 Body-centered cubic is just like simple cubic. But there's a sphere in the center, too. So since these count for one, and this is one, and there's two, so body-centered cubic, for example, is uranium. Metal forms that structure. And then this one is kind of harder to see, but it's face-centered cubic. There's nothing in the center, like in this one. But there's ones on each of the six faces of the cube, as well. So there's one here, here, here, here, here, and so forth. And then there's the eight original ones. And if we fill it in, it looks like this. This is closest-packed structure. This is as many as you can jam in there. And therefore, usually, metals that have a face-centered cubic structure are very dense. And gold is an example. Gold is extremely dense material of 19 grams per cubic centimeter. Much denser than water, for example. Each kind has a certain number of atoms per unit cell. And the way you count them is you count one-eighth if it's on a corner. You count one-quarter if it's on an edge. And you count one-half if it's on a face. And any atom that's entirely inside the unit cell is one, because it's on the inside. Every time you copy the unit cell, you're going to copy that whole atom. If it's on the corner, you're going to have to copy it eight times there, one, two, three, four, five, six, seven, eight, before you get the whole thing. And so for any one of them, you count one-eighth. Body-centered, there's one in the middle, plus the eight times one-eighth. So there are two atoms per unit cell for body-centered cubic. And face-centered cubic has the same eight times one-eighth on the corners. They're always there. Corners are always there. And then we have six times a half because when they're in the middle of the face, they're only shared by two unit cells, so we count each one-half. We put together two cells, we get one atom. And since there are six times a half, that's three. Three and one is four. We have four atoms per unit cell. This should influence the density. And therefore, if you know the density of a material, you can kind of guess what kind of unit cell it might have. If the density is low, it's probably not face-centered cubic. And here's a picture again just reiterating what I said earlier. You can see that in order to get the entire purple sphere, I've got to have eight boxes. So therefore, for any particular box, I should count one-eighth. And for here, on the corner here, I have to share it by four. And on a face in the center, I have to share it by two. This one's right in the middle here on the corner, so it's shared between all those. Four. Okay, let's have a practice problem. Sodium, very reactive metal, easy to melt. Sodium has a density of 0.987 grams per cubic centimeter and a cubic unit cell that has an edge length of 0.429 nanometers or 429 cubic meters. How many atoms are in the unit cell? This is kind of a reverse engineering problem. We can give you the unit cell and ask what element. We can give you the element and ask what unit cell and so on. You have to be able to go either way. And I will show you how to do this almost without thinking. It's dangerous to do things without thinking, but sometimes it's better if you're playing tennis and you start thinking about what you're doing with your left hand as you toss the ball up to serve, you'll probably double fault. In fact, in the good old days, that was a way of throwing your opponent off. That's called gamesmanship. Say, have you changed your toss and then you put their mind there and they start thinking, what do I do here which they never paid attention to and then they get it wrong. And you can over concentrate on things and get them wrong. But if you're going to do things by remote control, as it were, you've got to have a system that checks. And the system that checks is to always check that the units work. In chemistry, the units are your friends because they guide you right to the answer. You don't need to actually know too much more than how to cancel units like fractions. Okay, first of all, the first physical principle is the density of the unit cell is the same as the density of sodium because sodium is just made up of an infinite number of copies of unit cells that all have the same number of atoms and the same volume. So therefore, if we use the density and the molar mass, we can find the number of atoms per unit cell. We have to look up the molar mass. We probably wouldn't know that. There's the molar mass. And look how I just love canceling those units out. I start with the density. Okay? If I happen to start with something upside down, I wait till I get to the end and then I turn both sides over. I don't just clear the deck and start over. I may start, if I don't know exactly how to do the problem or I'm a beginner, I may start wrong. But the units are still going to lead me to the answer. I start with 0.987 grams per cubic centimeter. I want to know how many moles there are per gram. So I convert to moles by using the molar mass. And then I want to know how many atoms I use Avogadro's number and I convert to atoms. And then I want to convert to nanometers. So I know how many atoms there are in a nanometer. So I don't, if I don't recall what the conversion is from centimeters to nanometers, I convert first centimeters to meters. And then I convert meters to nanometers. Now I've got atoms per cubic