reaction kinetics and reactor design

Wewill ®nd that the normal dependence of reaction rate on temperature is one of themost intractable of nonlinearities in nature, providing at the same time many of thediculties and many

Reaction kinetics and reactor design / J.B Butt ± 2nd ed., rev and expanded.

p cm ± (Chemical industries ; v 79)

Includes index

ISBN 0±8247±7722±0 (alk paper)

1 Chemical kinetics 2 Chemical reactors I Title II Series

QD502 B87 1999

The ®rst edition was published by Prentice-Hall, Inc., 1980

This book is printed on acid-free paper

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Copyright ## 2000 by Marcel Dekker, Inc All Rights Reserved

Neither this book nor any part may be reproduced or transmitted in any form or by any means,electronic or mechanical, including photocopying, micro®lming, and recording, or by anyinformation storage and retrieval system, without permission in writing from the publisher.Current printing (last digit):

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A lot of things have happened in chemical reaction engineering since the ®rst edition

of this book, and this second version is very much di€erent from the ®rst Theremarks here are con®ned mostly to pointing out the changes

The ®rst edition was designed to combine a thorough description of the originand application of fundamental chemical kinetics all the way through to realisticreactor design problems This is continued, although the present e€ort increases theapplications to a much larger range of reactor design problems The discussion ofkinetics has been expanded in Chapter 1 to include material on chemical thermo-dynamics related to reaction systems and additional discussion of chain and poly-merization reactions New material on microbial and enzyme kinetics, and onadsorption±desorption theory, is given in Chapter 3 At the same time, some ofthe more deadly derivations arising from the kinetic theory of gases given in theoriginal Chapter 2 have been streamlined and the presentation simpli®ed in thepresent version Hopefully this material, which is very important to the philosophy

of this book, will now be more digestible

Chapters 4±6 retain the general organization of the ®rst edition, although thepresentation has been expanded and brought up to date After the ®rst part ofChapter 7 on transport e€ects in catalytic reactions, the present book goes its ownway A discussion of gas±solid noncatalytic reactions is given, and the development

of two-phase reactor theory based on plug ¯ow, mixing cell, and dispersion models(including transport e€ects and nonisothermality) rounds out the presentation ofthe chapter This then leads naturally into the detailed multiphase reactor design/analysis considerations of Chapter 8, which treats some speci®c types of multiphasereactors such as ¯uid beds and trickle beds This material is generally orientedtoward design considerations, but care has been taken to relate the presentation tothe pertinent theoretical developments given in earlier chapters

Finally, Chapter 9deals with unsteady-state ®xed-bed problems, includingcatalyst deactivation phenomena, adsorption, ion-exchange and chromatograto-graphic reactors This chapter probably has less to do with chemical reactions per

se than the others, but the foundations of the ®xed-bed analyses employed are ®rmlyestablished in the prior material, and it seems a waste not to make use of it here

iii

So much for the menu The ®rst edition was designed primarily as a text for anundergraduate course This edition, as one might expect from the above, goes con-siderably beyond that According to the desires of the instructor, portions can beused for undergraduate courses, other parts for graduate courses, or special topicsselected for seminar or discussion purposes.

There are a number of major changes, in addition to theextent of coverage, to be found here A major new feature is theinclusion of a large number of worked-out illustrations in thetext as the narrative continues These can be fairly comprehen-sive and are not always easy; hopefully they are indicative ofwhat is to be expected in the ``Exercises'' at the end of eachchapter We also have a house pet mouse, ``Horatio,'' whoappears at the end of almost all of the illustrations, askingfurther questions regarding the illustrationÐessentially givinganother set of exercises at the choice of the instructor They aree€ective in providing additional experience with speci®c topics

as desired Horatio also appears in several locations asking thereader to clarify for himself various points in the text Thanks

to Mr David Wright for permission to use this ®gure.Many references are given in the text, as we go along, to pertinent backgroundmaterial It has been an objective here to go back to sources that presented the early,developmental work on a given topic, and to supplement those references with laterones presenting signi®cant advances A consequence of this is that one should not besurprised to see references from the 1950s, or even much earlier, from time to time.Another procedure that has been followed is to use the notation as generallyemployed in the source material to make it easier in reading those references As aresult, there is no set of standard notation in the book; however, all the notation used

in each chapter is summarized as completely as possible, alphabetically, at the end ofthat chapter

Computer literacy is assumed, and there are many problems that require puter solution, particularly as one becomes involved with nonisothermal reactors,boundary-value dispersion problems, the more advanced ®xed-bed problems, andinterpretation of kinetic data We have not tried to get into the software businesshere, in view of the continuing rapid evolution of various aspects of that ®eld Wehave yielded to the temptation in a couple of instances to suggest, in outline, somealgorithms for speci®c problems, but in general this is left up to the reader.The indebtedness to colleagues, teachers, and students mentioned in thepreface to the ®rst edition remains unchanged In addition, I would like to thankProfessor James J Carberry for his comments on the preparation of this secondedition

com-John B Butt

All I know is just what I read in the papers

Ð Will Rogers

It is probably obvious even to the beginning student that much of chemical ing is centered on problems involving chemical transformation, that is, chemicalreaction It is probably not so obvious, at least in the beginning, that the rate atwhich such transformations occur is the determining factor in a great number of theprocesses that have been developed over the years to produce that vast array ofgoods that we consider an integral part of contemporary life The study, analysis,and interpretation of the rates of chemical reactions is, itself, a legitimate ®eld ofendeavor It ranges in scope from those problems concerned with the fundamentals

engineer-of detailed mechanisms engineer-of chemical transformation and the associated rates to blems that arise during the development and implementation of procedures forchemical reactor and process design on a large scale If we must give names tothese two extremes, we might call the ®rst ``chemical kinetics,'' and the second

pro-``chemical reaction engineering.''

