of around 100 K. There is, however, another serious obstacle to calculations for low temperatures, even for the energy-transfer mechanism alone, and that is the state of the numerical art of finding the eigenvalues (and eigenvectors) of large very ill-conditioned matrices. The spectrum of eigenvalues at various temperatures spans a range which is extremely temperature-dependent, roughly as follows : 599

T/K 10 000 5000 2000

lo00 500 300 100

-do ............... -d,-Js-l 1013 ............... 109 1013

............... 10s

1013 ...............107 1013 ............... 107 1013 ............... 106 1 0 1 3 ............... 106 1013 ............... 105

-4 - &f lo8 1og 10-1 10-'0 10-35

10-200

when predissociation of the quasi-bound states through the rotational pseudo-potential barrier is included (as it must -be at low temperatures). In other words, all of the (several hundred) eigenvalues with the exception of dn-l change only weakly with temperature, but da-l changes very markedly indeed because it contains implicitly the equilibrium constant K, for the reaction. If it were possible to reformulate the problem in such a way that do to dn-2were unchanged, but the last eigenvdue became KCf d,,-l ( ~ 1 0 - l ~ cm3molecule-' s- l, almost independent of temperature) the numerical problem would be no more difficult than we have already solved:55hat the moment, it would seem that this projection could only be achieved with l i 2 D. 0.Ham, D. W. Trainor, and F. Kaufman, J. Chem. Phys., 1970,53,4395; 1973, 58, 4599.

288

Reaction Kinetics

prior knowledge of the eigenvector associated with dfl-l. Hence, we therefore have to contemplate a brute-force approach which makes us dependent on the (empirical) fact that the best accuracy one can achieve for dfl-l is only a little better than one would get by subtracting the sum of all the other eigenvalues from the trace of the matrix: thus, using IBM double-length arithmetic of 16 decimal places, the best one could hope for at 2000 K would be 2 or 3 decimal places, and the best one can get is 3 or 4 decimal places. To get the eigenvalue corresponding to dissociation - and therefore to recombination - at 100 K, one would need to use about 200-decimal-place arithmetic in the calculation : consequently, even for only one of the nuclear spin states of H2, the storage requirements for the full relaxation matrix would be extravagant. In principle, Table 2 and Figure 2 demonstrate that we do not need to consider the full matrix, but only a banded approximation to it, and by so doing could reduce the storage requirement to within reasonable bounds. Unfortunately, the only methods for reducing banded matrices, whilst maintaining the banded are very slow for the kind of bandwidths we need; the Householder or Givens processes are much faster, but they require one to store the full matrix, as the banded structure is not maintained. What is needed, and what does not exist, is an algorithm for ‘cutting’ a band matrix into two, i.e.

the second alternative being preferable as it maintains the symmetry of the two factors, thereby easing further processing, but this is not essential: a neat advantage would be that most of the important internal relaxation eigenvalueswould end up in one of the factors, and dfl-lin the other; thus one would only need to process one half of the eigenvalue problem in multiplelength arithmetic, once a suitable factoring had been achieved. There is another approach to this problem which could be made to work at very low temperatures: that one does not symmetrize the relaxation matrix in the conventional 15* because it is almost Hessenberg in form before symmetrization;I7* the same arithmetical accuracy would be required, but the reduction to true Hessenberg form might be sufficiently fast to be acceptable. Successful calculations at these low temperatures would also require greater exponent lengths than are commonly available on commercial computing machines at the present time. 81p

173

174

J . H. Wilkinson, ‘The Algebraic Eigenvalue Problem’, Oxford University Press, Oxford, 1965. T. Carrington, unpublished comments.

Network Eflects in Dissociation and Recombination of a Diatomic Gas

289

Network Effectsin Chemical Physics.-The emergence of well-defined network effects in these calculations should warn us not to overlook their importance in other areas of chemical physics which are, as we noted earlier, rather closely related to this one. We have seen that the experimental rate of dissociation for H2 can be reproduced quite easily and naturally using a full rotation-vibration-dissociation coupling model. On the other hand, if one reduces the order of the problem from about 170 equations to 16 equations by approximating it to only a vibration-dissociation model, the rate of dissociation falls considerably, using the same schemes to assign transition p r o b a b i l i t i e ~ . ~ ~ ~ ~82 It is nevertheless possible to reproduce the experimental rate of dissociation and the vibrational relaxation time at the 16equation level simply by distorting the bound-continuum transition probabilities by a few powers of ten.8 Moreover we can simplify the model even further to 3 equations-what is usually called the two-state model and which at one time was often used in qualitative discussions of experimental shockwave dissociation results,175as well as Even at this level of approximation,it is perfectly possible to imitate the observed reaction rate and induction period by suitable choice of the transition probabilities between the two bound states themselves, and between them and the continuum: after all, if we only observe two characteristic times in the experiment (the induction time and the reaction rzte) we only need three eigenvalues (one of which is zero to conserve the mass of the system) and therefore only three equations to reproduce these two observations. However, in the transition from the fifteen-state to the two-state model, the whole relaxation behaviour has to be turned inside out and loses all physical reality - now, the induction time is determined by the bound-continuum transition probabilities and the reaction rate is determined by the transition probabilities assigned to the internal 177 The conclusion therefore is that whilst it is perfectly possible to mimic the experimentally observed behaviour (to the accuracy with which the observations can be made) using the two-state model, there can be no physical significance whatsoever attached to the transition probabilities needed to bring the two-state model calculations into coincidence with the experimental rates. The same conclusions would apply to the recent suggestion 178 that the thermal shock-wave isomerization of cyclopropane can be described by a threelstate model - so it can, but again, no physical significance can be attached to the set of ‘rate constants’ which must -be chosen to enable the model to mimic the observed rates. Presumably, it should be possible to throw some light on the breakdown of the physical reality in these over-simplified models in terms of Boyd’s lumping analysis for coupled relaxation processe~:~*~~ 17s however, we will probably find that 55h9

177 178

I79

789

E. S. Fishburne, J . Chem. Phys., 1966, 45, 4053. L. P. Leibowitz, Phys. Fluids, 1973, 16, 59. D. L. S. McElwain, L. Wagschal, and H. 0. Pritchard, Phys. Fluids, 1970, 13, 2200. E. A. Dorko,q.W. Crossley, U. W. Grimm, G. W. Mueller, and K. Scheller, J . Phys. Chem., 1973, 77, 143. R. K. Boyd, Canad. J . Chem., 1971,49, 1401.

290

Reaction Kinetics

acceptable physical reality of the models will emerge only as the number of states becomes sufficiently large, and concomitantly the problem becomes illconditioned in the sense that it is no longer possible to derive information about individual transition probabilities from the observed relaxation times.g2 We note here also a recent similar warning on the use of simple analytic approximations in a related field, that of nucleation of water droplets on ions.33 Another area in which we should proceed warily is in the interpretation of laser relaxation processes - both in the description of the relaxation processes within the laser system and also in the relaxation of object systems excited by laser radiation. It is common to discuss such relaxations in terms of rather few levels which one knows are definitely involved in the relaxation p r o c e s ~ , ~and ~ ~at- the ~ ~ same ~ time to superimpose on such models, aq we have already noted, assumptions of nearest-neighbour-only transitions and conservation of vibrational l a 4 Our experience in this work would suggest that all three of these approximations are likely to be dangerous, although the first is likely to be the most troublesome, and evidence of this trouble is beginning to appear.185 As we have just seen, the whole relaxation behaviour can be misinterpreted by using too simple a model with too few states, or else distorted in devious ways if one truncates a system of levels carelessly, even though the point of truncation is far away from the levels of interest :55j the examples to which we have referred involve only simple translation to internal energy-transfer processes, but one must also consider that at reasonable pressures some molecules may rapidly achieve very high energies in relatively few V-V or VR-VR l a 6at the same time, as we saw in Table 3, different assumptions in respect of the nature of the T-VR transition probabilities themselves have subtle effects on the calculated relaxation behaviour. Consequently, as in the ,dissociation case which is by now fairly well understood, we may well be’in danger of being able to mimic laser and laser-induced relaxation processes very accurately - with ‘lumped’ transition rates which, although related to, are very unlike the real rates of the processes they are supposed to represent; it is probably true in fact that the only case in which one can cavalierly ignore levels above those one knows to be involved in the relaxation is in the treatment of ultrasonic rotational relaxation in gaseous hydrogen, and then only at low temperatures.

180

J. I. Steinfeld, 1. Burak, D. G . Sutton, and A. V. Novak, J. Chem. Phys., 1970, 52, 5421.

C. T. Hsu and L. D. McMillen, J. Chem. Phys., 1972,56, 5327. le2 P. F. Zittel and C. B. Moore, J. Chem. Phys., 1973, 58, 2922. l S 3 J . Stone, E. Thiele, and M. F. Goodman, J. Chem. Phys., 1973, 59, 2909. lS4 B. A. Ridley and I. W. M. Smith, J.C.S. Faraday ZZ, 1972, 68, 1231. lS5 J. T. Knudston and G . W. Flynn, J. Chem. Phys., 1973, 58, 1467. lS6 S. H. Bauer, D. M. Lederman, E. L. Resler, and E. R. Fisher, Znternaf. J . Chem. Kinetics, 1973, 5, 93.

7 Recent Advances in the Analysis of Kinetic Data BY A. JONES

1 Introduction The object of much experimentation in chemical kinetics is to build a mechanistic model of a reaction process with, usually, the additional aim of determining more or less precise estimates of the unknown reaction rate constants. Over the recent past there have been significant developments in the tools available for data analysis, and in considering the impact of these developments on chemical kinetics it is useful to formalize the modelbuilding process. In describing this process we will separate the activities of the chemist, e.g. experimentation, and the activities of the data analyst, e.g. solving a set of ordinary differential equations. In practice, of course, all the activities may have to be carried out by the same person. The Model-building Process.-The model-building process (Figure) starts with the chemist postulating a chemical model for a particular phenomenon. Typically this will consist of a set of reaction equations together with a range of experimental conditions over which the model is expected to be valid. In particular, the parameters in this model (e.g. reaction rate constants) should remain constant when the experimental conditions are changed. At this stage the data analyst can be brought in and, together with the chemist, he can now formulate the equivalent mathematical model. They can take the opportunity of computing some predictions from this mathematical model to see whether they are reasonable and in accord with prior knowledge. These calculations are frequently useful in showing up gross model deficiencies. The second job of these two people is to define clearly the purpose of the next stage in the process, which is normally the design of an experiment to test the adequacy of the model. We do not distinguish, at this stage, between the chemical model and the mathematical model, as normally these terms can be treated as synonymous. However, before the advent of highspeed, general-purpose computers these models were often very different. The lack of suitable data-analysis techniques often meant that the mathematical model had to be simplified to make it tractable. Nowadays this is only rarely necessary and we are able to deal with more complicated models that include many more variables. In those cases where simplification is still necessary (in one sense it is always necessary - we can never include all the possible chemical reactions and species for instance) the appropriate changes 29 1

292

Recrction Kinetics

rc

EXPERIMENTER

AND DATA ANALYST

DEVISE NEW MODEL AND EXPERIMEN TAL DESIGN

MODEL

t L

MODEL TO DATA ESTIMATION OF PARAMETERS

t INVESTIGATE DEVIATIONS O F MODEL PREDICTIONS FROM EX PE RIMENT

1 MODEL I NADEQUATE

OBTAIN PRECISE ESTIMATES OF PARAMETERS I N FINAL MODEL

Figure Flow diagram for model development

should be made to the chemical model in order that their physical significance can be seen clearly. The data analyst and chemist can now decide the number of experiments required to test the adequacy of the model, the experimental conditions under which they are to be performed, and, if relevant, the order of testing. The next stage in the process is to ‘fit’ the model to the experimental data. In all models there are unknown or imprecisely known parameters and the best we can do is to choose values for these parameters that allow the model to fit the data as closely as possible. This is usually done by minimizing the sum of squares of the deviations of the predicted values from the corresponding experimental values. The final step in the model development process is the test of the adequacy of the model in explaining the phenomenon. Information about the inadequacy of the model can be obtained by analysing the deviations of the model predictions from the experimental results, Any relationships between the deviations and the experimental variables can prove useful in pinpointing

Recent Advances in the Analysis of Kinetic Data

293

deficiencies in the chemical model. Another useful procedure is to study the parameter estimates obtained under differing experimental conditions. For instance, a particular rate constant may have been assumed to have a simple Arrhenius temperature dependence. This assumption can be tested by determining the activation energy and the pre-exponential factor over several different temperature ranges and checking the parameters for any further temperature dependence. Inadequacies found in this last stage will normally lead to a new model, which has to be tested with a new set of experimental data. Usual changes introduced at this stage are the introduction or deletion of species and/or reactions from the mechanism. Other assumptions about physical effects can also be inserted or deleted at this stage. The cycle of model postulation, experimentation, data analysis, and final adequacy testing is repeated until the experimenter is satisfied that the model is sufficiently well developed for his purpose. As with all applications of the scientific method, no model can be proved to be correct; the most one can say is that a particular model explains the set of experimental data to within deviations which can be ascribed to experimental error, If the purpose of the experimentation was to elucidate the chemical mechanism the job is finished. However, if the purpose was to obtain precise estimates of certain imprecisely known parameteFs a further stage of data analysis is required. This can involve fitting the model to all the data obtained during the course of the work and may involve further experimentation to improve the precision of these parameters. It is important that the precision of the parameters is calculated and is reported unambiguously. Having described the process in some detail we can identify the techniques of data analysis that are required. These techniques come from the two fields of numerical analysis and statistics. There are differing opinions about whether a particular technique is statistical or numerical in nature and that is the reason why the phrase ‘Analysis of Kinetic Data’ has been used in the title for this Report. Thus there is a problem for the chemist who wants to keep abreast of recent developments, for he has to look at the literature in both areas and each one of these fields is probably as big as his own special interests in chemistry. Computing Predictions from Rate Equations.-The first requirement that we can identify is the ability to compute the predictions of the mathematical model. Since chemical kinetic models very often lead to sets of ordinary differential equations which describe the dependence of species concentration on time we can identify this requirement with the ability to solve these differential equations quickly and with reasonable accuracy. It is sad that only the differential equation representing the simplest reaction mechanisms can be solved analytically. For the most part we are forced to use numerical techniques. Further, most chemists who have attempted to use classical techniques such as Runge-Kutta methods to integrate their rate equation

294

Reaction Kinetics

will have encountered the phenomenon aptly described as the ‘stiffness’ problem; its manifestation is an inordinately large amount of computer time to solve a relatively straightforward problem. In the past some workers have overcome the stiffness problem by identifying certain species in the reaction scheme which could be considered as being in steady state; this identification was performed manually or sometimes automatically in a computer program. Computing the concentrations of these species from the non-linear equations rather than from the corresponding differential equation removed the difficulty and computing times fell to more usual values. However, this device cannot be recommended as a general-purpose procedure because it introduces the assumption that species in steady state at some time in the reaction process remain in steady state, an assumption that is not true in general. Further, the development of new implicit techniques fbr solving ordinary differential equations has removed the need to invoke devices of this sort. The new techniques appear to be able to solve large sets of differential equations rapidly and with reasonable accuracy. Various algorithms based on implicit methods have been published, some of which are readily available as computer subroutines. Fitting Models to Data.-The second easily identified requirement is an algorithm for fitting models to experimental data by adjusting the parameters in the model. For some years several algorithms have been available for computing least-squares solutions to these problems, and in the special case where the predictions of the model depend linearly on the unknown parameters, i.e. when the derivatives of the model predictions with respect to the unknown parameters do not depend on the parameter values, the procedure has been known for a long time - it is multiple regression. In the non-linear case recourse has to be made to iterative techniques and perhaps the most well-known algorithm that has been in use over the past ten years is that due to Marquardt.’ When the algorithm worked it worked well and reasonably quickly. However, for certain model/data combinations the algorithm did not work in the sense that convergence was very slow, and moreover alternative algo ithms that were available behaved in a similar manner. The difficulty could be traced to a lack of information in the experimental data or, to put it another way, to an attempt to find an estimated value for a parameter, or a function of the parameters, which had no influence on the predicted values as far as that set of experimental data was concerned. All these parameter-estimation techniques were based on the idea of making successive linear approximations to the model about a set of estimated parameter values and then using a multiple regression approach to compute adjustments to the parameter values. The lack of information in the data caused the matrix in the multiple regression to become almost singular and thus lead to very inaccurate adjusted parameter values. Some recent developments for handling singular matrices have led to new ideas for algorithms D. W. Marquardt, J . SOC.Indust. Appl. Math., 1963, 11, 431.

Recent Advances in the Analysis of Kinetic Data

295

that do not suffer from these deficiencies. Many authors have sketched algorithms based on the same idea, and some computer subroutines are available. The above two requirements are discussed in greater detail in Sections 2 and 3. We conclude this introduction with some brief remarks on sensitivity analysis, confidence limits, and model discrimination. Sensitivity Analysis.-In the initial stages of model development it is important to obtain a measure of the sensitivity of each dependent variable of the model to changes in the parameters. For example, this analysis can help the experimenter decide which concentration profiles he should monitor in order to be able to estimate the rate constants of interest. It also helps in pinpointing those rate constants whose values do not affect the concentration profiles in the experimental region and so may be held fixed at reasonable values in all the computations. Even with the aid of high-speed computers this work can become very time-consuming and tedious, but it appears that useful results can be obtained with the aid of Fourier analysis. The basic idea is to modify each parameter by a sinusoidal component of a chosen frequency. The rate equations are then integrated and the sensitivities can be determined from the Fourier analysis of the solutions. The sensitivity with respect to a particular parameter, averaged over both the linear and non-linear effects of the other parameters, is directly related to the corresponding Fourier amplitude. Details of this technique are given by Cukier et aZ.,2and an example of its use on a kinetic problem is given by Schaibly and S h ~ l e r .A~ similar idea has been reported by Steel.4 Confidence Limits.-If we estimate a single parameter in a model from a set of experimental data it is important to estimate at the same time its precision, for if we estimate the parameter again with a different set of experimental data we shall almost certainly obtain a different value. However, the precision estimate will help in deciding whether the difference between the two is significant. The usual precision estimate is the 95 % confidence limit; this may be interpreted as the interval within which the true value lies with probability 0.95, i.e. there is only a 1 in 20 chance that the true value lies outside this interval. (It is annoying but true that the 1 in 20 chance occurs about once in twenty times, however much we believe that it will never happen to us.) In the case when two parameters are estimated from a set of data the confidence limits must be replaced by a two-dimensional confidence region. Normally it is possible to compute an approximation to this region which turns out to be an ellipse. For more than two parameters the region is an R. I. Cukier, C. M. Fortuin, K. E. Shuler, A. G. Petschek, and J. H. Schaibly, J. Chem. Phys., 1973, 59, 3873. J. H. Schaibly and K. E. Shuler, J. Chem. Phys., 1973, 59, 3879. J. H. SteeIe, Nature, 1974, 248, 83.

296

Reaction Kinetics

ellipsoid in the appropriate number of dimensions. For two parameters a plot of the confidence ellipse can be useful but in higher dimensions it is not worth the trouble to plot projections of the multi-dimensional ellipsoid on all two-dimensional planes. In this case it is better to describe the region by computing the lengths and, perhaps, the directions of the principal axes of the ellipsoid. The size of the ellipse (or hyper-volume of the ellipsoid) depends not only on the experimental values but on the experimental conditions, i.e. the independent variables of the experiment. In designing an experiment to find the best, i.e. most precise, estimates of the parameters it is best to pick those experimental conditions that minimize the volume of the ellipse. Further, by changing the parameters to be estimated by a simple transformation, it is sometimes possible to change the shape and size of the confidence ellipse. Further information on these topics can be found in Draper and Smith’s book.6 Model Discrimination.-The model-development procedure outlined here assumes that there is only one model under consideration. However, very frequently an experimenter may have two or more models in mind from the start. Now the accent changes from testing the adequacy of a single model to discriminating among competing models - conceptually perhaps a simpler problem. Box and Hill have tackled this problem using Shannon’s concept of the entropy of information. They attach to each model the probability that it is the correct model; at the start these probabilities will be the same for all the models unless prior information is available, Then, using Bayes’ theorem, they compute updated probabilities from the original probabilities and the results from a set of experiments. They now seek to minimize the entropy by a suitable choice of a new set of experiments. The process is repeated until one of the models is clearly the winner. A comprehensive discussion of model building and the design of experiments can be found in M. J. Box’s thesis.’

2 Computing Predictions from Models Represented by Ordinary Differential Equations Integration Methods.-In discussing techniques for the numerical integration of the ordinary differential equations representing chemical reactions, it will help if we fix ideas by considering a simple reaction mechanism. The simplest consecutive reaction mechanism is the decay of a reactant A to a product C via an intermediate B : ki

kz

A-tB-tC

(1)

N. R. Draper and H. Smith, ‘Applied Regression Analysis’, Wiley, New York, 1967. G. E. P. Box and W. J. Hill, Technometrics, 1967, 9, 57. M. J. Box, D.Phil. Thesis, University of London, 1969.

Recent Advances in the Analysis of Kinetic Data the species having the following concentrations at time t

297 = 0:

[A1 = “0 [B] = 0 [C] = 0

The corresponding mathematical model is da/dt = -k,a dbldt = k,a - k2b dc/dt = k2b with, at time t = 0, a = l b=O c

(3)

=o

where we have introduced the dimensionless variables a b c

[AI/[Alo IBl/lAlo = [Cl/“*

= =

In this case we can obtain an analytic solution for the species concentrations at time t as

a b c

= exp( -k,t) =

=

kl(exp( -k,t) - exp( -kzt)}/(k2 - k,) 1 - (kz exp( -k,t) - k, exp( -k,t) >,i(kz- k,)

(4)

with the mass balance equation, a+b+c=l automatically satisfied. Suppose that we attempt to solve the differential equations (3) using a numerical integration procedure. It can be shown that in this case each numerical integration procedure is equivalent to evaluating the expressions (4) with the exponential terms approximated by some function p. The form of the function cp depends on the integration method; for example, for the fourth-order Runge-Kutta method (see below) we have exp( -klh)

E

y(k,h)

=

1 - (k,h) f $(k,h)2 - +(k1h)3 +

(5)

where h represents the time step. If a sequence of n time steps is taken, each of length h, the numerical approximation at time t = nh for the concentration of species A is given by a(nh) = exp( -k,nh) E cp(k,h)” (6) The approximation ( 5 ) is a reasonably accurate representation of exp( -k,h) for k,h < 0.1. Thus, if both rate constants in (1) are of the same order of magnitude (say k, = k , = k), then accurate solutions for a, b, and c can be obtained by taking time steps h which satisfy

h < 1/10k

298

Reaction Kinetics

Integration Step Size for Fast Processes.-Suppose now that the rate constant k z is very much larger than k,; then during the induction period, when both exponentials need to be approximated accurately, the size of the numerical integration step is governed by the faster process, i.e. h < l/lOk, However, after the induction period is over, we know that the exponential term representing the fast process is insignificant compared with the term representing the slow process and so we would like to use a step size governed only by the slow process. However, we can only do this if the approximation for the exponential terms representing the fast process decays for a large number of steps; this requires that

IY(hkz)l < f otherwise the term in (6) becomes very large. Now, from the form of cp for the fourth-order Runge-Kutta, it can be shown that this condition holds only if hk, < 2.0 Thus, even though the actual value of exp( - k d ) is small in this postinduction region, the approximation as calculated by the Runge-Kutta method is not necessarily small and in fact is increasing in magnitude unless the step size satisfies the above condition. The effect of these considerations is that the differential equations representing chemical reactions can be solved using the fourth-order Runge-Kutta but o ~ l yusing an integration time step the size of which is controlled by the fast reaction even in the post-induction period when the fast reaction is relatively unimportant from the point of view of the chemistry. In practice this means that the integration time step that is used is essentially the same over the whole of the reaction time and this can lead to an enormous and often impracticable number of integration steps. Now let us examine the effect of using the trapezium rule for integrating the differential equations (3). In this case the form of the function cp is given by I -- t h k T(hk) = 1 3hk

+

i.e. cp is the ratio of two first-degree polynomials. During the induction period it can be shown that the trapezium rule is equivalent to the fourthorder Runge-Kutta in efficiency. From the form of 9 we can see that as the step size becomes large the value of cp remains bounded. In fact, for large h,

T(hk) 21 -1

+ (4/hk)

so that in the post-induction period there is no restriction imposed on the step size because of the nature of the approximation function. The integration time step can be increased to a size that is appropriate to the accuracy required to follow the slow process.

299

Recent Advances in the Analysis of Kinetic Data

Implicit Integration Methods.-All explicit methods* for solving differential equations of this type suffer from the same disadvantage as the fourth-order Runge-Kutta. This is because they all have approximating functions ‘p which are polynominals. Implicit methods on the other hand have functions ‘p which are ratios of two polynominals and so behave in roughly the same manner as the trapezium rule. In order to explain the difference between explicit and implicit methods let us consider the solution of the single differential equation dridt = f ( Y ) (7) with y =yoatt = O

Suppose that we have computed the solution up to lime t = nh and that y(nh) = y , ; then an explicit one-step method gives the solution at time t = (n 4- l)h in terms of y , andf(y,). For example, the Euler method is

and the fourth-order Runge-Kutta is yn +

1 =

y,

+ 6(ko + 2k, + 2kz + k3)

where ko = hf(y,) kl = hf(Y%+ +ko) kz = M Y , + W I ) k , = hf(y, kz)

+

An implicit method gives the solution at t = n 1 h in terms of y,, f ( y , ) , and the unknown value f ( y , + t). For example, the simplest implicit method is the Implicit Euler, Y, +

1 =

Y,

l-

hf(& +

I)

while the trapezium ruIe is

Now in these cases we cannot evaluate the expressions directly because we do not know the value ofJ’(y, + 1), so that the equation only implicitly defines the value of y,, + 1. When the function f is linear in y the equation can be rearranged and solved for y ; otherwise iterative methods must be used to find the value of + 1. In the context of chemical rate equations we have a number of differential equations to solve simultaneously, each of which may be non-linear in the concentration variables. Thus the use of implicit methods implies the solution of a set of simultaneous non-linear equations at each integration

*

Further details o f most o f the methods described here can be found in ref. 8.

C. W. Gear, ‘Numerical Initial Value Problems in Ordinary Differential Equations’, Prentice-Hall, New Jersey, 1971.

300

Reaction Kinetics

step. Although this means that the cost of computing is high for each time step, the greatly increased size that can be used with implicit methods more than compensates for this disadvantage. In fact, practical problems have been solved recently that could not have been attempted without the use of implicit methods (see, for examples, refs. 9 and 10). Prothero and Robinson l 1 have described a useful method which is based on the following variant of the trapezium rule: In this case for large h the function CF tends to the limit ( p - l)/p, and for = 0.55 this gives a value of 911 1. From the value of this limit, which is smaller in magnitude than the corresponding limit for the trapezium rule (B = 0.5), and from equation ( 6 ) it can be shown that this method can use larger steps in the post-induction period for those cases where the ratio of the largest to the smallest rate constant is large. This method is almost as accurate as the trapezium rule during the induction period but it can allow larger steps to be taken during the post-induction period. Halstead, Prothero, and Quinn have made use of this algorithm in their work on cool flames.

p

The Phenomenon of Stiffness.-The phenomenon that has been described here is generally known as stiffness. Its manifestation is that when the differential equations representing chemical reaction* are solved using classical explicit methods the computer times seem inordinately long o r the computer programs get ‘hung-up’ and the integration process does not move forward in time. These are the symptoms of stiffness when the step size of the integration process is being controlled by reference to an accuracy criterion. When no check is made in accuracy and a constant step size is used almost anything can happen (and frequently does !) ; in particular, widely oscillating and/or infinite solutions can be obtained. Stiffness was first referred to as long ago as 1952 by Curtiss and Hirschfelder,12 who also diagnosed that the source of trouble lay in the large differences that frequently exist between the rates of the fast radical reactions and the relatively slow overall reaction rate. Since that time a large number of papers have been written about the subject and for a long time the only really successful method for dealing with very stiff problems was to make use of steady-state approximations. One such method was the basis of a computer program published in the very useful series of books with the title ‘Computer Programs for Chernistry’.l3

* The phenomenon also occurs in control theory and other areas where the behaviour of a dynamical system is represented by a set of differential equations. lo

l1 l2 l3

M. P. Halstead, A. Prothero, and C. P. Quinn, Proc. Roy. Soc., 1971, A322, 377. J. S. Chang, A. C. Hindmarsh, and N. K. Madsen, Proceedings of the International Symposium on Stiff Differential Equations (IBM Germany), 1973, to be published. A. Prothero and A. Robinson, Math. Comp., 1974, 28, 145. C. F. Curtiss and J. 0. Hirschfelder, Proc. Nar. Acad. Sci. U . S . A . , 1952, 33, 235. ‘Computer Programs for Chemistry’, ed. D. F. De Tar, W. A. Benjamin, New York, Vol. I, 196‘,’, Vol. 11, 1969.

Recent Advances in the Analysis of Kinetic Data

30 1

In this program the user has to decide a priori which species are to be considered as being in steady state. The differential equations for these species are replaced by algebraic equations obtained by equating the rates of their formation and removal. The set of differential equations can now be solved together with the algebraic equations by standard methods. This technique eliminates the fast processes from the system and with them goes much of the stiffness inherent in the use of explicit methods. Two other methods have been published which do not suffer from the disadvantage of having to make the steady-state assumption over the whole reaction time. The integration of the differential equations proceeds normally during the induction period and thereafter species are identified as being in steady state when the concentration flux satisfies either an analytical condition l4 or a numerical condition.16 Both these methods give complete numerical solutions in a reasonable time. However, the methods are based on the assumption that a species once in steady state remains in steady state. It does not seem possible to guarantee that this should be so, e.g. in branchedchain reactions l6and for models including bulk effects (heat/mass transfer for instance). This general point has been supported by Gelinas l 7 and in particular by Edelson,18who point out that once steady-state procedures are invoked chemical balance restrictions are lost, and although the errors in stoicheiometry are small the error in a particular component concentration may be significant, and this can then propagate throughout the system. Steady-state methods then cannot be recommended for general use. They may still have a part to play in solving particular problems but the assumptions and the implications of the assumptions need to be clearly stated.

A Multi-step Implicit Integration Method.-The answer to the stiffness problem appears to lie in the use of general-purpose implicit integration procedures, thus eliminating the need to make steady-state assumptions in cases where their validity is not fully justified. A major advance in the development of implicit methods was made in 1968 by Gear when he published a new method which was based in part on earlier work by Nordsieck.20 Gear’s original paper was presented at the 1968 Congress of the International Federation for Information Processing at Edinburgh and since that time has received much attention from numerical analysts and program writers and users. The method is a multi-step predictor-corrector, i.e. it relies on retaining values of y and f ( y ) for a number of successive steps and uses one formula to predict the solution y R+ 1, which is then ‘corrected’ l4

F. A. Melamed, L. S. Polak, and Yu. L. Khait, Trans. Faraday SOC.,1968,64, 1877.

l5

R. H. Snow, J . Phys. Chem., 1966,70,2780.

l6

2o

0. K. Rice, J . Phys. Chem., 1960,64, 1851. R. J. Gelinas, J. Computational Phys., 1972, 9, 222. D. Edelson, J. Computational Phys., 1973, 11, 455. C. W. Gear, ‘Information Processing 68’, ed. Morrell, North Holland, Amsterdam Vol. I, p. 187. A. Nordsieck, Math. Comp., 1962, 16, 22.

