Question

When an electric discharge is passed through hydrogen gas, the hydrogen molecules dissociate to produce excited hydrogen atoms, these excited atoms emit electromagnetic radiation of discrete frequencies which can be given by the general formula
 $ \overline \nu \, = \,109677\left[ {\dfrac{1}{{{n_i}^2}} - \dfrac{1}{{{n_f}^2}}} \right] $
(i)What points of bohr's model of an atom can be used to arrive at this formula?
(ii) Based on these points derive the above formula giving a description of each step and each term.

Answer 1

Hint :Before knowing the Bohor’s model, it is very important to have an idea of the atoms and their related theories. The first of all atomic theory was proposed by J. Dalton in 1808 in which he gave an introductory idea of the atom, but the main disadvantage of the theory was that, at that time electrons, protons, neutrons and other subatomic particles were not discovered.

Complete Step By Step Answer:
N. Bohr in 1913 gave his model in order to have a better understanding of the structure of an atom. His theory includes the following statements: -
An atom is subdivided into smaller particles named as electrons, protons and neutrons.
The electrons are revolving around the nucleon (neutrons and protons).
The electrons are revolving in a particular orbit with some sort of fixed energy. Hence called stationary states.
The electron never loses any sort of energy in revolutions.
Only those orbits are said to exist in which the angular momentum is an integral part of $ \dfrac{h}{{2\pi }} $ .
The change in energy only takes place when an electron jumps from one state to another.
(i)The electrons are revolving around the nucleon (neutrons and protons). The electrons are revolving in a particular orbit with some sort of fixed energy.
(ii) The gain or loss of energy will only take place when the electron jumps from one stationary state to another. The derivation is as follows: -
 $ {E_n} = \dfrac{{ - 2{\pi ^2}m{e^4}}}{{{n^2}{h^2}}} $
 $ \forall $ $ m $ is mass of an electron i.e., $ 9.1 \times {10^{ - 31}}\,Kg $ . $ e $ is charge of an electron i.e., $ - 1 $ . $ h $ is Planck's constant i.e., $ 6.62 \times {10^{ - 34}} $ . and $ n $ is the orbit number.
 $ \vartriangle E = {E_2} - {E_1} = \dfrac{{ - 2{\pi ^2}m{e^4}}}{{{h^2}}} - \left( {\dfrac{1}{{{n_1}^2}} - \dfrac{1}{{{n_2}^2}}} \right) $
But we know that
 $ \overline \nu = \dfrac{{\vartriangle E}}{{hc}} $
Then now substituting the values,
 $ \overline \nu = \dfrac{{\vartriangle E}}{{hc}} = \dfrac{{2{\pi ^2}m{e^4}}}{{{h^2}}} - \left( {\dfrac{1}{{{n_1}^2}} - \dfrac{1}{{{n_2}^2}}} \right) $
Putting the real values of all the constants, we get
 $ \overline \nu \, = \,109677\left[ {\dfrac{1}{{{n_i}^2}} - \dfrac{1}{{{n_f}^2}}} \right] $ .

Note :
Though the Bohor’s model was kept to be an ideal model, however this model fails in explaining the, uncertainty principle, cannot predict about the spectral lines of toms other than hydrogen, fails in explaining the Zeeman effect, the Stark effect, the Raman effect and the Rutherford effect ( production of the $ \beta $ rays).