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 --> 00:00:16 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16 --> 00:00:21 at ocw.mit.edu. 9 00:00:21 --> 00:00:25 PROFESSOR: Let's do a one minute review, and then move 10 00:00:25 --> 00:00:27 onto the Clausius-Clapeyron equation and see how 11 00:00:27 --> 00:00:29 far we can go on that. 12 00:00:29 --> 00:00:33 So, reminding you then, what you learned last time about 13 00:00:33 --> 00:00:42 phase transitions is, you drew a phase diagram in temperature 14 00:00:42 --> 00:00:45 pressure space here. 15 00:00:45 --> 00:00:49 And then you had a triple point somewhere, which 16 00:00:49 --> 00:00:51 was a unique point. 17 00:00:51 --> 00:00:56 The triple point temperature, triple point pressure. 18 00:00:56 --> 00:01:05 Then there was a gas, solid, call it coexistence line. 19 00:01:05 --> 00:01:08 Which kept on going ad infinitum. 20 00:01:08 --> 00:01:13 And the a gas liquid coexistent line. 21 00:01:13 --> 00:01:20 With a critical point. 22 00:01:20 --> 00:01:23 So that allowed you to go around. 23 00:01:23 --> 00:01:24 You wanted to go from liquid to gas, you could actually go 24 00:01:24 --> 00:01:27 around this critical point, and never actually see a 25 00:01:27 --> 00:01:30 phase transition. 26 00:01:30 --> 00:01:35 Then there was a solid liquid coexistence line. 27 00:01:35 --> 00:01:41 Which usually has a negative slope, except for the two most 28 00:01:41 --> 00:01:47 important substances on Earth, which are water and silicon. 29 00:01:47 --> 00:01:56 So for water and silicon, we have positive slope. 30 00:01:56 --> 00:01:59 H2O and silicon. 31 00:01:59 --> 00:02:07 This is why there's life on Earth. 32 00:02:07 --> 00:02:13 Did you hear the story of why there's life on Earth? 33 00:02:13 --> 00:02:15 This is the secret of life. 34 00:02:15 --> 00:02:19 Without this, this is why we're here. 35 00:02:19 --> 00:02:22 Because of this property. 36 00:02:22 --> 00:02:31 So, the reason why these slopes come from the, so I'll tell 37 00:02:31 --> 00:02:32 you the secret of life. 38 00:02:32 --> 00:02:38 But first let me remind you what the coexistence 39 00:02:38 --> 00:02:42 curve is. dp/dT. 40 00:02:42 --> 00:02:49 Coexistence is delta S over delta V, or delta 41 00:02:49 --> 00:02:54 H over T delta V. 42 00:02:54 --> 00:03:00 And that's the Clapeyron equation. 43 00:03:00 --> 00:03:05 Now, the slope of the curves points us by, look at this 44 00:03:05 --> 00:03:10 solid liquid line, delta S, for solid liquid. 45 00:03:10 --> 00:03:11 I looked at it up here. 46 00:03:11 --> 00:03:17 It's going to be S liquid minus S solid. 47 00:03:17 --> 00:03:19 And you know that's greater than zero because the entropy 48 00:03:19 --> 00:03:22 in the liquid state is much bigger than it is in 49 00:03:22 --> 00:03:22 the solid state. 50 00:03:22 --> 00:03:25 You have a crystal, entropy's very small. 51 00:03:25 --> 00:03:28 Liquid state, much more disorder. 52 00:03:28 --> 00:03:30 Got a positive sign here. 53 00:03:30 --> 00:03:31 Yes. 54 00:03:31 --> 00:03:31 Questions? 55 00:03:31 --> 00:03:40 STUDENT: [INAUDIBLE] 56 00:03:40 --> 00:03:44 PROFESSOR: So, this is almost everything. 57 00:03:44 --> 00:03:46 It has this slope here. 58 00:03:46 --> 00:03:51 Which is a negative slope. 59 00:03:51 --> 00:03:55 And water has a positive slope. 60 00:03:55 --> 00:03:56 Right? 61 00:03:56 --> 00:04:01 Is that wrong? 62 00:04:01 --> 00:04:03 It's wrong. 63 00:04:03 --> 00:04:04 Let's go through the argument. 64 00:04:04 --> 00:04:05 All right. 65 00:04:05 --> 00:04:06 All right. 66 00:04:06 --> 00:04:10 Let's go through the argument. 67 00:04:10 --> 00:04:18 So, then you have delta V, and delta V is V solid minus V 68 00:04:18 --> 00:04:21 liquid, this is per mole. 69 00:04:21 --> 00:04:29 Now, almost every substance has V solid less than V liquid. 70 00:04:29 --> 00:04:31 So this is negative. 71 00:04:31 --> 00:04:33 So almost every substance, you have delta S divided 72 00:04:33 --> 00:04:35 by delta V is negative. 73 00:04:35 --> 00:04:41 A negative slope. 74 00:04:41 --> 00:04:42 You're right. 75 00:04:42 --> 00:04:44 So I have it backwards. 76 00:04:44 --> 00:04:47 I do have it backwards. 77 00:04:47 --> 00:04:49 OK. 78 00:04:49 --> 00:04:52 See, I was really trying to make it so that my 79 00:04:52 --> 00:04:53 drawing was right. 80 00:04:53 --> 00:04:55 Which is why I inverted the solid and liquid here. 81 00:04:55 --> 00:05:00 So I really wanted to be right, but I ended up being wrong. 82 00:05:00 --> 00:05:05 Liquid is less than solid. 83 00:05:05 --> 00:05:09 Well this is water. 84 00:05:09 --> 00:05:11 Why do I have it backwards in my notes. 85 00:05:11 --> 00:05:17 I've got to correct that. 86 00:05:17 --> 00:05:19 So this is most substances, is positive. 87 00:05:19 --> 00:05:30 OK so, and most substances, OK? 88 00:05:30 --> 00:05:33 Except for water and silicon. 89 00:05:33 --> 00:05:39 Because for water and silicon this is reversed. 90 00:05:39 --> 00:05:42 Because the molar volume of solid for water is bigger 91 00:05:42 --> 00:05:44 than that for liquid. 92 00:05:44 --> 00:05:46 So this is bigger than here. 93 00:05:46 --> 00:05:47 Thank you. 94 00:05:47 --> 00:05:47 OK. 95 00:05:47 --> 00:05:49 Thanks for catching it. 96 00:05:49 --> 00:05:53 OK, so what happens if the molar volume for the liquid 97 00:05:53 --> 00:05:56 is bigger than the molar volume of the solid. 98 00:05:56 --> 00:06:01 The density of water which is the inverse of the mole volume, 99 00:06:01 --> 00:06:05 so the density, the molar density, is equal to one 100 00:06:05 --> 00:06:07 over the molar volume. 101 00:06:07 --> 00:06:11 So you've got the molar density for liquid is then smaller than 102 00:06:11 --> 00:06:14 the molar density for the solid. 103 00:06:14 --> 00:06:16 It's one over, right? 104 00:06:16 --> 00:06:18 So that's why ice floats. 105 00:06:18 --> 00:06:23 Because it's less dense than liquid water. 106 00:06:23 --> 00:06:28 So what do you think would happen in the winter if 107 00:06:28 --> 00:06:33 ice didn't float, in the Charles River. 108 00:06:33 --> 00:06:34 It would go to the bottom. 109 00:06:34 --> 00:06:38 It would freeze, solid and you would have no fish. 110 00:06:38 --> 00:06:39 You would have no life. 111 00:06:39 --> 00:06:45 It would freeze solid and that would be the end of life. 112 00:06:45 --> 00:06:47 And you know, five hundred million years ago there 113 00:06:47 --> 00:06:49 would be no fish. 114 00:06:49 --> 00:06:51 There would be no dinosaurs. 115 00:06:51 --> 00:06:53 There would be no anything, there would be no humans. 116 00:06:53 --> 00:06:55 That's the secret of life. 117 00:06:55 --> 00:06:58 And the fact that this is also the case is very 118 00:06:58 --> 00:07:06 useful for making silicon. 119 00:07:06 --> 00:07:10 For processing silicon. 120 00:07:10 --> 00:07:11 Another the secret of life. 121 00:07:11 --> 00:07:13 Where would we be without silicon? 122 00:07:13 --> 00:07:16 Our civilization would be, we'd still be using 123 00:07:16 --> 00:07:18 stones and things, right? 