Analysis of the Possible Mechanisms for a Catalytic Reaction System

ADVANCES IN CATALYSIS, VOLUME 32

Analysis of the Possible Mechanisms for a Catalytic Reaction System JOHN HAPPEL Department of Chemical Engineering and Applied Chemistry Columbia University New York, New York AND

PETER H. SELLERS The Rockefeller University New York, New York 1. Introduction . . . . . . . . . . . . . . 11. The Structure of a Chemical System . . . . . . . A. The S-Dimensional Space of All Mechanisms and the Q-Dimensional Space of All Reactions . . B. The &‘-Dimensional Subspace of All Steady-State Mechanisms and the &Dimensional Subspace of All Overall Reactions . . . . . . . . . C. Direct Mechanisms and Simple Reactions . . . . 111. General Formulas for Mechanisms and Reactions . . . A. A Change of Basis for the Mechanism Space. . . . B. Basic Overall Reactions, Steady-State Mechanisms, and Cycles . . . . . . . . . . . . . C. Algebraic Formulas. . . . . . . . . . . D. Multiple Overall Reactions. . . . . . . . . IV. A Procedure for Finding Every Direct Mechanism . . . A. The Cycle-Free Subsystem . . . . . . . . . B. The Direct Mechanisms . . . . . . . . . V. Systems with a Simple Overall Reaction . . . . . . Example I . Sulfur Dioxide Oxidation (No Cycles) . . . Example 2. The Hydrogen Electrode Reaction ( I Cycle) . Example 3. Ammonia Synthesis (2 Cycles) . . . . . Example 4. Dehydrogenation of I-Butene to 1,3-Butadiene (3 Cycles) . . . . . . . Example 5. Hydrogenation of Isooctenes (4 Cycles). . . VI. Overall Reactions with a Multiplicity Greater Than One. . Example 6. Ethylene Oxide Synthesis (NoCycles) . . . Example 7. Ethylene Oxide Synthesis ( I Cycle) . . . . Example 8. Isomerization of Butenes (1 Cycle) . . . .

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273 Copyright 0 1983 by Academic Press. Inc. All rights o f reproduction in any form reserved.

ISBN 0-12-007832-5

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Example9. n-Butane Dehydrogenation (3 Cycles) . Example 10. Methanation of Synthesis Gas (3 Cycles) VII. Discussion. . . . . . . . . . . . . A. Thermodynamics . . . . . . . . . B. Kinetics . . . . . . . . . . . . VIII. List of Symbols . . . . . . . . . . . References . . . . . . . . . . . . .

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Introduction

Complex chemical reactions proceed through a network of intermediates that are connected by elementary reaction steps. A reaction mechanism is defined as a combination of specific elementary steps which can be taken in appropriate proportions to produce the degrees of change of terminal species in a reacting system. Usually more elementary steps may be considered possible than are required to produce any single mechanism accounting for the observed overall reaction. It is thus necessary to consider how steps may be combined, given an appropriate initial choice of species and steps. The purpose of this article is to review methods for accomplishing this and to present detailed information on a new method with this objective developed by Happel and Sellers (I). For our purpose elementary steps can be chosen to include any reaction that cannot be further broken down so as to involve reactions in which the specified intermediates are produced or consumed. Ideally, elementary steps should consist of irreducible molecular events, usually with a molecularity no greater than two. Such steps are amenable to treatment by fundamental chemical principles such as collision and transition state theories. Often such a choice is not feasible because of lack of knowledge of the detailed chemistry involved. Each of these elementary reactions, even when carefully chosen, may itself have a definite mechanism, but theory may be unable to elucidate this finer detail [Moore (2)]. Regardless of how the possible intermediates and elementary steps are selected, the procedure given in this article presents a method for the unambiguous enumeration of all possible minimal reaction mechanisms that will generate the observed overall consumption and production of terminal species under given conditions of temperature, pressure and concentration. In developing this procedure two basic assumptions are made. The first is concerned with the fact that possible mathematical networks of reaction steps can be decomposed into two sets of steps, namely those that can be combined to give the observed overall production and consumption of chemical species and those that form cycles resulting in no net change in

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terminal species. Since the latter would correspond to no change in free energy, steps in such paths would be at equilibrium. According to the principle of microscopic reversibility (2),such cyclic paths would not occur physically. Their mathematical properties are important, as discussed in Section 111. The second assumption employed in this article is that all species designated as intermediates-those that do not enter into a given system as either terminal reactants or products--will be present at constant concentrations. This includes stationary systenis that can be described by a unique steady state rather than those which exhibit transient or oscillatory behavior. The general subject of enumerating reaction mechanisms has been considered previously in reviews that have appeared in this series over the past 30 years [see the reviews by Christiansen ( 3 ) ,Horiuti and Nakamura ( 4 ) , and Temkin ( 5 ) ] . Each of these successive contributions presented developments based on concepts developed earlier, as does the present treatment. Studies conducted by those authors are of special interest here since the subject matter is largely in the field of heterogeneous catalysis. Christiansen’s kinetic treatment of sequences forms the basis of much of a chapter in Boudart’s (6) excellent elementary treatment of the kinetics of chemical processes. A key concept is the recognition that there are two distinct types of sequences leading from reactants to products through active centers. One of these is an open sequence, in which an active center or site is not reproduced in any other step of the sequence. (One should not confuse this terminology with open systems referring to the continuous passage of reactants over a catalyst.) The second type is a closed sequence, in which an active center is reproduced so that a cyclic reaction pattern repeats itself and a large number of molecules of products can be made from only one active center. As Boudart notes, a closed sequence constitutes in some sense a good definition of catalysis. Although all reactions showing a closed sequence could be considered to be catalytic, there is a difference between those in which the entity of the active site is preserved by a catalyst and those in which it survives for only a limited number of cycles. In the first category are the truly catalytic reactions, whereas the second comprises the chain reactions. Both types can be considered by means of the steady-state approximation, as in Christiansen’s treatment. This important development dates to 1919 (6) when Christiansen as well as Herzfeld and Polanyi independently proposed an explanation of the kinetics of the reaction between hydrogen and bromine reported earlier by Bodenstein and Lind. In treating catalytic sequences of elementary steps, Christiansen adopted the simplification that each elementary step is first order in both directions with respect to concentration of a single active species. The resulting rates

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of production of terminal species could then be expressed quite generally by formulas involving a matrix whose elements are products of reaction velocity constants and concentrations of species. Such a treatment is especially useful for considering simple reactions such as isomerizations. Christiansen also noticed that some closed sequences would not yield an overall reaction and appropriately called such sequences cyclic. He was among the first to advance the viewpoint that the only possible stationary value for flow in such a sequence is zero and identified this with the principle of microscopic reversibility. Horiuti and his school, although not referring directly to the research of Christiansen, were no doubt familiar with it since Horiuti studied with Polanyi. Horiuti’s studies are summarized in a recent monograph devoted to the use of tracers in heterogeneous catalysis (7). He made an important advance in considering the velocities of mechanistic steps directly instead of relating them to reaction velocity constants as Christiansen had done. He showed that a chemical reaction mechanism can be described by specifying the number of occurrences of the elementary steps for each single occurrence of a given simple overall reaction. In Horiuti’s terminology these occurrences are the stoichiometric numbers associated with the specified reaction mechanism. Thus, the choice of possible reaction mechanisms reduces to the selection of appropriate stoichiometric numbers. Horiuti accomplished this selection by employing a step-by-species matrix in which the elements are the stoichiometric numbers and obtained the linearly independent sets. He described these sets as reaction routes. This method was generalized by Horiuti to systems in which more than one simple overall reaction could occur. Its implications are discussed further in Section II,C. The mathematical concept of linear independence introduced by Horiuti, although correct, will allow mechanisms to be combined in ways resulting in the cancellation of segments of the elementary reaction steps. Horiuti’s procedure in the case of simple overall reactions places a lower bound on the number of possible reaction mechanisms but does not furnish a complete listing. This limitation was recognized by Milner ( B ) , who introduced the concept of direct paths, each of which is unique in the sense that it cannot be considered to result from the superposition of any other member of the set of elementary reactions. Milner applied this idea to the enumeration of mechanisms for a number of simple overall reactions involving electrochemistry. He arrived at the rule that for such a reaction the number of nonzero stoichiometric numbers specifying a direct path can be no more than one greater than the number of intermediates. By a trial-and-error procedure he was able to count all mechanisms consistent with a given choice of possible unit steps. Sellers (9) developed a theory of chemical reaction networks for the

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generation of all possible direct mechanisms involving a simple overall reaction. In the present article this theory is developed more completely than in our earlier paper (1). Temkin (5, 10, 11)presented additional studies extending the original ideas of Horiuti to establish the number of routes or mathematically independent mechanisms consistent with a given initial choice of elementary steps. He showed that the algebra of reaction routes was consistent with the specification of the dimension of the space of such routes and that such a basis could include empty routes for which no net reaction occurs. However, instead of using such empty routes or cycles to generate the complete list of direct mechanisms as discussed in Section III,A, he assumed that such cycles could be disregarded in their effect on reaction mechanisms but not on kinetics. This is inconsistent with the treatment in this article since we assume that such cycles would not occur. A recent development aimed at classifying types of chemical mechanisms has been presented by Sinanoglu and co-workers (12, 13). It is an enormous classification problem, which they simplify by using a graph theoretic model. We have not adopted any such simplifying assumptions. Instead we require that a particular chemical system be given, and then we introduce procedures for analyzing that system, however complicated it may be from a combinatorial viewpoint, rather than undertaking to classify all chemical mechanisms in a general sense. The procedure presented here places no limit on the number of reactants in an elementary reaction or the number of elementary reactions which have a reactant in common. We use an incidence matrix to define the combinatorial structure of a system, and we analyze it by the methods of linear algebra. The graph theoretic model corresponds to a special type of incidence matrix (where each row has only two nonzero entries). Our procedure is applicable not only to this type, but to any matrix of integers. It is an algorithm, which could be carried out by computer, for finding all chemical mechanisms in a fixed context. In the following sections of this article we first define the terms necessary to identify a chemical system. After this, the use of an algebraic technique is developed for the expression of general reaction mechanisms and is compared with the previous treatments just mentioned. Next, a combinatorial method is used to determine all physically acceptable reaction mechanisms. This theoretical treatment is followed by a series of examples of increasing complexity. These examples have been chosen to illustrate the technique and for comparison with previous studies. They do not constitute a survey of all the most significant studies concerned with the mechanisms illustrated. Finally, a discussion is presented of the relationship of the present treatment to studies concerned with thermodynamics, and of the relationship between kinetics and mechanisms.

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JOHN HAPPEL A N D PETER H . SELLERS

II. The Structure of a Chemical System Let us view a chemical system as a network of elementary chemical reactions linked to one another by common reactants. The broad question of what kinds of network configurations are possible for chemical systems can be answered mathematically if it is stated in the following way: Given a hypothetical finite list of elementary chemical reactions, determine all the ways the reactions (or a subset of them) can be combined to form a specified overall reaction, or, more generally, to form any member of a specified family of overall reactions. The first stage in a mathematical solution of this problem is to define a chemical system formally. The mathematics we shall need is confined to the properties of vector spaces in which the scalar values are real numbers. From a mathematical viewpoint the whole discussion will take place in the context of two vector spaces, an S-dimensional space of chemical mechanisms and a Q-dimensional space of chemical reactions, which are related to each other by the fact that each mechanism m is associated with a unique reaction R(m) which it produces. The function R is a transformation of mechanisms to reactions which is linear by virtue of the fact that reactions are additive in a chemical system and that the reaction associated with combined mechanisms m, + m2 is R(m,) + R(m,). All mechanisms are combinations of a simplest kind of mechanism, called a step, which ideally consists of a one-step molecular interaction. Each step produces one of the elementary reactions which form a basis for the space of all reactions. For instance, let step s, be a collision process between a , and a, to form a3, and let step s2 be an isomerization of a3 to a4. Then R(s,) is a vector a4, and R(s, + s2)is a vector which - a , - a 2 + a3, R(s,) is a vector -a3 equals the sum -a, - a, + a4 of the first two vectors. This illustrates the linearity of R, which may be expressed in general by the linearity equation R(s, + s2) = R(s,) + R(s,). In particular, ifs is repeated cr times, the linearity equation becomes R(crs) = crR(s). This principle can be extended to all real values of cr, where cr is regarded as the rate of reaction, including the possibility of a negative value for cr to express the possibility of a reverse reaction. For simplicity we speak of a mechanism or a reaction, rather than a mechanism vector or reaction vector. The distinction lies in the fact that a reaction r (or mechanism) is essentially the same whether its rate of advancement is p or u, whereas pr and crr are different vectors (for p # 0 ) .Therefore, a reaction could properly be defined as a one-dimensional vector space which contains all the scalar multiples of a single reaction vector, but the mathematical development is simpler if a reaction is defined as a vector. This leaves open the question of when two reactions, or two mechanisms, are “essentially different” from a chemical viewpoint, which will be taken up

