Calculate the diffusion coefficient (D) using the Stokes-Einstein equation from particle radius, temperature, and fluid viscosity. Also calculate diffusion distance from Fick's second law. Includes presets for water, glycerol, and common particle types.
Einstein, A. (1905) — Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung
Original derivation of the diffusion-viscosity relationship for Brownian particles. Annalen der Physik, 17, 549.
🔬
NIST Thermophysical Properties of Fluid Systems
Reference viscosity data for water and common fluids. webbook.nist.gov
Stokes-Einstein Equation:
D = kₐT / (6πηr)
kₐ = 1.380649⋅10⁻²³ J/K | T in Kelvin | η in Pa·s | r in meters Fick's 2nd Law (diffusion distance):
1D: x ≈ 2√(Dt) | 3D: x ≈ √(6Dt)
What is the Diffusion Coefficient? Stokes-Einstein Explained
The diffusion coefficient D quantifies how fast a particle or molecule moves through a medium by random thermal (Brownian) motion. It appears in both of Fick's laws as the proportionality constant linking concentration gradient to flux. A higher D means faster diffusion — small, hot particles in low-viscosity fluids have the highest D values.
The Stokes-Einstein equation — derived by Albert Einstein in 1905 — gives the diffusion coefficient for a spherical particle in a fluid: D = kₐT/(6πηr), where kₐ is Boltzmann's constant, T is absolute temperature in Kelvin, η is the dynamic viscosity of the fluid, and r is the hydrodynamic radius of the particle. This equation bridges thermodynamics (kₐT = thermal energy) and fluid mechanics (the Stokes drag on a sphere).
Typical Diffusion Coefficients in Water at 25°C
Species / Particle
Approx. Radius
D (m²/s)
Category
Water molecule (self-diffusion)
~1.5 Å
2.3 ⋅ 10⁻⁹
Solvent
Na⁺ ion
~1.8 Å
1.3 ⋅ 10⁻⁹
Small ion
Glucose (C₆H₁₂O₆)
~3.6 Å
6.7 ⋅ 10⁻¹⁰
Small molecule
Lysozyme (14 kDa protein)
~2 nm
1.0 ⋅ 10⁻¹⁰
Small protein
Serum albumin (67 kDa)
~3.6 nm
6.1 ⋅ 10⁻¹¹
Medium protein
IgG antibody (150 kDa)
~5.5 nm
4.0 ⋅ 10⁻¹¹
Large protein
Gold nanoparticle (10 nm radius)
10 nm
2.4 ⋅ 10⁻¹¹
Nanoparticle
Gold nanoparticle (50 nm radius)
50 nm
4.8 ⋅ 10⁻¹²
Nanoparticle
How Temperature Affects Diffusion
Diffusion coefficient increases with temperature for two reasons: thermal energy (kₐT) increases linearly with T, and water viscosity drops significantly. At 25°C, water η = 0.890 mPa·s; at 37°C it drops to 0.692 mPa·s. Combined, a 10°C increase in temperature increases D by roughly 30–40% in aqueous systems.
Fick's Second Law and Diffusion Distance
Fick's second law predicts how diffusion changes concentration over time: ∂C/∂t = D ∂²C/∂x². For free diffusion from a point source, the mean squared displacement is:
1D: ⟨x²⟩ = 2Dt, so typical distance x ≈ 2√(Dt)
3D: ⟨r²⟩ = 6Dt, so x ≈ √(6Dt)
Example: a small protein (D = 10⁻¹⁰ m²/s) in 3D over 1 second: x = √(6 ⋅ 10⁻¹⁰ ⋅ 1) = √(6⋅10⁻¹⁰) ≈ 24.5 μm. Over 1 hour (3600 s): x ≈ √(6⋅10⁻¹⁰⋅3600) ≈ 1.47 mm.
💡 Applications of Diffusion Coefficient:
Drug delivery: predicts how fast drugs diffuse through tissue
DLS (Dynamic Light Scattering): measures D to determine particle size
NMR spectroscopy: DOSY NMR measures D to identify molecular size
Semiconductor fab: controls dopant diffusion depth in silicon
Membrane science: predicts flux through dialysis/filtration membranes
Frequently Asked Questions
The diffusion coefficient D quantifies how fast a particle moves through a medium by random Brownian motion. It has units of m²/s. Higher D = faster diffusion. D depends on particle size (inversely proportional to radius), temperature (proportional to T), and fluid viscosity (inversely proportional to η).
D = kₐT / (6πηr), where kₐ = 1.381⋅10⁻²³ J/K, T = temperature in Kelvin, η = dynamic viscosity in Pa·s, and r = hydrodynamic radius in meters. It applies to spherical particles in dilute solution.
D increases with temperature: directly through kₐT and indirectly because water viscosity drops. A 10°C increase roughly doubles D in aqueous systems. Always specify temperature when reporting D values.
J = -D ⋅ (dC/dx). Flux J (mol/m²/s) is proportional to concentration gradient dC/dx. The negative sign means diffusion occurs from high to low concentration. For steady-state diffusion where concentration doesn't change with time.
∂C/∂t = D ⋅ ∂²C/∂x². Concentration changes over time due to diffusion. For free diffusion: typical 1D distance x ≈ 2√(Dt); 3D distance x ≈ √(6Dt). This gives the Einstein relation for Brownian motion.
Water self-diffusion at 25°C is ~2.3⋅10⁻⁹ m²/s. Small ions (Na⁺): ~1.3⋅10⁻⁹ m²/s. Glucose: ~6.7⋅10⁻¹⁰ m²/s. Proteins depend on size — albumin (~3.6 nm): ~6.1⋅10⁻¹¹ m²/s.
D is inversely proportional to radius: D = kₐT/(6πηr). Doubling particle radius halves D. An ion (r ~1 Å) has D ~10⁻⁹ m²/s; a nanoparticle (r = 50 nm) has D ~5⋅10⁻¹² m²/s — roughly 200x slower in water at 25°C.
In water at 25°C: ions and small molecules: 10⁻⁹–10⁻¹⁰ m²/s. Proteins: 10⁻¹¹–10⁻¹⁰ m²/s. Nanoparticles (100 nm): ~10⁻¹² m²/s. In gases: ~10⁻⁴–10⁻⁵ m²/s. In solids: can be as low as 10⁻²⁰ m²/s.
Dynamic viscosity η measures fluid resistance to flow. D is inversely proportional to η. Water at 25°C: η = 0.89 mPa·s. Glycerol: η = 1490 mPa·s. Diffusion in glycerol is ~1670x slower than in water. Always lower viscosity = faster diffusion.
Brownian motion is the individual random thermal motion of particles. Diffusion is the net statistical result: the mean-squared displacement ⟨r²⟩ = 6Dt in 3D. D quantifies the rate of Brownian motion. Both are the same physical phenomenon at the molecular level.