`
`—
`Chemical
`Kmetlcs
`
`I 7
`
`I
`
`a
`
`I
`
`
`
`ll Dynamics
`
`‘
`I
`Jeffrey I. Steinfeld
`Joseph S. Francisco
`
`William L. Hase
`
`
`
`'
`
`Sl‘x
`
`-
`
`
`
`.....jillifififiifilifilifllfl
`
`1%!
`9
`w
`
`A
`
`
`
`:;
`
`q!
`
`i! 3.
`M
`ii!
`i:
`,3
`[M
`.3
`NH
`x’
`t;
`I!
`J u
`[H
`
`1ii
`
`IIII
`
`
`_7Page 1 of38 Samsung Exhibit1018
`
`
`
`
`
`
`
`
`
`Page 1 of 38
`
`Samsung Exhibit 1018
`
`

`

`Library of Congress Cataloging-in-Publication Data
`Steinfeld, Jeffrey I.
`Chemical kinetics and dynamics/ Jeffrey Steinfeld , Joseph
`Francisco, William Hase.
`p.
`cm.
`Bibliography.
`Includes index.
`ISBN 0-13-129479-2
`l . Chemical reaction, Rate of. 2. Molecular dynamics.
`I. Francisco, Joseph
`II. Hase, William.
`III. Title.
`QD502.S74 1989
`88-21842
`54l.3'94--dcl9
`CIP
`
`Editorial/production supervision: Karen Winget/Wordcrafters
`Cover design: Joel Mitnick Design, Inc.
`Manufacturing buyer: Paula Massenaro
`
`© 1989 by Prentice-Hall, Inc.
`A Division of Simon & Schuster
`Englewood Chffs, New Jersey 07632
`
`•
`
`All rights reserved. No part of this book may be
`reproduced, in any form or by any means,
`without permission in writing from the publisher.
`
`Printed in the United States of America
`1 098765432
`
`ISBN 0-13 -129 479 -2
`
`PRENTICE-HALL INTERNATIONAL (UK) LIMITED, London
`PRENTICE-HALL OF AUSTRALIA PTY. L IMITED, Sydney
`PRENTICE-HALL CANADA INC . • Toronto
`PRENTICE-HALL HISPANOAMERICANA, S.A . , Mexico
`PRENTICE-HALL OF INDIA PRIVATE LIMITED, New Delhi
`PRENTICE-HALL OF JAPAN, INC., Tokyo
`SIMON & SCHUSTER ASIA PTE. LTD .• Singapore
`EDITORA PRENTICE-HALL Do BRASIL, LTDA . , Rio de Janeiro
`
`Page 2 of 38
`
`

`

`Contents
`
`PREFACE
`
`xi
`
`Chapter 1
`BASIC CONCEPTS OF KINETICS
`
`1
`
`I. I Definition of the Rate of a Chemical Reaction, I
`1.2 Order and Molecularity of a Reaction, 3
`I .3 Elementary Reaction Rate Laws, 6
`1.4 Determination of Reaction Order: Reaction Half Lives, 13
`1.5 Temperature Dependence of Rate Constants: The Arrhenius Equation, 14
`1.6 Reaction Mechanisms, Molecular Dynamics, and the Road Ahead, 16
`References, I 7
`Bibliography, 17
`Problems, 18
`
`Chapter 2
`COMPLEX REACTIONS
`
`21
`
`2. I Exact Analytic Solutions for Complex Reactions, 21
`2.2 Approximation Methods, 38
`2.3 Example of a Complex Reaction Mechanism: The Hydrogen + Halogen Reaction , 4I
`2.4 Laplace Transform Method, 48
`
`v
`
`Page 3 of 38
`
`

`

`2.5 Determinant (Matrix) Methods , 54
`2.6 Numerical Methods, 56
`2. 7 Stochastic Method, 67
`References, 73
`Bibliography, 74
`Appendix 2. l The Laplace Transform, 75
`Appendix 2.2 Numerical Algorithms for Differential Equations, 92
`Appendix 2.3 Stochastic Numerical Simulation of Chemical Reactions, 97
`Problems, 103
`
`Chapter 3
`KINETIC MEASUREMENTS
`
`109
`
`3.1 Introduction, 109
`3 .2 Techniques for Kinetic Measurements, 111
`3.3 Treatment of Kinetic Data, 133
`References, 150
`Appendix 3.1 Least Square Method in Matrix Form, 152
`Problems, 154
`
`Chapter 4
`REACTIONS IN SOLUTION
`
`156
`
`4.1 General Properties of Reactions in Solution, 156
`4.2 Phenomenological Theory of Reaction Rates, 157
`4.3 Diffusion-Limited Rate Constant, 161
`4.4 Slow Reactions, 163
`4.5 Effect of Ionic Strength on Reactions Between Ions, 164
`4.6 Linear Free-Energy Relationships, 169
`4.7 Relaxation Methods for Fast Reactions, 171
`References, 174
`Bibliography, 175
`Problems, 175
`
`Chapter 5
`CATALYSIS
`
`178
`
`5. l Catalysis and Equilibrium, 178
`5.2 Homogeneous Catalysis, 180
`5.3 Autocatalysis and Oscillating Reactions, 182
`5 .4 Enzyme-Catalyzed Reactions, 190
`5. 5 Heterogeneous Catalysis and Gas-Surface Reactions, 194
`References, 198
`Problems, 199
`
`vi
`
`Content5
`
`Page 4 of 38
`
`

