WEBVTT 00:00:00.060 --> 00:00:01.780 The following content is provided 00:00:01.780 --> 00:00:04.019 under a Creative Commons license. 00:00:04.019 --> 00:00:06.870 Your support will help MIT OpenCourseWare continue 00:00:06.870 --> 00:00:10.730 to offer high quality educational resources for free. 00:00:10.730 --> 00:00:13.330 To make a donation or view additional materials 00:00:13.330 --> 00:00:17.217 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.217 --> 00:00:17.842 at ocw.mit.edu. 00:00:20.570 --> 00:00:23.670 PROFESSOR: And to this purpose, we 00:00:23.670 --> 00:00:28.470 need to calculate some thermodynamics. 00:00:28.470 --> 00:00:31.160 And we usually do that in statistical mechanics 00:00:31.160 --> 00:00:35.450 by calculating for some kind of a partition function. 00:00:35.450 --> 00:00:37.320 And we saw last time that it would 00:00:37.320 --> 00:00:42.180 be useful to calculate this grand partition function, which 00:00:42.180 --> 00:00:44.400 is the ensemble where you specify 00:00:44.400 --> 00:00:48.920 the temperature, the chemical potential, and the volume 00:00:48.920 --> 00:00:51.250 of the gas. 00:00:51.250 --> 00:00:55.120 And in this ensemble, your task is 00:00:55.120 --> 00:01:00.060 to look at within this box of volume V 00:01:00.060 --> 00:01:04.690 the possibility of there being any number of particles. 00:01:04.690 --> 00:01:09.400 So you have to sum over all possible N particle states. 00:01:09.400 --> 00:01:12.640 The contribution of each N particle states 00:01:12.640 --> 00:01:17.710 is exponentially related to the number of particles 00:01:17.710 --> 00:01:19.980 through the chemical potential. 00:01:19.980 --> 00:01:24.730 And then given that you are in a segment that has N particles, 00:01:24.730 --> 00:01:27.690 you have to look at all possible configurations 00:01:27.690 --> 00:01:30.890 of those particles and integrate over 00:01:30.890 --> 00:01:32.790 all of those possibilities. 00:01:32.790 --> 00:01:36.350 And that amounts to calculating the partition function. 00:01:36.350 --> 00:01:40.540 So for the partition function of an N particle system, 00:01:40.540 --> 00:01:43.880 you have to integrate over all of the momenta. 00:01:43.880 --> 00:01:47.020 Integration over each component of the momentum 00:01:47.020 --> 00:01:52.340 gives you a factor of 1 over lambda, where lambda again 00:01:52.340 --> 00:01:58.880 was related to the mass of the particle and temperature 00:01:58.880 --> 00:02:02.440 through this formula with h what we 00:02:02.440 --> 00:02:07.640 use to make these integrations over pq combinations 00:02:07.640 --> 00:02:09.120 dimensionless. 00:02:09.120 --> 00:02:11.860 There are 3N such integrations. 00:02:11.860 --> 00:02:16.680 So that's a contribution from the momenta. 00:02:16.680 --> 00:02:19.320 And then we have to do the integration 00:02:19.320 --> 00:02:21.470 over all of the coordinates. 00:02:21.470 --> 00:02:25.970 And as long as these particles are identical, 00:02:25.970 --> 00:02:30.140 we decided to divide by the number of permutations. 00:02:30.140 --> 00:02:33.150 Because we cannot tear them apart. 00:02:33.150 --> 00:02:37.710 So having done that, I need to now integrate over all of the N 00:02:37.710 --> 00:02:43.700 particles spanning a box of size V 00:02:43.700 --> 00:02:47.180 so the integration is within the box. 00:02:47.180 --> 00:02:51.080 And when I have interactions, then I 00:02:51.080 --> 00:02:54.200 have to worry about the Boltzmann weight that 00:02:54.200 --> 00:02:55.550 comes from the interaction. 00:02:55.550 --> 00:02:58.630 So here I should really put some kind of e 00:02:58.630 --> 00:03:02.920 to the minus beta times the interaction U. 00:03:02.920 --> 00:03:05.480 And what we did was we said that let's assume 00:03:05.480 --> 00:03:09.290 that this interaction comes from pairs of particles. 00:03:09.290 --> 00:03:13.640 And so this U is the sum over all possible pairs 00:03:13.640 --> 00:03:16.750 of particles, which when exponentiated 00:03:16.750 --> 00:03:24.790 will then give you a product over all possible pairs j, k 00:03:24.790 --> 00:03:29.450 and a factor that is related to the potential interaction 00:03:29.450 --> 00:03:30.980 between these. 00:03:30.980 --> 00:03:35.350 And we found it useful to write that factor as 1 00:03:35.350 --> 00:03:45.840 plus fjk, where this fjk stood for e to the minus 00:03:45.840 --> 00:03:53.150 beta of V qj minus qk minus 1. 00:03:53.150 --> 00:03:55.950 So basically if I add the 1 to that, 00:03:55.950 --> 00:03:59.190 I just get the exponentiated interaction potential. 00:03:59.190 --> 00:04:03.150 And then I have a forum such as this. 00:04:03.150 --> 00:04:05.780 So basically, this is the quantity 00:04:05.780 --> 00:04:07.980 that we wanted to compute. 00:04:07.980 --> 00:04:11.390 And so then we said, well, let's imagine 00:04:11.390 --> 00:04:17.250 expanding these factors of 1 plus f1, 2, 1 plus f1, 3, 00:04:17.250 --> 00:04:19.910 1 plus f2, 3, all of these factors, 00:04:19.910 --> 00:04:24.880 and organize them according to the powers of f that they have. 00:04:24.880 --> 00:04:27.990 So the leading term would be taking 1 from everybody. 00:04:27.990 --> 00:04:29.970 So that would give me V to the N, which 00:04:29.970 --> 00:04:32.730 would be the 0-th order partition 00:04:32.730 --> 00:04:35.000 function, the non-interacting system. 00:04:35.000 --> 00:04:37.590 And then I would start to get corrections 00:04:37.590 --> 00:04:41.080 where there's order of f integrated, order of f squared 00:04:41.080 --> 00:04:44.070 integrated, all kinds of things. 00:04:44.070 --> 00:04:49.330 And that being a somewhat difficult object to look at, 00:04:49.330 --> 00:04:53.860 we said, let's imagine graphically what we would get. 00:04:53.860 --> 00:04:58.800 And the typical contribution that we would get to this 00:04:58.800 --> 00:05:04.640 would involve having to iterate over all of these N particles. 00:05:04.640 --> 00:05:10.560 So we have to somehow imagine that we have particles 1 to N. 00:05:10.560 --> 00:05:16.370 And then for a particular term, we either pick 1's-- and there 00:05:16.370 --> 00:05:19.770 will be some points that are not connected to any f that is 00:05:19.770 --> 00:05:21.410 integrated. 00:05:21.410 --> 00:05:24.590 There will be a bunch of things that will be connected 00:05:24.590 --> 00:05:28.260 to things where there are pair of f's. 00:05:28.260 --> 00:05:30.100 There will be things later on maybe 00:05:30.100 --> 00:05:35.450 where there are triplets of f, and so forth and so on. 00:05:35.450 --> 00:05:40.500 And then we said that when I do the integrations that 00:05:40.500 --> 00:05:44.180 correspond to this, what do I get? 00:05:44.180 --> 00:05:48.140 I will get the contribution that comes from one particle 00:05:48.140 --> 00:05:50.650 by itself integrated. 00:05:50.650 --> 00:05:56.940 Let's call that b1 to the power of n1. 00:05:56.940 --> 00:06:02.620 I will get the contribution from this pair integrated. 00:06:02.620 --> 00:06:07.290 I will get b2 to the power of n2. 00:06:07.290 --> 00:06:10.820 I will get the contributions from these entities. 00:06:10.820 --> 00:06:14.310 And in general, I said, well, OK, somewhere in this, 00:06:14.310 --> 00:06:16.730 I will get bl to the power of nl. 00:06:19.418 --> 00:06:22.650 Now of course, I have a big constraint here 00:06:22.650 --> 00:06:30.080 that is that the sum over l of lnl 00:06:30.080 --> 00:06:34.750 has to be the total number of points 1 to n. 00:06:34.750 --> 00:06:40.330 So however I partition this, I will have for each graph 00:06:40.330 --> 00:06:43.090 that particular constraint acting. 00:06:46.180 --> 00:06:53.320 We said that clearly there's a lot of graphs and combinations 00:06:53.320 --> 00:06:56.710 that give you precisely this same factor. 00:06:56.710 --> 00:07:00.010 But all I had to do was to sort of rearrange 00:07:00.010 --> 00:07:02.120 the numbers and ordering, et cetera, 00:07:02.120 --> 00:07:04.750 and I would get all of this. 00:07:04.750 --> 00:07:07.520 So it would be very nice if I could figure out 00:07:07.520 --> 00:07:09.320 what the overall factor is out here. 00:07:12.450 --> 00:07:18.680 So we said that the factor is something like N factorial. 00:07:18.680 --> 00:07:22.500 Because what I can do is I can permute all of these numbers, 00:07:22.500 --> 00:07:25.950 and I would get exactly the same thing. 00:07:25.950 --> 00:07:29.600 But then I have to make sure that I don't over count. 00:07:29.600 --> 00:07:35.350 And not over counting required me to divide 00:07:35.350 --> 00:07:37.350 by the number of permutations that I 00:07:37.350 --> 00:07:40.050 have within each subgroup. 00:07:40.050 --> 00:07:43.380 So I have bl to the power of n. 00:07:43.380 --> 00:07:51.360 I have l factorial to the power of nl. 00:07:51.360 --> 00:07:55.340 And then I have the change. 00:07:55.340 --> 00:07:57.370 Let's say this is 1, 2, this is 3, 4. 00:07:57.370 --> 00:08:01.170 I could have called one of them 3, 4, the other one 1, 2. 00:08:01.170 --> 00:08:06.925 So basically I will have nl factorial from the permutations 00:08:06.925 --> 00:08:07.835 within each. 00:08:12.390 --> 00:08:16.950 But I would have gotten exactly the same numerical factor 00:08:16.950 --> 00:08:23.510 out front if I had the same configuration 00:08:23.510 --> 00:08:25.070 but I had this diagram. 00:08:27.770 --> 00:08:29.860 I would have gotten exactly the same factor. 00:08:29.860 --> 00:08:33.080 I would call the contribution to this b3. 00:08:33.080 --> 00:08:36.559 I didn't say currently what b3 is. 00:08:36.559 --> 00:08:39.640 I would have gotten exactly the same factor. 00:08:39.640 --> 00:08:44.080 So maybe then what I did was to sort of group 00:08:44.080 --> 00:08:46.080 all of those things that would come 00:08:46.080 --> 00:08:50.100 with the same numerical factor into this [INAUDIBLE] sum 00:08:50.100 --> 00:08:51.020 that I call bl. 00:08:51.020 --> 00:09:00.770 So I call bl to be the sum over all l-particle clusters. 00:09:09.250 --> 00:09:12.730 And then, of course here, I have to sum over 00:09:12.730 --> 00:09:15.610 all configurations of nl that are 00:09:15.610 --> 00:09:19.122 consistent with this constraint that I have to put up there. 00:09:24.430 --> 00:09:28.260 So the rest of it then was algebra. 