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 lattice diffusion, is commonly regarded to occur by two distinct mechanisms,^[3] interstitial and substitutional or vacancy diffusion. The former mechanism describes diffusion as the motion of the diffusing atoms between interstitial sites in the lattice of the solid it is diffusion 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 of vacancy 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]

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 co-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 to Revised 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

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

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.