Video: Quantum Numbers

Video Transcript

In this video, we will learn about the quantum numbers that are used to describe an electron in an atom or ion.

Firstly, let’s define what a quantum number is. The word sounds quite strange to begin with. On the surface, a quantum number is simply any number in a series of numbers that are all defined by the same rule. For instance, the positive integers are all round counting numbers, like one, two, three, four, and so on. But the numbers in between, like 2.5 and 𝜋, are not positive integers, so they’re not quantum numbers within this rule.

We can also think of quantum numbers as quantized numbers. There is an infinite number of numbers between zero and 10. But we could quantize the range by moving in increments of two, having zero, two, four, six, eight, and 10 only. Many things are quantized like shoe sizes. We could get a 9, 9 and half, or 10, but it’ll be very hard to get a 9.2 shoe.

Now, how do quantum numbers relate to electrons? Electrons behave in a way that is very complex. A lot of maths is required to model how an electron behaves. It would take many years to discuss all the maths that goes into quantum physics. However, there are a few simple parts that we can break down together. Imagine a row of hotels on a street where each hotel is unique. You can think of quantum numbers like the door number on the street, the room number in the hotel, and so on. Of course, the number doesn’t tell you every detail of the hotel, but you do need to understand the meaning of the number to know where to go. It’s the same with quantum numbers for electrons. Each quantum number is a part of a much more complicated equation. If we know the meaning of the quantum number, we can use it to determine the shell, subshell, orbital, and spin of an electron.

Let’s imagine we have an atom of helium with two electrons in the 1s subshell. On the surface, we can’t distinguish between the two electrons in the 1s subshell. However, on closer inspection, they are slightly different. Electrons have a property called spin. The spin of an electron is either up or down. It’s beyond the scope of this video to explain exactly what spin is. But you can remember that a spin up electron is represented by an upward pointing arrow and a spin down electron is represented by a downward pointing arrow. It’s convention to use a fishhook arrowhead when representing electrons.

So, let’s return to our helium atom. We can represent the orbital in the 1s subshell with a line or with a box. By convention, the first electron in the orbital is spin up and the second one is spin down. The electrons in the 1s orbital can now be distinguished. They’re in the same shell, same subshell, and the same orbital, but one is spin up and the other is spin down. That solves the problem of describing electrons in the same orbital. But what about electrons in different orbitals but in the same subshell?

Here is the electron configuration for an atom of neon, and there are six electrons in the 2p subshell. How do we go about labeling these electrons uniquely? To start with, the 2p subshell consists of three separate orbitals. They’re called the 2p 𝑥, 2p 𝑦, and the 2p 𝑧 orbitals. So, we can describe each electron in a unique way. For instance, an electron in the 2p subshell could be in the 2p 𝑥 orbital and be spin up.

The other pieces of information we need are already given to us. The two in 2p indicates we’re looking at the second electron shell, and the p tells us we’re in a p-type subshell. Therefore, the full address of this electron is spin-up, 2p 𝑥 orbital, 2p subshell, second electron shell. This is a great format for a chemist. We talk about this type of thing a lot, but it’s hard for a physicist or mathematician to plug these into an equation. This is where quantum numbers come in.

The first quantum number we need, which is equivalent of the number of the hotel, is called the principal quantum number and has the symbol 𝑛. The principal quantum number describes which electron shell the electron is in. The rule for the principal quantum numbers is they are all positive integers. Just like addresses, the lowest principal quantum number is number one. After that, we count up using just the integers. The second quantum number is called the subsidiary quantum number, and it has the symbol 𝑙. It’s also known as the as azimuthal quantum number, the orbital angular momentum quantum number, or the subshell quantum number. It’s the subsidiary quantum number that tells us which subshell an electron is in. The rule for 𝑙 actually depends on which subshell the electron is in. So, it depends on the value of 𝑛.

For the first electron shell, where 𝑛 is one, the value of 𝑙 can only be zero. For the second electron shell, 𝑙 can be either zero or one. And from the third electron shell, we start to see a pattern. 𝑙 can be zero, one, or two. If we extend this to any value of the principal quantum number 𝑛, the values of 𝑙 we can have are zero, one, two, and so on until we reach the value 𝑛 minus one.

