1.21.4: Thermodynamic Potentials
The following important equations describe changes in thermodynamic energy, enthalpy, Helmholtz energy and Gibbs energy of a closed system.
\[p\mathrm{dU}=\mathrm{T} \, \mathrm{dS}-\mathrm{p} \, \mathrm{dV}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mu_{\mathrm{j}} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}} \nonumber \]
\[\mathrm{dH}=\mathrm{T} \, \mathrm{dS}+\mathrm{V} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mu_{\mathrm{j}} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}} \nonumber \]
\[\mathrm{dF}=-\mathrm{S} \, \mathrm{dT}-\mathrm{p} \, \mathrm{dV}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mu_{\mathrm{j}} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}} \nonumber \]
\[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mu_{\mathrm{j}} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}} \nonumber \]
These four differential equations relate, for example, the change in \(\mathrm{U}, \mathrm{~H}, \mathrm{~F} \text { and } \mathrm{G}\) with the change in amount of each chemical substance, \(\mathrm{dn}_{j}\). These four equations are integrated [1] to yield the following four equations.
\[\mathrm{U}=\mathrm{T} \, \mathrm{S}-\mathrm{p} \, \mathrm{V}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}} \nonumber \]
\[\mathrm{H}=\mathrm{T} \, \mathrm{S} +\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}} \nonumber \]
\[\mathrm{F}=-\mathrm{p} \, \mathrm{V}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}} \nonumber \]
\[G=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}} \nonumber \]
The latter equation is particularly useful because it signals that the total Gibbs energy of a system is given by the sum of the products of amounts and chemical potentials of all substances in the system. In the case of an aqueous solution containing \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of chemical substance \(j\), the Gibbs energy of the solution is given by equation (i).
\[\mathrm{G}(\mathrm{aq})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
In conjunction with equation (i) we do not have to attach the phrase ‘at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\)’. Similarly the volume of the solution is given by equation (j) where \(\mathrm{V}_{1}(\mathrm{aq})\) and \(\mathrm{V}_{j}(\mathrm{aq})\) are the partial molar volumes of solvent and solute respectively.
\[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
The same argument applies in the case of a solution prepared using \(\mathrm{n}_{1}\) moles of solvent water, \(\mathrm{n}_{\mathrm{x}}\) moles of solute \(\mathrm{X}\) and \(\mathrm{n}_{\mathrm{y}}\) moles of solute \(\mathrm{Y}\). Then, for example,
\[\mathrm{G}(\mathrm{aq})=\mathrm{n}_{1} \, \mu_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{x}} \, \mu_{\mathrm{x}}(\mathrm{aq})+\mathrm{n}_{\mathrm{y}} \, \mu_{\mathrm{y}}(\mathrm{aq}) \nonumber \]
The analogue of equation (j) also follows but only if \(\mathrm{n}_{\mathrm{x}}\) and \(\mathrm{n}_{\mathrm{y}}\) are independent of pressure. If these two solutes are in chemical equilibrium [eg. \(\mathrm{X}(\mathrm{aq}) \Leftrightarrow \mathrm{Y}(\mathrm{aq})\), amounts \(\mathrm{n}_{\mathrm{x}}^{\mathrm{eq}}\) and \(\mathrm{n}_{\mathrm{y}}^{\mathrm{eq}}\) respectively], then account must be taken of the dependences of \(\mathrm{n}_{\mathrm{x}}^{\mathrm{eq}}\) and \(\mathrm{n}_{\mathrm{y}}^{\mathrm{eq}}\) on pressure at fixed temperature and (with reference to entropies and enthalpies) on temperature at fixed pressure.
The simple form of equations (i) and (k) emerge from equation (h) because other than the composition variables, the other differential terms \(\mathrm{dT}\) and \(\mathrm{dp}\) in equation (d) refer to change in intensive variables. For this reason chemists find it advantageous to describe chemical properties in the \(\mathrm{T}-\mathrm{p}\)-composition domain.
The relationships between thermodynamic potentials are described as Legendre transforms [2]. The product term \(\mathrm{T} \, \mathrm{S}\) may be called bounded energy. Then the Helmholtz energy (\(\mathrm{F}=\mathrm{U}-\mathrm{T} \, \mathrm{S}\)) is the free internal energy and the Gibbs energy (\(\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S}\)) is the free enthalpy. These comments help to understand the old designations of free Helmholtz energy (together with symbol \(\mathrm{F}\)), free Gibbs energy and the still currently used (in the French scientific literature) free enthalpy.
Footnotes
[1] The term “integrated” in this context deserves comment. Within the set of variables, \(\mathrm{p}-\mathrm{V}-\mathrm{T}-\mathrm{S}\), \(\mathrm{p}\) and \(\mathrm{T}\) are intensive whereas \(\mathrm{V}\) and \(\mathrm{S}\) are extensive variables. Similarly \(\mu_{j}\) is intensive whereas \(\mathrm{n}_{j}\) is extensive. These are the conditions for operating Euler’s integration method. Still the word “integrate” in the present context has been used in subtle arguments when Euler’s theorem is not invoked.
E. F. Caldin [Chemical Thermodynamics, Oxford, 1958, p. 166] identifies \(\mathrm{T}, \mathrm{~p} \text { and } \mu_{j}\) as intensive and then integrates by gradual increments of the amount of each chemical substance, keeping the relative amounts constant.
K. Denbigh [The Principles of Chemical Equilibrium, Cambridge, 1971, 3rd edn. p. 93] uses a similar argument, but comments that development of the equations (e) to (h) is not mathematical in the sense that the variables are simple. Rather we use our physical knowledge in that intensive variables do not depend on the state of the system.
E. A. Guggenheim [Thermodynamics, North-Holland, Amsterdam, 1950, 2nd edn. p. 23] states that the equations (a) to (d) can be integrated by following the artifice when \(\mathrm{dT} = 0, \mathrm{~dp = 0\) and each \(\mathrm{n}_{j}\) is changed by the same proportions as are the extensive variables \(\mathrm{S}\) and \(\mathrm{V}\).
{The term artifice is used here to mean a ‘device’, skill rather than “trickery” or “something intended to deceive”; Pocket Oxford Dictionary, Oxford 1942, 4th edn. and Cambridge International Dictionary of English, Cambridge, 1995.
[2] There is a pleasing internal consistency between the definitions advanced at this stage
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\[\mathrm{U}=\mathrm{T} \, \mathrm{S}-\mathrm{p} \, \mathrm{V}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}} \nonumber \]
Then (cf. definition of \(\mathrm{G}\))
\[\mathrm{G}=\mathrm{U}-\mathrm{T} \, \mathrm{S}+\mathrm{p} \, \mathrm{V} \nonumber \]
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\[F=-p \, V+\sum_{j=1}^{j=k} n_{j} \, \mu_{j} \nonumber \]
Then from (a),
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\[\mathrm{H}=\mathrm{T} \, \mathrm{S}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}} \nonumber \]
Then from (a),
- Combining equations. relating \(\mathrm{G}\) and \(\mathrm{U}\) to \(\mathrm{H}\) yields \(\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S}\). Similarly the connections of \(\mathrm{G}\) and \(\mathrm{F}\) to \(\mathrm{U}\) give \(\mathrm{G} = \mathrm{~F} + \mathrm{~p} \, \mathrm{~V}\)