### Video Transcript

In this video, we’re going to be talking about electron energy level transitions. Specifically, we’re going to be taking a look at how we can represent the different amounts of energy that the electrons in atoms can have. We’ll also look at how we can calculate the frequency of the light that’s emitted or absorbed when an electron gains or loses energy. But first, let’s just remind ourselves of some of the key principles behind electron behavior.

We know that electrons are negatively charged particles which surround the nuclei of atoms. These electrons can have different amounts of energy. And, crucially, the amounts of energy that it’s possible for an electron to have are discreet. This means that the electrons in atoms can only have certain specific amounts of energy. We can represent this using an energy level diagram, which looks like this. Here, we have an energy axis pointing upwards, and these horizontal lines are positioned at specific points on the energy axis to show the specific amounts of energy that it’s possible for an electron within a specific atom to have.

So, this diagram shows us that within a specific atom an electron’s energy could take any one of these values. But it couldn’t take any of the values in between. The allowed amounts of energy represented by the black lines are often referred to as energy levels. And when an electron has a certain amount of energy, we say that it occupies that energy level. We can show which energy levels are occupied by electrons by drawing electrons into our energy level diagram, like this. So, now, our diagram shows us that there’s one electron in the atom. And because this electron is on the lowest possible line, that means that it has the lowest possible amount of energy that an electron can have in that atom.

We can represent the energy level that an electron is in using what’s known as the principal quantum number represented by 𝑛. 𝑛 just takes a whole number value to tell us which energy level an electron occupies from lowest to highest. So, for example, an electron in the lowest energy level would have 𝑛 equals one. An electron in the second lowest energy level would have 𝑛 equals two and so on. So, although the value of 𝑛 doesn’t directly tell us the amount of energy that an electron has, we can still see that greater values of 𝑛 correspond to greater amounts of energy.

So, for example, an electron in the 𝑛 equals three energy level would have more energy than an electron in the 𝑛 equals two energy level. If an electron in an atom gains energy, then we say that it transitions to a higher energy level. Similarly, if an electron loses energy, then it transitions to a lower energy level. We can represent electron energy level transitions by drawing arrows on our energy level diagram.

For example, if we had an electron in the 𝑛 equals one energy level and it transition to the 𝑛 equals three energy level by gaining energy. Then we could represent this transition by drawing an arrow from the 𝑛 equals one energy level to the 𝑛 equals three energy level on our diagram, like this. If the electron then transitioned into the 𝑛 equals two energy level, which it could do by losing energy, we could represent this by drawing an arrow going from the 𝑛 equals three to the 𝑛 equals two energy levels, like this.

Note that when an electron transitions into a different energy level, it doesn’t necessarily transition into an adjacent energy level. For example, as shown by this arrow in our diagram, an electron can transition straight from 𝑛 equals one to 𝑛 equals three. It doesn’t necessarily need to transition from 𝑛 equals one to 𝑛 equals two and then from 𝑛 equals two to 𝑛 equals three. It’s also important to note that it’s impossible for an electron in the 𝑛 equals one energy level to lose energy because there aren’t any lower energy levels for it to transition into. Just like if we drop a ball, it will fall under gravity until it hits the ground and then stop.

An electron can lose energy only until it reaches the 𝑛 equals one energy level. For this reason, the 𝑛 equals one energy level is also referred to as the ground state. Electrons in the ground state are the most strongly bound to the nucleus of the atom. If an electron gains energy and moves to a higher energy level, then it’s less strongly bound to the nucleus. So, as the energy of an electron increases and the value of 𝑛 increases, then the force binding it to the nucleus becomes weaker and weaker.

We also find that as we look at higher energy levels, the gaps between adjacent energy levels get smaller and smaller. For example, the gap between the 𝑛 equals four and the 𝑛 equals three energy levels is smaller than the gap between the 𝑛 equals three and the 𝑛 equals two energy levels. This means that there’s a smaller energy difference between 𝑛 equals four and 𝑛 equals three than there is between 𝑛 equals three and 𝑛 equals two.

So, moving from the 𝑛 equals two energy level to the 𝑛 equals three energy level would represent a larger increase in energy than moving from the 𝑛 equals three to the 𝑛 equals four energy level. And moving from the 𝑛 equals four to the 𝑛 equals five energy level would represent an even smaller increase in energy than that. All atoms actually have an infinite number of energy levels. However, because they get closer and closer together, as the energy increases, we find that there’s actually a maximum amount of energy that an electron in an atom can have.

This is the energy level for which 𝑛 equals infinity. And once an electron reaches this energy level, it will no longer be bound to the atom. The fact that 𝑛 is infinitely large at this point just represents the fact that there’re infinitely many energy levels between this point and the ground state. It doesn’t mean that an infinite amount of energy is required to remove an electron from the atom. In fact, the amount of energy required to free a ground-state electron from the atom, known as the ionization energy of that atom, is actually well known for virtually all elements.

