Solid oxide cells are promising energy conversion devices that enable the generation of power from diverse fuel sources in fuel cell operation and the production of hydrogen and syngas fuels when operating as electrolyzers. Accelerated degradation has been observed for electrolytic operation in comparison to fuel cell operation. Of particular note, oxygen electrode delamination has been observed for solid oxide electrolyzers. It has been proposed that this mechanical failure mode arises from adverse gradients in the oxygen chemical potential across the electrode-electrolyte interface. A multiphysics model of solid oxide cell operation has been developed for mass and charge transport in the presence of a distributed electrochemical reactions. Both 1D and 2D transport within the cross-section of a planar positive electrode-electrolyte-negative electrode (PEN) unit cell are addressed. Mass transport in the porous composite electrodes is treated using the dusty-gas model. Charge transport is cast in terms of ion and electron electrochemical potentials, a feature that facilitates analysis of mixed ion and electron conduction. The present work focuses on analyzing oxygen chemical potential gradients within the solid electrolyte and the porous composite electrodes, with an emphasis on gradients near interfaces between the PEN components. Two general modeling cases are compared: one neglecting electronic conduction within the solid electrolyte and one accounting for small yet present electronic conduction.  Electrochemical potential distributions are assessed across the cell structure over a range of electrolytic and galvanic operating conditions. These distributions are used to assess the likelihood of adverse oxygen chemical potential gradients within the cell. The effects of interconnect geometry for planar cells are also considered. It is found that variations in electrochemical potential arise within the PEN structure, a phenomenon that may drive mechanical failure. These gradients may increase the likelihood of electrolyzer failure for certain combinations of electrode and interconnect geometry. The use of device scale geometric design is discussed as a means of mitigating these gradients and reducing the likelihood of cell mechanical failure from adverse distributions of reactant chemical potentials.