Diffusion Coefficient Calculator — Stokes-Einstein Equation
Compute how fast a spherical particle diffuses in a solvent using the Stokes-Einstein equation — enter temperature, solvent viscosity, and particle radius.
K
mPa·s
nm
μm²/s (Stokes-Einstein, spherical particle)
- 1
Numerator: k_B × T × 10²⁴
1.381 × 10⁻²³ × 298.15 × 10²⁴ = 4,116.405Scaling by 10²⁴ keeps intermediate values readable; it cancels in the final division. - 2
Denominator: 6π × η (mPa·s) × r (nm)
6π × 0.89 × 5 = 83.8805 - 3
D (μm²/s) = Numerator ÷ Denominator
4,116.405 ÷ 83.8805 = 49.0746
How does this calculator work?
D = k_BT/(6πηr) gives the diffusion coefficient of a spherical particle in a liquid. Enter temperature T (K), dynamic viscosity η (mPa·s), and hydrodynamic radius r (nm) to get D in m²/s and μm²/s. Higher temperature, lower viscosity, or smaller radius all increase D. The Boltzmann constant k_B = 1.381 × 10⁻²³ J/K.
Formula
How this is calculated
The Stokes-Einstein equation (Einstein 1905) relates the diffusion coefficient D of a spherical particle in a liquid to absolute temperature T, solvent dynamic viscosity η, and the hydrodynamic radius r of the particle. The denominator 6πηr is the Stokes drag coefficient (friction coefficient) ζ — the force per unit velocity opposing the particle motion. Thermal energy k_BT drives random fluctuations, while ζ resists them; a larger particle or more viscous solvent slows diffusion, while higher temperature speeds it up.
The mean squared displacement (MSD) for 3-D Brownian motion is ⟨r²⟩ = 6Dt, so after 1 second a particle diffuses an RMS distance of √(6D) in SI units. In biological contexts this is often expressed in μm²/s — a typical globular protein (~5 nm radius) in water at 25°C has D ≈ 40–100 μm²/s, meaning it diffuses ~1 μm in a few milliseconds.
The equation assumes a hard sphere with no-slip boundary conditions at its surface, Newtonian solvent, and that the particle is much larger than solvent molecules. It works well for globular proteins, nanoparticles, and colloids in aqueous or organic solvents, but overestimates D when the particle is comparable in size to solvent molecules, or in crowded/viscoelastic media such as the cell interior.
Frequently asked questions
D = k_BT/(6πηr) — it gives the diffusion coefficient of a spherical particle as the ratio of thermal energy k_BT to the Stokes hydrodynamic friction 6πηr. It was derived by Einstein in 1905 as part of his explanation of Brownian motion.
The hydrodynamic radius r (also called the Stokes radius) is the radius of a hard sphere that would diffuse at the same rate as the actual particle in that solvent. For non-spherical or solvated particles it differs from the geometric size measured by X-ray crystallography.
D scales linearly with T, but solvent viscosity η also decreases with temperature, so diffusion speeds up faster than linearly in practice. For water, going from 25°C to 37°C (physiological) increases D by roughly 40% due to both effects.
Also known as
TG we-Calculate Editorial Team. (2026). Diffusion Coefficient Calculator — Stokes-Einstein Equation [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/diffusion-coefficient-calculator
TG we-Calculate Editorial Team. "Diffusion Coefficient Calculator — Stokes-Einstein Equation." TG we-Calculate. 2026. https://we-calculate.com/calculator/diffusion-coefficient-calculator.
TG we-Calculate Editorial Team, "Diffusion Coefficient Calculator — Stokes-Einstein Equation," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/diffusion-coefficient-calculator
@misc{wecalculate_diffusion_coefficient_calculator, title = {Diffusion Coefficient Calculator — Stokes-Einstein Equation}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/diffusion-coefficient-calculator}}, year = {2026}, note = {TG we-Calculate} }
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