Micron et al. Ex.1018 p.1
`Micron et al. Ex.1018 p.1
`
`

`
`Library of Cangry.“ Cululugiug-in-Publication Dnla
`Slcinfcld. Jeffrey I.
`Chemical kinetics and dynamics/Jcffmy Sleinfrld, Joseph
`Fraticisco. William Hasc.
`p.
`cm.
`Bibliography.
`Includes index.
`ISBN 0-I3-129479-2
`2. Mulccular d
`I. Chemical reaction. Ralc of.
`1. Francisco, Joseph
`II. Hase. William.
`I
`QD502.S74
`I989
`38-21842
`54! .3'94--dcl9
`CIP
`
`nmics.
`. Tide.
`
`Editorial/production supervision: Karen Wingcllwonlcruftets
`Cover design: luel Mimick Design. Inc.
`Munufacuirlng buyer Paula Massenam
`
`63 I989 by Prexnice-Hull, Inc.
`A Division of Simtm & Schnslcr
`Englewoud Cliffs, New Jersey 07632
`
`All righls nuurved. No part of this haul: may be
`repmduccd, in any form ur by any means.
`without pemiission in willing fmm the publishet.
`
`Primed in the United States of Amcrica
`I0
`9
`B
`7
`6
`5
`4
`3
`2
`I
`
`ISBN U-1.3-lE"lH?‘|-E
`
`PR1-LN'l‘lClJ*lALL IN‘l‘EINA'l‘lUNAL (UK) Lmmzv. London
`Pxmrion-mu or-' AUSTRALIA Pn-. Lmmm. Sydncy
`PR!-.‘N'!1L‘B«HAu. CANADA
`Toronto
`PRENUCE-HAU. HISPANOAMERICANA. S.A.. Mexico
`PIEN'ncE.HALL or lnum PRIVATE LIMITED. Nm Delhi
`PIu='N'nCr.~HAu. or JAPAN. INC, Tr-kyn
`Smart 5: Scuus11—'.u ASIA Pn~.. Lm., Singapore
`liurruu PR1-.NTl('l£-HALL Do BRASIL. L‘l‘DA.. Rio de ./unchu
`
`Micron et al. Ex.1018 p.2
`
`

`
`
`
`
`Contents
`
`PIIEFJIGE
`
`
`II
`
`
`
`
`Chapter 1
`EASIC CONCEPT! OF KINETICS
`
`
`
`
`
`
`1
`
`
`
`
`
`
`
`1.1 Definition of the Rate of: Chemical Reaction. 1
`
`
`
`
`
`
`
`
`1.3 Order and Moleeularity of a Rclictian, 3
`
`
`
`
`
`
`1.3 Elementary Reaction Rate Laws. 6
`
`
`
`
`
`
`
`1.4 Deletrnirullion of Reaction Order: Reaction Hill!‘ Lives. I3
`
`
`
`
`
`
`
`
`
`1.5 Temperatme Dependence nfflate Constants: The Arrhenius Equation. 14
`
`
`
`
`
`
`
`
`
`
`1.6 Reaction Mechanisms. Molecular Dynamics. and the Road Ahead. I6
`
`
`
`
`
`
`
`
`
`
`References. 1'?
`
`
`Bibliography. I7
`
`
`Probtams. 18
`
`
`
`
`
`‘Chapter 2
`COMPLEX REIIIITIDPIS
`
`
`
`31
`
`
`
`2.1 Exact Analytic Solutions for Complex Reactions. 21
`
`
`
`
`
`
`
`2.2 Approximation Meihods. 38
`
`
`
`
`
`2.3 Example oi‘ as Cnntplex Reaction Mechanism The Hydrogen + Halogen Reaction. 41
`
`
`
`
`
`
`
`
`
`
`2.4 Laplace Transform Method. 43
`
`
`
`
`
`
`
`
`
`
`
`
`Micron et al. Ex.1018 p.3
`Micron et al. Ex.1018 p.3
`
`

