### Kinetic Theory of Ideal Gases

German scientists R.Classius and J.C. Maxwell had propounded the kinetic theory of gases. According to this model, every gas is made up of very fine particles called molecules. All the molecules of a gas are similar in all properties. We know that $1 cm^{3}$ of water at $100^{\circ}C$ temperature and $1$ atmosphere pressure produces $1671 cm$ of water vapour. From this, it is known that the volume actually occupied by the molecules in $1671 cm^{3}$ of water vapour is only $1 cm^{3}$ and the remaining $1670 cm^{3}$ of volume is empty. In other words, we can say that in the gaseous state of matter, there is a lot of free space between the molecules. This fact is true for all gases. The molecules of gases are always moving randomly in all possible directions. During motion, these molecules collide with each other and with the walls of the vessel in which the gas is kept. After each collision, both the direction and speed of motion of these molecules keep changing.

All collisions between molecules and molecules or between molecules and the walls are elastic. This implies that total kinetic energy and momentum are conserved.

Assumptions of Kinetic Theory of Gases

There are the following assumptions of the kinetic theory of gases.

1. The molecules of a gas are very small, rigid, spherical and completely elastic. The volume of the molecules is the volume in which the gas is present, is negligible compared to the volume of the gas.

2. Although the molecules of a gas can move with every possible velocity in every direction, but the number of molecules per unit volume or molecular density remains the same.

3. There is no force of attraction or repulsion between the molecules, as a result of which they do not have potential energy, their entire energy is kinetic energy.

4. When the molecules of a gas come very close to each other, there is a repulsive force between them, due to which their speed and direction of motion change. This phenomenon is called a 'collision' between two molecules. Between two consecutive collisions, the molecules move in a straight line with a constant speed. The distance travelled by the molecule between two consecutive collisions is called the free path and its average is called the mean free path.

5. The mutual collision between the molecules and the collision between the molecule and the wall is perfectly elastic, that is, the kinetic energy of the molecules is conserved.

6 The time of collision between the molecules is negligible compared to the distance travelled freely by the molecule.

7. Since the amount of molecules is negligible and the velocity is high, therefore there is no effect of gravity on the motion of the molecules.

### Numerical Aperture and Acceptance Angle of the Optical Fibre

Angle of Acceptance → If incident angle of light on the core for which the incident angle on the core-cladding interface equals the critical angle then incident angle of light on the core is called the "Angle of Acceptance. Transmission of light when the incident angle is equal to the acceptance angle If the incident angle is greater then the acceptance angle i.e. $\theta_{i}>\theta_{0}$ then the angle of incidence on the core-cladding interface will be less than the critical angle due to which part of incident light is transmitted into cladding as shown in the figure below Transmission of light when the incident angle is greater than the acceptance angle If the incident angle is less then the acceptance angle i.e. $\theta_{i}<\theta_{0}$ then the angle of incidence on the core-cladding interface will be greater than the critical angle for which total internal reflection takes place inside the core. As shown in the figure below Transmission of light w

### Fraunhofer diffraction due to a single slit

Let $S$ be a point monochromatic source of light of wavelength $\lambda$ placed at the focus of collimating lens $L_{1}$. The light beam is incident normally from $S$ on a narrow slit $AB$ of width $e$ and is diffracted from it. The diffracted beam is focused at the screen $XY$ by another converging lens $L_{2}$. The diffraction pattern having a central bright band followed by an alternative dark and bright band of decreasing intensity on both sides is obtained. Analytical Explanation: The light from the source $S$ is incident as a plane wavefront on the slit $AB$. According to Huygens's wave theory, every point in $AB$ sends out secondary waves in all directions. The undeviated ray from $AB$ is focused at $C$ on the screen by the lens $L_{2}$ while the rays diffracted through an angle $\theta$ are focussed at point $p$ on the screen. The rays from the ends $A$ and $B$ reach $C$ in the same phase and hence the intensity is maximum. Fraunhofer diffraction due to

### Particle in one dimensional box (Infinite Potential Well)

Let us consider a particle of mass $m$ that is confined to one-dimensional region $0 \leq x \leq L$ or the particle is restricted to move along the $x$-axis between $x=0$ and $x=L$. Let the particle can move freely in either direction, between $x=0$ and $x=L$. The endpoints of the region behave as ideally reflecting barriers so that the particle can not leave the region. A potential energy function $V(x)$ for this situation is shown in the figure below. Particle in One-Dimensional Box(Infinite Potential Well) The potential energy inside the one -dimensional box can be represented as $\begin{Bmatrix} V(x)=0 &for \: 0\leq x \leq L \\ V(x)=\infty & for \: 0> x > L \\ \end{Bmatrix}$ $\frac{d^{2} \psi(x)}{d x^{2}}+\frac{2m}{\hbar^{2}}(E-V)\psi(x)=0 \qquad(1)$ If the particle is free in a one-dimensional box, Schrodinger's wave equation can be written as: $\frac{d^{2} \psi(x)}{d x^{2}}+\frac{2mE}{\hbar^{2}}\psi(x)=0$ \$\frac{d^{2} \psi(x)}{d x