 Alright, so we have this equation, the Langmuir isotherm equation that tells us the surface coverage as a function of pressure, if we are adsorbing molecules onto a surface given their binding energy, the temperatures. So I'll show you a different way to obtain that same equation, derive that same equation. We derive this one using a little bit of statistical mechanics, starting with the lattice model and using some statistical mechanics. There's another way to get that same equation that's actually a little bit simpler where we can get by with just using equilibrium and thermodynamics without having to use partition functions. So I'll remind you the details of this model. We've got molecules up in the gas phase, we've got molecules that are adsorbed onto the surface, we've got an equilibrium between the two. Let's, in order to think about this from an equilibrium point of view, let's think about this in a little more detail. These molecules that are stuck to the surface attached to some binding site, there's some chemical structure of the binding site they're attached to, they might be bonded to one atom or bridging across two atoms, but there's some structure that they're bonding to on the surface. We might have sites or positions on the surface that are not bound to any adsorbed species, so we have some unoccupied surface sites and some occupied surface sites. So the O's that I've drawn here are the adsorbed species, the X's that I've drawn are either the surface molecules that adsorbed species are bound to or surface species that are not, that are empty, don't have anything adsorbed onto them. So we could, if we think about doing an equilibrium thermodynamic reaction and the chemical reaction that we're talking about is gas phase species in equilibrium with adsorbed species. If we want to think about it in terms of the two reactants that need to encounter each other and form a bond in order to create an adsorbed species, it doesn't require just a gas phase molecule, it requires one of these surface sites as well. So when a gas phase molecule encounters an empty surface site, then they have the possibility of binding, reacting, and becoming an adsorbed species. So this equilibrium process has an equilibrium constant. The equilibrium constant for that reaction, we can write that as a Kp or a Kn, we can write it in lots of different ways, but if I write that as molecule numbers, that'll be number of adsorbed species divided by number of these unoccupied empty surface sites, X, and number of gas phase species. Alright, that we can make look a little more useful if we write it not in terms of these extensive total number of species, but in terms of surface coverage. So remember, surface coverage is the fraction of the sites that are occupied by molecules, so that would be the total number of adsorbed species over the total number of sites. I could think of the total number of sites as the number of empty sites plus the number of adsorbed sites if I want to, adsorbed plus empty, or probably easier just to say it's the total number of adsorbed sites divided by N total, the total number of sites on the surface, or we've actually used the variable M to talk about the total number of sites onto which molecules can absorb. So theta, surface coverage is the adsorbed sites divided by the total, 1 minus theta would be the empty sites, the number of empty sites divided by the total number of places you can absorb. So if I rewrite this equation, instead of thinking about it as N over these Ns, if I divide on the top by an M, so it's the number of adsorbed over M on the top, on the bottom I'll leave Nx, I'll divide that by an M also, and I'll leave gas phase species alone. So all I've done is divide on top and bottom by M so that didn't change the value of this fraction. But it does allow me to say adsorbed over total is equal to theta, in the denominator unoccupied over M is equal to 1 minus theta, so I can write this as theta over 1 minus theta, and I have a 1 over N, that's this value of the equilibrium constant K, number of molecules in the gas phase. So we know pressure is Nkt over V, since we're talking about a gas, if it's a low pressure gas that behaves relatively ideally, I can rewrite the number of molecules as a pressure by just multiplying by some Ks and Ts and Vs in the right way. So I can say this is at least proportional to theta over 1 minus theta, not 1 over N but 1 over P. So instead of actually doing the multiplying by Kt over V, I've just said this quantity is proportional, 1 over N is proportional to 1 over P. And in fact maybe an easier way to think about it would be to say this value of K is equal to theta over 1 minus theta 1 over N, a different value of K, still going to be a constant with a different numerical value, so you can call that K prime or I've just written it as a different color K is equal to theta over 1 minus theta times 1 over P. So that result, let me rearrange that just slightly and write that as my new K times P is equal to theta over 1 minus theta. If I do a little bit of algebra, I won't walk through the algebra steps because it's very similar to something we did when we derived this equation via stat mech. Well, I might as well write it out. If I multiply by Kp by this 1 minus theta, I'll get Kp times 1 minus Kp times theta and on the left side I'll have theta. So theta ends up being equal to Kp, so I bring the Kp over here, theta times 1 plus Kp is equal to K times P over here and now if I've kept all that straight what I find is that theta is equal to Kp over this quantity in parentheses 1 plus K times P. Alright, so if I kept my color scheme straight, this equation that we just obtained is exactly the same as this equation that we obtained perhaps with a little less effort when we think about it from an equilibrium point of view than from the stat mech point of view that led to this result. The downside of doing it with thermodynamics is this K, this equilibrium constant that we've just determined, we don't know as much about this value of K as we did about this value of K. In principle, if we've studied the problem from a stat mech point of view, knowing something about the binding energy of the molecules to the surface, we can calculate the value of this K. From an equilibrium point of view, it's much more difficult to calculate the value of K to understand how this value of K depends on temperature whereas in this equation we know a lot more about how this K depends on the temperature. So we got to the same result via a simpler process but we've lost a little bit of information about where this K came from. Alright, so we've seen two different ways of obtaining the same equation, this Langmuir isotherm equation. It turns out there's yet a third way to look at the equation and that's what we'll do next is take a look at that third way.