Last updated: July 3, 2026
Diffusion Coefficient Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
This diffusion coefficient calculator uses the Stokes-Einstein equation D = k_B T/(6πηr) to estimate the Brownian diffusion coefficient of a spherical particle in a liquid. Enter absolute temperature in Kelvin, liquid viscosity in Pa·s, and particle radius in metres; the calculator returns D in m²/s using k_B = 1.380649 × 10⁻²³ J/K.
The diffusion coefficient of a spherical particle is D equals Boltzmann's constant times absolute temperature divided by six pi times viscosity times particle radius. Use Kelvin, pascal seconds, and metres to get square metres per second.
Key Takeaways
- The Stokes-Einstein equation for a spherical particle is D = k_B T/(6πηr).
- Use SI units: Kelvin, Pa·s, metres, and m²/s.
- Diffusion increases with temperature and decreases with viscosity and particle radius.
- The radius should be hydrodynamic radius, especially for solvated particles and proteins.
- The equation is best for dilute spherical particles in Newtonian continuum liquids.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
D = (k_B × T) / (6 × π × η × r)
Where:
- D=Diffusion coefficient(m²/s)
- k_B=Boltzmann constant(J/K)
- T=Absolute temperature(K)
- η=Dynamic viscosity of the liquid(Pa·s)
- r=Particle radius(m)
Worked Examples
1 nm particle in water at 25 °C
A nanoscale spherical particle diffusing through water at room temperature.
- 1Use k_B = 1.380649 × 10⁻²³ J/K and T = 298.15 K.
- 2Numerator: k_B × T = 1.380649e-23 × 298.15 ≈ 4.1159e-21 J.
- 3Denominator: 6π × η × r = 6π × 0.00089 × 1e-9 ≈ 1.67707e-11 Pa·s·m.
- 4D = 4.1159e-21 ÷ 1.67707e-11 ≈ 2.454e-10 m²/s.
5 nm particle in low-viscosity liquid
A larger particle diffusing at 310 K in a liquid less viscous than water at room temperature.
- 1Multiply Boltzmann's constant by temperature: 1.380649e-23 × 310 ≈ 4.2800e-21.
- 2Compute the drag term: 6π × 0.0007 × 5e-9 ≈ 6.5973e-11.
- 3Divide numerator by denominator.
- 4D ≈ 4.2800e-21 ÷ 6.5973e-11 ≈ 6.49e-11 m²/s.
2 nm particle in cold water
A small particle in a more viscous liquid near 0 °C.
- 1Numerator: 1.380649e-23 × 273.15 ≈ 3.7712e-21.
- 2Denominator: 6π × 0.00179 × 2e-9 ≈ 6.7476e-11.
- 3Apply D = k_B T/(6πηr).
- 4D ≈ 3.7712e-21 ÷ 6.7476e-11 ≈ 5.59e-11 m²/s.
Introduction
The diffusion coefficient describes how quickly particles spread by Brownian motion. For a spherical particle in a continuum liquid, the Stokes-Einstein equation D = k_B T/(6πηr) connects thermal energy to viscous drag. This calculator is useful for colloids, nanoparticles, proteins, micelles, and tracer particles when you know the absolute temperature, liquid viscosity, and hydrodynamic radius. It complements concentration tools such as the molarity calculator and electrochemical transport checks such as the Nernst equation calculator. The equation uses SI units and the exact SI value of the Boltzmann constant maintained by NIST.
What is a diffusion coefficient?
A diffusion coefficient D is the proportionality constant in Fick's laws that relates concentration gradients to molecular flux. In one dimension, a larger D means a substance spreads faster and covers a given distance in less time. For Brownian particles in liquids, D is usually reported in m²/s; nanoscale particles often fall between 10⁻¹² and 10⁻⁹ m²/s.
D has SI units of square metres per second.
Small molecules in water often diffuse near 10⁻⁹ m²/s.
Large nanoparticles, vesicles, or proteins diffuse more slowly.
Diffusion data are often combined with concentration calculations such as the concentration calculator.
The Stokes-Einstein equation
The equation used here is D = k_B T/(6πηr). The numerator k_B T is the thermal energy scale per particle, while the denominator 6πηr is the Stokes drag coefficient for a sphere with no-slip boundary conditions. Increasing temperature raises D; increasing viscosity or radius lowers D. The relationship was established in Einstein's 1905 Brownian-motion theory and is summarized in many physical chemistry references, including LibreTexts physical chemistry.