+

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

279

separately. It has not been universally recognized that, even though a onedimensional space contains essentially one reaction or mechanism, an N-dimensional space (for N > 1) generally contains more than N essentially different reactions or mechanisms, where the number varies from case to case and is not a function of N alone. SPACEOF ALL MECHANISMS A. THES-DIMENSIONAL AND THE Q-DIMENSIONAL SPACEOF ALLREACTIONS Let us begin here with a formalization of some of the ideas which have already been introduced. A chemical system contains species which will be denoted by a , , a,, . . ., aA and elementary reactions among these species which will be denoted by the S vectors in Eqs. ( l ) , rl = u l l a l r, = a Z l a l

+ a12a2+ . . . + ulAaA + a2,a2 + . . . + u2AaA

rs

+ us2a2 + . . + C(SAaA

=

aSlal

(1)

'

in which the a's are stoichiometric coefficients. Ordinarily, each elementary reaction will have one or two positive coefficients, one or two negative coefficients, and the remainder equal zero, but more nonzero coefficients are possible. Let us assume, however, that there is at least one positive coefficient and at least one negative coefficient. Any reaction in Eqs. (1) may be written as a conventional chemical equation by setting it equal to zero and transposing the negative terms to the other side of the equation. This notation has been discussed by Aris (14). Chemical equality, denoted by the symbol e, has been shown by Sellers (15) to be a group equivalence, thus satisfying ordinary rules of mathematical equality. Except when specific reservations are stated, every reaction is assumed to be reversible, that is, to be capable of any real rate of advancement, positive or negative. The elementary reactions in Eqs. (1) are not necessarily linearly independent, and, accordingly, let Q denote the maximum number of them in a linearly independent subset. This means that the set of all linear combinations of them defines a Q-dimensional vector space, called the reaction space. In matrix language Q is the rank of the S x A matrix ( 2 ) of stoichiometric coefficients which appear in the elementary reactions ( 1):

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JOHN HAPPEL AND PETER H . SELLERS

Next, let us define the space of all chemical mechanisms in the chemical system under consideration. Let step si denote the molecular interaction which produces the reaction denoted by ri or R(si). Let a mechanism m be any linear combination of steps of the following form: m

=

alsl

+ a2sz +

*

+ asss

(3)

where each coefficient oi is a real number, describing the rate of occurrence of si. The set of all such mechanisms constitutes an S-dimensional vector space, called the mechanism space. The reaction r produced by m is found by applying the linear transformation R to Eq. (3), which gives the following: r

=

a,rl

+ a2r2+ . . . + asrs

(4)

Since we have explicit expressions (1) for each f i rthey can be substituted into expression (4),so as to express the reaction r as the following explicit linear combination :

thus, obtaining a general expression for any reaction r in the system. It is usual in algebra to express vectors by linear combinations, but it is conventional also, particularly in working examples, to use matrix notation, where only the scalar coefficients are written. Thus, m would be expressed by (al* . . as)and Eq. ( 5 ) by the following matrix equation :

B. THEP-DIMENSIONAL SUBSPACE OF ALLSTEADY-STATE MECHANISMS AND THE R-DIMENSIONAL SUBSPACE OF ALLOVERALL REACTIONS To define a system in a steady state it is necessary to distinguish two kinds of species, the intermediates a , , . ,,a, and the terminal species a,, ,. . . , a , + T , where I + T = A. In such a system a steady-state mechanism is one whose reaction only involves terminal species. The net rate of production of each intermediate in such a mechanism is zero, which is equivalent to saying that the first I coefficients are zero in the general expression (5) for a reaction. This gives us the characterization, introduced by Horiuti ( 4 , 7 ) ,for a steady-

.

,

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

28 1

state mechanism as one whose coefficients p l , . . ., ps satisfy the I linear equations expressed by the following matrix equation: (PI . ’ ’ P S I

i“‘ us1

... @;I) ..*

= (O...O)

(7)

us1

If H denotes the rank of the S x I matrix in Eq. (7), then the dimension P of the space of all steady-state mechanisms equals S - H , and the dimension R of the space of all reactions which they produce equals Q - H . Let us describe the reactions in this R-dimensional space as the overall reactions, their essential property being that they involve terminal species only. Horiuti calls H the “number of independent intermediates.” Temkin (10) describes the equation P = S - H as Horiuti’s rule, and the equation R = Q - H as expressing the “number of basic overall equations.” To avoid confusion, let us confine the term basis and the concept of linear independence to sets of vectors, and let numbers such as H,P, Q, R, S be understood as dimensions of vector spaces. This makes it simple to determine their values and the relations among them, as will be done in Section 111. The dimension of a space equals the number of elements in a basis, which is defined as a set of elements such that every element in the space is equal to a unique linear combination of them. Therefore, P steady-state mechanisms can be chosen in terms of which all others can be uniquely expressed. This gives us a unique way to symbolize each steady-state mechanism and its overall reaction, but it does not provide a classification system for them which is valid from a chemical viewpoint, because the choice of a basis is arbitrary and is not dictated, in general, by any consideration of chemistry. A classification system for mechanisms is our next topic. C. DIRECT MECHANISMS AND SIMPLE REACTIONS

In a chemical system there is a unique collection of mechanisms, called the direct mechanisms of the system, which will be shown to be the fundamental constituents of any mechanism. Milner (8) called them “direct paths” and Sellers (9)-“cycle-free mechanisms.” Let m be any mechanism, and let r be the reaction which it produces; then m is defined as direct if it is minimal in the sense that, if one step is omitted, then there is no mechanism for r which can be formed from any linear combination of the remaining steps. Similarly, we can define r as a simple reaction if it is minimal in the sense that, if one of its reactants is omitted, then no reaction in the system involves only the remaining reactants. Let us use the word multiple as an antithesis to direct or simple. As we shall

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JOHN HAPPEL AND PETER H. SELLERS

see in the examples of Sections V and VI, direct mechanisms for both simple and multiple reactions are of major interest. The set of all direct mechanisms in a system contains within it a basis for the vector space of all mechanisms. In general, there are more direct mechanisms than basis elements, which means that there can exist linear dependence relations among direct mechanisms but, even so, they differ chemically. That is, a direct mechanism with a given step omitted cannot be considered to result from a combination of two other mechanisms in which that step is assumed to occur. In the latter case the net velocity of zero for that step would result from a cancellation of equal and opposite net velocities rather than from the complete absence of the step. The set of all direct mechanisms (unlike a basis) is a uniquely defined attribute of a chemical system. In fact what we have called a direct mechanism is what is usually called a mechanism in chemical literature, even though the definition may be implicit. There are special cases where the direct mechanisms are linearly independent and constitute a basis. If all the direct mechanisms for a particular reaction r are disjoint, in the sense of containing no steps in common, then they are obviously linearly independent, or if there is only one direct mechanism for r, it is independent. This last case suggests a way of finding all the direct mechanisms in a chemical system. If we can find a subsystem which contains at most one mechanism m for any reaction r, then m is direct. In other words, m is a direct mechanism if S = Q in the chemical system, consisting just of the steps in m. Accordingly, to find all the direct mechanisms in a system where Q < S, we may consider each of the (i)subsystems, where

Hence in the entire system there are at most (i)direct mechanisms for r, but usually there are many less than this, not only because of the excluded subsystems, but also because different subsystems can contain the same direct mechanism for a particular r. This approach to finding direct mechanisms is implemented in Section IV, where an efficient procedure is given for making a complete list of all the direct steady-state mechanisms for a given reaction, simple or multiple. All mechanisms can be classified in terms of direct mechanisms. In this article we consider the problem of listing all direct mechanisms in a given system, but we do not undertake to list all combinations which consist of two or more direct mechanisms, advancing simultaneously at independent rates. Each direct mechanism contains a minimal number of elementary steps. Combinations of direct mechanisms in which additional steps must

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

283

appear may also be termed mechanisms over the allowable range of such combinations without cycle formation. Such combinations are unique if they are composed of only two direct mechanisms. Combinations involving more than two direct mechanisms are not necessarily unique, in the sense that a given rate of reaction can no longer be represented by combinations of only those direct mechanisms. The combinations of increasing numbers of direct mechanisms will finally include all steps and thus constitute the most general mechanism consistent with the initial choice of elementary steps. The extent to which any given direct mechanisms may be combined without cycle formation can be determined by noting whether such combinations contain irreducible cycles. The latter are the cycles with a minimal number of steps which characterize a given system. They can be determined by a procedure that is analogous to that for finding direct mechanisms [Sellers (9a)I.For a multiple overall reaction, the relative degrees of advancement for each of the simple overall reactions chosen as a basis introduce additional restrictions on the allowable cycle free combinations) [Sellers (9b)I. Except for modeling isomerization systems involving multiple overall reactions, it is generally assumed that a single predominant direct mechanism is sufficient to characterize a given system. Usually the further simplifying assumption is made that there is a single rate-controlling step, other steps in a mechanism being taken at quasi-equilibrium. Another simplification is the assumption of unidirectional steps for reactions that are far from equilibrium. 111.

General Formulas for Mechanisms and Reactions

Equation (7) characterizes a steady-state mechanism algebraically, but it does not provide an explicit formula for any such mechanism. Therefore, using only the matrix (2) of elementary reaction coefficients and the knowledge of which columns in the matrix correspond to terminal species, let us derive a general formula for any steady-state mechanism. A. A

CHANGE OF

BASISFOR

THE

MECHANISM SPACE

The basis s l , . . ., ss for the mechanism space will be changed to one which contains H non-steady-state mechanisms and P steady-state mechanisms. The latter will be what Temkin (16) calls a “basis for all routes.” A route is what we are calling a steady-state mechanism, and a basis is a maximal linearly independent set of them. The basis is “stoichiometric” in Temkin’s

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JOHN HAPPEL AND PETER H. SELLERS

terminology when it is selected as follows: R of the P basis elements are mechanisms whose reactions constitute a basis for the space of overall reactions, and the remaining C basis elements are what Temkin (10) calls “empty routes,” each of which has a reaction equal to zero. Let us use the mathematical term cycle to describe any element m of the mechanism space such that R(m) = 0, instead of “empty route.” The C elements mentioned are a basis for the space of all cycles. In algebra this space is known as the “kernel of R.” To construct the new basis, described above, we will change the basis s l , . . ., ss of the S-dimensional space of all mechanisms into a basis (9), which separates into 3 parts, ml,...,mH;

mH+,,...,mQ;

mQ+,,...,mS

(9)

where each mi is a linear combination of the original basis elements, such that (mQ+ , . . ., m,) is a basis for the subspace all cycles; (mH+ . . ., mQ) is a set of steady-state mechanisms no linear combination of which is a cycle, and (ml , . . ., mH) is a set of mechanisms no linear combination of which is a steady-state mechanism. To get basis (9), start with following matrix :

,

ss \us,

.’ ’

as1

%,I+,

’

.’

%,I

+TJ

which is the same as matrix ( 2 )except that we now require the first I columns to correspond to intermediates and the remaining columns to correspond to terminal species. Furthermore, the rows are labeled by the steps which correspond to them [i.e., aij is the stoichiometric coefficient of species a j in R(s,)]. Next, perform elementary row operations on matrix (lo), so as to put it in the diagonal form shown in Fig. 1. This will require interchanging some pairs of columns (but a column corresponding to an intermediate is never interchanged with one corresponding to a terminal species). Every time a row operation is performed it is also applied to the column of basis elements at the left of the matrix, with the result that the entries in this column become linear combinations of steps which will achieve form (9) when the diagonalization is complete. For instance, if the first row operation is to replace row j by row j minus row i, then the column entry sj is replaced by ( s j - si).There is no need to describe the diagonalization procedure in detail, except to say that it must be performed in such a way as to insure that the combinations m, , . . ., m, are linearly independent. This will be achieved if the elementary row operations are confined to changing a row by adding to

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

285

FIG. I . This diagonalized matrix ( Pij) is formed from matrix ( a i j )by elementary row operations and column permutations. It has the same rank as (aij).

or subtracting from it one of the rows above. The only row permutation needed is moving a row of zeros to the bottom of the matrix. B.

