In the Bohr atom model, the frequency of transitions is given by the following expression `v=Rc(1/n^(2)-1/m^(2))`, where `nltm`, Consider the following transitions:

Show that the frequency of these transitions obey sum rule (which is known as Ritz combination principle)

Explanation

Ritz combination principle states that the spectral lines of any element include frequencies that are sum of the frequencies of two other lines.
An atom can make transition from 3 to 1 (first state) either directly (i.e. `3to1`) or in two steps `3to2and2to1`. Energy is conserved so the two frequencies (wave number) of later add to the frequency of the first way `(3to1)`
According to Bohr
`upsilon=RC(1/n^(2)-1/m^(2))`
`upsilon_(3to2)=RC(1/2^(2)-1/3^(2))=5/36`
`upsilon_(2to1)=RC(1/1^(2)-1/2^(2))=3/4`
`upsilon_(3to1)=RC(1/1^(2)-1/3^(2))=8/9`
Hence,
`upsilon_(3to1)=v_(3to2)+v_(2to1)rArr8/9=5/36+3/4=32/36=8/9`