1 00:00:00,000 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,830 Commons license. 3 00:00:03,830 --> 00:00:06,840 Your support will help MIT OpenCourseWare continue to 4 00:00:06,840 --> 00:00:10,510 offer high quality educational resources for free. 5 00:00:10,510 --> 00:00:13,390 To make a donation or view additional materials from 6 00:00:13,390 --> 00:00:17,150 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,150 --> 00:00:21,490 ocw.mit.edu. 8 00:00:21,490 --> 00:00:25,600 PROFESSOR: Let's do a one minute review, and then move 9 00:00:25,600 --> 00:00:27,940 onto the Clausius-Clapeyron equation and see how far we 10 00:00:27,940 --> 00:00:29,250 can go on that. 11 00:00:29,250 --> 00:00:33,390 So, reminding you then, what you learned last time about 12 00:00:33,390 --> 00:00:42,860 phase transitions is, you drew a phase diagram in temperature 13 00:00:42,860 --> 00:00:45,600 pressure space here. 14 00:00:45,600 --> 00:00:49,910 And then you had a triple point somewhere, which was a 15 00:00:49,910 --> 00:00:51,830 unique point. 16 00:00:51,830 --> 00:00:56,590 The triple point temperature, triple point pressure. 17 00:00:56,590 --> 00:01:05,200 Then there was a gas, solid, call it coexistence line. 18 00:01:05,200 --> 00:01:08,510 Which kept on going ad infinitum. 19 00:01:08,510 --> 00:01:13,080 And the a gas liquid coexistent line. 20 00:01:13,080 --> 00:01:20,450 With a critical point. 21 00:01:20,450 --> 00:01:23,000 So that allowed you to go around. 22 00:01:23,000 --> 00:01:24,860 You wanted to go from liquid to gas, you could actually go 23 00:01:24,860 --> 00:01:27,850 around this critical point, and never actually see a phase 24 00:01:27,850 --> 00:01:30,060 transition. 25 00:01:30,060 --> 00:01:35,040 Then there was a solid liquid coexistence line. 26 00:01:35,040 --> 00:01:41,920 Which usually has a negative slope, except for the two most 27 00:01:41,920 --> 00:01:47,230 important substances on Earth, which are water and silicon. 28 00:01:47,230 --> 00:01:56,580 So for water and silicon, we have positive slope. 29 00:01:56,580 --> 00:01:59,650 H2O and silicon. 30 00:01:59,650 --> 00:02:07,930 This is why there's life on Earth. 31 00:02:07,930 --> 00:02:13,740 Did you hear the story of why there's life on Earth? 32 00:02:13,740 --> 00:02:15,350 This is the secret of life. 33 00:02:15,350 --> 00:02:19,040 Without this, this is why we're here. 34 00:02:19,040 --> 00:02:22,000 Because of this property. 35 00:02:22,000 --> 00:02:31,290 So, the reason why these slopes come from the, so I'll 36 00:02:31,290 --> 00:02:32,250 tell you the secret of life. 37 00:02:32,250 --> 00:02:42,040 But first let me remind you what the coexistence curve is. 38 00:02:42,040 --> 00:02:42,890 dp/dT. 39 00:02:42,890 --> 00:02:52,000 Coexistence is delta S over delta V, or delta H over T 40 00:02:52,000 --> 00:03:00,640 delta V. And that's the Clapeyron equation. 41 00:03:00,640 --> 00:03:05,190 Now, the slope of the curves points us by, look at this 42 00:03:05,190 --> 00:03:10,030 solid liquid line, delta S, for solid liquid. 43 00:03:10,030 --> 00:03:11,790 I looked at it up here. 44 00:03:11,790 --> 00:03:17,710 It's going to be S liquid minus S solid. 45 00:03:17,710 --> 00:03:19,810 And you know that's greater than zero because the entropy 46 00:03:19,810 --> 00:03:21,890 in the liquid state is much bigger than it is 47 00:03:21,890 --> 00:03:22,990 in the solid state. 48 00:03:22,990 --> 00:03:25,690 You have a crystal, entropy's very small. 49 00:03:25,690 --> 00:03:28,060 Liquid state, much more disorder. 50 00:03:28,060 --> 00:03:30,200 Got a positive sign here. 51 00:03:30,200 --> 00:03:31,360 Yes. 52 00:03:31,360 --> 00:03:31,740 Questions? 53 00:03:31,740 --> 00:03:40,320 STUDENT: [INAUDIBLE] 54 00:03:40,320 --> 00:03:44,310 PROFESSOR: So, this is almost everything. 55 00:03:44,310 --> 00:03:46,880 It has this slope here. 56 00:03:46,880 --> 00:03:51,060 Which is a negative slope. 57 00:03:51,060 --> 00:03:55,890 And water has a positive slope. 58 00:03:55,890 --> 00:03:56,170 Right? 59 00:03:56,170 --> 00:04:01,380 Is that wrong? 60 00:04:01,380 --> 00:04:03,990 It's wrong. 61 00:04:03,990 --> 00:04:04,900 Let's go through the argument. 62 00:04:04,900 --> 00:04:05,460 All right. 63 00:04:05,460 --> 00:04:06,420 All right. 64 00:04:06,420 --> 00:04:10,230 Let's go through the argument. 65 00:04:10,230 --> 00:04:18,250 So, then you have delta V, and delta V is V solid minus V 66 00:04:18,250 --> 00:04:21,360 liquid, this is per mole. 67 00:04:21,360 --> 00:04:24,790 Now, almost every substance has V 68 00:04:24,790 --> 00:04:29,070 solid less than V liquid. 69 00:04:29,070 --> 00:04:31,660 So this is negative. 70 00:04:31,660 --> 00:04:34,410 So almost every substance, you have delta S divided by delta 71 00:04:34,410 --> 00:04:35,120 V is negative. 72 00:04:35,120 --> 00:04:41,850 A negative slope. 73 00:04:41,850 --> 00:04:42,630 You're right. 74 00:04:42,630 --> 00:04:44,650 So I have it backwards. 75 00:04:44,650 --> 00:04:47,230 I do have it backwards. 76 00:04:47,230 --> 00:04:49,750 OK. 77 00:04:49,750 --> 00:04:52,135 See, I was really trying to make it so that 78 00:04:52,135 --> 00:04:53,070 my drawing was right. 79 00:04:53,070 --> 00:04:55,310 Which is why I inverted the solid and liquid here. 80 00:04:55,310 --> 00:04:57,580 So I really wanted to be right, but I 81 00:04:57,580 --> 00:05:00,850 ended up being wrong. 82 00:05:00,850 --> 00:05:05,180 Liquid is less than solid. 83 00:05:05,180 --> 00:05:09,210 Well this is water. 84 00:05:09,210 --> 00:05:11,069 Why do I have it backwards in my notes. 85 00:05:11,069 --> 00:05:17,330 I've got to correct that. 86 00:05:17,330 --> 00:05:19,510 So this is most substances, is positive. 87 00:05:19,510 --> 00:05:30,750 OK so, and most substances, OK? 88 00:05:30,750 --> 00:05:33,360 Except for water and silicon. 89 00:05:33,360 --> 00:05:39,520 Because for water and silicon this is reversed. 90 00:05:39,520 --> 00:05:42,670 Because the molar volume of solid for water is bigger than 91 00:05:42,670 --> 00:05:44,210 that for liquid. 92 00:05:44,210 --> 00:05:46,550 So this is bigger than here. 93 00:05:46,550 --> 00:05:47,356 Thank you. 94 00:05:47,356 --> 00:05:47,760 OK. 95 00:05:47,760 --> 00:05:49,500 Thanks for catching it. 96 00:05:49,500 --> 00:05:54,040 OK, so what happens if the molar volume for the liquid is 97 00:05:54,040 --> 00:05:56,470 bigger than the molar volume of the solid. 98 00:05:56,470 --> 00:06:00,740 The density of water which is the inverse of the mole 99 00:06:00,740 --> 00:06:05,270 volume, so the density, the molar density, is equal to one 100 00:06:05,270 --> 00:06:07,060 over the molar volume. 101 00:06:07,060 --> 00:06:09,710 So you've got the molar density for liquid is then 102 00:06:09,710 --> 00:06:14,980 smaller than the molar density for the solid. 103 00:06:14,980 --> 00:06:16,640 It's one over, right? 104 00:06:16,640 --> 00:06:18,510 So that's why ice floats. 105 00:06:18,510 --> 00:06:23,710 Because it's less dense than liquid water. 106 00:06:23,710 --> 00:06:28,540 So what do you think would happen in the winter if ice 107 00:06:28,540 --> 00:06:33,320 didn't float, in the Charles River. 108 00:06:33,320 --> 00:06:34,660 It would go to the bottom. 109 00:06:34,660 --> 00:06:38,240 It would freeze, solid and you would have no fish. 110 00:06:38,240 --> 00:06:39,140 You would have no life. 111 00:06:39,140 --> 00:06:45,560 It would freeze solid and that would be the end of life. 112 00:06:45,560 --> 00:06:48,010 And you know, five hundred million years ago there would 113 00:06:48,010 --> 00:06:49,910 be no fish. 114 00:06:49,910 --> 00:06:51,210 There would be no dinosaurs. 115 00:06:51,210 --> 00:06:53,750 There would be no anything, there would be no humans. 116 00:06:53,750 --> 00:06:55,810 That's the secret of life. 117 00:06:55,810 --> 00:07:02,280 And the fact that this is also the case is very useful for 118 00:07:02,280 --> 00:07:06,470 making silicon. 119 00:07:06,470 --> 00:07:10,700 For processing silicon. 120 00:07:10,700 --> 00:07:11,750 Another the secret of life. 121 00:07:11,750 --> 00:07:13,710 Where would we be without silicon? 122 00:07:13,710 --> 00:07:17,550 Our civilization would be, we'd still be using stones and 123 00:07:17,550 --> 00:07:18,190 things, right? 124 00:07:18,190 --> 00:07:25,310 OK, so it's super super important. 125 00:07:25,310 --> 00:07:29,280 Alright, so this is a Clapeyron story. 126 00:07:29,280 --> 00:07:33,720 Which we ended up getting right. 127 00:07:33,720 --> 00:07:37,090 Even though I insisted on getting it wrong. 128 00:07:37,090 --> 00:07:40,910 So what happens now is Mr. Clausius came around and he 129 00:07:40,910 --> 00:07:43,360 realized that you can make some approximations that are 130 00:07:43,360 --> 00:07:44,760 very useful. 131 00:07:44,760 --> 00:07:48,380 So the first approximation is, you can realize that the molar 132 00:07:48,380 --> 00:07:53,260 volume of the gas is always much bigger than the volume of 133 00:07:53,260 --> 00:07:58,620 the solid or the liquid. 134 00:07:58,620 --> 00:08:03,250 And so as a result whenever you have these delta V's, for 135 00:08:03,250 --> 00:08:15,460 sublimation, or delta V for vaporization, which is the 136 00:08:15,460 --> 00:08:18,230 volume the gas minus the volume of the solid. 137 00:08:18,230 --> 00:08:21,790 Or the volume of the gas minus the volume of the liquid, you 138 00:08:21,790 --> 00:08:23,120 might as well ignore the volume of the 139 00:08:23,120 --> 00:08:24,490 liquid or of the solid. 140 00:08:24,490 --> 00:08:27,740 And this can just be equal to roughly the volume of the gas. 141 00:08:27,740 --> 00:08:30,090 So this is the first approximation he made. 142 00:08:30,090 --> 00:08:33,445 So now, if you go back to the to the Clapeyron equation up 143 00:08:33,445 --> 00:08:39,710 here, with this approximation then dp/dT, the coexistence 144 00:08:39,710 --> 00:08:48,070 line, is delta H divided by T V gas. 145 00:08:48,070 --> 00:08:51,480 For sublimation. 146 00:08:51,480 --> 00:08:52,700 Or vaporization. 147 00:08:52,700 --> 00:08:56,400 You don't have the delta V down there any more. 148 00:08:56,400 --> 00:09:00,180 Then the next thing that, next assumption that he realized 149 00:09:00,180 --> 00:09:04,920 you can make was, well, all these gases could, they're 150 00:09:04,920 --> 00:09:05,910 like ideal gases. 151 00:09:05,910 --> 00:09:10,515 So we can, instead of having the volume of the gas here, we 152 00:09:10,515 --> 00:09:11,960 can use the ideal gas law. 153 00:09:11,960 --> 00:09:16,350 This is all, let's do all this per mole. 154 00:09:16,350 --> 00:09:21,360 So V is equal to RT over p. 155 00:09:21,360 --> 00:09:24,400 You can plug this back in here. 156 00:09:24,400 --> 00:09:28,150 And then you end up with the Clausius-Clapeyron equation. 157 00:09:28,150 --> 00:09:29,600 So this is approximation now. 158 00:09:29,600 --> 00:09:31,610 We put in RT over p for here. 159 00:09:31,610 --> 00:09:39,550 We have p delta H, where this is either sublimation or 160 00:09:39,550 --> 00:09:51,700 vaporization divided by RT squared. dp/dT, sublimation or 161 00:09:51,700 --> 00:10:00,930 vaporization, and this is the Clausius-Clapeyron equation. 162 00:10:00,930 --> 00:10:04,250 Two important approximations that go in here. 163 00:10:04,250 --> 00:10:08,650 And it's not valid for solids or liquid. 164 00:10:08,650 --> 00:10:11,310 You have to have a gas in there because of the ideal gas 165 00:10:11,310 --> 00:10:17,370 for the approximation that goes in here. 166 00:10:17,370 --> 00:10:22,020 And once you have that, you can integrate both sides. 167 00:10:22,020 --> 00:10:24,620 You've got to put the temperatures on one side. 168 00:10:24,620 --> 00:10:29,140 Put the pressures on the other side. 169 00:10:29,140 --> 00:10:33,190 I'll go ahead and cover this up here. 170 00:10:33,190 --> 00:10:38,900 So you have one over p dp/dT is equal to 171 00:10:38,900 --> 00:10:43,070 delta H over RT squared. 172 00:10:43,070 --> 00:10:47,540 One over p dp/dT that's just like d log p. 173 00:10:47,540 --> 00:10:50,200 And that's another form that you're going to see often for 174 00:10:50,200 --> 00:10:54,760 the Clausius-Clapeyron approximation is d log p / dT 175 00:10:54,760 --> 00:10:58,930 is equal to delta H over RT squared. 176 00:10:58,930 --> 00:11:01,110 And I'm freely dropping the sublimation and the 177 00:11:01,110 --> 00:11:03,350 vaporization because we know that's what I mean here. 178 00:11:03,350 --> 00:11:06,420 It's only valid for those two lines. 179 00:11:06,420 --> 00:11:10,490 So that's another form that you'll see. 180 00:11:10,490 --> 00:11:16,710 And that's not dT here, that's d log p, d log p dp. 181 00:11:16,710 --> 00:11:20,330 Yeah, that's right, d log p / dT. 182 00:11:20,330 --> 00:11:21,620 And so now you can take the dT to the 183 00:11:21,620 --> 00:11:23,510 other side and integrate. 184 00:11:23,510 --> 00:11:28,970 From point one to point two d log p, is from point one to 185 00:11:28,970 --> 00:11:37,720 point two, delta H over RT squared dT. 186 00:11:37,720 --> 00:11:40,290 And you make the usual approximation, which we've 187 00:11:40,290 --> 00:11:41,570 made before. 188 00:11:41,570 --> 00:11:47,620 That delta H is roughly independent of temperature, or 189 00:11:47,620 --> 00:11:51,200 very slowly changing with temperature. 190 00:11:51,200 --> 00:11:52,670 And that's fine as long as you're in a 191 00:11:52,670 --> 00:11:54,290 narrow temperature range. 192 00:11:54,290 --> 00:11:56,720 Which is usually the case when you have these 193 00:11:56,720 --> 00:11:58,770 Clausius-Clapeyron equation problems. 194 00:11:58,770 --> 00:12:01,425 So you can take the delta H out of the integral, the R out 195 00:12:01,425 --> 00:12:01,900 of the integral. 196 00:12:01,900 --> 00:12:03,760 And you can integrate both sides. 197 00:12:03,760 --> 00:12:16,720 To give you log p2 minus log p1, is delta H times R, 198 00:12:16,720 --> 00:12:26,560 divided by R, delta H divided by R. Delta H over R times T2 199 00:12:26,560 --> 00:12:30,250 minus T1 over T1 T2. 200 00:12:30,250 --> 00:12:33,900 It looks a lot like and the temperature dependence 201 00:12:33,900 --> 00:12:36,910 equilibrium constant. 202 00:12:36,910 --> 00:12:40,080 And in fact, but it's not, right? 203 00:12:40,080 --> 00:12:43,320 And some people use that equation here instead of 204 00:12:43,320 --> 00:12:44,190 equilibrium constant. 205 00:12:44,190 --> 00:12:45,880 Temperature dependence. 206 00:12:45,880 --> 00:12:48,460 That's, can give rise to problems. 207 00:12:48,460 --> 00:12:50,940 Because obviously they're not the same equation. 208 00:12:50,940 --> 00:12:56,690 But roughly has the same form. 209 00:12:56,690 --> 00:13:01,980 So, another way that you'll also find this written is very 210 00:13:01,980 --> 00:13:05,770 often p1 and T1 will be some reference. 211 00:13:05,770 --> 00:13:08,890 Pressures and temperature, maybe one bar, let's say. 212 00:13:08,890 --> 00:13:11,600 T1, 298 degrees Kelvin. 213 00:13:11,600 --> 00:13:13,640 That's your reference point and you want to find out the 214 00:13:13,640 --> 00:13:21,000 pressure temperature dependence in an equation. 215 00:13:21,000 --> 00:13:24,410 So if you rearrange your equations so p1, T1 are now 216 00:13:24,410 --> 00:13:25,370 some reference point. 217 00:13:25,370 --> 00:13:26,140 They become numbers. 218 00:13:26,140 --> 00:13:29,480 So you have log of a reference point, T1 here is a number. 219 00:13:29,480 --> 00:13:36,400 So you often see this equation rewritten then as log p is 220 00:13:36,400 --> 00:13:43,080 equal to delta H over RT. 221 00:13:43,080 --> 00:13:45,360 And I feel like I've got a minus sign missing somewhere. 