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, its energy will be given by (n is an integer)

Options

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))`

Answer

D

Explanation

`m_(1)r_(1)=m_(2)r_(2)`
`r_(1)+r_(2)=r`
`:. r_(1)=(m_(2)r)/(m_(1)+m_(2))`
`:. epsilon=(1)/(2)omega^(2)`
`=(1)/(2)(m_(1)r_(1)^(2)+m_(2)r_(2)^(2)).omega^(2)`.......(i)
`mvr=(h)/(2pi)=Iomega`
`omega=(nh)/(2piI)`
`:. epsilon=(1)/(2)I. (n^(2)h^(2))/(4pi^(2)I^(2))=(n^(2)h^(2))/(8pi^(2))(1)/((m_(1)r_(1)^(2)+m_(2)r_(2)^(2)))`
`=(n^(2)h^(2))/(8pi^(2))(1)/((,_(2)^(2)r_(0)^(2))/((m_(1)+m_(2))^(2)))+m_(2)(m_(1)^(2)r^(2))/((m_(1)+m_(2))^(2))`
`=(n^(2)h^(2))/(8pi^(2)r^(2))((m_(1)+m_(2))^(2))/(m_(1)m_(2)(m_(1)+m_(2)))=((m_(1)+m_(2))n^(2)h^(2))/(8pi^(2)r^(2)m_(1)m_(2))`