1 00:00:00,030 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,860 Your support will help MIT OpenCourseWare continue to 4 00:00:06,860 --> 00:00:10,510 offer high quality educational resources for free. 5 00:00:10,510 --> 00:00:13,390 To make a donation or view additional materials from 6 00:00:13,390 --> 00:00:17,600 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,600 --> 00:00:20,650 ocw.mit.edu. 8 00:00:20,650 --> 00:00:25,200 PROFESSOR: Today I want to start us off on a 9 00:00:25,200 --> 00:00:26,820 somewhat new topic. 10 00:00:26,820 --> 00:00:28,530 Which is phase equilibria. 11 00:00:28,530 --> 00:00:30,810 So, recently you've been looking at chemical 12 00:00:30,810 --> 00:00:31,330 equilibria. 13 00:00:31,330 --> 00:00:33,870 What happens when substances change their chemical 14 00:00:33,870 --> 00:00:37,270 structure and interchange, and you've figured out how to 15 00:00:37,270 --> 00:00:39,480 calculate where equilibrium lies in the 16 00:00:39,480 --> 00:00:41,200 case of chemical reactions. 17 00:00:41,200 --> 00:00:43,830 Calculate equilibrium constants, and so forth. 18 00:00:43,830 --> 00:00:47,060 Now we'll introduce phase equilibria. 19 00:00:47,060 --> 00:00:48,340 So this is a little bit different. 20 00:00:48,340 --> 00:00:51,100 You still have a lot of molecules in one state 21 00:00:51,100 --> 00:00:53,480 changing to another state, but they're not changing their own 22 00:00:53,480 --> 00:00:54,320 molecular structure. 23 00:00:54,320 --> 00:00:56,470 They're changing what phase of matter they're in. 24 00:00:56,470 --> 00:00:58,580 From liquid to solid to gas. 25 00:00:58,580 --> 00:01:00,900 And so the treatment is a little bit different. 26 00:01:00,900 --> 00:01:04,010 And there's some new results that are somewhat different 27 00:01:04,010 --> 00:01:06,250 from what you've seen in the case of chemical reactions. 28 00:01:06,250 --> 00:01:15,830 So we'll start with phase equilibria in just one 29 00:01:15,830 --> 00:01:26,190 component system. 30 00:01:26,190 --> 00:01:27,900 So in other words, we're going to start with a single 31 00:01:27,900 --> 00:01:31,830 substance and see how it goes from solid to liquid to gas. 32 00:01:31,830 --> 00:01:34,550 And we'll from there see what happens in 33 00:01:34,550 --> 00:01:35,990 multiple component systems. 34 00:01:35,990 --> 00:01:38,240 Where, for example, then we'll be able to talk about 35 00:01:38,240 --> 00:01:41,370 solubility and mixtures of liquids and so forth. 36 00:01:41,370 --> 00:01:44,470 And into which phase different constituents in a 37 00:01:44,470 --> 00:01:46,030 mixture will go. 38 00:01:46,030 --> 00:01:47,330 But let's start here. 39 00:01:47,330 --> 00:01:58,480 So, let's say we just have two phases at equilibrium. 40 00:01:58,480 --> 00:02:07,350 And this could just look like ice and water. 41 00:02:07,350 --> 00:02:14,180 You could have a little bit of ice floating in water. 42 00:02:14,180 --> 00:02:23,820 So in this case you have solid and liquid. 43 00:02:23,820 --> 00:02:25,380 That could be in equilibrium. 44 00:02:25,380 --> 00:02:33,070 And this is going to be at some pressure and temperature. 45 00:02:33,070 --> 00:02:39,130 Or, you could have liquid and gas in equilibrium. 46 00:02:39,130 --> 00:02:41,530 So let's put a lid on our container. 47 00:02:41,530 --> 00:02:45,690 Still have some specified pressure and temperature. 48 00:02:45,690 --> 00:02:49,430 And now there's a liquid at the bottom. 49 00:02:49,430 --> 00:02:50,720 And a gas at the top. 50 00:02:50,720 --> 00:02:59,490 So, you could have equilibrium between liquid and gas phases. 51 00:02:59,490 --> 00:03:02,110 And in some cases you have equilibrium between solid and 52 00:03:02,110 --> 00:03:22,150 gas phases, depending on the pressure. 53 00:03:22,150 --> 00:03:24,030 And and a little piece of solid could sit at the bottom 54 00:03:24,030 --> 00:03:28,220 and there's equilibrium with the gas above it. 55 00:03:28,220 --> 00:03:32,050 And so, to start, what quantity are we going to need 56 00:03:32,050 --> 00:03:36,100 to look at to tell whether the different phases are in 57 00:03:36,100 --> 00:03:38,660 equilibrium, or if they're not in equilibrium, in which 58 00:03:38,660 --> 00:03:40,210 direction are things going to go? 59 00:03:40,210 --> 00:03:43,870 Are things going to change so that the ice will melt and 60 00:03:43,870 --> 00:03:45,970 we'll have water or, so the water will freeze 61 00:03:45,970 --> 00:03:46,590 and we'll have ice. 62 00:03:46,590 --> 00:03:51,290 What quantity is going to tell us about all that? 63 00:03:51,290 --> 00:03:54,000 Who thinks they can guess? 64 00:03:54,000 --> 00:03:55,500 Give you a hint. 65 00:03:55,500 --> 00:04:00,750 We're specifying pressure and temperature. 66 00:04:00,750 --> 00:04:04,420 Nobody really knows? 67 00:04:04,420 --> 00:04:07,800 What quantity is it that you've seen from which you can 68 00:04:07,800 --> 00:04:11,020 determine everything. 69 00:04:11,020 --> 00:04:12,170 Yes. 70 00:04:12,170 --> 00:04:25,770 The Gibbs free energy is going to tell us everything. 71 00:04:25,770 --> 00:04:28,230 And we're going to frame the discussion in terms of the 72 00:04:28,230 --> 00:04:29,320 chemical potential. 73 00:04:29,320 --> 00:04:43,150 That is, the Gibbs free energy for a mole. 74 00:04:43,150 --> 00:04:46,350 And since there's only one constituent, one component at 75 00:04:46,350 --> 00:04:47,930 this point, we just have n. 76 00:04:47,930 --> 00:04:51,100 We don't have different mole numbers for different 77 00:04:51,100 --> 00:04:54,430 constituents, we'll see that later. 78 00:04:54,430 --> 00:04:59,380 OK, so what's going to happen is, the value of the chemical 79 00:04:59,380 --> 00:05:02,000 potential in the different phases is going to tell us 80 00:05:02,000 --> 00:05:04,060 everything. 81 00:05:04,060 --> 00:05:07,430 Because you know that at equilibrium, the chemical 82 00:05:07,430 --> 00:05:10,320 potential needs to be the same in all of these. 83 00:05:10,320 --> 00:05:13,350 If not, let's say the potential energy here is lower 84 00:05:13,350 --> 00:05:17,930 in the solid than in the gas, nothing's going to stop the 85 00:05:17,930 --> 00:05:19,780 molecules in the gas phase from going 86 00:05:19,780 --> 00:05:20,640 into the solid phase. 87 00:05:20,640 --> 00:05:24,150 That's what they'll do. 88 00:05:24,150 --> 00:05:26,750 If the chemical potential of the water is lower than the 89 00:05:26,750 --> 00:05:29,850 chemical potential of the ice, then the ice is going to melt. 90 00:05:29,850 --> 00:05:32,770 Because the molecules are going to want to go toward 91 00:05:32,770 --> 00:05:37,000 lowest possible accessible chemical potential. 92 00:05:37,000 --> 00:05:38,200 And there's the liquid right there. 93 00:05:38,200 --> 00:05:41,850 They have access to it, so they can, the ice will melt. 94 00:05:41,850 --> 00:05:48,570 And you'll have just water. 95 00:05:48,570 --> 00:06:03,760 So the main point here is that at equilibrium, mu is 96 00:06:03,760 --> 00:06:11,240 identical everywhere. 97 00:06:11,240 --> 00:06:18,960 By everywhere, I mean in all phases that are at 98 00:06:18,960 --> 00:06:22,800 equilibrium. 99 00:06:22,800 --> 00:06:27,550 And so if multiple phases are present, mu has to be the same 100 00:06:27,550 --> 00:06:28,960 in all the phases. 