`

`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
`10 987 654 32
`
`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
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`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
`
`

`

`

`

`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
`
`

`

`

`

`

`

`

`

`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
`
`

`

`

`

`

`

`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
`as
`and
`three-body
`recombination
`process,
`such
`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
`
`

`

`

`

`

`

`

`

`

`

`(1 - 68)
`
`k (T) = A exp( - Eac1/ RT)
`This relationship is now known as the Arrhenius equation, and a plot of ln k (or
`log10 k) vs. 1/T is called an Arrhenius plot. In the Arrhenius equation, the tempera(cid:173)
`ture dependence comes primarily from the exponential term, although the quantity
`A, referred to as the pre-exponential or the frequency factor, may have a weak tem(cid:173)
`perature dependence, no more than some fractional power of T. The units of A are
`the same as the units of the rate constant k, since the exponential term has no units.
`In the case of a first-order reaction, A has units of sec- 1
`; for reactions of higher or(cid:173)
`der, A has units (concentration) 1-Psec- 1
`• The temperature Tis, of course, in absolute
`or Kelvin units (degrees Celsius + 273.16).
`The key quantity in the Arrhenius equation is the activation energy Eact· The
`activation energy can be thought of as the amount of energy which must be supplied
`to the reactants in order to get them to react with each other. Since this is a positive
`energy quantity, the majority of reactions have k increasing with temperature. For
`some reactions, however, the rate decreases with temperature, implying a negative
`activation energy. Such reactions are generally complex, involving the formation of
`a weakly bound intermediate species. An example is the recombination of iodine
`atoms in the presence of a molecular third body M, which proceeds via the follow(cid:173)
`ing steps: 6
`
`I +M ~ IM
`
`IM +I ~ h+M
`
`The IM species is a van der Waals complex whose stability decreases with increasing
`temperature.
`The standard method for obtaining Eact is to graph experimental rate constant
`data on an Arrhenius plot, i.e., log10k vs. 1/T. The slope gives Eac1/2.303R, where
`R = 8.3145 J mo1- 1K- 1
`• The units of Eact are thus J mo1- 1
`, but since the magnitudes
`of activation energies are typically in the range of a few to several hundred thousand
`J mo1- 1
`, it is customary to report their values in kJ mo1- 1
`• The older (non-SI) unit of
`calories or kcal mo1- 1 is still often encountered, but should be discouraged. The
`conversion factor7 is 1 calorie = 4.184 J.
`The origin of the activation energy is a barrier on the potential energy surface
`between the reactants and products; this is discussed in detail in chapters 7 and 10.
`For the time being, the enthalpy diagram shown in Figure 1-3 may be instructive.
`This is simply a sketch of the thermodynamic energies associated with the reactants,
`the products, and a (for the moment) hypothetical transition state connecting the
`two. The energy difference between the reactants and products is the difference in
`their heats of formation and is given by
`llH'ieaction = llHJ(products) -
`A reaction which is highly endothermic, that is, which has a large positive llH'ieaction,
`is not likely to proceed spontaneously except at very high temperatures. A highly
`exothermic reaction, however, may do so unless the activation energy required to
`reach the transition state is very high; in that case, the reaction will be slow at other
`than very high temperatures. The calculation of reaction exothermicities is often a
`
`llHJ(reactants)
`
`(1 - 69)
`
`Sec. 1.5 Temperature Dependence of Rate Constants: The Arrhenius Equation 15
`
`----
`
`Page 23 of 38
`
`

