Chemistry: Atomic Orbital Shapes, Quantum Numbers

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Let's take one last look at the Bohr atom before abandoning this model for good. Although the Bohr atom ultimately was unsuccessful, one of the things that it showed us was how we could mathematically describe having quantized energies. So I return back to this formula, and again the important part of this is this term n^2. n^2, again, was a positive integer and what it did for us mathematically is by n being only 1, 2, 3, or 4, so on, only our integer value, it quantized the resulting energy states, because n could not be 1.2 or 1.3. Again, we had a finite number of solutions in the Bohr atom. This same basic idea then is going to be expected as part of the outcome of the Schrödinger equation. In other words, part of the mathematical solution to the Schrödinger equation are quantum numbers and in the case of Schrödinger equation, there are in fact three quantum numbers that we need to worry about.
So let's look at those for a moment. We have as part of the Schrödinger equation three quantum numbers which will ultimately help us describe these different solutions to what state an electron can be in. There is the n quantum number or the principle quantum number. In fact, this n turns out to be the exact same n that we saw in the Bohr atom. This relates to the energy of the electron and, in fact, is no different than before. n can take on a value of 1, 2, 3, 4, again, any positive integer. The next quantum number we run into is l or the angular quantum number. l can take on values anywhere from 0 up through n - 1. And finally the third is the magnetic quantum number, m[l]. A magnetic quantum number can take on values anywhere from l all the way down to -l.
So how does this work in practice? Well, let's take the simplest case of n=1. If n is equal to 1, then our requirement for l is that it is anywhere between 0 and 1 - 1, or 0. So the value for l then must be 0. Our value for m[l] must also be 0 because it can only range from l to -l and in this case again, l is 0. And so we have one unique state corresponding to l = 1. That's the orbital we called 1s. For n = 2, we can have, once again, the value of l=0. That will describe once again an s orbital. In fact, let me go ahead and write that down. Anytime l is equal to 0, no matter what the orbital is, if l is equal to 0, it is an s orbital. That's a historical name that's applied to this. We know what the shape is of an s orbital and so from now on, we will associate that term, any S orbital, with it having an angular quantum number equal to 0. So again, the 2s orbital we've seen before and that corresponds to l is equal to 0.
But what about l is equal to 1? What does that mean? Well, right away we know that if l is equal to 1, m[l] could have 3 different values: 1, 0 or -1. So there are three other orbitals corresponding to an angular quantum number of 1 that we haven't been introduced to yet.
So let's look at that now. Meet the p orbital. A p orbital as you can see right away looks very different from an s orbital. The most important difference is that it is no longer spherical but is has directionality to it. Some people describe this orbital as dumbbell shaped. Well, if you look at where you're most likely to find electron density, one thing in common with an s orbital is as you start at the nucleus and move away, at least in this direction, we reach a maximum of electron density and then it tapers off again as we get further and further away from the nucleus. So as I've shaded in here more, we have more electron density here than here or here, for instance. But also again, you'll note the obvious difference in the shape.
Now, we described that difference in shape as an angular node. Why on earth do we have that term angular node? Well, suppose that we're worried about this electron being right here in some instant in time. If the electron were here, we could describe that electron as being 30 away from the z-axis. In other words, the direction that the orbital is pointing. If the electron were down here though in this plane, that angle would be 90. And the fact that we don't have any electron density in this place indicates that we have an angular node. In other words, when this angle is 90, we have no probability of finding the electron there. So a fundamental features of any p orbital is that it has one angular node. In other words, that it has this dumbbell shape. One plane is which we cannot find the electron density.
So once again we've introduced this term angular node and, in fact l = 1 indicates that we have one angular node. The important thing to remember about this is not so much the term "angular node," but just simply any time l is equal to 1, it has a shape that's very different than the shape of an s orbital.
