Hello, this is another lecture topic video discussing quantum numbers. So remember from the Schrodinger equation, we get this really cool wave function with the symbol. And what does that tell us?
It tells us a lot about the electron, particularly the location of the electron. Now, let's remember that because of the Heisenberg uncertainty principle, we cannot know, all right, the position and the velocity of an electron simultaneously to any sort of precision at the same time. OK, but this is starting to tell us where the electron is.
And the cool thing about it is if we square this function, what do we get? OK, we get. probabilities of where the electrons are likely to be found. All right, and so what this is doing is this is describing essentially orbitals, all right, not to be confused with orbits, all right, so that's when we're talking about Bohr's model of the atom. We're talking about orbitals, and what are orbitals?
Orbitals, excuse me, they describe the distribution of electron density, okay, electron density. So we get increased electron density. We get an increase, okay, in probability of being able to find a particular electron, okay? So these are probability densities, okay?
And again, it's an orbital, all right? It's describing an orbital. And orbitals have characteristic energy. And they also have this characteristic distribution of electron density.
So... When we're talking about orbitals, how can we start to describe these things? All right.
And that's where quantum numbers come in. All right. These are numbers that are used to describe the arrangements of electrons in an atom.
And there are four of them. So we have four of these guys that we're going to talk about. And so the first one is this principal quantum number. It is designated as n. All right.
So this describes to us. the shell of electron. So this is the shell of an electron.
So maybe a bad analogy, but I like to think about it this way. Think about your atom as an onion, okay? You move from the center of the onion outward, all right, as n increases.
So you're adding more and more and more layers or more and more and more shells, okay? So that's the principal quantum number. What is that number telling us? is telling us something about the size and the energy of a particular orbital. And it can have any positive integer value from one and theoretically all the way up through infinity.
Because again, these are things that are coming out of the Schrodinger equation. All right. So we can push that equation to the max. And who knows, maybe some decade down the road, someone will try and. model something that's way out here past what we would actually see on the periodic table.
But again, we have these shells that are described by these principal quantum numbers, and these shells are telling us, or they're relating to the size and the energy of a particular orbital. So as we increase in our principal quantum number, we're not only increasing in energy, but the electrons within those orbitals, on average, spend... their time further away from the nucleus.
So electrons are further away from the nucleus as we increase our principal quantum number, which also means that these orbitals are progressively getting bigger. Okay, so that's our principal quantum number. The next one is the angular momentum, all right, and it has this kind of weird scripted L, okay, and what is this one telling us? This is telling us something about this shape. Okay, the shape of an atomic orbital.
And it can adopt any integral value from zero to n minus one. So again, if n equals three, all right, our angular momentum quantum numbers can be L equals zero, one, or two. Okay, and so what are these describing? These are our subshells, okay? So these are our sub...
Sub... All right. And each of these angular momentum quantum numbers, 0, 1, 2, and even...
three have these letter designations, S for zero. When L equals one, that is our P orbital designation. When L equals two, that is our D orbital designation.
And when L equals three, that is our F orbital designation. Now these may seem like pretty obscure naming, all right, but they do stand for things. S stands for shark.
P stands for principal, D stands for diffuse, and F stands for fundamental, right? Now, we're never going to ask you this, but this is where these letter designations come from, all right? And so, we get these shells defined by our principal quantum number.
We get these sub-shells, all right, defined by our angular momentum. These things have orbital designation or letter designations. All right.
And this is telling us about a particular shape of the atomic orbital that takes us to the magnetic quantum number and that's M sub L. All right. And what this is doing is it's telling us about the orientation of a given orbital.
Okay. So if principal quantum number represents shells, angular momentum corresponds to subshells. The magnetic quantum number is talking essentially about the orbitals themselves, okay, and how they're oriented in three-dimensional space around the nucleus of an atom, okay.
And so these quantum numbers can have integral values that fall in between L and minus L. So let's take our example up here. We had an L equals two. So if we say n equals 3 and our l values are l equals 0, 1, and 2, what would this orbital designation be for our angular momentum when it equals 2?
All right. Well, it would be 3d. Okay.
3. coming from our principal quantum number, telling us again about the size and the energy of that particular orbital, and d telling us about essentially this shape, okay, of this particular set of orbitals. Now, how would this look in terms of our magnetic quantum numbers? Well, remember, we can go from l through minus l, or minus l through l, and it also includes zero.
Don't ever forget this. So I always like to draw it out like minus L to zero to L. All right, so if we're at L equals two, what can we get for our M sub Ls, our magnetic quantum numbers?