`

`

`

`plex reaction is the assembly of individual steps that make it up. One of the principal
`aims of experimental studies in chemical kinetics is to elucidate mechanisms and to
`describe the overall observed reaction process in terms of the elementary reaction
`steps which constitute the mechanism. In chapter 2, we consider various types of
`complex reactions together with methods for analyzing their kinetics. Chapter 3
`deals with some of the experimental methods that can be used to carry out rate mea(cid:173)
`surements on complex reaction systems, with emphasis on gas-phase reactions.
`Chapter 4 then goes on to treat reactions in liquid solutions, and in chapter 5 we
`consider catalyzed reactions, including those which occur at the gas-solid interface.
`Chapters 2 through 5 deal primarily with chemical kinetics, that is, the phe(cid:173)
`nomenological behavior of reactions. In chapters 6 through 13, we turn our attention
`to chemical dynamics, the description of chemical reactions at the molecular level.
`The approach there will be to attempt to isolate elementary reaction steps experi(cid:173)
`mentally, to examine them in great detail, and to relate the findings to microscopic
`molecular properties. Finally, chapters 14 and 15 present some examples of kinetic
`systems, that is, large coupled reaction sets which describe complex real-world phe(cid:173)
`nomena.
`
`REFERENCES
`
`1. K. J. Laidler, Pure App. Chem. 53, 753 (1981).
`2. G. Kornfeld and E. Klinger, Z. physik. Chem. B4, 37 (1929).
`3. G. G. Hammes, Principles of Chemical Kinetics (New York: Academic Press, 1978).
`4. S. W. Benson, Foundations of Chemical Kinetics (New York: McGraw-Hill, 1960).
`5. S. Arrhenius, Z. physik. Chem. 4, 226 (1889).
`6. G. Porter and J. A. Smith, Proc. Roy. Soc. A261, 28 (1961).
`7. I. Mills, ed., Quantities, Units, and Symbols in Physical Chemistry (Oxford: Blackwell,
`1988).
`
`BIBLIOGRAPHY
`
`AMDUR, I., and HAMMES, G. G. Chemical Kinetics: Principles and Selected Topics. New
`York: McGraw-Hill, 1966.
`CAPELLOS, C., and BIELSKI, B. Kinetic Systems. New York: Wiley, 1972.
`EsPENSON, J. H. Chemical Kinetics and Reaction Mechanisms. New York: McGraw-Hill,
`1981.
`EYRING, H., and EYRING, E. M. Modern Chemical Kinetics. New York: Reinhold, 1963.
`FLECK, G. M. Chemical Reaction Mechanisms. New York: Holt, Rinehart and Winston, 1971.
`FROST, A. A., and PEARSON, R. G. Kinetics and Mechanism. 2d ed. New York: Wiley, 1961.
`GARDINER, W. C., JR. Rates and Mechanisms of Chemical Reactions. Menlo Park, CA: W. A.
`Benjamin, 1969.
`GLASSTONE, S., LAIDLER, K. J., and EYRING, H. The Theory of Rate Processes. New York:
`McGraw-Hill, 1941.
`
`Chap. 1
`
`Bibliography
`
`17
`
`Page 25 of 38
`
`