124 00:07:18 --> 00:07:25 OK, so it's super super important. 125 00:07:25 --> 00:07:29 Alright, so this is a Clapeyron story. 126 00:07:29 --> 00:07:33 Which we ended up getting right. 127 00:07:33 --> 00:07:37 Even though I insisted on getting it wrong. 128 00:07:37 --> 00:07:40 So what happens now is Mr. Clausius came around and he 129 00:07:40 --> 00:07:43 realized that you can make some approximations 130 00:07:43 --> 00:07:44 that are very useful. 131 00:07:44 --> 00:07:48 So the first approximation is, you can realize that the molar 132 00:07:48 --> 00:07:53 volume of the gas is always much bigger than the volume 133 00:07:53 --> 00:07:58 of the solid or the liquid. 134 00:07:58 --> 00:08:03 And so as a result whenever you have these delta V's, for 135 00:08:03 --> 00:08:15 sublimation, or delta V for vaporization, which is the 136 00:08:15 --> 00:08:18 volume the gas minus the volume of the solid. 137 00:08:18 --> 00:08:21 Or the volume of the gas minus the volume of the liquid, you 138 00:08:21 --> 00:08:24 might as well ignore the volume of the liquid or of the solid. 139 00:08:24 --> 00:08:27 And this can just be equal to roughly the volume of the gas. 140 00:08:27 --> 00:08:30 So this is the first approximation he made. 141 00:08:30 --> 00:08:33 So now, if you go back to the to the Clapeyron equation up 142 00:08:33 --> 00:08:39 here, with this approximation then dp/dT, the coexistence 143 00:08:39 --> 00:08:48 line, is delta H divided by T V gas. 144 00:08:48 --> 00:08:51 For sublimation. 145 00:08:51 --> 00:08:52 Or vaporization. 146 00:08:52 --> 00:08:56 You don't have the delta V down there any more. 147 00:08:56 --> 00:09:00 Then the next thing that, next assumption that he realized you 148 00:09:00 --> 00:09:04 can make was, well, all these gases could, they're 149 00:09:04 --> 00:09:05 like ideal gases. 150 00:09:05 --> 00:09:10 So we can, instead of having the volume of the gas here, we 151 00:09:10 --> 00:09:11 can use the ideal gas law. 152 00:09:11 --> 00:09:16 This is all, let's do all this per mole. 153 00:09:16 --> 00:09:21 So V is equal to RT over p. 154 00:09:21 --> 00:09:24 You can plug this back in here. 155 00:09:24 --> 00:09:28 And then you end up with the Clausius-Clapeyron equation. 156 00:09:28 --> 00:09:29 So this is approximation now. 157 00:09:29 --> 00:09:31 We put in RT over p for here. 158 00:09:31 --> 00:09:39 We have p delta H, where this is either sublimation or 159 00:09:39 --> 00:09:51 vaporization divided by RT squared. dp/dT, sublimation or 160 00:09:51 --> 00:10:00 vaporization, and this is the Clausius-Clapeyron equation. 161 00:10:00 --> 00:10:04 Two important approximations that go in here. 162 00:10:04 --> 00:10:08 And it's not valid for solids or liquid. 163 00:10:08 --> 00:10:11 You have to have a gas in there because of the ideal gas 164 00:10:11 --> 00:10:17 for the approximation that goes in here. 165 00:10:17 --> 00:10:22 And once you have that, you can integrate both sides. 166 00:10:22 --> 00:10:24 You've got to put the temperatures on one side. 167 00:10:24 --> 00:10:29 Put the pressures on the other side. 168 00:10:29 --> 00:10:33 I'll go ahead and cover this up here. 169 00:10:33 --> 00:10:39 So you have one over p dp/dT is equal to delta 170 00:10:39 --> 00:10:43 H over RT squared. 171 00:10:43 --> 00:10:47 One over p dp/dT that's just like d log p. 172 00:10:47 --> 00:10:50 And that's another form that you're going to see often for 173 00:10:50 --> 00:10:54 the Clausius-Clapeyron approximation is d log p / dT 174 00:10:54 --> 00:10:58 is equal to delta H over RT squared. 175 00:10:58 --> 00:11:01 And I'm freely dropping the sublimation and the 176 00:11:01 --> 00:11:03 vaporization because we know that's what I mean here. 177 00:11:03 --> 00:11:06 It's only valid for those two lines. 178 00:11:06 --> 00:11:10 So that's another form that you'll see. 179 00:11:10 --> 00:11:16 And that's not dT here, that's d log p, d log p dp. 180 00:11:16 --> 00:11:20 Yeah, that's right, d log p / dT. 181 00:11:20 --> 00:11:21 And so now you can take the dT to the other 182 00:11:21 --> 00:11:23 side and integrate. 183 00:11:23 --> 00:11:28 From point one to point two d log p, is from point one 184 00:11:28 --> 00:11:37 to point two, delta H over RT squared dT. 185 00:11:37 --> 00:11:40 And you make the usual approximation, which 186 00:11:40 --> 00:11:41 we've made before. 187 00:11:41 --> 00:11:46 That delta H is roughly independent of temperature, 188 00:11:46 --> 00:11:51 or very slowly changing with temperature. 189 00:11:51 --> 00:11:52 And that's fine as long as you're in a narrow 190 00:11:52 --> 00:11:54 temperature range. 191 00:11:54 --> 00:11:56 Which is usually the case when you have these 192 00:11:56 --> 00:11:58 Clausius-Clapeyron equation problems. 193 00:11:58 --> 00:12:01 So you can take the delta H out of the integral, the 194 00:12:01 --> 00:12:01 R out of the integral. 195 00:12:01 --> 00:12:03 And you can integrate both sides. 196 00:12:03 --> 00:12:17 To give you log p2 minus log p1, is delta H times R, divided 197 00:12:17 --> 00:12:21 by R, delta H divided by R. 198 00:12:21 --> 00:12:30 Delta H over R times T2 minus T1 over T1 T2. 199 00:12:30 --> 00:12:33 It looks a lot like and the temperature dependence 200 00:12:33 --> 00:12:36 equilibrium constant. 201 00:12:36 --> 00:12:40 And in fact, but it's not, right? 202 00:12:40 --> 00:12:43 And some people use that equation here instead of 203 00:12:43 --> 00:12:44 equilibrium constant. 204 00:12:44 --> 00:12:45 Temperature dependence. 205 00:12:45 --> 00:12:48 That's, can give rise to problems. 206 00:12:48 --> 00:12:50 Because obviously they're not the same equation. 207 00:12:50 --> 00:12:56 But roughly has the same form. 208 00:12:56 --> 00:13:01 So, another way that you'll also find this written 209 00:13:01 --> 00:13:05 is very often p1 and T1 will be some reference. 210 00:13:05 --> 00:13:08 Pressures and temperature, maybe one bar, let's say. 211 00:13:08 --> 00:13:11 T1, 298 degrees Kelvin. 212 00:13:11 --> 00:13:13 That's your reference point and you want to find out 213 00:13:13 --> 00:13:21 the pressure temperature dependence in an equation. 214 00:13:21 --> 00:13:24 So if you rearrange your equations so p1, T1 are 215 00:13:24 --> 00:13:25 now some reference point. 216 00:13:25 --> 00:13:26 They become numbers. 217 00:13:26 --> 00:13:29 So you have log of a reference point, T1 here is a number. 218 00:13:29 --> 00:13:36 So you often see this equation rewritten then as log p is 219 00:13:36 --> 00:13:43 equal to delta H over RT. 220 00:13:43 --> 00:13:45 And I feel like I've got a minus sign missing somewhere. 221 00:13:45 --> 00:13:46 Here. 222 00:13:46 --> 00:13:49 Oh yeah, when you integrate this, the one over T squared, 223 00:13:49 --> 00:13:50 there's a minus sign. 224 00:13:50 --> 00:13:53 Oh, I took care of that by doing T2 minus T1 here. 225 00:13:53 --> 00:13:56 I think there's still a minus sign problem somewhere. 226 00:13:56 --> 00:13:59 I think there's a minus sign problem. 227 00:13:59 --> 00:14:06 Let me check up the notes here. 228 00:14:06 --> 00:14:10 Minus one over T1 minus T2 over T1 T2. 229 00:14:10 --> 00:14:27 Where's my minus sign? 230 00:14:27 --> 00:14:29 T2 minus T1. 231 00:14:29 --> 00:14:49 OK, so I got minus T1, minus T2 over T1 T2. 232 00:14:49 --> 00:14:49 T2 minus T1. 233 00:14:49 --> 00:14:51 This is fine, right. 234 00:14:51 --> 00:14:53 There's no minus sign problem. 235 00:14:53 --> 00:14:56 Plus a constant. 236 00:14:56 --> 00:15:08 OK, so you'll often see it written like that. 237 00:15:08 --> 00:15:10 And that gives you a relationship between the 238 00:15:10 --> 00:15:16 pressure and the temperature then, for a substance where 239 00:15:16 --> 00:15:20 all the reference point information is contained 240 00:15:20 --> 00:15:22 in this constant, C. 241 00:15:22 --> 00:15:27 Which is where the T1's and the p's come from. 242 00:15:27 --> 00:15:31 OK, any questions? 243 00:15:31 --> 00:15:39 We'll do a little example here of how this could be used. 244 00:15:39 --> 00:15:44 So let's switch to an example now. 245 00:15:44 --> 00:15:46 Let's go to the example, let's first do the example which is 246 00:15:46 --> 00:15:53 at the very end of the notes. 247 00:15:53 --> 00:16:00 So this is an example of RDX. 248 00:16:00 --> 00:16:08 RDX is a famous explosive, as I'm sure you know. 249 00:16:08 --> 00:16:14 And it's plastic explosive. 250 00:16:14 --> 00:16:18 It's got a melting point of 204 degrees Celsius. 251 00:16:18 --> 00:16:21 481 degrees Kelvin. 252 00:16:21 --> 00:16:28 And it's a problem for, so it's a very powerful explosive and 253 00:16:28 --> 00:16:30 you can't see it in X-rays. 254 00:16:30 --> 00:16:32 So it's a problem at airports. 255 00:16:32 --> 00:16:34 You've got be able to find if somebody's carrying a little 256 00:16:34 --> 00:16:37 piece of RDX in their luggage. 257 00:16:37 --> 00:16:40 And so, the question becomes what do you need to 258 00:16:40 --> 00:16:41 know to detect it? 259 00:16:41 --> 00:16:45 You want to find the vapor pressure. 260 00:16:45 --> 00:16:48 You've gotta have, and the machine which basically is 261 00:16:48 --> 00:16:53 sensitive enough to detect molecules of RDX that are in 262 00:16:53 --> 00:16:57 vapor at room temperature. 263 00:16:57 --> 00:17:02 So the way that you do that is, you do an experiment. 264 00:17:02 --> 00:17:05 Where you measure the pressure, the vapor pressure, as a 265 00:17:05 --> 00:17:07 function of temperature. 266 00:17:07 --> 00:17:10 And room temperature, it turns out if it's not volatile at 267 00:17:10 --> 00:17:13 all, if the vapor pressure is very, really tiny. 268 00:17:13 --> 00:17:17 And so in order to get accurate numbers, you actually do your 269 00:17:17 --> 00:17:20 measurements at significantly higher temperatures 270 00:17:20 --> 00:17:22 than room temperature. 271 00:17:22 --> 00:17:25 They do it close to the melting point, so you're 272 00:17:25 --> 00:17:28 actually looking at the sublimation of RDX. 273 00:17:28 --> 00:17:33 You do it, let's say, at 400 degrees Kelvin, where the 274 00:17:33 --> 00:17:35 melting point is at 481 degrees Kelvin. 275 00:17:35 --> 00:17:37 So slightly below the melting point you're looking 276 00:17:37 --> 00:17:38 at sublimation. 277 00:17:38 --> 00:17:40 And you plot. 278 00:17:40 --> 00:17:49 Log p versus one over T. 279 00:17:49 --> 00:17:52 And according to Clausius-Clapeyron, that should 280 00:17:52 --> 00:17:53 give you a straight line. 281 00:17:53 --> 00:17:56 And in fact, that's exactly what you see when you take 282 00:17:56 --> 00:17:58 RDX and do that experiment. 283 00:17:58 --> 00:18:02 You end up with a bunch of data points. 284 00:18:02 --> 00:18:08 That's fall nicely on this straight line. 285 00:18:08 --> 00:18:11 And then you can extrapolate to room temperature. 286 00:18:11 --> 00:18:15 Somewhere here, let's say, 300 degrees Kelvin. 287 00:18:15 --> 00:18:21 And find out what log p is, at 300 degrees Kelvin. 288 00:18:21 --> 00:18:23 And what you find is that, and that's the graph that's on the 289 00:18:23 --> 00:18:29 last page of the lecture notes, that the vapor pressure of RDX 290 00:18:29 --> 00:18:35 at room temperature is 10 to the minus 11 bar, That's 291 00:18:35 --> 00:18:37 ten parts per trillion. 292 00:18:37 --> 00:18:40 So it tells you that if you want to detect RDX with a 293 00:18:40 --> 00:18:44 sniffer machine at the airport, that machine better be able to 294 00:18:44 --> 00:18:49 tell you, find one molecule of RDX out of a tenth 295 00:18:49 --> 00:18:52 of a trillion other molecules, basically. 296 00:18:52 --> 00:18:58 Which is a really hard thing to do. 297 00:18:58 --> 00:19:02 So this gives you a design rule for these pieces of equipment. 298 00:19:02 --> 00:19:06 And why they're so expensive. 299 00:19:06 --> 00:19:12 OK, before we do the next example, let's see 300 00:19:12 --> 00:19:15 if any questions on Clausius-Clapeyron. 301 00:19:15 --> 00:19:15 Yeah. 302 00:19:15 --> 00:19:27 STUDENT: [INAUDIBLE] 303 00:19:27 --> 00:19:31 PROFESSOR: So, p as some exponential here? 304 00:19:31 --> 00:19:33 You know, I don't recall seeing it as an exponential form. 305 00:19:33 --> 00:19:35 But that doesn't mean it's not used the exponential form. 306 00:19:35 --> 00:19:38 This is an easy way to do it, because then you've 307 00:19:38 --> 00:19:40 got a linear relationship. 308 00:19:40 --> 00:19:43 And generally we like to see straight lines. 309 00:19:43 --> 00:19:49 So this gives you a nice straight line. 310 00:19:49 --> 00:19:57 OK, so now let's do a slightly more complicated example. 311 00:19:57 --> 00:20:05 Which is, so most of the time you don't have pure material. 312 00:20:05 --> 00:20:07 You have a mixture of some sort. 313 00:20:07 --> 00:20:11 For instance, if I have a glass of water on the 314 00:20:11 --> 00:20:13 table at room temperature. 315 00:20:13 --> 00:20:22 Above the glass of water, I have air, there's my H2O here. 316 00:20:22 --> 00:20:24 And then I've got some air. 317 00:20:24 --> 00:20:26 Above the glass of water. 318 00:20:26 --> 00:20:28 And the air's inert to the water. 319 00:20:28 --> 00:20:29 It's not reacting with the water. 320 00:20:29 --> 00:20:34 So it's like an inert gas sitting on top of the water. 321 00:20:34 --> 00:20:37 Then I want to ask the question, what is the vapor 322 00:20:37 --> 00:20:40 pressure of the water? 323 00:20:40 --> 00:20:43 In the presence of this inert gas, the air. 324 00:20:43 --> 00:20:47 It's not really the same as the problem that we've 325 00:20:47 --> 00:20:49 been looking at up here. 326 00:20:49 --> 00:20:50 Right? 327 00:20:50 --> 00:20:55 Because this gas liquid coexistence line, this diagram, 328 00:20:55 --> 00:21:01 and this Clapeyron equation, is all done for a pure substance. 329 00:21:01 --> 00:21:04 And at room temperature, at room temperature, we're sitting 330 00:21:04 --> 00:21:07 squarely in the liquid phase. 331 00:21:07 --> 00:21:09 We're not on the coexistence line. 332 00:21:09 --> 00:21:13 At room temperature and one bar pressure. 