WEBVTT 00:00:00.000 --> 00:00:05.467 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:05.467 --> 00:00:10.333 align:middle line:90% 00:00:10.333 --> 00:00:12.000 align:middle line:84% GIAN PAOLO BERETTA: OK, so good morning. 00:00:12.000 --> 00:00:15.030 align:middle line:90% 00:00:15.030 --> 00:00:20.860 align:middle line:84% Well, we finally got to the heat interaction. 00:00:20.860 --> 00:00:26.470 align:middle line:84% So we are almost comfortable with the old ideas 00:00:26.470 --> 00:00:33.700 align:middle line:84% of thermodynamics being the theory of heat and work. 00:00:33.700 --> 00:00:35.300 align:middle line:90% We did temperature. 00:00:35.300 --> 00:00:38.490 align:middle line:90% 00:00:38.490 --> 00:00:40.320 align:middle line:84% Usually, there is also another thing 00:00:40.320 --> 00:00:42.663 align:middle line:84% that maybe I didn't stress enough. 00:00:42.663 --> 00:00:46.440 align:middle line:90% 00:00:46.440 --> 00:00:50.730 align:middle line:84% Usually, you are taught that thermodynamics 00:00:50.730 --> 00:00:56.655 align:middle line:84% deals with large amounts of substances, many particle 00:00:56.655 --> 00:00:57.155 align:middle line:90% systems. 00:00:57.155 --> 00:01:00.040 align:middle line:90% 00:01:00.040 --> 00:01:04.409 align:middle line:84% But you see that, if you went to the seminar yesterday, 00:01:04.409 --> 00:01:05.379 align:middle line:90% I suggested-- 00:01:05.379 --> 00:01:11.050 align:middle line:84% if you are going to the seminar by Professor Howard Stone, who 00:01:11.050 --> 00:01:14.210 align:middle line:84% is a very important name in fluid mechanics, 00:01:14.210 --> 00:01:21.250 align:middle line:84% and you'll see today that much of current interest 00:01:21.250 --> 00:01:23.920 align:middle line:90% is in small devices-- 00:01:23.920 --> 00:01:33.040 align:middle line:84% microfluidics, bacteria floating on solutions, trying 00:01:33.040 --> 00:01:34.920 align:middle line:90% to find food or other things-- 00:01:34.920 --> 00:01:38.590 align:middle line:90% 00:01:38.590 --> 00:01:41.940 align:middle line:84% few particles, in some extremes, like 00:01:41.940 --> 00:01:44.010 align:middle line:84% in the quantum thermodynamics, only very 00:01:44.010 --> 00:01:49.230 align:middle line:84% few particles, maybe one, whereas in the fluid mechanics 00:01:49.230 --> 00:01:55.150 align:middle line:84% of small systems, it's few, but not so few. 00:01:55.150 --> 00:01:57.650 align:middle line:90% Nevertheless, it's not many. 00:01:57.650 --> 00:01:59.540 align:middle line:84% So these are not macroscopic systems. 00:01:59.540 --> 00:02:04.990 align:middle line:84% So you would wonder, well, does thermodynamics apply? 00:02:04.990 --> 00:02:07.570 align:middle line:84% OK, this is precisely the reason why 00:02:07.570 --> 00:02:13.890 align:middle line:84% I'm postponing the part that deals with many particles. 00:02:13.890 --> 00:02:18.190 align:middle line:84% In other words, all that we have done so far, if you noticed, 00:02:18.190 --> 00:02:22.910 align:middle line:84% didn't require any assumption about the number of particles. 00:02:22.910 --> 00:02:26.390 align:middle line:84% So all the things that we do also today, 00:02:26.390 --> 00:02:28.420 align:middle line:90% until maybe the next-- 00:02:28.420 --> 00:02:35.260 align:middle line:84% or the next, other lecture does apply 00:02:35.260 --> 00:02:41.180 align:middle line:84% to microscopic and macroscopic systems, all right? 00:02:41.180 --> 00:02:46.090 align:middle line:84% So this is important because if you work downstairs 00:02:46.090 --> 00:02:49.000 align:middle line:84% in the Hatsopoulos Microfluidics Lab, 00:02:49.000 --> 00:02:54.140 align:middle line:84% or if you do research in biology, at some point, 00:02:54.140 --> 00:02:56.680 align:middle line:90% you're going to have to-- 00:02:56.680 --> 00:02:58.550 align:middle line:84% even if you are an experimentalist, 00:02:58.550 --> 00:03:01.900 align:middle line:84% you're going to have to make a model of your experiments. 00:03:01.900 --> 00:03:04.570 align:middle line:84% If you are a theorist, you start from the model. 00:03:04.570 --> 00:03:09.820 align:middle line:84% If you are an experimentalist, you want to correlate your data. 00:03:09.820 --> 00:03:13.090 align:middle line:84% And you need a model usually to do that. 00:03:13.090 --> 00:03:16.420 align:middle line:84% And therefore, what we are doing, 00:03:16.420 --> 00:03:21.980 align:middle line:84% which is, of course, a lot of theory, a lot of equations, but 00:03:21.980 --> 00:03:26.620 align:middle line:84% is exactly what you need to do this modeling. 00:03:26.620 --> 00:03:30.380 align:middle line:84% So that's the motivation for this course. 00:03:30.380 --> 00:03:37.720 align:middle line:84% It's timely also because it's one 00:03:37.720 --> 00:03:41.660 align:middle line:84% of the few ways of presenting thermodynamics in such a way 00:03:41.660 --> 00:03:46.400 align:middle line:84% that it starts from generalities that hold not 00:03:46.400 --> 00:03:48.070 align:middle line:90% only for macroscopic systems. 00:03:48.070 --> 00:03:50.840 align:middle line:90% 00:03:50.840 --> 00:03:57.020 align:middle line:84% OK, let me just review quickly what we did last time. 00:03:57.020 --> 00:04:00.560 align:middle line:90% We defined heat interactions. 00:04:00.560 --> 00:04:05.540 align:middle line:84% So a heat interaction was defined as one 00:04:05.540 --> 00:04:07.880 align:middle line:90% in which a system-- 00:04:07.880 --> 00:04:15.350 align:middle line:84% two systems, A and B-- initially at stable equilibrium states 00:04:15.350 --> 00:04:18.140 align:middle line:90% exchange energy and entropy. 00:04:18.140 --> 00:04:25.640 align:middle line:84% But they do so in such a way that even if you put in between 00:04:25.640 --> 00:04:30.410 align:middle line:84% them some machinery that tries to separate the energy 00:04:30.410 --> 00:04:37.920 align:middle line:84% transferred into two streams, one of which is work-- 00:04:37.920 --> 00:04:41.570 align:middle line:84% so it's pure energy and no entropy-- 00:04:41.570 --> 00:04:43.600 align:middle line:90% that machine will not succeed. 00:04:43.600 --> 00:04:47.410 align:middle line:84% So no fraction of the energy exchanged 00:04:47.410 --> 00:04:50.440 align:middle line:90% can be recognized as work. 00:04:50.440 --> 00:04:59.510 align:middle line:84% So heat is the extreme, the farthest non-work interaction, 00:04:59.510 --> 00:05:04.410 align:middle line:84% farthest from work, most distinguishable from work. 00:05:04.410 --> 00:05:13.270 align:middle line:84% And that definition leads to the condition that the two systems, 00:05:13.270 --> 00:05:15.340 align:middle line:84% to have a heat interaction, must be 00:05:15.340 --> 00:05:19.930 align:middle line:84% at almost the same temperature, if not the same temperature, 00:05:19.930 --> 00:05:23.200 align:middle line:84% because only in that limit the Carnot ratio between the two 00:05:23.200 --> 00:05:27.430 align:middle line:84% temperatures, which represents the fraction of the energy that 00:05:27.430 --> 00:05:33.480 align:middle line:84% could be identified as work, goes to 0. 00:05:33.480 --> 00:05:36.230 align:middle line:90% 00:05:36.230 --> 00:05:42.770 align:middle line:84% We also mentioned that, in spite of the usual understanding that 00:05:42.770 --> 00:05:46.940 align:middle line:84% heat is the energy transferred between bodies at different 00:05:46.940 --> 00:05:54.800 align:middle line:84% temperature, and therefore seems to contradict our definition, 00:05:54.800 --> 00:06:00.590 align:middle line:84% we noticed that, if the standard model that we adopt in heat 00:06:00.590 --> 00:06:08.540 align:middle line:84% transfer is a local equilibrium or quasi-equilibrium model, 00:06:08.540 --> 00:06:16.730 align:middle line:84% whereby the system is divided in smaller systems that are modeled 00:06:16.730 --> 00:06:19.700 align:middle line:90% as being near equilibrium-- 00:06:19.700 --> 00:06:25.850 align:middle line:84% not at equilibrium, like highlighted here-- 00:06:25.850 --> 00:06:27.030 align:middle line:90% near equilibrium. 00:06:27.030 --> 00:06:32.790 align:middle line:84% And therefore, in that way, first of all, 00:06:32.790 --> 00:06:35.720 align:middle line:84% you could assign the temperature of the equilibrium state 00:06:35.720 --> 00:06:39.140 align:middle line:84% that these elements would reach if you isolate them 00:06:39.140 --> 00:06:42.920 align:middle line:84% from the rest and just let them relax to equilibrium. 00:06:42.920 --> 00:06:52.880 align:middle line:84% And second, on the E versus S diagram-- 00:06:52.880 --> 00:06:58.280 align:middle line:84% we notice also this is important and we'll return to this-- 00:06:58.280 --> 00:07:05.930 align:middle line:84% the heat interaction on one side pulls the system 00:07:05.930 --> 00:07:08.880 align:middle line:90% in a direction that-- 00:07:08.880 --> 00:07:12.650 align:middle line:84% So on the E versus S diagram, if I'm here 00:07:12.650 --> 00:07:19.640 align:middle line:84% and I'm going to inject both energy and entropy, 00:07:19.640 --> 00:07:24.200 align:middle line:84% that's the direction in which my state will move. 00:07:24.200 --> 00:07:28.550 align:middle line:84% And if another interaction injects 00:07:28.550 --> 00:07:34.100 align:middle line:90% the same energy but takes away-- 00:07:34.100 --> 00:07:36.540 align:middle line:84% I'm sorry, takes away energy, same energy, 00:07:36.540 --> 00:07:42.700 align:middle line:84% but takes away more entropy, then it goes that way. 00:07:42.700 --> 00:07:44.660 align:middle line:90% And the two are not aligned. 00:07:44.660 --> 00:07:49.750 align:middle line:84% So in order to align them to keep the state at steady state 00:07:49.750 --> 00:07:53.800 align:middle line:84% so that the state doesn't move, you need also something. 00:07:53.800 --> 00:07:59.390 align:middle line:84% And that something is another vector, 00:07:59.390 --> 00:08:03.130 align:middle line:84% which is representing the entropy produced 00:08:03.130 --> 00:08:06.700 align:middle line:90% by irreversibility. 00:08:06.700 --> 00:08:11.650 align:middle line:84% So you need irreversibility to keep a fluid element 00:08:11.650 --> 00:08:15.340 align:middle line:84% or a solid element in steady state 00:08:15.340 --> 00:08:18.340 align:middle line:84% when subjected to a gradient in temperature, that 00:08:18.340 --> 00:08:22.480 align:middle line:84% means a temperature difference between one 00:08:22.480 --> 00:08:25.490 align:middle line:90% side and the other side. 00:08:25.490 --> 00:08:28.382 align:middle line:90% OK. 00:08:28.382 --> 00:08:32.080 align:middle line:90% So let's do something new now. 00:08:32.080 --> 00:08:36.370 align:middle line:90% Not very new, but it's another-- 00:08:36.370 --> 00:08:40.870 align:middle line:84% it's similar to what we have done 00:08:40.870 --> 00:08:44.410 align:middle line:84% when we proved the conditions for mutual equilibrium 00:08:44.410 --> 00:08:46.240 align:middle line:90% between systems. 00:08:46.240 --> 00:08:51.550 align:middle line:84% This time, though, we are not applying the maximum entropy 00:08:51.550 --> 00:08:54.550 align:middle line:84% principle, but we are applying the principle 00:08:54.550 --> 00:08:57.560 align:middle line:84% of entropy nondecrease in weight processes. 00:08:57.560 --> 00:09:04.430 align:middle line:90% 00:09:04.430 --> 00:09:06.620 align:middle line:90% And let me say it again. 00:09:06.620 --> 00:09:10.460 align:middle line:84% The principle of entropy nondecrease 00:09:10.460 --> 00:09:14.998 align:middle line:84% is for weight processes or for systems that are isolated. 00:09:14.998 --> 00:09:16.790 align:middle line:84% In other words, it's impossible to decrease 00:09:16.790 --> 00:09:21.680 align:middle line:84% the entropy of an isolated system or a system that 00:09:21.680 --> 00:09:25.880 align:middle line:90% undergoes only a weight process. 00:09:25.880 --> 00:09:29.270 align:middle line:84% Otherwise, it is possible to decrease the entropy, 00:09:29.270 --> 00:09:31.400 align:middle line:90% like we do here. 00:09:31.400 --> 00:09:33.710 align:middle line:84% Like, the heat interaction is precisely 00:09:33.710 --> 00:09:36.830 align:middle line:84% the kind of interaction that allows to extract entropy 00:09:36.830 --> 00:09:37.645 align:middle line:90% from a system. 00:09:37.645 --> 00:09:40.770 align:middle line:90% 00:09:40.770 --> 00:09:42.330 align:middle line:90% All right. 00:09:42.330 --> 00:09:44.430 align:middle line:84% So here, the idea is that we start 00:09:44.430 --> 00:09:51.490 align:middle line:84% from a composite system of two systems, A and B, 00:09:51.490 --> 00:09:54.460 align:middle line:90% at different temperatures. 00:09:54.460 --> 00:09:59.680 align:middle line:90% This time, they are different. 00:09:59.680 --> 00:10:03.620 align:middle line:84% And we ask ourselves, is it possible-- 00:10:03.620 --> 00:10:13.090 align:middle line:84% when is it possible for them to exchange energy and volume 00:10:13.090 --> 00:10:17.540 align:middle line:90% through a movable partition? 00:10:17.540 --> 00:10:20.120 align:middle line:90% 00:10:20.120 --> 00:10:23.420 align:middle line:90% So here is the initial state. 00:10:23.420 --> 00:10:27.540 align:middle line:84% And on the E versus S diagram for the composite system, 00:10:27.540 --> 00:10:30.800 align:middle line:90% I denote it by this point here. 00:10:30.800 --> 00:10:32.750 align:middle line:90% And here is the final state-- 00:10:32.750 --> 00:10:35.450 align:middle line:90% same energy and same volume. 00:10:35.450 --> 00:10:39.030 align:middle line:84% So it's on the same energy line for the composite system. 00:10:39.030 --> 00:10:44.120 align:middle line:84% But internally, some energy has gone from A to B. 00:10:44.120 --> 00:10:51.140 align:middle line:84% So for example, if dE is greater than 0, the energy of A 00:10:51.140 --> 00:10:51.960 align:middle line:90% is increased. 00:10:51.960 --> 00:10:56.780 align:middle line:84% So the energy has flown from B to A. 00:10:56.780 --> 00:11:02.630 align:middle line:84% Or if dV is greater than 0, the volume has increased. 00:11:02.630 --> 00:11:08.660 align:middle line:84% So again, system B has lost volume 00:11:08.660 --> 00:11:15.850 align:middle line:84% for the gain of volume of system A. 00:11:15.850 --> 00:11:18.790 align:middle line:84% What we apply here is the condition that, 00:11:18.790 --> 00:11:22.090 align:middle line:84% since this system is isolated-- this is the particular case 00:11:22.090 --> 00:11:26.380 align:middle line:84% of a weight process with no external effects-- 00:11:26.380 --> 00:11:31.510 align:middle line:84% the entropy can only increase or remain 0 00:11:31.510 --> 00:11:34.420 align:middle line:90% if the process is reversible. 00:11:34.420 --> 00:11:41.410 align:middle line:84% So the final entropy of the composite 00:11:41.410 --> 00:11:43.540 align:middle line:84% must be greater than the initial. 00:11:43.540 --> 00:11:45.610 align:middle line:84% And so we have this inequality here-- 00:11:45.610 --> 00:11:48.340 align:middle line:84% S2 greater than S1, which we write this way, which 00:11:48.340 --> 00:11:52.220 align:middle line:90% we write this way. 00:11:52.220 --> 00:11:56.580 align:middle line:84% Then the tools are the same that we already used. 00:11:56.580 --> 00:11:58.400 align:middle line:90% We use additivity. 00:11:58.400 --> 00:12:02.933 align:middle line:90% We use the fundamental relation. 00:12:02.933 --> 00:12:07.610 align:middle line:90% 00:12:07.610 --> 00:12:12.080 align:middle line:84% Remember now, here your systems, system A 00:12:12.080 --> 00:12:14.960 align:middle line:84% starts from a stable equilibrium state. 00:12:14.960 --> 00:12:23.270 align:middle line:84% We want to extract or to, let's say, change with a negative dE 00:12:23.270 --> 00:12:25.940 align:middle line:90% the energy. 00:12:25.940 --> 00:12:28.770 align:middle line:84% And so at the new level of energy, 00:12:28.770 --> 00:12:33.410 align:middle line:84% there are many states that can be obtained. 00:12:33.410 --> 00:12:37.310 align:middle line:84% So I'm trying to explain the reason for this less than 00:12:37.310 --> 00:12:38.780 align:middle line:90% or equal to. 00:12:38.780 --> 00:12:46.740 align:middle line:84% The change in entropy for system A is either this one, 00:12:46.740 --> 00:12:48.650 align:middle line:90% but it can also be less-- 00:12:48.650 --> 00:13:00.410 align:middle line:84% I mean, I'm sorry, more and the same for B. 00:13:00.410 --> 00:13:05.450 align:middle line:84% So we end up with this relation here. 00:13:05.450 --> 00:13:10.490 align:middle line:84% And so these things must be greater than 0 00:13:10.490 --> 00:13:12.030 align:middle line:90% because these are all connected. 00:13:12.030 --> 00:13:14.920 align:middle line:90% 00:13:14.920 --> 00:13:19.960 align:middle line:84% So now, we can still choose some special cases. 00:13:19.960 --> 00:13:27.260 align:middle line:84% For example, suppose we don't change the volume. 00:13:27.260 --> 00:13:30.250 align:middle line:90% So the piston is-- 00:13:30.250 --> 00:13:35.960 align:middle line:84% it's blocked by a stop or there is no piston. 00:13:35.960 --> 00:13:38.770 align:middle line:90% This is just a wall. 00:13:38.770 --> 00:13:40.610 align:middle line:90% Then dV is equal to 0. 00:13:40.610 --> 00:13:44.110 align:middle line:84% So this contribution isn't there. 00:13:44.110 --> 00:13:49.480 align:middle line:84% And suppose we want to transfer energy from B to A so that dE 00:13:49.480 --> 00:13:51.270 align:middle line:90% is greater than 0. 00:13:51.270 --> 00:13:53.620 align:middle line:90% Here it is. 00:13:53.620 --> 00:13:58.210 align:middle line:84% So if this is greater than 0, then this term in parentheses 00:13:58.210 --> 00:13:59.325 align:middle line:90% must be greater than 0. 00:13:59.325 --> 00:14:02.420 align:middle line:90% 00:14:02.420 --> 00:14:09.320 align:middle line:84% And that hopefully will say that T1B must be greater than T1A 00:14:09.320 --> 00:14:15.090 align:middle line:84% if in a process without external effects, 00:14:15.090 --> 00:14:22.115 align:middle line:84% some energy is transferred in the direction from B to A. 00:14:22.115 --> 00:14:28.100 align:middle line:84% We say this in words by saying that the temperature-- 00:14:28.100 --> 00:14:34.040 align:middle line:84% or better somehow or equivalently-- the minus-- 00:14:34.040 --> 00:14:38.210 align:middle line:84% the negative of the inverse of the temperature 00:14:38.210 --> 00:14:42.620 align:middle line:84% can be understood as an escaping tendency for energy. 00:14:42.620 --> 00:14:44.990 align:middle line:84% So if you have two systems that have 00:14:44.990 --> 00:14:48.140 align:middle line:84% different escaping tendencies, the one 00:14:48.140 --> 00:14:51.050 align:middle line:84% who manages to give energy to the other 00:14:51.050 --> 00:14:54.820 align:middle line:84% is the hotter one, the one that has a higher escaping tendency. 00:14:54.820 --> 00:15:02.850 align:middle line:90% 00:15:02.850 --> 00:15:05.310 align:middle line:84% Suppose we choose that the two systems, A and B, 00:15:05.310 --> 00:15:08.220 align:middle line:84% are initially at the same temperature. 00:15:08.220 --> 00:15:11.285 align:middle line:84% So this term in parentheses isn't there. 00:15:11.285 --> 00:15:14.240 align:middle line:90% 00:15:14.240 --> 00:15:19.220 align:middle line:84% And we want system A to acquire volume 00:15:19.220 --> 00:15:21.900 align:middle line:84% at the expense of system B. All right, 00:15:21.900 --> 00:15:27.800 align:middle line:84% so the denominators here are equal. 00:15:27.800 --> 00:15:34.730 align:middle line:84% And so we find that p1A must be greater than p2B-- 00:15:34.730 --> 00:15:37.940 align:middle line:90% sorry, this should be also 1. 00:15:37.940 --> 00:15:39.500 align:middle line:90% These should be 1's. 00:15:39.500 --> 00:15:40.525 align:middle line:90% I have to correct that. 00:15:40.525 --> 00:15:47.930 align:middle line:90% 00:15:47.930 --> 00:15:52.940 align:middle line:84% So you can interpret pressure, as we all 00:15:52.940 --> 00:15:58.030 align:middle line:84% have the typical idea as a capturing tendency for volume. 00:15:58.030 --> 00:16:03.190 align:middle line:90% 00:16:03.190 --> 00:16:08.350 align:middle line:84% Now, we can do the same when instead of exchanging 00:16:08.350 --> 00:16:12.100 align:middle line:84% energy and volume, these two systems 00:16:12.100 --> 00:16:16.330 align:middle line:90% exchange energy and particles. 00:16:16.330 --> 00:16:21.300 align:middle line:84% And let's say particles of one type. 00:16:21.300 --> 00:16:24.950 align:middle line:84% I see here again another 2 that slipped. 00:16:24.950 --> 00:16:28.170 align:middle line:84% Fine, I'll need to remember to correct that. 00:16:28.170 --> 00:16:32.020 align:middle line:90% 00:16:32.020 --> 00:16:34.660 align:middle line:84% So it's the same story, only that instead 00:16:34.660 --> 00:16:38.830 align:middle line:84% of energy and volume, it's now energy and amount 00:16:38.830 --> 00:16:42.160 align:middle line:90% of constituent i. 00:16:42.160 --> 00:16:43.640 align:middle line:90% Still, the system is isolated. 00:16:43.640 --> 00:16:45.580 align:middle line:90% The entropy can only increase. 00:16:45.580 --> 00:16:50.950 align:middle line:84% And we end up with these inequalities. 00:16:50.950 --> 00:16:57.580 align:middle line:84% The first one still confirms that the minus 1 over T 00:16:57.580 --> 00:17:00.520 align:middle line:90% is escaping tendency for energy. 00:17:00.520 --> 00:17:03.050 align:middle line:84% Regarding chemical potentials, again, 00:17:03.050 --> 00:17:09.650 align:middle line:84% suppose that you choose the initial temperatures T1A 00:17:09.650 --> 00:17:13.310 align:middle line:90% and T1B to be equal. 00:17:13.310 --> 00:17:17.510 align:middle line:84% So the denominators here become equal. 00:17:17.510 --> 00:17:21.300 align:middle line:90% And this is going to be 0. 00:17:21.300 --> 00:17:22.819 align:middle line:90% So the only term is this. 00:17:22.819 --> 00:17:24.150 align:middle line:90% It must be greater than 0. 00:17:24.150 --> 00:17:26.869 align:middle line:90% So And also, this dV-- 00:17:26.869 --> 00:17:29.910 align:middle line:90% sorry for the slide. 00:17:29.910 --> 00:17:32.120 align:middle line:90% This was a cut and paste. 00:17:32.120 --> 00:17:40.740 align:middle line:84% This should be a dni and also this one. 00:17:40.740 --> 00:17:41.960 align:middle line:90% I wanted to see if you are-- 00:17:41.960 --> 00:17:46.240 align:middle line:90% 00:17:46.240 --> 00:17:48.880 align:middle line:90% if you are attentive. 00:17:48.880 --> 00:17:57.010 align:middle line:84% Well, anyway, the result is that if dni must be greater 00:17:57.010 --> 00:18:04.510 align:middle line:84% than 0, if you want, therefore that system A acquires particles 00:18:04.510 --> 00:18:09.190 align:middle line:84% of type i through this semi-permeable membrane that 00:18:09.190 --> 00:18:13.900 align:middle line:84% allows it to get it from system B, 00:18:13.900 --> 00:18:20.470 align:middle line:84% you need the chemical potential of A 00:18:20.470 --> 00:18:28.390 align:middle line:84% to be greater than the chemical potential of B. 00:18:28.390 --> 00:18:29.220 align:middle line:90% Is that correct? 00:18:29.220 --> 00:18:32.080 align:middle line:90% 00:18:32.080 --> 00:18:37.340 align:middle line:84% I'm not so sure anymore because of all these mistakes. 00:18:37.340 --> 00:18:42.760 align:middle line:90% And in fact, it shouldn't. 00:18:42.760 --> 00:18:46.910 align:middle line:84% It should be because there's a minus sign here. 00:18:46.910 --> 00:18:49.780 align:middle line:84% And so I either put a minus sign there 00:18:49.780 --> 00:18:53.840 align:middle line:84% or I invert A and B, the positions of A and B 00:18:53.840 --> 00:18:58.360 align:middle line:84% because, in order for constituent i 00:18:58.360 --> 00:19:04.300 align:middle line:84% to flow from B to A, I need that the chemical potential 00:19:04.300 --> 00:19:09.760 align:middle line:84% of constituent i in system B be higher than that in A 00:19:09.760 --> 00:19:11.680 align:middle line:90% because chemical potential. 00:19:11.680 --> 00:19:14.290 align:middle line:90% At least the title I got right. 00:19:14.290 --> 00:19:19.250 align:middle line:84% It's an escaping tendency for constituents. 00:19:19.250 --> 00:19:22.650 align:middle line:90% 00:19:22.650 --> 00:19:27.240 align:middle line:84% So this gets a feel of what the chemical potentials are. 00:19:27.240 --> 00:19:29.400 align:middle line:84% If you have a feel of what the temperature is, 00:19:29.400 --> 00:19:35.220 align:middle line:84% you also have a feel of what chemical potentials are. 00:19:35.220 --> 00:19:37.980 align:middle line:90% Temperature works for energy. 00:19:37.980 --> 00:19:43.990 align:middle line:84% Chemical potentials works for molecules and atoms and so on. 00:19:43.990 --> 00:19:48.280 align:middle line:90% 00:19:48.280 --> 00:19:52.720 align:middle line:84% And in fact, again, if you visited the talk today 00:19:52.720 --> 00:19:58.330 align:middle line:84% by Professor Stone, you'll see a lot of chemical potentials. 00:19:58.330 --> 00:20:03.320 align:middle line:84% The chemical gradients, they call them, for short. 00:20:03.320 --> 00:20:06.610 align:middle line:84% Instead of saying chemical potential gradient or gradient 00:20:06.610 --> 00:20:10.270 align:middle line:84% in chemical potential, they simply say chemical gradient. 00:20:10.270 --> 00:20:16.300 align:middle line:84% And since the chemical potential goes 00:20:16.300 --> 00:20:21.730 align:middle line:84% with the logarithm of the concentration, we'll see-- 00:20:21.730 --> 00:20:25.210 align:middle line:84% so a gradient in the chemical potential 00:20:25.210 --> 00:20:30.940 align:middle line:84% goes with the gradient of the log of C. 00:20:30.940 --> 00:20:40.220 align:middle line:84% And that's 1 over C gradient of C. Usually, 00:20:40.220 --> 00:20:46.520 align:middle line:84% people don't talk a lot about this 1 over C in front. 00:20:46.520 --> 00:20:49.890 align:middle line:84% And they simply say, well, for example, 00:20:49.890 --> 00:20:56.390 align:middle line:84% diffusion processes depend on the gradient in concentrations. 00:20:56.390 --> 00:20:57.950 align:middle line:90% True. 00:20:57.950 --> 00:21:01.230 align:middle line:84% They also depend on the gradient in concentration. 00:21:01.230 --> 00:21:05.360 align:middle line:84% But there is this C at the denominator. 00:21:05.360 --> 00:21:09.360 align:middle line:84% And when you have a number at the denominator and it is small, 00:21:09.360 --> 00:21:12.320 align:middle line:90% this enhances things. 00:21:12.320 --> 00:21:19.970 align:middle line:84% So the gradient in chemical potential has two contributions. 00:21:19.970 --> 00:21:25.400 align:middle line:84% One is also this C at the denominator. 00:21:25.400 --> 00:21:30.600 align:middle line:84% So for dilute systems, this gets amplified. 00:21:30.600 --> 00:21:35.660 align:middle line:90% 00:21:35.660 --> 00:21:43.460 align:middle line:84% And it's very nice to see how many experiments they 00:21:43.460 --> 00:21:47.170 align:middle line:84% are able to do in which this is the driving force. 00:21:47.170 --> 00:21:49.890 align:middle line:90% 00:21:49.890 --> 00:21:58.620 align:middle line:84% But it's also very nice to see how the modeling part is still 00:21:58.620 --> 00:22:04.440 align:middle line:90% a bit, I would say, primordial. 00:22:04.440 --> 00:22:12.480 align:middle line:84% And very few people really care about checking 00:22:12.480 --> 00:22:16.050 align:middle line:84% if some of the non-equilibrium phenomena that we will talk 00:22:16.050 --> 00:22:19.210 align:middle line:84% in particular Onsager reciprocity, 00:22:19.210 --> 00:22:22.470 align:middle line:84% cross-coupling effects that we will discuss 00:22:22.470 --> 00:22:25.830 align:middle line:84% in the third part of the course, whether they do 00:22:25.830 --> 00:22:29.760 align:middle line:84% play a role or not, whether they are small or not. 00:22:29.760 --> 00:22:33.750 align:middle line:84% Usually, they are neglected, but just 00:22:33.750 --> 00:22:36.400 align:middle line:84% neglected because you want a simple model, 00:22:36.400 --> 00:22:39.395 align:middle line:84% not because you know the effect isn't there. 00:22:39.395 --> 00:22:41.930 align:middle line:90% 00:22:41.930 --> 00:22:44.900 align:middle line:84% So that's why there is still a lot of research. 00:22:44.900 --> 00:22:48.350 align:middle line:84% And you find lots of very interesting papers 00:22:48.350 --> 00:22:50.090 align:middle line:90% on these subjects. 00:22:50.090 --> 00:22:56.390 align:middle line:84% OK, talking about experiments, last time, we also 00:22:56.390 --> 00:23:03.560 align:middle line:84% talked about-- we made a quick survey of the methods 00:23:03.560 --> 00:23:06.680 align:middle line:84% that we have in order to measure properties of substances 00:23:06.680 --> 00:23:09.960 align:middle line:90% or properties of systems. 00:23:09.960 --> 00:23:12.930 align:middle line:84% So one is measuring the temperature 00:23:12.930 --> 00:23:21.470 align:middle line:84% with the thermometer, measuring the pressure with the manometer, 00:23:21.470 --> 00:23:24.630 align:middle line:84% measuring the chemical potentials 00:23:24.630 --> 00:23:29.040 align:middle line:84% through a measurement of the partial pressure 00:23:29.040 --> 00:23:32.570 align:middle line:90% of a constituent. 00:23:32.570 --> 00:23:36.200 align:middle line:84% When this constituent is in mutual equilibrium 00:23:36.200 --> 00:23:41.720 align:middle line:84% with the system of interest through a semi-permeable 00:23:41.720 --> 00:23:45.170 align:middle line:84% membrane, permeable only to that constituent, when 00:23:45.170 --> 00:23:47.120 align:middle line:84% you reach chemical equilibrium that 00:23:47.120 --> 00:23:50.560 align:middle line:84% defines the partial pressure of the constituent in the mixture. 00:23:50.560 --> 00:23:53.090 align:middle line:90% 00:23:53.090 --> 00:23:54.480 align:middle line:90% And that's how you measure that. 00:23:54.480 --> 00:24:01.660 align:middle line:90% 00:24:01.660 --> 00:24:06.400 align:middle line:84% And I skipped all the other slides, but just to remind you 00:24:06.400 --> 00:24:12.580 align:middle line:84% that we said that, eventually, if you are an experimentalist 00:24:12.580 --> 00:24:15.590 align:middle line:84% and you like to characterize properties of substances, 00:24:15.590 --> 00:24:17.260 align:middle line:84% or maybe you discover a new substance 00:24:17.260 --> 00:24:20.080 align:middle line:84% and you want to characterize it, what you have to measure 00:24:20.080 --> 00:24:25.060 align:middle line:84% in the lab, and what you have to publish and eventually gets 00:24:25.060 --> 00:24:31.510 align:middle line:84% listed in like the Handbook of Chemical Physics 00:24:31.510 --> 00:24:36.700 align:middle line:84% or the NIST website, where you can 00:24:36.700 --> 00:24:39.400 align:middle line:90% find properties of substances-- 00:24:39.400 --> 00:24:42.670 align:middle line:84% in the old times, you would buy the book, the CRC Handbook 00:24:42.670 --> 00:24:43.850 align:middle line:90% of Chemical Physics. 00:24:43.850 --> 00:24:50.680 align:middle line:90% Now, books are disappearing. 00:24:50.680 --> 00:24:52.400 align:middle line:90% So you're all virtual. 00:24:52.400 --> 00:24:56.890 align:middle line:84% And so you go to the websites and look for these properties. 00:24:56.890 --> 00:24:58.336 align:middle line:90% Or you buy softwares. 00:24:58.336 --> 00:25:04.040 align:middle line:90% 00:25:04.040 --> 00:25:06.830 align:middle line:84% You probably use some software that 00:25:06.830 --> 00:25:14.400 align:middle line:84% has built-in correlations for many substances. 00:25:14.400 --> 00:25:20.850 align:middle line:84% And these correlations give you values for these quantities. 00:25:20.850 --> 00:25:24.200 align:middle line:84% The specific-- I'm sorry, yeah, the heat capacities 00:25:24.200 --> 00:25:33.110 align:middle line:84% of your substance, the compressibility, kappa, 00:25:33.110 --> 00:25:36.530 align:middle line:84% isothermal, or sometimes also another kind 00:25:36.530 --> 00:25:40.670 align:middle line:84% of-- the isentropic compressibility, 00:25:40.670 --> 00:25:44.480 align:middle line:84% which is equivalent or connected to this one. 00:25:44.480 --> 00:25:47.420 align:middle line:84% And also here, it's either the heat capacity 00:25:47.420 --> 00:25:50.930 align:middle line:84% at constant pressure or the heat capacity at constant volume 00:25:50.930 --> 00:25:56.390 align:middle line:84% because, through the Mayer relation, they are related, 00:25:56.390 --> 00:26:03.300 align:middle line:84% and the coefficient of expansion of your substance. 00:26:03.300 --> 00:26:08.790 align:middle line:84% For example, the compressibility, 00:26:08.790 --> 00:26:12.210 align:middle line:84% if you are in solid mechanics, you 00:26:12.210 --> 00:26:18.240 align:middle line:84% remember the Young modulus of elasticity. 00:26:18.240 --> 00:26:24.330 align:middle line:84% Well, this is a sort of Young modulus in the bulk. 00:26:24.330 --> 00:26:29.190 align:middle line:84% So in elasticity, you do stretching in one direction. 00:26:29.190 --> 00:26:31.420 align:middle line:84% If you do it in all three directions, 00:26:31.420 --> 00:26:37.710 align:middle line:84% you get the compressibility coefficient. 00:26:37.710 --> 00:26:38.210 align:middle line:90% No? 00:26:38.210 --> 00:26:41.230 align:middle line:90% 00:26:41.230 --> 00:26:47.470 align:middle line:84% That's because V, for a specimen of-- 00:26:47.470 --> 00:26:55.580 align:middle line:84% like a parallelepiped is given by that. 00:26:55.580 --> 00:27:04.630 align:middle line:84% And if you have 1 over V times the partial of V with respect 00:27:04.630 --> 00:27:10.300 align:middle line:84% to p at constant T times minus, that's 00:27:10.300 --> 00:27:17.190 align:middle line:84% kappa T. That's the isothermal compressibility. 00:27:17.190 --> 00:27:23.190 align:middle line:84% So you could write that as minus the partial of the log of V 00:27:23.190 --> 00:27:24.370 align:middle line:90% with respect to p. 00:27:24.370 --> 00:27:26.980 align:middle line:90% 00:27:26.980 --> 00:27:31.780 align:middle line:84% And because of the additivity of logarithms, this, 00:27:31.780 --> 00:27:37.630 align:middle line:84% you could write it as the partial of L1 00:27:37.630 --> 00:27:42.430 align:middle line:84% with respect to p times partial of L2. 00:27:42.430 --> 00:27:50.300 align:middle line:90% 00:27:50.300 --> 00:27:51.025 align:middle line:90% Is that correct? 00:27:51.025 --> 00:27:56.820 align:middle line:90% 00:27:56.820 --> 00:28:08.820 align:middle line:84% Yeah, no, plus, not minus, not minus, minus L3. 00:28:08.820 --> 00:28:13.110 align:middle line:90% OK, so this. 00:28:13.110 --> 00:28:17.820 align:middle line:84% And that's also-- sorry, oh boy, that's 00:28:17.820 --> 00:28:21.780 align:middle line:90% more than a senior moment. 00:28:21.780 --> 00:28:24.690 align:middle line:90% Let's do it again. 00:28:24.690 --> 00:28:37.380 align:middle line:84% Log L1 partial dT minus log L2 and so on. 00:28:37.380 --> 00:28:41.830 align:middle line:84% And each one of these is the Young modulus. 00:28:41.830 --> 00:28:43.290 align:middle line:90% It's 1 over L dL over p. 00:28:43.290 --> 00:28:50.430 align:middle line:84% So it's the percentage stretching, change in distance, 00:28:50.430 --> 00:28:54.155 align:middle line:90% for an increase in pressure. 00:28:54.155 --> 00:28:57.500 align:middle line:90% 00:28:57.500 --> 00:29:00.770 align:middle line:84% And that's the thing that you measure with the compression 00:29:00.770 --> 00:29:01.410 align:middle line:90% machine. 00:29:01.410 --> 00:29:06.320 align:middle line:84% It's the Young modulus in that direction. 00:29:06.320 --> 00:29:11.120 align:middle line:84% All right, what we proved is that once 00:29:11.120 --> 00:29:17.030 align:middle line:90% you have these functions here-- 00:29:17.030 --> 00:29:20.600 align:middle line:84% you don't just need one point, you need the entire function, 00:29:20.600 --> 00:29:23.420 align:middle line:84% so how they vary in temperature and pressure 00:29:23.420 --> 00:29:25.100 align:middle line:90% at given composition-- 00:29:25.100 --> 00:29:32.540 align:middle line:84% then, using these formulas, you can integrate these formulas 00:29:32.540 --> 00:29:35.990 align:middle line:84% and obtain the values for energy and entropy. 00:29:35.990 --> 00:29:38.600 align:middle line:84% And therefore, you can reconstruct 00:29:38.600 --> 00:29:40.320 align:middle line:90% the fundamental relation. 00:29:40.320 --> 00:29:43.820 align:middle line:84% So that's how you measure the fundamental relation, 00:29:43.820 --> 00:29:49.260 align:middle line:84% somewhat indirectly, through its derivatives, temperature, 00:29:49.260 --> 00:29:51.710 align:middle line:84% pressure, and derivatives of derivatives, 00:29:51.710 --> 00:29:57.510 align:middle line:84% like the specific heat or the heat capacities. 00:29:57.510 --> 00:30:00.270 align:middle line:84% So that introduces to the subject 00:30:00.270 --> 00:30:06.450 align:middle line:84% that I started, just barely started last time, which 00:30:06.450 --> 00:30:12.900 align:middle line:84% is, again, a bit boring because it's 00:30:12.900 --> 00:30:19.740 align:middle line:84% like a mathematical game played on the fundamental relation. 00:30:19.740 --> 00:30:25.170 align:middle line:84% But it is part of the traditional topics 00:30:25.170 --> 00:30:27.280 align:middle line:90% that we do in thermodynamics. 00:30:27.280 --> 00:30:32.010 align:middle line:84% It's certainly things that are needed when you do measurements 00:30:32.010 --> 00:30:33.940 align:middle line:84% and you want to correlate the data, 00:30:33.940 --> 00:30:40.830 align:middle line:84% you want to make sure that these relations are satisfied. 00:30:40.830 --> 00:30:42.630 align:middle line:90% So let's do it. 00:30:42.630 --> 00:30:45.830 align:middle line:84% I'll do it as quickly as possible. 00:30:45.830 --> 00:30:50.010 align:middle line:90% 00:30:50.010 --> 00:30:54.420 align:middle line:84% So the idea of a Legendre transform 00:30:54.420 --> 00:31:01.680 align:middle line:84% is to describe a curve, this curve, 00:31:01.680 --> 00:31:07.560 align:middle line:84% instead as the direct description, F equals F of y-- 00:31:07.560 --> 00:31:10.620 align:middle line:90% so this is F and y-- 00:31:10.620 --> 00:31:15.900 align:middle line:84% in a more indirect description, in terms 00:31:15.900 --> 00:31:23.940 align:middle line:84% of the family of the tangent lines to the curve so that 00:31:23.940 --> 00:31:29.105 align:middle line:84% the curve becomes the envelope of this family of lines. 00:31:29.105 --> 00:31:33.390 align:middle line:90% 00:31:33.390 --> 00:31:38.340 align:middle line:84% And this family of lines is described in such a way 00:31:38.340 --> 00:31:43.650 align:middle line:84% that you choose a value for the slope of the line. 00:31:43.650 --> 00:31:46.800 align:middle line:84% And then you shift the line until you 00:31:46.800 --> 00:31:49.260 align:middle line:84% find the point in which your curve is 00:31:49.260 --> 00:31:52.230 align:middle line:84% tangent to that line with that slope. 00:31:52.230 --> 00:31:56.190 align:middle line:84% And then you mark the value of the intercept at y 00:31:56.190 --> 00:31:57.600 align:middle line:90% equals 0 here. 00:31:57.600 --> 00:32:00.450 align:middle line:90% We call it L of lambda. 00:32:00.450 --> 00:32:05.880 align:middle line:84% So this L of lambda is the Legendre transform 00:32:05.880 --> 00:32:07.250 align:middle line:90% of your original function. 00:32:07.250 --> 00:32:10.840 align:middle line:90% 00:32:10.840 --> 00:32:15.700 align:middle line:84% When I have more time, but I don't here, 00:32:15.700 --> 00:32:17.600 align:middle line:90% I like to play this game. 00:32:17.600 --> 00:32:19.405 align:middle line:84% This is a game in which you can-- 00:32:19.405 --> 00:32:22.720 align:middle line:90% 00:32:22.720 --> 00:32:27.600 align:middle line:84% the Legendre transform has applications also in mechanics. 00:32:27.600 --> 00:32:32.880 align:middle line:84% So for example, it allows you to design, in a bridge-- 00:32:32.880 --> 00:32:37.340 align:middle line:84% you know those bridges with lots of wires? 00:32:37.340 --> 00:32:41.110 align:middle line:84% You want to have the wires be distributed in such a way 00:32:41.110 --> 00:32:43.210 align:middle line:90% that they make this envelope. 00:32:43.210 --> 00:32:46.450 align:middle line:84% And you want to decide which shape of envelope you 00:32:46.450 --> 00:32:50.290 align:middle line:90% want, for example. 00:32:50.290 --> 00:32:52.650 align:middle line:90% So here is my bridge. 00:32:52.650 --> 00:32:55.810 align:middle line:90% 00:32:55.810 --> 00:33:03.870 align:middle line:84% And suppose I want the envelope of these things 00:33:03.870 --> 00:33:05.790 align:middle line:90% that sustain the bridge-- 00:33:05.790 --> 00:33:09.540 align:middle line:90% 00:33:09.540 --> 00:33:13.290 align:middle line:84% from a distance, you want it to have 00:33:13.290 --> 00:33:17.760 align:middle line:84% a certain shape, maybe a circle, or maybe an ellipse, 00:33:17.760 --> 00:33:18.940 align:middle line:90% or whatever. 00:33:18.940 --> 00:33:21.210 align:middle line:90% So you fix that shape. 00:33:21.210 --> 00:33:25.270 align:middle line:84% And then by doing the Legendre transform, 00:33:25.270 --> 00:33:29.040 align:middle line:84% you find the points where you have 00:33:29.040 --> 00:33:34.370 align:middle line:90% to pull or attach your wires. 00:33:34.370 --> 00:33:42.612 align:middle line:90% 00:33:42.612 --> 00:33:48.470 align:middle line:84% So by construction here, essentially L-- 00:33:48.470 --> 00:33:50.030 align:middle line:84% if you look at this triangle, you 00:33:50.030 --> 00:33:57.320 align:middle line:84% realize that L is simply F minus y times the tangent 00:33:57.320 --> 00:34:00.290 align:middle line:90% of this angle, which is lambda. 00:34:00.290 --> 00:34:10.260 align:middle line:84% So L is F minus lambda y, F minus lambda y. 00:34:10.260 --> 00:34:12.190 align:middle line:90% So that's how you construct it. 00:34:12.190 --> 00:34:14.790 align:middle line:90% 00:34:14.790 --> 00:34:17.230 align:middle line:84% You can play-- these are some examples. 00:34:17.230 --> 00:34:21.360 align:middle line:84% For example, e to the y, the Legendre transform 00:34:21.360 --> 00:34:23.535 align:middle line:84% is lambda minus lambda log lambda. 00:34:23.535 --> 00:34:26.075 align:middle line:90% 00:34:26.075 --> 00:34:33.280 align:middle line:84% Or the Legendre transform of 1/2 of y squared, like the parabola, 00:34:33.280 --> 00:34:42.020 align:middle line:84% is exactly the same function and so for quadratic forms. 00:34:42.020 --> 00:34:45.739 align:middle line:84% If you put the 1/2, the 1/2 is important. 00:34:45.739 --> 00:34:48.320 align:middle line:90% What is also interesting is-- 00:34:48.320 --> 00:34:52.590 align:middle line:84% if you like this game, I'm not going into the details, 00:34:52.590 --> 00:34:57.530 align:middle line:84% but here is the proof that when you do this kind 00:34:57.530 --> 00:35:00.470 align:middle line:84% of transformation-- and this is why Legendre does it-- 00:35:00.470 --> 00:35:03.445 align:middle line:84% you don't lose information about your original function. 00:35:03.445 --> 00:35:06.830 align:middle line:90% 00:35:06.830 --> 00:35:11.300 align:middle line:84% And so if you do again the Legendre 00:35:11.300 --> 00:35:14.610 align:middle line:84% transform of the Legendre transform, 00:35:14.610 --> 00:35:16.585 align:middle line:84% you get back your original function. 00:35:16.585 --> 00:35:22.800 align:middle line:90% 00:35:22.800 --> 00:35:28.740 align:middle line:84% And this is similar to what we have done here because we 00:35:28.740 --> 00:35:40.730 align:middle line:84% describe this function here in a smart way, 00:35:40.730 --> 00:35:57.460 align:middle line:84% which was devised by Gibbs in the late 1800s, by tangents. 00:35:57.460 --> 00:36:01.190 align:middle line:84% You describe this function by its partial derivatives. 00:36:01.190 --> 00:36:04.180 align:middle line:84% So temperature is the slope in one direction 00:36:04.180 --> 00:36:06.650 align:middle line:84% and pressure is the slope in the other direction. 00:36:06.650 --> 00:36:11.080 align:middle line:84% So this is why this transformation matters. 00:36:11.080 --> 00:36:16.000 align:middle line:84% And here, in fact, is that transformation 00:36:16.000 --> 00:36:23.020 align:middle line:84% done, starting, for example, from the fundamental relation 00:36:23.020 --> 00:36:29.380 align:middle line:84% based on energy, in energy form, for which, you remember, 00:36:29.380 --> 00:36:31.490 align:middle line:84% this is the form of the differential. 00:36:31.490 --> 00:36:39.620 align:middle line:84% So dE is TdS minus pdV plus chemical potentials times dn. 00:36:39.620 --> 00:36:43.560 align:middle line:90% 00:36:43.560 --> 00:36:47.650 align:middle line:84% And this line reminds you of what we just saw. 00:36:47.650 --> 00:36:50.130 align:middle line:90% So it's the definition of-- 00:36:50.130 --> 00:36:53.460 align:middle line:84% the generic definition of Legendre transform 00:36:53.460 --> 00:36:56.070 align:middle line:90% for a function F of y. 00:36:56.070 --> 00:37:00.150 align:middle line:90% So L is F minus lambda y. 00:37:00.150 --> 00:37:02.280 align:middle line:90% And here is the application. 00:37:02.280 --> 00:37:04.546 align:middle line:90% So if I take that function-- 00:37:04.546 --> 00:37:07.120 align:middle line:90% 00:37:07.120 --> 00:37:09.765 align:middle line:84% now, it is a function of many variables. 00:37:09.765 --> 00:37:11.140 align:middle line:84% So I have to choose one at a time 00:37:11.140 --> 00:37:13.460 align:middle line:84% and do a Legendre transform of 1. 00:37:13.460 --> 00:37:21.900 align:middle line:84% For example, I want to substitute S with T. 00:37:21.900 --> 00:37:24.300 align:middle line:84% So I want to find an equivalent function that 00:37:24.300 --> 00:37:30.450 align:middle line:84% contains the same information of the fundamental relation, 00:37:30.450 --> 00:37:32.160 align:middle line:90% but in which the independent-- 00:37:32.160 --> 00:37:41.630 align:middle line:84% I change the independent variable S to its slope, T. 00:37:41.630 --> 00:37:47.510 align:middle line:84% So it's F, the new function, the Legendre transform, 00:37:47.510 --> 00:37:52.500 align:middle line:84% is the original function minus the product of lambda times y. 00:37:52.500 --> 00:37:56.000 align:middle line:84% So lambda, in this case, is T and y 00:37:56.000 --> 00:38:01.180 align:middle line:84% is S. This function is called the Helmholtz free energy. 00:38:01.180 --> 00:38:07.670 align:middle line:90% 00:38:07.670 --> 00:38:11.980 align:middle line:84% And you can also write very easily the-- 00:38:11.980 --> 00:38:19.230 align:middle line:90% 00:38:19.230 --> 00:38:20.870 align:middle line:90% let's do it only-- 00:38:20.870 --> 00:38:23.980 align:middle line:90% 00:38:23.980 --> 00:38:25.420 align:middle line:90% forget about the amounts. 00:38:25.420 --> 00:38:34.320 align:middle line:90% 00:38:34.320 --> 00:38:38.310 align:middle line:90% So we have this one up there. 00:38:38.310 --> 00:38:43.920 align:middle line:84% Now, we have defined a function F equals E minus TS. 00:38:43.920 --> 00:38:45.420 align:middle line:90% We want to compute dF. 00:38:45.420 --> 00:38:49.260 align:middle line:90% 00:38:49.260 --> 00:38:55.820 align:middle line:90% So this is dE minus d of TS. 00:38:55.820 --> 00:38:59.060 align:middle line:84% And then we can use the, I think it's called Leibniz rule, 00:38:59.060 --> 00:39:02.540 align:middle line:90% that this is equal to-- 00:39:02.540 --> 00:39:05.340 align:middle line:90% so for dE, we have it up there. 00:39:05.340 --> 00:39:10.010 align:middle line:90% It's TdS minus pdV. 00:39:10.010 --> 00:39:15.745 align:middle line:84% And here, this is minus TdS minus SdT. 00:39:15.745 --> 00:39:19.500 align:middle line:90% 00:39:19.500 --> 00:39:23.310 align:middle line:84% So one of these will cancel that. 00:39:23.310 --> 00:39:30.390 align:middle line:84% And we are left with minus SdT minus pdV, which I 00:39:30.390 --> 00:39:32.540 align:middle line:90% hope is what is written there. 00:39:32.540 --> 00:39:35.630 align:middle line:90% Yeah, OK. 00:39:35.630 --> 00:39:41.460 align:middle line:90% 00:39:41.460 --> 00:39:44.080 align:middle line:84% So in a sense, from the practical point of view, 00:39:44.080 --> 00:39:45.220 align:middle line:90% what have we done here? 00:39:45.220 --> 00:39:48.510 align:middle line:84% We gained the fact that we have a new function that 00:39:48.510 --> 00:39:50.040 align:middle line:90% contains the same information. 00:39:50.040 --> 00:39:54.065 align:middle line:84% And it is in terms of properties easily measurable-- temperature, 00:39:54.065 --> 00:39:54.565 align:middle line:90% volume. 00:39:54.565 --> 00:40:00.880 align:middle line:90% 00:40:00.880 --> 00:40:05.870 align:middle line:84% I could have done it also with respect to the second variable. 00:40:05.870 --> 00:40:09.190 align:middle line:84% So change the volume with pressure. 00:40:09.190 --> 00:40:10.880 align:middle line:90% It's the conjugate variable. 00:40:10.880 --> 00:40:17.290 align:middle line:84% So the slope in that direction is the conjugate value 00:40:17.290 --> 00:40:18.103 align:middle line:90% to the volume. 00:40:18.103 --> 00:40:19.770 align:middle line:84% It's the pressures-- minus the pressure. 00:40:19.770 --> 00:40:22.910 align:middle line:90% 00:40:22.910 --> 00:40:28.910 align:middle line:84% Then the Legendre transform is E, the original function, 00:40:28.910 --> 00:40:35.450 align:middle line:84% minus the slope, which is minus p, times V. So that's E plus pV. 00:40:35.450 --> 00:40:38.450 align:middle line:90% That's called the enthalpy. 00:40:38.450 --> 00:40:41.640 align:middle line:84% And you have used it for bulk flow interactions. 00:40:41.640 --> 00:40:50.810 align:middle line:84% We'll review that extremely quickly, probably next time. 00:40:50.810 --> 00:40:54.620 align:middle line:90% No, later because it needs to-- 00:40:54.620 --> 00:40:59.960 align:middle line:84% it needs the macroscopic, the simple system model. 00:40:59.960 --> 00:41:04.320 align:middle line:90% All right, so you take this one. 00:41:04.320 --> 00:41:07.190 align:middle line:90% It still has the-- 00:41:07.190 --> 00:41:08.990 align:middle line:84% see, this is now a function-- becomes 00:41:08.990 --> 00:41:10.580 align:middle line:90% a function of S, p, and n. 00:41:10.580 --> 00:41:15.470 align:middle line:84% You can compute, as we did, the differential. 00:41:15.470 --> 00:41:20.120 align:middle line:90% And it still has entropy. 00:41:20.120 --> 00:41:24.950 align:middle line:84% So now, we can take the Legendre transform of this one 00:41:24.950 --> 00:41:27.650 align:middle line:84% with respect to entropy, so substitute S 00:41:27.650 --> 00:41:34.490 align:middle line:90% with its conjugate, which is T. 00:41:34.490 --> 00:41:37.910 align:middle line:84% It's not always that T is the conjugate of S 00:41:37.910 --> 00:41:41.930 align:middle line:84% because previously here you prove, 00:41:41.930 --> 00:41:47.300 align:middle line:84% by looking at the differential, that T is also the partial of H 00:41:47.300 --> 00:41:52.110 align:middle line:84% with respect to S. So by doing that differential, 00:41:52.110 --> 00:41:59.090 align:middle line:84% we found that T, which is from the fundamental relation 00:41:59.090 --> 00:42:03.560 align:middle line:84% up there, is the partial of E with respect to S 00:42:03.560 --> 00:42:09.000 align:middle line:84% when the other variable is V. I forget the n for the moment. 00:42:09.000 --> 00:42:11.520 align:middle line:90% Now, we also proved-- 00:42:11.520 --> 00:42:13.000 align:middle line:90% well, we didn't prove it. 00:42:13.000 --> 00:42:16.020 align:middle line:84% But that form of differential says 00:42:16.020 --> 00:42:21.720 align:middle line:84% that this is the partial of H. What is it? 00:42:21.720 --> 00:42:28.755 align:middle line:84% Partial of H with respect to S when the other variable is p. 00:42:28.755 --> 00:42:32.832 align:middle line:90% 00:42:32.832 --> 00:42:35.940 align:middle line:84% So I do the Legendre transform of this one. 00:42:35.940 --> 00:42:39.500 align:middle line:84% So H minus the slope, which is T times 00:42:39.500 --> 00:42:43.500 align:middle line:84% the original variable, independent variable. 00:42:43.500 --> 00:42:45.800 align:middle line:90% So that's H minus TS. 00:42:45.800 --> 00:42:48.870 align:middle line:90% That's E minus TS plus pV. 00:42:48.870 --> 00:42:51.586 align:middle line:84% That's the Gibbs free energy, very important. 00:42:51.586 --> 00:42:56.610 align:middle line:90% 00:42:56.610 --> 00:43:04.890 align:middle line:84% And you compute this differential. 00:43:04.890 --> 00:43:07.470 align:middle line:84% Yeah, because, you see, by looking at-- 00:43:07.470 --> 00:43:10.200 align:middle line:90% 00:43:10.200 --> 00:43:14.590 align:middle line:84% by looking at the form of a differential, like this one, 00:43:14.590 --> 00:43:20.270 align:middle line:84% and look at the coefficients, it is clear 00:43:20.270 --> 00:43:26.830 align:middle line:84% that this that I circled is the partial derivative 00:43:26.830 --> 00:43:32.020 align:middle line:84% of F with respect to T when V is the other variable. 00:43:32.020 --> 00:43:34.810 align:middle line:84% So from this one, I immediately can 00:43:34.810 --> 00:43:39.730 align:middle line:84% see that minus S is the partial of F with respect 00:43:39.730 --> 00:43:45.490 align:middle line:84% to T when V is the other variable. 00:43:45.490 --> 00:43:49.840 align:middle line:84% And the other says that minus p is 00:43:49.840 --> 00:43:56.160 align:middle line:84% the partial of F with respect to V when T is the other variable. 00:43:56.160 --> 00:44:03.762 align:middle line:90% 00:44:03.762 --> 00:44:06.350 align:middle line:90% We could also go on. 00:44:06.350 --> 00:44:13.400 align:middle line:84% Usually, nobody does this because you're not 00:44:13.400 --> 00:44:16.490 align:middle line:90% interested in small systems. 00:44:16.490 --> 00:44:20.330 align:middle line:90% But we do it because we are. 00:44:20.330 --> 00:44:22.730 align:middle line:84% So if I take the Gibbs free energy 00:44:22.730 --> 00:44:27.950 align:middle line:84% and transform with respect to one of the amounts-- 00:44:27.950 --> 00:44:31.280 align:middle line:84% so I want to substitute now amounts (I meant, the amount) 00:44:31.280 --> 00:44:35.630 align:middle line:84% ni with its conjugate, which is the chemical potential, 00:44:35.630 --> 00:44:43.040 align:middle line:84% the slope because the chemical potential is 00:44:43.040 --> 00:44:47.390 align:middle line:84% the partial of G with respect to mu at constant n. 00:44:47.390 --> 00:44:51.830 align:middle line:90% 00:44:51.830 --> 00:44:58.950 align:middle line:84% All right, so I have G minus n times mu, ni mu i. 00:44:58.950 --> 00:45:05.810 align:middle line:90% 00:45:05.810 --> 00:45:09.930 align:middle line:84% I was short of letters for this game. 00:45:09.930 --> 00:45:14.030 align:middle line:84% So since eventually the last one I 00:45:14.030 --> 00:45:21.500 align:middle line:84% want to call Euler free energy, and this one maybe 00:45:21.500 --> 00:45:25.820 align:middle line:84% could be called osmotic free energy because it plays a role 00:45:25.820 --> 00:45:30.800 align:middle line:84% and because some colleagues who are working 00:45:30.800 --> 00:45:37.970 align:middle line:84% on nanothermodynamics are beginning 00:45:37.970 --> 00:45:41.610 align:middle line:90% to introduce these names. 00:45:41.610 --> 00:45:44.460 align:middle line:90% So I just call it Eu sub i. 00:45:44.460 --> 00:45:47.150 align:middle line:90% 00:45:47.150 --> 00:45:50.820 align:middle line:84% And if I do this not just for one but for all-- 00:45:50.820 --> 00:45:59.010 align:middle line:84% so I keep going until I've exhausted all the constituents-- 00:45:59.010 --> 00:46:04.920 align:middle line:84% I have this function that I call the Euler relation-- 00:46:04.920 --> 00:46:06.040 align:middle line:90% Euler free energy. 00:46:06.040 --> 00:46:10.235 align:middle line:84% It's G, the Gibbs free energy, minus mu times n. 00:46:10.235 --> 00:46:32.830 align:middle line:90% 00:46:32.830 --> 00:46:36.541 align:middle line:90% If you remember-- yeah? 00:46:36.541 --> 00:46:38.020 align:middle line:90% STUDENT: Oh, sorry. 00:46:38.020 --> 00:46:39.730 align:middle line:90% GIAN PAOLO BERETTA: No, no. 00:46:39.730 --> 00:46:42.140 align:middle line:84% STUDENT: Just, I might have missed this. 00:46:42.140 --> 00:46:47.510 align:middle line:84% Should the F in the Gibbs free energy row be something else, 00:46:47.510 --> 00:46:49.510 align:middle line:90% or is it meant to be F? 00:46:49.510 --> 00:46:51.346 align:middle line:90% GIAN PAOLO BERETTA: The F? 00:46:51.346 --> 00:46:57.050 align:middle line:84% STUDENT: In the second example row, yeah. 00:46:57.050 --> 00:46:58.680 align:middle line:90% F equals F T, V, n. 00:46:58.680 --> 00:47:01.992 align:middle line:90% 00:47:01.992 --> 00:47:02.950 align:middle line:90% GIAN PAOLO BERETTA: Ah. 00:47:02.950 --> 00:47:08.545 align:middle line:90% 00:47:08.545 --> 00:47:12.400 align:middle line:84% Yeah, this is also probably a cut and paste problem, I guess. 00:47:12.400 --> 00:47:14.570 align:middle line:90% Oh, no, no, no. 00:47:14.570 --> 00:47:15.230 align:middle line:90% No, no. 00:47:15.230 --> 00:47:21.770 align:middle line:84% This is still OK because if you start from the Helmholtz 00:47:21.770 --> 00:47:28.280 align:middle line:84% free energy and you transform, you get rid of the volume 00:47:28.280 --> 00:47:33.830 align:middle line:84% to get pressure, then you also get the Gibbs free energy 00:47:33.830 --> 00:47:37.220 align:middle line:84% because, for the Gibbs free energy, 00:47:37.220 --> 00:47:40.940 align:middle line:84% you transform with respect to S and V. 00:47:40.940 --> 00:47:44.870 align:middle line:84% And you can do it doing first the S 00:47:44.870 --> 00:47:49.070 align:middle line:84% and then the V or first the V and then the S. 00:47:49.070 --> 00:47:50.412 align:middle line:90% And you get the same result. 00:47:50.412 --> 00:47:52.370 align:middle line:84% STUDENT: Sorry, I think I was getting confused. 00:47:52.370 --> 00:47:55.232 align:middle line:90% So F is Helmholtz free energy. 00:47:55.232 --> 00:47:56.690 align:middle line:84% GIAN PAOLO BERETTA: F is the letter 00:47:56.690 --> 00:47:57.835 align:middle line:90% for the Helmholtz free energy. 00:47:57.835 --> 00:47:58.525 align:middle line:90% STUDENT: Oh, OK. 00:47:58.525 --> 00:47:58.940 align:middle line:90% All right. 00:47:58.940 --> 00:48:00.940 align:middle line:84% That's what was confusing me because it was also 00:48:00.940 --> 00:48:05.520 align:middle line:84% used as the general function, I guess. 00:48:05.520 --> 00:48:07.470 align:middle line:84% GIAN PAOLO BERETTA: Ah, the F up there. 00:48:07.470 --> 00:48:08.280 align:middle line:90% You're right. 00:48:08.280 --> 00:48:09.910 align:middle line:90% Yeah. 00:48:09.910 --> 00:48:10.890 align:middle line:90% Yeah, yeah. 00:48:10.890 --> 00:48:15.510 align:middle line:90% It's difficult to find letters. 00:48:15.510 --> 00:48:19.000 align:middle line:84% Even if you use the Greek alphabet and so on, 00:48:19.000 --> 00:48:23.040 align:middle line:90% there's still not enough. 00:48:23.040 --> 00:48:26.325 align:middle line:84% OK, I wanted to emphasize this one. 00:48:26.325 --> 00:48:28.060 align:middle line:90% We'll go back to this one. 00:48:28.060 --> 00:48:34.240 align:middle line:84% And I call it Euler because some of you might object, well, 00:48:34.240 --> 00:48:35.030 align:middle line:90% but this is 0. 00:48:35.030 --> 00:48:37.590 align:middle line:90% 00:48:37.590 --> 00:48:41.490 align:middle line:84% And the answer is yes, this will be 00:48:41.490 --> 00:48:45.120 align:middle line:90% 0 for the macroscopic systems. 00:48:45.120 --> 00:48:49.890 align:middle line:84% But for microscopic systems, it is not. 00:48:49.890 --> 00:48:54.240 align:middle line:84% And that's why I want to give it a name 00:48:54.240 --> 00:48:58.590 align:middle line:84% and let it play some role until we introduce 00:48:58.590 --> 00:49:07.200 align:middle line:84% the idea of many particles because this-- so in a sense, 00:49:07.200 --> 00:49:17.360 align:middle line:84% we will see that, in the limit of many particles, which means 00:49:17.360 --> 00:49:25.340 align:middle line:84% negligible wall rarefaction effects because the key will 00:49:25.340 --> 00:49:29.270 align:middle line:90% be that, this will turn to 0. 00:49:29.270 --> 00:49:30.320 align:middle line:90% Yeah? 00:49:30.320 --> 00:49:33.860 align:middle line:84% STUDENT: Does that apply to the microscopic systems? 00:49:33.860 --> 00:49:39.410 align:middle line:90% 00:49:39.410 --> 00:49:41.900 align:middle line:84% GIAN PAOLO BERETTA: This one is not 0. 00:49:41.900 --> 00:49:44.630 align:middle line:84% And therefore, it's something that you can measure. 00:49:44.630 --> 00:49:49.040 align:middle line:84% It will depend on how many particles you have. 00:49:49.040 --> 00:49:50.670 align:middle line:90% STUDENT: I just mean like for-- 00:49:50.670 --> 00:49:53.405 align:middle line:84% because Gibbs-Duhem relation is [INAUDIBLE]. 00:49:53.405 --> 00:49:59.630 align:middle line:90% 00:49:59.630 --> 00:50:09.800 align:middle line:84% GIAN PAOLO BERETTA: Gibbs-Duhem is in this realm. 00:50:09.800 --> 00:50:13.090 align:middle line:84% It starts from the Euler relation. 00:50:13.090 --> 00:50:20.740 align:middle line:84% It requires the condition that Eu equals 0. 00:50:20.740 --> 00:50:24.520 align:middle line:84% And so you get all those, like the phase rule that comes out 00:50:24.520 --> 00:50:28.870 align:middle line:90% of the Gibbs-Duhem relation. 00:50:28.870 --> 00:50:35.650 align:middle line:84% That applies in this domain of macroscopic systems. 00:50:35.650 --> 00:50:43.675 align:middle line:84% But here, let's say that it's still a bit open. 00:50:43.675 --> 00:51:02.530 align:middle line:84% There is a book by Terrell Hill, Thermodynamics of Small Systems, 00:51:02.530 --> 00:51:03.945 align:middle line:90% long ago-- 00:51:03.945 --> 00:51:06.045 align:middle line:90% I forget now the exact date-- 00:51:06.045 --> 00:51:08.980 align:middle line:90% 00:51:08.980 --> 00:51:11.460 align:middle line:90% who did care about this thing. 00:51:11.460 --> 00:51:14.240 align:middle line:90% 00:51:14.240 --> 00:51:18.530 align:middle line:84% But I'm not sure that he went far enough 00:51:18.530 --> 00:51:25.350 align:middle line:84% to talk about phase transitions in that domain. 00:51:25.350 --> 00:51:30.580 align:middle line:84% So this is still a pretty open subject. 00:51:30.580 --> 00:51:35.160 align:middle line:84% But it's becoming important because of the enormous advances 00:51:35.160 --> 00:51:40.740 align:middle line:84% in, I would say, experimental technologies that 00:51:40.740 --> 00:51:49.350 align:middle line:84% allow you, with microfluidics or quantum dots and all that stuff, 00:51:49.350 --> 00:51:53.740 align:middle line:84% to deal with very small systems, few particles. 00:51:53.740 --> 00:51:59.470 align:middle line:90% 00:51:59.470 --> 00:52:03.370 align:middle line:90% Well, math, and math, and math. 00:52:03.370 --> 00:52:08.150 align:middle line:84% All right, I told you, when we have a function, 00:52:08.150 --> 00:52:13.500 align:middle line:84% we can play as much math as we want forever and ever. 00:52:13.500 --> 00:52:16.780 align:middle line:90% 00:52:16.780 --> 00:52:21.130 align:middle line:84% So I'm concerned here to tell you only about the useful math. 00:52:21.130 --> 00:52:25.090 align:middle line:84% And apparently, also the second derivatives 00:52:25.090 --> 00:52:31.150 align:middle line:84% are useful, at least if our project is 00:52:31.150 --> 00:52:39.640 align:middle line:84% that of measuring properties because, again, we 00:52:39.640 --> 00:52:43.390 align:middle line:84% don't have the entropy meter and we don't have the energy meter. 00:52:43.390 --> 00:52:49.740 align:middle line:84% So we have to rely on these tangents to measure properties. 00:52:49.740 --> 00:52:52.200 align:middle line:84% So each one of the functions that we 00:52:52.200 --> 00:52:56.190 align:middle line:84% have defined through the Legendre transform 00:52:56.190 --> 00:53:01.870 align:middle line:84% has first order derivatives, but also second order derivatives, 00:53:01.870 --> 00:53:08.250 align:middle line:84% for which we have the Schwarz theorem of symmetry that 00:53:08.250 --> 00:53:11.030 align:middle line:84% says that the order of derivation doesn't matter. 00:53:11.030 --> 00:53:13.980 align:middle line:90% 00:53:13.980 --> 00:53:17.640 align:middle line:84% So for example, if I take the second derivative 00:53:17.640 --> 00:53:24.120 align:middle line:84% of just the fundamental relation in energy form with respect 00:53:24.120 --> 00:53:29.970 align:middle line:84% to S and V, that's equal if I do first S 00:53:29.970 --> 00:53:32.620 align:middle line:90% and after V and vice versa. 00:53:32.620 --> 00:53:37.560 align:middle line:84% So since the partial of E with respect to V 00:53:37.560 --> 00:53:40.260 align:middle line:84% is minus the pressure, this one can also 00:53:40.260 --> 00:53:43.620 align:middle line:84% be viewed as minus the derivative of minus the pressure 00:53:43.620 --> 00:53:50.350 align:middle line:84% with respect to the entropy when the other variables are V and n. 00:53:50.350 --> 00:53:53.160 align:middle line:84% The right-hand side can be viewed as the partial of E 00:53:53.160 --> 00:53:56.970 align:middle line:84% with respect to S. That's T, the temperature. 00:53:56.970 --> 00:53:59.250 align:middle line:84% So the partial of the temperature with respect 00:53:59.250 --> 00:54:00.530 align:middle line:90% to V at constant-- 00:54:00.530 --> 00:54:04.390 align:middle line:84% I mean, when S and n are the other variables. 00:54:04.390 --> 00:54:06.095 align:middle line:90% So we get a relation here. 00:54:06.095 --> 00:54:09.940 align:middle line:90% 00:54:09.940 --> 00:54:16.270 align:middle line:84% This relation is still not fully practical because, yes, it does 00:54:16.270 --> 00:54:23.230 align:middle line:84% have measurable properties almost all places except entropy 00:54:23.230 --> 00:54:24.465 align:middle line:90% here and entropy there. 00:54:24.465 --> 00:54:27.130 align:middle line:90% 00:54:27.130 --> 00:54:29.410 align:middle line:90% Let's do the same game. 00:54:29.410 --> 00:54:33.660 align:middle line:84% These are called Maxwell relations, by the way. 00:54:33.660 --> 00:54:35.910 align:middle line:90% And there are many. 00:54:35.910 --> 00:54:38.670 align:middle line:84% Anything that derives from the Schwarz theorem 00:54:38.670 --> 00:54:42.480 align:middle line:90% becomes a Maxwell relation. 00:54:42.480 --> 00:54:48.970 align:middle line:84% If you play that on the Helmholtz free energy, 00:54:48.970 --> 00:54:51.760 align:middle line:90% here is what we get. 00:54:51.760 --> 00:54:55.010 align:middle line:84% This is a bit more practical because, you see, 00:54:55.010 --> 00:54:57.010 align:middle line:90% we have just the entropy here. 00:54:57.010 --> 00:55:00.440 align:middle line:84% And the rest is easy, measurable stuff-- 00:55:00.440 --> 00:55:02.800 align:middle line:90% pressure, volume, temperature. 00:55:02.800 --> 00:55:06.190 align:middle line:84% So that means that if you take your substance 00:55:06.190 --> 00:55:09.850 align:middle line:84% and want to know how much the entropy varies 00:55:09.850 --> 00:55:14.770 align:middle line:84% with volume at constant temperature, 00:55:14.770 --> 00:55:17.620 align:middle line:84% you could measure it indirectly using this relation, 00:55:17.620 --> 00:55:18.940 align:middle line:90% by measuring how much-- 00:55:18.940 --> 00:55:22.930 align:middle line:84% by taking the substance, increasing the temperature 00:55:22.930 --> 00:55:26.050 align:middle line:84% a little bit, seeing how the pressure increases 00:55:26.050 --> 00:55:28.450 align:middle line:90% at constant volume. 00:55:28.450 --> 00:55:32.530 align:middle line:84% And once you've measured that, you've measured that indirectly. 00:55:32.530 --> 00:55:34.240 align:middle line:90% So this is one way to-- 00:55:34.240 --> 00:55:37.420 align:middle line:90% it's like an entropy meter. 00:55:37.420 --> 00:55:39.940 align:middle line:84% It's an indirect measurement of entropy. 00:55:39.940 --> 00:55:43.260 align:middle line:84% You could build an entropy meter based on this relation. 00:55:43.260 --> 00:55:45.850 align:middle line:90% 00:55:45.850 --> 00:55:52.370 align:middle line:84% If you do the same with the enthalpy, you get this one-- 00:55:52.370 --> 00:55:57.680 align:middle line:84% not very useful because it has S on both sides. 00:55:57.680 --> 00:56:01.256 align:middle line:84% It's not easy to keep S constant because keep S constant-- 00:56:01.256 --> 00:56:04.010 align:middle line:90% 00:56:04.010 --> 00:56:09.020 align:middle line:84% yes, you could do a weight process. 00:56:09.020 --> 00:56:13.310 align:middle line:84% And you could make sure that it is reversible. 00:56:13.310 --> 00:56:16.910 align:middle line:84% But usually, reversible processes are not easy to do. 00:56:16.910 --> 00:56:20.060 align:middle line:84% And often, you have to extrapolate 00:56:20.060 --> 00:56:24.990 align:middle line:84% to some zero velocities or some very high velocities, 00:56:24.990 --> 00:56:27.635 align:middle line:90% so not easy to make sure that-- 00:56:27.635 --> 00:56:33.280 align:middle line:84% to be sure in the lab that your process is reversible. 00:56:33.280 --> 00:56:37.240 align:middle line:84% The last one, if you play the Maxwell relations 00:56:37.240 --> 00:56:41.270 align:middle line:84% game on the Gibbs free energy, is also useful. 00:56:41.270 --> 00:56:43.720 align:middle line:84% It could also be a basis for an entropy method 00:56:43.720 --> 00:56:47.260 align:middle line:84% because it has entropy only on one side. 00:56:47.260 --> 00:56:49.780 align:middle line:84% So you can see how the entropy changes 00:56:49.780 --> 00:56:54.400 align:middle line:84% as a function of pressure at constant temperature 00:56:54.400 --> 00:56:59.260 align:middle line:84% by measuring how the volume changes 00:56:59.260 --> 00:57:01.845 align:middle line:84% if you change the temperature at constant pressure. 00:57:01.845 --> 00:57:09.478 align:middle line:90% 00:57:09.478 --> 00:57:12.370 align:middle line:84% You can also have Maxwell relations 00:57:12.370 --> 00:57:15.230 align:middle line:84% when you take some derivatives, second order-- 00:57:15.230 --> 00:57:22.540 align:middle line:84% I mean, second-- yeah, second order derivatives with respect 00:57:22.540 --> 00:57:26.650 align:middle line:84% to a pair of variables, of which one is 00:57:26.650 --> 00:57:32.370 align:middle line:90% the amount of one constituent. 00:57:32.370 --> 00:57:38.640 align:middle line:84% So if I do that, temperature and one constituent, 00:57:38.640 --> 00:57:39.515 align:middle line:90% here is the relation. 00:57:39.515 --> 00:57:42.220 align:middle line:90% 00:57:42.220 --> 00:57:51.190 align:middle line:84% The derivative of G with respect to n is the chemical potential. 00:57:51.190 --> 00:57:56.770 align:middle line:84% So this is the derivative-- is minus the chemical potential. 00:57:56.770 --> 00:57:59.920 align:middle line:90% No, sorry. 00:57:59.920 --> 00:58:02.010 align:middle line:84% No, it is the chemical potential. 00:58:02.010 --> 00:58:06.230 align:middle line:84% Then I moved the minus sign on the other side. 00:58:06.230 --> 00:58:11.990 align:middle line:84% So this one is the derivative of mu with respect to T, 00:58:11.990 --> 00:58:16.660 align:middle line:84% whereas the derivative of G with respect to T is minus S. 00:58:16.660 --> 00:58:22.420 align:middle line:84% And so this is the derivative of S with respect to n. 00:58:22.420 --> 00:58:26.650 align:middle line:84% When the other variables are T, p, and the other amounts 00:58:26.650 --> 00:58:29.640 align:middle line:90% of constituents. 00:58:29.640 --> 00:58:40.260 align:middle line:84% This is important because, as we do mixtures starting soon, 00:58:40.260 --> 00:58:43.140 align:middle line:84% the second part of the course, we 00:58:43.140 --> 00:58:50.760 align:middle line:84% will call this the partial entropy of constituent i 00:58:50.760 --> 00:58:51.395 align:middle line:90% in the mixture. 00:58:51.395 --> 00:58:56.310 align:middle line:90% 00:58:56.310 --> 00:59:01.530 align:middle line:84% In the limit of large amounts, the partial entropy 00:59:01.530 --> 00:59:05.790 align:middle line:84% contributes to the properties of the mixture in proportion 00:59:05.790 --> 00:59:07.150 align:middle line:90% to the amount. 00:59:07.150 --> 00:59:11.670 align:middle line:84% So you multiply the amount times the partial entropy, sum up, 00:59:11.670 --> 00:59:14.010 align:middle line:84% and you get the entropy for your mixture. 00:59:14.010 --> 00:59:17.670 align:middle line:84% That's why it's called partial entropy. 00:59:17.670 --> 00:59:25.290 align:middle line:84% This is true only in the limit of Euler going to 0. 00:59:25.290 --> 00:59:26.700 align:middle line:90% You do the same-- 00:59:26.700 --> 00:59:29.620 align:middle line:84% instead of T, you do it for the pressure. 00:59:29.620 --> 00:59:31.890 align:middle line:84% And you get this other derivative 00:59:31.890 --> 00:59:36.750 align:middle line:84% of the chemical potential with respect to p. 00:59:36.750 --> 00:59:41.670 align:middle line:84% And this is what we will call the partial volume 00:59:41.670 --> 00:59:45.310 align:middle line:84% of the constituent in the mixture. 00:59:45.310 --> 00:59:48.430 align:middle line:84% Again, if you take this one, multiply it by the amount, 00:59:48.430 --> 00:59:58.950 align:middle line:84% sum over all constituents, that gives you the volume in general, 00:59:58.950 --> 01:00:00.860 align:middle line:90% as long as Euler is equal to 0. 01:00:00.860 --> 01:00:03.690 align:middle line:90% 01:00:03.690 --> 01:00:07.890 align:middle line:84% You could also take the double derivative with respect 01:00:07.890 --> 01:00:12.630 align:middle line:84% to two different amounts of constituents. 01:00:12.630 --> 01:00:18.572 align:middle line:84% And you have this sort of reciprocal relation 01:00:18.572 --> 01:00:26.520 align:middle line:84% that says that the derivative of potential of constituent 1 01:00:26.520 --> 01:00:28.980 align:middle line:84% with respect to the amount of constituent 2 01:00:28.980 --> 01:00:33.840 align:middle line:84% is equal to the derivative of potential of 2 with respect 01:00:33.840 --> 01:00:35.345 align:middle line:90% to the amount of constituent 1. 01:00:35.345 --> 01:00:38.380 align:middle line:90% 01:00:38.380 --> 01:00:45.550 align:middle line:84% We will use this in one of the proofs related 01:00:45.550 --> 01:00:49.353 align:middle line:90% to the Duhem-Margules relation. 01:00:49.353 --> 01:00:54.450 align:middle line:90% 01:00:54.450 --> 01:00:56.825 align:middle line:90% Yeah, you could also-- 01:00:56.825 --> 01:00:59.890 align:middle line:90% 01:00:59.890 --> 01:01:08.264 align:middle line:84% play the same game starting with the relation-- 01:01:08.264 --> 01:01:11.770 align:middle line:90% 01:01:11.770 --> 01:01:12.610 align:middle line:90% I don't have it. 01:01:12.610 --> 01:01:16.330 align:middle line:84% Well, starting not with the fundamental relation in energy 01:01:16.330 --> 01:01:21.360 align:middle line:90% form, but in entropy form. 01:01:21.360 --> 01:01:23.080 align:middle line:84% And here is the result of the game. 01:01:23.080 --> 01:01:25.770 align:middle line:90% Let me skip the details. 01:01:25.770 --> 01:01:29.030 align:middle line:84% One of the functions is called the Massieu free energy. 01:01:29.030 --> 01:01:31.770 align:middle line:90% 01:01:31.770 --> 01:01:38.360 align:middle line:90% It's S minus E over T. 01:01:38.360 --> 01:01:44.930 align:middle line:84% This entropy-- or say, entropic formulation-- 01:01:44.930 --> 01:01:47.520 align:middle line:90% becomes more and more important. 01:01:47.520 --> 01:01:54.440 align:middle line:84% And so you will find in the literature often if you go-- 01:01:54.440 --> 01:02:01.550 align:middle line:84% yeah, people that do small systems or quantum 01:02:01.550 --> 01:02:07.730 align:middle line:84% thermodynamics, S minus E over T, you find it a lot. 01:02:07.730 --> 01:02:12.000 align:middle line:84% Some of them recognize that this is due to Massieu. 01:02:12.000 --> 01:02:18.020 align:middle line:84% Some just call it free something or free entropy. 01:02:18.020 --> 01:02:21.720 align:middle line:90% So this is-- following them-- 01:02:21.720 --> 01:02:23.810 align:middle line:84% I'll call it Massieu free entropy, 01:02:23.810 --> 01:02:31.550 align:middle line:84% although I think the term might be a bit confusing because I'm 01:02:31.550 --> 01:02:34.090 align:middle line:84% not sure what free entropy means. 01:02:34.090 --> 01:02:38.360 align:middle line:90% 01:02:38.360 --> 01:02:41.960 align:middle line:84% Sometimes, the wrong words are confusing 01:02:41.960 --> 01:02:44.490 align:middle line:90% and cause a lot of damage. 01:02:44.490 --> 01:02:47.870 align:middle line:84% So if I do that, I'm sorry about this, 01:02:47.870 --> 01:02:56.750 align:middle line:84% but I needed a name because instead of giving it 01:02:56.750 --> 01:02:58.760 align:middle line:84% this kind of name, for example, in the book 01:02:58.760 --> 01:03:03.560 align:middle line:84% with Professor Gyftopoulos, we were 01:03:03.560 --> 01:03:07.410 align:middle line:84% very careful not to use names that were misleading. 01:03:07.410 --> 01:03:10.470 align:middle line:84% And so we called them characteristic functions, 01:03:10.470 --> 01:03:13.300 align:middle line:84% characteristic stable equilibrium state functions. 01:03:13.300 --> 01:03:23.030 align:middle line:84% It's a more aseptic way of defining the same thing. 01:03:23.030 --> 01:03:27.350 align:middle line:90% Planck also has its own. 01:03:27.350 --> 01:03:34.820 align:middle line:84% And again, if I do the transform with respect 01:03:34.820 --> 01:03:46.400 align:middle line:84% to all the variables, including also the-- 01:03:46.400 --> 01:03:50.090 align:middle line:84% yeah, including also all the amounts, 01:03:50.090 --> 01:03:58.165 align:middle line:84% I end up with the equivalent of the Euler relation here-- 01:03:58.165 --> 01:04:02.090 align:middle line:90% 01:04:02.090 --> 01:04:03.270 align:middle line:90% how did I call it-- 01:04:03.270 --> 01:04:10.130 align:middle line:84% Su, which is equal to minus Eu over T. 01:04:10.130 --> 01:04:13.820 align:middle line:84% Notice that also the others are very-- 01:04:13.820 --> 01:04:18.010 align:middle line:84% for example, Planck is minus G over T. 01:04:18.010 --> 01:04:21.020 align:middle line:84% So it's like a Gibbs free energy divided by the temperature. 01:04:21.020 --> 01:04:27.920 align:middle line:84% And Massieu is Helmholtz over T. So they are related. 01:04:27.920 --> 01:04:31.220 align:middle line:84% And of course, you can play also the game 01:04:31.220 --> 01:04:33.950 align:middle line:90% of double derivatives here. 01:04:33.950 --> 01:04:38.750 align:middle line:84% Well, by the same token, these derivatives 01:04:38.750 --> 01:04:41.810 align:middle line:84% are useful if you want to construct an energy meter. 01:04:41.810 --> 01:04:44.370 align:middle line:90% 01:04:44.370 --> 01:04:47.850 align:middle line:84% The first one, not so much because you 01:04:47.850 --> 01:04:49.630 align:middle line:90% have energy on both sides. 01:04:49.630 --> 01:04:55.420 align:middle line:84% But this one, yes, I can measure how the energy varies 01:04:55.420 --> 01:04:57.820 align:middle line:84% with respect to volume at constant temperature 01:04:57.820 --> 01:05:00.820 align:middle line:84% by measuring pressure, temperature, 01:05:00.820 --> 01:05:04.000 align:middle line:84% and how the pressure varies as a function of temperature 01:05:04.000 --> 01:05:06.900 align:middle line:90% at constant volume. 01:05:06.900 --> 01:05:12.070 align:middle line:84% So you could build an energy meter based on this equation. 01:05:12.070 --> 01:05:16.630 align:middle line:84% This one, no, because it has both sides. 01:05:16.630 --> 01:05:23.920 align:middle line:84% But the last one, yes, although it's not so simple. 01:05:23.920 --> 01:05:26.710 align:middle line:84% You have to realize that you have 01:05:26.710 --> 01:05:32.620 align:middle line:84% to here change the temperature of your system 01:05:32.620 --> 01:05:39.520 align:middle line:84% and see how the volume will change while you keep p over T 01:05:39.520 --> 01:05:40.600 align:middle line:90% constant. 01:05:40.600 --> 01:05:42.290 align:middle line:84% So if you double the temperature, 01:05:42.290 --> 01:05:44.740 align:middle line:84% you also have to double the pressure in order 01:05:44.740 --> 01:05:46.870 align:middle line:90% to keep that ratio constant. 01:05:46.870 --> 01:05:49.210 align:middle line:84% And then you measure how the energy varies 01:05:49.210 --> 01:05:51.640 align:middle line:90% with this function of p. 01:05:51.640 --> 01:06:02.080 align:middle line:84% All right, so we have ways to measure stuff 01:06:02.080 --> 01:06:06.290 align:middle line:84% and, therefore, to characterize the stable equilibrium, 01:06:06.290 --> 01:06:10.516 align:middle line:90% stable states of substances. 01:06:10.516 --> 01:06:14.950 align:middle line:90% 01:06:14.950 --> 01:06:19.870 align:middle line:84% All right, so now, we go back to one 01:06:19.870 --> 01:06:29.230 align:middle line:84% of the characteristic questions for which thermodynamics 01:06:29.230 --> 01:06:36.350 align:middle line:84% was born, how much useful things or how much work, 01:06:36.350 --> 01:06:38.470 align:middle line:84% now that we've defined it, can we 01:06:38.470 --> 01:06:46.960 align:middle line:84% do by exploiting a change of some system from one state 01:06:46.960 --> 01:06:53.520 align:middle line:84% to another-- for example, by cooling a hot object. 01:06:53.520 --> 01:06:58.800 align:middle line:84% Or also, what is the minimum amount of work 01:06:58.800 --> 01:07:04.680 align:middle line:84% that is necessary in order to obtain a certain change 01:07:04.680 --> 01:07:07.610 align:middle line:90% in the state of some system? 01:07:07.610 --> 01:07:12.540 align:middle line:84% For example, in material processing, I transform-- 01:07:12.540 --> 01:07:15.360 align:middle line:84% I've got a bunch of raw materials coming in. 01:07:15.360 --> 01:07:20.790 align:middle line:84% And I want to produce my product, like steel. 01:07:20.790 --> 01:07:23.050 align:middle line:90% Steel is a product. 01:07:23.050 --> 01:07:29.340 align:middle line:84% You have to put in iron ore and carbon and lots 01:07:29.340 --> 01:07:31.560 align:middle line:90% of other stuff, energy. 01:07:31.560 --> 01:07:36.510 align:middle line:84% So what's the minimum requirement 01:07:36.510 --> 01:07:41.370 align:middle line:84% to produce a unit of your product? 01:07:41.370 --> 01:07:50.280 align:middle line:84% That is also the typical question. 01:07:50.280 --> 01:07:57.160 align:middle line:84% We have already seen the answer for the case 01:07:57.160 --> 01:08:03.820 align:middle line:90% of a system that varies in-- 01:08:03.820 --> 01:08:05.960 align:middle line:90% now, select your system. 01:08:05.960 --> 01:08:10.750 align:middle line:84% You want to have a change from A1 to A2, 01:08:10.750 --> 01:08:15.280 align:middle line:84% which could be heating or cooling. 01:08:15.280 --> 01:08:19.149 align:middle line:84% If it is cooling, it means that you are extracting 01:08:19.149 --> 01:08:21.859 align:middle line:90% energy out of your system. 01:08:21.859 --> 01:08:25.270 align:middle line:84% And then maybe you want to know how much of that energy 01:08:25.270 --> 01:08:27.710 align:middle line:84% can be used in order to lift the weight. 01:08:27.710 --> 01:08:30.399 align:middle line:90% So how much is it useful? 01:08:30.399 --> 01:08:37.750 align:middle line:84% Or it could be instead an increase in energy. 01:08:37.750 --> 01:08:40.630 align:middle line:90% You want to heat up your system. 01:08:40.630 --> 01:08:46.750 align:middle line:84% And then maybe you want to know how much 01:08:46.750 --> 01:08:51.439 align:middle line:90% do I need to lower my weight. 01:08:51.439 --> 01:08:53.340 align:middle line:90% So what's the minimum lowering? 01:08:53.340 --> 01:08:57.229 align:middle line:84% So what's the minimum work I need to spend in order 01:08:57.229 --> 01:08:59.720 align:middle line:90% to heat up something? 01:08:59.720 --> 01:09:03.649 align:middle line:84% And here, we do it with a reservoir. 01:09:03.649 --> 01:09:05.580 align:middle line:84% If we have a reservoir available, 01:09:05.580 --> 01:09:10.550 align:middle line:84% we can do a lot of stuff that we cannot do in an adiabatic 01:09:10.550 --> 01:09:14.029 align:middle line:84% or a weight process without the reservoir. 01:09:14.029 --> 01:09:18.310 align:middle line:84% So the reservoir plays a role, as we know. 01:09:18.310 --> 01:09:21.700 align:middle line:90% 01:09:21.700 --> 01:09:25.479 align:middle line:84% So all these questions about maximum and minimum work 01:09:25.479 --> 01:09:31.160 align:middle line:84% that can be done, useful stuff that can be done get solved. 01:09:31.160 --> 01:09:34.420 align:middle line:84% And this is the basis of all the homework problems 01:09:34.420 --> 01:09:38.620 align:middle line:84% that you have in our book, but also in any course 01:09:38.620 --> 01:09:40.420 align:middle line:90% in thermodynamics. 01:09:40.420 --> 01:09:46.600 align:middle line:84% You solve these problems by, one, defining the system, 01:09:46.600 --> 01:09:49.090 align:middle line:84% choosing a system for which to make an energy 01:09:49.090 --> 01:09:52.470 align:middle line:90% balance and an entropy balance. 01:09:52.470 --> 01:09:57.720 align:middle line:84% And the choice of your system depends 01:09:57.720 --> 01:09:59.330 align:middle line:90% on the question of the problem. 01:09:59.330 --> 01:10:03.370 align:middle line:90% 01:10:03.370 --> 01:10:07.610 align:middle line:90% The energy balance here is easy. 01:10:07.610 --> 01:10:12.160 align:middle line:84% And so it's the difference between the final 01:10:12.160 --> 01:10:16.090 align:middle line:84% and the initial energy on the left-hand side 01:10:16.090 --> 01:10:21.980 align:middle line:84% must be equal to the net energy input of your system. 01:10:21.980 --> 01:10:29.020 align:middle line:84% So here, since I'm interested in the work done in order 01:10:29.020 --> 01:10:30.580 align:middle line:90% to lift the weight-- 01:10:30.580 --> 01:10:37.275 align:middle line:84% or maybe, the work received if the weight goes instead down-- 01:10:37.275 --> 01:10:40.180 align:middle line:90% 01:10:40.180 --> 01:10:42.250 align:middle line:84% the system I'm doing the energy balance 01:10:42.250 --> 01:10:52.220 align:middle line:84% for is the composite system, A and R. The entropy balance says 01:10:52.220 --> 01:10:56.990 align:middle line:84% the final entropy, which is the sum of the entropies of the two 01:10:56.990 --> 01:11:02.420 align:middle line:84% systems, minus the initial entropy 01:11:02.420 --> 01:11:05.540 align:middle line:84% must be equal to the entropy exchanged plus the entropy 01:11:05.540 --> 01:11:07.960 align:middle line:90% generated within the system. 01:11:07.960 --> 01:11:10.620 align:middle line:84% But since this is a work interaction only, 01:11:10.620 --> 01:11:12.030 align:middle line:90% there is no entropy exchange. 01:11:12.030 --> 01:11:13.550 align:middle line:90% So there is 0 here. 01:11:13.550 --> 01:11:19.170 align:middle line:84% And then there is entropy generation. 01:11:19.170 --> 01:11:23.850 align:middle line:84% And then I also have to remember that, at least 01:11:23.850 --> 01:11:30.210 align:middle line:84% for the reservoir, I know an easy-- 01:11:30.210 --> 01:11:33.780 align:middle line:84% I mean, the fundamental relation is easy enough 01:11:33.780 --> 01:11:36.370 align:middle line:84% that I can write it as the equation of a straight line, 01:11:36.370 --> 01:11:38.940 align:middle line:84% so a linear relation between the change in energy 01:11:38.940 --> 01:11:41.550 align:middle line:84% and the change in entropy, the slope 01:11:41.550 --> 01:11:44.890 align:middle line:84% being the temperature of the reservoir. 01:11:44.890 --> 01:11:50.020 align:middle line:90% 01:11:50.020 --> 01:11:53.160 align:middle line:84% All right, so now you solve that. 01:11:53.160 --> 01:11:55.180 align:middle line:90% So we have three equations. 01:11:55.180 --> 01:11:58.330 align:middle line:84% You can use them to eliminate two variables. 01:11:58.330 --> 01:12:01.470 align:middle line:84% So let's eliminate the change in energy for the reservoir 01:12:01.470 --> 01:12:07.990 align:middle line:84% and the change in entropy for the reservoir. 01:12:07.990 --> 01:12:09.970 align:middle line:90% You combine. 01:12:09.970 --> 01:12:12.260 align:middle line:90% And this is what you obtain. 01:12:12.260 --> 01:12:18.790 align:middle line:84% You are left with one equation, the initial energy 01:12:18.790 --> 01:12:23.440 align:middle line:84% of A minus final energy of A minus the temperature 01:12:23.440 --> 01:12:25.870 align:middle line:84% of the reservoir times initial entropy of A, 01:12:25.870 --> 01:12:36.480 align:middle line:84% minus final entropy of A minus S generated multiplied by TR. 01:12:36.480 --> 01:12:44.940 align:middle line:84% So you could write this as Wrev minus TR S generated 01:12:44.940 --> 01:12:51.920 align:middle line:84% because when S generated is equal to 0, 01:12:51.920 --> 01:12:56.685 align:middle line:84% the work corresponds to the reversible process. 01:12:56.685 --> 01:12:59.830 align:middle line:90% 01:12:59.830 --> 01:13:05.740 align:middle line:84% You could also write this in terms of what we already 01:13:05.740 --> 01:13:10.630 align:middle line:84% defined as available energy, so the work 01:13:10.630 --> 01:13:14.300 align:middle line:90% done in the reversible case. 01:13:14.300 --> 01:13:19.540 align:middle line:84% So this first part of the right-hand side 01:13:19.540 --> 01:13:23.590 align:middle line:84% is equal to the difference of the available energy 01:13:23.590 --> 01:13:30.320 align:middle line:84% of system A in the initial state and that in the final state. 01:13:30.320 --> 01:13:32.320 align:middle line:84% And this is the formula for the available energy 01:13:32.320 --> 01:13:35.700 align:middle line:84% that we have seen just after the definition of entropy. 01:13:35.700 --> 01:13:39.250 align:middle line:84% If you go back, you can find that. 01:13:39.250 --> 01:13:41.866 align:middle line:84% OK, there is still another interesting way of writing this, 01:13:41.866 --> 01:13:42.408 align:middle line:90% by defining-- 01:13:42.408 --> 01:13:48.420 align:middle line:90% 01:13:48.420 --> 01:13:51.695 align:middle line:84% We can call it an availability function, another function. 01:13:51.695 --> 01:13:54.480 align:middle line:90% 01:13:54.480 --> 01:13:58.200 align:middle line:84% I'll call it canonical availability function, 01:13:58.200 --> 01:14:04.860 align:middle line:84% just because later we need to define other similar things. 01:14:04.860 --> 01:14:06.490 align:middle line:90% And we need the names. 01:14:06.490 --> 01:14:09.660 align:middle line:84% But soon, we will get short of letters and names 01:14:09.660 --> 01:14:12.210 align:middle line:90% also in this business here. 01:14:12.210 --> 01:14:16.270 align:middle line:84% In any case, we call gamma E minus TRS. 01:14:16.270 --> 01:14:20.030 align:middle line:90% 01:14:20.030 --> 01:14:24.140 align:middle line:84% And notice, therefore, that the available energy 01:14:24.140 --> 01:14:30.590 align:middle line:84% is equal to gamma minus gamma R. Gamma R 01:14:30.590 --> 01:14:36.620 align:middle line:84% is the evaluation of this function at the state AR, where 01:14:36.620 --> 01:14:40.470 align:middle line:84% the system A is in mutual stable equilibrium with the reservoir. 01:14:40.470 --> 01:14:45.190 align:middle line:90% 01:14:45.190 --> 01:14:47.980 align:middle line:84% So the available energy is the difference 01:14:47.980 --> 01:14:51.250 align:middle line:84% between the availability function 01:14:51.250 --> 01:14:54.640 align:middle line:84% at the generic state of your system 01:14:54.640 --> 01:14:58.360 align:middle line:84% and the value of that availability 01:14:58.360 --> 01:15:00.400 align:middle line:84% function at the mutual equilibrium 01:15:00.400 --> 01:15:01.610 align:middle line:90% state with the reservoir. 01:15:01.610 --> 01:15:04.130 align:middle line:90% 01:15:04.130 --> 01:15:10.970 align:middle line:84% Notice that when I am in equilibrium with the reservoir, 01:15:10.970 --> 01:15:14.554 align:middle line:90% the temperature of-- 01:15:14.554 --> 01:15:18.450 align:middle line:84% by the conditions for mutual equilibrium-- 01:15:18.450 --> 01:15:20.300 align:middle line:84% the temperature of my system, A, is 01:15:20.300 --> 01:15:23.600 align:middle line:84% equal to the temperature of the reservoir. 01:15:23.600 --> 01:15:27.980 align:middle line:84% And so this TR, in that state, is also 01:15:27.980 --> 01:15:31.450 align:middle line:90% the temperature of your system. 01:15:31.450 --> 01:15:36.830 align:middle line:84% So in that mutual stable equilibrium state, 01:15:36.830 --> 01:15:42.580 align:middle line:84% gamma R takes up the value of one of those Legendre 01:15:42.580 --> 01:15:45.980 align:middle line:84% transforms that we call the Helmholtz free energy. 01:15:45.980 --> 01:15:52.850 align:middle line:90% We called it F. It's E minus TS. 01:15:52.850 --> 01:15:57.140 align:middle line:84% But this function is not E minus TS for other states. 01:15:57.140 --> 01:15:59.350 align:middle line:90% It's E minus TRS. 01:15:59.350 --> 01:16:00.390 align:middle line:90% So it's different. 01:16:00.390 --> 01:16:04.610 align:middle line:90% 01:16:04.610 --> 01:16:09.640 align:middle line:84% We can do the representation on the E versus S diagram. 01:16:09.640 --> 01:16:16.550 align:middle line:90% 01:16:16.550 --> 01:16:23.540 align:middle line:90% So gamma is E minus TRS. 01:16:23.540 --> 01:16:27.830 align:middle line:90% So let's see. 01:16:27.830 --> 01:16:37.920 align:middle line:84% E is this much, from here to here. 01:16:37.920 --> 01:16:42.040 align:middle line:90% 01:16:42.040 --> 01:16:43.450 align:middle line:90% No, sorry. 01:16:43.450 --> 01:16:49.870 align:middle line:84% Let's say that we set this is equal to 0 is here, the origin. 01:16:49.870 --> 01:16:53.200 align:middle line:90% So E1 is this length. 01:16:53.200 --> 01:16:57.790 align:middle line:84% And then I need to subtract TR times S1. 01:16:57.790 --> 01:17:00.350 align:middle line:90% So TR is the slope of this line. 01:17:00.350 --> 01:17:05.170 align:middle line:84% So I take a line with a slope equal to TR 01:17:05.170 --> 01:17:07.720 align:middle line:90% starting from the origin. 01:17:07.720 --> 01:17:14.260 align:middle line:84% So TR times S1 is this segment dashed. 01:17:14.260 --> 01:17:21.880 align:middle line:84% So gamma is represented by this distance, between this and that, 01:17:21.880 --> 01:17:25.575 align:middle line:84% because it's this length minus this length. 01:17:25.575 --> 01:17:28.720 align:middle line:90% 01:17:28.720 --> 01:17:32.170 align:middle line:84% This is for the generic state, A1. 01:17:32.170 --> 01:17:34.060 align:middle line:84% For the state in which the system is 01:17:34.060 --> 01:17:36.080 align:middle line:84% in mutual equilibrium with the reservoir, 01:17:36.080 --> 01:17:41.440 align:middle line:84% its temperature is TR, we are here. 01:17:41.440 --> 01:17:44.500 align:middle line:84% And that same construction, so the distance 01:17:44.500 --> 01:17:51.250 align:middle line:84% from the line with slope TR originating at the origin, 01:17:51.250 --> 01:17:54.490 align:middle line:90% is this much. 01:17:54.490 --> 01:18:00.440 align:middle line:90% So this is gamma R. 01:18:00.440 --> 01:18:08.660 align:middle line:84% And now, you may see that this gamma R is the minimum value 01:18:08.660 --> 01:18:11.590 align:middle line:84% that that characteristic function can take-- 01:18:11.590 --> 01:18:14.320 align:middle line:90% 01:18:14.320 --> 01:18:18.820 align:middle line:84% I'm sorry, the availability function can take. 01:18:18.820 --> 01:18:23.390 align:middle line:84% Any other state, whether it is stable equilibrium-- 01:18:23.390 --> 01:18:27.850 align:middle line:84% so if you go along this curve, either side of AR, 01:18:27.850 --> 01:18:32.650 align:middle line:84% or if you go inside the region of non-equilibrium-- 01:18:32.650 --> 01:18:36.850 align:middle line:84% of course, here, we are talking about fixed values of n and V-- 01:18:36.850 --> 01:18:42.600 align:middle line:84% the values of gamma are greater than gamma R. 01:18:42.600 --> 01:18:44.295 align:middle line:90% So gamma R is minimum. 01:18:44.295 --> 01:18:49.180 align:middle line:90% 01:18:49.180 --> 01:18:57.130 align:middle line:84% So that means that if I start from here and go anywhere else, 01:18:57.130 --> 01:18:59.810 align:middle line:90% I must decrease gamma-- 01:18:59.810 --> 01:19:01.185 align:middle line:84% I'm sorry, I must increase gamma. 01:19:01.185 --> 01:19:05.770 align:middle line:90% 01:19:05.770 --> 01:19:10.965 align:middle line:84% So I start from state AR, go to any other state. 01:19:10.965 --> 01:19:14.140 align:middle line:90% 01:19:14.140 --> 01:19:18.560 align:middle line:84% Then the delta, so the change in gamma, is positive. 01:19:18.560 --> 01:19:21.620 align:middle line:90% 01:19:21.620 --> 01:19:29.980 align:middle line:84% That condition gives rules about-- 01:19:29.980 --> 01:19:35.420 align:middle line:90% 01:19:35.420 --> 01:19:43.880 align:middle line:84% it's the generating inequality for inequalities regarding 01:19:43.880 --> 01:19:45.450 align:middle line:90% the second order derivatives. 01:19:45.450 --> 01:19:51.170 align:middle line:84% And therefore, they have to do with stability and the concavity 01:19:51.170 --> 01:19:53.065 align:middle line:90% or convexity of curves. 01:19:53.065 --> 01:19:56.270 align:middle line:90% 01:19:56.270 --> 01:19:59.450 align:middle line:84% In particular, this is, in a sense, a review 01:19:59.450 --> 01:20:04.070 align:middle line:84% because we already encountered these particular ones. 01:20:04.070 --> 01:20:07.320 align:middle line:84% But this is just to give the idea. 01:20:07.320 --> 01:20:12.740 align:middle line:84% So I apply this principle that, as I move away from this state, 01:20:12.740 --> 01:20:17.110 align:middle line:84% no matter where I go, I always increase gamma. 01:20:17.110 --> 01:20:22.160 align:middle line:90% 01:20:22.160 --> 01:20:26.570 align:middle line:84% So let me apply it to one particular change. 01:20:26.570 --> 01:20:29.930 align:middle line:90% So I decide to go to-- 01:20:29.930 --> 01:20:36.440 align:middle line:84% just a small change, a small amount of entropy change, 01:20:36.440 --> 01:20:41.330 align:middle line:90% dS, d for infinitesimal. 01:20:41.330 --> 01:20:46.190 align:middle line:84% And also, I choose to move along the stable equilibrium state 01:20:46.190 --> 01:20:46.730 align:middle line:90% curve. 01:20:46.730 --> 01:20:48.980 align:middle line:90% So I'm going this side. 01:20:48.980 --> 01:20:50.760 align:middle line:90% But dS could also be negative. 01:20:50.760 --> 01:20:53.630 align:middle line:84% So I can go the other side, but along the stable equilibrium 01:20:53.630 --> 01:20:54.205 align:middle line:90% state curve. 01:20:54.205 --> 01:20:57.490 align:middle line:90% 01:20:57.490 --> 01:20:59.380 align:middle line:84% So if I change the entropy and I move 01:20:59.380 --> 01:21:03.400 align:middle line:84% along the stable equilibrium state curve, 01:21:03.400 --> 01:21:06.465 align:middle line:84% the energy changes according to the fundamental relation. 01:21:06.465 --> 01:21:09.880 align:middle line:90% 01:21:09.880 --> 01:21:17.650 align:middle line:84% So I move from ER to another energy, 01:21:17.650 --> 01:21:19.540 align:middle line:84% which is given by the fundamental relation. 01:21:19.540 --> 01:21:21.660 align:middle line:90% It's given by this one. 01:21:21.660 --> 01:21:24.480 align:middle line:84% And I can use a Taylor series expansion 01:21:24.480 --> 01:21:27.793 align:middle line:90% around the original point. 01:21:27.793 --> 01:21:36.050 align:middle line:90% 01:21:36.050 --> 01:21:43.020 align:middle line:90% And it's there. 01:21:43.020 --> 01:21:44.940 align:middle line:90% Yeah, OK. 01:21:44.940 --> 01:21:47.040 align:middle line:84% So the Taylor series expansion has 01:21:47.040 --> 01:21:50.250 align:middle line:84% a value, which is equal to AR, which cancels with that. 01:21:50.250 --> 01:21:52.920 align:middle line:84% And I have a first order term, which 01:21:52.920 --> 01:21:55.800 align:middle line:84% is the derivative of E with respect to S, 01:21:55.800 --> 01:22:02.280 align:middle line:84% which is the temperature, which, at the point where I started, 01:22:02.280 --> 01:22:09.000 align:middle line:84% is the temperature TR times dS plus the second order 01:22:09.000 --> 01:22:13.080 align:middle line:90% terms in the Taylor expansion. 01:22:13.080 --> 01:22:16.350 align:middle line:84% And remember that here, V and n are fixed. 01:22:16.350 --> 01:22:20.580 align:middle line:84% So this, we could call it the second differential of at fixed 01:22:20.580 --> 01:22:28.430 align:middle line:90% n and V. So that's delta E. 01:22:28.430 --> 01:22:34.730 align:middle line:84% Now, I substitute into delta gamma to compute-- 01:22:34.730 --> 01:22:37.460 align:middle line:90% so delta gamma is going to be-- 01:22:37.460 --> 01:22:40.190 align:middle line:84% since gamma is E minus TRS, delta gamma 01:22:40.190 --> 01:22:45.980 align:middle line:84% is delta E minus TR delta S. So I substitute. 01:22:45.980 --> 01:22:52.140 align:middle line:84% And this TR delta S cancels with that one. 01:22:52.140 --> 01:22:54.625 align:middle line:84% So I'm left with the second order differential only. 01:22:54.625 --> 01:22:57.180 align:middle line:90% 01:22:57.180 --> 01:23:00.750 align:middle line:84% And since these plus dots, dots, dots must be strictly 01:23:00.750 --> 01:23:08.300 align:middle line:84% greater than 0, it requires that this term must 01:23:08.300 --> 01:23:10.880 align:middle line:90% be greater than or equal to 0. 01:23:10.880 --> 01:23:17.560 align:middle line:90% So that gives us one condition. 01:23:17.560 --> 01:23:26.090 align:middle line:84% I could also have chosen to move to a state in which I instead I 01:23:26.090 --> 01:23:26.700 align:middle line:90% change-- 01:23:26.700 --> 01:23:29.840 align:middle line:84% I decide how much I change the energy. 01:23:29.840 --> 01:23:32.490 align:middle line:84% I still move along the stable equilibrium state curve. 01:23:32.490 --> 01:23:34.820 align:middle line:84% But now, I decide the value of dE. 01:23:34.820 --> 01:23:37.380 align:middle line:90% 01:23:37.380 --> 01:23:42.030 align:middle line:84% Well, then the Taylor series expansion gives me this. 01:23:42.030 --> 01:23:49.260 align:middle line:84% And the delta gamma becomes minus the second-- 01:23:49.260 --> 01:23:51.730 align:middle line:84% Lots of things cancel, as before. 01:23:51.730 --> 01:23:57.180 align:middle line:84% And so I'm left with delta gamma is minus 01:23:57.180 --> 01:24:01.790 align:middle line:90% the second differential of S. 01:24:01.790 --> 01:24:09.170 align:middle line:84% And for this to be greater than 0, it requires that to be-- 01:24:09.170 --> 01:24:10.920 align:middle line:90% oops, this is wrong. 01:24:10.920 --> 01:24:12.320 align:middle line:90% This should be-- 01:24:12.320 --> 01:24:13.710 align:middle line:90% I have to correct that, too. 01:24:13.710 --> 01:24:16.300 align:middle line:84% This should be less than or equal to 0. 01:24:16.300 --> 01:24:20.540 align:middle line:90% 01:24:20.540 --> 01:24:21.960 align:middle line:90% I'll correct it in the slides. 01:24:21.960 --> 01:24:25.730 align:middle line:90% 01:24:25.730 --> 01:24:29.550 align:middle line:84% So remember when we talked about concavity, 01:24:29.550 --> 01:24:35.840 align:middle line:84% we had quickly passed this slide where 01:24:35.840 --> 01:24:40.850 align:middle line:84% we took the fundamental relation and took, 01:24:40.850 --> 01:24:48.890 align:middle line:84% in full glory, the matrix of second order derivatives, 01:24:48.890 --> 01:24:53.830 align:middle line:84% which is the Hessian matrix of the relation. 01:24:53.830 --> 01:25:01.440 align:middle line:90% 01:25:01.440 --> 01:25:05.550 align:middle line:84% If you keep fixed some of these variables, 01:25:05.550 --> 01:25:08.500 align:middle line:84% it becomes a function of fewer variables. 01:25:08.500 --> 01:25:10.990 align:middle line:90% So the Hessian becomes smaller. 01:25:10.990 --> 01:25:19.860 align:middle line:84% And if I keep n and V, as we did in the previous thing, fixed, 01:25:19.860 --> 01:25:21.480 align:middle line:90% this becomes just-- 01:25:21.480 --> 01:25:25.750 align:middle line:84% the Hessian becomes just the second order derivative. 01:25:25.750 --> 01:25:28.680 align:middle line:90% 01:25:28.680 --> 01:25:31.325 align:middle line:84% So let's call it the partial Hessian. 01:25:31.325 --> 01:25:35.090 align:middle line:90% 01:25:35.090 --> 01:25:48.570 align:middle line:84% And what we proved is that d2S should be-- 01:25:48.570 --> 01:25:49.760 align:middle line:90% and here it is correct-- 01:25:49.760 --> 01:25:51.370 align:middle line:90% less than or equal to 0. 01:25:51.370 --> 01:25:57.218 align:middle line:90% 01:25:57.218 --> 01:26:08.140 align:middle line:84% So d2S is equal to the second derivative times 01:26:08.140 --> 01:26:18.880 align:middle line:84% dE squared because here d2S, in general, 01:26:18.880 --> 01:26:22.480 align:middle line:84% is the Hessian matrix that gets multiplied on the left 01:26:22.480 --> 01:26:25.280 align:middle line:90% and also on the right by the dE. 01:26:25.280 --> 01:26:28.290 align:middle line:84% So you get the dE squared and all the combinations. 01:26:28.290 --> 01:26:29.660 align:middle line:90% This is in general. 01:26:29.660 --> 01:26:33.250 align:middle line:84% In the particular case, this is just dE squared. 01:26:33.250 --> 01:26:35.980 align:middle line:84% And since it is squared, that simply 01:26:35.980 --> 01:26:42.380 align:middle line:84% requires that this derivative be negative. 01:26:42.380 --> 01:26:51.940 align:middle line:84% And therefore, we get the concavity of the S versus E 01:26:51.940 --> 01:26:58.180 align:middle line:84% relation, which justifies our drawing the E versus S 01:26:58.180 --> 01:26:59.385 align:middle line:90% diagram that way. 01:26:59.385 --> 01:27:04.910 align:middle line:90% 01:27:04.910 --> 01:27:12.200 align:middle line:84% But for example, recalling that the partial of S with respect 01:27:12.200 --> 01:27:18.800 align:middle line:84% to E is 1 over the temperature, this derivative 01:27:18.800 --> 01:27:23.860 align:middle line:84% can be written this way because it's the derivative with respect 01:27:23.860 --> 01:27:26.630 align:middle line:90% to E of 1 over the temperature. 01:27:26.630 --> 01:27:31.120 align:middle line:84% So it's minus 1 over T squared times derivative of T 01:27:31.120 --> 01:27:32.680 align:middle line:90% with respect to E. 01:27:32.680 --> 01:27:35.300 align:middle line:84% And if you go back to what we defined, 01:27:35.300 --> 01:27:39.060 align:middle line:84% we called heat capacity at constant volume-- 01:27:39.060 --> 01:27:43.340 align:middle line:90% this is just the inverse of CV. 01:27:43.340 --> 01:27:46.310 align:middle line:90% 01:27:46.310 --> 01:27:49.820 align:middle line:84% So this second derivative is equal to minus 1 01:27:49.820 --> 01:27:52.940 align:middle line:90% over T squared over CV. 01:27:52.940 --> 01:27:56.240 align:middle line:84% And that requires therefore that CV-- 01:27:56.240 --> 01:27:59.780 align:middle line:84% since it has to be negative, it requires that CV 01:27:59.780 --> 01:28:01.045 align:middle line:90% is greater than or equal to 0. 01:28:01.045 --> 01:28:03.670 align:middle line:90% 01:28:03.670 --> 01:28:10.900 align:middle line:84% So for your models, if you have a substance with a negative CV, 01:28:10.900 --> 01:28:12.410 align:middle line:90% you are in trouble. 01:28:12.410 --> 01:28:13.563 align:middle line:90% You have no substance. 01:28:13.563 --> 01:28:14.355 align:middle line:90% You made a mistake. 01:28:14.355 --> 01:28:22.280 align:middle line:90% 01:28:22.280 --> 01:28:26.240 align:middle line:84% Also from the other equation that we have-- 01:28:26.240 --> 01:28:30.700 align:middle line:84% condition that we have here, d2E greater than or equal to 0, 01:28:30.700 --> 01:28:35.420 align:middle line:84% we can remember that the partial of E with respect to S 01:28:35.420 --> 01:28:36.650 align:middle line:90% is the temperature. 01:28:36.650 --> 01:28:40.070 align:middle line:84% This is also written, also equal to the partial of T 01:28:40.070 --> 01:28:48.290 align:middle line:84% with respect to S, which is greater than 0. 01:28:48.290 --> 01:28:54.560 align:middle line:84% And this also is related to the one above. 01:28:54.560 --> 01:29:08.190 align:middle line:84% It gives the same practical result. 01:29:08.190 --> 01:29:18.510 align:middle line:84% These inequalities give an interesting-- 01:29:18.510 --> 01:29:23.850 align:middle line:84% give rise or are manifestations of a general principle that 01:29:23.850 --> 01:29:27.000 align:middle line:84% is called the Le Chatelier or Le Chatelier-Braun 01:29:27.000 --> 01:29:36.450 align:middle line:84% theorem or principle that, if we combine it with the idea 01:29:36.450 --> 01:29:40.980 align:middle line:84% that we have seen today, that temperature is 01:29:40.980 --> 01:29:48.250 align:middle line:84% an escaping tendency for energy, you see what this says. 01:29:48.250 --> 01:29:56.320 align:middle line:84% If, at fixed n and V, I give some energy to my system, 01:29:56.320 --> 01:29:58.900 align:middle line:90% it increases its temperature. 01:29:58.900 --> 01:30:00.760 align:middle line:90% Well, we all know that. 01:30:00.760 --> 01:30:05.680 align:middle line:84% Did we need so many lectures to get to this result? 01:30:05.680 --> 01:30:06.910 align:middle line:90% All right. 01:30:06.910 --> 01:30:11.110 align:middle line:84% This also says that if you give some entropy, 01:30:11.110 --> 01:30:14.380 align:middle line:90% the temperature will increase. 01:30:14.380 --> 01:30:19.220 align:middle line:84% But the temperature is the escaping tendency for energy. 01:30:19.220 --> 01:30:25.240 align:middle line:84% So that means that if I give energy to my system, in a sense, 01:30:25.240 --> 01:30:33.460 align:middle line:84% it reacts by heighering its willingness to give away energy. 01:30:33.460 --> 01:30:38.350 align:middle line:84% So it is as if it counteracts-- it wants to counteract 01:30:38.350 --> 01:30:39.440 align:middle line:90% what you're doing. 01:30:39.440 --> 01:30:45.250 align:middle line:84% So if you externally heat up the system, you give energy, 01:30:45.250 --> 01:30:52.540 align:middle line:84% and then it reacts by shaping things, so as soon as it can, 01:30:52.540 --> 01:30:55.330 align:middle line:90% it gives that energy away. 01:30:55.330 --> 01:30:58.390 align:middle line:84% So for example, if you start with your system 01:30:58.390 --> 01:31:02.500 align:middle line:84% in mutual equilibrium with your thermal reservoir 01:31:02.500 --> 01:31:06.530 align:middle line:84% and externally give some energy to your system, 01:31:06.530 --> 01:31:12.070 align:middle line:84% it will get off equilibrium, off mutual equilibrium, 01:31:12.070 --> 01:31:14.380 align:middle line:84% higher temperature than the reservoir. 01:31:14.380 --> 01:31:18.850 align:middle line:84% That higher temperature will make up 01:31:18.850 --> 01:31:22.030 align:middle line:84% a difference in temperature that will drive an energy 01:31:22.030 --> 01:31:25.990 align:middle line:90% exchange back to the reservoir. 01:31:25.990 --> 01:31:34.100 align:middle line:84% Thereby, your system will try to reestablish equilibrium. 01:31:34.100 --> 01:31:37.640 align:middle line:84% So you can see this natural tendency 01:31:37.640 --> 01:31:42.680 align:middle line:84% of reestablishing equilibrium here 01:31:42.680 --> 01:31:53.480 align:middle line:84% as a theorem that also emerges from the stability 01:31:53.480 --> 01:31:57.370 align:middle line:84% conditions and the concavity of the fundamental relation. 01:31:57.370 --> 01:32:00.450 align:middle line:90% 01:32:00.450 --> 01:32:05.490 align:middle line:84% The Le Chatelier-Braun theorem, of course, is more general. 01:32:05.490 --> 01:32:10.790 align:middle line:84% And we'll see a little more the next time about this. 01:32:10.790 --> 01:32:14.700 align:middle line:84% It's not just when you do this with respect to these variables, 01:32:14.700 --> 01:32:17.910 align:middle line:84% but also when you do it with respect to many other variables. 01:32:17.910 --> 01:32:23.870 align:middle line:84% And also, it has an additional little other twist 01:32:23.870 --> 01:32:29.610 align:middle line:84% that we will see when we do two variables. 01:32:29.610 --> 01:32:32.330 align:middle line:90% 01:32:32.330 --> 01:32:38.600 align:middle line:84% But for today, I think that, yeah, we are done. 01:32:38.600 --> 01:32:41.210 align:middle line:90% And we are just in time. 01:32:41.210 --> 01:32:44.860 align:middle line:90% So I'll see you next time. 01:32:44.860 --> 01:32:47.000 align:middle line:90%