BASICOVERALL REACTIONS, STEADY-STATE MECHANISMS, AND CYCLES

An explicit basis for the space of overall reactions is characterized by rows H + 1 through Q of the diagonalized matrix of Fig. 1. If the matrix is expressed by (Pij), then the desired basis is R(mH+I) = P H + l , l + l a l + l + ." + P H + l . A a A R ( m ~ + 2=) B H + ~ , I + ~ ~ I *+" I -I PH+2,AaA R(mQ)

=

DQ,l+laI+l

+

* * '

+

( 1 1)

PQAaA

A basis for the space of all steady-state mechanisms is (mH+1 , . . ., ms). Since mi is a linear combination of steps, this basis has the following form: mH+l

=

YH+1,lSl

+

" '

+ YH+l,SSS

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JOHN HAPPEL A N D PETER H. SELLERS

The rows in (12) from m?+ through ms are a basis for the space of all cycles, and the coefficients in these rows form a C x S matrix (yij), which will be needed in Section IV for the construction of the unique set of all direct steady-state mechanisms.

C. ALGEBRAIC FORMULAS

Every element of a space is a unique linear combination of its basis elements. Therefore, a general expression for any steady-state mechanism m, including cycles, has the following form: (13) =pH+lmH+l + C1H+2mH+2 + " ' + h m S where the coefficients are any real numbers. A general expression for any overall reaction r is obtained by applying the function R to Eq. (13), which gives the following: = pH+

lR(mH+1)

+

' * *

+ PQR(mQ)

(14)

These expressions for m and r are made explicit by substituting into them the values for mi and R(mi)stated in bases (1 1) and (12).

D. MULTIPLE OVERALL REACTIONS

If R = 1 in a chemical system, it means that all steady-state mechanisms [i.e., all m which can be obtained by assigning particular numerical values to p l , . . ., ps in Eq. (13)] will have the same overall reaction r or a multiple of it, because then Eq. (14) reduces to r = pH+lR(mH+]). In this case the system is said to have a simple overall reaction, and, when we come to list all the direct mechanisms for r, there is no loss of generality if we take the multiple pH+ to be unity. If, however, R > 1, then the general formula (14) for an overall reaction r involves two or more independent parameters and is said to be a multiple overall reaction. In such a case each direct mechanism for r must also involve these parameters, unless we are prepared to choose particular overall reactions and determine a list of direct mechanisms for each. However, if we take a basic set of overall reactions and combine the direct mechansims of all of them, the process of combining them will lead to nondirect mechanisms. Accordingly, to generate all the direct mechanisms in a system, we must

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ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

find them for the general overall reaction (14). This is contrary to the approach suggested by Lee and Sinanoglu (13). IV.

A Procedure for Finding Every Direct Mechanism

A procedure, which was introduced by Happel and Sellers (I) for finding all direct mechanisms for a given reaction will be demonstrated here from the standpoint of how to apply it in practice. We demonstrate it by applying it to an arbitrary S-step chemical system, as defined in Section 11, to find all the direct mechanisms for the general overall reaction (14) derived in Section 111. A. THECYCLE-FREE SUBSYSTEM In a chemical system with S steps and a maximum of Q linearly independent elementary reactions every set of Q steps whose reactions are linearly independent constitutes a cycle+ee subsystem. It is apparent that, if R(sl),. . ., R(s,) are linearly independent, then no cycle can be formed with the steps s l , . . ., sQ(unless all the coefficients are zero). Accordingly, a laborious way ways in to find all the cycle-free subsystems would be to consider all the which Q rows can be chosen from the S x A matrix (2) and select those which are row independent. Since each row corresponds to a step, each selection of Q independent rows gives a cycle-free subsystem. The same result can be achieved by considering columns C at a time in the following smaller C x S matrix: . . . YQ+ 1.S ?Q+ 1 . 1 ?Q+ 1 . 2

(z)

?Q+2.1

YQ+2.2

Ys2

...

".

YQ + 2,s

Yss

called the cycle matrix, whose entries are defined by the basic cycles constructed in Section III,B and given explicitly in the last C rows of basis (12). Each column of this matrix corresponds to a step in the system, and each linearly independent set of C columns corresponds to a set of steps which, if removed from the system, would leave a Q-step cycle-free subsystem. This procedure for finding the Q-step cycle-free subsystems was introduced by Happel and Sellers ( I ) , and is equivalent mathematically to the procedure described in the last paragraph, but it takes advantage of the fact that the

288

JOHN HAPPEL AND PETER H. SELLERS

basic cycles for the system have already been determined by diagonalizing matrix (2).

B. THEDIRECTMECHANISMS Let us find every direct mechanism for a given overall reaction r. Assume r to be of the general form given in Eq. (14) and of multiplicity R ( R = Q - H), which means that an expression for it contains R parameters. Any mechanism for r is of the general form given in Eq. (13) and depends not only on the R parameters in its reaction, but on C additional parameters, where C is the dimension of the space of all cycles ( R C = S - H). Therefore, to determine a unique mechanism for r, we need to determine C parameters, and they can be chosen to make it a direct mechanism by the following reasoning: (i) Every direct mechanism belongs to a Q-step cycle-free subsystem and (ii) every Q-step cycle-free subsystem is obtained, as shown in Section IV,A, removing C appropriate steps from the system as a whole (Q = S - C ) . Taking (i) and (ii) together, we obtain a direct mechanism for r by rewriting the general mechanism (1 3) as a linear combination of steps, then setting C coefficients equal to zero, where they are the coefficients of C steps whose removal defines a cycle-free system. Let us clarify this procedure by applying it to four cases of increasing complexity: (1) C = 0; (2) C = 1, R = I ; (3) C>l,R=l;(4)C>l,R>l. Case I . If C = 0, there are no cycles in the system. For any reaction r in the system there is one mechanism, and it is direct. This applies to Example 1 in Section V and Example 6 in Section VI. Case 2. If C = 1 and R = 1, then the general mechanism, as given by To simplify the notation, Eq. (14), takes the form pH+lm,+l + pH+2mH+2. let this be written as pm + 4n, where m is a mechanism, n is a cycle, and the coefficients p and 4 are unrestricted. The overall reaction depends only on p and is a simple reaction. Ordinarily, with simple reactions the scalar coefficient is omitted, since r and pr are essentially the same reaction. Therefore, in the present case the general mechanism is taken to be of the form m + 4n. It will be recalled that m and n are fixed linear combinations of steps, which result from the diagonalization procedure described in Section III,A. Accordingly, m + 4 n can be expanded to a linear combination of steps, whose coefficients depend on 4 or else are constant. This becomes a direct mechanism if any single nonconstant coefficient is set equal to zero. In the first place this removes one step, which means that the mechanism belongs to a cycle-free subsystem and thus is direct. Second, setting a nonconstant coefficient equal to zero allows us to solve for 4, which means that all other coefficients are known, and the direct mechanism is completely

+

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

289

determined. Furthermore, we get every possible direct mechanism if we go through the above procedure for every nonconstant coefficient. The above procedure applies to Example 2 of Section V, where it is carried out in full detail. Case3. If C > 1 and R = 1, then the general mechanism, as given by Eq. (1 4), takes the form p H+ mH+ + . . . + psms, in which mH+ is a mechanism and the other m’s are cycles. Rewrite it as pm + 41nl + . . . + &n,, where m is a mechanism, n, through n, are cycles and the coefficients are unrestricted, which means, as in Case 2, that we may set p equal to unity without loss of generality. To find each direct mechanism, expand m + 4 ,n, + . . . + &nc to a linear combination of steps, and set C coefficients equal to zero, where the choice of coefficients is made as follows: The cycles are of the following form:

,

n,

=

vllsl

+ . . . + vlsss

n,

=

vclsl

+ + vcsss

(16)

* * *

The C steps whose coefficients are set equal to zero and must correspond to C columns in (16) whose C x C matrix of coefficients is nonsingular. This choice will guarantee that the resulting mechanism is direct. Furthermore, each coefficient which we set equal to zero gives rise to a linear equation in the variables 41,. . ., 4,. We have C such equations, which are solvable because of the nonsingularity of the C x C matrix of coefficients. Accordingly, for each nonsingular C x C matrix of coefficients in (1 6), there is a complete solution for 41,.. ., &-,which means that the direct mechanism is completely determined. Furthermore, we get every possible direct mechanism, if we go through this procedure for every nonsingular C x C matrix of coefficients in (16). The above procedure applies to Examples 3, 4, and 5 of Section V, and it is carried out in full detail in Example 3. Case 4. If C > 1 and R > 1, then we have the general situation described in Section II1,B. Using Eqs. (12) and (13) from that section, we arrive at an explicit expression for the general mechanism as the sum of all of the following expressions : pH+1YH+1,ls1

+

~H+RYH+R.ISI

+ ... + P H + R Y H + R . S S S

PSYSlSl

” *

+ pH+lYH+l,SSS

+ . ’ . CCSYSSSS

The first R rows add up to a mechanism for the general overall reaction, and the remaining C rows are cycles. As in Case 3, our object is to choose coefficients for the cycles such that C of the columns in (1 7) add up to zero. The

290

JOHN HAPPEL A N D PETER H . SELLERS

C columns we choose must have the property that matrix ( y i j ) from these columns and from the last C rows of (17) form a nonsingular C x C matrix M. For simplicity of exposition let us suppose that the first C columns are the ones chosen, then M = (yij) where H + R + 1 5 i IS and 1 Ij IH . Our object now is to have the first C columns of (17) add up to zero. dSss,which is a Denote the sum of the first R rows of (17) by glsl * * mechanism for the overall reaction. Then the statement that the first C columns of (23) add up to zero is equivalent to the following matrix equation:

+

( O l * * ' O R )f (p(H+R+I"'pS)M

=

+

(18)

whose solution is (pH+R+I

" ' p S ) = (-Ol

* "

- CTR)M-

(19)

In other words, if the C coefficients P,,+~+ . . . p s are given the values determined by Eq. (19), then the total of the expressions in (17) will be a direct mechanism. Furthermore, if we go through this procedure for every selection of C columns in (1 7) such that the C x C matrix M is nonsingular, then we get every direct mechanism for the overall reaction (14). Altogether there are R + C undetermined coefficients pH+ 1 , . . ., ps in (17), the last C of which are determined for each direct mechanism. The remaining R parameters pH+ . ., P,,+~ are in the expression (14) for the overall reaction, which is of multiplicity R. Similarly each direct mechanism must be a function of these R parameters. The above procedure is carried out in the treatment of Example 9 in Section VI to obtain an initial direct mechanism. In all of the cases treated above the set of direct steady-state mechanisms which has been generated is exhaustive. However, it is possible for repetitions to occur among the mechanisms, but we can eliminate the possibility of repetitions in the following manner. Each time C values for pH + R + . . ., ps are determined, substitute them in the general mechanism, express it as the following linear combination of steps : ZilSl

+

..*

+

ZiSSS

(20)

and put its coefficients in the last row of Table I, which is a display of all direct mechanisms found, so far. To get the next line, choose a set of C columns from (17) which has not yet been considered. There are ( 5 ) ways of choosing columns which must be considered, but the computation (19) does not have to be carried out if either of the following conditions holds: (i) There is a row in Table I which already has zeros in the C columns we are considering. (ii) The matrix M determined by these columns is singular. These precautions will eliminate repetitions among the rows of Table I

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

29 1

TABLE I

more simply than if they are thrown out after they are found. This saves unnecessary matrix inversions. Each of the (): column selections must be tested. Any systematic way of ordering these selections so that none are omitted is acceptable. This additional procedure to avoid repetitions is carried out in the last part of Example 9 in Section VI. V.

