Note on the calculation of the second osmotic virial coefficient in stable and metastable liquid states

The second osmotic virial coefficient is calculated from analytical equations of state as illustrated with the van der Waals two-component equation. It is shown that when the fixed solvent chemical potential or pressure at which the virial coefficient is calculated is taken to be that of the pure solvent in coexistence with its vapor, as in a recent report, the liquid solution is in a metastable state. When, by contrast, that fixed chemical potential or pressure is that of the pure solvent in its one-phase liquid state, the solution, with increasing solute concentration, is initially in a stable state; then, on crossing the liquid-vapor equilibrium line, it becomes metastable and ultimately approaches a spinodal and incipient instability. Nevertheless, in practice, as seen in a numerical illustration for a hydrocarbon dissolved in water, there is scarcely any difference in the virial coefficient calculated with the fixed solvent chemical potential or pressure of the pure solvent at its vapor pressure (metastable states of the solution) or at 1 bar (initially stable states). It is also seen in that example that the virial coefficient may be reliably calculated only for solute concentrations that are neither too small nor too large; typically only for mole fractions roughly from 10(-7) to 10(-3.5).