m

Unimolecular Reactions P. J. ROBINSON Chemistry Department The University of Manchester Institute of Science and Technology K. A. HOLBROOK Chemistry Department University of Hull

AO Bibliothek

FB7

Physikol. Chemie / Chem. Technologis

Technische Hochschule Dcrmstodt

WILEY-INTERSCIENCE A division of John Wiley & Sons Ltd London • New York • Sydney • Toronto

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Contents i

CHAPTER 1 INTRODUCTION A N D E A R L Y THEORIES . . . GENERAL INTRODUCTION 1.1 Experimental Study of Unimolecular Reactions . . 1.1.1 E x p e r i m e n t a l m e t h o d s . . . . . . 1.1.2 D e d u c t i o n s c o n c e r n i n g m e c h a n i s m . . . . 1.1.3 H e t e r o g e n e o u s processes . . . . . 1.2 Potential Energy Surfaces, Activated Complexes and Absolute R a t e Theory 1.3 Basic Theories of Unimolecular Reactions . . . 1.3.1 T h e L i n d e m a n n t h e o r y . . . . . . . 1.3.2 C o m p a r i s o n o f L i n d e m a n n t h e o r y w i t h e x p e r i m e n t . 1.3.3 T h e H i n s h e l w o o d m o d i f i c a t i o n . . . . 1.3.4 C o m p a r i s o n o f H i n s h e l w o o d - L i n d e m a n n t h e o r y w i t h experiment . . . . . . . . 1.4 The Further Development o f Unimolecular Reaction R a t e Theories References . . . . . . . . . . CHAPTER 2 T H E SLATER THEORY INTRODUCTION

2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.3 2.3.1 2.3.2 2.3.3

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T h e Vibrational Analysis o f Polyatomic Molecules Development of the secular equation . . . Normal coordinates . . . . . Application t o t w o independent oscillators . Application t oa diatomic molecule . . . . Application t o a linear triatomic molecule . M o r e complicated molecules . . . . . T h e results o f vibrational analysis . . . . Slater's Harmonic Theory T h e general-pressure rate-constant . . . . T h e limiting forms a t high a n d l o w pressures . T h e t h e o r e t i c a l fall-off c u r v e T h e change i nactivation energy with pressure . T h e Assumptions of Slater Theory . . . . T h e harmonic assumption . . . . . T h e random-gap assumption . . . . . T h e strong-collision assumption . . . . xiii

1 3 3 4 6 7 13 15 18 19 23 25 27 .

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28 28 31 31 32 33 36 37 38 41 43 43 44 44 45 46 46

XIV

CONTENTS

2.4 T h e Application of Slater Theory References

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CHAPTER 3 T H E RICE-RAMSPERGER-KASSEL THEORIES INTRODUCTION

3.1 T h e Kassel Theories 3.2 Application of the R R K Theories References . . . . . . . CHAPTER 4 R INTRODUCTION

RKM

(MARCUS-RICE) THEORY

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46 50

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53 56 63

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64

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52

4.1 4.2 4.3 4.4 4.5 4.5.1 4.5.2 4.5.3 4.5.4 4.5.5 4.6 4.7 4.8 4.9 4.10 4.10.1 4.10.2 4.10.3 4.10.4

The R R K M Reaction Scheme Classification o f Energies and D e g r e e s o f F r e e d o m Terminology for Energies . . . . E x p r e s s i o n for &k1(E,^B,+SE,)lk2[ . . . . E x p r e s s i o n for ka(E*) E v a l u a t i o n o f k+(x) Evaluation of ([A+]/[A* ])eqm E x p r e s s i o n f o r N+C(x) . . . . . . R e s u l t f o r ka(E*) T w o m o d i f i c a t i o n s , a n d final r e s u l t f o r ka(E*) . R R K M E x p r e s s i o n for kuni T h e High-pressure L i m i t . . . . . . T h e Low-pressure L i m i t . . . . . . Statistical F a c t o r s Improved T r e a t m e n t of Adiabatic Rotations . Basic treatment First approximation . . . . . . Further approximations . . . . . . Conclusions

4.11

The Quantities £

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64 66 67 69 70 71 72 73 74 74 75 76 78 80 85 86 88 90 92

E+

P(£+r) andN*(E*)

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9 3

•pi

4.11.1 4.11.2 4.12 4.12.1 4.12.2 4.12.3 4.12.4 4.12.5

S u m of states,

£ P(EY)

93

D e n s i t y o f q u a n t u m s t a t e s , N(E)

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Assumptions of the Basic RRKM Theory . Free exchange of energy between oscillators Strong collisions The 'equilibrium hypothesis' R a n d o m lifetimes Continuous distribution function N*{E*) .

R e f e r e n c e s .

