The van der Waals equation: analytical and approximate solutions

The thermodynamic properties, enthalpy of vaporization, entropy, Helmholtz function, Gibbs function, but especially the heat capacity at constant volume of a van der Waals gas (and liquid) at the phase transition are examined in two different limit approximations. The first limit approximation is at the near-critical temperatures, i.e., for T/Tc→ 1, where Tc is the critical temperature, the other limit approximation is at the near-zero temperatures, T→ 0. In these limits, the analytical equations for liquid and gas concentrations at saturated conditions were obtained. Although the heat capacities at constant volume of a van der Waals gas and liquid do not depend on the volume, they have different values and their change during the phase transition was calculated. It should be noticed that for real substances the equations obtained at the near-zero temperature are only valid for T >  Ttriple   point and T ≪ Tc, which means that found equations can be used only for substances with Ttriple   point ≪ Tc.