Library / Foundations: Physics / FPH-49
Foundations: Physics
Root-mean-square speed of gas molecules
Typical speed of molecules in an ideal gas at absolute temperature T, from the molar mass.
- Ref
- FPH-49
- Rev
- v1
- Field
- Foundations: Physics
- Result
- vrms [m/s]
For an ideal gas the mean translational kinetic energy per molecule is 3/2 kBT, so the root-mean-square speed is vrms = √(3RT/M) with the molar mass M in kg/mol, or √(3kBT/m) with the molecular mass m in kg. Light molecules move faster at the same temperature. T is in kelvin and R = 8.3145 J/(mol K).
Typical speed of molecules in an ideal gas at absolute temperature T, from the molar mass. Lighter molecules and hotter gas move faster.
Calculator
- Pressure of a gas of density 1.16 kg/m³ moving at this speed, ρvrms²/3103,300 Pa
- Mean translational energy per molecule, 3/2 kBT6.213×10−21 J
- Speed ratio to hydrogen (M = 0.002016 kg/mol) at the same temperature0.2683
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Nitrogen (M = 0.028 kg/mol) at 300 K has an rms speed of 517 m/s, close to the speed of sound in air (about 347 m/s at this temperature), which is the same order because sound is passed on by molecular collisions.
- Hydrogen, with about one fourteenth of the molar mass, moves 3.73 times faster at the same temperature, which is why light gases escape from planets more readily.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| M Molar mass | 0.002 to 0.2 kg/mol | 193.4 | 1,934 | 1,741 | +10% | −4.65 % |
| T Absolute temperature | 50 to 1,500 K | 211 | 1,156 | 944.9 | +10% | +4.88 % |
Sweep of Absolute temperature
| Absolute temperature (K) | Root-mean-square speed (m/s) |
|---|---|
| 50 | 211 |
| 231.3 | 453.9 |
| 412.5 | 606.2 |
| 593.8 | 727.3 |
| 775 | 830.9 |
| 956.3 | 923 |
| 1,138 | 1,007 |
| 1,319 | 1,084 |
| 1,500 | 1,156 |
Sweep one variable
Variables and how each one contributes
Root-mean-square speed
What the equation gives you. Everything else feeds it.
Absolute temperature
The absolute temperature of the gas, in kelvin. Raises the rms speed with the square root of T: quadrupling the temperature doubles the speed. Raising it by 25 K (to 325 K) moves the result by +4.08 %, from 517 to 538.1 m/s.
Molar mass
The molar mass of the gas, in kg/mol (the molar mass in g/mol divided by 1000). Nitrogen is 0.028. Lowers the rms speed with the inverse square root of M: heavier molecules move more slowly. Raising it by 0.005 (to 0.033) moves the result by −7.89 %, from 517 to 476.2 m/s.
Worked example with the values above
- Numerator: 3RT = 3 × 8.3145 × 300 = 7,483 J/mol.
- Divide by the molar mass: 7,483 / 0.028 = 267,300 m²/s².
- vrms = √267,300 = 517 m/s.
Assumptions and limits
- An ideal gas: negligible molecular volume and no intermolecular forces except in collisions.
- Random, isotropic translational motion, with the gas at a single absolute temperature in kelvin.
Common mistakes
- Using the molar mass in g/mol instead of kg/mol, which makes the speed wrong by a factor of about 32.
- Using degrees Celsius for T.
Statistics and uncertainty
vrms depends on √T, so a 2 % error in T gives a 1 % error in the speed. The molar mass is known very accurately for pure gases, so temperature is the main source of error.
Where it is used
- Comparing how fast different gases move.
- Estimating gas speeds in flow, escape and mixing problems.
Maths you need
- Ideal gas: molecules with negligible size and no attraction
- Absolute temperature: kelvin scale, starting at absolute zero