View source for Mass diffusivity

{{Short description|Proportionality constant in some physical laws}}
{{About|the coefficient of molecular diffusion of mass||Diffusivity}}

'''Diffusivity''', '''mass diffusivity''' or '''diffusion coefficient''' is usually written as the proportionality constant between the [[Flux#Chemical diffusion|molar flux]] due to molecular [[diffusion]] and the negative value of the gradient in the concentration of the species. More accurately, the diffusion coefficient times the local concentration is the proportionality constant between the negative value of the mole fraction gradient and the molar flux. This distinction is especially significant in gaseous systems with strong temperature gradients. Diffusivity derives its definition from [[Fick's law]] and plays a role in numerous other equations of [[physical chemistry]].

The diffusivity is generally prescribed for a given pair of species and [[Pairwise comparison (psychology)|pairwise]] for a multi-species system. The higher the diffusivity (of one substance with respect to another), the faster they diffuse into each other. Typically, a compound's diffusion coefficient is ~10,000× as great in air as in water. Carbon dioxide in air has a diffusion coefficient of 16&nbsp;mm<sup>2</sup>/s, and in water its diffusion coefficient is 0.0016&nbsp;mm<sup>2</sup>/s.<ref>[http://www.crcpress.com/product/isbn/9781439820773 CRC Press Online: CRC Handbook of Chemistry and Physics, Section 6, 91st Edition]</ref><ref>[http://www.cco.caltech.edu/~brokawc/Bi145/Diffusion.html Diffusion<!-- Bot generated title -->]</ref>

Diffusivity has dimensions of length<sup>2</sup> / time, or m<sup>2</sup>/s in [[SI units]] and cm<sup>2</sup>/s in [[CGS units]].

==Temperature dependence of the diffusion coefficient==

=== Solids ===
The diffusion coefficient in solids at different temperatures is generally found to be well predicted by the [[Arrhenius equation]]:

<math display="block">D = D_0 \exp\left(-\frac{E_\text{A}}{RT}\right)</math>

where
*''D'' is the diffusion coefficient (in m<sup>2</sup>/s),
*''D''<sub>0</sub> is the maximal diffusion coefficient (at infinite temperature; in m<sup>2</sup>/s),
*''E''<sub>A</sub> is the [[activation energy]] for diffusion (in J/mol),
*''T'' is the absolute temperature (in K),
*''R'' ≈ 8.31446{{nbsp}}J/(mol⋅K) is the [[gas constant|universal gas constant]].
Diffusion in crystalline solids, termed [[Lattice diffusion coefficient|lattice diffusion]], is commonly regarded to occur by two distinct mechanisms,<ref name=":0">{{Cite book |last1=Callister |first1=William D. |title=Fundamentals of materials science and engineering: an integrated approach |last2=Rethwisch |first2=David G. |date=2012 |publisher=Wiley |isbn=978-1-118-06160-2 |edition=4 |location=Hoboken, NJ}}</ref> [[Interstitial site|interstitial]] and [[Substitutional alloy|substitutional]] or [[Lattice diffusion coefficient#Vacancy Diffusion|vacancy diffusion]]. The former mechanism describes diffusion as the motion of the diffusing atoms between [[Interstitial site|interstitial sites]] in the lattice of the solid it is diffusing into, the latter describes diffusion through a mechanism more analogue to that in liquids or gases: Any crystal at nonzero temperature will have a certain number of [[Vacancy defect|vacancy defects]] (i.e. empty sites on the lattice) due to the random vibrations of atoms on the lattice, an atom neighbouring a vacancy can spontaneously "jump" into the vacancy, such that the vacancy appears to move. By this process the atoms in the solid can move, and diffuse into each other. Of the two mechanisms, interstitial diffusion is typically more rapid.<ref name=":0" /> 

=== Liquids ===
An approximate dependence of the diffusion coefficient on temperature in liquids can often be found using [[Stokes–Einstein equation]], which predicts that

