WEBVTT 00:00:00.000 --> 00:00:01.940 align:middle line:90% [SQUEAKING] 00:00:01.940 --> 00:00:04.365 align:middle line:90% [RUSTLING] 00:00:04.365 --> 00:00:07.760 align:middle line:90% [CLICKING] 00:00:07.760 --> 00:00:11.460 align:middle line:90% 00:00:11.460 --> 00:00:12.510 align:middle line:90% GIAN PAOLO BERETTA: OK. 00:00:12.510 --> 00:00:14.160 align:middle line:90% Good afternoon. 00:00:14.160 --> 00:00:18.660 align:middle line:90% So what have we done so far? 00:00:18.660 --> 00:00:23.140 align:middle line:90% 00:00:23.140 --> 00:00:26.300 align:middle line:90% What is thermodynamics about? 00:00:26.300 --> 00:00:32.530 align:middle line:84% Remember you told us that it's about heat and work. 00:00:32.530 --> 00:00:39.910 align:middle line:84% But to date, we haven't defined heat yet, 00:00:39.910 --> 00:00:43.240 align:middle line:84% or we spoke about work, although indirectly, 00:00:43.240 --> 00:00:47.160 align:middle line:90% with the weight processes. 00:00:47.160 --> 00:00:55.620 align:middle line:84% But we have done four hours, and we'll do another two today 00:00:55.620 --> 00:01:00.260 align:middle line:90% without talking about heat. 00:01:00.260 --> 00:01:03.720 align:middle line:84% Not saying that thermodynamics is not about heat. 00:01:03.720 --> 00:01:07.070 align:middle line:90% It is also about heat. 00:01:07.070 --> 00:01:15.050 align:middle line:84% But you can go a long way without and before defining 00:01:15.050 --> 00:01:21.770 align:middle line:84% the heat interaction, which will be in the topic of next lecture 00:01:21.770 --> 00:01:22.300 align:middle line:90% on Friday. 00:01:22.300 --> 00:01:25.830 align:middle line:90% 00:01:25.830 --> 00:01:29.670 align:middle line:84% So how could we go that long, all right? 00:01:29.670 --> 00:01:33.670 align:middle line:84% We just review briefly what we've done. 00:01:33.670 --> 00:01:39.700 align:middle line:84% We define a few words, like system, property, state, 00:01:39.700 --> 00:01:44.830 align:middle line:84% what we mean by law or principle, which 00:01:44.830 --> 00:01:51.160 align:middle line:84% is, in fact, a theorem of the equation of motion 00:01:51.160 --> 00:01:52.200 align:middle line:90% of your model. 00:01:52.200 --> 00:01:56.340 align:middle line:90% 00:01:56.340 --> 00:02:00.520 align:middle line:84% Then we introduced the first law, 00:02:00.520 --> 00:02:05.580 align:middle line:84% which is a statement of existence, 00:02:05.580 --> 00:02:08.820 align:middle line:84% or actually, the first assertion is a statement 00:02:08.820 --> 00:02:14.700 align:middle line:84% of existence of certain class of processes, namely those 00:02:14.700 --> 00:02:20.160 align:middle line:84% that allow you to interconnect any two states of any system. 00:02:20.160 --> 00:02:23.610 align:middle line:84% And remember, we mean a well-defined system, 00:02:23.610 --> 00:02:30.530 align:middle line:84% separable in the sense that we have made very explicit. 00:02:30.530 --> 00:02:36.330 align:middle line:84% The second assertion allows us to introduce 00:02:36.330 --> 00:02:42.690 align:middle line:84% a property, a general property for any of these systems 00:02:42.690 --> 00:02:49.890 align:middle line:84% and for any of its states by supporting the measurement 00:02:49.890 --> 00:02:50.616 align:middle line:90% procedure. 00:02:50.616 --> 00:02:53.920 align:middle line:90% 00:02:53.920 --> 00:03:00.380 align:middle line:84% That implies finding a weight process 00:03:00.380 --> 00:03:03.030 align:middle line:90% that interconnects two states. 00:03:03.030 --> 00:03:08.890 align:middle line:84% That way you can measure the energy difference 00:03:08.890 --> 00:03:10.160 align:middle line:90% between those two states. 00:03:10.160 --> 00:03:16.300 align:middle line:84% And that is what defines the property energy. 00:03:16.300 --> 00:03:18.760 align:middle line:84% And then this property can be proved 00:03:18.760 --> 00:03:26.600 align:middle line:84% to be additive, to be transferable between systems 00:03:26.600 --> 00:03:33.400 align:middle line:84% as a result of interactions, and also 00:03:33.400 --> 00:03:40.190 align:middle line:84% to be conserved, which takes us to the idea of making 00:03:40.190 --> 00:03:47.640 align:middle line:84% a balance of this property, checking in a given process 00:03:47.640 --> 00:03:53.970 align:middle line:84% that the change in the value of the energy, so the difference 00:03:53.970 --> 00:03:56.760 align:middle line:84% between the final and the initial energy, 00:03:56.760 --> 00:04:03.660 align:middle line:84% that's the change in energy, be equal to the energy that 00:04:03.660 --> 00:04:10.920 align:middle line:84% has been exchanged during this process between the system 00:04:10.920 --> 00:04:14.475 align:middle line:84% and the environment, positive if received. 00:04:14.475 --> 00:04:17.880 align:middle line:90% 00:04:17.880 --> 00:04:21.360 align:middle line:84% And this is, of course, the hinge 00:04:21.360 --> 00:04:25.980 align:middle line:84% of the solution of many problems that you have certainly 00:04:25.980 --> 00:04:29.280 align:middle line:84% seen in the undergraduate or previous graduate courses 00:04:29.280 --> 00:04:30.910 align:middle line:90% on thermodynamics. 00:04:30.910 --> 00:04:37.440 align:middle line:84% So I rely upon your experience here because this 00:04:37.440 --> 00:04:40.830 align:middle line:84% is, as you know, not a course about applications or problem 00:04:40.830 --> 00:04:41.880 align:middle line:90% solving. 00:04:41.880 --> 00:04:45.660 align:middle line:84% I assume that you have some experience of that. 00:04:45.660 --> 00:04:51.660 align:middle line:84% Because the more experience you have, the more your intuition 00:04:51.660 --> 00:04:58.880 align:middle line:84% and your sense of things will allow 00:04:58.880 --> 00:05:06.020 align:middle line:84% you to pick up the meaning of the underlying theory 00:05:06.020 --> 00:05:08.120 align:middle line:84% and assumptions that you are actually 00:05:08.120 --> 00:05:12.550 align:middle line:84% making when you perform, for example, an energy balance. 00:05:12.550 --> 00:05:16.500 align:middle line:90% 00:05:16.500 --> 00:05:19.360 align:middle line:84% Then we moved on to the second law. 00:05:19.360 --> 00:05:25.790 align:middle line:84% Also, the second law is made up of two assertions. 00:05:25.790 --> 00:05:28.640 align:middle line:84% One is a statement of existence again. 00:05:28.640 --> 00:05:30.610 align:middle line:84% But now, this time it's the existence 00:05:30.610 --> 00:05:35.000 align:middle line:84% of the stable equilibrium states of a system. 00:05:35.000 --> 00:05:47.340 align:middle line:84% And that by itself, together with the definition of stable 00:05:47.340 --> 00:05:52.530 align:middle line:84% equilibrium state, remember, it's a state that cannot be 00:05:52.530 --> 00:05:59.340 align:middle line:84% changed without leaving some net external effects to your-- 00:05:59.340 --> 00:06:02.860 align:middle line:84% so in the environment of your system. 00:06:02.860 --> 00:06:05.290 align:middle line:84% So if your system is in a stable equilibrium state, 00:06:05.290 --> 00:06:08.450 align:middle line:84% in order to change it to another state, 00:06:08.450 --> 00:06:10.100 align:middle line:90% you have to spend something. 00:06:10.100 --> 00:06:16.360 align:middle line:84% You have to leave some net effect in the environment. 00:06:16.360 --> 00:06:17.780 align:middle line:90% Otherwise, it is impossible. 00:06:17.780 --> 00:06:26.380 align:middle line:84% And this impossibility is what allows 00:06:26.380 --> 00:06:29.770 align:middle line:84% us to prove the impossibility of a perpetual motion 00:06:29.770 --> 00:06:33.670 align:middle line:84% machine of the second kind, which is the Kelvin-Planck 00:06:33.670 --> 00:06:37.880 align:middle line:90% statement of the second law. 00:06:37.880 --> 00:06:45.400 align:middle line:84% The second assertion supports the definition of a property 00:06:45.400 --> 00:06:49.310 align:middle line:84% that now we know, because we've defined it the last time, 00:06:49.310 --> 00:06:52.540 align:middle line:90% is entropy. 00:06:52.540 --> 00:06:55.900 align:middle line:90% So it assures that-- 00:06:55.900 --> 00:07:06.840 align:middle line:84% it says, well, you can always make a reversible weight 00:07:06.840 --> 00:07:12.110 align:middle line:84% process, which takes your system from any state it is in, 00:07:12.110 --> 00:07:14.030 align:middle line:90% to a stable equilibrium state. 00:07:14.030 --> 00:07:15.740 align:middle line:84% But not only a stable equilibrium state. 00:07:15.740 --> 00:07:20.330 align:middle line:84% A stable equilibrium state that, for example, 00:07:20.330 --> 00:07:24.350 align:middle line:84% could be in mutual equilibrium with your reservoir. 00:07:24.350 --> 00:07:30.370 align:middle line:84% So in this picture, I jumped to this conclusion 00:07:30.370 --> 00:07:37.950 align:middle line:84% that we have used and proved as a consequence of the assertion 00:07:37.950 --> 00:07:41.520 align:middle line:84% two, the second law, and that we used 00:07:41.520 --> 00:07:45.570 align:middle line:84% as a means to define the measurement 00:07:45.570 --> 00:07:51.780 align:middle line:84% procedures for a property that we call temperature 00:07:51.780 --> 00:07:52.600 align:middle line:90% of a reservoir. 00:07:52.600 --> 00:07:56.840 align:middle line:90% 00:07:56.840 --> 00:07:59.600 align:middle line:90% I'm not going to repeat that. 00:07:59.600 --> 00:08:03.950 align:middle line:84% But as you have seen in the first homework set, 00:08:03.950 --> 00:08:10.530 align:middle line:84% I am asking you to repeat it in a short video. 00:08:10.530 --> 00:08:20.300 align:middle line:84% And we used that same process, reversible weight process 00:08:20.300 --> 00:08:22.130 align:middle line:84% between the system and the reservoir, 00:08:22.130 --> 00:08:28.770 align:middle line:84% in order to measure the property entropy, which 00:08:28.770 --> 00:08:32.280 align:middle line:90% is defined by this ratio. 00:08:32.280 --> 00:08:37.380 align:middle line:84% It's the ratio between the change 00:08:37.380 --> 00:08:41.460 align:middle line:84% in energy of the reservoir that occurs when 00:08:41.460 --> 00:08:48.360 align:middle line:84% you manage to make this reversible weight process that 00:08:48.360 --> 00:08:50.310 align:middle line:90% interconnects-- 00:08:50.310 --> 00:08:54.060 align:middle line:84% in which your system goes between the two states for which 00:08:54.060 --> 00:08:58.580 align:middle line:84% you want to measure the entropy difference, divided 00:08:58.580 --> 00:09:02.280 align:middle line:84% by the temperature of the reservoir, 00:09:02.280 --> 00:09:06.180 align:middle line:84% which was defined by the previous measurement procedure. 00:09:06.180 --> 00:09:16.240 align:middle line:84% This ratio turns out to be independent of the reservoir 00:09:16.240 --> 00:09:19.270 align:middle line:84% that you have chosen and used as therefore 00:09:19.270 --> 00:09:25.520 align:middle line:84% just an auxiliary system, auxiliary to this measurement 00:09:25.520 --> 00:09:26.280 align:middle line:90% procedure. 00:09:26.280 --> 00:09:30.280 align:middle line:84% But the result is independent of that system. 00:09:30.280 --> 00:09:34.780 align:middle line:84% Therefore, it is a property for just system A, something 00:09:34.780 --> 00:09:38.970 align:middle line:84% that depends on your system A, that we call the entropy. 00:09:38.970 --> 00:09:42.470 align:middle line:90% 00:09:42.470 --> 00:09:51.310 align:middle line:84% We also defined a property called available energy 00:09:51.310 --> 00:09:55.150 align:middle line:90% with respect to the reservoir. 00:09:55.150 --> 00:09:57.200 align:middle line:90% It is an additive property. 00:09:57.200 --> 00:10:04.010 align:middle line:84% It can be expressed in terms of energy and entropy. 00:10:04.010 --> 00:10:07.910 align:middle line:84% And it allows us to give an engineering interpretation 00:10:07.910 --> 00:10:15.200 align:middle line:84% of entropy as being, apart from a constant value, which 00:10:15.200 --> 00:10:17.510 align:middle line:90% is this group here. 00:10:17.510 --> 00:10:21.320 align:middle line:84% It's proportional through the inverse 00:10:21.320 --> 00:10:23.180 align:middle line:84% of the temperature of the reservoir 00:10:23.180 --> 00:10:27.590 align:middle line:84% to the energy that is not available. 00:10:27.590 --> 00:10:31.190 align:middle line:84% So it's not transferable to a weight in a weight 00:10:31.190 --> 00:10:37.160 align:middle line:84% process for the system when in combination with the reservoir 00:10:37.160 --> 00:10:40.230 align:middle line:90% with the temperature. 00:10:40.230 --> 00:10:45.230 align:middle line:84% So entropy is the nonavailable energy. 00:10:45.230 --> 00:10:49.010 align:middle line:90% 00:10:49.010 --> 00:10:50.660 align:middle line:90% And it is additive. 00:10:50.660 --> 00:10:55.460 align:middle line:84% The additivity, remember we also had another property, which 00:10:55.460 --> 00:11:00.920 align:middle line:84% is adiabatic availability which contains much of the information 00:11:00.920 --> 00:11:06.140 align:middle line:84% and much of the value in terms of, 00:11:06.140 --> 00:11:11.630 align:middle line:84% what's the best I can do with a given system in a given state 00:11:11.630 --> 00:11:15.540 align:middle line:84% if I constrain myself on just weight processes? 00:11:15.540 --> 00:11:22.160 align:middle line:84% But the disadvantage of the adiabatic availability, 00:11:22.160 --> 00:11:25.410 align:middle line:84% is the fact that it is not additive. 00:11:25.410 --> 00:11:28.170 align:middle line:90% 00:11:28.170 --> 00:11:32.760 align:middle line:84% Whereas entropy allows to compute the same things, 00:11:32.760 --> 00:11:36.090 align:middle line:84% and it is added, which is much more convenient. 00:11:36.090 --> 00:11:40.050 align:middle line:90% 00:11:40.050 --> 00:11:42.840 align:middle line:84% And then we moved, we also proved, 00:11:42.840 --> 00:11:50.670 align:middle line:84% or said that we can prove, this theorems about criteria 00:11:50.670 --> 00:11:56.330 align:middle line:84% to determine whether a weight process is reversible. 00:11:56.330 --> 00:11:59.960 align:middle line:84% And remember the definition of reversibility, 00:11:59.960 --> 00:12:02.450 align:middle line:84% a process is reversible if we can 00:12:02.450 --> 00:12:08.072 align:middle line:84% undo not only what happens to the system, 00:12:08.072 --> 00:12:09.780 align:middle line:84% but also what happens to the environment. 00:12:09.780 --> 00:12:13.190 align:middle line:84% So we have to be able to find some other process, not 00:12:13.190 --> 00:12:17.340 align:middle line:84% necessarily tracing back, going backwards, the original process, 00:12:17.340 --> 00:12:26.810 align:middle line:84% but one in which you can bring your system back 00:12:26.810 --> 00:12:33.710 align:middle line:84% to its initial state while also you bring the environment back 00:12:33.710 --> 00:12:35.980 align:middle line:90% to its initial state. 00:12:35.980 --> 00:12:40.940 align:middle line:84% That definition is very solid and conceptually valid, 00:12:40.940 --> 00:12:42.740 align:middle line:90% but it is also very abstract. 