# Diffusion Coefficient Calculator (Stokes–Einstein)

URL: https://speedycalculator.com/calculators/chemistry/diffusion-coefficient/
Last updated: 2026-10-30

> The diffusion coefficient calculator uses the Einstein relation D = k_B·T/ξ with friction coefficients for spheres (Stokes–Einstein), disks and elongated ellipsoids to find how fast a particle or molecule diffuses in a liquid, plus its typical displacement over a chosen time.

## Inputs

- Particle shape
- Temperature (K, °C, °F)
- Solvent viscosity η (mPa·s (cP), Pa·s, P (poise), µPa·s) — Water: 1.002 mPa·s at 20 °C, 0.890 at 25 °C, 0.692 at 37 °C
- Radius a (semi-major axis for an ellipsoid) (nm, m, cm, mm, µm, Å, pm)
- Semi-minor axis b (nm, m, cm, mm, µm, Å, pm)
- Friction coefficient ξ (kg/s, g/s, N·s/m)
- Time for displacement (s, min, h, days, years, ms)

## Results

- Friction coefficient ξ
- Diffusion coefficient D
- Thermal energy k_BT
- RMS displacement along one axis √(2Dt)
- RMS distance in 3-D √(6Dt)
- Time to diffuse 1 µm (3-D, on average)

## Formula

```
D = k_B·T / ξ · sphere: ξ = 6πηa (Stokes–Einstein) · disk: ξ = 16ηa (face first), 32ηa/3 (edge first), 12ηa (random) · prolate ellipsoid (a ≫ b): ξ = 4πηa/(ln(2a/b) − ½), 8πηa/(ln(2a/b) + ½), 6πηa/ln(2a/b) · ⟨x²⟩ = 2Dt (1-D), ⟨r²⟩ = 6Dt (3-D)
```

## Explanation

Diffusion is the random, thermally driven wandering of particles that slowly evens out differences in concentration. Its speed is captured by the diffusion coefficient D, in m²/s: Fick's first law says the flux is −D times the concentration gradient, and a particle's mean squared displacement grows as 2Dt along any one direction.

Einstein showed in 1905 that D is the thermal energy k_BT divided by the friction coefficient ξ – the drag force per unit velocity the particle feels as it moves through the fluid. For a sphere in a viscous liquid Stokes' law gives ξ = 6πηa, which leads to the Stokes–Einstein equation D = k_BT/(6πηa). Flat disks and long rod-like ellipsoids have different friction depending on whether they move along or across their long axis; a tumbling particle averages over all orientations.

The equation is used in reverse to size nanoparticles and proteins from dynamic light scattering (the "hydrodynamic radius"), to estimate how long a neurotransmitter needs to cross a synapse, and to judge whether mixing in a microfluidic channel is limited by diffusion. Small molecules in water have D ≈ 10⁻⁹ m²/s; proteins about 10⁻¹⁰–10⁻¹¹ m²/s.

### How to use the Diffusion Coefficient Calculator

- **Particle shape**
- **Temperature**
- **Solvent viscosity η** — Water: 1.002 mPa·s at 20 °C, 0.890 at 25 °C, 0.692 at 37 °C
- **Radius a (semi-major axis for an ellipsoid)**
- **Semi-minor axis b**
- **Friction coefficient ξ**
- **Time for displacement**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Pick the particle shape, enter the temperature, the solvent viscosity and the size (radius, or semi-axes for an ellipsoid), or enter a known friction coefficient. The calculator computes ξ for that shape, divides k_BT by it and shows how far the particle typically moves in the time you enter.

### Worked example

A spherical nanoparticle of radius 2 nm in water at 25 °C (η = 0.890 mPa·s): ξ = 6π × 0.00089 × 2×10⁻⁹ = 3.355 × 10⁻¹¹ kg/s, k_BT = 4.116 × 10⁻²¹ J, so D = **1.227 × 10⁻¹⁰ m²/s** (122.7 µm²/s). In one second it moves √(6Dt) = 27.1 µm on average in 3-D, and covers 1 µm in about 1.36 ms.

A rod-like virus with semi-axes 115 nm and 78 nm tumbling in water at 90 °C (η = 0.315 mPa·s): ln(2a/b) = 1.0813, ξ = 6π × 0.000315 × 115×10⁻⁹ / 1.0813 = 6.31 × 10⁻¹⁰ kg/s, D = 7.94 × 10⁻¹² m²/s.

### Viscosity of common solvents

SolventTemperatureη (mPa·s)Water20 °C1.002Water25 °C0.890Water37 °C0.692Ethanol25 °C1.074Methanol25 °C0.544Acetone25 °C0.306Hexane25 °C0.300

### Tips

- Keep temperature in kelvin and pressure in consistent units.
- For gases, ideal-gas behaviour is a good approximation near room conditions.
- Half-life and rate constant are linked by t½ = ln 2 / k for first-order processes.
- Everything is calculated in your browser.

