12.5: Summary, the Maxwell Relations, and the Gibbs-Helmholtz Relations

\[d U =T d S-P d V+\sum X d Y\]

\[d H =T d S+V d P+\sum X d Y\]

\[d A =-S d T-P d V+\sum X d Y\]

\[d G =-S d T+V d P+\sum X d Y\]

If the only reversible work done on or by a system is PdV work of expansion or compression, we have the more familiar forms

\[d U =T d S-P d V\]

\[d H =T d S+V d P\]

\[d A =-S d T-P d V\]

\[d G =-S d T+V d P\]

All four thermodynamic functions are functions of state (and hence their differentials are exact differentials) and therefore

\[\left(\frac{\partial U}{\partial S}\right)_{V}=T \quad\left(\frac{\partial U}{\partial V}\right)_{S}=-P\]

\[\left(\frac{\partial H}{\partial S}\right)_{P}=T \quad\left(\frac{\partial H}{\partial P}\right)_{S}=V\]

\[\left(\frac{\partial A}{\partial T}\right)_{V}=-S \quad\left(\frac{\partial A}{\partial V}\right)_{T}=-P\]

\[\left(\frac{\partial G}{\partial T}\right)_{P}=-S \qquad\left(\frac{\partial G}{\partial P}\right)_{T}=V\]

Further, by equating the mixed second derivatives, we obtain the four Maxwell Thermodynamic Relations:

\[\left(\frac{\partial T}{\partial V}\right)_{S}=-\left(\frac{\partial P}{\partial S}\right)_{V}\]

\[\left(\frac{\partial T}{\partial P}\right)_{S}=\left(\frac{\partial V}{\partial S}\right)_{P}\]

\[\left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial P}{\partial T}\right)_{V}\]

\[\left(\frac{\partial S}{\partial P}\right)_{T}=-\left(\frac{\partial V}{\partial T}\right)_{P}\]

The Gibbs-Helmholtz Relations are trivially found from A = U − TS and together with equations 12.6.11a and 12.6.12a. G = H − TS They are

\[U=A-T\left(\frac{\partial A}{\partial T}\right)_{V}\]

\[H=G-T\left(\frac{\partial G}{\partial T}\right)_{P}\]