In our learning objective number nine, we're going to start looking at the shapes of these orbitals. So I want you to know the shapes of the s, the d, the p. And I don't know why I said it in that order. The s, the p, the d orbital, the f orbital. And we are going to be looking at the energies of these orbitals for a multi-electron system. So we know that the orbitals are defined by our Schrödinger equations. They are defining regions in space that have a high probability of locating an electron. This picture here represents an s orbital. So it's shaped like a sphere. When l equals zero, we have got the s sub-level. And for that sub-level, we have an m sub l, which could be anywhere from a negative l up to a positive l. So it only has one choice. There's only one orientation in space of an s orbital, and that's this sphere. It doesn't matter how you look at a sphere, it looks the same. So, if you were dealing with the 1s sub-shell, you would have an orbital that looks like this, that would be a certain distance away from the nucleus. You could also have a 2s subshell. In that 2s subshell you would have a orbital that looks like this, only it would be larger. It would be further away from the nucleus. Now the p orbitals, we know that when p, is a subshell that the l value is one. And when l is 1, the n sub l's can be a negative 1, 0, and 1. Now what I told you in our lessons of these quantum numbers is that this tells you that there are three orientations in space of the p orbitals. So let's look at those three orientations. The first orientation is along the x axis, so it is where you have a density about the orbit. So the electrons can be located here. That probability drops off as you approach the edge of this, okay? So it's just diminishing as you go by, at some point, you say, okay, we'll call the boundary here. Where there is a beyond that there's a diminished probability out here in this space, a diminished probability of finding electron, but a high probability of finding it in here. And then they typically just kind of clear it up and just give it a instead of seeing that diminishing, it's getting less and less probable as you move further away, they just choose to draw this. And they call it a dumbbell shape, though I don't know if dumbbells are really shaped like that. But thinking about weights that you're lifting up. So we've got the px, which is oriented along the x axis. Well there's three of these, so the other one would be the py, would be oriented along the y axis, and pz, where it's oriented along the z axis. So those are your three orientations of space of those three orbitals. Now you need to understand that these are overlapping each other. So let's look at this space down here in the bottom right hand corner. And we will overlap these orbitals the way they really are. You've got an orbital along the x axis. You've got an orbital along the y axis. And you've got one coming up and below the axis along the z axis. So those are the three orbitals overlapping their space. So if an electron has the right values of m, l and m sub l, they would be located in, you know, maybe this ori, orientation. They'd be the x ax, the xp orbital. We call it p sub x, okay? Now there is also, you know, overlapping all of this space, an s orbital. So this might be the 2s. It is the 2s orbital. This would be the 2px, the 2py and the 2pz. And the electrons move about in their space where they will have a high probability, but there's a lot of overlapping space that they are occupying. Let's move on to the ds. Now when l is equal to d, to 2, we are talking about d subshell. That is the d subshell. These define the five orientations in space. The five different ways that you will find these orbitals. So let's look at each one of them. This is the dyz. We have got our, lobes in-between the y and the z axes. M'kay, so look at your y and your z, and we see that those lobes are in between those axes. And we are representing that over here in that more solid shape look. Now we see areas where you have nothing. Okay, there's nothing here. There's nothing here. There's no probability of finding those electrons, okay. We can kind of draw a little plane right there, a little plane right here. Here you're not going to find the electrons. There's a very low probability of finding those, actually, it's where the probably for electron density, according to the equations, drops all the way to zero. So don't think that this is an orbital. That's not the orbital. The whole thing is the orbital. And an electron can move about anywhere in that space if it's got the right energy to be located in that orbital. Somehow it travels about this orbital without occupying the spaces in between. Or at least, according to the equations, the probability drops down to 0. Now, that's one. Here is the other one. This is the second one. We have the dxz. Now I don't expect you to learn those numbers, those letters associated with them, but the dxz is like the yz except that it is in between the node, the planes. It's on the xz plane as we see here. The next one is the dxy. Now this picture is not very good, because it is showing them on the axes, and they should actually in, be in between the axes, okay? So don't focus on this. We see the axes running through here, and those are in between them. Then there is this one. It's very, very similar to the dxy. It's called the x squared minus y squared but those nodes, lobes are actually located on the x and y planes. Okay, so you actually see them there on those planes. I mean, on the axes. So those are the first four. You ought to recognize a d orbital when you see one. So when you see these four lobes like this, as opposed the two lobes of the p orbital, you see these four lobes you know it's a d orbital. But there's one oddball, and this oddball is called the dz squared. It's along the z axis. It has no nodal plane associated with it at all, that is the dz squared. So there's the five orbitals that are the d orbitals. Let's look at the f orbitals. When l equals 3 we know that the m sub l's are negative 3 up to a positive 3. There are seven numbers so that means there are seven orientations in space. I'm going to pop all seven up here at once. Four of them look very similar. They have these eight lobes associated with them just oriented differently amou, among the axis. Three of them have this, a different shape along the x, y, and z plane. It's kind of similar to our oddball d orbital except there is a nodal plane between them so in between them, the axis we can see that there is a nodal plane separating the two halves from each other. And those are your seven f orbitals. So I expect you to recognize an orbital if you were to see a picture of that orbital. Now I want to briefly mention about phases of orbitals. The phase would be the sine of the amplitude of the wave. And if you were to have a two dimensional