The Bjerrum length (after Danish chemist Niels Bjerrum 1879–1958 ) is the separation at which the

electrostatic Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for am ...

interaction between two

elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a funda ...

s is comparable in magnitude to the thermal energy scale,

k_\text T

, where

k_\text

is the

Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas consta ...

and

T

is the

absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic ...

kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ph ...

s. This length scale arises naturally in discussions of electrostatic, electrodynamic and electrokinetic phenomena in electrolytes,

polyelectrolyte Polyelectrolytes are polymers whose repeating units bear an electrolyte group. Polycations and polyanions are polyelectrolytes. These groups dissociate in aqueous solutions (water), making the polymers charged. Polyelectrolyte properties are th ...

s and colloidal dispersions. In standard units, the Bjerrum length is given by

\lambda_\text = \frac,

where

e

is the

\varepsilon_r

is the relative

dielectric constant The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insulat ...

of the medium and

\varepsilon_0

is the

vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...

. For water at room temperature

\varepsilon_r \approx 80

, so that In

Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs uni ...

4\pi\varepsilon_0 = 1

and the Bjerrum length has the simpler form Bjerrum length in water in nanometers

\lambda_\text = \frac.

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.

References

{{Reflist Physical chemistry Colloidal chemistry

See also

References