Ind. Eng. Chem. Res. 1990,29, 1057-1064 1057 Herskowitz, M.; Carbonell, R. G.; Smith, J. M. Effectiveness Factor and Mass Transfer in Trickle-Bed Reactors. AIChE J. 1979,25, Holah, D. G.; Hoodless, I. M.; Hughes, A. N.; Sedor, L. J. Kinetics of Liquid-Phase Hydrogenation of 1-Alkenes over a Partially Hydrogenated Nickel Boride and the Effects of Catalyst Poisons upon These Hydrocarbons. J. Catal. 1979,60, 148-155. Madon, R. J.; O’Connell, J.; Boudart, M. Catalytic Hydrogenation of Cyclohexane. 11: Liquid Phase Reaction on Supported Plati- num in a Gradientless Slurry Reactor. AZChE J. 1978, 24, Niiyama, H.; Smith, J. M. Adsorption of Nitric Oxide in Aqueous Slurries of Activated Carbon: Transport Rates by Moment 272-283. 904-91 1. Analysis of Dynamic Data. AZChE J. 1976,22, 961-970. Oudar, J. Sulfur Adsorption and Poisoning of Metallic Catalyst. Catal. Rev. Sci. Eng. 1980,22, 171-195. Satterfield, C. N; Ma, Y. H.; Sherwood, T. K. The Effectiveness Factor in a Liquid-Filled Porous Catalyst. Znst. Chem. Eng., Symp. Ser. 1968,28, 22-29. Zwich, J. J.; Gut, G. Kinetics, Poisoning and Mass Transfer Effecta in Liquid-Phase Hydrogenations of Phenolic Compounds over a Palladium Catalyst. Chem. Eng. Sci. 1978, 33, 1363, 1369. Received for review May 3, 1989 Revised manuscript received September 6, 1989 Accepted September 25, 1989 Mechanistic Study of Chemical Reaction Systems John Happel* Department of Chemical Engineering and Applied Chemistry, Columbia University, New York, New York 10027 Peter H. Sellers The Rockefeller University, 1230 York Avenue, New York, New York 10021 Masood Otarod Department of Mathematicslcomputer Science, University of Scranton, Scranton, Pennsylvania 18510 This paper presents a method for characterizing complex chemical systems often encountered in studies of homogeneous, heterogeneous catalytic, and enzyme reactions. From a list of chosen elementary steps, it is shown how all possible mechanisms and corresponding overall chemical equations can be enumerated. This furnishes a valuable tool for the further consideration of desired reactions and elucidation of their kinetics. Illustrative examples are given, including problems of current interest. A computer program is available to implement the procedures required. In the study of a chemical reaction, the first step is often the proposal of appropriate mechanisms to show how el- ementary steps may be combined to produce observed overall reactions. We have shown that for heterogeneous catalytic systems (Happel and Sellers, 1982,1983;Happel, 1986)it is possible to determine a unique set of mechanisms corresponding to a system specified by an initial choice of elementary reactions, where reactants are specified as being either intermediates or terminal species. These mechanisms, which we have termed direct, demonstrate the various ways that the stoichiometric equations involving only terminal species can break down into elementary reactions. Each such direct mechanism is irreducible in the sense that it cannot be separated into submechanisms, each of which produces the same overall reaction. What we call a direct mechanism is a formalization of what is usually called simply a mechanism in the chemical literature without an explicit definition being given. A further simplification, useful for development of rate equations based on appropriate mechanisms, is the ex- tension of the concept of directness to the enumeration of stoichiometricequations corresponding to given systems. Such direct overall reactions avoid repetition of species among reactions, a concept that is also used by chemists and engineers without formal identification. The concept of directness as applied to both mechanisms and reactions has now been developed in detail (Happel and Sellers, 1989). In a chemical system involving terminal species, intermediate species, and elementary reactions, we can ask, fmt, what are all the possible mechanisms and, second, what are all the corresponding overall reactions. The way we answer these questions is to list only the 0888-5885/90/ 2629-1057$02.50 JO mechanisms that