Mass diffusivity

Diffusivity, mass diffusivity or diffusion coefficient is usually written as the proportionality constant between the 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 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 mm²/s, and in water its diffusion coefficient is 0.0016 mm²/s.[1][2]

Diffusivity has dimensions of length² / time, or m²/s in SI units and cm²/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:

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

where

D is the diffusion coefficient (in m²/s),
D₀ is the maximal diffusion coefficient (at infinite temperature; in m²/s),
E_A is the activation energy for diffusion (in J/mol),
T is the absolute temperature (in K),
R ≈ 8.31446 J/(mol⋅K) is the universal gas constant.

Liquids

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

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

where

D is the diffusion coefficient,
T₁ and T₂ are the corresponding absolute temperatures,
μ is the dynamic viscosity of the solvent.

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%):[3]

D={\frac {AT^{\frac {3}{2}}}{p\sigma _{12}^{2}\Omega }}{\sqrt {{\frac {1}{M_{1}}}+{\frac {1}{M_{2}}}}},

A={\frac {3}{8}}k_{b}^{\frac {3}{2}}{\sqrt {\frac {N_{A}}{2\pi }}}

where

D is the diffusion coefficient (cm²/s),[3][4]
A is approximately ${\textstyle 1.859\times 10^{-3}\mathrm {{\frac {atm\cdot \mathrm {\AA} ^{2}\cdot {cm}^{2}}{K^{3/2}\cdot s}}{\sqrt {\frac {g}{mol}}}} }$ (with Boltzmann constant ${\textstyle k_{b}}$ , and Avogadro constant ${\textstyle N_{A}}$ )
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),
${\textstyle \sigma _{12}={\frac {1}{2}}(\sigma _{1}+\sigma _{2})}$ is the average collision diameter (the values are tabulated[5] page 545) (Å),
Ω is a temperature-dependent collision integral (the values are tabulated[5] but usually of order 1) (dimensionless).

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:[3]

{\frac {D_{P1}}{D_{P2}}}={\frac {\rho _{P2}}{\rho _{P1}}},

where

D is the diffusion coefficient,
ρ is the gas mass density,
P₁ and P₂ are the corresponding pressures.

Population dynamics: dependence of the diffusion coefficient on fitness

In population dynamics, 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:

D=D_{0}e^{-\alpha r},

where $D_{0}$ 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.[6] 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_e, is estimated as follows:

D_{\text{e}}={\frac {D\varepsilon _{t}\delta }{\tau }},

where

D is the diffusion coefficient in gas or liquid filling the pores,
ε_t 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.

Values of diffusion coefficients (gas)[3]
Species pair		Temperature (°C)	D (cm²/s)
Solute	Solvent	Temperature (°C)	D (cm²/s)
Water (g)	Air (g)	25	0.260
Oxygen (g)	Air (g)	25	0.176

Values of diffusion coefficients (liquid)[3]
Species pair		Temperature (°C)	D (cm²/s)
Solute	Solvent	Temperature (°C)	D (cm²/s)
Acetone (dis)	Water (l)	25	1.16×10⁻⁵
Air (dis)	Water (l)	25	2.00×10⁻⁵
Ammonia (dis)	Water (l)	12	1.64×10⁻⁵
Argon (dis)	Water (l)	25	2.00×10⁻⁵
Benzene (dis)	Water (l)	25	1.02×10⁻⁵
Bromine (dis)	Water (l)	25	1.18×10⁻⁵
Carbon monoxide (dis)	Water (l)	25	2.03×10⁻⁵
Carbon dioxide (dis)	Water (l)	25	1.92×10⁻⁵
Chlorine (dis)	Water (l)	25	1.25×10⁻⁵
Ethane (dis)	Water (l)	25	1.20×10⁻⁵
Ethanol (dis)	Water (l)	25	0.84×10⁻⁵
Ethylene (dis)	Water (l)	25	1.87×10⁻⁵
Helium (dis)	Water (l)	25	6.28×10⁻⁵
Hydrogen (dis)	Water (l)	25	4.50×10⁻⁵
Hydrogen sulfide (dis)	Water (l)	25	1.41×10⁻⁵
Methane (dis)	Water (l)	25	1.49×10⁻⁵
Methanol (dis)	Water (l)	25	0.84×10⁻⁵
Nitrogen (dis)	Water (l)	25	1.88×10⁻⁵
Nitric oxide (dis)	Water (l)	25	2.60×10⁻⁵
Oxygen (dis)	Water (l)	25	2.10×10⁻⁵
Propane (dis)	Water (l)	25	0.97×10⁻⁵
Water (l)	Acetone (l)	25	4.56×10⁻⁵
Water (l)	Ethyl alcohol (l)	25	1.24×10⁻⁵
Water (l)	Ethyl acetate (l)	25	3.20×10⁻⁵

Values of diffusion coefficients (solid)[3]
Species pair		Temperature (°C)	D (cm²/s)
Solute	Solvent	Temperature (°C)	D (cm²/s)
Hydrogen	Iron (s)	10	1.66×10⁻⁹
Hydrogen	Iron (s)	100	124×10⁻⁹
Aluminium	Copper (s)	20	1.3×10⁻³⁰

References

CRC Press Online: CRC Handbook of Chemistry and Physics, Section 6, 91st Edition
Diffusion
Cussler, E. L. (1997). Diffusion: Mass Transfer in Fluid Systems (2nd ed.). New York: Cambridge University Press. ISBN 0-521-45078-0.
Welty, James R.; Wicks, Charles E.; Wilson, Robert E.; Rorrer, Gregory (2001). Fundamentals of Momentum, Heat, and Mass Transfer. Wiley. ISBN 978-0-470-12868-8.
Hirschfelder, J.; Curtiss, C. F.; Bird, R. B. (1954). Molecular Theory of Gases and Liquids. New York: Wiley. ISBN 0-471-40065-3.
Grathwohl, P. (1998). Diffusion in natural porous media: Contaminant transport, sorption / desorption and dissolution kinetics. Kluwer Academic. ISBN 0-7923-8102-5.

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[1] CRC Press Online: CRC Handbook of Chemistry and Physics, Section 6, 91st Edition

[2] Diffusion

[Cussler-3] Cussler, E. L. (1997). Diffusion: Mass Transfer in Fluid Systems (2nd ed.). New York: Cambridge University Press. ISBN 0-521-45078-0.

[Welty-4] Welty, James R.; Wicks, Charles E.; Wilson, Robert E.; Rorrer, Gregory (2001). Fundamentals of Momentum, Heat, and Mass Transfer. Wiley. ISBN 978-0-470-12868-8.

[Hirschfelder-5] Hirschfelder, J.; Curtiss, C. F.; Bird, R. B. (1954). Molecular Theory of Gases and Liquids. New York: Wiley. ISBN 0-471-40065-3.

[Grathwohl-6] Grathwohl, P. (1998). Diffusion in natural porous media: Contaminant transport, sorption / desorption and dissolution kinetics. Kluwer Academic. ISBN 0-7923-8102-5.

Mass diffusivity

Temperature dependence of the diffusion coefficient

Solids

Liquids

Gases

Pressure dependence of the diffusion coefficient

Population dynamics: dependence of the diffusion coefficient on fitness

Effective diffusivity in porous media

Example values

See also

References