Korea Institute of Civil Engineering and Building Technology (KICT) researchers have introduced a new cubic equation of state that could reshape how engineers predict the behavior of liquids and gases. The model offers a physical explanation for a mathematical structure used in chemical and petroleum engineering for more than 50 years, while also improving predictions of liquid density across a broad range of substances. The work, led by Dr. Jai-Yeop Lee of KICT’s Department of Environmental Research Division, connects modern thermodynamic modeling with an overlooked idea proposed in 1913 by Dutch physicist Hugo Tetrode.
Equations of state are mathematical tools used to describe how the volume, pressure, and temperature of a fluid are related. They are central to the design and operation of distillation columns, refrigeration systems, natural-gas processing facilities, pipelines, chemical reactors, and energy-conversion equipment. In practical engineering, the equations must predict both vapor and liquid behavior, including what happens near a fluid’s critical point, where the distinction between the two phases disappears. Even small errors in predicted liquid volume can affect equipment sizing, heat-transfer calculations, safety margins, and the efficiency of industrial processes.
Among the most widely used models are the Soave–Redlich–Kwong, or SRK, equation, the Peng–Robinson equation, and the Patel–Teja equation. These models belong to the family of cubic equations of state because their mathematical formulation produces a cubic equation in molar volume. Their relative simplicity allows engineers to solve them quickly, even when modeling large industrial systems containing many chemical components. Yet their success has depended partly on an empirical modification of the attractive-force term, usually written in a quadratic form. The structure works remarkably well, but its physical origin has remained uncertain.
Lee’s new approach proposes that the quadratic form is not simply a product of trial and error. Instead, it can be traced to a correction associated with molecular vibrations. Tetrode’s early twentieth-century concept treated a fluid not only as a collection of particles moving freely through space, but also as a system containing vibrating molecular oscillators. By incorporating this vibrational contribution into a cubic equation of state through a new parameter called d, the researchers derived a mathematical structure that provides a physical basis for the familiar attractive term.
The model is designed to satisfy three requirements at the same time. First, it must reduce correctly to the ideal-gas law when the density becomes sufficiently low, because gas molecules then behave as though they interact only weakly. Second, it must remain mathematically solvable as a cubic equation, preserving the computational convenience that made conventional equations of state so valuable to industry. Third, it must retain enough flexibility to reproduce the critical compressibility factor of each substance. The critical compressibility factor describes how much a real fluid deviates from ideal-gas behavior at its critical point, and it varies significantly among different chemical compounds.
The researchers tested the new equation against high-accuracy reference data from NIST REFPROP for 76 fluids. The collection included simple, weakly interacting gases such as argon and methane, as well as strongly interacting substances including water and ammonia. The comparison focused particularly on saturated-liquid volume, a demanding property because liquid density is strongly influenced by molecular attractions and packing effects. In a fully predictive mode, the new model required only basic critical properties for each substance and did not use an adjustable volume-translation correction, a common empirical modification applied to improve liquid-density predictions.
Under those conditions, the new equation produced an average saturated-liquid-volume error of 4.0 percent across the 76 fluids. That result compared with 4.6 percent for the volume-translated Patel–Teja model, 5.5 percent for the standard Patel–Teja equation, 7.2 percent for Peng–Robinson, and 13.7 percent for SRK. The difference is significant because the competing methods often rely on additional fitted corrections when accurate liquid properties are required. A model that delivers comparable or better accuracy without those extra adjustments could make thermodynamic calculations more transparent and easier to apply when experimental data are limited.
The parameter d also appeared to carry information about molecular interactions rather than functioning as a purely numerical fitting constant. Across the tested fluids, its values showed a strong correlation, with an R² of approximately 0.93, with an empirical constant used in vapor-pressure correlations. That constant has traditionally been useful in engineering calculations but has lacked a clear theoretical interpretation. The researchers also found that the parameter grouped the 76 fluids into four distinct chemical families. When scaled according to molecular size, the correction increased from weakly interacting argon toward strongly hydrogen-bonded water, suggesting that it reflects the intensity of intermolecular forces.
This result could be especially relevant to industries working with fluids whose properties are difficult to measure or model. Hydrogen, carbon dioxide, ammonia, and other substances are central to emerging energy and climate technologies, including hydrogen storage and transport, carbon capture and utilization, refrigeration, and clean-ammonia fuels. In such systems, engineers must often predict fluid behavior across wide ranges of pressure and temperature, including conditions where conventional models can become unreliable. A physically grounded cubic equation could help connect fundamental molecular behavior with the fast calculations needed for process simulation and equipment design.
The study does not eliminate the need for experimental validation, nor does it suggest that one equation can perfectly represent every fluid under every condition. However, it offers a new explanation for why a core feature of cubic equations of state works so effectively. By linking Tetrode’s vibrational correction to the quadratic attractive term used in modern models, the research provides a route toward equations that are both computationally practical and physically interpretable. Published in Chemical Engineering Science, the work may encourage a broader re-examination of empirical formulas that have guided industrial thermodynamics for decades.
Subject of Research:
A theoretically grounded cubic equation of state for predicting fluid behavior, including vapor–liquid properties and saturated-liquid volume.
Article Title:
A theoretically grounded cubic equation of state: justifying quadratic attractive terms via Tetrode’s vibrational correction
Web References:
Korea Institute of Civil Engineering and Building Technology (KICT): https://www.kict.re.kr/eng/
DOI: https://doi.org/10.1016/j.ces.2026.124562
References:
Chemical Engineering Science, DOI: 10.1016/j.ces.2026.124562
NIST REFPROP reference data
Image Credits:
Korea Institute of Civil Engineering and Building Technology
Keywords
cubic equation of state, thermodynamics, fluid density, molecular interactions, Tetrode correction, chemical engineering, petroleum engineering, liquid volume prediction, hydrogen, carbon capture, ammonia, refrigeration, process modeling
Tags: advancements in petroleum engineeringcubic equation of statedistillation column designequations of state in energy systemsfluid property modeling for pipelineshistorical development of equations of stateHugo Tetrode’s physics contributionsindustrial process safety marginsliquid and gas phase behavior predictionliquid density prediction across substancesphase transition near critical pointthermodynamic modeling in chemical engineering
