Fallibilities Inherent in the Wilson Equation Applied to Systems Having a Negative Excess Gibbs Energy Andre J. Ladurelli,’ Claude H. Eon,* and Georges Guiochon Laboratoire de Chimie Anaiytique Physique. Ecoie Polytechnique. 75005 Paris, France
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I t is shown that the optimization of the coefficients of the Wilson equation to fit a set of experimental results can give rise to several different mathematical solutions when applied to mixtures exhibiting negative excess Gibbs energy. A s a result several pairs of Wilson coefficients give almost an equally good fit of experimental data. The analysis of 12 binary mixtures suggests a rule for choosing the meaningful roots from the knowledge of the relative size of the molecules. Furthermore, a new interpretation of the Wilson parameters given in terms of segment theory proves to be more suitable in predicting interaction parameters than its classical counterparts.
In spite of its lack of theoretical consistency, the Wilson equation (Wilson, 1964) has been shown to provide a suitable method for the calculation of vapor-liquid equilibria of miscible mixtures (Holmes, 1970; Neretnieks, 1968; Prausnitz, 1965, 1967). Its applicability has been well tested for systems of interest in engineering chemistry which, mostly exhibit a positive excess Gibbs energy. The results have been generally very good. It is not yet as clear, however, whether the Wilson equation is as valuable in the case of mixtures having a negative excess Gibbs energy. It is the aim of this paper to show that the determination of the Wilson coefficients is not always straightforward for such mixtures although the Wilson equation itself is above criticism.
I. Definition of the Problem In 1964, Wilson proposed an equation accounting for the variation of the excess Gibbs energy as a function of the composition of miscible mixtures. For a binary solution, it is
2 RT
= -xi In (xi + A12x2)- x, In ( x 2 + A21xJ (1)
where x l stands for the mole fraction of the species i and the Ai j ’ s are dimensionless parameters. By proper derivation eq 1 leads to activity coefficients of compounds 1 and 2. Considering solute 2, its activity coefficient in the mixture turns out to be (Orye and Prausnitz, 1965) In y2 = - In ( x ,
+
AZixi) -
Thus determining an isotherm with the help of the Wilson equation requires that we first determine both A12 and Azl, which in turn have to be estimated from the measurements corresponding to a part of the isotherm. At the limit, the measurement of the two activity coefficients a t infinite dilution (if it were easier) would appear to be the simplest alternative as proposed by Bruin (1970),for A12 and A21 could then be calculated from
- In A i ,
- AZ1 + 1 In yZm = - In A,, - Ai, + 1
In yim =
(3) The validity of this procedure is asserted by the work of Schreiber and Eckert (1971), who reported a fair agreement between calculated and experimental results for 1
Cenlre de Recherches E L. F . , 69360 Solaize, France.
mixtures having mostly a positive excess Gibbs energy. Whether it could have been different for mixtures exhibiting negative excess Gibbs energy is certainly the first question that the title of this paper brings to mind. To make this point clear let us combine the eq 3 t o eliminate one of the two parameters, A21 for example. This gives In
~1~
= - In Ai,
+ 1-
1 O)
72
exp(1
-
Ai,)
(4)
A plot of In
71” vs. In A12 for various values of yz” is given in Figure 1. This shows that eq 3 has a unique solution for mixtures having activity coefficients larger than unity while it has three possible solutions in the range of activity coefficient smaller than unity. Unless we are able to distinguish, once and for all, the proper solution, it is hardly possible to calculate the isotherm only from the two limiting activity coefficients. For the following discussion it is appropriate to define three domains, A, B, and C, in the ( y l ” , A121 space as shown in Figure 1, corresponding to each of the possible solutions. These domains are obtained by drawing the locus of the minima and maxima of the curves ylm( A d . The problem now is to find which one of the three possible solutions, if any, has a physical meaning. First it certainly helps to visualize the three solutions, as it sometimes will allow the elimination of 1 (or even 2) of the solutions. Figure 2 shows the isotherms corresponding to the three solutions obtained in the hypothetical case where the two limiting activity coefficients are ylm = 0.37 and y2” = 0.5, respectively. It will be observed that one of the Atj’s is very small in both solutions A and C. Due to its very unusual shape, solution C can probably be eliminated. The same thing could be said about isotherm A although to a lesser extent, so isotherm B looks somewhat more reasonable than the other two and therefore would be chosen by most people as the very probable solution. Without drawing any conclusion from this fact as neither the A nor the C isotherms are impossible, from mere thermodynamic grounds i t is worth pointing out that in most cases similar calculations carried out for other pairs of yt”’s lead to isotherms B having a most conventional “good shape”, whereas isotherms A and C exhibit very steep decrease in the activity coefficient with increasing concentration of the dilute species. There are cases, however, where either the A or the C solution is quite acceptable and even worse, there are cases where solutions A and B (or B and C) induce almost similar isotherms! This obviously corre-
Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1975
191
1
0
0.5
1
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Figure 3. Comparison of the solubility isotherms of pentane in squalane a t 30°C derived from the A and B roots. The solid points represent the values obtained by Ashworth using a static method. Figure 1. Variation of the activity coefficient y l m of species 1 a t infinite dilution in species 2 as function of A i 2 for various values of 7 2 ” : 1, 7 2 “ = 0.2; 2, 72” = 0.3; 3, 72” = 0.4; 4, 7 2 “ = 0.6; 5, 72” = 1; 6, 72 = 4;7 , 72 = m .