Each value of 𝑙 corresponds to a certain type of subshell. A value of zero for the subsidiary quantum number indicates an s-type subshell. One corresponds to a p-type subshell, two to a d-type subshell, and three to an f-type subshell. We could go higher, but it’s not that relevant. The letters s, p, d, and f correspond to features of a spectrum that correspond to these subshells. The descriptions are sharp, principal, diffuse, and fundamental. It’s not really important to know what s, p, d, and f mean, but it is interesting.

We can see from the general rule that the higher the value of 𝑛, the more types of subshell they’ll be in that electron shell. We can see from the pattern that the number of types of subshell in an electron shell is equal to 𝑛. The first electron shell has one type of subshell, the s shell, and the third electron shell has three types of subshell: an s, a p, and a d. We’ll look in detail about the nature of these subshells when we get to the magnetic quantum number.

The third quantum number is called the magnetic quantum number, and it has the symbol 𝑚 subscript 𝑙. You could call it the orbital quantum number since it identifies the orbital our electron is in. The potential values of the magnetic quantum number depend on the value of 𝑙, the subsidiary quantum number, which gives us the subshell. If 𝑙 is zero, then 𝑚 𝑙 can be only zero. If 𝑙 is one, 𝑚 𝑙 can be negative one, zero, or positive one. If 𝑙 is two, then 𝑚 𝑙 can be negative two, negative one, zero, positive one, or positive two. So, our general rule for the value of 𝑚 𝑙, the magnetic quantum number, is that it’s equal to zero, plus or minus one, plus or minus two all the way up to plus or minus 𝑙.

For each magnetic quantum number we throw into our formulas, we’d get an orbital. Unlike with 𝑙, it’s harder to connect magnetic quantum numbers to specific features of specific orbitals. But we can say for a p-type subshell where the value of the subsidiary quantum number is one, there are three different orbitals. For a d-type subshell, there are five orbitals. And for a generic value of 𝑙, the number of orbitals we get is two 𝑙 plus one. The important thing is to remember the general formula so we can apply it to any value of 𝑙.

Now we’ve got down to the orbital level, the last thing we need to account for is spin. The fourth and final quantum number relative to electrons is called the spin quantum number, and it has the symbol 𝑚 subscript 𝑠. This quantum number identifies whether the electron is spin up or spin down. There are only two potential values for the spin quantum number in this context, positive half or negative half. This allows for the fact that when we see an orbital, the maximum number electrons we can see in that orbital is two. That’s four quantum numbers that uniquely identify an electron in an atom or ion. Let’s do a quick run through of more.

At the top, we have the principal quantum number, the shell number. For the first electron shell, there’s only one value of the subsidiary quantum number allowed and that’s zero. And for the first electron shell and the s-type subshell, we can only have one value of the magnetic quantum number and that value is zero. And finally, the value of the spin quantum number can either be negative a half or positive a half. This allows for two electrons, one spin up and one spin down, in one orbital in the single s-type subshell of the first electron shell, which gives us two electrons in total in the first electron shell.

Let’s move on to the second shell. The values of 𝑙 allowable for the second shell are zero and one since we’re counting up to 𝑛 minus one. Just like in the first shell, when 𝑙 equals zero, 𝑚 𝑙 can only equal zero. But for 𝑙 equals one, 𝑚 𝑙 can equal negative one, zero, or positive one. And, as always, 𝑚 𝑠 is either equal to positive a half or negative a half. So, the s-type subshell in the second electron shell contains a maximum of two electrons, and the p-type subshell contains a maximum of six electrons. Altogether, the second electron shell can contain a maximum of eight electrons.

Let’s do one more, the third electron shell. For 𝑛 equals three, we can have 𝑙 equal to zero, one, or two. As the value of the subsidiary quantum number 𝑙 increases, the number of values we can get for the magnetic quantum number 𝑚 𝑙 increases as well. But the spin quantum number is reliable as always. So, the s-type subshell in the third electron shell can contain a maximum of two electrons. The p-type subshell can contain a maximum of six electrons. And the d-type subshell can contain a maximum of 10, giving us a grand total of 18 electrons for the maximum occupancy of the third electron shell.

