= Solution
The zero-temperature <internal energy> is
$$
E=\int_0^{E_F}\varepsilon g(\varepsilon)\,d\varepsilon
=\frac{VE_F^4}{4\pi^2\hbar^3c^3}
=\frac34NE_F.
$$
Using part (i),
$$
p=\frac{NE_F}{4V}
=\frac D4N^{4/3}V^{-4/3}.
$$
Consequently the <adiabatic equation of state> is
$$
\boxed{
pV^a=K,
\qquad
a=\frac43,
\qquad
K=\frac{\hbar c(3\pi^2)^{1/3}}4N^{4/3}
}.
$$
The pressure is <degeneracy pressure>: it remains nonzero at zero temperature because of the <Pauli exclusion principle>. By comparison, a classical ultrarelativistic <ideal gas> at fixed temperature obeys $pV=Nk_BT$, so its corresponding volume exponent is one and its pressure vanishes as $T\to0$.
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