Maxwell-Boltzmann Speed Distribution
The Maxwell-Boltzmann distribution gives the probability of gas molecule speeds at a given temperature.
Derives most probable, mean, and RMS speeds.
The Formula
The Maxwell-Boltzmann distribution was derived independently by James Clerk Maxwell (1860) and Ludwig Boltzmann (1872). It describes the probability distribution of speeds in an ideal gas at thermal equilibrium. Not all molecules move at the same speed — there is a spread of speeds, and this formula describes that spread.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| f(v) | Probability density — fraction of molecules with speed near v | s/m |
| v | Molecular speed | m/s |
| m | Mass of one molecule | kg |
| k_B | Boltzmann constant = 1.381 × 10−23 J/K | J/K |
| T | Absolute temperature | Kelvin (K) |
Three characteristic speeds can be derived:
- Most probable speed v_p = √(2k_BT/m) — the peak of the distribution
- Mean speed v¯ = √(8k_BT/πm) — the arithmetic average
- RMS speed v_rms = √(3k_BT/m) — related to kinetic energy
The ordering is always: v_p < v¯ < v_rms, with ratio 1 : 1.128 : 1.225
Example 1 — Nitrogen Gas at 300 K
Calculate v_p, v̄, and v_rms for nitrogen gas (N&sub2;, molar mass 28 g/mol) at T = 300 K.
m = 28 × 10−3 / 6.022 × 10²³ = 4.651 × 10−²&sup6; kg per molecule
k_BT = 1.381 × 10−²³ × 300 = 4.143 × 10−²¹ J
v_p = √(2 × 4.143 × 10−²¹ / 4.651 × 10−²&sup6;) = √(1.782 × 10&sup5;) = 422 m/s
v̄ = v_p × √(4/π) = 422 × 1.128 = 476 m/s
v_rms = v_p × √(3/2) = 422 × 1.225 = 517 m/s — nitrogen molecules average about 1,700 km/h!
Example 2 — Effect of Temperature on Speed
How much faster do hydrogen molecules move than nitrogen molecules at the same temperature?
v_rms ∝ √(1/m), so the ratio of speeds = √(m_N2 / m_H2)
ratio = √(28/2) = √14 ≈ 3.74
Hydrogen molecules move 3.74× faster than nitrogen molecules at the same temperature. This is why hydrogen escapes Earth's atmosphere — it can exceed escape velocity more easily.
When to Use It
Use the Maxwell-Boltzmann distribution when:
- Calculating gas molecule speeds and kinetic energies at a given temperature
- Understanding why lighter gases diffuse faster (Graham's law)
- Analyzing reaction rates in chemistry — only molecules with sufficient energy react (Arrhenius equation)
- Studying atmospheric escape — planets lose their atmospheres when molecules reach escape velocity
- Designing vacuum systems and gas-phase chemical reactors
The Maxwell-Boltzmann distribution is the classical (non-quantum) limit. At very low temperatures or very high densities, quantum effects become important and the Fermi-Dirac or Bose-Einstein distributions must be used instead.