# Wavelength of Photon emitted in De – Excitation

## Wavelength of Photon emitted in De – Excitation

According to Bohr, when an atom makes a transition from a higher energy level to a lower level, it emits a photon with energy equal to the energy difference between the initial and final levels. If Et is the initial energy of the atom before such a transition, Ef is the final energy after the transition, and the photon’s energy is hn = hc/ λ, then conservation of energy gives:

$$h\nu \,\,=\,\,\frac{hc}{\lambda }\,\,={{E}_{i}}-{{E}_{f}}$$ … (1)

Johann Balmer, a Swiss teacher found a formula that gives the wavelengths of these lines. This is now called the Balmer series. Balmer’s formula is:

$$\frac{1}{\lambda }\,\,=\,\,R\left( \frac{1}{{{2}^{2}}}-\frac{1}{{{n}^{2}}} \right)$$ … (2)

Where, n = 3, 4, 5 … etc.

R = Rydberg constant = 1.097 x 10⁷/ m,

λ = Wavelength of light/photon emitted during transition.

For n = 3, we obtain the wavelength of Hα line.

Similarly, for n = 4, we obtain the wavelength of Hβ line. For n = ∞, the smallest wavelength (= 3646⁰A) of this series is obtained. Using the relation $$E=\frac{hc}{\lambda }$$, we can find the photon energies corresponding to the wavelengths of the Balmer series.

$$E=\frac{hc}{\lambda }=hcR\left( \frac{1}{{{2}^{2}}}-\frac{1}{{{n}^{2}}} \right)=\frac{Rhc}{{{2}^{2}}}-\frac{Rhc}{{{n}^{2}}}$$.

This formula suggests that:

$${{E}_{n}}\,\,=\,\,\frac{Rhc}{{{n}^{2}}}$$, n = 1, 2, 3 … (3)

The wavelengths corresponding to other spectral series can be represented by formula similar to Balmer’s formula:

Lymen series: $$\frac{1}{\lambda }=R\left( \frac{1}{{{1}^{2}}}-\frac{1}{{{n}^{2}}} \right),\,\,n=2,3,4,…..$$.

Paschen series: $$\frac{1}{\lambda }=R\left( \frac{1}{{{3}^{2}}}-\frac{1}{{{n}^{2}}} \right),\,\,n=4,5,6,…..$$.

Bracket series: $$\frac{1}{\lambda }=R\left( \frac{1}{{{4}^{2}}}-\frac{1}{{{n}^{2}}} \right),\,\,n=5,6,7,……$$.

Pfund series: $$\frac{1}{\lambda }=R\left( \frac{1}{{{5}^{2}}}-\frac{1}{{{n}^{2}}} \right),\,\,n=6,7,8.$$.

The Lymen series is in the ultraviolet and the Paschen, Brackett and Pfund series are in the infrared region.