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, $k_B T$, where $k_B$ is the Boltzmann constant and $T$ is the absolute temperature in Kelvin. 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

$\lambda_B = \frac{e^2}{4\pi \varepsilon_0 \varepsilon_r \ k_B T},$

where $e$ is the elementary charge, $\varepsilon_r$ is the relative dielectric constant of the medium and $\varepsilon_0$ is the vacuum permittivity. For water at room temperature ($T = 300 \mbox{ K}$), $\varepsilon_r \approx 80$, so that $\lambda_B \approx 0.7 \mbox{nm}$.

In Gaussian units, $4\pi\varepsilon_0 = 1$ and the Bjerrum length has the simpler form

$\lambda_B = \frac{e^2}{\varepsilon_r k_B T}.$

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.