The radius should be the hydrodynamic radius, which includes solvation and surface effects, not necessarily the dry geometric radius.
How to calculate D step by step
Use strict SI units before calculating:
Convert temperature to Kelvin: T(K) = °C + 273.15.
Enter viscosity in Pa·s; 1 cP = 0.001 Pa·s.
Enter radius in metres; 1 nm = 1e-9 m.
Compute k_B × T.
Compute 6 × π × η × r.
Divide to obtain D in m²/s.
Typical diffusion coefficients
Diffusion varies strongly with particle size and medium viscosity. The table below gives approximate room-temperature values for orientation, not universal constants.
| Particle or solute | Approximate radius | Typical D in water |
|---|---|---|
| Small ions | 0.1–0.3 nm | ~1–2 × 10⁻⁹ m²/s |
| Small organic molecules | 0.3–0.7 nm | ~3–10 × 10⁻¹⁰ m²/s |
| Globular proteins | 2–5 nm | ~5 × 10⁻¹¹ to 1 × 10⁻¹⁰ m²/s |
| 100 nm nanoparticle | 50 nm | ~4 × 10⁻¹² m²/s |
Assumptions and limitations
Stokes-Einstein assumes a spherical particle, continuum hydrodynamics, dilute conditions, and a Newtonian liquid with known viscosity. It can be inaccurate for very small solutes comparable to solvent molecules, non-spherical particles, crowded media, porous gels, or systems with slip at the particle surface. For electrolyte systems, transport and activity may also require ionic-strength corrections; see the ionic strength calculator.
Where diffusion coefficients are used
Diffusion coefficients support design and interpretation in nanoparticle synthesis, drug delivery, dynamic light scattering, membrane transport, chromatography, electrochemistry, and environmental fate models. In electroanalytical chemistry, diffusion-limited currents depend directly on D; in colloid science, D can be inverted to estimate hydrodynamic size. Reference constants and viscosity data are available from NIST Chemistry WebBook and similar databases.
For dynamic light scattering, check whether your instrument reports hydrodynamic diameter; divide by two before entering radius.
Quick Reference Card
Diffusion Coefficient — Quick Reference
Quick reference • Diffusion Coefficient Calculator
D = k_B T/(6πηr), with k_B = 1.380649 × 10⁻²³ J/KValid range: T > 0 K, η > 0 Pa·s, r > 0 m; best for dilute spherical particles in Newtonian liquids
Common Values
⚠ Watch Out
- •Do not enter Celsius; convert to Kelvin first.
- •Do not enter viscosity in cP unless you convert to Pa·s.
- •Use radius, not diameter; divide a reported diameter by two.
- •The equation can fail for non-spherical particles, crowded media, and molecular-scale solutes.
- •Zero or negative temperature, viscosity, or radius makes the calculation undefined.
Pro Tips
- →For water, look up viscosity at the exact temperature rather than assuming 0.001 Pa·s.
- →If using dynamic light scattering data, check whether the size is radius or diameter.
- →Report D with two to three significant figures unless inputs are very precise.
- →For comparison across experiments, keep temperature and solvent composition fixed.
- →Use hydrodynamic radius when solvation layers or surface coatings matter.
FAQs
What units should I use in the diffusion coefficient calculator?
Use Kelvin for temperature, pascal-seconds for dynamic viscosity, and metres for particle radius. The result is returned in m²/s.
Why does a larger particle have a smaller diffusion coefficient?
A larger radius increases viscous drag. In the Stokes-Einstein equation D is inversely proportional to r, so doubling the hydrodynamic radius halves D if temperature and viscosity stay the same.
Does higher temperature always increase diffusion?
In the formula, D increases linearly with absolute temperature. In real liquids, viscosity usually also changes with temperature, often decreasing as temperature rises, which can amplify the increase in diffusion.
Is viscosity in centipoise the same as Pa·s?
No. 1 centipoise (cP) equals 0.001 Pa·s. Water near room temperature is about 0.89 cP, or 0.00089 Pa·s.
Can I use this for proteins or nanoparticles?
Yes, if the particle can be approximated as a sphere in a dilute Newtonian liquid and you use the hydrodynamic radius. Non-spherical proteins and crowded biological fluids may require experimental calibration.
What is the Boltzmann constant used here?
The calculator uses the exact SI-defined value k_B = 1.380649 × 10⁻²³ J/K.