Activity coefficients in solid solutions

Most of these complications are considered in some detail by Spear (1995, Chapter 7). 

Both these alternatives have drawbacks, but are very commonly used in the Earth sciences, where complex solutions at high temperatures and pressures are commonly of interest. The problem is that determining activity coefficients is difficult, and for the most part confined to binary and ternary systems. In solid solutions, it can be done in various ways, such as 

We will have a brief look at each of these methods. In the process, we will see the pervasive use of regular solution and Margules equations. 

After equilibrating the phases at T and P, the garnet is analyzed, and the ideal Raoultian activity of Ca3Al2Si30,2 is calculated from the thermodynamic mole reaction as in Equations (14.8), i.e.. 

This study has been included partly because it illustrates the usefulness of the buffering concept, which is the basis of the method. The actual value of the interaction coefficients in garnets are still a subject of some debate, as discussed by Newton and Haselton (1981). 

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Tao] Calculation Modeling of C activity coefficient in solid solutions... 

The level of impurity uptake can be considered to depend on the thermodynamics of the system as well as on the kinetics of crystal growth and incorporation of units in the growing crystal. The kinetics are mainly affected by the residence time which determines the supersaturation, by the stoichiometry (calcium over sulfate concentration ratio) and by growth retarding impurities. The thermodynamics are related to activity coefficients in the solution and the solid phase, complexation constants, solubility products and dimensions of the foreign ions compared to those of the ions of the host lattice [2,3,4]. 

The main difficulty in predicting the solid solubility in a mixed solvent consists in calculating the activity coefficient of a solute in a ternary mixture In this paper, the Kirkwood-Buff (KB) theory of solutions (or fluctuation theory) (Kirkwood and Buff, 1951) is employed to analyze the solid (particularly drug) solubility in mixed (mainly aqueous) solvents. The analysis is based on results obtained previously regarding the composition derivatives of the activity coefficients in ternary solutions (Ruckenstein and Shulgin, 2001). These equations were successfully applied to gas solubilities in mixed solvents (Ruckenstein and Shulgin, 2002 Shulgin and Ruckenstein, 2002). 

The activity coefficient has to be estimated for nonideal solutions. There is no general method for predicting activity coefficients of solid solutes in liquid solvents. For nonpolar solutes and solvents, however, a reasonable estimate can frequently be made with the regular solution theory, or the Scatchard-Hildebrand relation. 

The above procedure for determining activity coefficients for solid solutes is often applied to electrolytes. This important class of solutes always behaves non-ideally. The properties of electrolyte solutions are considered in detail in chapter 3. 

Satisfactory generalized equations for the calculation of activity coefficients in solid, liquid, and gaseous solutions under geological conditions will probably remain an important research goal for many years to come. 

Like activity coefficients in aqueous solutions, solid activity coefficients are concentration dependent. As more phosphate ions mix on the surface of Al(OH)3, for example, the hydroxide structure becomes more and more strained. The strain is re-... 

Thus, the solubility in a liquid of a solid at its melting point is equal to the reciprocal of its activity coefficient in the solute-solvent mixture. 

Generally, the activity coefficients are < 1 when polar interactions are important, with a resulting increase in solubility of compounds compared with the ideal solubility. The opposite is often true in nonpolar systems where dispersion forces are important, with the activity coefficients being > 1. A variety of methods are used to calculate activity coefficients of solid solutes in solution. A frequently used method is that of Scatchard-Hildebrand, which is also known as regular solution theory (Prausnitz et al. 1999). 

Phase relations in the system MgO-FeO-SiOj were first worked out in detail by Bowen and Schairer in a classic paper in 1935. Dozens, if not hundreds, of studies have been conducted on minerals and mineral assemblages in this system since then. A generalized subsolidus section is shown in Figure 14.5. There are three important solid solution series in this system, two of which, the orthopyroxenes and olivines, are common rock-forming minerals. To examine the determination of activity coefficients in these solutions we will use... 

Usually, however, the distribution coefficients determined experimentally are not equal to the ratios of the solubility product because the ratio of the activity coefficients of the constituents in the solid phase cannot be assumed to be equal to 1. Actually observed D values show that activity coefficients in the solid phase may differ markedly from 1. Let us consider, for example, the coprecipitation of MnC03 in calcite. Assuming that the ratio of the activity coefficients in the aqueous solution is close to unity, the equilibrium distribution may be formulated as (cf. Eq. A.6.11)... 

By examining the compositional dependence of the equilibrium constant, the provisional thermodynamic properties of the solid solutions can be determined. Activity coefficients for solid phase components may be derived from an application of the Gibbs-Duhem equation to the measured compositional dependence of the equilibrium constant in binary solid solutions (10). 

