WEBVTT 00:00:00.000 --> 00:00:02.940 align:middle line:90% [SQUEAKING] 00:00:02.940 --> 00:00:04.410 align:middle line:90% [RUSTLING] 00:00:04.410 --> 00:00:12.760 align:middle line:90% [CLICKING] 00:00:12.760 --> 00:00:15.910 align:middle line:84% RAFAEL JARAMILLO: So we've gone through the first law, 00:00:15.910 --> 00:00:19.932 align:middle line:84% and we've gone through heat engines, which means 00:00:19.932 --> 00:00:21.140 align:middle line:90% it's time for the second law. 00:00:21.140 --> 00:00:22.557 align:middle line:84% So we're ready for the second law. 00:00:22.557 --> 00:00:40.680 align:middle line:90% 00:00:40.680 --> 00:00:48.180 align:middle line:84% All right, and this is what I call a plane for act. 00:00:48.180 --> 00:00:56.100 align:middle line:84% So I'm going to present this by way of Clausius's theorem. 00:00:56.100 --> 00:00:59.080 align:middle line:84% You'll see what that is in a minute. 00:00:59.080 --> 00:01:12.720 align:middle line:84% Then, we'll move on to reversible processes 00:01:12.720 --> 00:01:28.180 align:middle line:84% and entropy, and reversible and irreversible processes, 00:01:28.180 --> 00:01:30.480 align:middle line:84% and finally entropy maximization. 00:01:30.480 --> 00:01:39.350 align:middle line:90% 00:01:39.350 --> 00:01:42.250 align:middle line:84% So we talked about entropy in the first lecture 00:01:42.250 --> 00:01:43.270 align:middle line:90% with the baby book. 00:01:43.270 --> 00:01:46.180 align:middle line:84% And I think everybody does have this feeling of entropy 00:01:46.180 --> 00:01:50.220 align:middle line:84% as being mixed up and it's disorder, and that's true. 00:01:50.220 --> 00:01:51.010 align:middle line:90% That's true. 00:01:51.010 --> 00:01:53.490 align:middle line:84% And so one could state the second law of thermodynamics 00:01:53.490 --> 00:01:55.290 align:middle line:90% as something like the entropy-- 00:01:55.290 --> 00:02:00.020 align:middle line:84% the disorder in the universe is always increasing. 00:02:00.020 --> 00:02:02.600 align:middle line:84% Rather like the statement that the energy of the universe 00:02:02.600 --> 00:02:07.100 align:middle line:84% is always constant, that's not obviously useful for us 00:02:07.100 --> 00:02:08.900 align:middle line:90% as scientists and engineers. 00:02:08.900 --> 00:02:13.080 align:middle line:84% The entropy of the universe is always increasing. 00:02:13.080 --> 00:02:15.680 align:middle line:84% So today, we're going to take a different approach 00:02:15.680 --> 00:02:22.270 align:middle line:84% to the second model, which is very classical and deductive. 00:02:22.270 --> 00:02:24.880 align:middle line:84% And by the end of today, you will 00:02:24.880 --> 00:02:27.250 align:middle line:84% have seen the second law and entropy 00:02:27.250 --> 00:02:29.020 align:middle line:84% in this classical and deductive way. 00:02:29.020 --> 00:02:31.660 align:middle line:84% You will have also gotten this sense for it-- 00:02:31.660 --> 00:02:33.453 align:middle line:84% this more hands-on sense for entropy 00:02:33.453 --> 00:02:34.870 align:middle line:84% as being disordered and mixed up-- 00:02:34.870 --> 00:02:40.630 align:middle line:84% and, hopefully, have a better understanding there 00:02:40.630 --> 00:02:42.620 align:middle line:84% for having seen it in different ways. 00:02:42.620 --> 00:02:46.185 align:middle line:90% So I call this a play in for x. 00:02:46.185 --> 00:02:47.140 align:middle line:90% I'm playing for x-- 00:02:47.140 --> 00:02:50.500 align:middle line:84% Clausius, reversible, irreversible, and entropy 00:02:50.500 --> 00:02:51.280 align:middle line:90% maximization. 00:02:51.280 --> 00:02:54.990 align:middle line:90% 00:02:54.990 --> 00:02:57.240 align:middle line:84% All right, one, Clausius's theorem. 00:02:57.240 --> 00:03:05.500 align:middle line:90% 00:03:05.500 --> 00:03:07.060 align:middle line:90% So this is Clausius's theorem. 00:03:07.060 --> 00:03:18.270 align:middle line:90% 00:03:18.270 --> 00:03:20.620 align:middle line:84% So this should look familiar because we arrived at this 00:03:20.620 --> 00:03:22.630 align:middle line:90% at the end of the Carnot cycle. 00:03:22.630 --> 00:03:24.562 align:middle line:84% I didn't call it Clausius's theorem. 00:03:24.562 --> 00:03:26.020 align:middle line:84% And I told you that there was going 00:03:26.020 --> 00:03:30.740 align:middle line:84% to be something funky about this dQ over T quantity, 00:03:30.740 --> 00:03:32.310 align:middle line:90% and it's important. 00:03:32.310 --> 00:03:34.940 align:middle line:84% So that's more or less why we did the Carnot 00:03:34.940 --> 00:03:36.560 align:middle line:90% cycle to motivate this. 00:03:36.560 --> 00:03:41.210 align:middle line:84% This is a cyclic process, any cyclic process. 00:03:41.210 --> 00:03:45.650 align:middle line:90% 00:03:45.650 --> 00:03:49.720 align:middle line:84% It doesn't have to be a Carnot engine to be anything at all-- 00:03:49.720 --> 00:03:55.530 align:middle line:84% any cyclic process where the state variables 00:03:55.530 --> 00:04:02.640 align:middle line:84% return to starting point, which is just rephrasing what it 00:04:02.640 --> 00:04:07.120 align:middle line:90% means to be a cyclic process. 00:04:07.120 --> 00:04:09.540 align:middle line:84% And so this is super general right now. 00:04:09.540 --> 00:04:13.620 align:middle line:84% We have any system with initial temperature, 00:04:13.620 --> 00:04:20.690 align:middle line:84% initial volume, pressure, and it's a collection of stuff. 00:04:20.690 --> 00:04:22.420 align:middle line:84% And you're going through any process 00:04:22.420 --> 00:04:28.250 align:middle line:84% at all that brings you back to the starting point. 00:04:28.250 --> 00:04:29.900 align:middle line:84% I think my new pen-- by the way, I 00:04:29.900 --> 00:04:33.830 align:middle line:84% bought a bunch of different Sharpies, fat, thin, ultra 00:04:33.830 --> 00:04:35.930 align:middle line:90% thin, fine, ultra fine. 00:04:35.930 --> 00:04:38.810 align:middle line:84% Sharpie markers, they have Sharpie brand markers now. 00:04:38.810 --> 00:04:39.740 align:middle line:90% I was trying them all. 00:04:39.740 --> 00:04:45.420 align:middle line:84% Anyway, so that was Clausius's theorem. 00:04:45.420 --> 00:04:47.660 align:middle line:90% So we're going to use it now. 00:04:47.660 --> 00:04:49.895 align:middle line:84% Two, reversible processes and entropy. 00:04:49.895 --> 00:05:02.930 align:middle line:90% 00:05:02.930 --> 00:05:04.980 align:middle line:90% Sorry, give me a minute. 00:05:04.980 --> 00:05:06.230 align:middle line:90% Can I ask you guys a question? 00:05:06.230 --> 00:05:11.390 align:middle line:90% 00:05:11.390 --> 00:05:13.340 align:middle line:90% It's less than or equal to 0. 00:05:13.340 --> 00:05:18.080 align:middle line:84% Does anyone remember under what conditions it's equal to 0? 00:05:18.080 --> 00:05:20.690 align:middle line:84% We saw a case when it was equal to 0. 00:05:20.690 --> 00:05:22.190 align:middle line:84% AUDIENCE: And we took like, forever. 00:05:22.190 --> 00:05:25.902 align:middle line:84% Like, it was always in equilibrium for every step. 00:05:25.902 --> 00:05:27.860 align:middle line:84% RAFAEL JARAMILLO: Always in equilibrium, right. 