1 00:00:00,000 --> 00:00:02,195 ANNOUNCER: The following content is provided under a 2 00:00:02,195 --> 00:00:03,830 creative commons license. 3 00:00:03,830 --> 00:00:06,850 Your support will help MIT Open Courseware continue to 4 00:00:06,850 --> 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 --> 00:00:17,170 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,170 --> 00:00:18,700 ocw.mit.edu. 8 00:00:18,700 --> 00:00:23,070 PROFESSOR: Last few lectures, and we started to see some of 9 00:00:23,070 --> 00:00:26,250 the consequences and looked at simple changes, 10 00:00:26,250 --> 00:00:29,460 transformations of expansion and mixtures 11 00:00:29,460 --> 00:00:31,020 of liquids and gases. 12 00:00:31,020 --> 00:00:34,470 And saw how in the framework of statistical mechanics, we 13 00:00:34,470 --> 00:00:38,310 could derive the thermodynamic results that you saw before, 14 00:00:38,310 --> 00:00:39,820 based on an empirical framework. 15 00:00:39,820 --> 00:00:41,400 On the thermodynamic framework we've been 16 00:00:41,400 --> 00:00:42,740 working with all term. 17 00:00:42,740 --> 00:00:45,940 But you saw them derived from a microscopic perspective. 18 00:00:45,940 --> 00:00:50,060 Now what I want to do is move to examples that are more 19 00:00:50,060 --> 00:00:52,000 commonly encountered in chemistry. 20 00:00:52,000 --> 00:00:55,860 One that we did last time actually was very common, or 21 00:00:55,860 --> 00:00:58,070 is at least a prototype for something common. 22 00:00:58,070 --> 00:01:02,275 That is we looked at what would be a simple model for a 23 00:01:02,275 --> 00:01:03,980 polymer with different configurations 24 00:01:03,980 --> 00:01:05,310 available to it. 25 00:01:05,310 --> 00:01:08,550 And just tried to understand what the basic thermodynamics 26 00:01:08,550 --> 00:01:09,240 is that it would exhibit. 27 00:01:09,240 --> 00:01:12,590 What the heat capacities would be in various limiting cases. 28 00:01:12,590 --> 00:01:14,320 High and low temperature. 29 00:01:14,320 --> 00:01:16,180 I want to continue that today. 30 00:01:16,180 --> 00:01:18,090 But for an even more common case. 31 00:01:18,090 --> 00:01:23,480 So the model that I want to lay out is indicated on your 32 00:01:23,480 --> 00:01:25,540 notes for today. 33 00:01:25,540 --> 00:01:28,440 And essentially, you could look at it as 34 00:01:28,440 --> 00:01:30,370 double stranded DNA. 35 00:01:30,370 --> 00:01:33,850 Any kind of polymer with a whole variety of 36 00:01:33,850 --> 00:01:34,950 configurations. 37 00:01:34,950 --> 00:01:39,150 Ultimately what's going to matter is what is the set of 38 00:01:39,150 --> 00:01:42,170 energy levels available to the system. 39 00:01:42,170 --> 00:01:44,480 And so here's what it looks like. 40 00:01:44,480 --> 00:01:48,700 The way I've tried to depict it. 41 00:01:48,700 --> 00:01:57,420 I'm imagining something like, covalently bonded chains that 42 00:01:57,420 --> 00:01:58,690 have links between them. 43 00:01:58,690 --> 00:02:03,250 So of course, it's reminiscent of DNA with hydrogen bonding 44 00:02:03,250 --> 00:02:05,770 between the strands. 45 00:02:05,770 --> 00:02:09,910 And the idea in this simple model is that you can have 46 00:02:09,910 --> 00:02:13,110 various numbers of these hydrogen bonds. 47 00:02:13,110 --> 00:02:15,890 So this is the case that would be the most 48 00:02:15,890 --> 00:02:16,980 energetically stable. 49 00:02:16,980 --> 00:02:20,010 That is, all the available hydrogen bonds are formed 50 00:02:20,010 --> 00:02:22,930 between neighboring pairs. 51 00:02:22,930 --> 00:02:26,570 So this would be our lowest energy. 52 00:02:26,570 --> 00:02:29,890 Now let's starts breaking some hydrogen bonds. 53 00:02:29,890 --> 00:02:34,020 And what I'm imagining is they break starting at one end and 54 00:02:34,020 --> 00:02:35,110 working their way down. 55 00:02:35,110 --> 00:02:52,370 So the next highest energy state would be like this. 56 00:02:52,370 --> 00:02:54,610 So now these are still hydrogen bonded. 57 00:02:54,610 --> 00:02:55,980 This is not. 58 00:02:55,980 --> 00:02:58,040 And there's a cost in energy for breaking 59 00:02:58,040 --> 00:03:01,140 that hydrogen bond. 60 00:03:01,140 --> 00:03:04,910 We'll call it epsilon zero. 61 00:03:04,910 --> 00:03:06,340 It's a positive number. 62 00:03:06,340 --> 00:03:09,100 So that this is higher energy than this. 63 00:03:09,100 --> 00:03:15,500 And now we'll go to the next one, which could have two 64 00:03:15,500 --> 00:03:17,440 broken bonds. 65 00:03:17,440 --> 00:03:26,980 Broken hydrogen bonds. 66 00:03:26,980 --> 00:03:27,580 And so on. 67 00:03:27,580 --> 00:03:29,990 And if we imagine that this is very long, there may be many 68 00:03:29,990 --> 00:03:31,320 many, members of the chain. 69 00:03:31,320 --> 00:03:35,030 Then of course this sequence of structures with 70 00:03:35,030 --> 00:03:39,360 corresponding energies might go on for a very long way. 71 00:03:39,360 --> 00:03:42,500 So this is a simple construction that gives us a 72 00:03:42,500 --> 00:03:47,670 set of states with distinct energies. 73 00:03:47,670 --> 00:03:56,340 And so we have a series of configurational energy levels. 74 00:03:56,340 --> 00:03:59,990 They're evenly spaced in this model. 75 00:03:59,990 --> 00:04:09,780 So there's zero, E zero, two E zero. 76 00:04:09,780 --> 00:04:15,530 And so on. 77 00:04:15,530 --> 00:04:18,630 So those are available energy states. 78 00:04:18,630 --> 00:04:21,920 And now, just based on that simple model, we should be 79 00:04:21,920 --> 00:04:23,570 able to figure out all the thermodynamics. 80 00:04:23,570 --> 00:04:25,620 We should should be able to figure out what the 81 00:04:25,620 --> 00:04:28,710 equilibrium energy is at a particular temperature. 82 00:04:28,710 --> 00:04:30,230 What the average energy is. 83 00:04:30,230 --> 00:04:33,330 And so forth. 84 00:04:33,330 --> 00:04:39,150 And we can do it in a straightforward way. 85 00:04:39,150 --> 00:04:47,440 So we'll start with our molecular partition 86 00:04:47,440 --> 00:04:50,230 function, as always. 87 00:04:50,230 --> 00:04:53,920 So it's q configurational. 88 00:04:53,920 --> 00:04:57,300 And it's just a sum over our available states with their 89 00:04:57,300 --> 00:04:59,240 associated Boltzmann factors. 90 00:04:59,240 --> 00:05:05,170 So sum over n of e to the minus En over kT. 91 00:05:05,170 --> 00:05:10,110 Now the way I've modeled this, there are no degeneracies. 92 00:05:10,110 --> 00:05:14,660 Each energy level has just one microscopic state of the 93 00:05:14,660 --> 00:05:16,280 molecule corresponding to it. 94 00:05:16,280 --> 00:05:20,240 So there are no molecular degeneracies. 95 00:05:20,240 --> 00:05:22,840 Turns out we'll be able to put this into a simple form. 96 00:05:22,840 --> 00:05:28,110 So the first thing is, let's approximate that we could take 97 00:05:28,110 --> 00:05:31,920 the sum -- not to some finite level, which would be the case 98 00:05:31,920 --> 00:05:34,900 if there's a finite number of elements in the chain. 99 00:05:34,900 --> 00:05:48,660 But all the way through infinity. 100 00:05:48,660 --> 00:05:52,560 And the idea here is the following. 101 00:05:52,560 --> 00:05:54,160 We'll assume that the chain at least has 102 00:05:54,160 --> 00:05:56,140 a great many members. 103 00:05:56,140 --> 00:05:58,590 Even if it's finite. 104 00:05:58,590 --> 00:06:04,140 And then at some point, this becomes a really small number. 105 00:06:04,140 --> 00:06:07,840 Even if epsilon naught, the energy of one of the bonds, is 106 00:06:07,840 --> 00:06:09,650 less than kT. 107 00:06:09,650 --> 00:06:13,530 Once we multiply it by a very large number n, it becomes 108 00:06:13,530 --> 00:06:15,320 very much greater than kT. 109 00:06:15,320 --> 00:06:19,200 So the underlying assumption here is, well there will not 110 00:06:19,200 --> 00:06:22,600 be a significant number of molecules in the highest 111 00:06:22,600 --> 00:06:24,790 energy state anyway. 