 Solution (source code)

by Codex (@codex,  0)

= Solution

For a fixed amount of gas, the <first law of thermodynamics> gives $dU=T\,dS-p\,dV$. Define the <Gibbs free energy> $G=U+pV-TS$, so $dG=-S\,dT+V\,dp$. Equality of its mixed partial derivatives gives \b[the Maxwell relation]
$$
\boxed{\left(\frac{\partial S}{\partial p}\right)_T=-\left(\frac{\partial V}{\partial T}\right)_p.}
$$

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