Question

The wavelength (in cm) of the second line in the Lyman series of the hydrogen atomic spectrum is (Rydberg constant \( R \, \text{cm}^{-1})\).

• A. \(\frac{8R}{9}\)
• B. \(\frac{9}{8R}\)
• C. \(\frac{4}{3R}\)
• D. \(\frac{3R}{4}\)

Hint:

The Lyman series represents transitions of electrons to the \( n_1 = 1 \) level. Use the Rydberg formula to calculate the spectral line wavelengths. The second line in the Lyman series corresponds to \( n_2 = 3 \).

Answer 1

The Correct Option is A

Step 1: Understanding the Lyman Series
The Lyman series corresponds to transitions of the electron in a hydrogen atom to the \( n_1 = 1 \) energy level. The second line in the Lyman series corresponds to the transition from \( n_2 = 3 \) to \( n_1 = 1 \).
Step 2: Using the Rydberg Formula
The wave number (\( \bar{\nu} \)) of the spectral line is given by the Rydberg formula: \[ \bar{\nu} = \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right), \] where:
\( R \) is the Rydberg constant,
\( n_1 = 1 \) (final energy level),
\( n_2 = 3 \) (initial energy level).
Step 3: Substituting the Values
Substitute \( n_1 = 1 \) and \( n_2 = 3 \) into the Rydberg formula: \[ \frac{1}{\lambda} = R \left( \frac{1}{1^2} - \frac{1}{3^2} \right) = R \left( 1 - \frac{1}{9} \right) = R \left( \frac{8}{9} \right). \]
Step 4: Matching with the Options
The calculated value is \( \frac{8R}{9} \), which corresponds to option (A). Final Answer: The wave number of the second line in the Lyman series is \(\frac{8R}{9}\).