222 00:13:45,360 --> 00:13:46,030 Here. 223 00:13:46,030 --> 00:13:49,420 Oh yeah, when you integrate this, the one over T squared, 224 00:13:49,420 --> 00:13:50,210 there's a minus sign. 225 00:13:50,210 --> 00:13:53,460 Oh, I took care of that by doing T2 minus T1 here. 226 00:13:53,460 --> 00:13:56,310 I think there's still a minus sign problem somewhere. 227 00:13:56,310 --> 00:13:59,420 I think there's a minus sign problem. 228 00:13:59,420 --> 00:14:06,150 Let me check up the notes here. 229 00:14:06,150 --> 00:14:10,830 Minus one over T1 minus T2 over T1 T2. 230 00:14:10,830 --> 00:14:27,030 Where's my minus sign? 231 00:14:27,030 --> 00:14:29,430 T2 minus T1. 232 00:14:29,430 --> 00:14:49,000 OK, so I got minus T1, minus T2 over T1 T2. 233 00:14:49,000 --> 00:14:49,900 T2 minus T1. 234 00:14:49,900 --> 00:14:51,170 This is fine, right. 235 00:14:51,170 --> 00:14:53,000 There's no minus sign problem. 236 00:14:53,000 --> 00:14:56,540 Plus a constant. 237 00:14:56,540 --> 00:15:08,030 OK, so you'll often see it written like that. 238 00:15:08,030 --> 00:15:10,160 And that gives you a relationship between the 239 00:15:10,160 --> 00:15:16,160 pressure and the temperature then, for a substance where 240 00:15:16,160 --> 00:15:20,425 all the reference point information is contained in 241 00:15:20,425 --> 00:15:24,220 this constant, C. Which is where the T1's and 242 00:15:24,220 --> 00:15:27,530 the p's come from. 243 00:15:27,530 --> 00:15:31,690 OK, any questions? 244 00:15:31,690 --> 00:15:39,940 We'll do a little example here of how this could be used. 245 00:15:39,940 --> 00:15:44,440 So let's switch to an example now. 246 00:15:44,440 --> 00:15:46,910 Let's go to the example, let's first do the example which is 247 00:15:46,910 --> 00:15:53,750 at the very end of the notes. 248 00:15:53,750 --> 00:16:00,960 So this is an example of RDX. 249 00:16:00,960 --> 00:16:08,780 RDX is a famous explosive, as I'm sure you know. 250 00:16:08,780 --> 00:16:14,820 And it's plastic explosive. 251 00:16:14,820 --> 00:16:18,440 It's got a melting point of 204 degrees Celsius. 252 00:16:18,440 --> 00:16:21,340 481 degrees Kelvin. 253 00:16:21,340 --> 00:16:28,450 And it's a problem for, so it's a very powerful explosive 254 00:16:28,450 --> 00:16:30,620 and you can't see it in X-rays. 255 00:16:30,620 --> 00:16:32,630 So it's a problem at airports. 256 00:16:32,630 --> 00:16:34,800 You've got be able to find if somebody's carrying a little 257 00:16:34,800 --> 00:16:37,210 piece of RDX in their luggage. 258 00:16:37,210 --> 00:16:40,930 And so, the question becomes what do you need to 259 00:16:40,930 --> 00:16:41,950 know to detect it? 260 00:16:41,950 --> 00:16:45,010 You want to find the vapor pressure. 261 00:16:45,010 --> 00:16:48,210 You've gotta have, and the machine which basically is 262 00:16:48,210 --> 00:16:53,520 sensitive enough to detect molecules of RDX that are in 263 00:16:53,520 --> 00:16:57,630 vapor at room temperature. 264 00:16:57,630 --> 00:17:02,230 So the way that you do that is, you do an experiment. 265 00:17:02,230 --> 00:17:05,000 Where you measure the pressure, the vapor pressure, 266 00:17:05,000 --> 00:17:07,180 as a function of temperature. 267 00:17:07,180 --> 00:17:10,980 And room temperature, it turns out if it's not volatile at 268 00:17:10,980 --> 00:17:13,900 all, if the vapor pressure is very, really tiny. 269 00:17:13,900 --> 00:17:16,930 And so in order to get accurate numbers, you actually 270 00:17:16,930 --> 00:17:20,070 do your measurements at significantly higher 271 00:17:20,070 --> 00:17:22,970 temperatures than room temperature. 272 00:17:22,970 --> 00:17:25,210 They do it close to the melting point, so you're 273 00:17:25,210 --> 00:17:28,150 actually looking at the sublimation of RDX. 274 00:17:28,150 --> 00:17:33,170 You do it, let's say, at 400 degrees Kelvin, where the 275 00:17:33,170 --> 00:17:35,890 melting point is at 481 degrees Kelvin. 276 00:17:35,890 --> 00:17:37,700 So slightly below the melting point you're looking at 277 00:17:37,700 --> 00:17:38,880 sublimation. 278 00:17:38,880 --> 00:17:40,610 And you plot. 279 00:17:40,610 --> 00:17:51,500 Log p versus one over T. And according to 280 00:17:51,500 --> 00:17:52,580 Clausius-Clapeyron, that should give 281 00:17:52,580 --> 00:17:53,890 you a straight line. 282 00:17:53,890 --> 00:17:56,820 And in fact, that's exactly what you see when you take RDX 283 00:17:56,820 --> 00:17:58,420 and do that experiment. 284 00:17:58,420 --> 00:18:02,530 You end up with a bunch of data points. 285 00:18:02,530 --> 00:18:08,150 That's fall nicely on this straight line. 286 00:18:08,150 --> 00:18:11,930 And then you can extrapolate to room temperature. 287 00:18:11,930 --> 00:18:15,790 Somewhere here, let's say, 300 degrees Kelvin. 288 00:18:15,790 --> 00:18:21,250 And find out what log p is, at 300 degrees Kelvin. 289 00:18:21,250 --> 00:18:23,700 And what you find is that, and that's the graph that's on the 290 00:18:23,700 --> 00:18:28,340 last page of the lecture notes, that the vapor pressure 291 00:18:28,340 --> 00:18:35,630 of RDX at room temperature is 10 to the minus 11 bar, That's 292 00:18:35,630 --> 00:18:37,540 ten parts per trillion. 293 00:18:37,540 --> 00:18:40,970 So it tells you that if you want to detect RDX with a 294 00:18:40,970 --> 00:18:43,670 sniffer machine at the airport, that machine better 295 00:18:43,670 --> 00:18:49,980 be able to tell you, find one molecule of RDX out of a tenth 296 00:18:49,980 --> 00:18:52,310 of a trillion other molecules, basically. 297 00:18:52,310 --> 00:18:58,550 Which is a really hard thing to do. 298 00:18:58,550 --> 00:19:00,080 So this gives you a design rule for 299 00:19:00,080 --> 00:19:02,300 these pieces of equipment. 300 00:19:02,300 --> 00:19:06,190 And why they're so expensive. 301 00:19:06,190 --> 00:19:12,520 OK, before we do the next example, let's see if any 302 00:19:12,520 --> 00:19:15,670 questions on Clausius-Clapeyron. 303 00:19:15,670 --> 00:19:15,940 Yeah. 304 00:19:15,940 --> 00:19:27,000 STUDENT: [INAUDIBLE] 305 00:19:27,000 --> 00:19:31,190 PROFESSOR: So, p as some exponential here? 306 00:19:31,190 --> 00:19:33,010 You know, I don't recall seeing it as 307 00:19:33,010 --> 00:19:33,940 an exponential form. 308 00:19:33,940 --> 00:19:35,940 But that doesn't mean it's not used the exponential form. 309 00:19:35,940 --> 00:19:39,250 This is an easy way to do it, because then you've got a 310 00:19:39,250 --> 00:19:40,850 linear relationship. 311 00:19:40,850 --> 00:19:43,940 And generally we like to see straight lines. 312 00:19:43,940 --> 00:19:49,780 So this gives you a nice straight line. 313 00:19:49,780 --> 00:19:57,800 OK, so now let's do a slightly more complicated example. 314 00:19:57,800 --> 00:20:05,690 Which is, so most of the time you don't have pure material. 315 00:20:05,690 --> 00:20:07,340 You have a mixture of some sort. 316 00:20:07,340 --> 00:20:11,840 For instance, if I have a glass of water on the table at 317 00:20:11,840 --> 00:20:13,840 room temperature. 318 00:20:13,840 --> 00:20:22,510 Above the glass of water, I have air, there's my H2O here. 319 00:20:22,510 --> 00:20:24,440 And then I've got some air. 320 00:20:24,440 --> 00:20:26,770 Above the glass of water. 321 00:20:26,770 --> 00:20:28,430 And the air's inert to the water. 322 00:20:28,430 --> 00:20:29,630 It's not reacting with the water. 323 00:20:29,630 --> 00:20:34,020 So it's like an inert gas sitting on top of the water. 324 00:20:34,020 --> 00:20:37,130 Then I want to ask the question, what is the vapor 325 00:20:37,130 --> 00:20:40,730 pressure of the water? 326 00:20:40,730 --> 00:20:43,220 In the presence of this inert gas, the air. 327 00:20:43,220 --> 00:20:47,980 It's not really the same as the problem that we've been 328 00:20:47,980 --> 00:20:49,520 looking at up here. 329 00:20:49,520 --> 00:20:50,050 Right? 330 00:20:50,050 --> 00:20:54,610 Because this gas liquid coexistence line, this 331 00:20:54,610 --> 00:20:57,860 diagram, and this Clapeyron equation, is all done for a 332 00:20:57,860 --> 00:21:01,140 pure substance. 