Integrable extended van der Waals model

TL;DR

This paper introduces a new four-parameter integrable extension of the van der Waals model, connecting nonlinear conservation laws with thermodynamics and phase transitions, and aligning well with empirical equations of state.

Contribution

The paper constructs a novel integrable multi-parameter van der Waals model using nonlinear conservation laws, compatible with empirical models and capable of describing complex phase transitions.

Findings

Extended model reproduces standard van der Waals at high temperatures.

Model aligns with Peng-Robinson and Redlich-Kwong equations.

Potential to describe nonclassical phase transitions.

Abstract

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.