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

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

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 ${\frac{\partial p}{\partial v}|}_{T=T_c}=0$, and ${\frac{{\partial }^{2}p}{\partial v^2}|}_{T=T_c}=0$, leading to

$T_c=\frac{8a}{27bR}$

$p_c=\frac{a}{27b^2}$

$V_c=3b$.

and

$\frac{p_c{V}_c}{T_c}=\frac{3R}{8}$

which then leads to

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

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

Virial form

One can re-write the van der Waals equation given above as a virial equation of state as follows:

$Z:=\frac{pV}{nRT}=\frac{1}{1-\frac{bn}{V}}-\frac{an}{RTV}$

Using the well known series expansion $(1-x{)}^{-1}=1+x+x^2+x^3+...$ one can write the first term of the right hand side as :

$\frac{1}{1-\frac{bn}{V}}=1+\frac{bn}{V}+{(\frac{bn}{V})}^2+{(\frac{bn}{V})}^3+...$

Incorporating the second term of the right hand side in its due place leads to:

$Z=1+(b-\frac{a}{RT})\frac{n}{V}+{(\frac{bn}{V})}^2+{(\frac{bn}{V})}^3+...$.

From the above one can see that the second virial coefficient corresponds to

$B_2(T)=b-\frac{a}{RT}$

and the third virial coefficient is given by

$B_3(T)=b^2$

Boyle temperature

The Boyle temperature of the van der Waals equation is given by

$B_2{|}_{T=T_B}=0=b-\frac{a}{R{T}_B}$

leading to

$T_B=\frac{a}{bR}$

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)