RUTHERFORD A R E
This paper conducts a tour through a maze that links the divergent fields of kinetics and mathematics
Mathematical Aspects of CHEMICAL REACTION t might be useful to introduce the term “formal chemkinetics” to denote that segment of natural philosophy that concerns itself with those aspects of chemical kinetics which are independent of the specific nature of the substances taking part in a reaction. T o do so is not an attempt to canonize the notorious A+B or the ubiquitous Z a Z j A , = 0, but is a recognition that the fundamental principles will become much clearer and important distinctions can be sharpened, when the subject is treated more abstractly. Such an abstraction has proved its worth repeatedly in diverse branches of science, as, for example, when the distinction among kinematical notions, dynamical laws, and constitutive relations allows the structure of fluid mechanics to be perceived. In an introduction to a recent ACS symposium of the above title ( I ) , I used Kuhn’s term “paradigm” ( 2 ) . This has been questioned (3) on the grounds that the subject has not yet assumed that framework of accepted ideas which a t once commands the attention of a sufficient number of workers and is sufficiently openended to leave them a number of problems to resolve. That there are quite a few people working in this area was clear from the interest in the symposium, and that there are any number of interesting problems to be solved has never been in doubt, but the relationship among the various ideas is well worth inquiring into, for what emerged at that meeting was an impression of the intricate web of cross-connections among the ideas of various scholars pursuing their studies quite independently. This essay is not a comprehensive review of the kind that this journal so usefully provides on various chemical engineering topics, though it will attempt to direct attention to as many references as possible. Nor is it a report on the symposium held a t the Illinois Institute of Technology, though it will refer to the papers given there, but it will be convenient to substantiate the claim
I ical
that there is a fabric of interconnections by attempting to chart those papers (4-16) in the following diagram. Such a diagram can only roughly indicate the relationships that we are trying to discern for these are naturally more subtle and tenuous than a block diagram. To take an illustration, the concept of reaction mechanism and the associated notion of the pseudo-steady state is a pervasive one. Seller’s combinatorial approach
Figure 1. Some interconnectiom among the papers of the 35th ACS December Symposium VOL. 61
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(6, 78) provides a way of relating the mechanism to the overall reaction and of describing the complete set of independent mechanisms. I t is algebraic and does not take directly into account the analytical aspects that come in when the stoichiometry or kinematics of a reaction system is invested with kinetic laws or a dynamics. The motivation behind the analysis of lumping by Wei and Kuo (77, 79, 79a) is similar to that of the kineticist in respect of the desirability of simplification of complex systems, and it clearly has a bearing on the notion of mechanism. Their interest, however, was first to discover the conditions under which a lumped system would remain monomolecular, whereas a reaction mechanism is almost always an example of what they would term improper lumping-z.e., it leads to a totally different kinetic expression. Zimmerman’s presentation of the kinetics of replicating macromolecules (72,20,27) paid some attention to the mechanism though it was primarily concerned with a very beautiful analysis involving stochastic and combinatorial ideas and the solution of the differential equations, as was Gavalas’ example of an autocatalytic reaction (9) though it principally featured the elegant methods of nonlinear differential operators. Similarly, Bartholomay’s valuable exposition of stochastic formulations of kinetics ( 4 ) )necessarily made passing reference to mechanism though it has had his more direct attention elsewhere (22). I t was in Silveston’s (75) and Higgins’ (76) works that the whole question of the pseudo-steadystate hypothesis was raised most directly. I n the first case by direct calculations intended to check out some criteria proposed for its validity; in the second, by showing the degree to which assumption would break down under certain frequencies of perturbation. Again, Higgins’ paper was related to the notion of lumping, and an extremely complex biological pathway was broken down into a smaller number of lumped steps. The notion of a reaction mechanism could scarcely fail to have a prominent role to play in any discussion of chemical reaction, but to illustrate the role of a less obvious concept we may turn to that of fading memory. In the mechanics of materials, it is now commonplace to recognize that the present state of stress can be a functional of the whole past history of strain. The mixtures considered by Bowen (5) [extending his earlier work (23)]and by Coleman (8, 24) are of materials with this memory. But Othmer and Scriven (73), in discussing the interaction of diffusion and reaction in surfaces, pointed out that at the limits of their analysis a hereditary or memory effect could not be ignored. Halsey (25) has used the phrase “short term memory” to describe nonequilibrium adsorption in catalysis, and it has been shown (26) that the same assumptions that justify the pseudo-steady-state hypothesis will obliviate the memory in a reaction rate functional. Here again is a link back to the notion of mechanism. With these indications of the intricate weave that we must expect to see in the fabric of formal chemical kinetics, let us attempt to sketch a more systematic cartoon. 18