It would be minus two, minus one, zero, one, and two. So in this case, we would have five orbitals, all right? And each of those five...
orbitals would have a different orientation around the nucleus of that particular atom. All right, so why don't we take a look at the allowed quantum number values for the first four energy levels of an electron. So when we're at n equals one, again this is our principle, okay, principle, talking about energy. and we're talking about size and this does essentially dictates our what shell we're in okay so when n equals one uh by definition all right l is equal to um zero through n minus one so the only um l our only principal quantum or angular momentum quantum number that we can have is zero okay And so if we go back, we can see that zero, all right, the letter designation for that is S, all right?
So our sublevel designation would be 1S. All right, so N, again, is the principal number, tells us about the energy and the size, tells us what shell we're in. L, okay, we're talking about the shape, okay?
Let's write this here just to remind us our shape, okay? And this is telling us which subshell. we are in.
All right. So this can be zero through n minus one. And our m sub l, all right, our magnetic quantum number can be minus l to zero to plus l. I guess I'll put this script on here. All right.
Well, l is zero. Then our magnetic quantum number is also zero. All right. In other words, we only have one orbital.
All right, and this designation is one S. All right, well, let's say we move up to two. All right, well, again, let's follow our rules. We can go to zero all the way up to N minus one. So we can have two designations for L, one being this two S.
Again, two coming from the principal quantum number, S coming from the letter designation for zero. And then we can have two P, all right? P coming again.
from this letter designation for when l equals one. All right, so just like we saw up here with n equals one, all right, our 2s also has a magnetic quantum number equal to zero, all right, which means it only has one orbital, okay. Whereas the 2p, all right, if we have l equals one, we can have minus one to zero to plus one. All right, so we get three orbitals in this case.
Now let's look at number three. In this case, L can be zero, one, or two. All right, again, let's follow our little rule here.
All right, zero corresponds to 3s. Again, three coming from the principal quantum number. All right, n equals three.
The one is p and the two corresponds to d. So we can go back here to this little table, all right? And how do our magnetic quantum numbers look?
Magnetic quantum number being zero, so we only have one orbital. The 3P, all right, just like what we saw up here with the 2P, we have minus one, zero, and one, so we got three orbitals, all right? Again, all three of those have a different orientation in space, so that's what we're talking about.
So this is orientation. and essentially number of orbitals. And now let's look at 3D. All right, so now since we have two for our angular momentum, we can have minus two, minus one, zero, one, and two. All right, so we can have five orbitals.
All right, and again, those are all gonna have a different orientation, okay? And then we can proceed through the principal quantum number of four. All right, you can see the same sort of pattern that arises with the 4s, the 4p, and the 4d, but now we have an angular momentum quantum number of three. All right, let's go back to our table, and three represents f.
All right, it's designated by the letter f, so this would be a 4f. All right, and then again, following our rules, we can go from minus three to zero, all the way to plus three, so we get seven different orbitals, and again, Each one of them are oriented differently in three-dimensional space. And again, because we've moved so far up in terms of our principal quantum number, energy is higher with these guys. The electrons within these orbitals are on average further away from the nucleus than those electrons that are in the earlier shells.
OK, and that also tells you that these orbitals are much more diffuse. So something you may be able to notice, all right, is that the number of orbitals is always equal to n squared. All right. So what do we do if we take one squared?
Well, the number of orbitals is one. Well, what if we take two squared? That equals four. How many do we have? We have four total orbitals.
Three squared. equals nine. We have five, six, seven, eight, nine. Okay.
Nine orbitals. What about four? That's 16. And how many do we have?
Seven plus five is 12, 13, 14, 15, 16. We got 16 orbitals. So that's a quick way to know how many particular orbitals there are in any given shell. So let's do an example. List all.
Allow quantum number values for the fifth energy level of an electron in an atom. All right. So the fifth energy level, that tells us that we are in equals five. As you guys will see on the periodic table, we have seven, actually, energy levels on the periodic table.
One for each of the periods. Okay. So remember, columns tell us something about the groups. All right. And if we're looking at the rows, those are called periods.
So there are seven periods. So we should have seven energy levels on the periodic table. So we're on n equals five. All right, n equals five. And what do we want to do?
We want to list all the allowed quantum numbers for that particular energy level. Well, if we're following our rules, all right, so if we're looking at our angular, momentum, quantum number. Again, something talks about the shape. All right.
