The ratio of the wavelengths of the light absorbed by a hydrogen atom when it undergoes and transitions, respectively, is [2025]

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Solution

Q.
The ratio of the wavelengths of the light absorbed by a hydrogen atom when it undergoes $n = 2 \to n = 3$ and $n = 4 \to n = 6$ transitions, respectively, is [2025]

1 $\frac{1}{9}$

2 $\frac{1}{4}$

3 $\frac{1}{36}$

4 $\frac{1}{16}$

Ans.
(2)

From Rydberg’s formula

$\frac{1}{λ} = R_{H} [\frac{1}{n_{i}^{2}} - \frac{1}{n_{f}^{2}}]$

For $n_{i} = 2, n_{f} = 3,$

$\frac{1}{λ_{1}} = R_{H} [\frac{1}{2^{2}} - \frac{1}{3^{2}}] = R_{H} [\frac{5}{36}]$ ...(i)

For $n_{i} = 4, n_{f} = 6,$

$\frac{1}{λ_{2}} = R_{H} [\frac{1}{4^{2}} - \frac{1}{6^{2}}] = R_{H} [\frac{1}{16} - \frac{1}{36}] = R_{H} [\frac{5}{144}]$ ...(ii)

On dividing eqn. (ii) by (i), we get

$\frac{λ_{1}}{λ_{2}} = R_{H} [\frac{5}{144}] \times \frac{1}{R_{H}} [\frac{36}{5}]$

$\frac{λ_{1}}{λ_{2}} = \frac{1}{4}$

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