Mass diffusivity

This article is about coefficient of molecular diffusion of mass. For other uses, see Diffusivity (disambiguation).

Diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is encountered in Fick's law and 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.

Diffusivity has an SI unit of m²/s (length² / time). In CGS units it is given in cm²/s.

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:

$D=D_{0}\,{\mathrm {e} }^{-E_{\mathrm {A} }/(kT)}$

where

D is the diffusion coefficient (m² / s),

D₀ is the maximum diffusion coefficient (at infinite temperature; m² / s),

E_A is the activation energy for diffusion in dimensions of (J atom⁻¹),

T is the absolute temperature (K),

k is the Boltzmann 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 (cm²/s),

T is the absolute temperature (K),

T₁ and T₂ denote temperatures 1 and 2, respectively,

μ is the dynamic viscosity of the solvent (Pa·s).

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%):

$D={\frac {1.858\cdot 10^{-3}T^{3/2}{\sqrt {1/M_{1}+1/M_{2}}}}{p\sigma _{12}^{2}\Omega }}$

where

D is the diffusion coefficient, which is expressed in cm²/s,

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),

$\sigma _{12}={\frac {1}{2}}(\sigma _{1}+\sigma _{2})$ is the average collision diameter (the values are tabulated) (Å),

Ω is a temperature-dependent collision integral (the values are tabulated 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:

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

where

D is the diffusion coefficient (m²/s),

P₁ and P₂ denote pressures 1 and 2, respectively,

ρ is the gas mass density (kg/m³).

Effective diffusivity in porous media

The effective diffusion coefficient describes diffusion through the pore space of porous media. 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:

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

where

D is the diffusion coefficient in gas or liquid filling the pores (m²s⁻¹),

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)

Species pair (solute-solvent)	Temperature (°C)	D (cm2/s)	Reference
Water (g) - Air (g)	25	0.282
Oxygen (g) - Air (g)	25	0.176

Values of diffusion coefficients (liquid)

Species pair (solute-solvent)	Temperature (°C)	D (cm2/s)	Reference
Acetone (dis) - Water (l)	25	1.16x10−5
Air (dis) - Water (l)	25	2.00x10−5
Ammonia (dis) - Water (l)	25	1.64x10−5
Argon (dis) - Water (l)	25	2.00x10−5
Benzene (dis) - Water (l)	25	1.02x10−5
Bromine (dis) - Water (l)	25	1.18x10−5
Carbon Monoxide (dis) - Water (l)	25	2.03x10−5
Carbon Dioxide (dis) - Water (l)	25	1.92x10−5
Chlorine (dis) - Water (l)	25	1.25x10−5
Ethane (dis) - Water (l)	25	1.20x10−5
Ethanol (dis) - Water (l)	25	0.84x10−5
Ethylene (dis) - Water (l)	25	1.87x10−5
Helium (dis) - Water (l)	25	6.28x10−5
Hydrogen (dis) - Water (l)	25	4.50x10−5
Hydrogen sulfide (dis) - Water (l)	25	1.41x10−5
Methane (dis) - Water (l)	25	1.49x10−5
Methanol (dis) - Water (l)	25	0.84x10−5
Nitrogen (dis) - Water (l)	25	1.88x10−5
Nitric oxide (dis) - Water (l)	25	2.60x10−5
Oxygen (dis) - Water (l)	25	2.10x10−5
Propane (dis) - Water (l)	25	0.97x10−5
Water (l) - Acetone (l)	25	4.56x10−5
Water (l) - Ethyl alcohol (l)	25	1.24x10−5
Water (l) - Ethyl acetate (l)	25	3.20x10−5

Values of diffusion coefficients (solid)

Species pair (solute-solvent)	Temperature (°C)	D (cm2/s)	Reference
Hydrogen - Iron (s)	10	1.66x10−9
Hydrogen - Iron (s)	100	124x10−9
Aluminium - Copper (s)	20	1.3x10−30

See also

• Atomic diffusion
• Effective diffusion coefficient
• Lattice diffusion coefficient
• Knudsen diffusion