Playground
Interactive Maxwell-Boltzmann speed distribution curve: adjust temperature and molecular mass to see the distribution reshape with marked most-probable, average, and RMS speeds.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Molecular speed Speed of an individual gas molecule | m/s | L·T⁻¹ | 0 – 2000 | |
| Molecular mass Mass of a single gas molecule | kg | M | 1e-27 – 1e-24 | |
| Temperature Absolute temperature of the gas | K | Θ | 50 – 5000 | |
| Boltzmann constant Relates average kinetic energy of particles to temperature | J/K | M·L²·T⁻²·Θ⁻¹ | 1.380649e-23 – 1.380649e-23 |
Deep dive
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) ∝ exp(−mv_i²/2k_BT). Converting to spherical coordinates in velocity space introduces the 4πv² factor (density of states), giving f(v) = 4πn(m/2πk_BT)^(3/2) v² exp(−mv²/2k_BT). Key speeds: v_mp = √(2k_BT/m), ⟨v⟩ = √(8k_BT/πm), v_rms = √(3k_BT/m).
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.
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 v² 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
Real-world applications
- Escape velocity of atmospheric gases — explains why Earth retains N₂ but loses H₂ 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)
Worked examples
Most probable speed of N₂ at 300 K
Given:
- m:
- 4.65e-26
- T:
- 300
- k_B:
- 1.380649e-23
Find: v_mp
Solution
v_mp = √(2k_BT/m) = √(2 × 1.380649×10⁻²³ × 300 / 4.65×10⁻²⁶) = 422 m/s
RMS speed of He at 300 K
Given:
- m:
- 6.646e-27
- T:
- 300
- k_B:
- 1.380649e-23
Find: v_rms
Solution
v_rms = √(3k_BT/m) = √(3 × 1.380649×10⁻²³ × 300 / 6.646×10⁻²⁷) = 1367 m/s
Scenarios
What if…
- scenario:
- What if you double the temperature?
- answer:
- The most probable speed increases by √2 ≈ 1.41× (since v_mp ∝ √T). The distribution broadens and shifts right, with more molecules in the high-speed tail. At 600 K, N₂ molecules would have v_mp ≈ 597 m/s.
- scenario:
- What if you compare H₂ and Xe at the same temperature?
- answer:
- H₂ (m = 3.35×10⁻²⁷ kg) moves ~8× faster than Xe (m = 2.18×10⁻²⁵ kg) because v_mp ∝ 1/√m. This is why hydrogen escapes Earth's atmosphere but xenon does not — the tail of H₂'s distribution exceeds escape velocity (11.2 km/s).
- scenario:
- What if temperature approaches absolute zero?
- answer:
- 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.
Limiting cases
- condition:
- T → 0
- result:
- Distribution collapses to delta at v = 0
- explanation:
- At absolute zero all molecules would be stationary (quantum effects intervene before this).
- condition:
- T → ∞
- result:
- Distribution flattens and shifts to high v
- explanation:
- Higher temperature spreads the distribution and increases the most probable speed.
- condition:
- m → 0 (light particles)
- result:
- Peak shifts to very high speeds
- explanation:
- Lighter molecules (H₂, He) move much faster than heavy ones (Xe) at the same temperature.
Context
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.
Hook
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.
Dimensions: [f(v)] = [n](([m]/[k_B][T])^(3/2))[v²]exp(−[mv²/2k_BT]) → m⁻³·(kg/(J/K·K))^(3/2)·(m/s)² = m⁻³·(s/m)³·(m²/s²) = m⁻³·s/m = s/m⁴ → probability density per unit speed per unit volume ✓
Validity: 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.