De Donder Relations in Mechanistic and Kinetic Analysis of

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Ind. Eng. Chem. Res. 2001, 40, 2416-2427

De Donder Relations in Mechanistic and Kinetic Analysis of Heterogeneous Catalytic Reactions Ilie Fishtik and Ravindra Datta* Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts 01609-2280

The equations relating the rates of the elementary reactions to their affinities, known as the De Donder relations, are utilized within the theory of direct reaction routes (RRs). It is shown that any substitution of the surface coverages of the intermediates with the affinities of the elementary reactions is equivalent to a RR. Two types of direct RRs are defined. One of these produces “intermediate” reactions involving the terminal species (reactants and products) and only one surface intermediate, while the other produces “overall” reactions involving only terminal species. This formalism provides simple relations between the affinities of the elementary reactions and affinities of the reactions produced by RRs. As a consequence, De Donder relations may be naturally partitioned into contributions coming from a finite and unique number of overall reactions produced by RRs. The results are applied to the particular case of rate-determiningstep approximation in kinetics. Introduction Provided that the set of elementary reactions along with their rate constants for a heterogeneous catalytic reaction is known, i.e., the microkinetic model,1-4 the numerical solution of the governing kinetic equations may, in principle, be easily obtained. This situation has become almost routine in modeling some gas-phase chemical processes that can include hundreds and thousands of elementary reactions, e.g., combustion or atmospheric pollution modeling.5,6 In heterogeneous catalysis, unfortunately, the lack of reliable rate constants makes such detailed kinetic modeling more difficult. Yet, the recent developments in microkinetic modeling of heterogeneous catalytic processes are very promising in that rate constants on different materials are becoming increasingly available.3,7,8 Boudart recently stated that “...the 21st century will be, for kinetics, the century of rate constants.”9 It may, thus, be expected that the progress in experimental techniques and quantum-chemical calculations in the coming years will provide databases comprising reliable rate constants for elementary reactions of many complex heterogeneous catalytic processes. The task is challenging, however, because of the many different catalytic materials that are possible. Nonetheless, the researchers will then face problems similar to those that now exist in the modeling of complex gas-phase processes, i.e., simplification, reduction, and rationalization of unwieldy mechanisms to facilitate comprehending, explaining, and predicting the kinetic behavior of complex reaction systems. The main approximations that conventionally are used to simplify the detailed chemistry are3,10 (a) the pseudo-steady-state (PSS) approximation, (b) quasiequilibrium (QE) and rate-determining-step (RDS) approximations, (c) the irreversible step (IS) approximation, and (d) the most abundant reactive intermediate (MARI) approximation. These approximations also provide incisive physicochemical insights into the mechanism, as described in two recent papers by Dumesic11 and Stoltze.3 Thus, as shown by Stoltze,3 within the RDS approximation many

characteristics of the microkinetic mechanisms, e.g., rate equations (De Donder relations), apparent activation energies, apparent reaction orders, etc., may be naturally partitioned into a sum of contributions associated with a special class of overall reactions involving only one surface intermediate. These reactions result when the QE conditions are algebraically solved for the surface coverages of the intermediates. Dumesic11 extended the idea of partitioning De Donder relations into contributions associated with overall reactions to general systems by expressing the surface coverages of the intermediates in terms of affinities of the elementary reactions. Remarkably, these simple transformations lead to several important insights into the kinetics of heterogeneous catalytic processes.3,11 In this work, we address an aspect of the approach outlined by Dumesic and Stoltze that has apparently been overlooked, namely, the arbitrariness of the partitioning of De Donder relations into contributions associated with different types of overall reactions. Thus, in complex heterogeneous catalytic reaction systems, the surface coverages of the intermediates may be written in terms of the affinities of the elementary reactions in many different (strictly speaking, in an infinite number of) ways. As a consequence, De Donder relations may be partitioned into contributions associated with arbitrarily selected overall reactions. Obviously, there is no mathematical error in this arbitrariness because De Donder relations are (and must be!) independent of the partitioning. However, an arbitrary set of De Donder relations is neither an essential feature of a chemical system nor an explanation of anything. This conclusion plainly compels us to revisit the partitioning procedure of De Donder relations and to look for its uniqueness. Notation and Definitions Consider the general case of a heterogeneous catalytic chemical reaction system. The species comprising the elementary reactions that describe the detailed chemistry of the catalytic process are explicitly divided into active sites on the surface of the catalyst S, intermedi-

10.1021/ie000871h CCC: $20.00 © 2001 American Chemical Society Published on Web 05/08/2001

Ind. Eng. Chem. Res., Vol. 40, No. 11, 2001 2417

ates (surface species) I1, I2, ..., Iq, and terminal (gasphase reactants and products) species T1, T2, ...,Tn. Thus, the process is described by the following set of p elementary reactions: q

sj ) Rj0S +

where the arrows f and r designate reactants and products, respectively. It is to be noted that, within this notation, all of the stoichiometric coefficients take on positive values. Then, the rates of the elementary reactions are given by

n

βjiTi ) 0; ∑ RjkIk + ∑ i)1

j ) 1, 2, ..., p (1)

rj ) b rj - a rj ) b r j[1 - exp(-Aj/RT)]

(6)

k)1

As usual, it is assumed that the stoichiometric coefficients Rj0, Rjs, and βji take positive values for products and negative values for reactants. The rank of the stoichiometric matrix is

and, consequently, the elementary reactions are not all linearly independent. Each of the elementary reactions is characterized by its affinity Aj (j ) 1, 2, ..., p) that is defined as usual as12

-

1 RT

where q

∏ k)1

b rj ) B k jθ0Rbj0

∑ Rjk ln θk +

k)1

n

βji ln Pi ∑ i)1

(2)

Here Kj is the equilibrium constant of the jth elementary reaction, θ0 is the fraction of the free (uncovered) surface of the catalyst, θk is the fraction of the surface covered by the intermediate Ik, and Pi is the partial pressure of the terminal species Ti. The coverages are subject to the surface site balance q

θ0 +

∑ θk ) 1

(3)

k)1

It may be noticed that eq 3 implies that the surface intermediates occupy one active site each on the surface. Further, by virtue of the site balance, eq 3, the active sites S and the surface intermediates I1, I2, ..., Iq are linearly dependent, i.e., q

Rj0 +

∑ Rjk ) 0;

j ) 1, 2, ..., p

(4)

k)1

This condition needs to be taken into account whenever the elementary reactions are linearly combined so as to eliminate the surface intermediates. In particular, when all of the surface intermediates are eliminated, then, by virtue of eq 3, the active site S is also eliminated. Similarly, a reaction involving only one surface intermediate will also involve the active site. The elementary reactions written in the form of eq 1 are suitable for a thermodynamic analysis. For kinetic considerations, however, it is useful to present them as q

bj0S + R

∑

k)1

n

bjkIk + R

∑ i)1

q

B β jiTi ) a Rj0S +

∑ RajkIk +

k)1 n

A β jiTi; ∑ i)1

j ) 1, 2, ..., p (5)

