A sample of hydrogen gas in its ground state is irradiated with photons of `10.02 eV` energies. The radiation from the above sample is used to irradiate two other sample of excited ionized `He^(+)` and excited ionized `Li^(2+)`, respectively. Both the ionized samples absorb the incident radiation.
How many spectral lines are obtained in the spectra of `Li^(2+)`?

Options

A. 10
B. 15
C. 20
D. 17

Answer

B

Explanation

Consider energy level diagram of hydrogen atom. After absorbing photon of energy `10.2 eV` , if would reach the first excited state of `- 3.4 eV` , since energy difference corresponding to `n = 1` to `n = 2` is `10.2 eV`. When this excited hydrogen atom de-excites, it would release `10.2 eV` energy , which is absorbed by `He^(+)` and `Li^(2 +)`.
Energy of the nth state of hydrogen - like atom with atomic number`Z` is given by
`E_(n) = (13.6 Z^(2))/(n^(2)) eV`
After absorbing `10.2 eV`, `He^(+)` electron move from `n = 2` to `n = 4` and `Li^(2 +)` electron moves from `n = 3` to `n = 6`.

In the spectrum of `He^(+)`, there would be `^(4) C_(2) = 6` lines.

`((lambda_(min)),(n = 4 to n = 1)) = (hc)/(Delta E) = (1242)/(13.6 [ 4 - (4)/(16)]] = 24.4 nm`

Similarly, in spectrum of `Li^(2 +)` there will be `^(6) C_(2) = 15lines`
`((lambda_(min)),(n = 6 to n = 1)) = (1242)/(13.6 [ 9 - (9)/(36)]] = 10.4 nm`