  We construct a novel approach, based on thermodynamic geometry, to
characterize first-order phase transitions from a microscopic perspective,
through the scalar curvature in the equilibrium thermodynamic state space. Our
method resolves key theoretical issues in macroscopic thermodynamic constructs,
and furthermore characterizes the Widom line through the maxima of the
correlation length, which is captured by the thermodynamic scalar curvature. As
an illustration of our method, we use it in conjunction with the mean field Van
der Waals equation of state to predict the coexistence curve and the Widom
line. Where closely applicable, it provides excellent agreement with
experimental data. The universality of our method is indicated by direct
calculations from the NIST database.
