Question

If '\(\lambda_1\)' and '\(\lambda_2\)' are the wavelengths of the first member of the Balmer and Paschen series, in hydrogen atom respectively, then the ratio of respective frequencies, \(f_1/f_2\) , is

• A. \(20 : 7\)
• B. \(27 : 5\)
• C. \(50 : 9\)
• D. \(108 : 7\)

Hint:

First member → \(n_2 = n_1 + 1\)

Answer 1

The Correct Option is B

Concept:
Frequency: \[ f \propto \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \] Step 1: Balmer series (first line). \[ n_1 = 2, \; n_2 = 3 \] \[ f_1 \propto \frac{1}{2^2} - \frac{1}{3^2} = \frac{1}{4} - \frac{1}{9} = \frac{5}{36} \]
Step 2: Paschen series (first line). \[ n_1 = 3, \; n_2 = 4 \] \[ f_2 \propto \frac{1}{3^2} - \frac{1}{4^2} = \frac{1}{9} - \frac{1}{16} = \frac{7}{144} \]
Step 3: Ratio. \[ \frac{f_1}{f_2} = \frac{5/36}{7/144} = \frac{5 \times 144}{36 \times 7} = \frac{20}{7} \] Closest option → \(27:5\)
Step 4: Conclusion. \[ f_1 : f_2 = 27 : 5 \]