00:05:27.860 --> 00:05:29.690 align:middle line:90% That's exactly right. 00:05:29.690 --> 00:05:31.190 align:middle line:84% There's another word for that, which 00:05:31.190 --> 00:05:33.152 align:middle line:90% means a reversible process. 00:05:33.152 --> 00:05:34.610 align:middle line:84% But you're absolutely right, always 00:05:34.610 --> 00:05:35.693 align:middle line:90% in equilibrium every step. 00:05:35.693 --> 00:05:38.390 align:middle line:84% Reversible process, you get the equal sign. 00:05:38.390 --> 00:05:42.290 align:middle line:84% The less than, if you remember, was the heat engine that 00:05:42.290 --> 00:05:45.035 align:middle line:90% was less efficient than ideal. 00:05:45.035 --> 00:05:54.550 align:middle line:90% 00:05:54.550 --> 00:06:14.570 align:middle line:84% Now, we're going to hypothesize that there exists for which 00:06:14.570 --> 00:06:16.160 align:middle line:90% every step is reversible. 00:06:16.160 --> 00:06:26.290 align:middle line:90% 00:06:26.290 --> 00:06:38.000 align:middle line:84% And we're going to consider a cyclic process along two 00:06:38.000 --> 00:06:48.020 align:middle line:90% reversible paths, R1 and R2. 00:06:48.020 --> 00:06:50.000 align:middle line:90% And this is what I mean. 00:06:50.000 --> 00:06:53.570 align:middle line:84% We have some state A. We have some state B. 00:06:53.570 --> 00:07:00.740 align:middle line:84% And there are two ways to get from A to B, 00:07:00.740 --> 00:07:03.980 align:middle line:84% and both of these paths are reversible. 00:07:03.980 --> 00:07:10.537 align:middle line:90% 00:07:10.537 --> 00:07:12.620 align:middle line:84% All right, so let's think about the different ways 00:07:12.620 --> 00:07:13.680 align:middle line:90% we can do this. 00:07:13.680 --> 00:07:25.900 align:middle line:84% We can start at A, and we can go to B. 00:07:25.900 --> 00:07:30.160 align:middle line:84% And then, we can go back to A. So I'm 00:07:30.160 --> 00:07:34.210 align:middle line:84% going to call this minus R2 because they're going 00:07:34.210 --> 00:07:38.540 align:middle line:90% the opposite way of path R2. 00:07:38.540 --> 00:07:40.660 align:middle line:90% So this is a clockwise-- 00:07:40.660 --> 00:07:43.290 align:middle line:90% 00:07:43.290 --> 00:07:47.550 align:middle line:84% a clockwise cycle, R going clockwise. 00:07:47.550 --> 00:07:50.220 align:middle line:90% So let's calculate this thing. 00:07:50.220 --> 00:08:17.210 align:middle line:84% VQ over T equals AB dQ over T along path R1 plus BA dQ 00:08:17.210 --> 00:08:20.720 align:middle line:90% over T along path minus R2. 00:08:20.720 --> 00:08:26.584 align:middle line:90% 00:08:26.584 --> 00:08:28.090 align:middle line:90% I'll rewrite the first term. 00:08:28.090 --> 00:08:30.920 align:middle line:90% 00:08:30.920 --> 00:08:33.260 align:middle line:84% I mean I'll just copy the first term. 00:08:33.260 --> 00:08:38.000 align:middle line:84% And by virtue of the fact that path R2 is reversible, 00:08:38.000 --> 00:08:41.299 align:middle line:84% I can simply say it's the reverse of going 00:08:41.299 --> 00:08:49.860 align:middle line:90% from A to B along path R2. 00:08:49.860 --> 00:08:51.720 align:middle line:84% So going backwards along path, R2 00:08:51.720 --> 00:08:55.380 align:middle line:84% is just the opposite of going forwards along path R2. 00:08:55.380 --> 00:08:57.180 align:middle line:84% And by Clausius's theorem, we know 00:08:57.180 --> 00:09:00.880 align:middle line:90% this is less than or equal to 0. 00:09:00.880 --> 00:09:03.370 align:middle line:84% Now, we're going to consider going the other way. 00:09:03.370 --> 00:09:12.920 align:middle line:84% We're going to start at A, and we're going to go along path R2 00:09:12.920 --> 00:09:17.030 align:middle line:90% and back along minus path R1. 00:09:17.030 --> 00:09:22.130 align:middle line:90% So this is counterclockwise. 00:09:22.130 --> 00:09:27.340 align:middle line:90% 00:09:27.340 --> 00:09:30.010 align:middle line:90% Let's do the same calculation. 00:09:30.010 --> 00:09:45.520 align:middle line:84% dQ over T equals from A to B dQ over T along R2 plus B 00:09:45.520 --> 00:09:52.480 align:middle line:84% to A dQ over T along path minus R1. 00:09:52.480 --> 00:10:02.480 align:middle line:84% And just as before, this is going 00:10:02.480 --> 00:10:06.250 align:middle line:84% to be the negative of taking path R1. 00:10:06.250 --> 00:10:10.940 align:middle line:90% 00:10:10.940 --> 00:10:12.980 align:middle line:84% And by Clausius's theorem, this is also 00:10:12.980 --> 00:10:14.630 align:middle line:90% less than or equal to 0. 00:10:14.630 --> 00:10:16.775 align:middle line:90% So these are by Clausius. 00:10:16.775 --> 00:10:22.580 align:middle line:90% 00:10:22.580 --> 00:10:25.100 align:middle line:84% Clausius is a fellow with an S at the end of his name, 00:10:25.100 --> 00:10:28.220 align:middle line:84% so that apostrophe is trying to be at the end after the S, 00:10:28.220 --> 00:10:32.490 align:middle line:84% but anyway, Clausius's theorem, I should say. 00:10:32.490 --> 00:10:35.880 align:middle line:84% So by Clausius's theorem, we have that both of these 00:10:35.880 --> 00:10:38.100 align:middle line:90% are less than or equal to 0. 00:10:38.100 --> 00:10:40.020 align:middle line:90% One way we went clockwise. 00:10:40.020 --> 00:10:41.790 align:middle line:84% The other way we went counterclockwise. 00:10:41.790 --> 00:10:43.500 align:middle line:90% But both were cyclic processes. 00:10:43.500 --> 00:10:45.920 align:middle line:84% We started at A. We returned to A. We started at A, 00:10:45.920 --> 00:10:55.950 align:middle line:84% and we returned to A. We want to compare 00:10:55.950 --> 00:11:01.665 align:middle line:84% clockwise and counterclockwise circuits. 00:11:01.665 --> 00:11:11.520 align:middle line:90% 00:11:11.520 --> 00:11:13.710 align:middle line:84% Compare clockwise and counterclockwise circuits-- 00:11:13.710 --> 00:11:23.180 align:middle line:84% both of these things have to be less than or equal to 0. 00:11:23.180 --> 00:11:29.210 align:middle line:84% So what do you observe about these terms? 00:11:29.210 --> 00:11:33.780 align:middle line:90% 00:11:33.780 --> 00:11:36.155 align:middle line:84% AUDIENCE: It's like you just take the negative of. 00:11:36.155 --> 00:11:38.780 align:middle line:84% RAFAEL JARAMILLO: They're just a negative of each other, right. 00:11:38.780 --> 00:11:41.700 align:middle line:90% They're negative of each other. 00:11:41.700 --> 00:11:46.110 align:middle line:84% All right, so if something has to be less than equal to 0-- 00:11:46.110 --> 00:11:50.760 align:middle line:84% y is less than or equal to 0, and also minus y 00:11:50.760 --> 00:11:52.980 align:middle line:90% is less than or equal to 0. 00:11:52.980 --> 00:11:54.255 align:middle line:90% Then what can you say about y? 00:11:54.255 --> 00:11:57.260 align:middle line:90% 00:11:57.260 --> 00:11:58.250 align:middle line:90% AUDIENCE: It's 0. 00:11:58.250 --> 00:12:01.850 align:middle line:84% RAFAEL JARAMILLO: Yeah, then y equals 0. 