 So it turns out that a key feature in using the topics we've been talking about to describe chemical systems is that molecules are very small, incredibly small. So any reasonably large real-world system that we describe is going to have very many, many molecules in it. So what that means is the microstates when we go to calculate microstates and microstates. The number of microstates that are accessible to a system that the system can have with so many possible different molecules, each one doing a different thing, the number of microstates is enormous and it turns out to be very hard to predict exactly what microstate the system is going to inhabit, but we don't really care which molecule is doing exactly which thing. Typically we're more interested in macrostates of the system and it turns out macrostates are much easier to understand and to predict. So we want to start using microstates and macrostates to be able to make real predictions about chemical systems. So to do that we'll start by thinking about how we'll do that for a problem like gases expanding. So let's say we have a box containing a gas, a box of some volume, and what we're going to do is we're going to represent that gas with something we call a lattice model. And what lattice models do is they divide the system up into a lattice or a grid. So let's say I've got 10 molecules in a box and I'm going to divide this box up into 20 smaller boxes. So I'm going to divide this large box into smaller boxes and the idea is I'm going to make the smaller boxes small enough that they're not much larger than an individual molecule so no more than one molecule can fit in each of these boxes at the same time. This is not how gases behave in the real world. This is just a model and approximation for how the gases behave. But I've divided my supposedly large, it's actually not all that large yet. Only 10 molecules in a volume as big as 20 molecules is not very large. But for this simple example, let's put 10 molecules into this large box. So that's 10 molecules that I've placed in a box containing 20 places for the molecules to exist. So what I've drawn here is one microstate. I've drawn one way for the molecules to inhabit the box and I've told you where every one of the individual molecules is. So that's a microstate. This is a description of the macrostate. As long as I put 10 molecules into my 20 grid positions, my 20 boxes, then that's a valid configuration of this individual macro-microstate. So if I tell you the number of molecules in the volume, then I've described the macroscopic properties. If I tell you where every individual molecule is, then I've described the microscopic properties. And of course we can calculate the number of microstates, the number of configurations that are consistent with this macrostate. If I have 20 boxes and I want to choose which 10 of them to occupy with molecules. So if I pull out a calculator and do that, turns out there's about 184,756. 184,756 different ways I could have drawn this diagram that have 10 molecules occupying 10 of the 20 grid sites. So far so good. I know how many microstates are consistent with this macrostate. Now let's consider what happens if I allow this box to get larger. If I, let's say, increase, so I'm going to draw a larger box. And I'm going to make that box intentionally twice as large, so I'm going to make it twice as large as the original box. So I'm going to have 40 lattice positions or 40 smaller boxes for the molecules to occupy. And I'm still going to have the same 10 molecules occupying the grid sites. So let's put 10 molecules into this box. One, two, three. Trying to do it somewhat randomly. Three, six, nine. I think that's 10 molecules. So that's again one microstate. I've chosen where to put the 10 molecules in that large box of size 40. So I've drawn one microstate out of the many possible microstates that are consistent with this macrostate. So when I say the gas expanded, it went from a volume of 20 to a volume of 40. So we can calculate how many ways there are to put 10 molecules in a box of size 40. And if I skip the intermediate step, that works out to be 847 million different microstates. So you can't see that number. I'll just write 8.6 times 10 to the 8th. A very large number of microstates are consistent with this macrostate. All right. So what does that mean? That means there's a lot more ways for this macrostate to exist than this macrostate. But what that means in particular is let's consider these macrostates in the larger box. So there's many of them that look somewhat like this. Some of them, coincidentally, might have all the molecules located in the bottom half of the box. If I threw darts at the box to put the molecules down, it might be true that all the molecules would end up in the lower half of the box, and the top half of the box would remain empty. So it's possible that all the molecules might spontaneously decide, not decide, not spontaneously find themselves occupying the lower half of this box. So what's the probability? So if I have a box that actually has a volume of 40, what is the probability that the molecules in the box will contract and occupy only the lower half of the box, occupying only the macrostate describing a volume of 20? There's 184,000 states consistent with a volume of 20 macrostate, but there's 864 million consistent with the larger box. So only this many occupy a volume of 20 out of this many that occupy the full volume of 40. So that number's very small, that's 0.02% of the microstates of this box are consistent with this macrostate instead of this macrostate. So what that means is if I place the molecules in the box randomly, then 0.02% of the time I'll find them occupying the one half of the box, maybe the lower half of the box. So the molecules in this box will spontaneously contract and occupy half the box only 0.02% of the time. 99.98% of the time they'll occupy a larger fraction of the box. So that's actually consistent with what we know about what gases do. If I have a gas and I raise the lid of the container doubling the volume, then the gas is going to expand and occupy the whole box. That's what we know about how gases behave. And that's what this says. It's quite likely 99.98% of the time the gas described by this model will expand and occupy the full box only 0.02% of the time will it not do so or will the box in a large volume spontaneously contract and occupy only half of the original volume. So there's several interesting things to point out about what we've just observed. Number one, we haven't said it's, well what we have said, the main thing we've said is only 0.02% of the time will a gas contract to occupy half its original volume. So it's quite unlikely to do that contraction on its own. Secondly, we haven't said that it's impossible, just like it's not impossible to win the lottery. It's not impossible that a gas will spontaneously contract down to occupy less than its original volume. It's just quite unlikely. The other thing that's significant is that we haven't described the properties of the gas in any way, any of the chemical properties of the gas. I haven't told you it's molecular weight. I haven't told you whether it behaves like oxygen or hydrogen or nitrogen or some other gas. I haven't told you anything about the chemistry of this model. All I had to assume was that there's molecules and they occupy certain positions on this lattice. So the lattice model is very simple. It doesn't involve any chemistry whatsoever and yet it's beginning to be capable of describing the properties of gases expanding. The next significant thing we can say about this model is that it begins to tell us not just numerical properties of the gas, but it begins to answer questions about why something happens. We can say why is it, why does a gas expand to fill its container? The real answer is just probability. It does so because it's more likely to do that than to do the opposite. It's quite unlikely that it spontaneously contracts. It's very likely that it spontaneously expands. And the last thing we want to say about this lattice model in particular is if I were to use a larger number than 20 molecules in a volume of 40, in a real world problem I don't have just 10 molecules in a volume of 20 or 10 molecules in a volume of 40. I might have Avogadro's number of molecules in a box that's many liters that could occupy many times Avogadro's number of molecular volume. So the ends we will be working with typically would be much larger than just 10 or 20. And imagine the smaller case where I have only, let's say, two molecules in a box containing four or a lot of sites, right? Four choose two. So four choose two is six, I believe. So out of the six possible orientations of the larger box, there's only one of them where both of the molecules are occupying the lower box. So in a very small box, one sixth of the time, we'll find these molecules randomly occupying the lower half of the box. So that's not particularly unusual. One time out of six is uncommon, but not terribly unusual. If I increase the volume to 20, then we're already seeing that some small fraction of a percent of the time molecules are occupying half the original box. If I continue to increase this value of n where it's not just 10 or 20, but thousands or millions or Avogadro's number of molecules, then what we see is that when n becomes very large, the probability becomes vanishingly small that a gas will ever spontaneously contract to occupy half the original volume. So the cases we're interested in in chemistry where we have very large numbers, these numbers are not just somewhat unlikely, but they're extremely unlikely to the point where it's a near certainty that the gas is going to expand and never be seen to contract. So we have two things we have to do next. Number one, we have to understand how to do these calculations when we're not just taking factorials of numbers like 10 and 20, but when we're taking factorials of large numbers like Avogadro's number. But before we get to that, we need to explore a few more of these different types of lattice models.