Research: Science & Education
Is the Reaction Equilibrium Composition in Non-Ideal Mixtures Uniquely Determined by the Initial Composition? ˇefˇ Ján S cík* Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455
A series of articles in this Journal offer the negative answer to the following question: Might there exist more than one physically meaningful solution to an equation describing a reaction equilibrium in a one-phase system at fixed temperature and pressure with a given initial composition? Various particular reaction systems were considered (1–8), but no attention was paid to systems exhibiting non-ideal mixing behavior. When there is a single reaction equilibrium among mixture components and the components form an ideal mixture, it was demonstrated by a simple, elegant mathematical argument that there is a unique composition satisfying the equilibrium equation (4 ). In the same place it was also suggested that this uniqueness argument, based on monotonicity of a certain function, can be extended to non-ideal mixtures by simply incorporating activity coefficients (4 ). However, the extension of this argument to non-ideal systems is not generally valid. Unlike ideal mixtures, non-ideal mixtures may undergo phase separation. While the Gibbs free energy function in ideal mixtures exhibits a unique critical point (a minimum), with the corresponding composition calculated from the equilibrium equation, the Gibbs free energy function in non-ideal mixtures may exhibit multiple critical points. In this paper we demonstrate that increasing non-ideality can result in nonmonotonicity of the function crucial for the simple uniqueness argument, and only later it leads to non-uniqueness and hence phase separation. Let us first go through the uniqueness argument as applied to an ideal binary mixture and then show on a simple example how non-ideality leads to non-monotonicity and finally to non-uniqueness. Consider a binary mixture of reactants A and B with the mole fractions xA0,x B0 = 1 – xA0 undergoing the reaction A B with the equilibrium constant Keq (at a given temperature and pressure). Let the fraction of mixture converted at equilibrium be ζ (i.e., the “dimensionless extent of reaction at equilibrium” equal to the extent of reaction at equilibrium divided by the total mixture concentration). Then the mole fractions of reactants at equilibrium are x A , eq = xA0 – ζ ,x B, eq + ζ If we assume that the mixture is an ideal one, we can express the equilibrium condition in terms of mole fractions as Keq = x B, eq /x A, eq = (xB0 + ζ )/(x A0 – ζ) or alternatively as f ( ζ ) = g (ζ )
(1)
where f ( ζ ) = Keq(xA0 – ζ) and g ( ζ ) = (x B0 + ζ ). The equilibrium equation (eq 1) has a unique solution in the region of *Present address: 139-74 Beckman Institute, Caltech, Pasadena, CA 91125.
non-negative mole fractions (i.e., ᎑ xB0 ≤ ζ ≤ xA0) if and only if the graphs of continuous functions f and g intersect just once in the allowed interval. Now we observe that f ( ᎑ x B0) > g(᎑ xB0), f (xA0) < g(x A0), and f monotonically decreases, while g monotonically increases in the interval [ ᎑ xB0 , xA0 ]. According to a theorem from the basic calculus functions f and g indeed intersect just once in the interval [ ᎑ xB0 , xA0 ], and thus the equilibrium composition of the binary reaction system is uniquely determined by the equilibrium equation above. The monotonicity properties of functions f and g are crucial for this particular uniqueness argument, which extends to any stoichiometry for a single reaction in an ideal, one-phase mixture (4). Let us further not assume that the mixture is ideal. Then the equilibrium condition includes the activity coefficients γ A ,γ B, which themselves depend on the mixture composition: K eq = γ B x B,eq / γ Ax A , eq = K γ( ζ)(xB0 + ζ)/(xA0 – ζ) or equivalently f ( ζ ) = g (ζ ) where f ( ζ ) = K eq(xA0 + ζ), g( ζ ) = K γ(ζ)(x B0 + ζ), and K γ(ζ) = γ B/γ A. It is obvious that unless the ratio of activity coefficients K γ is a monotonically increasing function of ζ, we do not have uniqueness of the solution to the equilibrium equation (eq 1) assured without further examination of properties of function K γ(ζ ) on the interval [᎑x B0 ,xA0]. For example, let us express the activity coefficients according to one-constant Margules equations1 (9) with an interaction parameter α: γA = e αx B, γB = e αx A 2
2
(2)
Negative interaction parameter α implies that heteromolecular (A–B) interactions are energetically more favorable than homomolecular (A–A or B–B) ones and activity coefficients of respective species decrease with their dilution (see Fig. 1 for the case α = ᎑1). In this case it is easy to see that the ratio of activity coefficients K γ( ζ) is a monotonically increasing function of ζ on the interval [᎑ x B0 ,x A0] and the uniqueness of the solution to the equilibrium equation is assured the same way as it is for the ideal mixture case. However, when the interaction parameter α is positive (A–B interactions are energetically less favorable than A–A or B–B), activity coefficients of respective species increase with their dilution (see Fig. 1 for the case α = 1). In this case the ratio of activity coefficients K γ(ζ) is a monotonically decreasing function of ζ on the interval [᎑x B0 ,x A0]. We will now demonstrate how the monotonicity argument for the uniqueness of the solution to the equilibrium equation (eq 1) breaks down when the interaction parameter α in the Margules equations (eq 2) gets sufficiently large while still being small enough that the mixture does not undergo phase separation.
