Cp - CV : general relation

3.30 \(C_p-C_V\): general relation

Starting from

\begin{equation*} dU= T dS - p dV \quad \mbox{i.e.} \quad \left(\frac{\partial U}{\partial V}\right)_T = T \left(\frac{\partial S}{\partial V}\right)_T - p = T \left(\frac{\partial p}{\partial T}\right)_V - p \label{eq:dpdT} \end{equation*}

(3.74)

where we have used the corresponding Maxwell relation, we get

\begin{equation*} \begin{split} C_p - C_V = &\;\; \left(\frac{\partial H}{\partial T}\right)_p - \left(\frac{\partial U}{\partial T}\right)_V \;\;= \;\; \left(\frac{\partial \left(U + p\,V\right)}{\partial T}\right)_p - \left(\frac{\partial U}{\partial T}\right)_V\\ = &\;\; \left(\frac{ \left(\frac{\partial U}{\partial T}\right)_V \partial T + \left(\frac{\partial U}{\partial V}\right)_T \partial V + p \partial V }{\partial T} \right)_p - \left(\frac{\partial U}{\partial T}\right)_V \\ = &\;\; \left(\frac{\partial U}{\partial T}\right)_V + \left[\left( \frac{\partial U }{\partial V} \right)_T + p \right] \left( \frac{\partial V }{\partial T} \right)_p - \left(\frac{\partial U}{\partial T}\right)_V \;\;=\;\; T \left(\frac{\partial p}{\partial T}\right)_V \left( \frac{\partial V }{\partial T} \right)_p\\ \end{split} \label{eq:Cp-CV_3} \end{equation*}

(3.75)

Using the definitions of the thermal expansion coefficient and the isothermal compressibility and applying the chain rule we find

\begin{equation*} \alpha = \frac{1}{V}\left( \frac{\partial V }{\partial T} \right)_p \qquad \kappa = - \frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_T \quad \mbox{and}\quad \left( \frac{\partial p }{\partial T} \right)_V = - \frac{\left( \frac{\partial V }{\partial T} \right)_p}{\left(\frac{\partial V}{\partial p}\right)_T} = \frac{\alpha}{\kappa} \label{eq:dp_dT} \end{equation*}

(3.76)

Including this in Eq. (3.75) we finally get

\begin{equation*} C_p - C_V = T \, V \, \frac{\alpha^2}{\kappa} \label{eq:Cp-CV_4} \end{equation*}

(3.77)