  Inspired by the recent developments in the study of the thermodynamics of van
der Waals fluids via the theory of nonlinear conservation laws and the
description of phase transitions in terms of classical (dissipative) shock
waves, we propose a novel approach to the construction of multi-parameter
generalisations of the van der Waals model. The theory of integrable nonlinear
conservation laws still represents the inspiring framework. Starting from a
macroscopic approach, a four parameter family of integrable extended van der
Waals models is indeed constructed in such a way that the equation of state is
a solution to an integrable nonlinear conservation law linearisable by a
Cole-Hopf transformation. This family is further specified by the request that,
in regime of high temperature, far from the critical region, the extended model
reproduces asymptotically the standard van der Waals equation of state. We
provide a detailed comparison of our extended model with two notable empirical
models such as Peng-Robinson and Soave's modification of the Redlich-Kwong
equations of state. We show that our extended van der Waals equation of state
is compatible with both empirical models for a suitable choice of the free
parameters and can be viewed as a master interpolating equation. The present
approach also suggests that further generalisations can be obtained by
including the class of dispersive and viscous-dispersive nonlinear conservation
laws and could lead to a new type of thermodynamic phase transitions associated
to nonclassical and dispersive shock waves.
