Quantized energy levels

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Created by Keller Porter

The nucleus of each every atom creates an electric field, and it is composed of different levels, or stationary orbits, and each one requires a different energy level for an electron to reside there. The electrons in this electric field are in a bound state, requiring energy to be removed from their current energy level. These energy levels are considered quantized. Quantization is a transition from a classical understanding of physical principles to a more modern understanding.

Background

In the 1814, Joseph von Fraunhofer and William Hyde Wollaston discovered that when viewed closely, the spectrum from sunlight contained dark lines. These lines represented wavelengths of sunlight that were not reaching us. These wavelengths were being absorbed by the sun's atmosphere.

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A typical absorption spectrum graph.
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An emission spectrum graph for hydrogen.

If light is shone through a gas, the gas will absorb the specific wavelengths characteristic of the atoms in the gas. If the light were to be put through a prism of light or a diffraction grating, then there would be absorption lines, or places where the wavelength of light had been absorbed into the gas. This process creates something called an absorption spectrum. Similarly, if this same gas was heated to the right temperature, it would emit the same wavelengths that it absorbed before. Putting this emitted light through a prism or diffraction grating would create an emission spectrum. This is the opposite of an absorption spectrum because it shows the emission lines from the gas instead of the absorption lines.

Atom Energy Stability

In 1911, Rutherford came up with his model for the atom. It used all the same components of an atom that we know exist today, but it had one glaring issue. His model lacked stability. Classical electromagnetic theory said that the electrons surrounding the nucleus would quickly collapse because they were emitting electromagnetic waves, causing them to lose energy. If this were true, then the atom as we know it would not be able to exist.

This is a model of the Bohr atom. It shows different levels for the electrons to orbit the nucleus.


Bohr's model of the atom solved this problem. He proposed that the laws of classical mechanics must be reconsidered. His model said that the electron cloud had stationary orbits, a specific set of orbits for electrons. This differed from the assumption that the electron cloud was just a continuum where the electrons were free to orbit the nucleus. His model was similar to the solar system in that electrons orbit the nucleus like planets orbit the sun. Electrons are held in place by electrostatic forces, and planets are held in place by gravitational forces. The base energy level, called the ground state, is the first stationary orbit. From there, there can be many more levels. The energy of each level can be denoted [math]\displaystyle{ E_n }[/math], [math]\displaystyle{ n=1,2,3... }[/math]. At the ground state, the energy required to free the electron is greatest. It requires a specific level of energy to excite an electron to another energy level, and energy can be released to bring the electron back down to the ground state.

Energy Level Calculations

The energy of each level can be found using the formula:
[math]\displaystyle{ E_n = \frac{-2\pi^2me^4Z^2}{n^2h^2} }[/math]
where [math]\displaystyle{ m }[/math] is the mass of an electron, [math]\displaystyle{ e }[/math] is the magnitude of the electric charge, [math]\displaystyle{ n }[/math] is the quantum number, [math]\displaystyle{ h }[/math] is Planck's constant, and [math]\displaystyle{ Z }[/math] is the atomic number of the atom. This gives the energy for a specific atom at each energy level. The value will always be negative because electrons in the electron cloud are in a bounded state, so the potential energy is negative.

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This represents the different levels that electrons can jump between provided they acquire energy from either another electron or a photon.


While the value for each energy level is set in place, the energy of each individual electron can change. This can happen either by an interaction with another electron or a photon. For an electron, they can jump from the ground state to the fourth level if [math]\displaystyle{ E_{particle} \ge E_4-E_1 }[/math]

Example: Hydrogen Model

Research done by Johannes Rydberg showed that hydrogen has a ground level energy of [math]\displaystyle{ E_R = 13.6eV }[/math]. With Rydberg's constant, the initial formula for energy levels can be altered to include this. This new formula is:
[math]\displaystyle{ E_n = \frac{-E_R}{n^2} }[/math]

Basic Level Calculation

Find the energy level for the level [math]\displaystyle{ n=6 }[/math].

Solution: [math]\displaystyle{ E_6 = \frac{-13.6}{6^2} }[/math]

Level Change Calculation

Will an electron with energy [math]\displaystyle{ E_{electron} = 10.4eV }[/math] be able to bump a hydrogen electron in the ground state to the second level?

Solution: [math]\displaystyle{ E_1 = \frac{-13.6}{1} = -13.6eV }[/math]
[math]\displaystyle{ E_2 = \frac{-13.6}{2^2} = \frac{-13.6}{4} = -3.4eV }[/math]
[math]\displaystyle{ E_2 - E_1 = -3.4eV - -13.6eV = 10.2eV }[/math], so it requires [math]\displaystyle{ 10.2eV }[/math] to bump an electron from the ground state to the second level.
[math]\displaystyle{ 10.4eV \gre 10.2eV }[/math]
Yes, the electron has enough energy to bump the electron from the ground state to the second energy level.

Implications

Until quantization of atomic energy levels occurred, there were a few different experiments that had already been performed that did not make sense. Once researchers discovered the actual way electrons make up the electron cloud as well as the mechanisms that help them jump from level to level, these experiments could be explained. Readings for these three experiments can be found below.

  1. Blackbody Radiation
  2. Photoelectric Effect
  3. Emission Spectra of Atoms

References

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