Question

ba mixture of hydrogen atoms in ground state and hydrogen like ions in their first excited state are being excited by same energy photons. After excitation,when they come directly to ground state wavelength of emitted light is in ratio 5 1 then find atomic number of ion and minimum value of energy of photon required to excite both hydrogen and hydrogen like ion

Answer 1

Dear Student,

$$\begin{gathered} \mathrm{We}\mathrm{know}\mathrm{that}, \\ {\mathrm{E}}_{\mathrm{n}}=-13.6\frac{{\mathrm{Z}}^2}{{\mathrm{n}}^2} \\ \mathrm{and},\mathrm{E}=\frac{\mathrm{hc}}{\mathrm{\lambda }} \\ \mathrm{So},\frac{1}{\mathrm{\lambda }}=\frac{-13.6}{\mathrm{hc}}\frac{{\mathrm{Z}}^2}{{\mathrm{n}}^2} \\ \mathrm{Let}{\mathrm{\lambda }}_1\mathrm{be}\mathrm{wavelength}\mathrm{emitted}\mathrm{by}\mathrm{H}-\mathrm{atom}\mathrm{and}{\mathrm{\lambda }}_2\mathrm{be}\mathrm{wavelength}\mathrm{emitted}\mathrm{by}\mathrm{H}-\mathrm{like}\mathrm{atomTherefore}, \\ \frac{{\mathrm{\lambda }}_1}{{\mathrm{\lambda }}_2}=\frac{{{\mathrm{n}}_1}^2}{{{\mathrm{Z}}_1}^2}.\frac{{{\mathrm{Z}}_2}^2}{{{\mathrm{n}}_2}^2} \\ \mathrm{n}\mathrm{will}\mathrm{also}\mathrm{be}\mathrm{same}\mathrm{as}\mathrm{both}\mathrm{are}\mathrm{excited}\mathrm{to}\mathrm{same}\mathrm{excited}\mathrm{state} \\ \mathrm{So}, \\ \frac{{\mathrm{\lambda }}_1}{{\mathrm{\lambda }}_2}=\frac{{{\mathrm{Z}}_2}^2}{{{\mathrm{Z}}_1}^2} \\ \mathrm{or},\frac{5}{1}=\frac{{{\mathrm{Z}}_2}^2}{1^1} \\ \mathrm{or},{{\mathrm{Z}}_2}^2=5 \\ \mathrm{or},{\mathrm{Z}}_2=2.2\approx 2 \\ \mathrm{For}\mathrm{minimum}\mathrm{value}\mathrm{of}\mathrm{energy},\mathrm{n}=2 \\ \mathrm{So},{\mathrm{E}}_1=-13.6\frac{1^2}{2^2}=-3.4\mathrm{eV} \\ {\mathrm{E}}_2=-13.6\frac{2^2}{2^2}=-13.6\mathrm{eV} \end{gathered}$$