1 00:00:00,030 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,860 Your support will help MIT OpenCourseWare continue to 4 00:00:06,860 --> 00:00:10,510 offer high quality educational resources for free. 5 00:00:10,510 --> 00:00:13,390 To make a donation or view additional materials from 6 00:00:13,390 --> 00:00:17,590 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,590 --> 00:00:20,290 ocw.mit.edu. 8 00:00:20,290 --> 00:00:24,660 PROFESSOR: And last time, you got to see how you can derive 9 00:00:24,660 --> 00:00:28,640 macroscopic thermodynamic results from the microscopic 10 00:00:28,640 --> 00:00:30,790 point of view of statistical mechanics. 11 00:00:30,790 --> 00:00:34,800 So in contrast to the way we'd gone through the course from 12 00:00:34,800 --> 00:00:38,030 the beginning, starting from empirical macroscopic 13 00:00:38,030 --> 00:00:41,920 observation and ideas through macroscopic laws and then 14 00:00:41,920 --> 00:00:44,330 working out the consequences of them, in statistical 15 00:00:44,330 --> 00:00:47,710 mechanics you start from a microscopic model, build up 16 00:00:47,710 --> 00:00:50,910 the energy levels that are going to give you the terms in 17 00:00:50,910 --> 00:00:53,310 the partition function, determine the partition 18 00:00:53,310 --> 00:00:56,910 function, and what you saw last time is that given that, 19 00:00:56,910 --> 00:00:58,190 you can calculate everything. 20 00:00:58,190 --> 00:01:00,730 You can calculate all the macroscopic properties that 21 00:01:00,730 --> 00:01:04,200 ordinarily come from the thermodynamic laws that were 22 00:01:04,200 --> 00:01:07,080 based on empirical macroscopic observation. 23 00:01:07,080 --> 00:01:09,810 So today, I just want to continue doing some 24 00:01:09,810 --> 00:01:11,360 statistical thermodynamics. 25 00:01:11,360 --> 00:01:14,330 Basically going through a few examples, just to see how it 26 00:01:14,330 --> 00:01:17,950 plays out when we calculate thermodynamic quantities, 27 00:01:17,950 --> 00:01:19,880 based on our microscopic picture. 28 00:01:19,880 --> 00:01:32,570 So let's just look at a few cases. 29 00:01:32,570 --> 00:01:35,640 And I'll start with a couple of examples that are entirely 30 00:01:35,640 --> 00:01:36,690 entropy driven. 31 00:01:36,690 --> 00:01:38,290 And we've seen them before from the 32 00:01:38,290 --> 00:01:39,760 macroscopic point of view. 33 00:01:39,760 --> 00:01:41,600 What we should hope, of course, is that we can derive 34 00:01:41,600 --> 00:01:43,910 those results from the partition functions that are 35 00:01:43,910 --> 00:01:45,010 appropriate here. 36 00:01:45,010 --> 00:01:57,100 So, we'll look at entropically driven processes. 37 00:01:57,100 --> 00:01:59,240 And the first one, the simplest one, maybe, to cover, 38 00:01:59,240 --> 00:02:01,280 is just free expansion of a gas. 39 00:02:01,280 --> 00:02:05,920 So what happens if we've got particles in a gas that are 40 00:02:05,920 --> 00:02:07,370 enclosed in a certain volume. 41 00:02:07,370 --> 00:02:09,890 And now we let the volume expand into vacuum on the 42 00:02:09,890 --> 00:02:10,370 other side. 43 00:02:10,370 --> 00:02:11,520 Of course, you know what'll happen. 44 00:02:11,520 --> 00:02:14,140 The volume will be filled and we've derived the 45 00:02:14,140 --> 00:02:15,710 thermodynamics for it before. 46 00:02:15,710 --> 00:02:17,830 So let's see what happens in a 47 00:02:17,830 --> 00:02:20,550 statistical mechanical treatment. 48 00:02:20,550 --> 00:02:27,320 So let's say, here's V1, there's our gas. 49 00:02:27,320 --> 00:02:29,830 There's vacuum. 50 00:02:29,830 --> 00:02:33,240 And now we'll open it up and just have the bigger volume, 51 00:02:33,240 --> 00:02:37,460 V2, and let the gas fill it. 52 00:02:37,460 --> 00:02:41,410 So I think last time you got introduced to basically a 53 00:02:41,410 --> 00:02:44,470 lattice model for translational motion. 54 00:02:44,470 --> 00:02:46,610 And I started in on this just a little bit 55 00:02:46,610 --> 00:02:48,060 the time before also. 56 00:02:48,060 --> 00:02:53,160 So this is a simple way to describe the statistical 57 00:02:53,160 --> 00:02:55,480 mechanics of filling an open volume. 58 00:02:55,480 --> 00:02:57,410 Filling a space with molecules. 59 00:02:57,410 --> 00:03:03,540 And all it does is say that we're going to divide up our 60 00:03:03,540 --> 00:03:06,700 available volume into little bits that are basically the 61 00:03:06,700 --> 00:03:09,930 size of an atom or a molecule, or whatever the particle is in 62 00:03:09,930 --> 00:03:11,370 the gas phase. 63 00:03:11,370 --> 00:03:14,160 And say, OK, there are maybe 10 of the 30th of 64 00:03:14,160 --> 00:03:15,340 those little volumes. 65 00:03:15,340 --> 00:03:17,750 The little volume elements are going to be on the order of an 66 00:03:17,750 --> 00:03:20,840 angstrom cubed if it's an atom, a little bit bigger if 67 00:03:20,840 --> 00:03:22,690 it's a molecule. 68 00:03:22,690 --> 00:03:26,100 And the available volume is on the order of meters cubed. 69 00:03:26,100 --> 00:03:30,810 So that works out, an angstrom is 10 to the minus 10 meters. 70 00:03:30,810 --> 00:03:33,170 And then the cube of that is 10 to the minus 30th. 71 00:03:33,170 --> 00:03:36,540 So if you look at the total number of little volume 72 00:03:36,540 --> 00:03:40,370 elements of this sort, it's on the order of 10 to the 30th. 73 00:03:40,370 --> 00:03:55,850 So, the total volume, capital V, let's make 74 00:03:55,850 --> 00:04:00,830 that distinction clear. 75 00:04:00,830 --> 00:04:04,230 And in this case, all the states of the system have 76 00:04:04,230 --> 00:04:06,570 equal energies. 77 00:04:06,570 --> 00:04:08,850 So in other words, it doesn't matter we're dividing our 78 00:04:08,850 --> 00:04:17,300 volume up into imagined little volume elements. 79 00:04:17,300 --> 00:04:22,080 And for both the molecular and the system states, the energy 80 00:04:22,080 --> 00:04:23,490 is the same. 81 00:04:23,490 --> 00:04:25,960 It doesn't matter whether the molecules are here, here, here 82 00:04:25,960 --> 00:04:27,160 and so forth. 83 00:04:27,160 --> 00:04:29,740 They're not interacting in this picture at all. 84 00:04:29,740 --> 00:04:32,440 So it doesn't matter how close or far they 85 00:04:32,440 --> 00:04:33,340 are from each other. 86 00:04:33,340 --> 00:04:35,590 They can't be in the same volume. 87 00:04:35,590 --> 00:04:37,310 So all the energies are the same. 88 00:04:37,310 --> 00:04:40,860 And what that means is, in the partition function, which is a 89 00:04:40,860 --> 00:04:43,880 sum over all these terms with these Boltzmann factors. 90 00:04:43,880 --> 00:04:45,920 That have e to the minus energy over kT. 91 00:04:45,920 --> 00:04:48,810 But energy's all the same in every one of the terms. 92 00:04:48,810 --> 00:04:51,200 So in order to determine the partition function, to 93 00:04:51,200 --> 00:04:54,890 determine Q, all we have to do is count up the total number 94 00:04:54,890 --> 00:04:58,170 of states that are available. 95 00:04:58,170 --> 00:05:03,740 So, our energy, our molecular translational energy, we'll 96 00:05:03,740 --> 00:05:05,270 just set it to zero. 97 00:05:05,270 --> 00:05:10,830 Same with our system translational energy. 98 00:05:10,830 --> 00:05:11,980 We'll set that to zero. 99 00:05:11,980 --> 00:05:16,320 All the microscopic available states, that is, if I take an 100 00:05:16,320 --> 00:05:18,630 individual particle and I say where it can be, all those 101 00:05:18,630 --> 00:05:20,410 states have the same energy. 102 00:05:20,410 --> 00:05:23,760 If I take a microstate of the system, you know, the whole 103 00:05:23,760 --> 00:05:27,290 collection of the particles, all of those states also have 104 00:05:27,290 --> 00:05:30,920 the same energy. 105 00:05:30,920 --> 00:05:35,490 So what that means is that little q is just equal to 106 00:05:35,490 --> 00:05:39,030 capital V over little v, right? 107 00:05:39,030 --> 00:05:40,320 How many states are there? 108 00:05:40,320 --> 00:05:44,450 Well, I've got my little volume, and then however many 109 00:05:44,450 --> 00:05:47,340 individual cells there are, that's the number of available 110 00:05:47,340 --> 00:05:49,090 states in this model. 111 00:05:49,090 --> 00:05:51,710 So it's big V over little v, on the order 112 00:05:51,710 --> 00:05:57,060 of 10 to the 30th. 113 00:05:57,060 --> 00:06:01,820 And then, big Q, the canonical partition function for the 114 00:06:01,820 --> 00:06:05,070 whole system, it's something that we've 115 00:06:05,070 --> 00:06:06,240 been through before. 116 00:06:06,240 --> 00:06:10,190 It's this process of saying, well, you take the first atom 117 00:06:10,190 --> 00:06:14,940 or molecule, and it has any one of these possible states. 118 00:06:14,940 --> 00:06:17,270 So it's on the order of 10 to the 30th possibilities. 119 00:06:17,270 --> 00:06:18,720 Then where does the second one go? 120 00:06:18,720 --> 00:06:20,490 Well, there are basically 10 to the 30th more 121 00:06:20,490 --> 00:06:21,360 possibilities. 122 00:06:21,360 --> 00:06:23,420 And the third one has 10 to the 30th. 123 00:06:23,420 --> 00:06:27,480 Since there are so many fewer atoms or molecules then there 124 00:06:27,480 --> 00:06:30,500 are volume elements, when we're dealing with the gas 125 00:06:30,500 --> 00:06:33,780 phase, we don't have to worry about the fact that, well the 126 00:06:33,780 --> 00:06:37,070 first million of the atoms filled some of the sites. 127 00:06:37,070 --> 00:06:40,160 So the next ones have fewer sites available to them. 128 00:06:40,160 --> 00:06:43,200 It's true, but it's such a small fraction that are ever 129 00:06:43,200 --> 00:06:47,070 filled that we don't need to worry about it. 130 00:06:47,070 --> 00:06:52,780 So Q is just little q to the capital N power, 131 00:06:52,780 --> 00:06:54,350 the number of particles. 132 00:06:54,350 --> 00:06:58,610 And then we've seen you have to divide by N factorial to 133 00:06:58,610 --> 00:07:02,900 avoid the overcounting of configurations that are in 134 00:07:02,900 --> 00:07:05,820 fact not distinguishable. 135 00:07:05,820 --> 00:07:09,190 The whole idea that maybe there's an atom here and 136 00:07:09,190 --> 00:07:10,420 another one here. 137 00:07:10,420 --> 00:07:12,990 That configuration is indistinguishable from the 138 00:07:12,990 --> 00:07:16,000 configuration with those two atoms reversed if we're 139 00:07:16,000 --> 00:07:21,780 dealing with a mole of identical atoms. 140 00:07:21,780 --> 00:07:22,640 So, OK. 141 00:07:22,640 --> 00:07:26,460 There's our capital Q. And that's also our system 142 00:07:26,460 --> 00:07:28,120 degeneracy. 143 00:07:28,120 --> 00:07:32,990 So the degeneracy, the little g for the molecular 144 00:07:32,990 --> 00:07:36,030 degeneracy, how many molecular states are there 145 00:07:36,030 --> 00:07:37,100 of a certain energy. 146 00:07:37,100 --> 00:07:39,780 Well, it's the same thing as q. 147 00:07:39,780 --> 00:07:42,130 And it's the same thing here for the system states. 148 00:07:42,130 --> 00:07:49,900 The capital Omega is just equal to that. 149 00:07:49,900 --> 00:07:55,400 Well, now let's look at what happens when we do this. 150 00:07:55,400 --> 00:07:59,400 When we expand from volume V1 to volume V2. 151 00:07:59,400 --> 00:08:03,270 So that capital V is going to change. 152 00:08:03,270 --> 00:08:10,230 So capital V1 goes to capital V2. 153 00:08:10,230 --> 00:08:11,780 And we know it's entropically driven. 154 00:08:11,780 --> 00:08:15,760 Let's calculate the change in entropy. 155 00:08:15,760 --> 00:08:26,160 So delta S is just k log capital Omega 2 minus k log 156 00:08:26,160 --> 00:08:27,540 capital Omega 1. 157 00:08:27,540 --> 00:08:30,720 Now, in the process of just seeing how the development 158 00:08:30,720 --> 00:08:34,330 goes, how you can derive all these thermodynamic quantities 159 00:08:34,330 --> 00:08:38,050 from the partition function, one of the really most central 160 00:08:38,050 --> 00:08:40,740 results concerns the entropy. 161 00:08:40,740 --> 00:08:45,040 How you can describe the entropy in terms of the 162 00:08:45,040 --> 00:08:48,760 probabilities that the different states are occupied. 163 00:08:48,760 --> 00:08:51,320 And how, in the case like this, where the states all 164 00:08:51,320 --> 00:08:54,800 have the same probability of being occupied, then you have 165 00:08:54,800 --> 00:08:57,030 this very simplified form for the entropy. 166 00:08:57,030 --> 00:09:00,680 Just, if S is k log capital Omega, where capital Omega is 167 00:09:00,680 --> 00:09:02,090 the degeneracy. 168 00:09:02,090 --> 00:09:06,220 The number of system states of that energy. 169 00:09:06,220 --> 00:09:08,960 An amazing result. 170 00:09:08,960 --> 00:09:10,440 You told them about Boltzmann's 171 00:09:10,440 --> 00:09:13,270 tombstone and so forth. 172 00:09:13,270 --> 00:09:15,650 So it's on there. 173 00:09:15,650 --> 00:09:20,440 In fact, the ill acceptance of that result kind of led to the 174 00:09:20,440 --> 00:09:23,460 tombstone being erected when it did. 175 00:09:23,460 --> 00:09:28,050 Boltzmann, depressed over the lack of acceptance of his 176 00:09:28,050 --> 00:09:30,270 theories, put himself to an early death. 177 00:09:30,270 --> 00:09:33,540 And this was off a big part of the reason. 178 00:09:33,540 --> 00:09:38,140 But we, generations later, have come to accept the 179 00:09:38,140 --> 00:09:41,560 results that concerned him so deeply. 180 00:09:41,560 --> 00:09:43,620 So this is our change in entropy, when we just have 181 00:09:43,620 --> 00:09:46,100 this expansion of gas from V1 to V2. 182 00:09:46,100 --> 00:09:52,220 So it's just k log omega 2, omega 1. 183 00:09:52,220 --> 00:09:55,850 So now let's just put in our results for the volumes. 184 00:09:55,850 --> 00:10:04,560 It's k log V2 over v, I'm exaggerating the size there. 185 00:10:04,560 --> 00:10:11,050 To the N, over N factorial, over V1 over v to the N 186 00:10:11,050 --> 00:10:13,530 divided by N factorial. 187 00:10:13,530 --> 00:10:16,180 So this is going to turn out to be a relatively simple 188 00:10:16,180 --> 00:10:18,840 case, because the factorials are going to cancel. 189 00:10:18,840 --> 00:10:28,700 So, then we just have N k, log V2 over V1. 190 00:10:28,700 --> 00:10:30,670 Terrific, right? 191 00:10:30,670 --> 00:10:34,410 Now, remember, k, the Boltzmann constant, is just 192 00:10:34,410 --> 00:10:39,260 the ideal gas constant per molecule rather than per mole. 193 00:10:39,260 --> 00:10:43,160 So this is the same thing as little N, the number of moles, 194 00:10:43,160 --> 00:10:49,090 times R, times log V2 over V1. 195 00:10:49,090 --> 00:10:51,060 And that should be a familiar result. 196 00:10:51,060 --> 00:10:53,870 That's the change in entropy in expansion, free expansion 197 00:10:53,870 --> 00:10:55,960 of a gas, from V1 to V2. 198 00:10:55,960 --> 00:10:58,500 So now we've been able to derive that just based on this 199 00:10:58,500 --> 00:11:01,960 really simple molecular picture. 200 00:11:01,960 --> 00:11:06,530 Based on that plus the result that Boltzmann fought and paid 201 00:11:06,530 --> 00:11:08,600 so dearly for, so that we would have it 202 00:11:08,600 --> 00:11:11,960 and understand it. 203 00:11:11,960 --> 00:11:14,200 As before, of course note it's greater than zero. 204 00:11:14,200 --> 00:11:17,300 If the volume, if we're expanding into a bigger volume 205 00:11:17,300 --> 00:11:18,490 than before. 206 00:11:18,490 --> 00:11:21,440 The entropy goes up. 207 00:11:21,440 --> 00:11:25,240 Also notice one of the other results that you've seen is 208 00:11:25,240 --> 00:11:28,420 that you could relate the partition 209 00:11:28,420 --> 00:11:30,930 function to A, right? 