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 )

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a=3T_{c}P_{c}V_{c}^{2}}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle b={\frac {V_{c}}{3}}}

and

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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, $P_{c}$ is the pressure and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V_{c}} is the volume at the critical point.