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
Axiornatics
Risible though it might appear to the experimental chemist, a strictly deductive theory of chemical kinetics is a worthy goal. Bunge has outlined this program for various branches of physics (27), and Basri has actually carried it out in detail for the fundamentally important theory of space and time (28). ’IVe are still a good deal farther away from this goal in chemical subjects than in physical, for chemistry has, on the whole, been much more resistant than physics to the incursions of the philosophy of science. The only attempt at axiomatization of kinetics is that of Wei (29), who presents the following simple and attractive set for a closed reaction system: 1. The total mass of the mixture is conserved 2. The masses of each species are never negative 3. T h e rates of change of each mass are smooth functions of the masses 4. The principle of microscopic reversibility obtains 5. There exists an appropriate Liapunov function assuring the stability of a unique equilibrium point I t may well be true that “Tot homines, quot propositiones per se notae, et de gustibus non est disputandum,” but it is interesting to inquire how primitive and complete this set may be. Thus, if we accept the continuum equations of continuity of distinguishable species we can replace item 1 above by two propositions which might be taken to be axiomatic : l a . Every property of the mean motion of a mixture is a mathematical consequence of the properties of the motions of its constituents l b . If all effects of diffusion are taken into account properly, the equations for the mean motion are the same as those governing the motion of a simple medium From these two postulates it follows as a theorem that if the mean motion of a heterogeneous medium is to satisfy the ordinary equation of continuity, then it is necessary and sufficient that the sum of the rates of change of the masses of the several components should be zero a t each point [cf. Truesdell (30)]. Thus, two propositions can be made to yield Wei’s first axiom as a theorem, though whether they are more reasonably “per se notae,” as the Doctor Communis would have put it, is open to discussion. O n the other hand, if we start with a proposition of structure, that the molecular species may be uniquely defined in terms of constituent atomic species, we may add a conservation law, that AUTHOR Rutherford Aris is Professor of Chemical Engineer-
ing at the University of Minnesota, Minneapolis, Minn. 55455. The author is indebted to Octave Levenspiel and Dan Luss for raising several points of interest in correspondence. In particular, the former points out that the term formal kinetics (la cinitique formelle) was used by Jungers to denote the area of the dzferential equations of kinetics and determination of rate constants and mechanisms. Since writing this paper the author has also seen a manuscript by F. J . Krambeck on ‘‘ The Structure of Chemical Kinetics” which develops an axiomatic treatment of kinetics paying particular attention to the thermodynamic basis,
the masses of the latter are all constant and obtain the equivalent of Wei’s first two axioms. Then Bowen has shown that in the absence of diffusion, or in a closed, uniform system, the changes in composition can be uniquely expressed in terms of a fixed number of independent reactions (31, Theorem 6). We can then impose Wei’s third axiom on the reaction rate expressions and, because the reactions are independent, the principle of microscopic reversibility emerges as a theorem. Wei shows that the existence of an equilibrium point is a consequence of his first three axioms and the fixed point theorem of Brouwer, but this does not ensure its uniqueness or stability. Likewise the principle of microscopic reversibility only ensures an ultimately monotonic approach to equilibrium by guaranteeing the symmetrizability of the local linearization (32). Wei’s fifth axiom is therefore necessary if the commonly observed uniqueness of equilibrium is to be reproduced in the mathematical system. This uniqueness is a feature of statically thermodynamic conditions (33) and so can be claimed for all mass-action kinetic expressions which are consistent with equilibrium (34, 35). This brief discussion does not exhaust the question of axiometic foundations but is intended to show that further work is needed along these lines. We shall allude to that question again in connection with Bartholomay’s stochastic formulation and very briefly in the next section. As indicated above, it needs to be related to the whole business of constructing deductive theories (cf. 27, 36-40) in the context of the physical and chemical sciences, but this is a large matter.
matrix of the products of the reaction (in fact, the steps of such a reaction can be written down as a sequence of such matrices), but this does not appear to be fruitful. An approach more directly related to truly topological notions is that of Lederberg (45-47). This has involved the classification of regular trivalent graphs (since the diagram of the organic molecule rarely has more than three branches at any one node) and the association of these with certain polyhedra. Nothing has been done to connect this theory with a representation of reaction. Topological notions have been introduced to elucidate complex molecular structures by Wasserman (48). Sellers’ treatment of chemical complexes is capable of sustaining a representation of structure (18)) but little of the detail has been worked out. Combinatorial Problems
The work of Sellers (6, 17, 78) represents a remarkable synthesis of combinatorial topology and chemistry, and its articulation should give lead to great advances in the art of discovering and proving the mechanism of a reaction. Though there is a need to soften the forbidding aspect that his principal account (78) may have for the
Structure
The mathematical representation of molecular structure is not our main concern here; but since it has a bearing on reaction we can scarcely avoid some mention of it, even though this connection has been little developed. J. J. Mulckhuyse has an axiomatic treatment of structure in organic chemistry (41)) but this does not seem to have been followed up. Spialter has devised a computer-oriented chemical nomenclature in which the names of the atoms are literal entries A , on the diagonal of a matrix, and the numerical off-diagonal elements b,, denote the number of bonds between A i and A , (4244). Since such a representation should be invariant with respect to permutations of rows and columns, Spialter suggest that the characteristic polynomial is a good representation. The difficulty here is that if there are two atoms of the same chemical element in the molecule and if they are not literally distinguished, then the matrix often cannot be uniquely reconstructed from its polynomial. O n the other hand, if every atom is distinguished only the three highest powers of the polynomial are needed. Moreover this representation does not seem to lend itself to the discussion of a reaction. It is true that a mixture of molecules can be represented by a partitioned matrix. If a matrix of the reactants is written down, a bond-breaking and -making matrix can be added to it, which will give the
Figure 2. Model of enzyme-catalyzed synthesis
nonmathematician, a brief summary of the representation of a reaction and its mechanism may be useful. Consider an enzyme-catalyzed synthesis of two substances, R and S, to form a product, P. If this is a balanced reaction the empirical formula for P must in fact be the sum of those for R and S and we might write P SOIR, when 01 represents a bond of some sort between the two parts of the molecule. Let the sides of the triangle, 2, in Figure 2 represent the species R, S, and P as shown. The triangle itself may represent the overall reaction which, since it is a combination of S and R , we write as S g 1 R where 8 1 is an abstract “product” of the two entities R and S. I n fact, we can write the chemical equation of the reaction in various ways. The stanS = P is easily seen to be equivalent to dard form R R S - P = 0, where all the species are written on
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one side of the equation and the product species is given a negative stoichiometric coefficient. Writing R S P = 0 in its equivalent form
+
R - SOlR
+S = 0
suggests a simple relation to the abstract product
S@ 1R I n fact, if we put an orientation on the face, 8,as shown by the anticlockwise arrow, we see that R - SOIR S is just the boundary of the face, the signs being given according to whether the orientation of the side is in the same (+) or the opposite (-) direction to that of the face. This may be written
+
d(SC31R) = R - SOlR
+S
where the rule for the boundary operator, b, is to delete the first term of the abstract product, subtract the species obtained by replacing @ I by the bond 0 1 , and finally add the species obtained by deleting the last factor of the product. This rule of deletion and changing of @ to 0 can be extended to any number of terms and corresponds to the geometric notion of taking the boundary. Because the face S@IR represents a properly balanced chemical reaction, its boundary is zero, for S O I R or P is just the sum of R and S. Xow suppose the reaction mechanism involves a catalyst, E , which first binds to R to form a complex, C, which then allows S and R to combine to P in a double complex, D, and finally releases itself from the catalyst. These steps might be written
S+C=D
R+E=C
D=P+E
+
and the sum of them is the overall reaction R S = P. But if we represent the binding to the catalyst by a second symbol 0 2 , then C is represented by ROzE and D by S 0 1 R 0 2 E . Thus, the overall reaction 2: R
- SOiR
+S = 0
has three steps
+E = 0 + ROzE = 0 II: E - SOlROzE + SOiR = 0 A:
I‘: R - ROzE S - SOlROzE
and clearly
~ : = r - - n + ~ KOWjust as 2 may be represented by the abstract product &‘@ 1R so that the boundary b(S@1R) is the chemical equation R - SOlR S = 0, so may r, A, and 11 be represented by R @ 2E, S@1R02E, and SOiR @ sE, re-
+
spectively. For example, in taking the boundary b ( S O I R @ p E )we , delete the first factor, SO& giving E, subtract the result of replacing 8 2 by 0 2 , namely, SOIR02E, and add SOlR obtained by deleting the second factor, E; thus, d(SO1R@?E),= E - SOiROzE SOiR. Now the four triangles 8, r, A, and II can be fitted together as shown in Figure 3, where the orientations of faces and edges are shown by arrows.
