The equations describing the dissolution and adsorption of the matter comprising the condensation nucleus in the droplet originating on this nucleus from supersaturated vapor were derived. The relaxation of the solution of the matter comprising the nucleus inside the droplet to chemical equilibrium with allowance made for its transport into the solution from the nucleus and its consumption by the adsorption surface layer of a droplet was studied. The dependences of the time of chemical relaxation of a solution on the sizes of the droplet and solid residue of the nucleus, the nucleus solubility, the rate of dissolution of the matter comprising nucleus, the coefficient of its diffusion in a solution, the adsorption, the coefficients of adsorption and desorption of this matter on the droplet surface, and the logarithmic derivative of adsorption with respect to solution concentration were revealed. The passage from the local (over the course of time) approximation where the adsorption is determined by the current value of the solution concentration at its boundary with the droplet surface layer was made for an arbitrary adsorption isotherm. Times of solution chemical relaxation under conditions typical of partial or complete dissolution of nucleus in a droplet were found in the analytical form. It was proved that the variation of droplet boundaries might be neglected at the initial (important for nucleation) stage of surmounting the activation barrier of nucleation by the droplets. It was shown that, at this stage, the state of the solution inside the droplets remains virtually constant. The hierarchy of the scale of kinetic times was disclosed, which enables us to understand the complex multistage process of overcoming the activation barrier of nucleation by the droplets. The applicability of thermodynamics to the kinetics of this process was also demonstrated.