# Transition of an Electron

## 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.