nanometer. But I know how many cubic nanometers are in a unit cell. Now I've got atoms per unit cell. It better come out to be an integer. Well, the numbers aren't all perfect. I didn't use Avogadro's number to six or seven digits. There are limits to how well you can weigh things after all 0.987. How are you going to get four digits on density? You have to be very, very careful to get four digits on anything, usually only voltage. And so we find about two atoms per unit cell. Knowing that and knowing that it's cubic, it's probably body centered cubic because that's the only one I know that has two. But the thing is if I write out and cancel all the units in problems of this type, you can just check you haven't made an error because if you end up with some funny units here that don't cancel or not what you want, you just say, hey, I need to know how many foot pounds is this or something. Yes. Oh, because I don't even believe the four. That's the fourth digit and I started with three. Okay. I don't really believe the four is significant. But it came out in the calculation. Of the three types of cubic unit cells then that we have, body centered cubic has two atoms. And so we would guess that sodium metal crystallizes into a body centered cubic lattice. And if you guessed that, you'd be right. And in fact, all the alkali metals prefer body centered cubic lattices. And therefore it probably has something to do with the structure of the valence electrons in those materials that they prefer that crystal structure. And here's one drawn smaller so that you can see the interior one and then the eight on the edges. This big one's kind of more realistic but a little harder to see sometimes what the unit cell is. And that's how it often transpires is that if you draw the molecule with the atoms the correct size, it's really hard to see what anything is because they all cover each other up. And if you make them translucent, it's still hard to see because there's too many of them going through each other. And so usually we draw the structure where we shrink them down for clarity. But don't get the idea that it's like that. They're really quite close and everybody's packed in there and there's hardly any wiggle room in most of these structures. Okay. Here's another one then. Let's go the other way. Now we have a soft reactive metal that has a face centered cubic crystal structure with a density of 1.54 grams per cubic centimeter. Boy that's a lot of clues right there. If the radius of the atom is 0.197 nanometers what is the metal? Okay the most important thing in a problem like this is to decide which breadcrumbs you have this time. We're going to have to try to identify the metal. The only way I know to identify the metal is to identify its molar mass and then look on the periodic table, cross my fingers and hope I find it there and hope that it is a metal and hope that it's soft and reactive because those were clues. If the metal turns out to be titanium I think probably not it. And therefore I need to figure out how I'm going to get the molar mass out of this gobbledygook. Well I have the density and I know the density of the unit cell is the same as the density of the entire material. Therefore if I knew the volume of the unit cell and I knew how many atoms were in the unit cell I'd know how many atoms there are per little volume. I could convert that to moles per bigger volume and then I could compare that with 1.54 grams and then I'd know how many grams was how many moles and that's the molar mass. And if I do that carefully I can find the molar mass and figure out what the element is. Let's see how we can go about it. Well the other little bit of information here is that it's face centered cubic crystal and the radius is 0.197 nanometers. Therefore I have to be able to take, I have to remember what face centered cubic crystal is and I have to take these spheres in my head and I have to figure out if this has a radius of 0.197 then what is this edge length of the cube? And if you do that incorrectly you just get completely cut off because it's very hard for example if you take three spheres and then you put the fourth sphere on top in the divot and then you ask okay how far down does it actually go compared to the plane of the top three? That problem's too hard for most people. They can't get it. I'm sure you can but you'd have to think about it. How far down as a percentage does that one tuck down? And you have to kind of have some geometric intuition about how to set up extra things where you can measure things. Okay? And I'll show you one way to do that right now. Okay. The density of the unit cell is the same as the density of the material so we need to identify how many atoms are in the unit cell and how large its volume will be. Well, we know how many atoms are in it. It's four because it's face centered cubic and we've had a course in chemistry. We have 8 times 1 eighth plus 6 times 1 half is 1 plus 3 and there it is that's the polonium one again. If you look this way and you say gee this one's jammed against this one and then there's a hole here and then this one's jammed, that way madness lies. You're never going to get the volume of the cube very quickly that way. It's very, very difficult to figure out how these things are going in because it's essentially the problem I told you about, the three and then