Mean Free Path Factors

Introduction

All matter around us is made up of molecules. Because these molecules are in motion, each molecule has kinetic energy and static energy due to the gravitational force of other molecules around it. The amount of kinetic energy and static energy is the internal energy of the molecules. When this internal energy flows out of an object, it is called thermal energy.

The matter is a term we use to refer to all things in the universe. Everything around us is matter. The air we breathe, the food we eat, the pen we write with, a cloud, stones, plants, animals, and all matter have two common characteristics, which are mass and the nature of filling space.

What is a collision?

Collision is a common occurrence around us from time to time. For example, in games like carrom, billiards, and volley, collisions between two objects can occur with or without contact. Linear momentum remains unchanged in all collisions. Total linear momentum and total energy never change in any collision process. Since the total kinetic energy need not always remain constant. Taking these effects into account we can categorize collisions as two types.

• Elastic Collision

• Inelastic Collision

Elastic Collision

When the initial and final overall kinetic energy are equal that collision is called Elastic collision that is, the overall kinetic energy before and after collision are equal.

Inelastic Collision

When the initial and final overall kinetic energy are not equal that collison is called inelastic collision, that is the overall kinetic energy before and after collision are not equal.

Free path

The kinetic gas theory is based on certain assumptions. Because these assumptions simplify the calculations. All gas molecules are in random motion. This motion of gas molecules is known as their free path. Gas collides with each other with the barriers of the enclosed container. Elastic collisions take place in these processes. Due to this collision, kinetic energy loss occurs. A gas molecule will be in motion with constant velocity in the middle of two successive collisions. Even when gas molecules are in irregular motion, they will fulfill Newton’s laws of motion. Gas molecules are all uniform, perfectly elastic spheres throughout.

Mean Free Path

Normally the mean speed of a gas molecule at room temperature is a few hundred per second. However, the scent of a genuine perfume in a room does not immediately reach us. The reason for this delay is that the scent molecules do not reach us in a strong line but collide with various close air molecules and travel in an irregular path to reach us. Thus the average distance moved by a molecule in the middle of two consecutive collisions is called the mean free path. The average collision distance is calculated based on kinetic theories.

According to kinetic gas theory, all gas molecules are in an irregular motion. And we know that they collide individually. These gas molecules follow a linear path with constant velocity. between two successive collisions This path is known as the mean free path. Assume a system composed of molecules of diameter ‘d’. It has some n molecules with mass. Let us consider that the combined molecule is in movement and all other molecules are at without motion as shown in the figure.

A molecule moving with an average velocity ‘v’ travels a distance of vt in time t. At this time the molecule is said to be moving inside an imaginary cylinder of volume π𝑑2vt. This molecule collides with all the molecules inside the cylinder. Hence the amount of molecules is equal to the number of collisions that occurred in the imaginary cylinder. It is equivalent to π𝑑2vt. The mean free path is the ratio of total path length and the number of collisions in time t.

$$\mathrm{Mean\:free\:path\:,\:\lambda\:=\:\frac{Distance\:travelled}{No\:of\:ollisions}}$$

We imagined that only a single molecule is in motion at a given time and all other molecules are not in motion. But practically all molecules are in disordered motion. So it is necessary to consider the average relative speed of a molecule here. After detailed calculations, the exact equation for the average collision distance is

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

From the above equation, the number density is inversely proportional to the mean free path. As the number density increases, so does the collision of molecules. Rewriting the above equation by the mass of the molecule (m).

$$\mathrm{\lambda\:=\:\frac{m}{\sqrt{2}\:\pi\:d^{2}mn}}$$

mn = density of the gas (ρ)

$$\mathrm{n\:=\:\frac{p}{KT}}$$

$$\mathrm{\lambda\:=\:\frac{KT}{\sqrt{2}\:\pi\:d^{2}P}}$$

Factors affect the mean free path

The mean free path rises when the temperature rises. Because the average speed of each molecule increases as the temperature increases. This process causes the smell of hot cooked food lasts longer than the smell of cold food. As the pressure of the gas decreases and the diameter of the gas molecule decreases, the mean free path velocity increases.

Solved Examples

Example 1 − An oxygen molecule at a temperature of 300 K and 1 atmospheric pressure travels through the air. Find the mean free path of an oxygen molecule if its diameter is $\mathrm{1.2\:\times\:10^{-10}m}$

Sol −

The Mean free path equation is, $\mathrm{\lambda\:=\:\frac{1}{\sqrt{2}\:n\pi\:d^{2}}}$

From the Ideal gas equation, we have to calculate the number density n,

$$\mathrm{n\:=\:\frac{P}{KT}\:=\:\frac{101.3\times\:10^{3}}{1.381\times\:10^{-23}\times\:300}}$$

$$\mathrm{=\:2.449\:\times\:10^{25}\:molecules\:/m^{3}}$$

$$\mathrm{\lambda\:=\:\frac{1}{\sqrt{2}\pi\:\times\:2.449\times\:10^{25}\times\:(1.2\times\:10{-10})^{2}}}$$

$$\mathrm{\lambda\:=\:\frac{1}{15.65\times\:10^{5}}}$$

$$\mathrm{\lambda\:=\:0.63\:\times\:10^{-6}m}$$

Conclusion

The matter is a term we use to refer to all things in the universe. Everything around us is matter. The total amount of kinetic energy and static energy is the internal energy of the molecules. When this internal energy flows out of an object, it is called thermal energy. Total linear momentum and total energy never change in any collision process. Since the total kinetic energy need not always remain constant. As the temperature rises, the mean free path rises. Because the average speed of each molecule increases as the temperature increases.

FAQs

1. What is Stefan-Boltzman’s law?

According to this rule energy emitted per second from the surface area of an entire black matter is directly proportional to the four times its temperature.

2. Define Evaporation and cooling

Evaporation is when a substance absorbs heat and changes from a liquid state to a gaseous state. Cooling is the process by which a substance in a gaseous state releases heat and turns into a liquid

3. What is convection?

Convection is the diffusion of heat by the media, particles moving from the hot terminal to the cooling terminal in the mat material.

4. What are the factors affecting Brownian motion?

The Brownian motion also increases as temperature increases. Brownian motion decreases as the bulk of liquid or gas particles increase due to high absorption and density.

5. Define the thermal effect of current

The potential difference between its two electrodes drives electrons through the resistor, creating a current. As current flows the battery must continue to expend its energy. Part of the energy is converted into useful work. The remaining energy is wasted by heating it.