101 00:06:28,960 --> 00:06:31,780 And if it isn't, then the molecules will go into the 102 00:06:31,780 --> 00:06:54,340 phase with the lower chemical potential. 103 00:06:54,340 --> 00:06:57,430 And this is what is going to guide everything in our 104 00:06:57,430 --> 00:07:01,340 consideration of phase equilibrium. 105 00:07:01,340 --> 00:07:03,940 So, OK, what's going to happen? 106 00:07:03,940 --> 00:07:08,780 Let's consider our ice water equilibrium. 107 00:07:08,780 --> 00:07:24,170 So of course, if mu of the solid is equal to mu of the 108 00:07:24,170 --> 00:07:31,190 liquid, which for water it should happen at zero degrees 109 00:07:31,190 --> 00:07:31,800 Centigrade. 110 00:07:31,800 --> 00:07:38,050 And pressure of one bar, for example, not the only place it 111 00:07:38,050 --> 00:07:38,490 could happen. 112 00:07:38,490 --> 00:07:43,110 But certainly at that condition. 113 00:07:43,110 --> 00:07:47,210 Then the ice and water will be in 114 00:07:47,210 --> 00:08:01,480 equilibrium, they'll co-exist. 115 00:08:01,480 --> 00:08:04,660 Now, what happens if you just want some ice water and you 116 00:08:04,660 --> 00:08:06,466 take some ice out of the freezer and put it in the cup, 117 00:08:06,466 --> 00:08:08,550 and you put some water in? 118 00:08:08,550 --> 00:08:09,540 And you see them both there. 119 00:08:09,540 --> 00:08:10,930 They're sitting there. 120 00:08:10,930 --> 00:08:13,940 Maybe for a while. 121 00:08:13,940 --> 00:08:16,060 And yet it's not at zero degrees 122 00:08:16,060 --> 00:08:17,420 Centigrade, at one bar. 123 00:08:17,420 --> 00:08:20,085 It may be at one bar, you might be out in the room, but 124 00:08:20,085 --> 00:08:24,050 it might be room temperature. 125 00:08:24,050 --> 00:08:27,440 Does that tell us that this isn't true? 126 00:08:27,440 --> 00:08:31,780 What's going to happen? 127 00:08:31,780 --> 00:08:32,700 The ice is going to melt. 128 00:08:32,700 --> 00:08:34,520 It may take a while. 129 00:08:34,520 --> 00:08:36,480 This doesn't tell us about the kinetics of it. 130 00:08:36,480 --> 00:08:38,880 And it could take a while, gradually, from the surface 131 00:08:38,880 --> 00:08:41,660 inward, for the ice to melt. 132 00:08:41,660 --> 00:08:44,020 So the fact that you might see something sitting there with 133 00:08:44,020 --> 00:08:47,130 two phases present, that alone of course, doesn't tell you 134 00:08:47,130 --> 00:08:48,970 the two phases are in equilibrium. 135 00:08:48,970 --> 00:08:57,480 And it may take time for equilibrium to occur. 136 00:08:57,480 --> 00:09:05,480 And what's going to happen if at some pressure and 137 00:09:05,480 --> 00:09:12,890 temperature, you have the mu s greater than mu of the liquid. 138 00:09:12,890 --> 00:09:14,070 Then which way are things going to go? 139 00:09:14,070 --> 00:09:19,200 What's going to happen? 140 00:09:19,200 --> 00:09:24,190 You've got to be able to figure this out. 141 00:09:24,190 --> 00:09:30,290 Of course. 142 00:09:30,290 --> 00:09:31,150 The ice is going to melt. 143 00:09:31,150 --> 00:09:33,380 The chemical potential of the liquid is lower. 144 00:09:33,380 --> 00:09:37,180 And so molecules are going to move toward that phase. 145 00:09:37,180 --> 00:09:48,340 And, of course, if mu solid less than mu liquid, then the 146 00:09:48,340 --> 00:09:50,650 water freezes. 147 00:09:50,650 --> 00:09:55,350 All this behavior is something that we can summarize in a 148 00:09:55,350 --> 00:10:03,310 phase diagram. 149 00:10:03,310 --> 00:10:09,660 So the phase diagram can tell us for all pressures and 150 00:10:09,660 --> 00:10:15,050 temperatures, what phase, or what phases, will be present 151 00:10:15,050 --> 00:10:15,850 in equilibrium. 152 00:10:15,850 --> 00:10:28,740 And so typically, phase diagram might look like this. 153 00:10:28,740 --> 00:10:30,790 Where here's our solid. 154 00:10:30,790 --> 00:10:32,410 Here's our liquid. 155 00:10:32,410 --> 00:10:33,930 Here's our gas phase. 156 00:10:33,930 --> 00:10:39,010 So let's just look at what's happening here. 157 00:10:39,010 --> 00:10:41,640 If we go to really low temperature, in 158 00:10:41,640 --> 00:10:42,790 general, what happens? 159 00:10:42,790 --> 00:10:45,310 What phase does stuff enter when you're at the lowest 160 00:10:45,310 --> 00:10:47,320 accessible temperatures? 161 00:10:47,320 --> 00:10:47,900 Solid, right? 162 00:10:47,900 --> 00:10:50,190 Everything eventually freezes. 163 00:10:50,190 --> 00:10:55,670 In fact, really I should make this, leave some room on my 164 00:10:55,670 --> 00:10:58,130 axis for that effect. 165 00:10:58,130 --> 00:10:59,780 So of course, low temperature you get solid. 166 00:10:59,780 --> 00:11:02,540 How about high pressure, what usually happens? 167 00:11:02,540 --> 00:11:05,660 You squeeze something real hard. 168 00:11:05,660 --> 00:11:08,900 Tends to go into the solid phase. 169 00:11:08,900 --> 00:11:11,880 Now let's warm stuff up. 170 00:11:11,880 --> 00:11:16,020 And depending on the pressure, you might go 171 00:11:16,020 --> 00:11:16,920 to the liquid phase. 172 00:11:16,920 --> 00:11:18,900 Certainly you'll go to the gas phase at high enough 173 00:11:18,900 --> 00:11:21,580 temperature. 174 00:11:21,580 --> 00:11:23,400 And that's what'll happen. 175 00:11:23,400 --> 00:11:26,880 And now, if you start changing the pressure, let's say you're 176 00:11:26,880 --> 00:11:30,350 at high temperature and you've got just the gas present, in 177 00:11:30,350 --> 00:11:32,140 some big container. 178 00:11:32,140 --> 00:11:34,380 Now you start squeezing it. 179 00:11:34,380 --> 00:11:37,100 You just apply more and more pressure to the gas at 180 00:11:37,100 --> 00:11:37,900 constant temperature. 181 00:11:37,900 --> 00:11:39,510 What's going to happen? 182 00:11:39,510 --> 00:11:40,660 Forget the phase diagram. 183 00:11:40,660 --> 00:11:42,400 You know what's going to happen. 184 00:11:42,400 --> 00:11:49,630 You keep squeezing, what will happen eventually? 185 00:11:49,630 --> 00:11:51,500 This isn't going to be part of your grade. 186 00:11:51,500 --> 00:11:53,810 You can speak up loud, and even if you're completely 187 00:11:53,810 --> 00:11:58,900 wrong, nothing bad is going to happen to you. 188 00:11:58,900 --> 00:11:59,430 Yeah. 189 00:11:59,430 --> 00:12:01,100 It's going to liquefy, of course. 190 00:12:01,100 --> 00:12:02,770 You're going to start getting liquid to form. 191 00:12:02,770 --> 00:12:03,600 It's essentially Le Chatelier's 192 00:12:03,600 --> 00:12:05,070 principle, of course. 193 00:12:05,070 --> 00:12:06,860 The liquid occupies less volume. 194 00:12:06,860 --> 00:12:08,600 The density of the liquid is much higher. 195 00:12:08,600 --> 00:12:11,290 So as you increase the pressure, the stuff wants to 196 00:12:11,290 --> 00:12:12,780 go from the gas phase to the liquid phase. 197 00:12:12,780 --> 00:12:12,980 Yeah. 198 00:12:12,980 --> 00:12:16,900 STUDENT: [INAUDIBLE] 199 00:12:16,900 --> 00:12:21,880 PROFESSOR: Well, with water you know that at ordinary 200 00:12:21,880 --> 00:12:25,860 pressure, you know where the gas-liquid 201 00:12:25,860 --> 00:12:28,100 coexistence would occur. 202 00:12:28,100 --> 00:12:29,220 So let's say you're there. 