1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:02 The following content is provided under a Creative 3 00:00:02 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:10 offer high quality educational resources for free. 6 00:00:10 --> 00:00:13 To make a donation or view additional materials from 7 00:00:13 --> 00:00:15 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:15 --> 00:00:20 at ocw.mit.edu. 9 00:00:20 --> 00:00:23 PROFESSOR: So. 10 00:00:23 --> 00:00:25 In the meantime, you've started looking at two 11 00:00:25 --> 00:00:27 phase equilibrium. 12 00:00:27 --> 00:00:29 So now we're starting to look at mixtures. 13 00:00:29 --> 00:00:32 And so now we have more than one constituent. 14 00:00:32 --> 00:00:35 And we have more than one phase present. 15 00:00:35 --> 00:00:36 Right? 16 00:00:36 --> 00:00:45 So you've started to look at things that look like this, 17 00:00:45 --> 00:00:51 where you've got, let's say, two components. 18 00:00:51 --> 00:01:02 Both in the gas phase. 19 00:01:02 --> 00:01:05 And now to try to figure out what the phase 20 00:01:05 --> 00:01:06 equilibria look like. 21 00:01:06 --> 00:01:09 Of course it's now a little bit more complicated than what you 22 00:01:09 --> 00:01:11 went through before, where you can get pressure temperature 23 00:01:11 --> 00:01:13 phase diagrams with just a single component. 24 00:01:13 --> 00:01:18 Now we want to worry about what's the composition. 25 00:01:18 --> 00:01:19 Of each of the components. 26 00:01:19 --> 00:01:20 In each of the phases. 27 00:01:20 --> 00:01:24 And what's the temperature and the pressure. 28 00:01:24 --> 00:01:26 Total and partial pressures and all of that. 29 00:01:26 --> 00:01:31 So you can really figure out everything about both phases. 30 00:01:31 --> 00:01:34 And there are all sorts of important reasons to do that, 31 00:01:34 --> 00:01:36 obviously lots of chemistry happens in liquid mixtures. 32 00:01:36 --> 00:01:37 Some in gas mixtures. 33 00:01:37 --> 00:01:40 Some where they're in equilibrium. 34 00:01:40 --> 00:01:41 All sorts of chemical processes. 35 00:01:41 --> 00:01:45 Distillation, for example, takes advantage of the 36 00:01:45 --> 00:01:49 properties of liquid and gas mixtures. 37 00:01:49 --> 00:01:52 Where one of them might be richer, will be richer, 38 00:01:52 --> 00:01:54 and the more volatile of the components. 39 00:01:54 --> 00:01:58 That can be used as a basis for purification. 40 00:01:58 --> 00:02:01 You mix ethanol and water together so you've got a 41 00:02:01 --> 00:02:03 liquid with a certain composition of each. 42 00:02:03 --> 00:02:05 The gas is going to be richer and the more volatile of 43 00:02:05 --> 00:02:07 the two, the ethanol. 44 00:02:07 --> 00:02:10 So in a distillation, where you put things up in the gas, 45 00:02:10 --> 00:02:13 more of the ethanol comes up. 46 00:02:13 --> 00:02:15 You could then collect that gas, right? 47 00:02:15 --> 00:02:18 And re-condense it, and make a new liquid. 48 00:02:18 --> 00:02:22 Which is much richer in ethanol than the original liquid was. 49 00:02:22 --> 00:02:25 Then you could make, then you could put some of them 50 00:02:25 --> 00:02:26 up into the gas phase. 51 00:02:26 --> 00:02:30 Where it will be still richer in ethanol. 52 00:02:30 --> 00:02:32 And then you could collect that and repeat the process. 53 00:02:32 --> 00:02:37 So the point is that properties of liquid gas, two-component 54 00:02:37 --> 00:02:40 or multi-component mixtures like this can be exploited. 55 00:02:40 --> 00:02:44 Basically, the different volatilities of the different 56 00:02:44 --> 00:02:48 components can be exploited for things like purification. 57 00:02:48 --> 00:02:52 Also if you want to calculate chemical equilibria in the 58 00:02:52 --> 00:02:53 liquid and gas phase, of course, now you've seen 59 00:02:53 --> 00:02:56 chemical equilibrium, so the amount of reaction depends 60 00:02:56 --> 00:02:58 on the composition. 61 00:02:58 --> 00:03:02 So of course if you want reactions to go, then this also 62 00:03:02 --> 00:03:05 can be exploited by looking at which phase might be richer in 63 00:03:05 --> 00:03:07 one reactant or another. 64 00:03:07 --> 00:03:10 And thereby pushing the equilibrium toward one 65 00:03:10 --> 00:03:12 direction or the other. 66 00:03:12 --> 00:03:12 OK. 67 00:03:12 --> 00:03:19 So. we've got some total temperature and pressure. 68 00:03:19 --> 00:03:29 And we have compositions. 69 00:03:29 --> 00:03:34 So in the gas phase, we've got mole fractions yA and yB. 70 00:03:34 --> 00:03:41 In the liquid phase we've got mole fractions xA and xB. 71 00:03:41 --> 00:03:49 72 00:03:49 --> 00:03:51 So that's our system. 73 00:03:51 --> 00:03:54 One of the things that you established last time is that, 74 00:03:54 --> 00:03:58 so there are the total number of variables including the 75 00:03:58 --> 00:04:00 temperature and the pressure. 76 00:04:00 --> 00:04:04 And let's say the mole fraction of A in each of the liquid 77 00:04:04 --> 00:04:06 and gas phases, right? 78 00:04:06 --> 00:04:07 But then there are constraints. 79 00:04:07 --> 00:04:11 Because the chemical potentials have to be equal, right? 80 00:04:11 --> 00:04:14 Chemical potential of A has to be equal in the liquid and gas. 81 00:04:14 --> 00:04:15 Same with B. 82 00:04:15 --> 00:04:19 Those two constraints reduce the number of 83 00:04:19 --> 00:04:20 independent variables. 84 00:04:20 --> 00:04:23 So there'll be two in this case rather than four 85 00:04:23 --> 00:04:25 independent variables. 86 00:04:25 --> 00:04:29 If you control those, then everything else will follow. 87 00:04:29 --> 00:04:33 What that means is if you've got a, if you control, if you 88 00:04:33 --> 00:04:36 fix the temperature and the total pressure, everything 89 00:04:36 --> 00:04:39 else should be determinable. 90 00:04:39 --> 00:04:43 No more free variables. 91 00:04:43 --> 00:04:48 And then, what you saw is that in simple or ideal liquid 92 00:04:48 --> 00:04:57 mixtures, a result called Raoult's law would hold. 93 00:04:57 --> 00:05:01 Which just says that the partial pressure of A is equal 94 00:05:01 --> 00:05:10 to the mole fraction of A in the liquid times the pressure 95 00:05:10 --> 00:05:12 of pure A over the liquid. 