React ion Kine tics

302

using an implicit formula. We can write down the structure of the method 21 for solving a single ordinary differential equation (7). Suppose we have the solution values y C = u(ih) at the times t = ih,wherei = n,n - 1, ..., n - p 1

+

Then the ‘prediction’ is given by Yn+

1

= aIYn

+ azYn - + - + 1

*

*

apyn

-

p+ 1

+

ap

+ lhf(Yn)

and the implicit corrector formula is Values of the coefficients a and B are given in ref. 19, together with the method for estimating the error in y involved in computing y , + from these formulae. Gear has published his method in algorithm form in FORT RAN.^^ The algorithm (DIFSUB)’ in fact contains two integration procedures, one designed for stiff systems of differential equations and the other designed for non-stiff systems. The main features of the algorithm are that it starts automatically, i.e. the user does not have to generate the first few solution points by another method, as is common with some predictor+orrector methods; it is of variable order, i.e. the number p of solution points that are retained is chosen within the routine and is changed when circumstances demand it; the step is variable and is controlled by the local accuracy required by the user. A view is sometimes expressed that variable step-size routines are the answer to stiff systems of differential equations ; the ideas- presented here should dispel this view : variable step-size routines are important for improving the efficiency of the solutions of stiff equations but only the implicit methods are efficient anyway. Another method similar to Gear’s has been published by Brayton et aZ.25 The method is described in flow-diagram form and so is not so conveniently available as is Gear’s. However, it is claimed to be better than Gear’s in certain respects. Recent work on the theory of methods for solving stiff equations has been reported by Prothero and Robinson.” Further, an international symposium was convened in Frankfurt by IBM Germany during the summer of 1973 and was devoted entirely to stiff equations. Several important papers were presented at this meeting and reference has already been made to the paper of

* Gear

has also described 23, 2 4 a comprehensive computer program package which is based on this algorithm. 21

l2 23

zp

25

ODESSY

C. W. Gear, Comm. A.C.M., 1971, 14, 176. G. W. Gear, Comm. A.C.M., 1971, 14, 185. ‘Mathematical Software’, ed. J. R. Rice, Academic Press, New York, 1971, p. 211. C. Dill, C. E. Ellis, C. W. Gear, and K. Ratcliff, University of Illinois, Dept. of Computing Science, Report 779, 1968. R. K. Brayton, F. G. Gustavson, and G. D. Hatchell, Proc. ZEEE, 1972,60,98.

Recent Advances in the Analysis of Kinetic Data

303

Chang et aZ.1° They used Gear’s method to simulate chemical kinetics/ transport phenomena in the stratosphere. Other papers of interest to chemists were also presented. Edsberg 26 described a computer package for integrating chemical rate equations based on an implicit mid-point rule with extrapolation due to Lindbe~-g.~’ This is one of the more useful packages for the chemist since only the reaction mechanism has to be supplied and the generation of the corresponding differentialequations is automatic. Gourlay and Watson 2 8 describe the implementation of Gear’s method in the IBM Continuous Systems Modelling Program, which will be useful for simulations of systems including kinetics. Lapidus et aZ.29also discuss the stiff methods available with particular reference to applications in kinetics, reactor dynamics, and transport phenomena. Hull’s paper 30 on a comparison of the latest methods of solving stiff systems shows that Gear’s method is among the best available.

3 Least-squares Fitting of Non-hear Models We now turn to the problem of fitting a postulated model to a set of experimental data.* For a first look at the problems involved, we will again take as an example the simple reaction scheme kr

ki

A-+B<-+C If we had available experimental values for the concentration of species A at various reaction times, we could make an estimate of the rate of the

first step in the reaction by plotting log [A] against time and fitting a straight line to the data. On the other hand, the experimental situation may be that the concentration of species B is measurable but that of A is not. Then, because of the way in which the concentration of B depends on the rate constants, there is no simple method of fitting the model to the data over the whole range of reaction times in order to estimate both rate constants. The difference between the two situations lies in the concept of linearity; it must be emphasized here that linearity in this context means whether a model can or cannot be expressed linearly in the parameter values, not in the values of the independent variable. In the first situation the model to be fitted would be [from equation (3)] In a

=

-klt

* An example of the use of a non-linear fitting technique may be found in Hurle, Jones, and Rosenfeld’s bination.

work on the estimation of rate constants for atomic hydrogen recom-

L. Edsberg, in ref. 10; Report NA 72.67, Department of Information Processing, Royal Institute of Technology, Stockholm, 1972. 27 B. Lindberg, in ref. 10. e8 A. R. Gourlay and H. D. D. Watson, in ref. 10. m L. Lapidus, R. Aiken, and Y . A. Liu, in ref. 10. 30 T. E. Hull, in ref. 10. s1 I. R. Hurle, A. Jones, and J. L. J. Rosenfeld, Proc. Roy. SOC.,1969, A310, 253. 26

304

Reaction Kinetics

By a transformation of the concentration variable, y = In a, we would have a straight line y = -klt to be fitted. Here the dependent variable y is a linear function of the unknown parameter k,. In the second situation the model is, from equations (4), b

=

kl[exp( -kit) - exp( -k2t)]/(k2 - k,)

(8)

Here the dependent variable b depends on both kl and kz in a non-linear fashion. Moreover, since the relation between kl and kz at constant b is non-linear, there is no transformation of the dependent variable, which will be a linear function of kl and k,. Techniques for fitting linear models involving one or several parameters have been available for a long time and are discussed in many standard statistical text books. Techniques for fitting non-linear models, however, only came into widespread use during the sixties, and they have received scant attention in the text books.* We shall examine some of the basic ideas in these newer techniques. We may write the model to be fitted in the form where y is the dependent variable to be experimentally observed, b is the vector of parameters to be estimated: bj ( j = 1 ., . q), and x is the independent variable (or a vector of independent variables if there are more than one). If we have a set of experimental observations(yi,xi),i = 1, ..., n, the parameters b may be estimated by use of the least-squares condition

cb c 1

c €2 n

n

s=

- f(b,x3I2 =

(10)

1

We may for the present purposes restrict ourselves to the simplest assumptions, that the experimental errors of the observations y r have Gaussian distributions and are independent of one another and that the error variance does not vary with the level of the observation. If the error variance could not be taken as constant, a weighted sum of squares would be required in place of S. The numerical methods available for locating the minimum of S all require that we can compute the model predictions f, [ = f(b,xi)], i = 1, ..., n for a given trial set of values of the parameters bj, j = 1, ...,q. Some of them also require values of the derivatives

However, it is important to note here that these requirements do not mean that the model must be given in the form of an explicit formula, as in equation (9). It is only necessary that some procedure is available for computing the * However, the discussion in Draper and Smith’s book,5 should be mentioned.

Recent Advances in the Analysis of Kinetic Data

305

predicted values. In chemical kinetics the usual case will be that the mathematical model is a set of differential equations for which we do not have an analytical solution, but for which the predictions can be obtained by the numerical integration procedures discussed in the previous section. The earlier methods used for finding the minimum of S were the Taylor series (or linearization) method and the method of steepest descent. The Taylor Series Method.-In the Taylor series method, the function f is expressed as a Taylor series expansion about the current estimates bo of the unknown parameters, and retaining only first-order terms : f(b0

+

Then the conditions for a stationary value of S, i.e.

as take the form

6

=

0

j

=

1, ...,q

/as\ We can substitute in this equation the expansion for f(b,xi)and so obtain a set of simultaneous linear equations for the vector of corrections, cM,to the current estimates bo of the parameters. We will express these equations in matrix form, using P for the matrix whose elements are

with 1" the transpose of P and g the vector whose elements are

Then the simultaneous equations take the form FP6b

=13E

=g

(1 7)

The procedure is to solve the equations (17) repeatedly, at each step obtaining from the current estimate 5, an improved estimate b , + 66. There are problems for which this procedure converges rapidly to the solution. Unfortunately occasions also arise when the solution of the linear equations gives new values of b which are so far away from 60 that they are outside the range of validity of the linear approximation (12). In such cases the sum of squares S at the new b may be larger than at b, and the method breaks down. Attempts to improve the procedure by adding a fraction of 66 often do no more than slow down the rate of divergence.

306

React ion Kine tics

The Steepest Descent Method.-A completely different approach to the problem of minimizing S is to move along the direction of steepest descent at the point bo. This direction is given by the vector g, which is normal to the contour of S at the point b,. It is always possible to find a reduced sum of squares along this direction unless b , is itself the point at which S is a minimum. Nevertheless algorithms based on this idea alone have not proved successful in practice, the reason being that the steps that can be taken along the steepest descent direction become so small that convergence is prohibitively slow. The restriction on the step that can be taken is a function of the nonlinear nature of the model. Marquardt’s Algorithm.-We turn now to the more recent methods for minimizing the sum of squares for non-linear models. Probably the most successful and widely used algorithm over the past ten years has been that due to Marquardt.l The main feature of his method wa3 to modify the linear equations for the correction vector 6b by adding constants to the main diagonal of the PP matrix. Writing A for PP in (16), we have the linear equations A86 = g (18) and in place of these Marquardt adds a multiple of the unit matrix I to give (A AZ) 6b = g (19)

+

Within an iteration the value of A is increased until the resulting value of the sum of squares S(bo 66) is smaller than S(bo). This has the effect of reducing the length of the correction vector 6b, and at the same time swinging the vector away from the Taylor series direction ( A = 0) towards the steepest descent direction ( A -+ 00). In starting a new iteration the value of A is reduced, and so A will be progressively reduced as long as the sum of squares continues to decrease. In this way the rapid convergence properties of the Taylor series method can be obtained once we are close enough to the minimum for the effects of non-linearity to be small. The publication of Marquardt’s algorithm, and of the computer program,3z was a significant advance in the techniques of parameter estimation since it increased the range of problems that could be solved. A class of problem that previously could be solved by the Taylor series method only if a close approximation to the solution was already known now became straightforward. Many variants of Marquardt’s method have been published (e.g. by Fletcher 33), and although some of these are computationally more efficient than the original algorithm, there was no great increase in the range of problems that could be solved. There have also been published algorithms based on different i d e a ~ , 35 ~ *but ~ again without marked effect on the range of problems.

+

32 33 34

35

D. W. Marquardt, Share Program Library # 3094, 1966. R. Fletcher, A.E.R.E. (Harwell) Report #R6799,1971. M. S. D. Powell, Computer J., 1965, 7 , 303. G. Peckham, Computer J., 1970, 13, 418.

Recent Advances in the Analysis of Kinetic Data

307

GeneraIized Inverses.-There is one recent computational advance which deserves special mention. It is based on the fact that a solution of the system of equations A6b=g (1 8) can be obtained directly by computing the least-squares solution of the overdetermined system of equations P6b=&

(20)

There are two computational advantages to be obtained from using this approach. First, as there is no need to compute the matrix product PP, there is a straight saving in computer time. Secondly, the accuracy of the solution to a set of linear equations depends on the ratio of the largest to the smallest singular value of the matrix of coefficients.* Since the singular values of PP are the squares of the singular values of P, the equatioiis (20) are better conditioned and can give numerical results of greater accuracy. In practice we have found that single-precision solutions to (20) are adequate for many problems that require double precision when solved using (18). The solution to equation (20) can be expressed as a sum of contributions, each of which depends on one only of the singular values. This sum takes the form

where the Mdare a set of matricest and the scalars Ad are the singular values of P. Clearly if any of the .Zi are very small then 6b becomes very large and the length of the Sb vector depends strongly on the size of the smallest 1;. Thus we have a direct method of controlling the size of the correction vector 66 by deleting from the summation all terms corresponding to singular values below a certain value. Although Golub and Reinsch36have published an algorithm for the solution of an over-determined system ofequations, this has not yet been incorporated into a generally available program for non-linear estimation. Similar ideas based on generalized inverses of rectangular matrices have been put forward by Marquardt s 7 and by Decell and Speed,38 who have published some results obtained with their computer program.

* Singular values of a matrix are defined in terms of the factorization of the rectangular matrix in the form UA where U and V are orthogonal matrices and A has non-negative elements only on its leading diagonal. These diagonal elements of A are the singular values of the original matrix. -f In matrix form we can write this as

66 = V A - l O ~ where A-' is a diagonal _matrix with entries hi-'. Only the non-zero &values are included here, The matrix VA-lcI is a generalized inverse of P. 36

37 38

C. H. Golub and C. Reinsch, Numer. Math., 1970, 14, 403. D. W. Marquardt, Technometrics, 1970, 12, 591. H. P. Decell, jun. and F. M. Speed, NASA Report CR-2119, 1972.

308

Reaction Kinetics

These look very promising and it is clear that new algorithms on these lines will become available in the near future and allow a further extension to the range of problems that may be tackled. Until that time Marquardt-type algorithms such as published by Fletcher 33 will continue to be the best available computer programs.

8 Ki netics of Oscil latory Reactions BY B. F. GRAY

1 Introduction

Over the past decade oscillating chemical reactions have been very actively researched, many times more publications appearing on the subject in this period than in the previous half century. The experimental characterization of easily reproducible oscillating reactions in closed systems independently in the fields of biochemistry on the one hand and combustion on the other, followed by work on the Zhabotinskii reaction, removed any remaining misgivings that these phenomena were artefacts due to the interaction of chemical kinetics with some non-chemicalrate-determining factors. Although many of these systems are still not fully understood from a mechanistic point of view, they are all known to possess features shown theoretically to be necessary for the occurrence of oscillations. Previous workers have reviewed oscillating reactions from various points of view. Hedges and Myers wrote a monograph in the 1920’s covering the early work in the field, including spatially distributed periodicities such as Liesegang rings. Higgins reviewed oscillating reactions from the standpoint of biochemistry, largely paying attention to feasibility questions, Hess and Boiteaux reviewed the experimental data on oscillating reactions collected from biochemical systems. Degn * presented a concise and elementary review of the chemical kinetic point of view, and Nicolis and Portnow have most recently written a very broad-ranging article including many non-kinetic and somewhat speculative aspects of oscillations, and some thermodynamic comments. The purpose of the present chapter is to deal with the purely chemical kinetic aspects of oscillatory reactions, both experimental and theoretical. Thus we shall first examine the experimental findings on individual reactions (and groups of reactions where they are sufficiently similar), aiming to extract as much kinetic information as can be formulated in terms of differential equations with a phenomenological basis. We shall then review the general theoretical problems presented by oscillating reactions and develop the mathematical techniques used in their analysis, i.e. the various forms of sta-

3

E. S. Hedges and J. E. Myers, ‘The Problem of Physicochemical Periodicity’, Arnold, London, 1926. J. Higgins, Ind. andEng. Chem., 1967, 59, 19. B. Hess and A. Boiteaux, Ann. Rev. Biochem., 1971,40, 237. H. Degn, J. Chem Educ, 1972,49, 302. G . Nicolis and J. Portnow, Chem. Rev., 1973,73, 356.

309

310

Reaction Kinetics

bility theory, largely developed by the Russian school of mathematicians and which chemists have been strangely slow to use. Finally the mathematical techniques presented will be applied to as many individual examples as have been sufficiently well characterized to make detailed analysis worthwhile at present. 2 The Experimental Facts The Decomposition of H202.-This reaction is probably the earliest reported allegedly homogeneous oscillating reaction.6 The original experiments were performed on a mixture of hydrogen peroxide, potassium iodate, and sulphuric acid at 60 " C ;the vessel was continually agitated by shdking, with a view to preventing the formation of supersaturated solutions by the oxygen evolved. Two global reactions taking place are

5H2O2+ I2 = 2HI03 + 4H20 and

+ 2H103

=

502

2H202

=

2H20

5H202

+ I2 + 6 H 2 0

the net result being

+

0 2

Caukins and Bray measured the rate of oxygen evolution and found it to oscillate, along with the iodine concentration. In an attempt to eliminate the bubbling of the gas from the solution they performed experiments at 25 "C, where the reaction rate is too slow to form bubbles and the gas presumably diffuses across the interface into the gas phase. The persistence of oscillations in the I2 concentration convinced them that supersaturation was not the cause of the oscillations. Earlier than the above work, Morgan reported oscillations in the rate of evolution of carbon monoxide from formic acid on dehydration with sulphuric acid, and also in the rate of evolution of carbon dioxide from the dissolution of calcium carbonate in acetic acid. He proposed that the former was due to a supersaturation effect as the oscillation disappeared on vigorous agitation, the monotonic rate being the mean value of the oscillatory rate. His explanation is supported by the observations of Findlay and King * on the relaxation of supersaturated aqueous solutions of C 0 2 . On relaxing the gas pressure to one atmosphere the solution degases in an oscillatory manner when unagitated, but monotonically when shaken. However, Morgan also found that the purer his formic acid, the less likely he was to get oscillations, although his highly pure acid could be induced to oscillate by impurities in general and alcohols in particular. This does not seem easy to reconcile with the super-

'

W. C. Bray, J . Amer. Chem. SOC.,1921, 43, 1262.

J. S. Morgan, J. Chem. SOC.,1916, 109,274.

A. Findlay and J. King, J. Chem. SOC.,1913, 103, 1175.

Kinetics of Oscillatory Reactions

311

saturation theory, although subtle surface tension effects cannot be ruled out. Equally the effect of agitation is difficult to explain on the basis of a homogeneous chemical theory (in fact impossible since the equations used to describe any theory of this type cannot include the effects of stirring since they implicitly assume this is being done perfectly in the first place). The above dilemma occurs frequently in systems of this sort and in particular it reappears in later work on the hydrogen peroxide decomposition. Rice and Rieff Oa were next to study this decomposition, under both catalysed and uncatalysed conditions. They synthesized their own hydrogen peroxide and found no oscillations in either the catalysed or uncatalysed reactions. The addition of small amounts of inhibitors produced oscillations, the oscillatory reaction being considerably slower than the non-oscillatory one. They suggested therefore that the oscillations were absent from the pure system and that when they occurred they did so as a result of inhibitors and heterogeneous reaction on dust particles. A significant similarity with Morgan's work is that in both cases highly pure systems did not oscillate and deliberate addition of impurities reproduced the oscillations. Peard and Cullis10 reopened the questions about the experimental behaviour in the post-war period. They paid considerable attention to the supersaturation question and remarked that oxygen has a strong tendency to supersaturate; they also gave references showing the inefficiency of rotatory stirring in eliminating supersaturation. They used a method which gives the reaction vessel a violent shock every second and showed that this gives results in agreement with those of vibratory agitation (this agreement is not in itself proof that the resulting solution is not supersaturated). They found oscillations in the gas evolution rate and in the iodine concentration (by visual observation) in the agitated solution, but in the unagitated case no oscillations appeared and the rate of gas evolution was much lower than the mean of the oscillatory value, in complete contrast to Morgan's results in the formic acid reaction. They pointed out that iodine is fairly volatile at 60 "Cand observed a significant amount in the gas phase above the solution after the maxima in the oxygen evolution rate. They suggested this as the cause of the periodicity, but this had already been ruled out by the work of Bray and Caukins, carried out at 25 "C,in so far as physical removal from the reactive phase is concerned. The increase in mean rate on agitation seems to point to a significant reverse reaction in solution involving oxygen itself. In 1967 Degn l1 performed some experiments on this reaction system. He used mechanical stirring in spite of earlier comments on its ineffectiveness in preventing supersaturation. He found that it did not influence the reaction, simply serving to remove oxygen bubbles from the walls of the vessel, thus preventing their interference with spectrophotometric measurements.

lo

l1

( a ) F. 0 . Rice and M. Rieff, 1.Phys. Chem., 1927,31, 1352; (b) D. N. Shaw and H. 0. Pritchard, ibid., 1968,72, 1403. M. G. Peard and C. F. Cullis, Truns. Furuduy Soc., 1951,47, 616. H. Degn, Acta Chem. Scand., 1967,21, 1057.

312

React ion Kinetics

Supersaturation cannot therefore be ruled out in Degn's experiments, which are in direct contradiction with Peard and Cullis, who found that agitation produced oscillations which were absent in its absence. Degn is also in direct contradiction with Rice and Rieff, in finding oscillations with normally purified hydrogen peroxide, and with Shaw and Pritchard 5ta in finding oscilla-. tions in the absence of light. Degn obtained oscillations using sodium peroxide in place of hydrogen peroxide, but did not discuss its quality or purity, nor did he go to the lengths of Rice and Rieff, who prepared their own hydrogen peroxide, fused the vessel surface to avoid surface reaction, and also took stringent precautions against dust particles. Logically, at the moment one must still accept that the results of Rice and Rieff, and the findings of Degn, made under less rigorous conditions, are not inconsistent in so far as Rice and Rieff found a small region of parameter space for this reaction where oscillations do not take place. The existence of this region is the question of importance and the fact that Degn's experiments did not enable him to enter it adds little to our present knowledge. Clearly, serious experimental work is required in this area, the first priority being to repeat the experiments of Rice and Rieff. The most recent work on the reaction 12, l3 has been in conditions where the oscillations occur, and shows that other observables also oscillate, in particular pH, [I-], and temperature. The temperature pulses had an amplitude of $0.1 "C and were in phase with the pulses in the rate of oxygen evolution. It is not surprising that a number of variables show oscillation in this case and the central problem remains - what is driving the oscillation? The Belousov-Zhabotinskii Reaction.-In 1959 Belousov l4 reported oscillations of almost constant amplitude in a homogeneous, liquid-phase, stirred reaction. He observed that the Ce4+/Ce3+ratio oscillated during the ceriumcatalysed oxidation of malonic acid by bromate ions in molar sulphuric acid (see Figure 1). Zhabotinskii 15-17 and others studied the reaction in more detail and varied a number of the components. In particular the reaction still oscillates when the malonic acid is replaced by other organic compounds with an active methylenic hydrogen atom, and the Ce4+/Ce3+couple is replaced by Mn3+/Mn2+or ferriin-ferroin. In the case of reactions involving Ce4+/Ce3+,KBrO,, and bromomalonic acid, in sulphuric acid solution, the oscillations commence immediately, whereas with malonic acid itself the oscillations only commence after an induction period of 5-20 min depending on the concentrations used. l2

l3

l4 l5

l6 l7

H. A. Liebhafsky and J. H. Woodson, Nature, 1969,224, 690. H. A. Liebhafsky, I. Matsuzaki, and J. H. Woodson, Bull. Chem. SOC.Japan, 1970,43, 3317. B. P. Belousov, Sbornik Ref. Radiat. Med., Medgiz, Moscow, 1959, Vol. I. A. M. Zhabotinskii Doklady Akad. Nauk S.S.S.R., 1964, 157, 392. A. M. Zhabotinskii, Biofizika, 1964, 157, 392. V. A. Vavilin, A. M. Zhabotinskii, and A. N. Zaikin, Russ. J. Phys. Chem., 1968, 42, 1649.

313

Kinetics of Oscillatory Reactions

This induction period can be shortened or removed by adding various amounts of bromomalonic acid, so an obvious interpretation involves the bromination of the original malonic acid by a reduction product (Br,) of the bromate. Clearly, the brominated acid is a key intermediate in the oscillation sequence. Zhabotinskii demonstrated that continuous addition of bromide ion to the reacting mixture suppressed the oscillations cornpletely. More recently, it has been shown l7 that U.V. light (A <3OOO &

Time/S

Figure 1 A typical chemical oscillation in a closed system, the bromate-ceriwnmalonic acid system (Reproduced by permission from J. Amer. Chem. Soc., 1972, 94, 8649)

modifies or completely suppresses the oscillations. A particularly interesting phenomenon observed in this system was the occurrence of intermittent oscillations, i.e. regular bursts of oscillation interspersed with regular periods of apparently non-oscillatory reaction (although clearly some non-monitored variables could still be oscillating during this period). The inhibition is consistent with the observation of Zhabotinskii on bromide ions, since the latter are formed photochemically from bromomalonic acid. However, there are also photosensitive compounds present capable of producing bromide ions. Degn l8 has shown that dibromomalonic acid occurs in the reaction mixture and is likely to decarboxylate to dibromoacetic acid, which itself is a precursor of tribromoacetic acid. The latter inhibits the oscillations if present in sufficient concentration. Kasperek and Bruice lS were the next to study the kinetics of this system, but they also made some interesting observations on physical factors as well. In particular they found that stirring gave oscillations which were highly regular (as indeed all previous workers had found) but on stopping stirring the amplitude became highly irregular and much more rapidly damped. On restarting stirring the behaviour returned to normal. Also it was coml8

H. Degn, Nature, 1967, 213, 589. G. J. Kasperek and T. C. Bruice, Znorg. Chem., 1971, 10, 382.

314

Reaction Kinetics

pletely unaffected by bubbling N2,02,or COz through the mixture as it is by the addition of ground glass and dust. They also found that the induction period was irreproducible, suggesting heterogeneity of the reactions presumably producing bromomalonic acid. Kasperek and Bruice concluded that the behaviour of the reaction can be explained by the formation and dissolution of colloidal particles or by the occurrence of a reaction whose rate is dependent on their surface area. The observations on the effect of stirring are interesting in so far as they parallel the observations of Griffiths, Gray, and Gray 2 o on an entirely different type of system, the gas-phase oxidation of propane. Here exactly the same effect is observed with respect to the effect of stirring on temperature oscillations in the reacting gas. The explanation in this case turns out to be due to the fact that in the unstirred case the reaction always commences at the top of the reaction vessel and propagates downwards past the thermocouple placed at the centre of the vessel, although to the naked eye it could be mistaken for a homogeneous reaction. In the stirred case the reaction is very nearly homogeneous and the oscillations are certainly temporal. Measurements of an average property over the reaction vessel such as are made in the Zhabotinskii reaction (e.g. optical density) would not reveal this type of behaviour, i.e. a very rapidly propagating reaction beginning for instance near the surface of the liquid. In this connection, Franck and Geiseler 21 and Busse 2 2 have independently reported temperature oscillations during the reaction of ca. 1 O. Nicolis and Portnow consider it unlikely that the heat of reaction is involved in the oscillatory mechanism and, in view of the fact that simple purely kinetic oscillatory mechanisms can be devised for this system (see below), this may be true. However, two facts should be borne in mind here: (i) The accepted kinetics 23-25 involve a steady-state concentration (of bromous acid) which can change almost discontinuously as a function of the bromide ion concentration. But, as the kinetic expression involves a number of elementary rate constants with different activation energies, it could equally well be an almost discontinuous function of the temperature of the solution also, although if temperature jumps of only 1 " occur one might expect that small variations in the ambient temperature would require considerable variations in the concentrations (in particular of bromide ion) to restore the system to an oscillating regime. (ii) The amplitude of the measured temperature oscillations may be considerably smaller than the actual temperature rise. Busse used a rubbercovered thermistor without commenting on its time constant, and his oscillations are not sharp, possibly showing only the time lag of the therm2o 21

28

23 24 25

J. F. Griffiths, B. F. Gray, and P. Gray, 13th Symposium (International) on Combustion, Combustion Institute, Pittsburgh, Pennsylvania, 1970. U. Franck and W. Geisler, Nuturwiss., 1970,58, 52. H. G. Busse, Nature Phys. Sci.,1971, 233, 137. G. E. Koros, Faraday Symposia of The Chemical Society, 1974, to be published. R. J. Field, E. Koros, and R. M. Noyes, J. Amer. Chem. SOC.,1972,94, 8649. R. M. Noyes, R. J. Field, and E. Koros, J . Amer. Chern. SOC.,1972, 94, 1394.

315

Kinetics of Oscillatory Reactions

istor. The temperature rises during the conversion of Ce3+ into Ce4+and drops during the reverse part of the cycle; Busse concludes (incorrectly) that this implies no overall heat production during a cycle, hence there must be very efficient conversion of chemical energy into mechanical energy (the formation of gaseous CO,). Unfortunately, Busse did not keep the surroundings of the reaction vessel at constant temperature and did not state whether the mean reaction temperature was above or below ambient. If the reaction mixture successively self-heated and self-cooled with respect to ambient, his remarks about net heat production would apply, but they would be inappropriate if a succession of exothermic pulses occurred, as is the case in oscillatory combusting systems. There is clearly a need in this area for experimental temperature measurements in a constant-temperature bath with rapid-response fine thermocouples to clarify the thermal properties of the system. Extensive investigations of the detailed kinetics of this system have recently been made.23-2s The results are based on following the concentrations of Ce4+and Br- ions potentiometrically. When the solution contains sufficient Br- ion, BrO; is reduced fo Br, by successiveoxygen atom transfers. The malonic acid is brominated by an enolization mechanism. When the concentration of Br- ion becomes too small to remove bromous acid sufficiently rapidly, the latter reacts with BrO; to form BrOz*radicals, which in turn oxidize Ce3+. The net result is that bromous acid is produced autocatalytically in the system at the expense of Ce3+ and BrO;. Eventually bromous acid is removed from the system by second-order disproportionation. The Ce4+ produced in the autocatalytic process oxidizes bromomalonic acid, liberating Br- ion, which can reduce BrO; to Br,, thus cutting off the autocatalytic production of bromous acid. The above summary contains the seeds of a purely kinetic explanation of the occurrence of oscillations and it is discussed in more detail in Section 3. Three overall reactions are involved: BrO; BrO;

+ 2Br- + 3CH,(C02H), + 3H+ + 4Ce3++ CH,(CO,H), + 5H+

-+

--+

3BrCH(CO,H), BrCH(CO,H),

+ 3H20 + 4Ce4+ + 3H@

4Ce4+

(1) (2)

+ BrCH(CO,H), + 2H20 +Br- + 4Ce3+ + HC0,H + 2C0, + 5H+ (3)

and all the reverse reactions are extremely slow. Bromate and malonic acid, two of the original reactants, are removed from the system irreversibly, whereas the other species, Ce3+, Ce4+,Br-, BrCH(C02H),, are produced by some reactions and consumed by others, hence they could achieve quasistationary concentrations or approach a nearly steady oscillation. Noyes e? al. believe that the overall stoicheiometry of the reaction is given by a combination of the above three processes:

316

Reaction Kinetics

3Br0,

+ SCH,(CO,H), + 3H+

+

3BrCH(C02H), 2HCOzH t KO2

-+

+ 5HsO

(4)

The kinetics of these overall processes will be governed by the appropriate rate-determining steps, overall stoicheiometries being incapable of giving any information about the possibility of oscillations. Reaction (1) is believed to consist of four elementary steps:

+ 2H+ HBrO, + HOBr HBrO, + Br- + H+ -+ 2HOBr HOBr + Br- t H+ Br, + H,O Br, + CH2(C02H)2-+ BrCH(CO,H), + Br- + H+ BrO; + Br-

--f

+

(5)

(6)

(7) (8)

The rates of these processes are known,z4925 and the rate-determining step turns out to be reaction ( 5 ) . Within reaction (l), bromous acid attains a steady-state value given by [HBr0211= k,[BrO;][H+]/k, When the bromide concentration is low, reaction (1) will be slow, and bromate ions can react with Ce3+and malonic acid according to reaction (2), whose elementary steps have been shown to be:

+ HBrO, + H+ -t2BrO2. + HzO BrO,* Ce3+ + H+ HBrO, + Ce4+ BrO,. + Ce4+ H 2 0 + BrO; + Ce3+ + 2H+ 2HBr02 + BrO; + HOBr t- H+

BrO;

-C

--t

HOBr + CH,(CO,H),

--f

BrCH(CO,H),

+ H,O

(9) (10)

(1 1) (12) (1 3

The rate-determining step in this group is now agreed to be reaction (9), and its rate constant is known, If this process alone is taking place, bromous acid, which is again involved as an intermediate, can achieve a steady state given by [HBrO,], = kg[BrO, ][H+]/2k12 and from the known rate constants, k6, k,, k12,and k, this value turns out to be lo5 times greater than [HBrO2IL.The correct steady-state value of [HBrOJ includes both processes, which are occurring simultaneously. The full steady-state value is easily shown to satisfy a[HBr02]i where a 1:

#?

