Wilson’s equation

Wilms tumor Wilson model Wilson plots Wilson s disease Wilson s equation... 

There are many simple two-parameter equations for Hquid mixture constituents, including the Wilson (25), Margules (2,3,18), van Laar (3,26), nonrandom two-Hquid (NRTI.v) (27), and universal quasichemical (UNIQUAC) (28) equations. In the case of the NRTL model, one of the three adjustable parameters has been found to be relatively constant within some homologous series, so NRTL is essentially a two-parameter equation. The third parameter is usually treated as a constant which is set according to the type of chemical system (27). A third parameter for Wilson s equation has also been suggested for use with partially miscible systems (29,30,31). These equations all require experimental data to fit the adjustable constants. Simple equations of this type have the additional attraction of being useful for hand calculations. 

The methods most generally used for the calculation of activity coefficients at intermediate pressures are the Wilson (1964) and UNIQUAC (Abrams and Prausnitz, 1975) equations. Wilson s equation was used by Sato et al. (1985) to predict the composition of fhe condensate gas stripped from a packed bed fermenter at 30°C, whilst Beck and Bauer (1989) used the UNIQUAC equation, with temperature-dependent parameters given by Kolbe and Gmehling (1985), for their model of an anaerobic gas-solid fluidized bed fermenter at 36°C. In this case it was necessary to go beyond the temperature range of fhe source data down to 16°C in order to predict the composition of the fluidizing gas leaving the condenser. 

As it is apparent from the above two categories, the first one is not useful for the prediction of the phase behavior of mixtures, specifically for mixtures of more than two species. Because of the time-consuming and expensive nature of binary interaction parameter evaluation of various chemical components, the first group of thermodynamic models (above) is not practical. As an example, the activity coefficient from Wilson s equation of state is found from [4] ... 

Wilson s equation of state is found from Equations (14) and (15). It can be seen that for obtaining the activity coefficient of a component 1 in a pure solvent 2, we need four interaction parameters (A12, A21, An a A22, which are temperature dependent. It is evident that for calculating the value of the binary interaction parameters, additional experimental data, such as molar volume is needed. Other models which belong to the first category have the same limitations as Wilson s method. The Wilson model was used in the prediction of various hydrocarbons in water in pure form and mixed with other solvents by Matsuda et al. [11], In order to estimate the pure properties of the species, the Tassios method [12] with DECHEMA VLE handbook [13] were used. Matsuda et al. also took some assumptions in the estimation of binary interactions (because of the lack of data) that resulted in some deviations from the experimental data. 

Wilson s equation and the modification proposed by Renon and Prausnitz (8) use the local mole fraction concept, produced because molecules in solution aggregate as a result of the variation in intermo-lecular forces. The local mole fraction concept results in a more useful description of the behavior of molecules in a non-ideal mixture. 

It is difficult to fit data with the local composition models due to their complex logarithmic forms. However, they are readily generalizable to multicomponent systems. Smith, van Ness, and Abbott and Prausnitz, Lichtenthaler, and Gomes de Azevedo present the Wilson and UNIQUAC models extended for multi-component solutions. They both employ the constants from binary data. However, the constants are not unique in the sense that valid, but different constants may be obtained from different sets of data. Wilson s equations cannot be employed for immiscible solutions, but the UNIQUAC model may be used to describe such solutions. However, Wilson s equations are good for polar or associating compounds. A compilation of Wilson parameters can be found by Hirata, Ohe, and Nagahama. ... 

Fig. 2 Isobaric vapor-liquid equilibrium of acetone and chlorobenzene at 1.01325 Bars. Solid lines are calculated from Wilson s equation. Dashed lines are from Raoult s law.

Wilson s equations Good for polar or associating compounds... 

Use Wilson s equation if neither plot produces a good fit to the data. 

It would be best to examine a plot of the activity coefficients versus xi first. If these plots exhibit a maxima or a minima, neither van Laar s nor Wilson s equations are useful. Furthermore, Wilson s equations are not suitable for use in systems of limited solubility. 

The activity coefficients can be calculated from Wilson s equations or from UNIFAC if the parameters of the models are known. There are some parameters for UNIFAC in Magnussen, Rasmussen, and Fredenslund, Gupte and Danner,and Hooper, Michel, and Prausnitz. These parameters are not as accurate as those for vapor-liquid equilibrium. 

