Diffusion Coefficient Calculator
The diffusion coefficient D governs how quickly atoms or ions migrate through a solid material. It follows the Arrhenius temperature dependence: D = D0 * exp(-Q / (R * T)), where D0 is the pre-exponential factor, Q is the activation energy for diffusion, R is the gas constant (8.314 J/(mol*K)), and T is absolute temperature in Kelvin. This relationship is fundamental to heat treatment processes including carburizing, nitriding, and diffusion bonding. Also calculated is the characteristic diffusion length sqrt(D * t), which approximates the penetration depth after time t.
Arrhenius diffusion formula
D = D0 * exp(-Q / (R * T))
R = 8.314 J/(mol*K); T(K) = T(C) + 273.15
Diffusion length = sqrt(D * t)
D0 is the pre-exponential diffusion coefficient (m^2/s), Q is the activation energy in J/mol (input in kJ/mol, converted), and T is absolute temperature in Kelvin. The diffusion length sqrt(D*t) (with t in seconds) gives the order-of-magnitude penetration depth.
Reference values for common diffusion systems
Carbon in austenite (FCC iron): D0 = 2.0e-5 m^2/s, Q = 142 kJ/mol. Carbon in ferrite (BCC iron): D0 = 6.2e-7 m^2/s, Q = 80 kJ/mol (much faster at same temperature). Nitrogen in austenite: D0 = 9.1e-5 m^2/s, Q = 168 kJ/mol. Self-diffusion in aluminum: D0 = 1.7e-4 m^2/s, Q = 142 kJ/mol. Source: ASM Handbook Vol. 4 and NIST diffusion database.
Diffusion coefficient: frequently asked questions
What is the diffusion coefficient?
The diffusion coefficient D (m^2/s) quantifies the rate at which atoms or molecules migrate through a material. It appears in Fick's first law (flux = -D * dC/dx) and Fick's second law. A higher D means faster diffusion. D depends strongly on temperature through the Arrhenius relationship.
What is the Arrhenius equation for diffusion?
D = D0 * exp(-Q / (R * T)), where D0 is the pre-exponential factor (m^2/s), Q is the activation energy (J/mol), R is the gas constant (8.314 J/(mol*K)), and T is absolute temperature (Kelvin). An Arrhenius plot of ln(D) vs. 1/T gives a straight line with slope -Q/R.
What are typical D0 and Q values for carbon in iron?
For carbon diffusion in austenite (gamma iron, FCC): D0 = 2.0 x 10^-5 m^2/s, Q = 142,000 J/mol. At 900 C (1,173 K): D = 2.0e-5 * exp(-142000 / (8.314 * 1173)) = 2.0e-5 * exp(-14.55) = 2.0e-5 * 4.8e-7 = 9.6e-12 m^2/s. This is a fundamental parameter in carburizing heat treatment design.
How is diffusion coefficient used in carburizing?
In case hardening (carburizing), carbon diffuses from the gas atmosphere into the steel surface. The case depth after time t is approximately proportional to sqrt(D*t). Knowing D at the carburizing temperature allows calculation of required time and temperature to achieve a specified case depth, using the complementary error function solution to Fick's second law.
Why is self-diffusion activation energy related to melting point?
Empirically, the activation energy for self-diffusion Q is approximately 18 times R times the absolute melting temperature (Q ~ 18 R Tm). This reflects that atom mobility requires overcoming lattice binding energy, which scales with Tm. This rule helps estimate Q when experimental data are unavailable.
Official sources
- NIST, Diffusion in Metals and Alloys Database: nist.gov.
- ASM International, "ASM Handbook Vol. 4A: Steel Heat Treating Fundamentals and Processes": asminternational.org.
- ASTM E1860, "Standard Test Method for Elapsed Time Calibration of Thermal Analyzers": astm.org.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.