210 00:11:30,930 --> 00:11:43,090 The Helmholtz free energy is minus k T log Q. And S is 211 00:11:43,090 --> 00:11:44,340 negative dA/dT. 212 00:11:50,610 --> 00:11:52,910 And so this would immediately give us the same result, 213 00:11:52,910 --> 00:11:56,170 because omega's going to go in here. 214 00:11:56,170 --> 00:12:00,210 So we could get this result from another pathway, too, but 215 00:12:00,210 --> 00:12:02,890 a more simple and direct way is just to start directly from 216 00:12:02,890 --> 00:12:07,330 the Boltzmann result for entropy. 217 00:12:07,330 --> 00:12:10,930 Any questions? 218 00:12:10,930 --> 00:12:15,080 Alright, let's go one step more complex. 219 00:12:15,080 --> 00:12:17,110 Let's now look at the entropy of mixing. 220 00:12:17,110 --> 00:12:19,050 So now we've got two species. 221 00:12:19,050 --> 00:12:21,900 Rather than one expanding into a free volume, and we're going 222 00:12:21,900 --> 00:12:23,860 to open a barrier between them and see them mix. 223 00:12:23,860 --> 00:12:25,930 And again, it's still entropically driven. 224 00:12:25,930 --> 00:12:27,880 And we should be able to calculate the entropy change 225 00:12:27,880 --> 00:12:31,100 that we saw before from a macroscopic perspective. 226 00:12:31,100 --> 00:12:39,000 So, let's take a look. 227 00:12:39,000 --> 00:12:48,300 So now, we have NA molecules of A in some volume VA. 228 00:12:48,300 --> 00:12:53,530 And NB molecules of B, in some volume VB. 229 00:12:53,530 --> 00:12:58,070 And then we're going to open it up. 230 00:12:58,070 --> 00:13:02,940 Then we've got capital N, NA plus NB. 231 00:13:02,940 --> 00:13:11,180 And capital V. VA plus VB, right? 232 00:13:11,180 --> 00:13:12,450 And the whole thing is 233 00:13:12,450 --> 00:13:19,290 happening at constant pressure. 234 00:13:19,290 --> 00:13:24,070 Well, then let's look at the initial and final expressions 235 00:13:24,070 --> 00:13:25,160 for the entropy. 236 00:13:25,160 --> 00:13:28,290 So, S1, at the beginning. 237 00:13:28,290 --> 00:13:44,600 It's just k log omega A plus k log omega B. And that's k log 238 00:13:44,600 --> 00:13:49,850 VA over little v to the NA power. 239 00:13:49,850 --> 00:13:58,730 Divided by NA factorial times the same thing for B. VB over 240 00:13:58,730 --> 00:14:05,770 little v to the NB power over NB factorial. 241 00:14:05,770 --> 00:14:07,640 So that's our initial entropy. 242 00:14:07,640 --> 00:14:11,230 The sum of the two entropy contributions, 243 00:14:11,230 --> 00:14:14,380 from the two sides. 244 00:14:14,380 --> 00:14:20,340 And S2, the final one, is just k log omega 245 00:14:20,340 --> 00:14:23,780 for the whole shebang. 246 00:14:23,780 --> 00:14:27,060 And that's k log. 247 00:14:27,060 --> 00:14:33,300 And now we have capital V over little v to the N power. 248 00:14:33,300 --> 00:14:35,870 And now we have a combinatorics result that I 249 00:14:35,870 --> 00:14:39,020 think probably you're all familiar with from one context 250 00:14:39,020 --> 00:14:39,890 or another. 251 00:14:39,890 --> 00:14:44,400 That is, now we have to divide by NA 252 00:14:44,400 --> 00:14:52,770 factorial times NB factorial. 253 00:14:52,770 --> 00:14:56,300 In other words, the amount that we need to divide by to 254 00:14:56,300 --> 00:14:59,680 avoid overcounting is the product of those two 255 00:14:59,680 --> 00:15:02,810 factorials. 256 00:15:02,810 --> 00:15:05,380 Just in case it's been a while since you've seen stuff like 257 00:15:05,380 --> 00:15:09,350 that, I've got a couple of pages further on the notes at 258 00:15:09,350 --> 00:15:10,590 the bottom of the page. 259 00:15:10,590 --> 00:15:13,860 I've just broken that out for a really simple example where 260 00:15:13,860 --> 00:15:16,680 there are just two molecules of A, and two molecules of B 261 00:15:16,680 --> 00:15:22,250 represented by different little balls in lattice sites. 262 00:15:22,250 --> 00:15:24,630 And I just worked it out for the total where there are only 263 00:15:24,630 --> 00:15:29,030 ten total in the mixture. 264 00:15:29,030 --> 00:15:33,150 The point is, now what happens is, when you start filling a 265 00:15:33,150 --> 00:15:37,560 lattice, but it's not all the same, right? 266 00:15:37,560 --> 00:15:39,430 So now you have A here. 267 00:15:39,430 --> 00:15:40,480 And A here. 268 00:15:40,480 --> 00:15:41,520 And B here. 269 00:15:41,520 --> 00:15:42,720 And B here. 270 00:15:42,720 --> 00:15:45,850 So, of course if you interchange A and B here, you 271 00:15:45,850 --> 00:15:49,180 don't have an identical state. 272 00:15:49,180 --> 00:15:51,410 An indistinguishable state. 273 00:15:51,410 --> 00:15:53,950 That's distinguishable and needs to be counted. 274 00:15:53,950 --> 00:15:57,470 It's only interchanging B with B that you need to correct 275 00:15:57,470 --> 00:16:01,770 for, and interchanging A with A. So of course you need to 276 00:16:01,770 --> 00:16:04,610 account for that in a way that's different from how it 277 00:16:04,610 --> 00:16:10,380 turns out if every one of the molecules is identical. 278 00:16:10,380 --> 00:16:13,680 So it's the product of these factorials. 279 00:16:13,680 --> 00:16:17,130 In the numerator. 280 00:16:17,130 --> 00:16:29,910 So now delta S is then just this minus this. 281 00:16:29,910 --> 00:16:48,390 So it's k log V over little v to the NA plus NB over VA 282 00:16:48,390 --> 00:16:55,710 divided by little v to the NA power VB divided by little v 283 00:16:55,710 --> 00:16:58,220 to the NB power. 284 00:16:58,220 --> 00:17:02,770 And, let's see, maybe I'd better back up. 285 00:17:02,770 --> 00:17:04,110 Let me know, that's all. 286 00:17:04,110 --> 00:17:06,780 And what's happening is, again, luckily, these 287 00:17:06,780 --> 00:17:10,850 factorials are canceling. 288 00:17:10,850 --> 00:17:25,910 So we just have k log V to the NA, V to the NB over VA to the 289 00:17:25,910 --> 00:17:33,340 NA VB to the NB. 290 00:17:33,340 --> 00:17:36,830 And now we're going to go back and use the fact that the 291 00:17:36,830 --> 00:17:40,840 pressures are the same, all the way through. 292 00:17:40,840 --> 00:17:43,850 And what that means is that the volumes must be in the 293 00:17:43,850 --> 00:17:50,020 same ratio as the number of molecules. 294 00:17:50,020 --> 00:17:56,300 And what that means is that, for example, VA over V is the 295 00:17:56,300 --> 00:18:01,380 same as NA over N. And that's just equal to XA, the mole 296 00:18:01,380 --> 00:18:07,780 fraction of A. And the same for B. So we 297 00:18:07,780 --> 00:18:11,680 can substitute that. 298 00:18:11,680 --> 00:18:21,360 And know our delta S just becomes k log, let's see, let 299 00:18:21,360 --> 00:18:27,700 me break this out from the beginning. 300 00:18:27,700 --> 00:18:39,930 And just take minus k log on XA to the NA power minus k log 301 00:18:39,930 --> 00:18:43,250 XB to the NB power. 302 00:18:43,250 --> 00:18:46,130 All I've done make these substitutions here. 303 00:18:46,130 --> 00:18:49,540 For the ratios of the volume. 304 00:18:49,540 --> 00:18:58,710 So it's just minus N k XA log XA plus XB log XB. 305 00:19:02,470 --> 00:19:04,640 Look familiar? 306 00:19:04,640 --> 00:19:07,120 Same thing we saw macroscopically before. 307 00:19:07,120 --> 00:19:09,960 So, again, we can still use our microscopic model. 308 00:19:09,960 --> 00:19:12,990 And continue to derive the macroscopic entropy changes. 309 00:19:12,990 --> 00:19:15,410 And, of course, for many of these we can still get our 310 00:19:15,410 --> 00:19:17,040 delta G and so forth. 311 00:19:17,040 --> 00:19:21,740 Of course, it's only entropy that's going to contribute. 312 00:19:21,740 --> 00:19:26,620 OK, let's look at just one more entropy driven problem. 313 00:19:26,620 --> 00:19:29,980 And that is, it's a little bit different from these. 314 00:19:29,980 --> 00:19:33,320 Let's look at the entropy mixing in a liquid. 315 00:19:33,320 --> 00:19:37,070 And the difference between the gas and the liquid is that in 316 00:19:37,070 --> 00:19:42,330 the case of the liquid, every single cell is filled. 317 00:19:42,330 --> 00:19:43,710 It's not like with the gas. 318 00:19:43,710 --> 00:19:45,815 And that makes a difference in how the 319 00:19:45,815 --> 00:19:47,950 states need to be counted. 