 We have an expression now for how the adsorbed volume of a gas depends on the pressure for an adsorberation system that obeys the BET model, the Brunauer-Emmett and Teller model. So next goal is to understand what that means, what this equation looks like if we were actually to plot a graph of adsorbed volume versus pressure and how that compares to the Langmuir model. Just as a reminder, this expression contains two constants. This constant K, which is essentially an equilibrium constant for multi-layer adsorption for a molecule from the gas phase adsorbing onto some layers that have been previously adsorbed onto the surface, just increasing the amount of the molecule on the surface. And then C is this ratio when a molecule adsorbs not onto itself but onto the bare substrate, C times K is the equilibrium constant. So that equilibrium constant for this reaction might be greater than or less than this equilibrium constant depending on the size of this C constant. Alright, so let's take a look at this expression and see if we can understand a little bit about it. The first thing that kind of jumps out about this expression is these quantities in the denominator that might cause the expression to blow up. So in particular, if it's the case that Kp is equal to 1, equilibrium constant is just some number. If the pressure is some number that causes K times P to be equal to 1, then this quantity in the denominator will be zero and the adsorption volume will approach infinity. So what does that mean and is that a problem with the model or what does it mean that the adsorbed volume should reach infinity? To make sense out of that, let's think a little harder about what this equilibrium constant K is for this reaction where a gas-phase molecule is adsorbing onto some other molecules of its own type. So that's a lot like a condensed phase, let's say a liquid phase or a solid phase onto which a gas-phase molecule is absorbing, just creating a thicker liquid phase. So once we've thought about it in those terms, that's just a phase change reaction. That's a gas-phase molecule condensing and becoming a liquid-phase molecule. So the equilibrium constant for that reaction products over reactants, if we are not concerned with the fact that this liquid just contains more layers than this liquid does than the liquid in the numerator of this equilibrium constant and the denominator of the equilibrium constant will cancel. The only thing that affects the equilibrium constant is the amount of gas-phase, since that's on the reactant side, it shows up in the denominator. So the equilibrium constant for this reaction is one over the pressure of the gas. In particular, it's one over the pressure of the gas under conditions where this reaction is in equilibrium because we have an equilibrium constant. So it's not just any old pressure, that's the equilibrium saturated vapor pressure of that gas. So P star will write for the vapor pressure. So the equilibrium constant is one over the vapor pressure of this particular adsorbate. So if I use the fact that K is equal to one over P star, the vapor pressure, then when K times P is equal to one, that means P times one over P star is equal to one or pressure is equal to the vapor pressure. So now macroscopically, that does make sense. If I increase the pressure to the point where I'm approaching the vapor pressure, then more and more molecules will adsorb onto the surface until the point where they're stacking up molecules on top of molecules. And by the time I get to the vapor pressure, molecules will be just as happy adsorbed onto the surface as in the gas phase. And if I exceed the vapor pressure, then all the molecules will fall out of the gas phase onto the surface. So what that means is wherever on this graph the vapor pressure is, this curve, whatever it does in between, we haven't discussed yet. But as the pressure increases, the amount of adsorbed species is going to climb, climb, climb, climb, and asymptotically approach infinity as we approach the vapor pressure. That just represents condensation of the gas phase into the liquid phase, molecules adsorbing onto themselves once we reach the vapor pressure. So that's not a problem with this model. That's actually a good prediction that the Langmuir model is unable to produce. The BET model is able to produce the fact that the gas will condense and become a liquid as it approaches the vapor pressure. All right, so, and I guess I'll keep referring back to this expression with the case in it, but we can rewrite this expression if we want the more conventional way of writing the Langmuir isotherm equation. Let's see, I'll put that down at the bottom of the screen. The adsorbed volume of the gas relative to the monolayer volume we can write instead of using these k's, I can write that as c times p over vapor pressure over, and now in the denominator, 1 minus p over vapor pressure. And the longer term is 1 minus p over vapor pressure plus c times p over vapor pressure. So that has removed these k's from