October 27, 2026
Van der Waals' Two Corrections
The ideal gas law treats molecules as points that never attract each other. That works at low pressure, but fails badly near condensation and at high pressure. In his 1873 doctoral thesis, Johannes van der Waals made two physical corrections. Molecules take up space, so the volume available to move in is reduced by a constant b. Molecules also attract each other, which lowers the pressure they exert on the walls by a term a/V². The result is (P + a/V²)(V − b) = RT.
The striking thing about the equation is that it predicts a phase transition. Below a certain temperature, its isotherms develop a loop, and the same equation describes both a dense liquid and a dilute vapour. Above that temperature the loop disappears, matching the existence of a critical point. Fitting a and b to the critical temperature and pressure gives a single equation that captures gas-liquid behaviour qualitatively, which earned van der Waals the 1910 Nobel Prize in Physics.
Three Roots and Which One to Use
Expanded, the van der Waals equation is cubic in volume, as are all its descendants. At a given temperature and pressure inside the two-phase region, it has three real roots. The smallest is the liquid volume, the largest is the vapour volume, and the middle one is physically meaningless, lying on the unstable part of the loop. Outside the two-phase region only one real root exists.
Which phase is actually stable at a given pressure is decided by fugacity: liquid and vapour coexist at the pressure where their fugacities are equal, which is the cubic-equation version of Maxwell's equal-area construction. This is exactly what a simulator does when it computes a vapour pressure or a flash from an equation of state, and it is why cubic equations are so convenient. A cubic has an analytical solution, so the roots are fast and reliable to find.
Redlich-Kwong, Soave, and Peng-Robinson
Quantitatively, van der Waals is poor. It predicts a critical compressibility factor Z_c of 0.375, while real fluids mostly fall between about 0.23 and 0.31. In 1949 Redlich and Kwong modified the attraction term, adding a temperature dependence and a different volume dependence, and greatly improved the description of gases. Their equation still predicted liquid-phase properties and vapour pressures poorly.
Giorgio Soave's 1972 modification fixed the vapour-pressure problem by making the attraction parameter a function of temperature through Pitzer's acentric factor, a measure of how non-spherical a molecule is. The Soave-Redlich-Kwong equation reproduced hydrocarbon vapour pressures well and quickly became an industry standard. In 1976 Ding-Yu Peng and Donald Robinson changed the volume dependence of the attraction term again, lowering the predicted Z_c to about 0.307 and improving liquid densities. SRK and Peng-Robinson remain the default property methods for hydrocarbon and gas processing work today.
Where Cubic Equations Run Out
Even Peng-Robinson typically misses liquid densities by several percent, because two constants cannot capture both the critical point and liquid packing accurately. Volume-translation corrections, which shift the computed volume by a constant without changing the phase equilibrium, are a common fix when densities matter.
The bigger limitation is with polar and hydrogen-bonding molecules like water, alcohols, and acids, whose association is not described by a simple attraction term. For such mixtures, simulators use activity-coefficient models such as NRTL or Wilson for the liquid phase, or switch to more advanced equations like CPA and SAFT that model association explicitly. Mixtures handled with cubic equations also need binary interaction parameters, k_ij, fitted to experimental data, and the quality of a simulation often depends more on those parameters than on the choice of equation.
References
- van der Waals, J. D. (1873). Over de Continuïteit van den Gas- en Vloeistoftoestand (doctoral thesis). Leiden: University of Leiden.
- Redlich, O., & Kwong, J. N. S. (1949). On the thermodynamics of solutions. V. An equation of state. Fugacities of gaseous solutions. Chemical Reviews, 44(1), 233–244. doi:10.1021/cr60137a013
- Soave, G. (1972). Equilibrium constants from a modified Redlich-Kwong equation of state. Chemical Engineering Science, 27(6), 1197–1203. doi:10.1016/0009-2509(72)80096-4
- Peng, D.-Y., & Robinson, D. B. (1976). A new two-constant equation of state. Industrial & Engineering Chemistry Fundamentals, 15(1), 59–64. doi:10.1021/i160057a011
- Smith, J. M., Van Ness, H. C., Abbott, M. M., & Swihart, M. T. (2018). Introduction to Chemical Engineering Thermodynamics (8th ed.), Chapter 3. New York: McGraw-Hill.