1 00:00:00,000 --> 00:00:01,690 The following content is provided 2 00:00:01,690 --> 00:00:03,820 under a Creative Commons license. 3 00:00:03,820 --> 00:00:06,770 Your support will help MIT OpenCourseWare continue 4 00:00:06,770 --> 00:00:10,520 to offer high quality educational resources for free. 5 00:00:10,520 --> 00:00:13,230 To make a donation or view additional materials 6 00:00:13,230 --> 00:00:16,590 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,590 --> 00:00:20,780 at ocw.mit.edu. 8 00:00:20,780 --> 00:00:22,280 PROFESSOR: So last time you learned 9 00:00:22,280 --> 00:00:30,570 about the ideal solution limits with Henry's law and Raoult's 10 00:00:30,570 --> 00:00:31,620 law. 11 00:00:31,620 --> 00:00:40,110 And I just want to start with one reminder of what 12 00:00:40,110 --> 00:00:41,310 this is all about. 13 00:00:41,310 --> 00:00:50,440 So let's say that you have a mixture of CS2 and acetone. 14 00:00:50,440 --> 00:00:56,680 So this is a review of what you saw last time. 15 00:00:56,680 --> 00:01:03,100 Acetone like this, CH3, CH3. 16 00:01:03,100 --> 00:01:15,770 And if you take the CS2 as the solvent and the acetone 17 00:01:15,770 --> 00:01:27,450 as a solute, then when you plot as a function of the amount 18 00:01:27,450 --> 00:01:30,730 of CS2 here as a solvent, then you 19 00:01:30,730 --> 00:01:36,800 have the limit where the mole fraction of CS2 goes to one, 20 00:01:36,800 --> 00:01:38,610 meaning that CS is the solvent. 21 00:01:38,610 --> 00:01:43,700 You expect Raoult to work if it's an ideal solution. 22 00:01:43,700 --> 00:01:46,720 So that the vapor pressure of CS2 23 00:01:46,720 --> 00:01:56,690 is related to the mole fraction of CS2 times the vapor 24 00:01:56,690 --> 00:01:59,460 pressure of CS2 if it was pure. 25 00:01:59,460 --> 00:02:02,520 So you're sitting here somewhere. 26 00:02:02,520 --> 00:02:07,490 This is the vapor pressure of CS2 pure 27 00:02:07,490 --> 00:02:10,060 and CS2 goes from zero to one. 28 00:02:10,060 --> 00:02:14,650 And you expect Raoult to work. 29 00:02:14,650 --> 00:02:18,740 So let's do Raoult. 30 00:02:18,740 --> 00:02:24,780 Like this. 31 00:02:24,780 --> 00:02:29,180 So in this area, if I were to plot the real curve, 32 00:02:29,180 --> 00:02:34,070 I would expect it to start out just like Raoult. 33 00:02:34,070 --> 00:02:36,160 And then there's going to be some deviation. 34 00:02:36,160 --> 00:02:42,370 Then if I take the other limit, where the mole fraction of CS2 35 00:02:42,370 --> 00:02:43,030 goes to zero. 36 00:02:43,030 --> 00:02:46,420 So now the CS2 is the solute. 37 00:02:46,420 --> 00:02:49,720 And the acetone is the solvent. what 38 00:02:49,720 --> 00:02:54,700 happens for this particular mixture is 39 00:02:54,700 --> 00:02:58,230 that Henry's law is no longer valid. 40 00:02:58,230 --> 00:03:01,860 Raoult's law is not valid here. 41 00:03:01,860 --> 00:03:03,890 Instead, you have to look at Henry's law. 42 00:03:03,890 --> 00:03:08,590 And Henry's law tells you that for the solute now, 43 00:03:08,590 --> 00:03:11,160 not the solvent, so the CS2 here is 44 00:03:11,160 --> 00:03:12,960 the solute, very small amount of it. 45 00:03:12,960 --> 00:03:15,090 Very small mole fraction. 46 00:03:15,090 --> 00:03:18,910 That the vapor pressure of the solute 47 00:03:18,910 --> 00:03:21,970 is related to the mole fraction in the solution. 48 00:03:21,970 --> 00:03:26,280 But instead of having the vapor pressure of the pure solute, 49 00:03:26,280 --> 00:03:27,280 you have another number. 50 00:03:27,280 --> 00:03:29,170 Another constant. 51 00:03:29,170 --> 00:03:32,790 Which we call K. Which is a Henry's law constant. 52 00:03:32,790 --> 00:03:39,460 And this constant depends on what else in the solution. 53 00:03:39,460 --> 00:03:45,190 It depends on the fact that you have acetone in there. 54 00:03:45,190 --> 00:03:47,140 You would have a different constant 55 00:03:47,140 --> 00:03:51,620 if the acetone were replaced with chloroform, let's say. 56 00:03:51,620 --> 00:04:00,940 So this constant here cares what the solvent is. 57 00:04:00,940 --> 00:04:03,080 And it cares what the temperature is, also. 58 00:04:03,080 --> 00:04:06,250 There's a lot of stuff that's built into this constant here. 59 00:04:06,250 --> 00:04:08,490 It's a phinemelogical constant. 60 00:04:08,490 --> 00:04:12,800 And in the case of the mixture of CS2 and acetone, 61 00:04:12,800 --> 00:04:17,430 the deviation from Raoult's law is positive. 62 00:04:17,430 --> 00:04:19,290 Meaning that the Henry's law constant 63 00:04:19,290 --> 00:04:28,910 is greater than the vapor pressure of the pure CS2. 64 00:04:28,910 --> 00:04:33,310 So that means that the molecules of CS2 65 00:04:33,310 --> 00:04:35,590 would rather go in the gas phase. 66 00:04:35,590 --> 00:04:37,800 If there's acetone around. 67 00:04:37,800 --> 00:04:41,210 That means they don't like to mix so much. 68 00:04:41,210 --> 00:04:43,050 They're not happy with each other. 69 00:04:43,050 --> 00:04:45,140 So the CS2 molecules would rather 70 00:04:45,140 --> 00:04:49,820 escape what would normally be a Raoult's law type of solution 71 00:04:49,820 --> 00:04:54,440 to go in the gas phase away from being close to the acetone. 72 00:04:54,440 --> 00:04:56,440 And as a result you get this positive deviation. 73 00:04:56,440 --> 00:05:02,450 You get a higher pressure than you expect from Raoult's law. 74 00:05:02,450 --> 00:05:13,350 So that the slope here This slope here is K CS2. 75 00:05:13,350 --> 00:05:18,820 And that slope here is p star CS2. 76 00:05:18,820 --> 00:05:20,520 This is a positive deviation. 77 00:05:20,520 --> 00:05:28,100 And you saw an example where you if you mix chloroform 78 00:05:28,100 --> 00:05:36,880 and acetone, you get a negative deviation. 79 00:05:36,880 --> 00:05:39,010 Chloroform and acetone are happy together 80 00:05:39,010 --> 00:05:44,280 because of the hydrogen bonding between the oxygen here 81 00:05:44,280 --> 00:05:47,420 and the hydrogen in chloroform. 82 00:05:47,420 --> 00:05:51,610 And you get a negative deviation. 83 00:05:51,610 --> 00:05:56,090 So let's start by doing an example with numbers. 84 00:05:56,090 --> 00:06:00,810 Of how this might come about. 85 00:06:00,810 --> 00:06:05,410 Any questions on this first? 86 00:06:05,410 --> 00:06:08,180 I always got confused with Henry's law and Raoult's law. 87 00:06:08,180 --> 00:06:10,385 And I always got confused as to when 88 00:06:10,385 --> 00:06:11,760 I was supposed to use Henry's law 89 00:06:11,760 --> 00:06:15,320 and when I was supposed to use Raoult's law. 90 00:06:15,320 --> 00:06:18,610 Very confusing to me, initially. 91 00:06:18,610 --> 00:06:20,410 The thing to remember is that if you're 92 00:06:20,410 --> 00:06:24,630 talking about something which in a very small amount, just 93 00:06:24,630 --> 00:06:29,995 the solute, like, well I'm going to give you an example which 94 00:06:29,995 --> 00:06:31,620 is a bad example, like sugar and water. 95 00:06:31,620 --> 00:06:33,036 Where sugar is clearly the solute, 96 00:06:33,036 --> 00:06:36,990 but sugar is not volatile clearly this is a bad example. 97 00:06:36,990 --> 00:06:40,850 But something which in a very small concentration, then 98 00:06:40,850 --> 00:06:42,830 Henry's law is going to apply. 99 00:06:42,830 --> 00:06:46,500 Raoult's law is only going to apply for the solvent. 100 00:06:46,500 --> 00:06:48,710 When you have a lot of stuff around. 