Einstein-Smoluchowski relationship
The Einstein-Smoluchowski relationship , also known as the Einstein equation , is a relationship in the field of kinetic gas theory that was first discovered by Albert Einstein (1905) and then by Marian Smoluchowski (1906) in his writings on Brownian motion . It links the diffusion coefficient with the mobility of the particles :
Inscribed therein
- the Boltzmann constant and
- the absolute temperature .
It is an early example of a fluctuation-dissipation relationship .
Diffusion of particles
In areas with a low Reynolds number , the mobility is the reciprocal of the flow coefficient :
The Stokes equation gives for spherical particles with radius :
where denotes the viscosity of the medium.
With this the Einstein equation can be transformed into:
This form is also called the Stokes-Einstein equation .
You can z. For example, be used to the diffusion coefficient of a globular protein in aqueous solution to determine if we have a density of about 1.2 × 10 3 accept kg / m³, we get a protein of 100 kDa : .
Electric conductivity
In relation to the electrical conductivity , one first defines the electron mobility :
in which
- is the electric field strength and
- the drift speed .
For a semiconductor with any density of states , one usually divides the right part of the equation by the charge of the charge carrier . Then the Einstein equation is
With
- the chemical potential and
- the number of particles .
This relationship also applies to the mobility of ions . The Einstein equation thus becomes the "Nernst-Einstein relationship":
literature
- A. Einstein: About the motion of particles suspended in liquids at rest as required by the molecular kinetic theory of heat . In: Annals of Physics . tape 322 , no. 8 , 1905, pp. 549-560 , doi : 10.1002 / andp.19053220806 (free full text).
- M. von Smoluchowski: On the kinetic theory of Brownian molecular motion and suspensions . In: Annals of Physics . tape 326 , no. 14 , 1906, pp. 756–780 , doi : 10.1002 / andp.19063261405 (free full text).