`

`

`

`

`

`

`

`and equal to the steady-state concentration [CH3],,. The second and third reactions
`in the mechanism given by equations 3-22 are the only significant reactions con(cid:173)
`suming CH3. Since the CH3 concentration is proportional to the CH3 ion signal, k1
`can be determined by fitting the half-life of the CH3 ion-count growth to its steady(cid:173)
`state value, as shown in Figure 3-15.
`
`3.3 TREATMENT OF KINETIC DATA
`
`3.3. 1 General Analysis of Kinetic Data
`
`3.3.1.1 Introduction. Two questions usually arise when rate data are ob(cid:173)
`tained experimentally. The first deals with the inherent errors in the data as con(cid:173)
`tributed by instrumental factors and experimental procedures; the second pertains to
`how these errors determine the uncertainty in the fitted rate constants. Correct esti(cid:173)
`mates of kinetic parameters and their uncertainties are fundamentally important for
`chemical kinetics, because such data are needed for assessing kinetic models and mi(cid:173)
`croscopic rate theories in chemical kinetics. These two questions are also important
`in assessing just how well correlated a postulated reaction mechanism is to the raw
`data. Since errors are inherent in any measurement correlations are never exact, and
`a method for judging whether a particular correlation is significant is needed to as(cid:173)
`sess the model. This section is given over to considering briefly some of these
`points.
`
`3.3.1.2 Types of measurement errors. No measurement can be made per(cid:173)
`fectly and with zero error; thus, it becomes important to know the uncertainty of a
`measurement and how different errors that arise from the experiment or the instru(cid:173)
`ment enter into it. Furthermore, once sources of errors are identified, ways of re(cid:173)
`ducing them may be undertaken. It is the duty of every experimental kineticist to
`take every precaution to minimize errors and to report the accuracy of the final re(cid:173)
`sult. The accuracy of kinetic results involves estimating the systematic errors in each
`observation in the experimental apparatus or procedure. Although this step is
`difficult to quantify, it is possible to correct for the errors incurred. The precision of
`the result depends on random or statistical errors, which are fluctuations in repeated
`measurements and are usually beyond the control of the experimentalist. To estimate
`such errors, we treat the experimental data statistically using standard methods to
`obtain the standard deviation in the results from the mean value. Knowing these un(cid:173)
`certainties in the experimental data permits the uncertainty in the measured rate
`constants to be estimated and also enables us to further understand how these
`sources of errors contribute to the overall uncertainty in the measured rate. In this
`section our purpose is not to provide the reader with an extensive review of statisti(cid:173)
`29
`cal methods of treating data (indeed, this has already been done by others21
`), but
`-
`rather to acquaint the would-be kineticist with the procedures for analyzing and re(cid:173)
`porting kinetic data together with appropriate error limits.
`
`3.3.1.2.1 Systematic Errors.
`Systematic errors in kinetic measurements
`0-33 In kinetic measurements
`have been discussed in some detail by Cvetanovic et al. 3
`there are four kinds of systematic errors: (1) instrumental (shortcomings of the in-
`
`Sec. 3.3
`
`Treatment of Kinetic Data
`
`133
`
`Page 29 of 38
`
`

`

`struments or the effects of the environment on the instrument or its user); (2) opera(cid:173)
`tional (the personal judgment of the kineticist, which enters into readings of the in(cid:173)
`strument, and errors in instrument calibration); (3) methodological (inaccuracies in
`the modeling equations, which are imperfect approximations to the true solutions);
`and (4) mechanistic (errors introduced by inaccurate representations of the underly(cid:173)
`ing chemical mechanism or some unrecognized chemical interference from second(cid:173)
`ary reactions or impurities in reagents). There are several ways of searching for sys(cid:173)
`tematic errors in measured kinetic parameters:
`
`1. Comparison of measured kinetic parameters under a variety of experimental
`conditions. Inconsistencies between measured rate coefficients greater than the
`cumulative random error indicate the presence of systematic errors. Caution
`must be exercised since apparently consistent results can conceal systematic er(cid:173)
`rors .
`2. Comparison of the absolute value of rate parameters obtained by different tech(cid:173)
`niques. Here, the importance of determining the same kinetic parameters by
`vastly different techniques cannot be overemphasized.
`3. Through insight and guidance from theoretical and semi-empirical methods
`such as those to be presented in later chapters.
`
`In chemical kinetics , an inadequate mechanism is frequently the largest source
`of systematic errors . If a mechanistic complication is the cause of systematic error,
`its magnitude and sign may be assessed by computer simulation. In this case, the re(cid:173)
`action or reactions that may be significant are included. Once upper and lower
`bounds of the systematic error are estimated, it is possible to evaluate their contribu(cid:173)
`tion to the overall uncertainty in the results of the measurements. When upper (e~)
`and lower (efl bounds of the systematic error are given, a correction for the error is
`given by the mean ,
`
`(e~ + e;)
`e, = - ---
`2
`
`In this case the remaining error becomes
`
`+( II + L)
`+( ) = - e,
`e,
`- e, max
`2
`
`(3-23)
`
`(3-24)
`
`An excellent discussion of other ways of estimating systematic errors is given by
`35
`Eisenhart. 34
`•
`
`3.3.1.2.2 Random Errors. Random errors are usually due to unknown
`causes and may occur even when all systematic errors have been accounted for. They
`also contribute to the uncertainty of the measured value. Because of their random(cid:173)
`ness , there is no known method of controlling or correcting these errors . The only
`way to offset them is to increase the number of repetitive measurements and use
`statistical means to obtain the estimate of the true value of the parameter under mea(cid:173)
`surement. For a large number of observations, the random errors have a Gaussian
`distribution;27
`36 that is, the probability of their occurrence is given by
`•
`
`134
`
`Kinetic Measurements
`
`Chap.3
`
`Page 30 of 38
`
`

