Statistical mechanics mixture theory

Fig. 10. Densities of coexisting phases for a fluid obeying the 6-12 potential according to a statistical mechanical perturbation theory developed by Barker and Henderson. The points are a mixture of machine calculations and actual experimental data (from Barker and Henderson, 1967).

The equilibrium between a pure solid and a gaseous mixture is one of very few classes of solution for which an exact treatment can be made by the methods of statistical mechanics. The earliest work on the theory of such solutions was based on empirical equations, such as those of van der Waals,45 of Keyes,44 and of Beattie and Bridgemann.3 However, the only equation of state of a gas mixture that can be derived rigorously is the virial expansion,46 66... 

One major question of interest is how much asphaltene will flocculate out under certain conditions. Since the system under study consist generally of a mixture of oil, aromatics, resins, and asphaltenes it may be possible to consider each of the constituents of this system as a continuous or discrete mixture (depending on the number of its components) interacting with each other as pseudo-pure-components. The theory of continuous mixtures (24), and the statistical mechanical theory of monomer/polymer solutions, and the theory of colloidal aggregations and solutions are utilized in our laboratories to analyze and predict the phase behavior and other properties of this system. 

Leland and Co-workers (8-10) have been able to re-derive the van der Waals mixing rules with the use of statistical mechanical theory of radial distribution functions. According to these investigators, for a fluid mixture with a pair intermolecular potential energy function,... 

The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

After the seminal work of Guggenheim on the quasichemical approximation of the lattice statistical-mechanical theory[l], various practical thermodynamic models such as excess Gibbs energies[2-3] and equations of state[4-5] were proposed. However, the quasichemical approximation of the Guggenheim combinatory yields exact solution only for pure fluid systems. Therefore one has to resort to numerical procedures to find the solution that is analytically applicable to real mixtures. Thus, in this study we present a new unified group contribution equation of state[GC-EOS] which is applicable for both pure or mixed state fluids with emphasis on the high pressure systems[6,7]. 

There have already appeared artides reviewing the progress achieved in the fields of intermolecular forces, methods of statistical mechanics, and the structures of gases, liqiuds, and their molecular mixtures, including the critical region. De Voe reviews the theory of the conformations of biological macromolecules in solution, and Yomosa that of charge transfer in the molecular compounds of biolo cal systems. 

Thermodynamic effects of directional forces in liquid mixtures.— The theory applied to pure liquids in the last two sections can be generalized to liquid mixtures and can be used to discuss the effects of directional forces on the thermodynamic functions of mixing. Classical statistical mechanics leads to a complete expression for the free energy of a multicomponent system in terms of the intermolecular energies Ust for all pairs of components s and t. Each Ust can be expanded in the general manner (2.1), so that it is separated into a spherically symmetric part and various directional terms. 

Most theoretical procedures for deriving expressions for AG iix start with the construction of a model of the mixture. The model is then analyzed by the techniques of statistical thermodynamics. The nature and sophistication of different models vary depending on the level of the statistical mechanical approach and the seriousness of the mathematical approximations that are invariably introduced into the calculation. The immensely popular Flory-Huggins theory, which was developed in the early 1940s, is based on the pseudolattice model and a rather low-level statistical treatment with many approximations. The theory is remarkably simple, explains correctly (at least qualitatively) a large number of experimental observations, and serves as a starting point for many more sophisticated theories. 

All three areas will be addressed here. The application of classical density functional theory has led to some of the most important recent theoretical advances in SFE and these have been the subject of several authoritative review articles [10-16]. On the other hand, we know of no recent comprehensive review addressing theoretical approaches other than density functional theories (DFT) and the other two subject areas, particularly the last one, and it was this that motivated us to write this chapter. We hope that the somewhat broader coverage of molecular modeling research in SFE given in this chapter will be of benefit to researchers new to the field. We should mention that this Chapter is written from a perspective that is more strongly influenced by liquid-state statistical mechanics than by solid-state theory. The interests of the authors in the problem at hand are an outgrowth of their previous work on phase equilibrium in fluids and fluid mixtures. 

