Diffusion Coefficient Calculator

Article — Diffusion Coefficient Calculator

Diffusion coefficient calculator: Stokes-Einstein and Graham law

A diffusion coefficient D measures how quickly molecules spread through a medium. The Stokes-Einstein equation D = k_BT/(6πηr) gives D for spherical particles in a liquid. Graham law r₁/r₂ = √(M₂/M₁) compares gas effusion rates. Small molecules in water sit near 10⁻⁹ m²/s; gases in air near 10⁻⁵ m²/s.

Einstein published the Stokes-Einstein result in 1905, the same year as his special relativity and photoelectric-effect papers. The work let Jean Perrin make the first absolute determination of Avogadro's number from observations of pollen grains in water — Brownian motion connected to molecular reality through a single equation.

What is a diffusion coefficient?

Diffusion is the random spreading of particles caused by thermal motion. The coefficient D quantifies the rate. By Fick's first law, the molar flux J equals −D times the concentration gradient: more gradient gives more flux, and the proportionality is D. Higher D means faster spreading.

The units are m²/s in SI, or cm²/s in older biochemistry literature. The reason for length-squared per time is geometric: in 3D random walks the mean squared displacement grows linearly with time, ⟨x²⟩ = 6Dt. A molecule with D = 10⁻⁹ m²/s spreads to about 77 µm RMS in 1 second and about 23 mm in a full day.

Stokes-Einstein diffusion formula

The Stokes-Einstein equation links diffusion to fluid viscosity, temperature, and particle size:

The equation assumes spherical rigid particles, dilute solutions, and laminar flow around each particle (Reynolds number well below 1). For typical molecular solutes in water, these conditions hold well. The hydrodynamic radius is not quite the molecular radius — it includes any hydration shell that moves with the particle — but the two are usually close for small uncharged molecules.

Graham law for gas diffusion

Gas molecules move so much faster than liquid molecules that a different framework applies. Graham's law, named for Thomas Graham (1846), relates effusion or diffusion rates of two gases to the inverse square root of their molar masses:

r₁ / r₂ = √(M₂ / M₁) = √(ρ₂ / ρ₁). For hydrogen versus oxygen, the ratio is √(32 / 2) = 4 — hydrogen diffuses through a porous barrier four times faster. Uranium hexafluoride enrichment exploits this scaling: ²³⁵UF₆ effuses 0.43% faster than ²³⁸UF₆, a tiny difference but enough to build a cascade.

Typical diffusion coefficient values

Diffusion coefficients span eight orders of magnitude across phases. Memorize these benchmarks to spot calculator errors:

• Gases in air 10⁻⁵ m²/s. O₂ in air: 2.0 × 10⁻⁵ at 25°C.
• Small molecules in water 10⁻⁹ m²/s. Glucose: 7 × 10⁻¹⁰. Caffeine: 6 × 10⁻¹⁰.
• Proteins in water 10⁻¹⁰ to 10⁻¹¹ m²/s. Lysozyme: 1.0 × 10⁻¹⁰. Hemoglobin: 7 × 10⁻¹¹.
• Ions in water 10⁻⁹ m²/s. H⁺ is unusually fast at 9.3 × 10⁻⁹ (Grotthuss mechanism).
• Solids 10⁻²⁰ m²/s or smaller. Carbon in iron at 25°C is essentially zero on human timescales.

How temperature changes diffusion

Temperature affects diffusion through two channels: the explicit T in the Stokes-Einstein numerator and the viscosity η in the denominator. In water, η drops from 1.79 mPa·s at 0°C to 0.89 mPa·s at 25°C to 0.55 mPa·s at 50°C — roughly halving over 50°C. The net effect is that D roughly doubles for every 25°C rise in water.

For gases, Chapman-Enskog theory predicts D ∝ T^1.5 to T^1.75. A 100°C temperature rise increases gas diffusion by 60–70%. This is much weaker than the liquid case because gas viscosity actually rises with temperature, partly canceling the kinetic-energy boost.

Diffusion coefficient pitfalls

Three common errors trip up new users:

• Mixing units — Stokes-Einstein needs SI throughout. Convert nm to m by 10⁻⁹, cP to Pa·s by 10⁻³, °C to K by adding 273.15.
• Using molecular radius instead of hydrodynamic radius — for hydrated species the hydrodynamic radius is 10–30% larger. NMR-DOSY measures the right value.
• Applying Stokes-Einstein to gases — it's valid only when the particle is much larger than solvent molecules. Use Chapman-Enskog or Graham law for gas-phase transport.

Measuring diffusion experimentally

Four common techniques span the practical range of diffusion measurement:

NMR-DOSY (Diffusion-Ordered Spectroscopy) applies pulsed magnetic field gradients to encode position; the signal decay vs. gradient strength yields D directly. Works for any NMR-active nucleus, with typical 5% accuracy. Used routinely for protein and polymer characterization.

Dynamic Light Scattering (DLS) measures intensity fluctuations of scattered laser light caused by Brownian motion. The autocorrelation function decay constant gives D; via Stokes-Einstein, this gives hydrodynamic radius. Standard tool for nanoparticle sizing.

Fluorescence Recovery After Photobleaching (FRAP) bleaches a spot in a fluorescent sample, then watches fresh fluorophores diffuse in. Recovery rate gives D. Essential for live-cell membrane and cytoplasm diffusion studies.

Taylor-Aris dispersion measures broadening of a tracer pulse flowing through a capillary at known Reynolds number. The bandwidth at the outlet gives D. Useful for liquid-phase measurements when NMR or fluorescence isn't practical.