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 --> 00:00:06,840 Your support will help MIT OpenCourseWare continue to 4 00:00:06,840 --> 00:00:10,520 offer high quality educational resources for free. 5 00:00:10,520 --> 00:00:13,390 To make a donation or view additional materials from 6 00:00:13,390 --> 00:00:17,580 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,580 --> 00:00:20,280 ocw.mit.edu. 8 00:00:20,280 --> 00:00:23,790 PROFESSOR: So let's start with some of the things we 9 00:00:23,790 --> 00:00:28,850 learned last time. 10 00:00:28,850 --> 00:00:49,280 So there are two things that were important. 11 00:00:49,280 --> 00:00:57,710 We learned that the chemical potential for a species is the 12 00:00:57,710 --> 00:01:00,140 Gibbs free energy for that species divided by the number 13 00:01:00,140 --> 00:01:02,800 of moles, this is Gibbs free energy per mole. 14 00:01:02,800 --> 00:01:06,050 We learned that the pressure dependence of the Gibbs free 15 00:01:06,050 --> 00:01:10,270 energy gives you the pressure dependence for 16 00:01:10,270 --> 00:01:15,070 the chemical potential. 17 00:01:15,070 --> 00:01:18,960 That it's equal to the chemical potential at one bar 18 00:01:18,960 --> 00:01:24,700 for an ideal gas plus RT log p. 19 00:01:24,700 --> 00:01:33,270 We also learned that a species will want to go to minimize 20 00:01:33,270 --> 00:01:37,280 its chemical potential, and we saw that for the cell bursting 21 00:01:37,280 --> 00:01:39,610 in salt, in distilled water. 22 00:01:39,610 --> 00:01:44,470 Or an ice cube melting at a temperature 23 00:01:44,470 --> 00:01:46,700 greater than 0 degrees. 24 00:01:46,700 --> 00:01:54,200 And what we want to prove right now is that if I take a 25 00:01:54,200 --> 00:02:00,430 species A, in a mixture, some temperature T, some pressure 26 00:02:00,430 --> 00:02:05,190 p, and I compare its chemical potential to the same species, 27 00:02:05,190 --> 00:02:09,760 A, when it's pure, the same temperature, the same 28 00:02:09,760 --> 00:02:15,590 pressure, what I want to argue is that the chemical potential 29 00:02:15,590 --> 00:02:17,830 in the mixture is always less than the chemical potential 30 00:02:17,830 --> 00:02:18,650 when it's pure. 31 00:02:18,650 --> 00:02:22,230 The same conditions of pressure and temperature. 32 00:02:22,230 --> 00:02:27,380 And, so we can do a little thought experiment. 33 00:02:27,380 --> 00:02:28,540 Let's do a little thought experiment. 34 00:02:28,540 --> 00:02:32,140 Let's make a box, and in our box we're 35 00:02:32,140 --> 00:02:40,410 going to have a partition. 36 00:02:40,410 --> 00:02:46,540 And a flexible membrane here. 37 00:02:46,540 --> 00:02:49,350 And on one side of the partition we're going to have 38 00:02:49,350 --> 00:02:54,760 a gas, B. A gas, A, on this side here. 39 00:02:54,760 --> 00:02:56,730 And gas A on this side here. 40 00:02:56,730 --> 00:03:03,630 So let me just red chalk for A, And I don't think I have 41 00:03:03,630 --> 00:03:04,670 any other colors listed. 42 00:03:04,670 --> 00:03:08,150 Yellow chalk for B here. 43 00:03:08,150 --> 00:03:09,850 And everything's one bar. 44 00:03:09,850 --> 00:03:10,700 Everything's one bar. 45 00:03:10,700 --> 00:03:13,520 So one bar B here. 46 00:03:13,520 --> 00:03:15,160 One bar A here, one bar A here . 47 00:03:15,160 --> 00:03:25,980 And this membrane here only lets A through. 48 00:03:25,980 --> 00:03:29,240 And this membrane is deformable, but it's elastic. 49 00:03:29,240 --> 00:03:31,440 You can't deform it forever. 50 00:03:31,440 --> 00:03:32,920 It has some strength to it, right? 51 00:03:32,920 --> 00:03:34,360 So if you push on, it'll push back. 52 00:03:34,360 --> 00:03:37,350 There'll be pressure associated with that. 53 00:03:37,350 --> 00:03:41,180 So the next thing I do, then, in my experiment, shouldn't 54 00:03:41,180 --> 00:03:43,010 have done it here. 55 00:03:43,010 --> 00:03:45,660 Next thing I do in my experiment is to break this 56 00:03:45,660 --> 00:03:46,940 partition here. 57 00:03:46,940 --> 00:03:51,920 I'm going to break the partition. 58 00:03:51,920 --> 00:03:55,550 And this will cause A and B to mix. 59 00:03:55,550 --> 00:04:07,760 So now in my box I have my partition, my membrane here. 60 00:04:07,760 --> 00:04:13,910 I've got A at one bar here, total pressure of one bar. 61 00:04:13,910 --> 00:04:19,970 And on the other side I have A plus B, with a total pressure 62 00:04:19,970 --> 00:04:28,450 of one bar. 63 00:04:28,450 --> 00:04:30,170 That's my initial point now. 64 00:04:30,170 --> 00:04:31,840 What's going to happen? 65 00:04:31,840 --> 00:04:36,860 What's going to happen is that molecules of A here are going 66 00:04:36,860 --> 00:04:39,950 to want to go through the membrane to go 67 00:04:39,950 --> 00:04:40,910 in this area here. 68 00:04:40,910 --> 00:04:42,230 And there's two ways to look at it. 69 00:04:42,230 --> 00:04:44,090 You can look at it the thermodynamic way, which is 70 00:04:44,090 --> 00:04:46,370 the way that we're going to want to be looking at it. 71 00:04:46,370 --> 00:04:49,460 Which is that, from what we're going to prove, is that the 72 00:04:49,460 --> 00:04:52,610 chemical potential in the mixture is always less than 73 00:04:52,610 --> 00:04:53,900 for the pure substance. 74 00:04:53,900 --> 00:04:55,600 Here we have a mixture. 75 00:04:55,600 --> 00:04:56,280 One bar. 76 00:04:56,280 --> 00:04:59,290 Here we have the pure substance at one bar. 77 00:04:59,290 --> 00:05:01,350 So these molecules are going to look around and say hey, 78 00:05:01,350 --> 00:05:03,680 you know, I'm much happier here. 79 00:05:03,680 --> 00:05:06,570 And they're all going to want to go in this area here. 80 00:05:06,570 --> 00:05:10,000 As a result, the volume of this area, if you want to keep 81 00:05:10,000 --> 00:05:14,310 the same pressure on both sides, you're going to deform 82 00:05:14,310 --> 00:05:16,440 the membrane. 83 00:05:16,440 --> 00:05:18,530 The membrane's going to get deformed. 84 00:05:18,530 --> 00:05:20,590 It's going to bloat on that side here. 85 00:05:20,590 --> 00:05:21,750 It's going to cause an increase in 86 00:05:21,750 --> 00:05:24,280 pressure on that side. 87 00:05:24,280 --> 00:05:28,210 And the increase in pressure from the membrane sort of 88 00:05:28,210 --> 00:05:31,600 deforming and pushing back, is going to increase until the 89 00:05:31,600 --> 00:05:36,160 partial pressure of A, here, equals the pressure of A here. 90 00:05:36,160 --> 00:05:38,980 At which point the flow of A from either side is going to 91 00:05:38,980 --> 00:05:41,550 be the same and you're going to be in equilibrium. 92 00:05:41,550 --> 00:05:45,180 And at that point the chemical potentials of both sides are 93 00:05:45,180 --> 00:05:49,460 going to be the same. 94 00:05:49,460 --> 00:05:52,700 So I sort of gave you the other way of looking at it. 95 00:05:52,700 --> 00:05:54,580 Which is just purely in terms of pressures. 96 00:05:54,580 --> 00:05:57,650 At equilibrium, the partial pressure of A here has to be 97 00:05:57,650 --> 00:05:59,630 the same as the pressure on this side here. 98 00:05:59,630 --> 00:06:03,000 So that the flow of A on either side of that membrane, 99 00:06:03,000 --> 00:06:05,580 going from right to left or left to right is the same. 100 00:06:05,580 --> 00:06:08,960 And that's going to cause this mixing to happen. 101 00:06:08,960 --> 00:06:11,940 If you look at it from partial pressure perspective. 102 00:06:11,940 --> 00:06:15,610 But really, it's a chemical potential idea. 103 00:06:15,610 --> 00:06:17,890 Now, the chemical potential, as we saw, was 104 00:06:17,890 --> 00:06:19,280 the Gibbs free energy. 