Bjerrum length

The Bjerrum length $(l_{B})$ is the distance for which the electrostatic potential energy between two charges, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 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