Other references in Table in discuss applications in precipitation of metal.compounds, gaseous reduction of metals from solution, equilibrium of copper in solvent extraction, electrolyte purification and solid-liquid equilibria in concentrated salt solutions. The papers by Cognet and Renon (25) and Vega and Funk (59) stand out as recent studies in which rational approaches have been used for estimating ionic activity coefficients. In general, however, few of the studies are based on the more recent developments in ionic activity coefficients. 

When = 0.90 this gives x2 = xCd = 3.1(10-4) while Eq. (A14) gives Xj = xHg = 0.9922. This is in agreement with the more exact computer calculations whose results are shown in Fig. 31. The activity coefficients in the metal-rich liquid in equilibrium with a solid solution with = 0.9 and at 673°K obtained from a computer calculation also agree to within 3% with the approximate values listed above. Thus the relative stability of CdTe(s) compared to HgTe(s) is a major factor in the tie-lines converging toward the Hg corner. The smallness of Q2/k5, which is determined in part by the large value of 244 for the activity coefficient of the CdTe liquid species, also enters and is less transparent. 

The case of binary solid-liquid equilibrium permits one to focus on liquid-phase nonidealities because the activity coefficient of solid component ij, Yjj, equals unity. Aselage et al. (148) investigated the liquid-solution behavior in the well-characterized Ga-Sb and In-Sb systems. The availability of a thermodynamically consistent data base (measurements of liquidus, component activity, and enthalpy of mixing) provided the opportunity to examine a variety of solution models. Little difference was found among seven models in their ability to fit the combined data base, although asymmetric models are expected to perform better in some systems. 

The quantities in brackets represent activies of ions in solution and of components in the solid phase. Application of the defining equation directly would require knowing activity coefficients for the ions in solution and also the Henry s law coefficient for the trace carbonate in solid solution. A practical approach is to rewrite equation (13) in terms of an effective or empirical distribution coefficient... 

Solubility of a Solid. For the solubilities of poorly soluble crystalline nonelectrolytes in a multicomponent mixed solvent, one can use the infinite-dilution approximation and consider that the activity coefficient of a solute in a mixed solvent is equal to the activity coefficient at infinite dilution. Therefore, one can write the following relations for the solubility of a poorly soluble crystalline nonelectrolyte in a ternary mixed solvent and in two of its binaries i2,i3... 

The present paper deals with the application of the fluctuation theory of solutions to the solubility of poorly soluble drugs in aqueous mixed solvents. The fluctuation theory of ternary solutions is first used to derive an expression for the activity coefficient of a solute at infinite dilution in an ideal mixed solvent and, further, to obtain an equation for the solubility of a poorly soluble solid in an ideal mixed solvent. Finally, this equation is adapted to the solubility of poorly soluble drugs in aqueous mixed solvents by treating the molar volume of the mixed solvent as nonideal and including one adjustable parameter in its expression. The obtained expression was applied to 32 experimental data sets and the results were compared with the three parameter equations available in the literature. 

As in a previous paper [Int. J. Pharm. 258 (2003) 193-201], the Kirkwood-Buff theory of solutions was employed to calculate the solubility of a solid in mixed solvents. Whereas in the former paper the binary solvent was assumed ideal, in the present one it was considered nonideal. A rigorous expression for the activity coefficient of a solute at infinite dilution in a mixed solvent [Int. J. Pharm. 258 (2003) 193-201] was used to obtain an equation for the solubility of a poorly soluble solid in a nonideal mixed solvent in terms of the solubilities of the solute in the individual solvents, the molar volumes of those solvents, and the activity coefficients of the components of the mixed solvent. 

In a previous paper (Ruckenstein and Shulgin, 2003), the Kirkwood-Buff theory of solutions (Kirkwood and Buff, 1951) was employed to obtain an expression for the solubility of a solid (particularly a drug) in binary mixed (mainly aqueous) solvents. A rigorous expression for the composition derivative of the activity coefficient of a solute in a ternary solution (Ruckenstein and Shulgin, 2001) was used to derive an equation for the activity coefficient of the solute at infinite dilution in an ideal binary mixed solvent and further for the solubility of a poorly soluble solid. By considering that the excess volume of the mixed solvent depends on composition, the above equation was modified empirically by including one adjustable parameter. The modified equation was compared with the other three-parameter equations available in the literature to conclude that it provided a better agreement. 

K2 are the activity coefficients of the solid solute in its samrated solutions in the cosolvent, water, and mixed solvent, respectively P) is the hypothetical fu-gacity of a solid as a (subcooled) liquid at a given pressure (P) and temperature (T) / is the fugac-ity of the pure solid component 2 and y indicates that the activity coefficients of the solute depend on composition. If the solubilities of the pure and mixed solvents in the solid phase are negligible, then the left hand sides of Eqs. (l)-(3) depend only on the properties of the solute. 

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