=

=

2k12,y

=

+ B[HBrO,],

-y =

0

k5[BrOJ[Br-][H+l2,and

k,[Br-][H+] k,[BrOJ[H+] (k10[Ce3+][H+] - kI1[Ce4+]}- k-l&ll[Ce4+J2 k,o[CeS+IIH+l4 kl,[Ce4+]

Kinetics of Oscillatory Reactions

317

When B > 0, the physically acceptable solution ( > 0 ) of the quadratic equation is [HBr021* Y / B but when p < 0 it is [HBrO,], 6 - B/a It must be emphasized that only one physically acceptable steady-state solution exists in this system for any given set of parameter values. Noyes et al. state that the concentration ‘of HBrO, changes almost discontinuously near #l = 0, implying that [HBrO$ can be treated as a step function. They claim, by inspection of the form of #l, that the ‘step’ can be triggered by a change in the Br- concentration or by a changedn the total cerium concentration or a change in temperature, or a combination of these. Little information is available about the activation energies of the reactions in this system so the effect of temperature on [HBrO& cannot be predicted, but Noyes et al. assume that the bromide ion concentration is the chief determinant of the sign of B. However (see Section 4), the concentration [HBrO& as given by the quadratic equation above in no way resembles a step function as claimed by Noyes et al., and regarded as a function of p it does not even have an inflection point. In such a complex system simplifications are necessary if any clear picture of the process is to be derived and also if a tractable mathematical model is to be set up to examine the system for the presence of oscillations. Noyes et al. have split the eleven chemical species present into three groups: (i) Major reactants, i.e. BrO;, H+, CH,(CO,H),, and BrCH(CO,H),. These are present in large concentrations relative to the other species present and one hopes their temporal variation will be small over the period of an oscillation, so that they may be treated as constants and absorbed into the appropriate rate constants. (ii) Derived reactants, i.e. HBrO,, HOBr, Br,, and BrO,.. These are quantities whose values cannot be varied independently of the major reactants and reference reactants owing to conservation conditions. (iii) Reference reactants, i.e. Br- and Ce4+, which are present in low concentrations and are measured directly by electrochemical methods. Noyes et al. suggest HOBr + Br, as a third reference reactant if one is needed, but one might also consider that HBrO, would be more convenient in view of its drastic variation as a function of one of the other reference reactants (Br-). Since the rate-determining steps for the overall processes in this system are known it should be possible to represent the experimental situation as now understood to a good degree of approximation by two or three rate equations for the reference reactants. These are easily seen to be, with 4 x = [Br-1, y = [HBrO,], and z = [ a 4 + ]

Reaction Kine tics

318

dY dt

dz dt

-

=

=

-ay2

kpy

-

+ k,x

c, - c,z

- y

-

-

kloz

c3z2

+ k,,z

k3z

where the k, now include the concentrations of the major and derived reactants. Here the C's are defined by C, = k9klOzT, C, = kgklo kll, C , = L I O k l l ,and zTis the total cerium concentration. Bowers et aLZ6replaced the malonic acid by pentane-2,4-dione, since no permanent gas is evolved in this case. They found that, in the presence of Mn", oscillations occur without induction period, and they damp away more rapidly if stirring is stopped. This effect was still present in a sealed system using degassed components, and the authors concluded that a heterogeneous step was involved.

+

Sodium Dithionite Decomposition.-In undergoes the overall reaction 2Na2S20,

+ H,O

--f

aqueous solution sodium dithionite 2NaHS0,

+ Na2S203

and the concentration of dithionite itself has been observed to undergo oscillation in a closed isothermal . ~ y s t e m . ~This ~ - ~example ~ is unique in so far as one of the initial reactants itself oscillates in time; in all other known cases the initial reactants decay monotonically,,although sometimes exhibiting sharp inflexions, The amplitude of the oscillations in this case is quite large, i.e. a significant fraction of the original concentration. The oscillations were not periodic and only two or three were observed; nevertheless, the effect is believed to be real, although little is known about the detailed kinetics of this system. Bischoff and Mason 30 have performed computations on a set of reactions which give oscillations resembling those observed, without making any attempt at chemical identification. The Oxidation of Carbon Monoxide.-In the gas-phase oxidation of carbon monoxide at high temperatures in closed reaction vessels, oscillations have been observed in the form of glows and temperature pulses in chloropicrinsensitized mixture^,^' 'wet' and 'dry' mixtures.33 Ashmore and Norrish 31 observed up to one hundred successively diminishing glows, whereas Dove 33 observed fifty-six maxima and also found that oscillation 26

P. G . Bowers, K. G. Caldwell, and D. F. Prendergast, J . Phys. Chem., 1972, 76,2185.

-"

S. Lynn, Ph.D. Thesis, Calif. Inst. of Tech., Pasadena, 1954.

28 29

30

31

32 33

R. G. Rinker, Ph.D. Thesis, Calif. Inst. of Tech., Pasadena, 1959. R. G. Rinker, S. Lynn, D. M. Mason, and W. H. Corcoran, Znd. and Eng. Chem. (Fundamentals), 1965, 4, 282. J . R. Bischoff and D. M. Mason, Chem. Eng. Sci., 1968,23, 447. P. G. Ashmore and R. G . W. Norrish, Nafure, 1951, 167, 390. J . W. Linnett, B. G . Reuben, and T. F. Wheatley, Combustion and Flame, 1968,12, 325. J . Dove, D.Phi1. Thesis, Oxford, 1959.

Kinetics of Oscillatory Reactions

319

was favoured by richness in CO/O,, no oscillations being observed in the reverse situation. On the other hand, McCaffreyand Berlad 34 have observed as many as two hundred oscillations before monotonic decay sets in, so undoubtedly this reaction rivals the Zhabotinskii reaction as a chemical oscillator. All workers are unanimous that these oscillations are isothermal, or very nearly so, as opposed to the oscillations observed in the gas-phase oxidations of all organic compounds. This system is typical of many chemically reacting systems which oscillate in that it shows other signs of structural instability in the form of well-defined explosion limits. The onset of explosive behaviour is not discussed here, and the reader is referred to a standard text.36 However, much of the stimulus to unravel the detailed kinetics in this system has come from attempts to explain the shape of the explosion-limitcurve. A branching chain mechanism is an obvious candidate and oxygen atoms are an obvious choice for chain carrier. Radiation is emitted from the system, and it has been suggested 36 that electronicallyexcited CO,, C 0 2 * ,is also involved, giving an energy chain

co + 0 -+ co,* co,* + 0, -+ CO, + 2 0 co,* + co co*+ CO, co* + 0, COa + 0 -+

+

The existence of the upper explosion limit, as in the case of the hydrogenoxygen system, requires a chain-termination process with a greater pressure dependence (i.e. order of reaction) than the branching process. The only possibility seems to be CO

+0 +M

+

CO,

+M

and in the region of the first (lower) limit the termination steps include CO,*

+

CO,

+ hv

0 + wall The above reactions are thought to explain the explosive behaviour in perfectly dry mixtures of oxygen and carbon monoxide, but such mixtures are difficult to obtain and the reaction is extremely sensitive to small traces of hydrogenous materials such as H,O or H2.36 This is easily explained on the basis of the branching reaction of oxygen atoms with either hydrogen or water : O+H2+OH+H OH

H

+ CO

.+CO,

+ 0,+ O H

+H +0

B. MacCaffrey and A. L. Berlad, private communication. sf, G. J. Minkoff and C. F. H. Tipper, ‘Chemistry of Combustion Reactions’, Butterworths, London, 1962. B. H. Mahan and R. B. Solo, J. Chem. Phys., 1962,37, 2669. 34

320

Reaction Kinetics

and this branching based on H, or H,O converts the system almost into the hydrogen-oxygen system. Linnett et aLa2have shown that the use of heavy water markedly raises the glow limit in the low-pressure region (below the first explosion limit) and that diffusion of radicals to the walls is not an important termination process. B r o k a ~ ,and ~ Dean and Kistiakow~ky,~~ have used the branching scheme based on H2or H,O to explain their shock tube data, which cannot be explained on the basis of the dry scheme alone. A few p.p.m. of water, however, enable them to fit the experimental data. Further data on this system indicate that if the mixture is brought to the explosion limit slowly by the heating method (i.e. admitted to the reaction vessel at a temperature well below the limit and slowly heated), then the explosion limit is considerably displaced towards higher temperatures, thus indicating some degree of product inhibition (incidentally, a type of kinetic scheme beloved of biochemists as an oscillator). A carbon suboxide radical is a possible candidate with the following reactions:

co* + co +-c,o + 0 c,o + 0 2co +

C20-+ inert products

I

i

E 1

D.

2

7

,ol '

curve

GLOW i !

I 450:

I

I

5c3

Ambient

,

1

4

553 temperature

1

1

600

/ 'C

+

Figure 2 The behaviour of dry mixtures of 2CO 0,as a function of total pressure and ambient temperature. The curves represent bifurcation loci (Reproduced by permission from Trans. Faraday Soc., 1954, 50, 37) 37

s8 3s

R. S. Brokaw, Eleventh International Symposium on Combustion, Combustion Institute, Pittsburgh, 1967, p. 1063. A. M. Dean and G. B. Kistiakowsky, J . Chem. Phys., 1970,53, 830. P. Harteck and S. Dondes, J. Chem. Phys., 1957, 27, 1419.

Kinetics of Oscillatory Reactions

321

but this aspect of the reaction is not at all well understood and deserves further investigation. Figure 2 shows the behaviour of dry mixtures of oxygen and carbon monoxide as a function of total pressure and ambient t e m p e r a t ~ r e . ~ ~ The Non-isothermal Oxidation of Organic Compounds.-This is a field in which many complex and extremely interesting phenomena occur, including oscillations both highly damped and hardly damped at all,41 explosion^,^^^ 4 2 oscillations followed by explo~ions,~~p44 discontinuities in reaction rate 45 regarded as a function of a smoothly varying parameter such as ambient temperature, and hysteresis phenomena, although the latter are more easily observed in open systems, and are discussed in the appropriate section below (p. 332). The most intensively studied system, as far as oscillations are concerned, is probably propane 42 but its behaviour is typical of most organic compounds which have been studied. The ignition diagram (Figure 3) shows clearly the regions in (P-To) space for fixed composition in which cool-flame oscillations can take place. The number of oscillations observed in systems 439

41p

\ \ \

\ \ I \

I I

Damped

I

I I

oscillations

I

I I I 1

/ / / / /

- # - d d

r

I 580

I

I 600

Ambient

I

I

620

temperature

I 640

I

I 660

/K

Figure 3 The behaviour of an equimolar propane-oxygen mixture under welt-stirred conditions in a closed vessel ‘O

D. E. Hoare and A. D. Walsh, Trans. Faraday SOC.,1954,50, 37. J. F. Griffiths, P. G. Felton, and P.Gray, 14th International Symposium on Combustion,

Combustion Institute Pittsburgh, 1973.

ra J. Chamboux and M. Lucquin, J. Chim. phys., 1962,59, 797. 4J

R. Ben-aim and M. Lucquin, Combustion and Oxidation Rev., 1965, 1, 7. M. B. Neumann and B. Aivazov, Nature, 1935,135,645. M. VanpQ, Fuel, 1955,34433.

I4

322

Reaction Kinetics

of this type is usually less than ten since a significant mount of the fuel is consumed in each oscillation. Similar, but less intensively studied, cool flames have been observed in 48 ethane,47 a ~ e t a l d e h y d e54 ,~~~ formaldehyde, butane, O 2-methylpentane, acetone, n-~entane?~ f ~ r m a l d e h y d e propionaldehyde ,~~ and b~tyraldehyde,~~ he~tane,~’ methyl ethyl s9 hexane,61162 heptene, octene, and isooctane,52 olefins in general,83-65side-chain 26$ aldehydes in 9

I

I

I

1

I

0

4

Time

1

I.

12

I

I

16

/S

Figure 4 ‘Undamped’ oscillations in an equimolar propane-oxygen mixture in a closed vessel under well-stirred conditions

p0 47 48 49

50

51 52

53 54 55

56 57

58 59

0o

62

63

64 65

M. VanpQ, Compt. rend., 1956,243, 804. J. H. Knox and R. G. W. Norrish, Trans. Faraday SOC.,1954,50, 928. J. Chamboux and M. Lucquin, J. Chim. phys., 1963, 60, 527. M. Vanpke, Compt. rend., 1955, 241, 951. M. Cherneskey and J. Bardwell, Canad. J. Chem., 1960,38,482. A. Fish, Proc. Roy. SOC.,1966, A293, 378. N. MacCormac and A. Townsend, J. Chem. SOC., 1938,238. M. V a n p k and F. Grard, Fuel, 1956, 34, 433. C. M. Newitt, L. M. Baxt, and V. V. Kelkar, J. Chem. SOC.,1939, 1703, F. E. Malherbe and A. D. Walsh, Trans. Faraday SOC.,1950,46, 835. R. Hughes and R. F. Simmons, 12th International Symposium on Combustion, Cornbustion Institute Pittsburgh, 1969, 449. B. H. Bonner and C. F. H. Tipper, 10th International Symposium on Combustion, Combustion Institute Pittsburgh, 1965, p. 145. J. Bardwell and C. N. Hinshelwood, Proc. Roy. SOC.,1951, A205, 375, A. R. Burgess and R. G. W. Laughlin, Chem. Comm., 1967, 769. J. Brown and C. F. H. Tipper, Proc. Roy. SOC.,1969, A312, 399. J. H. Burgoyne, T. L. Tang, and D. Newitt, Proc. Roy. SOC.,1940, A174, 379. E. Frehling, Rev. Znst. Franc. Petrole Ann. Combust. Liquides, 1956, 11, 134. D. T. A. Townsend and K. Spence, Colloq. Int. Sur la Comb., C.N.R.S., Paris, 1948, p. 113. S. A. Miller and J. R. Abbott, Nature, 1951, 168, 474. A. C. G. Egerton and J. H. Burgoyne, Proc. Roy. SOC.,1940, A174, 394.

Kinetics of Oscillatory Reactions

323

general,a2amines,66etc. A comprehensivelist of organic compounds showing oscillatory cool-flame behaviour has been given by Ben-aim and L u c q ~ i n . ~ ~ The most important general characteristic of all these reactions is the fact that they are strongly non-isothermal, and with extremely fine thermocouples with a very short response time *l the temperature of the reacting gases can be followed quite accurately. Temperature oscillations with amplitudes of up to 180 "C are common. The gas temperature can be measured at various points in the reaction vessel and as a result of this it was shown that,41under the conditions of most of the work quoted above, there is a considerable movement of gas in the reaction vessel owing to convection currents above a certain critical pressure.B7 It has been suggested that the oscillations were entirely an artefact, due to the influence of convection.68 It is now possible, after overcoming mechanical difficulties, to stir these hot reacting mixtures with a magnetically driven, ceramic coated, stainless steel 4i Under these conditions the oscillations appear much more regular and less well damped, these effects being attributable to the removal of convective gas motion and inhomogeneous cool-flame propagation through the mixture. In Figure 4 highly non-linear temperature oscillations in propane are shown; they approximate limit cycle behaviour, the fourth peak being considerably diminished owing to consumption of the reacting mixture. This 'damping' can be completely eliminated by flowing sufficient fuel into the reactor to give genuine limit cycle behaviour (see Section on open systems, p. 332). On the other hand, the damping shown in Figure 5 is inherent in so far as

a0 c L 0 0

P

s

40

t

Time/ s

Figure 5 Damped oscillations in an equimolar propane-oxygen mixture in a closed vessel under well-stirred conditions 66

*'

C. F. Cullis and D. J. Waddington, 5th International Symposium on Combustion, Rheinhold, New York, 1955, p. 545. P. G. Ashmore, B. J. Tyler, and T. A. B. Wesley, 11th International Symposium on Combustion, Combustion Institute, Pittsburgh, 1967, p. 1133.

324

Reaction Kinetics

the amplitude tends to zero whilst the reaction rate (as measured by the temperature excess) is still large. Replenishing the fuel in this case does not prolong the oscillations and a steady state is ultimately reached. The rate of heat release in these systems is so large that it is extremely difficult to produce a spatially homogeneous system, the temperature profile in a typical stirred system being shown in Figure 6. It will be seen to be a reasonable

Diameter

/ cm

Figure 6 Spatial temperature profile in a well-stirred closed non-isothermal reactor

approximation to the hypothetical Semenov type where a constant temperature is assumed across the reactor, but there is a discontinuous drop down to ambient within the vessel wall. This makes the theoretical treatment of well-stirred, self-heating systems reasonably tractable since ordinary (as opposed to partial) differential equations occur. Experimental problems still remain, however, since by reference to Figure 6 it is possible to conclude that a significant volume of gas (around the walls) is reacting at a temperature some 20-30 "C lower than the gas in the hotter central region, so although the present techniques of stirring remove the uncertainties associated with convection, etc., they do not solve the problems of chemical analysis as most sampling techniques will remove a large proportion of gases from the cooler shell. However, the oscillations are now known not to be artefacts of convective gas motion, and we can be sure we are dealing with a genuine thermochemical oscillator; thermochemical because the heat release rate is so large that it can couple with the reaction rates themselves, the latter usually having exponentially growing temperature coefficients, thus giving ample scope for a positive or negative feedback process. Indeed, this positive feedback process, if monotonic, will lead to thermal explosion and not oscillaA. Melvin, Combustion and Flame, 1969, 13, 438.

P. Gray and P. R. Lee, Combustion and Oxidation Rev., 1967, 2, 1.

Kinetics of Oscillatory Reactions

325

tion. That this does not occur in cool-flame processes is partly due to another unusual phenomenon - the decrease in reaction rate with increase in temperature over a small temperature range. The Negative Temperature Cueficient. In the temperature range of 350450 "C many of the compounds showing oscillatory cool-flame behaviour also exhibit a region, usually very close to the oscillatory region, where the reaction rate and heat release rate drop with increasing temperature, showing the so-called negative temperature coefficient (NTC). Pease 7 0 was the first to draw attention to this phenomenon, and he stated a connection between the NTC and the periodicity of the cool flames. Since then a number of examples have been found where a quantity, proportional to the reaction rate, decreases with increasing temperature. These quantities include the temperature excess above ambient, the rate of pressure rise in the reactor, the rate of consumption of fuel in the reactor, and the actual rate of heat release in the reactor. Neumann and Aivazov 44 found a very marked NTC in n-pentane as shown in Figure 7, where there is almost a discontinuous drop in the maximum

Ambient

temperature To

Figure 7 The negative temperature coeficient in the non-isothermal oxidation of pentane at three different total pressures as a function of ambient temperature

reaction rate as a function of ambient temperature. Knox and Norrish 71 measured a decrease in the temperature excess above ambient, with increasing ambient temperature, in the slow oxidation region of ethane. More recently,2 the actual maximum rate of heat production has been measured for propaneoxygen mixtures under various conditions, as shown in Figure 8. The negative temperature coefficient of reaction rate can now be regarded O0 71

R. N. Pease, J. Amer. Chem. SOC.,1940,62,2234. J. H. Knox, in 'Photochemistry and Reaction Kinetics', ed. P. G. Ashmore, F. S. Dainton, and T. M. Sugden, C.U.P. Cambridge, 1967.

326

Reaction Kine tics

590

610

650

630 Temperature

/

670

690

K

Figure 8 The negative temperature coeficient in heat-release rate in a well-stirred reactor, as a function of reactant temperature, with various total pressures of equimolar mixtures of propane and oxygen

as a firmly established fact in the slow oxidation region of organic compounds; in addition, it nearly always occurs in the region of oscillatory cool flames. We may thus expect a connection between oscillations and NTC (a) on purely empirical grounds, and (b) on the basis of the qualitative argument that product inhibition can give rise to an oscillatory process. In this case, the product is thermal energy and the inhibition occurs because the reaction rate decreases with the resultant increase in temperature. These expectations of a connection ’are fully confirmed on theoretical grounds (see below), At this point it is convenient to draw attention to a large group of compounds which exhibit NTC’s, namely the enzymes. Although the thermal effects in enzyme-catalysed reactions are not so marked as in the case of gas-phase oxidations, one may expect thermally coupled oscillations there also. 7 2 Isothermal Kinetics. It is not proposed here to deal exhaustively with the

isothermal kinetic schemes which have been proposed to explain many of the features of low-temperature oxidations in so far as the primary interest 72

B. F. Gray, P.Gray, and N. A. Kirwan, Combustion and Flame, 1972,18,439.

327

Kinetics of Oscillatory Reactions

is the existence of oscillation in these systems, and these oscillations are strongly thermally coupled, i.e. thermokinetic rather than purely kinetic. We would not expect to find sources of oscillation by examiningthe isothermal kinetics alone, as we may in some of the solution reactions which oscillate, but on the other hand we cannot construct theoretical models of. the feedback between energy release and reaction rates unless we have some knowledge of the kinetics of the system. For this reason we shall discuss briefly the kinetics of low-temperature oxidation in general, followed by a discussion of some of the simplest systems where the kinetics are fairly well understood. The alkylperoxy radical isomerization theory represents one attempt to write down a general mechanism for these oxidations. Since the oscillation and NTC characteristics are so invariant over wide ranges of compounds we have reason to expect that there must be important common features in all organic low-temperature oxidations, except in the lowest members of given homologous series, which often behave abnormally. In general, the 74 scheme can be represented as

+ R. + HO2* Re + 0,+ RO,. R 0 2 *-+ R-’ + product R*’ + RH + R*+ R’H RH

0 2

-+

ieomerization

This is seen to be a chain reaction, but in this form not a branched-chain or autocatalytic reaction. The degenerate branching step occurs if the peroxy radical abstracts a hydrogen and the resulting hydroperoxide splits thus: ROz*+ RH R02H R*

+

-+

R* + O2 += R02* R02H + RO* + OH* As the temperature is increased, intramolecular hydrogen abstraction becomes increasingly important due to its high activation energy, and it competes with the formation of alkyl monohydroperoxide HR02. *OH

+ RH

R-

+ O2

-+

-ROOH -+inert

-+

R. + H2O

+-

RO,.

+ *OH

Here the decomposition of the hydroperoxy radical is not a branching process as it is in the intermolecular mechanism, and Fish 73 believes this is responsible for the onset of the NTC as the temperature rises and net branching decreases. 74

A. Fish, Angew. Chem. Internat. Edn., 1968,7, 45. J. Cartlidge and C. F. H. Tipper, Proc. Roy. SOC.,1961, A261, 388.

328

Reaction Kinetics

However, a decrease in the rate of branching does not necessarily imply a NTC for the reaction ; a decreasing steady-state radical concentration is necessary for this. Thus, for a chain reaction involving a single intermediate, the overall reaction rate is proportional to the steady-state radical concentration, and the latter is unaffected by the value of the propagation rate constant. However, for two-centre chains it can be shown that if the propagation reaction removing the branching centres has a sufficiently high activation energy, then an NTC can result. Thus in the region of NTC, *OH radicals will be the main propagating species, and these are highly reactive and unselective, in accord with experiment. A difficulty with this theory is that the rearranged radical gR0,H would also be expected to add on 0, and hence give a dihydroperoxide H02R02H. An alternative theory, the alkene theory, makes use of the fact that there is considerable evidence for the production of olefins in the early stages of oxidation of the lower alkanes.’l This observation was largely a result of improved analytical techniques, and the first two steps are thought to be HR’-Re R’=R

+ 0, -+ R’=R + HO,* + 0, oxygenated products -+

the oxygenated products being largely aldehydes and ketones. Since olefins are rather more reactive than alkanes, their observation as early intermediates must have an important bearing on the path of the subsequent reaction. On this basis the chain would be

HR’-RH

+

+ HO2’ + HO2* -+ HR’-R* +

HR’--R*

0 2

+ R’=R

H202

hydrogen abstraction by HO,* being an important part of the scheme. This certainly does not occur in the hydrogen-oxygen system at temperatures up to 500 “C, and hydrocarbon oxidations proceed fairly rapidly as low as 300 “C. This would require RH bond dissociation energies -= 320 kJ, and it is difficult to accept this step occurring rapidly in alkanes at 300 “C. This theory also requires a process for converting the HO,. radical into an OHradical as the NTC region is approached and the abstraction becomes unselective. Knox 71 postulates

R=R

+ H02. + .R’R02H

.R’R02H .O,R’R02H

+ 0, -+ -O2R’RO2H

+ HO,.

-+

HOZR’R02H + 0

H02R’R02H+ 2 0 H -

2

+ RO + R’O

where RO and R’O will be aldehydes or ketones. The above scheme gives a straight chain, without any branching or degenerate branching. The latter can occur from the carbonyl compounds by decomposition into acyl radicals.

Kinetics of Oscillatory Reactions

329

The occurrence of the NTC is explained in this case by stating that the conversion of HOy into OH. becomes less efficient as the temperature rises, and this depends on the equilibrium between olefin, HOs-, and hydroperoxy radicals. The kinetics of three atypical compounds have now been reasonably well sorted out as a result of the efforts of many workers. These are methane,75 formaldehyde, and acetaldehyde. The first two compounds have not been studied in an oscillatory regime (although Vanpee 45 has reported a single sharp pressure pulse) so their discussion is hardly relevant here, although one should be optimistic about the appearance of oscillations under the right conditions. In the case of formaldehyde, difficulties with polymerization in the colder parts of the system occur, and with methane high temperatures and pressures are required to produce cool-flame behaviour. In the case of acetaldehyde the kinetics (isothermal) are well and have been subjected to a computational thermokinetic treatment 76 for conditions simulating a static system. The experimental data on this system42do not show clear oscillatory behaviour since no more than two cool flames were observed. These results have been confirmed,” and it is clear that this compound is not conveniently studied in a static system since such a large proportion of acetaldehyde is used up in each cool flame that. more than two oscillations would not be expected. The relative amplitudes of these two pulses are also very sensitive to stirring and this system would be very much more conveniently studied using a stirred flow reactor, bking the most promising thermokinetic system of this type for complete elucidation. It must be concluded that the general understanding of low-temperature oxidations still eludes us as far as the elementary kinetics is concerned, and more experimental information of a type different from that gathered classically is required, such as the rate constants for radical reactions with hydrocarbons measured by Baldwin et aZ.,78and reported in Chapter 4 of this volume. Fortunately, for present purposes, there is enough circumstantial evidencs for the occurrence of general features which favour oscillation to make it worthwhile theorizing a bout these systems. The occurrence of autocatalysis (branching), strong feedback of the negative variety (heat released and NTC), and other critical (bifurcation) phenomena such as explosions and discontinuities in rate can now be regarded as strong enough evidence for a priori prediction of oscillations under appropriate conditions for a system where oscillations have not yet ‘been observed. This is particularly interesting as most of the latter are kinetically the simplest systems, such as CHI and CH,O. However, the experimental difficulties with these two compounds 75 76

77 78

J. N. Knox, Adv. Chem. Ser., No. 76, 1968. M. P. Halstead, A. Prothero, and C. P. Quinn. Proc. Roy. Soc., 1971, A322, 377. J. F. Griffiths, private communication. R. R. Baldwin, D. H. Langford, M. J. Malchan, R. W. Walker, and D. A. Yorke, 13th International Symposium on Combustion, Combustion Institute, Pittsburgh, 1971, p. 251.

3 30

Reaction Kinetics

are somewhat greater than with the better studied compounds such as propane. Pressure oscillations have been reported in the gas-phase thermal decomposition of octafluorocyclobutane in the presence of toluene, but no attempt has been made to establish the kinetics in this system.79 Biochemical Systems, Closed and Open.-As many results in this area have been obtained in both closed and open systems, it is convenient to discuss them together. Appropriate distinction will be made in discussing theoretical models. Glycolysis is the process in which oscillations at the metabolic level are most easily observed.80 The first reported oscillations in the concentration of NADH were in suspensions of intact yeast cells,s1 but of more direct physicochemical interest were the oscillations reported later in cell-free extracts. 8 2 * 83 In these systems all metabolites with concentrations between and mol 1-1 as do pH and the production of C02.85 Comparison of the oscillations in cell-free extracts with those in yeast cells indicates that the basic oscillator is the same in both cases, although the cellular systems tend to oscillate with a higher frequency. The oscillations can be initiated by the addition of maltose to the cell-free but not by the addition of other sugars except trehalose. The complete kinetic scheme in glycolysis is extremely complicated, involving a large number of enzymes and intermediates and it is obviously a matter of great interest to isolate the subsection, if such exists, which actually generates the oscillations; many of the other concentration oscillations could then be regarded as driven or forced oscillations. A number of analytical studies with this aim in mind have been Closed systems were used (spectrophotometer cuvettes) and the reaction was monitored largely through absorption due to pyridine nucleotide. As many as eleven full cycles were observed with periods ca. 10 min. Addition of glucose-6-P caused a momentary increase in amplitude and also reactivated oscillations which had almost been damped out. Sinusoidal and non-sinusoidal (highly non-linear or spiky) oscillations can be observed in these dell-free yeast extracts, but only the sinusoidal variety appear in vivo. In the spiky oscillations the pyridine nucleotide remains oxidized during the flat portion of the curve and reduced during the ‘spikes’. Another interesting observation in this system is that the phase of the oscillations is advanced by addition of ADP (adenosine 79 80

82

83 84

a5 86

87

B. F. Gray and H. 0. Pritchard, J . Chem. SOC.,1956, 208, 1002. B. Hess and A. Boiteaux. Ann. Rev. Biochem., 1971, 40, 240. B. Chance, R. W. Estabrook, and A. Ghosh, Proc. Nat. Acad. Sci. U.S.A., 1964, 51, 1244. B. Chance, B. Hess, and A. Bertz, Biochem. Biophys. Res. Comm., 1961, 16, 182. K. Pye and B. Chance, Proc. Nat. Acad. Sci. U.S.A., 1966,55, 888. B. Hess, A. Boiteaux, and J. Kruger, Adv. Enzyme Reg., 1969, 7 , 149. B. Hess and A. Boiteaux, in ‘Regulatory Functions of Biological Membranes’, ed. J. Jahrnefelt, Elsevier, Amsterdam, 1968, p. 148. B. Chance, B. Schoener, and S. Elsaesser, Proc. Nut. Acad. Sci. U.S.A., 1964,52, 337. B. Chance, B. Schoener, and S. Elsaesser, J. Biol. Chem., 1965,240, 3170.

