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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_c} is the pressure at the critical point, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_c } is the critical volume per mol.

At the critical point one has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.\frac{\partial p}{\partial v}\right|_{T=T_c}=0 } , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.\frac{\partial^2 p}{\partial v^2}\right|_{T=T_c}=0 } , which leads to

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b= v_c - \frac{RT_c}{4P_c}}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c= \frac{3RT_c}{8P_c}-v_c}

For details see the Mathematica printout produced by Dr. John L. Hardwick.