1 00:00:00,030 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,810 Commons license. 3 00:00:03,810 --> 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 --> 00:00:17,180 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,180 --> 00:00:20,720 ocw.mit.edu. 8 00:00:20,720 --> 00:00:23,250 PROFESSOR: So. 9 00:00:23,250 --> 00:00:26,180 In the meantime, you've started looking at two phase 10 00:00:26,180 --> 00:00:27,300 equilibrium. 11 00:00:27,300 --> 00:00:29,970 So now we're starting to look at mixtures. 12 00:00:29,970 --> 00:00:32,560 And so now we have more than one constituent. 13 00:00:32,560 --> 00:00:35,960 And we have more than one phase present. 14 00:00:35,960 --> 00:00:36,930 Right? 15 00:00:36,930 --> 00:00:45,750 So you've started to look at things that look like this, 16 00:00:45,750 --> 00:00:51,800 where you've got, let's say, two components. 17 00:00:51,800 --> 00:01:02,680 Both in the gas phase. 18 00:01:02,680 --> 00:01:05,580 And now to try to figure out what the phase 19 00:01:05,580 --> 00:01:06,870 equilibria look like. 20 00:01:06,870 --> 00:01:09,530 Of course it's now a little bit more complicated than what 21 00:01:09,530 --> 00:01:11,070 you went through before, where you can get pressure 22 00:01:11,070 --> 00:01:13,910 temperature phase diagrams with just a single component. 23 00:01:13,910 --> 00:01:18,050 Now we want to worry about what's the composition. 24 00:01:18,050 --> 00:01:19,620 Of each of the components. 25 00:01:19,620 --> 00:01:20,980 In each of the phases. 26 00:01:20,980 --> 00:01:24,110 And what's the temperature and the pressure. 27 00:01:24,110 --> 00:01:26,930 Total and partial pressures and all of that. 28 00:01:26,930 --> 00:01:31,430 So you can really figure out everything about both phases. 29 00:01:31,430 --> 00:01:34,030 And there are all sorts of important reasons to do that, 30 00:01:34,030 --> 00:01:36,820 obviously lots of chemistry happens in liquid mixtures. 31 00:01:36,820 --> 00:01:37,900 Some in gas mixtures. 32 00:01:37,900 --> 00:01:40,010 Some where they're in equilibrium. 33 00:01:40,010 --> 00:01:41,780 All sorts of chemical processes. 34 00:01:41,780 --> 00:01:45,100 Distillation, for example, takes advantage of the 35 00:01:45,100 --> 00:01:49,580 properties of liquid and gas mixtures. 36 00:01:49,580 --> 00:01:52,400 Where one of them might be richer, will be richer, and 37 00:01:52,400 --> 00:01:54,820 the more volatile of the components. 38 00:01:54,820 --> 00:01:58,270 That can be used as a basis for purification. 39 00:01:58,270 --> 00:02:01,120 You mix ethanol and water together so you've got a 40 00:02:01,120 --> 00:02:03,060 liquid with a certain composition of each. 41 00:02:03,060 --> 00:02:05,750 The gas is going to be richer and the more volatile of the 42 00:02:05,750 --> 00:02:07,400 two, the ethanol. 43 00:02:07,400 --> 00:02:10,420 So in a distillation, where you put things up in the gas, 44 00:02:10,420 --> 00:02:13,150 more of the ethanol comes up. 45 00:02:13,150 --> 00:02:15,190 You could then collect that gas, right? 46 00:02:15,190 --> 00:02:18,330 And re-condense it, and make a new liquid. 47 00:02:18,330 --> 00:02:20,410 Which is much richer in ethanol than the original 48 00:02:20,410 --> 00:02:22,750 liquid was. 49 00:02:22,750 --> 00:02:25,410 Then you could make, then you could put some of them up into 50 00:02:25,410 --> 00:02:26,590 the gas phase. 51 00:02:26,590 --> 00:02:30,230 Where it will be still richer in ethanol. 52 00:02:30,230 --> 00:02:32,250 And then you could collect that and repeat the process. 53 00:02:32,250 --> 00:02:36,740 So the point is that properties of liquid gas, 54 00:02:36,740 --> 00:02:38,920 two-component or multi-component mixtures like 55 00:02:38,920 --> 00:02:40,820 this can be exploited. 56 00:02:40,820 --> 00:02:44,180 Basically, the different volatilities of the different 57 00:02:44,180 --> 00:02:48,510 components can be exploited for things like purification. 58 00:02:48,510 --> 00:02:52,100 Also if you want to calculate chemical equilibria in the 59 00:02:52,100 --> 00:02:53,820 liquid and gas phase, of course, now you've seen 60 00:02:53,820 --> 00:02:57,100 chemical equilibrium, so the amount of reaction depends on 61 00:02:57,100 --> 00:02:58,910 the composition. 62 00:02:58,910 --> 00:03:01,990 So of course if you want reactions to go, then this 63 00:03:01,990 --> 00:03:05,110 also can be exploited by looking at which phase might 64 00:03:05,110 --> 00:03:07,460 be richer in one reactant or another. 65 00:03:07,460 --> 00:03:10,220 And thereby pushing the equilibrium toward one 66 00:03:10,220 --> 00:03:12,120 direction or the other. 67 00:03:12,120 --> 00:03:12,950 OK. 68 00:03:12,950 --> 00:03:19,290 So. we've got some total temperature and pressure. 69 00:03:19,290 --> 00:03:29,350 And we have compositions. 70 00:03:29,350 --> 00:03:34,020 So in the gas phase, we've got mole fractions yA and yB. 71 00:03:34,020 --> 00:03:41,580 In the liquid phase we've got mole fractions xA and xB. 72 00:03:49,020 --> 00:03:51,300 So that's our system. 73 00:03:51,300 --> 00:03:54,190 One of the things that you established last time is that, 74 00:03:54,190 --> 00:03:58,010 so there are the total number of variables including the 75 00:03:58,010 --> 00:04:00,070 temperature and the pressure. 76 00:04:00,070 --> 00:04:03,950 And let's say the mole fraction of A in each of the 77 00:04:03,950 --> 00:04:06,670 liquid and gas phases, right? 78 00:04:06,670 --> 00:04:07,870 But then there are constraints. 79 00:04:07,870 --> 00:04:09,660 Because the chemical potentials have 80 00:04:09,660 --> 00:04:11,330 to be equal, right? 81 00:04:11,330 --> 00:04:13,320 Chemical potential of A has to be equal in 82 00:04:13,320 --> 00:04:14,010 the liquid and gas. 83 00:04:14,010 --> 00:04:18,270 Same with B. Those two constraints reduce the number 84 00:04:18,270 --> 00:04:20,840 of independent variables. 85 00:04:20,840 --> 00:04:23,590 So there'll be two in this case rather than four 86 00:04:23,590 --> 00:04:25,750 independent variables. 87 00:04:25,750 --> 00:04:29,380 If you control those, then everything else will follow. 88 00:04:29,380 --> 00:04:33,050 What that means is if you've got a, if you control, if you 89 00:04:33,050 --> 00:04:36,960 fix the temperature and the total pressure, everything 90 00:04:36,960 --> 00:04:39,650 else should be determinable. 91 00:04:39,650 --> 00:04:43,200 No more free variables. 92 00:04:43,200 --> 00:04:48,150 And then, what you saw is that in simple or ideal liquid 93 00:04:48,150 --> 00:04:57,800 mixtures, a result called Raoult's law would hold. 94 00:04:57,800 --> 00:05:01,090 Which just says that the partial pressure of A is equal 95 00:05:01,090 --> 00:05:10,150 to the mole fraction of A in the liquid times the pressure 96 00:05:10,150 --> 00:05:12,830 of pure A over the liquid. 