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 giving a critical compressibility factor of

$Z_c=\frac{p_c{v}_c}{R{T}_c}=\frac{3}{8}=0.375$

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^[?]

$p=\frac{RT}{v}(1+\frac{9}{128}\frac{p{T}_c}{p_{c}T}(1-\frac{6T_c^2}{T^2}))$