Now that we know everything there is to know about the allowed energy levels for atoms, let's move on to molecules and start with the simplest kinds of molecules, diatomic molecules. So just as we thought about how energy is stored in an atom, let's think about how you can store energy in a molecule. Exactly as was the case in an atom, we can still have electronic energy. So this is kinetic and potential energy associated with one or more electrons, usually more than one in a molecule. and it's their attraction to the two nuclei in a diatomic, as well as their repulsion from one another that dictates their potential energy, and then the kinetic energy of the electrons moving about the nuclei. Similarly, there is a translational energy that the molecule, as a whole, this diatomic molecule, is moving in space. And there's kinetic energy associated with that. We get this, both of these energies, in the same way as we do for atoms. So there is no exact solution of the Schrodinger equation for a many electron diatomic, just as there wasn't for a many electron atom. But we can, do calculations or we can look up energetics to get electronic energies. The translational energy doesn't depend on whether something's an atom or a molecule or a baseball for that matter. Once you know the mass and you know the size of a box, you've got a particle-in-a-box and you can get the solutions for the translational energy levels. However, translational energy is kinetic energy. There are two other kinds of kinetic energy available to a molecule that are not available to an atom. The first is rotational energy. So the entire molecule can rotate in space about various axes. And the relevant Schrodinger equation that we need to solve in order to get the allowed energy levels is called the rigid-rotator equation. In addition, if I have two atoms connected by a bond, their motion relative to one another, a vibration is a place where energy can be stored. And the relevant Schrodinger equation to understand the energy levels available for vibration is called the Quantum-mechanical harmonic oscillator equation. So each of those we'll take a look at to see what their, what the available energy levels are. Let's start with rotational energy levels. So here's a little diagram showing a diatomic molecule, which can be thought of as two masses and they have some distance between them. There is a center of mass, which need not be the midpoint if they are not equal masses. And so we can define certain constants within the system. We can find a center of mass. We could define the distance of each of our masses, or I'll call them atoms because we're doing chemistry, from that center of mass. The sum of those two distances is the bond distance. And from those data, we can compute the moment of inertia. So moment of inertia is the, the equation is simply the mass times the square of the distance of that mass from the center of mass. And so that has appropriate units of mass times distance squared. And when we solve the Schrodinger equation associated with rotational motion, we discover that the allowed energy levels have associated with them a quantum number, usually written as a capital J. This is the first time we've seen a quantum number series start with zero. Zero is an okay quantum number. Any integer in principal is okay. Zero or one is typically where quantum numbers start from. And the energy levels are given by h bar squared. So h, remember is Planck's constant, and h bar means divided by 2 pi. So we see that quantization constant creeping in. The moment of inertia appears in the denominator of the energy value. So the larger and larger the, the moment of inertia, the smaller and smaller the energy levels. And they are spaced by J times J plus 1. So if we just think about J equals zero for a moment, we would multiply zero times 1 is zero, times a bunch of constants is zero. So the ground state energy shown here is zero. And if you like, that's not rotating. So there is no energy associated with rotation. And it's, if you're not rotating. So that's why zero is a quantum number there. The next level up, when J is equal to 1 would give 1 times 1 plus 1 is 2. So 1 times 2 is 2. That cancels this two in the denominator. I'll get h bar squared over I. And so the next level up, shown here for J equals 1, has that energy. And then I'll get 2 times 3 is 6 divided by 2. I get a factor of 3 and then a factor of 6. I'll let you do the simple algebra if you like. so we see this spacing, not unlike in the hydrogen atom. The spacing gets sorry, exactly unlike in the hydrogen atom, the spacing gets bigger and bigger as you go up. Remember, in the hydrogen atom for electronic energy, the spacing got smaller and smaller. So that's an interesting difference. And the degeneracy of the levels, g sub J, is given by 2J plus 1. So the ground state where J is 0 has no degeneracy. If you like, there's only one way not to rotate. when you go to the first level, it has degeneracy 2 times 1 is 2 plus 1 is 3. And as a handy pneumonic to sort of remember this, a handy way to think about that is if you think of there being three axes in space, a molecule can sort of rotate about each of those axes, and that would give rise to a degeneracy of three. That doesn't actually explain the degeneracy here, but it's a great thing to keep in your mind to remember this rule for degeneracies, because you'll immediately come up with 2J plus 1. If you remember, the ground state is, is one, and the first state is three. So let's take a moment then, and you can play a bit with computation of a moment of inertia. [SOUND] Now let's move on, to look at the last place where energy can be stored in a diatomic molecule, and that is, in the vibrations. So, we'll look at the vibrational energy levels. The vibration is associated with the two atoms moving in and out relative to one another's positions. And one can think of