PiBβ ∏ i)1

(7)

ji

In what follows, eqs 6 and 7 are referred to as the De Donder relations. In general, a linear combination of the elementary reactions s1, s2, ..., sp that eliminate a specified number of species (either intermediates or terminal species) is called a reaction route (RR), or a mechanism. In the most general sense, a RR may be expressed as p

F)

q

Aj ) -ln Kj + Rj0 ln θ0 +

n

θ0Rbjk

σjsj ∑ j)1

(8)

where σ1, σ2, ..., σp is a set of real numbers, called stoichiometric numbers. Each RR produces an overall reaction (OR) that may be obtained by substituting the elementary reactions, eq 1, into eq 8 p

F)(

q

p

n

p

σjRj0)S + ∑ (∑σjRjk)Ik + ∑(∑σjβji)Ti ) 0 ∑ j)1 k)1 j)1 i)1 j)1

(9)

The above definition of RRs is more general than that used in the literature. Conventionally, a RR is defined as a linear combination of elementary reactions that eliminates all of the intermediates.13-16 The reason for this generalization will become clear later on. Direct Reaction Routes A RR can be derived requiring a specified set of species to vanish in eq 9. Because the total number of given elementary reactions p normally exceeds the total number of species q + n, there is no unique solution to stoichiometric numbers and the RRs may thus be generated in an infinite number of ways. That is, the RRs can be derived arbitrarily. It is, of course, most desirable for many purposes to limit the number of RRs to a finite and unique one. It was Milner17 who first noticed this aspect of the problem. He, thus, introduced the concept of the directness, or uniqueness, of RRs by simply requiring a RR to involve a minimum number of elementary reactions while eliminating all of the intermediates. Milner showed that the minimum number of elementary reactions necessary to eliminate q intermediates does not exceed q + 1, which is equal to the number of algebraic relations resulting from the specification that all q intermediates are eliminated from an OR along with eq 4. For this reason, a RR in which the minimum number of involved elementary steps exceeds by one the number of eliminated species may be called a Milner RR, or in short MRR. More recently,18-21 others have considered RRs that are the same as MRRs.

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Ind. Eng. Chem. Res., Vol. 40, No. 11, 2001

A substantial further development of the theory of RRs is due to Happel and Sellers.22-25 They observed that a unique and finite set of RRs may be generated not only by requiring a minimal number of elementary reactions to be involved in a RR but also by alternatively requiring that the ORs produced from these RRs involve a minimal number of species. In other words, Happell and Sellers extended the concept of directness not only with respect to RRs but also with respect to ORs. According to Happel and Sellers, thus, an OR is a direct one if it does not reduce to the sum of two distinct overall subreactions in the system. Alternatively, if one of the species is omitted from a direct OR, there is no reaction in the system involving only the remaining species. We have recently shown26 that the direct ORs of Happel and Sellers are essentially the same as the so-called response reactions that were deduced by us from chemical thermodynamics.27 A Happel-Sellers RR (HSRR) is, thus, defined as one that produces a response reaction by a linear combination of a set of m linearly independent elementary reactions.26 On the other hand, a MRR limits the number of elementary reactions to q + 1 but stipulates no limit on the stoichiometry of the OR. Our approach described here is general and is valid for any type of direct RR. For the sake of brevity and clarity, however, we discuss here in detail only the case of MRRs. A detailed analysis of the interrelation between the MRRs and HSRRs will be presented elsewhere.28

Rj1,1σ1 + Rj2,1σ2 + ... + Rjq,1σq ) 0 Rj1,2σ1 + Rj2,2σ2 + ... + Rjq,2σq ) 0 ... Rj1,k-1σ1 + Rj2,k-1σ2 + ... + Rjq,k-1σq ) 0 Rj1,k+1σ1 + Rj2,k+1σ2 + ... + Rjq,k+1σq ) 0 ... Rj1,qσ1 + Rj2,qσ2 + ... + Rjq,qσq ) 0

(11)

As shown in Appendix A, the solution of this system of homogeneous linear equations is

Substituting σ1, σ2, ..., σq into eq 10, followed by a few simple transformations based on the properties of the determinants, we obtain

Intermediate Reaction Routes We define two types of direct RRs. The first one involves q elementary reactions while eliminating q 1 intermediates, i.e., all of the intermediates but one. Such a RR is appropriately termed an intermediate RR (IRR). Let the q elementary reactions involved in an IRR be sj1, sj2, ..., sjq where j1, j2, ..., jq is an ordered q-tuple set of integers satisfying the condition 1 e j1 < j2 < ... < jq e p. Let further I1, I2, ..., Ik-1, Ik+1, ..., Iq be the q 1 intermediates that are eliminated and, hence, T1, T2, ..., Tn, S, Ik are the species that are involved in the IRR. Then, an IRR may be denoted either in terms of the eliminated species, i.e., F(sj1,sj2,...,sjq,I1,I2,...,Ik-1,Ik+1,...,Iq) or in terms of the species that are involved in the OR produced by the IRR, i.e., F(sj1,sj2,...,sjq,T1,T2,...,Tk-1,Ik+1,...,Iq). We shall use the second option. Further, because the terminal species T1, T2, ..., Tn and the active site S are involved in all IRRs, they can be dropped from the notation to keep it concise, so that an IRR may be denoted simply as F(sj1,sj2,...,sjq,Ik) specifying the elementary reactions and the single intermediate involved. By definition, eq 8, the general equation of an IRR is

F(sj1,sj2,...,sjq,Ik) ) σ1sj1 + σ2sj2 + ... + σqsjq (10) where the stoichiometric numbers σ1, σ2, ..., σq need to be selected so as to eliminate the q - 1 intermediates I1, I2, ..., Ik-1, Ik+1, ..., Iq, i.e.

The reaction produced by an IRR is called an intermediate reaction (IR). To obtain the IR, one needs to substitute sj1, sj2, ..., sjq, eq 1, into eq 13. Then, using the properties of the determinants, we have


It is seen that whenever l ) 1, 2, ..., k - 1, k + 1, ..., q two columns in the second determinant are equal and, consequently, the determinant is equal to zero, i.e., the intermediates I1, I2, ..., Ik-1, Ik+1, ..., Iq are eliminated. Taking into account eq 4, the general equation of an IR may be presented in a more compact form as n

F(sj1,sj2,...,sjq,Ik) )

γkiTi - γS + γIk ) 0 ∑ i)1

(14)

simply as R(sj1,sj2,...,sjq,sjq+1). The general equation of the ORR is

R(sj1,sj2,...,sjq,sjq+1) ) σ1sj1 + σ2sj2 + ... + σqsjq + σjq+1sjq+1 (20) where, now, σ1, σ2, ..., σq, σq+1 are selected so as to eliminate all of the intermediates. This gives the following system of homogeneous linear equations:

aj1,1σ1 + aj2,1σ2 + ... + ajq,1σq + ajq+1,1σq+1 ) 0

where

... aj1,2σ1 + aj2,2σ2 + ... + ajq,2σq + ajq+1,2σq+1 ) 0 ... aj1,qσ1 + aj2,qσ2 + ... + ajq,qσq + ajq+1,qσq+1 ) 0

and

Following the procedure described in Appendix A, it may be shown that the solution is

It should be noted that in eq 14 it is assumed that γ * 0. The IRs for which the determinant γ, eq 16, is equal to zero are not relevant to our analysis and are simply disregarded. By analogy, the affinities and the equilibrium constants of the IRs are related to the affinities and equilibrium constants of the elementary reactions via

In turn, the affinities and equilibrium constants of the IRs are interrelated through the conventional thermodynamic relation

-(1/RT)A(sj1,sj2,...,sjq,Ik) ) -ln K(sj1,sj2,...,sjq,Ik) n

γ ln θ0 + γ ln θk +

∑ i)1

γki ln Pi (19)

It may be noticed that the stoichiometric number σq+1 of the elementary reaction sjq+1 is

Overall Reaction Routes The second type of RR is defined as one that involves no more than q + 1 elementary reactions and eliminates all of the q intermediates I1, I2, ..., Iq (and, in view of eq 4, also the active site S). Alternatively, this type of RR involves only terminal species T1, T2 , ..., Tn and, thus, may be called an overall reaction route (ORR). Let the q + 1 elementary reactions involved in an ORR be sj1, sj2, ..., sjq, sjq+1 where j1, j2, ..., jq, jq+1 is an ordered (q + 1)-tuple set of integers satisfying the condition 1 e j1 < j2 < ... < jq, jq+1 e p. Then, by analogy, an ORR may be denoted as R(sj1,sj2,...,sjq,sjq+1,T1,T2,...,Tn), thus specifying the elementary reactions involved in an ORR and the species involved in the produced overall reaction. Because the list of terminal species is fixed, this may be dropped from the notation so that an ORR is denoted

and is independent of the choice of sjq+1. Substituting σ1, σ2, ..., σq, σq+1 into eq 20 gives

Taking into account the stoichiometry of the elementary reactions sj1, sj2, ..., sjq, sjq+1, eq 1, and using the properties

2420


of the determinants, we can present the equation of the OR produced by an ORR as

∑ i)1

γ1iTi - γS + γI1 ) 0 ∑ i)1

F(sj1,sj2,...,sjq,I2) )

γ2iTi - γS + γI2 ) 0 ∑ i)1

-ajq+1,1

n

n

R(sj1,sj2,...,sjq,sjq+1) )

n

F(sj1,sj2,...,sjq,I1) )

νiTi ) 0

(22)

-ajq+1,2

... where

n

F(sj1,sj2,...,sjq,Iq) )

γqiTi - γS + γIq ) 0 ∑ i)1 q

sjq+1 ) ajq+1,0S +

∑ Rj k)1

-ajq+1,q

n

,I + q+1 k k

βj ∑ i)1

,iTi ) 0

q+1

γ

n

Net: R(sj1,sj2,...,sjq,sjq+1) ) Similarly, the affinities and the equilibrium constants of the ORs are given by

νiTi ) 0 ∑ i)1

(27)

The complete enumeration of reduced RRs may be achieved by explicitly considering all of the possible choices of q + 1 elementary reactions sj1, sj2, ..., sjq, sjq+1 from a total of p. IRs, ORs, and De Donder Relations

and

A complete enumeration of IRs and ORRs along with the corresponding IRs and ORs may, in principle, be achieved by considering all of the appropriate choices of q and q + 1 elementary reactions from the total of p in the given set (eq 1). This method, however, is computationally quite laborious. A more effective method of enumeration is described elsewhere.28

As shown above, starting from a set of elementary reactions comprising the detailed chemistry of the system, one can define and derive two types of RRs, namely, IRRs and ORRs, thus producing two types of reactions, namely, IRs and ORs. We are now in a position to show that these purely stoichiometric considerations are intimately related to De Donder relations. First we observe that the IRs provide an easy way to express the surface concentrations of the intermediates through the affinities of IRs and the partial pressures of the terminal species. Indeed, eq 19 may be solved simultaneously with the site balance, eq 3, to obtain

{

q

θ0 ) 1/ 1 +

n

K(sj ,sj ,...,sj ,Ik)1/γ ∑ ∏ k)1 i)1

[

exp -

Reduced RRs Clearly, owing to the linear dependence of chemical reactions in a complex system, the elementary reactions are interrelated with the IRs and ORs. This interrelationship may be formulated as follows: q

γsjq+1 )

∑ Rj k)1

1

F(sj1,sj2,...,sjq,Ik) + R(sj1,sj2,...,sjq,sjq+1)

q+1,k

(26)

The proof of this result is given in Appendix B. We further observe that this relation is, in fact, a reduced RR that shows how a set of IRs may be linearly combined with an elementary reaction so as to eliminate all of the intermediates and to obtain an OR. Alternatively, eq 26 may be written in the more conventional format of a RR

2

q

γRT

θk ) K(sj1,sj2,...,sjq,Ik)1/γ

[

exp -

]∏

A(sj1,sj2,...,sjq,Ik)

n

γRT

i)1

q

n

[

1

2

Pi-(γki/γ)

(28)

Pi-(γki/γ)/ 1 +

K(sj ,sj ,...,sj ,Ik)1/γ ∑ ∏ k)1 i)1 exp -

] } {

A(sj1,sj2,...,sjq,Ik)

q

] }

A(sj1,sj2,...,sjq,Ik) γRT

Pi-(γki/γ)

(29)

As can be seen, the surface concentrations of the free sites and intermediate may be partitioned into contributions coming from IRs. Now, it remains to find the explicit interrelationship between the affinities of the elementary reactions with the affinities of the IRs and ORs. The latter follows from eq 26:


Ajq+1 )

q

1

∑ Rj γk)1

1 + A(sj1,sj2,...,sjq,sjq+1) γ (30)

A(sj1,sj2,...,sjq,Ik) q+1,k

Substituting eqs 28-30 into eqs 6 and 7, we ultimately obtain

[

(

)]

+ A(sj1,sj2,...,sjq,sjq+1)