320 00:19:47,950 --> 00:19:50,160 Because here, remember how I said when we were 321 00:19:50,160 --> 00:19:51,280 filling this lattice. 322 00:19:51,280 --> 00:19:53,170 Well, the first molecules goes somewhere. 323 00:19:53,170 --> 00:19:54,740 And there are 10 to the 30th possibilities. 324 00:19:54,740 --> 00:19:56,880 And the second molecule goes somewhere and there are 10 to 325 00:19:56,880 --> 00:19:58,750 the 30th possibilities. 326 00:19:58,750 --> 00:20:01,630 And there are always 10 to the 30th possibilities because 327 00:20:01,630 --> 00:20:04,480 there's so few molecules that you never have to worry about 328 00:20:04,480 --> 00:20:06,360 the fact that it's getting filled up, so there would 329 00:20:06,360 --> 00:20:12,040 become fewer possibilities for that last molecule. 330 00:20:12,040 --> 00:20:14,020 And the reason again is because there are so few 331 00:20:14,020 --> 00:20:19,640 molecules that essentially, all the cells are open and 332 00:20:19,640 --> 00:20:22,250 available even to the last molecule. 333 00:20:22,250 --> 00:20:23,880 Because maybe there's a mole of molecules. 334 00:20:23,880 --> 00:20:26,940 Maybe there are 10 to the 24 molecules. 335 00:20:26,940 --> 00:20:30,150 And there are 10 to the 30th lattice sites. 336 00:20:30,150 --> 00:20:32,140 So there might be one in a million lattice sites 337 00:20:32,140 --> 00:20:35,370 occupied, even at the very end of the procedure. 338 00:20:35,370 --> 00:20:37,470 But of course for the liquid, that's 339 00:20:37,470 --> 00:20:39,990 definitely not the case. 340 00:20:39,990 --> 00:20:42,070 All the available volume is filled. 341 00:20:42,070 --> 00:20:44,940 So by the time you get to that last molecule, it has one and 342 00:20:44,940 --> 00:20:47,580 only one space it can go into. 343 00:20:47,580 --> 00:20:51,290 So you have to count for the diminishment of the available 344 00:20:51,290 --> 00:20:56,660 sites as molecules are placed in them. 345 00:20:56,660 --> 00:20:57,470 OK. 346 00:20:57,470 --> 00:21:01,670 So it's a different kind of combinatorics 347 00:21:01,670 --> 00:21:04,020 problem that results. 348 00:21:04,020 --> 00:21:07,190 So, OK. 349 00:21:07,190 --> 00:21:12,620 Let's look at liquid mixture. 350 00:21:12,620 --> 00:21:22,140 So now, it's going to look like, and 351 00:21:22,140 --> 00:21:29,550 they're all filled up. 352 00:21:29,550 --> 00:21:39,865 I sure used fewer sites maybe, but that'll be easier here. 353 00:21:39,865 --> 00:21:43,910 We've got open circles. 354 00:21:43,910 --> 00:21:46,020 So these two are going to mix. 355 00:21:46,020 --> 00:21:52,950 And now we're going to have filled mixture. 356 00:21:52,950 --> 00:21:55,350 So we're going to have a bigger volume, but it's still 357 00:21:55,350 --> 00:21:58,480 going to be filled and all the positions are random. 358 00:21:58,480 --> 00:22:00,310 So we have to consider all the possible 359 00:22:00,310 --> 00:22:02,640 configurations that we have. 360 00:22:02,640 --> 00:22:03,510 OK. 361 00:22:03,510 --> 00:22:06,060 So we still should go through the 362 00:22:06,060 --> 00:22:08,110 same part of the procedure. 363 00:22:08,110 --> 00:22:16,710 So let's call this A. And this, B. So at first, already 364 00:22:16,710 --> 00:22:18,100 the situation is different. 365 00:22:18,100 --> 00:22:24,520 If we say what's the entropy of system A. Well, in this 366 00:22:24,520 --> 00:22:30,550 simple model the entropy of the pure liquid is zero. 367 00:22:30,550 --> 00:22:32,600 Now, of course, that's not realistic. 368 00:22:32,600 --> 00:22:34,650 But it's going to turn out to be suitable. 369 00:22:34,650 --> 00:22:37,430 Because the change in entropy that we go 370 00:22:37,430 --> 00:22:38,630 from here to the mixture. 371 00:22:38,630 --> 00:22:42,120 The other contributions to entropy are going to more or 372 00:22:42,120 --> 00:22:44,480 less be the same from beginning to end. 373 00:22:44,480 --> 00:22:47,680 What's going to be significantly different is 374 00:22:47,680 --> 00:22:51,040 simply the fact that in the mixture you have the random 375 00:22:51,040 --> 00:22:53,100 positions that can be occupied. 376 00:22:53,100 --> 00:22:55,310 So that's going to be the term that matters, the contribution 377 00:22:55,310 --> 00:22:56,570 that matters. 378 00:22:56,570 --> 00:23:02,420 So entropy is just k log Omega A. There's 379 00:23:02,420 --> 00:23:03,760 only one system state. 380 00:23:03,760 --> 00:23:06,110 All of the lattice sites are filled, and they're all filled 381 00:23:06,110 --> 00:23:08,840 with A. So there's only one way to do it. 382 00:23:08,840 --> 00:23:10,090 And of course, it doesn't matter if 383 00:23:10,090 --> 00:23:10,740 you interchange them. 384 00:23:10,740 --> 00:23:13,740 They're indistinguishable. 385 00:23:13,740 --> 00:23:16,650 So this is zero. 386 00:23:16,650 --> 00:23:18,320 This is one. 387 00:23:18,320 --> 00:23:19,820 There's only one state. 388 00:23:19,820 --> 00:23:27,430 Same, of course, for B. 389 00:23:27,430 --> 00:23:33,420 So that's our initial entropy in this simple model. 390 00:23:33,420 --> 00:23:37,800 Well then, delta S of the mixture is 391 00:23:37,800 --> 00:23:39,900 just S of the mixture. 392 00:23:39,900 --> 00:23:45,220 We don't need to worry about the original parts. 393 00:23:45,220 --> 00:23:51,080 Now, this is going to be k log of N factorial over NA 394 00:23:51,080 --> 00:23:55,460 factorial NB factorial, like we've seen before. 395 00:23:55,460 --> 00:23:58,220 That's not different from before. 396 00:23:58,220 --> 00:24:02,940 Except that unlike before, we don't have the convenient and 397 00:24:02,940 --> 00:24:06,440 simple cancellation of the factorials. 398 00:24:06,440 --> 00:24:09,050 That happened before, because they were there. 399 00:24:09,050 --> 00:24:11,290 In other words, they were there in the entropy 400 00:24:11,290 --> 00:24:15,880 contribution in the gas phase as well, in the pure gas. 401 00:24:15,880 --> 00:24:19,160 They're only there here in the liquid. 402 00:24:19,160 --> 00:24:21,290 And that means we have to deal with them. 403 00:24:21,290 --> 00:24:24,610 But it turns out it's not too bad to deal with them. 404 00:24:24,610 --> 00:24:31,230 And we're going to use this Stirling's approximation that 405 00:24:31,230 --> 00:24:32,810 I introduced before. 406 00:24:32,810 --> 00:24:42,240 So the log of N factorial is approximately equal to N log N 407 00:24:42,240 --> 00:24:48,700 minus N. So let's just break that out then, and use the 408 00:24:48,700 --> 00:24:53,620 Stirling's approximation for each of the factorial terms. 409 00:24:53,620 --> 00:25:03,420 So then we have S for the mixture is N k 410 00:25:03,420 --> 00:25:09,230 log N minus N k minus. 411 00:25:09,230 --> 00:25:11,440 And now we'll do the bottom, the numerator. 412 00:25:11,440 --> 00:25:20,460 NA k log NA minus NA times k, that's this part. 413 00:25:20,460 --> 00:25:22,310 And now we'll do this one. 414 00:25:22,310 --> 00:25:31,010 So it's plus NB k log NB minus NB times k. 415 00:25:31,010 --> 00:25:43,340 But NA and NB, of course, are just N. So these will cancel. 416 00:25:43,340 --> 00:25:51,050 So then we're left with NA plus NB times k 417 00:25:51,050 --> 00:25:55,870 log N minus NA -- 418 00:25:55,870 --> 00:25:58,200 And I just want to write this this way so that I can then 419 00:25:58,200 --> 00:26:00,760 separate the terms. 420 00:26:00,760 --> 00:26:11,530 Minus NA k log NA minus NB k log NB. 421 00:26:11,530 --> 00:26:16,430 And now I just want to combine the easily combined terms. 422 00:26:16,430 --> 00:26:30,330 So it's NA k log N over NA plus NB k log N over NB. 423 00:26:30,330 --> 00:26:32,980 Looks like we're home. 424 00:26:32,980 --> 00:26:45,520 Now we just have N k XA log XA plus XB log XB right? 425 00:26:45,520 --> 00:26:47,990 This NA over N is just XA. 426 00:26:47,990 --> 00:26:52,270 NB over is XB. 427 00:26:52,270 --> 00:26:55,200 And then I've divided the same thing to make XA's over here. 428 00:26:55,200 --> 00:26:58,760 And take the total number of moles out. 429 00:26:58,760 --> 00:26:59,880 Alright. 