<math display="block">\frac {D_{T_1}} {D_{T_2}} = \frac {T_1} {T_2} \frac {\mu_{T_2}} {\mu_{T_1}},</math>

where
* ''D'' is the diffusion coefficient,
* ''T''<sub>1</sub> and ''T''<sub>2</sub> are the corresponding absolute temperatures,
* ''μ'' is the [[dynamic viscosity]] of the solvent.
The description of diffusion coefficients in liquid mixtures is more difficult. They can be, for example, modeled using entropy scaling.<ref>{{Cite journal |last1=Schmitt |first1=Sebastian |last2=Hasse |first2=Hans |last3=Stephan |first3=Simon |date=2025-03-17 |title=Entropy scaling for diffusion coefficients in fluid mixtures |journal=Nature Communications |language=en |volume=16 |issue=1 |page=2611 |doi=10.1038/s41467-025-57780-z |issn=2041-1723 |pmc=11914492 |pmid=40097384|bibcode=2025NatCo..16.2611S }}</ref>

=== Gases ===
The dependence of the diffusion coefficient on temperature for gases can be expressed using [[Chapman–Enskog theory]] (predictions accurate on average to about 8%):<ref name="Cussler"/>

<math display="block">D = \frac{A T^\frac{3}{2}}{p\sigma_{12}^{2}\Omega}\sqrt{\frac{1}{M_1} + \frac{1}{M_2}},</math>
<math display="block">A = \frac{3}{8} k_b^\frac{3}{2} \sqrt{\frac{N_A}{2\pi}}</math>

where
* ''D'' is the diffusion coefficient (cm<sup>2</sup>/s),<ref name="Cussler"/><ref name="Welty">
{{cite book
 |first1=James R.    |last1=Welty
 |first2=Charles E. |last2=Wicks
 |first3=Robert E.  |last3=Wilson
 |first4=Gregory    |last4=Rorrer
 |title=Fundamentals of Momentum, Heat, and Mass Transfer
 |publisher=Wiley |year=2001
 |isbn=978-0-470-12868-8
}}</ref>
* ''A'' is approximately <math display="inline">1.859 \times 10^{-3} \mathrm{\frac{atm \cdot \AA^2 \cdot {cm}^2}{K^{3/2} \cdot s} \sqrt{\frac{g}{mol}}}</math> (with Boltzmann constant <math display="inline">k_b</math>, and Avogadro constant <math display="inline">N_A</math>)
* 1 and 2 index the two kinds of molecules present in the gaseous mixture,
* ''T'' is the absolute temperature (K),
* ''M'' is the molar mass (g/mol),
* ''p'' is the pressure (atm),
* <math display="inline">\sigma_{12} = \frac{1}{2}(\sigma_1 + \sigma_2)</math> is the average collision diameter (the values are tabulated<ref name="Hirschfelder">{{cite book |first1=J. |last1=Hirschfelder |first2=C. F. |last2=Curtiss |first3=R. B. |last3=Bird |title=Molecular Theory of Gases and Liquids |url=https://archive.org/details/moleculartheoryo0000hirs |url-access=registration |publisher=Wiley |location=New York |year=1954 |isbn=0-471-40065-3 }}</ref> page 545) (Å),
* Ω is a temperature-dependent collision integral (the values tabulated for some [[Pair potential|intermolecular potentials]],<ref name="Hirschfelder"/> can be computed from correlations for others,<ref>{{Cite journal |date=2014 |title=К юбилею Г.И. Канеля |journal=Теплофизика высоких температур |volume=52 |issue=4 |pages=487–488 |doi=10.7868/s0040364414040279 |issn=0040-3644 |language=ru}}</ref> or must be evaluated numerically.) (dimensionless).
The relation 

<math display="block">D \sim \frac{T^{3/2}}{p \Omega(T)}</math>

is obtained when inserting the ideal gas law into the expression obtained directly from [[Chapman–Enskog theory|Chapman-Enskog theory]],<ref>{{Cite book |last1=Chapman |first1=Sydney |title=The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases |last2=Cowling |first2=Thomas George |last3=Burnett |first3=David |date=1990 |publisher=Cambridge university press |isbn=978-0-521-40844-8 |edition=3rd |series=Cambridge mathematical library |location=Cambridge New York Port Chester [etc.]}}</ref> which may be written as

