Question

For hydrogen atom, $\lambda_1$ and $\lambda_2$ are the wavelengths corresponding to the transitions 1 and 2 respectively as shown in figure. The ratio of $\lambda_1$ and $\lambda_2$ is $\frac{x}{32}$. The value of $x$ is

Answer 1

$ \frac{1}{\lambda}= Rz ^2\left[\frac{1}{ n _1^2}-\frac{1}{ n _2^2}\right]$
$\frac{1}{\lambda_1}= Rz ^2\left[\frac{1}{1^2}-\frac{1}{3^2}\right]=\frac{8}{9} Rz ^2 $.....(1)
$ \frac{1}{\lambda_2}= Rz ^2\left[\frac{1}{1^2}-\frac{1}{2^2}\right]=\frac{3}{4} Rz ^2$......(2)
$1 / 2 \Rightarrow \frac{\lambda_2}{\lambda_1}=\frac{8}{9} \times \frac{4}{3}=\frac{32}{27} $
$\frac{\lambda_1}{\lambda_2}=\frac{27}{32}$
$=27$