Clausius equation of state

The Clausius equation of state, proposed in 1880 by Rudolf Julius Emanuel Clausius is given by (Equations 3 and 4 in )

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

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 ${\frac{\partial p}{\partial v}|}_{T=T_c}=0$, and ${\frac{{\partial }^{2}p}{\partial v^2}|}_{T=T_c}=0$, which leads to

$a=v_c-\frac{R{T}_c}{4P_c}$

$b=\frac{3R{T}_c}{8P_c}-v_c$

and

$c=\frac{27R^2{T}_c^2}{64P_c}$