WEBVTT 00:00:00.000 --> 00:00:02.398 align:middle line:90% [SQUEAKING] 00:00:02.398 --> 00:00:02.898 align:middle line:90% 00:00:02.898 --> 00:00:03.847 align:middle line:90% [RUSTLING] 00:00:03.847 --> 00:00:04.347 align:middle line:90% 00:00:04.347 --> 00:00:07.728 align:middle line:90% [CLICKING] 00:00:07.728 --> 00:00:11.320 align:middle line:90% 00:00:11.320 --> 00:00:16.239 align:middle line:84% GIAN PAOLO BERETTA: OK, so good morning. 00:00:16.239 --> 00:00:21.200 align:middle line:90% We're back to 2.43. 00:00:21.200 --> 00:00:24.080 align:middle line:84% So let me just very briefly review 00:00:24.080 --> 00:00:28.630 align:middle line:90% what we've done the last time. 00:00:28.630 --> 00:00:31.210 align:middle line:90% We are in part 1. 00:00:31.210 --> 00:00:36.190 align:middle line:84% And we bring you some of the basic definitions up 00:00:36.190 --> 00:00:37.970 align:middle line:84% to the statement of the second law, 00:00:37.970 --> 00:00:40.080 align:middle line:84% and we have to pick it up from there. 00:00:40.080 --> 00:00:43.700 align:middle line:90% 00:00:43.700 --> 00:00:45.420 align:middle line:90% Just a quick reminder. 00:00:45.420 --> 00:00:48.480 align:middle line:84% We said what we mean by the word system. 00:00:48.480 --> 00:00:52.010 align:middle line:84% And we explained that the word system is loaded with meaning. 00:00:52.010 --> 00:00:55.120 align:middle line:84% It's not just that anything can be a system. 00:00:55.120 --> 00:00:57.650 align:middle line:90% It has to qualify. 00:00:57.650 --> 00:01:00.110 align:middle line:84% In particular, there is a constraint 00:01:00.110 --> 00:01:07.650 align:middle line:84% on how the external forces should be mounted in your model. 00:01:07.650 --> 00:01:10.650 align:middle line:84% They must not depend on coordinates 00:01:10.650 --> 00:01:13.196 align:middle line:84% of external constituents of your system. 00:01:13.196 --> 00:01:17.660 align:middle line:90% 00:01:17.660 --> 00:01:21.740 align:middle line:84% Later on, we'll see that there is also another constraint 00:01:21.740 --> 00:01:25.130 align:middle line:90% relating to correlations. 00:01:25.130 --> 00:01:34.850 align:middle line:84% Also, this means that your system should not depend on, 00:01:34.850 --> 00:01:37.890 align:middle line:84% or be correlated with, some other system. 00:01:37.890 --> 00:01:39.950 align:middle line:84% If it is, you have to consider also 00:01:39.950 --> 00:01:42.740 align:middle line:84% the other system in order for it to be 00:01:42.740 --> 00:01:56.660 align:middle line:84% called a system in our little business, in our little theory. 00:01:56.660 --> 00:02:03.710 align:middle line:84% Then we define a property as being a physical observable, so 00:02:03.710 --> 00:02:10.190 align:middle line:84% something that you would obtain by probing your system 00:02:10.190 --> 00:02:13.310 align:middle line:84% in some way according to some recipe, some measurement 00:02:13.310 --> 00:02:14.270 align:middle line:90% procedure. 00:02:14.270 --> 00:02:17.990 align:middle line:84% And measurement procedures are what define these properties. 00:02:17.990 --> 00:02:21.830 align:middle line:84% And we gave the measurement procedure for energy, 00:02:21.830 --> 00:02:24.780 align:middle line:90% for example. 00:02:24.780 --> 00:02:27.730 align:middle line:84% We talked about what we mean by the word, state. 00:02:27.730 --> 00:02:35.770 align:middle line:84% It's the list of numbers that give 00:02:35.770 --> 00:02:39.165 align:middle line:84% the values of the properties at one instant of time. 00:02:39.165 --> 00:02:43.260 align:middle line:90% 00:02:43.260 --> 00:02:48.790 align:middle line:84% And the time evolution is described by the equation 00:02:48.790 --> 00:02:55.770 align:middle line:90% of motion in a detailed model. 00:02:55.770 --> 00:03:00.300 align:middle line:84% But it can also be described by just an initial state, 00:03:00.300 --> 00:03:05.460 align:middle line:84% a final state, and a description of the effects 00:03:05.460 --> 00:03:09.540 align:middle line:84% that occurred outside of your system 00:03:09.540 --> 00:03:11.680 align:middle line:90% during that time interval. 00:03:11.680 --> 00:03:14.640 align:middle line:90% 00:03:14.640 --> 00:03:20.780 align:middle line:84% In either way, we have a few things to check. 00:03:20.780 --> 00:03:24.560 align:middle line:84% In particular, the important business of thermodynamics 00:03:24.560 --> 00:03:28.760 align:middle line:84% is that it has to-- you have to check two things. 00:03:28.760 --> 00:03:32.690 align:middle line:84% One, the consistency with the first law, 00:03:32.690 --> 00:03:37.730 align:middle line:84% which wouldn't be a theorem in the framework in which you start 00:03:37.730 --> 00:03:39.060 align:middle line:90% from the equation of motion. 00:03:39.060 --> 00:03:41.652 align:middle line:84% Theorem of the equation of motion. 00:03:41.652 --> 00:03:45.690 align:middle line:84% And you also have to check the second law, which 00:03:45.690 --> 00:03:50.465 align:middle line:84% would be another theorem of the dynamical equation. 00:03:50.465 --> 00:03:54.910 align:middle line:90% 00:03:54.910 --> 00:03:58.660 align:middle line:84% The idea of process, already said, 00:03:58.660 --> 00:04:03.050 align:middle line:84% is three things, initial state, final state, 00:04:03.050 --> 00:04:06.760 align:middle line:84% and effects on the environment of the system. 00:04:06.760 --> 00:04:11.780 align:middle line:84% And how do we account for effects in the environment? 00:04:11.780 --> 00:04:16.664 align:middle line:84% Well, the most we can say is how its state changes. 00:04:16.664 --> 00:04:21.130 align:middle line:90% 00:04:21.130 --> 00:04:26.250 align:middle line:84% Then by defining a weight process, namely 00:04:26.250 --> 00:04:30.660 align:middle line:84% one in which the only external effect is the highering-- 00:04:30.660 --> 00:04:34.320 align:middle line:84% lifting or lowering of a weight in a gravity field, which 00:04:34.320 --> 00:04:38.560 align:middle line:84% is a prototype of the mechanical effects, 00:04:38.560 --> 00:04:42.410 align:middle line:84% we could introduce the statement of the first law. 00:04:42.410 --> 00:04:45.350 align:middle line:90% 00:04:45.350 --> 00:04:47.220 align:middle line:84% It is an ingredient of that statement. 00:04:47.220 --> 00:04:50.560 align:middle line:84% It says that you take any system and any-- 00:04:50.560 --> 00:04:54.410 align:middle line:84% pick any two states of that system. 00:04:54.410 --> 00:04:59.790 align:middle line:84% They can always be interconnected 00:04:59.790 --> 00:05:01.040 align:middle line:90% by means of a weight process. 00:05:01.040 --> 00:05:03.590 align:middle line:90% 00:05:03.590 --> 00:05:08.360 align:middle line:84% You-- not always you can do it in the direction you want, 00:05:08.360 --> 00:05:10.710 align:middle line:84% but if you cannot do it in one direction, 00:05:10.710 --> 00:05:12.103 align:middle line:90% you can do it in the other. 00:05:12.103 --> 00:05:16.180 align:middle line:90% 00:05:16.180 --> 00:05:22.390 align:middle line:84% And that gives grounds for the definition -- 00:05:22.390 --> 00:05:25.690 align:middle line:84% support for the measurement procedure 00:05:25.690 --> 00:05:31.150 align:middle line:84% that we introduce to define the property energy. 00:05:31.150 --> 00:05:35.350 align:middle line:84% Without the first law, you couldn't 00:05:35.350 --> 00:05:40.760 align:middle line:84% assert that by making this procedure, 00:05:40.760 --> 00:05:45.110 align:middle line:84% you get a property, a value of a property. 00:05:45.110 --> 00:05:48.106 align:middle line:84% So essentially, this is the main-- 00:05:48.106 --> 00:05:51.830 align:middle line:84% the most important consequence of the first law 00:05:51.830 --> 00:05:57.120 align:middle line:84% is the existence of property energy, which yes, then, it 00:05:57.120 --> 00:06:01.220 align:middle line:84% is additive, it can be exchanged between systems, 00:06:01.220 --> 00:06:03.720 align:middle line:90% and it's conserved. 00:06:03.720 --> 00:06:06.500 align:middle line:90% 00:06:06.500 --> 00:06:12.640 align:middle line:84% And that gives rise to the energy balance equation 00:06:12.640 --> 00:06:17.090 align:middle line:84% with this symbol, which you have applied 00:06:17.090 --> 00:06:20.980 align:middle line:84% in your undergraduate courses or previous-- 00:06:20.980 --> 00:06:26.190 align:middle line:84% We know every time we perform a model, 00:06:26.190 --> 00:06:29.750 align:middle line:84% we perform an energy balance to account 00:06:29.750 --> 00:06:34.430 align:middle line:84% for to check that the difference between the final and initial 00:06:34.430 --> 00:06:42.420 align:middle line:84% value of property energy is equal to the energy that-- 00:06:42.420 --> 00:06:45.960 align:middle line:84% the net energy that is being received by the system 00:06:45.960 --> 00:06:48.680 align:middle line:90% during that time interval. 00:06:48.680 --> 00:06:53.430 align:middle line:84% So that's a statement of conservation of energy. 00:06:53.430 --> 00:06:55.990 align:middle line:84% So some people call this, "I write the first law," 00:06:55.990 --> 00:06:58.550 align:middle line:84% but I prefer to say "I write the energy balance." 00:06:58.550 --> 00:07:02.800 align:middle line:90% 00:07:02.800 --> 00:07:04.930 align:middle line:84% Then we headed towards the second law 00:07:04.930 --> 00:07:10.390 align:middle line:84% by introducing this idea of classification of states 00:07:10.390 --> 00:07:13.510 align:middle line:84% based on how they change in time and why 00:07:13.510 --> 00:07:16.600 align:middle line:84% they change in time, particularly focused 00:07:16.600 --> 00:07:19.220 align:middle line:90% on equilibrium for the moment. 00:07:19.220 --> 00:07:22.220 align:middle line:84% But later on, we will focus on nonequilibrium. 00:07:22.220 --> 00:07:24.810 align:middle line:90% 00:07:24.810 --> 00:07:27.190 align:middle line:84% And there is equilibrium and equilibrium. 00:07:27.190 --> 00:07:29.730 align:middle line:84% And we defined what we mean by stable equilibrium 00:07:29.730 --> 00:07:31.770 align:middle line:90% and metastable equilibrium. 00:07:31.770 --> 00:07:35.970 align:middle line:84% The stable equilibrium state is most important 00:07:35.970 --> 00:07:38.670 align:middle line:84% because it enters the statement of the second law. 00:07:38.670 --> 00:07:43.960 align:middle line:90% And it is defined as a state-- 00:07:43.960 --> 00:07:45.350 align:middle line:90% an equilibrium state. 00:07:45.350 --> 00:07:48.920 align:middle line:84% So that means it doesn't move, it doesn't change in time, 00:07:48.920 --> 00:07:51.200 align:middle line:90% it doesn't move in state space. 00:07:51.200 --> 00:07:53.890 align:middle line:90% So it doesn't change with time. 00:07:53.890 --> 00:07:57.110 align:middle line:90% 00:07:57.110 --> 00:08:04.570 align:middle line:84% And you cannot change it without paying a cost, 00:08:04.570 --> 00:08:07.990 align:middle line:84% without adding some net external effects. 00:08:07.990 --> 00:08:10.930 align:middle line:84% You need to spend something in order 00:08:10.930 --> 00:08:14.080 align:middle line:84% to change a stable equilibrium state. 00:08:14.080 --> 00:08:21.260 align:middle line:84% You cannot do it for free, so to speak. 00:08:21.260 --> 00:08:23.140 align:middle line:84% And here is the statement of the second law. 00:08:23.140 --> 00:08:26.150 align:middle line:84% It is also made up of two assertions. 00:08:26.150 --> 00:08:28.230 align:middle line:90% Let's start first with the-- 00:08:28.230 --> 00:08:31.150 align:middle line:84% we'll focus first on assertion 1. 00:08:31.150 --> 00:08:33.730 align:middle line:90% 00:08:33.730 --> 00:08:39.280 align:middle line:84% It responds to the question, how many stable equilibrium states 00:08:39.280 --> 00:08:45.610 align:middle line:84% does a system have, in a way that is apparently contradicting 00:08:45.610 --> 00:08:53.110 align:middle line:84% what we know from mechanics, because it says that if you fix 00:08:53.110 --> 00:08:56.110 align:middle line:84% for your system the amounts of constituents, so that means 00:08:56.110 --> 00:08:58.990 align:middle line:84% the number of particles of each kind, 00:08:58.990 --> 00:09:00.910 align:middle line:84% and the values of the parameters, that 00:09:00.910 --> 00:09:06.820 align:middle line:84% is all the volume of the system, or the other things that 00:09:06.820 --> 00:09:12.640 align:middle line:84% describe the external forces, or the things that you can control 00:09:12.640 --> 00:09:13.760 align:middle line:90% from the external. 00:09:13.760 --> 00:09:16.670 align:middle line:90% So you fix these. 00:09:16.670 --> 00:09:20.550 align:middle line:84% Then for each value of the energy, 00:09:20.550 --> 00:09:23.890 align:middle line:84% there is one, and only one, stable equilibrium state. 00:09:23.890 --> 00:09:30.480 align:middle line:84% That is the statement introduced by, I think, 00:09:30.480 --> 00:09:34.700 align:middle line:84% by George Hatsopoulos when he was collaborating with Joe 00:09:34.700 --> 00:09:40.158 align:middle line:84% Keenan in the '60s in this department. 00:09:40.158 --> 00:09:43.060 align:middle line:90% 00:09:43.060 --> 00:09:45.705 align:middle line:84% We'll talk about this, the second assertion, in a moment. 00:09:45.705 --> 00:09:50.530 align:middle line:90% 00:09:50.530 --> 00:09:55.510 align:middle line:84% And we will see also, as we go on, 00:09:55.510 --> 00:09:59.380 align:middle line:84% that in fact, there is no contradiction as long as we 00:09:59.380 --> 00:10:04.060 align:middle line:84% understand, and we convince ourselves, that thermodynamics 00:10:04.060 --> 00:10:06.520 align:middle line:84% considers a set of states that is 00:10:06.520 --> 00:10:10.880 align:middle line:84% broader than the set of states considered in mechanics. 00:10:10.880 --> 00:10:13.000 align:middle line:84% So within the set of mechanics, there 00:10:13.000 --> 00:10:14.870 align:middle line:84% is only one stable equilibrium state, 00:10:14.870 --> 00:10:18.650 align:middle line:84% which is the lowest energy state for the given 00:10:18.650 --> 00:10:20.810 align:middle line:90% values of n and beta. 00:10:20.810 --> 00:10:27.282 align:middle line:84% But in the more broader set considered by thermodynamics, 00:10:27.282 --> 00:10:30.320 align:middle line:84% there is one equilibrium state, stable equilibrium state, 00:10:30.320 --> 00:10:35.990 align:middle line:90% for every value of the energy. 00:10:35.990 --> 00:10:40.370 align:middle line:84% And this has consequences, important consequences, 00:10:40.370 --> 00:10:45.370 align:middle line:90% that we will discuss today. 00:10:45.370 --> 00:10:51.280 align:middle line:90% 00:10:51.280 --> 00:10:56.510 align:middle line:84% Yeah, so one would ask, well, why-- 00:10:56.510 --> 00:10:59.110 align:middle line:84% I mean, we already have statements of the second law. 00:10:59.110 --> 00:11:01.950 align:middle line:84% Why do you accept this other one? 00:11:01.950 --> 00:11:07.620 align:middle line:84% Well, that's because it is more general. 00:11:07.620 --> 00:11:12.170 align:middle line:84% And it implies as consequences, as theorems, 00:11:12.170 --> 00:11:15.540 align:middle line:84% the most famous statements of the second law, 00:11:15.540 --> 00:11:21.900 align:middle line:84% which are the Kelvin-Planck statement, called also 00:11:21.900 --> 00:11:23.730 align:middle line:84% impossibility of a perpetual motion 00:11:23.730 --> 00:11:28.140 align:middle line:84% machine of the second kind, this is the only thing we will prove 00:11:28.140 --> 00:11:34.050 align:middle line:84% today in some fashion because it's easy, 00:11:34.050 --> 00:11:36.650 align:middle line:84% is a direct consequence of the first assertion 00:11:36.650 --> 00:11:39.330 align:middle line:90% of the second law. 00:11:39.330 --> 00:11:45.600 align:middle line:84% Later on, not today, probably next time, 00:11:45.600 --> 00:11:49.290 align:middle line:84% we prove the other statement of the second law, which 00:11:49.290 --> 00:11:52.990 align:middle line:84% is Clausius statement, the one that 00:11:52.990 --> 00:11:59.780 align:middle line:84% says that heat does not go from cold to hot spontaneously, 00:11:59.780 --> 00:12:01.820 align:middle line:90% but vice versa. 00:12:01.820 --> 00:12:04.790 align:middle line:84% That requires that we introduce-- 00:12:04.790 --> 00:12:08.540 align:middle line:84% we define what we mean by heat, what we mean by temperature. 