wave, okay, that wave could be above the plane like we see here, and it would have a plus sign. If you have a node in that, and there's a node where it's zero, you can have above and below, of the plane, and so we'd have a plus and a minus. Well, a three dimensional wave would have that very similar thing. So you're going to see when you've got these multi lobes, you might see pictures in the future in which you see multi colors. Those signs may come into play when we start bringing orbitals together to make bonds between atoms. But for now, just know that they do have those signs associated with them, and we don't need to really worry about those signs at this time. I want you to be familiar with the shapes of the various orbitals. And so I'm going to show them to you and move them about so you can see them from different vantage points. You have them in your notes and you can look at them there, but this is an s orbital. It's a sphere, there's only one orientation of a sphere, and we know from the m sub l value that there's only one s orbital. The next one is the p. And because of our m sub l's, we have three of them, negative 1, 0, and 1, there are three orientations of the p orbital. And here they are, so we have one along the x axis, we have one along the y axis, and one along the z axis. Now which one's which, I don't know, they usually call this one the z axis. But we see that we have them in the, the three orientations. Now these you need to imagine, in an atom, are actually all over top of each other so they exist in the same, on the same axis, but there are three of them and that's what a p orbital looks like. So these are our five d orbitals, and we know that n sub l gives us five different values for the d subshell, and this is how they're oriented. So four of them look very similar. They have these four lobes that we see here. In the x-y plane, these two are different in that these are between the axes, and this is on the axes, so we see that being different here. This is along the x-z plane, and they're perpendicular to each other, so that's four of them. The fifth one is very unusual. It doesn't look like any of the others. Now remember, the mathematical equations, the Schrödinger equations are what define these shapes. And so with the right quantum numbers, you end up with this three-dimensional shape. So the five d orbitals are four of them that look very similar with the four lobes, and then the fifth one which is very different. So now we have our f orbitals. And again, you just need to recognize their, their shape. There are four that are similar and then three others that have a similar look to each other. These first four here have eight lobes associated with it, so you'll recognize them by their eight lobes. And they're just a little bit oriented. In the x-y-z plane of, a little different from each other. So that's the four that look similar. And then we have the other three. And these other three kind of remind me of the d in that they have, the, well, it's just a similar shape, but there is a no plane between the two sides here. And so this is what the remaining three f orbitals look like, and they're arranged along the x, y, and z plane like we saw the p orbitals aligned along the x, y, and z plane. So remember with the f orbitals, we have got seven, because we would, m sub l values gives me seven values for the f subshell. And four of them have got the eight lobes, and then we have this general shape here. Now we've got a couple diagrams here. I want to focus on the left-hand side for right now. This is what the hydrogen atom would be. So we have the 1s orbital. Okay, now what is that, that we're in the first, we're in the first shell. The one. There is one orbital, in the, I mean there's one shell, it's called the s subshell. It has one orbital, and I've represented that orbital with a line. In a hydrogen atom when you go to the second shell, it's higher in energy, there exists the s and the p subshell in there. And there's the orbitals of the s and the p. Then we move up from there. It's not quite a big a climb as from one to two as you go from two to three. In the third shell there's the s orbital, the p orbital, and the d orbitals. There's one s, there are three p orbitals, and there are five d orbitals, and those are represented by the lines. Then we move a little further up in energy, and we have our s, our p, our d, and there are fs, and how many fs would there be? There would be seven of them, so I could draw seven lines out here beyond this, but I ran out of space and I didn't draw the lines, okay? So there does exist the 4f orbitals as well, and they'd be seven lines associated with them. But in the hydrogen atom, where we only have one electron, we say that all of those orbitals, and let's just focus on the d right now, all of those orbitals are said to be degenerate. They have exactly the same energy. All of them. And this is why hydrogen is so much easier. And when we did the electrons, calculations and we did them for hydrogen atoms, we had nice, simple equations. As we move over to the right side of this page, we see that we're dealing with an elec, a multi-electron atom. As soon as you put a second electron in there, because their spins in a, their interactions of their spins the orbitals don't stay, all of them, the dinner, within a shell. What happens is, the p's raise and have higher energy then the s's have. And the d's are higher then the p's, so here's a 3s. The 3p is a little bit higher. The 3d is higher yet. Then we have a the force, we have the s, the p, the d, and then the f's would be way up high, somewhere up here, and we'd have the 4f's. So they spread out. Now within a shell, I mean a subshell, okay there's the 2p subshell. Within a 2p subshell, all the orbitals have exactly the same energy. They're degenerate. Okay, they have exactly the same energy within a sub-level. But you don't keep the energies within the level all the same. So as soon as you put two electrons in there, as soon as there is more than one, then the energy of the s orbitals will be lower than the energy of the p orbitals of a subshell, which is lower than a d, which is lower than the f. And we have this spreading that occurs. So eventually we need to start putting electrons into these orbitals and assigning them their location in an atom. And that's what our next learning objective would be. But for now we've seen what they look like and we've seen their energies from the lowest energy up to the highest energy. Both within the, let me change the color of my pen. So both within a shell we see that we're going from a 1s. Higher in energy are the twos, higher in the energy are the threes. And then we see also within the actual sub-levels, that we have this increasing of energy from s to p and p to d. Okay, we'll keep that in mind as we move forward to our next learning objectives.
