Number of variables and the phase rule

The most broadly recognized theorem of chemical thermodynamics is probably the phase rule derived by Gibbs in 1875 (see Guggenheim, 1967 Denbigh, 1971). Gibbs phase rule defines the number of pieces of information needed to determine the state, but not the extent, of a chemical system at equilibrium. The result is the number of degrees of freedom Np possessed by the system. 

The phase rule says that for each phase beyond the first that occurs at equilibrium in a system, N. decreases by one. Expressed in general form, the phase rule is, 

The proof of the phase rule is actually implicit in the derivation of the governing equations (Eqns. 3.32-3.35), and is not repeated here. It is interesting, nonetheless, to compare this well-known result with the governing equations, if only to demonstrate that we have reduced the problem to the minimum number of independent variables. 

Since the gases are buffered independently, each counts as a separate phase. 

The phase rule (Eqn. 3.52), then, predicts that our system has N = Ni degrees of freedom. In other words, given a constraint on the concentration or activity of each basis species, we could determine the system s equilibrium state. To constrain the governing equations, however, we need Nc pieces of information, somewhat more than the degrees of freedom predicted by the phase rule. 

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