101 00:06:48,710 --> 00:06:50,930 On this side here is Raoult's law. 102 00:06:50,930 --> 00:06:56,500 On this side here is Henry's law. 103 00:06:56,500 --> 00:06:58,530 Henry's law, the deviation's from Raoult's law, 104 00:06:58,530 --> 00:07:02,210 are going to happen where the molecules in the solution 105 00:07:02,210 --> 00:07:07,696 mostly see molecules that are not like themselves. 106 00:07:07,696 --> 00:07:09,570 If they mostly see molecules like themselves, 107 00:07:09,570 --> 00:07:11,310 since it's an ideal solution, then they 108 00:07:11,310 --> 00:07:14,940 don't know that there's other stuff around. 109 00:07:14,940 --> 00:07:18,360 In this case here, the CS2 molecules don't really know. 110 00:07:18,360 --> 00:07:20,950 Most of them don't know that there's any acetone around. 111 00:07:20,950 --> 00:07:23,450 They couldn't care less. 112 00:07:23,450 --> 00:07:25,060 So it obeys an ideal solution. 113 00:07:25,060 --> 00:07:27,610 Here, they don't see any other CS2 molecules. 114 00:07:27,610 --> 00:07:29,190 They mostly see acetone. 115 00:07:29,190 --> 00:07:31,859 And so Raoult's law is going to fall apart. 116 00:07:31,859 --> 00:07:33,400 Because Raoult's law assumed that you 117 00:07:33,400 --> 00:07:37,720 have a single component. 118 00:07:37,720 --> 00:07:41,440 Or mostly one component. 119 00:07:41,440 --> 00:07:41,940 OK. 120 00:07:41,940 --> 00:07:45,990 Let's do an example. 121 00:07:45,990 --> 00:07:49,176 So in this case here, we're going 122 00:07:49,176 --> 00:07:52,100 to work the example out here. 123 00:07:52,100 --> 00:08:03,390 So let's take two components, A and B. Where 124 00:08:03,390 --> 00:08:07,670 B is, and let's look at something which 125 00:08:07,670 --> 00:08:14,970 is a dilute solution and non-ideal. 126 00:08:14,970 --> 00:08:18,620 So there's going to be some deviation from Raoult's law. 127 00:08:18,620 --> 00:08:20,800 And what we're told is at 50 degrees 128 00:08:20,800 --> 00:08:32,050 Celsius, that the vapor pressure of pure A is equal to 0.67 bar. 129 00:08:32,050 --> 00:08:41,880 And the vapor pressure of pure B is equal to 1.2 bar. 130 00:08:41,880 --> 00:09:02,250 So B is clearly the more volatile. 131 00:09:02,250 --> 00:09:09,660 And we're also told, then, that at 50 degrees 132 00:09:09,660 --> 00:09:17,350 C we have a molar fraction of A in the liquid phase is 0.9. 133 00:09:17,350 --> 00:09:22,620 And the molar fraction of B in the liquid phase is 0.1. 134 00:09:22,620 --> 00:09:30,860 And the total pressure above the solution is 0.7 bar. 135 00:09:30,860 --> 00:09:35,100 And eventually what we're going to try to find out is, 136 00:09:35,100 --> 00:09:39,580 we're going to change these concentrations. 137 00:09:39,580 --> 00:09:41,680 Are we going to try to figure out 138 00:09:41,680 --> 00:09:44,930 how the total pressure above the solution changes. 139 00:09:44,930 --> 00:09:48,420 That's maybe the goal of the problem. 140 00:09:48,420 --> 00:09:51,205 When you change this to, eventually we'll 141 00:09:51,205 --> 00:09:54,940 change it to 0.95 and 0.05, and we'll 142 00:09:54,940 --> 00:09:58,020 want to know how does that change the total pressure. 143 00:09:58,020 --> 00:10:03,860 So immediately we see that this is a solvent. 144 00:10:03,860 --> 00:10:04,700 This is a solute. 145 00:10:04,700 --> 00:10:09,670 This is volatile. 146 00:10:09,670 --> 00:10:13,170 So, we're close to where this is zero. 147 00:10:13,170 --> 00:10:15,850 So if we're going to apply any, sort of, Raoult or Henry's law 148 00:10:15,850 --> 00:10:18,810 problem to this, this is probably 149 00:10:18,810 --> 00:10:23,860 going to be Henry's law for B and Raoult's law for A. 150 00:10:23,860 --> 00:10:27,080 We're going to be close to this area here for B. 151 00:10:27,080 --> 00:10:32,360 We're going to be close to this area for A. 152 00:10:32,360 --> 00:10:32,960 OK. 153 00:10:32,960 --> 00:10:35,430 So ah the first question is, what's 154 00:10:35,430 --> 00:10:45,840 the composition in the gas phase. 155 00:10:45,840 --> 00:10:53,020 What is y sub B. And it looks like this particular set 156 00:10:53,020 --> 00:10:59,060 of notes doesn't have the calculations. 157 00:10:59,060 --> 00:12:01,230 I'm going to go to my other set of notes here that has them. 158 00:12:01,230 --> 00:12:03,610 So, because I'm going to make a mistake if I 159 00:12:03,610 --> 00:12:06,390 go through it without my notes, I'm 160 00:12:06,390 --> 00:12:09,650 going to leave it up to you to do part a. 161 00:12:09,650 --> 00:12:11,427 That's a good way out of making a mess. 162 00:12:11,427 --> 00:12:13,010 I'm just going to tell you the answer. 163 00:12:13,010 --> 00:12:15,520 Because the actual, the interesting part, 164 00:12:15,520 --> 00:12:16,740 is part b of the problem. 165 00:12:16,740 --> 00:12:18,260 Part a is the easy part. 166 00:12:18,260 --> 00:12:20,920 So this is something that you can do on your own. 167 00:12:20,920 --> 00:12:24,310 So yB is equal to 0.139. 168 00:12:24,310 --> 00:12:29,420 Is going to be answer to part b, given these two molar 169 00:12:29,420 --> 00:12:31,400 fractions and the total pressure. 170 00:12:31,400 --> 00:12:33,010 So now, what we're going to do is, 171 00:12:33,010 --> 00:12:35,680 we're going to change the molar fractions 172 00:12:35,680 --> 00:12:37,470 and we're going to find out how that 173 00:12:37,470 --> 00:12:44,980 changes the total pressure. 174 00:12:44,980 --> 00:12:51,310 So, second part is we're still at 50 degrees Celsius. 175 00:12:51,310 --> 00:12:55,970 And we're going to change the molar fraction of A to 0.95. 176 00:12:55,970 --> 00:13:02,210 And the molar fraction of B to 0.05. 177 00:13:02,210 --> 00:13:09,270 And we're going to ask, what is p total now. 178 00:13:09,270 --> 00:13:13,110 So we write what we know. 179 00:13:13,110 --> 00:13:15,742 Which is that we expect p total to be 180 00:13:15,742 --> 00:13:21,590 the sum of the partial pressures of A and B. A is the solvent, 181 00:13:21,590 --> 00:13:24,050 B is the solute. 182 00:13:24,050 --> 00:13:28,160 It's a dilute non-ideal solution. pA 183 00:13:28,160 --> 00:13:31,002 is going to be proportional to Raoult's law, 184 00:13:31,002 --> 00:13:31,960 because it's a solvent. 185 00:13:31,960 --> 00:13:33,600 It's in large quantities. 186 00:13:33,600 --> 00:13:37,870 So it's going to be xA pA star. 187 00:13:37,870 --> 00:13:40,800 And xB is going, and pB is going to be 188 00:13:40,800 --> 00:13:45,240 proportional to the mole fraction again. 189 00:13:45,240 --> 00:13:50,370 But because it's a solute, it's seen most the other molecules 190 00:13:50,370 --> 00:13:52,200 of A around, it's not necessarily 191 00:13:52,200 --> 00:13:54,030 going to be obeying Raoult's law. 192 00:13:54,030 --> 00:13:56,045 It's going to obey Henry's law. 193 00:13:56,045 --> 00:13:57,972 It's going to be close to the zero point. 194 00:13:57,972 --> 00:14:00,430 There's going to be a deviation because of the interactions 195 00:14:00,430 --> 00:14:02,950 between A and B. Which dominate now. 196 00:14:02,950 --> 00:14:06,220 So instead of pA star, we have KB here, 197 00:14:06,220 --> 00:14:07,920 where this is the Henry's law constant. 198 00:14:07,920 --> 00:14:11,064 Which depends on the fact that A is around. 199 00:14:11,064 --> 00:14:13,480 We get a different constant if we have a different solvent 200 00:14:13,480 --> 00:14:16,350 around. 201 00:14:16,350 --> 00:14:19,500 So we know this. 202 00:14:19,500 --> 00:14:22,780 We know this because that was given to us at the beginning. 