A particular hydrogen like ion emits the radiation of frequency Hz when it makes transition from to . The frequency of radiation emitted in transition from to is Hz, when

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Solution

Q.
A particular hydrogen like ion emits the radiation of frequency $3 \times 10^{15}$ Hz when it makes transition from $n = 2$ to $n = 1$ . The frequency of radiation emitted in transition from $n = 3$ to $n = 1$ is $\frac{x}{9} \times 10^{15}$ Hz, when $x =$ __________ . [2024]

Ans.
(32) $E = - 13.6 Z^{2} (\frac{1}{n_{i}^{2}} - \frac{1}{n_{f}^{2}})$

   $E = C (\frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}})$

$h ν = C [\frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}}]$

   $\frac{ν_{1}}{ν_{2}} = \frac{{[\frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}}]}_{2 - 1}}{{[\frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}}]}_{3 - 1}}$

   $= \frac{[\frac{1}{1} - \frac{1}{4}]}{[\frac{1}{1} - \frac{1}{9}]} = \frac{3 / 4}{8 / 9} = \frac{3}{4} \times \frac{9}{8}$

$\frac{ν_{1}}{ν_{2}} = \frac{27}{32}$

   $ν_{2} = \frac{32}{27} ν_{1} = \frac{32}{27} \times 3 \times 10^{15} Hz = \frac{32}{9} \times 10^{15} H z$

   $Given, f_{2} = \frac{x}{9} \times 10^{15} Hz$

   $\Rightarrow x = 32$

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