γRT

(31)

where q

(

n

∏∏K(s ,s ,...,s ,I )

exp -

; γRT j ) q + 1, q + 2, ..., p (34)

n

K(sj ,sj ,...,sj ,Ik)Rb /γPi(βB γ-Rb γ )/γ ∏ ∏ k)1 i)1 jk

∑ Rj k)1

b r jq+1 ) B k jq+1

)]

A(sj1,sj2,...,sjq,sj)

1

A(sj1,sj2,...,sjq,Ik) q+1,k

exp -

(

[

b rj ) b r j 1 - exp q

r jq+1 1 rjq+1 ) b q

In this case eqs 30 and 31 take the form

j1

k)1 i)1

j2

jq

q

n

bjq+1/γ R

[

)

bjq+1,kA(sj1,sj2,...,sjq,Ik) R γRT

k

Pi(βBjq+1,i-Rjq+1,kγki)/γ/ 1 +

∏∏K(s ,s ,...,s ,I )

(

j1

k)1 i)1

exp -

j2

jq

k

1/γ

) ]

A(sj1,sj2,...,sjq,Ik) γRT

∆R bjq+1

-γki/γ

Pi

(32)

and q

Rjq+1 + ∆R bjq+1 ) b

∑ Rbj k)1

q+1,k

(33)

We thus conclude that De Donder relations may be partitioned into contributions coming from IRs and ORs. The above relations may be utilized within the PSS formalism to calculate affinities, surface concentrations, and rates, as shown by Dumesic.2 However, in what follows we demonstrate their utility for the simple case of the RDS approximation.

kj b rj ) B

[

q

1+

2

ji

jk ki

q

n

∏ ∏K(sj ,sj ,...,sj ,Ik) k)1 i)1 1

2

q

1/γ

]

(35)

bj -γki/γ ∆R

Pi

thus providing the generic LHHW expression for an RDS. It can be seen that, up to a constant, which is just the stoichiometric number of the RDS, the affinities of the RDS are equal to the affinities of the ORs. As a result, De Donder relations may be presented in terms of contributions coming from the equilibrium constants of the IRs and affinities of ORs. Now, it is easy to realize that within the RDS approximation the above partitioning of De Donder relations is nothing but the explicit mathematical formulation of the following Boudart statement:29 All equilibrated steps following a RDS involving the MARI as the product can be combined in a single overall equilibrium that regulates the concentration of the MARI. Vice versa, all equilibrated steps preceding a RDS involving the MARI as the reactant can be combined in a similar overall equilibrium. Indeed, the interrelation between the partition of De Donder relations and the Boudart statement is readily deduced by observing that Boudart’s “overall equilibrium” is essentially equilibrated IRs while the interrelation between the RDS and equilibrated IRs and ORs is expressed through a reduced RR, eq 27. An Example with a Single ORR: Ammonia Synthesis Consider the sequence of elementary reactions involved in the ammonia synthesis:2

s1 ) -N2 - 2S + 2NS ) 0 RDS Approximation An important tool for the understanding and simplification of the kinetics of complex heterogeneous catalytic reactions is the concept of the rate-determining step (RDS). As is well-known, according to this approximation, the elementary reactions with a fast time scale may be considered to be at QE while the elementary reactions with a slow time scale are RDSs. In heterogeneous catalytic reactions, normally, the number of QE elementary reactions is assumed to be equal to the number of intermediates, i.e., q. The affinities of the QE elementary reactions are then set equal to zero, thus providing a set of q equations that can be solved simultaneously with the site balance to eliminate θ0 and θk. For simplicity, let the first q elementary reactions sj1, sj2, ..., sjq (j1 ) 1, j2 ) 2, ..., jq ) q) be at QE while the remaining p - q elementary reactions sj (j ) q + 1, q + 2, ..., p) are RDS. IRs are linear combinations of the QE elementary reactions, and, hence, the affinities of IRs are equal to zero

A(sj1,sj2,...,sjq,Ik) ) 0; k ) 1, 2, ..., q

s2 ) -H2 - 2S + 2HS ) 0 s3 ) -NS - HS + NHS + S ) 0 s4 ) -NHS + HS + NH2S + S ) 0 s5 ) -NH2S + HS + NH3S + S ) 0 s6 ) -NH3S + NH3 + S ) 0 For this system we have q ) 5 intermediates species (NS, HS, NHS, NH2S, and NH3S) and n ) 3 terminal species (H2, N2, and NH3). The rank of the stoichiometric matrix

2422


is m ) 6 and, hence, the given elementary reactions are linearly independent. By definition, an ORR may be derived from q + 1 ) 5 + 1 ) 6 elementary reactions. Hence, there is only one ORR for this system. According to eq 21 the only ORR is

first five elementary reactions, i.e., F(s1,s2,s3,s4,s5,NS), according to eq 13 is

As can be seen, the first elementary reaction is concomitantly an IR. The next four IRRs for NS, respectively the IRs, are stoichiometrically identical, i.e., while according to eqs 22 and 23, the OR produced by this ORR is

F(s1,s2,s3,s4,s6,NS) ) F(s1,s2,s3,s5,s6,NS) ) F(s1,s2,s4,s5,s6,NS) ) F(s1,s3,s4,s5,s6,NS) ) s1 ) -N2 - 2S + 2NS ) 0 The last IRR and IR, F(s2,s3,s4,s5,s6,NS), are stoichiometrically distinct

For the affinity and equilibrium constant of this IR, we obtain The affinity and the equilibrium constant of the OR are related to the affinities and the equilibrium constants of the elementary reactions via similar relations

A(s1,s2,s3,s4,s5,s6) ) 2(A1 + 3A2 + 2A3 + 2A4 + 2A5 + 2A6) ln K(s1,s2,s3,s4,s5,s6) ) 2(ln K1 + 3 ln K2 + 2 ln K3 + 2 ln K4 + 2 ln K5 + 2 ln K6) ) 2 ln K1K23K32Kk2K52K62 Consider next the derivation and enumeration of the possible IRRs and IRs. By definition, an IRR is obtained from q ) 5 elementary reactions. These may be selected in the following six different ways: 1. {s1, s2, s3, s4, s5} 2. {s1, s2, s3, s4, s6} 3. {s1, s2, s3, s5, s6} 4. {s1, s2, s4, s5, s6} 5. {s1, s3, s4, s5, s6} 6. {s2, s3, s4, s5, s6} Hence, one can expect a maximum of six IRRs for each intermediate. In reality, not all of the IRRs are stoichiometrically distinct. This means that the stoichiometric numbers of some of the elementary reactions are equal to zero, thus resulting in stoichiometrically equivalent IRRs. For example, the IRR for NS obtained from the

