  With the use of thermodynamics and general equilibrium conditions only, we
study the entropy of a fluid in the vicinity of the critical point of the
liquid-vapor phase transition. By assuming a general form for the coexistence
curve in the vicinity of the critical point, we show that the functional
dependence of the entropy as a function of energy and particle densities
necessarily obeys the scaling form hypothesized by Widom. Our analysis allows
for a discussion on the properties of the corresponding scaling function, with
the interesting prediction that the critical isotherm has the same functional
dependence, between the energy and particles densities, as the coexistence
curve. In addition to the derivation of the expected equalities of the critical
exponents, the conditions that lead to scaling also imply that while the
specific heat at constant volume can diverge at the critical point, the
isothermal compressibility must do so.
