Paragraph: The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular…
Paragraph:
The key feature of Bohr's theory of spectrum of hydrogen atom is the quantization of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition.Question:
A diatomic molecule has moment of inertia I. By Bohr's quantization condition its rotational energy in the $n$th level ( $n=0$ is not allowed) is
$\frac{1}{n^2}\left(\frac{h^2}{8 \pi^2 I}\right)$
$\frac{1}{n}\left(\frac{h^2}{8 \pi^2 I}\right)$
$n\left(\frac{h^2}{8 \pi^2 I}\right)$
$n^2\left(\frac{h^2}{8 \pi^2 I}\right)$
Solution
$
\begin{aligned}
L=I \omega & =\frac{n h}{2 \pi} \therefore \omega=\frac{n h}{2 \pi I} \\
K & =\frac{1}{2} I \omega^2=\frac{1}{2} I\left(\frac{n h}{2 \pi I}\right)^2 \\
& =\frac{n^2 h^2}{8 \pi^2 I}
\end{aligned}
$
$\therefore$ The correct answer is (d).