What is the mean free path in the Kinetic Theory of Gases

The kinetic theory of gases was introduced to understand the structure and composition of atoms and molecules concerning the sub-microscopic particles. This theory is based on the collision and constant movements that take place in between the sub-microscopic particles due to an increment in the pressure level. 

First, let’s understand the definition of the mean free path, which explains the behaviour of gases. It discusses the constant motion of molecules of gases. Further, in this article, we will discuss the formula of mean free path and examples of the mean free path. 

Definition of mean free path 

The mean free path can be defined as the distance between two successive collisions. From the kinetic theory of gases, it is known that the molecules of gases are always in constant motion. The molecules collide with each other along with the containers’ walls. 

Let’s presume the first molecule is considered molecule one and the second molecule is molecule two, and so on. When molecule one comes into collision with molecule two, the process will be known as the first collision. When it comes into collision with molecule three, the same process is known as the second collision. And this process continues and so on. 

The distance between the first and the second collision is called the free path. 

This free path is denoted by λ1. The collusive distance between the second and the third collisions will be known as λ2. This process continues like this. It has been found that there isn’t any collision between successive collisions at all. So, this particular path is known as the free path (λ1, λ2, λ3, etc.). 

So, the mean free path is defined as the average path between successive collisions by the molecules of the gases. It is known that different free paths have different path lengths. 

Here some explanation is given below 

First free path = λ1 

Second free path = λ2 

Third free path = λ3 

nth free path = λn 

The average path length of these mentioned path lengths is the mean free path. The mean free path is denoted by λ. Therefore, it is calculated as,

 λ = (λ1+λ2+λ3+⋯+λn)/n 

The formula of mean free path

Let’s assume by taking an example of a single molecule with a diameter of d. When it moves through the other molecules and imagines that the other molecules are not polluting each other. Then the molecules will cover a particular distance in a cylindrical form. Then the cross-sectional area of that form will be πd². After calculating, the volume of the cylindrical form will be πd²*Vt, where v is considered the velocity for the molecule and T is the time. N/V is the number of molecules per unit volume. The formula of the mean free path can be written as,

 λ = Length of the mean free path 

λ=Vtπd2vt NV

λ=Vπd2 N

The time between the collisions,

T=1nvtπd2

The mean free path is considered based on some factors. The factors are 

• Density 
• The radius of the molecules 
• Number of the molecules 
• Temperature, pressure, etc. 

An example of the mean free path is given below to understand the theory better. 

What will be the mean free path in CO2 at 27 °C when a 10-9 bar pressure is applied?

(Given, R =253J mol−1K−1, 2=1.4, 227, NA=6×1023)

The number of molecules per unit volume 

N=nVNA=PRTNA=10-91.013250.08213006.0231023

= 2.41013molecules/dm3

= 2.41010 molecules/cm3

The mean free path is 12π2N=11.4143.14(50010-2)22.41010=3.788103CM

Things to Remember

• The free path is a straight path between two successive collisions, as there will be no velocity change, and the molecules will not face any force on each other except at the collision moment.
• The journey path of any molecule is generally made up of a succession of small zigzag paths of different lengths.
• The aim to quantify the mean free path is difficult, as the measurement or characterization of the random motion of gas molecules is not possible in an easy manner.

• Conclusion

  The mean free path can be defined as the distance between two successive collisions. From the kinetic theory of gases, it is known that the molecules of gases are always in constant motion. The molecules collide with each other along with the containers’ walls. The distance between the first and the second collision is called the free path. The average of the distances between each collision is known as the mean free path. This means the free path is determined based on some factors. Particularly they are density, the radius of the molecules, number of the molecules, temperature, pressure, etc.