Isothermal compressibility from chemical-potential response (${\kappa }_T=n^{-2}(\partial n/\partial \mu {)}_T$)

For a homogeneous extensive single-component thermodynamic phase, the Gibbs-Duhem equation at fixed temperature gives $dp=nd\mu$. With number density $n=N/V$, fixed $N$ gives $dn=-(n/V)dV$. The chain rule therefore givesThe derivatives are evaluated within the same smooth equation of state; an interface contribution or phase coexistence can invalidate this simple homogeneous derivation.