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Energy Levels and Transitions

For an isolated atom, the electrons are in various orbitals. The atom as a whole has a certain energy, depending on what orbitals are occupied by electrons. Quantum theory predicts that only certain energies are allowed for a given atom.
For instance, a hydrogen atom can have only energies corresponding to the equation


where n is a positive integer. There is no such thing as an isolated hydrogen atom with energy –1.5 x 10–18 J or with energy –1.1 x 10–18 J, since they do not fit into the formula. This can be shown graphically on an energy level diagram (Figure 16-1), in which the vertical axis represents energy and the horizontal axis does not represent anything. Zero energy corresponds to the state in which the proton and electron are infinitely separated. Bringing the proton and electron together releases energy, so the energies for all the other states are negative.

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Figure 16-1


Even though the atom cannot have energies between the allowed energy states shown on the diagram, the atom can jump, or make a transition, from one state to another. It does this either by colliding with another atom or by absorbing or releasing a photon. The state corresponding to the lowest energy is called the ground state. Other states are excited states.
In previous chapters we discussed light in terms of a wave of disturbance of electric and magnetic fields. Quantum theory maintains that these disturbances come in little packets called photons. If there are a large number of photons, then they act like the classical field discussed in Chapter 13. At any rate, there is a connection between the frequency of the light and the energy of a photon of that light, so that

where h = 6.63 x 10–34 J/Hz is a constant of nature, Planck's constant.
If a photon is emitted or absorbed by an atom, then the photon has an energy given by the difference of the energies of the atomic states. Several examples will make this more clear.



A beam of laser light is incident on a sample of hydrogen gas. Most of the gas in the sample is in the ground state (lowest energy state). To what frequency should the laser be tuned so that the hydrogen atoms absorb the light and end up in the second excited state?



The ground state corresponds to n = 1, so its energy is


E1 = –2.18 x 10–18 J

The second excited state must have n = 3, so its energy is


E3 = –0.24 x 10–18 J

The energy of the photons in the laser are (–0.24 x 10–18 J) – (–2.18 x 10–18 J) = 1.94 x 10–18 J. Thus the frequency of the photons is


Transitions are shown in an energy level diagram by arrows from one line to another. Figure 16-1 shows the transition for Example 1.
Several things can happen after an atom has absorbed a photon. It can decay back to its previous state, emitting a photon of (almost) the same energy. This process, called scattering, is shown in Figure 16-2, in which the up arrow represents absorption and the down arrow, subsequent emission. Figure 16-3 shows the process differently, where absorption occurs between steps 1 and 2 and emission between steps 2 and 3.

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Figure 16-2

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Figure 16-3

An atom which has absorbed a photon may emit a photon of a different energy by making a transition to a new state. This is shown in Figure 16-4. The upward arrow represents the absorption of the original photon. The downward arrows represent emitted photons, so the atom emits two photons for each one it absorbs. This phenomenon, in which a substance absorbs one frequency of light and emits light of different frequencies, is called fluorescence.

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Figure 16-4

Do not get thrown off by notation. Sometimes the ground state energy is labeled E0 and sometimes E1. Also, sometimes we say the ground state has zero energy, but sometimes zero energy corresponds to complete ionization and the ground state energy is negative. It is a matter of convention which the problem will specify. The ground state is always the lowest-energy state.

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