By combining the hypothesis that the thermodynamic curvature is the correlation volume with the hypothesis that the free energy is the inverse of the correlation volume, I propose that the thermodynamic curvature is proportional to the inverse of the free energy near the critical point. This hypothesis leads to a partial differential geometric equation for the free energy which a generalized homogeneous function reduces to a third-order nonlinear ordinary differential equation whose solution is consistent with two-scale factor universality. The resulting scaled equation of state is, overall, in very good agreement with mean-field theory, the three-dimensional Ising model, and experiment for the pure fluid. Universal ratios among the critical amplitudes are also in good agreement with known values. For the non-mean-field theory exponents, the solution considered here is not analytic in the whole one-phase region; the second derivative of the free energy suffers a discontinuity.

• Received 17 June 1991

DOI:https://doi.org/10.1103/PhysRevA.44.3583