203 00:14:22,780 --> 00:14:24,020 But we don't know this here. 204 00:14:24,020 --> 00:14:27,670 We don't know what the Henry's law constant here is. 205 00:14:27,670 --> 00:14:29,160 But we can get it. 206 00:14:29,160 --> 00:14:32,050 We can get it from part a here. 207 00:14:32,050 --> 00:14:35,850 Because we have all the information we need there. 208 00:14:35,850 --> 00:14:42,380 So from part a, we already have a mixture 209 00:14:42,380 --> 00:14:44,810 where we're given the total pressure. 210 00:14:44,810 --> 00:14:53,840 So from part a, KB is pB over xB. 211 00:14:53,840 --> 00:15:00,990 And we calculated what yB is over there. 212 00:15:00,990 --> 00:15:01,680 On part a. 213 00:15:01,680 --> 00:15:04,130 So we know what the partial pressure of B is. 214 00:15:04,130 --> 00:15:07,500 It's the molar fraction of B in the gas 215 00:15:07,500 --> 00:15:09,040 phase times the total pressure. 216 00:15:09,040 --> 00:15:18,270 That's Dalton's law. 217 00:15:18,270 --> 00:15:21,780 Then we divide by xB. 218 00:15:21,780 --> 00:15:25,840 And then we plug in the number 0.139 times 219 00:15:25,840 --> 00:15:28,990 the total pressure up there is 0.7. 220 00:15:28,990 --> 00:15:31,780 Divided by 0.1. 221 00:15:31,780 --> 00:15:40,350 And we get that this is 0.973 bar. 222 00:15:40,350 --> 00:15:49,410 So now we can plug this in to our equation here. 223 00:15:49,410 --> 00:15:55,450 We know xA, we know xB, we know Raoult's law, the 0.67 pA star. 224 00:15:55,450 --> 00:15:56,240 We know KB here. 225 00:15:56,240 --> 00:15:58,240 We plug in all the numbers, and we 226 00:15:58,240 --> 00:16:04,890 get that the total pressure is now 0.685 bar. 227 00:16:04,890 --> 00:16:07,900 This is a pretty typical example of a problem that 228 00:16:07,900 --> 00:16:13,510 combines Raoult's law, Henry's law and Dalton's law together. 229 00:16:13,510 --> 00:16:18,630 And keeping track of these laws takes a little bit of thinking. 230 00:16:18,630 --> 00:16:23,400 And it's very, very common on problem sets or exams 231 00:16:23,400 --> 00:16:27,110 to mix these up. 232 00:16:27,110 --> 00:16:29,360 So it's good to go through examples like this 233 00:16:29,360 --> 00:16:34,247 and really keep track of where each law applies. 234 00:16:34,247 --> 00:16:35,830 Dalton's law applies in the gas phase. 235 00:16:35,830 --> 00:16:39,180 It's partial pressure in the gas phase. 236 00:16:39,180 --> 00:16:40,850 Henry's law is the pressure. 237 00:16:40,850 --> 00:16:44,670 If you know what's happening in the liquid phase, for a solute. 238 00:16:44,670 --> 00:16:47,080 Raoult's law is the partial pressure in the gas phase, 239 00:16:47,080 --> 00:16:49,160 if you know what's happening in the liquid phase. 240 00:16:49,160 --> 00:16:51,910 So Raoult and Henry tell you something about the pressure 241 00:16:51,910 --> 00:16:54,040 if you know the liquid. 242 00:16:54,040 --> 00:16:59,950 Dalton only deals with what's happening in the gas phase. 243 00:16:59,950 --> 00:17:06,640 OK, any questions on the problem? 244 00:17:06,640 --> 00:17:07,140 Alright. 245 00:17:07,140 --> 00:17:09,870 So we did this problem here, and now let's 246 00:17:09,870 --> 00:17:16,810 draw the phase diagram for our solute here, or for B, rather. 247 00:17:16,810 --> 00:17:18,860 Zero to one. 248 00:17:18,860 --> 00:17:23,110 And let's take a vote here. 249 00:17:23,110 --> 00:17:31,160 So if Raoult's law worked for B, for the mixture of A and B, 250 00:17:31,160 --> 00:17:34,360 then we would have pB star here. 251 00:17:34,360 --> 00:17:35,760 And everything would be great. 252 00:17:35,760 --> 00:17:39,220 But Raoult's law doesn't work. 253 00:17:39,220 --> 00:17:41,360 Close to molar fraction of zero. 254 00:17:41,360 --> 00:17:42,580 Instead, Henry's law. 255 00:17:42,580 --> 00:17:44,750 There's a deviation, due to the interactions 256 00:17:44,750 --> 00:17:47,760 between the molecules A and B here. 257 00:17:47,760 --> 00:17:50,070 So we have two choices for the deviation. 258 00:17:50,070 --> 00:17:55,230 We can have a positive deviation, 259 00:17:55,230 --> 00:17:58,670 or we could have a negative deviation. 260 00:17:58,670 --> 00:18:02,150 Positive deviation, if the molecules of B 261 00:18:02,150 --> 00:18:04,070 would rather be in the gas phase, 262 00:18:04,070 --> 00:18:06,570 than be in the mixture in the liquid phase. 263 00:18:06,570 --> 00:18:09,540 Meaning they don't like molecules of A. 264 00:18:09,540 --> 00:18:12,460 They don't like to be mixed with molecules 265 00:18:12,460 --> 00:18:23,090 of A. The energy of interaction is not favorable. 266 00:18:23,090 --> 00:18:25,970 The enthalpy is not favorable, for mixing 267 00:18:25,970 --> 00:18:28,245 A and B. In this case here, the enthalpy 268 00:18:28,245 --> 00:18:32,920 is favorable for mixing A and B. 269 00:18:32,920 --> 00:18:37,100 So in this problem here, which do you think it is? 270 00:18:37,100 --> 00:18:39,650 The blue one or the green one. 271 00:18:39,650 --> 00:18:40,900 Let's raise hands here. 272 00:18:40,900 --> 00:18:47,070 How many people think it's the blue one here. 273 00:18:47,070 --> 00:18:47,770 Anybody else? 274 00:18:47,770 --> 00:18:50,270 The right answer, we have four people. 275 00:18:54,260 --> 00:18:57,680 How many people think it's the green one? 276 00:18:57,680 --> 00:18:58,640 Four. 277 00:18:58,640 --> 00:18:59,140 Five. 278 00:18:59,140 --> 00:19:00,120 Six. 279 00:19:00,120 --> 00:19:05,290 You didn't vote twice, did you? 280 00:19:05,290 --> 00:19:08,110 Vote early, vote often, right? 281 00:19:08,110 --> 00:19:11,570 Mayor Daley, the old Mayor Daley in Chicago. 282 00:19:11,570 --> 00:19:15,350 And if you know somebody who's dead, vote for them. 283 00:19:15,350 --> 00:19:18,370 OK, so this is the, I've got about five people here. 284 00:19:18,370 --> 00:19:20,640 And a bunch of people didn't vote. 285 00:19:20,640 --> 00:19:22,190 So it's a tie. 286 00:19:22,190 --> 00:19:27,751 It's basically a tie here. 287 00:19:27,751 --> 00:19:29,250 I think you should think about this. 288 00:19:29,250 --> 00:19:30,833 Maybe talk to each other a little bit. 289 00:19:30,833 --> 00:19:36,470 See if you can figure out, given all the numbers 290 00:19:36,470 --> 00:19:38,430 that are in this problem here, you 291 00:19:38,430 --> 00:19:46,240 know what, the pure pressure on the solvent is. 292 00:19:46,240 --> 00:19:50,530 We change concentrations here. 293 00:19:50,530 --> 00:19:53,240 Got Henry's law constant out. 294 00:19:53,240 --> 00:19:54,615 And Henry's law constant is going 295 00:19:54,615 --> 00:19:56,906 to tell you whether or not that enthalpy of interaction 296 00:19:56,906 --> 00:20:02,636 is favorable or not favorable. 297 00:20:02,636 --> 00:20:04,177 It's going to be a positive deviation 298 00:20:04,177 --> 00:20:34,760 or a negative deviation. 299 00:20:34,760 --> 00:20:36,940 OK, shall we vote again. 300 00:20:36,940 --> 00:20:41,420 How many people say it's the blue one? 301 00:20:41,420 --> 00:20:44,130 How many people say it's the green one. 302 00:20:44,130 --> 00:20:45,640 Thank you. 303 00:20:45,640 --> 00:20:47,480 Alright. 304 00:20:47,480 --> 00:20:48,100 Good. 305 00:20:48,100 --> 00:20:48,600 Alright. 306 00:20:48,600 --> 00:20:52,230 We have K sub B is less than p star, 307 00:20:52,230 --> 00:20:55,410 therefore it's a negative deviation. 308 00:20:55,410 --> 00:20:57,540 The pressure is less than you'd expect. 309 00:20:57,540 --> 00:21:00,040 The molecules like each other better than you expect. 