the physical system that decribes that as being two masses connected by a spring. So this is a nice classic physical system. the masses correspond to the atomic masses. And the distance between them, when the spring is at its equilibrium length, that is it's neither stretched so that there's a restoring force, nor is it compressed so there's a different restoring force. It's just at rest. That defines a distance R called the equilibrium distance. So if we solve the quantum mechanical Schrodinger equation, then what we discover is that the allowed energy levels for the harmonic oscillator are given by this expression here, E the energy level indexed by a quantum number v. So v for vibration, if you like. And the quantum number, just like in rotation, it starts at 0. Goes 0, 1, 2, and so on. The energy is v plus a half times Planck's constant, there's Planck's constant again, times nu, the frequency. That is, the frequency of time associated with a vibration. So this equation, it's not my fault that physicists over many, many years decided, and chemists, too, to use v and nu in the same equation, because it takes a really good font to distinguish those in a substantial way, and you just have to know from context which one is which. V is the quantum number. Nu is the frequency. So, we'll try to be very careful and always keep track, but it's just something to be aware of. So if I go and look at these energy levels, and, what's shown here is the potential energy curve, it's a parabola. And that's what the potential energy looks like for a spring. As you stretch a spring, or as you compress a spring, the energy goes up as the square of the stretch or the compression. And how quickly it does so is given by the force constant of the spring. In a molecular system, that'd be like the strength of the bond. But in any case and that strength dictates the frequency. So the stronger the bond, the higher the frequency of the vibration. The really weak bond, big amplitude, slow vibration, low frequency. But irrespective of that, the spacing between the levels, because it's linear in the quantum number v, is constant. So this is yet another way that levels can be spaced, that we haven't seen yet. Constant spacing between all the levels. So here's the ground state, the first excited state, the second, and so on. So, the degeneracy of all these levels is one. There are no degeneracies for the quantum mechanical harmonic oscillator. The spacing in between them, the variation, if I start at 0 and I get a half, and I go to 1 and so I'll have 3 halves, the difference between 3 half and 1 half is 1. And so the spacing between all of them is h nu. So Planck's constant times the frequency of the vibration. Notice that the ground state energy is above the bottom of the potential energy curve. So that's an interesting phenomenon. That phenomenon is known as zero-point energy. So it's energy that the system is storing, even in the ground state. Were I to take the atoms completely apart, I could recapture that energy in principle. But they are they're not allowed to go down to the bottom. And that's associated with certain features of the quantum mechanical nature of this particular problem. So knowing what we know about the quantum mechanical harmonic oscillator, let's move from the, the model system of two masses on a spring and talk about a real chemical bond. So a bond does not go up a potential curve indefinitely. Instead, the atoms cease to interact with one another at a certain point and the bond breaks. So a more accurate curve for chemical bonding is shown here. When I bring, so this axis is distance. When I bring two atoms very close together, they repel each other strongly and the potential goes up, up, up, up, up. There is some equilibrium distance where the potential is at its lowest. And there will be a ground state vibrational energy level with a average distance somewhere near that lowest energy distance. And then as I pull the atoms apart, they go up for a while, looking like a parabola, but ultimately they just stop interacting. And that defines the zero of energy. They don't interact with each other anymore. And so, two important quantities on this curve. DE, so that is the separation between the bottom of the potential and zero. And D0, the separation between the ground state vibrational level, and zero. Those are different, and they differ by the zero-point energy, which is one half h nu. So the relationship I, I just described here, and I'll just show it in an equation. Namely, the negative of the ground state electronic energy. So here is an electronic energy, De, defined for a diatomic, by calling zero separated. And De the bottom of the potential is equal to D0, so that's dropping down to the ground-state vibrational level plus the zero-point vibration. So this is referred to as the dissociation energy. It's how much energy we really have to provide to rip the, the diatomic molecule apart. But the electronic energy has this additional h nu over 2 component. And so just as an example, hydrogen gas has a, a bond dissociation energy of 432 kilojoules per mole. If we measure the frequency of vibration of hydrogen with an infra-red spectrum, for instance, it's 4, 401 wave numbers. And if you plug that in to the relevant expression that we saw some time ago, to relate wave numbers to energy, that would be 52 kilojoules per mole. One half of that would be 26. And the difference between these two, the electronic energy and the bond association energy, is that 26. All right, so that completes the necessary quantum mechanical background to understand the energy levels of diatomic molecules. All that's really left now that we've done atoms and diatomic molecules, is polyatomic molecules. So that'll be the subject of the next video, and then we'll have wrapped up the necessary underpinnings of quantum mechanics to advance on to considering thermodynamics.