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

$k_{B}T=\frac{e^2}{4\pi {\epsilon }_0{\epsilon }_r}\frac{1}{|{\mathbf{r}}_1-{\mathbf{r}}_2|}$

leading to

$l_B=\frac{e^2}{4\pi {\epsilon }_0{\epsilon }_r{k}_{B}T}\approx \frac{1.671\times 10^{-5}}{{\epsilon }_{r}T}$

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 ${\epsilon }_r\approx 78$ at 298K one arrives at a value of around $l_B\approx 0.72$ nm.

See also

• Debye length