 So when we discuss the Equal Partition Theorem, we use this term degree of freedom and it turns out to be very crucial to be able to identify degrees of freedom and count how many degrees of freedom of different types you have. So let's make sure that concept is clear. So this idea of a degree of freedom, all that means is just a variable that you can change the value of completely independently. So some examples, hopefully we'll make that clear. So let's say I'm just talking about a single atom. A monatomic molecule, just a single atom, single helium atom, for example. So here's an atom. The question is, when I'm describing that atom, let's say I'm describing the position of that atom. How many different variables can I choose independently to specify the position of the atom? I can change its x-coordinate. I can change its y-coordinate. I can change its z-coordinate simultaneously. Changing one of those doesn't prevent me from changing any of the others. So we'd say that molecule has three degrees of freedom. Three variables that I can change independently to describe its position. It would also have three velocity degrees of freedom that I could use to specify its velocity. If I compare that to a diatomic molecule, here's a diatomic molecule. There's a couple different ways we can think about how to count the degrees of freedom to specify the orientation, the position, the configuration of that molecule. We can do it just like we did for the atom. This atom in the diatomic molecule, I can specify its x and y and z positions independently of one another. So that would be three degrees of freedom for x and y and z of atom number one. I can also specify the x and the y and the z coordinates of atom number two. And describing one of those doesn't prevent me from changing the values of any of the others as well. So they're all independent of one other. We would say that that diatomic molecule has six total degrees of freedom. There's six variables that I need to specify in order to describe the complete configuration of the molecule. That's usually not the way we have been describing the geometry of a diatomic molecule. If we instead think of that molecule as located at some position in space, it has some center of mass. Maybe one of the atoms is heavier than the other. So the center of mass is a little off-center. So I can specify the coordinates of the center of mass. I can tell you the x and the y and the z coordinates of the center of mass. That's like specifying the position of the molecule rather than just a single atom in the molecule. And then in addition to that, I can tell you the bond length in the molecule. And I can tell you the orientation of the molecule, which we do with the angular coordinates theta and phi. So those are the coordinates we've been using to describe the rigid rotor and the harmonic oscillator. And the center of mass coordinates are what we use to describe it as a particle in a box. So in this diagram, there's a theta degree of freedom and a phi degree of freedom that describe the orientation. So we've got two different angular coordinates to describe the rotation. One that describes the bond length or the vibration of the molecule. And three that describe the position of the molecule, or its translational coordinates. So first I'll point out that that is also six variables. If we wanted to go to the work, we can rewrite each of these coordinates in terms of these x, y, and z's. Or vice versa, I can describe the x, y, z coordinates of each atom as a function of the center of mass and the orientation and the bond length of the molecule, whichever way I choose to describe it. There's only six variables, no more, no less, than six variables needed to describe the geometry of the molecule, regardless of whether I'm thinking about it in internal coordinates. Describe its bond length and orientation. Or Cartesian coordinates that describe the position of every atom. If we categorize these degrees of freedom, let's do that over here. This diatomic molecule has translational degrees of freedom. It has rotational degrees of freedom. It has vibrational degrees of freedom. So let's count how many of each of those there are. The three center of mass coordinates, the x, y, z position of the center of mass, those are the translational degrees of freedom that describe the translation or the position of the molecule inside the box. The rotational degrees of freedom are the ones that describe its orientation. Those will be theta and phi, so there's two of those. And the degree of freedom that describes the vibrational motion of the molecule, that's describing the stretch of the bond. The bond length is only a single coordinate. So not surprisingly, that adds up to six. There's six total degrees of freedom in this molecule, three of which are translational, two of which are rotational, one of which is vibrational. And this illustrates why it's useful to think about the molecule's coordinates in internal coordinates rather than Cartesian coordinates, because these variables make it very clear how many of the degrees of freedom are rotational or vibrational or translational. If we back up a step and do this same sort of analysis for a single atom, I'm on atomic molecule. And if I ask how many translational, rotational, or vibrational degrees of freedom there are, and how many in total, looking back at this molecule, this single atom, the only three degrees of freedom are