One-constant Margules equation

Since the two activity coefficients are so close, and the azeotrope occurs near 0.5 mole fraction, so I will use the one-constant Margules equation. 

These relations, known as the one-constant Margules equations, are plotted in Fig. 

Two interesting features of the one-constant Margules equations are apparent from this figure. First, the two species activity coefficients are mirror images of each other as a function of the composition. This is not a general result, but follows from the choice of a symmetric function in the compositions for G . Second, yi 1 as Xj —> 1,... 

Figure 9.5-3 The activity coefficients for the one-constant Margules equation with A/RT) = 1.0.

The one-constant Margules equation provides a satisfactory representation for activity coefficient behavior only for liquid mixtures containing constituents of similar size, shape, and chemical nature. For more complicated systems, particularly mixtures of dissimilar molecules, simple relations such as Eq. 9.5-1 or 9.5-5 are not valid. In particular, the excess Gibbs energy of a general mixture will not be a symmetric function of the mole fraction, and the activity coefficients of the two species in a mixture should not be expected to be mirror images. One possible generalization of Eq. 9.5-1 to such cases is to set... 

Figure 9,5-4 Experimental activity coefficient data for the benzene-2,2,4-trimethyl pentane mixture and the correlation of these data obtained using the one-constant Margules equation.

The points in Figs. 9.5-4 and 9.5-5 represent smoothed values of the activity coefficients for both species in a benzene-2,2.4-trimethyl pentane mi.xture at 55 C taken from the vapor-liquid equilibrium measurements of Weissman and Wood (see Illustration 10.2-4). Test the accuracy of the one-constant Margules equation and the van Laar equations in correlating these data. 

At T = 60°C the vapor pressure of methyl acetate is 1.1260 bar, and the vapor pressure of methanol is 0.8465 bar. Their mixtures can be described by the one-constant Margules equation... 

Equations 11.2-2 can be used with experimental phase equilibrium data to calculate the activity coefficient of a species in one phase from its known value in the second phase or, with Eqs. 11.2-3 and experimental activity coefficient data or appropriate solution models, to compute the compositions of both coexisting liquid phases (see Illustration 11.2-2). For example, using the one-constant Margules equation to represent the activity coefficients, we obtain from Eq. 11.2-2 the following relationship between the phase compositions ... 

Equations 11.2-11 through 11.2-14 are specific to the choice of the one-constant Margules equation to represent the excess Gibbs energy of the mixmre. The use of more realistic models for G will lead to other predictions for phase separation, such as the limit of stability at the upper consolute occurring away from X] = 0.5 (Problem... 

Saxena and Nehru (1975) Independent equations of state for diopside and orthoenstatite solutions are assumed to obey regular solution models (one-constant Margules equations where Wqi= Wg2) Tbe W parameter for each solution is calculated from assumed differences in the standard free energies of fonnation for end-member phases and published experimental data on the compositions of coexisting phases. Temperatures for natural assemblages of enstatite and diopside are estimated from Margules expressions for homogeneous mixtures in which activities are presumed to be related to individual site occupancies. 

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