430 00:26:59,880 --> 00:27:02,480 Look familiar? 431 00:27:02,480 --> 00:27:04,800 Again, the same thing derived now in a way that's a bit 432 00:27:04,800 --> 00:27:06,730 different from what we did in the gas phase. 433 00:27:06,730 --> 00:27:09,620 So now I've got the entropy of mixing even in a condensed 434 00:27:09,620 --> 00:27:10,580 phase the liquid. 435 00:27:10,580 --> 00:27:14,670 And again, the reason this simple model works is because 436 00:27:14,670 --> 00:27:18,020 although a real liquid, certainly a pure liquid has a 437 00:27:18,020 --> 00:27:21,910 finite entropy, a substantial amount of disorder, that's 438 00:27:21,910 --> 00:27:24,590 present, that kind of disorder. 439 00:27:24,590 --> 00:27:26,770 In other words, the fact that you don't really have the 440 00:27:26,770 --> 00:27:28,750 molecules sitting in sites on a 441 00:27:28,750 --> 00:27:30,430 lattice but there's disorder. 442 00:27:30,430 --> 00:27:35,190 There's also rotational disorder if it's a molecule. 443 00:27:35,190 --> 00:27:37,660 There are various contributions to entropy. 444 00:27:37,660 --> 00:27:40,410 But those are all comparable in the 445 00:27:40,410 --> 00:27:43,180 starting and final states. 446 00:27:43,180 --> 00:27:46,320 The thing that's changed is simply the interchanging of A 447 00:27:46,320 --> 00:27:47,320 and B molecules. 448 00:27:47,320 --> 00:27:50,490 That's introduced a new kind of disorder that didn't used 449 00:27:50,490 --> 00:27:52,080 to be present. 450 00:27:52,080 --> 00:27:56,110 A very big difference in the number of states at the end 451 00:27:56,110 --> 00:27:57,350 from the beginning. 452 00:27:57,350 --> 00:27:59,480 And that's what we've calculated and captured here. 453 00:27:59,480 --> 00:28:04,520 And that's why such a simple model still works OK. 454 00:28:04,520 --> 00:28:09,860 Any questions about these entropy driven cases? 455 00:28:09,860 --> 00:28:11,350 OK. 456 00:28:11,350 --> 00:28:15,430 Now let's move on and talk about the cases which are more 457 00:28:15,430 --> 00:28:17,670 commonly encountered, where the states aren't 458 00:28:17,670 --> 00:28:19,290 all the same energy. 459 00:28:19,290 --> 00:28:21,970 Of course, here in the liquid, just like in the gas, we 460 00:28:21,970 --> 00:28:25,110 haven't treated interactions between the molecules. 461 00:28:25,110 --> 00:28:28,010 We've assumed that they're equal between A and B, as they 462 00:28:28,010 --> 00:28:31,640 are between A and A and B and B. So in this case then all 463 00:28:31,640 --> 00:28:33,965 the energy to the states are the same and it's just a 464 00:28:33,965 --> 00:28:34,900 counting problem. 465 00:28:34,900 --> 00:28:37,700 How many states are there available. 466 00:28:37,700 --> 00:28:40,450 As soon as the energies are different, then of course we 467 00:28:40,450 --> 00:28:43,550 need to account for all those Boltzmann factors. 468 00:28:43,550 --> 00:28:47,700 Those e to the minus energy over kT factors become part of 469 00:28:47,700 --> 00:28:49,940 the, they weight the counting. 470 00:28:49,940 --> 00:29:15,330 And we have to do that. 471 00:29:15,330 --> 00:29:20,510 So, what I want to do is go back to this simple polymer 472 00:29:20,510 --> 00:29:25,040 configurational model that we introduced before. 473 00:29:25,040 --> 00:29:29,355 And this sort of model, in one form or another, we're going 474 00:29:29,355 --> 00:29:34,190 to see a few times in the rest of this treatment. 475 00:29:34,190 --> 00:29:38,680 The reason, really, is because it's a simple and also 476 00:29:38,680 --> 00:29:45,120 realistic way of portraying a system that has a finite 477 00:29:45,120 --> 00:29:49,090 number of well-defined energies. 478 00:29:49,090 --> 00:29:50,520 If you don't have that, of course you 479 00:29:50,520 --> 00:29:53,810 could do the treatment. 480 00:29:53,810 --> 00:29:57,540 In a, sort of, classical mechanic sense, a continuum of 481 00:29:57,540 --> 00:30:00,050 states, all with different energies. 482 00:30:00,050 --> 00:30:01,570 It's hard. 483 00:30:01,570 --> 00:30:05,370 Because then those sum in the partition function become 484 00:30:05,370 --> 00:30:08,040 integrals over all the different configurations and 485 00:30:08,040 --> 00:30:08,530 possibilities. 486 00:30:08,530 --> 00:30:10,720 It's much more complicated to do. 487 00:30:10,720 --> 00:30:14,380 It's much more straightforward if you can identify discrete 488 00:30:14,380 --> 00:30:17,210 states and add over them. 489 00:30:17,210 --> 00:30:21,660 Now, someday you'll take quantum mechanics. 490 00:30:21,660 --> 00:30:25,220 And then you'll see that the states are discrete, even if 491 00:30:25,220 --> 00:30:27,190 we're talking about translation or 492 00:30:27,190 --> 00:30:29,500 rotation and so forth. 493 00:30:29,500 --> 00:30:31,980 So you'll have discrete quantum mechanical energy 494 00:30:31,980 --> 00:30:34,050 levels and so forth. 495 00:30:34,050 --> 00:30:35,770 You haven't had that yet. 496 00:30:35,770 --> 00:30:38,320 And so that's not the starting point that we're going to use. 497 00:30:38,320 --> 00:30:40,670 Instead, we're going to use a starting point that goes 498 00:30:40,670 --> 00:30:44,290 through the exact same formalism, which is just the 499 00:30:44,290 --> 00:30:47,970 discrete states available in molecules that have multiple 500 00:30:47,970 --> 00:30:50,900 configurations. 501 00:30:50,900 --> 00:31:11,220 So we're going to have unequal energy states. 502 00:31:11,220 --> 00:31:25,530 And what we're envisioning is molecular or polymer 503 00:31:25,530 --> 00:31:28,640 configurations. 504 00:31:28,640 --> 00:31:42,320 So here we have our states. 505 00:31:42,320 --> 00:31:46,130 And just like before, the idea is that if you have two 506 00:31:46,130 --> 00:31:49,310 sub-units that are in proximity, there's some kind 507 00:31:49,310 --> 00:31:50,520 of favorable interaction. 508 00:31:50,520 --> 00:31:52,450 It could be hydrogen bonding. 509 00:31:52,450 --> 00:31:53,290 Could be different. 510 00:31:53,290 --> 00:31:57,120 But the point is, there's some sort of interaction there that 511 00:31:57,120 --> 00:32:12,320 reduces the energy. 512 00:32:12,320 --> 00:32:14,490 And then there are other configurations that 513 00:32:14,490 --> 00:32:16,490 just don't have that. 514 00:32:16,490 --> 00:32:32,650 Because the sub-units aren't in proximity to each other. 515 00:32:32,650 --> 00:32:34,960 And it's not hard to work out that in this very simple 516 00:32:34,960 --> 00:32:36,550 model, these are the only 517 00:32:36,550 --> 00:32:39,530 configurations that are available. 518 00:32:39,530 --> 00:32:42,020 The only distinct configurations. 519 00:32:42,020 --> 00:32:50,090 So then, our molecular energy, E, we can define as zero here. 520 00:32:50,090 --> 00:32:53,530 Before we defined it as minus E int, but I'm going to, it's 521 00:32:53,530 --> 00:32:56,460 a little more convenient to make this the zero of energy, 522 00:32:56,460 --> 00:33:02,950 and then this is plus the interaction for 523 00:33:02,950 --> 00:33:04,830 each of these states. 524 00:33:04,830 --> 00:33:09,870 Of course, we can put the zero of energy wherever we prefer. 525 00:33:09,870 --> 00:33:15,220 And then we have the degeneracy is one. 526 00:33:15,220 --> 00:33:21,780 And in this case it's three. 527 00:33:21,780 --> 00:33:28,060 So now we can just write out the configurational partition 528 00:33:28,060 --> 00:33:33,040 function for the molecules and also the canonical partition 529 00:33:33,040 --> 00:33:34,770 function for the system. 530 00:33:34,770 --> 00:33:39,470 So q configurational. 531 00:33:39,470 --> 00:33:40,980 And we're just going to sum over the states. 532 00:33:40,980 --> 00:33:50,360 So it's e to the zero over kT, for the lowest state. 533 00:33:50,360 --> 00:33:52,180 And then there are three states that are going to have 534 00:33:52,180 --> 00:33:58,410 this term, e to the minus E interaction over kT, where E 535 00:33:58,410 --> 00:34:00,510 int is a positive number. 