Van der Waals equation of state

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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.

1 Equation of state
2 Critical point
3 Dimensionless formulation
4 Maxwell's equal area construction
5 Interesting reading
6 References

[edit] Equation of state

The van der Waals equation of state can be written as

$\left. p = \frac{ n R T}{V - n b } - a \left( \frac{ n}{V} \right)^2 \right.$ .

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.

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

$b= \frac{RT_c}{8P_c}$

[edit] 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

$\left.v_c\right.=3b$ .

[edit] 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: