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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 ______.

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answer is 27.

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Detailed Solution

Wavelength corresponding to a transition from n₁ to n₂ is given by

$\frac{1}{λ} = R_{H} \times Z^{2} (\frac{1}{{n_{1}}^{2}} - \frac{1}{{n_{2}}^{2}})$

therefore, for the given question

$\frac{1}{λ_{1}} = R \times [\frac{1}{1^{2}} - \frac{1}{3^{2}}] = \frac{8}{9} R$

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

$\Rightarrow \frac{λ_{1}}{λ_{2}} = \frac{27}{32}$

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