 OK, so let's continue talking about lattice models and see what we can learn from them. And it turns out that lattice models are not only useful for modeling gases, but we can also use them to model liquids as well, or fluids. So let's consider what it might look like if we use a lattice model to describe a liquid. So a liquid, as you know, is something that occupies the entire volume, rather than expanding to fill its volume like a gas, it occupies the lower portion of the gas, and molecules are not separated from one another. So let's say we have a lattice model for a fluid. If I'm going to take eight molecules, then they're going to occupy, in this small example, they're going to occupy all the eight lattice sites in this model. And I don't want to include vacant sites down at the bottom. So the fluid's going to fill the container from the bottom up. So if I want to talk about fluids mixing, which is what I do want to talk about, then I need to be able to describe, let's say, I put that beaker of a fluid next to a different beaker with the same size. But in this beaker, I'll put a different fluid. I'll put molecules of a different type in this beaker. And let's say I've got an equal size beaker with eight of these molecules that I'm calling b, molecules of b, rather than molecules of a. So these green molecules. So now what I want to do to understand what happens when the fluid's mixed or don't mix is, let's say, pour those molecules into the same container. So now I have a total of volume of 16. And really what I've got is, if I want to describe, so here's the microstate of the system. I've told you where every individual molecule is in that combined volume. The macrostate of the system, the way I've described it, is I've still got eight molecules on the left side of type A. And on the right side, I've got eight molecules of type B. So the macrostate would be eight of these on the left and none of them on the right. So that's enough for you to put the molecules down the same way I have here. The multiplicity, how many ways are there that I could have drawn microstates consistent with this macrostate? There's only a way to put eight molecules of the red molecules on the left, orange molecules on the left, and eight green ones on the right. But mathematically, that's like saying, how many ways are there to choose which eight of the spots to occupy with eight molecules on the left? And so I'm multiplying how many ways are there to choose which zero of the states on the right to occupy with the orange molecules. So eight choose eight is one. Eight choose zero is one. That product just comes out to one. There's only one way of drawing a microstate, this one, that is consistent with that description. So what we want to understand is what this lattice muzzle has to say about whether fluids mix or don't. So I can draw other configurations that show the fluids mixing partially or fully. And the multiplicity, the probabilities will tell us how likely those particular outcomes are. So let's consider a state. So I've still got this combined volume of 16 grid positions. And let's say I want to let the molecules partially mix. I want to find out how likely it is that most of the molecules have stayed separated, but one molecule has moved over to the other side of the box. So my macrostate would be seven of these on the left and one of them on the right. So that's, again, my macrostate. I can draw an individual microstate just so it's clear what I'm talking about. I still have eight of each type of particles, so I'm gonna fill every grid position in the box. And I can ask, how likely is it that this, or what's the multiplicity? How many different microstates could I have drawn that are consistent with this macrostate? How many ways do I have of choosing which seven spots on the left to occupy with orange molecules and which one spot on the right to occupy with green molecules? You might ask why I don't also include a binomial coefficient here for describing how the green molecules behave, but once I've told you which seven spots are occupied by orange molecules on the left, there's no more choices to be made. All the green molecules are specified. Same thing on the right. If I've told you which one spot is occupied, all the other ones have to be occupied by green molecules. So eight to seven is eight. Eight to one is also eight, so the product there is 64. So it turns out that the multiplicity for this partially mixed system where one molecule is crossed over to the other side is 64 times more likely. There's 64 ways to draw this. There's only one way to draw this. So it's 64 times more likely that the fluids mix at least a little bit. But of course that's not the most probable thing that will happen. I can ask questions about any degree of mixing I want, but it turns out not surprisingly at this point perhaps. If I ask the question, how many microstates are there that can be described by four orange molecules on the left and four of them on the right? So if that's my microstate, I want to put four molecules down on the left somehow and then four of them on the right somehow, the rest of them green molecules, four on the left and four on the right for the green molecules, then the multiplicity will be how do I choose which four of eight spots to occupy with orange molecules on the left and how do I choose which four of eight spots to occupy with orange molecules on the right? Eight choose four is 70. So 70 times 70, 4,900 is the numerical answer we get here. So even for this small system of only eight molecules in each beaker that I've mixed together, turns out that the mixed system, where they're fully mixed half and half, that is almost 5,000 times more likely than having the system remain completely unmixed. So that tells us two things. Tells us an answer to the question, why it is that fluids mixed together? When I mix two fluids together, why do they not stay separated? Why do they mix together? The real answer to that question is because it's much more probable that they'll find a mixed configuration than they'll find this rare configuration where the fluids are separated. The other thing it tells us is, excuse me, that again, it's not just, it is only overwhelmingly likely that the fluids will mix together. It's not guaranteed that the fluids will mix together. That it is possible to find a configuration that represents a separated configuration. But again, the larger the system gets, the larger these values of n that I'm taking factorials of, the larger this number will get and the more overwhelmingly likely it will be that the fluids mix. And it's also worth pointing out that, again, because we haven't built any chemistry into this model, we haven't said whether these are molecules or water molecules or methanol molecules or oil molecules or so on. For this simple case, we get the same answer for every pair of fluids. We don't have the ability yet to distinguish why water and methanol mix together fairly well, but water and oil don't mix together. So once we start building chemistry into the problem, then we'll begin to see some nuances in these answers.