chemical kinetics from molecular structure to chemical reactivity

mis-In this first phase of development, the theories of chemical kinetics tried to resolve theproblem of the calculation of the pre-exponential factor and activation energy in the Arrhen

Chemical Kinetics From Molecular Structure

to Chemical Reactivity

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Preface xi

1 Introduction 1

1.1 Initial Difficulties in the Development of Chemical Kinetics in the Twentieth Century 2

1.2 Chemical Kinetics: The Current View 4

References 14

2 Reaction Rate Laws 15

2.1 Reaction Rates 15

2.2 Factors that Influence the Velocities of Reactions 17

2.2.1 Nature of the reagents 17

2.2.2 Reactant concentration 19

2.2.3 Temperature 25

2.2.4 Light 26

2.2.5 Catalysts 29

2.2.6 Reaction medium 30

References 32

3 Experimental Methods 33

3.1 Application of Conventional Techniques to Study Reactions 34

3.1.1 First-order reactions 34

3.1.2 Second-order reactions 36

3.1.3 Complex reactions 39

3.1.4 Activation energy 41

3.1.5 Dependence of light intensity 43

3.1.6 Enzyme catalysis 46

3.1.7 Dependence on ionic strength 47

3.2 Application of Special Techniques for Fast Reactions 50

3.2.1 Flow methods 51

3.2.2 Relaxation methods 52

3.2.3 Competition methods 56

3.2.4 Methods with enhanced time resolution 61

References 75

4 Reaction Order and Rate Constants 77

4.1 Rates of Elementary Reactions 77

4.1.1 First-order reactions 77

4.1.2 Second-order reactions 80

4.1.3 Zero-order reactions 82

4.1.4 Third-order reactions 83

4.2 Rates of Complex Reactions 84

4.2.1 Parallel first-order reactions 85

4.2.2 Consecutive first-order reactions 86

4.2.3 Reversible first-order reactions 88

4.3 Methods for Solving Kinetic Equations 89

4.3.1 Laplace transforms 89

4.3.2 Matrix method 94

4.3.3 Runge–Kutta method 97

4.3.4 Markov chains 99

4.3.5 Monte Carlo method 103

4.4 Simplification of Kinetic Schemes 106

4.4.1 Isolation method 106

4.4.2 Pre-equilibrium approximation 107

4.4.3 Steady-state approximation 108

4.4.4 Rate-determining step of a reaction 111

References 113

5 Collisions and Molecular Dynamics 115

5.1 Simple Collision Theory 117

5.2 Collision Cross Section 122

5.3 Calculation of Classical Trajectories 128

5.4 PES Crossings 135

5.5 Molecular Dynamics 137

References 142

6 Reactivity in Thermalised Systems 143

6.1 Transition-State Theory 143

6.1.1 Classical formulation 144

6.1.2 Partition functions 147

6.1.3 Absolute rate calculations 149

6.1.4 Statistical factors 151

6.1.5 Beyond the classical formulation 154

6.2 Semi-Classical Treatments 156

6.2.1 Kinetic isotope effects 156

6.2.2 Tunnel effect 160

6.3 Intersecting-State Model 167

6.3.1 Activation energies 170

6.3.2 Classical rate constants 176

6.3.3 Absolute semi-classical rates 180

6.3.4 Relative rates 183

References 187

7 Relationships between Structure and Reactivity 189

7.1 Quadratic Free-Energy Relationships (QFER) 189

7.2 Linear Free-Energy Relationships (LFER) 193

7.2.1 Brönsted equation 194

7.2.2 Bell–Evans–Polanyi equation 196

7.2.3 Hammett and Taft relationships 196

7.3 Other Kinds of Relationships between Structure and Reactivity 202

7.3.1 The Hammond postulate 202

7.3.2 The reactivity–selectivity principle (RSP) 203

7.3.3 Relationships of the electronic effect: equation of Ritchie 205

7.3.4 An empirical extension of the Bell–Evans–Polanyi relationship 205

References 207

8 Unimolecular Reactions 209

8.1 Lindemann–Christiansen Mechanism 209

8.2 Hinshelwood’s Treatment 212

8.3 Rice–Rampsberger–Kassel–Marcus (RRKM) Treatment 215

8.4 Local Random Matrix Theory (LRMT) 218

8.5 Energy Barriers in the Isomerisation of Cyclopropane 220

References 222

9 Elementary Reactions in Solution 223

9.1 Solvent Effects on Reaction Rates 223

9.2 Effect of Diffusion 225

9.3 Diffusion Constants 229

9.4 Reaction Control 235

9.4.1 Internal pressure 237

9.4.2 Reactions between ions 240

9.4.3 Effect of ionic strength 244

9.4.4 Effect of hydrostatic pressure 246

References 249

10 Reactions on Surfaces 251

10.1 Adsorption 251

10.2 Adsorption Isotherms 256

10.2.1 Langmuir isotherm 256

10.2.2 Adsorption with dissociation 257

10.2.3 Competitive adsorption 258

10.3 Kinetics on Surfaces 259

10.3.1 Unimolecular surface reactions 259

10.3.2 Activation energies of unimolecular surface reactions 260

10.3.3 Reaction between two adsorbed molecules 261

10.3.4 Reaction between a molecule in the gas phase and an adsorbed molecule 263

10.4 Transition-State Theory for Reactions on Surfaces 263

10.4.1 Unimolecular reactions 263

10.4.2 Bimolecular reactions 265

10.5 Model Systems 268

10.5.1 Langmuir–Hinshelwood mechanism 268

10.5.2 Eley–Rideal mechanism 270

References 271

11 Substitution Reactions 273

11.1 Mechanisms of Substitution Reactions 273

11.2 SN2 and SN1 Reactions 274

11.3 Langford–Gray Classification 276

11.4 Symmetrical Methyl Group Transfers in the Gas-Phase 280

11.5 State Correlation Diagrams of Pross and Shaik 282

11.6 Intersecting-State Model 285

11.7 Cross-Reactions in Methyl Group Transfers in the Gas Phase 288

11.8 Solvent Effects in Methyl Group Transfers 289

References 294

12 Chain Reactions 295

12.1 Hydrogen–Bromine Reaction 295

12.2 Reaction between Molecular Hydrogen and Chlorine 298

12.3 Reaction between Molecular Hydrogen and Iodine 300

12.4 Calculation of Energy Barriers for Elementary Steps in Hydrogen–Halogens Reactions 301

12.5 Comparison of the Mechanisms of the Hydrogen–Halogen Reactions 303

12.6 Pyrolysis of Hydrocarbons 305

12.6.1 Pyrolysis of ethane 306

12.6.2 Pyrolysis of acetic aldehyde 308

12.6.3 Goldfinger–Letort–Niclause rules 309

12.7 Explosive Reactions 310

12.7.1 Combustion between hydrogen and oxygen 310

12.7.2 Thermal explosions 314

12.7.3 Combustion of hydrocarbons 316

12.8 Polymerisation Reactions 317

References 320

13 Acid–Base Catalysis and Proton-Transfer Reactions 321

13.1 General Catalytic Mechanisms 321

13.1.1 Fast pre-equilibrium: Arrhenius intermediates 322

13.1.2 Steady-state conditions: van’t Hoff intermediates 324

13.2 General and Specific Acid–Base Catalysis 326

13.3 Mechanistic Interpretation of the pH Dependence of the Rates 329

13.4 Catalytic Activity and Acid–Base Strength 338

13.5 Salt Effects 342

13.6 Acidity Functions 343

13.7 Hydrated Proton Mobility in Water 345

13.8 Proton-Transfer Rates in Solution 350

13.8.1 Classical PT rates 351

13.8.2 Semiclassical absolute rates 356

References 358

14 Enzymatic Catalysis 361

14.1 Terminology 361

14.2 Michaelis–Menten Equation 363

14.3 Mechanisms with Two Enzyme–Substrate Complexes 368

14.4 Inhibition of Enzymes 370

14.5 Effects of pH 373

14.6 Temperature Effects 375

14.7 Molecular Models for Enzyme Catalysis 376

14.8 Isomerisation of Dihydroxyacetone Phosphate to Glyceraldehyde 3-Phosphate Catalysed by Triose-Phosphate 379

14.9 Hydroperoxidation of Linoleic Acid Catalysed by Soybean Lipoxygenase-1 381

References 383

15 Transitions between Electronic States 385

15.1 Mechanisms of Energy Transfer 385

15.2 The “Golden Rule” of Quantum Mechanics 391

15.3 Radiative and Radiationless Rates 395

15.4 Franck–Condon Factors 400