112 00:06:24,790 --> 00:06:30,150 In other words, the terms higher than a finite but large 113 00:06:30,150 --> 00:06:34,200 value of n between there and infinity are going to be so 114 00:06:34,200 --> 00:06:38,585 small that those terms will contribute negligibly to the 115 00:06:38,585 --> 00:06:40,990 sum anyway. 116 00:06:40,990 --> 00:06:43,280 And on that basis, we could make this approximation. 117 00:06:43,280 --> 00:06:45,730 And the incentive to make this approximation and carry the 118 00:06:45,730 --> 00:06:48,860 sum to infinity is that then we can put this in what turns 119 00:06:48,860 --> 00:06:52,060 out to be a very simple closed form. 120 00:06:52,060 --> 00:06:56,280 So now if we just write out a few of the individual terms in 121 00:06:56,280 --> 00:06:59,450 the sum, it's one plus e to the minus 122 00:06:59,450 --> 00:07:02,330 epsilon zero over kT. 123 00:07:02,330 --> 00:07:08,240 Plus e to the minus epsilon zero over kT. 124 00:07:08,240 --> 00:07:10,520 I'll write this as squared. e to the minus two 125 00:07:10,520 --> 00:07:12,670 epsilon zero over k t. 126 00:07:12,670 --> 00:07:13,830 And so on. 127 00:07:13,830 --> 00:07:19,330 So this has the form of a sum that looks like one plus x, 128 00:07:19,330 --> 00:07:21,970 plus x squared, and so on. 129 00:07:21,970 --> 00:07:25,450 And because the sum goes to infinity, we can simply write 130 00:07:25,450 --> 00:07:30,630 it as one over one minus x. 131 00:07:30,630 --> 00:07:32,590 Sort of a very simple result. 132 00:07:32,590 --> 00:07:38,550 And now putting this back into x, it's one over one minus e 133 00:07:38,550 --> 00:07:43,290 to the minus epsilon zero over kT. 134 00:07:43,290 --> 00:07:45,450 That's our q zero. 135 00:07:45,450 --> 00:07:51,140 Our molecular partition function. 136 00:07:51,140 --> 00:07:53,860 So this model yields a particularly simple 137 00:07:53,860 --> 00:07:55,740 result for q zero. 138 00:07:55,740 --> 00:07:59,360 And then everything else follows from there. 139 00:07:59,360 --> 00:08:12,430 So then we can write the canonical partition function 140 00:08:12,430 --> 00:08:13,420 capital Qn. 141 00:08:13,420 --> 00:08:21,100 It's just q configurational to the capital Nth power, where 142 00:08:21,100 --> 00:08:25,690 capital N is the total number of these molecules. 143 00:08:25,690 --> 00:08:29,640 So it's just one over one minus e to the minus epsilon 144 00:08:29,640 --> 00:08:35,930 over kT to the Nth power. 145 00:08:35,930 --> 00:08:38,190 And then we can start writing out the results for the 146 00:08:38,190 --> 00:08:39,940 various thermodynamic properties. 147 00:08:39,940 --> 00:08:54,140 So A configurational is minus NkT, log q configurational. 148 00:08:54,140 --> 00:09:03,150 That is, remember, A is minus kT log capital Q. And this is 149 00:09:03,150 --> 00:09:05,410 just going to come straight out. 150 00:09:05,410 --> 00:09:10,850 So it's minus NkT log of little q. 151 00:09:10,850 --> 00:09:17,240 Minus NkT, log of one over one minus e to the minus 152 00:09:17,240 --> 00:09:19,950 epsilon over kT. 153 00:09:19,950 --> 00:09:26,880 Or positive NkT, log of one minus e to the minus 154 00:09:26,880 --> 00:09:30,140 epsilon over kT. 155 00:09:30,140 --> 00:09:35,010 So a pretty simple form for the Helmholtz free energy. 156 00:09:35,010 --> 00:09:37,790 And I'm not going to write out all of the individuals 157 00:09:37,790 --> 00:09:40,510 thermodynamic terms, but I'll write a few of them. 158 00:09:40,510 --> 00:09:46,485 So mu, the chemical potential for these configurations, is 159 00:09:46,485 --> 00:09:47,735 just dA/dN. 160 00:09:53,100 --> 00:09:54,190 With T and V constant. 161 00:09:54,190 --> 00:09:57,030 And so the only capital N dependence, the only 162 00:09:57,030 --> 00:10:00,490 dependence on the number of molecules is multiplicative. 163 00:10:00,490 --> 00:10:03,320 Right here. 164 00:10:03,320 --> 00:10:08,750 So this just gives us kT log of one minus e to the minus 165 00:10:08,750 --> 00:10:10,820 epsilon naught over kT. 166 00:10:10,820 --> 00:10:12,980 And the important point to realize here that we mentioned 167 00:10:12,980 --> 00:10:15,830 last time too, in the examples we considered then, especially 168 00:10:15,830 --> 00:10:21,750 the last one, is that because the only place that N figures 169 00:10:21,750 --> 00:10:26,050 in is as a multiplicative factor. 170 00:10:26,050 --> 00:10:29,490 What this is telling us is that we just have a chemical 171 00:10:29,490 --> 00:10:33,180 potential, of Helmholtz free energy per molecule. 172 00:10:33,180 --> 00:10:35,910 The molecules are all independent. 173 00:10:35,910 --> 00:10:38,140 The total energy doesn't depend on any interaction 174 00:10:38,140 --> 00:10:42,530 between this molecule and its neighbor somewhere else. 175 00:10:42,530 --> 00:10:45,750 So terms like this are simply additive. 176 00:10:45,750 --> 00:10:48,420 So if we work out the chemical potential, it's just one over 177 00:10:48,420 --> 00:10:56,120 N times A. 178 00:10:56,120 --> 00:11:04,410 And I'll just write the result for u configurational. 179 00:11:04,410 --> 00:11:08,040 Because I do want to look at the heat capacity. 180 00:11:08,040 --> 00:11:22,670 So it's Nk T squared, d log of little q configurational / dT. 181 00:11:22,670 --> 00:11:25,590 With N and V held constant. 182 00:11:25,590 --> 00:11:31,310 And it turns out to just be N epsilon zero. 183 00:11:31,310 --> 00:11:40,770 One over E zero over kT minus one. 184 00:11:40,770 --> 00:11:43,930 So now we can look at the heat capacity. 185 00:11:43,930 --> 00:11:49,030 Cv configurational. 186 00:11:49,030 --> 00:11:54,590 So it's du configurational / dT. 187 00:11:54,590 --> 00:11:59,420 With constant N and V. And taking this derivative with 188 00:11:59,420 --> 00:12:02,600 respect to temperature. 189 00:12:02,600 --> 00:12:05,160 Gives us an expression of the following form. 190 00:12:05,160 --> 00:12:16,920 It's Nk epsilon zero over k T squared. e to the E zero over 191 00:12:16,920 --> 00:12:24,460 kT, over e to the E zero over kT minus one squared. 192 00:12:24,460 --> 00:12:27,480 Just taking the derivative in the usual way. 193 00:12:27,480 --> 00:12:31,530 Now, this looks kind of complicated 194 00:12:31,530 --> 00:12:33,360 for the heat capacity. 195 00:12:33,360 --> 00:12:36,330 But let's take a look, just like we did last time for the 196 00:12:36,330 --> 00:12:39,130 simpler case that we treated then, let's take a look at the 197 00:12:39,130 --> 00:12:42,800 high and low temperature limits of what happens to the 198 00:12:42,800 --> 00:12:44,010 heat capacity. 199 00:12:44,010 --> 00:12:59,740 So at high temperature, well we can start by just looking 200 00:12:59,740 --> 00:13:01,100 at the energy. 201 00:13:01,100 --> 00:13:03,570 That has a form that's fairly simple. 202 00:13:03,570 --> 00:13:07,790 So when temperature is large, this is small, and we can 203 00:13:07,790 --> 00:13:09,950 Taylor expand it. 204 00:13:09,950 --> 00:13:12,130 So then we're just going to have one plus epsilon zero 205 00:13:12,130 --> 00:13:14,010 over kT minus one. 206 00:13:14,010 --> 00:13:16,160 Which means the ones will cancel. 207 00:13:16,160 --> 00:13:18,450 And we end up with a fairly simple result. 208 00:13:18,450 --> 00:13:24,830 So u configurational, in that case, is just NkT. 209 00:13:28,180 --> 00:13:31,580 The epsilon zeroes are going to cancel also. 210 00:13:31,580 --> 00:13:34,200 And what that says is that the heat capacity -- and of course 211 00:13:34,200 --> 00:13:36,720 we could take the limit of this, but we can just take the 212 00:13:36,720 --> 00:13:38,090 derivative of this with respect to 213 00:13:38,090 --> 00:13:40,850 temperature more easily. 