`
`2.5 Derennintuit (Matrix) Methods. 54
`2.6 Numerical Methods. 50
`2.7 Stochastic Method, 67
`References, 73
`
`Bibliography. 74
`Appendix 2.| The Laplace Transform. 75
`Appendix 2.2 Numerical Algorithms for Differential Equations. 92
`Appendix 2.3 Stochastic Numerical Simulation of Chemical Reactions, 97
`Problems. I03
`
`Chapter 3
`KINETIC MEASUREMENTS
`
`108
`
`Introduction, I09
`3.l
`3.2 Techniques for Kinetic Measurements. III
`3.3 Treatment of Kinetic Data, I33
`References. I50
`
`Appendix 3.1
`Problems. I54
`
`Least Square Method in Matrix Form. I52
`
`Chapter 4
`REACTIONS IN SOLUTION
`
`‘I56
`
`4.] General Properties of Reactions in Solution. 156
`4.2 Phenomenological Theory of Reaction Rates. 157
`4.3 Diffusion-Limited Rate Constant. I61
`4.4 Slow Reactions. I63
`
`4.5 Effect of Ionic Strength on Reactions Between lens. 164
`4.6 Linear Free-Energy Relationships. I69
`4.7 Relaxation Methods for Fast Reactions. I71
`References. 174
`
`Bibliography. I75
`Problems. I75
`
`Chapter 5
`OATALYSIS
`
`1 78
`
`5.1 Catalysis and Equilibrium. I73
`5.2 Homogeneous Catalysis. 180
`5.3 Autocatalysis und Oscillating Reactions. I82
`5.4 Enzyme-Cntnlyzcd Reactions. NO
`5.5 Heterogeneous Catalysis and Gas-Surface Reactions. 194
`References. 198
`Problems, I99
`
`Vi
`
`Contents
`
`Micron et al. Ex.1018 p.4
`
`

`
`Chapter 8
`THE TRANSITION FROM THE MACROROOPIO
`TO THE MIDROSGOPIC LEVEL
`20!
`
`6.1 Rotation Between Cross Section and Rate Coefficient. 202
`
`6.2 Microscopic Reversibility and Detailed Balancing. ‘EDS
`6.3 The Microscopic-Macroscopic Connection. 206
`References, 208
`
`Chapter 7
`POTENTIAL ENERGY SURFACES
`
`209
`
`Long-Range Potentials. 2I0
`7.1
`Empirical lntennoleculnr Potentials. 213
`7.2
`7.3 Mulocular Bonding Potentials. 2l6
`7.4
`lnvemnl Coordinates and Normal Modes of Vibration. 217
`
`7.5
`1.6
`7.7
`7.8
`7.9
`
`Potential Energy Surihccs, 220
`Ah lnitio Calculation of Potential Energy Surfaces. 225
`Analytic Polcntial Energy Functions, 23!
`Experimental Determination of Potential Energy Surface Properties, 235
`Details of the Reaction Path, 236
`
`7.10 Potential Energy Surfaces of Electronically Excited Molecules. 27
`References, 240
`
`Bibliography. 242
`Problems. 244
`
`Chapter 8
`DYNAMICS OF BIMOLECULAII COLLISIONS
`
`248
`
`Simple Collision Models. 246
`8.l
`8.2 Two»Body Classical Scattering. 251
`8.3 Complex Scattering Processes. 261
`References. 276
`Problems. 277
`
`Chapter 9
`EXPERIMENTAL DETERMINATION OF NEIN
`KINETIC PARAMETERS
`282
`
`9.1 Molecular Beam Scattering. 282
`9.2 State-Resolved Spectroscopic Tochniqucs. 290
`9.3 An Example of Stan:-to-State Kinetics: The F + H; Reaction. 293
`9.4 Some General Principles Concerning Energy Disposition in Chemical Reactions, 297
`9.5 Detailed Balance Revisited. 299
`
`Contents
`
`Micron et al. Ex.1018 p.5
`
`