Systems with a Simple Overall Reaction

Most systems treated in the literature exhibit a simple overall reaction, which can be uniquely represented by a conventional chemical equation. In addition, the elementary reactions are usually selected so that all of them must be combined to form the overall reaction, which means that the system is cycle free and that there is no mathematical distinction between an elementary reaction and the step which produces it. Often the combination of steps giving the overall reaction is such that each intermediate is produced by exactly one step and consumed by exactly one step. The following example illustrates such a system. 1 . SULFURDIOXIDE OXIDATION (No CYCLES) EXAMPLE

The important commercial process of sulfur dioxide oxidation has been studied by a number of investigators. A set of steps that has been proposed for both platinum and vanadium oxide-based catalysis by Horiuti (7) for the overall reaction 2 S 0 2 + O2 + 2 S 0 3 is as follows: sl:

s,:

0,+ 2 1 s 2 0 1

so, + 1 ==S 0 , I

sj: S 0 , I s4:

+ 01=

S0,l

so,/= so, + I

+1

This is the form in which steps will be listed in all our examples-a

symbol

29 2

JOHN HAPPEL AND PETER H. SELLERS TABLE I1 OI

S0,I

2 0

0

SI s2

s, s4

-I

-I

I 0

0

so,/ 0 0 1 -1

I

0,

-2

-1

-1 1

I

so, -I

0 0 0

0

so,

0

0 0 0 I

so,

0

TABLE I11

SI s2

SI

SI

+ 2s, + 2s,

+ 2s, + 2s, + 2s,

01

S0,I

SO&

I

0,

2 0 0

0 I 0

0 0 2

-2

-1

0

0

0

-1

-2 0

-I

0

-1

0

so,

-1

-2

0 0 0

-2

2

for the step, followed by a chemical equation for the elementary reaction produced by that step. In this example, but not in the subsequent ones, let us rewrite the elementary reactions in the following more formal vector notation: R(s,) = - 0 2 - 21 201

+ R(s2) = -SO2 - 1 + S 0 2 f R(s3) = - S 0 2 1 - 01 + S03f + 1 R(s4)= - S 0 3 1 + SO3 + 1

(22)

This displays the convention, tacitly assumed later, that the positive direction of a step corresponds to the advancement from left to right of the stated chemical equation. The matrix of stoichiometric coefficients for these reactions is shown in Table 11. The diagonalization of the matrix in Table I1 gives the matrix in Table 111, from which the steady-state mechanism is s1 + 2s, + 2s3 + 2s4. In Horiuti’s terminology the “stoichiometric numbers’’ are 1 for s1 and 2 for s 2 , s 3 ,and s4. Often, even in the case of simple reactions, it is possible to encounter systems with cycles. The following is an illustration of this situation.

EXAMPLE 2. THEHYDROGEN ELECTRODE REACTION ( 1 CYCLE) This system has received considerable attention. Milner (8) includes it in a study of several electrode reactions, and Horiuti (7) uses it as the basis of a general discussion.

293

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

TABLE IV H

H,

Ht

*

The overall reaction of the system is given as 2H+ + 2eH,, and 3 steps are postulated whose elementary reactions are as follows : s,: H t + H + e - = H 2 s,:

+ e- + H +HeH,

H+

s3: H

Since H + and e- are always together, let us regard H+ + e - as a single component and write it simply as H + . The matrix of stoichiometric coefficients is given in Table IV, the diagonalization of which gives the matrix in Table V. From this we conclude that the general steady-state mechanism is as follows :

+ ~ 2 +) 4 ( ~ -1 sz

( ~ 1

-

(1

~ 3= )

+

+ (1

4)~1

-

4 1 ~ 2- 4

~

3

(24)

where the coefficient 4 is unrestricted. Following the method of Case 2 in Section IV,B, we find each direct mechanism by setting one of the coefficients equal to zero in the right-hand side of Eq. (24). This gives three possible values - 1, 1,0 for 4. Putting them in Eq. (24), we get all the possible direct mechanisms as follows : m,

=

2s,

+ s3

m,

=

2s,

-

m3 = s1

s3

(25)

+ s2

This result may be tabulated as shown in Table VI. In subsequent examples this sort of table will be the principal way of listing direct mechanisms. There are three elementary steps and one independent intermediate, so TABLE V

s2

H

H,

1

1 0

H+ -I

294

JOHN HAPPEL A N D PETER H . SELLERS T A B L E VI

m,

m2

m,

0 2 I

2 0 1

-I

I 0

that there are 3 - I = 2 reaction routes according to Horiuti’s rule. Miyahara ( 1 7 ) and Horiuti (7) noted that any two of the three direct mechanisms could be combined algebraically to obtain the third. However, they are distinct from each other chemically since any two of them contain a step that is not involved at all in the third.

EXAMPLE 3. AMMONIA SYNTHESIS (2 CYCLES)

*

The reaction N, + 3H2 2NH, has been studied extensively from a mechanistic viewpoint. Horiuti (7) and Temkin (11) have proposed entirely different mechanisms for this reaction. Recognizing all steps in both mechanisms as possibilities, we find that there are in all 6 direct mechanisms, including the proposed ones, all of which produce the same overall reaction. In the following system for ammonia synthesis steps s l , s2,s 3 ,and s, were proposed by Temkin and s4, s5, s6, s8, and sg by Horiuti :

+ I* N,I N,I + H, * N,H,I N,H,I + I * 2NHI N, + 2/* 2NI NI + HI*NHI + I NHI + HI* NH,I + I NHI + H, * NH, + I H , + 21* 2HI NH21 + H I + NH, + 21

s , : N, s2:

s,: s4:

s5: s6: s,:

sg: sg:

The symbol I in system (26) refers to an active surface site on the catalyst. Every species with I in it is an intermediate and the rest are terminal species. For the purpose of our analysis we could omit I wherever it appears alone as a reactant. Notice that by including it as an intermediate, we get a case of a system where H < I , or in Horiuti’s terminology, the intermediates are not all independent. By diagonalizing the matrix of stoichiometric coefficients, we obtain the matrix given in Table VII. The seventh row shows that there is a simple

295

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

TABLE VII N21 N,H21 NHI N I SI s2

+ s1

s3

s2

+ + s1

s4

+ sq - s3 - s2 - s, - 2s, - s4 + 2s3 + 2s, + 2s1 2% + s3 + s2 + s , sg + 2s, + s4 - s3 - s2 - S I + - s, + S 6 2s, 2s,

sg

sg

HI

NH21

I

N2

H2

-1 -I

-I

I 0

0 1

0

0

0

0

-

I

0

0

0

0

-

I

0 0 0 0

0 0 0 0

2 0 0 0

0 0 0 2 0 0 0 - 2 0 0 0 2

0

0

0

0

0

0 0

0

0 0

0 0

0 0

NH3

0

2 2 2 -2

-I -I

-I

-2

0

0

0

-I

-3

2

0 0

0

0 0

0 0

-I 0

0

0

1

0 ~

0

0

0 0

overall reaction, and the last two rows show that the space of all cycles is of dimension 2. The general mechanism is given' by the following equation: 2S7

+ + s2 + + s3

s1

&s8

+ 2 S 5 + s4 -

s3

where

4)(s1

+ $(s9 + s8 - s7 + s6) +

- s2 - s1)

+ s2 + s3) + &s4 + 2s5) + $ ( s 6 + ( 2 - $ b 7 f (6+ 9 b 8

= (I -

s9)

(27)

and $ are unrestricted. Notice that the combinations of steps (sg + sg) may be treated as single steps. The cycle matrix is given in Table VIII. There are two singular 2 x 2 submatrices in Table VIII, consisting of columns 1 and 2 and columns 3 and 4. The remaining eight 2 x 2 submatrices are nonsingular. This means, according to the Case 3 in Section IV,B, that each direct mechanism is obtainable by setting a pair of coefficients equal to zero in the second line of Eq. (27) and solving for 4 and $. This leads to six distinct solutions, as shown in Table IX, which considers every pair of coefficients. Cases 1 and 8, which correspond to singular 2 x 2 submatrices of Table VIII do not have solutions, and cases 7 and 9 are repetitions of previous solutions. Accordingly, we are left with six pairs of values for 4 and $, which may be substituted in Eq. (27) to get the direct mechanisms given in the following list: (sl

CI#

+ s2 + s3), (s4 + 2s,), and

+ 2s5) + 2s7 + s8 m2 = (s4 + 2s5) + 2(s6 + s9) + 3s8 (Horiuti) m3 = (s4 + 2s5) + s9) + 3s7 m4 = ( s , + s2 + s3) + 2s7 (Temkin) m5 = (sl + s2 + s3) + 2(s6 + s9) + 2.5, m6 = 3(s, + s2 + sj) - 2(s4 + 2 4 + 2(s6 + s9)

m, = (s4

- (sg

and tabulated in Table X.

0 0 0

296

JOHN HAPPEL AND PETER H. SELLERS

4 9

I 0

-I 0

0 I

-I

0

1 1

TABLE IX Case

Pairs of coefficients

1-4.4 I -4.G 1-4,2-9 1 -l#J,4+9 4, 4*2- 9 4.4 t G 9.2 - 9 9.4 + 9 2 - 9.4 t J,

I 2 3

*

4

5 6

I 8 9 10

Solutions none fj=I,+=O

4=1.+=2 +=I,$= -1 fj = 0, I j = 0 4=0,$=2

+=o,g=o 4 4

= =

none 0,9 = 0

-2.9

=

2

TABLE X SI

m1 m2 m3

m4

m, m6

+ s2 + sj 0 0 0 I I 3

s4

+ 2s,

S6

+ sg

I

0 2

1

I 0

0 -2

-I

0 2 2

s,

sg

2 0 3 2 0 0

1 3 0 0 2 0

m, and m4 are the mechanisms proposed by Horiuti and Temkin, respectively. m3 and m6 might be omitted on the grounds that some of their steps proceed in the wrong direction, but m, and m5 remain for consideration. Rate equations for simple reversible reactions are often developed from mechanistic models on the assumption that the kinetics of elementary steps can be described in terms of rate constants and surface concentrations of intermediates. An application of the Langmuir adsorption theory for such development was described in the classic text by Hougen and Watson (18), and was used for constructing rate equations for a number of heterogeneous catalytic reactions. In their treatment it was assumed that one step would be rate-controlling for a unique mechanism with the other steps at equilibrium.

297

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

Such rate expressions are often termed Langmuir-Hinshelwood-HougenWatson (LHHW) equations and are widely used in chemical engineering [see Froment and Bischoff (19)].The usual procedure is to postulate plausible mechanisms without considering cycles, as in Example 1. In such cases it may be desirable to develop the complete list of possible direct mechanisms even if further considerations can rule out some as being unlikely. The following example illustrates a typical case. EXAMPLE 4. DEHYDROGENATION OF 1-BUTENE TO 1,3-BUTADIENE (3 CYCLES) Froment and Bischoff (19) report a study of the dehydrogenation of 1-butene to butadiene on a chromia-alumina catalyst. Neglecting isomerization of 1-butene, the following steps are postulated:

+ I * C,H,I + I*H21 H,I + I = 2HI H, + 2/* 2HI C,H,I + I * C,H,I + HI C,H,I + I* C,H,I + HI C,H,I * C4H6 + I C4H,I + I* C,H,I + H,I C,H,I + 21* C,H6/ + 2HI

s, : C,H, s,:

s3: s,: s5 :

s6: s, : s,:

s9:

H,

The overall reaction is

TABLE XI

'6 s5

s3 s2

s2

s, s, s*

+ s, + s5 - s3 - s2 + s, -

sg - s,

- S6 -

s9 - S6 -

s5 s5

+ s3

1 0 0 0 0

-

1

I 0 0 0

0

0

0 0 0

0 0 0

1

1 2

-

0 0

0 0 0 - 1 1 0 1 0 0 1

0

0 0 0 0

- 1 - 1 - 1 - 1 - 1 - 1

0 0 0 0

0 0 0

0 0 0 0

0 - 1

0 0 0

0 0 0

0 0

0

0 0 0 1 0

I

I

0 0 0

0 0 0

0 0 -

298

JOHN HAPPEL AND PETER H. SELLERS

The diagonalization of the matrix of stoichiometric coefficients is simplified in this case ifthe rows are not ordered as in steps (35).The result of diagonalizing is given in Table XI. Then, using the methods of Section IV,B, we find all the direct mechanisms of Table XI1 for the overall reaction

+ C,H,

C4H,

+ H,

Examination of the matrix in Table XI1 shows that 6 mechanisms are possible on the basis of the steps proposed by Bischoff and Froment. They identified m , , m2, m4, m5, and m6, and developed 15 rate equations corresponding to various choices of rate-controlling steps. After a set of experiments involving sequential testing and model discrimination, they retained mechanism m4 with step s1 as the rate-controlling step. According to the present procedure m, might be an additional mechanism to consider. A scheme which was considered by Hougen and Watson (18)and is slightly more complicated is the hydrogenation of isooctene codimer. This illustrates a reaction with a large number of cycles compared with the number of intermediates.