00:12:42.740 --> 00:12:49.030 align:middle line:84% It doesn't correspond to a practicable procedure 00:12:49.030 --> 00:12:52.880 align:middle line:84% to determine whether a process is reversible or not. 00:12:52.880 --> 00:12:54.920 align:middle line:84% Because if you go by that definition, 00:12:54.920 --> 00:12:57.550 align:middle line:84% you are supposed to try all possible processes 00:12:57.550 --> 00:13:02.210 align:middle line:84% until you are really sure that you tried them all. 00:13:02.210 --> 00:13:05.830 align:middle line:84% And it's impossible to, or possible 00:13:05.830 --> 00:13:11.090 align:middle line:84% to bring back those states to their originals. 00:13:11.090 --> 00:13:14.160 align:middle line:84% And so that's clearly impractical. 00:13:14.160 --> 00:13:21.430 align:middle line:84% Instead, property adiabatic availability, or entropy, 00:13:21.430 --> 00:13:29.050 align:middle line:84% or also available energy allow you to, 00:13:29.050 --> 00:13:32.680 align:middle line:84% if you know how to measure it, and we 00:13:32.680 --> 00:13:37.060 align:middle line:84% know because now we have the measurement procedures, 00:13:37.060 --> 00:13:42.130 align:middle line:84% they allow you to have a quantitative criterion 00:13:42.130 --> 00:13:46.540 align:middle line:90% to decide on the reversibility. 00:13:46.540 --> 00:13:52.540 align:middle line:84% And actually, you can also quantify the irreversibility 00:13:52.540 --> 00:13:58.660 align:middle line:84% in terms of how much these properties that 00:13:58.660 --> 00:14:03.550 align:middle line:90% appear in the criteria change. 00:14:03.550 --> 00:14:07.600 align:middle line:84% So in a reversible process, and remember, here 00:14:07.600 --> 00:14:11.320 align:middle line:84% we are talking about weight process, just weight process. 00:14:11.320 --> 00:14:15.310 align:middle line:84% In a reversible weight process, the entropy remains constant. 00:14:15.310 --> 00:14:20.610 align:middle line:90% 00:14:20.610 --> 00:14:25.266 align:middle line:84% In an irreversible weight process, the entropy increases. 00:14:25.266 --> 00:14:31.410 align:middle line:90% 00:14:31.410 --> 00:14:35.170 align:middle line:84% That, together with the fact that entropy is additive, 00:14:35.170 --> 00:14:40.370 align:middle line:84% that entropy can be exchanged via interactions, 00:14:40.370 --> 00:14:42.870 align:middle line:84% you can transfer entropy from one system to the other. 00:14:42.870 --> 00:14:44.680 align:middle line:90% It's something that you do. 00:14:44.680 --> 00:14:48.640 align:middle line:84% Remember the example of the can of Coke in the refrigerator? 00:14:48.640 --> 00:14:51.780 align:middle line:84% Well, when it cools down, there is not 00:14:51.780 --> 00:14:53.970 align:middle line:84% only energy that goes out of the can 00:14:53.970 --> 00:14:56.140 align:middle line:84% into the air of the refrigerator, 00:14:56.140 --> 00:14:59.110 align:middle line:90% but also, entropy flows. 00:14:59.110 --> 00:15:01.830 align:middle line:90% And we will get to that. 00:15:01.830 --> 00:15:08.220 align:middle line:84% And together with that, and with the principle 00:15:08.220 --> 00:15:12.330 align:middle line:84% of entropy non-decrease, which is in weight processes, which 00:15:12.330 --> 00:15:17.340 align:middle line:84% is just what we said about criteria for reversibility 00:15:17.340 --> 00:15:20.670 align:middle line:84% or irreversibility, you can write the entropy balance 00:15:20.670 --> 00:15:27.380 align:middle line:84% equation, which contains three concepts. 00:15:27.380 --> 00:15:30.470 align:middle line:84% On the left-hand side, the change 00:15:30.470 --> 00:15:33.350 align:middle line:84% in entropy during a given process for your system A, 00:15:33.350 --> 00:15:37.430 align:middle line:84% so the difference between the final and the initial entropy. 00:15:37.430 --> 00:15:41.720 align:middle line:84% On the right-hand side, you have two contributions 00:15:41.720 --> 00:15:45.350 align:middle line:84% that must balance the left-hand side. 00:15:45.350 --> 00:15:51.520 align:middle line:84% The first contribution is how much entropy 00:15:51.520 --> 00:15:57.610 align:middle line:84% you have exchanged between your system and its environment 00:15:57.610 --> 00:15:58.550 align:middle line:90% during the process. 00:15:58.550 --> 00:16:03.290 align:middle line:84% And that is positive if received. 00:16:03.290 --> 00:16:05.940 align:middle line:84% So you have to put it positive if received. 00:16:05.940 --> 00:16:08.900 align:middle line:84% That's the meaning we've seen of the arrow on the symbol 00:16:08.900 --> 00:16:10.195 align:middle line:90% on the energy transfer. 00:16:10.195 --> 00:16:12.910 align:middle line:90% 00:16:12.910 --> 00:16:16.270 align:middle line:84% And the second term is the entropy 00:16:16.270 --> 00:16:22.860 align:middle line:84% that has been generated by irreversibility. 00:16:22.860 --> 00:16:29.795 align:middle line:84% So it's the contribution of the irreversibility. 00:16:29.795 --> 00:16:32.295 align:middle line:84% So if the process is reversible, that term is equal to zero. 00:16:32.295 --> 00:16:36.910 align:middle line:90% 00:16:36.910 --> 00:16:41.340 align:middle line:84% Of course, we will return to energy and entropy balances. 00:16:41.340 --> 00:16:44.790 align:middle line:84% And as you know, if you have done 00:16:44.790 --> 00:16:49.380 align:middle line:84% in previous courses, or maybe in energy conservation, 00:16:49.380 --> 00:16:55.635 align:middle line:84% or energy technologies courses, if you've done exergy, 00:16:55.635 --> 00:17:00.750 align:middle line:84% you know that the combination of the energy and entropy balance 00:17:00.750 --> 00:17:03.540 align:middle line:90% gives you exergy. 00:17:03.540 --> 00:17:11.200 align:middle line:84% Exergy, meaning the maximum useful effect 00:17:11.200 --> 00:17:20.510 align:middle line:84% that you can obtain under given conditions 00:17:20.510 --> 00:17:25.270 align:middle line:84% from a system in a given state, or vice versa. 00:17:25.270 --> 00:17:29.270 align:middle line:84% It's also, so by the combined energy and entropy balance, 00:17:29.270 --> 00:17:35.720 align:middle line:84% you can compute, what is the minimum amount of energy, 00:17:35.720 --> 00:17:39.610 align:middle line:84% so valuable resources, that you will have to spend in order 00:17:39.610 --> 00:17:44.090 align:middle line:84% to obtain a given effect, so a given change, 00:17:44.090 --> 00:17:51.560 align:middle line:84% for example, to produce or to heat up something, water, steel. 00:17:51.560 --> 00:17:54.220 align:middle line:90% 00:17:54.220 --> 00:17:54.720 align:middle line:90% All right. 00:17:54.720 --> 00:18:00.050 align:middle line:84% And again, we'll come back when we talk about, 00:18:00.050 --> 00:18:04.100 align:middle line:84% in a couple of lectures, we will have one review 00:18:04.100 --> 00:18:05.785 align:middle line:90% of these exergy concepts. 00:18:05.785 --> 00:18:08.850 align:middle line:90% 00:18:08.850 --> 00:18:18.270 align:middle line:84% Then we moved on to focus on the equilibrium states, 00:18:18.270 --> 00:18:19.770 align:middle line:84% but not only the equilibrium states. 00:18:19.770 --> 00:18:27.260 align:middle line:84% Because here we have the maximum entropy principle 00:18:27.260 --> 00:18:32.810 align:middle line:84% that says that if you have your system, 00:18:32.810 --> 00:18:37.760 align:middle line:84% and you fix the values of the amounts of constituents, 00:18:37.760 --> 00:18:40.890 align:middle line:84% of the parameters of the external forces, 00:18:40.890 --> 00:18:45.150 align:middle line:84% such as the volume, but not necessarily only the volume. 00:18:45.150 --> 00:18:50.550 align:middle line:84% And you also fix the value of the energy, that 00:18:50.550 --> 00:18:56.070 align:middle line:84% fixes only a short part of the long list of values 00:18:56.070 --> 00:19:01.050 align:middle line:84% of properties that altogether form what we 00:19:01.050 --> 00:19:05.850 align:middle line:90% call the state of the system. 00:19:05.850 --> 00:19:11.920 align:middle line:84% So by fixing this short partial part of the list, 00:19:11.920 --> 00:19:15.280 align:middle line:84% you're leaving free the rest of the properties. 00:19:15.280 --> 00:19:19.590 align:middle line:84% So there are many states that belong 00:19:19.590 --> 00:19:26.060 align:middle line:84% to that subset, so that share the same values of the amounts, 00:19:26.060 --> 00:19:28.810 align:middle line:90% the parameters and the energy. 00:19:28.810 --> 00:19:36.960 align:middle line:84% And among all these in this subset, 00:19:36.960 --> 00:19:39.870 align:middle line:84% there is one and only one stable equilibrium state, 00:19:39.870 --> 00:19:44.490 align:middle line:84% that is a consequence of the second law. 00:19:44.490 --> 00:19:50.360 align:middle line:84% But-- and another theorem is also 00:19:50.360 --> 00:19:56.090 align:middle line:84% that that stable equilibrium state has the highest 00:19:56.090 --> 00:19:57.500 align:middle line:90% value of the entropy. 00:19:57.500 --> 00:20:00.290 align:middle line:84% No other state has the same value. 00:20:00.290 --> 00:20:06.205 align:middle line:84% It has the strictly highest value within that subset. 00:20:06.205 --> 00:20:12.400 align:middle line:90% 00:20:12.400 --> 00:20:21.800 align:middle line:84% And finally, we introduced the state principle, saying that, 00:20:21.800 --> 00:20:28.320 align:middle line:84% well, if it is true, like the second law says, 00:20:28.320 --> 00:20:34.280 align:middle line:84% that if I fix the values of amounts, parameters, and energy, 00:20:34.280 --> 00:20:41.010 align:middle line:84% and these values point to a unique, stable equilibrium 00:20:41.010 --> 00:20:49.070 align:middle line:84% state, it means that these values fix all 00:20:49.070 --> 00:20:53.090 align:middle line:90% the properties of that state. 00:20:53.090 --> 00:20:55.490 align:middle line:84% Because fixing the state means fixing the values 00:20:55.490 --> 00:20:56.620 align:middle line:90% of all the properties. 00:20:56.620 --> 00:21:01.610 align:middle line:84% So it means that for the category, 00:21:01.610 --> 00:21:05.540 align:middle line:84% for the very particular class of states, which 00:21:05.540 --> 00:21:08.060 align:middle line:84% are the stable equilibrium states, 00:21:08.060 --> 00:21:15.690 align:middle line:84% the properties are all functions of these three groups, amounts, 00:21:15.690 --> 00:21:19.850 align:middle line:84% parameters, and energy of these three properties, 00:21:19.850 --> 00:21:24.920 align:middle line:90% or it's actually r. 00:21:24.920 --> 00:21:31.370 align:middle line:84% If you have r constituencies, r values, it's as many parameters 00:21:31.370 --> 00:21:33.760 align:middle line:90% as you need, plus the energy. 00:21:33.760 --> 00:21:38.910 align:middle line:84% So r plus s plus 1 values fix the value 00:21:38.910 --> 00:21:43.230 align:middle line:84% of any other property of the stable equilibrium state, 00:21:43.230 --> 00:21:45.750 align:middle line:90% including the entropy. 00:21:45.750 --> 00:21:51.480 align:middle line:84% So that gives a proof of the existence for stable equilibrium 00:21:51.480 --> 00:21:55.990 align:middle line:84% states of this function, which we 00:21:55.990 --> 00:21:57.435 align:middle line:90% call the fundamental relation. 00:21:57.435 --> 00:22:01.630 align:middle line:90% 00:22:01.630 --> 00:22:07.630 align:middle line:84% And which, if it is sufficiently regular 00:22:07.630 --> 00:22:13.190 align:middle line:84% or well behaved so that you can take derivatives, 00:22:13.190 --> 00:22:17.120 align:middle line:84% and these derivatives are not zero or infinite, 00:22:17.120 --> 00:22:23.390 align:middle line:90% you can invert this function. 00:22:23.390 --> 00:22:27.180 align:middle line:84% So instead of writing S as a function of E, n and beta, 00:22:27.180 --> 00:22:31.530 align:middle line:84% you can write it as E is a function of S, n and beta. 00:22:31.530 --> 00:22:35.350 align:middle line:90% 00:22:35.350 --> 00:22:41.800 align:middle line:84% Whereas the fundamental relation in entropy form, this one, 00:22:41.800 --> 00:22:51.200 align:middle line:84% is always there for all systems, the E 00:22:51.200 --> 00:22:55.790 align:middle line:84% versus S, so the energy form, is there only 00:22:55.790 --> 00:23:04.970 align:middle line:84% for systems for which these derivatives don't go to zero. 00:23:04.970 --> 00:23:10.160 align:middle line:84% And therefore, we will see in a moment 00:23:10.160 --> 00:23:15.110 align:middle line:84% that there are some exceptions, particular systems that 00:23:15.110 --> 00:23:21.470 align:middle line:84% are important in the quantum world for which there is 00:23:21.470 --> 00:23:24.867 align:middle line:90% an upper bound in the energy. 00:23:24.867 --> 00:23:37.465 align:middle line:90% 00:23:37.465 --> 00:23:48.230 align:middle line:84% Now about taking derivatives, when I have a function, 00:23:48.230 --> 00:23:50.660 align:middle line:84% I can write a differential of the function. 00:23:50.660 --> 00:24:00.300 align:middle line:84% We just do it just for one variable, or say, two variables. 00:24:00.300 --> 00:24:09.680 align:middle line:84% So z is a function of x and y, and therefore, the differential 00:24:09.680 --> 00:24:15.860 align:middle line:84% is the partial of f with respect to x times 00:24:15.860 --> 00:24:22.315 align:middle line:84% dx, plus the partial of f with respect to y in the dy. 00:24:22.315 --> 00:24:26.968 align:middle line:90% 00:24:26.968 --> 00:24:35.160 align:middle line:84% Now we apply here the idea to the function 00:24:35.160 --> 00:24:38.280 align:middle line:84% to the entropy form of the fundamental relation, 00:24:38.280 --> 00:24:44.360 align:middle line:90% and so written this way. 00:24:44.360 --> 00:24:47.480 align:middle line:84% Really, here what I have is that the partial derivative 00:24:47.480 --> 00:24:51.350 align:middle line:84% of S with respect to E that multiplies dE, 00:24:51.350 --> 00:24:56.810 align:middle line:84% plus the partial derivative of S with respect to V, 00:24:56.810 --> 00:25:01.640 align:middle line:84% the second variable, times dV, plus the partial derivative of S 00:25:01.640 --> 00:25:10.900 align:middle line:84% with respect to n or ni times dni, 00:25:10.900 --> 00:25:12.540 align:middle line:84% and so on for all the other variables. 00:25:12.540 --> 00:25:18.170 align:middle line:90% 00:25:18.170 --> 00:25:22.030 align:middle line:84% We have defined those partial derivatives. 00:25:22.030 --> 00:25:23.280 align:middle line:90% Look at this one. 00:25:23.280 --> 00:25:25.130 align:middle line:84% The partial of S with respect to E, 00:25:25.130 --> 00:25:31.140 align:middle line:84% we called it 1 over T because we will identify. 00:25:31.140 --> 00:25:33.960 align:middle line:84% So the inverse of that partial derivative 00:25:33.960 --> 00:25:39.040 align:middle line:84% is the temperature of the stable equilibrium state. 00:25:39.040 --> 00:25:41.800 align:middle line:90% 00:25:41.800 --> 00:25:47.220 align:middle line:84% This property, notice that is defined based 00:25:47.220 --> 00:25:49.270 align:middle line:90% on the fundamental relation. 00:25:49.270 --> 00:25:51.480 align:middle line:84% And the fundamental relation is defined only 00:25:51.480 --> 00:25:53.350 align:middle line:84% for the stable equilibrium state. 00:25:53.350 --> 00:25:59.265 align:middle line:84% So temperature is defined only for the stable equilibrium 00:25:59.265 --> 00:25:59.765 align:middle line:90% states. 