Second osmotic virial coefficient from the two-component van der Waals equation of state

The second osmotic virial coefficient is in principle obtained from the second-order term in the expansion of the osmotic pressure Π or solute activity z(2) in powers of the solute density ρ(2) at fixed solvent activity z(1) and temperature T. It is remarked that the second-order terms in the analogous expansions at fixed pressure p or at liquid-vapor coexistence instead of at fixed z(1) also provide measures of the effective, solvent-mediated solute-solute interactions, but these are different measures. It is shown here how the function z(2)(ρ(2), z(1), T) required to obtain the second osmotic virial coefficient B from an expansion in ρ(2) may be obtained from an equation of state of the form p = p(ρ(1), ρ(2), T) with ρ(1) the solvent density, and also how the analogous coefficient B' in the fixed-p expansion may be so obtained. The magnitude of the difference B - B' is often much smaller than that of B and B' separately, so B' is sometimes an acceptable approximation to B. That is not true of the analogous coefficient in the expansion at liquid-vapor coexistence. These calculations are illustrated with the van der Waals two-component equation of state and applied to solutions of propane in water as an example.