<math display="block">D = D_0 \frac{T^{1/2}}{n \Omega(T)}</math>

where <math>n</math> is the molar density (mol / m<sup>3</sup>) of the gas, and 

<math display="block">D_0 = \frac{3 }{8 \sigma_{12} ^2 } \sqrt{\left(\frac{R}{2 \pi}\right)\left(\frac{1}{M_1} + \frac{1}{M_2}\right)}, </math>

with <math>R = k_B N_A</math> the universal gas constant. At moderate densities (i.e. densities at which the gas has a non-negligible [[Excluded volume|co-volume]], but is still sufficiently dilute to be considered as gas-like rather than liquid-like) this simple relation no longer holds, and one must resort to [[Revised Enskog theory|Revised Enskog Theory]].<ref>{{Cite journal |last=Cohen |first=E. G. D. |date=1993-03-15 |title=Fifty years of kinetic theory |journal=Physica A: Statistical Mechanics and Its Applications |volume=194 |issue=1 |pages=229–257 |doi=10.1016/0378-4371(93)90357-A |bibcode=1993PhyA..194..229C |issn=0378-4371}}</ref> Revised Enskog Theory predicts a diffusion coefficient that decreases somewhat more rapidly with density, and which to a first approximation may be written as

<math display="block">D = D_0 \frac{T^{1/2}}{n g(\sigma) \Omega(T)}</math>

where <math>g(\sigma)</math> is the [[radial distribution function]] evaluated at the [[Kinetic diameter|contact diameter]] of the particles. For molecules behaving like [[Hard spheres|hard, elastic spheres]], this value can be computed from the [[Hard spheres#The Carnahan-Starling Equation of State|Carnahan-Starling Equation]], while for more realistic intermolecular potentials such as the [[Mie potential]] or [[Lennard-Jones potential]], its computation is more complex, and may involve invoking a [[thermodynamic perturbation theory]], such as [[Statistical associating fluid theory|SAFT]].

==Pressure dependence of the diffusion coefficient==
For self-diffusion in gases at two different pressures (but the same temperature), the following empirical equation has been suggested:<ref name="Cussler">{{cite book |first=E. L. |last=Cussler |title=Diffusion: Mass Transfer in Fluid Systems |edition=2nd |publisher=Cambridge University Press |location=New York |year=1997 |isbn=0-521-45078-0 }}</ref>
<math display="block">\frac {D_{P1}} {D_{P2}} = \frac {\rho_{P2}} {\rho_{P1}}, </math>
where
* ''D'' is the diffusion coefficient,
* ''ρ'' is the gas mass density,
* ''P''<sub>1</sub> and ''P''<sub>2</sub> are the corresponding pressures.

== Population dynamics: dependence of the diffusion coefficient on fitness==

In population dynamics, [[Kinesis (biology)#Basic model of kinesis|kinesis]] is the change of the diffusion coefficient in response to the change of conditions. In models of purposeful kinesis, diffusion coefficient depends on fitness (or reproduction coefficient) ''r'':
<math display="block">D = D_0 e ^{-\alpha r},</math>

where <math>D_0</math> is constant and ''r'' depends on population densities and abiotic characteristics of the living conditions. This dependence is a formalisation of the simple rule: Animals stay longer in good conditions and leave quicker bad conditions (the  "Let well enough alone" model).

==Effective diffusivity in porous media==

The effective diffusion coefficient describes diffusion through the pore space of [[porous media]].<ref name="Grathwohl">{{cite book |first=P. |last=Grathwohl |title=Diffusion in natural porous media: Contaminant transport, sorption / desorption and dissolution kinetics |publisher=Kluwer Academic |year=1998 |isbn=0-7923-8102-5 }}</ref> It is [[macroscopic]] in nature, because it is not individual pores but the entire pore space that needs to be considered. The effective diffusion coefficient for transport through the pores, ''D''<sub>e</sub>, is estimated as follows:
<math display="block"> D_\text{e} = \frac{D\varepsilon_t \delta}{\tau},</math>
where
*''D'' is the diffusion coefficient in gas or liquid filling the pores,
*''ε<sub>t</sub>'' is the [[porosity]] available for the transport (dimensionless),
*''δ'' is the [[constrictivity]] (dimensionless),
*''τ'' is the [[tortuosity]] (dimensionless).
The transport-available [[porosity]] equals the total porosity less the pores which, due to their size, are not accessible to the diffusing particles, and less dead-end and blind pores (i.e., pores without being connected to the rest of the pore system).  The constrictivity describes the slowing down of diffusion by increasing the [[viscosity]] in narrow pores as a result of greater proximity to the average pore wall. It is a function of pore diameter and the size of the diffusing particles.