00:25:59.765 --> 00:26:02.610 align:middle line:90% 00:26:02.610 --> 00:26:07.910 align:middle line:84% Even for the reservoir, we define temperature 00:26:07.910 --> 00:26:10.440 align:middle line:90% in that particular way. 00:26:10.440 --> 00:26:15.200 align:middle line:84% And it's the temperature of the stable equilibrium 00:26:15.200 --> 00:26:16.949 align:middle line:90% state of the reservoir. 00:26:16.949 --> 00:26:22.220 align:middle line:90% 00:26:22.220 --> 00:26:27.740 align:middle line:84% Then, we give also names, for example, the partial derivative 00:26:27.740 --> 00:26:31.860 align:middle line:84% of S with respect to the volume, we 00:26:31.860 --> 00:26:45.090 align:middle line:84% call it p over T, where p will be identified as the pressure. 00:26:45.090 --> 00:26:51.400 align:middle line:84% And the pressure, I think I will not go into this proof, 00:26:51.400 --> 00:26:53.200 align:middle line:90% but you can prove it. 00:26:53.200 --> 00:26:57.750 align:middle line:84% And again, you find the proof and all details in the book. 00:26:57.750 --> 00:27:01.130 align:middle line:84% If you haven't seen it, it's useful to go and look it up. 00:27:01.130 --> 00:27:06.470 align:middle line:90% 00:27:06.470 --> 00:27:08.960 align:middle line:84% The pressure,-- when a system is-- 00:27:08.960 --> 00:27:12.930 align:middle line:84% first of all, is defined only for stable equilibrium state. 00:27:12.930 --> 00:27:17.880 align:middle line:84% And you say, well, look, if I have a wall, 00:27:17.880 --> 00:27:21.990 align:middle line:84% and I put a pressure sensor on the wall, 00:27:21.990 --> 00:27:24.070 align:middle line:84% I can always measure the pressure, 00:27:24.070 --> 00:27:27.870 align:middle line:84% even if my system is not in a stable equilibrium state. 00:27:27.870 --> 00:27:29.040 align:middle line:90% Well, no. 00:27:29.040 --> 00:27:32.820 align:middle line:84% What you're measuring then is the force 00:27:32.820 --> 00:27:36.610 align:middle line:90% per unit area on that location. 00:27:36.610 --> 00:27:39.060 align:middle line:84% So the force per unit area, you can always 00:27:39.060 --> 00:27:42.510 align:middle line:84% measure with the sensor that, we shouldn't call it 00:27:42.510 --> 00:27:43.540 align:middle line:90% pressure sensor. 00:27:43.540 --> 00:27:48.720 align:middle line:84% It's called force per unit area sensor. 00:27:48.720 --> 00:27:53.550 align:middle line:84% It turns out that when the system is 00:27:53.550 --> 00:27:57.390 align:middle line:84% in a stable equilibrium state, that force per unit area 00:27:57.390 --> 00:28:03.340 align:middle line:84% corresponds to the pressure we find this way. 00:28:03.340 --> 00:28:08.300 align:middle line:84% But, for example, in a non-equilibrium state, 00:28:08.300 --> 00:28:10.370 align:middle line:84% when you measure force per unit area 00:28:10.370 --> 00:28:14.420 align:middle line:84% in different locations of your system, 00:28:14.420 --> 00:28:18.500 align:middle line:90% you may find different values. 00:28:18.500 --> 00:28:21.990 align:middle line:84% Whereas if your system is in a stable equilibrium 00:28:21.990 --> 00:28:28.310 align:middle line:84% state, wherever on the surface of the confining surface 00:28:28.310 --> 00:28:30.980 align:middle line:84% of your system you measure force per unit area, 00:28:30.980 --> 00:28:35.220 align:middle line:90% you find always the same value. 00:28:35.220 --> 00:28:37.750 align:middle line:84% And that's-- which corresponds to the pressure, 00:28:37.750 --> 00:28:40.930 align:middle line:90% so to give that-- to fix ideas. 00:28:40.930 --> 00:28:41.945 align:middle line:90% Suppose I have a box. 00:28:41.945 --> 00:28:44.920 align:middle line:90% 00:28:44.920 --> 00:28:50.710 align:middle line:84% And in my box I got some particles. 00:28:50.710 --> 00:29:02.680 align:middle line:84% And suppose that the particles are initially all very 00:29:02.680 --> 00:29:08.760 align:middle line:84% regularly distributed only in this central part of the box. 00:29:08.760 --> 00:29:12.330 align:middle line:84% And they all move in one direction initially. 00:29:12.330 --> 00:29:14.930 align:middle line:84% So they're going to bounce on this wall, 00:29:14.930 --> 00:29:19.820 align:middle line:90% and then go back, and so on. 00:29:19.820 --> 00:29:23.720 align:middle line:84% But, of course, due to some randomness 00:29:23.720 --> 00:29:30.420 align:middle line:84% that results from their collisions with the wall, 00:29:30.420 --> 00:29:37.740 align:middle line:84% eventually they will scatter around and fill the entire box. 00:29:37.740 --> 00:29:42.790 align:middle line:84% But initially, if you measure force per unit area 00:29:42.790 --> 00:29:49.660 align:middle line:84% here when the particles are hitting this wall, or here, 00:29:49.660 --> 00:29:52.920 align:middle line:84% you're going to find a very different value 00:29:52.920 --> 00:29:59.410 align:middle line:84% from what you do if you measure your pressure there. 00:29:59.410 --> 00:30:04.870 align:middle line:84% And by the way, what do you mean by measuring pressure? 00:30:04.870 --> 00:30:10.110 align:middle line:84% A pressure measurement is simply that you are conceptually 00:30:10.110 --> 00:30:14.890 align:middle line:84% breaking the wall, substituting it with a movable piston, 00:30:14.890 --> 00:30:20.400 align:middle line:84% and then find what force you need 00:30:20.400 --> 00:30:26.860 align:middle line:84% on that little piston of area a, small a, a small area, 00:30:26.860 --> 00:30:31.310 align:middle line:84% in order to keep it in the same place where the wall is. 00:30:31.310 --> 00:30:35.720 align:middle line:84% Once you find this value of force, 00:30:35.720 --> 00:30:40.010 align:middle line:84% then the force per unit area is Fa over a. 00:30:40.010 --> 00:30:48.470 align:middle line:84% And that's the result of your force per unit area 00:30:48.470 --> 00:30:51.720 align:middle line:90% gauge or measurement device. 00:30:51.720 --> 00:30:56.900 align:middle line:84% But if you do that here, well, there 00:30:56.900 --> 00:30:58.970 align:middle line:84% are no particles sitting there, you 00:30:58.970 --> 00:31:03.290 align:middle line:90% find zero force per unit area. 00:31:03.290 --> 00:31:07.550 align:middle line:84% And as the particles scatter around and distribute, 00:31:07.550 --> 00:31:14.950 align:middle line:84% and become all randomly directed, then eventually, 00:31:14.950 --> 00:31:18.900 align:middle line:84% you will reach the stable equilibrium state. 00:31:18.900 --> 00:31:20.980 align:middle line:84% And in a stable equilibrium state, 00:31:20.980 --> 00:31:24.900 align:middle line:84% you will find the same values, no matter where 00:31:24.900 --> 00:31:26.670 align:middle line:84% on the wall of your system you're 00:31:26.670 --> 00:31:28.650 align:middle line:84% going to measure the force per unit area. 00:31:28.650 --> 00:31:32.740 align:middle line:84% And in that limit, so for the stable equilibrium state, 00:31:32.740 --> 00:31:38.580 align:middle line:84% you will find that F over a at stable equilibrium 00:31:38.580 --> 00:31:44.840 align:middle line:84% is equal to this p, The thing that we 00:31:44.840 --> 00:31:47.600 align:middle line:84% call pressure of the stable equilibrium state. 00:31:47.600 --> 00:31:50.940 align:middle line:90% 00:31:50.940 --> 00:31:53.940 align:middle line:84% And these other derivatives, partial derivatives 00:31:53.940 --> 00:31:57.090 align:middle line:84% of the fundamental relation, but partial of S with respect 00:31:57.090 --> 00:32:03.160 align:middle line:84% to the amounts of constituents, are, 00:32:03.160 --> 00:32:05.660 align:middle line:84% if you change sign and divide by T, 00:32:05.660 --> 00:32:16.060 align:middle line:84% that's the chemical potential of that particular constituent. 00:32:16.060 --> 00:32:21.610 align:middle line:84% We'll return to that, and also to a measurement device 00:32:21.610 --> 00:32:25.900 align:middle line:84% that is based, again, on measuring force per unit area, 00:32:25.900 --> 00:32:35.340 align:middle line:84% but not directly like this, but indirectly 00:32:35.340 --> 00:32:40.810 align:middle line:84% by putting a semipermeable, substituting 00:32:40.810 --> 00:32:43.510 align:middle line:84% the wall with a semipermeable membrane, 00:32:43.510 --> 00:32:56.250 align:middle line:84% and then measuring the force per unit area 00:32:56.250 --> 00:33:00.650 align:middle line:84% here, where just particles of one type 00:33:00.650 --> 00:33:03.000 align:middle line:84% can cross the semipermeable membrane. 00:33:03.000 --> 00:33:05.960 align:middle line:84% That will be a measurement of what we will call 00:33:05.960 --> 00:33:09.270 align:middle line:84% and what defines the partial pressure of constituent i. 00:33:09.270 --> 00:33:12.550 align:middle line:90% 00:33:12.550 --> 00:33:17.950 align:middle line:84% You measure it, again, by using this pressure 00:33:17.950 --> 00:33:20.955 align:middle line:90% of force per unit area gauge. 00:33:20.955 --> 00:33:25.220 align:middle line:90% 00:33:25.220 --> 00:33:31.640 align:middle line:84% So back to the differential of the fundamental relation. 00:33:31.640 --> 00:33:33.630 align:middle line:84% So essentially, it's this formula, 00:33:33.630 --> 00:33:38.210 align:middle line:84% except that for the partial derivatives, we've given names. 00:33:38.210 --> 00:33:41.420 align:middle line:84% And we'll see at the end of today 00:33:41.420 --> 00:33:44.520 align:middle line:84% why we give names to these partial derivatives, 00:33:44.520 --> 00:33:54.000 align:middle line:84% because they enter into the necessary conditions that must 00:33:54.000 --> 00:33:56.490 align:middle line:90% be satisfied in order for-- 00:33:56.490 --> 00:33:59.230 align:middle line:84% when two systems are in mutual equilibrium. 00:33:59.230 --> 00:34:03.480 align:middle line:90% 00:34:03.480 --> 00:34:05.580 align:middle line:84% And these conditions will involve 00:34:05.580 --> 00:34:08.790 align:middle line:90% these partial derivatives. 00:34:08.790 --> 00:34:15.389 align:middle line:84% And these conditions will be consequences 00:34:15.389 --> 00:34:20.000 align:middle line:84% of the principle of maximum entropy that we just reviewed. 00:34:20.000 --> 00:34:25.210 align:middle line:90% 00:34:25.210 --> 00:34:32.020 align:middle line:84% Now if you do the same development, 00:34:32.020 --> 00:34:35.139 align:middle line:84% so you write the differential of the fundamental relation, 00:34:35.139 --> 00:34:37.780 align:middle line:84% not in entropy form, but in energy form, 00:34:37.780 --> 00:34:46.000 align:middle line:84% so the same expression that you sort of invert, 00:34:46.000 --> 00:34:48.980 align:middle line:84% then you write the differential in this way. 00:34:48.980 --> 00:34:54.949 align:middle line:90% 00:34:54.949 --> 00:35:01.330 align:middle line:84% You can see it directly by solving the above equation 00:35:01.330 --> 00:35:03.830 align:middle line:90% for dE. 00:35:03.830 --> 00:35:08.090 align:middle line:84% So you take dE on the left-hand side 00:35:08.090 --> 00:35:16.580 align:middle line:84% and put the rest on the right-hand side, 00:35:16.580 --> 00:35:22.720 align:middle line:84% and you can obtain this equation. 00:35:22.720 --> 00:35:25.690 align:middle line:90% It's just algebra. 00:35:25.690 --> 00:35:29.700 align:middle line:84% In this particular form, which is called the Gibbs relation, 00:35:29.700 --> 00:35:33.810 align:middle line:84% you see that temperature, pressure, 00:35:33.810 --> 00:35:41.190 align:middle line:84% the negative of the pressure, and the chemical potentials 00:35:41.190 --> 00:35:45.860 align:middle line:90% are the partial derivatives. 00:35:45.860 --> 00:35:49.990 align:middle line:90% Why do I say that? 00:35:49.990 --> 00:35:57.510 align:middle line:84% That's because the expression for the differential 00:35:57.510 --> 00:36:01.740 align:middle line:90% geometrically is related to-- 00:36:01.740 --> 00:36:11.590 align:middle line:84% if I have a surface and I focus on one point. 00:36:11.590 --> 00:36:14.360 align:middle line:84% So suppose this is an equilibrium state 00:36:14.360 --> 00:36:16.610 align:middle line:90% and I'm talking about a surface. 00:36:16.610 --> 00:36:24.220 align:middle line:84% So if this is x, y, and z, and this surface has 00:36:24.220 --> 00:36:28.420 align:middle line:90% a function f of x, y equals z. 00:36:28.420 --> 00:36:32.260 align:middle line:84% So now I'm at a particular point, 00:36:32.260 --> 00:36:41.860 align:middle line:84% let's say, with x tilde for variable x, 00:36:41.860 --> 00:36:43.760 align:middle line:90% and y tilde for variable y. 00:36:43.760 --> 00:36:49.540 align:middle line:84% And therefore, that would be z tilde, so 00:36:49.540 --> 00:36:54.150 align:middle line:84% that particular stable equilibrium state. 00:36:54.150 --> 00:37:01.800 align:middle line:84% Now writing the differential means 00:37:01.800 --> 00:37:08.260 align:middle line:84% writing the equation of the plane which 00:37:08.260 --> 00:37:11.310 align:middle line:84% is tangent to the surface at that point. 00:37:11.310 --> 00:37:14.690 align:middle line:90% 00:37:14.690 --> 00:37:20.160 align:middle line:84% These partial derivatives are the slopes 00:37:20.160 --> 00:37:24.070 align:middle line:84% of that plane in the directions of x and y. 00:37:24.070 --> 00:37:27.595 align:middle line:90% 00:37:27.595 --> 00:37:36.130 align:middle line:84% Now if I rewrite this function as a function, so I rewrite it 00:37:36.130 --> 00:37:42.430 align:middle line:84% in terms of changing dependent variables, so I solve it, 00:37:42.430 --> 00:37:44.030 align:middle line:90% make x explicit. 00:37:44.030 --> 00:37:49.990 align:middle line:84% So it becomes x going to be a function, say, of g and z. 00:37:49.990 --> 00:37:53.990 align:middle line:84% I'm sorry, a function g of z and y. 00:37:53.990 --> 00:38:00.340 align:middle line:84% So y remains an independent property, and now z becomes. 00:38:00.340 --> 00:38:02.780 align:middle line:84% So it's a different function, of course. 00:38:02.780 --> 00:38:07.610 align:middle line:84% That's why I write f and g, but it's the same surface. 00:38:07.610 --> 00:38:10.930 align:middle line:84% So we can represent the same surface either in this way 00:38:10.930 --> 00:38:12.540 align:middle line:90% or in that way. 00:38:12.540 --> 00:38:15.210 align:middle line:84% And the tangent plane, therefore, 00:38:15.210 --> 00:38:17.610 align:middle line:84% hasn't changed, doesn't depend on how 00:38:17.610 --> 00:38:23.610 align:middle line:84% I decide to consider x, y, or z with a pair of them 00:38:23.610 --> 00:38:26.598 align:middle line:84% independent and then the other independent. 00:38:26.598 --> 00:38:35.530 align:middle line:90% 00:38:35.530 --> 00:38:38.050 align:middle line:84% So the expression for the differentials 00:38:38.050 --> 00:38:44.430 align:middle line:84% should be the same geometrically, 00:38:44.430 --> 00:38:48.415 align:middle line:84% in the sense that they should represent the same. 00:38:48.415 --> 00:38:52.720 align:middle line:90% 00:38:52.720 --> 00:39:01.370 align:middle line:84% Therefore, if I write the differential of this function 00:39:01.370 --> 00:39:05.700 align:middle line:84% here, dx, the partial of g with respect 00:39:05.700 --> 00:39:12.480 align:middle line:84% to z in dz, plus the partial of g with respect to y in dy. 00:39:12.480 --> 00:39:16.230 align:middle line:90% 00:39:16.230 --> 00:39:21.160 align:middle line:84% But I can solve this equation and write it 00:39:21.160 --> 00:39:28.180 align:middle line:84% as dz equals 1 divided by the partial of g 00:39:28.180 --> 00:39:34.660 align:middle line:84% with respect to z times dx plus-- 00:39:34.660 --> 00:39:42.170 align:middle line:84% actually, minus the partial of g with respect to y, 00:39:42.170 --> 00:39:49.310 align:middle line:84% divided by the partial of g with respect to z times dy. 00:39:49.310 --> 00:39:57.870 align:middle line:84% So I found either this or that represent this tangent plane. 