It is found that the excitation frequency from ground to the first excited state of rotation for the CO molecule is close to Then the moment of inertia of the CO molecule about its center of mass is close to [2010]

Question

The key feature of Bohr's theory of the spectrum of the hydrogen atom is the quantization of angular momentum when an electron revolves around a proton. We will extend this to a general rotational motion to find the quantized rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantization condition. [2010]

Q. It is found that the excitation frequency from ground to the first excited state of rotation for the CO molecule is close to $\frac{4}{π} \times 10^{11} Hz .$ Then the moment of inertia of the CO molecule about its center of mass is close to (Take $h = 2 π \times 10^{- 34} J s$ )

Smart Achievers · Accepted Answer

(2)

$Rotational kinetic energy K . E ._{rot} = \frac{1}{2} I ω^{2} = \frac{1}{2} I {(\frac{L}{I})}^{2}$ $[∵ L = I ω]$

And according to Bohr's quantisation principle, $L = \frac{n h}{2 π}$

$∴$ $K . E ._{rot} = \frac{1}{2} \frac{L^{2}}{I} = \frac{1}{2 I} \times \frac{n^{2} h^{2}}{4 π^{2}} = n^{2} [\frac{h^{2}}{8 π^{2} I}]$ $\dots (i)$

From ground state $(n = 1)$ to first excited state $(n = 2)$ ,

Energy given = change in kinetic energy

$h ν = K_{f} - K_{i} = \frac{h^{2}}{8 π^{2} I} (2^{2} - 1^{2})$ $[From Eq. (i)]$

$h ν = \frac{3 h^{2}}{8 π^{2} I}$ $\Rightarrow I = \frac{3 h}{8 π^{2} ν} = \frac{3 \times 2 π \times 10^{- 34}}{8 π^{2} \times \frac{4}{π} \times 10^{11}} = \frac{3}{16} \times 10^{- 45}$

$I = 1.87 \times 10^{- 46} {kg m}^{2}$

Solution

1 $2.76 \times 10^{- 46} {kg m}^{2}$

2 $1.87 \times 10^{- 46} {kg m}^{2}$

3 $4.67 \times 10^{- 47} {kg m}^{2}$

4 $1.17 \times 10^{- 47} {kg m}^{2}$

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Solution

1 2.76×10-46kg m2

2 1.87×10-46kg m2

3 4.67×10-47kg m2

4 1.17×10-47kg m2

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1 $2.76 \times 10^{- 46} {kg m}^{2}$

2 $1.87 \times 10^{- 46} {kg m}^{2}$

3 $4.67 \times 10^{- 47} {kg m}^{2}$

4 $1.17 \times 10^{- 47} {kg m}^{2}$