1 00:00:00,000 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 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 hundred of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,580 --> 00:00:20,740 ocw.mit.edu. 8 00:00:20,740 --> 00:00:22,540 PROFESSOR: Last time, you talked about 9 00:00:22,540 --> 00:00:24,270 the Gibbs free energy. 10 00:00:24,270 --> 00:00:26,570 And the fundamental equations. 11 00:00:26,570 --> 00:00:31,820 And how powerful the fundamental equations were in 12 00:00:31,820 --> 00:00:35,410 being able to calculate anything from pressure, 13 00:00:35,410 --> 00:00:40,310 volume, temperature data. 14 00:00:40,310 --> 00:00:44,350 And you saw that the Gibbs free energy was especially 15 00:00:44,350 --> 00:00:51,730 important for everyday sort of processes. 16 00:00:51,730 --> 00:00:56,250 Because of the constant pressure constraint. 17 00:00:56,250 --> 00:00:59,590 And the fact that the intrinsic variables are 18 00:00:59,590 --> 00:01:02,910 pressure and temperature. 19 00:01:02,910 --> 00:01:05,060 Well, it turns out that the Gibbs free energy is even more 20 00:01:05,060 --> 00:01:05,850 important than that. 21 00:01:05,850 --> 00:01:08,580 And this is something that it took me a while to learn. 22 00:01:08,580 --> 00:01:10,690 I had to take thermodynamics many times to really 23 00:01:10,690 --> 00:01:13,770 appreciate how important that was. 24 00:01:13,770 --> 00:01:18,480 Even when I was doing research, at the beginning, I 25 00:01:18,480 --> 00:01:21,770 was a theorist, and I was trying to calculate different 26 00:01:21,770 --> 00:01:25,440 quantities of liquids and polymers and in these papers 27 00:01:25,440 --> 00:01:27,225 the first thing they did was to calculate the Gibbs free 28 00:01:27,225 --> 00:01:29,530 energy and I didn't quite appreciate why 29 00:01:29,530 --> 00:01:31,050 they were doing that. 30 00:01:31,050 --> 00:01:32,640 And the reason is, is if you've got the Gibbs free 31 00:01:32,640 --> 00:01:36,320 energy, you got really everything you need to know. 32 00:01:36,320 --> 00:01:39,000 Because you can get everything from the Gibbs free energy. 33 00:01:39,000 --> 00:01:42,880 And it really becomes the fundamental quantity that you 34 00:01:42,880 --> 00:01:44,890 want to have. 35 00:01:44,890 --> 00:01:49,620 So let me give you an example of how important that is, if 36 00:01:49,620 --> 00:01:52,660 you have an equation that describes the Gibbs free 37 00:01:52,660 --> 00:01:55,780 energy as a function of pressure and temperature, 38 00:01:55,780 --> 00:01:59,450 number of moles of different things. 39 00:01:59,450 --> 00:02:00,570 Different things you can calculate. 40 00:02:00,570 --> 00:02:03,290 So let's just start from the fundamental equation for the 41 00:02:03,290 --> 00:02:12,970 Gibbs free energy, dG is minus S dT plus V dp. 42 00:02:12,970 --> 00:02:19,330 And let's say that you've gotten some expression, G, as 43 00:02:19,330 --> 00:02:22,670 a function of temperature and pressure for your system. 44 00:02:22,670 --> 00:02:24,700 It could be, as we're going to see today that we're going to 45 00:02:24,700 --> 00:02:27,580 increase the number of variables here, by making the 46 00:02:27,580 --> 00:02:28,670 system more complicated. 47 00:02:28,670 --> 00:02:31,100 So what I'm saying, now what I'm going to say now, is going 48 00:02:31,100 --> 00:02:35,010 to be more general than just these two variables here. 49 00:02:35,010 --> 00:02:37,110 So you've gotten this. 50 00:02:37,110 --> 00:02:38,280 So you have this. 51 00:02:38,280 --> 00:02:40,070 You've got the fundamental equation. 52 00:02:40,070 --> 00:02:41,920 You've got all the other fundamental equations, and 53 00:02:41,920 --> 00:02:43,700 from there you can calculate all these quantities. 54 00:02:43,700 --> 00:02:47,270 For instance, you can calculate an expression for S. 55 00:02:47,270 --> 00:02:50,320 Because you know that S, from the fundamental equation, is 56 00:02:50,320 --> 00:02:56,920 just the derivative of G, with respect to T, keeping the 57 00:02:56,920 --> 00:02:58,330 pressure fixed. 58 00:02:58,330 --> 00:03:01,440 So you've got your equation for G. That translates into an 59 00:03:01,440 --> 00:03:07,360 equation for S. You can get volume, volume is not one of 60 00:03:07,360 --> 00:03:08,330 the variables. 61 00:03:08,330 --> 00:03:10,340 Temperature and pressure are the two knobs. 62 00:03:10,340 --> 00:03:13,030 But you can get volume out, because volume is the 63 00:03:13,030 --> 00:03:17,540 derivative of G, with respect to pressure. 64 00:03:17,540 --> 00:03:20,420 Keeping the temperature constant. 65 00:03:20,420 --> 00:03:24,090 And you've got S now, you got volume. 66 00:03:24,090 --> 00:03:26,410 Do you know where you, how did you define G 67 00:03:26,410 --> 00:03:27,380 in the first place? 68 00:03:27,380 --> 00:03:32,030 We define G as H minus dS. 69 00:03:32,030 --> 00:03:35,020 One of the many definitions of g. 70 00:03:35,020 --> 00:03:40,530 Reverse that, you've got now H as a function of G and 71 00:03:40,530 --> 00:03:44,350 temperature and entropy. 72 00:03:44,350 --> 00:03:48,820 Well, you've got an expression for G, we just calculated, we 73 00:03:48,820 --> 00:03:51,110 just showed we could get an expression for S, which is 74 00:03:51,110 --> 00:03:52,410 sitting right here. 75 00:03:52,410 --> 00:03:55,170 Temperature is a variable here, so now we have an 76 00:03:55,170 --> 00:04:00,490 expression for H. And you can go on like that with every 77 00:04:00,490 --> 00:04:06,230 variable that you've learned in this class already. 78 00:04:06,230 --> 00:04:12,260 For instance, u is H minus pV. 79 00:04:12,260 --> 00:04:15,200 Well, there's the H here. 80 00:04:15,200 --> 00:04:19,060 We have that equation for H. We have an equation for V, 81 00:04:19,060 --> 00:04:20,730 coming from here. 82 00:04:20,730 --> 00:04:23,860 So there's nothing unknown here. 83 00:04:23,860 --> 00:04:29,600 If we have an equation for G here in terms of temperature 84 00:04:29,600 --> 00:04:31,740 and pressure. 85 00:04:31,740 --> 00:04:33,740 Same thing for the Helmholtz free energy. 86 00:04:33,740 --> 00:04:36,830 You can even get the heat capacities out. 87 00:04:36,830 --> 00:04:42,010 Every single one of these interesting, useful quantities 88 00:04:42,010 --> 00:04:45,590 that one would want to calculate falls out from the 89 00:04:45,590 --> 00:04:51,380 Gibbs free energy here. 90 00:04:51,380 --> 00:04:56,730 Any questions on that important step? 91 00:04:56,730 --> 00:04:59,240 And, really, I can't believe how clueless I was when I 92 00:04:59,240 --> 00:05:00,650 started doing research. 93 00:05:00,650 --> 00:05:02,970 Because I would go through the process of calculating G, and 94 00:05:02,970 --> 00:05:05,340 getting G, and et cetera, et cetera. 95 00:05:05,340 --> 00:05:08,270 I wrote papers, you know, G equals blah blah blah. 96 00:05:08,270 --> 00:05:11,040 And I didn't realize that that's why people wanted to 97 00:05:11,040 --> 00:05:15,780 know G. Anyway, I know better now. 98 00:05:15,780 --> 00:05:21,750 So, there are a few things we can say about G that are 99 00:05:21,750 --> 00:05:24,430 fairly easy to calculate. 100 00:05:24,430 --> 00:05:29,980 For instance, if I look at liquids or solids. 