Table of Contents

**Introduction: Mean Free Path**

In physics, the mean free path is the average distance over which a moving particle (such as an atom, molecule, or photon) significantly changes its direction or energy (or, in some cases, other properties), typically as a result of one or more successive collisions with other particles.

**Definition:**

The average distance travelled by a moving particle between successive collisions or impacts is referred to as the mean free path.

Furthermore, successive collisions change the energy or direction of the moving particle, as well as any other particle properties. Furthermore, the moving particle can be a molecule, an atom, or a photon.

In radiography, the pencil beam of a mono-energetic photon must have a mean free path. Furthermore, the mean free path is the average distance travelled by a photon between collisions with the atoms of the target material. Furthermore, the free path is affected by both the material and the energy of the photons. In electronics, the mean free path of electrons in a metal of a charge carrier is proportional to its electrical mobility. In addition, electrical mobility is a value that is directly related to electrical conductivity.

**Concept of the mean free path:**

Consider the motion of a gas molecule within an ideal gas. A typical molecule within an ideal gas will abruptly change direction and speed as it collides elastically with other molecules of the same gas. Though the molecule will move in a straight line at a constant speed between collisions, this is true for all molecules in the gas.

As the name implies, is the average distance travelled by any molecule between collisions. We expect it to vary inversely with N/V, which is the number of molecules per unit volume or the density of molecules because the more molecules there are, the more likely they will collide with each other, reducing the mean free path, and also inversely proportional to the diameter d of the molecules because if the molecules were point masses, they would never collide. Because it is difficult to measure or describe the random motion of gas molecules, we attempt to calculate their mean free path.

**Overview of Mean Free Path**

They collide with each other as well as the container’s walls. Assume that molecule 1 collided with molecule 2, then with molecule 3, and so on. The first collision occurs when molecule 1 collides with molecule 2. When it collides with molecule 3, it is referred to as the second collision, and so on. The distance between the first and second collisions is denoted as 1, the distance between the second and third collisions is denoted as 2, and so on. Because there is no collision between any two consecutive collisions, the path is referred to as a free path (1, 2, 3, etc).

To explain the structure and composition of molecules in relation to submicroscopic particles, the kinetic theory was introduced. The theory discusses the pressure increase caused by the constant movement and collision of submicroscopic particles. It also discusses other gas properties like temperature, pressure, volume, viscosity, diffusion, thermal conductivity, and so on. The theory establishes a link between microscopic particles and macroscopic properties. The molecule of gas is constantly in motion and collides with each other and the container’s walls; in such a case, learning the dynamics of the gases is both difficult and important.

Consider the motion of a gas molecule contained within an ideal gas. Furthermore, a typical molecule inside an ideal gas will abruptly change its direction and speed. This is due to the fact that it collides with other molecules of the same gas in an elastic manner. The molecule between the collisions must move in a straight line at a constant speed. Most importantly, this holds true for all of the molecules in the gas. Measuring the random movement of gas molecules is undoubtedly difficult. As a result, one must attempt to calculate its mean free path. The symbol represents the average distance travelled by a molecule between collisions. Furthermore, it varies inversely with N/V, which is the number of molecules per unit volume or the density of the molecules.

**Derivation of Mean Free Path:**

Certain assumptions will be used in the equation’s derivation. Assume the molecule is spherical. Furthermore, the collision occurs when one molecule collides with another. Furthermore, the emphasis here is on the moving molecule, while the others are stationary. Assume that the diameter of a single molecule is d. Consider the single-molecule moving through the gas. As a result, a short cylinder will be swept out. Furthermore, this short cylinder has a cross-section area of **π****d²**.

It will move the distance between successive collisions for time t. In this case, v is the molecule’s velocity. Most notably, sweeping this cylinder would result in a volume of **π****d²v t**. As a result, the number of collisions that the molecule may have can be determined by the number of point molecules contained within this volume. Without a doubt, N/V is the number of molecules per unit volume. As a result, the number of molecules in the cylinder is N/V multiplied by the volume of the cylinder,** π****d²****v t**.

**Formula of Mean Free Path**

As a result, the mean free path can be calculated as follows:

**λ= length of the path during the time t/ number of collision in time**

**r≈ v t / (πd²v t N/V) =1/πd²N/V**

The equation has been approximated because it has been assumed that all particles are stationary in relation to the particle under consideration. Most notably, all of the molecules are moving relative to one another. In the preceding equation, two velocities have been cancelled. Furthermore, v in the numerator denotes the average velocity, whereas v in the denominator denotes the relative velocity. As a result, there is a factor 2– the difference between the two.

**λ=1 / ( √2 πd²N/V)**

At sea level, the mean free path is 0.1 micrometres.

**Mean Free Path Factors:**

Density: As density increases, molecules get closer together, increasing the number of collisions and decreasing the mean free path.

Number of molecules: As the number of molecules increases, so does the probability of collision, and thus the mean free path.

The radius of the molecule: As the radius of the molecule increases, the space between the molecules shrinks, increasing the number of collisions and thus decreasing the mean free path.

Pressure, temperature, and other physical factors all influence gas density and thus the mean free path.

**Horizontal projectile motion:**

The horizontal and vertical components of projectile motion are separated in physics. The vertical component of projectile motion is caused by gravity in the majority of cases. The gravitational force causes a constant acceleration of 32.2 ft/s² or 9.8 m/s² towards the Earth for all objects. Horizontal motion is defined as a projectile motion in a horizontal plane that is affected by a force. A projectile’s vertical and horizontal components are perpendicular and independent of each other for a short distance.

The horizontal component of a projectile’s speed remains constant throughout its flight. This is due to the fact that no horizontal force acts on the projectile after it has been launched. As a result, the projectile moves horizontally at a constant speed. The following equation is used to calculate the distance covered by a projectile: Distance is equal to speed multiplied by time.

**d =v t**

A projectile must be launched in a straight line, not at an angle, to achieve horizontal motion. The projectile’s velocity varies, but the direction in which it is launched should be perpendicular to the Earth’s surface. The projectile is launched by applying a constant vertical force of gravity that is independent of the horizontal force. This means that the projectile’s total flight time will always be the same. By varying the initial velocity and the force used to launch the projectile, the projectile can be made to travel longer or shorter distances in the same amount of time. A projectile must be launched at a specific angle for long-distance travel, such as that of a missile, and the horizontal and vertical components must be determined to make the projectile travel a longer distance. Motion in a plane is also known as two-dimensional motion. Circular motion and projectile motion are two-dimensional motion examples.

m is a reference point taken at the origin of two coordinate axes, namely the X-axis and the Y-axis, for the analysis of two-dimensional projectile motion.

Also read: **Degrees of freedom**

**FAQs on Mean Free path **

##### What is the relationship between temperature and the mean free path?

The mean free path and temperature are related in that there is a display of linear proportionality from the mean free path to the temperature.

##### What exactly is the mean free path in electronics?

In electronics, the mean free path of a charge carrier in metal is proportional to its electrical mobility.

**Question 3:What are the Different Types of Projectile Motion?**

**Answer:** There are two distinct rectilinear motions:

- Along the x-axis: the axis of uniform velocity, which is responsible for the particle’s horizontal (forward) motion.
- Along the y-axis: the axis of uniform acceleration, which is responsible for the particle’s vertical (downward) motion.