FUNDAMENTAL CHEMICAL KINETICS An Explanatory Introduction to the Concepts "Talking of education, people have now a-days" (said he) "got a strange opinion that every thing should be taught by lectures. Now, I cannot see that lectures can do so much good as reading the books from which the lectures are taken. I know nothing that can be best taught by lectures, except where experiments are to be shewn. You may teach chymestry by lectures - You might teach making of shoes by lectures!" James Boswell: Life ofS amuel Johnson, 1766 ABOUT THE AUTHOR Margaret Robson Wright graduated with first class honours in Chemistry from the University of Glasgow in 1960, and then moved to Oxford University to continue her interests in chemical kinetics under the, then, Mr R.P. Bell, FRS, graduating with a D. Phil in 1963. While in Oxford she embarked on her teaching career by tutoring in Chemistry for her college, Somerville. She then moved as a Research Fellow to the Department of Chemistry of Queen's College, Dundee, then part of the University of St. Andrews. After two years she won the prestigious Shell Research Fellowship, being the first person, along with the late Dr. M.A.A. Clyne, to be awarded this honour. During her tenure of these two fellowships she continued her teaching career, lecturing, tutoring and demonstrating at all levels of the four year Scottish degree. In 1965, she married a colleague, Dr. Patrick Wright, a fellow physical chemist with a distinguished research output who also shared her commitment to excellence in teaching. After the birth of their first child in 1967, she became a part-time lecturer concentrating totally on teaching. She held this post until 1982 when, as a result of the first set of university "cuts", she was forced, along with other part-time lecturers, into voluntary redundancy. She then became the first University Fellow at Dundee University, an honorary position awarded to academic staff who, during their time in the university, performed outstanding teaching or research. During her seven years in the wilderness of unemployment, she maintained her interests in teaching through tutoring S-grade, H-grade and A-level school pupils and at one stage prepared students for the Cambridge University Entrance Examination. During this period she also wrote her first book The Nature of Electrolyte Solutions published in 1988. In 1989 she became part-time temporary lecturer in chemistry at the University of St. Andrews, becoming part-time Teaching Fellow in Chemistry in 1992, a post she held until her retirement. During this time she lectured and tutored on most of the topics in mainstream physical chemistry, covering all aspects of kinetics, thermodynamics, spectroscopy, equilibria and electrolyte solutions at all levels of the four year Scottish degree. She has written a wide variety of informal study guides over a considerable range of topics, produced problem booklets with extensive and detailed explanatory answers and a series of booklets on help in mathematics for chemists with a weak background in mathematics. Being solely available at St Andrews, these were voted as being very popular and extremely helpful to students. After retirement she published two further books, An Introduction to Chemical Kinetics and An Introduction to Aqueous Electrolyte Solutions. It is with this wide range of teaching experience that she has written this text, which reflects her total commitment to the teaching of university students. Margaret Wright is married with two sons and a daughter, all now grown up. FUNDAMENTAL CHEMICAL KINETICS An Explanatory Introduction to the Concepts Margaret Robson Wright, BSc, DPhil formerly Lecturer in Chemistry, The School Of Chemistry St. Andrews University, Scotland WP WOODHEAD PUBLISHING Oxford Cambridge Philadelphia New Delhi Published by Woodhead Publishing Limited, 80 High Street, Sawston, Cambridge CB22 3HJ, UK www.woodheadpublishing.com Woodhead Publishing, 1518 Walnut Street, Suite 1100, Philadelphia, PA 19102-3406, USA Woodhead Publishing India Private Limited, G-2, Vardaan House, 7/28 Ansari Road, Daryaganj, New Delhi – 110002, India www.woodheadpublishingindia.com First published by Horwood Publishing Limited, 1999 Reprinted by Woodhead Publishing Limited, 2011 © M. R. Wright, 1999 The author has asserted her moral rights This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials. Neither the author nor the publisher, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited. The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing Limited for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 978-1-898563-60-0 Printed by Lightning Source DEDICATION Dedicated to the memory of my mother, Anne, with much love, affection and deep gratitude and to my husband Patrick: Anne, Edward, Andrew and the cats. PREFACE This book contains many important topics in modem theoretical studies in chemical kinetics, and discusses