Chapter 29 Lecture Notes

For all of these lecture notes files, I’m going to be using the associated chapter resource files on the 10164 website as a general outline. Our coverage of Chapter 29 will be short. We will briefly mention topics in 29.2, 29.4 and 29.5 and focus most of our coverage on 29.3.

Section 29.2 discusses blackbody radiation, which we will not cover, but it is a useful concept for Astronomers. It gives us a way to measure the temperature of an object by measuring the distribution of light energy (the spectrum) being emitted by that object. It also happens to prove the quantum nature of light (the wave-particle duality I talked about in Chapter 27).

Section 29.3 covers the photoelectric effect, which was another key milestone in proving that light behaves as a particle sometimes, not just a wave. Read through this section and practice on the three examples in section 29.3 (although the math is the last example is portrayed in a complicated, all-at-once sort of calculation, which I would instead do step-by-step). I also recommend the Interactive LearningWare 29.1, which is found directly after example 1.

Section 29.4 discusses in detail how it was discovered that particles of light have a momentum given by h/λ, where h is Planck’s constant (see page 1 of your formula sheet). We will use the momentum formula in a simple example, but I do not plan to cover it any further or ask any exam questions about photon momentum.

Section 29.5 recreates the two-slit experiment I discussed back in the Chapter 27 lecture notes, but sadly, the book doesn’t take it into the interesting, reality-bending direction that occurs when you start playing around with the setup, like putting a detector over one slit to change the behavior of light or electrons passing through the other slit, etc. Other than that, there is nothing in this section directly relevant to the homework or exam 4.

Once you have read through these sections of the book and practiced a bit with section 29.3, please proceed.

The photoelectric effect is a phenomenon occurs when a metal containing free electrons is illuminated by a light source, as in the diagram below.

In this case, the metal is Potassium. The electrons inside the metal are free to move around within the metal. However, they are NOT free to simply drift away outside of the metal. There is an energy barrier, called the Work Function, that prevents this.

Imagine you are in a fenced-in yard. In order to escape the yard, you would need to be able to climb the fence. If the fence has a height h, then in order to climb the fence, you would have to do work. How much work is needed to climb the fence? If we take ground level to be zero energy, then your gravitational potential energy at the top of the fence would be given by mgh.

So if you do +mgh work and gravity does -mgh work as you climb the fence, then you will make it to the top of the fence and get out. We call this kind of situation a “potential well.” There is an energy barrier you have to overcome in order to get out of the fenced-in yard. Electrons have this same problem inside a metal. There is a fence that has a height that is equivalent to a small amount of energy.

For Potassium, the energy barrier (the Work Function) is 2 eV, so if something gives electrons 2 eV energy or more, then an electron right at the surface of the metal would be able to “jump the fence.” Electrons deeper inside the metal would need a little more energy to get out.

So if you were to give EVERY electron in the metal, say, 4 eV, then the electrons right at the surface would get out and still have 2 eV energy left over (as kinetic energy). Electrons further inside the metal might only have 1.5 eV or 0.8 eV or 0.1 eV left over when they get out.

We can express this situation as follows: To solve for the MAXIMUM Kinetic Energy an electron can have upon escape the metal, we say:

(KE)max = Energy input - Work function

In the case of the photoelectric effect, the energy input is given by a photon of energy Eɣ = hc/λ or hf.

The key point here is that the energy of a photon is given by its wavelength. The amount of energy received by the electrons in the metal is NOT related to the intensity of the light. That’s because light is not behaving strictly like a wave here. Since light is interacting with individual particles, light acts like a particle (or wave packet).

In the previous diagram, for example, the red light shining on the Potassium surface has a photon energy of 1.77 eV, which is less than the Work Function of the Potassium. Since an electron can only absorb one photon at a time, and that’s not enough energy to get out, no electrons can escape from the Potassium, even if the intensity if turned way up.

But if we shine blue light on the Potassium, even with a very low intensity, electrons will escape with a variety of energies. For simplicity, we will focus only on the electrons that escape with the maximum amount of KE, the ones that were right on the surface of the metal (instead of deep inside the metal).

Once the input energy is larger than the Work Function, light will escape from the metal with a maximum KE directly proportional to the difference between the input energy and the work function, making a simple linear graph.

The frequency where this line intersects the x-axis, f0, is called the cut-off frequency. Any light with a frequency below the cut-off frequency (or conversely above the cut-off wavelength) will not have enough energy to allow electrons to escape from the metal.

