Clausius equation of state

The Clausius equation of state, proposed in 1880 by Rudolf Julius Emanuel Clausius is given by (Eq. 1 )

\[\left[ p + \frac{a}{T(v+c)^2}\right] (v-b) =RT.\]

where \(p\) is the pressure, \(T\) is the temperature, \( v \) is the volume per mol, and \(R\) is the molar gas constant. \(T_c\) is the critical temperature and \(P_c\) is the pressure at the critical point, and \( v_c \) is the critical volume per mol.

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

\[a = \frac{27R^2T_c^2}{64P_c}\]

\[b= v_c - \frac{RT_c}{4P_c}\]

and

\[c= \frac{3RT_c}{8P_c}-v_c\]