1 00:00:00,000 --> 00:00:01,990 ANNOUNCER: The following content is provided under a 2 00:00:01,990 --> 00:00:03,820 creative commons license. 3 00:00:03,820 --> 00:00:06,840 Your support will help MIT Open Courseware continue to 4 00:00:06,840 --> 00:00:10,520 offer high quality educational resources for free. 5 00:00:10,520 --> 00:00:13,390 To make a donation or view additional materials from 6 00:00:13,390 --> 00:00:15,990 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:15,990 --> 00:00:19,920 ocw.mit.edu. 8 00:00:19,920 --> 00:00:21,670 PROFESSOR: So you've been learning 9 00:00:21,670 --> 00:00:23,985 about statistical mechanics. 10 00:00:23,985 --> 00:00:29,930 The microscopic underpinnings of thermodynamics. 11 00:00:29,930 --> 00:00:40,560 And last time we ended up working with the canonical 12 00:00:40,560 --> 00:00:43,530 partition function and showing that once you have the 13 00:00:43,530 --> 00:00:46,490 canonical position function, you have basically every 14 00:00:46,490 --> 00:00:48,210 thermodynamic quantity that you've learned how to 15 00:00:48,210 --> 00:00:52,280 calculate this far in the course. 16 00:00:52,280 --> 00:00:58,590 But before I start, on the lecture notes 25, there were a 17 00:00:58,590 --> 00:01:00,570 couple typos in there that actually didn't make any 18 00:01:00,570 --> 00:01:02,870 difference to the final result because they 19 00:01:02,870 --> 00:01:04,310 cancel each other out. 20 00:01:04,310 --> 00:01:08,460 But it's been corrected on the web version. 21 00:01:08,460 --> 00:01:12,590 And near the bottom of the page -- because we're going to 22 00:01:12,590 --> 00:01:14,060 be mostly using these notes today. 23 00:01:14,060 --> 00:01:19,570 Near the bottom of the page, where it says A is equal to u 24 00:01:19,570 --> 00:01:23,240 minus TS is equal to u. 25 00:01:23,240 --> 00:01:27,040 That should be a plus here. dA/dT, volume. 26 00:01:31,020 --> 00:01:33,650 And then further down the next line, blah 27 00:01:33,650 --> 00:01:34,600 blah blah blah blah. 28 00:01:34,600 --> 00:01:36,500 Minus u over T squared. 29 00:01:36,500 --> 00:01:37,680 And this should be a minus here. 30 00:01:37,680 --> 00:01:46,110 One over T. dA/dT, V. And et cetera. 31 00:01:46,110 --> 00:01:49,010 So luckily these two typos cancel each other out the 32 00:01:49,010 --> 00:01:49,940 result is correct. 33 00:01:49,940 --> 00:01:53,150 But there it is. 34 00:01:53,150 --> 00:01:57,180 OK. 35 00:01:57,180 --> 00:02:02,980 So last time, then, you saw how from the canonical 36 00:02:02,980 --> 00:02:06,230 partition function, you could get something like the energy. 37 00:02:06,230 --> 00:02:07,710 You wrote down an equation. 38 00:02:07,710 --> 00:02:16,460 The energy is equal to k, T squared, d log Q / dT. 39 00:02:16,460 --> 00:02:19,180 Under constant volume. 40 00:02:19,180 --> 00:02:25,100 And number of particles. 41 00:02:25,100 --> 00:02:32,260 And then you notice that the important variables are the 42 00:02:32,260 --> 00:02:35,820 volume, the number of particles, and the 43 00:02:35,820 --> 00:02:37,540 temperature. 44 00:02:37,540 --> 00:02:41,920 And we know that every thermodynamic quantity has a 45 00:02:41,920 --> 00:02:46,900 set of natural variables. 46 00:02:46,900 --> 00:02:49,230 For the Gibbs free energy it was the pressure and the 47 00:02:49,230 --> 00:02:52,070 temperature, the number of particles. 48 00:02:52,070 --> 00:02:55,510 And for the volume of the number of particles and the 49 00:02:55,510 --> 00:03:00,830 temperature, we know that that's the Helmholtz energy. 50 00:03:00,830 --> 00:03:03,795 So the natural variable that we would associate -- the 51 00:03:03,795 --> 00:03:06,130 natural thermodynamic variable we would associate with that 52 00:03:06,130 --> 00:03:08,510 set of constraints is the Helmholtz free energy. 53 00:03:08,510 --> 00:03:12,000 So it becomes interesting, then, to figure out, how can 54 00:03:12,000 --> 00:03:15,320 we write the Helmholtz free energy in terms of the 55 00:03:15,320 --> 00:03:16,660 canonical partition function? 56 00:03:16,660 --> 00:03:21,340 They seem to have the same set of natural variables. 57 00:03:21,340 --> 00:03:24,930 And that's what you started doing and what we'll do today. 58 00:03:24,930 --> 00:03:29,280 And that's where the typo comes in. 59 00:03:29,280 --> 00:03:33,046 So let's write what we know about the Helmholtz free 60 00:03:33,046 --> 00:03:40,600 energy in terms of the energy u. 61 00:03:40,600 --> 00:03:43,180 I already wrote it up there. u minus TS. 62 00:03:43,180 --> 00:03:45,860 That's the definition of the Helmholtz free energy. 63 00:03:45,860 --> 00:03:54,100 And from the fact that dA is equal to minus p dV, minus S 64 00:03:54,100 --> 00:04:00,170 dT, plus mu dN. 65 00:04:00,170 --> 00:04:04,153 We can read out what S, here, is in terms of the Helmholtz 66 00:04:04,153 --> 00:04:05,100 free energy. 67 00:04:05,100 --> 00:04:13,430 S is just dA/dT, constant V and N. So we can plug that in 68 00:04:13,430 --> 00:04:20,160 here. u minus T dA, with a minus sign there, so that 69 00:04:20,160 --> 00:04:21,000 becomes a plus. 70 00:04:21,000 --> 00:04:29,730 And hence the typo. dT, constant V, and N. OK. 71 00:04:29,730 --> 00:04:32,820 So now we have an equation that relates u and the 72 00:04:32,820 --> 00:04:34,420 partition function. 73 00:04:34,420 --> 00:04:37,240 We want an equation that relates A and 74 00:04:37,240 --> 00:04:39,580 the partition function. 75 00:04:39,580 --> 00:04:42,780 If we rearrange this slightly, we can get that u, then, is 76 00:04:42,780 --> 00:04:53,520 equal to A minus T, dA/dT, constant V and N. So the 77 00:04:53,520 --> 00:04:58,830 question that we could ask ourselves is -- is there a 78 00:04:58,830 --> 00:05:02,320 function of A that kind of looks like that? 79 00:05:02,320 --> 00:05:05,280 And I know the answer, so I'm going to give it to you. 80 00:05:05,280 --> 00:05:07,800 A function of A that looks like the sum of A minus 81 00:05:07,800 --> 00:05:09,640 something times the derivative, with respect to 82 00:05:09,640 --> 00:05:11,550 that something. 83 00:05:11,550 --> 00:05:16,490 We should try to look at something like the derivative 84 00:05:16,490 --> 00:05:22,200 d(A/T)/dT, constant V and N. So if you take that 85 00:05:22,200 --> 00:05:35,780 derivative, you end up with one over T, dA/dT, minus A 86 00:05:35,780 --> 00:05:41,040 over T squared. 87 00:05:41,040 --> 00:05:45,550 But if you look at this result, here, and you multiply 88 00:05:45,550 --> 00:05:50,000 by T squared, there's the A, and there's the T times dA/dT. 89 00:05:50,000 --> 00:05:52,910 There's the A that -- we just have the sign wrong. 90 00:05:52,910 --> 00:05:56,700 So we have -- multiply this by minus T squared. 91 00:05:56,700 --> 00:06:02,510 Minus T squared, times minus T squared. 92 00:06:02,510 --> 00:06:06,560 And you have the same thing as here. 93 00:06:06,560 --> 00:06:14,210 So that tells us then, that u can be written as minus T 94 00:06:14,210 --> 00:06:22,420 squared, d(A/T)/dT, constant volume. 95 00:06:22,420 --> 00:06:27,260 It's a nice way to relate those two energies. 96 00:06:27,260 --> 00:06:29,230 And we have an expression for u in terms of 97 00:06:29,230 --> 00:06:32,900 the canonical function. 98 00:06:32,900 --> 00:06:41,390 And we can then replace it in here. 99 00:06:41,390 --> 00:06:52,000 And that gets us, then, that minus T squared, d(A/T)/dT, 100 00:06:52,000 --> 00:06:55,410 constant number and volume is equal to u. 101 00:06:55,410 --> 00:07:02,630 And u, we saw, was equal to k T squared d log Q / dT. 102 00:07:08,920 --> 00:07:11,830 V and N fixed. 