101 00:05:29,980 --> 00:05:36,890 And I want to know how G changes with pressure. 102 00:05:36,890 --> 00:05:49,860 So, I know that the volume here dG/dp, that dG/dp is the 103 00:05:49,860 --> 00:05:50,400 volume here. 104 00:05:50,400 --> 00:05:54,220 So if I look under constant temperature, I pick my 105 00:05:54,220 --> 00:05:56,830 fundamental equation under constant temperature, and I 106 00:05:56,830 --> 00:05:59,270 want to know how G is changing, I integrate. 107 00:05:59,270 --> 00:06:04,570 So I have dG is equal to V dp. 108 00:06:04,570 --> 00:06:09,620 So if I change my, and I look at per mole, and if I change 109 00:06:09,620 --> 00:06:14,680 my pressure from p1 to p2, I integrate from p1 to p2, p1 to 110 00:06:14,680 --> 00:06:23,600 p2 here, final state minus the initial state is equal to the 111 00:06:23,600 --> 00:06:31,920 integral from p1 to p2, V dp per mole. 112 00:06:31,920 --> 00:06:35,000 And what can I say, for a liquid or a solid, the volume 113 00:06:35,000 --> 00:06:38,960 per mole, over a liquid or a solid, is small and it doesn't 114 00:06:38,960 --> 00:06:41,560 change very much. 115 00:06:41,560 --> 00:06:47,410 So V is small. 116 00:06:47,410 --> 00:06:51,320 And usually these solids and liquids, you can assume to be 117 00:06:51,320 --> 00:06:53,860 incompressible. 118 00:06:53,860 --> 00:06:55,765 Meaning, as you change the pressure, the 119 00:06:55,765 --> 00:06:57,700 volume doesn't change. 120 00:06:57,700 --> 00:07:00,670 It's a good approximation. 121 00:07:00,670 --> 00:07:05,310 So when you do your integral here, you get that G at the 122 00:07:05,310 --> 00:07:11,080 new pressure is G at the old pressure, then if this isn't 123 00:07:11,080 --> 00:07:13,190 changing very much with pressure, or not at all, then 124 00:07:13,190 --> 00:07:13,850 you can take it out. 125 00:07:13,850 --> 00:07:16,430 It's just a constant. 126 00:07:16,430 --> 00:07:21,900 Plus V times p2 minus p1. 127 00:07:21,900 --> 00:07:23,940 And so, this is the incompressible 128 00:07:23,940 --> 00:07:24,800 part, you take it out. 129 00:07:24,800 --> 00:07:28,190 The fact that it's small means that you can assume that this 130 00:07:28,190 --> 00:07:31,620 is zero, this whole thing is zero, that it's small enough. 131 00:07:31,620 --> 00:07:35,630 And then you see that G, approximately doesn't change 132 00:07:35,630 --> 00:07:37,830 with pressure. 133 00:07:37,830 --> 00:07:43,820 Tells you that G, for a liquid or solid, most of the time you 134 00:07:43,820 --> 00:07:46,230 can assume that it's just a function of temperature. 135 00:07:46,230 --> 00:07:49,310 Just like we saw for an ideal gas, that the energy and the 136 00:07:49,310 --> 00:07:51,280 enthalpy were just functions of temperature. 137 00:07:51,280 --> 00:07:55,430 And that's a useful approximation. 138 00:07:55,430 --> 00:07:57,640 It's useful, but it's not completely true. 139 00:07:57,640 --> 00:08:01,330 And if it were true, then there would not be any 140 00:08:01,330 --> 00:08:03,170 pressure dependents to phase transitions. 141 00:08:03,170 --> 00:08:04,620 And we know that's not the case. 142 00:08:04,620 --> 00:08:11,730 We know that if you press on water when it's close to the 143 00:08:11,730 --> 00:08:21,200 water liquid-solid transition, that you can lower the melting 144 00:08:21,200 --> 00:08:22,590 point of ice. 145 00:08:22,590 --> 00:08:25,120 You press on ice, and you press hard enough, and ice 146 00:08:25,120 --> 00:08:29,030 will melt, the temperature is closer to melting point. 147 00:08:29,030 --> 00:08:30,110 And we'll go through that. 148 00:08:30,110 --> 00:08:33,450 So, that means that there has to be some sort of pressure 149 00:08:33,450 --> 00:08:34,890 dependence, eventually. 150 00:08:34,890 --> 00:08:36,380 And we'll see that. 151 00:08:36,380 --> 00:08:39,210 This is just an approximation. 152 00:08:39,210 --> 00:08:39,940 What else can we do? 153 00:08:39,940 --> 00:08:44,750 We can calculate, also, for an ideal gas. 154 00:08:44,750 --> 00:08:50,910 Liquid and solid, we can do an ideal gas. 155 00:08:50,910 --> 00:08:53,990 So for an ideal gas, again, starting from the fundamental 156 00:08:53,990 --> 00:08:59,990 equation, we have dG equals V dp. 157 00:08:59,990 --> 00:09:04,780 We can do it per mole. 158 00:09:04,780 --> 00:09:10,530 So integrate both sides, G(T, p2) is equal to G at the 159 00:09:10,530 --> 00:09:13,630 initial pressure, plus the integral from 160 00:09:13,630 --> 00:09:15,890 p1 to p2, the volume. 161 00:09:15,890 --> 00:09:18,080 So instead of putting the volume, this is an ideal gas 162 00:09:18,080 --> 00:09:20,220 now, we can put the ideal gas law. 163 00:09:20,220 --> 00:09:22,870 So V is really RT over p. 164 00:09:22,870 --> 00:09:25,890 RT over p dp. 165 00:09:25,890 --> 00:09:27,730 We can integrate this. 166 00:09:27,730 --> 00:09:31,180 Get a log term out. 167 00:09:31,180 --> 00:09:42,080 G(T, p1) plus RT log p2 over p1. 168 00:09:42,080 --> 00:09:46,410 And then it's very useful to reference everything to the 169 00:09:46,410 --> 00:09:51,040 center state. p1 is equal to one bar, let's say. 170 00:09:51,040 --> 00:09:58,680 So if you take p1 equals one bar as our reference point, 171 00:09:58,680 --> 00:10:02,460 and get rid of the little subscript two here, we can 172 00:10:02,460 --> 00:10:07,170 write G of T at some pressure p, then is G and the little 173 00:10:07,170 --> 00:10:14,610 naught on top here means standard state one bar plus RT 174 00:10:14,610 --> 00:10:20,730 log p divided by one bar. 175 00:10:20,730 --> 00:10:26,350 And I put in a little dotted line here because very often 176 00:10:26,350 --> 00:10:28,550 you just write it without the one bar and bar. 177 00:10:28,550 --> 00:10:31,640 And you know that there has to be a one bar, because inside a 178 00:10:31,640 --> 00:10:33,500 log you can't have something with units. 179 00:10:33,500 --> 00:10:34,360 It has to be unitless. 180 00:10:34,360 --> 00:10:36,420 So you know if you have something with bar here, 181 00:10:36,420 --> 00:10:38,100 you've got to divide with something with bar, and there 182 00:10:38,100 --> 00:10:40,940 happens to be one bar here. 183 00:10:40,940 --> 00:10:46,850 So pressure p is G at its standard state plus RT log p. 184 00:10:46,850 --> 00:10:50,680 And this becomes a very useful, very 185 00:10:50,680 --> 00:10:56,060 useful, quantity to know. 186 00:10:56,060 --> 00:11:00,190 OK, so G is so important. 187 00:11:00,190 --> 00:11:06,430 And G per mole is so fundamental, that we're going 188 00:11:06,430 --> 00:11:09,470 to give it a special name. 189 00:11:09,470 --> 00:11:11,610 Not to make your life more complicated but just because 190 00:11:11,610 --> 00:11:12,410 it's just so important. 191 00:11:12,410 --> 00:11:17,040 We're going to call it the chemical potential. 192 00:11:17,040 --> 00:11:23,100 So G per mole, we're going to call mu. 193 00:11:23,100 --> 00:11:24,580 And that's going to be a chemical potential. 194 00:11:24,580 --> 00:11:25,750 We're going to do a lot more with the 195 00:11:25,750 --> 00:11:30,390 chemical potential today. 