Gas Molecule Speed and Mean Free Path Calculator (Kinetic Theory)
How fast the molecules of a gas move at a given temperature, three ways of averaging it, and how far a molecule travels between collisions.
Results
The dashed line marks the value you entered.
What this tool does
Temperature is molecular motion: the root-mean-square speed √(3RT ÷ M) follows directly from the mean kinetic energy 3/2 kT. At room temperature nitrogen molecules move at about 515 m/s, faster than sound in air, which travels at about three quarters of their mean speed. Between collisions each molecule travels only some 70 nm at atmospheric pressure, roughly 200 molecular diameters, and suffers billions of collisions a second; the mean free path grows in inverse proportion to the pressure. The three speeds differ because the Maxwell–Boltzmann distribution is lopsided: a few fast molecules pull the averages above the most probable value.
Formula
v_rms = √(3RT ÷ M) · v̄ = √(8RT ÷ πM) · v_p = √(2RT ÷ M) · λ = k_B T ÷ (√2 π d² p)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
mw | Molar mass of the gas | g/mol |
t | Temperature | °C |
p | Pressure | kPa |
dm | Molecular diameter (kinetic diameter) | pm |
VR | Root-mean-square speed | m/s |
VM | Mean speed | m/s |
VP | Most probable speed | m/s |
MFP | Mean free path | nm |
CF | Collisions per molecule per second | ×10⁹ /s |
ND | Molecules per cubic metre | ×10²⁵ /m³ |
Worked example
- Molar mass of the gas28.0134 g/mol
- Temperature25 °C
- Pressure101.325 kPa
- Molecular diameter (kinetic diameter)364 pm
- Root-mean-square speed515.2 m/s
- Mean speed474.7 m/s
- Most probable speed420.7 m/s
- Mean free path69.0 nm
- Collisions per molecule per second6.88 ×10⁹ /s
- Molecules per cubic metre2.461 ×10²⁵ /m³
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- The default values are typical reference figures, not measurements of your situation. Replace them with your own data whenever you have it.