Diffusion coefficient

Diffusion coefficient - a quantitative characteristic of the diffusion rate equal to the amount of substance (in mass units) passing per unit time through a unit area (for example, 1 m²) as a result of thermal motion of molecules with a concentration gradient equal to one (corresponding to a change of 1 mol / l → 0 mol / L per unit length). The diffusion coefficient is determined by the properties of the medium and the type of diffusing particles.

The temperature dependence of the diffusion coefficient in the simplest case is expressed by the Arrhenius law :

D=D_{0}\exp \left(-{\frac {E_{a}}{RT}}\right),

{\ displaystyle D = D_ {0} \ exp \ left (- {\ frac {E_ {a}} {RT}} \ right),}

{\ displaystyle D = D_ {0} \ exp \ left (- {\ frac {E_ {a}} {RT}} \ right),}

Where $D$ ${\ displaystyle D}$ $D$ - diffusion coefficient [m² / s]; $E_{a}$ ${\ displaystyle E_ {a}}$ $E_ {a}$ - activation energy [J]; $R$ ${\ displaystyle R}$ $R$ - universal gas constant [J / K]; $T$ ${\ displaystyle T}$ $T$ - temperature [K].

Content

In liquids

The approximate dependence of the diffusion coefficient on temperature in liquids in the absence of turbulence can be found using the Stokes-Einstein equation , according to the formula:

{\frac {D_{T_{1}}}{D_{T_{2}}}}={\frac {T_{1}}{T_{2}}}{\frac {\mu _{T_{2}}}{\mu _{T_{1}}}},

{\ displaystyle {\ frac {D_ {T_ {1}}} {D_ {T_ {2}}}} = {\ frac {T_ {1}} {T_ {2}}} {\ frac {\ mu _ { T_ {2}}} {\ mu _ {T_ {1}}}},}

{\ displaystyle {\ frac {D_ {T_ {1}}} {D_ {T_ {2}}}} = {\ frac {T_ {1}} {T_ {2}}} {\ frac {\ mu _ { T_ {2}}} {\ mu _ {T_ {1}}}},}

Where

D

{\ displaystyle D}

D

- diffusion coefficient,

T_{1}

{\ displaystyle T_ {1}}

T_ {1}

and

T_{2}

{\ displaystyle T_ {2}}

T_ {2}

absolute temperatures

\mu

{\ displaystyle \ mu}

\ mu

- dynamic viscosity of the solvent.

In gases

The temperature dependence of the diffusion coefficient for gases in the absence of turbulence can be expressed using the Chapman-Enskog theory (with an average accuracy of about 8%) according to the formula:

D={\frac {A\cdot T^{3/2}{\sqrt {1/M_{1}+1/M_{2}}}}{p\sigma _{12}^{2}\Omega }},

{\ displaystyle D = {\ frac {A \ cdot T ^ {3/2} {\ sqrt {1 / M_ {1} + 1 / M_ {2}}}} {p \ sigma _ {12} ^ {2 } \ Omega}},}

{\ displaystyle D = {\ frac {A \ cdot T ^ {3/2} {\ sqrt {1 / M_ {1} + 1 / M_ {2}}}} {p \ sigma _ {12} ^ {2 } \ Omega}},}

Where

D

{\ displaystyle D}

D

- diffusion coefficient ^[1] (cm ² / s),

A is an empirical coefficient equal to

1.859\times 10^{-3}

{\ displaystyle 1.859 \ times 10 ^ {- 3}}

{\ displaystyle 1.859 \ times 10 ^ {- 3}}

atm

\cdot

{\ displaystyle \ cdot}

\ cdot

Å ²

\cdot

{\ displaystyle \ cdot}

\ cdot

cm ²

\cdot {\sqrt {{\text{г}}/{\text{моль}}}}\cdot

{\ displaystyle \ cdot {\ sqrt {{\ text {g}} / {\ text {mol}}}} \ cdot}

{\ displaystyle \ cdot {\ sqrt {{\ text {g}} / {\ text {mol}}}} \ cdot}

K- ³ / ² / s.

1 and 2 are the indices of two types of molecules present in the gas mixture,

T

{\ displaystyle T}

T

- absolute temperature (K),

M

{\ displaystyle M}

M

- the molar mass of the molecules that make up the gas mixture (g / mol),

p

{\ displaystyle p}

p

- pressure (atm),

\sigma _{12}={\frac {1}{2}}(\sigma _{1}+\sigma _{2})

{\ displaystyle \ sigma _ {12} = {\ frac {1} {2}} (\ sigma _ {1} + \ sigma _ {2})}

{\ displaystyle \ sigma _ {12} = {\ frac {1} {2}} (\ sigma _ {1} + \ sigma _ {2})}

effective collision diameter, Å (values are given in the form of a table in ^[2] , p.545),

\Omega

{\ displaystyle \ Omega}

\ Omega

the dimensionless value of the collision integral as a function of temperature (the values are given in the form of a table in ^[2] , but, as a rule, of the order of 1).

Notes

↑ Welty, James R. Fundamentals of Momentum, Heat, and Mass Transfer / James R. Welty, Charles E. Wicks, Robert E. Wilson ... [and others. ] . - Wiley, 2001 .-- ISBN 978-0-470-12868-8 .
↑ ¹ ² Hirschfelder, J. Molecular Theory of Gases and Liquids / J. Hirschfelder, CF Curtiss, RB Bird. - New York: Wiley, 1954. - ISBN 0-471-40065-3 .

Diffusion coefficient

Content

In liquids

In gases

Notes

See also

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