Quantisation of Energy-Numericals Solved

List of Some Important Questions with solutions.


1.What is the shortest wavelength present in the Paschen series of spectral lines?
ans:
Rydberg’s formula is given as:

where,
h = Planck’s constant = 6.6 × 10^−34 Js
c = Speed of light = 3 × 108 m/s
 (n1 and n2 are integers)
The shortest wavelength present in the Paschen series of the spectral lines is given for
values n1 = 3 and n2 = ∞


2.The ground state energy of hydrogen atom is −13.6 eV. What are the kinetic and potential energies of the electron in this state?
Ans
Ground state energy of hydrogen atom, E = − 13.6 eV
This is the total energy of a hydrogen atom. Kinetic energy is equal to the negative of the total energy.
Kinetic energy = − E = − (− 13.6) = 13.6 eV
 Potential energy is equal to the negative of two times of kinetic energy.
Potential energy = − 2 × (13.6) = − 27 .2 eV

3.A hydrogen atom initially in the ground level absorbs a photon, which excites it to the 4 level. Determine the wavelength and frequency of the photon?
Ans:
For ground level, n1 = 1 Let E1 be the energy of this level. It is known that E1 is related with n1 as:

The atom is excited to a higher level, n2 = 4
Let E2 be the energy of this level.

The amount of energy absorbed by the photon is given as: E = E2 − E1


For a photon of wavelengthλ, the expression of energy is written as:

Where, h = Planck’s constant = 6.6 × 10−34 Js c = Speed of light = 3 × 108 m/s


Hence, the wavelength of the photon is 97 nm while the frequency is 3.1*10^15Hz.

4.Using the Bohr’s model calculate the speed of the electron in a hydrogen atom in the n=
1, 2, and 3 levels. (b) Calculate the orbital period in each of these levels.
Ans:
Let ν1 be the orbital speed of the electron in a hydrogen atom in the ground state level, = 1. For charge (e) of the electron,v1 is given by the relation


Where, e = 1.6 × 10−19 C
∈0 = Permittivity of free space = 8.85 × 10
 h = Planck’s constant = 6.62 × 10^-34Js

For level n2 = 2, we can write the relation for the corresponding orbital speed as:

Hence, the speed of the electron in a hydrogen atom in n = 1, n=2, and n=3 is 2.18 × 106 m/s, 1.09 × 106 m/s, 7.27 × 105 m/s respectively.
 Let T1 be the orbital period of the electron when it is in level n1 = 1.
 Orbital period is related to orbital speed as:

Where,
 r1 = Radius of the orbit


h = Planck’s constant = 6.62 × 10−34 Js
e = Charge on an electron = 1.6 × 10−19 C
∈0 = Permittivity of free space = 8.85 × 10−12 N−1 C2 m−2
m = Mass of an electron = 9.1 × 10−31 kg



For level n2 = 2, we can write the period as:

 Where, r2 = Radius of the electron in n2 = 2

Where, r3 = Radius of the electron in n3 = 3


5. The radius of the innermost electron orbit of a hydrogen atom is 5.3 ×10−11 m. what are the radii of n=2 and n=3 orbits?
Ans:
The radius of the innermost orbit of a hydrogen atom, r1 = 5.3 × 10−11 m.
Let r2 be the radius of the orbit at as:

 For n = 3, we can write the corresponding electron radius as:

 Hence, the radii of an electron for 10−10 m respectively.

6.A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. What series of wavelengths will be emitted?




7.Refer Principle of Physics chapter22:
Ans:

8.Refer Principle of Physics chapter22: