Closed-form answer to the molar volume of cubic equation of state.

The complete analytical solutions to the roots of the cubic function (of one independent variable) are tabulated. For an equation of state in which the intermolecular attractive forces are described by quadratic trinomials, the derived analytical expressions for the roots of the cubic equation can be adopted to obtain the molar volumes (liquid and vapour) at a given state of temperature and pressure. The molar volumes calculated in this work are compared with the experimental data as well as other equations of state. The Guevara-Rodríguez equation of state describes the experimental data better than the other models. The equation of state has three unequal real roots of the molar volume under the extremely low pressure and temperature. However, the smallest positive root is taken as the molar volume of the liquid phase. The equation of state for the ideal gas (the ideal-gas law) is applicable only for the pressure p ≤ 20,000,000 Pa and temperature T ≤ 253 K. Compared with the experimental values, the Guevara-Rodríguez equation of state for any substance, which includes the gas, liquid and solid, is appropriate for the extra low pressure 0.000000356 ≤ p ≤ 114,000 Pa and temperature 87.8 ≤ T ≤ 266.3 K.