Mean free path

The concept of mean free path describes the total distance between two successive collisions of molecules in continuous motion. The collisions can result in a change in the direction or shape of the molecules. The molecule can be an atom or a photon. The concept of the mean free path is based upon the kinetic theory. The mean free path depends upon the energy distribution of the particles of the medium. The gas molecules are in a continuous state of random motion; thus, they undergo perfectly elastic collisions. The free path between two successive collisions is the linear path with invariant velocity, as the molecules exert no force on one another except during the collision. The successive collision with different molecules shall be named as 1st, 2nd, 3rd, or an nth number of collisions. Simultaneously, their mean free path will be λ1, λ2, λ3, or λn. The mean free path depends both upon the material as well as the energy of the molecules. 

Kinetic theory of gases

The basic postulates of the kinetic theory of gases are as follows:

• All gases are composed of minute particles in large numbers
• They do exert pressure on the objects
• They attract molecules of one another
• The absolute temperature is directly proportional to this theory (kinetic theory of gases)
• The actual or real volume of the gaseous molecule is very minute
• The gravitational force has a great influence on the movement of the gaseous molecule

Derivation of mean free path

For the derivation of the mean free path, assume the shape of the molecule as spherical. The collision will take place when one molecule strikes the other during motion. The focus is only on the moving molecules rather than the molecules in a stationary position. If the diameter of the molecule is ‘d’, it moves in a gaseous medium. In such a way that it shall sweep out a cylinder. 

As we know, the area of the short cylinder will be πd^2.

During the successive collisions, the molecules will cover a distance ‘vt’ in time ‘t’. 

Where v= velocity of the molecule and t=time

We must know that if we sweep this cylinder we might get a volume of πd^2*vt. Therefore, the number of collisions will depend upon the number of point molecules inside this volume.

Number of molecules per unit volume= N/V

The number of molecules in the cylinder=N/V multiplied with the volume of a cylinder=πd^2vt

Therefore, the derivation of the mean free path is as follows:

λ= length of path during the time tnumber of collision in time r

λ= vtπd^2vtNV

λ=1πd^2NV

During the calculation, we assume that all other particles stay stationary concerning the particle under consideration. The movement of molecules is relative to each other.

The v in the numerator represents the average velocity, whereas V in the denominator represents the relative velocity. The difference between them shall be √2. 

The final equation of the mean free path will be as follows:

λ= 1√2πd^2NV.

Motion of molecules 

The molecules will possess random motion if there are enough mean free paths. If the mean free path is to be created, then the molecules will show diffusive motion. Diffusion is the net movement of molecules from a higher concentration towards a lower concentration. Thus, the gas molecules constantly move in a zig-zag manner. If the molecules are tightly packed, then they won’t be able to move freely, such as in solids.

Factors affecting the mean free path

The mean free path is affected by different factors, such as density, number of molecules, the radius of the molecules, temperature, pressure, etc. These mean free path factors are as follows:

• Density: Density is inversely proportional to the mean free path. On increasing the density, the molecules come closer and start colliding more often. Thus, it decreases the mean free path. In the same way, on decreasing the density, the collision decreases. This leads to an increase in the mean free path
• Number of molecules: The number of molecules is also inversely proportional to the mean free path. Increasing the number of molecules increases the collision. This causes a decrease in the mean free path
• The radius of the molecule: The increase in radius of the molecule is inversely proportional to the mean free path. The increase in radius increases the surface area of the molecule. Due to it occupying the space, it can touch the neighbouring molecules and decrease the effect on the mean free path
• Pressure: The mean free path is inversely proportional to the pressure, the physical factor affecting it. In other simpler terms, on increasing the pressure, the mean free path eventually decreases
• Volume: The mean free path is inversely proportional to the volume too. On increasing the volume, the mean free path decreases
• Temperature: The temperature is directly proportional to the concept of mean free path. As soon as we increase the temperature, the kinetic energy also increases. This leads to the faster motion of molecules

Conclusion 

The concept of Mean free path describes the total distance between two successive collisions of molecules in continuous motion. The collisions can result in a change in the direction or shape of the molecules. The molecule can be an atom or a photon. The kinetic theory of gas is the principle for the concept of mean free path. The derivation of mean free path results in this λ= 1√2πd^2NV., as the final formula. There are several factors on which the mean free path depends, such as density, number, and radius of molecules, pressure, temperature, and also the number of molecules. These molecules move in a zig-zag manner. Thus, we can say that the concept of mean free path holds great importance in the field of physics.