 All right, if we're going to make any progress describing the physical and chemical properties of solutions, mixtures of two different or more different compounds in the liquid phase, we're going to need to do the type of thing that we often do in physical chemistry, which is to define a simple model that describes the system as simply as we can. And kind of like we've done for the ideal gas, when we talked about gases, we describe the system as a gas with as simple a model as we can and see what the repercussions of that model are, even though it's not an absolutely correct version of describing a real world physical gas. It gives us a starting place in some things we can correct when talking about real gases. So we're going to do the same thing for solutions and define what we call the ideal solution model. And again, it'll be the simplest possible model we can construct of a solution. It may not be right. It may not perfectly describe real solutions, but it gives us a starting point. So we'll start with a sketch of what a solution looks like. So I've got a solution, and we're going to zoom down to the microscopic level and look at individual, at least a cartoon of individual molecules. Let's say we've got some two different species, mole fraction of A and mole fraction of B in this solution. And I'll draw my A molecules as one type of circle and my B molecules as a different type of circle. So I'll just draw a whole bunch of molecules in this box and some of them will be A's and some of them will be B's. So that goes on in all directions. So I've drawn those in a grid like we would need to if we're going to study this system with the lattice model of the type that we got used to when we first started talking about the probabilistic character or the entropy and the multiplicity, for example, of systems at the microscopic level. So if I think about a lattice model for a solution, I just put one molecule in each one of these boxes. None of the boxes are left empty. So the question becomes, what do we assume about this simple model to make it a good model for a solution? We can't do what we did for the ideal gas. We can't start by saying the molecules don't interact with each other at all. That was our assumption for the ideal gas. In fact, for a solution, they have to interact with each other. They have to interact strongly enough that they've condensed out of the gas phase. When we talk about Van der Waals gas model, for example, having that interaction between molecules is key to making the molecules condense and form the liquid phase. So we have to have some interaction between A and B molecules or between A molecules and themselves. But what we can assume at the macroscopic thermodynamic level, I can put the molecules down in the arrangement that I've just drawn them. That's, of course, only one of many different configurations I could have drawn. What I'm going to assume is that whichever configuration I draw, it's going to have the same energy. So if I rearrange this and make this molecule an A molecule instead of a B and put a B molecule there instead, if I rearrange the molecules in some way, then the energy is no different as long as I haven't changed the mole fractions of the solution. So every different rearrangement doesn't lead to a change in the internal energy or equivalently the enthalpy for that system. That would be a macroscopic way of understanding the system. If I want to think about it more at the microscopic level, the individual atomistic or molecular level, an equivalent statement to that requirement would be to say the total energy of the system, the total potential energy of the system is identical for every microstate. Every way I have of placing those molecules down on that lattice model, they all give me the same energy. In order for that to be true, so that's the assumption I'm making. One thing I could require in order to guarantee that this requirement is met, an equivalent statement would be what if I have the potential energy of interaction between an A molecule and an A molecule, this AA interaction? What if I require that that is exactly the same as a BB interaction and is exactly the same as an AB interaction? If a molecule doesn't care who its neighbors are, it has the same potential of the energy of interaction if it's surrounded by A molecules or by B molecules, then if I reshuffle those molecules, if I replace its neighbors by a different type of neighbors, then that's not going to end up changing its energy. This is certainly a good enough, a sufficient condition to guarantee that the energy doesn't change when I rearrange the molecules. It turns out that's actually a little bit stricter than I need. I won't go into the details too much, but it's actually good enough to require the heterogeneous interaction, the interaction between an A molecule and a B molecule. As long as that's equal to the average of the two different homogeneous interactions, as long as that's equal to the average of the AA and the BB interactions, that's also a good enough condition to guarantee that the energy won't change. Essentially the reason behind that is if I pick up a molecule and I exchange it with another one, I'm going to gain as many of one type of interactions as I lose of others. It turns out that this is also just as good a requirement. Whatever one of these we think about, the purpose of making this assumption at the microscopic level is to guarantee that the energy doesn't change for each one of my microstates. All those microstates are degenerate, the same energy as one another. Whichever one of these you want to think about, these are all equivalent assumptions for an ideal solution. What we would normally do at this point, or at least what we've done in the past, once we've defined a model, I've defined some energy of interaction for the system. We could write down a partition function for the system and use some thermodynamic connection formulas to derive entropies and energies, internal energies, for example, free energies and so on. And we'd be on the path toward calculating the thermodynamic properties of an ideal solution that obeys these assumptions. We've actually already done some of that already. We have written in particular for a lattice model involving fluids. We've already done the work of driving what is the entropy of mixing when I take a pure A liquid and a pure B liquid and I mix them together in a lattice model. We've seen already that that is equal to, for this two-component solution, so negative Boltzmann's constant times mole fraction, log mole fraction, summed up over each of the two different components in that solution. So like I said, we've already made a little bit of progress up toward the thermodynamic level and it's much easier to use thermodynamic relations to connect the different properties of a solution than it is to use the microscopic statistical mechanical equations. So we're going to start with this entropy of mixing at the thermodynamic level to describe this ideal solution and we'll see what we can learn about the thermodynamics of solutions from there and that's what we'll do next.