🌡️ Maxwell-Boltzmann Distribution

N elastic hard-sphere molecules in a box. The live speed histogram builds up and converges to the theoretical Maxwell-Boltzmann distribution f(v) = 4π(m/2πkT)^(3/2) v² exp(−mv²/2kT).

Gas Parameters

Temperature T 400 K Particle mass m 4 u (He) Particle count N 150

Statistics

v_p (most prob.)–

⟨v⟩ (mean)–

v_rms–

⟨KE⟩ per mol–

Pressure P–

Presets

What It Demonstrates

The Maxwell-Boltzmann distribution is the probability distribution of molecular speeds in an ideal gas at thermal equilibrium. It arises from the statistical mechanics of many interacting particles and defines three characteristic speeds: v_p (most probable), ⟨v⟩ (mean), and v_rms (root mean square). Higher temperature or lower mass shifts the distribution to higher speeds.

How to Use

Raise Temperature T and watch the histogram broaden and shift right
Increase mass m (heavier gas) and watch the distribution narrow and move left
The red curve is the theoretical Maxwell-Boltzmann; the blue bars are the live measured histogram
Try the presets to compare H₂ vs Pb — the 300× mass difference is dramatic

Did You Know?

At room temperature (300 K), hydrogen molecules (H₂) have a mean speed of ~1,700 m/s — fast enough to escape Earth's gravity over geological time, which is why the atmosphere has so little hydrogen. Oxygen molecules (O₂) move at ~480 m/s, well below escape velocity.