Bjerrum length: Difference between revisions
Carl McBride (talk | contribs) (Created page with "The '''Bjerrum length''' <math>(l_B)</math> is the distance for which the electrostatic potential energy between two charges, <math>e</math>, is equal to the thermal energy sc...") |
Carl McBride (talk | contribs) (Added an example (water)) |
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leading to | leading to | ||
:<math>l_B = \frac{e^2}{4\pi\varepsilon_0 \varepsilon_r k_BT} </math> | :<math>l_B = \frac{e^2}{4\pi\varepsilon_0 \varepsilon_r k_BT} \approx \frac{1.671 \times 10^{-5}}{\varepsilon_r T}</math> | ||
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 <math>\varepsilon_r \approx 78</math> at 298K <ref>[http://dx.doi.org/10.1063/1.555977 D. P. Fernández, Y. Mulev, A. R. H. Goodwin and J. M. H. Levelt Sengers "A Database for the Static Dielectric Constant of Water and Steam", Journal of Physical and Chemical Reference Data '''24''' pp. 33-69 (1995)]</ref> one arrives at a value of around <math>l_B \approx 0.72</math> nm. | |||
==See also== | ==See also== | ||
*[[Debye length]] | *[[Debye length]] | ||
==References== | ==References== | ||
<references/> | <references/> | ||
Revision as of 19:19, 25 March 2014
The Bjerrum length 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)} is the distance for which the electrostatic potential energy between two charges, 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 e} , is equal to the thermal energy scale 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 k_BT} . Using Coulomb's law, one sets
- 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 k_BT = \frac{e^2}{4\pi\varepsilon_0 \varepsilon_r} \frac{1}{|\mathbf{r}_1 - \mathbf{r}_2|}}
leading to
- 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 = \frac{e^2}{4\pi\varepsilon_0 \varepsilon_r k_BT} \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 [1] 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.