Mean Free Path Factors

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Concept of Mean Free Path

According to Classius, every gas comprises an infinite number of molecules that are perfectly elastic spheres.

The size of the gas molecules is tiny as compared to the distance between them.

Further, the molecules of a gas in the state of never-ending, speedy, and random motion; undergo perfectly elastic collisions with one another.

As the molecules exert no force on one another except during collision, the free path traveled between two successive collisions will be a linear path with invariant velocity.

Therefore, the path of a single molecule consists of a series of short zig-zag paths of different lengths.

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These paths of varying lengths are called free paths of the molecules, and their mean is called the mean free path.

Let’s talk about the free path of molecules.


Free Path

We presume in the kinetic theory of gases that the gas molecules are in constant motion, which means they are colliding with each other and with the walls of the container. This type of collision is elastic by nature.

Let’s say, in Fig, there are ‘n’ number of molecules.

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Now, focus on molecule A inside the gas, which is in a random motion and continuously colliding with all the other molecules one by one.

Molecule A collides with another molecule B, with C and then with D.

When molecule A collides with molecule B, then it is the first collision. Thereafter, it collides with C, which is the second collision, then with D, which is the third collision, and then with E, which is the fourth collision. Thus, it keeps on colliding with all the molecules.

Consider the distance between the first and the second collision be λ1 and the distance between the second and the third collision be  λ2, then the distance between the third and fourth collision  

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Here, λ1, λ2, and λ3 are free paths i.e., the path between each collision is free.

So, at λ1, molecule A after colliding with molecule B got the first free path, at  λ2, after colliding with molecule C, it got another free path and so.

Since no collisions occur between any two collisions, the path is free between each collision.

Therefore, we can say that the free path is the distance between the two consecutive collisions like λ1, which is the free path. Similarly, λ2 is another free path and so.


Mean Free Path 

We just understood the concept of a free path of molecules. Now, if we wish to find the average of these free paths, we get the mean of the free paths, which is represented as:

      λ1 = First free path

      λ2   = Second free path

        λ3    = Third free path

      λn = nth free path

So,

Mean free path formula is:      

    λmean = ( λ1+  λ2 +  λ3 +.....+ λn)/n 


Define Mean Free Path  

We can define the mean free path as the average distance between the two consecutive collisions.

Here, the distance between the molecules may vary, sometimes they would be separated at a large distance, sometimes close to each other, sometimes collision will happen early, sometimes collisions may take time. We need to an average of the overall conditions.


Mean Free Path in Physics

Consider a gas comprising ‘n’ number of molecules scattered inside the container. 

Assume molecule A, colliding with all other molecules.

Now, construct a cylinder of length ‘1 m’, and, imagine, whichever molecule, a molecule A collides with, is inside this cylinder.

This means molecule A doesn’t collide with all the molecules, rather only with those packed in the cylinder.

Let us assume that all the molecules travel in a straight line, including the molecule A.

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As this molecule A starts moving in a straight line, it won’t collide with the molecules in and around the cylinder shown in Fig

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Now, a question comes in our mind; if the distance between the two molecules is larger, a molecule A would pass between them without a collision. However, if it passes between the molecules that are close to each other, there will be a collision.


Let’s Understand This in a Better Way

Consider the diameter of molecule A as ‘d’,and the space required for it to pass through the molecules inside the cylinder without any collision between the molecules is also ‘d’.

Now, it is for sure, if the molecules inside the cylinder are close to each other, then, definitely, there will be a collision between A and these molecules; however, molecules completely outside this imaginary cylinder of diameter ‘2d’ won’t collide with A.

Now, the number of molecules per unit volume = n

(n molecules in 1 m3 volume)

No of molecules inside the cylinder = πd2 x 1= πd2 * 1 * n

No of collisions in traveling distance of 1 m  = πd2 * n

So, in 1 collision = 1/πd2 * n distance, which is the mean free path.

FAQ (Frequently Asked Questions)

1. How Do You Increase the Mean Free Path?

The mean free path is the average distance a molecule travels between collisions, which means the farther the molecules are, the more will be the free path. If the density of the gas increases, the molecules run into each other, decreasing the free path.Thus, it’s necessary to keep the molecules apart to increase the mean free path.

The mean free path formula is given by,

                 λ = RT/√2πd2NAP

2. How Does Temperature Affect the Mean Free Path?

According to the kinetic theory of gases, on increasing the temperature, molecules run fastly; however, the distance or the mean free path remains constant, and only the meantime of collision decreases.

Therefore, we can say that the mean free path is independent of the temperature.

3. Which Has the Maximum Value of the Mean Free Path?

The molecule possessing a small size will have a maximum mean free path.

4. What Is Meant by Collision Frequency?

Collision frequency is the repetitive time interval in which on average molecular collisions take place. It is denoted by letter Z. Its formula is given as:

Z = 1/f = T