In the following we shall range from one limit to the other, although ourprimary objective is the understanding of kinetic principles and their application

to engineering problems What will we ®nd? For one thing, we will ®nd that chemicalreactions are not simple things; those ®ne, balanced equations which everyone hasused in solving stoichiometry problems ordinarily represent only the sum of manyindividual steps We will ®nd that the rates of chemical transformation, particularly

in engineering application, are often a€ected by rates of other processes, such as thetransport of heat or mass, and cannot be isolated from the physical environment Wewill ®nd that the normal dependence of reaction rate on temperature is one of themost intractable of nonlinearities in nature, providing at the same time many of thediculties and many of the challenges in the analysis of chemical rates We will ®ndthat often it is not the absolute rate of a single reaction but the relative rates of two

or more reactions that will be important in determining a design We will ®nd thatspace as well as time plays an important role in reaction engineering, and in thetreatment of such problems it will be necessary to develop some facility in the use ofrational mathematical models Finally, we will ®nd that the artful compromise is asimportant, if not more so, in our applications of reaction kinetics as it is in all theother areas of chemical engineering practice

v

This may seem like a very short list of what is to be found if the topic is truly asimportant as we have indicated It is intentionally short, because the essence shouldnot be submerged in detail quite so soon, or, to paraphrase Thomas aÁ Kempis, it

is better not to speak a word at all than to speak more words than we should.The material of this text is intended primarily to provide instruction at theundergraduate level in both chemical kinetics and reactor design Of particular con-cern has been the detailing of reaction kinetics beyond phenomonological descrip-tion The rationale for the Arrhenius equation was a personal mystery to the author

in earlier years, who hopes an appropriate solution is revealed in Chapter 2.Numerous other aspects of classical theories of chemical kinetics are assembled

in Chapter 2 and Chapter 3 to give some perception of the origin of gical rate laws and an understanding of the di€ering types of elementary reactionsteps In Chapter 4 we swap the beret of the theoretician for the hard hat of theengineer, in pursuit of means for developing rational chemical reactor design andanalysis models A parallelism between mixing models and reactor models has beenmaintained in order to demonstrate clearly how reaction kinetic laws ®t into reactordesign Chapters 4 through 6 are based on homogeneous models, and proceed fromstandard plug ¯ow and stirred tank analysis to description of nonideal behavior viadispersion, segregated ¯ow, mixing cell and combined model approaches.Phenomena associated with reaction in more than one phase are treated inChapter 7 but no attempt is made to develop multiphase reactor models The factthat reaction selectivity as well as reaction rate is an important and often determiningfactor in chemical reaction or reactor analysis is kept before the eyes of the readerthroughout the text

phenomonolo-The exercises are an intentionally well-mixed bag phenomonolo-They range from simpleapplications of equations and concepts developed in the text to relatively open-ended situations that may require arbitrary judgement and, in some instances,have no unique answer The units employed are equally well-mixed Historically,multiple systems of measure have been a curse of the engineering profession and such

is the case here particularly, where we range from the scienti®c purity of Planck'sconstant to the ultimate practicality of a barrel of oil The SI system will eventuallyprovide standardization, it is to be hoped, but this is not a short-term proposition.Because both author and reader must continue to cope with diverse sets of units, noattempt at standardization has been made here

Symbols are listed in alphabetical order by the section of the chapter in whichthey appear Only symbols which have not been previously listed or which are used

in a di€erent sense from previous listings are included for each section Symbols used

in equations for simpli®cation of the form are generally de®ned immediately after and are not listed in the Notation section found at the end of each chapter.Each chapter is divided into more or less self-contained modules dealing with auni®ed concept or a group of related concepts Similarly, the exercises and notationare keyed to the individual modules, so that a variety of possibilities exist for pursuit

there-of the material presented

Acknowledgment must be made to teachers and colleagues who, over the years,have had in¯uence in what is to be found in this text I am grateful to the late Charles

E Littlejohn and R Harding Bliss, to Professors C A Walker, H M Hulburt, and

R L Burwell, Jr., and especially to Professor C O Bennett, who o€ered manyconstructive and undoubtedly kind comments during preparation of this manuscript

Thanks also to R Mendelsohn, D Casleberry and J Pherson for typing varioussections of the manuscript, and to the Northwestern chemical engineeringstudents for detecting unworkable problems, inconsistent equations, and all theother gremlins waiting to smite the unwary author.

John B Butt

Preface to the Second Edition iii

ix

3.3 Surface Reactions and Nonideal Surfaces 194

4.7 Other Fixed Beds and Other Waves: Ion Exchange and Adsorption 308

5.2 Modeling of Nonideal Flow or Mixing E€ects on Reactor

1.1 Mass Conservation and Chemical Reaction

Certainly the most fundamental of laws governing the chemical transformations andseparations with which chemical reaction engineering is involved is that of conserva-tion of mass Although this is surely not new to the readers of this text, it is worth thetime here to revisit a simple example to make clear what speci®c functions in atypical mass balance might arise as a result of chemical reaction The examplemay seem very elementary, but it is important that we all start at the same point.Consider then the steady-state separation process depicted in Figure 1.1 A stream,

L, mass/time, containing two components, A and B, is fed to the process, whichdivides it into two product streams, V and W, mass/time, also containing compo-nents A and B The mass fractions of components A and B in L, V, and W are given

as xA, xB, yA, yB, and zA, zB, respectively Mass is conserved in this separation; wemay express this mathematically with the following simple relations:

Now, since we also know that each stream consists of the sum of its parts, then

This, in turn, means that only two of the three mass balance relationships (1-1)

to (1-3) are independent and can be used to express the law of conservation of massfor the separation We are then left with a system of ®ve equations and nine potentialunknowns such that if any four are speci®ed, the remaining ®ve may be determined

Of course, all we have done is to say:

Total mass in=time ˆ total mass out=time

Mass A in=time ˆ mass A out=time

Mass B in=time ˆ mass B out=time

Since the uniform time dimension divides out of each term of these equations, ourresult is the direct mass conservation law

Now let us consider a slightly di€erent situation in which the process involved

is not a separation but a chemical transformation In fact, we shall simplify thesituation to a single input and output stream as in Figure 1.2 with the feed streamconsisting of component A alone However, within the process a chemical reactionoccurs in which B is formed by the reaction A ! B If the reaction is completedwithin the process, all the A reacts to form B and mass conservation requires that themass of B produced equal the mass of A reacted The material balance is trivial:

What happens, though, if not all of the A reacts to form B in the process?Then, obviously, the mass of B leaving is not equal to the mass of A entering, butrather

Comparison of the two processes illustrates in a simple but direct way the generalconcerns of this whole text These are

1 To determine zA and zB given a certain type and size of reaction process

2 To determine the type and size of reaction process needed to produce aspeci®ed zA and zB

Two factors enter into this problem The ®rst is the stoichiometry of the action transforming A to B Chemical equations as normally written express the

re-Figure 1.2 Chemical reaction process

relationship between molal quantities of reactants and products, and it is necessary totransform these to the mass relationship in problems of mass conservation involvingchemical reaction The details of this are clear from the familiar combustion/massbalance problems, which seem so vexing the ®rst time they are encountered In oursimple example reaction the stoichiometric relationship is 1 : 1, so mass conservationrequires the molecular weight of B to equal that of A, and thus the mass of B producedequals the mass of A reacted This particular type of transformation is called anisomerization reaction and is common and important in industrial applications.The second factor is the rate at which A reacts to form B Consider the problem

in which we wish to determine zAand zB, given the type and size of process used inFigure 1.2 For the mass balance on component A, we want to write

and for B (recalling that there is no B in the feed)

Mass B out=time ˆ mass B formed=time ˆ mass A reacted=time …1-10†The last two terms of equation (1-10) incorporate the information concerning thestoichiometric relationship involved in the chemical reaction, since we have alreadyseen that the mass relationship as well as the molal relationship in this particularexample is 1 : 1 We can also paraphrase the statement of equation (1-10) to say thatthe rate at which B passes out of the system is equal to the rate at which it is formed,which is also equal to the rate at which A reacts Thus, the rate of reaction is closelyinvolved in this mass balance relationshipÐexactly how is what we are to learnÐsofor the moment our simple example must remain unsolved

The most pressing matter at this point is to de®ne what we mean by rate ofreaction This is by no means trivial The de®nition of a reference volume for the ratesometimes gives trouble here It should be clear that the magnitude of the ratesinvolved in equation (1-10) depend on the magnitude of the process, so in principle

we could have an in®nite number of processes, as in Figure 1.2, each of di€erent sizeand each with a di€erent rate Clearly this is an undesirable way to go about things,but if we de®ne a reaction rate with respect to a unit volume of reaction mixture, thesize dependency will be removed Also, we should remember that the rate of reactionmust be speci®ed with respect to a particular component, reactant, or product Aslong as the ratio of stoichiometric coecients is unity, this presents no problem;however, if our example reaction were A ! 2B, then the rate (in mols/time) of Awould always be one-half that of B, regardless of the reference volume employed.Finally, we must de®ne the rate of reaction so that it will re¯ect the in¯uence of statevariables such as composition and temperature but will not be dependent on theparticular process or reactor in which the reaction takes place In accord with this,

where i is reactant or product, V the volume of the reaction mixture and M and N aremass and mols, respectively The rate is positive for change in product and negativefor change in reactant Most of our applications will deal with the use of molalquantities and, for the convenience of working with positive numbers, the rate ofreaction of a reactant species is often encountered as … ri†.

There are a number of traps involved in even this straightforward de®nition Theuse of the reference volume as that of the reaction mixture is necessary to account forthe fact that in some cases the total volume will change in proportion to the molalbalance between reactants and products If the reaction were A ! 2B and involvedideal gases with no change in temperature or total pressure, the volume of product atthe completion of the reaction would be twice that of initial reactant To conserve ourde®nition of reaction rate independent of the size of the system, the volume changemust be accounted for This is not dicult to do, as will be shown a little later Theproblem is that one often sees reaction rates written in the following form:

where concentration Ci(mols/volume) is used in the rate de®nition Equation (1-12)

is in some sense embedded in the history of studies of chemical rates of reaction anddates from the very early physicochemical studies These were mostly carried out inconstant-volume batch reactors, for which, as we shall shortly see, equation (1-12) isvalid If one wishes to use concentrations (normally the case) but still conform to thede®nitions of equations (1-11), then

where Niˆ CiV The problem with the use of equation (1-12) as a de®nition of rate

is that it envisions no change in the reference volume as the reaction proceeds.Equations (1-11) and (1-13) resolve that particular diculty, but we are still leftwith the fact that the proper formulation of a rate de®nition depends on the experi-mental system used to measure the rate, so there is really no single de®nition that isboth convenient and universal [for further reading on the matter see A.E Cassano,Chem Eng Educ., 14, Winter Issue (1980)]