105 00:06:19,280 --> 00:06:26,990 And the Gibbs free energy, you can write it as H minus TS. 106 00:06:26,990 --> 00:06:32,410 So basically, in this process that I described, the 107 00:06:32,410 --> 00:06:33,730 enthalpy's not doing anything. 108 00:06:33,730 --> 00:06:34,730 These are ideal gases. 109 00:06:34,730 --> 00:06:37,370 They're not interacting with each other. 110 00:06:37,370 --> 00:06:40,980 The only thing that's changing, that's driving the 111 00:06:40,980 --> 00:06:43,650 chemical potential, which is basically Gibbs free energy 112 00:06:43,650 --> 00:06:46,310 per mole, the only thing that's driving the chemical 113 00:06:46,310 --> 00:06:49,850 potential to be lower on this side here, is 114 00:06:49,850 --> 00:06:51,440 that entropy term. 115 00:06:51,440 --> 00:06:53,690 It's the entropy of mixing. 116 00:06:53,690 --> 00:06:56,790 So entropy of mixing is really super important. 117 00:06:56,790 --> 00:07:01,650 When we're talking about systems where you have 118 00:07:01,650 --> 00:07:03,910 multiple components. 119 00:07:03,910 --> 00:07:05,470 That's going to drive a lot of things. 120 00:07:05,470 --> 00:07:07,350 And in fact that's going to drive equilibrium, as we're 121 00:07:07,350 --> 00:07:11,310 going to see a little bit later today. 122 00:07:11,310 --> 00:07:16,100 Alright, let's quickly go through the math 123 00:07:16,100 --> 00:07:18,370 and prove this here. 124 00:07:18,370 --> 00:07:23,660 So our goal, then, is to have a mixture, chemical potential 125 00:07:23,660 --> 00:07:25,710 of the mixture on one side, and the chemical potential of 126 00:07:25,710 --> 00:07:29,310 the pure material on the other side. 127 00:07:29,310 --> 00:07:32,990 And so we're going to start by sort of a similar thing here. 128 00:07:32,990 --> 00:07:39,090 We're going to have a box, let me redo my box here, we're 129 00:07:39,090 --> 00:07:42,080 going to have a box with, let me get rid of 130 00:07:42,080 --> 00:07:44,490 these one bars here. 131 00:07:44,490 --> 00:07:50,750 We're going to put our membrane in the box. 132 00:07:50,750 --> 00:07:52,590 And we're going to have a pressure, pA, 133 00:07:52,590 --> 00:07:55,250 on this side here. 134 00:07:55,250 --> 00:07:57,470 And we're going to have a pressure pA prime for the 135 00:07:57,470 --> 00:08:01,790 partial pressure of A. And pB prime for the partial pressure 136 00:08:01,790 --> 00:08:12,130 of B. And the p total is going to be pA prime plus pB prime. 137 00:08:12,130 --> 00:08:14,030 And I'm going to see at equilibrium, I'm going to 138 00:08:14,030 --> 00:08:17,890 write everything I know about equilibrium. 139 00:08:17,890 --> 00:08:21,470 At equilibrium, I know that the partial pressure of A on 140 00:08:21,470 --> 00:08:25,530 that side here has to be equal to the pressure of A here. 141 00:08:25,530 --> 00:08:28,100 The partial pressure, the pressure is basically the 142 00:08:28,100 --> 00:08:32,130 force of these molecules hitting that membrane per unit 143 00:08:32,130 --> 00:08:35,280 time, times the number of molecules hitting. 144 00:08:35,280 --> 00:08:39,110 So we have the same flux of molecules going this way, is 145 00:08:39,110 --> 00:08:41,720 equal to the flux of molecules going the other way. 146 00:08:41,720 --> 00:08:50,180 So at equilibrium, pA prime equals pA. 147 00:08:50,180 --> 00:08:50,860 So. 148 00:08:50,860 --> 00:08:53,060 What else can I say? 149 00:08:53,060 --> 00:08:55,980 At equilibrium, in terms of the chemical potentials, I 150 00:08:55,980 --> 00:09:01,430 know that the chemical potential of the mixture, mu A 151 00:09:01,430 --> 00:09:07,470 in the mixture, temperature under pressure p total. 152 00:09:07,470 --> 00:09:12,740 Is equal to the chemical potential of the pure system, 153 00:09:12,740 --> 00:09:15,570 same temperature under pressure is p 154 00:09:15,570 --> 00:09:16,850 sub a on that side. 155 00:09:16,850 --> 00:09:21,250 These are the two things that I know at equilibrium. 156 00:09:21,250 --> 00:09:24,260 So let's start to turn the crank. 157 00:09:24,260 --> 00:09:28,150 And see if we can come up with, well, we basically 158 00:09:28,150 --> 00:09:30,320 already have this. 159 00:09:30,320 --> 00:09:35,180 If we can massage the right side of our equation so that 160 00:09:35,180 --> 00:09:39,480 the pressure term, pA, is p total. 161 00:09:39,480 --> 00:09:45,690 And then we'll have an equation that will compare the 162 00:09:45,690 --> 00:09:48,320 chemical potential of the mixture under the same 163 00:09:48,320 --> 00:09:50,740 conditions as the chemical potential of 164 00:09:50,740 --> 00:09:54,030 A in the pure state. 165 00:09:54,030 --> 00:09:55,590 So, what can we use? 166 00:09:55,590 --> 00:10:01,600 We can use Dalton's law here, which tells us that pA prime 167 00:10:01,600 --> 00:10:04,380 is equal to xA p total. 168 00:10:04,380 --> 00:10:08,700 That's from Dalton. 169 00:10:08,700 --> 00:10:10,780 And now, pA prime is equal to pA. 170 00:10:10,780 --> 00:10:12,710 So this is also just pA. 171 00:10:12,710 --> 00:10:18,780 And we can plug that in here, and suddenly we've got p total 172 00:10:18,780 --> 00:10:22,450 included in here. 173 00:10:22,450 --> 00:10:25,630 Let's just pass these out. 174 00:10:25,630 --> 00:10:30,160 If you want to, thank you. 175 00:10:30,160 --> 00:10:40,450 Thank you. 176 00:10:40,450 --> 00:10:50,820 So, this is equal to mu A pure temperature xA p total. 177 00:10:50,820 --> 00:10:51,730 So what else do I know? 178 00:10:51,730 --> 00:10:56,280 I know that I've written something here, that mu is 179 00:10:56,280 --> 00:10:59,610 equal to mu naught, to temp, so this is at one bar. 180 00:10:59,610 --> 00:11:02,660 RT log p. 181 00:11:02,660 --> 00:11:11,960 So I'm going to rewrite this as mu A pure 182 00:11:11,960 --> 00:11:14,230 temperature T at one bar. 183 00:11:14,230 --> 00:11:19,940 Plus RT log xA p total. 184 00:11:19,940 --> 00:11:23,340 Now, the log of xA times p total is the log of xA plus 185 00:11:23,340 --> 00:11:25,220 the log of p total. 186 00:11:25,220 --> 00:11:34,240 It's equal to plus RT log p total plus RT log xA. 187 00:11:34,240 --> 00:11:40,470 So I can lump these two things together. 188 00:11:40,470 --> 00:11:45,650 I have mu A pure T plus RT log p. 189 00:11:45,650 --> 00:11:52,480 Well, that's just the chemical potential of A, in the pure 190 00:11:52,480 --> 00:11:55,520 state, at temperature T and pressure pT. 191 00:11:58,170 --> 00:12:01,220 That's the equation here to relate pressure, at some 192 00:12:01,220 --> 00:12:04,650 variable pressure p, to what it would be at one bar. 193 00:12:04,650 --> 00:12:06,170 Which is that. 194 00:12:06,170 --> 00:12:08,640 Then we have the plus RT log xA sitting here. 195 00:12:08,640 --> 00:12:12,490 Plus RT log xA. 196 00:12:12,490 --> 00:12:13,310 So we're done. 197 00:12:13,310 --> 00:12:16,820 We're done because on that side here I have mu A, the 198 00:12:16,820 --> 00:12:21,600 chemical potential of A in the mixture, temperature T, 199 00:12:21,600 --> 00:12:23,220 pressure p total. 200 00:12:23,220 --> 00:12:26,270 I've got mu A pure temperature T pressure p 201 00:12:26,270 --> 00:12:29,530 total, plus a term. 202 00:12:29,530 --> 00:12:32,420 Plus RT log xA. 203 00:12:32,420 --> 00:12:37,100 Can xA be bigger than one? xA's the mole fraction. xA's 204 00:12:37,100 --> 00:12:38,150 always less than one. 205 00:12:38,150 --> 00:12:41,050 Log xA is always less than zero. 206 00:12:41,050 --> 00:12:48,920 This term here is always less than or equal to zero. 207 00:12:48,920 --> 00:12:54,870 Therefore, this term is always less than that term. 208 00:12:54,870 --> 00:12:57,850 Therefore the chemical potential in the mixture is 209 00:12:57,850 --> 00:13:02,950 always less than the chemical potential 210 00:13:02,950 --> 00:13:09,320 inside the pure material. 