Maxwell-Boltzmann Speed Distribution

Equivalent forms

f(v) = \left(\frac{m}{2\pi k_B T}\right)^{3/2} 4\pi v^2 \exp\left(-\frac{mv^2}{2k_BT}\right)

v_{\text{mp}} = \sqrt{\frac{2k_BT}{m}}

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

The bridge between the invisible chaos of individual molecules and the calm predictability of temperature and pressure — statistics taming randomness.

Symbol	Name	SI unit	Dimension	Range
$v$	Molecular speed Speed of an individual gas molecule	m/s	L·T⁻¹	0 – 2000
$m$	Molecular mass Mass of a single gas molecule	kg	M	1e-27 – 1e-24
$T$	Temperature Absolute temperature of the gas	K	Θ	50 – 5000
$k_B$	Boltzmann constant Relates average kinetic energy of particles to temperature	J/K	M·L²·T⁻²·Θ⁻¹	1.380649e-23 – 1.380649e-23

Unit systems

Symbol	SI	CGS	Imperial
$v$	m/s	cm/s	ft/s
$m$	kg	g	lb
$T$	K	K	°R
$k_B$	J/K	erg/K	ft·lbf/(°R)

Where it holds

Valid for classical ideal gases in thermal equilibrium. Breaks down for quantum gases (Bose-Einstein or Fermi-Dirac statistics required at very low T or high density), relativistic speeds, and strongly interacting systems.

Dimensional analysis

$[f(v)] = [n](([m]/[k_B][T])^(3/2))[v^{2}]\exp (-[mv^{2}/2k_BT]) \to m^{-3}\cdot (kg/(J/K\cdot K))^(3/2)\cdot (m/s)^{2} = m^{-3}\cdot (s/m)^{3}\cdot (m^{2}/s^{2}) = m^{-3}\cdot s/m = s/m^{4}$ → probability density per unit speed per unit volume $\checkmark$

Discovery

James Clerk Maxwell · 1860

Maxwell derived this distribution at age 29, founding statistical mechanics. Boltzmann later generalized it to all energy forms, creating the Boltzmann distribution.

Try this

In a room full of air, are all molecules moving at the same speed?

Find the most probable speed of nitrogen molecules (m = 4.65×10⁻²⁶ kg) at 300 K using k_B = 1.380649×10⁻²³ J/K.

Research status: stable

Real-world applications

Escape velocity of atmospheric gases — explains why Earth retains $N_{2} but$ loses $H_{2} and$ He
Reaction rate theory (Arrhenius equation) — only molecules in the high-speed tail have enough energy to react
Semiconductor physics — electron energy distributions in non-degenerate semiconductors
Gas effusion rates through small orifices (Graham's law)

Common misconceptions

The most probable speed, mean speed, and RMS speed are three different values — v_mp < ⟨v⟩ < v_rms
The distribution is not symmetric — it has a long tail toward high speeds (right-skewed) because the $v^{2}$ factor grows faster than the exponential decays at moderate v
This distribution applies only to classical ideal gases — quantum gases (fermions and bosons) follow Fermi-Dirac or Bose-Einstein distributions respectively

Experimental verification

Stern's molecular beam experiment (1920) and Zartman-Ko experiment (1930) directly measured molecular speed distributions by spinning slotted drums. Modern laser spectroscopy and time-of-flight mass spectrometry confirm the distribution to high precision.

Derivation

Maxwell derived the speed distribution by assuming: (1) velocity components v_x, v_y, v_z are independent, (2) the distribution is isotropic, (3) the distribution depends only on speed |v|.

These constraints uniquely determine a Gaussian for each component:

f(v_i) \propto \exp (-mv_i^{2}/2k_BT)

.

Converting to spherical coordinates in velocity space introduces the

4\pi v^{2}

factor (density of states), giving

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

.

Key speeds:

v_mp = \sqrt{2k_BT/m}

, ⟨v⟩

= \sqrt{8k_BT/\pi m}

,

v_rms = \sqrt{3k_BT/m}

.

Limiting cases

T \to 0

⟶ Distribution collapses to delta at

v = 0

At absolute zero all molecules would be stationary (quantum effects intervene before this).

T \to \infty

⟶ Distribution flattens and shifts to high vHigher temperature spreads the distribution and increases the most probable speed.

m \to 0

(light particles)⟶ Peak shifts to very high speedsLighter molecules

(H_{2}

, He) move much faster than heavy ones (Xe) at the same temperature.

What if…

What if you double the temperature?

The most probable speed increases $by \sqrt{2} \approx 1.41\times$ (since $v_mp \propto \sqrt{T})$ . The distribution broadens and shifts right, with more molecules in the high-speed tail. At 600 K, $N_{2}$ molecules would have $v_mp \approx 597\,\mathrm{m/s}$ .

What if you compare

H_{2} and Xe

at the same temperature?

$H_{2} (m = 3.35\times 10^{-27} kg)$ moves $\sim 8\times$ faster than $Xe (m = 2.18\times 10^{-25} kg)$ because $v_mp \propto 1/\sqrt{m}$ . This is why hydrogen escapes Earth's atmosphere but xenon does not — the tail of $H_{2}'s$ distribution exceeds escape velocity (11.2 km/s).

What if temperature approaches absolute zero?

The distribution collapses to a spike near $v = 0$ . All molecules would be nearly stationary. In reality, quantum effects take over before $T = 0$ — Bose-Einstein condensation occurs for bosonic atoms, and the classical Maxwell-Boltzmann picture fails.

1

Most probable speed of N₂ at 300 K

Given ·

m:: 4.65e-26
T:: 300
k B:: 1.380649e-23

Find · v_mp

Steps

$Use v_mp = \sqrt{2k_BT/m}$
Numerator: $2 \times 1.380649\times 10^{-23} \times 300 = 8.28389\times 10^{-21} J$
$v_mp = \sqrt{8.28389\times 10^{-21} / 4.65\times 10^{-26}} = \sqrt{178148} = 422\,\mathrm{m/s}$

Result ·

v_mp = \sqrt{2k_BT/m} = \sqrt{2 \times 1.380649\times 10^{-23} \times 300 / 4.65\times 10^{-26}} = 422\,\mathrm{m/s}

2

RMS speed of He at 300 K

Given ·

m:: 6.646e-27
T:: 300
k B:: 1.380649e-23

Find · v_rms

Steps

$Use v_rms = \sqrt{3k_BT/m}$
Numerator: $3 \times 1.380649\times 10^{-23} \times 300 = 1.24258\times 10^{-20} J$
$v_rms = \sqrt{1.24258\times 10^{-20} / 6.646\times 10^{-27}} = \sqrt{1.8696\times 10^{6}} = 1367\,\mathrm{m/s}$

Result ·

v_rms = \sqrt{3k_BT/m} = \sqrt{3 \times 1.380649\times 10^{-23} \times 300 / 6.646\times 10^{-27}} = 1367\,\mathrm{m/s}

Ideal Gas Law→Entropy Change→Stefan-Boltzmann Law→