97 00:05:12,830 --> 00:05:18,140 And so what this gives you is a diagram 98 00:05:18,140 --> 00:05:19,080 that looks like this. 99 00:05:19,080 --> 00:05:24,490 If we plot this versus xB, this is mole fraction of B in 100 00:05:24,490 --> 00:05:28,540 the liquid going from zero to one. 101 00:05:28,540 --> 00:05:39,050 Then we could construct a diagram of this sort. 102 00:05:39,050 --> 00:05:46,440 So this is the total pressure of A and B. The partial 103 00:05:46,440 --> 00:05:53,490 pressures are given by these lines. 104 00:05:53,490 --> 00:05:59,390 So this is our pA star and pB star. 105 00:05:59,390 --> 00:06:04,060 The pressures over the pure liquid A and B at the limits 106 00:06:04,060 --> 00:06:07,830 of mole fraction of B being zero and one. 107 00:06:07,830 --> 00:06:11,050 So in this situation, for example, A is the more 108 00:06:11,050 --> 00:06:12,640 volatile of the components. 109 00:06:12,640 --> 00:06:15,770 So it's partial pressure over its pure liquid. 110 00:06:15,770 --> 00:06:17,580 At this temperature. 111 00:06:17,580 --> 00:06:20,280 Is higher than the partial pressure of B 112 00:06:20,280 --> 00:06:21,410 over its pure liquid. 113 00:06:21,410 --> 00:06:24,440 A would be the ethanol, for example and B the water in 114 00:06:24,440 --> 00:06:30,140 that mixture. 115 00:06:30,140 --> 00:06:33,320 OK. 116 00:06:33,320 --> 00:06:38,340 Then you started looking at both the gas and the liquid 117 00:06:38,340 --> 00:06:40,600 phase in the same diagram. 118 00:06:40,600 --> 00:06:44,520 So this is the mole fraction of the liquid. 119 00:06:44,520 --> 00:06:47,810 If you look and see, well, OK now we should be able to 120 00:06:47,810 --> 00:06:51,310 determine the mole fraction in the gas as well. 121 00:06:51,310 --> 00:06:54,520 Again, if we note total temperature and pressure, 122 00:06:54,520 --> 00:06:56,900 everything else must follow. 123 00:06:56,900 --> 00:07:05,090 And so, you saw this worked out. 124 00:07:05,090 --> 00:07:15,060 Relation between p and yA, for example. 125 00:07:15,060 --> 00:07:23,940 The result was p is pA star times pB star over pA star 126 00:07:23,940 --> 00:07:31,250 plus pB star minus pA star times yA. 127 00:07:31,250 --> 00:07:34,260 And the point here is that unlike this case, where you 128 00:07:34,260 --> 00:07:38,680 have a linear relationship, the relationship between the 129 00:07:38,680 --> 00:07:42,730 pressure and the liquid mole fraction isn't linear. 130 00:07:42,730 --> 00:07:45,770 We can still plot it, of course. 131 00:07:45,770 --> 00:07:51,240 So if we do that, then we end up with a diagram that looks 132 00:07:51,240 --> 00:08:02,320 like the following. 133 00:08:02,320 --> 00:08:12,140 Now I'm going to keep both mole fractions, xB and yB, 134 00:08:12,140 --> 00:08:14,830 I've got some total pressure. 135 00:08:14,830 --> 00:08:21,720 I still have my linear relationship. 136 00:08:21,720 --> 00:08:30,610 And then I have a non-linear relationship between the 137 00:08:30,610 --> 00:08:36,260 pressure and the mole fraction in the gas phase. 138 00:08:36,260 --> 00:08:49,180 So let's just fill this in. 139 00:08:49,180 --> 00:08:55,260 Here is pA star still. 140 00:08:55,260 --> 00:08:56,220 Here's pB star. 141 00:08:56,220 --> 00:08:59,060 Of course, at the limits they're still, both mole 142 00:08:59,060 --> 00:09:05,220 fractions they're zero and one. 143 00:09:05,220 --> 00:09:07,120 OK. 144 00:09:07,120 --> 00:09:10,290 I believe this is this is where you ended up at the end 145 00:09:10,290 --> 00:09:11,440 of the last lecture. 146 00:09:11,440 --> 00:09:14,740 But it's probably not so clear exactly how you read 147 00:09:14,740 --> 00:09:15,640 something like this. 148 00:09:15,640 --> 00:09:16,650 And use it. 149 00:09:16,650 --> 00:09:18,200 It's extremely useful. 150 00:09:18,200 --> 00:09:23,340 You just have to kind of learn how to follow what happens in 151 00:09:23,340 --> 00:09:24,500 a diagram like this. 152 00:09:24,500 --> 00:09:26,640 And that's what I want to spend some of today doing. 153 00:09:26,640 --> 00:09:30,940 Is just, walking through what's happening physically, 154 00:09:30,940 --> 00:09:32,960 with a container with a mixture of the two. 155 00:09:32,960 --> 00:09:35,590 And how does that correspond to what gets read off the 156 00:09:35,590 --> 00:09:39,630 diagram under different conditions. 157 00:09:39,630 --> 00:09:42,500 So. 158 00:09:42,500 --> 00:09:55,660 Let's just start somewhere on a phase diagram like this. 159 00:09:55,660 --> 00:10:03,310 Let's start up here at some point one, so we're in the 160 00:10:03,310 --> 00:10:07,050 pure - well, not pure, you're in the all liquid phase. 161 00:10:07,050 --> 00:10:08,120 It's still a mixture. 162 00:10:08,120 --> 00:10:20,000 It's not a pure substance. pA star, pB star. 163 00:10:20,000 --> 00:10:21,430 There's the gas phase. 164 00:10:21,430 --> 00:10:33,460 So, if we start at one, and now 165 00:10:33,460 --> 00:10:34,950 there's some total pressure. 166 00:10:34,950 --> 00:10:37,640 And now we're going to reduce it. 167 00:10:37,640 --> 00:10:38,440 What happens? 168 00:10:38,440 --> 00:10:42,190 We start with a pure - with an all-liquid mixture. 169 00:10:42,190 --> 00:10:43,790 No gas. 170 00:10:43,790 --> 00:10:46,890 And now we're going to bring down the pressure. 171 00:10:46,890 --> 00:10:52,270 Allowing some of the liquid to go up into the gas phase. 172 00:10:52,270 --> 00:10:56,130 So, we can do that. 173 00:10:56,130 --> 00:11:02,530 And once we reach point two, then we find 174 00:11:02,530 --> 00:11:04,290 a coexistence curve. 175 00:11:04,290 --> 00:11:09,190 Now the liquid and gas are going to coexist. 176 00:11:09,190 --> 00:11:11,020 So this is the liquid phase. 177 00:11:11,020 --> 00:11:17,270 And that means that this must be xB. 178 00:11:17,270 --> 00:11:21,490 And it's xB at one, but it's also xB at two, and I want to 179 00:11:21,490 --> 00:11:24,410 emphasize that. 180 00:11:24,410 --> 00:11:30,110 So let's put our pressure for two. 181 00:11:30,110 --> 00:11:38,490 And if we go over here, this is telling us about the mole 182 00:11:38,490 --> 00:11:41,310 fraction in the gas phase. 183 00:11:41,310 --> 00:11:45,140 That's what these curves are, remember. 184 00:11:45,140 --> 00:11:49,030 So this is the one that's showing us the mole fraction 185 00:11:49,030 --> 00:11:49,550 in the liquid phase. 186 00:11:49,550 --> 00:11:52,800 This nonlinear one in the gas phase. 187 00:11:52,800 --> 00:11:57,050 So that means just reading off it, this is xB, that's the 188 00:11:57,050 --> 00:11:58,540 liquid mole fraction. 189 00:11:58,540 --> 00:12:01,220 Here's yB. 190 00:12:05,930 --> 00:12:07,750 The gas mole fraction. 191 00:12:07,750 --> 00:12:10,180 They're not the same, right, because of course the 192 00:12:10,180 --> 00:12:14,500 components have different volatility. 193 00:12:14,500 --> 00:12:17,110 A's more volatile. 194 00:12:17,110 --> 00:12:21,360 So that means that the mole fraction of B in the liquid 195 00:12:21,360 --> 00:12:24,900 phase is higher than the mole fraction of 196 00:12:24,900 --> 00:12:28,460 B in the gas phase. 