1 00:00:00,030 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,810 Commons license. 3 00:00:03,810 --> 00:00:06,860 Your support will help MIT OpenCourseWare continue to 4 00:00:06,860 --> 00:00:10,510 offer high quality educational resources for free. 5 00:00:10,510 --> 00:00:13,390 To make a donation or view additional materials from 6 00:00:13,390 --> 00:00:17,590 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,590 --> 00:00:20,270 ocw.mit.edu. 8 00:00:20,270 --> 00:00:23,450 PROFESSOR: And today I'll continue with both ideal and 9 00:00:23,450 --> 00:00:25,830 non-ideal, or real, liquid mixtures. 10 00:00:25,830 --> 00:00:28,850 And let me just say the reason we're in some sense lavishing 11 00:00:28,850 --> 00:00:31,670 so much attention on this topic is because, after all, 12 00:00:31,670 --> 00:00:33,340 there's just an enormous amount of chemistry that 13 00:00:33,340 --> 00:00:34,690 happens in liquid mixtures. 14 00:00:34,690 --> 00:00:36,070 Awful lot of biology. 15 00:00:36,070 --> 00:00:38,890 Awful lot of just processing of stuff. 16 00:00:38,890 --> 00:00:40,240 Distillation and everything else. 17 00:00:40,240 --> 00:00:42,940 So there's so much chemistry that takes place in liquid 18 00:00:42,940 --> 00:00:46,300 mixtures that it is really important to have a sense of 19 00:00:46,300 --> 00:00:49,375 what the free energy and chemical potential of each of 20 00:00:49,375 --> 00:00:51,110 the species is doing in there. 21 00:00:51,110 --> 00:00:53,250 Because of course this is what directly guides the chemistry 22 00:00:53,250 --> 00:00:54,140 that happens. 23 00:00:54,140 --> 00:00:58,970 So I just want to start by finishing up something, a 24 00:00:58,970 --> 00:01:02,880 couple of things, from the topics from the last lecture. 25 00:01:02,880 --> 00:01:07,140 One of them is, I derived an expression for the lever rule. 26 00:01:07,140 --> 00:01:08,900 And I just wanted to make a little more 27 00:01:08,900 --> 00:01:09,850 explicit the result. 28 00:01:09,850 --> 00:01:11,650 It may not have been completely clear. 29 00:01:11,650 --> 00:01:18,750 So, I just want to go back to that. 30 00:01:18,750 --> 00:01:33,810 We had looked at what happens if you start at some 31 00:01:33,810 --> 00:01:39,710 particular point, labeled one, and work your way up. 32 00:01:39,710 --> 00:01:41,300 So we're raising the pressure. 33 00:01:41,300 --> 00:01:47,730 So we're in the all gas region to start. 34 00:01:47,730 --> 00:01:50,230 All gas region of the phase diagram. 35 00:01:50,230 --> 00:01:53,920 This is xB and yB. 36 00:01:53,920 --> 00:01:58,240 The liquid and gas mole fractions of B. And the idea 37 00:01:58,240 --> 00:02:02,290 is that here is our initial value. 38 00:02:02,290 --> 00:02:03,670 Now we're only in the gas phase. 39 00:02:03,670 --> 00:02:04,520 There is no liquid. 40 00:02:04,520 --> 00:02:08,050 So this is yB of one. 41 00:02:08,050 --> 00:02:10,460 There is no xB at this point. 42 00:02:10,460 --> 00:02:12,360 We raise the pressure. 43 00:02:12,360 --> 00:02:18,470 And at some point, depending on where we end, we then look 44 00:02:18,470 --> 00:02:22,750 to both coexistence curves to determine the compositions of 45 00:02:22,750 --> 00:02:25,400 both the liquid and the gas phases. 46 00:02:25,400 --> 00:02:32,620 So when we look out here, at the liquid phase. 47 00:02:32,620 --> 00:02:35,220 That gives us xB. 48 00:02:35,220 --> 00:02:40,030 At where I labeled point two, so there's two after we raise 49 00:02:40,030 --> 00:02:40,450 the pressure. 50 00:02:40,450 --> 00:02:44,530 We started at one. 51 00:02:44,530 --> 00:02:48,670 And then over here, we look at the gas phase coexistence 52 00:02:48,670 --> 00:02:55,190 curve. so there's our yB value at two, right? 53 00:02:55,190 --> 00:03:00,620 And I derived an expression for the ratio of the number of 54 00:03:00,620 --> 00:03:03,130 moles in the gas and liquid phases. 55 00:03:03,130 --> 00:03:06,340 Because the idea here is, let's say you're working your 56 00:03:06,340 --> 00:03:07,840 way up on the curve. 57 00:03:07,840 --> 00:03:11,820 If you just barely get to the coexistence curve, of course 58 00:03:11,820 --> 00:03:14,230 then you could see how much material you've 59 00:03:14,230 --> 00:03:15,780 got in the two phases. 60 00:03:15,780 --> 00:03:17,910 But there's very little material still. 61 00:03:17,910 --> 00:03:20,910 You've still got almost everything in the gas phase 62 00:03:20,910 --> 00:03:22,370 when you first reach here. 63 00:03:22,370 --> 00:03:24,580 And as you work your way up in pressure, you'd have more and 64 00:03:24,580 --> 00:03:25,200 more liquid. 65 00:03:25,200 --> 00:03:27,250 Of course, if you were to just keep going, you'd get into the 66 00:03:27,250 --> 00:03:28,340 pure liquid phase. 67 00:03:28,340 --> 00:03:30,190 Typically, though, you might stop somewhere in the middle 68 00:03:30,190 --> 00:03:32,110 and have some reasonable amount of material in both 69 00:03:32,110 --> 00:03:33,210 phases, and you want to find out the 70 00:03:33,210 --> 00:03:34,930 composition in each phase. 71 00:03:34,930 --> 00:03:37,560 And also, want to know how much material 72 00:03:37,560 --> 00:03:39,700 there is in each phase. 73 00:03:39,700 --> 00:03:45,100 So the result that I derived was n gas at two 74 00:03:45,100 --> 00:03:46,620 over n liquid at two. 75 00:03:46,620 --> 00:03:51,040 The ratio of total moles in the gas to the liquid. 76 00:03:51,040 --> 00:03:58,820 What I derived was xA of two minus yA of one. 77 00:03:58,820 --> 00:04:01,410 Over yA of one. 78 00:04:01,410 --> 00:04:07,000 Minus yA of two, and I just want to, in some sense, finish 79 00:04:07,000 --> 00:04:10,170 up by making the obvious substitution, which is, of 80 00:04:10,170 --> 00:04:11,300 course these are mole fractions. 81 00:04:11,300 --> 00:04:15,660 So I can just write xA is one minus xB. 82 00:04:15,660 --> 00:04:20,540 And yA is one minus yB. 83 00:04:20,540 --> 00:04:22,990 And I just want to put this in the terms that are written 84 00:04:22,990 --> 00:04:25,200 here in the phase diagram. 85 00:04:25,200 --> 00:04:29,480 When I got to this result I pointed out that therefore you 86 00:04:29,480 --> 00:04:30,590 can use the lever rule. 87 00:04:30,590 --> 00:04:33,110 And I just want to make that a little bit more explicit. 88 00:04:33,110 --> 00:04:37,950 So using that relation, then we can see that ng of two over 89 00:04:37,950 --> 00:04:42,360 nl of two, total number of moles in the gas to the liquid 90 00:04:42,360 --> 00:04:48,280 is, now I'm going to substitute in to get yB of one 91 00:04:48,280 --> 00:04:57,860 minus xB of two over yB of two minus yB of one. 92 00:04:57,860 --> 00:05:02,300 So all I've done here is substitute in these 93 00:05:02,300 --> 00:05:04,200 expressions. 