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Structure of Atom — Chemistry STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceChemistryStructure of Atom5 Marks
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.
$\bar{\text{v}}=109677\frac{1}{\text{n}^2_\text{i}}-\frac{1}{\text{n}_\text{f}^2}$
What points of Bohr’s model of an atom can be used to arrive at this formula? Based on these points derive the above formula giving description of each step and each term.
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Answer

The following point of Bohr's model of an atom can be used to arrive at the given formula:
  1. Electrons revolve around the nucleus in circular orbits with fixed values of energy.
  2. When electron jumbs from one orbit to another, energy is emitted of absorbed.
Derivation of the given formula: The energy of electron in the $n^{th}$ stationary state is given by,
Where, $\text{E}_\text{n}=\frac{-2\pi^2\text{me}^4}{\text{n}^2\text{h}^2}$
m = mass of electron
e = Charge on electron
h = Planck's constant
When electron jumps from outer $n_2$ to inner orbit $n_1$ the difference of energy $(\Delta\text{E})$ is emitted.
$\Delta\text{E}=\text{E}_2-\text{E}_1=\frac{-2\pi^2\text{me}^4}{\text{n}_2^2\text{h}^2}-\Big(\frac{-2\pi^2\text{me}^4}{\text{n}_1^2\text{h}^2}\Big)$
$=\frac{-2\pi^2\text{me}^4}{\text{h}^2}-\Big(\frac{1}{\text{n}_1^2}-\frac{1}{\text{n}_2^2}\Big)$
$\bar{\text{v}}=\frac{\Delta\text{E}}{\text{hc}}=\frac{2\pi^2\text{me}^4}{\text{ch}^3}\Big(\frac{1}{\text{n}_1^2}-\frac{1}{\text{n}_2^2}\Big)$
By putting values of $\pi,$ m, c, h and e we get,
$\bar{\text{v}}=109677\Big(\frac{1}{\text{n}^2_\text{i}}-\frac{1}{\text{n}_\text{f}^2}\Big)$

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