196 00:11:30,390 --> 00:11:33,590 And the reason why we call potential is because we 197 00:11:33,590 --> 00:11:40,240 already saw that if you've got something under constant 198 00:11:40,240 --> 00:11:48,640 pressure, temperature, that you want to use G as the 199 00:11:48,640 --> 00:11:50,290 variable to tell you whether something is going to be 200 00:11:50,290 --> 00:11:51,950 spontaneous or not. 201 00:11:51,950 --> 00:11:55,190 So you want G to go downhill. 202 00:11:55,190 --> 00:11:57,940 And so, we're going to be talking 203 00:11:57,940 --> 00:11:59,520 about chemical species. 204 00:11:59,520 --> 00:12:04,410 And instead of having a car up and down mountains, trying to 205 00:12:04,410 --> 00:12:07,060 go down to the valleys, we're going to have chemical species 206 00:12:07,060 --> 00:12:09,330 trying to find the valleys. 207 00:12:09,330 --> 00:12:12,850 The potential valleys. 208 00:12:12,850 --> 00:12:14,890 To get to equilibrium. 209 00:12:14,890 --> 00:12:16,870 And so we're going to be looking at the Gibbs free 210 00:12:16,870 --> 00:12:19,030 energy, or the Gibbs free energy per mole at that 211 00:12:19,030 --> 00:12:22,220 particular species, and it's going to want to be as small 212 00:12:22,220 --> 00:12:24,580 as possible. 213 00:12:24,580 --> 00:12:26,410 We're going to want to minimize the chemical 214 00:12:26,410 --> 00:12:27,600 potentials. 215 00:12:27,600 --> 00:12:29,370 And that's why it's called potential. 216 00:12:29,370 --> 00:12:34,800 It's like an energy. 217 00:12:34,800 --> 00:12:45,900 So, that's the end of the one component, thermodynamic 218 00:12:45,900 --> 00:12:48,170 background, before we get to multi-components. 219 00:12:48,170 --> 00:12:50,950 So it's a good time to stop again and see 220 00:12:50,950 --> 00:12:52,950 if there's any questions. 221 00:12:52,950 --> 00:13:01,850 Any issues. 222 00:13:01,850 --> 00:13:02,610 OK. 223 00:13:02,610 --> 00:13:06,120 So, so far we've done everything with one species. 224 00:13:06,120 --> 00:13:09,180 One ideal gas, one liquid, one solid. 225 00:13:09,180 --> 00:13:12,380 We haven't done anything with mixtures, except for maybe 226 00:13:12,380 --> 00:13:15,210 looking at the entropy of mixing. 227 00:13:15,210 --> 00:13:17,990 We saw the entropy of mixing was really important, because 228 00:13:17,990 --> 00:13:24,470 it drove processes where energy was constant. 229 00:13:24,470 --> 00:13:27,410 But most of what we care about in chemistry, at least in 230 00:13:27,410 --> 00:13:31,790 chemical reactions, species change. 231 00:13:31,790 --> 00:13:32,660 They get destroyed. 232 00:13:32,660 --> 00:13:36,120 New species get created. 233 00:13:36,120 --> 00:13:37,120 There are mixtures. 234 00:13:37,120 --> 00:13:38,440 It's pretty complicated. 235 00:13:38,440 --> 00:13:42,350 For instance, if I take a reaction of hydrogen gas plus 236 00:13:42,350 --> 00:13:49,630 chlorine gas to form two moles of HCl gas, I'm destroying 237 00:13:49,630 --> 00:13:52,200 hydrogen, I'm destroying chlorine, I'm making HCl in 238 00:13:52,200 --> 00:13:54,330 the gas phase. 239 00:13:54,330 --> 00:13:55,630 I get a big mixture at the end. 240 00:13:55,630 --> 00:13:59,040 I get three different kinds of species at the end. 241 00:13:59,040 --> 00:14:01,550 So the fundamental equations that I've been talking about, 242 00:14:01,550 --> 00:14:04,350 that we've been talking about, they're too 243 00:14:04,350 --> 00:14:06,400 simple for such a system. 244 00:14:06,400 --> 00:14:09,990 Because they all care about one species. 245 00:14:09,990 --> 00:14:13,480 Even more complicated, for instance, if I take hydrogen 246 00:14:13,480 --> 00:14:22,310 gas and oxygen gas and I mix them together to make water, 247 00:14:22,310 --> 00:14:26,540 liquid, for instance, not only do I have species that are 248 00:14:26,540 --> 00:14:31,240 changing, that are getting destroyed or created, in this 249 00:14:31,240 --> 00:14:34,250 case here the total number of moles is changing. 250 00:14:34,250 --> 00:14:37,690 And the phase is changing. 251 00:14:37,690 --> 00:14:40,450 Got all sorts of changes going on here. 252 00:14:40,450 --> 00:14:43,390 And so if I want to understand equilibrium, if I want to 253 00:14:43,390 --> 00:14:46,790 understand the direction of time for these more 254 00:14:46,790 --> 00:14:49,330 complicated processes, I have to be able to take into 255 00:14:49,330 --> 00:14:55,590 account, in an easy way, these mixing processes, these phase 256 00:14:55,590 --> 00:15:03,730 changes, these changes in the number of moles. 257 00:15:03,730 --> 00:15:07,000 And that's what we're going to talk about today. 258 00:15:07,000 --> 00:15:10,010 We're going to try to change our fundamental equations to 259 00:15:10,010 --> 00:15:12,140 make them a little bit more complicated so that we can 260 00:15:12,140 --> 00:15:13,500 deal with these sorts of problems. 261 00:15:13,500 --> 00:15:15,070 Because those are the real problems we need 262 00:15:15,070 --> 00:15:19,950 to keep track of. 263 00:15:19,950 --> 00:15:23,540 And the ultimate goal, then, of changing our fundamental 264 00:15:23,540 --> 00:15:27,320 questions is to derive equilibrium from first 265 00:15:27,320 --> 00:15:28,320 principles. 266 00:15:28,320 --> 00:15:31,190 To really understand chemical equilibrium, which you've all 267 00:15:31,190 --> 00:15:31,680 seen before. 268 00:15:31,680 --> 00:15:35,410 You've all used the chemical equilibrium constant K, you've 269 00:15:35,410 --> 00:15:36,770 done problems. 270 00:15:36,770 --> 00:15:40,400 But you've been given, basically, the equilibrium 271 00:15:40,400 --> 00:15:43,870 constant, and not really derived it, understood where 272 00:15:43,870 --> 00:15:48,790 it came from. 273 00:15:48,790 --> 00:15:53,800 OK, another simple example here, which is actually the 274 00:15:53,800 --> 00:15:57,820 one that we're going to be looking at in the first case. 275 00:15:57,820 --> 00:16:02,520 Where there's a change going on, is just to 276 00:16:02,520 --> 00:16:04,360 look at a phase change. 277 00:16:04,360 --> 00:16:07,840 H2O liquid going to H2O solid. 278 00:16:07,840 --> 00:16:10,040 There's a phase change, you can think of it as one 279 00:16:10,040 --> 00:16:12,590 species, the H2O liquid sort of changing 280 00:16:12,590 --> 00:16:14,710 into an H2O a solid. 281 00:16:14,710 --> 00:16:17,340 It's the same chemical in this case here, there's no change 282 00:16:17,340 --> 00:16:20,890 in the molecules. 283 00:16:20,890 --> 00:16:27,640 But it's still a change that we have to account for. 284 00:16:27,640 --> 00:16:32,910 Another example that's also simple like this, that you 285 00:16:32,910 --> 00:16:41,990 all, I'm sure, have seen before, suppose I take a cell. 286 00:16:41,990 --> 00:16:44,780 My cell here, full of water. 287 00:16:44,780 --> 00:16:50,930 And then I put my cell, let's say I take a human cell. 288 00:16:50,930 --> 00:16:53,000 My skin or something. 289 00:16:53,000 --> 00:17:03,350 And I take it and I put it in distilled water. 290 00:17:03,350 --> 00:17:07,600 What's going to happen to the cell? 291 00:17:07,600 --> 00:17:10,520 Is it going to be happy? 292 00:17:10,520 --> 00:17:15,580 What's going to happen to it? 293 00:17:15,580 --> 00:17:17,330 It's going to burst, right? 294 00:17:17,330 --> 00:17:18,470 Why is it going to burst? 295 00:17:18,470 --> 00:17:26,060 Anybody have an idea why it's going to burst? 296 00:17:26,060 --> 00:17:26,460 Yes. 297 00:17:26,460 --> 00:17:38,260 STUDENT: [INAUDIBLE] 298 00:17:38,260 --> 00:17:38,780 PROFESSOR: That's right. 299 00:17:38,780 --> 00:17:42,960 So the water wants to go from, you're completely right. 300 00:17:42,960 --> 00:17:46,330 But let me rephrase it in a thermodynamic language here. 301 00:17:46,330 --> 00:17:48,770 The water is going to go from a place of high chemical 302 00:17:48,770 --> 00:17:52,720 potential to low chemical potential. 303 00:17:52,720 --> 00:17:56,650 And the cell can't take all that water in there. 304 00:17:56,650 --> 00:17:59,290 The membrane's going to try to swell. 305 00:17:59,290 --> 00:18:00,700 And eventually burst, right? 306 00:18:00,700 --> 00:18:05,680 Same thing if you if you take a, go fishing, go to the 307 00:18:05,680 --> 00:18:09,040 Atlantic Ocean and then get a nice cod or something. 308 00:18:09,040 --> 00:18:13,950 Bring it back and on your sailboat, you dump it in a tub 309 00:18:13,950 --> 00:18:15,770 of fresh water. 310 00:18:15,770 --> 00:18:18,380 Is that cod going to be happy? 311 00:18:18,380 --> 00:18:22,390 It's not going to be happy at all, right, because its 312 00:18:22,390 --> 00:18:26,910 biology is geared towards living in salt water. 313 00:18:26,910 --> 00:18:31,170 And turns out that the chemical potential of water, 314 00:18:31,170 --> 00:18:32,860 in salt water, is lower than the chemical 315 00:18:32,860 --> 00:18:34,830 potential of pure water. 316 00:18:34,830 --> 00:18:37,590 And so when you put the cod in there, the chemical potential 317 00:18:37,590 --> 00:18:40,340 of the water and the cod, is lower than the chemical 318 00:18:40,340 --> 00:18:42,480 potential of the fresh water you have on the outside. 319 00:18:42,480 --> 00:18:44,180 And the fresh water wants to be at a 320 00:18:44,180 --> 00:18:44,965 lower chemical potential. 321 00:18:44,965 --> 00:18:48,410 It rushes into the cod, and well, the cod does what the 322 00:18:48,410 --> 00:18:50,960 cell does, when you put it in distilled water. 323 00:18:50,960 --> 00:18:53,980 It sort of bloats. 324 00:18:53,980 --> 00:18:57,920 It isn't very happy. 325 00:18:57,920 --> 00:19:01,150 OK, so but all these things are basically 326 00:19:01,150 --> 00:19:02,250 the same idea here. 327 00:19:02,250 --> 00:19:05,220 Where you have a complicated change, where species are 328 00:19:05,220 --> 00:19:07,320 mixing, and things like this. 329 00:19:07,320 --> 00:19:09,500 And it turns out the chemical potential is going to tell us 330 00:19:09,500 --> 00:19:12,430 all about how to think about that. 331 00:19:12,430 --> 00:19:16,650 That's why the chemical potential is so important. 332 00:19:16,650 --> 00:19:19,070 So we're going to go back to these two 333 00:19:19,070 --> 00:19:23,940 examples here many times. 334 00:19:23,940 --> 00:19:26,000 So let's take the simplest example here. 335 00:19:26,000 --> 00:19:32,900 Let's go back and derive some equations. 336 00:19:32,900 --> 00:19:37,280 Let's take our simplest example that's not a one 337 00:19:37,280 --> 00:19:41,000 species system, but has two species. 338 00:19:41,000 --> 00:19:45,820 Species 1 and 2. 339 00:19:45,820 --> 00:19:50,450 And n1 and n2 are the number of moles of species 1 and 2. 340 00:19:50,450 --> 00:19:52,490 And then we're going to see if I make a 341 00:19:52,490 --> 00:19:55,150 perturbation in my system. 342 00:19:55,150 --> 00:19:57,940 I change the number of moles of 1, or the 343 00:19:57,940 --> 00:19:58,850 number of moles of 2. 344 00:19:58,850 --> 00:20:03,030 How does this affect the Gibbs free energy? 345 00:20:03,030 --> 00:20:05,730 That's the question we're going to post. 346 00:20:05,730 --> 00:20:08,880 And our goal is to find a new fundamental equation for G 347 00:20:08,880 --> 00:20:12,670 that includes the number of moles of the different species 348 00:20:12,670 --> 00:20:13,730 as they change. 349 00:20:13,730 --> 00:20:16,730 Because in chemistry they're going to be changing. 350 00:20:16,730 --> 00:20:19,330 They're not going to be fixed. 351 00:20:19,330 --> 00:20:23,560 So what we want is just purely mathematically formally, take 352 00:20:23,560 --> 00:20:27,200 the differential of the Gibbs free energy, which we know is 353 00:20:27,200 --> 00:20:33,060 dG/dT, keeping pressure, the number of moles of 1, the 354 00:20:33,060 --> 00:20:35,910 number of 2 constant, dT. 355 00:20:35,910 --> 00:20:43,820 That is, dG/dp constant temperature, n1 and n2 dp. 356 00:20:43,820 --> 00:20:48,920 And then we have our two more variables now, dG/dn1, 357 00:20:48,920 --> 00:20:53,570 remember this is just a formal statement keeping temperature 358 00:20:53,570 --> 00:21:03,000 and pressure and n2 constant. dn1 plus dG/dn2, dn2 keeping 359 00:21:03,000 --> 00:21:07,090 temperature and pressure and n1 constant here. 360 00:21:07,090 --> 00:21:08,910 I'm not writing anything new here, I'm just telling you 361 00:21:08,910 --> 00:21:13,890 what the definition of the differential is here, for G. 362 00:21:13,890 --> 00:21:16,950 We already know what some of these quantities are. 363 00:21:16,950 --> 00:21:23,410 We know that this is the entropy, minus the entropy. 364 00:21:23,410 --> 00:21:28,930 This here is the volume. 365 00:21:28,930 --> 00:21:31,440 And I know the answer already. 366 00:21:31,440 --> 00:21:34,180 But I'm going to define it anyways. 367 00:21:34,180 --> 00:21:37,710 And we're going to prove it. 368 00:21:37,710 --> 00:21:41,610 I'm going to define this as the chemical 369 00:21:41,610 --> 00:21:46,210 potential for species 1. 370 00:21:46,210 --> 00:21:51,640 I'm going to define this, I'm going to give it a symbol, 371 00:21:51,640 --> 00:21:56,960 chemical potential mu, for species 2. 372 00:21:56,960 --> 00:22:02,490 So that I can write my new fundamental equation as dG as 373 00:22:02,490 --> 00:22:04,890 minus S dT. 374 00:22:04,890 --> 00:22:11,570 Plus V dp plus, and if I have more than two species present, 375 00:22:11,570 --> 00:22:17,060 the sum of all species in my mixture times the chemical 376 00:22:17,060 --> 00:22:21,380 potential of that species, dni. 377 00:22:21,380 --> 00:22:27,510 The change in the number of moles of that species. 378 00:22:27,510 --> 00:22:36,500 So, this quantity mu, that I've just defined, dG/dni, 379 00:22:36,500 --> 00:22:39,700 keeping the temperature, the pressure and all the n's, 380 00:22:39,700 --> 00:22:44,220 except for the i'th one constant, that is 381 00:22:44,220 --> 00:22:45,180 an intensive quantity. 382 00:22:45,180 --> 00:22:53,250 Because G scales with size, scales, with size of system. 383 00:22:53,250 --> 00:22:56,470 G is intensive. n, obviously, scales with 384 00:22:56,470 --> 00:22:59,800 the size of the system. 385 00:22:59,800 --> 00:23:04,000 Also intensive, and you take the ratio of two extensive 386 00:23:04,000 --> 00:23:05,640 variables, you get an intensive variable which 387 00:23:05,640 --> 00:23:08,610 doesn't care about the size of the system. 388 00:23:08,610 --> 00:23:09,890 Which is a good thing. 389 00:23:09,890 --> 00:23:12,730 For what we've been talking about. 390 00:23:12,730 --> 00:23:15,670 Intensive. 