The Phase Rule is Simply Counting Variables

Having made up one such phase in a piston and cylinder, we now make up additional phases, each in its own cylinder, until we have a total of M such piston and cylinders, each containing one phase with A identifiable chemical species. We further specify that the M piston and cylinders each contain different phases for example one contains a gas, another one liquid, another a solid, another some other solid, and so on. 

Next we place all the pistons and cylinders together in a pile and surround them by an adiabatic, constant-volume cover. In making up this pile of pistons and cylinders we have been able to arbitrarily set M[2 + (A— 1)] variables. 

Now suppose that we wish to make up this pile in such a way that, after we have assembled it, we can remove the pistons and cylinders and find that the phases are all already in a state of chemical and physical equilibrium with all the other phases they wUl contact when the pistons and cylinders are removed. In this case, how many of these M[2 + (A 1)] variables can we arbitrarily set The answer must be M [2 + (A — 1)] minus the number of relations that must exist among the phases if they are to be at equilibrium. We know that for these phases to be in physical equilibrium 

FIGURE 15.4 Another of our piston and cylinder arrangements, showing how we make up some phase. 

Each of these rows is M — 1) equations, and there are (2 + AO rows. Thus, there are (M— 1) (2 + N) independent relations among the M— )[2+ (A — 1)] variables, which must be satisfied if we are to have physical equilibrium among all the phases. Therefore, we could set 

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