 Okay, so time to summarize what we've done with the Langmuir adsorption model and explain why it's not always as useful as we'd like it to be. So I've put up here a few of the key features of the model. We've got molecules adsorbing and desorbing off the surface. They have some binding energy going on the surface, no energy in the gas. That model leads to this Langmuir isotherm, which tells us that the surface coverage will increase as a function of pressure, saturating at a monolayer coverage, covering one full layer on the surface when the pressure gets large. So it turns out not every adsorption system obeys this Langmuir isotherm, and that's because this model is a little bit too simple in a lot of cases. So this model is kind of like the ideal gas analogous to the ideal gas model. So the ideal gas is a nice simple model. It gives us equations we can use. But real gases aren't ideal gases. Real adsorbate molecules and their interactions with surfaces aren't exactly like the way we portrayed them in this Langmuir adsorption model. So to see where the model breaks down, let's remind ourselves what some of the assumptions were that we used in deriving the model. So we've assumed a number of things. At some point, we said the system's in equilibrium. What we did operationally is we said, okay, the chemical potential in the gas is equal to that of the adsorbed species. So we've assumed the system is in equilibrium. Another assumption we made is that among these M total sites on which molecules can adsorb, we can't ever put more than one molecule on a site. That may seem like a fairly harmless assumption. We can make those boxes small enough or those chemical binding sites small enough that certainly no more than one molecule could fit. But what that accounts for is the fact that I can fill those lattice sites with one molecule each, but I'm not going to allow other molecules to stack on top of the molecules that are already here. So we're assuming that we can never have more than a single monolayer of adsorption onto a surface. So a third assumption that we made is this value of epsilon. Certainly molecules bind to the surface with some binding energy. That binding energy we can represent by some constant. But we've assumed that that binding energy is in fact a constant. That the binding energy right here is the same as the binding energy right here. Meaning that if I adsorb a molecule, another one of these pink molecules next to an existing molecule, it only interacts with the surface. And the fact that it's next to another pink molecule, it doesn't have any favorable interactions from being close to another molecule of the same species. It doesn't have any steric interactions or repulsive interactions with another molecule of the same species. So we've assumed that each of these binding events is independent. And that the adsorbate species don't interact with each other. So that's the assumption that we made by assuming that epsilon was a constant instead of allowing epsilon to depend on where I bind on the surface or whether I have another neighbor next to me when I bind. And lastly, one of the assumptions that we made in doing the math that led to this Langmuir isotherm is we used the sterling approximation. We assumed and we expanded the log of m factorial, log of n factorial both using the sterling approximation. So that's only going to be true if the number of binding sites is much larger than one and the number of molecules on the surface is much larger than one. So that assumption that we've made in that case is that the surface is relatively large. And large is a relative term, of course. Large just means the number of binding sites larger than one. So since the binding sites are small and molecularly sized, it doesn't have to be macroscopically super large. It just has to be large enough to hold many molecules, which is often not a terribly stringent approximation. So these are the approximations we've made. We can ask ourselves, what are the conditions we might expect those approximations to fail to fall apart? So I'll run through those one at a time. The equilibrium approximation can fall apart. So let me make another column here. This assumption will fall apart when the system is not in equilibrium. One circumstance that can be true is when we don't have chemisorption. I'm sorry, when we don't have physisorption, but we do have chemisorption. So process I've been talking about so far is a physisorption process. Molecules bind and then reversibly they can desorb or come off the surface. If I have some surface, let's say my species that's adsorbing out of the surface is water. So water molecules can bind to the surface. If they remain intact as a water molecule, then they can later desorb off the surface. But if when they bind to the surface, they covalently dissociate and become an OH bound to the surface and an H bound to the surface. That's something that happens on a titanium surface, for example. Titanium will catalyze the breaking of that HO bond. Number one, that process is much less reversible. That's a more irreversible process. And number two, even if the process were reversible, this OH stays next to each other, the reverse process can happen. But what actually happens is often this H molecule will bind, the OH will attach to the surface. The H is fairly mobile depending on the temperature. It can diffuse or hop around the surface. And if it diffuses far away from this OH surface species, then there's no way for the reverse reaction to happen in the same way as the forward reaction happens. So the desorption process is not just the reverse of the adsorption process. And so these chemisorption cases don't end up following the Langmuir isotherm, like we might hope that they would. So chemisorption applies to that sort of situation. Monolayer adsorption, the type of picture we've been drawing up until now, where I don't ever put more than one molecule in a box, where that would fail, would be a case where, so here's some molecules that are adsorbed onto a surface. As I absorb more and more molecules onto the surface approaching a full monolayer, one thing that can happen is not only can I fill up the rest of the surface, but in fact adsorbate molecules can begin to adsorb on top of other adsorbate molecules rather than on top of the surface sites. That can happen in particular when the interactions between these gas-phase molecules are stronger than the interaction between the gas-phase molecule and the surface. In fact, they may prefer to form a little droplets on the surface and adsorb in multiple layers on the surface rather than a single flat monolayer. So some adsorbates will absorb in a monolayer fashion and won't adsorb on top of themselves. Other species adsorb with multi-layer adsorption rather than monolayer adsorption. And that will completely violate the assumption that each one of these lattice sites can only hold one molecule. It can hold two or perhaps three or very many molecules in these multi-layer adsorption cases. So if you have multi-layer adsorption, again the Langmuir isotherm model won't be correct. This third assumption that the binding is independent. Let me draw a picture like this to talk about that. So here are my binding sites on the surface. For a small number of adsorbed species, I've got one here, one here, one over here, perhaps. As I absorb more and more molecules onto the surface, it may in fact be true that a molecule adsorbing right next to another molecule, if they have a favorable interaction, that may be more probable than adsorbing far away and not interacting with other species. So I may find that what I get is not random adsorption at random sites on the surface, but they may tend to cluster. Once I've got one molecule adsorbed in a particular spot, I tend to get more molecules adsorbing nearby because it's a favorable interaction. So that's an example of what we can call cooperative adsorption or cooperative binding. So one molecule adsorbing helps another molecule adsorbing nearby, makes it happen with a higher probability because of that cooperative adsorption phenomenon. You can also have an anti-cooperative binding if these molecules have an unfavorable interaction energy. In either one of those cases, again, the Langmuir isotherm model won't necessarily be very accurate. The last case is less common. It's not that common that your surface is not large enough to hold at least 100 molecules, for example, but if you have extremely small surfaces, then the approximations we made by assuming we had a large number of molecules using Strowing's approximation, those approximations can break down. So if your surface is incredibly small, then you won't obey the Langmuir isotherm model as well. So this sounds kind of depressing. We've got all these different mechanisms by which the Langmuir adsorption model can fail. Some of these are very common. Multi-layer adsorption is very common for a species like water that has favorable interactions with itself. Cooperative adsorption is also very common depending on the sites to which the molecules can bind on the surface. So in fact, a large number of adsorption cases don't obey Langmuir isotherm model. So the next question is what would we do if we have a situation where we wouldn't expect the system to obey the Langmuir model and we can solve some of these problems and develop a more accurate model that will help us understand adsorption in particular for the case where we have multi-layer adsorption. So we'll talk about that one next.