391 00:23:15,670 --> 00:23:19,500 If I'm talking about putting a freshwater fish and dumping it 392 00:23:19,500 --> 00:23:22,870 in the, putting it in the Atlantic Ocean, the chemical 393 00:23:22,870 --> 00:23:25,750 potential of the water in the Atlantic ocean better not care 394 00:23:25,750 --> 00:23:29,440 whether the Atlantic Ocean is huge or even huger. 395 00:23:29,440 --> 00:23:31,760 It just cares about the local environment. 396 00:23:31,760 --> 00:23:39,150 Just cares that it that wants to be in that freshwater fish. 397 00:23:39,150 --> 00:23:42,720 So the chemical potential is intensive. 398 00:23:42,720 --> 00:23:47,690 Just as we've written it down here for a 399 00:23:47,690 --> 00:23:50,150 single species system. 400 00:23:50,150 --> 00:23:51,710 It's the Gibbs energy, free energy per mole. 401 00:23:51,710 --> 00:23:53,330 Which we haven't proven yet. 402 00:23:53,330 --> 00:23:54,880 We haven't proven yet here. 403 00:23:54,880 --> 00:23:57,490 We've just defined it this way and we're going to prove that 404 00:23:57,490 --> 00:24:01,940 in fact the mu's are the Gibbs free energies per mole for 405 00:24:01,940 --> 00:24:07,820 each of the species in our system. 406 00:24:07,820 --> 00:24:14,240 OK, so this is now our first new fundamental equation. 407 00:24:14,240 --> 00:24:17,840 All we did was to add the sum here. 408 00:24:17,840 --> 00:24:19,970 Now, we started the lecture by saying that if you have the 409 00:24:19,970 --> 00:24:24,170 Gibbs free energy, you've got everything. 410 00:24:24,170 --> 00:24:30,220 And we wrote equations that are covered, were we can get 411 00:24:30,220 --> 00:24:35,160 S, we can get V, we can get H, we can get u, we can get A. So 412 00:24:35,160 --> 00:24:37,380 now that we have a fundamental question for G, we've got our 413 00:24:37,380 --> 00:24:40,470 new fundamental equations for everything else. 414 00:24:40,470 --> 00:24:45,290 Without really thinking too much. 415 00:24:45,290 --> 00:24:50,260 We go back to our definitions. 416 00:24:50,260 --> 00:24:54,840 Enthalpy is G minus TS. 417 00:24:54,840 --> 00:25:06,550 So dH is dG minus d of TS. dG, we've got our new fundamental 418 00:25:06,550 --> 00:25:09,450 equation for G. We plug it in here. 419 00:25:09,450 --> 00:25:12,440 Expand things out with a T dS here, we can write immediately 420 00:25:12,440 --> 00:25:19,840 a fundamental equation for H. dH is T dS. 421 00:25:19,840 --> 00:25:21,210 The beginning is going to look just like 422 00:25:21,210 --> 00:25:22,310 what you've seen before. 423 00:25:22,310 --> 00:25:23,540 Plus V dp. 424 00:25:23,540 --> 00:25:28,630 Plus an extra term, which is exactly the same extra term 425 00:25:28,630 --> 00:25:30,100 that we had in the fundamental equation for 426 00:25:30,100 --> 00:25:33,550 G. Exactly the same. 427 00:25:33,550 --> 00:25:37,090 And every one of the other fundamental questions can be 428 00:25:37,090 --> 00:25:40,600 derived in the similar way from G. And they're going to 429 00:25:40,600 --> 00:25:50,280 be what you had before minus S dT plus minus p dV plus the 430 00:25:50,280 --> 00:25:59,800 sum of the mu i's, dni, and du is T dS minus p 431 00:25:59,800 --> 00:26:05,420 dV plus i mu i dni. 432 00:26:08,940 --> 00:26:13,320 So immediately we can see that this mu, this quantity mu that 433 00:26:13,320 --> 00:26:16,930 we've defined as the derivative of G with respect 434 00:26:16,930 --> 00:26:22,080 to the n's, we can write many other equations for mu. 435 00:26:22,080 --> 00:26:24,880 There are many other ways to derive it. 436 00:26:24,880 --> 00:26:29,020 Because this is the differential for H. This is 437 00:26:29,020 --> 00:26:31,820 the first derivative of H with respect to entropy. 438 00:26:31,820 --> 00:26:34,290 This is the derivative of H with respect to pressure. 439 00:26:34,290 --> 00:26:36,840 And this is the derivative of H with respect to n. 440 00:26:36,840 --> 00:26:38,320 Just formally, that's what it is. 441 00:26:38,320 --> 00:26:41,020 When you write a differential. 442 00:26:41,020 --> 00:26:52,430 So formally, this is also mu i, is dH/dni, keeping, now, we 443 00:26:52,430 --> 00:26:56,590 have to be very careful, keeping the entropy and the 444 00:26:56,590 --> 00:26:57,410 pressure constant. 445 00:26:57,410 --> 00:26:59,900 Because those are the variables. 446 00:26:59,900 --> 00:27:03,430 Keeping the entropy, and the pressure, and all the other 447 00:27:03,430 --> 00:27:08,890 n's, constant. 448 00:27:08,890 --> 00:27:16,190 We can also write it as dA/dni, keeping the 449 00:27:16,190 --> 00:27:18,520 temperature and the volume constant. 450 00:27:18,520 --> 00:27:27,570 And all the other n's, or we can write it as du/dni, 451 00:27:27,570 --> 00:27:34,400 keeping that this entropy and the volume, and all the other 452 00:27:34,400 --> 00:27:35,420 n's, constant. 453 00:27:35,420 --> 00:27:42,060 So we have many ways to write the chemical potential. 454 00:27:42,060 --> 00:27:51,460 They give you all the same result. 455 00:27:51,460 --> 00:27:53,900 So this is the formal. 456 00:27:53,900 --> 00:27:59,640 Sort of the formal part of the chemical potential. 457 00:27:59,640 --> 00:28:01,530 Now, what we really want to show is that the chemical 458 00:28:01,530 --> 00:28:03,240 potential really is connected to the Gibbs 459 00:28:03,240 --> 00:28:05,060 free energy per mole. 460 00:28:05,060 --> 00:28:08,490 That's going to be the useful part. 461 00:28:08,490 --> 00:28:27,740 Let me get rid of this here. 462 00:28:27,740 --> 00:28:32,610 So I said earlier, at the beginning of the lecture, that 463 00:28:32,610 --> 00:28:34,880 the Gibbs free energy per mole was so important, we were 464 00:28:34,880 --> 00:28:37,010 going to call it the chemical potential. 465 00:28:37,010 --> 00:28:38,510 And I said that here. 466 00:28:38,510 --> 00:28:39,950 And then I said, well, we're going to define 467 00:28:39,950 --> 00:28:40,690 this here, the chemical. 468 00:28:40,690 --> 00:28:44,450 But I haven't equated the two yet. 469 00:28:44,450 --> 00:28:46,710 I haven't proven to you that in fact this quantity here, 470 00:28:46,710 --> 00:28:49,610 which we've formally defined as the derivative of G with 471 00:28:49,610 --> 00:28:53,045 respect to n, is the Gibbs free energy per 472 00:28:53,045 --> 00:28:54,930 mole, for this species. 473 00:28:54,930 --> 00:29:03,010 In fact, what we want to show is that if I take the sum of 474 00:29:03,010 --> 00:29:08,260 all the chemical potentials, times the number of moles per 475 00:29:08,260 --> 00:29:15,410 species, that that is the total Gibbs free energy. 476 00:29:15,410 --> 00:29:19,480 In other words, that the chemical potential for one 477 00:29:19,480 --> 00:29:25,390 species in the mixture is the Gibbs free energy per mole for 478 00:29:25,390 --> 00:29:29,960 that species. 479 00:29:29,960 --> 00:29:34,280 Once we have that idea, then we'll be able to talk about 480 00:29:34,280 --> 00:29:38,450 the concept of chemical potential as this thing that 481 00:29:38,450 --> 00:29:40,360 we can use to look at equilibrium. 482 00:29:40,360 --> 00:29:42,430 To look at going downhill for the species. 483 00:29:42,430 --> 00:29:45,340 To see why the cell bursts and all these things. 484 00:29:45,340 --> 00:29:47,420 Because now we understand that Gibbs free energy is so 485 00:29:47,420 --> 00:29:49,170 important for equilibrium. 