are not separable into submechanisms for the given system and to list only the overall reactions that are not separable into subreactions of the system. Directness is often system dependent for overall reac- tions. In some circumstances, one may assume that “all possible molecular transformations can occur among a set of species” (Aris and Mah, 1963; Amundson, 1966). In this case, the atom-by-species matrix gives a valid way to de- termine the maximum number of linearly independent overall reactions in a system. For such a system, the definition of direct reactions requires that, if any one of the components in a given reaction is deleted, a reaction among the remaining components no longer exists. Such reactions have been termed single (Petho and Kumer, 1985). For systems in which all possible transformations do not occur, the number of direct overall reactions will be fewer. In all cases, the sets generated are finite, per- mitting a logical scheme for selection of appropriate kinetic relationships among terminal species. For simplicity, it is assumed that a single direct mech- anism will characterize a system. This is not always the case. For example, it is sometimes possible for a homo- geneous reaction mechanism to exist in parallel with that promoted by a heterogeneous catalyst. It is possible to develop the criteria for combining direct mechanisms such that certain combinations will also be cycle-free (Sellers, 1984). The situation is different for direct overall reactions. The initial step in the calculation procedure for a chosen system generates a single mechanism and a corresponding set of overall reactions which, though linearly independent, may not all be direct. The number of such overall reactions obtained is, however, unique for the given system. From 0 1990 American Chemical Society
Transcript
Ind. Eng. Chem. Res. 1990,29, 1057-1064 1057
Herskowitz, M.; Carbonell, R. G.; Smith, J. M. Effectiveness Factor and Mass Transfer in Trickle-Bed Reactors. AIChE J. 1979,25,
Holah, D. G.; Hoodless, I. M.; Hughes, A. N.; Sedor, L. J. Kinetics of Liquid-Phase Hydrogenation of 1-Alkenes over a Partially Hydrogenated Nickel Boride and the Effects of Catalyst Poisons upon These Hydrocarbons. J. Catal. 1979,60, 148-155.
Madon, R. J.; O’Connell, J.; Boudart, M. Catalytic Hydrogenation of Cyclohexane. 11: Liquid Phase Reaction on Supported Plati- num in a Gradientless Slurry Reactor. AZChE J. 1978, 24,
Niiyama, H.; Smith, J. M. Adsorption of Nitric Oxide in Aqueous Slurries of Activated Carbon: Transport Rates by Moment
272-283.
904-91 1.
Analysis of Dynamic Data. AZChE J. 1976,22, 961-970. Oudar, J. Sulfur Adsorption and Poisoning of Metallic Catalyst.
Catal. Rev. Sci. Eng. 1980,22, 171-195. Satterfield, C. N; Ma, Y. H.; Sherwood, T. K. The Effectiveness
Factor in a Liquid-Filled Porous Catalyst. Znst. Chem. Eng., Symp. Ser. 1968,28, 22-29.
Zwich, J. J.; Gut, G. Kinetics, Poisoning and Mass Transfer Effecta in Liquid-Phase Hydrogenations of Phenolic Compounds over a Palladium Catalyst. Chem. Eng. Sci. 1978, 33, 1363, 1369.
Received for review May 3, 1989 Revised manuscript received September 6 , 1989
Accepted September 25, 1989
Mechanistic Study of Chemical Reaction Systems
John Happel* Department of Chemical Engineering and Applied Chemistry, Columbia University, New York, New York 10027
Peter H. Sellers The Rockefeller University, 1230 York Avenue, New York, New York 10021
Masood Otarod Department of Mathematicslcomputer Science, University of Scranton, Scranton, Pennsylvania 18510
This paper presents a method for characterizing complex chemical systems often encountered in studies of homogeneous, heterogeneous catalytic, and enzyme reactions. From a list of chosen elementary steps, i t is shown how all possible mechanisms and corresponding overall chemical equations can be enumerated. This furnishes a valuable tool for the further consideration of desired reactions and elucidation of their kinetics. Illustrative examples are given, including problems of current interest. A computer program is available to implement the procedures required.
In the study of a chemical reaction, the first step is often the proposal of appropriate mechanisms to show how el- ementary steps may be combined to produce observed overall reactions.
We have shown that for heterogeneous catalytic systems (Happel and Sellers, 1982,1983; Happel, 1986) it is possible to determine a unique set of mechanisms corresponding to a system specified by an initial choice of elementary reactions, where reactants are specified as being either intermediates or terminal species. These mechanisms, which we have termed direct, demonstrate the various ways that the stoichiometric equations involving only terminal species can break down into elementary reactions. Each such direct mechanism is irreducible in the sense that it cannot be separated into submechanisms, each of which produces the same overall reaction. What we call a direct mechanism is a formalization of what is usually called simply a mechanism in the chemical literature without an explicit definition being given.
A further simplification, useful for development of rate equations based on appropriate mechanisms, is the ex- tension of the concept of directness to the enumeration of stoichiometric equations corresponding to given systems. Such direct overall reactions avoid repetition of species among reactions, a concept that is also used by chemists and engineers without formal identification.
The concept of directness as applied to both mechanisms and reactions has now been developed in detail (Happel and Sellers, 1989). In a chemical system involving terminal species, intermediate species, and elementary reactions, we can ask, fmt, what are all the possible mechanisms and, second, what are all the corresponding overall reactions. The way we answer these questions is to list only the
0888-5885/90/ 2629-1057$02.50 JO
mechanisms that are not separable into submechanisms for the given system and to list only the overall reactions that are not separable into subreactions of the system.
Directness is often system dependent for overall reac- tions. In some circumstances, one may assume that “all possible molecular transformations can occur among a set of species” (Aris and Mah, 1963; Amundson, 1966). In this case, the atom-by-species matrix gives a valid way to de- termine the maximum number of linearly independent overall reactions in a system. For such a system, the definition of direct reactions requires that, if any one of the components in a given reaction is deleted, a reaction among the remaining components no longer exists. Such reactions have been termed single (Petho and Kumer, 1985). For systems in which all possible transformations do not occur, the number of direct overall reactions will be fewer. In all cases, the sets generated are finite, per- mitting a logical scheme for selection of appropriate kinetic relationships among terminal species.
For simplicity, it is assumed that a single direct mech- anism will characterize a system. This is not always the case. For example, it is sometimes possible for a homo- geneous reaction mechanism to exist in parallel with that promoted by a heterogeneous catalyst. It is possible to develop the criteria for combining direct mechanisms such that certain combinations will also be cycle-free (Sellers, 1984).
The situation is different for direct overall reactions. The initial step in the calculation procedure for a chosen system generates a single mechanism and a corresponding set of overall reactions which, though linearly independent, may not all be direct. The number of such overall reactions obtained is, however, unique for the given system. From
0 1990 American Chemical Society
1058 Ind. Eng. Chem. Res., Vol. 29, No. 6, 1990
this set, it is possible to determine the finite set of direct overall reactions, and they may be combined into appro- priate subsets. The number of reactions contained in each of these subsets is still the same as that originally calcu- lated for the system. Each such subset is tested for linear independence, and those not meeting this requirement are discarded. In this way, the final result consists of the minimal number of unique combinations necessary to describe the system. Overall reactions in each of these sets constitute the simplest ways to express the overall reac- tions among terminal species. If necessary, these direct overall reaction sets can each be modified by means of linear combinations of them that still retain linear inde- pendence.
This basic mathematical structure of chemical systems has been developed in the context of heterogeneous cata- lysis, and a further body of mathematical background is available (Sellers, 1971,1972, 1984, 1989). In this paper, we will emphasize applications including examples of ho- mogeneous and enzyme reactions.
In homogeneous reactions, if the velocity constants for reactions of a species are sufficiently high, its concentration during steady-state reaction will be negligible, and it can be considered as an intermediate, whose rate of formation and reaction are equal. In heterogeneous catalysis, ad- sorbed intermediates will, of course, also attain a constant concentration.