`
`9.6 Chemical Lasers. 301
`9.7 State-to-sum: Chemical Kinoncs can Be Huanious no your Henlthl. 301
`References. 302
`Problems. 303
`
`Chapter 10
`STATISTICAL APPROACH TO REACTION OVNAMIOS:
`TRANSITION STATE THEORY
`308
`
`10.1 Motion on the Potential Surfrwe, 308
`10.2 Basic Posuilntes and Standard Derivation of Transition State Theory. 311)
`10.3 Dynamical Derivntion of Transition Stntc Thoory. 315
`10.4 Quantum Mechanical Effects in Transition State Theory. 3111
`10.5 Thermodynamic Fwnmulntion of Ttansition State Theory, 321
`10.6 Applications of Transition State Theory. 323
`10.7 Micrucanonical 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 1 1
`UMIMOLEGIJLAII REACTION DYNAMICS
`
`342
`
`Formation of Energized Molecules. 344
`11.1
`Sum and Density of States. 347
`11.2
`Lindermmt-Hinshelwood Theory of Thennal Unimolecular Reactions. 352
`11.3
`Statistical Energy-Dependent Rate Constant HE). 357
`11.4
`RRK Theory. 358
`11.5
`RRKM Theory. 362
`11.6
`Application of RRKM Thecry to '1'het-ma! Activation. 368
`11.7
`11.8 Measurement of 11(5), 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 lntramolecular Motion and Unimolecular Decom-
`position. 382
`11.13 Mode Specificity. 385
`References. 389
`Blb1i08"3Ph)'. 392
`Problems, :93
`
`VIII
`
`Micron et al. Ex.1018 p.6
`
`

`
`
`
`Chapter" 1 E
`DYNAMICS BEYOND THE GAB PHASE
`
`
`
`
`
`
`40!
`
`
`
`12.!
`‘Transition Stale Theory of Solution Reactions, 403
`
`
`
`
`
`
`
`
`12,2 Kramerri‘ Theory and
`410
`
`
`
`
`
`
`l'.}us~SI.1rfaoe Reaction 415
`12.3
`
`
`
`
`
`References, 42?
`
`
`Bibliography. 428
`
`
`Pmhlqerlis, 429
`
`
`
`
`
`Chapter 13
`INFORMATION-THECIRETIEAL APPROACH TO
`
`
`
`STATE-TO-STATE DYNAMICS
`£31
`
`
`
`
`Introduction. 431
`I‘3.|
`
`
`
`
`13.2 The Mnximakflntropy Poslulnhe. 431
`
`
`
`
`
`13.3 Surprise] Analysis. and Synthesis: Product Slate Distribution in Exoll1ermic- Reactions. -438-
`
`
`
`
`
`
`
`
`
`
`
`
`I3.4 Informational-Theareiieal Analysis of Energy Transfer Process. 445
`
`
`
`
`
`
`
`
`I15 Surprise! Symhesis. 464
`
`
`
`
`l3.6 Conclusion. 470
`
`
`
`References. 470
`
`
`Bibliography". 472
`
`
`Problems. 473
`
`
`
`
`
`Chapter 14
`mmrsus as IIULTILEVEL numsm: svsreus
`
`
`
`
`
`
`
`an
`
`
`
`14.!
`Iutmduction. 4'34
`
`
`
`14.2 The Master Equation, 474
`
`
`
`
`
`Inforrraalionmaeurelicnl Treatment of the Master Equation, 479
`14.3
`
`
`
`
`
`
`
`14.4
`some Applications of Ma;sler—Equation Modeling, 433
`
`
`
`
`
`
`
`Ilefueiwes, 593
`
`
`Problems, 494
`
`
`
`
`
`
`
`Chaptor 15
`KINETIGS OF MIJLTIGOMPONENT SYSTEMS
`
`
`
`
`
`
`493
`
`
`
`I11 Atmospheric Chemistry. 496
`
`
`
`
`15.2 The Hydrogen-Oxygen Reaction. an Explosive Combustion Process. 509
`
`
`
`
`
`
`
`
`
`I5..'i The Methane Conlbuslinn Process. 516
`
`
`
`
`
`15.4 Conclusion. 52-1-
`
`
`
`References, 525
`
`
`Problen1s. 525
`
`
`
`
`
`Contents
`
`
`
`In
`
`
`
`Micron et al. Ex.1018 p.7
`Micron et al. Ex.1018 p.7
`
`

`
`Appendix 1
`QUANTUM STATISTICAL MECHANICS
`
`527
`
`Appendix 2
`CLASSICAL STATISTICAL MECHANICS
`
`Sum and Density of Slales. 528
`A2.|
`A12 Partition Function and Bollzmnn Distribution. 532
`
`Bibliography. 533
`Problems. 534
`
`Appendix 3
`DATA BABES IN CHEMICAL KINETICS
`
`Contents
`
`Micron et al. Ex.1018 p.8
`
`

