WEBVTT 00:00:00.000 --> 00:00:01.932 align:middle line:90% [SQUEAKING] 00:00:01.932 --> 00:00:04.347 align:middle line:90% [RUSTLING] 00:00:04.347 --> 00:00:07.728 align:middle line:90% [CLICKING] 00:00:07.728 --> 00:00:12.580 align:middle line:90% 00:00:12.580 --> 00:00:16.360 align:middle line:84% PROFESSOR: On Friday, we had a very almost philosophical 00:00:16.360 --> 00:00:17.500 align:middle line:90% lecture. 00:00:17.500 --> 00:00:20.440 align:middle line:84% And the second law can get like that. 00:00:20.440 --> 00:00:23.560 align:middle line:84% It gets sometimes a little bit head in the clouds. 00:00:23.560 --> 00:00:26.500 align:middle line:84% So today is going to be pretty mathematical. 00:00:26.500 --> 00:00:28.810 align:middle line:84% But, actually, it's very practical. 00:00:28.810 --> 00:00:33.520 align:middle line:84% We're going to start moving from this very highfalutin concept 00:00:33.520 --> 00:00:40.105 align:middle line:84% of the second law towards useful equations 00:00:40.105 --> 00:00:43.610 align:middle line:84% that we can use, things that we can use to make predictions. 00:00:43.610 --> 00:00:57.100 align:middle line:84% So I'm going to tell you about thermodynamic potentials 00:00:57.100 --> 00:00:58.180 align:middle line:90% and equilibrium. 00:00:58.180 --> 00:01:05.170 align:middle line:90% 00:01:05.170 --> 00:01:07.280 align:middle line:84% And I'm going to start off just by listing them. 00:01:07.280 --> 00:01:10.450 align:middle line:84% These are going to be presented without proof or motivation 00:01:10.450 --> 00:01:13.010 align:middle line:90% [INAUDIBLE] we'll discuss. 00:01:13.010 --> 00:01:17.230 align:middle line:84% So the first one is what we already know now. 00:01:17.230 --> 00:01:31.110 align:middle line:84% For fixed U entropy is max at equilibrium. 00:01:31.110 --> 00:01:32.810 align:middle line:90% So you already know that. 00:01:32.810 --> 00:01:33.780 align:middle line:90% But there are others. 00:01:33.780 --> 00:01:35.780 align:middle line:84% There are other conditions and other potentials. 00:01:35.780 --> 00:01:59.990 align:middle line:84% So for fixed entropy and volume, the internal energy U 00:01:59.990 --> 00:02:01.680 align:middle line:90% is minimum at equilibrium. 00:02:01.680 --> 00:02:04.580 align:middle line:84% So this is a condition that you encounter 00:02:04.580 --> 00:02:07.990 align:middle line:84% in your introductory physics classes. 00:02:07.990 --> 00:02:11.200 align:middle line:84% They don't really state that, but that's the condition. 00:02:11.200 --> 00:02:15.400 align:middle line:90% 00:02:15.400 --> 00:02:17.830 align:middle line:84% Is there ever really a way to fix the entropy? 00:02:17.830 --> 00:02:19.315 align:middle line:90% Is there a way? 00:02:19.315 --> 00:02:20.440 align:middle line:90% How can we fix the entropy? 00:02:20.440 --> 00:02:23.070 align:middle line:90% 00:02:23.070 --> 00:02:24.960 align:middle line:90% There's no entropy reservoir. 00:02:24.960 --> 00:02:27.540 align:middle line:84% We can't connect the system to an entropy reservoir 00:02:27.540 --> 00:02:29.790 align:middle line:84% like we can with a pressure reservoir or a temperature 00:02:29.790 --> 00:02:30.420 align:middle line:90% reservoir. 00:02:30.420 --> 00:02:34.590 align:middle line:90% 00:02:34.590 --> 00:02:37.230 align:middle line:84% So I'll tell you-- it's a little bit of an aside. 00:02:37.230 --> 00:02:40.130 align:middle line:84% But the only way to fix entropy is to be at 0 Kelvin. 00:02:40.130 --> 00:02:43.650 align:middle line:84% When you're at absolute zero, there's no entropy. 