96 00:05:12 --> 00:05:19 And so what this gives you is a diagram that looks like this. 97 00:05:19 --> 00:05:24 If we plot this versus xB, this is mole fraction of B in the 98 00:05:24 --> 00:05:28 liquid going from zero to one. 99 00:05:28 --> 00:05:39 Then we could construct a diagram of this sort. 100 00:05:39 --> 00:05:45 So this is the total pressure of A and B. 101 00:05:45 --> 00:05:53 The partial pressures are given by these lines. 102 00:05:53 --> 00:05:59 So this is our pA star and pB star. 103 00:05:59 --> 00:06:04 The pressures over the pure liquid A and B at the limits 104 00:06:04 --> 00:06:07 of mole fraction of B being zero and one. 105 00:06:07 --> 00:06:11 So in this situation, for example, A is the more 106 00:06:11 --> 00:06:12 volatile of the components. 107 00:06:12 --> 00:06:15 So it's partial pressure over its pure liquid. 108 00:06:15 --> 00:06:17 At this temperature. 109 00:06:17 --> 00:06:20 Is higher than the partial pressure of B over 110 00:06:20 --> 00:06:21 its pure liquid. 111 00:06:21 --> 00:06:23 A would be the ethanol, for example and B the 112 00:06:23 --> 00:06:30 water in that mixture. 113 00:06:30 --> 00:06:33 OK. 114 00:06:33 --> 00:06:38 Then you started looking at both the gas and the liquid 115 00:06:38 --> 00:06:40 phase in the same diagram. 116 00:06:40 --> 00:06:44 So this is the mole fraction of the liquid. 117 00:06:44 --> 00:06:47 If you look and see, well, OK now we should be able to 118 00:06:47 --> 00:06:51 determine the mole fraction in the gas as well. 119 00:06:51 --> 00:06:54 Again, if we note total temperature and pressure, 120 00:06:54 --> 00:06:56 everything else must follow. 121 00:06:56 --> 00:07:05 And so, you saw this worked out. 122 00:07:05 --> 00:07:15 Relation between p and yA, for example. 123 00:07:15 --> 00:07:25 The result was p is pA star times pB star over pA star plus 124 00:07:25 --> 00:07:31 pB star minus pA star times yA. 125 00:07:31 --> 00:07:34 And the point here is that unlike this case, where you 126 00:07:34 --> 00:07:38 have a linear relationship, the relationship between 127 00:07:38 --> 00:07:42 the pressure and the liquid mole fraction isn't linear. 128 00:07:42 --> 00:07:45 We can still plot it, of course. 129 00:07:45 --> 00:07:51 So if we do that, then we end up with a diagram that 130 00:07:51 --> 00:08:02 looks like the following. 131 00:08:02 --> 00:08:12 Now I'm going to keep both mole fractions, xB and yB, I've 132 00:08:12 --> 00:08:14 got some total pressure. 133 00:08:14 --> 00:08:21 I still have my linear relationship. 134 00:08:21 --> 00:08:30 And then I have a non-linear relationship between the 135 00:08:30 --> 00:08:36 pressure and the mole fraction in the gas phase. 136 00:08:36 --> 00:08:49 So let's just fill this in. 137 00:08:49 --> 00:08:55 Here is pA star still. 138 00:08:55 --> 00:08:56 Here's pB star. 139 00:08:56 --> 00:08:59 Of course, at the limits they're still, both mole 140 00:08:59 --> 00:09:05 fractions they're zero and one. 141 00:09:05 --> 00:09:07 OK. 142 00:09:07 --> 00:09:10 I believe this is this is where you ended up at the 143 00:09:10 --> 00:09:11 end of the last lecture. 144 00:09:11 --> 00:09:14 But it's probably not so clear exactly how you 145 00:09:14 --> 00:09:15 read something like this. 146 00:09:15 --> 00:09:16 And use it. 147 00:09:16 --> 00:09:18 It's extremely useful. 148 00:09:18 --> 00:09:23 You just have to kind of learn how to follow what happens 149 00:09:23 --> 00:09:24 in a diagram like this. 150 00:09:24 --> 00:09:26 And that's what I want to spend some of today doing. 151 00:09:26 --> 00:09:30 Is just, walking through what's happening physically, 152 00:09:30 --> 00:09:32 with a container with a mixture of the two. 153 00:09:32 --> 00:09:36 And how does that correspond to what gets read off the diagram 154 00:09:36 --> 00:09:39 under different conditions. 155 00:09:39 --> 00:09:42 So. 156 00:09:42 --> 00:09:55 Let's just start somewhere on a phase diagram like this. 157 00:09:55 --> 00:10:03 Let's start up here at some point one, so we're in the 158 00:10:03 --> 00:10:07 pure - well, not pure, you're in the all liquid phase. 159 00:10:07 --> 00:10:08 It's still a mixture. 160 00:10:08 --> 00:10:20 It's not a pure substance. pA star, pB star. 161 00:10:20 --> 00:10:21 There's the gas phase. 162 00:10:21 --> 00:10:34 So, if we start at one, and now there's some total pressure. 163 00:10:34 --> 00:10:37 And now we're going to reduce it. 164 00:10:37 --> 00:10:38 What happens? 165 00:10:38 --> 00:10:42 We start with a pure - with an all-liquid mixture. 166 00:10:42 --> 00:10:43 No gas. 167 00:10:43 --> 00:10:46 And now we're going to bring down the pressure. 168 00:10:46 --> 00:10:52 Allowing some of the liquid to go up into the gas phase. 169 00:10:52 --> 00:10:56 So, we can do that. 170 00:10:56 --> 00:11:02 And once we reach point two, then we find a 171 00:11:02 --> 00:11:04 coexistence curve. 172 00:11:04 --> 00:11:09 Now the liquid and gas are going to coexist. 173 00:11:09 --> 00:11:11 So this is the liquid phase. 174 00:11:11 --> 00:11:17 And that means that this must be xB. 175 00:11:17 --> 00:11:20 And it's xB at one, but it's also xB at two, and 176 00:11:20 --> 00:11:24 I want to emphasize that. 177 00:11:24 --> 00:11:30 So let's put our pressure for two. 178 00:11:30 --> 00:11:38 And if we go over here, this is telling us about the mole 179 00:11:38 --> 00:11:41 fraction in the gas phase. 180 00:11:41 --> 00:11:45 That's what these curves are, remember. 181 00:11:45 --> 00:11:49 So this is the one that's showing us the mole fraction 182 00:11:49 --> 00:11:49 in the liquid phase. 183 00:11:49 --> 00:11:52 This nonlinear one in the gas phase. 184 00:11:52 --> 00:11:56 So that means just reading off it, this is xB, that's 185 00:11:56 --> 00:11:58 the liquid mole fraction. 186 00:11:58 --> 00:12:01 Here's yB. 187 00:12:01 --> 00:12:05 188 00:12:05 --> 00:12:07 The gas mole fraction. 189 00:12:07 --> 00:12:09 They're not the same, right, because of course 190 00:12:09 --> 00:12:14 the components have different volatility. 191 00:12:14 --> 00:12:17 A's more volatile. 