1.14.53: Phase Rule
According to the Gibbs-Duhem Equation, the properties of a single phase at equilibrium containing i chemical substances are related; we divide the Gibbs-Duhem Equation by the total amount in the system such that \(\mathrm{x}_{j}(\alpha)\) is the mole fraction of substance \(j\) in the \(\alpha\) phase. The Gibbs-Duhem Equation. requires that
\[0=\mathrm{S}_{\mathrm{m}}(\alpha) \, \mathrm{dT}-\mathrm{V}_{\mathrm{m}}(\alpha) \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mathrm{x}_{\mathrm{j}}(\alpha) \, \mathrm{d} \mu_{\mathrm{j}}(\alpha) \nonumber \]
Within this phase, the definition of mole fraction means that over all \(\mathrm{i}\)-chemical substances,
\[\sum_{j=1}^{j=i} x_{j}(\alpha)=1 \nonumber \]
The number of independent intensive variables is \([\mathrm{P} \,(\mathrm{C}-1)+2]\) where \(\mathrm{C}\) is the number of independent chemical substances in phase \(\alpha\). The additional two variables refer to the intensive temperature and pressure. We consider the case where the closed system contains \(\mathrm{P}\) phases. Therefore we can set down \(\mathrm{P}\) equations of the form shown in equation (a). With reference to the chemical potential of substance \(j\), the overall equilibrium condition requires that the chemical potentials of this substance over all phases ( i.e. \(\alpha_{1}, \alpha_{2}, \alpha_{3}, \ldots \ldots \alpha_{p}\)) are equal.
\[\mu_{j}\left(\alpha_{1}\right)=\mu_{j}\left(\alpha_{2}\right)=\mu_{j}\left(\alpha_{3}\right)=\ldots \ldots \ldots . .=\mu_{j}\left(\alpha_{p}\right) \nonumber \]
Hence with reference to the intensive chemical potentials there are (\(\mathrm{P} - 1\)) constraints. Therefore the number of independent intensive variables for this system comprising \(\mathrm{i}\) chemical substances distributed through \(\mathrm{P}\) phases, namely \(\mathrm{F}\), equals \((\mathrm{C} −1) + 2 − (\mathrm{P} −1)\). Therefore
\[\mathrm{P}+\mathrm{F}=\mathrm{C}+2 \nonumber \]
The latter is the Phase Rule. This equation is possibly the most elegant and practical equation in chemistry.