The van der Waals equation of state around the van Laar point

Abstract

The discovery of van Laar that the van der Waals equation of state for binary mixtures can be solved analytically at the double point on the critical line, provided one introduces the geometric-mean condition on the interaction parameters, is used to obtain the explicit expression for the critical line for this case. The critical line is expressed as a function of the density variables, and the origin is shifted to this double point: the van Laar point. By doing so it is possible to show that this double point is also a tricritical point. This is different from the lattice gas, where the double point is always a tricritical point. Small deviations from this point in parameter space induce very different phase diagrams. The influence of excursions from the van Laar point is expressed as a function of the state variables. Both the k factor (the deviation from the geometric-mean rule) and the ‘‘asymmetry’’ coefficient e (the deviation from the crossing point) are introduced. The results are given in the form of polynomials in local coordinates in density space. Conditions under which the double point is maintained are given and the differences between the Scott and van Konynenburg classes II, III, IV, and IV* are explained.