  Here we investigate a limiting case of the theory for aggregation and
gelation in the electrical double layer (EDL) of ionic liquids (ILs). The
limiting case investigated only accounts for ion pairs, ignoring the
possibility of larger clusters and a percolating ionic network. This
simplification, however, permits analytical solutions for the properties of the
EDL. The resulting equations demonstrate the competition between the free
energy of an association and the electrostatic potential in the EDL. For small
electrostatic potentials and large negative free energies of associations, the
ion pairs dominate in the EDL. Whereas, for electrostatic potential energies
larger than the free energy of an association, electric-field-induced cracking
of ion pairs occurs. The differential capacitance for this consistent ion
pairing theory has a propensity to have a ``double hump camel'' shape. We
compare this theory against previous free ion approaches, which do not
consistently treat the reversible associations in the EDL.
