Calculator
Example data
These sample inputs show typical scales for aqueous solutions.
| Scenario | T (K) | η (mPa·s) | D (µm²/s) | Rh (nm) |
|---|---|---|---|---|
| Small molecule in water | 298.15 | 1.00 | 500 | 0.44 |
| Protein-sized particle | 298.15 | 1.00 | 50 | 4.37 |
| Nanoparticle (slower diffusion) | 298.15 | 1.00 | 5 | 43.7 |
Values are illustrative and assume spherical behavior.
Formula used
The calculator uses the Stokes–Einstein relation for dilute suspensions:
- Diffusion coefficient:
D = kB·T / (6·π·η·Rh) - Hydrodynamic radius:
Rh = kB·T / (6·π·η·D) - Viscosity:
η = kB·T / (6·π·D·Rh) - Temperature:
T = (6·π·η·D·Rh) / kB
Assumes Newtonian fluid, low Reynolds number, and effective spherical drag.
How to use this calculator
- Select what you want to compute: Rh, D, η, or T.
- Enter the required inputs and pick units from the dropdowns.
- Click Submit to display the results above the form.
- Use Download CSV or Download PDF to save a clean summary.
Use careful inputs to estimate particle size confidently today.
Professional notes
1) What the hydrodynamic radius represents
The hydrodynamic radius (Rh) is an effective size inferred from how a particle moves through a fluid. It reflects drag and can include hydration or solvation layers, so Rh may differ from a geometric radius.
2) Stokes–Einstein in one line
The model used is D = kB·T / (6π·η·Rh), suited to dilute motion in a Newtonian continuum.
Rearranging lets you compute Rh, D, η, or T. The calculator uses kB = 1.380649×10⁻²³ J/K in SI.
3) Typical scales
In water near room temperature, small molecules can show D ≈ 100–1000 µm²/s (sub‑nanometer Rh). Many proteins sit around D ≈ 30–100 µm²/s (Rh of a few nanometers). Larger nanoparticles can drop to single‑digit µm²/s, yielding Rh in the tens of nanometers.
4) Temperature effects
Diffusion rises with temperature through kB·T. Holding viscosity fixed, a 10 K increase near 300 K changes kB·T by about 3%. Use a viscosity value matched to the same temperature.
5) Viscosity drives uncertainty
When solving for Rh, a 5% error in η produces roughly a 5% error in Rh. Use the viscosity of your exact solvent and temperature (water is ~1.00 mPa·s near 20°C and ~0.89 mPa·s near 25°C).
6) Unit conversion pitfalls
The calculator converts inputs to SI before solving. Watch diffusion units: cm²/s and µm²/s differ by 10⁸. For viscosity, 1 cP equals 1 mPa·s. Unit mismatches are the most common cause of unrealistic radii.
7) When the estimate can break down
Stokes–Einstein assumes spherical drag at low Reynolds number. Deviations appear for non‑spherical particles, crowded solutions, viscoelastic media, or strong interactions. In those cases Rh is an effective value.
8) Reporting and exporting
Report the measured quantity (D or Rh), temperature, viscosity source, and method notes (e.g., DLS or NMR diffusion). Use CSV for spreadsheets and PDF to archive a single calculation snapshot with units and significant figures.
Accurate inputs produce dependable diffusion-based size estimates.
FAQs
1) Is hydrodynamic radius the same as physical radius?
No. Rh reflects drag in a fluid and can include hydration layers, polymer coils, and shape effects, so it may be larger (or sometimes smaller) than a geometric radius from microscopy.
2) Which mode should I use if I measured D?
Select “Compute hydrodynamic radius (Rh)” and enter T, η, and D. The calculator converts units to SI and returns Rh in your chosen output unit.
3) How accurate is the result?
Accuracy depends on input quality. Uncertainty in viscosity and temperature directly propagates. For Rh from D, a 5% error in η or D typically produces about a 5% error in Rh.
4) Can I use this for non-spherical particles?
You can, but Rh becomes an effective radius representing equivalent drag. Rods, disks, and aggregates may deviate from spherical behavior, so compare with complementary sizing methods when possible.
5) What temperature should I enter?
Use the measurement temperature in Kelvin. If you collected D at 25°C, enter 298.15 K. Pair it with viscosity for the same temperature and solvent composition.
6) Why does changing units change my result?
It shouldn’t, if values are entered correctly. Large changes usually mean a unit mismatch, such as cm²/s vs µm²/s for diffusion, or entering mPa·s when Pa·s was intended.
7) What do the CSV and PDF exports include?
They include T, η, D, Rh in SI units, the constant kB, and the model name. This makes it easy to paste results into reports and preserve calculation context.