333 00:21:13 --> 00:21:17 So instead of having this system here, I looked at the 334 00:21:17 --> 00:21:22 system where I had the water, H2O, with nothing on top. 335 00:21:22 --> 00:21:24 With no air on top. 336 00:21:24 --> 00:21:25 Except the cylinder. 337 00:21:25 --> 00:21:31 Now, I'd better make the surface of my water a 338 00:21:31 --> 00:21:33 straight line, otherwise I'm going to get in trouble. 339 00:21:33 --> 00:21:35 There's the water sitting right here. 340 00:21:35 --> 00:21:38 And I put a cylinder with one bar pressure on top. 341 00:21:38 --> 00:21:41 My cylinder's going to sit squarely on the stump of 342 00:21:41 --> 00:21:46 the surface of the water. 343 00:21:46 --> 00:21:49 This is not going to be any water vapor at one bar 344 00:21:49 --> 00:21:52 pressure on that cylinder. 345 00:21:52 --> 00:21:54 At one bar, we're way up in the liquid phase. 346 00:21:54 --> 00:21:59 I would have to decrease the pressure on that cylinder down 347 00:21:59 --> 00:22:05 to, I don't know, whatever the water vapor pressure is at 348 00:22:05 --> 00:22:07 room temperature here on this coexistence curve. 349 00:22:07 --> 00:22:09 0.1 bar or something. 350 00:22:09 --> 00:22:11 0.05 bar. 351 00:22:11 --> 00:22:14 Something pretty small. 352 00:22:14 --> 00:22:16 So that's what we've been working on. 353 00:22:16 --> 00:22:17 This pure system. 354 00:22:17 --> 00:22:20 And now we're going to be working on this system 355 00:22:20 --> 00:22:24 here, where we have air. 356 00:22:24 --> 00:22:28 With a cylinder on top, one bar. 357 00:22:28 --> 00:22:33 And we want to know what is the vapor pressure of 358 00:22:33 --> 00:22:40 H2O in this system here. 359 00:22:40 --> 00:22:45 Is it zero? 360 00:22:45 --> 00:22:49 Anybody want to guess, if it's zero? 361 00:22:49 --> 00:22:52 We know there's vapor pressure there, right? 362 00:22:52 --> 00:22:55 And a good guess is that that vapor pressure here, it's 363 00:22:55 --> 00:23:00 pretty close to the pressure that you would guess by using 364 00:23:00 --> 00:23:05 this diagram with the pure water. 365 00:23:05 --> 00:23:08 But the key word is, it's pretty close. 366 00:23:08 --> 00:23:12 It's not exactly the same. 367 00:23:12 --> 00:23:13 So, let me ask you this. 368 00:23:13 --> 00:23:21 Without looking at the notes, now is p H2O in this problem 369 00:23:21 --> 00:23:30 here greater than, less than, or the same as p H2O in 370 00:23:30 --> 00:23:37 the pure Clapeyron case? 371 00:23:37 --> 00:23:41 How many people think that it's the same? 372 00:23:41 --> 00:23:44 How about smaller than? 373 00:23:44 --> 00:23:46 Smaller than? 374 00:23:46 --> 00:23:47 Few people think that it's smaller than. 375 00:23:47 --> 00:23:50 How about greater than? 376 00:23:50 --> 00:23:54 A bunch of people don't know what to answer. 377 00:23:54 --> 00:23:55 OK. 378 00:23:55 --> 00:23:58 I think that if you think about it, what the difference is 379 00:23:58 --> 00:24:04 between here and there, you could probably reason 380 00:24:04 --> 00:24:06 which way it should go. 381 00:24:06 --> 00:24:09 Which way the partial pressure of the water should go under 382 00:24:09 --> 00:24:11 one bar pressure, room temperature. 383 00:24:11 --> 00:24:14 Compared to what you might expect it to be at room 384 00:24:14 --> 00:24:19 temperature with a diagram. 385 00:24:19 --> 00:24:21 So, I'm going to give you a little bit of time 386 00:24:21 --> 00:24:22 just to think about it. 387 00:24:22 --> 00:24:24 Without looking at the notes, what you think 388 00:24:24 --> 00:24:25 the right sign is here. 389 00:24:25 --> 00:24:33 Then we'll do the problem and then we'll figure out if 390 00:24:33 --> 00:25:08 you were right or wrong. 391 00:25:08 --> 00:25:09 OK. 392 00:25:09 --> 00:25:14 How many people say it's greater than? 393 00:25:14 --> 00:25:16 One, two. 394 00:25:16 --> 00:25:18 Two brave people here. 395 00:25:18 --> 00:25:20 Two. 396 00:25:20 --> 00:25:23 How many people say it's smaller than? 397 00:25:23 --> 00:25:24 OK. 398 00:25:24 --> 00:25:25 Smaller than. 399 00:25:25 --> 00:25:27 So I'm going to say a large number, OK? 400 00:25:27 --> 00:25:29 Many. 401 00:25:29 --> 00:25:33 How many people think it's equal? 402 00:25:33 --> 00:25:36 One person thinks it's equal. 403 00:25:36 --> 00:25:41 Alright, let's do it out now and see how it comes out. 404 00:25:41 --> 00:25:46 And then we can reason why it came out the way it came out. 405 00:25:46 --> 00:25:48 Logically, without actually doing math. 406 00:25:48 --> 00:25:51 Because there's a way to reason it out that you 407 00:25:51 --> 00:25:57 should be able to do. 408 00:25:57 --> 00:25:59 OK, so this is what we have here. 409 00:25:59 --> 00:26:05 Then the system is, we have this container. 410 00:26:05 --> 00:26:07 Let me redo it instead of writing air. 411 00:26:07 --> 00:26:11 So we have the liquid state here. 412 00:26:11 --> 00:26:13 Some substance, and it doesn't have to be water. 413 00:26:13 --> 00:26:16 That's a liquid, at some pressure p. 414 00:26:16 --> 00:26:18 And I'm going to say P capital, that's going to 415 00:26:18 --> 00:26:19 be the total pressure. 416 00:26:19 --> 00:26:23 And the total pressure is given by the piston 417 00:26:23 --> 00:26:26 here. p total, p T. 418 00:26:26 --> 00:26:29 And on top here I'm going to have two substances. 419 00:26:29 --> 00:26:33 I'm going to have A, in the gas phase, that's the water vapor. 420 00:26:33 --> 00:26:37 Some partial pressure, p sub A, temperature T. 421 00:26:37 --> 00:26:42 And I'm going to have the inert gas, with some 422 00:26:42 --> 00:26:45 partial pressure p, inert. 423 00:26:45 --> 00:26:49 Such that the sum of the partial pressure of the inert 424 00:26:49 --> 00:26:53 gas, plus the partial pressure of A, is the total pressure. 425 00:26:53 --> 00:26:54 One bar, in our case. 426 00:26:54 --> 00:26:57 With the water with the air on top. 427 00:26:57 --> 00:27:01 And the question, oh, let me make a few more definitions. 428 00:27:01 --> 00:27:12 We're going to define p0 as the vapor pressure of pure 429 00:27:12 --> 00:27:14 vapor pressure of pure A. 430 00:27:14 --> 00:27:19 So that's what you would read off the diagram here. 431 00:27:19 --> 00:27:26 And what we want, the question we ask is, what is the partial 432 00:27:26 --> 00:27:34 pressure of A as a function of the total pressure? 433 00:27:34 --> 00:27:36 That's what we're asking. 434 00:27:36 --> 00:27:39 As I'm increasing the total pressure, meaning I'm putting 435 00:27:39 --> 00:27:43 more and more inert gas in there, which way is this going? 436 00:27:43 --> 00:27:45 Is it going up, down, or staying the same? 437 00:27:45 --> 00:27:46 And we've got a bunch of votes up here. 438 00:27:46 --> 00:27:49 Most people think that it's going to go down. 439 00:27:49 --> 00:27:53 Because I put more gas in there. 440 00:27:53 --> 00:28:02 OK, I know that if P, capital P, is equal to pA, then 441 00:28:02 --> 00:28:05 pA is equal to p0. 442 00:28:05 --> 00:28:05 Right? 