JEE Main & Advanced Physics Atomic Physics Transition of Electron

When an electron makes transition from higher energy level having energy \[{{E}_{2}}({{n}_{2}})\] to a lower energy level having energy \[{{E}_{1}}({{n}_{1}})\] then a photon of frequency n is emitted

(1) Energy of emitted radiation

\[\Delta E={{E}_{2}}-{{E}_{1}}\]\[=\frac{-Rc\,h\,{{Z}^{2}}}{n_{2}^{2}}-\left( -\frac{Rch\,{{Z}^{2}}}{n_{1}^{2}} \right)\]

\[=13.6{{Z}^{2}}\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)\]

(2) Frequency of emitted radiation                        

\[\Delta E=h\nu \]\[\Rightarrow \]\[\nu =\frac{\Delta E}{h}=\frac{{{E}_{2}}-{{E}_{1}}}{h}=Rc\,{{Z}^{2}}\,\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)\]

(3) Wave number/wavelength

Wave number is the number of waves in unit length \[\bar{\nu }=\frac{1}{\lambda }=\frac{\nu }{c}\]\[\Rightarrow \frac{1}{\lambda }=R{{Z}^{2}}\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)=\frac{13.6{{Z}^{2}}}{hc}\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)\]

(4) Number of spectral lines : If an electron jumps from higher energy orbit to lower energy orbit it emits raidations with various spectral lines.

If electron falls from orbit \[{{n}_{2}}\] to \[{{n}_{1}}\] then the number of spectral lines emitted is given by

\[{{N}_{E}}=\frac{({{n}_{2}}-{{n}_{1}}+1)({{n}_{2}}-{{n}_{1}})}{2}\]

If electron falls from \[{{n}^{th}}\] orbit to ground state (i.e. \[{{n}_{2}}=n\] and \[{{n}_{1}}=1\]) then number of spectral lines emitted \[{{N}_{E}}=\frac{n\,(n-1)}{2}\]

(5) Recoiling of an atom : Due to the transition of electron, photon is emitted and the atom is recoiled Recoil momentum of atom = momentum of photon \[=\frac{h}{\lambda }=hR{{Z}^{2}}\,\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)\]

Also recoil energy of atom \[=\frac{{{p}^{2}}}{2m}\]\[=\frac{{{h}^{2}}}{2m{{\lambda }^{2}}}\]   (where \[m=\] mass of recoil atom)