A(s2,s3,s4,s5,s6,NS) ) 3A2 + 2A3 + 2A4 + 2A5 + 2A6 K(s2,s3,s4,s5,s6,NS) ) K23 K32 K42 K52 K62 A complete list of IRs along with their affinities and equilibrium constants may be readily derived by repeating this procedure over all possible choices of five elementary reactions. The results are summarized in Table 1. Thus, equipped with a complete list of IRs and OR (Table 1), one can enumerate the reduced RRs and, hence, De Donder relations. First, it may be mentioned that IRs involve only one intermediate. As a result, any set of corresponding IRs from Table 1 may, in principle, be used to equivalently express the surface concentrations of the intermediates through the affinities and equilibrium constants of the IRs. Further, according to eq 26 each of the elementary reactions may be written in terms of the IRs and ORs, thus resulting in a reduced RR. Concomitantly, a reduced RR provides a relation between the affinity of an elementary reaction and the affinities of IRs and ORs. Consider, for instance, the derivation of the reduced RR for the first elementary reaction s3

s3 ) -NS - HS + NHS + S ) 0 To eliminate the intermediates (NS, HS, and NHS) from this elementary reaction, we need to use the IRs obtained from the set of elementary reactions {s1, s2,

Ind. Eng. Chem. Res., Vol. 40, No. 11, 2001 2423 Table 1. Intermediate and Overall Reactions for Ammonia Synthesis NS F(s1,s2,s3,s4,s5,NS) ) F(s1,s2,s3,s4,s6,NS) ) F(s1,s2,s3,s5,s6,NS) ) F(s1,s2,s4,s5,s6,NS) ) F(s1,s3,s4,s5,s6,NS) ) s1 ) -N2 - 2S + 2NS ) 0 A(s1,s2,s3,s4,s5,NS) ) A(s1,s2,s3,s4,s6,NS) ) A(s1,s2,s3,s5,s6,NS) ) A(s1,s2,s4,s5,s6,NS) ) A(s1,s3,s4,s5,s6,NS) ) A1 K(s1,s2,s3,s4,s5,NS) ) K(s1,s2,s3,s4,s6,NS) ) K(s1,s2,s3,s5,s6,NS) ) K(s1,s2,s4,s5,s6,NS) ) K(s1,s3,s4,s5,s6,NS) ) K1 θNS ) θ0K(s1,s2,s4,s5,s6,NS)1/2 exp(-[A(s1,s3,s4,s5,s6,NS)]/2RT)PN21/2 F(s2,s3,s4,s5,s6,NS) ) 3s2 + 2s3 + 2s4 + 2s5 + 2s6 ) -2NS - 3H2 + 2NH3 + 2S ) 0 A(s2,s3,s4,s5,s6,NS) ) 3A2 + 2A3 + 2A4 + 2A5 + 2A6 K(s1,s2,s4,s5,s6,NS) ) K23K32K42K52K62 θNS ) θ0K(s2,s3,s4,s5,s6,NS)-1/2 exp([A(s2,s3,s4,s5,s6,NS)]/2RT)PNH3PH2-3/2 HS F(s1,s2,s3,s4,s5,HS) ) F(s1,s2,s3,s4,s6,HS) ) F(s1,s2,s3,s5,s6,HS) ) F(s1,s2,s4,s5,s6,HS) ) F(s2,s3,s4,s5,s6,HS) ) s2 ) -H2 - 2S + 2HS ) 0 A(s1,s2,s3,s4,s5,HS) ) A(s1,s2,s3,s4,s6,HS) ) A(s1,s2,s3,s5,s6,HS) ) A(s1,s2,s4,s5,s6,HS) ) A(s2,s3,s4,s5,s6,HS) ) A2 K(s1,s2,s3,s4,s5,HS) ) K(s1,s2,s3,s4,s6,HS) ) K(s1,s2,s3,s5,s6,HS) ) K(s1,s2,s4,s5,s6,HS) ) K(s2,s3,s4,s5,s6,HS) ) K2 θHS ) θ0K(s1,s2,s4,s5,s6,HS)1/2 exp(-[A(s1,s3,s4,s5,s6,HS)]/2RT)PH21/2 F(s1,s3,s4,s5,s6,HS) ) s1 + 2s3 + 2s4 + 2s5 + 2s6 ) -6HS - N2 + 2NH3 + 6S ) 0 A(s1,s3,s4,s5,s6,HS) ) A1 + 2A3 + 2A4 + 2A5 + 2A6 K(s1,s2,s4,s5,s6,HS) ) K1K23K32K52K62 θHS ) θ0K(s1,s2,s4,s5,s6,HS)-1/6 exp(-[A(s1,s3,s4,s5,s6,HS)]/6RT)PN2-1/6PNH31/3 