Quantization of Energy

Bohr’s Postulates

• An electron revolving round the nucleus has angular momentum equal to integral multiple of h/2π, i.e.

• An electron revolving round the nucleus doesn’t emit the energy continuously. It emit energy when it jumps from higher orbit to lower orbit and absorb energy when it jumps from higher to lower orbit.

Bohr’s Theory of Hydrogen Atom

Suppose an electron is revolving round the nucleus in an orbit of radius r_n with velocity v_n.

The electrostatic force between electron and nucleus provides centripetal force.

Electrostatic force = centripetal force

From Bohr’s postulate:

Putting value of v_n in equation (1):

Velocity of Electron in nth Orbit

Putting r_n value in equation (2):

Expression for Energy of Electron in nth Orbit

An electron revolving around the nucleus possesses K.E. due to its motion and P.E. due to force of attraction between electron and nucleus. So, total energy of electron is the sum of its K.E. and P.E.

And,

Therefore,

where R is Rydberg’s constant.

Spectral Series

1) Lyman Series

This series is obtained when an electron jumps from higher energy state to ground state (n = 1).

For this series n₁ = 1, n₂ = 2, 3, 4, … ∞

2) Balmer Series

This series is obtained when an electron jumps from higher energy state to first excited state (n = 2).

For this series n₁ = 2, n₂ = 3, 4, 5, …

3) Paschen Series

This series is obtained when an electron jumps from higher energy state to second excited state (n = 3).

For this series n₁ = 3, n₂ = 4, 5, 6, …

4) Brackett Series

This series is obtained when an electron jumps from higher energy state to third excited state (n = 4).

For this series n₁ = 4, n₂ = 5, 6, 7, …

5) Pfund Series

This series is obtained when an electron jumps from higher energy state to fourth excited state.

For this series n₁ = 5, n₂ = 6, 7, 8, …

de-Broglie Theory

It states that a particle moving in an orbit has both wave and particle nature. The wave associated with matter is called matter wave or de-Broglie wave.

The wavelength of matter wave is given by:

Heisenberg’s Uncertainty Principle

It states that it is impossible to measure position of the particle and momentum, energy and time accurately and simultaneously.

If Δx and Δp are uncertainty in position and momentum, then:

And if ΔE and Δt are uncertainty in energy and time, then:

X-Rays

X-rays are electromagnetic radiation having very short wavelength in the order of 1 Å to 100 Å. X-rays are produced when fast moving electrons strike a metal having high atomic no and high melting point.

Properties

• They are electromagnetic radiation having very short wavelength.
• They are chargeless.
• They are not deflected in electric and magnetic field.
• They have high penetrating power.
• They can ionize the gas through which they pass.
• They can cast shadow of an object on which they fall.

Production of X-Rays (Coolidge Tube)

The experimental arrangement is as shown in fig. It consists of an evacuated glass tube having a filament (F) connected to low tension battery and a target metal (T) connected to a copper tube in which water is circulating continuously. A high tension battery is connected between filament and target metal.

The filament gets heated due to low tension battery and electrons are emitted. The emitted electrons are accelerated by high tension battery…

…and strike the target metal. On striking almost 98% of energy of electrons is converted into heat and remaining energy is converted into X-rays.

Diffraction of X-Ray (Laue’s Experiment)

The experimental arrangement is as shown in figure. The X-rays produced from X-ray source are collimated by using slits and are allowed to strike ZnS crystal. When photographic plate is sufficiently exposed to X-rays, a pattern is observed with central bright spot and many faint spots. This concluded that X-rays show diffraction and prove wave nature of X-rays.

Bragg’s Law

It states that the intensity of reflected X-rays from different layers will be maximum if path difference between the waves is integral multiple of wavelength of X-ray.

Consider two beams of X-rays are incident on two layers of crystal as shown in figure. The path difference between waves is MB′ + B′N.

For maximum intensity:

Diffraction

It is the phenomenon of the bending of light around the corners of an obstacle if the size is comparable to the wavelength of light.

Types of Diffraction

Fresnel Diffraction: The source and screen are at finite distance from the obstacle. In general one or both of the incident wavefront is spherical or cylindrical.

Fraunhofer Diffraction: The source and screen are at infinite distance. Lenses are used to make the incident light parallel and the wavefront is plane.

Fraunhofer Diffraction at Single Slit

Consider a plane wavefront AB is incident on a slit of width d. According to Huygens’s theory each point on AB act as source of disturbance.

The point O is middle of wavefront whose perpendicular centre of screen is zero, so central maxima is obtained.

Let P is any point at angle θ at which position of maxima or minima is to be determined.

The path difference between waves from A and B on reaching P is BM.

In ΔABM:

Position of Minima

Position of Maxima

Width of Central Maxima

The separation between two 1st order minima gives width of central maxima.

Since, for 1st minima:

And from figure:

From (1) and (2):

Diffraction Grating

It is an arrangement having large number of equal sized slits separated by equal sized opaque space. If a represent size of slit and b represent size of opaque space and d is called grating element.

No. of lines per unit length of grating is given by N:

In diffraction grating maxima is obtained if:

n = order of diffraction.

Numerical Problems

1)

2)

Using d sinθ = nλ:

For maximum order, θ = 90°:

3)

4)

Additional Numerical