The energy-level scheme for the hypothetical one- electron...

The energy-level scheme for the hypothetical one- electron element Searsium is shown in Fig. E39.30. The potential energy is taken to be zero for an electron at an infinite distance from the nucleus. (a)How much energy (in electron volts) does it take to ionize an electron from the ground level? (b) An 18 -eV photon is absorbed by a Searsium atom in its ground level. As the atom returns to its ground level, what possible energies can the emitted photons have? Assume that there can be transitions between all pairs of levels. (c) What will happen if a photon with an energy of 8 eV strikes a Searsium atom in its ground level? Why? (d) Photons emitted in the Searsium transitions $n=3 \rightarrow n=2$ and $n=3 \rightarrow n=1$ will eject photoelectrons from an unknown metal, but the photon emitted from the transition $n=4 \rightarrow n=3$ will not. What are the limits (maximum and minimum possible values) of the work function of the metal?

The energy-level scheme for the hypothetical one- electron element Searsium is shown in Fig. E39.30. The potential energy is taken to be zero for an electron at an infinite distance from the nucleus.
(a)How much energy (in electron volts) does it take to ionize an electron from the ground level?
(b) An 18 -eV photon is absorbed by a Searsium atom in its ground level. As the atom returns to its
ground level, what possible energies can the emitted photons have? Assume that there can be transitions between all pairs of levels.
(c) What will happen if a photon with an energy of 8 eV strikes a Searsium atom in its ground level? Why? (d) Photons emitted in the Searsium transitions $n=3 \rightarrow n=2$ and $n=3 \rightarrow n=1$ will eject photoelectrons from an unknown metal, but the photon emitted from the transition $n=4 \rightarrow n=3$ will not. What are the limits (maximum and minimum possible values) of the work function of the metal?

Key Concepts

Photoelectric Effect and Work Function

The photoelectric effect involves the ejection of electrons from a material when it is exposed to light. The work function is the minimum energy needed to remove an electron from the surface of the material. In experiments involving the photoelectric effect, understanding the work function is crucial, as only photons whose energy exceeds this threshold can successfully liberate electrons, a concept that has important implications in both atomic and solid-state physics.

Energy Conservation in Atomic Transitions

Energy conservation in atomic transitions ensures that the total energy before and after a photon absorption or emission event remains constant. The energy absorbed by an electron must equal the difference in energy between the initial and excited states, and similarly, the energy of an emitted photon equals the energy drop when the electron transitions to a lower level. This principle is fundamental to analyzing atomic processes.

Atomic Energy Levels

Atomic energy levels are the quantized states that electrons can occupy in an atom. Each level corresponds to a specific energy, and the energy difference between these levels represents the amount of energy an electron must absorb or emit to transition between them. This quantization is a cornerstone of quantum mechanics and explains the discrete spectral lines observed in atomic spectra.

Ionization Energy

Ionization energy refers to the minimum amount of energy required to remove an electron from an atom or ion, taking it from a bound (negative energy) state to a free (zero or positive energy) state. It is a fundamental concept in atomic physics that is tied to the potential energy reference level, which is often set at zero when the electron is at an infinite distance from the nucleus.

Photon Absorption and Emission

Photon absorption occurs when an atom takes in a photon whose energy matches the gap between two energy levels, causing an electron to jump to a higher, excited state. Conversely, photon emission happens when an electron returns to a lower energy level, releasing a photon whose energy equals the difference between the two levels. These processes are governed by the conservation of energy and result in the characteristic spectra of elements.

Transcript

00:01 In this problem, we're going to talk about atomic transition.

00:03 So consider that we have an atom that has several possible energy lines, energy levels, i'm sorry.

00:11 So i'm going to just write two energy levels here, e1 and e2, but this can be extended to more energy levels.

00:22 And let's say that the atom is initially in the ground state, that is the smallest energy, and that is e1.

00:29 Now, a transition occurs in the atom when the atom absorbs a photon, and with that, the energy is increased.

00:43 So the atom must absorb a photon that has an energy, gamma, that is equal to the difference between two energy levels, e2 minus e1, in this case.

00:54 Okay.

00:57 And in order to transition back to the ground state, the atom must emit a photon that has this same energy here as before.

01:07 E gamma.

01:10 Also, in order to completely ionize the atom, we need the final energy of the electron, the electron that is ionized, that is thrown out of the atom, to be at least equal to zero.

01:25 So the final energy of the atom is equal to the initial energy, that's e1, plus the energy of the incoming photon.

01:34 This must be greater than zero, so e gamma must be greater than minus e1.

01:43 So now we can move on to our exercise.

01:45 So what we have is an atom that has these energy levels here.

01:54 And in question a, we have to find what must be the energy of a photon.

02:00 Such that an atom initially at the ground state is ionized.

02:06 I'll notice that since egama must be greater than minus e1, greater or equal than actually, then igama must be greater than the ground state energy level, that's 20 electron volts.

02:24 So the minimum energy is 20 electron volts.

02:28 That's the answer to question a.

02:31 In question b, we have, a photon, a photon that has an energy of 18 electron volts.

02:47 And the atom is initially in the ground state.

02:51 So e1 is equal to minus 20 electron.

02:56 In this case, notice that the final energy is equal to the initial energy, e1 plus the energy of the photon.

03:04 And this is equal to minus 20 plus 18.

03:08 So this is minus two electron volts.

03:13 Now minus two electron volts corresponds to n equals 4.

03:18 So this is e4.

03:20 So the atom is now at the n equals 4 energy level.

03:26 And we want the atom to transition back to the ground state.

03:29 So from n equals 4, we want the atom to go to n equals 1.

03:35 And we want to know what energies can be emitted by the fold.

03:40 In this transition.

03:43 So notice that for the atom to go from n equals 4 to n equals 1, there are several possible steps.

03:50 You can go from 4 to 3 to 2 to 1.

03:54 You can go from 4 to 3 to 1, from 4 to 2 to 1, or from 4 directly to 1.

04:05 So notice that the energies, the possible energies emitted by the photon, i'm sorry, the possible energies of the photon emitted by the atom is equal to e4 minus e3...

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