 Solution

At fixed $N$, the Fermi energy scales as $E_{\mathrm{F}}\propto (N/V{)}^{2/3}$, so $\partial E_{\mathrm{F}}/\partial V=-(2/3)E_{\mathrm{F}}/V$. From $U=3N{E}_{\mathrm{F}}/5$,This is the nonrelativistic electron degeneracy pressure. Because $n{E}_{\mathrm{F}}\propto V^{-5/3}$,The units are energy per volume, hence pressure. Converting the Fermi energies to joules gives the ideal Fermi gas bulk modulusThese are free-electron model values. The complete mechanical bulk modulus of a real metal also involves the ionic lattice, electronic bands and interactions; the model's positive electron pressure does not by itself explain the equilibrium crystal volume.