one on top, how does it go in? Therefore, we don't focus on that. We only focus where the spheres touch. Where they just kiss against each other and then we say look, this is R, this is R, this is R, this is R, uh-huh, I have the diagonal of the face. But if you don't happen to think of doing it that way, you never get this problem solved. You just go around in circles. Well, it's not that hard to think of that and I drew it now as 2D and I only care about the length of the unit cell. I know these continue on out but I don't care. I have R, R, R, R, 4R and then I have L for this side and L for that side and I happen to believe the Pythagorean theorem interestingly enough in the dark ages, they apparently forgot the 3, 4, 5 right triangle. The Greeks knew it, the library at Alexandria burns down, people forget, they can't even make a right angle anymore because that's how they used to do surveying. You make a string of length 3, string of length 4, string of length 5, stretch them out, you got a right angle. Now you've got North, South, East, West, you can survey land. You can't make a right angle, your property lines are all crooked, then you have a lot of war. This is my land, oh no it isn't, my land. And that's a famous example in which we went backwards for a long time and it can happen again. Knowledge is like a flame. When I die it's going to extinguish with me unless I've lit it up in you, it's gone. Everything I knew is gone. Nobody else will necessarily know it but I hope that won't happen. Well okay, back to this, 2L squared Pythagoras is 4R squared and boy do I like putting parentheses because I always put parentheses. Before I square things, boy I square everything in there because I've made that mistake too many times. Instead of 4 quantity R squared, I write 4R squared, I get it wrong and then I get a zero on the exam because it's totally and miserably wrong. It's not right, it's not part right. It's completely wrong and that's how I will grade it. I'm careful, I square that, I get 2L squared is 16R squared or L is the square root of 8R. Now I have L, the volume of the face center cubic cell is L cubed obviously since it's a cube and that's 8 root 8 and that's 22.6274 roundabout and then as we know what the radius of the atom is, the volume turns out to be 0.173 cubic nanometers. That's the volume of the unit cell. I know it has four of these soft reactive atoms in it because it's face center cubic. Therefore, I know how many moles there are per cubic nanometer because I have four atoms, that's a certain number of moles. It's very tiny but I'm going to convert that. Okay, four atoms in 0.173 cubic nanometers, convert to centimeters cubed and then to moles and we know how many moles are in a cubic centimeter. Okay, I start with four atoms per 0.173 cubic nanometers. I convert nanometers to meters because I'm more comfortable doing that. Fine if you want to, you know, take a shorthand and convert straight to centimeters but be careful. Sometimes people put 10 to the 11 instead of 10 to the 7 or something funny and they get it wrong. It doesn't take too long, 10 to the 9 cubed, 1 meter per 100 centimeters, total cubed, 1 mole per 6 times 10 to the 23 atoms, boom, boom, boom, boom, boom, I find my units left over are 0.0384 moles per cubic centimeter but I know how many grams there are per cubic centimeter because they told me. Therefore, I know how many grams there are in a mole. All I need to do is divide the number of grams by the number of moles and I get grams per mole. Well, that's pretty easy to do. I take 1.54 grams and I divide it by 0.0384 moles and I get 40.1 grams per mole and 40.1 I look on the periodic table and I just say yes, calcium, soft reactive metal. Of course, the calcium you take in food is not a soft reactive metal because it's the calcium 2 plus ion, of course, that you take in food and that is used to make your teeth and your bones. You can take extra calcium if you think you don't have enough but unless you've actually figured out how much calcium you've got and how much you need, it's a very bad idea to just chuck in extra chemicals just in case. Vitamins and things like that are not an insurance policy unless you actually know what you're doing, I wouldn't do that. All your ancestors just ate food and you're here. That means they survived and they reproduced. You're the end stage survivor of a huge long chain of people who didn't screw anything up. If you do what they did, you'll probably be a survivor. If you do something different than what they did, you're conducting a new experiment on yourself. It might be great or it might backfire and if it backfires and something happens to you, that's the end of that chain. Eventually, the only people left will be people who don't take vitamins if they turn out to be very bad for your health. Okay, let's do a problem now about inferring stoichiometry. If we have a clear enough picture of the unit cell and boy, do I mean clear enough, it really helps to have software that'll rotate it for you in this case. If we have a clear enough picture, we can infer the molecular formula for the compound. You just have to be careful to count the spheres on the faces and corners with the right fractional weight. Usually the only kind of problem you're given like this has two different atoms and they want to know, is it NaCl or is it NaCl2 or NaCl3, something like that. It's usually not more complex than that because if you have three different atoms and they're all colored in different sizes, it's extremely hard to see. So even if you know how to do it, it's very hard to actually do it. It's just very hard. Now let's try a practice problem. The unit cell for niobium oxide, I'm deliberately leaving off the oxidation state of the niobium which can have several oxidation states because if I gave you the oxidation state of the niobium, I'd be giving you the answer to the problem. Therefore, this nomenclature is not right. I should give the oxidation state. I should say niobium-3 oxide, for example, if that's what it is. Based on the picture of the unit cell, what's the molecular formula for niobium oxide? Well, let's look at the unit cell. Okay, this one's more interesting. The niobium atoms are blue and the oxygen atoms are red. Anybody want to venture a guess by looking? I want to know how many niobium and how many oxygen make up the molecular formula. People are saying three and three. Anybody else? Here's how I would look at it. These are clearly cut in half, the big blue ones. They're on the face. They're cut in half. They're six of them. Again, you see it's hard to see. One, two, three, four, five, six, but there's six of them. Therefore, six halves is three. Good. Then we have to count these. These are not on the corners, but they're on the middle here and they're cut into fourths. They've got four there. One, two, three, four there. And then I can see this one. Boy, that's just peeking out there. But by symmetry, I know there's four there. Not fair to have one missing on the corner that you can't see. And therefore, it would be three and three, but then I would write the empirical formula as niobium, one, not three, oxygen, one. And when it's one, we leave it off. So I just write niobium and oxygen, okay? That would be the formula, then, that I would get. And I've just written this out so that you'll have a record of it if you aren't here right now in the room or if you forget. Here it is again. And there are beautiful, one advantage of the web is that there are beautiful things that are rendered with software that is tough on purpose and sometimes very expensive or you just don't have it. And you can see these beautiful pictures of crystal structures. And you can really appreciate how beautiful things are by going to some of these websites. And here I went through it. I said it appears to be, I put nickel here that I made a mistake, niobium. I'll switch that. On niobium three, oxygen three which we just write niobium oxide. And in fact, there are several other oxidation states of niobium. This would be niobium two oxide. There's also NB02 that would have a completely different crystal structure. And NB205, that would be probably more complex because when I have even an odd, then it usually gets, the cell gets bigger before it starts repeating. And sometimes the problem is the unit cell is this enormous thing and then it repeats. And for protein crystallography, the unit cell usually has one or two copies of the whole protein in it somehow. And it's very, very complex to figure out where all the atoms are. And the way they do that is by x-ray diffraction. You know that x-rays tend to scatter off areas of, oh, excuse me, yeah. Well, you don't, that is the empirical formula. But in a crystal it's very hard to tell if, put it this way, if you actually did have three niobiums, let's say in a triangle, and then these oxygens hanging off in the wings, then I would say for sure, okay, that's niobium three, oxygen three. If it were like that, if I looked at it, if it's like this and they're just all packed in there, then I see no reason to pick niobium three, oxygen three rather than niobium seven, oxygen seven. And so in that case, I just reduce it down to one. But it is a little bit of a subtlety depending on how it looks. Usually, in a solid, it can be difficult to tell who is bonded to whom if they're all ions because they tend to just assort and pack. Okay, what I was going to say is that you know from your own experience, if you had a mishap as a kid, that x-rays scatter off regions of high electron density. And that's why you can see your bones with x-ray film because the x-rays get scattered by the calcium in your bones, among other things. And if they take x-rays at two different energies or wavelengths, they can figure out how much calcium is in your bones. They can give you a dexa scan and they can say you've got osteopenia, which will progress if you let it go to osteoporosis, and then you will look like a giant question mark. And finally, you'll be staring at your shoes all day long because you'll have lost all the integrity in your bones. If you want to keep the integrity of your bones strong, then lift weights and do explosive things like jump. The reason why men tend not to get osteoporosis quite as often is because men tend to carry the groceries to the car still. Women do less of that, especially with their arms and so your neck can get to be very, very thin and weak and your forearms and then you slip and they just snap like that. In fact, in some cases, you step off the curve, your femur breaks right at the ball as you step off because it's so weak and then you fall after that. You don't fall first, you fall after. If you doubt what I say, look at what happens to people who go into space. Those guys are big and strong and they come back and they're in a wheelchair because their bones are balsa wood and their muscles are just nothing because without the resistance of gravity, your body just says, hey, what's the point of having all these bones and all these muscles? It's easy to move around up here and just off loads everything. You have to force your body and your mind to improve. They won't do it on their own. On their own, they both go down the tube. Okay. Therefore, X-rays will scatter off electron density in a periodic lattice. Now, this is different than just making like what a radiologist does to look at your bones. They aren't looking at the crystal structure of your bones. They're looking at how much calcium is in there or something like that. But if we have a periodic lattice, then we can get diffractions. We can get diffraction spots and we can measure the periodicity of the lattice. You might say, why can't we do that with visible light? And the answer is visible light just has too long a wavelength, much, much too long. It's like saying, why can't I thread a needle with a gigantic piece of rope? It's the wrong size. X-rays are the right wavelength to have atomic dimensions. And the father-son team of the Braggs are known for the particular equation relating the scattering angle, which is conventionally called theta to the distance d between the crystal planes. Here's a kind of a cryptic thing you'll see. And I'll admit to you, when I was a student, this thing was never convincing to me. I accepted it because I wanted to pass the course. But it was never convincing to me and I was never given a proper argument as to what this had to do with anything. But let me give you the correct argument. First of all, this is at right angles to this vector here. And therefore the difference in the path length between just ricocheting off here and going down here is just this extra distance here. Because we measure at right angles to where we are. You have a fault starting a race if when I look at right angles, you're sneaking over the starting line. These are going to go the same distance back here. So therefore, once they're like this, if they're going together, they're all going to be going together like that because they'll go over the same distance. Now they aren't at the same place in space, but we'll get to that. And then we have another piece down here, which is this same distance. And the difference in path length is those two pieces. I have to figure out how those two pieces relate to the spacing between the periodicity and the electron density, which is just centered around each atom. Because each atom has a lot of electrons sitting there on it. The x-rays bounce off that. Now, why doesn't it matter that these are not actually on top of each other? The answer is that the kind of detector I use is not of angstrom scale. It's a detector that's bigger than that. And therefore, if all the x-rays coming into the detector are all on a line like this and they come in like a big wave breaking across the whole shore, then I get a lot of energy coming in at once and I get an event. The film develops, the CCD triggers, I get a count. But if the x-rays come in and one's going up and one's going down, then they tend to cancel out. They just cancel each other out. And the difference between waves and particles is that particles can never be negative. If I shoot cannonballs, I can't cancel one cannonball with another. But if I take waves, if I take a string and you can try this and one guy goes like this, the other guy on the other end goes like that and you make two traveling waves when they get to each other, they just disappear for a second and then they come out on the other side. Waves can and often do cancel each other because they can be plus or minus. But therefore, if they don't all add up when they get to the detector, I'm not going to be able to get a big signal and therefore I won't get a spot on the film. I don't think anybody uses film anymore. And the condition then is that the path difference should be an integral number of the wavelengths. That means I have to use x-rays that have the same wavelength. They can't be just random x-rays like white light. They've got to be like this laser pointer. It's got a specific color. And I use that specific color to shine in. Often it might be something like copper K alpha. Here's what I think is a little bit clearer drawing. I redid it. Here's the thing, the angle of incidence, the angle of reflection, those are theta. And how can I show that this little angle in here, how can I show that it is theta? Well, these are at 90 degrees, this thing that I drew because I said that's how you measure the race, the distance. And they're at 90 degrees. If they're at 90 degrees and this parts theta and this on the other side then it's 90 minus theta because this is 90 degrees. Ah-ha, but if this is 90 minus theta then this is obviously 90 degrees between horizontal and vertical. If this is 90 minus theta then this guy is theta. And then I have a right triangle, this is 90, so this is 90 minus theta as well. So I can label them all, just go through and label them all without knowing anything. And then I do need to know one thing. If this angles theta and this is the hypotenuse, kind of turn this triangle on