Deriving Second Osmotic Virial Coefficients from Equations of State and from Experiment

The osmotic virial coefficients, which are measures of the effective interactions between solute molecules in dilute solution, may be obtained from expansions of the osmotic pressure or of the solute activity in powers of the solute concentration. In these expansions, the temperature is held fixed, and one additional constraint is imposed. When the additional constraint is that of fixed chemical potential of the solvent, the coefficient of the second-order term yields directly the second osmotic virial coefficient itself. Alternative constraints, such as fixed pressure, fixed solvent density, or the specification of liquid-vapor equilibrium, yield alternative measures of the solute-solute interaction, different from but related to the osmotic virial coefficient. These relations are summarized and, where new, are derived here. The coefficient in question may be calculated from equations of state in which the parameters have been obtained by fitting to other experimental properties. Alternatively, the coefficients may be calculated from direct experimental measurements of the deviations from Henry's law based on measurements of the activity of the solute in a coexisting gas phase. It is seen for propane in water as a test case that with the latter method, even with what appear to be the best available experimental data, there are still large uncertainties in the resulting second osmotic virial coefficient. With the former method, by contrast, the coefficient may be obtained with high numerical precision but then depends for its accuracy on the quality of the equation of state from which it is derived.