Some new partial differntial equations are developed to describe the diffusion of a dilute solute in a linear elastic solid supporting a static deformation. The analysis is based exclusively on the use of the conservation laws of mass and momentum, as well as, on the introduction of an internal diffusive force to evaluate the diffusion effects. The interesting result is that steady states are insensitive to possible viscous behavior of the diffusing species. In addition, previous theories dealing with such problems are recovered and generalized. Finally, the important experimental observation that implies a linear dependence of the diffusion coefficient on the hydrostatic pressure is theoretically established.