When doing calculations with energy and light, if you use the equation E = hc/λ, where h and c are constants found on our formula sheet and λ is the wavelength of the light in meters, then you will get an Energy in mks units (Joules). Since photons in this type of experiment (and Work Functions) are typically very small, we usually work instead in units of eV.

We will first practice with energy and intensity of light, doing a worksheet problem somewhat similar to the problem you should have already worked through with Interactive LearningWare 29.1. The concepts in Example 1 should also be helpful for this kind of problem.

Please attempt worksheet 29.1 now. Once you have finished it or at least seriously attempted it and gotten stuck, please proceed to the next page to see my detailed solution.

If you have any questions about the way I solved worksheet 29.1, please post to the “Chapter 29-32 worksheets” discussion forum on the Physics 10164 course shell. I will be checking this forum often to answer questions.

There is similarly a forum for questions about the “Chapter 29-30 Homework” for the same purpose, and I plan to have two forums for each of the six remaining homework assignments this semester, shared by all three classes, on the same course shell.

—That first equation I refer to in the solution, the “counting equation” is not an equation found on any formula sheet. It is more of a definition or a common-sense equation that applies under many different circumstances.

For example, what is the total mass of all of the students in a classroom? Let N be the number of students and m(student) by the average mass of a student.

M(total) = N(students)*m(student)

How many atoms of Carbon is present in a 1.5 gram sample of Carbon-12?

M(total) = N(carbon)*m(carbon)

or N(carbon) = M(total)/m(carbon)

If you know the mass of a single atom of Carbon-12 in grams (which we can look up … more on this in Chapter 30), then you can solve for N.

—Next, I would like for you to try to do a simple example of a photoelectric effect problem. The main consideration here is to make sure that all of your energies are in the same units, whether that is Joules or eV. Please try worksheet 29.2 and note that I use the symbol Φ to represent the Work Function.

Once you have finished it or at least seriously attempted it and gotten stuck, please proceed to the next page to see my detailed solution.

If you have any questions about worksheet 29.2, please post to the appropriate “Chapter 29-30 worksheets” forum on d2l.tcu.edu, and I will respond there (or send me an email).

—If you would like more practice with photoelectric effect and photon energy problems, I have some videos available from old exam questions:

https://www.youtube.com/watch?v=VCfUtRy-IB4https://www.youtube.com/watch?v=Fpi6XhU-zPQhttps://www.youtube.com/watch?v=Du9yF6AVINEhttps://www.youtube.com/watch?v=_4eLQYwTTsEhttps://www.youtube.com/watch?v=pAgCt2mr7wg

—For the last worksheet problem in this chapter, I am giving you another counting-type problem which uses the formula for the momentum of a photon.

Remember for this problem that the momentum of a macroscopic object (like a ping pong ball) is given by the formula p = mv, and for power, we can use

Power (Watts) = Energy (Joules) / time (sec)

Try worksheet 29.3 on your own and see if you can master these concepts. Once you have finished it or at least seriously attempted it and gotten stuck, please proceed to the next page to see my detailed solution.

If you have any questions about worksheet 29.3, please post to the appropriate “Chapter 29-32 worksheets” forum on d2l.tcu.edu, and I will respond there (or send me an email).

—

Chapter 30 Lecture Notes

We are only covering sections 30.2 and 30.3 in this chapter, although 30.1 is an interesting read (and we will do an example based on this later on in Chapter 31).

I encourage you first to read through Chapter 30.2, and try to use the formula for calculating the wavelengths of light emitted from or absorbed by Hydrogen atoms to successfully work through example 2. You will find this formula on the last page of your formula sheet.

In section 30.3, animated figure 30.5 is a helpful little 2 minute video that shows the model of the atom we will be using. This model is completely fake, of course, as it tries to overlay a classical view on to what’s happening inside an atom. But it does give an accurate set of equations for describing the behavior of the atom, despite being conceptually wrong.

For that reason, I do not think it is worthwhile for us to delve into the derivation of the Bohr model and where our equation for the Hydrogen spectrum originates. Therefore, I encourage you to skip the part of the chapter subtitled “The energies and radii of the Bohr orbits” and go to the next subsection entitled “Energy level diagrams” which makes use of the final equation in the skipped section, equation 30.13.

