Van der Waals equation of state

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

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

where:

• $p$ is the pressure,

• $V$ is the volume,

• $n$ is the number of moles,

• $T$ is the absolute temperature,

• $R$ is the so-called 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.

$a=\frac{27}{64}\frac{R^2{T}_c^2}{P_c}$

$b=\frac{R{T}_c}{8P_c}$

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

$v_c=3b$.

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)