# Van der Waals equation of state

The van der Waals equation of state, developed by Johannes Diderik van der Waals [1] [2], takes into account two features that are absent in the ideal gas equation of state; the parameter ${\displaystyle b}$ introduces somehow the repulsive behavior between pairs of molecules at short distances, it represents the minimum molar volume of the system, whereas ${\displaystyle a}$ 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

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

where:

• ${\displaystyle p}$ is the pressure,
• ${\displaystyle V}$ is the volume,
• ${\displaystyle n}$ is the number of moles,
• ${\displaystyle T}$ is the absolute temperature,
• ${\displaystyle R}$ is the molar gas constant; ${\displaystyle R=N_{A}k_{B}}$, with ${\displaystyle N_{A}}$ being the Avogadro constant and ${\displaystyle k_{B}}$ being the Boltzmann constant.
• ${\displaystyle a}$ and ${\displaystyle b}$ 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 ${\displaystyle \left.{\frac {\partial p}{\partial v}}\right|_{T=T_{c}}=0}$, and ${\displaystyle \left.{\frac {\partial ^{2}p}{\partial v^{2}}}\right|_{T=T_{c}}=0}$, leading to

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

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

${\displaystyle \left.v_{c}\right.=3b}$

along with a critical point compressibility factor of

${\displaystyle {\frac {p_{c}v_{c}}{RT_{c}}}={\frac {3}{8}}=0.375}$

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

${\displaystyle b={\frac {RT_{c}}{8p_{c}}}}$

## Virial form

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

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

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

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

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

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

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

${\displaystyle B_{2}(T)=b-{\frac {a}{RT}}}$

and the third virial coefficient is given by

${\displaystyle B_{3}(T)=b^{2}}$

## Boyle temperature

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

${\displaystyle B_{2}\vert _{T=T_{B}}=0=b-{\frac {a}{RT_{B}}}}$

${\displaystyle T_{B}={\frac {a}{bR}}}$

## Dimensionless formulation

If one takes the following reduced quantities

${\displaystyle {\tilde {p}}={\frac {p}{p_{c}}};~{\tilde {V}}={\frac {V}{V_{c}}};~{\tilde {t}}={\frac {T}{T_{c}}};}$

one arrives at

${\displaystyle {\tilde {p}}={\frac {8{\tilde {t}}}{3{\tilde {V}}-1}}-{\frac {3}{{\tilde {V}}^{2}}}}$

The following image is a plot of the isotherms ${\displaystyle T/T_{c}}$ = 0.85, 0.90, 0.95, 1.0 and 1.05 (from bottom to top) for the van der Waals equation of state:

## Critical exponents

The critical exponents of the Van der Waals equation of state place it in the mean field universality class.

2. This expansion is valid as long as ${\displaystyle -1, which is indeed the case for ${\displaystyle bn/V}$