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 travelled 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 is ‘n’ number of molecules.
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Now, focus on molecule A inside the gas, which is in a random motion and continuously collides 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 on.
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 collisions will happen early, and sometimes collisions may take time. We need 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 is 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
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.
Factors Affecting the Mean Free Path
The different factors affecting the mean free path are density, number of molecules, the radius of the molecules and other physical factors including temperature, pressure and all.
Density - It simply measures the fraction of the mass of an object to its volume. As the density increases, molecules come closer and collide, thus reducing the mean free path. Similarly, as the density decreases, the molecules loosen up and more mean free paths appear. Therefore, density is inversely proportional to the mean free path. For eg, when air is compressed, the air molecules only get less space to roam around which will cause a collision.
The number of molecules - An increase in the number of molecules can result in more collisions and will decrease the mean free path.
The radius of the molecule - A rise in the radius of the molecules leads to the extension of the area it occupies and can touch the neighbouring molecules giving decreasing effect on the mean free path. For eg, heating pushes an increase in the kinesthetic energy of molecule particles and will steer to an expansion in the radius of the molecules creating collisions followed by a decrease in free mean paths.
Other physical factors - Pressure is one important factor that can regulate the free mean path. If the pressure among the molecules increases, the average speed between the molecules increases and thus collision happens. So the increase in pressure will decrease the free mean path.
Every factor is an integral of the free mean path and the interconnections between them stem the outcomes.
Here pressure and temperature have a great relationship concerning the mean free path.
As you have studied from the 11th NCERT Texts, Gay Luccas’ Law explains that the pressure of a given amount of gas is directly proportional to its’ temperature,
interconnection between the factors of the mean free path.
Motion of Molecules
If there are enough mean free paths, then the moment of the gas molecules will be fast and free, ie; it will possess random motion. However, if a mean free path is being created by releasing any of the factors, then the molecules would show diffusive motion since the molecules are getting rapid freedom and the moment is from an area of higher concentration to lower concentration. Diffusion is basically the intermixing of particles or molecules. This is the reason why keeping the volume constant. This law is self-sufficient to showcase the bond between temperature and pressure, thereby giving an idea about the i we get the cooking smell from the kitchen. This can be in contrast with another motion called ballistic motion. The molecules or particles mostly tend to move in a zig-zag motion.
Imagine what would have happened if these molecules were tightly packed? They would have never had any motions! Because of the distance that the molecules have before their successive collisions, motions like diffusion could occur. Not only that, they in turn are helpful in the transportation of matter, thermal energy and much more.
Mean Free Path in Ions
It is important to know the relative distance between ions to understand and calculate various physical properties and spatial information respectively. So let us take a brief look at the relationship between ions and the mean free path.
Ions are a bunch of atoms that take shape when an atom loses its stability by the loss or gain of proton or electron. Ions are used in water purification and many other things. Mean free path helps us to analyse the distance from which negative ions can be extracted and the relative prominence of several collisions. There are many more crucial findings and concepts pertaining to the mean free path. You will study them in higher classes.
Different matters will have different mean free paths, depending on their type of gas, density, volume etc. For eg, the mean free path in hydrogen is longer than in deuterium. Thus, the chaotic motions of these spherical and elastic molecules in gas can actually teach and be useful for many purposes.
FAQs on Mean Free Path Factors
1. What is meant by the mean free path of a gas molecule?
The mean free path (λ) is a fundamental concept in the kinetic theory of gases. It represents the average distance a molecule travels between two successive collisions with other molecules. Since molecules in a gas are in constant, random motion, they collide frequently. The path travelled between these collisions is a straight line, and the average of these path lengths gives the mean free path.
2. What are the main factors that affect the mean free path?
The mean free path is primarily influenced by the physical characteristics of the gas. The key factors include:
- Number Density (n): The number of molecules per unit volume. A higher density leads to a shorter mean free path as molecules are more crowded.
- Molecular Diameter (d): Larger molecules present a bigger target for collisions, which decreases the mean free path.
- Pressure (P): At constant temperature, increasing the pressure increases the density, thus decreasing the mean free path.
- Temperature (T): If pressure is kept constant, increasing the temperature will increase the mean free path. However, in a container of fixed volume, the mean free path is considered independent of temperature.
3. What is the formula for calculating the mean free path?
The mean free path (λ) can be calculated using the following formula, which is derived from the kinetic theory of gases:
λ = 1 / (√2 * π * d² * n)
Where:
- λ is the mean free path.
- d is the diameter of the gas molecule.
- n is the number of molecules per unit volume (number density).
4. How does increasing the pressure on a gas affect its mean free path?
Increasing the pressure on a gas (while keeping the temperature constant) forces the molecules closer together. This increases the number density (n) of the gas. With more molecules packed into the same space, the frequency of collisions rises, and the average distance a molecule can travel before hitting another one decreases. Therefore, the mean free path is inversely proportional to the pressure.
5. Does changing the temperature of a gas alter its mean free path?
According to the standard model for a gas in a container of fixed volume, the mean free path is considered independent of temperature. While increasing the temperature makes the molecules move faster and collide more frequently, it does not change the average distance between them. The increased speed simply means they cover the mean free path distance in a shorter amount of time.
6. How can the mean free path of a gas be increased?
To increase the mean free path, you need to reduce the probability of molecular collisions. This can be achieved by:
- Decreasing the Pressure: Lowering the pressure reduces the gas's density, spacing out the molecules.
- Reducing the Number of Molecules: Removing gas from a container decreases the number density.
- Using a Gas with Smaller Molecules: Gases with a smaller molecular diameter (d) will have a longer mean free path, as they present a smaller target for collisions.
7. Why is the concept of mean free path important for understanding diffusion in gases?
Diffusion is the process of particles mixing due to their random motion. The mean free path directly impacts the rate of diffusion. A long mean free path means molecules can travel farther without being deflected, allowing for faster mixing and a higher rate of diffusion. In contrast, in a dense gas with a short mean free path, molecules undergo constant collisions, following a zig-zag path which slows down their net movement from one area to another, resulting in slower diffusion.
8. If gas molecules travel at high speeds, why does it take time to smell a perfume from across a room?
This is a classic example explained by the mean free path. While individual perfume molecules do travel at very high speeds, they don't travel in a straight line to your nose. They continuously collide with billions of air molecules (Nitrogen, Oxygen, etc.). At atmospheric pressure, the mean free path is extremely short (around 68 nanometres). These collisions cause the perfume molecules to follow a random, zig-zag path. Their net movement, or diffusion, across the room is therefore much slower than their actual speed.
9. How would the mean free path differ for a light gas like Hydrogen compared to a heavier gas like Carbon Dioxide?
Assuming the temperature, pressure, and number density are the same for both gases, the mean free path would be longer for Hydrogen. The formula for the mean free path (λ) is inversely proportional to the square of the molecular diameter (d²). Hydrogen molecules (H₂) are significantly smaller in diameter than Carbon Dioxide molecules (CO₂). Because they present a smaller cross-sectional area for collisions, they can travel further on average before colliding with another molecule.