`
`CHAPTER 1
`
`Basic: Concepts of Kinetics
`
`Among the most familiar characteristics of a material system is its capacity for chem-
`ical change. In achcrrristry lecture, the demonstrator mixes two clear liquids and oh»
`tains a colored solid precipitate. Living orynisms 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 ferntoseconds (10"’ sec) to geologic times 00’ years, or 10*“ 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-
`proach the equilibrium state. Chemical kinetics deals with changes in chemical prop-
`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 dyrumrics. 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 -
`sential in our technological society.
`We begin our study of chemical kinetics with definitions of the basic observ-
`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 RECTION
`
`Broadly speaking, chemical kinetics may be described as the study of chemical sys-
`tems whose composition changcs 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
`ofthelatteristhereactionofagasndsorbedonthesuriirceofasolid.
`
`Micron et al. Ex.1018 p.9
`
`

`
`The chemical change that takes place in any reaction may he represented by a
`sroicltiornerric equation such as
`
`aA + bB ——>
`
`cC + 11!)
`
`(1-1)
`
`where a and b denote the number of moles of reactants A and B that react to yield r.-
`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-
`gen and oxygen may be written as the balanced, irreversible chemical reaction
`
`2H; + 0; —-9 2H;O
`
`(1-2)
`
`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-
`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
`
`Hz+lz : 2H1
`
`(1-3)
`
`While each of equations l-2 and l—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 ream-
`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; intuitivdy, 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
`
`O + H: —+ OH + H
`
`while in the hydrogen-iodine reaction it is
`
`21 + H1 **? H11 ‘i’ 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 (I-hi); this is often the case with elementary reactions.
`The change in composition of the reaction mixture with time is the rate of re-
`action. R. For reaction l- I. the rate of consumption of reactants is
`
`1 dim
`1 d[A]
`"‘"ZT=‘?5‘E‘
`
`"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
`
`Basic concepts of Kinetics
`
`chap. 1
`
`Micron et al. Ex.1018 p.10
`
`

`
`(1-5)
`
`The [actors a. b. c, and din equations I-4 and 1-5 are referred to as the stoi-
`chiomelric coefilcients for the chemical entities taking part in the reaction. Since the
`concentrations of‘reactants and products are related by equation l——l. measurement
`of the rate of change of any one of the reactants or products would suffice to deter-
`mine the rate of reaction R. In tlte reaction 1-2. the rate of reaction would be
`
`—____=_fl)J= ldmlol
`“'9
`R‘
`2
`dt
`dt
`+2
`as
`A number of different units have been used for the reaction rate. The dimen-
`sionality of R is
`
`[amount of materiall[volume]"[tirne]"
`
`[concentration][time]"
`
`The standard Sl unit of concentration is moles per cubic decimeter. abbreviated
`mol din".
`In the older literature on kinetics, one frequently finds the units
`mol liter" for reactions in solution and moi cm" for gas phase reactions. The S1
`unit is. preferred, and should be used consistently. Multiplying moles cm" by
`Avogadro's Number (6.022 X lo”) gives the units molecules cm". which is still ex-
`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
`clmrnicnl kinetics.‘ We have attempted to follow the subcommittee's recommenda-
`tions in this text.
`
`1.2 ORDER AND MOLECULARITY OF A REACTION
`
`In virtually all chemical reactions that have been studied experimentally, the reac-
`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(lA]~lBl)
`
`(1-7)
`
`In some cases the reaction rate also depends on the concentration of one or more in-
`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-]; 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..
`
`R °‘ lAl"'I3|"
`
`(1-8)
`
`see. 1.2
`
`Order and Molecutarity of a Reaction
`
`Micron et al. Ex.1018 p.11
`
`