103 00:07:11,830 --> 00:07:15,130 And then we can start -- 104 00:07:15,130 --> 00:07:18,050 The t squareds disappear here. 105 00:07:18,050 --> 00:07:21,070 And then we have d/dT on this side and d/dT. 106 00:07:21,070 --> 00:07:22,370 Let's just take the integral. 107 00:07:22,370 --> 00:07:27,760 You take the integral of both sides, and that gets us that 108 00:07:27,760 --> 00:07:40,420 then A over T is equal to k, log Q, plus the constant of 109 00:07:40,420 --> 00:07:41,580 integration. 110 00:07:41,580 --> 00:07:43,260 And we can take that constant of integration to 111 00:07:43,260 --> 00:07:45,380 be whatever we want. 112 00:07:45,380 --> 00:07:47,810 Energy is all relative to some reference point. 113 00:07:47,810 --> 00:07:52,600 We can take it to be zero. 114 00:07:52,600 --> 00:07:59,500 A is equal to k T log Q. 115 00:07:59,500 --> 00:08:01,230 That's a pretty neat result. 116 00:08:01,230 --> 00:08:05,850 There's the microscopic underpinning of things. 117 00:08:05,850 --> 00:08:08,480 Where we know about atoms, and energies, and states, and even 118 00:08:08,480 --> 00:08:09,610 quantum mechanics, and all sorts of 119 00:08:09,610 --> 00:08:11,600 things goes into here. 120 00:08:11,600 --> 00:08:14,460 All the microscopic information goes in here. 121 00:08:14,460 --> 00:08:17,290 And there's a thermodynamic variable that only cares about 122 00:08:17,290 --> 00:08:18,870 the macroscopic state of matter. 123 00:08:18,870 --> 00:08:22,060 It doesn't care that there are atoms there. 124 00:08:22,060 --> 00:08:23,670 It just cares that you know the pressure, the volume, the 125 00:08:23,670 --> 00:08:25,970 temperature, or any couple variables. 126 00:08:25,970 --> 00:08:30,270 And you can get a direct equality without any 127 00:08:30,270 --> 00:08:31,990 derivatives or anything. 128 00:08:31,990 --> 00:08:34,480 Between the macroscopic and the microscopic. 129 00:08:34,480 --> 00:08:41,390 So this is really pretty remarkable. 130 00:08:41,390 --> 00:08:43,970 And once you have A, you have everything. 131 00:08:43,970 --> 00:08:46,710 Just like before, we said once you have G, that Gibbs free 132 00:08:46,710 --> 00:08:48,610 energy, when we were talking about things that depend on 133 00:08:48,610 --> 00:08:49,800 pressure and temperature. 134 00:08:49,800 --> 00:08:50,460 You have everything. 135 00:08:50,460 --> 00:08:51,390 It's the same thing here. 136 00:08:51,390 --> 00:08:52,960 We've got A, we've got u. 137 00:08:52,960 --> 00:08:54,320 We have everything. 138 00:08:54,320 --> 00:08:56,400 We can calculate every single thermodynamic 139 00:08:56,400 --> 00:08:58,080 variable from then on. 140 00:08:58,080 --> 00:09:02,900 For instance, if we want to have the entropy, S is equal 141 00:09:02,900 --> 00:09:08,190 to minus A over T, minus u over T. Where did I get that? 142 00:09:08,190 --> 00:09:09,480 I got that from way up here. 143 00:09:09,480 --> 00:09:10,920 A is equal to u minus TS. 144 00:09:10,920 --> 00:09:13,400 I solved for S in terms of A and u. 145 00:09:13,400 --> 00:09:15,290 I've got expressions in terms of the canonical 146 00:09:15,290 --> 00:09:16,960 function for A and u. 147 00:09:16,960 --> 00:09:18,600 Plug that in there. 148 00:09:18,600 --> 00:09:32,860 Get k log Q, plus k T, d log Q / dT, constant number and 149 00:09:32,860 --> 00:09:35,180 volume. et cetera. 150 00:09:35,180 --> 00:09:36,310 You can get the pressure. 151 00:09:36,310 --> 00:09:41,780 You can get the pressure from the fact that the pressure up 152 00:09:41,780 --> 00:09:45,890 here is a derivative of A with respect to volume. 153 00:09:45,890 --> 00:09:49,150 So you take the derivative of A with respect to volume here. 154 00:09:49,150 --> 00:09:50,960 You get the pressure. 155 00:09:50,960 --> 00:09:52,430 If you want the chemical potential. 156 00:09:52,430 --> 00:09:55,300 The chemical potential from the fundamental equation up 157 00:09:55,300 --> 00:09:57,610 here is the derivative of A with respect to 158 00:09:57,610 --> 00:09:59,320 the number of particles. 159 00:09:59,320 --> 00:10:01,790 You take the derivative of A with respect to the number of 160 00:10:01,790 --> 00:10:03,920 particles, you get the chemical potential in terms of 161 00:10:03,920 --> 00:10:09,780 the canonical partition function. 162 00:10:09,780 --> 00:10:10,840 So you've got everything. 163 00:10:10,840 --> 00:10:11,370 You've got u. 164 00:10:11,370 --> 00:10:12,590 You've got A. You've got p. 165 00:10:12,590 --> 00:10:15,620 You've got S. You've got H. You've got G. Name your 166 00:10:15,620 --> 00:10:17,530 valuable, you've got it. 167 00:10:17,530 --> 00:10:18,890 You want the heat capacity? 168 00:10:18,890 --> 00:10:21,820 It can get you the heat capacity in terms of the 169 00:10:21,820 --> 00:10:25,600 partition function. 170 00:10:25,600 --> 00:10:29,550 Any questions? 171 00:10:29,550 --> 00:10:34,150 Alright so let's go on. 172 00:10:34,150 --> 00:10:39,050 Let's look a little bit closer at the entropy, in terms of 173 00:10:39,050 --> 00:10:46,650 the microscopic theory here. 174 00:10:46,650 --> 00:10:53,370 So let's start with S is A minus u over T. And let's see 175 00:10:53,370 --> 00:10:56,270 how far we can go. 176 00:10:56,270 --> 00:10:59,160 Alright. 177 00:10:59,160 --> 00:11:02,660 Let's write it in terms of -- let's write A and u in terms 178 00:11:02,660 --> 00:11:05,230 of things that we know. 179 00:11:05,230 --> 00:11:09,070 And let me just go to the end to tell you where I'm going, 180 00:11:09,070 --> 00:11:10,950 and why I'm going to make certain 181 00:11:10,950 --> 00:11:13,010 changes in my math here. 182 00:11:13,010 --> 00:11:15,920 So what we're going to get at the end is that S is this very 183 00:11:15,920 --> 00:11:23,730 nice quantity, which is minus k, pi log pi. 184 00:11:23,730 --> 00:11:27,050 Where the p's are the microstate probabilities. 185 00:11:27,050 --> 00:11:30,270 The probability that your state is in a particular -- 186 00:11:30,270 --> 00:11:30,640 yes? 187 00:11:30,640 --> 00:11:33,640 STUDENT: [UNINTELLIGIBLE] 188 00:11:33,640 --> 00:11:41,100 PROFESSOR: S is u minus A. Yes. u minus A. Thank you. 189 00:11:41,100 --> 00:11:48,000 I should read my notes. 190 00:11:48,000 --> 00:11:48,550 OK. 191 00:11:48,550 --> 00:11:52,000 So we're going to equate S here in terms of the 192 00:11:52,000 --> 00:11:54,760 probabilities of microstates. 193 00:11:54,760 --> 00:11:58,095 And that's going to be -- remember how we talked about S 194 00:11:58,095 --> 00:12:03,580 is related to concepts of randomness, 195 00:12:03,580 --> 00:12:05,440 or order, or disorder. 196 00:12:05,440 --> 00:12:09,550 So the number of possible microstates is related to the 197 00:12:09,550 --> 00:12:12,740 number of amount of disorder that you might have. 198 00:12:12,740 --> 00:12:15,370 If you have a pure crystal, and every atom is in its 199 00:12:15,370 --> 00:12:18,290 place, then the number of microstates at zero degree 200 00:12:18,290 --> 00:12:20,130 Kelvin is one. 201 00:12:20,130 --> 00:12:23,600 So the probability of being in that microstate is one. 202 00:12:23,600 --> 00:12:25,040 And the probability of being in every other 203 00:12:25,040 --> 00:12:27,270 microstate is zero. 204 00:12:27,270 --> 00:12:29,855 Alright, so there's a relationship that we're going 205 00:12:29,855 --> 00:12:31,780 to have here, which is going to be interesting. 