00:12:08.540 --> 00:12:13.105 align:middle line:84% It takes a while, not long, but another lecture. 00:12:13.105 --> 00:12:15.960 align:middle line:90% 00:12:15.960 --> 00:12:18.480 align:middle line:84% But the beauty of this construction 00:12:18.480 --> 00:12:21.750 align:middle line:84% is that you get to that result as a theorem 00:12:21.750 --> 00:12:25.110 align:middle line:84% on-- of the combined first and second law. 00:12:25.110 --> 00:12:27.690 align:middle line:90% 00:12:27.690 --> 00:12:29.910 align:middle line:84% There is also another important statement 00:12:29.910 --> 00:12:34.290 align:middle line:84% by Carathéodory, which is more mathematical, though, which 00:12:34.290 --> 00:12:36.075 align:middle line:90% can also be proved. 00:12:36.075 --> 00:12:37.940 align:middle line:84% But we will not do it in this course. 00:12:37.940 --> 00:12:41.010 align:middle line:90% 00:12:41.010 --> 00:12:43.740 align:middle line:84% OK, so what is the perpetual-- so now, 00:12:43.740 --> 00:12:48.060 align:middle line:84% I want to talk about the Kelvin-Planck statement. 00:12:48.060 --> 00:12:52.505 align:middle line:84% What is a perpetual motion machine of the second kind? 00:12:52.505 --> 00:12:55.470 align:middle line:90% 00:12:55.470 --> 00:12:59.250 align:middle line:84% Alright, it is any device, essentially, 00:12:59.250 --> 00:13:03.480 align:middle line:84% a system in our language, that would 00:13:03.480 --> 00:13:08.540 align:middle line:84% start from a stable equilibrium state. 00:13:08.540 --> 00:13:13.040 align:middle line:84% Remember, the state that cannot be changed without leaving some 00:13:13.040 --> 00:13:19.630 align:middle line:84% net external effects, and ends in another state, 00:13:19.630 --> 00:13:24.300 align:middle line:90% a different state, while-- 00:13:24.300 --> 00:13:30.270 align:middle line:84% in a weight process in which the weight goes up, 00:13:30.270 --> 00:13:34.740 align:middle line:84% so in which you have transferred some energy out of your system 00:13:34.740 --> 00:13:37.760 align:middle line:90% and into lifting the weight. 00:13:37.760 --> 00:13:41.820 align:middle line:84% If you managed to do that, it's a perpetual motion machine 00:13:41.820 --> 00:13:42.790 align:middle line:90% of the second kind. 00:13:42.790 --> 00:13:46.000 align:middle line:90% 00:13:46.000 --> 00:13:51.830 align:middle line:84% It's funny because it has been very long 00:13:51.830 --> 00:13:56.630 align:middle line:84% since I have not received any emails 00:13:56.630 --> 00:13:59.360 align:middle line:90% from these crazy inventors. 00:13:59.360 --> 00:14:06.660 align:middle line:84% But perhaps because I mentioned it, or maybe Google listens, 00:14:06.660 --> 00:14:11.790 align:middle line:84% or maybe that's because I'm at MIT, but on Wednesday, 00:14:11.790 --> 00:14:13.050 align:middle line:90% I received one. 00:14:13.050 --> 00:14:14.880 align:middle line:84% I would be happy to show it to you. 00:14:14.880 --> 00:14:18.890 align:middle line:84% Very funny, somebody who wants to put a container in the sea 00:14:18.890 --> 00:14:19.730 align:middle line:90% and-- 00:14:19.730 --> 00:14:23.540 align:middle line:84% it's-- yeah, it's not even a perpetual machine of the second 00:14:23.540 --> 00:14:24.060 align:middle line:90% kind. 00:14:24.060 --> 00:14:25.890 align:middle line:84% It's of the first kind, I would say. 00:14:25.890 --> 00:14:29.820 align:middle line:84% And I think the guy needs to learn some fluid mechanics, 00:14:29.820 --> 00:14:37.940 align:middle line:84% but in the old days, it was a continuous flow of applications 00:14:37.940 --> 00:14:41.510 align:middle line:90% for patents in-- 00:14:41.510 --> 00:14:47.430 align:middle line:84% violating the impossibility of the same. 00:14:47.430 --> 00:14:54.140 align:middle line:84% Until in the '60s or the '70s, I forget, the Patent Office, 00:14:54.140 --> 00:14:58.510 align:middle line:84% US Patent Office, decided that anything that violates 00:14:58.510 --> 00:15:01.300 align:middle line:84% the second law, or the principle, 00:15:01.300 --> 00:15:05.680 align:middle line:84% of this principle or impossibility of a PMM2 is 00:15:05.680 --> 00:15:09.235 align:middle line:84% rejected immediately without going to a referee. 00:15:09.235 --> 00:15:11.750 align:middle line:90% 00:15:11.750 --> 00:15:15.460 align:middle line:84% That's because, as I may have told you, 00:15:15.460 --> 00:15:19.050 align:middle line:84% I forget the last time, but I remember 00:15:19.050 --> 00:15:21.990 align:middle line:84% when I was a student with Professor Gyftopoulos, 00:15:21.990 --> 00:15:29.340 align:middle line:84% he would be so well educated to reply 00:15:29.340 --> 00:15:33.600 align:middle line:84% in return and in full length to any letter 00:15:33.600 --> 00:15:38.370 align:middle line:84% that he would receive, which is a principle that I still have. 00:15:38.370 --> 00:15:43.620 align:middle line:84% But he would send, like, 20 pages to explain line 00:15:43.620 --> 00:15:46.010 align:middle line:84% by line why you were wrong and why-- 00:15:46.010 --> 00:15:49.650 align:middle line:84% and often, these crazy inventors make 00:15:49.650 --> 00:15:53.700 align:middle line:84% very complicated and convoluted reasoning 00:15:53.700 --> 00:16:00.540 align:middle line:84% that makes it very hard to dismantle and reject. 00:16:00.540 --> 00:16:05.880 align:middle line:84% But he was doing that consistently whether you-- 00:16:05.880 --> 00:16:10.800 align:middle line:84% whether it was asked by the Patent Office or by the inventor 00:16:10.800 --> 00:16:13.080 align:middle line:90% directly. 00:16:13.080 --> 00:16:18.010 align:middle line:84% Fortunately, we don't have that sort of spam any longer. 00:16:18.010 --> 00:16:20.060 align:middle line:90% So how do we prove this? 00:16:20.060 --> 00:16:26.070 align:middle line:84% Well, one way, one simple way is to prove it 00:16:26.070 --> 00:16:28.460 align:middle line:84% if my system can be divided into two parts. 00:16:28.460 --> 00:16:32.770 align:middle line:84% So let's reason this so that we see, also, 00:16:32.770 --> 00:16:40.260 align:middle line:84% what kind of proofs, logical proofs, we are talking about. 00:16:40.260 --> 00:16:44.700 align:middle line:90% And it is a proof ab absurdum. 00:16:44.700 --> 00:16:48.690 align:middle line:84% So that means we assume that the PMM2 is possible 00:16:48.690 --> 00:16:53.710 align:middle line:84% and show that it leads to a contradiction. 00:16:53.710 --> 00:16:57.130 align:middle line:84% So let us show that our system, which 00:16:57.130 --> 00:16:59.170 align:middle line:84% can be divided in two parts, that also 00:16:59.170 --> 00:17:01.400 align:middle line:84% is an additional assumption can be removed, 00:17:01.400 --> 00:17:03.700 align:middle line:84% but it makes things more complicated. 00:17:03.700 --> 00:17:07.579 align:middle line:84% Let's assume that it does start from a stable equilibrium state 00:17:07.579 --> 00:17:10.569 align:middle line:84% and it does end in a different state, A1, 00:17:10.569 --> 00:17:16.702 align:middle line:84% with as external effect the lifting of a weight. 00:17:16.702 --> 00:17:17.359 align:middle line:90% All right. 00:17:17.359 --> 00:17:20.810 align:middle line:84% If that's true, if that happened, 00:17:20.810 --> 00:17:23.690 align:middle line:84% then you can take the energy of the weight 00:17:23.690 --> 00:17:30.620 align:middle line:84% back and use it to take one part of the system 00:17:30.620 --> 00:17:32.415 align:middle line:84% and accelerate it to some velocity. 00:17:32.415 --> 00:17:35.150 align:middle line:84% So give it in form of kinetic energy. 00:17:35.150 --> 00:17:36.830 align:middle line:84% Take the potential energy of the weight 00:17:36.830 --> 00:17:41.030 align:middle line:84% and give it back as kinetic energy of the second part 00:17:41.030 --> 00:17:41.700 align:middle line:90% of the system. 00:17:41.700 --> 00:17:44.630 align:middle line:90% 00:17:44.630 --> 00:17:47.860 align:middle line:90% So what have we achieved? 00:17:47.860 --> 00:17:50.950 align:middle line:84% So look at this state and the initial state. 00:17:50.950 --> 00:17:55.820 align:middle line:90% 00:17:55.820 --> 00:17:59.140 align:middle line:90% Why is this not possible? 00:17:59.140 --> 00:18:04.150 align:middle line:84% Because the final state is different from the initial one, 00:18:04.150 --> 00:18:11.070 align:middle line:84% and the initial one was a stable equilibrium state. 00:18:11.070 --> 00:18:13.950 align:middle line:84% And there are no external effects because the weight went 00:18:13.950 --> 00:18:15.780 align:middle line:90% up, but then it also went down. 00:18:15.780 --> 00:18:18.450 align:middle line:90% So the net external effect is 0. 00:18:18.450 --> 00:18:22.060 align:middle line:84% Yet, we managed to start from a stable equilibrium state 00:18:22.060 --> 00:18:27.770 align:middle line:84% and change it to an entirely-- to a different state. 00:18:27.770 --> 00:18:32.390 align:middle line:84% That contradicts the definition of stable equilibrium. 00:18:32.390 --> 00:18:35.280 align:middle line:90% OK, all right. 00:18:35.280 --> 00:18:40.200 align:middle line:84% So this is the proof of Kelvin-Planck's statement 00:18:40.200 --> 00:18:41.050 align:middle line:90% of the second law. 00:18:41.050 --> 00:18:45.430 align:middle line:90% 00:18:45.430 --> 00:18:48.920 align:middle line:84% Now, in order to go on the second aspect 00:18:48.920 --> 00:18:52.890 align:middle line:84% of this second law, the second assertion, 00:18:52.890 --> 00:18:56.015 align:middle line:84% we left behind one of the definitions that 00:18:56.015 --> 00:18:57.860 align:middle line:90% are peculiar to thermodynamics. 00:18:57.860 --> 00:19:00.540 align:middle line:84% And it's that of reversible process. 00:19:00.540 --> 00:19:02.815 align:middle line:84% What do we mean by reversible process? 00:19:02.815 --> 00:19:06.940 align:middle line:90% 00:19:06.940 --> 00:19:09.610 align:middle line:84% OK, the definition is as follows. 00:19:09.610 --> 00:19:12.290 align:middle line:84% You have a process, first of all. 00:19:12.290 --> 00:19:14.680 align:middle line:84% So it means you have an initial and final state. 00:19:14.680 --> 00:19:21.550 align:middle line:84% And you have an external effect, B1 goes to B2. 00:19:21.550 --> 00:19:26.260 align:middle line:84% The state of the environment changes so much. 00:19:26.260 --> 00:19:30.820 align:middle line:84% OK, the process is, we call it reversible if it is possible, 00:19:30.820 --> 00:19:36.340 align:middle line:84% not necessarily retracing backwards exactly 00:19:36.340 --> 00:19:41.590 align:middle line:84% the same forward path of your initial process. 00:19:41.590 --> 00:19:44.230 align:middle line:84% But in any other way, if you find 00:19:44.230 --> 00:19:52.370 align:middle line:84% a way to bring the state of your system back to its initial, 00:19:52.370 --> 00:19:55.160 align:middle line:84% and also the state of the environment 00:19:55.160 --> 00:19:59.630 align:middle line:84% back to its initial state, then and only then, 00:19:59.630 --> 00:20:02.940 align:middle line:90% we call the process reversible. 00:20:02.940 --> 00:20:09.280 align:middle line:84% OK, so that means you have to try hard all possibilities. 00:20:09.280 --> 00:20:13.500 align:middle line:84% So this is a very abstract definition, 00:20:13.500 --> 00:20:16.860 align:middle line:90% but it's also very precise. 00:20:16.860 --> 00:20:21.290 align:middle line:84% So I usually am not going to do it because they-- 00:20:21.290 --> 00:20:25.700 align:middle line:84% I want to make dust, but I usually take a piece of chalk 00:20:25.700 --> 00:20:30.470 align:middle line:84% and throw it to the wall, it breaks, and ask you, 00:20:30.470 --> 00:20:32.420 align:middle line:90% is that process-- 00:20:32.420 --> 00:20:35.340 align:middle line:84% was that process reversible or not? 00:20:35.340 --> 00:20:36.563 align:middle line:90% What would you say? 00:20:36.563 --> 00:20:37.562 align:middle line:90% It is not. 00:20:37.562 --> 00:20:46.200 align:middle line:90% Fine. 00:20:46.200 --> 00:20:52.100 align:middle line:90% But how did you decide? 00:20:52.100 --> 00:20:53.640 align:middle line:90% I know it's intuitive. 00:20:53.640 --> 00:21:08.574 align:middle line:90% 00:21:08.574 --> 00:21:09.290 align:middle line:90% Good. 00:21:09.290 --> 00:21:13.370 align:middle line:84% OK, so you use your physical intuition. 00:21:13.370 --> 00:21:15.080 align:middle line:84% That's because you have a background. 00:21:15.080 --> 00:21:17.870 align:middle line:84% You have already seen the idea of reversibility 00:21:17.870 --> 00:21:20.060 align:middle line:90% and irreversibility. 00:21:20.060 --> 00:21:27.590 align:middle line:84% But in our game, if you accept the rules of our game, 00:21:27.590 --> 00:21:29.870 align:middle line:84% what should you have done to decide? 00:21:29.870 --> 00:21:33.930 align:middle line:84% You should have tried to reconstruct the state 00:21:33.930 --> 00:21:35.990 align:middle line:84% from the piece of chalk to see if it 00:21:35.990 --> 00:21:38.520 align:middle line:90% is possible to place it back. 00:21:38.520 --> 00:21:42.090 align:middle line:84% And then also, whatever it did to the environment 00:21:42.090 --> 00:21:44.560 align:middle line:90% must be undone. 00:21:44.560 --> 00:21:47.110 align:middle line:84% And you have to try all possible ways. 00:21:47.110 --> 00:21:49.720 align:middle line:84% So how could you be so quick in answering? 00:21:49.720 --> 00:21:55.890 align:middle line:84% OK, so correct answer, but you didn't apply our rules. 00:21:55.890 --> 00:22:01.330 align:middle line:84% This is to just underline, this is a very abstract concept, 00:22:01.330 --> 00:22:03.310 align:middle line:90% and impossible in practice. 00:22:03.310 --> 00:22:06.310 align:middle line:84% It doesn't correspond to a procedure 00:22:06.310 --> 00:22:09.640 align:middle line:84% so that you can verify if a process is irreversible or not. 00:22:09.640 --> 00:22:11.940 align:middle line:90% But it's a concept. 00:22:11.940 --> 00:22:15.880 align:middle line:84% And soon, we will find a method, and of course, 00:22:15.880 --> 00:22:17.370 align:middle line:84% as soon as we have defined entropy, 00:22:17.370 --> 00:22:21.120 align:middle line:84% that will be the method to compute, 00:22:21.120 --> 00:22:25.920 align:middle line:84% to calculate entropy, measure entropy, and see how it changes. 00:22:25.920 --> 00:22:29.740 align:middle line:84% And from that, from the entropy balance equation, 00:22:29.740 --> 00:22:33.520 align:middle line:84% we can decide whether a process is reversible or not. 00:22:33.520 --> 00:22:37.830 align:middle line:84% So we develop a practical method to measure 00:22:37.830 --> 00:22:41.270 align:middle line:90% this very abstract concept. 00:22:41.270 --> 00:22:45.920 align:middle line:84% So why is reversible-- the concept of reversible process 00:22:45.920 --> 00:22:48.080 align:middle line:90% important in the second law? 00:22:48.080 --> 00:22:52.610 align:middle line:84% Because the second statement, second assertion, 00:22:52.610 --> 00:23:00.810 align:middle line:84% it says, you start from any state of your system 00:23:00.810 --> 00:23:09.428 align:middle line:84% and you can always go to a stable equilibrium state 00:23:09.428 --> 00:23:12.680 align:middle line:84% to bring your system from that state 00:23:12.680 --> 00:23:21.730 align:middle line:84% to a stable equilibrium state in a weight process which 00:23:21.730 --> 00:23:22.750 align:middle line:90% is reversible. 00:23:22.750 --> 00:23:25.280 align:middle line:90% 00:23:25.280 --> 00:23:28.110 align:middle line:84% So it's all our concepts together. 00:23:28.110 --> 00:23:30.710 align:middle line:90% That's why we find them before. 00:23:30.710 --> 00:23:33.950 align:middle line:84% Now, we put them all together there. 00:23:33.950 --> 00:23:38.300 align:middle line:84% But if you go slowly and you mumble about it, 00:23:38.300 --> 00:23:41.270 align:middle line:90% that's what it says. 00:23:41.270 --> 00:23:42.930 align:middle line:90% We defined all the terms. 00:23:42.930 --> 00:23:44.480 align:middle line:90% Start from an initial state. 00:23:44.480 --> 00:23:50.790 align:middle line:84% You can always get your system to a stable equilibrium state 00:23:50.790 --> 00:23:53.760 align:middle line:90% by means of a weight process-- 00:23:53.760 --> 00:23:55.240 align:middle line:90% a reversible weight process. 