`
`The exponents m and :1 may be integer. fractional. or negative. This proportionality
`can be converted to an equation by inserting a proportionality constant k, thus:
`
`R = It [A]"'[l3]"
`
`(l -9)
`
`This equation is called a mic equation or rate expression. The exponent m is the or-
`der of the reaction with respect to reactant A. and n is the order with respect to re-
`actant B. The proportionality constant k is called the rate constant. The overall or«
`der of the reaction is simply p = m + n. A generalized expression for the rate of a
`reaction involving K components is
`
`A
`
`R = /(Herr
`1-!
`
`(1-10)
`
`The product is taken over the concentrations of each of the K components of the rc~
`action. The reaction order with respect to the i th component is n,. p = f2, m is the
`overall order of the reaction, and It is the rate constant.
`In equation 1-10. It must have the units
`
`[concentration|""""|tirne] '
`
`so for a second—ordcr reaction. l.e., m = n = I in equation l~8, the units would be
`[concentration]
`'[time] ' ', or dm" mol“ sec“ in SI units. Note that the units of liter
`mol" sec” are frequently encountered in the older solution-kinetics literature. and
`cm’ mol“ sec" or cm’ molecule“ sec ‘ are still encountered in the gas-kinetics lit-
`erature.
`
`Elementary reactions may be described by their molcculurily. which specifies
`the number of reactants that are involved in the reaction step. If a reactant sponta-
`neously decomposes to yield products in a single reaction step, given by the equation
`
`A ——->
`
`products
`
`(1-11)
`
`the reaction is termed unimalecular. An example of a unimolecular reaction is the
`dissociation of N;O4, represented by
`
`N20.
`
`‘—> 2N0;
`
`If two reactants A and B react with each other to give products, i.e.,
`
`A + B —> products
`
`(1-12)
`
`the reaction is termed bimolecular. An example of a bimolecular reaction would be
`a metathetical atom—transfer reaction such as
`
`0+-H7 —-—> OH+H
`
`F+Hz —> 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
`
`Micron et al. Ex.1018 p.12
`
`

`
`Micron et al. Ex.1018 p.1331.D.8101.xENtenor
`.mM
`
`

`
`by fitting the rate expression to experimental data. Such coetiicients are more prop
`erly termed mt: caefficients. rather than rate constants. The latter term should be re-
`served for the coefficicnts 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-
`mined by integrating the rate law for a particular rate expression.
`
`1.3.! Zorn-Order Reaction
`
`The rate law for a reaction that is zero order is
`
`R = -% = k[A]" = It
`:1:
`
`(1-19)
`
`Zero-order reactions are most often encountered in heterogeneous reactions on sur-
`faces (see chapter 5). The rate of reaction for this case is independent of the concen-
`tration of the reacting substance. To find the time behavior of the reaction. equation
`1-19 is put into the differential form
`
`d[A] = -kd!
`
`(1-20)
`
`and then integrated over the boundary limits 1, and 1;. Assuming that the concentra-
`tion of A at t. = 0 is Min. and at tz = tis [A].. equation 1-20 becomes
`ML
`t2-I
`
`Hence.
`
`I d[A] = -1:
`
`lflla
`
`9"‘?
`
`dt
`
`[AL * [Ala = -140 - 0)
`
`(I-21)
`
`(1-22)
`
`Consequently, the integrated form of the rate expression for the zero-order reaction
`is
`
`A plot of [A] versus time should yield a straight line with intercept [A]o and slope 1:.
`
`[A], = [Ala - kl
`
`(1-23)
`
`1.3.2 Fin:-Order Reaction:
`
`A tirstrorcler reaction is one in which the rate of reaction depends only on one reac-
`tant. For example. the isomerlzation of methyl isocyanide, Cl-I3NC, is a first-order
`unimolecular reaction:
`
`CHaNC —- CHaCN
`
`(1-24)
`
`Basic Concepts of Kinetics
`
`Chap. 1
`
`Micron et al. Ex.1018 p.14
`
`

`
`Micron et al. Ex.1018 p.15
`Micron et al. Ex.1018 p.15
`
`

`
`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 cortsmnl -r which is called the decay time of the reaction. This
`quantity is defined as the time required for the concentration to decrease to 1/3 of its
`initial value [A]o, where e =5 2.7183 is the base of the natural logarithm. The time 7
`is given by
`
`(I-33)
`
`1k
`
`'r =
`
`In experimental determinations of the rate constant k, the integrated form of the rate
`law is often written in decimal logarithms as
`
`log.o[A], = log.u[A]n -
`
`kl
`2.303
`
`(1-34)
`
`and a semilog plot of [A]. versus twill yield a straight line with I:/2.303 as slope and
`[Mo as intercept.
`
`1.3.3 Soconrbordor Reactions
`
`There are two cases of second-order kinetics. The lirst is a reaction between two
`
`identical species. viz..
`
`The rate expression for this case is
`
`A + A —-» products
`
`__ _ld[AJ =
`R ——
`2 —-dt
`
`,
`
`k[A]
`
`(I-35)
`
`(I-36)
`
`The second case is an overall second-order reaction between two unlike species.
`given by
`
`A + B —> products
`
`(I-37)
`
`In this case. the reaction is first order in each of the reactants A and B and the rate
`expression is
`
`JIM _
`= T - k[A][Bl
`
`(I-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 l—35 and 1-36. Although not
`an elementary reaction. the disproportionation of H1 (equation 1-3) is a reaction
`which is exactly second order in a single reactant. Another example is the recombi-
`nation of two identical radicals, such as two methyl radicals:
`
`ZCH: —+ C:Ho
`
`(I-39)
`
`We integrate the rate law. equation I-36, to obtain
`
`Basic Concepts of Kinetics
`
`Chap. 1
`
`Micron et al. Ex.1018 p.16
`
`

`
`
`
`II;
`in-ll, _
`" at‘:
`J?” [A]: - lit
`
`
`
`
`
`
`
`
`
`
`which gives
`
`
`
`1
`
`[AL
`
`
`I
`
`[A10
`
`
`
`
`Zkt‘
`
`
`
`
`
`(1-40)
`
`
`
`
`ll—4lJ
`
`
`
`
`
`
`
`
`
`
`
`
`A plot of the inverse concentration of A ([A]' '} versus time should yield a straight
`
`
`
`
`
`
`
`
`
`
`line with sttope equal to 2k and intercept lfifitlu.
`
`
`
`
`it is
`To integrate the rate law for the second case, equations I—3'.-' and l—v38,
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`convenient to define it progress variable x which measures the progress of the reac-
`tion to products as
`
`
`
`
`
`
`
`I = (lAlu — IAI.) = (min — IBIJ
`
`
`
`
`
`
`
`
`(I-42)
`
`
`
`
`
`
`
`
`
`
`
`where [Ala and IB].; are the initial concentrations. The rate expression given by
`
`
`
`
`
`
`
`
`
`
`
`
`equation 1-38 can then be rewritten in terms of I as
`
`
`
`
`
`
`
`
`
`i
`
`xiii}
`
`
`(EAL. ~ xl(|B]u ~ xi
`
`g = mA]., ~ x)([B]u — x_}
`
`
`
`
`To find the time behavior. we integrate equation l--43 thus:
`
`
`
`
`
`
`
`
`
`
`hill
`1
`dx
`
`T = k
`
`
`
`
`
`(1.43)
`
`
`
`i
`
`1_44
`
`
`
`’
`
`
`
`’
`
`
`
`d
`
`
`
`
`
`To solve the integral on the left—hand side of equation 1-44, we separate the vari-
`
`
`
`
`
`
`
`
`
`
`
`
`
`ables and use the rnethorl of partial fractions:
`
`
`
`
`
`
`
`
`
`
`I .___£=___
`
`_f
`‘
`
`(mt. - x)(|Biu — x)
`
`
`
`
`
`_
`
`
`
`(1-45)
`
`
`
`_
`dx
`dx
`
`
`(mt. ~ i|3]uii[A]u — xi
`uA1..- |B]t.Ji|Biu — xi
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`Solving the righthiand side of equation 1-45 and equating it to the left-hand side of
`equation I-44. we obtain, as the solution to the rate expression for the second case,
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`“ 46'
`(|A1n— IB1u)'"(1AioIBi.
`
`_
`IBMAI.) :
`I
`
`
`In this case the experimental data may be plotted in the form of the left-hand side of
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`the equation against I.
`
`
`
`
`
`
`
`
`
`
`
`’”
`
`
`
`
`
`
`
`
`
`1.3.4 Third-Order Reactions
`
`
`
`
`
`From the definition of overall reaction order in equation i- I0, we see that there are
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`three possible types of third—ortier reactions:
`(1') SA —-r products;
`ii!) EA + B
`
`
`
`
`
`
`
`
`
`
`
`
`-4- 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
`
`
`
`Micron et al. Ex.1018 p.17
`Micron et al. Ex.1018 p.17
`
`

