KinexiaLEARN PHYSICS

RMS Speed — Helium Atoms at Room Temperature (Worked Example)

Medium
Problem

Find the root-mean-square speed of helium atoms (molar mass 4.00 g/mol) in a balloon at 300 K.

Given

  • T = 300 K

  • M = 4.00 g/mol = 0.00400 kg/mol

  • R = 8.314 J/(mol·K)

Find

v_rms.

Principle

Kinetic theory links temperature to average molecular kinetic energy: ½m⟨v²⟩ = ³⁄₂kT, giving v_rms = √(3RT/M).

Formula

Solution

  1. Write the formula: vrms=3RT/Mv_{\mathrm{rms}} = \sqrt{3RT/M}

  2. Substitute (M in kg/mol!): 3×8.314×300/0.004\sqrt{3 \times 8.314 \times 300 / 0.004}

  3. Compute inside: 3×8.314×300=7482.63 \times 8.314 \times 300 = 7482.6; 7482.6/0.004=1,870,6507482.6/0.004 = 1{,}870{,}650

  4. Square root: 1,870,650≈1368 m/s\sqrt{1{,}870{,}650} \approx 1368\ \mathrm{m/s}

≈ 1.37 × 10³ m/s

Unit check

√(J/mol / (kg/mol)) = √(J/kg) = √((kg·m²/s²)/kg) = m/s — correct.

What this means physically

Helium atoms at room temperature move at nearly 1.4 km/s — four times the speed of sound. Light molecules are speedy, which is why helium balloons deflate through tiny pores and your voice rises after inhaling it.

Common mistake

Forgetting to convert molar mass to kg/mol. Using 4.00 instead of 0.00400 inflates the denominator by 1000× and gives a nonsense speed 31.6× too small.