some of the advances in experimental techniques which have enabled these theories to be tested and validated. As the title indicates the primary aim is to help students to understand modem thinking about "how a chemical reaction occurs" by giving a slow, explanatory 'student-friendly' yet rigorous introduction to such thinking. Many students veer rapidly away from topics which can be labelled theoretical or conceptual because of a near involuntary and automatic assumption that they will be incapable of understanding. This book attempts to allay these fears by guiding the student through these modem topics in a step-by-step development which explains both the logic and the reasoning. There are two major problems for students to overcome before they can understand a theory: firstly, the concepts, and then the mathematics used to quantify them. In the author's experience students rarely manage to deduce the chemical concepts from the mathematical development. These concepts need to be fully and explicitly explained first before the student can start to understand 'how chemical reaction occurs'. It is the author's firm belief, based on years of teaching, that students can gain a considerable understanding and knowledge of what is involved in a theoretical discussion of chemical reaction without delving into the mathematics first. Once the concepts have been assimilated it is then time to move on to the mathematical presentation, which in this book is developed with the same full explanatory detail. It is unfair to expect students to work all this out for themselves, over and above the very necessary learning of facts. It is the teacher's duty to show them how to achieve understanding and to explain how to think scientifically. The philosophy behind this book is that this is best done by detailed explanation and guidance. It is understanding and confidence which help to stimulate and sustain interest. This book attempts to do precisely that. For a subject so wide-ranging as chemical kinetics, and in a book of this length, it is impossible to cover all theoretical topics, especially if explanation is the key aim. For this reason, the book is not comprehensive, and only the more important topics are covered. ACKNOWLEDGEMENTS This book is the result of the accumulated experience of nearly forty very stimulating years of teaching students at all levels. During these years I learned that being happy to help, being prepared to give extra explanation and to spend extra time on a topic could soon clear up problems and difficulties which many students thought they would never understand. Too often teachers forget that there were times when they themselves could not understand, and when a similar explanation and preparedness to give time were welcome. If, through the written word, I can help students to understand and to feel confident in their ability to learn, and to teach them in a book in a manner which gives them the feeling of direct contact with the teacher, then this book will not have been written in vain. To all the many students who have provided the stimulus and enjoyment of teaching I give my grateful thanks. I would also like to thank Ellis Horwood of Horwood Publishing for his encouragement to present this book for publication, and for his belief in the method of approach. His continuing encouragement, enthusiasm and help in preparing the manuscript is very gratefully acknowledged and appreciated. My thanks are due to Mrs Jill Blyth for a first-rate rendering of the hand-written manuscript into a working typescript. My especial and very grateful thanks are given to Mrs Rosmary Harris of Rosetec, Worthing, who made the publication of this book possible by her expert and speedy conversion of the initial typescript into camera ready form. Her help with the. technical aspects was vital and indispensible, and I have been totally dependent on her skills. My thanks are also due to Mrs Shirley Fox of the Department of Mathematics in the University of Dundee who typed the mathematical parts of the book. To my mother, Mrs Anne Robson, I have a very deep sense of gratitude for all the help she gave me in her lifetime in furthering my academic career. I owe her an enormous debt for her invaluable, excellent and irreplaceable help with my children when they were young and I was working part-time during the teaching terms of the academic year. Without her help and her loving care of my children I would never have gained my continued experience in teaching, and I would never have written this book. My deep and most grateful thanks are due to her. My husband, Patrick, has throughout my teaching career and throughout the thinking about and writing of this book, been a source of constant support, help and encourage ment. His very high intellectual calibre and wide-ranging knowledge and understanding have provided many fruitful and interesting discussions. He has been an excellent