`
`This rate law can be integrated readily to obtain the solution
`
`1
`
`1
`
`’i(iT1= ml
`
`1
`
`=
`
`3'“
`
`Rearranging gives
`
`l
`1
`——- = —-..
`{Ar
`[Ala " ‘W
`
`I-4
`
`9’
`
`‘
`
`A plot of the inverse squared concentration of A ([A]") with time should yield
`slope 6/: and intercept l/[A]3.
`The second case,
`
`which is second order in reactant A and first order in reactant B. has the overall or-
`der 3. The rate law for this reaction is
`
`2A+B ——> C
`
`(I-50)
`
`R *5
`
`2
`_ 1 d[Al _
`2-—-m k[A] [B]
`
`(l 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 ([8] }> [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 it -%i%]— = k'[A]‘
`so that the third—order expression reduces to a "pseudo second-or
`The solution for this case is equation l—41, i.e.,
`
`(I-52)
`" expression.
`
`1
`
`1
`[Ti ——
`
`,
`2k 1
`
`(I-53)
`
`A plot of IA)" vs. time, for a fixed [B], should then yield slope 21:’ and intercept
`I/lA]n; but note that the resulting rate coefficient is a function of the concentration
`of B. that is,
`
`k’ = us]
`
`(1-54)
`
`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
`three—body
`recombination
`process,
`such
`as
`I + I + M —’ 12 + M and
`0 +02 +M->0a+ M.lnthesecasesthe thirdbodyactstoremovetheexcess.
`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-
`
`10
`
`Basic Concepts of Kinetics
`
`Chap. 1
`
`Micron et al. Ex.1018 p.18
`
`

`
`nient to introduce a progress variable as we did in solving the second-order reaction
`case. Accordingly, we define a progress variable y by
`
`and
`
`[A]: = [Ala - 2)’
`
`[B]: = [Bio - y
`
`and with this condition we can rewrite equation l—5l in terms of y an
`
`iv
`dt
`
`= k(lAlo ~ 2y)’(lB]o ~ yl
`
`(I-55)
`
`Upon rearranging. we obtain
`
`___"Y___ =
`(min - 2y)*(tB1o — y)
`
`I: lit
`
`( l -56)
`
`This equation can be integrated by the method of partial fractions to yield
`
`um. — 21:31..) (moi mi.) J’ (mo — zine)‘ '“(iA|..ini.
`jlj L-_]_ __1_ [A]5m°) =
`
`"’
`
`" 57’
`_
`
`An example of such a reaction is the gas phase reaction between nitric oxide and
`oxygen
`
`2N0+ 0; —-> 2N0:
`
`(1-58)
`
`The third type of third—order reaction is first order in three different compo-
`nents. i.e..
`
`A + B + C ——>
`The rate law for this reaction is
`
`product
`
`R = %’:‘—' = klAllBllCl
`
`(1-59)
`
`(I-60)
`
`To solve for the integrated rate low 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 Ordnr
`
`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-
`eral solution for a reaction which is nth order in one reactant, for in equal to any in-
`teger or noninteger value. The rate expression for such a reaction is
`
`d[A]
`= 7 = k[A]"
`
`U-61)
`
`A simple integration of this expression yields the result
`
`Sec. 1.3
`
`Elementary Reaction Rate Lawn
`
`Micron et al. Ex.1018 p.19
`
`

`
`l
`
`I
`
`(n — 1) (iA1."”' - IAN") ‘ '“
`
`l
`
`which can be rewritten as
`
`1
`1
`[Air-' ‘EA? ‘ (" ‘ "'“
`
`‘M3’
`
`Equation 1-63 is valid for any value of n except n = l, in which case it is undefined
`and equation I-32 must be used instead. Figure 1-1 shows several plots of concen-
`tration vs. time for various values of n.
`
`E 5
`
`Time I
`
`Figure 1-]. Plot of [A], versus time for reaction of general order. The plot shows
`various functional behaviors for II = 2. 2.5. 3. 3.5. and 4. Concentration and time
`are in nrbitary 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 tirst- and second-order cases. When the
`order n is unknown. at van't Hoff plot can be constructed as an aid to deducing the
`order of the reaction. In a vsn’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 I-63 on log-log graph paper. The slope of such a plot gives the or-
`tier 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
`
`Micron et al. Ex.1018 p.20
`
`