The last thing we’re going to study is how to convert between quantum numbers and the notation we use too. This is the electron configuration of a fluorine atom. We can imagine that the electrons occupy the two p orbitals 2p 𝑥, 2p 𝑦, and 2p 𝑧 according to this pattern. And we’re going to figure out the quantum numbers that describe this electron. We could do it in any order, but I’m going to do it 𝑛, 𝑙, 𝑚 𝑙, 𝑚 𝑠.

We can find the value of 𝑛 by looking at the number in the subshell notation. 2p means that 𝑛 is equal to two. We’re in the second shell. What about 𝑙? In order to work out 𝑙, we need to convert p back to 𝑙 notation. p corresponds to a value of the subsidiary quantum number of one. Now the magnetic quantum number. Determining the magnetic quantum number from the orbital itself is a lot more complicated. It just so happens that the 2p 𝑧 orbital corresponds with a magnetic quantum number of zero. And lastly, we have the spin quantum number 𝑚 𝑠. The electron is spin up. Therefore, the spin quantum number for that electron is positive half. Normally, you wouldn’t need to know beyond 𝑛 and 𝑙, but it’s interesting to know the others as well. So, it’s about time we had some practice.

How many electrons in total can have the quantum numbers 𝑛 equals two and 𝑙 equals one?

Quantum numbers are numbers we assign to electrons to describe where they are, shell, subshell, and so on. 𝑛 is the symbol given to the principal quantum number. If 𝑛 equals one, we’re looking at the first electron shell. If it equals two, we’re looking at the second. In this example, 𝑛 equals two. So, we’re looking at the second electron shell. 𝑙 is the symbol used for the subsidiary quantum number, also known as the orbital angular momentum quantum number or the subshell number. Each value of 𝑙 corresponds with a type of subshell, like an s-type or a p-type subshell. With 𝑙 equal to one, we’re dealing with a p-type subshell.

The question is asking us, how many electrons in total can have these particular quantum numbers? The key here is that if we’re looking at the same atom or ion, no two electrons can have exactly the same set of quantum numbers. So, is the answer one, since we have quantum numbers 𝑛 and 𝑙 fixed at values of two and one, respectively? Well, no, there are two other quantum numbers that can uniquely define an electron.

There’s another quantum number that defines the orbital the electron is in, and that’s known as the magnetic quantum number. The possible values of the magnetic quantum number are zero, plus or minus one, plus or minus two until we reach plus or minus 𝑙, which for a value of one would give the values of 𝑚 𝑙 of negative one, zero, and one. This means we have three orbitals in total.

The fourth quantum number we need is the spin quantum number 𝑚 𝑠, which can either be positive a half or negative a half. This gives us two possible spin states. We can now calculate the number of possible sets of quantum numbers and therefore the number of electrons we can possibly have. There are two possible values for the spin quantum number, three values for the magnetic quantum numbers, so that’s two electrons per orbital, giving us six electrons in total. You could have done this question simply by remembering that a p-type subshell contains a maximum of six electrons. But this way, we’ve proved it.

Let’s finish off with the key points. There are four quantum numbers that uniquely describe electrons on the same atom or ion. The first, the principal quantum number, has the symbol 𝑛, and it describes the shell. The principal quantum number can take the value of any positive integer, starting with the number one. Next is the subsidiary quantum number, also known as the orbital angular momentum quantum number, which uniquely identifies the subshell. 𝑙 can take a value of zero, one, two up until 𝑛 minus one. s subshells match up with the value of 𝑙 of zero, p one, d two, and f three.

Meanwhile, magnetic quantum numbers of a subshell correspond roughly with the orbitals. 𝑚 𝑙 can take any integer value between negative 𝑙 and positive 𝑙. And lastly, we have the spin quantum number 𝑚 𝑠. The value of the spin quantum number can either be positive a half or negative a half corresponding to spin up or spin down. Together, these four quantum numbers allow us to efficiently describe the state of an electron.