197 00:12:28,460 --> 00:12:30,370 Because A is the more volatile component. 198 00:12:30,370 --> 00:12:33,990 So more, relatively more, of A, the mole fraction of A is 199 00:12:33,990 --> 00:12:35,730 going to be higher up in the gas phase. 200 00:12:35,730 --> 00:12:38,230 Which means the mole fraction of B is 201 00:12:38,230 --> 00:12:43,840 lower in the gas phase. 202 00:12:43,840 --> 00:12:58,490 So, yB less than xB if A is more volatile. 203 00:12:58,490 --> 00:13:04,140 OK, so now what's happening physically? 204 00:13:04,140 --> 00:13:09,910 Well, we started at a point where we only 205 00:13:09,910 --> 00:13:11,970 had the liquid present. 206 00:13:11,970 --> 00:13:23,790 So at our initial pressure, we just have all liquid. 207 00:13:23,790 --> 00:13:29,800 There's some xB at one. 208 00:13:29,800 --> 00:13:34,060 That's all there is, there isn't any gas yet. 209 00:13:34,060 --> 00:13:37,110 Now, what happened here? 210 00:13:37,110 --> 00:13:41,200 Well, now we lowered the pressure. 211 00:13:41,200 --> 00:13:45,710 So you could imagine, well, we made the box bigger. 212 00:13:45,710 --> 00:13:49,800 Now, if the liquid was under pressure, being squeezed by 213 00:13:49,800 --> 00:13:52,530 the box, right then you could make the box 214 00:13:52,530 --> 00:13:53,270 a little bit bigger. 215 00:13:53,270 --> 00:13:54,880 And there's still no gas. 216 00:13:54,880 --> 00:13:58,540 That's moving down like this. 217 00:13:58,540 --> 00:14:03,220 But then you get to a point where there's just barely any 218 00:14:03,220 --> 00:14:04,660 pressure on top of the liquid. 219 00:14:04,660 --> 00:14:06,680 And then you keep expanding the box. 220 00:14:06,680 --> 00:14:09,780 Now some gas is going to form. 221 00:14:09,780 --> 00:14:13,350 So now we're going to go to our case two. 222 00:14:13,350 --> 00:14:16,740 We've got a bigger box. 223 00:14:16,740 --> 00:14:18,855 And now, right around where this was, this 224 00:14:18,855 --> 00:14:22,610 is going to be liquid. 225 00:14:22,610 --> 00:14:26,850 And there's gas up here. 226 00:14:26,850 --> 00:14:30,330 So up here is yB at pressure two. 227 00:14:30,330 --> 00:14:35,900 Here's xB at pressure two. 228 00:14:35,900 --> 00:14:40,590 Liquid and gas. 229 00:14:40,590 --> 00:14:49,110 So that's where we are at point two here. 230 00:14:49,110 --> 00:14:52,760 Now, what happens if we keep going? 231 00:14:52,760 --> 00:14:56,560 Let's lower the pressure some more. 232 00:14:56,560 --> 00:15:00,610 Well, we can lower it and do this. 233 00:15:00,610 --> 00:15:04,020 But really if we want to see what's happening in each of 234 00:15:04,020 --> 00:15:08,490 the phases, we have to stay on the coexistence curves. 235 00:15:08,490 --> 00:15:12,260 Those are what tell us what the pressures are. 236 00:15:12,260 --> 00:15:13,520 What the partial pressure are going to be 237 00:15:13,520 --> 00:15:14,670 in each of the phases. 238 00:15:14,670 --> 00:15:18,240 In each of the two, in the liquid and the gas phases. 239 00:15:18,240 --> 00:15:20,930 So let's say we lower the pressure a little more. 240 00:15:20,930 --> 00:15:25,120 What's going to happen is, then we'll end up 241 00:15:25,120 --> 00:15:28,870 somewhere over here. 242 00:15:28,870 --> 00:15:32,270 In the liquid, and that'll correspond to something over 243 00:15:32,270 --> 00:15:33,220 here in the gas. 244 00:15:33,220 --> 00:15:37,980 So here's three. 245 00:15:37,980 --> 00:15:46,570 So now we're going to have, that's going to be xB at 246 00:15:46,570 --> 00:15:48,320 pressure three. 247 00:15:48,320 --> 00:16:01,800 And over here is going to be yB at pressure three. 248 00:16:01,800 --> 00:16:04,840 And all we've done, of course, is we've just 249 00:16:04,840 --> 00:16:07,090 expanded this further. 250 00:16:07,090 --> 00:16:12,250 So now we've got a still taller box. 251 00:16:12,250 --> 00:16:14,960 And the liquid is going to be a little lower because some of 252 00:16:14,960 --> 00:16:21,170 it has evaporated, formed the gas phase. 253 00:16:21,170 --> 00:16:23,710 So here's xB at three. 254 00:16:23,710 --> 00:16:32,530 Here's yB at three, here's our gas phase. 255 00:16:32,530 --> 00:16:38,740 Now we could decrease even further. 256 00:16:38,740 --> 00:16:42,120 And this is the sort of thing that you maybe 257 00:16:42,120 --> 00:16:42,980 can't do in real life. 258 00:16:42,980 --> 00:16:45,590 But I can do on a blackboard. 259 00:16:45,590 --> 00:16:49,190 I'm going to give myself more room on this curve, to finish 260 00:16:49,190 --> 00:16:59,810 this illustration. 261 00:16:59,810 --> 00:17:01,810 There. 262 00:17:01,810 --> 00:17:02,990 Beautiful. 263 00:17:02,990 --> 00:17:06,030 So now we can lower a little bit further, and what I want 264 00:17:06,030 --> 00:17:09,450 to illustrate is, if we keep going down, eventually we get 265 00:17:09,450 --> 00:17:14,980 to a pressure where now if we look over in the gas phase, 266 00:17:14,980 --> 00:17:17,860 we're at the same pressure, mole fraction that we had 267 00:17:17,860 --> 00:17:21,340 originally in the liquid phase. 268 00:17:21,340 --> 00:17:27,380 So let's make four even lower pressure. 269 00:17:27,380 --> 00:17:29,990 What does that mean? 270 00:17:29,990 --> 00:17:35,560 What it means is, we're running out of liquid. 271 00:17:35,560 --> 00:17:39,770 So what's supposed to happen is A is the 272 00:17:39,770 --> 00:17:41,620 more volatile component. 273 00:17:41,620 --> 00:17:47,580 So as we start opening up some room for gas to form, you get 274 00:17:47,580 --> 00:17:50,430 more of A in the gas phase. 275 00:17:50,430 --> 00:17:56,390 But of course, and the liquid is richer in B. But of course, 276 00:17:56,390 --> 00:17:58,020 eventually you run out of liquid. 277 00:17:58,020 --> 00:18:05,810 You make the box pretty big, and you run out, or you have 278 00:18:05,810 --> 00:18:08,670 the very last drop of liquid. 279 00:18:08,670 --> 00:18:12,150 So what's the mole fraction of B in the gas phase? 280 00:18:12,150 --> 00:18:14,640 It has to be the same as what it started in 281 00:18:14,640 --> 00:18:15,370 in the liquid phase. 282 00:18:15,370 --> 00:18:17,720 Because after all the total number of moles of A and B 283 00:18:17,720 --> 00:18:18,950 hasn't changed any. 284 00:18:18,950 --> 00:18:21,900 So if you take them all from the liquid and put them all up 285 00:18:21,900 --> 00:18:25,080 into the gas phase, it must be the same. 286 00:18:25,080 --> 00:18:30,350 So yB of four. 287 00:18:30,350 --> 00:18:38,730 Once you just have the last drop. 288 00:18:38,730 --> 00:18:45,250 So then yB of four is basically equal to xB of one. 289 00:18:45,250 --> 00:18:49,110 Because everything's now up in the gas phase. 290 00:18:49,110 --> 00:18:56,390 So in principle, there's still a tiny, tiny bit of xB at 291 00:18:56,390 --> 00:19:00,030 pressure four. 292 00:19:00,030 --> 00:19:01,500 Well, we could keep lowering the pressure. 