the expression, and now I can just write the adsorbed volume as a function of the pressure and the vapor pressure, which is a known quantity, as well as this BET constant called c. All right, so now let's pay a little more attention to what's going on, not at high pressures on this graph, but at lower pressures. So if the pressure is low, in particular, if the pressure is low enough that this kp quantity is not one, not even close to one, it's much smaller than one, I'm going to end up ignoring these kp terms compared to one. Physically what that means, since k is equal to 1 over p star, that means pressure is much, much less than the vapor pressure. So we're not talking about this limit, we're talking about the limit down here at low pressure, so when the pressure is very low compared to the vapor pressure, kp can be ignored compared to one, so this term is small compared to one, this term is also small compared to one, so I can pretend those terms aren't there, and then I can say the adsorbed volume relative to the molar volume is going to look like ckp in the numerator. I can't throw that away because it's not small compared to zero, which would be left if I threw it away. In the denominator, I've just got one times second term in parentheses one plus ckp. So I could, if I take that c and k and I bundle them into a second constant, if I just call that, let's say, k prime times the pressure, so I've just let c times k be equal to k prime, writing it that way makes me realize that's just the Langmuir expression. So constant times pressure over one plus a constant times the pressure, that's exactly the Langmuir expression. So in the limit of low pressures down here on this side of the curve, we're behaving exactly like the Langmuir model predicts we should behave, but when we get to pressures that are too high, we begin to behave a little differently, and the ways we can behave differently are we can exceed one monolayer of coverage, and as we approach the vapor pressure, we'll predict condensation. All right, so that gives us the general shape of this curve. That's what we would get if we plotted this curve. Qualitatively, the way this curve looks is a little bit different for different values of this constant c. In particular, let's take the case where, actually let's start with the opposite case first. Let's take the case where c is less than one. So this is a letter c, and that's a less than sign. When c is less than one, remember what that means is this equilibrium constant is less than this equilibrium constant. So the binding to the surface to create the first layer of adsorption is weaker than the binding of the molecule to itself. The molecule interacts more favorably with itself than it does with the surface. So I'll say that's weak adsorption to surface and stronger adsorption of the molecule to itself. What that's going to look like, the molecule, so again, the graph is going to have the same general shape. But because the molecule adsorbs relatively weakly to the surface, that constant, the rate at which the volume adsorbs at low pressure, that's going to be lower. So it's going to start out increasing relatively slowly. Eventually, as it approaches the vapor pressure, it's going to condense. But basically, that's condensation because it prefers to be in the liquid phase rather than the gas phase. It doesn't have any particularly strong attachment to the surface in particular. So it doesn't adsorb very well to the surface until it just falls out of the gas phase. On the other hand, in the opposite limit, if that constant is bigger than one, that means this equilibrium constant is larger than this equilibrium constant. Binding onto the surface is more favorable than binding of the molecule onto itself. So that's a relatively strong adsorption to the surface. That's a case where the molecule does bind relatively easily to the surface, more easily than it binds to itself. So in the beginning, this looks a lot like Langmuir adsorption. The curve's going to go up. It's going to bend over as if multilayer adsorption were not going to happen. And it couldn't exceed one monolayer of coverage because molecules fall down first and just fill up the surface sites. Once there's very little place to go other than binding onto more molecules of themselves, eventually it will increase across this single monolayer coverage and increase to see condensation when we approach the vapor pressure. And so that's the case when we have relatively strong adsorption to the surface, binds relatively quickly to the surface and then takes a little while after that for the molecules to bind to themselves. So we can get lots of different shapes for these curves depending on the value of C, depending on the value of the vapor pressure. And the BET model is able to capture all this behavior ranging from molecules that exhibit monolayer or near monolayer adsorption at low pressures all the way up to the vapor pressure at which they'll condense and adsorb many layers onto the surface at once.