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

Given

• T = 300 K

• M = 2.02 g/mol = 0.00202 kg/mol

• R = 8.314 J/(mol·K)

Find

v_rms and the ratio to escape velocity.

Principle

v_rms = √(3RT/M). A fraction of molecules in the Maxwell–Boltzmann tail exceed escape velocity and leak away over geological time.

Formula

Solution

1. $v_{\mathrm{rms}} = \sqrt{3 \times 8.314 \times 300 / 0.00202}$

2. Inside: $7482.6/0.00202 \approx 3{,}704{,}257$

3. Root: $\sqrt{3{,}704{,}257} \approx 1925\ \mathrm{m/s}$

4. Ratio to escape: $11{,}200/1925 \approx 5.8$

Unit check

m/s — correct for speed. Inside the root: (J/(mol·K) × K) / (kg/mol) = (kg·m²/s²)/kg = m²/s², whose square root is m/s.

What this means physically

The average H₂ molecule moves at ~1.9 km/s, and the fastest few percent exceed 11.2 km/s and escape to space. Over billions of years this bled Earth's primordial hydrogen away — one reason our air is nitrogen and oxygen, not hydrogen.