Einstein's ratio

In physics (mainly in molecular kinetic theory ), the Einstein relation (also called the Einstein – Smoluchowski relation ) is an expression connecting the mobility of a molecule (molecular parameter) with the diffusion coefficient and temperature (macro parameters). It was independently discovered by Albert Einstein in 1905 and Marian Smoluchowski (1906) in the course of his work on the study of Brownian motion :

D=\mu _{p}k_{B}T,

{\ displaystyle D = \ mu _ {p} k_ {B} T,}

{\ displaystyle D = \ mu _ {p} k_ {B} T,}

Where $D$ ${\ displaystyle D}$ $D$ - diffusion coefficient, $\mu _{p}$ ${\ displaystyle \ mu _ {p}}$ $\ mu _ {p}$ - particle mobility, $k_{B}$ ${\ displaystyle k_ {B}}$ $k_B$ - Boltzmann constant , and $T$ ${\ displaystyle T}$ $T$ - absolute temperature .

Mobility size $\mu _{p}$ ${\ displaystyle \ mu _ {p}}$ $\ mu _ {p}$ determined by the ratio

\mu _{p}=V/F,

{\ displaystyle \ mu _ {p} = V / F,}

{\ displaystyle \ mu _ {p} = V / F,}

Where $V$ ${\ displaystyle V}$ $V$ - stationary velocity of a particle in a viscous medium under the action of a force $F$ ${\ displaystyle F}$ $F$ .

This equation is a particular consequence of the fluctuation-dissipation theorem .

Stokes-Einstein formula

The magnitude of the mobility is not always easy to determine, so if we assume that the Reynolds numbers are small, then for the resistance force experienced by the macroscopic ball (particle), you can use the Stokes formula

F=6\pi \eta rV,

{\ displaystyle F = 6 \ pi \ eta rV,}

Where $\eta$ ${\ displaystyle \ eta}$ - fluid viscosity $r$ ${\ displaystyle r}$ - particle radius.

Thus, the expression is obtained:

D={\frac {k_{B}T}{6\pi \eta r}},

{\ displaystyle D = {\ frac {k_ {B} T} {6 \ pi \ eta r}},}

called the relation (formula) of Stokes - Einstein .

It should be noted that the use of a macroscopic approximation to describe the molecular characteristics of motion gives only estimated results. In practical applications, sometimes a factor of 4 is used instead of 6. It is also often assumed that the viscosity characteristic of microscopic movements is lower than that measured in macroscopic experiments. Nevertheless, the Stokes – Einstein formula gives an estimate of the diffusion coefficient that is correct in order of magnitude.

For the magnitude of the rotational diffusion coefficient , the expression is as follows:

D_{\mathrm {rot} }={\frac {k_{B}T}{8\pi \eta r^{3}}}.

{\ displaystyle D _ {\ mathrm {rot}} = {\ frac {k_ {B} T} {8 \ pi \ eta r ^ {3}}}.}

Einstein's ratio

Stokes-Einstein formula

See also

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