# Mass diffusivity

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.

It is generally prescribed for a given pair of species. For a multi-component system, it is prescribed for each pair of species in the system.

The higher the diffusivity (of one substance with respect to another), the faster they diffuse into each other.

This coefficient has an SI unit of m2/s (length2/time).

## Temperature dependence of the diffusion coefficient

Typically, a compound's diffusion coefficient is ~10,000× as great in air than in water. Carbon dioxide in air has a diffusion coefficient of 16 mm2/s, and in water its coefficient is 0.0016 mm2/s.[1][2]

The diffusion coefficient in solids at different temperatures is often found to be well predicted by

$D = D_0 \, {\mathrm e}^{-E_{\mathrm A}/RT},$

where

• $\, D$ is the diffusion coefficient
• $\, D_0$ is the maximum diffusion coefficient (at infinite temperature)
• $\, E_A$ is the activation energy for diffusion in dimensions of [energy (amount of substance)−1]
• $\, T$ is the temperature in units of [absolute temperature] (kelvins or degrees Rankine)
• $\, R$ is the gas constant in dimensions of [energy temperature−1 (amount of substance)−1]

An equation of this form is known as the Arrhenius equation.

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

$\frac {D_{T1}} {D_{T2}} = \frac {T_1} {T_2} \frac {\mu_{T2}} {\mu_{T1}}$

where:

T1 and T2 denote temperatures 1 and 2, respectively
D is the diffusion coefficient (cm2/s)
T is the absolute temperature (K),
μ is the dynamic viscosity of the solvent (Pa·s)

The dependence of the diffusion coefficient on temperature for gases can be expressed using the Chapman–Enskog theory (predictions accurate on average to about 8%)[3]:

$D=\frac{1.86 \cdot 10^{-3}T^{3/2}\sqrt{1/M_1+1/M_2}}{p\sigma_{12}^2\Omega}$

where:

• 1 and 2 index the two kinds of molecules present in the gaseous mixture
• T – temperature (K)
• M – molar mass (g/mol)
• p – pressure (atm)
• $\sigma_{12}=\frac{1}{2}(\sigma_1+\sigma_2)$ – the average collision diameter (the values are tabulated[4]) (Å)
• Ω – a temperature-dependent collision integral (the values are tabulated[4] but usually of order 1) (dimensionless).
• D – diffusion coefficient (which is expressed in cm2/s when the other magnitudes are expressed in the units as given above[3]).

## 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:

P1 and P2 denote pressures 1 and 2, respectively
D is the diffusion coefficient (m2/s)
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.[5] 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, De, 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)
• εt 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.

## References

1. ^ CRC Press Online: CRC Handbook of Chemistry and Physics, Section 6, 91st Edition
2. ^ Diffusion
3. ^ a b c Cussler, E. L. (1997). Diffusion: Mass Transfer in Fluid Systems (2nd ed.). New York: Cambridge University Press. ISBN 0521450780.
4. ^ a b Hirschfelder, J.; Curtiss, C. F.; Bird, R. B. (1954). Molecular Theory of Gases and Liquids. New York: Wiley. ISBN 0471400653.
5. ^ Grathwohl, P. (1998). Diffusion in natural porous media: Contaminant transport, sorption / desorption and dissolution kinetics. Kluwer Academic. ISBN 0792381025.

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