00:23:55.240 --> 00:24:04.860 align:middle line:90% 00:24:04.860 --> 00:24:19.120 align:middle line:84% It turns out that if you try other ways to change your state 00:24:19.120 --> 00:24:24.450 align:middle line:84% to a stable equilibrium state, other ways that are not 00:24:24.450 --> 00:24:32.760 align:middle line:84% reversible, you would lift the weight less than 00:24:32.760 --> 00:24:34.577 align:middle line:90% when it is reversible. 00:24:34.577 --> 00:24:37.320 align:middle line:90% 00:24:37.320 --> 00:24:40.540 align:middle line:90% So that means this is a theorem. 00:24:40.540 --> 00:24:44.740 align:middle line:84% And yeah, it's proved in the usual book. 00:24:44.740 --> 00:24:52.480 align:middle line:84% And the theorem says that the maximum lift that you 00:24:52.480 --> 00:24:56.420 align:middle line:84% can give to a weight in a weight process, 00:24:56.420 --> 00:25:00.720 align:middle line:84% with your system starting in any state, 00:25:00.720 --> 00:25:05.760 align:middle line:84% is obtained when your system ends in a stable equilibrium 00:25:05.760 --> 00:25:08.355 align:middle line:84% state and when the process is reversible. 00:25:08.355 --> 00:25:11.750 align:middle line:90% 00:25:11.750 --> 00:25:16.270 align:middle line:84% And that final stable equilibrium state 00:25:16.270 --> 00:25:21.940 align:middle line:84% is uniquely fixed by your initial state. 00:25:21.940 --> 00:25:23.680 align:middle line:84% Only one stable equilibrium state 00:25:23.680 --> 00:25:26.050 align:middle line:84% that can be reached in a reversible weight 00:25:26.050 --> 00:25:29.195 align:middle line:90% process from state A1. 00:25:29.195 --> 00:25:33.270 align:middle line:90% 00:25:33.270 --> 00:25:37.190 align:middle line:84% So this is enough to say that the amount of energy 00:25:37.190 --> 00:25:40.280 align:middle line:84% that you transferred to the weight in these conditions 00:25:40.280 --> 00:25:41.330 align:middle line:90% is a property. 00:25:41.330 --> 00:25:44.370 align:middle line:90% Depends only on state A1. 00:25:44.370 --> 00:25:51.170 align:middle line:84% So it depends only on the time t1, 00:25:51.170 --> 00:25:55.370 align:middle line:84% because the final state is a function of A1. 00:25:55.370 --> 00:26:01.430 align:middle line:84% So it also depends on t1, on the state at t1. 00:26:01.430 --> 00:26:04.100 align:middle line:84% And we call this property adiabatic availability 00:26:04.100 --> 00:26:05.865 align:middle line:90% of the system in state A1. 00:26:05.865 --> 00:26:10.560 align:middle line:90% 00:26:10.560 --> 00:26:13.430 align:middle line:90% It's the maximal-- 00:26:13.430 --> 00:26:16.380 align:middle line:84% I don't want to call it, but we could also 00:26:16.380 --> 00:26:18.180 align:middle line:84% call it work because, I mean, eventually 00:26:18.180 --> 00:26:20.590 align:middle line:90% that's a work interaction. 00:26:20.590 --> 00:26:22.440 align:middle line:84% So it's a maximum work you can extract 00:26:22.440 --> 00:26:28.500 align:middle line:84% from your system in the given state. 00:26:28.500 --> 00:26:34.520 align:middle line:84% And if you are an engineer and you're trying to make, 00:26:34.520 --> 00:26:48.470 align:middle line:84% devices to exploit natural resources like energy resources 00:26:48.470 --> 00:26:56.390 align:middle line:84% like coal, natural gas, nuclear, whatever, solar, you-- 00:26:56.390 --> 00:26:58.730 align:middle line:84% The main question that thermodynamics 00:26:58.730 --> 00:27:02.000 align:middle line:84% was set for 200 years ago by Carnot 00:27:02.000 --> 00:27:08.010 align:middle line:84% is, well, what's the most I can do out of this situation? 00:27:08.010 --> 00:27:12.113 align:middle line:84% And adiabatic availability is an answer. 00:27:12.113 --> 00:27:15.360 align:middle line:90% 00:27:15.360 --> 00:27:19.440 align:middle line:84% Of course, it gives an answer under certain conditions. 00:27:19.440 --> 00:27:21.860 align:middle line:84% You change the conditions, you get different answers. 00:27:21.860 --> 00:27:25.930 align:middle line:84% As we will see, we will have more availabilities. 00:27:25.930 --> 00:27:27.410 align:middle line:90% It depends on the condition. 00:27:27.410 --> 00:27:29.680 align:middle line:84% Here, the condition is in a weight process. 00:27:29.680 --> 00:27:33.680 align:middle line:84% So if I have my system, and I confine myself 00:27:33.680 --> 00:27:38.790 align:middle line:84% to just a weight process, that's the amount, the maximum amount 00:27:38.790 --> 00:27:41.760 align:middle line:84% of energy I can extract from my system. 00:27:41.760 --> 00:27:45.560 align:middle line:90% The most I can do out of it. 00:27:45.560 --> 00:27:53.340 align:middle line:84% If it weren't for an important detail, 00:27:53.340 --> 00:27:56.340 align:middle line:84% we would be satisfied that the adiabatic availability is all 00:27:56.340 --> 00:27:58.800 align:middle line:84% that we need in order to describe our engineering 00:27:58.800 --> 00:28:00.580 align:middle line:90% applications. 00:28:00.580 --> 00:28:04.442 align:middle line:84% The detail is that this property is not additive. 00:28:04.442 --> 00:28:07.260 align:middle line:84% And therefore, it's not very practical 00:28:07.260 --> 00:28:16.800 align:middle line:84% because when you have a system made out of two subsystems, 00:28:16.800 --> 00:28:18.450 align:middle line:84% and you measure the availability, 00:28:18.450 --> 00:28:21.000 align:middle line:84% the adiabatic availability of system A 00:28:21.000 --> 00:28:26.040 align:middle line:84% and then the adiabatic availability of system B, 00:28:26.040 --> 00:28:29.050 align:middle line:84% well, if you want to consider the combined system, 00:28:29.050 --> 00:28:30.750 align:middle line:84% you have to make another measurement 00:28:30.750 --> 00:28:36.690 align:middle line:84% to measure the adiabatic availability of the system C. 00:28:36.690 --> 00:28:40.040 align:middle line:84% The simplest example that I can think of 00:28:40.040 --> 00:28:46.200 align:middle line:84% is, suppose you have a glass of water, which is hot 00:28:46.200 --> 00:28:48.613 align:middle line:84% and a glass of water which is cold. 00:28:48.613 --> 00:28:52.380 align:middle line:90% 00:28:52.380 --> 00:28:56.240 align:middle line:84% So you measure the adiabatic availability of this one. 00:28:56.240 --> 00:28:58.700 align:middle line:84% And if it is hot, but in a stable equilibrium 00:28:58.700 --> 00:29:04.930 align:middle line:84% state, how much is the adiabatic availability? 00:29:04.930 --> 00:29:07.720 align:middle line:84% It's 0 because otherwise, it would be a PMM2. 00:29:07.720 --> 00:29:11.540 align:middle line:90% 00:29:11.540 --> 00:29:14.730 align:middle line:84% I mean, if you could-- if you started from a stable 00:29:14.730 --> 00:29:17.070 align:middle line:84% equilibrium state, you cannot lift a weight. 00:29:17.070 --> 00:29:20.040 align:middle line:84% So a system in a stable equilibrium state 00:29:20.040 --> 00:29:22.200 align:middle line:90% has 0 adiabatic availability. 00:29:22.200 --> 00:29:25.830 align:middle line:84% So the adiabatic availability of, we use 00:29:25.830 --> 00:29:32.430 align:middle line:84% a letter psi, so of this glass of water is 0. 00:29:32.430 --> 00:29:36.100 align:middle line:84% Also of that glass of water is 0. 00:29:36.100 --> 00:29:41.600 align:middle line:84% But if I consider the composite system, of course, 00:29:41.600 --> 00:29:48.380 align:middle line:84% we all know that, for example, with the Carnot engine, 00:29:48.380 --> 00:29:52.540 align:middle line:84% you can extract work out of this situation. 00:29:52.540 --> 00:29:57.520 align:middle line:84% So you see that 0 plus 0 does not make a number of different 00:29:57.520 --> 00:29:58.300 align:middle line:90% from 0. 00:29:58.300 --> 00:30:01.660 align:middle line:84% So we lack additivity of this property. 00:30:01.660 --> 00:30:06.130 align:middle line:84% And this is why it takes the next half hour 00:30:06.130 --> 00:30:10.240 align:middle line:90% to build up another property. 00:30:10.240 --> 00:30:15.130 align:middle line:84% It will be entropy, which will contain features 00:30:15.130 --> 00:30:18.190 align:middle line:84% so that we can compute the adiabatic availability 00:30:18.190 --> 00:30:20.090 align:middle line:90% and it will be additive. 00:30:20.090 --> 00:30:22.600 align:middle line:90% We have to build additivity on-- 00:30:22.600 --> 00:30:25.600 align:middle line:90% around adiabatic availability. 00:30:25.600 --> 00:30:31.100 align:middle line:90% 00:30:31.100 --> 00:30:38.030 align:middle line:84% There is also this other theorem that I'm not proving. 00:30:38.030 --> 00:30:43.090 align:middle line:84% But I'm pointing you at this proof. 00:30:43.090 --> 00:30:48.640 align:middle line:90% It's on page 76 of the book. 00:30:48.640 --> 00:30:50.780 align:middle line:84% Again, so since I'm not doing it, 00:30:50.780 --> 00:30:55.010 align:middle line:84% I'm not asking that you study it. 00:30:55.010 --> 00:30:58.995 align:middle line:84% But if you're curious and just go and look. 00:30:58.995 --> 00:31:02.060 align:middle line:90% 00:31:02.060 --> 00:31:07.460 align:middle line:84% It says that if you consider a weight process, so now, 00:31:07.460 --> 00:31:17.870 align:middle line:84% we confine ourselves to a weight process, between any two states, 00:31:17.870 --> 00:31:23.280 align:middle line:90% so clearly, if you-- 00:31:23.280 --> 00:31:25.800 align:middle line:84% since adiabatic availability is a property, 00:31:25.800 --> 00:31:29.070 align:middle line:90% you can measure it for state A1. 00:31:29.070 --> 00:31:32.640 align:middle line:84% You could also measure it for state A2. 00:31:32.640 --> 00:31:34.740 align:middle line:84% Like, energy, the property, so you 00:31:34.740 --> 00:31:38.980 align:middle line:84% can measure it for A1 and you can measure it for A2. 00:31:38.980 --> 00:31:43.670 align:middle line:84% So you give me two states of system A. 00:31:43.670 --> 00:31:46.550 align:middle line:84% And I can measure for each state its energy 00:31:46.550 --> 00:31:48.891 align:middle line:90% and its adiabatic availability. 00:31:48.891 --> 00:31:53.130 align:middle line:90% 00:31:53.130 --> 00:31:56.010 align:middle line:84% So it turns out that this theorem 00:31:56.010 --> 00:32:01.050 align:middle line:84% says if you have a necessary and sufficient condition 00:32:01.050 --> 00:32:05.380 align:middle line:84% for the process to be reversible, 00:32:05.380 --> 00:32:10.930 align:middle line:84% if the difference between energy and adiabatic availability, 00:32:10.930 --> 00:32:15.370 align:middle line:84% so you could interpret that as the energy that is not 00:32:15.370 --> 00:32:21.726 align:middle line:84% adiabatically available at the end, 00:32:21.726 --> 00:32:24.820 align:middle line:84% is equal to the energy not adiabatically 00:32:24.820 --> 00:32:26.690 align:middle line:90% available at the beginning. 00:32:26.690 --> 00:32:29.600 align:middle line:90% 00:32:29.600 --> 00:32:38.200 align:middle line:84% So in a reversible process, the energy not adiabatically 00:32:38.200 --> 00:32:42.400 align:middle line:90% available is conserved. 00:32:42.400 --> 00:32:52.420 align:middle line:84% Whereas in an irreversible process, it increases. 00:32:52.420 --> 00:32:56.340 align:middle line:84% Very similar, in fact, also entropy will play the same-- 00:32:56.340 --> 00:32:58.990 align:middle line:90% will play in a similar way. 00:32:58.990 --> 00:33:03.160 align:middle line:84% So this energy that is not adiabatically available 00:33:03.160 --> 00:33:09.110 align:middle line:90% is a witness of reversibility. 00:33:09.110 --> 00:33:13.985 align:middle line:84% It's the basis for measuring in practice whether a process is 00:33:13.985 --> 00:33:15.200 align:middle line:90% reversible or not. 00:33:15.200 --> 00:33:20.000 align:middle line:84% All you need to do, according to-- if you trust this theorem, 00:33:20.000 --> 00:33:23.030 align:middle line:84% is measure energy and measure adiabatic availability 00:33:23.030 --> 00:33:27.590 align:middle line:84% with the tools we developed and then check this inequality. 00:33:27.590 --> 00:33:30.750 align:middle line:84% If it is an equality, the process is reversible, 00:33:30.750 --> 00:33:34.100 align:middle line:84% the reverse-- sorry, the weight process is reversible. 00:33:34.100 --> 00:33:41.240 align:middle line:84% If it is an inequality like this one, 00:33:41.240 --> 00:33:45.950 align:middle line:90% the process is irreversible. 00:33:45.950 --> 00:33:48.240 align:middle line:84% If it's the other way around, it's impossible. 00:33:48.240 --> 00:33:50.420 align:middle line:90% You made be something wrong. 00:33:50.420 --> 00:33:57.170 align:middle line:84% It is impossible to decrease the amount of energy that 00:33:57.170 --> 00:34:01.050 align:middle line:90% is not adiabatically available. 00:34:01.050 --> 00:34:05.071 align:middle line:90% 00:34:05.071 --> 00:34:07.960 align:middle line:84% So this is a precursor of the principle 00:34:07.960 --> 00:34:11.860 align:middle line:84% that we will call of entropy non decrease-- we could call 00:34:11.860 --> 00:34:15.949 align:middle line:84% it the principle of non decrease of the energy, 00:34:15.949 --> 00:34:18.530 align:middle line:84% nonadiabatic-- adiabatically non available. 00:34:18.530 --> 00:34:21.909 align:middle line:90% 00:34:21.909 --> 00:34:33.270 align:middle line:84% OK, so now, we try to make or build up this property. 00:34:33.270 --> 00:34:38.230 align:middle line:84% There has-- that builds additivity 00:34:38.230 --> 00:34:41.290 align:middle line:90% into adiabatic availability. 00:34:41.290 --> 00:34:44.260 align:middle line:84% So we are heading towards entropy. 00:34:44.260 --> 00:34:48.679 align:middle line:90% 00:34:48.679 --> 00:34:53.630 align:middle line:84% In order to do that, we introduce some other concepts. 00:34:53.630 --> 00:34:57.470 align:middle line:84% The first one is the concept of mutual stable equilibrium, 00:34:57.470 --> 00:35:00.975 align:middle line:84% or mutual equilibrium, between two systems. 00:35:00.975 --> 00:35:04.250 align:middle line:90% 00:35:04.250 --> 00:35:06.450 align:middle line:84% So two systems are in mutual equilibrium, 00:35:06.450 --> 00:35:07.710 align:middle line:90% like the hot and cold. 00:35:07.710 --> 00:35:10.280 align:middle line:90% 00:35:10.280 --> 00:35:14.620 align:middle line:84% These are not in mutual equilibrium 00:35:14.620 --> 00:35:18.970 align:middle line:84% because the composite system is not 00:35:18.970 --> 00:35:21.220 align:middle line:90% in a stable equilibrium state. 00:35:21.220 --> 00:35:23.890 align:middle line:84% The fact that it has adiabatic availability, 00:35:23.890 --> 00:35:25.720 align:middle line:84% because that's an if-and-only-if theorem, 00:35:25.720 --> 00:35:30.850 align:middle line:84% so the fact that it has non-zero adiabatic availability is 00:35:30.850 --> 00:35:34.250 align:middle line:84% equivalent to not being a stable equilibrium state. 00:35:34.250 --> 00:35:37.860 align:middle line:90% 00:35:37.860 --> 00:35:41.360 align:middle line:84% So these two, they are individually 00:35:41.360 --> 00:35:43.870 align:middle line:90% in stable equilibrium. 00:35:43.870 --> 00:35:48.200 align:middle line:84% But the composite is not in a stable equilibrium state. 00:35:48.200 --> 00:35:51.655 align:middle line:84% So they are not in mutual stable equilibrium. 00:35:51.655 --> 00:35:56.080 align:middle line:84% In another situation in which instead, they are individually, 00:35:56.080 --> 00:36:00.020 align:middle line:84% and also the composite system has zero adiabatic availability, 00:36:00.020 --> 00:36:07.180 align:middle line:84% so that means the composite is in stable equilibrium, 00:36:07.180 --> 00:36:09.910 align:middle line:84% then you say that the two systems 00:36:09.910 --> 00:36:12.010 align:middle line:90% are in mutual equilibrium. 00:36:12.010 --> 00:36:15.860 align:middle line:90% That's the concept. 00:36:15.860 --> 00:36:23.250 align:middle line:84% The other concept is that of thermal reservoir. 00:36:23.250 --> 00:36:29.745 align:middle line:84% A thermal reservoir is a particular system, I would say, 00:36:29.745 --> 00:36:38.870 align:middle line:84% a limiting system, that has this feature that I 00:36:38.870 --> 00:36:41.120 align:middle line:84% will describe with a practical system 00:36:41.120 --> 00:36:44.550 align:middle line:90% that we can make in any lab. 00:36:44.550 --> 00:36:47.800 align:middle line:84% You take a substance, like water, 00:36:47.800 --> 00:36:57.250 align:middle line:84% in a fixed volume container and you produce a combination 00:36:57.250 --> 00:36:59.030 align:middle line:90% that we call triple point. 