1.8.3: Enthalpy- Thermodynamic Potential
The enthalpy \(\mathrm{H}\) of a closed system is related by definition to the thermodynamic energy \(\mathrm{U}\); \(\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V}\). But
\[\mathrm{dH}=\mathrm{q}+\mathrm{V} \, \mathrm{dp} \nonumber \]
From the second law of thermodynamics,
\[\mathrm{T} \, \mathrm{dS}=\mathrm{q}+\mathrm{A} \, \mathrm{d} \xi ; \quad \mathrm{A} \, \mathrm{d} \xi \geq 0 \nonumber \]
Then
\[\mathrm{dH}=\mathrm{T} \, \mathrm{dS}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq 0 \nonumber \]
Thus all spontaneous processes at constant entropy and pressure (i.e. isentropic and isobaric) lower the enthalpy of a closed system. This conclusion finds application in acoustics where the changes in a system perturbed by a travelling sound wave are discussed in terms of changes in enthalpy at constant entropy and pressure. Confining our attention to systems either at equilibrium (i.e. \(\mathrm{A} = 0\)) or at fixed \(\xi\), two key relationships follow from equation (c).
\[\mathrm{T}=(\partial \mathrm{H} / \partial \mathrm{S})_{\mathrm{p}} \nonumber \]
and
\[\mathrm{V}=(\partial \mathrm{H} / \partial \mathrm{p})_{\mathrm{S}} \nonumber \]
In these terms the extensive variable, volume, is given by the
- isentropic differential dependence of enthalpy on pressure, and
- isothermal differential dependence of Gibbs energy on pressure.