The recoil speed of a hydrogen atom after it emits a photon in going from n=5 state to n=1 state will be _________m/s.  Given, h=  $6.63\times {10}^{-34}\mathrm{Js}$, R=$1.097\times {10}^7{\mathrm{m}}^{-1}$.

(Round off to the nearest integer)

$\left(\mathrm{\Delta E}\right)\text{ Releases when photon going from }$$n=5\text{ to }n=1$

$\mathrm{\Delta }E=(13.6-0.54)\mathrm{eV}=13.06\mathrm{eV}$

$P_i=P_f\text{ (By linear momentum conservation) }$

$\begin{array}{l}0=\frac{\mathrm{h}}{\lambda }-\mathrm{Mv}\\ ={\mathrm{V}}_{\text{Recoil }}=\frac{\mathrm{h}}{\lambda \mathrm{M}}\cdots .\left(\mathrm{i}\right)\\ \mathrm{\&} \mathrm{\Delta E}=\frac{\mathrm{hc}}{\lambda }=\frac{\mathrm{hc}}{\lambda \mathrm{M}}\times \mathrm{M}\\ \Rightarrow {\mathrm{\Delta E}=\mathrm{McV}}_{\text{Recoil }}\\ \Rightarrow {\mathrm{V}}_{\text{Recoil }}=\frac{\mathrm{\Delta E}}{\mathrm{Mc}}\\ =\frac{13.06\times 1.6\times {10}^{-19}}{1.67\times {10}^{-27}\times 3\times {10}^8}\\ =4.17\mathrm{m}/\sec \end{array}$