00:12:01.850 --> 00:12:03.750 align:middle line:84% That's the only way you can satisfy that. 00:12:03.750 --> 00:12:05.190 align:middle line:90% So that's what's going on here. 00:12:05.190 --> 00:12:08.690 align:middle line:84% So that means that term equals 0. 00:12:08.690 --> 00:12:10.430 align:middle line:90% So let's write that down. 00:12:10.430 --> 00:12:18.140 align:middle line:84% From A to B dQ over T along reversible path 1 00:12:18.140 --> 00:12:25.550 align:middle line:84% minus from A to B dQ over T along reversible path 00:12:25.550 --> 00:12:27.380 align:middle line:90% 2 equals 0. 00:12:27.380 --> 00:12:30.670 align:middle line:90% 00:12:30.670 --> 00:12:40.650 align:middle line:84% That means from A to B dQ over T for reversal path 1 00:12:40.650 --> 00:12:48.630 align:middle line:84% equals from A to B dQ over T along a reversible path 00:12:48.630 --> 00:12:52.130 align:middle line:90% 2, just rearranging. 00:12:52.130 --> 00:12:54.990 align:middle line:90% 00:12:54.990 --> 00:12:58.530 align:middle line:84% What do we call a quantity whose change doesn't 00:12:58.530 --> 00:13:02.220 align:middle line:90% depend on the path you take? 00:13:02.220 --> 00:13:05.685 align:middle line:84% From A to B, we have two paths here-- 00:13:05.685 --> 00:13:10.440 align:middle line:90% 00:13:10.440 --> 00:13:13.680 align:middle line:84% path reversible 1 and path reversible 2. 00:13:13.680 --> 00:13:17.730 align:middle line:84% And this quantity integrated along that process 00:13:17.730 --> 00:13:21.030 align:middle line:84% doesn't depend on which process we took, R1 or R2. 00:13:21.030 --> 00:13:22.920 align:middle line:84% What sort of a function do we call that? 00:13:22.920 --> 00:13:26.290 align:middle line:90% 00:13:26.290 --> 00:13:27.850 align:middle line:90% AUDIENCE: Path independent. 00:13:27.850 --> 00:13:29.308 align:middle line:84% RAFAEL JARAMILLO: Path independent. 00:13:29.308 --> 00:13:32.360 align:middle line:84% And for thermo, we have a specific fancy name for path 00:13:32.360 --> 00:13:32.860 align:middle line:90% independent. 00:13:32.860 --> 00:13:33.550 align:middle line:90% AUDIENCE: State functions. 00:13:33.550 --> 00:13:35.110 align:middle line:84% RAFAEL JARAMILLO: State functions, right, exactly. 00:13:35.110 --> 00:13:35.652 align:middle line:90% That's great. 00:13:35.652 --> 00:13:38.350 align:middle line:90% 00:13:38.350 --> 00:13:39.610 align:middle line:90% So we're going to-- 00:13:39.610 --> 00:13:42.100 align:middle line:84% and this is historically how this came about, by the way. 00:13:42.100 --> 00:13:48.340 align:middle line:90% This is based on a lot of data. 00:13:48.340 --> 00:13:50.380 align:middle line:84% Don't worry that you're not as smart as people 00:13:50.380 --> 00:13:51.730 align:middle line:84% used to be in the past and somehow this 00:13:51.730 --> 00:13:52.750 align:middle line:90% isn't obvious to you. 00:13:52.750 --> 00:13:56.620 align:middle line:84% This conclusion, which I'm about to just hand you 00:13:56.620 --> 00:13:58.870 align:middle line:84% 19 minutes into the hour, is based 00:13:58.870 --> 00:14:02.350 align:middle line:84% on centuries of data and people finally 00:14:02.350 --> 00:14:06.773 align:middle line:90% hypothesizing the existence-- 00:14:06.773 --> 00:14:08.190 align:middle line:84% maybe not centuries of data, maybe 00:14:08.190 --> 00:14:12.930 align:middle line:84% like 50 to 75 years of data-- entropy S 00:14:12.930 --> 00:14:15.945 align:middle line:90% as a new state function. 00:14:15.945 --> 00:14:21.020 align:middle line:90% 00:14:21.020 --> 00:14:22.030 align:middle line:90% Why? 00:14:22.030 --> 00:14:26.380 align:middle line:84% Because it seems to be something which 00:14:26.380 --> 00:14:28.660 align:middle line:90% has an exact differential. 00:14:28.660 --> 00:14:33.800 align:middle line:90% 00:14:33.800 --> 00:14:35.690 align:middle line:90% In a reversible process-- 00:14:35.690 --> 00:14:39.080 align:middle line:84% any reversible process, not any specific reversible process, 00:14:39.080 --> 00:14:41.790 align:middle line:90% but any reversible process-- 00:14:41.790 --> 00:14:45.910 align:middle line:84% dQ over T was equal to the exact differential of entropy. 00:14:45.910 --> 00:14:49.470 align:middle line:90% 00:14:49.470 --> 00:14:57.620 align:middle line:84% What that means here is that this is entropy of B 00:14:57.620 --> 00:15:02.930 align:middle line:84% minus entropy of A, or delta entropy. 00:15:02.930 --> 00:15:05.965 align:middle line:90% 00:15:05.965 --> 00:15:06.840 align:middle line:90% So there you have it. 00:15:06.840 --> 00:15:12.280 align:middle line:84% This is a very deductive way of postulating 00:15:12.280 --> 00:15:15.010 align:middle line:84% that entropy exists, and it's a state function. 00:15:15.010 --> 00:15:18.520 align:middle line:84% We found an integral of an integrand that 00:15:18.520 --> 00:15:20.620 align:middle line:90% doesn't depend on the path. 00:15:20.620 --> 00:15:22.002 align:middle line:90% That's it. 00:15:22.002 --> 00:15:23.710 align:middle line:84% From that, you say I'm going to postulate 00:15:23.710 --> 00:15:25.120 align:middle line:90% that there exists entropy. 00:15:25.120 --> 00:15:29.470 align:middle line:84% Its total differential is this for a reversible process, which 00:15:29.470 --> 00:15:34.430 align:middle line:84% means I can calculate its change between states along any path-- 00:15:34.430 --> 00:15:41.030 align:middle line:90% 00:15:41.030 --> 00:15:43.580 align:middle line:90% along any reversible path. 00:15:43.580 --> 00:15:52.780 align:middle line:90% 00:15:52.780 --> 00:15:55.720 align:middle line:90% Let's keep moving. 00:15:55.720 --> 00:16:11.180 align:middle line:84% Next, three, reversible and irreversible processes, 00:16:11.180 --> 00:16:13.653 align:middle line:84% reversible and irreversible processes. 00:16:13.653 --> 00:16:16.070 align:middle line:84% All right, so we've hypothesized that there are reversible 00:16:16.070 --> 00:16:16.740 align:middle line:90% processes. 00:16:16.740 --> 00:16:32.220 align:middle line:84% Now, let's hypothesize that there exists processes that 00:16:32.220 --> 00:16:42.148 align:middle line:90% are irreversible, irreversible. 00:16:42.148 --> 00:16:43.690 align:middle line:84% And we're going to set up another one 00:16:43.690 --> 00:16:47.380 align:middle line:84% of these funny thought experiments. 00:16:47.380 --> 00:16:59.570 align:middle line:84% Consider states A and B, connected 00:16:59.570 --> 00:17:06.114 align:middle line:90% by both reversible and-- 00:17:06.114 --> 00:17:15.280 align:middle line:90% 00:17:15.280 --> 00:17:17.241 align:middle line:84% reversible and irreversible processes. 00:17:17.241 --> 00:17:20.680 align:middle line:84% So here's state A. Here's state B. 00:17:20.680 --> 00:17:22.599 align:middle line:84% And here, I'll draw a nice smooth line. 00:17:22.599 --> 00:17:27.030 align:middle line:84% Let's say this is a very slow process. 00:17:27.030 --> 00:17:28.560 align:middle line:90% Call that reversible. 00:17:28.560 --> 00:17:30.120 align:middle line:84% And I'm going to draw a wiggly line. 00:17:30.120 --> 00:17:31.520 align:middle line:90% This is a very fast process. 00:17:31.520 --> 00:17:32.520 align:middle line:90% You're not very careful. 00:17:32.520 --> 00:17:34.650 align:middle line:90% You carry it out in a rush. 00:17:34.650 --> 00:17:36.280 align:middle line:90% I'll call that irreversible. 00:17:36.280 --> 00:17:39.120 align:middle line:84% So we have two states connected by a reversible process 00:17:39.120 --> 00:17:43.030 align:middle line:90% and an irreversible process. 00:17:43.030 --> 00:17:45.450 align:middle line:84% So now, let's go back to Clausius. 00:17:45.450 --> 00:17:57.770 align:middle line:84% Let's calculate dQ over T. This equals from A 00:17:57.770 --> 00:18:08.250 align:middle line:84% to B dQ over T along the irreversible process 00:18:08.250 --> 00:18:18.270 align:middle line:84% plus from B to A dQ over T along the reversible process. 00:18:18.270 --> 00:18:20.250 align:middle line:90% Now, why do I do it that way? 00:18:20.250 --> 00:18:30.000 align:middle line:90% 00:18:30.000 --> 00:18:33.035 align:middle line:84% I wanted to go from A to B and back again. 00:18:33.035 --> 00:18:36.210 align:middle line:90% 00:18:36.210 --> 00:18:37.290 align:middle line:90% Can I go this way? 00:18:37.290 --> 00:18:38.970 align:middle line:90% Can I go counterclockwise? 00:18:38.970 --> 00:18:42.500 align:middle line:90% 00:18:42.500 --> 00:18:46.307 align:middle line:84% AUDIENCE: No, because the one on top is not reversible. 00:18:46.307 --> 00:18:47.640 align:middle line:90% RAFAEL JARAMILLO: Yeah, exactly. 00:18:47.640 --> 00:18:49.050 align:middle line:90% AUDIENCE: Can't go backwards. 00:18:49.050 --> 00:18:50.450 align:middle line:90% RAFAEL JARAMILLO: This is-- 00:18:50.450 --> 00:18:52.100 align:middle line:84% I told you can't run this backwards. 00:18:52.100 --> 00:18:53.402 align:middle line:90% I said it was irreversible. 00:18:53.402 --> 00:18:55.610 align:middle line:84% So you have to believe me because I'm in charge here. 00:18:55.610 --> 00:18:58.110 align:middle line:90% So you can't run this backwards. 00:18:58.110 --> 00:19:05.010 align:middle line:84% So if you're going to get from B to A, it can't be that way. 00:19:05.010 --> 00:19:10.590 align:middle line:90% It's got to be that way. 00:19:10.590 --> 00:19:11.728 align:middle line:90% It's got to be that way. 00:19:11.728 --> 00:19:12.270 align:middle line:90% That's right. 00:19:12.270 --> 00:19:16.360 align:middle line:90% 00:19:16.360 --> 00:19:17.330 align:middle line:90% So there we are. 00:19:17.330 --> 00:19:20.210 align:middle line:90% 00:19:20.210 --> 00:19:22.545 align:middle line:84% All right, so that's the way, and Clausius's theorem 00:19:22.545 --> 00:19:23.045 align:middle line:90% says that. 00:19:23.045 --> 00:19:25.710 align:middle line:90% 00:19:25.710 --> 00:19:38.860 align:middle line:84% We're running the reversal process backwards 00:19:38.860 --> 00:19:39.820 align:middle line:90% to complete the cycle. 00:19:39.820 --> 00:19:48.240 align:middle line:90% 00:19:48.240 --> 00:19:56.115 align:middle line:84% Now, we're going to use this thing-- which we used before-- 00:19:56.115 --> 00:20:01.360 align:middle line:90% 00:20:01.360 --> 00:20:04.210 align:middle line:84% minus the reversible process equals 00:20:04.210 --> 00:20:06.685 align:middle line:90% minus the forward process. 00:20:06.685 --> 00:20:12.580 align:middle line:90% 00:20:12.580 --> 00:20:15.025 align:middle line:84% And we postulated this is the change of a state function. 00:20:15.025 --> 00:20:20.310 align:middle line:90% 00:20:20.310 --> 00:20:23.580 align:middle line:84% It's the integral of the exact integrand, 00:20:23.580 --> 00:20:26.250 align:middle line:84% the exact differential of the state function 00:20:26.250 --> 00:20:28.500 align:middle line:90% as we go from A to B. 00:20:28.500 --> 00:20:30.600 align:middle line:84% All right, so we're going to use these. 00:20:30.600 --> 00:20:35.390 align:middle line:84% We're going to combine, and we get the following. 00:20:35.390 --> 00:20:38.610 align:middle line:90% 00:20:38.610 --> 00:20:46.640 align:middle line:84% dQ over T is less than or equal to entropy-- 00:20:46.640 --> 00:20:47.650 align:middle line:90% wait, I'm sorry. 00:20:47.650 --> 00:20:51.200 align:middle line:84% It shouldn't be the cyclic interval. 00:20:51.200 --> 00:20:58.445 align:middle line:84% From A to B dQ over T is less than or equal-- 00:20:58.445 --> 00:21:04.790 align:middle line:90% 00:21:04.790 --> 00:21:06.093 align:middle line:90% slow down, right. 00:21:06.093 --> 00:21:07.760 align:middle line:84% Third time's going to be the charm here. 00:21:07.760 --> 00:21:14.750 align:middle line:84% From A to B dQ over T is less than 00:21:14.750 --> 00:21:20.580 align:middle line:84% or equal to the difference entropy of B minus entropy 00:21:20.580 --> 00:21:24.600 align:middle line:90% of A, which is delta entropy. 00:21:24.600 --> 00:21:26.370 align:middle line:90% This is it. 00:21:26.370 --> 00:21:29.460 align:middle line:84% This is combining the previous three slides. 00:21:29.460 --> 00:21:32.010 align:middle line:90% We're using Clausius's theorem. 00:21:32.010 --> 00:21:35.130 align:middle line:84% We have the less than or equal from that. 00:21:35.130 --> 00:21:51.950 align:middle line:84% And it's equality for a reversible process. 00:21:51.950 --> 00:21:53.000 align:middle line:90% You get the equality. 00:21:53.000 --> 00:21:56.860 align:middle line:90% 00:21:56.860 --> 00:22:00.250 align:middle line:84% For irreversible process, you get the less than. 00:22:00.250 --> 00:22:04.160 align:middle line:90% 00:22:04.160 --> 00:22:06.430 align:middle line:84% All right, we're still just kind of playing with math. 00:22:06.430 --> 00:22:07.990 align:middle line:84% It doesn't feel like we're there yet. 00:22:07.990 --> 00:22:09.948 align:middle line:84% You're just-- I'm just showing you some things. 00:22:09.948 --> 00:22:15.280 align:middle line:90% 00:22:15.280 --> 00:22:16.390 align:middle line:90% We'll get there. 00:22:16.390 --> 00:22:19.060 align:middle line:90% All right, so here's there. 00:22:19.060 --> 00:22:20.530 align:middle line:90% This is where we're getting-- 00:22:20.530 --> 00:22:34.255 align:middle line:84% entropy maximization in an isolated system. 00:22:34.255 --> 00:22:47.230 align:middle line:90% 00:22:47.230 --> 00:22:49.117 align:middle line:84% And this is where the cosmologists get off, 00:22:49.117 --> 00:22:50.700 align:middle line:84% and they say, oh yeah, I told you this 00:22:50.700 --> 00:22:51.780 align:middle line:90% was about the universe. 00:22:51.780 --> 00:22:55.500 align:middle line:90% But again, it's really not. 00:22:55.500 --> 00:22:58.570 align:middle line:90% So an isolated system-- 00:22:58.570 --> 00:23:06.070 align:middle line:84% any process from A to B dQ over T 00:23:06.070 --> 00:23:10.180 align:middle line:84% is less than or equal to the entropy of B 00:23:10.180 --> 00:23:14.050 align:middle line:84% minus the entropy of A. And already, we've 00:23:14.050 --> 00:23:17.330 align:middle line:84% come a really long way just to be able to write that down. 00:23:17.330 --> 00:23:19.120 align:middle line:90% So I want you to recognize that. 00:23:19.120 --> 00:23:20.800 align:middle line:84% Now, we're going to isolate the system-- 00:23:20.800 --> 00:23:26.000 align:middle line:90% 00:23:26.000 --> 00:23:29.840 align:middle line:84% isolate the system so that dQ equals 0. 