`

`f (E) =
`
`)
`1
`O" \,12; exp
`(
`
`(-(x - µ)2
`
`)
`
`2u2
`
`(3-25)
`
`where e is the magnitude of the random error given by x - µ,andµ, and u are mo(cid:173)
`ments of the distribution. The quantity µ, is the mean of the population, and <r is the
`standard deviation of the population-
`it measures true spread of the individual ob(cid:173)
`servations about the mean. The term x - µ, represents the magnitude of the random
`error in an individual measurement.
`3.3.J .2.3 Propagation of Errors. Rate constants of chemical reactions are
`calculated from experimental measurements of quantities such as gas pressure, in(cid:173)
`tensity alteration, intensity of chemiluminescence or fluorescence, etc. Conse(cid:173)
`quently, the uncertainty in the rate constant is dependent on the individual uncer(cid:173)
`tainties
`in
`these measured parameters. For example, consider a resonance
`fluorescence experiment done in a flow system. The critical parameters affecting the
`results of measurement are (1) gas flow , (2) pressure, (3) temperature, (4) distance,
`and (5) light intensity. Table 3-1 provides a list of typical uncertainty ranges for
`these parameters. 37 Uncertainties in these values together with scatter in the kinetic
`data ultimately determine the precision of reported results. Using the propagation(cid:173)
`28
`36 the overall uncertainty in the measured parameter is given
`of-errors method,27
`•
`•
`by
`
`af )2]112
`<Fk = ~ a<fJ; m
`(
`[
`where the <F; are the uncertainties in the measurement parameters and the partial
`derivatives (a f / a<fJ,) are measured at the sample mean of the measurement parame(cid:173)
`ters.
`
`(3-26)
`
`TABLE 3-1 RANDOM ERROR SOURCES AND MAGNITUDES IN EXPERIMENTAL
`FLOW-TUBE MEASUREMENTS (FRO M REF. 37)
`
`Measurement
`(units)
`
`Gas flow, u
`(mliter (STP)sec- 1)
`Pressure, P (torr)
`
`Temperature, T (0C)
`
`Distance, z (cm)
`Light intensity
`/(mV)
`F(mV)
`fcL(pA)
`
`Device/method used
`
`Range used
`
`Uncertainty range•
`
`Volume displacement
`Critical flow orifices
`Closed-end manometers
`Bourdon gauges
`U-tube manometer
`Chromel-alumel thermocouple
`Pt vs. Pt-Rh thermocouple
`Centimeter scale
`
`Absorption-monochromator/PMT
`Fluorescence-light filter/PMT
`Chemiluminescence-calibrated
`monochromator/PMT
`
`0.05-250
`5-500
`1-1000
`1-50
`0.5-20
`- 80-200
`300- 2000
`0-100
`
`10-100
`1-20
`0.1 - 500
`
`0.01 - 15
`1.0-5.0
`0.1 - 5.0
`0.1 -2.5
`0.05-1.0
`0.1
`10-60
`0.05
`
`0.5- 2.0
`0.1 - 1.0
`0.02-50b
`
`•values correspond to uncertainties at extremes of range; units are the same as measurement units .
`bfocludes contribution of scatter in detection system calibration (=±5%) used to obtain absolute in(cid:173)
`tensities.
`
`Sec. 3.3
`
`Treatment of Kinetic Data
`
`135
`
`Page 31 of 38
`
`

`

`

`

`M
`
`(3-32)
`
`(3-33)
`
`mathematical form to the actual experimental rate data. Since there are errors in the
`measurements and perhaps some inaccuracies in the model, it is often difficult to ob(cid:173)
`tain an exact fit. Consequently, it become