\
o.2i I
‘t
Figure 2. Example of the three isotherms derived from the two limiting activity coefficients (71” = 0.37; 7 2 - = 0.50): A, -112 = 0.0403, A21 = 5.2; B, ;412 = 1, A21 = 2; C, A12 = 7.36, A21 = 3.4 x 10-3.
tion of the data. The coefficients are, respectively: A12 = 0.0817, A21 = 4.04 for solution A and A12 = 0.623, Azl = 2.35 for solution B. Solution C can be eliminated in this case as it does not fit the experimental results at finite concentrations. The two corresponding activity coefficients A and B are derived from eq 2 and reported in Figure 3. For comparison purposes, the static data of Ashworth et al. (1960) are also reported. Keeping in mind that the value of the coefficient at infinite dilution is granted and that chromatography does not permit measurements at very large concentration of solute, it is obvious that the two curves A and B fit the experimental data equally well. The only appreciable difference occurs at large values of the solute concentration where no significant measurement can be carried out by chromatography. Thus, the same question again arises: which solution should be chosen? This is immaterial if only a good fit of the experimental data is looked for, but not if some physical interpretation of the coefficients is considered. The key point in this matter is the presence of a maximum in the A curve which in turn induces a minimum in the corresponding y1 curve. Considering the well-known relationship between the activity coefficient and the molar excess Gibbs energy of the binary solution, we have RT In y1 =
sponds to conditions around the extrema of the curves in Figure 1. A typical example of this situation is the case of the system pentane-squalane (2, 6, 10, 15, 19, 23 hexamethyltetracosane) as studied via gas-liquid chromatography. The system has been studied at 30°C using the step and pulse method recently developed by Valentin and Guiochon (1975) to carry out the measurements. The application of this method to the determination of various isotherms will be published shortly (Ladurelli and Guiochon, 1975) and it is beyond the scope of this paper to give more details about it. It will be sufficient to say that it can only be applied for concentrations of solutes up to about 0.7 mole fraction in the stationary liquid due to flooding phenomena. When applied to the system pentane-squalane for which the activity coefficient of pentane is 0.62 at infinite dilution, proper calculations lead to two pairs of Wilson coefficients equally probable from the statistical evalua192
Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1975
g“ + (1 -
X )
dx dgE
and it is obvious that a minimum of the function y1 = f(xl), for x 1 differing from unity, implies that d2gE/dx12 = 0 a t this composition. Thus the presence or the absence of a point of inflection on the curve gE(xl) allows unambiguous choice of the right set of parameters. In absence of data regarding gE in the range of interest it might sometimes be useful to consider the shape of the calorigram as well, although it is clear that a point of inflection on hE does not always imply an inflection of gE, The calorigram determined for the system under study with a Picker calorimeter (Picker et al., 1971) is given in Figure 4 as a function of the volume fraction, which offers the advantage of giving more importance to the part of interest. The strongly S-shaped curve observed in the weak squalane range supports solution A. For such simple mixtures SE is positive and likely to vary almost linearily. This example throws into relief that the ambiguity does not arise from the Wilson’s equation itself but rather from
the consequence of having spare experimental data and an incomplete information regarding the isotherm. As this situation is often met in common practice, the question deserves a careful thermodynamic analysis. It is, for example, illustrative to compare the Wilson equation to the lattice theory of mixtures. 11. T h e Wilson Equation a n d t h e Lattice Theory of Mixtures
According to Wilson, the A parameters are related to molecular interaction parameters A i j through (Wilson, 1964) I
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where V stands for the molar volume and A i j = X j i . The elegance of this equation is immediately recognized, for eq 1 becomes identical with the well-known Flory-Huggins equation (1942) when the interaction parameters A i r , h j j , and A i j are identical. A trivial analysis of the derivative of eq 4 with respect to A12 shows that both domains A and C (cf. Figure 1) correspond to a product A l z A z l smaller than unity (notice that C is the convex of A) while the domain B corresponds to A l z h z l larger than unity. In terms of interaction parameters this means that the term AX (Ah = 2x12 - A 1 1 - A22) is positive in the first case (domains A and C) and negative in the second case (domain B). Although the exact physical meaning of the A's is not yet clear, we can assume that they are somehow related to the potential energy parameters as used in the lattice theory of mixtures. This last approach considers a liquid (made up of molecules not differing in size) to be solid like, in a quasi-crystalline state. The theory has been described by Guggenheim (1952, 1966); it involves sophisticated developments of statistical thermodynamics that are beyond the scope of this work; only the result is of importance. It leads to activity coefficients defined by
W'
In y , = - xZ2 RT where W , the interchange energy, is a function of the interaction parameters r r j and the coordination number 2
Comparing the results of the two theories shows that in both cases, y1 and yz are unity when A l z and r1z are the arithmetic average of the corresponding parameters of the pure species. Within the limits of the assumptions in which these theoretical models are valid we can also conclude that an activity coefficient smaller than unity implies that the interchange energy W is negative, which in turn suggests that AA would be negative. As a result, solution B is the only one that reconciles the Wilson theory with the lattice theory, whether the activity coefficients are smaller or greater than unity. It should be pointed out that this conclusion is valid only when the molecules are similar in size. When it is not so, the classical combination entropy has to be taken into account in the expression of the activity coefficients. It can be convenient to consider the infinite activity coefficient of the smaller solute through the extended Flory-Huggins mixture theory. Its expression becomes (Prausnitz, 1967)
The Flory interaction parameter
x
can be written in
,O,
a5
0
-
Y 1
Figure 4. Calorigram of mixing of pentane in squalane at 30°C AH ( < 0) is reported in arbitrary units versus the volume fraction of squalane.
Table I. Comparison of the Signs of L A and W for Some Systems Exhibiting a Negative gE Ah
System, 1-2
w
(solution)
(i) 1 and 2 of the Same Size Chloroform acetone" (0 (B) Benzene-chloroform' < O (B) Ethyl acetate-chloroform" O ( A ) >O S.iY-Dimethylaniline+O~ < O (B)
