Maxwell Speed Distribution (3D)

Equivalent forms

v_{mp} = \sqrt{\frac{2k_B T}{m}}

\langle v \rangle = \sqrt{\frac{8k_B T}{\pi m}}

v_{rms} = \sqrt{\frac{3k_B T}{m}}

The three characteristic speeds v_mp < ⟨v⟩ < v_rms are in the ratio 1 : √(4/π) : √(3/2) ≈ 1 : 1.128 : 1.225.

Symbol	Name	SI unit	Dimension	Range
$f(v)$	Speed distribution functionoutput Probability density for molecular speed v	s/m	LT⁻¹ × L⁻¹	0 – 0.001
$T$	Temperature Absolute temperature	K	Θ	50 – 10000
$m$	Molecular mass Mass of one molecule; N₂ = 4.65e-26 kg, He = 6.64e-27 kg	kg	M	6.64e-27 – 2e-25

Unit systems

SI:: v in m/s, m in kg, T in K
chemistry:: $m = M_r / N$ _A where M_r is molar mass in g/mol
astrophysics:: Jeans escape: v_escape $\approx 10\,\mathrm{v}_rms$ sets atmospheric retention

Where it holds

Ideal gas in thermal equilibrium. Breaks down at high density (collisions change distribution), near walls, or for quantum gases.

Discovery

James Clerk Maxwell · 1860

Maxwell derived the velocity distribution in 1860 using symmetry arguments alone, before Boltzmann's full statistical mechanics. It was the first time probability entered physics at a fundamental level.

Try this

What fraction of air molecules are moving faster than a bullet right now?

Room-temperature nitrogen molecules have an rms speed of 515 m/s — faster than a rifle bullet (900 m/s) for roughly 1-in-200 molecules. The Maxwell speed distribution tells you exactly how many, and why supersonic molecules exist at any temperature above absolute zero.

Research status: stable

Common misconceptions

v_mp (most probable) < ⟨v⟩ (mean) < v_rms because the distribution is skewed right. The rms speed (3k_BT/m)^{1/2} is used for kinetic energy, the mean speed

(8k_BT/\pi m)^{1/2}

for flux calculations, and v_mp for the peak.

Derivation

From the 3D velocity distribution f(v_x,v_y,

v_z) \propto \exp (-mv^{2}/2k_BT)

, integrate over all directions:

f(v) = 4\pi v^{2} \times (m/2\pi k_BT)^{3/2} \exp (-mv^{2}/2k_BT)

.

The

4\pi v^{2} is

the volume element in spherical velocity space.

Limiting cases

v \to 0

⟶

f(v) \to 0 (v^{2}

factor suppresses low-speed region)

v \to \infty

⟶

f(v) \to 0

exponentially (Boltzmann suppression dominates)

T \to \infty

⟶ Distribution broadens,

v_mp \to \infty

; all speeds become equally likely in relative sense

Maxwell-Boltzmann Speed Distribution→Equipartition Theorem→Ideal Gas Law→