IEMDaily - Video Lecture Notes

Law

Describe the shortcomings of Bohr's Theory

Shortcomings of Bohr Atomic Model Theory are as follows:

Bohrs Model could not explain the spectra of atoms containing more than one electron.
It could not explain the Zeeman effect.
It could not explain the Stark effect.
It violates the Heisenberg's Uncertainty principle.

Definition

Describe the Sommerfeld atom model

In order to explain the observed fine structure of spectral lines, Sommerfeld introduced two main modifications in Bohrs theory, which are as follows:
(i) According to Sommerfeld, the path of an electron around the nucleus, in general, is an ellipse with the nucleus at one of its foci.
(ii) The velocity of the electron moving in an elliptical orbit varies at different parts of the orbit. This causes the relativistic variation in the mass of the moving electron.

Definition

Ionization energy of an atom

The minimum energy required to free an electron from the ground state of the hydrogen atom is called the ionisation energy of the hydrogen atom.
It is given by

E_{i} = 13.6 \frac{Z^{2}}{n^{2}}

.
For hydrogen atom

Z = 1

Hence ionisation energy is

E_{i} = \frac{13.6}{n^{2}}

Definition

Energy levels for stationary states of a hydrogen atom

The total energy of the electron in the stationary states of the hydrogen atom can be obtained by substituting the value of orbital radius in

E_{n} = - \frac{e^{2}}{8 π ϵ_{0} r}

\Rightarrow E_{n} = - \frac{m e^{4}}{8 n^{2} ϵ_{0}^{2} h^{2}} = - \frac{13.6}{n^{2}} e V

Example

Problem based on energy difference between various levels of hydrogen spectrum

Ratio of difference in energy between the energy levels with

n = 3

and

n = 4

and between the energy levels with

n = 8

and

n = 9

for a
hydrogen like atom or ion is:
Energy difference between n

= 3

and n

= 4

E_{1} = - 13.6 [\frac{1}{3^{2}} - \frac{1}{4^{2}}]

Energy difference between n

= 8

and n

= 9

E_{2} = - 13.6 [\frac{1}{8^{2}} - \frac{1}{9^{2}}]

Ratio of energy difference :

= \frac{E_{1}}{E_{2}} = \frac{\frac{1}{9} - \frac{1}{16}}{\frac{1}{64} - \frac{1}{81}} = \frac{7 \times 64 \times 81}{9 \times 16 \times 17}

= 14.8235

Result

Franck and Herts Experiment

Franck and Hertz's original experiment used a heated vacuum tube containing a drop of mercury; they reported a tube temperature of 115

^{0}

C, at which the vapor pressure of mercury is about 100 pascals (and far below atmospheric pressure). It is fitted with three electrodes: an electron-emitting, hot cathode; a metal mesh grid; and an anode. The grid's voltage is positive relative to the cathode, so that electrons emitted from the hot cathode are drawn to it. The electric current measured in the experiment is due to electrons that pass through the grid and reach the anode. The anode's electric potential is slightly negative relative to the grid, so that electrons that reach the anode have at least a corresponding amount of kinetic energy after passing the grid.
The graphs published by Franck and Hertz show the dependence of the electric current flowing out of the anode upon the electric potential between the grid and the cathode.

Definition

Conclusions from the plot of Franck and Hertz experiment

Some conclusions that can be drawn from the graph plotted in the Franck and Hertz experiment are as follows:
1. Rising curve in the current v/s accelerating plot corresponds to region where the electron gain kinetic energy due to excitation potential but not enough to ionize the medium (mercury).
2. Decaying curve corresponds to the region where the medium is ionized and hence energy is lost in ionisation.
3. Maxima corresponds to the point when the energy of the electrons is just enough to ionize the medium.
4. Minima corresponds to the point where electrons start to gain energy from the applied accelerating potential.
5. Ideally, the distance between two maxima is constant an equals the excitation potential of the medium. However, mercury has more than one excitation and ionization potential which makes the second and third peaks of the curve complicated.
6. The above observations suggest that the electrons give energy to the atoms in only discrete levels.