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Figure 3. “Cutal3.zution model’’
Consider now the double “product” S@ 1R@SE, and apply the extension of the rule for the boundary
d(S@lR@pE)= R@zE - SOiRC3zE S@IR
=
+ SBiROzE -
r -n+A
- 2
Thus, just as the faces, which are two-dimensional triangles, have boundaries which are one-dimensional edges, so the three-dimensional tetrahedron has a boundary consisting of the faces. Moreover, just as the chemical equation for the reaction associated with a face is obtained by setting its boundary equal to zero, so the mechanism is obtained by setting the boundary of the tetrahedron equal to zero. For ~ ( S @ D R @ , = E )r - II A -Z = 0 can be written Z = F - II A , just as above. Sellers has called the three-dimensional complex, here a tetrahedron, a “catalyzation.” The boundary of a catalyzation is a signed collection of reactions or a mechanism. The boundary of a reaction is a signed collection of chemical species or a reaction equation. This is a bald description of Sellers’ concepts, but we hope it shows that the notions of chemical reaction and mechanism can be tied in with those of combinatorial topology. The power of these ideas arises from the fact that the resources of combinatorial mathematics are now harnessed to the tasks of the kineticist, such as that of computing the full independent set of possible mechanisms of a reaction. Another place where combinatorial considerations must be entertained has been shown by Zimmerman (72, 20, 21) in his work on the kinetics of polymerization. If growth starts at two centers on a long molecule, the possibility of interference between independently propagating chains must be considercd. In fact, all stochastic formulations have a combinatorial element since the basic probabilities are usually calculated in this fashion.
+
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Stochastic Formulations of Chemical Kinetics
That the course of a chemical reaction can be regarded as a stochastic process has been the observation
of several authors (22, 49-67). This formulation has the advantage over the deterministic approach of giving a measure of the intrinsic variability of the process. For example, if the irreversible first-order reaction A --c B is considered deterministically, the number of molecules of A that time, t, would satisfy are expressed by
dn _ -- -kn dt or
n(t) =
Algebraic Structure of Systems of Kinetics Equations
nee-"
O n the other hand, suppose that (no - n) molecules have decomposed in the time interval (0,t) and that the probability of a single decomposition in (t, t 4- st) is knst o ( 6 t ) and that the probability of more than one decomposition is o ( 6 t ) . Then, the probability of going from a state with i molecules to one withj( 0 except a t an equilibrium state. I n a strictly dissipative material, asymptotic stability at constant strain and temperature is equivalent to the statement that $(F*, e*, cy) > +*(F*’ e*’ cy*) in some neighborhood of the equilibrium state. I t is such results that Bowen has refined and extended (5, 772). He distinguishes between (a) Weak equilibrium, where reaction rates vanish, (b) Classical equilibrium, where affinities vanish, and (c) Strong equilibrium, where both vanish. Bowen shows that weak equilibrium implies the classical if baf is nonsingular and classical equilibrium implies the weak if ?),& is nonsingular. He has demonstrated some interesting results for waves propagation into a region of weak equilibrium. Coleman and Mizel (8, 773) have addressed themselves to the question of relating the thermodynamic condition for stability (that some free energy or equilibrium response function should be minimum) to the dynamical conditions. They have solved this problem completely for systems whose state can be described by a finite dimensional vector, which, however, may be governed by functional-differential equations. If, on the one hand, the continuum mechanics approach has enlarged the theoretical understanding of the nature of equilibrium, on the other, the resources of linear algebra and optimization techniques have been used to prove the uniqueness of equilibrium compositions and to calculate them. Shapiro’s work (7, 774-777) has led the way here to show very clearly the conditions under which the minimum of the free energy functional as constrained by the mass balance equations is unique (cf. 34, 35). These questions are important in view of the increasingly complex systems that are being considered, particularly in biological contexts, and the need to harness the techniques of optimization and machine computation to the efficient calculation of equilibria (778-789). These considerations lie very close to others of a purely stoichiometric nature, for Sellers has pointed out that stoichiometric equations are mathematical equations (790), and stoichiometric restrictions on equilibria have long been seen to be important (797, 792). Some discussion has centered on the form of the stoichiometric equation, which for a set of r reactions between s species, A,, can be written
