Mean Free Path

Introduction

We know that the gas’s large number of molecules are known as particles. The particles move randomly here and there without any alignment and due to this random motion, they strike each other and change their path. When it is not easy to understand the actual paths of moving particles, then we use the concept of mean free path. In the mean free path, the moving particle refers to an atom, a photon, or a molecule.

What is a collision?

In any instance, when two or more bodies hit each other by exerting force in a short period is called a collision. For example, when a bee settles on the petal of a flower, its legs collide with the petal. The term collision explains the following things:

• The measure of the force applied and the change in the motion of the colliding bodies (change in velocity).

• It means the magnitude of the velocity of the bodies before the collision is different from the velocity after the collision.

• The bodies involved in collision follow the law of conservation of momentum. Therefore, every collision conserves momentum.

There are different types of collisions depending upon the conservation of energy (mechanical or kinetic energy). The two types of collisions are as follows −

• Elastic collision − During collision when the entire kinetic energy of the system is conserved is called an elastic collision. It is also known as a perfectly elastic collision.

• Inelastic collision − During collision when a portion of kinetic energy is transferred into some other kind of energy and the total kinetic energy of the system is not conserved is called an inelastic collision

Relaxation Time

In physics, relaxation means the recoiling of an unstable system into equilibrium. All relaxation processes can be specified by relaxation time.

• The period between the succeeding collisions of two free electrons in a conductor is called the relaxation time.

• It is denoted by the letter 𝜏 (tau). The relaxation time for electrons is 10−14 seconds.

• Its formula is $\mathrm{\tau\:=\:v\frac{n}{e}\:E}$

where 𝑣 is the drift velocity of electrons

𝑛 is the no of electrons

𝑒 is the charge of an electron

𝐸 is the electric field

• Any material’s conductivity depends on its relaxation time. The most conductive material will have a longer relaxation time.

Mean Free Path

According to the kinetic theory of gas, the gas molecules are continuously moving in all directions at different speeds and collide with each other by exerting force.

• After a collision, there is a change in their direction and speed. They don't apply any force on each other but exert force only at collision.

• Due to this, they start moving at the same speed in a straight line between two successive collisions.

• If a single moving molecule is observed, it is found that it has short zig-zag paths of various lengths. These short zig-zag paths of the molecule are known as free paths and their mean is known as the mean free path.

• In radiography, the beam of mono-energetic photons, and in electronics, the charge carrier electrons have the mean free path.

Let’s assume a ball moving in a room as a moving molecule. The ball hits the walls of the room many times and after every collision, there is a change in the direction of the ball. There are four collisions between the ball and the wall. Between each one of the two successive collisions, the ball travels a particular path. It means, the ball travels three paths in which each path has a definite distance d between four collisions. The mean free path of the ball is the average length of three paths.

Therefore, $\mathrm{\lambda\:=\:\frac{d_{1}\:+\:d_{2}\:+\:d_{3}}{3}}$

Derivation of Mean Free Path

Let’s assume a moving spherical molecule in an ideal gas. The molecules of the gas are colliding with each other but we will study the moving molecule that we have considered and the rest of the molecules of the gas are stationary.

Let’s think that the moving molecule has a diameter, d. This particular molecule moves through gas and sweeps out a short cylinder whose area of cross-section is $\mathrm{\pi\:d^{2}}$

In between the successive collisions, if this molecule moves for a small time t with the velocity v, it will move a distance of vt. Now the volume $\mathrm{\pi\:d^{2}\times\:vd}$ is obtained if the cylinder is swept.

Thus, the number of collisions of the moving molecule can be obtained by the number of point molecules present in this volume.

Hence, the number of molecules per unit volume is $\mathrm{\frac{N}{V}}$

So, the number of molecules in the cylinder is $\mathrm{\pi\:d^{2}\:vd}\times\:\frac{N}{V}$

The mean free path will be as follows

$$\mathrm{\lambda\:=\:\frac{total\:distance\:covered\:in\:time\:t}{number\:of\:collisions\:in\:time\:t}\:\thickapprox\:\frac{vt}{\pi\:d^{2}vt\frac{N}{V}}\thickapprox\:\frac{1}{\pi\:d^{2}\frac{N}{V}}}$$

The derived equation of mean free path is taken approximately because at the start of the derivation it is assumed that only the molecule that is studied is moving and at rest of the molecules of the gas are stationary. In the above equation, the two velocities in the numerator (average velocity) and denominator (relative velocity) are canceled with each other due to which there is a difference of factors √2 between each other.

So, the final equation is

$\mathrm{\lambda\:=\frac{1}{\sqrt{2}\pi\:d^{2}\frac{N}{V}}}$

Conclusion

Collision is the hitting of two or more bodies with each other in a short time by applying a force that results in a change in motion of these bodies. Two types of collisions are elastic and inelastic collisions. Relaxation time is the time gap between the collision of free electrons in a conductor. It is denoted by 𝜏. The mean free path is the average distance traveled by moving molecules between successive collisions. The equation means free path is $\mathrm{\lambda\:=\frac{1}{\sqrt{2}\pi\:d^{2}\frac{N}{V}}}$

FAQs

1. Write the factors on which the mean free path depends.

The mean free path depends on

• Number of molecules

• Density

• Radius of molecule

• Pressure, temperature

2. Define relative velocity

Relative velocity is the time rate of change of relative position of an object regarding another object.

3. Give one example of each- elastic and inelastic collisions.

An example of elastic collision is the collision between atoms and an example of inelastic collision is a bus hitting a tree.

4.How do the two bodies move after an inelastic collision?

In inelastic collisions, there is no corresponding motion between the two colliding bodies. Hence, these bodies move together as a single body.

5. Write the other name for inelastic collision

The other name for inelastic collision is a plastic collision.

6. What is average velocity?

The total displacement covered by an object in total time t is called average velocity.

7. Name the conservation law which is applied to the elastic collision but not to an inelastic collision.

The law of conservation of kinetic energy is applied to the elastic collision but not to an inelastic collision.