Equation Wilson

This equation (Wilson, 1964) was developed on the basis of a liquid molecular model which assumes that interactions between molecules of two different components depend on the volume fraction of each component in the vicinity of a molecule of a given component. The volume fractions are determined from the probability of finding a molecule of one component or another in the vicinity of a molecule of the given component. The probabilities, and hence the volume fractions, are expressed through Boltzmann distribution functions of energy in terms of binary interaction parameters. The result for volume fractions is 

The equation is generalized to multi-component mixtures in terms of binary 

The Wilson equation is capable of representing both polar and non-polar molecules in multi-component mixtures using only binary parameters. It cannot, however, represent liquid-liquid equilibrium systems. The activity coefficients in a multi-component mixture are calculated by the Wilson equation as follows (Prausnitz et al., 1967)  

Note that Xy = X, and A = 1. Equation 1.36a reduces to Equation 1.36 for a binary system. 

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

D. The next card supplies the solvation and association parameters, and the third parameter for either the UNIQUAC, NRTL, or Wilson equation, if this parameter is not being fitted. 

SUFFIX MARGULES EQUATION UNIOUAC EQUATION NRTL EGU AT ION WILSON 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...

In a broad sense, one may include the Free-Wilson equation within the class of linear free energy relationships (LFER). It is also subjected to the assumption of additivity of the contributions to the biological activity by substituent groups at different substitution sites. The assumption requires, for example, that there is no hydrogen bonding interaction between the various substitution groups. 

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. 

The most important aspect of the simulation is that the thermodynamic data of the chemicals be modeled correctly. It is necessary to decide what equation of state to use for the vapor phase (ideal gas, Redlich-Kwong-Soave, Peng-Robinson, etc.) and what model to use for liquid activity coefficients [ideal solutions, solubility parameters, Wilson equation, nonrandom two liquid (NRTL), UNIFAC, etc.]. See Sec. 4, Thermodynamics. It is necessary to consider mixtures of chemicals, and the interaction parameters must be predictable. The best case is to determine them from data, and the next-best case is to use correlations based on the molecular weight, structure, and normal boiling point. To validate the model, the computer results of vapor-liquid equilibria could be checked against experimental data to ensure their validity before the data are used in more complicated computer calculations. 

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