For catalysis with enzymes, either immobilized or ho- mogeneous, the enumeration of mechanisms is complicated by the possibility of a very large number of alternatives. In studies of metabolic mechanisms, it is customary (Orten and Neuhaus, 1970) to identify reaction pathways in which a single precursor passes through a series of reactions to form a target product. Each of these reactions may be catalyzed by a different enzyme, and additional agents termed cofactors may also be required. Thus, each main reaction step is itself composed of a set of simpler reac- tions. The concept of pathways can also be effectively applied in our procedure, noting that such pathways often involve steps that follow a carbon balance, even though other atomic species may not balance in a step-to-step progression. Thus, it is possible to write the enzyme re- actions as pseudoequations involving carbon only in the transformations of key species. Such sets can be examined by our procedure to enumerate alternative mechanisms.
Even with these simplifications, it often happens that there will still be a large number of possibilities in many systems. Further reduction of choices may be desirable before detailed modeling of experimental data. The net velocity of all elementary steps may be taken to occur in either direction or not a t all. The mathematical procedure used here assumes initially that all elementary steps occur in both directions. The net directions of overall observed reactions among terminal species under given operating conditions will be dictated by thermodynamics so that their directions will often be known and required net di- rections of mechanistic steps can be established. We have employed single arrows to designate the directions of such steps, including not only irreversible steps but those with the indicated net directions. In addition, aside from their exact magnitudes, it may be possible to predict directions of elementary steps and, thus, to narrow the choice of possible viable mechanisms.
We have treated a number of complex reaction systems in heterogeneous catalysis (Happel and Sellers, 1983), methanol synthesis, and anodic oxidation of zinc (Valdes et al., 1985). More recently, these procedures have been discussed by Ostrovskii et al. (1986, 1989) as applied to
81 -1 92 1 93 -2
1 -1 0 -1 1 0
s.2: H+ + e- --p. H 53: H + H + H ,
Ind. Eng. Chem. Res., Vol. 29, No. 6, 1990 1059
advance by noting that the three mechanisms are distinct from each other chemically since for any two of them each one contains a step that is not involved in the other one. This illustrates the distinction between the number of algebraically independent elementary reactions and the number of direct reaction mechanisms, which is not always realized (Miyahara, 1969).
Suppose for purpose of discussion that a system follows mechanism ml which does not involve 9,. Choosing a linear combination of m2 and m3 such that step s1 always cancels out would not be an appropriate way to model mechanism m,. This basic distinction applies throughout all the systems considered in this paper.
Ammonia Synthesis. The reaction N2 + 3H2 = 2NH3 has been studied extensively from a mechanistic viewpoint. Horiuti (1973) and Temkin (1973) proposed entirely dif- ferent mechanisms. A reasonable approach, if we regard the opinions of both investigators as valid, is to inquire what the possibilities are, assuming all mechanistic steps proposed by both could occur as follows: 9,:
s2: N21+ H2 NzH21 53: N2H21 + 1 2NH1
s4:
N2 + 1 * N21
N2 + 21 + 2N1
s5: s6:
sa: H2 + 21 2H1
The symbol 1 in system (4) refers to an active surface site on the catalyst. Every species containing 1 is an inter- mediate, and the rest are terminal species.
