Ncert Exercises & Solutions

Question 7. 20. The oxygen molecule has a mass of 5.30 x 10^-26 kg and a moment of inertia of 1.94 x 10^-46 kg m² about an axis through its center perpendicular to the lines joining the two atoms. Suppose the mean speed of such a molecule in a gas is 500 m/s and that its kinetic energy of rotation is two-thirds of its kinetic energy of translation. Find the average angular velocity of the molecule.

L e t m i s t h e m a s s o f o x y g e n m o l e c u l e .

S o, m a s s o f e a c h o x y g e n a t o m = \frac{m}{2}

I = \frac{m}{2} r^{2} + \frac{m}{2} r^{2} = m r^{2}

\Rightarrow r = \sqrt{\frac{I}{m}}

\Rightarrow r = \sqrt{\frac{1.94 \times 10^{- 46}}{5.36 \times 10^{- 26}}} = 0.06 \times 10^{- 10} m

I t i s g i v e n t h a t,

K E_{r o t a t i o n a l} = \frac{2}{3} K E_{t r a n s l a t i o n a l}

\Rightarrow \frac{1}{2} I ω^{2} = \frac{2}{3} \times \frac{1}{2} m v^{2}

\Rightarrow ω = \sqrt{\frac{2}{3}} \times \frac{v}{r} = \sqrt{\frac{2}{3}} \times \frac{500}{0.60 \times 10^{- 10}}

= 6.80 \times 10^{12} r a d / s

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