The van Laar Equation

For a binary system whose components are not too dissimilar chemically but have different molar volumes, we can assume that interaction coefficients involving more than two molecules can be neglected. Then, the Wohl expression becomes  

From this equation, the following expressions are obtained for yj and y2  

Actually van Laar - a student and, later, colaborator of van der Waals - developed his expression early in the 20th century starting with the van der Waals equation of state (see Prausnitz). 

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Application of the algorithm for analysis of vapor-liquid equilibrium data can be illustrated with the isobaric data of 0th-mer (1928) for the system acetone(1)-methanol(2). For simplicity, the van Laar equations are used here to express the activity coefficients. 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

For the estimation of activity coefficients the Van Laar equation is used. 

For systems that are only partially miscible in the liquid state, the activity coefficient in the homogeneous region can be calculated from experimental values of the mutual solubility limits. The methods used are described by Reid et al. (1987), Treybal (1963), Brian (1965) and Null (1970). Treybal (1963) has shown that the Van-Laar equation should be used for predicting activity coefficients from mutual solubility limits. 

Null (1970) gives a computer program for the calculation of ternary diagrams from vie data, using the Van-Laar equation. 

A popular model to describe the activity coefficients is the van Laar equation... 

Calculate the coefficients of Van Laar equations and the three-suffix Redlich-Kister equations from experimental solubility data at 70°C (158°F, or 343 K) for the water(l)/trichloroethylene(2) system. The Van Laar equations are... 

Related Calculations. These calculations show how to use vapor-liquid equilibrium data to obtain parameters for activity-coefficient correlations such as those of Van Laar and Wilson. (Use of liquid-liquid equilibrium data for the same purpose is shown in Example 1.20.) If the system forms an azeotrope, the parameters can be obtained from a single measurement of the azeotropic pressure and the composition of the constant boiling mixture. If the activity coefficients at infinite dilution are available, the two parameters for the Van Laar equation are given directly, and the two in the case of the Wilson equation can be solved for as shown in the example. 

In principle, the parameters can be evaluated from minimal experimental data. If vapor-liquid equilibrium data at a series of compositions are available, the parameters in a given excess-free-energy model can be found by numerical regression techniques. The goodness of fit in each case depends on the suitability of the form of the equation. If a plot of GE/X X2RT versus X is nearly linear, use the Margules equation (see Section 3). If a plot of Xi X2RT/GE is linear, then use the Van Laar equation. If neither plot approaches linearity, apply the Wilson equation or some other model with more than two parameters. 

The results obtained in the solution of a sample problem are summarized here to illustrate the application of the method. An extractive distillation problem from Oliver (6) was used in which methylcyclo-hexane is separated from toluene by adding phenol. The column contains 11 stages (including the reboiler and condenser) and has a feed of 0.4 moles/unit time of methylcyclohexane and 0.6 moles/unit time of toluene to the fourth stage from the reboiler and 4.848 moles/unit time of phenol to the fourth stage from the condenser. We used the same physical property correlations as Oliver. The activity coefficients were obtained from a multicomponent form of the Van Laar Equation (7). 

The Redlich/Kister expansion, the Margules equations, and the van Laar equations are all special cases of a general treatment based on rational functions, i.e., on equations for G /x X2RT given by ratios of polynomials. They provide great flexibility in the fitting of VLE data for binary systems. However, they have scant theoretical foundation, and therefore fail to admit a rational basis for extension to multicomponent systems. Moreover, they do not incorporate an explicit temperature dependence for the parameters, though this can be supplied on an ad hoc basis. 

Equation (14.60) may be directly applied with these data to estimate In and In for temperatures greater tlian 363.15 K (90°C). The van Laar equations [Eqs. (12.17)] are well suited to this system, and the parameters for this equation are given as... 

The small value of RMS %6P showirfor 363.15 K (90°C) iirdicates both the suitability of the van Laar equatioir for correlation of the VLE data and the capability of the equation of state to reproduce the data. A direct fit of these data with the van Laar equation by the ganmra/phi procedure yields RMS % 6 P = 0.19. The results at 423.15 to473.15K(150 and 200°C) are based oirly on vapor-pressuredata for the pure species and on mixture data at lower temperatures. Tlre quality of prediction is indicated by the P-x-y diagram of Fig. 14.10, wlrich reflects the uncertainty of the data as well. 

For ideal systems these equations reduce to the van Laar equations (18.33) and (18.34). 

Derive expressions for log 7, where 7 is the activity coefficient a/N, on the basis of (i) the simple Margules equation (35.9), (ii) the van Laar equation (35.10). Show that for a liquid mixture exhibiting positive deviations from Raoult s law, the activity coefficient of each constituent, on the basis of the usual standard state, must be greater than unity, whereas for negative deviations it must be less than unity. 

The foregoing method involves a single constant and can be used when the deviations from Raoult s law are not too great. Better results are obtained by using the van Laar equations (35.10). Thus, the activities of mercury (component 1) in liquid mixtures with tin (component 2) at 323 C, determined by vapor pressure measurements, can be expressed by... 

Utilize the results in Exercise 5 to express log yi (for the mercury) as a function of the mole fraction by means of the van Laar equation. Hence derive an expression for log 72 (for the bismuth) and check the values obtained by the graphical integration. 

Following Fullarton and Schliinder we use the Van Laar equation for the evaluation of In %... 

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