211 00:13:09,320 --> 00:13:12,660 This is going to be important for the next part of the 212 00:13:12,660 --> 00:13:14,010 presentation. 213 00:13:14,010 --> 00:13:17,230 Any questions? 214 00:13:17,230 --> 00:13:20,985 This is what drives the death of saltwater fish in fresh 215 00:13:20,985 --> 00:13:22,250 water, right? 216 00:13:22,250 --> 00:13:25,500 Because osmotic pressure is basically given by 217 00:13:25,500 --> 00:13:27,010 this basic idea here. 218 00:13:27,010 --> 00:13:38,050 And it's all driven by entropy of mixing. 219 00:13:38,050 --> 00:13:38,960 Sure you don't have any questions? 220 00:13:38,960 --> 00:13:44,740 Speak up. 221 00:13:44,740 --> 00:13:47,810 So now we have all the tools we need to look at chemical 222 00:13:47,810 --> 00:13:53,070 equilibrium. 223 00:13:53,070 --> 00:13:57,250 When we have a mixture, and we're going to start with 224 00:13:57,250 --> 00:13:58,080 ideal gases. 225 00:13:58,080 --> 00:14:01,550 But everything I'm going to say about equilibrium and 226 00:14:01,550 --> 00:14:05,700 ideal gases is valid for solutions. 227 00:14:05,700 --> 00:14:07,220 An ideal gas, and we're going to be 228 00:14:07,220 --> 00:14:08,580 talking about ideal solutions. 229 00:14:08,580 --> 00:14:11,500 And molecules in an ideal solution, an ideal solvent, 230 00:14:11,500 --> 00:14:13,680 are not very different than molecules in an ideal gas. 231 00:14:13,680 --> 00:14:15,320 They don't interact with each other. 232 00:14:15,320 --> 00:14:18,190 You use concentration instead of partial pressures, but 233 00:14:18,190 --> 00:14:20,810 pretty much all the ideas are the same. 234 00:14:20,810 --> 00:14:22,035 All the concepts are the same. 235 00:14:22,035 --> 00:14:24,170 The equations are basically the same. 236 00:14:24,170 --> 00:14:27,110 You just do a little bit of replacement of variables. 237 00:14:27,110 --> 00:14:28,800 But it pretty much is the same thing. 238 00:14:28,800 --> 00:14:31,810 It's easier to think about it, to learn first in terms of the 239 00:14:31,810 --> 00:14:36,830 ideal gas, but it applies equally well to what you're 240 00:14:36,830 --> 00:14:37,840 more likely to use. 241 00:14:37,840 --> 00:14:42,040 Which is solutions. 242 00:14:42,040 --> 00:14:48,180 So let's look at the prototypical gas phase 243 00:14:48,180 --> 00:14:51,900 reaction that everybody writes down when they first do this 244 00:14:51,900 --> 00:14:52,580 sort of problems. 245 00:14:52,580 --> 00:14:55,980 The Haber process. 246 00:14:55,980 --> 00:14:59,310 Gas, temperature T, you take nitrogen. 247 00:14:59,310 --> 00:15:05,940 You react it with hydrogen, gas, temperature T, p. 248 00:15:05,940 --> 00:15:09,200 And you make ammonia. 249 00:15:09,200 --> 00:15:13,740 NH3 gas T, p. 250 00:15:13,740 --> 00:15:18,900 And we're going to ask the question, so I take nitrogen, 251 00:15:18,900 --> 00:15:21,470 I take some, hydrogen, I mix them together in a container. 252 00:15:21,470 --> 00:15:22,710 And I make ammonia. 253 00:15:22,710 --> 00:15:27,340 And I'm going to ask, after I reach equilibrium, what is the 254 00:15:27,340 --> 00:15:36,810 partial pressure of nitrogen hydrogen and ammonia? 255 00:15:36,810 --> 00:15:40,320 Standard equilibrium problem. 256 00:15:40,320 --> 00:15:42,740 You all, I'm sure you've all seen equations about 257 00:15:42,740 --> 00:15:43,560 equilibrium. 258 00:15:43,560 --> 00:15:47,460 Equilibrium constant K, log delta G of the reaction, et 259 00:15:47,460 --> 00:15:48,700 cetera, et cetera. 260 00:15:48,700 --> 00:15:51,330 But you probably don't have a really good intuition as to 261 00:15:51,330 --> 00:15:54,350 why it is what it is. 262 00:15:54,350 --> 00:15:59,100 So the point of the class here is not to relearn log K is 263 00:15:59,100 --> 00:16:02,880 equal to minus delta G, reaction divided by RT. 264 00:16:02,880 --> 00:16:05,150 The point is to learn how we get there. 265 00:16:05,150 --> 00:16:07,260 How we get to that equation. 266 00:16:07,260 --> 00:16:09,480 And what parts are important. 267 00:16:09,480 --> 00:16:13,130 Specifically, how entropy of mixing really becomes key to 268 00:16:13,130 --> 00:16:15,170 equilibrium. 269 00:16:15,170 --> 00:16:17,260 But before we get there, let me give you a little bit of 270 00:16:17,260 --> 00:16:20,340 history as to why this is such an important reaction. 271 00:16:20,340 --> 00:16:25,870 This reaction, which you see in every chemistry class, was 272 00:16:25,870 --> 00:16:28,360 developed by Mr. Haber and Mr. Bosch. 273 00:16:28,360 --> 00:16:33,410 Mr. Bosch made it large-scale around 1910. 274 00:16:33,410 --> 00:16:38,210 And it's the reaction that you use to make fertilizer. 275 00:16:38,210 --> 00:16:40,550 Ammonia is the feed stock to make nitrogen-based 276 00:16:40,550 --> 00:16:44,430 fertilizers. 277 00:16:44,430 --> 00:16:47,780 So today, there are a hundred million tons of fertilizer, of 278 00:16:47,780 --> 00:16:49,930 nitrogen fertilizer made, using, 279 00:16:49,930 --> 00:16:51,200 essentially, the Haber process. 280 00:16:51,200 --> 00:16:53,820 It's a huge, huge, commodity. 281 00:16:53,820 --> 00:16:57,520 1% of the world's energy is taken up 282 00:16:57,520 --> 00:17:00,640 to make this reaction. 283 00:17:00,640 --> 00:17:02,750 Almost 1% is about 3/4 of the world's 284 00:17:02,750 --> 00:17:06,510 energy is used on this. 285 00:17:06,510 --> 00:17:12,120 In World War 1, Germany was making explosives out of 286 00:17:12,120 --> 00:17:13,120 nitrogen feed stock. 287 00:17:13,120 --> 00:17:17,060 And it was getting its feed stock from Chile. 288 00:17:17,060 --> 00:17:18,530 From saltpeter mines in Chile. 289 00:17:18,530 --> 00:17:21,160 Chile was under British hands at the time. 290 00:17:21,160 --> 00:17:24,590 Well, the British didn't let Chile sell 291 00:17:24,590 --> 00:17:26,220 saltpeter to Germany. 292 00:17:26,220 --> 00:17:30,190 Germany had to find another way to make ammunitions. 293 00:17:30,190 --> 00:17:34,210 And the Haber process, which had just been invented by 294 00:17:34,210 --> 00:17:43,900 Bosch-Haber, became the way that Germany made explosives. 295 00:17:43,900 --> 00:17:48,700 Without this, Germany would have stopped the war in 1916. 296 00:17:48,700 --> 00:17:49,580 Or even before then. 297 00:17:49,580 --> 00:17:55,170 Way before then. 298 00:17:55,170 --> 00:18:03,460 And this process was basically, and Haber and Bosch 299 00:18:03,460 --> 00:18:05,020 got the Nobel Prize, essentially, for this process. 300 00:18:05,020 --> 00:18:08,390 For showing how to take chemistry, and doing chemistry 301 00:18:08,390 --> 00:18:10,870 and large scale processes under high pressure and high 302 00:18:10,870 --> 00:18:12,180 temperature conditions. 303 00:18:12,180 --> 00:18:17,130 Haber got the Nobel Prize in 1918 and Bosch got it in 1931. 304 00:18:17,130 --> 00:18:21,300 This process, arguably, you could say, was the birth of 305 00:18:21,300 --> 00:18:24,720 the dominance of Germany as a chemical industry. 306 00:18:24,720 --> 00:18:29,360 Based on that, and the fact that they had to make liquid 307 00:18:29,360 --> 00:18:30,940 fuels out of coal. 308 00:18:30,940 --> 00:18:34,750 The syn-gas, or the syn-fuel process. 309 00:18:34,750 --> 00:18:38,200 Also invented during World War 1, where they had to use, they 310 00:18:38,200 --> 00:18:43,060 couldn't find any oil and they had to use their coal mines. 