192 00:12:17 --> 00:12:21 So that means that the mole fraction of B in the liquid 193 00:12:21 --> 00:12:28 phase is higher than the mole fraction of B in the gas phase. 194 00:12:28 --> 00:12:30 Because A is the more volatile component. 195 00:12:30 --> 00:12:34 So more, relatively more, of A, the mole fraction of A is going 196 00:12:34 --> 00:12:35 to be higher up in the gas phase. 197 00:12:35 --> 00:12:43 Which means the mole fraction of B is lower in the gas phase. 198 00:12:43 --> 00:12:58 So, yB less than xB if A is more volatile. 199 00:12:58 --> 00:13:04 OK, so now what's happening physically? 200 00:13:04 --> 00:13:10 Well, we started at a point where we only had 201 00:13:10 --> 00:13:11 the liquid present. 202 00:13:11 --> 00:13:23 So at our initial pressure, we just have all liquid. 203 00:13:23 --> 00:13:29 There's some xB at one. 204 00:13:29 --> 00:13:34 That's all there is, there isn't any gas yet. 205 00:13:34 --> 00:13:37 Now, what happened here? 206 00:13:37 --> 00:13:41 Well, now we lowered the pressure. 207 00:13:41 --> 00:13:45 So you could imagine, well, we made the box bigger. 208 00:13:45 --> 00:13:49 Now, if the liquid was under pressure, being squeezed by the 209 00:13:49 --> 00:13:53 box, right then you could make the box a little bit bigger. 210 00:13:53 --> 00:13:54 And there's still no gas. 211 00:13:54 --> 00:13:58 That's moving down like this. 212 00:13:58 --> 00:14:03 But then you get to a point where there's just barely any 213 00:14:03 --> 00:14:04 pressure on top of the liquid. 214 00:14:04 --> 00:14:06 And then you keep expanding the box. 215 00:14:06 --> 00:14:09 Now some gas is going to form. 216 00:14:09 --> 00:14:13 So now we're going to go to our case two. 217 00:14:13 --> 00:14:16 We've got a bigger box. 218 00:14:16 --> 00:14:18 And now, right around where this was, this 219 00:14:18 --> 00:14:22 is going to be liquid. 220 00:14:22 --> 00:14:26 And there's gas up here. 221 00:14:26 --> 00:14:30 So up here is yB at pressure two. 222 00:14:30 --> 00:14:35 Here's xB at pressure two. 223 00:14:35 --> 00:14:40 Liquid and gas. 224 00:14:40 --> 00:14:49 So that's where we are at point two here. 225 00:14:49 --> 00:14:52 Now, what happens if we keep going? 226 00:14:52 --> 00:14:56 Let's lower the pressure some more. 227 00:14:56 --> 00:15:00 Well, we can lower it and do this. 228 00:15:00 --> 00:15:04 But really if we want to see what's happening in each of the 229 00:15:04 --> 00:15:08 phases, we have to stay on the coexistence curves. 230 00:15:08 --> 00:15:12 Those are what tell us what the pressures are. 231 00:15:12 --> 00:15:13 What the partial pressure are going to be in 232 00:15:13 --> 00:15:14 each of the phases. 233 00:15:14 --> 00:15:18 In each of the two, in the liquid and the gas phases. 234 00:15:18 --> 00:15:20 So let's say we lower the pressure a little more. 235 00:15:20 --> 00:15:25 What's going to happen is, then we'll end up 236 00:15:25 --> 00:15:28 somewhere over here. 237 00:15:28 --> 00:15:32 In the liquid, and that'll correspond to something 238 00:15:32 --> 00:15:33 over here in the gas. 239 00:15:33 --> 00:15:37 So here's three. 240 00:15:37 --> 00:15:46 So now we're going to have, that's going to be xB 241 00:15:46 --> 00:15:48 at pressure three. 242 00:15:48 --> 00:16:01 And over here is going to be yB at pressure three. 243 00:16:01 --> 00:16:05 And all we've done, of course, is we've just expanded 244 00:16:05 --> 00:16:07 this further. 245 00:16:07 --> 00:16:12 So now we've got a still taller box. 246 00:16:12 --> 00:16:15 And the liquid is going to be a little lower because some of it 247 00:16:15 --> 00:16:21 has evaporated, formed the gas phase. 248 00:16:21 --> 00:16:23 So here's xB at three. 249 00:16:23 --> 00:16:32 Here's yB at three, here's our gas phase. 250 00:16:32 --> 00:16:38 Now we could decrease even further. 251 00:16:38 --> 00:16:42 And this is the sort of thing that you maybe 252 00:16:42 --> 00:16:42 can't do in real life. 253 00:16:42 --> 00:16:45 But I can do on a blackboard. 254 00:16:45 --> 00:16:48 I'm going to give myself more room on this curve, to 255 00:16:48 --> 00:16:59 finish this illustration. 256 00:16:59 --> 00:17:01 There. 257 00:17:01 --> 00:17:02 Beautiful. 258 00:17:02 --> 00:17:06 So now we can lower a little bit further, and what I want to 259 00:17:06 --> 00:17:09 illustrate is, if we keep going down, eventually we get to a 260 00:17:09 --> 00:17:15 pressure where now if we look over in the gas phase, we're at 261 00:17:15 --> 00:17:18 the same pressure, mole fraction that we had originally 262 00:17:18 --> 00:17:21 in the liquid phase. 263 00:17:21 --> 00:17:27 So let's make four even lower pressure. 264 00:17:27 --> 00:17:29 What does that mean? 265 00:17:29 --> 00:17:35 What it means is, we're running out of liquid. 266 00:17:35 --> 00:17:40 So what's supposed to happen is A is the more 267 00:17:40 --> 00:17:41 volatile component. 268 00:17:41 --> 00:17:47 So as we start opening up some room for gas to form, you get 269 00:17:47 --> 00:17:50 more of A in the gas phase. 270 00:17:50 --> 00:17:55 But of course, and the liquid is richer in B. 271 00:17:55 --> 00:17:58 But of course, eventually you run out of liquid. 272 00:17:58 --> 00:18:05 You make the box pretty big, and you run out, or you have 273 00:18:05 --> 00:18:08 the very last drop of liquid. 274 00:18:08 --> 00:18:12 So what's the mole fraction of B in the gas phase? 275 00:18:12 --> 00:18:14 It has to be the same as what it started in 276 00:18:14 --> 00:18:15 in the liquid phase. 277 00:18:15 --> 00:18:17 Because after all the total number of moles of A and 278 00:18:17 --> 00:18:18 B hasn't changed any. 279 00:18:18 --> 00:18:21 So if you take them all from the liquid and put them all 280 00:18:21 --> 00:18:25 up into the gas phase, it must be the same. 281 00:18:25 --> 00:18:30 So yB of four. 282 00:18:30 --> 00:18:38 Once you just have the last drop. 283 00:18:38 --> 00:18:45 So then yB of four is basically equal to xB of one. 284 00:18:45 --> 00:18:49 Because everything's now up in the gas phase. 285 00:18:49 --> 00:18:50 So in principle, there's still a tiny, tiny bit 286 00:18:50 --> 00:19:00 of xB at pressure four. 287 00:19:00 --> 00:19:01 Well, we could keep lowering the pressure. 288 00:19:01 --> 00:19:03 We could make the box a little bigger. 289 00:19:03 --> 00:19:06 Then the very last of the liquid is going to be gone. 290 00:19:06 --> 00:19:09 And what'll happen then is, we're all here. 291 00:19:09 --> 00:19:10 There's no more liquid. 292 00:19:10 --> 00:19:13 We're not going down on the coexistence curve any more. 293 00:19:13 --> 00:19:15 We don't have a liquid gas coexistence any more. 294 00:19:15 --> 00:19:17 We just have a gas phase. 295 00:19:17 --> 00:19:20 Of course, we can continue to lower the pressure. 296 00:19:20 --> 00:19:24 And then what we're doing is just going down here. 297 00:19:24 --> 00:19:26 So there's five. 298 00:19:26 --> 00:19:32 And five is the same as this only bigger. 299 00:19:32 --> 00:19:34 And so forth. 300 00:19:34 --> 00:19:40 OK, any questions about how this works? 301 00:19:40 --> 00:19:45 It's really important to just gain facility in reading these 302 00:19:45 --> 00:19:51 things and seeing, OK, what is it that this is telling you. 303 00:19:51 --> 00:19:54 And you can see it's not complicated to do it, but it 304 00:19:54 --> 00:19:58 takes a little bit of practice. 305 00:19:58 --> 00:19:59 OK. 306 00:19:59 --> 00:20:02 Now, of course, we could do exactly the same thing 307 00:20:02 --> 00:20:04 starting from the gas phase. 308 00:20:04 --> 00:20:06 And raising the pressure. 309 00:20:06 --> 00:20:10 And although you may anticipate that it's kind of pedantic, I 310 00:20:10 --> 00:20:12 really do want to illustrate something by it. 311 00:20:12 --> 00:20:14 So let me just imagine that we're going to do that. 312 00:20:14 --> 00:20:34 Let's start all in the gas phase. 313 00:20:34 --> 00:20:49 Up here's the liquid. pA star, pB star. 314 00:20:49 --> 00:20:55 And now let's start somewhere here. 315 00:20:55 --> 00:20:58 So we're down somewhere in the gas phase with 316 00:20:58 --> 00:20:59 some composition. 317 00:20:59 --> 00:21:02 So it's the same story, except now we're starting here. 318 00:21:02 --> 00:21:02 It's all gas. 319 00:21:02 --> 00:21:04 And we're going to start squeezing. 320 00:21:04 --> 00:21:06 We're increasing the pressure. 321 00:21:06 --> 00:21:13 And eventually here's one, will reach two, so of 322 00:21:13 --> 00:21:18 course here's our yB. 323 00:21:18 --> 00:21:23 We started with all gas, no liquid. 324 00:21:23 --> 00:21:24 So this is yB of one. 325 00:21:24 --> 00:21:27 It's the same as yB of two, I'm just raising the pressure 326 00:21:27 --> 00:21:32 enough to just reach the coexistence curve. 327 00:21:32 --> 00:21:40 And of course, out here tells us xB of two, right? 328 00:21:40 --> 00:21:42 So what is it saying? 329 00:21:42 --> 00:21:45 We've squeezed and started to form some liquid. 330 00:21:45 --> 00:21:50 And the liquid is richer in component B. 331 00:21:50 --> 00:21:51 Maybe it's ethanol water again. 332 00:21:51 --> 00:21:54 And we squeeze, and now we've got more water in the liquid 333 00:21:54 --> 00:21:56 phase than in the gas phase. 334 00:21:56 --> 00:21:58 Because water's the less volatile component. 335 00:21:58 --> 00:22:04 It's what's going to condense first. 336 00:22:04 --> 00:22:08 So the liquid is rich in the less volatile 337 00:22:08 --> 00:22:11 of the components. 338 00:22:11 --> 00:22:14 Now, obviously, we can continue in doing exactly the reverse 339 00:22:14 --> 00:22:15 of what I showed you. 340 00:22:15 --> 00:22:18 But all I want to really illustrate is, this is a 341 00:22:18 --> 00:22:24 strategy for purification of the less volatile component. 342 00:22:24 --> 00:22:28 Once you've done this, well now you've got some liquid. 343 00:22:28 --> 00:22:33 Now you could collect that liquid in a separate vessel. 344 00:22:33 --> 00:22:46 So let's collect the liquid mixture with xB of two. 345 00:22:46 --> 00:22:48 So it's got some mole fraction of B. 346 00:22:48 --> 00:22:50 So we've purified that. 347 00:22:50 --> 00:22:52 But now we're going to start, we've got pure liquid. 348 00:22:52 --> 00:22:54 Now let's make the vessel big. 349 00:22:54 --> 00:23:00 So it all goes into the gas phase. 350 00:23:00 --> 00:23:04 Then lower p. 351 00:23:04 --> 00:23:06 All gas. 352 00:23:06 --> 00:23:12 So we start with yB of three, which equals xB of two. 353 00:23:12 --> 00:23:16 In other words, it's the same mole fraction. 354 00:23:16 --> 00:23:32 So let's reconstruct that. 355 00:23:32 --> 00:23:35 So here's p of two. 356 00:23:35 --> 00:23:41 And now we're going to go to some new pressure. 357 00:23:41 --> 00:23:47 And the point is, now we're going to start, since the mole 358 00:23:47 --> 00:23:49 fraction in the gas phase that we're starting from is the 359 00:23:49 --> 00:23:51 same number as this was. 360 00:23:51 --> 00:23:54 So it's around here somewhere. 361 00:23:54 --> 00:24:01 That's yB of three equals xB of two. 362 00:24:01 --> 00:24:04 And we're down here. 363 00:24:04 --> 00:24:08 In other words, all we've done is make the container big 364 00:24:08 --> 00:24:11 enough so the pressure's low and it's all in the gas phase. 365 00:24:11 --> 00:24:15 That's all we have, is the gas. 366 00:24:15 --> 00:24:17 But the composition is whatever the composition is that we 367 00:24:17 --> 00:24:20 extracted here from the liquid. 368 00:24:20 --> 00:24:25 So this xB, which is the liquid mole fraction, is now yB, 369 00:24:25 --> 00:24:26 the gas mole fraction. 370 00:24:26 --> 00:24:27 Of course, the pressure is different. 371 00:24:27 --> 00:24:30 Lower than it was before. 372 00:24:30 --> 00:24:31 Great. 373 00:24:31 --> 00:24:34 Now let's increase. 374 00:24:34 --> 00:24:37 So here's three. 375 00:24:37 --> 00:24:41 And now let's increase the pressure to four. 376 00:24:41 --> 00:24:45 And of course what happens, now we've got coexistence. 377 00:24:45 --> 00:24:48 So here's liquid. 378 00:24:48 --> 00:24:49 Here's gas. 379 00:24:49 --> 00:24:57 So, now we're over here again. 380 00:24:57 --> 00:25:02 There's xB at pressure four. 381 00:25:02 --> 00:25:08 Pure still in component B. 382 00:25:08 --> 00:25:10 We can repeat the same procedure. 383 00:25:10 --> 00:25:12 Collect it. 384 00:25:12 --> 00:25:15 All liquid, put it in a new vessel. 