486 00:29:49,170 --> 00:29:50,680 We don't understand that quite yet, with 487 00:29:50,680 --> 00:29:51,510 the chemical potential. 488 00:29:51,510 --> 00:29:54,690 So we got to make that relation here. 489 00:29:54,690 --> 00:29:58,120 We need to go from the formal definition to a relation that 490 00:29:58,120 --> 00:30:00,170 we can understand better, because it includes the Gibbs 491 00:30:00,170 --> 00:30:02,810 free energy. 492 00:30:02,810 --> 00:30:07,670 OK, so that's our goal now. 493 00:30:07,670 --> 00:30:08,790 So let's see. 494 00:30:08,790 --> 00:30:09,920 Let's formally do this now. 495 00:30:09,920 --> 00:30:15,360 Let's define, let's derive this. 496 00:30:15,360 --> 00:30:15,670 OK. 497 00:30:15,670 --> 00:30:21,890 So remember, our goal in this derivation is to show that 498 00:30:21,890 --> 00:30:24,620 this is true. 499 00:30:24,620 --> 00:30:27,150 Or that this is true, here. 500 00:30:27,150 --> 00:30:28,770 And again, we're going to start with the 501 00:30:28,770 --> 00:30:30,010 simplest system possible. 502 00:30:30,010 --> 00:30:33,890 We're going to start with a two component system. 503 00:30:33,890 --> 00:30:46,300 And we can easily generalize to multi-component. 504 00:30:46,300 --> 00:30:50,310 And in our derivation, what we're going to be after is, 505 00:30:50,310 --> 00:30:51,750 we're going to start with the Gibbs free energy, because 506 00:30:51,750 --> 00:30:53,640 that's where we always start with. 507 00:30:53,640 --> 00:30:57,360 And we're going to remember that by definition, mu i is 508 00:30:57,360 --> 00:31:04,640 dG/dni, So if somehow in our derivation dG/dni falls out, 509 00:31:04,640 --> 00:31:05,640 that would be great. 510 00:31:05,640 --> 00:31:08,120 Because we'll be able to replace this derivative with 511 00:31:08,120 --> 00:31:09,460 the chemical potential. 512 00:31:09,460 --> 00:31:11,650 So the goal was to find something where this falls 513 00:31:11,650 --> 00:31:17,360 out, so we can replace it with the chemical potential. 514 00:31:17,360 --> 00:31:18,700 We're going to start with the fact that G is 515 00:31:18,700 --> 00:31:21,050 an extensive variable. 516 00:31:21,050 --> 00:31:24,380 So if I take G at a temperature and pressure times 517 00:31:24,380 --> 00:31:28,800 some scaling factor for the size of my system, lambda, 518 00:31:28,800 --> 00:31:31,770 number of moles of n1, lambda times the number of moles of 519 00:31:31,770 --> 00:31:35,250 n2, if I double the size of the system, 520 00:31:35,250 --> 00:31:36,290 lambda is equal to 2. 521 00:31:36,290 --> 00:31:38,820 If I half it, lambda is equal to 1/2. 522 00:31:38,820 --> 00:31:41,290 Because it's extensive, this is the same thing as lambda 523 00:31:41,290 --> 00:31:46,600 times G of temperature and pressure, n1, n2. 524 00:31:46,600 --> 00:31:50,190 Just rewriting the fact that Gibbs free energy is an 525 00:31:50,190 --> 00:31:52,390 extensive property. 526 00:31:52,390 --> 00:31:54,470 And lambda is an arbitrary number here. 527 00:31:54,470 --> 00:31:56,480 Arbitrary variable. 528 00:31:56,480 --> 00:31:58,420 Now I'm going to take the derivative of both sides with 529 00:31:58,420 --> 00:32:00,420 respect to lambda. 530 00:32:00,420 --> 00:32:04,550 So I'm going to take d d lambda of this side here. 531 00:32:04,550 --> 00:32:12,010 And d d lambda of that side here. 532 00:32:12,010 --> 00:32:14,070 Now, lambda here is inside the variable here. 533 00:32:14,070 --> 00:32:16,900 So I'm going to have to use the chain rule. 534 00:32:16,900 --> 00:32:18,760 To do this properly. 535 00:32:18,760 --> 00:32:30,290 So this is going to be dG/d lambda n1, there's lambda 536 00:32:30,290 --> 00:32:32,510 sitting in the variable lambda n1 here, times 537 00:32:32,510 --> 00:32:37,620 d lambda n1 d lambda. 538 00:32:37,620 --> 00:32:47,950 Plus dG/d lambda n2 times d lambda n2 d lambda. 539 00:32:47,950 --> 00:32:49,470 Using the chain rule. 540 00:32:49,470 --> 00:32:50,930 And on this side here, lambda's sitting 541 00:32:50,930 --> 00:32:51,820 straight out here. 542 00:32:51,820 --> 00:32:53,370 So this is very easy. 543 00:32:53,370 --> 00:32:55,020 This is G(T, p, n1, n2). 544 00:33:06,390 --> 00:33:07,130 Now, this is good. 545 00:33:07,130 --> 00:33:09,340 Because this is what I'm looking for. 546 00:33:09,340 --> 00:33:11,850 I'm looking for the derivative of g with respect to the 547 00:33:11,850 --> 00:33:12,850 number of moles. 548 00:33:12,850 --> 00:33:14,290 Because that's the chemical potential. 549 00:33:14,290 --> 00:33:17,470 That was my goal up here, to make sure in the derivation, 550 00:33:17,470 --> 00:33:19,880 somehow, this came out. 551 00:33:19,880 --> 00:33:24,820 And so it's coming out right there. 552 00:33:24,820 --> 00:33:28,460 Right here and right here. 553 00:33:28,460 --> 00:33:31,800 Since the number of moles is lambda n1, that first 554 00:33:31,800 --> 00:33:33,650 derivative here is just the chemical potential 555 00:33:33,650 --> 00:33:35,450 of species 1 there. 556 00:33:35,450 --> 00:33:39,670 So mu 1, then we have d lambda n1 d lambda. 557 00:33:39,670 --> 00:33:43,170 Well, d lambda n1 d lambda, that's just n1. 558 00:33:43,170 --> 00:33:47,230 It's lambda times the number n1 that doesn't have anything 559 00:33:47,230 --> 00:33:48,070 to do with lambda. 560 00:33:48,070 --> 00:33:50,680 So this is n1. 561 00:33:50,680 --> 00:33:53,900 This is the chemical potential of species 2. 562 00:33:53,900 --> 00:33:56,080 Again, the derivative of lambda n2 with respect to 563 00:33:56,080 --> 00:33:58,400 lambda is just n2. 564 00:33:58,400 --> 00:34:04,830 And there is G here. 565 00:34:04,830 --> 00:34:07,260 It's a fairly simple derivation, but it gets us 566 00:34:07,260 --> 00:34:08,750 exactly what we want. 567 00:34:08,750 --> 00:34:13,280 An association between this formal definition of mu, up 568 00:34:13,280 --> 00:34:17,340 here, directly from taking the differential. 569 00:34:17,340 --> 00:34:20,480 How much more formal can you be, mathematically, here? 570 00:34:20,480 --> 00:34:24,120 To associating this formal definition to 571 00:34:24,120 --> 00:34:27,540 the Gibbs free energy. 572 00:34:27,540 --> 00:34:34,880 Number of moles times mu 1, number of moles times mu 2, 573 00:34:34,880 --> 00:34:40,630 this is the Gibbs free energy per mole of species 1. 574 00:34:40,630 --> 00:34:45,410 Gibbs free energy per mole of species 2. 575 00:34:45,410 --> 00:34:49,040 The sum of all the species of the Gibbs free energy per mole 576 00:34:49,040 --> 00:34:52,200 of species i times the number of moles of species i 577 00:34:52,200 --> 00:34:57,850 is G. Or, mu i. 578 00:34:57,850 --> 00:35:00,600 Voila, we've done it. 579 00:35:00,600 --> 00:35:02,500 This is what we wanted. 580 00:35:02,500 --> 00:35:06,360 The chemical potential is the Gibbs free energy per mole. 581 00:35:06,360 --> 00:35:09,120 And in the mixture, it's the Gibbs free energy per mole of 582 00:35:09,120 --> 00:35:11,800 the individual species in that mixture. 583 00:35:11,800 --> 00:35:13,730 And if you want to know what the total Gibbs free energy 584 00:35:13,730 --> 00:35:16,320 is, because if you have an equilibrium, what you care 585 00:35:16,320 --> 00:35:18,240 about is the total Gibbs free energy. 