Berthelot equation of state

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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[edit]

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)