Check out figure 30.8 and the associated discussion, then skip example 3 and go straight to “The Line Spectra of the Hydrogen Atom.” Example 4 will be useful for you to work through as well, and then that’s all we are doing with this chapter.

When you have finished this coverage of the chapter and done some of the practice WileyPlus problems recommended in the Chapter 29-30 Resources file, proceed to the next page where I will summarize what you need to know and make some comments.

Remember our discussion of a “potential well?” That’s what the electron is stuck in when it is in a Hydrogen atom. The positively charged nucleus provides an electric force that keeps the electron nearby. If the electron wants to move further away from the nucleus or escape the nucleus altogether, we will need to find a way to give the electron energy so that it can climb up out of this potential well.

This is analogous to humans on the surface of the Earth. You can also say we are in a potential well. If we want to move upwards, away from the center of the Earth, we need to find a way to add energy to ourselves so that we can do that, either by jumping or climbing or riding on some kind of propulsion device.

We have a negative potential energy, from a Newtonian perspective, and if we want to escape from the Earth altogether, someone will need to give us enough energy so that our total positive energy cancels out the negative potential energy we have on the ground, and then we will have escaped.

If we want to just get a little further from the surface of the Earth, we could gain some amount of energy +mgh and move to a higher altitude, further from the Earth’s center. We would say we now have LESS negative energy. Likewise, if electrons get a little bit of energy, they can move a little further away from the nucleus. Their energy is a little LESS negative as they get further away from the nucleus.

The difference is that while we can exist at any distance from the Earth’s center without difficulty, electrons can only exist on specific distances from the nucleus, or specific “energy levels”, which are defined by the equation 30.13:

Z represents the atomic number of the atom, the number of protons in its nucleus. For Hydrogen, Z = 1.

An electron moving up from level n = 1 to n = 2 would start with an energy E1 = -13.6 eV and end with E2 = -13.6/4 = -3.4 eV. And it would therefore gain E2 - E1 = -3.4 - (-13.6) = +10.2 eV. A lot of negative signs here, sorry about that, not my idea.

In order to move up, the electron would have to receive energy either through a collision of some kind or by absorbing a photon, a particle of light, with an energy of 10.2 eV. We could find the wavelength of that photon by converting the energy into Joules and then solving E = hc/λ for λ. I’m hoping you have been practicing that already while working through the chapter.

As you solve worksheets and homework problems, some key points to remember:

1. Higher energy photons have shorter wavelengths. 2. Electrons move down to lower energy levels by emitting light.3. The highest energy level in an atom is infinity. If an

electrons absorbs enough energy to move past the highest level, we say the atom is ionized.

So, for example, an electron is in energy level n = 3 and I ask what is the smallest wavelength of light that can be absorbed by the electron for which the electron stays in the atom. Smallest wavelength = highest energy. “Absorbed” = electron moves up to higher levels. So what’s the highest level the electron can move up to? Infinity. So I need to find the energy difference between level 3 and infinity and/or the wavelength of light associated with that transition.

Now it is time for the last worksheet for this relatively short part of the course. Please attempt both parts of worksheet 30.1 (hint: Paschen series is associated with level n = 3). Once you have finished it or at least seriously attempted it and gotten stuck, please proceed to the next page to see my detailed solution.

If you have any questions about worksheet 30.1, please post to the appropriate “Chapter 29-32 worksheets” forum on d2l.tcu.edu, and I will respond there (or send me an email).

—If you would like further practice with atomic absorption and emission for Hydrogen atoms, there are some video solutions from old exams you can work with:

https://www.youtube.com/watch?v=tHfAxtLp3zkhttps://www.youtube.com/watch?v=bcD6wxcOR0Qhttps://www.youtube.com/watch?v=ggAyb4BuGE4https://www.youtube.com/watch?v=5axS7T5AiKk

And that’s all we are doing for Chapters 29 and 30. I am trying to cover the parts of these last 4 chapters that are most directly relevant, that you may need familiarity with in order to grasp other concepts in Physics and other disciplines, and so we will be skipping a lot of material that’s really best suited for Physics majors who are going on to upper division or graduate level work.

At this point, you should get started on the Ch 29/30 homework, which is due on Tue Apr 28 at 11:59pm CDT.

As I say for every chapter, I hope you will solve or seriously attempt each problem before asking for help. I will be checking the “Chapter 29-30 homework” forum on our course shell occasionally and contributing helpful comments in response to any questions you may have about the homework.