Regular Solution model application

Ghiorso M. S. and Carmichael L S. E. (1980). A regular solution model for meta-aluminous silicate liquids Applications to geothermometry, immiscibility, and the source regions of basic magma. Contrib. Mineral Petrol, 71 323-342. 

Further elaboration requires a model. We shall consider the Bragg-Williams approximation (sec. I.3.8d) in which only the enthalpic part of G is accounted for, the entropy is assumed to remain ideal. For gas adsorbates this leads to the FFG isotherm II.3.8.17] and A1.5a] and in solutions it gives rise to the Regular Solution model, both models being fairly widely applicable. For this approximation, for a binary solution we derived I.3.8.25]... 

If applicable, the regular solution model provides values for and Ag. The following example calculation involving regular solutions is based on Appelo and Postma (1993). 

Comment on the applicability of the regular solution model to these two systems. 

If above relatiOTi can not be determined, it is sometimes possible to deduce activity coefficient values for components in solid phase based on thermodynamic solid solutimi model. For example, if symmetrical solvus (Fig. 1.2) exists for a binary system, regular solution model could be applicable to the estimation of activity coefficients and other thermodynamic parameters values of solid solution... 

This model was first described by JH Hildebrand in 1929, and is called the regular solution model [4]. Solutions with free energies of mixing that are well described by Equation (15.14) are called regular solutions. Here s an application of Equation (15.14). 

The regular solution model approach is very similar to the UNIFAC model developed by Fredenslund et al. [23,24], where the interactions between molecules are estimated on the basis of the groups present in each molecule. Extensive tables of interaction parameters [24,25] for vapor-liquid equilibrium data prediction are weii deveioped, and the method has found applicability in the modeling of deodorizer performance [26],... 

Figure 10,2 Deviations from Nernst s law in crystal-aqueous solution equilibria, as obtained from application of various thermodynamic models. (A and B) Regular solution (liyama, 1974). (C) Two ideal sites model (Roux, 1971a). (D) Model of local lattice distortion (liyama, 1974). Reprinted from Ottonello (1983), with kind permission of Theophrastus Publishing and Proprietary Co.

A key feature of this model is that no data for mixtures are required to apply the regular-solution equations because the solubility parameters are evaluated from pure-component data. Results based on these equations should be treated as only qualitative. However, mixtures of nonpolar or slightly polar, nonassociating chemicals, can sometimes be modeled adequately (1,3,18). Applications of this model have been limited to hydrocarbons (qv) and a few gases associated with petroleum (qv) and natural gas (see Gas, natural) processing, such as N2, H2, C02, and H2S. Values for 8 and JV can be found in many references (1—3,7). 

When gas solubility data are lacking or are unavailable at the desired temperature, they can be estimated using available models. The method of Prausnitz and Shair (1961), which is based on regular solution theory and thus has the limitations of that theory. The applicability of regular solution theory is covered in detail by Hildebrand et al. (1970). A more recent model, now widely used, is UNIFAC, which is based on structural contributions of the solute and solvent molecular species. This model is described by Fredenslund et al. (1977) and extensive tabulations of equilibrium data, based on UNIFAC, have been published by Hwang et al. (1992) for aqueous systems where the solute concentrations are low and the solutions depart markedly from thermodynamic equilibrium. 

The Chao-Seader and the Grayson-Streed methods are very similar in that they both use the same mathematical models for each phase. For the vapor, the Redlich-Kwong equation of state is used. This two-parameter generalized pressure-volume-temperature (P-V-T) expression is very convenient because only the critical constants of the mixture components are required for applications. For the liquid phase, both methods used the regular solution theory of Scatchard and Hildebrand (26) for the activity coefficient plus an empirical relationship for the reference liquid fugacity coefficient. Chao-Seader and Grayson-Streed derived different constants for these two liquid equations, however. 

Regular solution theory, the solubility parameter, and the three-dimensional solubility parameters are commonly used in the paints and coatings industry to predict the miscibility of pigments and solvents in polymers. In some applications quantitative predictions have been obtained. Generally, however, the results are only qualitative since entropic effects are not considered, and it is clear that entropic effects are extremely important in polymer solutions. Because of their limited usefulness, a method using solubility parameters is not given in this Handbook. Nevertheless, this approach is still of some use since solubility parameters are reported for a number of groups that are not treated by the more sophisticated models. 

