1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:02 The following content is provided under a Creative 3 00:00:02 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:10 offer high quality educational resources for free. 6 00:00:10 --> 00:00:13 To make a donation or view additional materials from 7 00:00:13 --> 00:00:17 hundred of MIT courses, visit MIT OpenCourseWare 8 00:00:17 --> 00:00:20 at ocw.mit.edu. 9 00:00:20 --> 00:00:22 PROFESSOR: Last time, you talked about the 10 00:00:22 --> 00:00:24 Gibbs free energy. 11 00:00:24 --> 00:00:26 And the fundamental equations. 12 00:00:26 --> 00:00:30 And how powerful the fundamental equations were 13 00:00:30 --> 00:00:35 in being able to calculate anything from pressure, 14 00:00:35 --> 00:00:40 volume, temperature data. 15 00:00:40 --> 00:00:45 And you saw that the Gibbs free energy was especially important 16 00:00:45 --> 00:00:51 for everyday sort of processes. 17 00:00:51 --> 00:00:56 Because of the constant pressure constraint. 18 00:00:56 --> 00:00:59 And the fact that the intrinsic variables are pressure 19 00:00:59 --> 00:01:02 and temperature. 20 00:01:02 --> 00:01:04 Well, it turns out that the Gibbs free energy is even 21 00:01:04 --> 00:01:05 more important than that. 22 00:01:05 --> 00:01:08 And this is something that it took me a while to learn. 23 00:01:08 --> 00:01:11 I had to take thermodynamics many times to really appreciate 24 00:01:11 --> 00:01:13 how important that was. 25 00:01:13 --> 00:01:18 Even when I was doing research, at the beginning, I was a 26 00:01:18 --> 00:01:22 theorist, and I was trying to calculate different quantities 27 00:01:22 --> 00:01:25 of liquids and polymers and in these papers the first thing 28 00:01:25 --> 00:01:28 they did was to calculate the Gibbs free energy and I 29 00:01:28 --> 00:01:31 didn't quite appreciate why they were doing that. 30 00:01:31 --> 00:01:34 And the reason is, is if you've got the Gibbs free energy, you 31 00:01:34 --> 00:01:36 got really everything you need to know. 32 00:01:36 --> 00:01:39 Because you can get everything from the Gibbs free energy. 33 00:01:39 --> 00:01:42 And it really becomes the fundamental quantity 34 00:01:42 --> 00:01:44 that you want to have. 35 00:01:44 --> 00:01:49 So let me give you an example of how important that is, if 36 00:01:49 --> 00:01:53 you have an equation that describes the Gibbs free energy 37 00:01:53 --> 00:01:56 as a function of pressure and temperature, number of 38 00:01:56 --> 00:01:59 moles of different things. 39 00:01:59 --> 00:02:00 Different things you can calculate. 40 00:02:00 --> 00:02:03 So let's just start from the fundamental equation for the 41 00:02:03 --> 00:02:12 Gibbs free energy, dG is minus S dT plus V dp. 42 00:02:12 --> 00:02:19 And let's say that you've gotten some expression, G, as 43 00:02:19 --> 00:02:22 a function of temperature and pressure for your system. 44 00:02:22 --> 00:02:24 It could be, as we're going to see today that we're going to 45 00:02:24 --> 00:02:27 increase the number of variables here, by making the 46 00:02:27 --> 00:02:28 system more complicated. 47 00:02:28 --> 00:02:31 So what I'm saying, now what I'm going to say now, is going 48 00:02:31 --> 00:02:35 to be more general than just these two variables here. 49 00:02:35 --> 00:02:37 So you've gotten this. 50 00:02:37 --> 00:02:38 So you have this. 51 00:02:38 --> 00:02:40 You've got the fundamental equation. 52 00:02:40 --> 00:02:41 You've got all the other fundamental equations, and 53 00:02:41 --> 00:02:43 from there you can calculate all these quantities. 54 00:02:43 --> 00:02:47 For instance, you can calculate an expression for S. 55 00:02:47 --> 00:02:50 Because you know that S, from the fundamental equation, is 56 00:02:50 --> 00:02:55 just the derivative of G, with respect to T, keeping 57 00:02:55 --> 00:02:58 the pressure fixed. 58 00:02:58 --> 00:03:00 So you've got your equation for G. 59 00:03:00 --> 00:03:04 That translates into an equation for S. 60 00:03:04 --> 00:03:08 You can get volume, volume is not one of the variables. 61 00:03:08 --> 00:03:10 Temperature and pressure are the two knobs. 62 00:03:10 --> 00:03:13 But you can get volume out, because volume is the 63 00:03:13 --> 00:03:17 derivative of G, with respect to pressure. 64 00:03:17 --> 00:03:20 Keeping the temperature constant. 65 00:03:20 --> 00:03:24 And you've got S now, you got volume. 66 00:03:24 --> 00:03:26 Do you know where you, how did you define G 67 00:03:26 --> 00:03:27 in the first place? 68 00:03:27 --> 00:03:32 We define G as H minus dS. 69 00:03:32 --> 00:03:35 One of the many definitions of g. 70 00:03:35 --> 00:03:40 Reverse that, you've got now H as a function of G and 71 00:03:40 --> 00:03:44 temperature and entropy. 72 00:03:44 --> 00:03:48 Well, you've got an expression for G, we just calculated, we 73 00:03:48 --> 00:03:51 just showed we could get an expression for S, which 74 00:03:51 --> 00:03:52 is sitting right here. 75 00:03:52 --> 00:03:55 Temperature is a variable here, so now we have 76 00:03:55 --> 00:03:57 an expression for H. 77 00:03:57 --> 00:04:03 And you can go on like that with every variable that you've 78 00:04:03 --> 00:04:06 learned in this class already. 79 00:04:06 --> 00:04:12 For instance, u is H minus pV. 80 00:04:12 --> 00:04:15 Well, there's the H here. 81 00:04:15 --> 00:04:17 We have that equation for H. 82 00:04:17 --> 00:04:20 We have an equation for V, coming from here. 83 00:04:20 --> 00:04:23 So there's nothing unknown here. 84 00:04:23 --> 00:04:29 If we have an equation for G here in terms of 85 00:04:29 --> 00:04:31 temperature and pressure. 86 00:04:31 --> 00:04:33 Same thing for the Helmholtz free energy. 87 00:04:33 --> 00:04:36 You can even get the heat capacities out. 88 00:04:36 --> 00:04:42 Every single one of these interesting, useful quantities 89 00:04:42 --> 00:04:45 that one would want to calculate falls out from the 90 00:04:45 --> 00:04:51 Gibbs free energy here. 91 00:04:51 --> 00:04:56 Any questions on that important step? 92 00:04:56 --> 00:04:59 And, really, I can't believe how clueless I was when I 93 00:04:59 --> 00:05:00 started doing research. 94 00:05:00 --> 00:05:02 Because I would go through the process of calculating G, 95 00:05:02 --> 00:05:05 and getting G, and et cetera, et cetera. 