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A diatomic molecule is made of two masse...

A diatomic molecule is made of two masses `m_(1) and m_(2)` which are separated by a distance `r` . If we calculate its rotational energy by applying Bohr's rule of angular momentum quantization it energy will be ( n is an integer )

A

`((m_(1)+m_(2))^(2)n^(2)h^(2))/(2m_(1)^(2)m_(2)^(2)r^(2))`

B

`(n^(2)h^(2))/(2(m_(1)+m_(2))r^(2))`

C

`(2n^(2)h^(2))/((m_(1)+m_(2))r^(2))`

D

`((m_(1)+m_(2))n^(2)h^(2))/(2m_(1)m_(2)r^(2))`

Text Solution

Verified by Experts

The correct Answer is:
D
Energy of the system of two atom of diatomic molecules
`E = 1/2 I omega^(2)`
I =Moment of inertia = `(m_(1)r_(1)^(2)+m_(2)r_(2)^(2))`
and `omega =L/I , L to ` angular momentum
` :. E = 1/2 (m_(1)r_(1)^(2)+m_(2)r_(2)^(2))omega^(2)`
`=1/2 (m_(1)r_(1)^(2)+m_(2)r_(2)^(2))(L^(2))/((m_(1)r_(1)^(2)+m_(2)r_(2)^(2))^(2))`
`= 1/2 (L^(2))/((m_(1)r_(1)^(2)+m_(2)r_(2)^(2)))= (n^(2)h^(2))/(8pi^(2)(m_(1)r_(1)^(2)+m_(2)r_(2)^(2)))`
`m_(1)r_(1)=m_(2)r_(2)`
`r_(1)+r_(2)=r`

` :. " "r_(1) =(m_(2)r)/(m_(1)+m_(2)),r_(2)=(m_(1)r)/(m_(1)+m_(2))`
`:. " " E = (n^(2)h^(2))/(8pi^(2)) . 1/(m_(1).(m_(2)^(2)r^(2))/((m_(1)+m_(2))^(2))+m_(2).(m_(1)^(2)r^(2))/((m_(1)+m_(2))^(2)))`
`= (n^(2)h^(2))/(8pi^(2)) ((m_(1)+m_(2)))/(m_(1)m_(2)r^(2))`
As `h = h(2pi) :. " " E = (n^(2)h^(2)(m_(1)+m_(2)))/(2m_(1)m_(2)r^(2))`
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