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The mass of the most common silicon atom is 4.646 * 10^-26 kg, and the mass of the most common oxygen atom is 2.656 * 10^-26 kg. When a molecule of silicon monoxide (SiO) makes a transition between the l = 1 and l = 0 rotational levels, it emits a photon of wavelength 6.882 mm. Find the distance between the Si and O nuclei.

Expert Solution

Step 1

$G i v e n d a t a : m a s s o f s i l i c o n, m_{s i} = 4.646 \times 10^{- 26} k g m a s s o f o x y g e n, m_{0} = 2.656 \times 10^{- 26} k g t r a n s i t i o n f r o m l = 1 t o l = 0 a n d w a v e l e n g t h λ = 6.882 m m = 6.882 \times 10^{- 3} m$

Step 2

$H e r e, r e d u c e d m a s s, μ = \frac{m_{s i} \times m_{o}}{m_{s i} + m_{0}} μ = \frac{4.646 \times 10^{- 26} \times 2.656 \times 10^{- 26}}{(4.646 + 2.656) \times 10^{- 26}} μ = 1.69 \times 10^{- 26} k g$

Step 3

$f o r m u l a f o r e n e r g y e i g e n v a l u e, E_{l} = \frac{h^{2}}{8 π^{2} I} l (l + 1) s o E_{1} = \frac{h^{2}}{8 π^{2} I} 1 (1 + 1) E_{1} = \frac{2 h^{2}}{8 π^{2} I} . . . . . . . . . . . . . . . . . . . e q (1) s i m i l l a r l y E_{0} = \frac{h^{2}}{8 π^{2} I} 0 (0 + 1) = 0 . . . . . . . . . . . . . . e q (2) c h a n g e i n e n e r g y, ∆ E = \frac{2 h^{2}}{8 π^{2} I} - 0 ∆ E = \frac{2 h^{2}}{8 π^{2} I} . . . . . . . . . . . . . . . . . . . . . . . . . . . . e q (3)$

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