NHS F(s1,s2,s3,s4,s5,NHS) ) F(s1,s2,s3,s4,s6,NHS) ) F(s1,s2,s3,s5,s6,NHS) ) s1 + s2 + 2s3 ) -N2 - H2 - 2S + 2NHS ) 0 A(s1,s2,s3,s4,s5,NHS) ) A(s1,s2,s3,s4,s6,NHS) ) A(s1,s2,s3,s5,s6,NHS) ) A1 + A2 + 2A3 K(s1,s2,s3,s4,s5,NHS) ) K(s1,s2,s3,s4,s6,NHS) ) K(s1,s2,s3,s5,s6,NHS) ) K1K2K32 θNHS ) θ0K(s1,s2,s4,s5,s6,NHS)1/2 exp(-[A(s1,s3,s4,s5,s6,NHS)]/2RT)PN21/2PH21/2 F(s1,s2,s4,s5,s6,NHS) ) F(s2,s3,s4,s5,s6,NHS) ) s2 + s4 + s5 + s6 ) -H2 + NH3 + S - NHS ) 0 A(s1,s2,s4,s5,s6,NHS) ) A(s2,s3,s4,s5,s6,NHS) ) A2 + A4 + A5 + A6 K(s1,s2,s4,s5,s6,NHS) ) K(s2,s3,s4,s5,s6,NHS) ) K2K4K5K6 θNHS ) θ0K(s1,s2,s4,s5,s6,NHS)-1 exp(-[A(s1,s3,s4,s5,s6,NHS)]/3RT)PH2-1PNH31/3 F(s1,s3,s4,s5,s6,NHS) ) s1 + 2s3 - s4 - s5 - s6 ) -N2 - NH3 - 3S + 3NHS ) 0 A(s1,s3,s4,s5,s6,NHS) ) A1 + 2A3 - A4 - A5 - A6 K(s1,s3,s4,s5,s6,NHS) ) K1K32K4-1K5-1K6-1 θNHS ) θ0K(s1,s2,s4,s5,s6,NH2S)1/3 exp(-[A(s1,s2,s3,s4,s5,NHS)]/3RT)PN21/2PNH31/3 NH2S F(s1,s2,s3,s4,s5,NH2S) ) F(s1,s2,s3,s4,s6,NH2S) ) s1 + 2s2 + 2s3 + 2s4 ) -N2 - 2H2 - 2S + 2NH2S ) 0 A(s1,s2,s3,s4,s5,NH2S) ) A(s1,s2,s3,s4,s6,NH2S) ) A1 + 2A2 + 2A3 + 2A4 K(s1,s2,s3,s4,s5,NH2S) ) K(s1,s2,s3,s4,s6,NH2S) ) K1K22K32K42 θNH2S ) θ0PK(s1,s2,s3,s4,s5,NH2S)1/2 exp(-[A(s1,s2,s3,s4,s5,NH2S)]/2RT)PN21/2PH2 F(s1,s2,s3,s5,s6,NH2S) ) F(s1,s2,s4,s5,s6,NH2S) ) F(s2,s3,s4,s5,s6,NH2S) ) s2 + 2s5 + 2s6 ) -H2 + 2NH3 + 2S - 2NH2S ) 0 A(s1,s2,s3,s5,s6,NH2S) ) A(s1,s2,s4,s5,s6,NH2S) ) A(s2,s3,s4,s5,s6,NH2S) ) A2 + 2A5 + 2A6 K(s1,s2,s3,s5,s6,NH2S) ) S(s1,s2,s4,s5,s6,NH2S) ) S(s2,s3,s4,s5,s6,NH2S) ) K2K52K52 -1/2 θNH2S ) θ0PK(s1,s2,s3,s5,s6,NH2S)-1/2 exp(-[A(s1,s2,s3,s5,s6,NH2S)]/2RT)PH PNH3 2 F(s1,s3,s4,s5,s6,NH2S) ) s1 + 2s3 + 2s4 - 4s5 - 4s6 ) -N2 - 4NH3 - 6S + 6NH2S ) 0 A(s1,s3,s4,s5,s6,NH2S) ) A1 + 2A3 + 2A4 - 4A5 - 4A6 K(s1,s3,s4,s5,s6,NH2S) ) K1K32K42K5-4K6-4 θNH2S ) θ0PK(r1,r3,r4,r5,r6,NH2S)1/6 exp(-[A(r1,r3,r4,r5,r6,NH2S)]/6RT)N21/6PNH32/3 NH3S F(s1,s2,s3,s4,s5,NH3S) ) s1 + 3s2 + 2s3 + 2s4 + 2s5 ) -N2 - 3H2 - 2S + 2NH3S ) 0 A(s1,s2,s3,s4,s5,NH3S) ) A1 + 3A2 + 2A3 + 2A4 + 2A5 K(s1,s2,s3,s4,s5,NH3S) ) K1K23K32K42K52 θNH3S ) θ0K(s1,s2,s3,s4,s5,NH3S)1/2 exp(-[A(s1,s2,s3,s4,s5,NH3S)]/2RT)PN21/2PH23/2 F(s1,s2,s3,s4,s6,NH3S) ) F(s1,s2,s3,s5,s6,NH3S) ) F(s1,s2,s4,s5,s6,NH3S) ) F(s1,s3,s4,s5,s6,NH3S) ) F(s2,s3,s4,s5,s6,NH3S) ) s6 ) NH3 + S - NH3S ) 0 A(s1,s2,s3,s4,s6,NH3S) ) A(s1,s2,s3,s5,s6,NH3S) ) A(s1,s2,s4,s5,s6,NH3S) ) A(s1,s3,s4,s5,s6,NH3S) ) A(s2,s3,s4,s5,s6,NH3S) ) A6 K(s1,s2,s3,s4,s6,NH3S) ) K(s1,s2,s3,s5,s6,NH3S) ) K(s1,s2,s4,s5,s6,NH3S) ) K(s1,s3,s4,s5,s6,NH3S) ) K(s2,s3,s4,s5,s6,NH3S) ) K6 θNH3S ) θ0K(s1,s2,s3,s4,s5,NH3S)-1 exp(-[A(s1,s2,s3,s4,s5,NH3S)]/RT)PNH3 OR R(s1,s2,s3,s4,s5,s6) ) s1 + 3s2 + 2s3 + 2s4 + 2s5 + 2s6 ) -N2 - 2H2 + 2NH3 ) 0 A(s1,s2,s3,s4,s5,s6) ) A1 + 3A2 + 2A3 + 2A4 + 2A5 + 2A6 K(s1,s2,s3,s4,s5,s6) ) K1K23K32K42K52K62

