Calculator
Formula used
This tool uses the Einstein relation that links diffusion to mobility under thermal equilibrium.
- μ is mobility, in m²/(V·s) or cm²/(V·s).
- D is diffusion coefficient, in m²/s or cm²/s.
- q is carrier charge in coulombs (C).
- T is absolute temperature in kelvin (K).
- kB is Boltzmann constant, 1.380649×10⁻²³ J/K.
How to use this calculator
- Select which variable you want to solve for.
- Enter the remaining values using consistent units.
- Choose units for D and μ using the dropdowns.
- Set charge using z·e or direct coulombs entry.
- Press Calculate to display results above the form.
- Use CSV or PDF buttons to export the summary.
Example data table
| Scenario | D (cm²/s) | T (K) | z | Computed μ (cm²/V·s) |
|---|---|---|---|---|
| Electron-like carrier | 35 | 300 | -1 | ≈ 1354 |
| Moderate ionic transport | 0.001 | 298 | +1 | ≈ 0.0389 |
| Slow diffusion in viscous medium | 0.00002 | 310 | +1 | ≈ 0.000748 |
Values are illustrative and assume equilibrium conditions.
Professional notes on mobility from diffusion
1) Transport link between diffusion and drift
Random thermal motion creates diffusion, while electric fields cause drift. The mobility parameter connects drift velocity to field strength, so converting diffusion data into mobility helps compare transport across materials, geometries, and experiments.
2) Einstein relation and equilibrium assumptions
The calculator applies the Einstein relation, μ = (q·D)/(kB·T). It is most reliable when carriers are near thermal equilibrium, scattering is reasonably steady, and the same carriers control both diffusion and conductivity in the measured regime. For strongly degenerate carriers, the proportionality can shift and a generalized relation may be required.
3) Useful constants and thermal voltage
Boltzmann’s constant is kB = 1.380649×10⁻²³ J/K and the elementary charge is e = 1.602176634×10⁻¹⁹ C. For |q| = e at 300 K, the thermal voltage magnitude |kBT/q| is about 25.85 mV, giving μ ≈ D/0.02585 when D is in m²/s.
4) Unit handling for labs and datasheets
Transport literature often mixes m²/s with cm²/s, and m²/(V·s) with cm²/(V·s). The form converts both ways so you can paste datasheet values directly and still export a clean, consistent record for reports. Remember: 1 cm²/s = 1×10⁻⁴ m²/s and 1 cm²/(V·s) = 1×10⁻⁴ m²/(V·s).
5) Semiconductor-scale reference values
At room temperature, electron and hole mobilities commonly range from ~100 to >1500 cm²/(V·s), depending on doping and crystal quality. The example row D = 35 cm²/s at 300 K with z = −1 produces μ ≈ 1354 cm²/(V·s), consistent with high-mobility scenarios.
6) Electrolytes and ion transport context
In liquids and polymers, diffusion can be much smaller. The table shows D = 0.001 cm²/s at 298 K with z = +1 giving μ ≈ 0.0389 cm²/(V·s). Multivalent ions (|z| > 1) scale mobility proportionally if other conditions stay comparable.
7) Temperature dependence and interpretation
Because μ ∝ 1/T for fixed D and q, increasing temperature reduces computed mobility if diffusion is held constant. In real systems, D often increases with temperature, so mobility trends must be interpreted with the measured D(T) rather than a single-point assumption. For comparisons, report both D and μ at the same stated temperature, commonly 300 K.
8) Practical workflow and uncertainty control
Use the “Solve for” selector to validate measurements by solving for a different variable and checking consistency. Propagate uncertainty by evaluating high and low bounds for D and T; the export buttons preserve your inputs, units, and constants for traceable documentation under stated conditions.
FAQs
1) Why is mobility negative for electrons?
Mobility carries the sign of charge when you enter a negative z or q. Many references report a positive magnitude using |q|, so you can interpret |μ| for comparison while keeping sign for drift-direction modeling.
2) Which units should I use for D and μ?
Either set works. Choose cm²/s and cm²/(V·s) for semiconductor-style datasheets, or m²/s and m²/(V·s) for SI reporting. The calculator converts and shows both in the results panel.
3) Do I have to use kelvin for temperature?
Yes. The Einstein relation uses absolute temperature. If you have Celsius data, convert using T(K) = T(°C) + 273.15 before entering it to avoid large systematic errors in mobility.
4) When should I enter q directly instead of z·e?
Use direct coulombs if you have an effective charge from fitting, fractional charges, or nonstandard carrier definitions. Use z·e for common electrons, holes, and simple ions where z is a small integer.
5) Is the Einstein relation always valid?
No. Strong nonequilibrium, energy-dependent scattering, degenerate statistics, or multiple carrier populations can break the simple proportionality. In those cases, mobility inferred from diffusion may differ from mobility measured electrically.
6) What does the thermal voltage value mean?
|kBT/q| is the scale that links diffusion to mobility via μ = D/|kBT/q| when using magnitude. At 300 K and |q| = e, it is ~25.85 mV, a common reference in transport calculations.
7) What do the CSV and PDF exports include?
Exports include the solved variable, μ, D, T, q, z = q/e, thermal voltage, and the constants used. This keeps your calculation auditable and easy to paste into lab notebooks or reports.