`

`Chapter 6
`THE TRANSITION FROM THE MACROSCOPIC
`TO THE MICROSCOPIC LEVEL
`202
`
`6. 1 Relation Between Cross Section and Rate Coefficient, 202
`6.2 Microscopic Reversibility and Detailed Balancing, 205
`6.3 The Microscopic-Macroscopic Connection, 206
`References, 208
`
`Chapter 7
`POTENTIAL ENERGY SURFACES
`
`209
`
`Long-Range Potentials, 210
`7. I
`Empirical Intermolecular Potentials, 213
`7 .2
`7.3 Molecular Bonding Potentials, 216
`7.4
`Internal Coordinates and Normal Modes of Vibration, 217
`Potential Energy Surfaces, 220
`7.5
`7 .6 Ab Initio Calculation of Potential Energy Surfaces, 225
`7. 7 Analytic Potential Energy Functions, 231
`7 .8
`Experimental Determination of Potential Energy Surface Properties, 235
`7. 9 Details of the Reaction Path, 236
`7.10 Potential Energy Surfaces of Electronically Excited Molecules, 237
`References , 240
`Bibliography, 242
`Problems, 244
`
`Chapter 8
`DYNAMICS OF BIMOLECULAR COLLISIONS
`
`246
`
`8.1 Simple Collision Models, 246
`8.2 Two-Body Classical Scattering, 251
`8.3 Complex Scattering Processes, 261
`References, 276
`Problems, 277
`
`Chapter 9
`EXPERIMENTAL DETERMINATION OF NEW
`KINETIC PARAMETERS
`282
`
`9. 1 Molecular Beam Scattering, 282
`9.2 State-Resolved Spectroscopic Techniques, 290
`9.3 An Example of State-to-State Kinetics: The F + H2 Reaction , 293
`9.4 Some General Principles Concerning Energy Disposition in Chemical Reactions, 297
`9.5 Detailed Balance Revisited , 299
`
`Contents
`
`vii
`
`Page 5 of 38
`
`

`

`9.6 Chemical Lasers, 301
`9.7 State-to-State Chemical Kinetics Can Be Hazardous to your Health!, 301
`References, 302
`Problems, 303
`
`Chapter 10
`STATISTICAL APPROACH TO REACTION DYNAMICS:
`308
`TRANSITION STATE THEORY
`
`10.1 Motion on the Potential Surface, 308
`10.2 Basic Postulates and Standard Derivation of Transition State Theory, 310
`10.3 Dynamical Derivation of Transition State Theory, 315
`10 .4 Quantum Mechanical Effects in Transition State Theory, 318
`10.5 Thermodynamic Formulation of Transition State Theory, 321
`10.6 Applications of Transition State Theory, 323
`10.7 Microcanonical Transition State Theory, 331
`10.8 Variational Transition State Theory, 333
`10.9 Critique of Transition State Theory, 336
`References, 337
`Bibliography, 338
`Problems, 339
`
`Chapter 11
`UNIMOLECULAR REACTION DYNAMICS
`
`342
`
`Formation of Energized Molecules, 344
`11.1
`Sum and Density of States, 347
`11 .2
`Lindemann-Hinshelwood Theory of Thermal Unimolecular Reactions, 352
`11.3
`Statistical Energy-Dependent Rate Constant k(E), 357
`11.4
`11.5 RRK Theory, 358
`11.6 RRKM Theory, 362
`11. 7 Application of RRKM Theory to Thermal Activation, 368
`11.8 Measurement of k(E), 370
`11. 9
`Intermolecular Energy Transfer, 374
`11.10 Product Energy Partitioning, 376
`11.11 Apparent and Intrinsic Non-RRKM Behavior, 379
`11 .12 Classical Mechanical Description of Intramolecular Motion and Unimolecular Decom-
`position, 382
`11.13 Mode Specificity, 385
`References, 389
`Bibliography, 392
`Problems, 393
`
`viii
`
`Contents
`
`Page 6 of 38
`
`

`

`Chapter 12
`DYNAMICS BEYOND THE GAS PHASE
`
`402
`
`12.1 Transition State Theory of Solution Reactions, 403
`12.2 Kramers' Theory and Friction, 410
`12.3 Gas-Surface Reaction Dynamics, 415
`References, 427
`Bibliography, 428
`Problems, 429
`
`Chapter 13
`INFORMATION-THEORETICAL APPROACH TO
`STATE-TO-STATE DYNAMICS
`431
`
`Introduction, 431
`13 .1
`13.2 The Maximal-Entropy Postulate, 431
`13.3 Surprisal Analysis and Synthesis: Product State Distribution in Exothermic Reactions, 438
`13.4
`Informational-Theoretical Analysis of Energy Transfer Process, 445
`13.5 Surprisal Synthesis, 464
`13.6 Conclusion, 470
`References, 470
`Bibliography, 472
`Problems, 473
`
`Chapter 14
`ANALYSIS OF MULTILEVEL KINETIC SYSTEMS
`
`474
`
`Introduction, 474
`14.1
`14.2 The Master Equation, 474
`Information-Theoretical Treatment of the Master Equation, 479
`14.3
`14.4 Some Applications of Master-Equation Modeling, 483
`References, 493
`Problems, 494
`
`Chapter 15
`KINETICS OF MULTICOMPONENT SYSTEMS
`
`496
`
`15.1 Atmospheric Chemistry, 496
`15.2 The Hydrogen-Oxygen Reaction, an Explosive Combustion Process, 509
`15.3 The Methane Combustion Process, 516
`15.4 Conclusion, 524
`References, 525
`Problems, 525
`
`Contents
`
`ix
`
`Page 7 of 38
`
`

`

`Appendix 1
`QUANTUM STATISTICAL MECHANICS
`
`Appendix 2
`CLASSICAL STATISTICAL MECHANICS
`
`527
`
`528
`
`A2. l Sum and Density of States, 528
`A2.2 Partition Function and Boltzman Distribution, 532
`Bibliography, 533
`Problems, 534
`
`Appendix 3
`DATA BASES IN CHEMICAL KINETICS
`
`535
`
`INDEX
`
`538
`
`x
`
`Contents
`
`Page 8 of 38
`
`

`

`CHAPTER 1
`Basic Concepts of Kinetics
`
`Among the most familiar characteristics of a material system is its capacity for chem(cid:173)
`ical change. In a chemistry lecture, the demonstrator mixes two clear liquids and ob(cid:173)
`tains a colored solid precipitate. Living organisms are born, grow, reproduce, and
`die. Even the formation of planetary rocks , oceans, and atmospheres consists of a set
`of chemical reactions. The time scale for these reactions may be anywhere from a
`few femtoseconds oo-ts sec) to geologic times (109 years, or 10+ 16 sec).
`The science of thermodynamics deals with chemical systems at equilibrium,
`which by definition means that their properties do not change with time. Most real
`systems are not at equilibrium and undergo chemical changes as they seek to ap(cid:173)
`proach the equilibrium state. Chemical kinetics deals with changes in chemical prop(cid:173)
`erties in time. As with thermodynamics, chemical kinetics can be understood in
`terms of a continuum model, without reference to the atomic nature of matter. The
`interpretation of chemical reactions in terms of the interactions of atoms and
`molecules is frequently called reaction dynamics. A knowledge of the dynamic basis
`for chemical reactions has, in fact, permitted us to design and engineer reactions for
`the production of an enormous number of compounds which we now regard as es(cid:173)
`sential in our technological society.
`We begin our study of chemical kinetics with definitions of the basic observ(cid:173)
`able quantities, which are the chemical changes taking place in a system, and how
`these changes depend on time.
`
`1.1 DEFINITION OF THE RATE OF A CHEMICAL REACTION
`
`Broadly speaking, chemical kinetics may be described as the study of chemical sys(cid:173)
`tems whose composition changes with time. These changes may take place in the
`gas, liquid, or solid phase of a substance. A reaction occurring in a single phase is
`usually referred to as a homogeneous reaction, while a reaction which takes place at
`an interface between two phases is known as a heterogeneous reaction. An example
`of the latter is the reaction of a gas adsorbed on the surface of a solid.
`
`1
`
`Page 9 of 38
`
`