00:02:43.650 --> 00:02:45.150 align:middle line:84% So the physicists like it that way. 00:02:45.150 --> 00:02:47.600 align:middle line:84% But for the rest of us, it's not terribly useful. 00:02:47.600 --> 00:02:51.745 align:middle line:90% 00:02:51.745 --> 00:03:04.540 align:middle line:84% For fixed S and P, new term here, the enthalpy H 00:03:04.540 --> 00:03:17.020 align:middle line:84% equals U plus PV is minimized at equilibrium. 00:03:17.020 --> 00:03:17.520 align:middle line:90% OK. 00:03:17.520 --> 00:03:19.900 align:middle line:90% So that's enthalpy. 00:03:19.900 --> 00:03:22.440 align:middle line:84% This condition, minimum enthalpy and equilibrium, 00:03:22.440 --> 00:03:24.990 align:middle line:90% is not terribly useful. 00:03:24.990 --> 00:03:26.970 align:middle line:90% But the enthalpy is useful. 00:03:26.970 --> 00:03:28.020 align:middle line:90% It's a useful thing. 00:03:28.020 --> 00:03:30.180 align:middle line:84% And one of the problems on the current p set 00:03:30.180 --> 00:03:35.760 align:middle line:84% is making connection between enthalpy 00:03:35.760 --> 00:03:37.620 align:middle line:90% and the heat of a process. 00:03:37.620 --> 00:03:39.720 align:middle line:84% Sometimes those terms enthalpy and heat 00:03:39.720 --> 00:03:42.300 align:middle line:84% are actually used interchangeably. 00:03:42.300 --> 00:03:44.580 align:middle line:84% That's not strictly correct, but you'll see why. 00:03:44.580 --> 00:03:48.140 align:middle line:90% 00:03:48.140 --> 00:03:56.115 align:middle line:84% More perfect temperature and volume-- 00:03:56.115 --> 00:03:57.990 align:middle line:84% so now we're getting a little more practical, 00:03:57.990 --> 00:03:58.650 align:middle line:90% fixed temperature. 00:03:58.650 --> 00:04:00.108 align:middle line:84% That's something we know how to do. 00:04:00.108 --> 00:04:01.170 align:middle line:90% We all have thermostats. 00:04:01.170 --> 00:04:04.170 align:middle line:84% That's what your thermostat does. 00:04:04.170 --> 00:04:27.690 align:middle line:84% The Helmholtz, F equals U minus TS is min at equilibrium. 00:04:27.690 --> 00:04:29.870 align:middle line:84% So I'll tell you, we spent almost no time 00:04:29.870 --> 00:04:31.610 align:middle line:84% at all on Helmholtz in this class 00:04:31.610 --> 00:04:35.240 align:middle line:84% because fixed temperature and volume are much less practical 00:04:35.240 --> 00:04:37.880 align:middle line:84% and useful than fixed temperature and pressure 00:04:37.880 --> 00:04:39.570 align:middle line:90% for materials processing. 00:04:39.570 --> 00:04:42.020 align:middle line:84% However, in other disciplines, especially 00:04:42.020 --> 00:04:44.282 align:middle line:84% in semiconductors and semiconductor physics, 00:04:44.282 --> 00:04:45.740 align:middle line:84% it's the Helmholtz free energy that 00:04:45.740 --> 00:04:50.210 align:middle line:84% matters because your device is typically operated at a fixed 00:04:50.210 --> 00:04:51.980 align:middle line:84% temperature, and your electrons are 00:04:51.980 --> 00:04:55.620 align:middle line:84% confined in the volume of the semiconductor device. 00:04:55.620 --> 00:04:58.280 align:middle line:84% So if we were to do things like calculate 00:04:58.280 --> 00:05:00.840 align:middle line:84% the theoretical limiting efficiency of a solar cell, 00:05:00.840 --> 00:05:02.960 align:middle line:84% we'd be worrying about Helmholtz. 00:05:02.960 --> 00:05:05.480 align:middle line:84% But as it is, you almost never hear us discuss that at all 00:05:05.480 --> 00:05:07.500 align:middle line:90% in the rest of the class. 