Kinetics of Oscillatory Reactions

331

diphosphate) and retarded by the addition of pyruvate. In this respect biochemical systems have an advantage over gas-phase oscillators in that perturbations of this sort can easily be carried out. All the chemicals in this system oscillate with the same frequency, but fall into two groups, all the chemicals in each group having the same phase! Glucose-6-P, fructose-6-P, and pH are each in phase and 180" out of phase with fructose diphosphate and glyceraldehyde phosphate. A second pair of groups, 180" out of phase with each other but a variable amount out of phase with the first two groups, also exists. These two groups contain compounds further along the glycolytic sequence of reactions and presumably the variable phase shift is a measure of their 'distance' from the oscillator in the chain. The existence of only two basic phases is encouraging in so far as it indicates that the fundamental oscillator is describable in terms of only two concentrations, i.e. in a phase plane as far as mathematical description goes. Oscillations in compounds further down the glycolytic pathway would then be regarded as driven, and it would not be necessary to consider them in the oscillatory model. A point worthy of note is that the glycolytic system produces a gas, COz, in common with many of the inorganic oscillating reactions. A further interesting point in this system is that whilst the reaction is oscillating, the relevant enzymes are operating at only a fraction of their maximum activity. 84 Many other types of oscillation can be observed in unicellular and multicellular systems at the molecular level, particularly in membrane processes 8 o and certain muscle contraction processes. However, these processes almost certainly involve coupling with non-physicochemical processes and will not be considered here. Another oscillation of physicochemical type occurs in various enzyme systems of the peroxidase type. Horseradish peroxidase has been observed to oxidize NADH in an oscillatory manner 88 and the lactoperoxidase system has also been shown to oscillate.sg This is an oxygen-consuming reaction, via the oxidation of reduced pyridine nucleotides. A trace of methylene blue is necessary for the oscillations to become stable whereupon the concentration of oxygen in the solution oscillates, being simultaneously replenished by bubbling into the solution. Degn has also studied this reaction and finds results in conflict with those of Yamazaki et a1.s8y8 9 Degn and Mayer have discussed the theory in terms of a Lotka model modified for an open system, but obviously experimental questions remain to be resolved. Wilson and Calvin 92 have reported oscillations during the dark cycle of photosynthesis and Chernavskaya and Chernavskii 93 have discussed an isothermal kinetic model involving autocatalysis.

91 92 y3

I. Yamazaki and K. Yokota, Biochem. Biophys. Acta, 1967, 132, 310. S. Nakamura, K. Yokota, and I. Yamazaki, Nature, 1969, 222, 794. H. Degn, Biochem. Biophys. Acta, 1969,180, 271. H. Degn and D. Mayer, Biochim. Biophys. Acta, 1969,180, 291. A. T. Wilson and M. Calvin, J . Amer. Chem. SOC.,1955, 77, 5948. N. M. Chernavskaya and D. S. Chernavskii, Sov. Phys. Clsp., 1961,4, 850.

332

Reaction Kinetics

Oscillations in Open Reactions.-In the study of chemical oscillations, open reacting systems, where reactive material is supplied and waste products removed, would seem to be ideal vehicles. Strangely enough they have hardly been studied to this end at all; traditional chemical methods using static closed vessels having been used extensively, particularly in the combustion field and for the Zhabotinskii reaction, although the latter has been studied in a flow system 94 as has the oxidation catalysed by horseradish peroxidase.95 Open reacting systems are the stock in trade of chemical engineers, however, and the continuous flow stirred tank reactor (CSTR) has been extensively The concept of a CSTR is an idealized one in so far as spatial gradients are assumed non-existent due to perfect stirring, which is assumed to be produced by turbulence, mechanical stirring, or both. Although this situation can never be realized in practice it can be approximated sufficiently closely to give meaningful results. Only ordinary differential equations, with time as the independent variable, are involved in the description and interpretation of the system, and real steady-states can be achieved as opposed to the quasi-steady states which occur in the theory of static reaction systems. Oscillations have been reported in systems of this type by Volter O 7 in the gas-phase polymerization of ethylene, Bush 9 8 in the vapour-phase chlorination of methyl chloride, and Gray and Felton 9 9 in the gas-phase o&ation of propane. The oscillations found by Volter and Bush are both of the type referred to by Frank-Kamenetskii l o o as ‘trivial relaxation oscillations’. These occur as a result of the conservation laws relating to the mass of the material in the reactor. The amplitude is clearly limited by the fact that the concentration of the material being consumed cannot become negative, and the frequency is determined in an equally trivial manner by the time taken for the initial reactant to refill the reaction vessel. Experimentally such oscillations can be characterized by large amplitudes, of the order in fact of the initial feed concentration. Thus, Bush observed oscillations in chlorine mole fraction of amplitude ca. 0.30 with an initial Jeed mole fraction of 0.40. Such oscillations have been extensively studied by chemical engineers from a theoretical point of view and these are discussed in the theoretical section on open systems 94

V. A. Vavilin, A. M. Zhabotinskii, and A. N. Zaikin. Russ. J. Phys. Chem., 1968, 42, 1649.

J. Yamazaki, K. Yokota, and R. Nakajima, Biochem. Biophys. Res. Comm., 1965,21, 582. 96 R. Aris, ‘Elementary Chemical Reactor Analysis’, Prentice-Hall, Englewood Cliffs, N.J., 1969. 9 7 B. Volter, ‘Conplex Putomation of the Chemical Industry’, Proceedings Moscow Inst. Chem. Machine Conf., 1963,25,48. g8 S. F. Bush, Proc. Roy. SOC., 1969, A309, 1. 99 B. F. Gray and P. G . Felton, Combustion and Flame, to be published. looD. A. Frank-Kamenetskii, ‘Diffusion and Heat Exchange in Chemical Kinetics’, Plenum Press, New York, 1969.

95

333

Kinetics of Oscillatory Reactions

(p. 357). Here we simply remark that criteria of oscillation have been formulated, originally because such oscillations were often regarded as detrimental to the performance of the reactor (which they usually are) largely due to the extreme temperatures reached. However, Renken lol has shown that enforced periodic concentration variations can be advantageous in certain isothermal reactors as far as yield and selectivity with respect to a certain desired product are concerned. For a single second-order reaction, steady-state operation represents the optimum, but for the scheme A+B+P P+B+R R 4- B + S , etc. where P is the desired product, selectivity and yield can be improved by oscillation. One is led to speculate here that a scheme which is inherently oscillatory can always be run more efficiently by driving the oscillation but no work appears to have been done on this point. The possible occurrence of trivial oscillations in a flow system does not preclude the study of more interesting kinetic or thermokinetic oscillations, i.e. oscillations which occur in a closed static vessel but are eventually damped out owing to exhaustion of the material or approach to equilibrium. The damping out can be nullified by flowing material in at the appropriate rate,O49 @ O thus giving stable kinetic or thermokinetic oscillations. This is particularly useful in systems where only a few oscillations can be observed before the material is used up (e.g.cool flames,where it is difficult to observe more than a few). The thermokineticoscillations(cool flames)in the propaneoxygen system can be prolonged indefinitely in a flow system, and damping due to fuel consumption easily distinguished from inherent damping, which is also present in this system. This is characterized by a region where a steady state is reached by oscillation, and on perturbation (e.g. of the flow rate) can be shown to return to equilibrium by a series of damped oscillations. On the other hand the limit cycle persists indefinitely. In this system, trivial relaxation oscillations also occur under the right parametric conditions. These consist of ignitions which rapidly consume all the reactive material, then repeat after the reaction vessel has filled sufficiently. They are easily distinguishable from the interesting oscillations and are best avoided in this system, frequently damaging thermocouples and reaction vessels.

3 Theory of Oscillating Reactions The Basic Equations.-In chemically reacting systems, one is dealing with the conservation principles - of mass and energy - and a number of phenomenological rate laws. The latter include, for instance, Fick's Law of lol

A. Renken, Chem. Eng. Sci., 1972,27, 1925.

334

Reaction Kinetics

Diffusion, Fourier’s Law of Heat Transport, and the rate laws of chemical reactions themselves. We shall not be concerned directly with the Laws of Fourier and Fick in so far as we shall confine ourselves to spatially homogeneous systems in our discussion of temporal oscillations. The reasons for this are twofold: (a) most oscillating systems can be stirred vigorously to produce homogeneity, or a good approximation to it in the case of non-isothermal systems; (b) a mathematical apparatus of considerable power exists for the discussion of ordinary autonomous differential equations such as arise in the homogeneous problems. The non-linear partial differential equations arising in the non-homogeneous case are hardly amenable to general treatment at the moment, although a method of investigating their stability in certain cases has been given.lo2 In the case of open systems the transport of matter and energy into or out of the system can be treated by allowing discontinuities in the temperature and concentration fields at the boundaries of the system and then applying empirical laws such as Newton’s Law of Cooling, etc., as was done originally by Semenov lo3in thermal explosion theory. We shall not be concerned with the theory of chemical kinetics at the molecular level but at the macroscopic level. For purposes of developing the general theory of oscillatory chemical reactions, it is necessary to treat reactions as elementary to allow us to devise rate laws using the Law of Mass Action, and indeed most authors have followed this procedure. On the other hand, detailed treatments of specific systems have made use of the experimental rate laws. Taking a general system containing n chemical components undergoing m elementary chemical reactions, simultaneously producing or absorbing heat, with fluxes of some or all of the components and energy across the boundary of the system, the equations assume the form

Components are labelled with a subscript j (1, .. .,n) and reactions with a subscript i (1, ..., m),forward reactions with a superscript (+) and reverse reactions with a superscript ( -). xjis the molar concentration of component lo?

V. Hlavacek, J. Sinkule, and M. Kubicek, J . Theoret.Biol., 1972, 36, 283. N. N. Semenov, ‘Chemical Kinetic and Chain Reactions’, Pergamon, Oxford, 1953.

lU3

Kinetics of Oscillatory Reactions

335

j and the k, are rate constants. v f is the order of reaction ‘i’ with respect to component in either the forward (+) or reverse ( -) direction, and vd, E vd; - v f . T is the absolute temperature and C is the average heat capacity of the reacting material. The .I,are the flux terms of components x,

across the boundary of the system, and in general they will depend on a number of external parameters a, B, y , . . . besides the concentrations x,. For example, the boundary of the system could well be a semi-permeable membrane and the parameters could be the external concentrations of certain components. In the case of the energy equation, where the h, are the molar heats evolved by the forward reaction i, the energy flux J E will depend on the ambient temperature To,which in most systemswill be a control variable, i.e. it can be varied at will, thus influencing the behaviour within the system. In an isolated system the 4’s and J E are zero. In a closed system the 4’s are zero but J i is allowed to be non-zero. Most organic non-isothermal oxidations come into this category, but the Zhabotinskii reaction also releases a significant amount of heat although, owing to the diluteness of the solutions involved, C is very large. In many isothermal (approximately) systems JE is taken to be zero and the Jj’s are non-zero. We shall assume that at all times the system is in mechanical equilibrium. The above equations can be written in a less detailed form more suitable for general discussion if we define

i.e. the rates of the forward and reverse ‘i’ th reactions. With this substitution they become

dT C--; dt The Conservation of Mass. The differential equations (19)-(21) by themselves are far too general to represent realistically a chemically reacting system. Even when their initial conditions are specified as physically realistic (i.e. all the x! are positive) a number of important auxiliary conditions are still needed, such as the principle of conservation of mass within the system if it is closed. This principle may also be applied to a large number of open reacting systems as well if their boundaries are stationary, i.e. they are not growing.

336

Reaction Kinetics

If the molecular weight of species j is M,, then the mass-conservation principle requires

which on substituting from equation (20) gives

which reduces to

The two terms in this equation represent the change in mass within the system due to chemical reaction and the total mass flux across the boundary, and these can clearly be separately equated to zero:

and since in general RB # &- we can write the second equation as n

1Mjv, o (i =

=

1, ..., rn)

(25)

j=1

i.e., m linear equations involving n quantities (the M j ) all known to be >O. In general, these equations will not be consistent unless a relation between vkjexists, which must be the case here since we know the M j exist (regarding them as the unknowns). Clearly, whether n 3 rn is important in deciding whether the system must show a relation between the vPkor whether they are independent.lo4

Detailed Balancing. In closed systems at equilibrium, the stationary values of the concentrations are not sufficient to define fully the state of the system in general. The principle of detailed balancing is also required, and in our notation it states

-

-

R,+ = R,

(i

=

l . . . rn)

(26)

where the bar denotes reaction rates evaluated at equilibrium. The principle of detailed balancing was first recognized as being necessary for the system

lo4

B. F. Gray, Trans. Furaduy. SOC.,1970, 66,363.

Kinetics of Oscillatory Reactions

337

If detailed balancing is not applied to this system a steady state with fluxes can occur, and the system.will tend to this state in an oscillatory manner. Thus application of detailed balancing in this case rules out oscillations, and this is true in general.1o4 For the triangular reaction system, if we take rates as being positive in the anticlockwise direction, we have at equilibrium

KA dt

=

-

-

vlAR1 - vaAR2= 0

gC vzcR2 - vSCR3 dt =

=

'IA

0

-'2A

-'IB

0

1

'3B '2c

=o

(28)

-'3c

K.

Detailed balancing is again needed as have a non-zero solution for the an extra postulate. The third and most common case is when n > m. The existence of nonzero solutions (Mj) of the equations of mass conservation now shows that all determinants of order m x rn are non-zero, in which case the steady-state

loo

T. Muir and W. H. Metzler, 'Theory of Determinants', published privately, 1930.

338

Reaction Kinetics

equations will not have a non-zero solution for the R,, and detailed balancing is not required as a separate postulate in this case. It is true then for closed systems in general, that at equilibrium

Ri

=

0 (i

=

1, ...,rn)

(30)

or

and these equations imply relations between the rate constants of the system. For the simple triangular system they imply the equation

The above discussion applies to closed systems, but equations such as (32) must hold regardless of whether the system is closed or open, since the chemical rate constants of the individual reactions can hardly depend on the openness of the system. Thus we would expect such equations to apply to the scheme C

where 0 A implies ‘A enters the Fystem from outside’ and B -+ 0 implies ‘B passes out of the system’. Clearly, detailed balancing cannot be imposed on this system in the steady state, nevertheless equation (32) still applies. Thus if an open system contains a kinetic scheme to which one would have to apply detailed balancing if the system were closed, any relations between rate constants can be used in the open systems. What constraints this will put on the behaviour of open systems of this type requires investigation. --f

Chemical Feasibility. The differential equations (14)-(17) describing the reacting system already incorporate the Law of Mass Action, but the equations are still too general to represent chemically reacting systems in sufficient detail. An obvious restriction is in the values of the exponents I$, the orders of individual reactions with respect to particular components. For elementary reactions these will be identical with the molecularities, hence values much greater than these will not be likely. Also, it is necessary to specify that for each elementary reaction i, substancej can appear only as a reactant or product, i.e. one of the pair vtf must b e zero. High values of the v$ favour oscillations and other interesting phenomena such as bifurcations, etc., so exact restrictions of this type are important. Other restrictions are also necessary, and although obvious in specific cases, difficult to characterize in general. For example, a zeroth-order term in any of the equations would have to have a positive sign [remembering that

339

Kinetics of Oscillatory Reactions

equations (14)-(17) represent the elementary reactions in a system with quasi-steady approximations for any of the components] corresponding to a constant feed-in of the.material. Motova lolla and Hanusse lOs* have made a start in this direction for a system with two variables and reactions with molecularities not greater than two. For example, Hanusse claims that any term in

x; in the expression for

9 must occur with a negative coefficient. dt

With a

number of assumptions of this sort Hanusse has proved that a limit cycle surrounding an unstable focus cannot exist in his twocomponent system. Further work in this direction is required, particularly with regard to surface reactions which may be present, as it is not difficult to envisage surface removal reactions which are zeroth order in the material being removed if the surface is near saturation, thus violating Hanusse’s assumptions. Further investigation is also needed where a number of chemicals of equal molecular weight appear in a kinetic scheme, particularly where oscillatory models are concerned, and the system is considered to be isothermal. The number of strictly degenerate isomers of a molecule is limited and schemes which employ three or more without the concomitant energy changes must be regarded as chemically unacceptable. An obvious but nevertheless highly important adjunct to equations (14)--(17) is the requirement that all the dependent variables be positive definite, i.e. >O, if they are to represent a chemically reacting system.

Mathematical Techniques.-The simplest possible case of chemical interest (i.e. that of two variables) will be discussed at some length. With only a single variable and real coefficients in the equation, oscillations are clearly impossible. Of the two variables considered, one may or may not be the temperature, so we shall use the notation x,y for these variables in the next section and discuss particular cases such as x = xl,y E x2 or x = xl, y = Tin later sections. The case of two variables is treated separately since much more can be said about the appearance of sustained oscillations in this case. In the transition to even three variables much qualitative knowledge is lost.

Two-variable Systems. Equations (14)-(17) the form

will be written in this case in

(33)

2 dt 106

=

P(x,y)

(a) M. I, Motova, in ‘Oscillatory Processes in Chemical and Biological Systems’, ed. E. E. Sel’kov, Puschino on Oka, 1973; (6) P. Hanusse, Compt. rend., 1972,274, C,

1245.

340

Reaction Kinetics

where P and Q will be assumed to possess the appropriate form to describe a chemical system and will be assumed to depend on a number of parameters (such as external concentrations etc.). A most important point is that these equations are autonomous, i x . they do not include the independent variable t on the right-hand side of the equation. Equations (33) can be replaced by the single differential equation

in the phase (xy) plane, formulations (33) and (34) being completely equivalent. Integrals of equation (34) are represented by curves in the (xy) phase plane and the motion of x and y with time will be described by movement of the point representing the state of the system in the xy plane. The curve along which the point moves is then referred to as a trajectory. Methods of obtaining information on the nature of the solutions of equations (33) and (34) are available in many texts 107-109 and are discussed only very briefly here, with a particular emphasis on oscillatory solutions. The steady states of equations (33) (equilibrium states in the case of a closed system) become singular points in the phase plane where dy/dx becomes indeterminate. Since both dy/dt and dx/dt -+ 0 as the singular point is approached, it can only be approached asymptotically in the phase plane if it is stable. Much information can be obtained about the behaviour of the system sufficiently close to a singular point and hence sometimes indirectly about the behaviour in the large. The co-ordinates of the singular points (there may be more than one, and usually are in interesting systems) will satisfy the equations

and it is convenient to take the particular singular point we are studying as the origin of co-ordinates, i.e. x’ = x - x,, y’ = y - yB. If we now perform a Taylor expansion of the functions P and Q about the steady states, equations (33) become dY’ dt

where P,

(a,> , i3P

E

etc., evaluated at the steady state.

U

lo’

ao8

Lee

N. Minorsky, ‘Nonlinear Oscillations’, D. Van Nostrand C o , Princeton, 1962. A. A. Andronov, A. A. Vitt, and S. E. Khaikin, ‘Theory of Oscillators’, Pergamon, Oxford, 1966. N. N. Krasovskii, ‘Stability of Motion’, Stanford U.P., 1953.

341

Kinetics of Oscillatory Reactions

The First Method of Liapounov. This method, by means of the First Stability Theorem, relates the behaviour of equations (36) in a sufficientlysmall region of the origin to the behaviour of the linear equations dY’ dt = P&’

+ P,y’

dx’ = Qs’ dt

+

I

(37)

QVY’

which are exactly soluble and hence their behaviour is completely known. The first stability theorem of Liapounov can be stated as follows: ‘In the neighbourhood of the steady states of the system (33) the stability characteristics of the solution are determined by the characteristic roots of equation (37) in the same sense as they characterize the solutions of equation (37) itself.’ If we adopt the notation A = P,, B = P,, C 3 Q,, D = Q,, the characteristic equation for equation (37) is

lBnA

c:A

I

=

A’

-

(B

+ C)A + BC

-

AD

=

0

(38)

+

Noting that the solutions of equations (37) are of the form Fe’ll‘ Ge’la‘, where A, and A, are the roots of equation (38) and F, G are arbitrary constants, the types of behaviour which can occur are as follows : (i) Both roots real and negative; in this case the system approaches the steady state asymptotically and monotonically, i.e. without oscillations; the steady state is a stable node in the phase plane. (ii) Both roots real and positive; in this case the steady state is unstable and the system moves away from it. Oscillation cannot be excluded in this case, as this system will spend most of its time outside a region near the origin; the steady state is an unstable node in the phase plane. (iii) Both roots real and differing in sign; again the steady state is unstable and the system will move away from it. This type of singularity is called a saddle point and it is less likely to occur in an oscillating system than an unstable node (case ii). In the phase plane, it is possible to draw a closed curve round an unstable node representing sustained oscillation, but not possible for a saddle point. (iv) Both roots conjugate complex numbers; if the sign of the real part of the roots is negative the steady state will be stable but approached in an oscillatory manner, i.e. a damped oscillation will be observed; in the phase plane the singular point will be a stable focus. If the real part of the roots is positive, the system will show oscillations diverging from the steady state and will be rather likely to settle down to sustained oscillations (around a limit cycle in the phase plane; see next section); the singular point will be an unstable focus in the phase plane. (v) Both roots pure imaginaries; in this case nothing can be deduced about

342

Reaction Kinetics

the behaviour, without further investigation. If the original equations are linear, the integral curves are closed curves encircling the singular point, thus giving oscillatory behaviour but of a non-physical kind in so far as the amplitude and period of the oscillations are not stable with respect to perturbations. When the original equations are non-linear the situation is more difficult and is discussed in detail by Andronov.108 The chances are that in this case the singular point will be a degenerate focus in the phase plane. It is clear then that for oscillatory behaviour near the steady states, whether damped or divergent, we must have complex roots of the characteristic equation. Since A2 - ( B + C)2 + B C - A D = O the roots are given by 21.2 =

B+C

-f 31/(B + C)' 2

B+C -f$ J ( B

2

-

C)'

+ 4AD

- 4BC

+ 4AD

(39)

so it is clearly necessary that AD < 0

for complex roots. So it is necessary that

(g)g (z)z x

< 0 and these

two terms must have opposite signs. However, if the oscillations are to grow to a finite amplitude, and the real parts of the roots are to be positive B+C>O

or

If x and y are both concentrations this inequality means that we have to have at least one reaction where one species is produced autocatalytically, or an exothermic reaction when T is one of the variables. Limit Cycles. A limit cycle is a closed curve in the phase plane and it is

called stable if all other trajectories in its neighbourhood are spirals winding themselves onto it. If they wind themselves away from it, it is called unstable (and in general will not be physically realized). If the trajectories in its neighbourhood wind themselves onto it from one side and away from it on the other then it is called semi-stable (and again will not be physically realized owing to perturbations). Limit cycles were discovered by Poincar6 and represent periodic phenomena in real time. These describe all known types of oscillations correctly in so far as the period and amplitude are independent

343

Kinetics of Oscillatory Reactions

of the initial conditions of the variables which are oscillating. This is also true of chemical oscillations and for this reason oscillations described by a singular point which is a centre (pure imaginary roots of the characteristic equation) cannot give a realistic description of such a system. The often-used Lotka Model (see p. 366) comes into this category. From simple topological considerations a limit cycle must contain at least one singular point inside it, and if the limit cycle is to be stable this must be an unstable one. It cannot be a saddle point lo8but either an unstable node or unstable focus can emanate trajectories which could wind onto a limit cycle, thus giving sustained oscillations. It is possible to state definitely when this will not happen (i.e. when limit cycles will not occur) simply from studying the differential equation in the phase plane over a finite region and not just in the neighbourhood of the singular points. Behaviour in t h e k r g e . We can rule out the occurrence of a limit cycle (or

centre) in any region of the phase! plane where

(;)>, + ($$does not

change sign (or vanishes identically). This is known lo* as the Negative Criterion of Bendixson. The converse is not true, i.e. if

($l+ (3#

does change sign in a region this does not imply the existence of a closed trajectory in that region of the phase plane. A more general negative criterion lo8is due to Dulac. It states that there are no closed curves in a domain if

a(W +WQ)does not change sign, where B is any continuous aY ax

function with continuous derivatives. For linear systems the quantity ap aQ - is constant, hence non-linearity is necessary for oscillation in a

ax + ay

two-component system. The Poincarh-Bendixson Theorem gives a positive result, i.e. the necessary and sufficient conditions that a closed curve exist in a given region. Unfortunately it requires some knowledge of the solutions of the differential equation rather than just the form of the differential equation itself. It states: ‘If a trajectory remains in a finite domain of the phase plane without approaching any singular points, it is either a closed trajectory or approaches such a trajectory.’ If one remembers that trajectories cannot cross each other in the phase plane except at singular points, this theorem becomes topologically obvious. Occasionally the form of the differential equations enables one to prove the existence of a limit cycle directly. Suppose we have an unstable singularity (e.g. a focus) such that all trajectories move out of its neighbourhood. Now suppose that at a finite distance from the singularity we can construct a closed curve enclosing the singularity along which we know the signs of both dx/dt and dyjdt. If these are such that the trajectories are pointing

344

Reaction Kinetics

t -+-&- \-

-#-I

k

-

d

k I

I

5

Figure 9 Topological conditions for existence of a limit cycle, encircling an unstable singularity

inwards at every point on this curve, then a stable limit cycle must exist somewhere within the region. In applications it is often convenient to choose the closed curve to be a rectangle (see Figure 9). This technique has been used by Bush g8 to show the existence of a limit cycle describing trivial relaxation oscillations in the chlorination of methyl chloride. Parasitic Oscillations. It may be that the differential equations have the form

>

where E is a small dimensionless parameter (1 E > 0). Systems of this sort can show discontinuous oscillations where the motion alternates between ‘slow’ motion, described very well by the ‘reduced‘ equations with E = 0

and rapid jumps along paths described by x = constant

&2 = P(x,y) dt

Kinetics of Oscillatory Reactions

345

It is usually good enough l o * to describe this segment of the oscillation by ‘infinitely rapid jumps’, i.e. e + 0, dy/dt + 00 in (44),so that equations (44)do not have to be solved at all. Many electrochemical oscillations have the appearance characteristic of this type of ‘parasitic’ oscillations l o * and Degn 110 has constructed a simple theoretical model which lends itself very well to this type of mathematical treatment although he did not use the already existing mathematical apparatus. This type of analysis has only been used occasionally to describe chemical oscillations (see later examples) and could be exploited further. It has also been used in discussion of the accuracy of the steady-state approximation in closed chemical systems ll1 and the occurrence of critical (bifurcation) points in thermally exploding systems.ll* Many chemical systems possess equations which can be cast in the form (42) when written in terms of dimensionless concentrations x / x o ,y / y o , E = y o / x o . Jf x is initially present in large excess, i.e. y o / x o 1, the limiting form of equation (43) would be a good description for the slow part of the cycle followed by a rapid jump following equations (44),i.e. x = constant. If y represents a radical or an ion and x is an initial reactant the limit E + 0, which can be treated exactly, would be a good approximation.

<

Piecewise Linear Systems. We have already seen that the Bendixson Negative Criterion shows the necessity of non-linearity for the manifestation of chemical oscillations. Unfortunately non-linearity is often also a suJgicient

criterion for the insolubility of the equations! We can get the best of both worlds if the non-linearity occurs only over a very narrow range of concentrations, then outside this range the equations will be linear. In the limit, if the range of non-linearity becomes infinitely narrow, the rate depends on concentration like a step function and the non-linearity is concentrated into a point (i.e. where the step is). Everywhere else the equations are linear and exactly soluble, all we have to do is match the solutions attained on each side of the discontinuity at the discontinuity. This involves the solution of an algebraic or transcendental equation, a relatively simple computational task. The oscillations (if they occur) survive the squashing of the nonlinearity into a point. An extra advantage of this method is that it reduces the dimensions of our space by one, since the variable with respect to which the discontinuity occurs is effectively eliminated. This method does not appear to have been exploited yet in the field of chemical oscillations, although it would appear to be a useful model in some cases. N-variable Systems. We will jump from two-variable systems to systems with N variables directly as the difficulties increase greatly in passing from N = 2 to N = 3, and in principle only slightly thereafter. The difficulties arise particularly with respect to the existence of closed curves and the

corresponding periodic trajectories. One can intuitively see that the topological no H. Degn, Trans. Faraday SOC.,1968, 64, 1348.

F. G. Heineken, H. M. Tsuchiya, and R. Aris, Math. Bioscience, 1967, 1, 95. B. F. Gray, Combustion and Flame, 1973, 20, 317.

ll1

346

Reaction Kinetics

arguments and negative criteria useful in the phase plane will break down when a third dimension is available, not to mention the added difficulty of a much greater variety of types of singular points. Hence most of the methods used are more formal than in the case of two dimensions. Much less can be said about the occurrence of sustained oscillations, but nevertheless a number of theorems are available about damped oscillations on the one hand and stability on the other. For convenience equations (14)-(17) are written out again :

dT

C-

42

=

h,(R'

-

R;)

i=1

+ JE = f f T ( x z.,.., x,,

T)

(47)

The steady states of the system are given by the equations

... x:, T 8 ) = 0 ... x i , T") = 0 .fi.(X! ... x:, T") = 0 fi(x: f,(x:

(48)

and the points satisfying these equations are singular points of the differential equations (45)-(47) in the (n 1)-dimensional space ( x l , ...,x,, 7'). To obtain the behaviour in a sufficiently small region of one of these singular points it is convenient to take the latter as the origin of co-ordinates and to define new co-ordinates xi = x1 - xfl, xi = x, - x:, T' = T - T8 (remembering that there will be a number of singular points and a different origin for each which is not reflected here in thepotation). For a sufficiently small region round the origin we can replace the functionsA in equations (45)--(47) by the expressions

+

and equations (49-447) become since dt

(

=

dt

347

Kinetics of Oscillatory Reactions

These linear equations can be solved exactly in terms of a sum of exponentials ex‘where the A are given as roots of the characteristic equation (here assumed distinct)

8x1

4

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

(!5)0

0

(2) 0

=0

+

(53)

The roots of this equation A,, (k = 1, ...,n 1) are called the characteristic exponents of the system. The following obvious results can be stated: (i) If the characteristic exponents of (53) all have negative real parts the steady state is asymptotically stable (i.e. for a sufficiently small perturbation from the steady state the system will return to it); (ii) If one (or more) characteristic exponents have a positive real part, the steady state will be unstable (i.e. it will not be attained physically); (iii) If all the characteristicexponents have negative real parts, but some of them are complex, then the steady state will be asymptotically stable but perturbations in some of the variables will exhibit damped oscillations during the return to the steady state. Such systems will not exhibit sustained oscillations in the region of the steady state under consideration but in other regions of parameter space may be likely to do so, although little can be said with certainty. If we are not to suffer from rigor mortis in this area we might state the following: If some of the characteristic exponents are complex with negative real parts in one region of parameter space, but with positive real parts in another region of parmeter space, then along the boundary between these two regions we would expect sustained oscillations to be initiated. In two dimensions some rigorous mathematical backing can be given to this statement, but this is not so for more than two. Nevertheless the statement is well supported by experimental results, particularly in the case of open reactions where stable steady states with complex characteristic exponents can be demonstrated by experimental perturbation. One can invariably fbd parameter values where these systems show sustained oscillations also. For the special case of isothermal systems, where the last differential equation can be dropped, a theorem of Liapounov gives us some information about the stability characteristics of the system ‘in the large’ and not just

Reaction Kinetics

348

in the region of the singular points. In this case, the equations describing the system can often be written in the form

where we have explicitly separated out the unimolecular reactions (the P,j may contain terms due to fluxes if the system is open) and the F, are all power series beginning with terms of degree at Zeast two. In practice there will rarely be terms with degree greater than three. The following results then hold.lo7 (i) If all the roots of the characteristic equation

IP - ill(=0

(55)

have negative real parts the steady state is asymptotically stable whatever the nature of the functions F,; (ii) If at least one of the roots has a positive real part, the steady state is unstable, whatever the nature of the functions F,; (iii) If no roots have zero real parts, the funct’ons Fi may influence stability, and nothing can be said on the basis of the linear equations. This critical case (iii) is rarely investigated in physical systems as it can usually be avoided by working in an appropriate area of parameter space; indeed, it is usually impossible to achieve it in parameter space in practice (for example it corresponds to sitting exactly on an explosion limit). Hurwitz and Similar Criteria. In the case of two-variable systems, once the characteristic equation is written down the stability problem is solved as it is simply a quadratic equation. In the case of an (n + 1)th degree equation this is not so since we cannot write down a formula for the roots explicitly. Fortunately there are various criteria for asymptotic stability (all roots < 0) and instability (at least one root > 0). The best known of these is the Hurwitz criterion, which gives a necessary and sufficientcondition for all the roots of the equation a O X alR-l ... + a, = 0 (56)

+

+

to be negative, in terms of a number of determinants built up from the coefficients ao, a,, ..., There is a similar criterion due to LiCnardChipart.l13 Of course, it should be remembered that sustained oscillation is not possible in a region where stability of a steady state can be proved. On the other hand, in a region where instability can be proved, sustained oscillation is possible, in fact likely if any of the characteristic exponents is llS

F. Porter, ‘Stability Criteria for Linear Dynamical Systems’, Oliver and Boyd, Mech. Eng. Monograph, 1968.

Kinetics of Oscillatory Reactions

349

complex. Instability criteria are not so well studied as stability criteria but a simple and well-known one derives from Descartes’s Rule of Signs: ‘The number of positive roots of an equation with real coefficients either equals the number of sign variations S in the coefficients or is less than S by an even integer, S - 2, S - 4, etc.’ In this form the rule does not give us what we need, i.e. a lower bound for the number of positive roots. However, if we apply it to the equation with A replaced by -A it will give us a maximum value for the number of negative roots of the original equation, and if this is less than r there must be at least one positive root, and the steady state will be unstable. The Second Method of Liapounov. Since stability and instability are directly relevant to the possibility of sustained oscillations, especially where these properties can be proved in a finite region, a method due to Liapounov which can be extremely powerful ll4*116 is mentioned here briefly. The aim of the method is to assess whether the system is stable or unstable without having to solve the system of differential equations, even in a small region. It relies on the possibility of finding a function, called a Liapounov function, which for two-variable systems becomes something like a potential energy surface upon which the system moves. For a region in which this function is concave upwards the system will be stable, very much as a ball will roll under gravity to the bottom of a hollow, and the system will move along a trajectory in the direction of decreasing V, if Y is the Liapounov function. In fact for mechanical systems the Liapounov function is the potential energy, but the mathematical method is inherently much more general. It has hardly been applied at all to systems of equations describing chemical change, but has already been used in chemical reaction theory by Aris,lls in explosive stability theory by Gray and Sherrington,l17~ 11*and by Shear in chemical equilibrium theory,ll8 which is discussed in the next section on closed systems.

Closed Systems: General Results.-Stability of Equilibrium. In closed systems we know from Boltzmann’s H Theorem that the system will eventually attain the equilibrium state, which will certainly be steady and which will therefore be represented by a singular point of the basic equations of the system (with all the J, = 0 for a closed system). The basic equations (19)(21) then have the form 11*

W. Hahn, ‘Theory and Application of Liapounov’s Direct Method’, Prentice-Hall, Englewood Cliffs, N.J., 1963.

ll6

J. S. La Salle and S. Lefschetz, ‘Stability by Liapounov’s Second Method, with

116 117 118 llV

Applications’, Academic Press, New York, 1961. R. Aris, ‘Elementary Chemical Reactor Analysis’, Prentice Hall, Englewood Cliffs, N.J., 1969. B. F. Gray and M. E. Sherrington, Combustion and Flame, 1972, 19,435. B. F. Gray and M. E. Sherrington, Combustion and Flame, 1972, 19,445. D.Shear, J , Theoret. Biol., 1967, 16, 212.

Reaction Kinetics

350

(57)

At equilibrium J E must also vanish and so

Also detailed balancing must hold, i.e. -

RT

=

7

R, (i = 1,

..., rn)

(63)

since this is an equilibrium state. In terms of the concentrations at equilibrium these equations become n

n

j=1

j=1

k d + n ~ > := k ; n z > ; ( i = I, ..., rn)

(64)

and it will be assumed that these equations determine a unique set of Zj, i.e. there is a unique equilibrium state. Application of Boltzmann’sH theorem shows that the system will approach this state asymptotically as time progresses, but it is interesting to see how this can be shown on the basis of the differential equations themselves for the isothermal case. These equations are the first n from equation (57) to (58), viz.,

Shear llS has shown that the function

is a Liapounov function for equations (65) and (66) provided detailed balance

Kinetics of Oscillatory Reactions

351

holds at equilibrium. Shear states in his paper that his proof holds for any stationary state regardless of detailed balancing, but this is incorrect as it would imply the impossibility of limit cycles in systems which are open, and many examples counter to this statement are known. However, for the equilibrium state, Shear’s proof implies that the function V acts like a ‘potential‘ for the system and adopts its minimum value at equilibrium. For V to be a Liapounov function it must be concave upwards everywhere, i.e. dV - < 0 everywhere as the system moves along a trajectory down to the dt minimum. In fact it is necessary to show that Y > 0, G 0 in a region for V dt to be a Liapounov function for the system (the equality signs can only hold at the origin). From equation (67)

=

c W,/.q c %5(Rd+ n

m

3=1

t=l

- R,)

(68)

using the differential equations (65) and (66) to evaluate dx* -.dt Shear shows that the expression (41) is always negative except at the origin (x, = Z5) where it is zero. The proof uses the fact that detailed balancing requires = and does not apply otherwise. Nevertheless, in a closed system, this shows that the equilibrium is stable for all positive values of the concentrations, and no matter what the initial conditions we will always finish up at the equilibrium point. This is equivalent to showing that the characteristic equation for this system [equation (55)] has roots with negative real parts. It does not say anything about the imaginary parts, i.e. whether they are zero or not. In other words, can a closed system approach equilibrium asymptotically but via damped oscillations? The answer to this question was given by Hearon 120 in 1953 for a system of unimolecular reactions and by Gray lo4 for the general system described by equations (45)-(47), i.e. for any combination of molecularities in non-isothermal conditions also. Detailed balancing is again an essential feature of the proof and therefore the theorem is restricted to the equilibrium state in closed systems (or isolated systems which are of little interest). The proof depends on a theorem stating that the eigenvalues of a symmetric matrix are real lo5 and then goes on to show under what conditions the secular equation for this system is symmetrical. The proof is illustrated with the simpler isothermal case. The characteristic equation for this case is equation (53) with the last row and column deleted, i.e. no J. Z .

Hearon, Bull. Math. Biophys., 1953, 15, 121.

3 52

Reaction Kinetics

=

0

(69)

where the derivatives are evaluated at equilibrium. It folfows from the rate equations themselves by differentiation that

It is easily seen by inspection that S,, equation can be written

=

S,, so that the characteristic

=

Multiplying the row j by (# 0) and the column j by determinant is transformed into the symmetrical form

0

(71)

<-* (# 0), this

=0

(72)

Kinetics of Oscillatory Reactions

353

The roots of (72) are real since it is a symmetrical determinant and they can be shown to be unchanged by the transformation from (71) into (72); therefore the roots of the former are also real, and approach to equilibrium must be monotonic. In the non-isothermal case the roots can be shown to be real in similar fashion, with the extra condition that the energy flux JB is independent of the xj, i.e.

5 = 0 for allj. axj

If JE represents black body radiation this will

be true, if not we are led to investigate a wider system than that represented by our basic equations (57j ( 5 9 ) and we would have to include another set of equations giving the variations with time of the photon densities in different parts of the spectrum. Such an investigation has not been undertaken. Damped Oscillations and Quasistationary States. Since oscillations do occur in closed systems, although they are never sustained, they are often quasisteady for long periods compared with the oscillation period. Examples of this are the Zhabotinskii reaction where ca. 100 cycles may be observed and the thermal oxidation of carbon monoxide where cu. 200 cycles have been observed. None of these oscillations encloses the final equilibrium point or is a damped approach to it, in accord with the theory discussed above. The question of how to describe these oscillations (other than by direct computation) is an extremely interesting one which is far from being fully understood. It is intimately connected with the steady-state hypothesis (SSH) often made in treating ordinary kinetics in closed systems. The justification of the SSH, or more importantly, the ability to predict when it will be jus;ified by inspection of the kinetic equations, has been studied only intermittently, originally by Hirschfe'der 121 and then by Giddings and Shin.122These two papers are now of doubtful value owing to mathematical inconsistencies as pointed out by Bowen, Acrivos, and O~penheim,'~~ who appear to be the first authors to apply the techniques of singular perturbation theory to this problem in a systematic manner. More recently, Heineken, Tsuchiya, and Aris ll1 have applied singular perturbation theory to the special case of Michaelis-Menten kinetics. They conclude that the sole requirement for the justification of the SSH is that the ratio of the initial concentrations of enzyme and substrate be small. This is correct here, but not true for other kinetic schemes with a greater mixture of reaction molecularities, when rateconstant ratios as well as initial concentration ratios also enter the picture. The question of the validity of the SSH is connected with (and indeed almost identical with) the question as to whether it is allowable to treat the concentration of a substance present in large excess as a constant. This is often assumed in the chemical literature but never justified in any mathematical sense. The clearest way to discuss the question of the meaning of a quasi-steady J. 0. Hirschfelder, J . Chem. Phys., 1957, 26, 271. J. C. Giddings and H. K. Shin, Trans. Faraday SOC.,1961,57,468. lza J. R. Bowen, A. Acrivas, and A. K. Oppenheim, Chem. Eng. Sci., 1963,18, 177. la*

354

Reaction Kinetics

state in a closed system is by using the simple example below (which does not oscillate, incidentally). But it is important to realize that, in closed systems which oscillate, there are two questions: (i) How can we define a quasi-steady state by the SSH, and under what conditions? (ii) Having defined and located the quasi-steady state we wish to examine its character, i.e. is it stable, unstable, approached by oscillation etc? This can be done by the standard methods discussed above using the characteristic equation. The question as to whether, or to what extent, the approximation made in using the SSH can affect the qualitative character of the quasi-steady state has hardly been investigated at all. It certainly can do so and if the SSH is made when it is not justified, the errors can result in the prediction of spurious oscillations, as has been shown by Gray 124 for the Lotka system (see also p. 366). Consider the simple system ki

k,

A-+X-+B

described by the differential equations da- -k,a dt dx dt

-=

kla

(73) -

k2x

Transforming to the dimensionless variables y1 = a/ao, y2 = x/xo, t A = k2/k, where x o and a. are initial concentrations, gives dY1 - = dY2 dz

=

klt,

(75)

dt

&-

(74)

= y1

-

A&y2

<

= a/A or in where e = xo/ao 1. From equation (74), the SSH gives im terms of the dimensionless variables

YZ. = Yz,/&l

(77)

This relationship can be derived in a rigorous way under some conditions from the differentialequations (75) and (76) by using the formulation developed for discontinuous oscillations which is outlined in the section above on parasitic oscillations in two-variable systems. If we compare equations (75) and (76) with equations (42), clearly we need to know under what conditions our equations can be written in the reduced form (43) in the regions of ‘SIOW’motion lo*and equations (44)for the jumps between the initial conditions and the region of ‘slow’ motion. We can use the treatment 120

B. F. Gray, Combustion and Flame, 1970, 14, 273.

Kinetics of Oscillatory Reactions

355

given in ref. 108 if our equation (76) is of the same form as the first of equations (42). If the right-hand side of equation (76) is independent of E, the slow equations will suffice to describe the system. For this to be so A must be of order c l , i.e. A = O(8-l) or A = a/&where a 1. The equations for 'slow' motion become N

0

= y1* - aY,*

(79)

which are identical with the equations obtained by making the SSH. However, we know the conditions under which (78) and (79) are an adequate approximation to the full equations (75) and (76) from the theory given in Andronov.108 It follows that the necessary conditions for the validity of the SSH are that A is of the order of e-l [A = O ( E - ~ or ) ] ,in terms of the original variables

The above discussion is entirely from the poi& of view of the SSH, but we can rewrite it in order to answer the question 'Under what conditions can we treat the concentration of A(y,) as constant and equal to its initial value if it is present in large excess?' To answer this we rewrite equations (75) and (76)

in terms of the new dimensionlesstime u

z =

-,which we may call a 'rapid' time.

We obtain

Clearly as E -+0 (a,, --t 00) equation (81) reduces to y1 = const = 1 from the initial conditions. Equation (82) would be the equation we would get in the usual procedure if A = 0(c1) again, i.e. the conditions for the validity of treating a substance present in excess are identical with those for the SSH to apply to the substance present in relatively small concentraion. The only difference is one of time-scale. As it is not always true that the SSH holds, it follows that one cannot always consistently treat an excess concentration as constant. The above results are peculiar to the particular example treated and no attempt appears to have been made to treat the general case in order to see what sort of kinetics are required for the SSH to be a good approximation. A case where oscillations occur in a closed system when the SSH is made is the classical system of Lotka,126which has kinetics 126

A. J. Lotka, J . Amer. Chem. SOC.,1920,42, 1595.

356

Reaction Kinetics kl

A+X--+2X+B

X

k2’

+ Y--+2Y

+B

k3‘

A+Y-+B The differential equations for x and y being dx

- -- kiax dt

dY - = k.ixy dt

-

kixy

-

kjay

If we introduce dimensionless variables r -- kkt, kl obtain

=

ki/k;, k3 = ki/k; we

dx -- klax - XY

dr

dY

- = xy

-

k,ay

-

k3ay

dr It is assumed that A is present in large excess so that a is treated as constant in these equations and no differential equation is written down for a. The usual treatment of equations (83) and (84) is well known to give sustained oscillations, albeit of a very peculiar and unphysical kind (conservative). What we are interested in are the questions (i) ‘Under what conditions do equations (83) and (84) constitute a good approximation to the full equations for the closed system including consumption of A?’ (ii) ‘If these conditions are not strictly satisfied will the system still oscillate?’ The full equations for the system are da - = - k,ax - k,ay dr

2 = xy dr

dx --klax - xy dr Introducing more dimensionless variables

(87)

t = UJ,

y’ = y/yo, E = xo/ao< 1 leads to

da‘ - -- --a‘(klx’ dz

dx’ -= dt

kla’x’

-

+ k3y’)

~x’y’

a’

=

a/ao,

X’ =

x/x0,

Kinetics of Oscillatory Reactions

357

Taking the limit E -+ 0, a’ = constant = 1 is immediately obtained from equation (88). But equations (89) and (90) reduce to dY- -k,a’y’ dt dx’ - = kla‘xr dt and these certainly do not oscillate nor do they resemble the equations normally used. Clearly, the condition E + 0, i.e. xo/uo 0, is not a suflcient condition simply to ignore a, replacing it by its initial value. It is also necessary that kl k 3 xo/aoas well if we are to recover equations (83) and (84) from the more rigorous equations (88)-(90). Thus, for example, if kl k , 1, the SSH for the Lotka system would produce spurious oscillations under certain conditions, but it is unknown how general this is, as no other oscillation mechanisms proposed for closed systems have been examined in this manner. However, it is quite likely that the region in parameter space where oscillations can occur will be considerably diminished. Rather difficult theoretical investigations are called for, particularly when highly non-linear thermokinetic oscillations are involved. This aspect is further discussed for thermokinetic oscillations in the appropriate section following, but the general conclusion is that it is far better to design experiments in open systems from the beginning, thus avoiding the difficult mathematics involved in treating closed systems, where the only rigorous steady state is the kinetically uninteresting one at equilibrium. --f

-

N

N

N

Open Systems: Genera1 Results.-The general results discussed above for closed systems depend very strongly on the principle of detailed balancing, and no such principle can be evoked in the case of open systems, although the consequences of the principle of detailed balancing with regard to rate constants still apply (i.e. there are relations between the rate constants for individual steps forming part of a closed loop in the kinetics). Nothing more can be said until the flux terms J1 in the basic equations are specifically discussed. As we are discussing homogeneous (stirred) systems only, in approximations to this state of affairs the flux terms will have to be represented by functions discontinuous at the boundary of the system, e.g. (93) J1 = aJx* - x:) where a, is a ‘permeability coefficient’ involving the physical dimensions of the system and xi is the concentration of species ‘i’external to the boundary of the system. In nearly all discussions, xi is treated as a constant, i.e. there is assumed to be an infinite reservoir of species ‘i’outside the system. This is strictly parallel to the infinite heat bath utilized when the energy flux JE is written according to Newton’s Law of Cooling

Reaction Kinetics

358

where Z involves the heat-transfer coefficient from the medium to the boundary of the system and also the physical dimensions of the latter. Expressions (93) and (94) are idealizations of Fick’s and Fourier’s Laws, respectively, necessary in order to evade the difficulty mentioned earlier of dealing with partial differential equations. A different form for the fluxes occurs when material enters the system as a result of direct mass flow in an open reactor. In this case J6 cc xi, the proportionality constant depending on the total mass flow rate. The general conclusion we can draw in this case as well as that described by equation dx4 (93) is that both fluxesgive a positive contribution to -and are thus formaZZy dt similar to an autocatalytic chemical step. From the discussion on p. 342, inequality (41), we see that this greatly favours the Occurrence of oscillations, and in fact ‘trivial relaxation oscillations’ discussed later can nearly always be produced in a flow reactor by manipulating the flow velocity so that the oscillation conditions are satisfied. Little else of generality can be said about open systems, except to point out again the fact that the singularities of the basic equations are exactly realized in these systems, hence they are easier to deal with theoretically and experimentally. Recently, Horn 12* has attempted to answer the question ‘When will an open isothermal system display “exotic” dynamics?’, where they do not define ‘exotic’ precisely but refer specifically to bistability and sustained oscillations. In view of the vagueness of the definition of ‘exotic’ it is surprising that the above authors produce a ‘theorem’ about this topic, albeit a negative one which delineates a large class of systems which cannot exhibit either of the above types of behaviour. Further work is clearly required in this area, particularly in improvingcommunication between physical chemistry and chemical engineering, Horn being particularly obscure in this respect. More definitive work is due to Hyver,12’who has shown that the characteristic roots of a linear chemical systems, whether open or not, cannot correspond to sustained oscillations. He does not assume detailed balancing but uses dx, not the condition that the coefficient of any term in the equation for dt involving x, (i.e. off diagonal) must have a positive sign. This means he is excluding zero-order reactions, which seems permissible if one is talking about elementary reactions. In a second paper, Hyver uses graph theory to show that a large class of non-linear open systems retain the property of non-oscillation.

126 lZ7

F. Horn, Proc. Roy. SOC.,1973, A334, 331. C. Hyver, J . Theoret. Biol., 1972, 36, 133; ibid., 1973, 42, 397.

Kinetics of Oscillatory Reactions

359

4 Specific Examples of Oscillating Reactions The Belousov-Zhabotinskii Reaction.-Zhabotinskii et a1.128*129 were the

first to propose a theoretical interpretation of the oscillations observed in this system. The equations proposed were not based on any experimental kinetic investigations and in view of more recent work they are not of great current interest. The experimental work of Noyes et 26 has now laid a firm foundation in this direction and we can safely assume that the equations describing the system are the following (x = [Br-1, y = [HBrO,], and z = [&*++I): aL2*9

i.e. the equations on p. 318, where the symbols are defined. In order to describe oscillations in a closed system, approximations are being made in using equations (95>--(97). In particular the 'rate constants' appearing in these equations include the concentrations of the primary reactants, [H+], [BrO;], etc., and it is assumed that these can be treated as constants as they are present in large excess. It was emphasized in the previous section that this is not always permissible unless the rate constants satisfy certain conditions, and clearly an investigation of this type, deriving equations (95)--(97) from the full set of equations for the system, including primary reactants, would be desirable. However, there will be some conditions where equations (95)-(97) will be valid, and it will be assumed that these are satisfied. Unfortunately some of the rate constants occurring in equation (96) are unknown, in particular klo and kll, but they are likely to be small. Neglecting these, as well as k-lo, considerable simplification occurs to give:

- = dy

dt

-2k12y2 + k,x -

and these equations have been derived by Field andl Noyes.lS0 Whilst these 1p8

12*

lSo

M. D. Korgukhim and A. M. Zhabotinskii, Mol. 3iophys. Nu& (Moscow), 1965,52. A. M. Zhabotinskii, A. M. Zhaikin, M. D. Korsukhim, and G. P. Kreitser, Kinetika i Kataliz, 1971, 12, 584. R. 1. Field and R. M. Noyes, J . Chem. Phys., 1974, in press.

Reaction Kinetics

360

equations are gratifyingly simple for such a kinetically complex system involving eleven species in all, they nevertheless contain three variables, and it has already been seen that much more can be said about the occurrence of oscillations in systems with two variables. It has been argued, on a qualitative basis, that bromous acid remains reasonably constant except when bromide ion reaches a critical concentration, whereupon it drops to a new (but almost constant) concentration. Field and Noyes plotted the steady-state bromous acid concentration as a function of bromide ion concentration. Unfortunately, they plotted a log-log curve, which gives the erroneous impression that the bromous acid concentration is independent of the bromide concentration except in a small region near the critical value, i.e. a step-like function. This error first appeared in ref. 25, with the statement: ‘Thus if the concentration of bromide ion is sufficiently great the residual rate of process B [equation (2)] is very small and independent of bromide concentration, while at very low concentrations of bromide ion the rate of process B is very much greater but is again independent of bromide concentration.’ This simple model suggests that the bromous acid concentration can be represented by a step function such that

where xcrit= k,/k, If this function is denoted by H(x - xWit)(it is unity minus the Heaviside unit-step function) the differentialequations become

-(z

or for the two regions of the xy plane (N= 1 or 0) dx dt -= for x < xcrit

+ k 5 ) x + k,z

Examination of the singularities of the equations for the region x < xCdt shows a singularity at the point

Kinetics of Oscillatory Reactions x,

361 kg2

= (k6k9

z,

=

f

2k5k12)

k g2

I I

J

2k12k3

Examination of the characteristic equation for this singularity shows it to be a stable node, and since it always exists (x, < xcrit),the system cannot oscillate as there are no other singularities in the region x > xcrit.Hence the qualitative arguments 25 put forward to describe the source of the oscillation in this system are incorrect. The lack of instability in equations (102)(105) arises because of the assumption that the bromous acid concentration is constant except near the critical bromide ion concentration, Using equations (95) and (97) again without this assumption, but with the steady-state approximation to equation (98), dx - -- -k6xyE dt

0

=

-2k&

-

kgx

+ k5x

+ k3z

-

kexy,

(95)

+ k9yE

(107)

and an instability can arise. Note that y, is now a function of x (not z), and this must be taken into account in evaluating the characteristic equation. The singularity in this case is at the point

z,

=

k9Y8 k3

Examination of the characteristic equation shows that this singularity can become unstable only if

($),is sufficiently large and negative. This does

not prove the existence of sustained oscillations, of course, but Field and Noyes 130 have solved equations (99, (97), and (107)numerically and obtained a limit cycle surrounding the unstable singularity. They also found that the instability only exists over a fairly small range of parameter values (see Figure 10). Field and Noyes have also treated equations (93, (97), and (98) numerically and obtained results very similar to those for the two-variable case. The results are shown as functions of time in Figure 1 1 . These results are sufficiently close to the observed oscillations to make it highly likely that the ‘Oregonator’, the name coined for the oscillator by Field and Noyes, is a very good theoretical description of the source of the oscillations in this

362

Reaction Kinetics

1.0

O: I

1

1

-4.0

I

1

-

I

I

-30 -20 -1.0 0.0 log T

I

I

1.0

2x)

I

I

30

LO

Figure 10 Limit-cycle behaviour calculated for the Zhabotinskii reaction; p dimensionless Ce4 concentration, 7 = dimensionless Br - concentration (Reproduced by permission from J. Chem. Phys., 1974,60. 1877)

=

+

-

3- f 21I

I

I

I

I

I

1

I

I

I

I

50

100

150

200

250

300

6.21 t / s

Figure 11 Concentration-time behaviour computed for the Zhabotinskii reaction (i) 10.30 log[HBrO,]/mol 1-1 (ii) 6.52 log[Br-]/mol 1-1 (iii) 7.62 log[Ce4+]/mol1-1 (Adapted by permission from J. Chem. Phys., 1974, 60, 1877)

+ +

+

363

Kinetics of Oscillatory Reactions

reaction. Field and Noyes also go on to discuss the nature of the roots of the characteristic equation for the three-variable case ahd plot out a region in parameter space where the singularity will be unstable (but not necessarily surrounded by a limit cycle). Recent work on spatial bands and similar non-homogeneous phenomena in this reaction is outside the scope of this ,chapter and will not be discussed here, although they are intimately connected with the homogeneous temporal oscillations. The Catalytic Decomposition of Hydrogen Peroxide.-In view of the experimental uncertainty still surrounding this reaction, as discussed in Section 2, appropriately little detailed theoretical work is available on this reaction. The only work assuming that the oscillations are of chemical origin is that of Degn l1 and Lindblad and Degn.131 They propose that the oscillations are of the relaxation type, iodine being produced by the reduction of iodate by hydrogen peroxide : and consumed by oxidation by hydrogen peroxide : 12

4- 5H202

=

2HI03 4- 4H20

Since the phase of the oscillation during which iodine is removed is accompanied by the evolution of oxygen, it is postulated that the oxidation of iodine is accompanied by the dismutation of hydrogen peroxide: 2H202

2H20

=

+

0 2

The last two reactions are claimed by Degn to show the properties of a branched chain reaction, and he proposes the following scheme for an open system where A is fed in continuously (N.B. most experimental work has been done on closed systems). ko -+

A

kbi

A+X-+2X kti

X --+

inert ki

reactant -+

X

kq

2 x -+

2Y

kb2

A +Y-+2X kt2

Y -+ lS1

inert

P. Lindblad and H. Degn, Acra. Chem. Scand., 1967, 21, 791.

364

Reaction Kinetics

There are two branching reactions, with rate constants k,, and kb2,and their appropriate termination reactions, with constants k,, and kt2. Quadratic chain transfer converts X into Y with rate constant k,, and it is assumed the Y propagates more efficientlythan X in the sense that ktl/kbl > kt2/kb2. It is then argued that, when A increases to a critical value, ktl/kbl,the chain reaction will begin branching, i.e. net branching will become positive. X then increases until the quadratic step becomes important. At a high concentration of X, there is a significant rate of conversion into Y and Degn considers the reaction to be governed by the Y branching in this region. The critical value of A for this chain is kt2/kb2and he postulates that the Y chain stops when A falls to this value. Since k,l/kbl > kt2/kb2, A will then be lower than the critical value for the X chain, so it will increase by replenishment, and repeat the cycle. Qualitative arguments of the above type cannot be accepted at face value, but numerical computations do produce oscillations which look qualitatively similar to the experimental ones in this system. However, a detailed analysis of the differential equations for this interesting scheme is desirable since it is not obvious why the system ever reaches the critical value for A(ktl/kbl)at the beginning of the cycle when X is small, since a steady state exists in this region, when the concentration (a) of A is less than this value. The differential equations involved are:

dx dt

--ki

+ kblax + 2kb2ay - k,,x

dY

-

2kqx2

(1 10)

+

- = -kbZaY - kt2y 2k,x2 (111) dt and in the early stages of the reaction where x and (therefore necessarily) y are small these equations clearly have a steady-state solution.

a,

= - kokt1

(k0 + ki)kbl

1

In general there are three steady states for this system, the values of x being given as the roots of the cubic zkq(kblkt2 - kb2ktl)x: + 2kqkb2(k0 + ki)-$

+

k0kb1kt2x kok,z(ko ki) = 0

+

(113)

When the coefficient of x: is negative, as assumed here, this equation has two positive roots and one negative root. One of the positive roots approximates

Kinetics of Oscillatory Reactions

365

+

(1 12) and the second approaches 00 as kblkt2 ensues and the limit will be given by

-+

kblktl;,i.e. a chain explosion

Whether the system would explode at this point depends on the stability characteristics and domain of attraction of the remaining steady state near (112), but if sustained oscillations are to be confirmed in this system, in that region one of the singularities must be unstable. The one which diverges is stable near the explosion limit (114) but its region of stability is not known. Clearly, in this system there is little scope yet for theoretical work until the basic reactions are understood at a level approaching that of the FKN mechanism for the Belousov-Zhabotinskii reaction. The Thermal Decomposition of Sodium Dithionite.-The only theory so far proposed for the interesting oscillations in this system (where the original reactant has periods of increasing concentration) is due to Bischoff and Mason.'' Again there is a great lack of detailed experimental work, but the following scheme is proposed: ki

A-+X

k2

X-+A k3

x-+Y

k4

Y

--+

inert

k5

Y-+A k6

A +X-+2X k7

X+Y-+2Y ks

A

+ Y -+

inert

Two cases are treated by computation : (i) k , = 0, which gives no oscillations in a itself, and (ii) k , = 0, k , # 0, which give oscillations in a itself. However, there is at the moment no chemical basis for this scheme (which is closely related to the Lotka mechanism, see below), and Mason and DePoy 132 have recently proposed a mechanism based on kinetics evaluated at temperatures where the oscillations do not take place. With certain modifications the mechanism can produce oscillations, and interestingly the production of colloidal sulphur has been invoked. lS2

D. M. Mason and P. E. DePoy, Faraday Symposia of the Chemical Society, 1974, to be published.

Reaction Kinetics

366

The Lotka Mechanism.-Lotka

125

showed that the reaction scheme kit

A +X-+2X X+Y---+2Y

k2' k3'

A

+ Y -+

inert

could give undamped oscillations in the concentrations of X and Y when the concentration of A was regarded as constant (i.e. the system was open to A). The differential equations studied were dx dt

- = kiax

-

k,xy

where a is constant. These equations have an infinite number of closed solutions around a singular point which is a centre. The amplitude and frequency of these oscillations depend strongly on the initial conditions, and this is not the case for any known sustained chemical and physical oscillations. A centre is a case of a critical singularity which is never realized in a physical system, it is structurally unstable,lo8and we have to look for stable limit cycles to describe sustained reliable oscillations. Nevertheless the Lotka mechanism has been extensively studied, particularly with respect to closed systems. Clearly, equations (115) can be regarded as approximations to an open system, into which A can diffuse: da _ -k,'ax dt -

dx

- kiay

- = k:ax

-

k;xy

dY -

-

kjay

dt

dt

=

k;xy

+ a'(a,

-

a)

1 J

and, in the limit a' --t 00, equations (1 15) are obtained. Here a, is the external concentration of A and a' the mass-transfer coefficient across the membrane defining the boundary of the system. If we write these equations in terms of the independent variable r = kit, with k, = k;/kg, etc. : da - -k,ax

-

dr

dx

- = k,ax dr

-

k,ay

xy

2 = xy - k,ay dr

+ .(a,

-

a)

1

Kinetics of Oscillatory Reactions

367

and for finite a those equations are not structurally unstable. They possess an unstable singularity and therefore the possibility of a structurally stable sustained oscillation exists. For example, with all the rate constants in equations (117) set equal to unity, with a, = 1, the singularity is at

and in its neighbourhood divergent oscillations certainly occur in x and y , but it remains to be shown that a limit cycle exists. The conditions under which equations (1 15) can legitimately be used to approximate a closed system (when a = 0) have been investigated in connection with combustion system^,^^^^ 134 and on p. 356 of this chapter, where it was shown that a sufficient condition for the equations in x and y only, i.e. dx - = k,ax dr

-

xy

<

to approximate the full equations (85)-(87) was that k , -h k , 6 xo/ao 1. The last inequality alone is not, as is commonly supposed, a sufficient condition for disregarding the variation of a. DePoy and Mason 134 have used a different criterion for the occurrence of damped oscillations in the closed system represented by equations (85)--(87). They require the frequency of oscillation of (83)--(84) to be much greater than the fractional rate of disappearance of a, i.e. very little a is consumed in one cycle. This gives klks 1, which would appear to be a necessary but not sufficient condition for the oscillations of (83)-(84) to be a good approximation to the full equations. An interesting, but so far unexploited, treatment of the full equations (85)--(87) has been given by Thomas.136 Owing to the homogeneous nature of the equations, by means of the transformation

<

ds --k,a, u dr

=

x/a, w = y / a

the system can be reduced exactZy to the two equations

-

ds

where y la'

=

l/kl and

=

=

ull

+u

- (7 - p)wJ

1

I

k3/k1. These equations can be treated rigorously

A. Perche, A. Perez, and M. Lucquin, Combustion and Flame, 1970,15, 89. P. E. DePoy and D. M. Mason, Combustion and Flame, 1973,20, 127. P. H. Thomas, private communication.

368

Reaction Kinetics

by phase-plane methods and they have two singularities, a saddle point at the origin and an unstable singularity in the physical region of the plane, which under certain conditions can be a focus, representing a divergent oscillation. The Carbon Monoxide 0sciUator.-This oscillator 136 is based on a kinetic scheme involving chain branching (or autocatalysis) with inhibition by an intermediate : kb

x-+2x kt 1

X -+

inert

kP

x-+X+Y

X

kq

+ Y -+ Y

kt2 -+

inert inert

where k, is the rate constant for branching, k,, and ktz are termination rate constants, k, is the propagation rate constant (propagation producing the inhibitor Y ) ,and k, the quadratic termination rate constant. The differential equations describing this system (with # = k, - ktl) are

and these have two singularities

For positive net branching (9 > 0) S, is a saddle point and Sz is stable, being either a node or a focus depending on whether

4$>
(123)

This system cannot exhibit limit cycle behaviour for $ < 0; the singularity at the origin is the only physical one and it represents a state of no reaction. This system can therefore show three distinct types of behaviour: (i) no reaction when 4 < 0; (ii) damped oscillation when q5 > 0 but less than a critical value; and (iii) explosive behaviour when q5 is greater than a critical value. The carbon monoxide-oxygen system shows all three types of behaviour experimentally, and Gray 136 has proposed this scheme to describe las

B. F. Gray, Trans.Faraday SOC.,1970, 66, 1118.

369

Kinetics of Oscillatory Reactions

the CO-O2 systems where X = 0.and Y = C,O. There is evidence that C,O inhibits the explosive reaction. YanglS7has also proposed this scheme to describe the C0-0, system, but with a different physical interpretation. Yang also assumes that X = 0but takes Y as excited carbon dioxide C02*; the analysis and results are identical with those already derived. Yang has also modified the above schemeto includea surface reaction, such that the surfacecan becomesaturated if the reaction rate becomes high. If wall-termination of oxygen atoms is assumed, with a saturation effect possible for this reaction, then limit cycles can be produced. In this case the equations (121) are modified to

dY dt

=

k,X - k t g - k,xy

dx k = (9.- -)x - k,xy dt 1+RX where #', k, and R are composite rate constants. For variations of temperature, the singularity S2 can be shown to undergo bifurcation, i.e. the appearance of a limit cycle surrounding an unstable focus from a stable focus. It is likely that the CO oscillations are described by a limit cycle rather than a stable focus, but further experimental work is required to clarify the physical interpretation of this model. End-product Inhibition.-This type of scheme has been discussed mainly in the biochemical area, e.g. by Walter.13* Many reactions are postulated which look strange to gas-phase kineticists, such as co-operative inhibition of an enzyme by many product molecules, giving individual reaction steps of high molecularity. A typical scheme involving n + 1 substrates (S,) is the following: EO El

so -+ s1-+

I

s, ............

-

where the system is assumed to be open to substrates So and S, + 1, and p molecules of S, + co-operatively inhibit the enzyme E,, (the minus sign indicating inhibition or negative feedback). The differential equations for this type of scheme are

137

C. H. Yang, Combustion and Flame, 1974, in press. C. F. Walter, J . Theoret. Biol., 1970, 27, 259.

370

Reaction Kinetics

The simplest case, with two variables, has been treated by Griffith.139 He showed the single singularity to be stable, and the Bendixson Negative Criterion indicates the impossibility of limit cycles. This finding is in conflict with results of G o o d ~ i nwho , ~ ~claimed ~ oscillations for p = 1 on the basis of analogue computer studies. Since Griffith's result is rigorous, sustained oscillations certainly cannot occur in this system. Griffith has also dealt with the case of three variables, being able to show for p < 8 that the only singularity is stable, but limit cycles may be possible for p > 8. Walter 141 claims that there are no limit cycles for n = 1, 2, or 3, but in another publication 142 he concludes that a limit cycle is possible for p > 8, n = 2 and p > 4, n = 3 on the basis of computer simulation. In view of the many parameters involved, these numerical results must be accepted with caution, and are of a rather speculative nature as far as physical chemistry is concerned at present. Further references to the literature in this area can be found in the review by Nicolis and P o r t n o ~ . ~ Glycolytia: Oscillations.-Higgins 143 was the first to attempt a mathematical model of the oscillations now easily produced in glycolysis in cell-free extracts. His model consisted of a first-order production reaction for the substrate, a product-activated enzyme-catalysed step, and an enzymic removal of the substrate. Steps of all these types are known experimentally in the system, for example the subset of reactions

F6P

GLU -+

F6P

+ El* -+-

Ej,*F6P

EI*F6P -+

+ ET -+El* FDP + Ez

El*

+ FDP

FDP

-+

E,FDP-+

EzFDP Ez

+ GAP

where El* is activated phosphofructokinase, E2 is a combination of aldolase and triose phosphate isomerase, GAP = glyceraldehyde phosphate, GLU = glucose, F6P = fructose-6-phosphate, FDP = fructose diphosphate, and E: is the inactive form of phosphofructokinase. The whole glycolytic chain is much more complicated than this, and metabolites besides the above also oscillate. However, if the above six reactions can generate oscillations, then driven oscillations will occur in chemicals further down the glycolyticpathway, with increasing phase differencesthe further they are from the basic oscillator. The two differential equations derived from this model are 139

140 141 142

143

J. S. Griffith, J. Theoret. Biol., 1968, 20, 202. B. Goodwin, Adv. Enzyme Regulation, 1965, 3, 425. C. F. Walter, J . Theoret. Biol., 1969, 23, 39. C. F. Walter, Biuphys. J., 1969, 9, 863. J. Higgins, Proc. Nat. Acad. Aci. U.S.A., 1964,51, 989.

Kinetics of Oscillatory Reactions

dY - = kp[ 1 dt

371

J

+ (1 + x)y -k3Y] XY

where x and y are the dimensionless concentrations of the substrate and product activator respectively (F6P and FDP). Sel’kov 144 has criticized this model strongly by applying the Bendixson Negative Criterion to equations (126), and thus showing that a closed cycle cannot exist in the region x, y > 0. Equations (126) are derived on the basis that the enzyme concentrations are far smaller than those of the two variables retained, x and y (SSH).’Higgins obtained limit cycles by varying the rate constants in his scheme arbitrarily, but departed from the realms of physical Sel’kov has proposed an alternative scheme for the source of the glycolytic oscillations:

-

ATP

+ El*

ATPE1*

ATPEl* + El* pADP

+ EC

+ ADP *

El*

where ATP = adenosine triphosphate and ADP = adenosine diphosphate. E l is again phosphofructokinase, and p is the number of product molecules necessary to activate the enzyme. Three of the five variables are removed from consideration by means of (SSH)and we are again left with a binary system

dY - = ky(xyP-1 - 1) dt

i

J

These equations have only one singularity at x , = ys = 1 (the variables are again dimensionlessconcentrations). It is easily shown that, for k > l / ( p - l), this singularity is unstable. This information, together with knowledge of the behaviour of the trajectories for large values of x and y, enabled Sel’kov to show that a limit cycle does indeed exist in this system. He confirmed this conclusion by numerical computation. The limit cycle encloses the singularity at (1,l) and trajectories from the latter wind outwards onto the limit cycle. This would correspond to oscillations of increasing amplitude in a physical system, the amplitude eventually decreasing if the system were closed. Recently, Goldbeter and Lefever 146 applied an analysis of an allosteric enzymemodel to phosphofructokinaseandobtained limit cycles on an analogue computer which agreed well with experimental results. They predict that sustained oscillations are favoured by substrate activation. 145

E. E. Sel’kov, EuropeanJ. Biochem., 1968, 4, 79. A. Goldbeter and R. Lefever, Biophys. J., 1972,12, 1302.

372

Reaction Kinetics

The Thermokinetic Oscillator of Sa1nikov.-We now turn to systems in which the temperature of the reacting mass has to be treated as one of the dependent variables. In principle this will always be the case, but often in dilute solution the heat changes due to the chemical reactions are absorbed by large quantities of solvent. This is not the case in gas-phase reactions, nor in reactions in any phase where large amounts of heat can be released in short periods. In these cases variations of temperature can be characterized by a similar timescale to that of the concentrations themselves, and on analysis it becomes clear that variations of temperature are very likely to produce oscillations in a kinetic scheme which could not oscillate isothermally. In a highly original paper, Salnikov 146 treated three simple kinetic schemes involving a single intermediate produced from a precursor whose concentration was assumed to be constant. We shall discuss the best known case, i.e. the scheme ki

kz

B

A-+X-+

where the concentration of the intermediate X is the only variable considered,

so

We now simultaneously consider the energy conservation equation, which is d-T - hk,x dt

-

I(T

-

To)

(129)

where I is proportional to the heat-transfer coefficient, T is the temperature of the reacting medium, To is the temperature of the surrounding heat bath, assumed constant, h = -AH2/pc where AH, is the enthalpy change associated with the second reaction, p is the mean density of the medium, and c is the mean heat capacity per unit volume of the medium. Now since k, = A2e-E@Tequation (129) is highly non-linear (even though in this simple model the heat release associated with the reaction A + X is assumed to be zero). Salnikov dealt with equations (128) and (129), using what are now standard phase-plane methods to obtain their qualitative behaviour. The singularities of these equations in the phase plane occur at

and, depending on the numerical values, the second of these equations may have one or three solutions (this is most easily shown graphically). From the characteristic equation for this system u6 I. E. Salnikov, Zhur. $2. Khim., 1949, 23, 258.

Kinetics of Oscillatory Reactions

373

it is easily shown that for an unstable focus to exist it is necessary that

and by further analysis along standard lines it can be shown that a limit cycle does indeed exist for this system under these conditions. Salnikov also treated schemes where X was produced autocatalytically and at constant rate from the precursor A. In all cases a general condition for oscillation in the system is that the temperature dependence of the second step be greater than that of the first step. In many reactions involvinga transient intermediate X this will not be the case, and so this system does not seem to have a direct experimental counterpart; however, when the first stage is simply physical feed-in of X (now the reactant) the same considerations apply and, since the latter is usually independent of temperature, the conditions for oscillation will be met. The equations are used in slightly modified form in the next section, on the stirred tank reactor. Thermokinetic Oscillations in a CSTR.-Oscillations in continuous stirred tank reactors have long been known to chemical engineers 14’ and are still provoking interest s8 at present. They can give rise to concentration oscillations under a very wide range of conditions owing to the interaction between the chemical rate process and the physical flow into the reaction. As far as the basic theme of this chapter is concerned these oscillations are trivial, in so far as the simplest possible first-order irreversible reaction can give rise to them, and indeed Frank-Kamenetskii l o o has suggested the title ‘trivial relaxation oscillations’ for them. Nevertheless, from the present standpoint it is important to understand both the phenomenology and mathematical origin of these oscillations if non-trivial chemical oscillators are to be studied in CSTRs, a procedure which has many advantages over classical procedures involving closed systems. The trivial oscillations are characterized by large variations in the concentration of the feed (reactant) species, and they are only bounded above by exhaustion of the latter and below by the feed-in rate (the an?logy of a dripping tap is a fairly exact one here). For a simple irreversible first-order reaction the mass and energy conservation equations are: lP7

R. Aris and N. R. Amundsen, Chem. Eng. Sci., 1958,7, 121; R. B. Warden, R. h i s , and N. R. Amundsen, ibid., 1964, 19, 149.

Reaction Kinetics

374 da _ dt

-u

(a,

-

a)

-

ka

(133)

dT 4 = hka - (- + 1)(T - To) dt where q is the volumetric flow rate through the reaction and u is its volume. The other quantities occurring are defined in the previous section. Equations (133) and (134) are clearly of the same mathematical form as equations (128) and (129), so limi t-cycle behaviour can arise. However, equations (133) and (1 34) contain an easily controlled parameter q, so a given chemical system can be made to oscillate at will. Examination of the singular points via the characteristic equation in the usual way shows an unstable singularity occurs when hE 2q < -R(T,)a, - (1 t k(T,)) (135) u RT,~ and when this is the only singularity (under certain conditions there may be three) it is easily shown that a limit cycle must surround this singularity. From the inequality (135) it is clear that, for a limit cycle, it is necessary that (i) h > 0, i.e. the reaction is exothermic, (ii) E is sufficiently large, and (iii) q is sufficiently small. These criteria reinforce the intuitive notion of relaxation oscillations, i.e. if the reaction is sufficiently accelerative and the flow rate is sufficiently small, then a series of exhaustions and refills of the reaction will result. The oscillations in the chlorination of methyl chloride described by Bush are of precisely this type; Bush was able to set up conditions for the reactor where inequality (1 35) was satisfied with relative ease. From the point of view of chemical oscillations an unexplored but very interesting problem arises when the feed material A can give rise to chemical oscillations itself, of the type which could occur in a closed system. Clearly, the oscillations will continue indefinitely in the CSTR and both theoretical and experimental analysis will be considerably simplified provided clear distinction can be made from the trivial relaxation oscillations. Such an experimental programme has already been carried out D g but the theoretical basis of the system, which will have two limit cycles, one due to the trivial oscillations and one due to the ‘interesting’oscillations, has yet to be investigated. Thermokinetic Models of Organic Oxidation Oscillations.-Oscillatory combustion (multiple cool flames) is a commonly observed phenomenon in non-isothermal organic oxidations, as outlined on p. 321, and the simplest theoretical thermokinetic model is that due to Salnikov (see p. 372). This model pointed the way to a slightly more complex 149 devised 148

lPB

C. H. Yang and B. F. Gray, Trans. Faraday SOC.,1969, 65, 1614.

C.H. Yang and B. F. Gray, J . Phys. Chem., 1969,73,3395; B. F. Gray, Trans.Faraday SOC.,1969,65,1603; C.H.Yang,J. Phys. Chem., 1969,73, 3407.

Kinetics of Oscillatory Reactions

375

to explain the often simultaneous occurrence of oscillations, a negative temperature coefficient in the reaction rate, and the existence of a lobe on the explosion-limit curve. The latter two phenomena are not of direct interest here, except that they are intimately connected with the bifurcations that can occur in this model. The kinetic model used by Yang and Gray for the source of the oscillations in this case is ki

A--+X kb

x-+2x kti

X -+

inert

X

kt2 -+

A

--+

inert

k

inert

and associated with each reaction is a corresponding quantity h = -AH/c, i.e. the heat evolved by each reaction in suitable units. Two termination reactions are necessary here and their rate constants are chosen so that the activation energies satisfy the inequality Et,

E b

Et,

(136)

for reasons which will become obvious later. If we write the net branching factor q5 = k, - k,, - kt2and define the thermal factor of the chain reaction 6 = kbhb + ktlhtl + kt2h,, then the differential equations for the model are: dx = ki dt

+ $X

dT - = go+ ex - i(T - T ~ ) dt gois the total. heat release rate of the initiation reaction k,and an alternative simultaneous non-chain reaction of the reactant A, which is assumed to be present in large excess, hence da/dt 0. The side-reactions goare not an essential feature of the oscillator, but are included for other reasons of little interest here. The singularities are given as solutions of

-

which gives on eliminating x,

Ok,

9

0

-d = f(Tu - To)

376

Reaction Kinetics

and this equation is best dealt with graphically by plotting the overall heatrelease rate at the steady state (the left-hand side) against the straight line given by the right-hand side. There can be up to five singularities if inequality (136) is satisfied, since this ensures that the heat-release function has a maximum and a minimum for finite values of the temperature, and hence a negative temperature coefficient over a certain limited temperature range. Investigation of the nature of the singularities for this system gives the condition for instability as

and the condition for a focus to be

where B(T) = 9?o(T) - ek,/#. Since we must have # (TJ < 0 as x, > 0, (142) will be greatly facilitated by the condition dB -d
(143)

so it becomes clear that the appearance of a limit cycle in this system is greatly favoured by the existence of a region of negative temperature coefficient in the reaction rate curve. Substituting into (141) for (dp/dT), and remembering that < 0, it is necessary that

+

dBo d8 - +x,-l>O dT dT for an unstable singularity to exist. Substituting similarly in (142), the condition that it also be a focus gives d9o dT

do

dx,

+xadT + e d- T- i < o

(145)

and these two inequalities together require

-dx,
dx, d > -{l(T dT dT

- - 8-

- To))

(147)

377

Kinetics of Oscillatory Reactions

Rate

Release Rate /Heat

Loss Rate

TEMPERATURE

Figure 12 Unstable singularities in a region of thermal stability in a non-isothermally reacting system. Tt, Tt are two ambient temperatures, T being the temperature of the reacting medium

since without the second term on the left-hand side this inequality is the familiar one much used in thermal explosion theory, where the only independent variable is the temperature. The second term on the left-hand side represents the effect of the chemistry on the thermal stability, and in normal circumstances when dx,/dT > 0, this effect will be stabilizing. However, in the oscillatory region, where dx,/dT < 0, the opposite is true and the singularities which occur i s the region of negative temperature coefficient (see Figure 12), and appear to be stable (d9/dT < 1) according to simple thermal theory, are unstable according to (147). This point has caused many difficulties in the past 74 in the interpretation of cool-flame phenomena on a ‘thermal switch’ basis. Equations (137) and (138), if used to describe a closed system, clearly do not include the variation of the concentration of the original material with time, and they should therefore be supplemented by a discussion along the lines for other systems of equations (see p. 354). For example, if the initiation and branching reactions were first order in A, we would have a system of three equations to replace (1 3 7 H 1 3 8 ) : da - = dt dx dt

-kia

- = k,a

-

kbaX

+ (kba

-

k,,

-

kt2)x

378

Reaction Kinetics

The conditions under which these equations can be meaningfully approximated by (137)-(138) have not yet been investigated in any rigorous fashion, although if they are taken pairwise the situation is reasonably well understood. For example, the conditions under which da/dt can be neglected with respect to dT/dt have been extensively investigated in connection with thermal explosion theory 11**150 and essentially this is permissible if the exothermicity and activation energy are sslfficiently large. Likewise neglect of da/dt with respect to dx/dt is discussed in this chapter for a number of systems, and usually results in a relationship between the ratios of the species X and A and certain rate constants. Generally, xo/aohas to be small and the rate constants for reactions not involving X (e.g. ki) also have to be small, i.e, xo/ao< 1 and kJk, 1 etc. Without a complete analysis one would expect that (137)(138) would give a good description of the system when the intermediate concentration is small and the exothermicity of the reaction is large. This is probably equivalent to the requirement of DePoy and Mason 34 that the amount of A consumed during one oscillation period is small. A different approach to cool flame oscillations is taken by Halstead, Prothero, and Quinn,lsl who state: ‘In order to make the best use of modern computers and computing techniques one must begin by constructing the most detailed model amenable to computation, so taking account of accumulated experimental evidence. It is only at a subsequent stage and in the light of the results of such detailed computations that any simplification of the model should be attempted.’ Few people would agree with the implication that the role of theoretical chemistry is simply curve-fitting, and of course the results of such computations are only as valid as the particular data used. For two reactions in the complete acetaldehyde scheme simulated, the rate constants used do not appear to be the best available. Thus for the reaction 11’9

<

CH,CO.

+M

--f

CH3-

+ CO + M

an activation energy of 45.2 kJ mol-1 given by Benson 152 rather than the more recent value of O’Neal and Benson lS3of 63 kJ mol-l is used. Similar comments apply to the reaction CH,C(O)OO* + CH,CHO

-+

CH,C(O)OOH

+ CH,CO*

where the authors have arbitrarily assumed a value of 42.0 kJ mol-l rather than the experimental value of 29.4 kJ although they may have used this value in a later paper. In this type of calculation, which undoubtedly can be useful, choices of this type have an effect on the results, which cannot be evaluated by a critical reader without performing the whole series lS0 151

lS2 153

15Q

B. F. Gray, Combustion and Flame, 1973, 21, 313, 317. M. P. Halstead, A. Prothero, and C. P. Quinn, Proc. Roy. SOC.,1971, A322, 377; Combustion and Flame, 1973, 20, 21 1. S. W. Benson, ‘The Foundation of Chemical Kinetics’, McGraw-Hill, New York, 1960. H. E. O’Neal and S . W. Benson, J . Chem. Phys., 1962,36,2196. C . A. McDowell and L. K. Sharples, Canad. J. Chem., 1958,36,268.

Kinetics ef Oscillarory Reactions

379

of computations repeatedly himself. From their computational results using nine simultaneous differential equations, Quinn et al. attempt to extract the basic oscillator. They eliminate five of the concentrations by holding them at their instantaneous steady-state values for any given temperature. The computed results are gratifyingly similar to those of the full system except in the lower temperature ranges, where the chemical approximations would be expected to break down. The three equations solved are of the general form of (148)-(150) (although the detailed kinetics are different), i.e. one equation is for fuel, one for the most important branching intermediate, and the third for the temperature. An attempt is then made to analyse these equations by the methods of stability theory as outlined in the earlier sections of this article. As the authors themselves point out, these three equations do not have any singularities except the trivial one at equilibrium. Nevertheless they define ‘quasi-singularities’which move according to the equation of fuel consumption, and which would be singularities exactly if fuel consumption were neglected. These ‘quasi-singularities’ change their character as they move (e.g. stable- unstable etc.) and the authors assume that the nature of the latter determines the nature of the trajectory of the system. Unfortunately the method appears to be mathematically unsound and a proper treatment would appear to warrant the application of the second method of Liapounov. During a period when the actual trajectory of the system undergoes two oscillations around the ‘quasi-singularity’,the ‘quasisingularity’ itself undergoes no less than five changes of characteristic, and one might be forgiven for asking ‘How quasi is quasi?’. The alternation of amplitude of successivecool flames which this work produces computationally has since been shown for the case of propane 41 to be an experimental artefact due to convective gas motion, disappearing completely in a well-stirred vessel. As the theoretical model assumes perfect mixing at all times, the computed effect must be in fortuitous agreement with the earlier unstirred experimental results. The acetaldehyde system, in spite of the attraction of its well-understood isothermal kinetics, is not a good system for studying oscillations in a closed system, since it rarely manages more than two pulses, and is very far from the region where fuel consumption can be either ignored or approximated by any ‘quasi-steady’ methods. Probably a necessary condition for such a method to be justifiable would be that the phase portrait retain a given characteristic for a period which is long compared with the period of the oscillation. The trajectory could then wind onto a limit cycle which was itself shrinking or growing slowly. In the acetaldehyde case the limit cycle is growing to a maximum, and shrinking to a point (in the stable period), in a period which is of the order of that of the oscillations themselves. Clearly a desirable goal would be to study this system in a flow reactor, where sustained oscillations could be achieved, but there are presently experimental difficulties due to the high reaction rate and relatively low volatility of acetaldehyde.

380

Reaction Kinetics

A recent discussion of oscillations involving coupling of the energy equation with a kinetics equation, which does rather less than justice to the literature, is due to Nitzan and Amongst others, they discuss the simplest possible reaction ki

A$B kz

but with photochemical removal of A present to break detailed balancing. The differential equations are da - = k2a0 - ( k , dt dT -= la +- h(kl + k,)a dt

+

-

hk2ao - l(T

-

To)

(152)

with the usual notation, I being a constant proportional to the light intensity and absorption coefficient. They derive the above equations by assuming, amongst other things, that light absorption by A and subsequent relaxation processes producing heat are much faster than the thermal reactions. It is not clear how equation (151) arises on this basis, with no removal term due to radiation. However, on the basis of these equations the authors present a rather lengthy discussion of the steady states, stability, hysteresis, etc., which can be shown to occur for this system; indeed, equations (151) and (152) are none other than a special case of equations (137) and (138), derived to describe thermokinetic oscillations.149The interpretation of the terms is different, but the discussion of stability, hysteresis, etc., follows automatically, in fact the scheme of Nitzan and Ross more nearly resembles the equations of Sdnikov, or their equivalent for a CSTR. Nitzan and Ross conclude that only damped oscillations are possible in this system, and then only when the forward reaction is endothermic (this is a special case of the result in ref. 144 that a negative temperature coefficient is required). A discussion of the isothermal triangular reaction system is given, and it is pointed out that if detailed balancing is broken by illumination, then oscillatory relaxation to the steady state can occur. Nitzan and Ross conclude that they have shown the possibility of oscillation, multistability, and hysteresis for the first time in a closed system where mass does not cross the boundary. This, of course, is incorrect as the whole of the combustion literature abounds with such examples, not to mention the Zhabotinskii reaction and others. The ‘Brusselator’.-This is a kinetic scheme devised by the Brussels school 156-168 (hence the name) as a chemical oscillator: A. Nitzan and J. Ross,J. Chem. Phys., 1973, 59, 241. R. Lefever, J . Chem. Phys., 1968, 49,4977. lJ7 I. Prigogine and R. Lefever, J. Chem. Phys., 1968, 48, 1695. lb8 B. Lavenda, G. Nicolis, and M. Herschkowitz-Kaufman,J. Theor. Biol., 1971.32.283. 155

lS6

38 1

Kinetics of Oscillatory Reactions kt

A--tX k2

B 2x

+ X -+

Y

+ inert

k3

+ Y-+3X k4

X -+

inert

where the concentrations of A and B are assumed constant in time, i.e. in contact with an infinite outside reservoir, and the system is open. This scheme looks somewhat implausible chemically, particularly the termolecular step ; however, Tyson l S a has replaced this by a sequence of enzyme reactions. The differential equations describing the original model are dx dt

- -- kl

-

(k2

+ k 4 ) +~ k3x2y

(1 53)

where the constant concentrations of A and B have been absorbed into the rate constants. The only singularity of (153)--(154) is x , = - k,

and the characteristic equation is kz - k4 - il

I

4

k2k 4 k&3

ys=-

k4

(1 5 5 )

k,kZlk%

2

from which we deduce the necessary condition for instability

Under certain conditions the unstable singularity can be shown to be surrounded by a stable limit cycle, thus giving sustained oscillations. It is interesting to enquire whether equations (154) and (155) represent a real chemical system without the unrealistic constraint that A and B, the original reactants, be held at constant concentration. If the system is closed, but A and B are initially in great excess over X and Y,the system will be described by four equations: da -- -kia dt

15*

J. J. Tyson, J. Chem. Phys., 1973, 58, 3919.

Reaction Kinetics

382

db dt -

5

-kibx

(159)

dx - -- k;ax dt

-

kibx

k;bx

-

k,x2y

dy

- 5

dt

+ k3x2y - k4x

where k;a = k , and kib = k,. If we introduce dimensionless concentrations a' = a/ao,b' = b/a,, x' = x/x;, xo/ao= E 1, and the dimensionless time z = k4t, these equations become

<

dx' dt

kia' k; - ao-b'x' ~k4 k4

k3 + E2aE-x'2y'

=-

dy' k6 = ao-bb'x' dt

k4

k4

- &ZQ2

k k4

0-x

'2

N

k4,k;

N

k4/ao,k ,

x'

Y'

and for these to approach equations (153)-(154) must have ki

-

N

in the limit as

k41e2$

E -+

0 we

(1 66)

and these conditions are necessary for A and B to be treated as constants, even when present in large excess. The instability condition (157) applied to (163) and (164) using equality signs in (166) when substituting leads to the absurdity 0 > 1. If some leeway is allowed in (166) it is possible that the instability condition may be just at the limit of the region where a and b can be treated as constants, but clearly the Brusselator, termolecular reactions apart, would not be likely to show oscillations in a closed system, after the manner of the Zhabotinskii reaction for instance. Its oscillatory properties appear t,o be intimately connected with the assumption that a and b can be maintained as constants. This necessitates that the system be open with respect to A and B, requiring a very peculiar flux term to maintain this condition. For A and B in an open system equations (158) and (159) have to be replaced by da -- -kia -k J A dt db -- -k@x -I-J B dt

Kinetics of Oscillatory Reactions

383

Thus (161) and (162) become

Conditions (166) are still necessary for (164) and (165) to be approximated well by equations (153) and (154), but from (169) and (170) we also need

which reduce to and if the fluxes are of the diffusive type JA

a,

-

a

N

=

aa(a, - a ) then we must have

k4xO/aA

(173)

and this of course is possible, but it implies careful balancing of the internal and external properties of the system, and one would not expect the resulting oscillations to be structurally stable over a significant region of parameter space. Further investigation is required along these lines, particularly with regard to other 'oscillators' where the reactants have been arbitrarily maintained at constant concentrations.leO Miscellaneous Oscillators.-Karfunkel and Seelig lel have discussed an interesting hypothetical model for enzyme catalysis involving inhibition of the enzyme and reactivation of it by the product of the reaction

S

+ E +ES

+E

+P

E+I+EI

nP

+ I + PJ

where I is an inhibitor and the product P acts as a reactivator of the enzyme E. This system shows multistability and relaxation (discontinuous) oscillations between the two stable steady states. Seelig le2 has also discussed a linear open reacting system involving a loop which he claims shows sinusoidal oscillations. Rossler has discussed a discontinuous oscillator (multivibrator) involving two variables (assuming constant pools for reactants) satisfying the equations ld0

lel le2 le3

M. Herschowitz-Kaufman and G. Nicolis, J. Chem.Phys.,1972,56, 1890. H. R. Karfunkel and F. E. Seelig, J. Theoret.Biol., 1972,36,237. F. F. Seelig, J. Theoret.Biol., 1970, 27, 197. 0. E. Rossler, J. Theoret.Biol., 1972, 36,413.