Various normalization schemes have been suggested (793-796), and Petho has developed some algebraic apparatus for achieving a simple form (797-203). From an abstract point of view, it is important to retain a sense of the arbitrariness of stoichiometry (3, 3 7 , 34, VOL. 6 1
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195, 204), for any nonsingular transformation of an independent set must be equivalent. The maximum number of independent reactions is governed by the rank of the structure matrix expressing the species A , in terms of their elements (34, 123, 204, 205). And here we return to some questions related to mechanism where linear combinations of the reactions are to be made, [the coefficients in this case are the stoichiometric numbers, as Horiuti (206) has defined them] and to count- I ing the possible mechanisms of reaction (78, 34, 207). Stoichiometric considerations have also been invoked in the discussion of thermodynamic coupling, the possibility that some reactions of a system may be absorbing the entropy generated by the others. Hooyman’s normalization (794) claims to have a certain “intrinsic” property alleged to be necessary in determining the existence of thermodynamic coupling (208) after Koenig and others had shown that system of reactions could always be uncoupled (209). Manes showed that near equilibrium reactions could be coupled or uncoupled by composition perturbations (270, 27 7) and the whole question is reminiscent of the arbitrariness surrounding the reciprocity relations of irreversible thermodynamics (212). It is very clear that algebraic artifice should not be allowed to replace real physical insight (213). Epilog
I n this circular tour of some of the mathematical aspects of chemical reaction it should be evident that there is no neat linear succession of ideas. Rather there is a web of interconnections between the several distinguishable areas of kinetics and mathematical disciplines. There is plenty of scope for investigation of the many avenues that can be opened up, and the contributions of workers with quite diverse backgrounds and intentions have many points of convergence. One of the earliest mathematicians to recognize the affinity between his work on invariants and (‘the new Atomic Theory, that sublime invention of KekulC” was J. J. Sylvester. Even if his ideas on the bearing of invariant theory (274) were to prove abortive (215) until transmuted into “something rich and strange” (cf. Weyl’s remarks, 216), his approach was consonant with the largeness of outlook proper to natural philosophy. Coming from an ampler age than ours, the Augustan measures of his prose may fall quaintly on our ears but it may be permissible to transpose one of his remarks to our subject. (‘The beautiful theory of atomicity”-or for that matter of formal chemical kinetics-“has its home in the attractive but somewhat misty border land lying between fancy and reality and cannot, I think, suffer from any not absolutely irrational guess which may assist the chemical enquirer to rise to a higher level of contemplation of the possibilities of his subject.” BIBLIOGRAPHY (1) Ark, R., IND.END.CXEM.60 (TI), 20 (1968). (2) Kuhn, T. S., “ T h e Structure cf Scientific Revolutians,” Chicago University Press, Chicago, Ill., 1962. (3) Feinberg, M., University of Rochenter, private communication, November 1968. Papers 4-16 were presented at the 35th ACS Chemical Engineering Symposium, Illinois Institute of Technology, December 1968
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(4) Bartholomay, A. F., “Stochastic Formulations in Chemical Kinetic?.” (5) Bowen, R . M., “ O n the Thermochemistry of Reacting Mixtures.” (6) Sellers, P. H., “Combinatorial Aspects of Chemical Reaction.” (7) Shapiro, N . Z . , “Towards an Axiomatization of Chemical Equilibria.” (8) Coleman, A. D., “ O n the Stability of Solutions of General Evolution Equations.” (9) Gavalas, G. R., “Nonlinear Phenomena in Glow Discharges and Other Autocatalytic Reactions.” (10) L,uss, D., and Lee, J. C . M. “Stability of a Chemical Reaction with Intraparticle Diffusion of Heat and Mass.” (11) Wei, J. and Kuo, J. C. W . , ” A Lumping Analysis in Monomolecular Reaction Systems.” (12) Zimmerman, J. M., “Replication Kinetics for Biological hfacromolccules.x (13) Othmer, H. G., and Scriven, L. E., “Interactions of Reaction and Diffusion in Open Systems.” (14) Strieder, W., “Knudsen Flow and Chemical Reaction in a Porous Catalyst,” (15) Silveston, P. 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