310 00:21:00,040 --> 00:21:04,130 There is a favorable enthalpy of interaction between A and B. 311 00:21:04,130 --> 00:21:05,360 They like to interact. 312 00:21:05,360 --> 00:21:06,940 It could be hydrogen bonding. 313 00:21:06,940 --> 00:21:10,900 Could be a lot of different things. 314 00:21:10,900 --> 00:21:16,570 OK, so now if I put the whole diagram together, 315 00:21:16,570 --> 00:21:18,720 which I shouldn't have erased here. 316 00:21:18,720 --> 00:21:20,580 Right, so there's my diagram here again. 317 00:21:20,580 --> 00:21:22,360 There's Raoult's law. 318 00:21:22,360 --> 00:21:24,660 There's the deviation from Raoult's law here. 319 00:21:24,660 --> 00:21:29,000 For one substance, let's say for B. There's 320 00:21:29,000 --> 00:21:35,010 Raoult's law for A. There's a deviation for A. It 321 00:21:35,010 --> 00:21:36,590 could be like this. 322 00:21:36,590 --> 00:21:39,920 And if I now add these two together, 323 00:21:39,920 --> 00:21:44,380 instead of having a diagram for, this 324 00:21:44,380 --> 00:21:55,300 is a pressure xB, yB diagram, pressure xB, yB diagram, 325 00:21:55,300 --> 00:22:00,490 instead of having Raoult's law sitting right here. 326 00:22:00,490 --> 00:22:05,990 And then the yB part down here below, your usual sort 327 00:22:05,990 --> 00:22:10,220 of ideal solution phase diagram with the high pressure 328 00:22:10,220 --> 00:22:13,940 of the liquid on top and the gas on the bottom. 329 00:22:13,940 --> 00:22:19,560 Now if you add the blue curve and the red curve together, 330 00:22:19,560 --> 00:22:21,340 you're not going to get a straight line 331 00:22:21,340 --> 00:22:22,600 like you did before. 332 00:22:22,600 --> 00:22:25,220 When Raoult's law applied throughout. 333 00:22:25,220 --> 00:22:26,850 Instead you're going to get a deviation 334 00:22:26,850 --> 00:22:28,620 from the straight line. 335 00:22:28,620 --> 00:22:31,140 You're going to get a curve that's 336 00:22:31,140 --> 00:22:33,000 always below the straight line. 337 00:22:33,000 --> 00:22:35,390 Because we have the negative deviation on both sides. 338 00:22:35,390 --> 00:22:38,350 So you end up with something that's 339 00:22:38,350 --> 00:22:41,070 curved and deviates below. 340 00:22:41,070 --> 00:22:45,930 And the same thing will happen for the bubble line. 341 00:22:45,930 --> 00:22:49,100 It will also deviate below. 342 00:22:49,100 --> 00:22:50,230 In some way. 343 00:22:50,230 --> 00:22:52,440 Maybe not as much, maybe. 344 00:22:52,440 --> 00:22:55,030 So this is what you usually see. 345 00:22:55,030 --> 00:22:58,280 You see deviation of that whole phase diagram, 346 00:22:58,280 --> 00:22:59,720 like you're pushing down on it. 347 00:22:59,720 --> 00:23:01,300 Or maybe it didn't pulled up. 348 00:23:01,300 --> 00:23:03,380 If you have a positive deviation, 349 00:23:03,380 --> 00:23:05,140 then it's the opposite. 350 00:23:05,140 --> 00:23:08,720 It gets, this is negative deviation. 351 00:23:08,720 --> 00:23:11,820 And if you have a positive deviation, 352 00:23:11,820 --> 00:23:14,550 instead of having a usual phase diagram, 353 00:23:14,550 --> 00:23:21,270 you're going to get something that's looking like this. 354 00:23:21,270 --> 00:23:22,890 Which is the Henry's law coming in. 355 00:23:22,890 --> 00:23:23,390 Yes. 356 00:23:23,390 --> 00:23:26,225 STUDENT: [INAUDIBLE] 357 00:23:26,225 --> 00:23:26,850 PROFESSOR: Yes. 358 00:23:26,850 --> 00:23:29,340 STUDENT: [INAUDIBLE] 359 00:23:29,340 --> 00:23:34,000 PROFESSOR: Doesn't matter. 360 00:23:34,000 --> 00:23:36,340 Good question, though. 361 00:23:36,340 --> 00:23:39,030 OK, so this is what it usually looks like. 362 00:23:39,030 --> 00:23:40,720 And everything you've learned so far 363 00:23:40,720 --> 00:23:43,190 can be applied to these diagrams here. 364 00:23:43,190 --> 00:23:46,460 There's one exception, though, which is an important one. 365 00:23:46,460 --> 00:23:50,430 Which is, sometimes, these non-ideal diagrams 366 00:23:50,430 --> 00:23:53,250 have a funny spot. 367 00:23:53,250 --> 00:23:56,900 Which is that if I draw them, so there's 368 00:23:56,900 --> 00:24:00,850 my xB, yB, on the bottom. 369 00:24:00,850 --> 00:24:02,150 Zero to one. 370 00:24:02,150 --> 00:24:04,540 I have pressure here. 371 00:24:04,540 --> 00:24:11,930 Sometimes, there's the dew line. 372 00:24:11,930 --> 00:24:13,540 And the bubble line. 373 00:24:13,540 --> 00:24:17,410 Sometimes, instead of having a nice separation 374 00:24:17,410 --> 00:24:19,380 between the bubble and the dew line, 375 00:24:19,380 --> 00:24:22,390 there's a special point which for some reason 376 00:24:22,390 --> 00:24:30,030 those two curves meet. 377 00:24:30,030 --> 00:24:31,460 Non-ideality. 378 00:24:31,460 --> 00:24:33,600 There's interaction between the molecules. 379 00:24:33,600 --> 00:24:37,140 In the gas phase and the liquid phase. 380 00:24:37,140 --> 00:24:41,750 This special point is an interesting one. 381 00:24:41,750 --> 00:24:46,880 Because, well, if I'm away from this special point, 382 00:24:46,880 --> 00:24:50,590 let's say I'm sitting to the left of the special point, 383 00:24:50,590 --> 00:24:52,680 right here. 384 00:24:52,680 --> 00:24:55,180 And I can distill this thing. 385 00:24:55,180 --> 00:24:57,860 I can take my gas phase here. 386 00:24:57,860 --> 00:25:00,080 And increase the pressure. 387 00:25:00,080 --> 00:25:02,220 Say I'm sitting here in the mixture, gas phase 388 00:25:02,220 --> 00:25:13,930 mixture with x prime B, y prime B. Sorry, x prime B. 389 00:25:13,930 --> 00:25:18,270 And I go up and I increase the pressure. 390 00:25:18,270 --> 00:25:20,960 Start making vapor. 391 00:25:20,960 --> 00:25:27,524 Start making vapor, I can find out what the, 392 00:25:27,524 --> 00:25:28,440 I start making liquid. 393 00:25:28,440 --> 00:25:30,523 I can find out where the composition in the liquid 394 00:25:30,523 --> 00:25:34,920 is by going over to the blue line here. 395 00:25:34,920 --> 00:25:37,170 That tells me the composition in the liquid. 396 00:25:37,170 --> 00:25:41,454 And reading off the composition of the liquid down here. 397 00:25:41,454 --> 00:25:43,620 So this is actually the composition in the gas phase 398 00:25:43,620 --> 00:25:47,620 is yB prime. yB prime is the gas phase. 399 00:25:47,620 --> 00:25:50,150 I'm starting here. 400 00:25:50,150 --> 00:25:52,190 I start to increase the pressure. 401 00:25:52,190 --> 00:25:54,880 I begin the distillation process. 402 00:25:54,880 --> 00:25:57,460 And I get a little bit of liquid out. 403 00:25:57,460 --> 00:26:00,330 And I can read off the composition in the liquid. xB 404 00:26:00,330 --> 00:26:00,840 prime. 405 00:26:00,840 --> 00:26:06,150 And I know from this diagram that xB prime is less then 406 00:26:06,150 --> 00:26:08,250 yB prime. 407 00:26:08,250 --> 00:26:12,460 So I'm enriching my liquid in A. I'm 408 00:26:12,460 --> 00:26:15,370 purifying my solution in A, right? 409 00:26:15,370 --> 00:26:17,690 I can get pure A out eventually. 410 00:26:17,690 --> 00:26:20,820 The fact of doing this, I can get pure A. 411 00:26:20,820 --> 00:26:23,520 So I decrease the pressure, I get the pure gas phase 412 00:26:23,520 --> 00:26:25,650 of my new solution here. 413 00:26:25,650 --> 00:26:27,157 And I increase the pressure. 414 00:26:27,157 --> 00:26:28,990 And go over and make a little bit of liquid. 415 00:26:28,990 --> 00:26:30,369 And capture that liquid. 416 00:26:30,369 --> 00:26:32,160 And repeat the process over and over again. 