293 00:19:01,500 --> 00:19:03,850 We could make the box a little bigger. 294 00:19:03,850 --> 00:19:06,240 Then the very last of the liquid is going to be gone. 295 00:19:06,240 --> 00:19:09,180 And what'll happen then is, we're all here. 296 00:19:09,180 --> 00:19:10,200 There's no more liquid. 297 00:19:10,200 --> 00:19:13,100 We're not going down on the coexistence curve any more. 298 00:19:13,100 --> 00:19:15,660 We don't have a liquid gas coexistence any more. 299 00:19:15,660 --> 00:19:17,640 We just have a gas phase. 300 00:19:17,640 --> 00:19:20,010 Of course, we can continue to lower the pressure. 301 00:19:20,010 --> 00:19:24,580 And then what we're doing is just going down here. 302 00:19:24,580 --> 00:19:26,370 So there's five. 303 00:19:26,370 --> 00:19:32,160 And five is the same as this only bigger. 304 00:19:32,160 --> 00:19:34,560 And so forth. 305 00:19:34,560 --> 00:19:40,050 OK, any questions about how this works? 306 00:19:40,050 --> 00:19:45,650 It's really important to just gain facility in reading these 307 00:19:45,650 --> 00:19:51,320 things and seeing, OK, what is it that this is telling you. 308 00:19:51,320 --> 00:19:54,730 And you can see it's not complicated to do it, but it 309 00:19:54,730 --> 00:19:58,880 takes a little bit of practice. 310 00:19:58,880 --> 00:19:59,880 OK. 311 00:19:59,880 --> 00:20:02,550 Now, of course, we could do exactly the same thing 312 00:20:02,550 --> 00:20:04,670 starting from the gas phase. 313 00:20:04,670 --> 00:20:06,830 And raising the pressure. 314 00:20:06,830 --> 00:20:10,020 And although you may anticipate that it's kind of 315 00:20:10,020 --> 00:20:12,560 pedantic, I really do want to illustrate something by it. 316 00:20:12,560 --> 00:20:14,960 So let me just imagine that we're going to do that. 317 00:20:14,960 --> 00:20:34,550 Let's start all in the gas phase. 318 00:20:34,550 --> 00:20:49,050 Up here's the liquid. pA star, pB star. 319 00:20:49,050 --> 00:20:55,480 And now let's start somewhere here. 320 00:20:55,480 --> 00:20:58,230 So we're down somewhere in the gas phase with some 321 00:20:58,230 --> 00:20:59,710 composition. 322 00:20:59,710 --> 00:21:02,030 So it's the same story, except now we're starting here. 323 00:21:02,030 --> 00:21:02,680 It's all gas. 324 00:21:02,680 --> 00:21:04,780 And we're going to start squeezing. 325 00:21:04,780 --> 00:21:06,780 We're increasing the pressure. 326 00:21:06,780 --> 00:21:14,530 And eventually here's one, will reach two, so of course 327 00:21:14,530 --> 00:21:18,580 here's our yB. 328 00:21:18,580 --> 00:21:23,590 We started with all gas, no liquid. 329 00:21:23,590 --> 00:21:24,980 So this is yB of one. 330 00:21:24,980 --> 00:21:27,880 It's the same as yB of two, I'm just raising the pressure 331 00:21:27,880 --> 00:21:32,530 enough to just reach the coexistence curve. 332 00:21:32,530 --> 00:21:40,810 And of course, out here tells us xB of two, right? 333 00:21:40,810 --> 00:21:42,180 So what is it saying? 334 00:21:42,180 --> 00:21:45,730 We've squeezed and started to form some liquid. 335 00:21:45,730 --> 00:21:50,650 And the liquid is richer in component B. Maybe it's 336 00:21:50,650 --> 00:21:51,590 ethanol water again. 337 00:21:51,590 --> 00:21:54,650 And we squeeze, and now we've got more water in the liquid 338 00:21:54,650 --> 00:21:56,280 phase than in the gas phase. 339 00:21:56,280 --> 00:21:58,100 Because water's the less volatile component. 340 00:21:58,100 --> 00:22:04,100 It's what's going to condense first. 341 00:22:04,100 --> 00:22:08,680 So the liquid is rich in the less volatile of the 342 00:22:08,680 --> 00:22:11,030 components. 343 00:22:11,030 --> 00:22:14,240 Now, obviously, we can continue in doing exactly the 344 00:22:14,240 --> 00:22:15,320 reverse of what I showed you. 345 00:22:15,320 --> 00:22:18,120 But all I want to really illustrate is, this is a 346 00:22:18,120 --> 00:22:24,200 strategy for purification of the less volatile component. 347 00:22:24,200 --> 00:22:28,880 Once you've done this, well now you've got some liquid. 348 00:22:28,880 --> 00:22:33,950 Now you could collect that liquid in a separate vessel. 349 00:22:33,950 --> 00:22:46,160 So let's collect the liquid mixture with xB of two. 350 00:22:46,160 --> 00:22:50,320 So it's got some mole fraction of B. So we've purified that. 351 00:22:50,320 --> 00:22:52,860 But now we're going to start, we've got pure liquid. 352 00:22:52,860 --> 00:22:54,880 Now let's make the vessel big. 353 00:22:54,880 --> 00:23:00,650 So it all goes into the gas phase. 354 00:23:00,650 --> 00:23:04,150 Then lower p. 355 00:23:04,150 --> 00:23:06,460 All gas. 356 00:23:06,460 --> 00:23:12,990 So we start with yB of three, which equals xB of two. 357 00:23:12,990 --> 00:23:16,290 In other words, it's the same mole fraction. 358 00:23:16,290 --> 00:23:32,510 So let's reconstruct that. 359 00:23:32,510 --> 00:23:35,810 So here's p of two. 360 00:23:35,810 --> 00:23:41,470 And now we're going to go to some new pressure. 361 00:23:41,470 --> 00:23:47,050 And the point is, now we're going to start, since the mole 362 00:23:47,050 --> 00:23:49,950 fraction in the gas phase that we're starting from is the 363 00:23:49,950 --> 00:23:51,550 same number as this was. 364 00:23:51,550 --> 00:23:54,460 So it's around here somewhere. 365 00:23:54,460 --> 00:24:01,490 That's yB of three equals xB of two. 366 00:24:01,490 --> 00:24:04,050 And we're down here. 367 00:24:04,050 --> 00:24:08,620 In other words, all we've done is make the container big 368 00:24:08,620 --> 00:24:11,820 enough so the pressure's low and it's all in the gas phase. 369 00:24:11,820 --> 00:24:15,180 That's all we have, is the gas. 370 00:24:15,180 --> 00:24:17,590 But the composition is whatever the composition is 371 00:24:17,590 --> 00:24:20,500 that we extracted here from the liquid. 372 00:24:20,500 --> 00:24:24,460 So this xB, which is the liquid mole fraction, is now 373 00:24:24,460 --> 00:24:26,290 yB, the gas mole fraction. 374 00:24:26,290 --> 00:24:27,900 Of course, the pressure is different. 375 00:24:27,900 --> 00:24:30,950 Lower than it was before. 376 00:24:30,950 --> 00:24:31,990 Great. 377 00:24:31,990 --> 00:24:34,460 Now let's increase. 378 00:24:34,460 --> 00:24:37,880 So here's three. 379 00:24:37,880 --> 00:24:41,840 And now let's increase the pressure to four. 380 00:24:41,840 --> 00:24:45,700 And of course what happens, now we've got coexistence. 381 00:24:45,700 --> 00:24:48,630 So here's liquid. 382 00:24:48,630 --> 00:24:49,940 Here's gas. 383 00:24:49,940 --> 00:24:57,180 So, now we're over here again. 384 00:24:57,180 --> 00:25:02,870 There's xB at pressure four. 385 00:25:02,870 --> 00:25:10,850 Pure still in component B. We can repeat the same procedure. 386 00:25:10,850 --> 00:25:12,060 Collect it. 387 00:25:12,060 --> 00:25:15,640 All liquid, put it in a new vessel. 