WEBVTT 00:00:00.000 --> 00:00:09.212 align:middle line:84% [SQUEAKING] [RUSTILING] [CLICKING] 00:00:09.212 --> 00:00:10.670 align:middle line:84% GIAN PAOLO BERETTA: Good afternoon. 00:00:10.670 --> 00:00:14.170 align:middle line:90% 00:00:14.170 --> 00:00:16.890 align:middle line:90% So here we are. 00:00:16.890 --> 00:00:19.420 align:middle line:90% 00:00:19.420 --> 00:00:21.830 align:middle line:90% And let's see where we are. 00:00:21.830 --> 00:00:27.670 align:middle line:84% We have introduced this energy versus entropy diagram 00:00:27.670 --> 00:00:30.610 align:middle line:90% to summarize concepts. 00:00:30.610 --> 00:00:35.410 align:middle line:84% And we have obtained these important inequalities 00:00:35.410 --> 00:00:42.860 align:middle line:84% that I called generically Clausius inequalities that 00:00:42.860 --> 00:00:46.640 align:middle line:90% essentially talk about-- 00:00:46.640 --> 00:00:49.760 align:middle line:84% say something about how much entropy 00:00:49.760 --> 00:00:54.140 align:middle line:84% must be exchanged between two systems 00:00:54.140 --> 00:00:59.690 align:middle line:84% if these two systems start in stable equilibrium states 00:00:59.690 --> 00:01:07.030 align:middle line:84% and they need to exchange energy with no other effects. 00:01:07.030 --> 00:01:11.770 align:middle line:84% So if this is the curve for the stable equilibrium 00:01:11.770 --> 00:01:14.870 align:middle line:84% states for system A and this is for system B, 00:01:14.870 --> 00:01:17.620 align:middle line:84% we suppose that the system A starts in this state 00:01:17.620 --> 00:01:22.990 align:middle line:84% and that we want to extract this much energy from it 00:01:22.990 --> 00:01:30.310 align:middle line:84% and give it to system B. We realize that there 00:01:30.310 --> 00:01:32.560 align:middle line:84% is no way to-- since there are no states 00:01:32.560 --> 00:01:38.070 align:middle line:84% below the initial state, we need to extract also some entropy. 00:01:38.070 --> 00:01:44.415 align:middle line:90% 00:01:44.415 --> 00:01:46.920 align:middle line:84% But the problem is that system B-- 00:01:46.920 --> 00:01:52.230 align:middle line:84% so if we extract entropy from system A, 00:01:52.230 --> 00:01:56.580 align:middle line:84% that entropy can only go into system B 00:01:56.580 --> 00:02:00.060 align:middle line:84% because there are no other external effects. 00:02:00.060 --> 00:02:03.900 align:middle line:84% But starting from this state, system B, 00:02:03.900 --> 00:02:09.690 align:middle line:84% and receiving this much energy, the amount of entropy 00:02:09.690 --> 00:02:12.250 align:middle line:84% that it can accommodate is only this much, 00:02:12.250 --> 00:02:14.020 align:middle line:90% which is called S max. 00:02:14.020 --> 00:02:19.500 align:middle line:84% So that's the maximum amount of entropy that B can accommodate. 00:02:19.500 --> 00:02:26.070 align:middle line:84% Whereas for system A, this is the minimum amount of entropy 00:02:26.070 --> 00:02:30.180 align:middle line:84% that it needs to give to system B 00:02:30.180 --> 00:02:34.920 align:middle line:84% in order to find a state with this lower energy level. 00:02:34.920 --> 00:02:39.460 align:middle line:84% So we are left with a possibility of possible states 00:02:39.460 --> 00:02:44.140 align:middle line:84% that are on these dashed lines for system A 00:02:44.140 --> 00:02:52.170 align:middle line:84% and for system B that correspond to entropy transfers 00:02:52.170 --> 00:02:56.430 align:middle line:84% within the range allowed by these constraints 00:02:56.430 --> 00:02:58.620 align:middle line:90% that we have described. 00:02:58.620 --> 00:03:03.540 align:middle line:84% So that turns out to be writeable as a couple 00:03:03.540 --> 00:03:05.970 align:middle line:90% of inequalities. 00:03:05.970 --> 00:03:08.670 align:middle line:84% The Clausius traditional statement 00:03:08.670 --> 00:03:12.630 align:middle line:84% is, as we have seen it, for infinitesimal amounts 00:03:12.630 --> 00:03:14.470 align:middle line:90% of energy transfers. 00:03:14.470 --> 00:03:15.960 align:middle line:90% This is for finite. 00:03:15.960 --> 00:03:19.320 align:middle line:90% But the concept is the same. 00:03:19.320 --> 00:03:24.600 align:middle line:84% So today, I wanted to talk about some traditional results. 00:03:24.600 --> 00:03:29.190 align:middle line:90% 00:03:29.190 --> 00:03:35.820 align:middle line:84% And we still need to define what we mean by heat. 00:03:35.820 --> 00:03:39.460 align:middle line:84% First of all, we define what we mean by work. 00:03:39.460 --> 00:03:43.080 align:middle line:84% So work is a mode of interaction in which 00:03:43.080 --> 00:03:47.850 align:middle line:84% there is no entropy transfer and no transfer of amounts 00:03:47.850 --> 00:03:49.060 align:middle line:90% of constituents. 00:03:49.060 --> 00:03:55.394 align:middle line:90% 00:03:55.394 --> 00:04:01.780 align:middle line:84% So that's, for example, what happens in a weight process. 00:04:01.780 --> 00:04:06.230 align:middle line:84% In a weight process, if during the entire process, 00:04:06.230 --> 00:04:11.740 align:middle line:84% there are no external effects, then the weight process 00:04:11.740 --> 00:04:14.110 align:middle line:90% experiences-- 00:04:14.110 --> 00:04:18.550 align:middle line:84% the system experiences a work interaction. 00:04:18.550 --> 00:04:22.690 align:middle line:84% As you may read here, it says that the weight process 00:04:22.690 --> 00:04:23.960 align:middle line:90% may be adiabatic. 00:04:23.960 --> 00:04:27.460 align:middle line:84% We call adiabatic a process in which there 00:04:27.460 --> 00:04:29.300 align:middle line:90% are only work interactions. 00:04:29.300 --> 00:04:32.450 align:middle line:84% And then we say, well, a weight process may be adiabatic. 00:04:32.450 --> 00:04:35.680 align:middle line:90% Why maybe and not is adiabatic? 00:04:35.680 --> 00:04:40.900 align:middle line:84% Well, because a weight process is a process 00:04:40.900 --> 00:04:44.800 align:middle line:84% in which the only net external effect 00:04:44.800 --> 00:04:52.430 align:middle line:84% is the lifting or lowering of a weight, the net effect. 00:04:52.430 --> 00:04:55.970 align:middle line:84% But you could have some temporary effects 00:04:55.970 --> 00:05:02.980 align:middle line:84% that are initially done and then undone, 00:05:02.980 --> 00:05:06.530 align:middle line:84% which may not be describable by lifting or lowering a weight. 00:05:06.530 --> 00:05:09.230 align:middle line:84% So in that case, it's still a weight process. 00:05:09.230 --> 00:05:14.080 align:middle line:84% But during the process, there were interactions 00:05:14.080 --> 00:05:17.530 align:middle line:90% that may not be work. 00:05:17.530 --> 00:05:23.530 align:middle line:84% But for simplicity, if you want to fix ideas, 00:05:23.530 --> 00:05:27.400 align:middle line:84% a good example of an adiabatic process 00:05:27.400 --> 00:05:34.030 align:middle line:84% is a weight process in which there are no entropy transfers 00:05:34.030 --> 00:05:35.385 align:middle line:90% all along the process. 00:05:35.385 --> 00:05:39.710 align:middle line:90% 00:05:39.710 --> 00:05:43.880 align:middle line:84% Anything that has an entropy transfer, we call it non-work. 00:05:43.880 --> 00:05:54.990 align:middle line:84% So non-work is a very broad class of processes, 00:05:54.990 --> 00:05:57.770 align:middle line:90% of types of interaction. 00:05:57.770 --> 00:06:02.690 align:middle line:84% And in this broad class, there is one limiting condition 00:06:02.690 --> 00:06:04.560 align:middle line:90% that we call heat interaction. 00:06:04.560 --> 00:06:08.600 align:middle line:84% So we will see in a moment what is the defining 00:06:08.600 --> 00:06:11.320 align:middle line:90% idea behind what we call heat. 00:06:11.320 --> 00:06:14.420 align:middle line:90% 00:06:14.420 --> 00:06:16.700 align:middle line:90% It is a limiting process. 00:06:16.700 --> 00:06:21.840 align:middle line:84% And it doesn't exhaust all the possible non-work interactions. 00:06:21.840 --> 00:06:27.400 align:middle line:84% So not all non-work interactions are heat. 00:06:27.400 --> 00:06:30.910 align:middle line:84% And in fact, I already anticipated the last time 00:06:30.910 --> 00:06:37.420 align:middle line:84% that, during a heat interaction, we 00:06:37.420 --> 00:06:42.700 align:middle line:84% have a very well-defined ratio between the energy that 00:06:42.700 --> 00:06:46.480 align:middle line:84% is exchanged, that we call heat, and the entropy that 00:06:46.480 --> 00:06:47.230 align:middle line:90% is exchanged. 00:06:47.230 --> 00:06:51.370 align:middle line:90% And the ratio is equal-- 00:06:51.370 --> 00:06:54.010 align:middle line:84% energy over entropy is the temperature 00:06:54.010 --> 00:07:00.210 align:middle line:84% at which that interaction occurs, so much that therefore, 00:07:00.210 --> 00:07:02.130 align:middle line:84% in order to characterize a heat interaction, 00:07:02.130 --> 00:07:08.780 align:middle line:84% we only need to say it's heat, how much energy is transferred, 00:07:08.780 --> 00:07:11.700 align:middle line:84% and what is the temperature at which it is transferred. 00:07:11.700 --> 00:07:16.020 align:middle line:90% 00:07:16.020 --> 00:07:21.950 align:middle line:84% But what's the defining criterion for heat? 00:07:21.950 --> 00:07:24.880 align:middle line:90% 00:07:24.880 --> 00:07:32.260 align:middle line:84% One way to put it is, we want to define an interaction, which 00:07:32.260 --> 00:07:35.920 align:middle line:84% is completely distinguishable from work, 00:07:35.920 --> 00:07:38.290 align:middle line:84% so an interaction in which these two 00:07:38.290 --> 00:07:46.030 align:middle line:84% systems, say system A and system B, exchange energy and entropy. 00:07:46.030 --> 00:07:49.300 align:middle line:84% And nothing, no fraction of the energy 00:07:49.300 --> 00:07:57.200 align:middle line:84% that is exchanged between the two, can be recognized as work. 00:07:57.200 --> 00:07:58.470 align:middle line:90% What does it mean? 00:07:58.470 --> 00:08:00.810 align:middle line:84% All right, we do it operationally. 00:08:00.810 --> 00:08:06.530 align:middle line:84% Let's say that we extract some energy from system A. 00:08:06.530 --> 00:08:10.020 align:middle line:84% And let's suppose that we intercept that energy. 00:08:10.020 --> 00:08:13.520 align:middle line:84% And before giving it to B, we feed it 00:08:13.520 --> 00:08:18.230 align:middle line:84% to some machinery, some machinery that 00:08:18.230 --> 00:08:24.620 align:middle line:84% will try, at its best, to split that energy into a part that 00:08:24.620 --> 00:08:27.445 align:middle line:84% is recognizable as work and the rest. 00:08:27.445 --> 00:08:32.740 align:middle line:90% 00:08:32.740 --> 00:08:34.539 align:middle line:84% Sometimes, I mean, traditionally you 00:08:34.539 --> 00:08:36.569 align:middle line:90% would call it a Carnot engine. 00:08:36.569 --> 00:08:38.950 align:middle line:84% But let's call it a cyclic machinery, 00:08:38.950 --> 00:08:45.170 align:middle line:84% cyclic because it has to return to its initial state. 00:08:45.170 --> 00:08:47.080 align:middle line:84% And so whatever it does, it's not 00:08:47.080 --> 00:08:51.815 align:middle line:84% because you consume the machine, but because of the operations 00:08:51.815 --> 00:08:52.440 align:middle line:90% of the machine. 00:08:52.440 --> 00:08:55.850 align:middle line:90% 00:08:55.850 --> 00:08:56.460 align:middle line:90% OK. 00:08:56.460 --> 00:09:00.120 align:middle line:84% So in this situation, suppose we extract this much energy 00:09:00.120 --> 00:09:01.130 align:middle line:90% and this much entropy. 00:09:01.130 --> 00:09:05.630 align:middle line:84% We feed it to the engine, to the machinery. 00:09:05.630 --> 00:09:08.110 align:middle line:90% How much work can we extract? 00:09:08.110 --> 00:09:11.690 align:middle line:84% Well, this is a standard problem in thermodynamics 00:09:11.690 --> 00:09:15.115 align:middle line:84% that gets solved by writing energy and entropy balances. 00:09:15.115 --> 00:09:18.710 align:middle line:90% 00:09:18.710 --> 00:09:23.720 align:middle line:84% So we write energy and entropy balances for system A and system 00:09:23.720 --> 00:09:35.150 align:middle line:84% B. We also assume that system A and system B begin and end 00:09:35.150 --> 00:09:36.820 align:middle line:90% in stable equilibrium states. 00:09:36.820 --> 00:09:41.110 align:middle line:90% 00:09:41.110 --> 00:09:44.380 align:middle line:84% And we write this bunch of equations. 00:09:44.380 --> 00:09:48.260 align:middle line:84% If you count the equal signs, it's how many equations? 00:09:48.260 --> 00:09:52.310 align:middle line:84% 1, 2, 3, 4, 5, 6, 7, 8 equations. 00:09:52.310 --> 00:09:55.900 align:middle line:90% 00:09:55.900 --> 00:09:58.210 align:middle line:84% Considering the initial temperatures 00:09:58.210 --> 00:10:02.320 align:middle line:84% as fixed parameters, the variables here 00:10:02.320 --> 00:10:06.840 align:middle line:84% are the various entropy exchanged 00:10:06.840 --> 00:10:12.600 align:middle line:84% and the energy and entropy changes for system A and system 00:10:12.600 --> 00:10:21.600 align:middle line:84% B. If you count them, I think there are nine variables, eight 00:10:21.600 --> 00:10:22.710 align:middle line:90% equations. 00:10:22.710 --> 00:10:24.160 align:middle line:90% You cannot solve the system. 00:10:24.160 --> 00:10:29.500 align:middle line:84% But you can solve it for a ratio of two of those variables. 00:10:29.500 --> 00:10:33.750 align:middle line:84% And we choose the ratio that we are interested in, 00:10:33.750 --> 00:10:38.970 align:middle line:84% namely the work that has been separated out 00:10:38.970 --> 00:10:45.480 align:middle line:84% of that stream over the inlet energy, inlet 00:10:45.480 --> 00:10:47.705 align:middle line:90% into the cyclic engine. 00:10:47.705 --> 00:10:51.720 align:middle line:84% So this ratio, as we all, because we've all 00:10:51.720 --> 00:10:54.270 align:middle line:84% taken thermodynamics before, it's 00:10:54.270 --> 00:11:00.960 align:middle line:84% the Carnot ratio between the two initial temperatures 00:11:00.960 --> 00:11:04.650 align:middle line:90% of these systems. 00:11:04.650 --> 00:11:12.630 align:middle line:84% So this says that if these two temperatures are different, 00:11:12.630 --> 00:11:15.780 align:middle line:90% this ratio is not 0. 00:11:15.780 --> 00:11:17.270 align:middle line:90% It's a definite number. 00:11:17.270 --> 00:11:19.830 align:middle line:90% 00:11:19.830 --> 00:11:26.100 align:middle line:84% Suppose the system B is at 300K and system A is at 600K. 00:11:26.100 --> 00:11:27.705 align:middle line:90% This is 300 over 600. 