Stokes–Einstein Relation formula
| Symbol | Quantity | SI unit | Meaning |
|---|---|---|---|
| D | Diffusion coefficient | m²/s | Translational diffusion coefficient of the particle |
| T | Temperature | K | Absolute temperature of the fluid |
| η | Dynamic viscosity | Pa·s | Dynamic viscosity of the fluid (water ≈ 0.89 mPa·s at 25 °C) |
| r | Hydrodynamic radius | m | Hydrodynamic (Stokes) radius of the particle |
What is Stokes–Einstein Relation?
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
The Stokes–Einstein equation links the random thermal motion of a particle to the drag it feels in a fluid. The numerator kBT is the thermal energy (Boltzmann constant kB = 1.380649 × 10⁻²³ J/K). The denominator 6πηr is the Stokes drag coefficient of a sphere.
The relation is the basis of dynamic light scattering (DLS) and nanoparticle tracking analysis. These instruments measure D and report a hydrodynamic diameter. It is also used to estimate diffusion of proteins, polymers, vesicles and colloids. It works best when the particle is much larger than the solvent molecules.
How to use this calculator
- Choose the quantity you want to find in the Calculate menu. It is highlighted as the result field.
- Enter the known values. Scientific notation such as
1.6e-11is accepted. - Select a unit for each value. Units are converted to SI automatically.
- The answer updates as you type. Open Show calculation steps to see the working, or use Copy result to share a link with your inputs.
Worked example
A 100 nm diameter nanoparticle (r = 50 nm) is suspended in water at 25 °C (η = 0.89 mPa·s).
- kBT = 1.380649 × 10⁻²³ × 298.15 = 4.116 × 10⁻²¹ J.
- 6πηr = 6π × 0.00089 × 50 × 10⁻⁹ = 8.388 × 10⁻¹⁰ kg/s.
- D = 4.116 × 10⁻²¹ / 8.388 × 10⁻¹⁰ = 4.91 × 10⁻¹² m²/s = 4.91 µm²/s.
Frequently asked questions
What is the hydrodynamic radius?
It is the radius of a hard sphere that diffuses at the same speed as your particle. It includes the solvation layer and any surface coating, so it is usually a bit larger than the size measured by electron microscopy.
Does Stokes–Einstein work for small molecules?
Only approximately. For solutes similar in size to the solvent, the "slip" boundary condition (4π instead of 6π) or empirical correlations like Wilke–Chang often fit better.
What viscosity should I use for water?
Water is about 1.79 mPa·s at 0 °C, 1.00 at 20 °C, 0.89 at 25 °C, 0.69 at 37 °C and 0.28 at 100 °C. Viscosity changes much faster with temperature than T does, so use the value at your actual temperature.