Two compositions coexist when one straight line is tangent to the free-energy density at both compositions. Equality of its slope gives equal chemical potentials, and equality of its intercept gives equal pressures.
The binodal is the coexistence boundary: its two branches give the compositions of phases joined by the common-tangent construction.
For a dilute-to-dense liquid coexistence problem, the dilute-phase coexistence concentration at specified conditions. It is a branch of the coexistence boundary, not generally the concentration at the critical endpoint.
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