So that's L. And remember that L can be anything from zero up to N minus one. So we can have a value of zero, one, two, three, four.
Okay. And what are the letter designations for these? The zero is always S. So this would be five S again. that five coming from our principal quantum number.
One would be five p, two would be five d, then five f, and now what about four? Well we left this off on the table, right? So this is purely hypothetical at this point, all right?
But as man discovers additional or creates additional elements pass our 118th element, all right, we're likely to discover that there are these additional subshells, all right? And if they do, what they're going to do is they would call this G, 5G. You guys will never need to know this for an exam, but interesting bit of trivia, okay?
So they're just going to continue with the alphabet. All right, so N equals 5. And then we see what our different values for our angular momentum quantum numbers can be. And now what are our magnetic quantum numbers going to be? So again, this is our M sub L, all right? So what's the rules for that?
We can go from minus L to zero plus L, all right? So for this one, clearly has to be zero and P, all right? it can be again here's l is equal to one so we can have minus one to zero plus one then we can go to minus two minus one zero one two and then we can go to minus three minus two minus one zero plus one plus two plus three and then minus four minus three minus two minus one zero one two three and four okay so again this is size energy this is our shell this is our sub shell and this is talking about shape And these are our orbitals. The numbers are telling us about the orientation in space. And so how many do we have?
One, two, three, four. So one plus three, that's four. Then we got five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen. 17, 18, 19, 20, 21, 22, 23, 24, 25. We have 25 total. Well, we're at n equals five.
Again, that's our principal quantum number. So five squared is equal to 25. So we have 25 orbitals within this particular shell. All right. So now here's a participation question for you.
Excuse me. Which of the following sets of quantum numbers are not allowed? Select all that apply.
So what I would do is I would start with your principal quantum number, figure out which angular momentum numbers are allowed for each given scenario. And then see if any of these are breaking the rules. Again, go back to the last couple of slides, review those rules.
Remember which quantum number talks about the shell, which one talks about the subshell. which one talks about orbitals, and what these different quantum numbers actually describe. OK, it's really, really important. So again, which sets of these numbers of these quantum numbers are not allowed? And again, this is a select all that apply.
All right. Select all that apply. OK, so I told you there are four quantum numbers.
Well, what was the last one? The last one is electron spin. OK, so it's our fourth quantum. And so experiments on these emission spectra, so go back a couple of lectures, all right, we talked about emission spectra of hydrogen and sodium, indicated that each line in the emission spectrum could be split into two lines by the application of an external magnetic field. Okay.
So in other words, what they could do is they could put a magnet kind of flanking this emission spectra. And what they always found is that there were two regions of electron density on the other end, okay, suggesting that some electrons were attracted to the positive pole of that magnet while the other were attracted to the other pole of the magnet, excuse me. And so the only way that that was possible is if these electrons were actually spinning in space, okay, because remember electrons carry this negative charge.
They're like a negative point charge. And going through physics, what you'll find is that spinning charges produce a magnetic moment. And the orientation of that spinning charge can produce an opposing magnetic moment.
So if we look at this particular spin, where ms, that's how we designate this quantum number, equals minus one half. Well, this electron is spinning in a clockwise orientation. And because it's carrying a point charge, what we do is we get this magnetic moment that's kind of being indicated by this green arrow.
Well, that's going to be attracted to one of the poles of that magnet that was put in place or flanking that beam of that emission spectrum. So whereas the other electron, okay, in this case, it is spinning. counterclockwise, all right, it's counterclockwise.
And so this is m sub s, all right, so our electron spin of plus one half, all right, and it has an equal but opposite magnetic moment, again, represented by this green arrow, right. And so when they saw that we had these two kind of regions of electron density that are coming from the submission spectra, all right, they were equal to each other. And the reason they're equal is because in this particular experiment, the two spins on these electrons were equally represented.
And because again, they're point charged, they're spinning, that gives them a magnetic moment. So they are attracted to another magnet if the poles line up properly. So we get this electron spin quantum number, this m sub s.
All right. And there's only two possible values for this can be minus one half or it can be plus one half. All right.
So that's our four quantum numbers that are describing. All right. Where these electrons are kind of hanging out.
All right. Around the nucleus of an atom. All right.
So we get the energy shell and the size from our principal quantum number. We get the shape. All right.
From the angular momentum. All right. And then from the magnetic quantum number. All right. We get the orientation and then we can have our electron spin.
All right. If you have any questions, don't hesitate to reach out. I'm happy to help. And I hope you have a great rest of your day.