Linearized planar screening solution ($\psi =\zeta e^{-\kappa x}$)

For $|e\psi |\ll k_{B}T$, $\psi \prime \prime ={\kappa }^2\psi$ with ${\kappa }^2=2e^2{n}_0/(\epsilon k_{B}T)$ for a symmetric monovalent salt with per-species reservoir density $n_0$. The half-space solution with $\psi (0)=\zeta$ and vanishing bulk potential is $\zeta e^{-\kappa x}$. If density denotes the sum of species, the written factor of two changes.

The Debye length depends on the density of every mobile ionic species, weighted by squared charge. For dilute ions at a common temperature in a uniform medium, ${\lambda }_D^{-2}={\sum }_i{n}_i{z}_i^2{e}^2/(ϵk_{B}T)$, where $n_i$ is the bulk number density of species $i$.

For a symmetric monovalent electrolyte with density $n$ of each sign, the sum gives $2n{e}^2/(ϵk_{B}T)$. If the symbol $n$ instead denotes the total ion density, the prefactor is $n{e}^2/(ϵk_{B}T)$. The physics is identical; silently changing the meaning of density creates a spurious factor of $\sqrt{2}$ in the screening length.