sounding board for many of the ideas and manner of presentation in this book. He has read in detail the whole manuscript and his clarity, insight and considerable knowledge of the subject matter has been of invaluable help. My debt to him is enormous, and my most grateful thanks are due to him. Finally, my thanks are due to my three children who have encouraged me throughout my career and, in particular, encouraged me to write this book. Their very fine and keen brains have always been a source of stimulus to me, and have always kept my interest in young people flourishing. Margaret Robson Wright School of Chemistry The University of St Andrews January, 1999 CONTENTS 1. Introduction: Historical Developments and Modern Kinetics 1.1 A historical development of concepts in kinetics I 1.2 Concepts involved in modem state to state kinetics 3 1.3 Collision processes and a master mechanism describing elementary reactions at the microscopic level 4 1.3. I Energy transfer processes in the master mechanism 4 1.3.2 Accumulation of energy in kinetics 5 1.3.3 Redistribution of accumulated energy 6 1.3 .4 Rate and mechanism of accumulation of energy 7 1.3.5 Life-times, intervals between collisions, and their implications for kinetics 7 1.3.6 Possibility of equilibrium between reactants and activated molecules 8 1.4 Kinetics of uni, bi and termolecular reactions using the master mechanism I 0 1.4.1 Comments on the table 16 1.5 A generalisation of these ideas 16 2. Basic Transition State Theory 18 2.1.1 Transition state theory and configuraion 18 2.1.2 Transition state theory and potential energy 18 2.2 Properties of the potential energy surface relevant to transition state theory 20 2.3 Major assumptions of transition state theory 25 2.4 Derivation of rate of reaction 26 2.4.1 Calculation of vlo= llT 26 2.4.2 Calculation of c~ and c~/cx Cyz 27 2.4.3 A statistical mechanical expression for the rate constant 28 2.4.4 Statistical mechanical quantities used in transition state theory 28 2.4.5 Relation between the molecular partition function,/, and the molecular partition function per unit volume 33 2.4.6 Contributions to the partition functions for polyatomic molecules and radicals 33 2.4.7 Contribution to the partition function for the activated complex 34 2.4.8 Simplification of the statistical mechanical transition state theory expression 34 2.4.9 Formulation of Q for reactants 35 2.4.10 Formulation of Q for the activated complex 37 2.5 Approximate calculations 38 2.5.1 Reaction of two atoms, A and 8, to give a linear diatomic activated complex 42 2.5.2 A more general reaction: two non-linear molecules giving a non-linear activated complex 42 2.5.3 Approximate transition state theory pre-exponential factors for typical reaction types 43 2.5.4 Temperature dependence of partition functions and A factors 45 2.6 Thermodynamic formulations of transition state theory 46 2.6. l Determination of thermodynamic functions for activation 47 2.6.2 Comparison and interpretations of the partition function form and the thermodynamic form of transition state theory and experimental data 49 2. 7 .1 Transition state theory applied to unimolecular reactions 53 2.7.2 Pre-exponential factors for reaction where the activated complex resembles molecule A 53 2.7.3 Pre-exponential factors for reaction where the activated complex differs considerably from reactant A 54 2.7.4 Transition state theory applied to termolecular reactions 55 3. Advanced Transition State Theory 57 3.1 Assumptions of transition theory reassessed 57 3.1.1 Assumption I 57 3.1.2 Assumption 2 58 3 .1.3 Assumption 3 62 3.1.4 Assumption 4 62 3.2 Symmetry numbers and statistical factors 63 3.2. l Symmetry numbers 63 3.2.2 Statistical factors 64 3.2.3 Equilibrium constants using statistical factors 66 3.2.4 Rate constants in terms of statistical factors 68 3.2.5 Use of statistical factors in the prediction of possible structures for the activated complex 71 3.3 Advances in transition state theory 74 3 .3. l Variational transition state theory 78 3.3.2 Microcanonical transition state theory 78 3.3.3 Canonical transition state theory 81 3 .4 Quantum transition state theory 82 3 .4.1 Quantum transition state theory 82 3.4.2 Fully quantum theories 82 3.43 Variational quantum theory 83 4. Basic Unimolecular Theory 89 4.1 Experimental observations and manipulation of experimental data 89 4.2 Significance and physical interpretation of the three rate constants in the general mechanism 91 4.3 Aims of any theory of unimolecular reactions 93 4.4 Aspects of vibrations used in unimolecular theory 94 4.4.1 Harmonic vibrations 94 4.4.2 Anharmonic vibrations 94 4.4.3 Energies in oscillators, and energy in bonds 94 4.5 Mathematical tools used in unimolecular theory 95 4.5.1 Equilibrium statistical mechanics 95 4.5.2 Calculation of kif k. 95 1 4.5.3 Summations and integrals 97 4.5.4 Probability theory 97 4.5.5 The actual Maxwell-Boltzmann calculations 97