= Heat-capacity and compressibility ratio identity
{title2=$C_p/C_V=\kappa_T/\kappa_S$}
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 $a,d\ne0$. 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 $C_p\geq C_V$ and $\kappa_T\geq\kappa_S$.
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