### Common mistakes to avoid

- Using the mass number instead of the molar mass in Graham's law.
- Mixing partial pressures and mole fractions.
- Forgetting that osmotic pressure uses the total particle concentration (van 't Hoff factor).

### Inputs explained

FieldWhat to enterParticle shapeChoose one: Sphere (Stokes–Einstein), Disk moving face first, Disk moving edge first, Disk, random orientation …TemperatureUnits: K, °C, °FSolvent viscosity ηUnits: mPa·s (cP), Pa·s, P (poise), µPa·s. Water: 1.002 mPa·s at 20 °C, 0.890 at 25 °C, 0.692 at 37 °CRadius a (semi-major axis for an ellipsoid)Units: nm, m, cm, mm, µm, Å …Semi-minor axis bUnits: nm, m, cm, mm, µm, Å …Friction coefficient ξUnits: kg/s, g/s, N·s/mTime for displacementUnits: s, min, h, days, years, ms

### What each result means

- **Diffusion coefficient D** = 1.380649e-23×[Temperature]÷[Friction coefficient ξ]
- **Thermal energy k_BT** = 1.380649e-23×[Temperature]×1e21
- **RMS displacement along one axis √(2Dt)** = sqrt(2×[Diffusion coefficient D]×[Time for displacement])
- **RMS distance in 3-D √(6Dt)** = sqrt(6×[Diffusion coefficient D]×[Time for displacement])
- **Time to diffuse 1 µm (3-D, on average)** = 1e-12÷(6×[Diffusion coefficient D])

## Worked example

Inputs — Particle shape: Sphere (Stokes–Einstein); Temperature: 298.15 K; Solvent viscosity η: 0.89 mPa·s (cP); Radius a (semi-major axis for an ellipsoid): 2 nm; Time for displacement: 1 s.

Results — Friction coefficient ξ = 3.3552e-11 kg/s; **Diffusion coefficient D = 1.2269e-10 m²/s**; Thermal energy k_BT = 4.1164 zJ; RMS displacement along one axis √(2Dt) = 15.6644 µm; RMS distance in 3-D √(6Dt) = 27.1315 µm; Time to diffuse 1 µm (3-D, on average) = 1.3585 ms.

## Key terms

- **Diffusion coefficient D**: Proportionality constant between flux and concentration gradient (m²/s).
- **Friction coefficient ξ**: Drag force per unit velocity on a particle moving through a fluid (kg/s).
- **Viscosity η**: A fluid's resistance to flow (Pa·s; 1 mPa·s = 1 cP).
- **Mean squared displacement**: Average of the squared distance travelled by a diffusing particle; grows linearly with time.

## FAQ

**What is the Stokes–Einstein equation?**
D = k_BT/(6πηa): the diffusion coefficient of a sphere of radius a in a liquid of viscosity η at absolute temperature T.

**What is the diffusion coefficient of a 2 nm particle in water?**
At 25 °C (η = 0.890 mPa·s): D = 1.380649×10⁻²³ × 298.15 / (6π × 0.00089 × 2×10⁻⁹) = 1.227 × 10⁻¹⁰ m²/s.

**What are typical diffusion coefficients?**
Gases in air about 10⁻⁵ m²/s; small molecules in water 10⁻⁹ m²/s (glucose 6.7×10⁻¹⁰); proteins 10⁻¹¹–10⁻¹⁰ m²/s; molecules in solids 10⁻¹⁴ m²/s or less.

**How does temperature affect diffusion?**
Directly through T and, much more, through viscosity: water at 37 °C is 22% less viscous than at 25 °C, so D rises about 34% between the two (the T factor adds 4%).

**What is a hydrodynamic radius?**
The radius of the sphere that diffuses at the same speed as your particle; dynamic light scattering instruments report it by inverting Stokes–Einstein.

**How far does a molecule diffuse in one second?**
For D = 10⁻⁹ m²/s the RMS 3-D distance is √(6Dt) ≈ 77 µm in 1 s – but it takes 10 000 times longer to go 100 times further.

**Why do the ellipsoid formulas need a ≫ b?**
They are slender-body approximations; for a/b close to 1 use the sphere formula with an equivalent radius instead.


## Sources

- [NIST – Fundamental physical constants (CODATA)](https://physics.nist.gov/cuu/Constants/)
- [LibreTexts Chemistry](https://chem.libretexts.org/)
- [Berg H. C., Random Walks in Biology (Princeton University Press)](https://press.princeton.edu/books/paperback/9780691000640/random-walks-in-biology)