= Isentropic temperature-volume derivative
{title2=$(\partial T/\partial V)_S=-(T/C_V)(\partial p/\partial T)_V$}
For a <simple compressible system> at fixed particle number, assume the <internal energy> has continuous second <partial derivatives> and $T$ is a locally invertible function of <entropy> at fixed <volume (thermodynamics)>. Its <exact differential> $dU=T\,dS-p\,dV$ gives the <Maxwell relation> $T_V|_S=-p_S|_V$. Meanwhile the <constant-volume heat capacity> is $C_V=T S_T|_V$. The <chain rule> therefore gives
$$
\left(\frac{\partial T}{\partial V}\right)_S=-\left(\frac{\partial p}{\partial T}\right)_V\left(\frac{\partial T}{\partial S}\right)_V=-\frac{T}{C_V}\left(\frac{\partial p}{\partial T}\right)_V.
$$
For one mole of an <ideal gas> with constant molar $C_V>0$, this becomes $T_V|_S=-RT/(C_VV)$, whose <integral curves> satisfy $TV^{R/C_V}=\text{constant}$. It expresses cooling on reversible <adiabatic expansion> in terms of measurable thermal and mechanical derivatives.
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