Okay, so, now that we’ve talked about the different kinds of energy transitions that electrons in atoms can make and how we can represent these in energy level diagrams, let’s quickly remind ourselves of the processes by which electrons can gain or lose energy. In order to gain energy and transition to a higher energy level, an electron must absorb a photon. Crucially, in order for the photon to be absorbed, facilitating an electron energy level transition, then the energy of the photon must be precisely the same as the energy gap between the two energy levels.

We can call this difference in energy 𝛥𝐸. And we can say that this has to be equal to the energy carried by the photon, which we can call 𝐸 p. Just like a dropped ball will naturally fall to the ground, electrons that occupy energy levels higher than the ground state will naturally tend to lose energy and fall back down to the ground state after some time. When an electron loses energy, it does so by emitting a photon. In this case, the energy of the emitted photon, which we can also call 𝐸 p, is exactly equal to the difference between the energy levels, 𝛥𝐸.

If we wanted to cause an electron to transition from the 𝑛 equals one to the 𝑛 equals three energy level, then we could do so by making sure a higher energy photon strikes the atom where its energy will be absorbed by the electron. After the electron has reached the 𝑛 equals three energy level, it will again naturally return to the ground state after some time. This can happen in two different ways.

We might find that it transitions back to the ground state in a single transition. In which case, it will emit a photon in the process with exactly the same amount of energy as the initial photon that the electron absorbed. However, we also might find that the electron first transitions from the 𝑛 equals three to the 𝑛 equals two energy level and then from the 𝑛 equals two to the 𝑛 equals one energy level. In this case, two photons will be emitted, one for each transition that the electron makes. The energies of each of these photons will correspond exactly.

So, the photon emitted when the electron moves from the 𝑛 equals three to the 𝑛 equals two energy level will have less energy than the photon which is emitted when the electron moves from the 𝑛 equals two energy level to the ground state. And in fact, we find that the energies of both of these photons add together exactly to the energy of the photon that was absorbed in the first place.

Now, there’s one other important thing to mention. And that is that the energy of a photon is closely related to its frequency. This is represented by the equation 𝐸 equals ℎ𝑓. The energy of a photon, 𝐸, is equal to the frequency of the photon, 𝑓, multiplied by a constant. This is Planck’s constant which we represent with the letter ℎ. This equation shows us that the energy a photon has is directly proportional to its frequency.

Because electrons in atoms can only absorb or emit photons with specific amounts of energy corresponding to the difference between energy levels, this means atoms can only absorb or emit photons with specific frequencies. And we can use this equation to calculate the frequency of light that’s either absorbed or emitted during a specific electron energy level transition. In order to do this, there’s one more thing we need to know. That is the actual amounts of energy that are involved in these electron transitions.

To find out the differences in energy between the energy levels, we need to know the specific energy values of each of the energy levels in our diagram. Now, each element actually has its own unique set of energy levels. So, let’s take a look at what the energy levels look like for the simplest element, hydrogen.

These values tell us the amount of energy that an electron would have in each energy level of a hydrogen atom. Since the energy values are related to the strength with which an electron is bound to the nucleus, they’re also known as the binding energy of the electrons in each energy level. So, for example, an electron in the ground state of a hydrogen atom has negative 2.2 times 10 to the power of negative 18 joules of energy.

The first thing you might have noticed about these values is that they’re all negative, except for this one, which is zero. While it is a bit weird to describe something as having a negative amount of energy, this is just a convention to show that we would have to give extra energy to an electron in order to free it from the atom. So, we say that an electron which is not bound to the atom — so, one that essentially has an infinitely high value of 𝑛 — has zero energy. And all the other energy levels are defined relative to this zero.

Since we know that these electrons in other energy levels which are bound to the atom have less energy than a freed electron, that means they must have negative energy. This convention is useful because it means we can easily find the amount of energy required to free any electron from an atom is just given by the magnitude of the energy level that the electron is in.

For example, consider an electron in the 𝑛 equals two energy level. If this electron has an energy of negative 5.4 times 10 to the power of minus 19 joules and a free electron has an energy of zero joules, then it becomes clear that if we wanted to give energy to the 𝑛 equals two electron in order to free it from the atom. Then we would need to give that electron 5.4 times 10 to the power of negative 19 joules of energy in order to increase its energy to zero and free it from the atom. In other words, the difference in energy between these energy levels, 𝛥𝐸, is 5.4 times 10 to the power of negative 19 joules. That is, it’s the difference between zero and negative 5.4 times 10 to the negative 19 joules.

We can find the energy associated with any other electron energy level transitions by just finding the difference between the energy level values. For example, let’s consider an electron in the 𝑛 equals one energy level transitioning to the 𝑛 equals three energy level. The difference in energy between these energy levels can be found by subtracting the energy of an electron with 𝑛 equals one, we’ll call this 𝐸 one, from the energy of an electron with 𝑛 equals three; we’ll call this 𝐸 three. So, that’s negative 2.4 times 10 to the negative 19 minus negative 2.2 times 10 to the negative 18, which gives us 1.96 times 10 to the power of negative 18 joules.