311 00:18:43,060 --> 00:18:45,960 And obviously this process, the syn-fuel process, is 312 00:18:45,960 --> 00:18:50,480 coming back in vogue with the energy crisis around now. 313 00:18:50,480 --> 00:18:56,770 So they developed a lot of knowledge about how to do 314 00:18:56,770 --> 00:18:58,090 large-scale chemical reactions. 315 00:18:58,090 --> 00:19:02,400 And that was the birth of, or the explosion of, companies 316 00:19:02,400 --> 00:19:08,210 like Merck and Bayer and Bosch, and BASF, and all these 317 00:19:08,210 --> 00:19:10,860 German companies that dominate, basically, the 318 00:19:10,860 --> 00:19:12,870 chemical industry. 319 00:19:12,870 --> 00:19:18,480 And when Germany lost the war, the US government confiscated 320 00:19:18,480 --> 00:19:23,390 the American divisions of these German companies. 321 00:19:23,390 --> 00:19:26,950 And I'm sure you've heard of Merck as the company. 322 00:19:26,950 --> 00:19:28,610 And you think of Merck as an American 323 00:19:28,610 --> 00:19:30,930 company that makes drugs. 324 00:19:30,930 --> 00:19:31,720 Well, that's true. 325 00:19:31,720 --> 00:19:34,240 It's a very large American company that makes drugs. 326 00:19:34,240 --> 00:19:37,110 But there's another Merck, which is the German Merck. 327 00:19:37,110 --> 00:19:39,610 Which also makes drugs. 328 00:19:39,610 --> 00:19:41,760 But it makes liquid crystals. 329 00:19:41,760 --> 00:19:43,092 And liquid crystals is where it makes all of 330 00:19:43,092 --> 00:19:45,110 its money right now. 331 00:19:45,110 --> 00:19:48,410 And it's very confusing because they're both Merck. 332 00:19:48,410 --> 00:19:52,040 And they both came from the Merck family from the 1600s 333 00:19:52,040 --> 00:19:54,470 that were pharmacists. 334 00:19:54,470 --> 00:20:02,420 But after World War 1, Merck Germany was allowed to use the 335 00:20:02,420 --> 00:20:07,060 name Merck everywhere in the world except for the US. 336 00:20:07,060 --> 00:20:09,980 In the US it's called EMD. 337 00:20:09,980 --> 00:20:13,380 Merck USA is allowed to use the name Merck in the US and 338 00:20:13,380 --> 00:20:16,820 outside of the US it's called, let me remind myself, it's 339 00:20:16,820 --> 00:20:18,460 called MSD. 340 00:20:18,460 --> 00:20:19,840 It's very confusing. 341 00:20:19,840 --> 00:20:20,850 Bayer is the same way. 342 00:20:20,850 --> 00:20:22,560 So Bayer was split up. 343 00:20:22,560 --> 00:20:26,260 And there was Buyer in Germany and there was Bayer in the US. 344 00:20:26,260 --> 00:20:30,570 You've got Bayer Aspirin from the US term Bayer. 345 00:20:30,570 --> 00:20:31,550 And you've got all the other 346 00:20:31,550 --> 00:20:33,770 pharmaceuticals from Bayer Germany. 347 00:20:33,770 --> 00:20:35,040 All the chemicals. 348 00:20:35,040 --> 00:20:39,450 And a few years ago, Buyer bought Bayer, and now we have 349 00:20:39,450 --> 00:20:44,750 one Buyer-Bayer, depending on where you live. 350 00:20:44,750 --> 00:20:47,060 And so it's very interesting to see the history of all 351 00:20:47,060 --> 00:20:48,240 these big chemical companies. 352 00:20:48,240 --> 00:20:51,690 And that's why Switzerland and Germany are still the home of, 353 00:20:51,690 --> 00:20:53,390 it's all due to that reaction. 354 00:20:53,390 --> 00:20:55,470 That's why you always see that reaction. 355 00:20:55,470 --> 00:20:58,520 Fertilizers, at the birth of the chemical company, and it's 356 00:20:58,520 --> 00:21:06,260 a great example for equilibrium. 357 00:21:06,260 --> 00:21:07,760 So that's the example. 358 00:21:07,760 --> 00:21:08,680 Let's generalize it. 359 00:21:08,680 --> 00:21:10,250 We're actually not going to work on this 360 00:21:10,250 --> 00:21:11,300 until the very end. 361 00:21:11,300 --> 00:21:12,940 And we'll see it again. 362 00:21:12,940 --> 00:21:14,040 But let's generalize. 363 00:21:14,040 --> 00:21:19,480 Let's just take a mixture of gases with stoichiometries nu 364 00:21:19,480 --> 00:21:27,970 A, a gas, pressure, temperature plus nu B, B, gas, 365 00:21:27,970 --> 00:21:35,946 pressure, temperature, goes to nu sub C moles of C, it's a 366 00:21:35,946 --> 00:21:38,930 gas, with a temperature and pressure, plus nu sub D moles 367 00:21:38,930 --> 00:21:41,620 of D, which is a gas, temperature, pressure. 368 00:21:41,620 --> 00:21:43,610 All ideal gases. 369 00:21:43,610 --> 00:21:48,430 For now. 370 00:21:48,430 --> 00:21:57,100 So this the setup here. 371 00:21:57,100 --> 00:22:04,730 Now, when we write that reaction up here, what we're 372 00:22:04,730 --> 00:22:11,930 really writing is, in terms of the process, is take a 373 00:22:11,930 --> 00:22:16,630 container that contains A, plus a container that contains 374 00:22:16,630 --> 00:22:21,950 B, and go to a container that contains C, plus a container 375 00:22:21,950 --> 00:22:28,970 that contains D. And when we write delta G for that 376 00:22:28,970 --> 00:22:34,000 reaction here, the process that we're talking about is 377 00:22:34,000 --> 00:22:40,150 taking the reactions separately from each other. 378 00:22:40,150 --> 00:22:41,830 That's the initial state. 379 00:22:41,830 --> 00:22:43,520 And the final state is the product. 380 00:22:43,520 --> 00:22:48,090 Separated in their own containers from each other. 381 00:22:48,090 --> 00:22:50,820 And when we say delta G is less than zero, for this 382 00:22:50,820 --> 00:22:53,770 process, which means it's spontaneous, we mean the 383 00:22:53,770 --> 00:22:56,300 process to go from the separated reactions to the 384 00:22:56,300 --> 00:22:59,000 separated products. 385 00:22:59,000 --> 00:23:01,310 But in reality, that's not what you do when you do an 386 00:23:01,310 --> 00:23:02,050 experiment. 387 00:23:02,050 --> 00:23:04,630 In reality, you take these two containers and 388 00:23:04,630 --> 00:23:08,020 you mix them together. 389 00:23:08,020 --> 00:23:11,250 A plus B. And you let that react. 390 00:23:11,250 --> 00:23:14,790 And at the end of the day, you have a big container with A 391 00:23:14,790 --> 00:23:20,430 plus B plus C plus D inside, all mixed up together in 392 00:23:20,430 --> 00:23:23,190 equilibrium. 393 00:23:23,190 --> 00:23:30,500 And this is not the same as that. 394 00:23:30,500 --> 00:23:32,030 It's not the same as that. 395 00:23:32,030 --> 00:23:35,070 Because you've got entropy of mixing 396 00:23:35,070 --> 00:23:37,760 happening in all of this. 397 00:23:37,760 --> 00:23:40,030 Forget about A and B interacting with each other. 398 00:23:40,030 --> 00:23:43,780 Entropy of mixing is going to dominate equilibrium. 399 00:23:43,780 --> 00:23:51,060 You've got this problem to deal with. 400 00:23:51,060 --> 00:23:54,310 So that means that we're going to have to worry about, if 401 00:23:54,310 --> 00:23:59,430 we're going to want to know at which state the process is in 402 00:23:59,430 --> 00:24:01,745 equilibrium, you're going to have to worry about this issue 403 00:24:01,745 --> 00:24:03,020 right here. 404 00:24:03,020 --> 00:24:06,470 It's not enough to know what delta G of the reaction is. 405 00:24:06,470 --> 00:24:10,920 So for instance, if I plot, as function of the reaction, I've 406 00:24:10,920 --> 00:24:14,810 got the reactants on that side here. 407 00:24:14,810 --> 00:24:18,830 And the products on this side here. 408 00:24:18,830 --> 00:24:22,570 And I want to plot delta G as a function of the reaction. 409 00:24:22,570 --> 00:24:28,190 Well, my initial delta G that I would write, to calculate 410 00:24:28,190 --> 00:24:32,050 delta G of the reaction, which is the delta G of the 411 00:24:32,050 --> 00:24:41,520 products, minus the delta G of the reactants, so initially I 412 00:24:41,520 --> 00:24:43,110 have delta G of the reactants. 413 00:24:43,110 --> 00:24:45,610 That's, in those two boxes, that aren't mixed. 