94 00:05:04,200 --> 00:05:07,500 And now I'm going to just switch both of these. 95 00:05:07,500 --> 00:05:10,750 And so I'm going to multiply both sides by negative one. 96 00:05:10,750 --> 00:05:19,340 So I can write xB of two minus yB of one over yB of 97 00:05:19,340 --> 00:05:23,980 one minus yB of two. 98 00:05:23,980 --> 00:05:27,390 So that's the result that I want. 99 00:05:27,390 --> 00:05:32,160 Because if I look at the two segments of this line, on 100 00:05:32,160 --> 00:05:35,930 either side of the pressure value I've reached, then what 101 00:05:35,930 --> 00:05:43,050 I see, of course, is that over here I've got xB of 102 00:05:43,050 --> 00:05:47,590 two minus yB of one. 103 00:05:47,590 --> 00:05:49,980 That's this. 104 00:05:49,980 --> 00:06:17,360 And this part is yB of one minus yB of two. 105 00:06:17,360 --> 00:06:19,450 That is, it's this part. 106 00:06:19,450 --> 00:06:21,290 And that's here. 107 00:06:21,290 --> 00:06:24,590 And so that's the point, is that the ratio of moles in the 108 00:06:24,590 --> 00:06:28,250 gas and liquid phases is just given by the two segments of 109 00:06:28,250 --> 00:06:29,420 this on either side. 110 00:06:29,420 --> 00:06:32,510 And so what this allows us to see very simply is alright, 111 00:06:32,510 --> 00:06:34,320 I'm going to raise the pressure at some point. 112 00:06:34,320 --> 00:06:37,340 And depending on how much material I feel like I need to 113 00:06:37,340 --> 00:06:40,530 collect in the new phase, I can always determine that 114 00:06:40,530 --> 00:06:45,290 simply by reading off the phase diagram what the ratio 115 00:06:45,290 --> 00:06:56,710 of these two segment lengths is. 116 00:06:56,710 --> 00:06:58,180 Now what I want to do. 117 00:06:58,180 --> 00:07:03,530 What what we've done mostly is just go through these phase 118 00:07:03,530 --> 00:07:05,630 diagrams, see how to read them. 119 00:07:05,630 --> 00:07:09,570 How to follow events on them as you change pressure in a 120 00:07:09,570 --> 00:07:10,610 diagram like this. 121 00:07:10,610 --> 00:07:13,110 Or change temperature in diagrams like I 122 00:07:13,110 --> 00:07:15,430 also showed last time. 123 00:07:15,430 --> 00:07:19,430 Now what I'd like to do is go a little bit further and just 124 00:07:19,430 --> 00:07:22,650 look at expressions for the chemical potential and the 125 00:07:22,650 --> 00:07:23,230 free energies. 126 00:07:23,230 --> 00:07:26,410 So that we can, a little more quantitatively, see 127 00:07:26,410 --> 00:07:29,260 what's going on. 128 00:07:29,260 --> 00:07:31,560 So let's just see what's happening. 129 00:07:31,560 --> 00:07:45,380 So let's look at the chemical potentials. 130 00:07:45,380 --> 00:07:51,900 In ideal liquid mixtures. 131 00:07:51,900 --> 00:08:00,730 So everything is derived from the fact that when we have any 132 00:08:00,730 --> 00:08:04,020 of the constituents in both phases, the chemical potential 133 00:08:04,020 --> 00:08:05,950 must be equal in both phases. 134 00:08:05,950 --> 00:08:06,950 Right? 135 00:08:06,950 --> 00:08:12,420 So, we can write for A, mu A of the liquid at some 136 00:08:12,420 --> 00:08:18,720 temperature and pressure must equal chemical potential of A 137 00:08:18,720 --> 00:08:22,010 for the gas. 138 00:08:22,010 --> 00:08:27,150 That's the partial pressure. 139 00:08:27,150 --> 00:08:31,315 So if we have an ideal gas, and certainly if we're going 140 00:08:31,315 --> 00:08:34,450 to assume an ideal liquid mixture, we can safely assume 141 00:08:34,450 --> 00:08:38,680 that it's an ideal gas above it. 142 00:08:38,680 --> 00:08:44,300 Then we can write mu A in the gas. 143 00:08:44,300 --> 00:08:46,960 Is just mu A naught. 144 00:08:46,960 --> 00:08:52,380 That's a function of the temperature plus RT 145 00:08:52,380 --> 00:08:56,800 log of pA over p0. 146 00:08:56,800 --> 00:08:59,430 Nothing new here, this is just our expression for the 147 00:08:59,430 --> 00:09:01,560 chemical potential in the gas, with reference 148 00:09:01,560 --> 00:09:04,080 to a standard potential. 149 00:09:04,080 --> 00:09:05,420 Usually one bar. 150 00:09:05,420 --> 00:09:07,190 At whatever the temperature is. 151 00:09:07,190 --> 00:09:10,350 And, of course, this is how it varies as the partial pressure 152 00:09:10,350 --> 00:09:14,340 of A in the gas phase varies. 153 00:09:14,340 --> 00:09:31,410 So then we can just write our expression for the liquid. 154 00:09:31,410 --> 00:09:38,220 At our temperature and pressure, it's given by mu 155 00:09:38,220 --> 00:09:42,370 naught in the gas phase. 156 00:09:42,370 --> 00:09:46,690 Plus RT log pA over p0. 157 00:09:46,690 --> 00:09:47,580 So it's the same thing. 158 00:09:47,580 --> 00:09:49,610 Because I'm just taking advantage of this equality. 159 00:09:49,610 --> 00:09:52,270 But of course, this is an obvious step, 160 00:09:52,270 --> 00:09:53,040 having written this. 161 00:09:53,040 --> 00:09:55,150 But is a super important step. 162 00:09:55,150 --> 00:09:57,980 That's what allows us to do this treatment in such a 163 00:09:57,980 --> 00:09:59,080 straightforward way. 164 00:09:59,080 --> 00:10:01,120 We know a lot about the chemical potential of 165 00:10:01,120 --> 00:10:02,230 something in the gas phase. 166 00:10:02,230 --> 00:10:04,800 Since the gas and liquid are in equilibrium, therefore we 167 00:10:04,800 --> 00:10:24,870 know the chemical potential in the liquid phase too. 168 00:10:24,870 --> 00:10:28,870 So we can rewrite this by recognizing that if we just go 169 00:10:28,870 --> 00:10:37,100 to the limit where we only have pure liquid A, so pure 170 00:10:37,100 --> 00:10:45,680 liquid A. Well, in that case, mu A naught or sorry, mu A in 171 00:10:45,680 --> 00:11:00,400 the liquid phase, mu A star is mu A naught in the gas phase. 172 00:11:00,400 --> 00:11:08,880 Plus RT log pA over p0. 173 00:11:08,880 --> 00:11:16,750 And so for the mixture, now all I'm going to do is just 174 00:11:16,750 --> 00:11:19,040 add and subtract terms. 175 00:11:19,040 --> 00:11:25,450 So we can write mu A in the liquid at T and p, right? 176 00:11:25,450 --> 00:11:28,560 So this is in the case of the limit of the pure liquid. 177 00:11:28,560 --> 00:11:33,030 Now we're going to the mixture. 178 00:11:33,030 --> 00:11:48,930 So it's just mu A naught plus RT log pA star over p0 minus 179 00:11:48,930 --> 00:11:54,230 RT log of pA star over p0. 180 00:11:59,090 --> 00:12:03,610 Wait a minute, lost a term. 181 00:12:03,610 --> 00:12:12,690 Plus RT log pA over p0. 