00:39:57.870 --> 00:40:00.520 align:middle line:90% 00:40:00.520 --> 00:40:05.380 align:middle line:84% But also, this other form represents the same tangent 00:40:05.380 --> 00:40:07.880 align:middle line:90% plane. 00:40:07.880 --> 00:40:13.490 align:middle line:84% Therefore, I can conclude that this coefficient 00:40:13.490 --> 00:40:16.730 align:middle line:84% of dx in the expression of these-- 00:40:16.730 --> 00:40:21.840 align:middle line:84% if you compare this and that, these coefficients are equal. 00:40:21.840 --> 00:40:25.370 align:middle line:84% So this is equal to the partial of f-- 00:40:25.370 --> 00:40:27.140 align:middle line:90% oops. 00:40:27.140 --> 00:40:32.990 align:middle line:84% The partial of f with respect to x. 00:40:32.990 --> 00:40:40.690 align:middle line:90% And this is equal to this. 00:40:40.690 --> 00:40:44.520 align:middle line:84% Same plane means same slopes, same coefficients. 00:40:44.520 --> 00:40:49.720 align:middle line:84% Therefore, this is the partial of f with respect to y. 00:40:49.720 --> 00:40:55.290 align:middle line:84% Of course, this is true only when 00:40:55.290 --> 00:40:58.620 align:middle line:84% f and g are related by the fact that one 00:40:58.620 --> 00:41:00.260 align:middle line:90% is the inverse of the other. 00:41:00.260 --> 00:41:05.880 align:middle line:90% 00:41:05.880 --> 00:41:12.420 align:middle line:84% And that's why-- that's what I've applied here 00:41:12.420 --> 00:41:27.830 align:middle line:84% to show that if that relation is 1 over T, 00:41:27.830 --> 00:41:31.896 align:middle line:90% its inverse is the derivative. 00:41:31.896 --> 00:41:36.190 align:middle line:90% 00:41:36.190 --> 00:41:40.140 align:middle line:84% It is equal to the inverse of the derivative 00:41:40.140 --> 00:41:46.790 align:middle line:84% of g, which is the energy form, with respect to entropy. 00:41:46.790 --> 00:41:52.280 align:middle line:84% So you go from here to there using that form. 00:41:52.280 --> 00:41:58.480 align:middle line:84% Now it immediately stands out, the fact 00:41:58.480 --> 00:42:05.350 align:middle line:84% that if you deal with equilibrium 00:42:05.350 --> 00:42:07.250 align:middle line:84% states, stable equilibrium states, 00:42:07.250 --> 00:42:10.990 align:middle line:84% and there is an entire long chapter about relations 00:42:10.990 --> 00:42:14.140 align:middle line:84% that you can prove between properties of stable equilibrium 00:42:14.140 --> 00:42:20.270 align:middle line:84% states, we just touch on this and it's not our main concern. 00:42:20.270 --> 00:42:26.000 align:middle line:84% But you may find lots of papers, lots of formulas. 00:42:26.000 --> 00:42:29.470 align:middle line:84% And we will use them only when we need them. 00:42:29.470 --> 00:42:32.950 align:middle line:84% But one thing I need to use because that would introduce 00:42:32.950 --> 00:42:41.720 align:middle line:84% and to, therefore, define, it's the symbol that we use. 00:42:41.720 --> 00:42:44.010 align:middle line:90% We use here a symbol. 00:42:44.010 --> 00:42:47.620 align:middle line:84% We just don't write only the partial derivative, 00:42:47.620 --> 00:42:51.700 align:middle line:84% but we write also parentheses and some subscripts. 00:42:51.700 --> 00:42:52.970 align:middle line:90% Why do we do that? 00:42:52.970 --> 00:42:59.970 align:middle line:90% 00:42:59.970 --> 00:43:10.900 align:middle line:84% We do that because it could happen that, for example, 00:43:10.900 --> 00:43:20.020 align:middle line:84% all right, we have the fundamental relation in energy 00:43:20.020 --> 00:43:21.810 align:middle line:90% form written this way. 00:43:21.810 --> 00:43:25.330 align:middle line:90% 00:43:25.330 --> 00:43:36.410 align:middle line:84% But, for example, later we will give a brief-- 00:43:36.410 --> 00:43:40.230 align:middle line:84% we'll touch on the subject of changing variables. 00:43:40.230 --> 00:43:44.240 align:middle line:90% 00:43:44.240 --> 00:43:47.060 align:middle line:84% For example, because maybe entropy 00:43:47.060 --> 00:43:53.090 align:middle line:84% is not that easily measurable in the lab, in the sense 00:43:53.090 --> 00:43:58.580 align:middle line:84% that we don't have an entropy meter, 00:43:58.580 --> 00:44:01.990 align:middle line:90% but we have a thermometer. 00:44:01.990 --> 00:44:11.470 align:middle line:84% So we measure entropy indirectly using the instruments that are 00:44:11.470 --> 00:44:13.270 align:middle line:90% simpler, like the thermometer. 00:44:13.270 --> 00:44:22.780 align:middle line:84% And the thermometer we will see measures the property 00:44:22.780 --> 00:44:27.300 align:middle line:84% temperature, which is the partial 00:44:27.300 --> 00:44:33.210 align:middle line:84% of this function with respect to the first variable, as written 00:44:33.210 --> 00:44:34.470 align:middle line:90% here. 00:44:34.470 --> 00:44:37.200 align:middle line:84% So why do I put the n and V there? 00:44:37.200 --> 00:44:42.426 align:middle line:84% OK, because, you see, the next thing I do, I can do 00:44:42.426 --> 00:44:47.100 align:middle line:84% is, for example, to write an expression 00:44:47.100 --> 00:44:51.330 align:middle line:90% of how the energy changes. 00:44:51.330 --> 00:44:58.450 align:middle line:84% Let me call it E tilde, as a function of T, V and n. 00:44:58.450 --> 00:45:04.990 align:middle line:90% 00:45:04.990 --> 00:45:05.490 align:middle line:90% See? 00:45:05.490 --> 00:45:11.110 align:middle line:90% 00:45:11.110 --> 00:45:14.350 align:middle line:84% So now I look at these two expressions. 00:45:14.350 --> 00:45:16.860 align:middle line:90% They're different functions. 00:45:16.860 --> 00:45:20.520 align:middle line:84% But, for example, suppose I'm interested in the pressure. 00:45:20.520 --> 00:45:23.250 align:middle line:90% I take the pressure. 00:45:23.250 --> 00:45:27.270 align:middle line:84% It's minus the E, the partial of E with respect 00:45:27.270 --> 00:45:31.500 align:middle line:84% to V of the fundamental relation of this formula. 00:45:31.500 --> 00:45:33.480 align:middle line:84% But now that I have another function that 00:45:33.480 --> 00:45:35.640 align:middle line:84% expresses the energy in terms of other variables 00:45:35.640 --> 00:45:38.250 align:middle line:84% in which I also have volume, I may 00:45:38.250 --> 00:45:42.720 align:middle line:84% be curious to see, well, how about doing 00:45:42.720 --> 00:45:49.370 align:middle line:84% minus the partial of E tilde with respect to V? 00:45:49.370 --> 00:45:52.530 align:middle line:90% Is that the pressure? 00:45:52.530 --> 00:45:54.030 align:middle line:90% Well, not necessarily. 00:45:54.030 --> 00:45:55.270 align:middle line:90% And actually, no. 00:45:55.270 --> 00:46:00.770 align:middle line:90% 00:46:00.770 --> 00:46:06.060 align:middle line:84% And sometimes if I'm sloppy and maybe I say, 00:46:06.060 --> 00:46:09.140 align:middle line:84% all right, this is the function I'm interested in. 00:46:09.140 --> 00:46:11.340 align:middle line:84% I don't want to carry around the tilde, 00:46:11.340 --> 00:46:19.020 align:middle line:84% so I just write it as E. Well, then I may get confused. 00:46:19.020 --> 00:46:23.080 align:middle line:84% Because when I look at this one or look at that one, 00:46:23.080 --> 00:46:24.960 align:middle line:90% which is which? 00:46:24.960 --> 00:46:30.810 align:middle line:84% So the idea is, if you want to be careful, 00:46:30.810 --> 00:46:35.400 align:middle line:84% is to carry with your partial derivatives also 00:46:35.400 --> 00:46:38.970 align:middle line:84% a precise memory or indication of what 00:46:38.970 --> 00:46:43.080 align:middle line:84% the other independent variables of your function are. 00:46:43.080 --> 00:46:49.830 align:middle line:84% So if I write here the other variables, so I write S and n, 00:46:49.830 --> 00:46:58.850 align:middle line:84% and here I write the other variables, which are T and n, 00:46:58.850 --> 00:47:01.070 align:middle line:84% there is no way I can get confused. 00:47:01.070 --> 00:47:03.560 align:middle line:84% Because by just looking at the formula 00:47:03.560 --> 00:47:06.600 align:middle line:84% for the partial derivative, I know also 00:47:06.600 --> 00:47:12.230 align:middle line:84% what were the original variables of the function of which I'm 00:47:12.230 --> 00:47:13.810 align:middle line:90% taking the derivative. 00:47:13.810 --> 00:47:23.520 align:middle line:84% So this is a notation to avoid to get lost and confused. 00:47:23.520 --> 00:47:26.770 align:middle line:90% 00:47:26.770 --> 00:47:36.610 align:middle line:84% And this kind of confusion is very common if you don't-- 00:47:36.610 --> 00:47:39.370 align:middle line:84% because many times, maybe you don't 00:47:39.370 --> 00:47:43.540 align:middle line:84% need this kind of pedantic way of proceeding 00:47:43.540 --> 00:47:48.910 align:middle line:84% because the context may be enough. 00:47:48.910 --> 00:47:54.180 align:middle line:84% If below is just below that, writing that partial derivative, 00:47:54.180 --> 00:47:55.930 align:middle line:84% you have a paragraph that says, well, this 00:47:55.930 --> 00:47:58.730 align:middle line:84% is the equation of which I'm taking the derivative, 00:47:58.730 --> 00:47:59.650 align:middle line:90% then you're OK. 00:47:59.650 --> 00:48:02.965 align:middle line:84% The context tells you what is the derivative of what. 00:48:02.965 --> 00:48:07.180 align:middle line:90% 00:48:07.180 --> 00:48:09.790 align:middle line:84% But if, in more complex situations, 00:48:09.790 --> 00:48:12.730 align:middle line:84% where you keep changing variables, then 00:48:12.730 --> 00:48:15.020 align:middle line:90% this is really essential. 00:48:15.020 --> 00:48:17.780 align:middle line:84% And that's why in the business of thermodynamics 00:48:17.780 --> 00:48:20.580 align:middle line:84% it has been introduced and it is standard. 00:48:20.580 --> 00:48:25.710 align:middle line:90% 00:48:25.710 --> 00:48:38.060 align:middle line:84% So also, to be sympathetic with mathematicians, this, 00:48:38.060 --> 00:48:41.970 align:middle line:84% you should not read this symbol as saying, 00:48:41.970 --> 00:48:44.610 align:middle line:84% the derivative of energy with respect 00:48:44.610 --> 00:48:49.090 align:middle line:84% to volume at constant entropy and amounts of constituents. 00:48:49.090 --> 00:48:53.150 align:middle line:90% 00:48:53.150 --> 00:48:55.100 align:middle line:90% Yeah, you can say it. 00:48:55.100 --> 00:48:56.410 align:middle line:90% Many people say it. 00:48:56.410 --> 00:49:03.390 align:middle line:84% Sometimes I even say it, but I don't mean it. 00:49:03.390 --> 00:49:05.340 align:middle line:90% Yeah. 00:49:05.340 --> 00:49:08.760 align:middle line:84% The real meaning of this is that these 00:49:08.760 --> 00:49:10.830 align:middle line:90% are the independent variables. 00:49:10.830 --> 00:49:14.490 align:middle line:84% So I'm taking the partial derivative of a function, 00:49:14.490 --> 00:49:16.470 align:middle line:84% the partial derivative with respect 00:49:16.470 --> 00:49:19.780 align:middle line:84% to volume of a function of energy, 00:49:19.780 --> 00:49:24.420 align:middle line:84% volume, entropy, and amounts of constituents. 00:49:24.420 --> 00:49:27.560 align:middle line:90% 00:49:27.560 --> 00:49:28.765 align:middle line:90% That's the proper language. 00:49:28.765 --> 00:49:42.460 align:middle line:90% 00:49:42.460 --> 00:49:46.690 align:middle line:84% Why do we give names to temperature, pressure, and then 00:49:46.690 --> 00:49:48.260 align:middle line:90% chemical potentials? 00:49:48.260 --> 00:49:55.780 align:middle line:84% Because they enter, and we will see it hopefully later today, 00:49:55.780 --> 00:49:57.910 align:middle line:90% not sure. 00:49:57.910 --> 00:50:00.550 align:middle line:84% Any way, because they enter in the conditions 00:50:00.550 --> 00:50:02.445 align:middle line:84% for mutual equilibrium between systems. 00:50:02.445 --> 00:50:05.380 align:middle line:90% 00:50:05.380 --> 00:50:08.810 align:middle line:84% So we'll return to these to make some proofs, 00:50:08.810 --> 00:50:18.020 align:middle line:84% but just to anticipate, when two systems, A and B, 00:50:18.020 --> 00:50:21.980 align:middle line:84% are in mutual equilibrium, it means that the overall system is 00:50:21.980 --> 00:50:23.260 align:middle line:90% in a stable equilibrium state. 00:50:23.260 --> 00:50:27.690 align:middle line:84% That's the definition of mutual equilibrium, remember. 00:50:27.690 --> 00:50:33.190 align:middle line:84% So we will prove and we will actually-- 00:50:33.190 --> 00:50:36.390 align:middle line:84% I want to go over these proofs because they're 00:50:36.390 --> 00:50:41.930 align:middle line:90% useful also for other reasons. 00:50:41.930 --> 00:50:44.910 align:middle line:90% 00:50:44.910 --> 00:50:49.630 align:middle line:84% It turns out that if the two systems, A and B, 00:50:49.630 --> 00:51:00.400 align:middle line:84% can exchange energy, then they must, 00:51:00.400 --> 00:51:02.020 align:middle line:84% if they are in mutual equilibrium, 00:51:02.020 --> 00:51:04.710 align:middle line:84% they must have the same temperature. 00:51:04.710 --> 00:51:07.500 align:middle line:84% So the temperature of system A of the stable equilibrium 00:51:07.500 --> 00:51:11.820 align:middle line:84% state in which system A is must be 00:51:11.820 --> 00:51:15.440 align:middle line:84% equal to the temperature of the stable equilibrium 00:51:15.440 --> 00:51:21.890 align:middle line:84% state in which system B is when they are in mutual equilibrium. 00:51:21.890 --> 00:51:26.480 align:middle line:84% They can also exchange volume, for example, 00:51:26.480 --> 00:51:33.380 align:middle line:84% by means of a movable piston or a partition. 00:51:33.380 --> 00:51:36.590 align:middle line:84% There is a system, as the piston moves, 00:51:36.590 --> 00:51:40.070 align:middle line:84% you see that, for example, it moves to the right. 00:51:40.070 --> 00:51:42.620 align:middle line:84% The volume of A increases, that of B 00:51:42.620 --> 00:51:44.310 align:middle line:90% decreases by the same amount. 00:51:44.310 --> 00:51:48.200 align:middle line:84% So you can say that the piston-- you can interpret it 00:51:48.200 --> 00:51:51.890 align:middle line:84% as the piston has allowed the systems to exchange volume. 00:51:51.890 --> 00:51:58.290 align:middle line:84% One gains the volume by subtracting it from the other. 00:51:58.290 --> 00:52:01.150 align:middle line:84% So it's like a transfer of volume. 00:52:01.150 --> 00:52:01.650 align:middle line:90% All right. 00:52:01.650 --> 00:52:05.930 align:middle line:84% In that situation, in order for the two systems 00:52:05.930 --> 00:52:11.720 align:middle line:84% to be in mutual equilibrium, you need also pressure equality. 00:52:11.720 --> 00:52:17.210 align:middle line:84% If, instead you have a porous wall, 00:52:17.210 --> 00:52:24.440 align:middle line:84% or a semipermeable membrane that allows the exchange of one 00:52:24.440 --> 00:52:31.480 align:middle line:84% of the constituents between system A and system B, 00:52:31.480 --> 00:52:36.460 align:middle line:84% well, the necessary condition for mutual equilibrium 00:52:36.460 --> 00:52:42.380 align:middle line:84% is equality of the chemical potentials of that constituent. 00:52:42.380 --> 00:52:46.870 align:middle line:84% So the chemical potential of constituent i 00:52:46.870 --> 00:52:50.140 align:middle line:84% measured in system A must be equal to the chemical potential 00:52:50.140 --> 00:52:54.250 align:middle line:84% of system B, measured in system B. 00:52:54.250 --> 00:52:57.470 align:middle line:84% So we will prove this because these are very important. 00:52:57.470 --> 00:53:01.100 align:middle line:84% And actually, regarding equality of chemical potentials, 00:53:01.100 --> 00:53:07.090 align:middle line:84% we will go a long way in finding very interesting results 00:53:07.090 --> 00:53:10.780 align:middle line:84% and important results for applications. 00:53:10.780 --> 00:53:16.000 align:middle line:84% But before we go and prove these conditions 00:53:16.000 --> 00:53:18.700 align:middle line:84% for mutual equilibrium, I promised 00:53:18.700 --> 00:53:25.210 align:middle line:84% that I would review what we have done so far. 00:53:25.210 --> 00:53:28.610 align:middle line:84% You may call it thermodynamics without heat 00:53:28.610 --> 00:53:40.640 align:middle line:84% so far by introducing this energy versus entropy diagram 00:53:40.640 --> 00:53:47.370 align:middle line:84% representation of states, which is so important that we put it 00:53:47.370 --> 00:53:50.585 align:middle line:90% in the cover of our book. 00:53:50.585 --> 00:53:55.320 align:middle line:90% 00:53:55.320 --> 00:53:58.420 align:middle line:90% OK, this is the surface. 00:53:58.420 --> 00:54:02.525 align:middle line:84% This is a fundamental relation for water. 00:54:02.525 --> 00:54:10.160 align:middle line:90% 00:54:10.160 --> 00:54:10.660 align:middle line:90% All right. 00:54:10.660 --> 00:54:15.090 align:middle line:84% So how do we get to construct these energy 00:54:15.090 --> 00:54:16.283 align:middle line:90% versus entropy diagrams? 00:54:16.283 --> 00:54:20.020 align:middle line:90% 00:54:20.020 --> 00:54:22.190 align:middle line:84% Again, we start from the idea of state. 00:54:22.190 --> 00:54:26.770 align:middle line:84% A state is a long list of numbers, usually very long. 00:54:26.770 --> 00:54:29.110 align:middle line:90% We define a few of-- 00:54:29.110 --> 00:54:31.307 align:middle line:84% now these numbers are the properties, 00:54:31.307 --> 00:54:33.890 align:middle line:84% are the amounts of constituents, the values of the parameters, 00:54:33.890 --> 00:54:42.690 align:middle line:84% but also, all the properties of our system 00:54:42.690 --> 00:54:44.280 align:middle line:90% at the given instant of time. 00:54:44.280 --> 00:54:48.370 align:middle line:84% Among these properties, we have energy and entropy, 00:54:48.370 --> 00:54:50.860 align:middle line:90% but we have also many other. 00:54:50.860 --> 00:54:55.100 align:middle line:84% So in order to have a complete representation, 00:54:55.100 --> 00:55:00.230 align:middle line:84% geometrical representation for a state of a system, 00:55:00.230 --> 00:55:07.990 align:middle line:84% you would need a multi-dimensional space 00:55:07.990 --> 00:55:12.740 align:middle line:84% in which we have one axis for each 00:55:12.740 --> 00:55:21.530 align:middle line:84% of the independent properties of your system. 00:55:21.530 --> 00:55:23.870 align:middle line:84% And since the independent properties of a system 00:55:23.870 --> 00:55:29.860 align:middle line:84% are generally very large, or even most of the times infinite, 00:55:29.860 --> 00:55:34.220 align:middle line:90% then you need to simplify. 00:55:34.220 --> 00:55:46.490 align:middle line:84% So what we do is to project onto a plane. 00:55:46.490 --> 00:55:51.770 align:middle line:84% But first, what we do is to make a cut 00:55:51.770 --> 00:55:54.770 align:middle line:84% of this multi-dimensional space, a cut 00:55:54.770 --> 00:55:58.940 align:middle line:84% at fixed values of the amounts of constituents 00:55:58.940 --> 00:56:01.560 align:middle line:90% and the parameters. 00:56:01.560 --> 00:56:03.820 align:middle line:84% When I say parameters, think of the volume. 00:56:03.820 --> 00:56:09.210 align:middle line:84% And that's the most common, at least when 00:56:09.210 --> 00:56:11.170 align:middle line:84% you study properties of substances. 00:56:11.170 --> 00:56:14.970 align:middle line:90% 00:56:14.970 --> 00:56:20.240 align:middle line:84% So if I fix the values of the amounts and parameters, 00:56:20.240 --> 00:56:23.690 align:middle line:90% I still have a large subset. 00:56:23.690 --> 00:56:29.210 align:middle line:84% I still need a multi-dimensional space, 00:56:29.210 --> 00:56:34.070 align:middle line:84% only its dimensions is the full thing 00:56:34.070 --> 00:56:39.770 align:middle line:84% minus the number of constituents and the number of parameters. 00:56:39.770 --> 00:56:41.050 align:middle line:90% Still very large. 00:56:41.050 --> 00:56:47.300 align:middle line:84% Then, you project all the states onto the plane 00:56:47.300 --> 00:56:51.530 align:middle line:84% made of the two variables that we most like, 00:56:51.530 --> 00:56:53.420 align:middle line:90% energy and entropy. 00:56:53.420 --> 00:56:58.330 align:middle line:84% And then you get a 2D representation, 00:56:58.330 --> 00:57:00.380 align:middle line:84% and it is going to look like this. 00:57:00.380 --> 00:57:03.160 align:middle line:90% 00:57:03.160 --> 00:57:05.620 align:middle line:90% Energy and entropy. 00:57:05.620 --> 00:57:12.310 align:middle line:84% You project-- So let's look at the generic point here. 00:57:12.310 --> 00:57:17.440 align:middle line:84% The generic point here, remember, is the projection 00:57:17.440 --> 00:57:20.260 align:middle line:90% onto that plane. 00:57:20.260 --> 00:57:27.300 align:middle line:84% Many states will project onto that plane 00:57:27.300 --> 00:57:31.560 align:middle line:84% because many states will share those same values of energy 00:57:31.560 --> 00:57:36.600 align:middle line:84% and entropy but differ in the values of other properties. 00:57:36.600 --> 00:57:40.140 align:middle line:84% So they are different states which, nevertheless, 00:57:40.140 --> 00:57:43.830 align:middle line:84% are represented by the same point 00:57:43.830 --> 00:57:45.820 align:middle line:90% as a result of the projection. 00:57:45.820 --> 00:57:53.210 align:middle line:90% 00:57:53.210 --> 00:57:59.370 align:middle line:84% Now that we have understood the geometry, 00:57:59.370 --> 00:58:05.130 align:middle line:84% let's see what the theorems that we have seen so far 00:58:05.130 --> 00:58:12.130 align:middle line:84% tell us about where these states can be in this diagram. 00:58:12.130 --> 00:58:19.510 align:middle line:84% So for example, let's start with a statement 00:58:19.510 --> 00:58:24.370 align:middle line:84% of the first assertion of the statement of the second law. 00:58:24.370 --> 00:58:28.600 align:middle line:84% It said, we'll consider all the states of the system 00:58:28.600 --> 00:58:30.650 align:middle line:84% with a given value of the energy. 00:58:30.650 --> 00:58:33.990 align:middle line:90% 00:58:33.990 --> 00:58:35.660 align:middle line:90% So fix the value of the energy. 00:58:35.660 --> 00:58:39.530 align:middle line:84% All the states will collapse or will 00:58:39.530 --> 00:58:45.600 align:middle line:84% project onto this dashed line, horizontal. 00:58:45.600 --> 00:58:49.950 align:middle line:90% 00:58:49.950 --> 00:58:58.160 align:middle line:84% And you see we have the maximum entropy principle that we just 00:58:58.160 --> 00:58:59.240 align:middle line:90% mentioned. 00:58:59.240 --> 00:59:02.840 align:middle line:84% It says that among all these states, 00:59:02.840 --> 00:59:06.530 align:middle line:84% there is a stable equilibrium state, which 00:59:06.530 --> 00:59:11.780 align:middle line:84% is the one that has the highest value of entropy. 00:59:11.780 --> 00:59:18.440 align:middle line:84% There are no other states with that value 00:59:18.440 --> 00:59:21.860 align:middle line:84% or even higher values of entropy. 00:59:21.860 --> 00:59:24.440 align:middle line:84% So that means that there is a maximum 00:59:24.440 --> 00:59:27.170 align:middle line:84% for the entropy for that given value for the energy. 00:59:27.170 --> 00:59:32.880 align:middle line:84% And corresponding to that maximum, there is one state. 00:59:32.880 --> 00:59:35.520 align:middle line:90% Notice it's only one state. 00:59:35.520 --> 00:59:42.460 align:middle line:84% So this particular point here is the projection 00:59:42.460 --> 00:59:45.610 align:middle line:84% of only one state, whereas here it's 00:59:45.610 --> 00:59:48.370 align:middle line:90% a projection of many states. 00:59:48.370 --> 00:59:53.710 align:middle line:84% When you go to equilibrium, there is only one state. 00:59:53.710 --> 01:00:00.760 align:middle line:84% Well, that's to say that geometrically, this curve, which 01:00:00.760 --> 01:00:04.360 align:middle line:84% is the locus of all the stable equilibrium states, 01:00:04.360 --> 01:00:10.110 align:middle line:84% is a sort of a wire curve in that multi-dimensional space. 01:00:10.110 --> 01:00:14.250 align:middle line:90% 01:00:14.250 --> 01:00:15.990 align:middle line:90% All right. 01:00:15.990 --> 01:00:19.050 align:middle line:84% So there are no states with higher entropy. 01:00:19.050 --> 01:00:23.580 align:middle line:84% So all the possible states are to the left 01:00:23.580 --> 01:00:30.780 align:middle line:84% with a lesser entropy than the stable equilibrium state. 01:00:30.780 --> 01:00:35.460 align:middle line:84% I do the same for another value of energy 01:00:35.460 --> 01:00:37.900 align:middle line:84% and I get another stable equilibrium state. 01:00:37.900 --> 01:00:45.110 align:middle line:90% 01:00:45.110 --> 01:00:51.810 align:middle line:84% I do it for the minimum value of energy 01:00:51.810 --> 01:00:59.280 align:middle line:84% compatible with the fixed values of amounts of constituents 01:00:59.280 --> 01:01:02.250 align:middle line:84% and parameters that we have chosen 01:01:02.250 --> 01:01:05.855 align:middle line:84% in order to make our cut of the multi-dimensional space. 01:01:05.855 --> 01:01:11.400 align:middle line:90% 01:01:11.400 --> 01:01:16.320 align:middle line:84% And we still have a stable equilibrium state. 01:01:16.320 --> 01:01:19.360 align:middle line:84% And that's the minimum energy stable equilibrium state. 01:01:19.360 --> 01:01:23.080 align:middle line:90% 01:01:23.080 --> 01:01:25.395 align:middle line:84% We'll come back to this in a moment. 01:01:25.395 --> 01:01:35.850 align:middle line:90% 01:01:35.850 --> 01:01:44.000 align:middle line:84% The slope of this curve, you see it? 01:01:44.000 --> 01:01:50.210 align:middle line:84% This curve represents the stable equilibrium state 01:01:50.210 --> 01:01:53.970 align:middle line:84% fundamental relation in energy form, 01:01:53.970 --> 01:02:02.380 align:middle line:84% because it's E is in the ordinates and S in the abscissa. 01:02:02.380 --> 01:02:07.270 align:middle line:84% And so this curve is the representation 01:02:07.270 --> 01:02:15.830 align:middle line:84% of E versus S at fixed the values of n and V. 01:02:15.830 --> 01:02:23.470 align:middle line:84% So the partial derivative of this function corresponds to-- 01:02:23.470 --> 01:02:24.370 align:middle line:90% it's this thing. 01:02:24.370 --> 01:02:31.470 align:middle line:84% It's the temperature, or actually, partial derivative 01:02:31.470 --> 01:02:39.210 align:middle line:84% is the slope of the tangent to the curve at a given stable 01:02:39.210 --> 01:02:41.400 align:middle line:90% equilibrium state. 01:02:41.400 --> 01:02:45.690 align:middle line:84% So the temperature is the angular coefficient 01:02:45.690 --> 01:02:52.171 align:middle line:84% of the tangent line, the slope, the temperature. 01:02:52.171 --> 01:02:56.890 align:middle line:90% 01:02:56.890 --> 01:03:02.410 align:middle line:84% So I can read the temperature, and I 01:03:02.410 --> 01:03:08.940 align:middle line:84% have built in this diagram also a result that we haven't proved 01:03:08.940 --> 01:03:19.170 align:middle line:84% yet, namely the concavity of this fundamental relation, 01:03:19.170 --> 01:03:25.270 align:middle line:84% implying that as you go high in energy, 01:03:25.270 --> 01:03:29.640 align:middle line:84% so you go stable equilibrium states with higher energy, 01:03:29.640 --> 01:03:31.560 align:middle line:90% the slope goes up. 01:03:31.560 --> 01:03:34.600 align:middle line:84% They also have higher temperature. 01:03:34.600 --> 01:03:40.470 align:middle line:84% So temperature is an increasing function of energy. 01:03:40.470 --> 01:03:45.360 align:middle line:84% But remember, temperature is defined 01:03:45.360 --> 01:03:50.460 align:middle line:84% only as being the slope of this curve, 01:03:50.460 --> 01:03:52.290 align:middle line:84% only for the stable equilibrium state 01:03:52.290 --> 01:03:56.570 align:middle line:90% for which there is a curve. 01:03:56.570 --> 01:03:58.730 align:middle line:90% You are inside here. 01:03:58.730 --> 01:04:00.540 align:middle line:84% There is no fundamental relation. 01:04:00.540 --> 01:04:03.570 align:middle line:84% There is no function to take derivatives. 01:04:03.570 --> 01:04:04.800 align:middle line:90% There is no temperature. 01:04:04.800 --> 01:04:11.060 align:middle line:90% 01:04:11.060 --> 01:04:13.970 align:middle line:84% Yeah, maybe you can compute things 01:04:13.970 --> 01:04:16.996 align:middle line:84% like the average kinetic energy of the particles. 01:04:16.996 --> 01:04:20.226 align:middle line:90% 01:04:20.226 --> 01:04:27.240 align:middle line:84% But whereas this is related to temperature 01:04:27.240 --> 01:04:34.470 align:middle line:84% at stable equilibrium, it is not temperature 01:04:34.470 --> 01:04:36.290 align:middle line:90% if you are off equilibrium. 01:04:36.290 --> 01:04:42.790 align:middle line:90% 01:04:42.790 --> 01:04:44.715 align:middle line:84% Of course, for stable equilibrium states-- 01:04:44.715 --> 01:04:48.720 align:middle line:90% 01:04:48.720 --> 01:04:49.740 align:middle line:90% all right. 01:04:49.740 --> 01:04:57.360 align:middle line:84% Yes, as I go down in energy, I reach the minimum energy 01:04:57.360 --> 01:04:58.870 align:middle line:90% stable equilibrium state. 01:04:58.870 --> 01:05:04.710 align:middle line:84% And these here, I have built in another feature that 01:05:04.710 --> 01:05:08.730 align:middle line:84% goes by the name of the third law of thermodynamics that 01:05:08.730 --> 01:05:13.500 align:middle line:84% says that the lowest energy stable equilibrium state 01:05:13.500 --> 01:05:18.460 align:middle line:90% has zero temperature. 01:05:18.460 --> 01:05:23.450 align:middle line:84% We will not prove that as this comes more 01:05:23.450 --> 01:05:31.070 align:middle line:84% from the models of statistical mechanics or quantum mechanics. 01:05:31.070 --> 01:05:41.700 align:middle line:84% So we have nothing to add to that important but very 01:05:41.700 --> 01:05:45.475 align:middle line:84% well known result. It comes from physics. 