 So let's take a step back and summarize what we've figured out so far for the three different quantum mechanical problems we've studied. The particle in a box, the rigid rotor, and the harmonic oscillator, all of which we've used to describe different properties of ideal gases. So with the 3D particle in a box, we, the thermodynamic properties we eventually discovered from the particle in a box were that the internal energy worked out to be three halves RT. So that was a nice, simple, satisfying result. The pressure of that ideal gas, actually first let's do the heat capacity. The, if we just take the temperature derivative of the energy, the constant volume heat capacity, temperature derivative of three halves RT is just three halves R. So the constant volume heat capacity of a 3D particle in a box is three halves R. And the pressure worked out to be NRT over V. So those are the properties that we found for the 3D particle in a box that we started considering as an ideal gas. And we eventually realized that really only described the properties of a monatomic ideal gas because diatomic gases can not only translate, which is what this 3D particle in a box tells us, but also have rotational motion and vibrational motion. So to consider the rotational motion, we talked about the rigid rotor model, when we calculated the thermodynamic energy, the internal energy, due to rotation of a diatomic molecule that worked out to be RT so that the heat capacity, constant volume heat capacity was R. And the pressure due to the rotations was zero. And then when we considered the harmonic oscillator, so now we allow the molecule not to rotate but to vibrate. It allows bond length to change the internal energy due to those vibrations. That was a little more complicated. That looked like 1 half h nu, the zero point energy for the harmonic oscillator. And h nu multiplying a rather complicated factor that tells us how many quanta of excitation the molecule experiences. So that full expression is the energy, the internal energy of due to vibration for a molecule behaving as a harmonic oscillator. To write down the heat capacity, I'd need to take the temperature derivative of this term. And we can do that. That would get a little bit complicated. But recognizing what we've seen that most of the diatomic molecules at temperatures near room temperature, this term is relatively small. There's very few excitations, very few average quanta of excitation. So under conditions where the temperature is quite small compared to the vibrational temperature, so that this exponent theta vibrational over T is a relatively large number. This quantity ends up being quite small. So the number of excitations is quite small. We can, under those conditions, say that the vibrational internal energy is pretty close to the zero point energy. Just the minimum amount of energy without much extra energy above that. And under those conditions, the temperature derivative of this term, these are just constants. There's no temperatures in there at all. So the temperature derivative of that internal energy just works out to be zero. So the heat capacity due to vibrations is either zero or quite small, which is different than the result for rotations or for vibrations. And then completing this little chart, the pressure due to vibrations is also zero. So if we combine these results and say for a full diatomic molecule, again, monatomic molecules can't do anything but translate. Diatomic molecules include all three of these terms because they can translate and rotate and vibrate. If we sum each of these three terms, we find that the total energy of the diatomic molecule is its zero point vibrational energy plus 5 halves RT. The temperature dependent component of that is just 5 halves R times the temperature. So the heat capacity of this diatomic molecule is 5 halves R. And the pressure, if I sum the contribution from translation and rotation and vibration, the pressure is the equation of state that we're used to, PV equals NRT. So this has begun to show us a couple of different things. First of all, you may remember from general chemistry, you may have learned that the heat capacity of a monatomic ideal gas is 3 halves R for CV and 5 halves R for CP. For a diatomic gas, it's 5 halves R for CV and 7 halves R for CP. And now we see why that's true. The constant volume heat capacity of a diatomic molecule, 3 halves of that 5 halves R come from the translational motion. One of the 5 halves R's come from the rotational motion. None of it comes from the vibrational motion. And those sum up to 5 halves R for CV or 1 factor of R larger for CP. So that extra factor of R comes from the PV work associated with constant pressure processes rather than constant volume processes. So we can see why diatomic molecules have different heat capacities than monatomic molecules. We can begin to see why the heat capacity for polyatomic molecules, a triatomic molecule like water, has even more vibrations, has even more types of rotations than a diatomic molecule does. We'll talk about that more in a future video. But now we have an explanation for why the heat capacities are different for different molecules, specifically different for monatomic than for diatomic ideal gas molecules. It also begins to raise a new question. These heat capacities seem to always be multiples of one half R. I've either got three times one half R or I've got twice one half R. Here I've got five times one half R, seven times one half R. Heat capacities keep being equal to one half times R times some integer, three or two or five or seven. So that's an interesting puzzle to think about why heat capacities would always be a multiple of this particular number one half R. So that's what we'll talk about in the next video lecture.