 Okay, so we have a relationship between the pressure, the temperature, and the coverage, the surface coverage for molecules adsorbing onto a surface, at least if there are being this model, the Langmuir model for which we've developed this equation. So let's see if we can understand this equation in a little more detail. First thing we can do is simplify it a little bit by getting rid of the natural logs here. We differentiate both sides, so e to the ln becomes just the argument of that ln pressure over pressure naught. On the other side, e to the ln of theta over 1 minus theta becomes just theta over 1 minus theta. And then, exponentiating this sum gives me the product of the two exponentials. So e to the epsilon over kt is multiplying. So sum became a product after I exponentiated. So there's maybe a simpler version of this equation to work with. If we're interested in the pressure, sort of the vapor pressure above the surface, surface coverage at some level, at some temperature with some binding energy, if we want to calculate the equivalent of a vapor pressure above that surface, that might be the most convenient form of the equation to use. Most often, though, we're going to control the pressure and the temperature, and we're going to be curious about what surface coverage that pressure induces, how much surface coverage is adsorbed onto the surface. So let's rearrange this to solve for theta. So let me get the denominator and the exponential over to this side. So I'll have p over p naught e to the negative epsilon over kt times 1 minus theta is equal to theta. And if I'm going to isolate the thetas, let's do that on the right-hand side. So on the right side, I've got theta times 1, and if I bring over theta times this pre-factor with that negative sign becoming a plus sign, it'll be p over p naught e to the minus epsilon over kt. And all I'm left with on the right side is this 1 times the pre-factor p over p naught, and then this Boltzmann factor e to the minus epsilon over kt. So now I can solve for the theta. Theta's going to be this left-hand side divided by the term in parentheses on the right-hand side, p over p naught and Boltzmann factor over 1 plus that same term, p over p naught Boltzmann factor. All right, so there's an expression that's in a convenient form to use if I want to solve for the surface coverage as a function of the pressure of the gas that's not adsorbed and the temperature of the system. Since I've got this collection of terms, 1 over p naught and then in a Boltzmann factor, if I define a constant, I'll choose to define that as a constant capital K. If I let 1 over p naught, 1 over standard pressure, multiplying this Boltzmann constant e to the minus binding energy over kt with a negative sign, if I call that thing K, that constant K shows up here, 1 over p naught exponential, 1 over p naught exponential. So I can rewrite my equation K times p in the numerator and 1 plus K times p in the denominator. So that's certainly simpler looking now that I've collected a bunch of these terms inside this constant capital K. This equation, that's an equation we're going to come back to several times. That equation is called the Langmuir isotherm and we'll see in just a second why isotherm is a good name for that equation called the Langmuir isotherm because it's the behavior that's predicted by using the Langmuir model that led to this particular equation. If we want to understand how that might work in a real experimental system, what we're asking about now is if I change the pressure, so remember the cartoon of our system is I have some surface, I have some molecules up in the gas phase, I have some molecules that are adsorbed onto the surface phase, I've got some equilibrium between those. If I change the pressure of the gas up in the gas phase that's going to change the fraction of the surface that's occupied the surface coverage. So if I want to know how specifically does the surface coverage depend on pressure given by this expression, we can say a couple of things just by looking at this equation. First of all notice in the limit of low pressures, if I go all the way down here to the low pressure limit, in particular if the pressure is small compared to 1 over k, if kp, that product is small compared to 1, that means this term in the denominator is small compared to the 1 so I could ignore that term in the denominator and the surface coverage will just be k times p divided by 1. So it'll increase linearly at low pressures with a slope equal to that constant that we've defined. On the other extreme at high pressures, in particular when this product k times p is much larger than 1 or when pressure is much larger than 1 over my constant k, then the opposite is true. kp is going to be much larger than 1 and if I ignore the plus 1 term in the denominator, then kp over kp is pretty close to 1. And that makes sense. In the limit of high pressures, the largest I can ever get the surface coverage, I can't with this model anyway, I can't ever get surface coverage greater than 1. High pressures is going to lead asymptotically in the long run to surface coverages that are approaching 1. In between, this is a monotonically increasing function so the surface coverage is just going to increase connecting that linear behavior at the beginning to flattening out and asymptotically approaching 100% surface coverage at high pressures. So this equation, this graph is described by this equation, this Langmuir isotherm. The reason we call this an isotherm is all the values on this curve were obtained with a particular value of k, which is true at a particular value of the temperature. If I were to do the same experiment and measure surface coverage as a function of pressure at a different temperature, so let's say this one is done at some temperature T, we can ask what would happen if I raise the temperature? If the temperature increases, temperature shows up here, so let's first talk about what happens to minus epsilon over kT. Here's the point where it's important to remember that that epsilon is a negative number. The binding energy molecules that are stuck onto the surface have an energy epsilon. That epsilon is a negative number, so negative epsilon is a positive number. That positive number divided by temperature, when the temperature goes up, this positive number is going down. And e to the minus epsilon over kT is also going down, and therefore k, this constant, is also going to be going down. So at high temperatures, if I increase the temperature to a hotter temperature than the one I already first did this experiment, that slope, the initial slope is going to be lower at the higher temperature. So the isotherm is still going to asymptotically approach 100% coverage, but it won't get there until higher pressure. So this would be at some temperature, I'll call it T sub-hot, a higher temperature than the one I thought about initially. In the opposite direction, if I had cooled the system, lowered the temperature, then exactly the opposite is going to be true. Negative epsilon over kT, negative epsilon is a positive number. If I divide by a smaller temperature, this positive number is going up. The exponential goes up, and the binding constant also goes up. So at a colder temperature, I get behavior that looks more like this, I'll call that T sub-cold. So I get one curve at this temperature, different curves at hotter or a colder temperature, so that's why we call these isotherms. Each one of these curves is appropriate for a particular temperature when I hold the temperature constant. The shapes of those curves make sense qualitatively. We'd certainly expect that if I have molecules adsorbed to a surface under a certain amount of pressure, if I raise the temperature, as you'd expect, because this binding is an exothermic process, Le Chatelier will tell you that increasing the temperature will drive molecules off the surface. So in other words, if I'm at this pressure and I increase the temperature, I get lower surface coverage. On the other hand, if I decrease the temperature, cool that system down, more molecules will precipitate out of the gas phase, they'll adsorb onto the solid phase, and I'll have a higher surface coverage at that colder temperature. So qualitatively, this makes sense. Quantitatively, we can predict exactly what's going on if we understand things like the binding energy and the temperature and the pressures of the system. So we could move on and do some practical examples and talk about how this is actually done experimentally. But before we do that, I'm actually going to show you a different way of approaching the problem and a different way of understanding where this Langmuir isotherm comes from. So that's coming up next.