Related to the question of how to de®ne the rate of chemical reaction is the use ofthe time derivative, dMi=dt or dNi=dt This implies that things are changing with timeand may tempt one to associate the appearance of reaction rate terms in massconservation equations with unsteady-state processes This is not necessarily true; ingeneral, one must make a distinction between the ongoing time of operation of someprocess (i.e., that measured by an observer) and individual phenomena such aschemical reactions that occur at a steady rate in an operation that does not vary withtime (steady-state) Later we will encounter both steady- and unsteady-state types ofprocesses involving chemical reaction, to be sure, but this depends on the process itself;the reaction rate de®nition has been made without regard to a particular process.Returning for a moment to the mass conservation equation, equation (1-9),

we may now restate it in terms of a working de®nition of the rate of reaction.For a speci®ed volume element,  of reaction mixture within the process andunder steady conditions:

…mass A into =time† ˆ …mass A out of =time ‡ mass A reacted in =time†

…1-14†

as shown in Figure 1.3(a) The reaction term in equation (1-14) now conforms to thereaction rate de®nition; the left side of the equality is the input term and the right sidethe output It is understood that A refers to a reaction or product species, not anelement.

In the event of unsteady-state operation, the means of incorporating the rateexpression in the mass balance are still the same, but the balance becomes

…Mass A into =time† …mass A out of =time† ‡ …mass A reacted in =time†

The input/output terms remain the same, but an accumulation term on the right siderepresents the time dependence of the mass balance

From equation (1-15) we can immediately derive expressions relating the tion rate to the type of process for two limiting cases First, if the volume element does not change as the reaction proceeds, and there is no ¯ow of reactant into or out

reac-of , equation (1-15) becomes (still in mass units)

The left-hand side is the de®nition of the reaction rate of A, rA0, and the right-handside (recall that  does not change) is given by the rate of change of concentration of

A, so

r0

This is the equation for a batch reactor, a constant volume of reaction mixture, and

we see that it corresponds to the rate de®nition given in equation (1-12) A similarexpression, of course, would apply for molal units

(a)

(b)Figure 1.3 (a) Mass balance with chemical reaction on a volume element of reactionmixture (b) Traveling batch reactor

In the second case we also assume that the volume element  does not changeand that there is no ¯ow of A into or out of the element However,  is now moving,with other like elements, in some environment such that the distance traversed from

a reference point is a measure of the time the reaction has been occurring, as shown

in Figure 1.3b Now, for this little traveling batch reactor we may rewrite the timederivative in terms of length and linear velocity For a constant velocity u, as thevolume element moves through the length dz

dt ˆ dz=u

and equation (1-13) becomes

This is the equation for a plug ¯ow reactor, or at least one form of it Again, though,

it is necessary to be careful The time derivative above is based on constant velocity,which would not be the case, for example, in a gas reaction with di€ering netstoichiometric coecients for reactants and products Further, equation (1-16) aswritten employs a mass average velocity; if the rate de®nition is in molal units, thenthe molal average velocity must be used

The batch reactor and plug ¯ow reactor are two di€erent types of reactors thatare extremely important in both the analysis and implementation of chemical reac-tion processes Much of the substance of what has been discussed above will berestated from a somewhat di€erent perspective in Chapter 4

1.2 Reaction Rate Equations: The Mass Action Law

Although a proper de®nition of the rate of reaction is necessary, we cannot do muchwith it until we ®nd how the rate depends on the variables of the system such astemperature, total pressure, and composition In general terms, we must set the ratede®nition equal to a mathematical expression that correlates properly the e€ects ofsuch variables That is,

r ˆ …Ci; Cj; ; T; P†

where  depends on the complexity of the reacting system For the present we shall

do this on the basis of observation and experience; good theoretical reasons for theresults will be given in Chapter 2

Again let us workthrough the example of a speci®c (but hypothetical) system,this time the irreversible reaction

disappearance of A, and use the de®nition of equation (1-12):

of experimental observation, including negative values

If the reaction is reversible, A, B, C, can be formed from the products

L, M, and a corresponding form of the law of mass action applies For thiscase equation (1-18) becomes

kf…CA†p…CB†q…CC†r‡ kr…CL†u…CM†wˆ rA …1-20†Here u and w are the orders of the reverse reaction with respect to L and M, kf theforward rate constant, and krthe reverse rate constant The same comments as givenearlier pertain to the relationship between the apparent orders and the stoichiometriccoecients l and m

The rate constants here have been de®ned in molal terms and we will followthis convention for the time being, although there is no reason that mass units couldnot be used, following equation (1-11) The rate expression for the appearance ordisappearance of a given component, however, implies a corresponding de®nition forthe rate constant In the preceding section we pointed out the general role of stoi-chiometric coecients in determining relative rates In more speci®c terms here, k inequation (1-18) de®nes the mols of A reacted per time per volume; whether the samenumber for k also gives mols of B reacted per time per volume depends on therelative values of the stoichiometric coecients a and b; yes only if a ˆ b Rewritethe reaction as

A ‡ …b=a†B ‡ …c=a†C


! …l=a†L ‡ …m=a†M ‡

Clearly …b=a† mols of B react for each mol of A, and so on The rate of reaction ofany constituent i is then related to the rate of reaction of A by

and the net rate the algebraic sum of the two:

or elementary steps that for one reason or another are not directly observable Insuch cases observation of the stationary condition expressed by equation (1-22)involves rate constants kf and kr that are combinations of the constants associatedwith elementary steps Further discussion of this and the relationship between ratesand thermodynamic equilibria is given later in this chapter

