The Maxwell-Boltzmann speed distribution

The construction rests on three facts about an equilibrium gas: each velocity component obeys ½m⟨vₓ²⟩ = ½kT, no component knows about the others, and no direction is preferred. Only a Gaussian in each component satisfies them all, so the probability of finding a molecule's velocity inside d³v goes as e^{(−m(vₓ² + v_y² + v_z²)/2kT)} d³v = e^{(−mv²/2kT)} d³v: the kinetic energy's Boltzmann factor, the same at every point of velocity space with one speed. Speed is a magnitude, so to count molecules with speed between v and v + dv you add that density over the spherical shell of radius v and thickness dv, whose volume is 4πv² dv. That is the origin of the v² in f(v): more distinct velocities exist at a larger speed, not more molecules wanting to move fast. So f(0) = 0, although the velocity density peaks there, and at large v the exponential wins. For nitrogen at 300 K the exponent at 1000 m s⁻¹ is −(1000/422)² = −5.6, so the Boltzmann factor is e^(−5.6) = 0.0037 while the shell factor has grown only 5.6-fold since the peak: the tail is exponentially thin.

A density is useful only once its integral is fixed. Write the shape as f(v) = A v² e^{(−v²/vₚ²)} with vₚ² = 2kT/m, and demand ∫₀^∞ f dv = 1. Substituting u = v/vₚ turns the integral into A vₚ³ ∫₀^∞ u² e^(−u²) du = A vₚ³ √π/4, so A = 4/(√π vₚ³), which is 4π (m/2πkT)³⁄² written out in m and T. Two things follow. First, the dimensions: vₚ³ carries m³ s⁻³, so f carries s m⁻¹ and f(v) dv is a pure number, a fraction, as it must be. Second, in reduced form the fraction in du is (4/√π) u² e^(−u²) du: one curve for every gas at every temperature, with m and T supplying only the scale vₚ. Using it is then a matter of reading the density. For nitrogen at 300 K, vₚ = 422 m s⁻¹ and the peak height is f(vₚ) = (4/√π)(1/e)/vₚ = 0.830/vₚ = 1.97 × 10⁻³ s m⁻¹, so a 10 m s⁻¹ band at the peak holds about 2.0 % of the molecules and the same band at 500 m s⁻¹ holds 1.8 %. A wide band needs the integral rather than f × Δv, because f changes across it.

The peak comes from df/dv = 0: d/dv[v² e^{(−v²/vₚ²)}] = 2v e^{(−v²/vₚ²)}(1 − v²/vₚ²), which vanishes at v = vₚ = √(2kT/m). The mean is the first moment, ⟨v⟩ = ∫₀^∞ v f dv = vₚ (4/√π) ∫₀^∞ u³ e^(−u²) du = vₚ (4/√π)(1/2) = 2vₚ/√π = 1.128 vₚ. The mean square is the second moment, ⟨v²⟩ = vₚ² (4/√π) ∫₀^∞ u⁴ e^(−u²) du = vₚ² (4/√π)(3√π/8) = (3/2) vₚ² = 3kT/m, so vᵣₘₛ = √(3/2) vₚ = 1.225 vₚ. That last line is the check on the whole construction, because it returns the ½m⟨v²⟩ = (3/2)kT that the kinetic model demanded before any distribution was written down. The order vₚ < ⟨v⟩ < vᵣₘₛ is not an accident of the numbers: the curve is bounded at v = 0 and unbounded on the right, so its long tail drags the mean past the peak, and the rms weights that tail by v² and is dragged further. For nitrogen at 300 K the three are 422, 476 and 517 m s⁻¹; only 42.8 % of the molecules are slower than vₚ, 53.3 % slower than ⟨v⟩ and 60.8 % slower than vᵣₘₛ.

Because f depends on m and T only through vₚ = √(2kT/m), every change of temperature or of gas is a rescaling of the speed axis. Quadrupling T doubles vₚ: the curve stretches to twice the width and, since the area under it must stay 1, its peak drops to half, because f(vₚ) = 0.830/vₚ is inversely proportional to vₚ. Swapping gases at fixed T does the same through the mass: helium (M = 4.00 g mol⁻¹) at 300 K has vₚ = 1117 m s⁻¹, 2.65 times nitrogen's 422 m s⁻¹, so its curve is 2.65 times broader and 2.65 times lower, while xenon (131.3 g mol⁻¹) is compressed by √(131.3/28.0) = 2.17 the other way. The rms speed inherits the scaling exactly: vᵣₘₛ = √(3RT/M) is 1.225 vₚ, so it too goes as √T and as 1/√M, and a plot of vᵣₘₛ against M at fixed T is one inverse-square-root curve that every gas sits on. One consequence deserves stating outright: helium at 42.9 K has the same vₚ as nitrogen at 300 K, and therefore the identical speed distribution, same peak, same mean, same rms, same tail, because T/M is all the curve knows.

The fraction faster than v₀ is the integral of the reduced density from x = v₀/vₚ to infinity. Integrating (4/√π) u² e^(−u²) by parts gives N(v > v₀)/N = (2x/√π) e^(−x²) + erfc(x); for x ≥ 2 the erfc term is small and the tail is ≈ (2x/√π) e^(−x²)(1 + 1/(2x²)). For nitrogen at 300 K, 1.06 % of molecules exceed 1000 m s⁻¹, where x = 2.37, and one in 73 000 exceeds 1500 m s⁻¹. Warm the gas to 600 K and the fraction above 1000 m s⁻¹ is 13.2 %, a factor of 12.5 for a factor of 2 in T, because the exponent x² = mv₀²/2kT halves, not the fraction. Evaporation lives here: only tail molecules can leave a liquid surface, so the rate rises steeply with T, and the leavers carry off more than the mean energy, cooling what remains. The same tail governs atmospheric escape, with v₀ the escape speed. At the top of the atmosphere, hydrogen atoms at 1000 K have x = 2.76 and 0.165 % of them exceed 11.2 km s⁻¹ at any instant; helium has x = 5.50 and a fraction 5 × 10⁻¹³; atomic oxygen has x = 11 and a fraction of order 10⁻⁵². Hydrogen leaks away fast, helium over the planet's lifetime, oxygen never.

Each assumption in the derivation is a check to make before quoting f(v). It is an equilibrium distribution, maintained by collisions that refill the tail as fast as escape or reaction empties it; where collisions are too rare for that, in the collisionless exosphere above about 500 km or in a beam leaving an oven, the speeds are not Maxwellian; the beam carries an extra factor of v because faster molecules reach the aperture more often. And it is classical. A gas is classical when the thermal de Broglie wavelength λ = h/√(2πmkT) is far smaller than the spacing between molecules; for nitrogen at 300 K and 1 atm, λ = 1.9 × 10⁻¹¹ m against a spacing n^(−1/3) = 3.4 × 10⁻⁹ m, a margin of 180, so Maxwell-Boltzmann is safe. For the conduction electrons in copper the ratio is inverted, λ some twenty times the spacing, and the speeds obey the Fermi-Dirac distribution instead. The correspondence runs the right way: in the classical limit both quantum distributions reduce to the Boltzmann factor, so Maxwell's curve is the low-density, high-temperature end of a more general result.