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
| Solvent | Temperature | η (mPa·s) |
|---|---|---|
| Water | 20 °C | 1.002 |
| Water | 25 °C | 0.890 |
| Water | 37 °C | 0.692 |
| Ethanol | 25 °C | 1.074 |
| Methanol | 25 °C | 0.544 |
| Acetone | 25 °C | 0.306 |
| Hexane | 25 °C | 0.300 |
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
| Field | What to enter |
|---|---|
| Particle shape | Choose one: Sphere (Stokes–Einstein), Disk moving face first, Disk moving edge first, Disk, random orientation … |
| Temperature | Units: K, °C, °F |
| Solvent viscosity η | Units: 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) | Units: nm, m, cm, mm, µm, Å … |
| Semi-minor axis b | Units: nm, m, cm, mm, µm, Å … |
| Friction coefficient ξ | Units: kg/s, g/s, N·s/m |
| Time for displacement | Units: 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])
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)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.
Frequently asked questions
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.