182 00:12:12,690 --> 00:12:15,670 And all I want to do now is combine terms. 183 00:12:15,670 --> 00:12:22,740 To write mu A star liquid temperature and pressure. 184 00:12:22,740 --> 00:12:30,730 Plus RT log of pA over pA star. 185 00:12:30,730 --> 00:12:35,900 So this is a very convenient form for it. 186 00:12:35,900 --> 00:12:40,570 And then, of course, we have an expression for pA, right? 187 00:12:40,570 --> 00:12:49,440 From Raoult's law. pA is just the mole 188 00:12:49,440 --> 00:12:55,870 fraction xA times pA star. 189 00:12:55,870 --> 00:12:59,880 That's just true for the ideal mixture. 190 00:12:59,880 --> 00:13:08,320 So now, finally, we can write that mu A in the liquid at T 191 00:13:08,320 --> 00:13:16,220 and p is just given by mu A star. 192 00:13:16,220 --> 00:13:18,400 So that's for the pure liquid at that 193 00:13:18,400 --> 00:13:19,530 temperature and pressure. 194 00:13:19,530 --> 00:13:25,970 Plus RT log of the mole fraction of A. So that's a 195 00:13:25,970 --> 00:13:33,240 very simple expression for the chemical potential of A. And 196 00:13:33,240 --> 00:13:36,000 of course the analogous expression will hold for any 197 00:13:36,000 --> 00:13:37,120 of the constituents. 198 00:13:37,120 --> 00:13:43,700 In an ideal liquid mixture. 199 00:13:43,700 --> 00:13:48,850 By the way, it's convenient because it looks just like the 200 00:13:48,850 --> 00:13:51,000 chemical potential in a mixture of ideal gases. 201 00:13:51,000 --> 00:13:59,210 Except that we have liquid instead of gases, right? 202 00:13:59,210 --> 00:14:03,100 Now, of course, since this is a mole fraction, it's always 203 00:14:03,100 --> 00:14:05,180 between zero and one. 204 00:14:05,180 --> 00:14:09,870 That means this is always a negative number. 205 00:14:09,870 --> 00:14:16,090 So what that means is that the chemical potential in the 206 00:14:16,090 --> 00:14:21,190 solution is always lower than the chemical potential of the 207 00:14:21,190 --> 00:14:57,650 pure liquid. 208 00:14:57,650 --> 00:15:00,430 Very important result. 209 00:15:00,430 --> 00:15:03,910 So it has all sorts of implications that we'll see. 210 00:15:03,910 --> 00:15:05,860 One of them is osmotic pressure. 211 00:15:05,860 --> 00:15:11,050 It means that if I have a biological cell, or some 212 00:15:11,050 --> 00:15:14,240 container with a membrane through which one of the 213 00:15:14,240 --> 00:15:17,520 constituents might pass. 214 00:15:17,520 --> 00:15:20,230 So in other words, the let's say, component A, maybe it's 215 00:15:20,230 --> 00:15:21,930 just water. 216 00:15:21,930 --> 00:15:25,520 It can pass freely, let's say, from the outside to the inside 217 00:15:25,520 --> 00:15:29,040 of the cell, or from one side to another of a membrane. 218 00:15:29,040 --> 00:15:31,570 And on one side I've just got pure water. 219 00:15:31,570 --> 00:15:34,340 And on the other I've got saline solution or whatever is 220 00:15:34,340 --> 00:15:36,650 in the cells, right? 221 00:15:36,650 --> 00:15:40,130 What's the water going to do in that situation? 222 00:15:40,130 --> 00:15:42,710 What's going to happen? 223 00:15:42,710 --> 00:15:44,970 Anybody know? 224 00:15:44,970 --> 00:15:50,380 So I've got, I just take fresh cells and plunk them into pure 225 00:15:50,380 --> 00:15:52,330 water solution. 226 00:15:52,330 --> 00:15:56,510 What happens? 227 00:15:56,510 --> 00:15:56,730 Yeah. 228 00:15:56,730 --> 00:15:58,020 They burst. 229 00:15:58,020 --> 00:16:00,610 Water rushes in. 230 00:16:00,610 --> 00:16:03,720 Because the chemical potential is lower inside. 231 00:16:03,720 --> 00:16:07,260 It's always lower in solution than outside. 232 00:16:07,260 --> 00:16:09,490 So water rushes in, and the membrane expands. 233 00:16:09,490 --> 00:16:11,800 Now, if the membrane is strong enough, at some 234 00:16:11,800 --> 00:16:14,300 point it may not burst. 235 00:16:14,300 --> 00:16:17,560 And the pressure might start to go up. 236 00:16:17,560 --> 00:16:19,400 Let's say the membrane is strong enough to 237 00:16:19,400 --> 00:16:22,260 resist and not burst. 238 00:16:22,260 --> 00:16:26,190 Eventually, the water won't keep going in indefinitely. 239 00:16:26,190 --> 00:16:30,040 That's because it won't be at the same temperature and 240 00:16:30,040 --> 00:16:30,730 pressure any more. 241 00:16:30,730 --> 00:16:34,510 In particular, the pressure will have changed. 242 00:16:34,510 --> 00:16:36,990 So you can build up some high pressure, what's called 243 00:16:36,990 --> 00:16:38,620 osmotic pressure. 244 00:16:38,620 --> 00:16:42,540 Because of the fact that at the same pressure the chemical 245 00:16:42,540 --> 00:16:45,730 potential of the water's lower inside the cell or inside the 246 00:16:45,730 --> 00:16:48,780 enclosure with the membrane. 247 00:16:48,780 --> 00:16:50,140 So water will keep filling. 248 00:16:50,140 --> 00:16:53,170 And at some point the chemical potentials will equalize 249 00:16:53,170 --> 00:16:54,880 because of the change in pressure. 250 00:16:54,880 --> 00:17:00,440 And at that point there'll be an equilibrium established. 251 00:17:00,440 --> 00:17:05,990 We'll see that quantitatively a little bit later. 252 00:17:05,990 --> 00:17:06,630 OK. 253 00:17:06,630 --> 00:17:13,160 The other thing to notice is, this is familiar from the 254 00:17:13,160 --> 00:17:15,820 expression for a gas mixture. 255 00:17:15,820 --> 00:17:18,850 What drives the gas mixture? 256 00:17:18,850 --> 00:17:21,840 Remember back when we discussed mixing of gases and 257 00:17:21,840 --> 00:17:24,220 the fact that they would mix at all. 258 00:17:24,220 --> 00:17:25,450 What makes that happen? 259 00:17:25,450 --> 00:17:28,070 What's driving it? 260 00:17:28,070 --> 00:17:30,050 Yeah, it's entropy, right? 261 00:17:30,050 --> 00:17:31,580 And that's what's happening here too. 262 00:17:31,580 --> 00:17:35,030 Of course, in the case of the ideal liquid mixture, there's 263 00:17:35,030 --> 00:17:37,070 no energetic interaction. 264 00:17:37,070 --> 00:17:40,440 The molecules are non-interacting in this case. 265 00:17:40,440 --> 00:17:43,220 But of course, entropy is going to want them to mix. 266 00:17:43,220 --> 00:17:44,950 And that's what's resulting in the 267 00:17:44,950 --> 00:17:50,070 decrease in chemical potential. 268 00:17:50,070 --> 00:17:53,440 Let's see that a little more explicitly by just calculating 269 00:17:53,440 --> 00:17:56,610 out the free energy change of mixing. 270 00:17:56,610 --> 00:18:02,510 So, delta G of mixing. 271 00:18:02,510 --> 00:18:15,580 So we're going to start with two separated liquids. 