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 )

\[a = 3 T_c p_c v_c^2\]

\[b= \frac{v_c}{3}\]

and giving a critical compressibility factor of

\[\frac{p_cv_c}{RT_c} = \frac{3}{8} = 0.375 \]

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^[?]

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