Show that for any given fluid [ C_{P}=V T(partial P / partial T)_{S}(partial P / partial T)_{V}

Question:

Show that for any given fluid

\[
C_{P}=V T(\partial P / \partial T)_{S}(\partial P / \partial T)_{V} \kappa_{T}
\]

and

\[
C_{V}=V T(\partial P / \partial T)_{S}(\partial P / \partial T)_{V \kappa_{S}}
\]

where the various symbols have their usual meanings. In the two-phase region, these formulae take the form

\[
C_{P}=V T(d P / d T)^{2} \kappa_{T} \quad \text { and } \quad C_{V}=V T(d P / d T)^{2} \kappa_{S}
\]

respectively. Using the last of these results, rederive equation (13.6.30) for $C_{V}$ at \(T