Question

The key feature of Bohr's theory of spectrum of hydrogen atom is the quantisation 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.
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

• A. $\frac{1}{n^{2}}\left(\frac{h^{2}}{8 \pi^{2} I}\right)$
• B. $\frac{1}{n}\left(\frac{h^{2}}{8 \pi^{2} I}\right)$
• C. $n\left(\frac{h^{2}}{8 \pi^{2} I}\right)$
• D. $n^{2}\left(\frac{h^{2}}{8 \pi^{2} I}\right)$

Answer 1

Answer: (D)

$I \omega=\frac{n h}{2 \pi}$
Rotational kinetic energy $=\frac{1}{2} I \omega^{2}=\frac{1}{2} \frac{n^{2} h^{2}}{4 \pi^{2} I}=\frac{n^{2} h^{2}}{8 \pi^{2} I}$