Video: The Kinetic Energy of Photoelectrons

In this video, we will learn how to calculate the maximum possible kinetic energy of electrons that are ejected from the surface of a metal due to the photoelectric effect.

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Video Transcript

In this video, we’re talking about the kinetic energy of photoelectrons. This energy plays into an important physical effect. But before we talk about that, let’s consider what photoelectrons are in the first place.

We know that electrons are the negatively charged objects in motion around atomic nuclei. We could say that the electrons are bound to the nucleus by some amount of energy. And for certain types of atoms, in particular for metals, this amount of energy can be very small for some of the outer valence electrons. This means that, from an energy perspective, it’s not unlikely that these outermost electrons in a metal will receive some energy from their environment and leave the atom entirely. It’s these mobile electrons that are able to form an electric current when many metal atoms are brought together in a material. But not only are electrons able to move within the material, but with the right amount of energy, they can be ejected from it.

In the early 20th century, experiments were performed that involved shining light on the surfaces of metals. In some cases, under some conditions, it was noted that this incident radiation led to the ejection of electrons from the metal. These electrons ejected from the surface were called photoelectrons. This prefix photo- explains how it is that the electrons were energized to leave the surface, by absorbing the energy of incident light.

Now, at this time, people thought of light primarily as a wave. Within this framework, the idea was, the way to add more energy to the surface of a metal, and thereby create more photoelectrons, was to increase the amplitude of the incoming wave. So, for example, if shining light like this on the surface wasn’t enough to create any photoelectrons, many scientists believed that the answer was to replace it with a wave like this, the same wavelength and frequency, but greater wave amplitude. And if that didn’t create any photoelectrons from the surface, that simply meant the thinking went that the amplitude should be increased even more. The trouble was, this expectation of how to create photoelectrons wasn’t backed up by experimental evidence.

Rather, researchers discovered that if the original wave didn’t lead to the creation of any photoelectrons, then neither would increasing that wave’s amplitude without changing its wavelength or its frequency. It turned out that rather than wave amplitude, it was the frequency of the incoming radiation that determined whether a photoelectron will be created or not. Once some minimum frequency value was achieved, then the incoming radiation would tend to lead to the creation of a photoelectron, regardless of the amplitude of that incoming light. This process of electrons within a material absorbing energy from incoming light and being ejected from the material can be called the photoelectric effect.

In order to explain this effect, the German physicist Albert Einstein said that we should start to think of light not only as a wave, but also as a particle. These particles are known as photons. And in order to explain the photoelectric effect, Einstein pointed to an equation generated by a colleague to describe the energy of photons. This equation says that the energy of a photon is equal to a constant value, called Planck’s constant, multiplied by the frequency of the photon. It was this energy, Einstein claimed, that was absorbed by electrons in a metal when light was incident on it that gave those electrons enough energy to escape the surface and become photoelectrons.

We see from this equation that photon energy depends on only one variable, the frequency of the radiation. This explains why an orange-colored wave at a lower frequency than our blue wave was never able to eject an electron from the surface, no matter how big its amplitude got. The energy absorbed by the electrons in the metal doesn’t have to do with amplitude, but instead frequency. Knowing this, let’s say that we try shining light of various frequencies on this metal surface. We would observe that the lower frequency radiation isn’t able to create photoelectrons because it doesn’t have enough energy to do so.

We would also notice though that once we pass a certain minimum frequency threshold, incoming light of that frequency and higher would create photoelectrons. But the photoelectrons these different incoming frequencies create are not the same. The higher the frequency of the incoming light, the more energetically the electrons would be ejected from the surface. And therefore, the more kinetic energy they would have.

From an energy perspective, here’s how we can think about this process. When a photon is incident on the surface of a metal, it brings with it an energy, which is equal to its frequency multiplied by Planck’s constant. When that photon is absorbed by an electron in the material, this amount of energy is transferred to the electron. At that moment, we could say that this is the amount of energy possessed by the electron. But remember we said that electrons are energetically bound to the atom or the material they’re a part of. This is why mobile electrons on the surface of a metal tend not to leave that surface. The energy that binds an electron to the surface of a metal is known as the work function of that metal. And it’s abbreviated using a capital 𝑊.

