  We studied ion concentration profiles and the charge density gradient caused
by electrode reactions in weak electrolytes by using the
Poisson--Nernst--Planck equations without assuming charge neutrality. In weak
electrolytes, only a small fraction of molecules is ionized in bulk. Ion
concentration profiles depend on not only ion transport but also the ionization
of molecules. We considered the ionization of molecules and ion association in
weak electrolytes and obtained analytical expressions for ion densities,
electrostatic potential profiles, and ion currents. We found the case that the
total ion density gradient was given by the Kuramoto length which characterized
the distance over which an ion diffuses before association. The charge density
gradient is characterized by the Debye length for 1:1 weak electrolytes. We
discuss the role of these length scales for efficient water splitting reactions
using photo-electrocatalytic electrodes.