96 00:05:05 --> 00:05:08 I wrote papers, you know, G equals blah blah blah. 97 00:05:08 --> 00:05:10 And I didn't realize that that's why people 98 00:05:10 --> 00:05:14 wanted to know G. 99 00:05:14 --> 00:05:15 Anyway, I know better now. 100 00:05:15 --> 00:05:21 So, there are a few things we can say about G that are 101 00:05:21 --> 00:05:24 fairly easy to calculate. 102 00:05:24 --> 00:05:29 For instance, if I look at liquids or solids. 103 00:05:29 --> 00:05:36 And I want to know how G changes with pressure. 104 00:05:36 --> 00:05:49 So, I know that the volume here dG/dp, that dG/dp 105 00:05:49 --> 00:05:50 is the volume here. 106 00:05:50 --> 00:05:54 So if I look under constant temperature, I pick my 107 00:05:54 --> 00:05:56 fundamental equation under constant temperature, and 108 00:05:56 --> 00:05:59 I want to know how G is changing, I integrate. 109 00:05:59 --> 00:06:04 So I have dG is equal to V dp. 110 00:06:04 --> 00:06:09 So if I change my, and I look at per mole, and if I change my 111 00:06:09 --> 00:06:14 pressure from p1 to p2, I integrate from p1 to p2, p1 to 112 00:06:14 --> 00:06:23 p2 here, final state minus the initial state is equal to the 113 00:06:23 --> 00:06:31 integral from p1 to p2, V dp per mole. 114 00:06:31 --> 00:06:35 And what can I say, for a liquid or a solid, the volume 115 00:06:35 --> 00:06:38 per mole, over a liquid or a solid, is small and it 116 00:06:38 --> 00:06:41 doesn't change very much. 117 00:06:41 --> 00:06:47 So V is small. 118 00:06:47 --> 00:06:50 And usually these solids and liquids, you can assume 119 00:06:50 --> 00:06:53 to be incompressible. 120 00:06:53 --> 00:06:55 Meaning, as you change the pressure, the 121 00:06:55 --> 00:06:57 volume doesn't change. 122 00:06:57 --> 00:07:00 It's a good approximation. 123 00:07:00 --> 00:07:05 So when you do your integral here, you get that G at the 124 00:07:05 --> 00:07:11 new pressure is G at the old pressure, then if this isn't 125 00:07:11 --> 00:07:13 changing very much with pressure, or not at all, 126 00:07:13 --> 00:07:13 then you can take it out. 127 00:07:13 --> 00:07:16 It's just a constant. 128 00:07:16 --> 00:07:21 Plus V times p2 minus p1. 129 00:07:21 --> 00:07:24 And so, this is the incompressible part, 130 00:07:24 --> 00:07:24 you take it out. 131 00:07:24 --> 00:07:28 The fact that it's small means that you can assume that this 132 00:07:28 --> 00:07:31 is zero, this whole thing is zero, that it's small enough. 133 00:07:31 --> 00:07:35 And then you see that G, approximately doesn't 134 00:07:35 --> 00:07:37 change with pressure. 135 00:07:37 --> 00:07:43 Tells you that G, for a liquid or solid, most of the time you 136 00:07:43 --> 00:07:46 can assume that it's just a function of temperature. 137 00:07:46 --> 00:07:49 Just like we saw for an ideal gas, that the energy and 138 00:07:49 --> 00:07:51 the enthalpy were just functions of temperature. 139 00:07:51 --> 00:07:55 And that's a useful approximation. 140 00:07:55 --> 00:07:57 It's useful, but it's not completely true. 141 00:07:57 --> 00:08:01 And if it were true, then there would not be any pressure 142 00:08:01 --> 00:08:03 dependents to phase transitions. 143 00:08:03 --> 00:08:04 And we know that's not the case. 144 00:08:04 --> 00:08:11 We know that if you press on water when it's close to the 145 00:08:11 --> 00:08:20 water liquid-solid transition, that you can lower the 146 00:08:20 --> 00:08:22 melting point of ice. 147 00:08:22 --> 00:08:25 You press on ice, and you press hard enough, and ice will melt, 148 00:08:25 --> 00:08:29 the temperature is closer to melting point. 149 00:08:29 --> 00:08:30 And we'll go through that. 150 00:08:30 --> 00:08:33 So, that means that there has to be some sort of pressure 151 00:08:33 --> 00:08:34 dependence, eventually. 152 00:08:34 --> 00:08:36 And we'll see that. 153 00:08:36 --> 00:08:39 This is just an approximation. 154 00:08:39 --> 00:08:39 What else can we do? 155 00:08:39 --> 00:08:44 We can calculate, also, for an ideal gas. 156 00:08:44 --> 00:08:50 Liquid and solid, we can do an ideal gas. 157 00:08:50 --> 00:08:53 So for an ideal gas, again, starting from the fundamental 158 00:08:53 --> 00:08:59 equation, we have dG equals V dp. 159 00:08:59 --> 00:09:04 We can do it per mole. 160 00:09:04 --> 00:09:10 So integrate both sides, G(T, p2) is equal to G at the 161 00:09:10 --> 00:09:14 initial pressure, plus the integral from p1 to 162 00:09:14 --> 00:09:15 p2, the volume. 163 00:09:15 --> 00:09:17 So instead of putting the volume, this is an ideal 164 00:09:17 --> 00:09:20 gas now, we can put the ideal gas law. 165 00:09:20 --> 00:09:22 So V is really RT over p. 166 00:09:22 --> 00:09:25 RT over p dp. 167 00:09:25 --> 00:09:27 We can integrate this. 168 00:09:27 --> 00:09:31 Get a log term out. 169 00:09:31 --> 00:09:42 G(T, p1) plus RT log p2 over p1. 170 00:09:42 --> 00:09:46 And then it's very useful to reference everything to the 171 00:09:46 --> 00:09:51 center state. p1 is equal to one bar, let's say. 172 00:09:51 --> 00:09:59 So if you take p1 equals one bar as our reference point, and 173 00:09:59 --> 00:10:03 get rid of the little subscript two here, we can write G of T 174 00:10:03 --> 00:10:08 at some pressure p, then is G and the little naught on top 175 00:10:08 --> 00:10:16 here means standard state one bar plus RT log p 176 00:10:16 --> 00:10:20 divided by one bar. 177 00:10:20 --> 00:10:26 And I put in a little dotted line here because very often 178 00:10:26 --> 00:10:28 you just write it without the one bar and bar. 179 00:10:28 --> 00:10:31 And you know that there has to be a one bar, because 180 00:10:31 --> 00:10:33 inside a log you can't have something with units. 181 00:10:33 --> 00:10:34 It has to be unitless. 182 00:10:34 --> 00:10:36 So you know if you have something with bar here, you've 183 00:10:36 --> 00:10:38 got to divide with something with bar, and there happens 184 00:10:38 --> 00:10:40 to be one bar here. 185 00:10:40 --> 00:10:46 So pressure p is G at its standard state plus RT log p. 186 00:10:46 --> 00:10:56 And this becomes a very useful, very useful, quantity to know. 187 00:10:56 --> 00:11:00 OK, so G is so important. 188 00:11:00 --> 00:11:06 And G per mole is so fundamental, that we're going 189 00:11:06 --> 00:11:09 to give it a special name. 190 00:11:09 --> 00:11:11 Not to make your life more complicated but just because 191 00:11:11 --> 00:11:12 it's just so important. 192 00:11:12 --> 00:11:17 We're going to call it the chemical potential. 193 00:11:17 --> 00:11:23 So G per mole, we're going to call mu. 194 00:11:23 --> 00:11:24 And that's going to be a chemical potential. 