333 00:21:01,140 --> 00:21:04,570 And at room temperature, at room temperature, we're 334 00:21:04,570 --> 00:21:07,130 sitting squarely in the liquid phase. 335 00:21:07,130 --> 00:21:09,710 We're not on the coexistence line. 336 00:21:09,710 --> 00:21:13,790 At room temperature and one bar pressure. 337 00:21:13,790 --> 00:21:17,160 So instead of having this system here, I looked at the 338 00:21:17,160 --> 00:21:22,820 system where I had the water, H2O, with nothing on top. 339 00:21:22,820 --> 00:21:24,150 With no air on top. 340 00:21:24,150 --> 00:21:25,300 Except the cylinder. 341 00:21:25,300 --> 00:21:31,500 Now, I'd better make the surface of my water a straight 342 00:21:31,500 --> 00:21:33,520 line, otherwise I'm going to get in trouble. 343 00:21:33,520 --> 00:21:35,580 There's the water sitting right here. 344 00:21:35,580 --> 00:21:38,980 And I put a cylinder with one bar pressure on top. 345 00:21:38,980 --> 00:21:41,370 My cylinder's going to sit squarely on the stump of the 346 00:21:41,370 --> 00:21:46,670 surface of the water. 347 00:21:46,670 --> 00:21:49,640 This is not going to be any water vapor at one bar 348 00:21:49,640 --> 00:21:52,090 pressure on that cylinder. 349 00:21:52,090 --> 00:21:54,710 At one bar, we're way up in the liquid phase. 350 00:21:54,710 --> 00:21:59,170 I would have to decrease the pressure on that cylinder down 351 00:21:59,170 --> 00:22:05,030 to, I don't know, whatever the water vapor pressure is at 352 00:22:05,030 --> 00:22:07,900 room temperature here on this coexistence curve. 353 00:22:07,900 --> 00:22:09,700 0.1 bar or something. 354 00:22:09,700 --> 00:22:11,120 0.05 bar. 355 00:22:11,120 --> 00:22:14,400 Something pretty small. 356 00:22:14,400 --> 00:22:16,870 So that's what we've been working on. 357 00:22:16,870 --> 00:22:17,620 This pure system. 358 00:22:17,620 --> 00:22:20,440 And now we're going to be working on this system here, 359 00:22:20,440 --> 00:22:24,610 where we have air. 360 00:22:24,610 --> 00:22:28,610 With a cylinder on top, one bar. 361 00:22:28,610 --> 00:22:34,320 And we want to know what is the vapor pressure of H2O in 362 00:22:34,320 --> 00:22:40,400 this system here. 363 00:22:40,400 --> 00:22:45,780 Is it zero? 364 00:22:45,780 --> 00:22:49,300 Anybody want to guess, if it's zero? 365 00:22:49,300 --> 00:22:52,050 We know there's vapor pressure there, right? 366 00:22:52,050 --> 00:22:55,630 And a good guess is that that vapor pressure here, it's 367 00:22:55,630 --> 00:23:00,000 pretty close to the pressure that you would guess by using 368 00:23:00,000 --> 00:23:05,050 this diagram with the pure water. 369 00:23:05,050 --> 00:23:08,060 But the key word is, it's pretty close. 370 00:23:08,060 --> 00:23:12,180 It's not exactly the same. 371 00:23:12,180 --> 00:23:13,450 So, let me ask you this. 372 00:23:13,450 --> 00:23:21,510 Without looking at the notes, now is p H2O in this problem 373 00:23:21,510 --> 00:23:31,100 here greater than, less than, or the same as p H2O in the 374 00:23:31,100 --> 00:23:37,340 pure Clapeyron case? 375 00:23:37,340 --> 00:23:41,320 How many people think that it's the same? 376 00:23:41,320 --> 00:23:44,060 How about smaller than? 377 00:23:44,060 --> 00:23:46,080 Smaller than? 378 00:23:46,080 --> 00:23:47,680 Few people think that it's smaller than. 379 00:23:47,680 --> 00:23:50,830 How about greater than? 380 00:23:50,830 --> 00:23:54,440 A bunch of people don't know what to answer. 381 00:23:54,440 --> 00:23:55,550 OK. 382 00:23:55,550 --> 00:23:58,590 I think that if you think about it, what the difference 383 00:23:58,590 --> 00:24:04,950 is between here and there, you could probably reason which 384 00:24:04,950 --> 00:24:06,030 way it should go. 385 00:24:06,030 --> 00:24:09,660 Which way the partial pressure of the water should go under 386 00:24:09,660 --> 00:24:11,700 one bar pressure, room temperature. 387 00:24:11,700 --> 00:24:14,830 Compared to what you might expect it to be at room 388 00:24:14,830 --> 00:24:19,610 temperature with a diagram. 389 00:24:19,610 --> 00:24:21,390 So, I'm going to give you a little bit of time just to 390 00:24:21,390 --> 00:24:22,370 think about it. 391 00:24:22,370 --> 00:24:24,670 Without looking at the notes, what you think the 392 00:24:24,670 --> 00:24:25,650 right sign is here. 393 00:24:25,650 --> 00:24:33,280 Then we'll do the problem and then we'll figure out if you 394 00:24:33,280 --> 00:25:08,210 were right or wrong. 395 00:25:08,210 --> 00:25:09,700 OK. 396 00:25:09,700 --> 00:25:14,140 How many people say it's greater than? 397 00:25:14,140 --> 00:25:16,360 One, two. 398 00:25:16,360 --> 00:25:18,610 Two brave people here. 399 00:25:18,610 --> 00:25:20,560 Two. 400 00:25:20,560 --> 00:25:23,350 How many people say it's smaller than? 401 00:25:23,350 --> 00:25:24,280 OK. 402 00:25:24,280 --> 00:25:25,500 Smaller than. 403 00:25:25,500 --> 00:25:27,640 So I'm going to say a large number, OK? 404 00:25:27,640 --> 00:25:29,720 Many. 405 00:25:29,720 --> 00:25:33,550 How many people think it's equal? 406 00:25:33,550 --> 00:25:36,710 One person thinks it's equal. 407 00:25:36,710 --> 00:25:41,890 Alright, let's do it out now and see how it comes out. 408 00:25:41,890 --> 00:25:46,530 And then we can reason why it came out the way it came out. 409 00:25:46,530 --> 00:25:48,270 Logically, without actually doing math. 410 00:25:48,270 --> 00:25:51,590 Because there's a way to reason it out that you should 411 00:25:51,590 --> 00:25:57,540 be able to do. 412 00:25:57,540 --> 00:25:59,100 OK, so this is what we have here. 413 00:25:59,100 --> 00:26:05,180 Then the system is, we have this container. 414 00:26:05,180 --> 00:26:07,970 Let me redo it instead of writing air. 415 00:26:07,970 --> 00:26:11,330 So we have the liquid state here. 416 00:26:11,330 --> 00:26:13,600 Some substance, and it doesn't have to be water. 417 00:26:13,600 --> 00:26:16,730 That's a liquid, at some pressure p. 418 00:26:16,730 --> 00:26:18,550 And I'm going to say P capital, that's going to be 419 00:26:18,550 --> 00:26:19,980 the total pressure. 420 00:26:19,980 --> 00:26:24,550 And the total pressure is given by the piston here. p 421 00:26:24,550 --> 00:26:28,710 total, p T. And on top here I'm going to have two 422 00:26:28,710 --> 00:26:29,300 substances. 423 00:26:29,300 --> 00:26:32,500 I'm going to have A, in the gas phase, 424 00:26:32,500 --> 00:26:33,950 that's the water vapor. 425 00:26:33,950 --> 00:26:37,610 Some partial pressure, p sub A, temperature T. And I'm 426 00:26:37,610 --> 00:26:42,970 going to have the inert gas, with some 427 00:26:42,970 --> 00:26:45,070 partial pressure p, inert. 428 00:26:45,070 --> 00:26:49,270 Such that the sum of the partial pressure of the inert 429 00:26:49,270 --> 00:26:53,560 gas, plus the partial pressure of A, is the total pressure. 430 00:26:53,560 --> 00:26:54,860 One bar, in our case. 431 00:26:54,860 --> 00:26:57,630 With the water with the air on top. 432 00:26:57,630 --> 00:27:01,370 And the question, oh, let me make a few more definitions. 433 00:27:01,370 --> 00:27:12,140 We're going to define p0 as the vapor pressure of pure 434 00:27:12,140 --> 00:27:16,110 vapor pressure of pure A. So that's what you would read off 435 00:27:16,110 --> 00:27:19,900 the diagram here. 436 00:27:19,900 --> 00:27:26,860 And what we want, the question we ask is, what is the partial 437 00:27:26,860 --> 00:27:34,730 pressure of A as a function of the total pressure? 438 00:27:34,730 --> 00:27:36,940 That's what we're asking. 439 00:27:36,940 --> 00:27:39,750 As I'm increasing the total pressure, meaning I'm putting 440 00:27:39,750 --> 00:27:42,580 more and more inert gas in there, 441 00:27:42,580 --> 00:27:43,440 which way is this going? 442 00:27:43,440 --> 00:27:45,030 Is it going up, down, or staying the same? 443 00:27:45,030 --> 00:27:46,500 And we've got a bunch of votes up here. 444 00:27:46,500 --> 00:27:49,120 Most people think that it's going to go down. 445 00:27:49,120 --> 00:27:53,940 Because I put more gas in there. 446 00:27:53,940 --> 00:28:03,080 OK, I know that if P, capital P, is equal to pA, then pA is 447 00:28:03,080 --> 00:28:05,130 equal to p0. 448 00:28:05,130 --> 00:28:05,640 Right? 449 00:28:05,640 --> 00:28:07,210 There's no inert gas. 450 00:28:07,210 --> 00:28:10,100 So that's the one extreme case here. 451 00:28:10,100 --> 00:28:12,460 There's no inert gas, then the partial pressure's going to be 452 00:28:12,460 --> 00:28:16,200 what you read off that graph here. 453 00:28:16,200 --> 00:28:19,020 So another way of saying what is p as a function of capital 454 00:28:19,020 --> 00:28:23,300 P, is to ask the question what is the slope? 455 00:28:23,300 --> 00:28:24,590 I've got one point here. 456 00:28:24,590 --> 00:28:27,290 If I can integrate the slope then I have this equation. 457 00:28:27,290 --> 00:28:35,210 So the other way to ask the question is, what is dpA/dp, 458 00:28:35,210 --> 00:28:37,720 at a constant temperature? 459 00:28:37,720 --> 00:28:39,260 Where I'm keeping the temperature constant. 460 00:28:39,260 --> 00:28:43,060 And that turns out to be easier to calculate. 461 00:28:43,060 --> 00:28:45,740 These slopes are always easier to calculate than 462 00:28:45,740 --> 00:28:51,870 the absolute things. 463 00:28:51,870 --> 00:28:53,560 So we're doing an equilibrium here. 464 00:28:53,560 --> 00:28:55,080 We have an equilibrium between the liquid 465 00:28:55,080 --> 00:28:56,870 state and the gas phase. 466 00:28:56,870 --> 00:28:59,180 And we want to know something about this equilibrium as we 467 00:28:59,180 --> 00:29:01,100 change a parameter. 468 00:29:01,100 --> 00:29:03,500 Equilibrium, what do we always start with when we have an 469 00:29:03,500 --> 00:29:03,770 equilibrium. 470 00:29:03,770 --> 00:29:06,240 What's the first thing we think about? 471 00:29:06,240 --> 00:29:08,280 We think about chemical potential. 472 00:29:08,280 --> 00:29:12,560 Chemical potentials of A, in the gas phase has to be the 473 00:29:12,560 --> 00:29:14,660 same as the chemical potential of A in the liquid phase. 474 00:29:14,660 --> 00:29:17,660 Well, I can't think of anything else to start with. 475 00:29:17,660 --> 00:29:19,300 So let me start with that. 476 00:29:19,300 --> 00:29:24,990 So let's start by writing the chemical potentials being 477 00:29:24,990 --> 00:29:26,000 equal to each other. 478 00:29:26,000 --> 00:29:30,070 Because we know that's true. 479 00:29:30,070 --> 00:29:37,570 So mu A, in the gas phase, temperature and with a partial 480 00:29:37,570 --> 00:29:45,350 pressure p sub A. Is equal to mu A in the liquid phase, 481 00:29:45,350 --> 00:29:50,040 where the pressure on the liquid is capital P. 482 00:29:50,040 --> 00:29:50,820 What else do we know? 483 00:29:50,820 --> 00:29:53,130 Well, we, what else are we going to need? 484 00:29:53,130 --> 00:29:57,720 Before we do that, since what we're after is the derivative, 485 00:29:57,720 --> 00:29:59,320 let's take the derivative of both sides. 486 00:29:59,320 --> 00:30:02,180 We're looking at a derivative with respect to pressure here. 487 00:30:02,180 --> 00:30:04,130 Let's take the derivative outside with pressure and see 488 00:30:04,130 --> 00:30:05,600 what happens. 489 00:30:05,600 --> 00:30:07,246 So let's take the derivative of this with 490 00:30:07,246 --> 00:30:08,220 respect to total pressure. 491 00:30:08,220 --> 00:30:09,860 Well, this isn't total pressure. 492 00:30:09,860 --> 00:30:12,290 This is the partial pressure of A. But it's a function of 493 00:30:12,290 --> 00:30:14,250 the total pressure. 494 00:30:14,250 --> 00:30:16,350 It's a function of the total pressure. 495 00:30:16,350 --> 00:30:19,690 Partial pressure of A is total pressure minus p inert. 496 00:30:19,690 --> 00:30:24,630 So, we have to use the chain rule. 497 00:30:24,630 --> 00:30:32,500 So on this side here, we have d mu A / dpA. 498 00:30:32,500 --> 00:30:40,840 So I'm taking d/dp of both sides. 499 00:30:40,840 --> 00:30:42,980 Then by chain rule, dpA/dP. 500 00:30:48,250 --> 00:30:49,660 Constant temperature. 501 00:30:49,660 --> 00:30:56,690 And then on the right-hand side, I have d mu A / dP, 502 00:30:56,690 --> 00:31:00,210 constant temperature. 503 00:31:00,210 --> 00:31:05,930 Well, d mu / dP, that may or may not seem familiar to you. 504 00:31:05,930 --> 00:31:08,550 I never remember these things, but I always remember that 505 00:31:08,550 --> 00:31:11,830 there's, that these relationships are interesting. 506 00:31:11,830 --> 00:31:13,890 Especially when you're talking about something like a 507 00:31:13,890 --> 00:31:15,930 chemical potential, which is really nothing but the Gibbs 508 00:31:15,930 --> 00:31:17,780 free energy. 509 00:31:17,780 --> 00:31:22,060 With respect to pressure, when we know that the variables 510 00:31:22,060 --> 00:31:23,990 that we use for Gibbs free energy are pressure and 511 00:31:23,990 --> 00:31:24,930 temperature. 512 00:31:24,930 --> 00:31:25,540 Right? 513 00:31:25,540 --> 00:31:28,000 So this is the variable that goes with Gibbs free energy. 514 00:31:28,000 --> 00:31:31,902 So this means that they were taking the derivatives of the 515 00:31:31,902 --> 00:31:34,280 Gibbs free energy with respect to one of its variables is 516 00:31:34,280 --> 00:31:36,010 something that we should know. 517 00:31:36,010 --> 00:31:38,200 So we go back to writing what the Gibbs free energy is. 518 00:31:38,200 --> 00:31:44,850 So dG is equal to V dp minus S dT. 519 00:31:44,850 --> 00:31:46,670 Which is the fundamental equation 520 00:31:46,670 --> 00:31:48,170 for Gibbs free energy. 521 00:31:48,170 --> 00:31:50,310 And this is the Gibbs free energy per mole. 522 00:31:50,310 --> 00:31:53,730 That's just the chemical potential. 523 00:31:53,730 --> 00:31:56,360 The same thing. 524 00:31:56,360 --> 00:31:58,950 And this is the derivative with respect to pressure. 525 00:31:58,950 --> 00:32:01,380 And this is the derivative with respect to temperature. 526 00:32:01,380 --> 00:32:05,890 So this thing here is d mu / dp. 527 00:32:05,890 --> 00:32:08,250 At constant temperature. 528 00:32:08,250 --> 00:32:09,570 It's just the volume. 529 00:32:09,570 --> 00:32:12,300 So now we have this equation, where we started out with the 530 00:32:12,300 --> 00:32:12,980 chemical potentials. 531 00:32:12,980 --> 00:32:15,490 And we've got these d mu A / dpA, we've 532 00:32:15,490 --> 00:32:17,310 got d mu A / dP here. 533 00:32:17,310 --> 00:32:22,160 That's just the molar volume of the gas. 534 00:32:22,160 --> 00:32:24,790 Right, because this is mu of the gas phase here. 535 00:32:24,790 --> 00:32:27,550 So this is the molar volume of the gas. 536 00:32:27,550 --> 00:32:30,310 And I have dpa/dP. 537 00:32:30,310 --> 00:32:32,510 And, this is very nice. 538 00:32:32,510 --> 00:32:34,790 Because this is actually what I'm trying to get. 539 00:32:34,790 --> 00:32:37,900 Trying to see how the partial pressure of a changes with the 540 00:32:37,900 --> 00:32:38,490 total pressure. 541 00:32:38,490 --> 00:32:41,730 That's what I'm trying to calculate here. 542 00:32:41,730 --> 00:32:42,630 I know what this is. 543 00:32:42,630 --> 00:32:45,590 This is just an experimental number. 544 00:32:45,590 --> 00:32:47,790 And then on this side here I have d mu A / dP. 545 00:32:47,790 --> 00:32:50,720 This is the liquid phase. 546 00:32:50,720 --> 00:32:53,350 Derivative with respect to total pressure, that's the 547 00:32:53,350 --> 00:32:57,460 molar volume of the liquid. 548 00:32:57,460 --> 00:33:00,200 The liquid. 549 00:33:00,200 --> 00:33:08,920 Great, I'm done. dpA/dP total, constant pressure, is the 550 00:33:08,920 --> 00:33:14,350 ratio of the molar volume of the liquid of A divided by the 551 00:33:14,350 --> 00:33:17,970 molar volume of the gas. 552 00:33:17,970 --> 00:33:21,590 What's the sign? 553 00:33:21,590 --> 00:33:23,130 Volumes are positive, right? 554 00:33:23,130 --> 00:33:25,060 This has to be positive. 