00:05:07.500 --> 00:05:13.530 align:middle line:84% This is the one that matters, perfect temperature 00:05:13.530 --> 00:05:15.420 align:middle line:90% and pressure. 00:05:15.420 --> 00:05:20.540 align:middle line:84% Anybody know what's the relevant potential for these conditions? 00:05:20.540 --> 00:05:21.380 align:middle line:90% Who's read ahead? 00:05:21.380 --> 00:05:25.250 align:middle line:90% 00:05:25.250 --> 00:05:26.520 align:middle line:90% AUDIENCE: Gibbs free energy. 00:05:26.520 --> 00:05:27.620 align:middle line:90% PROFESSOR: Gibbs, right. 00:05:27.620 --> 00:05:44.370 align:middle line:90% 00:05:44.370 --> 00:05:49.290 align:middle line:84% So this is the one that matters for material science. 00:05:49.290 --> 00:05:51.510 align:middle line:84% And one way to see the rest of this class, 00:05:51.510 --> 00:05:54.450 align:middle line:84% actually the rest of the term, is we're 00:05:54.450 --> 00:05:57.075 align:middle line:84% just calculating the Gibbs for different scenarios. 00:05:57.075 --> 00:05:59.580 align:middle line:84% And we're doing price shopping for Gibbs, I call it. 00:05:59.580 --> 00:06:02.040 align:middle line:84% We calculate the Gibbs free energy for different scenarios. 00:06:02.040 --> 00:06:06.300 align:middle line:84% And we choose the lowest one as the equilibrium case. 00:06:06.300 --> 00:06:08.680 align:middle line:84% So that is thermodynamic potentials. 00:06:08.680 --> 00:06:13.123 align:middle line:84% Now these are presented without proof. 00:06:13.123 --> 00:06:14.790 align:middle line:84% It's not worth the time here to motivate 00:06:14.790 --> 00:06:16.260 align:middle line:90% each and every one of those. 00:06:16.260 --> 00:06:18.360 align:middle line:84% On the piece that you're going to convince 00:06:18.360 --> 00:06:24.970 align:middle line:84% yourselves that given this, then this. 00:06:24.970 --> 00:06:29.020 align:middle line:84% So one of the exercises is going through a form 00:06:29.020 --> 00:06:29.770 align:middle line:90% of proof of this. 00:06:29.770 --> 00:06:32.230 align:middle line:84% And that's useful because Gibbs is the one we care about. 00:06:32.230 --> 00:06:38.620 align:middle line:90% 00:06:38.620 --> 00:06:53.620 align:middle line:84% Now I have to make some mathematical asides because 00:06:53.620 --> 00:06:58.480 align:middle line:84% without this, my math note, a lot of what comes 00:06:58.480 --> 00:07:00.310 align:middle line:90% is kind of weird and confusing. 00:07:00.310 --> 00:07:03.060 align:middle line:90% 00:07:03.060 --> 00:07:20.120 align:middle line:84% So in 3.020, state functions and state variables are functions 00:07:20.120 --> 00:07:33.240 align:middle line:84% of at most three independent variables and both stay 00:07:33.240 --> 00:07:34.230 align:middle line:90% independent variables. 00:07:34.230 --> 00:07:37.630 align:middle line:84% And this is where that comes from. 00:07:37.630 --> 00:07:43.860 align:middle line:84% It comes from-- sorry, this pen is kind of smushy. 00:07:43.860 --> 00:07:50.862 align:middle line:90% 00:07:50.862 --> 00:07:52.070 align:middle line:90% It doesn't come from nowhere. 00:07:52.070 --> 00:07:53.612 align:middle line:84% It comes from the combined statement. 00:07:53.612 --> 00:07:56.790 align:middle line:84% So it comes from physical reasoning. 00:07:56.790 --> 00:08:03.920 align:middle line:90% dU equals TdS minus PdV plus mu. 00:08:03.920 --> 00:08:06.700 align:middle line:90% Yeah. 00:08:06.700 --> 00:08:14.360 align:middle line:84% This implies the existence of a state function. 