00:09:28.260 --> 00:09:35.080 We said that if I constrain the total number, 00:09:35.080 --> 00:09:36.800 it's difficult for me to do. 00:09:36.800 --> 00:09:41.310 That's why I didn't go and calculate the partition 00:09:41.310 --> 00:09:44.880 function and switch to the grand partition function. 00:09:44.880 --> 00:09:48.550 Because in the grand partition, I can essentially 00:09:48.550 --> 00:09:54.910 make this N that constrains the values of these nl's 00:09:54.910 --> 00:09:57.260 to be all over the place. 00:09:57.260 --> 00:10:02.870 And therefore summing over things with nl unconstrained 00:10:02.870 --> 00:10:07.480 is equivalent to summing over terms with nl constrained, 00:10:07.480 --> 00:10:12.010 and then summing over whatever the final constraint is. 00:10:12.010 --> 00:10:16.310 So once I did that, I was liberated from this constraint. 00:10:16.310 --> 00:10:21.360 I could do the sum for each value of nl separately. 00:10:21.360 --> 00:10:24.380 And the thing nicely broke up into pieces. 00:10:24.380 --> 00:10:27.970 And so then what I could do is I could 00:10:27.970 --> 00:10:32.360 show that each term in the sum, there 00:10:32.360 --> 00:10:36.880 would be a product over different contributions l. 00:10:36.880 --> 00:10:43.290 For each l, I could sum over nl running from 0 to infinity. 00:10:43.290 --> 00:10:49.840 I had the 1 over nl factorial from out here. 00:10:49.840 --> 00:10:53.320 I had an e to the beta mu divided 00:10:53.320 --> 00:10:58.730 by lambda cubed from the combination of these things 00:10:58.730 --> 00:11:02.060 raised to the power of lnl, which 00:11:02.060 --> 00:11:05.700 is how big N would have been composed. 00:11:05.700 --> 00:11:08.550 So I would have here lnl. 00:11:08.550 --> 00:11:17.980 And I also had an l factorial raised to the power of nl. 00:11:17.980 --> 00:11:19.952 So I can write things in this fashion. 00:11:22.840 --> 00:11:25.740 So this thing then became the same thing 00:11:25.740 --> 00:11:31.795 as an exponential of a sum over l running from 1 to infinity. 00:11:35.160 --> 00:11:37.380 1 over l factorial, of if you like 00:11:37.380 --> 00:11:43.220 e to the beta mu over lambda cubed to the power of l 1 00:11:43.220 --> 00:11:44.920 over l factorial. 00:11:44.920 --> 00:11:46.610 And then I had bl. 00:11:52.690 --> 00:11:58.610 So it was this very nice result somehow summing 00:11:58.610 --> 00:12:03.820 over all kinds of things, and then taking the logarithm. 00:12:03.820 --> 00:12:06.732 The logarithm really depends only 00:12:06.732 --> 00:12:10.480 on the contributions of single clusters. 00:12:10.480 --> 00:12:12.250 And again, the reason it had to be 00:12:12.250 --> 00:12:15.090 that way is because the ultimate thing 00:12:15.090 --> 00:12:18.170 that I calculated in this ensemble 00:12:18.170 --> 00:12:22.960 is that the answer should be e to the beta V 00:12:22.960 --> 00:12:23.940 times the pressure. 00:12:26.470 --> 00:12:30.950 And so the expression that we have over here better 00:12:30.950 --> 00:12:36.890 have terms which are all proportional to volume. 00:12:36.890 --> 00:12:40.460 Sorry, they're all made extensive by proportionality 00:12:40.460 --> 00:12:41.690 to volume. 00:12:41.690 --> 00:12:45.380 And indeed, when I do the integrations over a single any 00:12:45.380 --> 00:12:48.620 cluster, there is one degree of freedom, if you like, 00:12:48.620 --> 00:12:52.080 associated with the center of mass of the cluster that 00:12:52.080 --> 00:12:54.790 can go and explore the entire volume. 00:12:54.790 --> 00:12:59.700 And so all of these things are in fact in the large end limit 00:12:59.700 --> 00:13:05.739 proportional to V and something that I call bl bar. 00:13:17.340 --> 00:13:24.610 So once I divide by this volume, the final outcome 00:13:24.610 --> 00:13:27.940 of my calculation is that I can calculate 00:13:27.940 --> 00:13:32.340 the pressure of an interacting gas 00:13:32.340 --> 00:13:41.860 by summing over a series whose terms are this e to the beta mu 00:13:41.860 --> 00:13:48.100 over lambda cubed raised to the power of l bl bar divided 00:13:48.100 --> 00:13:50.280 by l factorial. 00:13:59.480 --> 00:14:05.710 OK, this is correct, but not particularly illuminating. 00:14:05.710 --> 00:14:11.370 Because the thing that we said we have some intuition for 00:14:11.370 --> 00:14:15.640 is that the pressure of a gas-- let's 00:14:15.640 --> 00:14:19.630 say if I look at it in terms of beta P, 00:14:19.630 --> 00:14:26.900 actually P is the density times kt, which is 1 over beta, 00:14:26.900 --> 00:14:28.210 if you like, or nkT. 00:14:28.210 --> 00:14:30.930 It doesn't matter. 00:14:30.930 --> 00:14:32.880 And then there will be corrections. 00:14:32.880 --> 00:14:35.270 There will be a term that is order of n squared. 00:14:37.980 --> 00:14:42.640 There will be a term that is order of n cubed. 00:14:42.640 --> 00:14:44.660 And there are these coefficients, 00:14:44.660 --> 00:14:49.100 which are functions of temperature, 00:14:49.100 --> 00:14:52.460 that are called the virial coefficient. 00:14:52.460 --> 00:14:55.220 And this is a virial expansion. 00:14:55.220 --> 00:15:00.370 Essentially what it is is a fitting 00:15:00.370 --> 00:15:02.240 of the form of the pressure of the gas 00:15:02.240 --> 00:15:04.190 as a function of density. 00:15:04.190 --> 00:15:08.545 In the very low density limit you get ideal gas result. 00:15:08.545 --> 00:15:11.110 And presumably because of these interactions, 00:15:11.110 --> 00:15:14.650 you will get corrections that presumably also 00:15:14.650 --> 00:15:17.740 know about the potential that went 00:15:17.740 --> 00:15:23.100 into the construction of all of these b bars, et cetera. 00:15:23.100 --> 00:15:28.990 So how do we relate these things that are, say, experimentally 00:15:28.990 --> 00:15:35.040 accessible to this expansion that we have over here? 00:15:35.040 --> 00:15:38.970 And the reason it is not obvious immediately 00:15:38.970 --> 00:15:46.010 is because this is an expansion in chemical potential, 00:15:46.010 --> 00:15:49.720 whereas over here, I have density. 00:15:49.720 --> 00:15:51.730 So what do I do? 00:15:51.730 --> 00:15:56.040 Well, I realize that the density can be obtained as follows. 00:15:56.040 --> 00:16:02.150 In the grand canonical ensemble, the number of particles 00:16:02.150 --> 00:16:06.810 is in principal a random variable. 00:16:06.810 --> 00:16:10.650 But that random variable is governed by this e 00:16:10.650 --> 00:16:14.350 to the beta mu N that controls how many you have. 00:16:14.350 --> 00:16:21.330 And if I take the log of Q with respect to beta mu, 00:16:21.330 --> 00:16:23.840 I will generate the average number, 00:16:23.840 --> 00:16:26.050 which in the thermodynamic limit we 00:16:26.050 --> 00:16:28.940 expect to be the same thing as what we thermodynamically 00:16:28.940 --> 00:16:30.750 would call the number. 00:16:30.750 --> 00:16:36.080 And this log Q we just established is beta PV. 00:16:36.080 --> 00:16:40.600 So what we have here is the derivative of beta VP 00:16:40.600 --> 00:16:43.270 with respect to beta mu. 00:16:43.270 --> 00:16:45.450 And if I do this at constant temperature, 00:16:45.450 --> 00:16:47.240 the betas disappear. 00:16:47.240 --> 00:16:50.870 This is the same thing as V dP by d 00:16:50.870 --> 00:16:54.190 mu at constant temperature. 00:16:54.190 --> 00:17:00.970 But also I can take the derivative right here. 00:17:00.970 --> 00:17:02.800 So what happens? 00:17:02.800 --> 00:17:10.660 I will find that the density which is N over V, 00:17:10.660 --> 00:17:14.660 is the derivative of this expression with respect 00:17:14.660 --> 00:17:16.609 to beta mu. 00:17:16.609 --> 00:17:20.520 I go and I find that there's a beta mu. 00:17:20.520 --> 00:17:24.060 If I take a derivative of this with respect to beta mu, 00:17:24.060 --> 00:17:27.300 really it's exponential of beta mu l. 00:17:27.300 --> 00:17:31.420 The derivative of it will give me l e to the beta mu l. 00:17:31.420 --> 00:17:37.400 So what I get is a sum over l e to the beta mu 00:17:37.400 --> 00:17:39.460 over lambda cubed. 00:17:39.460 --> 00:17:42.300 l gets repeated. 00:17:42.300 --> 00:17:45.590 And then I will have l times this. 00:17:45.590 --> 00:17:50.300 So I will have vl bar divided by l minus 1 factorial. 00:17:53.790 --> 00:17:56.370 So the series for density is very 00:17:56.370 --> 00:17:58.680 much like the series for pressure, 00:17:58.680 --> 00:18:03.980 except that you replace 1 over l factorial with 1 over l 00:18:03.980 --> 00:18:05.009 minus 1 factorial. 00:18:09.410 --> 00:18:12.690 So now my task is clear. 00:18:12.690 --> 00:18:18.350 What I should do is I should solve 00:18:18.350 --> 00:18:24.090 given a particular density for what mu is. 00:18:24.090 --> 00:18:27.100 Once I have mu as a function of density, 00:18:27.100 --> 00:18:28.900 I can substitute it back here. 00:18:28.900 --> 00:18:32.860 And I will have pressure as a function of density. 00:18:32.860 --> 00:18:36.090 Of course it's clear that the right variable to look at 00:18:36.090 --> 00:18:43.710 is not mu, but x, which is e to the beta mu over lambda cubed. 00:18:43.710 --> 00:18:45.150 This actually has a name. 00:18:45.150 --> 00:18:46.960 It's sometimes called fugacity. 00:18:54.230 --> 00:18:58.130 So the second equation is telling me 00:18:58.130 --> 00:19:05.700 that the density I can write in terms of the fugacity as a sum 00:19:05.700 --> 00:19:13.140 over l x to the l bl bar divided by l minus 1 factorial, which 00:19:13.140 --> 00:19:16.860 if I were to sort of write in its full details, 00:19:16.860 --> 00:19:24.430 it starts with b1 bar x plus b2 bar x squared. 00:19:24.430 --> 00:19:28.890 Because there I have 0 factorial or 1 factorial respectively. 00:19:28.890 --> 00:19:33.940 The next term will be b3 bar over 2 x cubed. 00:19:33.940 --> 00:19:36.110 And this will go on and on. 00:19:41.880 --> 00:19:45.780 Also, let me remind you what b1 bar is. 00:19:45.780 --> 00:19:51.740 b1 bar, for every one of these b1 bars, 00:19:51.740 --> 00:19:55.140 I have to divide by 1 over V because of this V. 00:19:55.140 --> 00:19:57.980 And then I have to do the integration that corresponds 00:19:57.980 --> 00:20:02.050 to the one cluster, which is essentially one cluster going 00:20:02.050 --> 00:20:05.250 over the entirety of space, which will give me V. 00:20:05.250 --> 00:20:09.230 So this is in fact 1. 