The analysis of the situation with the mathematical description of mass transfer processes in solid substances in connection with the discovery of a complete solution of Fick's second law is made. This solution, in contrast to routinely used, can be used for the description of the mass transfer in the three media, not only in liquids or gases. The fact is confirmed that at the moment to describe the mass transfer in solids (even using commonly understood Fick's second law) approach is used that does not correspond to the real situation. The differences that must be considered in the mathematical description of the processes of heat transfer and mass transfer are shown. The continued development of a way of explaining the Kirkendall effect shows that Darken approach is flawed because it contains statements that are contrary to the basic tenets of physical theory. Calculations show that the experimental data obtained by Kirkendall, are normally described in terms of the resulting new solutions, and shift of markers is quantitatively explained by the compression of the sample due to the nature of the diffusion fronts in solids.