00:08:14.360 --> 00:08:20.670 align:middle line:90% U is a function of S, V, and N. 00:08:20.670 --> 00:08:24.060 align:middle line:90% This is a critical link here. 00:08:24.060 --> 00:08:26.400 align:middle line:84% If you write down a differential form, 00:08:26.400 --> 00:08:27.900 align:middle line:84% we're going to learn how to identify 00:08:27.900 --> 00:08:30.810 align:middle line:84% the independent variables, the coefficients, 00:08:30.810 --> 00:08:32.549 align:middle line:90% and the dependent variable. 00:08:32.549 --> 00:08:35.100 align:middle line:84% And the existence of this differential form 00:08:35.100 --> 00:08:36.360 align:middle line:90% implies a state function. 00:08:36.360 --> 00:08:40.090 align:middle line:90% 00:08:40.090 --> 00:08:58.350 align:middle line:84% Now, in most cases, we may never know this, but can-- 00:08:58.350 --> 00:09:00.020 align:middle line:90% we can still use this. 00:09:00.020 --> 00:09:02.630 align:middle line:90% 00:09:02.630 --> 00:09:05.900 align:middle line:84% That is, we may never know the closed form of the state 00:09:05.900 --> 00:09:06.500 align:middle line:90% function. 00:09:06.500 --> 00:09:10.350 align:middle line:84% But we can still use its differential form. 00:09:10.350 --> 00:09:13.160 align:middle line:84% In fact, outside of the ideal gas situation, 00:09:13.160 --> 00:09:17.220 align:middle line:84% we almost never write down any state functions in this class. 00:09:17.220 --> 00:09:19.470 align:middle line:84% You can't find a complete state function for something 00:09:19.470 --> 00:09:23.972 align:middle line:84% useful like a high-entropy alloy. 00:09:23.972 --> 00:09:24.680 align:middle line:90% It doesn't exist. 00:09:24.680 --> 00:09:28.220 align:middle line:90% It's too complicated. 00:09:28.220 --> 00:09:39.350 align:middle line:84% So a corollary of this is that thermodynamic potentials have 00:09:39.350 --> 00:09:50.150 align:middle line:84% what I'll call natural independent variables, which 00:09:50.150 --> 00:09:53.510 align:middle line:84% we will see aplenty in the case of Gibbs. 00:09:53.510 --> 00:10:04.143 align:middle line:84% So for example, entropy is a function of energy and volume. 00:10:04.143 --> 00:10:05.810 align:middle line:84% If you remember, at equilibrium, entropy 00:10:05.810 --> 00:10:10.330 align:middle line:84% was max for fixed energy and fixed volume. 00:10:10.330 --> 00:10:12.820 align:middle line:84% That tells you that those are its natural variables. 00:10:12.820 --> 00:10:15.790 align:middle line:84% At equilibrium, internal energy was maximum 00:10:15.790 --> 00:10:19.930 align:middle line:90% for fixed entropy and volume. 00:10:19.930 --> 00:10:22.000 align:middle line:90% Those are its natural variables. 00:10:22.000 --> 00:10:27.340 align:middle line:84% Enthalpy was equilibrium for a minimum at equilibrium 00:10:27.340 --> 00:10:29.980 align:middle line:90% for fixed entropy and pressure. 00:10:29.980 --> 00:10:32.570 align:middle line:90% Those are its natural variables. 00:10:32.570 --> 00:10:35.750 align:middle line:84% For Helmholtz, its natural variables 00:10:35.750 --> 00:10:45.720 align:middle line:84% are T and V. And for Gibbs, its natural variables are T and P. 00:10:45.720 --> 00:10:52.720 align:middle line:84% Before you ding me here, I'm not writing 00:10:52.720 --> 00:10:55.270 align:middle line:90% the ends for compactness. 00:10:55.270 --> 00:11:01.030 align:middle line:90% 00:11:01.030 --> 00:11:04.180 align:middle line:84% So everything is also a function of the size of a system. 