Einstein-Smoluchowski relationship

This article explains the Einstein-Smoluchowski relationship in kinetic gas theory. For more meanings - see Einstein equation .

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 : ${\displaystyle D}$ ${\displaystyle \mu }$

${\displaystyle D=\mu \cdot k_{\mathrm {B} }\cdot T}$

Inscribed therein

• ${\displaystyle k_{\mathrm {B} }}$the Boltzmann constant and
• ${\displaystyle T}$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 : ${\displaystyle \gamma }$

${\displaystyle \mu ={\frac {1}{\gamma }}}$

The Stokes equation gives for spherical particles with radius : ${\displaystyle r}$

${\displaystyle \gamma =6\pi \cdot \eta \cdot r}$

where denotes the viscosity of the medium. ${\displaystyle \eta }$

With this the Einstein equation can be transformed into:

${\displaystyle \Rightarrow D={\frac {k_{\mathrm {B} }\cdot T}{6\pi \cdot \eta \cdot r}}}$

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 : . ${\displaystyle D\approx 10^{-10}\,\mathrm {m^{2}/s} }$

Electric conductivity

In relation to the electrical conductivity , one first defines the electron mobility :

${\displaystyle \mu ={\frac {v_{\mathrm {d} }}{E}}}$

in which

• ${\displaystyle E}$is the electric field strength and
• ${\displaystyle v_{\mathrm {d} }}$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 ${\displaystyle q}$

${\displaystyle D={\frac {\mu \cdot p}{q\cdot {\frac {\mathrm {d} p}{\mathrm {d} \eta }}}}}$

With

• the chemical potential and ${\displaystyle \eta }$
• the number of particles . ${\displaystyle p}$

This relationship also applies to the mobility of ions . The Einstein equation thus becomes the "Nernst-Einstein relationship":

${\displaystyle \Rightarrow D={\frac {\mu \cdot k_{\mathrm {B} }\cdot T}{q}}}$

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).