Subjects and lessons to explore.

Law

General Chemistry I · Gases

Molecular Speeds and the Maxwell-Boltzmann Distribution

3 min read
SourcesReport an error
Want it in plain words first? Jump to Eli explains — the same idea, no jargon.
On this page 7 sections
  1. In 30 seconds
  2. Why this matters
  3. The college version
  4. Eli explains
  5. Key takeaway
  6. Study tools
  7. Sources & references

In 30 seconds

Gas particles at a given temperature do not all move at one speed — they have a distribution of speeds. A useful single number is the root-mean-square speed, u_rms = √(3RT/M). At a fixed temperature, lighter molecules move faster on average than heavier ones, and raising the temperature shifts the entire distribution to higher speeds.

Why this matters

Molecular speeds determine how fast gases diffuse and effuse, how quickly reactions occur, and whether a planet can hold onto an atmosphere. For example, Earth's atmosphere retains heavy N₂ and O₂ but not the very light H₂, whose molecules at our temperatures can reach escape velocity. u_rms also lets you compare the thermal motion of different gases directly.

The college version

Key Ideas

  • Distribution of speeds: molecules have a range of speeds, described by the Maxwell-Boltzmann distribution.
  • Root-mean-square speed: u_rms = √(3RT/M).
  • Units matter: use R = 8.314 J/(mol·K) and M in kg/mol to get u_rms in m/s.
  • Lighter = faster: at the same T, lower molar mass gives higher u_rms.
  • Hotter = faster: raising T raises the average speed (u_rms ∝ √T).

Equations and Variables

  • u_rms = √(3RT/M) — u_rms = root-mean-square speed (m/s), R = 8.314 J/(mol·K), T = kelvin, M = molar mass in kg/mol.
  • Related measures (same gas, same T): most probable speed u_mp = √(2RT/M); average speed u_avg = √(8RT/πM).
  • Ranking: u_mp < u_avg < u_rms.
  • Temperature dependence: u_rms ∝ √T; molar-mass dependence: u_rms ∝ 1/√M.

How It Works

  1. Kinetic energy and temperature are linked by KE_avg = (3/2)RT.
  2. Because KE = ½mv² per particle (or ½M·u² per mole), setting them equal and solving for the speed gives u_rms = √(3RT/M).
  3. At any temperature the molecules actually have a spread of speeds — a few slow, a few fast, most near the middle.
  4. Lighter molecules must move faster to have the same average kinetic energy as heavier ones at the same temperature.
  5. Increasing the temperature broadens the distribution and shifts its peak to higher speeds.

Worked Example

Calculate the rms speed of N₂ molecules (M = 28.02 g/mol) at 25 °C. Convert: M = 28.02 g/mol × (1 kg / 1000 g) = 0.02802 kg/mol; T = 25 + 273.15 = 298.15 K. u_rms = √(3RT/M) = √[(3)(8.314 J/mol·K)(298.15 K) / 0.02802 kg/mol] = √(265,000 m²/s²) = 515 m/s. For comparison, the lighter H₂ (M = 0.002016 kg/mol) at the same temperature: u_rms = √[(3)(8.314)(298.15) / 0.002016] = 1920 m/s.

Common Confusions

  • "All molecules move at the same speed at a fixed T" — they have a distribution of speeds; u_rms is only an average.
  • "Using g/mol for M is fine" — with R = 8.314 J/(mol·K) you must use kg/mol, or the speed comes out wrong by √1000.
  • "Heavier gases are faster" — at fixed temperature, heavier gases are slower.
  • "Doubling T doubles u_rms" — u_rms scales with √T, so doubling T raises it only by √2.
Eli, the EliExplains learning guide

Eli explains

The same idea, in plain words

Explain it like I’m 10

Think of a highway where every car drives at a different speed — some crawl, some fly, most are around the speed limit. A gas is the same: its molecules have a whole range of speeds. The "rms speed" is like a single speed that summarizes the whole highway's energy. And just like kids on a playground, the lighter molecules (the "helium kids") sprint around much faster than the heavier ones (the "xenon adults"). The analogy's limit: cars have engines and drivers, while gas speeds come purely from temperature and mass — there's no "choice" involved.

Key takeaways

  • u_rms = √(3RT/M); M in kg/mol for m/s.
  • u_rms ∝ √T and u_rms ∝ 1/√M.
  • At the same T, lighter gases move faster.
  • Speeds follow a distribution (Maxwell-Boltzmann), not a single value.
  • Ordering at fixed T: u_mp < u_avg < u_rms.
  • N₂ at 25 °C: u_rms ≈ 515 m/s.
  • u_rms = √(3RT/M); use M in kg/mol → m/s.
  • Lighter molecules → faster; hotter → faster.
  • Speeds have a Maxwell-Boltzmann distribution.
  • u_mp < u_avg < u_rms at fixed T.
  • N₂ at 25 °C ≈ 515 m/s.

Keep learning

Ready to build on this? Continue to the next lesson.

Practice General Chemistry I

This lesson has no separate scored set. Practice draws from the subject’s question bank.

Study tools & related lessonsYou’ll learn to · Related

You’ll learn to

  • Compute the root-mean-square (rms) speed of gas molecules.
  • Explain why molar mass must be in kg/mol for speeds in m/s.
  • Describe the spread of molecular speeds (Maxwell-Boltzmann distribution).
  • Predict how temperature and molar mass shift the speed distribution.

Sources & references

  1. OpenStax, "9.5 The Kinetic-Molecular Theory," Chemistry 2e.
  2. Petrucci et al., "6.8 Gas Properties Relating to the Kinetic-Molecular Theory," Chemistry LibreTexts.
  3. NIST CODATA, "molar gas constant."

This lesson was adapted from the open educational references above; their licenses and attributions are preserved. See Copyright & Licensing.

Educational content only. It is not medical, legal or professional advice. Found an error? Tell us.