Physics - Modern Physics Question with Solution | TestHub

The wavelength of a spectral line in the hydrogen atom is given by the Rydberg formula:

$\frac{1}{\lambda }=R\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$

For the Lyman series, the lower energy level is $n_1=1$.

The maximum wavelength ${\lambda }_{\text{max}}$ corresponds to the minimum energy transition, which occurs from $n_2=2$ to $n_1=1$.

$\frac{1}{{\lambda }_{\text{max}}}=R\left(\frac{1}{1^2}-\frac{1}{2^2}\right)=R\left(1-\frac{1}{4}\right)=\frac{3R}{4}$

${\lambda }_{\text{max}}=\frac{4}{3R}$

The minimum wavelength ${\lambda }_{\text{min}}$ corresponds to the maximum energy transition, which occurs from $n_2=\infty$ to $n_1=1$.

$\frac{1}{{\lambda }_{\text{min}}}=R\left(\frac{1}{1^2}-\frac{1}{{\infty }^2}\right)=R(1-0)=R$

${\lambda }_{\text{min}}=\frac{1}{R}$

The ratio of the maximum and minimum wavelength is:

$\frac{{\lambda }_{\text{max}}}{{\lambda }_{\text{min}}}=\frac{\frac{4}{3R}}{\frac{1}{R}}=\frac{4}{3}$

Answer: $4:3$