= Solution
For an <ideal gas> with constant <specific-heat ratio> $\gamma$, the specific <internal energy> is $u=c_VT$ and $P=\rho(c_P-c_V)T=(\gamma-1)\rho u$. Hence
$$
\boxed{\int P\,dV=(\gamma-1)U,\qquad U=\int u\,dm.}
$$
The <stellar virial theorem> consequently reads $\ddot I/2=2\mathcal T+3(\gamma-1)U-3P_SV+\Omega$. For a static nonrotating star with negligible surface pressure, $\mathcal T=\ddot I=P_S=0$, so $\Omega=-3(\gamma-1)U$.
Fully ionized nonrelativistic ions and electrons form a monatomic <ideal gas>, with $\gamma=5/3$, provided <electron degeneracy pressure> and <radiation pressure> are negligible. Thus
$$
\boxed{\Omega=-2U,\qquad E=U+\Omega=-U=\Omega/2.}
$$
Being steady alone does not eliminate rotation or other macroscopic <kinetic energy>; those are additional assumptions needed for this relation.
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