Thermodynamics of droplet formation around a soluble condensation nucleus in the atmosphere of a solvent vapor

An expression for the work of formation of a spherical droplet condensing on a soluble condensation nucleus out of a solvent vapor is derived. The dependence of the formation work on the solvent vapor chemical potential and the droplet and the nucleus residue sizes is analyzed. The balance of the solute matter between the liquid film and the nucleus residue and the effect of overlapping the surface layers of the thin film have been taken into account. It is shown that the equations of the chemical equilibrium of a solute and a solvent in the droplet, resulting from the generating properties of the formation work, coincide with the generalized Gibbs-Kelvin-Köhler and Ostwald-Freundlich equations. The numerical solution of these equations at a fixed number of molecules of the nucleus matter (at an initial size of the nucleus specified) has been performed in the case of the solvent vapor undersaturated over the bulk liquid solvent phase. The solution links the equilibrium sizes of the droplet and the soluble nucleus residue with the chemical potential or the pressure of the solvent vapor saturated over the droplet. It also determines the limiting sizes of the droplet with small nucleus residue above which the chemical equilibrium of the residue surface and the solution film does not exist. The existence of the limiting sizes is responsible for the specific behavior of the droplet thermodynamic characteristics and the work of droplet formation at deliquescence transition from the droplet state with a partly dissolved nucleus to the state of complete dissolution of the nucleus.