## How Gibbs derived the Phase Rule

I knew I’d been forgetting something about the end of summer. I’m embarrassed again it was Peter Mander’s Carnot Cycle blog resuming its discussions of thermodynamics.

The September article is about Gibbs’s phase rule. Gibbs here is Josiah Willard Gibbs, who established much of the mathematical vocabulary of thermodynamics. The phase rule here talks about the change of a substance from one phase to another. The classic example is water changing from liquid to solid, or solid to gas, or gas to liquid. Everything does that for some combinations of pressure and temperature and available volume. It’s just a good example because we can see those phase transitions happen whenever we want.

The question that feels natural to mathematicians, and physicists, is about degrees of freedom. Suppose that we’re able take a substance and change its temperature or its volume or its pressure. How many of those things can we change at once without making the material different? And the phase rule is a way to calculate that. It’s not always the same number because at some combinations of pressure and temperature and volume the substance can be equally well either liquid or gas, or be gas or solid, or be solid or liquid. These represent phase transitions, melting or solidifying or evaporating. There’s even one combination — the triple point — where the material can be solid, liquid, or gas simultaneously.

Carnot Cycle presents the way that Gibbs originally derived his phase rule. And it’s remarkably neat and clean and accessible. The meat of it is really a matter of counting, keeping track of how much information we have and how much we want and looking at the difference between the things. I recommend reading it even if you are somehow not familiar with differential forms. Simply trust that a “d” followed by some other letter (or a letter with a subscript) is some quantity whose value we might be interested in, and you should follow the reasoning well.

The Phase Rule formula was first stated by the American mathematical physicist Josiah Willard Gibbs in his monumental masterwork *On the Equilibrium of Heterogeneous Substances (1875-1878)*, in which he almost single-handedly laid the theoretical foundations of chemical thermodynamics.

In a paragraph under the heading *“On Coexistent Phases of Matter”*, Gibbs gives the derivation of his famous formula in just 77 words. Of all the many Phase Rule proofs in the thermodynamic literature, it is one of the simplest and shortest. And yet textbooks of physical science have consistently overlooked it in favor of more complicated, less lucid derivations.

To redress this long-standing discourtesy to Gibbs, CarnotCycle here presents Gibbs’ original derivation of the Phase Rule in an up-to-date form. His purely prose description has been supplemented with clarifying mathematical content, and the outmoded symbols used in the single equation to which he refers have been replaced with their…

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