414 00:24:45,610 --> 00:24:47,930 So I'm up here somewhere. 415 00:24:47,930 --> 00:24:55,470 Delta G of the reactants. 416 00:24:55,470 --> 00:24:57,950 And at the end of the process, in these two boxes, when I 417 00:24:57,950 --> 00:25:00,500 calculate this delta G naught reaction, I'm sitting 418 00:25:00,500 --> 00:25:01,290 somewhere here. 419 00:25:01,290 --> 00:25:03,590 Let's say it's a downhill process. 420 00:25:03,590 --> 00:25:06,480 Delta G naught of the products. 421 00:25:06,480 --> 00:25:09,900 Well, the first thing that happens, when I take these two 422 00:25:09,900 --> 00:25:13,760 boxes and mix them together, is delta G naught of the 423 00:25:13,760 --> 00:25:15,630 reaction's going to go up or going to go 424 00:25:15,630 --> 00:25:19,740 down, or stay the same. 425 00:25:19,740 --> 00:25:24,480 Anybody want to guess? 426 00:25:24,480 --> 00:25:28,030 When I go from here to here, before there's any reaction 427 00:25:28,030 --> 00:25:30,330 that happens. 428 00:25:30,330 --> 00:25:32,570 Shall we vote? 429 00:25:32,570 --> 00:25:36,580 How many say that it's going to go down? 430 00:25:36,580 --> 00:25:39,670 How many say that it's going to go up? 431 00:25:39,670 --> 00:25:42,830 How many say that it's going to stay the same? 432 00:25:42,830 --> 00:25:45,910 We have about three people that said it's going to go up. 433 00:25:45,910 --> 00:25:48,640 And a lot of people that say it's going to go down. 434 00:25:48,640 --> 00:25:52,850 And a few people that are abstaining. 435 00:25:52,850 --> 00:25:57,130 OK. 436 00:25:57,130 --> 00:25:59,140 We're mixing things. 437 00:25:59,140 --> 00:26:02,660 Anybody want to change their votes? 438 00:26:02,660 --> 00:26:05,170 Something is happening here. 439 00:26:05,170 --> 00:26:08,730 And this entropy term here, when A and B come together and 440 00:26:08,730 --> 00:26:14,660 start mixing up. 441 00:26:14,660 --> 00:26:15,680 You're changing your vote. 442 00:26:15,680 --> 00:26:17,200 Alright. 443 00:26:17,200 --> 00:26:18,610 It's going to go down. 444 00:26:18,610 --> 00:26:21,380 Now we have near-unanimity. 445 00:26:21,380 --> 00:26:22,850 So the first thing that happens is, you're 446 00:26:22,850 --> 00:26:24,140 going to go down here. 447 00:26:24,140 --> 00:26:26,200 You're going to have delta G naught of the 448 00:26:26,200 --> 00:26:32,870 reactants in the mixture. 449 00:26:32,870 --> 00:26:37,770 And if it were to go all the way, if A and B were to 450 00:26:37,770 --> 00:26:40,400 disappear, it still would be mixed. 451 00:26:40,400 --> 00:26:42,700 And so the end product here would actually be down here. 452 00:26:42,700 --> 00:26:44,110 It wouldn't be up in the products. 453 00:26:44,110 --> 00:26:48,890 It would be the mixture. 454 00:26:48,890 --> 00:26:50,050 So that's the first thing we know. 455 00:26:50,050 --> 00:26:52,160 That this delta G naught reaction is 456 00:26:52,160 --> 00:26:53,340 not the full story. 457 00:26:53,340 --> 00:26:57,560 And along the way, here I have two species to begin with. 458 00:26:57,560 --> 00:26:59,140 I've got two species to end with. 459 00:26:59,140 --> 00:27:04,380 But I've got three different species here in the middle. 460 00:27:04,380 --> 00:27:12,150 So as soon as I form a little bit of products, in this case 461 00:27:12,150 --> 00:27:14,370 here, or if I start from products and go the other way 462 00:27:14,370 --> 00:27:17,000 in the reaction and form the reactants, the first thing 463 00:27:17,000 --> 00:27:19,770 that's going to happen is I'm going to decrease the delta G. 464 00:27:19,770 --> 00:27:22,510 Just from the entropy of mixing. 465 00:27:22,510 --> 00:27:27,940 And so if I plot my delta G as a function of reaction 466 00:27:27,940 --> 00:27:34,070 conditions, I'm going to get a bowing curve like that. 467 00:27:34,070 --> 00:27:38,050 If the entropy wasn't there, then it would just be a 468 00:27:38,050 --> 00:27:40,890 straight line from one to the other. 469 00:27:40,890 --> 00:27:42,660 The entropy of mixing of reactants and 470 00:27:42,660 --> 00:27:43,510 products wasn't there. 471 00:27:43,510 --> 00:27:45,170 Actually, let me put it up here. 472 00:27:45,170 --> 00:27:48,420 If entropy of mixing wasn't there, I would 473 00:27:48,420 --> 00:27:50,990 start from up here. 474 00:27:50,990 --> 00:27:54,140 And as my stoichiometry changed, I would have a linear 475 00:27:54,140 --> 00:27:57,520 curve from here to there as a function of 476 00:27:57,520 --> 00:27:58,590 the process of reaction. 477 00:27:58,590 --> 00:28:01,410 But the entropy of mixing is causing my initial state and 478 00:28:01,410 --> 00:28:02,750 my final state to go down. 479 00:28:02,750 --> 00:28:05,010 And it's causing a bow in this here. 480 00:28:05,010 --> 00:28:08,330 Because if I have equal amounts of A, B, C, and D, 481 00:28:08,330 --> 00:28:12,400 that's a lot of entropy of mixing there. 482 00:28:12,400 --> 00:28:14,990 So equilibrium is actually somewhere down here. 483 00:28:14,990 --> 00:28:20,680 It's where delta G of the mixture is at its lowest. 484 00:28:20,680 --> 00:28:25,060 Delta G of the mixture is at its lowest. 485 00:28:25,060 --> 00:28:31,650 Any questions? 486 00:28:31,650 --> 00:28:33,250 Now we going to do the math. 487 00:28:33,250 --> 00:28:42,030 We're going to see how that comes about. 488 00:28:42,030 --> 00:28:52,510 Let me do it here. 489 00:28:52,510 --> 00:28:55,780 So the question we're going to ask is, suppose that I've got 490 00:28:55,780 --> 00:28:56,910 my mixture here. 491 00:28:56,910 --> 00:29:02,290 And I'm sitting somewhere on this curve here. 492 00:29:02,290 --> 00:29:07,240 So I've got pA for partial pressure of A, partial 493 00:29:07,240 --> 00:29:10,420 pressure of B, partial pressure of C, and partial 494 00:29:10,420 --> 00:29:12,640 pressure of D in my mixture. 495 00:29:12,640 --> 00:29:14,470 And I want to know, I've got this mixture of 496 00:29:14,470 --> 00:29:16,760 reactants and products. 497 00:29:16,760 --> 00:29:21,260 Which way is the reaction going to go? 498 00:29:21,260 --> 00:29:22,880 Is it going to go towards the products? 499 00:29:22,880 --> 00:29:24,420 Is it going to go towards the reactants? 500 00:29:24,420 --> 00:29:26,910 Or is it at equilibrium? 501 00:29:26,910 --> 00:29:31,120 And to answer that question, I'm going to let the reaction 502 00:29:31,120 --> 00:29:34,200 react a little bit more to create a 503 00:29:34,200 --> 00:29:36,870 little bit more products. 504 00:29:36,870 --> 00:29:38,930 Remove a little bit of reactants. 505 00:29:38,930 --> 00:29:42,500 And see what the sign of delta G is for that process. 506 00:29:42,500 --> 00:29:51,100 So I'm going to go from a moles of A, b moles of B, c 507 00:29:51,100 --> 00:29:55,720 moles of C and d moles of D in my mixture, which is that 508 00:29:55,720 --> 00:29:57,260 point on my graph here. 509 00:29:57,260 --> 00:29:59,270 Partial pressure A, partial pressure of B, partial 510 00:29:59,270 --> 00:30:02,160 pressure of C, and partial pressure D, and I'm going to 511 00:30:02,160 --> 00:30:13,360 react it a little bit more. a plus, a minus epsilon times nu 512 00:30:13,360 --> 00:30:17,830 A, where epsilon is a very small number. 513 00:30:17,830 --> 00:30:25,000 Of A, b minus epsilon times nu B times B. And I'm going to 514 00:30:25,000 --> 00:30:29,520 have, now, products being formed. c moles plus epsilon 515 00:30:29,520 --> 00:30:33,020 times nu C, I've going to have the stoichiometry in there. 516 00:30:33,020 --> 00:30:37,870 For every nu A moles of A that I lose, I create mu C moles of 517 00:30:37,870 --> 00:30:42,550 C. Times epsilon as the scaling factor. 518 00:30:42,550 --> 00:30:48,340 And d plus epsilon times nu D moles of D. That's going to be 519 00:30:48,340 --> 00:30:50,240 my new mixture. 520 00:30:50,240 --> 00:30:53,700 And I'm going to ask, as I go from this initial state to 521 00:30:53,700 --> 00:30:57,380 that final state, I'm sitting on that curve, which sign is 522 00:30:57,380 --> 00:31:01,300 delta G. How do I know what the sign of delta G is? 523 00:31:01,300 --> 00:31:04,470 What is delta G for this process? 524 00:31:04,470 --> 00:31:07,890 Not the reaction with the isolated reactants and 525 00:31:07,890 --> 00:31:10,390 products, but the reaction where 526 00:31:10,390 --> 00:31:11,870 everything is mixed together. 527 00:31:11,870 --> 00:31:13,470 Where I'm going to have to worry about the chemical 528 00:31:13,470 --> 00:31:15,680 potentials of mixtures. 529 00:31:15,680 --> 00:31:18,240 And I'm going to go from one mixture to another mixture, 530 00:31:18,240 --> 00:31:20,960 and that's going to be the key to telling me if I'm in 531 00:31:20,960 --> 00:31:24,420 equilibrium or not. 532 00:31:24,420 --> 00:31:25,690 But we know how to do this. 533 00:31:25,690 --> 00:31:28,050 We know how to do this. 534 00:31:28,050 --> 00:31:35,330 So I want to calculate delta G for the process is delta G 535 00:31:35,330 --> 00:31:39,030 after minus delta G before. 536 00:31:39,030 --> 00:31:46,340 That's delta G after minus delta G before. 537 00:31:46,340 --> 00:31:53,390 And I know that delta G, rather, let's call it G after 538 00:31:53,390 --> 00:31:59,330 minus G before, the Gibbs free energy after minus the Gibbs 539 00:31:59,330 --> 00:32:01,150 free energy before. 540 00:32:01,150 --> 00:32:05,460 And I know the Gibbs free energy is just the sum of the 541 00:32:05,460 --> 00:32:06,600 chemical potentials, right? 542 00:32:06,600 --> 00:32:09,550 You take all the species, and you take all the chemical 543 00:32:09,550 --> 00:32:10,070 potentials. 544 00:32:10,070 --> 00:32:12,550 You add it up together times the stoichiometry. 545 00:32:12,550 --> 00:32:14,420 And that gives you the Gibbs free energy. 546 00:32:14,420 --> 00:32:15,820 That's what we learned last time. 547 00:32:15,820 --> 00:32:18,820 So if I want to find what the Gibbs free energy at the end 548 00:32:18,820 --> 00:32:22,840 here is, I look at the chemical potentials of A, B, 549 00:32:22,840 --> 00:32:26,080 and C and D times their number of moles. 550 00:32:26,080 --> 00:32:30,310 So I look at a minus epsilon times nu A times the chemical 551 00:32:30,310 --> 00:32:36,500 potential of A, plus b minus epsilon times nu B times the 552 00:32:36,500 --> 00:32:41,400 chemical potential of B plus c, plus epsilon times nu C, 553 00:32:41,400 --> 00:32:45,860 the chemical potential at C, plus d, plus epsilon times nu 554 00:32:45,860 --> 00:32:53,510 D, the chemical potential of D. 555 00:32:53,510 --> 00:32:59,660 And then I subtract what G was before. 556 00:32:59,660 --> 00:33:03,620 This infinitesimally small process. 557 00:33:03,620 --> 00:33:17,050 Was a times mu A plus b times mu B plus c times mu C, plus d 558 00:33:17,050 --> 00:33:25,150 times mu D. And we're assuming that my small change, the 559 00:33:25,150 --> 00:33:28,890 small amount of A and B that get destroyed to form C and D 560 00:33:28,890 --> 00:33:31,070 is small enough that the chemical potential basically 561 00:33:31,070 --> 00:33:34,270 stays the same during this infinitesimally small change. 562 00:33:34,270 --> 00:33:36,250 That's why I can use the same chemical 563 00:33:36,250 --> 00:33:41,260 potentials before and after. 564 00:33:41,260 --> 00:33:44,150 OK, a lot of things drop out. 565 00:33:44,150 --> 00:33:46,500 This term drops out from that term. 566 00:33:46,500 --> 00:33:48,060 This term drops out from this term. 567 00:33:48,060 --> 00:33:50,240 This term drops out from this term. 568 00:33:50,240 --> 00:33:54,060 This is a minus, no, this is all plus. 569 00:33:54,060 --> 00:33:55,880 That's fine. 570 00:33:55,880 --> 00:33:57,240 There's the minus sign right here. 571 00:33:57,240 --> 00:34:00,990 So then this term drops out from that term. 572 00:34:00,990 --> 00:34:09,940 And I am left with epsilon times nu C mu C, the chemical 573 00:34:09,940 --> 00:34:16,970 potentials of the products minus the sum of the chemical 574 00:34:16,970 --> 00:34:29,790 potentials of the reactants, nu A, mu A plus nu B mu B. 575 00:34:29,790 --> 00:34:37,070 That's the delta G for this small change. 576 00:34:37,070 --> 00:34:40,030 Now, you remember way back, maybe from even just last 577 00:34:40,030 --> 00:34:43,980 semester if you've taken 5.112 last semester, or from last 578 00:34:43,980 --> 00:34:46,770 year, or from high school, that the partial pressure is 579 00:34:46,770 --> 00:34:49,130 going to the equilibrium constant. 580 00:34:49,130 --> 00:34:50,420 Somehow we're going to have to get partial 581 00:34:50,420 --> 00:34:52,300 pressures in there. 582 00:34:52,300 --> 00:34:54,170 But we know now how to go from chemical 583 00:34:54,170 --> 00:34:57,610 potentials to partial pressures. 584 00:34:57,610 --> 00:34:59,500 It's written right here. 585 00:34:59,500 --> 00:35:03,910 The chemical potentials of the, and we also know how to 586 00:35:03,910 --> 00:35:08,530 go from the chemical potential in the mixed species, in the, 587 00:35:08,530 --> 00:35:12,900 mixture to the chemical potential in a pure. 588 00:35:12,900 --> 00:35:18,150 We saw that mu A in the mixture, temperature, pressure 589 00:35:18,150 --> 00:35:24,710 was equal to mu A pure temperature, 590 00:35:24,710 --> 00:35:29,110 pressure plus RT log xA. 591 00:35:29,110 --> 00:35:32,470 So those are things that we're going to be using, to go from 592 00:35:32,470 --> 00:35:35,150 something that has chemical potential to something where 593 00:35:35,150 --> 00:35:38,390 we'll be able to get back pressures, partial pressures, 594 00:35:38,390 --> 00:35:41,510 and delta G of the reaction, because we're going to need 595 00:35:41,510 --> 00:35:45,730 the chemical potential of the pure stuff. 596 00:35:45,730 --> 00:35:51,970 So let me go forward just a little bit. 597 00:35:51,970 --> 00:35:57,360 Remind you, if I look at the delta G of the reaction, delta 598 00:35:57,360 --> 00:36:00,360 G of the reaction, in terms of chemical potentials. 599 00:36:00,360 --> 00:36:04,630 Delta G naught of the reaction. 600 00:36:04,630 --> 00:36:09,970 This is the delta G of the products minus the delta G of 601 00:36:09,970 --> 00:36:13,610 the reactants when they're pure, not when they're mixed. 602 00:36:13,610 --> 00:36:24,900 So delta G naught of the reaction is nu C mu C at one 603 00:36:24,900 --> 00:36:34,200 bar pure plus nu D mu D pure. 604 00:36:34,200 --> 00:36:45,740 Minus nu A mu A pure minus nu B mu B in the pure state. 605 00:36:45,740 --> 00:36:48,010 That's what delta G naught of the reaction is in terms of 606 00:36:48,010 --> 00:36:51,190 the chemical potentials of all these species. 607 00:36:51,190 --> 00:36:54,990 Everything's at one bar, and everything is pure. 608 00:36:54,990 --> 00:37:06,360 Somehow this is going to have to come out of this. 609 00:37:06,360 --> 00:37:09,240 So let's keep going. 610 00:37:09,240 --> 00:37:17,910 And see how it falls out. 611 00:37:17,910 --> 00:37:20,595 So for each one of these chemical potentials, I'm going 612 00:37:20,595 --> 00:37:22,310 to write it in terms of the pressure. 613 00:37:22,310 --> 00:37:24,550 What I just covered up here. 614 00:37:24,550 --> 00:37:33,290 Mu(T, p) is mu naught of T times RT log p. 615 00:37:33,290 --> 00:37:39,130 So now, delta G is going to be equal to epsilon, and I'm 616 00:37:39,130 --> 00:37:41,300 going to do a little massaging quickly. 