Zeroth Law of Thermodynamics Reino W. Hakalal Howard University Washington, D. C.
and Gibbs Phase Rule A statistical derivation
The present derivation parallels an earlier one of the Boltzmann distribution law1 whence it is not essential to include as much theoretical and mathematical background material as in the earlier derivation since the latter may be consulted for these details when necessary. Consider a system, at equilibrium, consisting of a different chemical species which are distributed in any manner among p phases. These species can be related to one another through one or more chemical reactions or they may be chemically independent. That is, the system may be of any composition and complexity whatsoever. In order for equilibrium to occur, there is of necessity an interaction among the various chemical species. We shall suppose this interaction to be sufficiently weak, as it is in a gas at very low pressure, that the probability of the system is given to a good degree of approximation by the product of the probabilities of the separate species in the various phases. For a simple system consisting of N molecules of a single species contained in a single phase, it was explained in the earlier derivation that the probability of the simple system is given by pnr = (N!/DNr!)m i
where pl is the probability of each possible separate arrangement of the N molecules, each arrangement being assumed a priori to be of equal probability, and N, is the number of molecules having the energy el. Consequently, according to the assumption of a sufficiently weak interaction, the probability of a system consisting of a species distributed among (p phases is given to a good degree of approximation by
where N,, is the total number of molecules of a particular species s in a particular phase f, N,,