203 00:12:29,220 --> 00:12:33,500 Let's say you're at the boiling point. 204 00:12:33,500 --> 00:12:38,620 And now, you already in some sense know that that's a point 205 00:12:38,620 --> 00:12:42,500 that's somewhere on the gas-liquid curve. 206 00:12:42,500 --> 00:12:43,970 So now what's going to happen? 207 00:12:43,970 --> 00:12:46,470 Well, if you from there keep that temperature constant. 208 00:12:46,470 --> 00:12:47,970 You're at the boiling point. 209 00:12:47,970 --> 00:12:50,180 And you could easily do this. 210 00:12:50,180 --> 00:12:51,970 You've got a pot that's covered, right? 211 00:12:51,970 --> 00:12:53,430 And it's boiling. 212 00:12:53,430 --> 00:12:56,070 So there's liquid down here and there's gas here. 213 00:12:56,070 --> 00:12:57,130 So what'll happen? 214 00:12:57,130 --> 00:12:59,890 As you start squeezing, literally what'll happen is 215 00:12:59,890 --> 00:13:02,070 you'll squeeze out all the room for the gas. 216 00:13:02,070 --> 00:13:05,300 And eventually you'll have all liquid. 217 00:13:05,300 --> 00:13:07,820 And in fact, you know what'll happen at that point, if you 218 00:13:07,820 --> 00:13:10,310 keep squeezing and you keep the heat on. 219 00:13:10,310 --> 00:13:13,280 The liquid will get hotter than the boiling point. 220 00:13:13,280 --> 00:13:15,710 Normally that doesn't happen, if you're just out at 221 00:13:15,710 --> 00:13:16,530 atmospheric pressure. 222 00:13:16,530 --> 00:13:18,630 Because it'll boil. 223 00:13:18,630 --> 00:13:22,360 And instead of getting hotter, molecules will simply leave 224 00:13:22,360 --> 00:13:25,520 the liquid and go into the gas phase. 225 00:13:25,520 --> 00:13:26,670 So that's what'll happen. 226 00:13:26,670 --> 00:13:30,860 If you increase the pressure, you'll liquefy all the gas. 227 00:13:30,860 --> 00:13:33,120 And obviously the opposite will also occur. 228 00:13:33,120 --> 00:13:37,510 If you just allow the volume to expand and keep expanding, 229 00:13:37,510 --> 00:13:39,140 maybe you don't have too much liquid there, you wouldn't 230 00:13:39,140 --> 00:13:40,600 have to expand it too much. 231 00:13:40,600 --> 00:13:43,390 You'll run out, all the liquid will vaporize. 232 00:13:43,390 --> 00:13:49,090 And you'll just have gas at 100 degrees C. You could heat 233 00:13:49,090 --> 00:13:53,710 it more if you wanted, but even there. 234 00:13:53,710 --> 00:13:57,630 So in other words, changing the pressure if you think 235 00:13:57,630 --> 00:14:00,120 about actually executing that in the lab, or in this case 236 00:14:00,120 --> 00:14:03,370 even in your kitchen, you'll get the result that you should 237 00:14:03,370 --> 00:14:05,060 see on the phase diagram. 238 00:14:05,060 --> 00:14:06,100 So let's just look at this. 239 00:14:06,100 --> 00:14:09,850 So we've already talked about boiling. 240 00:14:09,850 --> 00:14:14,000 That's what's happening along this line. 241 00:14:14,000 --> 00:14:19,470 The line, and I'll explain the end of this in a while, all 242 00:14:19,470 --> 00:14:28,070 these lines are coexistence curves. 243 00:14:28,070 --> 00:14:30,440 That is, these are where you have 244 00:14:30,440 --> 00:14:33,380 equilibrium between two phases. 245 00:14:33,380 --> 00:14:38,360 They coexist in equilibrium. and it could happen for water, 246 00:14:38,360 --> 00:14:43,400 for that case, for 100 degrees C and one bar. 247 00:14:43,400 --> 00:14:44,840 But that's not the only place. 248 00:14:44,840 --> 00:14:46,250 It could happen it a whole bunch of 249 00:14:46,250 --> 00:14:47,150 temperatures and pressure. 250 00:14:47,150 --> 00:14:50,750 Because as you vary the pressure, you're almost surely 251 00:14:50,750 --> 00:14:53,850 familiar with this, the boiling point changes. 252 00:14:53,850 --> 00:14:57,300 So the boiling point up in Denver at high elevation is 253 00:14:57,300 --> 00:14:59,800 not the same as the boiling point here in Massachusetts 254 00:14:59,800 --> 00:15:01,020 right at sea level. 255 00:15:01,020 --> 00:15:08,710 Because the pressure's lower up in Denver. 256 00:15:08,710 --> 00:15:09,770 Let's just look at the others. 257 00:15:09,770 --> 00:15:12,910 So, of course, this is going from solid to liquid. 258 00:15:12,910 --> 00:15:16,990 So we're looking at melting. 259 00:15:16,990 --> 00:15:19,680 And then there is solid-gas equilibria in many cases. 260 00:15:19,680 --> 00:15:29,450 And this is sublimation. 261 00:15:29,450 --> 00:15:38,750 And this endpoint is called the critical point. 262 00:15:38,750 --> 00:15:43,730 And there's one point where all three phases are in 263 00:15:43,730 --> 00:15:51,740 equilibrium, called the triple point. 264 00:15:51,740 --> 00:15:54,010 So let's look at the various cases. 265 00:15:54,010 --> 00:15:56,280 For example, let's look at the melting line. 266 00:15:56,280 --> 00:15:58,190 It could be water, it could be water and ice. 267 00:15:58,190 --> 00:16:02,900 Solid going to a liquid. 268 00:16:02,900 --> 00:16:06,420 Let's just look at them one at a time. 269 00:16:06,420 --> 00:16:09,900 What you have there is mu of the solid at some temperature 270 00:16:09,900 --> 00:16:10,990 and pressure. 271 00:16:10,990 --> 00:16:15,650 Is equal to mu of the liquid at the same 272 00:16:15,650 --> 00:16:18,270 temperature and pressure. 273 00:16:18,270 --> 00:16:22,660 Now, just if we back off and look at this essentially 274 00:16:22,660 --> 00:16:25,260 mathematically, there's an equation here. 275 00:16:25,260 --> 00:16:26,830 And there are two free variables. 276 00:16:26,830 --> 00:16:28,020 Temperature and pressure. 277 00:16:28,020 --> 00:16:32,170 The equation puts a constraint on them. 278 00:16:32,170 --> 00:16:34,950 So there are going to be a bunch of solutions along some 279 00:16:34,950 --> 00:16:37,660 line or some curve. 280 00:16:37,660 --> 00:16:44,800 So in other words, it's one equation in two unknowns. 281 00:16:44,800 --> 00:16:53,330 Temperature and pressure. 282 00:16:53,330 --> 00:17:01,160 So the solution is a line or a curve. 283 00:17:01,160 --> 00:17:08,710 In other words, we could solve for pressure as a function of 284 00:17:08,710 --> 00:17:13,820 temperature, or temperatures as a function of pressure, 285 00:17:13,820 --> 00:17:18,780 along the coexistence curve. 286 00:17:18,780 --> 00:17:24,600 And we'll do that shortly. 287 00:17:24,600 --> 00:17:32,740 Now, if we're in one of these regions, then there aren't any 288 00:17:32,740 --> 00:17:33,630 equalities. 289 00:17:33,630 --> 00:17:36,510 There aren't any constraints. 290 00:17:36,510 --> 00:17:40,890 Because we don't have multiple chemical potentials that are 291 00:17:40,890 --> 00:17:45,510 equal to each other. 292 00:17:45,510 --> 00:17:48,990 Right here we have three phases in chemical 293 00:17:48,990 --> 00:17:49,980 equilibrium. 294 00:17:49,980 --> 00:17:53,620 Solid, liquid and gas. 295 00:17:53,620 --> 00:18:00,370 So let's just look at that triple point. 296 00:18:00,370 --> 00:18:12,470 Mu solid equals mu liquid equals mu of the gas. 297 00:18:12,470 --> 00:18:14,390 We have two unknowns, or two variables, 298 00:18:14,390 --> 00:18:15,880 temperature and pressure. 299 00:18:15,880 --> 00:18:18,560 And we have two equations. 300 00:18:18,560 --> 00:18:21,040 What that means is there's only one solution, and that's 301 00:18:21,040 --> 00:18:23,860 what the phase diagram shows. 