Parameter Margules Equation
Upon differentiating Eq. (85.5) as indicated by Eqs. (85.3) and (85.4), we find
With these relations we can now calculate activity coefficients 71 and 72 at any desired x even though experimental data were obtained only at one point, namely, X\ = x2 = i
This simplified example illustrates how the concept of excess function, coupled with the GibbsDuhem equation, can be used to interpolate or extrapolate experimental data with respect to composition. Unfortunately, the GibbsDuhem equation tells nothing about interpolating or extrapolating such data with respect to temperature or pressure.
Equations (85.2) to (85.4) indicate the intimate relation between activity coefficients and excess Gibbs energy GE. Many expressions relating gE (per mole of mixture) to composition have been proposed, and a few are given in Table 83. All these expressions contain adjustable constants which, at least in principle, depend on temperature. That dependence may in some cases be neglected, especially if the temperature interval is not large. In practice, the number of adjustable constants per binary is typically two or three; the larger the number of constants, the better the representation of the data but, at the same time, the larger the number of reliable experimental data points required to determine the constants. Extensive and highly accurate experimental data are required to justify more than three empirical constants for a binary mixture at a fixed temperature.f
For many moderately nonideal binary mixtures, all equations for gE containing two (or more) binary parameters give good results; there is little reason to choose one over another except that the older ones (Margules, van Laar) are mathematically easier to handle than the newer ones (Wilson, NRTL, UNIQUAC). The twosuffix (oneparameter) Margules equation is applicable only to simple mixtures where the components are similar in chemical nature and in molecular size.
For strongly nonideal binary mixtures, e.g., solutions of alcohols with hydrocarbons, the equation of Wilson is probably the most useful because, unlike the NRTL equation, it contains only two adjustable parameters and it is mathematically simpler than the UNIQUAC equation. For such mixtures, the threesuffix Margules equation and the van Laar equation are likely to represent the data with significantly less success, especially in the region dilute with respect to alcohol, where the Wilson equation is particularly suitable.
The foursuffix (threeparameter) Margules equation has no significant advantages over the threeparameter NRTL equation.
tThe models shown in Table 83 are not applicable to solutions of electrolytes; such solutions are not considered here. However, brief attention is given to aqueous solutions of volatile weak electrolytes in a later section of this chapter.
Numerous articles in the literature use the RedlichKister expansion [see Eq. (89.20)] for gE. This expansion is mathematically identical to the Margules equation.
The Wilson equation is not applicable to a mixture which exhibits a miscibility gap; it is inherently unable, even qualitatively, to account for phase splitting. Nevertheless, Wilson's equation may be useful even for those mixtures where miscibility is incomplete provided attention is confined to the onephase region.
Unlike Wilson's equation, the NRTL and UNIQUAC equations are applicable to both vaporliquid and liquidliquid equilibria.t Therefore, mutual solubility data [see Sec. 810] can be used to determine NRTL or UNIQUAC parameters but not Wilson parameters. While UNIQUAC is mathematically more complex than NRTL, it has three advantages: (1) it has only two (rather than three) adjustable parameters, (2) UNIQUAC's parameters often have a smaller dependence on temperature, and (3) because the primary concentration variable is a surface fraction (rather than mole fraction), UNIQUAC is applicable to solutions containing small or large molecules, including polymers.
Simplifications: oneparameter equations
It frequently happens that experimental data for a given binary mixture are so fragmentary that it is not possible to determine two (or three) meaningful binary parameters; limited data can often yield only one significant binary parameter. In that event, it is tempting to use the twosuffix (oneparameter) Margules equation, but this is usually an unsatisfactory procedure because activity coefficients in a real binary mixture are rarely symmetric with respect to mole fraction. In most cases better results are obtained by choosing the van Laar, Wilson, NRTL, or UNIQUAC equation and reducing the number of adjustable parameters through reasonable physical approximations.
To reduce the van Laar equation to a oneparameter form, for mixtures of nonpolar fluids, the ratio A/B can often be replaced by the ratio of molar liquid volumes: A/B = Vi/V^. This simplification, however, is not reliable for binary mixtures containing one (or two) polar components.
To simplify the Wilson equation, we first note that tWilson [130] has given a threeparameter form of his equation which is applicable also to liquidliquid equilibria; the molecular significance of the third parameter has been discussed by Renon and Prausnitz [106]. The threeparameter Wilson equation has not received much attention, primarily because it is not readily extended to multicomponent systems.
Name 
sE 
Binary parameters 
In 7, and In 72 
Twosuffix' Margules 
gE = Ax,x2 
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