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

Answer: 27

$\frac{1}{\lambda }=R{z}^2\left[\frac{1}{n_1^2}-\frac{1}{n_2^2}\right]$
$\frac{1}{{\lambda }_1}=R{z}^2\left[\frac{1}{1^2}-\frac{1}{3^2}\right]=\frac{8}{9}R{z}^2$.....(1)
$\frac{1}{{\lambda }_2}=R{z}^2\left[\frac{1}{1^2}-\frac{1}{2^2}\right]=\frac{3}{4}R{z}^2$......(2)
$1/2\Rightarrow \frac{{\lambda }_2}{{\lambda }_1}=\frac{8}{9}\times \frac{4}{3}=\frac{32}{27}$
$\frac{{\lambda }_1}{{\lambda }_2}=\frac{27}{32}$
$=27$