00:11:04.180 --> 00:11:06.640 align:middle line:84% I'm just leaving that off to keep the lines short. 00:11:06.640 --> 00:11:12.467 align:middle line:90% 00:11:12.467 --> 00:11:13.050 align:middle line:90% One more note. 00:11:13.050 --> 00:11:17.280 align:middle line:84% Where on Earth does this stuff come from? 00:11:17.280 --> 00:11:19.590 align:middle line:84% When I learned thermo, these potentials 00:11:19.590 --> 00:11:22.818 align:middle line:84% were just told to me the way I just told them to you. 00:11:22.818 --> 00:11:24.610 align:middle line:84% And I think in the back of my mind, I said, 00:11:24.610 --> 00:11:25.350 align:middle line:90% where does that come from? 00:11:25.350 --> 00:11:26.642 align:middle line:90% And then I forgot the question. 00:11:26.642 --> 00:11:28.920 align:middle line:84% And I didn't come back to it for years. 00:11:28.920 --> 00:11:30.602 align:middle line:90% It does come from somewhere. 00:11:30.602 --> 00:11:40.250 align:middle line:90% 00:11:40.250 --> 00:11:54.600 align:middle line:84% The potentials are related by a change of variables 00:11:54.600 --> 00:11:56.460 align:middle line:90% via Legendre transforms. 00:11:56.460 --> 00:12:03.790 align:middle line:90% 00:12:03.790 --> 00:12:06.360 align:middle line:84% So this is something we don't have time for in this class, 00:12:06.360 --> 00:12:10.650 align:middle line:84% and probably I wouldn't be able to teach very well anyway. 00:12:10.650 --> 00:12:13.200 align:middle line:90% This is the math concept. 00:12:13.200 --> 00:12:15.020 align:middle line:84% And so for those of you who have seen 00:12:15.020 --> 00:12:17.855 align:middle line:84% Legendre transforms in your math classes, that's great. 00:12:17.855 --> 00:12:19.230 align:middle line:84% For those of you who are curious, 00:12:19.230 --> 00:12:21.627 align:middle line:90% you can start with a wiki entry. 00:12:21.627 --> 00:12:23.210 align:middle line:84% We're not going to cover it in detail. 00:12:23.210 --> 00:12:26.360 align:middle line:84% You can think of it as just one more type of a transform, 00:12:26.360 --> 00:12:30.890 align:middle line:84% like a Fourier transform or Laplace transform. 00:12:30.890 --> 00:12:33.650 align:middle line:84% Fourier transform, for example, changes your natural variables 00:12:33.650 --> 00:12:35.810 align:middle line:90% from time to frequency. 00:12:35.810 --> 00:12:37.095 align:middle line:90% That's an example. 00:12:37.095 --> 00:12:40.730 align:middle line:84% Legendre transform will change natural variables, for example, 00:12:40.730 --> 00:12:51.700 align:middle line:84% from U as a function of S and V to H as a function of S and P, 00:12:51.700 --> 00:13:02.960 align:middle line:84% exchanging V for P. And if you'll remember, 00:13:02.960 --> 00:13:05.340 align:middle line:90% H equals U plus PV. 00:13:05.340 --> 00:13:06.590 align:middle line:90% And this comes from somewhere. 00:13:06.590 --> 00:13:08.540 align:middle line:90% This is all beyond-- 00:13:08.540 --> 00:13:16.450 align:middle line:90% 00:13:16.450 --> 00:13:18.010 align:middle line:90% this is beyond 020. 00:13:18.010 --> 00:13:20.480 align:middle line:84% I just want you to know it comes from somewhere. 00:13:20.480 --> 00:13:22.240 align:middle line:90% It comes from somewhere. 00:13:22.240 --> 00:13:24.370 align:middle line:84% For most of the class, you can just memorize. 