Diffusion Coefficient Calculator
Use this diffusion coefficient calculator to estimate how quickly particles diffuse under Einstein-Smoluchowski or Stokes-Einstein assumptions.
--
Run the calculator.
What This Diffusion Coefficient Calculator Helps You Do
This page combines the two practical diffusion-coefficient workflows people actually use: direct Einstein relation from a known friction coefficient, and the spherical-particle shortcut built from viscosity and particle size. That keeps the calculator useful for both textbook work and quick physical estimates.
The result is shown in both m²/s and µm²/s so you can move easily between SI transport calculations and smaller-scale intuition.
How to Calculate Diffusion Coefficient Calculator
- Choose the diffusion model: Use spherical-particle mode when you know the solvent viscosity and particle radius, or use friction mode when the friction coefficient is already known.
- Enter temperature as a physical state variable: The calculator converts Celsius to Kelvin because the diffusion relation uses absolute temperature.
- Calculate the friction term if needed: In spherical-particle mode, the friction coefficient is built from viscosity and radius with the Stokes expression.
- Interpret the magnitude: Higher temperatures raise D, while larger particles and more viscous solvents lower D.
Diffusion Coefficient Calculator Formula
| Variable | Meaning | Unit |
|---|---|---|
| D | Diffusion coefficient | m²/s |
| kB | Boltzmann constant | 1.380649 × 10^-23 J/K |
| T | Absolute temperature | K |
| ξ | Friction coefficient | kg/s |
| η | Dynamic viscosity | Pa·s |
Use the worked examples below to check how the formula behaves with real values. If the result looks unexpected, verify the unit assumptions and the meaning of each variable before interpreting the answer.
Worked Examples
- Temperature: 25 C
- Viscosity: 0.89 mPa·s
- Radius: 2 nm
Result: D is 1.2269 × 10^-10 m²/s.
Small particles diffuse quickly in low-viscosity solvents.
- Temperature: 25 C
- Viscosity: 5.0 mPa·s
- Radius: 10 nm
Result: D is much smaller than in water.
Increasing size and viscosity both increase friction and suppress diffusion.
- Temperature: 20 C
- Friction coefficient: 4.0 × 10^-11 kg/s
Result: D is 1.01 × 10^-10 m²/s.
When friction is known from another model, the diffusion coefficient follows directly from temperature and ξ.
- Temperature: 40 C
- Viscosity: 0.65 mPa·s
- Radius: 2 nm
Result: D rises relative to the 25 C case.
Higher temperature and lower viscosity both move D upward.
How to Interpret Your Results
| Range | Meaning | Action |
|---|---|---|
| Above 10^-9 m²/s | Very fast diffusion for small molecules in favorable conditions. | Check whether the assumptions still match a simple particle-scale model. |
| Around 10^-10 to 10^-12 m²/s | Typical nanoparticle or colloidal diffusion scale. | Use the result for order-of-magnitude transport estimates. |
| Below 10^-12 m²/s | Slow diffusion from large particles or strong drag. | Expect transport to be friction-limited and verify viscosity and size inputs carefully. |
Frequently Asked Questions
References
Last reviewed: March 2026