EXAMPLE 5. HYDROGENATION OF ISOOCTENES (4 CYCLES) A supported nickel catalyst was used to study the reaction in which mixed isooctenes, commercially known as codimer, are hydrogenated in the vapor phase to the corresponding isooctanes. Neglecting isomerization, the following steps are assumed to occur: s , : C,H,,

s,:

H,

+ I*

s3: C,H,,/

+ I*

C,H,,I

H,I

+HJe

s4: C , H , , I ~ C , H , , s5:

H,

+I

+ 2 / * 2HI + 2 H I e C,H,,/ + 21

C,H,,I s,: C,H,, s,:

C,H,,I +I

s,:

C,H,,

sg:

H,

+ H,/* C,H,,I + 2 H I e C,H,,/ + /

+ C , H , , I ~C,H,,/

The overall reaction is It is not necessary to know in advance what the overall reaction is (Hougen and Watson assumed that it might also occur as an uncatalyzed reaction in the gas phase). For our purposes it is enough to know what the terminal

299

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

TABLE XI1 s1

+ s,

s,

I I I

m1 m2 m 3 .

m4

-1 -1 -1

I

m6

ss

s4

-I -1 -I 0 0 0

0 0 1 0

0 0 0

1

m5

s3

-I -1

+

s*

sg

0

0

1

1

0 1 I

0 0 0

0 0

1 0

S6

0 0 0 0

species are, since the overall reaction among them is furnished by the diagonalization (see Table XIII) of the matrix of stoichiometric coefficients in steps 30. Using the methods of Section IV,B, we find all the direct mechanisms (see Table XIV) in this system for the overall reaction H2

+ CSH,, * CSH,,

The matrix given in Table XIV shows that there are nine direct mechanisms. Five of these, namely, m2, m5, m7, m,, and m,, were identified by Hougen and Watson. Seventeen different mechanisms with single ratecontrolling steps were modeled and tested for agreement with observed kinetic data. The model corresponding to m7 with s, as the rate-controlling step was chosen as the recommended rate equation. The mechanisms obtained by our procedure in addition to those developed by Hougen and Watson are m, , m3, m4, and m6. The large number of these mechanisms is related to the fact that four cycles appear in the diagonalized matrix. These models have unusual features that would probably not be noticed unless a procedure like this were employed. For example, in mechanism m, neither isooctane nor hydrogen adsorbs directly on the TABLE XI11

-1

s3 S2

SS SS

+ sg + ss s1 + s, + -s2 + s5 + s2 - ss + s, S) + sg s4

s3

sg -

S)

S8

-s2

-

S6

s*

0 0 0 0 0 0 0 0

I

0 -1 0

-I

0 0 -1 0

0

0

-I

-1

0 0 0 0

0 0 0 0

-1 1

1

0

0

0

0 0 0 0 0 0 0

0 0 0 0 0 0

1

-2

2

1 -1 -2

0 0 0 0 1 0 0 0 0

~

0 0 0 0

0

0 0 0

300

JOHN HAPPEL AND PETER H . SELLERS

TABLE XIV

m, m2 m , m 4 m5

m,

m, mR m,

'I

'2

'3

0 0

0 0 0 0 I I 0 0 I

0 0 -1 -1 0 1 0 0 1

O

0 0 0 I I I

' 5

'6

1 1 1 1 1

0 1 0

-I

I

0

1 1 1

0 1 0

'4

'8

'9

0 1

1 1 0

I

I

0 . 0

0

1

0 0 1 0

0

0 0 0

I 0

'7

0 1

0 0

0 1 0 0

0

0

0

-I

0

1 0 1

catalyst, but instead reacts with adsorbed species. In this mechanism HI is not produced from the initial reactants and is recycled. If it is assumed that the reaction will be characterized by a single direct mechanism as well as a single rate-controlling step, one possible model is exhibited for every nonzero entry in the matrix of Table XIV. Tracer techniques with high-speed computers are useful in relaxing the requirement of a single rate-controlling step. Direct mechanisms can also be combined, if care is taken to avoid the possible occurrence of cycles. In Example 5 all nine direct mechanisms can be combined without cycle formation, though this is not always the case [Sellers (9a)I. VI.

Overall Reactions with a Multiplicity Greater Than One

Each system considered in this section has a space of overall reactions whose dimension exceeds one. In many industrial reactions involving organic substances a major product is formed, but a side reaction contributes to loss in selectivity or yield of the desired product. Such cases may be said to exhibit a multiple overall reaction, unless the ratio of desired product to by-product remains constant over a range of operating conditions, so that a simple chemical equation might be employed to express the stoichiometry. It is important to note that in these cases one cannot add up the separate direct mechanisms for all the simple reactions which add up to the overall reaction and expect, in general, to get direct mechanisms for the overall reaction, unless there are no common steps in the mechanisms of the simple reactions that form a basis for the system. The direct mechanisms for the reaction systems are, however, unique and any observed rates of appearance or disappearance of terminal species can

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

30 1

be expressed in terms of these mechanisms. If one such mechanism predominates, it will be possible to express the rates of changes of terminal species in terms of step velocities for that mechanism. A typical process is the oxidation of ethylene over silver catalyst with a side reaction to produce undesirable carbon dioxide. Miyahara (17) was among the first to appreciate the problem of assigning unique mechanisms to this system and demonstrated that an appropriate set of steps could be chosen that corresponds to a single mechanism. In this mechanism, discussed by Happel and Sellers (Z), there are seven elementary steps and five independent intermediates, and following Horiuti’s rule there are 7 - 5 = 2 independent “routes.” This is also equal to the number of independent reactions and there are no cycles. Miyahara and Yokohama (20)considered this reaction further, employing a different mechanism with four steps and two intermediates and again arrived at a single unique mechanism for the system. Both these mechanisms followed the original view that dissociative adsorption of oxygen occurred and that the epoxide was formed as a result of interaction of ethylene with one adsorbed oxygen atom. However, later results [see Patterson (21)]suggest that diatomic oxygen is involved in the formation of the epoxide. The following two examples employ such mechanisms for the purpose of illustration, although a recent survey by Sachtler et al. (22)indicates that both views are still tenable.

EXAMPLE 6. ETHYLENE OXIDESYNTHESIS (No CYCLES) The following steps are postulated for the oxidation of ethylene:

sg:

+ 0,I + C,H, * C,H,O + OI C,H4 + 601 * 2C0, + 2H,O + 61

s4:

201*

s , : 0, I*O,I s,:

0,

(31)

+ 21

where all species involving the symbol I are intermediates and the rest are terminal species. From the diagonalization of the matrix of stoichiometric coefficients given in Table XV, the general mechanism (32) and the general overall reaction (33) can be found:

+

+ + ~ ( 2 +~ 21 ~ 2+

~ ( 6 ~ 61 ~ 2 sj) P(-7CzH4

-

~ 4 )

6 0 2 + 6 C 2 H 4 0+ 2 C 0 2 + 2 H 2 0 )

+ a(-2C2H4

-

0,

+ 2C2H40)

(32) (33)

Since there are no cycles in the system, the general mechanism (32) is a direct mechanism for a multiple overall reaction (33), where p and a are unrestricted

302

JOHN HAPPEL AND PETER H. SELLERS

TABLE X V

6s, 2s,

+ 6s, + s3 + 2s, + s4

0,I

01

0 0

0 0

I

0 0

CO,

C,H4

C,H40

2 0

-1

6

-2

2

H,O

0,

2

-6

0

-I

values in both expressions. The form in which the multiple reaction (33) is written suggests that it is made up of two specific reactions operating independently, but this separation is not inherent in the chemical system. Any two basis elements for the reaction space could be used to construct an expression identical to (33). What is inherent in the system is the fact that there are four simple reactions in the space of all overall reactions. Their equations are

+ 0, * 2C,H40 + 4H,O * 2C,H40 + 5 0 , + 2H,O * C,H4 + 3 0 ,

2C,H4 4C0,

2C0,

2C0,

(34)

+ 2H,O + 5C,H4 * 6C,H40

having been determined by a procedure which is analogous to the procedure for finding direct mechanisms. Any two of them constitute a basis for the space of overall reactions and could have been used to express the general overall reaction. Notice that the first reaction, used in (331, is not simple. Let us collect terms and rewrite the general mechanism (32) and its overall reaction (33) in a more conventional way:

+ 2 0 ) ~ i+ (6p + 2 0 ) ~ 2+ + (35) ( 7 p + 20)C2H4 + (6p + 0)Oz * (6p + 20)C2H40 + 2pC02 + 2 p H 2 0 (6p

ps3

0 ~ 4

If the ratio p/rr remains constant, the last equation can be expressed with the ratio as a selectivity so that only a single free parameter remains, as recently suggested by Temkin (23). Examination of the matrix given in Table XV brings up an item of special interest. If the combination s4 of atomic oxygen were assumed not to occur, we would still be able to produce ethylene oxide by a combination of the first three steps. This scheme could place a lower limit on the selectivity at 6:7 or 85.7%, corresponding to a simple overall reaction rather than a multiple overall reaction. This serves to illustrate that we get fewer overall reactions than would be predicted by considering only the atom-by-species matrix, as a result of a more restricted choice of possible steps.

303

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

EXAMPLE 7. ETHYLENE OXIDE SYNTHFSIS ( 1 CYCLE) Let us consider a more complicated case, similar to Temkin’s proposed system (5) for ethylene oxide formation, but without any prior assumption about the direction of any step in it. The overall reaction space remains the same as in Example 6, but there are additional intermediates. In particular, acetaldehyde (CH3CHO)is an intermediate which is not bound to the catalyst. Its role still requires clarification, as indicated in recent studies by Wachs and Chersick (24, but, whether or not Temkin’s scheme proves to be correct, it illustrates our method. The steps are as follows: sl:

0,+ I* 0,I

*0,I + 1 + C,H, * 01 + CH,CHO

s, : 201

s,: s,:

0,I

+ I + C,H,O/ + C,H, * C,H,O + 01 + CH,CHO * 5 0 1 + 2C0, + 2H,O C2H,01 * 01 + C,H,

(36)

C,H,O

s s : O,/ s6: SO,/

s7:

Diagonalization of the matrix of stoichiometric coefficients gives Table XVI, from which we read off the general steady-state mechanism (37) and its overall reaction (38), where p , a, and C#J are unrestricted:

+ 3 4 s , + ( p + 30 + + d s 3 + + (2p + + 4(s4 + s,) p ( - 0, - 2C2H4 + 2C2H40)+ a( - 3 0 2 - C2H4+ 2 C 0 2 + 2 H 2 0 )

(p

$ 0 2

(37)

4)Sg

S6)

(38)

Following the method given in Section IV,B for determining those values of which make (37) into a direct mechanism, we arrive at the three direct mechanisms for the overall reaction (38), which are given by the matrix in Table XVII. Temkin (5)gave mechanism m3, but m, and m, would be equally valid choices, if all steps can occur in both directions. In interpreting his

4

TABLE XVI 0,I SI

-

s2

+ s1 + s2

2s, sA

SI

SI

+ s* + 2s,

3SI

+ 3s, + s, + S6

S,+S,+S,+S,

1 0

0 0

01 CH,CHO

-2

C,H,O/

0

0

0 0

0 2 0

0 0 0 I

1

C,H,O

-I

0 0 0

2 0 -1

-I

0

0

0

0

0 0

0 0

0 0

2 0

0

0

0

0

0

0

C,H,

0, C O , H,O

0 0 -2 0

-I

-2

-I -3

-1

0

I

-I 0

0

0 0 0 0

0

0 2

0 2

0

0

0 0 0

304

JOHN HAPPEL A N D PETER H . SELLERS

TABLE XVll SI

m, m2

p p p

m3

s2

+ 3a + 30 + 3a

’3

0

+ 3a + 3a

-p p

+ ‘h

SS

U

p-3a

U

0

U

2P

s4

+Sl

-p-3a

- 2P 0

results, Temkin concluded that step s7 should be included in a rate expression although it does not occur in mechanism m3. From the viewpoint developed in this article, we would conclude that if s, is to be employed, it must occur in the negative direction to accommodate the only possible other direct mechanisms m, and m 2 . Combinations of these three direct mechanisms which are cycle-free are discussed by Sellers (9b). There would be no essential change in the above results if we had diagonalized the matrix differently. If line 6 in the matrix of Table XVI were replaced by twice line 6 minus line 5, the matrix would still be in the appropriate diagonal form, but the general mechanism (37) would appear as (37’) and the overall reaction (38) would appear as (38’):

+ 5 0 ’ ) s , + ( p ’ + 50‘ + + 20’(s, + Sg) + 4”s4 + s7) + (2p’ - 20’ + 4 ’ ) S S p’( -02 - 2C2H4 + 2C,H40) + - 2C2H4O + 4 c o 2 + 4H2O)

(p’

4’)SZ

(37‘) (38’)

a’(-502

The matrix given in Table XVII would take the form given in Table XVIII. Each of the three mechanisms listed in Tables XVII and XVIII corresponds to elimination of the same step. The mechanisms in Table XVIII correspond to 0 = 20’ and p = p‘ - of, so we have simply changed the way of writing the two arbitrary advancement parameters without altering the mechanisms. One other item is worth noting in this example. Since s4 and s7 appear only in the cycle, to omit either one from the choice of possible steps would reduce the system to a unique direct mechanism with a multiplicity of two. But it would not be possible to eliminate further steps and still obtain a reaction among all the terminal species as we were able to do in Example 5. TABLE XVIII SI

m, m2 m3

p’ p’ p’

+ 5a’ + 5a’

+ 5a‘

s2

0 -p‘ p’

+ 7a’ +5d

s.3

+ Sh

s5

2a’

p, - 7a’

2a’

0 2p’ - 20’

2a’

s4

+ s7

- p ( - 5ar

2a’ - 2p‘ 0

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

305

Another type of reaction that has received considerable study is that of hydrocarbon isomerizations [see Pines (25)l. These and similar multicomponent systems have been the subject of an elegant kinetic analysis by Wei and Prater (26).In this method pseudomonomolecular reactions following mass action laws are assumed. With these simplifications they treat the general mechanism which includes the occurrence of all possible steps without cycle formation. Data are obtained for concentration changes of reacting components versus time and individual rate constants are calculated by a novel method of integration of the differential equations that model the systems. The Wei and Prater method has been applied to n-butene isomerizations, as well as to several other systems [see Christoffel(27)l.The following example illustrates a different way of considering such systems. In this example models are first generated in which all possible elementary step velocities may not occur.

