Maxwell–Boltzmann Distribution — Speed & Energy of Gas Molecules

1. Kinetic Theory: Molecules & Temperature

In kinetic theory an ideal gas is a collection of N point particles bouncing inside a container. Four key assumptions define the model:

• Molecules are identical, perfectly elastic spheres (no energy loss on collision).
• The volume of the molecules is negligible compared to the container volume.
• The only interactions are instantaneous elastic collisions; there are no long-range forces.
• Collisions are statistically uncorrelated — the molecular chaos assumption.

With these assumptions the pressure–volume–temperature relationship emerges purely from Newton's laws. More importantly, the equipartition theorem links absolute temperature to the average kinetic energy per degree of freedom:

where k_B = 1.380 649 × 10⁻²³ J/K is Boltzmann's constant and m is the molecular mass. Temperature is therefore a macroscopic measure of microscopic kinetic energy.

2. Gaussian Velocity Components

Because the gas is in thermal equilibrium and there is no preferred direction, each Cartesian velocity component is independent and identically distributed. Combining the equipartition result with the Boltzmann factor for a single component gives a one-dimensional Gaussian:

This is the probability density for the x-component of velocity. The mean is zero (no net drift) and the distribution widens with temperature. The same form holds for v_y and v_z.

In 3D, the velocity vector (v_x, v_y, v_z) follows a spherically symmetric multivariate Gaussian — a product of three independent one-dimensional Gaussians.

3. The Speed Distribution f(v)

The speed v = |v| (a scalar magnitude, v ≥ 0) has a different distribution than each component. To find it, we integrate the 3D Gaussian over all angles, treating the velocity space as a set of thin spherical shells of radius v and thickness dv. The surface area of such a shell is 4πv², giving:

The three factors have intuitive interpretations:

• 4πv² — the volume of the spherical shell in velocity space (more ways to reach high speed).
• (m / 2πk_BT)^(3/2) — normalisation constant.
• exp(−mv² / 2k_BT) — Boltzmann factor suppressing high energies.

The competition between the v² term (growing) and the Boltzmann suppression (falling) creates a peak at a finite speed, giving the characteristic right-skewed bell shape.

4. Characteristic Speeds: v_p, ⟨v⟩, v_rms

Three speeds characterise the distribution:

Because the distribution is positively skewed, the RMS speed lies above the mean, which lies above the most-probable speed. For all three, speed scales as √T (doubling temperature increases speeds by ≈ 41 %) and as 1/√m (lighter molecules move faster at the same temperature).

5. Energy Distribution

Substituting E = ½mv² (so dE = mv dv, v = √(2E/m)) into the speed distribution gives the distribution over kinetic energies:

Key moments of the energy distribution:

• Peak (most probable energy): E_p = ½ k_BT
• Mean energy: ⟨E⟩ = (3/2) k_BT (from equipartition)
• Variance: Var(E) = (3/2)(k_BT)²

The energy distribution is universal in the sense that it depends on temperature only through the ratio E/k_BT — the Boltzmann factor again.

6. Temperature Dependence

As temperature increases the distribution broadens and its peak shifts to the right. Qualitatively:

• At low T the molecules are slow and tightly clustered around v_p.
• At high T the distribution spreads out — more molecules at both very low and very high speeds.
• The area under the curve is always 1 (normalisation), so as it widens it also flattens.

Heavier molecules have smaller v_p at the same temperature. Hydrogen (H₂, M = 2 g/mol) at room temperature has v_p ≈ 1578 m/s — nearly four times that of oxygen (O₂, M = 32 g/mol) at ≈ 394 m/s. This mass dependence underpins Graham's law of effusion and isotope separation by gaseous diffusion.

The High-Speed Tail

The exponential tail means there is always some probability of extremely fast molecules. This tail matters enormously:

• Atmospheric escape: molecules in the high-speed tail of Earth's exosphere can exceed escape velocity (≈ 11.2 km/s). The lightweight hydrogen and helium leak away over geological time.
• Nuclear fusion: the Sun's core temperature (≈ 1.5 × 10⁷ K) is too low for classical fusion — only the high-speed tail enables quantum tunnelling through the Coulomb barrier (Gamow peak).

