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Physics · 2024 · 3 marks
CBSE 2024 · Region 1 · Set 3 · Q27
State the three postulates of Bohr's theory of hydrogen atom. A hydrogen atom de-excites from level n to level ( n -1 ). Show that, according to…
State the three postulates of Bohr's theory of hydrogen atom. A hydrogen atom de-excites from level n to level ( $\displaystyle \mathrm{n}-1$ ). Show that, according to Bohr's theory, the frequency of radiation emitted $\displaystyle v \cong \frac{\alpha}{\mathrm{n}^{3}}$, for large values of n , where $\displaystyle \alpha$ is a constant. This result exactly agrees with that obtained from classical physics - one of the successes of Bohr's theory.
Marking-scheme solution
Bohr's postulates:-
(i)
An electron in an atom could revolve in certain stable orbits without the emission of radiant energy.
(ii)
The electron revolves around the nucleus only in those orbits for which the angular momentum is some integral multiple of $\displaystyle h/2\pi$
(iii)
An electron might make a transition from one of its specified non-radiating orbits to another of lower energy. When it does so, a photon is emitted having energy equal to the energy difference between the initial and final states.
\[h\nu = E_i - E_f \qquad \text{------- (i)} \]
\[E_n \propto \frac{1}{n^2} \]
\[E_n - E_{n-1} = k\left[\frac{1}{(n-1)^2} - \frac{1}{n^2}\right] \]
\[= k\,\frac{n^2 - \left[n^2 + 1 - 2n\right]}{n^4 - 2n^3 + n^2} \]
\[= k\,\frac{2n - 1}{n^4 - 2n^3 + n^2} \]
For large n , 2n-$\displaystyle 1$ $\displaystyle \cong$ 2n
$\displaystyle n^2\,[n^2 - 2n + 1] \cong n^4$
\[E_n - E_{n-1} \cong k\,\frac{2n}{n^4} \cong \frac{2k}{n^3} \qquad \text{----- (ii)} \]
From equation (i) and (ii)
\[\nu \cong \frac{\alpha}{n^3} \]
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