272 00:18:15,580 --> 00:18:18,050 And then we'll remove the barrier. 273 00:18:18,050 --> 00:18:26,940 And we'll have the two mixed together. 274 00:18:26,940 --> 00:18:31,900 So of course, the free energy in either case is just the sum 275 00:18:31,900 --> 00:18:34,670 of the number of moles of each times the chemical 276 00:18:34,670 --> 00:18:35,710 potential of each. 277 00:18:35,710 --> 00:18:37,910 We have expressions for that. 278 00:18:37,910 --> 00:18:42,160 So starting G1 in this case. 279 00:18:42,160 --> 00:18:46,030 It's n of A, number of moles of A. Times 280 00:18:46,030 --> 00:18:49,930 the chemical potential. 281 00:18:49,930 --> 00:18:58,780 So it's xA mu A star in the liquid. 282 00:18:58,780 --> 00:19:08,830 Plus nB xB B mu B star of the liquid. 283 00:19:08,830 --> 00:19:12,300 Here, G of two. 284 00:19:12,300 --> 00:19:14,590 After we let them mix. 285 00:19:14,590 --> 00:19:21,270 Then it's n -- wait a minute. 286 00:19:21,270 --> 00:19:26,110 It's just n. 287 00:19:26,110 --> 00:19:29,220 I think I've got that wrong in the notes also. 288 00:19:29,220 --> 00:19:31,860 Right, it's just a number of moles times the chemical 289 00:19:31,860 --> 00:19:33,410 potential in each case. 290 00:19:33,410 --> 00:19:37,880 So it's the number of moles times xA. 291 00:19:37,880 --> 00:19:43,600 Times mu A in the mixture. 292 00:19:43,600 --> 00:19:52,660 Plus n xB mu B in the mixture. 293 00:19:52,660 --> 00:19:55,620 But we've just figured it out our expressions for mu A and 294 00:19:55,620 --> 00:19:58,930 mu B, our chemical potentials of each constituent in the 295 00:19:58,930 --> 00:20:02,040 ideal liquid mixture. 296 00:20:02,040 --> 00:20:11,690 So this is just n xA. 297 00:20:11,690 --> 00:20:13,940 And here's our expression. 298 00:20:13,940 --> 00:20:15,250 And of course these are going to cancel. 299 00:20:15,250 --> 00:20:16,780 Let's write it out. 300 00:20:16,780 --> 00:20:27,910 So it's mu A star plus RT log xA. 301 00:20:27,910 --> 00:20:37,820 And then, mu B star plus RT log xB. 302 00:20:37,820 --> 00:20:39,890 So our delta G of mixing is just the difference 303 00:20:39,890 --> 00:20:44,670 between these two. 304 00:20:44,670 --> 00:20:58,730 So it's n RT xA log xA plus xB log xB. 305 00:20:58,730 --> 00:21:03,570 Where have you seen that before? 306 00:21:03,570 --> 00:21:03,720 Yeah. 307 00:21:03,720 --> 00:21:07,710 The same thing that you had for the delta G of mixing for 308 00:21:07,710 --> 00:21:10,330 an ideal gas mixture. 309 00:21:10,330 --> 00:21:10,700 Why? 310 00:21:10,700 --> 00:21:13,890 Because we're not accounting for any interactions between 311 00:21:13,890 --> 00:21:16,640 the molecules in either of the phases. 312 00:21:16,640 --> 00:21:22,690 And if the molecules aren't interacting, it's all entropy. 313 00:21:22,690 --> 00:21:26,770 And the entropy term has the same form in either case. 314 00:21:26,770 --> 00:21:29,780 In microscopic terms, it's just measuring the fact that 315 00:21:29,780 --> 00:21:31,460 there's more disorder in the mixture 316 00:21:31,460 --> 00:21:32,860 than in the pure liquids. 317 00:21:32,860 --> 00:21:45,955 Just the same it was in the gas phase. 318 00:21:45,955 --> 00:21:49,710 OK. 319 00:21:49,710 --> 00:21:55,490 We can see this even more explicitly if we just recall 320 00:21:55,490 --> 00:22:00,980 that G is V dp minus S dT. 321 00:22:00,980 --> 00:22:02,710 So we can just explicitly calculate 322 00:22:02,710 --> 00:22:05,060 the entropy of mixing. 323 00:22:05,060 --> 00:22:13,110 Delta S of mixing is just the partial of delta G of mixing. 324 00:22:13,110 --> 00:22:17,360 Negative partial. 325 00:22:17,360 --> 00:22:19,130 With respect to temperature. 326 00:22:19,130 --> 00:22:20,700 At constant pressure. 327 00:22:20,700 --> 00:22:30,180 So it's just minus n R xA log xA plus xB log xB. 328 00:22:32,980 --> 00:22:36,930 So that's our entropy of mixing. 329 00:22:36,930 --> 00:22:41,470 We can calculate our enthalpy of mixing. 330 00:22:41,470 --> 00:22:44,550 Just delta G of mixing. 331 00:22:44,550 --> 00:22:48,380 Plus T delta S of mixing. 332 00:22:48,380 --> 00:22:50,630 But it's immediately apparent that these are just going to 333 00:22:50,630 --> 00:22:53,630 cancel when we multiply this by T. So there's 334 00:22:53,630 --> 00:22:55,970 no enthalpy of mixing. 335 00:22:55,970 --> 00:22:58,220 Just as we expect when there are no 336 00:22:58,220 --> 00:22:59,560 energetic terms involved. 337 00:22:59,560 --> 00:23:06,170 It's all entropy that's driving the mixture. 338 00:23:06,170 --> 00:23:08,320 One more detail, it's straightforward to see that 339 00:23:08,320 --> 00:23:14,120 there's no volume change. 340 00:23:14,120 --> 00:23:17,320 That is, if we take the derivative of delta G of 341 00:23:17,320 --> 00:23:22,210 mixing the partial derivative with respect to pressure, at 342 00:23:22,210 --> 00:23:24,840 constant temperature, of course, there's no explicit 343 00:23:24,840 --> 00:23:26,820 pressure dependence. 344 00:23:26,820 --> 00:23:27,870 This is zero. 345 00:23:27,870 --> 00:23:32,820 So again, in the ideal liquid mixture case, their molecules 346 00:23:32,820 --> 00:23:33,580 aren't interacting. 347 00:23:33,580 --> 00:23:36,460 So there's no reason, when I open that barrier, that the 348 00:23:36,460 --> 00:23:46,640 amount of volume they occupy altogether is going to change. 349 00:23:46,640 --> 00:24:00,130 Any questions, so far? 350 00:24:00,130 --> 00:24:01,110 OK. 351 00:24:01,110 --> 00:24:26,150 Then, let's move on to non-ideal solutions. 352 00:24:26,150 --> 00:24:30,190 By the way, just to return briefly to this topic of 353 00:24:30,190 --> 00:24:34,050 osmotic pressure, I just want to emphasize that result 354 00:24:34,050 --> 00:24:36,950 didn't need any kind of energy of mixing, either, right? 355 00:24:36,950 --> 00:24:39,850 Just from the entropy term you would burst 356 00:24:39,850 --> 00:24:41,060 the cell or do whatever. 357 00:24:41,060 --> 00:24:43,710 You end up with a larger pressure. 358 00:24:43,710 --> 00:24:55,300 So in other words, if you have a situation where, here is A, 359 00:24:55,300 --> 00:24:57,650 pure liquid. 360 00:24:57,650 --> 00:25:07,740 And here is A plus B liquid, so mu A star is greater than 361 00:25:07,740 --> 00:25:13,610 mu A. We know that in all cases at the same pressure, 362 00:25:13,610 --> 00:25:16,480 the chemical potential of the mixture is lower than the 363 00:25:16,480 --> 00:25:18,230 chemical potential of the pure liquid. 364 00:25:18,230 --> 00:25:19,600 What's going to happen? 