Alright, it's time to dig in. So, we are going to take a look at atomic energy levels. What are the allowed energies in an atomic system? And in order to answer that question, first we have to ask ourselves, how is energy stored within an atom? So, when we connect, macroscopic thermodynamics, to a molecular understanding, we need to understand that energy distribution on a microscopic scale. And if we think about atoms, as the most fundamental particles, there are two ways to store energy in an atom. One is the electronic energy. So, the electrons, which are distributed about a nucleus, will have associated with them kinetic energy, they're in motion, and potential energy. They are attracted to the positive nucleus, and they are repelled from one another. An atom will also as a physical object, have translational energy, that is, as an atom moves through space, it has kinetic energy, associated with it's velocity and it's mass. So that's kinetic energy only, unlike the electronic energy, which is potential as well as kinetic. One of the simplest systems chemically, simplest system in the universe, I suppose, is the hydrogen atom. So the hydrogen atom is one electron surrounding one proton. It turns out that the Schrodinger equation that describes the motion of an electron about a proton can be solved exactly, analytically, and from that solution, we learn the following. There are quantized energy levels, one, two, three, four, up to an infinite number actually. And, in terms of terminology, we refer to the lowest allowed energy level as the ground state of a system. So, a ground state hydrogen atom has an energy, binding the electron to the proton of minus 2.18 times 10 to the minus 18th joules. The next allowed energy level up, n equal two, has an energy of minus 5.44 times 10 to the minus 19th joules. And there's another level after that, number three, and then number four, and you'll notice that spacing between these levels is getting closer, and closer. So they're not equally spaced, and if we look, at the physical properties hydrogen atoms, in these various energy levels. We discover that's, what is happening is that the electron is moving further and further away, in terms of the average distance from the proton. And so for the ground state, remember that's the lowest quantum number, so n is called a quantum number, it is the integer that indexes what level, what of the allowed energy levels is occupied? The distance, the average distance, this is actually an expectation value, is what we would call that in quantum mechanics. So what is the most likely distance you would get after averaging over a large number of experiments, is 5.3 times 10 to the minus 11th meters. And so, if you remember an angstrom is about 10 to the minus 10th meters. Its about half an angstrom. This unit of distance actually defines the atomic unit of distance which is called the Bohr. So one Bohr is defined by the hydrogen atom in its ground state and as I raise the energy higher higher higher, ultimately I get the electron to the infinitely separated from the proton. That is there is no interaction between them and we define that to be zero there is no interaction so the energy is zero. And that's why we see here at n equals infinity that zero, that would be an ionized hydrogen atom. And relative to that zero, that allows us to assign this number to the ground state. Now if you focus on these energy levels, you would note that they have a certain progression which can be well described by a simple equation. It says that the energy of a given allowed level indexed by its quantum number is equal to some constant divided by the square of the quantum number. So, if n is one, then n squared is one and we would get the ground state energy, which, sure enough, is right here, it just isn't expanded to as many digits, so rounded to 2.18. We can also convert joules to a different unity of energy. Here's the wave number. So if you prefer reciprocal centimeters, it's 109,680 divided by n squared reciprocal centimeters. So, this simple formula to compute the energy levels would also let us compute the difference in energy between different levels. And the energy required to ionize the hydrogen atom, by letting n go to infinity at which point energy is zero. So I'll let you take a moment here to actually do a little self assessment, making use of this equation. And then we'll come back to continue to look at atomic energy levels. Most of you will certainly have seen this analysis of hydrogen before, and probably, the way you've seen it is in the context of hydrogenic orbitals. So, what are the wave functions associated with these allowed energy levels? Well, the ground state energy level, there is one way to function with that energy and we usually call that the 1S orbital. So, the next level up n equals two, has four different wave functions. All of which have the same energy. That's called degeneracy. So when there are multiple solutions to the Shrodinger equation, all with a common energy, we say that those solutions are degenerate. So given that there are four possible solutions, the degeneracy is four. And we often index degeneracy by g, and again a subscript to say which level. If we go up to the third allowed energy level or the third quantum level sometimes we would say now it turns out that there are nine orbitals or wave functions that satisfy the Schrodinger equation that all have that energy. So the degeneracy is now nine. And looking at these orbitals of course you recognize here's the classic one s. Here's a two s, it has has a node somewhere in it but we're drawing it solid so we can't see it. And here are the two p orbitals. Here's a three s, two nodes hidden in there somewhere. Here are the two p's and for some of them, this one for instance, you can actually see the nodal structure. And here you see it as well for the three p orbitals. And these are the three d orbitals, there are five of those. And these are the classic atomic orbits, more carefully, hydrogenic orbitals that one is presented with very early in the study of chemistry. Now, some of you may be experiencing a moment cognitive dissonance perhaps because you've seen these orbitals arranged in energy ordering before and someone has told you that the two s is lower in energy than the two p's. Or the three s's is lower than the three p's is lower than the three d's. That's actually not true, it's not true for this system. As I've shown it, these degeneracies are correct. All nine of the solutions for the third quantum level, the three s, the three p, the three d, they all have the same energy. In a one electron atom, normally we don't work with just one electron. It's not just hydrogen in an excited state. It is a more complicated atom. It turns out once you add additional electrons to the system, that changes the energies of s and p and d solutions. So everything that you've been taught hasn't been a lie. But at least for the hydrogenic system, the degeneracy as shown here is what it is, and you've probably noticed, it looks like it's n squared, right. For n equal one, the degeneracy is one, for n equal two it's four, three goes to nine, and sure