00:36:59.030 --> 00:37:02.650 align:middle line:84% So a situation in which you have the coexistence 00:37:02.650 --> 00:37:09.970 align:middle line:84% in stable equilibrium of ice, liquid water, and water vapor. 00:37:09.970 --> 00:37:14.171 align:middle line:84% Right here, ice, liquid water, and water vapor. 00:37:14.171 --> 00:37:22.840 align:middle line:84% OK, then you find another system, let's call it 00:37:22.840 --> 00:37:27.200 align:middle line:84% C, that is in mutual equilibrium with it. 00:37:27.200 --> 00:37:29.960 align:middle line:84% For example, it could be another copy 00:37:29.960 --> 00:37:34.490 align:middle line:84% of another piece of water at the triple point, 00:37:34.490 --> 00:37:37.390 align:middle line:90% but not necessarily. 00:37:37.390 --> 00:37:41.490 align:middle line:84% The important thing that defines a reservoir, 00:37:41.490 --> 00:37:45.680 align:middle line:84% a thermal reservoir, is that if you 00:37:45.680 --> 00:37:54.810 align:middle line:84% change the energy of this particular system, for example, 00:37:54.810 --> 00:37:56.730 align:middle line:84% I consider a state with more energy. 00:37:56.730 --> 00:38:01.490 align:middle line:84% So it will have lesser ice and maybe a little more water 00:38:01.490 --> 00:38:05.440 align:middle line:90% vapor or less liquid water. 00:38:05.440 --> 00:38:11.240 align:middle line:84% In any case, so the proportions of the three phases changed. 00:38:11.240 --> 00:38:15.750 align:middle line:84% So these are two different states with different energies. 00:38:15.750 --> 00:38:28.420 align:middle line:84% But they are both in mutual equilibrium with the system C. 00:38:28.420 --> 00:38:33.070 align:middle line:84% Now, why did I choose triple-point water? 00:38:33.070 --> 00:38:40.400 align:middle line:84% That's because it is easy to prepare in any lab. 00:38:40.400 --> 00:38:44.940 align:middle line:90% 00:38:44.940 --> 00:38:50.420 align:middle line:84% And as you know, we haven't defined what the temperature is. 00:38:50.420 --> 00:38:53.070 align:middle line:90% In a moment, we'll do it. 00:38:53.070 --> 00:38:56.020 align:middle line:84% But just because you know what temperature is, 00:38:56.020 --> 00:39:03.910 align:middle line:84% a triple point is at a low pressure like 0.6 kilopascal 00:39:03.910 --> 00:39:08.390 align:middle line:90% and at a temperature of 273.16. 00:39:08.390 --> 00:39:13.210 align:middle line:90% So it's 0.01 Celsius. 00:39:13.210 --> 00:39:24.630 align:middle line:84% OK, now that we have the reservoir, 00:39:24.630 --> 00:39:28.230 align:middle line:84% we apply again the second assertion 00:39:28.230 --> 00:39:33.410 align:middle line:84% of the second law, which says we can always 00:39:33.410 --> 00:39:36.870 align:middle line:84% start from a system in any state, 00:39:36.870 --> 00:39:41.830 align:middle line:84% and reach a stable equilibrium state in a reversible weight 00:39:41.830 --> 00:39:42.330 align:middle line:90% process. 00:39:42.330 --> 00:39:45.090 align:middle line:90% 00:39:45.090 --> 00:39:48.800 align:middle line:84% This is an important loaded statement. 00:39:48.800 --> 00:39:53.930 align:middle line:84% It allows-- I mean, how do I know that it always exists? 00:39:53.930 --> 00:39:57.050 align:middle line:90% It's very abstract. 00:39:57.050 --> 00:40:01.400 align:middle line:84% But it's very similar to the statement that says in the first 00:40:01.400 --> 00:40:04.710 align:middle line:84% law, the second assertion, that says yeah, 00:40:04.710 --> 00:40:07.490 align:middle line:84% you can always interconnect the two-weight processes-- 00:40:07.490 --> 00:40:09.810 align:middle line:84% I mean two states by means of a weight process. 00:40:09.810 --> 00:40:14.100 align:middle line:84% But no matter what weight process you choose, 00:40:14.100 --> 00:40:17.670 align:middle line:84% it's always the same energy that you transfer out to the weight. 00:40:17.670 --> 00:40:19.580 align:middle line:90% So how do you know that by-- 00:40:19.580 --> 00:40:25.080 align:middle line:84% because I can invent many different ways to do that job. 00:40:25.080 --> 00:40:28.810 align:middle line:84% And how do I know that they will all get the same outcome? 00:40:28.810 --> 00:40:32.090 align:middle line:84% OK, that also is a very abstract idea. 00:40:32.090 --> 00:40:35.060 align:middle line:84% But it works because energy has-- 00:40:35.060 --> 00:40:37.910 align:middle line:84% after all, has been very successful in describing stuff 00:40:37.910 --> 00:40:38.910 align:middle line:90% in physics. 00:40:38.910 --> 00:40:40.970 align:middle line:84% And we believe in it so much to be 00:40:40.970 --> 00:40:43.100 align:middle line:84% one of those great conservation principles 00:40:43.100 --> 00:40:44.640 align:middle line:90% that we talked about last time. 00:40:44.640 --> 00:40:46.880 align:middle line:90% And here is the same thing. 00:40:46.880 --> 00:40:50.330 align:middle line:84% We have a very abstract idea about the existence 00:40:50.330 --> 00:40:51.060 align:middle line:90% of something. 00:40:51.060 --> 00:40:56.210 align:middle line:84% It guarantees that now, we define a measurement procedure 00:40:56.210 --> 00:40:57.730 align:middle line:90% based on this one. 00:40:57.730 --> 00:41:00.040 align:middle line:84% If we didn't have this statement, 00:41:00.040 --> 00:41:05.830 align:middle line:84% we couldn't trust our measurement procedure. 00:41:05.830 --> 00:41:12.000 align:middle line:84% And the measurement procedure is based 00:41:12.000 --> 00:41:17.890 align:middle line:84% on this ingredient, which I repeat here, 00:41:17.890 --> 00:41:19.930 align:middle line:90% with a little more detail. 00:41:19.930 --> 00:41:24.800 align:middle line:84% So I take the system and the reservoir. 00:41:24.800 --> 00:41:31.300 align:middle line:84% And they are in some initial states, A1 and R1. 00:41:31.300 --> 00:41:34.575 align:middle line:84% The reservoir starts in a stable equilibrium state, 00:41:34.575 --> 00:41:36.030 align:middle line:90% like water at the triple point. 00:41:36.030 --> 00:41:40.980 align:middle line:90% 00:41:40.980 --> 00:41:46.290 align:middle line:84% Then I do a reversible weight process for this combination, 00:41:46.290 --> 00:41:50.580 align:middle line:84% for this combined system, composite system, 00:41:50.580 --> 00:41:57.700 align:middle line:84% which has as only external effect the lifting of a weight. 00:41:57.700 --> 00:42:02.682 align:middle line:84% And it's maximal lifting because the process is reversible-- 00:42:02.682 --> 00:42:05.130 align:middle line:84% because the weight process is reversible. 00:42:05.130 --> 00:42:07.020 align:middle line:90% So that's the maximum. 00:42:07.020 --> 00:42:08.790 align:middle line:84% We are applying essentially the theorem 00:42:08.790 --> 00:42:10.780 align:middle line:90% of adiabatic availability here. 00:42:10.780 --> 00:42:18.010 align:middle line:84% But to the composite system, A and R. 00:42:18.010 --> 00:42:18.510 align:middle line:90% All right. 00:42:18.510 --> 00:42:27.930 align:middle line:84% If this listing is maximal, then-- and I 00:42:27.930 --> 00:42:29.320 align:middle line:90% think I jumped a little bit. 00:42:29.320 --> 00:42:34.980 align:middle line:84% But maybe I could say that if the final state-- in order 00:42:34.980 --> 00:42:38.040 align:middle line:84% to apply the theorem of adiabatic availability, 00:42:38.040 --> 00:42:44.410 align:middle line:84% the final state would have to be a stable equilibrium state 00:42:44.410 --> 00:42:47.810 align:middle line:90% for the composite system now. 00:42:47.810 --> 00:42:50.340 align:middle line:84% And by our definition of mutual equilibrium, 00:42:50.340 --> 00:42:53.600 align:middle line:84% it would mean that since the composite is 00:42:53.600 --> 00:42:59.250 align:middle line:84% in stable equilibrium, the two parts are in mutual equilibrium. 00:42:59.250 --> 00:43:04.230 align:middle line:84% It means that the final state A2, the system A 00:43:04.230 --> 00:43:07.020 align:middle line:84% is in mutual equilibrium with the reservoir. 00:43:07.020 --> 00:43:11.730 align:middle line:90% 00:43:11.730 --> 00:43:16.580 align:middle line:84% Here, I want to do it a little more general. 00:43:16.580 --> 00:43:24.740 align:middle line:84% And you see I'm applying that theorem twice 00:43:24.740 --> 00:43:26.750 align:middle line:90% because I'm going-- 00:43:26.750 --> 00:43:35.080 align:middle line:84% in order to go from A1 to A2, I could-- 00:43:35.080 --> 00:43:40.065 align:middle line:84% in which my reservoir is in R1 and here is an R2. 00:43:40.065 --> 00:43:46.910 align:middle line:90% 00:43:46.910 --> 00:43:52.940 align:middle line:84% So now, I do consider the composite of the two systems. 00:43:52.940 --> 00:43:58.160 align:middle line:90% 00:43:58.160 --> 00:44:04.680 align:middle line:84% The theorem of adiabatic availability says that if I go, 00:44:04.680 --> 00:44:07.530 align:middle line:90% let's say our-- 00:44:07.530 --> 00:44:11.830 align:middle line:90% let's call it RR. 00:44:11.830 --> 00:44:15.370 align:middle line:84% OK, so if I go from here to here, 00:44:15.370 --> 00:44:16.870 align:middle line:84% and this one is a stable equilibrium 00:44:16.870 --> 00:44:19.390 align:middle line:84% state for the composite, then I have 00:44:19.390 --> 00:44:21.420 align:middle line:90% extracted as much as I could. 00:44:21.420 --> 00:44:24.620 align:middle line:90% 00:44:24.620 --> 00:44:27.070 align:middle line:84% Also, I can do it starting from state two. 00:44:27.070 --> 00:44:29.710 align:middle line:90% 00:44:29.710 --> 00:44:33.770 align:middle line:84% So here, I would extract the adiabatic availability, 00:44:33.770 --> 00:44:39.550 align:middle line:84% let's call it psi 11 of the composite system, AR. 00:44:39.550 --> 00:44:46.516 align:middle line:84% And here, I would have extracted the adiabatic availability 00:44:46.516 --> 00:44:52.970 align:middle line:84% of the system in this other state, 22. 00:44:52.970 --> 00:44:59.000 align:middle line:84% OK, but since in doing this extraction, 00:44:59.000 --> 00:45:02.750 align:middle line:84% I'm using reversible weight processes, 00:45:02.750 --> 00:45:09.000 align:middle line:90% I can also reverse the arrow. 00:45:09.000 --> 00:45:11.120 align:middle line:90% So I can go this way. 00:45:11.120 --> 00:45:14.300 align:middle line:90% 00:45:14.300 --> 00:45:20.400 align:middle line:84% And of course, I undo the effect on the weight. 00:45:20.400 --> 00:45:22.440 align:middle line:90% So as I go from here-- 00:45:22.440 --> 00:45:25.350 align:middle line:84% so that means I can go from here to there 00:45:25.350 --> 00:45:27.940 align:middle line:84% by means of that reversible weight process. 00:45:27.940 --> 00:45:30.750 align:middle line:84% And the amount of energy I transferred to the weight 00:45:30.750 --> 00:45:33.580 align:middle line:90% is a psi 1 minus psi 2. 00:45:33.580 --> 00:45:36.454 align:middle line:90% 00:45:36.454 --> 00:45:40.890 align:middle line:90% OK, however, in this-- 00:45:40.890 --> 00:45:44.700 align:middle line:84% the ingredient-- yes, so we know how much energy 00:45:44.700 --> 00:45:51.860 align:middle line:84% we transfer to the weight in that process on the slide. 00:45:51.860 --> 00:45:53.420 align:middle line:84% But for the measurement procedures 00:45:53.420 --> 00:45:59.010 align:middle line:84% that we are going to introduce in a moment, what's important 00:45:59.010 --> 00:46:06.080 align:middle line:84% is the change in energy of the reservoir. 00:46:06.080 --> 00:46:09.020 align:middle line:84% How much the energy of the reservoir 00:46:09.020 --> 00:46:15.800 align:middle line:84% changed during this process, because that's something 00:46:15.800 --> 00:46:17.900 align:middle line:90% that we can measure. 00:46:17.900 --> 00:46:24.210 align:middle line:84% You do that process, so you give me a system A, 00:46:24.210 --> 00:46:26.880 align:middle line:90% you choose two states A1 and A2. 00:46:26.880 --> 00:46:30.300 align:middle line:84% The second law guarantees that you can do this process 00:46:30.300 --> 00:46:31.832 align:middle line:90% with any reservoir. 00:46:31.832 --> 00:46:34.930 align:middle line:90% 00:46:34.930 --> 00:46:37.880 align:middle line:84% And then once you've done it, you 00:46:37.880 --> 00:46:42.572 align:middle line:84% can measure energy because we defined property energy. 00:46:42.572 --> 00:46:44.980 align:middle line:84% And so measure it for the reservoir 00:46:44.980 --> 00:46:48.670 align:middle line:84% and make the difference of the final energy of the reservoir 00:46:48.670 --> 00:46:51.330 align:middle line:90% and its initial energy. 00:46:51.330 --> 00:46:56.850 align:middle line:84% Now, this symbol here is complicated 00:46:56.850 --> 00:47:02.030 align:middle line:84% because I want to emphasize that, yes, we are measuring 00:47:02.030 --> 00:47:06.830 align:middle line:84% an energy difference for the reservoir, 00:47:06.830 --> 00:47:09.320 align:middle line:90% but it's more than that. 00:47:09.320 --> 00:47:13.540 align:middle line:84% We have to do it in those particular conditions. 00:47:13.540 --> 00:47:17.870 align:middle line:84% So it says that system A starts like this and so on, 00:47:17.870 --> 00:47:23.140 align:middle line:84% that the process is a weight process and that it is rigorous. 00:47:23.140 --> 00:47:26.950 align:middle line:84% You do it under other conditions, it's another story. 00:47:26.950 --> 00:47:28.070 align:middle line:90% OK, all right. 00:47:28.070 --> 00:47:31.070 align:middle line:90% So this is the ingredient. 00:47:31.070 --> 00:47:34.010 align:middle line:90% 00:47:34.010 --> 00:47:37.670 align:middle line:84% OK, with that ingredient, here is the measurement procedure 00:47:37.670 --> 00:47:42.260 align:middle line:84% that defines a property that we will call 00:47:42.260 --> 00:47:43.950 align:middle line:90% temperature of the reservoir. 00:47:43.950 --> 00:47:47.000 align:middle line:90% 00:47:47.000 --> 00:47:53.220 align:middle line:84% And that will turn out to be a constant for a given reservoir. 00:47:53.220 --> 00:47:57.685 align:middle line:84% So equal for all of its stable equilibrium states. 00:47:57.685 --> 00:48:02.240 align:middle line:90% 00:48:02.240 --> 00:48:07.340 align:middle line:84% We already know for-- we already know that for the triple-point 00:48:07.340 --> 00:48:16.850 align:middle line:84% water, it's 273.16K or 0.01 Celsius. 00:48:16.850 --> 00:48:22.190 align:middle line:90% And in fact it is-- 00:48:22.190 --> 00:48:26.720 align:middle line:84% for as long as you-- in the box, you have all three phases, 00:48:26.720 --> 00:48:29.960 align:middle line:84% you can put in energy, take energy out, and so on. 00:48:29.960 --> 00:48:34.910 align:middle line:84% As long as you have at least some ice, some water, 00:48:34.910 --> 00:48:40.870 align:middle line:84% and some water vapor, that temperature doesn't change. 00:48:40.870 --> 00:48:44.250 align:middle line:84% So it's equal for all the states of that system. 00:48:44.250 --> 00:48:46.930 align:middle line:90% 00:48:46.930 --> 00:48:48.640 align:middle line:90% OK, but how do we measure it? 00:48:48.640 --> 00:48:51.950 align:middle line:84% OK, so now for this measurement procedure, 00:48:51.950 --> 00:48:54.730 align:middle line:84% the object of measurement is the reservoir. 00:48:54.730 --> 00:48:57.590 align:middle line:84% You come to my lab with your reservoir, 00:48:57.590 --> 00:49:01.610 align:middle line:84% and I am supposed to measure this property 00:49:01.610 --> 00:49:02.720 align:middle line:90% for your reservoir. 00:49:02.720 --> 00:49:05.410 align:middle line:90% So here's the object. 00:49:05.410 --> 00:49:08.550 align:middle line:90% 00:49:08.550 --> 00:49:12.685 align:middle line:84% In order to do the measurement, I take an auxiliary system. 00:49:12.685 --> 00:49:17.340 align:middle line:84% So I choose whatever system I want 00:49:17.340 --> 00:49:23.940 align:middle line:84% and any arbitrary two states for that system. 00:49:23.940 --> 00:49:30.320 align:middle line:84% And I do my reversible weight process 00:49:30.320 --> 00:49:36.490 align:middle line:84% for the combination that will lift the weight by somehow, 00:49:36.490 --> 00:49:38.460 align:middle line:90% some amount. 00:49:38.460 --> 00:49:43.900 align:middle line:84% And I measure how much your reservoir changes in energy. 00:49:43.900 --> 00:49:48.640 align:middle line:90% And I put that in the numerator. 00:49:48.640 --> 00:49:52.390 align:middle line:84% In the denominator, I put the result 00:49:52.390 --> 00:49:54.730 align:middle line:84% of the similar measurement that I 00:49:54.730 --> 00:49:59.530 align:middle line:84% have done once and for all for a reference reservoir, 00:49:59.530 --> 00:50:03.570 align:middle line:84% which I happen to like to choose the triple-point water. 