 All right, so we've been talking quite a bit about ideal gases, and just to refresh your memory, we've used the term ideal gas as a shorthand to talk about the three-dimensional particle-in-a-box problems, specifically in the case where we've taken the classical limit, where we can assume the system doesn't really need to worry about quantum mechanics. We can think about the system classically and get the same result. So although we talk about quantum mechanical 3D particle-in-a-boxes under classical limit, or whether we talk about ideal gas, that's the same thing. So the purpose of the next few minutes is to make some connections between what we've been saying about ideal gases and what you may already know about ideal gases from earlier chemistry courses like general chemistry. So the assumptions that we used in deriving the ideal gas model via the 3D particle-in-a-box, remember, anytime we have a particle-in-a-box, that's the system for which the potential energy is zero when the molecule is inside the box, the energy is zero, and it's not allowed to be outside the box. The other important set of assumptions that we made related to this assumption that we're in the classical limit, and that is true anytime either the spacing between the energy levels, the quantum mechanical energy levels, is small enough that we don't really have to worry about them being discreetly spaced when that energy level spacing is very small compared to kT. Then there's many, many of those states that are occupied. That means the partition function is going to be typically much larger than one. And another way that we've expressed that is to say that the volume of the system is very big compared to lambda cubed, where lambda is the thermal de Broglie wavelength of the particles in the system. Under different circumstances it might be easier to think about different versions of this assumption, but these all basically boil down to the same thing. Either we need very small spacing between the energy levels, a large partition function, or molecular sizes that are very small compared to the box that we can find them into. So those set of assumptions are what define our ideal gas model and led to our result, for example, that PV equals nRT or PV equals nKT. Those set of assumptions are equivalent, as it turns out, to assumptions that you might have heard about in a different context. So one condition for a molecule behaving well as an ideal gas is that it shouldn't have any intermolecular interactions. If there's strong intermolecular interactions between two molecules, then they won't behave ideally. And a truly ideal gas is one that has no intermolecular interactions at all. And the second condition is the molecules can't take up any space. So typically in a general chemistry course, the way the assumptions behind the ideal gas model would be phrased is a gas will behave ideally if the molecules don't interact at all and if the molecules themselves are so tiny that we can totally ignore the fact that they take up any space. These conditions are equivalent to the conditions we've imposed more mathematically when we derive the 3D particle in a box. For example, no intermolecular interactions, that's equivalent to saying the potential energy is zero. If the molecules were in fact interacting, if molecule A were attracting molecule B, then that would be some potential energy that was non-zero, and we'd have to have included potential energy as well as kinetic energy when we solved the Schrodinger equation. The assumption of no molecular volume, the volume of each individual molecule is zero, that's equivalent to this expression that the thermal de Broglie wavelength, the size of the individual molecules needs to be small compared to the size of the box that they are confined in. So these are just more mathematical ways of expressing what may be somewhat familiar assumptions about the ideal gas. Another common way of thinking about the ideal gas law is to say that we know that an ideal gas obeys pv equals nrt, and that's true under certain conditions about the temperature and the pressure. Also learned or memorized when you took general chemistry that you can use the ideal gas law as long as the temperature is high enough and the pressure is low enough, and if you go to conditions where the temperature is too cold, then the gas will behave not ideally, or if you go to conditions where the pressure is too high, then the gas will behave non-ideally. And again, those conditions can be related to what we've now discovered for our version of the ideal gas law, saying that the temperature must be high as equivalent to this condition, where we say k times t has to be bigger than some inherent property of the molecules. Whatever the energy level spacing in the molecules is for the particle in a box model, we need a temperature such that kt ends up quite a bit bigger than that. So that's our definition of needing a high enough temperature. Pressure being low, that's like saying the molecular volume is high, the volume occupied by the volume of the container divided by the number of molecules is a large number, and that's again equivalent to this expression where if the container volume is very large compared to the size of the molecules themselves, then the ideal gas approximation works just fine. And again, assumptions of no molecular volume or high molar volume or low pressure, those are all equivalent to this condition. So that all works fine as long as these conditions are satisfied. And in practice for molecule-sized things inside of box-sized boxes, then these conditions typically hold pretty well. The volume of the molecules is indeed very, very negligible due to the volume of the box itself. But it's not correct to say that the interaction between the molecules is zero, that there's no potential energy at all between the molecules in the gas. So for a real gas as opposed to an ideal gas, it's typical that the potential energy might not be zero. And that would be a case where we wouldn't be able to use the ideal gas law perhaps. Likewise, since we can't necessarily assume that the ideal gas law assumptions hold for real-world gases, if we go to low enough temperatures or high enough pressures, in other words, the opposite of these conditions, then instead of the ideal gas law being obeyed, we'll find that the pressure times volume is not exactly equal to nRT or nKT. And instead, we're going to need to come up with some different equations to describe how the pressure and the volume of a gas depend on its number of molecules and the temperature. So that'll be what we investigate next is what to do for real gases under conditions that don't obey the ideal gas law.