Chemical potential: the slope, or the intercept?
A free-energy curve is familiar enough that it is easy to read too much into its slope. For a binary molar Gibbs free energy \(g(x_B)\), at fixed temperature and pressure, the slope is the difference between the two chemical potentials:
The tangent drawn at the composition of interest intersects the vertical lines at \(x_B=0\) and \(x_B=1\) at the values \(\mu_A\) and \(\mu_B\), respectively:
This is why a common tangent gives equality of both component chemical potentials between two phases. Its intercepts refer to chemical potentials at the tangent composition, not necessarily to the pure-component free energies.
There is a useful subtlety here. In phase-field models, the quantity called a chemical potential may instead be a variational derivative with respect to a conserved field. With a binary composition constraint it is often an exchange or diffusion potential. Calling that a free-energy derivative is appropriate; identifying it with one individual component’s partial molar Gibbs free energy in the given phase, or simply its chemical potential is the source of confusion. If the plotted curve is only the mixing free energy, its intercepts give chemical-potential changes relative to the chosen pure-component references.
A question for discussion: Which free energy is on the vertical axis, which composition variable is on the horizontal axis, and what is held fixed when we differentiate?
Further reading: Caltech’s binary phase-equilibria tutorial.
Draw the distinction.
Two illustrative molar free-energy branches, \(g^\alpha=(x-0.2)^2\) and \(g^\beta=(x-0.8)^2\), share the same energy reference. Move the slope: both tangents remain parallel. At slope zero, they coincide as a common tangent.
| Phase | Touching composition | μA: intercept at x = 0 | μB: value at x = 1 |
|---|
The colors distinguish phases α and β. The circles mark tangent contacts; squares mark tangent values at x = 0 and x = 1. Energies are illustrative dimensionless molar values. These parabolas demonstrate the geometry, not a fitted alloy model.