Bjerrum length

The Bjerrum length $(l_{B})$ is the distance for which the electrostatic potential energy between two charges, $e$, is equal to the thermal energy scale $k_{B}T$. Using Coulomb's law, one sets

$k_{B}T={\frac {e^{2}}{4\pi \varepsilon _{0}\varepsilon _{r}}}{\frac {1}{|\mathbf {r} _{1}-\mathbf {r} _{2}|}}$

leading to

$l_{B}={\frac {e^{2}}{4\pi \varepsilon _{0}\varepsilon _{r}k_{B}T}}\approx {\frac {1.671\times 10^{-5}}{\varepsilon _{r}T}}$

The charges could come in the form of monovalent ions in a solvent. Thus for distances greater than the Bjerrum length, where thermal fluctuations become stronger than electrostatic interactions, it becomes reasonable to introduce a continuum mean-field representation. For water, whose relative permittivity is $\varepsilon _{r}\approx 78$ at 298K one arrives at a value of around $l_{B}\approx 0.72$ nm.

See also

• Debye length