Phase coexistence means that distinct equilibrium phases can persist together under common external conditions. For exchanging material, the phases have equal temperature and equal chemical potentials of each exchanging species. A flat unstressed interface also requires equal pressure; curvature changes the mechanical balance through capillary pressure.
A sealed pure fluid below its liquid-gas critical point can contain both liquid and vapor. Changing the total amount or volume can change their proportions without changing their equilibrium densities at fixed temperature. In a binary mixture, the two phases can instead have different compositions. The common-tangent construction determines their coexistence compositions by enforcing the appropriate chemical and mechanical equalities. Coexistence concerns competing phases with matching intensive conditions, rather than equality of their total extensive free energies for arbitrary sample amounts.
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