Mass action forms of rate correlation, often referred to as power law tions, are widely applied, particularly for reactions in homogeneous phases Table 1.1gives a representative selection of correlations for various reactions It is seen inseveral of the examples that the reaction orders are not those to be expected onthe basis of the stoichiometric coecients Since these orders are normally estab-lished on the basis of experimental observation, we may consider them ``correct'' asfar as the outside world is concerned, and the fact that they do not correspond tostoichiometry is a sure indication that the reaction is not proceeding the way we havewritten it on paper Thus we will maintain a further distinction between the elemen-tary steps of a reaction and the overall reaction under consideration The directapplication of the law of mass action where the orders and the stoichiometric coe-cients correspond will normally pertain only to the elementary steps of a reaction, aswill the dependence of rate on temperature to be discussed in the next section Also,Table 1.1 Mass Action Law Rate Equations (II)

the theoretical background to be given in Chapter 2 concerning pressure, tion, and temperature dependence of rate pertains to these elementary steps Finally,for the forward and reverse steps of a reaction that is an elementary step, theprinciple of microscopic reversibility applies This states that the reaction pathwaymost probable in the forward direction is most probable in the reverse direction aswell In terms of energy, one may say that the forward and reverse reactions of theelementary step are confronted with the same energy barrier.

concentra-1.3 Temperature Dependence of Reaction Rate

Mass action law rate equations are sometimes referred to as separable forms becausethey can be written as the product of two factors, one dependent on temperature andthe other not This can be illustrated by writing equation (1-18) as

where the rate constant k…T† is indicated to be a function of temperature and theconcentration terms, aside from possible gas law dependencies, are independent ofthat variable The possible dependence of p, q, and r on temperature is small andnormally arises from factors associated with the ®tting of rate forms to kineticdataÐnot the problem we are concerned with here

The temperature dependence of k associated with the rate of an elementary step

is almost universally given by the awkward exponential form called the Arrheniusequation:

where E is the activation energy of the reaction, T the absolute temperature, andk8 the preexponential factor In equation (1-25) k8 is written as independent oftemperature; in fact, this may not be so, but the dependence is weakand the expo-nential term is by far the predominant one Figure 1.4 illustrates the general form ofdependence of reaction-rate constants obeying the Arrhenius law Note that over avery wide range of temperatures this relation is sigmoid The commonly expectedexponential dependence of the rate constant on temperature is found only in a

Figure 1.4 Reaction-rate and rate-constant dependence on temperature according to theArrhenius law

certain regionÐdepending on the value of the activation energyÐand at very hightemperature the response can be even less than linear.

Illustration 1.1

The statement is sometimes made that the velocity of a reaction doubles for each108C rise in temperature If this were true for the temperatures 298 K and 308 K,what would be the activation energy of the reaction? Repeat for 373 and 383 K Bywhat factor will the rate constant be increased between 298 and 308 K if the activa-tion energy is 40,000 cal/gmol?

equation (1-25) Taking logarithms of both sides:

Thus, a plot of the logarithm of the rate constant versus the reciprocal of thetemperature is linear with slope …E=R† and intercept ln…k8†, as shown in Figure1.5 In some cases the linear correlation may not be obtained This may be due toseveral factors, the most frequent of which are:

1 The mechanism of the reaction changes over the temperature rangestudied

2 The form of rate expression employed does not correspond to the reactionoccurring (i.e., some composite rate is being correlated)

3 Other rate processes, such as mass di€usion, are suciently slow topartially obscure the reaction rates

4 The temperature dependence of k8 becomes important (see problem 10)

In general, the Arrhenius correlation is probably one of the most reliable to be found

in the kinetic repertoire if properly applied There are sound theoretical reasons forthis, as will be seen in Chapter 2 A problem of particular interest in chemicalreaction engineering is that listed as factor 3 above; this shall be treated inChapter 7

Stated from a somewhat di€erent point of view, the Arrhenius equation serves

to de®ne the activation energy E:

The form of equation (1-26) is reminiscent of the van't Ho€ relationship for thetemperature dependence of the equilibrium constant:

where
H8 is the standard state heat of reaction Consider the reaction

A ‡ B $ L ‡ M, which is an elementary step Then the ratio of forward to reverse

Figure 1.5 Form of Arrhenius plot for determination of activation energy

rate constants does give the equilibrium constant K, and we may write

1.4 Rate Laws and Integrated Forms for Elementary Steps

1.4.1 Individual and Overall Reactions

We have made a distinction between an overall reaction and its elementary steps indiscussing the law of mass action and the Arrhenius equation Similarly, the basickinetic laws treated in this section can be thought of as applying primarily to ele-mentary steps What relationships exist between these elementary steps and theoverall reaction? In Table 1.1 we gave as illustrations the rate laws that have beenestablished on the basis of experimental observations for several typical reactions Aclose look, for example, at the ammonia synthesis result is enough to convince onethat there may be real diculties with mass action law correlations This situationcan extend even to those cases in which there is apparent agreement with the massaction correlation but other factors, such as unreasonable values of the activationenergy, appear Let us consider another example from Table 1.1, the decomposition

A little chemical intuition is helpful here; let us start by writing a simple sional structural formula for the ether:

HjCjHÐH

and askourselves what must happen for the reaction to occur as in (III) Sinceethane is a major product of the reaction, then a hydrogen would have to be shiftedfrom one side of the molecule to the other at the same time that a C±O bond is beingbroken Thus the single-step reaction of (III) implies a concerted action involvingthree bonds: breaking of one C±O and one C±H and formation of another C±H Forthe suspicious among us it is not necessary to delve more deeply into the chemistrythan this A large number of events have to happen at essentially the same time(implying a substantial activation energy for the reaction, at least) so the question iswhether it is not possible that some more chemically plausible, and energeticallyfavorable, pathway might exist Indeed it is the actual sequence of the reaction is

