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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_BT = \frac{e^2}{4\pi\varepsilon_0 \varepsilon_r} \frac{1}{|\mathbf{r}_1 - \mathbf{r}_2|}}

leading to

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_B = \frac{e^2}{4\pi\varepsilon_0 \varepsilon_r k_BT} \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon_r \approx 78} at 298K one arrives at a value of around Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_B \approx 0.72} nm.

See also

• Debye length