Chemistry: Atomic Orbital Size

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So our search for the ellusive electron continues. The Bohr atom was a very appealing model in that it played to our general understanding of a planet in orbit around the sun, but it's ultimately wrong because it implies that we know exactly the position and momentum of the electron, which we can't by the Heisenburg uncertainty principle. So it's not until we understand the wave properties of matter, and in particular, of electrons, that we're starting to be able to really come to an answer here.
Now, notice we're going to go through a subtle change. Our quest for the location of the electron is going to end because we know that that's something we'll never be able to find. But now we're going to ask where are we most likely to find the electron? So instead of looking for exact location, we're looking for probability.
Now, it's Schrödinger who brings all of these ideas together, knowing about the potential energy and kinetic energy of our system, again, I'm talking about the attraction of the electron and the nucleus as well as the speed and mass of the electron, along with the notion that the electron has waves properties, we arrive at a series of solutions to this problem called wave functions. These come out of what's called the Schrödinger equation. These wave functions have embedded in them information talking about not only the energy of the electrons in the hydrogen atom, but also where we're most likely to find the electron.
So we're going to take some time now and look in depth at what these physically are. For each solution of the Schrödinger equation, we refer to a state, a state of being, if you will, of the electron. Now, what we want to do now is understand more of the physical properties of each one of these states. Each state, in fact, will be characterized by an energy but also by a probability. In other words, a shape, if you will, of where we're likely to find the electron density.
So we're going to start out by becoming more familiar with the lowest energy state of the hydrogen atom, what we refer to as the 1s orbital - and the names will make more sense soon. The 1s orbital - if we take , which is again this mathematical expression describing the electron and we square it. Now, you might ask why square it? We'll get to this a little later why we're going to do this. But right now I just simply need to tell you that by squaring , that is what corresponds to the probability of finding the electron in space. And that's what we're trying to answer right now. If I take squared and look at that function as a function of distance from the nucleus, r, what I find is I have the highest probability of finding the electron very close into the nucleus. And as I move away from the nucleus, it exponentially decays as I get further and further away.
Now, remember that this is a three-dimensional object we're talking about and I'm only looking at what happens as a function of r going in any direction. So I have some sense of if I could look at this orbital, and that's the nucleus, as I get further and further away in any direction from that nucleus, my probability of finding the electron decreases.
Now, what if I ask where am I most likely to find the electron as a function of distance from the nucleus? In other words, at what distance do I have the greatest chance of finding the electron? Well, the quick answer that you might think of is at the nucleus. Didn't you just say that? But, in fact, I'm asking a slightly different question. I'm asking what would be the distance that I'd have to go to have the greatest chance in any direction now of finding that electron. In fact, what that's going to look like is somewhat different. In fact, I'll have very little probability, actually zero probability, of finding the electron right at the nucleus, but my probability will increase, and it reaches a maximum at about a half an angstrom. What's going on? It sounds like I'm asking the same thing but I'm giving you two different answers.
To understand this, let me give you an analogy. Suppose that you have a radio station and you located at this position in town, and let's suppose that each one of these squares corresponds to one city block, running this way and this way. Now, suppose I ask at what distance are you going to have the most people that's listening to your radio station? Well, the first answer you might give is, "I'm going to have the most people listening to my radio station at the very closest distance away from where my radio station is located, because I know as I get further and further from my radio station the signal gets weaker and weaker. So I have the greatest chance of someone hearing my radio station if I'm very close to the radio station. But you also might be thinking, now wait a minute. As I get further and further away, the number of people that are at that distance increases. In other words, there's a lot more people at this distance than there are at this distance. And likewise, there's even more people way out here. So while the signal gets weaker and weaker going away from the radio station, the number of people increases, simply because we're getting into a larger and larger circle in this example. That's basically the idea of why we need to look at this other graph.
What it's saying is although you have the greatest chance of finding the electron right at the nucleus, if you go away in any particular direction, if you instead ask where, at what distance are you most likely to find an electron, taking into account the fact that as you move further and further away, you have a greater and greater volume, then what we find is that point comes right here at a half an angstrom. Now, understand this does not mean that that's the edge of the atom. The atom is indeed a very fuzzy, fuzzy thing. It's not like a beach ball where there's a defined end to it. But, there is a maximum of probability, and then that probability decreases as we get further and further away from the nucleus. Likewise, it decreases as we get closer to the nucleus overall. So again, this graph is commonly used by chemists, and it tells us that this is where you're most likely to find that electron.
So that's the 1s orbital. What about the 2s orbital? Again, let's look at the radial distribution function I've just drawn for you again what we just talked about the 1s orbital. And again, this is the position where we would have the greatest probability of finding the electron. The 2s orbital, by comparison, is a little bit more complicated function. In fact, you'll notice here that the probability actually goes to zero again, and we call that a radial node. But what I want to point out to you right now is that the greatest probability of finding the electron is, in fact, at a further distance than it is for the 1s orbital. And when we go down to a 3s orbital, again, we see it's even more complicated. In fact, we have two radial nodes now, but the highest probability of finding the electron is even further away. So what this is saying is that on average the electron is moving further and further away from the nucleus as we go from a 1s state to a 2s state to a 3s state. And by the way, I'm using the term "orbital" now. Notice that that is different than the word "orbit." We're not talking about the electron orbiting the nucleus. Unfortunately, the term that's been chosen is very much like it and it's an easy place to make a mistake. We're talking about orbitals and an orbital is state of the electron. As we go from, again, 1s to 2s to 3s, what we're seeing is that the size of the orbital is increasing. And so if we were to actually be able to look at these orbitals without perturbing them, we would be going from here to here to here. This is generally what we'd see.
You'll notice that again the edges are not well defined. They're very fuzzy and, again, that means that our probability is decreasing but it's not zero. It just fades off as we get further and further away from the nucleus. We talk about this a lot as an electron cloud, or again electron density or electron probability density. All of these terms are used to describe again the chances of finding the electron at some point in space.
So what have we learned? We've learned that a series of solutions come out of the Schrödinger equation, which tell us information about energy, and about size, and ultimately even about the shape. The shape, there's not much of a story yet, because these orbitals are all spherical. But as we'll see soon, orbitals of electrons can take on very different shapes than what you see here. []
Modern Atomic Theory
Atomic Orbitals
Atomic Orbital Size Page [1 of 2]