00:11:27.705 --> 00:11:28.930 align:middle line:90% That Is 1/2. 00:11:28.930 --> 00:11:32.940 align:middle line:90% So 1 minus 1/2 is 50%. 00:11:32.940 --> 00:11:35.970 align:middle line:84% So that means that the cyclic engine here 00:11:35.970 --> 00:11:39.090 align:middle line:84% could split 50% of the received energy from A 00:11:39.090 --> 00:11:42.910 align:middle line:84% and make it go into the work channel. 00:11:42.910 --> 00:11:45.535 align:middle line:90% 00:11:45.535 --> 00:11:47.170 align:middle line:90% Sure. 00:11:47.170 --> 00:11:54.130 align:middle line:84% If we want to complete the job once, we have split this energy 00:11:54.130 --> 00:11:56.950 align:middle line:84% and give it to the weight, then we take the weight back 00:11:56.950 --> 00:11:58.870 align:middle line:84% and give that energy to system B. 00:11:58.870 --> 00:12:00.830 align:middle line:90% So the end result is the same. 00:12:00.830 --> 00:12:05.390 align:middle line:84% We transfer the energy from A to B with no external effects. 00:12:05.390 --> 00:12:09.070 align:middle line:84% But in between, we manage to recognize 00:12:09.070 --> 00:12:13.640 align:middle line:84% a part of what flows from one system to the other as work. 00:12:13.640 --> 00:12:15.570 align:middle line:84% That's not what we want to call heat. 00:12:15.570 --> 00:12:18.700 align:middle line:90% 00:12:18.700 --> 00:12:20.800 align:middle line:84% We will call heat only in the limit 00:12:20.800 --> 00:12:25.600 align:middle line:90% in which this ratio goes to 0. 00:12:25.600 --> 00:12:28.450 align:middle line:84% And that goes to 0 only in the limit 00:12:28.450 --> 00:12:33.815 align:middle line:84% in which the two temperatures get equal or very close 00:12:33.815 --> 00:12:34.440 align:middle line:90% to one another. 00:12:34.440 --> 00:12:39.210 align:middle line:90% 00:12:39.210 --> 00:12:44.740 align:middle line:84% And in terms of that train of inequalities that we have seen, 00:12:44.740 --> 00:12:48.150 align:middle line:84% the Clausius inequality last time, 00:12:48.150 --> 00:12:52.150 align:middle line:84% that interval of values, if the two temperatures are equal, 00:12:52.150 --> 00:12:54.120 align:middle line:90% shrinks to a single value. 00:12:54.120 --> 00:13:01.620 align:middle line:84% And therefore, we get a definite entropy transfer 00:13:01.620 --> 00:13:03.810 align:middle line:90% for a given energy transfer. 00:13:03.810 --> 00:13:06.870 align:middle line:84% And that's how we get the famous equation 00:13:06.870 --> 00:13:11.470 align:middle line:84% that delta S, the entropy exchanged, 00:13:11.470 --> 00:13:19.370 align:middle line:84% is the energy exchanged over T, so Q over T. 00:13:19.370 --> 00:13:23.190 align:middle line:84% And of course, I have stressed it so many times. 00:13:23.190 --> 00:13:26.900 align:middle line:84% But we noticed that we have defining heat here using 00:13:26.900 --> 00:13:28.610 align:middle line:90% entropy. 00:13:28.610 --> 00:13:30.870 align:middle line:84% Usually, it's the other way around. 00:13:30.870 --> 00:13:33.200 align:middle line:84% The traditional expositions start 00:13:33.200 --> 00:13:41.360 align:middle line:84% from some hand-waving definition of heat, 00:13:41.360 --> 00:13:44.430 align:middle line:84% and then some definition of temperature. 00:13:44.430 --> 00:13:48.950 align:middle line:84% And then Q over T becomes the entropy. 00:13:48.950 --> 00:13:50.090 align:middle line:90% And that's the-- 00:13:50.090 --> 00:13:56.630 align:middle line:84% I call it the hand-waving heuristic definition of entropy 00:13:56.630 --> 00:13:59.960 align:middle line:84% that, to be frank with you, is what 00:13:59.960 --> 00:14:04.340 align:middle line:84% got me started with this subject because I was so unhappy 00:14:04.340 --> 00:14:06.710 align:middle line:84% of the way I was explained the subject 00:14:06.710 --> 00:14:09.380 align:middle line:90% that I wanted to study it more. 00:14:09.380 --> 00:14:10.830 align:middle line:90% It's funny. 00:14:10.830 --> 00:14:14.190 align:middle line:84% You hate the subject because you don't understand it. 00:14:14.190 --> 00:14:17.360 align:middle line:84% And then you become to love it because you 00:14:17.360 --> 00:14:18.720 align:middle line:90% want to understand it. 00:14:18.720 --> 00:14:21.260 align:middle line:90% 00:14:21.260 --> 00:14:22.400 align:middle line:90% All right. 00:14:22.400 --> 00:14:28.280 align:middle line:84% But how does this definition match 00:14:28.280 --> 00:14:30.120 align:middle line:90% with our usual understanding? 00:14:30.120 --> 00:14:33.360 align:middle line:84% Because if I ask you what's heat, well, 00:14:33.360 --> 00:14:35.570 align:middle line:84% heat is the energy that is transferred 00:14:35.570 --> 00:14:39.860 align:middle line:84% between systems at different temperature, you would say. 00:14:39.860 --> 00:14:42.770 align:middle line:84% Well, we just said, no, heat is the energy 00:14:42.770 --> 00:14:48.570 align:middle line:84% that's transferred between systems at the same temperature. 00:14:48.570 --> 00:14:52.440 align:middle line:90% So is there something wrong? 00:14:52.440 --> 00:14:54.120 align:middle line:90% In fact, there isn't. 00:14:54.120 --> 00:14:57.240 align:middle line:84% Because, think of the typical model 00:14:57.240 --> 00:15:00.670 align:middle line:90% that we adopt in heat transfer. 00:15:00.670 --> 00:15:03.840 align:middle line:84% Suppose we have two systems at different temperatures. 00:15:03.840 --> 00:15:07.260 align:middle line:90% And we put them in contact. 00:15:07.260 --> 00:15:18.920 align:middle line:84% In the region of contact, their initial equilibrium, 00:15:18.920 --> 00:15:20.980 align:middle line:84% if they had a temperature initially, 00:15:20.980 --> 00:15:24.230 align:middle line:84% as soon as they get in contact, in the region of contact, 00:15:24.230 --> 00:15:27.010 align:middle line:84% you get a non-equilibrium situation. 00:15:27.010 --> 00:15:38.930 align:middle line:84% But we may describe this system as a composite of many smaller 00:15:38.930 --> 00:15:40.672 align:middle line:90% systems. 00:15:40.672 --> 00:15:42.830 align:middle line:84% In fluid mechanics, you would call them 00:15:42.830 --> 00:15:44.705 align:middle line:90% infinitesimal fluid elements. 00:15:44.705 --> 00:15:47.470 align:middle line:90% 00:15:47.470 --> 00:15:50.210 align:middle line:84% In a two-dimensional problem, you 00:15:50.210 --> 00:15:58.550 align:middle line:84% could have infinitesimal slices of your solid or body. 00:15:58.550 --> 00:16:01.280 align:middle line:90% And then you may assume that-- 00:16:01.280 --> 00:16:04.740 align:middle line:84% we will return, when we do non-equilibrium, 00:16:04.740 --> 00:16:07.040 align:middle line:84% we return very much to this idea-- 00:16:07.040 --> 00:16:13.670 align:middle line:84% we subdivide our system in such a way 00:16:13.670 --> 00:16:19.880 align:middle line:84% that each local subsystem is still 00:16:19.880 --> 00:16:23.570 align:middle line:90% in a non-equilibrium state. 00:16:23.570 --> 00:16:26.570 align:middle line:84% But it is not too far from equilibrium. 00:16:26.570 --> 00:16:30.090 align:middle line:84% So it turns out that we can assume 00:16:30.090 --> 00:16:37.990 align:middle line:84% that this is what we do when we describe a temperature 00:16:37.990 --> 00:16:42.577 align:middle line:90% distribution in our solid. 00:16:42.577 --> 00:16:44.410 align:middle line:84% Temperature distribution means that there is 00:16:44.410 --> 00:16:46.220 align:middle line:90% a temperature at each location. 00:16:46.220 --> 00:16:50.380 align:middle line:84% So that means that each infinitesimal element 00:16:50.380 --> 00:16:53.080 align:middle line:84% is close enough to a stable equilibrium state 00:16:53.080 --> 00:16:54.670 align:middle line:84% that you can assign the temperature 00:16:54.670 --> 00:16:56.085 align:middle line:84% of that stable equilibrium state. 00:16:56.085 --> 00:17:02.106 align:middle line:90% 00:17:02.106 --> 00:17:05.250 align:middle line:84% And the fact that the temperature distribution 00:17:05.250 --> 00:17:12.390 align:middle line:84% is typically, except extreme cases, continuous 00:17:12.390 --> 00:17:16.890 align:middle line:84% means that if you look at it from the small partitioning 00:17:16.890 --> 00:17:22.050 align:middle line:84% point of view, each element has a temperature, which 00:17:22.050 --> 00:17:27.359 align:middle line:84% differs only slightly from the adjacent element, and so on, 00:17:27.359 --> 00:17:29.010 align:middle line:90% by continuity. 00:17:29.010 --> 00:17:34.500 align:middle line:84% So each lamina, or each element, is at a temperature 00:17:34.500 --> 00:17:40.800 align:middle line:84% very close to the temperature of the near, of the neighbor. 00:17:40.800 --> 00:17:43.710 align:middle line:84% And therefore, the interaction between them 00:17:43.710 --> 00:17:46.320 align:middle line:84% is an interaction in which there is exchange 00:17:46.320 --> 00:17:51.690 align:middle line:84% of energy and entropy and in which the two adjacent elements 00:17:51.690 --> 00:17:54.810 align:middle line:84% have almost the same temperature, 00:17:54.810 --> 00:17:58.150 align:middle line:84% which fulfills our definition of a heat interaction. 00:17:58.150 --> 00:18:00.780 align:middle line:84% So locally, it is a heat interaction 00:18:00.780 --> 00:18:02.820 align:middle line:90% at the local temperature. 00:18:02.820 --> 00:18:04.560 align:middle line:84% And then with the other element, it's 00:18:04.560 --> 00:18:08.220 align:middle line:84% another interaction at a lower temperature or a higher 00:18:08.220 --> 00:18:09.430 align:middle line:90% temperature and so on. 00:18:09.430 --> 00:18:12.776 align:middle line:90% 00:18:12.776 --> 00:18:20.800 align:middle line:84% This describes what I just said for, say, a solid. 00:18:20.800 --> 00:18:23.410 align:middle line:90% 00:18:23.410 --> 00:18:28.830 align:middle line:84% So suppose we concentrate on these infinitesimal elements 00:18:28.830 --> 00:18:34.940 align:middle line:84% of solid or continuum in which the coordinate is only, 00:18:34.940 --> 00:18:38.740 align:middle line:84% let's say, one dimensional, coordinate is between x and x 00:18:38.740 --> 00:18:39.520 align:middle line:90% plus dx. 00:18:39.520 --> 00:18:43.030 align:middle line:90% So it's infinitesimally thin. 00:18:43.030 --> 00:18:47.380 align:middle line:84% And it is in contact with the element to the left 00:18:47.380 --> 00:18:49.900 align:middle line:90% and the element to the right. 00:18:49.900 --> 00:18:56.020 align:middle line:84% And for simplicity, let's just study the steady state situation 00:18:56.020 --> 00:19:00.405 align:middle line:84% here and make the energy balance and the entropy balance. 00:19:00.405 --> 00:19:03.290 align:middle line:90% 00:19:03.290 --> 00:19:09.350 align:middle line:84% So if I make the energy balance to this lamina, 00:19:09.350 --> 00:19:11.570 align:middle line:84% I should put on the left hand-side 00:19:11.570 --> 00:19:16.850 align:middle line:84% the rate of change of the energy of the lamina 00:19:16.850 --> 00:19:21.050 align:middle line:84% and, on the right hand side, the net contributions 00:19:21.050 --> 00:19:23.910 align:middle line:84% of the interactions to which that lamina is subjected. 00:19:23.910 --> 00:19:29.360 align:middle line:90% So it receives Q dot. 00:19:29.360 --> 00:19:31.770 align:middle line:84% Now, I'm assuming it is a heat interaction. 00:19:31.770 --> 00:19:35.060 align:middle line:84% So it receives Q dot from the left. 00:19:35.060 --> 00:19:38.630 align:middle line:90% And it gives Q dot to the right. 00:19:38.630 --> 00:19:43.130 align:middle line:84% If what comes into the left and goes out to the right 00:19:43.130 --> 00:19:49.390 align:middle line:84% is balanced, same rates, the energy balance 00:19:49.390 --> 00:19:53.740 align:middle line:84% says that the rate of change or the rate of accumulation 00:19:53.740 --> 00:19:55.650 align:middle line:90% of energy of my system is 0. 00:19:55.650 --> 00:19:58.845 align:middle line:84% So that property doesn't change with time. 00:19:58.845 --> 00:20:02.440 align:middle line:90% 00:20:02.440 --> 00:20:09.040 align:middle line:84% For entropy, the entropy balance should be written this way. 00:20:09.040 --> 00:20:12.680 align:middle line:84% Left-hand side, the rate of change of the property, 00:20:12.680 --> 00:20:15.730 align:middle line:84% so the rate of change of the entropy of-- 00:20:15.730 --> 00:20:19.090 align:middle line:90% 00:20:19.090 --> 00:20:22.930 align:middle line:90% this is our system, the lamina. 00:20:22.930 --> 00:20:24.830 align:middle line:84% On the right-hand side, I have two terms. 00:20:24.830 --> 00:20:30.910 align:middle line:84% The terms having to do with the net entropy received 00:20:30.910 --> 00:20:33.880 align:middle line:84% by interaction and the last term is 00:20:33.880 --> 00:20:37.540 align:middle line:84% the entropy produced by irreversibility 00:20:37.540 --> 00:20:38.715 align:middle line:90% within the system. 00:20:38.715 --> 00:20:43.010 align:middle line:90% 00:20:43.010 --> 00:20:48.730 align:middle line:84% So from the left, the entropy that is received, 00:20:48.730 --> 00:20:52.530 align:middle line:84% the rate with which entropy is received is Q dot over T. 00:20:52.530 --> 00:20:57.414 align:middle line:84% And T here is Tx, T of location x. 00:20:57.414 --> 00:21:00.640 align:middle line:84% From the right there is an outflow 00:21:00.640 --> 00:21:05.170 align:middle line:84% of entropy, which is Q over T. But here, the temperature 00:21:05.170 --> 00:21:05.780 align:middle line:90% is different. 00:21:05.780 --> 00:21:07.150 align:middle line:90% It's T at x plus dx-- 00:21:07.150 --> 00:21:10.300 align:middle line:90% 00:21:10.300 --> 00:21:13.330 align:middle line:84% one with a plus because it's entering, one with the minus 00:21:13.330 --> 00:21:15.460 align:middle line:90% because it's leaving. 00:21:15.460 --> 00:21:17.180 align:middle line:90% And that's the entropy balance. 00:21:17.180 --> 00:21:21.580 align:middle line:84% So if we assume a steady state, steady state 00:21:21.580 --> 00:21:25.720 align:middle line:84% means the state doesn't change with time. 00:21:25.720 --> 00:21:27.080 align:middle line:90% The state is what? 00:21:27.080 --> 00:21:28.480 align:middle line:90% It's the properties. 00:21:28.480 --> 00:21:31.270 align:middle line:84% So none of the properties changes 00:21:31.270 --> 00:21:32.810 align:middle line:90% with time, including entropy. 00:21:32.810 --> 00:21:37.660 align:middle line:84% So steady state implies that dS/dt is equal to 0. 