For hydrogen atom, and are the wavelengths corresponding to the transitions 1 and 2 respectively as shown in figure. The ratio of and is . The value of is _________. [2023]

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Solution

Q.
For hydrogen atom, $λ_{1}$ and $λ_{2}$ are the wavelengths corresponding to the transitions 1 and 2 respectively as shown in figure. The ratio of $λ_{1}$ and $λ_{2}$ is $\frac{x}{32}$ . The value of $x$ is _________. [2023]

Ans.
(27)

$\frac{1}{λ} = R z^{2} [\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}]$

$\frac{1}{λ_{1}} = R z^{2} [\frac{1}{1^{2}} - \frac{1}{3^{2}}] = \frac{8}{9} R z^{2} \dots (1)$

$\frac{1}{λ_{2}} = R z^{2} [\frac{1}{1^{2}} - \frac{1}{2^{2}}] = \frac{3}{4} R z^{2} \dots (2)$

$\frac{1}{2} \Rightarrow \frac{λ_{2}}{λ_{1}} = \frac{8}{9} \times \frac{4}{3} = \frac{32}{27} \Rightarrow \frac{λ_{1}}{λ_{2}} = \frac{27}{32}$

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