Question

The ratio of the wavelength of the last line of Paschen series to that of Balmer series is

• A. \(\frac{9}{4}\)
• B. \(\frac{3}{2}\)
• C. \(\frac{2}{3}\)
• D. \(\frac{4}{9}\)

Hint:

Last line → \(n_2 = \infty\)

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The "last line" or series limit of a spectral series corresponds to a transition from an energy level at infinity (\(n = \infty\)) to the base level of that series.
The wavelength is calculated using the Rydberg formula.
Step 2: Key Formula or Approach:
Rydberg formula: \(\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\).
For the last line, \(n_2 = \infty\), so \(\frac{1}{\lambda} = \frac{R}{n_1^2} \implies \lambda = \frac{n_1^2}{R}\).
Step 3: Detailed Explanation:
For Paschen series: \(n_1 = 3\).
Wavelength of the last line of Paschen series (\(\lambda_P\)): \[ \lambda_P = \frac{3^2}{R} = \frac{9}{R} \] For Balmer series: \(n_1 = 2\).
Wavelength of the last line of Balmer series (\(\lambda_B\)): \[ \lambda_B = \frac{2^2}{R} = \frac{4}{R} \] Calculate the ratio: \[ \frac{\lambda_P}{\lambda_B} = \frac{9/R}{4/R} = \frac{9}{4} \] Step 4: Final Answer:
The ratio of the wavelengths is \(9 : 4\).