617 00:37:41,300 --> 00:37:45,030 And I'll let you take this, go home and see how I went from 618 00:37:45,030 --> 00:37:45,910 one state to the other. 619 00:37:45,910 --> 00:37:50,260 The secret is to put in here the pressure dependence. 620 00:37:50,260 --> 00:38:05,480 Nu C mu C naught, plus nu D mu D naught, minus nu A mu A 621 00:38:05,480 --> 00:38:19,870 naught, plus nu B mu B naught, plus RT log pC to the nu C, pD 622 00:38:19,870 --> 00:38:28,820 to the nu D, divided by pA to the nu A, pB to the nu B. 623 00:38:28,820 --> 00:38:34,190 These log partial pressures all come from expanding out 624 00:38:34,190 --> 00:38:57,910 the chemical potential as mu naught plus RT log p. 625 00:38:57,910 --> 00:39:06,650 And then we recognize that this part right here, this 626 00:39:06,650 --> 00:39:16,250 part right here is delta G naught reaction. 627 00:39:16,250 --> 00:39:22,120 So I have delta G now, for this process of taking 628 00:39:22,120 --> 00:39:25,360 reactants to products just a little bit, let me take it as 629 00:39:25,360 --> 00:39:27,740 a function of epsilon here. 630 00:39:27,740 --> 00:39:36,070 Delta G naught of the reaction plus RT log of this ratio of 631 00:39:36,070 --> 00:39:37,100 partial pressures. 632 00:39:37,100 --> 00:39:41,410 I'm going to call that Q. I'm going to call this thing here 633 00:39:41,410 --> 00:39:44,480 Q, which you've seen before. 634 00:39:44,480 --> 00:39:51,730 The reaction quotient. 635 00:39:51,730 --> 00:39:56,050 And this tells me that for this very small process, 636 00:39:56,050 --> 00:40:03,310 epsilon, very small, if delta G naught, delta G, of this 637 00:40:03,310 --> 00:40:09,130 process, is less than zero, then the reaction will keep 638 00:40:09,130 --> 00:40:09,690 going forward. 639 00:40:09,690 --> 00:40:12,090 It means that I'll be on the side of the curve here. 640 00:40:12,090 --> 00:40:15,720 I'm going to go down a little bit. delta G naught is going 641 00:40:15,720 --> 00:40:18,690 to be great, it's going to be spontaneous. 642 00:40:18,690 --> 00:40:20,630 I'm not in equilibrium. 643 00:40:20,630 --> 00:40:25,810 I'm going to go towards the products. 644 00:40:25,810 --> 00:40:30,960 If delta G is equal to zero, then I'm at the bottom of this 645 00:40:30,960 --> 00:40:32,350 curve here. 646 00:40:32,350 --> 00:40:33,210 I'm here. 647 00:40:33,210 --> 00:40:36,570 If I go forward a little bit, the slope is zero. 648 00:40:36,570 --> 00:40:42,890 Delta G is zero, I'm in equilibrium. 649 00:40:42,890 --> 00:40:50,380 And if delta G is greater than zero, then I'm going to back 650 00:40:50,380 --> 00:40:54,800 to reactants, and I'm sitting, then, on that part of the 651 00:40:54,800 --> 00:40:56,320 curve here. 652 00:40:56,320 --> 00:40:58,600 And if I try to make more products I'm just going uphill 653 00:40:58,600 --> 00:41:02,780 a little bit. 654 00:41:02,780 --> 00:41:06,790 So now, I've done this all for epsilon is 655 00:41:06,790 --> 00:41:07,900 equal to very small. 656 00:41:07,900 --> 00:41:10,490 What you usually find written in books is where the epsilon 657 00:41:10,490 --> 00:41:11,260 equals to one. 658 00:41:11,260 --> 00:41:14,420 Basically taking one mole of this process. 659 00:41:14,420 --> 00:41:18,840 So you'll see it per-mole of this reaction. 660 00:41:18,840 --> 00:41:19,820 But it's the same idea. 661 00:41:19,820 --> 00:41:22,140 It's the same thing. 662 00:41:22,140 --> 00:41:24,560 And in fact it doesn't really matter, because the quantity 663 00:41:24,560 --> 00:41:27,710 that matters is this delta G naught reactions plus RT log 664 00:41:27,710 --> 00:41:30,900 Q. It's the sign of this quantity here. 665 00:41:30,900 --> 00:41:32,630 This epsilon is just an arbitrary number. 666 00:41:32,630 --> 00:41:36,040 I can pick it, really whatever I want. 667 00:41:36,040 --> 00:41:42,010 So it's the sign of this thing that's important. 668 00:41:42,010 --> 00:41:46,670 Let me go back. 669 00:41:46,670 --> 00:41:50,680 So what you'll see, then, is delta G is equal to delta G 670 00:41:50,680 --> 00:41:58,030 naught reaction plus RT log Q. Basically, taking epsilon is 671 00:41:58,030 --> 00:41:59,790 going to zero. 672 00:41:59,790 --> 00:42:12,980 As determining where the equilibrium is going to be. 673 00:42:12,980 --> 00:42:20,570 OK, any questions? 674 00:42:20,570 --> 00:42:22,810 All driven by entropy of mixing. 675 00:42:22,810 --> 00:42:26,350 Without entropy of mixing, we would be sitting 676 00:42:26,350 --> 00:42:28,720 on this curve here. 677 00:42:28,720 --> 00:42:31,350 Delta G naught of the reaction would tell us that everything 678 00:42:31,350 --> 00:42:33,170 should go to completion. 679 00:42:33,170 --> 00:42:36,010 Things don't go to completion, and that's a good thing. 680 00:42:36,010 --> 00:42:38,940 Otherwise there would not be life on Earth. 681 00:42:38,940 --> 00:42:43,110 We're basically a set of equilibria in a big membrane. 682 00:42:43,110 --> 00:42:44,840 Which is our skin, right? 683 00:42:44,840 --> 00:42:46,390 All these biochemical cycles are in 684 00:42:46,390 --> 00:42:47,460 equilibrium with each other. 685 00:42:47,460 --> 00:42:49,330 And it's a complicated process. 686 00:42:49,330 --> 00:42:52,880 And it's all driven by entropy. 687 00:42:52,880 --> 00:42:55,570 Ultimately. 688 00:42:55,570 --> 00:42:58,560 And other things, but entropy is very important. 689 00:42:58,560 --> 00:42:59,140 Alright, so equilibrium now. 690 00:42:59,140 --> 00:43:02,060 Equilibrium is when we have this delta G equal to zero. 691 00:43:02,060 --> 00:43:10,520 That's when delta G naught of the reaction equals RT log Q. 692 00:43:10,520 --> 00:43:15,045 And at that point, we replace Q with to equilibrium, and we 693 00:43:15,045 --> 00:43:19,390 call that the equilibrium constant. 694 00:43:19,390 --> 00:43:21,940 And we're going to put a little p here, because it's in 695 00:43:21,940 --> 00:43:25,520 terms of the partial pressures. 696 00:43:25,520 --> 00:43:32,900 And it's equal to the partial pressures of the products 697 00:43:32,900 --> 00:43:36,980 raised to their stoichiometry. 698 00:43:36,980 --> 00:43:40,560 And divided by the partial pressures of the reactants, 699 00:43:40,560 --> 00:43:45,390 raised to the power of their stoichiometry. 700 00:43:45,390 --> 00:43:48,380 And this, you've seen before, I'm sure. 701 00:43:48,380 --> 00:43:54,140 At equilibrium. 702 00:43:54,140 --> 00:43:58,130 And there's a minus sign somewhere here. 703 00:43:58,130 --> 00:44:00,730 Because it's this plus that that's equal to zero, and 704 00:44:00,730 --> 00:44:04,160 there's the minus sign that I forgot to write down. 705 00:44:04,160 --> 00:44:08,120 So we define, then, equilibrium constant this way. 706 00:44:08,120 --> 00:44:15,370 And one of the things to note is that Kp as written here is 707 00:44:15,370 --> 00:44:16,340 actually unitless. 708 00:44:16,340 --> 00:44:18,310 It doesn't look like it. 709 00:44:18,310 --> 00:44:19,680 The way that I've written it. 710 00:44:19,680 --> 00:44:25,670 Because I did a shorthand way of writing the pressures, when 711 00:44:25,670 --> 00:44:33,400 I wrote RT log p here. 712 00:44:33,400 --> 00:44:35,560 There's the assumption, when you have the log of the 713 00:44:35,560 --> 00:44:39,450 pressure, that there's always one bar sitting behind there. 714 00:44:39,450 --> 00:44:40,370 Underneath. 715 00:44:40,370 --> 00:44:43,080 It's always referenced to a reference pressure. 