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: It is found that the excitation frequency from ground to the first excited state of rotation for the $\mathrm{CO}$ molecule is close to $\frac{4}{\pi} \times 10^{11} \mathrm{~Hz}$. Then the moment of inertia of CO molecule about its centre of mass is close to (Take $h=2 \pi \times 10^{-34} J-s$ )
  1. $2.76 \times 10^{-46} \mathrm{~kg}-\mathrm{m}^2$
  2. $1.87 \times 10^{-46} \mathrm{~kg}-\mathrm{m}^2$
  3. $4.67 \times 10^{-47} \mathrm{~kg}-\mathrm{m}^2$
  4. $1.17 \times 10^{-47} \mathrm{~kg}-\mathrm{m}^2$

Solution

$ \begin{gathered} h v=K_2-K_1=\frac{3 h^2}{8 \pi^2 I} \\ \therefore I=\frac{3 h}{8 \pi^2 f}=\frac{3 \times 2 \pi \times 10^{-34} \times \pi}{8 \times \pi^2 \times 4 \times 10^{11}} \end{gathered} $ $ =1.87 \times 10^{-46} \mathrm{~kg}-\mathrm{m}^2 $ $\therefore$ The correct answer is (b). `

Asked in: JEE Advanced 2010 (Paper 2)

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