RMS Speed — Hydrogen, the Fastest Gas (Worked Example)

Given

• T = 300 K

• M = 0.002 kg/mol

• R = 8.314 J/(mol·K)

Find

v_rms.

Principle

The lightest molecule is the fastest: v_rms = √(3RT/M).

Formula

Solution

1. Substitute: $v_{rms} = \sqrt{3 \times 8.314 \times 300/0.002}$

2. Inside: $3{,}741{,}300$

3. Square root: $v_{rms} = 1934\ \mathrm{m/s}$

Unit check

√(J/(mol·K) × K ÷ kg/mol) = m/s. The units confirm a molecular speed.

What this means physically

1934 m/s — over 5× the speed of sound. Fast H₂ molecules at the top of the atmosphere can even reach escape velocity, which is why Earth has almost no hydrogen left.