Question

A hydrogen atom is in the excited state of the principal quantum number $n$. It emits a photon of wavelength ${}^{\prime }{\lambda }^{\prime }$ when returns to ground state. The value of $n$ is

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

Answer 1

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

Energy of photon, $E=Rhc(1-\frac{1}{n^2})$

$\Rightarrow \frac{hc}{\lambda }=Rhc(1-\frac{1}{n^2})$

$\Rightarrow \frac{1}{\lambda }=R(1-\frac{1}{n^2})$

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

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

$\Rightarrow n=\sqrt{\frac{\lambda R}{(\lambda R-1)}}$