302 00:18:23,860 --> 00:18:27,810 There's a unique solution, the triple point temperature and 303 00:18:27,810 --> 00:18:30,520 the triple point pressure. 304 00:18:30,520 --> 00:18:48,480 So this has only one solution. 305 00:18:48,480 --> 00:18:52,520 If we summarize, how many degrees of freedom, or how 306 00:18:52,520 --> 00:18:57,610 many free variables, we have, anywhere here, we can do that 307 00:18:57,610 --> 00:18:58,640 in the following way. 308 00:18:58,640 --> 00:19:03,060 We can write F is 3 minus p. 309 00:19:03,060 --> 00:19:16,850 This is the number of degrees of freedom. 310 00:19:16,850 --> 00:19:22,880 Or independent variables. 311 00:19:22,880 --> 00:19:33,200 And this is the number of phases in equilibrium. 312 00:19:33,200 --> 00:19:37,020 So, what's it telling us? 313 00:19:37,020 --> 00:19:41,270 If we have just one phase in equilibrium, or way over in 314 00:19:41,270 --> 00:19:43,550 the solid or just gas or just liquid part 315 00:19:43,550 --> 00:19:45,670 of the phase diagram. 316 00:19:45,670 --> 00:19:46,610 This is one. 317 00:19:46,610 --> 00:19:49,310 We have two independent variables. 318 00:19:49,310 --> 00:19:51,240 Which is exactly what we know to be the case. 319 00:19:51,240 --> 00:19:54,200 In other words, in those regions, temperature and 320 00:19:54,200 --> 00:19:59,980 pressure can vary freely without constraints. 321 00:19:59,980 --> 00:20:04,120 If we go to any of the coexistence curves, now we 322 00:20:04,120 --> 00:20:07,210 have two phases in equilibrium. 323 00:20:07,210 --> 00:20:11,570 And then only one variable can be changed freely. 324 00:20:11,570 --> 00:20:15,700 If we want to stay on the coexistence curve, let's say 325 00:20:15,700 --> 00:20:19,410 we've got gas and liquid in equilibrium. 326 00:20:19,410 --> 00:20:21,130 It's boiling. 327 00:20:21,130 --> 00:20:24,800 We could change the pressure some, but if we want to keep 328 00:20:24,800 --> 00:20:27,500 them in equilibrium, if we've changed the pressure, we'd 329 00:20:27,500 --> 00:20:29,920 better change the temperature accordingly to stay on the 330 00:20:29,920 --> 00:20:30,980 coexistence curve. 331 00:20:30,980 --> 00:20:33,230 Otherwise everything will go up into the liquid or 332 00:20:33,230 --> 00:20:35,460 into the gas phase. 333 00:20:35,460 --> 00:20:38,940 Because you'll go off of the coexistence curve if you were 334 00:20:38,940 --> 00:20:42,480 to change one of these variables and not the other. 335 00:20:42,480 --> 00:20:44,390 In other words, there's a constraint. 336 00:20:44,390 --> 00:20:46,430 So only one can vary freely. 337 00:20:46,430 --> 00:20:48,890 And if all three phases are in equilibrium at the triple 338 00:20:48,890 --> 00:20:51,600 point, then you have no degrees of freedom. 339 00:20:51,600 --> 00:20:53,210 Nothing can vary and there's only one 340 00:20:53,210 --> 00:20:59,910 place where that happens. 341 00:20:59,910 --> 00:21:04,960 Now let's see if we can understand also, why does the 342 00:21:04,960 --> 00:21:07,830 whole thing have the overall structure that it does. 343 00:21:07,830 --> 00:21:12,070 Why are some of the slopes generally steeper than others. 344 00:21:12,070 --> 00:21:15,220 Turns out, we can get a qualitative understanding of 345 00:21:15,220 --> 00:21:20,360 phase equilibria in a pretty simple way. 346 00:21:20,360 --> 00:21:41,850 So in other words, can we understand how p and T change 347 00:21:41,850 --> 00:21:44,870 with respect to each other along any one of these 348 00:21:44,870 --> 00:21:46,700 coexistence curves? 349 00:21:46,700 --> 00:21:49,440 Can we understand the qualitative features of the 350 00:21:49,440 --> 00:21:54,820 phase diagrams? 351 00:21:54,820 --> 00:22:03,900 And what we'll see is, we can get an equation for dp/dT at 352 00:22:03,900 --> 00:22:04,760 coexistence. 353 00:22:04,760 --> 00:22:07,650 That's, in a sense, our objective here in order to get 354 00:22:07,650 --> 00:22:14,010 the sort of qualitative picture. 355 00:22:14,010 --> 00:22:16,870 So let's say we are somewhere here. 356 00:22:16,870 --> 00:22:19,140 It could be the gas-solid part. 357 00:22:19,140 --> 00:22:29,130 And we have some T. And now we go to T plus dT, and out here 358 00:22:29,130 --> 00:22:30,800 is pressure. 359 00:22:30,800 --> 00:22:37,300 How much does the pressure change when we stay on the 360 00:22:37,300 --> 00:22:44,790 coexistence curve? 361 00:22:44,790 --> 00:22:46,250 Well, let's go through this. 362 00:22:46,250 --> 00:22:48,830 Let's just consider any two phases, alpha and beta. 363 00:22:48,830 --> 00:23:04,900 They could be solid, liquid, gas. 364 00:23:04,900 --> 00:23:07,830 And the main condition, if we're going to be on the 365 00:23:07,830 --> 00:23:10,000 coexistence curve, is that the chemical 366 00:23:10,000 --> 00:23:22,480 potentials have to be equal. 367 00:23:22,480 --> 00:23:25,190 Well, that's fine. 368 00:23:25,190 --> 00:23:45,050 Now let's change the conditions a little bit. 369 00:23:45,050 --> 00:23:48,870 Staying on the coexistence curve. 370 00:23:48,870 --> 00:23:56,546 So, let's start again. mu s, or mu alpha of T and p is 371 00:23:56,546 --> 00:24:04,620 equal to mu beta at T and p. 372 00:24:04,620 --> 00:24:12,260 Now let's let temperature go to T plus dT, and pressure go 373 00:24:12,260 --> 00:24:15,480 to p plus dp. 374 00:24:15,480 --> 00:24:19,220 But since we're staying on the coexistence curve still, at 375 00:24:19,220 --> 00:24:21,570 the new temperature and pressure, mu alpha and beta 376 00:24:21,570 --> 00:24:23,600 have to be equal to each other. 377 00:24:23,600 --> 00:24:30,390 So mu alpha is going to go to mu alpha plus d mu alpha. 378 00:24:30,390 --> 00:24:33,790 There'll be some change in it. 379 00:24:33,790 --> 00:24:40,690 Mu beta is going to go to mu beta plus d mu beta. 380 00:24:40,690 --> 00:24:43,980 But we already have that mu alpha and mu beta at the 381 00:24:43,980 --> 00:24:46,360 beginning were equal to each other. 382 00:24:46,360 --> 00:24:48,260 And these have to be equal to each other. 383 00:24:48,260 --> 00:24:49,930 Because we've specified that we're staying on the 384 00:24:49,930 --> 00:24:52,070 coexistence curve. 385 00:24:52,070 --> 00:25:03,600 So what that says is d mu alpha equals d mu beta. 386 00:25:03,600 --> 00:25:12,370 And now, remember in general d mu is dG per mole, so our 387 00:25:12,370 --> 00:25:19,840 fundamental equation for G, it's minus S per mole dT plus 388 00:25:19,840 --> 00:25:22,490 molar volume dp. 389 00:25:25,660 --> 00:25:38,190 So this equality is telling us that minus S alpha dT plus V 390 00:25:38,190 --> 00:25:51,300 alpha dp is equal to minus S beta dT plus V beta dp. 391 00:25:55,470 --> 00:25:58,100 So we're just applying the fundamental relation to the 392 00:25:58,100 --> 00:26:00,510 changes in each of the phases. 393 00:26:00,510 --> 00:26:05,580 But since the changes in mu are equal, 394 00:26:05,580 --> 00:26:07,230 so are these changes. 395 00:26:07,230 --> 00:26:09,130 And now I'm just going to rearrange these. 396 00:26:09,130 --> 00:26:19,320 In other words, S beta minus S alpha dT is equal to V theta 397 00:26:19,320 --> 00:26:22,690 minus V alpha dp. 398 00:26:25,560 --> 00:26:27,120 And that immediately is going to give us 399 00:26:27,120 --> 00:26:29,580 our derivative dp/dT. 400 00:26:45,190 --> 00:26:56,740 So, dp/dT, staying the coexistence curve, is just S 401 00:26:56,740 --> 00:27:04,960 beta minus S alpha over V beta minus V alpha. 402 00:27:04,960 --> 00:27:13,190 Which is to say it's just, we can say delta S over delta V. 403 00:27:13,190 --> 00:27:18,820 Going from alpha to beta. 404 00:27:18,820 --> 00:27:21,340 So we have a result that can guide our thinking into what 405 00:27:21,340 --> 00:27:24,060 the slopes of those coexistence lines 406 00:27:24,060 --> 00:27:25,960 are going to be. 407 00:27:25,960 --> 00:27:30,200 And it's pretty useful to look at that. 408 00:27:30,200 --> 00:27:33,920 By the way, we can actually express this in 409 00:27:33,920 --> 00:27:35,490 another way if we want. 410 00:27:35,490 --> 00:27:38,930 And in some cases it can be advantageous. 411 00:27:38,930 --> 00:27:46,710 And that is, we know that of course G is H minus TS. 412 00:27:49,530 --> 00:27:55,685 So our mu alpha equals mu beta condition, or a d mu alpha is 413 00:27:55,685 --> 00:28:00,310 d mu beta condition, can be applied that way. 414 00:28:00,310 --> 00:28:07,200 That is, H alpha minus TS alpha equals H 415 00:28:07,200 --> 00:28:11,840 beta minus TS beta. 416 00:28:11,840 --> 00:28:13,780 So we can do the same rearrangements. 417 00:28:13,780 --> 00:28:23,340 H beta minus H alpha equals T S beta minus S alpha. 418 00:28:23,340 --> 00:28:25,980 And what that's saying, in other words, is that we can 419 00:28:25,980 --> 00:28:33,700 substitute for delta S, delta H over T. So we can write our 420 00:28:33,700 --> 00:28:36,810 relation for dp/dT in two ways. 421 00:28:36,810 --> 00:28:45,070 We can also write dp/dT at coexistence is equal to delta 422 00:28:45,070 --> 00:29:00,730 H over T delta V. Going from alpha to beta. 423 00:29:00,730 --> 00:29:11,630 These are both forms of what's called the Clapeyron equation. 424 00:29:11,630 --> 00:29:12,930 And these are always valid. 425 00:29:12,930 --> 00:29:14,800 We haven't made any approximations. 426 00:29:14,800 --> 00:29:16,370 Yet. 427 00:29:16,370 --> 00:29:18,380 Later, in going to the gas phase, we'll make some 428 00:29:18,380 --> 00:29:20,870 approximations about having an ideal gas. 429 00:29:20,870 --> 00:29:25,220 We'll be able to treat this delta V in a simple way using 430 00:29:25,220 --> 00:29:26,370 the ideal gas law. 431 00:29:26,370 --> 00:29:29,500 But for now, these are exact. 432 00:29:29,500 --> 00:29:31,970 OK, so now let's look at what it tells us. 433 00:29:31,970 --> 00:29:42,140 Let's look at the different pieces of the phase diagram 434 00:29:42,140 --> 00:29:56,660 and see what we can conclude. 435 00:29:56,660 --> 00:30:00,120 Essentially we should be able to construct qualitatively, 436 00:30:00,120 --> 00:30:05,520 the features of the phase diagram using this relation. 437 00:30:05,520 --> 00:30:10,340 So let's try to do that. 438 00:30:10,340 --> 00:30:13,850 Here's p and T. We're going to start with an empty slate. 439 00:30:13,850 --> 00:30:20,690 And now let's look at the liquid gas case. 440 00:30:20,690 --> 00:30:25,800 So, let's look at our delta S delta V, going 441 00:30:25,800 --> 00:30:28,430 from liquid to gas. 442 00:30:28,430 --> 00:30:35,230 Well, one thing you know, when you go from liquid to gas, 443 00:30:35,230 --> 00:30:40,740 what happens to the molar volume? 444 00:30:40,740 --> 00:30:42,310 It increases. 445 00:30:42,310 --> 00:30:45,110 By a little, by a lot? 446 00:30:45,110 --> 00:30:46,440 By a lot, right? 447 00:30:46,440 --> 00:30:49,270 The molar volume in the gas is enormous compared to the molar 448 00:30:49,270 --> 00:30:51,560 volume in the liquid. 449 00:30:51,560 --> 00:30:56,890 So delta V is positive and very big. 450 00:30:56,890 --> 00:31:01,160 Now let's think about delta S. What's it going to do, going 451 00:31:01,160 --> 00:31:04,610 from liquid to gas? 452 00:31:04,610 --> 00:31:05,830 It's going to increase? 453 00:31:05,830 --> 00:31:07,550 The disorder increases. 454 00:31:07,550 --> 00:31:10,390 Now, it increases significantly. 455 00:31:10,390 --> 00:31:12,970 But the liquid already is a disordered phase. 456 00:31:12,970 --> 00:31:14,510 And the molecules and the liquid already have 457 00:31:14,510 --> 00:31:16,200 considerable freedom. 458 00:31:16,200 --> 00:31:18,350 Certainly not near as much as they have in the gas phase. 459 00:31:18,350 --> 00:31:19,800 But still considerable. 460 00:31:19,800 --> 00:31:22,910 So, to be sure, delta S positive. 461 00:31:22,910 --> 00:31:28,180 And it's not small. 462 00:31:28,180 --> 00:31:32,520 So there's your condition. 463 00:31:32,520 --> 00:31:35,245 It's not nearly as big as delta V is going 464 00:31:35,245 --> 00:31:37,300 to turn out to be. 465 00:31:37,300 --> 00:31:40,600 So what's going to happen here is, if we look at the slope 466 00:31:40,600 --> 00:31:45,430 going from liquid to gas, it's positive. 467 00:31:45,430 --> 00:31:46,560 But it's not enormous. 468 00:31:46,560 --> 00:31:48,380 It's not very steep. 469 00:31:48,380 --> 00:32:02,490 So dp/dT, for liquid-gas coexistence right, greater 470 00:32:02,490 --> 00:32:07,420 than zero, but small. 471 00:32:07,420 --> 00:32:14,220 Not steep slope. 472 00:32:14,220 --> 00:32:19,970 So somewhere, we're going to put our liquid-gas curve. 473 00:32:19,970 --> 00:32:22,400 And I, where exactly we can worry about it. 474 00:32:22,400 --> 00:32:24,080 We haven't put any numbers on this scale. 475 00:32:24,080 --> 00:32:26,200 But the point is, wherever it is, it's not going to be a 476 00:32:26,200 --> 00:32:30,430 very steep slope. 477 00:32:30,430 --> 00:32:35,180 How about solid to gas? 478 00:32:35,180 --> 00:32:42,650 Well, so what's delta V going from the solid to the gas? 479 00:32:42,650 --> 00:32:49,960 What happens to the molar volume? 480 00:32:49,960 --> 00:32:54,100 What happens to the molar volume, going from solid? 481 00:32:54,100 --> 00:32:56,520 Of course, by a lot, by a little? 482 00:32:56,520 --> 00:32:57,030 By a lot. 483 00:32:57,030 --> 00:32:57,990 It's an enormous increase. 484 00:32:57,990 --> 00:33:01,010 Even bigger than before. 485 00:33:01,010 --> 00:33:06,420 How about delta S, the entropy change? 486 00:33:06,420 --> 00:33:07,290 Yeah, it increases. 487 00:33:07,290 --> 00:33:09,920 And also by a lot, because now you're going from an ordered 488 00:33:09,920 --> 00:33:13,650 crystalline solid, usually, so you have a very, very, low 489 00:33:13,650 --> 00:33:15,260 entropy there. 490 00:33:15,260 --> 00:33:17,440 Into a disordered phase. 491 00:33:17,440 --> 00:33:19,010 Not just the liquid, but the gas phase. 492 00:33:19,010 --> 00:33:23,150 So this is also very big. 493 00:33:23,150 --> 00:33:28,170 Well, ok, so now we've got delta S over delta V. There's 494 00:33:28,170 --> 00:33:30,090 going to be a steeper slope. 495 00:33:30,090 --> 00:33:40,610 Because this is bigger then for liquid to gas. 496 00:33:40,610 --> 00:33:46,330 So typically, the solid-liquid part will have a steeper slope 497 00:33:46,330 --> 00:33:55,550 then the liquid to gas part. 498 00:33:55,550 --> 00:33:57,760 Now, wait. 499 00:33:57,760 --> 00:33:59,850 Sorry, sorry, this isn't right. 500 00:33:59,850 --> 00:34:06,710 And that's because I've really badly misplaced things here. 501 00:34:06,710 --> 00:34:08,570 That was solid to gas that we just discussed. 502 00:34:08,570 --> 00:34:13,730 It has the steepest slope relative to liquid-gas. 503 00:34:13,730 --> 00:34:23,840 OK, now let's think about solid to liquid. 504 00:34:23,840 --> 00:34:30,670 Delta V for going from solid to a liquid. 505 00:34:30,670 --> 00:34:31,670 What is it? 506 00:34:31,670 --> 00:34:35,340 What sign is it? 507 00:34:35,340 --> 00:34:36,200 How big is it? 508 00:34:36,200 --> 00:34:38,590 Let's forgot about the sign for a minute. 509 00:34:38,590 --> 00:34:41,180 How much does the volume, molar volume, change going 510 00:34:41,180 --> 00:34:42,830 from solid to liquid? 511 00:34:42,830 --> 00:34:44,290 Yeah, not a very lot, right? 512 00:34:44,290 --> 00:34:45,220 You melt the solid. 513 00:34:45,220 --> 00:34:48,330 There's still a little piece of stuff down there. 514 00:34:48,330 --> 00:34:50,060 Didn't increase in volume enormously. 515 00:34:50,060 --> 00:34:52,860 Usually, it increases. 516 00:34:52,860 --> 00:35:00,600 So generally delta V is greater than, or sort 517 00:35:00,600 --> 00:35:01,490 of equal to, zero. 518 00:35:01,490 --> 00:35:05,750 It's not a big amount. 519 00:35:05,750 --> 00:35:06,540 How about entropy? 520 00:35:06,540 --> 00:35:10,400 Going from solid to liquid? 521 00:35:10,400 --> 00:35:11,860 It increases, by a good amount. 522 00:35:11,860 --> 00:35:14,690 You're going from an ordered to a disordered phase. 523 00:35:14,690 --> 00:35:19,800 So delta S is greater than zero. 524 00:35:19,800 --> 00:35:23,750 And so the result here is, now, if you look at dS/dV, or 525 00:35:23,750 --> 00:35:27,610 delta S over delta V, delta S is positive and you a pretty 526 00:35:27,610 --> 00:35:28,970 good amount. 527 00:35:28,970 --> 00:35:31,510 Delta V is small. 528 00:35:31,510 --> 00:35:33,690 The molar volume hardly changed. 529 00:35:33,690 --> 00:35:35,665 And so what that's going to tell us is the slope is going 530 00:35:35,665 --> 00:35:37,420 to be steep. 531 00:35:37,420 --> 00:35:39,400 And generally positive. 532 00:35:39,400 --> 00:35:47,620 So now, we're going to really be on the increase. 533 00:35:47,620 --> 00:35:51,060 Typical kind of phase diagram and 534 00:35:51,060 --> 00:35:59,280 individual coexistence curve. 535 00:35:59,280 --> 00:36:02,100 And all these things, again, you could understand in terms 536 00:36:02,100 --> 00:36:04,360 of Le Chatelier's principle. 537 00:36:04,360 --> 00:36:09,530 In terms of going from, say, solid to liquid if you know 538 00:36:09,530 --> 00:36:15,330 think about two pressures. 539 00:36:15,330 --> 00:36:18,980 Go from p1 to p2, some higher pressure. 540 00:36:18,980 --> 00:36:19,760 Where are you? 541 00:36:19,760 --> 00:36:23,910 Well, if you want to stay on the coexistence curve you need 542 00:36:23,910 --> 00:36:29,020 to raise the temperature quite a bit in order to do that. 543 00:36:29,020 --> 00:36:33,700 Because, basically, to keep on there, you know what's going 544 00:36:33,700 --> 00:36:36,350 to happen if you have, say, solid and liquid. 545 00:36:36,350 --> 00:36:37,570 It's ice and water. 546 00:36:37,570 --> 00:36:39,950 You know the freezing point, it does depend on the 547 00:36:39,950 --> 00:36:44,670 pressure, of course. 548 00:36:44,670 --> 00:36:49,320 But since the change in the volume isn't that big, what 549 00:36:49,320 --> 00:36:54,260 happens is, a modest change, if you change the temperature 550 00:36:54,260 --> 00:36:56,720 by a little bit, maybe easier to think about it that way, 551 00:36:56,720 --> 00:36:59,750 you've got to change the pressure by a lot to stay on 552 00:36:59,750 --> 00:37:01,830 the coexistence curve. 553 00:37:01,830 --> 00:37:04,550 Otherwise it'll have a great tendency to simply go into one 554 00:37:04,550 --> 00:37:10,100 phase or the other. 555 00:37:10,100 --> 00:37:14,190 Now we've, in a sense, reconstructed the coexistence 556 00:37:14,190 --> 00:37:21,220 curve by our consideration of the Clapeyron equation. 557 00:37:21,220 --> 00:37:24,490 There's another way we can usefully understand the 558 00:37:24,490 --> 00:37:28,070 qualitative features of the phase diagram. 559 00:37:28,070 --> 00:37:30,610 And that's just by looking at mu itself. 560 00:37:30,610 --> 00:37:33,750 Looking at the free energy itself, at different 561 00:37:33,750 --> 00:37:36,210 temperature. 562 00:37:36,210 --> 00:37:37,470 So let's just do that. 563 00:37:37,470 --> 00:37:49,390 Let's just consider mu of T at some fixed pressure. 564 00:37:49,390 --> 00:37:51,230 Just qualitatively, what will happen? 565 00:37:51,230 --> 00:38:02,110 So let's go back to d mu is minus S dT plus V dp. 566 00:38:02,110 --> 00:38:05,030 And just look at the derivatives. 567 00:38:05,030 --> 00:38:11,910 So d mu / dT at constant pressure is minus s. 568 00:38:11,910 --> 00:38:13,380 Minus the molar volume. 569 00:38:13,380 --> 00:38:15,320 This is the one we're really going to look at. 570 00:38:15,320 --> 00:38:21,000 Of course, we also could look at d mu / dp at constant 571 00:38:21,000 --> 00:38:24,870 temperature, and just the molar volume. 572 00:38:24,870 --> 00:38:29,700 Let's just look at this. 573 00:38:29,700 --> 00:38:32,840 We know that entropy is always a positive number, right? 574 00:38:32,840 --> 00:38:36,710 Entropy at the lowest, in a perfect crystal, at zero 575 00:38:36,710 --> 00:38:40,490 Kelvin, could be what value? 576 00:38:40,490 --> 00:38:42,650 Entropy of a perfect pure crystal at 577 00:38:42,650 --> 00:38:44,920 zero degrees Kelvin. 578 00:38:44,920 --> 00:38:45,980 Zero, right? 579 00:38:45,980 --> 00:38:49,090 And it's only going up from there. 580 00:38:49,090 --> 00:38:59,370 So entropy is always positive. 581 00:38:59,370 --> 00:39:06,510 So what that tells us is that, then, if we just look at mu 582 00:39:06,510 --> 00:39:10,960 versus T, it's negative S, for whatever phase. 583 00:39:10,960 --> 00:39:24,490 So it's always negative. 584 00:39:24,490 --> 00:39:26,840 So in some sense, you say why? 585 00:39:26,840 --> 00:39:29,950 Well, it's because we're keeping pressure constant and 586 00:39:29,950 --> 00:39:33,290 we're raising the temperature and essentially the effect of 587 00:39:33,290 --> 00:39:35,040 the entropy is more important. 588 00:39:35,040 --> 00:39:37,210 Is going up. 589 00:39:37,210 --> 00:39:41,490 You could see it just by saying well, G is H minus TS. 590 00:39:41,490 --> 00:39:46,900 And as you raise temperature, the product TS is increasing. 591 00:39:46,900 --> 00:39:48,980 Which means the G is decreasing, because 592 00:39:48,980 --> 00:39:50,230 it's H minus TS. 593 00:39:52,750 --> 00:39:54,340 So it's always a negative slope. 594 00:39:54,340 --> 00:39:56,670 Let's look at it in the different phases. 595 00:39:56,670 --> 00:40:00,470 So we have entropy of the gas, that's greater than the 596 00:40:00,470 --> 00:40:01,550 entropy of the liquid. 597 00:40:01,550 --> 00:40:05,930 Which is greater than the entropy of the solid. 598 00:40:05,930 --> 00:40:09,680 So that immediately tells us the relative slopes for the 599 00:40:09,680 --> 00:40:11,870 three phases. 600 00:40:11,870 --> 00:40:25,720 So d mu gas / dT at constant pressure is minus S gas. d mu 601 00:40:25,720 --> 00:40:42,720 liquid / dT is minus S of the liquid. d mu solid / dT is 602 00:40:42,720 --> 00:40:46,170 minus S of the solid. 603 00:40:46,170 --> 00:40:48,680 And the magnitude is much bigger in the gas than the 604 00:40:48,680 --> 00:40:50,590 liquid than the solid. 605 00:40:50,590 --> 00:40:54,500 And so the same goes for the slope. 606 00:40:54,500 --> 00:41:10,880 So now let's sketch this. 607 00:41:10,880 --> 00:41:12,300 There's mu. 608 00:41:12,300 --> 00:41:13,890 And there's temperature. 609 00:41:13,890 --> 00:41:18,380 So what we've seen is for the gas, first of all, the slope's 610 00:41:18,380 --> 00:41:20,000 always negative. 611 00:41:20,000 --> 00:41:22,590 And it's steepest by far, or steepest 612 00:41:22,590 --> 00:41:27,760 substantially, for the gas. 613 00:41:27,760 --> 00:41:35,590 Next steepest for the liquid. 614 00:41:35,590 --> 00:41:46,340 And least steep of all for the solid. 615 00:41:46,340 --> 00:41:48,020 Now, there can be variation. 616 00:41:48,020 --> 00:41:49,635 Because, of course, these do change as 617 00:41:49,635 --> 00:41:51,830 a function of pressure. 618 00:41:51,830 --> 00:41:55,060 But this, certainly the slope, the magnitudes of the slopes 619 00:41:55,060 --> 00:41:56,720 are always going to be like this. 620 00:41:56,720 --> 00:41:58,600 What could change as a function of pressure is where 621 00:41:58,600 --> 00:42:01,340 these happen to lie. 622 00:42:01,340 --> 00:42:02,840 But this is certainly typical. 623 00:42:02,840 --> 00:42:04,000 So what does it mean? 624 00:42:04,000 --> 00:42:09,140 So, it means, remember, the stuff is always going to be in 625 00:42:09,140 --> 00:42:16,360 the phase with the lowest value of mu. 626 00:42:16,360 --> 00:42:20,780 So at low temperature, down here, that's where the solid 627 00:42:20,780 --> 00:42:32,950 phase has the lowest chemical potential. 628 00:42:32,950 --> 00:42:34,200 Now, let's keep going up. 629 00:42:34,200 --> 00:42:35,320 Eventually they cross. 630 00:42:35,320 --> 00:42:40,130 This is where, for this particular pressure, this is 631 00:42:40,130 --> 00:42:42,450 where the liquid and the solid are in equilibrium. 632 00:42:42,450 --> 00:42:44,260 In other words, we've started, we've got the solid and we're 633 00:42:44,260 --> 00:42:44,910 raising the pressure. 634 00:42:44,910 --> 00:42:46,600 Now it's going to melt. 635 00:42:46,600 --> 00:42:51,930 Here's our melting point at this particular pressure. 636 00:42:51,930 --> 00:42:54,750 And now we've got, if we keep raising the temperature, we're 637 00:42:54,750 --> 00:42:58,520 no longer in equilibrium, and we'll be in the one phase part 638 00:42:58,520 --> 00:42:59,430 of the phase diagram. 639 00:42:59,430 --> 00:43:04,530 It's just the liquid. 640 00:43:04,530 --> 00:43:07,950 Because that's the lowest chemical potential. 641 00:43:07,950 --> 00:43:09,930 Let's keep raising the temperature. 642 00:43:09,930 --> 00:43:12,140 Down here, of course, the chemical potential of the gas 643 00:43:12,140 --> 00:43:14,710 is way up there, right? 644 00:43:14,710 --> 00:43:17,350 So it's not going to come into play now, though they're going 645 00:43:17,350 --> 00:43:18,750 to be in equilibrium. 646 00:43:18,750 --> 00:43:24,400 Here is the boiling point. 647 00:43:24,400 --> 00:43:26,250 So now let's keep raising the temperature. 648 00:43:26,250 --> 00:43:34,210 And we'll be in just the gas part of the curve. 649 00:43:34,210 --> 00:43:40,440 And of course that'll go on forever. 650 00:43:40,440 --> 00:43:43,520 So that can just help us understand, in addition to the 651 00:43:43,520 --> 00:43:47,955 slopes, the positions, the relative positions, of where 652 00:43:47,955 --> 00:43:50,710 these equilibria occur. 653 00:43:50,710 --> 00:43:54,710 Now if we start changing the pressure in particular, we 654 00:43:54,710 --> 00:43:55,890 could move this. 655 00:43:55,890 --> 00:43:58,230 So that you could have a sublimation point where the 656 00:43:58,230 --> 00:44:00,790 gas and the solid are in equilibrium because this has 657 00:44:00,790 --> 00:44:06,930 moved over to here. 658 00:44:06,930 --> 00:44:10,600 Any questions about the overall structure of the phase 659 00:44:10,600 --> 00:44:15,400 diagrams and these phase equilibria? 660 00:44:15,400 --> 00:44:20,230 OK, let me just end with a few comments about one part of the 661 00:44:20,230 --> 00:44:26,370 phase diagram that we haven't talked about yet. 662 00:44:26,370 --> 00:44:27,580 This. 663 00:44:27,580 --> 00:44:31,250 What's going on there? 664 00:44:31,250 --> 00:44:34,560 If we were to change, of course, I've got a finite 665 00:44:34,560 --> 00:44:37,030 amount of space on the blackboard, so I haven't drawn 666 00:44:37,030 --> 00:44:39,700 T and p out to infinity. 667 00:44:39,700 --> 00:44:46,190 But if I could, this line would keep going forever. 668 00:44:46,190 --> 00:44:48,900 There would always be some pressure and temperature at 669 00:44:48,900 --> 00:44:50,830 which you would have equilibrium between two 670 00:44:50,830 --> 00:44:54,790 different, distinct solid and liquid phases. 671 00:44:54,790 --> 00:44:59,230 Turns out, that's not true for the gas and the liquid. 672 00:44:59,230 --> 00:45:00,740 And there are a bunch of ways to see this. 673 00:45:00,740 --> 00:45:04,100 But one is you could think of the symmetry difference. 674 00:45:04,100 --> 00:45:08,320 When you go from a crystal, a crystal and solid, to a liquid 675 00:45:08,320 --> 00:45:11,270 or a gas, there's no question that you've changed phase. 676 00:45:11,270 --> 00:45:14,150 Because there's a symmetry in the solid that's no longer 677 00:45:14,150 --> 00:45:18,080 maintained in the isotropic liquid or gas. 678 00:45:18,080 --> 00:45:20,690 So clearly there's never going to be a point where these two 679 00:45:20,690 --> 00:45:23,470 phases become the same thing. 680 00:45:23,470 --> 00:45:25,860 Become somehow indistinguishable. 681 00:45:25,860 --> 00:45:29,430 But the gas and the liquid, I don't know, what's the 682 00:45:29,430 --> 00:45:30,260 difference between them? 683 00:45:30,260 --> 00:45:31,930 The density, right? 684 00:45:31,930 --> 00:45:33,200 The gas is more dense. 685 00:45:33,200 --> 00:45:34,930 It's condensed. 686 00:45:34,930 --> 00:45:36,370 Sorry, the liquid is condensed. 687 00:45:36,370 --> 00:45:39,570 The gas isn't. 688 00:45:39,570 --> 00:45:44,400 But that starts to change when you adjust the pressure and 689 00:45:44,400 --> 00:45:46,690 temperature accordingly. 690 00:45:46,690 --> 00:45:48,930 And you can sort of see it coming. 691 00:45:48,930 --> 00:45:54,120 What'll happen is, you change the temperature, the molar 692 00:45:54,120 --> 00:45:57,430 volumes of the liquid and the gas start to get 693 00:45:57,430 --> 00:45:59,290 equal to each other. 694 00:45:59,290 --> 00:46:01,730 That's what'll end up happening. 695 00:46:01,730 --> 00:46:05,080 And at that point, what's the difference between them? 696 00:46:05,080 --> 00:46:06,230 There is no difference. 697 00:46:06,230 --> 00:46:07,840 There's no symmetry difference. 