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In the Bohr atom model, the frequency of...

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)

Text Solution

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In the Bohr atom model, the frequency of transition
`upsilon = R_(c) ((1)/(n^(2)) - (1)/(m^(2))) n lt m `
`I^(st)` transition, m = 3 and n = 2
`upsilon_(3) to 2 = R_(c) ((1)/(2^(2)) - (1)/(3^(2))) = R_(c) ((1)/(4)- (1)/(9)) `
`= R_(c) ((9-4)/(36)) = R_(c) ((5)/(36))`
`II^(nd)` transition , m = 2 and n = 1
`upsilon _(2) to 1 R_(c) ((1)/(1^(2))-(1)/(2^(2))) = R_(c) (1 - (1)/(4)) = R_(c) ((3)/(4))`
`III^(rd)` transition m = 3 and n = 1
`upsilon _(3) to1 = R_(c) ((1)/(1^(2)) - (1)/(3^(2))) = R_(c)(1-(1)/(9)) = R_(c) ((8)/(9))`
According to Ritz combination principle, the frequency transition of single step is the sum of frequency transition in two steps.
`upsilon_((3) to 2) + upsilon_((2) to 1) = upsilon_((3) to 1)`
`R_(c) ((5)/(36)) + R_(c) ((3)/(4)) = R_(c) ((8)/(9))`
`R_(c) ((8)/(9)) = R_(c) ((8)/(9))`
`upsilon_((3) to 2) + upsilon_((2) to 1) = upsilon_((3) to 1)`
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