Transition of an Electron
When an electron makes transition from higher energy level having energy \({{E}_{2}}\,\left( {{n}_{2}} \right)\) to a lower energy level having energy \({{E}_{1}}\,\left( {{n}_{1}} \right)\) then a photon of frequency \(\nu \) is emitted.
1) Energy of emitted radiation:
\(\Delta E={{E}_{2}}-{{E}_{1}}=\frac{-Rch{{Z}^{2}}}{n_{2}^{2}}-\left( -\frac{Rch{{Z}^{2}}}{n_{1}^{2}} \right)=-13.6{{Z}^{2}}\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)\),
\(\Delta E=-13.6{{Z}^{2}}\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)\).
2) Frequency of emitted radiation:
\(\Delta E=h\nu \Rightarrow \nu =\frac{\Delta E}{h}=\frac{{{E}_{2}}-{{E}_{2}}}{h}=Rc{{Z}^{2}}\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)\).
3) Wave number/Wavelength:
Wave number is the number of waves in unit length, \(\overline{\nu }=\frac{1}{\lambda }=\frac{\nu }{c}\),
\(\Rightarrow \frac{1}{\lambda }=R{{z}^{2}}\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)=\frac{13.6{{Z}^{2}}}{hc}\left( \frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}} \right)\).
4) Number of spectral lines:
If an electron jumps from higher energy orbit to lower energy orbit it emits radiations with various spectral lines. If electron falls from orbit \({{n}_{2}}\) to \({{n}_{1}}\) then the number of spectral lines emitted is given by:
\({{N}_{E}}=\frac{\left( {{n}_{2}}-{{n}_{1}}+1 \right)\left( {{n}_{2}}-{{n}_{1}} \right)}{2}\),
If electron from \({{n}^{th}}\) orbit to ground state ( i.e. \({{n}_{2}}=n\) and \({{n}_{1}}=1\)) then the number of spectral lines emitted is: \({{N}_{E}}=\frac{n\left( n-1 \right)}{2}\).
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 an atom = \(\frac{{{p}^{2}}}{2m}=\frac{{{h}^{2}}}{2m{{\lambda }^{2}}}\) ; where, \(m=\)Mass of recoil atom.