Other molecular properties have been also proposed to model the hydrophobic interactions. The parachor, which is related to the surface tension of a compound (139, 140) represents mainly the intermolecular interactions in a liquid. The Hildebrand-Scott solubility parameter, 6, (141) is related to intermolecular van der Waals forces and the closely related molar attraction constant, F, is obtained by multiplying 6 by the molar volume (142). The partition coefficient between two solvents can be obtained from the solubility parameters and the molar volumes of the solute and the solvents (193). This relationship is based on regular solution theory (194) and the assumption that the partial molar volumes of the solute is not different from its molar volume. Recently this has been criticized and a new derivation was proposed (195) in which the partial molar volumes are taken into account. The molar refractivity, MR, is related to dispersion forces and can be obtained as a sum of the partial molar refractivi-ties assigned to atoms and bonds (140, 143). These parameters have been compared (144) to establish their relative applicability to correlations with biological activity. The conclusion was that logP and molecular refractivity were the best parameters. Parameters obtained from high pressure liquid chromatography (144,... 

Owing to their polymeric character, silicate melts belong to the solutions of type II, which do not follow Raoult s law. The classic regular solution approach is not applicable, since the limiting laws are not obeyed. The Temkin s model of ideal ionic solution, which has been widely applied in molten salt systems, cannot be used, since the real anionic composition, owing to a broad polyanionic distribution, is not known a priori. 

This is the case for the C6H6 - CC14 system, but not for the C6H6 - CS2 system. Therefore, regular solution theory is not applicable to the C6H6-CS2 system. To test the Hildebrand-Scatchard model we use... 

For chromatographic applications, the most useful models of solvent properties are the solubility parameter concept, Snyder s solvent strength and selectivity parameters, solvatochromic parameters and the system constants of the solvation parameter model for gas to liquid transfer. The Hildebrand solubility parameter, 8h (total solubility parameter), is a rough measure of solvent strength, and is easily caleulated from the physical properties of the pure solvent. It is equivalent to the square root of the solvent vaporization energy divided by its molar volume. The original solubility parameter concept was developed from assumptions of regular solution behavior in which the principal intermolecular interactions were dominated by dispersion forces. 

Because of the importance of distillation processes, first it was the objective to develop models only for the prediction of VLE. The first predictive model with a wide range of applicability was developed by Hildebrand and Scatchard [48]. The so-called regular solution theory is based on considerations of van Laar, who was a student of van der Waals and used the van der Waals equation of state to derive an expression for the excess Gibbs energy [49]. Since the two parameters a and b of the van der Waals equation of state can be obtained from critical data, it should be possible to calculate the required activity coefficients using critical data. However, the results were strongly dependent on the mixing rules applied. 

Actually, perfluorocarbons behave in many ways quite differently from hydrocarbons, as already established in the literature. For example, then-behaviour in the context of thermodynamic models differs significantly from all other compounds, including hydrocarbons (Figure 3.12). Moreover, in the context of the regular solution/solubility parameter theory (see below), solutions containing fluorocarbons show deviations not shared by other mixtures of non-polar compounds, which constitutes the main area of applicability of regular solution/ solubility parameter theory (Prausnitz, Lichtenthaler and de Azevedo, 1999). 

One model that has found applicability in the lipids area is the regular solution theory developed by Hildebrand and Scott [9] and Scatchard [10]. Incorporating a partial molar entropy of mixing term [11-14] into the regular solution theory yields the following expression for the activity of a component in a liquid mixture ... 

Our laboratory has planned the theoretical approach to those systems and their technological applications from the point of view that as electrochemical systems they have to follow electrochemical theories, but as polymeric materials they have to respond to the models of polymer science. The solution has been to integrate electrochemistry and polymer science.178 This task required the inclusion of the electrode structure inside electrochemical models. Apparently the task would be easier if regular and crystallographic structures were involved, but most of the electrogenerated conducting polymers have an amorphous and cross-linked structure. 

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