3 84

Reaction Kinetics dx _ - klx dt

2 dt

=

k,x

-

-

key

If y is treated as a parameter, (174) has three steady states, one being unstable, and can obviously exhibit hysteresis. If y is allowed to vary slowly - the sole purpose of equation (175) - then discontinuous oscillations between these steady states can occur. This work is of little chemical interest as no attempt is made to ensure that the equations can represent a kinetic scheme. Degn 110 has discussed electrochemical oscillations of current at constant external potential, which are found with many electrode systems. It is interesting that his model has a negative slope in the current-potential curve and the oscillations are of the relaxation discontinuous type. A number of papers giving similar discussions at a phenomenological level, as well as experimental results, have been reviewed very briefly by Nicolis and port no^.^ Further work in this area, involving the concept of bistability, is that of Epelboin et al.ls4 The relaxations between activity and passivity on the part of the electrode have not as yet been given a kinetic interpretation, and it is not clear at the moment how important chemical reaction rates are in this field. Many oscillations occur in biological membrane systems, whole cells, muscle fibres, and so on. Their connection with oscillatory chemical reactions is somewhat tenuous and they will not be discussed here, nor will the circadian rhythms now believed to be present in practically all living species, and discussed from a biological point of view elsewhere.lss 5 Conclusions The subject of oscillatory chemical reactions has moved, in a little over ten years, from being a chemical curiosity with some doubts as to its legitimacy to being a fairly intensively studied area with some examples now well understood. The situation in closed systems (the type traditionally used by the chemist) is now well understood from an experimental and theoretical point of view, and there is no cause for discussion as to the legitimate existence of oscillations in such systems, provided it is realized that the oscillations cannot enclose the equilibrium situation. They are in the region of the ‘quasistationary state’ and whilst some work has appeared in an attempt to quantify this by using singular perturbation theory, much remains to be done when the system has the ability to oscillate. This problem could be side-stepped completely if the traditional chemical approach of using closed 164

165

I. Epelboin, C. Gabrielli, M. Keddam, and H. Takenouti, Compt. rend., 1973, 276, C, 145; I. Epelboin, C. Gabrielli, M. Keddam, J. Lestrade, and H. Takenouti, J . Electrochem. Sac., 1972, 119, 1632.

‘Circadian Clocks’, ed. J. Aschoff, North-Holland, Amsterdam, 1965.

Kinetics c f l Oscillatory Reactions

385

vessels were abandoned and continuous flow systems used wherever possible. Parameters such as frequency, amplitude, and phase would then be wellcharacterized and presumably sensitive functions of the rate constants involved in the oscillator. Perhaps in the near future the decomposition of hydrogen peroxide will be exhaustively studied from an experimental point of view, as the existence of a well-understood model physicochemical oscillator is essential if the more complex biochemical oscillators are to be understood on a firm (as opposed to the presently speculative)basis. Questions such as the interaction of chemical oscillators, entrainment, frequency shifting, and so on, which are of great interest in biochemical systems, can hardly be studied profitably from the chemical kinetic point of view until a number of relatively simple oscillators have been fully analysed, the Zhabotinskii reaction being the furthest advanced at the moment from this point of view. Little attention has been paid to the coupling of the energy conservation equation via the temperature to the rate equations except in combustion systems, presumably on the grounds that in the fairly dilute liquid systems often used temperature changes are very small. Nevertheless they are there, and may not always be driven by an isothermal oscillator as is presently assumed, since the energy equation lends itself very well to the production of oscillations due to the highly non-linear character of the rate equations when temperature is regarded as a variable. The possible uses to the organism of biochemical oscillations and the uses, if any, to which simpler chemical oscillators can be put are fascinating questions which are very far from being answered at present. On the mathematical side, oscillators with two variables can be handled analytically and visualized elegantly by using the methods of the phase plane. Any given kinetic scheme involving only two dependent variables can be examined for the absence of oscillatory characteristics using well-known negative criteria and, as a result, general positive criteria for oscillations have emerged for such systems. For systems with more than two variables the situation is not so well characterized since, although in certain situations sustained oscillations can be ruled out (e.g. when Hurwitz stability criteria are satisfied), little else can be said about the likely occurrence of oscillations except that they will be closely associated with the occurrence of instability and this in turn, in a chemical system, will be due to positive feedback via autocatalysis or self heating. Beyond this, resort to computation is often necessary, but this itself appears to present few problems, and it is not heavy compared with other branches of theoretical chemistry. However, it is well to remember that numerical computations are very specific and highly dependent on the numerical values of the rate constants used and, particularly with non-linear systems, it is impossible to draw general conclusions from numerical results. On the analytical side, one very quickly reaches the boundaries of available mathematics when more than two variables are involved, and it appears that oscillating chemical reactions will simply have to join the queue of subjects waiting for new results in non-linear mathematics. In a slightly different direction, one would expect to see treatments of

386

Reaction Kinetics

oscillating reactions involving controlled variables appearing in the near future, particularly from the biochemical side, where the possibilities of highly selective synthesis using the concepts of chemical oscillators would seem a fruitful area for research.

Author Index

Abbott, J. R., 322 Abraham, F. F., 245 Abrahamson, E. W., 23 Abramson, A. S., 104 Acrivas, A., 353 Acuna, A. U., 18,46, 124 Aiken, R., 303 Airey, J. R., 27 Aivazov, B., 321 Agarunov, M. J., 234 Akbai, M., 164 Akimoto, H., 124 Akins, D. L., 264 Albers, E. A., 174 Albritton, D. L., 84 Alfassi, Z. B., 128, 171, 172, 190

Allera, D. L., 194 Allred, E. L., 149 Alonso, J. H., 132 Amundsen, N. R., 373 Anderson, D., 164 Anderson, F. W., 158 Anderson, J. B., 96 Anderson, J. D., 245 Anderson, K. H., 185 Anderson, R. J., 57 Andlauer, B., 110 Andrews, G. D., 142, 146 Andronov, A. A., 340 Anlauf, K. G., 24 Appleby, W.G., 172 Appleton, J. P., 30, 254, 268,269

Arai, M., 132, 134, 146 Archie, W. C., 138 Aris, R., 332, 345, 349, 373 Arnoldi, D., 54 Armstrong, D. A., 32, 127, 157

Ashmore, P. G., 318, 323 Ashton, J., 31, 251, 277 Atherton, J. G., 203 Atkinson, R., 21, 23, 27, 59, 120

Atton, D., 220 Augustin, S. D., 47 Austad, T., 155 Avery, N. L., 101 Avery, W. H., 172 Babayan, V. I., 96 Babich, E. D., 212 Back, M. H., 177 Bailey, H. E., 254 Bailey, W. J., 212 Baird, R. B., 231 Baker, R. R., 165,185,201, 203

Balakhuin, V. P., 54 Baldwin, J. C., 234 Baldwin, J. E., 122, 134 135,. 139,, 142, 146 Baldwin, R. R., 163, 165, 166, 174, 183, 185, 188, 192, 193, 201, 203, 329 Ballash, N. M., 32 Barassin, J., 53 Barat, P., 204, 206 Bardwell, J., 322 Barker, J. R., 35, 98 Barnard, J. A., 133, 164, 197 Barnes, G. R., 158 Baronnet, F., 175 Barroeta, N., 154 Barton, T. J., 212 Basco, M., 50, 53, 161 Bashkerova, S. A., 221 Bass, A. M., 39, 76, 178, 181 Bass, H. E., 246 Basu, R., 96 Bates, D. R., 245 Batt, L., 230 Bauer, E., 35, 51, 56 Bauer, H. J., 246 Bauer, S. H., 133, 149,243, 248,290 Bauer, W., 142 Baulch, D. L., 29, 163, 244 Baumgardner, C. L., 127 B a t , L. M., 322 Bayrakceken, F., 178 Bazant, V., 217 Beadle. P. C .., 120,. 154._ 157,‘167 Beard, C. D., 226 Becker. K. H.. 40 Bell, J.’A., 125 Bell, T. N., 214, 216 Belles, F. E., 176 Bel’ferman. G. L.. 158 Belford, R: L.,274 Bellobono, I. R., 146 Belloli, R., 96 Belluman, R., 275 Belausov, B. P., 312 Beltrame, P., 146 Bemand, F. F.,17, 50, 74, 78 Ben-aim, R., 321 Bender, C. F., 155 Benes, V. E., 246 Bennet, J. E., 29, 184 Bennett, S. W., 218, 240 Benson, H. S., 146

387

Benson, S. W., 94,96, 120, 126, 159, 170, 178. 204;

128, 163, 171, 180.

151, 154, 167, 168, 172, 173, 181. 185. 263,‘ 378 Berend, G. C., 263

157, 169, 177, 190,

Berger,. M., 42 Bergman, R. G. 132, 134 Berkley, R. E., 187, 216, 217

Berlad, A. L., 319 Berry, M. J., 21, 32, 73 Berry, R. S., 82 Berry, J., 164, 202 Berson, J. A., 135, 142 Bernstein, R. B., 30,33,87, 249, 253

Bertrand, M., 112 Bertz, A., 330 Bevan, M. J., 46 Bevan, W. A., 240 Bevan, W. I., 238 Beyer, T., 94 Beynon, J. H., 112 Bichlmeir, B., 238 Bickel, A. F., 164 Biernbaum, M. S., 238 Bigley, D. B., 154 Billups, W. E., 132 Binenboym, J., 227 Birchall, J. M., 158 Bird, P. F., 55, 284 Birks, J., 24, 244 Birladeanu, L., 135 Bischoff, J. R., 318 Bishop, D. J., 157 Black, G., 43, 46, 69 Blackmore, D. R., 29 Blades, A. T., 137

Blake, P. G., 151 Blauer, J., 266 Boiteaux, A., 309, 330 Boitnott, C. A,, 279 Boak, D. S., 219 Bonanno, R. J., 27 Bone, S. A., 220 Bonner, B. H., 322 Booth, D., 155 Bott, J. F., 127, 157 Boudart, M., 247 Boudjouk, P., 213, 238 Bowen, J. R., 353 Bowers, P. G., 318 Bowles, A. J., 219 Bowman, C. T., 177 Bowman, D. F., 184 Bowrey, M., 158, 221 Box, G. E. P., 296

388 Box, M. J., 296 Boxall, C., 57, 66 Boyd, R. K., 257, 268, 289

Bozzelli. J. W., 75 Brabbs,'T. A.,'176 Bradley, D., 23 Bradley, J. N., 50,133, 146, 155. 177. 243

Brau,'C. A., 269 Brauman, J. I., 138, 142, 146

Bra& W., 16, 18, 22, 35,

39, 60, 64, 76, 178, 181 Brav. W. C.. 310 Braiton, R. 'K., 302 Breckenridge, W. H., 66 Breen, J. E., 262 Breiland, W. G., 227 Breshears, W. D., 55, 284 Breslow, R., 142 Brewer, L., 18 Brieland, W. G., 125 Briggs, P. R., 234 Broadbent, T. W., 274 Broida, H. P., 35, 38 Brokaw, R. S., 173, 176, 320 Brook, A. G., 238 Brooks, C. T., 157 Brooks, P. R., 22, 33 Brophy, J. H., 178 Brown, A., 90 Brown, A. J., 203 Brown, D. M., 184 Brown, J., 322 Brown, J. M., 146 Brown, R. L., 43 Browne, W. G., 182 Bruice, J. C., 313 Bryant, J. T., 118 Buger, P. A., 49 Buenker, R. J., 28, 138 Bullock, G. E., 5 5 Bumgardner, C. C., 151 Bunker, D. L., 94, 97, 103, 113, 253, 262 Burak, I., 270, 290 Burcat, A., 155, 166 Burgess, A. R., 322 Burgoyne, J. H., 322 Burlingame, A. L., 110 Burns, G., 80, 245, 248, 252, 257, 262 Bush, S. F., 332 Bush, Y. A., 84 Buss, J. H., 169 Busse, H. G., 314 Butcher, R. J., 65, 78 Bykhovskii, V. K., 88

Cabell, M., 149 Cadman, P., 78, 150, 159, 218, 219 Cain, E. N., 142 Cala, F. R., 124

Caldwell, K. G., 318 Caledonia, G. E., 246 Callear, A. B., 24, 46, 61,

64, 65, 85, 86, 161, 178, 181, 259, 263,274

Author Index Calvert, J. G., 164, 184, 202

Calvin, M., 331 Campbell, I. M., 22 Campbell, M. M., 142 Caponecchi, A. J., 133 Carless, H. A. J., 118, 137 Carmichael, H., 121, 127, 151

Carr, R. W., 119, 124 Carrier, G. F., 253 Carrington, A., 74 Carrington, T., 3 1,96,244, 260,261, 288

Carroll, H. F., 120 Carter, C. F., 15, 49 Cartlidge, J., 327 Cattania, M. G., 146 Cavazza, M., 154 Center, R. E., 55, 246, 262 Ciabattoni, J., 149 Chamboux, J., 321, 322 Champion, R. L., 282 Chan, E. IS.,221 Chan, S. C., 118 Chance, B., 330 Chang, H. W., 71,80, 122, 245

Chang, J. S., 300 Chao, K.-J., 98 Chappell, G. A., 175 Charters, P. E., 24 Chen, H. L., 21, 264 Chen, M. Y. D., 264 Cheng, T. M. H., 236, 237 Chernavskaya, N. M., 331 Chernavskii, D. S., 331 Chernesky, M., 322 Chernyshev, E. A., 221 Cheung, J. T., 109 Chmurny, A. B., 134 Choo, K. Y., 218, 227 Chong, S.-L., 107 Christie, M. I., 244 Chuchani, G., 154 Chukka, W. A., 110 Chvalovsky, V., 226 Clark, A. H., 182 Clark, A. P., 261 Clark, T. C., 179 Clarke. A. G.. 80.252. 267 Claridge, R. F. C:, 23; 86 Cleugh, C. J., 166 Clifford, R. P., 132, 228 Clough, P. N., 49, 86 Clyne, M. A. A., 17,22,23, 50, 74, 76, 78, 80 Cocks, A. T., 96, 132, 137, 142, 146, 149, 151 Cohen. A.. 197 Coleman. M.L.. 83 Collier, S. S., 184 Combe, A., 174 Combourieu, J., 53, 77 Come. G. M.. 175 Congin, L., 154 Conlin, R. T., 225 Connor, J., 61 Connor, J. H., 86 Contineanu, M. A., 217 Conway, D. C., 112

Cook, G. R., 55 Cook, M. A., 238 Cooks, R. G., 112 Cool, J. A,, 264 Coombe, R. D., 21, 71, 72 Cooper, R., 55 Corcoran, W. H., 318 Corney, A., 69 Cosa, J. J., 118, 127, 132, 149

Cottrell, T. L., 260 Cox, R. A., 158 Coxon, J. A., 50, 84 Coy, C. A., 171 Craig, J. C., 226 Craig, N. L., 122 Cram, D. J., 134 Cramp, J. H. W., 55, 262 Crawford, R. J., 132, 134, 146, 157

Cross, J. B., 26 Crossley, R. W., 133, 155, 289

Cruikshank, F. R., 169, 171. 172

Cruse; H. W., 15, 23, 37 Cukier, R. I., 295 Cullis, C. F., 164,202,204, 206,311,323

Cummins. C. P. R.. 157 Curry, C.'L., 182 ' Curtiss, C. F., 30,253, 300 Cvetanovii, R. J., 23, 24, 27, 51, 57, 95

Czajkowski, M., 88 Dabbagh, A. M., 142 Daby, E. E., 182 Dagdigian, P. J., 15 Dakabu, M., 151 Dalgleish, A., 67 Daly, N. J., 149, 154 D'Amore, M. B., 142 Danby, C. J., 111 Daniel, S. H., 32, 225,226, 235

Dasch, C., 133, 149 Dastoor, P. N., 151 Datz, S., 88 Davalt, M., 142, 146 David, M. P., 154 Davidson, N. R., 244,262 Davidson, I. M. T., 157, 212, 214, 220, 228, 229, 230, 231, 233, 234 T. A,. 89 B., 27 K., 254 D., 16, 18, 22, 39, 51, 53, 60, 63, 64, 76, 191, 192,283

Davidson. Davies, PI Davies, P. Davis, D.

Day, M.,is0 Day, M. J., 182 Dean, A. M., 166,173,284, 326.

Deakin, J. J., 65, 78 Decell, H. P., jun., 307 Decelle. R.. 151 de Chang, S. P., 154 Dedinas, J., 235

389

Author Index Degn, H., 309, 311, 313, 331, 345, 363 DeGraff, B. A.. 80 de Maas, N., 26, 53, 161, 162, 192 Delf, M. E., 214 Del Re, G., 138 De Meiiere. A.. 141 De Moie, W. B., 58 DePoy, P., 365, 367 Derran, P. B., 135 Derrick, P. J., 110 Derwent, B. de B., 187 de Santis, V., 154 de Sorgo, M., 66 Devore, J. A., 172 Dewar, M. J. S., 135, 138 Dickinson, A. S., 261 Dill, C., 302 Dillon, J. A., 263 Ding, A. M. G., 24 Dinghra, A. K., 123 Dittmar, W., 132 di Valentin, M. A., 179 Dixon, D. A., 49 Dixon, D. J., 174 Dixon-Lewis, G., 167, 182 Doemeny, L. J., 88 Doering, W. von E., 101, 12< 1JJ

Dogra, S. K., 50 Dolbier, W. R., 132 Dondes, S., 320 Donovan, R. J., 14,16, 19, 47,60,61,64,65,66,67, 68. 77, 78 Dorer, F. H., 104 Dorfman, L. M., 27 Dorko, E. A., 133, 288 Douglas, D. J., 77 Dove, J. E., 179, 190, 254, 255, 318 Doverspike, L. D., 282 Draper, N. R., 296, 304 Drew, C. M., 187 Drummond, D. L., 24 Drysdale, D. D., 29, 163, 244 Dubac, J., 226 Dubrin, J. W., 83 Dulwer, W. H., 73, 84 Duewei, W. M., 84 DunlaD. L. H., 142 Dunn,-M, R., 25 Dunn, 0. J., 21 Dunning, F. B., 282 Durden, D. A., 177 Duthler, C. J., 35 Dutton, M. L., 97 Dzierzynski, M., 175 Eaborn, C., 218, 230, 233, 234,238,240,241 Earl, B. L., 35 East, R. L., 183 Eastes, W., 262 Eckstrom, D. J., 55, 262 Edelson, D., 301 Edsberg, L., 303 Egerton, A. C. G., 322

Egger, K. W., 96, 132, 138, 142, 149, 151, 154 Egorov, V. J., 54 Eguchi, S., 132, 134, 146 Eigenmann, H. K., 169 Eisch, J. J., 238 Eland, J. H. D., 111 Elliott, L. E., 232 Ellis, C. E., 302 Ellsworth, R. L., 57 Elsaesser, S., 330 Elsenaar, R. J., 35 Elwood, J. P., 259 Emmanuel, G., 94 Emmons, H. W., 255 Emovon, E. U., 151 Endrenyi, L., 186 Engel, P. S., 157 Engleman, V. S., 266 Entemann, E. A., 26 Epelboin, I., 384 Epstein, M. J., 145 Estabrook, R. W., 330 Estacio, P., 221, 232 Euker, C. A., 183 Everett, C. J., 165, 185 Eyre, J. A., 27 Eyring, H., 94, 252 Fair, R. W., 60, 61 Falconer, J. W., 164 Falick, A. M., 110 Farmer, J. B., 174 Farneth, W. E., 142 Fara, A., 154 Fehsenfeld, F. C., 84 Feldman, E. V., 157 Felder, W., 60 Felton, P. G., 321, 332 Ferguson, K. C., 170 Ferrero, J. C., 132 Field, R. J., 314, 359 Field, R. W., 38 Figuera, J. M., 96, 124 Filseth, S. V., 21, 69 Findlay, A., 310 Fink, E. H., 264 Fink. R. D.. 178 Finzi, J., 263 Fish,A., 163,164,165,322, 327 Fishburne. E. S.. 289 Fisher, E. 'R.,35, 56, 290 Fisk, G. A., 106 Fite, W. L., 36, 35 Fleming. R. H.. 135 Fletcher; R., 306 Flowers, M. C., 122, 141, 149, 181, 212 Flowers, W. T., 142 Fluendy, M. A. D., 22, 89 Flvnn. G. W.. 290 Fdon,-R., 75 . Foner, S. N., 174 Fontain, C. M., 295 Fontijn, A., 43, 52, 82 Ford, P. W., 134 Ford, W. G. F., 122 Forst. W.. 93. 94. 95 Fortin, C.'J.,'56 . Fowler, R. H., 248

Franck, U., 314 Frank-Kamenetskii, D. G.,

Frend, M.A., 133, 155 Frey, H. M., 95, 119, 122, 137, 139, 142, 146, 149 159, 163 Freund, S. M., 106 Fricke, J., 35 Friedman, L., 105 Friedrich, L. E., 149 Fristrom, R. M., 167 Frost, J. J., 222 Fujimoto, T., 118, 155 Fukui, K., 96 Fuller, A. R., 166, 193,203 Fuller, M., 174, 188 Fullerton, D. C., 108 Furinsky, E., 185 Furuyama, S., 171 Fushiki, Y., 23, 87 Gabbott, R. E., 154 Gabrielli, C., 384 Gac, N. A., 151, 157, 167 Gagneja, G. L., 234 Gaidis, J. M., 234 Gajewski, J. J., 135 Gale, D. M., 137, 142 Gallagher, A., 24, 34 Gardiner, W. C., 173 Garland, J. K., 124 Gaspar, P. P., 218, 222, 225, 227,236,237 Gaydon, A. G., 243 Gear, C. W., 299, 301, 302 Gebelin, H., 245 Gedanken, A., 83 Geddes, J., 26, 49 Geiseler, G., 157 Geisler, W., 3 14 Gelbart, W. M., 245 Gelinas, R. J., 301 Gemmer, R. V., 146 Gennaro, G. P., 225, 235 Georgakakos, J. H., 115, 120 George, T. F., 96 Gerardo, J. B., 83 Gerber, G., 83 Gerjouy, E., 261 Ghormley, J. A., 57, 181 Ghosh, A., 330 Giachardi, D. J., 56 Gibbons, A. R., 122 Gibbs, R., 157 Giddings, J. C., 353 Gilbert, J. C., 101, 132 Gilbert, R. G., 94, 96, 245 Gillan, J., 184 Gilmore, F. R., 35 Gilray, K. M., 157 Gird, S. R., 221 Gislason E. A., 107 GlHnzer,'K., 126, 127, 157 Glasgow, L. C., 217 Glass, G. P., 27, 54, 262

390 Harker, A. B., 53 Glasstone, S., 252 Harris, D. O., 38 Gleiter, R,,135 Harris. H. H.. 97. 113 Godfroid, M., 151 Harteck, P., 320 ' Goldbeter A., 371 Golde. M. F.. 53 Hartig, R., 151, 157, 175 Hartley, D. B., 27 Golden, D. M., 94, 120, 126, 128, 151, 154, 157, Harwood, B. A., 164 Hase, W. L., 94, 113, 124, 167, 169, 170, 171, 172, 178, 180, 181, 194 125, 227 Golding. B. T.. 146 Haskell, W. W., 164 Haszeldine, R. N., 142, Goldst& J. M.,146 Golino, C'. M., 213 157, 158, 220,238,239 Golub, C. H., 307 Hatchell, G. D., 302 Hathorn, F. G. M., 78 Goodman, M. F., 96,290 Hatzenbuhler, D. A., 264 Goodwin, B., 370 Haugen, G. R., 169 Gordon. A. S.. 154. 187 Haugh, M. J., 84 Gorse, R. A., 191 ' Hay, A. J., 274 Gourlay, A. R., 303 Hayashi, M., 217 Gowenlack, B. G., 79,219, 228, 234 Hearon, J. Z., 351 Hedges, E. S., 309 Graham, D., 239 Hedges, R. E. M., 24 Grand, F., 322 Gray, B. F., 314, 326, 330, Heicklen, J., 56, 57, 90, 332, 336, 345, 349, 354, 149, 183, 191, 192 Heidner, R. F., tert., 17, 56 368, 374, 378 Heineken, F. G., 345 Gray, C. N., 43 Hemminger, J. C., 104 Gray, D. N., 221 Gray, P., 163, 182, 189, Hendry, G., 194 Henfling, D., 132 314, 321, 324, 326 Henis, J. M. S., 236, 237 Greenberg, R. J., 57 Herm, R. R., 35, 36, 37, Greene, E. F.. 243 106 Greig, G., 61 . Herman, R., 259 Greiner, N. R., 161, 166 Grice, R., 15, 33, 49 Herman, Z., 107, 110 Griffith. J. S.. 370 Herod, A. A., 163, 189 Grlffiths. J. F.._166. _ 314. . Herron, J. T., 18, 22, 35, 51, 63, 188, 283 321, 329 Herschbach, D. R., 15, 22, G r i m , V. W., 133,289 Gross, R. W. F., 245 25,49,105,106,108, 109 Herschkowitz-Kaufman, Grosser, J., 25 M., 380, 383 Grotewold, J., 181 Herzberg, G., 28, 244, 248 Groth, W., 40 Herzfeld, K. F., 260 Growcock, F. B., 124, 125 Hess, B., 309, 330 Grunwald, E., 190 Heydtmann, H., 122, 240 Gsponer, H. E., 118, 127, Hiatt, R., 168, 178 149 Hierl, P., 107 Guggenheim, E. A., 248 Higgin, R. M. R., 166 Gunning, H. E., 24,60,61, Higgins, J., 309, 370 69,86,187,215,216,217 Higley, D. P., 132 Gusel'nikov, L. E., 212, Hikida. T.. 27 213 Hill, T: L.;256 Gustavson, F. G., 302 Hill, W. J., 296 Gutman, D., 51 Hillman, J. S., 174 Hindmarsh, A. C., 300 Haberland, H., 25 Hinshelwood. C. N.. 322 Hahn, W., 349 Hippler, H., 80, 81 . Haizlip, A. D., 218 Heischfelder, J. O., 300, Hajiev, S. H., 234 353 Hall, H. K., 137, 142 Hisatsune, I. C.. 149 Hall, J. G., 245 Halstead, M. P.,. 180,_300, . Hirokami; S.-I.,-51 Hlavacek, V., 334 329, 378 Hoare, D. E., 164,167,192 Haluk Arican, M., 188 321 Ham. D. 0.. 29. 287 Hoare, M. R., 247 Hammond, K.,86, 88 Hobson, R. M., 245 Hammond, R., 83 Hochanadel, C. J., 57, 181 Hampson, R. F., jun., 59 Hoffman, M. Z., 90 Hanusse, P., 339 Hoffman, R., 135 Hardin, D. R., 33 Holbrook, K. A., 93, 132, Hardwidge, E. A., 102,171 149, 151 Hare, N. E., 134

A uthor Index Holmes, B. E., 121 Holmes, J. L., 151 Holton, J. D., 222 Homann, K. H., 60 Homer, J. B., 245 Hong, J.-H., 86 Honig, M. L., 139 Hopf, H., 122, 137 Hopkins, B. M., 264 Hopkins, D. E., 165 Hopkins, R. G., 95, 146, 149 Horgan, G. P., 25 Horn, F., 358 Horne, D. G., 29, 163 Horne, D. S., 24 Hornig, D. F., 245 Horowitz, A., 159 Hosomi, A., 218, 238 Hotop, H., 83, 84, 85 Houghton, W., 173 Howard, A. B., 157 Howard, A. V., 229 Howard, J. S., 85 Howe, I., 110 Howerd, J. A., 184 Hsu, C. J., 38, 290 Hsu, J. N. C., 135 Hsu, K., 138 Hudson, A., 219 Hudson, R. L., 174 Hughes, R., 164, 322 Huie, R. E., 18,22, 51, 63, 188, 283 Hull, L. A., 149 Hull, T. E., 303 Hunt, L. P., 234 Hunt, W. J., 36 Hunter, G., 283 Huntress, W. T., 107 Hurle. I. R.. 243. 262, 265.