00:21:37.660 --> 00:21:45.010 align:middle line:84% Then using this expression, we can compute the entropy 00:21:45.010 --> 00:21:47.510 align:middle line:90% produced by irreversibility. 00:21:47.510 --> 00:21:50.390 align:middle line:90% 00:21:50.390 --> 00:22:00.040 align:middle line:84% And we recognize here Tx and Tx plus dx. 00:22:00.040 --> 00:22:02.960 align:middle line:84% If I divide that by dx, that's an incremental ratio. 00:22:02.960 --> 00:22:06.975 align:middle line:84% So that's the derivative of T with respect to x. 00:22:06.975 --> 00:22:10.940 align:middle line:90% 00:22:10.940 --> 00:22:16.350 align:middle line:84% T squared at the denominator-- so this is the result. 00:22:16.350 --> 00:22:18.980 align:middle line:84% It says that the entropy produced by irreversibility is 00:22:18.980 --> 00:22:24.290 align:middle line:84% proportional to the heat that goes through divided by T 00:22:24.290 --> 00:22:27.420 align:middle line:90% squared and multiplied by the-- 00:22:27.420 --> 00:22:33.440 align:middle line:84% dT/dx is the gradient in temperature in the x direction 00:22:33.440 --> 00:22:37.120 align:middle line:90% with the minus sign in front. 00:22:37.120 --> 00:22:39.520 align:middle line:90% So this tells us a few things. 00:22:39.520 --> 00:22:45.630 align:middle line:84% First of all, it tells us in which 00:22:45.630 --> 00:22:52.410 align:middle line:84% direction Q can exist for a given temperature distribution 00:22:52.410 --> 00:22:55.800 align:middle line:84% because the entropy produced by irreversibility 00:22:55.800 --> 00:23:02.700 align:middle line:84% can only be positive or 0 if we were in a reversible case. 00:23:02.700 --> 00:23:05.670 align:middle line:90% But it cannot be negative. 00:23:05.670 --> 00:23:08.110 align:middle line:90% So it means that Q dot-- 00:23:08.110 --> 00:23:09.600 align:middle line:90% here, I have a T squared. 00:23:09.600 --> 00:23:14.310 align:middle line:84% So it means that Q dot must have opposite sign 00:23:14.310 --> 00:23:16.090 align:middle line:90% of the gradient in temperature. 00:23:16.090 --> 00:23:19.770 align:middle line:84% So the Q dot flows in the direction 00:23:19.770 --> 00:23:24.640 align:middle line:84% of decreasing temperature, which is, of course, what 00:23:24.640 --> 00:23:31.875 align:middle line:84% we knew, what Clausius told us in another way the other day. 00:23:31.875 --> 00:23:36.050 align:middle line:90% 00:23:36.050 --> 00:23:38.960 align:middle line:90% The point, though, is that-- 00:23:38.960 --> 00:23:42.740 align:middle line:84% let's look at what kind of model we're 00:23:42.740 --> 00:23:45.280 align:middle line:84% doing by assigning a temperature here. 00:23:45.280 --> 00:23:48.380 align:middle line:90% 00:23:48.380 --> 00:23:51.680 align:middle line:84% It's really an approximation that we are making. 00:23:51.680 --> 00:23:56.340 align:middle line:84% So in the whole of heat transfer and fluid mechanics, 00:23:56.340 --> 00:24:00.740 align:middle line:84% when we make the local equilibrium assumption, 00:24:00.740 --> 00:24:09.100 align:middle line:84% it is an approximation, which is good in order 00:24:09.100 --> 00:24:12.880 align:middle line:84% to assign local properties like temperature, pressure, chemical 00:24:12.880 --> 00:24:16.770 align:middle line:84% potentials to the fluid elements. 00:24:16.770 --> 00:24:20.400 align:middle line:84% But we should not take it too literally 00:24:20.400 --> 00:24:27.930 align:middle line:84% because, if we were truly in an equilibrium state, 00:24:27.930 --> 00:24:30.220 align:middle line:90% these things couldn't happen. 00:24:30.220 --> 00:24:33.150 align:middle line:90% And look at it this way. 00:24:33.150 --> 00:24:37.530 align:middle line:84% Let's say that this is the E versus S diagram of my system 00:24:37.530 --> 00:24:39.755 align:middle line:84% here, the lamina, the infinitesimal element. 00:24:39.755 --> 00:24:43.380 align:middle line:90% 00:24:43.380 --> 00:24:48.570 align:middle line:84% Now, yeah, I think I have it bigger here. 00:24:48.570 --> 00:24:56.240 align:middle line:90% Now, so the state-- 00:24:56.240 --> 00:24:59.930 align:middle line:84% I draw the state of this lamina just 00:24:59.930 --> 00:25:03.200 align:middle line:84% a bit off the stable equilibrium state curve. 00:25:03.200 --> 00:25:06.070 align:middle line:90% 00:25:06.070 --> 00:25:08.690 align:middle line:84% And let's see what happens there. 00:25:08.690 --> 00:25:09.810 align:middle line:90% So this is the state. 00:25:09.810 --> 00:25:12.580 align:middle line:90% 00:25:12.580 --> 00:25:19.140 align:middle line:84% The lamina receives both energy and entropy from the left. 00:25:19.140 --> 00:25:22.050 align:middle line:84% So that energy and entropy from the left 00:25:22.050 --> 00:25:24.065 align:middle line:90% that received from this side-- 00:25:24.065 --> 00:25:28.238 align:middle line:90% 00:25:28.238 --> 00:25:30.280 align:middle line:84% so the left-hand stuff-- from the left-hand side, 00:25:30.280 --> 00:25:35.190 align:middle line:84% there is a pull that tries to pull the state up, 00:25:35.190 --> 00:25:37.800 align:middle line:84% because it wants to receive energy, 00:25:37.800 --> 00:25:42.870 align:middle line:84% and to the right, because it receives an entropy given by Q 00:25:42.870 --> 00:25:50.280 align:middle line:84% dot over T. The net result is that the state is 00:25:50.280 --> 00:25:51.970 align:middle line:90% pulled in this direction. 00:25:51.970 --> 00:25:53.105 align:middle line:90% It wants to move there. 00:25:53.105 --> 00:25:56.930 align:middle line:90% 00:25:56.930 --> 00:26:01.350 align:middle line:84% On the other hand, there is also an interaction to the left-- 00:26:01.350 --> 00:26:05.360 align:middle line:84% I'm sorry, to the right, which takes things away. 00:26:05.360 --> 00:26:08.660 align:middle line:90% It takes energy down. 00:26:08.660 --> 00:26:11.990 align:middle line:84% So it pulls down the same amount. 00:26:11.990 --> 00:26:13.310 align:middle line:90% So these are balanced. 00:26:13.310 --> 00:26:18.770 align:middle line:84% And therefore, the state will remain at this level. 00:26:18.770 --> 00:26:27.200 align:middle line:84% But it pulls to the left a different value 00:26:27.200 --> 00:26:31.820 align:middle line:84% because the temperature is different. 00:26:31.820 --> 00:26:35.840 align:middle line:84% And here, at x plus dx, the temperature, 00:26:35.840 --> 00:26:43.280 align:middle line:84% I assume it's smaller, so this ratio, since it's a denominator. 00:26:43.280 --> 00:26:46.650 align:middle line:84% The smaller at the denominator means bigger. 00:26:46.650 --> 00:26:53.540 align:middle line:84% So it means that this pull to the left is higher than the pull 00:26:53.540 --> 00:26:54.340 align:middle line:90% to the right. 00:26:54.340 --> 00:26:57.830 align:middle line:90% So there is an unbalance. 00:26:57.830 --> 00:27:03.440 align:middle line:84% And if there weren't entropy produced by irreversibility, 00:27:03.440 --> 00:27:08.180 align:middle line:84% this state would follow the unbalance and go backwards. 00:27:08.180 --> 00:27:17.840 align:middle line:84% So that is to say that I need this irreversibility to be 00:27:17.840 --> 00:27:20.930 align:middle line:84% produced locally in order to maintain the steady state 00:27:20.930 --> 00:27:22.495 align:middle line:90% of that fluid element. 00:27:22.495 --> 00:27:25.400 align:middle line:90% 00:27:25.400 --> 00:27:29.100 align:middle line:84% But in order to produce entropy by irreversibility, 00:27:29.100 --> 00:27:32.750 align:middle line:84% I must not be at stable equilibrium 00:27:32.750 --> 00:27:35.300 align:middle line:84% because, at stable equilibrium, the system is happy. 00:27:35.300 --> 00:27:41.300 align:middle line:84% And it doesn't rearrange things in order-- it already 00:27:41.300 --> 00:27:42.870 align:middle line:90% is in the maximum entropy. 00:27:42.870 --> 00:27:47.600 align:middle line:84% There is no tendency to produce more. 00:27:47.600 --> 00:27:54.530 align:middle line:84% So this is to say that there is a bit of a contradiction, 00:27:54.530 --> 00:27:58.040 align:middle line:84% or at least that we have to be careful in understanding what 00:27:58.040 --> 00:28:00.140 align:middle line:90% the actual assumptions are. 00:28:00.140 --> 00:28:03.120 align:middle line:84% This state is a non-equilibrium state. 00:28:03.120 --> 00:28:06.360 align:middle line:84% Otherwise, it couldn't produce entropy by irreversibility. 00:28:06.360 --> 00:28:09.420 align:middle line:84% But it is close to a stable equilibrium state. 00:28:09.420 --> 00:28:16.100 align:middle line:84% And therefore, we can use the idea of temperature 00:28:16.100 --> 00:28:18.280 align:middle line:84% and assign it to the fluid element. 00:28:18.280 --> 00:28:20.960 align:middle line:84% But indeed, what it is is the temperature that 00:28:20.960 --> 00:28:23.330 align:middle line:84% would be reached if I took that fluid element 00:28:23.330 --> 00:28:26.330 align:middle line:84% and isolate it from the rest and let 00:28:26.330 --> 00:28:32.600 align:middle line:84% it relax to equilibrium because if I shut the interactions left 00:28:32.600 --> 00:28:35.270 align:middle line:84% and right, this non-equilibrium state 00:28:35.270 --> 00:28:40.940 align:middle line:84% will spontaneously move to stable equilibrium. 00:28:40.940 --> 00:28:45.290 align:middle line:84% And then it will have a temperature. 00:28:45.290 --> 00:28:47.600 align:middle line:84% And that's the temperature that we assign 00:28:47.600 --> 00:28:49.200 align:middle line:90% to this non-equilibrium state. 00:28:49.200 --> 00:28:52.700 align:middle line:90% 00:28:52.700 --> 00:28:57.770 align:middle line:84% This is interesting because it is the local equilibrium 00:28:57.770 --> 00:29:04.200 align:middle line:84% model that we are used to using in fluid mechanics. 00:29:04.200 --> 00:29:08.990 align:middle line:90% But you can improve. 00:29:08.990 --> 00:29:14.180 align:middle line:84% And people are doing extensions of fluid mechanics treatment 00:29:14.180 --> 00:29:20.420 align:middle line:84% in which you deal with this little difference in entropy 00:29:20.420 --> 00:29:24.540 align:middle line:84% because maybe you can assign a temperature, 00:29:24.540 --> 00:29:26.840 align:middle line:84% but the entropy of this state is slightly 00:29:26.840 --> 00:29:32.210 align:middle line:84% less than the entropy of a truly equilibrium state 00:29:32.210 --> 00:29:33.390 align:middle line:90% at that temperature. 00:29:33.390 --> 00:29:40.730 align:middle line:84% And that difference in entropy does sometimes-- 00:29:40.730 --> 00:29:48.990 align:middle line:84% has an importance for certain applications. 00:29:48.990 --> 00:29:51.790 align:middle line:90% 00:29:51.790 --> 00:29:52.290 align:middle line:90% OK. 00:29:52.290 --> 00:29:57.730 align:middle line:90% 00:29:57.730 --> 00:30:05.670 align:middle line:84% So now I want to talk about these very traditional ideas 00:30:05.670 --> 00:30:10.530 align:middle line:84% of heat engines, heat pumps, and refrigeration units, 00:30:10.530 --> 00:30:12.850 align:middle line:84% and see the energy and entropy that flows. 00:30:12.850 --> 00:30:17.220 align:middle line:84% And also, it gives us the opportunity 00:30:17.220 --> 00:30:20.940 align:middle line:84% to define what we mean by first and second law efficiency 00:30:20.940 --> 00:30:22.450 align:middle line:90% of devices like these. 00:30:22.450 --> 00:30:28.120 align:middle line:90% 00:30:28.120 --> 00:30:34.040 align:middle line:84% So let's start from what we are seeing. 00:30:34.040 --> 00:30:38.590 align:middle line:90% So let's do it our own way. 00:30:38.590 --> 00:30:43.180 align:middle line:84% Suppose we take the Clausius inequalities 00:30:43.180 --> 00:30:46.690 align:middle line:84% that we have seen the other day and remember 00:30:46.690 --> 00:30:56.770 align:middle line:84% that these subscripts over the extreme inequalities 00:30:56.770 --> 00:31:00.010 align:middle line:84% indicate what the condition should 00:31:00.010 --> 00:31:03.830 align:middle line:84% be for the strict equality to hold. 00:31:03.830 --> 00:31:11.260 align:middle line:84% And here, if you remember, we have a strict equality 00:31:11.260 --> 00:31:17.200 align:middle line:84% if the system is in a stable equilibrium state-- 00:31:17.200 --> 00:31:19.210 align:middle line:84% ends in a stable equilibrium state and it 00:31:19.210 --> 00:31:24.280 align:middle line:84% is a reservoir, so that the second derivative is equal to 0. 00:31:24.280 --> 00:31:28.270 align:middle line:84% So let's consider our system to be 00:31:28.270 --> 00:31:32.210 align:middle line:90% a reservoir at temperature TA. 00:31:32.210 --> 00:31:35.050 align:middle line:90% 00:31:35.050 --> 00:31:38.890 align:middle line:84% I dropped the subscript 1 because, for a reservoir, 00:31:38.890 --> 00:31:43.600 align:middle line:84% there is always one temperature for all states. 00:31:43.600 --> 00:31:46.780 align:middle line:84% And let's assume that also system B is a reservoir. 00:31:46.780 --> 00:31:57.490 align:middle line:84% So also for B here, I have an equal sign 00:31:57.490 --> 00:32:00.610 align:middle line:84% because I also assume that the final states are stable 00:32:00.610 --> 00:32:02.560 align:middle line:90% equilibrium. 00:32:02.560 --> 00:32:03.140 align:middle line:90% All right. 00:32:03.140 --> 00:32:06.460 align:middle line:84% So then either this or that inequality-- 00:32:06.460 --> 00:32:09.050 align:middle line:84% if the two temperatures are different-- 00:32:09.050 --> 00:32:11.950 align:middle line:84% and here, we would like to consider reservoirs 00:32:11.950 --> 00:32:16.820 align:middle line:84% at different temperatures, even large differences-- 00:32:16.820 --> 00:32:19.840 align:middle line:90% 00:32:19.840 --> 00:32:25.390 align:middle line:84% there is no way that this left extreme can 00:32:25.390 --> 00:32:28.090 align:middle line:90% be equal to the right extreme. 00:32:28.090 --> 00:32:38.200 align:middle line:84% And therefore, somewhere, there must be some irreversibility, 00:32:38.200 --> 00:32:42.100 align:middle line:84% either in system A or in system B, 00:32:42.100 --> 00:32:51.100 align:middle line:84% so as to make this train of equalities and inequalities 00:32:51.100 --> 00:32:52.210 align:middle line:90% satisfied. 00:32:52.210 --> 00:32:57.830 align:middle line:84% So it is necessary-- if I transfer energy 00:32:57.830 --> 00:33:00.230 align:middle line:84% from a reservoir to another reservoir 00:33:00.230 --> 00:33:02.900 align:middle line:84% at a different temperature, it is necessary 00:33:02.900 --> 00:33:08.055 align:middle line:84% that some entropy be generated by irreversibility in either one 00:33:08.055 --> 00:33:09.805 align:middle line:84% because there are no other systems around. 