On the osmotic and activity coefficients of Na2WO4(aq) and Na2MoO4(aq) at the temperature 298.15 K, and the relations between mean activity coefficients of solutes under isopiestic conditions

Abstract

Wu et al. (J. Chem. Thermodynamics1990, 22, 35) reported osmotic coefficients for Na2WO4(aq) and Na2MoO4(aq) at T = 298.15 K which were calculated from their isopiestic measurements. They analyzed their results with a variant of the Gibbs-Duhem equation, and from those calculations obtained mean activity coefficients for the two electrolytes. Osmotic coefficients for both have highly anomalous dependences on molality, but no mention of this was made by the authors. The osmotic coefficients exhibit double minima followed by an approach to nearly constant values above molalities m ≈ 2 mol·kg-1. These unusual features appear to be due to the onset of hydrolytic polymerization as the pHs of the solutions decrease as they are diluted, followed by depolymerization at low molalities. A detailed re-examination of these osmotic coefficients indicates that the mean activity coefficients of Wu et al. were incorrectly calculated and are too large by a factor of 1.5. In addition, their calculations involved integration of a function that diverges to -∞ as m→0, which makes values of the integral highly uncertain. Furthermore, the unusual molality dependences of the osmotic coefficient preclude any meaningful extrapolations to infinite dilution, as are require for the calculation of reliable activity coefficients. A detailed analysis is given of the functional relations between the mean activity coefficients of two different solutes in separate solutions at isopiestic equilibrium. It is shown that the integral which appears in one equation, in the form in which it is usually written, has an integrand that diverges to +∞ or -∞ as m→0 for solutes of different charge types. However, that integral can be recast into a different form which has no divergences.