385 00:25:15 --> 00:25:18 Expand it, lower the pressure, all goes back 386 00:25:18 --> 00:25:20 into the gas phase. 387 00:25:20 --> 00:25:21 Do it all again. 388 00:25:21 --> 00:25:24 And the point is, what you're doing is walking along here. 389 00:25:24 --> 00:25:25 Here to here. 390 00:25:25 --> 00:25:28 Then you start down here, and go from here to here. 391 00:25:28 --> 00:25:29 From here to here. 392 00:25:29 --> 00:25:31 And you can purify. 393 00:25:31 --> 00:25:36 Now, of course, the optimal procedure, you have to 394 00:25:36 --> 00:25:36 think a little bit. 395 00:25:36 --> 00:25:40 Because if you really do precisely what I said, you're 396 00:25:40 --> 00:25:43 going to have a mighty little bit of material each 397 00:25:43 --> 00:25:44 time you do that. 398 00:25:44 --> 00:25:47 So yes it'll be the little bit you've gotten at the end is 399 00:25:47 --> 00:25:52 going to be really pure, but there's not a whole lot of it. 400 00:25:52 --> 00:25:56 Because, remember, what we said is let's raise the pressure 401 00:25:56 --> 00:26:00 until we just start being on the coexistence curve. 402 00:26:00 --> 00:26:03 So we've still got mostly gas. 403 00:26:03 --> 00:26:05 Little bit of liquid. 404 00:26:05 --> 00:26:07 Now, I could raise the pressure a bit higher. 405 00:26:07 --> 00:26:11 So that in the interest of having more of the liquid, when 406 00:26:11 --> 00:26:15 I do that, though, the liquid that I have at this higher 407 00:26:15 --> 00:26:20 pressure won't be as enriched as it was down here. 408 00:26:20 --> 00:26:21 Now, I could still do this procedure. 409 00:26:21 --> 00:26:24 I could just do more of them. 410 00:26:24 --> 00:26:27 So it takes a little bit of judiciousness to figure 411 00:26:27 --> 00:26:29 out how to optimize that. 412 00:26:29 --> 00:26:34 In the end, though, you can continue to walk your way down 413 00:26:34 --> 00:26:38 through these coexistence curves and purify repeatedly 414 00:26:38 --> 00:26:42 the component B, the less volatile of them, and end 415 00:26:42 --> 00:26:43 up with some amount of it. 416 00:26:43 --> 00:26:46 And there'll be some balance between the amount that you 417 00:26:46 --> 00:26:48 feel like you need to end up with and how pure 418 00:26:48 --> 00:26:54 you need it to be. 419 00:26:54 --> 00:26:58 Any questions about how this works? 420 00:26:58 --> 00:27:16 So purification of less volatile components. 421 00:27:16 --> 00:27:20 Now, how much of each of these quantities in 422 00:27:20 --> 00:27:21 each of these phases? 423 00:27:21 --> 00:27:25 So, pertinent to this discussion, of course 424 00:27:25 --> 00:27:26 we need to know that. 425 00:27:26 --> 00:27:30 If you want to try to optimize a procedure like that, of 426 00:27:30 --> 00:27:32 course it's going to be crucial to be able to understand and 427 00:27:32 --> 00:27:40 calculate for any pressure that you decide to raise to, just 428 00:27:40 --> 00:27:43 how many moles do you have in each of the phases? 429 00:27:43 --> 00:27:47 So at the end of the day, you can figure out, OK, now when I 430 00:27:47 --> 00:27:49 reach a certain degree of purification, here's how much 431 00:27:49 --> 00:27:52 of the stuff I end up with. 432 00:27:52 --> 00:27:55 Well, that turns out to be reasonably 433 00:27:55 --> 00:27:57 straightforward to do. 434 00:27:57 --> 00:28:00 And so what I'll go through is a simple mathematical 435 00:28:00 --> 00:28:01 derivation. 436 00:28:01 --> 00:28:05 And it turns out that it allows you to just read right off the 437 00:28:05 --> 00:28:09 diagram how much of each material you're going 438 00:28:09 --> 00:28:11 to end up with. 439 00:28:11 --> 00:28:15 So, here's what happens. 440 00:28:15 --> 00:28:21 This is something called the lever rule. 441 00:28:21 --> 00:28:39 How much of each component is there in each phase? 442 00:28:39 --> 00:28:52 So let's consider a case like this. 443 00:28:52 --> 00:28:56 Let me draw yet once again, just to get the 444 00:28:56 --> 00:29:00 numbering consistent. 445 00:29:00 --> 00:29:07 With how we'll treat this. 446 00:29:07 --> 00:29:08 So we're going to start here. 447 00:29:08 --> 00:29:09 And I want to draw it right in the middle, so I've 448 00:29:09 --> 00:29:14 got plenty of room. 449 00:29:14 --> 00:29:26 And we're going to go up to some pressure. 450 00:29:26 --> 00:29:29 And somewhere out there, now I can go to my 451 00:29:29 --> 00:29:31 coexistence curves. 452 00:29:31 --> 00:29:33 Liquid. 453 00:29:33 --> 00:29:34 And gas. 454 00:29:34 --> 00:29:40 And I can read off my values. 455 00:29:40 --> 00:29:44 So this is the liquid xB. 456 00:29:44 --> 00:29:52 So I'm going to go up to some point two, here's xB of two. 457 00:29:52 --> 00:29:57 Here's yB of two. 458 00:29:57 --> 00:30:00 Great. 459 00:30:00 --> 00:30:10 Now let's get these written in. 460 00:30:10 --> 00:30:17 So let's just define terms a little bit. nA, nB. 461 00:30:17 --> 00:30:28 Or just our total number of moles. ng and n liquid, of 462 00:30:28 --> 00:30:38 course, total number of moles. 463 00:30:38 --> 00:30:40 In the gas and liquid phases. 464 00:30:40 --> 00:30:44 So let's just do the calculation for each 465 00:30:44 --> 00:30:45 of these two cases. 466 00:30:45 --> 00:30:46 We'll start with one. 467 00:30:46 --> 00:30:47 That's the easier case. 468 00:30:47 --> 00:30:51 Because then we have only the gas. 469 00:30:51 --> 00:30:57 So at one, all gas. 470 00:30:57 --> 00:30:59 It says pure gas in the notes, but of course 471 00:30:59 --> 00:31:00 that isn't the pure gas. 472 00:31:00 --> 00:31:04 It's the mixture of the two components. 473 00:31:04 --> 00:31:07 So. 474 00:31:07 --> 00:31:09 How many moles of A? 475 00:31:09 --> 00:31:12 Well it's the mole fraction of A in the gas. 476 00:31:12 --> 00:31:17 Times the total number of moles in the gas. 477 00:31:17 --> 00:31:19 Let me put one in here. 478 00:31:19 --> 00:31:25 Just to be clear. 479 00:31:25 --> 00:31:28 And since we have all gas, the number of moles in the gas is 480 00:31:28 --> 00:31:30 just the total number of moles. 481 00:31:30 --> 00:31:37 So this is just yA at one times n total. 