Omitting the matrix representation of (41, which would be analogous to Table I, we go directly to Table IV, which is analogous to Table 11. In this table, steps sl, s2, s3, and s7 were proposed by Temkin and steps s4, s5, 96, and sg by Horiuti. The seventh row shows that there is only one overall reaction, and one of its mechanisms is given in the left hand column. It is the mechanism corresponding to Temkin's steps, but the diagonalization produced could have resulted in other possible mechanisms. The lower two rows show that there are two cycles or null reactions, as contrasted to the hydrogen electrode example in which only one such cycle mechanism existed. By writing the steps corresponding to the last three rows, we obtain one representation of the general mechanism, as follows:
Table I exhibits these three reactions in matrix form where the reactions are regarded as vectors, and the minus signs denote reactants on the left side of the equation. In the top row, the intermediate is listed first a t the left and a vertical line separates intermediates from terminal species. Since H+ and e- are always present together, let us regard H+ + e- as a single component and write it simply as H+. This matrix can be manipulated by simple row operations to obtain the diagonalized form shown in Table I1 (Happel and Sellers, 1989). In Table 11, zeros appear in the lower and left-hand portion. A column at the left records the combinations of steps required to accomplish this. It is seen that the second row in Table I1 represents one way to obtain the overall reaction. The third row is a cycle, having the same reactants as products, and would not exist a t steady state. We can write the general mechanism, including both steady and unsteady states, as follows: (81 + 82) + 4b1 - 92 - 83) = (1 + $)s1 + (1 - d s , - 483
(2) The coefficient 4 is unrestricted in general. To generate mechanisms at steady state, we must eliminate the for- mation of a cycle. This corresponds to setting one of the coefficients on the right-hand side of eq 2 equal to zero. Such a procedure gives three possible values of 4, namely -1, 1, and 0. Setting these values into eq 2, we obtain all the possible direct mechanisms as follows:
m, = 2s2 + s3 m2 = 2s, - s3 m3 = 8 , + s2 (3)
This result is given in Table I11 in matrix form. Each mechanism in the table exhibits a zero entry for one of the three steps, s,, s2, or s3, respectively. This implies that for each proposed direct mechanism one possible step does not exist. I t does not mean that the step omitted in each case can occur with balanced equal forward and reverse rates and zero net velocity. Horiuti and Nakamura (1957) showed that any two of these three mechanisms could be combined algebraically to obtain the third. Following this important observation, Milner (1964) contributed a further
Table IV. Diagonalized Matrix: Ammonia intermediate species terminal species
m1 0 x 0 2 1 m2 0 1 2 0 3 m3 0 i I 3 0 m4 I u 0 2 0 m6 1 0 2 0 2 m6 3 -2 2 0 0
cycles. We can avoid cycle formation by choosing values of 4 and $ such that mechanisms are formed in which one step is missing from each of the cyclic mechanisms so that it cannot occur.
The choice is more complicated than in the case of the hydrogen electrode. It could be accomplished by a series of systematic trials of values for pairs of choices of 4 and $, but a mathematical formalism is more convenient, es- pecially in more complicated cases. In this case, there are six such combinations that are not repetitive for the pairs (@,$I, namely (LO), ( W , (l,-l), (O,O), (0,2), and (-22). These pairs may be substituted into eq 5 to exhibit the corresponding mechanisms shown in Table V. m2 and m4 are the mechanisms proposed by Horiuti and Temkin, respectively. m3 and m6 might be considered unlikely on the grounds that in each case one of their steps proceeds in a direction opposite to what would be expected for a series of hydrogenation reactions. However. both m, and m5 then remain for consideration.
Other considerations in the study of mechanisms for this reaction have been discussed (Boudart and Djega-Maria- dassou, 1984; Happel, 1986).
Ethylene Oxide Synthesis. The oxidation of ethylene to ethylene oxide introduces another feature often en- countered in industrial reactions where along with the formation of the desired product other undesired species may be formed. For illustration purposes, we have chosen a system similar to that proposed by Temkin (1979) but without any prior assumption about the direction of in- dividual steps. The steps are as follows: 81: 0 2 + 1 0 2 1
201 + 0 2 1 + 1
C2H40 + 1 + C2H401
s2: 93: 021 + C2H4 F= 01 + CHSCHO
s4:
95:
sc 0 2 1 + C2H4 === C2H40 + 01
5021 + CH3CHO F= 501 + 2C02 + 2H20 57: C2H401+ 01 + C2H4 (6) Note that in this sequence acetaldehyde (CH3CHO) is taken as an intermediate, although the overall reaction involves a solid catalyst. Diagonalization of the matrix of stoichiometric coefficients corresponding to eq 6 results in the matrix shown in Table VI.
