1.11.1: Gibbs-Duhem Equation
This equation is at the heart of chemical thermodynamics. The Gibbs energy of a closed system can be expressed as follows.
\[\mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{\mathrm{i}}\right] \nonumber \]
Here \(\mathrm{n}_{\mathrm{i}}\) represents the amounts of all chemical substances in the system. Then for a system containing two chemical substances, 1 and 2,
\[\mathrm{G}=\mathrm{G}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right] \nonumber \]
If we prepare a system containing \(\mathrm{k} \, \mathrm{n}_{1}\) and \(\mathrm{k} \, \mathrm{n}_{2}\) moles of the two chemical substances [cf. Euler’s Theorem], the Gibbs energy increases by a factor, \(\mathrm{k}\). Hence,
\[\mathrm{G}=\mathrm{n}_{1} \, \mu_{1}+\mathrm{n}_{2} \, \mu_{2} \nonumber \]
In general [1],
\[\mathrm{G}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}} \nonumber \]
Equation (c) is differentiated [2].
\[\mathrm{dG}=\mathrm{n}_{1} \, \mathrm{d} \mu_{1}+\mu_{1} \, \mathrm{dn} \mathrm{n}_{1}+\mathrm{n}_{2} \, \mathrm{d} \mu_{2}+\mu_{2} \, \mathrm{dn}_{2} \nonumber \]
But [3],
\[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}+\mu_{1} \, \mathrm{dn}_{1}+\mu_{2} \, \mathrm{dn}_{2} \nonumber \]
Therefore, combining equations (c) and (f),
\[-S \, d T+V \, d p-n_{1} \, d \mu_{1}-n_{2} \, d \mu_{2}=0 \nonumber \]
Therefore, for a system held at fixed \(\mathrm{T}\) and \(\mathrm{p}\) [4],
\[\mathrm{n}_{1} \, \mathrm{d} \mu_{1}+\mathrm{n}_{2} \, \mathrm{d} \mu_{2}=0 \nonumber \]
Equation (h) expresses the ‘communication’ between the two chemical substances in the system. In some senses equation (h) is a Happy Family Equation. If the chemical potential of substance 1, \(\mu_{1}\) increases [i.e. \(\mathrm{d} \mu_{1}> 0\)], the chemical potential of chemical substance 2 decreases, [i.e. \(\mathrm{d} \mu_{2}<0\)], in order to hold the condition expressed by equation (h) [5].
Footnotes
[1] Similarly \(\mathrm{U}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}+\mathrm{T} \, \mathrm{S}-\mathrm{p} \, \mathrm{V}\) \(\mathrm{H}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{n}_{\mathrm{j}} \, \mu_{\mathrm{j}}+\mathrm{T} \, \mathrm{S}\) and \(F=\sum_{j=1}^{j=i} n_{j} \, \mu_{j}-p \, V\)
[2] Generally, \(\mathrm{dG}=\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}}\left[\mathrm{n}_{\mathrm{j}} \, \mathrm{d} \mu_{\mathrm{j}}+\mu_{\mathrm{j}} \, \mathrm{dn} \mathrm{n}_{\mathrm{j}}\right]\)
[3] Generally \(\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mu_{\mathrm{j}} \, \mathrm{dn}_{\mathrm{j}}\(
[4] Generally \(\sum_{j=1}^{j=i} n_{j} \, d \mu_{j}=0\)
[5] If one member of a family is sad for some reason, other members of the family say something along the lines, ‘cheer up --it is not as bad as all that’. If another member of the family becomes over-excited, the family says something along the lines, ‘calm down’. Similar things happen to chemical substances in a closed system at fixed \(\mathrm{T}\) and \(\mathrm{p}\).