What this calculator does
The Stokes-Einstein equation gives the diffusion coefficient of a sphere as thermal energy divided by the drag it experiences. A 5 nm particle in water at room temperature diffuses at about 4.9×10⁻¹¹ m²/s.
The inverse relationship with radius is gentler than intuition suggests. A particle ten times larger diffuses ten times slower, not a hundred or a thousand times, because drag scales with radius rather than with volume. That is why proteins of quite different sizes have diffusion coefficients within an order of magnitude of each other.
The formula
The diffusion coefficient is the Boltzmann constant times absolute temperature, divided by six pi times the viscosity times the particle radius. The denominator is the Stokes drag coefficient for a sphere. The equation assumes a rigid sphere much larger than the solvent molecules moving through a continuous fluid, which is a reasonable approximation for proteins and colloids and a poor one for small molecules.
| Term | Meaning |
|---|---|
| Diffusion coefficient (D) | How quickly a particle spreads by random motion, in m²/s. |
| Stokes drag | 6πηa, the resistance a sphere feels moving through a viscous fluid. |
| Hydrodynamic radius | The effective radius including bound solvent, which is what the equation really uses. |
| Boltzmann constant | 1.380649×10⁻²³ J/K, exact by definition since 2019. |
The inputs explained
| Field | What to enter |
|---|---|
| Temperature (K) | Temperature in kelvin. |
| Fluid viscosity η (Pa·s) | Fluid viscosity in Pa·s. Water at 25 °C is 0.00089. |
| Particle radius (nm) | Particle radius in nanometres. This should be the hydrodynamic radius, which exceeds the dry radius because of bound solvent. |
When to use it
Estimating protein mobility
Diffusion coefficients determine how fast molecules reach their targets in solution and in cells.
Interpreting light scattering data
Dynamic light scattering measures D and inverts this equation to report a hydrodynamic radius.
Designing a microfluidic device
Mixing in small channels happens by diffusion rather than turbulence, so D sets the timescale.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does particle size affect diffusion?
A range of particle radii in the same fluid.
| Particle radius | D (m²/s) | D (cm²/s) | Particle radius used |
|---|---|---|---|
| 1 nm | 2.4537e-10 m²/s | 2.4537e-6 cm²/s | 1.00 nm |
| 5 nm | 4.9075e-11 m²/s | 4.9075e-7 cm²/s | 5.00 nm |
| 50 nm | 4.9075e-12 m²/s | 4.9075e-8 cm²/s | 50.00 nm |
| 500 nm | 4.9075e-13 m²/s | 4.9075e-9 cm²/s | 500.00 nm |
Questions
What is the hydrodynamic radius?
The effective radius of the particle including the layer of solvent that moves with it. It is always larger than the dry radius from a crystal structure, often by a nanometre or more for a protein, and it is what the equation actually requires.
Does this work for small molecules?
Only roughly. The derivation assumes the particle is much larger than the solvent molecules and moves through a continuous fluid. For a molecule comparable in size to water, that assumption fails and the equation typically underestimates the diffusion coefficient.
How does temperature affect diffusion?
Twice over, and in the same direction. Higher temperature increases thermal energy in the numerator and reduces viscosity in the denominator. Since water viscosity falls steeply with temperature, the viscosity effect usually dominates.
How far does a particle diffuse in a given time?
The mean squared displacement is 6Dt in three dimensions, so distance grows with the square root of time. A particle diffusing 1 micrometre in one second takes 100 seconds to travel 10 micrometres, which is why diffusion is useless over large distances.
For the gas-phase equivalent, see the Graham’s law calculator. For capillary pressure, see the Young-Laplace calculator.