00:50:03.570 --> 00:50:07.360 align:middle line:90% 00:50:07.360 --> 00:50:11.390 align:middle line:84% Of course, when I need these measurements, 00:50:11.390 --> 00:50:16.680 align:middle line:84% I use the same system A and the same two states A1 and A2. 00:50:16.680 --> 00:50:23.990 align:middle line:84% OK, and I measure how much energy changes this reference 00:50:23.990 --> 00:50:25.040 align:middle line:90% reservoir. 00:50:25.040 --> 00:50:28.240 align:middle line:90% And I take this ratio. 00:50:28.240 --> 00:50:33.700 align:middle line:84% So this ratio is a measurement by comparison. 00:50:33.700 --> 00:50:43.170 align:middle line:84% It's exactly like when you say, well, how long is this thing? 00:50:43.170 --> 00:50:49.440 align:middle line:84% OK, you compare it with the meter or the inch. 00:50:49.440 --> 00:50:53.710 align:middle line:84% And the meter is another object that is maybe 00:50:53.710 --> 00:50:55.250 align:middle line:90% in some museum in Paris. 00:50:55.250 --> 00:50:58.160 align:middle line:84% But and you see how many times this thing stays. 00:50:58.160 --> 00:51:02.200 align:middle line:84% So take the ratio of this length to the length of the meter. 00:51:02.200 --> 00:51:04.240 align:middle line:90% You get a dimensionless number. 00:51:04.240 --> 00:51:07.520 align:middle line:90% It's exactly like here. 00:51:07.520 --> 00:51:10.310 align:middle line:90% It's a dimensionless number. 00:51:10.310 --> 00:51:12.650 align:middle line:84% As you know, when you have measurements like that, 00:51:12.650 --> 00:51:15.650 align:middle line:84% you can introduce a unit of measure. 00:51:15.650 --> 00:51:24.566 align:middle line:84% You call the meter 1 meter, but you could have called it, 00:51:24.566 --> 00:51:29.260 align:middle line:84% I don't know, how many inches are in a meter, 00:51:29.260 --> 00:51:35.320 align:middle line:84% but you could call it like 40 something inches. 00:51:35.320 --> 00:51:38.190 align:middle line:90% 00:51:38.190 --> 00:51:50.610 align:middle line:84% OK, in this case, we take the ratio, we multiply-- 00:51:50.610 --> 00:51:56.820 align:middle line:84% we take as a unit, not 1 Gibbs or 1 Hatsopoulos, 00:51:56.820 --> 00:51:59.260 align:middle line:84% but we call it-- since Kelvin was the first. 00:51:59.260 --> 00:52:04.950 align:middle line:84% We call it-- and he didn't put 1, but he put 273.16. 00:52:04.950 --> 00:52:05.760 align:middle line:90% Alright. 00:52:05.760 --> 00:52:08.310 align:middle line:90% So that's our basis. 00:52:08.310 --> 00:52:13.050 align:middle line:84% But we put here this number in the result. 00:52:13.050 --> 00:52:17.160 align:middle line:84% So you multiply by that number, this dimensionless ratio, 00:52:17.160 --> 00:52:22.140 align:middle line:84% and you get this property temperature in kelvin. 00:52:22.140 --> 00:52:26.950 align:middle line:90% 00:52:26.950 --> 00:52:31.250 align:middle line:84% Of course, there is an important theorem. 00:52:31.250 --> 00:52:36.720 align:middle line:84% Again, it's proved there, that says, first of all, 00:52:36.720 --> 00:52:40.580 align:middle line:84% that this ratio is positive, that it 00:52:40.580 --> 00:52:47.248 align:middle line:84% is independent of the initial states of the reservoirs, 00:52:47.248 --> 00:52:52.530 align:middle line:84% that it is independent of your choice of auxiliary system. 00:52:52.530 --> 00:52:55.620 align:middle line:84% And once you've chosen that system, 00:52:55.620 --> 00:52:59.240 align:middle line:84% independent of the initial and final 00:52:59.240 --> 00:53:01.105 align:middle line:90% states A1 and A2 of that system. 00:53:01.105 --> 00:53:04.700 align:middle line:90% 00:53:04.700 --> 00:53:09.960 align:middle line:84% And also that it is equal for all stable equilibrium 00:53:09.960 --> 00:53:13.450 align:middle line:90% states of a reservoir. 00:53:13.450 --> 00:53:17.670 align:middle line:84% So that is a property having a constant value. 00:53:17.670 --> 00:53:19.760 align:middle line:84% We call it a temperature of the reservoir. 00:53:19.760 --> 00:53:23.060 align:middle line:84% Notice, we define just the temperature of the reservoir. 00:53:23.060 --> 00:53:25.880 align:middle line:84% We haven't defined yet the temperature 00:53:25.880 --> 00:53:29.650 align:middle line:84% of a stable equilibrium state of a general system, not yet. 00:53:29.650 --> 00:53:34.820 align:middle line:90% 00:53:34.820 --> 00:53:37.435 align:middle line:84% But this is enough for our purposes. 00:53:37.435 --> 00:53:40.560 align:middle line:90% 00:53:40.560 --> 00:53:44.410 align:middle line:84% Also, notice that, by the way we define the property, so 00:53:44.410 --> 00:53:46.690 align:middle line:84% this formula, you can rewrite that formula 00:53:46.690 --> 00:53:50.080 align:middle line:84% by taking the things with the subscript R 00:53:50.080 --> 00:53:54.070 align:middle line:84% on the left-hand side and subscript 00:53:54.070 --> 00:53:56.800 align:middle line:90% R0 on the right-hand side. 00:53:56.800 --> 00:54:01.650 align:middle line:90% You get this other equality. 00:54:01.650 --> 00:54:06.720 align:middle line:90% 00:54:06.720 --> 00:54:10.020 align:middle line:84% From this equality, you can infer what, 00:54:10.020 --> 00:54:17.440 align:middle line:84% that if I measure this ratio for reservoir R, 00:54:17.440 --> 00:54:19.850 align:middle line:84% it's the same as for the reference reservoir. 00:54:19.850 --> 00:54:24.215 align:middle line:84% It's the same for all reservoirs. 00:54:24.215 --> 00:54:27.320 align:middle line:84% So it's independent of the reservoir. 00:54:27.320 --> 00:54:33.800 align:middle line:84% So this ratio, although we use the reservoir to measure it, 00:54:33.800 --> 00:54:35.460 align:middle line:90% is independent of the reservoir. 00:54:35.460 --> 00:54:37.020 align:middle line:90% So what does it depend on? 00:54:37.020 --> 00:54:40.430 align:middle line:84% It depends only on the system A and the auxiliary system 00:54:40.430 --> 00:54:45.040 align:middle line:84% that you have used, and the two states. 00:54:45.040 --> 00:54:47.540 align:middle line:84% This is why this is very important, 00:54:47.540 --> 00:54:54.640 align:middle line:84% because essentially, this leads us to the definition of entropy. 00:54:54.640 --> 00:55:03.360 align:middle line:84% That ratio, we call it the entropy difference between two 00:55:03.360 --> 00:55:10.880 align:middle line:84% states A1 and A2 of our system A. 00:55:10.880 --> 00:55:12.780 align:middle line:84% If you want to put it in another way, 00:55:12.780 --> 00:55:16.130 align:middle line:84% what's the measurement procedure for entropy? 00:55:16.130 --> 00:55:21.030 align:middle line:84% OK, so now, you come to my lab and bring your system 00:55:21.030 --> 00:55:24.300 align:middle line:84% and bring the system in a given state 00:55:24.300 --> 00:55:28.350 align:middle line:84% on which you are interested in measuring the entropy. 00:55:28.350 --> 00:55:31.900 align:middle line:84% Again, I tell you, look, I just can measure entropy differences. 00:55:31.900 --> 00:55:37.570 align:middle line:84% So I will pick a reference state A0, 00:55:37.570 --> 00:55:42.210 align:middle line:84% give it whatever value you want for the entropy of that one. 00:55:42.210 --> 00:55:44.700 align:middle line:84% I'll measure only the difference. 00:55:44.700 --> 00:55:47.550 align:middle line:84% And how do I measure the difference? 00:55:47.550 --> 00:55:51.620 align:middle line:90% Well, I run-- 00:55:51.620 --> 00:55:55.800 align:middle line:84% I find a reversible weight process 00:55:55.800 --> 00:56:00.140 align:middle line:84% for your system and my reservoir. 00:56:00.140 --> 00:56:05.510 align:middle line:84% For example, I could choose water at the triple point, 00:56:05.510 --> 00:56:11.540 align:middle line:84% but not necessarily, and measure the energy 00:56:11.540 --> 00:56:16.070 align:middle line:84% change of the reservoir using this process 00:56:16.070 --> 00:56:19.620 align:middle line:90% and put it at the numerator. 00:56:19.620 --> 00:56:23.400 align:middle line:84% In the denominator, I put the temperature of the reservoir 00:56:23.400 --> 00:56:27.750 align:middle line:84% that I have previously measured with that other measurement 00:56:27.750 --> 00:56:30.290 align:middle line:90% procedure. 00:56:30.290 --> 00:56:34.835 align:middle line:84% And this ratio we just proved is independent of the reservoir 00:56:34.835 --> 00:56:37.650 align:middle line:90% I had chosen here. 00:56:37.650 --> 00:56:42.040 align:middle line:90% So I add this ratio to-- 00:56:42.040 --> 00:56:44.440 align:middle line:84% so this ratio essentially is the answer. 00:56:44.440 --> 00:56:48.165 align:middle line:84% That's the difference in entropy between these two states. 00:56:48.165 --> 00:56:52.420 align:middle line:90% 00:56:52.420 --> 00:56:52.940 align:middle line:90% All right. 00:56:52.940 --> 00:56:59.440 align:middle line:84% So we have an explicit measurement procedure 00:56:59.440 --> 00:57:03.260 align:middle line:90% that gives us entropy. 00:57:03.260 --> 00:57:07.100 align:middle line:84% And the advantage of this property 00:57:07.100 --> 00:57:09.260 align:middle line:84% with respect to adiabatic availability 00:57:09.260 --> 00:57:13.190 align:middle line:90% is that it is additive. 00:57:13.190 --> 00:57:20.910 align:middle line:84% You do it for a system A, you do it for a system B, 00:57:20.910 --> 00:57:25.560 align:middle line:84% and then you do it for a system composite AB, 00:57:25.560 --> 00:57:29.190 align:middle line:90% well the entropy is the sum. 00:57:29.190 --> 00:57:31.461 align:middle line:84% The entropy differences are additive. 00:57:31.461 --> 00:57:34.290 align:middle line:90% 00:57:34.290 --> 00:57:43.295 align:middle line:84% In a moment, we will see also, a practical engineering meaning 00:57:43.295 --> 00:57:45.440 align:middle line:90% of entropy. 00:57:45.440 --> 00:57:52.050 align:middle line:84% But let me also define this other property, which we 00:57:52.050 --> 00:57:55.630 align:middle line:90% essentially already did here. 00:57:55.630 --> 00:57:58.120 align:middle line:84% When you have a system in combination with a reservoir, 00:57:58.120 --> 00:58:00.530 align:middle line:84% like here, a system and a reservoir, 00:58:00.530 --> 00:58:03.193 align:middle line:84% they start in some initial states 00:58:03.193 --> 00:58:07.045 align:middle line:84% the reservoir in a stable equilibrium state system 00:58:07.045 --> 00:58:11.090 align:middle line:84% A in any state, equilibrium or non-equilibrium doesn't matter. 00:58:11.090 --> 00:58:14.160 align:middle line:90% 00:58:14.160 --> 00:58:16.410 align:middle line:84% I can apply the theorem of adiabatic availability 00:58:16.410 --> 00:58:24.330 align:middle line:84% to the composite system and find that the maximum weight lift 00:58:24.330 --> 00:58:28.620 align:middle line:84% that I can obtain from this composite system 00:58:28.620 --> 00:58:33.550 align:middle line:90% in those initial states, 00:58:33.550 --> 00:58:35.680 align:middle line:84% is the adiabatic availability, is 00:58:35.680 --> 00:58:41.790 align:middle line:84% obtained when we make the weight process reversible. 00:58:41.790 --> 00:58:47.110 align:middle line:90% 00:58:47.110 --> 00:58:51.210 align:middle line:84% And it also ends in a stable equilibrium 00:58:51.210 --> 00:58:52.360 align:middle line:90% for the composite system. 00:58:52.360 --> 00:58:53.580 align:middle line:90% So when the two-- 00:58:53.580 --> 00:58:56.880 align:middle line:84% when A and R are in mutual equilibrium. 00:58:56.880 --> 00:59:01.080 align:middle line:84% So here, system A ends in a state 00:59:01.080 --> 00:59:05.250 align:middle line:84% that I call A sub R just to remind us 00:59:05.250 --> 00:59:08.100 align:middle line:84% that it is the state in which system 00:59:08.100 --> 00:59:10.776 align:middle line:84% A is in mutual equilibrium with the reservoir. 00:59:10.776 --> 00:59:14.350 align:middle line:90% 00:59:14.350 --> 00:59:17.680 align:middle line:84% And the final state of the reservoir 00:59:17.680 --> 00:59:28.670 align:middle line:84% is also uniquely fixed by the initial states of A and R. 00:59:28.670 --> 00:59:31.240 align:middle line:90% OK, this is a new property. 00:59:31.240 --> 00:59:32.630 align:middle line:90% You can view it-- 00:59:32.630 --> 00:59:36.890 align:middle line:84% if you fix the reservoir or with respect to a given reservoir, 00:59:36.890 --> 00:59:43.390 align:middle line:84% this is another property for system A. 00:59:43.390 --> 00:59:48.020 align:middle line:84% It is another availability, another maximum work 00:59:48.020 --> 00:59:53.720 align:middle line:84% in other conditions when the system can also interact, or has 00:59:53.720 --> 00:59:56.455 align:middle line:84% a reservoir available with which to interact. 00:59:56.455 --> 00:59:59.100 align:middle line:90% 00:59:59.100 --> 01:00:07.930 align:middle line:84% We call it available energy with respect to the reservoir R. 01:00:07.930 --> 01:00:11.030 align:middle line:90% We can make-- so we call omega. 01:00:11.030 --> 01:00:15.800 align:middle line:84% Omega is the energy that goes through the weight. 01:00:15.800 --> 01:00:22.980 align:middle line:84% We can make-- you can write an energy balance for that process 01:00:22.980 --> 01:00:25.330 align:middle line:84% because now, we know how to compute the energy. 01:00:25.330 --> 01:00:28.730 align:middle line:84% So we can compute the energy difference 01:00:28.730 --> 01:00:36.820 align:middle line:84% for system A. It goes from A1 to AR. 01:00:36.820 --> 01:00:44.150 align:middle line:84% So the energy that goes out of system A is E1 minus ER. 01:00:44.150 --> 01:00:49.350 align:middle line:84% The energy that goes out of the reservoir is E-- 01:00:49.350 --> 01:00:54.930 align:middle line:84% the energy of R at state S1 minus the energy of R 01:00:54.930 --> 01:00:56.080 align:middle line:90% in this final state. 01:00:56.080 --> 01:01:00.900 align:middle line:90% 01:01:00.900 --> 01:01:02.940 align:middle line:84% And since the energy is conserved, 01:01:02.940 --> 01:01:05.340 align:middle line:84% these are the things that go out. 01:01:05.340 --> 01:01:11.850 align:middle line:84% They sum up to what is received by the weight. 01:01:11.850 --> 01:01:15.160 align:middle line:90% So this is the energy balance. 01:01:15.160 --> 01:01:19.060 align:middle line:84% But now, if you remember how we defined 01:01:19.060 --> 01:01:26.260 align:middle line:84% the entropy of system A, just previous slide, 01:01:26.260 --> 01:01:30.890 align:middle line:84% it's the ratio of the energy change of the reservoir. 01:01:30.890 --> 01:01:33.750 align:middle line:90% So it's this much. 01:01:33.750 --> 01:01:36.010 align:middle line:84% The change is actually final minus initial. 01:01:36.010 --> 01:01:39.400 align:middle line:84% So it's minus this much, divided by the temperature. 01:01:39.400 --> 01:01:41.280 align:middle line:90% So that's the entropy. 01:01:41.280 --> 01:01:47.100 align:middle line:84% So this ratio-- we could substitute this part using 01:01:47.100 --> 01:01:50.460 align:middle line:84% this equation and make an expression 01:01:50.460 --> 01:01:56.310 align:middle line:84% for the available energy in terms of the entropy. 01:01:56.310 --> 01:02:02.260 align:middle line:90% I hope you see the algebra here. 01:02:02.260 --> 01:02:08.770 align:middle line:84% So this is the working equation for computing available energy 01:02:08.770 --> 01:02:11.010 align:middle line:90% of a given system. 01:02:11.010 --> 01:02:15.990 align:middle line:84% All you need to know, available energy with respect 01:02:15.990 --> 01:02:19.140 align:middle line:84% to the reservoir R, you need to know 01:02:19.140 --> 01:02:24.660 align:middle line:84% the initial energy and the initial entropy E1 and S1. 01:02:24.660 --> 01:02:29.230 align:middle line:84% You need to know the energy and the entropy 01:02:29.230 --> 01:02:33.010 align:middle line:84% that the system has once it gets into mutual equilibrium 01:02:33.010 --> 01:02:34.180 align:middle line:90% with the reservoir. 01:02:34.180 --> 01:02:38.650 align:middle line:84% And you need to know the temperature of that reservoir. 01:02:38.650 --> 01:02:40.160 align:middle line:90% And this is a combination. 