The paper describes a mathematical analysis of multi-component diffusion with chemical reaction in the porous materials of high-temperature solid oxide fuel cells. The objectives are to clarify the underlying physics, to investigate different modeling approaches and to establish expressions for the cell voltage loss. The description proceeds from the simplest non-reactive binary diffusion process, through a multi-component analysis with non-reactive diluent gases present, to diffusion in the presence of the water-gas shift chemical reaction. Using a single average diffusion coefficient, an analytical solution can be found, not only for the limiting cases of frozen and equilibrium water-gas shift chemistry but also for the general non-equilibrium situation. A Damköhler number is identified and it is shown that shift equilibrium is not necessarily preserved in the anode flow. The non-equilibrium analysis also reveals unusual behavior whereby the molar fluxes become discontinuous in the equilibrium limit while the mole fractions and cell voltage loss approach the limit continuously. A physically more realistic model based on two diffusion coefficients provides a more detailed description for frozen and equilibrium chemistry but does not yield an explicit non-equilibrium solution. In all, the analysis provides fundamental insight and quantitative predictions for many of the flow phenomena occurring in the porous materials of SOFCs.