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 )

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

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

$a=v_{c}-{\frac {RT_{c}}{4P_{c}}}$

$b={\frac {3RT_{c}}{8P_{c}}}-v_{c}$

and

$c={\frac {27R^{2}T_{c}^{2}}{64P_{c}}}$

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.