206 00:12:31,780 --> 00:12:34,710 Let's derive it. 207 00:12:34,710 --> 00:12:38,230 That means that we're going to want to have this somehow pop 208 00:12:38,230 --> 00:12:40,960 out of the equation. 209 00:12:40,960 --> 00:12:42,760 We're going to want to have this pop out of the equation. 210 00:12:42,760 --> 00:12:59,350 If you remember, pi is the -- pi is e to the minus the Ei 211 00:12:59,350 --> 00:13:08,640 over kT, divided by the partition function Q. So 212 00:13:08,640 --> 00:13:11,150 somehow we are going to have to get this to come out. 213 00:13:11,150 --> 00:13:13,660 We're going to have to have these e to the minus Ei over 214 00:13:13,660 --> 00:13:15,250 kT's come out. 215 00:13:15,250 --> 00:13:15,450 OK. 216 00:13:15,450 --> 00:13:18,860 So let's try to get them to come out right away. 217 00:13:18,860 --> 00:13:19,770 We know u. 218 00:13:19,770 --> 00:13:22,880 A way to write u is the average energy. 219 00:13:22,880 --> 00:13:29,760 Which means let's take one over kT out here. 220 00:13:29,760 --> 00:13:33,300 And there's a -- let's divided by k here. 221 00:13:33,300 --> 00:13:35,970 Let's get the k here. kT here. 222 00:13:35,970 --> 00:13:37,610 We're going to want to have a k come out here. 223 00:13:37,610 --> 00:13:38,700 There's the k here. 224 00:13:38,700 --> 00:13:40,950 So that's one way of getting it to come out. 225 00:13:40,950 --> 00:13:49,770 And then the u is going to be the average energy. e to the 226 00:13:49,770 --> 00:13:52,330 minus Ei, minus kT. 227 00:13:52,330 --> 00:13:56,760 That's just writing the average energy. 228 00:13:56,760 --> 00:13:58,800 So the energy times the probability of having that 229 00:13:58,800 --> 00:14:03,000 energy divided by the normalization. 230 00:14:03,000 --> 00:14:10,880 Plus, well, A is just log Q. A over kT is just log Q. So 231 00:14:10,880 --> 00:14:16,670 we've managed to extract this guy out here. 232 00:14:16,670 --> 00:14:17,530 OK. 233 00:14:17,530 --> 00:14:21,880 Now, this sum, here, this is sum over i. 234 00:14:21,880 --> 00:14:25,850 I'd really like to have this sum come all the way out. 235 00:14:25,850 --> 00:14:28,930 So I've got to find a way to do that. 236 00:14:28,930 --> 00:14:36,430 And it would be nice if I could find a sum here. 237 00:14:36,430 --> 00:14:37,930 Maybe if I multiply by one. 238 00:14:37,930 --> 00:14:41,470 If I write one in a funny way. 239 00:14:41,470 --> 00:14:42,430 Get a log here. 240 00:14:42,430 --> 00:14:44,900 You know, if I've got a couple logs here, maybe I can combine 241 00:14:44,900 --> 00:14:46,740 them to get a ratio. 242 00:14:46,740 --> 00:14:51,920 So let's rewrite this Ei in a funny way, here. 243 00:14:51,920 --> 00:14:54,562 Ei is -- 244 00:14:54,562 --> 00:14:57,710 I'm just rewriting it, but in a strange way. 245 00:14:57,710 --> 00:15:03,490 Log e to the minus Ei over kT. 246 00:15:03,490 --> 00:15:06,110 So if I take the log of e to the minus the Ei over kT. 247 00:15:06,110 --> 00:15:08,220 I get minus Ei over kT. 248 00:15:08,220 --> 00:15:09,800 The kT's disappear. 249 00:15:09,800 --> 00:15:11,820 So I just get Ei is equal to Ei. 250 00:15:11,820 --> 00:15:16,830 I'm just writing something that's pretty obvious here. 251 00:15:16,830 --> 00:15:18,950 And then we're going to take that expression, 252 00:15:18,950 --> 00:15:22,090 and put it in here. 253 00:15:22,090 --> 00:15:29,270 So now I'm going to be able to write S over k is minus -- 254 00:15:29,270 --> 00:15:34,580 So the kT here cancels out this kT here. 255 00:15:34,580 --> 00:15:42,890 Sum over i. e to the minus Ei over kT. 256 00:15:42,890 --> 00:15:45,810 Over Q. That's that term right here. 257 00:15:45,810 --> 00:15:50,210 And then I have the Ei, which is log e to the 258 00:15:50,210 --> 00:15:53,600 minus Ei over kT. 259 00:15:53,600 --> 00:15:55,990 This whole thing is in this parenthesis here. 260 00:15:55,990 --> 00:15:57,210 And then I have the plus log 261 00:15:57,210 --> 00:16:17,770 Q. Plus log Q. OK. 262 00:16:17,770 --> 00:16:20,460 So I have this nice thing here. e to the minus Ei over 263 00:16:20,460 --> 00:16:23,770 kT divided by Q. Well that's looking an awful lot 264 00:16:23,770 --> 00:16:26,390 like this pi here. 265 00:16:26,390 --> 00:16:28,280 Which is what I'm trying to get out. 266 00:16:28,280 --> 00:16:30,200 I'm trying to get these pi's coming out. 267 00:16:30,200 --> 00:16:32,320 So that's a nice thing to have here. 268 00:16:32,320 --> 00:16:35,110 Now if I could only have a pi coming out here, somehow, that 269 00:16:35,110 --> 00:16:36,330 would be great too. 270 00:16:36,330 --> 00:16:39,816 And if I also got a sum here, that's a sum over i I could 271 00:16:39,816 --> 00:16:41,490 sort of combine everything together. 272 00:16:41,490 --> 00:16:45,640 So I'm going to write one in a funny way. 273 00:16:45,640 --> 00:16:50,890 One is equal to the sum of all probabilities. 274 00:16:50,890 --> 00:16:53,100 That's obvious. 275 00:16:53,100 --> 00:16:55,790 And I'm going to write this pi here in a form 276 00:16:55,790 --> 00:16:58,880 that looks like this. 277 00:16:58,880 --> 00:17:05,300 Sum over all i, e to the minus Ei over kT, divided by Q. Just 278 00:17:05,300 --> 00:17:06,830 writing one. 279 00:17:06,830 --> 00:17:09,870 The sum of all probability is equal to one. 280 00:17:09,870 --> 00:17:14,220 And I'm going to take this one, here, and I'm going to 281 00:17:14,220 --> 00:17:16,180 put it right in here. 282 00:17:16,180 --> 00:17:19,940 Now log Q doesn't care on i, so it's just a number. 283 00:17:19,940 --> 00:17:23,120 So that allows me to rewrite -- 284 00:17:23,120 --> 00:17:32,170 S over k is minus the sum over i. e to the minus Ei over kT, 285 00:17:32,170 --> 00:17:35,470 divided by the partition function. 286 00:17:35,470 --> 00:17:41,930 Times the log of e to the minus Ei over kT. 287 00:17:41,930 --> 00:17:56,190 Plus sum over i, e to the minus Ei over kT over Q. Times 288 00:17:56,190 --> 00:17:56,430 log 289 00:17:56,430 --> 00:17:59,920 Q. OK good. 290 00:17:59,920 --> 00:18:01,830 I'm going to take these summations. 291 00:18:01,830 --> 00:18:03,340 Now everything is over the sum of Ei. 292 00:18:03,340 --> 00:18:03,840 This is great. 293 00:18:03,840 --> 00:18:06,000 And there's this factor here. 294 00:18:06,000 --> 00:18:07,705 E to the minus Ei over kT divided by 295 00:18:07,705 --> 00:18:08,500 the partition function. 296 00:18:08,500 --> 00:18:09,364 That appears in both. 297 00:18:09,364 --> 00:18:12,740 I can factor that out. and. 298 00:18:12,740 --> 00:18:14,360 Then I have these two logs. 299 00:18:14,360 --> 00:18:18,390 Log of this and log of that. 300 00:18:18,390 --> 00:18:20,800 That I can also combine together. 301 00:18:20,800 --> 00:18:22,140 This is the log of this. 302 00:18:22,140 --> 00:18:24,260 And then there's a minus sign. 303 00:18:24,260 --> 00:18:27,860 So if I take the -- it's going to be the log of this minus 304 00:18:27,860 --> 00:18:28,610 the log of that,. 305 00:18:28,610 --> 00:18:35,190 It's going to end up with a ratio. 306 00:18:35,190 --> 00:18:41,430 S over k is equal to minus Ei. 307 00:18:41,430 --> 00:18:48,280 Taking the summation out. e to the minus Ei over kT, over Q. 308 00:18:48,280 --> 00:18:50,710 That's this term. 309 00:18:50,710 --> 00:18:52,440 That term. 310 00:18:52,440 --> 00:18:55,210 And then I have the logs. 311 00:18:55,210 --> 00:18:58,740 Log of e to the minus Ei over kT. 312 00:18:58,740 --> 00:19:00,520 This is a minus sign. 313 00:19:00,520 --> 00:19:01,430 This is a plus sign. 314 00:19:01,430 --> 00:19:08,170 That means I divide here by Q. This is great. 