586 00:35:18,240 --> 00:35:21,750 It's not the Gibbs free energy for one particular species. 587 00:35:21,750 --> 00:35:24,020 What's going to tell you whether you have a minimum or 588 00:35:24,020 --> 00:35:29,220 not in your system, whether you're at equilibrium, where 589 00:35:29,220 --> 00:35:32,500 you're at the lowest state possible, is the total Gibbs 590 00:35:32,500 --> 00:35:35,350 free energy. 591 00:35:35,350 --> 00:35:37,010 Now we'll be able to manipulate chemical 592 00:35:37,010 --> 00:35:40,120 potentials, of the individual species, to get 593 00:35:40,120 --> 00:35:45,810 at this number here. 594 00:35:45,810 --> 00:35:46,820 Any questions? 595 00:35:46,820 --> 00:35:50,810 This is really, we're going to see this over 596 00:35:50,810 --> 00:35:51,500 and over again now. 597 00:35:51,500 --> 00:35:52,030 This chemical potential. 598 00:35:52,030 --> 00:35:53,100 This idea. 599 00:35:53,100 --> 00:36:04,460 And it's not an easy concept. 600 00:36:04,460 --> 00:36:06,560 OK, let me give you an example, then, 601 00:36:06,560 --> 00:36:13,140 of the water melting. 602 00:36:13,140 --> 00:36:17,350 And how the chemical potential comes in, now, instead of 603 00:36:17,350 --> 00:36:18,470 using the chemical potential. 604 00:36:18,470 --> 00:36:30,160 Instead of the Gibbs free energy. 605 00:36:30,160 --> 00:36:31,290 This is the phase transition. 606 00:36:31,290 --> 00:36:35,190 But it's not very different than the cell bursting when 607 00:36:35,190 --> 00:36:39,410 you put it in distilled water. 608 00:36:39,410 --> 00:36:44,570 So, we take a glass of water with an ice cube in it. 609 00:36:44,570 --> 00:36:47,580 H2O liquid. 610 00:36:47,580 --> 00:36:50,640 This is H2O solid. 611 00:36:50,640 --> 00:36:53,920 And I'm looking at the melting process. 612 00:36:53,920 --> 00:36:58,470 I'm looking at a process where I take a small number of 613 00:36:58,470 --> 00:37:01,950 molecules of water from the solid phase. 614 00:37:01,950 --> 00:37:04,550 And I bring them to the liquid phase. 615 00:37:04,550 --> 00:37:10,570 And I want to know, is this process spontaneous or in 616 00:37:10,570 --> 00:37:17,970 equilibrium, or not possible? 617 00:37:17,970 --> 00:37:19,100 Is this process going to go on? 618 00:37:19,100 --> 00:37:21,010 Is the direction of time that this is melting. 619 00:37:21,010 --> 00:37:24,890 And I want to do this formally thermodynamically. 620 00:37:24,890 --> 00:37:27,370 In terms of the chemical potentials. 621 00:37:27,370 --> 00:37:30,750 That's going to be what we're going to be talking about. 622 00:37:30,750 --> 00:37:35,020 So, formally then, what's going on is, I'm taking nl 623 00:37:35,020 --> 00:37:41,960 moles of liquid water, H2O liquid, which is in here. 624 00:37:41,960 --> 00:37:47,520 Plus ns moles of solid water. 625 00:37:47,520 --> 00:37:50,670 And I'm, this is my initial state. 626 00:37:50,670 --> 00:37:59,440 My final state is nl, plus a small number of moles, dn, of 627 00:37:59,440 --> 00:38:06,900 H2O liquids, H2O liquid, plus ns minus dn, there's a 628 00:38:06,900 --> 00:38:09,150 conservation of the number of molecules here. 629 00:38:09,150 --> 00:38:10,880 Whatever I add to the liquid has to come 630 00:38:10,880 --> 00:38:13,040 from the solid here. 631 00:38:13,040 --> 00:38:19,440 Of H2O solid, of ice. 632 00:38:19,440 --> 00:38:23,210 And to know if this is spontaneous or not, if this is 633 00:38:23,210 --> 00:38:26,540 done under constant temperature and pressure, what 634 00:38:26,540 --> 00:38:31,070 variable should we look at? 635 00:38:31,070 --> 00:38:31,900 G, right. 636 00:38:31,900 --> 00:38:33,640 We want to look at the Gibbs free energy. 637 00:38:33,640 --> 00:38:38,260 So what is G doing during this process here? 638 00:38:38,260 --> 00:38:42,530 What is delta G here? 639 00:38:42,530 --> 00:38:45,010 Well, we have a way of doing it now, in terms of the 640 00:38:45,010 --> 00:38:45,630 chemical potentials. 641 00:38:45,630 --> 00:38:51,720 Because we've just shown that this is the case here. 642 00:38:51,720 --> 00:38:56,140 So G is the sum of the chemical potentials times the 643 00:38:56,140 --> 00:38:58,020 number of moles in the species. 644 00:38:58,020 --> 00:39:07,290 Therefore, delta G is going to be equal to mu for the number 645 00:39:07,290 --> 00:39:09,240 of moles of l. 646 00:39:09,240 --> 00:39:20,320 Liquid, dn, number of moles of liquid, plus mu s dns. 647 00:39:20,320 --> 00:39:22,840 So the change in G is going to be equal to the chemical 648 00:39:22,840 --> 00:39:25,700 potential times the change in the species, which is in a 649 00:39:25,700 --> 00:39:28,780 liquid form, plus the chemical potential of the solid. 650 00:39:28,780 --> 00:39:32,520 Times the change in the species in the solid form. 651 00:39:32,520 --> 00:39:37,700 Now, dns is equal to minus dnl. 652 00:39:37,700 --> 00:39:39,300 This is what we did right here. 653 00:39:39,300 --> 00:39:41,880 You take a certain number of moles from the solid form. 654 00:39:41,880 --> 00:39:42,940 You put it to the liquid form. 655 00:39:42,940 --> 00:39:44,490 That the dn up here. 656 00:39:44,490 --> 00:39:46,970 And you've got to have the negative of it, up here. 657 00:39:46,970 --> 00:39:49,250 So dns is minus dnl. 658 00:39:49,250 --> 00:39:53,380 Which is minus dn. 659 00:39:53,380 --> 00:40:11,510 Delta G is dn times mu l minus mu solid. 660 00:40:11,510 --> 00:40:16,800 So now we can rephrase this, it's all rephrasing. 661 00:40:16,800 --> 00:40:18,300 It's all basically the same thing. 662 00:40:18,300 --> 00:40:22,490 But, we can rephrase this process by asking the 663 00:40:22,490 --> 00:40:28,890 question, is the chemical potential of the liquid 664 00:40:28,890 --> 00:40:33,150 greater than, equal to, or less than the chemical 665 00:40:33,150 --> 00:40:36,320 potential of the solid? 666 00:40:36,320 --> 00:40:37,640 Of the water in the solid. 667 00:40:37,640 --> 00:40:40,530 So the chemical potential of the water in the liquid phase 668 00:40:40,530 --> 00:40:44,160 is greater than the chemical potential of the water in the 669 00:40:44,160 --> 00:40:49,040 solid phase, mu l is greater than mu s, then delta G 670 00:40:49,040 --> 00:40:52,560 becomes positive. 671 00:40:52,560 --> 00:40:59,320 In that case, delta G is greater than zero. 672 00:40:59,320 --> 00:41:03,150 And that's not going to happen. 673 00:41:03,150 --> 00:41:06,840 On the other hand, if the chemical potential of the 674 00:41:06,840 --> 00:41:11,520 water molecules in the liquid phase is smaller than the 675 00:41:11,520 --> 00:41:16,450 chemical potential of the water in the solid phase, mu s 676 00:41:16,450 --> 00:41:19,870 is bigger than mu l, this becomes a negative number. 677 00:41:19,870 --> 00:41:24,730 Delta G is less than zero. 678 00:41:24,730 --> 00:41:27,440 And this will happen spontaneously. 679 00:41:27,440 --> 00:41:30,200 So that illustrates this idea, that the chemical potential of 680 00:41:30,200 --> 00:41:33,890 a species will want to go, so the species will want to go, 681 00:41:33,890 --> 00:41:37,410 where it can minimize its chemical potential. 