Mixing ideal, binary, ternary multicomponents

The prediction of the solubility of poorly soluble substances of environmental significance in multicomponent (ternary and higher) aqueous mixed solvents is a difficult task because it requires the knowledge of the activity coefficient of a solute in a multicomponent mixed solvent. The method most often used for the solubility of a solid in ternary and multicomponent mixed solvents is the combined nearly ideal binary solvent/Redlich - Kister equation (33). That equation was applied to the solubility of a solid in ternary... 

The present paper is concerned with mixtures composed of a highly nonideal solute and a multicomponent ideal solvent. A model-free methodology, based on the Kirkwood—Buff (KB) theory of solutions, was employed. The quaternary mixture was considered as an example, and the full set of expressions for the derivatives of the chemical potentials with respect to the number of particles, the partial molar volumes, and the isothermal compressibility were derived on the basis of the KB theory of solutions. Further, the expressions for the derivatives of the activity coefficients were applied to quaternary mixtures composed of a solute and an ideal ternary solvent. It was shown that the activity coefBcient of a solute at infinite dilution in an ideal ternary solvent can be predicted in terms of the activity coefBcients of the solute at infinite dilution in subsystems (solute + the individual three solvents, or solute + two binaries among the solvent species). The methodology could be extended to a system formed of a solute + a multicomponent ideal mixed solvent. The obtained equations were used to predict the gas solubilities and the solubilities of crystalline nonelectrolytes in multicomponent ideal mixed solvents. Good agreement between the predicted and experimental solubilities was obtained. 

First, a rigorous expression for the activity coefficient of a solid solute at infinite dilution in an ideal multicomponent solvent was derived using the fluctuation theory of solution. Second, the obtained expression was used to express the solubility of a poorly soluble solid in an ideal multicomponent solvent in terms of the solubilities of this solid in two subsystems of the multicomponent solvent and their molar volumes. Finally, the developed procedure was used to predict the drug solubilities in ternary and quaternary aqueous mixed solvents using the drug solubilities in the constituent binary aqueous mixed solvents. The predicted solubilities were compared with the experimental ones and good agreement was found. 

The paper is organized as follows first, the thermodynamic relations for the solubility of poorly soluble solids in pure and multicomponent mixed solvents are written. Second, an equation for the activity coefficient of a solute at infinite dilution in a binary nonideal mixed solvent [23) is employed to derive an expression for its solubility in terms of the properties of the mixed solvent. Third, various expressions for the activity coefficients of the cosolvents, such as Margules and Wilson equations [19), are inserted into the above equation for the solubility. The obtained equations are used to correlate the HOP solubilities in binary aqueous mixed solvents and the results are compared with experiment. Finally, the case of an ideal multicomponent solvent is considered and used for ternary and higher mixed solvents. 

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