1.13.2: Equilibrium- Isochoric and Isobaric Paramenters
In a description of a given closed system we define two extensive state variables, the Gibbs energy \(\mathrm{G}\) and the Helmholtz energy \(\mathrm{F}\).
\[\mathrm{G}=\mathrm{U}+\mathrm{p} \, \mathrm{V}-\mathrm{T} \, \mathrm{S} \nonumber \]
\[\mathrm{F}=\mathrm{U}-\mathrm{T} \, \mathrm{S} \nonumber \]
Hence,
\[\mathrm{G}=\mathrm{F}+\mathrm{p} \, \mathrm{V} \nonumber \]
The latter interesting equation links two practical thermodynamic potentials;
- \(\mathrm{G}\) for processes at fixed \(\mathrm{T}\) and \(\mathrm{p}\),
- \(\mathrm{F}\) for processes at fixed \(\mathrm{T}\) and \(\mathrm{V}\).
The dependence of \(\mathrm{G}\) on extent of reaction at constant temperature and pressure is related to the differential dependence of \(\mathrm{F}\) on \(\xi\) at fixed temperature and pressure.
\[\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}=\left(\frac{\partial \mathrm{F}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}+\mathrm{p} \,\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \nonumber \]
At equilibrium where \(\mathrm{A} = 0\), \(\xi = \xi^{\mathrm{eq}\) and the Gibbs energy is a minimum [i.e. \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}=0\)],
\[\left(\frac{\partial F}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{A}=0}=\mathrm{p} \,\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{A}=0} \nonumber \]
In other words the differential dependence of the Helmholtz energy on extent of reaction at equilibrium (at constant \(\mathrm{T}\) and \(\mathrm{p}\)) is related to the volume of reaction. We rewrite equation (c);
\[\mathrm{F}=\mathrm{G}-\mathrm{p} \, \mathrm{V} \nonumber \]
At constant temperature and volume,
\[\left(\frac{\partial \mathrm{F}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{V}}=\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{V}}-\mathrm{V} \,\left(\frac{\partial \mathrm{p}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{V}} \nonumber \]
At equilibrium (at constant \(\mathrm{T}\) and \(\mathrm{V}\)) where the Helmholtz energy \(\mathrm{F}\) is a minimum, clearly the Gibbs energy is not at a minimum. The dependence of both \(\mathrm{G}\) and \(\mathrm{F}\) on temperature at equilibrium can be expressed using two Gibbs - Helmholtz equations. Thus,
\[\left[\frac{\partial(\Delta \mathrm{G} / \mathrm{T})}{\partial(1 / \mathrm{T})}\right]_{\mathrm{p}, \mathrm{A}=0}^{\mathrm{eq}}=\Delta \mathrm{H}^{\mathrm{eq}} \nonumber \]
\[\left[\frac{\partial(\Delta \mathrm{F} / \mathrm{T})}{\partial(1 / \mathrm{T})}\right]_{\mathrm{V}, \mathrm{A}=0}^{\mathrm{eq}}=\Delta \mathrm{U}^{\mathrm{eq}} \nonumber \]
From a practical standpoint, determination of \(\Delta\mathrm{H}^{\mathrm{eq}}\) is reasonably straightforward because over a range of temperatures the isobaric condition is readily satisfied. Thus we probe this differential dependence at a series of defined temperatures at fixed pressure; i.e. over the range \(\mathrm{T}-\delta \mathrm{T}\) to \(\mathrm{T}+\delta \mathrm{T}\) about \(\mathrm{T}\) for a number of temperatures.
The condition ‘at constant volume’ presents problems. In principle we change the pressure to hold \(\mathrm{V}\) constant over a range of temperatures. Then we probe the differential dependence of \((\Delta \mathrm{F} / \mathrm{T})\) at a series of fixed temperatures; e.g. over the range \(\mathrm{T}-\Delta \mathrm{T}\) to \(\mathrm{T}-\Delta \mathrm{T}\) about a given temperature T. If the range of temperatures is large, there is a high probability that very high pressures will be required to hold the global isochoric condition.
Another approach probes the dependence of \((\Delta \mathrm{F} / \mathrm{T})\) on temperature at a series of temperatures where volume \(\mathrm{V}\) is held constant by changing the pressure over the range
\[\mathrm{T}_{\mathrm{i}}-\delta \mathrm{T}\) to
\[\mathrm{T}_{\mathrm{i}}-\delta \mathrm{T} \nonumber \]
about \(\mathrm{T}_{\mathrm{i}}\). Volume \(\mathrm{V}_{\mathrm{i}}\) is constant over a small range of temperature. Here the isochoric condition is local to temperature \(\mathrm{T}\); thus \(\Delta\mathrm{U}\) is obtained at \(\mathrm{T}_{\mathrm{i}}\) and \(\mathrm{V}_{\mathrm{i}}\). Under these circumstances, comparison of derived \(\Delta \mathrm{U}\) - quantities as a function of temperature is not straightforward.
Interestingly the solvent water presents pairs of temperatures either side of the TMD where molar volume of water is the same at, for example, ambient pressure. It might be possible to explore this feature by assuming that the volumes of two very dilute solutions are also identical at matched pairs of temperatures.