443 00:28:05 --> 00:28:07 There's no inert gas. 444 00:28:07 --> 00:28:10 So that's the one extreme case here. 445 00:28:10 --> 00:28:12 There's no inert gas, then the partial pressure's going to be 446 00:28:12 --> 00:28:16 what you read off that graph here. 447 00:28:16 --> 00:28:19 So another way of saying what is p as a function of capital 448 00:28:19 --> 00:28:23 P, is to ask the question what is the slope? 449 00:28:23 --> 00:28:24 I've got one point here. 450 00:28:24 --> 00:28:27 If I can integrate the slope then I have this equation. 451 00:28:27 --> 00:28:35 So the other way to ask the question is, what is dpA/dp, 452 00:28:35 --> 00:28:37 at a constant temperature? 453 00:28:37 --> 00:28:39 Where I'm keeping the temperature constant. 454 00:28:39 --> 00:28:43 And that turns out to be easier to calculate. 455 00:28:43 --> 00:28:45 These slopes are always easier to calculate than 456 00:28:45 --> 00:28:51 the absolute things. 457 00:28:51 --> 00:28:53 So we're doing an equilibrium here. 458 00:28:53 --> 00:28:55 We have an equilibrium between the liquid state 459 00:28:55 --> 00:28:56 and the gas phase. 460 00:28:56 --> 00:28:59 And we want to know something about this equilibrium as 461 00:28:59 --> 00:29:01 we change a parameter. 462 00:29:01 --> 00:29:03 Equilibrium, what do we always start with when 463 00:29:03 --> 00:29:03 we have an equilibrium. 464 00:29:03 --> 00:29:06 What's the first thing we think about? 465 00:29:06 --> 00:29:08 We think about chemical potential. 466 00:29:08 --> 00:29:12 Chemical potentials of A, in the gas phase has to be the 467 00:29:12 --> 00:29:14 same as the chemical potential of A in the liquid phase. 468 00:29:14 --> 00:29:17 Well, I can't think of anything else to start with. 469 00:29:17 --> 00:29:19 So let me start with that. 470 00:29:19 --> 00:29:24 So let's start by writing the chemical potentials being 471 00:29:24 --> 00:29:26 equal to each other. 472 00:29:26 --> 00:29:30 Because we know that's true. 473 00:29:30 --> 00:29:37 So mu A, in the gas phase, temperature and with a 474 00:29:37 --> 00:29:40 partial pressure p sub A. 475 00:29:40 --> 00:29:46 Is equal to mu A in the liquid phase, where the pressure 476 00:29:46 --> 00:29:50 on the liquid is capital P. 477 00:29:50 --> 00:29:50 What else do we know? 478 00:29:50 --> 00:29:53 Well, we, what else are we going to need? 479 00:29:53 --> 00:29:57 Before we do that, since what we're after is the derivative, 480 00:29:57 --> 00:29:59 let's take the derivative of both sides. 481 00:29:59 --> 00:30:02 We're looking at a derivative with respect to pressure here. 482 00:30:02 --> 00:30:03 Let's take the derivative outside with pressure 483 00:30:03 --> 00:30:05 and see what happens. 484 00:30:05 --> 00:30:07 So let's take the derivative of this with respect 485 00:30:07 --> 00:30:08 to total pressure. 486 00:30:08 --> 00:30:09 Well, this isn't total pressure. 487 00:30:09 --> 00:30:11 This is the partial pressure of A. 488 00:30:11 --> 00:30:14 But it's a function of the total pressure. 489 00:30:14 --> 00:30:16 It's a function of the total pressure. 490 00:30:16 --> 00:30:19 Partial pressure of A is total pressure minus p inert. 491 00:30:19 --> 00:30:24 So, we have to use the chain rule. 492 00:30:24 --> 00:30:32 So on this side here, we have d mu A / dpA. 493 00:30:32 --> 00:30:40 So I'm taking d/dp of both sides. 494 00:30:40 --> 00:30:42 Then by chain rule, dpA/dP. 495 00:30:42 --> 00:30:48 496 00:30:48 --> 00:30:49 Constant temperature. 497 00:30:49 --> 00:30:56 And then on the right-hand side, I have d mu A / dP, 498 00:30:56 --> 00:31:00 constant temperature. 499 00:31:00 --> 00:31:05 Well, d mu / dP, that may or may not seem familiar to you. 500 00:31:05 --> 00:31:08 I never remember these things, but I always remember 501 00:31:08 --> 00:31:11 that there's, that these relationships are interesting. 502 00:31:11 --> 00:31:14 Especially when you're talking about something like a chemical 503 00:31:14 --> 00:31:15 potential, which is really nothing but the 504 00:31:15 --> 00:31:17 Gibbs free energy. 505 00:31:17 --> 00:31:22 With respect to pressure, when we know that the variables that 506 00:31:22 --> 00:31:24 we use for Gibbs free energy are pressure and temperature. 507 00:31:24 --> 00:31:25 Right? 508 00:31:25 --> 00:31:28 So this is the variable that goes with Gibbs free energy. 509 00:31:28 --> 00:31:31 So this means that they were taking the derivatives of the 510 00:31:31 --> 00:31:34 Gibbs free energy with respect to one of its variables is 511 00:31:34 --> 00:31:36 something that we should know. 512 00:31:36 --> 00:31:38 So we go back to writing what the Gibbs free energy is. 513 00:31:38 --> 00:31:44 So dG is equal to V dp minus S dT. 514 00:31:44 --> 00:31:48 Which is the fundamental equation for Gibbs free energy. 515 00:31:48 --> 00:31:50 And this is the Gibbs free energy per mole. 516 00:31:50 --> 00:31:53 That's just the chemical potential. 517 00:31:53 --> 00:31:56 The same thing. 518 00:31:56 --> 00:31:58 And this is the derivative with respect to pressure. 519 00:31:58 --> 00:32:01 And this is the derivative with respect to temperature. 520 00:32:01 --> 00:32:05 So this thing here is d mu / dp. 521 00:32:05 --> 00:32:08 At constant temperature. 522 00:32:08 --> 00:32:09 It's just the volume. 523 00:32:09 --> 00:32:12 So now we have this equation, where we started out with 524 00:32:12 --> 00:32:12 the chemical potentials. 525 00:32:12 --> 00:32:15 And we've got these d mu A / dpA, we've got 526 00:32:15 --> 00:32:17 d mu A / dP here. 527 00:32:17 --> 00:32:22 That's just the molar volume of the gas. 528 00:32:22 --> 00:32:24 Right, because this is mu of the gas phase here. 529 00:32:24 --> 00:32:27 So this is the molar volume of the gas. 530 00:32:27 --> 00:32:30 And I have dpa/dP. 531 00:32:30 --> 00:32:32 And, this is very nice. 532 00:32:32 --> 00:32:34 Because this is actually what I'm trying to get. 533 00:32:34 --> 00:32:37 Trying to see how the partial pressure of a changes 534 00:32:37 --> 00:32:38 with the total pressure. 535 00:32:38 --> 00:32:41 That's what I'm trying to calculate here. 536 00:32:41 --> 00:32:42 I know what this is. 537 00:32:42 --> 00:32:45 This is just an experimental number. 538 00:32:45 --> 00:32:47 And then on this side here I have d mu A / dP. 539 00:32:47 --> 00:32:50 This is the liquid phase. 540 00:32:50 --> 00:32:53 Derivative with respect to total pressure, that's the 541 00:32:53 --> 00:32:57 molar volume of the liquid. 542 00:32:57 --> 00:33:00 The liquid. 543 00:33:00 --> 00:33:10 Great, I'm done. dpA/dP total, constant pressure, is the ratio 544 00:33:10 --> 00:33:14 of the molar volume of the liquid of A divided by the 545 00:33:14 --> 00:33:17 molar volume of the gas. 546 00:33:17 --> 00:33:21 What's the sign? 547 00:33:21 --> 00:33:23 Volumes are positive, right? 548 00:33:23 --> 00:33:25 This has to be positive. 549 00:33:25 --> 00:33:27 There's no choice. 550 00:33:27 --> 00:33:29 It can't even be zero. 551 00:33:29 --> 00:33:30 It has to be positive, right? 