JChemEd.chem.wisc.edu • Vol. 75 No. 5 May 1998 • Journal of Chemical Education
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Research: Science & Education 4
1.4
3.5
1.2
γA (α=1)
γΒ (α=1)
1
2.5
f(ζ), g(ζ)
γA(ζ),γB(ζ)
3
2 1.5
f
g(α=0.5)
0.8 0.6
g(α=2)
0.4
1 0.5 0
Keq =1.2
γB (α=-1) 0
0.2
0.2
γA (α=-1) 0.4
0.6
0.8
0
1
g(α=3) 0
ζ + xB0
0.2
0.4
0.6
ζ + xB0
0.8
1
Figure 1. Activity coefficients of components A and B are functions of the mixture composition. α is the parameter in the Margules equations (eq 2).
Figure 2. The left-hand side f (eq 3) and the right-hand side g (eq 4, for the indicated values of parameter α) of the equilibrium equation (eq 1).
Using the Margules equations (eq 2) we can express the ratio of activity coefficients as
that the uniqueness is somehow assured, one can here still use an iterative procedure to calculate the equilibrium composition as suggested previously for single (4) or multiple (8) reaction systems. For α above the critical value α * the equilibrium equation has three distinct solutions in the region of non-negative mole fractions (e.g., α = 3 for Keq = 1.2; see Fig. 2). Then the solutions of the equilibrium equation correspond to critical points (two minima separated by a maximum) of the Gibbs free energy function and the equilibrium state involves multiple phases. However, one would need to inspect the actual Gibbs free energy function in order to decide which solution of the equilibrium equation corresponds to the equilibrium state.
K γ = γ B/γA = e α(x 2A – x 2B) = e α(xA + xB)(xA – xB) = e α(xA – xB) where we used an algebraic identity x 2A – x 2B = (x A + xB)(xA – xB), taking into account that x A + xB = 1. At the equilibrium K γ( ζ) = e α(xA , eq – xB,eq ) = e α(xA 0 – xB0 – 2ζ ) and so we get f ( ζ ) = Keq(x A0 – ζ) (3) g ( ζ ) = (xB0 + ζ) e α(xA 0 – xB0 – 2ζ )
(4)
While f (᎑xB0) > g(᎑xB0), f (xA0) < g(x A0), and f is a monotonically decreasing function of ζ, g increases monotonically in the interval [᎑xB0 ,x A0] if and only if α ≤ 0.5. To show this, we examine the derivative of the g function dg( ζ )/d ζ = [1 – 2α (x B0 + ζ)]e α(xA 0 – xB0 – 2ζ ) Since the exponential part is always positive, the sign of the derivative of g is determined by the sign of [1 – 2α (x B0 + ζ)]. This expression is positive; that is, the function g is monotonically increasing in the interval [ ᎑ x B0 , x A0 ] if and only if α ≤ 0.5. When α > 0.5, the function g is monotonically increasing in the interval [ ᎑ xB0 ,᎑ x B0 + 1/(2α) ], going through a maximum at ζ = ᎑ x B0 + 1/(2 α), and monotonically decreasing in the interval [᎑ xB0 + 1/(2α),x A 0 ]. There is always a unique solution to the equilibrium equation (eq 1) for α below the critical value α* (α * = 2 for K eq = 1; otherwise, α* is a solution of a nonalgebraic equation). However, for α ∈(0.5,α *), one cannot use the monotonicity argument to prove the uniqueness, since the function g is not monotonic in the interval [ ᎑ xB0 , xA0 ] for this range of parameter α values (e.g., α = 2 for Keq = 1.2, see Fig. 2). Provided
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Note 1. Margules equations represent the simplest one-parameter model for the excess Gibbs free energy of mixing (9 ). Anyone can substitute a favorite expression for activity coefficients and work out the result.
Literature Cited 1. Ludwig, O. G. J. Chem. Educ. 1983, 60, 547. 2. Corao, E.; Morales, D.; Araujo, O. J. Chem. Educ. 1986, 63, 693– 694. 3. Smith, W. R.; Missen, R. W. J. Chem. Educ. 1989, 66, 489–490. 4. Weltin, E. J. Chem. Educ. 1990, 67, 548. 5. Weltin, E. J. Chem. Educ. 1992, 69, 393–396. 6. Ludwig, O. G. J. Chem. Educ. 1992, 69, 884. 7. Weltin, E. J. Chem. Educ. 1993, 70, 568–571. 8. Weltin, E. J. Chem. Educ. 1995, 72, 508–511. 9. Sandler, S. I. Chemical and Engineering Thermodynamics, 2nd ed.; Wiley: New York, 1989; p 323.
Journal of Chemical Education • Vol. 75 No. 5 May 1998 • JChemEd.chem.wisc.edu
