RMS Speed — Helium Escape (Worked Example)

Medium

Problem

Helium (M = 0.004 kg/mol) fills a party balloon at 300 K. What is the rms speed of the helium atoms? (R = 8.314 J/(mol·K))

Given

T = 300 K
M = 0.004 kg/mol

Find

The rms speed v_rms.

Principle

Light molecules move faster at the same temperature: v_rms = √(3RT/M) with small M.

Formula

v_{rms} = \sqrt{\frac{3RT}{M}}

v_rms = √(3RT/M)

Solution

Substitute: $v_{rms} = \sqrt{3 \times 8.314 \times 300/0.004}$
Inside: $7482.6/0.004 = 1{,}870{,}650$
Square root: $v_{rms} = 1368\ \mathrm{m/s}$

1368 m/s

Unit check

√(J/(mol·K) × K ÷ kg/mol) = √(m²/s²) = m/s. A speed — dimensionally correct.

What this means physically

1368 m/s — nearly 3× nitrogen's speed. Such fast atoms leak through balloon rubber quickly, which is why helium balloons sag overnight while air-filled ones last days.

Common mistake

Thinking heavier means faster. v_rms ∝ 1/√M — lighter molecules are faster at the same temperature.