388 00:25:15,640 --> 00:25:18,000 Expand it, lower the pressure, all goes back 389 00:25:18,000 --> 00:25:20,400 into the gas phase. 390 00:25:20,400 --> 00:25:21,800 Do it all again. 391 00:25:21,800 --> 00:25:24,810 And the point is, what you're doing is walking along here. 392 00:25:24,810 --> 00:25:25,830 Here to here. 393 00:25:25,830 --> 00:25:28,600 Then you start down here, and go from here to here. 394 00:25:28,600 --> 00:25:29,020 From here to here. 395 00:25:29,020 --> 00:25:31,870 And you can purify. 396 00:25:31,870 --> 00:25:36,440 Now, of course, the optimal procedure, you have to think a 397 00:25:36,440 --> 00:25:36,860 little bit. 398 00:25:36,860 --> 00:25:40,890 Because if you really do precisely what I said, you're 399 00:25:40,890 --> 00:25:43,480 going to have a mighty little bit of material 400 00:25:43,480 --> 00:25:44,540 each time you do that. 401 00:25:44,540 --> 00:25:47,800 So yes it'll be the little bit you've gotten at the end is 402 00:25:47,800 --> 00:25:52,670 going to be really pure, but there's not a whole lot of it. 403 00:25:52,670 --> 00:25:55,710 Because, remember, what we said is let's raise the 404 00:25:55,710 --> 00:25:58,340 pressure until we just start being on 405 00:25:58,340 --> 00:26:00,130 the coexistence curve. 406 00:26:00,130 --> 00:26:03,150 So we've still got mostly gas. 407 00:26:03,150 --> 00:26:05,430 Little bit of liquid. 408 00:26:05,430 --> 00:26:07,810 Now, I could raise the pressure a bit higher. 409 00:26:07,810 --> 00:26:11,340 So that in the interest of having more of the liquid, 410 00:26:11,340 --> 00:26:15,400 when I do that, though, the liquid that I have at this 411 00:26:15,400 --> 00:26:20,060 higher pressure won't be as enriched as it was down here. 412 00:26:20,060 --> 00:26:21,890 Now, I could still do this procedure. 413 00:26:21,890 --> 00:26:24,560 I could just do more of them. 414 00:26:24,560 --> 00:26:28,060 So it takes a little bit of judiciousness to figure out 415 00:26:28,060 --> 00:26:29,660 how to optimize that. 416 00:26:29,660 --> 00:26:34,060 In the end, though, you can continue to walk your way down 417 00:26:34,060 --> 00:26:38,810 through these coexistence curves and purify repeatedly 418 00:26:38,810 --> 00:26:42,760 the component B, the less volatile of them, and end up 419 00:26:42,760 --> 00:26:43,920 with some amount of it. 420 00:26:43,920 --> 00:26:46,550 And there'll be some balance between the amount that you 421 00:26:46,550 --> 00:26:48,730 feel like you need to end up with and how pure 422 00:26:48,730 --> 00:26:54,470 you need it to be. 423 00:26:54,470 --> 00:26:58,280 Any questions about how this works? 424 00:26:58,280 --> 00:27:16,210 So purification of less volatile components. 425 00:27:16,210 --> 00:27:20,290 Now, how much of each of these quantities in 426 00:27:20,290 --> 00:27:21,920 each of these phases? 427 00:27:21,920 --> 00:27:25,490 So, pertinent to this discussion, of course we need 428 00:27:25,490 --> 00:27:26,860 to know that. 429 00:27:26,860 --> 00:27:30,010 If you want to try to optimize a procedure like that, of 430 00:27:30,010 --> 00:27:32,110 course it's going to be crucial to be able to 431 00:27:32,110 --> 00:27:35,650 understand and calculate for any pressure that you decide 432 00:27:35,650 --> 00:27:42,720 to raise to, just how many moles do you have in each of 433 00:27:42,720 --> 00:27:43,690 the phases? 434 00:27:43,690 --> 00:27:47,290 So at the end of the day, you can figure out, OK, now when I 435 00:27:47,290 --> 00:27:49,620 reach a certain degree of purification, here's how much 436 00:27:49,620 --> 00:27:52,520 of the stuff I end up with. 437 00:27:52,520 --> 00:27:55,550 Well, that turns out to be reasonably 438 00:27:55,550 --> 00:27:57,570 straightforward to do. 439 00:27:57,570 --> 00:28:00,120 And so what I'll go through is a simple mathematical 440 00:28:00,120 --> 00:28:01,170 derivation. 441 00:28:01,170 --> 00:28:05,160 And it turns out that it allows you to just read right 442 00:28:05,160 --> 00:28:09,170 off the diagram how much of each material you're going to 443 00:28:09,170 --> 00:28:11,420 end up with. 444 00:28:11,420 --> 00:28:15,680 So, here's what happens. 445 00:28:15,680 --> 00:28:21,830 This is something called the lever rule. 446 00:28:21,830 --> 00:28:39,880 How much of each component is there in each phase? 447 00:28:39,880 --> 00:28:52,630 So let's consider a case like this. 448 00:28:52,630 --> 00:28:58,550 Let me draw yet once again, just to get the numbering 449 00:28:58,550 --> 00:29:00,260 consistent. 450 00:29:00,260 --> 00:29:07,220 With how we'll treat this. 451 00:29:07,220 --> 00:29:08,330 So we're going to start here. 452 00:29:08,330 --> 00:29:09,760 And I want to draw it right in the middle, so I've 453 00:29:09,760 --> 00:29:14,090 got plenty of room. 454 00:29:14,090 --> 00:29:26,550 And we're going to go up to some pressure. 455 00:29:26,550 --> 00:29:29,360 And somewhere out there, now I can go to 456 00:29:29,360 --> 00:29:31,410 my coexistence curves. 457 00:29:31,410 --> 00:29:33,190 Liquid. 458 00:29:33,190 --> 00:29:34,630 And gas. 459 00:29:34,630 --> 00:29:40,460 And I can read off my values. 460 00:29:40,460 --> 00:29:44,280 So this is the liquid xB. 461 00:29:44,280 --> 00:29:52,060 So I'm going to go up to some point two, here's xB of two. 462 00:29:52,060 --> 00:29:57,100 Here's yB of two. 463 00:29:57,100 --> 00:30:00,120 Great. 464 00:30:00,120 --> 00:30:10,310 Now let's get these written in. 465 00:30:10,310 --> 00:30:17,060 So let's just define terms a little bit. nA, nB. 466 00:30:17,060 --> 00:30:28,900 Or just our total number of moles. ng and n liquid, of 467 00:30:28,900 --> 00:30:38,040 course, total number of moles. 468 00:30:38,040 --> 00:30:40,990 In the gas and liquid phases. 469 00:30:40,990 --> 00:30:44,010 So let's just do the calculation for each 470 00:30:44,010 --> 00:30:45,820 of these two cases. 471 00:30:45,820 --> 00:30:46,670 We'll start with one. 472 00:30:46,670 --> 00:30:47,790 That's the easier case. 473 00:30:47,790 --> 00:30:51,040 Because then we have only the gas. 474 00:30:51,040 --> 00:30:57,720 So at one, all gas. 475 00:30:57,720 --> 00:31:00,020 It says pure gas in the notes, but of course that 476 00:31:00,020 --> 00:31:00,850 isn't the pure gas. 477 00:31:00,850 --> 00:31:04,400 It's the mixture of the two components. 478 00:31:04,400 --> 00:31:07,760 So. 479 00:31:07,760 --> 00:31:09,750 How many moles of A? 480 00:31:09,750 --> 00:31:12,760 Well it's the mole fraction of A in the gas. 481 00:31:12,760 --> 00:31:17,830 Times the total number of moles in the gas. 482 00:31:17,830 --> 00:31:19,630 Let me put one in here. 483 00:31:19,630 --> 00:31:25,300 Just to be clear. 484 00:31:25,300 --> 00:31:28,350 And since we have all gas, the number of moles in the gas is 485 00:31:28,350 --> 00:31:30,290 just the total number of moles. 486 00:31:30,290 --> 00:31:37,570 So this is just yA at one times n total. 