Root Mean Square Speed
Average molecular speed in gases
Understanding RMS Speed
Root mean square (RMS) speed is a type of average molecular speed derived from the kinetic molecular theory of gases. Unlike arithmetic mean speed, RMS speed accounts for the distribution of molecular velocities in a gas sample, providing a measure directly related to kinetic energy. It represents the square root of the average of squared speeds of all molecules.
According to kinetic molecular theory, gas molecules are in constant random motion, colliding with container walls and each other. These collisions create pressure, and the average kinetic energy of molecules is directly proportional to absolute temperature. The RMS speed quantifies this molecular motion and varies with both temperature and molecular mass.
The relationship vrms = √(3RT/M) shows that lighter molecules move faster than heavier ones at the same temperature, explaining phenomena like gas effusion rates and why helium balloons deflate faster than air-filled ones.
Formula
vrms = √(3RT / M)
- vrms = root mean square speed (m/s)
- R = gas constant (8.314 J·molâ»Â¹Â·Kâ»Â¹)
- T = absolute temperature (K)
- M = molar mass (kg/mol)
Unit Consistency:
Ensure M is in kg/mol, not g/mol. Convert by dividing grams by 1000. Temperature must be in Kelvin (K = °C + 273.15).
Detailed Example: O₂ at 25°C
Given: M(Oâ‚‚) = 32 g/mol = 0.032 kg/mol, T = 298 K.
Step 1: Convert units: 32 g/mol ÷ 1000 = 0.032 kg/mol
Step 2: Apply formula: vrms = √[(3 × 8.314 × 298) / 0.032]
Step 3: Calculate numerator: 3 × 8.314 × 298 = 7,434 J/mol
Step 4: Divide: 7,434 / 0.032 = 232,312 m²/s²
Step 5: Take square root: √232,312 ≈ 482 m/s
Answer: vrms ≈ 482 m/s (about 1,730 km/h)
This is roughly the speed of sound in air!
Types of Molecular Speeds
Most Probable Speed (vmp)
vmp = √(2RT/M) — The peak of Maxwell-Boltzmann distribution; most molecules have this speed.
Average Speed (vavg)
vavg = √(8RT/πM) — Arithmetic mean of all molecular speeds.
Root Mean Square Speed (vrms)
vrms = √(3RT/M) — Highest of the three; directly related to average kinetic energy.
Relationship: vmp < vavg < vrms (ratio ≈ 1 : 1.13 : 1.22)
Temperature and Mass Effects
Effect of Temperature
vrms ∠√T — Doubling absolute temperature increases speed by factor of √2 (≈1.41). Higher T = faster molecular motion.
Effect of Molar Mass
vrms ∠1/√M — Lighter molecules move faster. H₂ molecules move 4× faster than O₂ at same temperature.
Real-World Applications
Gas Separation
Uranium isotope separation (U-235/U-238) exploits small mass differences affecting diffusion rates based on molecular speeds.
Atmospheric Science
Explains why hydrogen and helium escape Earth's atmosphere while heavier gases like Nâ‚‚ and Oâ‚‚ remain gravitationally bound.
Vacuum Technology
Determines pumping speed requirements and gas flow characteristics in high-vacuum systems.
Chemical Kinetics
Collision frequency and reaction rates depend on molecular speeds predicted by kinetic theory.
Comparative Speeds at 25°C
| Gas | M (g/mol) | vrms (m/s) | Relative Speed |
|---|---|---|---|
| H₂ | 2 | 1,920 | 4.0× |
| He | 4 | 1,360 | 2.8× |
| N₂ | 28 | 515 | 1.1× |
| O₂ | 32 | 482 | 1.0× (reference) |
| CO₂ | 44 | 410 | 0.85× |
Key Takeaways
- Lighter molecules move faster at same temperature.
- Higher temperature = higher vrms (proportional to √T).
- Related to kinetic energy: KEavg = ½ m vrms² = (3/2) kBT.
- RMS speed is always greater than average speed and most probable speed.
- Kinetic molecular theory assumptions (ideal gas) work best at low pressure and high temperature.
- At same temperature, all gases have same average kinetic energy but different speeds.