Now, remember we had, in fact, 3 different solutions. m[l] could be 1, 0 or -1, and indeed that means there are three different p orbitals that we would call 2p orbitals, for instance. Here's the p[Z] orbital again, and I show you this one as well as the p[X] orbital and the p[Y] orbital. Let me just caution you. Notice a lot of textbooks will draw a finite boundary to these things. Just remember that just like the s orbital, these are really fuzzy boundaries. There is not any finite distance that you're going to find the electron. We have to describe it always in terms of probability of finding the electron. So it's better to think of it as kind of this fuzzy diagram here again with a maximum of probability here and here.
By the way, how does the electron get from here to here if there's a nodal plane and the electron can't pass through that plane? If that's bugging you, the problem is that you're living in a classical world. I'm living in a classical world, too. We don't have a good conceptual understanding of quantum mechanics, but we're trying to think of the electron again as a particle. If we accept the fact that it has also these wave properties, the electron can exist simultaneously both above and below that plane. So once again, we talk in terms of electron density or the chances of finding the electron being either here or here, but again not here.
Now let's return to our quantum numbers and see what's next on the list. When we get to the n = 3 level, now we have three possibilities for l. It can be either 0, 1 or 2. We know that anytime l is equal to 0 it's an s orbital. We can include in our definitions now any time l is equal to 1 that'll correspond to a p orbital, whether it's a 2p or 3p or 4p. And by the way, what would the difference be between a 2p orbital and a 3p orbital and a 4p orbital? The size just like it was in the case of s. As we go from 2p to 3p to 4p, the orbital gets bigger. The electrons get on average further from the nucleus.
Now what about this guy, l is equal to 2? Well, that tells us that there are two angular nodes. What does that look like? We'll come back to that in a moment. And m[l] could be 2, 1, 0, -1 or -2. So 5 different possibilities now that we would call a d orbital. Once again, anytime l is equal to 2, we refer to that as a d orbital.
So let's look at the d orbitals. Here they are. Five different d orbitals depending on what the value of m[l] is. That is, the d[yz], the d[xz], the d[x]^2[ -y]^2[ ], the d[yz] and the d[z]^2 . Who came up with these names? I don't know. But these are the 5 d orbitals. Notice that they all have two angular nodes. It's not necessarily easy to notice, but we can understand that here, one of the nodes corresponds to this plane, one of the nodes corresponds to this plane, so those are the two angular nodes. What about this guy down here? Well, that's a harder one to see. We no longer have a nodal plane. In fact, we'd have a nodal cone, if you would. There are two angular nodes here though. One of them is that angle where you won't find any electron density. The other is this angle where you won't find electron density. So a common feature of all d orbitals is that they have two angular nodes, and that there is a total of five of them.
Now we can play this game all day long. Let's go one more example. If n is equal to 4, we have now four possibilities for l, the s orbitals, the p orbitals, the d orbitals and there is 1, 3 and 5 of those respectively, and now l = 3. I think you're catching on now. If l is equal to 3, there are three angular nodes. I'm not even going to begin to try to draw what that would look like, but I have the help of our content box. These are the f orbitals and, again, you'll notice that m[l] now can be 3, 2, 1, 0, -1, -2 or -3. So a total of seven f orbitals now. Any time we have f orbitals, there's a total of seven of them, but the first time that we ever can have f orbitals is not until the n is equal to 4 level. So the 4f orbitals will be the first f orbitals we encounter. There will be no such thing as 3f or 2f, because quantum mechanics doesn't recognize that. Those are not part of the Schrödinger solutions.
So what have we learned? Well, we said that the Schrödinger equation in addition to telling us information about the energy of the electron, and that's coming. That information is embedded in what the principle quantum is. We have information about the shape of the orbital. In other words, where are we mostly likely to find the electron in different orbitals. The angular quantum number tells us how many angular nodes and that defines the shape, whether it's an s orbital with no angular nodes, so it's a perfect sphere, p orbitals with one angular node so it's a dumbbell-shaped thing, a d orbital with two angular nodes, an f orbital with three angular nodes, a g orbital with four angular nodes. It goes on and on.
Now, it turns out electrons are not in g orbitals, for the elements that we know of at least, the only ones that we've been able to actually observe so far. So we don't talk much about g orbitals, but they're there. They just don't have any electrons in them. []
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Atomic Orbital Shapes and Quantum Numbers Page [2 of 2]