15.5 Radiationless Transitions within a Molecule 407

15.6 Triplet-Energy (or Electron) Transfer between Molecules 410

15.7 Electronic Coupling 421

15.8 Triplet-Energy (and Electron) Transfer Rates 430

References 434

16 Electron Transfer Reactions 437

16.1 Rate Laws for Outer-Sphere Electron Exchanges 437

16.2 Theories of Electron-Transfer Reactions 440

16.2.1 The classical theory of Marcus 440

16.2.2 Solute-driven and solvent-driven processes 443

16.2.3 Critique of the theory of Marcus 445

16.2.4 ISM as a criterion for solute-driven electron transfers 449

16.3 ISM and Electron-Transfer Reactions 452

16.3.1 Representing ET reactions by the crossing of two potential-energy curves 452

16.3.2 Adiabatic self-exchanges of transition-metal complexes 454

16.3.3 Outer-sphere electron transfers with characteristics of an inner-sphere mechanism 456

16.4 Non-Adiabatic Self-Exchanges of Transition-Metal Complexes 458

16.4.1 A source of non-adiabaticity: orbital symmetry 458

16.4.2 Electron tunnelling at a distance 458

16.4.3 Non-adiabaticity due to spin forbidden processes 459

16.5 Electron Self-Exchanges of Organic Molecules 460

16.6 Inverted Regions 462

16.7 Electron Transfer at Electrodes 469

16.7.1 The Tafel equation 469

16.7.2 Calculations of rate constants 475

16.7.3 Asymmetry in Tafel plots 478

16.7.4 Electron transfer at surfaces through a blocking layer 479

References 482

Appendix I: General Data 485

Appendix II: Statistical Thermodynamics 487

Appendix III: Parameters Employed in ISM Calculations 495

Appendix IV: Semi-classical Interacting State Model 499

IV.1 Vibrationally Adiabatic Path 499

IV.2 Tunnelling Corrections 502

IV.3 Semi-classical Rate Constants 503

References 504

Appendix V: The Lippincott–Schroeder Potential 505

V.1 Lippincott—Schroeder (LS) Potential 505

V.2 The LS–ISM Reaction Path 508

V.3 Rate Constants for Proton Transfer along an H-bond 508

References 509

Appendix VI: Problems 511

Subject Index 543

Chemical kinetics is the area of science devoted to the study of the rates as well as themechanisms of reactions Its applications range from the understanding of the interplaybetween metabolic processes, where the intricate control of the rates of enzymaticprocesses is fundamental for the overall wellbeing of biological systems, throughindustrial synthesis of both fine and heavy chemicals to the long-term geological andatmospheric changes occurring on our planet since the evolution of the Universe andthose expected to occur in future At the economic level, the overwhelming majority ofindustrial chemical syntheses involves heterogeneous or homogeneous catalysis, and

an understanding of the inherent processes and interactions is fundamental for the optimisation of reaction conditions Moreover, a kinetic and mechanistic understanding

of the complex series of interrelated reactions occurring between molecules such asoxygen, carbon dioxide, hydrogen, nitrogen and its oxides in the stratosphere and thestudy of processes induced by the absorption of light or high-energy radiation is fun-damental to our appreciation of effects such as global warming or the depletion of theozone layer The timescales involved in these dynamic processes vary by many orders

of magnitude, from less than the time of vibration of a chemical bond up to the age ofthe Universe

All textbooks in physical chemistry have sections dedicated to kinetics However, erally, owing to space constraints, they cannot treat the topic in the depth that is necessaryfor its full appreciation, and frequently, they treat its mechanics rather than its practicalapplications or its relations to the other areas of physical sciences such as thermodynam-ics and structural studies Further, although a number of excellent student texts (at theundergraduate as well as postgraduate levels) are devoted to this topic, some of the mostimportant ones were published several decades ago and cannot be expected to reflect thenumerous significant research advances that have been acknowledged by the award ofmany Nobel prizes and other important distinctions in this area

gen-This book aims to provide a coherent, extensive view of the current situation in the field

of chemical kinetics Starting from the basic theoretical and experimental background, itgradually moves into specific areas such as fast reactions, heterogeneous and homoge-neous catalysis, enzyme-catalysed reactions and photochemistry It also focusses onimportant current problems such as electron-transfer reactions, which have implications atthe chemical as well as biological levels The cohesion between all these chemicalprocesses is facilitated by a simple, user-friendly model that is able to correlate the kineticdata with the structural and the energetic parameters

While the book is primarily meant for chemists, we feel that it can also be useful to dents and research workers in related disciplines in the physical sciences, the biological andbiomedical areas and in the earth and atmospheric sciences It is hoped that this text will bebeneficial to students at the undergraduate as well as postgraduate levels In addition, theprograms available free of cost at a dedicated website (http://www.ism.qui.uc.pt:8180/ism/)will be valuable to many research workers whose investigations necessitate the use of thetools of chemical kinetics.

stu-The task of compiling this book would have been impossible without the excellent laboration of many of our colleagues and co-workers, whose studies have been citedthroughout the text The feedback on the earlier versions of this text from our students atthe University of Coimbra have contributed greatly to the improvement of the same Veryspecial thanks are due to Dr Carlos Serpa and Dr Monica Barroso for their contribution

col-to the design of experiments and models that have helped us col-to understand the relationshipbetween chemical structure and reactivity

Luis ArnautSebastiao FormosinhoHugh Burrows

– 1 – Introduction

It is easy with the hindsight of the twenty-first century to think that chemical kinetics hasdeveloped in a logical and coherent fashion But this was far from the case However, anunderstanding of the way we achieved our present ideas on chemical kinetics is a very goodbasis for truly understanding the subject In the first chapter we start by looking at some ofthe milestones and pitfalls in the development of chemical kinetics We then consider therelationship between kinetics and thermodynamics and finally, we consider the relationshipbetween the macroscopic world we live in and the microscopic world of molecules.The great success of Newtonian mechanics in the areas of mechanics and astronomy,which involved the idea of explaining phenomena by simple forces acting between particles,led scientists in the nineteenth century to try to introduce such a mechanical explanation

to all areas involving natural phenomena In chemistry, for example, these concepts wereapplied to interpret “chemical affinity”, leading to the so-called “chemical mechanics” Wewill see that this is not far removed from many of our modern ideas in this area, and we willdevelop our understanding of chemical kinetics within this context

In this chapter, we will see that the concepts of chemical kinetics evolved relatively late

in terms of the overall studies of reactions and reactivity The study of chemical kinetics can

be traced back to Ludwig Wilhelmy [1], who carried out in 1850 the first study of the sion of cane sugar (sucrose) in the presence of acids that he formulated in terms of a first-order mathematical expression to interpret the progress of the reaction Unfortunately, thiswork went unrecognised until Ostwald [2] drew attention to it some 34 years later It mayseem strange today that such an idea of studying the variation of “chemical affinity” withtime had not occurred earlier Farber [3] had tried to explain this and has shown that, in fact,there were some earlier attempts to study the time evolution of reactions, even beforeWilhelmy, but that these tended to be isolated observations Most probably, the practicalimportance of such studies did not exist at the end of the eighteenth century, and it was onlywith the advent of the chemical industry at the beginning of the nineteenth century thatchemists, rather late, needed to consider this problem Eventually, this became of greatimportance for the development of industrial research at the end of that century An excel-

inver-lent discussion of this problem is given by Christine King [4–6] in her studies on the History

of Chemical Kinetics, where she analyses the impact of the various theoretical,

experimen-tal and conceptual works of Berthelot and Péan de St Giles [7–9], Guldberg and Waage [10]and Harcourt and Essen [11–14] These researchers can truly be considered to be thefounders of this new branch of chemistry, chemical kinetics

1.1 INITIAL DIFFICULTIES IN THE DEVELOPMENT OF CHEMICAL

KINETICS IN THE TWENTIETH CENTURY

One of the major difficulties in the development of chemical kinetics stemmed from thelack of mathematical preparation of chemists of that period For example, Morris Travers[4,5,15] in his biography of William Ramsey noted that his lack of mathematical prepara-tion was the determining factor that made him decide not to become a physicist Harcourtalso notes his own mathematical weakness and his inability to understand many of themathematical treatments that were made on his experimental data on chemical reactions.These were due to the mathematician Esson, professor of geometry at Oxford Such devel-opments were sufficiently complex that they were not even understood by many of hiscontemporary mathematicians, let alone by the chemists of the period Also, the work ofGuldberg and Waage in this area resulted from a collaboration between a professor ofapplied mathematics and a chemist, while the extremely promising work of Berthelot andPéan de St Gilles on kinetics was finally abandoned by the premature death of the latterscientist at the age of 31

Berthelot and St Giles, in their kinetic study of esterification reactions, showed that theamount of ester formed at each instant was proportional to the product of the “activemasses” of the reactants and inversely proportional to the volume Rather inexplicably,these authors did not take into account the role of these factors in defining the rate law ofthe reaction [4,5,15] A possible explanation for this can be seen in a note on the life andwork on Marcelin Berthelot [16] In this work, indications are given of Berthelot’s under-standing of the role of mathematics in chemistry: “the mathematicians make an incoher-ent block out of physical and chemical phenomena For better or for worse, they force us

to fit our results to their formulae, assuming reversibility and continuity on all sides,which, unfortunately, is contradicted by a large number of chemical phenomena, in par-ticular the law of definite proportions.”

Guldberg and Waage arrived at the concept of chemical equilibrium during 1864–1867through the laws of classical mechanics: that there are two opposing forces, one owing tothe reactants and the other to the products, which act during a chemical reaction to achieveequilibrium In an analogy with the theory of gravity, such forces will be proportional to themasses of the different substances; actually, they established two separate laws, one relat-ing to the effect of masses and the other to that of volume, and it was only later that theywere combined into a single law, involving concentrations or “active masses”

Guldberg and Waage also initially experienced difficulties in finding the proper nents involved in the description of the variations in the concentrations of the differentsubstances; this problem was resolved in 1887, in terms of molecular kinetic theory.However, far more importantly, these authors did not manage to distinguish the rate laws(what we would call today the initial conditions) from the derivatives of the equilibriumconditions This considerably complicated and delayed the future development of chemi-cal kinetics The dynamic nature of chemical equilibrium was never in doubt However,the complexity of the systems was far from being considered and the link between equi-librium and kinetics was weak The works of Harcourt and Esson are models of meticu-lous experimental and theoretical work, but on reading them, it is also obvious that theseauthors had to confront many conceptual and technical problems Their kinetic studies

expo-needed fairly slow reactions that could be started and stopped quickly and easily The tions that best satisfied these experimental conditions were, in fact, fairly complex inmechanistic terms In spite of the fact that Harcourt knew that such reactions did not hap-pen in a single step, he was far from being able to recognise all their complexities It wasthis difficulty in seeing simplicity in the macroscopic observations and extending it to thecorresponding microscopic interpretation that became one of the main obstacles to theproper development of chemical kinetics.

reac-Another area of chemical kinetics that has been the focus of various historical studies,involved the interpretation of the effect of temperature on the rates of chemical reactions.For rates measured under standard concentration conditions, Arrhenius expressed thiseffect by the equation

(1.1)

where k is the rate under standard conditions and A and Eaconstants, which are practically

independent of temperature A is called the frequency factor or pre-exponential factor and

Eathe activation energy.

The Arrhenius law took a long time to become accepted; many other expressions werealso proposed to explain the dependence of rate on temperature [17–19] However, theArrhenius expression eventually became dominant, as it was the model that was the easi-est to relate to in terms of physical significance Nevertheless, its acceptance did not comequickly, and was compounded by great difficulties in scientific communication at the time,with lack of interaction between different research groups often carrying out similar, andoften parallel studies, instead of drawing on the progress that had already been achieved

in this area

Many of these conceptual and experimental difficulties would disappear with the liant work of van’t Hoff [20], who introduced the concept of order of reaction and, through

bril-it, the possibility of knowing the mechanism of a chemical reaction just on the basis of

chemical kinetics [21] In fact, van’t Hoff used the term molecularity for what we would call today reaction order (the power to which a concentration of a component enters into the rate equation) When referring to the actual concept of molecularity, this author used

the explicit expression “the number of molecules that participate in the reaction” [6] The

term order is due to Ostwald Van’t Hoff received the first Nobel Prize in 1901 for his

dis-covery of the laws of chemical dynamics

During this period, interest in chemical kinetics remained fairly high until 1890, andthen declined “due to the lack of stimulus from kinetic theories which could suggest appro-priate experiments, sufficient to stimulate a discussion” [22] and, in essence, it neededsomething to allow a connection between molecular structure and chemical reactivity This

is true, not just of chemical kinetics: all areas of science suffer in the absence of ate theories, which help to guide development of experiments

appropri-A revival of interest in this area began around 1913 with the “radiation hypothesis”, due

to M Trautz, Jean Perrin and William Lewis The particular challenge they tackled wouldprobably have escaped notice of the scientific community, except for the fact that Perrinand Lewis were two highly respected scientists Their developments required a strongmathematical preparation Lewis was the first chemist to develop a theory of chemical

k=Ae− (Ea RT)

kinetics based on statistical mechanics and quantum theory It is clear that Lewis was anexception in terms of mathematical background to the majority of British chemists, whoeven in the 1920s had a mathematical background that was insufficient to address, or evenunderstand such problems.

The “radiation theory”, which was received with enthusiasm, was later seen to be taken However, it was important as it stirred up a lively debate that greatly contributed tothe development of the correct theories of chemical kinetics

mis-In this first phase of development, the theories of chemical kinetics tried to resolve theproblem of the calculation of the pre-exponential factor and activation energy in the

Arrhenius equation The difficulties in calculating A stemmed in large part from the

con-fusion that had existed ever since the first quarter of the nineteenth century over the role

of molecular collisions on the rates of reaction Today, we know that molecular collisionslead to the distribution of energy between molecules, but the rate of chemical reactions isdetermined both by the frequency of these collisions and the factors associated with thedistribution of energy

Max Trautz in 1916 and William Lewis in 1918 developed mathematical expressionsthat allowed the formulation of a collision theory for pre-exponential factors In 1936Henry Eyring, and almost independently, Michael Polanyi and M.G Evans came todevelop the transition state theory, having as its bases thermodynamics and statisticalmechanics

The concept of potential energy surface (PES) was developed to calculate the activationenergy Based on quantum mechanics, the first PES was constructed, at the start of the1930s by Eyring and Polanyi for the reaction H ⫹ H2 But the concept of PES is muchmore comprehensive because it allows the dynamic study of the rates of elementary reac-tions This is based on the study of the forces that cause molecular motions that will lead

to chemical reaction

Chemistry is concerned with the study of molecular structures, equilibria between thesestructures and the rates with which some structures are transformed into others The study

of molecular structures corresponds to study of the species that exist at the minima of dimensional PESs, and which are, in principle, accessible through spectroscopic measure-ments and X-ray diffraction The equilibria between these structures are related to thedifference in energy between their respective minima, and can be studied by thermochem-istry, by assuming an appropriate standard state The rate of chemical reactions is a mani-festation of the energy barriers existing between these minima, barriers that are not directlyobservable The transformation between molecular structures implies varying times for thestudy of chemical reactions, and is the sphere of chemical kinetics The “journey” from oneminimum to another on the PES is one of the objectives of the study of molecular dynam-ics, which is included within the domain of chemical kinetics It is also possible to classifynuclear decay as a special type of unimolecular transformation, and as such, nuclear chem-istry can be included as an area of chemical kinetics Thus, the scope of chemical kineticsspans the area from nuclear processes up to the behaviour of large molecules

multi-The range of rates of chemical reactions is enormous Figure 1.1 gives a generalpanorama of the variety of reaction rates of processes in the world around us Nucleartransformations and geological processes can be considered to be some of the slowestreactions that we come across The corrosion of some metals frequently takes place dur-ing the life expectancy of a human (80 years ⫽ 2.5⫻109sec) The time of cooking food isreadily measurable simply by visual observation, and extends from minutes to hours Wecan contrast this with the case of reactions such as the precipitation of a salt or neutralisa-tion of an acid that occur in ⬍0.1 sec, because visually we can no longer distinguishimages on this timescale There are, however, special techniques which allow muchshorter time resolution in our observation window, and which allow study of extremelyrapid reactions The limit of time resolution of interest for chemical kinetics is defined bythe movement of nuclei in molecules in their vibrational or rotational motion.

The chemical reaction can be considered as a voyage on a multi-dimensional PES(Figure 1.2) The definition of the PES has its origin in the separation of the movement ofelectrons and nuclei This separation is justified on the basis of the difference in massbetween an electron and a proton (the mass of the former is 1/1800 times the rest mass ofthe second), which means that the movement of electrons is much more rapid than that of

the nuclei Because of this, the electrons can be considered to re-adjust instantaneously to

each of the geometries that the nuclei might adopt The PES results from solving theSchrödinger equation for each of the possible nuclear geometries The sum of the elec-tronic energy and the nuclear repulsion governs the movement of the nuclei Ideally, theSchrödinger equation must be solved for a great number of nuclear geometries using onlythe laws of quantum mechanics and the universal constants, which are given in Appendix 1.From this, a set of points will be obtained, and the energy determined for each of the

possible geometries This type of calculation, known as ab initio, is very time consuming

Figure 1.1 Range of rates of chemical reactions.

and difficult for polyatomic systems As a consequence, many PESs include experimentalinformation and are described by more or less complex functions, which are fitted to

results of ab initio calculations and experimental information on the system.

In a hypersurface of a polyatomic system there can exist a number of more or less ble structures, which correspond to deeper or shallower potential wells The separationbetween these wells is made up of hills, that is, potential barriers with variable heights Theheight of the potential barriers determines the energy necessary to convert from one struc-ture into another, that is, for a chemical reaction to occur In the passage from a reactantwell, or valley, to that of products, there is normally one that goes by a path, whose point

sta-of highest energy is termed the saddle point, given the topographic similarity to the saddle

of a horse A saddle point corresponds to a maximum energy on the route that leads fromreactants to products, but a minimum one on the direction orthogonal to this The reaction

pathway, which goes through the lowest energy path, is called the minimum energy path.

It is natural that a chemical reaction, which occurs on a single PES will follow tially the minimum energy pathway or route The surface orthogonal to the minimumenergy pathway between reactants and product and which contains the saddle point, cor-

preferen-responds to a set of nuclear configurations that is designated the transition state Its

exis-tence can be considered as virtual or conceptual, because the transition state corresponds

to a region on the potential energy hypersurface from which the conversion of reactants toproducts leads to a decrease in the potential energy of the system The transition state is,therefore, intrinsically unstable

The minimum energy pathway for a given reaction can be defined by starting from thetransition state as being the path of the largest slope that leads to the reactants valley on

Figure 1.2 PES for collinear approach of atom A to the diatomic molecule BC in the triatomic

sys-tem A ⫹BC → AB ⫹C, with the most important topographical regions: reactant valley (A⫹BC), transition state (‡), product valley (AB ⫹C), dissociation plateau for all bonds (A⫹B⫹C) and low- est energy pathway from reactants to products (dashed line).

one side and the product valley on the other This minimum energy pathway is shown inFigure 1.2 for a typical reaction that involves breaking one bond in the reactants and form-ing a new bond in the products Normally, the PES for a chemical system cannot be deter-

mined accurately since for a molecule containing N atoms the PES is a function of 3N

nuclear coordinates Some of these coordinates can be separated, in particular the threecoordinates which describe translational motion, given the conservation of the movement

of the centre of mass, and the coordinates corresponding to rotational motion, given theconservation of angular momentum After separation of these motions, the PES will be a

function of 3N-5 internal, interdependent coordinates for linear configurations and 3N-6

for non-linear configurations The complexity of the PES for polyatomic systems justifiesthe use of simplified models that simulate the variation of the potential energy of the sys-tem as a function of a reaction coordinate of the reaction The reaction coordinate starts tohave a particular significance for each model that represents the variation of the energy ofthe system on the conversion of reactants to products Given that these models are simpli-fied representations of the PES, the reaction coordinate given by a model may not corre-spond to the minimum energy pathway

Figure 1.2 shows the case of a very simple PES In fact, the topography of the PESs can

be very diverse Figure 1.3 shows an example of a PES, where instead of a maximum dle point), there is a minimum (intermediate) in the middle of the minimum energy pathway.The movement of atoms across the reaction coordinate can, in an elementary approxi-mation, be compared with that of atoms in a bond with a low-frequency vibration Thevibrational frequency  of a bond between atoms A and B is characteristic of the AB bond

Figure 1.3 Reaction occurring on a surface with a potential well separating reactants and products,

and corresponding to the formation of a reactive intermediate The reaction is exothermic.

and depends, to a first approximation, on the force constant of the bond f and the reduced

mass of its atoms, 

(1.2)where the reduced mass is given by

(1.3)

The energies where vibrations of AB are expected to occur lie between 300 and 3000 cm⫺1(4–40 kJ mol⫺1) such that they can be seen in the infrared These energies can be related tothe corresponding vibrational frequencies by the Planck equation

where v is the vibrational quantum number, the minimum distortion that a diatomic

mol-ecule suffers relative to its equilibrium bond length, req, can be calculated by consideringthat the bond is in its lowest vibrational energy, v ⫽ 0, and using a harmonic oscillator asthe model of the variation of energy with the distortion (Figure 1.4) The variation of theenergy with the distortion is given by the equation of a parabola

(1.6)

where f is the force constant characteristic of the oscillator By substituting eq (1.2) into

expression (1.5) with v ⫽ 0, and equating to (1.6), we obtain

(1.7)

