# Berthelot equation of state

The Berthelot equation of state [1][2] 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 [3][4])

$a = 3 T_c p_c v_c^2$

$b= \frac{v_c}{3}$

and giving a critical compressibility factor of

$Z_c = \frac{p_cv_c}{RT_c} = \frac{3}{8} = 0.375$

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[?]

$p = \frac{RT}{v} \left( 1 + \frac{9}{128} \frac{pT_c}{p_c T} \left( 1- \frac{6T_c^2}{T^2} \right) \right)$