482 00:31:37 --> 00:31:38 Let's just write that in. 483 00:31:38 --> 00:31:43 And of course n total is equal to nA plus nB. 484 00:31:43 --> 00:31:47 485 00:31:47 --> 00:31:51 So now let's look at condition two. 486 00:31:51 --> 00:31:54 Now we have to look a little more carefully. 487 00:31:54 --> 00:32:03 Because we have a liquid gas mixture. 488 00:32:03 --> 00:32:11 So nA is equal to yA at pressure two. 489 00:32:11 --> 00:32:19 Times the number of moles of gas at pressure two. 490 00:32:19 --> 00:32:23 Plus xA, at pressure two, times the number of moles 491 00:32:23 --> 00:32:38 of liquid at pressure two. 492 00:32:38 --> 00:32:40 Now, of course, these things have to be equal. 493 00:32:40 --> 00:32:45 The total number of moles of A didn't change, right? 494 00:32:45 --> 00:32:52 So those are equal. 495 00:32:52 --> 00:32:58 Then yA of two times ng of two. 496 00:32:58 --> 00:33:07 Plus xA of two times n liquid of two, that's equal to 497 00:33:07 --> 00:33:13 yA of one times n total. 498 00:33:13 --> 00:33:19 Which is of course equal to yA of one times n gas at 499 00:33:19 --> 00:33:24 two plus n liquid at two. 500 00:33:24 --> 00:33:27 I suppose I could be, add that equality. 501 00:33:27 --> 00:33:28 Of course, it's an obvious one. 502 00:33:28 --> 00:33:30 But let me do it anyway. 503 00:33:30 --> 00:33:32 The total number of moles is equal to nA plus nB. 504 00:33:32 --> 00:33:40 But it's also equal to n liquid plus n gas. 505 00:33:40 --> 00:33:46 And that's all I'm taking advantage of here. 506 00:33:46 --> 00:33:47 And now I'm just going to rearrange the terms. 507 00:33:47 --> 00:33:58 So I'm going to write yA at one minus yA at two, times ng at 508 00:33:58 --> 00:34:03 two, is equal to, and I'm going to take the other terms, the xA 509 00:34:03 --> 00:34:18 term. xA of two minus yA of one times n liquid at two. 510 00:34:18 --> 00:34:21 So I've just rearranged the terms. 511 00:34:21 --> 00:34:30 And I've done that because now, I think I omitted something 512 00:34:30 --> 00:34:37 here. yA of one times ng. 513 00:34:37 --> 00:34:43 514 00:34:43 --> 00:34:51 No, I forgot a bracket, is what I did. yA of one there. 515 00:34:51 --> 00:34:53 And I did this because now I want to do is look at 516 00:34:53 --> 00:34:57 the ratio of liquid to gas at pressure two. 517 00:34:57 --> 00:35:09 So, ratio of I'll put it gas to liquid, that's ng of 518 00:35:09 --> 00:35:15 two over n liquid at two. 519 00:35:15 --> 00:35:26 And that's just equal to xA of two minus yA at one minus 520 00:35:26 --> 00:35:37 yA at one minus yA at two. 521 00:35:37 --> 00:35:38 So what does it mean? 522 00:35:38 --> 00:35:45 It's the ratio of these lever arms. 523 00:35:45 --> 00:35:46 That's what it's telling me. 524 00:35:46 --> 00:35:51 I can look, so I raise the pressure up to two. 525 00:35:51 --> 00:35:55 And so here's xB at two, here's yB at two. 526 00:35:55 --> 00:35:57 And I'm here somewhere. 527 00:35:57 --> 00:36:05 And this little amount and this little amount, 528 00:36:05 --> 00:36:09 that's that difference. 529 00:36:09 --> 00:36:13 And it's just telling me that ratio of those arms is the 530 00:36:13 --> 00:36:16 ratio of the total number of moles of gas to liquid. 531 00:36:16 --> 00:36:18 And that's great. 532 00:36:18 --> 00:36:21 Because now when I go back to the problem that we were just 533 00:36:21 --> 00:36:24 looking at, where I say, well I'm going to purify the less 534 00:36:24 --> 00:36:27 volatile component by raising the pressure until I'm at 535 00:36:27 --> 00:36:29 coexistence starting in the gas phase. 536 00:36:29 --> 00:36:31 Raise the pressure, I've got some liquid. 537 00:36:31 --> 00:36:34 But I also want some finite amount of liquid. 538 00:36:34 --> 00:36:37 But I don't want to just, when I get the very, very first drop 539 00:36:37 --> 00:36:39 of liquid now collected, of course it's enriched in the 540 00:36:39 --> 00:36:40 less volatile component. 541 00:36:40 --> 00:36:43 But there may be a minuscule amount, right? 542 00:36:43 --> 00:36:47 So I'll raise the pressure a bit more. 543 00:36:47 --> 00:36:48 I'll go up in pressure. 544 00:36:48 --> 00:36:53 And now, of course, when I do that the amount of enrichment 545 00:36:53 --> 00:36:57 of the liquid isn't as big as it was if I just raised it up 546 00:36:57 --> 00:36:59 enough to barely have any liquid. 547 00:36:59 --> 00:37:01 Then I'd be out here. 548 00:37:01 --> 00:37:06 But I've got more material in the liquid phase to collect. 549 00:37:06 --> 00:37:08 And that's what this allows me to calculate. 550 00:37:08 --> 00:37:13 Is how much do I get in the end. 551 00:37:13 --> 00:37:16 So it's very handy. 552 00:37:16 --> 00:37:22 You can also see, if I go all the way to the limit where the 553 00:37:22 --> 00:37:27 mole fraction in the liquid at the end is equal to what it was 554 00:37:27 --> 00:37:32 in the gas when I started, what that says is that there's 555 00:37:32 --> 00:37:34 no more gas left any more. 556 00:37:34 --> 00:37:39 In other words, these two things are equal. 557 00:37:39 --> 00:37:44 If I go all the way to the point where I've got all the, 558 00:37:44 --> 00:37:49 this is the amount I started with, in the pure gas phase, 559 00:37:49 --> 00:37:51 now I keep raising it all the way. 560 00:37:51 --> 00:37:54 Until I've got the same mole fraction in the liquid. 561 00:37:54 --> 00:37:56 Of course, we know what that really means. 562 00:37:56 --> 00:37:59 That means that I've gone all the way from pure 563 00:37:59 --> 00:38:00 gas to pure liquid. 564 00:38:00 --> 00:38:03 And the mole fraction in that case has to be the same. 565 00:38:03 --> 00:38:07 And what this is just telling us mathematically is, when 566 00:38:07 --> 00:38:08 that happens this is zero. 567 00:38:08 --> 00:38:14 That means I don't have any gas left. 568 00:38:14 --> 00:38:15 Yeah. 569 00:38:15 --> 00:38:26 STUDENT: [INAUDIBLE] 570 00:38:26 --> 00:38:26 PROFESSOR: No. 571 00:38:26 --> 00:38:31 Because, so it's the mole fraction in the gas phase. 572 00:38:31 --> 00:38:35 But you've started with some amount that it's only going 573 00:38:35 --> 00:38:38 to go down from there. 574 00:38:38 --> 00:38:41 STUDENT: [INAUDIBLE] 575 00:38:41 --> 00:38:44 PROFESSOR: Yeah. 576 00:38:44 --> 00:38:48 Yeah. 577 00:38:48 --> 00:38:53 Any other questions? 