How fast does a molecule wander through a liquid? The Stokes–Einstein equation answers from first principles: thermal energy drives the wandering, and solvent drag resists it, with the balance set by the particle’s size. This Diffusion Coefficient Calculator takes temperature, solvent viscosity and hydrodynamic radius, and returns the diffusion coefficient D in m²/s and cm²/s — for molecules, nanoparticles and colloids in solution.
Worked Example: A Small Molecule in Water at 25 °C
With T = 298.15 K, η = 0.89 mPa·s and r = 0.5 nm, the calculator gives D = kT/(6πηr) = (1.380649×10⁻²³ × 298.15) / (6π × 0.00089 × 0.5×10⁻⁹) = 4.91×10⁻¹⁰ m²/s (4.91×10⁻⁶ cm²/s). That figure sits exactly in the measured band for small molecules in water — glucose diffuses at roughly 6×10⁻¹⁰ m²/s, sucrose near 5×10⁻¹⁰ — which is the Stokes–Einstein equation performing at the edge of its comfort zone, since a 0.5 nm solute is barely larger than the solvent molecules the continuum drag law imagines. Translate D into a timescale with the diffusion length x ≈ √(2Dt): crossing a 10 µm cell takes this molecule about (10⁻⁵)²/(2 × 4.91×10⁻¹⁰) ≈ 0.10 s, while crossing 1 mm takes about 1,000 s — diffusion is swift at cellular scales and hopeless at macroscopic ones, the single fact that explains why cells are small and why stirred tanks exist.
The Equation and Its Two Levers
Stokes–Einstein merges two classic results: Stokes’ drag on a sphere (F = 6πηrv, the source of the 6π and the inverse-radius law) and Einstein’s fluctuation–dissipation relation (D = kT/friction coefficient). The consequences are all ratios, easy to reason with: D is proportional to absolute temperature, inversely proportional to viscosity, and inversely proportional to radius. Warm water from 25 °C to 37 °C and two effects compound — T rises 4% and water’s viscosity falls from 0.89 to about 0.69 mPa·s — so D grows by roughly a third, a correction physiological calculations ignore at their peril. Size runs the other way through biology: a protein of radius 3 nm in water lands near 8×10⁻¹¹ m²/s, a particle of radius 100 nm near 2.5×10⁻¹² — two orders of magnitude of mobility lost to radius alone, which is why nanoparticle formulations rely on Brownian stability and why ultracentrifuges exist. The hydrodynamic radius is the operative size: it includes the solvation shell and any coating, so a PEGylated particle diffuses as its dressed radius, not its core’s.
Where D Is the Design Number
Dynamic light scattering measures D and inverts this very equation to size nanoparticles and proteins — the calculator run backwards is the instrument. Drug delivery times release from gels and across membranes with D at its center (and hindered diffusion through pores lowers the effective value below this free-solution figure). Electrochemistry feeds D into the Cottrell and Randles–Ševčík equations to read concentrations from currents. Environmental modeling spreads contaminant plumes with molecular D in groundwater’s still zones, before dispersion takes over. In each case the honest caveat travels with the number: Stokes–Einstein assumes a spherical particle much larger than the solvent molecules, in a continuum Newtonian fluid, at infinite dilution. Small solutes diffuse somewhat faster than predicted (the boundary condition slides from stick toward slip, 6π toward 4π), crowded cytoplasm slows everything several-fold, and charged species drag their ion atmospheres with them.
Common Mistakes
Entering diameter where radius belongs — a clean factor-of-two error in D. Using viscosity at the wrong temperature (water at 20 °C is 1.00 mPa·s, not 0.89 — and the calculator trusts your η input at your stated temperature, it does not look water up for you). Mixing unit systems mid-equation: the SI set (kelvin, Pa·s, metres) is what the calculator converts into from the friendly inputs shown. Treating D as a constant of the molecule rather than of the molecule–solvent–temperature triple. And applying the free-solution D inside pores, gels or cells without the hindrance and crowding corrections those environments demand.
The Random Walk Behind the Number
D is also the exchange rate of Einstein’s random walk: mean-square displacement grows as ⟨x²⟩ = 2Dt in one dimension and 6Dt in three. For this page’s molecule, the three-dimensional RMS displacement in a single second is √(6 × 4.907×10⁻¹⁰ × 1) ≈ 5.4×10⁻⁵ m — about 54 µm per second of wander, executed as billions of nanometre-scale steps in random directions. That is the microscopic picture behind every figure the calculator prints: not a drift, but a stagger whose statistics the diffusion coefficient summarizes.
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Frequently Asked Questions
What is the Stokes-Einstein equation?
D = kT / (6πηr): the diffusion coefficient equals Boltzmann’s constant times absolute temperature, divided by six pi times solvent viscosity times hydrodynamic radius. This calculator implements it with unit conversion built in.
What D do small molecules have in water?
Roughly 5×10⁻¹⁰ m²/s at 25 °C — the worked example’s 4.91×10⁻¹⁰ m²/s for a 0.5 nm radius is representative; glucose measures near 6×10⁻¹⁰ and sucrose near 5×10⁻¹⁰ m²/s.
How long does diffusion take over a distance?
Use t ≈ x²/(2D) for one dimension. The example molecule crosses 10 µm in about 0.1 s but needs about 1,000 s for 1 mm — the quadratic scaling that makes diffusion a short-range courier only.
Why does temperature change D so much?
Twice over: D is directly proportional to absolute temperature, and viscosity usually falls steeply as temperature rises. Water from 25 °C to 37 °C gains roughly a third in D through both effects combined.
What is hydrodynamic radius?
The radius of the equivalent hard sphere that diffuses at the same rate — including solvation shell, shape effects and coatings. It is what light scattering reports, and it is larger than a dry or core radius.
Does Stokes-Einstein work for tiny solutes?
Approximately, with a known bias: solutes comparable in size to solvent molecules tend to diffuse somewhat faster than the 6π stick-boundary prediction (slip conditions approach 4π). Treat small-molecule results as good estimates, not exact values.