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 \fracx9 \times 10^15 ~Hz , when x=\underline\hspace2cm .

Atoms and Nuclei PYQ — JEE Main 2024 · 1 Feb · Shift 2 · Q90 with Solution

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Correct answer: 32

For a hydrogen-like ion, the frequency emitted in a transition from $n_i$ to $n_f$ is

\nu = RcZ^2\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)

where $R$ is the Rydberg constant and $Z$ is the atomic number of the ion.

\nu_{21} = RcZ^2\left(1-\frac{1}{4}\right)=RcZ^2\cdot \frac{3}{4}

Given:

\nu_{21}=3\times 10^{15}\,\text{Hz}

So,

RcZ^2\cdot \frac{3}{4}=3\times 10^{15}

RcZ^2=4\times 10^{15}

\nu_{31}=RcZ^2\left(1-\frac{1}{9}\right)=RcZ^2\cdot \frac{8}{9}

Substitute $RcZ^2=4\times 10^{15}$ :

\nu_{31}=4\times 10^{15}\cdot \frac{8}{9}=\frac{32}{9}\times 10^{15}\,\text{Hz}

\nu_{31}=\frac{x}{9}\times 10^{15}\,\text{Hz}

Thus,

x=32