578 00:38:53 --> 00:38:55 OK. 579 00:38:55 --> 00:39:01 Well, now what I want to do is just put up a slightly 580 00:39:01 --> 00:39:05 different kind of diagram, but different in an important way. 581 00:39:05 --> 00:39:09 Namely, instead of showing the mole fractions as a 582 00:39:09 --> 00:39:11 function of the pressure. 583 00:39:11 --> 00:39:14 And I haven't written it in, but all of these are at 584 00:39:14 --> 00:39:16 constant temperature, right? 585 00:39:16 --> 00:39:19 I've assumed the temperature is constant in all these things. 586 00:39:19 --> 00:39:22 Now let's consider the other possibility, the other simple 587 00:39:22 --> 00:39:25 possibility, which is, let's hold the pressure constant 588 00:39:25 --> 00:39:26 and vary the temperature. 589 00:39:26 --> 00:39:29 Of course, you know in the lab, that's usually 590 00:39:29 --> 00:39:31 what's easiest to do. 591 00:39:31 --> 00:39:35 Now, unfortunately, the arithmetic gets 592 00:39:35 --> 00:39:36 more complicated. 593 00:39:36 --> 00:39:41 It's not monumentally complicated, but here in this 594 00:39:41 --> 00:39:45 case, where you have one linear relationship, 595 00:39:45 --> 00:39:46 which is very convenient. 596 00:39:46 --> 00:39:48 From Raoult's law. 597 00:39:48 --> 00:39:51 And then you have one non-linear relationship 598 00:39:51 --> 00:39:54 there for the mole fraction of the gas. 599 00:39:54 --> 00:39:59 In the case of temperature, they're both, neither 600 00:39:59 --> 00:40:01 one is linear. 601 00:40:01 --> 00:40:03 Nevertheless, we can just sketch what the 602 00:40:03 --> 00:40:04 diagram looks like. 603 00:40:04 --> 00:40:06 And of course it's very useful to do that, and 604 00:40:06 --> 00:40:10 see how to read off it. 605 00:40:10 --> 00:40:13 And I should say the derivation of the curves isn't 606 00:40:13 --> 00:40:14 particularly complicated. 607 00:40:14 --> 00:40:16 It's not particularly more complicated than what 608 00:40:16 --> 00:40:19 I think you saw last time to derive this. 609 00:40:19 --> 00:40:21 There's no complicated math involved. 610 00:40:21 --> 00:40:24 But the point is, the derivation doesn't yield a 611 00:40:24 --> 00:40:28 linear relationship for either the gas or the liquid part of 612 00:40:28 --> 00:40:54 the coexistence curve. 613 00:40:54 --> 00:40:57 OK, so we're going to look at temperature and mole 614 00:40:57 --> 00:41:04 fraction phase diagrams. 615 00:41:04 --> 00:41:06 Again, a little more complicated mathematically but 616 00:41:06 --> 00:41:20 more practical in real use. 617 00:41:20 --> 00:41:24 And this is T. 618 00:41:24 --> 00:41:33 And here is the, sort of, form that these things take. 619 00:41:33 --> 00:41:35 So again, neither one is linear. 620 00:41:35 --> 00:41:38 Up here, now, of course if you raise the temperatures, that's 621 00:41:38 --> 00:41:41 where you end up with gas. 622 00:41:41 --> 00:41:42 If you lower the temperature, you condense and 623 00:41:42 --> 00:41:48 get the liquid. 624 00:41:48 --> 00:41:55 So, this is TA star. 625 00:41:55 --> 00:42:00 TB star. 626 00:42:00 --> 00:42:07 So now I want to stick with A as the more volatile component. 627 00:42:07 --> 00:42:10 At constant temperature, that meant that pA star 628 00:42:10 --> 00:42:12 is bigger than pB star. 629 00:42:12 --> 00:42:16 In other words, the vapor pressure over pure liquid A 630 00:42:16 --> 00:42:21 is higher than the vapor pressure over pure liquid B. 631 00:42:21 --> 00:42:24 Similarly, now I've got constant pressure and really 632 00:42:24 --> 00:42:26 what I'm looking at, let's say I'm at the limit where 633 00:42:26 --> 00:42:30 I've got the pure liquid. 634 00:42:30 --> 00:42:32 Or the pure A. 635 00:42:32 --> 00:42:34 And now I'm going to, let's say, raise the temperature 636 00:42:34 --> 00:42:38 until I'm at the liquid-gas equilibrium. 637 00:42:38 --> 00:42:42 That's just the boiling point. 638 00:42:42 --> 00:42:45 So if A is the more volatile component, it has the 639 00:42:45 --> 00:42:47 lower boiling point. 640 00:42:47 --> 00:42:49 And that's what this reflects. 641 00:42:49 --> 00:42:54 So higher pB star A corresponds to lower TA star A. 642 00:42:54 --> 00:42:59 Which is just the boiling point of pure A. 643 00:42:59 --> 00:43:11 So, this is called the bubble line. 644 00:43:11 --> 00:43:14 That's called the dew line. 645 00:43:14 --> 00:43:17 All that means is, let's say I'm at high temperature. 646 00:43:17 --> 00:43:19 I've got all gas. 647 00:43:19 --> 00:43:21 Right no coexistence, no liquid yet. 648 00:43:21 --> 00:43:23 And I start to cool things off. 649 00:43:23 --> 00:43:25 Just to where I just barely start to get liquid. 650 00:43:25 --> 00:43:29 What you see that as is, dew starts forming. 651 00:43:29 --> 00:43:30 A little bit of condensation. 652 00:43:30 --> 00:43:32 If you're outside, it means on the grass a little 653 00:43:32 --> 00:43:34 bit of dew is forming. 654 00:43:34 --> 00:43:38 Similarly, if I start at low temperature, all liquid now I 655 00:43:38 --> 00:43:41 start raising the temperature until I just start to boil. 656 00:43:41 --> 00:43:46 I just start to see the first bubbles forming. 657 00:43:46 --> 00:43:51 And so that's why these things have those names. 658 00:43:51 --> 00:43:56 So now let's just follow along what happens when I do the same 659 00:43:56 --> 00:43:57 sort of thing that I illustrated there. 660 00:43:57 --> 00:44:00 I want to start at one point in this phase diagram. 661 00:44:00 --> 00:44:04 And then start changing the conditions. 662 00:44:04 --> 00:44:17 So let's start here. 663 00:44:17 --> 00:44:19 So I'm going to start all in the liquid phase. 664 00:44:19 --> 00:44:22 That is, the temperature is low. 665 00:44:22 --> 00:44:23 Here's xB. 666 00:44:23 --> 00:44:26 667 00:44:26 --> 00:44:27 And my original temperature. 668 00:44:27 --> 00:44:34 Now I'm going to raise it. 669 00:44:34 --> 00:44:39 So if I raise it a little bit, I reach a point at which 670 00:44:39 --> 00:44:41 I first start to boil. 671 00:44:41 --> 00:44:47 Start to find some gas above the liquid. 672 00:44:47 --> 00:44:51 And if I look right here, that'll be my composition. 