A hydrogen -like ion `He^(o+) (Z = 2)` is expased to electromagnetic waves of `256.4 Å` The excited electron gives out induced radiation .Find the wavelength of the induced radiation when the electron de-excetes back to the ground state `(R = 109677 cm^(-1))`

Explanation

`He^(o+)` ion containts only one electron, it disorts a photon of wavelength `lambda=256.4Å` Assume the electron to be in the ground state initialy .Let it jump to an excited state `a_(2)` Use the following relaation to find `n_(2)`
`barv = (1)/(lambda) = RZ^(2)((1)/(n_(1)^(2)) - (1)/(n_(2)^(2)))`
Substituting for `lambda = 256.4 Å = 256.4 xx 10^(-8) cm`
`R = 1096777 Z = 2` for `He^(o+)` ion `n_(1) = 1`
Now find `n_(2)`
`(1)/(256.4 xx 10^(-6)) = 109677 xx 10^(7) xx (2)^(2) ((1)/(n_(1)^(2)) - (1)/(n_(2)^(2)))`
`rArr n_(2) =3`
From `n = 3` , the electron can fall back in the ground state in three possibie ways (transition)
`3 rarr 1, 3 rarr 2, 2 rarr 1 `
Hence three possible radiation are emitted find the wavelengths coresponding to these transition
The wavelength `(lambda)` for the transition `3 rarr 1 ` will be same i.e. `256.4 Å` find `lambda` for `3 rarr 2 and 2 rarr 1 ` using the same relation
`lambda (3 rarr 1) = 256.4 Å lambda (3 rarr2) = 1641.3 Å`
`lambda( 2 rarr 1) = 303.9 Å`
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