555 00:33:25,060 --> 00:33:27,370 There's no choice. 556 00:33:27,370 --> 00:33:29,200 It can't even be zero. 557 00:33:29,200 --> 00:33:30,970 It has to be positive, right? 558 00:33:30,970 --> 00:33:36,470 So that means the slope is a positive slope. 559 00:33:36,470 --> 00:33:44,770 As I increase the total pressure, p is going to go up. 560 00:33:44,770 --> 00:33:49,680 If I integrate now, this, starting at, so the total 561 00:33:49,680 --> 00:33:54,490 pressure is equal to pA, I'm going to get a curve that 562 00:33:54,490 --> 00:33:58,610 looks like this. 563 00:33:58,610 --> 00:34:02,410 OK, if I write on this axis p inert, so when p inert is 564 00:34:02,410 --> 00:34:05,180 equal to zero, total pressure is pA. 565 00:34:05,180 --> 00:34:07,550 I'm going to start at p0. 566 00:34:07,550 --> 00:34:11,700 And I'm going to go up from there, positive slope. 567 00:34:11,700 --> 00:34:13,400 So. 568 00:34:13,400 --> 00:34:16,790 Who was right? 569 00:34:16,790 --> 00:34:17,910 A couple people are right. 570 00:34:17,910 --> 00:34:20,340 So let's think about it in a different way. 571 00:34:20,340 --> 00:34:23,750 Let's think about it in a different way. 572 00:34:23,750 --> 00:34:26,260 So we did the math, and we found that actually it's, and 573 00:34:26,260 --> 00:34:28,590 that turns out to be a very small amount. 574 00:34:28,590 --> 00:34:32,960 You do a very good job by just taking this curve here. 575 00:34:32,960 --> 00:34:34,180 And using the pure substance. 576 00:34:34,180 --> 00:34:37,880 In fact, in the notes we have an example of how much it 577 00:34:37,880 --> 00:34:39,390 changes for mercury. 578 00:34:39,390 --> 00:34:45,630 So for mercury, at 100 degrees C, for pure mercury, the vapor 579 00:34:45,630 --> 00:34:48,730 pressure is 0.27 bar. 580 00:34:48,730 --> 00:34:53,170 If you add enough inert gas to make it 1 bar for the total 581 00:34:53,170 --> 00:34:56,720 pressure, you go from 0.27 to 0.2701. 582 00:34:56,720 --> 00:34:57,680 Not much of a change. 583 00:34:57,680 --> 00:35:00,680 They do a really good job of just picking this. 584 00:35:00,680 --> 00:35:04,300 And if you go to 100 bar, then you increase it to 0.28. 585 00:35:04,300 --> 00:35:07,130 So it takes a lot of pressure on top, a lot of extra inert 586 00:35:07,130 --> 00:35:10,550 gas to really make much of a difference. 587 00:35:10,550 --> 00:35:13,470 But, let's think about why it would make a 588 00:35:13,470 --> 00:35:19,180 difference, just logically. 589 00:35:19,180 --> 00:35:22,310 So in the absence of this inert gas here, I just have 590 00:35:22,310 --> 00:35:25,420 the pure substance. 591 00:35:25,420 --> 00:35:30,980 I add some inert gas, what happens to the entropy on top? 592 00:35:30,980 --> 00:35:33,660 It's increasing, right? 593 00:35:33,660 --> 00:35:36,790 Do systems like to have more entropy? 594 00:35:36,790 --> 00:35:38,630 Yeah. 595 00:35:38,630 --> 00:35:43,520 So, I have a lot of inert gas here. 596 00:35:43,520 --> 00:35:48,460 A small amount of A. What if I had a little bit more A, in 597 00:35:48,460 --> 00:35:49,510 the presence of the inert gas? 598 00:35:49,510 --> 00:35:54,050 Which way would the entropy go? 599 00:35:54,050 --> 00:35:55,060 It would go up a little bit more, 600 00:35:55,060 --> 00:35:57,040 you'd have more disorder. 601 00:35:57,040 --> 00:36:00,370 If I have nothing, if I have no A here, up here and just 602 00:36:00,370 --> 00:36:02,020 the inert gas, then there's no entropy of mixing. 603 00:36:02,020 --> 00:36:04,615 I add a little bit of inert gas, there's a little bit more 604 00:36:04,615 --> 00:36:05,720 entropy of mixing. 605 00:36:05,720 --> 00:36:08,890 Eventually, I add more and more, entropy of mixing goes 606 00:36:08,890 --> 00:36:11,330 up and up and up, and eventually it comes back down 607 00:36:11,330 --> 00:36:14,150 to where I have pure A on top. 608 00:36:14,150 --> 00:36:21,960 So the entropy of mixing up here is driving the increase 609 00:36:21,960 --> 00:36:27,090 in the pressure of, in the partial pressure of A. Entropy 610 00:36:27,090 --> 00:36:29,120 is this amazingly important term. 611 00:36:29,120 --> 00:36:29,750 That's what we saw. 612 00:36:29,750 --> 00:36:33,710 Entropy of mixing was responsible for equilibrium. 613 00:36:33,710 --> 00:36:36,310 Without entropy of mixing, then everything would go 614 00:36:36,310 --> 00:36:37,610 directly to the products. 615 00:36:37,610 --> 00:36:39,630 You'd never have an equilibrium. 616 00:36:39,630 --> 00:36:43,290 Another secret of life. 617 00:36:43,290 --> 00:36:46,870 So entropy of mixing here is driving this positive slope. 618 00:36:46,870 --> 00:36:48,400 That's another way of thinking about it. 619 00:36:48,400 --> 00:36:52,660 And that's why you can guess ahead of time what the signs 620 00:36:52,660 --> 00:36:54,540 of these things ought to be. 621 00:36:54,540 --> 00:36:58,320 Without doing the math. 622 00:36:58,320 --> 00:37:06,450 Alright, any questions? 623 00:37:06,450 --> 00:37:06,930 OK. 624 00:37:06,930 --> 00:37:14,130 So there's another sample problem with the notes, which 625 00:37:14,130 --> 00:37:16,100 is sort of like a standard problem. 626 00:37:16,100 --> 00:37:25,300 Where you're given a Clausius-Clapeyron equation 627 00:37:25,300 --> 00:37:29,640 for a substance in this form. 628 00:37:29,640 --> 00:37:39,200 So if you look at it, log p is minus some energy divided by 629 00:37:39,200 --> 00:37:43,080 RT plus some number. 630 00:37:43,080 --> 00:37:46,290 In this case, it's negative, the delta H is negative for 631 00:37:46,290 --> 00:37:48,210 the substance that you have there. 632 00:37:48,210 --> 00:37:50,640 And you're given it for the liquid phase, for the vapor 633 00:37:50,640 --> 00:37:52,100 pressure above the liquid and the vapor 634 00:37:52,100 --> 00:37:54,200 pressure above the solid. 635 00:37:54,200 --> 00:37:56,510 You can, because the only way it doesn't work is if you're 636 00:37:56,510 --> 00:37:58,250 looking at solid liquid. 637 00:37:58,250 --> 00:38:02,130 So, and then you're asked to manipulate this equation to 638 00:38:02,130 --> 00:38:03,120 find different properties. 639 00:38:03,120 --> 00:38:04,480 For instance the triple point. 640 00:38:04,480 --> 00:38:06,670 It's a very common thing to get, the 641 00:38:06,670 --> 00:38:10,530 triple point from this. 642 00:38:10,530 --> 00:38:13,800 The delta H, it's a way to get delta H. Just like we saw for 643 00:38:13,800 --> 00:38:14,910 the RDX, right? 644 00:38:14,910 --> 00:38:17,690 The data points formed a straight line. 645 00:38:17,690 --> 00:38:20,860 And the slope gave you delta H, for RDX. 646 00:38:20,860 --> 00:38:23,240 Delta H of sublimation for RDX. 647 00:38:23,240 --> 00:38:27,040 So this is the way that this equation is used for problems. 648 00:38:27,040 --> 00:38:28,110 I'm not going to do this problem. 649 00:38:28,110 --> 00:38:30,020 Instead I'm going to go ahead and forge 650 00:38:30,020 --> 00:38:33,590 ahead to the next topic. 651 00:38:33,590 --> 00:38:34,520 And start it. 652 00:38:34,520 --> 00:38:41,200 This is the topic we're going to do it after spring break. 653 00:38:41,200 --> 00:38:48,610 And and it really is a topic that follows this pretty 654 00:38:48,610 --> 00:38:49,890 straightforwardly. 655 00:38:49,890 --> 00:38:52,900 Well, it's more complicated, but it's similar to this 656 00:38:52,900 --> 00:38:54,890 problem here. 657 00:38:54,890 --> 00:38:59,340 So we saw that in real life yes, question? 658 00:38:59,340 --> 00:39:06,100 STUDENT: [INAUDIBLE] 659 00:39:06,100 --> 00:39:07,110 PROFESSOR: The exam. 660 00:39:07,110 --> 00:39:08,770 Oh there's an exam is, which is, oh I 661 00:39:08,770 --> 00:39:13,120 forgot about the exam. 662 00:39:13,120 --> 00:39:13,610 So let's see. 663 00:39:13,610 --> 00:39:16,150 There's a problem set that's due today. 664 00:39:16,150 --> 00:39:19,530 The exam will be up to the problem set that's due today. 