Berthelot equation of state

The Berthelot equation of state can be written as

$RT=\left(p+{\frac {a}{Tv^{2}}}\right)\left(v-b\right)$.

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

which leads to (Eqs. 4.1 - 4.3 )

$a=3T_{c}p_{c}v_{c}^{2}$

$b={\frac {v_{c}}{3}}$

and giving a critical compressibility factor of

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

where $p$ is the pressure, $T$ is the temperature and $R$ is the molar gas constant. $T_{c}$ is the critical temperature, $p_{c}$ is the pressure and $v_{c}$ is the volume at the critical point.

Low pressure variant

Berthelot also proposed an equation of state for use at low pressures:

$p={\frac {RT}{v}}\left(1+{\frac {9}{128}}{\frac {pT_{c}}{p_{c}T}}\left(1-{\frac {6T_{c}^{2}}{T^{2}}}\right)\right)$