Gibbs phase rule proof

For the purpose of classifying heterogeneous equilibria we shall make use of a very general law, called the Phase Rule of Willard Gibbs (1876), the proof of which is deferred to a later chapter. 

An alternative, and equally satisfactory, proof of the Phase Rule emerges in Frame 50, section 50.3 using the Gibbs-Duham Equation as the starting point. 

The Gibbs-Duhem equation (50.6), derived below, proves to be a useful starting point for an alternative derivation of the Clausius Claperyron equation to that offered in Frame 26) and offers an alternative proof of the Phase Rule to that given in Frame 30. 

The first important contribution to atomic stoichiometry in this century seems to be provided by Brinkley (1946). He has shown the importance of the rank of the atomic matrix and presented a proof of the phase rule of Gibbs (1876). A systematic outline of stoichiometry was presented by Petho (sometimes Petheo) and Schay (Petheo Schay, 1954 Schay, Petho, 1962). They gave a necessary and sufficient condition for the possibility of calculating an unknown reaction heat from known ones based upon the rank of the stoichiometric matrix. They introduced the notion of independence of components and of elementary reactions, the completeness of a complex chemical reaction (see the Exercises and Problems) and gave a method to generate a complete set of independent elementary reactions with as many zeros in the stoichiometric matrix as possible (see Petho, 1964). 

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