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:

Find the wavelength of the first line of each series using the Rydberg formula; frequency is inversely proportional to wavelength.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
\(\frac{1}{\lambda}=R\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)\) and frequency \(f=\frac{c}{\lambda}\) is proportional to \(\frac{1}{\lambda}\).

Step 2: Balmer first line
\(n_1=2,\ n_2=3\): \(\frac{1}{\lambda_1}=R\left(\frac14-\frac19\right)=\frac{5R}{36}\).

Step 3: Paschen first line
\(n_1=3,\ n_2=4\): \(\frac{1}{\lambda_2}=R\left(\frac19-\frac1{16}\right)=\frac{7R}{144}\).

Step 4: Ratio
\(\frac{f_1}{f_2}=\frac{5/36}{7/144}=\frac{5\times144}{36\times7}=\frac{20}{7}\).

Step 5: Compare the options
The ratio is \(20:7\), the first option.

Final Answer:
The ratio \(f_1:f_2\) is \(20:7\). \[ \boxed{\text{(A) }20:7} \]