214 00:13:40,850 --> 00:13:45,230 So Cv configurational. 215 00:13:45,230 --> 00:13:47,340 It's just N times k. 216 00:13:47,340 --> 00:13:50,450 In other words, the heat capacity in the high 217 00:13:50,450 --> 00:13:53,480 temperature limit is a constant. 218 00:13:53,480 --> 00:13:56,430 Last time we treated a simpler case, where there were only 219 00:13:56,430 --> 00:14:01,930 four configurations altogether available, to this sort of 220 00:14:01,930 --> 00:14:04,490 simple polymer model that we drew. 221 00:14:04,490 --> 00:14:08,330 Unlike this present model where we're saying there are 222 00:14:08,330 --> 00:14:12,510 essentially infinite number of configurations and different 223 00:14:12,510 --> 00:14:14,240 energies available. 224 00:14:14,240 --> 00:14:17,420 So many that the highest ones we're never even going to 225 00:14:17,420 --> 00:14:21,420 access, because they'll be much higher than kT. 226 00:14:21,420 --> 00:14:23,760 Here it's different, so now we have -- 227 00:14:23,760 --> 00:14:27,620 You know before, when we had a limited number of total 228 00:14:27,620 --> 00:14:31,980 states, remember what happened in the heat capacity at high 229 00:14:31,980 --> 00:14:32,420 temperature. 230 00:14:32,420 --> 00:14:34,910 What was the limiting case? 231 00:14:34,910 --> 00:14:36,920 Finite number of states. 232 00:14:36,920 --> 00:14:41,710 What's the heat capacity at high temperature? 233 00:14:41,710 --> 00:14:45,180 This is going to be on the exam. 234 00:14:45,180 --> 00:14:47,750 What's the heat capacity at high temperature, if there's a 235 00:14:47,750 --> 00:14:49,540 finite number of states available? 236 00:14:49,540 --> 00:14:50,660 STUDENT: Zero. 237 00:14:50,660 --> 00:14:52,310 PROFESSOR: It's zero. 238 00:14:52,310 --> 00:14:57,550 What was the low temperature limit of the heat capacity? 239 00:14:57,550 --> 00:14:57,670 STUDENT: Zero. 240 00:14:57,670 --> 00:14:58,700 PROFESSOR: It was also zero. 241 00:14:58,700 --> 00:14:59,180 Right. 242 00:14:59,180 --> 00:15:05,980 And the idea was for a system with a 243 00:15:05,980 --> 00:15:11,510 finite number of states. 244 00:15:11,510 --> 00:15:15,010 The idea was -- so let's say, the way we had it before, 245 00:15:15,010 --> 00:15:16,970 there was one state with energy zero. 246 00:15:16,970 --> 00:15:20,290 And we had three states with some energy epsilon zero. 247 00:15:20,290 --> 00:15:21,840 And that was it. 248 00:15:21,840 --> 00:15:24,500 Those were all the molecular states available. 249 00:15:24,500 --> 00:15:27,960 So we looked at the two cases. 250 00:15:27,960 --> 00:15:31,130 In one case, when the high temperature limit where kT is 251 00:15:31,130 --> 00:15:33,640 much bigger than any of this stuff. 252 00:15:33,640 --> 00:15:38,990 In that limit, the molecules are just equally likely to be 253 00:15:38,990 --> 00:15:40,560 in any of these states. 254 00:15:40,560 --> 00:15:43,260 Because there's much, much more thermal energy than this 255 00:15:43,260 --> 00:15:44,130 energy difference. 256 00:15:44,130 --> 00:15:45,860 And the molecules are constantly getting kicked 257 00:15:45,860 --> 00:15:48,960 around by the available thermal energy. 258 00:15:48,960 --> 00:15:51,940 So if you raise the temperature a little bit more, 259 00:15:51,940 --> 00:15:53,320 it doesn't make any difference. 260 00:15:53,320 --> 00:15:54,800 The molecules already are evenly 261 00:15:54,800 --> 00:15:57,260 distributed among the states. 262 00:15:57,260 --> 00:16:00,080 There's no additional configurational energy. 263 00:16:00,080 --> 00:16:03,480 So du/dT is zero. 264 00:16:03,480 --> 00:16:06,640 It can't increase any more. 265 00:16:06,640 --> 00:16:09,090 So in that case, the high temperature limiting heat 266 00:16:09,090 --> 00:16:11,950 capacity is zero. 267 00:16:11,950 --> 00:16:19,240 And in the other case -- 268 00:16:19,240 --> 00:16:21,640 In the low temperature limit, now let's say 269 00:16:21,640 --> 00:16:22,640 kT is really low. 270 00:16:22,640 --> 00:16:24,700 It's much less than epsilon zero. 271 00:16:24,700 --> 00:16:26,900 So now we'll redraw this as zero. 272 00:16:26,900 --> 00:16:36,940 And put the epsilon zero states up here. 273 00:16:36,940 --> 00:16:39,740 And kT is here. 274 00:16:39,740 --> 00:16:43,710 Well when it's like this, the temperature is so low that 275 00:16:43,710 --> 00:16:46,240 there's not nearly enough thermal energy to populate any 276 00:16:46,240 --> 00:16:47,420 of these states. 277 00:16:47,420 --> 00:16:49,320 And if you change the temperature by a little bit, 278 00:16:49,320 --> 00:16:51,700 it's still not enough thermal energy to populate any of 279 00:16:51,700 --> 00:16:52,740 these states. 280 00:16:52,740 --> 00:16:55,400 So once again, du/dT is zero. 281 00:16:55,400 --> 00:16:57,980 You change the temperature and the configurational energy 282 00:16:57,980 --> 00:16:59,230 doesn't change. 283 00:16:59,230 --> 00:17:02,810 So the heat capacity is zero again. 284 00:17:02,810 --> 00:17:05,360 This is why it's so informative to measure heat 285 00:17:05,360 --> 00:17:06,110 capacities. 286 00:17:06,110 --> 00:17:10,040 Because you can learn an awful lot about the intrinsic 287 00:17:10,040 --> 00:17:12,000 structure of the material. 288 00:17:12,000 --> 00:17:14,260 What are the energy levels available to it? 289 00:17:14,260 --> 00:17:15,890 What do they do? 290 00:17:15,890 --> 00:17:18,930 You can learn a tremendous amount about that by making 291 00:17:18,930 --> 00:17:20,960 measurements of the heat capacity. 292 00:17:20,960 --> 00:17:24,690 Well, now we're in a different case. 293 00:17:24,690 --> 00:17:30,350 We have a whole set of evenly spaced energy levels. 294 00:17:30,350 --> 00:17:33,550 Never ends. 295 00:17:33,550 --> 00:17:36,680 So now let's look at the high temperature limit. 296 00:17:36,680 --> 00:17:39,170 But we're never in a limit that's higher than the highest 297 00:17:39,170 --> 00:17:43,430 level, because we're assuming it goes on forever. 298 00:17:43,430 --> 00:17:45,420 So it's up here somewhere. 299 00:17:45,420 --> 00:17:47,400 There's kT. 300 00:17:47,400 --> 00:17:48,320 So what happens? 301 00:17:48,320 --> 00:17:52,220 Well if you raise the temperature, there still are 302 00:17:52,220 --> 00:17:54,970 higher lying levels that can be populated, and that will 303 00:17:54,970 --> 00:17:56,190 get populated. 304 00:17:56,190 --> 00:17:59,210 So du/dT isn't going to be zero in the high temperature 305 00:17:59,210 --> 00:18:00,550 limit, in this case. 306 00:18:00,550 --> 00:18:04,660 But it stops changing at some point. 307 00:18:04,660 --> 00:18:08,060 Because nothing's very different about this than 308 00:18:08,060 --> 00:18:12,480 having kT be, let's say, up here or up here. 309 00:18:12,480 --> 00:18:15,580 So in the high temperature limit, yes, the energy does 310 00:18:15,580 --> 00:18:17,920 change with temperature. 311 00:18:17,920 --> 00:18:22,270 But it changes in the same way at any temperature. 312 00:18:22,270 --> 00:18:25,950 In other words, the energy is just linearly varying with 313 00:18:25,950 --> 00:18:26,320 temperature. 314 00:18:26,320 --> 00:18:30,910 And the heat capacity is a constant. du/dT doesn't change 315 00:18:30,910 --> 00:18:34,880 anymore, once you're in the high temperature limit. 316 00:18:34,880 --> 00:18:37,760 Now without me writing any expression on the board, 317 00:18:37,760 --> 00:18:40,510 what's the low temperature limit of the heat capacity 318 00:18:40,510 --> 00:18:45,240 going to be in this case? 319 00:18:45,240 --> 00:18:46,480 Going to be on the exam. 320 00:18:46,480 --> 00:18:48,490 STUDENT: Zero? 321 00:18:48,490 --> 00:18:51,690 PROFESSOR: Zero, that's still going to be the same, right? 