Cubic equations of state

Cubic equations of state are a specific class of thermodynamic models for modeling the pressure of a gas as a function of temperature and density and which can be rewritten as a cubic function of the molar volume.

Equations of state are generally applied in the fields of physical chemistry and chemical engineering, particularly in the modeling of vapor–liquid equilibrium and chemical engineering process design.

The van der Waals equation of state may be written as

where ${\displaystyle T}$ is the absolute temperature, ${\displaystyle p}$ is the pressure, ${\displaystyle V_{\text{m}}}$ is the molar volume and ${\displaystyle R}$ is the universal gas constant. Note that ${\displaystyle V_{\text{m}}=V/n}$, where ${\displaystyle V}$ is the volume, and ${\displaystyle n=N/N_{\text{A}}}$, where ${\displaystyle n}$ is the number of moles, ${\displaystyle N}$ is the number of particles, and ${\displaystyle N_{\text{A}}}$ is the Avogadro constant. These definitions apply to all equations of state below as well.

Proposed in 1873, the van der Waals equation of state was one of the first to perform markedly better than the ideal gas law. In this equation, usually ${\displaystyle a}$ is called the attraction parameter and ${\displaystyle b}$ the repulsion parameter (or the effective molecular volume). While the equation is definitely superior to the ideal gas law and does predict the formation of a liquid phase, the agreement with experimental data for vapor-liquid equilibria is limited. The van der Waals equation is commonly referenced in textbooks and papers for historical and other reasons, but since its development other equations of only slightly greater complexity have been since developed, many of which are far more accurate.

The van der Waals equation may be considered as an ideal gas law which has been "improved" by the inclusion of two non-ideal contributions to the equation. Consider the van der Waals equation in the form

as compared to the ideal gas equation

The form of the van der Waals equation can be motivated as follows: