A temperature is a distribution

What does thermal speed actually mean?

What does thermal speed actually mean?

  1. Compare 4 g/mol and 28 g/mol at 300 K.
  2. Increase temperature and watch the histogram broaden.
  3. Compare speed ratios with the square-root law.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

Each Cartesian molecular velocity component follows a Gaussian distribution at equilibrium. Speed is the length of the three-component vector, so its distribution is not Gaussian. The most probable speed, mean speed and RMS speed are three different numbers. The RMS value connects directly to kinetic energy. A heavier gas at the same temperature has lower characteristic speeds because the mean kinetic energy per particle is the same. The three-dimensional ensemble uses the same seeded normal samples as the visible speed histogram, making changes reproducible.

f(v)∝v² exp(−Mv²/2RT) · ½m⟨v²⟩=3kBT/2

Units and parameter definitions are shown beside every laboratory control and live reading.

Build the argument

  1. Specify the ensemble, particle mass, temperature and boundary conditions.
  2. Average over many particles before comparing microscopic and macroscopic quantities.
  3. For independent isotropic diffusion, each coordinate has variance 2Dt, giving mean squared displacement 6Dt in three dimensions.

Worked example 1

At 300 K and 28 g/mol, vᵣₘₛ≈517 m/s.

Worked example 2

Quadrupling temperature doubles RMS speed.

Applications

Microscopic statistics connect to pressure, diffusion and transport. The visible particles represent an ensemble; their rendered size and playback speed are illustrative.

Common misconception

Temperature is a distribution of molecular kinetic energies, not the speed of every molecule. A stochastic trajectory cannot predict a particular future collision.

A further investigation

At equal temperature, a heavier molecule has…

Check your understanding

At equal temperature, a heavier molecule has…

  • Lower RMS speed
  • Higher mean kinetic energy
  • Zero motion
Explanation

Equipartition fixes mean kinetic energy; speed scales as M⁻¹/².

Model limits

Classical dilute equilibrium gas; finite representative ensemble, not a many-body collision solver.

OpenStax · Molecular model of an ideal gas