Examination of Table VI shows that two overall reac- tions are involved but only one cycle. The general mech-
anism involves unrestricted values for the two overall re- actions as well as the unrestricted value of @ for the cycle. It may be written as follows: ( p + 3a)q + ( p + 3a + 4)s, + a(s3 + 86) +
(2P + @)s5 + 4494 + 87) (7 ) where the degrees of advancement of the two overall re- actions given in rows 5 and 6 of Table VI are p and a, respectively, corresponding to arbitrary degrees of ad- vancement of the two corresponding reactions: p(-0, - 2C2H4 + 2C2H40) +
The two reactions shown in expression 8 happened to occur in the course of diagonalization. Their number, in this case, also corresponds to what would be obtained by an atom-by-species matrix as discussed earlier (A r i s and Mah, 1963; Amundson, 1966). It is also a coincidence that they are direct overall reactions, but they are not the complete list of direct overall reactions, which can be obtained by matrix operations similar to those involved in determining direct mechanisms. The complete set of direct overall reactions, as shown by Happel and Sellers (1989), is as follows: dl: 2C2H4 + O2 = 2C2H40
dq: 6C2H40 = 2C02 + 2H20 + 5C2H4 (9) It can be seen that they have the characteristic property of direct reactions, that for any two of them each has a reactant not in the other. Thus, Table VI shows dl and d3 corresponding to eq 8. There are six possible combi- nations of the four reactions taken two at a time shown in eq 9. If desired, the other five combinations could be developed into matrices similar to Table VI. Any mech- anism developed from Table VI can be expressed in terms of these six possible combinations of direct overall reac- tions.
As for the mechanisms themselves, a procedure similar to that employed for the two previous examples serves to generate the appropriate mechansims expressed in terms of dl and d3 as shown in Table VII. Similar results would be obtained in terms of the other five possible combina- tions of direct reactions. The construction of each mech- anism in Table VI1 requires eliminating at least one step in the cycle while at the same time retaining sufficient steps to construct the two overall reactions.
Choosing fewer elementary steps (Happel and Sellers, 1983) can result in a simpler mechanism and only a single corresponding overall reaction such as the following:
7 0 2 + 4C2H4 = 2C2H40 + 4C0, + 4H2O (10) This serves as an example in which the atom-by-species matrix does not serve to predict the number of direct overall reactions.
.__ Table VI. Diagonalized Matrix: Ethylene Oxide terminal species - intermediate species
&5: CHO + CH3 - CO + CH4 (11) Additional information beyond the listing of steps can
be obtained by using our procedure. We will first assume that all reactions may be considered reversible and omit steps s9 and sll since they will not influence the mechanism for CH30H production. Formaldehyde (CH20) will be considered an intermediate because the experimental data show it to be present in only trace amounts. There are then 13 elementary steps, 9 intermediates, and 5 terminal species. The corresponding diagonalized matrix is given in Table VIII, omitting the detailed steps corresponding to each row.
This matrix shows two overall reactions among terminal species and two cycles. Further calculation shows that there are six possible mechanisms, but before listing them, we wish to consider the appropriate choice of overall re- actions to use.
We find that there are four possible direct overall re- actions as given by the matrix shown in Table IX. We have written these reactions in a direction so that methane is consumed and CH30H is produced. Note that each reaction is missing one species that occurs in all the others. Since the system has been shown to be characterized by two independent overall reactions, we could write mech- anisms in terms of any desired combination of two of the four direct reactions. Only one of these reactions, d4, consumes methane and oxygen to produce methanol, so a reasonable choice would include this reaction.