`

`The chemical change that takes place in any reaction may be represented by a
`stoichiometric equation such as
`aA + bB ~ cC + dD
`where a and b denote the number of moles of reactants A and B that react to yield c
`and d moles of products C and D. Various symbols are used in the expression which
`relates the reactants and products. For example, the formation of water from hydro(cid:173)
`gen and oxygen may be written as the balanced, irreversible chemical reaction
`(1 -2)
`
`(1 - 1)
`
`In this simple example, the single arrow is used to indicate that the reaction proceeds
`from the left (reactant) side to the right {product) side as written: water does not
`spontaneously decompose to form hydrogen and oxygen. A double arrow in the stoi(cid:173)
`chiometric equation is often used to denote a reversible reaction, that is , one which
`can proceed in either the forward or the reverse direction; an example is
`
`(1-3)
`
`While each of equations 1-2 and 1-3 describes an apparently simple chemical
`reaction, it so happens that neither of these reactions proceeds as written. Instead,
`the reactions involve the formation of one or more intermediate species, and include
`several steps. These steps are known as elementary reactions . An elementary reac(cid:173)
`tion is one in which the indicated products are formed directly from the reactants ,
`for example, in a direct collision between an A and a B molecule; intuitively, they
`correspond to processes occurring at the molecular level. In the hydrogen-oxygen
`reaction, a key elementary reaction is the attack of oxygen atoms on hydrogen
`molecules given by
`
`while in the hydrogen-iodine reaction it is
`
`0 +Hz ~ OH + H
`
`21 + Hz ~ Hzl + I
`The details of these reactions are discussed in sections 15.2 and 2.3.2, respectively.
`In the meantime, note here that they involve atoms (0, I), free radicals (OH), and/or
`unstable intermediates (H2I); this is often the case with elementary reactions.
`'
`The change in composition of the reaction mixture with time is the rate of re(cid:173)
`action, R. For reaction 1-1, the rate of consumption of reactants is
`
`(1-4)
`
`A standard convention in chemical kinetics is to use the chemical symbol enclosed in
`brackets for species concentration; thus, [X] denotes the concentration of X. The
`negative signs in equation 1-4 indicate that during the course of the reaction the
`concentration of reactants decreases as the reactants are consumed; conversely, a
`positive sign indicates that the concentration of products increases as those species
`are formed. Consequently, the rate of formation of products C and D can be written
`as
`
`2
`
`Basic Concepts of Kinetics
`
`Chap. 1
`
`Page 10 of 38
`
`

`

`R = +! d[C] = +! d[D]
`cdt
`ddt
`
`(1 - 5)
`
`The factors a, b, c, and din equations 1-4 and 1-5 are referred to as the stoi(cid:173)
`chiometric coefficients for the chemical entities taking part in the reaction. Since the
`concentrations of reactants and products are related by equation 1- 1, measurement
`of the rate of change of any one of the reactants or products would suffice to deter(cid:173)
`mine the rate of reaction R. In the reaction 1- 2, the rate of reaction would be
`
`(1 - 6)
`
`A number of different units have been used for the reaction rate. The dimen(cid:173)
`sionality of R is
`
`[amount of material][volumer 1[timer 1
`
`or
`
`[concentration] [time 1- 1
`The standard SI unit of concentration is moles per cubic decimeter, abbreviated
`mol dm- 3
`• In the older literature on kinetics, one frequently finds the units
`mol liter- 1 for reactions in solution and mol cm- 3 for gas phase reactions. The SI
`unit is preferred, and should be used consistently. Multiplying moles cm- 3 by
`Avogadro's Number (6.022 x 1023
`) gives the units molecules cm- 3
`, which is still ex(cid:173)
`tensively used and, indeed, is convenient for gas phase reactions.
`A subcommittee of the International Union of Pure and Applied Chemistry
`chaired by Laidler has attempted to standardize units, terminology, and notation in
`chemical kinetics.' We have attempted to follow the subcommittee's recommenda(cid:173)
`tions in this text.
`
`1.2 ORDER AND MOLECULARITY OF A REACTION
`
`In virtually all chemical reactions that have been studied experimentally, the reac(cid:173)
`tion rate depends on the concentration of one or more of the reactants. In general,
`the rate may be expressed as a function f of these concentrations,
`R = f ([A],[B])
`In some cases the reaction rate also depends on the concentration of one or more in(cid:173)
`termediate species, e.g., in enzymatic reactions (see chapter 5). In other cases the
`rate expression may involve the concentration of some species which do not appear
`in the stoichiometric equation 1- 1; such species are known as catalysts, and will be
`discussed in chapter 5. In still other cases, the concentration of product molecules
`may appear in the rate expression.
`The most frequently encountered functional dependence given by equation 1-7
`is the rate's being proportional to a product of algebraic powers of the individual
`concentrations, i.e.,
`
`(1 - 7)
`
`Sec. 1.2
`
`Order and Molecu larity of a Reaction
`
`(1-8)
`
`3
`
`Page 11 of 38
`
`