01:05:45.475 --> 01:05:49.390 align:middle line:90% 01:05:49.390 --> 01:05:52.390 align:middle line:84% In a moment, we will also see, but I 01:05:52.390 --> 01:06:02.770 align:middle line:84% can anticipate that the entropy takes a minimum value. 01:06:02.770 --> 01:06:13.530 align:middle line:84% And the minimum value is for those particular non-equilibrium 01:06:13.530 --> 01:06:22.680 align:middle line:84% states for which the adiabatic availability is 01:06:22.680 --> 01:06:29.370 align:middle line:84% equal to the energy, or better to the difference between energy 01:06:29.370 --> 01:06:34.710 align:middle line:84% and the minimum energy allowed for that given 01:06:34.710 --> 01:06:38.850 align:middle line:84% combination of amounts of constituents and volume. 01:06:38.850 --> 01:06:47.400 align:middle line:84% So if I have a state here, this state-- from this state, 01:06:47.400 --> 01:06:52.330 align:middle line:84% you can transfer in a weight process to the weight, 01:06:52.330 --> 01:06:55.590 align:middle line:84% you can extract out of your system in a weight process 01:06:55.590 --> 01:06:59.550 align:middle line:84% all the energy of the system, meaning 01:06:59.550 --> 01:07:05.610 align:middle line:84% that you can take that state and bring the system down 01:07:05.610 --> 01:07:07.720 align:middle line:84% to this state, the minimum energy. 01:07:07.720 --> 01:07:11.080 align:middle line:90% 01:07:11.080 --> 01:07:14.750 align:middle line:84% You can do this here and you extract this much energy. 01:07:14.750 --> 01:07:19.650 align:middle line:90% 01:07:19.650 --> 01:07:21.840 align:middle line:84% Whereas in a moment, you'll see that if you 01:07:21.840 --> 01:07:26.860 align:middle line:84% do that in a non-equilibrium state, 01:07:26.860 --> 01:07:30.690 align:middle line:84% the adiabatic availability corresponds to this much. 01:07:30.690 --> 01:07:36.000 align:middle line:84% You can go at most down to this stable equilibrium state. 01:07:36.000 --> 01:07:38.860 align:middle line:84% And if you start from a stable equilibrium state, 01:07:38.860 --> 01:07:43.620 align:middle line:84% remember what the theorem about impossibility 01:07:43.620 --> 01:07:49.380 align:middle line:84% of perpetual motion machine of the second kind says. 01:07:49.380 --> 01:07:52.050 align:middle line:84% There's no way you can start from an equilibrium-- 01:07:52.050 --> 01:07:56.100 align:middle line:84% from a stable equilibrium state and extract energy 01:07:56.100 --> 01:07:58.980 align:middle line:90% in a weight process. 01:07:58.980 --> 01:08:00.250 align:middle line:90% And here it is. 01:08:00.250 --> 01:08:02.880 align:middle line:90% 01:08:02.880 --> 01:08:09.290 align:middle line:84% You cannot go down in a weight process. 01:08:09.290 --> 01:08:11.960 align:middle line:84% You see in a weight process the feature 01:08:11.960 --> 01:08:16.290 align:middle line:84% is that if you want to go down in energy here, 01:08:16.290 --> 01:08:19.319 align:middle line:84% you need to also reduce the entropy. 01:08:19.319 --> 01:08:21.805 align:middle line:84% But you cannot reduce the entropy in a weight process. 01:08:21.805 --> 01:08:25.284 align:middle line:90% 01:08:25.284 --> 01:08:29.190 align:middle line:84% We'll return, of course, into these ideas. 01:08:29.190 --> 01:08:34.130 align:middle line:90% 01:08:34.130 --> 01:08:35.960 align:middle line:84% Here I just wanted to say that there 01:08:35.960 --> 01:08:40.930 align:middle line:84% are some special systems for which there is 01:08:40.930 --> 01:08:42.599 align:middle line:90% an upper bound to the energy. 01:08:42.599 --> 01:08:45.960 align:middle line:90% 01:08:45.960 --> 01:08:47.880 align:middle line:84% These systems are those that do not 01:08:47.880 --> 01:08:55.250 align:middle line:84% have translational, or rotational, or vibrational 01:08:55.250 --> 01:08:56.335 align:middle line:90% degrees of freedom. 01:08:56.335 --> 01:09:02.359 align:middle line:90% 01:09:02.359 --> 01:09:04.920 align:middle line:84% Because if you have a translational degree of freedom, 01:09:04.920 --> 01:09:07.680 align:middle line:90% a particle can move. 01:09:07.680 --> 01:09:11.550 align:middle line:84% If it can move, you can accelerate it 01:09:11.550 --> 01:09:15.420 align:middle line:90% to as much velocity as you wish. 01:09:15.420 --> 01:09:21.000 align:middle line:84% Yeah, there is the limit of the speed of light, and so on. 01:09:21.000 --> 01:09:23.359 align:middle line:90% Maybe that is an upper bound. 01:09:23.359 --> 01:09:26.460 align:middle line:84% I've never really thought about that. 01:09:26.460 --> 01:09:28.710 align:middle line:84% But here we are talking about systems 01:09:28.710 --> 01:09:34.340 align:middle line:90% like the spin of an electron. 01:09:34.340 --> 01:09:39.260 align:middle line:84% So if you want to concentrate on just the spin 01:09:39.260 --> 01:09:45.260 align:middle line:84% feature of an electron without all the other details 01:09:45.260 --> 01:09:49.890 align:middle line:84% so that the electron must not be bound in an atom and so on, 01:09:49.890 --> 01:09:54.800 align:middle line:84% but if you can define a system in which the only aspect 01:09:54.800 --> 01:09:57.800 align:middle line:84% that you are interested in is the spin, 01:09:57.800 --> 01:09:59.360 align:middle line:84% or if you don't like the spin, it 01:09:59.360 --> 01:10:05.000 align:middle line:84% could be the polarization of photons. 01:10:05.000 --> 01:10:11.450 align:middle line:84% It's very typical of quantum optics 01:10:11.450 --> 01:10:17.810 align:middle line:84% because with lasers and so on you can produce very diluted 01:10:17.810 --> 01:10:22.770 align:middle line:84% beams with photons, like you can treat one photon one at a time. 01:10:22.770 --> 01:10:27.620 align:middle line:84% And you can, using particular filters, 01:10:27.620 --> 01:10:29.930 align:middle line:84% when you produce these photons, you 01:10:29.930 --> 01:10:33.530 align:middle line:84% can have them polarized in various ways. 01:10:33.530 --> 01:10:42.450 align:middle line:84% But these various ways are finite. 01:10:42.450 --> 01:10:46.860 align:middle line:84% So the spin or maybe even an atom 01:10:46.860 --> 01:10:55.060 align:middle line:84% can be modeled with a finite number of energy levels. 01:10:55.060 --> 01:11:02.770 align:middle line:84% And if it's finite, it means that the energy cannot go above 01:11:02.770 --> 01:11:06.160 align:middle line:90% the upper energy level. 01:11:06.160 --> 01:11:13.870 align:middle line:84% Well, in this situation, the stable equilibrium state 01:11:13.870 --> 01:11:18.990 align:middle line:84% fundamental relation looks like this when you project 01:11:18.990 --> 01:11:20.535 align:middle line:90% on an E versus S diagram. 01:11:20.535 --> 01:11:25.920 align:middle line:90% 01:11:25.920 --> 01:11:28.360 align:middle line:84% This is the fundamental relation that you obtain. 01:11:28.360 --> 01:11:34.435 align:middle line:90% 01:11:34.435 --> 01:11:37.930 align:middle line:84% That function that you are representing 01:11:37.930 --> 01:11:44.860 align:middle line:84% there, let me write it as S is a function of E. 01:11:44.860 --> 01:11:48.070 align:middle line:90% So this is E minimum. 01:11:48.070 --> 01:11:49.485 align:middle line:90% This is E maximum. 01:11:49.485 --> 01:11:52.030 align:middle line:90% 01:11:52.030 --> 01:12:03.650 align:middle line:84% And the fundamental relation S as a function of E plus 01:12:03.650 --> 01:12:09.680 align:middle line:84% whatever other parameters and amounts are, looks like that. 01:12:09.680 --> 01:12:15.350 align:middle line:90% 01:12:15.350 --> 01:12:15.850 align:middle line:90% All right. 01:12:15.850 --> 01:12:20.700 align:middle line:90% So yeah, nothing more. 01:12:20.700 --> 01:12:25.950 align:middle line:84% For systems with translational degrees of freedom, S versus E 01:12:25.950 --> 01:12:28.565 align:middle line:90% instead, there is a minimum. 01:12:28.565 --> 01:12:31.470 align:middle line:90% 01:12:31.470 --> 01:12:34.320 align:middle line:90% Particles are not moving. 01:12:34.320 --> 01:12:39.960 align:middle line:84% So there is no kinetic energy, and therefore, they 01:12:39.960 --> 01:12:42.870 align:middle line:90% are at minimum. 01:12:42.870 --> 01:12:47.030 align:middle line:84% And this corresponds to the zero temperature stable equilibrium 01:12:47.030 --> 01:12:47.530 align:middle line:90% state. 01:12:47.530 --> 01:12:56.350 align:middle line:84% And then this is, well, the fundamental relation, 01:12:56.350 --> 01:13:02.950 align:middle line:84% the one that we have seen just in the previous diagram. 01:13:02.950 --> 01:13:09.660 align:middle line:84% But if there is an upper bound, you reach this value here 01:13:09.660 --> 01:13:12.840 align:middle line:90% and you will go down. 01:13:12.840 --> 01:13:15.350 align:middle line:84% So look at what happens to the slopes. 01:13:15.350 --> 01:13:20.450 align:middle line:90% 01:13:20.450 --> 01:13:25.990 align:middle line:90% Here the slope is infinite. 01:13:25.990 --> 01:13:31.120 align:middle line:84% And remember that the partial of S with respect to E 01:13:31.120 --> 01:13:37.310 align:middle line:84% is the inverse of the temperature, 1 over T. 01:13:37.310 --> 01:13:41.790 align:middle line:84% So if 1 over T is infinite, the temperature here is 0. 01:13:41.790 --> 01:13:45.990 align:middle line:90% 01:13:45.990 --> 01:13:48.360 align:middle line:84% Then it reaches a maximum, the entropy 01:13:48.360 --> 01:13:51.845 align:middle line:84% reaches a maximum for a value of the energy. 01:13:51.845 --> 01:13:54.660 align:middle line:90% 01:13:54.660 --> 01:13:59.500 align:middle line:84% So we are moving-- this is 0 temperature 01:13:59.500 --> 01:14:01.125 align:middle line:90% reaches a maximum here. 01:14:01.125 --> 01:14:03.940 align:middle line:90% 01:14:03.940 --> 01:14:08.260 align:middle line:84% Here, the inverse of the temperature is equal to 0 01:14:08.260 --> 01:14:10.280 align:middle line:90% because the slope is equal to 0. 01:14:10.280 --> 01:14:12.620 align:middle line:84% So that means the temperature here is infinite. 01:14:12.620 --> 01:14:19.370 align:middle line:90% 01:14:19.370 --> 01:14:22.530 align:middle line:84% And then you go on the other side here 01:14:22.530 --> 01:14:26.000 align:middle line:90% and the entropy goes down. 01:14:26.000 --> 01:14:32.640 align:middle line:84% So you get another state with zero entropy, 01:14:32.640 --> 01:14:35.880 align:middle line:84% another stable equilibrium state with zero entropy corresponding 01:14:35.880 --> 01:14:38.430 align:middle line:90% to the maximum energy. 01:14:38.430 --> 01:14:40.800 align:middle line:84% The second law is still satisfied. 01:14:40.800 --> 01:14:45.720 align:middle line:84% For every value of the energy, there is one and only 01:14:45.720 --> 01:14:47.210 align:middle line:90% one stable equilibrium state. 01:14:47.210 --> 01:14:52.130 align:middle line:90% 01:14:52.130 --> 01:14:55.110 align:middle line:84% You cannot do it the other way around. 01:14:55.110 --> 01:15:00.430 align:middle line:84% In other words, if I fix the value of the entropy, 01:15:00.430 --> 01:15:04.670 align:middle line:84% so the minimum energy principle instead 01:15:04.670 --> 01:15:06.610 align:middle line:90% doesn't apply in this case. 01:15:06.610 --> 01:15:07.930 align:middle line:90% In this case, it does. 01:15:07.930 --> 01:15:10.730 align:middle line:84% I fixed the value of the entropy. 01:15:10.730 --> 01:15:13.790 align:middle line:84% And the stable equilibrium state corresponds 01:15:13.790 --> 01:15:19.520 align:middle line:84% to the minimum energy because the possible states are only 01:15:19.520 --> 01:15:20.910 align:middle line:90% in this area here. 01:15:20.910 --> 01:15:22.760 align:middle line:90% And that's the minimum energy. 01:15:22.760 --> 01:15:26.450 align:middle line:84% But for a special system like the one 01:15:26.450 --> 01:15:30.570 align:middle line:84% that we're talking about, I fixed the value of the entropy. 01:15:30.570 --> 01:15:33.120 align:middle line:84% And there are two stable equilibrium states, 01:15:33.120 --> 01:15:38.420 align:middle line:84% one with minimum energy and one with maximum energy. 01:15:38.420 --> 01:15:44.080 align:middle line:84% Except here, I fixed the value of the entropy. 01:15:44.080 --> 01:15:45.980 align:middle line:84% There is only one stable equilibrium state. 01:15:45.980 --> 01:15:49.560 align:middle line:90% 01:15:49.560 --> 01:15:56.860 align:middle line:84% And if I fix a higher value of entropy than this one, 01:15:56.860 --> 01:15:57.695 align:middle line:90% there are no states. 01:15:57.695 --> 01:16:02.690 align:middle line:90% 01:16:02.690 --> 01:16:06.140 align:middle line:84% So also, this last stage, so like the third law 01:16:06.140 --> 01:16:12.260 align:middle line:84% says that there is 0 temperature for the minimum energy 01:16:12.260 --> 01:16:17.180 align:middle line:84% stable equilibrium state, it also says that-- 01:16:17.180 --> 01:16:23.870 align:middle line:84% implies, I mean, that the temperature 01:16:23.870 --> 01:16:27.715 align:middle line:84% of the highest energy stable equilibrium state is 0. 01:16:27.715 --> 01:16:35.070 align:middle line:90% 01:16:35.070 --> 01:16:39.810 align:middle line:84% However, here you have a little problem somehow of discontinuity 01:16:39.810 --> 01:16:40.715 align:middle line:90% because you go-- 01:16:40.715 --> 01:16:44.060 align:middle line:90% 01:16:44.060 --> 01:16:46.970 align:middle line:84% yeah, I forgot to say one important thing. 01:16:46.970 --> 01:16:50.780 align:middle line:84% The slope on this side of the curve 01:16:50.780 --> 01:16:56.290 align:middle line:84% is positive, whereas on this other side of the curve, 01:16:56.290 --> 01:16:58.860 align:middle line:90% the slope is negative. 01:16:58.860 --> 01:17:07.440 align:middle line:84% So part of all these stable equilibrium states 01:17:07.440 --> 01:17:10.710 align:middle line:90% have negative temperatures. 01:17:10.710 --> 01:17:14.970 align:middle line:84% Negative temperatures are not so esoteric. 01:17:14.970 --> 01:17:18.170 align:middle line:84% They were discovered in, I think it 01:17:18.170 --> 01:17:26.530 align:middle line:84% was 1956, that's when I was born, by Ramsey at Harvard, 01:17:26.530 --> 01:17:27.710 align:middle line:90% just next door. 01:17:27.710 --> 01:17:35.510 align:middle line:90% 01:17:35.510 --> 01:17:38.420 align:middle line:84% Actually, his daughter was the wife 01:17:38.420 --> 01:17:44.070 align:middle line:84% of Professor Keck, who was downstairs, and with whom 01:17:44.070 --> 01:17:46.520 align:middle line:90% I have done my master's thesis. 01:17:46.520 --> 01:17:53.330 align:middle line:90% 01:17:53.330 --> 01:17:57.450 align:middle line:84% All right, so negative temperatures. 01:17:57.450 --> 01:18:05.220 align:middle line:84% We will see that the idea of negative temperature, the name 01:18:05.220 --> 01:18:08.920 align:middle line:90% is not really very good. 01:18:08.920 --> 01:18:10.090 align:middle line:90% As you say, negative. 01:18:10.090 --> 01:18:11.170 align:middle line:90% It's below zero. 01:18:11.170 --> 01:18:13.440 align:middle line:84% Well, really, actually, if temperature 01:18:13.440 --> 01:18:15.910 align:middle line:84% measures the tendency of a system, 01:18:15.910 --> 01:18:23.620 align:middle line:84% as we will see, to yield energy, to give out energy, 01:18:23.620 --> 01:18:27.670 align:middle line:84% these negative temperature equilibrium states 01:18:27.670 --> 01:18:35.980 align:middle line:84% have more tendency to give out energy 01:18:35.980 --> 01:18:38.975 align:middle line:84% than the positive temperature equilibrium states. 01:18:38.975 --> 01:18:42.810 align:middle line:90% 01:18:42.810 --> 01:18:49.786 align:middle line:84% They are hard to keep also because we-- 01:18:49.786 --> 01:18:54.830 align:middle line:84% I don't know if we can make in a lab something that 01:18:54.830 --> 01:18:58.260 align:middle line:84% has the properties of that we could call a reservoir. 01:18:58.260 --> 01:19:01.370 align:middle line:84% Although I'm not sure, but that we 01:19:01.370 --> 01:19:06.650 align:middle line:84% could call a reservoir with a negative temperature. 01:19:06.650 --> 01:19:11.000 align:middle line:84% I'm not sure it can be done in the lab 01:19:11.000 --> 01:19:15.350 align:middle line:84% simply because most of the times reservoirs 01:19:15.350 --> 01:19:19.250 align:middle line:84% have been constructed by using systems 01:19:19.250 --> 01:19:21.600 align:middle line:84% that have translational degrees of freedom. 01:19:21.600 --> 01:19:24.890 align:middle line:84% And therefore, they have no negative temperatures. 01:19:24.890 --> 01:19:32.350 align:middle line:84% But maybe it is possible also to do a reservoir 01:19:32.350 --> 01:19:34.280 align:middle line:90% with negative temperatures. 01:19:34.280 --> 01:19:38.450 align:middle line:84% Anyway, so we have a problem here 01:19:38.450 --> 01:19:43.310 align:middle line:84% that the temperature goes from zero positive values 01:19:43.310 --> 01:19:45.980 align:middle line:90% to plus infinity here. 01:19:45.980 --> 01:19:49.910 align:middle line:84% Then, immediately after it switches to minus infinity 01:19:49.910 --> 01:19:52.650 align:middle line:90% and it goes down to zero again. 01:19:52.650 --> 01:19:55.350 align:middle line:84% So there is some sort of discontinuity 01:19:55.350 --> 01:20:00.460 align:middle line:84% if we insist in talking of temperature. 01:20:00.460 --> 01:20:06.060 align:middle line:84% But if instead we look at a scale in which we, instead 01:20:06.060 --> 01:20:19.460 align:middle line:84% of temperature, we do minus 1 over temperature, this, 01:20:19.460 --> 01:20:23.360 align:middle line:84% which can be interpreted again as the tendency 01:20:23.360 --> 01:20:26.630 align:middle line:84% and escaping tendency for energy. 01:20:26.630 --> 01:20:28.370 align:middle line:90% It still measures the energy-- 01:20:28.370 --> 01:20:32.870 align:middle line:84% the tendency of a system to, in a stable equilibrium state, 01:20:32.870 --> 01:20:37.100 align:middle line:90% to give up energy. 01:20:37.100 --> 01:20:44.900 align:middle line:84% And you see then this property here for this curve 01:20:44.900 --> 01:20:51.230 align:middle line:84% would go from minus infinity in the minimum energy, 01:20:51.230 --> 01:20:56.170 align:middle line:90% minus infinity, then goes to 0. 01:20:56.170 --> 01:20:58.810 align:middle line:84% And from 0 without discontinuities, 01:20:58.810 --> 01:21:02.740 align:middle line:90% it goes to plus infinity. 01:21:02.740 --> 01:21:05.680 align:middle line:90% So that is also a better scale. 01:21:05.680 --> 01:21:12.760 align:middle line:84% So it also shows you that this property is continuously 01:21:12.760 --> 01:21:17.300 align:middle line:84% increasing from minus infinity, through 0 to plus infinity. 01:21:17.300 --> 01:21:22.060 align:middle line:84% And these ones that have high value for this property 01:21:22.060 --> 01:21:28.180 align:middle line:84% are the hotter, if we understand as hot something that 01:21:28.180 --> 01:21:31.830 align:middle line:84% has a strong tendency to give out energy. 01:21:31.830 --> 01:21:34.520 align:middle line:90% 01:21:34.520 --> 01:21:43.820 align:middle line:84% So this much for the special systems. 01:21:43.820 --> 01:21:47.300 align:middle line:84% Now let's go back to our E versus S diagram 01:21:47.300 --> 01:21:51.320 align:middle line:84% and try to visualize the idea of adiabatic availability. 01:21:51.320 --> 01:21:53.030 align:middle line:84% Remember, the adiabatic availability 01:21:53.030 --> 01:21:58.020 align:middle line:84% is the property that measures the maximum amount of energy 01:21:58.020 --> 01:22:00.390 align:middle line:84% that you can extract out of a system in a given 01:22:00.390 --> 01:22:03.960 align:middle line:90% state in a weight process. 01:22:03.960 --> 01:22:07.590 align:middle line:84% And you manage to extract it when 01:22:07.590 --> 01:22:12.410 align:middle line:90% you manage to take your system. 01:22:12.410 --> 01:22:13.980 align:middle line:90% Yeah, there is a y there. 01:22:13.980 --> 01:22:15.300 align:middle line:90% That's a wrong spelling. 01:22:15.300 --> 01:22:16.643 align:middle line:90% That's Italian. 01:22:16.643 --> 01:22:19.830 align:middle line:90% 01:22:19.830 --> 01:22:21.430 align:middle line:90% It's also wrong in Italian. 01:22:21.430 --> 01:22:23.520 align:middle line:90% Anyway, I'll correct it. 01:22:23.520 --> 01:22:29.270 align:middle line:84% It has to finish to end up in the stable equilibrium state. 01:22:29.270 --> 01:22:32.610 align:middle line:90% 01:22:32.610 --> 01:22:35.015 align:middle line:84% And that process must be reversible. 01:22:35.015 --> 01:22:39.630 align:middle line:90% 01:22:39.630 --> 01:22:42.750 align:middle line:84% OK, so what do we know of a reversible process 01:22:42.750 --> 01:22:46.100 align:middle line:84% or reversible weight process in terms of entropy? 01:22:46.100 --> 01:22:50.490 align:middle line:90% What did we learn? 01:22:50.490 --> 01:22:53.930 align:middle line:84% We call it the principle of entropy non-decrease. 01:22:53.930 --> 01:22:58.040 align:middle line:84% But we call it also criterion for reversibility of processes. 01:22:58.040 --> 01:23:03.270 align:middle line:84% If a process, a weight process is reversible, 01:23:03.270 --> 01:23:05.390 align:middle line:90% the entropy stays constant. 01:23:05.390 --> 01:23:08.930 align:middle line:90% So let's do this in a plot. 01:23:08.930 --> 01:23:20.300 align:middle line:84% I start from my system in some state here, generic state A1. 01:23:20.300 --> 01:23:25.310 align:middle line:84% I make a process in which I want to take energy out 01:23:25.310 --> 01:23:26.160 align:middle line:90% of the system. 01:23:26.160 --> 01:23:30.750 align:middle line:84% So the energy goes down because I'm taking out of the system. 01:23:30.750 --> 01:23:36.290 align:middle line:84% It goes down because I want to give it to the weight. 01:23:36.290 --> 01:23:41.030 align:middle line:84% I want the process to be not only weight process, 01:23:41.030 --> 01:23:43.030 align:middle line:90% but reversible. 01:23:43.030 --> 01:23:45.440 align:middle line:84% So in a reversible weight process, 01:23:45.440 --> 01:23:47.290 align:middle line:90% the entropy doesn't change. 01:23:47.290 --> 01:23:50.180 align:middle line:90% So I go down this way. 01:23:50.180 --> 01:23:52.770 align:middle line:90% 01:23:52.770 --> 01:23:55.545 align:middle line:84% I go down until there are states. 01:23:55.545 --> 01:23:58.420 align:middle line:90% 01:23:58.420 --> 01:24:03.040 align:middle line:84% Eventually, I reach the minimum energy stable equilibrium 01:24:03.040 --> 01:24:05.350 align:middle line:90% state with that entropy. 01:24:05.350 --> 01:24:08.090 align:middle line:84% And there are no more states below that. 01:24:08.090 --> 01:24:11.870 align:middle line:84% So that's all I can do in a reversible weight process. 01:24:11.870 --> 01:24:14.935 align:middle line:84% This is as much as I can go down. 01:24:14.935 --> 01:24:17.640 align:middle line:90% 01:24:17.640 --> 01:24:21.810 align:middle line:84% So the difference between the original energy and the energy 01:24:21.810 --> 01:24:29.130 align:middle line:84% of the lowest energy stable equilibrium 01:24:29.130 --> 01:24:35.020 align:middle line:84% state with the initial entropy is the adiabatic availability. 01:24:35.020 --> 01:24:38.510 align:middle line:84% So this is the visualization of the adiabatic availability. 01:24:38.510 --> 01:24:41.950 align:middle line:84% It's the distance in that diagram 01:24:41.950 --> 01:24:47.530 align:middle line:84% between your state and the stable equilibrium curve 01:24:47.530 --> 01:24:48.310 align:middle line:90% vertically. 01:24:48.310 --> 01:24:51.460 align:middle line:90% 01:24:51.460 --> 01:24:53.650 align:middle line:90% So, for example, if I consider-- 01:24:53.650 --> 01:25:01.430 align:middle line:84% if my initial state were here at 0 entropy, 01:25:01.430 --> 01:25:11.860 align:middle line:84% you see I can go down and meet the zero entropy stable 01:25:11.860 --> 01:25:18.450 align:middle line:84% equilibrium state, which is the minimum energy 01:25:18.450 --> 01:25:24.920 align:middle line:84% stable equilibrium state of mechanics, 01:25:24.920 --> 01:25:31.590 align:middle line:84% or actually, regarding this area, it's used. 01:25:31.590 --> 01:25:33.070 align:middle line:90% It's discussed in the book. 01:25:33.070 --> 01:25:34.620 align:middle line:84% I don't want to go into the details, 01:25:34.620 --> 01:25:40.820 align:middle line:84% but somehow the stable equilibrium state 01:25:40.820 --> 01:25:44.120 align:middle line:90% is not necessarily 0 entropy. 01:25:44.120 --> 01:25:47.060 align:middle line:84% It's 0 temperature by the third law, 01:25:47.060 --> 01:25:51.800 align:middle line:84% but it could have a finite entropy 01:25:51.800 --> 01:25:59.030 align:middle line:84% equal to the logarithm of the degeneracy of that lowest 01:25:59.030 --> 01:26:00.660 align:middle line:90% energy or ground state. 01:26:00.660 --> 01:26:05.900 align:middle line:90% 01:26:05.900 --> 01:26:09.190 align:middle line:84% Nevertheless, the amount of the adiabatic availability 01:26:09.190 --> 01:26:16.090 align:middle line:84% for a state with 0 entropy is equal to the difference 01:26:16.090 --> 01:26:24.460 align:middle line:84% between its energy and the lowest energy for those values 01:26:24.460 --> 01:26:27.680 align:middle line:90% of amounts and parameters. 01:26:27.680 --> 01:26:31.400 align:middle line:84% So this is the visualization of the adiabatic availability. 01:26:31.400 --> 01:26:34.100 align:middle line:90% 01:26:34.100 --> 01:26:36.380 align:middle line:84% For a given energy, the more I go 01:26:36.380 --> 01:26:42.490 align:middle line:84% to states that have higher entropy, the less adiabatic 01:26:42.490 --> 01:26:46.750 align:middle line:84% availability they have, until I reach the stable equilibrium 01:26:46.750 --> 01:26:50.560 align:middle line:84% state that has 0 adiabatic availability. 01:26:50.560 --> 01:26:51.880 align:middle line:90% I'm already there. 01:26:51.880 --> 01:26:56.200 align:middle line:84% It cannot go down vertically because there are no states 01:26:56.200 --> 01:26:59.170 align:middle line:90% available there. 01:26:59.170 --> 01:27:03.400 align:middle line:90% We talked about reservoirs. 01:27:03.400 --> 01:27:08.210 align:middle line:84% Remember, reservoirs have a constant temperature. 01:27:08.210 --> 01:27:14.610 align:middle line:84% Their stable equilibrium states have all the same temperature, 01:27:14.610 --> 01:27:15.110 align:middle line:90% TR. 01:27:15.110 --> 01:27:19.200 align:middle line:90% 01:27:19.200 --> 01:27:23.940 align:middle line:84% For as long as the system behaves as a reservoir, because, 01:27:23.940 --> 01:27:28.050 align:middle line:84% as we said, if I make a reservoir with-- 01:27:28.050 --> 01:27:33.060 align:middle line:84% yeah, if it's done like water at the triple point, 01:27:33.060 --> 01:27:35.910 align:middle line:84% as long as I have all the three phases coexisting 01:27:35.910 --> 01:27:39.110 align:middle line:84% that behaves as a reservoir in stable equilibrium states. 01:27:39.110 --> 01:27:42.210 align:middle line:90% 01:27:42.210 --> 01:27:45.010 align:middle line:84% But if I put too much energy, for example, 01:27:45.010 --> 01:27:49.080 align:middle line:84% and one of the phrases disappears, 01:27:49.080 --> 01:27:54.340 align:middle line:84% then that system doesn't behave as a reservoir anymore. 01:27:54.340 --> 01:27:57.640 align:middle line:84% So it's in a limited range of states 01:27:57.640 --> 01:28:00.600 align:middle line:84% the system can behave as a thermal reservoir 01:28:00.600 --> 01:28:06.095 align:middle line:84% or can approximate, because that is an approximation practice. 01:28:06.095 --> 01:28:08.910 align:middle line:90% 01:28:08.910 --> 01:28:18.230 align:middle line:84% Because rigorously speaking, a true reservoir 01:28:18.230 --> 01:28:21.420 align:middle line:90% doesn't really exist. 01:28:21.420 --> 01:28:24.770 align:middle line:84% I know I'm confusing you, but sometimes it's 01:28:24.770 --> 01:28:27.110 align:middle line:90% good to be confused. 01:28:27.110 --> 01:28:35.670 align:middle line:84% Because a reservoir is a very limiting system. 01:28:35.670 --> 01:28:38.455 align:middle line:90% 01:28:38.455 --> 01:28:43.080 align:middle line:84% And in fact, if you think deeply enough, 01:28:43.080 --> 01:28:47.420 align:middle line:84% and you'll find also this discussed in the book, 01:28:47.420 --> 01:28:51.070 align:middle line:84% you'll see that, strictly speaking, 01:28:51.070 --> 01:28:56.140 align:middle line:84% if a reservoir satisfied its defining features, 01:28:56.140 --> 01:28:58.220 align:middle line:84% it would be violating the second law. 01:28:58.220 --> 01:29:00.950 align:middle line:90% 01:29:00.950 --> 01:29:04.850 align:middle line:84% So we are using that as a limiting system. 01:29:04.850 --> 01:29:10.080 align:middle line:84% And this, in fact, allows me to say 01:29:10.080 --> 01:29:16.260 align:middle line:84% that in the years after publishing that book, 01:29:16.260 --> 01:29:20.980 align:middle line:84% we've been together with Professor 01:29:20.980 --> 01:29:25.340 align:middle line:84% Zanchini of the University of Bologna in Italy, 01:29:25.340 --> 01:29:27.955 align:middle line:84% who contributed also in some of the proofs of the book. 01:29:27.955 --> 01:29:33.310 align:middle line:90% 01:29:33.310 --> 01:29:37.690 align:middle line:84% We went on and tried to find an alternative way of defining 01:29:37.690 --> 01:29:44.110 align:middle line:90% entropy, without the reservoir. 01:29:44.110 --> 01:29:53.080 align:middle line:84% Because the reservoir is some sort of a macroscopic object. 01:29:53.080 --> 01:29:55.420 align:middle line:84% And we wanted to do it more generally 01:29:55.420 --> 01:29:57.850 align:middle line:84% so that our definition of entropy 01:29:57.850 --> 01:30:03.440 align:middle line:84% could extend also to microscopic situations, 01:30:03.440 --> 01:30:10.110 align:middle line:84% but also in an operational, therefore, 01:30:10.110 --> 01:30:18.800 align:middle line:84% laboratory applicable measurement procedure. 01:30:18.800 --> 01:30:20.060 align:middle line:90% So we did. 01:30:20.060 --> 01:30:25.410 align:middle line:84% We have some papers around that concept. 01:30:25.410 --> 01:30:33.810 align:middle line:84% And also, a famous mathematician, Lieb, 01:30:33.810 --> 01:30:41.190 align:middle line:84% later on after our book was published, 01:30:41.190 --> 01:30:43.070 align:middle line:84% alternative definitions of entropy 01:30:43.070 --> 01:30:44.940 align:middle line:84% valid for non-equilibrium states. 