We might consider d3 as a second independent reaction, but the six mechanisms expressed in terms of d3 and d4 all require step sl0 (CH3 + OH - CH30H) in eq 11 to proceed in the negative direction, which is assumed to be unlikely. Another choice for a second independent reaction is offered by the nondirect reaction ro = d3/2 + d4/2: 2CH4 + 202 = CH30H + CO + 2H20. In this case, two of the six mechanisms wil l not contain steps in the negative direction, following the assumption by Yarlagadda et al. (1988).
Table X lists the six reaction mechanisms for the com- bination of ro and d4 with all steps taken aq reversible. The number of times each of the assumed steps must be taken to arrive at one occurrence of ro and d4 is specified in appropriate rows for each mechanism.
CH30 - CH20 + H does not appear in any of the mechanisms. It is therefore not a viable choice in conjunction with the assumed steps. If we assume that all elementary reactions occur in the direction taken by Yarlagadda et al. (1988), mechanisms m3 and m4 are the only ones allowed.
Ethylene oxide synthesis reactions and mechanisms are not simple since commercial production involves the use of chlorine compounds as promotors, as discussed by Berty (1989). Other mechanistic considerations are discussed by Happel (1988) and Haul and Neubauer (1988).
Conversion of Methane to Methanol. The partial oxidation of methane to methanol is a process of consid- erable economic interest in view of recent discoveries of large resources of natural gas and the possibility of using methanol for motor fuel. Heterogeneous catalytic studies of this reaction have been reviewed by Pitchai and Klier (1986), but no commercially viable system has been de- veloped. An experimental program has recently been conducted by Yarlagadda et al. (1988) with the objective of selectively converting methane to methanol by homo- geneous gas-phase oxidation. Experiments were performed in a high-pressure tubular reactor under various process conditions. The experimental data indicated that meth- anol selectivities of 70-80% could be achieved at methane conversions of 8-10%.
A proposed mechanism suggested by Yarlagadda et al. (1988) consists of the following listing without further elaboration about how the steps might be combined and taking them as unidirectional: 81: CH4 + 02 4 CH3 + HOz
Choosing other elementary steps might also be consid- ered in deducing other possible mechanisms. Data pres- ented by Pitchai and Klier (1986) show that production of COz is substantial, especially a t high oxygen concen- trations in the feed. This would favor the water gas shift reaction. Yarlagadda et al. (1988) also suggest that wall effects such as might be experienced in the stainless steel reactor would be important in reducing selectivity for methanol production. Metals, of course, catalyze the water gas shift.
Conversion of Glucose to Pyruvate. As an example of the kind of information in the gene library, Sagan (1980) has illustrated the first steps in the metabolism of glucose. Recently an interesting study of this complicated process has been presented by Seressiotis and Bailey (1988) using an artificially intelligent software system for the analysis and synthesis of metabolic pathways. This development is termed MPS (metabolic pathway synthesis). MPS consists of data bases for enzymes and substrates and a search and screening algorithm. The enzyme data base categorizes associated catalyzed reactions as reversible or irreversible in vivo. The search and screening algorithm has been written in Common Lisp using a IBM-AT, equipped with 3.5 Mbytes of CPU memory.
The algorithm is not completely described, but it pro- poses to distinguish between the traditional point of view based on linear independence of reactions and those cat- alyzed by enzymes by the fact that the latter are conducted in vivo instead of in vitro.
In the following analysis of their example, we show that it is not necessary to invoke the concept of in vivo reactions in order to deal with these enzyme reactions using our procedure. As stated in the introduction to this paper, the concept of linear independence introduced by Horiuti is not a sufficient criterion to characterize direct reaction mechanisms.