EXAMPLE8.

ISOMERIZATION OF

BUTENES (1

CYCLE)

Isomerizations (39) among the species 1-butene, trans-2-butene, and cis-2butene are postulated: s, : C4H8-1

+ I*C4H8-II

s*: C4H8-II= C4H8-2CI

s, : C4H8-lI* C4H8-2TI s4 : s5 : S6

:

C,H8-2CI= C4H8-2TI

(39)

* C4H8-2C + I C4H8-2TI * C4H8-2T + I C4H8-2CI

Let the isomers be denoted by 1,2C, and 2T. Then diagonalizing the matrix of stoichiometric coefficients gives matrix of Table XIX. From the matrix of Table XIX we obtain the general steady-state mechanism (40) with p and a unrestricted: P(S,

+ s5) +

+ S 6 ) + ( P + 4)sz + (a - 41% + 4 s 4

(40)

whose overall reaction is as follows : p( -(C4H8 - I)

+ (C4H8- 2C)) + a( -(C4H8 - 1) + (C4H8- 2T))

(41)

By the method of Section IV,B we arrive at three direct mechanisms tabulated in the matrix of Table XX. All six elementary steps assumed possible for any mechanism are shown in Fig. 2,

306

JOHN HAPPEL AND PETER H . SELLERS

TABLE XIX 11 S1

SI

SI

+ s2 + s3

S]

+ s2 + ss + s3 + s,

s2

-

S]

s3

+ s4

2CI

2Tl

I

2C

2T

I

-I -I -I

0

0

0 0

0 0

-I -I

I

0

0

1

0

0

0 0 I

0 0

0 0

0 0

0 0

I 0

0 I

0

0

0

0

0

0

-I

-1

-I

0

TABLE XX

Wei and Prater, in treating this system, assumed in effect that the surface species would be in equilibrium with butenes in the gas phase so that only reactions s2, s 3 , and s4 were employed, corresponding to six first-order rate constants. We have used the more general form shown in Fig. 2. The three direct mechanisms in the matrix of Table XX can be diagrammed as shown in Fig. 3. The arrows in Fig. 3 show the net direction the reaction velocity steps would take for the case where a mixture of reactants is employed such that 1-butene would produce both of the 2-butenes. The steps are not assumed to

C4Hg - 1

1lS1

C4Hg-2CP

551t

C4Hg- 2C FIG.

54

C4Hg-2TP

C4Hg - 2 T

2. Possible elementary steps for isomerization of butenes. This diagram corresponds

to listing (39).

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

307

R

FIG.3. Direct mechanisms for isomerization of butenes.

be unidirectional. The directions of these two independent reactions can be calculated from thermodynamic data. If experimental data cannot be satisfactorily modeled by one of the three direct mechanisms, a combination of m, and m, will accommodate any possible values for step s4, except when s4 does not occur in either direction, but that case would have already been ruled out since sq is not contained in the direct mechanism m3. If adsorption steps s l , s 5 , and s6 were considered to be at equilibrium, as assumed by Wei and Prater, the situation would be considerably simplified. Since the 2-butenes are connected by a direct reaction, it would be possible from thermodynamics to calculate the direction of the reaction connecting them. Thus, it would be necessary at the outset to model only two direct mechanisms, namely either m, or m2, and m3. If neither of these direct mechanisms were capable of modeling the data, a combination of the two would be sufficient to evaluate all six reaction velocity constants. Such modeling would, of course, be strengthened by supplementing the usual overall reaction rate experiments by tracer data. Recent studies of the kinetics and mechanism of n-butene isomerization over lanthanum oxide by Rosynek et al. (28) indicate that for this catalyst interconversion of the two 2-butene isomers (s4 in Example 8) is very slow and in that case the system could be described by mechanism m3. Studies by Goldwasser and Hall (29) indicate that as temperature is increased, there is appreciable direct conversion via s4 so that one or both of the other two direct mechanisms may be involved. These authors suggest that further studies with all three isomers, at several temperatures and with tracers, would be desirable. The monomolecular conversion of three components has also been considered in some detail by Kallo (30). Rate equations based on Langmuir adsorption were developed assuming a number of different mechanistic schemes including steps in which surface adsorption was not at equilibrium. Since the rate equations developed became complicated, practical application was devoted to cases in which only initial reaction rates were observed, so

308

JOHN HAPPEL AND PETER H. SELLERS

that the final steps involving product formation could be considered unidirectional. The scheme shown in Fig. 2 is one of a number of alternatives considered by Kallo (his scheme VII), and that corresponding to s4 = 0 in Fig. 3 was also identified by him, but not the alternatives with s2 = 0 and s3 = 0. A number of additional alternates could be developed following our procedure, consistent with an appropriate choice of possible elementary steps. Dehydrogenation, hydrogenation, and aromatization of hydrocarbons have also been widely studied [see Pines (25)]and applied industrially. Since olefinic compounds produced by such reactions can simultaneously react to form isomers, it is of interest to explore effects of such combined reactions on the mechanisms involved. Model studies of hydrogenation-dehydrogenation of the n-butenex-butene:hydrogensystem by Happel et al. (31)and of the isobutane:isobutene:hydrogensystem by Happel et al. (32) showed that they are governed by different kinetics. This seems to be due to the occurrence of isomerization reactions in the former case. Studies by Hnatow (33) indicated that in the case of n-butane dehydrogenation 1-butene is the primary product. However, the 2-butenes hydrogenate more rapidly than 1-butene. The following example illustrates how mechanisms can be developed to reflect these observations.

EXAMPLE 9.

BUTANE DEHYDROGENATION (3 CYCLES)

Following a variation of the well-known Horiuti-Polanyi mechanism, we consider the following steps as possible for the system n-butane-n-buteneshydrogen over chromia-alumina catalyst : C4Hl0

+ I*

C4Hl0I

+ I*H,I H21 + I* 2HI C4Hl0I + I* C4H9-II + HI C4HloI + I* C4H9-2/+ HI C4H9-11 + I*C4H,-II + HI C4H9-21 + I = C4H,-2CI + HI C4H9-21+ I * C4H8-2TI + HI H,

C4H8-II* C4H,-2CI C,H,-II* C,H,-ZTI* C4H8-2CI C4H,-11* C4H8-2CI

C4H8-2TI C4H8-2T

* C4H,-2C C4H8-I

+I +I

+I

= C,H,-ZTI

TABLE XXI C,H8-2Tl

C4H,-2CI

C,H,-I1

C4H,-21 -1 -1

se

+ s5 + S8 + S I 1 + s5 + + sl, + s4 + Sb + SI3 - s5 + - S) + s9

S)

-

S8

- s,

'7

-

'8

+ '14

s1 - s, - s3 s1 - sz - s3 S6

S)

-- 1

1

0

+ SIO

0 0 0

0 0

0

0 0

0

0

0

0

0 0

0 0

0 0 0

0 0

0

0 0

0 0 0 0 0 s1 - s1 - s3

C4H9-II

0

0

1

H,I 0 0 0 0 0

C4HlOI HI 0 0 0 -1 -1

I

C4H8-2T

1 1

-1 -1

1

-I

1 1

-1 -1

-1

C4H,-2C

C4H,, 0 0 0 0 0

C4H,-I 0 0 0 0 0 0 0 0

H, 0 0

0 0 0 -1

0

1

0

0

0

0

0

I 0

0 0 2

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

1

0

-1

0

1

0 0

1

-1 -1

0 1

1 1

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0

0 0 0

0 0 0

0

-1 -2

-1

0

0

0

0 0 0

0 0 0

0 -I

310

JOHN HAPPEL A N D PETER H . SELLERS

Diagonalization of the matrix of stoichiometric coefficients in (42) gives the matrix of Table XXI, from which we read off the steady-state mechanism (43) and its overall reaction (44), which has a multiplicity of three:

+ + s g + + a(s1 - s2 - + + + + T(s1 - s2 + + + + 4(s4 - + - + + s9 + + - f =(P + + T)(S1 - S2 psi1 + as12 + T S 1 3 ( P f a - 4 ) S g + (T 4 x S 4 + (a - 4 -t X +

p ( s , - s2 - s3

- s3

x(s7

s3

s11)

sg

sg

$(s7

slO)

s8 -

s13)

s6

s8

s5

s7

s12)

s5

s6

s7

sg)

s14)

S3)

Sg)

+(P - x

-

Il/ls8

$Is7

+ (4 - d S 9 + X s 1 0

(43)

f $s14

+ a(-C,Hio + (C4H8 - 2C) + + ~ ( - C 4 H l o+ (C4H8 - 1) + H2)

P ( - C ~ H , O+ (C4Hg - 2T) +

H2)

H2)

(44)

Now let us determine all the direct mechanisms for reaction (44) in detail to illustrate the method introduced in Case 4 of Section IV,B. Each direct mechanism for reaction (44) must depend on the parameters p, a, and T , because they appear in the reaction (44), but 4, X, and $ must be evaluated, which is done by means of formula (19) in Section IV,B. Start with the general mechanism (43) and extract from it the cycle the matrix of Table XXII and one mechanism ( 4 9 , written as a row vector: TABLE XXll

4 x

*

0

0 0 p+(T+T

0 0 0 /3

0 0 0

0 0 0

-1

T

p+a

0 0

1

0 0 't

-I

I

-I

0

1 - 1

a

p

1

-1 0

0

0

0

I

0

0

1

0

0

(45) The cycle matrix of Table XXll is a tabulation of mechanism (43) with y = 0, a = 0, and T = 0, and the row vector (51) consists of the coeEients in (43) with 4 = 0, x = 0, and Il/ = 0. Any three independent cycles could have been chosen to generate Table XXII and any mechanism for the overall reaction could have been chosen to establish the row vector (45).The choices we made are arbitrary and depend on the diagonalization procedure used to find the matrix of Table XXI, which is far from unique. The important point is that the list of direct mechanisms we are looking for is unique and independent of how the above choices are made. Starting from the left side of the matrix of Table XXII, choose the first three columns which constitute a nonsingular 3 x 3 matrix, which will be

31 1

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

those headed by ss, s7, and s,. In these three columns of Table XXII and row vector ( 4 9 , we find matrix (46) and row vector (47):

I/ = (

p

+ 0,0,0)

(47)

Apply formula (19) to these to get an evaluation for (4,x, $) as follows:

\

0

-1

1/

+ 6,p + 0,-0) (48) Put these values, that is, 4 = p + 0,x = p + 0,and $ = -6, in the general =(

p

mechanism (43), which gives the first direct mechanism. It appears as the first row of the matrix in Table XXIII. The subsequent rows are determined in the same way with an additional procedure to avoid repetitions, which can be explained by first looking at the row m3 in Table XXIII, which is the direct mechanism corresponding to the columns s 5 , s,, s 1 4 .Since s I 4 is the last column, the next three columns to consider are s s , s8, s,. Before testing the 3 x 3 submatrix M of the matrix of Table XXII with those columns for nonsingularity, look for any mechanisms already listed in Table XXIII which have zeros in those columns. If any are found, dismiss that choice of columns. Mechanism m, has zeros in columns ss, s8, s9, wfiich means that this choice of columns would lead to the same mechanism as m , , unless M was singular, in which case we would also want to dismiss this choice. Thus, we avoid the test for nonsingularity in a case such as this. The next two choices would be s 5 , s 8 ,s l 0 and s S r sg, s14,which would be dismissed for the same reason. The next choice is ss, sg, s l 0 ,for which matrix M is singular. Finally, the next choice s 5 , s,, s14is accepted, and it determines the direct mechanism m4. A continuation of this procedure gives the complete set of direct mechanisms through m18. Mechanism m18 corresponds to the case where the isomerization steps s9, s l 0 , and s14are assumed to be very slow compared with steps of hydrogenation and dehydrogenation. In that case, if p and r~ were taken as negative and 5 as positive, we would model a situation in which n-butane was dehydrogenating to produce l-butene and at the same time the 2-butenes were being hydrogenated to produce n-butane. This qualitatively follows the observations of Hnatow (33). This mechanism is of interest because it calls attention to the limitation of a rule of thumb sometimes employed in catalytic research. It has at times

TABLE XXIII (si

-

'2

-

'3)

'11

'12

'13

'5.

