The ratio of the wavelengths of the light absorbed by a Hydrogen atom when it undergoes n = 2 → n = 3

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

(1) \(\frac{1}{36}\)

(2) \(\frac{1}{16}\)

(3) \(\frac{1}{9}\)

(4) \(\frac{1}{4}\)

Answer 1

Correct option is: (4) \(\frac{1}{4}\)

\(\frac{1}{\lambda} = RZ^2 \left( \frac{1}{n_1 ^2} -\frac{1}{n_2 ^2} \right)\)

For \(2 \rightarrow 3 \Rightarrow \frac{1}{\lambda_1} = R \times (1)^2 \left( \frac{1}{2^2} - \frac{1}{3^2} \right)\)  ...(1)

For \(4 \rightarrow 6 \Rightarrow \frac{1}{\lambda_2} = R \times (1)^2 \left( \frac{1}{4^2} - \frac{1}{6^2} \right)\) ...(2)

Divide (1) \(\&\) (2)

\(\frac{\lambda_2}{\lambda_1} = \frac{\frac{1}{2^2} - \frac{1}{3^2}}{\frac{1}{4^2} - \frac{1}{6^2}}\)

\(\frac{\lambda_2}{\lambda_1} = \frac{4}{1}\)

\(\therefore \) \(\frac{\lambda_1}{\lambda_2} = \frac{1}{4}\)