A numerical problem for mass and energy transport considering thermodynamic equilibrium is solved around a spherical particle in the absence of hydrodynamic effects in a binary solution. The analysis includes the radial convective term generated due to the differences on density between the solid and liquid phases. Because the transport of mass and energy compete in the process, how the rate of dissolution is affected by compositional diffusivity or thermal diffusivity is distinguished. The partial differential equations are discretized with the finite difference method in space, and the resulting set of ordinary differential equations in time is solved by the method of lines. The numerical solution for the thermal dissolution of a spherical particle in a binary melt is compared with heat-balance integral method for small times of the process. The solutions are found to agree for conditions close to the dissolution regime.