Let's expand the region of two phase equilibrium in terms of our pressure volume space. What we have here is the region of two phase equilibrium. Remember, the first line that we drew on the diagram that showed us the two points that determine, at a given pressure and a given temperature, the volumes of the liquid and the volumes of the vapor phase that are present at that temperature T1. And again, we defined thepressure to be the saturated vapor pressure at temperature T1. Now when we go slightly above and slightly below in terms of pressure, we are going to generate a curve, and we'll speak more about this curve as we go. But what this curve is describing for us is the boundary that separates the single phase field from the liquid plus vapor phase field. In other words all around that dome is the vapor phase. So we're coming along T1 and we're doing a process like the isothermal compression. So that isotherm that's moving along here is decreasing the volume. And because we know what the relationship between pressure and volume is in this system, we can continue along with the decreased volume, determine what the actual pressure is inside a cylinder. And here's the example of the cylinder that we're going to be dealing with. It's a transparence cylinder, it has a certain volume. And that volume contains only water vapor. And we have a plunger at the end and that plunger is going to wind up being pushed from right to left. And so this a way for us to compress the material that's in that volume. And it's moving along that red line in terms of the behavior on that particular isotherm. Now, when we get to the boundary of that isotherm, something begins to happen. We've now moved into the two phase field. We are still operating at temperature T1, so we're still at the same temperature, except now what's happened is we begin to see the appearance of a second phase coming out on the surface of the cylinder. And this is just nothing more than liquid droplets that are condensing on the outer surface. Now, it's very easy to understand why this is happening. First of all, as we reduce the volume of the system and we can now begin to form. Because we're in that two phase field, we can now begin to form liquid which has a much smaller volume. So as that push rod goes in, we're going to be reducing the total volume that we have available, and as a result, the amount of vapor we have in the system is decreasing while the amount of liquid in the system is increasing. Now, if we continue on, what we're going to see is, once we wind up going through this two phase field, we now are going to embark with continued decrease in volume, on a very steep rise with respect to pressure. And this is because in effect, this material is incompressible. Because now we have moved out of the liquid plus vapor phase field and we're in a single field. Namely, we're talking about the presence of a liquid. And that liquid is going to rise up straight, and that gives us the boundary now between the liquid and the vapor phase. With further decreases in volume, we are rapidly increasing the pressure. And so, as a result, we have that large increase in the pressure in the system. And so, once we recognize that, all the vapor that was present in that, as a result of this compression, winds up turning into liquid at this particular temperature. Now, there's something that's rather interesting about what happens in this particular system. What we see is that if we look at two different temperatures, temperature T1 and temperature T2, they move us up inside of that boundary for two phase equilibrium, and we see the two isotherms on the pressure-volume diagram. And what we see is that as a result of the increase in pressure. And the corresponding increase in temperature as we go from T1 to T2. What we're finding is the difference between the volume of the liquid and the volume of the vapor are reducing. And if we continue to go up in temperature, we would ultimately reach a point that we would refer to as the critical point. And that critical point Is the point at which we begin to not be able to recognize the difference between the liquid phase and the vapor phase. They're appearing very similar in terms of the characteristic that we're using. The volume of the two phases, and they are coming close together. And when they reach that critical point they are the same value. Now let's think about what that all means with respect to forming a path and using a system like water. So let's start by beginning at some temperature and pressure. In this case we're going to begin at pressure P1 and we're going to keep the temperature constant at T2. So if we follow Path 1, that is what we're doing is, we're doing an iso thermal process where we're taking our vapor phase crossing the liquid vapor phase boundary, and, going up into the all single-phase field. So, what's happened here, is a result of the path that we've taken. We have a continuous change of the volume of the material, up until we reach the value of the pressure at P2. And, once we reach that particular value, a new phase appears, and that new phase is the liquid phase. And when we continue up, the vapor phase has disappeared and all we know have is the liquid phase. So that's Path 1. Now what we could also do is, we know that the enthalpy Changes, as we increase the temperature. And it also discontinuously changes at the phase boundary. And it is discontinuously changing at that phase boundary because we have a difference with respect to the volume of the material. That arch, the volume of the material that's transforming into another material as well as the entropy of the two. So, at that particular point there is a transformation enthalpy that's associated with going from vapor to liquid. So that is one of our paths that we can take. The other path is indicated by Path 2, and what we can do is altogether avoid the transformation. We avoid the transformation by following that path. And as a result, what we're doing is, we are going from state one, which is a vapor phase. We're following that vapor phase, and eventually it becomes another phase. And when we get to that position at the end It is the same as it was that we followed path one. But now, we have done that without going across the phase boundary, and having a discontinuous change that gives us enthalpy. Now, we do have, and we do recognize that our locations, regardless of whether Path 1 or Path 2 is followed. What we know is that the total integrated enthalpy has got to be the same. Because, in fact, the enthalpy's a state function. So, delta h is going to be the same for Path 1 as it is for Path 2. The only difference is whether it happens as we go around the transformation or, alternatively, if we go through the transformation. Thank you.