Van Laar Equation Constants

These are in the log form. We also often see this equation in the In form (In -yi = etc.) for which the constants A and B are 2.303 times as large as those shown here. We must always check to see which form the reported constants correspond to. 

Most of the values in this table are for data at a constant pressure of 760torr. Some are constant temperatures, as shown. As discussed in Chapter 9, there should not be much difference between the constants obtained either way, which is observed for most of the pairs in this table for which both forms are shown. 

Some pairs, such as acetone-water, are shown twice, once with acetone as component 1 and once with water as component 1. The reader may check to see that this simply interchanges the values of A and B. 

Source Taken ftom Holmes, M. J., and M. van Winkle, Prediction of ternary vapOT-liquid from binary data. Ind. Eng. Chem. 62 21-31 (1970).  

8 ENTHALPIES AND GIBBS ENERGIES OF and25°C (aq),, an ideal 1-molal solution of that substance in  

---

Coefficient in Van Laar equation Constant in the vapour pressure relation Distillate rate... 

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... 

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. 

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... 

Calculate the activity coefficients, bubble point pressure, and the vapor composition as a function of liquid composition for the hydrofluoric acid— water binary at 120°C. Use the van Laar equation with the constants calculated in Problem 1.10. Vapor pressure data may also be obtained from Problem 1.10. Assume ideal gas behavior in the vapor phase. 

A stream of hydrofluoric acid (1) in water (2) at 120°C and 200 kPa contains 12% mole hydrofluoric acid (HF). It is proposed to concentrate the HF in solution by partial vaporization in a single stage, by means of temperature and pressure control. Calculate the resulting liquid composition and the fraction vaporized at 120°C and 135 kPa. Can this process be used to concentrate the liquid for any starting composition Use the van Laar equation for liquid activity coefficients and assume ideal gas behavior in the vapor phase. The vapor pressures of HF and water at 120°C are 1693 and 207 kPa, respectively, and the van Laar constants are Ajj = -6.0983, A2] = -6.9658 (see Problems 1.8 and 1.9). 

The separation of benzene (1) and cyclohexane (2) by distillation is complicated due to the formation of an azeotrope (Example 2.5). Vapor-liquid equilibrium data for this binary are required for the design of a workable separation process. As a first step, find the azeotropic composition and temperature at 100 kPa pressure. Use the van Laar equation for activity coefficients with parameters A12 = 0.147, A21 = 0.165. The computations can be made with assumptions consistent with low pressure conditions. The vapor pressures of benzene and cyclohexane can be represented by the Antoine Equation 2.19 with the following constants Al = 13.88, Bl = 2788.5, Q = -52.36, A2 = 13.74, = 2766.6,... 

Compare the van Laar and Wilson equations by solving the problem in Example 2.7A using the van Laar equation for activity coefficients. Ideal gas behavior may be assumed for the vapor phase. Use the feed composition and vapor pressure data from Example 2.7A, and calculate the temperature and phase compositions at the pressure and vapor fraction in that example. The following are the binary van Laar constants ... 

It is required to determine the effect of the furfural rate on the separation. The multi-component van Laar equation is used to calculate the liquid activity coefficients, and the vapor phase is assumed to behave as an ideal solution. Solvent rates of 0, 100, 200, and 400 kmol/h are to be considered. The vapor pressure data are calculated using the Antoine Equation 2.19, with the constants given below (p,° in kPa and T in K) ... 

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. 

Develop expressions for y, and y° using each of the following the one-constant and two-constant Margules equations, the van Laar equation, regular solution theory, and the UNIFAC model. 

Use the van Laar equations to estimate the compositions of the coexisting liquid phases in an isobutane-furfurai mixture at 37.8°C and a pressure of 5 bar. (You may assume that the van Laar constants for this system given in Table 9.5-1 are applicable at this pressure.)... 