365 00:25:19,600 --> 00:25:26,980 Well, A is going to, oh, sorry, A over here is 366 00:25:26,980 --> 00:25:28,240 going to rush in. 367 00:25:28,240 --> 00:25:30,970 It's going to get through this, if I've got a 368 00:25:30,970 --> 00:25:34,800 semi-permeable membrane, it's permeable to A but not to B. 369 00:25:34,800 --> 00:25:36,240 Common situation, of course. 370 00:25:36,240 --> 00:25:38,440 That's certainly the case for the cell membrane. 371 00:25:38,440 --> 00:25:42,080 Water may flow easily in through the membrane, but all 372 00:25:42,080 --> 00:25:45,590 of the stuff, all the salt and everything else that's inside 373 00:25:45,590 --> 00:25:48,320 the cell generally won't. 374 00:25:48,320 --> 00:25:52,400 So of course A then flows through this, what's called a 375 00:25:52,400 --> 00:26:01,980 semi-permeable membrane. 376 00:26:01,980 --> 00:26:09,450 And you'll wind up with either a burst membrane, or something 377 00:26:09,450 --> 00:26:10,390 that looks like this. 378 00:26:10,390 --> 00:26:18,170 Where now there is additional pressure here. 379 00:26:18,170 --> 00:26:21,270 And you'll have some different mole fraction than you started 380 00:26:21,270 --> 00:26:29,220 with before. 381 00:26:29,220 --> 00:26:54,650 Now let's go over to non-ideal solutions. 382 00:26:54,650 --> 00:26:58,760 So let's just think microscopically for a moment 383 00:26:58,760 --> 00:27:01,720 about how this is going to work. 384 00:27:01,720 --> 00:27:05,320 You know, normally, always, there are interactions between 385 00:27:05,320 --> 00:27:06,730 the molecules and the liquid. 386 00:27:06,730 --> 00:27:08,120 The liquid is a condensed phase. 387 00:27:08,120 --> 00:27:11,520 The molecules are in immediate proximity to each other. 388 00:27:11,520 --> 00:27:14,250 So it's very different from the gas phase, where it can be 389 00:27:14,250 --> 00:27:16,330 a pretty realistic approximation to say, well, 390 00:27:16,330 --> 00:27:19,750 the molecules are essentially non-interacting. 391 00:27:19,750 --> 00:27:22,120 The ideal gas law may turn out to be a very good 392 00:27:22,120 --> 00:27:24,080 approximation. 393 00:27:24,080 --> 00:27:26,930 In the liquid, really, it's never the case that you don't 394 00:27:26,930 --> 00:27:28,650 have interactions. 395 00:27:28,650 --> 00:27:43,770 So if we just sketch that. 396 00:27:43,770 --> 00:27:50,890 If we start with a bunch of molecules in A, they're 397 00:27:50,890 --> 00:27:53,160 interacting. 398 00:27:53,160 --> 00:27:58,780 So there's some interaction energy. 399 00:27:58,780 --> 00:28:01,630 I'll call it AA, it's the interaction between two 400 00:28:01,630 --> 00:28:09,630 molecules of A. In most cases it'll be less than zero. 401 00:28:09,630 --> 00:28:13,350 That is, there'd be a weak attraction. 402 00:28:13,350 --> 00:28:26,120 Same thing between molecules of B. 403 00:28:26,120 --> 00:28:29,620 So that's my situation in the separated liquids. 404 00:28:29,620 --> 00:28:32,020 I've got molecules in each container. 405 00:28:32,020 --> 00:28:32,850 They're interacting. 406 00:28:32,850 --> 00:28:38,530 There's some energy associated with that. 407 00:28:38,530 --> 00:28:50,490 So here's the pure separated liquids. 408 00:28:50,490 --> 00:28:53,850 And now, I'll open up the barrier and 409 00:28:53,850 --> 00:28:54,820 let the liquids mix. 410 00:28:54,820 --> 00:28:57,430 And now suddenly, of course, A and B are going to interact 411 00:28:57,430 --> 00:29:00,320 with each other as well as other molecules 412 00:29:00,320 --> 00:29:02,470 of their own kind. 413 00:29:02,470 --> 00:29:11,040 So now, suddenly, there's going to be some interaction 414 00:29:11,040 --> 00:29:12,440 energy. uAB. 415 00:29:31,390 --> 00:29:36,240 And very simplistically, we can envision that for some of 416 00:29:36,240 --> 00:29:38,470 the molecules, essentially, there would be 417 00:29:38,470 --> 00:29:40,230 exchange of this sort. 418 00:29:40,230 --> 00:29:44,390 So, one neighbor of this molecule and one neighbor of 419 00:29:44,390 --> 00:29:47,640 this molecule will wind up exchanged for some pairs of 420 00:29:47,640 --> 00:29:51,440 molecules with the unlike species. 421 00:29:51,440 --> 00:29:56,000 And essentially, likes interactions will be replaced 422 00:29:56,000 --> 00:30:00,780 by unlike interactions. 423 00:30:00,780 --> 00:30:04,680 So there's some now change in energy involved. 424 00:30:04,680 --> 00:30:08,250 There's a delta u. 425 00:30:08,250 --> 00:30:15,590 And as outlined in this simple picture, it's just 2 uAB minus 426 00:30:15,590 --> 00:30:18,230 uAA plus uBB. 427 00:30:23,530 --> 00:30:26,910 Real liquids are very complicated. 428 00:30:26,910 --> 00:30:31,140 For liquids with relatively simple non-directional 429 00:30:31,140 --> 00:30:34,000 interactions, things like organic liquids that might 430 00:30:34,000 --> 00:30:37,900 interact van der Waal's interactions. 431 00:30:37,900 --> 00:30:42,630 This might be a reasonable starting point. 432 00:30:42,630 --> 00:30:46,440 And in general, if we're going to deal with relatively small 433 00:30:46,440 --> 00:30:51,320 deviations from the ideal liquid mixture, the ideal 434 00:30:51,320 --> 00:30:56,250 solution, then we can start with a model like this. 435 00:30:56,250 --> 00:31:00,800 And this difference is what's going to determine how far the 436 00:31:00,800 --> 00:31:05,950 liquid mixture will deviate from the ideal solution case. 437 00:31:05,950 --> 00:31:08,380 Of course, we could always just write a term like this. 438 00:31:08,380 --> 00:31:13,920 It may or may not be easily expressed in this sort of way. 439 00:31:13,920 --> 00:31:16,020 So let's just think about what the possibilities are. 440 00:31:16,020 --> 00:31:18,010 Of course, simply, there are two. 441 00:31:18,010 --> 00:31:21,090 The sign could be positive or negative. 442 00:31:21,090 --> 00:31:25,270 And what that means is, you could have situations where 443 00:31:25,270 --> 00:31:28,680 it's positive because basically there's an energy of 444 00:31:28,680 --> 00:31:30,310 mixing now in these cases. 445 00:31:30,310 --> 00:31:35,320 If that's not zero, unlike the ideal case, now unlike before, 446 00:31:35,320 --> 00:31:38,560 where we just had entropy driving the mixture, now 447 00:31:38,560 --> 00:31:40,870 there's an energy of mixing. 448 00:31:40,870 --> 00:31:43,180 Could be positive or negative. 449 00:31:43,180 --> 00:31:47,260 If it's positive, that means energetically speaking, the 450 00:31:47,260 --> 00:31:49,210 mixture is unfavorable. 451 00:31:49,210 --> 00:31:54,080 In other words, the intermolecular interactions 452 00:31:54,080 --> 00:31:59,650 between like molecules are more favorable then the 453 00:31:59,650 --> 00:32:02,290 interactions between the unlike 454 00:32:02,290 --> 00:32:11,350 molecules, in that case. 455 00:32:11,350 --> 00:32:25,430 So, delta u is greater than zero. 456 00:32:25,430 --> 00:32:34,680 And our delta H of mixing is going to be close 457 00:32:34,680 --> 00:32:35,880 to delta u of mixing. 458 00:32:35,880 --> 00:32:38,640 That is, we're going to figure that delta pV is not going to 459 00:32:38,640 --> 00:32:43,070 be something large. 460 00:32:43,070 --> 00:32:52,020 And in that case, we can say delta G of mixing is 1/4 n. 461 00:32:52,020 --> 00:32:57,180 And that's only because there are 4 species involved here. 462 00:32:57,180 --> 00:32:58,380 Times delta u. 463 00:32:58,380 --> 00:33:00,120 In other words, I'm multiplying this 464 00:33:00,120 --> 00:33:02,360 molecular delta u. 465 00:33:02,360 --> 00:33:04,870 This is the change in energy for this 466 00:33:04,870 --> 00:33:07,050 collection of four molecules. 467 00:33:07,050 --> 00:33:11,760 Or of four molecular pairs. 468 00:33:11,760 --> 00:33:15,690 Plus the other stuff that we've seen for the ideal case. 469 00:33:15,690 --> 00:33:20,270 That is, the entropy term, of course, is still there. xA log 470 00:33:20,270 --> 00:33:29,500 xA plus xB log xB. 471 00:33:29,500 --> 00:33:33,820 And because we're treating the case where this is positive, 472 00:33:33,820 --> 00:33:47,730 that means this is bigger than in the ideal solution case. 473 00:33:47,730 --> 00:33:50,270 So of course there are lots of examples of each of these. 474 00:33:50,270 --> 00:33:53,570 In fact, this is the more common deviation. 475 00:33:53,570 --> 00:33:55,630 So, and lots of examples. 476 00:33:55,630 --> 00:33:59,820 An easy one is acetone. 477 00:33:59,820 --> 00:34:02,960 And carbon disulfide. 478 00:34:02,960 --> 00:34:09,920 So, if we look at that, and just see what the phase 479 00:34:09,920 --> 00:34:38,940 diagram is going to look like, it's the following. 480 00:34:38,940 --> 00:34:43,670 So let's start with the ideal case. 481 00:34:43,670 --> 00:34:47,090 So I'm going to make B the more volatile component. 482 00:34:47,090 --> 00:34:49,560 That is, B is going to be CS2. 483 00:34:49,560 --> 00:35:00,110 So here's p star CS2, which I'll call p star B. And 484 00:35:00,110 --> 00:35:11,470 somewhere down here will be p star for A, that is, p star 485 00:35:11,470 --> 00:35:15,400 for acetone. 486 00:35:15,400 --> 00:35:18,100 And this is going to be the mole fraction of CS2. 487 00:35:18,100 --> 00:35:24,870 Or xB. 488 00:35:24,870 --> 00:35:29,660 So let's start with the ideal case. 489 00:35:29,660 --> 00:35:32,760 Not steep enough. 490 00:35:32,760 --> 00:35:33,680 Pretty good. 491 00:35:33,680 --> 00:35:45,630 OK. 492 00:35:45,630 --> 00:35:47,110 And now let's look at what's going to 493 00:35:47,110 --> 00:35:49,510 happen in the real mixture. 494 00:35:49,510 --> 00:35:54,140 Well, because we've got a positive deviation, what's 495 00:35:54,140 --> 00:35:58,790 going to happen is the mixing is unfavorable. 496 00:35:58,790 --> 00:36:00,630 The molecules aren't that happy any more 497 00:36:00,630 --> 00:36:02,960 about being in solution. 498 00:36:02,960 --> 00:36:06,250 So what that means is, relative to the ideal 499 00:36:06,250 --> 00:36:09,520 solution, they'd rather go up into the gas phase. 500 00:36:09,520 --> 00:36:12,610 They don't like being in the mixture. 501 00:36:12,610 --> 00:36:16,900 Compared to what they would feel in the ideal solution, 502 00:36:16,900 --> 00:36:17,930 where there are no interactions. 503 00:36:17,930 --> 00:36:20,310 Because the interactions between unlike molecules are 504 00:36:20,310 --> 00:36:22,850 unfavorable. 505 00:36:22,850 --> 00:36:27,540 So what's going to happen, then, is 506 00:36:27,540 --> 00:36:33,510 the pressure is higher. 507 00:36:33,510 --> 00:36:36,550 You have deviations from the ideal case, 508 00:36:36,550 --> 00:36:38,300 in the total pressure. 509 00:36:38,300 --> 00:36:53,790 And in each of the individual partial pressures. 510 00:36:53,790 --> 00:36:58,400 That's the total pressure. pA plus pB. 511 00:36:58,400 --> 00:37:03,000 In the ideal case it's this straight line. 512 00:37:03,000 --> 00:37:06,340 In the non-ideal case, though, it's bigger than 513 00:37:06,340 --> 00:37:07,900 the straight line. 514 00:37:07,900 --> 00:37:10,120 Because more stuff wants to get up into the gas phase. 515 00:37:10,120 --> 00:37:12,930 And it's the same for each individual component. 516 00:37:12,930 --> 00:37:18,720 So here's pA. 517 00:37:21,980 --> 00:37:23,230 And there's pB. 518 00:37:27,580 --> 00:37:32,700 Any questions? 519 00:37:32,700 --> 00:37:36,440 OK, so it's obvious just from looking at this that each of 520 00:37:36,440 --> 00:37:38,870 the partial pressures is bigger than it is 521 00:37:38,870 --> 00:37:40,300 in the ideal case. 522 00:37:40,300 --> 00:37:43,490 And so, so is the total pressure. 523 00:37:43,490 --> 00:37:46,910 So in other words, if I say what's the partial pressure of 524 00:37:46,910 --> 00:37:50,520 CS2, well, since it's not the simple ideal case, actually I 525 00:37:50,520 --> 00:37:51,990 don't know what it is in general. 526 00:37:51,990 --> 00:37:55,260 It's not easy to calculate a priori what it's going to do 527 00:37:55,260 --> 00:37:58,940 all the way across the phase diagram for any mole fraction. 528 00:37:58,940 --> 00:38:01,510 What I do know, though, is it's bigger than it would be 529 00:38:01,510 --> 00:38:03,950 for the ideal solution case. 530 00:38:03,950 --> 00:38:11,530 In other words, it's bigger than x CS2 times p star CS2, 531 00:38:11,530 --> 00:38:15,580 the pressure over the pure liquid CS2. 532 00:38:15,580 --> 00:38:19,650 Same for acetone. 533 00:38:19,650 --> 00:38:29,280 It's bigger than x acetone p star acetone. 534 00:38:29,280 --> 00:38:31,090 And since both partial pressures are bigger, the 535 00:38:31,090 --> 00:38:32,260 total pressure has to be bigger. 536 00:38:32,260 --> 00:38:34,770 And of course, that's something you can read right 537 00:38:34,770 --> 00:38:36,420 off the diagram as well. 538 00:38:36,420 --> 00:38:49,390 In other words, the total pressure is bigger than in the 539 00:38:49,390 --> 00:38:55,710 ideal solution case. 540 00:38:55,710 --> 00:39:00,320 Any questions? 541 00:39:00,320 --> 00:39:02,460 OK. 542 00:39:02,460 --> 00:39:05,115 So the negative case, of course, 543 00:39:05,115 --> 00:39:07,390 is exactly the opposite. 544 00:39:07,390 --> 00:39:17,440 So I'll just go through it quickly. 545 00:39:17,440 --> 00:39:23,240 So, remember positive deviations came, let's put 546 00:39:23,240 --> 00:39:27,090 this back up there. 