The great appeal of the virial equations derives from their interpretations in terms of molecular theory. Virial coefficients can he calculated from potential fonctions describing interactions among moleculas. More importantly, statistical mechanics provides rigorous expressions for the composition dependeace of ihe virial coefficients. Thus, the nth virial coefficient of a mixture is nth order in the mole fractions ... 

We will consider only one additional activity coefficient equation here, the UNI-QUAC (universal quasichemical) model of Abrams and Prausnitz. This model, based on statistical mechanical theory, allows local compositions to result from both the size and energy differences between the molecules in the mixture. The result is the expression... 

Most of the recent theories of liquid solution behavior have been based on well-defined thermodynamic or statistical mechanical assumptions, so that the parameters that appear can be related to the molecular properties of the species in the mixture, and the resulting models have some predictive ability. Although a detailed study of the more fundamental approaches to liquid solution theory is beyond the scope of this book, we consider two examples here the theory of van Laar, which leads to regular solution theory and the UNIFAC group contribution model, which is based on the UNIQUAC model introduced in the previous section. Both regular solution theory and the UNIFAC model are useful for estimating solution behavior in the absence of experimental data. However, neither one is considered sufficiently accurate for the design of a chemical process. 

Another theory for has been proposed by Adelman and Chen. Their approach differs from the theories described above in that they do not rely completely on statistical mechanical methods, but attempt to solve generalized Poisson-Boltzmann equations for ion-solvent mixtures. The approximation obtained predicts that increases with p+ for all con-... 

The mathematical and physical theory of equilibrium cooperative phenomena in crystals has been reviewed by Newell and Montroll, and Domb, and the basic statistical mechanics is reviewed in Hill s monograph. Rowlinson has given a very thorough discussion of the classical thermodynamics of the coexistence curve and the critical region, and has also appraised much of the better data on equUibrium properties (of liquids and hquid mixtures). Rice > has several times reviewed the field of critical phenomena. 

Every serious student of fluids will own a copy of Rowlinson s book on liquids and liquid mixtures, and there is no warrant for a repetition of his scholarly and lucid exposition of the classical thermodynamics of the critical state. But a few of the important points must be brought to mind. Consider the classical isotherm portrayed in Fig. 1. The solid fine represents the observed pressure of a system in its most stable state of volume V. Between points 1 and 4 the compressibility of the fluid is infinite, although approximate statistical-mechanical theories, when based on the canonical ensemble, give a loop between points 1 and 4 and... 

At first sight this looks like nothing more than a polynomial expansion of the ideal gas law. However, it turns out to have real physical significance, and the form of the equation follows directly from statistical mechanics. The details can be found in most textbooks on statistical mechanics (see, for example, Mayer and Mayer, 1940 Hill, 1960, Chapter 15). We will outline the underlying theory very briefly here because virial equations (or similar approaches) appear several times in this book—see for example the discussion of the Pitzer equations for the non-ideal properties of salt solutions in Chapter 17 and Chapter 16 on gas mixtures. 

We have so far described a statistical mechanics of molecular liquids, implying that a system includes only one chemical species. However, in ordinary chemistry, a system contains more than one component, and major and minor components in the mixture are conventionally called solvent and solute , respectively. The vanishing limit of solute concentration, or infinite dilution, is of particular interest because it purely reflects the nature of solute-solvent interactions. The word solvation is most commonly used for describing properties concerning solute-solvent interactions at the infinite dilution limit. Here, we provide a brief outline of the way to obtain solvation properties, solvation structure and thermodynamics, from the RISM theory described in the previous sections [3]. It is straightforward to generalize the RISM equation to a mixture of different molecular species. The equation for a mixture can be written in a matrix notation as... 

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