7. JavaScript: Box–Muller Sampling

To simulate a gas, we need to sample from the Maxwell–Boltzmann distribution. The cleanest approach uses the Box–Muller transform to produce standard-normal samples for each velocity component, then scales by the thermal velocity σ = √(k_BT/m).

// Box–Muller transform: two uniform → two standard-normal samples
function boxMuller() {
  const u1 = Math.random();      // uniform in (0, 1]
  const u2 = Math.random();
  const r  = Math.sqrt(-2 * Math.log(u1));
  return [
    r * Math.cos(2 * Math.PI * u2),
    r * Math.sin(2 * Math.PI * u2),
  ];
}

// Sample a single particle velocity (vx, vy, vz) in m/s
function sampleVelocity(T, m) {
  const kB    = 1.380649e-23;            // Boltzmann constant J/K
  const sigma = Math.sqrt(kB * T / m);   // thermal speed (std dev per component)
  const [z0, z1] = boxMuller();
  const [z2]     = boxMuller();
  return {
    vx: sigma * z0,
    vy: sigma * z1,
    vz: sigma * z2,
  };
}

// Compute the speed probability density at speed v (m/s) for a molecule of mass m
function maxwellBoltzmannPDF(v, T, m) {
  const kB = 1.380649e-23;
  const a  = m / (2 * kB * T);
  return 4 * Math.PI * Math.pow(a / Math.PI, 1.5)
       * v * v
       * Math.exp(-a * v * v);
}

// Example: initialise N₂ molecules at 300 K
const N  = 500;
const T  = 300;                        // Kelvin
const mN2 = 28 * 1.6605e-27;          // 28 u in kg

const particles = Array.from({length: N}, () => sampleVelocity(T, mN2));

// Compute speed of each and histogram
const speeds = particles.map(p =>
  Math.hypot(p.vx, p.vy, p.vz)
);

const vp = Math.sqrt(2 * 1.380649e-23 * T / mN2);  // ~422 m/s
const vrms = Math.sqrt(3 * 1.380649e-23 * T / mN2); // ~517 m/s

Scaling for a Canvas Simulation

For a 2D browser simulation the velocity vector is reduced to (v_x, v_y). Since we cannot use SI units directly, we scale velocities to pixel-per-frame units: choose a reference speed V₀ (in pixels/frame) corresponding to v_p at the desired temperature, then multiply samples by V₀/v_p. The shape of the distribution is identical; only the units change.

8. Applications

Graham's Law of Effusion

When a gas escapes through a tiny hole into vacuum, faster molecules effuse more frequently. The effusion rate is proportional to the mean speed, giving:

Uranium hexafluoride isotopes ²³⁵UF₆ (M ≈ 349) and ²³⁸UF₆ (M ≈ 352) differ by only √(352/349) ≈ 1.004 per stage. Cascading thousands of stages allows isotope enrichment for nuclear reactors and weapons.

Arrhenius Equation & Reaction Rates

The rate constant k(T) is proportional to the Boltzmann factor — the fraction of molecular collisions with enough energy to surmount the activation barrier. This explains why raising T by 10 °C roughly doubles most reaction rates.

Doppler Broadening of Spectral Lines

The thermal motion of atoms in a hot gas shifts each atom's emitted/absorbed photon frequency by the Doppler effect. Since v_x follows a Gaussian, the resulting spectral line has a Gaussian profile with width Δν ∝ √(T/M). Astronomers use this broadening to measure the temperature and turbulent velocity of stellar atmospheres.

Jeans Escape & Planetary Atmospheres

At the top of an atmosphere (exosphere) molecules that reach escape velocity are lost to space. The fraction above v_escape is the tail of the Maxwell–Boltzmann distribution. Hydrogen and helium are so light that even at Earth's exosphere temperature (≈ 1000 K) an appreciable fraction exceeds v_esc ≈ 11.2 km/s — explaining why the Earth has retained its nitrogen-oxygen atmosphere but lost its primordial hydrogen.