00:33:09.805 --> 00:33:12.380 align:middle line:90% 00:33:12.380 --> 00:33:22.845 align:middle line:84% But suppose we interpose a machinery between that exchange. 00:33:22.845 --> 00:33:25.510 align:middle line:90% 00:33:25.510 --> 00:33:32.770 align:middle line:84% Well, in that case, it is possible to have things-- 00:33:32.770 --> 00:33:36.610 align:middle line:84% to operate without irreversibility, 00:33:36.610 --> 00:33:38.726 align:middle line:90% so without entropy production. 00:33:38.726 --> 00:33:44.950 align:middle line:90% 00:33:44.950 --> 00:33:49.240 align:middle line:84% This is the entropy balance for the machinery. 00:33:49.240 --> 00:33:55.900 align:middle line:84% It says that, well, if the machinery produces some entropy 00:33:55.900 --> 00:34:01.060 align:middle line:84% by irreversibility, it must be equal to the difference 00:34:01.060 --> 00:34:05.205 align:middle line:84% between what it puts out from what it receives. 00:34:05.205 --> 00:34:08.030 align:middle line:90% 00:34:08.030 --> 00:34:11.219 align:middle line:84% And this is the energy balance for the machinery. 00:34:11.219 --> 00:34:19.320 align:middle line:84% It says that it can produce work if it 00:34:19.320 --> 00:34:28.139 align:middle line:84% can manage to give to reservoir B less energy 00:34:28.139 --> 00:34:33.679 align:middle line:84% than it receives from reservoir A. 00:34:33.679 --> 00:34:45.540 align:middle line:84% So we can assume that everything-- it is 00:34:45.540 --> 00:34:48.120 align:middle line:84% possible to assume that everything has 00:34:48.120 --> 00:34:51.480 align:middle line:84% no entropy produced by irreversibility, 00:34:51.480 --> 00:34:59.520 align:middle line:84% so no delta S irr in system, no delta S irr in system B, 00:34:59.520 --> 00:35:04.350 align:middle line:84% and also no delta S irr in system X. 00:35:04.350 --> 00:35:09.010 align:middle line:84% In that case, from the entropy point of view, 00:35:09.010 --> 00:35:11.400 align:middle line:84% the machinery simply passes the entropy that 00:35:11.400 --> 00:35:17.560 align:middle line:84% comes from reservoir A and gives it to reservoir B. 00:35:17.560 --> 00:35:22.900 align:middle line:84% But it also recognizes that the machinery is made there just 00:35:22.900 --> 00:35:30.220 align:middle line:84% to recognize that, in order to pass that entropy to system B, 00:35:30.220 --> 00:35:34.000 align:middle line:84% it doesn't need to give it all the energy it received 00:35:34.000 --> 00:35:42.100 align:middle line:84% from A. There is an unbalance, the difference, which can be 00:35:42.100 --> 00:35:47.290 align:middle line:84% transferred out by means of an interaction in which there 00:35:47.290 --> 00:35:49.245 align:middle line:90% is no entropy transfer, work. 00:35:49.245 --> 00:35:52.790 align:middle line:90% 00:35:52.790 --> 00:36:02.430 align:middle line:84% OK, so that's the idea of the heat engine, machinery 00:36:02.430 --> 00:36:06.700 align:middle line:84% that receives heat from a reservoir at high temperature. 00:36:06.700 --> 00:36:09.270 align:middle line:84% This is a heat engine between thermal reservoirs because you 00:36:09.270 --> 00:36:12.120 align:middle line:84% could have also heat engines between systems 00:36:12.120 --> 00:36:14.910 align:middle line:90% that are not reservoirs. 00:36:14.910 --> 00:36:21.910 align:middle line:84% I take a big piece of steel out of a steelmaking plant. 00:36:21.910 --> 00:36:24.660 align:middle line:84% And I want to recuperate the energy that it 00:36:24.660 --> 00:36:27.060 align:middle line:90% has before it cools down. 00:36:27.060 --> 00:36:33.390 align:middle line:84% That's a way of recovering energy 00:36:33.390 --> 00:36:34.990 align:middle line:90% from industrial processes. 00:36:34.990 --> 00:36:39.660 align:middle line:84% And of course, as I call it down, the piece of steel 00:36:39.660 --> 00:36:41.590 align:middle line:90% decreases its temperature. 00:36:41.590 --> 00:36:45.470 align:middle line:90% And so it's not a reservoir. 00:36:45.470 --> 00:36:49.620 align:middle line:84% That doesn't mean we cannot compute also for that situation 00:36:49.620 --> 00:36:52.680 align:middle line:84% what is the maximum work that we can extract. 00:36:52.680 --> 00:36:56.660 align:middle line:90% But that's another situation. 00:36:56.660 --> 00:36:58.490 align:middle line:84% And for example, the maximum work 00:36:58.490 --> 00:37:04.850 align:middle line:84% we can extract in that case is not the Carnot coefficient. 00:37:04.850 --> 00:37:09.030 align:middle line:84% Whereas here, if you do energy and entropy balance-- 00:37:09.030 --> 00:37:11.090 align:middle line:90% and here they are. 00:37:11.090 --> 00:37:13.530 align:middle line:90% I'm going quickly here. 00:37:13.530 --> 00:37:15.620 align:middle line:84% And I'm not going too into the details, 00:37:15.620 --> 00:37:19.640 align:middle line:84% just give you some ideas by following 00:37:19.640 --> 00:37:21.650 align:middle line:90% the flows of the entropy. 00:37:21.650 --> 00:37:25.430 align:middle line:84% But here are the full glory of the balance 00:37:25.430 --> 00:37:33.110 align:middle line:84% equations for the machinery, from which 00:37:33.110 --> 00:37:36.235 align:middle line:90% you can compute the work. 00:37:36.235 --> 00:37:38.990 align:middle line:90% 00:37:38.990 --> 00:37:42.650 align:middle line:84% Here is the result. So the work that the machinery-- the useful 00:37:42.650 --> 00:37:44.790 align:middle line:84% effect that the machinery can produce 00:37:44.790 --> 00:37:49.980 align:middle line:84% is equal to a fraction of the heat 00:37:49.980 --> 00:37:52.650 align:middle line:84% that it receives from the hot reservoir. 00:37:52.650 --> 00:37:56.820 align:middle line:84% And that fraction is the Carnot coefficient 00:37:56.820 --> 00:38:00.950 align:middle line:90% between those two temperatures. 00:38:00.950 --> 00:38:05.790 align:middle line:84% And then there is also a term which detracts. 00:38:05.790 --> 00:38:07.820 align:middle line:84% And that is proportional to the entropy 00:38:07.820 --> 00:38:09.320 align:middle line:84% that is produced by irreversibility 00:38:09.320 --> 00:38:11.600 align:middle line:90% within the machinery. 00:38:11.600 --> 00:38:12.840 align:middle line:90% It has a minus sign. 00:38:12.840 --> 00:38:17.420 align:middle line:84% So if the machinery does produce entropy, 00:38:17.420 --> 00:38:21.080 align:middle line:84% that will detract and will allow you to get 00:38:21.080 --> 00:38:23.065 align:middle line:90% less work than the maximum. 00:38:23.065 --> 00:38:34.630 align:middle line:90% 00:38:34.630 --> 00:38:40.630 align:middle line:84% Let's view it from the point of view of entropy flows. 00:38:40.630 --> 00:38:49.960 align:middle line:84% Actually, maybe we can also do it 00:38:49.960 --> 00:38:54.690 align:middle line:90% in terms of E versus S diagrams. 00:38:54.690 --> 00:38:58.850 align:middle line:90% 00:38:58.850 --> 00:39:06.680 align:middle line:84% So this is system A, reservoir A. And this is reservoir B. 00:39:06.680 --> 00:39:09.770 align:middle line:90% And I'm extracting some energy. 00:39:09.770 --> 00:39:25.830 align:middle line:84% So I want to take this much energy out of system A. 00:39:25.830 --> 00:39:28.810 align:middle line:84% Clearly, there are no states here. 00:39:28.810 --> 00:39:34.565 align:middle line:84% So I need to extract also some entropy, at least this much. 00:39:34.565 --> 00:39:37.380 align:middle line:90% 00:39:37.380 --> 00:39:42.730 align:middle line:84% But in order to extract that entropy, I need another system-- 00:39:42.730 --> 00:39:46.260 align:middle line:84% that's why I need the reservoir B-- 00:39:46.260 --> 00:39:52.000 align:middle line:90% that picks up that entropy. 00:39:52.000 --> 00:39:53.550 align:middle line:90% And so it will end up here. 00:39:53.550 --> 00:39:59.050 align:middle line:90% 00:39:59.050 --> 00:40:03.280 align:middle line:90% It will end up here, provided-- 00:40:03.280 --> 00:40:06.850 align:middle line:84% since there are no states to the right of the maximum entropy, 00:40:06.850 --> 00:40:08.840 align:middle line:84% I need also to provide some energy. 00:40:08.840 --> 00:40:12.730 align:middle line:90% And that energy is this much. 00:40:12.730 --> 00:40:24.850 align:middle line:84% It's Q received by B. So the difference between this and that 00:40:24.850 --> 00:40:25.825 align:middle line:90% is the work. 00:40:25.825 --> 00:40:29.230 align:middle line:90% 00:40:29.230 --> 00:40:37.540 align:middle line:84% If I draw a line with the slope of reservoir A here, 00:40:37.540 --> 00:40:43.970 align:middle line:84% this is the work that I have extracted. 00:40:43.970 --> 00:40:47.380 align:middle line:84% Actually, this is W rev because I 00:40:47.380 --> 00:40:55.810 align:middle line:84% have assumed that system B takes just barely the entropy 00:40:55.810 --> 00:40:57.940 align:middle line:90% that we need to-- 00:40:57.940 --> 00:41:02.140 align:middle line:84% that is being received by system A. So it flows-- 00:41:02.140 --> 00:41:08.795 align:middle line:84% the entropy comes from A. The machinery doesn't 00:41:08.795 --> 00:41:11.480 align:middle line:84% add because it's reversible, it doesn't 00:41:11.480 --> 00:41:13.200 align:middle line:90% produce additional entropy. 00:41:13.200 --> 00:41:20.570 align:middle line:84% It simply has to transfer it to system B. 00:41:20.570 --> 00:41:29.000 align:middle line:84% And the energy that is required to transfer that to system B 00:41:29.000 --> 00:41:31.020 align:middle line:90% is less than QA. 00:41:31.020 --> 00:41:32.790 align:middle line:84% And therefore, there is a difference. 00:41:32.790 --> 00:41:35.900 align:middle line:84% And that difference can be transferred 00:41:35.900 --> 00:41:39.110 align:middle line:84% in an interaction where there is no entropy transfer, which 00:41:39.110 --> 00:41:40.280 align:middle line:90% is work. 00:41:40.280 --> 00:41:43.260 align:middle line:84% So this is the reversible operation. 00:41:43.260 --> 00:41:46.710 align:middle line:84% If we have an irreversible operation, 00:41:46.710 --> 00:41:49.045 align:middle line:84% so if the machinery is irreversible, 00:41:49.045 --> 00:41:52.580 align:middle line:84% like most practical machineries, and therefore it 00:41:52.580 --> 00:41:55.670 align:middle line:84% produces some entropy, that entropy will also 00:41:55.670 --> 00:42:01.880 align:middle line:84% need to be disposed of into the environmental reservoir. 00:42:01.880 --> 00:42:09.780 align:middle line:84% And therefore, it will require more energy to follow. 00:42:09.780 --> 00:42:14.790 align:middle line:84% So in the system B, I need to add not only this entropy, 00:42:14.790 --> 00:42:19.240 align:middle line:84% but also the entropy that is produced by irreversibility. 00:42:19.240 --> 00:42:23.220 align:middle line:84% So that is S irr produced in the machinery. 00:42:23.220 --> 00:42:29.975 align:middle line:84% And the machinery is called M. So system B ends up here. 00:42:29.975 --> 00:42:34.120 align:middle line:90% 00:42:34.120 --> 00:42:39.820 align:middle line:90% The amount of Q is higher. 00:42:39.820 --> 00:42:46.300 align:middle line:84% And therefore, the amount left, the difference, 00:42:46.300 --> 00:42:50.910 align:middle line:84% which is what will be transferred as work, is less. 00:42:50.910 --> 00:42:55.420 align:middle line:90% 00:42:55.420 --> 00:42:57.610 align:middle line:90% Good. 00:42:57.610 --> 00:43:00.410 align:middle line:84% So we have seen it in various ways. 00:43:00.410 --> 00:43:03.966 align:middle line:84% Now, typically, one defines the efficiency. 00:43:03.966 --> 00:43:07.170 align:middle line:90% 00:43:07.170 --> 00:43:14.070 align:middle line:84% Actually, most of you probably remember 00:43:14.070 --> 00:43:19.080 align:middle line:84% that this ratio is not called the Carnot coefficient, 00:43:19.080 --> 00:43:20.140 align:middle line:90% like I'm calling it. 00:43:20.140 --> 00:43:23.580 align:middle line:84% But it's called Carnot efficiency, 00:43:23.580 --> 00:43:30.720 align:middle line:84% which is true in the sense that I can define an efficiency-- 00:43:30.720 --> 00:43:33.756 align:middle line:84% we are going to call it here first-law efficiency-- 00:43:33.756 --> 00:43:36.490 align:middle line:90% 00:43:36.490 --> 00:43:38.570 align:middle line:84% as the ratio of the useful work-- 00:43:38.570 --> 00:43:46.180 align:middle line:84% so the work-- divided by the energy of the resource 00:43:46.180 --> 00:43:47.420 align:middle line:90% that you have used. 00:43:47.420 --> 00:43:50.680 align:middle line:84% In this case, the resource is the hot reservoir, 00:43:50.680 --> 00:43:53.050 align:middle line:84% so the energy extracted from the hot reservoir. 00:43:53.050 --> 00:43:58.720 align:middle line:84% So this ratio is the first-law efficiency. 00:43:58.720 --> 00:44:01.150 align:middle line:84% And the result of the energy and entropy balance 00:44:01.150 --> 00:44:06.460 align:middle line:84% shows that the Carnot coefficient 00:44:06.460 --> 00:44:12.400 align:middle line:84% is the upper limit of this efficiency, which is reached 00:44:12.400 --> 00:44:15.610 align:middle line:90% when S here is equal to 0. 00:44:15.610 --> 00:44:20.290 align:middle line:84% So the Carnot coefficient is equal to the first law 00:44:20.290 --> 00:44:23.610 align:middle line:84% efficiency for the reversible engine. 00:44:23.610 --> 00:44:27.990 align:middle line:90% 00:44:27.990 --> 00:44:39.360 align:middle line:84% What is somehow inconvenient of first-law efficiencies 00:44:39.360 --> 00:44:48.870 align:middle line:84% is that usually when you say, something is 50% efficient, 00:44:48.870 --> 00:44:52.620 align:middle line:84% I would say, well, why aren't you doing better? 00:44:52.620 --> 00:44:56.260 align:middle line:84% There is 50% room for improvement. 00:44:56.260 --> 00:44:59.490 align:middle line:90% 00:44:59.490 --> 00:45:03.540 align:middle line:84% Not here though, because there is no way-- 00:45:03.540 --> 00:45:09.760 align:middle line:84% suppose you have TA is equal to 600K and TB is 300K. 00:45:09.760 --> 00:45:12.960 align:middle line:84% So the Carnot coefficient is 1/2, 00:45:12.960 --> 00:45:16.990 align:middle line:84% 50% That's the maximum you can do. 00:45:16.990 --> 00:45:19.060 align:middle line:90% There is no improvement. 00:45:19.060 --> 00:45:21.840 align:middle line:90% You cannot improve. 00:45:21.840 --> 00:45:29.210 align:middle line:84% So it would be better to have to call efficiency or to-- 00:45:29.210 --> 00:45:31.980 align:middle line:84% so people have introduced the second-law efficiency, 00:45:31.980 --> 00:45:37.460 align:middle line:84% which compares what you actually obtained and put it 00:45:37.460 --> 00:45:42.470 align:middle line:84% in a numerator to the maximum that you could have obtained, 00:45:42.470 --> 00:45:46.340 align:middle line:84% given the constraints and the physical laws, 00:45:46.340 --> 00:45:47.875 align:middle line:90% so without violating anything. 