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Energy Levels in an Atom for AP Physics B

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Practice problems for these concepts can be found at:

Atomic and Nuclear Physics Practice Problems for AP Physics B

The electrons in an atom can only have certain, specific amounts of energy. This is an unusual concept—if an electron is whirling around an atom with a given amount of energy, couldn't you just nudge it slightly to give it a tiny bit more energy? No, you can't. To understand what we mean, look at Figure 25.1.

In Figure 25.1, we've drawn a few of the energy levels of a hypothetical atom. Let's start by looking at E1. This is the lowest energy level, and it is called the ground state energy of the atom. When an electron is sitting as close to the nucleus as possible, and when it's completely unexcited, it will have the energy of E1, -10 eV.

To increase the energy of the electron—to move the electron into a higher energy level—energy must somehow be transferred to the electron. This energy transfer is done by a photon. To jump up in energy, an electron absorbs a photon, and to drop down in energy, an electron emits a photon.

The energy diagram in Figure 25.1 tells us that to get to E2 from the ground state, an electron must absorb a photon carrying 3 eV—and only 3 eV!—of energy. If a photon with an energy of 2.9 eV comes along and knocks into the electron, nothing will happen. Similarly, if a photon with an energy of 3.1 eV comes along, nothing will happen. It's all or nothing; either the electron gets just the right amount of energy to go from one energy level to the next, or it doesn't.

How does the electron get from the ground state to E3? There are two ways. The electron could first absorb a photon with an energy of 3 eV, taking it from E1 to E2, and then it could absorb another photon with an energy of 1 eV, taking it from E2 to E3. Or, the electron could start in E1 and simply absorb a photon with an energy of 4 eV.

We've been talking about photons having certain energies, but we haven't yet told you how to figure out the energy of a photon. Here's the formula:

This formula tells us that the energy of a photon is equal to Planck's constant, h, which is 6.63 × 10-34 J·s (this value is given to you on the constants sheet), multiplied by the frequency of the photon. You should remember from Chapter 23 that the frequency of a wave is related to the wavelength by the formula

v = λf

For light, the velocity is c, or 3 . 108 m/s, so we can instead write

c = λf

This means that we can rewrite the equation for the energy of a photon to read

These formulas tell us that a photon with a high frequency, and therefore with a small wavelength, is higher in energy than a photon with a low frequency and long wavelength. So gamma rays, for example, are a lot higher energy than radio waves because gamma rays have a higher frequency.

Okay, let's try these formulas with a problem.

This is a simple plug-and-chug problem. Figure 25.1 gives us the energy levels, and we just need to use our formula to find the wavelength of the absorbed photon. But what value do we use for the energy? -10 eV? -7 eV? Wrong… the value we use is the energy of the jump, not the energy of one of the states. To get from E1 to E2, an electron must absorb 3 eV of energy, so that's the value we use in the formula:

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