716 00:44:43,080 --> 00:44:47,990 There's always a reference pressure. p naught dividing it 717 00:44:47,990 --> 00:44:51,120 by, because you don't want to have any units inside the log. 718 00:44:51,120 --> 00:44:53,410 And this reference pressure, we took as one bar. 719 00:44:53,410 --> 00:44:57,010 And it's pretty common to just ignore the fact that you've 720 00:44:57,010 --> 00:44:59,940 got one bar sitting in the denominator. 721 00:44:59,940 --> 00:45:04,760 And so, actually, all these pressures here are divided by 722 00:45:04,760 --> 00:45:10,720 p reference divided by p reference, divided by p 723 00:45:10,720 --> 00:45:13,590 reference, divided by p reference, which 724 00:45:13,590 --> 00:45:15,740 happens to be 1 bar. 725 00:45:15,740 --> 00:45:17,450 But it's there. 726 00:45:17,450 --> 00:45:20,200 And that means that the bars on top and the bars on the 727 00:45:20,200 --> 00:45:22,030 bottom cancel out. 728 00:45:22,030 --> 00:45:25,790 Which means that K sub p doesn't have any units. 729 00:45:25,790 --> 00:45:27,040 It is unitless. 730 00:45:27,040 --> 00:45:28,640 It's a number. 731 00:45:28,640 --> 00:45:30,200 Straight number. 732 00:45:30,200 --> 00:45:35,880 Very common mistake to make, to write it and forget that 733 00:45:35,880 --> 00:45:37,530 there's one bar sitting on the bottom here. 734 00:45:37,530 --> 00:45:40,440 And you take all the bars to the new powers on top, and the 735 00:45:40,440 --> 00:45:42,350 bars to the new powers on the bottom. 736 00:45:42,350 --> 00:45:45,450 And make units there. 737 00:45:45,450 --> 00:45:49,090 And get Kp is equal to some number, to the, times bar to 738 00:45:49,090 --> 00:45:49,960 some power. 739 00:45:49,960 --> 00:45:54,550 That would be wrong. 740 00:45:54,550 --> 00:45:56,920 I know I've made that mistake before. 741 00:45:56,920 --> 00:45:59,750 But you are not going to make that mistake, right? 742 00:45:59,750 --> 00:46:01,720 Because you've been warned. 743 00:46:01,720 --> 00:46:03,280 That this is a common mistake. 744 00:46:03,280 --> 00:46:07,810 This is unitless. 745 00:46:07,810 --> 00:46:11,270 Now, you can invert this to get Kp as a function of delta 746 00:46:11,270 --> 00:46:16,750 G. e to the minus delta G naught of the 747 00:46:16,750 --> 00:46:20,380 reaction divided by RT. 748 00:46:20,380 --> 00:46:22,980 And those are the things that you use to go back and forth 749 00:46:22,980 --> 00:46:27,240 between the thermodynamic quantities, like G that you 750 00:46:27,240 --> 00:46:31,440 calculate, to equilibrium quantities like the K, to 751 00:46:31,440 --> 00:46:34,610 finding equilibria between something 752 00:46:34,610 --> 00:46:36,830 like the Haber process. 753 00:46:36,830 --> 00:46:39,840 Now, there's other equilibrium constant that is used a lot. 754 00:46:39,840 --> 00:46:42,200 Which is the form of the equilibrium constant not in 755 00:46:42,200 --> 00:46:44,800 terms of the partial pressures, but in terms of the 756 00:46:44,800 --> 00:46:45,920 mole fraction. 757 00:46:45,920 --> 00:46:50,950 That's also an important one when you're looking at 758 00:46:50,950 --> 00:46:54,290 solution cases. 759 00:46:54,290 --> 00:46:56,800 Because we can rewrite the partial pressures using 760 00:46:56,800 --> 00:47:03,610 Dalton's law. pC is equal to the mole fraction of species C 761 00:47:03,610 --> 00:47:06,850 times the total pressure, I'll call it p. 762 00:47:06,850 --> 00:47:09,710 So if I replace every one of these partial pressures using 763 00:47:09,710 --> 00:47:19,120 Dalton's law, I get K sub p is equal to xC pC times p to the 764 00:47:19,120 --> 00:47:30,710 nu C, times xD pD to the nu D, divided by xA pA - no, not pA, 765 00:47:30,710 --> 00:47:32,150 total pressure. 766 00:47:32,150 --> 00:47:33,350 Dalton's law. 767 00:47:33,350 --> 00:47:39,550 To the nu A, xB p to the nu B. 768 00:47:39,550 --> 00:47:42,930 OK, so I've got p, p, p, p. 769 00:47:42,930 --> 00:47:44,420 They all come out. 770 00:47:44,420 --> 00:47:48,240 And that's p to the minus delta nu. 771 00:47:48,240 --> 00:47:51,230 Where delta nu is the difference in the number of 772 00:47:51,230 --> 00:47:55,370 moles of reactants minus the number of moles of products. 773 00:47:55,370 --> 00:48:06,710 So delta nu is nu C plus nu D minus nu A minus nu B. It's 774 00:48:06,710 --> 00:48:10,540 the moles of products minus moles of reactants. 775 00:48:10,540 --> 00:48:16,900 And then I have xA to the nu A, xC to the nu C, xD to the 776 00:48:16,900 --> 00:48:26,560 nu D, xA to the nu A, xB to the nu B. And that ratio, 777 00:48:26,560 --> 00:48:31,170 which is in terms of mole fractions, we call K sub x. 778 00:48:31,170 --> 00:48:32,680 The mole fractions. 779 00:48:32,680 --> 00:48:36,380 Because p, to the minus delta nu, times Kx. 780 00:48:40,430 --> 00:48:44,700 Now, this was unitless. 781 00:48:44,700 --> 00:48:47,430 This pressure to the minus delta nu sitting here, this 782 00:48:47,430 --> 00:48:50,900 has bars to the minus delta nu. 783 00:48:50,900 --> 00:48:53,420 Kx has units. 784 00:48:53,420 --> 00:48:55,420 It may not look like it because it's a bunch of mole 785 00:48:55,420 --> 00:48:58,840 fractions, and it certainly doesn't look 786 00:48:58,840 --> 00:48:59,830 like it has any units. 787 00:48:59,830 --> 00:49:11,250 Mole fractions or ratio, but it's got units. 788 00:49:11,250 --> 00:49:20,790 So let's rewrite Kx in terms of Kp. 789 00:49:20,790 --> 00:49:28,090 Rewrite Kx as equal to p to the minus delta nu, K sub p. 790 00:49:28,090 --> 00:49:34,240 And the units are in terms of bars, let's say, bar to the 791 00:49:34,240 --> 00:49:37,670 minus delta nu. 792 00:49:37,670 --> 00:49:39,890 Always want to check your units at the end of the day. 793 00:49:39,890 --> 00:49:41,710 If your units don't work out after you've done a 794 00:49:41,710 --> 00:49:48,950 calculation, you're in trouble. 795 00:49:48,950 --> 00:49:49,710 Any questions? 796 00:49:49,710 --> 00:49:51,490 Because we're going to end today, at here. 797 00:49:51,490 --> 00:49:51,830 Yes. 798 00:49:51,830 --> 00:49:57,070 STUDENT: [INAUDIBLE] 799 00:49:57,070 --> 00:49:58,680 PROFESSOR: In the notes it says Kp 800 00:49:58,680 --> 00:50:00,030 and Kx are both unitless. 801 00:50:00,030 --> 00:50:01,970 That is a mistake. 802 00:50:01,970 --> 00:50:05,480 I'm going to fix that, and put it on the Web. 803 00:50:05,480 --> 00:50:09,050 So it may be that there's a special case where the number 804 00:50:09,050 --> 00:50:12,140 of moles before and after are the same. 805 00:50:12,140 --> 00:50:15,370 But generally that is not true. 806 00:50:15,370 --> 00:50:19,390 Where did I say that? 807 00:50:19,390 --> 00:50:21,590 Oh, yeah. 808 00:50:21,590 --> 00:50:24,600 Both unitless. 809 00:50:24,600 --> 00:50:37,420 Well, that is actually true. 810 00:50:37,420 --> 00:50:38,570 Let me think through this. 811 00:50:38,570 --> 00:50:42,410 STUDENT: [INAUDIBLE] 812 00:50:42,410 --> 00:50:43,900 PROFESSOR: About one bar, one bar, one bar. 813 00:50:43,900 --> 00:50:44,630 That is actually true. 814 00:50:44,630 --> 00:50:49,270 They're both, that is true. 815 00:50:49,270 --> 00:50:53,750 They are both unitless. 816 00:50:53,750 --> 00:50:56,460 Well, this is a big boob on my part here. 817 00:50:56,460 --> 00:50:57,960 Because you are absolutely right. 818 00:50:57,960 --> 00:51:01,280 Because there's one bar sitting here. one bar, one 819 00:51:01,280 --> 00:51:05,720 bar, one bar, one bar, and the bars cancel out. 820 00:51:05,720 --> 00:51:08,850 Good catch. 821 00:51:08,850 --> 00:51:10,250 OK, the notes are right. 822 00:51:10,250 --> 00:51:13,060 My notes are right. 823 00:51:13,060 --> 00:51:14,500 Alright. 824 00:51:14,500 --> 00:51:16,440 Next time we'll talk about the temperature dependence and the 825 00:51:16,440 --> 00:51:19,260 pressure dependence of equilibrium constants.