Margules expansion

For those dilute mixtures where the solute and the solvent are chemically very different, the activity coefficient of the solute soon becomes a function of solute mole fraction even when that mole fraction is small. That is, if solute and solvent are strongly dissimilar, the relations valid for an infinitely dilute solution rapidly become poor approximations as the concentration of solute rises. In such cases, it is necessary to relax the assumption (made by Krichevsky and Kasarnovsky) that at constant temperature the activity coefficient of the solute is a function of pressure but not of solute mole fraction. For those moderately dilute mixtures where the solute-solute interactions are very much different from the solute-solvent interactions, we can write the constant-pressure activity coefficients as Margules expansions in the mole fractions for the solvent (component 1), we write at constant temperature and at reference pressure Pr ... 

To illustrate such a calculation, Balder (Bl) considered a simple case wherein he assumed that the (symmetric) excess Gibbs energy of the ternary system is given by a two-suffix Margules expansion ... 

Since the Margules expansions represent a convergent power series in the mole fractions,8 they can be summed selectively to yield closed-form model equations for the adsorbate species activity coefficients. A variety of two-parameter models can be constructed in this way by imposing a constraint on the empirical coefficients in addition to the Gibbs-Duhem equation. For example, a simple interpolation equation that connects the two limiting values of f (f°° at infinite dilution and f = 1.0 in the Reference State) can be derived after imposing the scaling constraint... 

To see this point in detail, one can express the activity coefficients in Eq. 5.33 with a third-order Margules expansion, as in Eq. 5.18 20... 

These equations can be solved for the coefficients c, d 0, c, and d 0 in terms of the infinite-dilution activity coefficients (which are binary-system properties), but the solution will not be unique. Equation 5.48d, connecting the ternary second-order Margules expansion coefficients to the binary infinite-dilution activity coefficients, shows that a constant (say, c0) can be added to any c and c[Pg.202]

J. Grover, Chemical mixing in multicomponent solutions. An introduction to the use of Margules and other thermodynamics excess functions to represent nonideal behavior, pp. 67-97 in Thermodynamics in Geology, ed. by D. G. Fraser, D. Reidel, Dordrecht, The Netherlands, 1977. It follows from Eqs. 5.17 and 5.19 that, in general, In fA = Xjat and In fBA = Xjbj. Equation 5.20a is a special case of this relation for a third-order Margules expansion. 

The purpose of this paper is to review two thermodynamic models for calculating aqueous electrolyte properties and give examples of parameter evaluations to high temperatures and pressures as well as applications to solubility calculations. The first model [the ion-interaction model of Pitzer (1) and coworkers] has been discussed extensively elsewhere (1-4) and will be reviewed only briefly here, while more detail will be given for an alternate model using a Margules expansion as proposed by Pitzer and Simonson (5). 

The Margules expansion model has been tested on some ionic systems over very wide ranges of composition, but over limited ranges of temperature and pressure (33,34). In this study, the model is applied over a wider range of temperature and pressure, from 25-350 C and from 1 bar or saturation pressure to 1 kb. NaCl and KCl are major solute components in natural fluids and there are abundant experimental data from which their fit parameters can be evaluated. Models based on the ion-interaction ajiproach are available for NaCl(aq) and KCl(aq) (8,9), but these are accurate only to about 6 molal. Solubilities of NaCl and KCl in water, however, reach 12 and 20 m, respectively, at 350 C, and ionic strengths of NaCl-KCl-H20 solutions reach more than 30 m at this temperature (35). The objective of this study is to describe the thermodynamic properties, particularly the osmotic and activity coefficients, of NaCl(aq) and KCl(aq) to their respective saturation concentrations in binary salt-H20 mixtures and in ternary NaCl-KCl-H20 systems, and to apply the Margules expansion model to solubility calculations to 350 C. 

Figure 4. Solubilities of halite (NaCl) in water to 350°C. The curve represents values calculated using the Margules expansion model for activity coefficients (infinite dilution reference state), and standard state Gibbs energies for NaCl(aq) derived from the equations of Pitzer et al. to 300°C, and of Tanger and Helgeson above 300 C.

Using the Margules expansion for the chemical potential of a component in a liquid mixture (see Sec. 11-6), we may write... 

In this section, we consider a real mixture of two components. We describe Margules expansions for the logarithms of the activity coefficients and investigate relations between the coefficients in the expansions of log/, and log/2 imposed by the Gibbs-Duhem equation. 

Thus, the Margules expansion contains no constants or first-degree terms. The first non-zero term is the second-degree one. 

Still with the aim of having mathematical expressions for the representation of the solution, Redlich and Kister offered a representation that provides an expansion of the excess Gibbs energy, a pure-substance reference in the same state of segregation as the solution (reference (I)), the equivalent of ihe Margules expansion for the activity coefficients. For a two-component solution, the pol5momial expansion up to order m is written ... 

Let us take the Margules expansion, limited to the first non-zero term. In... 

This model, which yields excellent results for polar and non-polar molecular liquids, is especially well suited for the study of liquid/ vapor equilibrium and the equilibrium between two liquids that are not completely miscible. Regardless of the number of components of the solution, the application of this model only requires the knowledge of two adjustment parameters per binary system, which can be deduced from the solution. The model is so widely applicable that it actually contains a number of previously classic models such as the models put forward by Van Laar, Wilson, Renon et al. (the NRTL - Non Random Two Liquids -model), Scatchard and Hildebrand, Flory and Huggins as special cases. In addition, it lends a physical meaning to the first three coefficients P, 5 and , in the Margules expansion (equation [2.1]). 

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