C2H5OC2H5 ! ECH3‡ ECH2OC2H5

ECH3‡ C2H5OC3H5 ! C2H6‡ ECH3‡ ECH2OC2H5

ECH2OC2H5 ! ECH3‡ CH3CHO

The ether decomposition is an example of a chain reaction in which elementarysteps such as (IVb) and (IVc), involving the active intermediates in the chain, pro-duce the ®nal products There are formal methods for treating the kinetics of suchchain reactions that we shall encounter later; the important point here is to note therelationship between the overall reaction and the elementary steps and to note thateven though the overall kinetics are apparently in accord with those for an elemen-tary step, reaction (III) is not one

As a second example consider a heterogeneous reactionÐone occurring in twophasesÐrather than the homogeneous cases we have been discussing so far Thewater gas reaction is a well-known example of one whose rate is in¯uenced by thepresence of a solid catalyst:

where the two phases involved are the gaseous reaction mixture and the solid lytic surface It is more apparent for (V) than for (III) that the overall reaction mustconsist of some sequence of elementary steps, since the scheme of (V) in no way

cata-accounts for the in¯uence of the catalyst on the reaction If we let S represent somechemically active site on the catalytic surface, then one could envision writing (V) as

H2O ‡ S ! H2‡ ‰SOŠ

where [SO] is an oxide complex on the surface and plays the role of the intermediate

in this reaction sequence

When we compare (IV) and (VI), it is clear that much more detail concerningelementary steps is given in the former The two reactions of (VI) provide a closerdescription of the water gas catalysis than does (V) but are not necessarily themselvesthe elementary steps This is important in the applications of kinetics that we areconcerned with, since frequently it is necessary to workwith only partial informationand knowledge of complicated reactions The two-step sequence of (VI) is moredesirable than (V), since it provides a means for incorporating the catalytic surfaceinto the reaction scheme even though the two steps involved may not be elementary

We may summarize this by saying that the two-step sequence provides for anessential feature of the reaction, the involvement of the surface, which is absent in theoverall reaction (V) Applied kinetics often requires the modeling of complicatedreactions in terms of individual steps that incorporate the essential features of theoverall reaction As one develops more detailed information concerning a reaction, itwill be possible to write in more depth the sequences of reaction steps involved,approaching (ideally) the actual elementary steps

Boudart (M Boudart, Kinetics of Chemical Processes, Prentice-Hall,Englewood Cli€s, N.J., 1968) has given a convenient means of classi®cation ofreaction sequences such as (IV) and (VI) The intermediates, such as ECH3, S, and

SO, are active centers, since the reaction proceeds via steps involving the reactivity ofthese species In some cases, such as (IV) and (VI), the active centers are reacted inone step and regenerated in another ‰ECH3 and ECH2OC2H5 in (IV), S and SO in(VI)], so that a large number of product molecules can be produced through theaction of a single active center This is termed a closed sequence In the case wherethis utilization and regeneration of active centers does not occur, the sequence istermed an open one and a single active center is associated with a single product-producing step The gas-phase decomposition of ozone is an example of such anopen sequence:

where the individual steps are

O3! O2‡ O

and oxygen atoms are the active centers for the reaction

We shall ultimately be concerned with methods for establishing the rate lawsfor overall reactions such as (IV), (VI), and (VII) from a knowledge of (or specula-tion as to) the elementary steps To do this we must certainly develop some facilitywith the simple rate laws that elementary reactions might be expected to obey This isthe topic of the following section

1.4.2 Rates and Conversions for Simple Reactions

In this section we must be careful to respect our prior concern about the de®nition ofrate with regard to the volume of reaction mixture involved Further, since we wish

to concentrate attention on the kinetics, we shall study systems in which the servation equation contains the reaction term alone, which is the batch reactor ofequation (1-12) It is convenient to view this type of reactor in a more general sense

con-as one in which all elements of the reaction mixture have been in the reactor for thesame length of time That is, all elements have the same age Since the reactions weare considering here occur in a single phase, the relationships presented below per-tain particularly to homogeneous batch reactions, and the systems are isothermal.The simplest of these is the class in which the reactions are irreversible and ofconstant volume The most important here are zero, ®rst, and second order withrespect to the reactant(s), respectively

Zero Order It may seem somewhat at odds with the law of mass action to talkabout rates that are independent of the concentration of reactant, but apparentzero-order reactions do occur, particularly in the description of the overall kinetics

of some closed reaction sequences We consider the model reaction,

First Order The ®rst-order case can be represented by the same model reaction,

A ! B, with the rate law

When the reactants A and B are initially present in stoichiometric proportion, that is

CA0 ˆ CB0, equation (1-42) becomes indeterminate The result, however, is given byequation (1-38), since the reaction stoichiometry requires that CAˆ CB for

CA0 ˆ CB0

Nonintegral Order A nonintegral-order rate equation, such as that for dehyde decomposition [the ®rst reaction of Table 1.1], does not ®t into the pat-tern expected for the rates of true elementary steps, but it is convenient toconsider it here in succession with the other simple-order rate laws Here wehave, for example A ! B, where the reaction is n-order with respect to A:

The integrated result for CA is

Half-Life, Conversion, and Volume Change Forms At this point we are going tobreakinto the narrative of rate forms to discuss three particular aspects of kineticformulations that pertain to most of the situations we discuss, at least for simplereactions.

