Question

If \(\lambda _{1}\) and \(\lambda _{2}\) are the wavelengths of the first members of the Lyman and Paschen series respectively then \(\lambda _{1}:\lambda _{2}\) is

• A. \(1:35\)
• B. \(1:30\)
• C. \(7:50\)
• D. \(7:108\)

Hint:

Use the Rydberg formula with n = 1 to 2 for Lyman and n = 3 to 4 for Paschen, then take the ratio of wavelengths.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The Rydberg formula is \(\dfrac{1}{\lambda} = R\left(\dfrac{1}{n_1^2} - \dfrac{1}{n_2^2}\right)\).
The first member of a series has \(n_2 = n_1 + 1\).

Step 2: Lyman Series, First Line
\(n_1 = 1,\ n_2 = 2\)
\(\dfrac{1}{\lambda_1} = R\left(1 - \dfrac{1}{4}\right) = \dfrac{3R}{4}\)

Step 3: Paschen Series, First Line
\(n_1 = 3,\ n_2 = 4\)
\(\dfrac{1}{\lambda_2} = R\left(\dfrac{1}{9} - \dfrac{1}{16}\right) = \dfrac{7R}{144}\)

Step 4: Ratio
\(\dfrac{\lambda_1}{\lambda_2} = \dfrac{4/3}{144/7} = \dfrac{28}{432} = \dfrac{7}{108}\)

Final Answer:
The ratio of the wavelengths is 7 to 108. \[ \boxed{\text{(D) }7:108} \]