Understanding the Radiation Emitted During Hydrogen Atom Transitions: Excited States and EM Spectrum

The behavior of hydrogen atoms under different energy transitions is a fascinating area of study in atomic physics. When a hydrogen atom moves from one excited state to another, it emits or absorbs photons. This article delves into the specifics of a transition from the 3rd excited state to the 2nd excited state, revealing the type of radiation emitted and its place within the electromagnetic spectrum.

Excited States in Hydrogen Atom

In a hydrogen atom, the energy levels are quantized; this means they are discrete and not continuously varying. Each excited state is defined by a principal quantum number (n). For instance, the first excited state corresponds to n 2, the second excited state to n 3, and the third excited state to n 4. When an electron transitions between these states, it either emits or absorbs a photon whose energy matches the difference between the states.

Photon Energy Calculation

When a hydrogen atom transitions from the 3rd excited state (n 4) to the 2nd excited state (n 3), it releases a photon. The energy of this photon can be calculated using the Rydberg formula:

[ E -13.6 , text{eV} left( frac{1}{n_f^2} - frac{1}{n_i^2} right) ]

Here, ( n_f ) is the final energy level (n 3) and ( n_i ) is the initial energy level (n 4).

For this transition:n_i  4, n_f  3E  -13.6 , text{eV} left( frac{1}{3^2} - frac{1}{4^2} right)end{code>

Calculating the fractions:

[frac{1}{9} - frac{1}{16} frac{16 - 9}{144} frac{7}{144}]

Now calculate the energy:

[begin{align*}E approx -13.6 , text{eV} times frac{7}{144} approx -0.66 , text{eV}end{align*}]

Wavelength Calculation

To determine the wavelength of the emitted photon, we use the equation:

[ E frac{hc}{lambda} ]

Where:

[begin{align*}h 4.1357 times 10^{-15} , text{eV·s} c 3 times 10^8 , text{m/s}end{align*}]

Solving for λ:

[lambda approx frac{4.1357 times 10^{-15} , text{eV·s} times 3 times 10^8 , text{m/s}}{0.66 , text{eV}} approx 1.88 times 10^{-6} , text{m} approx 1880 , text{nm}]

This wavelength falls within the infrared region of the electromagnetic spectrum.

Conclusion

The transition from the 3rd excited state to the 2nd excited state in a hydrogen atom emits infrared radiation. This process showcases the direct relationship between energy differences within a hydrogen atom and the resultant electromagnetic radiation, highlighting the principles of quantized energy and the electromagnetic spectrum.

Keywords: hydrogen atom, excited states, electromagnetic spectrum, radiation, UV-Vis