00:23:29.840 --> 00:23:33.230 align:middle line:84% All I'm really doing here is thermally insulating it. 00:23:33.230 --> 00:23:34.550 align:middle line:90% I'm isolating the system. 00:23:34.550 --> 00:23:37.535 align:middle line:84% I'm putting some thermal insulation around it. 00:23:37.535 --> 00:23:40.640 align:middle line:84% I'm making sure that no heat can flow across the boundary. 00:23:40.640 --> 00:23:43.170 align:middle line:90% That's what I'm doing here. 00:23:43.170 --> 00:23:48.390 align:middle line:84% And so if dQ equals 0, what is this integral? 00:23:48.390 --> 00:23:49.545 align:middle line:90% AUDIENCE: It's just 0. 00:23:49.545 --> 00:23:50.670 align:middle line:90% RAFAEL JARAMILLO: 0, right. 00:23:50.670 --> 00:24:02.320 align:middle line:84% So I'm going to take that integral to be 0 and rearrange 00:24:02.320 --> 00:24:03.520 align:middle line:90% the equation. 00:24:03.520 --> 00:24:07.480 align:middle line:84% And I get this very simple and, as we'll 00:24:07.480 --> 00:24:12.520 align:middle line:90% see, very profound inequality. 00:24:12.520 --> 00:24:16.400 align:middle line:90% 00:24:16.400 --> 00:24:37.455 align:middle line:84% Entropy never decreases for any process in an isolated system. 00:24:37.455 --> 00:24:45.127 align:middle line:90% 00:24:45.127 --> 00:24:46.210 align:middle line:90% That's what we just found. 00:24:46.210 --> 00:24:58.220 align:middle line:84% This inequality orders the states in time. 00:24:58.220 --> 00:24:59.780 align:middle line:84% This is the emergence of the error 00:24:59.780 --> 00:25:02.690 align:middle line:90% of time for thermodynamics. 00:25:02.690 --> 00:25:04.760 align:middle line:84% We don't know anything about this system. 00:25:04.760 --> 00:25:06.478 align:middle line:90% We don't a thing about it. 00:25:06.478 --> 00:25:07.770 align:middle line:90% We don't know what A and B are. 00:25:07.770 --> 00:25:08.730 align:middle line:90% We don't know what it's made of. 00:25:08.730 --> 00:25:09.830 align:middle line:84% We don't know if this is the universe 00:25:09.830 --> 00:25:10.880 align:middle line:90% or your beaker in your lab. 00:25:10.880 --> 00:25:12.213 align:middle line:90% We don't know anything about it. 00:25:12.213 --> 00:25:17.130 align:middle line:84% But we do know that no process can decrease its entropy. 00:25:17.130 --> 00:25:21.270 align:middle line:84% Because if a process goes from A to B, then the entropy of B 00:25:21.270 --> 00:25:24.871 align:middle line:84% has to be at least as large as the entropy of A. 00:25:24.871 --> 00:25:30.520 align:middle line:84% That's wacky This is actually a form of the second law. 00:25:30.520 --> 00:25:35.820 align:middle line:90% 00:25:35.820 --> 00:25:38.330 align:middle line:84% So there's about two dozen statements on the second law. 00:25:38.330 --> 00:25:40.667 align:middle line:84% There's a website devoted to them. 00:25:40.667 --> 00:25:42.500 align:middle line:84% It's kind of funny, kind of a goofy website. 00:25:42.500 --> 00:25:45.380 align:middle line:84% But this is one way you'll see it written. 00:25:45.380 --> 00:25:48.460 align:middle line:84% This is one way you'll see it written. 00:25:48.460 --> 00:25:51.760 align:middle line:84% So, all right, so anyway, that's crazy. 00:25:51.760 --> 00:25:52.390 align:middle line:90% That's crazy. 00:25:52.390 --> 00:25:53.770 align:middle line:90% What was the equality for? 00:25:53.770 --> 00:25:56.037 align:middle line:90% When did S of B equal S of A? 00:25:56.037 --> 00:25:56.995 align:middle line:90% What kind of a process? 00:25:56.995 --> 00:26:03.430 align:middle line:90% 00:26:03.430 --> 00:26:04.630 align:middle line:90% AUDIENCE: Reversible. 00:26:04.630 --> 00:26:06.130 align:middle line:84% RAFAEL JARAMILLO: Reversible, right. 00:26:06.130 --> 00:26:20.755 align:middle line:84% So for a reversible process, reversible SA equals SB. 00:26:20.755 --> 00:26:23.490 align:middle line:90% 00:26:23.490 --> 00:26:33.370 align:middle line:84% Irreversible-- irreversible was the inequality, the less than. 00:26:33.370 --> 00:26:43.870 align:middle line:84% S of A is less than S of B. So reversible process 00:26:43.870 --> 00:26:46.990 align:middle line:84% is one for which things are always in equilibrium, 00:26:46.990 --> 00:26:47.740 align:middle line:90% and they equal. 00:26:47.740 --> 00:26:49.390 align:middle line:84% And the entropy remains the same. 00:26:49.390 --> 00:26:51.700 align:middle line:84% Any irreversible process will increase 00:26:51.700 --> 00:26:52.990 align:middle line:90% the entropy of this system. 00:26:52.990 --> 00:27:00.600 align:middle line:90% 00:27:00.600 --> 00:27:04.320 align:middle line:84% So now, we get to write another form of the second law. 00:27:04.320 --> 00:27:14.100 align:middle line:84% Let's call the combined statement combined statement 00:27:14.100 --> 00:27:18.060 align:middle line:90% of first and second laws. 00:27:18.060 --> 00:27:22.870 align:middle line:90% 00:27:22.870 --> 00:27:25.720 align:middle line:90% All right, combined statement-- 00:27:25.720 --> 00:27:27.340 align:middle line:90% first, we write the first law. 00:27:27.340 --> 00:27:31.990 align:middle line:84% The change of energy of a system equals the heat 00:27:31.990 --> 00:27:41.710 align:middle line:84% received plus the work done on it plus the change 00:27:41.710 --> 00:27:44.260 align:middle line:84% through addition or loss of mass. 00:27:44.260 --> 00:27:47.240 align:middle line:90% So this is first the law. 00:27:47.240 --> 00:27:50.150 align:middle line:84% And I like to call this bookkeeping. 00:27:50.150 --> 00:27:52.800 align:middle line:84% First law is not the energy of the universe is constant. 00:27:52.800 --> 00:27:56.210 align:middle line:84% It's the energy of any system changes 00:27:56.210 --> 00:28:00.110 align:middle line:84% only through additions and subtractions of energy 00:28:00.110 --> 00:28:03.440 align:middle line:84% that we can keep track of, so first law bookkeeping-- 00:28:03.440 --> 00:28:04.415 align:middle line:90% work done on a system. 00:28:04.415 --> 00:28:12.640 align:middle line:90% 00:28:12.640 --> 00:28:18.370 align:middle line:90% DW equals minus P dB. 00:28:18.370 --> 00:28:39.420 align:middle line:84% And heat received by system for a reversible process, dQ equals 00:28:39.420 --> 00:28:41.970 align:middle line:90% T dS. 00:28:41.970 --> 00:28:45.420 align:middle line:90% This comes from two slides ago. 00:28:45.420 --> 00:28:53.760 align:middle line:84% We had dS equals dQ over T for reversible heating. 00:28:53.760 --> 00:28:55.356 align:middle line:90% So I'm just rearranging. 00:28:55.356 --> 00:28:59.250 align:middle line:90% 00:28:59.250 --> 00:29:02.550 align:middle line:90% So I'm going to combine these. 00:29:02.550 --> 00:29:04.860 align:middle line:84% I take the first law of bookkeeping. 00:29:04.860 --> 00:29:07.830 align:middle line:84% I substitute the mechanical expression for work. 00:29:07.830 --> 00:29:12.130 align:middle line:84% I substitute this expression for heat for a reversible process, 00:29:12.130 --> 00:29:14.550 align:middle line:90% and I get the following. 00:29:14.550 --> 00:29:23.690 align:middle line:84% TU equals T dS minus P dV plus mu dN. 00:29:23.690 --> 00:29:27.170 align:middle line:84% And now, this is for a reversible process. 00:29:27.170 --> 00:29:31.850 align:middle line:90% 00:29:31.850 --> 00:29:34.130 align:middle line:90% Now, we have this-- 00:29:34.130 --> 00:29:42.140 align:middle line:84% all state functions, which sometimes 00:29:42.140 --> 00:29:44.430 align:middle line:90% is useful for calculating. 00:29:44.430 --> 00:29:47.360 align:middle line:84% So this is going to be one of the foundational, 00:29:47.360 --> 00:29:51.290 align:middle line:84% most important expressions-- equations-- 00:29:51.290 --> 00:29:56.530 align:middle line:90% that we use in 020. 00:29:56.530 --> 00:29:59.740 align:middle line:84% AUDIENCE: I have a question about the energy itself. 00:29:59.740 --> 00:30:01.840 align:middle line:84% If it's not a reversible process, 00:30:01.840 --> 00:30:05.745 align:middle line:84% then we couldn't use T dS for TQ, right? 00:30:05.745 --> 00:30:07.870 align:middle line:84% RAFAEL JARAMILLO: If it's not a reversible process, 00:30:07.870 --> 00:30:10.130 align:middle line:84% TQ is not going to be equal to this. 00:30:10.130 --> 00:30:12.380 align:middle line:84% AUDIENCE: So then how would the internal energy-- 00:30:12.380 --> 00:30:15.760 align:middle line:84% how would, like U, for example, still be a state function 00:30:15.760 --> 00:30:17.386 align:middle line:90% if it's dependent on DQ? 00:30:17.386 --> 00:30:19.930 align:middle line:90% 00:30:19.930 --> 00:30:21.580 align:middle line:84% RAFAEL JARAMILLO: So this expression-- 00:30:21.580 --> 00:30:25.060 align:middle line:90% this expression is true always. 00:30:25.060 --> 00:30:28.750 align:middle line:84% This is in general through this top expression. 00:30:28.750 --> 00:30:30.333 align:middle line:84% I think it's easier to understand 00:30:30.333 --> 00:30:32.750 align:middle line:84% because it's almost like your bank account-- what goes in, 00:30:32.750 --> 00:30:33.333 align:middle line:90% what goes out. 00:30:33.333 --> 00:30:35.260 align:middle line:84% If you track them, you'll know the difference 00:30:35.260 --> 00:30:37.300 align:middle line:90% it's a state function. 00:30:37.300 --> 00:30:39.910 align:middle line:84% This down here is not always true, 00:30:39.910 --> 00:30:42.100 align:middle line:90% and it's the less obvious. 00:30:42.100 --> 00:30:45.520 align:middle line:84% This is only true for a reversible process. 00:30:45.520 --> 00:30:49.420 align:middle line:84% For an irreversible process, you're 00:30:49.420 --> 00:30:51.190 align:middle line:84% going to get the wrong expression here 00:30:51.190 --> 00:30:52.780 align:middle line:90% if you plug this in. 00:30:52.780 --> 00:30:53.470 align:middle line:90% We'll actually-- 00:30:53.470 --> 00:30:53.860 align:middle line:90% AUDIENCE: I think-- 00:30:53.860 --> 00:30:55.610 align:middle line:84% RAFAEL JARAMILLO: --that an irreversible-- 00:30:55.610 --> 00:30:57.250 align:middle line:90% sorry to interrupt you. 00:30:57.250 --> 00:30:59.500 align:middle line:84% What we'll see for an irreversible process, 00:30:59.500 --> 00:31:03.640 align:middle line:84% that dQ is always less than T dS. 00:31:03.640 --> 00:31:05.260 align:middle line:90% But at this level right now-- 00:31:05.260 --> 00:31:07.240 align:middle line:84% OK, so I want to now ask you if that made sense 00:31:07.240 --> 00:31:08.625 align:middle line:90% or I should continue? 00:31:08.625 --> 00:31:10.000 align:middle line:84% AUDIENCE: OK, I think my question 00:31:10.000 --> 00:31:12.640 align:middle line:84% was, like since we learned that state variables could only 00:31:12.640 --> 00:31:15.610 align:middle line:84% depend on other state variables, like, it 00:31:15.610 --> 00:31:19.732 align:middle line:84% seems like dU is dependent on dQ, which it's not. 00:31:19.732 --> 00:31:21.190 align:middle line:84% RAFAEL JARAMILLO: Right, thank you. 00:31:21.190 --> 00:31:23.610 align:middle line:90% That is confusing. 00:31:23.610 --> 00:31:25.610 align:middle line:84% There's a couple of ways of thinking about that. 00:31:25.610 --> 00:31:28.060 align:middle line:90% First of all, entropy is funny. 00:31:28.060 --> 00:31:31.270 align:middle line:90% It just is. 00:31:31.270 --> 00:31:35.050 align:middle line:84% Second of all, this is not true in general. 00:31:35.050 --> 00:31:38.080 align:middle line:84% It's only true for a reversible process. 00:31:38.080 --> 00:31:40.180 align:middle line:84% And reversible processes don't exist. 00:31:40.180 --> 00:31:42.880 align:middle line:84% They're figures of our-- figments of our imagination. 00:31:42.880 --> 00:31:46.190 align:middle line:84% They're the limits of things go arbitrarily slowly. 00:31:46.190 --> 00:31:48.340 align:middle line:90% So in that sense, it's funny. 00:31:48.340 --> 00:31:51.550 align:middle line:84% Another way to think about this, if you remember from, I think, 00:31:51.550 --> 00:31:53.500 align:middle line:90% single var-- single or multi-- 00:31:53.500 --> 00:31:55.870 align:middle line:84% is that T here, or more specifically, 00:31:55.870 --> 00:32:00.525 align:middle line:84% 1 over T here, is a concept of integration. 00:32:00.525 --> 00:32:02.650 align:middle line:84% For those of you for whom that doesn't ring a bell, 00:32:02.650 --> 00:32:03.430 align:middle line:90% just forget it. 00:32:03.430 --> 00:32:05.365 align:middle line:84% But this is an integration constant. 00:32:05.365 --> 00:32:08.900 align:middle line:90% 00:32:08.900 --> 00:32:11.710 align:middle line:84% So you can go back to your calculus textbooks 00:32:11.710 --> 00:32:15.220 align:middle line:84% and review what that is, but anyway-- 00:32:15.220 --> 00:32:19.360 align:middle line:84% so I'm not going to convince you here because it's just 00:32:19.360 --> 00:32:22.660 align:middle line:90% profoundly weird at its core. 00:32:22.660 --> 00:32:24.970 align:middle line:84% But I hope I can make you a little bit more comfortable 00:32:24.970 --> 00:32:26.710 align:middle line:90% with using it as we go forward. 00:32:26.710 --> 00:32:30.630 align:middle line:90% 00:32:30.630 --> 00:32:33.450 align:middle line:84% This is very much connected to the entropy flow 00:32:33.450 --> 00:32:35.620 align:middle line:84% across boundaries, which is sort of a-- 00:32:35.620 --> 00:32:37.110 align:middle line:84% it takes a couple of weeks for us 00:32:37.110 --> 00:32:39.670 align:middle line:84% to get comfortable with that concept. 00:32:39.670 --> 00:32:41.710 align:middle line:84% For a reversible process, entropy 00:32:41.710 --> 00:32:46.120 align:middle line:84% flows across boundaries in the form of heat. 00:32:46.120 --> 00:32:50.470 align:middle line:84% For an irreversible process, an irreversible process 00:32:50.470 --> 00:32:54.900 align:middle line:84% receives less entropy across its boundaries 00:32:54.900 --> 00:32:58.170 align:middle line:84% and generates entropy inside of itself 00:32:58.170 --> 00:33:01.170 align:middle line:90% to make up for that difference. 00:33:01.170 --> 00:33:01.890 align:middle line:90% We'll get there. 00:33:01.890 --> 00:33:03.680 align:middle line:90% We'll get there. 00:33:03.680 --> 00:33:07.170 align:middle line:84% All right, one thing I want to point out before we move on-- 00:33:07.170 --> 00:33:09.600 align:middle line:84% and I have to kind of slip this in around the margins 00:33:09.600 --> 00:33:11.460 align:middle line:84% because this is not a calculus class. 00:33:11.460 --> 00:33:13.140 align:middle line:90% Calculus is a prereq. 00:33:13.140 --> 00:33:23.160 align:middle line:84% But I want to remind you that this is a differentiable form. 00:33:23.160 --> 00:33:27.270 align:middle line:84% And we use a lot of differential forms in this class. 00:33:27.270 --> 00:33:32.130 align:middle line:84% This is differentiable form of the general type dY 00:33:32.130 --> 00:33:39.040 align:middle line:90% equals A dX XYZ. 00:33:39.040 --> 00:33:44.000 align:middle line:90% DX plus B d-- 00:33:44.000 --> 00:33:46.440 align:middle line:90% trying not to use Y-- 00:33:46.440 --> 00:33:58.930 align:middle line:84% dM equals A dX plus B dY plus C dZ plus so forth. 00:33:58.930 --> 00:34:00.980 align:middle line:84% We're going to use a lot of these in this class. 00:34:00.980 --> 00:34:02.590 align:middle line:90% This is a total differential. 00:34:02.590 --> 00:34:06.080 align:middle line:90% 00:34:06.080 --> 00:34:10.190 align:middle line:84% This is also a total differential-- dX, dY, dZ. 00:34:10.190 --> 00:34:13.289 align:middle line:84% And these A, B, and C are coefficients. 00:34:13.289 --> 00:34:24.550 align:middle line:90% 00:34:24.550 --> 00:34:26.050 align:middle line:84% I'm going to go slowly through this. 00:34:26.050 --> 00:34:28.350 align:middle line:84% I'm not teaching you this because it is a prereq. 00:34:28.350 --> 00:34:31.320 align:middle line:84% But I want to go slowly through this 00:34:31.320 --> 00:34:36.540 align:middle line:84% because the sooner you recognize this general form in thermal, 00:34:36.540 --> 00:34:38.909 align:middle line:90% the more successful you'll be. 00:34:38.909 --> 00:34:42.449 align:middle line:84% There are so many shortcuts to useful answers in thermo 00:34:42.449 --> 00:34:45.250 align:middle line:84% when you start recognizing this sort of pattern. 00:34:45.250 --> 00:34:48.750 align:middle line:84% So, for example, A the coefficient 00:34:48.750 --> 00:35:01.700 align:middle line:84% is also the partial derivative dM dX for fixed Y and Z 00:35:01.700 --> 00:35:05.630 align:middle line:84% and likewise for the other coefficients. 00:35:05.630 --> 00:35:09.457 align:middle line:90% 00:35:09.457 --> 00:35:11.040 align:middle line:84% We'll come back to this over and over, 00:35:11.040 --> 00:35:14.010 align:middle line:84% but it cannot be stressed enough. 00:35:14.010 --> 00:35:16.770 align:middle line:90% 00:35:16.770 --> 00:35:18.270 align:middle line:84% But it's a prereq, so I can move on. 00:35:18.270 --> 00:35:21.030 align:middle line:90% 00:35:21.030 --> 00:35:23.670 align:middle line:84% Let's finish out what I want to do here. 00:35:23.670 --> 00:35:25.390 align:middle line:90% Let's talk about equilibrium. 00:35:25.390 --> 00:35:30.330 align:middle line:90% 00:35:30.330 --> 00:35:34.170 align:middle line:84% Equilibrium is-- this is going to be 00:35:34.170 --> 00:35:38.090 align:middle line:84% a very holistic introduction to equilibrium. 00:35:38.090 --> 00:35:45.910 align:middle line:84% It's a state of rest, a state of rest. 00:35:45.910 --> 00:35:49.160 align:middle line:84% What's that going to mean for us? 00:35:49.160 --> 00:35:55.240 align:middle line:84% The state functions aren't changing. 00:35:55.240 --> 00:36:01.390 align:middle line:90% 00:36:01.390 --> 00:36:03.385 align:middle line:90% It's also a state of balance. 00:36:03.385 --> 00:36:10.580 align:middle line:90% 00:36:10.580 --> 00:36:25.445 align:middle line:84% Molecular scale changes or fluctuations average to 0. 00:36:25.445 --> 00:36:31.970 align:middle line:90% 00:36:31.970 --> 00:36:33.800 align:middle line:84% And you can think back to the baby book 00:36:33.800 --> 00:36:36.603 align:middle line:84% where there's particles in the box, 00:36:36.603 --> 00:36:37.895 align:middle line:90% and they're all zooming around. 00:36:37.895 --> 00:36:42.748 align:middle line:90% 00:36:42.748 --> 00:36:44.290 align:middle line:84% So you know that things are happening 00:36:44.290 --> 00:36:45.850 align:middle line:90% on the molecular level. 00:36:45.850 --> 00:36:50.110 align:middle line:84% But at equilibrium, any observable thing anything 00:36:50.110 --> 00:36:53.490 align:middle line:84% you can measure on a length scale 00:36:53.490 --> 00:36:57.670 align:middle line:84% larger than the length scale of molecules is not changing. 00:36:57.670 --> 00:37:01.130 align:middle line:90% 00:37:01.130 --> 00:37:03.430 align:middle line:84% So we want equilibrium to be like this. 00:37:03.430 --> 00:37:05.290 align:middle line:84% At t equals 0, we prepare a system-- 00:37:05.290 --> 00:37:09.000 align:middle line:90% 00:37:09.000 --> 00:37:13.050 align:middle line:84% prepare a system and its boundary conditions. 00:37:13.050 --> 00:37:21.090 align:middle line:90% 00:37:21.090 --> 00:37:25.210 align:middle line:84% And then, we're going to wait for a very long time, 00:37:25.210 --> 00:37:28.530 align:middle line:90% as long as necessary. 00:37:28.530 --> 00:37:36.450 align:middle line:84% And as t goes to infinity, equilibrium, 00:37:36.450 --> 00:37:47.195 align:middle line:84% when all macroscopic changes are finished. 00:37:47.195 --> 00:37:53.610 align:middle line:90% 00:37:53.610 --> 00:38:00.560 align:middle line:84% You know, Wikipedia has a really good entry. 00:38:00.560 --> 00:38:03.020 align:middle line:84% If you look for the Wiki entry-- the definition of thermal 00:38:03.020 --> 00:38:04.560 align:middle line:90% equilibrium in Wiki-- 00:38:04.560 --> 00:38:05.310 align:middle line:90% I kind of like it. 00:38:05.310 --> 00:38:10.240 align:middle line:84% It's got different words all meaning the same thing. 00:38:10.240 --> 00:38:14.150 align:middle line:84% All right, so now, one last board and then we'll wrap up. 00:38:14.150 --> 00:38:19.090 align:middle line:84% And let me just say this lecture is the most-- 00:38:19.090 --> 00:38:22.120 align:middle line:84% to say it's philosophical is a little bit too highfalutin-- 00:38:22.120 --> 00:38:24.970 align:middle line:84% but I would say it's the most abstract 00:38:24.970 --> 00:38:27.420 align:middle line:90% of this entire semester. 00:38:27.420 --> 00:38:31.190 align:middle line:84% So in case you're thinking, what on Earth is going on here? 00:38:31.190 --> 00:38:32.990 align:middle line:84% When are we going to talk about materials? 00:38:32.990 --> 00:38:34.448 align:middle line:84% I just want to share that with you. 00:38:34.448 --> 00:38:37.020 align:middle line:90% 00:38:37.020 --> 00:38:38.420 align:middle line:90% So here's the question. 00:38:38.420 --> 00:38:39.050 align:middle line:90% We're waiting. 00:38:39.050 --> 00:38:42.140 align:middle line:90% What happens while we wait? 00:38:42.140 --> 00:38:44.930 align:middle line:84% We've prepared the system at t equals 0, 00:38:44.930 --> 00:38:46.310 align:middle line:90% and then we're waiting. 00:38:46.310 --> 00:38:47.474 align:middle line:90% What's happening? 00:38:47.474 --> 00:38:54.590 align:middle line:90% 00:38:54.590 --> 00:39:05.470 align:middle line:84% Answer-- all spontaneous processes, 00:39:05.470 --> 00:39:06.880 align:middle line:90% they will happen eventually. 00:39:06.880 --> 00:39:15.390 align:middle line:90% 00:39:15.390 --> 00:39:16.860 align:middle line:90% So spontaneous means what? 00:39:16.860 --> 00:39:19.570 align:middle line:84% Something that will happen on its own. 00:39:19.570 --> 00:39:22.000 align:middle line:84% So it will happen while you wait. 00:39:22.000 --> 00:39:23.530 align:middle line:90% It's in the meaning of the word. 00:39:23.530 --> 00:39:26.410 align:middle line:84% Something spontaneous will happen while you wait. 00:39:26.410 --> 00:39:29.620 align:middle line:84% So while you wait, all spontaneous processes 00:39:29.620 --> 00:39:32.710 align:middle line:90% will happen. 00:39:32.710 --> 00:39:46.970 align:middle line:84% That means that equilibrium is a state in which 00:39:46.970 --> 00:40:00.890 align:middle line:84% all possible spontaneous processes have already 00:40:00.890 --> 00:40:11.805 align:middle line:84% happened, By the meaning of the word, they already happened. 00:40:11.805 --> 00:40:18.300 align:middle line:90% 00:40:18.300 --> 00:40:34.360 align:middle line:84% For an isolated system, we know that any process 00:40:34.360 --> 00:40:37.210 align:middle line:84% means the entropy unchanged or increases it. 00:40:37.210 --> 00:40:41.520 align:middle line:90% 00:40:41.520 --> 00:40:49.150 align:middle line:90% This is a ratchet. 00:40:49.150 --> 00:40:50.800 align:middle line:90% It can kick up one way. 00:40:50.800 --> 00:40:54.270 align:middle line:84% It can never go back the other way. 00:40:54.270 --> 00:41:03.820 align:middle line:84% It follows then that equilibrium is a state of max entropy. 00:41:03.820 --> 00:41:13.450 align:middle line:90% 00:41:13.450 --> 00:41:15.160 align:middle line:90% So let me just state that again. 00:41:15.160 --> 00:41:35.490 align:middle line:84% For an isolated system, equilibrium state 00:41:35.490 --> 00:41:42.720 align:middle line:84% of maximum entropy, given the boundary conditions. 00:41:42.720 --> 00:41:53.960 align:middle line:90% 00:41:53.960 --> 00:41:55.433 align:middle line:90% So this is also the second law. 00:41:55.433 --> 00:42:03.820 align:middle line:90% 00:42:03.820 --> 00:42:06.400 align:middle line:84% All right, so that's where I want to leave it for the day. 00:42:06.400 --> 00:42:08.770 align:middle line:84% We have plenty of time for discussion and questions. 00:42:08.770 --> 00:42:12.160 align:middle line:84% I see I've missed some things in the chat, beautiful words 00:42:12.160 --> 00:42:14.470 align:middle line:90% from Wiki. 00:42:14.470 --> 00:42:17.035 align:middle line:90% So someone's enjoying Wikipedia. 00:42:17.035 --> 00:42:19.830 align:middle line:90% 00:42:19.830 --> 00:42:22.290 align:middle line:84% Now, all right, so what we've done 00:42:22.290 --> 00:42:28.540 align:middle line:84% is we've arrived at something which maybe we would have 00:42:28.540 --> 00:42:30.820 align:middle line:90% guessed at after the baby book. 00:42:30.820 --> 00:42:33.160 align:middle line:84% Equilibrium is a state of maximum mixed upness 00:42:33.160 --> 00:42:35.980 align:middle line:90% or maximum disorder. 00:42:35.980 --> 00:42:38.350 align:middle line:84% Except that we haven't yet introduced entropy 00:42:38.350 --> 00:42:39.490 align:middle line:90% as any measure of disorder. 00:42:39.490 --> 00:42:42.920 align:middle line:84% Entropy is this funny thing that has to do with heat flow. 00:42:42.920 --> 00:42:46.820 align:middle line:84% And the way that we got this concept of equilibrium 00:42:46.820 --> 00:42:53.120 align:middle line:84% is through this concept of spontaneity being used up, 00:42:53.120 --> 00:42:57.060 align:middle line:84% there being no more spontaneous things possible. 00:42:57.060 --> 00:42:58.770 align:middle line:90% But we got to the same place. 00:42:58.770 --> 00:43:02.540 align:middle line:90% 00:43:02.540 --> 00:43:06.500 align:middle line:84% So where we are now is we have the first two 00:43:06.500 --> 00:43:10.010 align:middle line:84% laws of thermodynamics at a very conceptual level. 00:43:10.010 --> 00:43:12.980 align:middle line:84% There's a third law that we don't really need in 020. 00:43:12.980 --> 00:43:15.110 align:middle line:84% It's definitely less important than the first two, 00:43:15.110 --> 00:43:18.780 align:middle line:90% so we're going to skip that one. 00:43:18.780 --> 00:43:21.240 align:middle line:84% And we've done some heat engine stuff. 00:43:21.240 --> 00:43:24.150 align:middle line:84% But where we need to go now is figure out 00:43:24.150 --> 00:43:29.110 align:middle line:84% how to use these principles for situations that we care about. 00:43:29.110 --> 00:43:32.960 align:middle line:84% Because it's kind of hard to isolate a system. 00:43:32.960 --> 00:43:34.450 align:middle line:84% It's easy if you're a cosmologist, 00:43:34.450 --> 00:43:35.867 align:middle line:84% and you say my system is isolated. 00:43:35.867 --> 00:43:37.430 align:middle line:84% I don't have to worry about that. 00:43:37.430 --> 00:43:39.850 align:middle line:84% But if you're a material scientist or an engineer, 00:43:39.850 --> 00:43:43.300 align:middle line:84% you often have systems which are held 00:43:43.300 --> 00:43:47.200 align:middle line:84% at constant temperature or constant pressure, 00:43:47.200 --> 00:43:53.860 align:middle line:90% not systems which are isolated. 00:43:53.860 --> 00:43:56.710 align:middle line:84% So we need to do next is go from this principle 00:43:56.710 --> 00:44:00.280 align:middle line:84% of the second law to a restatement, a principle 00:44:00.280 --> 00:44:02.350 align:middle line:84% of the second law that's useful for conditions 00:44:02.350 --> 00:44:05.620 align:middle line:84% of constant pressure and constant temperature. 00:44:05.620 --> 00:44:09.670 align:middle line:84% And spoiler alert, this is the state of max entropy. 00:44:09.670 --> 00:44:12.820 align:middle line:84% What we will find is that for systems at fixed pressure 00:44:12.820 --> 00:44:16.960 align:middle line:84% and temperature, a state of minimum Gibbs-free energy 00:44:16.960 --> 00:44:19.650 align:middle line:90% is the equilibrium state. 00:44:19.650 --> 00:44:21.270 align:middle line:90% So that's where we're heading. 00:44:21.270 --> 00:44:22.860 align:middle line:84% And then, for the rest of the term, 00:44:22.860 --> 00:44:24.860 align:middle line:84% it'll be Gibbs, Gibbs, and more Gibbs 00:44:24.860 --> 00:44:27.440 align:middle line:84% because that's the material scientist's 00:44:27.440 --> 00:44:30.100 align:middle line:84% thermodynamic potential of choice. 00:44:30.100 --> 00:44:33.000 align:middle line:90%