Mass diffusivity

Diffusivity, mass diffusivity or diffusion coefficient is usually written as the proportionality constant between the

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

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:

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

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.

Diffusion in crystalline solids, termed

Liquids

An approximate dependence of the diffusion coefficient on temperature in liquids can often be found using

Stokes–Einstein equation

, which predicts that

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

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%):^[4]

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

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

where

• D is the diffusion coefficient (cm²/s),^[4][5]
• 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^[6] page 545) (Å),
• Ω is a temperature-dependent collision integral (the values tabulated for some intermolecular potentials,^[6] can be computed from correlations for others,^[7] or must be evaluated numerically.) (dimensionless).

The relation

${\displaystyle D\sim {\frac {T^{3/2}}{p\Omega (T)}}}$

is obtained when inserting the ideal gas law into the expression obtained directly from Chapman-Enskog theory,^[8] which may be written as

${\displaystyle D=D_{0}{\frac {T^{1/2}}{n\Omega (T)}}}$

where ${\displaystyle n}$ is the molar density (mol / m${\displaystyle ^{3}}$) of the gas, and

${\displaystyle D_{0}={\frac {3}{8\sigma _{12}^{2}}}{\sqrt {\left({\frac {R}{2\pi }}\right)\left({\frac {1}{M_{1}}}+{\frac {1}{M_{2}}}\right)}}}$,

with ${\displaystyle R=k_{B}N_{A}}$ the universal gas constant. At moderate densities (i.e. densities at which the gas has a non-negligible

where ${\displaystyle g(\sigma )}$ is the radial distribution function evaluated at the contact diameter of the particles. For molecules behaving like hard, elastic spheres, this value can be computed from the 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 pertubation theory, such as 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:^[4]

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

• 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:

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

where ${\displaystyle D_{0}}$ 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

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

• 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.

See also

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

References