552 00:33:30 --> 00:33:36 So that means the slope is a positive slope. 553 00:33:36 --> 00:33:44 As I increase the total pressure, p is going to go up. 554 00:33:44 --> 00:33:49 If I integrate now, this, starting at, so the total 555 00:33:49 --> 00:33:54 pressure is equal to pA, I'm going to get a curve 556 00:33:54 --> 00:33:58 that looks like this. 557 00:33:58 --> 00:34:02 OK, if I write on this axis p inert, so when p inert is equal 558 00:34:02 --> 00:34:05 to zero, total pressure is pA. 559 00:34:05 --> 00:34:07 I'm going to start at p0. 560 00:34:07 --> 00:34:11 And I'm going to go up from there, positive slope. 561 00:34:11 --> 00:34:13 So. 562 00:34:13 --> 00:34:16 Who was right? 563 00:34:16 --> 00:34:17 A couple people are right. 564 00:34:17 --> 00:34:20 So let's think about it in a different way. 565 00:34:20 --> 00:34:23 Let's think about it in a different way. 566 00:34:23 --> 00:34:26 So we did the math, and we found that actually it's, 567 00:34:26 --> 00:34:28 and that turns out to be a very small amount. 568 00:34:28 --> 00:34:32 You do a very good job by just taking this curve here. 569 00:34:32 --> 00:34:34 And using the pure substance. 570 00:34:34 --> 00:34:37 In fact, in the notes we have an example of how much 571 00:34:37 --> 00:34:39 it changes for mercury. 572 00:34:39 --> 00:34:45 So for mercury, at 100 degrees C, for pure mercury, the 573 00:34:45 --> 00:34:48 vapor pressure is 0.27 bar. 574 00:34:48 --> 00:34:53 If you add enough inert gas to make it 1 bar for the total 575 00:34:53 --> 00:34:56 pressure, you go from 0.27 to 0.2701. 576 00:34:56 --> 00:34:57 Not much of a change. 577 00:34:57 --> 00:35:00 They do a really good job of just picking this. 578 00:35:00 --> 00:35:04 And if you go to 100 bar, then you increase it to 0.28. 579 00:35:04 --> 00:35:06 So it takes a lot of pressure on top, a lot of extra 580 00:35:06 --> 00:35:10 inert gas to really make much of a difference. 581 00:35:10 --> 00:35:14 But, let's think about why it would make a difference, 582 00:35:14 --> 00:35:19 just logically. 583 00:35:19 --> 00:35:22 So in the absence of this inert gas here, I just 584 00:35:22 --> 00:35:25 have the pure substance. 585 00:35:25 --> 00:35:30 I add some inert gas, what happens to the entropy on top? 586 00:35:30 --> 00:35:33 It's increasing, right? 587 00:35:33 --> 00:35:36 Do systems like to have more entropy? 588 00:35:36 --> 00:35:38 Yeah. 589 00:35:38 --> 00:35:43 So, I have a lot of inert gas here. 590 00:35:43 --> 00:35:46 A small amount of A. 591 00:35:46 --> 00:35:48 What if I had a little bit more A, in the presence 592 00:35:48 --> 00:35:49 of the inert gas? 593 00:35:49 --> 00:35:54 Which way would the entropy go? 594 00:35:54 --> 00:35:57 It would go up a little bit more, you'd have more disorder. 595 00:35:57 --> 00:36:00 If I have nothing, if I have no A here, up here and just the 596 00:36:00 --> 00:36:02 inert gas, then there's no entropy of mixing. 597 00:36:02 --> 00:36:04 I add a little bit of inert gas, there's a little bit 598 00:36:04 --> 00:36:05 more entropy of mixing. 599 00:36:05 --> 00:36:09 Eventually, I add more and more, entropy of mixing goes up 600 00:36:09 --> 00:36:11 and up and up, and eventually it comes back down to where 601 00:36:11 --> 00:36:14 I have pure A on top. 602 00:36:14 --> 00:36:21 So the entropy of mixing up here is driving the increase 603 00:36:21 --> 00:36:26 in the pressure of, in the partial pressure of A. 604 00:36:26 --> 00:36:29 Entropy is this amazingly important term. 605 00:36:29 --> 00:36:29 That's what we saw. 606 00:36:29 --> 00:36:33 Entropy of mixing was responsible for equilibrium. 607 00:36:33 --> 00:36:36 Without entropy of mixing, then everything would go 608 00:36:36 --> 00:36:37 directly to the products. 609 00:36:37 --> 00:36:39 You'd never have an equilibrium. 610 00:36:39 --> 00:36:43 Another secret of life. 611 00:36:43 --> 00:36:46 So entropy of mixing here is driving this positive slope. 612 00:36:46 --> 00:36:48 That's another way of thinking about it. 613 00:36:48 --> 00:36:52 And that's why you can guess ahead of time what the signs 614 00:36:52 --> 00:36:54 of these things ought to be. 615 00:36:54 --> 00:36:58 Without doing the math. 616 00:36:58 --> 00:37:06 Alright, any questions? 617 00:37:06 --> 00:37:06 OK. 618 00:37:06 --> 00:37:13 So there's another sample problem with the notes, 619 00:37:13 --> 00:37:16 which is sort of like a standard problem. 620 00:37:16 --> 00:37:25 Where you're given a Clausius-Clapeyron equation 621 00:37:25 --> 00:37:29 for a substance in this form. 622 00:37:29 --> 00:37:37 So if you look at it, log p is minus some energy divided 623 00:37:37 --> 00:37:43 by RT plus some number. 624 00:37:43 --> 00:37:46 In this case, it's negative, the delta H is negative for the 625 00:37:46 --> 00:37:48 substance that you have there. 626 00:37:48 --> 00:37:50 And you're given it for the liquid phase, for the vapor 627 00:37:50 --> 00:37:52 pressure above the liquid and the vapor pressure 628 00:37:52 --> 00:37:54 above the solid. 629 00:37:54 --> 00:37:56 You can, because the only way it doesn't work is if you're 630 00:37:56 --> 00:37:58 looking at solid liquid. 631 00:37:58 --> 00:38:02 So, and then you're asked to manipulate this equation to 632 00:38:02 --> 00:38:03 find different properties. 633 00:38:03 --> 00:38:04 For instance the triple point. 634 00:38:04 --> 00:38:07 It's a very common thing to get, the triple 635 00:38:07 --> 00:38:10 point from this. 636 00:38:10 --> 00:38:12 The delta H, it's a way to get delta H. 637 00:38:12 --> 00:38:14 Just like we saw for the RDX, right? 638 00:38:14 --> 00:38:17 The data points formed a straight line. 639 00:38:17 --> 00:38:20 And the slope gave you delta H, for RDX. 640 00:38:20 --> 00:38:23 Delta H of sublimation for RDX. 641 00:38:23 --> 00:38:27 So this is the way that this equation is used for problems. 642 00:38:27 --> 00:38:28 I'm not going to do this problem. 643 00:38:28 --> 00:38:30 Instead I'm going to go ahead and forge ahead 644 00:38:30 --> 00:38:33 to the next topic. 645 00:38:33 --> 00:38:34 And start it. 646 00:38:34 --> 00:38:41 This is the topic we're going to do it after spring break. 647 00:38:41 --> 00:38:48 And and it really is a topic that follows this pretty 648 00:38:48 --> 00:38:49 straightforwardly. 649 00:38:49 --> 00:38:52 Well, it's more complicated, but it's similar to 650 00:38:52 --> 00:38:54 this problem here. 651 00:38:54 --> 00:38:59 So we saw that in real life yes, question? 652 00:38:59 --> 00:39:06 STUDENT: [INAUDIBLE] 653 00:39:06 --> 00:39:07 PROFESSOR: The exam. 654 00:39:07 --> 00:39:13 Oh there's an exam is, which is, oh I forgot about the exam. 655 00:39:13 --> 00:39:13 So let's see. 656 00:39:13 --> 00:39:16 There's a problem set that's due today. 657 00:39:16 --> 00:39:19 The exam will be up to the problem set that's due today. 