Mean Free Path of Gas Molecules

The mean free path (λ) is the average distance a gas molecule travels between successive collisions.

Formula:

λ = kB × T / (√2 × π × d² × P)

Where:

• kB = 1.380649 × 10⁻²³ J/K (the Boltzmann constant, which since 2019 is a defined exact value rather than a measured one)
• T = temperature (Kelvin)
• d = molecular diameter (meters)
• P = pressure (Pascals)

Collision frequency (z), meaning collisions per second:

z = v_avg / λ

where v_avg = √(8kBT / πm) = average molecular speed, and m = molecular mass in kg.

Approximate mean free paths at 25°C, 1 atm:

Effect of pressure: λ ∝ 1/P, so halving the pressure doubles the mean free path. At 0.1 mPa (high vacuum): λ ≈ 68 m for N₂, so molecules cross the chamber and hit a wall long before they meet each other. This is why vacuum systems are described by whether the mean free path exceeds the chamber size, not by pressure alone.

Knudsen number: Kn = λ / L (where L is characteristic system length)

• Kn « 1: continuum flow (many collisions, classical fluid dynamics applies)
• Kn » 1: molecular flow (molecules collide with walls more than each other)
• Kn ≈ 1: transitional regime

The calculator works Kn out for a 10 cm chamber, which is a reasonable stand-in for a small bell jar or a benchtop vacuum system. Scale it yourself for anything else: Kn is directly proportional to λ and inversely proportional to the size of whatever the gas is inside, so a 1 mm channel has a Knudsen number a hundred times larger than a 10 cm chamber at the same pressure.

Applications:

• Vacuum technology (mean free path determines vacuum quality)
• Semiconductor manufacturing (deposition processes require Kn > 1)
• Atmospheric science (transport coefficients)
• Thermal conductivity and viscosity of gases