Phase rule and the azeotrope — A critique and a new interpretation

The differences between the modern version of the phase rule and the one originally proposed by Gibbs are pointed out. The local analysis implied in Gibbs's approach to the phase rule is carried forward to its logical conclusion using the implicit function theorem. The results of the analysis are used to resolve the apparent contradictions in the interpretation of the phase rule, using the Gibbs–Duhem equations, for a system exhibiting an azeotrope. Specifically, the pitfalls in treating the differentials in the Gibbs–Duhem equations as variations are demonstrated. The critical role played by the rank of a submatrix in the coefficient matrix of the Gibbs–Duhem equations is highlighted. A hierarchy in the application of the phase rule is pointed out and the need for a unified framework for interpreting the phase rule is indicated.