698 00:46:07,840 --> 00:46:11,380 So you can actually, instead of going through a phase 699 00:46:11,380 --> 00:46:14,220 transition, where there's a meniscus, you can see the top 700 00:46:14,220 --> 00:46:15,460 of the liquid when you heat it up. 701 00:46:15,460 --> 00:46:19,650 There's boiling and then there's the gas. 702 00:46:19,650 --> 00:46:23,630 You can actually go from the gas, let's go the other way, 703 00:46:23,630 --> 00:46:28,400 from the liquid to the gas phase, without ever boiling. 704 00:46:28,400 --> 00:46:30,790 If you change the pressure as well. 705 00:46:30,790 --> 00:46:33,470 So that you can actually walk all the way around the 706 00:46:33,470 --> 00:46:34,530 critical point. 707 00:46:34,530 --> 00:46:37,450 What happens, is up here you don't have two distinguishable 708 00:46:37,450 --> 00:46:39,540 phases any more. 709 00:46:39,540 --> 00:46:44,510 And actually, material out at this region has a bunch of 710 00:46:44,510 --> 00:46:45,980 very unusual behaviors. 711 00:46:45,980 --> 00:46:47,400 Very useful behaviors, actually. 712 00:46:47,400 --> 00:46:50,390 Chemical behavior can be really very, very, different 713 00:46:50,390 --> 00:46:52,100 in what's called supercritical, the 714 00:46:52,100 --> 00:46:56,470 supercritical region beyond the critical point. 715 00:46:56,470 --> 00:46:58,670 Water behaves really unusually there. 716 00:46:58,670 --> 00:47:02,440 Or it turns out, in that region, it turns out to be a 717 00:47:02,440 --> 00:47:05,620 terrific solvent for organic stuff. 718 00:47:05,620 --> 00:47:07,490 Organic molecules which normally don't 719 00:47:07,490 --> 00:47:09,690 dissolve well in water. 720 00:47:09,690 --> 00:47:13,440 And it doesn't dissolve salts very well. 721 00:47:13,440 --> 00:47:15,630 That's pretty unusual. 722 00:47:15,630 --> 00:47:20,100 Carbon dioxide, CO2, turns out to be a pretty good solvent 723 00:47:20,100 --> 00:47:22,290 for organics in the supercritical region. 724 00:47:22,290 --> 00:47:25,720 That's what dry cleaning is. 725 00:47:25,720 --> 00:47:30,940 Very nice, because know CO2, apart from its slow effect on 726 00:47:30,940 --> 00:47:32,730 our atmosphere, it's not poisonous. 727 00:47:32,730 --> 00:47:33,650 It's not toxic. 728 00:47:33,650 --> 00:47:38,880 So you can do dry cleaning without organic solvents. 729 00:47:38,880 --> 00:47:41,520 All sorts of unusual behavior happens above 730 00:47:41,520 --> 00:47:42,200 the critical point. 731 00:47:42,200 --> 00:47:44,070 Beyond the critical point. 732 00:47:44,070 --> 00:47:48,900 There's also a whole range of behavior at 733 00:47:48,900 --> 00:47:49,830 the critical point. 734 00:47:49,830 --> 00:47:53,490 Right at the critical point that's extremely unusual. 735 00:47:53,490 --> 00:47:56,535 That you could think of as a consequence of the fact that 736 00:47:56,535 --> 00:47:58,600 the molar volumes are getting to be almost 737 00:47:58,600 --> 00:48:00,980 equal to each other. 738 00:48:00,980 --> 00:48:03,420 That's unusual in a lot of ways. 739 00:48:03,420 --> 00:48:06,950 And what ends up happening is, you can see if you go online, 740 00:48:06,950 --> 00:48:11,040 actually, if you just Google critical point, you can see 741 00:48:11,040 --> 00:48:13,730 amazing movies of stuff that's at the critical point. 742 00:48:13,730 --> 00:48:16,230 Because what happens is, so you've got the liquid. 743 00:48:16,230 --> 00:48:18,290 And you've got a meniscus, and you've got the gas. 744 00:48:18,290 --> 00:48:18,990 And you can tell. 745 00:48:18,990 --> 00:48:19,700 You look at it. 746 00:48:19,700 --> 00:48:20,430 There's the liquid. 747 00:48:20,430 --> 00:48:22,070 There's the gas. 748 00:48:22,070 --> 00:48:26,200 But if you go real close to the critical point, the stuff 749 00:48:26,200 --> 00:48:27,990 doesn't really know what phase it's in. 750 00:48:27,990 --> 00:48:29,640 Because it's losing the distinction 751 00:48:29,640 --> 00:48:30,840 between the two phases. 752 00:48:30,840 --> 00:48:33,750 You start seeing globules of liquid up 753 00:48:33,750 --> 00:48:34,970 there in the gas phase. 754 00:48:34,970 --> 00:48:37,690 And globules of the gas down in the liquid. 755 00:48:37,690 --> 00:48:39,960 Because their densities - I mean, the only reason the 756 00:48:39,960 --> 00:48:41,580 liquid's at the bottom normally is because the 757 00:48:41,580 --> 00:48:43,500 density is higher. 758 00:48:43,500 --> 00:48:48,430 If that stops being the case, it has no more right to occupy 759 00:48:48,430 --> 00:48:51,050 the bottom part than the gas. 760 00:48:51,050 --> 00:48:54,580 And so they sort of start interchanging. 761 00:48:54,580 --> 00:48:57,200 And if you get even closer, of course, what eventually 762 00:48:57,200 --> 00:48:59,190 happens is, there's just one phase. 763 00:48:59,190 --> 00:49:02,450 You don't see distinct meniscus's any more. 764 00:49:02,450 --> 00:49:08,460 But right near there, you see enormous fluctuations in the 765 00:49:08,460 --> 00:49:11,780 local density and in the location. 766 00:49:11,780 --> 00:49:14,870 The physical locations of different phases. 767 00:49:14,870 --> 00:49:17,510 And we'll see that come in another context later. 768 00:49:17,510 --> 00:49:20,520 Because when we talk about two phase equilibria solutions. 769 00:49:20,520 --> 00:49:24,330 You get this with you oil and water, you get critical 770 00:49:24,330 --> 00:49:27,970 mixtures also, where right at the special point, the 771 00:49:27,970 --> 00:49:31,650 meniscus goes away and the oil and water start interchanging. 772 00:49:31,650 --> 00:49:36,210 And in those situations, a tiny amount of force can give 773 00:49:36,210 --> 00:49:38,480 you a really big response. 774 00:49:38,480 --> 00:49:41,650 Because the two things are just teetering in the balance. 775 00:49:41,650 --> 00:49:43,620 So a little bit of force pushes things 776 00:49:43,620 --> 00:49:45,140 one way or the other. 777 00:49:45,140 --> 00:49:47,000 That's actually useful in a whole bunch 778 00:49:47,000 --> 00:49:47,820 of different cases. 779 00:49:47,820 --> 00:49:51,770 Because that critical behavior happens in all kinds of phase 780 00:49:51,770 --> 00:49:52,470 transitions. 781 00:49:52,470 --> 00:49:54,570 Magnetic phase transitions. 782 00:49:54,570 --> 00:49:55,960 Ferroelectric phase transitions. 783 00:49:55,960 --> 00:49:59,430 And in those situations, a tiny little magnetic field, 784 00:49:59,430 --> 00:50:02,170 for example, gives you a really big change in the 785 00:50:02,170 --> 00:50:02,930 magnetization. 786 00:50:02,930 --> 00:50:05,440 If the alignment of spins. 787 00:50:05,440 --> 00:50:06,460 And that's very useful. 788 00:50:06,460 --> 00:50:09,000 You can get a very big material response and change. 789 00:50:09,000 --> 00:50:12,630 You can switch something, for example, with a really small 790 00:50:12,630 --> 00:50:14,690 external field. 791 00:50:14,690 --> 00:50:18,060 And in these cases, you can change the density really a 792 00:50:18,060 --> 00:50:21,990 lot by only a small change in pressure. 793 00:50:21,990 --> 00:50:26,040 OK, next time we'll extend the Clapeyron equation and also go 794 00:50:26,040 --> 00:50:27,800 to two component systems. 795 00:50:27,800 --> 00:50:29,840 So you actually, Professor Blendi will take 796 00:50:29,840 --> 00:50:31,260 over again on Friday. 797 00:50:31,260 --> 00:50:33,100 And we'll see you then.