= Solution
For fixed <Electron> number $N$, the specified <electron degeneracy pressure> scales as $P\propto V^{-5/3}$. Differentiating at fixed low <temperature> gives the <degeneracy-pressure bulk modulus>,
$$
\boxed{K_T=-V\frac{\partial P}{\partial V}=\frac53P}.
$$
A monovalent sample has one conduction <Electron> per atom. Its <carrier density> is
$$
n=\frac{862\,\mathrm{kg\,m^{-3}}}{0.039\,\mathrm{kg\,mol^{-1}}}N_A
=1.331\times10^{28}\,\mathrm{m^{-3}},
$$
using the <Avogadro constant> $N_A=6.02214076\times10^{23}\,\mathrm{mol^{-1}}$. Substitution with <reduced Planck constant> $\hbar=1.05457\times10^{-34}\,\mathrm{J\,s}$ and <electron mass> $m_e=9.10938\times10^{-31}\,\mathrm{kg}$ gives
$$
P=1.747\times10^9\,\mathrm{Pa},\qquad
\boxed{K_T=2.91\times10^9\,\mathrm{Pa}\simeq2.9\,\mathrm{GPa}}.
$$
Equivalently, $P=2nE_F/5$ and $K_T=2nE_F/3$ for this <free electron model>. This is the assumed electronic contribution, not a calculation of the complete equation of state of real potassium: ionic cohesion and interaction corrections have been excluded as instructed. The supplied pressure is the zero-temperature form, appropriate to leading order for $T\ll T_F$.
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