00:45:47.875 --> 00:45:50.990 align:middle line:90% 00:45:50.990 --> 00:45:57.180 align:middle line:84% So in that case, this ratio, for the reversible case, is 1. 00:45:57.180 --> 00:46:03.810 align:middle line:84% So it's 100% And so if the efficiency is 100%, 00:46:03.810 --> 00:46:05.370 align:middle line:90% there's no room for improvement. 00:46:05.370 --> 00:46:08.940 align:middle line:90% 00:46:08.940 --> 00:46:16.170 align:middle line:84% If it is not 100% or 1, it means that there 00:46:16.170 --> 00:46:19.320 align:middle line:90% is something that reduces that. 00:46:19.320 --> 00:46:22.950 align:middle line:84% And that something is the entropy 00:46:22.950 --> 00:46:24.610 align:middle line:90% produced by irreversibility. 00:46:24.610 --> 00:46:31.440 align:middle line:84% So by thinking in these terms, this second-law efficiency 00:46:31.440 --> 00:46:42.960 align:middle line:84% allows you to identify the actual places in your machinery 00:46:42.960 --> 00:46:52.140 align:middle line:84% or power plant that are the sites where entropy gets 00:46:52.140 --> 00:46:56.340 align:middle line:84% produced by irreversibility because the higher the entropy 00:46:56.340 --> 00:47:00.220 align:middle line:84% produced by irreversibility, the lesser this efficiency, 00:47:00.220 --> 00:47:03.780 align:middle line:84% so the more room for improvement. 00:47:03.780 --> 00:47:10.300 align:middle line:84% And in fact, a lot of energy technology engineering 00:47:10.300 --> 00:47:16.070 align:middle line:84% is into improving this efficiency. 00:47:16.070 --> 00:47:19.240 align:middle line:84% That means identifying where the irreversibilities are 00:47:19.240 --> 00:47:22.580 align:middle line:84% and removing them or reducing them, optimizing 00:47:22.580 --> 00:47:23.580 align:middle line:90% from that point of view. 00:47:23.580 --> 00:47:28.256 align:middle line:90% 00:47:28.256 --> 00:47:36.930 align:middle line:84% So let's do, even more quickly, the same idea, 00:47:36.930 --> 00:47:38.695 align:middle line:90% but for a refrigerator unit. 00:47:38.695 --> 00:47:41.250 align:middle line:90% 00:47:41.250 --> 00:47:44.970 align:middle line:84% It's again a situation in which I have two reservoirs, TA 00:47:44.970 --> 00:47:47.130 align:middle line:90% and TB. 00:47:47.130 --> 00:47:58.015 align:middle line:84% And I identify the environment as the system 00:47:58.015 --> 00:47:59.760 align:middle line:90% at a higher temperature now. 00:47:59.760 --> 00:48:02.740 align:middle line:90% 00:48:02.740 --> 00:48:09.460 align:middle line:84% And the other system, at the lower temperature 00:48:09.460 --> 00:48:16.270 align:middle line:84% is, for example, the refrigerator cell, the room 00:48:16.270 --> 00:48:18.530 align:middle line:84% that you want to keep refrigerated. 00:48:18.530 --> 00:48:21.620 align:middle line:84% And you want to keep it refrigerated. 00:48:21.620 --> 00:48:25.810 align:middle line:84% So since typically there are heat losses 00:48:25.810 --> 00:48:29.720 align:middle line:84% from the walls of this refrigerator, 00:48:29.720 --> 00:48:34.060 align:middle line:84% you need to compensate by removing heat. 00:48:34.060 --> 00:48:39.550 align:middle line:84% So the unit wants to extract heat from a low temperature, 00:48:39.550 --> 00:48:41.530 align:middle line:84% from a temperature from a reservoir which 00:48:41.530 --> 00:48:44.266 align:middle line:84% is at lower temperature than the environment. 00:48:44.266 --> 00:48:49.050 align:middle line:90% 00:48:49.050 --> 00:48:50.915 align:middle line:90% So if we go on our-- 00:48:50.915 --> 00:48:56.390 align:middle line:90% 00:48:56.390 --> 00:49:04.790 align:middle line:84% do again the construction here, now I 00:49:04.790 --> 00:49:12.320 align:middle line:84% want my system B, which is the refrigerated room-- 00:49:12.320 --> 00:49:18.560 align:middle line:90% you want to extract energy. 00:49:18.560 --> 00:49:22.710 align:middle line:84% And as usual, in order to extract energy, 00:49:22.710 --> 00:49:25.010 align:middle line:84% you need to extract also entropy. 00:49:25.010 --> 00:49:28.130 align:middle line:90% So this is QB out. 00:49:28.130 --> 00:49:37.510 align:middle line:90% And this is SB out. 00:49:37.510 --> 00:49:43.280 align:middle line:84% So if I have this entropy going somewhere, 00:49:43.280 --> 00:49:46.515 align:middle line:84% I need to put it to another system. 00:49:46.515 --> 00:49:50.400 align:middle line:90% 00:49:50.400 --> 00:49:57.850 align:middle line:84% Now, this system A, which is my environment, is there. 00:49:57.850 --> 00:50:07.680 align:middle line:84% And I would like to give it this much entropy. 00:50:07.680 --> 00:50:11.310 align:middle line:84% But in order to give it, I also need 00:50:11.310 --> 00:50:21.070 align:middle line:84% to give some energy, which is actually higher than this one 00:50:21.070 --> 00:50:25.880 align:middle line:84% because the environment is hotter. 00:50:25.880 --> 00:50:27.490 align:middle line:90% So it has a higher slope. 00:50:27.490 --> 00:50:36.200 align:middle line:84% So if I plot here the slope of system B, 00:50:36.200 --> 00:50:45.995 align:middle line:84% this is the energy that I extract from system B. 00:50:45.995 --> 00:50:49.940 align:middle line:84% But I need to give to A more energy in order 00:50:49.940 --> 00:50:52.280 align:middle line:84% for it to accommodate the entropy that I 00:50:52.280 --> 00:50:59.060 align:middle line:84% must extract from B. So this much is the work 00:50:59.060 --> 00:51:00.385 align:middle line:90% that I need to spend. 00:51:00.385 --> 00:51:10.345 align:middle line:90% 00:51:10.345 --> 00:51:14.380 align:middle line:84% And of course, if the machinery is irreversible 00:51:14.380 --> 00:51:17.420 align:middle line:84% and it produces also some entropy by irreversibility, 00:51:17.420 --> 00:51:21.580 align:middle line:84% not only I must accommodate this much entropy, 00:51:21.580 --> 00:51:25.630 align:middle line:84% but I must accommodate even more. 00:51:25.630 --> 00:51:34.400 align:middle line:84% That's the irreversibility of the machinery, this much. 00:51:34.400 --> 00:51:36.860 align:middle line:90% So system A must go up there. 00:51:36.860 --> 00:51:40.650 align:middle line:84% And therefore, I need to spend more work. 00:51:40.650 --> 00:51:49.590 align:middle line:84% So this was W rev. And this match is W, say, irreversible. 00:51:49.590 --> 00:51:55.191 align:middle line:90% 00:51:55.191 --> 00:52:03.030 align:middle line:84% Now also here, the Carnot coefficient or similar 00:52:03.030 --> 00:52:06.900 align:middle line:90% does play a role. 00:52:06.900 --> 00:52:11.010 align:middle line:84% Usually, in practice, when you buy a refrigerator, 00:52:11.010 --> 00:52:17.760 align:middle line:84% you want to know what the COP, the coefficient of performance, 00:52:17.760 --> 00:52:21.240 align:middle line:90% is, which is a first-law ratio. 00:52:21.240 --> 00:52:26.280 align:middle line:84% It's the ratio of the useful, the cooling effect, which 00:52:26.280 --> 00:52:30.780 align:middle line:84% is the energy you extract from the low temperature cell, 00:52:30.780 --> 00:52:35.760 align:middle line:84% to the work, typically electrical work, 00:52:35.760 --> 00:52:41.375 align:middle line:84% that you need to spend in order to do this. 00:52:41.375 --> 00:52:46.130 align:middle line:90% 00:52:46.130 --> 00:52:54.530 align:middle line:84% And this COP has a highest value. 00:52:54.530 --> 00:52:57.930 align:middle line:84% And then there is a term, which is always positive. 00:52:57.930 --> 00:53:03.630 align:middle line:84% And that is bigger and bigger the more 00:53:03.630 --> 00:53:05.460 align:middle line:90% irreversible is your machinery. 00:53:05.460 --> 00:53:09.938 align:middle line:90% 00:53:09.938 --> 00:53:15.365 align:middle line:84% Also here, COP is typically a number greater than 1. 00:53:15.365 --> 00:53:18.322 align:middle line:90% 00:53:18.322 --> 00:53:21.680 align:middle line:84% It doesn't give you an indication of what 00:53:21.680 --> 00:53:26.510 align:middle line:90% the room for improvement is. 00:53:26.510 --> 00:53:31.250 align:middle line:84% The second-law efficiency instead does. 00:53:31.250 --> 00:53:37.730 align:middle line:84% And again, we can define it as the actual useful effect divided 00:53:37.730 --> 00:53:44.780 align:middle line:84% by the best useful effect, the maximum useful effect you 00:53:44.780 --> 00:53:49.870 align:middle line:84% could obtain with a reversible machinery. 00:53:49.870 --> 00:53:58.570 align:middle line:84% That too has a maximum value equal to 100%. 00:53:58.570 --> 00:54:01.180 align:middle line:84% In that case, there is no room for improvement. 00:54:01.180 --> 00:54:03.740 align:middle line:90% 00:54:03.740 --> 00:54:07.630 align:middle line:84% And it's less than 1, due to the entropy produced 00:54:07.630 --> 00:54:08.870 align:middle line:90% by irreversibility. 00:54:08.870 --> 00:54:12.700 align:middle line:90% 00:54:12.700 --> 00:54:22.330 align:middle line:84% The last of this series of traditional machineries 00:54:22.330 --> 00:54:25.360 align:middle line:90% is the heat pump. 00:54:25.360 --> 00:54:29.950 align:middle line:84% The heat pump is very similar to a refrigerator, more or less 00:54:29.950 --> 00:54:36.190 align:middle line:84% the same equations, except that now what switches 00:54:36.190 --> 00:54:39.010 align:middle line:84% is the position of the environment, 00:54:39.010 --> 00:54:43.000 align:middle line:84% of your environment, the environmental reservoir. 00:54:43.000 --> 00:54:50.530 align:middle line:84% Whereas for the refrigerator you were extracting energy 00:54:50.530 --> 00:54:53.830 align:middle line:84% from something which is colder than the environment, 00:54:53.830 --> 00:54:58.900 align:middle line:84% in the heat pump, you want to put energy 00:54:58.900 --> 00:55:03.650 align:middle line:84% in a reservoir which is hotter than the environment. 00:55:03.650 --> 00:55:06.230 align:middle line:90% 00:55:06.230 --> 00:55:10.760 align:middle line:84% So now, the environment is system B. 00:55:10.760 --> 00:55:16.100 align:middle line:84% But the idea is more or less the same, 00:55:16.100 --> 00:55:20.570 align:middle line:84% except that the useful effect now is not the cooling of B, 00:55:20.570 --> 00:55:23.960 align:middle line:84% but is the heating of A. You want to heat up something, 00:55:23.960 --> 00:55:28.470 align:middle line:84% like the house, using a heat pump, 00:55:28.470 --> 00:55:33.140 align:middle line:84% which is nowadays been done worldwide, 00:55:33.140 --> 00:55:38.310 align:middle line:84% at least in places where you need heating in the winter time. 00:55:38.310 --> 00:55:43.490 align:middle line:90% And so the useful effect is QA. 00:55:43.490 --> 00:55:46.520 align:middle line:84% Therefore, you define a different COP. 00:55:46.520 --> 00:55:49.460 align:middle line:84% It's called COP, but it's different, 00:55:49.460 --> 00:55:51.026 align:middle line:90% a different definition. 00:55:51.026 --> 00:55:54.110 align:middle line:90% 00:55:54.110 --> 00:56:00.830 align:middle line:84% And this time, you put in the numerator 00:56:00.830 --> 00:56:06.470 align:middle line:84% the useful effect, which is the heat given 00:56:06.470 --> 00:56:13.850 align:middle line:84% to the hot system divided by the work 00:56:13.850 --> 00:56:19.020 align:middle line:84% that you have spent in order to do it. 00:56:19.020 --> 00:56:25.040 align:middle line:84% And again, the second-law efficiency 00:56:25.040 --> 00:56:31.970 align:middle line:84% is the ratio between the useful effect of your actual machinery 00:56:31.970 --> 00:56:35.180 align:middle line:84% and the useful effect of the reversible machinery, 00:56:35.180 --> 00:56:40.920 align:middle line:84% so that it would be equal to 1 for a reversible machinery. 00:56:40.920 --> 00:56:44.360 align:middle line:84% And it's less than 1 if there is entropy produced 00:56:44.360 --> 00:56:46.750 align:middle line:90% by your machinery here. 00:56:46.750 --> 00:56:56.239 align:middle line:90% 00:56:56.239 --> 00:57:04.740 align:middle line:84% So since now we are talking about some practical aspects, 00:57:04.740 --> 00:57:12.930 align:middle line:84% let's talk about another chapter of thermodynamics that 00:57:12.930 --> 00:57:17.250 align:middle line:84% has to do with establishing the properties of substances 00:57:17.250 --> 00:57:20.250 align:middle line:90% at stable equilibrium. 00:57:20.250 --> 00:57:22.880 align:middle line:84% And let's see, this is an important chapter. 00:57:22.880 --> 00:57:27.660 align:middle line:90% 00:57:27.660 --> 00:57:31.050 align:middle line:84% Many scientists have devoted their lives 00:57:31.050 --> 00:57:35.016 align:middle line:84% in constructing the properties of substances. 00:57:35.016 --> 00:57:38.770 align:middle line:90% 00:57:38.770 --> 00:57:51.230 align:middle line:84% For example, to stay here, Professor Keenan, 00:57:51.230 --> 00:57:57.650 align:middle line:84% in 1940 or maybe earlier, spent a lot 00:57:57.650 --> 00:58:01.130 align:middle line:84% of time developing the steam tables, which 00:58:01.130 --> 00:58:05.910 align:middle line:90% is the properties of water. 00:58:05.910 --> 00:58:09.500 align:middle line:84% So how do we measure properties of substances 00:58:09.500 --> 00:58:11.280 align:middle line:90% in stable equilibrium? 00:58:11.280 --> 00:58:15.580 align:middle line:90% 00:58:15.580 --> 00:58:22.180 align:middle line:84% Well, much of the tools that we have 00:58:22.180 --> 00:58:28.060 align:middle line:84% is related to the conditions for mutual stable equilibrium 00:58:28.060 --> 00:58:33.050 align:middle line:84% that we have developed, like equality of temperature, 00:58:33.050 --> 00:58:38.830 align:middle line:84% equality of pressure, equality of chemical potentials. 00:58:38.830 --> 00:58:42.680 align:middle line:84% The good thing is that, if you have a thermometer-- 00:58:42.680 --> 00:58:45.700 align:middle line:90% 00:58:45.700 --> 00:58:46.840 align:middle line:90% what is a thermometer? 00:58:46.840 --> 00:58:52.030 align:middle line:84% A thermometer is a system in which, in some way, 00:58:52.030 --> 00:58:58.720 align:middle line:84% you have put a scale in which you can read 00:58:58.720 --> 00:59:01.700 align:middle line:90% the value of the temperature. 00:59:01.700 --> 00:59:04.660 align:middle line:84% Then what you do, you stick it in contact 00:59:04.660 --> 00:59:07.230 align:middle line:84% to the system you want to measure. 