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Hydrogen atom from excited state comes to the ground state by emitting a photon of wavelength $λ .$ The value of principal quantum number ‘n’ of the excited state will be (R : Rydberg constant)

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$\sqrt{\frac{λ R^{2}}{λ R - 1}}$

$\sqrt{\frac{λ}{λ R - 1}}$

$\sqrt{\frac{λ R}{λ R - 1}}$

$\sqrt{\frac{λ R}{λ - 1}}$

answer is B.

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Detailed Solution

$Z = 1, n_{1} = 1, n_{2} = n$

$\frac{1}{λ} = R Z^{2} [\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}]$

$\frac{1}{λ} = R [1 - \frac{1}{n^{2}}]$

$\frac{1}{λ R} = 1 - \frac{1}{n^{2}} \Rightarrow \frac{1}{n^{2}} = 1 - \frac{1}{λ R}$

$\frac{1}{n^{2}} = \frac{λ R - 1}{λ R}$

$n^{2} = \frac{λ R}{λ R - 1} \Rightarrow n = \sqrt{\frac{λ R}{λ R - 1}}$

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