We will apply our method to show that a more complete picture of the possible reaction mechanisms emerges. Let us first write the assumed species that appear in the metabolic chart shown as Figure I in Seressiotis and Bailey (1988) using the following letter abbreviations to facilitate later discussion: C, carbon dioxide; D, dihydroxyacetone P; E, erythrose 4-P; F, fructose 6-P; G, glucose 6-P; K, 2-keto-3-deoxy 6-P gluconate; L, ribulose 5-P; N, 6-P gluconate; P, pyruvate; R, ribose 5-P S, sedoheptulose 7-P;
X, xylulose 5-P; Y, glyceraldehyde 3-P.
mechanistic steps: SI: R + X = S + Y
These species are used as follows to designate the
s2:
s3:
L = R N - L + C
Sq: G - N 95: F - D + Y Sg: G + F s7: D = Y Sg: N - K
s10: E + X = Y + F Sg: L = X
s11: Y - P 512: D - P s13: K - Y + P
%4: S + Y = E + F (12) These steps are the same as those used by Seressiotis and Bailey except for s3 where COz was added to obtain a carbon balance. Then we can write the elementary steps in a simple form as pseudoequations.
With these conventions, we may write the diagonalized matrix consisting of 14 elementary steps, 10 intermediates, and 3 terminal species as shown in Table XI.
This matrix results in 2 overall reactions and 12 possible mechanisms, assuming that all elementary steps are taken to be reversible.
A list of the three possible direct overall reactions is given in Table XII. Algebraically any two of these direct reactions suffice to generate all possible overall reactions in this system. It is seen that, in general, if all elementary steps are assumed to be possible, there are two overall reactions that can occur simultaneously, rather than only one. The two overall reactions can include linear com- binations of the direct reactions shown in Table XII. One of those chosen by Seressiotis and Bailey was dl in Table XII. Another of their choices is obtained by the following combination, designated as ro, which is not a direct overall reaction: 3dl: 3G = 6P dz: P = 3co2 net: ro: 3G = 5P + 3C02
Table XII. Direct Overall Reactions: Glycolysis G P C
dl -1 2 0 dz 0 -1 3
The 12 possible mechanisms can be expressed in terms of these 2 reactions:
dS -1 0 6
dl:
ro:
G = 2P
3G = 5P + 3C as shown in Table XIII.
Each of the 12 mechanisms shown in Table XI11 lists the steps occurring with the two reactions occurring si- multaneously in the direction of glucose 6-P conversion. There are six different ways in these mechanisms that reaction dl occurs and nine different ways of ro occurrence.
Taking into account the steps that are assumed to be unidirectional, mechanisms m2, m3, m4, and m12 are ruled out so that eight combined mechanisms remain. In this situation, there are five different ways that dl can occur and five different ways for r,. Seressiotis and Bailey (1988)
listed in their Table I possible pathways for each of the separate reactions. Their reaction dl (G = 2P) is listed as present in 5 pathways, namely, their pathways 2 ,7 ,8 , 9 and 10. However, they list ro (3G = 5P + 3C) as oc- curring by only 3 pathways, namely their pathways 1,3, and 4. Our method shows from Table XI11 that five dif- ferent mechanisms exist for ro. Since complete details of MPS and a listing of the steps in the 10 pathways shown in their Table I have not been reported, it is difficult to consider the reasons for this discrepancy. It is easy to verify by direct calculation that mechanisms developed both by our procedure and theirs are direct.
Conclusion The method presented here enables the listing of all
possible direct reaction mechaniims consistent with a given choice of reaction steps for a chemical reaction system. This information should be valuable as a guide for addi- tional theoretical and experimental research on mecha- nisms. As is evident from the examples, the approach used is concerned with real systems as the chemist visualizes these elementary reaction steps. It does not provide a procedure for predicting elementary steps. Pitfalls in many
mechanistic studies can be avoided by a clear development of the conclusions that can be drawn from a given choice of mechanistic steps. Such information also furnishes a logical basis for the further development of kinetics for systems treated.
Acknowledgment
John Happel is g-rateful to Hugh Hulburt for stimulating discussions on chemical kinetics beginning over 40 years ago in the context of the mechanisms of the ammonia synthesis reaction. We are grateful for helpful discussions with M. A. Hnatow during the course of the research. Partial support of this research by the National Science Foundation under Grant CBT-87-00711 is appreciated.
Literature Cited Amundson, N. R. Mathematical Methods in Chemical
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Received for review December 8, 1988 Revised manuscript received November 2, 1989