00:20:09.230 --> 00:20:14.290 So basically this coefficient here is 1. 00:20:14.290 --> 00:20:16.100 And then I will have the corrections. 00:20:20.400 --> 00:20:23.600 So you say, well, I have n as a function of x. 00:20:23.600 --> 00:20:26.890 I want x as a function of n. 00:20:26.890 --> 00:20:29.360 And I say, OK, that's not that difficult. 00:20:29.360 --> 00:20:34.340 I will write that x equals n minus b2 bar 00:20:34.340 --> 00:20:40.690 x squared minus b3 bar x cubed over 2, and so forth. 00:20:40.690 --> 00:20:41.860 And I have now x. 00:20:46.910 --> 00:20:50.560 Maybe a few of you are skeptical. 00:20:50.560 --> 00:20:52.330 Some of you don't seem to be bothered. 00:20:55.440 --> 00:20:58.810 OK, so my claim is that this is indeed 00:20:58.810 --> 00:21:04.860 a systematic way of solving a series in which when 00:21:04.860 --> 00:21:10.475 the density goes to 0, you expect x to also go to 0. 00:21:10.475 --> 00:21:14.500 And to lowest order, x will be of the order of density. 00:21:14.500 --> 00:21:18.450 And these will be higher order powers in density. 00:21:18.450 --> 00:21:24.620 So to get a systematic series in density, all you need to do 00:21:24.620 --> 00:21:29.150 is to sort of work with a series such as this keeping 00:21:29.150 --> 00:21:33.030 in mind what order you have solved things to. 00:21:33.030 --> 00:21:35.900 So I claim that to lowest order, this really 00:21:35.900 --> 00:21:37.960 just says that x is n. 00:21:37.960 --> 00:21:42.780 And then there are corrections that are order of n squared. 00:21:42.780 --> 00:21:46.770 And to get the next order term, all I need to do 00:21:46.770 --> 00:21:50.310 is to substitute the lower order in this equation. 00:21:50.310 --> 00:21:55.300 So basically if I substitute x equals n in this equation, 00:21:55.300 --> 00:22:00.900 I will get n minus b2 n square. 00:22:00.900 --> 00:22:04.990 If I substitute x over n here, it will be a higher order term. 00:22:04.990 --> 00:22:06.970 And I claim that this is the correct result 00:22:06.970 --> 00:22:08.380 to order of n cubed. 00:22:11.940 --> 00:22:15.250 And then to get the result to order of n 00:22:15.250 --> 00:22:20.420 cubed, I substitute this back into the original equation. 00:22:20.420 --> 00:22:26.050 So I start with n minus b2 bar, the square 00:22:26.050 --> 00:22:29.750 of the previous solution at the right order. 00:22:29.750 --> 00:22:34.630 So squaring this, I will get n squared minus twice 00:22:34.630 --> 00:22:38.030 b2 bar n cubed. 00:22:38.030 --> 00:22:40.660 And the next term that would be the square of this, which 00:22:40.660 --> 00:22:44.210 is order of n to the fourth, I don't write down. 00:22:44.210 --> 00:22:49.200 And then I have minus b3 bar over 2, 00:22:49.200 --> 00:22:51.670 the cube of the previous solution. 00:22:51.670 --> 00:22:53.600 And at the right order that I have, 00:22:53.600 --> 00:22:58.610 that's the same thing as b3 bar over 2 n cubed 00:22:58.610 --> 00:23:02.700 and order of n to the fourth. 00:23:02.700 --> 00:23:05.750 And all I really need to do at this order 00:23:05.750 --> 00:23:08.870 is to recognize that I have two terms that 00:23:08.870 --> 00:23:10.500 are order of n cubed. 00:23:10.500 --> 00:23:16.590 Putting them together, I have n minus b2 bar n squared. 00:23:16.590 --> 00:23:22.370 And then I have plus 2b2 bar squared 00:23:22.370 --> 00:23:31.180 minus b3 bar over 2 n cubed and order of n to the fourth. 00:23:41.000 --> 00:23:42.180 So now we erase this. 00:23:53.420 --> 00:23:58.700 OK, so now I have solved for x in a power series in n. 00:23:58.700 --> 00:24:00.910 All I need to do is to substitute 00:24:00.910 --> 00:24:03.160 in the power series for p. 00:24:03.160 --> 00:24:04.840 So let's write that down. 00:24:04.840 --> 00:24:11.720 The power series for beta p is b1 bar times 00:24:11.720 --> 00:24:16.810 the density b2 bar squared over 2 density 00:24:16.810 --> 00:24:23.670 square, b3 bar over 3 factorial, which 00:24:23.670 --> 00:24:27.480 is 6 n cubed, and so forth. 00:24:27.480 --> 00:24:30.190 But I only calculated things to order of n cubed. 00:24:32.930 --> 00:24:36.740 Also, b1 bar is the same thing as 1. 00:24:36.740 --> 00:24:40.740 So basically that's what I should be working with. 00:24:40.740 --> 00:24:44.900 And so all I need to do is to substitute-- oops, 00:24:44.900 --> 00:24:46.420 except that these are all x's. 00:24:54.170 --> 00:24:56.890 The series here for beta p that I 00:24:56.890 --> 00:25:01.600 have is in powers of this quantity x. 00:25:01.600 --> 00:25:07.450 And it is x that I had calculated as a function of n. 00:25:07.450 --> 00:25:15.430 To the order that I calculated, it is n minus b2n squared. 00:25:15.430 --> 00:25:26.360 And then I have 2b2 squared minus b3 over b3. 00:25:26.360 --> 00:25:30.850 Somehow this sounds incorrect to me. 00:25:30.850 --> 00:25:33.222 Nope, that's fine. 00:25:33.222 --> 00:25:35.010 Because here I have 2. 00:25:35.010 --> 00:25:37.020 So that's correct. 00:25:37.020 --> 00:25:43.970 b3 over 2 n cubed-- so essentially, 00:25:43.970 --> 00:25:47.910 I just substituted for x here. 00:25:47.910 --> 00:25:51.630 The next order term is going to be 00:25:51.630 --> 00:25:56.280 b2 over 2 times the square of x. 00:25:56.280 --> 00:26:00.550 The square of x will give me a term that is n squared, 00:26:00.550 --> 00:26:07.110 a term that is from here 2b2n cubed. 00:26:07.110 --> 00:26:11.800 Order of n to the fourth I haven't calculated correctly. 00:26:11.800 --> 00:26:17.440 Order of n cubed I have b3 over 6. 00:26:17.440 --> 00:26:21.850 And just x cubed is the same thing as n cubed to this order. 00:26:21.850 --> 00:26:26.562 And I have not calculated anything at the next order. 00:26:29.830 --> 00:26:33.544 So let's see what we have. 00:26:33.544 --> 00:26:36.120 We have n. 00:26:36.120 --> 00:26:40.550 At the next order, there are two terms that are n squared. 00:26:40.550 --> 00:26:43.570 There is this term, and there is that term. 00:26:43.570 --> 00:26:49.040 Putting them together, I will get minus b2 over 2 n squared. 00:26:52.030 --> 00:26:55.140 At the next order, at the order of n cubed, 00:26:55.140 --> 00:26:57.540 I have a bunch of terms. 00:26:57.540 --> 00:27:03.470 First of all, there is this 2b2 squared. 00:27:03.470 --> 00:27:09.810 But then multiplying this with this will subtract one b2. 00:27:09.810 --> 00:27:14.500 So I'm going to be left with b2 squared. 00:27:14.500 --> 00:27:20.030 And then I have minus 1/2 b3. 00:27:20.030 --> 00:27:22.320 So that's one term. 00:27:22.320 --> 00:27:26.140 And then I have plus 1/6 b3. 00:27:26.140 --> 00:27:30.560 Minus 1/2 plus 1/6 is minus 1/3. 00:27:30.560 --> 00:27:35.210 So this is minus b3 over 3 n cubed. 00:27:35.210 --> 00:27:39.325 And I haven't calculated order of n to the fourth. 00:27:42.170 --> 00:27:44.095 And this is the formula for BP. 00:27:51.680 --> 00:27:55.865 So lowest order, I have the ideal gas result. 00:27:59.170 --> 00:28:01.570 Actually, let me, for simplicity, 00:28:01.570 --> 00:28:05.445 define the virial coefficients in this fashion. 00:28:08.030 --> 00:28:13.410 The second order, I will get a correction B2, 00:28:13.410 --> 00:28:16.210 which is minus 1/2 of b2 bar. 00:28:18.755 --> 00:28:24.990 So this is minus 1/2 of the diagram that 00:28:24.990 --> 00:28:30.640 corresponds to essentially one of these lines 00:28:30.640 --> 00:28:32.950 that I have up here. 00:28:32.950 --> 00:28:34.770 And so what is it? 00:28:34.770 --> 00:28:42.140 It is minus 1/2 the integral over the relative coordinate 00:28:42.140 --> 00:28:45.900 of e to the minus beta v as a function 00:28:45.900 --> 00:28:47.940 of the relative coordinate minus 1. 00:28:52.340 --> 00:28:57.580 Now earlier, we had in fact calculated this result directly 00:28:57.580 --> 00:28:59.510 through the partition function. 00:28:59.510 --> 00:29:04.300 I did a calculation in which I calculated the first term 00:29:04.300 --> 00:29:06.740 in the other way of looking at things, 00:29:06.740 --> 00:29:12.060 in the cumulant expansion, as a function of expansions 00:29:12.060 --> 00:29:13.280 in the potential. 00:29:13.280 --> 00:29:15.110 We saw that there was a term that all 00:29:15.110 --> 00:29:16.840 was order of density squared. 00:29:16.840 --> 00:29:20.140 We summed all of the terms to get this factor. 00:29:20.140 --> 00:29:23.270 And there was precisely a factor of minus 1/2 00:29:23.270 --> 00:29:25.710 as a correction to pressure once we 00:29:25.710 --> 00:29:27.772 took the derivative of the partition function 00:29:27.772 --> 00:29:28.730 with respect to volume. 00:29:31.340 --> 00:29:38.600 OK, so the thing that is new is really the next order term, B3, 00:29:38.600 --> 00:29:44.360 which is b2 squared minus b3 bar over 3. 00:29:44.360 --> 00:29:48.230 OK, so what is that? 00:29:48.230 --> 00:29:52.340 Diagrammatically, B2 was this factor 00:29:52.340 --> 00:29:58.480 that we calculated above and is the square of this pair. 00:29:58.480 --> 00:30:03.360 And then I have to subtract from that 1/3 of whatever 00:30:03.360 --> 00:30:06.290 goes into B3. 00:30:06.290 --> 00:30:10.770 Now remember, we said that B3, I have to pick three points 00:30:10.770 --> 00:30:14.740 and make sure I link them in all possible ways. 00:30:14.740 --> 00:30:19.910 So the diagrams that go into B3, one of them is this. 00:30:19.910 --> 00:30:22.160 But then there are three other ways 00:30:22.160 --> 00:30:26.743 of making a linked object, which is these things. 00:30:31.950 --> 00:30:35.150 We also saw that if I were to calculate 00:30:35.150 --> 00:30:39.010 the contribution of any one of these objects, 00:30:39.010 --> 00:30:43.480 I can very easily choose to measure my coordinates 00:30:43.480 --> 00:30:47.690 with respect to, say, the point that is at their apex. 00:30:47.690 --> 00:30:49.730 And then I would have one variable 00:30:49.730 --> 00:30:51.710 which is this distance, one variable which 00:30:51.710 --> 00:30:53.000 is that distance. 00:30:53.000 --> 00:30:55.150 And independently, each one of them 00:30:55.150 --> 00:31:00.030 would give me the factor that I calculated before. 00:31:00.030 --> 00:31:06.050 In essence, all of these one particle reducible graphs 00:31:06.050 --> 00:31:09.362 give a contribution, which is the product 00:31:09.362 --> 00:31:13.690 of these single line graphs. 