`

`The exponents m and n may be integer, fractional, or negative. This proportionality
`can be converted to an equation by inserting a proportionality constant k, thus:
`R = k [A]'"[B]"
`This equation is called a rate equation or rate expression . The exponent m is the or(cid:173)
`der of the reaction with respect to reactant A, and n is the order with respect to re(cid:173)
`actant B. The proportionality constant k is called the rate constant. The overall or(cid:173)
`der of the reaction is simply p = m + n. A generalized expression for the rate of a
`reaction involving K components is
`
`(1-9)
`
`R = k CT cl';
`
`K
`
`i = I
`
`(1-10)
`
`The product is taken over the concentrations of each of the K components of the re(cid:173)
`action. The reaction order with respect to the ith component is n;, p = ~~= in; is the
`overall order of the reaction, and k is the rate constant.
`In equation 1-10, k must have the units
`[ concentration] - (p- O[ time 1- 1
`so for a second-order reaction, i.e., m = n = 1 in equation 1-8, the units would be
`, or dm3 mo1 - 1 sec- 1 in SI units. Note that the units of liter
`[concentrationr 1[timer 1
`mo1- 1 sec- 1 are frequently encountered in the older solution-kinetics literature, and
`cm3 mo1- 1 sec- 1 or cm3 molecule- 1 sec- 1 are still encountered in the gas-kinetics lit(cid:173)
`erature.
`Elementary reactions may be described by their molecularity, which specifies
`the number of reactants that are involved in the reaction step. If a reactant sponta(cid:173)
`neously decomposes to yield products in a single reaction step, given by the equation
`
`A ~ products
`
`(1 - 11)
`
`the reaction is termed unimolecular. An example of a uni molecular reaction is the
`dissociation of N104, represented by
`
`N104 ~ 2N02
`
`If two reactants A and B react with each other to give products, i.e.,
`A + B ~ products
`the reaction is termed bimolecular. An example of a bimolecular reaction would be
`a metathetical atom-transfer reaction such as
`0 + H1 ~ OH+ H
`
`(1 - 12)
`
`or
`
`F + H1 ~ HF+ H
`Both of these reactions are discussed in subsequent chapters.
`Three reactants that come together to form products constitute a termolecular
`reaction. In principle, one could go on to specify the molecularity of four, five, etc.,
`reactants involved in an elementary reaction, but such reactions have not been en-
`
`4
`
`Basic Concepts of Kinetics
`
`Chap. 1
`
`Page 12 of 38
`
`

`

`(1-13)
`
`countered in nature. The situation reflects the molecular basis of elementary reac(cid:173)
`tions. A single, suitably energized molecule can decompose according to equation
`1-11; such unimolecular processes are discussed in chapter 11 . A collision between
`two molecules can lead to a bimolecular reaction according to equation 1-12; this is
`further discussed in chapters 8 and 10. At moderate to high gas pressures, termolec(cid:173)
`ular processes can occur, such as three-body recombination, i.e.,
`A + B + M ---7 AB + M
`However, physical processes involving simultaneous interaction of four or more in(cid:173)
`dependent particles are so rare in chemical kinetics as to be completely negligible.
`An elementary reaction is one in which the molecularity and the overall order
`of the reaction are the same. Thus, a bimolecular elementary reaction is second or(cid:173)
`der, a termolecular reaction third order, and so on. The reverse is not always true,
`however. For example, the hydrogen-iodine reirction 1-3 is second order in both di(cid:173)
`rections, but bimolecular reactions between H2 and lz, and between two HI
`molecules, are thought not to occur. Instead, the reaction consists of several uni(cid:173)
`molecular, bimolecular, and possibly termolecular steps (see chapter 2).
`A further distinction between molecularity and reaction order is that, while
`molecularity has only the integer values 0, 1, 2, and 3, order is an experimentally
`determined quantity which can take on noninteger values. In principle, these values
`could be any number between -oo and +oo, but values between -2 and 3 are usually
`encountered in practice. Negative orders imply that the component associated with
`that order acts to slow down the reaction rate; such a component is termed an in(cid:173)
`hibitor for that reaction. Fractional values of the reaction order always imply a com(cid:173)
`plex reaction mechanism (see section 1. 6). An example of a fractional-order reaction
`is the thermal decomposition of acetaldehyde given by
`300-800°C
`CH4 + CO
`---7
`
`CH3 CHO
`
`(1 - 14)
`
`which has a ~ reaction order, i.e.,
`
`d[CH4]
`~ = (constant)[CH3 CH0]312
`
`Similarly, under certain conditions the reaction of hydrogen with bromine
`Hz + Br2 ~ 2HBr
`has a ~ reaction order, first order in [Hz] and ~ order in [Br2]:
`
`d[HBr]
`~ = (constant)[H2][Br2]1l 2
`
`(1 - 15)
`
`(1 - 16)
`
`(1 - 17)
`
`Under other conditions, reaction 1-16 can display an even more complicated behav-
`ior, viz.,
`
`(constant)[H2][Br2]112
`d[HBr]
`~~- = ...;_~~__;_:'---~___;;~
`1 + (constant')[HBr]
`dt
`The constants in equations 1- 15, 1- 17, and 1- 18 are clearly not identifiable
`with an elementary reaction, but instead are phenomenological coefficients obtained
`
`(1-18)
`
`Sec. 1.2
`
`Order and Molecularity of a Reaction
`
`5
`
`Page 13 of 38
`
`

