 So the heat capacities for ideal gases, at least the ideal gases that we've been able to understand using the 3D particle in a box model, have turned out to be fairly simple. So the internal energy of an ideal gas as predicted by the 3D particle in a box model is three-halves RT. The enthalpy per mole of that same gas would be five-halves RT. If I take the derivative with respect to temperature to find the heat capacities, the constant volume heat capacity, three-halves RT becomes three-halves R. The constant pressure heat capacity, if I take the HDT, five-halves RT becomes five-halves R. So those are the expressions we've seen for the constant volume molar heat capacity and constant pressure molar heat capacity for a gas predicted by the 3D particle in a box. Those expressions are fairly remarkable. They make a prediction that says it doesn't matter what the gas is, doesn't matter whether it's argon gas or nitrogen gas or water, steam, vapor, doesn't matter what the gas is, doesn't matter what the temperature is, these are the heat capacities of those gases. If I want a numerical value, if I take the gas constant 8.314 and I multiply by one-and-a-half, predicts that the heat capacity, the constant volume heat capacity of every gas is 12-and-a-half joules per mole Kelvin. If I multiply the gas constant by five-and-a-half, that I'll have to remind myself of what the calculator tells me. That's 20.79 joules per mole Kelvin for the heat capacity. So again, same prediction for every gas. Does it actually work that way? We can see. What's if I bring up a table of numbers here that give us the heat capacities for a bunch of different gases? Here are the experimentally determined heat capacities, molar heat capacities for a bunch of different gases. Do they all equal 20.79? Well, some of them do. Helium, neon, argon, the heat capacities of all those gases are exactly 20.79. So the 3D particle in a box is making pretty good predictions. And I can color code these values to make it clear what the patterns are. So clearly the first three, helium, neon, argon, the prediction is pretty good for the 3D particle in a box model. But the rest of these are not. 29, these numbers I've colored orange, values near 29, that's nothing like 20.79. CO2 and H2O, those are even worse. Those values are much higher than 20.79. And we can see some patterns here. We can see that for the monatomic gases, helium, neon, argon, are just gases that are composed of a single atom. Those predictions are perfect. Carbon monoxide, hydrogen gas, nitrogen gas, oxygen gas, those are diatomic gases. The prediction is much less good for them. But all of those values are pretty close to each other. So it seems like there's a different value that's a reasonable estimate for diatomic gases. For these triatomic gases, carbon dioxide and water, again, it's a completely different value. And this prediction is even worse. So clearly something has gone wrong with the 3D particle in a box model. And it's actually even worse than what these data show. Because the heat capacity here, again, doesn't depend on what the molecule is. Also, it doesn't depend on the conditions like the temperature. So we would make the prediction that these heat capacities are the same regardless of what the conditions are. And we can see whether that's true or not if we pull up a graph over here. So what I'm plotting now is the heat capacity of the gas as a function of the temperature. So room temperature is down here somewhere around 300 Kelvin. And for argon gas, again, monatomic ideal gas, monatomic gas that behaves fairly ideally, things are fine. The value is 20.79 exactly as we predicted with the 3D particle in a box model. And that value does not change very much at all as the temperature changes. But if I pull up the data for another gas, like nitrogen gas, we can see that the value at room temperature, which we saw was about 29 and changed at room temperature. If I change the temperature, the prediction gets even worse. So at 500 Kelvin or 1,000 Kelvin or 2,000 Kelvin, the value is not well predicted by the 3D particle in a box model. And it also keeps changing. So pretty clearly, the heat capacity for some gases, at least, is dependent on the temperature, has some dependence on the temperature. So that's going to complicate things a little bit. We can take two approaches. We can take the empirical approach and say, well, now that we know that heat capacities are different for every gas, we can just look up those values or look up a graph of those values and see how they vary. And one way to do that would be to say, for every gas, to handle the temperature dependence, a pretty common thing to do is to use an empirical equation like this one. In fact, this equation has a name. We use the Schoemait equation to come up with an equation. So essentially, this is a polynomial that describes how the heat capacity is depending on the temperature. So with this linear term, quadratic term, cubic term, and a 1 over T squared term, that allows us to do a pretty good job of predicting the variation in the heat capacity as the temperature goes on at the cost of, however, having these five empirical parameters. So we could look up the values of A and B and C and D and E for a gas like nitrogen and use that equation to predict the heat capacity. And sometimes we'll have occasion to do that. In fact, as we go forward, we'll have better ways to predict, more accurately, exactly what these coefficients should be or exactly what the heat capacity of a gas should be under different conditions. But for now, the important point is the 3D particle in a box model that has been working so well for us so far works great for monatomic ideal gases. It's pretty clear that it's left something out for the diatomic and triatomic gases. And so one approach and the one that we'll take next is to take a step backwards, look at the quantum mechanics again, and say, OK, what have we left out of our model for a diatomic molecule that we didn't include in the 3D particle in a box model that might help us make a better prediction about properties like the heat capacity? So that's what we'll tackle next.