417 00:26:32,160 --> 00:26:33,490 And eventually I have pure A. 418 00:26:33,490 --> 00:26:41,190 Now, suppose I'm sitting here at this special point. 419 00:26:41,190 --> 00:26:49,420 And I'm starting with yB And let me call it a here. yB of a. 420 00:26:49,420 --> 00:26:51,087 I'm sitting right here. 421 00:26:51,087 --> 00:26:52,420 And I'm increasing the pressure. 422 00:26:52,420 --> 00:26:56,630 I go all the way up here. 423 00:26:56,630 --> 00:27:00,230 And I start making liquid. . 424 00:27:00,230 --> 00:27:02,000 What's the composition of liquid, compared 425 00:27:02,000 --> 00:27:04,670 to the composition of the gas phase? 426 00:27:04,670 --> 00:27:05,460 What's y? 427 00:27:05,460 --> 00:27:13,140 What's xB compared to yB of this point up here? 428 00:27:13,140 --> 00:27:15,420 It's the same. 429 00:27:15,420 --> 00:27:17,530 So I'm making liquid. 430 00:27:17,530 --> 00:27:19,600 I got the same composition. 431 00:27:19,600 --> 00:27:22,150 Doesn't help me. 432 00:27:22,150 --> 00:27:25,390 If I make a gas again, I still have the same composition. 433 00:27:25,390 --> 00:27:27,020 I can't distill. 434 00:27:27,020 --> 00:27:28,860 I can't purify it. 435 00:27:28,860 --> 00:27:35,350 So if I have this special mixture here, of A and B, 436 00:27:35,350 --> 00:27:46,442 at this point here, I form what's called an azeotrope. 437 00:27:46,442 --> 00:27:47,900 And then you can't separate it out. 438 00:27:47,900 --> 00:27:51,690 You can't separate out the mixtures into A and B. 439 00:27:51,690 --> 00:27:55,860 You can't distill it at that point here. 440 00:27:55,860 --> 00:28:02,420 Now, you can draw the same diagram in, instead of 441 00:28:02,420 --> 00:28:05,610 pressure, mole fraction, you can draw the same diagram 442 00:28:05,610 --> 00:28:07,090 in terms of temperature. 443 00:28:07,090 --> 00:28:08,560 And the same thing happens. 444 00:28:08,560 --> 00:28:16,050 You can have deviations from, this 445 00:28:16,050 --> 00:28:19,860 is the extra figures that are on the bottom page 446 00:28:19,860 --> 00:28:22,220 of the notes from last time here. 447 00:28:22,220 --> 00:28:24,630 That we've replaced here. 448 00:28:24,630 --> 00:28:29,480 You can have a positive deviation. 449 00:28:29,480 --> 00:28:31,430 With a positive azeotrope. 450 00:28:31,430 --> 00:28:34,660 And your T p diagram, T x diagram. 451 00:28:34,660 --> 00:28:37,720 Or you can have, depending on the interactions 452 00:28:37,720 --> 00:28:39,650 between the two molecules, you can 453 00:28:39,650 --> 00:28:43,100 have something that has a negative deviation. 454 00:28:43,100 --> 00:28:48,060 And a negative azeotrope here. 455 00:28:48,060 --> 00:28:52,630 So in this case here, this is a particularly awful situation. 456 00:28:52,630 --> 00:28:56,940 Because there's your liquid phase down here. 457 00:28:56,940 --> 00:28:59,210 There's the gas phase here. 458 00:28:59,210 --> 00:29:04,260 And now, suppose that you're doing 459 00:29:04,260 --> 00:29:06,660 some sort of distillation. 460 00:29:06,660 --> 00:29:12,420 So you start here in the liquid phase, with some composition. 461 00:29:12,420 --> 00:29:15,791 You make a gas. 462 00:29:15,791 --> 00:29:17,790 Let's say you're doing a fractional distillation 463 00:29:17,790 --> 00:29:19,850 of some sort. 464 00:29:19,850 --> 00:29:23,610 Your gas now has a new mixture. 465 00:29:23,610 --> 00:29:26,760 So this is the composition in the gas phase. 466 00:29:26,760 --> 00:29:30,920 You take that gas phase mixture, and you condense it 467 00:29:30,920 --> 00:29:31,670 in your condenser. 468 00:29:31,670 --> 00:29:34,630 In your fractional distillation condenser. 469 00:29:34,630 --> 00:29:37,270 So now I've condensed it. 470 00:29:37,270 --> 00:29:40,400 And I heat it up again. 471 00:29:40,400 --> 00:29:42,990 I've condensed it. 472 00:29:42,990 --> 00:29:44,300 So I'm here somewhere. 473 00:29:44,300 --> 00:29:47,470 And cool it down so that I have a liquid. 474 00:29:47,470 --> 00:29:48,910 And I condense it. 475 00:29:48,910 --> 00:29:51,890 And slowly I move over to the azeotrope point. 476 00:29:51,890 --> 00:29:54,990 And once I get there, I'm stuck. 477 00:29:54,990 --> 00:29:57,410 And the same thing happens if I'm on the other side. 478 00:29:57,410 --> 00:30:04,600 I'm on this point here, I keep my solution, my mixture. 479 00:30:04,600 --> 00:30:07,010 I make a gas, I read off the composition of the gas phase 480 00:30:07,010 --> 00:30:07,510 here. 481 00:30:07,510 --> 00:30:11,180 I take my gas phase composition out and I re-liquefy it. 482 00:30:11,180 --> 00:30:12,900 So I'm here, down here somewhere, again. 483 00:30:12,900 --> 00:30:14,010 And I heat it up again. 484 00:30:14,010 --> 00:30:18,570 And and I move to the azeotrope point. 485 00:30:18,570 --> 00:30:21,470 That's what happens if you have a mixture of benzene and water, 486 00:30:21,470 --> 00:30:25,240 for instance. 487 00:30:25,240 --> 00:30:28,430 You can't distill it. 488 00:30:28,430 --> 00:30:30,439 You work your way down to this point here. 489 00:30:30,439 --> 00:30:31,980 You can't use fractional distillation 490 00:30:31,980 --> 00:30:34,510 until you've separated it out. 491 00:30:34,510 --> 00:30:36,520 So this is a real life problem. 492 00:30:36,520 --> 00:30:40,250 And you have to do things like change the pressure. 493 00:30:40,250 --> 00:30:43,150 Do your distillation under high pressure or low pressure. 494 00:30:43,150 --> 00:30:46,340 To try to get this point to separate out. 495 00:30:46,340 --> 00:30:48,320 So you can get past this point. 496 00:30:48,320 --> 00:30:52,860 Or to try to get the deviation to become of the opposite sign. 497 00:30:52,860 --> 00:30:55,940 Where you can start to do some distillation. 498 00:30:55,940 --> 00:30:58,330 Any questions on azeotropes, before we 499 00:30:58,330 --> 00:31:04,970 get to colligative properties? 500 00:31:04,970 --> 00:31:06,990 If you're doing organic chemistry, 501 00:31:06,990 --> 00:31:08,590 and you're trying to purify things, 502 00:31:08,590 --> 00:31:10,660 this is a real pain, right? 503 00:31:10,660 --> 00:31:11,710 Pain in the neck. 504 00:31:11,710 --> 00:31:21,982 So you've got to know, which things form azeotropes. 505 00:31:21,982 --> 00:31:23,940 So we're going to start colligative properties. 506 00:31:23,940 --> 00:31:25,523 There are four colligative properties, 507 00:31:25,523 --> 00:31:28,570 and you already know what they are. 508 00:31:28,570 --> 00:31:32,970 And you've even seen the math behind them. 509 00:31:32,970 --> 00:31:35,540 And in fact, you've seen the math for one of them 510 00:31:35,540 --> 00:31:41,461 already pretty clearly. 511 00:31:41,461 --> 00:31:42,960 And these colligative properties are 512 00:31:42,960 --> 00:31:50,650 properties that you use all the time in your day to day life. 513 00:31:50,650 --> 00:31:54,690 At least, three of them. 514 00:31:54,690 --> 00:32:27,630 OK what are the four colligative properties? 515 00:32:27,630 --> 00:32:29,940 There is vapor pressure lowering. 516 00:32:29,940 --> 00:32:34,130 If you have a mixture and the vapor pressure of the mixture 517 00:32:34,130 --> 00:32:37,470 is less than the vapor pressure of the pure material, 518 00:32:37,470 --> 00:32:43,570 there's the boiling point elevation. 519 00:32:43,570 --> 00:32:50,270 If you add a large amount of insoluble - not insoluble, 520 00:32:50,270 --> 00:32:55,720 non-volatile solute to a solution, you will depress, 521 00:32:55,720 --> 00:32:59,220 you will increase the boiling point of that solution like 522 00:32:59,220 --> 00:33:01,184 the example of the salt and water that I gave. 523 00:33:01,184 --> 00:33:02,850 So if you need a lot of salt to increase 524 00:33:02,850 --> 00:33:05,030 the boiling point of water. 