Beginning with the isotropic form of the thermoelectric equations of Onsager's irreversible thermodynamics as treated by Callen and deGroot, a direct transcription is made to the case of inhomogeneous and anisotropic media. The "emf" of Kelvin, Bridgman, Ehrenfest, and Rutgers is the gradient of the electrochemical potential and a general formula is given for this gradient in anisotropic media. Both the energy density vector and the electrochemical potential gradient are expressed in terms of a "transport entropy matrix" ${S_{\mathrm{ij}}}^{*}$. The theory leads in a natural and straightforward way to the postulated expression of Ehrenfest and Rutgers. The formulation is in a form easily adaptable to given experimental boundary conditions and, therefore, brings out the limited validity of the Kelvin symmetry relations, as first pointed out clearly by Kohler. Finally, the theory includes the irreversible effects of heat conduction and Joulean heating, as well as the reversible thermoelectric phenomena and is, therefore, more realistic and complete than the theory of Ehrenfest and Rutgers.

• Received 30 July 1953

DOI:https://doi.org/10.1103/PhysRev.92.877