`

`by fitting the rate expression to experimental data. Such coefficients are more prop(cid:173)
`erly termed rate coefficients, rather than rate constants. The latter term should be re(cid:173)
`served for the coefficients in rate expressions for elementary reactions, which follow
`a rate expression having the form of equation 1-10.
`
`1.3 ELEMENTARY REACTION RATE LAWS
`
`Thus far, we have defined the rate of reaction in terms of concentrations, orders, and
`reaction rate constants. Next, we consider the time behavior of the concentration of
`reactants in elementary reactions with simple orders. The time behavior is deter(cid:173)
`mined by integrating the rate law for a particular rate expression.
`
`1.3. 1 Zero-Order Reaction
`
`The rate law for a reaction that is zero order is
`
`R = - d[A] = k[A] 0 = k
`dt
`
`(1-19)
`
`J[A],
`
`[A]o
`
`dt
`
`t1 ~ 0
`
`(1 -20)
`
`(1-21)
`
`Zero-order reactions are most often encountered in heterogeneous reactions on sur(cid:173)
`faces (see chapter 5). The rate of reaction for this case is independent of the concen(cid:173)
`tration of the reacting substance. To find the time behavior of the reaction, equation
`1-19 is put into the differential form
`d[A] = -kdt
`and then integrated over the boundary limits t1 and t1. Assuming that the concentra(cid:173)
`tion of A at ti = 0 is [A]o, and at ti = tis (A],, equation 1-20 becomes
`f'2 ~ 1
`d[A] = -k
`
`Hence,
`
`(A]o = -k(t - 0)
`Consequently, the integrated form of the rate expression for the zero-order reaction
`is
`
`(1-22)
`
`[A], -
`
`(A], = (A]o - kt
`A plot of [A] versus time should yield a straight line with intercept [A]o and slope k.
`
`(1-23)
`
`1.3.2 First-Order Reactions
`
`A first-order reaction is one in which the rate of reaction depends only on one reac(cid:173)
`tant. For example, the isomerization of methyl isocyanide, CH3NC, is a first-order
`unimolecular reaction:
`
`(1-24)
`
`6
`
`Basic Concepts of Kinetics
`
`Chap. 1
`
`Page 14 of 38
`
`

`

`This type of equation can be represented symbolically as
`
`and the rate of disappearance of A can be written as
`
`A ~ B
`
`R = _ _!_ d[A] = k[A]1
`a dt
`
`(1 -25)
`
`Note that the reaction is of order one in the reactant A. Thus, since only one A
`molecule disappears to produce one product B molecule, a = 1 and equation 1-25
`becomes
`
`- d[A] = k[A]
`dt
`
`(1 - 26)
`
`Integration of equation 1-26 leads to
`
`J
`
`dt
`
`J d[A]
`[A] = k
`- ln[A] = kt + constant
`If the boundary conditions are such that at t = 0 the initial value of A is [A]o, the
`constant of integration in equation 1-27 can be eliminated if we integrate over the
`boundary limits as follows:
`
`-
`
`-
`
`f'
`
`JIAJ, d[A]
`
`!Alo
`
`--=k dt
`(A]
`o
`
`(1 -27)
`
`(1 -28)
`
`(1 -29)
`
`(1 - 30)
`
`(1 -3 1)
`
`This gives
`
`and hence
`
`- (ln[A], -
`
`ln[A]o) = kt
`
`- ln(A], = kt -
`
`ln[A]o
`
`Thus, the constant in equation 1-27 is just
`
`constant = - ln[A]o
`
`Equation 1- 30 can be written in various forms. Some that are commonly used
`
`are
`
`and
`
`I ([A],) = -kt
`n [A]o
`[A], = [A]oe -k'
`
`[A], _
`-kr
`[A]o - e
`
`Sec. 1.3
`
`Elementary Reaction Rate Laws
`
`(1 -32a)
`
`(1 - 32b)
`
`(l - 32c)
`
`7
`
`Page 15 of 38
`
`