195 00:11:24 --> 00:11:26 We're going to do a lot more with the chemical 196 00:11:26 --> 00:11:30 potential today. 197 00:11:30 --> 00:11:33 And the reason why we call potential is because we already 198 00:11:33 --> 00:11:40 saw that if you've got something under constant 199 00:11:40 --> 00:11:49 pressure, temperature, that you want to use G as the variable 200 00:11:49 --> 00:11:50 to tell you whether something is going to be 201 00:11:50 --> 00:11:51 spontaneous or not. 202 00:11:51 --> 00:11:55 So you want G to go downhill. 203 00:11:55 --> 00:11:59 And so, we're going to be talking about chemical species. 204 00:11:59 --> 00:12:04 And instead of having a car up and down mountains, trying to 205 00:12:04 --> 00:12:07 go down to the valleys, we're going to have chemical species 206 00:12:07 --> 00:12:09 trying to find the valleys. 207 00:12:09 --> 00:12:12 The potential valleys. 208 00:12:12 --> 00:12:14 To get to equilibrium. 209 00:12:14 --> 00:12:16 And so we're going to be looking at the Gibbs free 210 00:12:16 --> 00:12:19 energy, or the Gibbs free energy per mole at that 211 00:12:19 --> 00:12:21 particular species, and it's going to want to be 212 00:12:21 --> 00:12:24 as small as possible. 213 00:12:24 --> 00:12:27 We're going to want to minimize the chemical potentials. 214 00:12:27 --> 00:12:29 And that's why it's called potential. 215 00:12:29 --> 00:12:34 It's like an energy. 216 00:12:34 --> 00:12:45 So, that's the end of the one component, thermodynamic 217 00:12:45 --> 00:12:48 background, before we get to multi-components. 218 00:12:48 --> 00:12:51 So it's a good time to stop again and see if 219 00:12:51 --> 00:12:52 there's any questions. 220 00:12:52 --> 00:13:01 Any issues. 221 00:13:01 --> 00:13:02 OK. 222 00:13:02 --> 00:13:06 So, so far we've done everything with one species. 223 00:13:06 --> 00:13:09 One ideal gas, one liquid, one solid. 224 00:13:09 --> 00:13:12 We haven't done anything with mixtures, except 225 00:13:12 --> 00:13:15 for maybe looking at the entropy of mixing. 226 00:13:15 --> 00:13:17 We saw the entropy of mixing was really important, because 227 00:13:17 --> 00:13:24 it drove processes where energy was constant. 228 00:13:24 --> 00:13:27 But most of what we care about in chemistry, at least in 229 00:13:27 --> 00:13:31 chemical reactions, species change. 230 00:13:31 --> 00:13:32 They get destroyed. 231 00:13:32 --> 00:13:36 New species get created. 232 00:13:36 --> 00:13:37 There are mixtures. 233 00:13:37 --> 00:13:38 It's pretty complicated. 234 00:13:38 --> 00:13:42 For instance, if I take a reaction of hydrogen gas plus 235 00:13:42 --> 00:13:49 chlorine gas to form two moles of HCl gas, I'm destroying 236 00:13:49 --> 00:13:51 hydrogen, I'm destroying chlorine, I'm making 237 00:13:51 --> 00:13:54 HCl in the gas phase. 238 00:13:54 --> 00:13:55 I get a big mixture at the end. 239 00:13:55 --> 00:13:59 I get three different kinds of species at the end. 240 00:13:59 --> 00:14:01 So the fundamental equations that I've been talking about, 241 00:14:01 --> 00:14:04 that we've been talking about, they're too simple 242 00:14:04 --> 00:14:06 for such a system. 243 00:14:06 --> 00:14:09 Because they all care about one species. 244 00:14:09 --> 00:14:13 Even more complicated, for instance, if I take hydrogen 245 00:14:13 --> 00:14:22 gas and oxygen gas and I mix them together to make water, 246 00:14:22 --> 00:14:26 liquid, for instance, not only do I have species that are 247 00:14:26 --> 00:14:31 changing, that are getting destroyed or created, in this 248 00:14:31 --> 00:14:34 case here the total number of moles is changing. 249 00:14:34 --> 00:14:37 And the phase is changing. 250 00:14:37 --> 00:14:40 Got all sorts of changes going on here. 251 00:14:40 --> 00:14:43 And so if I want to understand equilibrium, if I want to 252 00:14:43 --> 00:14:47 understand the direction of time for these more complicated 253 00:14:47 --> 00:14:51 processes, I have to be able to take into account, in an easy 254 00:14:51 --> 00:14:56 way, these mixing processes, these phase changes, these 255 00:14:56 --> 00:15:03 changes in the number of moles. 256 00:15:03 --> 00:15:07 And that's what we're going to talk about today. 257 00:15:07 --> 00:15:10 We're going to try to change our fundamental equations to 258 00:15:10 --> 00:15:12 make them a little bit more complicated so that we can deal 259 00:15:12 --> 00:15:13 with these sorts of problems. 260 00:15:13 --> 00:15:15 Because those are the real problems we need 261 00:15:15 --> 00:15:19 to keep track of. 262 00:15:19 --> 00:15:23 And the ultimate goal, then, of changing our fundamental 263 00:15:23 --> 00:15:27 questions is to derive equilibrium from 264 00:15:27 --> 00:15:28 first principles. 265 00:15:28 --> 00:15:30 To really understand chemical equilibrium, which 266 00:15:30 --> 00:15:31 you've all seen before. 267 00:15:31 --> 00:15:35 You've all used the chemical equilibrium constant K, 268 00:15:35 --> 00:15:36 you've done problems. 269 00:15:36 --> 00:15:40 But you've been given, basically, the equilibrium 270 00:15:40 --> 00:15:43 constant, and not really derived it, understood 271 00:15:43 --> 00:15:48 where it came from. 272 00:15:48 --> 00:15:53 OK, another simple example here, which is actually the 273 00:15:53 --> 00:15:57 one that we're going to be looking at in the first case. 274 00:15:57 --> 00:16:02 Where there's a change going on, is just to 275 00:16:02 --> 00:16:04 look at a phase change. 276 00:16:04 --> 00:16:07 H2O liquid going to H2O solid. 277 00:16:07 --> 00:16:10 There's a phase change, you can think of it as one species, the 278 00:16:10 --> 00:16:14 H2O liquid sort of changing into an H2O a solid. 279 00:16:14 --> 00:16:16 It's the same chemical in this case here, there's no 280 00:16:16 --> 00:16:20 change in the molecules. 281 00:16:20 --> 00:16:27 But it's still a change that we have to account for. 282 00:16:27 --> 00:16:34 Another example that's also simple like this, that you all, 283 00:16:34 --> 00:16:41 I'm sure, have seen before, suppose I take a cell. 284 00:16:41 --> 00:16:44 My cell here, full of water. 285 00:16:44 --> 00:16:50 And then I put my cell, let's say I take a human cell. 286 00:16:50 --> 00:16:53 My skin or something. 287 00:16:53 --> 00:17:03 And I take it and I put it in distilled water. 288 00:17:03 --> 00:17:07 What's going to happen to the cell? 289 00:17:07 --> 00:17:10 Is it going to be happy? 290 00:17:10 --> 00:17:15 What's going to happen to it? 291 00:17:15 --> 00:17:17 It's going to burst, right? 