Abstract

The quest for a generalized equation of state (EOS) for molecular fluids remains a central challenge bridging fundamental statistical mechanics and applied chemical engineering. The conceptual foundation for understanding fluid non-ideality was established more than a century ago by the van der Waals (vdW) theory, which identified excluded volume and intermolecular cohesion as the principal factors governing fluid phase behavior [1]. Despite well-known theoretical limitations, particularly in the treatment of short-range repulsion, vdW-type cubic equations of state (CEOS), such as the Soave-Redlich-Kwong and Peng-Robinson models, remain the standard industrial tools for process simulation [2,3]. Their continuing success is largely due to their mathematical simplicity, computational robustness, and effective empirical parameterization of attractive interactions [2].

In contrast, modern statistical mechanics seeks to establish a rigorous connection between macroscopic thermodynamic behavior and microscopic intermolecular forces through thermodynamic perturbation theory (TPT). Traditional TPT approaches employ a purely repulsive hard-sphere (HS) reference system and therefore require higher-order perturbation terms to achieve quantitative accuracy. Recent developments, however, demonstrate the advantages of non-HS reference systems, in which the short-range attractive contribution is incorporated directly into the reference state. Such range-based decompositions substantially reduce the perturbation magnitude and restore the viability of highly accurate analytical first-order TPT descriptions [4,5].

At the same time, translating microscopically rigorous EOS models into general engineering practice remains challenging. Advanced non-cubic EOS formulations, including Statistical Associating Fluid Theory (SAFT)-type approaches, provide improved predictive capabilities for complex and associating fluids, but may also exhibit non-physical mathematical artifacts, such as fictitious critical points or anomalous multiple-volume roots, which complicate reliable industrial implementation [6]. In parallel, alternative interpretations of supercritical fluid behavior based on higher-order percolation transitions have recently been proposed, suggesting the existence of a mesophase separating liquid-like and gas-like states [7].

This lecture will review the historical evolution of EOS development and analyze the structural trade-offs between empirical simplicity and theoretical rigor. Particular emphasis will be placed on recent non-HS perturbation approaches developed in our work, which aim to reconcile molecular realism with analytical tractability. By combining modern molecular simulation results, range-separated perturbation schemes, and rigorous analysis of mathematical artifacts, we outline a pathway toward a new generation of microscopically grounded EOS models suitable for both fundamental studies and industrial chemical process design [8].

References

[1] J.O. Valderrama, J. Supercritical Fluids, 55, 415 (2010). [2] G.M. Kontogeorgis, R. Privat and J.N. Jaubert, J. Chem. Eng. Data, 64, 4619 (2019). [3] S. Gupta, J.R. Elliott, A. Anderko, J. Crosthwaite, W.G. Chapman and C.T. Lira, Ind. Eng. Chem. Res., 62, 3394 (2023). [4] I. Nezbeda, R. Melnyk and A. Trokhymchuk, J. Supercritical Fluids, 55, 448 (2010). [5] A. Trokhymchuk, V. Hordiichuk, R. Melnyk and I. Nezbeda, arXiv:2604.25882 (2026). [6] N.M. Alsaifi and J.R. Elliott, Ind. Eng. Chem. Res., 61, 15661 (2022). [7] J.F. Maguire and L.V. Woodcock, J. Molec. Liq., 373, 121199 (2023). [8] J.M. Kontogeorgis, R. Dohrn, I.G. Economou, J.C. de Hemptinne, A. ten Kate, S. Kuitunen, M. Mooijer, L.F. Zilnik and V. Vesovic, Ind. Eng. Chem. Res., 60, 4987 (2021).