Why parallel tangents are not enough
We sometimes invoke a parallel-tangent construction when equilibrium actually requires a common tangent. The algebra behind the distinction is simple:
For example, 5 − 3 = 8 − 6, but 5 ≠ 8 and 3 ≠ 6. Equal differences do not imply equality of the individual quantities.
For two phases, parallel tangents establish only
At ordinary binary phase equilibrium, with both components free to exchange at the same temperature and pressure, we instead require
A common tangent has the same slope and the same intercepts on the two molar Gibbs free-energy curves. Parallel tangents may have different intercepts: they fix the difference between the chemical potentials without fixing either one separately. For stable equilibrium, the common tangent must also support the lower free-energy envelope, rather than merely touch two metastable branches.
A question for discussion: If two tangents are parallel but vertically displaced, which equilibrium condition have we checked, and which have we missed? This discussion assumes ordinary bulk phase equilibrium; additional constraints or coherent elastic contributions require the appropriate constrained thermodynamic formulation.
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.