A particular hydrogen - like ion emits the radiation of frequency 3 × 1 0 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 9

Question

A particular hydrogen - like ion emits the radiation of frequency 3 × 1 0 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 9 x ​ × 1 0 15 Hz , when x = _____.

TestHub · Accepted Answer

For Hydrogen like atom, energy emitted is $E = - 13.6 Z^{2} (\frac{1}{n _{i}^{2}} - \frac{1}{n _{f}^{2}})$

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

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

Therefore, the ratio $\frac{v _{1}}{v _{2}} = \frac{[ \frac{1}{n _{f}^{2}} - \frac{1}{n _{i}^{2}} ] _{2 - 1}}{[ \frac{1}{n _{f}^{2}} - \frac{1}{n _{i}^{2}} ] _{3 - 1}}$

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

$= \frac{3}{4} \times \frac{9}{8}$

$\Rightarrow \frac{v _{1}}{v _{2}} = \frac{27}{32}$

$\Rightarrow v_{2} = \frac{32}{27} v_{1} = \frac{32}{27} \times 3 \times 1 0^{15} Hz = \frac{32}{9} \times 1 0^{15} Hz$

Hence, $x = 32$ .

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