 So another thing that pressure-temperature phase diagrams are useful in illustrating is the concept of thermodynamic degrees of freedom. So we've talked about degrees of freedom before. These are a different type of degrees of freedom. These are not to be confused with the mechanical or molecular degrees of freedom that we talked about when we discussed the equipartition theorem. A thermodynamic degree of freedom is the number of thermodynamic variables that I can independently change of one another. So that's number of thermodynamic variables that can be changed independently. And we'll do a few examples to make sure that idea is clear. For example, let's suppose we have liquid water. So all I've told you is that we have some liquid water. Imagine we have a container, a jar containing water. It's liquid, so it must be in this portion of the phase diagram somewhere. Might be here, here, here. It might be somewhere in the liquid portion of the phase diagram. But if we're at 298 Kelvin, one atmosphere pressure, for example, we're solidly in the liquid part of the phase diagram. If I think about how many variables, so these thermodynamic variables are things like the temperature, the pressure, the volume, the free energy, any of the thermodynamic variables we've discussed so far. How many of those can I modify independently, freely varying some, while still being able to choose any value I want? So for example, am I able to modify the temperature of my water? Can I heat water up? Sure I can. I can heat it up, I can cool it down, and it will remain liquid. So I can modify the temperature of a liquid within certain bounds. I can't change it beyond the boiling point or it will become a gas. But within certain bounds, I can certainly modify the temperature of the liquid. I can also modify the pressure of the liquid, as is made clear from this phase diagram. I can do both at the same time. I can take the liquid, I can heat it up while increasing the pressure, and it will still remain a liquid. But those are the only two degrees of freedom that I have. So we've seen, for example, that if I know the temperature and pressure, I can calculate the free energy from them. We've seen equations of state that tell me I can calculate the volume. We've seen them more often for gases than for liquids, but there are equations of state that tell me how to calculate the volume, the molar volume, as a function of the pressure and the temperature. So once I've calculated two of these thermodynamic variables or set to these thermodynamic variables, the temperature and the pressure, all the others are determined for me, and I can't vary them independently. So I can choose two of them, for example, temperature and pressure. Likewise, let's talk about a gas. I could talk about water gas. I could stick on the same phase diagram and talk about steam being over here in the gaseous portion of the phase diagram. Or if you want a gas that you can picture a little more easily at room temperature and pressure, let's say the nitrogen gas that makes up most of the air in this room right now. So if this phase diagram were not for water, but for nitrogen, on the nitrogen phase diagram, nitrogen is solidly over here in the gaseous part of the phase diagram. So at 298 Kelvin and 1 atmosphere pressure on nitrogen's phase diagram, we're in the gas portion, or technically the supercritical fluid region of the phase diagram, but the term won't make sense for a few more lectures. So we'll say that we're in the gaseous portion of the phase diagram, but I can still independently modify the temperature and the pressure, whether it's steam or whether it's nitrogen gas, I can change the temperature, I can change the pressure without having the substance change into a different phase. So there's also two degrees of freedom for a gas. Things get a little more interesting when our system is composed of a mixture of two phases, however. Let's say I have a system that has both liquid and gaseous water in equilibrium with each other. So let's say, for example, I have an open container of water, a glass of water sitting on the counter. It's in equilibrium with the atmosphere, so there's water vapor in the atmosphere in equilibrium with the water liquid in the container, so I have both liquid and gas in equilibrium with each other. I know if I have liquid and gas in coexistence, I must be somewhere on this liquid gas coexistence curve. In fact, for water, if I'm at 298 Kelvin in this room, there's only one point on this curve, on the coexistence curve that occurs at a temperature of 298 Kelvin, so once I've told you that I have water liquid and water vapor in equilibrium with each other at 298 Kelvin, there is exactly one pressure that the water vapor must have in this room, so the vapor pressure of water is determined, is a constant, and the pressure of the water in this system is going to be the vapor pressure. So in this case, I only have one degree of freedom. I can tell you that the temperature is 298 Kelvin, and then I can't independently say the pressure is two atmospheres or half an atmosphere. The pressure is exactly roughly 24 Tor, 25 Tor, the vapor pressure of water at room temperature. I can modify the temperature. I can take my liquid vapor equilibrium. I can heat the system up to, let's say, 310 Kelvin or some other temperature. When I do that, I can't keep the pressure where it was initially in order to keep liquid and vapor in coexistence with each other. The vapor pressure increases, so the pressure of the water vapor in the room will increase also. So by constraining myself to be on this liquid vapor coexistence line, if I insist on having two phases in equilibrium with each other, I've reduced to only one degree of freedom that's possible. Once I give you the temperature, you can tell me the pressure, or vice versa. If I give you the pressure, you can tell me what the temperature must be. Let's consider one more example, even more constrained. Let's say I have solid and liquid and vapor all in coexistence with each other, in equilibrium with each other as well. So how can I prepare a system that has solid and liquid and gas? Well, that would, solid and liquid is easy. I can make a glass of ice water, ice cubes floating in a glass of water. If I have that in an open container, allow water to evaporate out of the liquid. So I've got water vapor above the liquid, liquid water, and ice cubes in the water. If I allow that system to come to equilibrium, and all the ice doesn't melt, and all the water doesn't evaporate, if I have an equilibrium system with all three of those in coexistence with each other at the same time, there's exactly one point on the phase diagram where I can have solid and liquid and gas all having the same free energy as each other. That's at the triple point. So for water, that's a hair above 273 Kelvin, and the pressure is down around 100th of an atmosphere. So if I tell you we have three phases in coexistence with each other, I'm not allowed to also tell you it's at 300 Kelvin or 200 Kelvin. It must be at the temperature of the triple point. It must be at the pressure of the triple point. There are no degrees of freedom. I don't have the freedom to specify any of the thermodynamic variables independently. Just specifying that I have three phases in equilibrium is enough to guarantee that we're sitting at the triple point. So what we can see and generalize from these examples is if we are somewhere in the middle of a region of the phase diagram that's a single phase, we're always going to have two degrees of freedom. We can modify the temperature and pressure independently. If I'm sitting on any one of these phase coexistence curves, it doesn't have to be the liquid gas coexistence curve. If I'm sitting on one of these curves, if I give you one degree of freedom, you can tell me the other one. So when I have two phases in coexistence, I'm always going to have one degree of freedom. If I have three phases in coexistence, that must be a point where three of these coexistence lines come together, and then I have no degrees of freedom. That pins it down to a specific point in temperature, pressure, space. So in general, we can say the number of degrees of freedom is going to be equal to 3 minus the number of phases that are in coexistence with each other. If I have only a single phase, 3 minus 1 gives me two degrees of freedom. A pair of phases, 3 minus 2 gives me a single degree of freedom. Three phases in coexistence, 3 minus 3 gives me zero thermodynamic degrees of freedom. So there's a rule that we can use to remind ourselves how many thermodynamic variables we're able to choose independently of one another if we have phases in equilibrium with each other as well. This is really just sort of a warm-up for a more complicated rule describing the number of thermodynamic degrees of freedom that will come in much more handy once we start talking about systems with more than just one component in them, when we start talking about multi-component solutions and liquid solutions, for example. So there's something called the Gibbs phase rule. This is not the Gibbs phase rule. This is sort of a precursor to the Gibbs phase rule that we'll see later on. So next thing we'll explore then is try to get a little more precise now that we've seen how interesting these phase coexistence curves are. Try to go a little more precise about understanding how exactly the pressure varies with temperature so we can predict the locations of these phase coexistence curves.