In $L i^{+ +}$ ion, electron in first Bohr orbit is excited to a level by a radiation of wavelength $λ$ . When the ion gets excited to the ground state in all possible ways (including intermediate emissions), a total of six spectral lines are observed. What is the value of $λ$ approximately? (atomic number of lithium is 3)
(Given : $h = 6.63 \times 10^{- 34} J s; c = 3 \times 10^{8} m s^{- 1}$ )

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11.4 nm

9.4 nm

10.8 nm

12.3 nm

answer is D.

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Detailed Solution

Spectral lines obtained on account of transition from n^th orbit to various lower orbits is $\frac{n (n - 1)}{2}$

$\Rightarrow 6 = \frac{n (n - 1)}{2} \Rightarrow n = 4 Δ E = \frac{h c}{λ} = \frac{- Z^{2}}{n^{2}} (13.6 e V) \Rightarrow \frac{1}{λ} = Z^{2} (\frac{13.6 e V}{h c}) (\frac{1}{n_{2}^{2}} - \frac{1}{n_{1}^{2}}) = (13.4) {(3)}^{2} [1 - \frac{1}{16}] e V \Rightarrow λ = \frac{1242 \times 16}{(13.4) \times (9) (15)} n m ≃ 10.8 n m$