Calculator
Formula used
When you know molar mass M (kg/mol), the RMS speed is:
vrms = √(3RT / M)
When you know the mass of one molecule m (kg), the equivalent form is:
vrms = √(3kT / m)
If enabled, the calculator also reports:
- v̄ = √(8RT / (πM)) or √(8kT / (πm))
- vmp = √(2RT / M) or √(2kT / m)
How to use this calculator
- Select the method that matches your mass data.
- Enter temperature and choose its unit.
- For molar method, enter molar mass or pick a gas.
- For molecular method, enter molecular mass in amu or kg.
- Select the output speed unit you want.
- Press Calculate to view results above.
- Use CSV or PDF buttons to save your summary.
Example data table
| Gas | Temperature (K) | Molar mass (g/mol) | RMS speed (m/s) | Average speed (m/s) | Most probable (m/s) |
|---|---|---|---|---|---|
| Nitrogen (N₂) | 300 | 28.0134 | 517.2 | 476.3 | 422.1 |
| Oxygen (O₂) | 300 | 31.9988 | 483.0 | 444.9 | 394.2 |
| Helium (He) | 300 | 4.0026 | 1368.0 | 1259.6 | 1116.3 |
| Dry air (approx.) | 288.15 | 28.965 | 496.6 | 457.2 | 405.2 |
Example values are rounded and assume ideal-gas behavior.
Article
1) What RMS speed represents
Root mean square speed, vrms, comes from averaging squared molecular speeds and taking a square root. Squaring weights fast molecules more, so vrms is larger than the most probable speed and usually slightly larger than the average speed. It links microscopic motion to macroscopic temperature in kinetic theory.
2) Temperature sets the kinetic energy
In thermal equilibrium, absolute temperature is proportional to mean translational kinetic energy. Higher temperature shifts the Maxwell–Boltzmann distribution toward larger speeds. Because speeds scale as √T, doubling T increases characteristic speeds by √2, not by two.
3) Mass controls how fast molecules move
At the same temperature, lighter molecules move faster with a 1/√mass dependence. A gas with four times the molecular mass has about half the RMS speed. This explains why helium is much faster than carbon dioxide at room conditions.
4) Two equivalent formulas, one output
With molar mass M (kg/mol), the calculator uses vrms = √(3RT/M). With per-molecule mass m (kg), it uses vrms = √(3kT/m). The forms match because R = NAk and M = NAm.
5) Typical numeric ranges you should expect
At 300 K, nitrogen is about 517 m/s, oxygen about 483 m/s, and dry air near 497 m/s. Helium can exceed 1.3 km/s. These ranges help you sanity-check inputs before exporting results.
6) Comparing RMS, average, and most probable speeds
The calculator can report three Maxwell–Boltzmann measures: vmp (distribution peak), v̄ (mean speed), and vrms (energy-weighted speed). Their ordering is vmp < v̄ < vrms.
7) Practical uses in labs and engineering checks
RMS speed supports quick comparisons for diffusion and effusion and gives a thermal velocity scale for gas dynamics. It also complements mean free path or collision-rate estimates in basic vacuum and transport calculations. In teaching, it converts temperature into an intuitive speed benchmark. For example, at 300 K, switching from nitrogen to helium raises RMS speed by roughly 2.6×, reflecting helium’s low molar mass.
8) Input tips and common mistakes to avoid
Always use absolute temperature; negative values are acceptable only before conversion. Keep mass definitions consistent: g/mol and kg/mol are per mole, while amu is per molecule. A 1000× mass unit error changes speed by about √1000 ≈ 31.6.
FAQs
1) Why is RMS speed higher than average speed?
RMS uses squared speeds, which weights faster molecules more strongly. After averaging, taking the square root still reflects that emphasis, so vrms exceeds the simple mean speed.
2) Can I use Celsius or Fahrenheit inputs?
Yes. Enter the temperature and select its unit. The calculator converts to Kelvin internally. The final speeds are computed from absolute temperature, so conversion is essential for correct physics.
3) Should I enter molar mass or molecular mass?
Use molar mass when you have g/mol or kg/mol data. Use molecular mass when you have the mass of one molecule in amu or kg. Both methods produce identical RMS speed when units are consistent.
4) What values are reasonable at room temperature?
Many common gases fall between 400 and 600 m/s near 300 K. Light gases are faster; helium can be around 1.3 to 1.4 km/s. Use these ranges as a quick sanity check.
5) Does pressure affect RMS speed?
For an ideal gas at equilibrium, RMS speed depends on temperature and molecular mass, not pressure directly. Pressure changes density, but thermal speed scales are still set mainly by temperature.
6) What is the difference between average and most probable speed?
Most probable speed is the peak of the Maxwell–Boltzmann distribution. Average speed is the mean of speeds. Because the distribution is skewed, the mean is higher than the mode, so v̄ exceeds vmp.
7) Why does a unit mistake change results so much?
Speed scales with the square root of inverse mass. If mass is off by 1000× (g vs kg), speed shifts by √1000 ≈ 31.6×. That is why unit selection and conversion are critical in kinetic theory calculations.