536 00:34:00,510 --> 00:34:02,470 In other words, the probability of any one of 537 00:34:02,470 --> 00:34:06,170 these states is a little bit lower than this state. 538 00:34:06,170 --> 00:34:09,320 Because this state has lower energy. 539 00:34:09,320 --> 00:34:12,580 Remember what I mentioned earlier, which is although the 540 00:34:12,580 --> 00:34:16,040 probability of any one of these states is lower than the 541 00:34:16,040 --> 00:34:21,740 probability of this state, the probability of this energy is 542 00:34:21,740 --> 00:34:24,300 likely to be higher than the probability of this energy. 543 00:34:24,300 --> 00:34:26,450 Because there's only one of these states and the 544 00:34:26,450 --> 00:34:28,250 degeneracy here is three. 545 00:34:28,250 --> 00:34:31,370 So there are three possibilities in which the 546 00:34:31,370 --> 00:34:33,380 molecule could have this energy, and only one this. 547 00:34:33,380 --> 00:34:39,880 So if, basically, kT is bigger than E int, so in other words, 548 00:34:39,880 --> 00:34:44,370 if this term isn't very much smaller than one, then of 549 00:34:44,370 --> 00:34:48,390 course this will be bigger than this. 550 00:34:48,390 --> 00:34:51,120 So of course this is one plus three e to the 551 00:34:51,120 --> 00:34:59,950 minus E int over kT. 552 00:34:59,950 --> 00:35:05,490 And now we have our capital Q, our canonical configurational 553 00:35:05,490 --> 00:35:07,380 partition function. 554 00:35:07,380 --> 00:35:09,530 And that's just little q 555 00:35:09,530 --> 00:35:12,160 configurational to the Nth power. 556 00:35:12,160 --> 00:35:16,140 And like you've seen so far, in various cases where the 557 00:35:16,140 --> 00:35:19,200 molecules are separate non-interacting molecules, 558 00:35:19,200 --> 00:35:24,950 this is just the molecular partition function taken 559 00:35:24,950 --> 00:35:26,730 capital N times. 560 00:35:26,730 --> 00:35:30,250 If all the molecules are behaving independently, then 561 00:35:30,250 --> 00:35:31,740 you sum over all of those states. 562 00:35:31,740 --> 00:35:33,570 And you see that each one of these is just 563 00:35:33,570 --> 00:35:34,830 taken again and again. 564 00:35:34,830 --> 00:35:36,580 As we've seen implicitly in all these 565 00:35:36,580 --> 00:35:38,600 treatments so far as well. 566 00:35:38,600 --> 00:35:40,440 Now, in the translational case where 567 00:35:40,440 --> 00:35:41,970 you interchange particles. 568 00:35:41,970 --> 00:35:45,140 Like particles, you have to divide by N factorial. . 569 00:35:45,140 --> 00:35:49,070 But the different configurations don't do that. 570 00:35:49,070 --> 00:35:53,220 If I've got a system where a molecule over here is in this 571 00:35:53,220 --> 00:35:56,210 configuration and a molecule somewhere else is in this one, 572 00:35:56,210 --> 00:35:58,470 and now they change configurations, that's a 573 00:35:58,470 --> 00:36:01,710 distinguishable state. 574 00:36:01,710 --> 00:36:04,240 So there's no N factorial involved here. 575 00:36:04,240 --> 00:36:07,670 In the configurational partition function. 576 00:36:07,670 --> 00:36:13,860 So, fine, then it's just one plus three e to the minus E 577 00:36:13,860 --> 00:36:17,770 int over kT to the Nth power. 578 00:36:17,770 --> 00:36:20,810 That's Q. And once we know Q, as you've seen, we know 579 00:36:20,810 --> 00:36:22,970 everything. 580 00:36:22,970 --> 00:36:26,550 And so we can immediately start in deriving all of the 581 00:36:26,550 --> 00:36:27,770 thermodynamics, right? 582 00:36:27,770 --> 00:36:33,300 And the place to start is A, Helmholtz free energy, or 583 00:36:33,300 --> 00:36:35,720 configurational free energy. 584 00:36:35,720 --> 00:36:40,410 Because that's the simplest relation, just as minus kT log 585 00:36:40,410 --> 00:36:46,550 of capital Q configurational. 586 00:36:46,550 --> 00:36:54,580 So it's minus N kT one plus three e to the 587 00:36:54,580 --> 00:37:21,180 minus E int over kT. 588 00:37:21,180 --> 00:37:25,950 I'm going to rewrite A configurational in terms of 589 00:37:25,950 --> 00:37:28,270 beta rather than kT, just because there are going to be 590 00:37:28,270 --> 00:37:29,930 a lot of these factors. 591 00:37:29,930 --> 00:37:38,715 So it's minus N kT one plus three e to the minus beta E 592 00:37:38,715 --> 00:37:39,900 interaction. 593 00:37:39,900 --> 00:37:43,190 And that's our A. Everything's going to follow from that. 594 00:37:43,190 --> 00:37:45,660 Before I write some of the other results, one thing to 595 00:37:45,660 --> 00:37:50,670 notice is, it has N in it. 596 00:37:50,670 --> 00:37:54,990 In other words, there's a free energy per molecule. 597 00:37:54,990 --> 00:38:01,430 The total free energy is just something times N. And that's 598 00:38:01,430 --> 00:38:03,800 because all the molecules in this model are 599 00:38:03,800 --> 00:38:06,290 independently behaving. 600 00:38:06,290 --> 00:38:11,390 So whatever their average free energy is, that's going to add 601 00:38:11,390 --> 00:38:13,940 up and give us the total. 602 00:38:13,940 --> 00:38:15,770 They're all acting independently. 603 00:38:15,770 --> 00:38:18,410 And in fact, we could have gotten this directly from 604 00:38:18,410 --> 00:38:24,380 minus kT log little q configurational. 605 00:38:24,380 --> 00:38:28,250 And if we look at energy, regular energy, u, 606 00:38:28,250 --> 00:38:35,040 configurational, it's minus d log capital Q 607 00:38:35,040 --> 00:38:39,630 with respect to beta. 608 00:38:39,630 --> 00:38:44,700 V and N. And I'll just write the result. 609 00:38:44,700 --> 00:38:54,720 It's N times three E int e to the minus beta E int over one 610 00:38:54,720 --> 00:39:00,890 plus three e to the minus beta E int. 611 00:39:00,890 --> 00:39:06,090 And once again, the feature I want to point out is that 612 00:39:06,090 --> 00:39:09,380 there's a factor of N here. 613 00:39:09,380 --> 00:39:13,930 Again, there's an energy per molecule. 614 00:39:13,930 --> 00:39:20,730 And so if we want, we can also just write the average E, 615 00:39:20,730 --> 00:39:25,740 little E, configurational. 616 00:39:25,740 --> 00:39:30,090 It's just u configurational over N. It's the same as this 617 00:39:30,090 --> 00:39:35,890 without the factor of N. Not only that, again, we could get 618 00:39:35,890 --> 00:39:40,530 this directly from the molecular 619 00:39:40,530 --> 00:39:42,750 partition function up there. 620 00:39:42,750 --> 00:39:46,220 From little q configurational. 621 00:39:46,220 --> 00:39:48,860 We'd have the same result. 622 00:39:48,860 --> 00:39:50,260 Exactly as it should be. 623 00:39:50,260 --> 00:39:55,610 So if we wrote E as we've seen in the past, it's just the sum 624 00:39:55,610 --> 00:40:03,680 over that energy times the probability for each state. 625 00:40:03,680 --> 00:40:09,010 And that's just the sum of Ei, e to the minus beta 626 00:40:09,010 --> 00:40:13,620 Ei over little q. 627 00:40:13,620 --> 00:40:19,050 And if you put in the terms, you immediately get this. 628 00:40:19,050 --> 00:40:22,400 Here's our little q, and this has the interaction energy 629 00:40:22,400 --> 00:40:29,820 brought out, multiplied here. 630 00:40:29,820 --> 00:40:33,000 So the point is, in a case like this, where you have a 631 00:40:33,000 --> 00:40:37,170 bunch of independently behaving particles, the totals 632 00:40:37,170 --> 00:40:41,030 for these, for quantities like energy, are simply for the 633 00:40:41,030 --> 00:40:44,530 system energy, it's just the average particle energy times 634 00:40:44,530 --> 00:40:46,420 the number of particles. 635 00:40:46,420 --> 00:40:48,460 The number of molecules. 636 00:40:48,460 --> 00:40:52,000 And remember, before, I spoke a little bit about the fact 637 00:40:52,000 --> 00:40:55,460 that, well, if you look at one individual molecule and 638 00:40:55,460 --> 00:40:57,330 another and another, the energy will fluctuate. 