682 00:41:37,410 --> 00:41:41,200 So in this case here, when we have the spontaneous process 683 00:41:41,200 --> 00:41:44,490 of the water, of the ice cube, melting, you can think of it 684 00:41:44,490 --> 00:41:47,240 as these water molecules that are in the ice 685 00:41:47,240 --> 00:41:49,350 phase looking around. 686 00:41:49,350 --> 00:41:51,000 They know what their chemical potential here 687 00:41:51,000 --> 00:41:51,670 is in the ice phase. 688 00:41:51,670 --> 00:41:53,290 They're looking around, they're looking to see the 689 00:41:53,290 --> 00:41:54,270 water phase. 690 00:41:54,270 --> 00:41:57,960 And they see that in the water phase, those water molecules 691 00:41:57,960 --> 00:41:59,730 have a smaller chemical potential. 692 00:41:59,730 --> 00:42:02,320 They're happier. 693 00:42:02,320 --> 00:42:08,680 And so these solid water molecules are jealous. 694 00:42:08,680 --> 00:42:11,500 And they want to go in the water phase. 695 00:42:11,500 --> 00:42:13,870 And the ice cube's going to melt. 696 00:42:13,870 --> 00:42:15,790 And it all has to do with this difference in chemical 697 00:42:15,790 --> 00:42:19,160 potentials for the water. 698 00:42:19,160 --> 00:42:22,060 And the same thing happens for the water molecules that are 699 00:42:22,060 --> 00:42:24,170 inside or outside of that cell that you put in 700 00:42:24,170 --> 00:42:25,590 the distilled water. 701 00:42:25,590 --> 00:42:27,490 The water molecules in the distilled water have a 702 00:42:27,490 --> 00:42:31,400 chemical potential which is higher than the water 703 00:42:31,400 --> 00:42:34,580 molecules inside the cell. 704 00:42:34,580 --> 00:42:36,220 And they don't want to be like that. 705 00:42:36,220 --> 00:42:39,740 They want to change, the system wants to change, until 706 00:42:39,740 --> 00:42:41,960 the water molecules couldn't care less whether they're in 707 00:42:41,960 --> 00:42:46,580 the water phase, or outside or inside the cell. 708 00:42:46,580 --> 00:42:51,120 The system is going to change until the water molecules have 709 00:42:51,120 --> 00:42:53,850 the same chemical potential everywhere. 710 00:42:53,850 --> 00:43:00,540 Where they don't have to choose one place or the other. 711 00:43:00,540 --> 00:43:02,880 And so that gives us, immediately, what we're going 712 00:43:02,880 --> 00:43:05,530 to need when we talk about equilibrium. 713 00:43:05,530 --> 00:43:09,450 Equilibrium, chemical equilibrium, is going to be 714 00:43:09,450 --> 00:43:13,680 where the chemical potential of a species is the same 715 00:43:13,680 --> 00:43:17,240 everywhere in the system. 716 00:43:17,240 --> 00:43:21,900 So at 0 degrees Celsius, one bar, which is the melting 717 00:43:21,900 --> 00:43:26,530 point of water, the chemical potential of a molecule of 718 00:43:26,530 --> 00:43:30,120 water in the ice phase and in the liquid phase is the same. 719 00:43:30,120 --> 00:43:31,620 That's the definition of the melting point. 720 00:43:31,620 --> 00:43:32,065 It doesn't care. 721 00:43:32,065 --> 00:43:33,460 It could go either way. 722 00:43:33,460 --> 00:43:34,950 It's an equilibrium. 723 00:43:34,950 --> 00:43:39,420 You take an ice cube, water, liquid water, 0 degrees 724 00:43:39,420 --> 00:43:40,950 Celsius, one bar. 725 00:43:40,950 --> 00:43:42,910 You come back three days later. 726 00:43:42,910 --> 00:43:43,900 It's the same. 727 00:43:43,900 --> 00:43:45,610 Come back a week later, it's the same. 728 00:43:45,610 --> 00:43:49,200 It's an equilibrium. 729 00:43:49,200 --> 00:43:52,050 Chemical potential of the water species is the same 730 00:43:52,050 --> 00:43:52,920 everywhere. 731 00:43:52,920 --> 00:43:54,000 It's an equilibrium. 732 00:43:54,000 --> 00:43:59,200 And I'm just repeating that because this is so important. 733 00:43:59,200 --> 00:44:04,810 OK, any questions? 734 00:44:04,810 --> 00:44:08,720 The last thing we're going to do is to illustrate also the 735 00:44:08,720 --> 00:44:19,150 importance of mixing to the chemical potential. 736 00:44:19,150 --> 00:44:27,210 So I'm going to set up sort of an arbitrary system here. 737 00:44:27,210 --> 00:44:32,390 This is kind of like the cell, or the fish, also, idea. 738 00:44:32,390 --> 00:44:35,790 I'm going to put a system where on one side I have a 739 00:44:35,790 --> 00:44:41,295 pure gas, A. On the other side, I have a mixture of A 740 00:44:41,295 --> 00:44:46,110 and B. And here is going to be a membrane that only allows A 741 00:44:46,110 --> 00:44:51,450 to go through. 742 00:44:51,450 --> 00:44:54,710 Only A can go through that membrane. 743 00:44:54,710 --> 00:44:57,215 They're going to be partial pressures in here, p prime B, 744 00:44:57,215 --> 00:45:01,930 and p prime A. For the gas pressures on that side, and 745 00:45:01,930 --> 00:45:05,340 pressure on that side is p sub A. 746 00:45:05,340 --> 00:45:11,240 And my goal in this example here is to show that if I 747 00:45:11,240 --> 00:45:16,970 compare the chemical potential of a species in a mixture, 748 00:45:16,970 --> 00:45:23,320 where the temperature and the pressure total are T and p, 749 00:45:23,320 --> 00:45:26,150 and I compare that to the chemical potential of the same 750 00:45:26,150 --> 00:45:31,040 species when it's pure, when it's not mixed with anything 751 00:45:31,040 --> 00:45:33,560 else, under the same temperature and pressure 752 00:45:33,560 --> 00:45:41,110 conditions, that, in fact that that equals sign is not there. 753 00:45:41,110 --> 00:45:42,240 That's not what I'm trying to show. 754 00:45:42,240 --> 00:45:43,230 I'm trying to show that there's a 755 00:45:43,230 --> 00:45:47,370 less-than sign right here. 756 00:45:47,370 --> 00:45:51,640 To show that if you take, again, this is the cell idea. 757 00:45:51,640 --> 00:45:55,050 If you take the water and the distilled water, under 758 00:45:55,050 --> 00:45:56,140 constant temperature and pressure 759 00:45:56,140 --> 00:45:59,400 conditions, it's pure. 760 00:45:59,400 --> 00:46:01,240 And it's looking at the cell. 761 00:46:01,240 --> 00:46:03,356 And inside the cell is there, boy, is there 762 00:46:03,356 --> 00:46:04,820 a mixture of things. 763 00:46:04,820 --> 00:46:06,730 There's salts, there are proteins, there are all sorts 764 00:46:06,730 --> 00:46:10,670 of things, right? 765 00:46:10,670 --> 00:46:12,350 It's a mixed system. 766 00:46:12,350 --> 00:46:14,230 The water in that mixed system, under the same 767 00:46:14,230 --> 00:46:16,910 temperature and pressure conditions, the chemical 768 00:46:16,910 --> 00:46:19,970 potential of that water molecule is less. 769 00:46:19,970 --> 00:46:23,560 And that's just an entropy thing. 770 00:46:23,560 --> 00:46:25,080 Entropy wants to increase. 771 00:46:25,080 --> 00:46:32,940 It just wants to be in a place with high energy. 772 00:46:32,940 --> 00:46:34,240 It's the Gibbs free energy. 773 00:46:34,240 --> 00:46:36,240 Gibbs free energy has enthalpy and entropy 774 00:46:36,240 --> 00:46:37,230 incorporated into it. 775 00:46:37,230 --> 00:46:39,860 The enthalpy's not doing anything. 776 00:46:39,860 --> 00:46:43,820 It's all driven by entropy. 777 00:46:43,820 --> 00:46:49,250 So this is what we're going to try to show. 778 00:46:49,250 --> 00:46:51,960 And I'm not going to get to it today. 779 00:46:51,960 --> 00:46:56,180 We'll start with it on Wednesday. 780 00:46:56,180 --> 00:47:01,250 And I'll let your ruminate on this for the next few days.