292 00:17:17 --> 00:17:18 Why is it going to burst? 293 00:17:18 --> 00:17:26 Anybody have an idea why it's going to burst? 294 00:17:26 --> 00:17:26 Yes. 295 00:17:26 --> 00:17:38 STUDENT: [INAUDIBLE] 296 00:17:38 --> 00:17:38 PROFESSOR: That's right. 297 00:17:38 --> 00:17:42 So the water wants to go from, you're completely right. 298 00:17:42 --> 00:17:46 But let me rephrase it in a thermodynamic language here. 299 00:17:46 --> 00:17:48 The water is going to go from a place of high chemical 300 00:17:48 --> 00:17:52 potential to low chemical potential. 301 00:17:52 --> 00:17:56 And the cell can't take all that water in there. 302 00:17:56 --> 00:17:59 The membrane's going to try to swell. 303 00:17:59 --> 00:18:00 And eventually burst, right? 304 00:18:00 --> 00:18:05 Same thing if you if you take a, go fishing, go to the 305 00:18:05 --> 00:18:09 Atlantic Ocean and then get a nice cod or something. 306 00:18:09 --> 00:18:12 Bring it back and on your sailboat, you dump it in 307 00:18:12 --> 00:18:15 a tub of fresh water. 308 00:18:15 --> 00:18:18 Is that cod going to be happy? 309 00:18:18 --> 00:18:22 It's not going to be happy at all, right, because its 310 00:18:22 --> 00:18:26 biology is geared towards living in salt water. 311 00:18:26 --> 00:18:31 And turns out that the chemical potential of water, in salt 312 00:18:31 --> 00:18:33 water, is lower than the chemical potential 313 00:18:33 --> 00:18:34 of pure water. 314 00:18:34 --> 00:18:37 And so when you put the cod in there, the chemical potential 315 00:18:37 --> 00:18:40 of the water and the cod, is lower than the chemical 316 00:18:40 --> 00:18:42 potential of the fresh water you have on the outside. 317 00:18:42 --> 00:18:44 And the fresh water wants to be at a lower chemical potential. 318 00:18:44 --> 00:18:48 It rushes into the cod, and well, the cod does what the 319 00:18:48 --> 00:18:50 cell does, when you put it in distilled water. 320 00:18:50 --> 00:18:53 It sort of bloats. 321 00:18:53 --> 00:18:57 It isn't very happy. 322 00:18:57 --> 00:19:02 OK, so but all these things are basically the same idea here. 323 00:19:02 --> 00:19:05 Where you have a complicated change, where species are 324 00:19:05 --> 00:19:07 mixing, and things like this. 325 00:19:07 --> 00:19:09 And it turns out the chemical potential is going to tell 326 00:19:09 --> 00:19:12 us all about how to think about that. 327 00:19:12 --> 00:19:16 That's why the chemical potential is so important. 328 00:19:16 --> 00:19:19 So we're going to go back to these two examples 329 00:19:19 --> 00:19:23 here many times. 330 00:19:23 --> 00:19:26 So let's take the simplest example here. 331 00:19:26 --> 00:19:32 Let's go back and derive some equations. 332 00:19:32 --> 00:19:38 Let's take our simplest example that's not a one species 333 00:19:38 --> 00:19:41 system, but has two species. 334 00:19:41 --> 00:19:45 Species 1 and 2. 335 00:19:45 --> 00:19:50 And n1 and n2 are the number of moles of species 1 and 2. 336 00:19:50 --> 00:19:54 And then we're going to see if I make a perturbation 337 00:19:54 --> 00:19:55 in my system. 338 00:19:55 --> 00:19:58 I change the number of moles of 1, or the number of moles of 2. 339 00:19:58 --> 00:20:03 How does this affect the Gibbs free energy? 340 00:20:03 --> 00:20:05 That's the question we're going to post. 341 00:20:05 --> 00:20:09 And our goal is to find a new fundamental equation for G that 342 00:20:09 --> 00:20:12 includes the number of moles of the different species 343 00:20:12 --> 00:20:13 as they change. 344 00:20:13 --> 00:20:16 Because in chemistry they're going to be changing. 345 00:20:16 --> 00:20:19 They're not going to be fixed. 346 00:20:19 --> 00:20:23 So what we want is just purely mathematically formally, take 347 00:20:23 --> 00:20:27 the differential of the Gibbs free energy, which we know is 348 00:20:27 --> 00:20:33 dG/dT, keeping pressure, the number of moles of 1, the 349 00:20:33 --> 00:20:35 number of 2 constant, dT. 350 00:20:35 --> 00:20:43 That is, dG/dp constant temperature, n1 and n2 dp. 351 00:20:43 --> 00:20:49 And then we have our two more variables now, dG/dn1, remember 352 00:20:49 --> 00:20:53 this is just a formal statement keeping temperature and 353 00:20:53 --> 00:21:03 pressure and n2 constant. dn1 plus dG/dn2, dn2 keeping 354 00:21:03 --> 00:21:07 temperature and pressure and n1 constant here. 355 00:21:07 --> 00:21:08 I'm not writing anything new here, I'm just telling you 356 00:21:08 --> 00:21:13 what the definition of the differential is here, for G. 357 00:21:13 --> 00:21:16 We already know what some of these quantities are. 358 00:21:16 --> 00:21:23 We know that this is the entropy, minus the entropy. 359 00:21:23 --> 00:21:28 This here is the volume. 360 00:21:28 --> 00:21:31 And I know the answer already. 361 00:21:31 --> 00:21:34 But I'm going to define it anyways. 362 00:21:34 --> 00:21:37 And we're going to prove it. 363 00:21:37 --> 00:21:41 I'm going to define this as the chemical 364 00:21:41 --> 00:21:46 potential for species 1. 365 00:21:46 --> 00:21:51 I'm going to define this, I'm going to give it a symbol, 366 00:21:51 --> 00:21:56 chemical potential mu, for species 2. 367 00:21:56 --> 00:22:00 So that I can write my new fundamental equation 368 00:22:00 --> 00:22:04 as dG as minus S dT. 369 00:22:04 --> 00:22:11 Plus V dp plus, and if I have more than two species present, 370 00:22:11 --> 00:22:17 the sum of all species in my mixture times the chemical 371 00:22:17 --> 00:22:21 potential of that species, dni. 372 00:22:21 --> 00:22:27 The change in the number of moles of that species. 373 00:22:27 --> 00:22:36 So, this quantity mu, that I've just defined, dG/dni, keeping 374 00:22:36 --> 00:22:40 the temperature, the pressure and all the n's, except for the 375 00:22:40 --> 00:22:45 i'th one constant, that is an intensive quantity. 376 00:22:45 --> 00:22:53 Because G scales with size, scales, with size of system. 377 00:22:53 --> 00:22:56 G is intensive. n, obviously, scales with 378 00:22:56 --> 00:22:59 the size of the system. 379 00:22:59 --> 00:23:04 Also intensive, and you take the ratio of two extensive 380 00:23:04 --> 00:23:06 variables, you get an intensive variable which doesn't care 381 00:23:06 --> 00:23:08 about the size of the system. 382 00:23:08 --> 00:23:09 Which is a good thing. 383 00:23:09 --> 00:23:12 For what we've been talking about. 384 00:23:12 --> 00:23:15 Intensive. 