A particular hydrogen - like ion emits the radiation of frequency \(3 \times 10^{15} \mathrm{~Hz}\) when it makes transition from \(n=2\) to \(n=1\). The frequency of radiation emitted in transition from \(n=3\) to \(n=1\) is \(\frac{x}{9} \times 10^{15} \mathrm{~Hz}\), when \(x=\) = _____.

\(\mathrm{E}=\mathrm{C}\left(\frac{1}{\mathrm{n}_{\mathrm{f}}^2}-\frac{1}{\mathrm{n}_{\mathrm{i}}^2}\right)\)

\(\mathrm{h} \nu=\mathrm{C}\left[\frac{1}{\mathrm{n}_{\mathrm{f}}^2}-\frac{1}{\mathrm{n}_{\mathrm{i}}^2}\right]\)

\(\frac{v_1}{v_2}=\frac{\left[\frac{1}{n_f^2}-\frac{1}{n_i^2}\right]_{2-1}}{\left[\frac{1}{n_f^2}-\frac{1}{n_i^2}\right]_{3-1}}\)

\(=\frac{\left[\frac{1}{1}-\frac{1}{4}\right]}{\left[\frac{1}{1}-\frac{1}{9}\right]}=\frac{3 / 4}{8 / 9}\)

\(=\frac{3}{4} \times \frac{9}{8}\)

\(\frac{v_1}{v_2}=\frac{27}{32}\)

\(v_2=\frac{32}{27} v_1=\frac{32}{27} \times 3 \times 10^{15} \mathrm{~Hz}=\frac{32}{9} \times 10^{15} \mathrm{~Hz}\)