1.8.1: Enthalpies and Gibbs Energies
By definition, the Gibbs energy,
\[\mathrm{G}=\mathrm{U}+\mathrm{p} \, \mathrm{V}-\mathrm{T} \, \mathrm{S} \nonumber \]
Enthalpy,
\[\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V} \nonumber \]
Combination of equations (a) and (b) yields an important equation relating Gibbs energy \(\mathrm{G}\) and enthalpy \(\mathrm{H}\).
\[\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S} \nonumber \]
Just as we can never know the thermodynamic energy of a system, so we can never know the enthalpy. Consequently analysis of enthalpies is more complicated than analysis of volumetric properties, bearing in mind that the density of a solution (liquid) can be accurately measured. Differences are therefore emphasised in the context of enthalpies.
A differential change in Gibbs energy at constant temperature is related to the changes in enthalpy \(\mathrm{dH}\) and entropy, \(\mathrm{dS}\).
\[\mathrm{dG}=\mathrm{dH}-\mathrm{T} \, \mathrm{dS} \nonumber \]
For an isothermal process from state I to state II, the change in Gibbs energy \(\Delta \mathrm{G}\) is given by equation (e).
\[\Delta \mathrm{G}=\Delta \mathrm{H}-\mathrm{T} \, \Delta \mathrm{S} \nonumber \]
Equation (e) signals how enthalpy and entropy changes determine the change in Gibbs energy.
A closed system at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) is prepared using \(\mathrm{n}_{1}\) moles of solvent (water) and \(\mathrm{n}_{j}\) moles of solute-\(j\). The system is at equilibrium such that the composition/organisation is represented by \(\xi^{\mathrm{eq}}\) and the affinity for spontaneous change is zero. Using an over-defined representation we define the system as follows.
\[\mathrm{G}^{\mathrm{eq}}=\mathrm{G}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right] \nonumber \]
Under such circumstances the Gibbs energy \(\mathrm{G}\) is a minimum \(\mathrm{G}^{\mathrm{eq}}\) when plotted as a function of \(\xi\). The enthalpy of this system can be defined using a similar equation.
\[\mathrm{H}^{\mathrm{eq}}=\mathrm{H}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right] \nonumber \]
It is unlikely that \(\mathrm{H}^{\mathrm{eq}}\) corresponds to a minimum in the plot of enthalpy \(\mathrm{H}\) against \(\xi\). Indeed the same comment applies to the entropy \(\mathrm{S}^{\mathrm{eq}}\);
\[\mathrm{S}^{\mathrm{eq}}=\mathrm{S}^{\mathrm{eq}}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}, \xi^{\mathrm{eq}}, \mathrm{A}=0\right] \nonumber \]
The plots showing the product \(\mathrm{T} \, \mathrm{S}\) and \(\mathrm{H}\) against \(\xi\) may not show extrema though taken together they produce a minimum in \(\mathrm{G}\) at \(\xi^{\mathrm{eq}}\).
\[\mathrm{G}^{e q}=\mathrm{H}^{e q}-\mathrm{T} \, \mathrm{S}^{e q} \nonumber \]