00:59:07.230 --> 00:59:11.750 align:middle line:90% 00:59:11.750 --> 00:59:18.350 align:middle line:84% And you leave enough time so that you convince yourself 00:59:18.350 --> 00:59:21.360 align:middle line:84% that the two systems have reached mutual equilibrium. 00:59:21.360 --> 00:59:24.000 align:middle line:84% And if you are convinced that they are in mutual equilibrium, 00:59:24.000 --> 00:59:27.210 align:middle line:84% then the theory says that the two temperatures are equal. 00:59:27.210 --> 00:59:33.600 align:middle line:84% So by reading the temperature of the thermometer, 00:59:33.600 --> 00:59:36.245 align:middle line:84% you have measured the temperature of system A, 00:59:36.245 --> 00:59:39.230 align:middle line:90% of which you may know nothing. 00:59:39.230 --> 00:59:43.975 align:middle line:84% So it's an indirect measurement of a property. 00:59:43.975 --> 00:59:46.740 align:middle line:90% 00:59:46.740 --> 00:59:54.000 align:middle line:84% Similarly, the manometer-- you can construct a manometer 00:59:54.000 --> 00:59:59.040 align:middle line:84% so that you can read out the scale, the value here. 00:59:59.040 --> 01:00:03.720 align:middle line:84% You put it in equilibrium, mutual equilibrium 01:00:03.720 --> 01:00:11.550 align:middle line:84% with your system by drilling a little hole in the boundary 01:00:11.550 --> 01:00:18.930 align:middle line:84% with either a little piston or some moving interface, 01:00:18.930 --> 01:00:19.680 align:middle line:90% like here. 01:00:19.680 --> 01:00:25.410 align:middle line:90% 01:00:25.410 --> 01:00:28.500 align:middle line:84% If you are convinced that they are in mutual equilibrium, 01:00:28.500 --> 01:00:30.750 align:middle line:84% reading the pressure of B gives you 01:00:30.750 --> 01:00:34.410 align:middle line:84% information about the pressure of system A. 01:00:34.410 --> 01:00:40.430 align:middle line:84% So this is one way of measuring it. 01:00:40.430 --> 01:00:47.460 align:middle line:84% Yeah, this is just to remind of that proof 01:00:47.460 --> 01:00:53.420 align:middle line:84% that I don't want to go into details of, 01:00:53.420 --> 01:01:02.610 align:middle line:84% but that shows why the pressure can be measured 01:01:02.610 --> 01:01:07.900 align:middle line:84% by measuring a force per unit area 01:01:07.900 --> 01:01:13.910 align:middle line:84% anywhere on the boundary of your system. 01:01:13.910 --> 01:01:15.680 align:middle line:84% So you take the boundary of your system 01:01:15.680 --> 01:01:19.840 align:middle line:84% when it is in stable equilibrium, drill a hole, 01:01:19.840 --> 01:01:24.640 align:middle line:84% and substitute that piece of wall with the movable wall. 01:01:24.640 --> 01:01:28.510 align:middle line:84% And then you find a force per unit area 01:01:28.510 --> 01:01:31.790 align:middle line:84% that keeps that wall where the original one was. 01:01:31.790 --> 01:01:36.020 align:middle line:84% Once you've found that, you have measured the force per unit area 01:01:36.020 --> 01:01:41.020 align:middle line:84% that balances the pressure of your system. 01:01:41.020 --> 01:01:43.220 align:middle line:84% And when you are in stable equilibrium, 01:01:43.220 --> 01:01:45.850 align:middle line:84% that force per unit area is equal no matter 01:01:45.850 --> 01:01:50.560 align:middle line:84% where you go and do this procedure, 01:01:50.560 --> 01:01:56.770 align:middle line:84% whereas off equilibrium, it can be-- 01:01:56.770 --> 01:02:00.050 align:middle line:84% you can always measure force per unit area. 01:02:00.050 --> 01:02:06.590 align:middle line:84% But it may be non-uniform, not equal everywhere. 01:02:06.590 --> 01:02:09.480 align:middle line:84% And there is no pressure because you are not in equilibrium. 01:02:09.480 --> 01:02:14.880 align:middle line:90% 01:02:14.880 --> 01:02:26.610 align:middle line:84% Another kind of measurement that we can do is the so-called Joule 01:02:26.610 --> 01:02:33.300 align:middle line:84% apparatus, in which you input some mechanical energy 01:02:33.300 --> 01:02:38.820 align:middle line:84% in your substance, or in a piston-- 01:02:38.820 --> 01:02:44.910 align:middle line:84% or in a variable volume, or in a constant volume container. 01:02:44.910 --> 01:02:50.610 align:middle line:84% You put some energy in, wait for the system 01:02:50.610 --> 01:02:53.310 align:middle line:84% to go to a new stable equilibrium, 01:02:53.310 --> 01:03:03.960 align:middle line:84% and measure how much the temperature 01:03:03.960 --> 01:03:07.110 align:middle line:90% has changed for your system. 01:03:07.110 --> 01:03:14.310 align:middle line:84% So Mg delta z represents the amount of energy 01:03:14.310 --> 01:03:17.340 align:middle line:84% that you have put in, in the form of work. 01:03:17.340 --> 01:03:20.010 align:middle line:84% So you produce a work interaction 01:03:20.010 --> 01:03:23.850 align:middle line:84% and put energy into your system, let 01:03:23.850 --> 01:03:27.580 align:middle line:84% it relax until it reaches equilibrium, so 01:03:27.580 --> 01:03:30.240 align:middle line:84% a new temperature, compute the difference 01:03:30.240 --> 01:03:32.830 align:middle line:84% between the new temperature and the initial temperature, 01:03:32.830 --> 01:03:36.750 align:middle line:84% delta T. This ratio turns out to be 01:03:36.750 --> 01:03:39.990 align:middle line:84% equal to the partial derivative of this property 01:03:39.990 --> 01:03:44.530 align:middle line:84% that we're going to define in a moment. 01:03:44.530 --> 01:03:47.970 align:middle line:84% And that's the property that we call heat capacity 01:03:47.970 --> 01:03:49.750 align:middle line:90% at constant pressure. 01:03:49.750 --> 01:03:52.500 align:middle line:84% It's a stable equilibrium state property. 01:03:52.500 --> 01:03:54.680 align:middle line:84% We obtain it from the fundamental relation. 01:03:54.680 --> 01:03:58.110 align:middle line:90% 01:03:58.110 --> 01:04:04.720 align:middle line:84% There is one heat capacity that is equal to the same ratio 01:04:04.720 --> 01:04:07.530 align:middle line:84% when you do the experiment at constant volume. 01:04:07.530 --> 01:04:10.140 align:middle line:84% And there is another heat capacity when 01:04:10.140 --> 01:04:12.410 align:middle line:90% you do it at constant pressure. 01:04:12.410 --> 01:04:17.160 align:middle line:90% 01:04:17.160 --> 01:04:19.500 align:middle line:84% So let's start from the constant pressure 01:04:19.500 --> 01:04:31.220 align:middle line:84% because this gives us also a procedure to define the enthalpy 01:04:31.220 --> 01:04:38.560 align:middle line:84% and to get to this, the derivative that turns out 01:04:38.560 --> 01:04:41.270 align:middle line:84% to represent the result of these measurements. 01:04:41.270 --> 01:04:44.470 align:middle line:84% So it's important because it is the way in which you 01:04:44.470 --> 01:04:45.720 align:middle line:90% measure that property. 01:04:45.720 --> 01:04:50.660 align:middle line:90% 01:04:50.660 --> 01:04:55.280 align:middle line:84% Let's do it starting from the fundamental relation 01:04:55.280 --> 01:04:58.160 align:middle line:90% in energy form. 01:04:58.160 --> 01:05:00.750 align:middle line:90% And this is its differential. 01:05:00.750 --> 01:05:03.600 align:middle line:90% 01:05:03.600 --> 01:05:12.410 align:middle line:84% So H-- in a moment, we will see that this H has 01:05:12.410 --> 01:05:18.500 align:middle line:84% a particular transform of the fundamental relation when 01:05:18.500 --> 01:05:19.410 align:middle line:90% we change variables. 01:05:19.410 --> 01:05:25.530 align:middle line:84% But for the moment, let's just take it as an abrupt definition. 01:05:25.530 --> 01:05:29.390 align:middle line:84% Let's say that we call H the sum of energy 01:05:29.390 --> 01:05:32.100 align:middle line:84% plus the product of pressure and volume. 01:05:32.100 --> 01:05:33.160 align:middle line:90% We call it enthalpy. 01:05:33.160 --> 01:05:38.750 align:middle line:90% 01:05:38.750 --> 01:05:43.520 align:middle line:84% Then by taking the differential of this expression 01:05:43.520 --> 01:05:46.220 align:middle line:84% and recalling that the differential of a product 01:05:46.220 --> 01:05:54.950 align:middle line:84% is equal to pdV plus Vdp, where I see dE, 01:05:54.950 --> 01:06:00.080 align:middle line:84% I use the differential that I have written up here. 01:06:00.080 --> 01:06:01.370 align:middle line:90% So I substitute. 01:06:01.370 --> 01:06:04.040 align:middle line:90% And this is what I'm left with. 01:06:04.040 --> 01:06:06.950 align:middle line:90% From this expression-- 01:06:06.950 --> 01:06:13.570 align:middle line:84% I can view this expression as the differential. 01:06:13.570 --> 01:06:19.990 align:middle line:84% So you see, the independent variables are under the d-- 01:06:19.990 --> 01:06:22.560 align:middle line:90% so dS, dp, dn. 01:06:22.560 --> 01:06:27.190 align:middle line:90% 01:06:27.190 --> 01:06:29.020 align:middle line:84% So this would be the differential 01:06:29.020 --> 01:06:34.160 align:middle line:84% of a function in which H is a function of S, p, and n. 01:06:34.160 --> 01:06:38.440 align:middle line:90% 01:06:38.440 --> 01:06:42.100 align:middle line:84% And also, the coefficients of the differential 01:06:42.100 --> 01:06:43.780 align:middle line:90% are the partial derivatives. 01:06:43.780 --> 01:06:47.230 align:middle line:84% So for example, the T is the partial 01:06:47.230 --> 01:06:51.820 align:middle line:84% of H with respect to S. The partial of what? 01:06:51.820 --> 01:06:55.200 align:middle line:84% The partial of this function, so the partial 01:06:55.200 --> 01:06:59.550 align:middle line:84% of a function in which the enthalpy is expressed 01:06:59.550 --> 01:07:03.780 align:middle line:90% as a function of S, p, and n. 01:07:03.780 --> 01:07:06.460 align:middle line:90% So that's the temperature. 01:07:06.460 --> 01:07:11.480 align:middle line:90% 01:07:11.480 --> 01:07:19.030 align:middle line:84% Now, in the business of measuring properties, 01:07:19.030 --> 01:07:21.460 align:middle line:90% we like to have instruments. 01:07:21.460 --> 01:07:26.320 align:middle line:84% So we have a thermometer that measures temperature. 01:07:26.320 --> 01:07:30.610 align:middle line:84% A manometer measures the pressure. 01:07:30.610 --> 01:07:33.700 align:middle line:90% There is no energy meter. 01:07:33.700 --> 01:07:37.090 align:middle line:90% There is no entropy meter. 01:07:37.090 --> 01:07:40.330 align:middle line:84% We measure energy and entropy indirectly 01:07:40.330 --> 01:07:48.765 align:middle line:84% through temperature and pressure and volume. 01:07:48.765 --> 01:07:53.470 align:middle line:90% 01:07:53.470 --> 01:07:55.850 align:middle line:84% So let's do that also mathematically. 01:07:55.850 --> 01:07:59.620 align:middle line:84% So you look at this equation here and this other equation, 01:07:59.620 --> 01:08:04.450 align:middle line:84% and consider it as a system of two equations in which you want 01:08:04.450 --> 01:08:09.910 align:middle line:84% to eliminate the property that is hardly measurable-- difficult 01:08:09.910 --> 01:08:12.610 align:middle line:84% to measure, hard to measure, which is the entropy. 01:08:12.610 --> 01:08:21.010 align:middle line:84% So for example, solve this expression, 01:08:21.010 --> 01:08:23.410 align:middle line:84% this function here as a function-- instead 01:08:23.410 --> 01:08:27.130 align:middle line:84% of T as a function of S, write it as S as a function of T. 01:08:27.130 --> 01:08:32.450 align:middle line:84% And then take that S and put it in the original function 01:08:32.450 --> 01:08:34.029 align:middle line:90% up there. 01:08:34.029 --> 01:08:39.060 align:middle line:84% You manage to find H as a function of T, p, n. 01:08:39.060 --> 01:08:41.830 align:middle line:84% So you change the independent variables 01:08:41.830 --> 01:08:45.790 align:middle line:84% to ones that can be easily measured, 01:08:45.790 --> 01:08:48.859 align:middle line:90% temperature and pressure. 01:08:48.859 --> 01:08:52.189 align:middle line:84% So this new function-- and this is the way you have obtained 01:08:52.189 --> 01:08:54.020 align:middle line:90% it-- so we have a precise way-- 01:08:54.020 --> 01:08:57.080 align:middle line:84% recipe to obtain that function, if we start 01:08:57.080 --> 01:08:59.989 align:middle line:90% from the fundamental relation-- 01:08:59.989 --> 01:09:03.720 align:middle line:84% is-- of this function, you can take derivatives. 01:09:03.720 --> 01:09:07.250 align:middle line:84% For example, the derivative of H with respect to T 01:09:07.250 --> 01:09:10.069 align:middle line:84% is what we call the specific heat-- 01:09:10.069 --> 01:09:14.250 align:middle line:84% sorry, the heat capacity at constant pressure. 01:09:14.250 --> 01:09:18.220 align:middle line:90% And it is a function of T and p. 01:09:18.220 --> 01:09:21.520 align:middle line:84% So pressure and temperature, we can measure. 01:09:21.520 --> 01:09:26.649 align:middle line:84% Heat capacity we can measure with this instrument here. 01:09:26.649 --> 01:09:29.970 align:middle line:84% And so this is a fully measurable function. 01:09:29.970 --> 01:09:35.840 align:middle line:90% 01:09:35.840 --> 01:09:42.260 align:middle line:84% A similar procedure we can follow to define the heat 01:09:42.260 --> 01:09:45.950 align:middle line:90% capacity at constant volume. 01:09:45.950 --> 01:09:48.535 align:middle line:84% So we start again from the fundamental relation. 01:09:48.535 --> 01:09:51.040 align:middle line:90% 01:09:51.040 --> 01:09:54.160 align:middle line:84% And notice that the temperature, which 01:09:54.160 --> 01:09:58.660 align:middle line:84% is the partial derivative of E with respect to S, 01:09:58.660 --> 01:10:02.020 align:middle line:84% is being a function of this variable-- 01:10:02.020 --> 01:10:04.300 align:middle line:84% I mean, a derivative of a function of these variables 01:10:04.300 --> 01:10:06.440 align:middle line:84% has the same independent variables. 01:10:06.440 --> 01:10:09.460 align:middle line:84% So it's also a function of S, V, and n. 01:10:09.460 --> 01:10:13.210 align:middle line:84% And also here, play the game of eliminating 01:10:13.210 --> 01:10:20.350 align:middle line:84% the not so easy to measure property, entropy. 01:10:20.350 --> 01:10:28.530 align:middle line:84% So solve this as S function of T and substitute up there 01:10:28.530 --> 01:10:30.780 align:middle line:90% to obtain that the energy-- 01:10:30.780 --> 01:10:33.690 align:middle line:84% a new function in which the energy 01:10:33.690 --> 01:10:36.490 align:middle line:84% is expressed in terms of the independent variables, 01:10:36.490 --> 01:10:37.850 align:middle line:90% temperature and volume. 