This is the amount of energy that we need to give an electron in the 𝑛 equals one energy level in order for it to get to the 𝑛 equals three energy level. And it’s also the amount of energy that would be released by an electron in the 𝑛 equals three energy level if it transitioned to the 𝑛 equals one energy level. In other words, this is the amount of energy that a photon would need to have in order to cause an electron to transition from the ground state to the 𝑛 equals three energy level.

This is exactly the same as the energy of the photon that would be released if the electron moved the other way. Because the energies involved in these transitions are quite small, it’s common to express them in electronvolts, rather than joules. Recall that one electron volt is equivalent to 1.60 times 10 to the negative 19 joules. So, to convert our values from joules into electronvolts, we just need to multiply them by 1.6 times 10 to the negative 19 joules.

So, the energy levels in a hydrogen atom take these values in electron volts. And the energy difference between the 𝑛 equals one and the 𝑛 equals three energy levels is 12.1 electronvolts, which we can also easily obtain by finding the difference between the energy levels when expressed in electronvolts. Now that we’ve found the amount of energy associated with the transition between the 𝑛 equals one and the 𝑛 equals three energy levels. We can use this equation 𝐸 equals ℎ𝑓 in order to find the frequency of the photon that’s either absorbed or released during such a process.

So, we’ll just clear some space on the left side of the screen and show how we can do this calculation. Okay, so, first of all, we can rearrange the equation to make frequency 𝑓 the subject, giving us 𝑓 equals 𝐸 over ℎ. Recall that 𝐸 is the energy of a photon and ℎ is a constant, known as Planck’s constant. The value of Planck’s constant is 6.63 times 10 to the negative 34 joule-seconds. But it can also be expressed in electronvolts seconds, in which case it takes a value of 4.14 times 10 to the negative 15.

The reason that we express Planck’s constant in different units is that we commonly use either joules or electronvolts to express energy. And the version of Planck’s constant that we use depends on whether we’re expressing energy in joules or electronvolts. So, in this equation, if we use a value of 𝐸 in joules, then we must also make sure that we use Planck’s constant expressed in joule-seconds. Conversely, if 𝐸 takes a value in electronvolts, then we need to make sure that we use Planck’s constant expressed in electronvolt seconds.

So, to find the frequency of the photon which is emitted when an electron moves from the 𝑛 equals three to the 𝑛 equals one energy level, we know that the energy of the photon 𝐸 is exactly equal to the difference between the two energy levels. Expressed in joules, that’s 1.96 times 10 to the negative 18 joules. And we divide that by Planck’s constant expressed in the corresponding units, in this case joule-second. Which gives us a frequency of 2.956 times 10 to the power of 15 hertz or rounded to one significant figure three times 10 to the power of 15 hertz. Let’s repeat this calculation, but this time we’ll express the energy in electronvolts instead of joules.

The energy of the photon expressed in electronvolts is 12.1. And since we’re now using electronvolts, we need to be careful to use the version of Planck’s constant that’s expressed in electronvolts seconds. If we type this into a calculator, we find that we get the same answer. This is the frequency of a photon which is emitted when an electron moves from the 𝑛 equals three to the 𝑛 equals one energy levels in a hydrogen atom. Equivalently, it’s the frequency of the photon which is absorbed when an electron moves from the 𝑛 equals one to the 𝑛 equals three energy level.

If we look at our energy level diagram for hydrogen, we can see that because there are an infinite number of energy levels, there are, theoretically at least, an infinite number of electron energy level transitions that can take place. We actually find that electron transitions most commonly occur within the lower energy levels. Because electron energy levels are unique to different elements and each possible transition produces a specific frequency of light, we find that we can actually identify different elements by looking at the frequencies of light that they emit.

The set of frequencies emitted by a certain element is known as its emission spectrum. The emission spectrum of hydrogen has been really well studied by physicists. And names have been given to different groups of frequencies, which were emitted. The different frequencies of the photons emitted when an electron falls into the ground state of the hydrogen atom are known as the Lyman series. The frequencies produced when an electron falls into the 𝑛 equals two energy level from a higher energy level are known as the Balmer series. And the frequencies produced when an electron falls into the 𝑛 equals three energy level from a higher energy level are known as the Paschen series.

Now, let’s summarize all the things we’ve learned about electron energy level transitions. Firstly, the energy level of an electron bound to a nucleus is denoted by the principal quantum number, 𝑛. The lowest possible energy level is 𝑛 equals one, known as the ground state. And higher values of 𝑛 mean the electron has more energy and is more weakly bound to the nucleus.

We’ve seen that electrons gain energy by absorbing photons, and they lose energy by emitting photons. And these photon frequencies correspond to the difference between the electron energy levels. And finally, we’ve seen that certain groups of electron energy level transitions have names, such as the Lyman, Balmer, and Paschen series.