Question

The largest wavelength in the ultraviolet region of the hydrogen spectrum is $122nm$. The smallest wavelength in the infrared region of the hydrogen spectrum (to the nearest integer) is:

• A. $802nm$
• B. $823nm$
• C. $1882nm$
• D. $1648nm$

Answer 1

The correct option is B $823nm$

$\frac{1}{\lambda }=R[\frac{1}{n_2^2}-\frac{1}{n_1^2}]$
The largest wavelength in the ultraviolet region of the hydrogen spectrum corresponds to the transition $n_1=2$ to $n_2=1$ (Lyman series).
Thus, $\frac{1}{122nm}=R[1-\frac{1}{4}]=\frac{3R}{4}$
Which gives $R=\frac{4}{3\times 122nm}$
The smallest wavelength $\lambda$ in the infrared region of the hydrogen spectrum corresponds to $n_1=\infty$ and $n_2=3$ (Paschen series).
Therefore, $\frac{1}{\lambda }=R[\frac{1}{3^2}-\frac{1}{\infty }]=\frac{R}{9}$$⟹\lambda =\frac{9}{R}=\frac{9\times 3\times 122nm}{4}=823nm$