665 00:39:19,530 --> 00:39:23,430 Which means that what I talked about today is not going to be 666 00:39:23,430 --> 00:39:24,060 on the exam. 667 00:39:24,060 --> 00:39:30,800 Because it's not covered in the problem set. 668 00:39:30,800 --> 00:39:35,150 So let's say it goes up to Wednesday's lecture. 669 00:39:35,150 --> 00:39:36,970 Which means you might still have a phase transition 670 00:39:36,970 --> 00:39:42,170 problem, right? 671 00:39:42,170 --> 00:39:44,270 In fact, you're very likely to get a 672 00:39:44,270 --> 00:39:47,040 phase transition problem. 673 00:39:47,040 --> 00:39:48,890 It's very likely to be the case. 674 00:39:48,890 --> 00:39:53,070 Because it's an easy problem to put together. 675 00:39:53,070 --> 00:39:54,590 But you don't have to worry about Clausius-Clapeyron. 676 00:39:54,590 --> 00:39:56,840 You do have to worry about Clapeyron, but not 677 00:39:56,840 --> 00:40:03,000 Clausius-Clapeyron, OK? 678 00:40:03,000 --> 00:40:07,400 So, for the exam, we're going to put on the Web last year's 679 00:40:07,400 --> 00:40:11,970 exam, and you guys are going to do reviews. 680 00:40:11,970 --> 00:40:13,070 The usual thing. 681 00:40:13,070 --> 00:40:13,340 Yeah? 682 00:40:13,340 --> 00:40:17,470 STUDENT: [INAUDIBLE] 683 00:40:17,470 --> 00:40:18,850 PROFESSOR: So that you don't see the answers. 684 00:40:18,850 --> 00:40:21,620 Yeah, absolutely this can easily be done. 685 00:40:21,620 --> 00:40:23,690 We'll do that. 686 00:40:23,690 --> 00:40:30,800 Any other questions, about the exam? 687 00:40:30,800 --> 00:40:31,490 Now, I don't remember. 688 00:40:31,490 --> 00:40:33,900 Is there a problem set that, so I'm not assigning a problem 689 00:40:33,900 --> 00:40:35,530 set today, right, then? 690 00:40:35,530 --> 00:40:35,840 Right. 691 00:40:35,840 --> 00:40:36,120 OK. 692 00:40:36,120 --> 00:40:40,510 Good. 693 00:40:40,510 --> 00:40:41,840 Because I hadn't loaded it up yet. 694 00:40:41,840 --> 00:40:45,300 And so now I won't load it up. 695 00:40:45,300 --> 00:40:48,630 OK. 696 00:40:48,630 --> 00:40:50,200 Anything else? 697 00:40:50,200 --> 00:40:51,620 OK, so let me tell you where we're going to go 698 00:40:51,620 --> 00:40:53,110 then, after the break. 699 00:40:53,110 --> 00:40:54,810 So after the break, we're going to be starting to talk 700 00:40:54,810 --> 00:40:58,260 about things that are called colligative properties. 701 00:40:58,260 --> 00:41:03,390 Colligative properties are the properties like the vapor 702 00:41:03,390 --> 00:41:08,650 pressure, lowering when you have a mixture in a liquid, 703 00:41:08,650 --> 00:41:10,920 instead of having a pure substance you have a mixture 704 00:41:10,920 --> 00:41:12,730 of substances here. 705 00:41:12,730 --> 00:41:16,440 And then the vapor pressure of both 706 00:41:16,440 --> 00:41:20,060 substances above gets lowered. 707 00:41:20,060 --> 00:41:23,140 Anybody want to guess why they got lowered? 708 00:41:23,140 --> 00:41:25,990 What's the magic word? 709 00:41:25,990 --> 00:41:27,580 Entropy of mixing, right. 710 00:41:27,580 --> 00:41:28,440 Right. 711 00:41:28,440 --> 00:41:30,350 There's more entropy of mixing if you've got a mixture in the 712 00:41:30,350 --> 00:41:32,790 liquid than if you have a pure gas up there. 713 00:41:32,790 --> 00:41:34,540 So usually, so let me back up. 714 00:41:34,540 --> 00:41:39,760 Usually it's when you have a solute, like salt in water. 715 00:41:39,760 --> 00:41:43,390 Where water is volatile and the salt is not. 716 00:41:43,390 --> 00:41:46,720 Right, a salt in water solution will have a lower 717 00:41:46,720 --> 00:41:50,450 vapor pressure than a solution of pure water. 718 00:41:50,450 --> 00:41:51,740 Because of the entropy of mixing in a 719 00:41:51,740 --> 00:41:53,350 salt and water solution. 720 00:41:53,350 --> 00:41:57,780 The salt and water solution we have a lowering, 721 00:41:57,780 --> 00:41:59,590 lower melting point. 722 00:41:59,590 --> 00:42:01,470 And that's significantly lower. 723 00:42:01,470 --> 00:42:03,510 Which is why we use it on roads, right? 724 00:42:03,510 --> 00:42:05,010 That's a colligative properties. 725 00:42:05,010 --> 00:42:06,120 Osmotic pressure. 726 00:42:06,120 --> 00:42:08,840 Which you've already sort of heard about. 727 00:42:08,840 --> 00:42:13,320 Why saltwater fish are unhappy in fresh water. 728 00:42:13,320 --> 00:42:14,550 That's a colligative property. 729 00:42:14,550 --> 00:42:16,660 So we're going to go through these colligative properties. 730 00:42:16,660 --> 00:42:19,000 And the mainstay of these colligative properties is that 731 00:42:19,000 --> 00:42:20,360 we're going to be talking about mixtures 732 00:42:20,360 --> 00:42:21,520 in the liquid phase. 733 00:42:21,520 --> 00:42:24,510 Here we talked about a mixture in the gas phase, changing 734 00:42:24,510 --> 00:42:25,600 some property. 735 00:42:25,600 --> 00:42:29,070 From now on, we're going to be talking about mixtures in the 736 00:42:29,070 --> 00:42:32,380 liquid phase, pretty much. 737 00:42:32,380 --> 00:42:36,510 And so our goal is going to be able to, is going to be 738 00:42:36,510 --> 00:42:42,000 predict the way that properties change as you have 739 00:42:42,000 --> 00:42:45,530 these mixtures. 740 00:42:45,530 --> 00:42:52,220 OK. so the standard mixture that we're going to have, is 741 00:42:52,220 --> 00:42:57,400 to have a binary liquid gas mixture. 742 00:42:57,400 --> 00:43:00,160 So that means we're going to have two substances. 743 00:43:00,160 --> 00:43:02,340 A and B. And at first, we're going to make it 744 00:43:02,340 --> 00:43:04,010 as general as possible. 745 00:43:04,010 --> 00:43:08,563 So we're going to have liquid phase of A, the liquid phase 746 00:43:08,563 --> 00:43:12,810 of B. So, for instance, this could be vodka, right? 747 00:43:12,810 --> 00:43:15,100 Water and ethanol. 748 00:43:15,100 --> 00:43:17,380 And water and ethanol are both volatile. 749 00:43:17,380 --> 00:43:25,580 So we're going to have water and ethanol in the gas phase. 750 00:43:25,580 --> 00:43:28,160 There, and it's a mixture, they're perfectly miscible in 751 00:43:28,160 --> 00:43:32,210 the liquid phase, A and B. And there's going to be some 752 00:43:32,210 --> 00:43:35,380 fraction, molar fraction, of these things. 753 00:43:35,380 --> 00:43:38,400 We're going to have a molar fraction in the liquid phase 754 00:43:38,400 --> 00:43:42,320 for A and B. And some molar fraction in the gas phase, for 755 00:43:42,320 --> 00:43:45,340 A and B. We're going to call it yA and yB, that's the molar 756 00:43:45,340 --> 00:43:47,190 fraction in the gas phase. 757 00:43:47,190 --> 00:43:50,465 And there's no reason why the molar fractions in the gas 758 00:43:50,465 --> 00:43:54,810 phrase should be the same as that in the liquid phase. 759 00:43:54,810 --> 00:43:55,410 And we know that. 760 00:43:55,410 --> 00:43:57,825 We know that ethanol is more volatile than water. 761 00:43:57,825 --> 00:44:02,260 And so if you have a gas of vodka, then you can smell the 762 00:44:02,260 --> 00:44:05,430 alcohol because it's coming out. 763 00:44:05,430 --> 00:44:07,750 But the water is, vapor pressure is, 764 00:44:07,750 --> 00:44:09,830 very slow, very small. 765 00:44:09,830 --> 00:44:11,190 It's the usual, it's pretty close to 766 00:44:11,190 --> 00:44:14,310 the usual vapor pressure. 767 00:44:14,310 --> 00:44:17,410 And our question is, what we're going to be asking is, 768 00:44:17,410 --> 00:44:20,260 how many, if I give you the total pressure and the 769 00:44:20,260 --> 00:44:23,960 temperature here, what else do I need to know to find out all 770 00:44:23,960 --> 00:44:28,940 these molar fractions? 771 00:44:28,940 --> 00:44:30,140 So let's first see what all the 772 00:44:30,140 --> 00:44:31,240 variables are that we have. 773 00:44:31,240 --> 00:44:32,650 Usually, we have two variables. 774 00:44:32,650 --> 00:44:34,390 The temperature and pressure, and that's enough to tell us 775 00:44:34,390 --> 00:44:35,820 everything. 