For practical applications, it is important that the van Laar equation correctly predict azeotrope formation. If activity coefficients are known or can be computed at the azeotropic composition, say from (5-21), (-y,x = P/Pf since Ki = 1.0), these coefficients can be used to determine the van Laar constants directly from the following equations obtained by solving (5-26) simultaneously for A12 and A21... 

The Wilson equation is readily extended to multicomponent mixtures. Like the van Laar equation (5-33), the following multicomponent Wilson equation involves only binary interaction constants. 

Brian showed that, when binary constants are derived from mutual solubility data, the van Laar equation behaves well over the entire single liquid-phase regions and is superior to the Margules and Scatchard-Hammer equations. 

From the experimental infinite-dilution activity coefficients given in Problem 5.7 for the benzene-cyclopentane system, calculate the constants in the van Laar equation. With these constants, use the van Laar equation to compute the activity coefficients... 

Solution. Because the pressure is likely to be low, the modified Raoult s law is suitable for computing X-values. Therefore, the bubble point can be obtained directly from (7-24). The activity coefficients calculated from the van Laar equation in Example 5.11 can be used. Vapor pressures of the pure species are computed from the Antoine relation (4-69) using constants from Appendix 1. Equation (7-24) applies to either the methanol-rich layer or the cyclohexane-rich layer, since from (4-31) yix = y- x . Results will differ depending on the accuracy of the activity coefficients. Choosing the methanol-rich layer, we find... 

A mixture of cyclohexane and cyclopentane is to be separated by liquid-liquid extraction at 25°C with methanol. Phase equilibria for this system may be predicted by the van Laar equation with constants given in Example 5.11. Calculate, by the ISR method, product rates and compositions and interstage flow rates and compositions for the conditions below with ... 

A li binary interaction constant in van Laar equation as defined by... 

At 45°C, the van Laar constants for two of the pairs in the ternary system n-bexane(l)-isohexane(2)-methyI alcohol(3) are Ayy = 1.3S, A3i = 2.36, A23 = 2.14, and Ayi - 2.22. Assume that the two hexane isomers form an ideal solution. Use (5-33), the multicomponent form of the van Laar equation, to predict the liquid-phase activity coefficients of an equimolal mixture of the three components at 45°C. It is possible that application of the van Laar equation to this system may result in erroneous prediction of two liquid phases. 

Notice that in this model the excess Gibbs free energy is not a polynomial in the mol fraction. The van Laar equation is a two-parameter model and the constants and Ba are related to the activity coefficients at infinite dilution ... 

The value of /if of a solute is a function of its activity coefficients in the two phases and of the solvent molar volumes [Eq. (5)]. Thus, K depends on the type of the solvent system as well as the solute concentration and temperature. For a given solvent-water and temperature, K is constant for a solute only at low-solute concentrations when y and V approach constant values. The variation of y with the solute concentration for the two components of a binary solution may be characteristically described by van Laar equations (72, 102, 103) ... 

Activity coefficients calculated by these methods agree fairly well for systems where the original equations, i.e., Eqs. (3.50) to (3.66), apply. Carlson and Colburn (5) and Colburn, Schoenborn, and Shilling (8) have shown that the van Laar constants cannot be calculated from solubility data for n-butanol-water and isobutanol-water, but the van Laar equations do not satisfactorily describe the activity coefficients obtained from vapor-liquid data in these systems either. 

Another two-constant equation for the representation of partial pressure data in a binary system is known by the name of van Laar. The equation was originally put forward as the result of a theory based on the van der Waals equation of state. This theory is probably erroneous, but the van Laar equation is nevertheless very useful for the empirical representation of partial pressure data. For the component whose mole fraction is x the equation is... 

This equation is also a solution to the Gibbs-Duhem equation. The constants may be calculated from infinite dilution activity coefficients or from a single vapor-liquid equilibrium data point. The multi-component van Laar equation takes the following form (Carlson et al., 1942 Wohl,... 

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. 

---