Question

If \(\lambda_1\) and \(\lambda_2\) are the wavelengths of the first spectral line of the Lyman and Paschen series respectively, then \(\lambda_1 : \lambda_2\) is

• A. \(1 : 3\)
• B. \(7 : 50\)
• C. \(1 : 30\)
• D. \(7 : 108\)

Hint:

Always identify \(n_1\) and \(n_2\) correctly for hydrogen spectral series before applying the Rydberg formula.

Answer 1

The Correct Option is D

Step 1: Use the Rydberg formula.
The wavelength of a spectral line is given by:
\[ \frac{1}{\lambda} = R\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \]
Step 2: First line of the Lyman series.
For Lyman series, \(n_1 = 1\) and first line corresponds to \(n_2 = 2\).
\[ \frac{1}{\lambda_1} = R\left(1 - \frac{1}{4}\right) = \frac{3R}{4} \] \[ \lambda_1 = \frac{4}{3R} \]
Step 3: First line of the Paschen series.
For Paschen series, \(n_1 = 3\) and first line corresponds to \(n_2 = 4\).
\[ \frac{1}{\lambda_2} = R\left(\frac{1}{9} - \frac{1}{16}\right) = R\left(\frac{7}{144}\right) \] \[ \lambda_2 = \frac{144}{7R} \]
Step 4: Calculate the ratio.
\[ \lambda_1 : \lambda_2 = \frac{4}{3R} : \frac{144}{7R} = \frac{4 \times 7}{3 \times 144} = \frac{28}{432} = 7 : 108 \]
Step 5: Conclusion.
The required ratio is \(7 : 108\).