enough that's correct. The degeneracy of hydrogenic solutions is n squared where n indexes the quantum level. So, what do we do with many electron atoms? Well, it turns out there's no simple formula for the electronic energy of such atoms. We can either do electronic structure calculations with a big digital computer. Or, you know atoms have been around a long time, and they have been studied pretty carefully. So you can look up in tabulated data, what are energy levels? So here's an example drawn from so called Moore's tables, which are these big tables of atomic energy levels that are available. And it's for the Sodium atom, and it's just showing you for a variety of different states, different ways to organize the electrons around the atom what's the degeneracy and what's the allowed energy associated with that state. If you look here you see that the ground state, which will be used to define zero here, you have to go up 16,956 it's a very precise .183 wave numbers to the first excited state. If you were to work out, I won't make you do a self assessment, but if you were to work out, what wavelength of light that is, you would discover it's sort of roughly yellowish. And that's why sodium, lighting, on highways for instance, or in cities. Often has kind of a yellowish tinge to it, that's sodium vapor that's been heated hot enough that it emits light from it's first excited state down to it's ground state, and the color of that light is dictated by the allowed energy level. Alright, happily, for most of the thermonamics that we will be working on we will not be at temperatures that are hot enough that these kinds of states are accessed very often. That will matter when we compute partition functions, which we'll be doing in not so long. But for now we will just recognize that complicated atoms or molecules, we may to just have to look up things. Translational energy, so that's the other energy available to an atom. It's moving through space, it has kinetic energy. So, how do we go about computing that? We construct a Schrodinger equation, and the equation has to do with a particle of mass M constrained in a box having variable side links. So in one dimension then, you would have a little particle here and it can move left or right and the box, it's only one dimension and we'll think of a one dimensional box, you're allowed to be anywhere from say zero to A, where A is the length. Particle can be anywhere in there and it can't be outside there. And if you solve the relevant trade in your equation, you will discover that the allowed energy levels, so indexed by a quantum number, and the number goes, one, two, three, dot dot dot. The degeneracy of every level is one, so every state is unique in its energy, goes as n squared, here's Planck's constant again, h squared. Divided by eight, times the mass, times the square of the length of the box. So, given that you would know those parameters ahead of time, the mass, the length of the box, poof. You have all of the allowed energy levels. If we go to three dimensions, it's a little bit more complicated. Obviously, our box can be a little more complicated. I've got a little particle here, it's bouncing around A for one length, b for another length, c for another length. I've drawn what looks like a q, but they don't have to all be equal. And when you do the solution, you find that it has roughly the same structure. There's still this h squared over 8m that you find here, h squared over 8m where m is the mass. But now, every dimension x, y, and z has its own quantum number associated with it, still depends on the square, still depends on the square of the lengths of the sides, but notice that states can be degenerate. That is, for instance, let's say that a and c just happen to be the same length. Well, in that case, quantum numbers, indexing a state, I could talk about state one one two which is just the list of the x y and z quantum numbers. If I consider a different state, two one one. Since a and c are the same value, it doesn't matter whether I take two squared here and one squared here, or vice versa. I'll get the same total energy. And so the degeneracy will just depend on the side lengths and the quantum numbers. It's hard to predict until you've given all of the data. So lets pause for a second and I'll let you play with those equations in a short self assessment and you can make sure that you've grasped that concept, of the quantum numbers in the particle in a box. Alright, we've completed, our examination of the available energy levels, for an atomic system. Next, we're going to move on to consider molecules and we'll consider the simplest molecule beyond an atom. We really shouldn't call an atom a molecule, but two atoms can be a molecule. So we will look at diatomics and the allowed energy levels in diatomic molecules. Before we move on to that lecture though, let's take a look at one more demonstration to illustrate some of the key principles of quantum mechanics, and in particular wave like behavior. In a prior demonstration, we saw that electronic energy levels in atoms are quantized, and I pointed out how unusual that seems compared to the continuum of energies that can be accessed by say, throwing a rock. However, we are all familiar with quantized phenomena at the macro-scopic level, although we don't necessarily think of them that way. For instance, an organ pipe of a given size, can only emit certain tones when air vibrates within it. If you want a different tone, you have to make a pipe of a different size. In this apparatus, I have a motor, that is going to drive this large string in such a way to generate a wave. The other end of my string is fixed, so that it must always be in the same position. A mathematician, or a physicist, might call this a boundary condition. That is, a condition that must be satisfied at the edge of a system. Such boundary conditions can often lead to Quantized Phenomena and are present in many ways at the microscopic level. But here, we'll see an illustration at the macroscopic level. So, when I turn the motor on, I can adjust the speed and notice that the motion of the string. [SOUND]. [NOISE] It's rather chaotic. Until I hit one certain point, I generate a so-called standing wave. The system is seemingly indefinitely stable in its behavior. Now, if I turn the speed up more [NOISE] the motion again becomes chaotic until a certain rate of speed at which I hit a new standing wave. [SOUND] See how this one has one position in the center where the band neither goes up nor down? That is called a node. And we can call this new wave an overtone or a first harmonic of the original wave that had no nodes. In general, in wave mechanics, the more nodes. [SOUND] The more energy, if we turn the speed up still higher, we may be able to hit the second overtone. [SOUND] And there it is, see how it adds two nodes. With a powerful enough motor we could continue to access higher, and higher over tones. But for now let's just take this as a another example of a quantized phenomenon. And note that in atoms and molecules, such phenomena will ultimately play an important role in how they store, and distribute energy.