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, ${\displaystyle k_{\text{B}}T}$, where ${\displaystyle k_{\text{B}}}$ 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

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

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 (${\displaystyle T\approx 293{\text{ K}}}$), ${\displaystyle \varepsilon _{r}\approx 80}$, so that ${\displaystyle \lambda _{\text{B}}\approx 0.71{\text{ nm}}}$.

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

Bjerrum length in water calculated as a function of temperature.

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

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.