Van der Waals equation of state

The van der Waals equation of state, developed by Johannes Diderik van der Waals, 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.

Equation of state

The van der Waals equation of state can be written as

$\left(p+{\frac {an^{2}}{V^{2}}}\right)\left(V-nb\right)=nRT$

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.
• $a$ and $b$ are constants that introduce the effects of attraction and volume respectively and depend on the substance in question.

Critical point

At the critical point one has $\left.{\frac {\partial p}{\partial v}}\right|_{T=T_{c}}=0$, and $\left.{\frac {\partial ^{2}p}{\partial v^{2}}}\right|_{T=T_{c}}=0$, leading to

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

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

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

and

${\frac {p_{c}V_{c}}{T_{c}}}={\frac {3R}{8}}$

which then leads to

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

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

Dimensionless formulation

If one takes the following reduced 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

• 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)

References

• J. D. van der Waals "Over de Continuiteit van den Gas- en Vloeistoftoestand", doctoral thesis, Leiden, A,W, Sijthoff (1873).

English translation:

• J. D. van der Waals "On the Continuity of the Gaseous and Liquid States", Dover Publications ISBN: 0486495930