To apply this equation to the real case of the 35Cl᎐35Cl bond, it is necessary to know itsreduced mass and vibrational frequency The reduced mass of Cl2 calculated from theatomic mass of chlorine and eq (1.3) leads to  ⫽ 2.905⫻10⫺26kg The Cl᎐Cl vibration

is seen at 559.71 cm⫺1with  ⫽ c–, where c ⫽ 2.998⫻108m sec⫺1is the speed of light invacuum and the frequency is 1.68⫻1013sec⫺1 Using eq (1.2), the force constant for this

bond is f⫽ 322.7 N m⫺1, since by definition 1 N ⫽ 1 kg m sec⫺2 Force constants are often

2 eq

Ev =⎛⎝⎜v+1⎞⎠⎟h v=

2 , 0 1 2 3, , , ,…

E=h

=+

1 2

 

expressed in mdyn Å⫺1, kcal mol⫺1Å⫺2or J mol⫺1pm⫺2, so that it is useful to know theconversion factors for these units:

1 mdyn Å⫺1⫽ 100 N m⫺1 ⫽ 143.8 kcal mol⫺1Å⫺2⫽ 60.17 J mol⫺1pm⫺2

Knowing the values of f and , eq (1.7) gives (req⫺r) ⫽ 5.87⫻10⫺12m As such, avibration goes through 1.17⫻10⫺11m in 5.95⫻10⫺14sec, or, in other words, the speed atwhich the atoms undergo vibrational movement is about 200 m sec⫺1(720 km h⫺1) in thefundamental vibrational level It should be noted that the Cl᎐Cl bond, whose equilibriumbond length is 1.99⫻10⫺10m, is distorted by about 3% of its length

Today, a technique called transition state spectroscopy that uses lasers with pulse widthsaround 10 fsec facilitates the detection of transient species with extremely short lifetimes Inthis time interval, a bond in its fundamental vibration covers a distance of only 2⫻10⫺12m

As such, this technique enables one to obtain a sequence of images of vibrational motion of

a chemical bond in the act of breaking However, it is worth remembering that owing to theHeisenberg uncertainty principle:

(1.8)observations on the time scale 10⫺14sec correspond to an uncertainty in energy of 3 kJmol⫺1 A better time resolution leads to greater uncertainties in energy, which will not be

of much use in chemical kinetics, given that, according to eq (1.1), an uncertainty of 3 kJmol⫺1in transition state energy leads to a factor of 3 in the rate of a reaction at 25 ⬚C

Δ ΔE t≥1

2

Figure 1.4 Harmonic oscillator with the characteristic behaviour of Cl2molecule.

Nevertheless, these ultrashort pulse techniques do find applications in areas of troscopy where one is dealing with broad bands in terms of frequency distribution, andspectral bandwidth is not the limiting factor.

spec-It is anticipated that the most rapid chemical reactions will be those that occur everytime there is a bond vibration, that is, when the energy barrier is equal or less than that ofthe vibrational energy This corresponds to a purely repulsive PES that could be obtained,for example, by electronic excitation Figure 1.5 gives a typical example of a reaction ofthis type For molecules with more than two atoms the situation becomes significantlymore complicated, because it is necessary to consider energy distribution between the var-ious bonds involved For a bimolecular reaction, the maximum rate will be achieved for

an exothermic reaction that occurs on every collision between reactant molecules Thislimit is reached in some reactions of free radicals in the gas phase that occur without anyactivation barrier In fact, however, even in some of these reactions potential wells are seeninstead of barriers separating reactants and products, as in the surface shown in Figure 1.3

In these cases, the rate of the reaction may be limited by the formation of a complex orintermediate with a finite lifetime In this case, the reaction is no longer elementary, andfollows a two-step mechanism: formation of an intermediate, followed by its decay Influid solutions the maximum rate of a bimolecular reaction is limited by the rate at whichthe reactants can diffuse in the medium to achieve the reaction radius

Beyond the above limiting situation, in a chemical reaction of the general type

(1.I)energy barriers are always found The simplest model for the origin of these energy barriersconsists in assuming that to break the B᎐C bond, we need to supply to the BC molecules anenergy equal to the energy of this bond However, frequently, the observed energy barriers

A+BC→AB C+

Figure 1.5 Reaction occurring on a barrier-free surface, obtained by electronic excitation of the

reactants.

of these reactions are only ca 10% of the energy of the bonds being broken As such, we can

see that, in general, the reaction cannot proceed in one step in which the B᎐C bond is ken followed by a subsequent and independent step in which there is formation of the A᎐Bbond In all the steps of the reaction there must be a strong correlation between the bondwhich is broken and that which is being formed The transition state appears to have an elec-tronic configuration that maximises the bonding in all of the parts (A᎐B and B᎐C) of which

bro-it is composed The energy barrier results from two opposing factors: on one hand, theapproach between the species A and BC allows the formation of a new bond, AB, which low-ers the energy of the system, while on the other, this approach results in an increase in theenergy of the system, given the repulsion between the molecules at short distances The totalenergy depends on the correlation between the breaking and formation of the bonds.The potential energy is a microscopic variable For any configuration of the reactivesystem, in principle it is possible to calculate a potential energy Knowing the potentialenergy along the minimum energy path, it is possible to define a continuous analyticalfunction that will describe the evolution of the system from reactants to products Theknowledge of the PES allows estimation of the potential energy of activation of a chemi-cal reaction, that is, following classical mechanics, the minimum energy necessary, for theisolated reactants to be transformed to isolated product molecules To make a comparisonbetween the potential energy of activation calculated from the PES and the experimentalactivation energy, it is necessary to make the change from the microscopic to the macro-scopic domain The energy of the system, which is observed macroscopically, is a ther-modynamic energy The energy differences between reactants and products in solution arenormally measured in terms of their equilibrium constants As the equilibrium constant of

a reaction is related to the free energy

(1.9)the barrier height to be surmounted in the course of a reaction must be expressed in macro-scopic terms by a free energy of activation The variation of potential energy calculated inthis way thus corresponds to the variation in free energy when the entropy differences arenegligible The relation between the microscopic models and the experimental macroscopicreality can be made through statistical mechanics In statistical terms, although a chemicalspecies is a group of particles with a determined range of properties, all the particles of onespecies have to have the same equilibrium configuration So, as many species exist as equi-librium configurations that can be statistically defined along the reaction coordinate

In this context, equilibrium configuration denotes a geometry in mechanical equilibrium,that is, a geometry corresponding to a point for which the derivative of the potential energyfunction is zero This derivative is zero for potential maxima and minima As such, alongthe reaction coordinate, we can define three configurations that fulfil these requisites: thereactants, the products and the transition state The first two correspond to minima and are

in stable equilibria, while the latter corresponds to a maximum along the reaction nate, and is in unstable equilibrium (or pseudo-equilibrium) Then, although the reactioncoordinate is continuous, the thermodynamic energy along it is discontinuous, containingonly three points Nevertheless, it is useful to formulate the variation of free energy as afunction of the reaction coordinate in terms of a continuous function It should be noted,

coordi-ΔG0 = −RTlnKeq

however, that the interpolated points between the equilibrium configurations do not haveany thermodynamic significance.

In the transformation from the microscopic world to the macroscopic one, we also need

to consider the effect of molecular collision on the distribution of molecular velocity orenergy in these systems The majority of molecules will have a velocity close to the meanvalue for the molecules, but there are always some molecules with velocity much greaterthan and others with velocity much lesser than the mean velocity The distribution ofvelocities of gas molecules was first described by Maxwell in 1860 The Maxwell distri-bution of velocities is given by

(1.10)The mean velocity can be calculated from the integral

(1.11)

where M is the molar mass of the molecules, and f(s) ds the fraction of molecules which have velocities between s and s ⫹ds Figure 1.6 illustrates the distribution of molecular

velocities of a gas at various temperatures

For the case of N2molecules at 298 K, using the constants from Appendix I, we obtain

a mean velocity of 475 m sec⫺1

Until now, we have only considered elementary reactions, that is, ones that occur in asingle step These reactions are observed between a certain number (that must be a wholenumber) of atoms, molecules or ions The number of species involved in an elementaryreaction is designated as the molecularity of an elementary reaction A chemical reaction

is generally described by an equation of the type

(1.12)

where vA, vB, vX, vY, … are the stoichiometric coefficients of the species A, B, X, Y, …These coefficients are also whole numbers, but as the overall reaction can occur with asequence of elementary steps of molecularity one, two or three, the stoichiometric coeffi-cients will only correspond to the molecularity of the reaction if this is an elementary reac-tion The sequence of these microscopic elementary steps is known as the mechanism ofthe reaction In any case, the balance of all the microscopic elementary steps has to result

in the macroscopic process described by the above general chemical equation, whose librium constant is

equi-(1.13)

where a i represents the activity of species i.

