An excited hydrogen atom emits a photon of wavelength ' $\lambda$ ' in returning to ground state. The…

An excited hydrogen atom emits a photon of wavelength ' $\lambda$ ' in returning to ground state. The quantum number ' $n$ ' of the excited state is ( $\mathrm{R}=$ Rydberg's constant)
  1. $\sqrt{\lambda \cdot R(\lambda R-1)}$
  2. $\sqrt{\frac{\lambda \mathrm{R}}{(\lambda \mathrm{R}-1)}}$
  3. $\sqrt{\frac{(\lambda \mathrm{R}-1)}{\lambda \mathrm{R}}}$
  4. $\sqrt{\frac{1}{\lambda \mathrm{R}(\lambda \mathrm{R}-1)}}$

Solution

Using Rydberg's formula $\frac{1}{\lambda}=\mathrm{R}\left[\frac{1}{1^2}-\frac{1}{\mathrm{n}^2}\right] \quad \ldots(\because \mathrm{m}=1)$ $\therefore \quad \frac{\mathrm{n}^2-1}{\mathrm{n}^2}=\frac{1}{\lambda \mathrm{R}} \Rightarrow 1-\frac{1}{\mathrm{n}^2}=\frac{1}{\lambda \mathrm{R}} \Rightarrow \frac{\lambda \mathrm{R}-1}{\lambda \mathrm{R}}=\frac{1}{\mathrm{n}^2}$ $\therefore \quad \mathrm{n}=\sqrt{\frac{\lambda \mathrm{R}}{\lambda \mathrm{R}-1}}$ :

Asked in: MHT CET 2023 (13 May Shift 2)

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