315 00:19:08,170 --> 00:19:08,740 Look. 316 00:19:08,740 --> 00:19:11,520 This e to the minus Ei over kT over Q. e to the minus Ei over 317 00:19:11,520 --> 00:19:13,690 kT over Q. What is that? 318 00:19:13,690 --> 00:19:19,320 That's just pi. 319 00:19:19,320 --> 00:19:20,710 This is pi. 320 00:19:20,710 --> 00:19:25,900 That's the probability of microstate i. 321 00:19:25,900 --> 00:19:40,390 And this is equal to minus sum over i, pi log pi. 322 00:19:40,390 --> 00:19:41,800 There's the k, here. 323 00:19:41,800 --> 00:19:55,460 S is equal to minus k, log sum over i, pi, log pi. 324 00:19:55,460 --> 00:19:59,820 Another great result. 325 00:19:59,820 --> 00:20:02,110 Now if you system is isolated. 326 00:20:02,110 --> 00:20:03,840 If you have an isolated system, that means 327 00:20:03,840 --> 00:20:05,610 that the energy -- 328 00:20:05,610 --> 00:20:07,170 You've got your boundary. 329 00:20:07,170 --> 00:20:10,080 The boundary doesn't let energy go in and out. 330 00:20:10,080 --> 00:20:13,350 Doesn't let the number of particles go in and out. 331 00:20:13,350 --> 00:20:17,120 Every single microstate is going to have the same energy. 332 00:20:17,120 --> 00:20:19,010 If this system is isolated. 333 00:20:19,010 --> 00:20:22,000 The only thing you're going to change is the 334 00:20:22,000 --> 00:20:23,720 positions of the particles. 335 00:20:23,720 --> 00:20:25,390 Or their vibrational energy. 336 00:20:25,390 --> 00:20:25,940 Or something. 337 00:20:25,940 --> 00:20:28,550 But let's just stick with translation. 338 00:20:28,550 --> 00:20:30,690 You're just going to change the positions. 339 00:20:30,690 --> 00:20:34,950 You're not going to change the energy. 340 00:20:34,950 --> 00:20:42,530 So, if the system is isolated, then the degeneracy of your 341 00:20:42,530 --> 00:20:45,260 energy is just a number of ways that you can flip the 342 00:20:45,260 --> 00:20:46,840 positions around. 343 00:20:46,840 --> 00:20:49,960 Indistinguishable ways. 344 00:20:49,960 --> 00:20:54,580 So the probability is just one over the number of possible 345 00:20:54,580 --> 00:21:00,060 ways of switching positions around for your particles. 346 00:21:00,060 --> 00:21:30,650 So for an isolated system, all microstates 347 00:21:30,650 --> 00:21:35,830 have the same energy. 348 00:21:35,830 --> 00:21:39,580 We can set that equal to zero as our reference point. 349 00:21:39,580 --> 00:21:44,090 And the probability of being in any one microstate is just 350 00:21:44,090 --> 00:21:45,600 one over the number of possible ways 351 00:21:45,600 --> 00:21:48,050 of rearranging things. 352 00:21:48,050 --> 00:21:53,420 So the probability of been in any one microstate is one over 353 00:21:53,420 --> 00:21:56,220 the number of my microstates. 354 00:21:56,220 --> 00:22:01,030 They all have the same energy. 355 00:22:01,030 --> 00:22:07,470 Where this is the degeneracy. 356 00:22:07,470 --> 00:22:14,760 So now when you plug that in here, S is minus k. 357 00:22:14,760 --> 00:22:23,460 Sum over all microstates from one to blah, one over blah, 358 00:22:23,460 --> 00:22:26,530 log of blah. 359 00:22:26,530 --> 00:22:27,510 This is a number. 360 00:22:27,510 --> 00:22:29,930 It can come out. 361 00:22:29,930 --> 00:22:33,260 The sum of one to omega. 362 00:22:33,260 --> 00:22:34,070 Of one over omega. 363 00:22:34,070 --> 00:22:35,040 Is omega times omega. 364 00:22:35,040 --> 00:22:37,930 It's one. 365 00:22:37,930 --> 00:22:40,520 The one over omega -- the log of one over omega is minus the 366 00:22:40,520 --> 00:22:41,070 log of omega. 367 00:22:41,070 --> 00:22:53,410 The negative signs cancel out. k log capital omega. 368 00:22:53,410 --> 00:23:04,930 For an isolated system. 369 00:23:04,930 --> 00:23:08,280 You've seen this before, probably. 370 00:23:08,280 --> 00:23:10,660 This is called the Boltzmann equation. 371 00:23:10,660 --> 00:23:13,940 And that is what is on his tombstone. 372 00:23:13,940 --> 00:23:16,960 If you go to - I think it's in Germany somewhere. 373 00:23:16,960 --> 00:23:17,520 Is it in Germany? 374 00:23:17,520 --> 00:23:18,560 Do you guys know? 375 00:23:18,560 --> 00:23:19,070 STUDENT: Austria. 376 00:23:19,070 --> 00:23:20,270 PROFESSOR: Austria, thank you. 377 00:23:20,270 --> 00:23:21,910 I knew it was that part of the world. 378 00:23:21,910 --> 00:23:25,520 If you go to Austria, to some famous cemetery, and go look 379 00:23:25,520 --> 00:23:27,990 for the tombstone that says S is equal to log omega. 380 00:23:27,990 --> 00:23:33,890 That's where Boltzmann is buried. 381 00:23:33,890 --> 00:23:34,730 OK. 382 00:23:34,730 --> 00:23:37,040 So this picture of -- yes question? 383 00:23:37,040 --> 00:23:44,000 STUDENT: Can you repeat the argument for making that one 384 00:23:44,000 --> 00:23:44,060 over omega go away? 385 00:23:44,060 --> 00:23:46,850 PROFESSOR: Making the one over omega go away here? 386 00:23:46,850 --> 00:23:50,660 Because you're taking the sum of all states from one, i 387 00:23:50,660 --> 00:23:52,550 equals one to omega. 388 00:23:52,550 --> 00:23:54,690 So it's one over our omega plus one over omega plus one 389 00:23:54,690 --> 00:23:57,510 over omega plus -- omega times. 390 00:23:57,510 --> 00:23:58,840 And this is just a number. 391 00:23:58,840 --> 00:23:59,980 So it comes out. 392 00:23:59,980 --> 00:24:01,750 It's not in the sum. 393 00:24:01,750 --> 00:24:13,310 So the sum, here, is all by itself. 394 00:24:13,310 --> 00:24:21,520 Any other questions? 395 00:24:21,520 --> 00:24:21,750 OK. 396 00:24:21,750 --> 00:24:30,140 That's why we talk about entropy as being this 397 00:24:30,140 --> 00:24:34,700 fundamental property that tells you about the number of 398 00:24:34,700 --> 00:24:35,910 available states. 399 00:24:35,910 --> 00:24:36,790 That's what it is. 400 00:24:36,790 --> 00:24:40,250 You've got this connection now between this variable, which 401 00:24:40,250 --> 00:24:43,400 is sort of hard to really intuitively understand, when 402 00:24:43,400 --> 00:24:45,480 you're talking about thermodynamics. 403 00:24:45,480 --> 00:24:47,690 And this is much easier to understand here, in terms of 404 00:24:47,690 --> 00:24:52,430 the available ways of distributing your energy, or 405 00:24:52,430 --> 00:24:54,430 your particles, in this case here. 406 00:24:54,430 --> 00:24:59,910 In different bins. 407 00:24:59,910 --> 00:25:06,450 OK any questions? 408 00:25:06,450 --> 00:25:13,850 So the next topic is we're going to work a little bit 409 00:25:13,850 --> 00:25:15,720 with the partition functions. 410 00:25:15,720 --> 00:25:21,190 And see how when you have systems that have multiple 411 00:25:21,190 --> 00:25:25,610 degrees of freedom, where each degree of freedom has a 412 00:25:25,610 --> 00:25:26,380 different kind of energy. 413 00:25:26,380 --> 00:25:30,200 Let's say translation, rotation, vibration. 414 00:25:30,200 --> 00:25:32,860 Then you can have a partition function for each of these 415 00:25:32,860 --> 00:25:34,630 degrees of freedom. 416 00:25:34,630 --> 00:25:38,570 And whereas the energies of the degrees of freedom add up, 417 00:25:38,570 --> 00:25:41,960 the partition functions get multiplied. 418 00:25:41,960 --> 00:25:45,950 So it's the separation of the partition functions into 419 00:25:45,950 --> 00:25:50,610 subsystem partition functions. 