Ideal gas and Maxwell–Boltzmann distribution

Temperature is the average kinetic energy of the particles. Let gas molecules collide at random and their speeds always settle into the same shape, the Maxwell–Boltzmann distribution, whatever speeds they started with.

Interactive, free and runs in your browser. Part of the Thermodynamics simulations.

What you will see

Particles bounce elastically around a box, coloured by speed from slow blue to fast orange. A histogram of their speeds builds up live and matches the Maxwell–Boltzmann curve. Raise the temperature and the peak moves to higher speeds while the distribution spreads out.

Try this

  1. Raise the temperature and watch the histogram peak shift right.
  2. Look for the long tail of fast particles: there are always a few.
  3. Compare the most likely speed with the average speed.

Common misconception

At a given temperature the particles do not all move at the same speed. Temperature fixes the average kinetic energy; individual speeds spread over a wide range.

Exam-style question

What is the root-mean-square speed of nitrogen molecules (m = 4.65 × 10⁻²⁶ kg) at 300 K?

c_rms = √(3kT/m) = √(3 × 1.38 × 10⁻²³ × 300 / 4.65 × 10⁻²⁶) ≈ √(2.67 × 10⁵) ≈ 517 m/s.

Key equations

  • pV = NkT
  • ½m⟨c²⟩ = (3/2)kT
  • c_rms = √(3kT / m)

Curriculum

AQA A-level physics 3.6.2, IB Physics B.1–B.4.

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