From Eq. (3-41) the partial molal change in internal energy is and on the basis of Van Laar's assumptions:

combining constants,

Scatchard and Hildebrand obtained similar expressions without the use of a van der Waals fluid. They recommend evaluating the expression for (A2?i)l as where u = internal energy of vaporization

V = molal volume This leads to equations of the same form as Eqs. (3-44) and (3-45), except that

A Vt

It has been pointed out by Cooper (Ref. 7) that the same relationship can be obtained more simply than the method employed by Van Laar. What is desired in calculating the activity coefficient is the difference between the partial molal free energy of mixing of an actual solution and that of an ideal solution. If the excess free energy of mixing per unit volume of mixture is assumed proportional to the product of the volume fractions of the two components, an expression identical in form to the Van Laar equation is obtained. Thus,

where AFe = excess free-energy change of mixing = actual free-energy change minus ideal free-energy change rii = mols of component 1 n2 = mols of component 2 Vi = molal volume of component 1 V2 == molal volume of component 2

These equations are identical with those given by Van Laar, but the assumptions are somewhat different. In the case of the Van Laar, Scatchard, and Hildebrand derivations, both A and B should be positive, while in Cooper's equation B could be either positive or negative.

On the basis of the derivations, the two constants of the Van Laar equations and the modifications of it are related to the physical properties of the pure components. When the best values of the constants are chosen to fit the data, they usually do not agree with the predicted values, although the trends are approximately the same. Generally the constants are chosen to agree with the data, and the equations are used empirically.

As was the case with the Margules equation, two constants are involved. For a binary mixture one vapor-liquid equilibrium point will give the activity coefficients of both components and thus define the whole equation. The form of the equations are significantly different from that of the Margules. It includes a temperature correction, and the value of the constants should be independent of temperature. Thus an experimental determination at one temperature should allow equilibrium data to be calculated at other temperatures. There are several ways in which Van Laar equations can be rearranged to plot as a straight line in order that the data for more than one determination can be easily correlated. One of the most convenient methods of making such a plot is to use the reciprocal of the square root of the temperature times the logarithm of the activity of component 1, 1 /{T In yi)^, vs. the ratio, xi/x2. A similar plot can be made for the other component. The plots should both be straight lines; the slopes and intercepts will be different but related because they are based on the same constants.

Clark Equation. For the vapor-liquid equilibria of a binary mixture at either constant pressure or constant temperature, Clark (Ref. 6) has suggested that the ratio of the mol fractions in one phase is a linear function of the ratio of the mol fractions in the other phase, when the ratios are utilized such that the component in largest amount appears in the numerator.

Thus, when component 1 is present in largest amount,

2/2 x2 v and, when component 2 is present in largest amount,

Clark uses the value xi/x2 = \Zafb/abf as the change-over point between the two equations. The use of these equations requires three experimental points. This greatly limits its utility.

Evaluation of Empirical Equations. Tucker (Ref. 24) and Mason (Ref. 18) have studied the agreement of the various equations with published experimental data. The data available were screened, and only those that gave good agreement with the Duhem-type equations were selected for the evaluation. The experimental data were plotted to evaluate the equation constants, and the average constants so obtained were used to recalculate the y,x curve. In distillation calculations the difference between the vapor and the liquid compositions gives a better indication of the ease of separation than the absolute value of the vapor composition. Mason therefore made the comparison on the basis of

As a qualitative standard, he classified average deviation of 0 to 5 per cent as good, 5 to 11 per cent as fair, and greater than 11 per cent as poor. Some of the results are given in Table 3-7. The table gives the average deviation and the maximum deviation. In certain cases, the agreement between the experimental and the calculated values is very good but very poor in other cases. It is difficult to determine any definite types of mixtures that give good or poor agreement. However, in all cases, mixtures approaching immiscibility, i.e., y}x curves that are nearly horizontal over an appreciable concentration region, gave poor agreement with the Margules and Van Laar equations. Good agreement was obtained with all the maximum boiling mixtures studied. These latter mixtures give negative values of the Van Laar

Scatchard

Van Laar

Clark

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Ethanol-water |

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