The mechanism of a reaction is normally determined experimentally, but although it ispossible to disprove a mechanism, it is impossible to confirm that a proposed mechanism

is correct simply on the basis of the experimental results available It is also possible that

a reaction can occur by two distinct mechanisms

In the above explanation, we did not consider the possibility of an elementary processhaving a molecularity greater than 3 The probability that elementary kinetic processesinvolve the simultaneous collision of four particles is negligible When more than threereactant molecules are involved, it is certain that the chemical transformation which occursdoes not take place in a single elementary step

Current practice in chemical kinetics tries to identify the one particular elementary stepthat has a very large effect on the overall reaction rate This elementary process is known

as the rate-determining step of the reaction This rate-determining step depends on the ative magnitudes of the energy barriers for each elementary reaction as well as on the con-centrations of the reactants and intermediates When such a step exists, the rate of theoverall reaction is either the value for this rate-determining step, or it is this value multi-plied by some equilibrium constants of steps preceding it (pre-equilibria)

rel-In Chapters 2 and 3, we will consider, respectively, the factors involved in determiningthe rates of chemical reaction and the techniques that allow the experimental study of theirkinetics In Chapter 4, we will start from empirical knowledge of the variation of concen-tration of reactants and products with time to establish the rate laws for the correspondingelementary reactions Chapters 5–8 will consider some theories that allow us to calculate

or rationalise the numerical values in the above rate laws Chapters 9–14 will discuss indetail some of the most important reactions that have been studied using chemical kinet-ics The last two chapters will focus on some less familiar topics in textbooks in this area,

but we believe they will lead to a conceptual awareness of the role of energy and electrontransfer in chemical kinetics.

REFERENCES

[1] L Wilhelmy, Pogg Ann 81 (1850) 423.

[2] W Ostwald, J Prakt Chem 29 (1884) 358.

[3] E Farber, Chymia 7 (1961) 135.

[4] MC King, Ambix 29 (1982) 49.

[5] MC King, Ambix 31 (1984) 16.

[6] MC King, KJ Laidler, Arch Hist Exact Sci 30 (1984) 45.

[7] M Berthelot, LP St Gilles, Ann Chim 65 (1862) 385.

[8] M Berthelot, LP St Gilles, Ann Chim 66 (1862) 5.

[9] M Berthelot, LP St Gilles, Ann Chim 68 (1863) 255.

[10] J Lamor, Manchester Mem 32 (1908) 10.

[11] AV Harcourt, W Esson, Chem News 10 (1864) 171.

[12] AV Harcourt, W Esson, Proc Roy Soc (London) 14 (1865) 470.

[13] AV Harcourt, W Esson, Philos Trans 156 (1866) 202.

[14] AV Harcourt, W Esson, Chem News 18 (1868) 13.

[15] MC King, Ambix 28 (1981) 70.

[16] E Jungfleische, Bull Soc Chem (1913) 102.

[17] EW Lund, J Chem Educ 45 (1968) 125.

[18] SR Logan, J Chem Educ 59 (1982) 279.

[19] KJ Laidler, J Chem Educ 61 (1984) 494.

[20] JH van’t Hoff, Études de Dynamique Chimique, Muller Amsterdam, 1884.

[21] A Findlay, T William, A Hundred Years of Chemistry, University Paperbacks, Methuen,

London, 1965.

[22] AJ Ihde, The Development of Modern Chemistry, Harper & Row, New York, 1964.

– 2 – Reaction Rate Laws

chem-extent of reaction 

The rate of conversion in a chemical reaction is defined as the variation of  with time, t,

(2.2)Using the definition of  from eq (2.1), we obtain

(2.3)

where, to simplify the notation, we will write n i instead of n i()

The reaction velocity is defined by

= 1 d

d

dd

dd





t

n t i i

= 1

r t

being given in dm3, the reaction velocity is given in mol dm⫺3sec⫺1 If the volume stays

constant during the reaction, it is more common to give the concentration of i, that is, [i] in

mol L⫺1(1 L = 1 dm3) in the determination of the rate of conversion of reactions Thus, for

a total constant volume V,

(2.5)

The experimental determination of the velocity of a reaction in solution is made by ing the change in the concentration of reactants or products with time, since the volume ofthe environment in which the reaction occurs does not vary appreciably during the reaction.This is also true of reactions in the gas phase when the reactor is kept at constant volume.Sometimes for a reaction in the gas phase at constant temperature and volume, it is more

measur-convenient to measure the partial pressure P i of one component i rather than its

concen-tration Assuming ideal mixing behaviour of the gases, we can express the rate of sion as a function of partial pressure of any one of the components

conver-(2.6)

When there is a change in the total number of moles of gas in the system, v i⫽ 0, the totalpressure will vary proportionally with the extent of reaction From eq (2.1), for the set ofall the components of the reaction we can write

0

 i n i  n i

dd

dd

P t

dd





t

V i t

As a consequence, where there is a change in the total number of moles of gas, the rate

of conversion for a reaction at constant volume and temperature will be given by

(2.12)

where P represents the total pressure of the reaction system.

From the perfect gas equation, if a gas phase reaction in which v i⫽ 0 occurs at stant temperature and pressure, the volume must change In this case, if all the componentsfollow ideal behaviour

con-(2.13)

where V() is the total volume of the system for an extent of reaction  Substituting into

eq (2.8) and considering that the volume remains constant, P i() = P i (0) = P,

(2.14)

and

(2.15)

It is important to prove that eq (2.2) defines the rate of conversion in the reaction, that

eq (2.4) defines the velocity of the reaction and that in the expressions (2.5), (2.6), (2.12)

and (2.15), the quantities (d[i]/dt) V,T , (dP i /dt) V,T , (dP/dt) V,T and (dV/dt) T,pare proportional

to the velocity of the reaction It must also be emphasised that, generally, the rate can bedefined in terms of any of the reactant or product molecules, provided that the stoichiom-etry of the reaction is included In other words, eq (2.16) is valid

(2.16)

2.2.1 Nature of the reagents

The velocities of elementary chemical reactions depend on a great number of factors, inparticular the nature of the reactants, concentrations or pressures, temperature, light, cata-lysts and the solvent used The great variation observed in reaction velocities will berelated first to the nature of the reagents Many reactions, such as those between oppositely

B B X X

Y

dd

dd

dd

dd





t

P RT

V t

( )= ( ) ( )

dd

dd





t

V RT

P t

charged ions in aqueous solution, which do not involve breaking chemical bonds are veryrapid at room temperature.