385 00:23:15 --> 00:23:19 If I'm talking about putting a freshwater fish and dumping it 386 00:23:19 --> 00:23:22 in the, putting it in the Atlantic Ocean, the chemical 387 00:23:22 --> 00:23:25 potential of the water in the Atlantic ocean better not care 388 00:23:25 --> 00:23:29 whether the Atlantic Ocean is huge or even huger. 389 00:23:29 --> 00:23:31 It just cares about the local environment. 390 00:23:31 --> 00:23:39 Just cares that it that wants to be in that freshwater fish. 391 00:23:39 --> 00:23:42 So the chemical potential is intensive. 392 00:23:42 --> 00:23:48 Just as we've written it down here for a single 393 00:23:48 --> 00:23:50 species system. 394 00:23:50 --> 00:23:51 It's the Gibbs energy, free energy per mole. 395 00:23:51 --> 00:23:53 Which we haven't proven yet. 396 00:23:53 --> 00:23:54 We haven't proven yet here. 397 00:23:54 --> 00:23:57 We've just defined it this way and we're going to prove that 398 00:23:57 --> 00:24:02 in fact the mu's are the Gibbs free energies per mole for each 399 00:24:02 --> 00:24:07 of the species in our system. 400 00:24:07 --> 00:24:14 OK, so this is now our first new fundamental equation. 401 00:24:14 --> 00:24:17 All we did was to add the sum here. 402 00:24:17 --> 00:24:19 Now, we started the lecture by saying that if you have 403 00:24:19 --> 00:24:24 the Gibbs free energy, you've got everything. 404 00:24:24 --> 00:24:30 And we wrote equations that are covered, were we can get S, we 405 00:24:30 --> 00:24:35 can get V, we can get H, we can get u, we can get A. 406 00:24:35 --> 00:24:37 So now that we have a fundamental question for G, 407 00:24:37 --> 00:24:40 we've got our new fundamental equations for everything else. 408 00:24:40 --> 00:24:45 Without really thinking too much. 409 00:24:45 --> 00:24:50 We go back to our definitions. 410 00:24:50 --> 00:24:54 Enthalpy is G minus TS. 411 00:24:54 --> 00:25:06 So dH is dG minus d of TS. dG, we've got our new 412 00:25:06 --> 00:25:07 fundamental equation for G. 413 00:25:07 --> 00:25:09 We plug it in here. 414 00:25:09 --> 00:25:12 Expand things out with a T dS here, we can write immediately 415 00:25:12 --> 00:25:19 a fundamental equation for H. dH is T dS. 416 00:25:19 --> 00:25:21 The beginning is going to look just like what 417 00:25:21 --> 00:25:22 you've seen before. 418 00:25:22 --> 00:25:23 Plus V dp. 419 00:25:23 --> 00:25:27 Plus an extra term, which is exactly the same extra 420 00:25:27 --> 00:25:31 term that we had in the fundamental equation for G. 421 00:25:31 --> 00:25:33 Exactly the same. 422 00:25:33 --> 00:25:36 And every one of the other fundamental questions 423 00:25:36 --> 00:25:40 can be derived in the similar way from G. 424 00:25:40 --> 00:25:45 And they're going to be what you had before minus S dT plus 425 00:25:45 --> 00:25:59 minus p dV plus the sum of the mu i's, dni, and du is T dS 426 00:25:59 --> 00:26:05 minus p dV plus i mu i dni. 427 00:26:05 --> 00:26:08 428 00:26:08 --> 00:26:13 So immediately we can see that this mu, this quantity mu that 429 00:26:13 --> 00:26:19 we've defined as the derivative of G with respect to the n's, 430 00:26:19 --> 00:26:22 we can write many other equations for mu. 431 00:26:22 --> 00:26:24 There are many other ways to derive it. 432 00:26:24 --> 00:26:28 Because this is the differential for H. 433 00:26:28 --> 00:26:31 This is the first derivative of H with respect to entropy. 434 00:26:31 --> 00:26:34 This is the derivative of H with respect to pressure. 435 00:26:34 --> 00:26:36 And this is the derivative of H with respect to n. 436 00:26:36 --> 00:26:38 Just formally, that's what it is. 437 00:26:38 --> 00:26:41 When you write a differential. 438 00:26:41 --> 00:26:52 So formally, this is also mu i, is dH/dni, keeping, now, we 439 00:26:52 --> 00:26:56 have to be very careful, keeping the entropy and 440 00:26:56 --> 00:26:57 the pressure constant. 441 00:26:57 --> 00:26:59 Because those are the variables. 442 00:26:59 --> 00:27:03 Keeping the entropy, and the pressure, and all 443 00:27:03 --> 00:27:08 the other n's, constant. 444 00:27:08 --> 00:27:17 We can also write it as dA/dni, keeping the temperature 445 00:27:17 --> 00:27:18 and the volume constant. 446 00:27:18 --> 00:27:27 And all the other n's, or we can write it as du/dni, 447 00:27:27 --> 00:27:33 keeping that this entropy and the volume, and all 448 00:27:33 --> 00:27:35 the other n's, constant. 449 00:27:35 --> 00:27:42 So we have many ways to write the chemical potential. 450 00:27:42 --> 00:27:51 They give you all the same result. 451 00:27:51 --> 00:27:53 So this is the formal. 452 00:27:53 --> 00:27:59 Sort of the formal part of the chemical potential. 453 00:27:59 --> 00:28:01 Now, what we really want to show is that the chemical 454 00:28:01 --> 00:28:03 potential really is connected to the Gibbs 455 00:28:03 --> 00:28:05 free energy per mole. 456 00:28:05 --> 00:28:08 That's going to be the useful part. 457 00:28:08 --> 00:28:27 Let me get rid of this here. 458 00:28:27 --> 00:28:32 So I said earlier, at the beginning of the lecture, that 459 00:28:32 --> 00:28:34 the Gibbs free energy per mole was so important, we were 460 00:28:34 --> 00:28:37 going to call it the chemical potential. 461 00:28:37 --> 00:28:38 And I said that here. 462 00:28:38 --> 00:28:39 And then I said, well, we're going to define 463 00:28:39 --> 00:28:40 this here, the chemical. 464 00:28:40 --> 00:28:44 But I haven't equated the two yet. 465 00:28:44 --> 00:28:46 I haven't proven to you that in fact this quantity here, which 466 00:28:46 --> 00:28:50 we've formally defined as the derivative of G with respect to 467 00:28:50 --> 00:28:54 n, is the Gibbs free energy per mole, for this species. 468 00:28:54 --> 00:29:03 In fact, what we want to show is that if I take the sum of 469 00:29:03 --> 00:29:08 all the chemical potentials, times the number of moles per 470 00:29:08 --> 00:29:15 species, that that is the total Gibbs free energy. 471 00:29:15 --> 00:29:19 In other words, that the chemical potential for one 472 00:29:19 --> 00:29:24 species in the mixture is the Gibbs free energy per 473 00:29:24 --> 00:29:29 mole for that species. 474 00:29:29 --> 00:29:34 Once we have that idea, then we'll be able to talk about the 475 00:29:34 --> 00:29:38 concept of chemical potential as this thing that we can 476 00:29:38 --> 00:29:40 use to look at equilibrium. 477 00:29:40 --> 00:29:42 To look at going downhill for the species. 478 00:29:42 --> 00:29:45 To see why the cell bursts and all these things. 