Mean Free Path

The mean free path or average distance between collisions for a gas molecule may be estimated from kinetic theory. Serway's approach is a good visualization - if the molecules have diameter d, then the effective cross-section for collision can be modeled by


using a circle of diameter 2d to represent a molecule's effective collision area while treating the "target" molecules as point masses. In time t, the circle would sweep out the volume shown and the number of collisions can be estimated from the number of gas molecules which were in that volume. The mean free path could then be taken as the length of the path divided by the number of collisions.


Refinement of mean free pathCalculationFrequency of collision
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Refinement of Mean Free Path

The intuitive development of the mean free path expression suffers from a significant flaw - it assumes that the "target" molecules are at rest when in fact they have a high average velocity. What is needed is the average relative velocity, and the calculation of that velocity from the molecular speed distribution yields the result

The resulting mean free path is

The number of molecules per unit volume can be determined from Avogadro's number and the Ideal gas law, leading to

Calculation

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Mean Free Path Calculation

The mean free path equation depends upon the temperature and pressure as well as the molecular diameter.

For pressure =mmHg =inHg =kPa

and temperature T=K = C =F,

Molecules of diameter x 10^-10 meters

should have a mean free path of = x 10^m

which is times the molecular diameter

and times the average molecular separation of x 10^m.


The values for pressure, temperature, and molecular diameter may be changed above to recalculate the mean free path. The pressure required for a given mean free path can be calculated by changing the value of the mean free path above. The calculation is not designed to allow any other changes (their values will be replaced unchanged by the calculation.)

From the mean free path and the average velocity, the mean time between collisions and the frequency of collision can be calculated.

Example of mean free path compared to molecular separation.
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Mean Free Path Perspective

You may be surprised by the length of the mean free path compared to the average molecular separation in an ideal gas. An atomic size of 0.3 nm was assumed to calculate the other distances.


Mean Free Path CalculationFrequency of collision
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Gas law concepts

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