00:31:13.690 --> 00:31:14.770 There are three of them. 00:31:14.770 --> 00:31:18.630 Minus 1/3 precisely cancels that. 00:31:18.630 --> 00:31:24.450 And so the calculation that goes into the third virial 00:31:24.450 --> 00:31:31.670 coefficient ultimately will only depend on this one graph. 00:31:31.670 --> 00:31:34.720 And we had anticipated this before. 00:31:34.720 --> 00:31:36.640 When we were doing things previously 00:31:36.640 --> 00:31:39.880 using the expansion of the partition function, 00:31:39.880 --> 00:31:43.270 we saw that in this cumulant expansion, 00:31:43.270 --> 00:31:46.210 only the cumulants were appearing. 00:31:46.210 --> 00:31:48.360 And for calculation of the cumulants, 00:31:48.360 --> 00:31:50.860 I had to do lots of subtractions. 00:31:50.860 --> 00:31:53.280 And those subtractions were genetically 00:31:53.280 --> 00:32:00.880 removing these one particle reducible graphs. 00:32:00.880 --> 00:32:04.230 And this continues to all orders. 00:32:04.230 --> 00:32:10.280 And the general formula is that the l-th contribution here 00:32:10.280 --> 00:32:17.060 will be a factor of l minus 1 over l factorial, which is not 00:32:17.060 --> 00:32:19.750 so difficult to get by following the procedures 00:32:19.750 --> 00:32:24.940 that I have described over here, and then the part of bl that 00:32:24.940 --> 00:32:28.150 is one particle irreducible. 00:32:33.360 --> 00:32:38.600 So this is the eventual result for how 00:32:38.600 --> 00:32:44.760 you would be able, given some particular form of the two body 00:32:44.760 --> 00:32:51.120 interaction, to calculate an expansion for pressure 00:32:51.120 --> 00:32:55.960 in powers of density, how the coefficients of that expansion 00:32:55.960 --> 00:32:59.261 are related to properties of this potential 00:32:59.261 --> 00:33:00.760 through this diagrammatic expansion. 00:33:10.160 --> 00:33:13.100 So this is formally correct. 00:33:13.100 --> 00:33:16.440 And then the next question is, is it practical? 00:33:16.440 --> 00:33:18.380 Is it useful? 00:33:18.380 --> 00:33:23.310 So we need to start computing things to see something 00:33:23.310 --> 00:33:27.524 about the usefulness of this series for some particular type 00:33:27.524 --> 00:33:28.065 of potential. 00:33:31.060 --> 00:33:34.790 So we are going to look at the kind of potential 00:33:34.790 --> 00:33:37.370 that I already described for you. 00:33:37.370 --> 00:33:43.800 That is, if I look at two particles in a gas that 00:33:43.800 --> 00:33:47.720 are separated by an amount r-- let's 00:33:47.720 --> 00:33:50.630 imagine the potential only depends 00:33:50.630 --> 00:33:54.580 on the relative distance, not orientation or anything else. 00:33:54.580 --> 00:33:57.610 We said that basically, at large distances, 00:33:57.610 --> 00:34:02.080 the potential is attractive because of van der Waals. 00:34:02.080 --> 00:34:05.890 At short distances, the potential is repulsive. 00:34:05.890 --> 00:34:09.350 And so you have a general form such as this. 00:34:12.900 --> 00:34:15.610 Now, if I want to do calculations, 00:34:15.610 --> 00:34:18.530 it would be useful for me to have something 00:34:18.530 --> 00:34:22.070 that I can do calculations exactly 00:34:22.070 --> 00:34:24.909 with and get an estimate. 00:34:24.909 --> 00:34:27.659 So what I'm going to do is to replace that potential, 00:34:27.659 --> 00:34:31.130 essentially, with a hard wall. 00:34:31.130 --> 00:34:38.469 So my approximation to the potential is that my v of r 00:34:38.469 --> 00:34:42.135 is infinite for distances, separations 00:34:42.135 --> 00:34:45.060 that are less than some r0. 00:34:45.060 --> 00:34:48.480 So basically, I define some kind of r0, 00:34:48.480 --> 00:34:52.389 which is the-- if you think of them as billiard balls, 00:34:52.389 --> 00:34:56.380 it's related to the closest distance of approach. 00:34:56.380 --> 00:34:59.740 This potential at large distances 00:34:59.740 --> 00:35:06.660 has typical van der Waals form, by which 00:35:06.660 --> 00:35:10.580 I mean it falls off as a function of separation 00:35:10.580 --> 00:35:13.180 with the sixth power. 00:35:13.180 --> 00:35:14.140 It is attractive. 00:35:14.140 --> 00:35:16.710 So I put a minus sign here. 00:35:16.710 --> 00:35:20.440 In order to make sure that I get eventually the dimensions 00:35:20.440 --> 00:35:25.820 right, the coefficient that goes here I write as r0 with a u0 00:35:25.820 --> 00:35:33.350 up here such that if I assume that the potential is precisely 00:35:33.350 --> 00:35:42.870 this for all r that are greater than r0, then 00:35:42.870 --> 00:35:47.410 I'm replacing the actual potential 00:35:47.410 --> 00:35:50.710 by something like this. 00:35:50.710 --> 00:35:54.550 And the minimum depth of potential 00:35:54.550 --> 00:35:58.010 will occur at this cusp over here at minus u0. 00:36:08.190 --> 00:36:13.090 So why did I do that? 00:36:13.090 --> 00:36:16.410 Because now I can calculate with that 00:36:16.410 --> 00:36:18.850 what the second virial coefficient is. 00:36:18.850 --> 00:36:21.870 So what is B2? 00:36:21.870 --> 00:36:31.060 B2 is minus 1/2 the integral of relative potential. 00:36:35.100 --> 00:36:40.620 Now this integral will take two parts 00:36:40.620 --> 00:36:44.800 from these two different contributions. 00:36:44.800 --> 00:36:50.050 One part is when I'm in the regime where the particles are 00:36:50.050 --> 00:36:52.570 excluded, the potential is infinity. 00:36:52.570 --> 00:36:57.430 And there, f is simply minus 1. 00:36:57.430 --> 00:37:02.180 So that minus 1 then gets integrated from 0 to r, 00:37:02.180 --> 00:37:04.700 giving me the volume that is excluded. 00:37:04.700 --> 00:37:07.750 So what I will get here is a minus 1 00:37:07.750 --> 00:37:11.400 times the excluded volume. 00:37:11.400 --> 00:37:16.680 That's the first part, where the volume is, of course, 00:37:16.680 --> 00:37:21.628 4 pi over 3 r0 cubed for the volume of the sphere. 00:37:24.700 --> 00:37:27.180 And then I have to add the part where 00:37:27.180 --> 00:37:32.000 I go from f0 all the way to infinity. 00:37:32.000 --> 00:37:35.110 The potential is vertically symmetric. 00:37:35.110 --> 00:37:41.930 So d cubed r becomes 4 pi r squared dr. 00:37:41.930 --> 00:37:49.390 And I have to integrate e to the minus 00:37:49.390 --> 00:37:52.700 beta times the attractive part of the potential. 00:37:52.700 --> 00:37:57.900 So it's beta u0 r0 over r to the sixth minus 1. 00:38:03.390 --> 00:38:06.780 So what I will do, I will additionally 00:38:06.780 --> 00:38:11.280 assume that I'm in the range of parameters 00:38:11.280 --> 00:38:16.200 where this beta u0 is much less than 1, 00:38:16.200 --> 00:38:19.840 that is, at temperatures that are higher 00:38:19.840 --> 00:38:22.880 compared to the depth of this potential converted 00:38:22.880 --> 00:38:24.910 to units of kT. 00:38:24.910 --> 00:38:29.010 And if that is the case, then I can expand this. 00:38:29.010 --> 00:38:31.700 And the expansion to the lowest order 00:38:31.700 --> 00:38:37.380 will simply give me beta u0 r0 over r to the sixth power. 00:38:37.380 --> 00:38:40.485 And I will ignore higher order terms. 00:38:40.485 --> 00:38:44.820 So this, if you like, is order of beta u0 squared. 00:38:49.570 --> 00:38:54.530 Now, having done that, then the second integral becomes simple. 00:38:54.530 --> 00:38:56.700 Because I have to integrate r squared 00:38:56.700 --> 00:39:01.330 divided by r to the fourth, which is r to the minus 4. 00:39:01.330 --> 00:39:09.320 Integral of r to the minus 4 gives me r to the minus 3. 00:39:09.320 --> 00:39:13.020 And then there's a factor of minus 1/3 from the integration. 00:39:13.020 --> 00:39:18.280 It has to be evaluated between 0 and infinity. 00:39:18.280 --> 00:39:26.040 And so the final answer, then, for B2 is minus 1/2. 00:39:26.040 --> 00:39:32.211 I have minus 4 pi over 3 r0 cubed from the excluded volume 00:39:32.211 --> 00:39:32.710 part. 00:39:35.850 --> 00:39:49.180 From here, I will get a 4 pi beta u0 r0 to the sixth. 00:39:52.700 --> 00:39:57.130 And then I will get this factor of r the minus 3 00:39:57.130 --> 00:39:59.500 over 3 evaluated between infinity, 00:39:59.500 --> 00:40:05.740 which gives me 0 and r0. 00:40:13.720 --> 00:40:16.530 So you can see that with this potential, actually 00:40:16.530 --> 00:40:22.900 both terms are proportional to 4 pi r0 cubed over 3. 00:40:22.900 --> 00:40:29.200 And so I can write the answer as minus omega over 2 00:40:29.200 --> 00:40:33.760 where I have defined omega to be this 4 pi r0 cubed 00:40:33.760 --> 00:40:36.505 over 3, which is the excluded volume. 00:40:39.240 --> 00:40:46.404 Basically, it's the volume that is excluded from exploration 00:40:46.404 --> 00:40:47.570 when you have two particles. 00:40:47.570 --> 00:40:52.900 Because their center of mass cannot come as close as r0. 00:40:52.900 --> 00:40:58.330 So that 1/2 appears here as 1. 00:40:58.330 --> 00:41:01.810 Because I already took care of that. 00:41:01.810 --> 00:41:05.400 This rest of the coefficients are proportional to this 00:41:05.400 --> 00:41:08.690 except with a factor of minus beta u0. 00:41:08.690 --> 00:41:12.280 So I get minus beta u0. 00:41:12.280 --> 00:41:19.351 So this is your second virial coefficient for this. 00:41:19.351 --> 00:41:21.100 AUDIENCE: It should be positive out front. 00:41:21.100 --> 00:41:21.849 PROFESSOR: Pardon? 00:41:21.849 --> 00:41:26.544 AUDIENCE: It should be positive out front, [INAUDIBLE]? 00:41:26.544 --> 00:41:28.516 PROFESSOR: Yes, because there was a minus, 00:41:28.516 --> 00:41:29.502 and there's a minus. 00:41:29.502 --> 00:41:30.488 There's a plus. 00:41:50.701 --> 00:41:51.687 Let's keep that. 00:42:00.620 --> 00:42:05.620 So what we have so far, we've said 00:42:05.620 --> 00:42:11.940 that beta times the pressure starts with ideal gas behavior 00:42:11.940 --> 00:42:13.644 n. 00:42:13.644 --> 00:42:17.440 And then I have the next correction, 00:42:17.440 --> 00:42:22.860 which is this B2 multiplying n squared. 00:42:22.860 --> 00:42:27.970 So I will have n squared times this coefficient 00:42:27.970 --> 00:42:31.070 that I have calculated for this type of gas, which 00:42:31.070 --> 00:42:34.710 is omega over 2 1 minus beta u0. 