479 00:29:45 --> 00:29:47 Because now we understand that Gibbs free energy is so 480 00:29:47 --> 00:29:49 important for equilibrium. 481 00:29:49 --> 00:29:50 We don't understand that quite yet, with the 482 00:29:50 --> 00:29:51 chemical potential. 483 00:29:51 --> 00:29:54 So we got to make that relation here. 484 00:29:54 --> 00:29:58 We need to go from the formal definition to a relation that 485 00:29:58 --> 00:30:00 we can understand better, because it includes the 486 00:30:00 --> 00:30:02 Gibbs free energy. 487 00:30:02 --> 00:30:07 OK, so that's our goal now. 488 00:30:07 --> 00:30:08 So let's see. 489 00:30:08 --> 00:30:09 Let's formally do this now. 490 00:30:09 --> 00:30:15 Let's define, let's derive this. 491 00:30:15 --> 00:30:15 OK. 492 00:30:15 --> 00:30:19 So remember, our goal in this derivation is to 493 00:30:19 --> 00:30:24 show that this is true. 494 00:30:24 --> 00:30:27 Or that this is true, here. 495 00:30:27 --> 00:30:29 And again, we're going to start with the simplest 496 00:30:29 --> 00:30:30 system possible. 497 00:30:30 --> 00:30:33 We're going to start with a two component system. 498 00:30:33 --> 00:30:46 And we can easily generalize to multi-component. 499 00:30:46 --> 00:30:50 And in our derivation, what we're going to be after is, 500 00:30:50 --> 00:30:51 we're going to start with the Gibbs free energy, 501 00:30:51 --> 00:30:53 because that's where we always start with. 502 00:30:53 --> 00:30:57 And we're going to remember that by definition, mu i is 503 00:30:57 --> 00:31:03 dG/dni, So if somehow in our derivation dG/dni falls 504 00:31:03 --> 00:31:05 out, that would be great. 505 00:31:05 --> 00:31:08 Because we'll be able to replace this derivative with 506 00:31:08 --> 00:31:09 the chemical potential. 507 00:31:09 --> 00:31:12 So the goal was to find something where this falls out, 508 00:31:12 --> 00:31:17 so we can replace it with the chemical potential. 509 00:31:17 --> 00:31:18 We're going to start with the fact that G is an 510 00:31:18 --> 00:31:21 extensive variable. 511 00:31:21 --> 00:31:25 So if I take G at a temperature and pressure times some scaling 512 00:31:25 --> 00:31:29 factor for the size of my system, lambda, number of moles 513 00:31:29 --> 00:31:34 of n1, lambda times the number of moles of n2, if I double 514 00:31:34 --> 00:31:36 the size of the system, lambda is equal to 2. 515 00:31:36 --> 00:31:38 If I half it, lambda is equal to 1/2. 516 00:31:38 --> 00:31:41 Because it's extensive, this is the same thing as lambda times 517 00:31:41 --> 00:31:46 G of temperature and pressure, n1, n2. 518 00:31:46 --> 00:31:50 Just rewriting the fact that Gibbs free energy is 519 00:31:50 --> 00:31:52 an extensive property. 520 00:31:52 --> 00:31:54 And lambda is an arbitrary number here. 521 00:31:54 --> 00:31:56 Arbitrary variable. 522 00:31:56 --> 00:31:58 Now I'm going to take the derivative of both sides 523 00:31:58 --> 00:32:00 with respect to lambda. 524 00:32:00 --> 00:32:04 So I'm going to take d d lambda of this side here. 525 00:32:04 --> 00:32:12 And d d lambda of that side here. 526 00:32:12 --> 00:32:14 Now, lambda here is inside the variable here. 527 00:32:14 --> 00:32:16 So I'm going to have to use the chain rule. 528 00:32:16 --> 00:32:18 To do this properly. 529 00:32:18 --> 00:32:30 So this is going to be dG/d lambda n1, there's lambda 530 00:32:30 --> 00:32:32 sitting in the variable lambda n1 here, times 531 00:32:32 --> 00:32:37 d lambda n1 d lambda. 532 00:32:37 --> 00:32:47 Plus dG/d lambda n2 times d lambda n2 d lambda. 533 00:32:47 --> 00:32:49 Using the chain rule. 534 00:32:49 --> 00:32:51 And on this side here, lambda's sitting straight out here. 535 00:32:51 --> 00:32:53 So this is very easy. 536 00:32:53 --> 00:32:55 This is G(T, p, n1, n2). 537 00:32:55 --> 00:33:06 538 00:33:06 --> 00:33:07 Now, this is good. 539 00:33:07 --> 00:33:09 Because this is what I'm looking for. 540 00:33:09 --> 00:33:11 I'm looking for the derivative of g with respect to 541 00:33:11 --> 00:33:12 the number of moles. 542 00:33:12 --> 00:33:14 Because that's the chemical potential. 543 00:33:14 --> 00:33:17 That was my goal up here, to make sure in the derivation, 544 00:33:17 --> 00:33:19 somehow, this came out. 545 00:33:19 --> 00:33:24 And so it's coming out right there. 546 00:33:24 --> 00:33:28 Right here and right here. 547 00:33:28 --> 00:33:31 Since the number of moles is lambda n1, that first 548 00:33:31 --> 00:33:33 derivative here is just the chemical potential 549 00:33:33 --> 00:33:35 of species 1 there. 550 00:33:35 --> 00:33:39 So mu 1, then we have d lambda n1 d lambda. 551 00:33:39 --> 00:33:43 Well, d lambda n1 d lambda, that's just n1. 552 00:33:43 --> 00:33:46 It's lambda times the number n1 that doesn't have 553 00:33:46 --> 00:33:48 anything to do with lambda. 554 00:33:48 --> 00:33:50 So this is n1. 555 00:33:50 --> 00:33:53 This is the chemical potential of species 2. 556 00:33:53 --> 00:33:55 Again, the derivative of lambda n2 with respect 557 00:33:55 --> 00:33:58 to lambda is just n2. 558 00:33:58 --> 00:34:04 And there is G here. 559 00:34:04 --> 00:34:07 It's a fairly simple derivation, but it gets 560 00:34:07 --> 00:34:08 us exactly what we want. 561 00:34:08 --> 00:34:12 An association between this formal definition of mu, 562 00:34:12 --> 00:34:17 up here, directly from taking the differential. 563 00:34:17 --> 00:34:20 How much more formal can you be, mathematically, here? 564 00:34:20 --> 00:34:24 To associating this formal definition to 565 00:34:24 --> 00:34:27 the Gibbs free energy. 566 00:34:27 --> 00:34:34 Number of moles times mu 1, number of moles times mu 2, 567 00:34:34 --> 00:34:40 this is the Gibbs free energy per mole of species 1. 568 00:34:40 --> 00:34:45 Gibbs free energy per mole of species 2. 569 00:34:45 --> 00:34:49 The sum of all the species of the Gibbs free energy per mole 570 00:34:49 --> 00:34:53 of species i times the number of moles of species i is G. 571 00:34:53 --> 00:34:57 Or, mu i. 572 00:34:57 --> 00:35:00 Voila, we've done it. 573 00:35:00 --> 00:35:02 This is what we wanted. 574 00:35:02 --> 00:35:06 The chemical potential is the Gibbs free energy per mole. 575 00:35:06 --> 00:35:09 And in the mixture, it's the Gibbs free energy per mole of 576 00:35:09 --> 00:35:11 the individual species in that mixture. 577 00:35:11 --> 00:35:13 And if you want to know what the total Gibbs free energy 578 00:35:13 --> 00:35:16 is, because if you have an equilibrium, what you care 579 00:35:16 --> 00:35:18 about is the total Gibbs free energy. 