Bjerrum length

The Bjerrum length (after Danish chemist Niels Bjerrum 1879–1958 [1]) is the separation at which the electrostatic interaction between two elementary charges is comparable in magnitude to the thermal energy scale, ${\displaystyle k_{\rm {B}}T}$, where ${\displaystyle k_{\rm {B}}}$ is the Boltzmann constant and ${\displaystyle T}$ is the absolute temperature in kelvins. This length scale arises naturally in discussions of electrostatic, electrodynamic and electrokinetic phenomena in electrolytes, polyelectrolytes and colloidal dispersions. [2]

In standard units, the Bjerrum length is given by

${\displaystyle \lambda _{\rm {B}}={\frac {e^{2}}{4\pi \varepsilon _{0}\varepsilon _{r}\ k_{\rm {B}}T}},}$

where ${\displaystyle e}$ is the elementary charge, ${\displaystyle \varepsilon _{r}}$ is the relative dielectric constant of the medium and ${\displaystyle \varepsilon _{0}}$ is the vacuum permittivity. For water at room temperature (${\displaystyle T=300{\mbox{ K}}}$), ${\displaystyle \varepsilon _{r}\approx 80}$, so that ${\displaystyle \lambda _{\rm {B}}\approx 0.7{\mbox{nm}}}$.

In Gaussian units, ${\displaystyle 4\pi \varepsilon _{0}=1}$ and the Bjerrum length has the simpler form

${\displaystyle \lambda _{\rm {B}}={\frac {e^{2}}{\varepsilon _{r}k_{\rm {B}}T}}.}$

References

1. http://www.rsc.org/delivery/_ArticleLinking/DisplayArticleForFree.cfm?doi=TF959550X001&JournalCode=TF
2. Russel, William B.; Saville, D. A.; Schowalter, William R. (1989). Colloidal Dispersions. New York: Cambridge University Press.