 Okay, so we understand the Langmuir isotherm and this form in which we've made the Langmuir isotherm linear, inverse surface coverage linearly depending on inverse pressure. But the surface coverage is not the most convenient property to measure experimentally. If you imagine how you'd go about measuring surface coverage experimentally, you need to somehow look at the surface and determine what fraction of it is covered by molecules. That's not an impossible thing to do. You can do imaging studies, scanning electron microscope images, TEM studies, for example, or SEM, and somehow after you've adsorbed the molecules onto the surface, if you're careful not to let any of them desorb, if you quench or bind those molecules to the surface, take it into your SEM or TEM and image the surface, you can count what fraction of the surface is occupied by molecule. That would be one way of measuring surface coverage, but it's a fairly elaborate, maybe a difficult procedure, more difficult than it needs to be. A simpler procedure, so remember what we're doing when we adsorb molecules onto the surface, we have a surface, we introduce molecules above the surface at some particular pressure and then they'll bind to the surface until we have an equilibrium between bound adsorbed species and gas-based species. So imagine we've got a tank of gas, so in this container we introduce from this large tank of gas, so here I'll just draw a reservoir with lots of gas-phase species, doesn't matter what pressure this gas is at. I've got a reservoir of gas, introduce a certain volume of gas into this system at whatever pressure we're interested in, let it reach equilibrium with the surface, I've got a certain amount of surface coverage. Now imagine that I take this surface out of the system, I remove the surface from the container leaving the gas in here, so now I've removed some of the gas, the original gas-phase molecules that are now adsorbed with the surface and then when I take this gas and I bring it back out of the system the total volume of gas that I removed from the system is smaller than the total volume of gas that I've introduced into the system even if I bring it back to the original pressure because I've lost some of the molecules that adsorbed to the species. So that difference in the volume, if I say what the volume be, the difference between the volume of gas I put in and the volume of gas I took out, that's the quantity we can call adsorbed volume, that's the volume of gas that adsorbed onto the surface. So adsorbed volume sounds a little bit strange, once the molecules are adsorbed we don't think of them as occupying volume, they might have a surface area, they might be a number of molecules or a number of moles attached to the surface, so when we say adsorbed volume what we really mean is the volume of gas that is equivalent to the number of adsorbed species attached to the surface, so that's the adsorbed volume. If we imagine doing that experiment that I just described, I'm going to now measure the adsorbed volume as a function of the pressure of the gas above the surface. So at a low pressure I might adsorb a relatively small number of molecules on the surface which I measure not as a surface coverage but as an adsorbed volume at a higher pressure, I might get more molecules on the surface and at a higher pressure more still, very high pressures that curve begins to plateau. This is a graph of the Langmuir isotherm, I'm just plotting volume as a function of pressure rather than surface coverage. So asymptotically it no longer approaches one as the surface coverage does. If I were plotting surface coverage versus pressure it absolutely would approach one as I get a full monolayer coverage on the surface, but since I'm measuring volume rather than surface coverage it's going to approach some volume that we'll call V sub m. You can think of it as Vm for max, it's the maximum amount of volume that will adsorb onto the surface or m for monolayer, it's the volume of gas that's equivalent to a full monolayer if all the surface sites were adsorbed occupied by adsorbed molecules. So by doing this experiment at a number of different pressures we can obtain the value of V sub m. If we want to connect these adsorbed volume measurements to surface coverage, what we know is that the surface coverage is one only when I've filled the whole monolayer, when I've put a molecule at every possible surface site. So when V is equal to Vm then we've got full coverage. In fact the surface coverage is the total volume relative to that maximum volume that we can get. So now once we've measured V sub m we can use the measurements of volume to determine the surface coverage. So we don't measure surface coverage directly we get it indirectly by measuring adsorbed volume at several different conditions. If we ask ourselves what does that say about this equation, if I go back to this equation and rewrite my linear form of the Langmuir isotherm equation, instead of writing 1 over theta I'll write 1 over this quantity, 1 over V over Vm is the upside down of that Vm over V. So that's the left side. On the right side I've still got 1 over k, 1 over p plus 1. If, let's see, if I take the V sub m from the left side over to the right side, so all I've left, all I've got left on the left is 1 over V. I'm dividing by Vm on both sides so I'll write 1 over k times Vm, 1 over p and then the 1 becomes a 1 over Vm. So that's also a linear equation in the form y is equal to m times x plus b. The m term with k's and Vm's, Vm is just a constant whether we've measured it yet or not it's just some particular number for a given system. Those k's and Vm's are constant, pressure is the independent variable that we vary. So if we want to make a graph that corresponds to the things we're actually measuring, again instead of plotting V versus p and getting this nonlinear curve, what's more convenient is to plot, as suggested by this equation, 1 over V as a function of 1 over p and again I'll get a straight line. My data will be on or close to that straight line. And now notice that we have a slope and we have an intercept. The slope of this curve is 1 over k times the monolayer volume or the maximum volume. The intercept now tells us something. The intercept is 1 over Vm. So rather than having to do the extrapolation I described here and go to very high pressures in order to measure the asymptotic limit of this behavior, we measure these values at whatever pressure we want including very low pressures if we're interested. And then fit the data with the straight line, the intercept of that curve gives us 1 over the maximum volume. So we don't have to saturate the surface covered all if we don't want to. The intercept of the curve tells us 1 over Vm so we can solve for the value of Vm from the intercept. Once we know the value of Vm, the slope tells us 1 over k Vm so we can obtain the value of k from the slope. So in this form of the linear equation we're learning something both from the intercept and from the slope. I should point out also that this equation where instead of just saying the monolayer volume, let's say the monolayer volume, the volume of gas that's required to completely saturate one monolayer, one single layer on the surface of this substrate is let's say 0.4 liters or something like that. That number of molecules will occupy all the surface sites. So in addition to just telling us the monolayer volume reaches a surface coverage of 1, the monolayer volume also tells us something about the number of surface sites, the variable that previously we called and the total number of surface sites. So if we can use the ideal gas lot to obtain how many molecules are in this monolayer volume then we know that that many molecules can bind to the surface when it's fully saturated. So knowing how many surface sites there are can in turn tell us something perhaps about the surface area or the size of each little lattice site occupied by individual molecule. So knowing this monolayer volume actually tells us quite a bit about the surface that we're binding to. Alright so we're done manipulating the Langmuir isotherm equation at this point. We've seen it in lots of different forms at this point. It's now time to confess that actually when you go into the lab and you do this sort of experiment unfortunately the data don't always fall along a straight line. Sometimes you measure the data and they don't plot them in this way and they don't fall in a straight line. So the next question will be why would that happen and what is it we would do if the data don't obey the Langmuir isotherm. That's coming up next.