525 00:33:05,030 --> 00:33:12,850 There's the freezing point depression. 526 00:33:12,850 --> 00:33:17,720 That's what we use to melt ice in the winter. 527 00:33:17,720 --> 00:33:19,410 By adding salt to it. 528 00:33:19,410 --> 00:33:23,470 Then there's the osmotic pressure. 529 00:33:23,470 --> 00:33:28,880 Which is the example of the freshwater fish in salt water, 530 00:33:28,880 --> 00:33:30,790 or the saltwater fish in fresh water. 531 00:33:30,790 --> 00:33:35,310 Or the cells bursting when you put them in. 532 00:33:35,310 --> 00:33:41,040 OK, so these colligative properties all come 533 00:33:41,040 --> 00:33:45,590 from the fact if I have a mixture, 534 00:33:45,590 --> 00:33:51,650 the chemical potential of a molecule in a mixture, 535 00:33:51,650 --> 00:33:55,245 in the liquid phase here, is less 536 00:33:55,245 --> 00:34:02,530 than the chemical potential of the pure guy. 537 00:34:02,530 --> 00:34:14,590 Where A is the solvent, and B is some solute. 538 00:34:14,590 --> 00:34:18,060 So this is always true if your number of moles of A 539 00:34:18,060 --> 00:34:21,620 is much larger than number of moles of B. Mix 540 00:34:21,620 --> 00:34:25,150 a little bit of B in there, and the chemical potential of A 541 00:34:25,150 --> 00:34:26,930 in the liquid is going to come down. 542 00:34:26,930 --> 00:34:28,350 Because of the mixing process. 543 00:34:28,350 --> 00:34:30,910 Because of the entropy of mixing. 544 00:34:30,910 --> 00:34:36,520 And that's going to give rise to these properties. 545 00:34:36,520 --> 00:34:41,290 And, to do these, one more thing before we start. 546 00:34:41,290 --> 00:34:47,410 We're going to use a special concentration for these guys. 547 00:34:47,410 --> 00:34:50,480 We're going to use, call something molality. 548 00:34:50,480 --> 00:34:52,600 Molality. 549 00:34:52,600 --> 00:34:58,310 Molality is the number of moles of solute 550 00:34:58,310 --> 00:35:07,050 divided by kilograms of solvent. 551 00:35:07,050 --> 00:35:12,370 So it's moles per weight of solvent. 552 00:35:12,370 --> 00:35:13,870 As opposed to moles per liter, which 553 00:35:13,870 --> 00:35:16,420 is something that we're more used to. 554 00:35:16,420 --> 00:35:18,000 Moles per kilograms of solvent. 555 00:35:18,000 --> 00:35:21,230 It's pretty close to moles for liter for water. 556 00:35:21,230 --> 00:35:23,890 Because a gram of water is a milliliter, roughly. 557 00:35:23,890 --> 00:35:28,320 The density of water's close to one. 558 00:35:28,320 --> 00:35:31,060 So let's start on these things. 559 00:35:31,060 --> 00:35:41,130 So we've already seen this. 560 00:35:41,130 --> 00:35:47,740 You've already seen vapor pressure lowering. 561 00:35:47,740 --> 00:35:50,690 It comes from Raoult's law. 562 00:35:50,690 --> 00:35:55,620 If I look at the change in the vapor pressure of A, pA 563 00:35:55,620 --> 00:36:04,790 minus pA star. pA is equal to xA pA star. xA pA star, that's 564 00:36:04,790 --> 00:36:06,060 Raoult's law. 565 00:36:06,060 --> 00:36:11,930 Minus pA star, that's equal to one minus xA. 566 00:36:11,930 --> 00:36:14,740 Minus pA star. 567 00:36:14,740 --> 00:36:21,170 Which is minus xB pA star. 568 00:36:21,170 --> 00:36:25,290 Which is less than zero. 569 00:36:25,290 --> 00:36:31,070 So, that means that the vapor pressure above the solution 570 00:36:31,070 --> 00:36:33,000 is lowered. 571 00:36:33,000 --> 00:36:39,290 So I should have said vapor pressure lowering. 572 00:36:39,290 --> 00:36:41,760 And we also drew a phase diagram to show 573 00:36:41,760 --> 00:36:44,490 how this vapor pressure lowering was 574 00:36:44,490 --> 00:36:50,600 going to imply freezing point depression and boiling point 575 00:36:50,600 --> 00:36:53,340 depression. 576 00:36:53,340 --> 00:37:01,260 So now let's go on and look at the freezing point depression. 577 00:37:01,260 --> 00:37:01,760 Number two. 578 00:37:01,760 --> 00:37:08,300 We're going to find that the change in the boiling point 579 00:37:08,300 --> 00:37:11,190 is going to be some constant K sub b, which is not 580 00:37:11,190 --> 00:37:13,140 to be confused with Henry's law's constant. 581 00:37:13,140 --> 00:37:14,790 It's completely different. 582 00:37:14,790 --> 00:37:16,480 This is a colligative property. 583 00:37:16,480 --> 00:37:19,490 Times the molality of B. That's going 584 00:37:19,490 --> 00:37:21,860 to be the answer we're going to get. 585 00:37:21,860 --> 00:37:26,020 Amount of temperature change depends linearly 586 00:37:26,020 --> 00:37:28,640 on the molality of B, with a constant 587 00:37:28,640 --> 00:37:30,990 that depends on what the mixture is. 588 00:37:30,990 --> 00:37:33,890 And what the pressure is. 589 00:37:33,890 --> 00:37:36,860 Whenever we derive anything connected to phase transitions, 590 00:37:36,860 --> 00:37:39,610 we usually always start with the fact 591 00:37:39,610 --> 00:37:42,290 that the chemical potential changes. 592 00:37:42,290 --> 00:37:43,965 And it has to be equal at equilibrium. 593 00:37:43,965 --> 00:37:45,340 So that's what we're going to do. 594 00:37:45,340 --> 00:37:51,370 We're going to start with the chemical potential equal 595 00:37:51,370 --> 00:37:53,980 of the liquid phase is equal to the chemical potential 596 00:37:53,980 --> 00:37:55,500 of the gas phase. 597 00:37:55,500 --> 00:37:59,790 For the solvent at equilibrium. 598 00:37:59,790 --> 00:38:05,330 And the liquid phase obeys Raoult's law. 599 00:38:05,330 --> 00:38:08,810 So it's going to be the chemical potential of the pure material. 600 00:38:08,810 --> 00:38:10,770 The pure liquid. 601 00:38:10,770 --> 00:38:15,040 Plus RT log xA. 602 00:38:15,040 --> 00:38:19,440 So this is where the mixing of the solute comes in. 603 00:38:19,440 --> 00:38:24,480 And on the gas phase side, mu A, it's 604 00:38:24,480 --> 00:38:26,944 going to be the same thing as the chemical potential 605 00:38:26,944 --> 00:38:28,360 of the pure A. Because we're going 606 00:38:28,360 --> 00:38:35,540 to take B to be non-volatile. 607 00:38:35,540 --> 00:38:37,060 So the only thing in the gas phase 608 00:38:37,060 --> 00:38:39,760 is A. Same chemical potential as if it were pure. 609 00:38:39,760 --> 00:38:42,200 Because it is pure. 610 00:38:42,200 --> 00:38:50,850 So now we solve for log xA is equal to one over RT mu 611 00:38:50,850 --> 00:38:58,810 A star in the gas phase minus mu A star in the liquid phase. 612 00:38:58,810 --> 00:39:00,480 And this is just the Gibbs free energy 613 00:39:00,480 --> 00:39:01,980 per mole of A in the gas phase. 614 00:39:01,980 --> 00:39:04,520 The Gibbs free energy per mole of A in the liquid phase. 615 00:39:04,520 --> 00:39:08,470 So we can replace this with the change in Gibbs free energy 616 00:39:08,470 --> 00:39:10,840 from going from gas to liquid. 617 00:39:10,840 --> 00:39:13,343 Per mole, which is just the Gibbs 618 00:39:13,343 --> 00:39:15,270 free energy of vaporization. 619 00:39:15,270 --> 00:39:19,460 We're going from liquid to gas. 620 00:39:19,460 --> 00:39:23,410 And I probably have a negative sign somewhere. 621 00:39:23,410 --> 00:39:25,160 No this is right, this is the final state. 622 00:39:25,160 --> 00:39:27,400 This is the initial state. 623 00:39:27,400 --> 00:39:32,030 Divided by RT. 624 00:39:32,030 --> 00:39:33,275 OK. 625 00:39:33,275 --> 00:39:34,900 That's not really what we want, though. 