Heat-capacity and compressibility ratio identity ($C_p/C_V={\kappa }_T/{\kappa }_S$)

In a smooth fixed-composition thermodynamic region, write $a=(\partial S/\partial T{)}_V$, $b=(\partial S/\partial V{)}_T$, $c=(\partial p/\partial T{)}_V$, and $d=(\partial p/\partial V{)}_T$, with $a,d\ne 0$. The chain rule gives $(\partial S/\partial T{)}_p=a-bc/d$ and $(\partial p/\partial V{)}_S=d-bc/a$. Multiplying these expressions by $d$ and $a$, respectively, gives the same Jacobian determinant $ad-bc$.

Using heat capacity at constant pressure $C_p=T(\partial S/\partial T{)}_p$, heat capacity at constant volume $C_V=Ta$, isothermal compressibility ${\kappa }_T=-1/(Vd)$ and isentropic compressibility ${\kappa }_S=-1/[V(\partial p/\partial V{)}_S]$ proves the displayed ratio. The derivation does not require division by thermal expansion, so it still applies when that derivative vanishes. For a stable single phase, the Maxwell relation $b=c$ also explains why $C_p\ge C_V$ and ${\kappa }_T\ge {\kappa }_S$.