658 00:39:19 --> 00:39:23 Which means that what I talked about today is not going 659 00:39:23 --> 00:39:24 to be on the exam. 660 00:39:24 --> 00:39:30 Because it's not covered in the problem set. 661 00:39:30 --> 00:39:35 So let's say it goes up to Wednesday's lecture. 662 00:39:35 --> 00:39:36 Which means you might still have a phase transition 663 00:39:36 --> 00:39:42 problem, right? 664 00:39:42 --> 00:39:47 In fact, you're very likely to get a phase transition problem. 665 00:39:47 --> 00:39:48 It's very likely to be the case. 666 00:39:48 --> 00:39:53 Because it's an easy problem to put together. 667 00:39:53 --> 00:39:54 But you don't have to worry about Clausius-Clapeyron. 668 00:39:54 --> 00:39:56 You do have to worry about Clapeyron, but not 669 00:39:56 --> 00:40:03 Clausius-Clapeyron, OK? 670 00:40:03 --> 00:40:07 So, for the exam, we're going to put on the Web last year's 671 00:40:07 --> 00:40:11 exam, and you guys are going to do reviews. 672 00:40:11 --> 00:40:13 The usual thing. 673 00:40:13 --> 00:40:13 Yeah? 674 00:40:13 --> 00:40:17 STUDENT: [INAUDIBLE] 675 00:40:17 --> 00:40:18 PROFESSOR: So that you don't see the answers. 676 00:40:18 --> 00:40:21 Yeah, absolutely this can easily be done. 677 00:40:21 --> 00:40:23 We'll do that. 678 00:40:23 --> 00:40:30 Any other questions, about the exam? 679 00:40:30 --> 00:40:31 Now, I don't remember. 680 00:40:31 --> 00:40:33 Is there a problem set that, so I'm not assigning a problem 681 00:40:33 --> 00:40:35 set today, right, then? 682 00:40:35 --> 00:40:35 Right. 683 00:40:35 --> 00:40:36 OK. 684 00:40:36 --> 00:40:40 Good. 685 00:40:40 --> 00:40:41 Because I hadn't loaded it up yet. 686 00:40:41 --> 00:40:45 And so now I won't load it up. 687 00:40:45 --> 00:40:48 OK. 688 00:40:48 --> 00:40:50 Anything else? 689 00:40:50 --> 00:40:51 OK, so let me tell you where we're going to go 690 00:40:51 --> 00:40:53 then, after the break. 691 00:40:53 --> 00:40:55 So after the break, we're going to be starting to talk about 692 00:40:55 --> 00:40:58 things that are called colligative properties. 693 00:40:58 --> 00:41:03 Colligative properties are the properties like the vapor 694 00:41:03 --> 00:41:08 pressure, lowering when you have a mixture in a liquid, 695 00:41:08 --> 00:41:10 instead of having a pure substance you have a mixture 696 00:41:10 --> 00:41:12 of substances here. 697 00:41:12 --> 00:41:16 And then the vapor pressure of both substances 698 00:41:16 --> 00:41:20 above gets lowered. 699 00:41:20 --> 00:41:23 Anybody want to guess why they got lowered? 700 00:41:23 --> 00:41:25 What's the magic word? 701 00:41:25 --> 00:41:27 Entropy of mixing, right. 702 00:41:27 --> 00:41:28 Right. 703 00:41:28 --> 00:41:30 There's more entropy of mixing if you've got a mixture in the 704 00:41:30 --> 00:41:32 liquid than if you have a pure gas up there. 705 00:41:32 --> 00:41:34 So usually, so let me back up. 706 00:41:34 --> 00:41:39 Usually it's when you have a solute, like salt in water. 707 00:41:39 --> 00:41:43 Where water is volatile and the salt is not. 708 00:41:43 --> 00:41:47 Right, a salt in water solution will have a lower vapor 709 00:41:47 --> 00:41:50 pressure than a solution of pure water. 710 00:41:50 --> 00:41:51 Because of the entropy of mixing in a salt 711 00:41:51 --> 00:41:53 and water solution. 712 00:41:53 --> 00:41:57 The salt and water solution we have a lowering, 713 00:41:57 --> 00:41:59 lower melting point. 714 00:41:59 --> 00:42:01 And that's significantly lower. 715 00:42:01 --> 00:42:03 Which is why we use it on roads, right? 716 00:42:03 --> 00:42:05 That's a colligative properties. 717 00:42:05 --> 00:42:06 Osmotic pressure. 718 00:42:06 --> 00:42:08 Which you've already sort of heard about. 719 00:42:08 --> 00:42:13 Why saltwater fish are unhappy in fresh water. 720 00:42:13 --> 00:42:14 That's a colligative property. 721 00:42:14 --> 00:42:16 So we're going to go through these colligative properties. 722 00:42:16 --> 00:42:19 And the mainstay of these colligative properties is that 723 00:42:19 --> 00:42:21 we're going to be talking about mixtures in the liquid phase. 724 00:42:21 --> 00:42:23 Here we talked about a mixture in the gas phase, 725 00:42:23 --> 00:42:25 changing some property. 726 00:42:25 --> 00:42:29 From now on, we're going to be talking about mixtures in the 727 00:42:29 --> 00:42:32 liquid phase, pretty much. 728 00:42:32 --> 00:42:38 And so our goal is going to be able to, is going to be predict 729 00:42:38 --> 00:42:45 the way that properties change as you have these mixtures. 730 00:42:45 --> 00:42:52 OK. so the standard mixture that we're going to have, 731 00:42:52 --> 00:42:57 is to have a binary liquid gas mixture. 732 00:42:57 --> 00:43:00 So that means we're going to have two substances. 733 00:43:00 --> 00:43:01 A and B. 734 00:43:01 --> 00:43:04 And at first, we're going to make it as general as possible. 735 00:43:04 --> 00:43:08 So we're going to have liquid phase of A, the 736 00:43:08 --> 00:43:08 liquid phase of B. 737 00:43:08 --> 00:43:12 So, for instance, this could be vodka, right? 738 00:43:12 --> 00:43:15 Water and ethanol. 739 00:43:15 --> 00:43:17 And water and ethanol are both volatile. 740 00:43:17 --> 00:43:25 So we're going to have water and ethanol in the gas phase. 741 00:43:25 --> 00:43:28 There, and it's a mixture, they're perfectly miscible in 742 00:43:28 --> 00:43:30 the liquid phase, A and B. 743 00:43:30 --> 00:43:34 And there's going to be some fraction, molar fraction, 744 00:43:34 --> 00:43:35 of these things. 745 00:43:35 --> 00:43:37 We're going to have a molar fraction in the liquid 746 00:43:37 --> 00:43:39 phase for A and B. 747 00:43:39 --> 00:43:42 And some molar fraction in the gas phase, for A and B. 748 00:43:42 --> 00:43:45 We're going to call it yA and yB, that's the molar 749 00:43:45 --> 00:43:47 fraction in the gas phase. 750 00:43:47 --> 00:43:50 And there's no reason why the molar fractions in the gas 751 00:43:50 --> 00:43:54 phrase should be the same as that in the liquid phase. 752 00:43:54 --> 00:43:55 And we know that. 753 00:43:55 --> 00:43:57 We know that ethanol is more volatile than water. 754 00:43:57 --> 00:44:02 And so if you have a gas of vodka, then you can smell 755 00:44:02 --> 00:44:05 the alcohol because it's coming out. 756 00:44:05 --> 00:44:08 But the water is, vapor pressure is, very 757 00:44:08 --> 00:44:09 slow, very small. 758 00:44:09 --> 00:44:11 It's the usual, it's pretty close to the 759 00:44:11 --> 00:44:14 usual vapor pressure. 760 00:44:14 --> 00:44:17 And our question is, what we're going to be asking is, how 761 00:44:17 --> 00:44:20 many, if I give you the total pressure and the temperature 762 00:44:20 --> 00:44:23 here, what else do I need to know to find out all 763 00:44:23 --> 00:44:28 these molar fractions? 764 00:44:28 --> 00:44:31 So let's first see what all the variables are that we have. 765 00:44:31 --> 00:44:32 Usually, we have two variables. 766 00:44:32 --> 00:44:34 The temperature and pressure, and that's enough to 767 00:44:34 --> 00:44:35 tell us everything. 768 00:44:35 --> 00:44:37 But here now we have a mixture. 