J . Phys. Chem. 1988, 92, 226-227
226
Entropy Production and the Ostwald Step Rulet William H. Casey Geochemistry Research, Sandia National Laboratories, Albuquerque, New Mexico 871 85 (Received: November 24, 1986; In Final Form: June 29, 1987)
Ostwald ( Z . Phys. Chem. 1879,22, 289) observed that crystallization from solution commonly proceeds via thermodynamically unstable phases. Linear irreversible thermodynamics has been used to examine this phenomenon. We show that, contrary to previous treatments, rates of entropy production are independent of the number of steps in a reaction sequence. Thus indirect reaction paths are not thermodynamically favored a priori relative to more direct formation of the stable product.
Introduction The theory of linear irreversible thermodynamics tells us that a disequilibrium system evolves toward a minimum in the rate of entropy production.2 This minimum corresponds to a steady state. For an isolated system, this steady state represents equilibrium. For open systems, a unique disequilibrium steady state is maintained by conditions at the system boundary, such as fixed chemical affinities. Previous workers3 have used this theory to show that, for some cases, the formation of intermediates (the Ostwald step rule) results in a lower rate of entropy production than direct formation of the stable product. In this paper we show that this result is misderived and that the number of intermediate steps of a given overall reaction bears no relation to the rate of entropy production. We begin this discussion by reviewing the concept of equivalent disequilibrium systems laid out by Prigogine2 (pp 41-44). Entropy Production and Intermediate Steps Consider two cases for the formation of product P from the reactant R: (i) direct formation without intermediates and (ii) formation via several intermediates. A general reaction scheme can be written:
discussed by Temkin4 and Boudart5 and maintain a simple formalism. It will become clear, however, that our conclusions are general for all combinations of parallel and sequential step reactions. Equation 2 can be expanded: exp(-A/RT) = 1
- (-A/RT)~
+ Ck
k!
(3)
where k = 1 , 2, 3, .... Close to equilibrium, all terms after k = 1 can be truncated, and the net rate can be approximated by a linear relation: Ji = F,Ai/RT
(4)
Equation 1 for entropy production can be rewritten for each reaction path:
We simplify these equations further by relating the chemical affinities for each case. In general Ad = E A , The rate of entropy production
is given by2 n Ai u = CJ,i T where n is the number of elementary step reactions, J, is the rate of step reaction i, and Ai is the chemical affinity of this step reaction in the forward direction. In subsequent discussion the subscript d is used to identify the case of direct formation of the stable product. We use a direct (meaning one step) path in the subsequent derivation only to simplify the formalism. The results can be generalized to an m-step process, as long as m < n. In all cases, we use the formalism of Prigogine,' which differs slightly from that that used by van Santem3 At a steady state, rates of reaction can be directly related to the chemical affinity via transition-state t h e ~ r y : ~ . ~ step reaction i Ji = Fj - Pi = F i ( l - exp(-Ai/RT)) (2) where F, and Fi are the forward and reverse reaction rates of step reaction i, and Ji is the net rate at a steady R and T identify the gas constant and temperature, respectively. In presenting eq 2 we ignore the average stoichiometric coefficient 'This work performed at Sandia National Laboratories supported by the
US.Department of Energy under Contract No. DE-AC04-76DP00789. 0022-365418812092-0226$01SO10
(7)
I
u
Following the lead of van Santeq3we conclude that the Ostwald step rule is general if it can be shown that the rate of entropy production in a steady-state reaction decreases with the number of intermediate steps. That is, ud > u. This criteria is expressed by rearranging eq 5 and 6:
or
(1) Ostwald, W. Z . Phys. Chem. 1879, 22, 289. (2) Prigogine, I. Introduction to Thermodynamics of Irreversible Pro-
cesses; Wiley: New York, 1967. (3) van Santen, R. A. J . Phys. Chem. 1984, 88, 5768. (4) Temkin, M. I. Int. Chem. Eng. 1971, I ! , 709. (5) Boudart, M . J . Phys. Chem. 1976,80, 2869. ( 6 ) Hollingsworth, C. A. J . Chem. Phys. 1957, 27, 1346. (7) Krupka, R. M.; Kaplan, H.; Laidler, K. J. Trans. Faraday SOC.1966, 62, 2754. (8) Moore, W. J. Physical Chemistry; Prentice-Hall: Englewood Cliffs, N J , 1972. (9) Hill, C. G. An Introduction to Chemical Engineering Kinetics and Reactor Design; Wiley: New York, 1977.
