Stokes-Einstein Relation Calculator
Enter temperature, viscosity and particle radius to compute the diffusion coefficient D=kT/(6πηr) and the drag coefficient γ=6πηr. Water at 300 K, r=1 μm → D≈2.19e-13 m²/s.
Input Data
Results
At a glance:The Stokes-Einstein relation connects the diffusivity of a spherical particle in a viscous fluid to the fluid's temperature and viscosity: D = k_B·T/(6π·η·r), where k_B=1.380649e-23 J/K is the Boltzmann constant, T the absolute temperature, η the dynamic viscosity and r the particle radius. The Stokes drag coefficient γ = 6πηr then gives the terminal velocity under an external force as v_t = F/γ, and the mean-square displacement of Brownian motion as ⟨x²⟩ = 2Dt (in one dimension). The relation follows from balancing thermal energy k_BT against viscous drag at low Reynolds number (Re≪1). It is the bridge between microscopic thermal motion and macroscopic diffusion.
Formula
Diffusion coefficient: D = k_B·T / (6π·η·r)
Drag coefficient: γ = 6π·η·r
Mean-square displacement: ⟨x²⟩ = 2Dt (1-D) or 6Dt (3-D)
Terminal velocity: v_t = F / (6π·η·r)
$$D = \frac{k_B T}{6\pi\eta r}, \quad \gamma = 6\pi\eta r, \quad \langle x^2 \rangle = 2Dt$$How to Use
- Enter temperature T (K), dynamic viscosity η (Pa·s) and particle radius r (m).
- The calculator returns D = kT/(6πηr) and γ = 6πηr.
- Typical values: water 300 K, r=1 μm → D=2.19e-13; r=1 nm → D=2.19e-10.
Case Studies
Protein diffusion in water
Serum at 37°C=310 K, η=0.69e-3 Pa·s; protein radius r=3 nm=3e-9 m.
D = 1.38e-23×310 / (6π×0.69e-3×3e-9) ≈ 4.28e-21 / 3.9e-11 ≈ 1.10e-10 m²/s.
The 1-second RMS displacement √(2Dt) = √(2.2e-10) ≈ 1.48e-5 m = 14.8 μm.
PM2.5 diffusion in air
Air η=1.8e-5 Pa·s, T=293 K; PM2.5 radius r=1.25 μm=1.25e-6 m.
D = 1.38e-23×293 / (6π×1.8e-5×1.25e-6) ≈ 4.04e-21 / 4.24e-10 ≈ 9.5e-12 m²/s.
One-hour displacement √(2Dt) ≈ 2.6e-4 m = 0.26 mm (Brownian motion is tiny; turbulent advection dominates atmospheric spread).
FAQ
Why do smaller particles diffuse faster?
Since D=kT/(6πηr) is inversely proportional to r, halving the radius doubles D. A 1 nm particle has 1000× larger D than a 1 μm particle — which is also why nanoscale drugs penetrate tissue faster.
How does temperature affect diffusion?
D∝T and η falls with temperature. From 0°C to 100°C, T rises 37% but η drops 84%, so D grows about 6-fold overall — the reason hot water dissolves and reacts faster.
What are the conditions for the Stokes-Einstein relation?
(1) the particle is a rigid sphere; (2) low Reynolds number (Re≪1, viscous-dominated); (3) the particle is much larger than the solvent molecules (continuum approximation); (4) dilute solution with no inter-particle interaction. Nanoscale particles may need slip-boundary corrections.
How is the diffusion coefficient measured?
Common methods: (1) dynamic light scattering (DLS) — autocorrelation of Brownian motion; (2) fluorescence correlation spectroscopy (FCS) — intensity fluctuations; (3) particle tracking under microscopy; (4) FRAP — fluorescence recovery rate. D is then inverted to estimate the particle radius (standard in DLS).
Why is intracellular diffusion slower than in water?
Cytoplasmic viscosity η≈2–5e-3 Pa·s (2–5× water) plus molecular crowding from macromolecules reduces the effective D to 1/3–1/10 of water. Large molecules (>500 kDa) barely diffuse and rely on active transport along microtubules.
Related Tools
References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.