547 00:39:27,090 --> 00:39:31,690 That came from the case where the unlike intermolecular 548 00:39:31,690 --> 00:39:34,180 interactions are not as favorable. 549 00:39:34,180 --> 00:39:38,710 There's less attraction, more repulsion than in this case. 550 00:39:38,710 --> 00:39:41,250 Of course, there are plenty of cases of the opposite sort. 551 00:39:41,250 --> 00:39:44,270 Where the unlike intermolecular interactions 552 00:39:44,270 --> 00:39:45,290 are attractive. 553 00:39:45,290 --> 00:39:50,340 More than the interactions between like molecules. 554 00:39:50,340 --> 00:39:54,570 And in that case, you have the opposite result. 555 00:39:54,570 --> 00:40:05,620 So you can have negative deviations. 556 00:40:05,620 --> 00:40:09,920 In other words, delta u is less than zero. 557 00:40:09,920 --> 00:40:10,970 Lots of examples. 558 00:40:10,970 --> 00:40:13,500 I've got one in the notes, it's just acetone and 559 00:40:13,500 --> 00:40:14,920 chloroform. 560 00:40:14,920 --> 00:40:22,020 So if we get rid of the carbon disulfide and put CHCl3 in 561 00:40:22,020 --> 00:40:25,020 there, that's more favorable. 562 00:40:25,020 --> 00:40:27,030 Because there's weak hydrogen bonding 563 00:40:27,030 --> 00:40:29,140 between the unlike species. 564 00:40:29,140 --> 00:40:31,450 And not between the like species. 565 00:40:31,450 --> 00:40:39,460 So what happens is, you have acetone, it's this. 566 00:40:39,460 --> 00:40:47,500 And you have chloroform. 567 00:40:47,500 --> 00:40:51,340 And there's weak hydrogen bonding between them. 568 00:40:51,340 --> 00:40:52,690 There's an attraction there. 569 00:40:52,690 --> 00:40:56,820 That doesn't exist between either 570 00:40:56,820 --> 00:40:59,610 case of the like molecules. 571 00:40:59,610 --> 00:41:03,570 So now there's stronger attraction between them than 572 00:41:03,570 --> 00:41:05,870 there is between molecules in either of 573 00:41:05,870 --> 00:41:08,840 the pure liquid cases. 574 00:41:08,840 --> 00:41:11,240 So, of course, you know that the opposite 575 00:41:11,240 --> 00:41:12,600 result is going to happen. 576 00:41:12,600 --> 00:41:18,010 In this case, now you mix them together, relative to the 577 00:41:18,010 --> 00:41:21,555 ideal solution where there are no interactions, now they want 578 00:41:21,555 --> 00:41:22,900 to be together. 579 00:41:22,900 --> 00:41:27,470 They're clinging to each other through these hydrogen bonds. 580 00:41:27,470 --> 00:41:30,890 What's that mean for the pressure up in the gas phase, 581 00:41:30,890 --> 00:41:32,950 in the solution? 582 00:41:32,950 --> 00:41:35,440 They don't want to go there any more. 583 00:41:35,440 --> 00:41:38,880 Stuff in the gas phase now wants to condense down into 584 00:41:38,880 --> 00:41:39,340 the liquid. 585 00:41:39,340 --> 00:41:43,600 Because there they have favorable interactions. 586 00:41:43,600 --> 00:41:45,650 So that's, of course, what'll happen. 587 00:41:45,650 --> 00:41:50,280 And so then you just see the opposite sort of deviation 588 00:41:50,280 --> 00:41:56,120 from what I just illustrated before. 589 00:41:56,120 --> 00:42:08,320 So then if we look at p star CHCl3, where 590 00:42:08,320 --> 00:42:13,820 are these lines going? 591 00:42:13,820 --> 00:42:18,760 Well, may have been more artistic. 592 00:42:18,760 --> 00:42:27,710 But I'm afraid I'm going to have to go with reality. 593 00:42:27,710 --> 00:42:28,670 p star for acetone. 594 00:42:28,670 --> 00:42:35,330 There's chloroform. 595 00:42:35,330 --> 00:42:38,360 And now our deviation is going to be in 596 00:42:38,360 --> 00:42:39,610 the opposite direction. 597 00:42:39,610 --> 00:42:41,880 So all the, both of the partial pressures and the 598 00:42:41,880 --> 00:42:55,840 total pressure are going to go lower than indicated here. 599 00:42:55,840 --> 00:43:03,180 So p total is now less than, for the real case, is less 600 00:43:03,180 --> 00:43:11,460 than p total for the ideal case. 601 00:43:11,460 --> 00:43:14,090 In the situation where you have attractive interactions 602 00:43:14,090 --> 00:43:17,110 between the unlike molecules. 603 00:43:17,110 --> 00:43:20,860 Any questions? 604 00:43:20,860 --> 00:43:22,650 OK. 605 00:43:22,650 --> 00:43:27,780 Now, in general, this is complicated. 606 00:43:27,780 --> 00:43:30,190 To describe quantitatively throughout the 607 00:43:30,190 --> 00:43:33,100 entire phase diagram. 608 00:43:33,100 --> 00:43:36,430 The interactions are complicated. 609 00:43:36,430 --> 00:43:38,480 Lots of times, molecules interact with 610 00:43:38,480 --> 00:43:40,750 more than one neighbor. 611 00:43:40,750 --> 00:43:45,770 So in general, we don't have a simple analytical expression 612 00:43:45,770 --> 00:43:49,950 for what the pressure is going to do as a function of mole 613 00:43:49,950 --> 00:43:52,230 fraction all the way from zero to one. 614 00:43:52,230 --> 00:43:54,380 Of course, we have it in very simple form 615 00:43:54,380 --> 00:43:55,960 for the ideal solution. 616 00:43:55,960 --> 00:43:58,940 Because everything's linear. 617 00:43:58,940 --> 00:44:01,700 So it's very simple. 618 00:44:01,700 --> 00:44:03,510 These are replaced by equalities and we know 619 00:44:03,510 --> 00:44:04,060 everything. 620 00:44:04,060 --> 00:44:05,550 Now we don't. 621 00:44:05,550 --> 00:44:08,520 But we can at least make some progress by looking at the 622 00:44:08,520 --> 00:44:10,080 limiting cases. 623 00:44:10,080 --> 00:44:13,730 Where you have just very dilute solutions. 624 00:44:13,730 --> 00:44:17,160 So in that case, you can anticipate 625 00:44:17,160 --> 00:44:18,030 what's going to happen. 626 00:44:18,030 --> 00:44:24,730 So let's say we mix A and B. And one of them is in very 627 00:44:24,730 --> 00:44:26,740 dilute concentration. 628 00:44:26,740 --> 00:44:32,010 For the other one, for B, most molecules of B are just going 629 00:44:32,010 --> 00:44:33,200 to see themselves. 630 00:44:33,200 --> 00:44:34,980 They hardly know that you've mixed any 631 00:44:34,980 --> 00:44:37,840 molecules of A in there. 632 00:44:37,840 --> 00:44:42,570 For them, it's pretty much an ideal solution. 633 00:44:42,570 --> 00:44:44,090 They think they're in an ideal solution. 634 00:44:44,090 --> 00:44:47,860 Which is to say there's an entropy change in mixing. 635 00:44:47,860 --> 00:44:51,400 But hardly any of them experience interactions with 636 00:44:51,400 --> 00:44:54,830 the unlike molecules. 637 00:44:54,830 --> 00:44:58,850 For the dilute molecules, there is a change. 