776 00:44:35,820 --> 00:44:37,550 But here now we have a mixture. 777 00:44:37,550 --> 00:44:40,380 So we're going to need more than that. 778 00:44:40,380 --> 00:44:41,620 We're going to need four variables. 779 00:44:41,620 --> 00:44:43,820 We're going to need the temperature. 780 00:44:43,820 --> 00:44:48,090 The pressure, and we're going to need the molar fraction of 781 00:44:48,090 --> 00:44:49,770 A in the liquid phase. 782 00:44:49,770 --> 00:44:52,750 And the molar fraction of A in the gas phase. 783 00:44:52,750 --> 00:44:55,160 We're not going to need any more than that. 784 00:44:55,160 --> 00:44:56,320 Actually we're going to need less than that. 785 00:44:56,320 --> 00:44:59,100 But those are the four variables that we can easily 786 00:44:59,100 --> 00:45:00,370 identify here. 787 00:45:00,370 --> 00:45:05,470 And we don't need yB and xB, because the sum of the molar 788 00:45:05,470 --> 00:45:13,800 fraction has to be equal to one. yB is one minus yA, xB is 789 00:45:13,800 --> 00:45:15,650 one minus xA. 790 00:45:15,650 --> 00:45:17,440 So we don't need y. 791 00:45:17,440 --> 00:45:20,120 So these are a priori the independent variables that we 792 00:45:20,120 --> 00:45:23,270 have to work with. 793 00:45:23,270 --> 00:45:26,380 Now, we have some constraints here. 794 00:45:26,380 --> 00:45:28,280 We have a constraint. 795 00:45:28,280 --> 00:45:30,970 Our constraint is that we are the coexistence point. 796 00:45:30,970 --> 00:45:35,210 There's a coexistence between the gas phase 797 00:45:35,210 --> 00:45:37,170 and the liquid phase. 798 00:45:37,170 --> 00:45:44,400 And what does it mean for A liquid to be coexistent with A 799 00:45:44,400 --> 00:45:46,980 in the gas phase? 800 00:45:46,980 --> 00:45:47,940 Chemical potential. 801 00:45:47,940 --> 00:45:50,160 The chemical potential of a molecule of A in the liquid 802 00:45:50,160 --> 00:45:52,266 phase here is the same as the chemical potential of 803 00:45:52,266 --> 00:45:52,670 A in the gas phase. 804 00:45:52,670 --> 00:45:53,830 So we have two constraints. 805 00:45:53,830 --> 00:46:00,330 We have mu A for the liquid phase is equal to mu A for the 806 00:46:00,330 --> 00:46:06,330 gas phase, and mu B for the liquid phase equal to mu B for 807 00:46:06,330 --> 00:46:06,520 the gas phase. 808 00:46:06,520 --> 00:46:09,650 So we have four variables, two constraints. 809 00:46:09,650 --> 00:46:13,180 That means that the total number of independent 810 00:46:13,180 --> 00:46:14,850 variables we have is just two. 811 00:46:14,850 --> 00:46:18,740 It's the number of variables four, minus the constraints. 812 00:46:18,740 --> 00:46:19,740 OK? 813 00:46:19,740 --> 00:46:22,850 Or you could say six variables if you include yB and xB, and 814 00:46:22,850 --> 00:46:24,110 four constraints. 815 00:46:24,110 --> 00:46:26,700 Whatever we accounted, you just have 816 00:46:26,700 --> 00:46:30,660 two degrees of freedom. 817 00:46:30,660 --> 00:46:37,530 So we have two degrees of freedom. 818 00:46:37,530 --> 00:46:43,290 Which means that if you know the temperature and pressure, 819 00:46:43,290 --> 00:46:45,840 then you still know everything. 820 00:46:45,840 --> 00:46:47,670 At the coexistence point. 821 00:46:47,670 --> 00:46:51,370 You still know what the molar fraction is of your two 822 00:46:51,370 --> 00:46:53,220 substances in both phases. 823 00:46:53,220 --> 00:46:57,050 So that's really powerful. 824 00:46:57,050 --> 00:47:01,690 Or if you know just the molar fraction of A, and the 825 00:47:01,690 --> 00:47:03,940 temperature, then you know the pressure above. 826 00:47:03,940 --> 00:47:06,690 And you know everything else. 827 00:47:06,690 --> 00:47:09,490 So in this very complicated mixture here, you still only 828 00:47:09,490 --> 00:47:11,460 need to know a couple of things to tell you everything 829 00:47:11,460 --> 00:47:16,240 about this mixture. 830 00:47:16,240 --> 00:47:22,750 And this example here is a specific example of something 831 00:47:22,750 --> 00:47:24,370 called the Gibbs phase rule. 832 00:47:24,370 --> 00:47:27,580 Which you've seen before. 833 00:47:27,580 --> 00:47:28,980 In this diagram. 834 00:47:28,980 --> 00:47:33,880 In this diagram that we wrote here. 835 00:47:33,880 --> 00:47:38,780 Where, let me write down what the Gibbs phase rule is first. 836 00:47:38,780 --> 00:47:41,410 The Gibbs phase rule tells you the number of independent 837 00:47:41,410 --> 00:47:43,710 variables given a number of constraints. 838 00:47:43,710 --> 00:47:56,930 So, it tells you that the number of degrees of freedom, 839 00:47:56,930 --> 00:47:59,290 the number of independent variables in a mixture, in a 840 00:47:59,290 --> 00:48:03,140 complicated system like this, is equal to C, which is the 841 00:48:03,140 --> 00:48:05,050 number of components. 842 00:48:05,050 --> 00:48:09,470 So in this case, it would be two components. 843 00:48:09,470 --> 00:48:11,500 You have A and B. The components is the number of 844 00:48:11,500 --> 00:48:14,910 independent species that you have, minus 845 00:48:14,910 --> 00:48:17,120 the number of phases. 846 00:48:17,120 --> 00:48:19,780 So here we have two phases, the gas phase and the water 847 00:48:19,780 --> 00:48:24,800 phase, plus two. 848 00:48:24,800 --> 00:48:29,340 So for this example here, then we would have two components, 849 00:48:29,340 --> 00:48:30,680 C equals two. 850 00:48:30,680 --> 00:48:33,480 Two phases liquid and gas. 851 00:48:33,480 --> 00:48:33,930 A and B are the components. 852 00:48:33,930 --> 00:48:34,950 And then plus two. 853 00:48:34,950 --> 00:48:40,220 So two minus two plus two, F equals to two. 854 00:48:40,220 --> 00:48:45,400 Two degrees of freedom, two independent variables. 855 00:48:45,400 --> 00:48:49,750 If I'm looking at this diagram here, in this diagram here I 856 00:48:49,750 --> 00:48:52,210 have C equals one. 857 00:48:52,210 --> 00:48:54,740 One component, right? 858 00:48:54,740 --> 00:48:59,560 If I'm sitting where there's only one phase, say, the 859 00:48:59,560 --> 00:49:04,930 liquid phase or the gas phase, then p equals one. 860 00:49:04,930 --> 00:49:08,190 Then F is one minus one plus two, is equal to two. 861 00:49:08,190 --> 00:49:09,620 Two degrees of freedom. 862 00:49:09,620 --> 00:49:11,800 And so in the pure phase, I am in a plane. 863 00:49:11,800 --> 00:49:14,610 The pressure and the temperature are my two degrees 864 00:49:14,610 --> 00:49:15,190 of freedom. 865 00:49:15,190 --> 00:49:17,430 I can freely move around the plane. 866 00:49:17,430 --> 00:49:20,370 At a coexistence line, then the number of 867 00:49:20,370 --> 00:49:25,130 phases equals to two. 868 00:49:25,130 --> 00:49:30,540 Then I have one minus two is minus one, plus two is one. 869 00:49:30,540 --> 00:49:33,350 The degrees of freedom is one. 870 00:49:33,350 --> 00:49:34,580 That's a line. 871 00:49:34,580 --> 00:49:35,520 I have one degree of freedom. 872 00:49:35,520 --> 00:49:38,750 I can move in the line, in the line in a plane. 873 00:49:38,750 --> 00:49:41,130 Then if I change to the triple point, where the number of 874 00:49:41,130 --> 00:49:45,510 phases is three, then the number of degrees of freedom 875 00:49:45,510 --> 00:49:49,140 is one minus three is minus two, two plus two is zero. 876 00:49:49,140 --> 00:49:50,950 I don't have any degrees of freedom, I'm 877 00:49:50,950 --> 00:49:54,230 stuck at one point. 878 00:49:54,230 --> 00:49:57,960 So this is an example of the Gibbs phase rule, specific 879 00:49:57,960 --> 00:50:00,180 where C happens to be equal to one. 880 00:50:00,180 --> 00:50:04,910 And then, this is a more generalized example. 881 00:50:04,910 --> 00:50:08,350 So next time, then, what we'll do is we'll start by deriving 882 00:50:08,350 --> 00:50:09,890 the Gibbs phase rule. 883 00:50:09,890 --> 00:50:11,220 Which is not so hard. 884 00:50:11,220 --> 00:50:14,010 And then start on our way to the colligative properties. 885 00:50:14,010 --> 00:50:15,900 And then you'll have an exam.