(145c) 2-Component Van Der Waals EoS and the Second Osmotic Virial Coefficient

The second osmotic virial coefficient of a solute in a solvent is a measure of the effective solvent-mediated solute-solute interactions.

Normally the properties of a two-component mixture are known from an equation of state in the form p=p(d1,d2,T) where d1 and d2 are the densities of the two components, but this contains the osmotic virial coefficients only implicitly. To obtain the osmotic coefficients requires transforming the equation of state from p=p(d1,d2,T) to z2=z2(z1,d2,T), where z1 and z2 are the activities of the solvent and solute, respectively. Then by the McMillan-Mayer solution theory, the second osmotic virial coefficient is related to and may be identified from the coefficient of the second-order term in the expansion of z2 in powers of d2 at fixed z1 and T. It is shown here how that transformation of the equation of state may be effected. The process is illustrated with the two-component van der Waals equation of state applied to obtain the second osmotic virial coefficient of propane in water. The resulting virial coefficient is compared with that obtained by integrating the solute-solute pair correlation function in the limit of infinite dilution as found by molecular dynamics computer simulations.