Work function is an amount of energy. It’s given in units of joules or in units of electron volts. And it varies from material to material. Different materials have different work functions. In other words, electrons have an easier or harder time escaping from their surfaces. We could think of the work function as an energy barrier that keeps electrons from escaping. But as the photoelectric effect demonstrates, if an electron receives enough energy from an incoming photon, then even if the work function energy is subtracted from it, there will still be some left over. Now, at this point, we should note that the work function for a material is the minimum amount of energy needed for an electron to escape that surface. This means for an electron to move away from the surface into the vacuum surrounding it.

Any of this ℎ times 𝑓 energy left over, so to speak, after the work function energy has been subtracted from it, is devoted to energy of motion of the electron, kinetic energy. So the energy of an incoming photon, which becomes the energy of an electron once that photon is absorbed by it, minus the work function energy of the particular material the electron is part of is equal to the maximum kinetic energy of a photoelectron. And we say that this is the maximum kinetic energy because again we’ve defined the work function to be the minimum amount of energy an electron needs to escape the surface.

This equation helps us understand why some electrons leaving the metal have higher energy than others. All the escaping electrons face the same energy barrier, the work function. But depending on the frequency of the incoming photons they absorb, they can overcome this energy barrier by a little, leading to a slow moving photoelectron, or by a lot, leading to a photoelectron with lots of kinetic energy. One way to represent this relationship between the frequency of an incoming photon and the energy of an ejected electron is by creating a graph.

Say that we create a graph where, on the vertical axis, we plot the kinetic energy of ejected electrons, in energy units of electron volts. And on the horizontal axis, we show the frequency of incoming radiation. Now, we know from experiment that until the frequency of the incoming radiation reaches a certain minimum value, no photoelectrons are created. Say that we start shining light at fairly low frequency on our metal surface. And as we increase that frequency, we see that still no photoelectrons are created. But then, at some critical frequency value, we start to see our curve turn up. This point where the graph bends shows us the minimum frequency needed to eject an electron. Then, as our frequency continues to go up and up, so does the kinetic energy of our photoelectrons.

Now, let’s say that, for this particular material, the photoelectron kinetic energy looked like this. This means that, for an incoming photon with frequency four times 10 to the 14th hertz, we would have a corresponding kinetic energy of the photoelectron of one electron volt. And likewise, if we dialed up the frequency of the incoming photons to five times 10 to the 14th hertz, that would eject an electron with the kinetic energy of two electron volts. A graph like this shows us the minimum frequency needed to create a photoelectron from a given material. And by looking at that point, which is this point right here, we can solve for that material’s work function. We do this by considering the work function expressed in this relationship.

Going back to our graph, at this bend in the graph, the point we’ve highlighted, this is where photoelectrons are just starting to be created. But their kinetic energy is zero. That is, the electron energy is just barely meeting the minimum value, the work function of the material, and escaping from the metal surface. These electrons, barely able to overcome the work function barrier, have no kinetic energy by the time they’re ejected. So at this particular frequency — and we’ll call this frequency 𝑓 sub zero — the energy of the incoming photon, and therefore the energy of the electron after it absorbs this photon, minus the work function is equal to zero.

And then, if we add the work function to both sides of this equation, we see that at this particular frequency, what we could call the threshold frequency for this material, our work function is equal to that frequency times Planck’s constant. Now, this isn’t the only way to solve for the work function from a graph like we’ve seen. We could’ve also picked a point on the graph, say, this one right here, where our frequency is five times 10 to the 14th hertz and our ejected electron energy is two electron volts. And we could substitute those values into this original equation.

We could say that Planck’s constant, which is known, multiplied by the frequency on our graph, five times 10 to the 14th hertz, minus the work function 𝑊 is equal to two electron volts, the kinetic energy of the ejected electrons. Then, if we subtract two electron volts from both sides of this equation and we add the work function to both sides. We now have an equation for the work function in terms of values we’ve read off the graph and a known constant.

One other point we can make about this equation is, because we’re talking about photons, objects that move at the speed of light, and these photons have wave-like properties. We can recall that, in general, the frequency of a wave is equal to the wave speed divided by the wavelength, which means we can replace the frequency in our equation with 𝑐, the speed of these photons, divided by the wavelength 𝜆. This is an equivalent expression for the energy of a photoelectron.