580 00:35:18 --> 00:35:21 It's not the Gibbs free energy for one particular species. 581 00:35:21 --> 00:35:24 What's going to tell you whether you have a minimum or 582 00:35:24 --> 00:35:29 not in your system, whether you're at equilibrium, where 583 00:35:29 --> 00:35:32 you're at the lowest state possible, is the total 584 00:35:32 --> 00:35:35 Gibbs free energy. 585 00:35:35 --> 00:35:38 Now we'll be able to manipulate chemical potentials, of 586 00:35:38 --> 00:35:45 the individual species, to get at this number here. 587 00:35:45 --> 00:35:46 Any questions? 588 00:35:46 --> 00:35:50 This is really, we're going to see this over 589 00:35:50 --> 00:35:51 and over again now. 590 00:35:51 --> 00:35:52 This chemical potential. 591 00:35:52 --> 00:35:53 This idea. 592 00:35:53 --> 00:36:04 And it's not an easy concept. 593 00:36:04 --> 00:36:13 OK, let me give you an example, then, of the water melting. 594 00:36:13 --> 00:36:17 And how the chemical potential comes in, now, instead of 595 00:36:17 --> 00:36:18 using the chemical potential. 596 00:36:18 --> 00:36:30 Instead of the Gibbs free energy. 597 00:36:30 --> 00:36:31 This is the phase transition. 598 00:36:31 --> 00:36:35 But it's not very different than the cell bursting when you 599 00:36:35 --> 00:36:39 put it in distilled water. 600 00:36:39 --> 00:36:44 So, we take a glass of water with an ice cube in it. 601 00:36:44 --> 00:36:47 H2O liquid. 602 00:36:47 --> 00:36:50 This is H2O solid. 603 00:36:50 --> 00:36:53 And I'm looking at the melting process. 604 00:36:53 --> 00:36:57 I'm looking at a process where I take a small number 605 00:36:57 --> 00:37:01 of molecules of water from the solid phase. 606 00:37:01 --> 00:37:04 And I bring them to the liquid phase. 607 00:37:04 --> 00:37:10 And I want to know, is this process spontaneous or in 608 00:37:10 --> 00:37:17 equilibrium, or not possible? 609 00:37:17 --> 00:37:19 Is this process going to go on? 610 00:37:19 --> 00:37:21 Is the direction of time that this is melting. 611 00:37:21 --> 00:37:24 And I want to do this formally thermodynamically. 612 00:37:24 --> 00:37:27 In terms of the chemical potentials. 613 00:37:27 --> 00:37:30 That's going to be what we're going to be talking about. 614 00:37:30 --> 00:37:35 So, formally then, what's going on is, I'm taking nl moles of 615 00:37:35 --> 00:37:41 liquid water, H2O liquid, which is in here. 616 00:37:41 --> 00:37:47 Plus ns moles of solid water. 617 00:37:47 --> 00:37:50 And I'm, this is my initial state. 618 00:37:50 --> 00:37:59 My final state is nl, plus a small number of moles, dn, of 619 00:37:59 --> 00:38:06 H2O liquids, H2O liquid, plus ns minus dn, there's a 620 00:38:06 --> 00:38:09 conservation of the number of molecules here. 621 00:38:09 --> 00:38:10 Whatever I add to the liquid has to come 622 00:38:10 --> 00:38:13 from the solid here. 623 00:38:13 --> 00:38:19 Of H2O solid, of ice. 624 00:38:19 --> 00:38:23 And to know if this is spontaneous or not, if this is 625 00:38:23 --> 00:38:27 done under constant temperature and pressure, what variable 626 00:38:27 --> 00:38:31 should we look at? 627 00:38:31 --> 00:38:31 G, right. 628 00:38:31 --> 00:38:33 We want to look at the Gibbs free energy. 629 00:38:33 --> 00:38:38 So what is G doing during this process here? 630 00:38:38 --> 00:38:42 What is delta G here? 631 00:38:42 --> 00:38:45 Well, we have a way of doing it now, in terms of the 632 00:38:45 --> 00:38:45 chemical potentials. 633 00:38:45 --> 00:38:51 Because we've just shown that this is the case here. 634 00:38:51 --> 00:38:56 So G is the sum of the chemical potentials times the number 635 00:38:56 --> 00:38:58 of moles in the species. 636 00:38:58 --> 00:39:07 Therefore, delta G is going to be equal to mu for the 637 00:39:07 --> 00:39:09 number of moles of l. 638 00:39:09 --> 00:39:20 Liquid, dn, number of moles of liquid, plus mu s dns. 639 00:39:20 --> 00:39:22 So the change in G is going to be equal to the chemical 640 00:39:22 --> 00:39:25 potential times the change in the species, which is in a 641 00:39:25 --> 00:39:28 liquid form, plus the chemical potential of the solid. 642 00:39:28 --> 00:39:32 Times the change in the species in the solid form. 643 00:39:32 --> 00:39:37 Now, dns is equal to minus dnl. 644 00:39:37 --> 00:39:39 This is what we did right here. 645 00:39:39 --> 00:39:41 You take a certain number of moles from the solid form. 646 00:39:41 --> 00:39:42 You put it to the liquid form. 647 00:39:42 --> 00:39:44 That the dn up here. 648 00:39:44 --> 00:39:46 And you've got to have the negative of it, up here. 649 00:39:46 --> 00:39:49 So dns is minus dnl. 650 00:39:49 --> 00:39:53 Which is minus dn. 651 00:39:53 --> 00:40:11 Delta G is dn times mu l minus mu solid. 652 00:40:11 --> 00:40:16 So now we can rephrase this, it's all rephrasing. 653 00:40:16 --> 00:40:18 It's all basically the same thing. 654 00:40:18 --> 00:40:24 But, we can rephrase this process by asking the question, 655 00:40:24 --> 00:40:30 is the chemical potential of the liquid greater than, equal 656 00:40:30 --> 00:40:36 to, or less than the chemical potential of the solid? 657 00:40:36 --> 00:40:37 Of the water in the solid. 658 00:40:37 --> 00:40:40 So the chemical potential of the water in the liquid phase 659 00:40:40 --> 00:40:44 is greater than the chemical potential of the water in the 660 00:40:44 --> 00:40:48 solid phase, mu l is greater than mu s, then delta 661 00:40:48 --> 00:40:52 G becomes positive. 662 00:40:52 --> 00:40:59 In that case, delta G is greater than zero. 663 00:40:59 --> 00:41:03 And that's not going to happen. 664 00:41:03 --> 00:41:07 On the other hand, if the chemical potential of the water 665 00:41:07 --> 00:41:11 molecules in the liquid phase is smaller than the chemical 666 00:41:11 --> 00:41:17 potential of the water in the solid phase, mu s is bigger 667 00:41:17 --> 00:41:19 than mu l, this becomes a negative number. 668 00:41:19 --> 00:41:24 Delta G is less than zero. 669 00:41:24 --> 00:41:27 And this will happen spontaneously. 670 00:41:27 --> 00:41:30 So that illustrates this idea, that the chemical potential of 671 00:41:30 --> 00:41:33 a species will want to go, so the species will want to 672 00:41:33 --> 00:41:37 go, where it can minimize its chemical potential. 