00:42:34.710 --> 00:42:37.540 And presumably there are higher order terms in this. 00:42:42.860 --> 00:42:47.840 So whenever you see something like this, 00:42:47.840 --> 00:42:51.760 then you have to start thinking about, 00:42:51.760 --> 00:42:55.680 is this a good expansion? 00:42:55.680 --> 00:42:59.690 So let's think about how this expansion could 00:42:59.690 --> 00:43:00.790 have become problematic. 00:43:03.690 --> 00:43:08.850 Actually, there is already one thing that I should have noted 00:43:08.850 --> 00:43:17.790 and I didn't, which is short versus long range potential. 00:43:21.890 --> 00:43:25.680 Now, it was nice for this potential 00:43:25.680 --> 00:43:29.460 that I got an answer that was in the form 00:43:29.460 --> 00:43:33.440 that I could write down, factor out omega. 00:43:33.440 --> 00:43:35.570 You can say, well, would you have always 00:43:35.570 --> 00:43:38.630 been able to do something similar to that? 00:43:38.630 --> 00:43:40.630 Because you see, ultimately what this 00:43:40.630 --> 00:43:47.950 says is that in order for all of these terms in this expansion 00:43:47.950 --> 00:43:54.560 to be dimensionally correct, the next term is a density, 00:43:54.560 --> 00:43:57.210 has dimensions of inverse volume, 00:43:57.210 --> 00:43:58.720 density squared compared to this. 00:43:58.720 --> 00:44:02.780 So it should be compensated by some factor of volume. 00:44:02.780 --> 00:44:05.530 And we can see that this factor of volume 00:44:05.530 --> 00:44:08.250 came from something that was of the order 00:44:08.250 --> 00:44:12.986 of this size of the molecule, this r0. 00:44:12.986 --> 00:44:16.860 OK, so it's interesting. 00:44:16.860 --> 00:44:20.830 It says that it's really the short-range part 00:44:20.830 --> 00:44:26.770 of the potential that seems to be setting the correction. 00:44:26.770 --> 00:44:29.500 Now, where did that come from? 00:44:29.500 --> 00:44:31.670 Well, there was one place that I had 00:44:31.670 --> 00:44:34.920 to do an integration over the potential. 00:44:34.920 --> 00:44:36.850 And I found that the integration was 00:44:36.850 --> 00:44:40.876 dominated by the lower range. 00:44:40.876 --> 00:44:43.710 That was over here. 00:44:43.710 --> 00:44:45.395 Where would this become difficult? 00:44:49.454 --> 00:44:52.300 AUDIENCE: In a small box? 00:44:52.300 --> 00:44:53.876 PROFESSOR: In a small box, but we 00:44:53.876 --> 00:44:56.050 are always taking the thermodynamic limit 00:44:56.050 --> 00:44:57.721 where V goes to infinity. 00:44:57.721 --> 00:44:59.685 AUDIENCE: Or inverse square log? 00:44:59.685 --> 00:45:01.790 PROFESSOR: Inverse square log, yes, that's right. 00:45:01.790 --> 00:45:05.090 So who says that this integral should converge? 00:45:05.090 --> 00:45:10.760 I'm doing an integral d cubed r, something like v of r. 00:45:10.760 --> 00:45:14.240 And convergence was the reason why this integral 00:45:14.240 --> 00:45:16.695 was dominated by short distance. 00:45:16.695 --> 00:45:22.490 If this potential goes like 1 over r cubed, 00:45:22.490 --> 00:45:26.050 then it will logarithmically depend on the size of the box. 00:45:26.050 --> 00:45:29.320 For Coulomb interaction, 1/r potential 00:45:29.320 --> 00:45:30.940 you can't even think about it. 00:45:30.940 --> 00:45:34.030 It's just too divergent. 00:45:34.030 --> 00:45:38.640 So this expansion will fail for potentials 00:45:38.640 --> 00:45:43.300 that have tails that are decaying as 1 00:45:43.300 --> 00:45:46.680 over r cubed or even slower. 00:45:46.680 --> 00:45:50.670 Fortunately, that's not the case for van der Waals' 00:45:50.670 --> 00:45:53.320 potential and typical potentials. 00:45:53.320 --> 00:45:56.100 But if you have a plasma, you have to worry about this. 00:45:56.100 --> 00:45:58.610 And that's why I was also saying last time 00:45:58.610 --> 00:46:01.597 that the typical expansions that you have to do for plasmas 00:46:01.597 --> 00:46:02.180 are different. 00:46:04.930 --> 00:46:10.050 So given that we are dominated by the short range, 00:46:10.050 --> 00:46:14.380 what is the correction that I've obtained 00:46:14.380 --> 00:46:16.510 compared to the first term? 00:46:16.510 --> 00:46:19.915 So essentially, I have calculated a second order term 00:46:19.915 --> 00:46:23.450 that is of the order of this divided by the first order 00:46:23.450 --> 00:46:25.940 term and pressure that was density. 00:46:25.940 --> 00:46:31.020 And we find that this is of the order of n omega. 00:46:31.020 --> 00:46:36.810 How many particles are within the range of interaction? 00:46:36.810 --> 00:46:44.720 And this would be kind of like the ratio 00:46:44.720 --> 00:46:53.390 of the density of liquid to the density of gas. 00:46:53.390 --> 00:46:59.648 Because the density of liquid would be related-- oh, 00:46:59.648 --> 00:47:00.731 it's the other way around. 00:47:04.030 --> 00:47:09.260 The density of liquid would be 1 over the volume 00:47:09.260 --> 00:47:10.860 that one particle occupies. 00:47:10.860 --> 00:47:15.410 So the density of liquid would go in the denominator. 00:47:15.410 --> 00:47:24.020 The density of gas is this n that I have over here. 00:47:24.020 --> 00:47:26.420 And again, for the gas in this room, 00:47:26.420 --> 00:47:29.980 this ratio is of the order of 10 to the minus 3. 00:47:29.980 --> 00:47:31.780 And we are safe. 00:47:31.780 --> 00:47:35.300 But if I were to start compressing this 00:47:35.300 --> 00:47:38.500 so that I go to higher and higher densities, 00:47:38.500 --> 00:47:41.690 ultimately I say that this second order term becomes 00:47:41.690 --> 00:47:43.300 of the order of the first order term. 00:47:43.300 --> 00:47:45.730 And then perturbation doesn't make sense. 00:47:45.730 --> 00:47:49.430 Naturally, the reason for that is I haven't calculated. 00:47:49.430 --> 00:47:51.530 But typically what you find is that if you 00:47:51.530 --> 00:47:53.360 go to higher and higher order terms 00:47:53.360 --> 00:47:57.400 in the series, in most cases, but clearly not in all cases, 00:47:57.400 --> 00:48:01.220 the ratio of successive terms is more or less 00:48:01.220 --> 00:48:03.990 set by the ratio of the first terms. 00:48:03.990 --> 00:48:06.770 It has to be dimensionally correct. 00:48:06.770 --> 00:48:09.170 And we've established that the typical dimension that 00:48:09.170 --> 00:48:15.450 is controlling everything is the volume of the particle. 00:48:15.450 --> 00:48:20.590 So this expansion will fail for long range potentials, 00:48:20.590 --> 00:48:23.980 for going to liquid-like densities. 00:48:23.980 --> 00:48:27.570 But also I made another thing, which 00:48:27.570 --> 00:48:30.540 is that I assumed that that could expand this exponential. 00:48:33.140 --> 00:48:35.820 And then in principle, there are other difficulties 00:48:35.820 --> 00:48:40.070 that could arise if this condition that I wrote here 00:48:40.070 --> 00:48:41.080 is violated. 00:48:41.080 --> 00:48:46.080 If beta u0 is greater than 1, then this coefficient by itself 00:48:46.080 --> 00:48:49.180 becomes large exponentially. 00:48:49.180 --> 00:48:53.160 And then you would expect that higher order terms will also 00:48:53.160 --> 00:48:56.350 get more factors of this exponential and potential. 00:48:56.350 --> 00:48:58.130 And things will blow up on you. 00:48:58.130 --> 00:49:01.390 So it will also have difficulties 00:49:01.390 --> 00:49:06.550 at low temperatures for attractive potentials. 00:49:10.400 --> 00:49:13.010 Again, the reason for that is obvious. 00:49:13.010 --> 00:49:15.320 If you have an attractive potential, 00:49:15.320 --> 00:49:17.380 you go to low enough temperature, 00:49:17.380 --> 00:49:20.330 and the ground state is everybody sticking together-- 00:49:20.330 --> 00:49:25.560 looks nothing like a gas, looks like a solid or something. 00:49:25.560 --> 00:49:28.170 So these are the kind of limitations 00:49:28.170 --> 00:49:29.667 that one has in this series. 00:49:54.060 --> 00:50:01.650 OK, let's be brave and do some more 00:50:01.650 --> 00:50:03.500 rearrangements of this equation. 00:50:03.500 --> 00:50:06.110 So I have that to this order, beta P 00:50:06.110 --> 00:50:10.930 is n plus n squared this excluded 00:50:10.930 --> 00:50:14.680 volume over 2 1 minus beta u0. 00:50:17.870 --> 00:50:22.430 And then there's higher order terms of course. 00:50:22.430 --> 00:50:25.800 And then I notice that there are two terms on this equation that 00:50:25.800 --> 00:50:27.860 are proportional to beta. 00:50:27.860 --> 00:50:30.680 And I say, why not put both of them together? 00:50:30.680 --> 00:50:32.650 So I will have beta. 00:50:32.650 --> 00:50:34.910 Bring that term to this side, and it 00:50:34.910 --> 00:50:42.800 becomes P plus n squared omega over 2 u0. 00:50:45.420 --> 00:50:48.510 And then what is left on the other side? 00:50:48.510 --> 00:50:49.950 Let's factor out the n. 00:50:49.950 --> 00:50:53.650 I expect this to be a series in higher and higher powers of n. 00:50:53.650 --> 00:50:56.420 And the first correction to 1 comes 00:50:56.420 --> 00:50:59.130 from here, which is n omega over 2. 00:50:59.130 --> 00:51:01.550 And I expect there to be higher order terms. 00:51:13.830 --> 00:51:18.660 Now again, to order of n squared that I have calculated things 00:51:18.660 --> 00:51:22.310 correctly, this expression is no different 00:51:22.310 --> 00:51:27.870 from the following expression-- 1 minus n omega over 2. 00:51:27.870 --> 00:51:31.910 Again, there will be higher order terms in n. 00:51:31.910 --> 00:51:35.100 But the order of n squared, both of these equations, 00:51:35.100 --> 00:51:36.325 expressions, are equivalent. 00:51:41.220 --> 00:51:46.740 So if I now ignore higher order terms, 00:51:46.740 --> 00:51:54.700 this whole thing is equivalent to P plus n squared u-- 00:51:54.700 --> 00:51:57.790 well, let's write this in the following fashion-- p 00:51:57.790 --> 00:52:05.220 plus u0 omega over 2 N over V, which is density squared. 00:52:07.790 --> 00:52:13.350 And actually this side, if I were to multiply by V, 00:52:13.350 --> 00:52:14.860 what do I get? 00:52:14.860 --> 00:52:20.055 This becomes numerator V, denominator V minus N 00:52:20.055 --> 00:52:21.710 omega over 2. 00:52:21.710 --> 00:52:27.840 Let's multiply by V minus N omega over 2. 00:52:27.840 --> 00:52:32.900 And the right-hand side will be-- 00:52:32.900 --> 00:52:36.810 AUDIENCE: Is that where it should be there? 00:52:36.810 --> 00:52:37.866 PROFESSOR: What happened? 00:52:37.866 --> 00:52:40.346 AUDIENCE: This one, [INAUDIBLE]. 00:52:45.802 --> 00:52:52.860 PROFESSOR: Yes, so if I multiply this, which is N over V, by V, 00:52:52.860 --> 00:52:56.920 I will get N, good, which removes 00:52:56.920 --> 00:52:58.490 the difficulty that I had. 00:52:58.490 --> 00:53:02.790 Because now I multiply by kT. 00:53:02.790 --> 00:53:05.460 And the left hand side disappears. 00:53:05.460 --> 00:53:07.650 And on the right hand side, I will get NkT. 00:53:13.940 --> 00:53:18.830 So we'll spend some time on this equation 00:53:18.830 --> 00:53:22.105 that you will likely recognize as the van der Waals equation. 00:53:28.790 --> 00:53:34.810 I kind of justify it by rearranging this series. 00:53:34.810 --> 00:53:37.390 But van der Waals himself, of course, 00:53:37.390 --> 00:53:40.890 had a different way of justifying it, 00:53:40.890 --> 00:53:45.080 which is that basically if we think about the ideal gas, 00:53:45.080 --> 00:53:51.260 and you have particles that are moving within some volume V, 00:53:51.260 --> 00:53:54.600 if you have excluded volume interactions, 00:53:54.600 --> 00:53:59.420 then some of this volume is no longer available. 00:53:59.420 --> 00:54:01.690 And so maybe what you should do is 00:54:01.690 --> 00:54:05.300 you should reduce the volume by an amount that 00:54:05.300 --> 00:54:07.470 is proportional to the number of particles. 00:54:10.320 --> 00:54:14.410 This factor of 1/2 is actually very interesting. 00:54:14.410 --> 00:54:16.510 Because it is correct. 00:54:16.510 --> 00:54:20.610 And I see that a number of people 00:54:20.610 --> 00:54:24.020 in well known journals, et cetera, 00:54:24.020 --> 00:54:27.950 write that the excluded volume should be essentially 00:54:27.950 --> 00:54:31.630 n times the volume that is excluded around each particle. 00:54:31.630 --> 00:54:32.560 It's not that. 00:54:32.560 --> 00:54:34.480 It is 1/2 of that. 00:54:34.480 --> 00:54:36.600 And I'll leave you to mull on that. 00:54:36.600 --> 00:54:39.790 Because we will justify it later on. 00:54:39.790 --> 00:54:41.990 But in the meantime, you can think 00:54:41.990 --> 00:54:46.860 about why the factor of 1/2 is there. 00:54:46.860 --> 00:54:50.450 The other issue is that-- so there 00:54:50.450 --> 00:54:54.100 has to be a correction to the volume. 00:54:54.100 --> 00:54:57.070 And the kind of hand-waving statement 00:54:57.070 --> 00:54:59.960 that you make about the correction to pressure 00:54:59.960 --> 00:55:05.730 is that if you think about the particle that is in the middle, 00:55:05.730 --> 00:55:10.190 it is being attracted by everybody, whereas when 00:55:10.190 --> 00:55:13.750 it comes to the surface, it is really being attracted 00:55:13.750 --> 00:55:17.760 by things that are half of the space. 00:55:17.760 --> 00:55:19.810 So there is an effective potential 00:55:19.810 --> 00:55:23.850 that the particles feel from the collective action 00:55:23.850 --> 00:55:27.790 of all the others, which is slightly less steep when 00:55:27.790 --> 00:55:31.140 you approach the boundaries. 00:55:31.140 --> 00:55:36.030 And therefore, you can either think that because of this, 00:55:36.030 --> 00:55:38.590 there's less density that you have at the boundary. 00:55:38.590 --> 00:55:41.050 Less density will give you less pressure. 00:55:41.050 --> 00:55:42.770 Or if you have a particle that is 00:55:42.770 --> 00:55:45.780 kind of moving towards the wall, it 00:55:45.780 --> 00:55:47.840 is being pulled back so it doesn't 00:55:47.840 --> 00:55:51.110 hit the wall as strongly as you would expect, 00:55:51.110 --> 00:55:53.440 that would give the pressure of the ideal gas. 00:55:53.440 --> 00:55:57.730 So there is a pressure that has to be reduced related 00:55:57.730 --> 00:56:00.910 to the strength of the potential and something that 00:56:00.910 --> 00:56:03.160 has to do with all of the other particles. 00:56:03.160 --> 00:56:06.470 And there's density squared will appear there. 00:56:06.470 --> 00:56:10.470 We will have a more full justification of this equation 00:56:10.470 --> 00:56:11.690 later on. 00:56:11.690 --> 00:56:16.140 But for the time being, let's sort of sit with this equation 00:56:16.140 --> 00:56:18.510 and think about its consequences for awhile. 00:56:25.200 --> 00:56:27.560 Because the thing that we would like to do 00:56:27.560 --> 00:56:35.060 is we have come from a perspective of looking 00:56:35.060 --> 00:56:39.710 at the ideal gas and how the pressure of the ideal gas 00:56:39.710 --> 00:56:43.610 starts to get corrected because of the interactions. 00:56:43.610 --> 00:56:46.340 Of course, things become interesting 00:56:46.340 --> 00:56:48.470 when you go to the dense limit. 00:56:48.470 --> 00:56:51.040 And then the gas becomes something like a liquid. 00:56:51.040 --> 00:56:53.830 And you have transitions and things like that. 00:56:53.830 --> 00:56:58.570 So really, it's the other limit, the dense, highly interacting 00:56:58.570 --> 00:57:00.740 limited that is interesting. 00:57:00.740 --> 00:57:05.640 And to get that, we have few choices. 00:57:05.640 --> 00:57:10.770 Either I have to somehow sum many, many terms in the series, 00:57:10.770 --> 00:57:14.700 which will be very difficult, and we can't do that, 00:57:14.700 --> 00:57:18.155 or you can make some kind of approximation, rearrangement, 00:57:18.155 --> 00:57:19.490 and a guess. 00:57:19.490 --> 00:57:24.100 And this is what the van der Waals equation is based on. 00:57:24.100 --> 00:57:27.950 I made the guess here by somewhat rearranging 00:57:27.950 --> 00:57:30.490 and re-summing the terms in this series. 00:57:30.490 --> 00:57:34.390 But I will give you shortly a different justification 00:57:34.390 --> 00:57:37.920 that is more transparent and tells you immediately 00:57:37.920 --> 00:57:39.740 what the limitations are. 00:57:39.740 --> 00:57:42.550 But basically that's why we are going 00:57:42.550 --> 00:57:45.110 to spend some time with this equation. 00:57:45.110 --> 00:57:48.640 Because ultimately, we are hoping to transition 00:57:48.640 --> 00:57:51.502 from the weakly interacting case to the strongly interacting 00:57:51.502 --> 00:57:52.002 case. 00:57:58.360 --> 00:57:59.277 Yes. 00:57:59.277 --> 00:58:00.152 AUDIENCE: [INAUDIBLE] 00:58:09.050 --> 00:58:12.190 PROFESSOR: No, no, no, no, omega was 00:58:12.190 --> 00:58:16.030 defined as the volume excluded around one particle. 00:58:16.030 --> 00:58:19.070 So if you're thinking about billiard balls, 00:58:19.070 --> 00:58:21.490 r0 is the diameter. 00:58:21.490 --> 00:58:23.320 It's not the radius. 00:58:23.320 --> 00:58:28.780 And 4 pi over 3 r0 cubed is 8 times the diameter 00:58:28.780 --> 00:58:30.980 of a single billiard ball. 00:58:30.980 --> 00:58:34.030 So the correction that we get, if you like, 00:58:34.030 --> 00:58:38.860 is 4 times the volume of a billiard ball multiplied 00:58:38.860 --> 00:58:40.472 by the number of billiard balls. 00:58:51.350 --> 00:58:52.249 Where was I? 00:59:10.840 --> 00:59:17.060 OK, so what this equation gives you, the van der Waals, 00:59:17.060 --> 00:59:20.350 is an expression for how the pressure behaves 00:59:20.350 --> 00:59:21.765 as a function of volume. 00:59:26.390 --> 00:59:31.970 Actually, it would be nicer if we were to sort of replace 00:59:31.970 --> 00:59:35.150 this by volume per particle, which 00:59:35.150 --> 00:59:36.670 would be the inverse density. 00:59:36.670 --> 00:59:39.980 But you can use one or the other. 00:59:39.980 --> 00:59:42.320 It doesn't matter. 00:59:42.320 --> 00:59:46.410 Now, what you find is that there is, first 00:59:46.410 --> 00:59:51.510 of all, a limitation to the volume. 00:59:51.510 --> 00:59:54.880 So basically, none of your cares are 00:59:54.880 --> 00:59:59.520 going to go to volumes that are lower than n omega over 2. 00:59:59.520 --> 01:00:04.040 So basically there's a barrier here that occurs over here. 01:00:04.040 --> 01:00:06.190 But if you go to the other limit, 01:00:06.190 --> 01:00:10.720 where you go to large volumes, you can ignore terms like this. 01:00:10.720 --> 01:00:15.610 And then you get back the kind of ideal gas behavior. 01:00:15.610 --> 01:00:19.060 So basically, in one limit, where 01:00:19.060 --> 01:00:22.720 you are either at high temperatures-- 01:00:22.720 --> 01:00:27.340 and at high temperatures, essentially, the correction 01:00:27.340 --> 01:00:30.660 here will also be negligible. 01:00:30.660 --> 01:00:34.090 Or you are at high values of the volume. 01:00:34.090 --> 01:00:38.160 You get isotherms that are very much like the isotherms 01:00:38.160 --> 01:00:41.180 that you have for the ideal gas, except that 01:00:41.180 --> 01:00:45.760 rather than asymptote to 0, PV going like NkT, 01:00:45.760 --> 01:00:50.910 you asymptote to this excluded volume. 01:00:50.910 --> 01:00:54.440 So this is for high T, T large. 01:00:57.170 --> 01:00:58.680 T larger than what? 01:00:58.680 --> 01:01:01.910 Well essentially, what happens is 01:01:01.910 --> 01:01:07.020 that if I look at the pressure, I can write it in this fashion 01:01:07.020 --> 01:01:07.520 also. 01:01:07.520 --> 01:01:15.060 It is NkT divided by V minus N omega over 2. 01:01:15.060 --> 01:01:18.180 So that's the term that dominates at high temperature. 01:01:18.180 --> 01:01:20.640 It's proportional to kT. 01:01:20.640 --> 01:01:26.850 But then there's the subtraction u0 omega over 2 N/V squared. 01:01:26.850 --> 01:01:30.270 And again, this term is not so important at large volume. 01:01:30.270 --> 01:01:32.300 Because at large volume, this 1/V 01:01:32.300 --> 01:01:35.350 is more dominant than 1 over V squared. 01:01:35.350 --> 01:01:38.880 But as you go to lower temperatures 01:01:38.880 --> 01:01:42.980 and intermediate volumes, then essentially you 01:01:42.980 --> 01:01:47.140 have potentially a correction that falls off as minus 1 01:01:47.140 --> 01:01:49.110 over V squared. 01:01:49.110 --> 01:01:53.620 And so this correction that falls off as 1 over V squared 01:01:53.620 --> 01:01:57.880 can potentially modify your curve, 01:01:57.880 --> 01:02:03.370 bring it down, and then give it a structure such as this. 01:02:03.370 --> 01:02:06.350 So this is T less than. 01:02:06.350 --> 01:02:10.320 And clearly, between these two types of behavior, 01:02:10.320 --> 01:02:14.100 where there is monotonic behavior 01:02:14.100 --> 01:02:16.880 or non-monotonic behavior, there has 01:02:16.880 --> 01:02:20.860 to be a limiting curve that, let's say, does something 01:02:20.860 --> 01:02:27.190 like this, comes tangentially to the horizontal axis, 01:02:27.190 --> 01:02:30.800 and then goes on like this. 01:02:30.800 --> 01:02:33.870 So this would be for T equals Tc. 01:02:33.870 --> 01:02:37.543 This is T greater than Tc, T less than Tc. 01:02:47.700 --> 01:02:50.140 OK, so that's fine, except that now we 01:02:50.140 --> 01:02:53.080 have encountered the difficulty. 01:02:53.080 --> 01:02:57.040 Because one of the things that we had established 01:02:57.040 --> 01:03:10.210 for thermodynamic stability was that delta P delta 01:03:10.210 --> 01:03:11.870 V had to be negative. 01:03:15.870 --> 01:03:20.690 And the ideal gas curve and portions 01:03:20.690 --> 01:03:26.480 of this curve which have a negative slope 01:03:26.480 --> 01:03:29.280 are all consistent with this. 01:03:29.280 --> 01:03:39.720 But this portion over here where dP by dV is positive, 01:03:39.720 --> 01:03:42.350 it kind of violates the condition 01:03:42.350 --> 01:03:48.380 that the compressibility kappa T, which is minus 1/V dV 01:03:48.380 --> 01:03:51.550 by dP at constant temperature, better be positive. 01:03:55.360 --> 01:04:02.830 And so clearly, the expression that one gets through this van 01:04:02.830 --> 01:04:07.850 der Waals equation has a limitation. 01:04:07.850 --> 01:04:11.460 And the most natural way about it 01:04:11.460 --> 01:04:15.320 is to say, well, the question that you wrote down 01:04:15.320 --> 01:04:19.214 is clearly incorrect. 01:04:19.214 --> 01:04:20.130 That's certainly true. 01:04:20.130 --> 01:04:22.710 Also, this equation is incorrect, 01:04:22.710 --> 01:04:26.856 except that this kind of reminds us 01:04:26.856 --> 01:04:34.780 with what actually happens if I look at the isotherms of a gas, 01:04:34.780 --> 01:04:38.660 such as gas in this room, or something that 01:04:38.660 --> 01:04:41.010 is more familiar, such as water. 01:04:41.010 --> 01:04:46.330 What you find is that at high temperatures, 01:04:46.330 --> 01:04:50.440 you indeed have curves that look like this. 01:04:53.130 --> 01:04:56.550 But at low temperatures, you liquidify. 01:04:56.550 --> 01:05:00.632 And what you have is a zone of coexistence, 01:05:00.632 --> 01:05:03.580 let's say something like this, by which I 01:05:03.580 --> 01:05:07.950 mean that the isotherm that you draw 01:05:07.950 --> 01:05:13.910 has a portion that lives in the gas phase 01:05:13.910 --> 01:05:18.630 and is a slightly modified version of the ideal gas. 01:05:18.630 --> 01:05:21.200 But then it has a portion that corresponds 01:05:21.200 --> 01:05:24.090 to the more or less incompressible liquid. 01:05:24.090 --> 01:05:28.870 Although this has clearly still some finite compressibility. 01:05:28.870 --> 01:05:33.100 And then in between, there is a region where if you have a box, 01:05:33.100 --> 01:05:36.090 part of your box would be liquid, and part of your box 01:05:36.090 --> 01:05:37.240 would be gas. 01:05:37.240 --> 01:05:42.100 And as you compress the box, the proportion of liquid and gas 01:05:42.100 --> 01:05:44.120 will change. 01:05:44.120 --> 01:05:48.140 And this happens for T less than some Tc. 01:05:48.140 --> 01:05:53.030 And once more, there is a curve that kind of 01:05:53.030 --> 01:05:58.660 looks like this at some intermediate temperature Tc. 01:06:01.550 --> 01:06:06.220 So you look at the comparison, you say, well, 01:06:06.220 --> 01:06:09.340 as long as I stay above Tc-- of course 01:06:09.340 --> 01:06:13.865 I don't expect this equation to give the right numbers 01:06:13.865 --> 01:06:15.750 for what Tc or whatever is. 01:06:15.750 --> 01:06:19.940 But qualitatively, I get topologies and behaviors 01:06:19.940 --> 01:06:22.970 at that T greater than Tc all the way up 01:06:22.970 --> 01:06:26.300 to Tc are not very different. 01:06:26.300 --> 01:06:29.660 But at T less than Tc, they also signal 01:06:29.660 --> 01:06:31.980 that something bad is happening. 01:06:31.980 --> 01:06:37.120 And somehow, the original description has to be modified. 01:06:37.120 --> 01:06:42.080 And so the thing that Maxwell and van der Waals and company 01:06:42.080 --> 01:06:50.520 did was to somehow convert this incorrect set of equations 01:06:50.520 --> 01:06:54.260 to something that resembles this. 01:06:54.260 --> 01:06:57.724 So let's see how they managed to do this. 01:07:23.960 --> 01:07:28.180 Actually, this may not be a bad thing to keep in mind. 01:07:30.750 --> 01:07:33.770 Let's have a thermodynamic-- well, 01:07:33.770 --> 01:07:40.200 I kind of emphasize that there is thermodynamic reason for why 01:07:40.200 --> 01:07:43.560 this is not valid. 01:07:43.560 --> 01:07:45.760 Always there's a corresponding reason 01:07:45.760 --> 01:07:48.100 if you look at things from the perspective 01:07:48.100 --> 01:07:49.920 of statistical mechanics. 01:07:49.920 --> 01:07:51.590 So it may be useful to sort of think 01:07:51.590 --> 01:07:54.820 about what's happening from that perspective. 01:07:54.820 --> 01:07:57.510 Let's imagine that we are in this grand canonical ensemble. 01:07:57.510 --> 01:08:02.040 In the grand canonical ensemble, the number of particles, 01:08:02.040 --> 01:08:05.760 as we discussed, is not fixed. 01:08:05.760 --> 01:08:10.800 But the mean number of particles is given by the expressions 01:08:10.800 --> 01:08:12.850 that we saw over here, is related 01:08:12.850 --> 01:08:17.700 to the pressure through dP by d mu at constant temperature. 01:08:22.560 --> 01:08:25.790 But there are fluctuations. 01:08:25.790 --> 01:08:32.340 So you would say that the fluctuations are related 01:08:32.340 --> 01:08:38.490 by taking a second derivative of this object. 01:08:38.490 --> 01:08:43.910 Because clearly, q, if I were to expand it in powers of beta mu, 01:08:43.910 --> 01:08:47.779 will generate various moments of n. 01:08:47.779 --> 01:08:50.880 Log of q will generate cumulants. 01:08:50.880 --> 01:08:54.399 So the variance would be one more derivative. 01:08:54.399 --> 01:08:59.880 And so that will amount to taking 01:08:59.880 --> 01:09:02.310 a derivative of the first derivative, which 01:09:02.310 --> 01:09:04.740 was the number itself. 01:09:04.740 --> 01:09:11.970 So I will get this to be dN by d mu at constant T, 01:09:11.970 --> 01:09:15.250 except that there's an additional factor of kT. 01:09:15.250 --> 01:09:18.529 So maybe I will write this carefully enough. 01:09:18.529 --> 01:09:28.020 I have d2 log Q with respect to beta mu squared 01:09:28.020 --> 01:09:30.632 will give me N squared. 01:09:30.632 --> 01:09:37.370 Now, the first derivative with respect to beta mu of log Q 01:09:37.370 --> 01:09:42.200 gave me the mean, which I'm actually thinking of as N 01:09:42.200 --> 01:09:43.870 itself. 01:09:43.870 --> 01:09:45.800 And then there's this factor of beta. 01:09:45.800 --> 01:09:50.955 So this is really kT dN by d mu. 01:09:50.955 --> 01:09:53.290 All of these are done at constant temperature. 01:10:04.680 --> 01:10:10.110 So the statistical analog of stability 01:10:10.110 --> 01:10:15.270 really comes down to variances being positive. 01:10:15.270 --> 01:10:21.320 So my claim is that this variance being positive 01:10:21.320 --> 01:10:26.630 is related to this stability condition. 01:10:26.630 --> 01:10:29.010 Let me do that in the following fashion. 01:10:29.010 --> 01:10:30.880 I divide these two expressions. 01:10:30.880 --> 01:10:33.470 I will get the average of N squared divided 01:10:33.470 --> 01:10:40.130 by average of N, which is really N, so the variance over N. 01:10:40.130 --> 01:10:41.290 I have kT. 01:10:44.350 --> 01:10:51.300 And then I have the ratio of two of these derivatives. 01:10:51.300 --> 01:10:54.090 And the reason I wanted to do that ratio 01:10:54.090 --> 01:10:56.690 was to get rid of the d mu. 01:10:56.690 --> 01:10:59.160 Because d mu is not so nice. 01:10:59.160 --> 01:11:01.645 If I take the ratio of those two derivatives, 01:11:01.645 --> 01:11:09.830 I will get dN by dP at constant T. 01:11:09.830 --> 01:11:13.410 But what I really wanted was to say something about dV 01:11:13.410 --> 01:11:17.800 by dP at constant T rather than dN by dP. 01:11:17.800 --> 01:11:24.750 So what I'm going to do is to somehow convert this dN by dP 01:11:24.750 --> 01:11:30.180 into dN by dV, and then dV by dP. 01:11:32.890 --> 01:11:36.250 And actually to do that, I need to use the chain rule. 01:11:36.250 --> 01:11:40.440 So this will be T, P. This will be T, N. 01:11:40.440 --> 01:11:44.590 And the chain rule will give me an additional minus sign. 01:11:44.590 --> 01:11:48.830 So this is a reminder of how your partial derivatives 01:11:48.830 --> 01:11:51.580 and the chain rule goes. 01:11:51.580 --> 01:11:58.460 Finally, dN by dV is none other than the density N/V. 01:11:58.460 --> 01:12:02.050 At constant temperature, N and V will be proportional. 01:12:02.050 --> 01:12:07.980 And what I have here is 1/V dV by dP, 01:12:07.980 --> 01:12:09.560 which is the compressibility. 01:12:09.560 --> 01:12:17.940 So the whole thing here is NkT times kappa of T. 01:12:17.940 --> 01:12:23.730 And so this being positive, this positivity of the variance, 01:12:23.730 --> 01:12:28.035 is-- the thermodynamic analog of it 01:12:28.035 --> 01:12:31.770 was from the stability, and something such as this. 01:12:31.770 --> 01:12:35.080 The statistical analog of it is that the probability 01:12:35.080 --> 01:12:39.710 distribution that I'm looking at in the grand canonical ensemble 01:12:39.710 --> 01:12:43.850 as a function of N has to be peaked around some region. 01:12:43.850 --> 01:12:45.720 The variance around that peak better 01:12:45.720 --> 01:12:51.360 be positive so that I'm looking in the vicinity of a maximum. 01:12:51.360 --> 01:12:53.250 If it was negative, it means that I'm 01:12:53.250 --> 01:12:55.620 looking in the vicinity of a minimum. 01:12:55.620 --> 01:12:58.580 And I'm looking at the least likely configuration. 01:12:58.580 --> 01:13:00.270 So it cannot be allowed. 01:13:00.270 --> 01:13:03.380 So that's the statistical reason. 