Mean Free Path

Mean free path refers to the average distance that a moving particle travels between successive collisions or impacts. Furthermore, the successive collisions result in modification of the moving particle’s energy or direction or any other particle properties. Moreover, the moving particle here refers to a molecule, an atom, or a photon.

Introduction to Mean Free Path

In the field of radiography, the pencil beam of mono-energetic photon shall have a mean free path. Furthermore, this mean free path is the average travelling distance of a photon between collisions with the target material’s atoms. Moreover, this means that the free path is dependent on the material as well as the energy of the photons.

In electronics, a charge carrier’s mean free path of electrons in a metal is proportional to the electrical mobility. Furthermore, electrical mobility is a value that has a direct relation to the electrical conductivity.

mean free path

Mean Free Path

How do we Measure Mean Free Path?

Let’s consider the motion of a gas molecule that is present inside an ideal gas. Furthermore, a typical molecule inside an ideal gas will change its direction and speed in an abrupt fashion. This is because its collision takes place elastically with other molecules of the same gas.

The molecule between the collisions shall move at a constant speed in a straight line. Most noteworthy, this is applicable for all the gas’s molecules.

Measuring this random motion of gas molecules is certainly difficult.  Therefore, one must try to measure its mean free path λ.

The symbol λ represents the average distance that a molecule travels between collisions. Furthermore, one can expect λ to vary inversely with N/V, which happens to be the number of molecules per unit volume or the density of the molecules.

Formula of Mean Free Path

Mathematically, representation of the mean free path can take place as follows:

λ = \(\frac{1}{\sqrt{2}\pi d^{2}\frac{N}{V}}\)

Derivation of the Formula of Mean Free Path

The derivation of the equation will make use of certain assumptions. Consider the molecule to be spherical. Furthermore, the collision takes place when one molecule hits the other. Moreover, the focus here is on the molecule that is in motion while the rest are stationary.

Assume the diameter of a single-molecule to be d. Furthermore, consider the single-molecule to move via the gas.

As such, it shall sweep out a short cylinder. Moreover, the cross-section area of this short cylinder is πd2.

Between the successive collisions, it will move a distance of vt for time t. Here, v happens to be the velocity of the molecule. Most noteworthy, one would attain the volume of πd2*vt on sweeping this cylinder.

Therefore, the number of collisions the molecule might have can be determined by the number of point molecules inside this volume.

Certainly, N/V is the number of molecules per unit volume. Therefore, the number of molecule in the cylinder will be N/V multiplied by the volume of cylinder i.e.πd2vt. As such, the derivation of mean free path can take place as follows,

λ = length of path during the time t/number of collision in time r ≈ \(\frac{vt}{\pi d^{2}vt\frac{N}{V}}\) = \(\frac{1}{\pi d^{2}\frac{N}{V}}\)

The approximation of the equation has taken place because it has been assumed that all the particles are stationary in relation to the particle under consideration. Most noteworthy, there is a movement of all the molecules relative to each other.

There has been the cancellation of two velocities in the above equation. Moreover, v in the numerator represents the average velocity while v in the denominator represents the relative velocity. As such, there is a difference between both with a factor \(\sqrt{2}\).

Most importantly, the final equation is,

λ = \(\frac{1}{\sqrt{2}\pi d^{2}\frac{N}{V}}\)

FAQs For Mean Free Path

Question 1: What is the relation between the mean free path and temperature?

Answer 1: The relation between the mean free path and temperature is that there is a display of linear proportionality from the mean free path to the temperature.

Question 2: Explain the mean free path in electronics?

Answer 2: In electronics, charge carrier’s mean free path in metal is proportional to the electrical mobility.

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