(2.II)(2.III)However, reactions involving structurally similar reactant molecules such as exchangereactions of electrons between two isotopically labelled transition metal complexes, whichalso do not involve breaking chemical bonds often show great differences in rates undersimilar conditions

(2.IV)(2.V)Reaction (2.IV) is 105times faster than reaction (2.V) In contrast, although reactions involv-ing bond breaking and bond formation are generally slow, there are some extremely fastreactions of this type such as the oxidation of iron (II) by permanganate ion in acid solution:

(2.VI)Thus, there appears to be a lack of even qualitative general rules to evaluate the effect ofthe nature of reactants on the rates of reactions The success of theoretical calculations ofreaction velocities in chemical kinetics without using adjustable parameters has been lim-ited Even for the simplest reactions in the gas phase, a large amount of computer time is

needed with ab initio quantum mechanical calculations to obtain detailed and precise

information on the potential energy surface of a reaction system, in addition to requiringthe calculation of a large number of trajectories before we can calculate the macroscopicvelocity of a reaction For more complex reactions, such calculations are prohibitive interms of both computer time and money In practice, it is more common to base the inter-pretation and prediction of reaction velocities as a function of the nature of reactants onthe development of simplified theoretical models that relate some properties of the reac-tants with the ease with which their bonds can be broken and/or new bonds formed in theproducts In Table 2.1 we give typical chemical reactions, with their respective activationenergies, and some appropriate parameters that have been used for the theoretical calcula-tion of reaction rates [2]

In this table, we stress the fact that if we keep a series of the parameters constant, it ispossible to attribute the change in activation energy, and consequently, the velocity of reac-tion (eq (1.1)), to the change in a specific parameter For example, in the reactions (i) and(ii) the increase in the sum of bond lengths in reactants and products leads to an increase inactivation energy With reactions (ii) and (iii), the predominant factor in the decrease in acti-vation energy appears to be the decrease in force constants Comparison between reactions(iii) and (iv) shows, as we may suspect intuitively, that the most exothermic reaction is thefastest one, from which we can say that the predominant factor in this case is the change inenthalpy of the reaction However, if we compare reactions (ii) and (v), we find a very sig-nificant decrease in activation energy, in contrast to what we would expect from the posi-tive enthalpy change, the increase in bond lengths or the decrease in force constants In this

5Fe(aq)2 MnO4 8H 5Fe Mn 4H O

3

(aq) 2 2

Co( ) *Co Co Co

* ( )

H(+aq)+OH(−aq)→H O2 ( )l

Ag(+aq)+Cl(−aq)→AgCl( )s

case, what appears to be the dominant factor is the difference in electronic structurebetween the chlorine atom and the methyl radical, represented here by their ionizationpotentials and electron affinities.

Within this framework, it is useful to define “families of reactions”, in the sense of aseries of reactions that occur via the same mechanism and under the same experimentalconditions, but where the reactants differ from each other by only small changes in struc-ture, such that they produce only small perturbations in the reaction centre These smallstructural changes are normally associated with minor changes in the bonding of sub-stituent groups in the region of the bonds that are broken or being formed during the reac-tion Many theoretical models have been suggested for such classes of reactions, leading

to empirical correlations between the activation energy and the change in a specific eter such as reaction enthalpy or electronic or steric parameters

param-2.2.2 Reactant concentration

Almost all reaction rates depend on the concentration of reagents, while for reversiblereactions they are also affected by those of products The mathematical expression thatrelates the reaction velocity and the concentration of species present is called the law ofreaction velocity, the kinetic law, or most simply, the rate law Assuming that the reactionbeing considered involves an elementary process,

(2.VII)the corresponding rate law can be written

cIonization energies and electron affinities of the radicals.

dExperimental activation energies.

where the powers a and b are known as the partial orders of reaction for components A and

B, and the proportionality constant k is called the specific rate constant for the reaction, or

simply the rate constant This constant is independent of the concentrations of the tants, but depends on the temperature, pressure and reaction medium The partial orderscan only be identified with the stoichiometric coefficients when we are dealing with an ele-mentary reaction The rate law of any reaction always has to be determined experimentally.The formulation of this principle is known as the Law of Mass Action, which can be statedthat in dilute solutions the velocity of each chemical reaction is proportional to the prod-uct of the concentration of reactants, is independent of the concentration of other speciesand of the presence of any other reactions

reac-Treating the case of an elementary reversible reaction

where Keqis the equilibrium constant for the reaction From eq (2.19), we can see that

only two of the three constants kd, kiand Keqare independent The same reasoning can beapplied to more complicated systems such as that described by the following mechanism:

2 2 2

3 3 3

[ ] [ ] [ ] [ ] [ ] [ ]

such that K1K2K3= 1, from which

(2.21)

In this scheme only five of the six rate constants are independent

At first sight, it may appear strange that the equilibrium constant expression onlydepends on the stoichiometry of a reaction when it stems from equating the rate laws forthe forward and reverse reactions, where these reactions have an empirical characterand, except for the case of elementary reactions, the rate expressions cannot be obtainedfrom the overall equation for the chemical reaction However, this observation has its

basis on an important physical principle, the principle of microscopic reversibility This

can be stated in the form that in the state of macroscopic equilibrium each elementaryprocess is in equilibrium, and is reversible at the microscopic level In other words, themechanism of a reversible reaction is the same in the forward and reverse directions.The mathematical basis of this principle comes from the fact that the equations ofmotion are symmetrical relative to time inversion, from which a particle which follows

a given trajectory in the time from 0 to t will follow the identical reverse trajectory in

the time from t to 0 We can see, in fact, that at equilibrium, the concentrations of tants and products are constant and do not oscillate about a mean value Thus, mecha-nism (2.IX) represents a possible chemical system, which is in agreement with theprinciple of microscopic reversibility and which will respond promptly to any perturba-tion from the equilibrium state The same is not true for mechanism (2.X), where thestep for formation of A from B implies an intermediate which is not involved in the for-mation of B from A, i.e the mechanism of formation of B from A is different from thatfor the formation of A from B

reac-(2.X)

The principle of microscopic reversibility can be applied to the explanation of reactionmechanisms and of rate laws As an example, we can take the disproportionation of thehypochlorite ion under equilibrium conditions

(2.XI)with the equilibrium constant given by

(2.22)

eq

ClO ClClO

and with a reaction mechanism involving the following steps in equilibrium, a slow firststep followed by a second fast one

ele-(2.XIII)

and considering the values for the change of reaction enthalpies of the elementary processes,

it would be expected that the first process would be the slower one, that is, that this will bethe rate-determining step of the reaction The rate law for the reaction written in the direc-tion of eqs (2.XIII) will then be

 

  ⌬ = kJ mol 1

kJ molClO Cl 2−+ − k− ClO ClO−+ − −

1

042

 

  ⌬ kJ mol−−1

Equating the rates of the reaction in the forward and reverse directions, we obtain

(2.28)which bears no relationship to the equilibrium constant, or to any power of it, and alsodoes not follow the stoichiometry of the reaction

(2.29)

For a system far from equilibrium, if the reverse reaction follows a distinct mechanismfrom the forward one, common sense tells us that the ratio of concentrations will be different from that given by the rate laws and the equilibrium constant The mecha-nisms of reaction in the forward and reverse directions can only be assumed to be thesame close to equilibrium From the above, it is clear that both the rate laws for thereaction must be obtained experimentally The relationship between the rate laws for reactions far from equilibrium is only equal to the equilibrium constant for ele-mentary reactions, or for systems where the forward and reverse reactions have thesame rate-determining step

As the partial orders of reaction are experimentally determined parameters, they cantake any value, and the positive integer values we have presented until now are only spe-cific cases We will show this with some experimental rate laws The exchange of iodineatoms between iodate ion and molecular iodine has been studied with radioisotopes

(2.XIV)and the following rate law obtained:

(2.30)where the partial orders are all fractions Another example is the following reaction inaqueous solution, which has a negative partial order

(2.XV)(2.31)indicating that the reaction rate decreases with increasing pH These laws also show thatthe velocity of a reaction can depend on other species apart from the reactants involved inthe chemical equation

Until now, all the rate laws we have given take the general form of eq (2.17) However,some reactions show very different forms for their rate laws One example is the forma-tion of hydrogen bromide from bromine and molecular hydrogen

whose rate law is

(2.32)

For this case, it is not possible to define reaction orders, except over limited concentrationregions In the first stages of reaction, when [HBr] is small, we can make the approximation

(2.33)and the previous expression simplifies to

(2.34)

with k = k /k In this example, the velocity depends on the concentration of a product of

the reaction, such that as the reaction progresses, [HBr] will increase and towards the end

of the reaction we arrive at the situation in which

(2.35)Under these conditions, eq (2.32) reduces to

(2.36)Both eqs (2.34) and (2.36) represent kinetic laws of the general form of eq (2.17), but theapparent reaction orders will vary with the extent of the reaction such that we cannotstrictly talk about reaction orders For example, the reaction order of bromine changesfrom 1/2 at the start of the reaction to 3/2 at the end

This type of kinetic law illustrates two important points that have to be taken intoaccount in the determination of reaction orders First, a kinetic law is only strictly validclose to the zone where it is studied experimentally; its extrapolation to other concentra-tion regions requires further experimental and theoretical studies Second, considering thevariation in reaction order during a reaction, from this it is important that studies of reac-tion orders must be limited to periods of relatively small extent of reaction (not ⬎10%),otherwise mean vales may be obtained which have no chemical meaning

Until now, we have only considered chemical reactions in homogeneous systems.However, the study of heterogeneous systems, in which more than one phase is present,are equally important, particularly in the areas of catalysis and corrosion For example, theoxidation of a metal is faster when its area exposed to the oxidising medium increases Ingeneral, it is seen that the rate of a heterogeneous reaction is directly proportional to the

contact area between the reactants, Sc Thus, in the rate law (2.17), these reactions are first

order relative to Sc

=k H2 Br2 3 2/ HBr 1

HBr[ ] [ ]Br2 >> ′′k

=k H2 Br2 1 2

HBr[ ] [ ]Br2 << ′′k

1 2

2

2.2.3 Temperature

van’t Hoff gave the first description of the effect of temperature on the equilibrium

con-stant of a reaction, Keq, with the relationship

(2.37)

where ⌬H0is the change in standard enthalpy for the reaction In contrast to the spirit ofthe era, in which attempts were made to relate equilibria and rates in terms of molecularmotions and Maxwell’s distribution law of molecular velocities, the Swedish chemistArrhenius considered that the effect of temperature on reaction rates is normally too great

to be explained simply in terms of its effect on the translational energy of molecules As

an alternative, he proposed that there will be equilibrium between “normal” molecules and

“reactive” ones, and that this has the same temperature dependence given by eq (2.37) It

is implicit in this work that the energy difference between molecules is independent oftemperature From this, Arrhenius was able to provide the first adequate description of therelationship between the rate constant and the temperature:

(2.38)

where A is the pre-exponential (or frequency) factor, Eathe activation energy and T the

absolute temperature Arrhenius received the Nobel Prize in 1903 However, it was not onthe basis of his work on chemical kinetics, and the merits of eq (2.38) remained contro-versial for many years

In the Arrhenius equation the temperature dependence comes basically from the nential term However, it should be noted that the pre-exponential factor can also have a

expo-weak temperature dependence, and it is more correct to assume that A is proportional to T m

where the temperature is in Kelvin, k and A are now taken, simply, as their numerical

val-ues, and the units in eq (2.38) cancel out

Assuming that Arrhenius behaviour is followed, for a first-order process a plot of ln(k) against 1/T leads to a straight line of slope ⫺Ea/R The extrapolation to 1/T = 0, which is

0 2

K T

H RT

normally significantly outside the temperature range studied and, as such, subject to

appre-ciable errors, gives us ln(A), where A is given in units (time⫺1) We must note that whenthe experimental results are only obtained over a limited temperature range, the accuracy

in the values of A and Eawill be low

The rate constants for second-order reactions can be expressed in dm3mol⫺1sec⫺1, or,

if the rate is measured from the change of gas pressure at constant volume, as described in

eq (2.12), in (pressure)⫺1/(time)⫺1 The conversion between units of pressure and those ofconcentration for gas phase reactions which are not first order, can be obtained from the per-

fect gas equation, from which, for the general case of a reaction of order n, the variation in the total gas pressure can be related to the change in the total gas concentration by p = cRT

(2.41)Thus, from the rate constants given in the units (pressure)⫺1(time)⫺1, to obtain the terms A and Eafor reactions other than the first-order case, we must start by converting all the rateconstants to be used with expression (2.41) into units of (concentration)⫺1(time)⫺1before wecan make an Arrenhius plot If these values were obtained from logarithmic plots of rateconstants given as (pressure)⫺1(time)⫺1against 1/T, we will get different numerical values for A and Ea This is obvious for A, which, for a second-order process, will be given in terms

of (pressure)⫺1(time)⫺1 However, it is equally true for Ea, and we need to take care to rect for the change in dimensionality to avoid the illusion that the values of activation energyare comparable between the two plots, since they are always given in kilo Joules per mole

cor-For reactions in solution the values of Eatypically lie between 40 and 120 kJ mol⫺1 As

a first approximation, these values correspond to an increase in reaction rate by a factor of2–5 for every 10 K rise in temperature The empirical rule often given is that the rate dou-bles for every 10 K rise in temperature

sug-to the darkening of silver salts through the reduction of Ag+ions to metallic silver

The action of light on a chemical reaction results essentially from the reactants in thepresence of light having the possibility of following a different mechanism that is not pos-sible in the dark For experimental reasons, the wavelengths () most commonly used forinitiating photochemical processes vary between the ultraviolet (200–250 nm) to the nearinfrared (750–800 nm) Light at these wavelengths has an energy, which is given by

(2.42)

Ephoton =h=h

kc=kp( )RT n− 1

and corresponds roughly to 600–150 kJ mol⫺1, which is very close to the energies of manychemical bonds This energy is much greater than the thermal energy at 25 ⬚C, which can

be estimated from the mean velocity given by eq (1.11) following the expression:

(2.43)

and is ~3.7 kJ mol⫺1

In the dark, the electrons in a molecule occupy the lowest energy electronic levels, ing rise to the so-called electronic ground state When irradiated by light whose wave-length corresponds to the energy difference between the ground state and an electronicallyexcited state, a molecule can absorb a photon and be transformed into this electronicallyexcited state This is a chemically distinct species, with its own structure, properties andreactivity

giv-The quantitative description of light absorption by a sample is based on the Beer–Lambert(or, more correctly, Beer–Lambert–Bouger) law In deriving this law, we consider that

incident monochromatic light of intensity I0crosses an infinitesimal thickness, dl, of an absorbing species of concentration c The decrease in light intensity, dI, is proportional to

the thickness of the sample, the concentration of the absorbing species and the incidentlight intensity

(2.44)

where a is a proportionality constant which depends on the sample and wavelength Integrating this expression over the whole optical path of the light in the sample, l, gives

(2.45)Expressing the above relationship in the most common form of logarithms to the base 10gives the normal form of the Beer–Lambert equation

(2.46)where  is the molar absorption coefficient (sometimes referred to as extinction coefficient

or molar absorptivity), which normally has the units L mol⫺1cm⫺1 The Beer–Lambert lawcan also be written as

(2.47)

where A is the absorbance, a dimensionless parameter.

The electronic distribution in the electronic excited state formed by absorption of violet or visible light is different from that of the ground state, but immediately after lightabsorption the nuclear configuration is identical to that of the ground state This is the basis

ultra-of the Franck–Condon principle The structural relaxation towards a new nuclear ration, corresponding to a potential energy minimum in the electronically excited state,

2 0