639 00:40:57,330 --> 00:40:59,540 It'll vary considerably. 640 00:40:59,540 --> 00:41:01,160 Of course, this is a simple case with only 641 00:41:01,160 --> 00:41:02,740 two possible energies. 642 00:41:02,740 --> 00:41:04,910 But in a case where there may be many more possible 643 00:41:04,910 --> 00:41:07,110 energies, the molecular energies 644 00:41:07,110 --> 00:41:09,440 may vary quite widely. 645 00:41:09,440 --> 00:41:11,400 Still, there will be a well-defined average. 646 00:41:11,400 --> 00:41:13,940 And then the system energy would simply be the total 647 00:41:13,940 --> 00:41:16,430 number of molecules times that. 648 00:41:16,430 --> 00:41:18,340 Where they're all independent. 649 00:41:18,340 --> 00:41:21,760 Where all the energies are independent of each other. 650 00:41:21,760 --> 00:41:25,390 And, of course, the system energy fluctuates a great deal 651 00:41:25,390 --> 00:41:30,660 less than the individual molecule energies. 652 00:41:30,660 --> 00:41:36,780 OK, so that's our energy term. 653 00:41:36,780 --> 00:41:39,240 And I think I won't write out the results. 654 00:41:39,240 --> 00:41:41,750 They're in your notes for entropy. 655 00:41:41,750 --> 00:41:45,270 For chemical potential. 656 00:41:45,270 --> 00:41:46,010 They're all there. 657 00:41:46,010 --> 00:41:48,260 And again, the same point would hold. 658 00:41:48,260 --> 00:41:53,900 They all scale with N. But I want to talk a little bit 659 00:41:53,900 --> 00:41:55,840 about the heat capacity. 660 00:41:55,840 --> 00:41:59,630 The expression for it isn't so simple. 661 00:41:59,630 --> 00:42:04,980 So let me just right Cv configurational. 662 00:42:04,980 --> 00:42:12,770 It's du configurational / dT, at constant V and N. And 663 00:42:12,770 --> 00:42:15,200 again, the details are worked out in the notes. 664 00:42:15,200 --> 00:42:20,740 So I won't write out the whole derivation of it. 665 00:42:20,740 --> 00:42:24,460 And even writing out the results, I'm almost 666 00:42:24,460 --> 00:42:26,130 reluctant to do it. 667 00:42:26,130 --> 00:42:27,700 But I'll put it up. 668 00:42:27,700 --> 00:42:32,470 Three E int over k T squared, because there are some things 669 00:42:32,470 --> 00:42:33,750 I want to point out about it. 670 00:42:33,750 --> 00:42:45,400 N, and then one plus three e to the minus beta E int minus 671 00:42:45,400 --> 00:42:52,630 E int e to the minus beta E int. 672 00:42:52,630 --> 00:43:02,610 Minus e to the minus beta, I have to get rid of this. 673 00:43:02,610 --> 00:43:14,380 Minus e to the minus beta E int times negative three E int 674 00:43:14,380 --> 00:43:19,370 e to the minus beta E int. 675 00:43:19,370 --> 00:43:30,380 All over one plus three e to the minus beta E int, that 676 00:43:30,380 --> 00:43:33,160 whole thing squared. 677 00:43:33,160 --> 00:43:36,180 And this is just, of course, the kind of messy results that 678 00:43:36,180 --> 00:43:39,160 comes from taking this derivative with respect to 679 00:43:39,160 --> 00:43:42,660 temperature and doing the chain rule to get it with 680 00:43:42,660 --> 00:43:44,400 respect to beta and so forth. 681 00:43:44,400 --> 00:43:48,490 So it looks like a little bit of an intractable, or at least 682 00:43:48,490 --> 00:43:50,380 a little bit of a complicated, function. 683 00:43:50,380 --> 00:43:52,890 And the detailed functional form is complicated. 684 00:43:52,890 --> 00:43:56,180 But what I want to emphasize is that it has simple limits 685 00:43:56,180 --> 00:43:58,780 that are very easy to understand physically. 686 00:43:58,780 --> 00:43:59,550 And that are important to 687 00:43:59,550 --> 00:44:01,800 understand for lots of systems. 688 00:44:01,800 --> 00:44:09,930 So I just want to look at the limits of the heat capacity at 689 00:44:09,930 --> 00:44:11,540 low and high temperature. 690 00:44:11,540 --> 00:44:15,980 And this is something that recurs in statistical 691 00:44:15,980 --> 00:44:18,760 mechanics, in an enormous number of systems where you 692 00:44:18,760 --> 00:44:20,570 have simplified limits. 693 00:44:20,570 --> 00:44:21,500 And they're really important. 694 00:44:21,500 --> 00:44:26,390 Because what's going to matter is this. 695 00:44:26,390 --> 00:44:30,690 Maybe I shouldn't have covered up those configurations. 696 00:44:30,690 --> 00:44:35,490 So there's a big difference in what happens when kT is much 697 00:44:35,490 --> 00:44:37,490 bigger than this energy. 698 00:44:37,490 --> 00:44:40,490 Where you know that under those conditions - let's say 699 00:44:40,490 --> 00:44:43,380 it's hot, and this is a small energy difference. 700 00:44:43,380 --> 00:44:46,530 Then the probabilities are essentially equal. 701 00:44:46,530 --> 00:44:48,300 That the molecules are in this state or in 702 00:44:48,300 --> 00:44:49,000 any of these states. 703 00:44:49,000 --> 00:44:50,850 Because the energy is so tiny compared 704 00:44:50,850 --> 00:44:51,720 to the thermal energy. 705 00:44:51,720 --> 00:44:54,070 The molecules are continually being kicked around between 706 00:44:54,070 --> 00:44:56,740 all the states, among all those states. 707 00:44:56,740 --> 00:45:00,070 So the high temperature limit, physically, is one that's 708 00:45:00,070 --> 00:45:01,400 simple to understand. 709 00:45:01,400 --> 00:45:04,510 Where now you're really just going to revert to the cases 710 00:45:04,510 --> 00:45:06,980 that we've treated before, where all the energies are 711 00:45:06,980 --> 00:45:08,770 effectively the same. 712 00:45:08,770 --> 00:45:11,860 Because compared to kT, they are. 713 00:45:11,860 --> 00:45:14,350 And in the low temperature limit, now go to 714 00:45:14,350 --> 00:45:15,200 the opposite limit. 715 00:45:15,200 --> 00:45:17,540 Let's say kT is much, much, smaller than 716 00:45:17,540 --> 00:45:19,380 the interaction energy. 717 00:45:19,380 --> 00:45:22,150 So that now this term is really small. 718 00:45:22,150 --> 00:45:24,280 Because this is much bigger than this. 719 00:45:24,280 --> 00:45:27,710 So, what it means physically is all the molecules are in 720 00:45:27,710 --> 00:45:28,580 the ground state. 721 00:45:28,580 --> 00:45:30,380 The probability of this is basically one. 722 00:45:30,380 --> 00:45:33,070 The probability of being in any of these states is zero. 723 00:45:33,070 --> 00:45:35,700 And that's also a simple result. 724 00:45:35,700 --> 00:45:38,940 And there are lots and lots of cases where one of those 725 00:45:38,940 --> 00:45:41,030 limits really obtains. 726 00:45:41,030 --> 00:45:44,600 If you look at molecules moving around in the gas phase 727 00:45:44,600 --> 00:45:46,110 or in a liquid. 728 00:45:46,110 --> 00:45:47,530 And you say, well, OK, let's think about 729 00:45:47,530 --> 00:45:49,290 their rotational motion. 730 00:45:49,290 --> 00:45:51,800 Rotational energies are small. 731 00:45:51,800 --> 00:45:54,700 At room temperature, kT's much bigger than them. 732 00:45:54,700 --> 00:45:57,220 You immediately go to the high temperature limit. 733 00:45:57,220 --> 00:45:59,340 Now let's go to a small molecule and look at the 734 00:45:59,340 --> 00:46:00,990 vibrational energy. 735 00:46:00,990 --> 00:46:03,750 In most cases, the vibrational frequency is pretty, molecules 736 00:46:03,750 --> 00:46:05,860 are pretty stiff. 737 00:46:05,860 --> 00:46:07,860 And in many cases you can just say, look, forget it. 738 00:46:07,860 --> 00:46:09,700 All the molecules are in the ground state. 739 00:46:09,700 --> 00:46:12,760 And again, in the opposite, but also simple 740 00:46:12,760 --> 00:46:14,430 limit ends up holding. 741 00:46:14,430 --> 00:46:17,340 Or molecular electronic states, right? 742 00:46:17,340 --> 00:46:19,970 If you've got benzene or hydrogen atoms in, say, at 743 00:46:19,970 --> 00:46:23,470 room temperature, how many hydrogen atoms thermally are 744 00:46:23,470 --> 00:46:26,920 going to be up in the n equals two state, the 2p 745 00:46:26,920 --> 00:46:28,080 state or the 2s state? 