Taking Another Look at the van der Waals Equation of State–Almost 150 Years Later

The van der Waals (vdW) equation of state has long fascinated researchers and engineers, largely because of its simplicity and engineering flexibility. At the same time as vdW and other equations of state (EoS) have been proposed, at first mostly used for petrochemical applications, solution theories in the form of activity coefficient models have been developed, focusing on the accurate representation of the liquid phase, of interest to the chemical industry. For both the van der Waals (and other cubic EoS) and the activity coefficient models we can identify size and energy terms, although different terminologies have been used (e.g., combinatorial-free volume, entropic, excess entropy or repulsive terms on one side and "residual", energetic, excess energy or attractive contributions on the other side). These definitions are not necessarily identical, as it will be shown here, and the identification of distinct separable contributions in thermodynamic models is not always straightforward. Moreover, the different traditions lead sometimes to confusion as to the actual range of applicability of these models, and it may be useful to consider them (EoS and activity coefficient models) together. While much has been written about the van der Waals equation of state, a particular insight is obtained when the model is expressed in terms of excess Gibbs energy and activity coefficient expressions. We show that such a transformation, analysis of the distinct size and energy terms of vdW and comparison to classical solution theories, provides insight into the physical meaning, capabilities, and limitations of the model, including the associated mixing and combining rules used. We will discuss how this analysis of the vdW and subsequent equations of state, such as the Soave-Redlich-Kwong and Peng-Robinson, have enhanced our understanding of the actual applicability range of the models in terms of size effects or excess properties. The analysis reveals that the capabilities of the vdW and in general of the vdW-type cubic equations of state are possibly more significant than traditionally considered, and that the task of more advanced models in trying to "beat them (the cubic EoS)", may be more difficult than previously anticipated.