Excess Gibbs energy Wilson

Activity Coefficient at Infinite Dilution. A procedure similar to that employed by Wilson will be used here to obtain an expression for the excess Gibbs energy. Wilson started from the Flory and Huggins expression" 2 for the excess free energy of athermal solutions, but expressed the volume fractions in terms of local molar fractions. We selected Wilson s approach from a number of approaches, because it provided a better description of phase equilibria and because the interactions that count the most are the local one, but started from the more... 

We can estimate the activity coefficients by using the excess Gibbs energy models. Based on the local composition concept, the Wilson, NRTL, and UNIQUAC models for excess Gibbs energy provide relations for activity coefficient... 

The excess Gibbs energy of the ternary mixture was expressed through the Wilson [38], NRTL [39] and Zielkiewicz [32] expressions. Because of the agreement between the latter two expressions, detailed results are presented only for the more simple NRTL expression. The parameters in the NRTL equation were found by htting x-P (the composition of liquid phase-pressure) experimental data [32]. The derivatives (9 i/9xi) c2 ( IX2/dx2)xi and (diX2/dxi)x2 in the ternary mixture were found by the analytical differentiation of the NRTL equation. The excess molar volume (V ) in the binary mixtures (i-j) was expressed via the Redlich-Kister equation... 

Fig. 1.11 Plot of the excess Gibbs energy of mixing for the carbon tetrachloride-acetonitrile system against the mole fraction of acetonitrile at 45°C. The points show the experimental results and the solid curve was calculated using equation (1.10.7) with the parameters given by Wilson [7] (see text).

The activity coefficients are typically computed from a model for the excess Gibbs energy g, as described in the thermodynamics chapter (Chapter 4). The most popular are the Wilson, NRTL, and Uniquac models, described in detail in many places [15, 36 0]. They contain two or three adjustable (and possibly temperature-dependent) parameters per binary. One cannot predict which model will be best for a given system however, the Wilson equation is incapable of describing LEE. 

Ignoring any excess energy contribution, Wilson s excess Gibbs energy is -Ts as follows ... 

Wilson" has proposed that the excess Gibbs energy of a multicomponent system is given by... 

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. 

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). 

Table E.3 Selected binary liquid mixtures in which the excess Gibbs energy can be approximately represented by Wilson equations (5.6.24) and (5.6.30) ...

The Wilson equation is based on local composition theories and accounts for inter-molecular interactions between a molecule and its immediate neighbors. The Wilson expression for the excess Gibbs energy and the resulting equations for the activity coefficients are... 

Figure 5.14 Concentration dependence of the activity coefficients and of the dimensionless excess Gibbs energy for the system ethanol (l)-water (2) at 70 C [8] — Wilson model.

Wilson started from a similar equation as Flory and Huggins for the derivation of his equation for arbitrary mixtures apart from polymers [13]. However, instead of the true volume fractions Wilson used the so-called local volume fraction in the expression for the excess Gibbs energy ... 

In turn, the different -models can be used to describe the contribution of the excess Gibbs energy or activity coefficients. An exception is the Wilson model. No miscibility gap can be represented by this equation because the Wilson equation describes a monotone behavior of the composition for each parameter combination, that is, (9 Ag/3x > 0) (see Appendix C, E3). 

It can be seen that the concentrations of the species 1 and 2 can be different in the areas around a molecule of type 1 and around a molecule of type 2, although the overall concentration in the mixture remains constant. Taking this into account, the excess Gibbs energy can according to Wilson [5] be assumed to be... 

More sophisticated models for have been developed from molecular principles. For example, the universal quasi-chemical theory, UNIQUAC, is an extension of the Wilson equation. It divides the excess Gibbs energy into two parts, one due to entropy, the combinatorial part, and one due to ener, the residual part ... 

The Wilson equation is commonly used for multicomponent mixtures. It is relatively simple, represents many systems well, and also just depends on binary pair data. For a system ofm components, the excess Gibbs energy is written as ... 

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,... 

Wilson presented the following expressions for the molar excess Gibbs free energy of a binary solution ... 

Wilson equation Interaction method for the excess Gibb s energy suitable for totally miscible systems, not applicable for systems with limited miscibility since only the binary parameters are used, applicable to multicomponent systems only valid for small and medium operating pressures Wilson, G.M., J. Am. Chem. Soc. 86 (1964) 127. 

The concept of local compositions revolutionized the whole approach to developing expressions for the excess Gibbs free energy and, consequently, the activity coefficient. Wilson used it in developing the equa-... 

The NRTL equation was developed by Renon and Prausnitz.54 The expression for the excess energy function gl (see Table 14-12) was developed on the basis of a two-liquid theory. In the theoretical development, the A,/s are the Gibbs interaction energies and the a,/s are the reciprocals of the lattice coordination numbers. In practice, the quantities (Ajt - A ), (A0 — A ), and atj are taken to be three adjustable parameters per binary pair in the mixture and are obtained by a regression of the equilibrium data. The NRTL equation has been used extensively. It is superior to the Wilson equation in that it can be used to represent liquid-liquid systems (al7 < 0.426). However, the NRTL equation requires three parameters per binary pair, whereas the Wilson equation requires only two parameters per binary pair. 

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