pure gaseous phase and pure liquid phase where no useful pressure data, except saturation pressure, are available. If only data in the gaseous phase were analyzed, the third virial coefficient becomes very small, because the PρT isotherm is almost linear in the gaseous phase. However, if data points in the pure liquid phase are included, a second-order regression would give a large third virial coefficient. The third virial coefficient thus derived would increase monotonically as the temperature is lowered from the critical point to the triple point. The expectation that the third virial coefficient is a monotonically increasing function of can be verified with equations of state which accurately predicted the PρT isotherms in the saturation region where gaseous and liquid phases coexist. Casting equations of the state into virial form Most equations of state can be reformulated and cast in virial equations to evaluate and compare their implicit second and third virial coefficients. The seminal Van der Waals equation of state was proposed in 1873: where is molar volume. It can be rearranged by expanding into a Taylor series: In the Van der Waals equation, the second virial coefficient has roughly the correct behavior, as it decreases monotonically when the temperature is lowered. The third and higher virial coefficients are independent of temperature, and are not correct, especially at low temperatures. Almost all subsequent equations of state derived from the Van der Waals equation, like those from Dieterici, Berthelot, Redlich-Kwong, Peng-Robinson, etc., suffer from the singularity introduced by 1/(v - b). Hence they cannot represent accurately the PρT isotherms at temperatures below critical temperature. Many of them produce adequate second virial coefficients, but most give incorrect third virial coefficients . Other equations of state, started by Beattie-Bridgeman, however, are more closely related to virial equations, and show to be more accurate in representing behavior of fluids in both gaseous and liquid phases. They can be reformulated into virial equations of state, and compared with one another. The Beattie-Bridgeman equation of state, proposed in 1928, where can be rearranged: This equation of state represents well the second virial coefficient. However, the third virial coefficient has the wrong sign. Thus, it fails to represent isotherms close to and below the critical temperature. The Benedict-Webb-Rubin equation of state of 1940 represents better isotherms below the critical temperature: More improvements are achieved by Starling in 1972: Following are plots of reduced second and