phase under saturation pressure , and the saturation temperature . Under the saturation pressure, the liquid phase has a molar volume of , and the gaseous phase has a molar volume of . The corresponding molar densities are and . These are the saturation properties needed to compute second and third virial coefficients. A valid equation of state must produce an isotherm which crosses the horizontal line of at and , on . Under and , gas is in equilibrium with liquid. This means that the PρT isotherm has three roots at . The cubic virial equation of state at is: It can be rearranged as: The factor is the volume of saturated gas according to the ideal gas law, and can be given a unique name : In the saturation region, the cubic equation has three roots, and can be written alternatively as: which can be expanded as: is a volume of an unstable state between and . The cubic equations are identical. Therefore, from the linear terms in these equations, can be solved: From the quadratic terms, B can be solved: And from the cubic terms, C can be solved: Since , and have been tabulated for many fluids with as a parameter, B and C can be computed in the saturation region of these fluids. The results are generally in agreement with those computed from Benedict-Webb-Rubin and Starling equations of state. However, the accuracy in B and C are critically dependent on the measurements of and , which are very difficult to measure accurately at low temperatures. The measurement errors thus introduced into B and C should be considered when one compares the values thus derived with those derived from second order regression analysis of PρT isotherms. Gas-liquid-solid equilibrium The cubic virial equation of state accurately represents the gas-liquid equilibrium of most substance from the critical point down to the triple point, where solid phase starts to appear. It can be extended to account for the gas-liquid-solid equilibrium: In this equation, the first term represents the pressure generated by kinetic energy of molecules. The second term represents long-range bimolecular attraction. The third term represents short-range tri-molecular repulsion. The second term pulls the PVT isotherm down as volume is reduced, while the third term pushes the isotherm up. When the temperature is below the critical point, the PVT isotherm thus has an S-shaped bent which allows a liquid