• Pixel-art sticker of Speedra, personifying RMS Speed of Gas Molecules, with the caption “Hotter Gas, Faster Crowd.”

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Speedra

RMS Speed of Gas Molecules Physics

\[v_{\text{rms}} = \sqrt{\frac{3k_BT}{m}}\]
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Meet Speedra, who runs hotter, runs faster: gas molecules move at a spread of speeds, and the typical speed, vrms = √(3kBT/m), grows with the square root of temperature. Nitrogen at 300 K has a vrms of about 517 m/s. Pixel-art kiss-cut sticker in 2, 3, 4 and 6 inches; equation, symbols and sources in the physics panel below.

Size
Surface

Flat $6 shipping worldwideMade to order, ships in 2–5 business days

Details

Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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Flat $6 shipping on every order, worldwide, whatever's in it.

Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

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The physics

Meet Speedra, a gas molecule zipping around at hundreds of metres per second.

The molecules of a gas move in all directions at a spread of speeds, described by the Maxwell–Boltzmann distribution, worked out by James Clerk Maxwell in 1860 and Ludwig Boltzmann. Their average kinetic energy is set by temperature: ½m⟨v²⟩ = (3/2)k_BT. The root-mean-square speed is therefore v_rms = √(3k_BT/m).

For nitrogen, the main gas in air, at 300 K, v_rms is about 517 metres per second, faster than the speed of sound. Lighter molecules move faster: hydrogen at the same temperature is nearly four times quicker. Raising the temperature 2.25 times makes the molecules 1.5 times faster, as the two curves in the sticker show.

These speeds explain why gas pressure exists, how quickly smells spread and why light gases like hydrogen and helium escape from Earth's atmosphere over time.

Individual molecules move at a wide range of speeds, and collisions make each one's path a zigzag, so a smell spreads far more slowly than 500 m/s.

Equations

\[v_{\text{rms}} = \sqrt{\frac{3k_BT}{m}}\]
SymbolMeaningUnit
\(v_{\text{rms}}\) root-mean-square speed of the molecules \(\mathrm{m/s}\)
\(k_B\) Boltzmann constant \(\mathrm{J/K}\)
\(T\) absolute temperature \(\mathrm{K}\)
\(m\) mass of one molecule \(\mathrm{kg}\)

Sources

  1. OpenStax University Physics Vol. 2 — Pressure, Temperature, and RMS Speed
  2. OpenStax University Physics Vol. 2 — Distribution of Molecular Speeds
  3. NIST CODATA — Boltzmann constant k
  4. Wikipedia — Maxwell–Boltzmann distribution

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