Physics - Modern Physics Question with Solution | TestHub

For the first member of the Lyman series, the transition is from $n=2$ to $n=1$. Using the Rydberg formula:

$\frac{1}{{\lambda }_1}=R\left(\frac{1}{1^2}-\frac{1}{2^2}\right)=R\left(1-\frac{1}{4}\right)=\frac{3R}{4}$

${\lambda }_1=\frac{4}{3R}$

For the first member of the Paschen series, the transition is from $n=4$ to $n=3$.

$\frac{1}{{\lambda }_2}=R\left(\frac{1}{3^2}-\frac{1}{4^2}\right)=R\left(\frac{1}{9}-\frac{1}{16}\right)=\frac{7R}{144}$

${\lambda }_2=\frac{144}{7R}$

Taking the ratio of the two wavelengths:

$\frac{{\lambda }_1}{{\lambda }_2}=\frac{\frac{4}{3R}}{\frac{144}{7R}}=\frac{4}{3R}\times \frac{7R}{144}=\frac{28}{432}=\frac{7}{108}$

Answer: $7:108$