673 00:41:37 --> 00:41:41 So in this case here, when we have the spontaneous process of 674 00:41:41 --> 00:41:45 the water, of the ice cube, melting, you can think of it as 675 00:41:45 --> 00:41:47 these water molecules that are in the ice phase 676 00:41:47 --> 00:41:49 looking around. 677 00:41:49 --> 00:41:51 They know what their chemical potential here 678 00:41:51 --> 00:41:51 is in the ice phase. 679 00:41:51 --> 00:41:54 They're looking around, they're looking to see the water phase. 680 00:41:54 --> 00:41:56 And they see that in the water phase, those water 681 00:41:56 --> 00:41:59 molecules have a smaller chemical potential. 682 00:41:59 --> 00:42:02 They're happier. 683 00:42:02 --> 00:42:08 And so these solid water molecules are jealous. 684 00:42:08 --> 00:42:11 And they want to go in the water phase. 685 00:42:11 --> 00:42:13 And the ice cube's going to melt. 686 00:42:13 --> 00:42:15 And it all has to do with this difference in chemical 687 00:42:15 --> 00:42:19 potentials for the water. 688 00:42:19 --> 00:42:22 And the same thing happens for the water molecules that are 689 00:42:22 --> 00:42:24 inside or outside of that cell that you put in the 690 00:42:24 --> 00:42:25 distilled water. 691 00:42:25 --> 00:42:27 The water molecules in the distilled water have a chemical 692 00:42:27 --> 00:42:31 potential which is higher than the water molecules 693 00:42:31 --> 00:42:34 inside the cell. 694 00:42:34 --> 00:42:36 And they don't want to be like that. 695 00:42:36 --> 00:42:39 They want to change, the system wants to change, until the 696 00:42:39 --> 00:42:42 water molecules couldn't care less whether they're in the 697 00:42:42 --> 00:42:46 water phase, or outside or inside the cell. 698 00:42:46 --> 00:42:50 The system is going to change until the water molecules 699 00:42:50 --> 00:42:53 have the same chemical potential everywhere. 700 00:42:53 --> 00:43:00 Where they don't have to choose one place or the other. 701 00:43:00 --> 00:43:02 And so that gives us, immediately, what we're 702 00:43:02 --> 00:43:05 going to need when we talk about equilibrium. 703 00:43:05 --> 00:43:09 Equilibrium, chemical equilibrium, is going to be 704 00:43:09 --> 00:43:13 where the chemical potential of a species is the same 705 00:43:13 --> 00:43:17 everywhere in the system. 706 00:43:17 --> 00:43:21 So at 0 degrees Celsius, one bar, which is the melting 707 00:43:21 --> 00:43:26 point of water, the chemical potential of a molecule of 708 00:43:26 --> 00:43:30 water in the ice phase and in the liquid phase is the same. 709 00:43:30 --> 00:43:31 That's the definition of the melting point. 710 00:43:31 --> 00:43:32 It doesn't care. 711 00:43:32 --> 00:43:33 It could go either way. 712 00:43:33 --> 00:43:34 It's an equilibrium. 713 00:43:34 --> 00:43:39 You take an ice cube, water, liquid water, 0 degrees 714 00:43:39 --> 00:43:40 Celsius, one bar. 715 00:43:40 --> 00:43:42 You come back three days later. 716 00:43:42 --> 00:43:43 It's the same. 717 00:43:43 --> 00:43:45 Come back a week later, it's the same. 718 00:43:45 --> 00:43:49 It's an equilibrium. 719 00:43:49 --> 00:43:52 Chemical potential of the water species is the same everywhere. 720 00:43:52 --> 00:43:54 It's an equilibrium. 721 00:43:54 --> 00:43:59 And I'm just repeating that because this is so important. 722 00:43:59 --> 00:44:04 OK, any questions? 723 00:44:04 --> 00:44:08 The last thing we're going to do is to illustrate also the 724 00:44:08 --> 00:44:19 importance of mixing to the chemical potential. 725 00:44:19 --> 00:44:27 So I'm going to set up sort of an arbitrary system here. 726 00:44:27 --> 00:44:32 This is kind of like the cell, or the fish, also, idea. 727 00:44:32 --> 00:44:34 I'm going to put a system where on one side I 728 00:44:34 --> 00:44:37 have a pure gas, A. 729 00:44:37 --> 00:44:42 On the other side, I have a mixture of A and B. 730 00:44:42 --> 00:44:45 And here is going to be a membrane that only 731 00:44:45 --> 00:44:51 allows A to go through. 732 00:44:51 --> 00:44:54 Only A can go through that membrane. 733 00:44:54 --> 00:44:56 They're going to be partial pressures in here, p 734 00:44:56 --> 00:44:59 prime B, and p prime A. 735 00:44:59 --> 00:45:02 For the gas pressures on that side, and pressure 736 00:45:02 --> 00:45:05 on that side is p sub A. 737 00:45:05 --> 00:45:11 And my goal in this example here is to show that if I 738 00:45:11 --> 00:45:16 compare the chemical potential of a species in a mixture, 739 00:45:16 --> 00:45:23 where the temperature and the pressure total are T and p, and 740 00:45:23 --> 00:45:28 I compare that to the chemical potential of the same species 741 00:45:28 --> 00:45:31 when it's pure, when it's not mixed with anything else, under 742 00:45:31 --> 00:45:37 the same temperature and pressure conditions, that, in 743 00:45:37 --> 00:45:41 fact that that equals sign is not there. 744 00:45:41 --> 00:45:42 That's not what I'm trying to show. 745 00:45:42 --> 00:45:47 I'm trying to show that there's a less-than sign right here. 746 00:45:47 --> 00:45:51 To show that if you take, again, this is the cell idea. 747 00:45:51 --> 00:45:55 If you take the water and the distilled water, under constant 748 00:45:55 --> 00:45:59 temperature and pressure conditions, it's pure. 749 00:45:59 --> 00:46:01 And it's looking at the cell. 750 00:46:01 --> 00:46:03 And inside the cell is there, boy, is there 751 00:46:03 --> 00:46:04 a mixture of things. 752 00:46:04 --> 00:46:06 There's salts, there are proteins, there are all 753 00:46:06 --> 00:46:10 sorts of things, right? 754 00:46:10 --> 00:46:12 It's a mixed system. 755 00:46:12 --> 00:46:14 The water in that mixed system, under the same temperature and 756 00:46:14 --> 00:46:17 pressure conditions, the chemical potential of that 757 00:46:17 --> 00:46:19 water molecule is less. 758 00:46:19 --> 00:46:23 And that's just an entropy thing. 759 00:46:23 --> 00:46:25 Entropy wants to increase. 760 00:46:25 --> 00:46:32 It just wants to be in a place with high energy. 761 00:46:32 --> 00:46:34 It's the Gibbs free energy. 762 00:46:34 --> 00:46:36 Gibbs free energy has enthalpy and entropy 763 00:46:36 --> 00:46:37 incorporated into it. 764 00:46:37 --> 00:46:39 The enthalpy's not doing anything. 765 00:46:39 --> 00:46:43 It's all driven by entropy. 766 00:46:43 --> 00:46:49 So this is what we're going to try to show. 767 00:46:49 --> 00:46:51 And I'm not going to get to it today. 768 00:46:51 --> 00:46:56 We'll start with it on Wednesday. 769 00:46:56 --> 00:47:00 And I'll let your ruminate on this for the next few days. 770 00:47:00 --> 00:47:03