s4, s5, s6}. From Table 1, we have

F(s1,s2,s4,s5,s6,NS) ) -N2 - 2S + 2NS ) 0

This stoichiometric relation represents the following reduced RR:

F(s1,s2,s4,s5,s6,NS) ) -N2 - 2S + 2NS ) 0

1

F(s1,s2,s4,s5,s6,HS) ) -H2 - 2S + 2HS ) 0

F(s1,s2,s4,s5,s6,HS) ) -H2 - 2S + 2HS ) 0

1

F(s1,s2,s4,s5,s6,NHS) ) -H2 + NH3 + S - NHS ) 0

s3 ) -NS - HS + NHS + S ) 0

2

According to eq 26, these IRs are interrelated with s3 and OR via

2s3 + F(s1,s2,s4,s5,s6,NS) + F(s1,s2,s4,s5,s6,HS) + 2F(s1,s2,s4,s5,s6,NHS) ) R(s1,s2,s3,s4,s5,s6)

F(s1,s2,s4,s5,s6,NHS) ) -NHS - H2 + NH3 + S ) 0 2 Net: R(s1,s2,s3,s4,s5,s6) ) -N2 - 3H2 + 2NH3 ) 0 Evidently, the affinities of these reactions are inter-

2424


neously with the surface site balance gives

Table 2. Reduced RRs for Ammonia Synthesis I. s1 + F(s2,s3,s4,s5,s6,NS) ) R(s1,s2,s3,s4,s5,s6) s1 ) -N2 - 2S + 2NS ) 0 F(s2,s3,s4,s5,s6,NS) ) -2NS - 3H2 + 2NH3 + 2S ) 0 Net: R(s1,s2,s3,s4,s5,s6) ) -N2 - 3H2 + 2NH3 ) 0 II. 3s2 + F(s1,s3,s4,s5,s6,HS) ) R(s1,s2,s3,s4,s5,s6) s2 ) -H2 - 2S + 2HS ) 0 F(s1,s3,s4,s5,s6,HS) ) -6HS - N2 + 2NH3 + 6S ) 0 Net: R(s1,s2,s3,s4,s5,s6) ) -N2 - 3H2 + 2NH3 ) 0 III. 2s3 + F(s1,s2,s4,s5,s6,NS) + F(s1,s2,s4,s5,s6,HS) + 2F(s1,s2,s4,s5,s6,NHS) ) R(s1,s2,s3,s4,s5,s6) F(s1,s2,s4,s5,s6,NS) ) -N2 - 2S + 2NS ) 0 F(s1,s2,s4,s5,s6,HS) ) -H2 - 2S + 2HS ) 0 s3 ) -NS - HS + NHS + S ) 0 F(s1,s2,s4,s5,s6,NHS) ) -NHS - H2 + NH3 + S ) 0 Net: R(s1,s2,s3,s4,s5,s6) ) -N2 - 3H2 + 2NH3 ) 0 IV. 2s4 + F(s1,s2,s3,s5,s6,HS) + F(s1,s2,s3,s5,s6,NHS) + F(s1,s2,s3,s5,s6,NH2S) ) R(s1,s2,s3,s4,s5,s6) F(s1,s2,s3,s5,s6,HS) ) -H2 - 2S + 2HS ) 0 F(s1,s2,s3,s5,s6,NHS) ) -N2 - H2 - 2S + 2NHS ) 0 s4 ) -NHS - HS + NH2S + S ) 0 F(s1,s2,s3,s5,s6,NH2S) ) -2NH2S - H2 + NH3 + 2S ) 0 Net: R(s1,s2,s3,s4,s5,s6) ) -N2 - 3H2 + 2NH3 ) 0 V. 2s5 + F(s1,s2,s3,s4,s6,HS) + F(s1,s2,s3,s4,s6,NH2S) + 2F(s1,s2,s3,s4,s6,NH3S) ) R(s1,s2,s3,s4,s5,s6) F(s1,s2,s3,s4,s6,HS) ) -H2 - 2S + 2HS ) 0 F(s1,s2,s3,s4,s6,NH2S) ) -N2 - 2H2 - 2S + 2NH2S ) 0 s5 ) -NH2S - HS + NH3S + S ) 0 F(s1,s2,s3,s4,s6,NH3S) ) -NH3S + NH3 + S ) 0 Net: R(s1,s2,s3,s4,s5,s6) ) -N2 - 3H2 + 2NH3 ) 0 VI. s6 + F(s1,s2,s3,s4,s5,NH3S) ) R(s1,s2,s3,s4,s5,s6) F(s1,s2,s3,s4,s5,NH3S) ) -N2 - 3H2 - S + NH3S ) 0 s6 ) -NH3S + NH3 + S ) 0 Net: R(s1,s2,s3,s4,s5,s6) ) -N2 - 3H2 + 2NH3 ) 0

1 1

θ0-1 ) 1 + K(s2,s3,s4,s5,s6,NS)-1/2 × exp

3 1

(

)

A(s2,s3,s4,s5,s6,NS) PNH3PH2-3/2 + 2RT

(

K(s2,s3,s4,s5,s6,HS)1/2

exp -

1 1 2 2

exp 1 1 2 1

exp

(

)

A(s2,s3,s4,s5,s6,HS) PH21/2 + 2RT

K(s2,s3,s4,s5,s6,NHS)-1

)

A(s2,s3,s4,s5,s6,NHS) PH2-1PNH3 + RT

(

K(s2,s3,s4,s5,s6,NH2S)-1/2

)

A(s2,s3,s4,s5,s6,NH2S) PH2-1/2PNH3 + 2RT K(s2,s3,s4,s5,s6,NH3S)-1

1 1 2 2

exp

(

)

A(s2,s3,s4,s5,s6,NH3S) PNH3 (37) RT

Further, from Table 2 we have 1 1

related via

A3 ) -1/2A(s1,s2,s4,s5,s6,NS) 1/2A(s1,s2,s4,s5,s6,HS) - A(s1,s2,s4,s5,s6,NHS) + 1/2A(s1,s2,s3,s4,s5,s6) A summary of such an analysis for all of the elementary reactions comprising the ammonia synthesis is presented in Table 2. The information provided in Tables 1 and 2 is, in fact, a complete enumeration of the partitions of De Donder relations into contributions coming from IRs and OR for ammonia synthesis. As an example, consider the partition of the De Donder relation for the elementary reaction s1

k 1θ02PN2[1 - exp(-A1/RT)] r ) r1 ) B In this case, the De Donder relation may be partitioned into contributions associated with the IRs obtained from the set of remaining elementary reactions, i.e., {s2, s3, s4, s5, s6}. According to Table 1, these IRs are

s1 + F(s2,s3,s4,s5,s6,NS) ) F(s1,s2,s3,s4,s5,s6) that gives

A1 ) -A(s2,s3,s4,s5,s6,NS) + A(s1,s2,s3,s4,s5,s6) (38) The final partition of the De Donder relation for s1 is obtained by substituting eqs 37 and 38 into eq 36. For the particular case when s1 is assumed to be the RDS and, hence, {s2, s3, s4, s5, s6} are at QE, the above equations are substantially simplified. Indeed, in this case the affinities of the IRs are equal to zero

A(s2,s3,s4,s5,s6,NS) ) A(s2,s3,s4,s5,s6,HS) ) A(s2,s3,s4,s5,s6,NHS) ) A(s2,s3,s4,s5,s6,NH2S) ) A(s2,s3,s4,s5,s6,NH3S) ) 0 thus leading to the well-known LHHW rate equation11

[

(

k 1θ02PN2 1 - exp r ) r1 ) B where

θ0-1 ) 1 + K(s2,s3,s4,s5,s6,NS)-1/2PNH3PH2-3/2 +

F(s2,s3,s4,s5,s6,NS) ) -2NS - 3H2 + 2NH3 + 2S ) 0

K(s2,s3,s4,s5,s6,HS)1/2PH21/2 +

F(s2,s3,s4,s5,s6,HS) ) -H2 - 2S + 2HS ) 0

K(s2,s3,s4,s5,s6,NHS)-1PH2-1PH3 +

F(s2,s3,s4,s5,s6,NHS) ) -H2 + NH3 + S - NHS ) 0

K(s2,s3,s4,s5,s6,NH2S)-1/2PH2-1/2PNH3 +

F(s2,s3,s4,s5,s6,NH2S) ) -H2 + 2NH3 + 2S 2NH2S ) 0 F(s2,s3,s4,s5,s6,NH3S) ) NH3 + S - NH3S ) 0 This set of IRs provides a set of equations for the surface coverages of the intermediates that solved simulta-

)]