The ®rst of these are the concepts of relaxation time and reaction half-life.The rate equations we have written incorporate the constant k to express theproportionality between the rate and the state variables of the system, and, as such, it

is some measure of a characteristic time constant for the reaction For ®rst-orderreactions in particular, a convenient association may be made owing to the logarith-mic time dependence of reactant concentration If we rewrite equation (1-35) inexponential form, we see that the e-folding time of …CA=CA0†, that is, the time atwhich kt ˆ 1 and where …CA=CA0† has decreased to 36.8% of its original value,occurs for t ˆ 1=k This value of time is referred to, for ®rst-order processes, as therelaxation time  and gives directly the inverse of the rate constant Since rate lawsother than ®rst order do not obey exponential relations, this interpretation of relaxa-tion time is strictly correct for ®rst-order reactions; however, a related concept, that

of reaction half-life, can be used more generally As indicated by the name, reactionhalf-life speci®es the time required for reaction of half the original reactant This isalso a function of the rate constant and is obtained directly from the integratedequations where CAˆ …1=2†CA0, and t ˆ t1=2 For the cases of the previous section:Zero Order: t1=2ˆCA0

2k

Second Order …2A ! C ‡ D†: t1=2ˆkC1

A0Second Order …A ‡ B ! C ‡ D†: t1=2ˆ 1

k…CA0 CB0†ln



…CB0†…2CB0 CA0†



Nonintegral Order: t1=2ˆ‰…1=2†1 n…n 1†k1Š…CA0†1 n

The second of these aspects has to do with whether we wish to write kinetic sions from the point of view of reactant remaining, as we have done so far, or fromthe point of view of product produced (reactant reacted) It is convenient in manyinstances to talkabout the conversion in a reaction, which de®nes the amount ofreactant consumed or product made For constant-volume systems that can berepresented in the rate equation as a concentration of material reacted, Cx,

expres-CAˆ CA0 Cx For a ®rst-order example, then

fractional-conversion de®nition in terms of concentration ratios applies only forconstant-volume reaction systems, so we will de®ne conversion more generally asthe ratio of mols reactant remaining to initial mols of reactant Hence

Equation (1-47) is one of the few that should be committed to memory immediately.There are, of course, corresponding de®nitions of conversion on the basis of massunits, and in some cases fractional conversion to product (mols product made permol reactant fed or reacted) is a useful measure The important thing is always to besure how the term conversion is de®ned The integrated forms of simple irreversiblerate laws written with conversion, x, from equation (1-47) are:

Zero Order: Cxˆ kt; x ˆCkt

A0First Order: ln 1 CCx

A0

ˆ ln…1 x† ˆ ktSecond Order …2A ! C ‡ D†: CCx

A0ˆ x ˆ1 ‡ ktCktCA0

A0Second Order …A ‡ B ! C ‡ D†: ln …1 C…1 Cx=CA0†

ˆ …1 x†1 n

An idea related to that of fractional conversion is the molar extent of reaction If wereturn for a moment to the general irreversible reaction of (I) and replace the arrowwith an equality, we can write

Ni;0 and mols at time t of reaction of Ni Then the molar extent of reaction will bede®ned by

Niˆ Ni;0‡ iX

or

A little thought about equation (1-50) will convince one that X de®ned in thisway has the same value for each species Thus, given Ni;0 and X, all values of Nimay be calculated directly Remember that the molar extent of reaction X is not thesame as conversion x, but it is also a number that is constrained to be between zeroand one The use of X is particularly convenient in dealing with the mathematics thatdescribe large numbers of reactions occurring simultaneously (see R Aris,Introduction to the Analysis of Chemical Reactors, Prentice-Hall, Inc., EnglewoodCli€s, NJ, 1965), and to problems in chemical equilibrium Since the convention offractional conversion is still most often encountered, however, we by and large use it

The third aspect of kinetic formulations is how to modify the rate equationswhen the reaction system volume is not constant Volume changes can occur whenthere is a net change in the number of mols as the reaction proceeds For mostreactions that occur in the liquid phase, such changes in volume are small andnormally can be neglected Conversely, in gas-phase reactions the volume/molalrelation is governed in the limit by the ideal gas law and volume changes must

be taken into account For the simple reactions we have been discussing, it is possible

to write the volume change directly in terms of the amount of reaction in aconvenient form using the fractional conversion First de®ne an expansion (orcontraction) factor , which represents the di€erence in reaction system volume atthe completion of the reaction and at the start of the reaction divided by theinitial value:

where V0is the initial volume and V1the ®nal volume The relationship between theamount of reactant consumed and the reaction volume can now be determined fromthe stoichiometry of the particular reaction As an example consider the irreversible

®rst-order reaction A ! B, where  is some stoichiometric coecient other thanunity and is a positive quantity for products of reaction When the volume/molalrelation is a direct proportionality, as in gas-phase reactions,

We can now use these relationships in the basic reaction-rate de®nition For the

®rst-order irreversible batch reaction,

…1=V†…dNA=dt† ˆ k…NA=V†

If the reaction is reversible, what is the limitingvalue of X based on the two ways of writing thestoichiometric coecients?

and on substitution for V and NA,

…1 x†n 1

…1 x†n dx ˆ …CA0†n 1ktThe integral in the last expression above is not a simple form and is best evaluated bynumerical means Use of the expansion factor is limited to reactions where there is alinear relationship between conversion and volume For reactions that have complexsequences of steps, a linear relationship may not be true Then we must rewrite therate de®nition for a batch reactor:

rAˆ …1=V†…dNA=dt† ˆ …1=V†‰d…CAV†=dtŠ

and the volume terms must be evaluated from the detailed stiochiometry of thereaction steps

1.4.3 Rates and Conversions for Some Reversible Reactions

The most important forms for reversible reactions are ®rst order, forward andreverse, and second order, forward and reverse If change in the number ofmols occurs and order follows stiochiometry, the reversible reactions can alsohave forward and reverse steps of di€erent order Since in the following presentation

we treat the reactions as elementary steps, the ratio of rate constants does de®ne theequilibrium constant for the reaction, K ˆ kf=kr

