= Solution
At fixed particle number, expand the <entropy> differential in $(T,p)$. The definition of <heat capacity at constant pressure> and the <Maxwell relation> $(\partial S/\partial p)_T=-(\partial V/\partial T)_p$ give
$$
\boxed{T\,dS=C_p\,dT-T\left(\frac{\partial V}{\partial T}\right)_pdp.}
$$
Similarly expansion in $(T,V)$, using <heat capacity at constant volume> and $(\partial S/\partial V)_T=(\partial p/\partial T)_V$, gives
$$
\boxed{T\,dS=C_V\,dT+T\left(\frac{\partial p}{\partial T}\right)_VdV.}
$$
At constant pressure, $dV=V\alpha\,dT$. Also $dV/V=\alpha\,dT-\kappa\,dp$, so $(\partial p/\partial T)_V=\alpha/\kappa$. Equating the two entropy expressions at constant pressure yields
$$
\boxed{C_p-C_V=\frac{TV\alpha^2}\kappa.}
$$
For a mechanically stable system with positive <isothermal compressibility>, this also gives $C_p\geq C_V$.
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