673 00:44:51 --> 00:44:53 Let me raise it a little farther, now that we've 674 00:44:53 --> 00:44:57 already seen the lever rule and so forth. 675 00:44:57 --> 00:44:58 I'll raise it up to here. 676 00:44:58 --> 00:45:11 And that means that out here, I suppose I should do here. 677 00:45:11 --> 00:45:18 So, here is the liquid mole fraction at temperature two. 678 00:45:18 --> 00:45:23 xB at temperature two. 679 00:45:23 --> 00:45:27 This is yB at temperature two. 680 00:45:27 --> 00:45:33 The gas mole fraction. 681 00:45:33 --> 00:45:38 So as you should expect, what's going to happen here is that 682 00:45:38 --> 00:45:42 the gas, this is going to be lower in B. 683 00:45:42 --> 00:45:45 A, that means that the mole fraction of A must be 684 00:45:45 --> 00:45:47 higher in the gas phase. 685 00:45:47 --> 00:45:49 That's one minus yB. 686 00:45:49 --> 00:46:03 So xA is one minus -- yA, which is one minus yB 687 00:46:03 --> 00:46:10 higher in gas phase. 688 00:46:10 --> 00:46:16 Than xA, which is one minus xB. 689 00:46:16 --> 00:46:20 In other words, the less volatile component is enriched 690 00:46:20 --> 00:46:39 up in the gas phase. 691 00:46:39 --> 00:46:42 Now, what does that mean? 692 00:46:42 --> 00:46:46 That means I could follow the same sort of procedure that I 693 00:46:46 --> 00:46:50 indicated before when we looked at the pressure mole 694 00:46:50 --> 00:46:51 fraction phase diagram. 695 00:46:51 --> 00:46:57 Namely, I could do this and now I could take the gas phase. 696 00:46:57 --> 00:46:58 Which has less of B. 697 00:46:58 --> 00:47:01 It has more of A. 698 00:47:01 --> 00:47:03 And I can collect it. 699 00:47:03 --> 00:47:05 And then I can reduce the temperature. 700 00:47:05 --> 00:47:06 So it liquefies. 701 00:47:06 --> 00:47:08 So I can condense it, in other words. 702 00:47:08 --> 00:47:11 So now I'm going to start with, let's say I lower the 703 00:47:11 --> 00:47:14 temperature enough so I've got basically pure liquid. 704 00:47:14 --> 00:47:17 But its composition is the same as the gas here. 705 00:47:17 --> 00:47:19 Because of course that's what that liquid is formed from. 706 00:47:19 --> 00:47:23 I collected the gas and separated it. 707 00:47:23 --> 00:47:25 So now I could start all over again. 708 00:47:25 --> 00:47:31 Except instead of being here, I'll be down here. 709 00:47:31 --> 00:47:33 And then I can raise the temperature again. 710 00:47:33 --> 00:47:35 To some place where I choose. 711 00:47:35 --> 00:47:37 I could choose here, and go all the way to hear. 712 00:47:37 --> 00:47:39 A great amount of enrichment. 713 00:47:39 --> 00:47:41 But I know from the lever rule that if I do that, I'm going to 714 00:47:41 --> 00:47:44 have precious little material over here. 715 00:47:44 --> 00:47:46 So I might prefer to raise the temperature a little more. 716 00:47:46 --> 00:47:50 Still get a substantial amount of enrichment. 717 00:47:50 --> 00:47:54 And now I've got, in the gas phase, I'll further 718 00:47:54 --> 00:47:55 enriched in component A. 719 00:47:55 --> 00:47:57 And again I can collect the gas. 720 00:47:57 --> 00:48:00 Condense it. 721 00:48:00 --> 00:48:03 Now I'm out here somewhere, I've got all liquid and I'll 722 00:48:03 --> 00:48:04 raise the temperature again. 723 00:48:04 --> 00:48:07 And I can again keep walking my way over. 724 00:48:07 --> 00:48:11 And that's what happens during an ordinary distillation. 725 00:48:11 --> 00:48:17 Each step of the distillation walks along in the phase 726 00:48:17 --> 00:48:23 diagram at some selected point. 727 00:48:23 --> 00:48:25 And of course what you're doing is, you're always 728 00:48:25 --> 00:48:27 condensing the gas. 729 00:48:27 --> 00:48:30 And starting with fresh liquid that now is enriched in more 730 00:48:30 --> 00:48:33 volatile of the components. 731 00:48:33 --> 00:48:36 So of course if you're really purifying, say, ethanol from 732 00:48:36 --> 00:48:39 an ethanol water mixture, that's how you do it. 733 00:48:39 --> 00:48:41 Ethanol is the more volatile component. 734 00:48:41 --> 00:48:42 So a still is set up. 735 00:48:42 --> 00:48:44 It will boil the stuff and collect the gas 736 00:48:44 --> 00:48:45 and and condense it. 737 00:48:45 --> 00:48:48 And boil it again, and so forth. 738 00:48:48 --> 00:48:50 And the whole thing can be set up in a very efficient way. 739 00:48:50 --> 00:48:53 So you have essentially continuous distillation. 740 00:48:53 --> 00:48:57 Where you have a whole sequence of collection and condensation 741 00:48:57 --> 00:49:00 and reheating and so forth events. 742 00:49:00 --> 00:49:04 So then, in a practical way, it's possible to walk quite far 743 00:49:04 --> 00:49:11 along the distillation, the coexistence curve, and distill 744 00:49:11 --> 00:49:16 to really a high degree of purification. 745 00:49:16 --> 00:49:22 Any questions about how that works? 746 00:49:22 --> 00:49:24 OK. 747 00:49:24 --> 00:49:29 I'll leave till next time the discussion of the 748 00:49:29 --> 00:49:30 chemical potentials. 749 00:49:30 --> 00:49:33 But what we'll do, just to foreshadow a little bit, what 750 00:49:33 --> 00:49:36 I'll do at the beginning of the next lecture is what's at 751 00:49:36 --> 00:49:37 the end of your notes here. 752 00:49:37 --> 00:49:40 Which is just to say OK, now if we look at Raoult's law, it's 753 00:49:40 --> 00:49:44 straightforward to say what is the chemical potential for 754 00:49:44 --> 00:49:47 each of the substances in the liquid and the gas phase. 755 00:49:47 --> 00:49:50 Of course, it has to be equal. 756 00:49:50 --> 00:49:53 Given that, that's for an ideal solution. 757 00:49:53 --> 00:49:55 We can gain some insight from that. 758 00:49:55 --> 00:49:58 And then look at real solutions, non-ideal solutions, 759 00:49:58 --> 00:50:00 and understand a lot of their behavior as well. 760 00:50:00 --> 00:50:03 Just from starting from our understanding of what the 761 00:50:03 --> 00:50:06 chemical potential does even in a simple ideal mixture. 762 00:50:06 --> 00:50:08 So we'll look at the chemical potentials. 763 00:50:08 --> 00:50:10 And then we'll look at non-ideal solution 764 00:50:10 --> 00:50:12 mixtures next time. 765 00:50:12 --> 00:50:13 See you then.