638 00:44:58,850 --> 00:45:01,690 You can't use Raoult's law any more. 639 00:45:01,690 --> 00:45:06,470 But at least we can look of the slope of the curve and 640 00:45:06,470 --> 00:45:09,580 say, OK, at least in the dilute solution case for a 641 00:45:09,580 --> 00:45:11,620 little while it's going to be linear. 642 00:45:11,620 --> 00:45:13,930 We can understand it. 643 00:45:13,930 --> 00:45:16,950 We can tabulate it. 644 00:45:16,950 --> 00:45:41,920 So let's look at how that works. 645 00:45:41,920 --> 00:45:43,310 Alright. 646 00:45:43,310 --> 00:45:46,300 That limit is something that has a name, that's called the 647 00:45:46,300 --> 00:45:48,860 ideal dilute solution. 648 00:45:48,860 --> 00:45:50,810 Not to be confused with the ideal solution. 649 00:45:50,810 --> 00:45:53,840 Where both constituents follow Raoult's law, 650 00:45:53,840 --> 00:45:55,490 as we've seen before. 651 00:45:55,490 --> 00:46:07,010 So ideal dilute solution. 652 00:46:07,010 --> 00:46:09,440 We're going to just look at these limits. 653 00:46:09,440 --> 00:46:17,360 So, if we start with the ideal case, so let's imagine we're 654 00:46:17,360 --> 00:46:22,510 going to look at carbon disulfide again. 655 00:46:22,510 --> 00:46:32,860 Here's p star CS2, which is our pB star. 656 00:46:32,860 --> 00:46:36,870 And this is x CS2, or xB. 657 00:46:40,020 --> 00:46:49,340 OK, well, what happens if we've got 658 00:46:49,340 --> 00:46:53,980 almost pure carbon disulfide. 659 00:46:53,980 --> 00:46:58,520 Well, what's going to happen is in this limit, like I was 660 00:46:58,520 --> 00:47:02,910 mentioning, up here the CS2 pretty much sees other CS2. 661 00:47:02,910 --> 00:47:06,770 It thinks it's in an ideal solution. 662 00:47:06,770 --> 00:47:11,130 It's going to look like that. 663 00:47:11,130 --> 00:47:15,920 Now, let's go to the other limit. 664 00:47:15,920 --> 00:47:30,320 So this is the limit x CS2 equals xB approaches one. 665 00:47:30,320 --> 00:47:33,250 Raoult's law. 666 00:47:33,250 --> 00:47:38,070 Two, the other limit. x CS2 approaches zero. 667 00:47:38,070 --> 00:47:39,920 We're down here. 668 00:47:39,920 --> 00:47:43,270 So it's definitely not going to be the same. 669 00:47:43,270 --> 00:47:47,300 Because now, every molecule, every CS2 molecule, is 670 00:47:47,300 --> 00:47:50,890 completely surrounded by acetone. 671 00:47:50,890 --> 00:47:54,250 So they all feel the energy of mixing. 672 00:47:54,250 --> 00:47:55,620 Because there are hardly any of them there. 673 00:47:55,620 --> 00:47:58,520 So every one of them is, it's very rare for one to find a 674 00:47:58,520 --> 00:48:00,870 neighbor of its own kind. 675 00:48:00,870 --> 00:48:03,820 And this is a case where there's a positive deviation 676 00:48:03,820 --> 00:48:04,550 from ideality. 677 00:48:04,550 --> 00:48:07,730 That is, the interactions are less favorable 678 00:48:07,730 --> 00:48:09,940 between unlike species. 679 00:48:09,940 --> 00:48:13,120 So, it's up here. 680 00:48:13,120 --> 00:48:14,990 Doesn't like that. 681 00:48:14,990 --> 00:48:18,190 It says, get me out here, I'd much rather be in the gas 682 00:48:18,190 --> 00:48:22,970 phase, compared to the ideal case, then surrounded by all 683 00:48:22,970 --> 00:48:26,470 these acetone molecules, right? 684 00:48:26,470 --> 00:48:27,520 So that's what'll happen. 685 00:48:27,520 --> 00:48:35,050 And now it's up to the artist. 686 00:48:35,050 --> 00:48:39,080 But in some way there's going to be a set 687 00:48:39,080 --> 00:48:41,610 of values in between. 688 00:48:41,610 --> 00:48:44,910 And the ideal dilute solution by itself isn't going to tell 689 00:48:44,910 --> 00:48:45,850 us about that. 690 00:48:45,850 --> 00:48:48,970 It's telling us about the two limits. 691 00:48:48,970 --> 00:48:51,100 Now, if we follow this, and I can see I'm 692 00:48:51,100 --> 00:48:52,650 going to be in trouble. 693 00:48:52,650 --> 00:48:58,200 But I can do a little creative bending of the line here. 694 00:48:58,200 --> 00:49:01,350 If I follow this slope up to here, 695 00:49:01,350 --> 00:49:03,660 it'll intercept somewhere. 696 00:49:03,660 --> 00:49:04,650 That has a name. 697 00:49:04,650 --> 00:49:07,620 It's called the Henry's law constant. 698 00:49:07,620 --> 00:49:10,020 I can write it as K CS2, or KB. 699 00:49:14,540 --> 00:49:18,910 So the point is, that at least for some range of 700 00:49:18,910 --> 00:49:21,380 concentrations, the variation is linear. 701 00:49:21,380 --> 00:49:25,060 And what that means is, if I know this Henry's law 702 00:49:25,060 --> 00:49:32,160 constant, for CS2 mixed with acetone specifically, it would 703 00:49:32,160 --> 00:49:32,690 be different. 704 00:49:32,690 --> 00:49:34,050 If it's mixed with something else, right? 705 00:49:34,050 --> 00:49:36,750 Other kinds of interactions with different energies will 706 00:49:36,750 --> 00:49:38,700 have a different slope. 707 00:49:38,700 --> 00:49:41,190 But if I know that number, it's still useful. 708 00:49:41,190 --> 00:49:47,260 Because at least for that pair of constituents, I know the 709 00:49:47,260 --> 00:49:52,450 dependents for some range of concentrations. 710 00:49:52,450 --> 00:50:02,000 So this is Henry's law. p CS -- 711 00:50:02,000 --> 00:50:15,720 Oops, CS2 is x CS2 times K CS2. 712 00:50:15,720 --> 00:50:27,690 Not quite as convenient as Raoult's law. 713 00:50:27,690 --> 00:50:30,010 That number, I don't need to know what the other 714 00:50:30,010 --> 00:50:31,660 constituent is. 715 00:50:31,660 --> 00:50:33,010 It's just a property of CS2. 716 00:50:33,010 --> 00:50:36,780 It depends on the temperature, but not the other member of 717 00:50:36,780 --> 00:50:37,510 the mixture. 718 00:50:37,510 --> 00:50:42,300 This one does, but still there's at least a substantial 719 00:50:42,300 --> 00:50:43,890 number of these things tabulated. 720 00:50:43,890 --> 00:50:46,650 Especially for important mixtures. 721 00:50:46,650 --> 00:50:49,540 So it becomes useful for dilute solutions. 722 00:50:49,540 --> 00:50:51,150 And there's an awful lot of chemistry. 723 00:50:51,150 --> 00:50:53,330 There's an awful lot of biology that happens in 724 00:50:53,330 --> 00:50:56,190 relatively dilute solutions. 725 00:50:56,190 --> 00:51:02,390 OK. 726 00:51:02,390 --> 00:51:04,060 I'll leave it at this point. 727 00:51:04,060 --> 00:51:06,980 Next time we'll look at the total phase diagram, just like 728 00:51:06,980 --> 00:51:10,330 we did last time looking at both the liquid and gas 729 00:51:10,330 --> 00:51:13,450 coexistence curves, and seeing what the behavior is as a 730 00:51:13,450 --> 00:51:15,120 function of mole fractions. 731 00:51:15,120 --> 00:51:17,580 And we'll talk about distillation and so forth in 732 00:51:17,580 --> 00:51:19,840 non-ideal liquids.