Phase Equilibria and the usefulness of μ

The Carnot Cycle blog for this month is about chemical potential. “Chemical potential” in thermodynamics covers a lot of interesting phenomena. It gives a way to model chemistry using the mechanisms of statistical mechanics. It lets us study a substance that’s made of several kinds of particle. This potential is written with the symbol μ, and I admit I don’t know how that symbol got picked over all the possible alternatives.

The chemical potential varies with the number of particles. Each different type of particle gets its own chemical potential, so there may be a μ₁ and μ₂ and μ₃ and so on. The chemical potential μ₁ is how much the free energy varies as the count of particles-of-type-1 varies. μ₂ is how much the free energy varies as the count of particles-of-type-2 varies, and so on. This might strike you as similar to the way pressure and volume of a gas depend on each other, or if you retained a bit more of thermodynamics how the temperature and entropy vary. This is so. Pressure and volume are conjugate variables, as are temperature and entropy, and so are chemical potential and particle number. (And for a wonder, “particle number” means exactly what it suggests: the number of particles of that kind in the system.)

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It was the American mathematical physicist Josiah Willard Gibbs who introduced the concepts of phase and chemical potential in his milestone monograph On the Equilibrium of Heterogeneous Substances (1876-1878) with which he almost single-handedly laid the theoretical foundations of chemical thermodynamics.

In a paragraph under the heading “On Coexistent Phases of Matter” Gibbs mentions – in passing – that for a system of coexistent phases in equilibrium at constant temperature and pressure, the chemical potential μ of any component must have the same value in every phase.

This simple statement turns out to have considerable practical value as we shall see. But first, let’s go through the formal proof of Gibbs’ assertion.

An important result

Consider a system of two phases, each containing the same components, in equilibrium at constant temperature and pressure. Suppose a small quantity dn_i moles of any component i is transferred from phase A in…

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