The frequency of light emitted for the transition `n = 4` to `n =2` of `He^+` is equal to the transition in `H` atom corresponding to which of the following ?

Options

A. `n = 3 "to" n = 1`
B. `n = 2 "to" n = 1`
C. `n = 3 "to" n = 2`
D. `n = 4 "to" n = 3`

Answer

B

Explanation

(b) `Delta E = hu = (2 pi^2 mZ^2 e^4 k^2)/(h^2) [(1)/(n_1^2)-(1)/(n_2^2)]`
If electron falls from `n_2 - "level to" n_1 - "level"`
`:.` In `He^+, "for" n_2 = 4 "to" n_1 = 2 transition`
`v_(He^+) = "constant" (4) [(1)/(2^2) -(1)/(4^2)][because Z_(He^2) = 2]`
=`"constant" xx 4 [(3)/(16)] = (3)/(4) xx "constant"`
`v(H) = "constant" (1) 2 [(1)/(n_1^2) - (1)/(n_2^2)]`
=`"constant" xx [(1)/(n_1^2) - (1)/(n_2^2)]`.
(a) For `n_2 = 3` and `n_1 = 1`
`v(H) = "constant" [(1)/(1) -(1)/(9)]`
=`(8)/(9) xx "constant" ne (3)/(4) xx "constant"`
(b) For `n_2 = 2` and `n_1 = 1`
`v(H) = "constant" xx [(1)/(1) -(1)/(4)]`
=`(3)/(4) xx "constant" = v(He^+)`
( c) For `n=3` and `n_1 = 2`
`v(H) = "constant" xx [(1)/(2^2) -(1)/(3^2)]`
=`(5)/(36) xx "constant" ne v(He^+)`
(d) For `n_2 = 4` and `n_1 = 3`
`v(H) = "constant" xx [(1)/(3^2) - (1)/(4^2)]`
=`(7)/(144) xx "constant" ne v(He^+)`.