To extend the activity-coefficient equation to partially miscible solutions, Renon and Prausnitz [8] introduced a factor to the exponential energy term in Wilson s equation. With a < 1, the effect is to suppress the preferential attraction of molecules to the central molecule. The local mole fraction of component 2 about component 1 in a binary solution is given by... 

Comparison with experimental data shows that the complete local-composition equation preserves the quality of Wilson s equation in describing vapor-liquid equilibrium of completely miscible systems. There are no more than slight differences between the complete equation and Wilson s equation in the fitting of data. But the complete local-composition (CLC) equation extends Wilson s local-composition equation to partially miscible solutions. Good predictions of the coexistent liquid compositions of ternary mixtures based on the binary parameters have been found for water + ethyl acetate + ethanol, for water + methyl acetate + acetone, and for water + acrylonitrile + acetonitrile. 

The fifth approach is more a field than a concise method, since it embodies so many theoretical concepts and associated methods. All are grouped together as adsorbed mixture models. Basically, this involves treating the adsorbed mixture in the same manner that the liquid is treated when doing VLE calculations. The major distinction is that the adsorbed phase composition cannot be directly measnred (i.e., it can only be inferred) hence, it is difficult to pursue experimentally. A mixture model is nsed to account for interactions, which may be as simple as Raoult s law or as involved as Wilson s equation. These correspond roughly to the Ideal Adsorbed Solution theory and Vacancy Solution model, respectively. Pure component and mixture equilibrium data are required. The unfortunate aspect is that they require iterative root-finding procedures and integration, which complicates adsorber simnlation. They may be the only route to acceptably accurate answers, however. It would be nice if adsorbents could be selected to avoid both aspects, but adsorbate-adsorbate interactions may be nearly as important and as complicated as adsorbate-adsorbent interactions. 

For mixtures that do not obey the Porter or Margules equations, additional high-order terms must be kept in the Redlich-Kister expansion hence, more parameters must be evaluated from experimental data. Alternatively, if we want to keep only two parameters, then we must abandon the Redlich-Kister expansion for some more complicated representation of g. Many functional forms have been proposed [1, 2], but here we restrict our attention to a useful expression proposed by Wilson in 1964 [14] and now identified as one of the class of "local-composition" models [2], For binary mixtures Wilson s equation takes the form... 

The Redlich-Kister expansion for the excess Gibbs energy provides no guidance about the temperature dependence of its parameters, and so temperature effects can only be obtained from experiment. In contrast, Wilson s equation is based on a theory that estimates the temperature dependence of the parameters. 

Note that the Ajy in (5.6.34) and the A jt in (5.6.35) are all binary parameters that is, their values are obtained from data for binary mixtures, and their temperature dependence is still usually assumed to be described by (5.6.30). Unlike the multicomponent versions of the Redlich-Kister expansion discussed in 5.6.4, the theoretical basis for (5.6.33) suggests that high-order multibody parameters are not needed in Wilson s equation in practice, this appears to be true for many mixtures. 

At 105°C mixtures of ethanol(l) and toluene(2) have activity coefficients at infinite dilution given approximately as y = 5.197 and y = 4.811. Compute and plot y vs. Xi using (a) the Margules equations and (b) Wilson s equations. 

The following tables provide values for parameters in models for the excess Gibbs energy of selected binary liquid mixtures. Table E.l contains values for the Porter equation ( 5.6.2), Table E.2 for the Margules equation ( 5.6.3), and Table E.3 for Wilson s equation ( 5.6.5). 

Wilson s equation corresponds to the special case of equation (41) with only one ij pair. The extended form has apparently never been used. Wilson s equations are useful as smoothing equations and because the parameters derived for binary mixtures can be applied directly to multicomponent mixtures. Their chief defect is their inability to give C large enough to lead to phase separation. This defect is usually remedied when the equations are used near a liquid-liquid region by including an arbitrary third parameter to give... 

A good account of the uses of Wilson s equations has been given by Prausnitz. G. M. Wilson, J. Amer. Chem. Soc., 1964, 86, 127. 

The initial surface appearance of the zinc deposited at 500 A/m for 90 kC/m (3 minutes) was observed by SEM. The structure of the deposited Zn was analyzed by X-ray diffraction analysis (XRD) and the crystal orientation index was calculated by Wilson s equation (5). 

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