The critical-point conditions for classical polydisperse fluids

The critical-point conditions for a polydisperse mixture are shown to be equivalent to those for the existence of nontrivial solutions to two homogeneous integral equations of the Fredholm type. This mathematically rigorous treatment is not dependent on the form of any particular model free energy and hence shows that there is no formal distinction between the critical-point conditions of a polydisperse fluid and those conditions derived by Gibbs for the critical point of a mixture with a finite number of components. Using the method of Fredholm, we express the critical-point conditions in terms of the zeros of two absolutely convergent expansions, and demonstrate how the expansions may be used to determine the shifts in critical density and temperature caused by changes in the composition of the fluid.