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 1, 1988 221
Entropy Production and the Ostwald Step Rule n
n
?d(cA1)’ 2 c ? l A ~ ~ I
(9)
I
We will show that eq 9 is always an equality by remembering that the derivation requires a steady state. The most meaningful case to examine is where the two reaction paths (direct and with intermediates) proceed with identical net rates: Jl = J / = J d (10) or n
? d ( C A , / R T ) = ?jA,/RT = FiA//RT
(11)
I
By reinserting eq 11 into eq 9 and eliminating common constants in the denominator, we can reexpress eq 9 in terms of a single forward reaction rate: n
n
n
n
?d(cAj)’ 2 C ~ I A I A = JC(iddCA/AI) I
I
I
(12)
/
Note that, for a given overall reaction at a steady state, the term ? d e ; A I is constant and can be removed from within the first summation on the right-hand side of eq 12. Thus ?d(iAI)’ I
= ?,jtA/iAl /
(13)
I
Equation 13 shows that the number of intermediate steps is immaterial to the rate of entropy production for a given overall reaction rate at a steady state. This derivation is a general treatment of Prigogine’s’ discussion of equivalent disequilibrium systems (pp 41-44). It is critical to note that this conclusion could be reached without employing irreversible thermodynamics. In our treatment, as well as van S a n t e n ’ ~all , ~ entropy production results from microscopically reversible reactions. Entropy is not produced, for example, by solute transport in addition to reaction. For this case, eq 13 simply reduces to an affirmation that entropy of reaction is a state variable.
Unequal Net Reaction Rates Our derivation in the previous section relied on the simplification that J , = Jd, which was not covered by van Santen.3 We can, however, show that the previous result is general by reconsidering a special case discussed by van S a n t e ~We ~ will show that his conclusions were based upon an incorrect assumption that, by examining entropy production, one may eliminate possible reaction pathways. Van Santenj considered the case where forward rates of the step reactions are equal to the direct case i,
?,j
Noting that eq 16 is less than eq 15 led him to conclude that the indirect reaction path is favored. However, this leads to a paradox that the slowest reaction path is favored. This conclusion was reached by improperly using entropy production to eliminate reaction paths from consideration. The correct approach is to express entropy production for an overall reaction in terms of all possible paths. For the above example, the total entropy production is the sum of that from the direct and indirect paths:
(14)
where the number of step reactions, n, is greater than one. The entropy production of the direct and indirect reaction paths are given by
Remembering our stipulation that Fi = i d , and that a steady state is required, allows us to rearrange eq 17 in terms of reaction rates:
and
As long as the overall reaction rate is constant ( J = Jd + Jid), the relative velocities of the indirect and direct paths are immaterial to entropy production. More importantly, note that eq 19 is simply a restatement of eq 1. Thus, entropy production, even for this special case, is independent of the reaction path for a given overall reaction rate. The Nature of the Ostwald Step Rule Van Santen was led to his incorrect conclusion because he used entropy production to eliminate from consideration the faster, and more entropic, of parallel reaction paths. It must be remembered, however, that all reaction paths contribute to the overall reaction rate and entropy production. Standard kinetic theory tells us that metastable intermediates appear in appreciable quantities if the indirect path of a transient reaction is fast relative to more direct formation of the stable product (see Moore, pp 346-347, and Hill, pp 143-151). Ostwald’ did not provide a rule for eliminating reaction paths but only observed that the precipitation pathways that form intermediates are commonly more rapid than direct crystallization of the stable product. To the limit that linear irreversible thermodynamics can be applied to this problem, no thermodynamic benefit is derived a priori from the abundant number of steps in an indirect reaction path.
