We present a thermodynamic treatment for obtaining the free energy surface as a function of droplet composition as well as the activation barrier for droplet formation in a multicomponent vapor mixture on a nucleating center which consists of both an insoluble core and some soluble species. An extension of Kuni's method <Kuni et al., 1993a> of studying the free energy of droplet formation to heterogeneous multicomponent condensation makes it possible to find all the main features of the free energy surface without explicitly knowing the free energy itself. Using that method, we obtain the inequalities that serve as criteria for whether heterogeneous condensation is barrierless or occurs in a fluctuational way or does not occur at all. An explicit expression for the height of the activation barrier for heterogeneous multicomponent condensation on mixed nuclei is proposed. The theoretical results are illustrated by numerical calculations for condensation on mixed nuclei in the water-methanol and water-sulfuric acid systems. ¿ 2001 American Geophysical Union