(s4

+ '6)

s7

'8

s9

'10

0

P + U

m1 m 2

U

m4

0

m 5

0 -'I

m 8

U

0

m9 m10 m11

-p

m12 ml 3

- 'I

0

-T

m14

0

m15

U

m16 m17 m18

0

-U

-T 0 - T -T

0 0 0

-U

0

P + T

0

-T

0 0 0

P + U

0

m 7

P

P

P + U

m6

-U

0

P + O

m 3

' 1 4

P -'I

0 0 0 0 P

0

P

0 0 0 -U

0 P

0 0

T

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

313

been suggested that a useful approach to improve catalyst development may be to study the formation of reactants from products instead of the desired forward reaction [see Bond (34)].This example shows how this idea may need to be modified for mechanisms involving isomer formation. Another point illustrated by Table XXIII is the need to carefully consider the effect of lumping isomers for convenience, when mechanistic models are generated. Thus, in Example 4, isomerization of 1-butene is neglected in selecting the elementary steps for butadiene production. In effect, it is assumed that all intermediates are indistinguishable whether 1-butene or a mixture of n-butenes reacts. If that scheme were used in the present case, we could consider a model in which only sl, s2,s3, s4, s6, and s I 3were retained, which would correspond to only a single direct mechanism. However, if instead we chose to retain all the elementary steps as possibilities except s l , and s12,we would obtain five direct mechanisms for a system producing only 1-butene (in which p = (T = 0). Reactions involving the catalytic hydrogenation of carbon monoxide to produce hydrocarbons and oxygenated products are important for chemical and fuel production from coal. The following example, in which the methanation of synthesis gas is simulated, illustrates a typical system.

EXAMPLE 10. METHANATION OF SYNTHESIS GAS(3

CYCLES)

A 16-step system for methanation over a nickel catalyst is selected that includes many features discussed in the literature :

+ I*CI + 01 + HI* CHI + I CHI + HI* CH,I + I CH,I + HI * CH,I + I CH,I + HI* CH, + 21 OH/ + HI* H,O + 21 co, + I* COJ co + I*COI COI CI

+ 21* 2HI C0,I + HI* CHOOI + I CHOOI + HI* CHOI + OH1 01 + HI* OH1 + I COI + OI* C0,I + I CHOOI + I * OH1 + COI CHOI + HI* CHI + OH1 COI + HI*CHO/+ I H,

(49)

314

JOHN HAPPEL A N D PETER H. SELLERS

It is by no means exhaustive but does exhibit the main possibilities for reaction mechanisms. The idea of a CHO,,, intermediate has been advocated by Pichler and Schultz (35),although other intermediates such as CHOH,,, could have been equally well postulated to take into account the hypothesis that C O dissociation might be assisted by hydrogen. The water gas shift reaction is considered to occur via the CHOO,,, intermediate according to studies of Oki et al. (36).The hydrogenation of carbidic species by atomic rather than molecular hydrogen follows the findings of Happel et a/. (37). By diagonalizing the matrix of stoichiometric coefficients in (49), we get the matrix of Table XXIV, from which we can read off the overall reaction

+ 0(-2CO

-

2Hz

+ CH4 + C 0 2 )

( 50)

which has a multiplicity of two. Then we can apply the methods of Section IV,B to determine all the direct mechanisms for r (shown in Table XXV) from which are omitted the following steps with constant coefficients:

+ 0)(s3 + s4 + +

+ ( p + 20)sS + (3p + 20)s9 (51) The maximum number of steps is obtained ( H + R = 13) for m,, m I 2 , (p

sg)

ps6

- cs7

m I 3 , m14, and m15.If the CHOI is thought to be unlikely [see Yates and Cavanagh (38)],we are left with only m a , m,,, and m, for consideration, so it can be seen that establishment of the presence or absence of such an intermediate is important. Mechanism ma is probably the simplest since it does not require the CHOOl intermediate that appears in the water gas shift reaction. However, if it were assumed that over certain catalysts the reaction of O/ with COI ( ~ 1 3 )could not occur, then mechanism m,, should be considered as a possibility. In this example we only listed the following two overall simple chemical reactions to represent the system: r,:

3H,

r,:

2H,

+ CO * CH, + H,O + 2CO * CH, + CO,

Altogether there are five simple overall reactions in the system, the others being as follows :

+ H,O * CO, + H, (water gas shift) CO, + 4H, * CH, + 2H,O (Sabatier) 4CO + 2H,O + CH, + 3C0, (Kolbel-Engelhardt)

r 3 : CO

r4: rs :

(53)

Any 2 of these 5 equations could be employed to represent the overall reaction with a multiplicity of 2 and the same 15 direct mechanisms would be obtained. In this example we expressed the general overall reaction as

I

l

l

oooooooo--m I

0 0 0 0 0 0 0 - 0 0 0

I

I

1

0 0 0 0 0 0 - 0 0 N N

m w I I I

I

- N

0 -

--

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0 0 - 0 0 0 0 0 0

- 0

0 0 0

I

3 0

I 0 0 0

I 3 0

0 0 0

I

3 0

0 0 0

I

3 0

0 0 0

I

I

0 - - 0 0 0 0 0 0 0 0

I

0 0 - - 0 0 0 0 0 0 0

I

0 0 0 - - 0 0 0 0 0 0

I

- 0 0 0 0 0 0 - 0 0 0

I

oooooooooom

3 0

0 0 0

- - - - N N - - N N m

O O O O O O O O O N O

D O

0 0 0

I

3 0

0 0 0

I

3 0

0 0 0

I

0 - - - - - 0 0 N 0 0

3 0

0 0 0

I

0 0 0 0 0 0 - 0 0 0 0

3 0

0 0 0

I

0 0 0 0 0 - 0 0 0 0 0

3 0

I

- - 0 0 0 0 0 0 0 0 0

I

m

m

-

0 0 0

N - 0 1-

-

3 0

- 0 0 0 0 0 0 0 0 0 0

m

N

c

m

N

mm

' D P

m v I

t +

v)N

m m

::

'1

m

N

m-

m-

m

o+

0 1

m-

I

I

p

+ " Z

m-

m N

N I

m-

lo+ v)-

n

b c

s

+

II

m

-

m

N

n m

v)

d

m

m

- + I f + my + + d, + + + 7 A: + - I y + 4

mo" vm)w N

mm

++,

++

t l + -

316

JOHN HAPPEL AND PETER H. SELLERS

TABLE XXV 6 1

+ s2) 0 0 0 0 0 0

m1 m2

%I

0 0

p+a

-a -a -a P + U

a

0 0 0

P + O

P

a

P + U

p + a m12

810

P+

--d

P fP

0 -a p + a P+a

0 0 P

0 0 P

tP

tP +a

mi4

/J + O

-0

-a

p+2a

-a

-a

'1.3

0 0 0 -p

- 2a

0 P

0 0 P+a

0 0

P+a p+2a

'14

s1s

0

P + U

a

-a -a

0

m13

ml5

s12

p

a

- p - a

0 0 0

-a

+ 20 a a a

P + U

0

-U

P+a fP +

0 0

0 - 2a 0 0 0 -P P

-- P

=

0 0 0 0

p+a p+a P + O

p + a P + a

P

0 P

0 0 0

PI2 0 -a

'16

P + O

0 p + a p 0 2a

+

0 P

0 0 0 0 -P

0 a

0

pr, + or,. If we had separately calculated direct mechanisms for r l and r2 instead of considering the combined reaction system, i.e., by leaving out the terminal species CO, for r1 and H,O for r,, we would have obtained 7 direct mechanisms for rl and 10 direct mechanisms for r,. Therefore, if we took m, as 1 of the 7 direct mechanisms for rl and m, as 1 of the 10 direct mechanisms for I-,, then there would be 70 combinations of the form pm, am2. However, if we proceeded to examine the result obtained in this manner, we would find that only 15 of them are direct mechanisms and the other 55 are combinations of 2 of the 15 without any canceled steps. Similarly, the numbers of direct mechanisms for r3, r4, and r5 are 7, 10, and 15, respectively. Obtaining the direct mechanisms for the multiple reaction at the outset is clearly a more effective procedure than combining results for separate individual simple reactions. In fact, without the complete list of possible steps for all reactions, it would not be possible to know whether the complete list of direct mechanisms for each had been derived. The reverse reaction, steam cracking of methane, involves the same elementary steps as the methanation reaction. The kinetics for that reaction have been developed for a single direct mechanism by Snagovskii and Ostrovskii (39). A recent comprehensive review of a set of elementary reactions that can be chosen to represent the Fischer-Tropsch synthesis has been presented by Rofer-De Poorter (40). Such sets of elementary reactions form the starting point for our treatment as discussed in Section 11.

+

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

VII.

317

Discussion

This survey has been concerned with the enumeration of all possible mechanisms for a complex chemical reaction system based on the assumption of given elementary reaction steps and species. The procedure presented for such identification has been directly applied to a number of examples in the field of heterogeneous catalysis. Application to other areas is clearly indicated. These would include complex homogeneous reaction systems, many of which are characterized by the presence of intermediates acting as catalysts or free radicals. Enzyme catalysis should also be amenable to this approach. The subject of reaction mechanism also has a bearing on other fundamental problems of physical chemistry. In the following two sections the relationship of the material presented here to thermodynamics and chemical kinetics is considered. A. THERMODYNAMICS In carrying out the procedure for determining mechanisms that is presented here, one obtains a set of independent chemical reactions among the terminal species in addition to the set of reaction mechanisms. This set of reactions furnishes a fundamental basis for determination of the components to be employed in Gibbs’ phase rule, which forms the foundation of thermodynamic equilibrium theory. This is possible because the specification of possible elementary steps to be employed in a system presents a unique a priori resolution of the number of components in the Gibbs sense. The number of comporienfs as defined for application in the phase rule is equal to the number of terminal species minus the number of independent chemical reactions and minus the number of any restrictive conditions, such as material balance or charge neutrality. The number of independent reactions includes only those that actually occur under the conditions in question. For example, Gibbs (41), in citing the case of equilibrium in a vessel containing water and free hydrogen and oxygen, states that we should be obliged to recognize three components in the gaseous part since the reaction H, + +02 H,O is assumed not to occur (i.e.,3 - 0). If, however, a suitable catalyst were present, or if the temperature were high enough for reaction (2),the system would reduce to 3 - 1 = 2 components. If we imposed the condition that all the H, and O2 came from dissociation of H 2 0 , we would have a system of only one component. Thus, Gibbs clearly recognized that it was the reactions that would actually occur that should be employed in thermodynamic calculations. A method for calculating the number of independent reactions discussed

