 So we've seen the Equal Partition Theorem tells us that for every quadratic degree of freedom, every degree of freedom that shows up in the energy as some variable squared, that degree of freedom in the classical limit is going to end up contributing a factor of 1 half r to the molar heat capacity, constant volume heat capacity. So let's see if we can use that to actually make some interesting predictions about the heat capacities of molecules. So for example, easiest example we can use is a monatomic ideal gas. In other words, an ideal gas is only consisting of one atom. If we start counting degrees of freedom so that we can use the Equal Partition Theorem, we know that since it's just one atom, it can translate. It's not allowed to rotate. It's not allowed to vibrate. So there are no rotational or vibrational degrees of freedom. If we think about what the energies are associated with each of these motions, we don't need to think about rotations and vibrations. The translational energy of the molecule, it has a kinetic energy, one half mv squared. We can ask is that energy quadratic, is the velocity a quadratic degree of freedom and since it shows up as something squared, that is indeed a quadratic degree of freedom. So therefore, how much is it going to contribute to the heat capacity? There's three of these degrees of freedom, the x, the y, and the z coordinates, each of which is quadratic, so that's going to contribute three times one half r. Each of those quadratic degrees of freedom is going to contribute one half r to the heat capacity. That's the only type of energy we have is the kinetic energy. We don't have to worry about rotations and vibrations. So if we sum up all the contributions, that's responsible for our prediction that matches what we've seen in the past, that the constant volume molar heat capacity for a monatomic ideal gas, for an ideal gas described only by the particle in a box, is three halves r. To see how that works for a more complicated case like a diatomic molecule, that would be a molecule that we need, not just the particle in a box model to describe, but also the rigid rotor to describe its rotations and the harmonic oscillator to describe its vibrations, we'll go through the same process. If we count the translational, rotational, vibrational degrees of freedom, we've seen that there's three translational degrees of freedom. The molecule can move in x, y, and z. It's a linear molecule, so there's only two ways it can rotate, the variables we've called theta and phi, and there's just one bond, so it has one vibrational degree of freedom. Diatomic molecule has three n, or six total degrees of freedom. So far so good. Now we have to count degrees of freedom. How many, so we've counted the number of degrees of freedom, how many of them are quadratic? So we can ask for the kinetic energy, go through the same process, the translational kinetic energy is always going to be one half mv squared. It's always going to be quadratic. In fact, we've seen for every type of kinetic energy, whether it's translational kinetic energy or rotational kinetic energy or vibrational kinetic energy, those are all quadratic. The kinetic energy of rotations looks like one half moment of inertia times an angular velocity squared, that's quadratic. Kinetic energy of vibration looks like one half a reduced mass times a bond velocity squared, that's quadratic. So each of these degrees of freedom are quadratic and we have varying numbers of them. So how much do they contribute to the heat capacity? Three quadratic degrees of freedom contribute three times one half r. Two rotational degrees of freedom, each contributing one half r. One vibrational kinetic energy degree of freedom contributes one half r. So we could sum those up and get a heat capacity, but we'd have left something out. We have to consider every contribution to the energy, not just the kinetic energies, but also the potential energies. And the reason I bring up potential energies now is because when we model the vibration of a dynamic molecule as a harmonic oscillator, we did include a potential energy term. The potential energy looks like one half kx squared. Is that a quadratic degree of freedom? Does it depend on some variable squared? It does. So it's also going to contribute, there's one vibrational degree of freedom. It also contributes quadratically to the potential energy. So there's an extra factor of one half r that it contributes to the heat capacity. So these and these are both contributions to the heat capacity. Now that I've brought up the idea of potential energy, do we also need to include the potential energy of translations and rotations? We can consider it, but it doesn't end up affecting anything. If a molecule is translating around in a room, its x, y, z positions don't affect the potential energy of the molecule. So the potential energy, the contribution of the translation to the potential energy is zero. That is not a quadratic degree of freedom. So it doesn't contribute to the heat capacity. Likewise for rotations, if a molecule rotates, it's not changing its potential energy. That is not a quadratic degree of freedom. So it's not contributing to the heat capacity. So now, if we combine all the contributions, we see three plus two plus one plus another one. That's a total of seven halves r for the heat capacity. But with one important caveat, if we look back at our definition of the equal partition theorem, each of these quadratic degrees of freedom is contributing one half r to the heat capacity if we're in the classical limit, if we're at a temperature high enough that those degrees of freedom are behaving classically. Remember that translations behave classically. Rotations usually behave classically. The vibrational temperature of diatomic molecules is usually well below room temperature. The vibrational, did I say vibrational? The rotational temperature is well below room temperature. The vibrational temperature of a molecule is usually several thousand degrees that's way above room temperature. So if we're at temperatures above the vibrational temperature, and the vibrations are behaving classically, then this would indeed be our prediction. But let me put these contributions in parentheses because under normal conditions, if the vibrational temperature of a molecule is 3,000 degrees and we're interested in properties at 300 degrees Kelvin, then we are not in the classical limit. The temperature is well below the vibrational temperature. And in that circumstance, if we add up the contributions that don't include the classical, don't include the vibrations because we can't use the classical limit, echopartition theorem says, if only the translations and rotations are classical, then 3 plus 2 would give me 5 halves times r if we're at relatively cold temperatures. So the echopartition theorem can make two different predictions. Prediction at cold temperatures, where translations and rotations can be treated classically, and if we get to high enough temperatures that the vibrations can be treated classically, it makes a different prediction. 7 halves r for the heat capacity. So we can see how that works if we bring up a graph here of what the heat capacity looks like for an actual diatomic molecule. This will be the prediction for any diatomic molecule. The data on the screen now is specifically for the case of nitrogen, which I don't think it says anywhere. So this graph is for diatomic nitrogen gas. What's being plotted on this graph is the heat capacity, the constant volume heat capacity in units of joules per Kelvin mole, so with numbers ranging from 21 up to 29. If we divide those numbers by r, so that's 5 halves r, this quantity 2.5 r. If the heat capacity is indeed 5 halves r, then the heat capacity would be on this line. And we see that down here at cold temperatures, it is indeed true that the heat capacity is much colder than the vibrational temperature, which is about 3,300, 3,400 degrees Kelvin. So at temperatures much colder than that, it is true that the heat capacity is pretty close to 5 halves r. As we get into a range where the temperature of 500 or 1,000 or 2,000 Kelvin is not small compared to 3,400 Kelvin, the heat capacity begins to rise. And then as we get the temperatures at or above the vibrational temperature, then the value that the heat capacity begins to plateau at is not exactly, but is fairly close to 7 halves r. So we can see that the Equal Partition Theorem, knowing nothing about the molecule other than that it's diatomic, it has three translations, two rotations, and one vibration. We can predict that at cold temperatures, the heat capacity is one value. At high temperatures, the heat capacity is a different value, and in between, it has some sort of transition between them. So that's actually fairly impressive that we can predict the properties of this molecule knowing nothing other than the types of motions it has. So if we use that to go on and predict some properties of larger molecules, we'll see how well that works. And then we'll save that for the next video.