Calculator Input Panel
Plotly Graph
This curve shows the Stokes-Einstein relationship between hydrodynamic radius and diffusion coefficient at the solved temperature and viscosity.
Example Data Table
These sample values show how the same formula behaves across changing temperature, viscosity, and diffusion inputs.
| Case | Temperature (K) | Viscosity (mPa·s) | Diffusion (m²/s) | Radius (nm) |
|---|---|---|---|---|
| Water-like medium | 298.15 | 0.89 | 2.2000e-10 | 1.1153 |
| Moderate viscosity | 298.15 | 1.5 | 1.4000e-10 | 1.0399 |
| Warm sample | 310.15 | 0.7 | 3.1000e-10 | 1.0469 |
| Slow diffuser | 293.15 | 2 | 6.5000e-11 | 1.6517 |
Formula Used
Primary equation:
Rh = kBT / (6πηD)
D = kBT / (6πηRh)
η = kBT / (6πRhD)
Where:
- Rh = hydrodynamic radius
- D = diffusion coefficient
- η = dynamic viscosity
- T = absolute temperature in kelvin
- kB = Boltzmann constant
This model is widely used for spherical particles moving in a continuum fluid. It is most reliable for dilute systems, stable temperature, and low interaction effects.
How to Use This Calculator
- Select what you want to solve: radius, diffusion coefficient, or viscosity.
- Enter temperature and choose the correct unit.
- Provide the known fluid viscosity and its unit unless viscosity is the target.
- Enter either diffusion coefficient or hydrodynamic radius depending on the selected mode.
- Choose display precision and click Calculate Now.
- Read the main result above the form, review unit conversions, inspect the graph, and export the report if needed.
Frequently Asked Questions
1) What is hydrodynamic radius?
Hydrodynamic radius is the effective size a particle shows while moving through a fluid. It includes solvent interaction effects, so it can differ from the particle’s dry geometric radius.
2) Which equation does this calculator use?
It uses the Stokes-Einstein relationship. That equation links diffusion coefficient, temperature, viscosity, and hydrodynamic radius for small particles in a fluid.
3) Why does viscosity matter so much?
A thicker fluid resists motion more strongly. For the same particle and temperature, higher viscosity lowers diffusion and increases the calculated resistance to movement.
4) Does higher temperature increase diffusion?
Usually yes. Higher temperature raises thermal energy, which tends to increase diffusion. The exact outcome still depends on how viscosity changes with temperature.
5) Is hydrodynamic radius the same as geometric radius?
Not always. Hydrodynamic radius reflects how the particle behaves in a fluid. Surface layers, shape effects, and solvent binding can make it larger than the dry radius.
6) Can this be used for non-spherical particles?
It can provide an effective equivalent radius, but accuracy drops for strongly non-spherical or flexible particles. In those cases, the value is best treated as an approximation.
7) What units are supported here?
The tool supports kelvin, Celsius, and Fahrenheit for temperature; Pa·s, mPa·s, and cP for viscosity; multiple diffusion units; and radius in meters, millimeters, micrometers, nanometers, and angstroms.
8) Can I estimate particle diameter from the result?
Yes. The calculator automatically reports particle diameter as twice the hydrodynamic radius. That value is useful for quick comparisons and visualization.