Isentropic compressibility on a valid local chart

Fix a constitution $c$, a positive-temperature local energy chart with entropy coordinate $s$ and physical volume $V>0$, and fixed amounts $n=(n_1,\dots ,n_r)$. The energy representation and pressure on this chart are those of Local energy representation and thermodynamic conjugates and the response-chart conventions are those of Thermodynamic response functions on valid local charts. Where $(s,V,n)\mapsto (s,p,n)$ has a C1 local inverse, define the isentropic compressibility of the whole system by

${\kappa }_S:=-\frac{1}{V}{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }p}\right)}_{s,n}\mathrm{.}$

The entropy $s$ is held fixed, as are all amounts $n_j$. The local energy representation gives $p=-(\mathrm{\partial }U\mathrm{/}\mathrm{\partial }V{)}_{s,n}$ (Local energy representation and thermodynamic conjugates). Since $U$ is in joules and $V$ in ${\mathrm{m}}^3$, the relation $\mathrm{J}{\mathrm{m}}^{-3}=\mathrm{P}\mathrm{a}$ in SI quantities and unit conventions gives $p$ in $\mathrm{P}\mathrm{a}$, and hence ${\kappa }_S$ in $\mathrm{P}{\mathrm{a}}^{-1}$. This definition asserts no sign, finiteness, or existence outside the specified inverse chart; in particular it does not assign a value at a singular or critical response point.