The Wilson Equation

Wilson postulated that the ratio of local compositions are related to the overall mole fractions through a Boltzmann type expression  

Using this relationship he arrived at expressions for the local volume fractions, which he combined with the Flory-Huggins equation for polymer solutions (Section 13.10.2) to develop an expression for the excess Gibbs free energy. 

From the latter and Eq. 11.13.6 he obtained the following expression for the activity coefficient of component i in a multicomponent mixture  

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Compilation of binary experimental data reduced with the Wilson equation and, for high pressures, with a modified Redlich-Kwong 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 ... 

The local-composition models have hmited flexibility in the fitting of data, but they are adequate for most engineering purposes. Moreover, they are implicitly generalizable to multicomponent systems without the introduction of any parameters beyond those required to describe the constituent binaiy systems. For example, the Wilson equation for multicomponent systems is written ... 

Table 13-1, based on the binary-system activity-coefficient-eqnation forms given in Table 13-3. Consistent Antoine vapor-pressure constants and liquid molar volumes are listed in Table 13-4. The Wilson equation is particularly useful for systems that are highly nonideal but do not undergo phase splitting, as exemplified by the ethanol-hexane system, whose activity coefficients are snown in Fig. 13-20. For systems such as this, in which activity coefficients in dilute regions may...

The Wilson equation is superior to the familiar Van-Laar and Margules equations (see Volume 2, Chapter 11) for systems that are severely non-ideal but, like other three suffix equations, it cannot be used to represent systems that form two phases in the concentration range of interest. 

A significant advantage of the Wilson equation is that it can be used to calculate the equilibrium compositions for multicomponent systems using only the Wilson coefficients obtained for the binary pairs that comprise the multicomponent mixture. The Wilson coefficients for several hundred binary systems are given in the DECHEMA vapour-liquid data collection, DECHEMA (1977), and by Hirata (1975). Hirata gives methods for calculating the Wilson coefficients from vapour liquid equilibrium experimental data. 

The use of a spreadsheet to solve the Wilson equation is illustrated in Example 8.156. The spreadsheet used was Microsoft Excel. Copies of the spreadsheet example can be downloaded from support material for this chapter given on the publisher s web site at bh.com/companions/075641428. 

Using the Wilson equation, calculate the activity coefficients for isopropyl alcohol (IPA) and water in a mixture of IPA, methanol, water, and ethanol composition, all mol fraction ... 

The NRTL equation developed by Renon and Prausnitz overcomes the disadvantage of the Wilson equation in that it is applicable to immiscible systems. If it can be used to predict phase compositions for vapour-liquid and liquid-liquid systems. 

The values of the activity coefficients determined at the azeotropic composition can be used to calculate the coefficients in the Wilson equation (or any other of the three-suffix equations) and the equation used to estimate the activity coefficients at other compositions. Horsley (1973) gives an extensive collection of data on azeotropes. 

The constants in any of the activity coefficient equations can be readily calculated from experimental values of the activity coefficients at infinite dilution. For the Wilson equation ... 

Table 4.8 Data for methanol (1) and water (2) for the Wilson equation at 1 atm6. ...

Table 4.10 Bubble-point calculation for a methanol-water mixture using the Wilson equation. 

Thus, the composition of the vapor phase at 1 atm is y = 0.7863, y2 = 0.2136 from the Wilson Equation. For this mixture, at these conditions, there is not much difference between the predictions of Raoult s Law and the Wilson equation, indicating only moderate deviations from ideality at the chosen conditions. 

Example 4.5 2-Propanol (isopropanol) and water form an azeotropic mixture at a particular liquid composition that results in the vapor and liquid compositions being equal. Vapor-liquid equilibrium for 2-propanol-water mixtures can be predicted by the Wilson equation. Vapor pressure coefficients in bar with temperature in Kelvin for the Antoine equation are given in Table 4.113. Data for the Wilson equation are given in Table 4.126. Assume the gas constant R = 8.3145 kJ-kmol 1-K 1. Determine the azeotropic composition at 1 atm. 

A model is needed to calculate liquid-liquid equilibrium for the activity coefficient from Equation 4.67. Both the NRTL and UNIQUAC equations can be used to predict liquid-liquid equilibrium. Note that the Wilson equation is not applicable to liquid-liquid equilibrium and, therefore, also not applicable to vapor-liquid-liquid equilibrium. Parameters from the NRTL and UNIQUAC equations can be correlated from vapor-liquid equilibrium data6 or liquid-liquid equilibrium data9,10. The UNIFAC method can be used to predict liquid-liquid equilibrium from the molecular structures of the components in the mixture3. 

