The behavior of the electric field together with the electron and ion densities in the vicinity of a non-emitting, plane anode is investigated. The selected approach involves non-linear analysis techniques on the continuum equations for steady-state, isothermal conditions where both ionization and two- body recombination are included. Ions, created through electron bombardment of neutral atoms, are repelled toward two stagnation regions: within or near the sheath boundary and near the plasma interface. These equilibria form as a result of the chemistry present: recombination establishes the latter while ionization stipulates the former. As presented, the sheath is fundamentally unstable - ions are driven toward the negative electrode. Using nitrogen data for a numeric example, the following observations are made: a sufficiently strong applied electric field pushes the ion density toward that of the electrons through a well - a constrictive phenomenon. Both a transition region, dominated by density gradients, and a diffusion-driven zone are found to move the system toward the plasma interface. The characteristics of this process are influenced by the applied electric field, but the instability of the chemistry-induced stagnation regions precludes numeric convergence. Insufficient dissipation may prevent the stability of the anode fall model as presented. Suggested improvements to the model descriptions include considering the effects of temperature gradients, magnetic fields, three-body recombination, diffusion written in terms of the electric field, multi-dimensionality and/or time-dependencies.