`

`These forms of the integrated rate expression for the first-order reaction are worth
`remembering. From the exponential form of equations l -32b and 1-32c, one can
`determine a time constant r which is called the decay time of the reaction . This
`quantity is defined as the time required for the concentration to decrease to l/e of its
`initial value [A]o, where e = 2.7183 is the base of the natural logarithm. The timer
`is given by
`
`1
`'T = -
`k
`
`(1-33)
`
`In experimental determinations of the rate constant k, the integrated form of the rate
`law is often written in decimal logarithms as
`
`log10 [A], = log10 [A]o -
`
`kt
`_
`303
`
`2
`
`(1 - 34)
`
`and a semilog plot of [A], versus twill yield a straight line with k/2 .303 as slope and
`[A]o as intercept.
`
`1.3.3 Second-Order Reactions
`
`There are two cases of second-order kinetics. The first is a reaction between two
`identical species, viz.,
`
`A + A
`The rate expression for this case is
`R = _! d[A] = k[A]2
`2 dt
`
`products
`
`-
`
`(1 - 35)
`
`(1 - 36)
`
`The second case is an overall second-order reaction between two unlike species,
`given by
`
`A + B
`products
`-
`In this case, the reaction is first order in each of the reactants A and B and the rate
`expression is
`
`(1-37)
`
`R = - d[A] = k[A][B]
`dt
`
`(1-38)
`
`Note the appearance of the stoichiometric coefficient~ in equation 1- 36, but not in
`equation 1-38.
`Let us consider the first case, given by equations 1- 35 and 1-36. Although not
`an elementary reaction , the disproportionation of HI (equation 1-3) is a reaction
`which is exactly second order in a single reactant. Another example is the recombi(cid:173)
`nation of two identical radicals, such as two methyl radicals:
`
`C2H6
`2CH3 -
`We integrate the rate law, equation 1-36, to obtain
`
`(1 -39)
`
`8
`
`Basic Concepts of Kinetics
`
`Chap. 1
`
`Page 16 of 38
`
`

`

`--
`
`-f [AJ, d[A] -
`
`[A] 2
`
`- 2k
`
`which gives
`
`[A]o
`
`ft
`
`0
`
`dt
`
`1
`1
`[A], = [A]o - 2kt
`
`(1-40)
`
`(1 - 41)
`
`A plot of the inverse concentration of A ([Ar 1) versus time should yield a straight
`
`line with slope equal to 2k and intercept 1/[A]0 •
`To integrate the rate law for the second case, equations 1-37 and 1- 38, it is
`convenient to define a progress variable x which measures the progress of the reac(cid:173)
`tion to products as
`
`x = ([A]o -
`
`[A],) = ([B]o -
`
`[B]i)
`
`(1-42)
`
`where [A]o and [B]o are the initial concentrations. The rate expression given by
`equation 1-38 can then be rewritten in terms of x as
`
`dx
`dt = k([A]o - x)([B]o - x)
`
`(1-43)
`
`To find the time behavior, we integrate equation 1-43 thus:
`
`f'
`
`dt
`
`(1-44)
`
`Jx(t)
`
`x(O)
`
`dx
`- - - - - - - - = k
`([A]o - x)([B]o - x)
`0
`To solve the integral on the left-hand side of equation 1-44, we separate the vari(cid:173)
`ables and use the method of partial fractions:
`
`I ([A]o - ~~[B]o - x)
`= I
`
`([A]o -
`
`- I
`
`([A]o -
`
`dx
`[B]o)([B]o - x)
`
`dx
`[B]o)([A]o - x)
`
`( 1- 45)
`
`Solving the right-hand side of equation 1- 45 and equating it to the left-hand side of
`equation 1- 44, we obtain, as the solution to the rate expression for the second case,
`
`1
`([A]o -
`
`[B]o)
`
`ln ([B]o[A],) _ k
`t
`[A] 0[B] 1
`
`-
`
`(1-46)
`
`In this case the experimental data may be plotted in the form of the left-hand side of
`the equation against t.
`
`1.3.4 Third-Order Reactions
`
`From the definition of overall reaction order in equation 1- 10, we see that there are
`three possible types of third-order reactions: (1) 3A ~ products; (2) 2A + B
`~products; and (3) A + B + C ~ products. In the first case, in which the rate
`law depends on the third power of one reactant, the rate expression is
`
`Sec. 1.3
`
`Elementary Reaction Rate Laws
`
`9
`
`Page 17 of 38
`
`

`

`R =- - ! d[A] = k[A]3
`3 dt
`
`This rate law can be integrated readily to obtain the solution
`
`Rearranging gives
`
`1 ( 1
`- l [A]2 -
`
`1 )
`[A]6 = 3kt
`
`1
`1
`[A]2 = [A]6 + 6kt
`
`(1-47)
`
`(1-48)
`
`(1-49)
`
`A plot of the inverse squared concentration of A ([Ar 2
`slope 6k and intercept 1/[A]5.
`The second case,
`
`) with time should yield
`
`2A+B ~ C
`
`(1-50)
`
`which is second order in reactant A and first order in reactant B, has the overall or(cid:173)
`der 3. The rate law for this reaction is
`R = - ! d[A] = k[A]2[B]
`2 dt
`
`(1 - 51)
`
`This rate expression can be integrated for two possible subcases. The first is when
`the concentration of B is so much greater than that of the reactant A ([B] P [A]),
`that the concentration of B does not change during the course of the reaction. Under
`this condition, the rate expression can be rewritten as
`
`R = -~d[A] = k'[A]2
`2 dt
`
`(1-52)
`
`so that the third-order expression reduces to a "pseudo second-order" expression.
`The solution for this case is equation 1-41, i.e.,
`
`I
`1
`k I
`[A] = [A]o - 2
`
`t
`
`(1-53)
`
`A plot of [Ar' vs. time, for a fixed [B], should then yield slope 2k' and intercept
`l/[A]o; but note that the resulting rate coefficient is a function of the concentration
`of B, that is,
`
`k' = k[B]
`Forgetting that this rate coefficient contains an added concentration term can lead to
`errors in interpretation of data. A simple example of this type of reaction is the
`I + I + M ---? Ii + M
`three-body
`recombination
`process,
`such
`as
`and
`0 + 02 + M---? 03 + M. In these cases the third body acts to remove the excess
`energy from the recombining reactants, thereby stabilizing the molecular products.
`The other instance in which equation 1-51 can be easily integrated is when the
`initial concentrations of the dissimilar reactants A and B are equal. In integrating the
`rate law for this case, subject to the stated initial conditions, it is once again conve-
`
`(1-54)
`
`10
`
`Basic Concepts of Kinetics
`
`Chap. 1
`
`Page 18 of 38
`
`

