Understanding the interaction of vibrations with diffusion, e.g. of carbon in fcc lattice, is still of practical relevance. An analytical framework is offered based on the mechanics and thermodynamics, where the evaluated total dissipation by diffusion allows calculation of damping of a vibrating rod from the Gibbs energy balance. Finally, a simple relation is found relating the diffusion coefficient, geometry and elastic properties of the rod with its damping coefficient. This method could be utilized for measurement of the diffusion coefficient.