Clausius equation of state

The Clausius equation of state, proposed in 1880 by Rudolf Julius Emanuel Clausius is given by (Eq. 3 )

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

where $p$ is the pressure, $T$ is the temperature, $v$ is the volume per mol, and $R$ is the molar gas constant. 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 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 $\left.{\frac {\partial ^{2}p}{\partial v^{2}}}\right|_{T=T_{c}}=0$, which leads to

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 a = \frac{27R^2T_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.