The model of isothermal diffusion in a polymeric medium derived by El Afil et at. [A. El Afif, M. Grmela, and G. Lebon, J. Non-Newtonian Fluid Mech. 86, 253 (1999)] is investigated in the absence of an overall flow and in mechanical equilibrium. First, we derive its more macroscopic reduced versions and compare them with the models introduced previously in the literature. Next, we investigate the wave propagation of disturbances in the solvent concentration. Subsequently, we specify the free energy and kinetic coefficients that appear in the general governing equations and solve (using both qualitative and numerical methods) the governing equations expressed in the material coordinates. In this way we obtain the time evolution of the solvent concentration, the diffusion flux, the swelling, the internal deformations and stresses, and the internal viscosity associated with the solvent penetration and the swelling. The governing equations involve three parameters that express the individual nature of the mixture: the relaxation time of the polymeric structure, the relaxation time of the diffusion flux, and one parameter that expresses coupling of the polymeric structure and the solvent concentration in free energy. As an illustration, we show that with these three characteristic parameters we can reproduce results of observations that we have selected from the literature [N. L. Thomas and A. H. Windle, Polymer 19, 255 (1978)]. In particular, we reproduce the case 11 type diffusion observed in the absence of a glass-rubber transition.