Thermodynamic Properties of the van der Waals Fluid

The van der Waals (vdW) theory of fluids is the first and simplest theory that takes into account interactions between the particles of a system that result in a phase transition versus temperature. Combined with Maxwell's construction, this mean-field theory predicts the conditions for equilibrium coexistence between the gas and liquid phases and the first-order transition between them. However, important properties of the vdW fluid have not been systematically investigated. Here we report a comprehensive study of these properties. Ambiguities about the physical interpretation of the Boyle temperature and the influence of the vdW molecular interactions on the pressure of the vdW gas are resolved. Thermodynamic variables and properties are formulated in reduced units that allow all properties to be expressed as laws of corresponding states that apply to all vdW fluids. Lekner's parametric solution for the vdW gas-liquid coexistence curve in the pressure-temperature plane and related thermodynamic properties [Am. J. Phys. 50, 161 (1982)] is explained and significantly extended. Hysteresis in the first-order transition temperature on heating and cooling is examined and the maximum degrees of superheating and supercooling determined. The latent heat of vaporization and the entropy change on crossing the coexistence curve are investigated. The temperature dependences of the isothermal compressibility, thermal expansion coefficient and heat capacity at constant pressure for a range of pressures above, at and below the critical pressure are systematically studied from numerical calculations including their critical behaviors and their discontinuities on crossing the coexistence curve. Joule-Thomson expansion of the vdW gas is investigated in detail and the pressure and temperature conditions for liquifying a vdW gas on passing through the throttle are determined.