Grahame equation ($\sigma =2\epsilon \kappa (k_{B}T/e)\sinh [e{\psi }_0/(2k_{B}T)]$)

For a planar surface contacting a symmetric monovalent ideal electrolyte on one side, the Poisson-Boltzmann equation is ${\psi }^{\prime \prime }=(2e{n}_{\infty }/\epsilon )\sinh (e\psi /(k_{B}T))$. Multiplication by ${\psi }^{\prime }$ and integration using the zero field and zero potential in the bulk givesThe solution decays towards the bulk, and the boundary condition $\sigma =-\epsilon {\psi }^{\prime }(0^{+})$ therefore yields the displayed charge-potential relation, with ${\kappa }^2=2e^2{n}_{\infty }/(\epsilon k_{B}T)$. Its small-potential limit is the planar Debye-Hückel double layer. It is nonlinear but still a mean-field model with point-like ions and constant permittivity; it does not include a compact interfacial layer or finite-ion-size saturation.

Differentiate the Grahame equation for a planar symmetric monovalent electrolyte with respect to surface potential at fixed reservoir concentration and temperature. The result is the displayed capacitance per unit area. It approaches $\epsilon {\kappa }_D$ in the Debye–Hückel approximation. A compact surface layer can contribute a separate series capacitance; point-ion packing assumptions also fail at sufficiently high near-surface concentration.^[ref]