its side like this, I have the hypotenuse, then the opposite side over the hypotenuse, that's the definition of the sine function, the sine of theta is the opposite over the hypotenuse, therefore X upon D is equal to sine theta, multiply by D, X is D sine theta. The path length I'm after is 2X and therefore 2X is 2D sine theta and that should be an integral number of wavelengths so that when one comes down the other comes down, they come back and then they are going up together so that they're going to build up into the detector and that's our condition then, Bragg's law, n lambda is 2D sine theta. And in freshman chemistry in my experience n is always 1 but in real life you have to know what you're doing because you get all kinds of reflections sort of like looking into a kaleidoscope. You get just all kinds of things and you have to sort them out, you have to figure out what's going on, it doesn't necessarily reveal itself to you. Okay, let's talk about three different types of solids. An atomic solid is interconnected identical atoms and here I've drawn a, I didn't draw it but I mugged it up, a piece of diamond in which you can see that each atom has this beautiful tetrahedral coordination of four other carbon atoms. Very, very, very hard material but there are materials nearly as hard as diamond. Why do you think the dentist uses a diamond drill instead of titanium carbide or something? Anybody know? Well, yeah, you wouldn't want rust in your mouth but you can have stainless steel, surgeons use stainless steel implements, cut you up with those. I mean, you'll never see a big drill bit like that because that's too big to do a drill of 2,000. No, it's because diamond conducts heat very effectively. If you ever wonder if you've been given a phony ring, I might get in trouble. You get a needle and you heat it up so it's red hot and then you take your supposed diamond and you put your tongue on one side of it and the needle carefully on the other side. You can use your lips too. Your lips and your tongue are extremely sensitive to temperature. I've seen people debug computer boards by pulling out the computer board and then going on his face and he says that's the bad chip. That's the one that's hot, pulls it out, puts it in. It's amazing. Sergeant, you're $5,000. Okay, sometimes knowledge is power or money and the diamond will conduct heat like crazy and sadly cubic zirconia is an insulator and so it won't conduct anything, so you could tell. And the dentist wants to pull the heat away because as they drill your tooth at that high, very high 450,000 RPM, they don't want your tooth to heat up and explode and they don't want you to smell like you're cooking a bug with a magnifying glass either. It's your enamel burns and that's why they use diamond. Sodium chloride is an ionic solid. It has the so-called rock salt structure. That's where the name came from. And then here's a molecular solid, ice, in which we have discrete molecules that are then locked into place and molecular solids have structures that depend a lot on the molecular geometry because these things are no longer spheres at all. These are funny shapes like boomerangs that have to pack together and they could pack together in all kinds of shapes. Metals have atoms on lattice points but then the atoms seem to have lost their identity when it comes to the valence electron. The valence electrons literally find themselves in an orbital that extends over the entire material. They're all in there together moving around but they aren't associated with any particular atom anymore like they would be in a vapor phase of the same material. The big positive charges stay put because they're the big heavy atomic nuclei and they have some inner core shell of electrons that usually makes them partially positively charged. Metals never like to be negatively charged. And then the rest of the electrons kind of go around in the spaces between the lattice and they can go through and electrons have a wave character to them and they can go right through and that's why most metals will conduct electricity well because if I inject electrons on this side that can go all the way through to the other side. Things where the electrons are stuck on each atom like plastic won't conduct electricity and therefore I use copper in the wire and then I use some kind of plastic around the wire to keep it from shortening to ground. Okay? Or having something happen to it. Okay. Let's talk about phases. A phase is a state of matter that is uniform in chemical composition and physical state. For example, liquid water is a phase. Solid water is a different phase. Solid water and liquid water don't have the same properties at all with respect to very important things. Same molecule but different physical state. Same chemical composition, different physical state equals different phase. And then I could have a solution of sodium chloride in water and that would be uniform throughout salt water. That's again one phase, a solution phase. You have to be a little bit careful because the question is how close do you look? And the answer is don't look too close. Don't get inside the sodium nucleus. That's getting too close. Now it doesn't look uniform. There's a bunch of protons and neutrons here and then a big space and then some electrons. Doesn't look uniform. Take a small representative chunk like