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

$R={\frac {8p_{c}v_{c}}{3T_{c}}}$

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