these X, Y, Z degrees of freedom, the Cartesian degrees of freedom. So what those are describing is not the orientation of the molecule, it's rotational degrees of freedom. It can't vibrate because there's no bonds to vibrate. There's just three translational degrees of freedom and no rotations and no vibrations. So all three of the degrees of freedom are translational for a single atom. What about for a more complicated molecule? So we'll do the general case. Maybe we're talking about a tetrahedral molecule or maybe there's more bonds than that. So some arbitrary molecule with a total of n different atoms in it. Again, we can think of the number of degrees of freedom in two ways. If we wanted to think of the internal coordinates, we could think of how many individual bonds there are to specify, how many individual bond angles we're free to specify, how many other types of internal degrees of freedom there are. And once I've specified some of those angles, does it restrict other angles? That's a little bit complicated. It involves more trigonometry than we wanna volunteer for. So it's a little bit easier just to say, since we're gonna get the same answer in either case, it's easier to think about it in Cartesian coordinates and say, certainly atom number one has an x and a y and a z. Atom number two has an x and a y and a z that we could specify all the way down to atom number n, which has its own x and y and z. So altogether, there must be a total of three times n degrees of freedom in this polyatomic molecule, an x and a y and a z for each of the n atoms in the molecule. So it's pretty easy to count the total numbers of degrees of freedom. If it's a triatomic molecule, it'll have nine degrees of freedom. If it's a 12 atom molecule, it's gonna have 36 degrees of freedom. If we want to categorize those into how many rotations, vibrations, translations, that's not too much more difficult now that we know how many total degrees of freedom there are. So we've just seen that there must be a total of three n, total degrees of freedom. We can ask how many of those degrees of freedom would be translational for the molecule as a whole. How many ways are there for me to specify its location in a box? Just like the monatomic and diatomic molecules, the center of mass of the molecule only has three coordinates that it needs to describe where that molecule is located. So no matter how complicated the molecule gets, describing the location of the molecule requires no more, no less than three degrees of freedom. For rotations, things get a little bit more complicated. You might think it's similar to the diatomic molecule. So the diatomic molecule, we needed two degrees of freedom, theta and phi to describe the rotation of the molecule. But for a more complicated nonlinear molecule, we can actually specify three different angles. So with my cartoon of the molecule over here, we can imagine we can rotate that angle, that molecule around the x-axis, we can rotate it around the y-axis, and we can additionally rotate it around the z-axis. So there's three different axes I can rotate this molecule around because it occupies a three-dimensional space. So in general, molecules have three different rotational coordinates, three ways they can rotate. The reason that was not true for this diatomic molecule, which after all still does occupy a three-dimensional space, is because the third coordinate, I can describe the orientation of this molecule by bending it down from the z-axis, describing how much rotation it has about the z-axis. The third coordinate would be, if I were to describe the rotation around that bond of the molecule, spinning it along its own bond. For a diatomic molecule, because it's cylindrically symmetrical, spinning it around that axis doesn't actually change the molecule. So I haven't changed it by performing that rotation. So this third rotation doesn't actually perturb the molecule. So there's only two rotations that modify any linear molecule. So here we have to have a special case. If we have a linear molecule, there's only two different rotations. If we have a non-linear molecule, it can rotate in three different ways. And the good news now is we can predict how many vibrational degrees of freedom that molecule is gonna have because we know it has to add up to three n. For a linear molecule, if there's three translations and two rotations, there have to be three n minus five that we would categorize as vibrational motions. For a non-linear molecule, that would be three n minus six. And notice that this holds true for the diatomic molecule. It's linear, so three n minus five gives me just one coordinate that is a vibrational coordinate because there's only one bond that can vibrate. For my more complicated molecule, it might be difficult to count the number of bonds or count the number of degrees of freedom that can be represented as vibrations, but through process of elimination, if we know how many atoms there are in the molecule multiplied by three to get the numbers of degrees of freedom, and that subtract either five or six, depending if we're a linear or non-linear molecule, and that tells us how many independent vibrations that molecule has. So now that we know how to count the number of translational, rotational, vibrational degrees of freedom in a molecule, we can use that together with the Echor-Partition Theorem to predict the heat capacities of a variety of molecules.