WEBVTT 00:00:00.000 --> 00:00:02.934 align:middle line:90% [SQUEAKING] 00:00:02.934 --> 00:00:04.401 align:middle line:90% [RUSTLING] 00:00:04.401 --> 00:00:07.335 align:middle line:90% [CLICKING] 00:00:07.335 --> 00:00:10.280 align:middle line:90% 00:00:10.280 --> 00:00:11.820 align:middle line:84% GIAN PAOLO BERETTA: Good afternoon. 00:00:11.820 --> 00:00:18.680 align:middle line:84% So the last time we have done the definition 00:00:18.680 --> 00:00:22.380 align:middle line:90% of a simple system model. 00:00:22.380 --> 00:00:26.370 align:middle line:84% It's very important because we acquire 00:00:26.370 --> 00:00:29.772 align:middle line:84% the so-called macroscopic limit which 00:00:29.772 --> 00:00:37.380 align:middle line:84% holds when we deal with many particles in a container 00:00:37.380 --> 00:00:41.910 align:middle line:84% and which is the framework in which most of thermodynamics 00:00:41.910 --> 00:00:48.910 align:middle line:84% has been developed until maybe 20 years ago. 00:00:48.910 --> 00:01:00.020 align:middle line:84% And so we will review just quickly what we've done. 00:01:00.020 --> 00:01:08.020 align:middle line:84% Remember that the issue here is that walls cause rarefaction 00:01:08.020 --> 00:01:13.210 align:middle line:84% effects because the density of particles 00:01:13.210 --> 00:01:16.500 align:middle line:90% must go to 0 at the wall. 00:01:16.500 --> 00:01:20.730 align:middle line:84% In most frameworks, we have seen some examples. 00:01:20.730 --> 00:01:27.060 align:middle line:84% And so these rarefaction effects are important. 00:01:27.060 --> 00:01:30.460 align:middle line:84% Like here, because when I remove a partition, 00:01:30.460 --> 00:01:38.940 align:middle line:84% they leave sort of a hole or a rarefaction area 00:01:38.940 --> 00:01:44.370 align:middle line:84% that will spontaneously be filled by the particles 00:01:44.370 --> 00:01:47.790 align:middle line:84% as they realize that the wall is not there anymore. 00:01:47.790 --> 00:01:51.000 align:middle line:84% And so if you remove the wall, you 00:01:51.000 --> 00:01:55.440 align:middle line:84% get a spontaneous and, therefore, irreversible process 00:01:55.440 --> 00:01:59.970 align:middle line:84% that takes you, as represented on the E versus S diagram, 00:01:59.970 --> 00:02:05.110 align:middle line:90% from this state to that state. 00:02:05.110 --> 00:02:10.570 align:middle line:84% And this is an irreversible spontaneous process. 00:02:10.570 --> 00:02:14.440 align:middle line:84% Later on today, we will talk about-- 00:02:14.440 --> 00:02:15.970 align:middle line:90% just in a few minutes-- 00:02:15.970 --> 00:02:20.200 align:middle line:90% about going also up this way. 00:02:20.200 --> 00:02:25.300 align:middle line:84% But for the moment, let me just recall 00:02:25.300 --> 00:02:29.950 align:middle line:84% that in the limit of many particles, 00:02:29.950 --> 00:02:35.740 align:middle line:84% the stable equilibrium state curve for the composite system 00:02:35.740 --> 00:02:40.390 align:middle line:84% divided into two partitions, gets 00:02:40.390 --> 00:02:45.490 align:middle line:84% closer and closer to the stable equilibrium state 00:02:45.490 --> 00:02:49.660 align:middle line:84% curve for the system with the same particles 00:02:49.660 --> 00:02:51.760 align:middle line:90% but no partitions. 00:02:51.760 --> 00:03:00.820 align:middle line:84% And therefore, also the rarefaction effect is smaller, 00:03:00.820 --> 00:03:05.830 align:middle line:84% and it becomes negligible as we have several particles. 00:03:05.830 --> 00:03:12.400 align:middle line:84% In fact-- yes, we have seen some examples in that paper 00:03:12.400 --> 00:03:15.040 align:middle line:84% that I pointed to you where there 00:03:15.040 --> 00:03:17.680 align:middle line:90% are 10 to the fifth particles. 00:03:17.680 --> 00:03:26.520 align:middle line:84% But that is at extremely cold temperatures, I believe. 00:03:26.520 --> 00:03:30.240 align:middle line:84% Whereas at normal temperatures, as soon 00:03:30.240 --> 00:03:34.260 align:middle line:84% as you have maybe 10, 20 particles, 00:03:34.260 --> 00:03:43.540 align:middle line:84% these rarefaction effects become already pretty negligible. 00:03:43.540 --> 00:03:50.870 align:middle line:84% And that's why we can use the many particle-- 00:03:50.870 --> 00:03:53.020 align:middle line:90% the simple system model-- 00:03:53.020 --> 00:03:59.140 align:middle line:84% for most systems, not only macroscopic, but also mesoscopic 00:03:59.140 --> 00:04:01.450 align:middle line:84% and even microscopic as long as you 00:04:01.450 --> 00:04:04.696 align:middle line:90% don't go to just few molecules. 00:04:04.696 --> 00:04:09.790 align:middle line:90% 00:04:09.790 --> 00:04:13.870 align:middle line:84% And the idea is that in that limit 00:04:13.870 --> 00:04:20.769 align:middle line:84% you can neglect essentially the effect of inserting or removing 00:04:20.769 --> 00:04:23.760 align:middle line:84% partitions from a stable equilibrium state. 00:04:23.760 --> 00:04:29.300 align:middle line:90% 00:04:29.300 --> 00:04:33.530 align:middle line:84% Mathematically, that is equivalent to assuming 00:04:33.530 --> 00:04:40.380 align:middle line:84% that the fundamental relation is a homogeneous function 00:04:40.380 --> 00:04:47.050 align:middle line:84% of first degree in all of its variables, meaning this. 00:04:47.050 --> 00:04:53.200 align:middle line:84% And as we said, from that mathematical condition which 00:04:53.200 --> 00:04:56.260 align:middle line:84% represents the physics that we said, 00:04:56.260 --> 00:05:01.870 align:middle line:84% we gain one relation, which is the Euler relation, which 00:05:01.870 --> 00:05:06.350 align:middle line:84% I was unable to prove with the previous viewgraph. 00:05:06.350 --> 00:05:09.870 align:middle line:84% So here is a new viewgraph which does a better job. 00:05:09.870 --> 00:05:13.835 align:middle line:90% 00:05:13.835 --> 00:05:18.060 align:middle line:90% This is the condition. 00:05:18.060 --> 00:05:26.160 align:middle line:84% And in order to prove it, yeah, here is a proof. 00:05:26.160 --> 00:05:29.550 align:middle line:84% You just have to take the partial derivative of both sides 00:05:29.550 --> 00:05:32.430 align:middle line:90% with respect to lambda. 00:05:32.430 --> 00:05:34.750 align:middle line:84% And so on the left-hand side, there is no lambda. 00:05:34.750 --> 00:05:36.990 align:middle line:90% So the partial derivative is 0. 00:05:36.990 --> 00:05:40.090 align:middle line:84% And on the right-hand side, it's lambda times U. 00:05:40.090 --> 00:05:45.990 align:middle line:84% So it's U plus lambda times the derivative 00:05:45.990 --> 00:05:49.260 align:middle line:84% of U with respect to the first variable, which 00:05:49.260 --> 00:05:54.960 align:middle line:84% is the temperature evaluated at the same independent variables, 00:05:54.960 --> 00:06:00.440 align:middle line:84% times the derivative of this argument here 00:06:00.440 --> 00:06:03.030 align:middle line:84% that is S over lambda, which does depend on lambda. 00:06:03.030 --> 00:06:07.910 align:middle line:84% So the derivative is S over minus lambda squared 00:06:07.910 --> 00:06:10.200 align:middle line:84% and so on for the other derivatives. 00:06:10.200 --> 00:06:11.750 align:middle line:90% And here is what you get. 00:06:11.750 --> 00:06:13.920 align:middle line:84% And now, in this expression here, 00:06:13.920 --> 00:06:24.120 align:middle line:84% if you set lambda equal to 1, out comes the Euler relation. 00:06:24.120 --> 00:06:26.850 align:middle line:84% You can also take from this the fact 00:06:26.850 --> 00:06:34.050 align:middle line:84% that the partial derivatives of the fundamental relation 00:06:34.050 --> 00:06:38.710 align:middle line:84% are homogeneous functions of degree 0. 00:06:38.710 --> 00:06:43.510 align:middle line:84% In other words, see, there is no lambda multiplying in front. 00:06:43.510 --> 00:06:47.550 align:middle line:84% So the temperature of a system with given S, V, 00:06:47.550 --> 00:06:50.860 align:middle line:84% and n is equal to the temperature-- of course, 00:06:50.860 --> 00:06:54.150 align:middle line:84% we're talking about stable equilibrium states-- 00:06:54.150 --> 00:06:58.260 align:middle line:84% for a system in which the S, V, and n's are 00:06:58.260 --> 00:07:01.260 align:middle line:90% divided by the same number. 00:07:01.260 --> 00:07:04.540 align:middle line:84% As long as it's the same number, it will be the same temperature. 00:07:04.540 --> 00:07:07.690 align:middle line:90% 00:07:07.690 --> 00:07:13.060 align:middle line:84% So that means small or large, as long as the relative amounts are 00:07:13.060 --> 00:07:15.520 align:middle line:84% equal, you get the same temperature 00:07:15.520 --> 00:07:18.600 align:middle line:84% for the stable equilibrium states. 00:07:18.600 --> 00:07:23.760 align:middle line:90% So that's the Euler relation. 00:07:23.760 --> 00:07:27.000 align:middle line:90% And we have proved that-- 00:07:27.000 --> 00:07:29.700 align:middle line:84% thanks to the Euler relation-- we can define specific 00:07:29.700 --> 00:07:35.410 align:middle line:84% properties, which, for example, is the specific energy-- 00:07:35.410 --> 00:07:38.100 align:middle line:90% is the energy divided by n-- 00:07:38.100 --> 00:07:42.430 align:middle line:90% 00:07:42.430 --> 00:07:46.300 align:middle line:84% And then, we can see how this specific property 00:07:46.300 --> 00:07:50.620 align:middle line:84% is a function of the other specific properties-- 00:07:50.620 --> 00:07:56.240 align:middle line:84% the specific entropy, specific volume, and the mole fractions, 00:07:56.240 --> 00:07:59.450 align:middle line:84% which are the specific amounts of constituents. 00:07:59.450 --> 00:08:03.800 align:middle line:84% And, as we see here, there is a lot of n dependence, 00:08:03.800 --> 00:08:06.220 align:middle line:84% but we proved that the partial derivative 00:08:06.220 --> 00:08:09.850 align:middle line:84% of U with respect to n is equal to minus 1 00:08:09.850 --> 00:08:13.760 align:middle line:84% over n squared times this quantity. 00:08:13.760 --> 00:08:16.570 align:middle line:84% And this quantity, you remember, is what we 00:08:16.570 --> 00:08:20.200 align:middle line:90% called the Euler free energy. 00:08:20.200 --> 00:08:22.390 align:middle line:84% It's the characteristic function. 00:08:22.390 --> 00:08:36.169 align:middle line:84% In fact-- And this is still valid also for small systems. 00:08:36.169 --> 00:08:41.929 align:middle line:84% Here is how we define the Euler free energy. 00:08:41.929 --> 00:08:47.020 align:middle line:84% And here is its differential from which, for example, we 00:08:47.020 --> 00:08:52.960 align:middle line:84% can see that the amount of one constituent 00:08:52.960 --> 00:08:56.050 align:middle line:84% is equal to the partial of the Euler free energy with respect 00:08:56.050 --> 00:08:59.830 align:middle line:84% to the chemical potential of that constituent 00:08:59.830 --> 00:09:03.730 align:middle line:84% when you hold fixed the temperature, the pressure, 00:09:03.730 --> 00:09:06.950 align:middle line:84% and the chemical potentials of all the other constituents. 00:09:06.950 --> 00:09:10.640 align:middle line:84% So if you have a reservoir that keeps temperature, pressure, 00:09:10.640 --> 00:09:13.120 align:middle line:90% and also chemical potentials-- 00:09:13.120 --> 00:09:19.420 align:middle line:84% except this one-- by changing the chemical potential 00:09:19.420 --> 00:09:24.040 align:middle line:84% and measuring the amounts, you can 00:09:24.040 --> 00:09:28.510 align:middle line:90% measure the Euler free energy. 00:09:28.510 --> 00:09:32.980 align:middle line:84% This might be a possible procedure 00:09:32.980 --> 00:09:35.350 align:middle line:84% to measure this for small systems 00:09:35.350 --> 00:09:37.573 align:middle line:90% using molecular dynamics. 00:09:37.573 --> 00:09:40.960 align:middle line:90% 00:09:40.960 --> 00:09:47.260 align:middle line:84% Well, here, what we have done in the previous slide in order 00:09:47.260 --> 00:09:49.240 align:middle line:84% to compute the derivative of energy 00:09:49.240 --> 00:09:50.990 align:middle line:90% with respect to the amounts-- 00:09:50.990 --> 00:09:53.050 align:middle line:84% the derivative of the specific energy 00:09:53.050 --> 00:09:54.910 align:middle line:90% with respect to the amounts-- 00:09:54.910 --> 00:10:00.070 align:middle line:84% at fixed specific entropy volume and mole fractions-- 00:10:00.070 --> 00:10:08.230 align:middle line:84% to find that it is minus 1 over n times the Euler free energy. 00:10:08.230 --> 00:10:12.850 align:middle line:84% We have done it also for these other properties. 00:10:12.850 --> 00:10:18.470 align:middle line:84% And those derivatives turn out to be all equal, 00:10:18.470 --> 00:10:23.050 align:middle line:90% at least for those that are-- 00:10:23.050 --> 00:10:25.120 align:middle line:84% for the specific properties that emerge 00:10:25.120 --> 00:10:32.560 align:middle line:84% from the Legendre transform of the energy version 00:10:32.560 --> 00:10:34.220 align:middle line:90% of the fundamental relation. 00:10:34.220 --> 00:10:39.520 align:middle line:84% They all come out to be Euler free energy divided 00:10:39.520 --> 00:10:44.520 align:middle line:90% by n squared negative. 00:10:44.520 --> 00:10:49.280 align:middle line:84% Whereas for the entropy, it's minus 1 00:10:49.280 --> 00:10:52.570 align:middle line:90% over T of the same result. 00:10:52.570 --> 00:11:00.500 align:middle line:84% So since in the limit of many particles, 00:11:00.500 --> 00:11:05.690 align:middle line:84% there may be also significance here that you are left with an n 00:11:05.690 --> 00:11:10.770 align:middle line:84% at the denominator, which for many particles grows up, 00:11:10.770 --> 00:11:16.170 align:middle line:84% and whereas this is the specific Euler free energy per particle. 00:11:16.170 --> 00:11:19.490 align:middle line:84% So this may be an indication already that this dependence 00:11:19.490 --> 00:11:21.950 align:middle line:90% goes down with n. 00:11:21.950 --> 00:11:27.920 align:middle line:84% In any case, again, these relations 00:11:27.920 --> 00:11:38.140 align:middle line:84% may be ways to measure the Euler free energy for small systems, 00:11:38.140 --> 00:11:38.640 align:middle line:90% of course. 00:11:38.640 --> 00:11:42.440 align:middle line:84% Because in the large system limit, 00:11:42.440 --> 00:11:47.690 align:middle line:90% the Euler free energy goes to 0. 00:11:47.690 --> 00:11:51.910 align:middle line:84% The Euler relation is equivalent to setting Eu equals 0. 00:11:51.910 --> 00:11:55.970 align:middle line:90% 00:11:55.970 --> 00:12:04.740 align:middle line:84% So now, let's go back to this issue of partitioning a system-- 00:12:04.740 --> 00:12:07.660 align:middle line:84% a small system, so a few particles. 00:12:07.660 --> 00:12:17.730 align:middle line:84% And we want to see if there is a cost in dividing 00:12:17.730 --> 00:12:22.260 align:middle line:84% the system, which is in a stable equilibrium state in this one, 00:12:22.260 --> 00:12:28.240 align:middle line:84% and producing another stable equilibrium 00:12:28.240 --> 00:12:35.500 align:middle line:84% state of the system with lambda partitions 00:12:35.500 --> 00:12:39.540 align:middle line:84% in each of which there is the same amount. 00:12:39.540 --> 00:12:46.020 align:middle line:84% It has the same volume and also the same entropy. 00:12:46.020 --> 00:12:49.740 align:middle line:84% And the overall entropy must be the same. 00:12:49.740 --> 00:12:52.440 align:middle line:84% If we do this, we can go from here 00:12:52.440 --> 00:12:55.940 align:middle line:84% to there in a reversible weight process. 00:12:55.940 --> 00:12:58.770 align:middle line:90% 00:12:58.770 --> 00:13:04.620 align:middle line:84% And if we go from partitioned to without partition, 00:13:04.620 --> 00:13:05.680 align:middle line:90% you see we go down. 00:13:05.680 --> 00:13:09.630 align:middle line:84% So we extract the adiabatic availability, essentially, 00:13:09.630 --> 00:13:14.310 align:middle line:84% of that state that is generated the moment we 00:13:14.310 --> 00:13:15.700 align:middle line:90% remove those partitions. 00:13:15.700 --> 00:13:18.240 align:middle line:84% The moment we remove these partitions, 00:13:18.240 --> 00:13:21.270 align:middle line:84% this is not anymore the stable equilibrium 00:13:21.270 --> 00:13:23.440 align:middle line:84% state curve because we've changed the system. 00:13:23.440 --> 00:13:26.250 align:middle line:84% So the stable equilibrium state curve becomes this one. 00:13:26.250 --> 00:13:31.480 align:middle line:84% So yes, we could let thing evolve spontaneously. 00:13:31.480 --> 00:13:33.690 align:middle line:84% So the system will evolve spontaneously 00:13:33.690 --> 00:13:39.550 align:middle line:84% at constant energy, generating entropy by irreversibility. 00:13:39.550 --> 00:13:45.270 align:middle line:84% But if we are a good enough and fast enough-- 00:13:45.270 --> 00:13:48.660 align:middle line:84% faster than irreversibility-- we could 00:13:48.660 --> 00:13:53.010 align:middle line:84% attempt to extract this adiabatic availability 00:13:53.010 --> 00:13:55.890 align:middle line:84% and go all the way down at constant entropy 00:13:55.890 --> 00:13:59.950 align:middle line:84% to reach this stable equilibrium state. 00:13:59.950 --> 00:14:02.540 align:middle line:90% 00:14:02.540 --> 00:14:06.500 align:middle line:84% But this also says that if I want to go from here 00:14:06.500 --> 00:14:12.550 align:middle line:84% and introduce the partitions in a weight process since I cannot 00:14:12.550 --> 00:14:18.220 align:middle line:84% go leftward in the weight process, 00:14:18.220 --> 00:14:22.780 align:middle line:84% the first time I reach the stable equilibrium state 00:14:22.780 --> 00:14:27.110 align:middle line:84% of the system curve of the system with partitions is here. 00:14:27.110 --> 00:14:32.410 align:middle line:90% So this is the first available-- 00:14:32.410 --> 00:14:35.870 align:middle line:84% I mean reachable-- partitioned state. 00:14:35.870 --> 00:14:38.470 align:middle line:90% Others could be up here. 00:14:38.470 --> 00:14:41.030 align:middle line:84% And so as I go from here to there, 00:14:41.030 --> 00:14:45.730 align:middle line:84% I need to spend at least this much work 00:14:45.730 --> 00:14:47.860 align:middle line:84% or energy from the weight in order 00:14:47.860 --> 00:14:50.290 align:middle line:90% to introduce those partitions. 00:14:50.290 --> 00:14:52.480 align:middle line:84% And physically, you can think of the idea 00:14:52.480 --> 00:14:56.140 align:middle line:84% that since there are refraction effects at the wall as you 00:14:56.140 --> 00:14:58.660 align:middle line:84% introduce the partition, the partition 00:14:58.660 --> 00:15:02.800 align:middle line:84% moves the particles away because it has to create that space. 00:15:02.800 --> 00:15:05.190 align:middle line:90% And that is work. 00:15:05.190 --> 00:15:10.840 align:middle line:90% 00:15:10.840 --> 00:15:19.040 align:middle line:84% Of course, the mathematics of the same idea 00:15:19.040 --> 00:15:24.970 align:middle line:84% is that you can compute the work to introduce those lambda 00:15:24.970 --> 00:15:29.350 align:middle line:84% partitions, which is the minimum work to go from 0 to-- 00:15:29.350 --> 00:15:30.960 align:middle line:90% from just 1 to lambda. 00:15:30.960 --> 00:15:33.910 align:middle line:90% 00:15:33.910 --> 00:15:38.080 align:middle line:84% It's the maximum you can extract if you go from lambda to 1. 00:15:38.080 --> 00:15:40.570 align:middle line:84% And it is the difference between the energy 00:15:40.570 --> 00:15:44.710 align:middle line:84% here, which is E lambda, which is 00:15:44.710 --> 00:15:49.430 align:middle line:84% lambda times the energy of each partition. 00:15:49.430 --> 00:15:51.586 align:middle line:90% So it's E1. 00:15:51.586 --> 00:15:54.340 align:middle line:90% E1 is the system with-- 00:15:54.340 --> 00:15:56.440 align:middle line:84% the fundamental relation of the system 00:15:56.440 --> 00:16:01.570 align:middle line:84% without partitions, that you evaluate 00:16:01.570 --> 00:16:09.310 align:middle line:84% at the values of the entropy, volume and amounts that 00:16:09.310 --> 00:16:14.300 align:middle line:90% are present in one partition. 00:16:14.300 --> 00:16:21.620 align:middle line:84% So E lambda minus E, and this is instead E1-- 00:16:21.620 --> 00:16:25.430 align:middle line:84% fundamental relation of the system without partitions 00:16:25.430 --> 00:16:28.790 align:middle line:84% evaluated at the value of the entropy 00:16:28.790 --> 00:16:32.340 align:middle line:84% of the overall system, which is the sum of all the entropies. 00:16:32.340 --> 00:16:36.380 align:middle line:84% It's essentially lambda times the entropy 00:16:36.380 --> 00:16:39.150 align:middle line:90% that is in each partition. 00:16:39.150 --> 00:16:41.220 align:middle line:90% So this is this difference. 00:16:41.220 --> 00:16:45.210 align:middle line:90% 00:16:45.210 --> 00:16:51.260 align:middle line:84% And, for example, we could see how it changes as we go from 00:16:51.260 --> 00:16:56.240 align:middle line:84% lambda to lambda plus 1, or-- which is essentially the same-- 00:16:56.240 --> 00:17:00.290 align:middle line:84% from lambda to lambda minus 1-- so 00:17:00.290 --> 00:17:07.359 align:middle line:84% minimum work to add or remove one partition. 00:17:07.359 --> 00:17:10.990 align:middle line:84% And notice that it's not exactly so obvious 00:17:10.990 --> 00:17:14.619 align:middle line:84% because when I add one lambda, it's not 00:17:14.619 --> 00:17:16.040 align:middle line:90% that I just have to push in. 00:17:16.040 --> 00:17:19.000 align:middle line:84% I have to also rearrange the particles in the other lambdas 00:17:19.000 --> 00:17:23.370 align:middle line:84% so that they are all divided in equal amounts. 00:17:23.370 --> 00:17:26.920 align:middle line:90% 00:17:26.920 --> 00:17:32.210 align:middle line:84% But with that understanding, if you see that this work-- 00:17:32.210 --> 00:17:35.170 align:middle line:84% so that's essentially the work of going to lambda plus 1 00:17:35.170 --> 00:17:39.650 align:middle line:84% minus the work of going to lambda partitions divided by 1, 00:17:39.650 --> 00:17:42.310 align:middle line:84% and this is a fancy way of writing 1. 00:17:42.310 --> 00:17:47.320 align:middle line:84% But to make this ratio look like the partial derivative of W 00:17:47.320 --> 00:17:52.442 align:middle line:84% with respect to lambda and since we have an expression for W, 00:17:52.442 --> 00:17:54.400 align:middle line:84% we can take the partial derivative with respect 00:17:54.400 --> 00:17:55.660 align:middle line:90% to lambda. 00:17:55.660 --> 00:17:59.470 align:middle line:84% Do the math, and out comes that that work 00:17:59.470 --> 00:18:11.570 align:middle line:84% is the Euler free energy of each partition 00:18:11.570 --> 00:18:16.470 align:middle line:90% in the lambda subdivided system. 00:18:16.470 --> 00:18:20.160 align:middle line:84% So that's another physical interpretation 00:18:20.160 --> 00:18:22.500 align:middle line:90% of the Euler free energy. 00:18:22.500 --> 00:18:25.320 align:middle line:90% OK, good. 00:18:25.320 --> 00:18:31.910 align:middle line:84% So now that we have the simple system model, 00:18:31.910 --> 00:18:38.240 align:middle line:84% there are a number of results that you 00:18:38.240 --> 00:18:41.790 align:middle line:84% have seen in previous courses of thermodynamics. 00:18:41.790 --> 00:18:47.900 align:middle line:84% So I will do a very brief review of those 00:18:47.900 --> 00:18:54.350 align:middle line:84% because it also helps brushing up a bit, 00:18:54.350 --> 00:18:59.880 align:middle line:84% considering that the other day you didn't remember 00:18:59.880 --> 00:19:01.980 align:middle line:90% the value of gamma for air. 00:19:01.980 --> 00:19:04.770 align:middle line:90% 00:19:04.770 --> 00:19:11.520 align:middle line:84% So the first idea is that of an extensive property. 00:19:11.520 --> 00:19:15.990 align:middle line:84% And here, notice I keep repeating these titles. 00:19:15.990 --> 00:19:19.800 align:middle line:84% This is now within the simple system model. 00:19:19.800 --> 00:19:22.330 align:middle line:84% What do we mean by an extensive property? 00:19:22.330 --> 00:19:27.160 align:middle line:84% For example, the energy is an extensive property. 00:19:27.160 --> 00:19:32.700 align:middle line:84% This is precisely the mathematical definition 00:19:32.700 --> 00:19:34.000 align:middle line:90% of a simple system model. 00:19:34.000 --> 00:19:37.770 align:middle line:84% Now, the homogeneity of first degree of the function 00:19:37.770 --> 00:19:38.910 align:middle line:90% that represents-- 00:19:38.910 --> 00:19:41.940 align:middle line:90% 00:19:41.940 --> 00:19:43.378 align:middle line:90% that was for the energy. 00:19:43.378 --> 00:19:43.920 align:middle line:90% This is for-- 00:19:43.920 --> 00:19:48.380 align:middle line:90% 00:19:48.380 --> 00:19:52.590 align:middle line:84% Any other function that has that property, we call it extensive. 00:19:52.590 --> 00:19:55.550 align:middle line:84% And you can prove that examples-- 00:19:55.550 --> 00:19:57.840 align:middle line:84% In the list of these extensive properties, 00:19:57.840 --> 00:19:59.760 align:middle line:90% you find energy, of course. 00:19:59.760 --> 00:20:00.810 align:middle line:90% We find entropy. 00:20:00.810 --> 00:20:03.930 align:middle line:84% We find volume, the amounts, the total amount, 00:20:03.930 --> 00:20:06.320 align:middle line:84% which is the sum of the amounts of the various different 00:20:06.320 --> 00:20:08.515 align:middle line:90% constituents, the enthalpy. 00:20:08.515 --> 00:20:12.020 align:middle line:90% 00:20:12.020 --> 00:20:15.470 align:middle line:84% Enthalpy is not an additive property. 00:20:15.470 --> 00:20:18.010 align:middle line:90% 00:20:18.010 --> 00:20:21.340 align:middle line:84% Because if you have two systems that 00:20:21.340 --> 00:20:24.960 align:middle line:90% are at different pressures-- 00:20:24.960 --> 00:20:26.724 align:middle line:90% enthalpy is U plus PV-- 00:20:26.724 --> 00:20:30.360 align:middle line:90% 00:20:30.360 --> 00:20:38.350 align:middle line:84% the P is different, and, therefore, U and V are additive. 00:20:38.350 --> 00:20:40.770 align:middle line:84% But the P that is different doesn't 00:20:40.770 --> 00:20:44.820 align:middle line:84% make that linear combination of U and V additive. 00:20:44.820 --> 00:20:56.340 align:middle line:84% But if you have a simple system in which you can assume that-- 00:20:56.340 --> 00:21:02.460 align:middle line:84% you see these independent properties are all 00:21:02.460 --> 00:21:06.510 align:middle line:84% divided by the same number, we have 00:21:06.510 --> 00:21:10.260 align:middle line:84% said before that the partial derivatives 00:21:10.260 --> 00:21:13.650 align:middle line:84% of the fundamental relation, including temperature 00:21:13.650 --> 00:21:17.760 align:middle line:84% and pressure, do not change if you 00:21:17.760 --> 00:21:21.310 align:middle line:84% change with this particular change of variables. 00:21:21.310 --> 00:21:27.000 align:middle line:84% And that is why you find the enthalpy here in this list. 00:21:27.000 --> 00:21:29.310 align:middle line:90% It is extensive. 00:21:29.310 --> 00:21:33.790 align:middle line:84% Also, the Gibbs free energy is missing in this list, 00:21:33.790 --> 00:21:38.690 align:middle line:84% but it is another example, as well as the specific heat 00:21:38.690 --> 00:21:41.095 align:middle line:90% capacities and the mass. 00:21:41.095 --> 00:21:44.150 align:middle line:90% 00:21:44.150 --> 00:21:50.630 align:middle line:84% Now, if I take any two extensive properties and take the ratio, 00:21:50.630 --> 00:21:58.010 align:middle line:90% I obtain specific properties. 00:21:58.010 --> 00:21:59.540 align:middle line:84% So, for example, the first line here 00:21:59.540 --> 00:22:04.010 align:middle line:84% represents the specific properties when I divide. 00:22:04.010 --> 00:22:09.360 align:middle line:84% So these are the molar specific processes because I divide by n. 00:22:09.360 --> 00:22:12.360 align:middle line:84% Or I could have the mass specific properties 00:22:12.360 --> 00:22:13.805 align:middle line:90% when I divide by m. 00:22:13.805 --> 00:22:16.470 align:middle line:90% 00:22:16.470 --> 00:22:19.990 align:middle line:84% Or I could have the volume specific properties when 00:22:19.990 --> 00:22:25.150 align:middle line:84% I divide by V. For example, the mass divided by the volume 00:22:25.150 --> 00:22:27.820 align:middle line:90% is the density. 00:22:27.820 --> 00:22:32.740 align:middle line:84% Of course, they are related between one another. 00:22:32.740 --> 00:22:38.020 align:middle line:84% And sometimes here I try to differentiate 00:22:38.020 --> 00:22:44.950 align:middle line:84% by using a different symbol like the star here is there just 00:22:44.950 --> 00:22:49.730 align:middle line:84% in case you need to make sure that you don't get confused. 00:22:49.730 --> 00:22:53.230 align:middle line:84% But in most of the treatments, once you 00:22:53.230 --> 00:22:56.110 align:middle line:84% set the definition and the context 00:22:56.110 --> 00:22:57.880 align:middle line:84% makes it clear that you're working 00:22:57.880 --> 00:23:03.880 align:middle line:84% with a mass specific properties, you just use a lowercase letter 00:23:03.880 --> 00:23:08.440 align:middle line:84% without the star, and you let the context 00:23:08.440 --> 00:23:12.080 align:middle line:90% define what you're doing. 00:23:12.080 --> 00:23:18.720 align:middle line:84% For example, the amounts if you divide them by n, 00:23:18.720 --> 00:23:22.040 align:middle line:90% you get the molar fraction. 00:23:22.040 --> 00:23:25.945 align:middle line:84% I'll use the letter y sub i, but not everybody uses that letters. 00:23:25.945 --> 00:23:28.840 align:middle line:90% 00:23:28.840 --> 00:23:36.350 align:middle line:84% If I have the mass-- so I take take the number of particles i 00:23:36.350 --> 00:23:40.040 align:middle line:84% multiplied by their molecular weight-- 00:23:40.040 --> 00:23:44.540 align:middle line:84% capital M i-- of particles i and divide 00:23:44.540 --> 00:23:50.600 align:middle line:84% by the total mass, that becomes the mass fraction 00:23:50.600 --> 00:23:52.186 align:middle line:90% of that constituent. 00:23:52.186 --> 00:23:55.490 align:middle line:84% If I divide the amount of constituents 00:23:55.490 --> 00:23:59.480 align:middle line:84% by the volume what I get is the concentration 00:23:59.480 --> 00:24:04.010 align:middle line:84% of those constituents, which in the chemistry business 00:24:04.010 --> 00:24:07.030 align:middle line:84% sometimes is also denoted this way. 00:24:07.030 --> 00:24:10.375 align:middle line:84% This is the volume concentration of particles of type i. 00:24:10.375 --> 00:24:13.170 align:middle line:90% 00:24:13.170 --> 00:24:20.880 align:middle line:84% So notation varies depending on the various fields in which you 00:24:20.880 --> 00:24:22.630 align:middle line:90% apply thermodynamics. 00:24:22.630 --> 00:24:28.220 align:middle line:90% 00:24:28.220 --> 00:24:33.770 align:middle line:84% Remember that we define extensive property 00:24:33.770 --> 00:24:37.260 align:middle line:84% when for the properties there is a relation like this, 00:24:37.260 --> 00:24:39.260 align:middle line:84% but there is a lambda here multiplying. 00:24:39.260 --> 00:24:45.580 align:middle line:84% If there is no lambda like as we have said for the partial 00:24:45.580 --> 00:24:47.840 align:middle line:84% derivatives of the fundamental relations-- so for T, 00:24:47.840 --> 00:24:55.180 align:middle line:84% P and the chemical potentials but also for all of the specific 00:24:55.180 --> 00:24:58.540 align:middle line:84% properties that we just defined-- 00:24:58.540 --> 00:25:04.330 align:middle line:84% you can prove that there is no lambda there, 00:25:04.330 --> 00:25:05.940 align:middle line:90% and this relation holds. 00:25:05.940 --> 00:25:09.110 align:middle line:84% We call these properties intensive properties. 00:25:09.110 --> 00:25:11.690 align:middle line:90% 00:25:11.690 --> 00:25:14.060 align:middle line:84% Also here, I have to warn you that I 00:25:14.060 --> 00:25:19.970 align:middle line:84% may be using a definition which might differ from what you've 00:25:19.970 --> 00:25:29.940 align:middle line:84% seen elsewhere, so just be careful when you see the word 00:25:29.940 --> 00:25:32.070 align:middle line:90% intensive property. 00:25:32.070 --> 00:25:37.380 align:middle line:84% Check how the author defines that intensive. 00:25:37.380 --> 00:25:40.260 align:middle line:90% This is our definition. 00:25:40.260 --> 00:25:41.590 align:middle line:90% There are many properties. 00:25:41.590 --> 00:25:46.800 align:middle line:84% So we said not just T, P and chemical potentials but also 00:25:46.800 --> 00:25:48.750 align:middle line:90% all the specific properties. 00:25:48.750 --> 00:25:50.400 align:middle line:84% All right, so if you take the list 00:25:50.400 --> 00:25:56.280 align:middle line:84% of the values of all these intensive properties, which 00:25:56.280 --> 00:26:00.180 align:middle line:84% is usually an infinite number of properties, 00:26:00.180 --> 00:26:04.620 align:middle line:84% but most of the times, there is a smaller set 00:26:04.620 --> 00:26:07.170 align:middle line:90% of independent properties. 00:26:07.170 --> 00:26:10.710 align:middle line:84% The list of all properties of this kind 00:26:10.710 --> 00:26:15.240 align:middle line:84% for a stable equilibrium state is called the intensive state. 00:26:15.240 --> 00:26:21.390 align:middle line:84% And you can show that if you have the intensive state 00:26:21.390 --> 00:26:27.240 align:middle line:84% and you also have the value for one extensive property, then 00:26:27.240 --> 00:26:29.225 align:middle line:84% you have completely fixed the state. 00:26:29.225 --> 00:26:31.750 align:middle line:90% 00:26:31.750 --> 00:26:35.350 align:middle line:84% So you must-- in order to get the state-- 00:26:35.350 --> 00:26:37.330 align:middle line:84% we're talking about stable equilibrium states 00:26:37.330 --> 00:26:39.130 align:middle line:90% within the simple system model-- 00:26:39.130 --> 00:26:44.290 align:middle line:84% if you want the state, you can start from the intensive state 00:26:44.290 --> 00:26:48.030 align:middle line:90% and add an extensive property. 00:26:48.030 --> 00:26:58.270 align:middle line:90% 00:26:58.270 --> 00:27:05.080 align:middle line:84% The other idea that is a very important consequence 00:27:05.080 --> 00:27:08.020 align:middle line:84% of the simple system model is that the fact 00:27:08.020 --> 00:27:09.940 align:middle line:84% that you can add and remove partitions 00:27:09.940 --> 00:27:15.850 align:middle line:84% allows you to describe a stable equilibrium states in which you 00:27:15.850 --> 00:27:21.250 align:middle line:84% have the coexistence of parts of the system that are 00:27:21.250 --> 00:27:23.200 align:middle line:90% in different intensive states. 00:27:23.200 --> 00:27:35.490 align:middle line:84% For example, think of a pot with water inside, 00:27:35.490 --> 00:27:38.680 align:middle line:90% and it's on the stove. 00:27:38.680 --> 00:27:43.660 align:middle line:84% And so you have heating below, and there is boiling. 00:27:43.660 --> 00:27:49.000 align:middle line:84% And as you know boiling will produce bubbles. 00:27:49.000 --> 00:27:55.523 align:middle line:84% These bubbles will go to the surface and may generate drops. 00:27:55.523 --> 00:27:58.940 align:middle line:90% 00:27:58.940 --> 00:28:03.500 align:middle line:84% If the boiling is sufficiently vigorous, it generates drops. 00:28:03.500 --> 00:28:05.700 align:middle line:90% So this is a-- 00:28:05.700 --> 00:28:08.730 align:middle line:90% 00:28:08.730 --> 00:28:10.810 align:middle line:84% it's not really a stable equilibrium state, 00:28:10.810 --> 00:28:14.450 align:middle line:84% but you can approximate the properties with those 00:28:14.450 --> 00:28:20.340 align:middle line:84% of a stable equilibrium state at least after a while 00:28:20.340 --> 00:28:22.960 align:middle line:84% that you have stopped the heating. 00:28:22.960 --> 00:28:24.270 align:middle line:90% So you don't have-- 00:28:24.270 --> 00:28:28.720 align:middle line:84% as you get a sort of uniform temperature, uniform pressure. 00:28:28.720 --> 00:28:32.580 align:middle line:84% But you still have some drops and bubbles-- 00:28:32.580 --> 00:28:36.240 align:middle line:84% drops in the vapor and bubbles in the liquid. 00:28:36.240 --> 00:28:40.960 align:middle line:84% So this is a complicated system to describe. 00:28:40.960 --> 00:28:44.700 align:middle line:84% But conceptually, the simple system model 00:28:44.700 --> 00:28:47.460 align:middle line:90% allows you to do the following. 00:28:47.460 --> 00:28:54.330 align:middle line:84% I can imagine to take another pot-- 00:28:54.330 --> 00:28:57.210 align:middle line:90% my model pot, so to speak. 00:28:57.210 --> 00:28:59.880 align:middle line:84% It's a mental image of that thing. 00:28:59.880 --> 00:29:04.320 align:middle line:84% And let me take a physical partition 00:29:04.320 --> 00:29:09.410 align:middle line:84% that runs at the interface between liquid and vapor. 00:29:09.410 --> 00:29:12.450 align:middle line:90% 00:29:12.450 --> 00:29:20.330 align:middle line:84% And, as you know, inserting that partition doesn't cost. 00:29:20.330 --> 00:29:23.450 align:middle line:84% So you already see that that doesn't cost 00:29:23.450 --> 00:29:26.420 align:middle line:84% means you are neglecting-- the simple system 00:29:26.420 --> 00:29:30.620 align:middle line:84% model is neglecting also the surface tension effects. 00:29:30.620 --> 00:29:37.070 align:middle line:84% Because that interface does matter, 00:29:37.070 --> 00:29:38.790 align:middle line:84% but not within the simple system model. 00:29:38.790 --> 00:29:45.740 align:middle line:84% So I can take this and think of putting it here. 00:29:45.740 --> 00:29:51.710 align:middle line:84% And then I take another bubble and-- 00:29:51.710 --> 00:29:55.540 align:middle line:90% sorry, let me put them up here. 00:29:55.540 --> 00:30:01.910 align:middle line:84% So I take all the bubbles, and I put them there. 00:30:01.910 --> 00:30:04.160 align:middle line:84% And I also need to change the shape a bit so 00:30:04.160 --> 00:30:06.300 align:middle line:90% that it fills the same space. 00:30:06.300 --> 00:30:10.580 align:middle line:84% And also here, this I can take larger areas-- 00:30:10.580 --> 00:30:12.380 align:middle line:84% so essentially, all the continuous area 00:30:12.380 --> 00:30:15.000 align:middle line:84% that doesn't contain drops-- and put it there. 00:30:15.000 --> 00:30:17.420 align:middle line:84% So eventually, what I managed to do 00:30:17.420 --> 00:30:23.455 align:middle line:84% is to put all the vapor here and all the liquid here. 00:30:23.455 --> 00:30:27.030 align:middle line:90% 00:30:27.030 --> 00:30:31.930 align:middle line:84% Usually we use these letters to represent that. 00:30:31.930 --> 00:30:38.600 align:middle line:84% So I've generated two regions in which 00:30:38.600 --> 00:30:41.810 align:middle line:90% there are no inhomogeneities. 00:30:41.810 --> 00:30:45.410 align:middle line:84% Whereas here, the inhomogeneity stems from the fact 00:30:45.410 --> 00:30:49.950 align:middle line:84% that, as you know, there is a huge difference in density, 00:30:49.950 --> 00:30:54.710 align:middle line:84% for example, which is one of the intensive properties-- 00:30:54.710 --> 00:30:58.100 align:middle line:84% like a factor of a thousand between the density 00:30:58.100 --> 00:31:00.450 align:middle line:84% of the liquid and the density of the vapor. 00:31:00.450 --> 00:31:10.430 align:middle line:84% So this is an inhomogeneous or heterogeneous situation 00:31:10.430 --> 00:31:16.460 align:middle line:84% that can be represented, certainly not 00:31:16.460 --> 00:31:21.630 align:middle line:90% in a uniform intensive state. 00:31:21.630 --> 00:31:24.740 align:middle line:84% But I can group the various intensive states 00:31:24.740 --> 00:31:28.250 align:middle line:84% that are present in my multiple phase state 00:31:28.250 --> 00:31:33.470 align:middle line:84% into just areas that we call phases 00:31:33.470 --> 00:31:35.850 align:middle line:90% that are locally homogeneous. 00:31:35.850 --> 00:31:48.060 align:middle line:84% So the f phase is the collection of all the parts of my system 00:31:48.060 --> 00:31:51.000 align:middle line:84% that have the same intensive state, 00:31:51.000 --> 00:31:55.110 align:middle line:90% like the liquid, high density. 00:31:55.110 --> 00:32:01.470 align:middle line:84% And this g is the collection of all the other parts that share 00:32:01.470 --> 00:32:04.650 align:middle line:90% this other intensive state. 00:32:04.650 --> 00:32:08.220 align:middle line:90% And if you have a triple point-- 00:32:08.220 --> 00:32:12.270 align:middle line:84% suppose you also have ice here because we 00:32:12.270 --> 00:32:16.125 align:middle line:90% are at the proper pressure. 00:32:16.125 --> 00:32:24.230 align:middle line:84% If I have also ice, I need also to produce a third phase. 00:32:24.230 --> 00:32:30.264 align:middle line:90% 00:32:30.264 --> 00:32:32.365 align:middle line:84% So that defines the idea of phase. 00:32:32.365 --> 00:32:35.170 align:middle line:90% 00:32:35.170 --> 00:32:43.730 align:middle line:84% And then, as you know, there is this famous Gibbs phase rule. 00:32:43.730 --> 00:32:46.810 align:middle line:90% And here is how it follows. 00:32:46.810 --> 00:32:54.430 align:middle line:90% 00:32:54.430 --> 00:33:01.620 align:middle line:84% All right, one way of looking at it is-- 00:33:01.620 --> 00:33:04.710 align:middle line:90% let's see what happens-- 00:33:04.710 --> 00:33:07.560 align:middle line:84% Let's choose this set of variables 00:33:07.560 --> 00:33:20.460 align:middle line:84% that are the variables that describe this equilibrium state, 00:33:20.460 --> 00:33:23.190 align:middle line:84% temperature and pressure I select them 00:33:23.190 --> 00:33:29.060 align:middle line:84% because I assume that all the various parts are in-- 00:33:29.060 --> 00:33:32.620 align:middle line:84% the various phases must be in mutual equilibrium. 00:33:32.620 --> 00:33:35.130 align:middle line:84% Therefore, they must be at the same temperature 00:33:35.130 --> 00:33:41.640 align:middle line:84% for the overall system to be in a stable equilibrium state. 00:33:41.640 --> 00:33:46.050 align:middle line:84% Because, as we said, we consider the overall system 00:33:46.050 --> 00:33:52.860 align:middle line:84% as a composite of the phases, and they 00:33:52.860 --> 00:33:54.570 align:middle line:90% are in mutual equilibrium. 00:33:54.570 --> 00:33:56.430 align:middle line:84% So the conditions for mutual equilibrium 00:33:56.430 --> 00:34:01.470 align:middle line:84% are equality of pressure because they can exchange volume, 00:34:01.470 --> 00:34:07.580 align:middle line:84% equality of temperature because they can exchange energy, 00:34:07.580 --> 00:34:09.806 align:middle line:84% and equality of chemical potentials. 00:34:09.806 --> 00:34:13.860 align:middle line:90% 00:34:13.860 --> 00:34:18.650 align:middle line:84% The intensive states that are present here-- 00:34:18.650 --> 00:34:21.159 align:middle line:90% 00:34:21.159 --> 00:34:24.929 align:middle line:84% the number of independent properties that define 00:34:24.929 --> 00:34:28.370 align:middle line:90% intensive state is this much. 00:34:28.370 --> 00:34:45.300 align:middle line:84% It's T, p, plus the composition of each of the phases because-- 00:34:45.300 --> 00:34:48.810 align:middle line:84% yeah, here I consider a simple situation in which 00:34:48.810 --> 00:34:50.650 align:middle line:90% the substance is only water. 00:34:50.650 --> 00:34:53.310 align:middle line:84% But you could have also a mixture. 00:34:53.310 --> 00:34:59.320 align:middle line:84% We will do that as we approach the second part of the course. 00:34:59.320 --> 00:35:01.670 align:middle line:84% This is very useful for that part. 00:35:01.670 --> 00:35:05.595 align:middle line:90% 00:35:05.595 --> 00:35:12.230 align:middle line:90% So these are the base variables. 00:35:12.230 --> 00:35:18.160 align:middle line:84% However, they are not all truly independent 00:35:18.160 --> 00:35:24.630 align:middle line:84% because, first of all, the mole fractions by obvious definition 00:35:24.630 --> 00:35:27.300 align:middle line:90% must sum up to 1. 00:35:27.300 --> 00:35:30.930 align:middle line:90% So for each of the phases-- 00:35:30.930 --> 00:35:35.100 align:middle line:84% for example, for the summation here 00:35:35.100 --> 00:35:40.560 align:middle line:84% I only have r minus 1 mole fractions that 00:35:40.560 --> 00:35:46.650 align:middle line:84% are independent because the last one can 00:35:46.650 --> 00:35:51.770 align:middle line:84% be computed as 1 minus the summation of the others. 00:35:51.770 --> 00:36:01.040 align:middle line:84% Plus, we also have the fact that the chemical potentials 00:36:01.040 --> 00:36:01.780 align:middle line:90% must be equal. 00:36:01.780 --> 00:36:04.650 align:middle line:90% 00:36:04.650 --> 00:36:07.980 align:middle line:84% So I need to have equality of chemical potentials 00:36:07.980 --> 00:36:12.240 align:middle line:84% between the phases for each of the constituents 00:36:12.240 --> 00:36:13.420 align:middle line:90% that can be exchanged. 00:36:13.420 --> 00:36:16.710 align:middle line:84% And here, there are no membranes or things 00:36:16.710 --> 00:36:21.850 align:middle line:84% that disallow the exchange of types of particles. 00:36:21.850 --> 00:36:23.940 align:middle line:90% They can all be exchanged. 00:36:23.940 --> 00:36:31.470 align:middle line:84% And therefore, I have to satisfy the chemical potential equality 00:36:31.470 --> 00:36:37.120 align:middle line:84% for mutual stable equilibrium for each type of constituents. 00:36:37.120 --> 00:36:41.420 align:middle line:90% 00:36:41.420 --> 00:36:43.460 align:middle line:84% You see, actually here, the reason 00:36:43.460 --> 00:36:51.950 align:middle line:84% why I have chosen as candidates for independent variables-- 00:36:51.950 --> 00:36:55.550 align:middle line:84% temperature, pressure, and the mole fractions-- 00:36:55.550 --> 00:36:59.780 align:middle line:84% because we proved that the chemical potentials are 00:36:59.780 --> 00:37:01.985 align:middle line:90% functions of this T, P, and y. 00:37:01.985 --> 00:37:06.090 align:middle line:90% 00:37:06.090 --> 00:37:09.660 align:middle line:84% And as we know, from the chemical 00:37:09.660 --> 00:37:12.600 align:middle line:90% potentials we can compute-- 00:37:12.600 --> 00:37:17.150 align:middle line:84% we will see that we can compute all the properties-- 00:37:17.150 --> 00:37:22.380 align:middle line:84% at least, yeah, all the properties-- 00:37:22.380 --> 00:37:24.890 align:middle line:90% 00:37:24.890 --> 00:37:29.480 align:middle line:84% because we proved that the chemical potentials are 00:37:29.480 --> 00:37:32.210 align:middle line:84% derivatives of the Gibbs free energy, which 00:37:32.210 --> 00:37:34.920 align:middle line:84% is one of the characteristic functions, 00:37:34.920 --> 00:37:36.940 align:middle line:84% Legendre transforms of the fundamental relation. 00:37:36.940 --> 00:37:39.620 align:middle line:90% 00:37:39.620 --> 00:37:43.320 align:middle line:84% All right, so now it's just a matter of counting. 00:37:43.320 --> 00:37:46.620 align:middle line:84% How many do I-- did I start with? 00:37:46.620 --> 00:37:48.830 align:middle line:90% So it's q times r-- 00:37:48.830 --> 00:37:54.230 align:middle line:84% is the mole fractions-- plus 2, temperature and pressure. 00:37:54.230 --> 00:37:59.120 align:middle line:84% Then I have to subtract q relations because 00:37:59.120 --> 00:38:00.800 align:middle line:90% of this condition here. 00:38:00.800 --> 00:38:06.120 align:middle line:84% And then, I have to subtract as many equations I have here. 00:38:06.120 --> 00:38:08.250 align:middle line:90% Now, the equal sign-- 00:38:08.250 --> 00:38:10.170 align:middle line:84% each equal sign counts for an equation. 00:38:10.170 --> 00:38:12.420 align:middle line:90% So I have q phases. 00:38:12.420 --> 00:38:14.040 align:middle line:90% So it's q minus 1. 00:38:14.040 --> 00:38:19.280 align:middle line:84% And I have r lines here because I have one line like this 00:38:19.280 --> 00:38:21.330 align:middle line:90% for each type of constituent. 00:38:21.330 --> 00:38:24.440 align:middle line:84% So it's q minus 1 times r And then, 00:38:24.440 --> 00:38:28.100 align:middle line:84% so if you subtract from that 2 plus rq-- 00:38:28.100 --> 00:38:33.300 align:middle line:84% subtract q and subtract q minus 1 times r-- 00:38:33.300 --> 00:38:39.530 align:middle line:84% you are left with this r plus 2 minus q which 00:38:39.530 --> 00:38:41.450 align:middle line:90% is the so-called variance. 00:38:41.450 --> 00:38:43.640 align:middle line:84% It's the real number of independently 00:38:43.640 --> 00:38:45.780 align:middle line:90% variable intensive properties. 00:38:45.780 --> 00:38:49.230 align:middle line:90% 00:38:49.230 --> 00:38:53.500 align:middle line:84% And this is assuming that we have no chemical reactions. 00:38:53.500 --> 00:38:57.440 align:middle line:84% If we also have chemical reactions, 00:38:57.440 --> 00:39:04.880 align:middle line:84% we will see that there is one additional equation, which 00:39:04.880 --> 00:39:07.460 align:middle line:84% we will call the chemical equilibrium 00:39:07.460 --> 00:39:14.850 align:middle line:84% condition for each independent reaction in play in your system. 00:39:14.850 --> 00:39:18.380 align:middle line:84% And so you will have to subtract also-- let's call it 00:39:18.380 --> 00:39:21.410 align:middle line:84% z, the number of independent chemical reactions. 00:39:21.410 --> 00:39:28.960 align:middle line:84% We have to subtract z to get the actual variance. 00:39:28.960 --> 00:39:34.673 align:middle line:84% For example, we take a pure substance like water. 00:39:34.673 --> 00:39:37.810 align:middle line:90% 00:39:37.810 --> 00:39:42.060 align:middle line:84% So pure substance means only one mole fraction. 00:39:42.060 --> 00:39:45.670 align:middle line:84% And, of course, one mole fraction is equal to 1. 00:39:45.670 --> 00:39:48.480 align:middle line:90% That's 100% of that substance. 00:39:48.480 --> 00:39:50.060 align:middle line:90% So that's not really a variable. 00:39:50.060 --> 00:39:53.670 align:middle line:90% 00:39:53.670 --> 00:39:58.200 align:middle line:84% The Gibbs phase rule says that the independent variables 00:39:58.200 --> 00:39:59.390 align:middle line:90% are 3 minus q. 00:39:59.390 --> 00:40:07.130 align:middle line:90% 00:40:07.130 --> 00:40:12.340 align:middle line:84% So, for example, if I have a single phase-- 00:40:12.340 --> 00:40:15.410 align:middle line:84% now, single phase means that it's all water, 00:40:15.410 --> 00:40:18.070 align:middle line:90% for example, homogeneous. 00:40:18.070 --> 00:40:23.630 align:middle line:84% It means that at every point in my stable equilibrium state, 00:40:23.630 --> 00:40:27.260 align:middle line:84% no matter where I go and make my little partition, 00:40:27.260 --> 00:40:30.340 align:middle line:84% I will find an intensive state that 00:40:30.340 --> 00:40:34.720 align:middle line:90% is equal to everywhere else. 00:40:34.720 --> 00:40:35.890 align:middle line:90% So it's homogeneous. 00:40:35.890 --> 00:40:38.020 align:middle line:90% Single phase is homogeneous-- 00:40:38.020 --> 00:40:42.660 align:middle line:90% same intensive state everywhere. 00:40:42.660 --> 00:40:48.640 align:middle line:84% So with q equals 1, one phase, the variance is 3 minus 1. 00:40:48.640 --> 00:40:50.580 align:middle line:90% So it's 2. 00:40:50.580 --> 00:40:59.670 align:middle line:84% That means that if I have a liquid state, 00:40:59.670 --> 00:41:03.660 align:middle line:84% and I want to change that stable equilibrium 00:41:03.660 --> 00:41:06.810 align:middle line:84% state to a neighboring stable equilibrium state, 00:41:06.810 --> 00:41:10.950 align:middle line:90% I can move in two directions. 00:41:10.950 --> 00:41:13.270 align:middle line:90% I can change two variables. 00:41:13.270 --> 00:41:16.260 align:middle line:84% So I can change, for example, the temperature and the pressure 00:41:16.260 --> 00:41:17.470 align:middle line:90% independently. 00:41:17.470 --> 00:41:20.250 align:middle line:90% 00:41:20.250 --> 00:41:26.140 align:middle line:90% If I have two phases like this-- 00:41:26.140 --> 00:41:28.240 align:middle line:90% well, without the ice-- 00:41:28.240 --> 00:41:32.530 align:middle line:84% and I want to move to another neighboring stable equilibrium 00:41:32.530 --> 00:41:34.630 align:middle line:84% state with the same two phases, so, 00:41:34.630 --> 00:41:37.210 align:middle line:90% for example, liquid and vapor. 00:41:37.210 --> 00:41:40.520 align:middle line:84% I want to move to an adjacent stable equilibrium state, 00:41:40.520 --> 00:41:43.990 align:middle line:90% again, with liquid and vapor. 00:41:43.990 --> 00:41:47.950 align:middle line:84% I cannot do it by arbitrarily changing both the temperature 00:41:47.950 --> 00:41:48.680 align:middle line:90% and the pressure. 00:41:48.680 --> 00:41:52.030 align:middle line:84% I can change only one, and the other must be tuned-- 00:41:52.030 --> 00:41:58.840 align:middle line:84% the change of the other must be tuned in a proper way 00:41:58.840 --> 00:42:01.690 align:middle line:84% because I need to follow the condition imposed 00:42:01.690 --> 00:42:09.660 align:middle line:84% by the chemical potential equality, which is written here. 00:42:09.660 --> 00:42:14.650 align:middle line:84% I need that chemical potential equality to remain satisfied. 00:42:14.650 --> 00:42:21.390 align:middle line:84% And, therefore, if I change T, that imposes a change in p. 00:42:21.390 --> 00:42:24.960 align:middle line:84% And, in fact, this relation between temperature and pressure 00:42:24.960 --> 00:42:29.650 align:middle line:84% for two-phase states is what we call the saturation relation. 00:42:29.650 --> 00:42:33.390 align:middle line:84% Pressure and temperature are related by saturation. 00:42:33.390 --> 00:42:41.540 align:middle line:84% If I have three phases, and I want to move from a stable 00:42:41.540 --> 00:42:45.260 align:middle line:84% equilibrium state with those three phases to another stable 00:42:45.260 --> 00:42:51.540 align:middle line:84% equilibrium state with the same three phases and different 00:42:51.540 --> 00:42:54.350 align:middle line:84% temperatures or pressure, I cannot. 00:42:54.350 --> 00:42:57.110 align:middle line:90% 00:42:57.110 --> 00:42:59.180 align:middle line:90% Because the variance is to 0. 00:42:59.180 --> 00:43:01.595 align:middle line:84% There are no neighboring states with different values 00:43:01.595 --> 00:43:02.095 align:middle line:90% of T and p. 00:43:02.095 --> 00:43:04.820 align:middle line:90% 00:43:04.820 --> 00:43:09.240 align:middle line:84% Because the equality of the chemical potential here, 00:43:09.240 --> 00:43:13.460 align:middle line:84% see I have two equalities and the two variables are T and p. 00:43:13.460 --> 00:43:16.550 align:middle line:84% So that fixes uniquely the value of T and p 00:43:16.550 --> 00:43:20.150 align:middle line:84% at which you can have those three phases. 00:43:20.150 --> 00:43:22.970 align:middle line:90% That's the triple point. 00:43:22.970 --> 00:43:26.480 align:middle line:84% This is better represented, for example-- 00:43:26.480 --> 00:43:29.305 align:middle line:84% this is the pressure temperature diagram for water. 00:43:29.305 --> 00:43:33.190 align:middle line:90% 00:43:33.190 --> 00:43:34.840 align:middle line:90% So if we have-- 00:43:34.840 --> 00:43:40.820 align:middle line:84% in this area-- this is the vapor area. 00:43:40.820 --> 00:43:45.890 align:middle line:84% And if we have some vapor in this state, 00:43:45.890 --> 00:43:51.080 align:middle line:84% I could move to adjacent conditions in which I still 00:43:51.080 --> 00:43:56.030 align:middle line:84% have vapor, but different pressure and temperature. 00:43:56.030 --> 00:43:58.150 align:middle line:84% And I can move in both directions here. 00:43:58.150 --> 00:44:01.610 align:middle line:84% So I have two directions in which I can move. 00:44:01.610 --> 00:44:03.450 align:middle line:90% That's the meaning of variance. 00:44:03.450 --> 00:44:06.390 align:middle line:90% Variance equals 2. 00:44:06.390 --> 00:44:11.780 align:middle line:84% Along here are the line of coexistence of liquid. 00:44:11.780 --> 00:44:13.940 align:middle line:84% So inside the liquid also the variance 00:44:13.940 --> 00:44:17.130 align:middle line:90% is 2 and also inside the solid. 00:44:17.130 --> 00:44:20.410 align:middle line:84% These are the homogeneous single phase states. 00:44:20.410 --> 00:44:25.510 align:middle line:84% Along this curve, which is represented by the P saturation 00:44:25.510 --> 00:44:29.590 align:middle line:84% relation between pressure and temperature, 00:44:29.590 --> 00:44:33.070 align:middle line:84% we have the coexistence of liquid and vapor. 00:44:33.070 --> 00:44:35.530 align:middle line:84% And as you see, the variance is 1. 00:44:35.530 --> 00:44:39.250 align:middle line:84% If I want to move to a neighboring equilibrium 00:44:39.250 --> 00:44:41.020 align:middle line:84% state with different pressure, I also 00:44:41.020 --> 00:44:43.450 align:middle line:84% have to adjust the change in temperature so as 00:44:43.450 --> 00:44:49.350 align:middle line:84% to follow this curve up to the critical point. 00:44:49.350 --> 00:44:52.980 align:middle line:84% Above that, there is no distinction between the liquid 00:44:52.980 --> 00:44:54.220 align:middle line:90% and the vapor phase. 00:44:54.220 --> 00:44:57.494 align:middle line:84% And therefore, it is possible, as you know, 00:44:57.494 --> 00:45:02.520 align:middle line:84% to go all the way from liquid to vapor and vice 00:45:02.520 --> 00:45:12.590 align:middle line:84% versa by going this path in which there is no sharp-- 00:45:12.590 --> 00:45:16.280 align:middle line:84% there is no two-phase conditions. 00:45:16.280 --> 00:45:23.180 align:middle line:84% So the liquid becomes smoothly vapor and vice versa 00:45:23.180 --> 00:45:26.360 align:middle line:90% without phase separation. 00:45:26.360 --> 00:45:32.850 align:middle line:84% But if you go from here to there at a certain stage, 00:45:32.850 --> 00:45:37.230 align:middle line:84% like we do when we boil, we boil at 00:45:37.230 --> 00:45:40.700 align:middle line:90% constant atmospheric pressure. 00:45:40.700 --> 00:45:43.240 align:middle line:84% So I think-- let's see where is it? 00:45:43.240 --> 00:45:47.910 align:middle line:84% There should be one somewhere for-- 00:45:47.910 --> 00:45:51.010 align:middle line:84% but one bar is atmospheric pressure more or less. 00:45:51.010 --> 00:45:58.650 align:middle line:84% So if I go from here, and then I heat up at constant pressure, 00:45:58.650 --> 00:46:00.760 align:middle line:84% I move to the right in the diagram. 00:46:00.760 --> 00:46:06.050 align:middle line:84% At a certain stage, I'll reach the two-phase condition in which 00:46:06.050 --> 00:46:08.340 align:middle line:90% the water begins to boil. 00:46:08.340 --> 00:46:13.080 align:middle line:84% And, as you know, the temperature 00:46:13.080 --> 00:46:17.340 align:middle line:84% gets stuck to that value until it's all boiled. 00:46:17.340 --> 00:46:20.580 align:middle line:84% And then, you can still heat up, and the temperature 00:46:20.580 --> 00:46:22.200 align:middle line:90% will begin to increase. 00:46:22.200 --> 00:46:28.770 align:middle line:84% So here, essentially when you have the two phases, 00:46:28.770 --> 00:46:34.140 align:middle line:84% the only way in which the system likes to stay in equilibrium 00:46:34.140 --> 00:46:39.020 align:middle line:84% is to separate into vapor and liquid. 00:46:39.020 --> 00:46:43.680 align:middle line:90% 00:46:43.680 --> 00:46:46.200 align:middle line:90% Triple point is this one. 00:46:46.200 --> 00:46:49.500 align:middle line:84% As you know, there is no adjacent values 00:46:49.500 --> 00:46:55.060 align:middle line:84% of temperature and pressure, but there are other triple points. 00:46:55.060 --> 00:47:03.710 align:middle line:84% If ice has many stable, but some are also metastable. 00:47:03.710 --> 00:47:06.220 align:middle line:84% For example, ice IV is not present 00:47:06.220 --> 00:47:07.850 align:middle line:90% here because it's metastable. 00:47:07.850 --> 00:47:13.330 align:middle line:84% But these are stable forms, and, therefore, 00:47:13.330 --> 00:47:17.290 align:middle line:84% you may find several triple points. 00:47:17.290 --> 00:47:22.480 align:middle line:84% You can have no more than three coexisting phases for a pure 00:47:22.480 --> 00:47:26.486 align:middle line:84% substance because the variance cannot go negative. 00:47:26.486 --> 00:47:33.746 align:middle line:90% 00:47:33.746 --> 00:47:43.550 align:middle line:84% Now, remember that we gave these relations that 00:47:43.550 --> 00:47:46.940 align:middle line:84% allow to reconstruct the fundamental relation 00:47:46.940 --> 00:47:51.080 align:middle line:84% and, in particular, energy, entropy, and enthalpy, which 00:47:51.080 --> 00:47:54.410 align:middle line:84% is useful for bulk flow applications. 00:47:54.410 --> 00:47:57.143 align:middle line:90% When you start from-- 00:47:57.143 --> 00:48:02.184 align:middle line:90% 00:48:02.184 --> 00:48:05.700 align:middle line:84% if you have the expressions of how 00:48:05.700 --> 00:48:13.230 align:middle line:84% the coefficient of thermal expansion-isobaric expansion, 00:48:13.230 --> 00:48:17.450 align:middle line:84% isothermal compressibility, heat capacity of constant pressure-- 00:48:17.450 --> 00:48:20.170 align:middle line:84% how these things depend on T and p. 00:48:20.170 --> 00:48:22.840 align:middle line:90% And these were the relations. 00:48:22.840 --> 00:48:30.740 align:middle line:84% If we have a pure substance, for example, 00:48:30.740 --> 00:48:39.070 align:middle line:84% we could simply divide by the mass that we have in our system. 00:48:39.070 --> 00:48:41.410 align:middle line:90% And so we can rewrite. 00:48:41.410 --> 00:48:47.730 align:middle line:84% So E becomes little e, which I write u 00:48:47.730 --> 00:48:52.500 align:middle line:84% because we said that we use the letter u for energy just 00:48:52.500 --> 00:48:55.550 align:middle line:84% to remind ourselves that we are within the simple system model. 00:48:55.550 --> 00:48:59.050 align:middle line:90% 00:48:59.050 --> 00:49:05.530 align:middle line:84% So the heat capacity becomes the specific heat or specific heat 00:49:05.530 --> 00:49:07.940 align:middle line:90% capacity. 00:49:07.940 --> 00:49:14.115 align:middle line:84% The volume becomes a specific volume here and here. 00:49:14.115 --> 00:49:16.050 align:middle line:84% And so you get these expression that 00:49:16.050 --> 00:49:21.730 align:middle line:84% allow you to compute for any substance-- 00:49:21.730 --> 00:49:24.550 align:middle line:84% any pure substance in the simple system 00:49:24.550 --> 00:49:27.970 align:middle line:90% model-- the various properties. 00:49:27.970 --> 00:49:35.140 align:middle line:84% And from that, we have some extreme, simple examples that-- 00:49:35.140 --> 00:49:37.950 align:middle line:84% simple behaviors that we call ideal behavior. 00:49:37.950 --> 00:49:41.030 align:middle line:90% 00:49:41.030 --> 00:49:46.100 align:middle line:84% The first one is the ideal incompressible solid or fluid 00:49:46.100 --> 00:49:53.850 align:middle line:84% model in which we assume that the specific volume is 00:49:53.850 --> 00:50:00.780 align:middle line:84% approximately constant because of the incompressibility. 00:50:00.780 --> 00:50:05.050 align:middle line:84% It is not strictly true because the solid, as you know, 00:50:05.050 --> 00:50:06.350 align:middle line:90% can be compressed. 00:50:06.350 --> 00:50:08.790 align:middle line:84% The Young's modulus of elasticity 00:50:08.790 --> 00:50:11.040 align:middle line:90% describes precisely that. 00:50:11.040 --> 00:50:18.360 align:middle line:84% But for some applications, for some purposes, 00:50:18.360 --> 00:50:24.010 align:middle line:84% for thermal aspects, you may neglect that change in volume. 00:50:24.010 --> 00:50:26.410 align:middle line:84% And if you can neglect that change in volume, 00:50:26.410 --> 00:50:34.560 align:middle line:84% then the compressibility and the coefficient of expansion 00:50:34.560 --> 00:50:35.980 align:middle line:90% are approximately zero. 00:50:35.980 --> 00:50:39.090 align:middle line:84% So if you take the limit and assume that they are exactly 00:50:39.090 --> 00:50:45.270 align:middle line:84% zero and that they go to zero in such a way that from the Mayer 00:50:45.270 --> 00:50:50.405 align:middle line:84% relation you also have that it doesn't-- 00:50:50.405 --> 00:50:54.650 align:middle line:90% 00:50:54.650 --> 00:50:56.450 align:middle line:84% so it is well behaved, because there is 00:50:56.450 --> 00:50:57.960 align:middle line:90% a ratio in the Mayer relation-- 00:50:57.960 --> 00:51:01.320 align:middle line:84% there is a ratio, where there is an alpha p, 00:51:01.320 --> 00:51:05.690 align:middle line:84% I think squared at the numerator divided by kappa T. 00:51:05.690 --> 00:51:10.060 align:middle line:84% And so you have a zero over zero. 00:51:10.060 --> 00:51:15.310 align:middle line:84% So if that limit is appropriate, then the specific heats 00:51:15.310 --> 00:51:16.305 align:middle line:90% become equal. 00:51:16.305 --> 00:51:19.200 align:middle line:90% 00:51:19.200 --> 00:51:22.410 align:middle line:84% And you can show that they are also 00:51:22.410 --> 00:51:26.480 align:middle line:90% just a function of temperature. 00:51:26.480 --> 00:51:28.330 align:middle line:90% How do you show that? 00:51:28.330 --> 00:51:29.830 align:middle line:90% Well, here it is. 00:51:29.830 --> 00:51:38.800 align:middle line:84% If kappa and alpha are zero, this coefficient 00:51:38.800 --> 00:51:40.220 align:middle line:90% here is always zero. 00:51:40.220 --> 00:51:44.980 align:middle line:84% So that means that the internal energy 00:51:44.980 --> 00:51:48.150 align:middle line:84% is a function of temperature but not a function of pressure. 00:51:48.150 --> 00:51:50.960 align:middle line:90% 00:51:50.960 --> 00:51:54.440 align:middle line:84% And since the specific heat at constant volume 00:51:54.440 --> 00:52:00.500 align:middle line:84% is the partial of the energy with respect to temperature, 00:52:00.500 --> 00:52:05.570 align:middle line:84% it is also a function of temperature. 00:52:05.570 --> 00:52:08.840 align:middle line:90% Since u is a function of T only. 00:52:08.840 --> 00:52:11.230 align:middle line:90% And if we go here for entropy-- 00:52:11.230 --> 00:52:13.790 align:middle line:90% 00:52:13.790 --> 00:52:15.200 align:middle line:90% let's do also for entropy. 00:52:15.200 --> 00:52:19.570 align:middle line:84% So if alpha is equal to zero, entropy 00:52:19.570 --> 00:52:23.020 align:middle line:84% is a function only of the temperature. 00:52:23.020 --> 00:52:25.510 align:middle line:84% It's not true for the enthalpy, though. 00:52:25.510 --> 00:52:27.790 align:middle line:84% If alpha is equal to zero, the enthalpy 00:52:27.790 --> 00:52:31.270 align:middle line:84% depends on both temperature and pressure. 00:52:31.270 --> 00:52:37.240 align:middle line:84% So for a liquid, enthalpy depends on T and p. 00:52:37.240 --> 00:52:41.020 align:middle line:84% And here is the expressions for the du, ds, dh, 00:52:41.020 --> 00:52:48.640 align:middle line:84% which come from these relations here 00:52:48.640 --> 00:52:52.980 align:middle line:84% when you substitute these assumptions for-- sorry. 00:52:52.980 --> 00:52:54.820 align:middle line:90% Whoops. 00:52:54.820 --> 00:52:55.330 align:middle line:90% Whoa. 00:52:55.330 --> 00:52:57.090 align:middle line:90% OK. 00:52:57.090 --> 00:52:58.480 align:middle line:90% All right. 00:52:58.480 --> 00:53:01.860 align:middle line:84% Of course, from these expressions, 00:53:01.860 --> 00:53:04.210 align:middle line:84% if you need to compute the differences, 00:53:04.210 --> 00:53:07.570 align:middle line:84% you just do the integral because c varies with temperature. 00:53:07.570 --> 00:53:10.410 align:middle line:90% Do some integrals. 00:53:10.410 --> 00:53:14.250 align:middle line:84% Sometimes you're lucky and the specific heats are not 00:53:14.250 --> 00:53:15.700 align:middle line:90% functions of temperature. 00:53:15.700 --> 00:53:18.550 align:middle line:84% Then in that case, if in the range of your application, 00:53:18.550 --> 00:53:21.090 align:middle line:84% you can assume constant specific heat. 00:53:21.090 --> 00:53:24.000 align:middle line:84% The integral is easy, and sometimes 00:53:24.000 --> 00:53:28.260 align:middle line:84% you also call that a perfect behavior. 00:53:28.260 --> 00:53:30.840 align:middle line:84% It's perfect only for the purpose of the calculation, 00:53:30.840 --> 00:53:32.380 align:middle line:90% because it makes it easy. 00:53:32.380 --> 00:53:35.640 align:middle line:84% Not that there is anything better 00:53:35.640 --> 00:53:38.500 align:middle line:90% from the physics point of view. 00:53:38.500 --> 00:53:43.300 align:middle line:84% Now, the ideal gas model is another, 00:53:43.300 --> 00:53:47.550 align:middle line:84% of course, very well-known and useful model. 00:53:47.550 --> 00:53:50.940 align:middle line:90% 00:53:50.940 --> 00:53:56.980 align:middle line:84% If you assume this relation for the equation of state. 00:53:56.980 --> 00:54:00.630 align:middle line:84% We already proved that kappa T is one over the pressure, 00:54:00.630 --> 00:54:02.500 align:middle line:84% alpha p is one over the temperature, 00:54:02.500 --> 00:54:10.890 align:middle line:84% and the Mayer relation becomes that cp is equal to cv plus R. 00:54:10.890 --> 00:54:23.000 align:middle line:84% In that case, let's go back once more here. 00:54:23.000 --> 00:54:27.940 align:middle line:84% Since alpha is 1 over T, this coefficient 00:54:27.940 --> 00:54:30.610 align:middle line:84% in front of the pressure variation 00:54:30.610 --> 00:54:34.300 align:middle line:84% is zero, so it means that the enthalpy is 00:54:34.300 --> 00:54:35.745 align:middle line:90% a function of temperature only. 00:54:35.745 --> 00:54:38.470 align:middle line:90% 00:54:38.470 --> 00:54:43.300 align:middle line:84% And here also for the energy, kappa is 1 over p. 00:54:43.300 --> 00:54:45.520 align:middle line:90% So p times 1 over p is 1. 00:54:45.520 --> 00:54:48.520 align:middle line:90% Alpha is 1 over T, so T minus-- 00:54:48.520 --> 00:54:50.990 align:middle line:84% T times 1 is 1, so 1 minus 1 is zero. 00:54:50.990 --> 00:54:54.260 align:middle line:84% So also the energy as a function of temperature only. 00:54:54.260 --> 00:54:59.080 align:middle line:84% Whereas for the entropy, this alpha is 1 over p. 00:54:59.080 --> 00:55:02.320 align:middle line:84% And from that, you see that the entropy, 00:55:02.320 --> 00:55:06.880 align:middle line:84% in the case of an ideal gas, is a function of both temperature 00:55:06.880 --> 00:55:08.650 align:middle line:90% and pressure. 00:55:08.650 --> 00:55:11.050 align:middle line:90% And here you can integrate-- 00:55:11.050 --> 00:55:15.010 align:middle line:84% I mean, this is how that formula appears-- 00:55:15.010 --> 00:55:16.900 align:middle line:90% comes out. 00:55:16.900 --> 00:55:24.130 align:middle line:84% And if you use the Mayer relation to switch from-- 00:55:24.130 --> 00:55:26.440 align:middle line:84% sometimes it is useful instead of working 00:55:26.440 --> 00:55:29.230 align:middle line:84% with the specific heat at constant pressure, 00:55:29.230 --> 00:55:37.210 align:middle line:84% using that of constant volume because maybe in your problem, 00:55:37.210 --> 00:55:39.890 align:middle line:84% one of these variables remains zero 00:55:39.890 --> 00:55:43.840 align:middle line:84% and so you can choose either of these alternatives. 00:55:43.840 --> 00:55:47.260 align:middle line:84% And again, if the specific heats are not 00:55:47.260 --> 00:55:54.910 align:middle line:84% function of temperature, which happens in some cases, 00:55:54.910 --> 00:55:56.750 align:middle line:90% then you can integrate easily. 00:55:56.750 --> 00:55:58.820 align:middle line:84% Otherwise you have to do the integrals. 00:55:58.820 --> 00:56:00.236 align:middle line:90% I didn't write them explicitly. 00:56:00.236 --> 00:56:05.840 align:middle line:90% 00:56:05.840 --> 00:56:11.090 align:middle line:84% Within the ideal gas model, you may 00:56:11.090 --> 00:56:17.360 align:middle line:84% realize that the way the molecules of the gas 00:56:17.360 --> 00:56:23.090 align:middle line:84% accumulate or carry the energy is mainly 00:56:23.090 --> 00:56:25.290 align:middle line:84% through their degrees of freedom. 00:56:25.290 --> 00:56:28.850 align:middle line:84% So certainly there is the translational 00:56:28.850 --> 00:56:31.430 align:middle line:84% and the rotational degrees of freedom. 00:56:31.430 --> 00:56:34.710 align:middle line:84% And then if you go to sufficiently high temperatures, 00:56:34.710 --> 00:56:37.700 align:middle line:84% you also have the vibrational degrees of freedom 00:56:37.700 --> 00:56:41.450 align:middle line:84% as well as the electronic degrees of freedom. 00:56:41.450 --> 00:56:50.130 align:middle line:84% So at relatively low temperatures, not too low, 00:56:50.130 --> 00:56:55.320 align:middle line:84% the vibrational and electronic degrees of freedom 00:56:55.320 --> 00:56:58.750 align:middle line:90% are less active. 00:56:58.750 --> 00:57:03.250 align:middle line:84% There is sufficient ability for your molecules 00:57:03.250 --> 00:57:05.620 align:middle line:84% to accumulate whatever energy has 00:57:05.620 --> 00:57:08.545 align:middle line:84% to be distributed within the system 00:57:08.545 --> 00:57:18.790 align:middle line:84% by using the translational and rotational structure. 00:57:18.790 --> 00:57:25.630 align:middle line:84% And each degree of freedom contributes to the specific heat 00:57:25.630 --> 00:57:29.590 align:middle line:90% a factor of 1/2. 00:57:29.590 --> 00:57:31.450 align:middle line:90% You remember the famous 1/2kT. 00:57:31.450 --> 00:57:33.160 align:middle line:84% That's the energy that is carried. 00:57:33.160 --> 00:57:35.380 align:middle line:84% It's the equipartition energy that 00:57:35.380 --> 00:57:40.840 align:middle line:84% is carried by each degree of freedom at equilibrium. 00:57:40.840 --> 00:57:45.490 align:middle line:84% So if you divide by T, if you take the derivative with respect 00:57:45.490 --> 00:57:48.010 align:middle line:84% to T, you are left with 1/2 of k. 00:57:48.010 --> 00:57:52.180 align:middle line:84% And k is the Boltzmann constant which, 00:57:52.180 --> 00:57:54.700 align:middle line:84% when you multiply it by the Avogadro's number, 00:57:54.700 --> 00:58:01.580 align:middle line:84% becomes the gas constant R. So for a gas that only has 00:58:01.580 --> 00:58:03.780 align:middle line:84% translational degrees of freedom, 00:58:03.780 --> 00:58:09.800 align:middle line:84% so the molecules are point-wise and can move in the three 00:58:09.800 --> 00:58:12.020 align:middle line:84% directions, you have the three translational degrees 00:58:12.020 --> 00:58:17.030 align:middle line:84% of freedom, so you have 1/2 of R contribution for each of them. 00:58:17.030 --> 00:58:24.620 align:middle line:84% So the specific heat contributed by those degrees of freedom is 00:58:24.620 --> 00:58:36.650 align:middle line:84% just 3/2 of R. And then if this is cv, then cp is cv plus R. 00:58:36.650 --> 00:58:39.440 align:middle line:90% So the 3/2 becomes 5/2. 00:58:39.440 --> 00:58:44.270 align:middle line:84% Gamma, this famous ratio of specific heats, 00:58:44.270 --> 00:58:57.790 align:middle line:84% is therefore 5 over 3, which is 1.67. 00:58:57.790 --> 00:59:02.950 align:middle line:84% Now other molecules, for example, oxygen or nitrogen, 00:59:02.950 --> 00:59:06.490 align:middle line:90% they're biatomic. 00:59:06.490 --> 00:59:11.120 align:middle line:84% And these molecules also have the ability 00:59:11.120 --> 00:59:19.240 align:middle line:84% to put energy, yes, in the vibration. 00:59:19.240 --> 00:59:22.350 align:middle line:84% But most importantly-- here, we said that we 00:59:22.350 --> 00:59:25.030 align:middle line:90% are interested in the rotation. 00:59:25.030 --> 00:59:27.380 align:middle line:90% So the rotation of the molecule. 00:59:27.380 --> 00:59:35.672 align:middle line:90% 00:59:35.672 --> 00:59:47.160 align:middle line:84% So if you look at the principal axis of this thing here, 00:59:47.160 --> 00:59:51.990 align:middle line:84% the moment of inertia for rotation 00:59:51.990 --> 00:59:58.800 align:middle line:84% is large in two directions and very small in the other one. 00:59:58.800 --> 01:00:02.370 align:middle line:84% Because when the molecule rotates along 01:00:02.370 --> 01:00:08.100 align:middle line:84% the axis of alignment of these two atoms, it doesn't have-- 01:00:08.100 --> 01:00:13.920 align:middle line:84% since the atoms are concentrated in very small radii, 01:00:13.920 --> 01:00:15.820 align:middle line:90% the mass is concentrated there. 01:00:15.820 --> 01:00:18.820 align:middle line:84% So the moment of inertia is small. 01:00:18.820 --> 01:00:31.420 align:middle line:84% So if you multiply the moment times the angular velocity, 01:00:31.420 --> 01:00:32.145 align:middle line:90% probably squared. 01:00:32.145 --> 01:00:35.740 align:middle line:90% 01:00:35.740 --> 01:00:41.350 align:middle line:84% Since that M is important only in the two directions 01:00:41.350 --> 01:00:47.020 align:middle line:84% where the molecule rotates perpendicular 01:00:47.020 --> 01:00:51.160 align:middle line:84% to the axis of alignment, and so you 01:00:51.160 --> 01:00:53.930 align:middle line:84% have only-- in this molecule with aligned atoms, 01:00:53.930 --> 01:00:58.000 align:middle line:84% you have only two additional degrees of freedom 01:00:58.000 --> 01:01:01.280 align:middle line:84% in which the molecule can carry the energy. 01:01:01.280 --> 01:01:05.845 align:middle line:84% And that's why you go from 3, plus 2, to 5. 01:01:05.845 --> 01:01:08.170 align:middle line:90% 5/2. 01:01:08.170 --> 01:01:11.250 align:middle line:90% And that makes gamma equals 1.4. 01:01:11.250 --> 01:01:14.505 align:middle line:84% This is typical of oxygen, nitrogen, and therefore air. 01:01:14.505 --> 01:01:17.840 align:middle line:90% 01:01:17.840 --> 01:01:30.170 align:middle line:84% More complex molecules like H2O where the atoms are not 01:01:30.170 --> 01:01:38.570 align:middle line:84% aligned but enjoy the possibility of storing energy 01:01:38.570 --> 01:01:42.590 align:middle line:84% also in the rotations, in the three principal axis 01:01:42.590 --> 01:01:45.980 align:middle line:84% of rotation, all three have a non-negligible moment 01:01:45.980 --> 01:01:49.880 align:middle line:90% of inertia. 01:01:49.880 --> 01:01:53.150 align:middle line:84% And therefore, you add another three. 01:01:53.150 --> 01:01:58.810 align:middle line:90% 3 plus 3 halves makes 6/2. 01:01:58.810 --> 01:02:01.810 align:middle line:90% That's gamma equals 1.33. 01:02:01.810 --> 01:02:06.350 align:middle line:84% For example, the products of combustion-- also H2O 01:02:06.350 --> 01:02:09.150 align:middle line:90% has a similar. 01:02:09.150 --> 01:02:11.070 align:middle line:84% And as the products of combustion 01:02:11.070 --> 01:02:15.510 align:middle line:90% are mainly done with the-- 01:02:15.510 --> 01:02:22.200 align:middle line:84% well, not exactly, because you have CO2-- 01:02:22.200 --> 01:02:23.190 align:middle line:90% but OK. 01:02:23.190 --> 01:02:33.570 align:middle line:84% Let's say that if you have water and carbon dioxide, 01:02:33.570 --> 01:02:38.160 align:middle line:84% then they both belong to these categories. 01:02:38.160 --> 01:02:40.830 align:middle line:84% If you have-- like most of the times, 01:02:40.830 --> 01:02:43.810 align:middle line:84% also air you are somewhere in between here and there. 01:02:43.810 --> 01:02:49.550 align:middle line:84% So it's a weighted sum of 1.4 and 1.33. 01:02:49.550 --> 01:02:53.570 align:middle line:84% So these numbers allow you, just by looking at the molecule, 01:02:53.570 --> 01:02:59.270 align:middle line:84% to get a good rough, immediate idea 01:02:59.270 --> 01:03:03.920 align:middle line:90% of what the specific heat is. 01:03:03.920 --> 01:03:09.830 align:middle line:84% But of course, as you go higher in temperatures, 01:03:09.830 --> 01:03:13.580 align:middle line:84% the various thresholds, you can activate also 01:03:13.580 --> 01:03:17.600 align:middle line:84% the various vibrational modes of the molecule. 01:03:17.600 --> 01:03:22.130 align:middle line:84% And even higher, you can also shift the electrons 01:03:22.130 --> 01:03:25.770 align:middle line:84% from the lower shell to the higher shells, 01:03:25.770 --> 01:03:29.780 align:middle line:90% and that adds to specific heat. 01:03:29.780 --> 01:03:33.860 align:middle line:84% So the specific heat, as we are interpreting it as, 01:03:33.860 --> 01:03:37.730 align:middle line:84% is the capacity, represents the capacity of our molecule 01:03:37.730 --> 01:03:41.725 align:middle line:90% to store the energy. 01:03:41.725 --> 01:03:45.740 align:middle line:90% 01:03:45.740 --> 01:03:49.940 align:middle line:84% Yeah, these are expressions that are often used when 01:03:49.940 --> 01:03:58.400 align:middle line:90% you assume isentropic change. 01:03:58.400 --> 01:04:02.630 align:middle line:84% So if the entropy doesn't change, 01:04:02.630 --> 01:04:05.300 align:middle line:84% you just set from those equations 01:04:05.300 --> 01:04:10.890 align:middle line:84% here that the entropy remains constant, 01:04:10.890 --> 01:04:13.490 align:middle line:90% then you get these expressions. 01:04:13.490 --> 01:04:19.420 align:middle line:84% And since R is cp minus cv, that's 01:04:19.420 --> 01:04:23.290 align:middle line:84% how you can express the setting equal to zero, 01:04:23.290 --> 01:04:26.540 align:middle line:84% these in terms of just the ratio of the heat capacities. 01:04:26.540 --> 01:04:27.040 align:middle line:90% Gamma. 01:04:27.040 --> 01:04:28.560 align:middle line:90% That's why gamma appears here. 01:04:28.560 --> 01:04:32.450 align:middle line:90% 01:04:32.450 --> 01:04:36.006 align:middle line:90% Now, going back to two phases. 01:04:36.006 --> 01:04:52.800 align:middle line:90% 01:04:52.800 --> 01:04:55.950 align:middle line:84% Well, actually, how do we prove this relation here 01:04:55.950 --> 01:05:01.230 align:middle line:84% that connects the pressure and the temperature? 01:05:01.230 --> 01:05:03.810 align:middle line:84% This is called the Clausius-Clapeyron relation, 01:05:03.810 --> 01:05:08.880 align:middle line:84% and it gives the slope of the saturation line on that pressure 01:05:08.880 --> 01:05:16.222 align:middle line:84% temperature diagram that we have shown before. 01:05:16.222 --> 01:05:22.020 align:middle line:84% It allows you to compute it using 01:05:22.020 --> 01:05:29.070 align:middle line:84% the ratio of the enthalpy of the change of phase 01:05:29.070 --> 01:05:33.880 align:middle line:84% and the volume of the change of phase and the temperature. 01:05:33.880 --> 01:05:37.720 align:middle line:90% 01:05:37.720 --> 01:05:43.300 align:middle line:84% And the way you obtain this relation is, well, it 01:05:43.300 --> 01:05:47.560 align:middle line:84% follows again from the condition of equality 01:05:47.560 --> 01:05:51.400 align:middle line:84% of chemical potentials of the single substance we have, 01:05:51.400 --> 01:05:54.490 align:middle line:90% water, in the two phases. 01:05:54.490 --> 01:05:57.700 align:middle line:90% So it's this relation here. 01:05:57.700 --> 01:06:03.350 align:middle line:84% So one way to write it is to take 01:06:03.350 --> 01:06:08.090 align:middle line:84% the differential of the left-hand side 01:06:08.090 --> 01:06:10.940 align:middle line:84% and put it equal to the differential 01:06:10.940 --> 01:06:13.040 align:middle line:90% of the right-hand side. 01:06:13.040 --> 01:06:17.795 align:middle line:84% So it's the d mu because we have-- 01:06:17.795 --> 01:06:21.240 align:middle line:90% 01:06:21.240 --> 01:06:27.665 align:middle line:84% the differential of d mu can be written if you-- 01:06:27.665 --> 01:06:30.560 align:middle line:90% 01:06:30.560 --> 01:06:33.240 align:middle line:84% this is not written in terms of the differential, 01:06:33.240 --> 01:06:36.140 align:middle line:84% but you can see that it also represents a differential. 01:06:36.140 --> 01:06:39.320 align:middle line:84% This is the Gibbs-Duhem relation. 01:06:39.320 --> 01:06:40.610 align:middle line:90% OK? 01:06:40.610 --> 01:06:46.350 align:middle line:84% But it's also viewed as the differential of the chemical 01:06:46.350 --> 01:06:50.280 align:middle line:84% potential as a function of temperature and pressure. 01:06:50.280 --> 01:06:52.170 align:middle line:84% And the Gibbs-Duhem relation shows 01:06:52.170 --> 01:06:57.120 align:middle line:84% that the partial derivatives of mu with respect to T is minus s 01:06:57.120 --> 01:07:03.310 align:middle line:84% and partial derivative of mu with respect to p is v. OK? 01:07:03.310 --> 01:07:07.540 align:middle line:84% So if I want to move from a state in which I have these two 01:07:07.540 --> 01:07:15.190 align:middle line:84% phases coexisting, one and two could be liquid and vapor 01:07:15.190 --> 01:07:21.060 align:middle line:84% or ice and water or vapor and ice. 01:07:21.060 --> 01:07:24.190 align:middle line:90% 01:07:24.190 --> 01:07:27.580 align:middle line:84% And I want to move to another state in which I 01:07:27.580 --> 01:07:30.410 align:middle line:84% change the temperature, therefore 01:07:30.410 --> 01:07:32.650 align:middle line:90% also change the pressure. 01:07:32.650 --> 01:07:37.480 align:middle line:84% Then it means that the two differentials-- also, the change 01:07:37.480 --> 01:07:42.520 align:middle line:84% in chemical potential must be the same so that if they were 01:07:42.520 --> 01:07:45.310 align:middle line:84% equal at the beginning, they are equal also after, 01:07:45.310 --> 01:07:51.100 align:middle line:84% and therefore the two phases are still in mutual equilibrium. 01:07:51.100 --> 01:07:53.850 align:middle line:90% So if you equate d mu y-- 01:07:53.850 --> 01:07:58.740 align:middle line:90% d mu 1 with d mu 2 here-- 01:07:58.740 --> 01:08:03.480 align:middle line:84% so eliminate these two because they must be equal-- 01:08:03.480 --> 01:08:08.160 align:middle line:84% and you see you get a relation between dT and dp 01:08:08.160 --> 01:08:12.750 align:middle line:84% that involves the differences in the specific entropies 01:08:12.750 --> 01:08:14.820 align:middle line:84% for the two phases and the differences 01:08:14.820 --> 01:08:20.130 align:middle line:84% in the specific volumes, which are the coefficients here. 01:08:20.130 --> 01:08:22.710 align:middle line:84% Then how do we go from here to there? 01:08:22.710 --> 01:08:27.330 align:middle line:84% Well, again, the equation, the equality 01:08:27.330 --> 01:08:31.020 align:middle line:84% for mutual equilibrium of the chemical potentials, 01:08:31.020 --> 01:08:39.750 align:middle line:84% remember that the chemical potential for a pure substance 01:08:39.750 --> 01:08:43.590 align:middle line:84% is equal to the Gibbs free energy. 01:08:43.590 --> 01:08:45.050 align:middle line:90% Specific Gibbs free energy. 01:08:45.050 --> 01:08:50.460 align:middle line:84% So it's h, which can also be written as enthalpy minus Ts. 01:08:50.460 --> 01:08:51.680 align:middle line:90% OK? 01:08:51.680 --> 01:08:57.960 align:middle line:84% This is true only for the pure substance. 01:08:57.960 --> 01:09:00.640 align:middle line:84% So if the chemical potentials are equal, 01:09:00.640 --> 01:09:04.560 align:middle line:84% it means that h minus Ts must be equal in the two phases, 01:09:04.560 --> 01:09:08.940 align:middle line:84% and therefore, in this equality I can extract T. 01:09:08.940 --> 01:09:11.670 align:middle line:84% And it's equal to the ratio of the enthalpy 01:09:11.670 --> 01:09:15.499 align:middle line:84% of the change of phase over the entropy of the change of phase. 01:09:15.499 --> 01:09:20.880 align:middle line:90% 01:09:20.880 --> 01:09:25.034 align:middle line:84% And so I can substitute-- instead of the entropy, 01:09:25.034 --> 01:09:38.124 align:middle line:84% I put the enthalpy of the change of phase divided by T. 01:09:38.124 --> 01:09:39.740 align:middle line:90% Let's be more specific. 01:09:39.740 --> 01:09:43.090 align:middle line:84% Suppose we just have liquid and vapor, and then 01:09:43.090 --> 01:09:47.424 align:middle line:84% how do we compute the properties of a system in which I 01:09:47.424 --> 01:09:49.370 align:middle line:90% have two phases. 01:09:49.370 --> 01:09:50.420 align:middle line:90% Or three. 01:09:50.420 --> 01:09:54.650 align:middle line:84% Well, as we said, thanks to the simple system model 01:09:54.650 --> 01:09:59.255 align:middle line:84% and this idea of inserting and removing partitions for free, 01:09:59.255 --> 01:10:03.830 align:middle line:84% we have represented this kind of complex situation 01:10:03.830 --> 01:10:05.480 align:middle line:90% into a simpler one. 01:10:05.480 --> 01:10:10.130 align:middle line:84% It is a composite system of as many subsystems 01:10:10.130 --> 01:10:11.005 align:middle line:90% as there are phases. 01:10:11.005 --> 01:10:13.532 align:middle line:90% 01:10:13.532 --> 01:10:17.480 align:middle line:84% In the slide here, we have only two. 01:10:17.480 --> 01:10:19.780 align:middle line:90% It's a composite system. 01:10:19.780 --> 01:10:23.080 align:middle line:84% So for the properties of the composite system, 01:10:23.080 --> 01:10:25.480 align:middle line:84% the additive properties, it's easy. 01:10:25.480 --> 01:10:28.400 align:middle line:90% For example, the volume. 01:10:28.400 --> 01:10:31.650 align:middle line:84% That's the volume of one plus the volume of the other. 01:10:31.650 --> 01:10:34.170 align:middle line:84% The energy is an additive property, 01:10:34.170 --> 01:10:38.660 align:middle line:84% so it's the energy of one phase plus the energy of the other. 01:10:38.660 --> 01:10:42.195 align:middle line:84% The entropy also is an additive property. 01:10:42.195 --> 01:10:44.070 align:middle line:84% The entropy of one plus entropy of the other. 01:10:44.070 --> 01:10:47.930 align:middle line:90% The enthalpy, here it's also-- 01:10:47.930 --> 01:10:50.430 align:middle line:84% in general, it is not an additive property, 01:10:50.430 --> 01:10:56.510 align:middle line:84% but here, because both systems have the same pressure, then 01:10:56.510 --> 01:10:58.806 align:middle line:90% it becomes additive. 01:10:58.806 --> 01:11:03.670 align:middle line:90% 01:11:03.670 --> 01:11:07.510 align:middle line:84% And then we would be done, except that 01:11:07.510 --> 01:11:13.110 align:middle line:84% in terms of expressing how much of the substance 01:11:13.110 --> 01:11:16.240 align:middle line:84% we have in one phase and how much we have in the other, 01:11:16.240 --> 01:11:19.740 align:middle line:90% sometimes it is often use-- 01:11:19.740 --> 01:11:26.370 align:middle line:84% I mean, sometimes it's useful to work in terms of fractions. 01:11:26.370 --> 01:11:31.325 align:middle line:84% So we call, for example, vapor fraction or steam quality. 01:11:31.325 --> 01:11:34.102 align:middle line:90% 01:11:34.102 --> 01:11:39.600 align:middle line:84% The ratio, the amount of mass that 01:11:39.600 --> 01:11:44.915 align:middle line:84% is in the vapor phase over the overall mass of our system. 01:11:44.915 --> 01:11:48.950 align:middle line:90% 01:11:48.950 --> 01:11:55.580 align:middle line:84% And by doing that and substituting in-- 01:11:55.580 --> 01:11:59.900 align:middle line:84% dividing here in these properties, 01:11:59.900 --> 01:12:04.810 align:middle line:84% the additive properties, we can compute the specific properties 01:12:04.810 --> 01:12:13.920 align:middle line:84% in terms of this mass fraction and of the specific properties 01:12:13.920 --> 01:12:19.440 align:middle line:84% that characterize each of the phases at that temperature 01:12:19.440 --> 01:12:22.480 align:middle line:90% and pressure. 01:12:22.480 --> 01:12:28.630 align:middle line:84% So these are typically called with the subscript g and f, 01:12:28.630 --> 01:12:33.580 align:middle line:84% and the difference is called the vaporization-- 01:12:33.580 --> 01:12:37.330 align:middle line:84% So for example, hfg is the enthalpy 01:12:37.330 --> 01:12:41.050 align:middle line:84% of vaporization. sfg is the entropy of vaporization. 01:12:41.050 --> 01:12:43.210 align:middle line:84% And if we go back to this expression 01:12:43.210 --> 01:12:46.990 align:middle line:84% here where 2 and 1 is f and g, this 01:12:46.990 --> 01:12:50.350 align:middle line:84% says that the temperature is equal to the ratio 01:12:50.350 --> 01:12:55.459 align:middle line:84% of the enthalpy of vaporization over the entropy vaporization. 01:12:55.459 --> 01:12:59.770 align:middle line:90% 01:12:59.770 --> 01:13:06.060 align:middle line:84% These properties vg and vf are listed 01:13:06.060 --> 01:13:11.370 align:middle line:84% in the steam tables for water or, in the equivalent 01:13:11.370 --> 01:13:15.470 align:middle line:84% of the steam tables for other substances. 01:13:15.470 --> 01:13:20.570 align:middle line:84% Or maybe they are represented by correlations. 01:13:20.570 --> 01:13:24.680 align:middle line:84% For example, the steam tables that were developed by Keenan 01:13:24.680 --> 01:13:32.360 align:middle line:84% and Keyes and in the '30s, 1930s, 01:13:32.360 --> 01:13:40.870 align:middle line:84% were based on correlations with some 40 parameters that were 01:13:40.870 --> 01:13:47.120 align:middle line:84% used to correlate the experimental data for water over 01:13:47.120 --> 01:13:50.290 align:middle line:90% the full range of applicable-- 01:13:50.290 --> 01:13:57.160 align:middle line:90% I mean-- important values. 01:13:57.160 --> 01:14:02.070 align:middle line:84% So we need a lot of experimental data. 01:14:02.070 --> 01:14:05.560 align:middle line:84% And with those 40 parameters, you 01:14:05.560 --> 01:14:09.585 align:middle line:84% can actually construct the entire steam tables. 01:14:09.585 --> 01:14:14.330 align:middle line:90% 01:14:14.330 --> 01:14:17.930 align:middle line:84% Now pictorially, you can represent 01:14:17.930 --> 01:14:19.740 align:middle line:90% properties of substances. 01:14:19.740 --> 01:14:23.410 align:middle line:90% 01:14:23.410 --> 01:14:29.510 align:middle line:84% Of course, yes, you could do it in at most a three 01:14:29.510 --> 01:14:34.750 align:middle line:90% dimensional visualization. 01:14:34.750 --> 01:14:40.090 align:middle line:84% And pictorially, this is what an energy 01:14:40.090 --> 01:14:44.470 align:middle line:84% versus volume versus entropy diagram would look. 01:14:44.470 --> 01:14:48.530 align:middle line:84% So it's like, remember the E versus S diagram? 01:14:48.530 --> 01:14:49.130 align:middle line:90% All right? 01:14:49.130 --> 01:14:51.580 align:middle line:90% Here we also add volume. 01:14:51.580 --> 01:14:57.040 align:middle line:84% So if I fixed volume, this curve here 01:14:57.040 --> 01:15:03.130 align:middle line:84% is the curve that we usually would represent on an E versus S 01:15:03.130 --> 01:15:06.085 align:middle line:84% diagram for that particular value of the volume. 01:15:06.085 --> 01:15:10.230 align:middle line:84% Here is for another value of the volume. 01:15:10.230 --> 01:15:13.590 align:middle line:84% We fix the entropy, then you have energy versus volume 01:15:13.590 --> 01:15:15.300 align:middle line:90% and you follow this line. 01:15:15.300 --> 01:15:20.150 align:middle line:84% The slope of this line is the negative of the pressure. 01:15:20.150 --> 01:15:22.840 align:middle line:90% 01:15:22.840 --> 01:15:27.050 align:middle line:84% Whereas the slope of that one is the temperature. 01:15:27.050 --> 01:15:31.400 align:middle line:90% 01:15:31.400 --> 01:15:40.930 align:middle line:84% Notice that when we have, for example, at the triple point, 01:15:40.930 --> 01:15:46.060 align:middle line:84% the triple point, which is a point on the pressure 01:15:46.060 --> 01:15:53.980 align:middle line:84% temperature diagram, becomes a planar 01:15:53.980 --> 01:15:58.707 align:middle line:84% region of triangular shape called the triple point triangle 01:15:58.707 --> 01:15:59.207 align:middle line:90% here. 01:15:59.207 --> 01:15:59.707 align:middle line:90% Because-- 01:15:59.707 --> 01:16:09.230 align:middle line:90% 01:16:09.230 --> 01:16:12.080 align:middle line:84% It's true that you can have the three phases only 01:16:12.080 --> 01:16:17.390 align:middle line:84% at one particular set of values of p and T, 01:16:17.390 --> 01:16:23.730 align:middle line:84% but you may have many different states like that. 01:16:23.730 --> 01:16:28.410 align:middle line:84% For example, you may have more ice, less water, and so on. 01:16:28.410 --> 01:16:34.480 align:middle line:84% So all the various combinations give you two mass fractions. 01:16:34.480 --> 01:16:35.970 align:middle line:84% So you got two degrees of freedom 01:16:35.970 --> 01:16:39.250 align:middle line:84% and you can span within that triangle. 01:16:39.250 --> 01:16:45.760 align:middle line:84% For example, here you have almost only vapor. 01:16:45.760 --> 01:16:50.540 align:middle line:84% Here you have almost only liquid and here almost only solid, 01:16:50.540 --> 01:16:56.350 align:middle line:84% and in between, you have all the possible combinations. 01:16:56.350 --> 01:17:00.350 align:middle line:84% This is the region for liquid and vapor coexistence. 01:17:00.350 --> 01:17:04.110 align:middle line:84% This is solid and vapor coexistence. 01:17:04.110 --> 01:17:08.260 align:middle line:84% This is the value of the critical isotherm. 01:17:08.260 --> 01:17:10.950 align:middle line:84% So this is points that are the same temperature 01:17:10.950 --> 01:17:13.830 align:middle line:84% equal to the critical temperature. 01:17:13.830 --> 01:17:18.405 align:middle line:84% And above that critical isotherm, 01:17:18.405 --> 01:17:20.320 align:middle line:90% we change name from vapor. 01:17:20.320 --> 01:17:21.670 align:middle line:90% We call it gas. 01:17:21.670 --> 01:17:24.430 align:middle line:84% There is no real physical distinction. 01:17:24.430 --> 01:17:29.490 align:middle line:84% But conventionally, we usually use the term gas 01:17:29.490 --> 01:17:34.120 align:middle line:84% where something like the ideal gas applies. 01:17:34.120 --> 01:17:38.060 align:middle line:84% Now, if you-- this is a pictorial representation. 01:17:38.060 --> 01:17:43.780 align:middle line:84% If you try to do it for water with the real property using 01:17:43.780 --> 01:17:48.850 align:middle line:84% the steam tables, this is how it looks. 01:17:48.850 --> 01:17:55.600 align:middle line:84% And you have to realize that since the specific volume spans 01:17:55.600 --> 01:18:01.720 align:middle line:84% over how many, maybe eight orders of magnitude 01:18:01.720 --> 01:18:08.140 align:middle line:84% in the interesting parts, not only because there is a factor 01:18:08.140 --> 01:18:13.900 align:middle line:84% of 1,000 between the density of the specific volume of the vapor 01:18:13.900 --> 01:18:19.480 align:middle line:84% and the liquid, but also, as you-- 01:18:19.480 --> 01:18:23.960 align:middle line:84% so that would account only for three orders of magnitude here. 01:18:23.960 --> 01:18:30.040 align:middle line:84% But at lower pressures, this difference 01:18:30.040 --> 01:18:33.790 align:middle line:90% becomes even more important. 01:18:33.790 --> 01:18:35.630 align:middle line:90% It's inconvenient. 01:18:35.630 --> 01:18:37.850 align:middle line:84% You cannot do this on a linear scale. 01:18:37.850 --> 01:18:44.360 align:middle line:84% So you see this, which is supposed to be straight lines, 01:18:44.360 --> 01:18:46.790 align:middle line:84% become distorted into exponentials 01:18:46.790 --> 01:18:49.940 align:middle line:84% because we have used a logarithmic scale 01:18:49.940 --> 01:18:53.225 align:middle line:90% for the specific volume. 01:18:53.225 --> 01:19:00.140 align:middle line:84% So this is still the triple point triangle. 01:19:00.140 --> 01:19:02.330 align:middle line:84% And excuse the Italian here, but that 01:19:02.330 --> 01:19:05.605 align:middle line:84% was done for a booklet I have in Italian. 01:19:05.605 --> 01:19:09.210 align:middle line:90% 01:19:09.210 --> 01:19:11.970 align:middle line:84% Obviously working with three dimensional diagrams 01:19:11.970 --> 01:19:19.860 align:middle line:84% is not useful, so most people who work with water 01:19:19.860 --> 01:19:26.540 align:middle line:84% have a diagram of this sort, which is called the Mollier 01:19:26.540 --> 01:19:31.030 align:middle line:84% diagram, in the form of a poster on the wall of their offices. 01:19:31.030 --> 01:19:36.950 align:middle line:84% And this is how they compute the properties of water. 01:19:36.950 --> 01:19:38.600 align:middle line:90% And this is how-- 01:19:38.600 --> 01:19:42.830 align:middle line:84% I'm sure you've probably done some homeworks perusing 01:19:42.830 --> 01:19:47.810 align:middle line:84% this diagram in the old days of your undergraduate 01:19:47.810 --> 01:19:49.140 align:middle line:90% thermodynamics. 01:19:49.140 --> 01:19:54.210 align:middle line:90% 01:19:54.210 --> 01:19:57.660 align:middle line:90% This is another view. 01:19:57.660 --> 01:19:59.295 align:middle line:84% It's like pressure versus volume. 01:19:59.295 --> 01:20:03.560 align:middle line:90% 01:20:03.560 --> 01:20:04.060 align:middle line:90% Yeah. 01:20:04.060 --> 01:20:10.860 align:middle line:84% So I should change this because I forgot to update this. 01:20:10.860 --> 01:20:14.332 align:middle line:90% 01:20:14.332 --> 01:20:18.100 align:middle line:84% The reason I'm mentioning this is because sometimes-- 01:20:18.100 --> 01:20:18.790 align:middle line:90% yes. 01:20:18.790 --> 01:20:23.560 align:middle line:84% So in order to do this calculation here, 01:20:23.560 --> 01:20:30.400 align:middle line:84% I needed all the 40 parameters that Keenan used for the steam 01:20:30.400 --> 01:20:34.390 align:middle line:84% tables to obtain the real properties of water 01:20:34.390 --> 01:20:37.400 align:middle line:84% in the full range, including drawing these lines. 01:20:37.400 --> 01:20:41.380 align:middle line:84% So this is the true thing for water. 01:20:41.380 --> 01:20:45.040 align:middle line:84% There is a much simpler model that 01:20:45.040 --> 01:20:49.510 align:middle line:84% covers, at least qualitatively, the liquid vapor 01:20:49.510 --> 01:20:59.340 align:middle line:84% state, which is the Van der Waals equation of state. 01:20:59.340 --> 01:21:03.750 align:middle line:90% Which looks like pv equals RT. 01:21:03.750 --> 01:21:06.330 align:middle line:84% If it weren't for this additional term 01:21:06.330 --> 01:21:14.680 align:middle line:84% here and this additional subtracting term here, 01:21:14.680 --> 01:21:19.080 align:middle line:84% this would be the ideal gas equation. 01:21:19.080 --> 01:21:24.330 align:middle line:84% So Van der Waals, with a simple modification 01:21:24.330 --> 01:21:27.270 align:middle line:84% of the equation of state for a gas, for a perfect gas, 01:21:27.270 --> 01:21:31.750 align:middle line:84% for an ideal gas, correcting the volume, 01:21:31.750 --> 01:21:34.090 align:middle line:90% we're subtracting the covolume. 01:21:34.090 --> 01:21:35.850 align:middle line:84% So it's essentially something that 01:21:35.850 --> 01:21:42.090 align:middle line:84% represents the volume not available to the particles, 01:21:42.090 --> 01:21:46.710 align:middle line:84% because the particles themselves occupy some volume. 01:21:46.710 --> 01:21:55.890 align:middle line:84% And this one that has to do with the attractive forces that 01:21:55.890 --> 01:21:56.490 align:middle line:90% happen-- 01:21:56.490 --> 01:21:57.610 align:middle line:90% that are important. 01:21:57.610 --> 01:21:59.320 align:middle line:84% And we'll return to this, for example, 01:21:59.320 --> 01:22:03.490 align:middle line:84% when we talk about the Lennard-Jones potential. 01:22:03.490 --> 01:22:07.350 align:middle line:90% 01:22:07.350 --> 01:22:12.120 align:middle line:84% The electrostatics of molecules make it such 01:22:12.120 --> 01:22:15.690 align:middle line:84% that the electrons are attracted by the nuclei 01:22:15.690 --> 01:22:18.970 align:middle line:84% of the other molecule, and so there are some attractive forces 01:22:18.970 --> 01:22:20.540 align:middle line:90% up to a certain stage. 01:22:20.540 --> 01:22:27.910 align:middle line:84% And so this makes an additional contribution here. 01:22:27.910 --> 01:22:32.350 align:middle line:84% So if you use this expression here, 01:22:32.350 --> 01:22:42.310 align:middle line:84% it gives you a curve which is similar to the actual curve 01:22:42.310 --> 01:22:45.170 align:middle line:90% for the actual substance. 01:22:45.170 --> 01:22:47.400 align:middle line:84% Of course, we cannot match everything. 01:22:47.400 --> 01:22:50.600 align:middle line:90% So one way is to choose. 01:22:50.600 --> 01:22:57.460 align:middle line:90% It does entail a critical point. 01:22:57.460 --> 01:23:00.220 align:middle line:84% And the critical point for the Van der 01:23:00.220 --> 01:23:04.390 align:middle line:84% Waals has these values that depends 01:23:04.390 --> 01:23:05.780 align:middle line:90% on the values of a and b. 01:23:05.780 --> 01:23:10.090 align:middle line:90% 01:23:10.090 --> 01:23:17.710 align:middle line:84% If I choose to select a and b using the actual values 01:23:17.710 --> 01:23:20.980 align:middle line:84% for water, for the critical pressure 01:23:20.980 --> 01:23:24.310 align:middle line:84% and the critical temperature, I get two values of a and b 01:23:24.310 --> 01:23:28.785 align:middle line:84% so I can have a representation of water. 01:23:28.785 --> 01:23:32.010 align:middle line:90% 01:23:32.010 --> 01:23:37.210 align:middle line:84% It's not exactly good because there is also 01:23:37.210 --> 01:23:39.670 align:middle line:84% an expression for the Van der Waals 01:23:39.670 --> 01:23:44.710 align:middle line:84% models of the specific volume at the critical point, which is 01:23:44.710 --> 01:23:47.080 align:middle line:90% also a combination of a and b. 01:23:47.080 --> 01:23:52.110 align:middle line:84% And if I use these values, it will not match. 01:23:52.110 --> 01:23:56.830 align:middle line:84% So that's already a first error that I do. 01:23:56.830 --> 01:24:02.830 align:middle line:84% But you see that sometimes simple models 01:24:02.830 --> 01:24:08.640 align:middle line:84% capture, at least qualitatively, part of the physics. 01:24:08.640 --> 01:24:11.960 align:middle line:90% 01:24:11.960 --> 01:24:17.725 align:middle line:84% It actually captures also an additional interesting part. 01:24:17.725 --> 01:24:22.480 align:middle line:90% 01:24:22.480 --> 01:24:26.036 align:middle line:84% These curves here are the isotherms. 01:24:26.036 --> 01:24:30.730 align:middle line:90% 01:24:30.730 --> 01:24:31.230 align:middle line:90% OK. 01:24:31.230 --> 01:24:34.100 align:middle line:84% So this is the critical isotherm. 01:24:34.100 --> 01:24:37.490 align:middle line:84% And these are the other isotherms. 01:24:37.490 --> 01:24:42.170 align:middle line:84% As you go farther away from the saturation region or the two 01:24:42.170 --> 01:24:48.590 align:middle line:84% phase region, this becomes an hyperbola, 01:24:48.590 --> 01:24:53.900 align:middle line:84% and that's the ideal gas behavior. 01:24:53.900 --> 01:24:57.160 align:middle line:84% But the interesting thing happens here inside. 01:24:57.160 --> 01:25:02.770 align:middle line:84% Somehow this model allows you to extrapolate also inside 01:25:02.770 --> 01:25:05.820 align:middle line:90% of the saturation dome. 01:25:05.820 --> 01:25:08.550 align:middle line:90% 01:25:08.550 --> 01:25:13.110 align:middle line:90% And so you may go this way. 01:25:13.110 --> 01:25:19.630 align:middle line:84% Then the model takes you up and then down again. 01:25:19.630 --> 01:25:24.890 align:middle line:84% So we go to a minimum and a maximum here. 01:25:24.890 --> 01:25:31.540 align:middle line:84% So that suggests that for as much as this equation represents 01:25:31.540 --> 01:25:39.370 align:middle line:84% some physics, it may be possible to have states this way. 01:25:39.370 --> 01:25:42.680 align:middle line:90% 01:25:42.680 --> 01:25:47.030 align:middle line:84% The only problem is that they are not stable equilibrium 01:25:47.030 --> 01:25:55.560 align:middle line:84% states because, you see, if you impose the equality of chemical 01:25:55.560 --> 01:26:02.100 align:middle line:84% potentials here, if you look at the chemical potential, 01:26:02.100 --> 01:26:06.270 align:middle line:84% the equality is obtained only if you stop here with the liquid 01:26:06.270 --> 01:26:09.450 align:middle line:84% and you start here with the vapor. 01:26:09.450 --> 01:26:17.550 align:middle line:84% And this is why when you do things in practice 01:26:17.550 --> 01:26:24.000 align:middle line:84% and you boil water here in noisy situation, 01:26:24.000 --> 01:26:26.315 align:middle line:84% this is exactly the path that is followed. 01:26:26.315 --> 01:26:29.160 align:middle line:90% 01:26:29.160 --> 01:26:33.400 align:middle line:84% The liquid realizes that there is 01:26:33.400 --> 01:26:36.370 align:middle line:84% a vapor with the same chemical potential in which it 01:26:36.370 --> 01:26:41.680 align:middle line:84% could boil and stay in mutual equilibrium with. 01:26:41.680 --> 01:26:44.920 align:middle line:90% 01:26:44.920 --> 01:26:54.380 align:middle line:84% But in a less noisy environment, you could also go down here. 01:26:54.380 --> 01:26:58.270 align:middle line:84% So it is possible to generate these 01:26:58.270 --> 01:27:06.480 align:middle line:84% that would be metastable states, states 01:27:06.480 --> 01:27:10.800 align:middle line:84% in which the liquid didn't realize yet 01:27:10.800 --> 01:27:14.806 align:middle line:84% that there is a vapor in which it could transform. 01:27:14.806 --> 01:27:18.010 align:middle line:90% 01:27:18.010 --> 01:27:20.440 align:middle line:84% And also going back to the other side, 01:27:20.440 --> 01:27:28.610 align:middle line:84% if you could go up this way, the vapor can go in metastable, 01:27:28.610 --> 01:27:29.930 align:middle line:90% supersaturated. 01:27:29.930 --> 01:27:39.110 align:middle line:84% It means it remains vapor even in the region where, 01:27:39.110 --> 01:27:41.103 align:middle line:84% in a noisy environment, it would condense. 01:27:41.103 --> 01:27:45.430 align:middle line:90% 01:27:45.430 --> 01:27:51.190 align:middle line:84% And this is actually experimentally achievable 01:27:51.190 --> 01:27:53.030 align:middle line:90% only up to this dotted line. 01:27:53.030 --> 01:27:57.540 align:middle line:84% This dotted line is called the spinodal curve. 01:27:57.540 --> 01:28:00.890 align:middle line:90% 01:28:00.890 --> 01:28:04.610 align:middle line:84% It is the locus of all the minima on this side 01:28:04.610 --> 01:28:07.550 align:middle line:90% and the maxima on that side. 01:28:07.550 --> 01:28:14.380 align:middle line:84% And this is because to the left-- 01:28:14.380 --> 01:28:19.780 align:middle line:84% so outside, in this region between the spinodal curve 01:28:19.780 --> 01:28:25.590 align:middle line:84% and the saturation dome, in this region, the conditions 01:28:25.590 --> 01:28:30.990 align:middle line:84% for stability, those second derivatives that we have 01:28:30.990 --> 01:28:36.040 align:middle line:90% developed, are still satisfied. 01:28:36.040 --> 01:28:38.655 align:middle line:90% So the liquid can exist there. 01:28:38.655 --> 01:28:42.400 align:middle line:90% 01:28:42.400 --> 01:28:52.340 align:middle line:84% And of course, it is not completely stable, 01:28:52.340 --> 01:28:53.480 align:middle line:90% so it's metastable. 01:28:53.480 --> 01:28:57.710 align:middle line:84% So if you give it a perturbance which is large enough, 01:28:57.710 --> 01:28:59.450 align:middle line:90% it will collapse. 01:28:59.450 --> 01:29:00.500 align:middle line:90% It will jump. 01:29:00.500 --> 01:29:04.900 align:middle line:84% The liquid will immediately flash into vapor 01:29:04.900 --> 01:29:13.590 align:middle line:84% or the vapor will condense explosively into liquid. 01:29:13.590 --> 01:29:16.140 align:middle line:90% 01:29:16.140 --> 01:29:18.466 align:middle line:90% In this flashing and-- 01:29:18.466 --> 01:29:22.380 align:middle line:90% 01:29:22.380 --> 01:29:24.212 align:middle line:90% oh, I don't know. 01:29:24.212 --> 01:29:27.210 align:middle line:90% 01:29:27.210 --> 01:29:28.130 align:middle line:90% Senior moment. 01:29:28.130 --> 01:29:32.475 align:middle line:84% I don't remember the name of the collapse. 01:29:32.475 --> 01:29:37.260 align:middle line:84% It's a typical phenomenon that can 01:29:37.260 --> 01:29:41.230 align:middle line:84% cause the erosion of, for example, 01:29:41.230 --> 01:29:44.460 align:middle line:90% blades of steam turbines. 01:29:44.460 --> 01:29:45.130 align:middle line:90% Cavitation. 01:29:45.130 --> 01:29:45.630 align:middle line:90% Thanks. 01:29:45.630 --> 01:29:48.660 align:middle line:90% 01:29:48.660 --> 01:29:49.160 align:middle line:90% All right. 01:29:49.160 --> 01:29:52.800 align:middle line:90% 01:29:52.800 --> 01:29:57.570 align:middle line:84% Because the vapor goes to this region and then 01:29:57.570 --> 01:30:01.210 align:middle line:84% it immediately realizes that it has to condense, 01:30:01.210 --> 01:30:04.370 align:middle line:90% and it does it in an abrupt way. 01:30:04.370 --> 01:30:06.890 align:middle line:90% 01:30:06.890 --> 01:30:11.805 align:middle line:84% That's a shock that creates a damage to the structure. 01:30:11.805 --> 01:30:12.305 align:middle line:90% OK. 01:30:12.305 --> 01:30:17.230 align:middle line:90% 01:30:17.230 --> 01:30:17.780 align:middle line:90% All right. 01:30:17.780 --> 01:30:21.650 align:middle line:90% 01:30:21.650 --> 01:30:22.410 align:middle line:90% Yeah. 01:30:22.410 --> 01:30:26.360 align:middle line:90% We still have a few moments. 01:30:26.360 --> 01:30:32.930 align:middle line:84% Maybe we can go ahead with what we-- 01:30:32.930 --> 01:30:36.690 align:middle line:84% the subject that we left on the side, but now we have to return. 01:30:36.690 --> 01:30:43.460 align:middle line:84% We did the first part, and today we'll do just a little bit more. 01:30:43.460 --> 01:30:48.620 align:middle line:84% Remember, we were talking about exergy and this second law 01:30:48.620 --> 01:30:49.410 align:middle line:90% efficiency. 01:30:49.410 --> 01:30:53.990 align:middle line:84% We talked about cogeneration the last time 01:30:53.990 --> 01:30:56.700 align:middle line:84% and you told me that MIT has cogeneration, 01:30:56.700 --> 01:31:02.700 align:middle line:84% which makes me very happy, and district heating. 01:31:02.700 --> 01:31:05.930 align:middle line:90% 01:31:05.930 --> 01:31:07.175 align:middle line:90% OK. 01:31:07.175 --> 01:31:11.980 align:middle line:84% So now that we have developed the simple system 01:31:11.980 --> 01:31:21.620 align:middle line:84% model and the bulk flow model for the exchange of mass, 01:31:21.620 --> 01:31:24.600 align:middle line:90% we can ask the question. 01:31:24.600 --> 01:31:29.570 align:middle line:84% If I have a certain flow at certain conditions, 01:31:29.570 --> 01:31:36.005 align:middle line:84% say, pressure, temperature, and therefore enthalpy, entropy, 01:31:36.005 --> 01:31:39.230 align:middle line:84% and possibly also in this case I forget 01:31:39.230 --> 01:31:43.975 align:middle line:84% about gravitational potential and kinetic energy. 01:31:43.975 --> 01:31:46.790 align:middle line:90% 01:31:46.790 --> 01:31:52.010 align:middle line:84% So we ask ourselves, what is the maximum work 01:31:52.010 --> 01:32:00.030 align:middle line:84% that I can obtain by using that flow in a given environment? 01:32:00.030 --> 01:32:03.020 align:middle line:90% 01:32:03.020 --> 01:32:06.610 align:middle line:84% And here I use the environment in two ways. 01:32:06.610 --> 01:32:09.120 align:middle line:90% 01:32:09.120 --> 01:32:11.790 align:middle line:84% The first way is that I want my machinery 01:32:11.790 --> 01:32:15.820 align:middle line:84% to process the flow without accumulating the material. 01:32:15.820 --> 01:32:20.400 align:middle line:84% So the mass that goes in must also go out 01:32:20.400 --> 01:32:23.650 align:middle line:84% and must be given to the environment. 01:32:23.650 --> 01:32:27.960 align:middle line:90% 01:32:27.960 --> 01:32:31.050 align:middle line:84% And I will give it to the environment at conditions 01:32:31.050 --> 01:32:38.905 align:middle line:84% where I have extracted as much as I could. 01:32:38.905 --> 01:32:41.420 align:middle line:84% So when it is mutual equilibrium with the environment, 01:32:41.420 --> 01:32:47.040 align:middle line:84% essentially, at least from the thermal point of view 01:32:47.040 --> 01:32:51.340 align:middle line:84% and typically also the pressure point of view. 01:32:51.340 --> 01:32:56.750 align:middle line:84% So the environment here provides also an ambient pressure. 01:32:56.750 --> 01:32:59.990 align:middle line:90% 01:32:59.990 --> 01:33:04.160 align:middle line:84% And the other way is that I need the environment 01:33:04.160 --> 01:33:09.000 align:middle line:84% as a dump for dumping the entropy, 01:33:09.000 --> 01:33:14.390 align:middle line:84% because if I have to change the state of my flow 01:33:14.390 --> 01:33:16.760 align:middle line:90% from one to this ambient-- 01:33:16.760 --> 01:33:20.840 align:middle line:84% to environmental conditions, most likely 01:33:20.840 --> 01:33:25.790 align:middle line:84% it will have to change entropy from s1 to sa. 01:33:25.790 --> 01:33:29.900 align:middle line:84% And that change of entropy, suppose 01:33:29.900 --> 01:33:32.990 align:middle line:90% it's a reduction in entropy. 01:33:32.990 --> 01:33:37.460 align:middle line:84% It means that I have to extract entropy from the flow. 01:33:37.460 --> 01:33:40.130 align:middle line:84% That's the purpose of the machinery. 01:33:40.130 --> 01:33:41.690 align:middle line:84% And of course, the machinery will 01:33:41.690 --> 01:33:44.150 align:middle line:84% have to give it somewhere because the machinery operates 01:33:44.150 --> 01:33:44.650 align:middle line:90% cyclically. 01:33:44.650 --> 01:33:46.790 align:middle line:90% It cannot go accumulate entropy. 01:33:46.790 --> 01:33:56.600 align:middle line:84% So the environment is there to accept at least the entropy 01:33:56.600 --> 01:33:59.970 align:middle line:84% that I must extract from the flow in order to exploit it. 01:33:59.970 --> 01:34:04.430 align:middle line:84% But it may also be there to dump also 01:34:04.430 --> 01:34:07.580 align:middle line:84% the entropy that may be produced by irreversibility, 01:34:07.580 --> 01:34:09.170 align:middle line:90% by the machinery. 01:34:09.170 --> 01:34:12.510 align:middle line:84% But if we are asking for the maximum we can do, 01:34:12.510 --> 01:34:14.660 align:middle line:84% we suppose the machinery is reversible so 01:34:14.660 --> 01:34:17.750 align:middle line:84% at least that contribution of the entropy produced 01:34:17.750 --> 01:34:19.820 align:middle line:90% by irreversibility is not there. 01:34:19.820 --> 01:34:23.450 align:middle line:90% And so this is the setup. 01:34:23.450 --> 01:34:26.060 align:middle line:84% As usual, you solve the problem by making an energy 01:34:26.060 --> 01:34:30.960 align:middle line:84% balance and an entropy balance and setting 01:34:30.960 --> 01:34:35.310 align:middle line:84% the entropy produced by irreversibility equal to zero. 01:34:35.310 --> 01:34:39.980 align:middle line:84% This is the result. So the maximum work 01:34:39.980 --> 01:34:43.190 align:middle line:84% is the linear combination of changing enthalpy and changing 01:34:43.190 --> 01:34:46.110 align:middle line:90% entropy represented here. 01:34:46.110 --> 01:34:52.370 align:middle line:84% This is called the exergy of the flow of that bulk flow state 01:34:52.370 --> 01:34:56.390 align:middle line:84% with respect to an environment temperature Ta 01:34:56.390 --> 01:35:02.600 align:middle line:84% and with outlet conditions that are ambient conditions denoted 01:35:02.600 --> 01:35:05.360 align:middle line:90% by a. 01:35:05.360 --> 01:35:14.070 align:middle line:84% Of course, I may also have the opposite problem. 01:35:14.070 --> 01:35:16.360 align:middle line:90% It's exactly the same situation. 01:35:16.360 --> 01:35:22.140 align:middle line:84% Simply that now I don't want to exploit a certain bulk flow 01:35:22.140 --> 01:35:25.140 align:middle line:84% state one, but I want to produce a bulk flow state which 01:35:25.140 --> 01:35:30.720 align:middle line:84% is at conditions different from mutual equilibrium 01:35:30.720 --> 01:35:31.660 align:middle line:90% with the environment. 01:35:31.660 --> 01:35:35.100 align:middle line:84% For example, this is material processing, 01:35:35.100 --> 01:35:39.420 align:middle line:84% or maybe I want to heat up some water. 01:35:39.420 --> 01:35:45.240 align:middle line:84% What is the minimum work that is needed by this machinery 01:35:45.240 --> 01:35:46.920 align:middle line:90% to do it? 01:35:46.920 --> 01:35:51.490 align:middle line:84% Well, again, we use the environment in two ways. 01:35:51.490 --> 01:35:54.720 align:middle line:84% We take the material from the environment. 01:35:54.720 --> 01:35:57.480 align:middle line:84% For example, we go to a mine for raw materials 01:35:57.480 --> 01:36:02.400 align:middle line:84% or we go to a lake or reservoir, the water reservoir 01:36:02.400 --> 01:36:08.680 align:middle line:84% for fresh water, and we also use the environment 01:36:08.680 --> 01:36:12.880 align:middle line:84% as a provider of the entropy that may be needed 01:36:12.880 --> 01:36:14.450 align:middle line:90% in order to achieve that task. 01:36:14.450 --> 01:36:19.870 align:middle line:84% Because if the bulk flow conditions one have more entropy 01:36:19.870 --> 01:36:22.660 align:middle line:84% than ambient conditions, then I need 01:36:22.660 --> 01:36:26.290 align:middle line:90% to give entropy to this flow. 01:36:26.290 --> 01:36:30.040 align:middle line:84% And that entropy, the best way to do it 01:36:30.040 --> 01:36:32.810 align:middle line:84% is not to generate it by irreversibility, 01:36:32.810 --> 01:36:37.870 align:middle line:84% but that is to get it from the environment. 01:36:37.870 --> 01:36:41.860 align:middle line:84% For example, when we heat up water from 20 degrees 01:36:41.860 --> 01:36:52.330 align:middle line:84% to, say, 60 degrees, I mean Celsius, yes, we could also 01:36:52.330 --> 01:36:53.900 align:middle line:90% do it without the environment. 01:36:53.900 --> 01:36:57.010 align:middle line:84% We just could use an electric boiler. 01:36:57.010 --> 01:37:00.010 align:middle line:84% The electric boiler would use some work, 01:37:00.010 --> 01:37:03.430 align:middle line:84% dissipates it by Joule effect so as 01:37:03.430 --> 01:37:10.690 align:middle line:84% to generate the entropy necessary to heat up the water. 01:37:10.690 --> 01:37:13.120 align:middle line:90% OK? 01:37:13.120 --> 01:37:14.860 align:middle line:90% Doesn't use the environment. 01:37:14.860 --> 01:37:17.560 align:middle line:84% Whereas this reversible situation 01:37:17.560 --> 01:37:21.367 align:middle line:84% is like a heat pump setup in which I 01:37:21.367 --> 01:37:27.370 align:middle line:84% do take the entropy from the environment and the environment 01:37:27.370 --> 01:37:32.600 align:middle line:84% gives me that entropy only if I take also some energy. 01:37:32.600 --> 01:37:37.570 align:middle line:84% And that energy gets pumped to my fluid, 01:37:37.570 --> 01:37:41.740 align:middle line:84% and that fluid will receive exactly the entropy 01:37:41.740 --> 01:37:47.200 align:middle line:84% and the energy needed in order to go from here to there. 01:37:47.200 --> 01:37:50.085 align:middle line:84% This is, again, the same exergy as before. 01:37:50.085 --> 01:37:53.840 align:middle line:90% 01:37:53.840 --> 01:37:54.350 align:middle line:90% OK. 01:37:54.350 --> 01:37:57.950 align:middle line:84% Well, I think time is up, isn't it? 01:37:57.950 --> 01:38:02.950 align:middle line:84% So we'll take it from here next time. 01:38:02.950 --> 01:38:11.000 align:middle line:90%