Molecular Collisions (How Far, How Fast, How Long)

Mean Free Path

The mean free path is the distance that a molecule travels between collisions.

The mean free path is determined by the criterion that there is one molecule within the "collision tube" that is swept out by a molecular trajectory.

The criterion is: λ (N/V) π r2 ≈ 1, where r is the radius of a molecule.

This gives:   λ = 1 / [(N/V) π r2]

Numerical estimates
Molecular cross-sectional area:   π r2 ≈ 10-19 m2
Molecular density (sea level):   N/V = P/kT ≈ 3 x 1025 molecules/m3
Mean free path (sea level):   &lambda ≈ 10-7m ≈ hundreds of molecular diameters

Mean free path for air at different altitudes:

Altitude Mean Free Path
Sea Level 0.1 μm
100 km16 cm
300 km20 km

Typical Molecular Speed

Determine the typical molecular speed by:

½ mv2 ≈ KT   →   v &asymp (kT/m)½ ≈ 500 m/s.

Typical Collision Time

Determine the typical collision time by:

τ = λ/v ≈ 10-10 sec.