Diffusion Coefficient Calculator - Stokes-Einstein D

A diffusion coefficient calculator estimates how quickly particles spread through a fluid from the Stokes-Einstein relationship or from a supplied mobility value. Use it when a chemistry assignment gives temperature, viscosity, and hydrodynamic radius; when a lab notebook records particle mobility; when a nanoparticle result needs a quick reasonableness check; or when a transport-property problem asks for D in SI units and in micrometer-scale units. The result is an estimate, not a replacement for measured diffusion data in a complex material.

• • Chemistry homework: Check whether a stated particle radius and fluid viscosity produce the expected order of magnitude for translational diffusion.
• • Nanoparticle screening: Compare how a larger hydrodynamic radius lowers D before deciding which samples deserve instrument time.
• • Protein and colloid work: Translate temperature and viscosity assumptions into a diffusion value that can be compared with DLS or fluorescence measurements.
• • Mobility conversion: Use D = mobility times kBT when your problem supplies a mobility coefficient instead of radius and viscosity.

Diffusion coefficient values are usually small in SI units, so this page reports D as both x 10^-10 m2/s and um2/s. The first output keeps the physics convention clear. The second output is easier to scan when you are dealing with micron-scale motion over seconds.

The calculator assumes random thermal motion in a continuous fluid. That assumption is reasonable for many dilute particle and molecule examples, but it can break down for crowded gels, charged surfaces, non-spherical aggregates, or fluids whose viscosity changes during the experiment.

If the same sphere is settling under gravity instead of spreading by Brownian motion, Stokes Law Calculator gives the adjacent drag workflow.

This diffusion coefficient calculator uses the Stokes-Einstein equation in sphere mode. Thermal energy, kBT, pushes random motion, while the friction term, 6 pi eta r, resists motion. A hotter system raises the numerator. A thicker fluid or larger hydrodynamic radius raises the denominator.

• D: Diffusion coefficient in m2/s. Higher D means faster spreading under Brownian motion.
• k_B: Boltzmann constant, fixed at 1.380649 x 10^-23 J/K in this calculator.
• T: Absolute temperature in kelvin. Celsius must be converted before entry.
• eta: Dynamic viscosity in Pa s after unit conversion from Pa s, mPa s, or cP.
• r: Hydrodynamic radius in meters after conversion from nm, um, or m.
• mu: Particle mobility in s/kg. It equals 1 divided by the friction coefficient.

For a 1 nm spherical particle in water-like viscosity at 298.15 K, the denominator is 6 pi x 0.0010016 Pa s x 1e-9 m. That gives a friction coefficient near 1.888 x 10^-11 kg/s and a mobility near 5.297 x 10^10 s/kg.

The mobility model skips the radius and viscosity denominator. It is useful when electrophoretic, sedimentation, or other analysis already produced a mobility value and you only need the Einstein relation step.

According to NIST, the revised SI value of the Boltzmann constant is 1.380649 x 10^-23 J K^-1.

According to LibreTexts, dynamic light scattering relates particle size, temperature, viscosity, and translational diffusion through the Stokes-Einstein equation.

When viscosity data arrives in older cgs units, Poise Stokes Converter helps convert the fluid-property input before you estimate D.

The calculator is short because the physics packs several ideas into a compact formula. These four terms are the ones worth checking before trusting the number.

Diffusion coefficient and thermal diffusivity use similar language but describe different transport processes. This page estimates particle or molecule spreading through a medium. Thermal diffusivity describes heat spreading through a material.

Unit discipline matters. A common error is entering 1 cP as 1 Pa s, which makes viscosity 1000 times too large and D 1000 times too small. The unit selector prevents that specific mistake.

For heat-spreading problems rather than particle-spreading problems, Thermal Diffusivity Calculator uses the related transport-property idea with thermal inputs.

Start with the model that matches your information. The diffusion coefficient calculator defaults to sphere mode for a textbook particle in a liquid. Mobility mode is only for cases where mobility is already known.

1. 1 Choose the model: Select the sphere equation for radius and viscosity inputs, or select mobility when the problem gives mobility directly.
2. 2 Enter temperature: Use kelvin. Convert Celsius by adding 273.15 before entry.
3. 3 Enter viscosity: Type the numeric viscosity and choose Pa s, mPa s, or cP. The calculator converts it to Pa s internally.
4. 4 Enter hydrodynamic radius: Use the particle radius, not diameter. Select nm, um, or m to match your source data.
5. 5 Read D and audit terms: Use the scaled SI result for reporting and the friction, mobility, and kBT outputs to check the path used to compute it.

For a dilute nanoparticle suspension, you might prepare the sample concentration separately, enter the measurement temperature and solvent viscosity here, then compare the estimated D against a DLS report. If the report lists diameter, divide by two before entering radius.

Before a lab comparison, Solution Dilution Calculator can help prepare the sample concentration that will later be paired with a diffusion measurement.

A compact diffusion estimate helps when the real task is judgment: deciding whether a number is physically plausible before spending more time on analysis.

• • Catches unit mistakes: The mPa s, cP, nm, and um selectors reduce the common 1000x and 1,000,000x errors that come from mixed unit systems.
• • Shows audit values: Friction coefficient, mobility, and kBT are displayed so you can trace the denominator and numerator instead of accepting a single black-box D.
• • Supports lab-scale units: The um2/s result is easier to compare with particle tracking and microscopy notes than a long decimal in m2/s.
• • Handles mobility data: When mobility is measured or supplied, the mobility model applies the Einstein relation directly without forcing a radius estimate.
• • Improves sample planning: Changing temperature, viscosity, or radius makes the trend visible before you design a dilution, DLS run, or teaching example.

The calculator is especially useful for sensitivity checks. Try doubling the radius, then doubling the viscosity. Each change halves the sphere-model D, which is the central intuition students and lab users usually need.

For protein or colloid samples, pair the result with concentration notes. A diffusion estimate may look reasonable while aggregation, crowding, or poor sample preparation still changes the measured value.

For protein diffusion experiments, Protein Concentration Calculator keeps concentration estimates separate from the transport-property estimate shown here.

The formula is simple, but the inputs can be subtle. Treat each input as a model assumption, then compare the estimate with measured data when the sample or fluid is complicated.

• • The sphere model assumes a dilute, spherical particle in a continuum fluid with no wall, crowding, or interparticle interaction corrections.
• • The calculator does not infer viscosity from solvent composition or temperature. Enter a viscosity value from your source, instrument method, or course problem.
• • For gases, membranes, polymers, or biological tissues, measured diffusion coefficients or model-specific correlations may be more appropriate than this liquid-particle estimate.

Use the output as an order-of-magnitude estimate unless your system matches the assumptions closely. In reports, list temperature, viscosity, radius source, and whether the radius is hydrodynamic or geometric.

The displayed SI value in this diffusion coefficient calculator uses a scaled unit because ordinary liquid-particle diffusion is often around 10^-11 to 10^-9 m2/s. The scaling is only for readability; the underlying D remains in m2/s.

According to NIST SP 811, a quantity value should be expressed as a numerical value multiplied by a unit, which supports reporting diffusion as m2/s rather than an unlabeled number.