HyGer, C., 358 Iden, C. R., 105 Illenberger, E., 83 Ingold, K. U., 184, 218 Inn, E. C. Y.,53 Intezarova, E. J., 54 Ip, J. K. K., 245 Ireton, R. C., 102 Isaacs, N. S., 149 Iseard, B. S., 234 Ishikawa, M., 225 Islamov, T. Kh., 213 Iyer, R. S., 75 Izawa, G., 124 Izod,T. P. J., 166, 167, 179 Jachimowski, C. J., 173 Jackson, D., 165, 174 Jackson, G. R., 119 Jackson, J. M., 261 Jackson, R. A., 212, 218, 219,240,241 Jackson, W. M., 187

Author Index Jacobs, T. A., 245 Jakubowski, E., 118,215 Jamamoto, K., 225 James, D. G. L., 161 James, F. C., 161 Janline, D. K., 32 Jarvie, A. M.P.,213 Jaster, W., 42 Jeffers, P., 129, 133, 149 Jeffers, P. M., 145, 146 Jenkins, R. L., 221 Jensen, R. J., 20 Jodham, A., 122 John, P., 79, 224 Johnson, A. L., 149 Johnson, A. W., 83 Johnson, C. A. F., 228, 234

Johnson, D. W., 133, 145 Johnson, N., 146 Johnson, R. L., 73, 125 Johnson, S. E., 43, 52 Johnson, S. G., 88 Johnston, H., 24 Johnston, H. S., 23, 53, 174,244

Joklik, J., 217 Jonah, C. D., 36 Jonathan, N., 71 Jondahl, T. P., 145 Jones, A., 163, 189, 253, 262, 303

Jones, C. R., 38 Jones, D. G., 254,255 Jones, E. G., 112 Jones, M., 225,230 Jones, S. H., 170 Jones, W. E., 75, 163, 179 Jortner, J., 83, 245 Jung, K. H., 127, 157 Kalos, F., 22 Kamaratos, E., 31, 220, 251,273

Kamil, M., 164 Kanofsky, J. R., 51 Kaplan, M. S., 139 Karfunkel, H. R., 383 Karl, G., 87 Karplus, M., 26, 96,261 Kasper, J. V. V., 22 Kasperek, G. J., 313 Kassal, T., 261 Kassel, L. S., 247 Kaufman, F., 23, 29, 46,

53, 163, 244, 287 Kaufman, M., 75 Kaufman, M. S., 212 Kaufmann, D., 141 Kebarle, P., 24, 69 Keck, J., 253 Keck, J. C., 30,253, 254 Keddam, M., 384 Kelkar, V. V., 322 Kelley, J. D., 259, 264 Kelly, C. C., 188 Kempter, V., 236 Kende, A. S., 135 Kennedy, G. J., 79 Kennedy, R. C., 157

391 Kerr, J. A., 151, 159, 163, 170, 177, 185, 201, 215, 216, Kerr, R. D., 258 Khaikin, S. E., 340 Khait, Yu. L., 301 Kichtin, N. N., 90 Kiefer, J. H., 259, 267 Kim, K. C., 73, 101, 121, 123 Kim, P., 27 King, J., 310 King, K. D., 120, 171 Kingston, A. E., 245 Kinnear, C. G., 209 Kinsey, J. L., 33 Kirsch, L. J., 14, 24, 40, 46, 47, 66 Kirschner, S., 138 Kirwen, N. A., 326 Kistiakowsky, G. B., 42, 124, 166, 173, 320 Kivel, B., 251 Klein, F. S., 72 Klemm, R. B., 60, 64 Klemperer, W., 106 Kley, D., 40 Kline, E., 212 Klochkova, T. A., 221 Klots, C. E., 106 Klunder, E. B., 124 Knecht, D. A., 146 Knoll, H., 157 Knox, J. H., 163, 164, 165, 171, 188, 194, 209, 322, 329

Knudston, J. T., 290 Kobrinsky, P. C., 124, 179 Kochi, J. K., 218 Kogure, T., 226 Kolb, C. E., 75 Kolker, H. J., 269 Kollmar, H., 134 Komalenkova, N. G., 221 Kompa, K. L., 21, 72, 75 Konar, R. S., 175, 180 Kondratiev, V. M., 163, 274

Konkoly, T. I., 96 Konstantatos, J., 68 Koob, R. D., 123 Korgukhim, M. D., 359 Kornornicki, A., 138 Koros, E., 314 Koros. G. E., 314 Kortzeborn, R. N., 245 Koski, W. S., 105 Koskikallio, J., 86 Koutkova, J., 226 Krasovskii, N. N., 340 Kraulinya, E. K., 88 Krause, H. F., 26, 35, 88 Krause, L., 24, 34, 86, 88, 90

Kreitser, G. P., 359 Krohn, K. A., 103 Krome. G.. 60 Kruger; J. ;330 Kruus, P., 87 Kubicek, M., 334 Kuiter, L., 96

Kulich, D. M., 164 Kumada, M., 225 Kuntz, P. J., 24 Kuntz, R. R., 157 Kurylo, M. J., 18, 22, 64, 181

Kurzel, R. B., 264 Kushina, I. D., 158 Kushner, R., 124 Kwart, H., 146, 154,238 Kwei, G. H., 26, 108 Labil, N. J., 251, 281 LaDato, V. A., 251 Laidler, K. J., 170, 171, 184, 185, 252

Lalonde, A. C., 157 Lam, E. Y.Y., 227 Lam, S. K., 282 Lambert, C. A., 214,228 Lambert, J. D., 55, 262 Lamont. A. M.. 139 Lampe, ' F. W.', 80, 220, 236, 237

Langer, S. H., 146, 165, 329

Lapidus, L., 303 Lappert, M. F., 234 Larkin, F. S.,29 Larsh, A. E., 244 Larson, C. W., 171 La Salle, J. S., 349 Lawley, K. P., 22, 89 Lawrence, T. R., 257 Lawson, D. R., 123 Lau, K. H., 106 Laughlin, R. G. W., 322 Lavenda, B., 380 Leathard, D. A., 180 Leavell, K. H., 132, 154 Le Bras, G., 77 LeBreton, P. R., 22,25 Lechtken, P., 142 Lederman, D. M., 290 Lee, A,, 107, 110 Lee, E., 158 Lee, E. K. C., 104, 118, 124, 137

Lee, J. H., 27 Lee, N. E., 104, 118 Lee, P. H., 35 Lee. P. R.. 324 Lee; Y. T.; 22,25, 76, 104, 107, 108, 119

Leermakers, P. A., 101 Lees. A. B.. 108 Lefever, R.; 371, 380 Leffler, J. E., 190 Lefohn, A. S., 23 Lefschetz, S., 349 Leggett, C., 183, 186 Lehmann, E. J., 157 Leibowitz, L. P., 289 Leinroth, J. P., 183 Leipunskii, 1. O., 75 Lemal. D. M.. 142 Lenzi,'M., 18' Leone, S. R., 77, 270 Lephardt, J., 53 Leroi, G. E., 264

Author Index

392 LeRoy, D. J., 26, 86, 88, 95, 186 LeRoy, R. J., 249 LeRoy, R. L., 110, 112 Lestrade, J., 384 Letort, M., 174 Levin, R. H., 225 Levine, R. D., 87 Levy, D. H., 74 Levy, J. B., 157 Levy, M. R., 15,49 Lewis, D., 129 Lewis, E. S., 132, 154 Liardon, R., 105 Liebhafsky, H. A., 312 Lifshitz, A., 166 Light, J. C., 94, 106 Lighthill, M. J., 247 Lijnse, P. L., 34, 35 Lin, C.-L., 23, 46, 58 Lin, J. L., 265 Lin, M. C., 171, 184 Lin, S. H., 94, 106 Lin, S.-M., 35, 36, 37 Lin, Y.N., 110, 118, 119 Lindblad, P., 363 Lindberg, B., 303 Linnett, J. W., 318 Lim, D;, 157 Lippiatt, J. H., 24, 257 Liskow, D. H., 155 Lissi,.E. A., 181 Litovitz. T. A.. 260 Little, D. J., 16, 60, 65, 67, 68.78 Littler, J. G. F., 91 Liu, M. T. H., 149, 157 LIU. Y. A.. 303 LJoid, A. C., 29, 163, 244 Lo, V. W. S., 26 Lombana, L., 154 Longthorn, D., 166, 193 Lotka, A. J., 355 Loucks, L. F., 57, 157 Lowe, R. P., 109 Lubin, T., 230 Lucquin, M., 321,322, 367 Luckett, G . A., 203 Luckraft, D. A., 132 Luther, K., 80, 81 Lutz, R. W., 262 Luu, S. H., 118 Lynn, S., 318 Lythgoe, S., 220 McAskill, N. A., 75 MacCaffery, B., 319 McCart, G., 277 Maccoll, A., 151, 154 MacCormac, N., 322 McCoubrey, J. C., 260 McCullaugh, E. A., 22 McCullough, D. W., 57 McDonald, J. D., 22, 25, 102, 109 McDonald, R. G., 24 McDowell, C. A., 174,378 McElwain, D. L. S., 31, 251, 258,267,289 McEwan, M. J., 23, 25, 86 McFadden, D. L.,22

MacFadden, K. 0.. 127, 157, 182 McFarland, M., 84, 173 McGee, T. H., 137 McGhee, D. B., 133 McGillis. D. A.. 34 McGrath, P. W.', 125, 227 McGrath, W. D., 57 McGurk, J. C., 24, 85 McIntosh, C. L., 212 McIver, J. W., 96, 138 Mack, G. P. R., 57 McKenney, D. J., 74 Mackey, P., 265 MacKnight, S. D., 163 McLafferty, F. W., 110 McLaren, T. J., 245, 269 McLean, J., 19 . McMillan, G. R., 164 McMillen. D. F.. 171. 180 McMillen; L. D.; 290McNeal, R. J., 55 McNesby, J. R., 181, 187 McOmje, J. F. W., 154 McWhIrter, R. P., 245 Madhavan. V.. 90 Madsen, N. K:, 300 Mahan, B. H., 105, 107, 319 Maier, H. N., 80 Mains, G. J., 86, 235 Malan, 0. G., 49 Malchan, M. J., 329 Malherbe, F. E., 322 Maloney, K. M., 155, 157 Margrave, J. L., 221 Marielle, R. P., 106 Markusch, P., 222 Marquardt, D. W., 294, 306, 307 Marshall, R., 244 Marshall, R. M., 175,180 Martens, G. J., 151 Martin, G., 154 Martin, I., 154 Martin, R., 175 Martin, R. M., 88 Mason, D. M., 318, 365, 367 Matchan, M. J., '65, 166 Matsuda, S., 166, 167 Matsumoto, H., 217 Matsuzaki, I., 312 Matthews, J. I., 220 Maurayannis, C., 46 Mayer, D., 331 Maylotte, D. H., 24 Mayo, F. R., 194, 210 Mazac, C. J., 125,227 Mazerolles, P., 226 Mazzali, R., 154 Meaburn, G. M., 40 Meagher, J. F., 98 Meerbott, W. K., 172 Melamed, F. A., 301 Melliar-Smith, C. M., 71 Melvin, A., 324 Menendez, V., 96, 124 Menzinger, M., 36 Metcalfe, J., 118, 137 Metzler, W. H., 337

Meyer, E., 173 Meyer, R. T., 24 Micha, D. A., 266 Michael, J. V., 29, 159 Michaelson, R. C., 157 Michaud, P., 24 Middleton, J., 238 Mies, F. H., 106 Mihelcic, D., 217 Mile, B., 184 Mill, T., 194, 210 Miller, S. A., 322 Miller, T. A., 74 Miller, W. B., 105, 108 Miller, W. H., 47, 82 Millward, G. E., 151, 157, 158 Milne, G. S., 163, 189, 192 Mims, C. A., 35, 36, 37 Minkoff, G. J., 163, 319 Minorsky, N., 340 Mintz, K. J., 95 Miralles, A., 154 Mirels, H., 245, 286 Mitchell, D. M., 26 Mitchell, R. L., 52 Mitsch, R. A., 149 Moehlmann, J. G., 102 Molina, M. J., 21, 112 Montague, D. C., 178 Montroll, E. W., 255, 259 Moore, C. B., 21, 112,263, 264, 270, 290 Moore, P. L.,49 Moos, H. W., 83 Moran, T. F., 108, 109 Morgan, J. E., 43 Morgan, J. S., 310 Morganroth, W. E., 184 Morgenstern, R., 83 Morgner, H., 83 Moringa, K., 173 Morosov, J. J., 75 Morris, E. D., 23, 182 Morse, R. D., 53 Morse, R. J., 263 Moser, .H. C., 122 Mosher, H. S., 238 Mosin, A. M., 221 Mott, N. F., 261 Muckerman, J. T., 30 Mueller, G. W., 133, 289 Muir, J., 337 Murrell, J. N., 96, 155 Muschlitz, E. E., 82 Musgrave, R. G., 171 Myers, J. E., 309 Myerson, A. L., 244 Nagai, Y.,217, 226 Nagra, S. N., 151 Nakajima, R., 332 Nakamura, S., 331 Nalbandyan, A. B., 193 Nametkin, N. S., 212, 213 Nangia, P. S., 171 Neely, B. D., 121 Nelson, R. L., 188 Neumann, M. B., 321 Neumann, M. G., 181 Neuvar, E. W., 149

Author Index Newitt, C. M., 322 Newman, R. H.,86 Nicholas, J. E.,178 Niclause, M.,174, 175 Nicolis, G.,309, 380, 383 Niehaus, A., 83, 84, 85 Nika, G.G.,258 Niki, H.,21, 22, 182 Nikitin, E. E.,55, 88,253, 264 Nitzan, A., 380 Noble, B., 151 Noble, N.,127 Nordsieck, A., 301 Norris, A. C., 165 Norrish, R. G. W., 244, 318,322 Norstrorn, R. J., 178 Noter, Y.,270 Novak, A. V., 290 Noxon, J. F.,23, 46 Noyes, R. M.,314, 359 Obenauf, R. H., 38 Obi, K.,124,217 O’Callaghan, W.B., 24,69 O’Deen, L. A.,123, 185 Odell, B. G.,135 Odiorne, T.J., 22 Ogren, P. J., 181 Ogryzlo, E. A.,40 Ohno, M.,132, 134, 146 Ohtsuki, M.,217 Okabe, H.,59 Okinoshima, H.,225 Okuda, S.,71 Olbrich, G.,217 Oldman, R. J., 46 Ollershaw, J., 122 Olschewski, H.A., 173 Olson, R. E.,59 O’NeaI, H.E.,96,132,159, 163, 169, 172, 204, 230, 232, 378 Oppenheim, A. K.,35? Oppenheim, I, 247, 254 Orchard, S. W.,118, 145 Oref, I., 102 Osborne, D.T.,29 Osipov, A. I., 253,259,261 Ottinger, Ch., 110 Owen, P. W.,226 Pacey, P. D., 24, 27, 157, 190 Pack, R. T., 30, 252 Padrick, T. D.,112 Pahari, M.B., 96 Painter, C. E.,133 Palmer, H.B., 38, 157 Pannell, K.H.,213 Panshin, Y. K.,151 Panshina, N. G.,151 Papic, M. M.,185 Paquette, L. A., 145 Parkes, D. A.,23 Parkes, N. J., 103 Parker, J. H.,21, 72 Parker, R. M.,149 Parkinson, C.,158,240

393 Parson, J. M., 76,104,107, 108 Parsonage, M.J., 163 Parrish, D. D.,49, 106, 108 Parry, K. A. W., 132, 151 Pascoe, J. D.,238 Pattengill, M.,94 Patton, J. E.,146 Paukert, T.T.,23, 174 Paul, D. M.,23 Paul, E.,256 Paulet, G.,77 Pavlou, S.,230 Pavlou, S. P., 118 Payne, W. A., 191, 192 Peacock, S. J., 157 Pearce, R., 218 Peard, M. G.,311 Pease, R. N.,325 Pearson, G.S., 167 Pearson, P. K.,47 Pearson, R. G.,96 Peatman, W.B., 83 Pechukas, P., 106 Peckharn, G., 306 Pedley, J. B., 234 Penny, D.E.,141, 149 Penzes, S., 86 Perche, A.,367 Perez, J. M.,96, 367 Perner, D.,40 Perona, M. J., 157, 167 Perry, D. S., 24 Persky, A., 72 Peterson, N. C.,64, 181 Petschek, A. G.,295 Peyerimhoff, S. D., 28, 138 Phaneuf. R. A.. 88.90 Phillips,‘L. F.,‘23,-25, 43, 86, 183 Pilling, M. J., 60, 178 Pimentel, G.C., 21,23,71, 72, 73, 112 Piper, L. G.,23, 84 PiDkin. 0.A.. 133 Piire, J., 86, 88 Platt, A. E.,216 Polak, L. S., 301 Poland, J. S., 234 Polanyi, J. C.,24, 27, 71, 73,77,78,86,87,96,253 Pollard, R. T.,203, 204, 206 Pollock, T. L.,75, 122 Pornmelet, J.-C., 149 Pope, R. L.,253 Porter, F., 348 Porter, G.,17, 244, 252 Porter, R. P., 182 Portnow, J., 309 Potter, E., 188 Potzinger, P., 217, 236 Powell, M.J. D., 306 Pozzoli. A. S.. 138 Prasil, Z.,94 Prendergast, D.F., 318 Present. R. D..261 Preston‘, K. E.: 57 Price, D., 24 . Price, S. J. W., 157

Prigogine, I., 380 Pritchard, G.O.,124, 179 Pritchard, H.O.,31, 155, 157, 158, 171, 243, 245, 248, 251, 258, 265, 266, 273, 277, 281, 283, 289, 311 Prochazka, M., 157 Prothero, A.,245,300,329, 378 Puentes, M.J., 232 Purnell, J. H.,158, 175, 180, 190,221,224 Pye, K., 330 Quane, D., 234 Quick, L. M.,57 Quinn, C. P., 23, 300, 329, 378 Quiring, W. J., 151 Quy, R. B., 262 Rabideau, S. W.,26 Rabinovitch, B. S., 94, 98, 102, 110, 115, 118, 119, 120. 122, 126. 155. 158. 163; 171- . . Raff, L. M., 279 Rajbenbach, L. A., 159 Rakita, P. E.,238 Ramik. K. M..235 Rankin, C.,106 Rapp, D., 261 Rastelli, A 138 Ratajczak,‘k, 142, 163 Ratcliff, K.,302 Ray, D.J. M., 166, 195 Raz, B., 83 Read, 1. A.,164 Rees, I. G.,94 Reid, G.P., 75 Reilly, J. P., 40 Reinsch, C., 307 Renken, A., 333 Rennekarnp, M. E.,112 Resler, E. L.,290 Reuben, B. G.,105, 318 Rice, F. O.,311 Rice, J. K.,178 Rice, 0.K.,94, 244, 245, 301 Rice, S. A., 76, 104, 108, 1 19, 245 Rice. W.W.,20 Rich; J. W., 245 Richard, J. P., 157 Richardson, W.C.,23,84 Richey, G. H.,132, 134 Richmond. A. B., 142 Ridley, B. ‘A.,26,-59, 290 Riecke, E. E.,135 Rieff, M.,31 1 Riley, S. J., 105 Ring, M. A.,221,222,230, 231, 232 Rinker, R. G.,318 Riola, J. P., 85 Ripley, D. L., 173 Roark, D.N., 213 Roberts, D.W.,158 Roberts, J. R., 213

Author Index

394 Roberts, R. E., 30, 253 Robinson, A., 300 Robinson, P. J., 93, 132, 134, 142, 158, 220, 239, 240 Robson, R. C., 23 Roche, A. E., 53 Rodgers, A. S., 96, 122, 169, 172 Root,.J. W., 103 Roscoe, J. M., 43 Rosenfeld, J. L. J., 253, 262,265, 273, 303 Rosenwaks. S.. 49 Ross, D. S.; 123 Ross, I. G., 96, 245 Ross, I.,22, 96, 380 Rossler, 0. E., 383 Rowland, C., 134 Rowland, F. S., 75, 103, 124 Rowley, R. J., 213 Ruis, S. P., 221 Ruiz Diaz, K., 166 Rulis, A. M., 33 Rumfeldt, R. C., 157 Rundel, R. D., 82, 85 Rush, D. G., 265 Russell, K. E., 244 Russo, A. L., 269 Ruthuen, D. M.,133 Ryba, O., 157 Rynbrandt, J. D., 98, 120 Saalfeld, F. E., 230 Saban, G. H., 109 Sachyan, G. A., 193 Sadie, F. G., 49 Sadler, I. H., 132 Saeed, M., 164, 206 Safarik, I., 216, 217 Safrany, D. R., 42 Safron, S. A., 105,106,108 St. John, G. A., 46 Sakurai, H., 218,238 Salnikov, J. E., 372 Salem, L., 134, 135 Sampson, R. J., 201 Sandhu. H. S., 118, 122, 137, 215,217Sandri, G., 253 Sarner, S. F., 137, 142, 154 Sarr. M.. 129 Sartor, A. F., 172 Sasaki, T.. 134, 146 Sato, Y . , 259 . Saunders, B. B., 124 Schacke, H., 54 Schaefer, H. F., tert., 36, 47, 82, 155 Schaeffer, R., 227 Schaibly, J. H., 295 Schallner. 0.. 141 Schaschel, E.-T., 221 Scheller, K., 133, 155, 166, 289 Scherszer, K., 157 Schiff, H. I., 43 Schindler, R. H., 217 Schlag, E. W., 119, 245, 259

Schliefer, A., 137 Schmatjko, K. J., 54 Schmeltekopf, A. L. S., 84 Schmidt, A. H., 142 Schoeller, W. W., 135 Schoener, B., 330 Schofield, K., 15 Schollkopf, U., 134 Schrieber, J. L., 24 Schuchmann, H. P., 170 Schuetzle, D., 102 Schug, K. P., 127, 158 Schultz, A., 37 Schuster, G. B., 149 Schweid, A. N., 80, 262 Scott, D. A., 149 Scott, P. M., 57 Secrest, D., 262 Seebohm, R. P., 133 Seelig, F. E., 383 Seery, D. J., 157, 177 Sefcik, M. D., 221, 222, 721 L.-r I

Seibt, D., 122, 240 Sekhai, M. V. C., 158 Sekikawa. N.. 217 Selinger, B. K.,104 Sel’kov, E. E., 371 Semenov, N. N., 334 Serres, B., 226 Setser, D. W., 23, 71, 73, 82, 84,93, 101, 112, 118, 121, 122, 123 Seyferth, D., 226 Shannon, T. W., 234 Sharma, R. D., 263 Sharples, L. K., 378 Sharples, R., 43 Shaw, D. H., 157 Shaw, D. N., 311 Shaw, H., 175, 179 Shaw, R., 123, 169, 182 Shchepinov, S. A., 221 Shear, D., 349 Sherrington, M. E., 349 Shevchuck, V. U., 158 Shibasaki, M.,155 Shields, F. D., 246 Shih. C. N.. 135 Shih’ H.-M‘ 226 Shin: H. K.’,’262, 264, 353 Shizgal, B., 261 Shobatake, K., 76, 104, 108, 119 Shortridge, R., 183 Shui, V. H., 30, 254 Shuler, K. E., 247, 255, 259.295 Siara.’ I. -N.. 24. 34 Siefert, E. I?., 1.18 Simm, I. G., 111 Simmie, J. M., 150, 230, 233. 234 Simmons, J. D., 39 Simmons, R. F., 157, 158, 164, 322 Simon, M.,177 Simonaitis, R., 56, 57, 90, 191, 192 Simonetta, M., 146 Simons, J., 244

Simons, J. P., 52, 57, 66, 87 Simons, J. W., 124, 125, 227 Simpson, J., 234 Simpson, J. M., 132, 134 Sinkule, J., 334 Sirtc, E., 234 Skardis, G., 88 Skell, P.S., 226 Skinner, G. B., 94, 122, 155, 166 Skirrow, G., 166, 174 Skonieczny, J., 90 Skrlac, W. J., 24 Slanger, T. G., 43,46, 69 Slater, D. A., 184 Slater, D. H., 71, 202 Sliepcevitch, C. M., 133 Sloan, J. J., 77 Sloane, T. M., 105 Slocomb, C. A., 82 Slutsky, J. 146, 154, 238 Smith, G. G., 154 Smith, H., 296, 304 Smith. I. W. M.. 49., 78,. 262; 290 Smith, W. D., 30, 2512 Snelling, D. R., 56 Snider, N. S., 269 Snow, R. H., 301 Snow, R. L., 30,252 Sobtsov, A. A., 213 Sokolov, N. D., 253 Solc, M:, 95 . Solly, R. K., 94, 142, 151, 157, 159, 170, 171, 172 Solo, R. B., 319 Solomon. W. C.. 266 Sommer,’L. H., 213, 218 Songstad, J., 155 Spalding, T. R., 234 Spangler, B., 145 Spangler, C . W., 145 Spears, K. G., 104 Speed, F. M., 307 Speis, A., 151 Spence, K., 322 Spencer, D. J., 245 Spier, L. D., 118 Spokes, G. N., 126, 167, 173 Srinivasan, R., 132, 135 Stanton, R. E., 96 Staricco, E. H.. 118. 127. 132, 149 Steacie, E. W. R., 162 St:tyngs, R. F., 82, 85, I

,

.

LOL

Stedman, D. H., 82, 84 Steele, J. H., 295 Steele, W. C., 230 Stein, S . E., 94 Steinfeld, J. I., 80,262,264, 290 Stephens, A., 215, 216 Stephens, R. R., 264 Stephenson, 1. L., 229 Stephenson,J. C., 263,264, 363 Stephenson, L. M.,146

Author Index Stepukhovitch, A. D., 96, 244

Stevenson, C. D., 78 Stevenson, R. J., 132 Stevenson, S.,228 Stewart, G. W., 236, 237 Stewart, J. A. G., 132 Stief, L. J., 191, 192 Stock, M. G., 16 Stockbauer, R., 111 Stofko, J. J., 146 Stone, A. J., 27 Stone, F. G. A., 230 Stone, J., 96, 290 Strain, R. H., 19, 65, 78 Strausz, 0. P., 24, 60, 61,

66,69,86, 118, 122, 187, 215,216,217

Stuhl, F., 21, 22, 69 Stupochenko, E. V., 253, 26 1

SU,Y.-Y., 64,225 Suart, R. D., 161 Suess, G. N., 29, 159 Sutton, D. G., 290 Svec, H. J., 230 Svoboda, P.,226 Swaminathan, S., 135 Swenton, J. S., 141 Swinehart, D. F., 94 Szabo, Z. G., 96 Szoke, A., 83,270 Szirovicza, L., 159 Taguchi, R. J., 86 Tai, H., 261 Tait, K. B., 75 Taiima. E.. 217 Tikacs; G.’ A., 27, 54 Takagi, K., 157 Takaoka, T., 225 Takayanagi, K., 262 Takenouti, K., 384 Takeyama, T., 173 Tal’Roze. V. L.,75 Tanaka, I., 124Tang, K. T., 26 Tang, T. L., 322 Tang, Y.-N., 32, 225, 226, 23 5

Tardif, A., 56

Tardy, D. C., 24,96 Tasker, P.W., 57,66, 87 Tassie, L. J., 253 Taylor, G. A., 238 Taylor, G. W., 23, 841,125 Tavlor. J. E.. 164 Tailor; R., 154 Tellinghuisen, J. B., 18 Teng,.L.! 163, 179 Teranishi, H., 171 Terashima, S., 155 Thiele, E.,96, 290 Thomas, P. H., 367 Thompson, J. F., 214 Thompson, K., 182 ThomDson. M.. 119 ‘Thompson; W: w., 159, 167, 172 Thompson, J. C., 221

395 Thrush, B. A., 15, 22, 23,

27, 29, 40, 51, 60, 118, 120. 145. 163 Thynne, J.-C. J., 181, 182, 193, 186 Tibbals, H. F., 230 Tilsley, G. M., 218, 21.9 Timlin, D. M., 71, 151, 185. 216 Tim&o&, R. B., 27 Timms. P. L.. 221 Ting, 6. T., 103 Tippei, C. F. H., 163, 166, 174,238,319, 322, 327 Toby, S., 124, 179 Toennies, J. P.,243 Tomlin. J. A.. 246 Tomlin; S. G.’,246 Topor, M. G., 119 Toriyama, K., 149 Torrie, G., 86 Townend, A., 322 Townend, D. T. A., 322 Trainor, D. W., 29, 287 Treanor, C. E., 245 Trenwith, A. B., 157, 158, 169 Treverton, J. A., 234 Trimm. D. L.,~164. 202. . 206 Tripodi, M. K.,236 Troe, J., 80, 81, 93, 118, 119, 126, 127, 1-173, 174. 175. 182. 244

Trotman-Dickenson, A. F., 150,159,163,171,177, 189,201,218,219

Truby, F. K., 178 Tsai. M. R.. 238 Tsang, W., ‘132, 133, 150, 158,180

Tsang, Y.-N., 104 Tschuikow-Roux, E., 127, 150,151,157,

158

Tsuchiya, H. M., 345 Tsuchiya, S., 23, 87, 259 Tullyj J. C., 106 Turro, N. J., 142 Tyerman, W. J. R., 24, 52, 61,64,65,69,259

Tyler, B. J., 155, 158, 323 Tyson, J. J., 381 Ulland, L. A., 218 Ullenius, C., 134 Umanski, S. Ya., 5 5 Umanskii, V. M.,244 Urch, D. S., 104 Ushakova, R. L., 212 Valance, W. G., 245, 259 Vallana, C. A., 118, 127, 149

Valsarova, V., 226 VanDalen, M. J., 234 Vardanyan, I. A., 193 van den Bergh, H. E., 161, 178, 181

Van Den Berg, P. J., 234 Van den Bogaerde, J., 23 Vanderpool, S., 132

Vanderwielen, A. J., 221 Van Itallie, F. J., 88 Vanpk, M., 321, 322 Van Volkenburgh, G., 31 van Roodselaar, 60, 61 Vavilin, V. A., 312, 332 Vdovin, V. M.,212, 213 Verbeyst, J., 151 Verlin, J. D., 182 Vikis, A. C., 86, 88, 122 Vinall, I. C., 95, 159 Vitt, A. A., 340 Volman, D. H., 191 Volter, B., 332 von Weyssenhoff, H.,119 Voorhees, K. J., 149, 154 Vorachek, J. H., 123 Waage, E. V., 126, 158, 163

Waddington, D. J., 166, 195,323

Waech. T. G.. 249 Wagner, H. Gg., 23, 60,

74,75, 93,127, 157,158, 173.175.244 Wagschal; ’L,289 Walker, B. F., 173 Walker,-R. F., 74, 76, 183 Walker, R. W., 163, 165, 166, 174, 183, 185, 188, 192, 193,201,203, 329 Waller, M. J., 134 Walsh, A. D., 321, 322 Walter, C. F., 369, 370 Walsh, R., 119, 139, 159, 163, 169, 171, 172, 214, 230 Walsingham, R. W., 240, 24 1 Walton. D. R. M..238 Wang, F.-M., 118; 155 Wang, H. Y.,27 Wang, I. S. Y.,96 Wanner, J., 75 Warden. R. B.. 373 Warder; R. C. ,.279 Ware, W., 104 Warnatz, J., 74, 75 Washida, N., 40, 51 Wasson, J. S., 135 Watanabe. H.. 226

Watkins, K. W., 122, 123,

159, 167, 172, 184, 185, 186

Watson, H. D. D., 303 Watson, R. T., 23, 50 Watt, W. S., 244 Watts, G. B., 218 Wayne, F. D., 27 Wayne, R. P.,56 Webster, N. J., 157 Webster, S. J., 165, 192 Weeks, R. W., 124 Weinstein, N. D., 106 Weinstock, B., 182 Weisenfeld, J. R., 65 Weiss, G. H., 247 Welge K. H., 21, 59, 69 wells,’c. H. J., 164 Wells, J. M.,230

Author Index Werner, A. S., 107 Wesley, T. A. B., 323 West, K. O., 146 West, M. A., 17 West, R., 238 Westenberg, A. A., 15, 23, 26,53,161,162,167,192

Weston, R. E., 35 Wexler, A., 141 Wharton, P. S., 145 Wheatley, T. F., 318 White, A. J., 159 White, M. L., 157 White, R. A., 106 Whitlock, P. A., 30 Whitson, M. E., jun., 55 Whittle, E., 170 Whytock, D. A., 50, 188 Widman, R. P., 80 Widom. B., 96, 245. 269 Wien, C. M.,154 . Wiesenfeld, J. R., 17, 18, 21, 23, 46, 56, 78

Wijnen, M. H. J., 188 Wilkinson. J. H.. 288 Williams, A., 166, 167 Williams, D. L., 221 Williams, G. J., 263 Williams, J. L., 32 Williams, 0. M., 69 Williams, R. L., 75 Williams, W. J., 158, 239

Wilson, A. T., 331 Wilson, P. W., 221 Wilson, W. E., 162 Winkler, C. A., 46 Winter, G. K., 104 Winter, R. E. K., 139 Winter, T. G., 279 Witiak, J. L., 212 Wodarczyk, F. J., 77 Wolf, A. P., 227 Wolfgang, R., 107, 110 Wolfrum, J., 54 Wollenberg, R. H., 96 Wong, N.-B., 226 Wong, W. H., 53, 80, 248,

Yarkony, D. R., 36 Yamazaki, J., 331, 332 Yang, C. H., 369, 379 Yates, J., 253 Yau, A. Y., 283 Yaurada, S. I., 155 Yee Quee, M. J., 186 Yip, C . K., 155, 157, I 59 Yokota, K., 331, 332 York, E. J., 132 Yorke, D. A., 165,166, 329 Young, A. W., 264 Young, J. C., 215,216, 240 Young, R. A., 21, 31, 46, 60,69

252

Yu, T.-Y., 236 Yu,W. H. S., 188

49,71

Zabel. F., 157 Zaikin, A. M., 312, 332 Zaleski, T. A., 50 Zare, R. N., 15, 36,37 Zeck, 0.F., 225 Zeelenberg, A. P., 164 Zetzsch. C.. 74.75 Zhabotinskii, A. M., 312,

Wood, P. M., 86, 262 Woodall, G. N. C., 187 Woodall, K. B., 15,24,33, Woodson, J. H., 312 Woodworth, 1. R., 83 Woon-Fat, A. R., 80 Wray, K. L., 157 Wren, D. J., 36 Wright, J. S., 135 Wu, D. T., 83 Wu, M. C. R., 96 Yakov, Y. B., 72 Yankee, E. W., 134

332, 359

Zhaikin, A. M., 359 Ziolkowski, F. J., 149, 154 Zittel. P. F.. 290 Zucker, U. F., 214,216