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

Question

The ratio of the wavelengths of the light absorbed by a Hydrogen atom when it undergoes n = 2 $\rightarrow$ n = 3 and n = 4 $\rightarrow$ n = 6 transitions, respectively, is:

Answer 1

Step 1: Define the Lyman Series
The Lyman series involves electron transitions in a hydrogen atom to the $ n_1 = 1 $ energy level. The second line corresponds to a transition from $ n_2 = 3 $ to $ n_1 = 1 $.
Step 2: Apply the Rydberg Formula
The Rydberg formula gives the wave number ($ \bar{u} $) of spectral lines:\[\bar{u} = \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right),\]with $ R $ as the Rydberg constant, $ n_1 = 1 $ (final level), and $ n_2 = 3 $ (initial level).
Step 3: Calculate the Wave Number
Substitute the values into the 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: Identify the Corresponding Option
The calculated wave number is $ \frac{8R}{9} $, matching option (A).Final Answer: The wave number for the second line of the Lyman series is $\frac{8R}{9}$.

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

Show Hint

The Correct Option is A

Solution and Explanation

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