We've reached the last members of the chemical menagerie. Now that we've taken care of atoms and diatomic molecules, let's talk about polyatomic molecules. Already shown you this slide, how is energy stored in a molecule? There's electronic energy, which is both potential and kinetic associated with the electrons, and then, there is kinetic energy of the molecule itself. There's the translational energy, which is movement of the entire molecule in space, comes from particle in a box solutions. There's the rotational energy of the molecule rotating about its center of mass that comes from the Rigid-rotator equation. And finally, there is vibrational energy associated with the nuclei moving relative to one another. It's handy to divide up the various contributions to the kinetic energy that comes from molecular motion into what are known as degrees of freedom. So, if I want to specify completely the position in space of a molecule that has n nuclei, I must need three n coordinates, right? There's an x, a y and a z coordinate for each atom, and if there are n atoms, that's 3 times n. So, we often will refer to that then as 3n degrees of freedom which dictate specification of every atomic position. And degrees of freedom can be thought of as being subdivided into translational, rotational, or vibrational components. And so, if I think about translation, that's really the motion of the center of mass in three-dimensional space, and so, there are three degrees of freedom. It can move in the x direction, it can move in the y direction, and it can move in the z direction. There's rotation, so that is movement about the center of mass. And for a linear molecule, if I imagine a linear molecule laid out along some axis, I can rotate this way. I guess that's end over end in a sense. I can rotate this way, end over end in a different plane. But, there is no rotation about the axis that's defined by the bond of the molecule. So there's two ways to rotate a diatomic. Once I am nonlinear, so that's really the, the issue there. A diatomic molecule per force is linear. Two points determine a line. once I go to three or more atoms, I could be linear. There are examples of linear molecules with more than two atoms, in which case, I still only have two ways to rotate, but once I'm nonlinear, now I can rotate about any of the Cartesian axes, so that's three degrees of freedom. Well, so all that's left then is vibrations, and when determines the number of vibrational degrees of freedom by taking the total 3n and subtracting out the translational, always 3 and the rotational, 2 for linear, 3 for nonlinear. And that gives you these these values, 3n minus 5 vibrations for a linear molecule, 3n minus 6 vibrations for a nonlinear molecule. And so, when we sum them together appropriately, always linear or always nonlinear, you always get 3n as you must. The diatomics, which we dealt with up till now, of course only have our per force linear and only have 3n minus 5 vibrations. So let's, let's just remember that. 3 times 2 would be 6, minus 5 is 1. One vibration. We looked at the vibration in a diatomic molecule. In a polyatomic, there would be more. So actually, let's pause here for a moment, and this is sort of trivial arithmetic and and thinking about the physics, but maybe take a moment and think about the numbers of degrees of freedom available in different molecules. Well, let's take a look now at rotational energy levels in polyatomic molecules. So it turns out, if the polyatomic molecule is indeed still linear, then, exactly the same Schrodinger equation applies. You get exactly the same solutions. Namely, that the allowed energy levels are given by this expression, with these quantum numbers, and this degeneracy. The only difference is the moment of inertia is defined by all of the atoms, not just by two atoms, and so this is simply a generalization of the formula for moment of inertia, namely that it's the sum over all the atoms, their mass times the square of their distance from the center of mass. Nonlinear molecules, things become a bit more complicated. So actually, there are three rotational axes in a nonlinear molecule, and for each of those axes, there is an associated moment of inertia. There are categories of molecules and the categories are named depending on the relationship between the moments of inertia. So for certain special molecules, all three moments of inertia along the different Cartesian directions are the same, such molecules are called spherical tops. And so, an example of a spherical top would be a baseball, that's not a molecule, but it's a macroscopic object. Or, in the chemical arena, methane is a molecule with three moments of inertia all equal to one another. The next step down, clearly, would be instead of having all three be equal to one another, have two be equal to one another, and so, such molecules are called symmetric tops. And again, a macroscopic object that has that characteristic would be an American football, and so its got one long axis. But otherwise, it looks pretty symmetric and that gives rise to two identical moments of inertial, but one that's different. and example in the case of a molecule would be ammonia. So ammonia, two equal moments of inertial. And then finally, obviously, the most general possible case would be all three moments of inertial are different from one another, that is called an asymmetric top. Seems, sort of odd maybe to keep using the word tops since we've removed any kind of agreement between moments of inertia, but that’s what the terminology is. So an example of an asymmetric top would be macroscopically, a boomerang and a molecule that looks a little bit like a boomerang water. So water with its three atoms bent with oxygen in the middle. When you take account of those those different