The vapor-liquid x-y diagram in Figures 4.6c and d can be calculated by setting a liquid composition and calculating the corresponding vapor composition in a bubble point calculation. Alternatively, vapor composition can be set and the liquid composition determined by a dew point calculation. If the mixture forms two-liquid phases, the vapor-liquid equilibrium calculation predicts a maximum in the x-y diagram, as shown in Figures 4.6c and d. Note that such a maximum cannot appear with the Wilson equation. 

Thus, if the saturated vapor pressure is known at the azeotropic composition, the activity coefficient can be calculated. If the composition of the azeotrope is known, then the compositions and activity of the coefficients at the azeotrope can be substituted into the Wilson equation to determine the interaction parameters. For the 2-propanol-water system, the azeotropic composition of 2-propanol can be assumed to be at a mole fraction of 0.69 and temperature of 353.4 K at 1 atm. By combining Equation 4.93 with the Wilson equation for a binary system, set up two simultaneous equations and solve Au and A21. Vapor pressure data can be taken from Table 4.11 and the universal gas constant can be taken to be 8.3145 kJ-kmol 1-K 1. Then, using the values of molar volume in Table 4.12, calculate the interaction parameters for the Wilson equation and compare with the values in Table 4.12. 

Jaques and Furter (7, 8) successfully fitted the T-P-X data of various isobaric systems saturated with a salt by treating the systems as pseudobinaries and using the Wilson equation for correlation. Rousseau et al. (9) used a similar approach in correlating the data of Johnson and Furter by means of the van Laar (10), Wilson (11) and NRTL (12) equations. 

The Wilson equation was considered first. Great problems were encountered with this equation. Negative values for one or both of the parameters A12 and A21 were often obtained by regressing the solvent-salt data. As it can be seen from Equation 3, a negative value for Aij is unacceptable. 

A method for interpolation of calculated vapor compositions obtained from U-T-x data is described. Barkers method and the Wilson equation, which requires a fit of raw T-x data, are used. This fit is achieved by dividing the T-x data into three groups by means of the miscibility gap. After the mean of the middle group has been determined, the other two groups are subjected to a modified cubic spline procedure. Input is the estimated errors in temperature and a smoothing parameter. The procedure is tested on two ethanol- and five 1-propanol-water systems saturated with salt and found to be satisfactory for six systems. A comparison of the use of raw and smoothed data revealed no significant difference in calculated vapor composition. 

The present author (4) has previously preferred to use raw isobaric data coupled with Barker s method (2) and the Wilson equation (5), but interpolation of the calculated discrete vapor composition values requires smoothing of boiling point-liquid composition data at some stage. 

The vapor pressures of the pure liquid components are replaced by the vapor pressures of the liquids saturated with salts. The Wilson equation (5) in its three-constant form is employed as the correlating equation. This yields values of A21, A12, and C and the corresponding vapor compositions. 

An accurate representation of the phase equilibrium behavior is required to design or simulate any separation process. Equilibrium data for salt-free systems are usually correlated by one of a number of possible equations, such as those of Wilson, Van Laar, Margules, Redlich-Kister, etc. These correlations can then be used in the appropriate process model. It has become common to utilize parameters from such correlations to obtain insight into the fundamentals underlying the behavior of solutions and to predict the behavior of other solutions. This has been particularly true of the Wilson equation, which is shown below for a binary system. 

An advantage of the Wilson equation is that it involves only two parameters per binary and may be extended, without further information, to estimate multi-component phase equilibrium behavior. 

An alternative approach is to estimate activity coefficients of the solvents from experimental data and correlate these coefficients using, for example, the Wilson equation. Rousseau et al. (3) and Jaques and Furter (4) have used the Wilson equation, as well as other integrated forms of the Gibbs-Duhem equation, to show the utility of this approach. These authors found it necessary, however, to modify the definitions of the solvent reference states so that the results could be normalized. 

To apply the binary form of the Wilson equation, Jaques and Furter (4,5) and Rousseau et al. (3) treated the salt-solvent systems as pseudobinaries by expressing the solvent compositions on a salt-free basis. In addition, reference fugacities were defined to adjust the vapor pressure of the liquids by an amount... 

In this study, a thermodynamic framework has been presented for the calculation of vapor-liquid equilibria for binary solvents containing nonvolatile salts. From an appropriate definition of a pseudobinary system, infinite dilution activity coefficients for the salt-containing system may be estimated from a knowledge of vapor pressure lowering, salt-free infinite dilution activity coefficients, and a single system-dependent constant. Parameters for the Wilson equation may be determined from the infinite dilution activity coefficients. 

TABLE 13.2. Activity Coefficients from Solubility Parameters and from the Wilson Equation... 

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