626 00:39:34,900 --> 00:39:37,010 What we want is delta T. Here I've 627 00:39:37,010 --> 00:39:39,180 got some complicated equation that has T in it, 628 00:39:39,180 --> 00:39:41,920 but not delta T. But before we do 629 00:39:41,920 --> 00:39:44,930 that, before we transfer and try to get a delta T out 630 00:39:44,930 --> 00:39:46,930 of this equation, let's see if we 631 00:39:46,930 --> 00:39:51,650 can do any sort of simplifications here. 632 00:39:51,650 --> 00:39:54,040 I don't like to carry logs around. 633 00:39:54,040 --> 00:39:56,930 There are no logs around here. 634 00:39:56,930 --> 00:40:03,230 So let's see if we can get rid of that log. 635 00:40:03,230 --> 00:40:08,780 What is log xA? 636 00:40:08,780 --> 00:40:12,640 Log xA, well, xA is close to one. 637 00:40:12,640 --> 00:40:13,700 It's connected to xB. 638 00:40:13,700 --> 00:40:15,640 And everything in here is connected 639 00:40:15,640 --> 00:40:18,030 to B. There's a molality of B here. 640 00:40:18,030 --> 00:40:21,060 The concentration of A doesn't come into play here. 641 00:40:21,060 --> 00:40:22,310 I don't want it there. 642 00:40:22,310 --> 00:40:27,880 So let me replace it with B. one minus xB. xB is a small number. 643 00:40:27,880 --> 00:40:31,180 It's a solute, it's in small amount. 644 00:40:31,180 --> 00:40:34,020 We're close to being a pure solvent here. 645 00:40:34,020 --> 00:40:36,520 So one minus something very small. 646 00:40:36,520 --> 00:40:37,940 The log of that. 647 00:40:37,940 --> 00:40:41,460 Well if you've worked with Taylor's expansions. 648 00:40:41,460 --> 00:40:46,320 You probably know that that's close to minus xB. 649 00:40:46,320 --> 00:40:49,330 The log of one minus x where x goes to zero 650 00:40:49,330 --> 00:40:51,820 is close to minus x. 651 00:40:51,820 --> 00:40:53,050 So that's minus xB. 652 00:40:53,050 --> 00:40:55,590 Now, minus xB, in terms of the moles, 653 00:40:55,590 --> 00:40:58,130 is the number of moles of B divided by the number of moles 654 00:40:58,130 --> 00:41:02,560 of B plus the number of moles of A, the total number of moles. 655 00:41:02,560 --> 00:41:04,660 The number of moles of B is very small, 656 00:41:04,660 --> 00:41:10,310 compared to the number of moles of A. So I can get rid of this. 657 00:41:10,310 --> 00:41:11,860 So, and there's a minus sign here. 658 00:41:11,860 --> 00:41:17,170 So this is approximately minus nB divided by nA. 659 00:41:17,170 --> 00:41:20,710 Now, the only I can do is, I'm trying to get at molality here. 660 00:41:20,710 --> 00:41:22,450 I said we're going to use molality. 661 00:41:22,450 --> 00:41:23,226 Got moles. 662 00:41:23,226 --> 00:41:24,850 I don't have kilograms of solvent here. 663 00:41:24,850 --> 00:41:25,990 I have moles of solvent. 664 00:41:25,990 --> 00:41:27,840 I'm not so happy with that. 665 00:41:27,840 --> 00:41:30,905 So, in order to get something that has kilograms out, 666 00:41:30,905 --> 00:41:37,170 let's multiply up and down by capital M where capital M is 667 00:41:37,170 --> 00:41:40,760 that total mass of A. Basically the number of kilograms 668 00:41:40,760 --> 00:41:41,680 of solvent. 669 00:41:41,680 --> 00:41:50,840 Total mass of A. So now, this fraction here, 670 00:41:50,840 --> 00:41:52,880 that's the molality. 671 00:41:52,880 --> 00:41:55,430 That's what I'm looking for. 672 00:41:55,430 --> 00:41:58,850 So this is minus the molality. 673 00:41:58,850 --> 00:42:03,330 Then I have the total mass of A divided 674 00:42:03,330 --> 00:42:08,010 by the number of moles of A. Well, that's 675 00:42:08,010 --> 00:42:09,296 the molecular weight of it. 676 00:42:09,296 --> 00:42:11,004 Total mass divided by the number of moles 677 00:42:11,004 --> 00:42:12,600 is the molecular weight. 678 00:42:12,600 --> 00:42:15,370 So this is the molality of B times the molecular weight 679 00:42:15,370 --> 00:42:18,790 of A. OK. 680 00:42:18,790 --> 00:42:22,820 So now I can plug this instead of log xA. 681 00:42:22,820 --> 00:42:29,930 And I get minus nB times nA is equal to delta 682 00:42:29,930 --> 00:42:38,830 G of vaporization divided by RT. 683 00:42:38,830 --> 00:42:42,360 Let's rearrange a little bit. 684 00:42:42,360 --> 00:42:46,690 And mB is equal to to delta G vaporization divided 685 00:42:46,690 --> 00:42:52,220 by nA times RT, with a negative sign. 686 00:42:52,220 --> 00:42:54,350 Well, I've eliminated the log at least, 687 00:42:54,350 --> 00:42:56,110 but I still don't have a delta T in there. 688 00:42:56,110 --> 00:42:58,590 I don't have a delta T. How am I going 689 00:42:58,590 --> 00:43:00,780 to get a delta T in there? 690 00:43:00,780 --> 00:43:03,770 Got to get the delta T. Everything's very small. 691 00:43:03,770 --> 00:43:06,350 So one way to get a small change in temperature 692 00:43:06,350 --> 00:43:08,640 is to take a derivative with respect to temperature. 693 00:43:08,640 --> 00:43:11,970 That's going to get a delta T in there. 694 00:43:11,970 --> 00:43:14,140 So I'm going to take the derivative of both sides 695 00:43:14,140 --> 00:43:16,930 with respect to T. 696 00:43:16,930 --> 00:43:18,550 And delta T is going to come out that. 697 00:43:18,550 --> 00:43:25,840 I'm going to take d nB / dT, constant pressure. 698 00:43:25,840 --> 00:43:36,560 Minus one over mA times R d delta G over T. This 699 00:43:36,560 --> 00:43:43,170 is delta G of vaporization. dT, constant pressure. 700 00:43:43,170 --> 00:43:47,070 And I wish I'd made my boards a little bit better. 701 00:43:47,070 --> 00:43:51,720 Because I have to cover things up. 702 00:43:51,720 --> 00:43:54,470 OK, now I have this derivative of the Gibbs free energy 703 00:43:54,470 --> 00:43:56,590 divided by the temperature. 704 00:43:56,590 --> 00:43:59,910 This turns out to be something that you've seen before. 705 00:43:59,910 --> 00:44:10,220 It's called the Gibbs-Helmholtz equation. 706 00:44:10,220 --> 00:44:13,940 And it relates the temperature change in the Gibbs free energy 707 00:44:13,940 --> 00:44:16,220 with the enthalpy change. 708 00:44:16,220 --> 00:44:23,490 The Gibbs-Helmholtz equation is that the d delta G over T dT 709 00:44:23,490 --> 00:44:28,050 is equal to minus delta H over T. 710 00:44:28,050 --> 00:44:31,720 And you can get that from just writing fundamental equations 711 00:44:31,720 --> 00:44:34,735 about G and H. And turning the crank. 712 00:44:34,735 --> 00:44:36,710 And we've done that before. 713 00:44:36,710 --> 00:44:39,250 And I'll leave that as an exercise for you 714 00:44:39,250 --> 00:44:41,460 to go back and figure out what, you can go back 715 00:44:41,460 --> 00:44:44,820 to your textbook and look at, you know under the index. 716 00:44:44,820 --> 00:44:49,000 Gibbs-Helmholtz, and figure out why this is the case. 717 00:44:49,000 --> 00:44:50,560 So now, we have this equation here. 718 00:44:50,560 --> 00:44:52,143 This is going to be nice, because it's 719 00:44:52,143 --> 00:44:53,890 going to get rid of the d/dT. 720 00:44:53,890 --> 00:44:56,200 On the right-hand side. 721 00:44:56,200 --> 00:44:58,960 And so now our equation becomes d 722 00:44:58,960 --> 00:45:09,360 mB / dT is equal to delta H vaporization divided by mA R T 723 00:45:09,360 --> 00:45:13,340 squared. 724 00:45:13,340 --> 00:45:17,170 Now, I can rewrite this in a slightly different way. 725 00:45:17,170 --> 00:45:20,220 This is delta m over delta T, basically. 726 00:45:20,220 --> 00:45:22,446 Delta the change in number of moles. 