01:02:40.160 --> 01:02:43.370 align:middle line:90% 01:02:43.370 --> 01:02:46.330 align:middle line:84% If you-- we call it available energy with respect 01:02:46.330 --> 01:02:51.300 align:middle line:84% to the thermal reservoir R. Later on, 01:02:51.300 --> 01:02:55.480 align:middle line:84% when we say a few words about exergy, 01:02:55.480 --> 01:02:58.320 align:middle line:90% and this is one form of exergy. 01:02:58.320 --> 01:03:01.440 align:middle line:84% Exergy is a word that was invented during 01:03:01.440 --> 01:03:07.210 align:middle line:90% the '70s or maybe earlier. 01:03:07.210 --> 01:03:09.670 align:middle line:84% Keenan and called it availability in 1941. 01:03:09.670 --> 01:03:12.000 align:middle line:90% So this concept is around for-- 01:03:12.000 --> 01:03:15.420 align:middle line:90% has been around for a while. 01:03:15.420 --> 01:03:21.615 align:middle line:84% But exergy, the ex means it's like the essential part 01:03:21.615 --> 01:03:27.050 align:middle line:84% of energy because essentially, exergy 01:03:27.050 --> 01:03:30.770 align:middle line:84% is the answer to the question, what's the maximum I can do 01:03:30.770 --> 01:03:32.070 align:middle line:90% for-- from a given situation? 01:03:32.070 --> 01:03:33.860 align:middle line:90% So there are many exergies. 01:03:33.860 --> 01:03:35.910 align:middle line:84% And we already have seen a couple. 01:03:35.910 --> 01:03:38.090 align:middle line:84% Adiabatic availability is an exergy 01:03:38.090 --> 01:03:41.335 align:middle line:84% that applies when the system undergoes just a weight process. 01:03:41.335 --> 01:03:44.000 align:middle line:90% 01:03:44.000 --> 01:03:46.020 align:middle line:90% Here, we have available energy. 01:03:46.020 --> 01:03:49.550 align:middle line:84% It's another exergy that applies when a system also 01:03:49.550 --> 01:03:52.580 align:middle line:90% has available a reservoir. 01:03:52.580 --> 01:03:55.580 align:middle line:84% It is the combination of the system and the reservoir, 01:03:55.580 --> 01:03:59.120 align:middle line:90% which undergo a weight process. 01:03:59.120 --> 01:04:00.986 align:middle line:90% So this is the formula. 01:04:00.986 --> 01:04:04.540 align:middle line:90% So let's play a game here. 01:04:04.540 --> 01:04:06.170 align:middle line:84% Let's play a bit with the formula 01:04:06.170 --> 01:04:12.180 align:middle line:84% and rewrite it in this way, putting in evidence the entropy 01:04:12.180 --> 01:04:13.340 align:middle line:90% of state one of a system. 01:04:13.340 --> 01:04:19.400 align:middle line:90% 01:04:19.400 --> 01:04:27.790 align:middle line:84% Yeah, before I do that, notice that except for the temperature 01:04:27.790 --> 01:04:31.920 align:middle line:84% of the reservoir, this formula doesn't 01:04:31.920 --> 01:04:35.260 align:middle line:84% have any other properties of the reservoir. 01:04:35.260 --> 01:04:40.020 align:middle line:84% So it doesn't have the difference in energy 01:04:40.020 --> 01:04:42.090 align:middle line:84% as has disappeared because we wrote that 01:04:42.090 --> 01:04:47.750 align:middle line:84% in terms of the entropy change for the system. 01:04:47.750 --> 01:04:52.015 align:middle line:84% OK, so we put in evidence of the entropy of system A state A1. 01:04:52.015 --> 01:04:54.800 align:middle line:90% 01:04:54.800 --> 01:05:01.580 align:middle line:84% And we notice that except for some things that 01:05:01.580 --> 01:05:13.290 align:middle line:84% are constant fixed by the chosen reservoir, 01:05:13.290 --> 01:05:17.630 align:middle line:84% the entropy is proportional to the difference between energy 01:05:17.630 --> 01:05:18.660 align:middle line:90% and available energy. 01:05:18.660 --> 01:05:27.160 align:middle line:84% So again, it is the energy that is not available with respect 01:05:27.160 --> 01:05:31.300 align:middle line:84% to the reservoir divided by the temperature of the reservoir. 01:05:31.300 --> 01:05:37.745 align:middle line:84% So the entropy is proportional to the non-available energy 01:05:37.745 --> 01:05:39.724 align:middle line:90% with respect to the reservoir. 01:05:39.724 --> 01:05:44.830 align:middle line:90% 01:05:44.830 --> 01:05:48.400 align:middle line:84% So the higher the entropy, the lesser the energy 01:05:48.400 --> 01:05:52.640 align:middle line:84% that can be transformed into lifting a weight, 01:05:52.640 --> 01:05:56.830 align:middle line:84% which is the prototype of a useful effect, 01:05:56.830 --> 01:05:58.382 align:middle line:90% mechanical effect. 01:05:58.382 --> 01:06:01.340 align:middle line:90% 01:06:01.340 --> 01:06:06.170 align:middle line:84% So that's why I call it engineering meaning of entropy. 01:06:06.170 --> 01:06:09.650 align:middle line:84% A lot of mechanical engineering, energy engineering, 01:06:09.650 --> 01:06:13.400 align:middle line:84% is related to trying to exploit as much 01:06:13.400 --> 01:06:17.730 align:middle line:84% as possible the availability of systems, and in other words, 01:06:17.730 --> 01:06:20.090 align:middle line:90% to-- 01:06:20.090 --> 01:06:25.330 align:middle line:90% and entropy plays against it. 01:06:25.330 --> 01:06:30.840 align:middle line:84% You let entropy increased, you lose ability 01:06:30.840 --> 01:06:33.900 align:middle line:90% to do useful effects. 01:06:33.900 --> 01:06:36.810 align:middle line:84% So that's why in engineering, we have 01:06:36.810 --> 01:06:41.980 align:middle line:84% to work in trying to improve the efficiency of stuff, 01:06:41.980 --> 01:06:45.760 align:middle line:90% of machinery, and so on. 01:06:45.760 --> 01:06:53.980 align:middle line:84% With the purpose of reducing the spontaneous generation 01:06:53.980 --> 01:06:59.020 align:middle line:84% of entropy because that will kill the available energy. 01:06:59.020 --> 01:07:05.000 align:middle line:90% 01:07:05.000 --> 01:07:10.560 align:middle line:84% OK, we said-- already said that the entropy is additive, 01:07:10.560 --> 01:07:14.690 align:middle line:84% that it can be proved that it is additive. 01:07:14.690 --> 01:07:22.360 align:middle line:84% And then, the theorem on the criteria for reversibility. 01:07:22.360 --> 01:07:25.370 align:middle line:90% 01:07:25.370 --> 01:07:28.250 align:middle line:84% We have already seen the first line here expressed 01:07:28.250 --> 01:07:31.030 align:middle line:84% in terms of adiabatic availability of the system. 01:07:31.030 --> 01:07:32.700 align:middle line:90% You can express the same. 01:07:32.700 --> 01:07:38.890 align:middle line:84% That theorem translates also for available energy. 01:07:38.890 --> 01:07:43.260 align:middle line:84% So the difference between energy and available energy 01:07:43.260 --> 01:07:47.670 align:middle line:84% in respect to any reservoir, any given reservoir, 01:07:47.670 --> 01:07:51.510 align:middle line:84% remains constant in a reversible process. 01:07:51.510 --> 01:07:54.150 align:middle line:84% So the unavailable energy remains 01:07:54.150 --> 01:07:58.590 align:middle line:84% constant in a reversible process while it increases 01:07:58.590 --> 01:08:01.740 align:middle line:90% in an irreversible process. 01:08:01.740 --> 01:08:03.725 align:middle line:90% And it cannot decrease. 01:08:03.725 --> 01:08:07.720 align:middle line:90% 01:08:07.720 --> 01:08:10.570 align:middle line:84% And since we just saw that this unavailable energy is 01:08:10.570 --> 01:08:15.570 align:middle line:84% proportional to the entropy, we can write this 01:08:15.570 --> 01:08:28.109 align:middle line:84% as S remains constant in a reversible weight process. 01:08:28.109 --> 01:08:32.250 align:middle line:84% Energy, I'm sorry, entropy increases 01:08:32.250 --> 01:08:34.605 align:middle line:90% if the process is irreversible. 01:08:34.605 --> 01:08:37.800 align:middle line:90% 01:08:37.800 --> 01:08:41.614 align:middle line:84% Entropy cannot decrease in a weight process. 01:08:41.614 --> 01:08:45.189 align:middle line:90% 01:08:45.189 --> 01:08:46.930 align:middle line:84% Careful here about the conditions 01:08:46.930 --> 01:08:53.210 align:middle line:84% that we made this statement in because it would 01:08:53.210 --> 01:08:55.860 align:middle line:90% be wrong to say, ah, energy-- 01:08:55.860 --> 01:08:57.710 align:middle line:84% I mean, the entropy cannot decrease. 01:08:57.710 --> 01:09:00.332 align:middle line:90% 01:09:00.332 --> 01:09:03.050 align:middle line:84% The entropy decreases all the times. 01:09:03.050 --> 01:09:08.640 align:middle line:84% Whenever you take a can of Coke and put it in your refrigerator, 01:09:08.640 --> 01:09:11.930 align:middle line:84% the entropy of that can of Coke decreases 01:09:11.930 --> 01:09:13.910 align:middle line:90% because you're cooling it down. 01:09:13.910 --> 01:09:17.100 align:middle line:90% That is not a weight process. 01:09:17.100 --> 01:09:20.399 align:middle line:84% What we're saying here that the entropy cannot decrease 01:09:20.399 --> 01:09:21.450 align:middle line:90% in a weight process. 01:09:21.450 --> 01:09:27.279 align:middle line:84% If the only external effect is only the changing-- 01:09:27.279 --> 01:09:31.410 align:middle line:84% I mean, the lifting or lowering of a weight, 01:09:31.410 --> 01:09:37.060 align:middle line:84% then you cannot decrease the entropy. 01:09:37.060 --> 01:09:39.090 align:middle line:84% But you can decrease the entropy of a system 01:09:39.090 --> 01:09:42.750 align:middle line:84% by simply taking making it flow out of your system 01:09:42.750 --> 01:09:48.225 align:middle line:84% because entropy is actually transferable. 01:09:48.225 --> 01:09:53.220 align:middle line:90% 01:09:53.220 --> 01:09:57.480 align:middle line:84% We prove that in the same way as we did for the transferability 01:09:57.480 --> 01:09:59.910 align:middle line:90% or exchangeability of energy. 01:09:59.910 --> 01:10:02.160 align:middle line:84% We consider a composite system, you apply-- 01:10:02.160 --> 01:10:06.310 align:middle line:90% 01:10:06.310 --> 01:10:08.710 align:middle line:84% if I do it for a reversible process, 01:10:08.710 --> 01:10:11.000 align:middle line:90% then entropy remains constant. 01:10:11.000 --> 01:10:16.580 align:middle line:84% So it's a principle of entropy conservation in a weight-- 01:10:16.580 --> 01:10:19.391 align:middle line:90% a reversible weight process. 01:10:19.391 --> 01:10:26.190 align:middle line:84% You combine it with additivity, you find that the entropy cannot 01:10:26.190 --> 01:10:28.530 align:middle line:90% disappear. 01:10:28.530 --> 01:10:30.870 align:middle line:84% If it goes out of system A, it means 01:10:30.870 --> 01:10:34.300 align:middle line:84% that you find it somewhere else in system B, 01:10:34.300 --> 01:10:38.090 align:middle line:90% in a reversible weight process. 01:10:38.090 --> 01:10:43.100 align:middle line:84% So in the same way as we decided to give 01:10:43.100 --> 01:10:46.610 align:middle line:84% a symbol to this amount of entropy that is transferred 01:10:46.610 --> 01:10:49.750 align:middle line:84% from one system to the other, we use a symbol that is similar. 01:10:49.750 --> 01:10:54.200 align:middle line:84% So S for entropy transferred from B to A 01:10:54.200 --> 01:10:56.400 align:middle line:90% in the direction of R arrow. 01:10:56.400 --> 01:11:00.440 align:middle line:84% So it's a positive number if the flow is 01:11:00.440 --> 01:11:02.520 align:middle line:90% in the direction of the arrow. 01:11:02.520 --> 01:11:06.320 align:middle line:90% Otherwise, it's a negative. 01:11:06.320 --> 01:11:14.630 align:middle line:84% And so we can write this as a balance equation, a balance 01:11:14.630 --> 01:11:16.460 align:middle line:90% equation. 01:11:16.460 --> 01:11:19.990 align:middle line:84% For a reversible process, you wouldn't have this power. 01:11:19.990 --> 01:11:24.220 align:middle line:84% But simply say if there is a change in entropy 01:11:24.220 --> 01:11:28.090 align:middle line:84% of your system, it is due to the amount 01:11:28.090 --> 01:11:31.302 align:middle line:84% that it has-- of entropy that has been transferred 01:11:31.302 --> 01:11:34.900 align:middle line:90% with this environment. 01:11:34.900 --> 01:11:41.310 align:middle line:84% But then entropy is not really always conserved. 01:11:41.310 --> 01:11:43.750 align:middle line:84% It's conserved only in a reversible process. 01:11:43.750 --> 01:11:48.510 align:middle line:84% In an irreversible process, it's not conserved. 01:11:48.510 --> 01:11:52.050 align:middle line:84% And if the process is a weight process, it can only increase. 01:11:52.050 --> 01:12:02.230 align:middle line:84% So that means that the entropy change of my system can be 01:12:02.230 --> 01:12:07.915 align:middle line:84% greater than the amount it received but cannot be lower. 01:12:07.915 --> 01:12:11.480 align:middle line:90% 01:12:11.480 --> 01:12:16.950 align:middle line:84% So essentially, we are writing an imbalanced equation, 01:12:16.950 --> 01:12:20.270 align:middle line:90% a balance equation if you want. 01:12:20.270 --> 01:12:26.290 align:middle line:84% It becomes a balanced equation if you put in the balance also 01:12:26.290 --> 01:12:30.930 align:middle line:84% the imbalance by calling it entropy generated, or produced 01:12:30.930 --> 01:12:33.517 align:middle line:90% by irreversibility. 01:12:33.517 --> 01:12:36.200 align:middle line:90% 01:12:36.200 --> 01:12:41.105 align:middle line:84% And this brings us to the entropy balance equation. 01:12:41.105 --> 01:12:46.860 align:middle line:90% 01:12:46.860 --> 01:12:54.130 align:middle line:84% I assume that you have played with this in solving problems 01:12:54.130 --> 01:12:59.270 align:middle line:84% in undergraduate thermodynamics and in your applications 01:12:59.270 --> 01:13:09.210 align:middle line:84% in energy or exergy analysis, any technological problem 01:13:09.210 --> 01:13:15.020 align:middle line:84% having to do with energy transformation of energy 01:13:15.020 --> 01:13:18.870 align:middle line:84% in various forms requires you to make not only an energy balance, 01:13:18.870 --> 01:13:20.660 align:middle line:90% but also an entropy balance. 01:13:20.660 --> 01:13:22.520 align:middle line:84% And the combination of the two is 01:13:22.520 --> 01:13:29.210 align:middle line:84% what allows you to determine things like how good am I doing? 01:13:29.210 --> 01:13:33.730 align:middle line:84% What's the efficiency of what I'm doing? 01:13:33.730 --> 01:13:39.410 align:middle line:84% So these balance equations, as we'll see more explicitly, 01:13:39.410 --> 01:13:43.360 align:middle line:84% are the fundamentals to define efficiencies 01:13:43.360 --> 01:13:53.150 align:middle line:84% of power plants i and any device that 01:13:53.150 --> 01:13:55.010 align:middle line:90% does transformation of energy. 01:13:55.010 --> 01:13:58.350 align:middle line:90% 01:13:58.350 --> 01:14:05.760 align:middle line:84% OK, this is also the version of the entropy balance, the rate 01:14:05.760 --> 01:14:07.200 align:middle line:90% version of the entropy balance. 01:14:07.200 --> 01:14:10.140 align:middle line:84% If you have the kind of description 01:14:10.140 --> 01:14:12.820 align:middle line:84% in which you can follow as a function of time, 01:14:12.820 --> 01:14:14.970 align:middle line:84% the state of the system, and therefore, you 01:14:14.970 --> 01:14:18.640 align:middle line:84% can measure as a function of time, the entropy, 01:14:18.640 --> 01:14:24.650 align:middle line:84% and also how much entropy is being exchanged, so 01:14:24.650 --> 01:14:29.350 align:middle line:90% the rate of entropy exchange. 01:14:29.350 --> 01:14:32.560 align:middle line:84% And remember that in an entropy balance, 01:14:32.560 --> 01:14:36.550 align:middle line:84% there are three very distinct elements. 01:14:36.550 --> 01:14:39.960 align:middle line:84% There are three parts of that equation. 01:14:39.960 --> 01:14:44.600 align:middle line:84% The left-hand side says how much the property 01:14:44.600 --> 01:14:47.500 align:middle line:84% changes from the final to the initial state. 01:14:47.500 --> 01:14:51.950 align:middle line:90% 01:14:51.950 --> 01:14:58.310 align:middle line:84% The transfer term says how much the system, the net entropy that 01:14:58.310 --> 01:15:00.950 align:middle line:84% has been received by your system, or exchanged, 01:15:00.950 --> 01:15:02.960 align:middle line:84% if it is negative, it's given out 01:15:02.960 --> 01:15:06.710 align:middle line:90% by a system due to interactions. 01:15:06.710 --> 01:15:10.375 align:middle line:84% It's a flow of entropy going in and out of your system. 01:15:10.375 --> 01:15:15.140 align:middle line:90% 01:15:15.140 --> 01:15:22.200 align:middle line:84% The last term instead is the generation. 01:15:22.200 --> 01:15:24.930 align:middle line:84% It occurs internally to your system. 