Berthelot equation of state

The Berthelot equation of state can be written as

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

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