`

`-
`
`nient to introduce a progress variable as we did in solving the second-order reaction
`case. Accordingly, we define a progress variable y by
`
`[A], = [A]o - 2y
`
`and
`
`[B], = [B]o - y
`and with this condition we can rewrite equation 1-51 in terms of y as
`
`dy
`dt = k([A]o - 2y) 2([B]o - y)
`
`Upon rearranging, we obtain
`
`dy
`([A]o - 2y )2([B]o - y)
`
`= kdt
`
`This equation can be integrated by the method of partial fractions to yield
`
`(1 -55)
`
`(1-56)
`
`1
`([A]o - 2[B]o)
`
`( 1
`[Ao] -
`
`1
`1 )
`([A],[B]o)
`[A], + ([A]o - 2[B]0) 2 In [A]o[B], = kt
`An example of such a reaction is the gas phase reaction between nitric oxide and
`oxygen
`
`(1 -57)
`
`(1-58)
`
`The third type of third-order reaction is first order in three different compo(cid:173)
`nents, i.e.,
`
`A + B + C ~ product
`The rate law for this reaction is
`
`R = d[A] = k(A][B][C]
`dt
`
`(1 -59)
`
`(1 -60)
`
`To solve for the integrated rate law expression in this case, we use the method of
`partial fractions as before. The solution is left as an exercise at the end of this
`chapter.
`
`1.3.5 Reactions of General Order
`
`There are no known examples of fourth- , fifth-, or higher order reactions in the
`chemical literature. The highest order which has been empirically encountered for
`chemical reactions is third order. Nevertheless, in this section we develop the gen(cid:173)
`eral solution for a reaction which is nth order in one reactant, for n equal to any in(cid:173)
`teger or noninteger value. The rate expression for such a reaction is
`
`R = - d[A] = k[A]"
`dt
`
`A simple integration of this expression yields the result
`
`Sec. 1.3
`
`Elementary Reaction Rate Laws
`
`(1 -61)
`
`11
`
`Page 19 of 38
`
`

`

`1
`(n - 1)
`
`1
`(
`[A];'- 1
`
`1
`) = kt
`[A]3- 1
`
`which can be rewritten as
`
`1
`[A];i- 1 -
`
`1
`[A]3- 1 = (n -
`
`l)kt
`
`(1 - 62)
`
`(1-63)
`
`Equation 1-63 is valid for any value of n except n = 1, in which case it is undefined
`and equation 1-32 must be used instead. Figure 1- 1 shows several plots of concen(cid:173)
`tration vs. time for various values of n.
`
`[A]o
`
`n = 4.0
`
`0
`
`Timet
`
`Figure 1-1. Plot of [A], versus time for reaction of general order. The plot shows
`various functional behaviors for n = 2, 2.5, 3, 3.5, and 4. Concentration and time
`are in arbitary units.
`
`In the general case there is no simple plot that can be constructed to test the
`order of the reaction, as can be done for the first- and second-order cases. When the
`order n is unknown, a van't Hoff plot can be constructed as an aid to deducing the
`order of the reaction. In a van't Hoff plot, the logarithm of the rate is plotted against
`the logarithm of the concentration of the reactant A. This is equivalent to making a
`plot of equation 1-63 on log-log graph paper. The slope of such a plot gives the or(cid:173)
`der of the reaction n. Examples of van't Hoff plots for several reaction orders are
`shown in Figure 1-2.
`
`12
`
`Basic Concepts of Kinetics
`
`Chap. 1
`
`Page 20 of 38
`
`

`

`n =Sn =4n=3 n =2
`
`n = 1
`
`logk
`
`log [A]
`Figure 1-2. Van' t Hoff plot of log ( - d~~l) versus log [A] for various reaction
`orders.
`
`1.4 DETERMINATION OF REACTION ORDER: REACTION
`HALF-LIVES
`