Van der Waals equation of state

The van der Waals equation of state, developed by Johannes Diderik van der Waals, can be written as

$\left.p={\frac {nRT}{V-nb}}-a\left({\frac {n}{V}}\right)^{2}\right.$.

where:

• $p$ is the pressure,

• $V$ is the volume,

• $n$ is the number of moles,

• $T$ is the absolute temperature,

• $R$ is the molar gas constant; $R=N_{A}k_{B}$, with $N_{A}$ being the Avogadro constant and $k_{B}$ being the Boltzmann constant.

The van der Waals equation of state takes into account two features that are absent in the ideal gas equation of state; the parameter $b$ introduces somehow the repulsive behavior between pairs of molecules at short distances, it represents the minimum molar volume of the system, whereas $a$ measures the attractive interactions between the molecules. The van der Waals equation of state leads to a liquid-vapor equilibrium at low temperatures, with the corresponding critical point.

$a={\frac {27}{64}}{\frac {R^{2}T_{c}^{2}}{P_{c}}}$

$b={\frac {RT_{c}}{8P_{c}}}$

Critical point

The critical point for the van der Waals equation of state can be found at

$T_{c}={\frac {8a}{27bR}}$,

$p_{c}={\frac {a}{27b^{2}}}$

and at

$\left.v_{c}\right.=3b$.

Dimensionless formulation

If one takes the following quantities

${\tilde {p}}={\frac {p}{p_{c}}};~{\tilde {v}}={\frac {v}{v_{c}}};~{\tilde {t}}={\frac {T}{T_{c}}};$

one arrives at

${\tilde {p}}={\frac {8{\tilde {t}}}{3{\tilde {v}}-1}}-{\frac {3}{{\tilde {v}}^{2}}}$

The following image is a plot of the isotherms $T/T_{c}$ = 0.85, 0.90, 0.95, 1.0 and 1.05 (from bottom to top) for the Van der Waals equation of state:

Plot of the isotherms T/T_c = 0.85, 0.90, 0.95, 1.0 and 1.05 for the Van der Waals equation of state

Interesting reading

• J. D. van der Waals "On the Continuity of the Gaseous and Liquid States", Dover Publications ISBN: 0486495930
• Johannes Diderik van der Waals "The Equation of State for Gases and Liquids", Nobel Lecture, December 12, 1910
• Luis Gonzalez MacDowell and Peter Virnau "El integrante lazo de Van der Waals", Anales de la Real Sociedad Española de Química 101 #1 pp. 19-30 (2005)