V. J. Lee University of Missouri Columbia, 65201
Generalization of Gibbs' Phase Rule for Heterogeneous Chemical Equilibrium
Gibbs' phase rule for heterogeneous systems contaiuing reactive chemical species has been presented by many well known textbooks on thermodynamics (1-5) in a variety of forms and definitions. It was usually introduced by a simple extension of the original Gibhs' phase rule without a fundamental analysis of the variants and conditions of equilibrium pertaining to the heterogeneous reaction system. For specific applications to particular systems correct results are given by most textbooks. This is achieved by stoichiometric considerations or by application of the conditions of equilibrium of the particular system concerned (1). This note presents a formulism of the generalized Gihhs' phase rule for reactive heterogeneous systems. The analysis is based on a detailed consideration of a heterogeneous system in which the number of reactive and/or inert components in each phase is assumed to be unequal as in most cases [see eqns. (3-5)]. This differs from Gibbs' original analysis and that in many textbooks where the analysis is based on a system with an equal number of components in every phase. Moreover, the generalized phase rule is made precise by introducing terms due to chemical invariant relations [esn. ( W I .
the R independent chemical reactions give the equilhrium relationships:
where 1 9 is the chemical potential of the species A,. For heterogeneous phase equilibrium with reactive species, the number of chemical species in each phase may not he equal. Let there be Cn species in the kth phase. Naturally, some of the Cn species in the kth phase also exist simultaneously in other phases. We suppose that there are altoghether M , species in the reactive system which exist simultaneously in any n phases. From these considerations, we have
and
Combining eqns. (3) and (4), and changing the dummy index of summation from n to k , we obtain
Analysis
Following J. W. Gihbs (6), we shaU consider the simplest case of heterogeneous phase equilibrium. Namely, the actions of gravitation and electric and/or magnetic field are neglected. The phases are in contact directly and the interphase curvature exerts no effect. The solid phases are strain free and the pressure is isotropic. The heterogeneous reactive system is "enclosed in a rigid and iixed envelope, which is impermeable to and unalterable by any of the substances enclosed and perfectly nonconducting to heat" (6). The system contains, altogether, C chemical species, some of which may be chemically nonreactive. Let there be R independent reactions and $ . phases at chemical and phase equilibrium. The chemical reactions are symbolically represented by:
I n heterogeneous chemical equilibrium we have, in addition to eqn. (2), the following equilibrium relationships between the phases: there are (4 - 1) equations each for both thermal and mechanical equilibriums, i.e., TI= T2= . . . = TQ, P , = P n = ... = P m ;
for interphase-mass-transfer equilibrium there are (n - 1) .M, equations for the M , species which exist simultaneously in any n phases, altogether there are
equations. Hence, the total number of equilibrium relationships (E.R.), which together determine the heterogeneous chemical equilibrium, is where a j l is the stoichiometric coefficient of A , in the ith chemical reaction. I n writing down eqn. (I), we have adopted the convention that ajr = 0, if the jth chemical species A,, is inert in the ith reaction; w j s > 0, if A , is a species among the product of the ith reaction; and a,, < 0, if A, is a species among the reactants of ith reaction. At reaction equilibrium, 164
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For the total number of variables (variants) of the reactive system, we note that there are C B chemical species or components in kth phase. Due to the chemical reactions and other conditions of the system, the number of species in any phase, say, the kth phase
may not be completely independent. Let there be In invariant linear combinations of concentrations due to stoichiometric and other conditions. Physically, the invariant relations are valid at all times and at any extent of chemical reaction. It is the same nature as the invariant relations of reactions (7). However, here the invariant relations must relate concentrations of different species which exist in any one phase, say the kth phase. For the conditions of invariant, let one of theinvariant relations in the kth phase be
the total number of equations. Hence, from eqns. (4), (6), and (lo),we have d
F = C - ~ I I - R + Z - +
(11)
k=l
Equation (11) is the generalized Gibbs' phase rule for heterogeneous chemical equilibrium. It reduces to Gibbs' phase rule for a non-reactive system a t phase equilibrium. I n this latter case, all the C species simultaneously exist in every phase as was first supposed by J. W. Gibbs (6). Hence, we have R = 0, In- = 0, Ck = C; the total number of variables is +.(C 1); the total number of equations is (+ 1 ) . (C 2). This gives the well known Gibbs' phase rule,
+ +
where p,, is a coefficient, cl(t) is the molar concentration of the species A, at any time t. Due to the R independent chemical reactions, c,(t) is given by:
where Xi is the extent of reaction or degree of advance ment of the ith reaction; it has the unit of concentration. By combining eqns. (7) and (8), we obtain
Illustrative Example
To illustrate the application of the phase rule, eqn. ( l l ) , the reduction of zinc oxide by carbon is taken as an example. This is chosen so that the following analysis may be compared with that described in reference ( I ) . The system at heterogeneous chemical equilibrium contains two solid phases, zinc oxide and carbon, and a gas phase. Three species, zinc vapor, CO, and COz, exist in the gas phase. Between the five species, there are two independent chemical reactions,
The reactions can be written in the same convention as that of eqn. (1) Equation (9) shows that the row vector of matrix (81,) and the column vectors of the matrix (q,) are orthogonal to each other. Consequently, the product matrix of the two matrices is the null matrix. Thus eqn. (9) can be considered as the criten'on of the invariant relationships. The integer Ik is the number of independent row vectors of the matrix (p,,) for the kth phase. I n actual applications, the invariant relations or their coefficient matrix (Pu) may be determined by stoichiometric considerations or by applying conservation laws, e.g., the conservation of electric neutrality for ionic species. Examples given in the last section will illustrate how this is done. The actual independent variable components of the kth phase is Cx - Ir because of the I x invariant relations between the concentrations of the Cx species. Furthermore, since the nature of the phase equilibrium properties is independent of the size of the phase, only Cr - In - 1 variables, such as mole fraction, mole ratio, or weight percentage, are sufficient to specify the material variables. Thus, including temperature and pressure, each phase is completely determined by Cn: - In 1 variables. The total number of variables is
+
The degree of freedom or variants, F, of the system at heterogeneous chemical equilibrium is defined as the difference between the total number of variables and
+ Zn(g) + CO(g) + 0 = 0 + 0 + Zn(g) - CO(g) + CO>(g) 0
-ZnO(s) - C(s) -ZnO(s)
(12') (13')
=
The coefficient matrix is:
The invariant relation between gaseous species may be obtained by overall balance on oxygen and zinc atoms: AZnO(s)
+ ACO(g) + 2ACOdg) = 0, AZnO(s) + AZn(g) = 0,
(15) (16)
where A Al denotes ej(t) - ej(o). Eliminating AZnO (s) from eqns. (15) and (16), one obtains AZn(g) - ACO(g) - 2ACO?(g) = 0
(17)
Hence, the coefficient matrix of the invariant relation between the gaseous species is: This can be verified by showing that the product of the matrices eqns. (14) and (18) is, indeed, the null matrix:
Since there is no invariant relation for the two solid Volume 44, Number
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phases, we have 2 I x = 1; by eqn. (11) with C = 5, = 2, @ = 3, the degree of freedom of the system is F = 5-1-2+2-3 = 1 (20)
R
Now, if a liquid phase of zinc metal also exists, the invariant relation, eqn. (17) is nullified. But the number of phases, 6, is now equal to four, and the system is still univariant at heterogeneous chemical equilibrium. Literature Cited (1) DENBIGH,K., "The Principles of Chemical Equilibrium," Cambridge University Press, London, 1955, pp. 180-192. (2) WEBER,15. C., A N D MEISSNER,H. P., "Thermodynamics for
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Chemical Engineers," (2nd. Edition), John Wiley & Sons, New York. 1957. DO. 38!&391. (3) CALLEN,H. B., "Thermodynamics," John Wiley & Sons, New York, 1963, pp. 206-208. (4) LEE, J. F., AND SEARS,F. W., "Thermodynamics," (2nd Edition), Addison-Wesley, Reading, Mass., 1963, pp. 50% ..A
Cn.4 ""=.
(5) COULL,J., AND STUART,E. B., "Equilibrium Thermodynamics," John Wiley & Sons, New York, 1964, pp. 226270. (6) GIBBS,J. W., "The Collected Works of J. Willard Gibbs," Vol. I, Longmans, Green, & Co., New Yark, 1928, pp. 56100. (7) ARE, R., "Introduction to the Analysis of Chemical Reactors," Prentice-Hall, Englewood Cliffs, New Jersey, 1965, pp. 24-25.
