Question

Electron of an excited hydrogen atom jumps to the ground state and emits a photon of wavelength $\lambda$. The principle quantum number $n$ of initial excited state in terms of $\lambda$ and $R$ (Rydberg constant) is:

• A. $\lambda R$
• B. $\frac{\lambda R}{\lambda R-1}$
• C. $\lambda \sqrt{R(R-1)}$
• D. $\sqrt{\frac{\lambda R}{\lambda R-1}}$

Answer 1

The correct option is D $\sqrt{\frac{\lambda R}{\lambda R-1}}$

We Know,

$\begin{array}{c}\frac{1}{\lambda }=R\left(1-\frac{1}{n^2}\right)\\ \Rightarrow \frac{1}{\lambda R}=1-\frac{1}{n^2}\\ \Rightarrow \frac{1}{n^2}=1-\frac{1}{\lambda R}\\ \Rightarrow n^2=\frac{\lambda R}{\lambda R-1}\\ \Rightarrow n=\sqrt{\frac{\lambda R}{\lambda R-1}}\end{array}$

$\therefore$ Option $D$ is correct.