Diffusivity,mass diffusivity ordiffusion coefficient is usually written as the proportionality constant between themolar flux due to moleculardiffusion 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 fromFick's law and plays a role in numerous other equations ofphysical chemistry.

The diffusivity is generally prescribed for a given pair of species andpairwise 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 inSI units and cm²/s inCGS units.

The diffusion coefficient in solids at different temperatures is generally found to be well predicted by theArrhenius 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 theactivation energy for diffusion (in J/mol),
• T is the absolute temperature (in K),
• R ≈ 8.31446 J/(mol⋅K) is theuniversal gas constant.

Diffusion in crystalline solids, termedlattice diffusion, is commonly regarded to occur by two distinct mechanisms,^[3]interstitial andsubstitutional orvacancy diffusion. The former mechanism describes diffusion as the motion of the diffusing atoms betweeninterstitial 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 ofvacancy 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.^[3]

An approximate dependence of the diffusion coefficient on temperature in liquids can often be found usingStokes–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 andT₂ are the corresponding absolute temperatures,
• μ is thedynamic viscosity of the solvent.

The dependence of the diffusion coefficient on temperature for gases can be expressed usingChapman–Enskog theory (predictions accurate on average to about 8%):^[4]

$${\displaystyle D=\frac{A{T}^{\frac{3}{2}}}{p{\sigma }_{12}^2\mathrm{\Omega }}\sqrt{\frac{1}{M_1}+\frac{1}{M_2}},}$$$${\displaystyle A=\frac{3}{8}{k}_b^{\frac{3}{2}}\sqrt{\frac{N_A}{2\pi }}}$$

where

• D is the diffusion coefficient (cm²/s),^[4][5]
• A is approximately$1.859\times {10}^{-3}\frac{\mathrm{a}\mathrm{t}\mathrm{m}\cdot {\AA }^2\cdot {\mathrm{c}\mathrm{m}}^2}{{\mathrm{K}}^{3/2}\cdot \mathrm{s}}\sqrt{\frac{\mathrm{g}}{\mathrm{m}\mathrm{o}\mathrm{l}}}$ (with Boltzmann constant$k_b$, and Avogadro constant$N_A$)
• 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),
• ${\sigma }_{12}=\frac{1}{2}({\sigma }_1+{\sigma }_2)$ is the average collision diameter (the values are tabulated^[6] page 545) (Å),
• Ω is a temperature-dependent collision integral (the values tabulated for someintermolecular potentials,^[6] can be computed from correlations for others,^[7] or must be evaluated numerically.) (dimensionless).

The relation

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

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

$${\displaystyle D=D_0\frac{T^{1/2}}{n\mathrm{\Omega }(T)}}$$

where${\displaystyle n}$ is the molar density (mol / m³) 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-negligibleco-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 toRevised Enskog Theory.^[9] Revised Enskog Theory predicts a diffusion coefficient that decreases somewhat more rapidly with density, and which to a first approximation may be written as

$${\displaystyle D=D_0\frac{T^{1/2}}{ng(\sigma )\mathrm{\Omega }(T)}}$$

where${\displaystyle g(\sigma )}$ is theradial distribution function evaluated at thecontact diameter of the particles. For molecules behaving likehard, elastic spheres, this value can be computed from theCarnahan-Starling Equation, while for more realistic intermolecular potentials such as theMie potential orLennard-Jones potential, its computation is more complex, and may involve invoking athermodynamic perturbation theory, such asSAFT.

For self-diffusion in gases at two different pressures (but the same temperature), the following empirical equation has been suggested:^[4]$${\displaystyle \frac{D_{P1}}{D_{P2}}=\frac{{\rho }_{P2}}{{\rho }_{P1}},}$$where

• D is the diffusion coefficient,
• ρ is the gas mass density,
• P₁ andP₂ are the corresponding pressures.

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 andr 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).

The effective diffusion coefficient describes diffusion through the pore space ofporous media.^[10] It ismacroscopic 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:$${\displaystyle D_{\text{e}}=\frac{D{\epsilon }_t\delta }{\tau },}$$where

• D is the diffusion coefficient in gas or liquid filling the pores,
• ε_t is theporosity available for the transport (dimensionless),
• δ is theconstrictivity (dimensionless),
• τ is thetortuosity (dimensionless).

The transport-availableporosity 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 theviscosity 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.

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

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

• Transport phenomena
• Diffusion

• CS1 Russian-language sources (ru)
• Articles with short description
• Short description matches Wikidata
• All articles with unsourced statements
• Articles with unsourced statements from April 2021