01:13:03.380 --> 01:13:06.690 OK, now I want to use that in connection 01:13:06.690 --> 01:13:11.260 with what we have over here and see 01:13:11.260 --> 01:13:12.985 what I can learn about this. 01:13:25.090 --> 01:13:28.680 I said that I wrote two expressions and divided them, 01:13:28.680 --> 01:13:32.810 because I didn't want to deal with the chemical potential. 01:13:32.810 --> 01:13:37.940 But let me redefine that and say the following, 01:13:37.940 --> 01:13:44.330 that N is V dP by d mu at constant T. 01:13:44.330 --> 01:13:48.290 And maybe I will start to gain some idea about chemical 01:13:48.290 --> 01:13:51.430 potential if I rearrange this as d mu 01:13:51.430 --> 01:14:02.500 is V/N dP along this surface of constant P at constant T. 01:14:02.500 --> 01:14:08.340 And then I can calculate mu as a function of P 01:14:08.340 --> 01:14:12.430 at some particular temperature minus mu 01:14:12.430 --> 01:14:18.810 at some reference point by integrating from the reference 01:14:18.810 --> 01:14:22.590 point to the pressure of interest 01:14:22.590 --> 01:14:30.020 the quantity V/N dP prime where V of P prime I 01:14:30.020 --> 01:14:32.360 take from this curve. 01:14:32.360 --> 01:14:37.950 So this curve gives me P as a function of V. 01:14:37.950 --> 01:14:43.420 But I can invert it and think of V as a function of P, 01:14:43.420 --> 01:14:47.310 put it here, and see what is happening with the chemical 01:14:47.310 --> 01:14:51.420 potential if I walk along one of these trajectories. 01:14:51.420 --> 01:14:58.250 And in particular, let's kind of draw one of these 01:14:58.250 --> 01:15:01.520 curves that I am unhappy with that 01:15:01.520 --> 01:15:06.620 have this kind of form in general 01:15:06.620 --> 01:15:11.230 and calculate what happens to the chemical potential 01:15:11.230 --> 01:15:14.060 if I, let's say, pick this reference point 01:15:14.060 --> 01:15:18.100 A, and the corresponding pressure PA, 01:15:18.100 --> 01:15:22.610 and go along this curve, which corresponds to some temperature 01:15:22.610 --> 01:15:27.350 T that is less than this instability temperature, 01:15:27.350 --> 01:15:29.485 and track the shape of the chemical potential. 01:15:32.460 --> 01:15:36.330 So this formula says that the change in chemical potential we 01:15:36.330 --> 01:15:41.000 obtained by calculating V-- divide by N, 01:15:41.000 --> 01:15:44.320 but our N is fixed-- as a function of P, 01:15:44.320 --> 01:15:47.560 and integrating as you go up in pressure. 01:15:47.560 --> 01:15:52.070 So essentially, I'm calculating the integral starting 01:15:52.070 --> 01:15:55.250 from point A as I go under this curve. 01:15:58.700 --> 01:16:04.120 So let's plot as a result of that integration how 01:16:04.120 --> 01:16:09.460 that chemical potential at P minus PA is going to look 01:16:09.460 --> 01:16:15.340 like as I go in P beyond PA. 01:16:15.340 --> 01:16:17.990 So at PA, that's my reference. 01:16:17.990 --> 01:16:21.230 The chemical potential is some value. 01:16:21.230 --> 01:16:26.960 As I go up and up, because of this area of this curve, 01:16:26.960 --> 01:16:29.810 I keep adding to the chemical potential 01:16:29.810 --> 01:16:34.420 until I reach the maximum here at this point C. 01:16:34.420 --> 01:16:43.380 So there is some curve that goes from this A to some point C. 01:16:43.380 --> 01:16:51.110 Now, the thing is that when I continue going along this curve 01:16:51.110 --> 01:16:56.900 down this potential all the way to the bottom of this, which 01:16:56.900 --> 01:17:02.840 I will call D, my DP's are negative. 01:17:02.840 --> 01:17:08.650 So I start subtracting from what I had before. 01:17:08.650 --> 01:17:12.351 And so then the curve starts to go down. 01:17:12.351 --> 01:17:16.600 So all the way to D, I'm proceeding 01:17:16.600 --> 01:17:19.070 in the opposite direction. 01:17:19.070 --> 01:17:24.370 Once I hit D, I start going all the way to the end of the curve 01:17:24.370 --> 01:17:26.340 however far I want to go. 01:17:26.340 --> 01:17:29.180 And I'm adding some positive area. 01:17:29.180 --> 01:17:37.415 So the next part I also have a variation in chemical potential 01:17:37.415 --> 01:17:38.620 that goes like this. 01:17:44.140 --> 01:17:51.384 So you ask, well, if I give you what the temperature is 01:17:51.384 --> 01:17:53.300 and what the pressure is-- and I have told you 01:17:53.300 --> 01:17:55.360 what the temperature and pressure are-- 01:17:55.360 --> 01:17:57.850 I should be able to calculate the chemical potential. 01:17:57.850 --> 01:17:59.800 It's an intensive function of the other two 01:17:59.800 --> 01:18:01.700 intensive functions. 01:18:01.700 --> 01:18:04.650 And this curve tells me that, except that there 01:18:04.650 --> 01:18:07.710 are regions where I don't know which one to pick. 01:18:11.060 --> 01:18:13.050 There are regions where it's obvious there's 01:18:13.050 --> 01:18:15.630 one value of the chemical potential. 01:18:15.630 --> 01:18:18.760 But there is in between, because of this instability, 01:18:18.760 --> 01:18:21.880 three possible values. 01:18:21.880 --> 01:18:27.570 And typically, if you have systems-- 01:18:27.570 --> 01:18:31.200 think about different chemicals that can go between each other. 01:18:31.200 --> 01:18:34.860 And there are potentials for chemical transformations. 01:18:34.860 --> 01:18:38.450 You say, go and pick the lowest chemical potential. 01:18:38.450 --> 01:18:41.900 The system will evolve onto the condition that 01:18:41.900 --> 01:18:44.410 has the lowest chemical potential. 01:18:44.410 --> 01:18:48.290 So that says that when you have a situation such as this, 01:18:48.290 --> 01:18:52.490 you really have to pick the lowest one. 01:18:52.490 --> 01:19:00.190 And so you have to bail out on your first curve at the point 01:19:00.190 --> 01:19:08.600 B, and on your second curve at a point E, where on this curve 01:19:08.600 --> 01:19:15.030 B and E are such that when you integrate this BDP all 01:19:15.030 --> 01:19:19.450 the way from B to E, you will get 0, which means that you 01:19:19.450 --> 01:19:27.280 have to find points B and E such that the integral here 01:19:27.280 --> 01:19:30.470 is the same as the integral here. 01:19:30.470 --> 01:19:33.416 And this is the so-called Maxwell construction. 01:19:42.200 --> 01:19:47.820 So what that means is that originally I was telling you 01:19:47.820 --> 01:19:54.700 over here that there is some portion of the curve that 01:19:54.700 --> 01:19:56.860 violates all kinds of thermodynamics. 01:19:56.860 --> 01:19:59.420 And you should not access it. 01:19:59.420 --> 01:20:03.500 And this other argument that we are pursuing here says that 01:20:03.500 --> 01:20:07.550 actually the regions that you cannot access are beyond that. 01:20:07.550 --> 01:20:10.200 There's a portion that extends in this direction, 01:20:10.200 --> 01:20:13.160 and a portion that extends in that direction. 01:20:13.160 --> 01:20:16.890 And really there's a portion that you can access up to here, 01:20:16.890 --> 01:20:19.960 and a portion that you can access up to here. 01:20:19.960 --> 01:20:22.010 And maybe you should sort of then 01:20:22.010 --> 01:20:25.210 join them by a straight line and make an analog 01:20:25.210 --> 01:20:28.980 with what we have over there. 01:20:28.980 --> 01:20:36.310 Now, in another thing, another way of looking at this 01:20:36.310 --> 01:20:41.370 is that the green portion is certainly unstable, 01:20:41.370 --> 01:20:44.950 violates all kinds of thermodynamic conditions. 01:20:44.950 --> 01:20:48.850 And people argue that these other portions that 01:20:48.850 --> 01:20:53.960 have the right slope are in some sense metastable. 01:20:53.960 --> 01:21:01.040 So this would be a stable equilibrium. 01:21:01.040 --> 01:21:04.540 This would be an unstable equilibrium. 01:21:04.540 --> 01:21:07.700 And a metastable equilibrium is that you are at a minimum, 01:21:07.700 --> 01:21:10.880 but there's a deeper minimum somewhere else. 01:21:10.880 --> 01:21:23.350 And the picture is that if you manage to take your gas 01:21:23.350 --> 01:21:27.270 and put additional pressure on it, 01:21:27.270 --> 01:21:33.190 there is a region where there is a coexistence with the liquid 01:21:33.190 --> 01:21:37.180 which is better in terms of free energy, et cetera. 01:21:37.180 --> 01:21:40.250 But there could be some kind of a kinetic barrier, 01:21:40.250 --> 01:21:44.990 like nucleation or whatever, that prevents you to go there. 01:21:44.990 --> 01:21:48.180 And this again is something that is experimentally observed. 01:21:48.180 --> 01:21:51.890 If you try to rapidly pressurize a gas, 01:21:51.890 --> 01:21:56.900 you could sometimes avoid and not make your transition 01:21:56.900 --> 01:21:59.680 to the liquid state at the right pressure. 01:21:59.680 --> 01:22:02.595 But if you were to do things sufficiently slowly, 01:22:02.595 --> 01:22:07.020 you would ultimately reach this point 01:22:07.020 --> 01:22:09.230 that corresponds to the true equilibrium 01:22:09.230 --> 01:22:13.660 pressure of making this transition. 01:22:13.660 --> 01:22:18.020 Now, the thing is that somehow this whole story 01:22:18.020 --> 01:22:21.690 is not very satisfactory. 01:22:21.690 --> 01:22:23.940 I started with some rearrangement 01:22:23.940 --> 01:22:27.840 of some perturbative to arrive at this equation. 01:22:27.840 --> 01:22:31.200 This equation has this unstable portion. 01:22:31.200 --> 01:22:34.900 And somehow I'm trying to relate something that makes absolutely 01:22:34.900 --> 01:22:37.880 no thermodynamic sense and try to gain 01:22:37.880 --> 01:22:43.545 from it some idea about what is happening in a real liquid gas 01:22:43.545 --> 01:22:45.250 system. 01:22:45.250 --> 01:22:48.580 So it would be good if all of this 01:22:48.580 --> 01:22:53.250 could be more formally described within a framework 01:22:53.250 --> 01:22:57.530 where all of the approximations, et cetera, are clear, 01:22:57.530 --> 01:22:59.410 and we know what's going on. 01:22:59.410 --> 01:23:02.575 And so next time, we will do that. 01:23:02.575 --> 01:23:06.910 We will essentially developed a much more systematic formalism 01:23:06.910 --> 01:23:11.550 for calculating the properties of an interacting gas, 01:23:11.550 --> 01:23:16.619 and see that we get this, and we also understand why it fails.