Main content

# Quantum numbers

## Video transcript

- [Voiceover] In the Bohr model of the hydrogen atom, the one electron of hydrogen is in orbit around the nucleus at a certain distance, r. So in the Bohr model, the electron is in orbit. In the quantum mechanics version of the hydrogen atom, we don't know exactly where the electron is, but we can say with high probability that the electron is in an orbital. An orbital is the region of space where the electron is most likely to be found. For hydrogen, imagine a sphere, a three-dimensional volume, a sphere, around the nucleus. Somewhere in that region of space, somewhere in that sphere, we're most likely to find the one electron of hydrogen. So we have these two competing visions. The Bohr model is classical mechanics. The electron orbits the nucleus like the planets around the sun, but quantum mechanics says we don't know exactly where that electron is. The Bohr model turns out to be incorrect, and quantum mechanics has proven to be the best way to explain electrons in orbitals. We can describe those electrons in orbitals using the four quantum numbers. Let's look at the first quantum number here. This is called the principal quantum number. The principal quantum number is symbolized by n. n is a positive integer, so n could be equal to one, two, three, and so on. It indicates the main energy level occupied by the electron. This tells us the main energy level. You might hear this referred to as a shell sometimes, so we could say what kind of shell the electron is in. As n increases, the average distance of the electron from the nucleus increases, and therefore so does the energy. For example, if this was our nucleus right here, and let's talk about n is equal to one. For n is equal to one, let's say the average distance from the nucleus is right about here. Let's compare that with n is equal to two. n is equal to two means a higher energy level, so on average, the electron is further away from the nucleus, and has a higher energy associated with it. That's the idea of the principal quantum number. You're thinking about energy levels or shells, and you're also thinking about average distance from the nucleus. All right, our second quantum number is called the angular momentum quantum number. The angular momentum quantum number is symbolized by l. l indicates the shape of the orbital. This will tell us the shape of the orbital. Values for l are dependent on n, so the values for l go from zero all the way up to n minus one, so it could be zero, one, two, or however values there are up to n minus one. For example, let's talk about the first main energy level, or the first shell. n is equal to one. There's only one possible value you could get for the angular momentum quantum number, l. n minus one is equal to zero, so that's the only possible value, the only allowed value of l. When l is equal to zero, we call this an s orbital. This is referring to an s orbital. The shape of an s orbital is a sphere. We've already talked about that with the hydrogen atom. Just imagine this as being a sphere, so a three-dimensional volume here. The angular momentum quantum number, l, since l is equal to zero, that corresponds to an s orbital, so we know that we're talking about an s orbital here which is shaped like a sphere. So the electron is most likely to be found somewhere in that sphere. Let's do the next shell. n is equal to two. If n is equal to two, what are the allowed values for l? l goes zero, one, and so on all the way up to n minus one. l is equal to zero. Then n minus one would be equal to one. So we have two possible values for l. l could be equal to zero, and l could be equal to one. Notice that the number of allowed values for l is equal to n. So for example, if n is equal to one, we have one allowed value. If n is equal to two, we have two allowed values. We've already talked about what l is equal to zero, what that means. l is equal to zero means an s orbital, shaped like a sphere. Now, in the second main energy level, or the second shell, we have another value for l. l is equal to one. When l is equal to one, we're talking about a p orbital. l is equal to one means a p orbital. The shape of a p orbital is a little bit strange, so I'll attempt to sketch it in here. You might hear several different terms for this. Imagine this is a volume. This is a three-dimensional region in here. You could call these dumbbell shaped or bow-tie, whatever makes the most sense to you. This is the orbital, this is the region of space where the electron is most likely to be found if it's found in a p orbital here. Sometimes you'll hear these called sub-shells. If n is equal to two, if we call this a shell, then we would call these sub-shells. These are sub-shells here. Again, we're talking about orbitals. l is equal to zero is an s orbital. l is equal to one is a p orbital. Let's look at the next quantum number. Let's get some more space down here. This is the magnetic quantum number, symbolized my m sub l here. m sub l indicates the orientation of an orbital around the nucleus. This tells us the orientation of that orbital. The values for ml depend on l. ml is equal to any integral value that goes from negative l to positive l. That sounds a little bit confusing. Let's go ahead and do the example of l is equal to zero. l is equal to zero up here. Let's go ahead and write that down here. If l is equal to zero, what are the allowed values for ml? There's only one, right? There's only one. The only possible value we could have here is zero. When l is equal to zero ... Let me use a different color here. If l is equal to zero, we know we're talking about an s orbital. When l is equal to zero, we're talking about an s orbital, which is shaped like a sphere. If you think about that, we have only one allowed value for the magnetic quantum number. That tells us the orientation, so there's only one orientation for that orbital around the nucleus. And that makes sense, because a sphere has only one possible orientation. If you think about this as being an xyz axis, (clears throat) excuse me, and if this is a sphere, there's only one way to orient that sphere in space. So that's the idea of the magnetic quantum number. Let's do the same thing for l is equal to one. Let's look at that now. If we're considering l is equal to one ... Let me use a different color here. l is equal to one. Let's write that down here. If l is equal to one, what are the allowed values for the magnetic quantum number? ml is equal to -- This goes from negative l to positive l, so any integral value from negative l to positive l. Negative l would be negative one, so let's go ahead and write this in here. We have negative one, zero, and positive one. So we have three possible values. When l is equal to one, we have three possible values for the magnetic quantum number, one, two, and three. The magnetic quantum number tells us the orientations, the possible orientations of the orbital or orbitals around the nucleus here. So we have three values for the magnetic quantum number. That means we get three different orientations. We already said that when l is equal to one, we're talking about a p orbital. A p orbital is shaped like a dumbbell here, so we have three possible orientations for a dumbbell shape. If we went ahead and mark these axes here, let's just say this is x axis, y axis, and the z axis here. We could put a dumbbell on the x axis like that. Again, imagine this as being a volume. This would be a p orbital. We call this a px orbital. It's a p orbital and it's on the x axis here. We have two more orientations. We could put, again, if this is x, this is y, and this is z, we could put a dumbbell here on the y axis. There's our second possible orientation. Finally, if this is x, this is y, and this is z, of course we could put a dumbbell on the z axis, like that. This would be a pz orbital. We could write a pz orbital here, and then this one right here would be a py orbital. We have three orbitals, we have three p orbitals here, one for each axis. Let's go to the last quantum number. The last quantum number is the spin quantum number. The spin quantum number is m sub s here. When it says spin, I'm going to put this in quotations. This seems to imply that an electron is spinning on an axis. That's not really what's happening, but let me just go ahead and draw that in here. I could have an electron ... Let me draw two different versions here. I could have an electron spin like a top, if you will, this way, or I could have an electron spin around that axis going this way. Again, this is not actually what's happening in reality. The electrons don't really spin on an axis like a top, but it does help me to think about the fact that we have two possible values for this spin quantum number. You could spin one way, so we could say the spin quantum number is equal to a positive one-half. Usually you hear that called spin up, so spin up, and we'll symbolize this with an arrow going up in later videos here. Then the other possible value for the spin quantum number, so the spin quantum number is equal to a negative one-half. You usually hear that referred to as spin down, and you could put an arrow going down. Again, electrons aren't really spinning in a physical sense like this, but, again, if you think about two possible ways for an electron to spin, then you get these two different, these two possible spin quantum numbers, so positive one-half or negative one-half. Those are the four quantum numbers, and we're going to use those to, again, think about electrons in orbitals.