318

JOHN HAPPEL A N D PETER H. SELLERS

by Gibbs has been called Gibbs’ rule of stoichiometry by Christiansen (3). Aris and Mah (42) have discussed this rule at length and stated that it gives an upper bound to the number of independent reactions. This method is based on determination of the number of independent reactions by means of an atom by species matrix, as also developed by Amundson (43).Such a method will give the maximum number of independent reactions, assuming no isomerizations, rather than those required by the phase rule itself. Aris and Mah developed an ingenious scheme whereby experimental observations of rates of change of species consumed and produced can be used to determine an observational matrix which will show how many independent reactions are actually accounting for the observed composition changes. They applied this procedure to an example given by Beek (44) in which ethylene was converted to ethylene oxide as well as to CO, and HzO, so that an atom-species matrix would correspond to at most two independent overall reactions, similar to Examples 6 and 7 in Section VI of this article. It was shown that the data indeed were consistent with two independent overall reactions, but this procedure may be difficult to apply practically given the usual uncertainties in kinetic data. Bjornbom (45)has discussed the relation between reaction mechanism and the stoichiometric behavior of chemical reactions. He pointed out that the set of linearly independent reactions which is obtained from the atom-by species matrix may be larger than the number of independent reactions required to describe an actual physical system because the latter must be related to its reaction mechanism. He gave several examples of complex homogeneous reaction systems including a trial-and-error procedure for hydroperoxide decomposition, based on data obtained by Hiatt and coworkers (46).The system considered was the metal catalyzed decomposition of secondary butyl hydroperoxide in n-pentene solution to give secondary ethyl butyl alcohol, methyl butyl ketone, water and oxygen. The atom-byspecies matrix analysis showed that the maximum number of independent reactions was two, but by forming matrices of the intermediates he showed that the number of independent reactions was actually equal to one. Then by trying linear combinations of the elementary reaction such that the intermediates cancel, he found the single reaction consistent with Hiatt’s experiments: 3C4H,00H

* 2C4H,0H + CH,COC,H, + 0, + H,O

(54)

The procedure described here is consistent with his approach but is simpler to use in more complicated cases. Often thermodynamic calculations for complex systems are made assuming that all chemical changes can take place that are allowed within the framework of the atomic material balances. This approximation may be appropriate at high temperatures but is often not true for catalytic systems.

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

319

Smith (47)and Bjornbom (48)have discussed the introduction of restrictions into equilibrium calculations in addition to the elemental abundance constraints. Often only a restricted set of terminal species is chosen, but it would seem logical in choosing such additional restrictions to revert to Gibbs’ original idea of specifying possible reactions as well as possible species. The choice of elementary reactions, rather than of overall reactions with complicated stoichiometry, would be simplified by modern developments in theoretical chemistry and surface science. B. KINETICS Our object has been to enumerate all sets of steps corresponding to possible direct mechanisms. Insight into how to choose the elementary steps themselves can often be obtained from physicochemical principles and experimental surface examination as well as from rate data. This information will also throw light on the most likely mechanisms from among those generated. The magnitudes of concentrations of intermediates and of step velocities appearing in these mechanisms are the parameters in kinetic models that form the next step for further discrimination. A detailed treatment of model building for this purpose is beyond the scope of this article. The subject is briefly discussed here in the context of the methods presented. At the outset it may also be advantageous to consider problems of structural identifiability and distinguishability. A model is not identifiable if in principle it is impossible to determine the desired parameters on the basis of proposed data to be obtained even if there is no experimental uncertainty or inadequacy in computer programming. Even when it is theoretically possible to determine the parameters for a given model, it may also be possible to compute those for a competing model from the same data and the models will then be indistinguishable. Resolution of these problems is often not simple. The subject is discussed in an advanced monograph by Walter (4Y) as well as in papers by Park and Himmelblau (49a)and Walter et al. (4%). Most standard chemical engineering tests on kinetics [see those of Carberry (50), Smith (52), Froment and Bischoff (19), and Hill (52)],omitting such considerations, proceed directly to comprehensive treatment of the subject of parameter estimation in heterogeneous catalysis in terms of rate equations based on LHHW models for simple overall reactions, as discussed earlier. The data used consist of overall reaction velocities obtained under varying conditions of temperature, pressure, and concentrations of reacting species. There seems to be no presentation of a systematic method for initial consideration of the possible mechanisms to be modeled. Details of the methodology for discrimination and parameter estimation among models chosen have been discussed by Bart (53)from a mathematical standpoint.

3 20

JOHN HAPPEL AND PETER H. SELLERS

Information on the steps in a reaction mechanism can be extended significantly by isotopic tracer measurements, especially by transient tracing [see Happel et al. (54,55)].Studies by Temkin and Horiuti previously referenced here have been confined to steady-state isotopic transfer techniques. Modeling with transient isotope data is often more useful since it enables direct determination of concentrations of intermediates as well as elementary step velocities. When kinetic rate equations alone are used for modeling, determination of these parameters is more indirect. The use of tracers in this manner has also been considered by Le Cardinal et al. (56), with special reference to homogeneous systems, and discussed by Happel (57) and Le Cardinal (58).Such an approach parallels the viewpoint of Aris and Mah (42) in which they distinguished between the kinematics and kinetics of overall reactions. Rates of change of species are considered without reference to their correlation in terms of rate equations related to particular physical conditions. To summarize, the type of information that has been presented in this review should be useful in furnishing a logical first step in comprehensive understanding of complex chemical reaction systems. Consideration of a chemical system in terms of unique direct reaction mechanisms required to produce observable rates of change of terminal species has distinct advantages, especially when multiple overall reactions are involved. The required necessary assumptions regarding possible elementary reaction steps are becoming increasingly accessible through modern tools for surface spectroscopy and fundamental theories of chemical kinetics of elementary reaction steps. A number of examples worked out in detail illustrate that the procedure can be rather readily followed even if one does not wish to go into mathematical details. In many cases insights are obtained that are not immediately obvious from more superficial considerations.

VIII. A

(4

C=S-Q

H I I M m m, N n,

P=S- H

List of Symbols

Number of species in chemical system. ith species in chemical system. Dimension of cycle space. Rank of matrix of stoichiometric coefficients of intermediates only. Number of intermediates in chemical system. Active site on a catalyst. C x C submatrix of matrix (oij). Mechanism. ith element in a basis for mechanism space. Dimension of unspecified vector space. ith cycle. Dimension of steady-state mechanism space.

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

Q

R = Q - H

R R(m) ri = R(si) S S Si

T = A - I V

aij Bij

Yij

Pi

v..

P U bi

T T..

i’ 4i x

*

32 1

Dimension of reaction space. Dimension of overall reaction space. Linear transformation of mechanism space to reaction space. Reaction produced by mechanism m. ith elementary reaction in chemical system. Dimension of mechanism space; also the number of steps (types of molecular interaction) in chemical system. Step in a mechanism. ith step (type of molecular interaction) in chemical system. Number of terminal spacies in chemical system. Row vector for a steady-state mechanism. Stoichiometric coefficient of species a j in elementary reaction T i . Stoichiometric coefficient of species a j in overall reaction R(mi). Net rate of advancement of step sj in steady-state mechanism mi. Coefficient of mi in a mechanism m. Net rate of advancement of step sj in cycle ni. Rate of advancement of a mechanism. Rate of advancement of a mechanism. Coefficient of step si in a mechanism m. Rate of advancement of a mechanism. Net rate of advancement of step sj in the ith direct mechanism. Rate of advancement of a cycle. Coefficient of cycle ni in a cycle. Rate of advancement of a cycle. Rate of advancement of a cycle. REFERENCEB

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JOHN HAPPEL AND PETER H. SELLERS

16. Temkin, M. I., I n / . Chrm. Eng. 16, 264 (1976). 17. Miyahara, K . , J. Res. Inst. Catal. Hokkaido Uniu. 17, 219 (1969). 18. Hougen, 0. A,, and Watson, K . M., “Chemical Process Principles,” Vol. 3. Wiley, New

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New York, 1979.

20. Miyahara, K.. and Yokohama, S., J . Res. Inst. Catal. Hokkaido Uniu. 19, 127 (1972). 21. Patterson, W. R., in “Catalysis and Chemical Processes” (R.Pearce and W. R. Patterson, 22. 23. 24. 25.

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44. 45. 46. 47. 48. 49. 49a. 496.

50.

eds.), p. 253. Blackie, London, 1981. Sachtler, W. M. H., Backx,C.,and Van Santen, R. A,, Catul. Rev. Sci. Eny. 23,127(1981). Temkin, M. I . , Kinet. C a d . Engl. Trans. 22,409 (1981). Wachs, I . E., and Chersick, C. C., J. Card. 72, 160 (1981). Pines, H., “The Chemistry of Catalytic Hydrocarbon Conversions.” Academic Press, New York, 1981. Wei, J., and Prater, C. D., Adu. Catal. 13, 206 (1962). Christoffel. E. G . . Catal. Reu. Sci. Eng. 24, 159 (1982). Rosynek, M. P., Fox, J. S., and Jensen, J. L., J . Catal. 71.64 (1981). Goldwasser, J., and Hall, W. K., J . Catal. 71, 53 (1981). Kallo, D., J. Catal. 66, 1 (1 980). Happel, J . , Hnatow, M. A., and Mezaki, R., Adu. Chem. Ser. (97). 92 (1970). Happel, J . , Kamholz, K., Walsh, D., and Strangio, V., I&EC Fundam. 12, 263 (1973). Hnatow, M. A., Ph.D. thesis, New York University, 1970. Bond, G . C., “Heterogeneous Catalysis-Principles and Applications,” p. I I I . Clarendon, Oxford, 1974. Pichler, H., and Schultz, H., Chrm. l n g . Tech. 42, I162 (1970). Oki, S., Happel, J., Hnatow, M. A., and Kaneko, Y., Proc. In!. Congr. Caral., 5th. p. 173 (1972). Happel, J., Fthenakis, V., Suzuki, I . , Yoshida, T., and Ozawa, S . , Proc. Inr. Congr. C u td ,, 7th, Tokyo p. 542 (1981). Yates, J. T., Jr., and Cavanagh. R.R., J . Caral. 74,97 (1982). Snagovskii. Y. S., and Ostrovskii, G . M., “Modeling the Kinetics of Heterogeneous Catalytic Processes” (in Russian). p. 232. Khimia, Moscow, 1976. Rofer-De Poorter, C. K., Chem. Rev. 81, 447 (1981). Gibbs, J. W., “The Scientific Papers of J. Willard Gibbs,” Vol. 1, Thermodynamics, pp, 63. 70, 138. Dover, New York, 1961. Aris. R.,and Mah, R. H. S . , Ind. Eng. Fundam. 2,90 (1963). Amundson, N.R.. “Mathematical Methods in Chemical Engineering,” p. SO. PrenticeHall, Englewood Cliffs, New Jersey, 1966. Beek, J . , A h ) . Chem. Eng. 3,204 (1962). Bjornbom, P. H., AIChEJ. 20,285 (1977). Hiatt, R., Irwin, K. C., and Gould, C. W., J . Org. Chem. 33, 1430 (1968). Smith, W. R.. Ind. Eng. Chem. Fundam. 19, l(1980). Bjornbom, P.. Ind. Eny. Chem. Fundam. 20, 161 (1981). Walter, E.. “Identifiability of State Space Models.” Lecture notes in “Biomathematics.” Vol. 46. Springer-Verlag, Berlin and New York, 1982. Park, S. W., and Himmelblau, D, M., Chem. Eng. J . 25, 163 (1982). Walter, E., Le Courtier, Y., and Happel, J., 1EEE Trans. Autom. Conrrol, in press (1983). Carberry, J. J., “Chemical and Catalytic Reaction Engineering.” McGraw-Hill, New York. 1976.

ANALYSIS OF MECHANISMS FOR REACTION SYSTEMS

323

51. Smith, J. M., “Chemical Engineering Kinetics,” 3rd ed. McGraw-Hill, New York, 1981. 52. Hill, C. G., Jr., “An Introduction to Chemical Engineering Kinetics and Reactor Design.” Wiley, New York, 1977. 53. Bard, G., “Nonlinear Parameter Estimation.” Academic Press, New York, 1974. 54. Happel, J., Suzuki, I., Kokayeff, P., and Fthenakis, V.,J. C u d . 65, 59 (1980). 55. Happel, J., Cheh, H. Y., Otarod, M., Ozawa, S., Severdia, A. J., Yoshida, T., and Fthenakis, V.,J. Curd. 75, 314 (1982). 56. Le Cardinal, G., Walter, E., Bertrand, P., Zoulalian, A,, and Gelas, M., Chem. Eng. Sci. 32, 733 (1977). 57. Happel, J., Chem. Eng. Sci. 33, 1567 (1978). 58. Le Cardinal, G., Chem. Eng. Sci. 33, 1568 (1978).