= Thermodynamic internal pressure
{title2=$\pi_T=(\partial U/\partial V)_T=T(\partial p/\partial T)_V-p$}
For a <simple compressible system> at fixed composition and particle number, the internal pressure is the <partial derivative> of <internal energy> with respect to <volume (thermodynamics)> at fixed <temperature>. From $dU=T\,dS-p\,dV$ and the <Maxwell relation> $(\partial S/\partial V)_T=(\partial p/\partial T)_V$, it has the measurable expression in the title. It is distinct from the mechanical <pressure> $p$.
For an <ideal gas>, $p=nRT/V$ gives $\pi_T=0$, so its <internal energy> depends only on <temperature> at fixed amount of gas. For the <Dieterici equation> $p=nRT\exp[-an/(VRT)]/(V-nb)$, direct <differentiation> gives
$$
\pi_T=\frac{an^2}{V(V-nb)}\exp\left(-\frac{an}{VRT}\right).
$$
Thus an <equation of state> determines this volume sensitivity even without determining the full caloric <function> $U(T,V)$.
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