01:15:24.930 --> 01:15:27.300 align:middle line:90% It's the dissipation. 01:15:27.300 --> 01:15:29.750 align:middle line:90% It's the waste and the curse. 01:15:29.750 --> 01:15:38.110 align:middle line:90% If you let things move around. 01:15:38.110 --> 01:15:43.500 align:middle line:84% For example, if you take Irish coffee, 01:15:43.500 --> 01:15:49.410 align:middle line:84% no, it's very nice if you-- until it is separate. 01:15:49.410 --> 01:15:53.470 align:middle line:84% The first time you see it, you are excited. 01:15:53.470 --> 01:15:57.220 align:middle line:84% If you stir it, you lose the beauty. 01:15:57.220 --> 01:15:58.610 align:middle line:90% No? 01:15:58.610 --> 01:16:01.710 align:middle line:84% And by stirring it, you have mixed up things. 01:16:01.710 --> 01:16:06.420 align:middle line:84% You have produced-- generated entropy. 01:16:06.420 --> 01:16:11.140 align:middle line:84% You've lost, I mean, the beauty because you 01:16:11.140 --> 01:16:12.470 align:middle line:90% are sort of a bit drunk. 01:16:12.470 --> 01:16:13.720 align:middle line:90% And so that's-- 01:16:13.720 --> 01:16:18.280 align:middle line:84% But this gives the idea of the fact 01:16:18.280 --> 01:16:27.040 align:middle line:84% that stirring is something that happens spontaneously. 01:16:27.040 --> 01:16:33.080 align:middle line:84% If you help it by stirring it, it goes faster. 01:16:33.080 --> 01:16:37.150 align:middle line:84% And it's something that you cannot undo. 01:16:37.150 --> 01:16:41.390 align:middle line:84% I mean, the Irish thing will not separate. 01:16:41.390 --> 01:16:45.570 align:middle line:90% Maybe it will but not enough-- 01:16:45.570 --> 01:16:52.630 align:middle line:84% not if you are talking about energy applications. 01:16:52.630 --> 01:17:00.810 align:middle line:84% So if you're not careful enough, and things get stirred up, 01:17:00.810 --> 01:17:03.930 align:middle line:84% you lose opportunities to convert the energy 01:17:03.930 --> 01:17:06.820 align:middle line:84% and to extract energy out of your source. 01:17:06.820 --> 01:17:08.180 align:middle line:90% And you've lost it forever. 01:17:08.180 --> 01:17:13.060 align:middle line:90% 01:17:13.060 --> 01:17:19.025 align:middle line:90% OK, luckily. 01:17:19.025 --> 01:17:21.650 align:middle line:90% 01:17:21.650 --> 01:17:23.590 align:middle line:90% I don't think this time-- 01:17:23.590 --> 01:17:26.950 align:middle line:84% I don't think today, but the next time, we 01:17:26.950 --> 01:17:32.660 align:middle line:84% will be able to introduce a graphical representation of all 01:17:32.660 --> 01:17:42.150 align:middle line:84% that we have said that is very useful to remember and to reason 01:17:42.150 --> 01:17:47.960 align:middle line:84% the various consequences and also to set up problem 01:17:47.960 --> 01:17:51.070 align:middle line:84% solving and answering additional questions. 01:17:51.070 --> 01:17:53.805 align:middle line:84% It will be the energy versus entropy diagram. 01:17:53.805 --> 01:17:57.300 align:middle line:90% 01:17:57.300 --> 01:18:00.210 align:middle line:84% But we are still building up material 01:18:00.210 --> 01:18:04.170 align:middle line:84% so that we can use that diagram to summarize 01:18:04.170 --> 01:18:07.390 align:middle line:90% what we have seen so far. 01:18:07.390 --> 01:18:09.820 align:middle line:84% And one of these ingredients that we still need 01:18:09.820 --> 01:18:13.750 align:middle line:84% is another theorem that is called the maximal entropy 01:18:13.750 --> 01:18:19.390 align:middle line:84% principle, which says that among all the states of a given 01:18:19.390 --> 01:18:26.880 align:middle line:84% system, once you fix the values of the amounts of constituents 01:18:26.880 --> 01:18:29.225 align:middle line:84% and the parameters of the external forces. 01:18:29.225 --> 01:18:31.860 align:middle line:90% 01:18:31.860 --> 01:18:36.190 align:middle line:84% And you consider-- of course, you 01:18:36.190 --> 01:18:40.580 align:middle line:84% remember that if I fix the amounts of constituents, 01:18:40.580 --> 01:18:51.360 align:middle line:84% let's call them n of t, let's forget the t, n, and beta, 01:18:51.360 --> 01:18:56.370 align:middle line:84% this doesn't fix the state because the state is 01:18:56.370 --> 01:19:04.780 align:middle line:84% these numbers plus the values of all the conceivable properties. 01:19:04.780 --> 01:19:10.920 align:middle line:84% Most of the times, this list is an infinite number of numbers. 01:19:10.920 --> 01:19:14.580 align:middle line:84% So fixing just these two leaves you, again, 01:19:14.580 --> 01:19:16.250 align:middle line:90% with an infinity of states. 01:19:16.250 --> 01:19:19.020 align:middle line:90% 01:19:19.020 --> 01:19:26.640 align:middle line:84% Now suppose I fix also the value of the energy. 01:19:26.640 --> 01:19:30.840 align:middle line:90% OK, so I fix this much. 01:19:30.840 --> 01:19:33.870 align:middle line:90% Still, I have a lot of states. 01:19:33.870 --> 01:19:38.610 align:middle line:84% They will differ in the values of all these other properties, 01:19:38.610 --> 01:19:40.240 align:middle line:90% but they have the same energy. 01:19:40.240 --> 01:19:40.740 align:middle line:90% OK? 01:19:40.740 --> 01:19:42.540 align:middle line:84% So if I consider all these states that 01:19:42.540 --> 01:19:45.060 align:middle line:84% have the same energy, amounts of constituents 01:19:45.060 --> 01:19:52.740 align:middle line:84% and parameters of the external forces, it's a set. 01:19:52.740 --> 01:19:56.370 align:middle line:84% In this set, the second law, first assertion 01:19:56.370 --> 01:19:59.350 align:middle line:84% says there is a unique stable equilibrium state. 01:19:59.350 --> 01:20:02.500 align:middle line:90% 01:20:02.500 --> 01:20:06.860 align:middle line:84% This theorem says that state has also the highest 01:20:06.860 --> 01:20:08.180 align:middle line:90% value for the entropy. 01:20:08.180 --> 01:20:14.290 align:middle line:90% 01:20:14.290 --> 01:20:18.380 align:middle line:90% And it's a strict inequality. 01:20:18.380 --> 01:20:21.000 align:middle line:84% It's the only state that has that value. 01:20:21.000 --> 01:20:23.750 align:middle line:84% All other states with the same energy and so 01:20:23.750 --> 01:20:27.660 align:middle line:90% on have lesser entropy. 01:20:27.660 --> 01:20:30.740 align:middle line:90% 01:20:30.740 --> 01:20:33.560 align:middle line:84% The stable equilibrium state, among these ones, 01:20:33.560 --> 01:20:37.700 align:middle line:84% is the one that achieves the maximum value. 01:20:37.700 --> 01:20:40.570 align:middle line:84% And it is the only one that achieves that value. 01:20:40.570 --> 01:20:43.195 align:middle line:84% This is called the maximum entropy principle. 01:20:43.195 --> 01:20:46.840 align:middle line:90% 01:20:46.840 --> 01:20:53.510 align:middle line:84% There is also a dual principle that 01:20:53.510 --> 01:20:55.691 align:middle line:84% is called the minimum energy principle. 01:20:55.691 --> 01:20:58.780 align:middle line:90% 01:20:58.780 --> 01:21:05.040 align:middle line:84% It's similar to this one, except that here, you put S. So 01:21:05.040 --> 01:21:07.260 align:middle line:84% if I consider all the states of a system that 01:21:07.260 --> 01:21:08.460 align:middle line:90% have the same given-- 01:21:08.460 --> 01:21:15.460 align:middle line:84% same values of n, beta and the same entropy, 01:21:15.460 --> 01:21:22.480 align:middle line:84% then among all these states, there is one stable equilibrium 01:21:22.480 --> 01:21:24.280 align:middle line:90% state only. 01:21:24.280 --> 01:21:25.820 align:middle line:90% And it is the lowest energy. 01:21:25.820 --> 01:21:29.630 align:middle line:90% 01:21:29.630 --> 01:21:38.550 align:middle line:84% When you fix the value of S to 0, 01:21:38.550 --> 01:21:41.456 align:middle line:84% that corresponds to the domain of mechanics. 01:21:41.456 --> 01:21:44.370 align:middle line:90% 01:21:44.370 --> 01:21:49.110 align:middle line:84% And that minimum energy stable equilibrium state 01:21:49.110 --> 01:21:50.170 align:middle line:90% is the only one. 01:21:50.170 --> 01:21:52.830 align:middle line:84% And so then we recover the minimal energy principle 01:21:52.830 --> 01:21:53.990 align:middle line:90% of mechanics. 01:21:53.990 --> 01:21:57.910 align:middle line:90% 01:21:57.910 --> 01:22:03.210 align:middle line:84% But thermodynamics is about states that have also entropies 01:22:03.210 --> 01:22:04.430 align:middle line:90% different from 0. 01:22:04.430 --> 01:22:08.060 align:middle line:90% 01:22:08.060 --> 01:22:13.475 align:middle line:84% And so we have a more general minimum entropy principle. 01:22:13.475 --> 01:22:13.975 align:middle line:90% Yes. 01:22:13.975 --> 01:22:30.040 align:middle line:90% 01:22:30.040 --> 01:22:31.330 align:middle line:90% Yes. 01:22:31.330 --> 01:22:42.290 align:middle line:84% If you start from the low-- a stable equilibrium state, 01:22:42.290 --> 01:22:45.960 align:middle line:84% like this one is a stable equilibrium state. 01:22:45.960 --> 01:22:53.720 align:middle line:84% If you spend some work, so if I go this direction 01:22:53.720 --> 01:22:56.210 align:middle line:84% and I spend some work, I can obtain 01:22:56.210 --> 01:23:00.980 align:middle line:84% any other state, any other state, 01:23:00.980 --> 01:23:07.690 align:middle line:84% but only those that have the same entropy as I started with. 01:23:07.690 --> 01:23:10.850 align:middle line:84% If the process is reversible, I-- 01:23:10.850 --> 01:23:12.860 align:middle line:84% the entropy will remain constant. 01:23:12.860 --> 01:23:18.560 align:middle line:84% So I can go from my minimum energy stable equilibrium state 01:23:18.560 --> 01:23:22.040 align:middle line:84% to any other state which has higher energy because I 01:23:22.040 --> 01:23:24.630 align:middle line:90% put in some work. 01:23:24.630 --> 01:23:31.530 align:middle line:84% But if it is reversible, it has the same entropy. 01:23:31.530 --> 01:23:35.520 align:middle line:84% If it is irreversible, it will have higher entropy. 01:23:35.520 --> 01:23:39.870 align:middle line:84% So it can go only, let's say, from here with energy, 01:23:39.870 --> 01:23:41.705 align:middle line:90% I can go up. 01:23:41.705 --> 01:23:43.120 align:middle line:90% And this is entropy. 01:23:43.120 --> 01:23:44.500 align:middle line:90% So this will-- 01:23:44.500 --> 01:23:45.000 align:middle line:90% OK. 01:23:45.000 --> 01:23:47.440 align:middle line:84% So I start with a certain value of the entropy, 01:23:47.440 --> 01:23:55.860 align:middle line:84% say SR. I can go up in a reversible process 01:23:55.860 --> 01:24:02.390 align:middle line:84% to values of the energy higher than I started with. 01:24:02.390 --> 01:24:05.820 align:middle line:90% So this is ER. 01:24:05.820 --> 01:24:14.730 align:middle line:84% Or I can also go to the right because I put in energy 01:24:14.730 --> 01:24:17.910 align:middle line:84% but if the process is irreversible, 01:24:17.910 --> 01:24:20.400 align:middle line:90% I generate some entropy. 01:24:20.400 --> 01:24:23.210 align:middle line:90% So the entropy adds. 01:24:23.210 --> 01:24:25.285 align:middle line:90% And I go to the right. 01:24:25.285 --> 01:24:29.375 align:middle line:90% 01:24:29.375 --> 01:24:30.270 align:middle line:90% So yes. 01:24:30.270 --> 01:24:33.320 align:middle line:84% Thanks because this anticipates essentially this energy 01:24:33.320 --> 01:24:36.020 align:middle line:84% versus entropy diagram, and which 01:24:36.020 --> 01:24:39.050 align:middle line:84% is useful because as you know, a picture 01:24:39.050 --> 01:24:40.530 align:middle line:90% is worth a thousand words. 01:24:40.530 --> 01:24:44.240 align:middle line:84% And this is the picture that allows you to answer 01:24:44.240 --> 01:24:46.750 align:middle line:90% these kind of questions. 01:24:46.750 --> 01:24:52.260 align:middle line:90% 01:24:52.260 --> 01:24:55.700 align:middle line:84% The last ingredient that we need in order 01:24:55.700 --> 01:24:58.850 align:middle line:84% to go to that graphical representation 01:24:58.850 --> 01:25:02.370 align:middle line:90% is called the state principle. 01:25:02.370 --> 01:25:05.310 align:middle line:90% 01:25:05.310 --> 01:25:08.940 align:middle line:84% Again, it is a consequence very directly 01:25:08.940 --> 01:25:14.300 align:middle line:84% from the first assertion of the second law, 01:25:14.300 --> 01:25:17.785 align:middle line:90% which says you fix n and beta. 01:25:17.785 --> 01:25:21.450 align:middle line:90% 01:25:21.450 --> 01:25:27.300 align:middle line:84% Fix n and beta, you fix the energy. 01:25:27.300 --> 01:25:29.900 align:middle line:90% 01:25:29.900 --> 01:25:34.940 align:middle line:84% And once these are fixed, there is a unique-- 01:25:34.940 --> 01:25:38.215 align:middle line:84% there exists a unique stable equilibrium state. 01:25:38.215 --> 01:25:45.485 align:middle line:90% 01:25:45.485 --> 01:25:49.540 align:middle line:90% Alright, the last S means state. 01:25:49.540 --> 01:25:50.780 align:middle line:90% So yes. 01:25:50.780 --> 01:25:53.580 align:middle line:84% It's a state which is stable equilibrium. 01:25:53.580 --> 01:25:56.110 align:middle line:90% So what does it mean, state? 01:25:56.110 --> 01:26:02.350 align:middle line:84% State means the values of all the properties. 01:26:02.350 --> 01:26:07.540 align:middle line:84% So that means that this thing fixes the values 01:26:07.540 --> 01:26:08.750 align:middle line:90% of all the properties. 01:26:08.750 --> 01:26:20.900 align:middle line:90% 01:26:20.900 --> 01:26:24.370 align:middle line:84% Of course, you're talking about of a stable equilibrium-- 01:26:24.370 --> 01:26:27.556 align:middle line:90% of a stable equilibrium state. 01:26:27.556 --> 01:26:34.720 align:middle line:84% So we-- so mathematically, that's exactly what we 01:26:34.720 --> 01:26:36.960 align:middle line:90% mean by a function. 01:26:36.960 --> 01:26:41.340 align:middle line:84% The properties, it means that these properties, any property, 01:26:41.340 --> 01:26:43.490 align:middle line:90% is a function of these numbers. 01:26:43.490 --> 01:26:46.290 align:middle line:84% If these numbers fix the value, that's 01:26:46.290 --> 01:26:49.380 align:middle line:84% what we mean by value of a function. 01:26:49.380 --> 01:26:54.850 align:middle line:84% So we can write a property of stable equilibrium, 01:26:54.850 --> 01:27:00.670 align:middle line:84% at stable equilibrium, as a function of the amounts, 01:27:00.670 --> 01:27:04.670 align:middle line:90% the parameters, and the energy. 01:27:04.670 --> 01:27:07.430 align:middle line:84% And since among the various properties 01:27:07.430 --> 01:27:10.670 align:middle line:84% we defined one is entropy, we can also 01:27:10.670 --> 01:27:11.990 align:middle line:90% write that for the entropy. 01:27:11.990 --> 01:27:15.180 align:middle line:84% So we can say that for the stable equilibrium states, 01:27:15.180 --> 01:27:19.350 align:middle line:84% the entropy, the value of the entropy, 01:27:19.350 --> 01:27:23.090 align:middle line:84% is fixed like all other properties 01:27:23.090 --> 01:27:26.400 align:middle line:84% by the amounts of constituents, the parameters of the forces, 01:27:26.400 --> 01:27:27.410 align:middle line:90% and the energy. 01:27:27.410 --> 01:27:31.250 align:middle line:84% This expression, this formula, this function 01:27:31.250 --> 01:27:33.590 align:middle line:84% is called the fundamental relation 01:27:33.590 --> 01:27:35.900 align:middle line:84% for the stable equilibrium states. 01:27:35.900 --> 01:27:42.180 align:middle line:84% In some old-fashioned books of thermodynamics, 01:27:42.180 --> 01:27:45.750 align:middle line:90% this function is page one. 01:27:45.750 --> 01:27:52.230 align:middle line:84% You take the book by Callen, Callen was a PhD from MIT also. 01:27:52.230 --> 01:27:54.780 align:middle line:90% Very nice contributions. 01:27:54.780 --> 01:27:58.500 align:middle line:84% But in his times, thermodynamics was only 01:27:58.500 --> 01:27:59.990 align:middle line:90% about stable equilibrium states. 01:27:59.990 --> 01:28:02.500 align:middle line:90% 01:28:02.500 --> 01:28:05.320 align:middle line:84% And so that's the starting point. 01:28:05.320 --> 01:28:11.580 align:middle line:84% And since I'm at it why not saying and reminding you 01:28:11.580 --> 01:28:19.650 align:middle line:84% that instead, we've seen a definition of entropy today 01:28:19.650 --> 01:28:22.210 align:middle line:84% in which you may or may have not noticed, 01:28:22.210 --> 01:28:27.250 align:middle line:84% but I will stress it anyway, that we never 01:28:27.250 --> 01:28:30.350 align:middle line:84% said anything about how many particles 01:28:30.350 --> 01:28:33.090 align:middle line:90% the system should have. 01:28:33.090 --> 01:28:35.820 align:middle line:84% In our game so far, we could have 01:28:35.820 --> 01:28:40.480 align:middle line:84% a system with one particle or 10 to the 23. 