420 00:25:50,610 --> 00:25:54,960 So so far, we've written for 421 00:25:54,960 --> 00:25:59,760 translation partition function. 422 00:25:59,760 --> 00:26:05,250 That the system partition function is the molecular 423 00:26:05,250 --> 00:26:09,970 partition function to the Nth power. 424 00:26:09,970 --> 00:26:13,880 If you have distinguishable particles. 425 00:26:13,880 --> 00:26:18,610 And you have to divide by N factorial. 426 00:26:18,610 --> 00:26:20,670 If you can swap particles without knowing the 427 00:26:20,670 --> 00:26:22,270 difference. 428 00:26:22,270 --> 00:26:25,550 This is the number of ways of swapping N particles with each 429 00:26:25,550 --> 00:26:40,340 other, so it's indistinguishable particles. 430 00:26:40,340 --> 00:26:40,660 OK. 431 00:26:40,660 --> 00:26:46,730 So now let's say that -- 432 00:26:46,730 --> 00:26:51,930 Let's just make sure that -- 433 00:26:51,930 --> 00:26:55,770 Let's say that your energy of your system -- yes? 434 00:26:55,770 --> 00:27:09,630 STUDENT: It says that when system's not isolated, 435 00:27:09,630 --> 00:27:09,900 [UNINTELLIGIBLE]. 436 00:27:09,900 --> 00:27:10,420 PROFESSOR: Where does it say that? 437 00:27:10,420 --> 00:27:11,240 In the notes? 438 00:27:11,240 --> 00:27:19,320 STUDENT: [UNINTELLIGIBLE] 439 00:27:19,320 --> 00:27:21,880 PROFESSOR: Let me see the notes here. 440 00:27:21,880 --> 00:27:45,140 What does it say? 441 00:27:45,140 --> 00:27:49,510 So that sentence has to do with this guy here. 442 00:27:49,510 --> 00:27:52,840 Basically it says that if you've got a huge number 443 00:27:52,840 --> 00:27:56,600 particles, the average energy is a given number. 444 00:27:56,600 --> 00:28:00,070 And the fluctuations around that average are very small. 445 00:28:00,070 --> 00:28:07,680 And so, the system behaves as if it's isolated. 446 00:28:07,680 --> 00:28:11,010 So when you have a system which is not isolated, then 447 00:28:11,010 --> 00:28:12,530 energy can come in and out of the system. 448 00:28:12,530 --> 00:28:15,370 So in principle, over time, you could have huge energy 449 00:28:15,370 --> 00:28:16,040 fluctuations. 450 00:28:16,040 --> 00:28:18,550 As energy comes out, or energy comes in. 451 00:28:18,550 --> 00:28:23,000 And if you have a countable number of molecules in your 452 00:28:23,000 --> 00:28:27,830 system, then if one molecule suddenly captures a lot of 453 00:28:27,830 --> 00:28:31,430 energy, then the whole system energy will go up a lot. 454 00:28:31,430 --> 00:28:34,620 But if you have ten to the 24th molecules, if one 455 00:28:34,620 --> 00:28:37,680 molecule suddenly gains a lot of energy, the system energy 456 00:28:37,680 --> 00:28:39,030 doesn't care. 457 00:28:39,030 --> 00:28:42,090 So small fluctuations -- or big fluctuations in the small 458 00:28:42,090 --> 00:28:43,890 number of molecules doesn't make any difference to the 459 00:28:43,890 --> 00:28:45,110 total energy. 460 00:28:45,110 --> 00:28:48,580 And so you can still use this then. 461 00:28:48,580 --> 00:28:49,430 It's good enough. 462 00:28:49,430 --> 00:28:57,440 STUDENT: So how long is that good enough [UNINTELLIGIBLE] 463 00:28:57,440 --> 00:29:00,340 PROFESSOR: Well if it's accountable, a handful of 464 00:29:00,340 --> 00:29:02,280 things, and it's not valid. 465 00:29:02,280 --> 00:29:04,260 If it's ten to the 24th and it's valid, and somewhere in 466 00:29:04,260 --> 00:29:05,200 between it breaks down. 467 00:29:05,200 --> 00:29:06,950 Then I don't know what the answer is. 468 00:29:06,950 --> 00:29:09,570 But usually if you have a thermodynamic system, then 469 00:29:09,570 --> 00:29:11,120 it's big enough. 470 00:29:11,120 --> 00:29:12,900 That's what thermodynamics is about. 471 00:29:12,900 --> 00:29:14,740 Where you don't really care that you have atoms there. 472 00:29:14,740 --> 00:29:16,540 You don't even know you have atoms there. 473 00:29:16,540 --> 00:29:23,090 So it's big enough. 474 00:29:23,090 --> 00:29:25,530 Good question. 475 00:29:25,530 --> 00:29:31,900 Alright, so now let's take our microstate energy here. 476 00:29:31,900 --> 00:29:35,870 And our microstate energy is the sum of all the molecular 477 00:29:35,870 --> 00:29:39,470 energies Ei. 478 00:29:39,470 --> 00:29:56,870 So it's the sum of all energies E sub, 479 00:29:56,870 --> 00:29:59,190 over all the atoms. 480 00:29:59,190 --> 00:30:02,580 And each one of these energies, if it's a molecular 481 00:30:02,580 --> 00:30:08,460 energies, can be indexed by a quantum number of some sort. 482 00:30:08,460 --> 00:30:13,760 So it would be the sum over all energies. 483 00:30:13,760 --> 00:30:18,760 So quantum number for particle one, n1 is some sort of 484 00:30:18,760 --> 00:30:21,660 quantum number. n2 is some sort of quantum number. n3 is 485 00:30:21,660 --> 00:30:23,190 some sort of quantum number. 486 00:30:23,190 --> 00:30:26,460 And then you have all the molecules. 487 00:30:26,460 --> 00:30:31,200 En1 plus En2 plus En3, et cetera. 488 00:30:31,200 --> 00:30:34,250 So this is the energy from molecule one, energy from 489 00:30:34,250 --> 00:30:36,430 molecule two, energy from molecule three, energy from 490 00:30:36,430 --> 00:30:37,390 molecule four. 491 00:30:37,390 --> 00:30:40,230 And that little n tells you which energy state that 492 00:30:40,230 --> 00:30:42,260 molecule is. 493 00:30:42,260 --> 00:30:47,050 And the sum of all these energies is 494 00:30:47,050 --> 00:30:49,310 your microstate energy. 495 00:30:49,310 --> 00:30:51,890 As long as you can write this this way, then you're allowed 496 00:30:51,890 --> 00:30:57,220 to write this this way. 497 00:30:57,220 --> 00:30:59,840 So that basically means that they're not interacting with 498 00:30:59,840 --> 00:31:00,700 each other. 499 00:31:00,700 --> 00:31:02,630 They're independent from each other. 500 00:31:02,630 --> 00:31:03,900 In this case here. 501 00:31:03,900 --> 00:31:09,900 So now if I write Q in terms of the sum over all 502 00:31:09,900 --> 00:31:16,550 microstates Ei. e to the minus Ei over kT. 503 00:31:16,550 --> 00:31:20,290 I'm going to replace this Ei here with the sum over all 504 00:31:20,290 --> 00:31:22,010 these energies here. 505 00:31:22,010 --> 00:31:24,830 And so the sum over all microstates, then, becomes the 506 00:31:24,830 --> 00:31:28,210 sum over all possible combinations of quantum 507 00:31:28,210 --> 00:31:37,090 numbers. n1, n2, n3, n4, et cetera. 508 00:31:37,090 --> 00:31:40,180 All the possible ways of getting 509 00:31:40,180 --> 00:31:41,830 molecule one in some state. 510 00:31:41,830 --> 00:31:44,330 All the possible ways of getting molecule two in some 511 00:31:44,330 --> 00:31:45,850 quantum number state. 512 00:31:45,850 --> 00:31:51,050 And then e to the minus -- and instead of capital Ei, I'm 513 00:31:51,050 --> 00:31:54,080 going to write the molecular energies. 514 00:31:54,080 --> 00:32:00,040 En1, plus En2, plus epsilon n3 plus et cetera. 515 00:32:00,040 --> 00:32:06,200 And then divide by kT. 516 00:32:06,200 --> 00:32:10,270 Basically I'm going to prove that this is a fine statement 517 00:32:10,270 --> 00:32:17,260 to make, as long as you can write the energy as a sum of 518 00:32:17,260 --> 00:32:26,380 component energies. 519 00:32:26,380 --> 00:32:29,840 OK so now this term here, e to the minus En1, only cares 520 00:32:29,840 --> 00:32:31,340 about this sum here. 521 00:32:31,340 --> 00:32:33,590 En2, that's molecule number two, only cares 522 00:32:33,590 --> 00:32:35,130 about this sum here. 523 00:32:35,130 --> 00:32:37,320 Molecule number three only cares about the sum over all 524 00:32:37,320 --> 00:32:38,790 possible quantum numbers connected to 525 00:32:38,790 --> 00:32:40,000 molecule number three. 526 00:32:40,000 --> 00:32:47,970 So I can factor out all these sums into a factor of sums. 527 00:32:47,970 --> 00:32:51,220 Is equal to the sum over quantum number n1. e to the 528 00:32:51,220 --> 00:32:58,780 minus epsilon of n1, divided by kT. 529 00:32:58,780 --> 00:33:10,104 Times n2 e to the minus epsilon n2 over kT, times n3 e 530 00:33:10,104 --> 00:33:13,200 to the minus epsilon n3 divided by kT, et cetera. 531 00:33:13,200 --> 00:33:15,750 And now each one of these is basically the molecular 532 00:33:15,750 --> 00:33:16,500 partition function. 533 00:33:16,500 --> 00:33:21,180 These are all the possible energies of that molecule. 534 00:33:21,180 --> 00:33:24,340 And the sum over all possible energies times e to the minus 535 00:33:24,340 --> 00:33:26,860 E over kT is the partition function for the molecule. 536 00:33:26,860 --> 00:33:31,190 So we have q for molecule one times q for molecule two. 537 00:33:31,190 --> 00:33:32,900 And they're all the same. 538 00:33:32,900 --> 00:33:34,630 There are N of them. 539 00:33:34,630 --> 00:33:42,190 Plus q to the N. So just, in a way, clarified that the reason 540 00:33:42,190 --> 00:33:46,170 why we're able to write this system partition function, in 541 00:33:46,170 --> 00:33:50,840 terms of the molecular partition functions, with N of 542 00:33:50,840 --> 00:33:53,410 them, to the Nth power, is because we were able to 543 00:33:53,410 --> 00:33:55,320 separate out the energy here. 544 00:33:55,320 --> 00:33:58,550 In terms of the independent molecular energies. 545 00:33:58,550 --> 00:34:01,280 Where this is saying the molecules don't interact with 546 00:34:01,280 --> 00:34:02,570 each other. 547 00:34:02,570 --> 00:34:04,410 And are independent from each other. 548 00:34:04,410 --> 00:34:07,700 And then the one over N factorial comes in, so that 549 00:34:07,700 --> 00:34:12,930 you don't over count, for translation, the positions. 550 00:34:12,930 --> 00:34:21,360 They're indistinguishable. 551 00:34:21,360 --> 00:34:22,070 OK. 552 00:34:22,070 --> 00:34:23,320 So now we can have -- 553 00:34:23,320 --> 00:34:25,100 Actually we're not -- 554 00:34:25,100 --> 00:34:30,560 This basic concept of the partition function multiplying 555 00:34:30,560 --> 00:34:36,310 each other, if the energies add, is not limited to going 556 00:34:36,310 --> 00:34:39,710 from the molecular partition functions to the system 557 00:34:39,710 --> 00:34:40,440 partition function. 558 00:34:40,440 --> 00:34:43,250 You can also look at the molecular 559 00:34:43,250 --> 00:34:45,090 partition function itself. 560 00:34:45,090 --> 00:34:49,120 And if the energy, the molecular energy, can be 561 00:34:49,120 --> 00:34:53,770 written in terms of a sum of energies of different degrees 562 00:34:53,770 --> 00:34:56,744 of freedom, for instance, the energy of a molecule could be 563 00:34:56,744 --> 00:34:58,560 the energy of the vibration, plus the energy of the 564 00:34:58,560 --> 00:35:01,685 translation, plus the energy of the rotation, plus the 565 00:35:01,685 --> 00:35:03,596 energy of the magnetic field, plus the energy of the 566 00:35:03,596 --> 00:35:04,410 electric field. et cetera. 567 00:35:04,410 --> 00:35:07,520 You have many energies that can add up with each other to 568 00:35:07,520 --> 00:35:10,640 create the molecular energy. 569 00:35:10,640 --> 00:35:14,320 And what we're going to be able to write, then, is that 570 00:35:14,320 --> 00:35:17,270 this molecular partition function itself can be written 571 00:35:17,270 --> 00:35:22,310 in terms of a product of partition functions for the 572 00:35:22,310 --> 00:35:25,680 sub parts of the molecular energy. 573 00:35:25,680 --> 00:35:40,520 So let me clarify that statement. 574 00:35:40,520 --> 00:35:46,350 So if I can write my molecular energy, epsilon, is equal to a 575 00:35:46,350 --> 00:35:51,790 translational energy, plus a vibrational energy, plus a 576 00:35:51,790 --> 00:35:55,090 rotational energy, plus every other little energies that you 577 00:35:55,090 --> 00:35:58,730 can think of that are independent of each other. 578 00:35:58,730 --> 00:36:03,640 Then using the same argument we used to show that Q is the 579 00:36:03,640 --> 00:36:07,550 multiplication of these molecular partition function, 580 00:36:07,550 --> 00:36:11,300 we can write that the molecular partition function, 581 00:36:11,300 --> 00:36:18,120 little q, is just the multiplication of the degree 582 00:36:18,120 --> 00:36:19,410 of freedom partition functions -- 583 00:36:19,410 --> 00:36:20,490 molecular partition functions. 584 00:36:20,490 --> 00:36:23,810 The translational partition function times the vibrational 585 00:36:23,810 --> 00:36:28,200 partition function, times the rotational partition 586 00:36:28,200 --> 00:36:29,670 function, et cetera. 587 00:36:29,670 --> 00:36:32,910 If the energies add, then the partition functions multiply 588 00:36:32,910 --> 00:36:36,190 each other. 589 00:36:36,190 --> 00:36:37,830 And that's going to be powerful because when we look 590 00:36:37,830 --> 00:36:43,220 at something like a polymer or DNA or protein or something, 591 00:36:43,220 --> 00:36:44,490 in solution. 592 00:36:44,490 --> 00:36:46,710 And we're going to be looking at the configurations possible 593 00:36:46,710 --> 00:36:52,970 for that polymer or that biopolymer, then we'll know 594 00:36:52,970 --> 00:36:56,938 that the energy of that polymer in solution is going 595 00:36:56,938 --> 00:37:00,010 to be -- we'll be able to approximate it as the energy 596 00:37:00,010 --> 00:37:03,653 of the configuration for that polymer. 597 00:37:03,653 --> 00:37:06,030 The different ways that you can fold the 598 00:37:06,030 --> 00:37:08,210 protein, for instance. 599 00:37:08,210 --> 00:37:10,050 Plus everything else. 600 00:37:10,050 --> 00:37:14,880 The energy of everything else. 601 00:37:14,880 --> 00:37:15,300 OK. 602 00:37:15,300 --> 00:37:18,530 So if the configurational energy can be separated from 603 00:37:18,530 --> 00:37:21,070 the sum of all vibration energies of all the bonds in 604 00:37:21,070 --> 00:37:22,380 that polymer. 605 00:37:22,380 --> 00:37:26,290 The way that polymer interacts with -- 606 00:37:26,290 --> 00:37:30,840 The way that the solution itself interacts with itself. 607 00:37:30,840 --> 00:37:34,780 Then if we can do this and we can do this approximation most 608 00:37:34,780 --> 00:37:38,000 of the time, then we'll be able to take the partition 609 00:37:38,000 --> 00:37:40,590 function for the polymer, and write it as the 610 00:37:40,590 --> 00:37:43,610 configurational partition function times the partition 611 00:37:43,610 --> 00:37:45,750 function for everything else. 612 00:37:45,750 --> 00:37:55,750 And we'll find that this part here will tend to factor out. 613 00:37:55,750 --> 00:37:58,660 We won't have to worry about it. 614 00:37:58,660 --> 00:38:01,380 And that this will carry all the important information that 615 00:38:01,380 --> 00:38:05,220 we'll need to know to see about changes in the system. 616 00:38:05,220 --> 00:38:07,055 Changes in Gibbs free energy, changes in 617 00:38:07,055 --> 00:38:08,110 the chemical potential. 618 00:38:08,110 --> 00:38:12,620 Everything will be related to this partition function. 619 00:38:12,620 --> 00:38:14,060 This subsystem. 620 00:38:14,060 --> 00:38:19,575 And because of the fact that you can factor them out, then 621 00:38:19,575 --> 00:38:21,770 this thing will end up dropping out. 622 00:38:21,770 --> 00:38:30,310 And this will become the important factor. 623 00:38:30,310 --> 00:38:30,970 OK. 624 00:38:30,970 --> 00:38:55,860 Let's do a quick example. 625 00:38:55,860 --> 00:38:56,100 OK. 626 00:38:56,100 --> 00:39:00,670 So this is the example of having a 627 00:39:00,670 --> 00:39:02,260 very very, short polymer. 628 00:39:02,260 --> 00:39:06,330 Containing three monomers. 629 00:39:06,330 --> 00:39:13,850 Which can be in two configurations. 630 00:39:13,850 --> 00:39:17,060 And the energies are the same for these two configurations. 631 00:39:17,060 --> 00:39:21,200 So the configurational partition function, which you 632 00:39:21,200 --> 00:39:26,690 would generally write as the sum of e to the minus Ei for 633 00:39:26,690 --> 00:39:29,680 that configuration, divided by kT, plus e to the -- 634 00:39:29,680 --> 00:39:34,500 So we would usually write it as e to the minus epsilon one 635 00:39:34,500 --> 00:39:38,373 over kT, plus e to the minus epsilon two 636 00:39:38,373 --> 00:39:41,100 over kT, plus et cetera. 637 00:39:41,100 --> 00:39:43,890 They're all the same energy. 638 00:39:43,890 --> 00:39:45,780 And there are two configurations. 639 00:39:45,780 --> 00:39:47,950 The degeneracy is two. 640 00:39:47,950 --> 00:39:50,640 So you can write this as the degeneracy of the 641 00:39:50,640 --> 00:39:55,820 configuration, times the energy of the configuration. 642 00:39:55,820 --> 00:40:00,070 And you can set your energy reference to be zero. 643 00:40:00,070 --> 00:40:01,870 You can choose whatever you want it to be. 644 00:40:01,870 --> 00:40:03,170 And zero is a good number. 645 00:40:03,170 --> 00:40:05,380 So that e to the zero is equal to one. 646 00:40:05,380 --> 00:40:07,980 So that configuration partition function is just the 647 00:40:07,980 --> 00:40:23,060 degeneracy, which is equal to two, in this case here. 648 00:40:23,060 --> 00:40:32,740 So now let's calculate the molecular and canonical 649 00:40:32,740 --> 00:40:38,760 partition functions for an ideal gas of 650 00:40:38,760 --> 00:40:41,540 these molecules here. 651 00:40:41,540 --> 00:40:44,600 And it's usually interesting to use a 652 00:40:44,600 --> 00:40:47,110 lattice model as a guide. 653 00:40:47,110 --> 00:40:50,960 And so in this lattice model here, you would divide space 654 00:40:50,960 --> 00:40:55,040 up into little cells. 655 00:40:55,040 --> 00:40:57,940 Pieces in two dimensions, but in reality it would be three 656 00:40:57,940 --> 00:40:58,800 dimensions. 657 00:40:58,800 --> 00:41:03,510 And then you place your molecules in lattice sites. 658 00:41:03,510 --> 00:41:06,470 Something like this. 659 00:41:06,470 --> 00:41:09,280 And then you end up counting the number of ways of 660 00:41:09,280 --> 00:41:12,810 arranging the molecules on the lattice. 661 00:41:12,810 --> 00:41:16,540 And let's say that we have N molecules that 662 00:41:16,540 --> 00:41:18,930 are in the gas phase. 663 00:41:18,930 --> 00:41:26,580 And the molecular volume, i.e. the size of the lattice site 664 00:41:26,580 --> 00:41:31,310 is v. That's the molecular volume. 665 00:41:31,310 --> 00:41:36,640 And N times v is the total volume. 666 00:41:36,640 --> 00:41:41,970 That's the total volume occupied by the particles. 667 00:41:41,970 --> 00:41:43,970 The total volume is the number of lattice sites. 668 00:41:43,970 --> 00:41:56,500 V is the total volume. 669 00:41:56,500 --> 00:42:01,620 And we're going to assume that all particles, all molecules 670 00:42:01,620 --> 00:42:03,890 have the same translation energy. 671 00:42:03,890 --> 00:42:05,850 It's an adequate approximation. 672 00:42:05,850 --> 00:42:08,790 We're going to set that equal to zero. 673 00:42:08,790 --> 00:42:12,770 So all molecules at any position here has the same E 674 00:42:12,770 --> 00:42:15,590 translation. 675 00:42:15,590 --> 00:42:20,590 And we're going to set that equal to zero. 676 00:42:20,590 --> 00:42:23,400 So the translational partition function. 677 00:42:23,400 --> 00:42:33,750 The molecular transitional partition function is -- 678 00:42:33,750 --> 00:42:35,420 Well there's only one energy. 679 00:42:35,420 --> 00:42:36,930 It's zero. 680 00:42:36,930 --> 00:42:40,020 So we only care about the degeneracy of that molecule. 681 00:42:40,020 --> 00:42:44,800 That molecule could be in here, or it could be here, it 682 00:42:44,800 --> 00:42:46,920 could be here, it could be here, it could 683 00:42:46,920 --> 00:42:48,370 be anywhere, right? 684 00:42:48,370 --> 00:42:52,910 The number of ways of putting that molecule on the lattice 685 00:42:52,910 --> 00:42:55,310 is the number of lattice sites available. 686 00:42:55,310 --> 00:42:58,200 Which is basically the molecular volume. 687 00:42:58,200 --> 00:42:59,750 Which is the total volume divided by 688 00:42:59,750 --> 00:43:02,720 the molecular volume. 689 00:43:02,720 --> 00:43:15,580 The total volume is -- right. 690 00:43:15,580 --> 00:43:23,870 So the total volume is the number of lattice sites times 691 00:43:23,870 --> 00:43:26,980 the volume of each lattice site. 692 00:43:26,980 --> 00:43:30,420 So the total volume divided by the small volume is the total 693 00:43:30,420 --> 00:43:32,030 number of lattice sites. 694 00:43:32,030 --> 00:43:34,630 And the number of choices of putting that one molecule is 695 00:43:34,630 --> 00:43:39,470 anywhere on the lattice. 696 00:43:39,470 --> 00:43:44,330 That's your degeneracy. 697 00:43:44,330 --> 00:43:50,810 So now if I look at the total molecular partition function, 698 00:43:50,810 --> 00:43:52,710 it's going to be the multiplication of the 699 00:43:52,710 --> 00:43:55,640 configurational partition function and the translational 700 00:43:55,640 --> 00:43:58,260 partition function. 701 00:43:58,260 --> 00:44:18,100 At each site, the molecule could have two configurations. 702 00:44:18,100 --> 00:44:22,280 So q, for the molecule, that's q translational. 703 00:44:22,280 --> 00:44:23,710 I'm going to ignore all vibrations, 704 00:44:23,710 --> 00:44:24,430 rotation, et cetera. 705 00:44:24,430 --> 00:44:26,470 I'm going to assume that there are two degrees of freedom. 706 00:44:26,470 --> 00:44:29,430 Translational one, which is basically the positional one. 707 00:44:29,430 --> 00:44:31,340 And then the configurational one, which is 708 00:44:31,340 --> 00:44:36,170 internal to the molecule. 709 00:44:36,170 --> 00:44:39,880 So this one is V over v. That's the degeneracy. 710 00:44:39,880 --> 00:44:42,400 Capital V over little v. The degeneracy of placing the 711 00:44:42,400 --> 00:44:45,580 molecule on the lattice. 712 00:44:45,580 --> 00:44:49,970 And the configuration is the degeneracy of how 713 00:44:49,970 --> 00:44:52,440 the molecule folds. 714 00:44:52,440 --> 00:44:52,750 Yes. 715 00:44:52,750 --> 00:45:01,730 STUDENT: [UNINTELLIGIBLE] 716 00:45:01,730 --> 00:45:05,980 PROFESSOR: The notes are wrong. 717 00:45:05,980 --> 00:45:10,390 So usually capital V is large and little v is small. 718 00:45:10,390 --> 00:45:12,920 So in the notes, if we have it in reverse, 719 00:45:12,920 --> 00:45:18,620 we should fix that. 720 00:45:18,620 --> 00:45:21,410 This is lecture 25, right? 