322 00:18:51,690 --> 00:18:55,650 Put kT way down here. 323 00:18:55,650 --> 00:18:57,780 That's just like this, right? 324 00:18:57,780 --> 00:19:01,340 Not near enough energy to populate even though lowest, 325 00:19:01,340 --> 00:19:04,320 you know, the first level above the ground state. 326 00:19:04,320 --> 00:19:06,800 Change the temperature a little bit, still not enough 327 00:19:06,800 --> 00:19:08,690 thermal energy to get up there. 328 00:19:08,690 --> 00:19:11,020 So the energy doesn't change with temperature in that low 329 00:19:11,020 --> 00:19:12,790 temperature limit. 330 00:19:12,790 --> 00:19:15,790 So you can immediately see what's going to happen at low 331 00:19:15,790 --> 00:19:33,440 temperature. 332 00:19:33,440 --> 00:19:35,020 Any questions? 333 00:19:35,020 --> 00:19:35,230 Yeah? 334 00:19:35,230 --> 00:19:42,902 STUDENT: [UNINTELLIGIBLE] will still be n k t, and 335 00:19:42,902 --> 00:19:43,320 then it's just -- 336 00:19:43,320 --> 00:19:43,760 [UNINTELLIGIBLE] 337 00:19:43,760 --> 00:19:45,425 PROFESSOR: So, of course, that's a 338 00:19:45,425 --> 00:19:48,300 limiting case here, right? 339 00:19:48,300 --> 00:19:51,640 That happened because in the limit of high temperature, 340 00:19:51,640 --> 00:19:54,000 then this exponent is really small, right? 341 00:19:54,000 --> 00:19:56,770 So then you can Taylor expand it. 342 00:19:56,770 --> 00:19:59,120 In the limit of low temperature, 343 00:19:59,120 --> 00:20:02,310 this is really big. 344 00:20:02,310 --> 00:20:04,770 This is nearly zero. 345 00:20:04,770 --> 00:20:09,050 So e epsilon zero is much bigger than kT in that case. 346 00:20:09,050 --> 00:20:10,170 This is big. 347 00:20:10,170 --> 00:20:13,070 This is negligible, right? 348 00:20:13,070 --> 00:20:14,370 Oh. 349 00:20:14,370 --> 00:20:19,470 Wait a minute. 350 00:20:19,470 --> 00:20:24,800 Something's making me real unhappy. 351 00:20:24,800 --> 00:20:42,890 I can't have this right. 352 00:20:42,890 --> 00:20:43,790 Ah. 353 00:20:43,790 --> 00:20:46,160 Yes. 354 00:20:46,160 --> 00:20:50,480 It needs to be zero, is what it needs to be. 355 00:20:50,480 --> 00:20:51,110 What am I thinking? 356 00:20:51,110 --> 00:20:51,430 Of course. 357 00:20:51,430 --> 00:20:52,560 It's one over a huge number. 358 00:20:52,560 --> 00:20:54,310 It's zero. 359 00:20:54,310 --> 00:20:55,480 OK. 360 00:20:55,480 --> 00:20:58,260 So of course you can't expanded it. 361 00:20:58,260 --> 00:21:01,000 It's just, this is much bigger than this. 362 00:21:01,000 --> 00:21:03,830 This is an enormous number at that point. 363 00:21:03,830 --> 00:21:07,220 So in other words, what it's saying is the configurational 364 00:21:07,220 --> 00:21:11,720 energy is zero, because everything is stuck in the 365 00:21:11,720 --> 00:21:13,660 ground state. 366 00:21:13,660 --> 00:21:15,960 And it stays zero if you vary the temperature. 367 00:21:15,960 --> 00:21:19,860 So you get the zero limiting value for the heat capacity, 368 00:21:19,860 --> 00:21:23,910 and the energy itself is also zero. 369 00:21:23,910 --> 00:21:26,490 Other questions? 370 00:21:26,490 --> 00:21:32,930 OK. 371 00:21:32,930 --> 00:21:37,030 Let me just make a few comments about the entropy. 372 00:21:37,030 --> 00:21:42,120 So I didn't write it out before. 373 00:21:42,120 --> 00:21:47,040 And I'm tempted not to do it again, but I 374 00:21:47,040 --> 00:21:57,260 suppose I'll do it. 375 00:21:57,260 --> 00:22:02,320 So it turns out to be Nk, minus log of one minus e to 376 00:22:02,320 --> 00:22:07,710 the minus epsilon zero over kT. 377 00:22:07,710 --> 00:22:10,570 Plus epsilon zero over kT. 378 00:22:10,570 --> 00:22:19,930 Over e to the epsilon zero over kT, minus one. 379 00:22:19,930 --> 00:22:24,320 And this comes from combining the terms for A and u. 380 00:22:24,320 --> 00:22:36,840 It comes from minus A over T. Plus u over T. And what I want 381 00:22:36,840 --> 00:22:41,040 to do is look at its limiting cases also. 382 00:22:41,040 --> 00:22:45,120 In particular, what happens here in the limit of high 383 00:22:45,120 --> 00:22:58,560 temperature. 384 00:22:58,560 --> 00:23:07,620 And what happens turns out to be k log kT over epsilon zero 385 00:23:07,620 --> 00:23:10,450 to the Nth power. 386 00:23:10,450 --> 00:23:16,010 Or put the N over here. 387 00:23:16,010 --> 00:23:19,910 OK. 388 00:23:19,910 --> 00:23:24,210 And what this is telling us about is the number of 389 00:23:24,210 --> 00:23:26,360 available states. 390 00:23:26,360 --> 00:23:31,140 Roughly, how many states are there that are accessible to 391 00:23:31,140 --> 00:23:34,860 the system at some temperature. 392 00:23:34,860 --> 00:23:38,470 So in other words, think of it as -- let's put the N back 393 00:23:38,470 --> 00:23:43,270 there. k log of kT over epsilon zero to the Nth power. 394 00:23:43,270 --> 00:23:47,260 And think of it as k log capital omega. 395 00:23:47,260 --> 00:23:49,880 Where that would be the degeneracy. 396 00:23:49,880 --> 00:23:52,810 Now all the states are in equal energy, but remember for 397 00:23:52,810 --> 00:23:57,070 the whole system, remember we discussed this before. 398 00:23:57,070 --> 00:24:00,960 How you have a very, very narrow distribution of system 399 00:24:00,960 --> 00:24:05,360 energy states at equilibrium. 400 00:24:05,360 --> 00:24:08,460 So you can think of this as the degeneracy of the system 401 00:24:08,460 --> 00:24:11,230 states that are actually going to exist at a particular 402 00:24:11,230 --> 00:24:16,090 temperature. 403 00:24:16,090 --> 00:24:28,670 So if we look at the limiting value of the partition 404 00:24:28,670 --> 00:24:34,780 function, it's just kT over omega. 405 00:24:34,780 --> 00:24:44,660 Or the same thing for capital Q. kT over 406 00:24:44,660 --> 00:24:46,110 epsilon zero, sorry. 407 00:24:46,110 --> 00:24:48,750 To the Nth power. 408 00:24:48,750 --> 00:24:51,140 So you have a very simple expression. 409 00:24:51,140 --> 00:24:54,500 And so again, what this is doing, is it's giving us a 410 00:24:54,500 --> 00:24:57,350 measure of about how many states are available. 411 00:24:57,350 --> 00:25:00,290 And it's particularly informative to look at that 412 00:25:00,290 --> 00:25:03,180 for the molecular partition function. 413 00:25:03,180 --> 00:25:13,110 What it's telling us is, if I look at kT over epsilon. 414 00:25:13,110 --> 00:25:15,160 And these levels, remember, are evenly spaced. 415 00:25:15,160 --> 00:25:19,730 So here's epsilon zero, two epsilon zero, three epsilon 416 00:25:19,730 --> 00:25:22,730 zero, and so on. 417 00:25:22,730 --> 00:25:28,060 It says, you know, if kT is about ten times bigger than 418 00:25:28,060 --> 00:25:30,780 epsilon zero. 419 00:25:30,780 --> 00:25:34,460 So this is ten. 420 00:25:34,460 --> 00:25:38,760 It's telling us, roughly how many states does the system 421 00:25:38,760 --> 00:25:40,840 have thermal access to. 422 00:25:40,840 --> 00:25:42,170 The molecular state. 423 00:25:42,170 --> 00:25:45,320 It has about ten states. 424 00:25:45,320 --> 00:25:50,920 So going over to our picture of the set of structures, you 425 00:25:50,920 --> 00:25:54,630 could have anywhere up to about ten bonds broken. 426 00:25:54,630 --> 00:25:59,520 Now, the individual molecules are going to be in a whole 427 00:25:59,520 --> 00:26:01,480 range of states. 428 00:26:01,480 --> 00:26:03,450 Some of them will have fewer than that, and some of them 429 00:26:03,450 --> 00:26:06,690 will have more than that number of bonds broken. 430 00:26:06,690 --> 00:26:12,040 But on average, it's going to be about that number. 431 00:26:12,040 --> 00:26:15,870 And then, if you look at the whole system, the number of 432 00:26:15,870 --> 00:26:19,510 states available, of course it's astronomical. 433 00:26:19,510 --> 00:26:23,995 Because you have to take each molecule, and say, well it 434 00:26:23,995 --> 00:26:26,540 could be in any one of something on the 435 00:26:26,540 --> 00:26:28,080 order of ten states. 436 00:26:28,080 --> 00:26:31,060 And then whole set of N other molecules can be in the states 437 00:26:31,060 --> 00:26:33,400 they might be in. 438 00:26:33,400 --> 00:26:34,790 Change the first one, and do it again. 439 00:26:34,790 --> 00:26:40,540 So you have this astronomical number of system states. 440 00:26:40,540 --> 00:26:44,840 But remember, like we discussed once before, it'll 441 00:26:44,840 --> 00:26:49,000 turn out that although the individual molecule states 442 00:26:49,000 --> 00:26:54,800 vary considerably with energy, the system states, which are 443 00:26:54,800 --> 00:26:57,350 averaging over some astronomical number of 444 00:26:57,350 --> 00:26:59,150 molecules, where capital N is something 445 00:26:59,150 --> 00:27:01,460 like ten to the 24th. 446 00:27:01,460 --> 00:27:06,490 Once you average over that many individual molecules, you 447 00:27:06,490 --> 00:27:08,540 find that there's very, very little 448 00:27:08,540 --> 00:27:10,110 fluctuation in the energy. 449 00:27:10,110 --> 00:27:12,000 In the system energy. 450 00:27:12,000 --> 00:27:16,050 The individual molecule energies vary considerably. 451 00:27:16,050 --> 00:27:21,370 Realistically the variation of the molecular energies -- that 452 00:27:21,370 --> 00:27:23,200 variation is going to be comparable 453 00:27:23,200 --> 00:27:25,210 to the energy itself. 454 00:27:25,210 --> 00:27:27,260 To the average energy. 455 00:27:27,260 --> 00:27:30,690 So if the average energy is roughly you know, the ten 456 00:27:30,690 --> 00:27:33,550 epsilon zero, you say okay, how much might it vary? 457 00:27:33,550 --> 00:27:35,550 Well, there are going to be some molecules that have only 458 00:27:35,550 --> 00:27:37,240 a couple of bonds broken, and some that might 459 00:27:37,240 --> 00:27:39,070 have 20 bonds broken. 460 00:27:39,070 --> 00:27:42,120 The variation will be on the same order of magnitude as the 461 00:27:42,120 --> 00:27:44,550 average itself. 462 00:27:44,550 --> 00:27:49,450 But then you average over capital N of them. 463 00:27:49,450 --> 00:27:51,840 And then you immediately discover that there's very, 464 00:27:51,840 --> 00:27:54,960 very, very little variation. 465 00:27:54,960 --> 00:28:03,870 And in particular, what happens then, is, you know, 466 00:28:03,870 --> 00:28:14,970 for molecular average energy, epsilon zero, and variance, or 467 00:28:14,970 --> 00:28:22,950 standard deviation about the same magnitude. 468 00:28:22,950 --> 00:28:29,830 System average energy is u, and it's going to 469 00:28:29,830 --> 00:28:31,940 be capital N times. 470 00:28:31,940 --> 00:28:35,490 Well let's say that -- average -- 471 00:28:35,490 --> 00:28:38,350 Oh sorry, let me not put the zero There. 472 00:28:38,350 --> 00:28:41,650 It's just average energy. 473 00:28:41,650 --> 00:28:44,620 And the system average energy is just N times 474 00:28:44,620 --> 00:28:48,820 the molecular energy. 475 00:28:48,820 --> 00:28:56,180 But the system variance is going to be on the order of 476 00:28:56,180 --> 00:29:02,300 the square root of N times epsilon. 477 00:29:02,300 --> 00:29:03,990 If you've done statistics, then you've seen 478 00:29:03,990 --> 00:29:05,050 that sort of result. 479 00:29:05,050 --> 00:29:09,740 You do a bunch of samplings a capital N number of samplings, 480 00:29:09,740 --> 00:29:12,180 and the variation looks like the square 481 00:29:12,180 --> 00:29:15,140 root of that number. 482 00:29:15,140 --> 00:29:17,240 So what ends up happening then, if you look at the 483 00:29:17,240 --> 00:29:19,900 relative variation, you might look and then say, well it's 484 00:29:19,900 --> 00:29:20,440 pretty big. 485 00:29:20,440 --> 00:29:21,720 This is ten to the 12, right? 486 00:29:21,720 --> 00:29:23,860 It's still a huge variance. 487 00:29:23,860 --> 00:29:47,140 But let's compare it to the average. 488 00:29:47,140 --> 00:29:50,150 So this is ten to the 24th. 489 00:29:50,150 --> 00:29:51,160 This is ten to the 12th. 490 00:29:51,160 --> 00:29:54,820 It's on the order of ten to the minus 12. 491 00:29:54,820 --> 00:29:57,840 An incredibly tiny fractional variation 492 00:29:57,840 --> 00:30:00,930 in the system energy. 493 00:30:00,930 --> 00:30:01,940 You could never -- 494 00:30:01,940 --> 00:30:04,540 There would be no practical way to measure it. 495 00:30:04,540 --> 00:30:06,800 And of course that is consistent with 496 00:30:06,800 --> 00:30:07,550 what you would expect. 497 00:30:07,550 --> 00:30:12,150 If you say let's measure the configurational energy of a 498 00:30:12,150 --> 00:30:15,090 bunch of molecules in a liquid solution, or molecules in a 499 00:30:15,090 --> 00:30:16,320 gas floating around. 500 00:30:16,320 --> 00:30:21,960 And it's a mole of them, that total average energy is not 501 00:30:21,960 --> 00:30:24,160 going to fluctuate significantly, even though 502 00:30:24,160 --> 00:30:27,570 individual molecules that you pick out of that whole system 503 00:30:27,570 --> 00:30:30,700 might have quite widely varying energies. 504 00:30:30,700 --> 00:30:34,250 And that's the point. 505 00:30:34,250 --> 00:30:37,740 So it's an incredibly small variation. 506 00:30:37,740 --> 00:30:39,500 Now you can derive this. 507 00:30:39,500 --> 00:30:40,870 I didn't derive this, of course. 508 00:30:40,870 --> 00:30:43,080 I asserted that this is the case. 509 00:30:43,080 --> 00:30:45,540 And it's probably familiar to some of you if you've seen 510 00:30:45,540 --> 00:30:47,270 some statistics. 511 00:30:47,270 --> 00:30:49,710 But the way you would derive it is, you know in addition to 512 00:30:49,710 --> 00:30:52,700 calculating the average energy, the average of E, you 513 00:30:52,700 --> 00:30:55,340 can also calculate the average of the energy squared. 514 00:30:55,340 --> 00:30:58,980 And you can calculate the standard deviation that way. 515 00:30:58,980 --> 00:31:01,590 You calculate the average of E squared. 516 00:31:01,590 --> 00:31:05,670 And then you minus the average of E, the quantity squared. 517 00:31:05,670 --> 00:31:06,390 Take the square root. 518 00:31:06,390 --> 00:31:09,180 That's your root mean square. 519 00:31:09,180 --> 00:31:13,040 And that's what leads to this result. 520 00:31:13,040 --> 00:31:17,400 So it's pretty straight forward to calculate it also. 521 00:31:17,400 --> 00:31:17,910 Alright. 522 00:31:17,910 --> 00:31:21,230 So any questions about just the extent of variation of the 523 00:31:21,230 --> 00:31:27,370 individual molecule energies or the system energies? 524 00:31:27,370 --> 00:31:28,330 Okay. 525 00:31:28,330 --> 00:31:42,250 Then, now what I'd like to do is look at another kind of 526 00:31:42,250 --> 00:31:46,450 energy that's going to turn out to have exactly the same 527 00:31:46,450 --> 00:31:48,650 set of levels that we just derived 528 00:31:48,650 --> 00:31:50,480 from this simple model. 529 00:31:50,480 --> 00:31:52,670 And some of you may have seen this before. 530 00:31:52,670 --> 00:31:55,010 Many of you may not have. 531 00:31:55,010 --> 00:31:57,350 And for right now, I'll just assert it. 532 00:31:57,350 --> 00:32:00,360 But it will illustrate why this is so useful. 533 00:32:00,360 --> 00:32:06,270 It turns out that if I've got molecular vibrations -- you 534 00:32:06,270 --> 00:32:08,310 know, there's nitrogen and oxygen in the air. 535 00:32:08,310 --> 00:32:11,640 If I look at those nitrogen vibrational energy levels. 536 00:32:11,640 --> 00:32:14,930 Or oxygen energy levels. 537 00:32:14,930 --> 00:32:17,280 They look like this. 538 00:32:17,280 --> 00:32:23,220 Evenly spaced, non degenerate energy levels. 539 00:32:23,220 --> 00:32:26,730 So the model that we've constructed here, based on 540 00:32:26,730 --> 00:32:34,230 this simple two-chain polymer, actually gives us a set of 541 00:32:34,230 --> 00:32:38,180 energy levels that maps directly onto the vibrational 542 00:32:38,180 --> 00:32:40,700 energy levels of a molecule. 543 00:32:40,700 --> 00:32:50,890 So all the results that we've just seen apply, not just for 544 00:32:50,890 --> 00:32:57,410 conformations of a polymer, but for 545 00:32:57,410 --> 00:32:59,220 vibrations of a molecule. 546 00:32:59,220 --> 00:33:04,200 So everywhere where it says configurational, you can just 547 00:33:04,200 --> 00:33:08,900 write in vibrational. 548 00:33:08,900 --> 00:33:11,040 And you'll still be right. because of course, what 549 00:33:11,040 --> 00:33:15,000 matters is what are the states that are available to the 550 00:33:15,000 --> 00:33:17,130 system, and what are their energies? 551 00:33:17,130 --> 00:33:19,880 After that, the formalism of statistical mechanics takes 552 00:33:19,880 --> 00:33:22,140 over, and calculates partition functions and 553 00:33:22,140 --> 00:33:24,610 thermodynamic functions. 554 00:33:24,610 --> 00:33:28,380 The input into that is states. 555 00:33:28,380 --> 00:33:29,880 And their energies. 556 00:33:29,880 --> 00:33:36,200 Well, we have the exact same set of states and energies. 557 00:33:36,200 --> 00:33:41,630 So we immediately arrive at a super important case and its 558 00:33:41,630 --> 00:33:46,030 results for molecular vibrations. 559 00:33:46,030 --> 00:33:51,590 And not only molecular vibrations, but vibrations of 560 00:33:51,590 --> 00:33:57,000 a crystal lattice. 561 00:33:57,000 --> 00:34:00,440 Acoustic vibrations of a glass. 562 00:34:00,440 --> 00:34:02,380 Or even a liquid. 563 00:34:02,380 --> 00:34:07,140 So there's an enormously wide ranging set of results that 564 00:34:07,140 --> 00:34:13,020 we've derived by starting at this simple picture. 565 00:34:13,020 --> 00:34:15,670 And when you take quantum mechanics, you'll see that 566 00:34:15,670 --> 00:34:19,250 indeed you do get this set of evenly spaced energy levels. 567 00:34:19,250 --> 00:34:21,530 You may well have seen the results before, and you'll see 568 00:34:21,530 --> 00:34:24,100 it derived there. 569 00:34:24,100 --> 00:34:25,180 OK. 570 00:34:25,180 --> 00:34:31,050 Given that, then I want to talk a little bit further 571 00:34:31,050 --> 00:34:33,620 about the heat capacity. 572 00:34:33,620 --> 00:34:39,360 So we've seen the limiting cases for the heat capacity. 573 00:34:39,360 --> 00:34:42,380 Namely, the low temperature limit is zero. 574 00:34:42,380 --> 00:34:45,150 That's the case whenever you have quantized levels. 575 00:34:45,150 --> 00:34:49,340 You can always get kT lower by far than the 576 00:34:49,340 --> 00:34:52,060 lowest excited state. 577 00:34:52,060 --> 00:34:54,550 So that everything is stuck in the ground state. 578 00:34:54,550 --> 00:34:56,770 How low that is depends on how far apart 579 00:34:56,770 --> 00:34:57,720 the states are spaced. 580 00:34:57,720 --> 00:35:00,350 But there's got to be some temperature somewhere that's 581 00:35:00,350 --> 00:35:02,490 down there. 582 00:35:02,490 --> 00:35:16,310 So we've seen the heat capacity limiting cases. 583 00:35:16,310 --> 00:35:36,670 Let me just sort of draw them again. 584 00:35:36,670 --> 00:35:41,200 So here's kT much less than epsilon zero. 585 00:35:41,200 --> 00:35:46,750 And up here, kT is much bigger than epsilon zero. 586 00:35:46,750 --> 00:35:50,510 Now I'm just going to sketch the full temperature 587 00:35:50,510 --> 00:35:50,990 dependence. 588 00:35:50,990 --> 00:35:53,130 In other words, I'm going to connect those two limits that 589 00:35:53,130 --> 00:35:56,130 we've seen. 590 00:35:56,130 --> 00:36:14,990 So here's what that looks like. 591 00:36:14,990 --> 00:36:20,600 And I'll write it as vibrational. 592 00:36:20,600 --> 00:36:25,200 So we know it's got to be zero at the low temperature limit. 593 00:36:25,200 --> 00:36:30,390 And we know that it's this constant value in the high 594 00:36:30,390 --> 00:36:31,550 temperature limit. 595 00:36:31,550 --> 00:36:33,110 It's Nk. 596 00:36:33,110 --> 00:36:40,790 And in some way, it smoothly connects. 597 00:36:40,790 --> 00:36:44,470 And if you look at the actual scale, here -- so here's kT 598 00:36:44,470 --> 00:36:47,340 over epsilon zero. 599 00:36:47,340 --> 00:36:48,920 Here's the limit of low temperature. 600 00:36:48,920 --> 00:36:52,910 Actually, you don't have to be very high above epsilon zero 601 00:36:52,910 --> 00:36:55,430 to be in in this high temperature limit. 602 00:36:55,430 --> 00:37:00,360 So if you look at the plot that's on your notes, this is 603 00:37:00,360 --> 00:37:02,790 already -- it's not quite leveling off precisely. 604 00:37:02,790 --> 00:37:04,940 But it's already pretty close. 605 00:37:04,940 --> 00:37:10,290 When kT is just two times epsilon zero. 606 00:37:10,290 --> 00:37:22,300 It's already almost at the limiting case. 607 00:37:22,300 --> 00:37:24,960 Now this has tremendous significance. 608 00:37:24,960 --> 00:37:30,230 So a long time before quantum mechanics was developed, 609 00:37:30,230 --> 00:37:31,760 people made measurements of the heat 610 00:37:31,760 --> 00:37:34,450 capacities of materials. 611 00:37:34,450 --> 00:37:37,830 And they were familiar with the fact that when you got to 612 00:37:37,830 --> 00:37:41,580 ordinary temperature -- room temperature, the heat capacity 613 00:37:41,580 --> 00:37:45,960 was temperature independent. 614 00:37:45,960 --> 00:37:48,500 And that was understandable for reasons 615 00:37:48,500 --> 00:37:49,990 we'll discuss shortly. 616 00:37:49,990 --> 00:37:55,150 Basically, in that case, the heat capacity -- it's not just 617 00:37:55,150 --> 00:37:58,720 Nk for one vibrational mode. 618 00:37:58,720 --> 00:38:01,600 Of course, that's what I describe for a single 619 00:38:01,600 --> 00:38:03,710 vibrational mode of, say, a molecule. 620 00:38:03,710 --> 00:38:06,120 If you have a crystal lattice with N atoms in it. 621 00:38:06,120 --> 00:38:08,440 Let's say it's an atomic crystal. 622 00:38:08,440 --> 00:38:10,610 Each atom has three degrees of freedom. 623 00:38:10,610 --> 00:38:14,550 It can move in each of three independent directions. 624 00:38:14,550 --> 00:38:15,930 And there are N of those atoms. 625 00:38:15,930 --> 00:38:18,080 There are 3N total degrees of freedom. 626 00:38:18,080 --> 00:38:23,570 That's how many vibrations the lattice has. 627 00:38:23,570 --> 00:38:29,070 So in fact, you have to multiply this by 3N. 628 00:38:29,070 --> 00:38:34,580 So what would be seen is, you'd have 3R, the gas 629 00:38:34,580 --> 00:38:38,680 constant, per mole, in other words. 630 00:38:38,680 --> 00:38:44,550 And if you looked at Cv over 3R, then this 631 00:38:44,550 --> 00:38:50,160 value would be one. 632 00:38:50,160 --> 00:38:53,290 Very useful to see that. 633 00:38:53,290 --> 00:38:55,250 To figure that out. 634 00:38:55,250 --> 00:38:56,970 And people understood it. 635 00:38:56,970 --> 00:39:03,190 What people didn't understand is this stuff. 636 00:39:03,190 --> 00:39:06,740 Why did it do that? 637 00:39:06,740 --> 00:39:08,830 That was a great mystery. 638 00:39:08,830 --> 00:39:13,110 Why did the heat capacity go to zero at low temperature? 639 00:39:13,110 --> 00:39:15,870 And the reason they didn't understand it is because the 640 00:39:15,870 --> 00:39:18,980 model for vibration was really simple. 641 00:39:18,980 --> 00:39:21,960 The lattice is a bunch of masses and springs. 642 00:39:21,960 --> 00:39:25,060 I know the vibrational energy. 643 00:39:25,060 --> 00:39:26,810 And I can calculate it. 644 00:39:26,810 --> 00:39:31,890 That is, I know the classical vibrational energy. 645 00:39:31,890 --> 00:39:35,880 Remember, the reason it goes to zero is this. 646 00:39:35,880 --> 00:39:39,150 It's all because of the fact that the energy levels are 647 00:39:39,150 --> 00:39:43,460 discrete, quantized levels, with gaps in between them. 648 00:39:43,460 --> 00:39:47,730 If I've got classical mechanics, then that means 649 00:39:47,730 --> 00:39:48,850 it's vibrational energy. 650 00:39:48,850 --> 00:39:51,770 The energy just gets bigger if the amplitude get bigger. 651 00:39:51,770 --> 00:39:54,420 It's not discrete, it's continuous. 652 00:39:54,420 --> 00:39:56,690 There are always energies in there. 653 00:39:56,690 --> 00:39:59,790 In that case, there's never a situation like this. 654 00:39:59,790 --> 00:40:03,600 Where kT is lower than the first available excited level, 655 00:40:03,600 --> 00:40:04,650 and everything's in the ground state. 656 00:40:04,650 --> 00:40:06,830 That never happens in classical mechanics, because 657 00:40:06,830 --> 00:40:09,390 there are always levels there. 658 00:40:09,390 --> 00:40:13,250 But it does happen in quantum mechanics. 659 00:40:13,250 --> 00:40:17,910 And Einstein recognized that this was a way to explain this 660 00:40:17,910 --> 00:40:20,800 low temperature limiting heat capacity. 661 00:40:20,800 --> 00:40:24,200 So actually if you look at early development of quantum 662 00:40:24,200 --> 00:40:25,990 mechanics, really it was all predicated 663 00:40:25,990 --> 00:40:28,780 on statistical mechanics. 664 00:40:28,780 --> 00:40:31,790 And the idea that, well, that you could then do the 665 00:40:31,790 --> 00:40:35,400 statistical mechanics with quantized levels, just the way 666 00:40:35,400 --> 00:40:36,260 we've done it. 667 00:40:36,260 --> 00:40:38,070 And what you immediately discover is, gee, 668 00:40:38,070 --> 00:40:39,330 it all makes sense. 669 00:40:39,330 --> 00:40:43,100 You go to low temperature, everything's stunk down here. 670 00:40:43,100 --> 00:40:45,160 And suddenly you've got zero heat capacity, because you 671 00:40:45,160 --> 00:40:46,890 changed the temperature a little bit, and everything is 672 00:40:46,890 --> 00:40:51,680 still stuck back here. 673 00:40:51,680 --> 00:40:56,800 So that's the vibrational heat capacity of a solid. 674 00:40:56,800 --> 00:40:57,890 That's what it looks like. 675 00:40:57,890 --> 00:41:01,230 And at moderate temperature, this doesn't have to be very 676 00:41:01,230 --> 00:41:03,280 much bigger than epsilon zero. 677 00:41:03,280 --> 00:41:05,410 You're already in, essentially, the limit. 678 00:41:05,410 --> 00:41:06,520 The high temperature limit. 679 00:41:06,520 --> 00:41:12,140 And now for molecules, that limit isn't usually reached at 680 00:41:12,140 --> 00:41:13,640 room temperature. 681 00:41:13,640 --> 00:41:20,570 Here's a kind of calibration. kT at room temperature is 682 00:41:20,570 --> 00:41:27,480 about equal to 200 wave numbers. 683 00:41:27,480 --> 00:41:32,020 So molecular vibrations, you know, you've taken IR spectra 684 00:41:32,020 --> 00:41:36,190 they're typically on the order of 1,000 wave numbers or so. 685 00:41:36,190 --> 00:41:40,540 kT isn't bigger than the vibrational energy. 686 00:41:40,540 --> 00:41:43,330 But a crystal lattice, you know the vibrations are the 687 00:41:43,330 --> 00:41:46,560 acoustics vibrations. 688 00:41:46,560 --> 00:41:48,770 And those are much lower in frequency. 689 00:41:48,770 --> 00:41:52,590 If you have an atomic crystal, it just has the sound 690 00:41:52,590 --> 00:41:54,750 vibrations at all the different 691 00:41:54,750 --> 00:41:56,830 wavelengths that are available. 692 00:41:56,830 --> 00:41:58,890 They're never very high. 693 00:41:58,890 --> 00:42:01,890 So it's easy to get into the high temperature 694 00:42:01,890 --> 00:42:04,700 limit, in that case. 695 00:42:04,700 --> 00:42:06,660 Where you basically see a temperature 696 00:42:06,660 --> 00:42:14,720 independent heat capacity. 697 00:42:14,720 --> 00:42:15,720 OK. 698 00:42:15,720 --> 00:42:18,440 By the way, this was actually used commonly to determine the 699 00:42:18,440 --> 00:42:20,860 molecular weights of molecular crystals. 700 00:42:20,860 --> 00:42:25,060 Because you've got a factor of the number of moles in there. 701 00:42:25,060 --> 00:42:26,670 If you ask how big is the heat capacity? 702 00:42:26,670 --> 00:42:28,270 Well it does depend on how many moles of 703 00:42:28,270 --> 00:42:29,590 material you have. 704 00:42:29,590 --> 00:42:31,360 Because it depends on how many atoms. 705 00:42:31,360 --> 00:42:34,130 That control how many modes there are in the crystal. 706 00:42:34,130 --> 00:42:35,510 How many vibrational modes. 707 00:42:35,510 --> 00:42:39,050 Because each atom has 3N degrees of freedom. 708 00:42:39,050 --> 00:42:41,830 So if you just weigh the whole crystal, and then you measure 709 00:42:41,830 --> 00:42:44,390 the heat capacity, you know how many moles there are, and 710 00:42:44,390 --> 00:42:45,830 now you know the weight, you can figure out 711 00:42:45,830 --> 00:42:49,660 the molecular weight. 712 00:42:49,660 --> 00:42:53,150 OK. 713 00:42:53,150 --> 00:42:54,860 So that's the low temperature limit. 714 00:42:54,860 --> 00:42:57,670 And now I want to talk a little bit further about the 715 00:42:57,670 --> 00:42:59,670 high temperature limit. 716 00:42:59,670 --> 00:43:04,510 And in particular, I want to talk about both the heat 717 00:43:04,510 --> 00:43:07,510 capacity and the energy that we've seen. 718 00:43:07,510 --> 00:43:11,060 Namely, this thing. 719 00:43:11,060 --> 00:43:14,070 Of course it's really the same result for the energy and the 720 00:43:14,070 --> 00:43:14,900 heat capacity. 721 00:43:14,900 --> 00:43:18,720 They're obviously intimately connected. 722 00:43:18,720 --> 00:43:22,340 So these are the high temperature limits. 723 00:43:22,340 --> 00:43:32,470 Now it turns out that that high temperature limit doesn't 724 00:43:32,470 --> 00:43:35,150 just obtain for vibrations. 725 00:43:35,150 --> 00:43:39,270 But it's also the case for molecular rotations, 726 00:43:39,270 --> 00:43:41,680 translations. 727 00:43:41,680 --> 00:43:46,860 All these low energy degrees of freedom. 728 00:43:46,860 --> 00:43:50,050 So let's see why. 729 00:43:50,050 --> 00:43:51,880 It has a name, that result. 730 00:43:51,880 --> 00:43:59,980 It's called the equipartition of energy. 731 00:43:59,980 --> 00:44:06,760 Sometimes it's called the classical equipartition of 732 00:44:06,760 --> 00:44:10,970 energy theorem. 733 00:44:10,970 --> 00:44:22,460 And what it says is one half kT per degree of freedom 734 00:44:22,460 --> 00:44:31,040 equals energy in the high T limit. 735 00:44:31,040 --> 00:44:38,440 And in particular, for translation, so E 736 00:44:38,440 --> 00:44:44,390 translational is 3/2. 737 00:44:44,390 --> 00:44:49,900 Well, NkT for N atoms. 738 00:44:49,900 --> 00:44:55,250 Or molecules. 739 00:44:55,250 --> 00:45:00,890 E rotational is, now it depends how many rotational 740 00:45:00,890 --> 00:45:02,770 degrees of freedom there are. 741 00:45:02,770 --> 00:45:05,170 If I've got a linear molecule, there are only two. 742 00:45:05,170 --> 00:45:08,320 It can rotate this way, and it can rotate this way. 743 00:45:08,320 --> 00:45:11,140 If I've got a non linear molecule, parts all over the 744 00:45:11,140 --> 00:45:14,410 place, it has three unique axes. 745 00:45:14,410 --> 00:45:15,890 It can also spin this way. 746 00:45:15,890 --> 00:45:17,880 And that's a rotational degree of freedom. 747 00:45:17,880 --> 00:45:19,510 Of course the linear molecule wasn't like that. 748 00:45:19,510 --> 00:45:23,340 Nothing is moving when you do that. 749 00:45:23,340 --> 00:45:30,050 So it's either NkT, linear. 750 00:45:30,050 --> 00:45:34,120 Or 3/2 NkT. 751 00:45:34,120 --> 00:45:34,770 Non linear. 752 00:45:34,770 --> 00:45:38,560 Great. 753 00:45:38,560 --> 00:45:49,500 And then E vibrational is NkT per vibrational mode. 754 00:45:49,500 --> 00:45:52,420 That's because vibrational energy is potential and 755 00:45:52,420 --> 00:45:56,390 kinetic energy, and it's 1/2 kT each. 756 00:45:56,390 --> 00:45:59,510 Why does all that stuff happen? 757 00:45:59,510 --> 00:46:10,040 Turns out, you can see why pretty easily. 758 00:46:10,040 --> 00:46:12,580 All those degrees of freedom. 759 00:46:12,580 --> 00:46:20,710 Those classical degrees of freedom. 760 00:46:20,710 --> 00:46:31,400 If you say, let's write out an expression for the energy. 761 00:46:31,400 --> 00:46:35,910 Classical. 762 00:46:35,910 --> 00:46:39,760 You know it's 1/2 m v squared. 763 00:46:39,760 --> 00:46:42,590 Or it's 1/2 k x squared. 764 00:46:42,590 --> 00:46:51,080 Or for rotational energy it's 1/2 I omega squared. 765 00:46:51,080 --> 00:46:52,850 You see a functional form emerging here? 766 00:46:52,850 --> 00:46:56,380 That looks similar in all these cases? 767 00:46:56,380 --> 00:46:59,070 You know, 1/2 times some constant, times 768 00:46:59,070 --> 00:47:03,130 some variable squared. 769 00:47:03,130 --> 00:47:11,820 Well, now let's look at our expressions for the energy. 770 00:47:11,820 --> 00:47:16,260 Average energy. 771 00:47:16,260 --> 00:47:21,650 We're going to sum over all the energies of epsilon i, e 772 00:47:21,650 --> 00:47:25,050 to the minus Ei over kT. 773 00:47:25,050 --> 00:47:30,420 Over sum over i, e to the minus Ei over kT. 774 00:47:30,420 --> 00:47:34,960 That's just how we originally derived the average of energy. 775 00:47:34,960 --> 00:47:37,800 In other words, it's the probability of each state 776 00:47:37,800 --> 00:47:39,340 times the energy of that state. 777 00:47:39,340 --> 00:47:44,320 Summed up over all the states. 778 00:47:44,320 --> 00:47:47,560 Well, now we have an expression for the energy. 779 00:47:47,560 --> 00:47:48,700 It's one of these things. 780 00:47:48,700 --> 00:47:53,670 It's something times a variable squared. 781 00:47:53,670 --> 00:48:01,210 So, we'll use some general functional form, a y squared. 782 00:48:01,210 --> 00:48:04,670 And now, let's assume we're in the high temperature limit. 783 00:48:04,670 --> 00:48:06,570 And we've seen what that means. 784 00:48:06,570 --> 00:48:11,060 It means kT is big, compared to the 785 00:48:11,060 --> 00:48:13,580 separation between the energies. 786 00:48:13,580 --> 00:48:15,450 When that's the case, there are lots of these 787 00:48:15,450 --> 00:48:16,580 terms in the sum. 788 00:48:16,580 --> 00:48:18,120 We can convert them to integrals. 789 00:48:18,120 --> 00:48:20,390 We can forget about the fact that the 790 00:48:20,390 --> 00:48:21,390 energies are discrete. 791 00:48:21,390 --> 00:48:22,400 We can say look, they're so close 792 00:48:22,400 --> 00:48:26,590 together, compared to kT. 793 00:48:26,590 --> 00:48:29,850 That we can turn the sums into integrals. 794 00:48:29,850 --> 00:48:36,680 So then we have integrals instead of sums. 795 00:48:36,680 --> 00:48:37,600 And here's our energy. 796 00:48:37,600 --> 00:48:39,070 It's a y squared. 797 00:48:39,070 --> 00:48:39,930 Whatever that is. 798 00:48:39,930 --> 00:48:41,680 It could be this, it could be this, it could be this. 799 00:48:41,680 --> 00:48:46,880 And it's not going to matter. e to the minus a y squared 800 00:48:46,880 --> 00:48:54,880 over kT over, integral over e to the minus a y squared over 801 00:48:54,880 --> 00:48:56,570 kT. dy, dy. 802 00:49:00,560 --> 00:49:03,650 OK. 803 00:49:03,650 --> 00:49:06,710 And here's what's going to happen. 804 00:49:06,710 --> 00:49:09,280 If you do this integral by parts. 805 00:49:09,280 --> 00:49:11,160 This one. 806 00:49:11,160 --> 00:49:16,800 What ends up happening is it gives you this integral times 807 00:49:16,800 --> 00:49:19,370 a certain number. 808 00:49:19,370 --> 00:49:23,200 This is going to come out of the integral. 809 00:49:23,200 --> 00:49:26,410 And so it will turn out. 810 00:49:26,410 --> 00:49:27,920 And it's straightforward to do it. 811 00:49:27,920 --> 00:49:28,720 It's in the notes. 812 00:49:28,720 --> 00:49:29,700 It goes through. 813 00:49:29,700 --> 00:49:31,680 But I'll just write the result here. 814 00:49:31,680 --> 00:49:35,850 The result is that you get, in this high temperature limit 815 00:49:35,850 --> 00:49:42,370 where you've gone to the integral form, you get 1/2 kT. 816 00:49:42,370 --> 00:49:43,790 And it will always be the case. 817 00:49:43,790 --> 00:49:47,530 All you need to know is the form of the energy. 818 00:49:47,530 --> 00:49:49,260 As long as that's the case -- 819 00:49:49,260 --> 00:49:52,920 Remember y is just a variable of integration here. 820 00:49:52,920 --> 00:49:56,460 That's not going to be preserved. 821 00:49:56,460 --> 00:50:00,070 This comes out because of what happens here. 822 00:50:00,070 --> 00:50:03,520 So what that's telling you is whenever you have an energy of 823 00:50:03,520 --> 00:50:07,320 this form, and you're in the high temperature limit, then 824 00:50:07,320 --> 00:50:11,860 you get this classical 825 00:50:11,860 --> 00:50:14,110 equipartition of energy result. 826 00:50:14,110 --> 00:50:16,610 So, for translation, of course there are three separate 827 00:50:16,610 --> 00:50:17,770 degrees of freedom. 828 00:50:17,770 --> 00:50:21,300 For velocity in the x, y, or z direction. 829 00:50:21,300 --> 00:50:23,640 For rotation, if it's a linear molecule, let's say there are 830 00:50:23,640 --> 00:50:24,740 two separate degrees of freedom. 831 00:50:24,740 --> 00:50:26,660 You have to keep track of how many degrees of 832 00:50:26,660 --> 00:50:27,630 freedom there are. 833 00:50:27,630 --> 00:50:29,710 But that's all you have to do. 834 00:50:29,710 --> 00:50:31,300 That's enormously powerful. 835 00:50:31,300 --> 00:50:35,590 It means that without doing anything, I know the average 836 00:50:35,590 --> 00:50:39,070 translational energy of the molecules in this room. 837 00:50:39,070 --> 00:50:40,270 Because of course I'm certainly in the high 838 00:50:40,270 --> 00:50:41,170 temperature limit. 839 00:50:41,170 --> 00:50:43,820 With respect to translational energy levels, they're really 840 00:50:43,820 --> 00:50:45,370 closely spaced. 841 00:50:45,370 --> 00:50:48,250 Same with rotations. 842 00:50:48,250 --> 00:50:51,210 Now at room temperature vibrations, forget it. 843 00:50:51,210 --> 00:50:53,950 I have essentially no vibrational energy. 844 00:50:53,950 --> 00:50:57,460 I'm at the low temperature limit for the molecular 845 00:50:57,460 --> 00:50:59,500 vibrations of nitrogen or oxygen. 846 00:50:59,500 --> 00:51:01,700 Those are high in frequency. 847 00:51:01,700 --> 00:51:04,570 So much higher than 200 wavenumbers. 848 00:51:04,570 --> 00:51:07,990 But for each molecule, I have 3/2 kT of 849 00:51:07,990 --> 00:51:09,680 translational energy. 850 00:51:09,680 --> 00:51:12,650 Linear molecules, I have kT of rotational energy. 851 00:51:12,650 --> 00:51:14,890 Without doing any work at all. 852 00:51:14,890 --> 00:51:18,400 And since that applies to most molecules at room temperature, 853 00:51:18,400 --> 00:51:22,390 it's an incredibly useful, very, very general result. 854 00:51:22,390 --> 00:51:22,860 OK. 855 00:51:22,860 --> 00:51:26,320 Next time we'll do a little bit of chemistry, and look at 856 00:51:26,320 --> 00:51:27,910 phase transformations.