746 00:46:28,080 --> 00:46:29,120 Forget it. 747 00:46:29,120 --> 00:46:31,530 There's not nearly enough thermal energy to do that. 748 00:46:31,530 --> 00:46:35,710 So these simpler limiting cases play a huge role in 749 00:46:35,710 --> 00:46:37,810 simplifying statistical mechanics and the calculations 750 00:46:37,810 --> 00:46:39,550 from them generally. 751 00:46:39,550 --> 00:46:42,400 OK, so let's just see what happens. 752 00:46:42,400 --> 00:46:54,660 So, this we want, so this we can move down. 753 00:46:54,660 --> 00:47:06,330 So our limiting cases, when T goes to zero, not going to 754 00:47:06,330 --> 00:47:09,040 work out what happens to all the betas, beta 755 00:47:09,040 --> 00:47:11,250 gets big in that case. 756 00:47:11,250 --> 00:47:18,130 But the result is that Cv configurational goes to zero. 757 00:47:18,130 --> 00:47:21,680 That's the limit where, like I've described, here is E 758 00:47:21,680 --> 00:47:24,810 interaction, or E int. 759 00:47:24,810 --> 00:47:29,050 Here is zero. 760 00:47:29,050 --> 00:47:33,000 And your kT is barely above zero. 761 00:47:33,000 --> 00:47:35,160 So when all the molecules are here, none of them has enough 762 00:47:35,160 --> 00:47:38,650 thermal energy to be up here. 763 00:47:38,650 --> 00:47:42,430 So why should the heat capacity be zero? 764 00:47:42,430 --> 00:47:45,340 The heat capacity is du/dT. 765 00:47:45,340 --> 00:47:48,750 It's zero because, here's kT. 766 00:47:48,750 --> 00:47:51,720 Let's say I increment kT up a little bit. 767 00:47:51,720 --> 00:47:55,190 I just heat the system a tiny, tiny bit. 768 00:47:55,190 --> 00:47:58,930 An infinitesimal amount, to look at the derivative. 769 00:47:58,930 --> 00:48:01,900 Once I do this, how many molecules are in this 770 00:48:01,900 --> 00:48:04,090 higher level now? 771 00:48:04,090 --> 00:48:05,280 Still zero, right? 772 00:48:05,280 --> 00:48:07,230 There's still not nearly enough thermal energy to have 773 00:48:07,230 --> 00:48:09,060 any molecules up here. 774 00:48:09,060 --> 00:48:11,920 So what, was the derivative of the energy with respect to the 775 00:48:11,920 --> 00:48:12,610 temperature? 776 00:48:12,610 --> 00:48:14,100 I changed the temperature. 777 00:48:14,100 --> 00:48:16,610 The energy didn't change a bit. 778 00:48:16,610 --> 00:48:22,470 And that means the heat capacity is zero. 779 00:48:22,470 --> 00:48:27,420 Now let's look at the other limit. 780 00:48:27,420 --> 00:48:29,160 High temperature limit. 781 00:48:29,160 --> 00:48:30,680 Really, it doesn't need to be infinity. 782 00:48:30,680 --> 00:48:34,870 It's really just T greater than, much bigger than, kT is 783 00:48:34,870 --> 00:48:36,810 much bigger than E interaction. 784 00:48:36,810 --> 00:48:39,425 And in this case, kT is much less than 785 00:48:39,425 --> 00:48:41,170 the interaction energy. 786 00:48:41,170 --> 00:48:44,180 So it doesn't need to be nearly that extreme. 787 00:48:44,180 --> 00:48:51,230 Well, what you find out is the heat capacity is zero. 788 00:48:51,230 --> 00:48:52,910 Now, it's zero. 789 00:48:52,910 --> 00:49:02,900 Because we're in the following limit. 790 00:49:02,900 --> 00:49:09,830 Now kT is way up here. 791 00:49:09,830 --> 00:49:15,490 Compared to kT, E interaction is way down here. 792 00:49:15,490 --> 00:49:16,870 So, what happens? 793 00:49:16,870 --> 00:49:21,660 It means that this is so hot, that term, The e to the minus 794 00:49:21,660 --> 00:49:26,540 E int over kT, forget it, it's one. 795 00:49:26,540 --> 00:49:28,960 And so the probability, in other words, that the 796 00:49:28,960 --> 00:49:30,440 molecules are in this state or this state 797 00:49:30,440 --> 00:49:31,710 are essentially equal. 798 00:49:31,710 --> 00:49:33,280 So now let's say I raise the temperature 799 00:49:33,280 --> 00:49:34,960 a little bit higher. 800 00:49:34,960 --> 00:49:36,550 What happens to those probabilities? 801 00:49:36,550 --> 00:49:38,140 What changes? 802 00:49:38,140 --> 00:49:38,360 Right. 803 00:49:38,360 --> 00:49:39,900 Nothing changes. 804 00:49:39,900 --> 00:49:43,760 So what happened to the energy when I raised the temperature? 805 00:49:43,760 --> 00:49:44,920 Of course, nothing happened to it. 806 00:49:44,920 --> 00:49:46,980 Which means the heat capacity is zero. 807 00:49:46,980 --> 00:49:52,430 Now, the first limit that I described, this 808 00:49:52,430 --> 00:49:56,080 one is almost universal. 809 00:49:56,080 --> 00:49:59,640 For any system where you have quantized level, you can 810 00:49:59,640 --> 00:50:04,180 always eventually get to a low enough temperature that you're 811 00:50:04,180 --> 00:50:05,130 in the first limit. 812 00:50:05,130 --> 00:50:07,680 Where the kT is lower than, by far, than the 813 00:50:07,680 --> 00:50:09,030 first excited level. 814 00:50:09,030 --> 00:50:11,600 Everything's in the ground state. 815 00:50:11,600 --> 00:50:14,440 And so you reach this limit. 816 00:50:14,440 --> 00:50:17,650 This one, it's only cases where you have a 817 00:50:17,650 --> 00:50:20,240 finite number of levels. 818 00:50:20,240 --> 00:50:22,980 Then you have the same high temperature limit to it. 819 00:50:22,980 --> 00:50:27,110 In other words, the reason this result happens is because 820 00:50:27,110 --> 00:50:30,350 there aren't a bunch of other levels up here that eventually 821 00:50:30,350 --> 00:50:33,100 get to be comparable to kT. 822 00:50:33,100 --> 00:50:35,830 And not all kinds of systems or degrees of 823 00:50:35,830 --> 00:50:37,340 freedom are like that. 824 00:50:37,340 --> 00:50:41,120 This one is, because there's a finite number, four in this 825 00:50:41,120 --> 00:50:44,170 case, of configurations. 826 00:50:44,170 --> 00:50:48,720 So there's just nothing above there. 827 00:50:48,720 --> 00:50:50,840 There's another really important kind of degree of 828 00:50:50,840 --> 00:50:51,540 freedom like that. 829 00:50:51,540 --> 00:50:54,030 And that's spin. 830 00:50:54,030 --> 00:50:55,350 Think of proton spins. 831 00:50:55,350 --> 00:50:58,060 It's plus 1/2 or it's minus 1/2, and that's it. 832 00:50:58,060 --> 00:51:02,190 You can't put any more spin energy into it. 833 00:51:02,190 --> 00:51:05,790 Just like you can't put any more configurational energy 834 00:51:05,790 --> 00:51:07,510 into the system than to be in this state. 835 00:51:07,510 --> 00:51:08,740 That's it. 836 00:51:08,740 --> 00:51:13,900 So in other words, the maximum possible energy is finite. 837 00:51:13,900 --> 00:51:16,060 Of course, lots of other degrees of freedom are 838 00:51:16,060 --> 00:51:17,280 different from that. 839 00:51:17,280 --> 00:51:19,450 If you think a molecule's rotating, they could always 840 00:51:19,450 --> 00:51:21,190 spin faster. 841 00:51:21,190 --> 00:51:22,760 Vibrating, they can always vibrate 842 00:51:22,760 --> 00:51:25,170 harder, translate faster. 843 00:51:25,170 --> 00:51:31,420 Those degrees of freedom won't have this limit. 844 00:51:31,420 --> 00:51:34,010 They also will have a simple high temperature limit, but 845 00:51:34,010 --> 00:51:36,550 not zero, because, of course, if there are always more 846 00:51:36,550 --> 00:51:39,720 levels, and I keep increasing kT, then I'll have thermal 847 00:51:39,720 --> 00:51:42,290 energy to go into those higher and higher levels. 848 00:51:42,290 --> 00:51:44,380 It'll still go up. 849 00:51:44,380 --> 00:51:48,280 But for systems with a finite number of possible levels, and 850 00:51:48,280 --> 00:51:51,700 a finite amount of total energy, for degrees of freedom 851 00:51:51,700 --> 00:51:55,400 like that, once I get to a temperature higher than any of 852 00:51:55,400 --> 00:51:57,210 that stuff, then forget it. 853 00:51:57,210 --> 00:51:59,160 You can't change the energy anymore thermally. 854 00:51:59,160 --> 00:52:01,040 So your heat capacity is zero. 855 00:52:01,040 --> 00:52:04,030 You can change the temperature and nothing further happens. 856 00:52:04,030 --> 00:52:06,840 OK, next time we'll see what happens when you do have 857 00:52:06,840 --> 00:52:10,280 continuing basically unbounded possible energies.