487 00:31:37,570 --> 00:31:38,610 Let's just write that in. 488 00:31:38,610 --> 00:31:43,220 And of course n total is equal to nA plus nB. 489 00:31:47,750 --> 00:31:51,990 So now let's look at condition two. 490 00:31:51,990 --> 00:31:54,820 Now we have to look a little more carefully. 491 00:31:54,820 --> 00:32:03,790 Because we have a liquid gas mixture. 492 00:32:03,790 --> 00:32:11,970 So nA is equal to yA at pressure two. 493 00:32:11,970 --> 00:32:19,190 Times the number of moles of gas at pressure two. 494 00:32:19,190 --> 00:32:23,322 Plus xA, at pressure two, times the number of moles of 495 00:32:23,322 --> 00:32:38,240 liquid at pressure two. 496 00:32:38,240 --> 00:32:40,780 Now, of course, these things have to be equal. 497 00:32:40,780 --> 00:32:45,230 The total number of moles of A didn't change, right? 498 00:32:45,230 --> 00:32:52,900 So those are equal. 499 00:32:52,900 --> 00:32:58,580 Then yA of two times ng of two. 500 00:32:58,580 --> 00:33:08,770 Plus xA of two times n liquid of two, that's equal to yA of 501 00:33:08,770 --> 00:33:13,520 one times n total. 502 00:33:13,520 --> 00:33:20,460 Which is of course equal to yA of one times n gas at two plus 503 00:33:20,460 --> 00:33:24,350 n liquid at two. 504 00:33:24,350 --> 00:33:27,380 I suppose I could be, add that equality. 505 00:33:27,380 --> 00:33:28,540 Of course, it's an obvious one. 506 00:33:28,540 --> 00:33:30,030 But let me do it anyway. 507 00:33:30,030 --> 00:33:32,770 The total number of moles is equal to nA plus nB. 508 00:33:32,770 --> 00:33:40,230 But it's also equal to n liquid plus n gas. 509 00:33:40,230 --> 00:33:46,050 And that's all I'm taking advantage of here. 510 00:33:46,050 --> 00:33:47,680 And now I'm just going to rearrange the terms. 511 00:33:47,680 --> 00:33:57,870 So I'm going to write yA at one minus yA at two, times ng 512 00:33:57,870 --> 00:34:02,120 at two, is equal to, and I'm going to take the other terms, 513 00:34:02,120 --> 00:34:13,490 the xA term. xA of two minus yA of one 514 00:34:13,490 --> 00:34:18,050 times n liquid at two. 515 00:34:18,050 --> 00:34:21,570 So I've just rearranged the terms. 516 00:34:21,570 --> 00:34:30,330 And I've done that because now, I think I omitted 517 00:34:30,330 --> 00:34:37,170 something here. yA of one times ng. 518 00:34:43,270 --> 00:34:51,700 No, I forgot a bracket, is what I did. yA of one there. 519 00:34:51,700 --> 00:34:53,910 And I did this because now I want to do is look at the 520 00:34:53,910 --> 00:34:57,040 ratio of liquid to gas at pressure two. 521 00:34:57,040 --> 00:35:10,750 So, ratio of I'll put it gas to liquid, that's ng of two 522 00:35:10,750 --> 00:35:15,870 over n liquid at two. 523 00:35:15,870 --> 00:35:26,990 And that's just equal to xA of two minus yA at one minus yA 524 00:35:26,990 --> 00:35:37,070 at one minus yA at two. 525 00:35:37,070 --> 00:35:38,420 So what does it mean? 526 00:35:38,420 --> 00:35:45,100 It's the ratio of these lever arms. 527 00:35:45,100 --> 00:35:46,570 That's what it's telling me. 528 00:35:46,570 --> 00:35:51,380 I can look, so I raise the pressure up to two. 529 00:35:51,380 --> 00:35:55,520 And so here's xB at two, here's yB at two. 530 00:35:55,520 --> 00:35:57,960 And I'm here somewhere. 531 00:35:57,960 --> 00:36:08,640 And this little amount and this little amount, that's 532 00:36:08,640 --> 00:36:09,440 that difference. 533 00:36:09,440 --> 00:36:13,020 And it's just telling me that ratio of those arms is the 534 00:36:13,020 --> 00:36:16,900 ratio of the total number of moles of gas to liquid. 535 00:36:16,900 --> 00:36:18,250 And that's great. 536 00:36:18,250 --> 00:36:21,000 Because now when I go back to the problem that we were just 537 00:36:21,000 --> 00:36:24,100 looking at, where I say, well I'm going to purify the less 538 00:36:24,100 --> 00:36:27,140 volatile component by raising the pressure until I'm at 539 00:36:27,140 --> 00:36:29,260 coexistence starting in the gas phase. 540 00:36:29,260 --> 00:36:31,700 Raise the pressure, I've got some liquid. 541 00:36:31,700 --> 00:36:34,030 But I also want some finite amount of liquid. 542 00:36:34,030 --> 00:36:36,990 But I don't want to just, when I get the very, very first 543 00:36:36,990 --> 00:36:39,840 drop of liquid now collected, of course it's enriched in the 544 00:36:39,840 --> 00:36:40,910 less volatile component. 545 00:36:40,910 --> 00:36:43,970 But there may be a minuscule amount, right? 546 00:36:43,970 --> 00:36:47,710 So I'll raise the pressure a bit more. 547 00:36:47,710 --> 00:36:48,890 I'll go up in pressure. 548 00:36:48,890 --> 00:36:53,900 And now, of course, when I do that the amount of enrichment 549 00:36:53,900 --> 00:36:57,710 of the liquid isn't as big as it was if I just raised it up 550 00:36:57,710 --> 00:36:59,110 enough to barely have any liquid. 551 00:36:59,110 --> 00:37:01,650 Then I'd be out here. 552 00:37:01,650 --> 00:37:06,700 But I've got more material in the liquid phase to collect. 553 00:37:06,700 --> 00:37:08,710 And that's what this allows me to calculate. 554 00:37:08,710 --> 00:37:13,720 Is how much do I get in the end. 555 00:37:13,720 --> 00:37:16,350 So it's very handy. 556 00:37:16,350 --> 00:37:22,420 You can also see, if I go all the way to the limit where the 557 00:37:22,420 --> 00:37:27,020 mole fraction in the liquid at the end is equal to what it 558 00:37:27,020 --> 00:37:32,170 was in the gas when I started, what that says is that there's 559 00:37:32,170 --> 00:37:34,960 no more gas left any more. 560 00:37:34,960 --> 00:37:39,490 In other words, these two things are equal. 561 00:37:39,490 --> 00:37:44,580 If I go all the way to the point where I've got all the, 562 00:37:44,580 --> 00:37:49,630 this is the amount I started with, in the pure gas phase, 563 00:37:49,630 --> 00:37:51,920 now I keep raising it all the way. 564 00:37:51,920 --> 00:37:54,950 Until I've got the same mole fraction in the liquid. 565 00:37:54,950 --> 00:37:56,770 Of course, we know what that really means. 566 00:37:56,770 --> 00:37:58,780 That means that I've gone all the way from 567 00:37:58,780 --> 00:38:00,310 pure gas to pure liquid. 568 00:38:00,310 --> 00:38:03,620 And the mole fraction in that case has to be the same. 569 00:38:03,620 --> 00:38:07,110 And what this is just telling us mathematically is, when 570 00:38:07,110 --> 00:38:08,280 that happens this is zero. 571 00:38:08,280 --> 00:38:14,660 That means I don't have any gas left. 572 00:38:14,660 --> 00:38:15,090 Yeah. 573 00:38:15,090 --> 00:38:26,390 STUDENT: [INAUDIBLE] 574 00:38:26,390 --> 00:38:26,980 PROFESSOR: No. 575 00:38:26,980 --> 00:38:31,580 Because, so it's the mole fraction in the gas phase. 576 00:38:31,580 --> 00:38:35,960 But you've started with some amount that it's only going to 577 00:38:35,960 --> 00:38:38,690 go down from there. 578 00:38:38,690 --> 00:38:41,940 STUDENT: [INAUDIBLE] 579 00:38:41,940 --> 00:38:44,290 PROFESSOR: Yeah. 580 00:38:44,290 --> 00:38:48,850 Yeah. 581 00:38:48,850 --> 00:38:53,310 Any other questions? 