721 00:45:21,410 --> 00:45:22,840 So we have q. 722 00:45:22,840 --> 00:45:23,930 No the notes seem to be right. 723 00:45:23,930 --> 00:45:31,720 Total volume is capital V, molecular volume is little v. 724 00:45:31,720 --> 00:45:33,170 Where is it that wrong in the notes? 725 00:45:33,170 --> 00:45:46,030 STUDENT: [UNINTELLIGIBLE] 726 00:45:46,030 --> 00:45:46,870 PROFESSOR: Oh look at that. 727 00:45:46,870 --> 00:45:51,130 My notes are different than yours. 728 00:45:51,130 --> 00:45:53,910 My notes are right. 729 00:45:53,910 --> 00:45:57,060 OK well it's obviously right on the web. 730 00:45:57,060 --> 00:45:58,720 Because this is the latest version. 731 00:45:58,720 --> 00:46:03,440 Alright so flip those big V's and little v's then. 732 00:46:03,440 --> 00:46:03,770 Huh. 733 00:46:03,770 --> 00:46:08,530 I thought they were from the same pile. 734 00:46:08,530 --> 00:46:08,860 OK. 735 00:46:08,860 --> 00:46:15,000 So this is your molecular partition function. 736 00:46:15,000 --> 00:46:20,420 And then when you look at the system, the system partition 737 00:46:20,420 --> 00:46:24,510 function can also be separated into a translation and the 738 00:46:24,510 --> 00:46:27,650 configuration for the system. 739 00:46:27,650 --> 00:46:31,020 We know what you need to do is take all the molecular 740 00:46:31,020 --> 00:46:36,360 partition functions, the transitional ones, and -- to 741 00:46:36,360 --> 00:46:37,010 the N factor. 742 00:46:37,010 --> 00:46:39,930 The number of particles. 743 00:46:39,930 --> 00:46:42,230 But now you have degeneracy. 744 00:46:42,230 --> 00:46:45,070 You've over counted. 745 00:46:45,070 --> 00:46:50,000 So you need to divide this N factorial. 746 00:46:50,000 --> 00:46:53,130 You also have the system partition function for the 747 00:46:53,130 --> 00:46:56,190 configurations. 748 00:46:56,190 --> 00:47:02,250 And that's q configuration to the N. Except here we don't 749 00:47:02,250 --> 00:47:04,280 need to divide by one over N factorial, because we're not 750 00:47:04,280 --> 00:47:07,390 over counting here. 751 00:47:07,390 --> 00:47:13,840 The over counting only happens when you're placing identical 752 00:47:13,840 --> 00:47:18,310 particles in a lattice, and you can swap them without 753 00:47:18,310 --> 00:47:19,950 making a difference. 754 00:47:19,950 --> 00:47:21,950 Here we talking about configurations. 755 00:47:21,950 --> 00:47:23,420 When we're talking about configurations, we're not 756 00:47:23,420 --> 00:47:26,660 talking about placing the identical particles in 757 00:47:26,660 --> 00:47:27,440 different spots. 758 00:47:27,440 --> 00:47:31,390 We're just looking at these two configurations here. 759 00:47:31,390 --> 00:47:33,590 And then the next particle is two configurations. 760 00:47:33,590 --> 00:47:37,550 The next particle is two configurations. 761 00:47:37,550 --> 00:47:38,680 So this is really important. 762 00:47:38,680 --> 00:47:41,200 That this N over factorial only comes into play when 763 00:47:41,200 --> 00:47:41,890 you're talking about the 764 00:47:41,890 --> 00:47:43,790 translational degree of freedom. 765 00:47:43,790 --> 00:47:46,380 Not the other degrees of freedom. 766 00:47:46,380 --> 00:47:48,880 And now the total system partition function is the 767 00:47:48,880 --> 00:47:51,080 multiplication of these two. 768 00:47:51,080 --> 00:47:57,290 It ends up being capital V over v, to the N power, over N 769 00:47:57,290 --> 00:47:59,820 factorial, times 770 00:47:59,820 --> 00:48:07,200 two to the N. OK. 771 00:48:07,200 --> 00:48:10,970 In general you could extend this analysis to include 772 00:48:10,970 --> 00:48:15,420 vibrations, rotations, energy in a magnetical field, 773 00:48:15,420 --> 00:48:18,510 electric field, et cetera. 774 00:48:18,510 --> 00:48:24,400 Any questions? 775 00:48:24,400 --> 00:48:26,520 Alright the next thing that you're going to do then is to 776 00:48:26,520 --> 00:48:31,480 use this concept, as a sort of example, as a way to begin to 777 00:48:31,480 --> 00:48:34,010 calculate things that you've already calculated before. 778 00:48:34,010 --> 00:48:38,570 For instance, if you look at an expansion of an ideal gas, 779 00:48:38,570 --> 00:48:41,740 can we now calculate the entropy change. 780 00:48:41,740 --> 00:48:45,120 Not based on thermodynamics, but based on 781 00:48:45,120 --> 00:48:46,330 the statistical mechanics. 782 00:48:46,330 --> 00:48:49,410 On the microscopic description that we've just gone through. 783 00:48:49,410 --> 00:48:53,110 And it turns out that that's what happens. 784 00:48:53,110 --> 00:48:54,030 That you can do that. 785 00:48:54,030 --> 00:48:55,090 You get the same answer. 786 00:48:55,090 --> 00:48:57,430 Thank god you get the same answer. 787 00:48:57,430 --> 00:49:00,680 Otherwise we'd be in big trouble. 788 00:49:00,680 --> 00:49:02,446 So I'm just going to set up the problem, because I won't 789 00:49:02,446 --> 00:49:03,570 have time to do it. 790 00:49:03,570 --> 00:49:06,010 And then you can do it next time, when Keith comes back. 791 00:49:06,010 --> 00:49:11,990 So the problem is going to be the usual problem of having a 792 00:49:11,990 --> 00:49:20,870 volume V1 of a gas on one side, and a vaccuum expanding 793 00:49:20,870 --> 00:49:23,420 to volume V2, gas. 794 00:49:23,420 --> 00:49:29,700 And asking what is the entropy change. 795 00:49:29,700 --> 00:49:37,000 And you know that from thermo, that delta S in this case here 796 00:49:37,000 --> 00:49:43,190 is nR log V2 over V1, when the temperature is constant. 797 00:49:43,190 --> 00:49:44,050 That's going to be our answer. 798 00:49:44,050 --> 00:49:45,580 It has to be our answer. 799 00:49:45,580 --> 00:49:47,340 But this time, instead of knowing the answer, we're 800 00:49:47,340 --> 00:49:51,040 going to calculate it from microscopics. 801 00:49:51,040 --> 00:49:54,550 So what you do is you start out with your initial state. 802 00:49:54,550 --> 00:50:01,080 You ask what is -- you write down the molecular volume V. 803 00:50:01,080 --> 00:50:02,330 The total volume here is V1. 804 00:50:10,960 --> 00:50:14,420 You assume that all molecules have the same 805 00:50:14,420 --> 00:50:15,830 translational energy. 806 00:50:15,830 --> 00:50:17,380 You set that equal to zero. 807 00:50:17,380 --> 00:50:20,700 The system translational energy is equal to zero. 808 00:50:20,700 --> 00:50:26,910 And so the entropy for this gas here is just the number of 809 00:50:26,910 --> 00:50:30,540 ways of placing the molecule in the lattice. 810 00:50:30,540 --> 00:50:34,730 This model that we have of space being separated into 811 00:50:34,730 --> 00:50:35,880 little cells. 812 00:50:35,880 --> 00:50:43,380 And so S is k log omega, where omega is the number of ways of 813 00:50:43,380 --> 00:50:45,480 placing the molecules in the lattice. 814 00:50:45,480 --> 00:50:50,230 Which is basically k log V, over v. Where this is the 815 00:50:50,230 --> 00:51:03,170 number of lattice sites. 816 00:51:03,170 --> 00:51:05,940 OK. 817 00:51:05,940 --> 00:51:07,370 And what you're going to do next time, then, 818 00:51:07,370 --> 00:51:10,250 is start from here. 819 00:51:10,250 --> 00:51:11,640 Calculate what it is before. 820 00:51:11,640 --> 00:51:15,020 Calculate what it is after, and turn the crank, and get to 821 00:51:15,020 --> 00:51:16,770 the right answer. 822 00:51:16,770 --> 00:51:19,070 Then you're going to do the same thing for liquids. 823 00:51:19,070 --> 00:51:25,590 And that'll be it for this simple statistical mechanics.