Arrhenius Equation for Diffusion formula
| Symbol | Quantity | SI unit | Meaning |
|---|---|---|---|
| D | Diffusion coefficient | m²/s | Diffusion coefficient at temperature T |
| D₀ | Pre-exponential factor | m²/s | Temperature-independent pre-exponential factor |
| Q | Activation energy | J/mol | Activation energy for diffusion (per mole or per atom) |
| T | Temperature | K | Absolute temperature (°C and °F are converted automatically) |
What is Arrhenius Equation for Diffusion?
Arrhenius Equation for Diffusion: Diffusion in solids is thermally activated: the diffusion coefficient increases exponentially with absolute temperature.
Atoms in a solid must jump over an energy barrier to move from one lattice site to the next. The fraction of atoms with enough thermal energy follows a Boltzmann factor, so the diffusion coefficient obeys an Arrhenius relation. D0 collects the jump frequency and geometry, and Q is the activation energy. R = 8.314 J/(mol·K) is the gas constant.
Because the dependence is exponential, small temperature changes matter a lot. For carbon in γ-iron, going from 900 °C to 1000 °C roughly triples D. A plot of ln D against 1/T gives a straight line with slope −Q/R, which is how D₀ and Q are measured.
If you know Q per atom in electron-volts, select eV/atom. The calculator multiplies by Avogadro's number for you (1 eV/atom = 96.485 kJ/mol).
How to use this calculator
- Choose the quantity you want to find in the Calculate menu. It is highlighted as the result field.
- Enter the known values. Scientific notation such as
1.6e-11is accepted. - Select a unit for each value. Units are converted to SI automatically.
- The answer updates as you type. Open Show calculation steps to see the working, or use Copy result to share a link with your inputs.
Worked example
Carbon in FCC iron has D0 = 2.3 × 10⁻⁵ m²/s and Q = 148 kJ/mol. Find D at 900 °C.
- T = 900 + 273.15 = 1173.15 K.
- Q/RT = 148 000 / (8.314 × 1173.15) = 15.17.
- D = 2.3 × 10⁻⁵ × e−15.17 = 5.9 × 10⁻¹² m²/s.
Frequently asked questions
How do I find Q from two measured diffusion coefficients?
Use Q = R · ln(D₁/D₂) / (1/T₂ − 1/T₁). Once you have Q, put either data point into this calculator with "D₀" as the unknown to get the pre-exponential factor.
Typical activation energies for diffusion?
Interstitial diffusion (C, N, H in metals) usually has Q of about 40–150 kJ/mol. Substitutional self-diffusion in metals is about 150–300 kJ/mol. Interstitial diffusion is therefore much faster.
Is the Arrhenius equation valid for liquids and gases?
Liquid diffusion often follows an Arrhenius-like trend over modest ranges, but the Stokes–Einstein relation (D ∝ T/η) is usually more physical. Gas diffusivity scales roughly with T^1.75 and inversely with pressure.