Arrhenius Diffusion Coefficient Calculator

Diffusion in solids follows an Arrhenius relationship: atoms need enough thermal energy to jump from one lattice site to another. Higher temperature means more jumps per second and faster diffusion.

The Arrhenius diffusion equation:

D = D0 x exp(-Q / (R x T))

where D is the diffusion coefficient in m²/s, D0 is the pre-exponential factor (m²/s), Q is the activation energy (J/mol), R is the gas constant (8.314 J/mol K), and T is temperature in Kelvin.

Diffusion distance. The root-mean-square diffusion distance in time t is:

x = sqrt(2 x D x t)

This tells you how far atoms have spread on average.

Representative values for carbon in iron:

• In austenite (FCC iron): D0 = 2.0×10⁻⁵ m²/s, Q = 142 kJ/mol
• In ferrite (BCC iron): D0 = 1.1×10⁻⁶ m²/s, Q = 80 kJ/mol

Carbon diffuses far faster in ferrite than in austenite at the same temperature: about 30x at 900°C and roughly 100x by 700°C, since the gap widens as things cool. The reason is that body-centred cubic iron is less densely packed than face-centred cubic (a packing factor of 0.68 against 0.74), so a carbon atom has a shorter, easier hop between sites and the activation energy is 80 kJ/mol rather than 142.

Its interstitial sites are smaller, though, not larger, and that is the other half of the story. Ferrite dissolves almost no carbon, about 0.02% at best, while austenite takes up to 2.1%. So carburizing is done in the austenite range despite the slower diffusion, because there is no point moving carbon quickly into a phase that will not hold it. Fast diffusion and high solubility pull in opposite directions here, and solubility wins.

Why Arrhenius? The probability of a thermal fluctuation providing energy Q follows the Boltzmann factor exp(-Q/kT). This same form appears throughout materials science, chemistry, and biology wherever rate processes must overcome an energy barrier.