582 00:38:53,310 --> 00:38:55,390 OK. 583 00:38:55,390 --> 00:39:01,980 Well, now what I want to do is just put up a slightly 584 00:39:01,980 --> 00:39:05,470 different kind of diagram, but different in an important way. 585 00:39:05,470 --> 00:39:09,830 Namely, instead of showing the mole fractions as a function 586 00:39:09,830 --> 00:39:11,640 of the pressure. 587 00:39:11,640 --> 00:39:14,680 And I haven't written it in, but all of these are at 588 00:39:14,680 --> 00:39:16,030 constant temperature, right? 589 00:39:16,030 --> 00:39:17,820 I've assumed the temperature is constant 590 00:39:17,820 --> 00:39:19,480 in all these things. 591 00:39:19,480 --> 00:39:22,410 Now let's consider the other possibility, the other simple 592 00:39:22,410 --> 00:39:25,960 possibility, which is, let's hold the pressure constant and 593 00:39:25,960 --> 00:39:26,960 vary the temperature. 594 00:39:26,960 --> 00:39:29,270 Of course, you know in the lab, that's usually what's 595 00:39:29,270 --> 00:39:31,920 easiest to do. 596 00:39:31,920 --> 00:39:35,350 Now, unfortunately, the arithmetic gets more 597 00:39:35,350 --> 00:39:36,600 complicated. 598 00:39:36,600 --> 00:39:41,070 It's not monumentally complicated, but here in this 599 00:39:41,070 --> 00:39:45,820 case, where you have one linear relationship, which is 600 00:39:45,820 --> 00:39:46,630 very convenient. 601 00:39:46,630 --> 00:39:48,080 From Raoult's law. 602 00:39:48,080 --> 00:39:52,740 And then you have one non-linear relationship there 603 00:39:52,740 --> 00:39:54,970 for the mole fraction of the gas. 604 00:39:54,970 --> 00:39:59,180 In the case of temperature, they're both, 605 00:39:59,180 --> 00:40:01,960 neither one is linear. 606 00:40:01,960 --> 00:40:03,730 Nevertheless, we can just sketch what the 607 00:40:03,730 --> 00:40:04,850 diagram looks like. 608 00:40:04,850 --> 00:40:07,160 And of course it's very useful to do that, and see how to 609 00:40:07,160 --> 00:40:10,370 read off it. 610 00:40:10,370 --> 00:40:13,370 And I should say the derivation of the curves isn't 611 00:40:13,370 --> 00:40:14,530 particularly complicated. 612 00:40:14,530 --> 00:40:16,850 It's not particularly more complicated than what I think 613 00:40:16,850 --> 00:40:19,600 you saw last time to derive this. 614 00:40:19,600 --> 00:40:21,600 There's no complicated math involved. 615 00:40:21,600 --> 00:40:24,730 But the point is, the derivation doesn't yield a 616 00:40:24,730 --> 00:40:28,900 linear relationship for either the gas or the liquid part of 617 00:40:28,900 --> 00:40:54,730 the coexistence curve. 618 00:40:54,730 --> 00:41:03,140 OK, so we're going to look at temperature and mole fraction 619 00:41:03,140 --> 00:41:04,270 phase diagrams. 620 00:41:04,270 --> 00:41:06,460 Again, a little more complicated mathematically but 621 00:41:06,460 --> 00:41:20,160 more practical in real use. 622 00:41:20,160 --> 00:41:31,570 And this is T. And here is the, sort of, form that these 623 00:41:31,570 --> 00:41:33,130 things take. 624 00:41:33,130 --> 00:41:35,680 So again, neither one is linear. 625 00:41:35,680 --> 00:41:38,430 Up here, now, of course if you raise the temperatures, that's 626 00:41:38,430 --> 00:41:41,120 where you end up with gas. 627 00:41:41,120 --> 00:41:42,310 If you lower the temperature, you 628 00:41:42,310 --> 00:41:48,790 condense and get the liquid. 629 00:41:48,790 --> 00:41:55,620 So, this is TA star. 630 00:41:55,620 --> 00:42:00,970 TB star. 631 00:42:00,970 --> 00:42:05,020 So now I want to stick with A as the 632 00:42:05,020 --> 00:42:07,590 more volatile component. 633 00:42:07,590 --> 00:42:11,030 At constant temperature, that meant that pA star is bigger 634 00:42:11,030 --> 00:42:12,200 than pB star. 635 00:42:12,200 --> 00:42:16,480 In other words, the vapor pressure over pure liquid A is 636 00:42:16,480 --> 00:42:23,150 higher than the vapor pressure over pure liquid B. Similarly, 637 00:42:23,150 --> 00:42:25,380 now I've got constant pressure and really what I'm looking 638 00:42:25,380 --> 00:42:26,990 at, let's say I'm at the limit where I've 639 00:42:26,990 --> 00:42:30,310 got the pure liquid. 640 00:42:30,310 --> 00:42:33,910 Or the pure A. And now I'm going to, let's say, raise the 641 00:42:33,910 --> 00:42:38,920 temperature until I'm at the liquid-gas equilibrium. 642 00:42:38,920 --> 00:42:42,500 That's just the boiling point. 643 00:42:42,500 --> 00:42:45,820 So if A is the more volatile component, it has the lower 644 00:42:45,820 --> 00:42:47,800 boiling point. 645 00:42:47,800 --> 00:42:49,370 And that's what this reflects. 646 00:42:49,370 --> 00:42:54,280 So higher pB star A corresponds to lower TA star 647 00:42:54,280 --> 00:42:59,230 A. Which is just the boiling point of pure A. 648 00:42:59,230 --> 00:43:11,690 So, this is called the bubble line. 649 00:43:11,690 --> 00:43:14,990 That's called the dew line. 650 00:43:14,990 --> 00:43:17,630 All that means is, let's say I'm at high temperature. 651 00:43:17,630 --> 00:43:19,370 I've got all gas. 652 00:43:19,370 --> 00:43:21,090 Right no coexistence, no liquid yet. 653 00:43:21,090 --> 00:43:23,130 And I start to cool things off. 654 00:43:23,130 --> 00:43:25,820 Just to where I just barely start to get liquid. 655 00:43:25,820 --> 00:43:29,230 What you see that as is, dew starts forming. 656 00:43:29,230 --> 00:43:30,570 A little bit of condensation. 657 00:43:30,570 --> 00:43:32,765 If you're outside, it means on the grass a little bit of dew 658 00:43:32,765 --> 00:43:34,440 is forming. 659 00:43:34,440 --> 00:43:38,300 Similarly, if I start at low temperature, all liquid now I 660 00:43:38,300 --> 00:43:41,660 start raising the temperature until I just start to boil. 661 00:43:41,660 --> 00:43:46,180 I just start to see the first bubbles forming. 662 00:43:46,180 --> 00:43:51,650 And so that's why these things have those names. 663 00:43:51,650 --> 00:43:56,090 So now let's just follow along what happens when I do the 664 00:43:56,090 --> 00:43:57,780 same sort of thing that I illustrated there. 665 00:43:57,780 --> 00:44:00,730 I want to start at one point in this phase diagram. 666 00:44:00,730 --> 00:44:04,890 And then start changing the conditions. 667 00:44:04,890 --> 00:44:17,610 So let's start here. 668 00:44:17,610 --> 00:44:19,330 So I'm going to start all in the liquid phase. 669 00:44:19,330 --> 00:44:22,200 That is, the temperature is low. 670 00:44:22,200 --> 00:44:23,970 Here's xB. 671 00:44:26,670 --> 00:44:27,690 And my original temperature. 672 00:44:27,690 --> 00:44:34,570 Now I'm going to raise it. 673 00:44:34,570 --> 00:44:39,590 So if I raise it a little bit, I reach a point at which I 674 00:44:39,590 --> 00:44:41,600 first start to boil. 675 00:44:41,600 --> 00:44:47,890 Start to find some gas above the liquid. 676 00:44:47,890 --> 00:44:51,880 And if I look right here, that'll be my composition. 