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99 99 103 104 105 106

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95

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1 0 6

CONTENTS

XV

CHAPTER 5 T H E EVALUATION OF SUMS AND DENSITIES OF MOLECULAR QUANTUM STATES INTRODUCTION

109

5.1

Separation of Vibrational and Rotational Degrees of Freedom 110 5.2 Classical Treatment of Rotational State Distributions . 112 5.2.1 Quantized rotations 112 5.2.2 Classical rotations . . . . . . . 1 1 4 5.2.3 Results for N*(E*), -£P(E+r), and kuni . . 115 5.2.4 Validity of the classical independent rotor treatment . 116 5.3 Direct Count of Vibrational States . . . . 1 1 9 5.3.1 T h e basic m e t h o d 119 5.3.2 Grouped-frequency models . . . . . 125 5.3.3 Commensurable-frequency models . . . . 1 2 6 5.3.4 Conclusions on the direct-count method . . . 128 5.4 Classical Treatment of Vibrational States, and Derived Semiclassical Approximations . . . . . 128 5.4.1 Treatment of classical harmonic vibrations . . 128 5.4.2 Semiclassical (Marcus-Rice) approximation . . 131 5.4.3 Whitten-Rabinovitch approximation . . . 131 5.4.4 Whitten-Rabinovitch treatment of vibrationalrotational systems . . . . . . . 1 3 7 5.5 Inverse Laplace Transformation of the Partition Function 138 5.5.1 Direct inversion of an approximate partition function 139 5.5.2 Inversion by complex integration: M e t h o d of residues 141 5.5.3 Evaluation of complex inversion integral by method of steepest descent . . . . . . . 1 4 4 5.6 Comparisons and Conclusions . . . . . 146 References . . . . . . . . . . 149 CHAPTER

6

N U M E R I C A L A P P L I C A T I O N OF THE

RRKM

THEORY

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151

INTRODUCTION

6.1 6.2 6.2.1 6.2.2 6.2.3 6.3

Calculation of Activation Parameters for a Postulated Model of the Reaction . . . . . . 1 5 1 Specification of a Model for the Reaction . . . 1 5 3 Treatment of rotational degrees of freedom; 'rigid' a n d 'loose' complexes . . . . . . . 1 5 4 Assignment of numerical properties t o the activated complex . . . . . . . . 156 Empirical approach t o specification of the activated complex . . . . . . . . 160 The RRKMIntegration 161

XVI

CONTENTS

6.4

Application of RRKM Theory to the Isomerization of 1,1-Dichlorocyclopropane . . . . . 1 6 5 6.5 Sensitivity of the RRKM Calculation to Computational Details and Features of the Model . . . . 1 7 1 6.5.1 Details of the computational procedure . . . 1 7 1 6.5.2 Variation of the model 173 References . . . . . . . . . . 182 CHAPTER

7 EXPERIMENTAL D A T A FOR THERMAL UNIMOLECULAR

REACTIONS

184

INTRODUCTION

7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.1.5 7.1.6 7.1.7 7.1.8 7.1.9 7.2 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 7.3 7.4

Unimolecular Reactions at High Pressures . . . Cyclopropane and its derivatives . . . . Cyclobutane and its derivatives . . . . Cyclobutene and its derivatives . . . . Polycyclic systems Olefins a n d polyolefins Heterocyclic compounds . . . . . . Alkylhalides Esters Other unimolecular reactions . . . . . Unimolecular Reactions at Low Pressures . . . The structural and geometrical isomerizations of cyclopropane and its derivatives at low pressures . The reactions of cyclobutane and its derivatives at low pressures . . . . . . . . The isomerizations of cyclobutene a n d its derivatives at low pressures . . . . . . . . The pyrolysis of alkyl halides at low pressures . . The isomerization ofisocyanides at low pressures. . Unimolecular Reactions Producing Free Radicals . Unimolecular Decomposition Reactions of Free Radicals

References

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CHAPTER 8 CHEMICAL ACTIVATION INTRODUCTION

8.1 Basic Principles . 8.1.1 The average rate constant 8.1.2 The form of the distribution function f(E) 8.1.3 The average energy. 8.2 Experimental Studies . References . . . . . . .

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1 8 4 185 1 9 3 1 9 6 198 206 214 217 226 2 3 1 238 243 247 249 250 253 255 260

263

268 271 271 272 274 275 284

CONTENTS CHAPTER 9

XV11 K I N E T I C ISOTOPE EFFECTS IN UNIMOLECULAR REACTIONS

286

INTRODUCTION

9.1 9.2 9.3

General Discussion of Isotope Effects . . . 2 8 6 Basis of Application to Unimolecular Reactions . . 290 Secondary Kinetic Isotope Effects on ka(E*) (Theory and Experiment) 291 9.4 Primary Kinetic Isotope Effects on ka(E*) (Theory and Experiment) 295 9.5 Isotope Effects in Thermal Unimolecular Reactions . 299 9.5.1 Theory 299 9.5.2 Experimental studies . . . . . . 3 0 2 References 306

CHAPTER 10

COLLISIONAL E N E R G Y TRANSFER IN

UNIMOLECULAR

REACTION SYSTEMS

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309

INTRODUCTION

10.1 General Equations . . . . . . . 10.2 Transition Probability Models 10.3 Chemically Activated and Photoactivated Systems 10.4 Thermal Unimolecular Reaction Systems . . References

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309 313 315 319 327

APPENDICES

1 2 3 4 5 6 7

Nomenclature . . . . . . . . Statistical Mechanics Computer Programs for the Direct Count of Vibrational Quantum States Classical Approximation to W(ET), the S u m of Rotational States . Partition Function of a System of Classical Rotors . . Classical Approximation to W{EY), the Sum of Vibrational States The Gamma Function T(n)

INDEX

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328 333 345 348 355 356 358 359