1 00:00:00,060 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:20,570 --> 00:00:23,670 PROFESSOR: And to this purpose, we 9 00:00:23,670 --> 00:00:28,470 need to calculate some thermodynamics. 10 00:00:28,470 --> 00:00:31,160 And we usually do that in statistical mechanics 11 00:00:31,160 --> 00:00:35,450 by calculating for some kind of a partition function. 12 00:00:35,450 --> 00:00:37,320 And we saw last time that it would 13 00:00:37,320 --> 00:00:42,180 be useful to calculate this grand partition function, which 14 00:00:42,180 --> 00:00:44,400 is the ensemble where you specify 15 00:00:44,400 --> 00:00:48,920 the temperature, the chemical potential, and the volume 16 00:00:48,920 --> 00:00:51,250 of the gas. 17 00:00:51,250 --> 00:00:55,120 And in this ensemble, your task is 18 00:00:55,120 --> 00:01:00,060 to look at within this box of volume V 19 00:01:00,060 --> 00:01:04,690 the possibility of there being any number of particles. 20 00:01:04,690 --> 00:01:09,400 So you have to sum over all possible N particle states. 21 00:01:09,400 --> 00:01:12,640 The contribution of each N particle states 22 00:01:12,640 --> 00:01:17,710 is exponentially related to the number of particles 23 00:01:17,710 --> 00:01:19,980 through the chemical potential. 24 00:01:19,980 --> 00:01:24,730 And then given that you are in a segment that has N particles, 25 00:01:24,730 --> 00:01:27,690 you have to look at all possible configurations 26 00:01:27,690 --> 00:01:30,890 of those particles and integrate over 27 00:01:30,890 --> 00:01:32,790 all of those possibilities. 28 00:01:32,790 --> 00:01:36,350 And that amounts to calculating the partition function. 29 00:01:36,350 --> 00:01:40,540 So for the partition function of an N particle system, 30 00:01:40,540 --> 00:01:43,880 you have to integrate over all of the momenta. 31 00:01:43,880 --> 00:01:47,020 Integration over each component of the momentum 32 00:01:47,020 --> 00:01:52,340 gives you a factor of 1 over lambda, where lambda again 33 00:01:52,340 --> 00:01:58,880 was related to the mass of the particle and temperature 34 00:01:58,880 --> 00:02:02,440 through this formula with h what we 35 00:02:02,440 --> 00:02:07,640 use to make these integrations over pq combinations 36 00:02:07,640 --> 00:02:09,120 dimensionless. 37 00:02:09,120 --> 00:02:11,860 There are 3N such integrations. 38 00:02:11,860 --> 00:02:16,680 So that's a contribution from the momenta. 39 00:02:16,680 --> 00:02:19,320 And then we have to do the integration 40 00:02:19,320 --> 00:02:21,470 over all of the coordinates. 41 00:02:21,470 --> 00:02:25,970 And as long as these particles are identical, 42 00:02:25,970 --> 00:02:30,140 we decided to divide by the number of permutations. 43 00:02:30,140 --> 00:02:33,150 Because we cannot tear them apart. 44 00:02:33,150 --> 00:02:37,710 So having done that, I need to now integrate over all of the N 45 00:02:37,710 --> 00:02:43,700 particles spanning a box of size V 46 00:02:43,700 --> 00:02:47,180 so the integration is within the box. 47 00:02:47,180 --> 00:02:51,080 And when I have interactions, then I 48 00:02:51,080 --> 00:02:54,200 have to worry about the Boltzmann weight that 49 00:02:54,200 --> 00:02:55,550 comes from the interaction. 50 00:02:55,550 --> 00:02:58,630 So here I should really put some kind of e 51 00:02:58,630 --> 00:03:02,920 to the minus beta times the interaction U. 52 00:03:02,920 --> 00:03:05,480 And what we did was we said that let's assume 53 00:03:05,480 --> 00:03:09,290 that this interaction comes from pairs of particles. 54 00:03:09,290 --> 00:03:13,640 And so this U is the sum over all possible pairs 55 00:03:13,640 --> 00:03:16,750 of particles, which when exponentiated 56 00:03:16,750 --> 00:03:24,790 will then give you a product over all possible pairs j, k 57 00:03:24,790 --> 00:03:29,450 and a factor that is related to the potential interaction 58 00:03:29,450 --> 00:03:30,980 between these. 59 00:03:30,980 --> 00:03:35,350 And we found it useful to write that factor as 1 60 00:03:35,350 --> 00:03:45,840 plus fjk, where this fjk stood for e to the minus 61 00:03:45,840 --> 00:03:53,150 beta of V qj minus qk minus 1. 62 00:03:53,150 --> 00:03:55,950 So basically if I add the 1 to that, 63 00:03:55,950 --> 00:03:59,190 I just get the exponentiated interaction potential. 64 00:03:59,190 --> 00:04:03,150 And then I have a forum such as this. 65 00:04:03,150 --> 00:04:05,780 So basically, this is the quantity 66 00:04:05,780 --> 00:04:07,980 that we wanted to compute. 67 00:04:07,980 --> 00:04:11,390 And so then we said, well, let's imagine 68 00:04:11,390 --> 00:04:17,250 expanding these factors of 1 plus f1, 2, 1 plus f1, 3, 69 00:04:17,250 --> 00:04:19,910 1 plus f2, 3, all of these factors, 70 00:04:19,910 --> 00:04:24,880 and organize them according to the powers of f that they have. 71 00:04:24,880 --> 00:04:27,990 So the leading term would be taking 1 from everybody. 72 00:04:27,990 --> 00:04:29,970 So that would give me V to the N, which 73 00:04:29,970 --> 00:04:32,730 would be the 0-th order partition 74 00:04:32,730 --> 00:04:35,000 function, the non-interacting system. 75 00:04:35,000 --> 00:04:37,590 And then I would start to get corrections 76 00:04:37,590 --> 00:04:41,080 where there's order of f integrated, order of f squared 77 00:04:41,080 --> 00:04:44,070 integrated, all kinds of things. 78 00:04:44,070 --> 00:04:49,330 And that being a somewhat difficult object to look at, 79 00:04:49,330 --> 00:04:53,860 we said, let's imagine graphically what we would get. 80 00:04:53,860 --> 00:04:58,800 And the typical contribution that we would get to this 81 00:04:58,800 --> 00:05:04,640 would involve having to iterate over all of these N particles. 82 00:05:04,640 --> 00:05:10,560 So we have to somehow imagine that we have particles 1 to N. 83 00:05:10,560 --> 00:05:16,370 And then for a particular term, we either pick 1's-- and there 84 00:05:16,370 --> 00:05:19,770 will be some points that are not connected to any f that is 85 00:05:19,770 --> 00:05:21,410 integrated. 86 00:05:21,410 --> 00:05:24,590 There will be a bunch of things that will be connected 87 00:05:24,590 --> 00:05:28,260 to things where there are pair of f's. 88 00:05:28,260 --> 00:05:30,100 There will be things later on maybe 89 00:05:30,100 --> 00:05:35,450 where there are triplets of f, and so forth and so on. 90 00:05:35,450 --> 00:05:40,500 And then we said that when I do the integrations that 91 00:05:40,500 --> 00:05:44,180 correspond to this, what do I get? 92 00:05:44,180 --> 00:05:48,140 I will get the contribution that comes from one particle 93 00:05:48,140 --> 00:05:50,650 by itself integrated. 94 00:05:50,650 --> 00:05:56,940 Let's call that b1 to the power of n1. 95 00:05:56,940 --> 00:06:02,620 I will get the contribution from this pair integrated. 96 00:06:02,620 --> 00:06:07,290 I will get b2 to the power of n2. 97 00:06:07,290 --> 00:06:10,820 I will get the contributions from these entities. 98 00:06:10,820 --> 00:06:14,310 And in general, I said, well, OK, somewhere in this, 99 00:06:14,310 --> 00:06:16,730 I will get bl to the power of nl. 100 00:06:19,418 --> 00:06:22,650 Now of course, I have a big constraint here 101 00:06:22,650 --> 00:06:30,080 that is that the sum over l of lnl 102 00:06:30,080 --> 00:06:34,750 has to be the total number of points 1 to n. 103 00:06:34,750 --> 00:06:40,330 So however I partition this, I will have for each graph 104 00:06:40,330 --> 00:06:43,090 that particular constraint acting. 105 00:06:46,180 --> 00:06:53,320 We said that clearly there's a lot of graphs and combinations 106 00:06:53,320 --> 00:06:56,710 that give you precisely this same factor. 107 00:06:56,710 --> 00:07:00,010 But all I had to do was to sort of rearrange 108 00:07:00,010 --> 00:07:02,120 the numbers and ordering, et cetera, 109 00:07:02,120 --> 00:07:04,750 and I would get all of this. 110 00:07:04,750 --> 00:07:07,520 So it would be very nice if I could figure out 111 00:07:07,520 --> 00:07:09,320 what the overall factor is out here. 112 00:07:12,450 --> 00:07:18,680 So we said that the factor is something like N factorial. 113 00:07:18,680 --> 00:07:22,500 Because what I can do is I can permute all of these numbers, 114 00:07:22,500 --> 00:07:25,950 and I would get exactly the same thing. 115 00:07:25,950 --> 00:07:29,600 But then I have to make sure that I don't over count. 116 00:07:29,600 --> 00:07:35,350 And not over counting required me to divide 117 00:07:35,350 --> 00:07:37,350 by the number of permutations that I 118 00:07:37,350 --> 00:07:40,050 have within each subgroup. 119 00:07:40,050 --> 00:07:43,380 So I have bl to the power of n. 120 00:07:43,380 --> 00:07:51,360 I have l factorial to the power of nl. 121 00:07:51,360 --> 00:07:55,340 And then I have the change. 122 00:07:55,340 --> 00:07:57,370 Let's say this is 1, 2, this is 3, 4. 123 00:07:57,370 --> 00:08:01,170 I could have called one of them 3, 4, the other one 1, 2. 124 00:08:01,170 --> 00:08:06,925 So basically I will have nl factorial from the permutations 125 00:08:06,925 --> 00:08:07,835 within each. 126 00:08:12,390 --> 00:08:16,950 But I would have gotten exactly the same numerical factor 127 00:08:16,950 --> 00:08:23,510 out front if I had the same configuration 128 00:08:23,510 --> 00:08:25,070 but I had this diagram. 129 00:08:27,770 --> 00:08:29,860 I would have gotten exactly the same factor. 130 00:08:29,860 --> 00:08:33,080 I would call the contribution to this b3. 131 00:08:33,080 --> 00:08:36,559 I didn't say currently what b3 is. 132 00:08:36,559 --> 00:08:39,640 I would have gotten exactly the same factor. 133 00:08:39,640 --> 00:08:44,080 So maybe then what I did was to sort of group 134 00:08:44,080 --> 00:08:46,080 all of those things that would come 135 00:08:46,080 --> 00:08:50,100 with the same numerical factor into this [INAUDIBLE] sum 136 00:08:50,100 --> 00:08:51,020 that I call bl. 137 00:08:51,020 --> 00:09:00,770 So I call bl to be the sum over all l-particle clusters. 138 00:09:09,250 --> 00:09:12,730 And then, of course here, I have to sum over 139 00:09:12,730 --> 00:09:15,610 all configurations of nl that are 140 00:09:15,610 --> 00:09:19,122 consistent with this constraint that I have to put up there. 141 00:09:24,430 --> 00:09:28,260 So the rest of it then was algebra. 142 00:09:28,260 --> 00:09:35,080 We said that if I constrain the total number, 143 00:09:35,080 --> 00:09:36,800 it's difficult for me to do. 144 00:09:36,800 --> 00:09:41,310 That's why I didn't go and calculate the partition 145 00:09:41,310 --> 00:09:44,880 function and switch to the grand partition function. 146 00:09:44,880 --> 00:09:48,550 Because in the grand partition, I can essentially 147 00:09:48,550 --> 00:09:54,910 make this N that constrains the values of these nl's 148 00:09:54,910 --> 00:09:57,260 to be all over the place. 