First-Order Forward and Reverse Here we have A $ B:

rAˆ …dCA=dt† ˆ kfCA‡ krCB

where x is the conversion of A

Equation (1-60) can be integrated to several possible forms; a convenient one is

and

The rate constant k0 is equal to the sum of the forward and reverse rate constants,

kf ‡ kr, and one must exercise care in treating the temperature dependence of k0,since it does not follow an Arrhenius law The usefulness of equation (1-63) depends

on the availability of data on equilibrium conversion

Second-Order Forward and Reverse The reaction in this case is A ‡ B $ C ‡ D and

to equation (1-63) may be written These are, for the case of no products presentinitially:

kf‰…1=x1† 1ŠCA0t

then the following reactions are included:

1.4.4 Summary: Simple Reactions

The examples given here provide only an indication of the variety of rate forms andconversion relationships that exist even for simple reactions Certainly there aremany more combinations of reaction order and reversibility that could be included,but the tools for dealing with them have been established here Some of these addi-tional forms are involved in subsequent exercises and discussion without individualderivation (See also J.W Moore and R.G Pearson, Kinetics and Mechanism, 3rdedition, John Wiley and Sons, New York, 1981, and S W Benson, The Foundations

of Chemical Kinetics, McGraw-Hill, New York, 1960)

Although the formulation and integration of simple rate equations is not cularly troublesome, re¯ection on the nature of nonintegral and higher order (largerthan unity) rate laws indicates much more dicult mathematics, since these are non-linear in the concentration variables Together with the exponential dependence ofrate, the nonlinear concentration dependence of rate in these cases means that evenapparently simple reactions may be dicult to analyze within the context of practicaloperation

parti-Illustration 1.3

An autocatalytic reaction is one in which a product of the reaction acts as a catalyst,

so that the forward rate of reaction depends on the concentration of both reactantsand products Consider the reaction:

A ! B ‡ C

with rate constant k, where C acts as a catalyst for the decomposition of A Derivethe appropriate form for the concentration of A as a function of time of reaction inthis isothermal, constant-volume reaction system Under initial conditions the con-centration of A is CA0 and that of C is CC0

Solution

Most autocatalytic reactions occur in liquid solutions where volume changes uponreaction are usually negligible, so the analysis using the constant-volume case here isreasonable The normal kinetic formulation for this reaction is then

rAˆ kCACC where CC0> 0

In terms of mols (for constant volume), let A be mols of reactant at time t, a be mols

of reactant reacted at t, and A0 initial mols of reactant Then

B ˆ B0‡ a

C ˆ C0‡ a

Since the system is constant-volume the concentrations can be replaced by molaramount in the rate equation, and

…dA=dt† ˆ …da=dt† ˆ k…A0 a†…C0‡ a†

Upon integration from 0 to a

‰1=…A0‡ C0†Š ln‰A0=…C0‡ a†Š=‰C0=…A0 a†Š ˆ kt

The following properties of this autocatalytic reaction are also of interest:

At time zero (initially)

rAˆ kA0C0 …positive and Enite†

At time 1 (completion of reaction)

rAˆ k…A0 A0†…C0‡ A0† ˆ 0

Thus, the rate must pass through a maximum value at some time between

0 < t > 1 At the maximum of rate:

…da=dt† ˆ 0 ˆ k…A0 a†…C0‡ a†

…d2a=dt2† ˆ 0 ˆ kC0‡ kA0 2ka

a ˆ …A0 C0†=2 at the maximum rate:

This can be considered as a kind of ®ngerprint of autocatalytic reactions in thatcertain concentration constraints will pertain at the maximum of reaction rate

1.5 Kinetics of ``Nearly Complex'' Reaction Sequences

The fact that the kinetics of an overall reaction normally represent the net e€ects

of the rates of a number of individual elementary steps means that one of ourmajor concerns in analysis must be how to assemble the individual steps into thewhole There are systematic procedures for doing this, some of which will bediscussed in detail for homogeneous reactions that occur by chain mechanisms.However, we have also pointed out that often, in the absence of detailed infor-mation, models that account for the essential features of the steps involved in anoverall reaction can be of great utility In this section we shall discuss three suchschemes that have found useful application in a wide variety of reaction systems

Watch out for autocatalysis in some biochemicalreactions Also, do not confuse product-catalyzedreactions with those requiring an initiator, as dis-cussed later

of the reaction paths that occur and are ordinarily based on some reactant or product-to-product relationship Here we shall de®ne yield as the frac-tion of reactant converted to a particular product, so if more than one product is ofinterest in a reaction, there can be several yields de®ned The yield of product j withrespect to reactant i, Yj, is

mols of i present initially

Overall selectivity is de®ned by the following:

Inserting the de®nitions for yield and conversion, we obtain

S0 ˆ overall selectivity ˆmols i reactedmols i ! j

For some types of reaction schemes this overall selectivity is a function of the degree

of conversion, so often it is convenient to de®ne in addition a point or di€erentialselectivity referring to a particular conversion level:

Sd ˆ differential selectivity ˆrate of production of jrate of reaction of i

Application of these de®nitions to the three reaction sequences mentioned above willresult in yield and selectivity values falling in the range from 0 to 1 If the stoichio-metric coecients of i and j di€er, values outside this range may be obtained In thiscase the preceding de®nitions can be normalized to the 0 to 1 range through multi-plication by the ratio of the appropriate coecients

One should be careful in using these de®nitions in comparison with quantitiesgiven the same name elsewhere, since there is a considerable variation in the nomen-clature found in the literature Further, the present de®nitions are dimensionless,whereas yields and selectivities reported in various sources, particularly those