Question

In the Bohr atom model, the frequency of transitions is given by the following expression

v = `"Rc" (1/"n"^2 - 1/"m"^2)`, where n < m

Consider the following transitions:

Transitions	m → n
1	3 → 2
2	2 → 1
3	3 → 1

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

Answer 1

In the Bohr atom model, the frequency of transition

v = `"Rc" (1/"n"^2 - 1/"m"^2)`   ...n < m

I^st transition, m = 3 and n = 2

`"v"_(3 -> 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^st transition, m = 2 and n = 1

`"v"_(2 -> 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

`"v"_(3 -> 1) = "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

`"v"_(3->2) + "v"_(2->1) = "v"_(3->1)`

`"R"_"c" (5/36) + "R"_"c" (3/4) = "R"_"c" (8/9)`

`= "R"_"c" (8/9) = "R"_"c" (8/9)`

`"v"_(3->2) + "v"_(2->1) = "v"_(3->1)`