Formula used
For a gas in thermal equilibrium, the Maxwell speed distribution is:
f(v) = 4\pi \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 \exp\left(-\frac{m v^2}{2 k_B T}\right)
Characteristic speeds are computed from temperature T and particle mass m:
- v_p = \sqrt{\frac{2 k_B T}{m}}
- \bar{v} = \sqrt{\frac{8 k_B T}{\pi m}}
- v_{rms} = \sqrt{\frac{3 k_B T}{m}}
If you enter molar mass M, the calculator uses m = M/N_A. For range probabilities, it evaluates the analytic cumulative form using an error-function approximation.
How to use this calculator
- Select what you want to solve for.
- Enter temperature and pick the correct temperature unit.
- Choose molar mass or particle mass, then enter a value.
- Pick the output speed unit for the reported speeds.
- Optionally enter a probe speed and a speed range.
- Press Calculate to view results above this form.
Example data table
| Gas | Molar mass (g/mol) | Temperature (K) | Expected vp (m/s) | Expected v̄ (m/s) | Expected vrms (m/s) |
|---|---|---|---|---|---|
| Nitrogen (N₂) | 28 | 300 | ~422 | ~476 | ~517 |
| Oxygen (O₂) | 32 | 300 | ~395 | ~446 | ~482 |
| Helium (He) | 4 | 300 | ~1116 | ~1258 | ~1366 |
Values are approximate and assume ideal behavior.