677 00:44:51,880 --> 00:44:53,430 Let me raise it a little farther, now that we've 678 00:44:53,430 --> 00:44:57,100 already seen the lever rule and so forth. 679 00:44:57,100 --> 00:44:58,140 I'll raise it up to here. 680 00:44:58,140 --> 00:45:11,790 And that means that out here, I suppose I should do here. 681 00:45:11,790 --> 00:45:18,730 So, here is the liquid mole fraction at temperature two. 682 00:45:18,730 --> 00:45:23,320 xB at temperature two. 683 00:45:23,320 --> 00:45:27,640 This is yB at temperature two. 684 00:45:27,640 --> 00:45:33,860 The gas mole fraction. 685 00:45:33,860 --> 00:45:37,840 So as you should expect, what's going to happen here is 686 00:45:37,840 --> 00:45:43,510 that the gas, this is going to be lower in B. A, that means 687 00:45:43,510 --> 00:45:45,570 that the mole fraction of A must be 688 00:45:45,570 --> 00:45:47,020 higher in the gas phase. 689 00:45:47,020 --> 00:45:49,700 That's one minus yB. 690 00:45:49,700 --> 00:46:05,340 So xA is one minus -- yA, which is one minus yB higher 691 00:46:05,340 --> 00:46:10,970 in gas phase. 692 00:46:10,970 --> 00:46:16,370 Than xA, which is one minus xB. 693 00:46:16,370 --> 00:46:20,480 In other words, the less volatile component is enriched 694 00:46:20,480 --> 00:46:39,100 up in the gas phase. 695 00:46:39,100 --> 00:46:42,320 Now, what does that mean? 696 00:46:42,320 --> 00:46:46,890 That means I could follow the same sort of procedure that I 697 00:46:46,890 --> 00:46:50,710 indicated before when we looked at the pressure mole 698 00:46:50,710 --> 00:46:51,930 fraction phase diagram. 699 00:46:51,930 --> 00:46:54,970 Namely, I could do this and now I could 700 00:46:54,970 --> 00:46:57,150 take the gas phase. 701 00:46:57,150 --> 00:47:01,730 Which has less of B. It has more of A. And 702 00:47:01,730 --> 00:47:03,780 I can collect it. 703 00:47:03,780 --> 00:47:05,210 And then I can reduce the temperature. 704 00:47:05,210 --> 00:47:06,310 So it liquefies. 705 00:47:06,310 --> 00:47:08,820 So I can condense it, in other words. 706 00:47:08,820 --> 00:47:11,820 So now I'm going to start with, let's say I lower the 707 00:47:11,820 --> 00:47:14,710 temperature enough so I've got basically pure liquid. 708 00:47:14,710 --> 00:47:17,500 But its composition is the same as the gas here. 709 00:47:17,500 --> 00:47:19,850 Because of course that's what that liquid is formed from. 710 00:47:19,850 --> 00:47:23,800 I collected the gas and separated it. 711 00:47:23,800 --> 00:47:25,830 So now I could start all over again. 712 00:47:25,830 --> 00:47:31,240 Except instead of being here, I'll be down here. 713 00:47:31,240 --> 00:47:33,060 And then I can raise the temperature again. 714 00:47:33,060 --> 00:47:35,020 To some place where I choose. 715 00:47:35,020 --> 00:47:37,350 I could choose here, and go all the way to hear. 716 00:47:37,350 --> 00:47:39,120 A great amount of enrichment. 717 00:47:39,120 --> 00:47:41,305 But I know from the lever rule that if I do that, I'm going 718 00:47:41,305 --> 00:47:44,460 to have precious little material over here. 719 00:47:44,460 --> 00:47:46,730 So I might prefer to raise the temperature a little more. 720 00:47:46,730 --> 00:47:50,940 Still get a substantial amount of enrichment. 721 00:47:50,940 --> 00:47:54,370 And now I've got, in the gas phase, I'll further enriched 722 00:47:54,370 --> 00:47:57,880 in component A. And again I can collect the gas. 723 00:47:57,880 --> 00:48:00,850 Condense it. 724 00:48:00,850 --> 00:48:03,280 Now I'm out here somewhere, I've got all liquid and I'll 725 00:48:03,280 --> 00:48:04,350 raise the temperature again. 726 00:48:04,350 --> 00:48:07,380 And I can again keep walking my way over. 727 00:48:07,380 --> 00:48:11,640 And that's what happens during an ordinary distillation. 728 00:48:11,640 --> 00:48:17,270 Each step of the distillation walks along in the phase 729 00:48:17,270 --> 00:48:23,530 diagram at some selected point. 730 00:48:23,530 --> 00:48:25,190 And of course what you're doing is, you're always 731 00:48:25,190 --> 00:48:27,520 condensing the gas. 732 00:48:27,520 --> 00:48:30,830 And starting with fresh liquid that now is enriched in more 733 00:48:30,830 --> 00:48:33,860 volatile of the components. 734 00:48:33,860 --> 00:48:36,680 So of course if you're really purifying, say, ethanol from 735 00:48:36,680 --> 00:48:39,710 an ethanol water mixture, that's how you do it. 736 00:48:39,710 --> 00:48:41,640 Ethanol is the more volatile component. 737 00:48:41,640 --> 00:48:42,870 So a still is set up. 738 00:48:42,870 --> 00:48:45,080 It will boil the stuff and collect the gas and and 739 00:48:45,080 --> 00:48:45,920 condense it. 740 00:48:45,920 --> 00:48:48,350 And boil it again, and so forth. 741 00:48:48,350 --> 00:48:50,600 And the whole thing can be set up in a very efficient way. 742 00:48:50,600 --> 00:48:53,420 So you have essentially continuous distillation. 743 00:48:53,420 --> 00:48:56,970 Where you have a whole sequence of collection and 744 00:48:56,970 --> 00:49:00,610 condensation and reheating and so forth events. 745 00:49:00,610 --> 00:49:03,930 So then, in a practical way, it's possible to walk quite 746 00:49:03,930 --> 00:49:10,870 far along the distillation, the coexistence curve, and 747 00:49:10,870 --> 00:49:16,630 distill to really a high degree of purification. 748 00:49:16,630 --> 00:49:22,270 Any questions about how that works? 749 00:49:22,270 --> 00:49:24,240 OK. 750 00:49:24,240 --> 00:49:29,720 I'll leave till next time the discussion of the chemical 751 00:49:29,720 --> 00:49:30,190 potentials. 752 00:49:30,190 --> 00:49:33,980 But what we'll do, just to foreshadow a little bit, what 753 00:49:33,980 --> 00:49:36,330 I'll do at the beginning of the next lecture is what's at 754 00:49:36,330 --> 00:49:37,280 the end of your notes here. 755 00:49:37,280 --> 00:49:40,710 Which is just to say OK, now if we look at Raoult's law, 756 00:49:40,710 --> 00:49:44,470 it's straightforward to say what is the chemical potential 757 00:49:44,470 --> 00:49:47,290 for each of the substances in the liquid and the gas phase. 758 00:49:47,290 --> 00:49:50,100 Of course, it has to be equal. 759 00:49:50,100 --> 00:49:53,450 Given that, that's for an ideal solution. 760 00:49:53,450 --> 00:49:55,490 We can gain some insight from that. 761 00:49:55,490 --> 00:49:57,930 And then look at real solutions, non-ideal 762 00:49:57,930 --> 00:50:00,860 solutions, and understand a lot of their behavior as well. 763 00:50:00,860 --> 00:50:03,160 Just from starting from our understanding of what the 764 00:50:03,160 --> 00:50:06,420 chemical potential does even in a simple ideal mixture. 765 00:50:06,420 --> 00:50:08,150 So we'll look at the chemical potentials. 766 00:50:08,150 --> 00:50:10,740 And then we'll look at non-ideal solution 767 00:50:10,740 --> 00:50:12,310 mixtures next time. 768 00:50:12,310 --> 00:50:13,770 See you then.