Question

The hydrogen atom is in excited state of principal quantum number (n), it emits a photon of wavelength ($\lambda$), when it returns to the ground state. The value of n is :

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

Answer 1

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

As $\frac{1}{\lambda }=R(\frac{1}{n_1^2}-\frac{1}{n_2^2})$
$\therefore \frac{1}{\lambda }=R(\frac{1}{1^2}-\frac{1}{n^2})$
Multiply both sides by $\lambda$
$1=\lambda R(1-\frac{1}{n^2})$ or $\frac{1}{\lambda R}=1-\frac{1}{n^2}$
or $\frac{1}{n^2}=1-\frac{1}{\lambda R}=\frac{\lambda R-1}{\lambda R}$
or $n=\sqrt{\frac{\lambda R}{\lambda R-1}}$