01:10:37.850 --> 01:10:42.420 align:middle line:84% These are things that are easy to measure because temperature 01:10:42.420 --> 01:10:48.100 align:middle line:84% you measure with the thermometer, volume you 01:10:48.100 --> 01:10:51.330 align:middle line:90% measure with the stick. 01:10:51.330 --> 01:10:54.010 align:middle line:90% 01:10:54.010 --> 01:10:56.950 align:middle line:84% So the first derivative of this function, 01:10:56.950 --> 01:10:59.200 align:middle line:84% the partial of E with respect to T, 01:10:59.200 --> 01:11:06.560 align:middle line:84% is called the heat capacity at constant volume. 01:11:06.560 --> 01:11:11.170 align:middle line:84% And it is the thing that you measure with that other Joule 01:11:11.170 --> 01:11:15.190 align:middle line:90% apparatus at constant volume. 01:11:15.190 --> 01:11:16.340 align:middle line:90% So we retain that. 01:11:16.340 --> 01:11:19.760 align:middle line:90% 01:11:19.760 --> 01:11:22.950 align:middle line:84% But we can also do this other procedure. 01:11:22.950 --> 01:11:26.190 align:middle line:84% We take the temperature, like before here, 01:11:26.190 --> 01:11:28.740 align:middle line:84% but now we make a system also with the pressure. 01:11:28.740 --> 01:11:33.530 align:middle line:84% The pressure is the derivative of E with respect to V. 01:11:33.530 --> 01:11:35.510 align:middle line:84% And being a derivative of a function 01:11:35.510 --> 01:11:37.170 align:middle line:84% with these independent variables, 01:11:37.170 --> 01:11:40.130 align:middle line:84% it also has the same independent variables. 01:11:40.130 --> 01:11:42.520 align:middle line:84% Eliminate entropy also from these two. 01:11:42.520 --> 01:11:45.230 align:middle line:90% 01:11:45.230 --> 01:11:50.130 align:middle line:84% That results in a relation between pressure, volume, 01:11:50.130 --> 01:11:53.930 align:middle line:84% and temperature, which we call the equation of state. 01:11:53.930 --> 01:11:55.760 align:middle line:84% Pressure, volume, and temperature 01:11:55.760 --> 01:11:59.510 align:middle line:84% are also easily measurable stuff. 01:11:59.510 --> 01:12:01.640 align:middle line:90% That's the equation of state. 01:12:01.640 --> 01:12:08.270 align:middle line:84% And the partial derivatives of this equation of state-- 01:12:08.270 --> 01:12:12.860 align:middle line:84% so the derivative of V with respect to T, 01:12:12.860 --> 01:12:18.510 align:middle line:84% says how much the volume of your substance 01:12:18.510 --> 01:12:23.450 align:middle line:84% changes as you vary the temperature, as you heat it up. 01:12:23.450 --> 01:12:27.550 align:middle line:84% So it's the coefficient of expansion. 01:12:27.550 --> 01:12:31.530 align:middle line:84% If you do it divided by V, it's the percentage increase 01:12:31.530 --> 01:12:33.870 align:middle line:90% in volume per given amount-- 01:12:33.870 --> 01:12:38.160 align:middle line:84% or given increase in temperature. 01:12:38.160 --> 01:12:44.190 align:middle line:84% And you can measure this when you do the experiment 01:12:44.190 --> 01:12:45.450 align:middle line:90% at constant pressure. 01:12:45.450 --> 01:12:48.690 align:middle line:84% Or the second, the other derivative, 01:12:48.690 --> 01:12:53.190 align:middle line:84% the derivative of V with respect to P, 01:12:53.190 --> 01:13:02.430 align:middle line:84% it says how much the substance gets reduced in volume if I 01:13:02.430 --> 01:13:05.430 align:middle line:90% increase the pressure. 01:13:05.430 --> 01:13:10.200 align:middle line:84% So the reduction is minus delta V 01:13:10.200 --> 01:13:12.670 align:middle line:84% per unit of increase in pressure. 01:13:12.670 --> 01:13:15.030 align:middle line:84% And again, if I divide by V, it's 01:13:15.030 --> 01:13:25.280 align:middle line:84% the percentage reduction in volume. 01:13:25.280 --> 01:13:30.080 align:middle line:84% So when you do the experiment, increase the pressure 01:13:30.080 --> 01:13:31.800 align:middle line:90% at constant temperature. 01:13:31.800 --> 01:13:35.000 align:middle line:84% This is the isothermal compressibility 01:13:35.000 --> 01:13:37.750 align:middle line:90% of the substance. 01:13:37.750 --> 01:13:39.950 align:middle line:84% Being derivatives of the equation of state, 01:13:39.950 --> 01:13:43.510 align:middle line:84% they are functions of T and P. And these are easily 01:13:43.510 --> 01:13:46.420 align:middle line:84% measured because it simply relates 01:13:46.420 --> 01:13:49.760 align:middle line:84% measuring changes in volume, measuring temperatures. 01:13:49.760 --> 01:13:54.570 align:middle line:90% 01:13:54.570 --> 01:13:57.210 align:middle line:84% So the differential of the equation 01:13:57.210 --> 01:14:08.660 align:middle line:84% of state done at fixed amounts has these two coefficients, 01:14:08.660 --> 01:14:11.510 align:middle line:84% isobaric expansion and isothermal compressibility 01:14:11.510 --> 01:14:14.200 align:middle line:90% appearing directly here. 01:14:14.200 --> 01:14:17.100 align:middle line:90% 01:14:17.100 --> 01:14:18.990 align:middle line:84% There are other things that you can 01:14:18.990 --> 01:14:21.400 align:middle line:90% measure in other experiments. 01:14:21.400 --> 01:14:25.470 align:middle line:84% For example, an interesting one is 01:14:25.470 --> 01:14:31.765 align:middle line:90% the speed of sound propagation. 01:14:31.765 --> 01:14:33.800 align:middle line:90% It should be an of here. 01:14:33.800 --> 01:14:40.370 align:middle line:84% Or the Joule-Thomson coefficient in the expansion 01:14:40.370 --> 01:14:45.440 align:middle line:84% of a gas in a throttle, which occurs at constant enthalpy. 01:14:45.440 --> 01:14:47.540 align:middle line:84% You measure the change in temperature 01:14:47.540 --> 01:14:50.720 align:middle line:90% over a drop in pressure. 01:14:50.720 --> 01:14:53.030 align:middle line:84% And it can be positive and negative. 01:14:53.030 --> 01:15:00.770 align:middle line:84% And that is related to the things, the properties 01:15:00.770 --> 01:15:02.870 align:middle line:84% that we have already defined before. 01:15:02.870 --> 01:15:08.060 align:middle line:84% And also, the Mayer relation shows 01:15:08.060 --> 01:15:16.580 align:middle line:84% that Cp, CV, alpha p, and kappa T are not independent. 01:15:16.580 --> 01:15:19.770 align:middle line:84% So you don't need to measure all four of them. 01:15:19.770 --> 01:15:21.280 align:middle line:90% You just need to measure three. 01:15:21.280 --> 01:15:34.270 align:middle line:90% 01:15:34.270 --> 01:15:37.640 align:middle line:84% Let me skip that one for a moment. 01:15:37.640 --> 01:15:39.260 align:middle line:90% No, no, I need it. 01:15:39.260 --> 01:15:39.810 align:middle line:90% I need it. 01:15:39.810 --> 01:15:42.850 align:middle line:84% In order to measure chemical potentials, 01:15:42.850 --> 01:15:46.780 align:middle line:84% I do need first to show that the chemical potentials can 01:15:46.780 --> 01:15:51.720 align:middle line:84% be written as functions of temperature and pressure. 01:15:51.720 --> 01:15:56.870 align:middle line:90% 01:15:56.870 --> 01:15:59.990 align:middle line:84% Because you see, the chemical potential 01:15:59.990 --> 01:16:02.870 align:middle line:84% is defined as the partial of E with respect 01:16:02.870 --> 01:16:09.200 align:middle line:84% to n at constant S and V, but that way, it 01:16:09.200 --> 01:16:13.430 align:middle line:84% isn't easy to measure because of the S. 01:16:13.430 --> 01:16:23.120 align:middle line:84% So mu i is the partial of E with respect to ni at constant S, V 01:16:23.120 --> 01:16:28.820 align:middle line:90% and all the other n's held-- 01:16:28.820 --> 01:16:32.660 align:middle line:84% well, I shouldn't say that, where the function that I am 01:16:32.660 --> 01:16:35.913 align:middle line:84% taking the derivative of is a function of S, V, 01:16:35.913 --> 01:16:36.580 align:middle line:90% and all the n's. 01:16:36.580 --> 01:16:39.470 align:middle line:90% 01:16:39.470 --> 01:16:43.740 align:middle line:84% However, when you view it this way, 01:16:43.740 --> 01:16:49.130 align:middle line:84% it's not easy to measure because I don't have an energy meter 01:16:49.130 --> 01:16:50.635 align:middle line:84% and I don't have an entropy meter. 01:16:50.635 --> 01:16:53.820 align:middle line:90% 01:16:53.820 --> 01:16:58.120 align:middle line:84% So the trick is to change variables. 01:16:58.120 --> 01:17:01.720 align:middle line:84% And one way to do it is to define a function, which is 01:17:01.720 --> 01:17:07.510 align:middle line:90% called the Gibbs free energy. 01:17:07.510 --> 01:17:10.970 align:middle line:84% For the moment, think of it as just an arbitrary definition. 01:17:10.970 --> 01:17:15.360 align:middle line:84% So I say, define this energy minus TS plus pV. 01:17:15.360 --> 01:17:18.430 align:middle line:90% 01:17:18.430 --> 01:17:20.650 align:middle line:90% Write the differential. 01:17:20.650 --> 01:17:22.990 align:middle line:84% Use the differential of a fundamental relation 01:17:22.990 --> 01:17:25.030 align:middle line:90% in order to express dE. 01:17:25.030 --> 01:17:28.510 align:middle line:84% And here is what you are left with. 01:17:28.510 --> 01:17:32.320 align:middle line:84% This expression shows that you can view this function 01:17:32.320 --> 01:17:36.670 align:middle line:84% G as a function of the independent variables that 01:17:36.670 --> 01:17:39.190 align:middle line:84% appear under the d of the differential-- so 01:17:39.190 --> 01:17:42.520 align:middle line:84% the temperature, the pressure, and the amounts. 01:17:42.520 --> 01:17:50.740 align:middle line:84% So G has as, say, natural variables, T, p and n. 01:17:50.740 --> 01:17:53.450 align:middle line:84% And since this is the differential of that function, 01:17:53.450 --> 01:17:57.670 align:middle line:84% the chemical potential can therefore 01:17:57.670 --> 01:18:04.440 align:middle line:84% be written now also as the partial of G with respect to ni. 01:18:04.440 --> 01:18:07.710 align:middle line:90% 01:18:07.710 --> 01:18:09.750 align:middle line:84% And since it is a partial of G, which 01:18:09.750 --> 01:18:11.700 align:middle line:84% is a function of T, p, and n, it's 01:18:11.700 --> 01:18:14.960 align:middle line:84% also a function of just T, p, and n. 01:18:14.960 --> 01:18:20.520 align:middle line:84% So the chemical potential, we would like to measure it. 01:18:20.520 --> 01:18:22.910 align:middle line:84% It is very important to determine the properties 01:18:22.910 --> 01:18:26.335 align:middle line:90% of multi-component systems. 01:18:26.335 --> 01:18:29.620 align:middle line:90% 01:18:29.620 --> 01:18:31.850 align:middle line:84% But it cannot be measured directly. 01:18:31.850 --> 01:18:38.320 align:middle line:84% There is no chemical potential meter. 01:18:38.320 --> 01:18:41.770 align:middle line:84% However, it can be measured indirectly 01:18:41.770 --> 01:18:44.920 align:middle line:84% via the definition of what we call the partial pressure 01:18:44.920 --> 01:18:48.580 align:middle line:90% of a component in the mixture. 01:18:48.580 --> 01:18:51.520 align:middle line:84% Here, the experimental setup is this. 01:18:51.520 --> 01:18:54.820 align:middle line:84% I have my mixture in which I have many components. 01:18:54.820 --> 01:18:58.360 align:middle line:84% And they are in a stable equilibrium state. 01:18:58.360 --> 01:19:01.610 align:middle line:84% They are at some pressure, some temperature, 01:19:01.610 --> 01:19:04.520 align:middle line:84% and they have their chemical potentials, and so on. 01:19:04.520 --> 01:19:07.120 align:middle line:90% 01:19:07.120 --> 01:19:12.340 align:middle line:84% So what I do is to drill a hole in the wall somewhere 01:19:12.340 --> 01:19:18.070 align:middle line:84% of my system and substitute the wall 01:19:18.070 --> 01:19:21.370 align:middle line:84% with a membrane, a semi-permeable or porous medium, 01:19:21.370 --> 01:19:26.320 align:middle line:84% or a semi-permeable membrane, where only molecules of type 01:19:26.320 --> 01:19:28.490 align:middle line:90% I can cross. 01:19:28.490 --> 01:19:30.150 align:middle line:90% The others don't. 01:19:30.150 --> 01:19:37.765 align:middle line:90% 01:19:37.765 --> 01:19:39.590 align:middle line:90% So I put on the other side-- 01:19:39.590 --> 01:19:47.765 align:middle line:84% I put on the other side molecules of substance i. 01:19:47.765 --> 01:19:50.410 align:middle line:84% And with a piston, I keep them there. 01:19:50.410 --> 01:19:52.390 align:middle line:84% And I measure the force per unit area 01:19:52.390 --> 01:20:01.040 align:middle line:84% that is necessary to keep them in mutual equilibrium. 01:20:01.040 --> 01:20:07.592 align:middle line:84% When I reach mutual equilibrium between this little system, ii, 01:20:07.592 --> 01:20:13.660 align:middle line:84% the double script i we use as a way to say 01:20:13.660 --> 01:20:15.470 align:middle line:90% that it is a pure substance. 01:20:15.470 --> 01:20:23.770 align:middle line:84% So in this little box, there is only substance i, 01:20:23.770 --> 01:20:26.800 align:middle line:84% whereas in the mixture there is substance i mixed 01:20:26.800 --> 01:20:29.610 align:middle line:90% with other stuff. 01:20:29.610 --> 01:20:32.430 align:middle line:84% When we are in mutual equilibrium 01:20:32.430 --> 01:20:37.740 align:middle line:84% across this membrane, we have proved also 01:20:37.740 --> 01:20:40.740 align:middle line:84% the other day that the conditions 01:20:40.740 --> 01:20:44.440 align:middle line:84% for mutual equilibrium are equality of temperature-- 01:20:44.440 --> 01:20:47.400 align:middle line:90% 01:20:47.400 --> 01:20:49.590 align:middle line:84% so the temperature of system ii must 01:20:49.590 --> 01:20:52.440 align:middle line:84% be equal to the temperature of the mixture. 01:20:52.440 --> 01:20:56.610 align:middle line:84% And the equality of chemical potentials for the substance 01:20:56.610 --> 01:20:58.406 align:middle line:90% that can be exchanged. 01:20:58.406 --> 01:21:01.620 align:middle line:90% 01:21:01.620 --> 01:21:03.870 align:middle line:90% So the substance is i. 01:21:03.870 --> 01:21:06.000 align:middle line:84% And so I need that the chemical potential 01:21:06.000 --> 01:21:11.070 align:middle line:84% of substance i of the mixture must 01:21:11.070 --> 01:21:15.420 align:middle line:84% be equal to the chemical potential of the pure substance 01:21:15.420 --> 01:21:19.670 align:middle line:84% i just on the other side of a semi-permeable membrane. 