In this lesson, we're going to talk about the one component phase diagram and in particular we're interested in talking about the phase diagram of water. When we look at the simple water diagram, we see lines on the diagram, and those lines have specific meanings. What they do is they separate boundaries with respect to phases. So each one of these lines on this one component phase diagram represents a boundary that separates two regions, and hence these lines are two phase fields. They describe the variation with respect to, in the case that we're working with water. They're describing the variation with respect to the solid phase, the liquid phase, and the vapor phase. And so, in those open regions where we have solid and liquid and vapor, those regions are single phase. That is, regardless of what the temperature and pressure are, we will find when we are inside of those single phase fields, the only phase that is present is the phase that's indicated. However when we come and approach those lines that separate the single phase, that represent single phase boundaries. Where we have a solid and a liquid phase in coincidence, then what we have is two phase equilibria, namely the solid and the liquid are in equilibrium. We can then begin to understand how all of these diagrams come together. So, the three lines that are indicated are the solid-liquid, liquid-vapor, and the solid-vapor equilibrium. We also have in addition to the single phase solids and the two phase regions of solid, vapor-solid liquid and liquid-vapor, we have a point that we refer to as a triple point. At that one point on this diagram it tells that only the solid phase, the liquid phase, and the vapor phase, can all co-exist together. And that has a fixed pressure, and it occurs at a fixed temperature. Now what we often do with respect to the water diagram. Is to consider what's happening at one atmosphere pressure. And we talk about one atmosphere pressure. What we're doing is we describe when we have the transformation to the solid phase to the liquid phase. What we do is we refer the temperature that that transformation is occurring. As the normal melting temperature or the normal freezing temperature because it's done at one atmosphere. As we continue along and we wind up increasing the temperature at one atmosphere, what we then do is we reach the boiling point and this is then defined as the normal boiling point. Now if we take a look at sum pressure P1. We extend it over to the phase boundary that separates the liquid phase from the vapor phase. What we know is, at that particular point on that boundary, we have defined immediately as a result of choosing the pressure. We have immediately fixed the temperature in the system. Once we say that we are in a two phase liquid-vapor field, we know that by fixing the pressure in that field, we automatically fix the temperature and that temperature then is given as T1. So when we look at the phase rule, which was written down by Gibbs in the early part of the 20th century. What it´s telling us is that we have a parameter that we refer to as f, and that´s the number of degrees of freedom. We have the number of components C. The number of phases that are present, plus 2. And in this case, the 2 corresponds to the 2 thermodynamic variables of pressure and temperature. Now let's go back and think about what we mean by the degrees of freedom. Let's say that what we have are a system of equations in x, y and z, and so it's a first order set of simultaneous equations. Now, if we have all three equations then what we know is there are no degrees of freedom, in that, the system x, y, and z that satisfies that set of equations is completely fixed. Now if we would take and erase one of those equations, then what would happen is we would lose one of our variables and what we would then need to do is to fix the next variable. So let's say we fixed the value of x, then when we put the value of x in the last two equations that we have, we would then be able to come up with the values for y and z. So we would say we have one degree of freedom, so we have the ability to choose one of the values of x, y, or z. Then when we come down to taking away two, then what we have to do is to talk about two degrees of freedom. In other words, before we can solve any one of those equations, we have to fix the values of x and y. And once we've done that, then what we can do is determine what the value of z is. So this is what we mean by this idea of the degrees of freedom. Let's return to the water diagram again. And this time we're going to think of the water diagram in terms of the Gibbs Phase Rule. Remember that the Gibbs Phase Rule says that we have the degrees of freedom, the number of components, the number of phases. And the pressure and temperature in the system. Now when we talk about all those open regions, which are the single phase field, we have two degrees of freedom. What that means is, I have to tell you both the pressure and the temperature for you to be able to determine where you are on that Phase diagram. So if I tell you that you are in a solid phase field, then I have to give you inside of that solid phase field, pressure and temperature. Now, that then describes the degrees of freedom for a single phase field. Now, when we look at the boundaries that separate these single face fields. These are the regions of two phase equilibrium, they now tell us that we have 1 degree of freedom. And what we mean here simply is that once we fix the pressure or the temperature. Along one of those lines we automatically have determined the thermodynamic variable of the other. In other words if we describe the pressure in the region where we have liquid vapor equilibrium and we know therefore once the pressure has been determined we know what exactly the temperature is. So that tells us that we have that single degree of freedom. Now when we look at the situation where we have three phases in equilibrium, that is F is now equal to 0, that tells us what happens at this particular point on the diagram. Once I tell you that you are at a triple point in a one component system. You automatically know where you are when I tell you the particular phases that are in equilibrium. In other words ice, water, and water vapor. So once that's been established we know that we are at that specific point. Let's continue our analysis of the diagram. Let's focus on the two phase region that's represented by the line describing the boundary between the liquid phase and the vapor phase. When we look at the liquid vapor line, we can describe, for example, two different points. One point that tells us a given pressure, namely pressure P1. And when we look at pressure P1, what we see is correspondingly, a temperature T1. So we then defined the relationship between P and Pressure and temperature for this system, and because we've fixed the pressure, we automatically know what the temperature is. Now this diagram that we're describing, is a diagram that talks about the two variables, the variable pressure and the variable temperature. There is, however, an additional plot that we can make where we describe the relationship between the pressure and the volume. And what we're going to do is come up with the pressure volume diagram and we're going to focus on the pressure at P1. So when we look at the pressure at P1 because we have fixed the pressure. We immediately know that we have a temperature, T1. And what we can say is, at T1, we have what we refer to as a saturated vapor pressure. Now, we know also that that horizontal line that describes the volume of the phases that are in equilibrium, when I look down along the X axis of that plot. My two phases that are in equilibrium are determined by the volume of the liquid and the volume of the vapor. So we have our saturated vapor pressure, we have our region of two phase equilibrium. And the boundary of the two phase equilibrium is the points that are indicated on the diagram. And depending upon where we lie between the volume of the liquid and the volume of the vapor, we're going to have different ratios of those two phases. Now we know that we are a long an isotherm because at pressure P1 there is a corresponding temperature T1. So these then give us a way for us to begin to think about the pressure temperature and now volume diagrams. One thing that we should point out is that pressure and temperature are both intensive variables, and do not depend on the size of the system. Where s volume is an extensive property and we need to specify how much of the material that we're actually looking at. Thank you.