Diffusion Calculators
Everything you need to calculate diffusion: flux, concentration profiles, diffusion coefficients, how far and how fast particles spread.
Fick's First Law Calculator
Diffusion flux from a concentration gradient
Fick's Second Law Calculator
Concentration vs. depth and time (erf solution)
Diffusion Coefficient vs Temperature (Arrhenius)
D = D₀·exp(−Q/RT) for solids
Stokes–Einstein Equation Calculator
Diffusion coefficient of a particle in a fluid
Diffusion Length & Time Calculator
Mean displacement L = √(2nDt)
Graham's Law of Effusion Calculator
Relative effusion/diffusion rates of gases
Membrane Diffusion & Permeability Calculator
Flux through a membrane with partition coefficient
Einstein Relation Calculator (Mobility ↔ Diffusivity)
D = μ·k_B·T / q for charge carriers and ions
Thermal Diffusivity Calculator
α = k / (ρ·cₚ) — how fast heat diffuses
Understanding diffusion
Diffusion is mass transport driven by random molecular motion. No stirring or pressure is needed — just a concentration difference and time. It is described by two equations published by Adolf Fick in 1855:
- Fick's first law gives the flux at steady state: J = −D·dC/dx.
- Fick's second law gives how concentration changes with time: ∂C/∂t = D·∂²C/∂x².
The diffusion coefficient D
The diffusion coefficient (diffusivity) measures how fast a species spreads through a medium. It has units of m²/s (often cm²/s) and spans an enormous range:
| System | Typical D (m²/s) | Model |
|---|---|---|
| Gases in air | 10⁻⁵ | Kinetic theory, Graham's law |
| Small molecules in water | 10⁻⁹ | Stokes–Einstein |
| Proteins in water | 10⁻¹¹ – 10⁻¹⁰ | Stokes–Einstein |
| Interstitial atoms in metals (hot) | 10⁻¹² – 10⁻⁹ | Arrhenius |
| Substitutional atoms in solids | 10⁻²⁰ – 10⁻¹⁴ | Arrhenius |
| Electrons in silicon | 3.6 × 10⁻³ | Einstein relation |
How far does diffusion go?
Diffusion distance grows with the square root of time, L ≈ √(2nDt). A molecule in water spreads about 80 µm in one second, but needs hours to cross a centimetre. Use the diffusion length calculator to estimate times and distances for your system.
Diffusion laws at a glance
The key equations behind every diffusion calculator.
Fick's First Law of Diffusion
The diffusion flux is proportional to the negative concentration gradient: matter flows from high to low concentration at a rate set by the diffusion coefficient. (Adolf Fick, 1855)
Open the Fick's First Law calculator →Fick's Second Law of Diffusion
The rate of change of concentration at a point equals the diffusion coefficient times the curvature of the concentration profile. (Adolf Fick, 1855)
Open the Fick's Second Law calculator →Arrhenius Equation for Diffusion
Diffusion in solids is thermally activated: the diffusion coefficient increases exponentially with absolute temperature.
Open the Diffusion Coefficient vs Temperature calculator →Stokes–Einstein Relation
A sphere undergoing Brownian motion in a viscous fluid has a diffusion coefficient equal to the thermal energy divided by its Stokes friction coefficient. (Albert Einstein, 1905; William Sutherland, 1905)
Open the Stokes–Einstein Equation calculator →Einstein–Smoluchowski Diffusion Relation
For a random walk, the mean squared displacement grows linearly with time: ⟨r²⟩ = 2nDt in n dimensions. (Einstein 1905, Smoluchowski 1906)
Open the Diffusion Length & Time calculator →Graham's Law of Effusion
The rate of effusion of a gas is inversely proportional to the square root of its molar mass. (Thomas Graham, 1848)
Open the Graham's Law of Effusion calculator →Membrane Permeation (Solution–Diffusion Model)
Steady-state flux through a membrane equals the permeability coefficient times the concentration difference across it, where permeability = D·K/L.
Open the Membrane Diffusion & Permeability calculator →Einstein Relation (Nernst–Einstein)
For particles in thermal equilibrium, the diffusion coefficient is proportional to the mobility, with the thermal voltage kT/q as the constant. (Albert Einstein, 1905)
Open the Einstein Relation calculator →Heat Equation (Heat Diffusion)
Temperature diffuses just like concentration: ∂T/∂t = α∇²T. The thermal diffusivity α plays the role of Fick's D. (Joseph Fourier, 1822)
Open the Thermal Diffusivity calculator →