01:28:40.480 --> 01:28:47.040 align:middle line:84% It could be just water in a box or a nuclear power plant, 01:28:47.040 --> 01:28:51.820 align:middle line:84% complex or simple, small or large. 01:28:51.820 --> 01:28:55.390 align:middle line:84% Plus, we didn't make any assumption 01:28:55.390 --> 01:28:56.960 align:middle line:90% about the kind of state. 01:28:56.960 --> 01:28:59.620 align:middle line:90% 01:28:59.620 --> 01:29:05.000 align:middle line:84% In fact, the most interesting states 01:29:05.000 --> 01:29:06.937 align:middle line:84% are those that are not stable equilibrium, 01:29:06.937 --> 01:29:09.020 align:middle line:84% at least from the point of view of making a weight 01:29:09.020 --> 01:29:13.600 align:middle line:84% process, because in a stable equilibrium state, 01:29:13.600 --> 01:29:15.830 align:middle line:84% we just have seen that the perpetual motion 01:29:15.830 --> 01:29:17.700 align:middle line:84% machine of the second kind is impossible. 01:29:17.700 --> 01:29:20.720 align:middle line:90% You cannot extract work. 01:29:20.720 --> 01:29:25.345 align:middle line:84% So not very useful from an engineering point of view. 01:29:25.345 --> 01:29:29.150 align:middle line:90% 01:29:29.150 --> 01:29:32.840 align:middle line:84% And that's why-- consider the ocean. 01:29:32.840 --> 01:29:35.990 align:middle line:84% The ocean as a thermal reservoir. 01:29:35.990 --> 01:29:38.140 align:middle line:84% Not the waves, that's a different story. 01:29:38.140 --> 01:29:40.110 align:middle line:90% But as a thermal reservoir. 01:29:40.110 --> 01:29:42.710 align:middle line:84% And if you forget also the fact that inside there 01:29:42.710 --> 01:29:46.655 align:middle line:84% are temperature gradients, and also salinity gradients, 01:29:46.655 --> 01:29:47.280 align:middle line:90% and so on. 01:29:47.280 --> 01:29:52.040 align:middle line:84% But if you think of it as a huge system, 01:29:52.040 --> 01:29:54.210 align:middle line:84% so it has a huge amount of energy. 01:29:54.210 --> 01:29:57.850 align:middle line:84% But that energy is essentially useless. 01:29:57.850 --> 01:30:01.190 align:middle line:84% The thermal energy is useless because it's 01:30:01.190 --> 01:30:02.940 align:middle line:90% in a stable equilibrium state. 01:30:02.940 --> 01:30:06.180 align:middle line:90% 01:30:06.180 --> 01:30:17.480 align:middle line:84% Whereas if I have a system in a non-equilibrium state, 01:30:17.480 --> 01:30:22.270 align:middle line:84% and I'm quick enough to extract its adiabatic availability, 01:30:22.270 --> 01:30:25.310 align:middle line:90% I can do something useful. 01:30:25.310 --> 01:30:31.800 align:middle line:84% For example, we will do later on in the second part, 01:30:31.800 --> 01:30:33.510 align:middle line:84% we will do, like, chemical reactions. 01:30:33.510 --> 01:30:39.370 align:middle line:84% You think of what happens inside your internal combustion engine. 01:30:39.370 --> 01:30:45.950 align:middle line:84% You've got the piston, you got a mixture of air and gasoline, 01:30:45.950 --> 01:30:49.720 align:middle line:90% and you light the spark. 01:30:49.720 --> 01:30:55.290 align:middle line:84% At that moment, you produced a non-equilibrium state, 01:30:55.290 --> 01:30:58.890 align:middle line:84% which very quickly in the matter of micro-- 01:30:58.890 --> 01:31:05.440 align:middle line:84% milliseconds, will interest the entire combustion chamber. 01:31:05.440 --> 01:31:09.570 align:middle line:90% Pressure will go up and so on. 01:31:09.570 --> 01:31:14.700 align:middle line:84% The internal combustion engine is an incredible device 01:31:14.700 --> 01:31:19.920 align:middle line:84% that has been there for a century now, maybe longer. 01:31:19.920 --> 01:31:22.950 align:middle line:84% And it's still powering our cars. 01:31:22.950 --> 01:31:30.770 align:middle line:84% And because it's a smart way to extract the availability. 01:31:30.770 --> 01:31:33.950 align:middle line:90% 01:31:33.950 --> 01:31:37.170 align:middle line:84% It doesn't do the entire job, though, 01:31:37.170 --> 01:31:40.910 align:middle line:84% because it loses the opportunity that you 01:31:40.910 --> 01:31:46.020 align:middle line:84% have from when you light the spark to when 01:31:46.020 --> 01:31:51.410 align:middle line:84% the flame is engulfed the entire combustion chamber. 01:31:51.410 --> 01:31:55.020 align:middle line:84% That is a very irreversible process. 01:31:55.020 --> 01:31:58.930 align:middle line:90% And that's where you lose the-- 01:31:58.930 --> 01:32:05.010 align:middle line:84% about, what is it, 60% of your availability, 01:32:05.010 --> 01:32:07.250 align:middle line:84% no, because the internal combustion engine 01:32:07.250 --> 01:32:09.980 align:middle line:90% is 40% efficient. 01:32:09.980 --> 01:32:14.020 align:middle line:84% So it means that I lost 60% somewhere. 01:32:14.020 --> 01:32:15.380 align:middle line:90% And that's where I lost it. 01:32:15.380 --> 01:32:23.740 align:middle line:90% 01:32:23.740 --> 01:32:26.140 align:middle line:90% So, non-equilibrium states. 01:32:26.140 --> 01:32:30.860 align:middle line:84% And so we define entropy also for non-equilibrium states, 01:32:30.860 --> 01:32:35.080 align:middle line:84% which is most important for the more modern applications 01:32:35.080 --> 01:32:36.237 align:middle line:90% of thermodynamics. 01:32:36.237 --> 01:32:37.320 align:middle line:90% And that's why we're here. 01:32:37.320 --> 01:32:40.410 align:middle line:90% 01:32:40.410 --> 01:32:49.030 align:middle line:84% Actually, I was very happy just yesterday to receive an email. 01:32:49.030 --> 01:32:52.140 align:middle line:84% I think it was sent to the entire community, not very 01:32:52.140 --> 01:32:52.720 align:middle line:90% special. 01:32:52.720 --> 01:32:56.340 align:middle line:84% But an email from the MIT president 01:32:56.340 --> 01:32:59.390 align:middle line:84% announcing this initiative about climate. 01:32:59.390 --> 01:33:05.120 align:middle line:90% 01:33:05.120 --> 01:33:10.490 align:middle line:84% A lot of the applications that will come out 01:33:10.490 --> 01:33:16.490 align:middle line:84% of that project in the future will 01:33:16.490 --> 01:33:21.610 align:middle line:84% have to do with energy and the best use of energy. 01:33:21.610 --> 01:33:29.270 align:middle line:84% OK, not only CO2 segregation, or greenhouse gases, 01:33:29.270 --> 01:33:34.580 align:middle line:84% but all the crazy ideas that people are talking about 01:33:34.580 --> 01:33:41.705 align:middle line:84% in trying to control, for example, the amount of heating 01:33:41.705 --> 01:33:44.863 align:middle line:90% that comes from the Sun. 01:33:44.863 --> 01:33:46.280 align:middle line:84% There was an article the other day 01:33:46.280 --> 01:33:48.410 align:middle line:84% in the New York Times talking about people 01:33:48.410 --> 01:33:53.330 align:middle line:84% that want to build shields in space, no, 01:33:53.330 --> 01:34:00.040 align:middle line:84% to reflect away some of the power, some of the photons, that 01:34:00.040 --> 01:34:04.530 align:middle line:84% come from the sun so that you could control temperature. 01:34:04.530 --> 01:34:07.630 align:middle line:84% Of course, you need a surface to, 01:34:07.630 --> 01:34:13.480 align:middle line:84% I think it was the surface of the entire Argentina, which 01:34:13.480 --> 01:34:16.930 align:middle line:84% is essentially half of South America. 01:34:16.930 --> 01:34:23.080 align:middle line:84% Somewhere put in a place, which they call it a Lagrange place, 01:34:23.080 --> 01:34:25.870 align:middle line:84% or a Lagrange spot where it would 01:34:25.870 --> 01:34:29.770 align:middle line:84% stay because the gravitational force on the sun is on the Earth 01:34:29.770 --> 01:34:30.730 align:middle line:90% is the same. 01:34:30.730 --> 01:34:35.090 align:middle line:90% All right, so yes, crazy ideas. 01:34:35.090 --> 01:34:41.350 align:middle line:84% But one tool with which you have to evaluate these ideas 01:34:41.350 --> 01:34:45.620 align:middle line:84% and see if they are feasible or not 01:34:45.620 --> 01:34:49.940 align:middle line:84% is precisely what we are doing here for any energy technology, 01:34:49.940 --> 01:34:55.850 align:middle line:84% or for any technology, you have to see that you're transforming 01:34:55.850 --> 01:34:59.740 align:middle line:84% forms of energy, or you're spending energy 01:34:59.740 --> 01:35:01.650 align:middle line:90% to recover another effect. 01:35:01.650 --> 01:35:04.040 align:middle line:90% You have to define efficiency. 01:35:04.040 --> 01:35:07.730 align:middle line:84% Feasibility and efficiency are very important, no, 01:35:07.730 --> 01:35:11.420 align:middle line:84% because if you-- if the amount of energy 01:35:11.420 --> 01:35:18.650 align:middle line:84% you're consuming for a certain task is more than the work, 01:35:18.650 --> 01:35:22.610 align:middle line:84% minimum work to do that task, it's worthless. 01:35:22.610 --> 01:35:24.050 align:middle line:90% You better not do it. 01:35:24.050 --> 01:35:25.490 align:middle line:90% Do it in another way. 01:35:25.490 --> 01:35:31.970 align:middle line:84% OK, so what I'm saying is that somehow, 01:35:31.970 --> 01:35:38.900 align:middle line:84% by learning these tools and ways to understand the equilibrium 01:35:38.900 --> 01:35:41.340 align:middle line:84% and the non-equilibrium for small-- 01:35:41.340 --> 01:35:47.210 align:middle line:84% and for large and small systems, you are in good position 01:35:47.210 --> 01:35:52.880 align:middle line:84% to go and contribute to this very bold MIT 01:35:52.880 --> 01:35:59.360 align:middle line:84% project, new MIT enterprise for the next 10 years. 01:35:59.360 --> 01:36:04.470 align:middle line:84% So that brings me to, yeah, one more thing. 01:36:04.470 --> 01:36:07.460 align:middle line:84% And then just we have three minutes. 01:36:07.460 --> 01:36:15.570 align:middle line:84% Once I have a function, like this function here, 01:36:15.570 --> 01:36:22.080 align:middle line:84% then I can apply all the tools of mathematics. 01:36:22.080 --> 01:36:25.140 align:middle line:84% For example, if I am interested in looking 01:36:25.140 --> 01:36:31.500 align:middle line:84% at how the function behaves in the neighborhood of a given 01:36:31.500 --> 01:36:35.370 align:middle line:84% point, so here's the function, and here, 01:36:35.370 --> 01:36:41.220 align:middle line:84% and I want to see how it behaves in the neighborhood 01:36:41.220 --> 01:36:43.090 align:middle line:84% by changing one of the variables, 01:36:43.090 --> 01:36:52.310 align:middle line:84% for example, the energy, I can write a series expansion 01:36:52.310 --> 01:36:53.395 align:middle line:90% of the function. 01:36:53.395 --> 01:36:57.710 align:middle line:84% I take derivatives, first, second order, 01:36:57.710 --> 01:37:01.580 align:middle line:84% if the function is regular enough so that you 01:37:01.580 --> 01:37:03.890 align:middle line:84% can take at least two-- the first two 01:37:03.890 --> 01:37:06.210 align:middle line:84% derivatives with respect to all the variables, 01:37:06.210 --> 01:37:09.080 align:middle line:90% you're in good shape. 01:37:09.080 --> 01:37:12.970 align:middle line:84% In practice, in most of the models, 01:37:12.970 --> 01:37:16.020 align:middle line:84% the function is analytic, so if there's all derivatives. 01:37:16.020 --> 01:37:19.930 align:middle line:90% 01:37:19.930 --> 01:37:25.470 align:middle line:84% Some of these partial derivatives 01:37:25.470 --> 01:37:38.680 align:middle line:84% will turn out to be useful and to be important in determining 01:37:38.680 --> 01:37:41.895 align:middle line:84% necessary conditions for mutual equilibrium between systems. 01:37:41.895 --> 01:37:44.610 align:middle line:90% 01:37:44.610 --> 01:37:46.900 align:middle line:84% And it turns out that all the first derivatives, 01:37:46.900 --> 01:37:51.100 align:middle line:84% the first partial derivatives of this function here, 01:37:51.100 --> 01:37:58.090 align:middle line:84% play a role in determining those mutual equilibrium conditions. 01:37:58.090 --> 01:38:02.110 align:middle line:90% And that's why we give names. 01:38:02.110 --> 01:38:07.250 align:middle line:84% For example, in the derivative of S with respect to E, 01:38:07.250 --> 01:38:09.440 align:middle line:84% is the inverse of what we will call 01:38:09.440 --> 01:38:15.015 align:middle line:84% the temperature of the stable equilibrium state of the system. 01:38:15.015 --> 01:38:19.955 align:middle line:90% 01:38:19.955 --> 01:38:22.990 align:middle line:84% Yeah, that's because I write it also 01:38:22.990 --> 01:38:28.680 align:middle line:84% this way as a derivative of S-- of E with respect to S 01:38:28.680 --> 01:38:34.485 align:middle line:84% because once you have a function, and if it is well 01:38:34.485 --> 01:38:40.030 align:middle line:84% behaved enough so that all the derivatives are not 01:38:40.030 --> 01:38:41.380 align:middle line:90% neither 0 or infinite. 01:38:41.380 --> 01:38:43.670 align:middle line:90% So you can invert the facts. 01:38:43.670 --> 01:38:45.700 align:middle line:84% OK, there are mathematical theorems 01:38:45.700 --> 01:38:47.020 align:middle line:90% for inverting functions. 01:38:47.020 --> 01:38:50.300 align:middle line:84% So if I have this function of S as a function of E, 01:38:50.300 --> 01:38:52.900 align:middle line:84% I can also most of the times write it 01:38:52.900 --> 01:38:57.890 align:middle line:84% as E is a function of S. If I can do that, 01:38:57.890 --> 01:39:00.710 align:middle line:84% not for all systems, we'll see you can do that. 01:39:00.710 --> 01:39:03.910 align:middle line:84% But if you can't do that, then you 01:39:03.910 --> 01:39:06.820 align:middle line:84% have another version of this fundamental relation, which 01:39:06.820 --> 01:39:08.890 align:middle line:84% is the energy form, because it makes 01:39:08.890 --> 01:39:12.340 align:middle line:84% energy explicit and entropy becomes a dependent-- 01:39:12.340 --> 01:39:15.220 align:middle line:90% an independent variable. 01:39:15.220 --> 01:39:18.350 align:middle line:84% And you can also take derivatives of this one. 01:39:18.350 --> 01:39:22.180 align:middle line:84% So the partial derivative of E with respect to S 01:39:22.180 --> 01:39:25.300 align:middle line:84% is going to be what we call the temperature 01:39:25.300 --> 01:39:27.760 align:middle line:90% of the stable equilibrium state. 01:39:27.760 --> 01:39:29.710 align:middle line:84% The partial of E with respect to n 01:39:29.710 --> 01:39:33.580 align:middle line:84% is going to be the total potential, or chemical 01:39:33.580 --> 01:39:35.920 align:middle line:84% potential, depending on situations. 01:39:35.920 --> 01:39:40.095 align:middle line:84% And if one of the parameters is the volume, 01:39:40.095 --> 01:39:43.150 align:middle line:84% the partial of the energy with respect to the volume 01:39:43.150 --> 01:39:45.700 align:middle line:84% is what we will call the pressure 01:39:45.700 --> 01:39:48.880 align:middle line:90% with the negative in front-- 01:39:48.880 --> 01:39:50.590 align:middle line:90% sign in front. 01:39:50.590 --> 01:39:57.310 align:middle line:84% So these partial derivatives are related 01:39:57.310 --> 01:39:59.500 align:middle line:84% to conditions for mutual equilibrium, 01:39:59.500 --> 01:40:04.540 align:middle line:84% and are also therefore related to instruments with which you 01:40:04.540 --> 01:40:12.340 align:middle line:84% measure them by exploiting the mutual equilibrium condition. 01:40:12.340 --> 01:40:22.510 align:middle line:84% So you can have the thermometer, the manometer, and the, again, 01:40:22.510 --> 01:40:26.110 align:middle line:84% a manometer that allows it to measure partial pressure so 01:40:26.110 --> 01:40:30.160 align:middle line:84% that you can compute the chemical potentials. 01:40:30.160 --> 01:40:30.660 align:middle line:90% All right. 01:40:30.660 --> 01:40:35.220 align:middle line:84% So we are out of time and we stop here. 01:40:35.220 --> 01:40:40.650 align:middle line:84% So next time, we go to the energy versus entropy diagrams 01:40:40.650 --> 01:40:45.280 align:middle line:84% to summarize all that we have seen, and go further. 01:40:45.280 --> 01:40:54.000 align:middle line:90%