⚛️ What’s a gas doing at the microscopic level?
Imagine a crowded playground where every child (molecule) runs around, bumping into each other and the walls. The motion of these “kids” determines the pressure, temperature and volume of the gas.
🚀 Key Formula
The root‑mean‑square speed is the square root of the average of the squares of all molecular speeds:
\$c_{\text{rms}}=\sqrt{\frac{3kT}{m}}\$
Where \$k\$ = Boltzmann constant, \$T\$ = absolute temperature (K), and \$m\$ = mass of one molecule (kg).
📐 Analogy
Think of a group of runners who finish a race at different speeds. If you square each speed, average those squares, and then take the square root, you get a number that’s always larger than the simple average speed. That’s exactly what \$c_{\text{rms}}\$ does for gas molecules.
Because faster molecules contribute more to the pressure, \$c_{\text{rms}}\$ is a better indicator of the “effective” speed that matters in gas behaviour.
| Gas | Molar Mass (g mol⁻¹) | Mass per Molecule (kg) | \$c_{\text{rms}}\$ at 300 K (m s⁻¹) |
|---|---|---|---|
| Nitrogen (N₂) | 28 | 4.65 × 10⁻²⁶ | ≈ 520 |
| Helium (He) | 4 | 6.63 × 10⁻²⁷ | ≈ 1 200 |
| Oxygen (O₂) | 32 | 5.31 × 10⁻²⁶ | ≈ 480 |
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Remember the conversion
When you see a molar mass in g mol⁻¹, first convert it to kg per molecule:
\$m = \frac{M}{N_A}\$
Where \$M\$ is molar mass and \$N_A\$ = Avogadro’s number.
⚠️ Don’t forget to keep the temperature in Kelvin!
🧠 Quick mental check: For a lighter gas (smaller \$m\$), \$c_{\text{rms}}\$ will be higher.
📌 If the question asks for the average speed, use:
\$c_{\text{avg}}=\sqrt{\frac{8kT}{\pi m}}\$
and remember that \$c{\text{rms}} > c{\text{avg}}\$.