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
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
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
where η is viscosity and ρ density.
REFERENCES
Kennard, E. H. (1938) Kinetic Theory of Gases. McGraw-Hill. New York.
References
- Kennard, E. H. (1938) Kinetic Theory of Gases. McGraw-Hill. New York.