Stokes-Einstein Diffusion Calculator

Calculate diffusion coefficient or hydrodynamic radius using the Stokes-Einstein equation.

Calculate

Temperature T (K) (K)

Viscosity eta (Pa.s) (Pa.s)

Water at 25C = 0.001

Radius r (nm) or D (m2/s) (nm or m2/s)

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What the Equation Describes

The Stokes–Einstein relation connects two things that seem unrelated: the random thermal motion of a particle, and the viscous drag opposing it. Einstein showed that the same molecular collisions that push a particle around also resist its motion, and the balance between them fixes the diffusion coefficient.

D = k_BT / (6πηr)

Term	Meaning	Value / units
D	Diffusion coefficient	m²/s
k_B	Boltzmann constant	1.381 × 10⁻²³ J/K
T	Absolute temperature	K — thermal driving energy
η	Solvent viscosity	Pa·s — about 0.00089 for water at 25°C
r	Hydrodynamic radius	m — the particle plus its solvation shell
6π	Stokes drag factor	For a sphere with no-slip boundary

The inverse relationship with radius has an important consequence: D scales with 1/r, not 1/r³. A particle eight times heavier is only twice as large in radius, so it diffuses only half as slowly. Diffusion is therefore far less size-discriminating than intuition suggests, which is why size-exclusion chromatography separates poorly on diffusion alone.

Hydrodynamic Radius

The r in this equation is not the crystallographic radius. It is the effective radius of the particle together with the solvent molecules that move with it. For proteins the hydrodynamic radius is typically 10–20% larger than the radius calculated from crystal structure, and for highly charged or extended molecules the difference is larger still.

This is why the equation is used in reverse so often. Dynamic light scattering measures D directly, and rearranging gives r — the standard method for sizing nanoparticles, micelles and protein aggregates in solution rather than in a crystal.

Worked Examples

Example 1: 5nm nanoparticle in water 25C

D=1.381e-23x298/(6pixe-3x5e-9)

Result: D=8.7e-12 m2/s

DLS measurement target

Example 2: Protein D=6e-11 m2/s in water

r=kBT/(6pi x 0.001 x 6e-11)

Result: r=3.6nm - approximately 220kDa globular

Stokes radius from FCS experiment

Example 3: Small molecule in water

Glucose, r ≈ 0.36 nm, water at 25°C

Result: D ≈ 6.8 × 10⁻¹⁰ m²/s

Close to the measured value of about 6.7 × 10⁻¹⁰. The relation works well for compact molecules in simple solvents.

Example 4: Effect of temperature

Same 5 nm particle at 25°C versus 50°C

Result: D rises about 1.8×

T rises only 8% in kelvin, but water’s viscosity falls about 40%. Most of the gain comes from the viscosity change, not the temperature term.

Example 5: Viscous solvent

5 nm particle in glycerol, η ≈ 1.4 Pa·s

Result: D ≈ 3.1 × 10⁻¹⁴ m²/s

Over 1,500 times slower than in water. Viscosity dominates the denominator, which is why diffusion in polymers and gels is so restricted.

Common Mistakes

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Using the crystallographic radius

The equation requires hydrodynamic radius, which includes the bound solvation shell. Using a crystal radius systematically overestimates D, typically by 10–20% for proteins.

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Applying it to non-spherical particles

The 6π factor assumes a sphere. Rods, discs and unfolded polymers deviate significantly, and the calculated radius becomes an equivalent-sphere approximation rather than a real dimension.

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Forgetting viscosity’s temperature dependence

Raising temperature increases D twice over — directly through T, and indirectly because η falls. Water’s viscosity drops roughly 2% per degree near room temperature, so the net effect is much larger than the T term alone.

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Using it in crowded or confined media

The relation assumes a dilute particle in a continuous solvent. Inside cells, or in gels and concentrated solutions, anomalous diffusion applies and Stokes–Einstein substantially overestimates mobility.

Frequently Asked Questions

Hydrodynamic radius vs crystallographic?

Hydrodynamic radius includes hydration shell and is measured dynamically. Larger than crystallographic radius. Non-spherical proteins give anomalously large apparent size.

Applications?

Dynamic light scattering (DLS), NMR diffusion, FCS, sedimentation analysis - all use Stokes-Einstein to convert D to particle size.

What is hydrodynamic radius?

The effective radius of a particle including the solvent molecules that move with it. It is typically 10–20% larger than the crystallographic radius for proteins.

Why does D depend on 1/r rather than 1/volume?

Because drag on a sphere is proportional to radius, not volume. An eight-fold increase in mass corresponds to only a doubling of radius, so diffusion slows by just half.

How is the equation used to size nanoparticles?

Dynamic light scattering measures D directly, and rearranging gives r = k_BT/(6πηD). This is the standard method for sizing particles, micelles and aggregates in solution.

When does the Stokes–Einstein relation fail?

In crowded environments such as cell interiors, in gels, for strongly non-spherical particles, and when the particle is comparable in size to the solvent molecules.

Why does warming a solution speed diffusion so much?

Two effects compound. D rises directly with absolute temperature, and solvent viscosity falls as temperature increases. For water the viscosity effect is the larger of the two.

Formula Explorer connections

Interpretation: This formula connects concentration, time, temperature or transport to the speed of a chemical process. Assumption: The reaction order and mechanism must match the model. Temperature, catalyst, mixing and mass-transfer limitations can alter the observed rate.

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