In this lecture, we're finally going to address the question of how electrons move. We know that the electrons move in the big empty space of the atom, outside of the tiny nucleus. We know that the electrons arrange themselves into groups that we call shells, that are varying distances away from the nucleus. We know that the electrons can have only certain specific energy levels, consistent with the atomic spectra that we looked at in the previous lecture. But we still don't have any idea of what the electrons actually do. We've drawn these pictures of electrons moving around the nucleus looking somewhat like the planets, moving around the sun and the solar system. In this lecture, we're going to find that that's actually not an accurate picture of how the electrons move. Let's remember what we know so far about atoms is that at least in a hydrogen atom we can describe the energy of the electron in any one of the given energy levels according to a quantum number n. And we can use this equation to calculate the energy. But it doesn't tell us why the electron has that particular energy. It must somehow or another be related to its potential energy, that is its attraction to the nucleus, and it must somehow be related to its kinetic energy, that is how rapidly it is moving. So what can we know about the motion of electrons? Well, one thing that we discover actually, is looking experimentally. So that if we take beams of electrons and we either defract them off of a gradient or we pass them through a grid, we discover we get these interesting and beautiful just, diffraction patterns which you can see in the slides which are here. Notice the very intense bright spots and notice the areas of dark between the bright spots? In both of these images, what do we learn from the fact that the electrons in fact will diffract when we pass them off of a grating or through a grid. Well, why do we have diffraction? Turns out diffraction arises from the interference of waves, it's actually a commonly observed phenomena. Hopefully most of you have seen diffraction of waves. Near a coastline where waves are, water waves are encountering, say, rocks. And we wind up with incident and outgoing waves or waves coming from different regions around the rocks interfering with each other. Different parts of the wave may reinforce as they collide, or they may subtract each other out as they collide. And as a consequence, we may wind up with peaks and we may wind up with valleys. What that suggests to us then is that electrons move as waves. That we can use waves to describe the motions of electrons. Well can we actually see electron waves? It turns out that the answer to that question is yes. That about 25 years ago or so, scientists at the Almaden lab of IBM used a scanning tunneling microscope to position the atoms on a surface and to then image those atoms. They actually showed that they could use the STM not only to image the atoms on a surface but also then to arrange the atoms. So here in this image we see atoms located variously around on a surface. Each one of these little bumps. That we're pointing to here. Let me draw a circle around one of them. That little bump right there, is, in fact, a single atom, sitting on a metal surface. Sitting on top of a metal surface. What you can see is that the scientists actually progressively arranged the atoms to fall into a, a sort of a semicircle pattern here. Those rural atoms which have been picked up and relocated. You notice here that they there slowing, but surely beginning to close the loop. And then in the final picture there, you see that they have successfully closed the loop. And notice what happens when they do so. In this region in the middle in here, we see a clear pattern of troughs and peaks, consistent with the idea of waves. Now, what are these waves of? They're not atoms, because the atoms are the larger dots that you see on the surface here. They're actually the motion of electrons moving on the surface of the sphere, trapped in what the scientists called corrals. And inside the corral we wind up with what looks like a standing wave. For electrons moving about apparently resulting from constructive and destructive combinations of the electron motion inside there. This relatively well establishes that electrons are actually moving as waves. And what are the consequences of saying that electrons are moving as wave? Well let's see what's, what can we say is different about wave motion versus particle motion? One thing we can say, is that with a particle we can actually know exactly what its location is, and we can know what its trajectory is going forward. We can know both where it is and how it travels. As a consequence we can localize it. And we can watch its path or its arc as it moves around. But a wave is not so. A wave is in fact delocalized. It doesn't actually sit in one particular location. And you can have a stationary wave, such as the waves that we're seeing in this diagram here. As a consequence, the primary thing that we learn from this is that we will not know the position and the motion of a wave simultaneously. We have to abandon the idea that we're going to know the orbit in which the electron moves around the nucleus. We cannot know -- much the same way that Earth moves around the sun, we cannot describe the way that the electron moves around the nucleus. As a consequence, there's this thing that we will call the uncertainly principle. The uncertainty principles says that the uncertainty in the position of the particle and the uncertainty in the momentum of the particle both cannot be 0. In fact, neither one of them can be 0 because the product of those two has to be greater than a non 0 value. You don't really have to worry about what the value of that is other than for electrons it is a significant number. As a consequence, we have to abandon the idea that we're going to know the path that the electron follows, or know its orbit. Then what can we know? The answer turns out to be that we can know a probability function that tells us where the electron might be. Before we actually figure out what this probability function is going to look like, we're going to do a brief exercise here. And I want you to actually think about this yourself, maybe even get out a piece of paper and follow along with it. Imagine that we were trying to draw a probability map for movement for the last week, say between home, and if you're traveling to a college campus, here are various sites around the campus. Here's the coffee shop on campus, here's the gym on campus, here is one of the academic buildings on campus, maybe where the classrooms are, and over here might be the library. This is wherever your home is. Imagine the way in which we try to map out the probability was to have a camera overhead looking down on your position. And every 5 seconds or so we took a picture of you and we put that on a map. So back over here, let's imagine now, here's the house. And over here for example, is some big academic building. And let's see, what do I have up here, is the gym. And over here is the coffee shop which I'll make look like a little cup. And down below here is the library. And where might I be found most of the time? Well it seems that I might be found near the home quite often. Because that might be in the morning hours or around lunchtime, or in the evenings hours. So I would expect actually that this camera that is taking pictures of me would put a lot of dots somewhere around the house. And if I have classes in this building over here, there'll be some dots back over here. Notice there won't be all that many dots in between because I don't really spend much time walking between the house. And the and the academic building. If like many of us I like to go to the coffee shop there'll be some dots here as the camera catches me in motion in the vicinity of the coffee shop. And if I'm good about studying there may be some dots over here by the library. And since I don't get to the gym nearly as often as I want maybe there's only a few dots over here. But notice there are large regions where there really are virtually no dots because there's very little probability of finding me here. You know that I do travel in these paths, you just don't know what the paths are that take me from location to location. Overall then, I don't necessarily know anything about the path of a person whose dots and probabilities are described by what we see here. But I do know a lot about their behavior. I know how much time they spend at home, I know how much time they spend in class, I know how much time they spend at the coffee shop, how little time they spend in they gym and so forth. And it turns out, that's the kind of information that we can know for a, an electron. We can actually draw a map of what we are now going to call the orbital. A kind of a take off on the word orbit that says we can't really know what the orbit of an electron is, but we can know something about its probability distribution. And here's what the graph actually looks like. And this image mimics something that is on a site I'm going to show in a minute called The Orbitron from Sheffield University. And what you see here is actually a series of dots. They tell me the probability for where the electron might be about the nucleus. The nucleus is actually represented right here in the origin of this particular xyz diagram. And notice that the probability gets very dense near the nucleus and much less dense out away from the nucleus. Let's actually take a closer look at this by going over to examine this particular site which is the site at Sheffield University, University of Sheffield. And here is, in fact, what the lowest energy probability distribution looks like for an electron in a hydrogen atom. In a little while, we're going to see that this is called the 1Sor, but we'll describe what that means in a moment. One of the nice things about this sign is, you can actually pick up the image and rotate it around. And you notice that we wind up with a probability distribution that looks roughly spherical. It doesn't matter which way we go out from the nucleus. The probability of being a certain distance away from the nucleus is the same in all directions. It's a spherical distribution. Notice that the density is intense near the middle, lots of probability of finding the electron near the nucleus. Not too much probability of finding the electron farther out, away from the nucleus. You can actually examine a number of these probability distributions. Let's look at, say one that was called the 2s here. To find these probability distributions, click on the little number at the top that says dots and hopefully here we go. Notice that the 2s again looks spherical and the nucleus is buried right about in there. Let's look at one that is called 2P and hopefully, well