 All right, so we have one more thing to investigate with this isotherm for the BET model. We've obtained this expression that's nice because it tells us how the adsorbed volume of a gas depends on the pressure of the gas. That curve, as we've seen, depends in a fairly nonlinear way on the pressure, blows up as we approach the vapor pressure and then increases monotonically but in a fairly complicated way as the pressure increases and then somewhere along here there's the monolayer volume. So it's not super convenient because it's a very nonlinear curve. Turns out we can rearrange this equation as usual to put it in a linear form so that when we're making measurements, we make experimental measurements that have some experimental error. Instead of having to fit this nonlinear expression, we can just use normal linear least squares regression to fit this equation. So our next task is to rearrange this equation and make it look like a straight line. So easiest way to start doing that since I've got a number of things in the denominator, I'd rather have the complicated stuff in the numerator. So let's flip both sides of this equation. So on the left I've got Vm over the adsorbed volume. On the right hand side now I've got these two quantities in the denominator are now up in the numerator. 1 minus p over vapor pressure and 1 minus p over vapor pressure plus c times p over vapor pressure. In the denominator I've got cp over p star. So rather than writing that in the denominator, I'll just flip it upside down and write cp under p star out here. And now, let's see, I'll leave the 1 over c out front. This p star over p, I'll distribute that into this first term in parentheses. So p star over p times 1 looks like p star over p. And then p star over p times p over p star, those cancel and I'm left with just 1. And then after I've done that multiplication I'm going to leave the last term alone for now. Alright, so what I'm after is a graph of something, maybe not the volume, but some function of the volume that's going to appear on my y-axis and maybe not the pressure but some function of the pressure that's going to appear on the x-axis. So I'm going to leave my adsorb volume over here on the left, certainly. That's going to be my independent variable. In fact, let me take away that equal sign. This term, p star over p minus 1, let me bring that over to the left-hand side. So when I do that, it appears in the denominator. So now on the right-hand side I've still got the 1 over c. I haven't written a numerator over here because I actually want to take this monolayer volume and move that over to the right-hand side as well. And then the only term left on the right-hand side is this longer term in parentheses. When I rewrite that, I'm going to write that as 1 minus both these, the second and third pieces of that term in parentheses, both involve a p over p star. So I'm going to collect those terms. p over p star is multiplying c and it's also multiplying in negative 1. So I'll combine those together and call it c minus 1 times p over p star. Alright, so that's beginning to look more like a linear equation on the right-hand side anyway. So if I now expand into this square bracketed term, 1 over c and 1 over v times each of these terms, I'm going to exchange their order to make it look like the y equals mx plus b equation. So the first one I'm going to write down is this, and actually I've gotten a sign wrong. I've got a plus c and a minus 1. So to get a plus c and a minus 1, I'll need a plus sign there. So I'm going to write this c minus 1 p over p star term first. I've got c minus 1 p over p star. I've got a divide by c and I've got a divide by vm. That takes care of the second term in the square brackets. I've also got the first term in the square brackets, this 1, so that's just 1 over c times vm. Alright, that's my right-hand side. Left-hand side is fine the way it is. Adsorbed volume p star over p minus 1 upside down. Okay, that doesn't look pretty, but it does look linear. If this is my independent variable, I've got some constants times p over p star. That's the thing I'm going to use as my horizontal axis. I've got more constants here that I'll call b, and then this whole big ugly thing is what I'll call y. So if I plot not adsorbed volume versus pressure, but if instead I plot this beast 1 over adsorbed volume with this function of the pressure down in the denominator as well, p star over p minus 1, that's my dependent variable that I'll plot on the y-axis. On the x-axis, I'll plot pressure relative to its vapor pressure. So here I'm plotting p over p star. So instead of this graph blowing up at when the pressure reaches the vapor pressure, it's going to blow up when the ratio of the pressure to the vapor pressure reaches 1. Alright, so next what I have is just a linear function. So in fact probably easier if I don't talk about how the graph is going to blow up at 1. This is a linear graph. This function is going to be linearly dependent on p over p star, and it's going to have a positive slope. So my experimental data are going to be scattered around a straight line. If I have the experimental data, I can fit those relatively easily with a straight line, and doing that fit tells me two things. It tells me the value of m, tells me the value of b. So the intercept of this curve, this straight line, tells me the value of 1 over c vm, 1 over the constant c times the monolayer volume of the gas. The slope of the graph, once I get it fit, the slope of the graph tells me c minus 1 over c times 1 over vm. So notice that I've got two constants, the slope and the intercept. If I get numerical values for the slope and the intercept, the two things I'm interested in learning about my gas are this BET constant, the value of c, and the monolayer volume, v sub m. So by solving those two equations and two unknowns, I can obtain the value of c and the value of vm from the value of the intercept and the value of the slope. That's useful primarily because the c constant is not uninteresting. It tells us how much more tightly the molecule binds to the second layers than to the first layers. More interesting, perhaps, is the monolayer volume. If I have, remember, we're talking about a process where molecules are adsorbing out of the gas phase and binding to the surface phase, they can bind in multiple layers. They don't have to be stuck in one monolayer. But if I know what the monolayer volume is, how much volume of gas it takes to cover exactly one monolayer of the surface by knowing the volume of the gas that tells me the number of molecules in the gas that it takes to cover the surface. If I also know the size of one of these molecules, depending on what gas I'm adsorbing under the surface, if I know the size of each one of those gas molecules, I know how many of them it takes to cover a monolayer of the surface, that will tell me in turn the surface area of the surface. So doing this BET adsorption analysis, learning what the monolayer volume is can lead us to a measurement, a direct measurement of what the surface area of the surface that we're doing the adsorption onto is. That may not be interesting. If I take a, you know, five centimeter by five centimeter slab of copper and I adsorb a gas onto it, I don't need someone to tell me what the surface area is. But if I have a more interesting substance like a porous material or a powdered material and I want to know what is the surface area of the powder or the surface area of the holes inside that porous material, I can do a BET adsorption experiment, measure the monolayer volume, use that to determine the surface area. So in fact, that's often the most common use to which this BET model is put is surface area analysis, instruments that measure the surface area of something often do that by adsorption and they use this BET model in their prediction of the surface area of the material.