Mean Free Path

In kinetic theory, the mean free path is defined as the mean distance travelled by a molecule between collision with any other molecule. For a dilute gas composed of hard spherical molecules of kinds A and B, the mean time τ_AB between successive collisions of a given A molecule with B molecules is given by

(1)

Here, d_AB is the mean diameter of molecules A and B, is the number density of B molecules, and c_AB is the mean relative speed of molecules A and B. According to the Maxwell-Boltzmann velocity distribution law, molecular velocities are not correlated and c_AB is therefore given in terms of the mean speeds of the individual molecules by c_AB = where, for molecules of type i with mass m_i, c_i = (8kT/πm_i)^1/2, where k is Boltzmann's constant and T is the absolute temperature.

The mean free path l_AB travelled by a given A molecule between successive collisions with B molecules is simply c_A τ_AB or

(2)

Real molecules interact through intermolecular forces which vary smoothly with distance. Consequently, there is no unique counterpart of the hard-sphere collision cross-section and no unique definition of what constitutes a collision. Often, d_AB is replaced by the separation σ_AB at which the A-B intermolecular potential energy crosses zero. Alternatively, d_AB may be determined from viscosity data by comparison with the theoretical predictions for hard spheres [Kennard (1938)]. These formulas are easily specialized to the case where A and B are identical and, if d is estimated from viscosity data, then

(3)

where η is viscosity and ρ density.