How this instrument works
Diffusion is the gradual spreading of particles from where they're concentrated toward where they're scarce, driven by nothing more than the random thermal jostling every molecule and particle experiences at any temperature above absolute zero. The diffusion coefficient, D, quantifies how fast that spreading happens for a given particle in a given fluid — a larger D means the particle wanders farther, faster, from random thermal collisions with the surrounding fluid molecules.
The Stokes-Einstein relation, D = kBT / (6πηr), connects that microscopic wandering to three measurable quantities: temperature (more thermal energy means more vigorous jostling), the fluid's viscosity (a thicker fluid resists the particle's motion more), and the particle's own radius (a bigger particle gets buffeted around less by comparison, so it diffuses more slowly). It's named for Albert Einstein, who derived the relationship in his 1905 work on Brownian motion, and George Gabriel Stokes, whose earlier work on drag on a sphere moving through a viscous fluid supplied the 6πηr term.
Two of the three relationships are simple: D scales directly with temperature (hotter means faster diffusion) and inversely with viscosity (thicker fluid means slower diffusion). The radius relationship is the one most often surprising at first — D scales inversely with radius, not with any power of it, so a particle ten times larger diffuses exactly ten times slower, all else held equal. This is why small ions and molecules diffuse through water dramatically faster than larger proteins or colloidal particles of the same charge and shape.
- Enter the fluid's absolute temperature into Temperature (K) — kelvin only, not Celsius or Fahrenheit.
- Enter the fluid's dynamic viscosity into Dynamic viscosity (Pa*s) — water near room temperature is about 0.001 Pa·s.
- Enter the diffusing particle's radius into Particle radius (m) — typical small ions and molecules run from about 10⁻¹⁰ to 10⁻⁹ m.
- Read the estimated diffusion coefficient off Diffusion coefficient, D (m^2/s).
- All three inputs must be greater than zero; the instrument will ask for a valid value if temperature, viscosity or radius is entered as zero or negative.
Worked example — a 1 nm particle diffusing in water
Enter 298 into Temperature (K), 0.001 into Dynamic viscosity (Pa*s), and 1e-9 into Particle radius (m) — room temperature, water's approximate viscosity near 25°C, and a 1-nanometre particle, roughly the size of a small protein or a large ion. Diffusion coefficient, D (m^2/s) reads 2.1827×10⁻¹⁰ m²/s.
By hand: D = (1.380649×10⁻²³ × 298) / (6 × π × 0.001 × 1×10⁻⁹) = 4.1143×10⁻²¹ / 1.8850×10⁻¹¹ ≈ 2.1827×10⁻¹⁰ m²/s. This sits squarely in the well-known order-of-magnitude range (roughly 10⁻⁹ to 10⁻¹⁰ m²/s) reported for small-molecule diffusion in water, confirming the formula's output against a figure a biochemist or physical chemist would recognize on sight.
Questions
Why does a bigger particle diffuse more slowly?
A larger particle presents more surface area to the fluid's random molecular collisions, so the countless tiny random pushes from all directions average out more completely, leaving less net random motion per unit time. The Stokes-Einstein relation captures this precisely: D is inversely proportional to radius r, so doubling a particle's radius exactly halves its diffusion coefficient, all else held equal.
What does the Boltzmann constant have to do with diffusion?
The Boltzmann constant, kB, converts temperature into a quantity of thermal energy per particle — it's the same constant that appears throughout statistical mechanics wherever microscopic, particle-level thermal motion needs to be tied to a macroscopic, measurable temperature. In the Stokes-Einstein relation, kBT represents the thermal energy driving a particle's random Brownian motion, which is exactly what produces diffusion in the first place.
Why does higher viscosity slow diffusion down?
Viscosity measures a fluid's internal resistance to flow, and that same resistance opposes a particle's random thermal motion through it — a thicker, more viscous fluid drags on a moving particle more strongly, damping the random walk that produces diffusion. That's why the same particle diffuses far more slowly through honey than through water, even at an identical temperature.
Does the Stokes-Einstein relation work for any particle shape?
Strictly, it assumes a spherical particle moving through a continuous fluid whose molecules are much smaller than the particle itself — conditions that hold well for roughly spherical proteins, colloids and nanoparticles in a simple solvent. Non-spherical particles, or particles comparable in size to the solvent molecules around them, deviate from this idealized formula and need a shape-corrected or more detailed treatment.
How accurate is this for real ions and small molecules?
Reasonably close for a first estimate — plugging in water's actual viscosity at 20°C (0.001002 Pa·s) and a typical small-ion radius of about 3×10⁻¹⁰ m gives a diffusion coefficient around 7.14×10⁻¹⁰ m²/s, consistent with measured aqueous diffusion coefficients for small ions. Real values can still differ somewhat because ions interact electrostatically with surrounding water molecules in ways the simple Stokes-Einstein model, which treats the fluid as a featureless continuum, doesn't capture.