 So we've seen this Langmuir isotherm from several different points of view, three points of view, statistical mechanics, thermodynamics, kinetics. Now it's time to think about how we use it a little more in an experimental context. So this equation or equivalently we've seen the same equation in the rearranged form. This one is useful if I want to solve for theta as a function of pressure. This one might be useful if I want to solve for either K or P as a function of theta. If we think about how we would do experiments involving adsorption, let's say I make a measurement, I introduce some gas above the surface at a particular pressure and I measure that. So at pressure P1 I can measure surface coverage theta 1 so this much of the surface was covered at that particular pressure. Turns out that's enough information to use this equation to solve for K and once I know K I know the whole curve. So essentially what that means is there's exactly one of these isotherms that passes through that point and that's convenient because all we need to do in principle is make one measurement and we know everything about the entire isotherm. So for example let's suppose, let's work an example, let's suppose that we introduce some gas phase species above the surface at a very low pressure of just a little below half a torr and that's enough that we can measure that the surface coverage is 30%. 30% of the surface is covered by these molecules adsorbed under the species, under these conditions at a particular temperature and then the question is if we want to let's say cover 90% of the surface what would we need to increase the pressure to to make sure we covered 90% of the surface so what would P2 be? So the way to proceed is since we know P1 and theta 1 we can use this expression to calculate K. So if I write K as 1 over P theta over 1 minus theta rearranging that equation just slightly I can calculate that to be if theta is 0.3 then 1 minus theta is 0.7. If I calculate that particular ratio 1 over 0.45 times 0.3 divided by 0.7 what I'll get is a value for K of 0.95, 1 over torr. So K needs to have units of 1 over pressure so that when I multiply it by pressure this quantity ends up unitless just like this side does. That's given me the value of K. Like I said that determines the entire isotherm knowing the value of K we can plot this entire isotherm. So now if we want to know either what would the surface coverage be at some P2 or equivalently in order to get some surface coverage what would P2 be? In this case we want to know what is P2 when I know theta 2 so again use this expression and say pressure is 1 over K again theta over 1 minus theta. For the conditions I'm interested in K is still the same 0.95 inverse torr. The theta I'm interested in is 0.9 so I've got 0.9 on top and 1 minus 0.9 on the bottom. So if I take 1 over 0.95 and multiply by this ratio of 9 I'll get a pressure of about 9.4 torr. So this particular so if I use P2 and theta 2 I'll have determined the pressure at which the surface coverage will reach 0.9 torr. So if I'm interested in up here at 0.9 torr that's the pressure at which I reach 0.9 torr. So far so good the equations nice and simple to use all we need is one point on this curve and we did discover the whole curve experimentally however usually you make more than just one measurement we might measure the surface coverage at a lot of different pressures at P1 at P2 at some P3 maybe some others and as usual there'll be some experimental error if I've some of these points I've drawn above and below the curve to represent some errors in my either measurement of the pressure or measurement of the surface coverage so with that error not all the points will fall on the line and we're typically interested in saying okay what is the isotherm that is the best fit to these points and of course we could do a best fit curve using the Langmuir isotherm but it's a little more convenient to do a best fit using least squares with a linear equation rather than a nonlinear equation like this one so turns out we can make this graph linear if we rearrange it just slightly instead of talking about how theta depends on pressure let me write down 1 over theta so if I take the inverse of the left side take the inverse of the right side gives me 1 plus Kp over Kp now that I've got the sum in the numerator instead of in the denominator I can break that up into 1 over Kp and then Kp over Kp is just 1 so 1 over theta is equal to 1 over K 1 over P plus 1 that looks like a linear equation y is equal to some slope m times some independent variable x and then some intercept B so now that I've thought of the equation in this way it tells me that if I rather than plotting surface coverage as a function of pressure if instead I plot 1 over surface coverage as a function of 1 over the pressure I'm going to get a straight line the intercept for that line is going to be at 1 the slope of that line is going to be 1 over K it's going to be a positive slope and now if I've made a bunch of experimental measurements at different pressures since I'm plotting 1 over P the low pressure measurements are going to be at high values of 1 over P so I've got some measurements if I had made those measurements and I didn't have the line yet then I just have to do a least squares fit and obtain the slope of this particular line the slope of the line in particular is going to tell me the value of 1 over K so this slope is 1 over K so if I've made the measurements fit the line obtain the slope of the line slope tells me 1 over K so that's the typical way we would measure the value of K or determine the shape of this Langmuir isotherm if we had made a bunch of experimental measurements in this particular case notice that the intercept of the equation is always 1 we get some information from the slope we don't get that all that much information from the intercept the intercept just tells us as 1 over P becomes very small as 1 over P approaches 0 so in other words as the pressure P becomes very large the surface coverage approaches 100% 1 over theta is 1 theta is 1 we get 100% surface coverage at high pressures which is something we already knew from the asymptotic behavior of this Langmuir isotherm so the slope is all that tells us anything. We have one more improvement to this equation that we can make because in fact the surface coverage although it's nice to be able to talk about the surface coverage theoretically qualitatively is the surface 30% covered 90% covered 100% covered by adsorbate molecules it's not the most easy quantity to measure experimentally so actually the form of this equation that we use most often is a slightly different form that we'll talk about next.