Paragraph: The key feature of Bohr's theory of spectrum of hydrogen...

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

Options

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 $n^2\left(\frac{h^2}{8 \pi^2 I}\right)$

Correct answer

D. $n^2\left(\frac{h^2}{8 \pi^2 I}\right)$

Step-by-step 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).

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