`
`fl=5fl=4I’1=3 H =2
`
`Figure In-2. Van‘! Hull‘ plot of log
`orders.
`
`versus log [A] for various reaction
`
`108 [A]
`
`1.4 DETERMINATION OF REACTION ORDER: REACTION
`HALF-LIVES
`
`Thus far, we have introduced the concept of a rate law and have shown that experi-
`mental rate laws can frequently be written as the product of concentrations of react-
`ing species raised to some power; the exponents in such a rate law then define the
`order of the particular reaction. These reaction orders are empirically determined
`and may be nonintegtnl. We have also considered some simple rate laws and. by in-
`tegmting them, have shown how experimental data can be plotted to enable the reac-
`tion order to be determined. The vun't Hofi' method has been introduced as a
`method for determining a -general reaction order.
`An alternative to the van't Hull’ method, and one of the more popular methods
`for determining reaction order, is the half-life method. The reaction half-life rm is
`defined as the period of time necessary for the concentration of a specified reactant
`to reach one-half of its initial concentration- Measurement of mg as a function of
`initial reactant concentration can help establish the order with respect to that reac-
`tant. Consider the simple first-order reaction 1- I l. The integrated rate equation is
`given by equation 1-32a,
`
`1I1(lAL/{Mo} = -16
`
`Sec. 1.4
`
`Determination of Reaction Order: Reaction Half-Lives
`
`Micron et al. Ex.1018 p.21
`
`

`
`By definition. at I = 1. ,1. [A]. = lAJo/2; therefore. the rate equation can be rewrit-
`ten as
`
`The half-life is then
`
`_
`
`[Ala/2
`
`'“( mo ) = kt-/2
`
`(1-64)
`
`(1455)
`
`Thus. for a first-order reaction, ti/2 is independent of concentration. On the
`other hand, for a reaction of order n > I in a single reactant. the reaction half-life
`is
`
`30/:
`
`=_
`
`(2~-* — 1)
`km — xnA1a~'
`
`(1-55)
`
`where k is the rate constant and [A10 is the initial concentration of the reactant.
`Thus, the half-life for orders it > l is a function of the initial concentration of the
`reactant; consequently. a plot of the logarithm of up against the logarithm of [Mo
`should enable one to determine the reaction order. To illustrate this. let us take the
`logrrithm of both sides of equation 1-66. We obtain
`(I-67)
`log 11;; = lg- - (n — l)log[A]o
`it is clear front this expression that the- plot will be linear with a slope equal to
`n - 1. from which the order can be determined. win: this information. and one or
`more absolute values of I./;. the rate constant can also be calculated. The reader is
`reminded that this procedure is valid only for reactions which are nth order in a sin-
`gle reactant.
`Other methods of determining reaction orders are discussed in some earlier ki-
`netics textbooks, including those by Hamrnes’ and Benson‘.
`
`1.5 TEMPEIMTURE DEPENDENCE OF RATE CONSTANTS:
`THE ARRNENIUS EQUATION
`
`We have seen that rate expressions are often simple functions of reactant concentra-
`tions with a characteristic rate constant k. If the rate expression is correctly fonnu-
`lated. the rate constant should indeed be a constant—that is, it should not depend on
`the concentrations of species appearing in the rate law, or of any other species which
`may be present in the reaction mixture." The rate constant should also be indepen-
`dent of time. lt does, however, depend strongly on temperature. This behavior was
`described by Svunte Arrhenius in 1889’ on the basis of numerous experimental rate
`measurements. Anhenins found that rate constants varied as the negative exponen-
`tial of the reciprocal absolute tetnperature. that is,
`
`‘The rate constant for reactions in liquid sohuions may depend on pH. that is. H’ concentration;
`this will he discussed in chapter 4.
`
`14
`
`Basic Concepts of Kinetics
`
`Chap. 1
`
`Micron et al. Ex.1018 p.22
`
`

`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`