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

References

Related 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)