01:30:44.940 --> 01:30:50.510 align:middle line:84% And initially, they didn't do it for non-equilibrium states. 01:30:50.510 --> 01:30:53.920 align:middle line:84% But then they realized that they could do it, 01:30:53.920 --> 01:30:59.530 align:middle line:84% and that we were sort of doing the same thing. 01:30:59.530 --> 01:31:04.480 align:middle line:84% So what I'm saying is that I'm telling you 01:31:04.480 --> 01:31:10.730 align:middle line:84% this definition of entropy, as it were an easy stuff, 01:31:10.730 --> 01:31:13.000 align:middle line:90% but it's not really very easy. 01:31:13.000 --> 01:31:22.230 align:middle line:84% Because conceptually, entropy has always 01:31:22.230 --> 01:31:30.410 align:middle line:84% been a difficult topic to understand. 01:31:30.410 --> 01:31:33.000 align:middle line:90% It has so many facets. 01:31:33.000 --> 01:31:34.600 align:middle line:90% One is the laboratory. 01:31:34.600 --> 01:31:36.590 align:middle line:90% The other is the statistical. 01:31:36.590 --> 01:31:40.140 align:middle line:84% And now we have also the quantum. 01:31:40.140 --> 01:31:47.430 align:middle line:84% So it has so many facets that you shouldn't say, yes, 01:31:47.430 --> 01:31:49.580 align:middle line:90% I understand what entropy is. 01:31:49.580 --> 01:31:53.300 align:middle line:84% On the other hand, you should say that you 01:31:53.300 --> 01:31:54.611 align:middle line:90% understand what entropy is. 01:31:54.611 --> 01:31:58.460 align:middle line:84% You know, I like contradicting a bit. 01:31:58.460 --> 01:32:03.730 align:middle line:84% Because if you want to go home now and say, well, 01:32:03.730 --> 01:32:08.370 align:middle line:84% what did you learn in the thermodynamics course so far? 01:32:08.370 --> 01:32:13.600 align:middle line:84% Well, I learned you could say that, yeah, there is energy. 01:32:13.600 --> 01:32:15.780 align:middle line:90% Everybody knows about energy. 01:32:15.780 --> 01:32:23.140 align:middle line:84% With energy, you can do useful things. 01:32:23.140 --> 01:32:30.595 align:middle line:84% But not all the energy is convertible into useful effects. 01:32:30.595 --> 01:32:34.430 align:middle line:90% 01:32:34.430 --> 01:32:41.140 align:middle line:84% In fact, entropy is a measurement of that part 01:32:41.140 --> 01:32:45.700 align:middle line:84% of the energy that cannot be converted into useful effects. 01:32:45.700 --> 01:32:49.030 align:middle line:84% So from an engineer point of view, 01:32:49.030 --> 01:32:53.530 align:middle line:84% that's a very clear statement, and it's true. 01:32:53.530 --> 01:32:55.560 align:middle line:84% There's no controversy about this. 01:32:55.560 --> 01:33:02.150 align:middle line:90% 01:33:02.150 --> 01:33:06.040 align:middle line:84% All right, so let's move on because we are running out 01:33:06.040 --> 01:33:06.740 align:middle line:90% of time. 01:33:06.740 --> 01:33:15.260 align:middle line:84% And so the reservoir has stable equilibrium states 01:33:15.260 --> 01:33:17.670 align:middle line:84% that all have the same temperature. 01:33:17.670 --> 01:33:20.450 align:middle line:84% So since temperature is the slope 01:33:20.450 --> 01:33:22.820 align:middle line:90% of the fundamental relation-- 01:33:22.820 --> 01:33:26.150 align:middle line:84% of the curve that represents the fundamental relation, 01:33:26.150 --> 01:33:30.770 align:middle line:84% the fundamental relation for a reservoir is a straight line. 01:33:30.770 --> 01:33:35.410 align:middle line:84% So the equation for a straight line is this one. 01:33:35.410 --> 01:33:42.290 align:middle line:84% Changes in energy are related to changes in entropy between two. 01:33:42.290 --> 01:33:44.570 align:middle line:84% If the system goes from one stable equilibrium 01:33:44.570 --> 01:33:46.590 align:middle line:84% state to another stable equilibrium state, 01:33:46.590 --> 01:33:51.800 align:middle line:84% and therefore, it moves along this curve, 01:33:51.800 --> 01:33:56.900 align:middle line:84% then the ratio of the energy change to the entropy change 01:33:56.900 --> 01:33:59.590 align:middle line:84% is equal to the temperature of the reservoir. 01:33:59.590 --> 01:34:04.650 align:middle line:90% 01:34:04.650 --> 01:34:09.740 align:middle line:90% Why we use the reservoir? 01:34:09.740 --> 01:34:16.060 align:middle line:84% Remember, we use it so that we could make that property 01:34:16.060 --> 01:34:17.470 align:middle line:90% additive. 01:34:17.470 --> 01:34:23.560 align:middle line:84% So we built additivity into the concept 01:34:23.560 --> 01:34:25.270 align:middle line:90% of adiabatic availability. 01:34:25.270 --> 01:34:28.300 align:middle line:84% And in fact, if we consider the adiabatic availability 01:34:28.300 --> 01:34:33.040 align:middle line:84% of a composite system, composite of a system 01:34:33.040 --> 01:34:39.490 align:middle line:84% like a normal system, and the reservoir, 01:34:39.490 --> 01:34:42.520 align:middle line:84% then the adiabatic availability of this situation 01:34:42.520 --> 01:34:46.275 align:middle line:84% is what we call available energy. 01:34:46.275 --> 01:34:50.920 align:middle line:90% 01:34:50.920 --> 01:34:56.400 align:middle line:84% It turns out to be a property of system A with respect to a given 01:34:56.400 --> 01:34:57.420 align:middle line:90% reservoir. 01:34:57.420 --> 01:34:59.550 align:middle line:84% It's called available energy of a system 01:34:59.550 --> 01:35:04.180 align:middle line:84% with respect to a given reservoir. 01:35:04.180 --> 01:35:08.010 align:middle line:84% And we have given the formal expression 01:35:08.010 --> 01:35:09.950 align:middle line:90% in terms of energy and entropy. 01:35:09.950 --> 01:35:16.610 align:middle line:84% So you can compute the available energy. 01:35:16.610 --> 01:35:19.690 align:middle line:84% The maximum work that you can extract from your system A 01:35:19.690 --> 01:35:22.870 align:middle line:84% when it has available the possibility 01:35:22.870 --> 01:35:30.050 align:middle line:84% to interact with a reservoir and exchange with it. 01:35:30.050 --> 01:35:34.140 align:middle line:84% So let's see how we could construct. 01:35:34.140 --> 01:35:36.520 align:middle line:84% See, this is the formula for available energy. 01:35:36.520 --> 01:35:40.030 align:middle line:90% 01:35:40.030 --> 01:35:43.470 align:middle line:84% And how can we construct that formula? 01:35:43.470 --> 01:35:46.600 align:middle line:90% 01:35:46.600 --> 01:35:56.820 align:middle line:84% So here is the E versus S diagram for our system A. Fixed 01:35:56.820 --> 01:36:02.570 align:middle line:84% values of n and V. Stable equilibrium state curve. 01:36:02.570 --> 01:36:06.070 align:middle line:84% Then we consider, again, state A1. 01:36:06.070 --> 01:36:09.180 align:middle line:90% 01:36:09.180 --> 01:36:20.190 align:middle line:84% If, for state A1, I make a reversible weight process, 01:36:20.190 --> 01:36:25.300 align:middle line:84% the entropy remains constant, and I can go down this much. 01:36:25.300 --> 01:36:30.623 align:middle line:84% And that's how I extract the adiabatic availability. 01:36:30.623 --> 01:36:34.490 align:middle line:90% 01:36:34.490 --> 01:36:38.450 align:middle line:84% When I want to extract the available energy here, remember 01:36:38.450 --> 01:36:44.820 align:middle line:84% in the end, in the final state, the composite system 01:36:44.820 --> 01:36:47.640 align:middle line:84% must be in a stable equilibrium state. 01:36:47.640 --> 01:36:51.330 align:middle line:84% It's only then-- it's only when I reach the stable equilibrium 01:36:51.330 --> 01:36:57.210 align:middle line:84% state that I cannot extract any more in a weight process. 01:36:57.210 --> 01:37:00.750 align:middle line:84% So stable equilibrium state for a combination 01:37:00.750 --> 01:37:01.990 align:middle line:90% means mutual equilibrium. 01:37:01.990 --> 01:37:05.970 align:middle line:84% So the final state of system A is in mutual equilibrium 01:37:05.970 --> 01:37:09.370 align:middle line:90% with the reservoir. 01:37:09.370 --> 01:37:16.785 align:middle line:84% If the reservoir is a temperature TR, when the system 01:37:16.785 --> 01:37:19.650 align:middle line:84% A is in mutual equilibrium with the reservoir, 01:37:19.650 --> 01:37:24.570 align:middle line:84% we said that we will prove that the two 01:37:24.570 --> 01:37:27.340 align:middle line:84% systems in mutual equilibrium must have the same temperature. 01:37:27.340 --> 01:37:32.040 align:middle line:84% So system A must end up in the stable equilibrium 01:37:32.040 --> 01:37:36.100 align:middle line:90% state that has a slope-- 01:37:36.100 --> 01:37:38.370 align:middle line:84% that has temperature equal to the temperature 01:37:38.370 --> 01:37:41.550 align:middle line:90% of the reservoir. 01:37:41.550 --> 01:37:43.710 align:middle line:84% So, if is this the temperature of the reservoir, 01:37:43.710 --> 01:37:49.530 align:middle line:84% I go and look for the point here on this curve 01:37:49.530 --> 01:37:51.900 align:middle line:90% where the slope is equal to TR. 01:37:51.900 --> 01:37:53.880 align:middle line:84% That is the state in which system 01:37:53.880 --> 01:37:56.220 align:middle line:84% A will be in mutual equilibrium with the reservoir. 01:37:56.220 --> 01:38:01.680 align:middle line:84% And that is going to be the final state of system A. 01:38:01.680 --> 01:38:02.200 align:middle line:90% You see? 01:38:02.200 --> 01:38:06.470 align:middle line:90% 01:38:06.470 --> 01:38:09.710 align:middle line:84% In order to go from the initial state of system A 01:38:09.710 --> 01:38:15.410 align:middle line:84% to this final state, I need to change 01:38:15.410 --> 01:38:17.890 align:middle line:90% the entropy of the system. 01:38:17.890 --> 01:38:21.700 align:middle line:90% 01:38:21.700 --> 01:38:25.840 align:middle line:84% In this case, since state AR is to the right 01:38:25.840 --> 01:38:33.430 align:middle line:84% in the diagram of state A1, the entropy SR is greater than S1. 01:38:33.430 --> 01:38:43.840 align:middle line:84% And one possibility is to let entropy increase 01:38:43.840 --> 01:38:47.350 align:middle line:84% by means of some spontaneous process that 01:38:47.350 --> 01:38:51.610 align:middle line:84% make a weight process an irreversible weight process. 01:38:51.610 --> 01:38:53.185 align:middle line:84% In an irreversible weight process, 01:38:53.185 --> 01:38:55.420 align:middle line:90% I can increase the entropy. 01:38:55.420 --> 01:38:59.950 align:middle line:84% And so I can go from here to there in a weight process. 01:38:59.950 --> 01:39:06.630 align:middle line:84% And the amount of energy I would extract in that weight process 01:39:06.630 --> 01:39:08.500 align:middle line:84% would be only the difference between 01:39:08.500 --> 01:39:11.140 align:middle line:90% the initial and final energy. 01:39:11.140 --> 01:39:14.220 align:middle line:90% That's the energy balance. 01:39:14.220 --> 01:39:15.930 align:middle line:84% So I would extract only this much. 01:39:15.930 --> 01:39:19.620 align:middle line:90% 01:39:19.620 --> 01:39:26.105 align:middle line:90% But another way is to-- 01:39:26.105 --> 01:39:28.740 align:middle line:90% 01:39:28.740 --> 01:39:33.240 align:middle line:84% since I need this entropy in order to go from here to there, 01:39:33.240 --> 01:39:36.000 align:middle line:90% I can ask it to the reservoir. 01:39:36.000 --> 01:39:41.310 align:middle line:84% The reservoir is there and interacts 01:39:41.310 --> 01:39:51.210 align:middle line:84% with my system in whatever way is allowed. 01:39:51.210 --> 01:39:55.525 align:middle line:84% So they can also exchange entropy. 01:39:55.525 --> 01:39:58.400 align:middle line:90% 01:39:58.400 --> 01:40:04.040 align:middle line:84% So if I ask the reservoir to give me this much entropy 01:40:04.040 --> 01:40:11.420 align:middle line:84% so that I can go from here to here in the entropy axis, 01:40:11.420 --> 01:40:14.390 align:middle line:84% the reservoir-- let's go back to the-- 01:40:14.390 --> 01:40:18.720 align:middle line:84% so if the reservoir has to give me some entropy, 01:40:18.720 --> 01:40:23.720 align:middle line:84% it means that it must go from here to the left. 01:40:23.720 --> 01:40:31.050 align:middle line:84% In order to give me some entropy, 01:40:31.050 --> 01:40:35.885 align:middle line:90% it can give me also some energy. 01:40:35.885 --> 01:40:39.930 align:middle line:90% 01:40:39.930 --> 01:40:44.700 align:middle line:84% So I say, all right, reservoir, while you are giving me 01:40:44.700 --> 01:40:48.330 align:middle line:84% this much entropy in going from here to here, 01:40:48.330 --> 01:40:51.135 align:middle line:84% you can give me also that energy. 01:40:51.135 --> 01:40:53.280 align:middle line:84% You can go from this stable equilibrium state 01:40:53.280 --> 01:40:56.100 align:middle line:84% to this lower energy stable equilibrium state. 01:40:56.100 --> 01:41:06.890 align:middle line:84% And that energy becomes available to go to the weight. 01:41:06.890 --> 01:41:15.290 align:middle line:84% So you take energy out of the reservoir 01:41:15.290 --> 01:41:17.990 align:middle line:84% and give that energy to the weight, 01:41:17.990 --> 01:41:20.200 align:middle line:84% and give the entropy to the system. 01:41:20.200 --> 01:41:24.900 align:middle line:90% 01:41:24.900 --> 01:41:35.460 align:middle line:84% And therefore, you can see this inclined line. 01:41:35.460 --> 01:41:41.100 align:middle line:84% You can view it also as the stable equilibrium state 01:41:41.100 --> 01:41:43.560 align:middle line:90% curve for the reservoir. 01:41:43.560 --> 01:41:56.480 align:middle line:84% So this much energy reduction for the reservoir 01:41:56.480 --> 01:41:59.940 align:middle line:90% implies this much energy. 01:41:59.940 --> 01:42:03.600 align:middle line:84% So this much entropy reduction implies this much energy 01:42:03.600 --> 01:42:04.240 align:middle line:90% reduction. 01:42:04.240 --> 01:42:10.920 align:middle line:84% So in addition to the energy that you can extract directly 01:42:10.920 --> 01:42:17.550 align:middle line:84% from system A going from here to here, the difference 01:42:17.550 --> 01:42:20.230 align:middle line:84% between initial and final energy of system A, 01:42:20.230 --> 01:42:26.670 align:middle line:84% you can add also this much as energy 01:42:26.670 --> 01:42:31.980 align:middle line:84% that is extracted from the reservoir. 01:42:31.980 --> 01:42:39.420 align:middle line:84% And therefore, geometrically the available energy 01:42:39.420 --> 01:42:42.360 align:middle line:84% is the sum of these two contributions, 01:42:42.360 --> 01:42:46.520 align:middle line:84% which is also the geometric representation of this formula. 01:42:46.520 --> 01:42:51.980 align:middle line:90% 01:42:51.980 --> 01:42:52.600 align:middle line:90% All right. 01:42:52.600 --> 01:42:58.000 align:middle line:84% Well, I think we will take it from here. 01:42:58.000 --> 01:43:00.420 align:middle line:84% I want to skip this for the moment. 01:43:00.420 --> 01:43:04.650 align:middle line:84% And so the next time I still have 01:43:04.650 --> 01:43:07.930 align:middle line:84% to do what I didn't manage to do today. 01:43:07.930 --> 01:43:12.870 align:middle line:84% It's the necessary conditions for mutual equilibrium. 01:43:12.870 --> 01:43:19.390 align:middle line:84% So I think, yes, time is up for today. 01:43:19.390 --> 01:43:23.750 align:middle line:90% So I'll see you on Friday. 01:43:23.750 --> 01:43:30.000 align:middle line:90%