== Example values ==
Gases at 1 atm., solutes in liquid at infinite dilution.  Legend: (s) – solid; (l) – liquid; (g) – gas; (dis) – dissolved.

{| class="wikitable sortable"
|+ Values of diffusion coefficients (gas)<ref name="Cussler"/>
|-
! colspan=2 | Species pair
! rowspan=2 | Temperature <br/>(°C)
! rowspan=2 | ''D'' <br/>(cm<sup>2</sup>/s)
|-
! Solute
! Solvent
|-
|Water (g) || Air (g) || 25 || 0.260
|-
|Oxygen (g) || Air (g) || 25 || 0.176{{citation needed|reason=Ref [5] does contain the value of 0.176, but the sources cited there don't actually support it; the oxygen/air pair was also removed in the more recent 3rd edition of Ref [5]|date=June 2025}}
|-
|}
{| class="wikitable sortable"
|+ Values of diffusion coefficients (liquid)<ref name="Cussler"/>
|-
! colspan=2 | Species pair
! rowspan=2 | Temperature <br/>(°C)
! rowspan=2 | ''D'' <br/>({{nowrap|{{10^|−5}} cm<sup>2</sup>/s}})
|-
! Solute
! Solvent
|-
| Acetone (dis) || Water (l) || 25 || 1.16
|-
| Air (dis) || Water (l) || 25 || 2.00
|-
| Ammonia (dis) || Water (l) || 12{{citation needed|date=April 2021}} || 1.64
|-
| Argon (dis) || Water (l) || 25 || 2.00
|-
| Benzene (dis) || Water (l) || 25 || 1.02
|-
| Bromine (dis) || Water (l) || 25 || 1.18
|-
| Carbon monoxide (dis) || Water (l) || 25 || 2.03
|-
| Carbon dioxide (dis) || Water (l) || 25 || 1.92
|-
| Chlorine (dis) || Water (l) || 25 || 1.25
|-
| Ethane (dis) || Water (l) || 25 || 1.20
|-
| Ethanol (dis) || Water (l) ||25 || 0.84
|-
| Ethylene (dis) || Water (l) ||25 || 1.87
|-
| Helium (dis) || Water (l) || 25 || 6.28
|-
| Hydrogen (dis) || Water (l) || 25 || 4.50
|-
| Hydrogen sulfide (dis) || Water (l) || 25 || 1.41
|-
| Methane (dis) || Water (l) ||25 || 1.49
|-
| Methanol (dis) || Water (l) ||25 || 0.84
|-
| Nitrogen (dis) || Water (l) || 25 || 1.88
|-
| Nitric oxide (dis) || Water (l) || 25 || 2.60
|-
| Oxygen (dis) || Water (l) ||25 || 2.10
|-
| Propane (dis) || Water (l) ||25 || 0.97
|-
| Water (l) || Acetone (l) || 25 || 4.56
|-
| Water (l) || Ethyl alcohol (l) || 25 || 1.24
|-
| Water (l) || Ethyl acetate (l) || 25 || 3.20
|}
{| class="wikitable sortable"
|+ Values of diffusion coefficients (solid)<ref name="Cussler"/>
|-
! colspan=2 | Species pair
! rowspan=2 | Temperature <br/>(°C)
! rowspan=2 | ''D'' <br/>(cm<sup>2</sup>/s)
|-
! Solute
! Solvent
|-
| Hydrogen || Iron (s) || 10 || 1.66×10<sup>−9</sup>
|-
| Hydrogen || Iron (s) || 100 || 124×10<sup>−9</sup>
|-
| Aluminium || Copper (s) || 20 || 1.3×10<sup>−30</sup>
|}

==See also==
*[[Atomic diffusion]]
*[[Effective diffusion coefficient]]
*[[Lattice diffusion coefficient]]
*[[Knudsen diffusion]]

==References==
{{Reflist}}

{{Authority control}}

{{DEFAULTSORT:Mass Diffusivity}}
[[Category:Transport phenomena]]
[[Category:Diffusion]]