 If we are going to get a little more quantitative about how to understand adsorption, molecules adsorbing onto a surface, we need to do what we typically do to understand a problem in PCM, which is to develop a model for that process, starting with an idealized model, a relatively simple model and then possibly progressing to more complicated models. So for adsorption, we'll start by developing the type of model that we've used previously, and that's a lattice model. So I've got, in general, I'm drawing a surface here, some sort of surface to which these molecules are going to adsorb. I might have some molecules up here in the gas phase or the solution phase that adsorb onto the surface. I might have some molecules on the surface that can desorb off of the surface, and then we'll break the surface itself up into a lattice. So these lattice sites can contain a molecule, or not the molecules can desorb off the surface once they've adsorbed or adsorb onto the surface from the gas or solution phase. So to define that model a little more carefully, that lattice model, I've got a certain number of lattice sites, let's say there's a total number of sites that we can give a label to. M lattice sites on the surface in the, let's go ahead and assume these are in the gas phase. For now we've got molecules up in the gas phase, and we've got the same molecule in the adsorbed phase when it's stuck to the surface of the solid down here. We can specify what the energies are of a molecule in the gas phase or a molecule in the adsorbed phase. Those energies will in general be different. Just to choose a scale for the energy, I'll say let's let the energy be zero. I'm going to use epsilon for the energy of these molecules. That energy, I'm sorry, I'm going to let the energy be capital E for the molecule in the gas phase, and for the adsorb phase I'm going to let that energy be some constant that I'll name epsilon. So when it's stuck to the surface, it has some energy in the gas phase, it has no energy, no potential energy. This energy, if it's adsorbed to the surface, if it's interacting strongly enough with the surface to stick to it, that energy epsilon is going to be a number less than zero. So I'll use epsilon as the label, epsilon the constant is a value that's a negative number. So keep that in mind when we get to the point of plugging numbers into these problems, remember that epsilon is going to be a negative number. Alright, what else do I need to know? In specifying this model, I've made two assumptions that I'll go ahead and write down and make explicit. We have said each one of these lattice sites, each one of these boxes into which I can put a molecule. Let's say the boxes are such that only one molecule of A can fit in one of those lattice sites, so the boxes are small enough, there's no room for two molecules in one lattice site. Also, by saying the energy of an adsorbed molecule is this number always, regardless of which lattice site it adsorbs to, also regardless of whether it's adsorbed next to another molecule or far away from other molecules, we've made the assumption that we have no interactions of the molecules on the surface. Each of these interactions might or might not be true for a real system and we'll eventually be able to talk about what happens if these assumptions are not true, but for our simple first model, we'll assume that the molecule sticks to the surface with some energy epsilon regardless of where it sticks. Every lattice site it binds with the same energy. And then I guess the only other thing we need to define is a count of how many of these M sites are full. If I have M total lattice sites, some fraction of them are going to be occupied by molecules. The total number of adsorbed molecules out of those M possible sites is going to be N. So those are my choice of the two variables we'll talk about for that system. Now that we've defined the problem, essentially everything up until here has been just defining the variables, assigning variables to different quantities in this model. Now that we've done that, we can do what we have typically done once we define a lattice model, which is to define a partition function. In particular this partition function, I'm going to get a value for this partition function that's going to depend on how many molecules are adsorbed to the surface. If I have N equals 5 molecules stuck to the surface, I'll get a different value for Q than I would get if I had 100 molecules adsorbed to the surface or a mole of molecules adsorbed to the surface. So I'm going to show Q depending on the value of N. In general, of course, a partition function is always just the sum of the Boltzmann factors. So I have e to the minus epsilon over kT for each one of the molecules on this surface. And if I have N molecules adsorbed onto the surface into a total of M different adsorption sites, there's a degeneracy. There's a number of ways I can have...actually, that's not true. This again should be a capital E. It's the total energy of the system. So the total energy of the system, I have some molecules with no energy. I have some molecules with epsilon amount of energy. The total amount of energy is going to be each of the N molecules adsorbed to the surface, has an energy epsilon. The molecules that are not adsorbed to the surface have an energy zero. So the total amount of energy is N times epsilon. If I have N molecules adsorbed to the surface, that's always going to be the energy. So this is always going to be the Boltzmann factor, but I have some degeneracy. I have a number of different ways I could have placed those N molecules onto the surface. The four molecules I've drawn on the surface, I could have put them here, here, here, and here, or I could have put them in four other sites or four other sites. The total number of places I can, total number of ways I can put N molecules into M sites is M choose N, which is M factorial over N factorial M minus N factorial. So combining those two pieces of information, I can write my partition function as these factorials, M factorial over N factorial M minus N factorial e to the minus N times epsilon over kT. All right, so that's an expression for the partition function of the system. If I have capital N molecules adsorbed onto the surface, that's a good starting point. That's essentially, we'll stop there because now we've defined the model. We know what the partition function is for this model. The next step will be to take a closer look at that partition function and do what we often do with partition functions, which is to try to understand the thermodynamics of the system via that partition function. Do a little bit of statistical mechanics. So that's what's coming up next.