727 00:45:22,446 --> 00:45:23,820 I mean, the molality of B divided 728 00:45:23,820 --> 00:45:25,028 by the change in temperature. 729 00:45:25,028 --> 00:45:26,540 I'm getting very close here. 730 00:45:26,540 --> 00:45:28,430 There's a change in temperature here. 731 00:45:28,430 --> 00:45:30,580 There's a change in molality here. 732 00:45:30,580 --> 00:45:33,400 I can rewrite this as delta T, then, 733 00:45:33,400 --> 00:45:37,380 is equal to molecular weight of A times 734 00:45:37,380 --> 00:45:46,530 RT squared divided by delta H vaporization times delta mB. 735 00:45:46,530 --> 00:45:49,610 Now, I'm making a very small change in B here. 736 00:45:49,610 --> 00:45:50,420 Starting at zero. 737 00:45:50,420 --> 00:45:53,530 Basically I'm taking pure A and I'm 738 00:45:53,530 --> 00:45:56,890 dumping in a little bit of B. That's 739 00:45:56,890 --> 00:45:58,990 what I'm doing when I experiment. 740 00:45:58,990 --> 00:46:01,519 I'm dumping in a little bit of B in my pure A, 741 00:46:01,519 --> 00:46:04,060 and I'm trying to figure out how the temperature is changing. 742 00:46:04,060 --> 00:46:08,690 So I'm doing delta m sub B is my small amount of B 743 00:46:08,690 --> 00:46:10,470 that I'm adding to the solution. 744 00:46:10,470 --> 00:46:13,160 Minus what I started out with, which was zero. 745 00:46:13,160 --> 00:46:18,114 So delta mB is basically m sub B. Right? 746 00:46:18,114 --> 00:46:19,280 That's basically what it is. 747 00:46:19,280 --> 00:46:21,630 So let's get rid of this delta here. 748 00:46:21,630 --> 00:46:22,930 What else can I say? 749 00:46:22,930 --> 00:46:25,490 Well, I can say that the temperature change that I'm 750 00:46:25,490 --> 00:46:27,019 looking at is also pretty small. 751 00:46:27,019 --> 00:46:28,560 Compared to the absolute temperature. 752 00:46:28,560 --> 00:46:30,380 This is a small change. 753 00:46:30,380 --> 00:46:33,350 I'm only adding a small amount of B in there. 754 00:46:33,350 --> 00:46:38,000 So this temperature up here is not going to change very much. 755 00:46:38,000 --> 00:46:40,530 Let's take it to be the temperature 756 00:46:40,530 --> 00:46:45,100 of the boiling point of the pure material. 757 00:46:45,100 --> 00:46:50,570 T sub B star, of the pure material. 758 00:46:50,570 --> 00:46:53,190 OK And there's my delta T here. 759 00:46:53,190 --> 00:46:56,025 So this is of the form delta T is 760 00:46:56,025 --> 00:47:03,290 equal to molecular weight of A. Times R. Times 761 00:47:03,290 --> 00:47:06,250 the boiling point of the pure material squared. 762 00:47:06,250 --> 00:47:08,250 Divided by delta H of vaporization 763 00:47:08,250 --> 00:47:10,750 of the pure solvent. 764 00:47:10,750 --> 00:47:19,140 Times the molality of B. And this combination of properties 765 00:47:19,140 --> 00:47:23,360 of A, the molecular weight of A, the boiling point of pure A. 766 00:47:23,360 --> 00:47:32,300 The heat of vaporization of pure A, this is K sub b up here. 767 00:47:32,300 --> 00:47:33,670 Which I covered up. 768 00:47:33,670 --> 00:47:44,700 It's of the form delta T is equal to K sub b times mB. 769 00:47:44,700 --> 00:47:50,580 And this turns out to be always greater than zero. 770 00:47:50,580 --> 00:47:52,530 Now. 771 00:47:52,530 --> 00:47:54,720 If you want to look at freezing point depression, 772 00:47:54,720 --> 00:47:57,200 it's really easy. 773 00:47:57,200 --> 00:48:07,830 In your equations, it's the same thermodynamic picture. 774 00:48:07,830 --> 00:48:10,730 All you need to do, in all your equations, 775 00:48:10,730 --> 00:48:15,590 you replace delta G of vaporization 776 00:48:15,590 --> 00:48:19,200 with minus delta G of freezing. 777 00:48:19,200 --> 00:48:24,410 When we did a vaporization, we're going from liquid to gas. 778 00:48:24,410 --> 00:48:26,710 We had delta G of vaporization. 779 00:48:26,710 --> 00:48:29,730 When we're doing freezing point depression, 780 00:48:29,730 --> 00:48:34,350 we're going from liquid to solid. 781 00:48:34,350 --> 00:48:40,450 And so we have to use minus delta G of fusion. 782 00:48:40,450 --> 00:48:42,674 Minus delta H of fusion. s to l would 783 00:48:42,674 --> 00:48:44,090 be the positive delta G of fusion. 784 00:48:44,090 --> 00:48:47,150 So we're going from liquid to solid. 785 00:48:47,150 --> 00:48:48,900 We have to replace delta G of vaporization 786 00:48:48,900 --> 00:48:50,910 with delta G of fusion. 787 00:48:50,910 --> 00:48:53,810 We replace delta H of vaporization, 788 00:48:53,810 --> 00:48:57,220 where minus delta H of fusion. 789 00:48:57,220 --> 00:49:01,380 Replace the boiling point with the freezing point. 790 00:49:01,380 --> 00:49:04,980 And replace K sub b with K sub f. 791 00:49:04,980 --> 00:49:08,740 And the math is exactly identical. 792 00:49:08,740 --> 00:49:13,110 It all starts from the chemical potentials being equal. 793 00:49:13,110 --> 00:49:15,220 And turning the crank at equilibrium, being 794 00:49:15,220 --> 00:49:17,560 equal between the solid phase and the liquid phase. 795 00:49:17,560 --> 00:49:18,620 And turning the crank. 796 00:49:18,620 --> 00:49:20,910 The solid phase is like the gas phase. 797 00:49:20,910 --> 00:49:24,680 It's a pure A, pretty much. 798 00:49:24,680 --> 00:49:27,380 Because the molecules of B are stuck. 799 00:49:27,380 --> 00:49:30,769 And A doesn't know that B's around, in the solid phase. 800 00:49:30,769 --> 00:49:32,810 They know that they're around in the liquid phase 801 00:49:32,810 --> 00:49:35,500 because the fusion fast enough. . 802 00:49:35,500 --> 00:49:37,600 And then we can write, if we turn the crank, 803 00:49:37,600 --> 00:49:39,590 we get an equation that looks like this. 804 00:49:39,590 --> 00:49:43,990 The freezing point depression is equal to minus mA 805 00:49:43,990 --> 00:49:48,810 R Tf star squared. 806 00:49:48,810 --> 00:49:53,670 Divided by delta H of fusion, for mole. 807 00:49:53,670 --> 00:49:56,240 Times the molality of B. And this number 808 00:49:56,240 --> 00:49:58,740 is always less than zero. 809 00:49:58,740 --> 00:50:01,510 There's the negative signs in here. 810 00:50:01,510 --> 00:50:03,090 OK. 811 00:50:03,090 --> 00:50:05,050 Always depressed. 812 00:50:05,050 --> 00:50:09,840 Any questions about those freezing point depression 813 00:50:09,840 --> 00:50:13,840 and boiling point elevation. 814 00:50:13,840 --> 00:50:14,820 Yes. 815 00:50:14,820 --> 00:50:20,000 STUDENT: [INAUDIBLE] PROFESSOR: Why did I drop it here? 816 00:50:20,000 --> 00:50:23,850 Because the small change that I'm making in nB 817 00:50:23,850 --> 00:50:27,840 is going from pure A, where nB is zero, 818 00:50:27,840 --> 00:50:31,330 to a very small amount of nB. 819 00:50:31,330 --> 00:50:34,550 That's why I'm allowed to say this. 820 00:50:34,550 --> 00:50:38,230 And then just replace delta nB with nB. 821 00:50:38,230 --> 00:50:40,020 I'm comparing every, my reference point 822 00:50:40,020 --> 00:50:41,550 is the pure material. 823 00:50:41,550 --> 00:50:44,850 I'm just adding a little bit to start with. 824 00:50:44,850 --> 00:50:45,910 It's a good question. 825 00:50:45,910 --> 00:50:50,730 Any other questions? 826 00:50:50,730 --> 00:50:51,300 OK. 827 00:50:51,300 --> 00:50:53,670 I didn't get to osmotic pressure. 828 00:50:53,670 --> 00:50:56,580 You'll get to that on Friday. 829 00:50:56,580 --> 00:50:58,190 And also start statistical mechanics. 830 00:50:58,190 --> 00:51:03,090 And after osmotic pressure, you'll be done with thermo. 831 00:51:03,090 --> 00:51:05,800 It'll be statistical mechanics and then kinetics.