Question

A single electron ion has charge+Ze where Z is atomic number and e is electronic charge. it requires 16.52eV to excite the electron from the second Bohr orbit to third Bohr orbit. find

- Atomic number of element

- The energy required for transition of electron from first to third orbit

- Wavelength of photon required to remove electron from first Bohr orbit to infinity

- The Kinetic Energy of electron in first Bohr orbit

Answer 1

Dear Student,

$$\begin{gathered} \left(\mathrm{a}\right)1\mathrm{eV}=1.602\times {10}^{-12}\mathrm{erg} \\ \mathrm{Also},∆\mathrm{E}=\frac{\mathrm{hc}}{\mathrm{\lambda }}={\mathrm{E}}_3-{\mathrm{E}}_2 \\ ={\mathrm{R}}_{\mathrm{H}}.\mathrm{c}.\mathrm{h}.{\mathrm{Z}}^2[\frac{1}{2^2}-\frac{1}{3^2}] \\ \mathrm{or},16.52\times 1.602\times {10}^{-12}=109678{\mathrm{Z}}^2[\frac{1}{2^2}-\frac{1}{3^2}]\times 3\times {10}^{10}\times 6.626\times {10}^{-27} \\ {\mathrm{Z}}^2=\frac{16.52\times 1.602\times {10}^{-12}}{109678\times 3\times {10}^{10}\times 6.626\times {10}^{-27}}\times \frac{36}{5} \\ {\mathrm{Z}}^2=8.7400 \\ \mathrm{or},\mathrm{Z}\approx 3 \\ \left(\mathrm{b}\right)∆\mathrm{E}={\mathrm{E}}_3-{\mathrm{E}}_1={\mathrm{R}}_{\mathrm{H}}.\mathrm{c}.\mathrm{h}.{\mathrm{Z}}^2[\frac{1}{1^2}-\frac{1}{3^2}] \\ =109678\times 3\times {10}^{10}\times 6.626\times {10}^{-27}\times 3^2\times \frac{8}{9} \\ =17.4\times {10}^{-11}\mathrm{erg} \\ \left(\mathrm{c}\right)\frac{1}{\mathrm{\lambda }}={\mathrm{R}}_{\mathrm{H}}.{\mathrm{Z}}^2[\frac{1}{1^2}-\frac{1}{{\infty }^2}] \\ \frac{1}{\mathrm{\lambda }}=109678\times 9 \\ \mathrm{or},\mathrm{\lambda }=1.01\times {10}^{-6}\mathrm{cm} \\ \left(\mathrm{d}\right)\mathrm{K}.\mathrm{E}.=\frac{1}{2}{\mathrm{mv}}^2 \\ =\frac{1}{2}\mathrm{m}\left(\frac{2{\mathrm{\pi Ze}}^2}{\mathrm{nh}}\right)=\frac{2{\mathrm{\pi }}^2{\mathrm{Z}}^2{\mathrm{e}}^4\mathrm{m}}{{\mathrm{n}}^2{\mathrm{h}}^2} \\ =\frac{2\times (3.14{)}^2\times 3^2\times (4.803\times {10}^{-10}{)}^4\times 9.108\times {10}^{-28}}{1^2\times (6.625\times {10}^{-27}{)}^2} \\ =2.17\times {10}^{-11}\mathrm{erg} \end{gathered}$$