On the shape of the coexistence curve and of the critical isotherm in a density-density plane

The problem, recently posed by Widom and Khosla, of providing a general classification of pairs of densities in the planes of which the coexistence curves and critical isotherms are tangent and of degree (1 − α)/β is solved in the framework of a general transformation theory in thermodynamics. As a general rule we find that a canonical pair of densities is obtained by taking both densities conjugate to field variables in the same canonical expression of the fundamental equation. Moreover, the predicted tangency of the coexistence curve and critical isotherm is shown to be a particular case of a general geometrical property of the field-density transformation.