A(s2,s3,s4,s5,s6) RT

K(s2,s3,s4,s5,s6,NH3S)-1PNH3 Discussion and Concluding Remarks The surface concentrations of the intermediates in De Donder relations can always be substituted with the affinities of the elementary reactions comprising the


detailed chemistry of a heterogeneous catalytic process. This new form of De Donder relations has been shown by Dumesic11 and Stoltze3 to be a useful means in microkinetic modeling. The conventional (purely mathematical) substitution procedure, however, is arbitrary, thus resulting in arbitrary forms of De Donder relations. Of course, there is nothing mathematically wrong with the arbitrariness of De Donder relations because for a given system De Donder relations are (and must be!) thermodynamically independent of any substitution procedure. The problem is rather one of understanding the physicochemical meaning of this substitution. In this work, we have addressed the problem of arbitrariness of De Donder relations from the viewpoint of the theory of RRs. Our starting point is the simple observation that the affinities follow the same linear transformation rules as chemical reactions. If so, the substitution of the intermediates in terms of the affinities of the elementary reactions is equivalent to the elimination of the intermediates by a linear combination of the elementary reactions. As is well-known, the elimination of the intermediates from the appropriate combination of elementary reactions is the premise of the theory of RRs. We, thus, conclude that there exists a direct interrelationship between De Donder relations and the theory of RRs. To be able to visualize this interrelationship, however, it is necessary to further extend the theory of RRs. The latter are normally defined as a linear combination of the elementary reactions that eliminates all of the intermediates. This conventional type of RRs produces ORs that involve only a specified number of terminal species. For our purposes, however, it is necessary to introduce a different type of RRs defined as a linear combination of a specified set of elementary reactions that eliminates all but one of the intermediates, thus producing a reaction that involves only one intermediate and a specified number of terminal species, i.e., IRs. In fact, the IRs represent the “chemical solution” of the substitution problem. In other words, the IRs involve only one intermediate and, hence, are a simple means to express the surface concentrations of the intermediates through the affinities of intermediate reactions. From general considerations, it is clear that there exists a certain interrelationship among elementary reactions, IRs and ORs. We have found this interrelationship to be expressed by very simple equations. Concomitantly, these equations offer an interesting interpretation of the linear interrelationship between the elementary, intermediate, and overall reactions. Namely, they may be treated as reduced RRs, showing how an elementary reaction may be linearly combined with IRs so as to eliminate the intermediates and arrive at an OR. By virtue of the fact that the affinities are a function of state, similar equations hold for the affinities, thus opening the door to the desired interrelationship between the affinities of the elementary reactions and the affinities of the IRs and ORs. The immediate consequence of the mechanistic formalism developed in this work is that De Donder relations are naturally partitioned into contributions coming from IRs and ORs. Because for a given mechanism both the IRs and ORs are unique and their number is finite, so is the partition of De Donder relations. In other words, for any set of elementary reactions comprising the detailed chemistry of a heterogeneous catalytic process, the enumeration of IRs, ORs, and reduced

RRs implies concomitantly the enumeration of a finite and unique set of De Donder relations. This, so far unnoticed, property of De Donder relations to be partitioned into contributions coming from IRs and ORs is a gateway to gain a deeper insight into many aspects of the microkinetics of heterogeneous catalytic reactions. In addition to the advantages provided by the conventional analysis of De Donder relations that have been outlined by Dumesic and Stoltze, there are several more. In this respect, we mention first that the formalism developed in this work is in close relation with the Boudart treatment of the kinetics of heterogeneous catalytic reactions according to which, for the particular case of the RDS approximation, the QE elementary reactions may be combined into “overall equilibria”. Then, a “catalytic sequence” of elementary reactions (i.e., a reaction route) may be substituted with a set of “overall equilibria” and a RDS. The RDS may be linearly combined with IRs so as to eliminate the intermediates, thus resulting in the overall reaction. It is quite transparent that Boudart’s “overall equilibria” are essentially IRs, while the interrelation between the “overall equilibria”, RDS, and ORs is nothing but a reduced RR. Further, the approach based on the theory of RRs allows a natural extension of this simple principle to non-RDS systems. Thus, all of these, so far qualitative interpretational considerations, are now formulated in a rigorous mathematical language. As a result, we are in a position to develop a computer software that may be effectively used for the enumeration, analysis, and discrimination among De Donder relations. When combined with the energetics of the elementary reactions estimated using either ab initio1 or UBI-QEP methods,30 this analysis might result in a powerful tool in microkinetic modeling.31 In this work we have mainly discussed the theoretical and mathematical aspects of the problem. The application of these results to the microkinetic analysis of specific systems is underway. Appendix A: Proof of Equation 12 Equation 11 represents a homogeneous system of q - 1 linear equations in q variables σ1, σ2, ..., σq. Let us solve it with respect to the first q - 1 variables σ1, σ2, ..., σq-1

aj1,1σ1 + aj2,1σ2 + ... + ajq-1,1σq-1 ) -Rjq,1σq aj1,2σ1 + aj2,2σ2 + ... + ajq-1,2σq-1 ) -Rjq,2σq ... aj1,k-1σ1 + aj2,k-1σ2 + ... + ajq-1,k-1σq-1 ) -Rjq,k-1σq aj1,k+1σ1 + aj2,k+1σ2 + ... + ajq-1,k+1σq-1 ) -Rjq,k+1σq aj1,qσ1 + aj2,qσ2 + ... + ajq-1,qσq-1 ) -Rjq,qσq (A1) Consider the solution for σ1. Employing the Cramer rule, we have

2426


where

Equation 12 is obtained by accepting

Observe that the determinant D may be also identically written as

Appendix B: Proof of Equation 26 From eq 14 we have

Ik ) S -

1

n

1

∑γkiTi +γF(sj ,sj ,...,sj ,Ik) γi)1 1

2

q

Substituting this equation into the elementary reaction and after transposition q

rjq+1 ) Rjq+1,0S +

q

∑

Rjq+1,kIk +

1

q

k)1

∑ βj

k)1

q+1,i

Ti

we have Equation A2 may be transformed, first, by permutation of the columns

rjq+1 ) q

(Rjq+1,0 +

∑ Rj k)1

)S q+1,k

∑ Rj γk)1

+

1N

(

q+1,k

q

∑ k)1 ∑ Rj

γi)1

and, then, by transposition

-

F(sj1,sj2,...,sjq,Ik) +

q+1,k

)

γki + γβjq+1,i Ti (B1)

The first term in this equation vanishes because of eq 4. Comparing eqs B1 and 26, we see that they coincide if q

-

∑ Rj

k)1

The same result may be written as

γki + γβjq+1,i ) νi

q+1,k

(B2)

for every value of jq+1. This identity may be proved as follows. Taking into account eq 15, the left-hand side of eq B2, denoted as L, may be written as

Similar relations may be easily derived for σ2, ..., σq-1

where, according to eq 16

The first term in eq B3 may be transformed as


while the second term may be written as

Thus

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Received for review October 11, 2000 Revised manuscript received February 8, 2001 Accepted February 21, 2001 IE000871H

De Donder Relations in Mechanistic and Kinetic Analysis of

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