The Stokes-Einstein equation
This gives the diffusion coefficient D (in m²/s) for a sphere drifting through a fluid under thermal motion. It holds for roughly spherical particles in dilute solution where flow is laminar (Reynolds number well below 1) — the standard case for molecules and proteins in water.
- kB — Boltzmann constant, 1.380649 × 10−23 J/K.
- T — absolute temperature in kelvin (not °C).
- η — dynamic viscosity of the fluid, in Pa·s. Water is about 0.001 Pa·s at 20 °C.
- r — hydrodynamic (Stokes) radius of the particle, in meters, including its solvent shell.
What moves the number
D climbs with temperature and drops with both viscosity and particle size. Viscosity and radius sit in the denominator, so doubling either roughly halves D. Only the numerator, temperature, pushes diffusion up.
| If this rises | D does | Relationship |
|---|---|---|
| Temperature T | increases | D ∝ T (plus viscosity falls too) |
| Viscosity η | decreases | D ∝ 1 ÷ η |
| Radius r | decreases | D ∝ 1 ÷ r |
Rule of thumb: raising temperature by 10 °C lifts D by about 20–30% in water, because warmer water is also less viscous. To rescale a known D: D₂ = D₁ × (T₂/T₁) × (η₁/η₂).
Typical magnitudes
| Medium | Diffusion coefficient |
|---|---|
| Small molecule in a gas | ~10−5 m²/s |
| Small molecule in water | ~10−9 m²/s |
| Protein in water | ~10−10–10−11 m²/s |
Results are often quoted in cm²/s: 1 m²/s = 10⁴ cm²/s, so a value of 1 × 10−9 m²/s is 1 × 10−5 cm²/s. For non-spherical molecules or organic solvents, use the empirical Wilke-Chang correlation instead.
Common questions
What is the Stokes-Einstein equation?
It gives the diffusion coefficient of a spherical particle in a fluid: D equals the Boltzmann constant times absolute temperature, divided by 6 times pi times the fluid viscosity times the particle radius. It applies to roughly spherical molecules in dilute solution at low Reynolds number.
Does diffusion go up or down with temperature and viscosity?
Diffusion rises with temperature and falls with viscosity. Warmer fluid means more thermal motion, so D increases; thicker fluid means more drag, so D decreases. The two are linked, since viscosity itself usually drops as temperature rises.
Why is diffusion so much slower in liquids than in gases?
A diffusion coefficient in a liquid is typically 10,000 to 100,000 times smaller than in a gas. In a gas molecules travel far between collisions; in a liquid they are packed tight and constantly jostled, so net spread is far slower.