01:21:19.670 --> 01:21:26.050 align:middle line:90% 01:21:26.050 --> 01:21:26.770 align:middle line:90% OK. 01:21:26.770 --> 01:21:30.430 align:middle line:84% Here, I also put the functional dependence 01:21:30.430 --> 01:21:35.140 align:middle line:84% because I used what we just proved before, that the chemical 01:21:35.140 --> 01:21:38.230 align:middle line:90% potential are functions-- 01:21:38.230 --> 01:21:42.670 align:middle line:84% can be written as functions of T and p. 01:21:42.670 --> 01:21:45.160 align:middle line:84% And so these are the functions that we are going 01:21:45.160 --> 01:21:47.210 align:middle line:90% to use for this measurement. 01:21:47.210 --> 01:21:52.500 align:middle line:90% 01:21:52.500 --> 01:21:59.460 align:middle line:84% So by looking at this equation, which 01:21:59.460 --> 01:22:06.840 align:middle line:84% is required by the condition for mutual equilibrium, 01:22:06.840 --> 01:22:13.140 align:middle line:84% we see that inside the independent variables are all 01:22:13.140 --> 01:22:17.680 align:middle line:90% easily measurable things-- 01:22:17.680 --> 01:22:24.700 align:middle line:84% temperature, pressure, and also this pii, 01:22:24.700 --> 01:22:29.260 align:middle line:84% it's the pressure of the system on the other side 01:22:29.260 --> 01:22:31.390 align:middle line:90% of the semi-permeable membrane. 01:22:31.390 --> 01:22:34.330 align:middle line:84% We call that pressure the partial pressure 01:22:34.330 --> 01:22:36.700 align:middle line:90% of constituent i in the mixture. 01:22:36.700 --> 01:22:39.550 align:middle line:90% It is a property of the mixture. 01:22:39.550 --> 01:22:42.730 align:middle line:90% And it can be measured. 01:22:42.730 --> 01:22:46.090 align:middle line:84% And by measuring it, and by knowing 01:22:46.090 --> 01:22:48.460 align:middle line:84% how the chemical potential of the pure substance 01:22:48.460 --> 01:22:52.490 align:middle line:84% depends on-- so if I know this for the pure substance, 01:22:52.490 --> 01:22:55.000 align:middle line:84% I can infer from that measurement 01:22:55.000 --> 01:22:59.030 align:middle line:84% the chemical potential of the constituent in the mixture. 01:22:59.030 --> 01:23:01.883 align:middle line:84% This is how we characterize mixtures. 01:23:01.883 --> 01:23:07.740 align:middle line:90% 01:23:07.740 --> 01:23:11.040 align:middle line:84% So here is a little summary of properties 01:23:11.040 --> 01:23:14.220 align:middle line:84% that we have talked about so far-- 01:23:14.220 --> 01:23:20.290 align:middle line:84% energy, entropy, also the volumes 01:23:20.290 --> 01:23:23.011 align:middle line:84% and the parameters that enter in-- 01:23:23.011 --> 01:23:26.020 align:middle line:84% the amounts of constituents that enter in the definition 01:23:26.020 --> 01:23:29.200 align:middle line:90% of our system. 01:23:29.200 --> 01:23:33.970 align:middle line:84% These properties are defined for all states, equilibrium 01:23:33.970 --> 01:23:35.150 align:middle line:90% and non-equilibrium. 01:23:35.150 --> 01:23:38.250 align:middle line:90% 01:23:38.250 --> 01:23:44.070 align:middle line:84% Then we defined some other properties 01:23:44.070 --> 01:23:47.070 align:middle line:84% based on the fundamental relation for stable equilibrium 01:23:47.070 --> 01:23:49.710 align:middle line:90% states. 01:23:49.710 --> 01:23:51.990 align:middle line:84% So clearly, these are defined only if you 01:23:51.990 --> 01:23:54.450 align:middle line:90% have a fundamental relation. 01:23:54.450 --> 01:23:58.230 align:middle line:84% So they are defined only for stable equilibrium states-- 01:23:58.230 --> 01:24:02.280 align:middle line:84% temperature, pressure, chemical potentials, the enthalpy, 01:24:02.280 --> 01:24:11.370 align:middle line:84% the Gibbs free energy, the enthalpy because it's E plus pV 01:24:11.370 --> 01:24:13.030 align:middle line:90% and p is the pressure. 01:24:13.030 --> 01:24:16.050 align:middle line:84% Pressure is defined only for a stable equilibrium. 01:24:16.050 --> 01:24:19.870 align:middle line:84% Or the Gibbs free energy has the p and the T also. 01:24:19.870 --> 01:24:24.710 align:middle line:84% So it also is defined only for stable equilibrium states. 01:24:24.710 --> 01:24:27.550 align:middle line:90% 01:24:27.550 --> 01:24:34.780 align:middle line:84% And then we also defined these four properties today-- 01:24:34.780 --> 01:24:41.290 align:middle line:84% the heat capacities, the coefficient of expansion, 01:24:41.290 --> 01:24:45.640 align:middle line:84% and the coefficient of compressibility. 01:24:45.640 --> 01:24:51.880 align:middle line:84% These ones also are restricted for equilibrium states. 01:24:51.880 --> 01:24:54.520 align:middle line:84% But they're even more restricted because some 01:24:54.520 --> 01:24:57.525 align:middle line:84% of those mathematical steps that-- 01:24:57.525 --> 01:25:00.190 align:middle line:90% 01:25:00.190 --> 01:25:07.060 align:middle line:84% for example, if I go back here, when 01:25:07.060 --> 01:25:11.080 align:middle line:84% I do this eliminating one variable, 01:25:11.080 --> 01:25:19.340 align:middle line:84% I can do that only if some derivative is not equal to 0. 01:25:19.340 --> 01:25:30.270 align:middle line:84% So for example, if the partial of T with respect to S is equal 01:25:30.270 --> 01:25:37.080 align:middle line:84% to 0, I cannot invert that function because it's not 01:25:37.080 --> 01:25:39.210 align:middle line:90% monotonic. 01:25:39.210 --> 01:25:44.440 align:middle line:84% And this is what happens, for example, 01:25:44.440 --> 01:25:50.400 align:middle line:84% when I have two-phase states, like liquid and vapor 01:25:50.400 --> 01:25:52.380 align:middle line:90% coexisting. 01:25:52.380 --> 01:25:58.470 align:middle line:84% In that case, some of these functions 01:25:58.470 --> 01:26:06.365 align:middle line:84% are not defined or, which is the same, take up infinite values. 01:26:06.365 --> 01:26:10.600 align:middle line:90% 01:26:10.600 --> 01:26:13.485 align:middle line:90% So we go back to here. 01:26:13.485 --> 01:26:16.430 align:middle line:90% 01:26:16.430 --> 01:26:29.180 align:middle line:84% Now, this also says that if we do our measurements 01:26:29.180 --> 01:26:32.030 align:middle line:84% and we can, for a given substance, 01:26:32.030 --> 01:26:38.030 align:middle line:84% or for a set of mixtures, develop or measure 01:26:38.030 --> 01:26:42.020 align:middle line:84% experimentally these functions, the alpha p 01:26:42.020 --> 01:26:45.500 align:middle line:84% as a function of T, p, and composition, kappa T, 01:26:45.500 --> 01:26:48.260 align:middle line:84% is a function of temperature, pressure and composition. 01:26:48.260 --> 01:26:49.820 align:middle line:90% The specific heat at constant-- 01:26:49.820 --> 01:26:52.268 align:middle line:84% I'm sorry, the heat capacity at constant pressure 01:26:52.268 --> 01:26:54.560 align:middle line:84% as a function of temperature, pressure, and composition 01:26:54.560 --> 01:26:57.870 align:middle line:84% and the chemical potentials as a function of temperature, 01:26:57.870 --> 01:26:59.100 align:middle line:90% pressure, and composition. 01:26:59.100 --> 01:27:05.690 align:middle line:84% If I have all these functions, then, with these functions, 01:27:05.690 --> 01:27:09.900 align:middle line:84% I can reconstruct the fundamental relation. 01:27:09.900 --> 01:27:11.720 align:middle line:84% So I know everything that I need to know 01:27:11.720 --> 01:27:15.800 align:middle line:84% about the stable equilibrium states of that substance. 01:27:15.800 --> 01:27:21.750 align:middle line:84% And actually, the reconstruction is done through these relations 01:27:21.750 --> 01:27:23.260 align:middle line:90% here, for example, the energy. 01:27:23.260 --> 01:27:27.150 align:middle line:90% 01:27:27.150 --> 01:27:30.780 align:middle line:84% These expressions for the energy here 01:27:30.780 --> 01:27:36.030 align:middle line:84% can be related to these properties in this way. 01:27:36.030 --> 01:27:41.400 align:middle line:84% So if I integrate from one temperature to another, 01:27:41.400 --> 01:27:45.450 align:middle line:84% from one pressure to another, this expression, 01:27:45.450 --> 01:27:49.980 align:middle line:90% I can get variations in energy. 01:27:49.980 --> 01:27:53.010 align:middle line:84% The entropy also is related this way. 01:27:53.010 --> 01:27:56.010 align:middle line:84% The enthalpy is related that way. 01:27:56.010 --> 01:28:00.510 align:middle line:84% And these are general forms that are 01:28:00.510 --> 01:28:03.630 align:middle line:90% valid for any pure substance. 01:28:03.630 --> 01:28:09.700 align:middle line:84% If you also need to consider mixtures, 01:28:09.700 --> 01:28:14.670 align:middle line:84% then you need to generalize this to use the expressions where 01:28:14.670 --> 01:28:17.593 align:middle line:90% also dn appears. 01:28:17.593 --> 01:28:20.390 align:middle line:90% 01:28:20.390 --> 01:28:32.770 align:middle line:84% So in this way, we have a way to measure the fundamental relation 01:28:32.770 --> 01:28:33.876 align:middle line:90% of substances. 01:28:33.876 --> 01:28:36.610 align:middle line:90% 01:28:36.610 --> 01:28:37.110 align:middle line:90% OK. 01:28:37.110 --> 01:28:42.078 align:middle line:90% 01:28:42.078 --> 01:28:46.045 align:middle line:90% Now, yeah. 01:28:46.045 --> 01:28:47.170 align:middle line:90% Yeah, we have five minutes. 01:28:47.170 --> 01:28:49.320 align:middle line:90% I can do this. 01:28:49.320 --> 01:28:56.370 align:middle line:84% I want to show that some of these function, 01:28:56.370 --> 01:29:03.420 align:middle line:84% like the enthalpy and the Gibbs free energy, 01:29:03.420 --> 01:29:08.790 align:middle line:84% are obtained as Legendre transforms 01:29:08.790 --> 01:29:10.780 align:middle line:90% of the fundamental relation. 01:29:10.780 --> 01:29:14.340 align:middle line:84% Legendre transform is a particular procedure 01:29:14.340 --> 01:29:19.980 align:middle line:90% that we can explain in this way. 01:29:19.980 --> 01:29:24.570 align:middle line:84% Let's start with the function of a single variable. 01:29:24.570 --> 01:29:27.990 align:middle line:84% So suppose you give me a function of a single variable, 01:29:27.990 --> 01:29:32.310 align:middle line:90% like any function. 01:29:32.310 --> 01:29:41.970 align:middle line:84% But to simplify, let me suppose that this function is either 01:29:41.970 --> 01:29:50.550 align:middle line:84% convex or concave and monotonic so that I can invert and play 01:29:50.550 --> 01:29:51.900 align:middle line:90% with it. 01:29:51.900 --> 01:29:54.700 align:middle line:84% So usually, we define a function-- 01:29:54.700 --> 01:30:05.040 align:middle line:84% I'll call it f of y, meaning that, for each value of y, 01:30:05.040 --> 01:30:06.305 align:middle line:90% I get a value of f. 01:30:06.305 --> 01:30:09.360 align:middle line:90% 01:30:09.360 --> 01:30:15.810 align:middle line:84% And as I move y, the function tells me how f moves. 01:30:15.810 --> 01:30:17.700 align:middle line:90% But there is an alternative. 01:30:17.700 --> 01:30:21.780 align:middle line:84% The idea of Legendre is that there 01:30:21.780 --> 01:30:26.940 align:middle line:84% is an alternative description because that function also has 01:30:26.940 --> 01:30:32.525 align:middle line:84% a tangent and has a line tangent to every point. 01:30:32.525 --> 01:30:35.910 align:middle line:90% 01:30:35.910 --> 01:30:44.130 align:middle line:84% So this tangent is characterized by two items, one 01:30:44.130 --> 01:30:51.630 align:middle line:84% that we will call lambda, which is the angular coefficient 01:30:51.630 --> 01:30:54.870 align:middle line:90% or the slope of the line. 01:30:54.870 --> 01:31:00.420 align:middle line:84% And this other value here, that we call L, which 01:31:00.420 --> 01:31:04.460 align:middle line:90% is the intercept at y equals 0. 01:31:04.460 --> 01:31:11.750 align:middle line:90% 01:31:11.750 --> 01:31:18.630 align:middle line:84% So the idea of Legendre is that you can-- 01:31:18.630 --> 01:31:23.370 align:middle line:84% if you draw, like it's sketched here, 01:31:23.370 --> 01:31:25.840 align:middle line:84% the slopes at every point of the line, 01:31:25.840 --> 01:31:31.080 align:middle line:84% you get a family of lines, the envelope of which 01:31:31.080 --> 01:31:32.510 align:middle line:90% is your original curve. 01:31:32.510 --> 01:31:37.160 align:middle line:90% 01:31:37.160 --> 01:31:44.750 align:middle line:84% Each line is characterized for a given lambda, for a given lambda 01:31:44.750 --> 01:31:49.840 align:middle line:90% by L. I change the point. 01:31:49.840 --> 01:31:51.760 align:middle line:90% So I take another lambda. 01:31:51.760 --> 01:31:56.370 align:middle line:84% Suppose I want this slope, this match. 01:31:56.370 --> 01:31:58.370 align:middle line:90% So I fix lambda. 01:31:58.370 --> 01:32:01.310 align:middle line:90% So say this is lambda 1. 01:32:01.310 --> 01:32:02.885 align:middle line:90% Now, I fix lambda 2. 01:32:02.885 --> 01:32:05.900 align:middle line:90% 01:32:05.900 --> 01:32:11.930 align:middle line:90% And that gives me L of 2. 01:32:11.930 --> 01:32:21.110 align:middle line:84% So selecting a value of lambda specifies, 01:32:21.110 --> 01:32:25.340 align:middle line:84% selects the point in which the tangent 01:32:25.340 --> 01:32:28.910 align:middle line:84% as that slope, of the original curve. 01:32:28.910 --> 01:32:33.860 align:middle line:84% And therefore, when you place your line there, 01:32:33.860 --> 01:32:38.400 align:middle line:90% it also selects an intercept. 01:32:38.400 --> 01:32:44.280 align:middle line:84% So there is a relation between lambda and L. 01:32:44.280 --> 01:32:47.430 align:middle line:84% And that relation is the Legendre transform 01:32:47.430 --> 01:32:48.305 align:middle line:90% of your function. 01:32:48.305 --> 01:32:52.780 align:middle line:90% 01:32:52.780 --> 01:33:01.150 align:middle line:84% The procedure runs this way, that you take the function, 01:33:01.150 --> 01:33:08.380 align:middle line:84% you write the equation for the partial, for the derivative. 01:33:08.380 --> 01:33:16.280 align:middle line:84% And then you define L by the geometrical construction. 01:33:16.280 --> 01:33:24.760 align:middle line:84% So it's the function f minus this distance here. 01:33:24.760 --> 01:33:35.680 align:middle line:84% And this distance here is equal to the horizontal segment 01:33:35.680 --> 01:33:41.050 align:middle line:84% times the slope, so the tangent of this angle, so lambda. 01:33:41.050 --> 01:33:48.705 align:middle line:84% So L is equal to f minus lambda times y. 01:33:48.705 --> 01:33:52.780 align:middle line:90% 01:33:52.780 --> 01:33:56.320 align:middle line:84% So it corresponds to that description. 01:33:56.320 --> 01:33:59.960 align:middle line:84% And I think the time is up for today. 01:33:59.960 --> 01:34:07.110 align:middle line:84% So we'll pick it up from here the next time. 01:34:07.110 --> 01:34:11.000 align:middle line:90%