149 00:09:57,260 --> 00:10:02,870 And therefore summing over things with nl unconstrained 150 00:10:02,870 --> 00:10:07,480 is equivalent to summing over terms with nl constrained, 151 00:10:07,480 --> 00:10:12,010 and then summing over whatever the final constraint is. 152 00:10:12,010 --> 00:10:16,310 So once I did that, I was liberated from this constraint. 153 00:10:16,310 --> 00:10:21,360 I could do the sum for each value of nl separately. 154 00:10:21,360 --> 00:10:24,380 And the thing nicely broke up into pieces. 155 00:10:24,380 --> 00:10:27,970 And so then what I could do is I could 156 00:10:27,970 --> 00:10:32,360 show that each term in the sum, there 157 00:10:32,360 --> 00:10:36,880 would be a product over different contributions l. 158 00:10:36,880 --> 00:10:43,290 For each l, I could sum over nl running from 0 to infinity. 159 00:10:43,290 --> 00:10:49,840 I had the 1 over nl factorial from out here. 160 00:10:49,840 --> 00:10:53,320 I had an e to the beta mu divided 161 00:10:53,320 --> 00:10:58,730 by lambda cubed from the combination of these things 162 00:10:58,730 --> 00:11:02,060 raised to the power of lnl, which 163 00:11:02,060 --> 00:11:05,700 is how big N would have been composed. 164 00:11:05,700 --> 00:11:08,550 So I would have here lnl. 165 00:11:08,550 --> 00:11:17,980 And I also had an l factorial raised to the power of nl. 166 00:11:17,980 --> 00:11:19,952 So I can write things in this fashion. 167 00:11:22,840 --> 00:11:25,740 So this thing then became the same thing 168 00:11:25,740 --> 00:11:31,795 as an exponential of a sum over l running from 1 to infinity. 169 00:11:35,160 --> 00:11:37,380 1 over l factorial, of if you like 170 00:11:37,380 --> 00:11:43,220 e to the beta mu over lambda cubed to the power of l 1 171 00:11:43,220 --> 00:11:44,920 over l factorial. 172 00:11:44,920 --> 00:11:46,610 And then I had bl. 173 00:11:52,690 --> 00:11:58,610 So it was this very nice result somehow summing 174 00:11:58,610 --> 00:12:03,820 over all kinds of things, and then taking the logarithm. 175 00:12:03,820 --> 00:12:06,732 The logarithm really depends only 176 00:12:06,732 --> 00:12:10,480 on the contributions of single clusters. 177 00:12:10,480 --> 00:12:12,250 And again, the reason it had to be 178 00:12:12,250 --> 00:12:15,090 that way is because the ultimate thing 179 00:12:15,090 --> 00:12:18,170 that I calculated in this ensemble 180 00:12:18,170 --> 00:12:22,960 is that the answer should be e to the beta V 181 00:12:22,960 --> 00:12:23,940 times the pressure. 182 00:12:26,470 --> 00:12:30,950 And so the expression that we have over here better 183 00:12:30,950 --> 00:12:36,890 have terms which are all proportional to volume. 184 00:12:36,890 --> 00:12:40,460 Sorry, they're all made extensive by proportionality 185 00:12:40,460 --> 00:12:41,690 to volume. 186 00:12:41,690 --> 00:12:45,380 And indeed, when I do the integrations over a single any 187 00:12:45,380 --> 00:12:48,620 cluster, there is one degree of freedom, if you like, 188 00:12:48,620 --> 00:12:52,080 associated with the center of mass of the cluster that 189 00:12:52,080 --> 00:12:54,790 can go and explore the entire volume. 190 00:12:54,790 --> 00:12:59,700 And so all of these things are in fact in the large end limit 191 00:12:59,700 --> 00:13:05,739 proportional to V and something that I call bl bar. 192 00:13:17,340 --> 00:13:24,610 So once I divide by this volume, the final outcome 193 00:13:24,610 --> 00:13:27,940 of my calculation is that I can calculate 194 00:13:27,940 --> 00:13:32,340 the pressure of an interacting gas 195 00:13:32,340 --> 00:13:41,860 by summing over a series whose terms are this e to the beta mu 196 00:13:41,860 --> 00:13:48,100 over lambda cubed raised to the power of l bl bar divided 197 00:13:48,100 --> 00:13:50,280 by l factorial. 198 00:13:59,480 --> 00:14:05,710 OK, this is correct, but not particularly illuminating. 199 00:14:05,710 --> 00:14:11,370 Because the thing that we said we have some intuition for 200 00:14:11,370 --> 00:14:15,640 is that the pressure of a gas-- let's 201 00:14:15,640 --> 00:14:19,630 say if I look at it in terms of beta P, 202 00:14:19,630 --> 00:14:26,900 actually P is the density times kt, which is 1 over beta, 203 00:14:26,900 --> 00:14:28,210 if you like, or nkT. 204 00:14:28,210 --> 00:14:30,930 It doesn't matter. 205 00:14:30,930 --> 00:14:32,880 And then there will be corrections. 206 00:14:32,880 --> 00:14:35,270 There will be a term that is order of n squared. 207 00:14:37,980 --> 00:14:42,640 There will be a term that is order of n cubed. 208 00:14:42,640 --> 00:14:44,660 And there are these coefficients, 209 00:14:44,660 --> 00:14:49,100 which are functions of temperature, 210 00:14:49,100 --> 00:14:52,460 that are called the virial coefficient. 211 00:14:52,460 --> 00:14:55,220 And this is a virial expansion. 212 00:14:55,220 --> 00:15:00,370 Essentially what it is is a fitting 213 00:15:00,370 --> 00:15:02,240 of the form of the pressure of the gas 214 00:15:02,240 --> 00:15:04,190 as a function of density. 215 00:15:04,190 --> 00:15:08,545 In the very low density limit you get ideal gas result. 216 00:15:08,545 --> 00:15:11,110 And presumably because of these interactions, 217 00:15:11,110 --> 00:15:14,650 you will get corrections that presumably also 218 00:15:14,650 --> 00:15:17,740 know about the potential that went 219 00:15:17,740 --> 00:15:23,100 into the construction of all of these b bars, et cetera. 220 00:15:23,100 --> 00:15:28,990 So how do we relate these things that are, say, experimentally 221 00:15:28,990 --> 00:15:35,040 accessible to this expansion that we have over here? 222 00:15:35,040 --> 00:15:38,970 And the reason it is not obvious immediately 223 00:15:38,970 --> 00:15:46,010 is because this is an expansion in chemical potential, 224 00:15:46,010 --> 00:15:49,720 whereas over here, I have density. 225 00:15:49,720 --> 00:15:51,730 So what do I do? 226 00:15:51,730 --> 00:15:56,040 Well, I realize that the density can be obtained as follows. 227 00:15:56,040 --> 00:16:02,150 In the grand canonical ensemble, the number of particles 228 00:16:02,150 --> 00:16:06,810 is in principal a random variable. 229 00:16:06,810 --> 00:16:10,650 But that random variable is governed by this e 230 00:16:10,650 --> 00:16:14,350 to the beta mu N that controls how many you have. 231 00:16:14,350 --> 00:16:21,330 And if I take the log of Q with respect to beta mu, 232 00:16:21,330 --> 00:16:23,840 I will generate the average number, 233 00:16:23,840 --> 00:16:26,050 which in the thermodynamic limit we 234 00:16:26,050 --> 00:16:28,940 expect to be the same thing as what we thermodynamically 235 00:16:28,940 --> 00:16:30,750 would call the number. 236 00:16:30,750 --> 00:16:36,080 And this log Q we just established is beta PV. 237 00:16:36,080 --> 00:16:40,600 So what we have here is the derivative of beta VP 238 00:16:40,600 --> 00:16:43,270 with respect to beta mu. 239 00:16:43,270 --> 00:16:45,450 And if I do this at constant temperature, 240 00:16:45,450 --> 00:16:47,240 the betas disappear. 241 00:16:47,240 --> 00:16:50,870 This is the same thing as V dP by d 242 00:16:50,870 --> 00:16:54,190 mu at constant temperature. 243 00:16:54,190 --> 00:17:00,970 But also I can take the derivative right here. 244 00:17:00,970 --> 00:17:02,800 So what happens? 245 00:17:02,800 --> 00:17:10,660 I will find that the density which is N over V, 246 00:17:10,660 --> 00:17:14,660 is the derivative of this expression with respect 247 00:17:14,660 --> 00:17:16,609 to beta mu. 248 00:17:16,609 --> 00:17:20,520 I go and I find that there's a beta mu. 249 00:17:20,520 --> 00:17:24,060 If I take a derivative of this with respect to beta mu, 250 00:17:24,060 --> 00:17:27,300 really it's exponential of beta mu l. 251 00:17:27,300 --> 00:17:31,420 The derivative of it will give me l e to the beta mu l. 252 00:17:31,420 --> 00:17:37,400 So what I get is a sum over l e to the beta mu 253 00:17:37,400 --> 00:17:39,460 over lambda cubed. 254 00:17:39,460 --> 00:17:42,300 l gets repeated. 255 00:17:42,300 --> 00:17:45,590 And then I will have l times this. 256 00:17:45,590 --> 00:17:50,300 So I will have vl bar divided by l minus 1 factorial. 257 00:17:53,790 --> 00:17:56,370 So the series for density is very 258 00:17:56,370 --> 00:17:58,680 much like the series for pressure, 259 00:17:58,680 --> 00:18:03,980 except that you replace 1 over l factorial with 1 over l 260 00:18:03,980 --> 00:18:05,009 minus 1 factorial. 261 00:18:09,410 --> 00:18:12,690 So now my task is clear. 262 00:18:12,690 --> 00:18:18,350 What I should do is I should solve 263 00:18:18,350 --> 00:18:24,090 given a particular density for what mu is. 264 00:18:24,090 --> 00:18:27,100 Once I have mu as a function of density, 265 00:18:27,100 --> 00:18:28,900 I can substitute it back here. 266 00:18:28,900 --> 00:18:32,860 And I will have pressure as a function of density. 267 00:18:32,860 --> 00:18:36,090 Of course it's clear that the right variable to look at 268 00:18:36,090 --> 00:18:43,710 is not mu, but x, which is e to the beta mu over lambda cubed. 269 00:18:43,710 --> 00:18:45,150 This actually has a name. 270 00:18:45,150 --> 00:18:46,960 It's sometimes called fugacity. 271 00:18:54,230 --> 00:18:58,130 So the second equation is telling me 272 00:18:58,130 --> 00:19:05,700 that the density I can write in terms of the fugacity as a sum 273 00:19:05,700 --> 00:19:13,140 over l x to the l bl bar divided by l minus 1 factorial, which 274 00:19:13,140 --> 00:19:16,860 if I were to sort of write in its full details, 275 00:19:16,860 --> 00:19:24,430 it starts with b1 bar x plus b2 bar x squared. 276 00:19:24,430 --> 00:19:28,890 Because there I have 0 factorial or 1 factorial respectively. 277 00:19:28,890 --> 00:19:33,940 The next term will be b3 bar over 2 x cubed. 278 00:19:33,940 --> 00:19:36,110 And this will go on and on. 279 00:19:41,880 --> 00:19:45,780 Also, let me remind you what b1 bar is. 280 00:19:45,780 --> 00:19:51,740 b1 bar, for every one of these b1 bars, 281 00:19:51,740 --> 00:19:55,140 I have to divide by 1 over V because of this V. 282 00:19:55,140 --> 00:19:57,980 And then I have to do the integration that corresponds 283 00:19:57,980 --> 00:20:02,050 to the one cluster, which is essentially one cluster going 284 00:20:02,050 --> 00:20:05,250 over the entirety of space, which will give me V. 285 00:20:05,250 --> 00:20:09,230 So this is in fact 1. 286 00:20:09,230 --> 00:20:14,290 So basically this coefficient here is 1. 287 00:20:14,290 --> 00:20:16,100 And then I will have the corrections. 288 00:20:20,400 --> 00:20:23,600 So you say, well, I have n as a function of x. 289 00:20:23,600 --> 00:20:26,890 I want x as a function of n. 290 00:20:26,890 --> 00:20:29,360 And I say, OK, that's not that difficult. 