Bjerrum length

{{short description|Comparative measure of electrostatic and thermal energy}}

The Bjerrum length (after Danish chemist Niels Bjerrum 1879–1958 {{Cite journal| url=http://www.rsc.org/delivery/_ArticleLinking/DisplayArticleForFree.cfm?doi=TF959550X001&JournalCode=TF|doi = 10.1039/TF959550X001 | title = Obituary: Professor Niels J. Bjerrum|journal = Transactions of the Faraday Society|year = 1959|volume = 55|pages = X001|url-access = subscription}}) is the separation at which the electrostatic interaction between two elementary charges is comparable in magnitude to the thermal energy scale, $k\_\backslash text\{B\}\; T$, where $k\_\backslash text\{B\}$ is the Boltzmann constant and $T$ 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.

{{cite book | last1=Russel | first1=William B. | first2=D. A. | last2=Saville | first3=William R. | last3=Schowalter |

title=Colloidal Dispersions | publisher=Cambridge University Press | location=New York | year=1989}}

In standard units, the Bjerrum length is given by

$$\backslash lambda\_\backslash text\{B\}\; =\; \backslash frac\{e^2\}\{4\backslash pi\; \backslash varepsilon\_0\; \backslash varepsilon\_r\; \backslash \; k\_\backslash text\{B\}\; T\},$$

where $e$ is the elementary charge, $\backslash varepsilon\_r$ is the relative dielectric constant of the medium and $\backslash varepsilon\_0$ is the vacuum permittivity.

For water at room temperature {{nowrap|($T\; \backslash approx\; 293\; \backslash text\{\; K\}$),}} $\backslash varepsilon\_r\; \backslash approx\; 80$, so that

{{nowrap|$\backslash lambda\_\backslash text\{B\}\; \backslash approx\; 0.71\; \backslash text\{\; nm\}$.}}

In Gaussian units, $4\backslash pi\backslash varepsilon\_0\; =\; 1$ and the Bjerrum length has the simpler form

File:Bjerrum length in water in nanometers.png

$$\backslash lambda\_\backslash text\{B\}\; =\; \backslash frac\{e^2\}\{\backslash varepsilon\_r\; k\_\backslash text\{B\}\; T\}.$$

The relative permittivity ε_r of water decreases so strongly with temperature that the product (ε_r·T) decreases. Therefore, in spite of the (1/T) relation, the Bjerrum length λ_B increases with temperature, as shown in the graph.