A particular hydrogen like ion emits radiation of frequency $2.92\times {10}^{15} \mathrm{Hz}$ when it makes transition from n = 3 to n = 1. The frequency in Hz of radiation emitted in transition from n = 2 to n = 1 will be

The frequency of the emitted radiation by a particular hydrogen like ion when it makes transition from ${\mathrm{n}}_1={\mathrm{n}}_{\mathrm{initial}}$ to ${\mathrm{n}}_2={\mathrm{n}}_{\mathrm{final}}$ is given by

        $\mathrm{f}={\mathrm{RZ}}^2\mathrm{c}\left(\frac{1}{{\mathrm{n}}_2}-\frac{1}{{\mathrm{n}}_1}\right)$

where, c = velocity of light

and R = Rydberg’s constant.

For H-atom, Z = 1

So, for transition from n = 3 to n = 1,

          ${\mathrm{f}}_1=\mathrm{Rc}\left(1-\frac{1}{3^2}\right) ...\left(\mathrm{i}\right)$

Similarly for transition from n = 2 to n = 1,

         ${\mathrm{f}}_2=\mathrm{Rc}\left(1-\frac{1}{2^2}\right) ...\left(\mathrm{ii}\right)$

Dividing Eq. (i) by Eq. (ii), we get

          $\frac{{\mathrm{f}}_1}{{\mathrm{f}}_2}=\frac{\mathrm{Rc} \left(1-\frac{1}{9}\right)}{\mathrm{Rc} \left(1-\frac{1}{4}\right)}=\frac{\frac{8}{9}}{\frac{3}{4}}=\frac{32}{27}$

$\begin{array}{l}\mathrm{Here}, {\mathrm{f}}_1=2.92\times {10}^{15}\mathrm{Hz}\\ \Rightarrow \frac{2.92\times {10}^{15}}{{\mathrm{f}}_2}=\frac{32}{27}\\ \Rightarrow {\mathrm{f}}_2=\frac{2.92\times {10}^{15}\times 27}{32} \left(\mathrm{given}\right)\\ =2.46\times {10}^{15}\mathrm{Hz}\end{array}$