Van Laar equations

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. 

The Wilson equation, like the Margules and van Laar equations, contains just two parameters for a binary system (A12 and A91), and is written ... 

Outlined below are the steps required for of a X T.E calciilation of vapor-phase composition and pressure, given the liquid-phase composition and temperature. A choice must be made of an equation of state. Only the Soave/Redlich/Kwong and Peng/Robinson equations, as represented by Eqs. (4-230) and (4-231), are considered here. These two equations usually give comparable results. A choice must also be made of a two-parameter correlating expression to represent the liquid-phase composition dependence of for each pq binaiy. The Wilson, NRTL (with a fixed), and UNIQUAC equations are of general applicabihty for binary systems, the Margules and van Laar equations may also be used. The equation selected depends on evidence of its suitability to the particular system treated. Reasonable estimates of the parameters in the equation must also be known at the temperature of interest. These parameters are directly related to infinite-dilution values of the activity coefficients for each pq binaiy. 

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

Van Laar equation 611, 614 Van t Hoff equation 56, 373 Vapour pressure 137 Variable... 

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. 

The Margules and van Laar equations apply only at constant temperature and pressure, as they were derived from equation 11.21, which also has this restriction. The effect of pressure upon y values and the constants and 2i is usually negligible, especially at pressures far removed from the critical. Correlation procedures for activity coefficients have been developed by Balzhiser et al.(ll Frendenslund et alSls>, Praunsitz et alS19>, Reid et al. 2 ) van Ness and Abbott(21) and Walas 22 and actual experimental data may be obtained from the PPDS system of the National Engineering Laboratory, UK1-23). When the liquid and vapour compositions are the same, that is xA = ya, point xg in... 

The activity coefficients were assumed to satisfy van Laar equations of the form... 

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

Only the three data points given below are available for a particular binary system of interest at temperature T. Determine whether these data are better represented by the Margules or van Laar equation at temperature T, where Pf = 21(psia) arid Pf1 - 47(psia). 

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. 

For this system, / " may be assumed to be the same as the vapor pressure (for a discussion of the grounds for this assumption, see Example 3.6). Activity coefficients can be calculated using the Wilson, Margules, or Van Laar equations (see Example 3.4). 

The Modified van Laar equations (I) have been used to calculate the liquid phase activity coefficients. Coefficients at three temperatures are given in Table III. These are used by the computer to calculate activity coefficients at any composition and temperature in the distillation column. 

From Equations 1 and 2, the phase equilibria depend upon knowing the pure component vapor pressures P 0, liquid phase activity coefficients ji and imperfection-pressure coefficients ft. The computer program which has been developed uses any of four different vapor pressure equations for providing Pi°. It uses the modified van Laar Equations (5) to give liquid phase activity coefficients and a Modified van der Waals Equation of State (4,6) to give imperfection-pressure coefficients ft. 

The Modified van Laar Equations can represent vapor-liquid and liquid-liquid equilibria. Accordingly, they can be used to predict three-phase equilibria when conditions allow two liquid phases as well as a vapor phase to exist. This might occur on the trays in the distillation column or at the condenser and accumulator for the overhead product from the azeotropic distillation column. 

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