RMS Speed — Oxygen Molecules at 300 K (Worked Example)

Given

• T = 300 K

• M = 32.0 g/mol = 0.0320 kg/mol

• R = 8.314 J/(mol·K)

Find

v_rms.

Principle

v_rms = √(3RT/M): at the same temperature, heavier molecules move slower by the square root of the mass ratio.

Formula

Solution

1. $v_{\mathrm{rms}} = \sqrt{3 \times 8.314 \times 300 / 0.032}$

2. Inside the root: $7482.6/0.032 = 233{,}831$

3. Square root: $\sqrt{233{,}831} \approx 484\ \mathrm{m/s}$

Unit check

m/s — correct for speed.

What this means physically

Oxygen molecules cruise at ~484 m/s, about 1.4× the speed of sound. Compare helium's 1368 m/s: the ratio 1368/484 ≈ 2.83 = √8, exactly the square root of the mass ratio — kinetic theory confirmed.