moments on inertia, you get complicated expressions for the energy levels. I'm not going to show you actual formulas. You can look up certain of them if you really needed them, but for the moment, we are going to focus on the more simple linear levels that provides enough basis to make progress. Now, what about the vibrations in a polyatomic molecule? So, we refer to the, the various vibrations in a molecule, a polyatomic, now that there's more than one, as soon as you go past a diatom, you have more than one vibration. So, each one of them has a, a characteristic motion and that motion is called a normal mode. So, the individual vibrations are also called the normal modes of the molecule. And so as an example, water has three normal modes. That is, it's non-linear, so the number of vibrational degrees of freedom will be 3n minus 6, 3 translations, 3 rotations, so 3 times 3 is 9 minus 6 is 3. Three normal modes, three vibrational modes. If we look at those modes, the actual vibrations themselves correspond to, and they're shown here with their corresponding vibrational frequencies. The lowest vibrational frequency is a bending motion and generations of physical chemists have done this, so bear with me. Imagine that my two fists are hydrogen atoms and my head is an oxygen atom. The lowest frequency is a bend, so it consists of the molecule doing something like this, just a little bit of a chicken move sort of, that's a bending motion. There's also a symmetric stretching motion at 3,686 wave numbers. And so, if we do the anthropomorphism of water again, that's the two hydrogen atoms moving outward from the oxygen in concert with one another. And then, finally, the last mode is the asymmetric stretch, and that's where one molecule, one hydrogen atom moves in while the other moves out. So here is the asymmetric stretch, and actually, the oxygen will move a bit so it's kind it's like doing a little exercise in the middle of a lecture video. Thank you for bearing with me for that, it's a, it's a classic routine. Coming back to the to the energetics, each of these normal modes act as an independent harmonic oscillator, so each one contributes. The normal harmonic oscillator contribution, but they've got associated with them these unique frequencies that are shown over on the right-hand side. So, I would index for vibration 1, Planck's constant times the vibrational frequency for vibration 1 times the quantum number, that describes which level the first normal mode is sitting in. And then I would add to that h times the frequency for two times its quantum number plus a half, and so on, so I just add them up. So what's the total energy then in a polyatomic molecule, or indeed, in, in an atom or a diatomic for that matter? Well, it's the sum of the energies overall the degrees of freedom. So remember, that, for a molecule in a three-dimensional box, you would get this from solving a particle in a box equation. So there are quantum numbers and box lengths, side lengths required, as well as a mass. Rotation for linear molecules is given by this rotational energy formula and these quantum numbers, and I won't write down as I said, the nonlinear, sometimes it has a nice form, sometimes less nice. We'll deal with that if we need to . The vibrations are determined from the quantum mechanical harmonic oscillator expression, these energy levels. One associated with each vibration, and finally, the electronic energy, which has a pretty simple form for the hydrogen atom. Hydrogen atom, not that exciting. We won't be doing a whole lot more with it. for diatomics, there's a number we can associate it with, with it, DE. Otherwise, it's probably something we're going to be looking up. And what I want to leave you with is a feeling for the spacing of these energy levels. How far apart are they? And that really matters in terms of how you can store energy in a system? So there's a general trend in energy spacing. And that is, that the electronic energy spacing is much, much greater than the vibrational spacing, which itself is greater than rotational, which itself is much, much greater than translational. So let me, rather than just memorizing words, let's actually have a picture at least to associate with that. So, shown here is a plot of two electronic states of a system. So here's some low energy potential associated with a, let's call it the ground state. And then, somewhere up here is an excited electronic state, a long way up in energy, this separation. I'll just, just dictate that. We'll call that big. Big relative to what? Everything else. So, within this potential is a series of vibrational levels. So they are much more closely spaced, one to another, than the bottom of the ground state electronic energy to the first excited state electronic energy. And so, if I zoom in on the vibration energy levels and move them over here. Now, I could see above each of the vibrational levels the rotational spacing. So the rotational levels are much closer to one another than the vibrational levels are. And finally, if I zoom in, yet again, just take this little red circle here, the first few rotational levels. And expand those a great deal, then, on top of each rotational level, superdense are the translational levels. So they're extremely close to one another. And keeping in mind that spacing will be useful as we proceed forward and think about what that means for how energy can partition in to these various kinds of motion. So, that wraps up what we're going to do in terms of studying the quantized energy levels of atoms and molecules. And indeed, we're almost done with the material for this week of the course. The next video, we'll try to provide a sort of review summary picture of what we've talked about that's important so far and that will complete Week 1.