Activity coefficient of a solute

The variable x is usually the mole fraction of the components. The last expression was first introduced by Guggenheim [5]. Equation (3.60) is a particular case of the considerably more general Taylor series representation of Y as shown by Lupis [6]. Let us apply a Taylor series to the activity coefficient of a solute in a dilute binary solution ... 

Worked Example 7.16 What is the activity coefficient of a solution of CuSC>4 of concentration 10 2 mol dm 3 ... 

The weight fraction activity coefficient of a solute i in a polymer is then calculated from the activity by dividing by its weight fraction in a manner analogous to mole fraction activity coefficients ... 

Equation [6.3] can be used for SI units (molm ), molarity (molL ) or molality (molkg ). In all cases, y is a dimensionless term, since a and [C] are expressed in the same units. The activity coefficient of a solute is effectively unity in dilute solution, decreasing as the solute concentration increases (Table 6.1). At high concentrations of certain ionic solutes, y may increase to become greater than unity. 

The theoretical expressions based on the Debye-Hiickel limiting law together with more empirical expressions are given in Table 3.3. In defining the mean activity coefficient of a solute, in the equations of Table 3.3 should be replaced by z+z, where the charges are taken without regard to sign. 

A similar procedure can be employed for the prediction of the activity coefficient of a solute in any ideal re-component mixed solvent, namely, (1) the re-component ideal mixed solvent can be represented by two (re — l)-component ideal mixed solvents or by one pure solvent and a (re — l)-component ideal mixed solvent, (2) the (re — l)-component ideal mixed solvent can be represented by two (re — 2)-component ideal mixed solvents or by one pure solvent and a (re — 2)-component ideal mixed solvent, and so on. 

One can see from eqs 17 and 20 that the activity coefficient of a solute at infinite dilution in an ideal ternary mixed solvent (yg ) can be calculated in terms of the activity coefficients of that solute at infinite dilution in any two binaries of the solvent and their molar volumes or in terms of the activity coefficients of the solute at infinite dilution in one binary solvent and in the remaining individual solvent and their molar volumes. The activity coefficients of a solute at infinite dilution in a binary mixed solvent can be obtained experimentally or calculated. For instance, they can be calculated using eq 9 and, in this case, the activity coefficient of a solute at infinite dilution in an ideal ternary mixed solvent (yg ) can be predicted from the... 

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... 

On the basis of the fluctuation theory, the following expression for the derivative of the activity coefficient of a solute iY2t) in a water (l)-solute (2) -cosolvent (3) mixture can be derived [19], which is valid for any kinds of solutes and cosolvents ... 

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. 

The activity coefficients of a solute in a mixed solvent could be also calculated by employing various well-known phase equilibria models, such as the Wilson, NRTL, Margules, etc., which using information for binary subsystems could predict the activity coefficients in ternary mixtures (Fan and Jafvert, 1997 Domanska, 1990). 

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 of a solute in a mixed solvent at infinite dilution... 

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. 

In the present paper, an equation for the activity coefficient of a solute at infinite dilution in a nonideal mixed solvent is used to derive expressions for its solubility in a nonideal binary mixed solvent. 

The paper is organized as follows first, an equation for the activity coefficient of a solute at infinite dilution in a binary nonideal mixed solvent (Ruckenstein and Shulgin, 2003) is employed to derive an expression for its solubility in terms of the properties of the mixed solvent. Second, various expressions for the activity coefficients of the cosolvents are inserted into the above equation. Finally, the obtained equations are used to correlate the drug solubilities in binary aqueous mixed solvents and the results are compared with experimental data and other models available in the literature. 

In contrast to previous papers (Ruckenstein and Shulgin, 2003a-d), the solubility of the drug in a binary solvent is considered to be finite, and the infinite dilution approximation is replaced by a more realistic one, the dilute solution approximation. An expression for the activity coefficient of a solute at low concentrations in a binary solvent was derived by combining the fluctuation theory of solutions (Kirkwood and Buff, 1951) with the dilute approximation. This procedure allowed one to relate the activity coefficient of a solute forming a dilute solution in a binary solvent to the solvent properties and some parameters characterizing the nonidealities of the various pairs of the ternary mixture. 

In this paper, a previously developed expression for the activity coefficient of a solute at infinite dilution in multi-component solutions [22—24) will be applied to the solubility of environmentally significantcompounds in aqueous solvent mixtu res. The above expression for the activity coefficient of a solute at infinite dilution in multicomponent solutions [22— 24) is based on the fluctuation theory of solutions [25). This model-free thermodynamic expression can be applied to both binary and multicomponent solvents. 

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