[en] We present a new approach for solving phase equilibria problems in multicomponent systems together with several applications. A mathematical framework is developed that provides a method for generalizing the thermodynamics of a finite-component system to that of a system with an infinite number of components: a polydisperse system. Two new functions, the mole fraction distribution function and the mole fraction density function, play a key role in our method. The phase equilibria conditions are written in terms of these functions and are formally solved. We illustrate the utility of our approach by solving, for a polydisperse generalization of the van der Waals model, three phase-equilibria problems: (1) the fractionation of a polydisperse impurity dissolved in a solvent; (2) the shift of the critical temperature and density due to the presence of a polydisperse impurity; (3) the calculation of the cloud-point surface and critical point of a completely polydisperse system