a cubic nanometer and look at it. If they all look about the same, it's one phase. If they look radically different, they are not one phase. For example, if I put oil and water like salad dressing, then I can see a phase boundary. I can see a region where the oil is floating on the water because it's less dense. Therefore the oil is one phase. If I take a little dropper, a little pipetman, I take part of that and I analyze it and then I go down into the water and do the same thing. They come out totally different. That's the key that there's two phase. I see a phase boundary. If I have ice cubes floating in water, I see where the solid ice is and the liquid water is. No question that they're different. Likewise with oil and water, that's a two phase system that we could find a blob of pure oil or a blob of pure water. Allotropes is the fancy name for different phases of the same material. For example, carbon is graphite or carbon as diamond. Graphite is the more stable form at atmospheric temperature and pressure. Therefore, thermodynamically, diamonds are not forever. They'll eventually degrade into graphite. However, it would take an awful lot of energy to take all those tetrahedral carbons locked in that big giant network and then suddenly change them into these hexagonal things like honey bee comb and at room temperature, there isn't that much energy. There certainly isn't enough energy to break bonds and therefore diamonds are metastable. They aren't strictly thermodynamically stable. They're doomed eventually but eventually is a very, very long time in this case and so there's still a reasonable investment from that standpoint. But they are a terrible investment. Okay, let's talk about phase diagrams. A phase diagram shows the thermodynamically most stable phase as a function of temperature and pressure or if there's more than one kind of chemical present, sometimes as a function of temperature and composition and it just depends or pressure and composition sometimes. We'll tend to focus on phase diagrams that just have a single pure material. Once we know the phase diagram, we can predict what we will eventually find in a container at the given temperature and pressure and keep in mind, we've got an imaginary container. I always imagine a cylinder and a piston and I can push on the piston and I can increase the pressure and I could put them flames from below and I can increase the temperature. But here's the thing, there's nothing else in there. If it's water, there's only water in there it's not a pot of water boiling on a stove in air. That's not the picture. It's just the pure material locked inside this thing. That's what you think about when you look at a phase diagram. Once you have water and air and stuff, that's more than one phase, more than one chemical in the soup although oftentimes the behavior is similar so it may not matter but it's important conceptually to keep them clear. There can be many phases, almost an endless number of them, especially once you get into solids or liquid crystals, you get many phases. All the displays you see for TV, they say LCD TV, liquid crystal, you have to first discover liquid crystal phases. Some guy was doing that, they said, what are you doing? I'm studying liquid crystal phases. That is so useless, man. Why don't you do something useful? Well, I'm interested in the properties of these smectic and cholesteric phases. What? That is totally useless. And then 30, 40 years later, every single device is using those molecules that change refractive index and so forth with an electric field. So you never know, it's important just to know what is the case in science. What's the truth? Don't worry yet if it's useful, if it's interesting. It will be useful, believe me. Here's the phase diagram for water and water is unique because this line goes backwards to the left if you plot pressure. Conventionally plot pressure on the Y and temperature on the X but not always and especially if you get some funky data off the web, you look closely at what the axes are and if the axes aren't labeled, that's very bad practice. Never give a graph where the axes aren't labeled with units. Here's a solid. What this says is if I start here and I put pressure on the solid, it turns into a liquid. Usually that's not how things work. That means the liquid is denser than the solid. Well, that's what we know to be true. That's why the ice cubes float. If you make an ice cube out of B2O as a joke, you can put it in a glass of water in the sink. People stare at it. It's just a nice soap. And if I start here and I heat it, the solid will melt and then the liquid will boil but if the pressure is very low, look what happens. The solid just sublines. It never becomes a liquid. And therefore I can purify things by distilling them under low pressure rather than just heating the stuff up and then it may actually have a side reaction that does something. Instead what I do is reduce the pressure and I can get it to boil or if I increase the pressure then I can keep it as a liquid well into the very high temperature range. And here it then is a so-called log lint. Engineers like to use log plots. Pressure here is in bar and temperature is in Kelvin. There's the triple point and look at all these different phases of solids way up high. They all have different crystal structures. Okay, we'll quit there.