Van Laar parameters

When there is significant random error in all the variables, as in this example, the maximum-likelihood method can lead to better parameter estimates than those obtained by other methods. When Barker s method was used to estimate the van Laar parameters for the acetone-methanol system from these data, it was estimated that = 0.960 and A j = 0.633, compared with A 2 0.857 and A2- = 0.681 using the method of maximum likelihood. Barker s method uses only the P-T-x data and assumes that the T and x measurements are error free. 

Figure 6-2. Confidence ellipses for van Laar parameters. Acetone(l)-methanol(2) system at 755 mm Hg (Othmer, 1928).

For the acetone-methanol data of Othmer, the correlation coefficient is -0.678, indicating a moderate degree of correlation between the two van Laar parameters. The elongated confidence ellipses shown in Figure 2 further emphasize this correlation. 

Van Laar parameters Aq and can be translated into the corresponding Mar-gules subregular parameters W12 and JV21, posing... 

Figure 3.9D shows the form of the curve of the excess Gibbs free energy of mixing obtained with Van Laar parameters variable with T. the mixture is subregular— i.e., asymmetric over the binary compositional field. 

EX3104 3.10.4 Van Laar parameter estimation (error-invariables method) M16,M18,M41,M45, M52... 

Error—in-variables estimation of van Laar parameters from vapor—liquid equilibrium data... 

Estimate the van Laar parameters of methanol (1) and 1,2-dichloro-ethane (2) from equilibria data obtained at T = 323.15 and shown in Table 3.9 if the Antoine parameters for these components are (ref. 52) Aj = 23.8843,... 

The van Laar parameters A = 5135.9 J/mol and = 43213.7 J/mol yield a good fit. The observed variables are only slightly corrected to satisfy the model equations. The quantity "equation error after correction" is expressed in Pascals, hence the above values are negligible small. 

From the isothermal vapor-liquid equilibrium data for the ethanol(l)/toluene(2) system given in Table 1.11, calculate (a) vapor composition, assuming that the liquid phase and the vapor phase obey Raoult s and Dalton s laws, respectively, (b) the values of the infinite-dilution activity coefficients, Y and y2°°, (c) Van Laar parameters using data at the azeotropic point as well as from the infinite-dilution activity coefficients, and (d) Wilson parameters using data at the azeotropic point as well as from the infinite-dilution activity coefficients. 

In these equations, y, and y2 are activity coefficients of components 1 and 2, respectively, GE is Gibbs molar excess free energy, A, 2 and A2j, are Van Laar parameters, G, 2 and G2j, are Wilson parameters, that is,... 

HVOS, MHV2, MHV1, and LCVM mixing rules combined with the van Laar excess free-energy model and the PRSV equation of state. The solid lines represent correlations with the van Laar parameters fit to experimental data, and the dashed lines show predictions with the van Laar parameter obtained from the DECHEMA tables at 298 K. The points are measured VLE data at 298 K from the DECHEMA Chemistry Data Series, GmehUng and Onken 1977, Vol. l,Pt. Lp. 238. 

With the known values for the van Laar parameters, these eight equations are solved simultaneously for the four Xs and four ys. The results are listed as follows ... 

Benzene (1) and water (2) form a heterogeneous azeotrope at 450 K and 2000 kPa. The benzene-rich liquid phase has 95.7% mole benzene, the water-rich phase has 97.7% mole water, and the vapor phase has 48% mole benzene. Use these data to determine the van Laar parameters for the benzene-rich and the water-rich phases. An approximation of ideal gas behavior in the vapor phase may be assumed. The vapor pressures at 450 K are 936 kPa for water and 978 kPa for benzene. 

This development provides both a justification for the van Laar equations and a method of estimating the van Laar parameters for liquid-phase activity coefficients from the parameters in the van der Waals equation of state. Since we know the van der Waals equation is not very accurate, it is not surprising that if a and P are treated as adjustable parameters, the correlative value of the van Laar equations is greater than when a and P are determined from Eqs. 9.6-6 (Problem 9.9). 

These values are in reasonable agreement with, but slightly different from, those found in the previous illustration. Using the values for the van Laar parameters, we obtain the y and P values in Table 10.2-3. Clearly, the agreement is excellent. ... 

Chemical stability, (9 lna,/9y,)j > 0 (/ = 1 or 2), imposes the following restriction on the van Laar parameters a and P if homogeneous single-phase behavior is favored ... 

Examples of Liquid-Liquid Phase Separation in Regular Solutions, van Laar parameters at ambient pressure are provided in Table 29-1 for three binary mixtures that exhibit concentration-dependent miscibility. The corresponding graphs of Agmixing VS. Composition at constant T and p are provided in Figures 29-1 and 29-2. There is a range of compositions where... 

Analogous Van Laar parameter In Helmholtz free energy Eqn., a-3... 

This equation is unique because the mixture rules are already defined. Also Van Laar parameters derived from fitting binary and multicomponent vapor-liquid equilibrium data relate directly to parameters in the equation of state. By simple substitution it can be shown that Ajj /RT in Equation (a-17) is related to in Equation (a-3) by the equation ... 

To this purpose all data are regressed to obtain the parameter values that best fit em. They can, of course, be also obtained from a pair of activity coefficient data by solving a system of two equations with two unknowns. Thus, if for a given the corresponding activity coefficients are Yi and Y2> we obtain the following expressions for the van Laar parameters ... 

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