769 00:44:37 --> 00:44:40 So we're going to need more than that. 770 00:44:40 --> 00:44:41 We're going to need four variables. 771 00:44:41 --> 00:44:43 We're going to need the temperature. 772 00:44:43 --> 00:44:47 The pressure, and we're going to need the molar fraction 773 00:44:47 --> 00:44:49 of A in the liquid phase. 774 00:44:49 --> 00:44:52 And the molar fraction of A in the gas phase. 775 00:44:52 --> 00:44:55 We're not going to need any more than that. 776 00:44:55 --> 00:44:56 Actually we're going to need less than that. 777 00:44:56 --> 00:44:58 But those are the four variables that we can 778 00:44:58 --> 00:45:00 easily identify here. 779 00:45:00 --> 00:45:05 And we don't need yB and xB, because the sum of the molar 780 00:45:05 --> 00:45:12 fraction has to be equal to one. yB is one minus yA, 781 00:45:12 --> 00:45:15 xB is one minus xA. 782 00:45:15 --> 00:45:17 So we don't need y. 783 00:45:17 --> 00:45:20 So these are a priori the independent variables that 784 00:45:20 --> 00:45:23 we have to work with. 785 00:45:23 --> 00:45:26 Now, we have some constraints here. 786 00:45:26 --> 00:45:28 We have a constraint. 787 00:45:28 --> 00:45:30 Our constraint is that we are the coexistence point. 788 00:45:30 --> 00:45:35 There's a coexistence between the gas phase 789 00:45:35 --> 00:45:37 and the liquid phase. 790 00:45:37 --> 00:45:43 And what does it mean for A liquid to be coexistent 791 00:45:43 --> 00:45:46 with A in the gas phase? 792 00:45:46 --> 00:45:47 Chemical potential. 793 00:45:47 --> 00:45:50 The chemical potential of a molecule of A in the liquid 794 00:45:50 --> 00:45:52 phase here is the same as the chemical potential 795 00:45:52 --> 00:45:52 of A in the gas phase. 796 00:45:52 --> 00:45:53 So we have two constraints. 797 00:45:53 --> 00:46:00 We have mu A for the liquid phase is equal to mu A for the 798 00:46:00 --> 00:46:04 gas phase, and mu B for the liquid phase equal to mu 799 00:46:04 --> 00:46:06 B for the gas phase. 800 00:46:06 --> 00:46:09 So we have four variables, two constraints. 801 00:46:09 --> 00:46:13 That means that the total number of independent variables 802 00:46:13 --> 00:46:14 we have is just two. 803 00:46:14 --> 00:46:18 It's the number of variables four, minus the constraints. 804 00:46:18 --> 00:46:19 OK? 805 00:46:19 --> 00:46:22 Or you could say six variables if you include yB and xB, 806 00:46:22 --> 00:46:24 and four constraints. 807 00:46:24 --> 00:46:30 Whatever we accounted, you just have two degrees of freedom. 808 00:46:30 --> 00:46:37 So we have two degrees of freedom. 809 00:46:37 --> 00:46:43 Which means that if you know the temperature and pressure, 810 00:46:43 --> 00:46:45 then you still know everything. 811 00:46:45 --> 00:46:47 At the coexistence point. 812 00:46:47 --> 00:46:51 You still know what the molar fraction is of your two 813 00:46:51 --> 00:46:53 substances in both phases. 814 00:46:53 --> 00:46:57 So that's really powerful. 815 00:46:57 --> 00:47:01 Or if you know just the molar fraction of A, and the 816 00:47:01 --> 00:47:03 temperature, then you know the pressure above. 817 00:47:03 --> 00:47:06 And you know everything else. 818 00:47:06 --> 00:47:09 So in this very complicated mixture here, you still only 819 00:47:09 --> 00:47:11 need to know a couple of things to tell you everything 820 00:47:11 --> 00:47:16 about this mixture. 821 00:47:16 --> 00:47:22 And this example here is a specific example of something 822 00:47:22 --> 00:47:24 called the Gibbs phase rule. 823 00:47:24 --> 00:47:27 Which you've seen before. 824 00:47:27 --> 00:47:28 In this diagram. 825 00:47:28 --> 00:47:33 In this diagram that we wrote here. 826 00:47:33 --> 00:47:38 Where, let me write down what the Gibbs phase rule is first. 827 00:47:38 --> 00:47:41 The Gibbs phase rule tells you the number of independent 828 00:47:41 --> 00:47:43 variables given a number of constraints. 829 00:47:43 --> 00:47:56 So, it tells you that the number of degrees of freedom, 830 00:47:56 --> 00:47:59 the number of independent variables in a mixture, in a 831 00:47:59 --> 00:48:03 complicated system like this, is equal to C, which is 832 00:48:03 --> 00:48:05 the number of components. 833 00:48:05 --> 00:48:09 So in this case, it would be two components. 834 00:48:09 --> 00:48:10 You have A and B. 835 00:48:10 --> 00:48:13 The components is the number of independent species that 836 00:48:13 --> 00:48:17 you have, minus the number of phases. 837 00:48:17 --> 00:48:19 So here we have two phases, the gas phase and the 838 00:48:19 --> 00:48:24 water phase, plus two. 839 00:48:24 --> 00:48:28 So for this example here, then we would have two 840 00:48:28 --> 00:48:30 components, C equals two. 841 00:48:30 --> 00:48:33 Two phases liquid and gas. 842 00:48:33 --> 00:48:33 A and B are the components. 843 00:48:33 --> 00:48:34 And then plus two. 844 00:48:34 --> 00:48:40 So two minus two plus two, F equals to two. 845 00:48:40 --> 00:48:45 Two degrees of freedom, two independent variables. 846 00:48:45 --> 00:48:49 If I'm looking at this diagram here, in this diagram 847 00:48:49 --> 00:48:52 here I have C equals one. 848 00:48:52 --> 00:48:54 One component, right? 849 00:48:54 --> 00:48:59 If I'm sitting where there's only one phase, say, the liquid 850 00:48:59 --> 00:49:04 phase or the gas phase, then p equals one. 851 00:49:04 --> 00:49:08 Then F is one minus one plus two, is equal to two. 852 00:49:08 --> 00:49:09 Two degrees of freedom. 853 00:49:09 --> 00:49:11 And so in the pure phase, I am in a plane. 854 00:49:11 --> 00:49:14 The pressure and the temperature are my two 855 00:49:14 --> 00:49:15 degrees of freedom. 856 00:49:15 --> 00:49:17 I can freely move around the plane. 857 00:49:17 --> 00:49:25 At a coexistence line, then the number of phases equals to two. 858 00:49:25 --> 00:49:30 Then I have one minus two is minus one, plus two is one. 859 00:49:30 --> 00:49:33 The degrees of freedom is one. 860 00:49:33 --> 00:49:34 That's a line. 861 00:49:34 --> 00:49:35 I have one degree of freedom. 862 00:49:35 --> 00:49:38 I can move in the line, in the line in a plane. 863 00:49:38 --> 00:49:41 Then if I change to the triple point, where the number of 864 00:49:41 --> 00:49:45 phases is three, then the number of degrees of freedom 865 00:49:45 --> 00:49:49 is one minus three is minus two, two plus two is zero. 866 00:49:49 --> 00:49:50 I don't have any degrees of freedom, I'm 867 00:49:50 --> 00:49:54 stuck at one point. 868 00:49:54 --> 00:49:57 So this is an example of the Gibbs phase rule, specific 869 00:49:57 --> 00:50:00 where C happens to be equal to one. 870 00:50:00 --> 00:50:04 And then, this is a more generalized example. 871 00:50:04 --> 00:50:08 So next time, then, what we'll do is we'll start by deriving 872 00:50:08 --> 00:50:09 the Gibbs phase rule. 873 00:50:09 --> 00:50:11 Which is not so hard. 874 00:50:11 --> 00:50:14 And then start on our way to the colligative properties. 875 00:50:14 --> 00:50:15 And then you'll have an exam. 876 00:50:15 --> 00:50:16