it's still asking me if I want to run and I do. And we'll see actually here a distribution that doesn't look spherical, the nucleus actually is about here, between what we see as this blue lobe and this red lobe. Lots of probability to be around here, lots of probability to be around here, almost no probability, in fact zero probability, it turns out, to be immediately adjacent to the nucleus. We can pick this image up, and notice it doesn't really seem to matter too much if we rotate it this way, but if we rotate it this way, it actually has sort of a cylindrical symmetry. As we look along the axis, between those two lobes. What have we learned, actually, from learning about these probability distributions. Go back and look here and the answer turns out to be, that we can describe these orbitals according to 3 different properties. The first has to do with the energy of the orbitals which actually is determined in large part by the size of the orbital. And it turns out that each hydrogen orbital can be described by the quantum number n, but we've see that before. We recall actually that the energy of a hydrogen orbital depends only upon the n quantum number and you remember the formula. We even showed it at the beginning of this lecture is, this is an n here, is minus hr 1 over n squared. n can have any integer value from 1 on up. And the higher value of n, the lower the energy that is the high, I'm sorry, the higher the energy, the closer towards 0 it is. Because the energies are always negative and the larger value of n corresponds to a smaller fraction here so the energy actually goes up as n goes up. So the energy's described by n. There's a shape to the orbital. You'll recall as we look back at these images here. We get a different shape to the orbital when we ran the 2P than when we ran the 2S and in fact the shapes of orbitals described by a quantum number this is actually an l. That's the closest font I could find to the script letter l, that's what I was trying to draw. And notice that the values of l can range from 0 up to the number, n minus 1. If n is equal to 1, then it's 0 or 0. So l must be 0 when n is equal to 1. When n is equal to 2, l can be 0 or 1. When n is equal to 3, l can be 0 or 1 or 2. And the difference between the different l quantum numbers tells us the different kinds of shapes. For example, here, this corresponds to an l equal to 1 orbital. If we look, instead, at an l equal to 0 orbital. Click here. We see that it has a spherical shape. So we learn something about the shapes by knowing what the l quantum number is. And then, finally, actually, there's a third quantum number back over here corresponding to the orientation for particular kinds of orbitals. If it is an orbital which is spherical, we would imagine it wouldn't have much of an orientation. In fact, there should be only one orientation because it's a sphere. And in fact, when l is equal to 0 that's the spherical orientation. And according to what is written here the quantum number m that tells me the orientation has to run between 0, this is, should actually say minus l and 0, and so it has to be 0. So, in fact there is only 1 orientation for a spherical orbital. How about for the p orbital that we looked where l was equal to 1? Then m could be minus 1 or 0, or 1. And there would be 3 orientations for the p orbital. If we go back and look at the p orbital again, in fact, there are 3 orientations. They are depicted here. And they correspond to each of the axes, x, y, and z. Or, alternatively stated, I can orient this along the x axis. I can orient it along the y axis or I could orient it along the z axis. So there'll be 3 values there. So now we can characterize those orbitals. The last thing that we're going to do today is simply indicate that there is a common chemistry nomenclature for describing the orbitals. When l is equal to 0 we call it an s orbital, just a name. When l equal to 1, we call it a p orbital. When l is equal to 2, we called it a d orbital, and when l is equal to 3, we call that an f orbital. And these are simply names that help us keep track, of what the probability distribution looks like. Remember, from what we saw before, when the, the n quantum number, the larger it is, the larger the radius of that orbital will be. The l quantum number tells us whether it's spherical or one of these dumbbell shapes, and the end quantum number tells us what the orientation is. What else can we get out of this? Well let's just check to see that we have the right nomenclature, so when we have a 1 s orbital, when we use that nomenclature, what we mean, the 1 number means n is equal to 1. The s means that l is equal to 0, a 2p orbital means that n, the n quantum number is equal to 2. Which will be higher energy than n equal to 1. And the l is equal to 1, because p means that the l is equal to 1. Notice that everything I talked about here in this lecture has to do with hydrogen atoms. All the different ways in which an electron can move when it is an electron in a hydrogen atom. Lots of different orbitals that it can be in characterized by these n, l, and m quantum numbers. And given names according to what the n, l, and m quantum numbers are. In the next lecture, we're going to talk about the ways in which the electrons move, when there are lots of electrons in the atom. In other words, for all the other atoms besides the hydrogen atom, and we'll pick that up next time.