 So we've spent a fair amount of time by this point talking about phase diagrams for solutions and mixtures, talking about vapor-liquid equilibrium, liquid-liquid phase equilibrium if I've miscible solvents into the vapor when we heat those up, but we haven't said anything yet about solid-liquid equilibrium. If I take that solution and I lower the temperature, freeze the solution for most solid, of course that is visible on a phase diagram as well, so that's what we'll talk about next. And we'll do that as usual with a temperature-composition phase diagram, so I'll put the mole fraction of one of the components on the horizontal axis and let's do a case that you've got some everyday experience with, sugar water. So let's say you're making sweet tea or mixing a glass of Kool-Aid or you're drinking a soda or something like that, all of those are sugar dissolved in water essentially. So if we want to understand what happens in that solution, we know a fair amount about that already. Let's say this is the mole fraction of sucrose. We know the boiling point, so we're going to talk about solid-liquid equilibrium. So up here is going to be the liquid, down below is going to be the solid phase. We know the temperature at which pure water will melt and turn from solid water or ice into liquid water. That happens at zero degrees Celsius. If you may or may not know the value for sucrose, that's 186 Celsius, but that's the melting point, so these are melting points of water and sucrose. If you're a cook, you like to cook desserts, you might know the melting point of sucrose in Fahrenheit instead of Celsius, that's 330-some Fahrenheit. But pure sucrose has a melting point, pure water has a melting point. What we're interested in is what that melting point looks like on the graph between these two when I've dissolved a certain amount of sucrose in water. So again, let's say we've made ourselves a sweetened beverage of some sort. I've put a fair amount of sugar into water. When we do that, I can prepare a solution, let's say, at this concentration. Dissolve a little bit of sugar into some water. And because the molar mass of sucrose is pretty large, it's a little bit difficult to get a very high mole fraction of sucrose dissolved in water. So I've dissolved, let's say, one or two percent mole fraction of sucrose in water. That would be a fairly sweet beverage. We know a few things about that as well from everyday usage of sugar water. If we throw a bottle of Gatorade or Kool-Aid or Coca-Cola or something into the freezer, you probably know that if I cool that down, it will freeze at a temperature significantly below zero degrees Celsius. Actually, not all that significantly below. One or two percent sugar water solution will freeze at about one degree Celsius, a couple degrees Fahrenheit below the freezing point of pure water. And we'll talk about that phenomenon in a little bit more detail coming up soon. But in general, when I dissolve sugar in water, the freezing point decreases. That's something you may have some experience with. Likewise, if I have sugar that's not pure sucrose, but sucrose with a little bit of water mixed in, its freezing point will also decrease as the purity moves away from pure sucrose. So those two curves are going to meet together in the middle somewhere. The other thing we know that tells us that must be true is if I take this solution I prepared with a couple percent sucrose by moles, if I try to add more and more sugar to that solution, if I try to make a three percent, five percent, ten percent, twenty percent mole fraction of sucrose, eventually I'll pour in enough teaspoons of sugar into my glass of water that there's no way I can dissolve any more. I will have encountered the boundary of this phase coexistence line. When I get to this point, what it's telling me is I can't exist. I can't prepare a solution in the liquid phase anymore. I've gotten to the point where I have phase coexistence. What you know from doing that is here's my glass of sugar water. I've got a fair amount of sucrose already dissolved in a bunch of water. If I take another teaspoon of sugar and try to pour that into the solution, what's going to happen is I won't be able to dissolve any more of it. It won't stay in solution. Instead I'll get sucrose precipitating out of the solution. So what that's told me is I have entered a phase coexistence region. I have two phases coexisting, the solid sucrose phase together with the liquid solution phase. If I try to make the solution too rich in sucrose. So if I continue these two curves both to the point where they meet, what I see is I can prepare solutions that freeze at freezing point significantly below the zero degree Celsius freezing point of pure water and certainly well below the 186 degree freezing point of sugar. So in fact it turns out that limit on the coldest I can make those freezing point of the solution that turns out to be roughly negative 14 degrees Celsius. If I combine a roughly 10% sugar, 90% water by moles, that particular mixture has a name that is called the eutectic mixture. So this is analogous to what we talked about for liquid vapor equilibrium where the minimum boiling composition of a solution is called the azeotropic composition. In this case we're talking about freezing solid, liquid freezing into solid or solid melting into liquid. That freezing point becomes as low as it can ever get when I'm at the eutectic composition of that particular mixture. So I can show you another phase diagram that's a lot like this but it's quantitatively correct instead of just a sketch. So I'll pull up a phase diagram over here now and this is a similar phase diagram in shape with a few extra details that we'll talk about. And this is a phase diagram between lead and tin. So I'll go ahead and write that up here somewhere. It's a lead, tin phase diagram. So both of which we think of as solids at room temperature. If I heat, let's see this is weight percent tin, not mole percent but weight percent. So over here is pure lead and over here is pure tin. If I heat lead up to a temperature of 327 Celsius it will melt and change from the solid phase to the liquid phase. So up here we have a liquid phase, lead and tin mixed together at various compositions. This alpha phase, this phase that's labeled alpha is if I were on this line with zero percent tin that would be pure lead. If I'm in this region I'm in a lead rich region. So I have a little bit of tin dissolved into a lot of lead but this is a solid phase. It's a solid phase with mostly lead and a little bit of tin. Likewise over here this beta phase, this phase that's labeled beta is a very tin rich phase. It's maybe 100% tin or maybe 99% tin if I'm in this little wedge over here. But these are the two solid phases, liquid phases up here and then just like these segments I have a mixture of liquid phase and solid phase coexisting so I could draw tie lines in this portion of the phase diagram representing the fact that it's a two phase diagram, two phase coexistence portion of the diagram so for example if I had tried to make a 30% tin by weight mixture and heat it up to 200 Celsius I'm going to be in this phase coexistence region. If I try to prepare this system at equilibrium it will be a mix of a solid phase that is very lead rich and a liquid phase that has a different composition. Likewise over here this is a two phase coexistence region. Down here is more interesting. This is both on this diagram and this diagram this is a mixture of two solid phases. The alpha phase, the lead rich solid phase mixed with the tin rich beta phase. So this we could call this a solid plus solid mixture. Down here this is not a pure solid phase that's a mixture of pure sugar and pure water the way I've drawn the phase diagram. So let me pause here and point out several features of this phase diagram that are new to us only because we haven't talked about solid liquid phase coexistence before but are very similar to what we've talked about for liquid vapor coexistence. So for example the melting point of a particular mixture let's say I prepare a solution with this 30%, roughly 30% by weight tin down here around room temperature that would be roughly here. If I heat that system up until it melts what this phase diagram is telling us is something very similar to what the liquid vapor phase diagram would tell us just with different phases. This temperature I enter a phase coexistence region solid solid and liquid phase coexistence. So when I get to this temperature of is it labeled 182 Celsius when I get to 182 Celsius I begin to see melting I begin to see the first amounts of liquid appear in my sample. When it does begin to melt this is the composition I've used this tie line to tell me that this is the composition of the liquid the first drop load of liquid that forms this would be the composition of the pure solid that it would be in equilibrium with as I proceed upward through this phase diagram I still have tie lines connecting solids at some composition with liquids at other composition eventually when I get to this line I've fully melted the system I don't have any solid left and over this range of temperatures between here and here I've had solid coexisting with liquid it doesn't fully melt until I get up to this temperature so and above that I have liquid that's at the same mixture that I prepared so I can melt the mixture of solids over a range of different temperatures rather than a sharp melting point except when I have a eutectic so just like the eutectic is the lowest melting point on this phase diagram this composition is also a eutectic for the lead tin phase diagram so that's not only the temperature at which the melting point has has reached the lowest value 182 Celsius but it's also a point where I have a sharp melting point if I prepare a 63 weight percent tin 37 weight percent lead mixture that's going to melt at a sharp melting point it'll be solid beneath that temperature liquid above that temperature it has a sharp melting point and in fact for lead and tin as you may know if you've ever played around with electronics at all if you've ever used a soldering iron what solder is the stuff that you use in your soldering iron is a mixture of lead and tin and we and it's usually at a concentration very close to this eutectic composition of 63 percent tin by weight because that means we don't have to heat it up we only have to heat it up to 182 Celsius in order to get it to melt and when it re solidifies you formed a little electrical connection between two elements of a circuit that's lower than if you used pure lead or if you use pure tin so the reason we use solder is because it's a particularly low melting solid material that's electrically conductive so that's one practical use we have of eutectics in these composition combinations of metals one last thing to point out about these phase diagrams that's comparable to what we've talked about for liquid vapor coexistence curves is we can use these phase diagrams and these this range of melting points to purify a substance so again let's say I prepared my 30% tin by weight mixture now heat it up when it begins to melt the at any of these points where I have coexistence between a liquid phase and a solid phase the solid phase it's in coexistence with the liquid because the liquid is enriched in tin the solid will be depleted in tin so I if I wanted if I had a mixture and I wanted to purify one of the two components I could heat it up melt it get the solid which was enriched in lead in this case and then cool it back down I could keep doing that over and over and work my way up to pure lead in a process it's reminiscent of what we call distillation for liquid vapor coexistence so there's all these similarities between how to use these phase diagrams for solid liquid phase diagrams that are similar to liquid vapor phase diagrams one difference in the solid phase that we're gonna have to talk about in these phase diagrams is a feature of solids which is that when I combine certain elements they form stoichiometric compounds lead and tin I can mix in just about any ratio I want but there's other substances that form stoichiometric compounds and they add some extra complexity to these phase diagrams so we'll take a look at that next