291 00:20:29,360 --> 00:20:34,340 I will write that x equals n minus b2 bar 292 00:20:34,340 --> 00:20:40,690 x squared minus b3 bar x cubed over 2, and so forth. 293 00:20:40,690 --> 00:20:41,860 And I have now x. 294 00:20:46,910 --> 00:20:50,560 Maybe a few of you are skeptical. 295 00:20:50,560 --> 00:20:52,330 Some of you don't seem to be bothered. 296 00:20:55,440 --> 00:20:58,810 OK, so my claim is that this is indeed 297 00:20:58,810 --> 00:21:04,860 a systematic way of solving a series in which when 298 00:21:04,860 --> 00:21:10,475 the density goes to 0, you expect x to also go to 0. 299 00:21:10,475 --> 00:21:14,500 And to lowest order, x will be of the order of density. 300 00:21:14,500 --> 00:21:18,450 And these will be higher order powers in density. 301 00:21:18,450 --> 00:21:24,620 So to get a systematic series in density, all you need to do 302 00:21:24,620 --> 00:21:29,150 is to sort of work with a series such as this keeping 303 00:21:29,150 --> 00:21:33,030 in mind what order you have solved things to. 304 00:21:33,030 --> 00:21:35,900 So I claim that to lowest order, this really 305 00:21:35,900 --> 00:21:37,960 just says that x is n. 306 00:21:37,960 --> 00:21:42,780 And then there are corrections that are order of n squared. 307 00:21:42,780 --> 00:21:46,770 And to get the next order term, all I need to do 308 00:21:46,770 --> 00:21:50,310 is to substitute the lower order in this equation. 309 00:21:50,310 --> 00:21:55,300 So basically if I substitute x equals n in this equation, 310 00:21:55,300 --> 00:22:00,900 I will get n minus b2 n square. 311 00:22:00,900 --> 00:22:04,990 If I substitute x over n here, it will be a higher order term. 312 00:22:04,990 --> 00:22:06,970 And I claim that this is the correct result 313 00:22:06,970 --> 00:22:08,380 to order of n cubed. 314 00:22:11,940 --> 00:22:15,250 And then to get the result to order of n 315 00:22:15,250 --> 00:22:20,420 cubed, I substitute this back into the original equation. 316 00:22:20,420 --> 00:22:26,050 So I start with n minus b2 bar, the square 317 00:22:26,050 --> 00:22:29,750 of the previous solution at the right order. 318 00:22:29,750 --> 00:22:34,630 So squaring this, I will get n squared minus twice 319 00:22:34,630 --> 00:22:38,030 b2 bar n cubed. 320 00:22:38,030 --> 00:22:40,660 And the next term that would be the square of this, which 321 00:22:40,660 --> 00:22:44,210 is order of n to the fourth, I don't write down. 322 00:22:44,210 --> 00:22:49,200 And then I have minus b3 bar over 2, 323 00:22:49,200 --> 00:22:51,670 the cube of the previous solution. 324 00:22:51,670 --> 00:22:53,600 And at the right order that I have, 325 00:22:53,600 --> 00:22:58,610 that's the same thing as b3 bar over 2 n cubed 326 00:22:58,610 --> 00:23:02,700 and order of n to the fourth. 327 00:23:02,700 --> 00:23:05,750 And all I really need to do at this order 328 00:23:05,750 --> 00:23:08,870 is to recognize that I have two terms that 329 00:23:08,870 --> 00:23:10,500 are order of n cubed. 330 00:23:10,500 --> 00:23:16,590 Putting them together, I have n minus b2 bar n squared. 331 00:23:16,590 --> 00:23:22,370 And then I have plus 2b2 bar squared 332 00:23:22,370 --> 00:23:31,180 minus b3 bar over 2 n cubed and order of n to the fourth. 333 00:23:41,000 --> 00:23:42,180 So now we erase this. 334 00:23:53,420 --> 00:23:58,700 OK, so now I have solved for x in a power series in n. 335 00:23:58,700 --> 00:24:00,910 All I need to do is to substitute 336 00:24:00,910 --> 00:24:03,160 in the power series for p. 337 00:24:03,160 --> 00:24:04,840 So let's write that down. 338 00:24:04,840 --> 00:24:11,720 The power series for beta p is b1 bar times 339 00:24:11,720 --> 00:24:16,810 the density b2 bar squared over 2 density 340 00:24:16,810 --> 00:24:23,670 square, b3 bar over 3 factorial, which 341 00:24:23,670 --> 00:24:27,480 is 6 n cubed, and so forth. 342 00:24:27,480 --> 00:24:30,190 But I only calculated things to order of n cubed. 343 00:24:32,930 --> 00:24:36,740 Also, b1 bar is the same thing as 1. 344 00:24:36,740 --> 00:24:40,740 So basically that's what I should be working with. 345 00:24:40,740 --> 00:24:44,900 And so all I need to do is to substitute-- oops, 346 00:24:44,900 --> 00:24:46,420 except that these are all x's. 347 00:24:54,170 --> 00:24:56,890 The series here for beta p that I 348 00:24:56,890 --> 00:25:01,600 have is in powers of this quantity x. 349 00:25:01,600 --> 00:25:07,450 And it is x that I had calculated as a function of n. 350 00:25:07,450 --> 00:25:15,430 To the order that I calculated, it is n minus b2n squared. 351 00:25:15,430 --> 00:25:26,360 And then I have 2b2 squared minus b3 over b3. 352 00:25:26,360 --> 00:25:30,850 Somehow this sounds incorrect to me. 353 00:25:30,850 --> 00:25:33,222 Nope, that's fine. 354 00:25:33,222 --> 00:25:35,010 Because here I have 2. 355 00:25:35,010 --> 00:25:37,020 So that's correct. 356 00:25:37,020 --> 00:25:43,970 b3 over 2 n cubed-- so essentially, 357 00:25:43,970 --> 00:25:47,910 I just substituted for x here. 358 00:25:47,910 --> 00:25:51,630 The next order term is going to be 359 00:25:51,630 --> 00:25:56,280 b2 over 2 times the square of x. 360 00:25:56,280 --> 00:26:00,550 The square of x will give me a term that is n squared, 361 00:26:00,550 --> 00:26:07,110 a term that is from here 2b2n cubed. 362 00:26:07,110 --> 00:26:11,800 Order of n to the fourth I haven't calculated correctly. 363 00:26:11,800 --> 00:26:17,440 Order of n cubed I have b3 over 6. 364 00:26:17,440 --> 00:26:21,850 And just x cubed is the same thing as n cubed to this order. 365 00:26:21,850 --> 00:26:26,562 And I have not calculated anything at the next order. 366 00:26:29,830 --> 00:26:33,544 So let's see what we have. 367 00:26:33,544 --> 00:26:36,120 We have n. 368 00:26:36,120 --> 00:26:40,550 At the next order, there are two terms that are n squared. 369 00:26:40,550 --> 00:26:43,570 There is this term, and there is that term. 370 00:26:43,570 --> 00:26:49,040 Putting them together, I will get minus b2 over 2 n squared. 371 00:26:52,030 --> 00:26:55,140 At the next order, at the order of n cubed, 372 00:26:55,140 --> 00:26:57,540 I have a bunch of terms. 373 00:26:57,540 --> 00:27:03,470 First of all, there is this 2b2 squared. 374 00:27:03,470 --> 00:27:09,810 But then multiplying this with this will subtract one b2. 375 00:27:09,810 --> 00:27:14,500 So I'm going to be left with b2 squared. 376 00:27:14,500 --> 00:27:20,030 And then I have minus 1/2 b3. 377 00:27:20,030 --> 00:27:22,320 So that's one term. 378 00:27:22,320 --> 00:27:26,140 And then I have plus 1/6 b3. 379 00:27:26,140 --> 00:27:30,560 Minus 1/2 plus 1/6 is minus 1/3. 380 00:27:30,560 --> 00:27:35,210 So this is minus b3 over 3 n cubed. 381 00:27:35,210 --> 00:27:39,325 And I haven't calculated order of n to the fourth. 382 00:27:42,170 --> 00:27:44,095 And this is the formula for BP. 383 00:27:51,680 --> 00:27:55,865 So lowest order, I have the ideal gas result. 384 00:27:59,170 --> 00:28:01,570 Actually, let me, for simplicity, 385 00:28:01,570 --> 00:28:05,445 define the virial coefficients in this fashion. 386 00:28:08,030 --> 00:28:13,410 The second order, I will get a correction B2, 387 00:28:13,410 --> 00:28:16,210 which is minus 1/2 of b2 bar. 388 00:28:18,755 --> 00:28:24,990 So this is minus 1/2 of the diagram that 389 00:28:24,990 --> 00:28:30,640 corresponds to essentially one of these lines 390 00:28:30,640 --> 00:28:32,950 that I have up here. 391 00:28:32,950 --> 00:28:34,770 And so what is it? 392 00:28:34,770 --> 00:28:42,140 It is minus 1/2 the integral over the relative coordinate 393 00:28:42,140 --> 00:28:45,900 of e to the minus beta v as a function 394 00:28:45,900 --> 00:28:47,940 of the relative coordinate minus 1. 395 00:28:52,340 --> 00:28:57,580 Now earlier, we had in fact calculated this result directly 396 00:28:57,580 --> 00:28:59,510 through the partition function. 397 00:28:59,510 --> 00:29:04,300 I did a calculation in which I calculated the first term 398 00:29:04,300 --> 00:29:06,740 in the other way of looking at things, 399 00:29:06,740 --> 00:29:12,060 in the cumulant expansion, as a function of expansions 400 00:29:12,060 --> 00:29:13,280 in the potential. 401 00:29:13,280 --> 00:29:15,110 We saw that there was a term that all 402 00:29:15,110 --> 00:29:16,840 was order of density squared. 403 00:29:16,840 --> 00:29:20,140 We summed all of the terms to get this factor. 404 00:29:20,140 --> 00:29:23,270 And there was precisely a factor of minus 1/2 405 00:29:23,270 --> 00:29:25,710 as a correction to pressure once we 406 00:29:25,710 --> 00:29:27,772 took the derivative of the partition function 407 00:29:27,772 --> 00:29:28,730 with respect to volume. 408 00:29:31,340 --> 00:29:38,600 OK, so the thing that is new is really the next order term, B3, 409 00:29:38,600 --> 00:29:44,360 which is b2 squared minus b3 bar over 3. 410 00:29:44,360 --> 00:29:48,230 OK, so what is that? 411 00:29:48,230 --> 00:29:52,340 Diagrammatically, B2 was this factor 412 00:29:52,340 --> 00:29:58,480 that we calculated above and is the square of this pair. 413 00:29:58,480 --> 00:30:03,360 And then I have to subtract from that 1/3 of whatever 414 00:30:03,360 --> 00:30:06,290 goes into B3. 415 00:30:06,290 --> 00:30:10,770 Now remember, we said that B3, I have to pick three points 416 00:30:10,770 --> 00:30:14,740 and make sure I link them in all possible ways. 417 00:30:14,740 --> 00:30:19,910 So the diagrams that go into B3, one of them is this. 418 00:30:19,910 --> 00:30:22,160 But then there are three other ways 419 00:30:22,160 --> 00:30:26,743 of making a linked object, which is these things. 420 00:30:31,950 --> 00:30:35,150 We also saw that if I were to calculate 421 00:30:35,150 --> 00:30:39,010 the contribution of any one of these objects, 422 00:30:39,010 --> 00:30:43,480 I can very easily choose to measure my coordinates 423 00:30:43,480 --> 00:30:47,690 with respect to, say, the point that is at their apex. 424 00:30:47,690 --> 00:30:49,730 And then I would have one variable 425 00:30:49,730 --> 00:30:51,710 which is this distance, one variable which 426 00:30:51,710 --> 00:30:53,000 is that distance. 427 00:30:53,000 --> 00:30:55,150 And independently, each one of them 428 00:30:55,150 --> 00:31:00,030 would give me the factor that I calculated before. 429 00:31:00,030 --> 00:31:06,050 In essence, all of these one particle reducible graphs 430 00:31:06,050 --> 00:31:09,362 give a contribution, which is the product 431 00:31:09,362 --> 00:31:13,690 of these single line graphs. 432 00:31:13,690 --> 00:31:14,770 There are three of them. 433 00:31:14,770 --> 00:31:18,630 Minus 1/3 precisely cancels that. 434 00:31:18,630 --> 00:31:24,450 And so the calculation that goes into the third virial 435 00:31:24,450 --> 00:31:31,670 coefficient ultimately will only depend on this one graph. 436 00:31:31,670 --> 00:31:34,720 And we had anticipated this before. 437 00:31:34,720 --> 00:31:36,640 When we were doing things previously 438 00:31:36,640 --> 00:31:39,880 using the expansion of the partition function, 439 00:31:39,880 --> 00:31:43,270 we saw that in this cumulant expansion, 440 00:31:43,270 --> 00:31:46,210 only the cumulants were appearing. 441 00:31:46,210 --> 00:31:48,360 And for calculation of the cumulants, 442 00:31:48,360 --> 00:31:50,860 I had to do lots of subtractions. 443 00:31:50,860 --> 00:31:53,280 And those subtractions were genetically 444 00:31:53,280 --> 00:32:00,880 removing these one particle reducible graphs. 445 00:32:00,880 --> 00:32:04,230 And this continues to all orders. 446 00:32:04,230 --> 00:32:10,280 And the general formula is that the l-th contribution here 447 00:32:10,280 --> 00:32:17,060 will be a factor of l minus 1 over l factorial, which is not 448 00:32:17,060 --> 00:32:19,750 so difficult to get by following the procedures 449 00:32:19,750 --> 00:32:24,940 that I have described over here, and then the part of bl that 450 00:32:24,940 --> 00:32:28,150 is one particle irreducible. 451 00:32:33,360 --> 00:32:38,600 So this is the eventual result for how 452 00:32:38,600 --> 00:32:44,760 you would be able, given some particular form of the two body 453 00:32:44,760 --> 00:32:51,120 interaction, to calculate an expansion for pressure 454 00:32:51,120 --> 00:32:55,960 in powers of density, how the coefficients of that expansion 455 00:32:55,960 --> 00:32:59,261 are related to properties of this potential 456 00:32:59,261 --> 00:33:00,760 through this diagrammatic expansion. 457 00:33:10,160 --> 00:33:13,100 So this is formally correct. 458 00:33:13,100 --> 00:33:16,440 And then the next question is, is it practical? 459 00:33:16,440 --> 00:33:18,380 Is it useful? 460 00:33:18,380 --> 00:33:23,310 So we need to start computing things to see something 461 00:33:23,310 --> 00:33:27,524 about the usefulness of this series for some particular type 462 00:33:27,524 --> 00:33:28,065 of potential. 463 00:33:31,060 --> 00:33:34,790 So we are going to look at the kind of potential 464 00:33:34,790 --> 00:33:37,370 that I already described for you. 465 00:33:37,370 --> 00:33:43,800 That is, if I look at two particles in a gas that 466 00:33:43,800 --> 00:33:47,720 are separated by an amount r-- let's 467 00:33:47,720 --> 00:33:50,630 imagine the potential only depends 468 00:33:50,630 --> 00:33:54,580 on the relative distance, not orientation or anything else. 469 00:33:54,580 --> 00:33:57,610 We said that basically, at large distances, 470 00:33:57,610 --> 00:34:02,080 the potential is attractive because of van der Waals. 471 00:34:02,080 --> 00:34:05,890 At short distances, the potential is repulsive. 472 00:34:05,890 --> 00:34:09,350 And so you have a general form such as this. 473 00:34:12,900 --> 00:34:15,610 Now, if I want to do calculations, 474 00:34:15,610 --> 00:34:18,530 it would be useful for me to have something 475 00:34:18,530 --> 00:34:22,070 that I can do calculations exactly 476 00:34:22,070 --> 00:34:24,909 with and get an estimate. 477 00:34:24,909 --> 00:34:27,659 So what I'm going to do is to replace that potential, 478 00:34:27,659 --> 00:34:31,130 essentially, with a hard wall. 479 00:34:31,130 --> 00:34:38,469 So my approximation to the potential is that my v of r 480 00:34:38,469 --> 00:34:42,135 is infinite for distances, separations 481 00:34:42,135 --> 00:34:45,060 that are less than some r0. 482 00:34:45,060 --> 00:34:48,480 So basically, I define some kind of r0, 483 00:34:48,480 --> 00:34:52,389 which is the-- if you think of them as billiard balls, 484 00:34:52,389 --> 00:34:56,380 it's related to the closest distance of approach. 485 00:34:56,380 --> 00:34:59,740 This potential at large distances 486 00:34:59,740 --> 00:35:06,660 has typical van der Waals form, by which 487 00:35:06,660 --> 00:35:10,580 I mean it falls off as a function of separation 488 00:35:10,580 --> 00:35:13,180 with the sixth power. 489 00:35:13,180 --> 00:35:14,140 It is attractive. 490 00:35:14,140 --> 00:35:16,710 So I put a minus sign here. 491 00:35:16,710 --> 00:35:20,440 In order to make sure that I get eventually the dimensions 492 00:35:20,440 --> 00:35:25,820 right, the coefficient that goes here I write as r0 with a u0 493 00:35:25,820 --> 00:35:33,350 up here such that if I assume that the potential is precisely 494 00:35:33,350 --> 00:35:42,870 this for all r that are greater than r0, then 495 00:35:42,870 --> 00:35:47,410 I'm replacing the actual potential 496 00:35:47,410 --> 00:35:50,710 by something like this. 497 00:35:50,710 --> 00:35:54,550 And the minimum depth of potential 498 00:35:54,550 --> 00:35:58,010 will occur at this cusp over here at minus u0. 499 00:36:08,190 --> 00:36:13,090 So why did I do that? 500 00:36:13,090 --> 00:36:16,410 Because now I can calculate with that 501 00:36:16,410 --> 00:36:18,850 what the second virial coefficient is. 502 00:36:18,850 --> 00:36:21,870 So what is B2? 503 00:36:21,870 --> 00:36:31,060 B2 is minus 1/2 the integral of relative potential. 504 00:36:35,100 --> 00:36:40,620 Now this integral will take two parts 505 00:36:40,620 --> 00:36:44,800 from these two different contributions. 506 00:36:44,800 --> 00:36:50,050 One part is when I'm in the regime where the particles are 507 00:36:50,050 --> 00:36:52,570 excluded, the potential is infinity. 508 00:36:52,570 --> 00:36:57,430 And there, f is simply minus 1. 509 00:36:57,430 --> 00:37:02,180 So that minus 1 then gets integrated from 0 to r, 510 00:37:02,180 --> 00:37:04,700 giving me the volume that is excluded. 511 00:37:04,700 --> 00:37:07,750 So what I will get here is a minus 1 512 00:37:07,750 --> 00:37:11,400 times the excluded volume. 513 00:37:11,400 --> 00:37:16,680 That's the first part, where the volume is, of course, 514 00:37:16,680 --> 00:37:21,628 4 pi over 3 r0 cubed for the volume of the sphere. 515 00:37:24,700 --> 00:37:27,180 And then I have to add the part where 516 00:37:27,180 --> 00:37:32,000 I go from f0 all the way to infinity. 517 00:37:32,000 --> 00:37:35,110 The potential is vertically symmetric. 518 00:37:35,110 --> 00:37:41,930 So d cubed r becomes 4 pi r squared dr. 519 00:37:41,930 --> 00:37:49,390 And I have to integrate e to the minus 520 00:37:49,390 --> 00:37:52,700 beta times the attractive part of the potential. 521 00:37:52,700 --> 00:37:57,900 So it's beta u0 r0 over r to the sixth minus 1. 522 00:38:03,390 --> 00:38:06,780 So what I will do, I will additionally 523 00:38:06,780 --> 00:38:11,280 assume that I'm in the range of parameters 524 00:38:11,280 --> 00:38:16,200 where this beta u0 is much less than 1, 525 00:38:16,200 --> 00:38:19,840 that is, at temperatures that are higher 526 00:38:19,840 --> 00:38:22,880 compared to the depth of this potential converted 527 00:38:22,880 --> 00:38:24,910 to units of kT. 528 00:38:24,910 --> 00:38:29,010 And if that is the case, then I can expand this. 529 00:38:29,010 --> 00:38:31,700 And the expansion to the lowest order 530 00:38:31,700 --> 00:38:37,380 will simply give me beta u0 r0 over r to the sixth power. 531 00:38:37,380 --> 00:38:40,485 And I will ignore higher order terms. 532 00:38:40,485 --> 00:38:44,820 So this, if you like, is order of beta u0 squared. 533 00:38:49,570 --> 00:38:54,530 Now, having done that, then the second integral becomes simple. 534 00:38:54,530 --> 00:38:56,700 Because I have to integrate r squared 535 00:38:56,700 --> 00:39:01,330 divided by r to the fourth, which is r to the minus 4. 536 00:39:01,330 --> 00:39:09,320 Integral of r to the minus 4 gives me r to the minus 3. 537 00:39:09,320 --> 00:39:13,020 And then there's a factor of minus 1/3 from the integration. 538 00:39:13,020 --> 00:39:18,280 It has to be evaluated between 0 and infinity. 539 00:39:18,280 --> 00:39:26,040 And so the final answer, then, for B2 is minus 1/2. 540 00:39:26,040 --> 00:39:32,211 I have minus 4 pi over 3 r0 cubed from the excluded volume 541 00:39:32,211 --> 00:39:32,710 part. 542 00:39:35,850 --> 00:39:49,180 From here, I will get a 4 pi beta u0 r0 to the sixth. 543 00:39:52,700 --> 00:39:57,130 And then I will get this factor of r the minus 3 544 00:39:57,130 --> 00:39:59,500 over 3 evaluated between infinity, 545 00:39:59,500 --> 00:40:05,740 which gives me 0 and r0. 546 00:40:13,720 --> 00:40:16,530 So you can see that with this potential, actually 547 00:40:16,530 --> 00:40:22,900 both terms are proportional to 4 pi r0 cubed over 3. 548 00:40:22,900 --> 00:40:29,200 And so I can write the answer as minus omega over 2 549 00:40:29,200 --> 00:40:33,760 where I have defined omega to be this 4 pi r0 cubed 550 00:40:33,760 --> 00:40:36,505 over 3, which is the excluded volume. 551 00:40:39,240 --> 00:40:46,404 Basically, it's the volume that is excluded from exploration 552 00:40:46,404 --> 00:40:47,570 when you have two particles. 553 00:40:47,570 --> 00:40:52,900 Because their center of mass cannot come as close as r0. 554 00:40:52,900 --> 00:40:58,330 So that 1/2 appears here as 1. 555 00:40:58,330 --> 00:41:01,810 Because I already took care of that. 556 00:41:01,810 --> 00:41:05,400 This rest of the coefficients are proportional to this 557 00:41:05,400 --> 00:41:08,690 except with a factor of minus beta u0. 558 00:41:08,690 --> 00:41:12,280 So I get minus beta u0. 559 00:41:12,280 --> 00:41:19,351 So this is your second virial coefficient for this. 560 00:41:19,351 --> 00:41:21,100 AUDIENCE: It should be positive out front. 561 00:41:21,100 --> 00:41:21,849 PROFESSOR: Pardon? 562 00:41:21,849 --> 00:41:26,544 AUDIENCE: It should be positive out front, [INAUDIBLE]? 563 00:41:26,544 --> 00:41:28,516 PROFESSOR: Yes, because there was a minus, 564 00:41:28,516 --> 00:41:29,502 and there's a minus. 565 00:41:29,502 --> 00:41:30,488 There's a plus. 566 00:41:50,701 --> 00:41:51,687 Let's keep that. 567 00:42:00,620 --> 00:42:05,620 So what we have so far, we've said 568 00:42:05,620 --> 00:42:11,940 that beta times the pressure starts with ideal gas behavior 569 00:42:11,940 --> 00:42:13,644 n. 570 00:42:13,644 --> 00:42:17,440 And then I have the next correction, 571 00:42:17,440 --> 00:42:22,860 which is this B2 multiplying n squared. 572 00:42:22,860 --> 00:42:27,970 So I will have n squared times this coefficient 573 00:42:27,970 --> 00:42:31,070 that I have calculated for this type of gas, which 574 00:42:31,070 --> 00:42:34,710 is omega over 2 1 minus beta u0. 575 00:42:34,710 --> 00:42:37,540 And presumably there are higher order terms in this. 576 00:42:42,860 --> 00:42:47,840 So whenever you see something like this, 577 00:42:47,840 --> 00:42:51,760 then you have to start thinking about, 578 00:42:51,760 --> 00:42:55,680 is this a good expansion? 579 00:42:55,680 --> 00:42:59,690 So let's think about how this expansion could 580 00:42:59,690 --> 00:43:00,790 have become problematic. 581 00:43:03,690 --> 00:43:08,850 Actually, there is already one thing that I should have noted 582 00:43:08,850 --> 00:43:17,790 and I didn't, which is short versus long range potential. 583 00:43:21,890 --> 00:43:25,680 Now, it was nice for this potential 584 00:43:25,680 --> 00:43:29,460 that I got an answer that was in the form 585 00:43:29,460 --> 00:43:33,440 that I could write down, factor out omega. 586 00:43:33,440 --> 00:43:35,570 You can say, well, would you have always 587 00:43:35,570 --> 00:43:38,630 been able to do something similar to that? 588 00:43:38,630 --> 00:43:40,630 Because you see, ultimately what this 589 00:43:40,630 --> 00:43:47,950 says is that in order for all of these terms in this expansion 590 00:43:47,950 --> 00:43:54,560 to be dimensionally correct, the next term is a density, 591 00:43:54,560 --> 00:43:57,210 has dimensions of inverse volume, 592 00:43:57,210 --> 00:43:58,720 density squared compared to this. 593 00:43:58,720 --> 00:44:02,780 So it should be compensated by some factor of volume. 594 00:44:02,780 --> 00:44:05,530 And we can see that this factor of volume 595 00:44:05,530 --> 00:44:08,250 came from something that was of the order 596 00:44:08,250 --> 00:44:12,986 of this size of the molecule, this r0. 597 00:44:12,986 --> 00:44:16,860 OK, so it's interesting. 598 00:44:16,860 --> 00:44:20,830 It says that it's really the short-range part 599 00:44:20,830 --> 00:44:26,770 of the potential that seems to be setting the correction. 600 00:44:26,770 --> 00:44:29,500 Now, where did that come from? 601 00:44:29,500 --> 00:44:31,670 Well, there was one place that I had 602 00:44:31,670 --> 00:44:34,920 to do an integration over the potential. 603 00:44:34,920 --> 00:44:36,850 And I found that the integration was 604 00:44:36,850 --> 00:44:40,876 dominated by the lower range. 605 00:44:40,876 --> 00:44:43,710 That was over here. 606 00:44:43,710 --> 00:44:45,395 Where would this become difficult? 607 00:44:49,454 --> 00:44:52,300 AUDIENCE: In a small box? 608 00:44:52,300 --> 00:44:53,876 PROFESSOR: In a small box, but we 609 00:44:53,876 --> 00:44:56,050 are always taking the thermodynamic limit 610 00:44:56,050 --> 00:44:57,721 where V goes to infinity. 611 00:44:57,721 --> 00:44:59,685 AUDIENCE: Or inverse square log? 612 00:44:59,685 --> 00:45:01,790 PROFESSOR: Inverse square log, yes, that's right. 613 00:45:01,790 --> 00:45:05,090 So who says that this integral should converge? 614 00:45:05,090 --> 00:45:10,760 I'm doing an integral d cubed r, something like v of r. 615 00:45:10,760 --> 00:45:14,240 And convergence was the reason why this integral 616 00:45:14,240 --> 00:45:16,695 was dominated by short distance. 617 00:45:16,695 --> 00:45:22,490 If this potential goes like 1 over r cubed, 618 00:45:22,490 --> 00:45:26,050 then it will logarithmically depend on the size of the box. 619 00:45:26,050 --> 00:45:29,320 For Coulomb interaction, 1/r potential 620 00:45:29,320 --> 00:45:30,940 you can't even think about it. 621 00:45:30,940 --> 00:45:34,030 It's just too divergent. 622 00:45:34,030 --> 00:45:38,640 So this expansion will fail for potentials 623 00:45:38,640 --> 00:45:43,300 that have tails that are decaying as 1 624 00:45:43,300 --> 00:45:46,680 over r cubed or even slower. 625 00:45:46,680 --> 00:45:50,670 Fortunately, that's not the case for van der Waals' 626 00:45:50,670 --> 00:45:53,320 potential and typical potentials. 627 00:45:53,320 --> 00:45:56,100 But if you have a plasma, you have to worry about this. 628 00:45:56,100 --> 00:45:58,610 And that's why I was also saying last time 629 00:45:58,610 --> 00:46:01,597 that the typical expansions that you have to do for plasmas 630 00:46:01,597 --> 00:46:02,180 are different. 631 00:46:04,930 --> 00:46:10,050 So given that we are dominated by the short range, 632 00:46:10,050 --> 00:46:14,380 what is the correction that I've obtained 633 00:46:14,380 --> 00:46:16,510 compared to the first term? 634 00:46:16,510 --> 00:46:19,915 So essentially, I have calculated a second order term 635 00:46:19,915 --> 00:46:23,450 that is of the order of this divided by the first order 636 00:46:23,450 --> 00:46:25,940 term and pressure that was density. 637 00:46:25,940 --> 00:46:31,020 And we find that this is of the order of n omega. 638 00:46:31,020 --> 00:46:36,810 How many particles are within the range of interaction? 639 00:46:36,810 --> 00:46:44,720 And this would be kind of like the ratio 640 00:46:44,720 --> 00:46:53,390 of the density of liquid to the density of gas. 641 00:46:53,390 --> 00:46:59,648 Because the density of liquid would be related-- oh, 642 00:46:59,648 --> 00:47:00,731 it's the other way around. 643 00:47:04,030 --> 00:47:09,260 The density of liquid would be 1 over the volume 644 00:47:09,260 --> 00:47:10,860 that one particle occupies. 645 00:47:10,860 --> 00:47:15,410 So the density of liquid would go in the denominator. 646 00:47:15,410 --> 00:47:24,020 The density of gas is this n that I have over here. 647 00:47:24,020 --> 00:47:26,420 And again, for the gas in this room, 648 00:47:26,420 --> 00:47:29,980 this ratio is of the order of 10 to the minus 3. 649 00:47:29,980 --> 00:47:31,780 And we are safe. 650 00:47:31,780 --> 00:47:35,300 But if I were to start compressing this 651 00:47:35,300 --> 00:47:38,500 so that I go to higher and higher densities, 652 00:47:38,500 --> 00:47:41,690 ultimately I say that this second order term becomes 653 00:47:41,690 --> 00:47:43,300 of the order of the first order term. 654 00:47:43,300 --> 00:47:45,730 And then perturbation doesn't make sense. 655 00:47:45,730 --> 00:47:49,430 Naturally, the reason for that is I haven't calculated. 656 00:47:49,430 --> 00:47:51,530 But typically what you find is that if you 657 00:47:51,530 --> 00:47:53,360 go to higher and higher order terms 658 00:47:53,360 --> 00:47:57,400 in the series, in most cases, but clearly not in all cases, 659 00:47:57,400 --> 00:48:01,220 the ratio of successive terms is more or less 660 00:48:01,220 --> 00:48:03,990 set by the ratio of the first terms. 661 00:48:03,990 --> 00:48:06,770 It has to be dimensionally correct. 662 00:48:06,770 --> 00:48:09,170 And we've established that the typical dimension that 663 00:48:09,170 --> 00:48:15,450 is controlling everything is the volume of the particle. 664 00:48:15,450 --> 00:48:20,590 So this expansion will fail for long range potentials, 665 00:48:20,590 --> 00:48:23,980 for going to liquid-like densities. 666 00:48:23,980 --> 00:48:27,570 But also I made another thing, which 667 00:48:27,570 --> 00:48:30,540 is that I assumed that that could expand this exponential. 668 00:48:33,140 --> 00:48:35,820 And then in principle, there are other difficulties 669 00:48:35,820 --> 00:48:40,070 that could arise if this condition that I wrote here 670 00:48:40,070 --> 00:48:41,080 is violated. 671 00:48:41,080 --> 00:48:46,080 If beta u0 is greater than 1, then this coefficient by itself 672 00:48:46,080 --> 00:48:49,180 becomes large exponentially. 673 00:48:49,180 --> 00:48:53,160 And then you would expect that higher order terms will also 674 00:48:53,160 --> 00:48:56,350 get more factors of this exponential and potential. 675 00:48:56,350 --> 00:48:58,130 And things will blow up on you. 676 00:48:58,130 --> 00:49:01,390 So it will also have difficulties 677 00:49:01,390 --> 00:49:06,550 at low temperatures for attractive potentials. 678 00:49:10,400 --> 00:49:13,010 Again, the reason for that is obvious. 679 00:49:13,010 --> 00:49:15,320 If you have an attractive potential, 680 00:49:15,320 --> 00:49:17,380 you go to low enough temperature, 681 00:49:17,380 --> 00:49:20,330 and the ground state is everybody sticking together-- 682 00:49:20,330 --> 00:49:25,560 looks nothing like a gas, looks like a solid or something. 683 00:49:25,560 --> 00:49:28,170 So these are the kind of limitations 684 00:49:28,170 --> 00:49:29,667 that one has in this series. 685 00:49:54,060 --> 00:50:01,650 OK, let's be brave and do some more 686 00:50:01,650 --> 00:50:03,500 rearrangements of this equation. 687 00:50:03,500 --> 00:50:06,110 So I have that to this order, beta P 688 00:50:06,110 --> 00:50:10,930 is n plus n squared this excluded 689 00:50:10,930 --> 00:50:14,680 volume over 2 1 minus beta u0. 690 00:50:17,870 --> 00:50:22,430 And then there's higher order terms of course. 691 00:50:22,430 --> 00:50:25,800 And then I notice that there are two terms on this equation that 692 00:50:25,800 --> 00:50:27,860 are proportional to beta. 693 00:50:27,860 --> 00:50:30,680 And I say, why not put both of them together? 694 00:50:30,680 --> 00:50:32,650 So I will have beta. 695 00:50:32,650 --> 00:50:34,910 Bring that term to this side, and it 696 00:50:34,910 --> 00:50:42,800 becomes P plus n squared omega over 2 u0. 697 00:50:45,420 --> 00:50:48,510 And then what is left on the other side? 698 00:50:48,510 --> 00:50:49,950 Let's factor out the n. 699 00:50:49,950 --> 00:50:53,650 I expect this to be a series in higher and higher powers of n. 700 00:50:53,650 --> 00:50:56,420 And the first correction to 1 comes 701 00:50:56,420 --> 00:50:59,130 from here, which is n omega over 2. 702 00:50:59,130 --> 00:51:01,550 And I expect there to be higher order terms. 703 00:51:13,830 --> 00:51:18,660 Now again, to order of n squared that I have calculated things 704 00:51:18,660 --> 00:51:22,310 correctly, this expression is no different 705 00:51:22,310 --> 00:51:27,870 from the following expression-- 1 minus n omega over 2. 706 00:51:27,870 --> 00:51:31,910 Again, there will be higher order terms in n. 707 00:51:31,910 --> 00:51:35,100 But the order of n squared, both of these equations, 708 00:51:35,100 --> 00:51:36,325 expressions, are equivalent. 709 00:51:41,220 --> 00:51:46,740 So if I now ignore higher order terms, 710 00:51:46,740 --> 00:51:54,700 this whole thing is equivalent to P plus n squared u-- 711 00:51:54,700 --> 00:51:57,790 well, let's write this in the following fashion-- p 712 00:51:57,790 --> 00:52:05,220 plus u0 omega over 2 N over V, which is density squared. 713 00:52:07,790 --> 00:52:13,350 And actually this side, if I were to multiply by V, 714 00:52:13,350 --> 00:52:14,860 what do I get? 715 00:52:14,860 --> 00:52:20,055 This becomes numerator V, denominator V minus N 716 00:52:20,055 --> 00:52:21,710 omega over 2. 717 00:52:21,710 --> 00:52:27,840 Let's multiply by V minus N omega over 2. 718 00:52:27,840 --> 00:52:32,900 And the right-hand side will be-- 719 00:52:32,900 --> 00:52:36,810 AUDIENCE: Is that where it should be there? 720 00:52:36,810 --> 00:52:37,866 PROFESSOR: What happened? 721 00:52:37,866 --> 00:52:40,346 AUDIENCE: This one, [INAUDIBLE]. 722 00:52:45,802 --> 00:52:52,860 PROFESSOR: Yes, so if I multiply this, which is N over V, by V, 723 00:52:52,860 --> 00:52:56,920 I will get N, good, which removes 724 00:52:56,920 --> 00:52:58,490 the difficulty that I had. 725 00:52:58,490 --> 00:53:02,790 Because now I multiply by kT. 726 00:53:02,790 --> 00:53:05,460 And the left hand side disappears. 727 00:53:05,460 --> 00:53:07,650 And on the right hand side, I will get NkT. 728 00:53:13,940 --> 00:53:18,830 So we'll spend some time on this equation 729 00:53:18,830 --> 00:53:22,105 that you will likely recognize as the van der Waals equation. 730 00:53:28,790 --> 00:53:34,810 I kind of justify it by rearranging this series. 731 00:53:34,810 --> 00:53:37,390 But van der Waals himself, of course, 732 00:53:37,390 --> 00:53:40,890 had a different way of justifying it, 733 00:53:40,890 --> 00:53:45,080 which is that basically if we think about the ideal gas, 734 00:53:45,080 --> 00:53:51,260 and you have particles that are moving within some volume V, 735 00:53:51,260 --> 00:53:54,600 if you have excluded volume interactions, 736 00:53:54,600 --> 00:53:59,420 then some of this volume is no longer available. 737 00:53:59,420 --> 00:54:01,690 And so maybe what you should do is 738 00:54:01,690 --> 00:54:05,300 you should reduce the volume by an amount that 739 00:54:05,300 --> 00:54:07,470 is proportional to the number of particles. 740 00:54:10,320 --> 00:54:14,410 This factor of 1/2 is actually very interesting. 741 00:54:14,410 --> 00:54:16,510 Because it is correct. 742 00:54:16,510 --> 00:54:20,610 And I see that a number of people 743 00:54:20,610 --> 00:54:24,020 in well known journals, et cetera, 744 00:54:24,020 --> 00:54:27,950 write that the excluded volume should be essentially 745 00:54:27,950 --> 00:54:31,630 n times the volume that is excluded around each particle. 746 00:54:31,630 --> 00:54:32,560 It's not that. 747 00:54:32,560 --> 00:54:34,480 It is 1/2 of that. 748 00:54:34,480 --> 00:54:36,600 And I'll leave you to mull on that. 749 00:54:36,600 --> 00:54:39,790 Because we will justify it later on. 750 00:54:39,790 --> 00:54:41,990 But in the meantime, you can think 751 00:54:41,990 --> 00:54:46,860 about why the factor of 1/2 is there. 752 00:54:46,860 --> 00:54:50,450 The other issue is that-- so there 753 00:54:50,450 --> 00:54:54,100 has to be a correction to the volume. 754 00:54:54,100 --> 00:54:57,070 And the kind of hand-waving statement 755 00:54:57,070 --> 00:54:59,960 that you make about the correction to pressure 756 00:54:59,960 --> 00:55:05,730 is that if you think about the particle that is in the middle, 757 00:55:05,730 --> 00:55:10,190 it is being attracted by everybody, whereas when 758 00:55:10,190 --> 00:55:13,750 it comes to the surface, it is really being attracted 759 00:55:13,750 --> 00:55:17,760 by things that are half of the space. 760 00:55:17,760 --> 00:55:19,810 So there is an effective potential 761 00:55:19,810 --> 00:55:23,850 that the particles feel from the collective action 762 00:55:23,850 --> 00:55:27,790 of all the others, which is slightly less steep when 763 00:55:27,790 --> 00:55:31,140 you approach the boundaries. 764 00:55:31,140 --> 00:55:36,030 And therefore, you can either think that because of this, 765 00:55:36,030 --> 00:55:38,590 there's less density that you have at the boundary. 766 00:55:38,590 --> 00:55:41,050 Less density will give you less pressure. 767 00:55:41,050 --> 00:55:42,770 Or if you have a particle that is 768 00:55:42,770 --> 00:55:45,780 kind of moving towards the wall, it 769 00:55:45,780 --> 00:55:47,840 is being pulled back so it doesn't 770 00:55:47,840 --> 00:55:51,110 hit the wall as strongly as you would expect, 771 00:55:51,110 --> 00:55:53,440 that would give the pressure of the ideal gas. 772 00:55:53,440 --> 00:55:57,730 So there is a pressure that has to be reduced related 773 00:55:57,730 --> 00:56:00,910 to the strength of the potential and something that 774 00:56:00,910 --> 00:56:03,160 has to do with all of the other particles. 775 00:56:03,160 --> 00:56:06,470 And there's density squared will appear there. 776 00:56:06,470 --> 00:56:10,470 We will have a more full justification of this equation 777 00:56:10,470 --> 00:56:11,690 later on. 778 00:56:11,690 --> 00:56:16,140 But for the time being, let's sort of sit with this equation 779 00:56:16,140 --> 00:56:18,510 and think about its consequences for awhile. 780 00:56:25,200 --> 00:56:27,560 Because the thing that we would like to do 781 00:56:27,560 --> 00:56:35,060 is we have come from a perspective of looking 782 00:56:35,060 --> 00:56:39,710 at the ideal gas and how the pressure of the ideal gas 783 00:56:39,710 --> 00:56:43,610 starts to get corrected because of the interactions. 784 00:56:43,610 --> 00:56:46,340 Of course, things become interesting 785 00:56:46,340 --> 00:56:48,470 when you go to the dense limit. 786 00:56:48,470 --> 00:56:51,040 And then the gas becomes something like a liquid. 787 00:56:51,040 --> 00:56:53,830 And you have transitions and things like that. 788 00:56:53,830 --> 00:56:58,570 So really, it's the other limit, the dense, highly interacting 789 00:56:58,570 --> 00:57:00,740 limited that is interesting. 790 00:57:00,740 --> 00:57:05,640 And to get that, we have few choices. 791 00:57:05,640 --> 00:57:10,770 Either I have to somehow sum many, many terms in the series, 792 00:57:10,770 --> 00:57:14,700 which will be very difficult, and we can't do that, 793 00:57:14,700 --> 00:57:18,155 or you can make some kind of approximation, rearrangement, 794 00:57:18,155 --> 00:57:19,490 and a guess. 795 00:57:19,490 --> 00:57:24,100 And this is what the van der Waals equation is based on. 796 00:57:24,100 --> 00:57:27,950 I made the guess here by somewhat rearranging 797 00:57:27,950 --> 00:57:30,490 and re-summing the terms in this series. 798 00:57:30,490 --> 00:57:34,390 But I will give you shortly a different justification 799 00:57:34,390 --> 00:57:37,920 that is more transparent and tells you immediately 800 00:57:37,920 --> 00:57:39,740 what the limitations are. 801 00:57:39,740 --> 00:57:42,550 But basically that's why we are going 802 00:57:42,550 --> 00:57:45,110 to spend some time with this equation. 803 00:57:45,110 --> 00:57:48,640 Because ultimately, we are hoping to transition 804 00:57:48,640 --> 00:57:51,502 from the weakly interacting case to the strongly interacting 805 00:57:51,502 --> 00:57:52,002 case. 806 00:57:58,360 --> 00:57:59,277 Yes. 807 00:57:59,277 --> 00:58:00,152 AUDIENCE: [INAUDIBLE] 808 00:58:09,050 --> 00:58:12,190 PROFESSOR: No, no, no, no, omega was 809 00:58:12,190 --> 00:58:16,030 defined as the volume excluded around one particle. 810 00:58:16,030 --> 00:58:19,070 So if you're thinking about billiard balls, 811 00:58:19,070 --> 00:58:21,490 r0 is the diameter. 812 00:58:21,490 --> 00:58:23,320 It's not the radius. 813 00:58:23,320 --> 00:58:28,780 And 4 pi over 3 r0 cubed is 8 times the diameter 814 00:58:28,780 --> 00:58:30,980 of a single billiard ball. 815 00:58:30,980 --> 00:58:34,030 So the correction that we get, if you like, 816 00:58:34,030 --> 00:58:38,860 is 4 times the volume of a billiard ball multiplied 817 00:58:38,860 --> 00:58:40,472 by the number of billiard balls. 818 00:58:51,350 --> 00:58:52,249 Where was I? 819 00:59:10,840 --> 00:59:17,060 OK, so what this equation gives you, the van der Waals, 820 00:59:17,060 --> 00:59:20,350 is an expression for how the pressure behaves 821 00:59:20,350 --> 00:59:21,765 as a function of volume. 822 00:59:26,390 --> 00:59:31,970 Actually, it would be nicer if we were to sort of replace 823 00:59:31,970 --> 00:59:35,150 this by volume per particle, which 824 00:59:35,150 --> 00:59:36,670 would be the inverse density. 825 00:59:36,670 --> 00:59:39,980 But you can use one or the other. 826 00:59:39,980 --> 00:59:42,320 It doesn't matter. 827 00:59:42,320 --> 00:59:46,410 Now, what you find is that there is, first 828 00:59:46,410 --> 00:59:51,510 of all, a limitation to the volume. 829 00:59:51,510 --> 00:59:54,880 So basically, none of your cares are 830 00:59:54,880 --> 00:59:59,520 going to go to volumes that are lower than n omega over 2. 831 00:59:59,520 --> 01:00:04,040 So basically there's a barrier here that occurs over here. 832 01:00:04,040 --> 01:00:06,190 But if you go to the other limit, 833 01:00:06,190 --> 01:00:10,720 where you go to large volumes, you can ignore terms like this. 834 01:00:10,720 --> 01:00:15,610 And then you get back the kind of ideal gas behavior. 835 01:00:15,610 --> 01:00:19,060 So basically, in one limit, where 836 01:00:19,060 --> 01:00:22,720 you are either at high temperatures-- 837 01:00:22,720 --> 01:00:27,340 and at high temperatures, essentially, the correction 838 01:00:27,340 --> 01:00:30,660 here will also be negligible. 839 01:00:30,660 --> 01:00:34,090 Or you are at high values of the volume. 840 01:00:34,090 --> 01:00:38,160 You get isotherms that are very much like the isotherms 841 01:00:38,160 --> 01:00:41,180 that you have for the ideal gas, except that 842 01:00:41,180 --> 01:00:45,760 rather than asymptote to 0, PV going like NkT, 843 01:00:45,760 --> 01:00:50,910 you asymptote to this excluded volume. 844 01:00:50,910 --> 01:00:54,440 So this is for high T, T large. 845 01:00:57,170 --> 01:00:58,680 T larger than what? 846 01:00:58,680 --> 01:01:01,910 Well essentially, what happens is 847 01:01:01,910 --> 01:01:07,020 that if I look at the pressure, I can write it in this fashion 848 01:01:07,020 --> 01:01:07,520 also. 849 01:01:07,520 --> 01:01:15,060 It is NkT divided by V minus N omega over 2. 850 01:01:15,060 --> 01:01:18,180 So that's the term that dominates at high temperature. 851 01:01:18,180 --> 01:01:20,640 It's proportional to kT. 852 01:01:20,640 --> 01:01:26,850 But then there's the subtraction u0 omega over 2 N/V squared. 853 01:01:26,850 --> 01:01:30,270 And again, this term is not so important at large volume. 854 01:01:30,270 --> 01:01:32,300 Because at large volume, this 1/V 855 01:01:32,300 --> 01:01:35,350 is more dominant than 1 over V squared. 856 01:01:35,350 --> 01:01:38,880 But as you go to lower temperatures 857 01:01:38,880 --> 01:01:42,980 and intermediate volumes, then essentially you 858 01:01:42,980 --> 01:01:47,140 have potentially a correction that falls off as minus 1 859 01:01:47,140 --> 01:01:49,110 over V squared. 860 01:01:49,110 --> 01:01:53,620 And so this correction that falls off as 1 over V squared 861 01:01:53,620 --> 01:01:57,880 can potentially modify your curve, 862 01:01:57,880 --> 01:02:03,370 bring it down, and then give it a structure such as this. 863 01:02:03,370 --> 01:02:06,350 So this is T less than. 864 01:02:06,350 --> 01:02:10,320 And clearly, between these two types of behavior, 865 01:02:10,320 --> 01:02:14,100 where there is monotonic behavior 866 01:02:14,100 --> 01:02:16,880 or non-monotonic behavior, there has 867 01:02:16,880 --> 01:02:20,860 to be a limiting curve that, let's say, does something 868 01:02:20,860 --> 01:02:27,190 like this, comes tangentially to the horizontal axis, 869 01:02:27,190 --> 01:02:30,800 and then goes on like this. 870 01:02:30,800 --> 01:02:33,870 So this would be for T equals Tc. 871 01:02:33,870 --> 01:02:37,543 This is T greater than Tc, T less than Tc. 872 01:02:47,700 --> 01:02:50,140 OK, so that's fine, except that now we 873 01:02:50,140 --> 01:02:53,080 have encountered the difficulty. 874 01:02:53,080 --> 01:02:57,040 Because one of the things that we had established 875 01:02:57,040 --> 01:03:10,210 for thermodynamic stability was that delta P delta 876 01:03:10,210 --> 01:03:11,870 V had to be negative. 877 01:03:15,870 --> 01:03:20,690 And the ideal gas curve and portions 878 01:03:20,690 --> 01:03:26,480 of this curve which have a negative slope 879 01:03:26,480 --> 01:03:29,280 are all consistent with this. 880 01:03:29,280 --> 01:03:39,720 But this portion over here where dP by dV is positive, 881 01:03:39,720 --> 01:03:42,350 it kind of violates the condition 882 01:03:42,350 --> 01:03:48,380 that the compressibility kappa T, which is minus 1/V dV 883 01:03:48,380 --> 01:03:51,550 by dP at constant temperature, better be positive. 884 01:03:55,360 --> 01:04:02,830 And so clearly, the expression that one gets through this van 885 01:04:02,830 --> 01:04:07,850 der Waals equation has a limitation. 886 01:04:07,850 --> 01:04:11,460 And the most natural way about it 887 01:04:11,460 --> 01:04:15,320 is to say, well, the question that you wrote down 888 01:04:15,320 --> 01:04:19,214 is clearly incorrect. 889 01:04:19,214 --> 01:04:20,130 That's certainly true. 890 01:04:20,130 --> 01:04:22,710 Also, this equation is incorrect, 891 01:04:22,710 --> 01:04:26,856 except that this kind of reminds us 892 01:04:26,856 --> 01:04:34,780 with what actually happens if I look at the isotherms of a gas, 893 01:04:34,780 --> 01:04:38,660 such as gas in this room, or something that 894 01:04:38,660 --> 01:04:41,010 is more familiar, such as water. 895 01:04:41,010 --> 01:04:46,330 What you find is that at high temperatures, 896 01:04:46,330 --> 01:04:50,440 you indeed have curves that look like this. 897 01:04:53,130 --> 01:04:56,550 But at low temperatures, you liquidify. 898 01:04:56,550 --> 01:05:00,632 And what you have is a zone of coexistence, 899 01:05:00,632 --> 01:05:03,580 let's say something like this, by which I 900 01:05:03,580 --> 01:05:07,950 mean that the isotherm that you draw 901 01:05:07,950 --> 01:05:13,910 has a portion that lives in the gas phase 902 01:05:13,910 --> 01:05:18,630 and is a slightly modified version of the ideal gas. 903 01:05:18,630 --> 01:05:21,200 But then it has a portion that corresponds 904 01:05:21,200 --> 01:05:24,090 to the more or less incompressible liquid. 905 01:05:24,090 --> 01:05:28,870 Although this has clearly still some finite compressibility. 906 01:05:28,870 --> 01:05:33,100 And then in between, there is a region where if you have a box, 907 01:05:33,100 --> 01:05:36,090 part of your box would be liquid, and part of your box 908 01:05:36,090 --> 01:05:37,240 would be gas. 909 01:05:37,240 --> 01:05:42,100 And as you compress the box, the proportion of liquid and gas 910 01:05:42,100 --> 01:05:44,120 will change. 911 01:05:44,120 --> 01:05:48,140 And this happens for T less than some Tc. 912 01:05:48,140 --> 01:05:53,030 And once more, there is a curve that kind of 913 01:05:53,030 --> 01:05:58,660 looks like this at some intermediate temperature Tc. 914 01:06:01,550 --> 01:06:06,220 So you look at the comparison, you say, well, 915 01:06:06,220 --> 01:06:09,340 as long as I stay above Tc-- of course 916 01:06:09,340 --> 01:06:13,865 I don't expect this equation to give the right numbers 917 01:06:13,865 --> 01:06:15,750 for what Tc or whatever is. 918 01:06:15,750 --> 01:06:19,940 But qualitatively, I get topologies and behaviors 919 01:06:19,940 --> 01:06:22,970 at that T greater than Tc all the way up 920 01:06:22,970 --> 01:06:26,300 to Tc are not very different. 921 01:06:26,300 --> 01:06:29,660 But at T less than Tc, they also signal 922 01:06:29,660 --> 01:06:31,980 that something bad is happening. 923 01:06:31,980 --> 01:06:37,120 And somehow, the original description has to be modified. 924 01:06:37,120 --> 01:06:42,080 And so the thing that Maxwell and van der Waals and company 925 01:06:42,080 --> 01:06:50,520 did was to somehow convert this incorrect set of equations 926 01:06:50,520 --> 01:06:54,260 to something that resembles this. 927 01:06:54,260 --> 01:06:57,724 So let's see how they managed to do this. 928 01:07:23,960 --> 01:07:28,180 Actually, this may not be a bad thing to keep in mind. 929 01:07:30,750 --> 01:07:33,770 Let's have a thermodynamic-- well, 930 01:07:33,770 --> 01:07:40,200 I kind of emphasize that there is thermodynamic reason for why 931 01:07:40,200 --> 01:07:43,560 this is not valid. 932 01:07:43,560 --> 01:07:45,760 Always there's a corresponding reason 933 01:07:45,760 --> 01:07:48,100 if you look at things from the perspective 934 01:07:48,100 --> 01:07:49,920 of statistical mechanics. 935 01:07:49,920 --> 01:07:51,590 So it may be useful to sort of think 936 01:07:51,590 --> 01:07:54,820 about what's happening from that perspective. 937 01:07:54,820 --> 01:07:57,510 Let's imagine that we are in this grand canonical ensemble. 938 01:07:57,510 --> 01:08:02,040 In the grand canonical ensemble, the number of particles, 939 01:08:02,040 --> 01:08:05,760 as we discussed, is not fixed. 940 01:08:05,760 --> 01:08:10,800 But the mean number of particles is given by the expressions 941 01:08:10,800 --> 01:08:12,850 that we saw over here, is related 942 01:08:12,850 --> 01:08:17,700 to the pressure through dP by d mu at constant temperature. 943 01:08:22,560 --> 01:08:25,790 But there are fluctuations. 944 01:08:25,790 --> 01:08:32,340 So you would say that the fluctuations are related 945 01:08:32,340 --> 01:08:38,490 by taking a second derivative of this object. 946 01:08:38,490 --> 01:08:43,910 Because clearly, q, if I were to expand it in powers of beta mu, 947 01:08:43,910 --> 01:08:47,779 will generate various moments of n. 948 01:08:47,779 --> 01:08:50,880 Log of q will generate cumulants. 949 01:08:50,880 --> 01:08:54,399 So the variance would be one more derivative. 950 01:08:54,399 --> 01:08:59,880 And so that will amount to taking 951 01:08:59,880 --> 01:09:02,310 a derivative of the first derivative, which 952 01:09:02,310 --> 01:09:04,740 was the number itself. 953 01:09:04,740 --> 01:09:11,970 So I will get this to be dN by d mu at constant T, 954 01:09:11,970 --> 01:09:15,250 except that there's an additional factor of kT. 955 01:09:15,250 --> 01:09:18,529 So maybe I will write this carefully enough. 956 01:09:18,529 --> 01:09:28,020 I have d2 log Q with respect to beta mu squared 957 01:09:28,020 --> 01:09:30,632 will give me N squared. 958 01:09:30,632 --> 01:09:37,370 Now, the first derivative with respect to beta mu of log Q 959 01:09:37,370 --> 01:09:42,200 gave me the mean, which I'm actually thinking of as N 960 01:09:42,200 --> 01:09:43,870 itself. 961 01:09:43,870 --> 01:09:45,800 And then there's this factor of beta. 962 01:09:45,800 --> 01:09:50,955 So this is really kT dN by d mu. 963 01:09:50,955 --> 01:09:53,290 All of these are done at constant temperature. 964 01:10:04,680 --> 01:10:10,110 So the statistical analog of stability 965 01:10:10,110 --> 01:10:15,270 really comes down to variances being positive. 966 01:10:15,270 --> 01:10:21,320 So my claim is that this variance being positive 967 01:10:21,320 --> 01:10:26,630 is related to this stability condition. 968 01:10:26,630 --> 01:10:29,010 Let me do that in the following fashion. 969 01:10:29,010 --> 01:10:30,880 I divide these two expressions. 970 01:10:30,880 --> 01:10:33,470 I will get the average of N squared divided 971 01:10:33,470 --> 01:10:40,130 by average of N, which is really N, so the variance over N. 972 01:10:40,130 --> 01:10:41,290 I have kT. 973 01:10:44,350 --> 01:10:51,300 And then I have the ratio of two of these derivatives. 974 01:10:51,300 --> 01:10:54,090 And the reason I wanted to do that ratio 975 01:10:54,090 --> 01:10:56,690 was to get rid of the d mu. 976 01:10:56,690 --> 01:10:59,160 Because d mu is not so nice. 977 01:10:59,160 --> 01:11:01,645 If I take the ratio of those two derivatives, 978 01:11:01,645 --> 01:11:09,830 I will get dN by dP at constant T. 979 01:11:09,830 --> 01:11:13,410 But what I really wanted was to say something about dV 980 01:11:13,410 --> 01:11:17,800 by dP at constant T rather than dN by dP. 981 01:11:17,800 --> 01:11:24,750 So what I'm going to do is to somehow convert this dN by dP 982 01:11:24,750 --> 01:11:30,180 into dN by dV, and then dV by dP. 983 01:11:32,890 --> 01:11:36,250 And actually to do that, I need to use the chain rule. 984 01:11:36,250 --> 01:11:40,440 So this will be T, P. This will be T, N. 985 01:11:40,440 --> 01:11:44,590 And the chain rule will give me an additional minus sign. 986 01:11:44,590 --> 01:11:48,830 So this is a reminder of how your partial derivatives 987 01:11:48,830 --> 01:11:51,580 and the chain rule goes. 988 01:11:51,580 --> 01:11:58,460 Finally, dN by dV is none other than the density N/V. 989 01:11:58,460 --> 01:12:02,050 At constant temperature, N and V will be proportional. 990 01:12:02,050 --> 01:12:07,980 And what I have here is 1/V dV by dP, 991 01:12:07,980 --> 01:12:09,560 which is the compressibility. 992 01:12:09,560 --> 01:12:17,940 So the whole thing here is NkT times kappa of T. 993 01:12:17,940 --> 01:12:23,730 And so this being positive, this positivity of the variance, 994 01:12:23,730 --> 01:12:28,035 is-- the thermodynamic analog of it 995 01:12:28,035 --> 01:12:31,770 was from the stability, and something such as this. 996 01:12:31,770 --> 01:12:35,080 The statistical analog of it is that the probability 997 01:12:35,080 --> 01:12:39,710 distribution that I'm looking at in the grand canonical ensemble 998 01:12:39,710 --> 01:12:43,850 as a function of N has to be peaked around some region. 999 01:12:43,850 --> 01:12:45,720 The variance around that peak better 1000 01:12:45,720 --> 01:12:51,360 be positive so that I'm looking in the vicinity of a maximum. 1001 01:12:51,360 --> 01:12:53,250 If it was negative, it means that I'm 1002 01:12:53,250 --> 01:12:55,620 looking in the vicinity of a minimum. 1003 01:12:55,620 --> 01:12:58,580 And I'm looking at the least likely configuration. 1004 01:12:58,580 --> 01:13:00,270 So it cannot be allowed. 1005 01:13:00,270 --> 01:13:03,380 So that's the statistical reason. 1006 01:13:03,380 --> 01:13:06,690 OK, now I want to use that in connection 1007 01:13:06,690 --> 01:13:11,260 with what we have over here and see 1008 01:13:11,260 --> 01:13:12,985 what I can learn about this. 1009 01:13:25,090 --> 01:13:28,680 I said that I wrote two expressions and divided them, 1010 01:13:28,680 --> 01:13:32,810 because I didn't want to deal with the chemical potential. 1011 01:13:32,810 --> 01:13:37,940 But let me redefine that and say the following, 1012 01:13:37,940 --> 01:13:44,330 that N is V dP by d mu at constant T. 1013 01:13:44,330 --> 01:13:48,290 And maybe I will start to gain some idea about chemical 1014 01:13:48,290 --> 01:13:51,430 potential if I rearrange this as d mu 1015 01:13:51,430 --> 01:14:02,500 is V/N dP along this surface of constant P at constant T. 1016 01:14:02,500 --> 01:14:08,340 And then I can calculate mu as a function of P 1017 01:14:08,340 --> 01:14:12,430 at some particular temperature minus mu 1018 01:14:12,430 --> 01:14:18,810 at some reference point by integrating from the reference 1019 01:14:18,810 --> 01:14:22,590 point to the pressure of interest 1020 01:14:22,590 --> 01:14:30,020 the quantity V/N dP prime where V of P prime I 1021 01:14:30,020 --> 01:14:32,360 take from this curve. 1022 01:14:32,360 --> 01:14:37,950 So this curve gives me P as a function of V. 1023 01:14:37,950 --> 01:14:43,420 But I can invert it and think of V as a function of P, 1024 01:14:43,420 --> 01:14:47,310 put it here, and see what is happening with the chemical 1025 01:14:47,310 --> 01:14:51,420 potential if I walk along one of these trajectories. 1026 01:14:51,420 --> 01:14:58,250 And in particular, let's kind of draw one of these 1027 01:14:58,250 --> 01:15:01,520 curves that I am unhappy with that 1028 01:15:01,520 --> 01:15:06,620 have this kind of form in general 1029 01:15:06,620 --> 01:15:11,230 and calculate what happens to the chemical potential 1030 01:15:11,230 --> 01:15:14,060 if I, let's say, pick this reference point 1031 01:15:14,060 --> 01:15:18,100 A, and the corresponding pressure PA, 1032 01:15:18,100 --> 01:15:22,610 and go along this curve, which corresponds to some temperature 1033 01:15:22,610 --> 01:15:27,350 T that is less than this instability temperature, 1034 01:15:27,350 --> 01:15:29,485 and track the shape of the chemical potential. 1035 01:15:32,460 --> 01:15:36,330 So this formula says that the change in chemical potential we 1036 01:15:36,330 --> 01:15:41,000 obtained by calculating V-- divide by N, 1037 01:15:41,000 --> 01:15:44,320 but our N is fixed-- as a function of P, 1038 01:15:44,320 --> 01:15:47,560 and integrating as you go up in pressure. 1039 01:15:47,560 --> 01:15:52,070 So essentially, I'm calculating the integral starting 1040 01:15:52,070 --> 01:15:55,250 from point A as I go under this curve. 1041 01:15:58,700 --> 01:16:04,120 So let's plot as a result of that integration how 1042 01:16:04,120 --> 01:16:09,460 that chemical potential at P minus PA is going to look 1043 01:16:09,460 --> 01:16:15,340 like as I go in P beyond PA. 1044 01:16:15,340 --> 01:16:17,990 So at PA, that's my reference. 1045 01:16:17,990 --> 01:16:21,230 The chemical potential is some value. 1046 01:16:21,230 --> 01:16:26,960 As I go up and up, because of this area of this curve, 1047 01:16:26,960 --> 01:16:29,810 I keep adding to the chemical potential 1048 01:16:29,810 --> 01:16:34,420 until I reach the maximum here at this point C. 1049 01:16:34,420 --> 01:16:43,380 So there is some curve that goes from this A to some point C. 1050 01:16:43,380 --> 01:16:51,110 Now, the thing is that when I continue going along this curve 1051 01:16:51,110 --> 01:16:56,900 down this potential all the way to the bottom of this, which 1052 01:16:56,900 --> 01:17:02,840 I will call D, my DP's are negative. 1053 01:17:02,840 --> 01:17:08,650 So I start subtracting from what I had before. 1054 01:17:08,650 --> 01:17:12,351 And so then the curve starts to go down. 1055 01:17:12,351 --> 01:17:16,600 So all the way to D, I'm proceeding 1056 01:17:16,600 --> 01:17:19,070 in the opposite direction. 1057 01:17:19,070 --> 01:17:24,370 Once I hit D, I start going all the way to the end of the curve 1058 01:17:24,370 --> 01:17:26,340 however far I want to go. 1059 01:17:26,340 --> 01:17:29,180 And I'm adding some positive area. 1060 01:17:29,180 --> 01:17:37,415 So the next part I also have a variation in chemical potential 1061 01:17:37,415 --> 01:17:38,620 that goes like this. 1062 01:17:44,140 --> 01:17:51,384 So you ask, well, if I give you what the temperature is 1063 01:17:51,384 --> 01:17:53,300 and what the pressure is-- and I have told you 1064 01:17:53,300 --> 01:17:55,360 what the temperature and pressure are-- 1065 01:17:55,360 --> 01:17:57,850 I should be able to calculate the chemical potential. 1066 01:17:57,850 --> 01:17:59,800 It's an intensive function of the other two 1067 01:17:59,800 --> 01:18:01,700 intensive functions. 1068 01:18:01,700 --> 01:18:04,650 And this curve tells me that, except that there 1069 01:18:04,650 --> 01:18:07,710 are regions where I don't know which one to pick. 1070 01:18:11,060 --> 01:18:13,050 There are regions where it's obvious there's 1071 01:18:13,050 --> 01:18:15,630 one value of the chemical potential. 1072 01:18:15,630 --> 01:18:18,760 But there is in between, because of this instability, 1073 01:18:18,760 --> 01:18:21,880 three possible values. 1074 01:18:21,880 --> 01:18:27,570 And typically, if you have systems-- 1075 01:18:27,570 --> 01:18:31,200 think about different chemicals that can go between each other. 1076 01:18:31,200 --> 01:18:34,860 And there are potentials for chemical transformations. 1077 01:18:34,860 --> 01:18:38,450 You say, go and pick the lowest chemical potential. 1078 01:18:38,450 --> 01:18:41,900 The system will evolve onto the condition that 1079 01:18:41,900 --> 01:18:44,410 has the lowest chemical potential. 1080 01:18:44,410 --> 01:18:48,290 So that says that when you have a situation such as this, 1081 01:18:48,290 --> 01:18:52,490 you really have to pick the lowest one. 1082 01:18:52,490 --> 01:19:00,190 And so you have to bail out on your first curve at the point 1083 01:19:00,190 --> 01:19:08,600 B, and on your second curve at a point E, where on this curve 1084 01:19:08,600 --> 01:19:15,030 B and E are such that when you integrate this BDP all 1085 01:19:15,030 --> 01:19:19,450 the way from B to E, you will get 0, which means that you 1086 01:19:19,450 --> 01:19:27,280 have to find points B and E such that the integral here 1087 01:19:27,280 --> 01:19:30,470 is the same as the integral here. 1088 01:19:30,470 --> 01:19:33,416 And this is the so-called Maxwell construction. 1089 01:19:42,200 --> 01:19:47,820 So what that means is that originally I was telling you 1090 01:19:47,820 --> 01:19:54,700 over here that there is some portion of the curve that 1091 01:19:54,700 --> 01:19:56,860 violates all kinds of thermodynamics. 1092 01:19:56,860 --> 01:19:59,420 And you should not access it. 1093 01:19:59,420 --> 01:20:03,500 And this other argument that we are pursuing here says that 1094 01:20:03,500 --> 01:20:07,550 actually the regions that you cannot access are beyond that. 1095 01:20:07,550 --> 01:20:10,200 There's a portion that extends in this direction, 1096 01:20:10,200 --> 01:20:13,160 and a portion that extends in that direction. 1097 01:20:13,160 --> 01:20:16,890 And really there's a portion that you can access up to here, 1098 01:20:16,890 --> 01:20:19,960 and a portion that you can access up to here. 1099 01:20:19,960 --> 01:20:22,010 And maybe you should sort of then 1100 01:20:22,010 --> 01:20:25,210 join them by a straight line and make an analog 1101 01:20:25,210 --> 01:20:28,980 with what we have over there. 1102 01:20:28,980 --> 01:20:36,310 Now, in another thing, another way of looking at this 1103 01:20:36,310 --> 01:20:41,370 is that the green portion is certainly unstable, 1104 01:20:41,370 --> 01:20:44,950 violates all kinds of thermodynamic conditions. 1105 01:20:44,950 --> 01:20:48,850 And people argue that these other portions that 1106 01:20:48,850 --> 01:20:53,960 have the right slope are in some sense metastable. 1107 01:20:53,960 --> 01:21:01,040 So this would be a stable equilibrium. 1108 01:21:01,040 --> 01:21:04,540 This would be an unstable equilibrium. 1109 01:21:04,540 --> 01:21:07,700 And a metastable equilibrium is that you are at a minimum, 1110 01:21:07,700 --> 01:21:10,880 but there's a deeper minimum somewhere else. 1111 01:21:10,880 --> 01:21:23,350 And the picture is that if you manage to take your gas 1112 01:21:23,350 --> 01:21:27,270 and put additional pressure on it, 1113 01:21:27,270 --> 01:21:33,190 there is a region where there is a coexistence with the liquid 1114 01:21:33,190 --> 01:21:37,180 which is better in terms of free energy, et cetera. 1115 01:21:37,180 --> 01:21:40,250 But there could be some kind of a kinetic barrier, 1116 01:21:40,250 --> 01:21:44,990 like nucleation or whatever, that prevents you to go there. 1117 01:21:44,990 --> 01:21:48,180 And this again is something that is experimentally observed. 1118 01:21:48,180 --> 01:21:51,890 If you try to rapidly pressurize a gas, 1119 01:21:51,890 --> 01:21:56,900 you could sometimes avoid and not make your transition 1120 01:21:56,900 --> 01:21:59,680 to the liquid state at the right pressure. 1121 01:21:59,680 --> 01:22:02,595 But if you were to do things sufficiently slowly, 1122 01:22:02,595 --> 01:22:07,020 you would ultimately reach this point 1123 01:22:07,020 --> 01:22:09,230 that corresponds to the true equilibrium 1124 01:22:09,230 --> 01:22:13,660 pressure of making this transition. 1125 01:22:13,660 --> 01:22:18,020 Now, the thing is that somehow this whole story 1126 01:22:18,020 --> 01:22:21,690 is not very satisfactory. 1127 01:22:21,690 --> 01:22:23,940 I started with some rearrangement 1128 01:22:23,940 --> 01:22:27,840 of some perturbative to arrive at this equation. 1129 01:22:27,840 --> 01:22:31,200 This equation has this unstable portion. 1130 01:22:31,200 --> 01:22:34,900 And somehow I'm trying to relate something that makes absolutely 1131 01:22:34,900 --> 01:22:37,880 no thermodynamic sense and try to gain 1132 01:22:37,880 --> 01:22:43,545 from it some idea about what is happening in a real liquid gas 1133 01:22:43,545 --> 01:22:45,250 system. 1134 01:22:45,250 --> 01:22:48,580 So it would be good if all of this 1135 01:22:48,580 --> 01:22:53,250 could be more formally described within a framework 1136 01:22:53,250 --> 01:22:57,530 where all of the approximations, et cetera, are clear, 1137 01:22:57,530 --> 01:22:59,410 and we know what's going on. 1138 01:22:59,410 --> 01:23:02,575 And so next time, we will do that. 1139 01:23:02,575 --> 01:23:06,910 We will essentially developed a much more systematic formalism 1140 01:23:06,910 --> 01:23:11,550 for calculating the properties of an interacting gas, 1141 01:23:11,550 --> 01:23:16,619 and see that we get this, and we also understand why it fails.