Thinking in terms of wavelength, rather than frequency like we did before, we’ll change the shape of a graph like this. If we have the kinetic energy of ejected photoelectrons on the vertical axis, like before, but now the wavelength of the incoming photons on the horizontal axis. The shape our graph will take will be something like this. Here, we see that as wavelength goes up, the kinetic energy of ejected electrons gets smaller and smaller until, above a certain wavelength value, that kinetic energy is zero. This means that no electrons are being ejected. The incoming photons aren’t high energy enough to do that.

One thing to keep in mind in a graph like this is that as wavelength increases, the photon energy decreases. The two quantities are inversely related. This explains why as photon wavelength gets smaller and smaller, the kinetic energy of our ejected electrons gets greater. Given a graph like this, we could once again use points on it to solve for the work function of the material, this time using this equation. Knowing all this, let’s get a bit of practice now through an example exercise.

A copper cathode in a vacuum chamber is illuminated with light from a laser, causing electrons to be emitted from the surface of the metal. The light has a frequency of 1.80 times 10 to the 15th hertz. The maximum kinetic energy of the ejected electrons is 2.80 electron volts. What is the work function of copper? Use a value of 4.14 times 10 to the negative 15th electron volt seconds for the value of the Planck constant. Give your answer in electron volts to three significant figures.

Okay, so in this exercise, we have a copper cathode. And let’s say that this is that cathode. It’s in a vacuum chamber being illuminated, which causes some electrons to be ejected from the copper. We’re told that the frequency of the incoming radiation, we’ll call it 𝐹, is 1.80 times 10 to the 15th hertz and that the maximum kinetic energy of the electrons ejected from the copper, we’ll call it 𝐾𝐸 sub 𝑚, is 2.80 electron volts. Recall that one electron volt is equal to the amount of energy a charge of one electron gets when it’s moved across the potential difference of one volt.

Knowing all this, we want to solve for the work function of copper. To figure this out, we can recall a relationship between frequency, 𝐹, maximum kinetic energy, 𝐾𝐸 sub 𝑚, and the work function of a material, called 𝑊. That relationship states that Planck’s constant ℎ multiplied by the frequency of incoming radiation minus the work function of a material is equal to the maximum kinetic energy of ejected electrons. Since we want to solve for the work function 𝑊, we’ll rearrange this equation by subtracting 𝐾𝐸 sub 𝑚 from both sides and adding the work function 𝑊 to both sides. When we do that, we find that 𝑊 is equal to ℎ times 𝑓 minus 𝐾𝐸 sub 𝑚.

Note that a work function 𝑊 is specific to a particular material. In our case, we’re solving for the work function of our copper cathode. To help us solve for it, we’re given the frequency 𝑓 of incoming radiation as well as the maximum kinetic energy of ejected electrons. We’re also told to treat Planck’s constant ℎ as 4.14 times 10 to the negative 15th electron volt seconds. So let’s substitute those three values into this equation. With those values plugged in, let’s take a look for a second at the units involved.

In our first term, we have electron volt seconds multiplied by hertz. But hertz, the number of cycles performed per second, can be replaced with one over seconds. And when we do this, we see that in multiplying these two values together, that factor of seconds will cancel out. So now, we have one number in electron volts being subtracted from another number in electron volts. Therefore, our final answer will be in these desired units. And when we calculate the value of this expression to three significant figures, we find a result of 4.65 electron volts. This is the work function of copper.

Let’s summarize now what we’ve learned about the kinetic energy of photoelectrons. Starting off, we saw that photoelectrons are electrons ejected from a surface after absorbing incident light energy. We saw further that ejecting electrons from the surface of a metal by shining light on the metal is called the photoelectric effect. This effect can be best understood by thinking of light as a particle rather than as a wave.

Furthermore, the work function of material, we learned, is the energy needed for an electron to escape the surface of that material. And lastly, the maximum kinetic energy of an electron ejected from a surface is equal to the energy of an incoming photon absorbed by the electron. This energy is ℎ times 𝑓 minus the work function 𝑊 of the material. And equivalently, this maximum kinetic energy is equal to ℎ times 𝑐 over 𝜆, the wavelength of the incoming photon, minus 𝑊. This is a summary of the kinetic energy of photoelectrons.

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