00:13:24.370 --> 00:13:31.500 align:middle line:90% 00:13:31.500 --> 00:13:32.500 align:middle line:90% Try a bit a little more. 00:13:32.500 --> 00:13:54.280 align:middle line:90% 00:13:54.280 --> 00:13:57.010 align:middle line:84% Let's start with the combined statement. 00:13:57.010 --> 00:14:07.600 align:middle line:84% dU equals T dS minus PdV plus mu. 00:14:07.600 --> 00:14:09.860 align:middle line:84% I'll let this be a multi-component system. 00:14:09.860 --> 00:14:11.110 align:middle line:90% So I'll keep the labels there. 00:14:11.110 --> 00:14:13.680 align:middle line:90% 00:14:13.680 --> 00:14:18.070 align:middle line:84% Somebody please remind me, what are the independent variables 00:14:18.070 --> 00:14:19.735 align:middle line:90% in this differential form? 00:14:19.735 --> 00:14:30.050 align:middle line:90% 00:14:30.050 --> 00:14:33.368 align:middle line:90% AUDIENCE: S, V, and I. 00:14:33.368 --> 00:14:34.160 align:middle line:90% PROFESSOR: Correct. 00:14:34.160 --> 00:14:41.910 align:middle line:90% 00:14:41.910 --> 00:14:44.310 align:middle line:84% Those are the independent variables. 00:14:44.310 --> 00:14:47.160 align:middle line:90% The other things we call-- 00:14:47.160 --> 00:14:48.900 align:middle line:90% having fun with color today. 00:14:48.900 --> 00:14:51.300 align:middle line:84% The other things we call coefficients. 00:14:51.300 --> 00:15:00.800 align:middle line:90% 00:15:00.800 --> 00:15:07.010 align:middle line:84% So the coefficients describe how the dependent variable varies 00:15:07.010 --> 00:15:10.630 align:middle line:90% with the independent variables. 00:15:10.630 --> 00:15:13.390 align:middle line:84% And we can write down by inspection coefficient 00:15:13.390 --> 00:15:14.710 align:middle line:90% relations. 00:15:14.710 --> 00:15:15.580 align:middle line:90% And here they are. 00:15:15.580 --> 00:15:24.190 align:middle line:90% 00:15:24.190 --> 00:15:27.490 align:middle line:90% dU dS. 00:15:27.490 --> 00:15:34.960 align:middle line:84% DU dS at fixed volume and fixed particle number equals what? 00:15:34.960 --> 00:15:37.960 align:middle line:90% What is partial U by S? 00:15:37.960 --> 00:15:39.660 align:middle line:90% You can pick it off of-- 00:15:39.660 --> 00:15:40.870 align:middle line:90% AUDIENCE: p? 00:15:40.870 --> 00:15:42.803 align:middle line:84% PROFESSOR: Yep, you just look at the equation. 00:15:42.803 --> 00:15:44.460 align:middle line:90% Put the terms apart. 00:15:44.460 --> 00:15:46.060 align:middle line:90% And pick these off. 00:15:46.060 --> 00:15:54.850 align:middle line:84% dU dV at fixed entropy and particle number equals minus p. 00:15:54.850 --> 00:16:06.000 align:middle line:84% And dU d Ni at fixed entropy and volume and particle number J 00:16:06.000 --> 00:16:10.540 align:middle line:90% not equal to i equals mu i. 00:16:10.540 --> 00:16:11.600 align:middle line:90% Where's your coefficient? 00:16:11.600 --> 00:16:14.910 align:middle line:90% 00:16:14.910 --> 00:16:15.960 align:middle line:90% Good call. 00:16:15.960 --> 00:16:18.510 align:middle line:84% You can them O. I just picked partial differentials. 00:16:18.510 --> 00:16:21.820 align:middle line:84% But they're called coefficient relations. 00:16:21.820 --> 00:16:24.520 align:middle line:84% A lot of problem solving in thermo 00:16:24.520 --> 00:16:28.419 align:middle line:84% comes down to calculating these coefficients. 00:16:28.419 --> 00:16:32.730 align:middle line:90% 00:16:32.730 --> 00:16:36.621 align:middle line:84% The whole chapter, actually, in-- 00:16:36.621 --> 00:16:39.390 align:middle line:90% get to chapter 10 of the book. 00:16:39.390 --> 00:16:41.070 align:middle line:84% The most useful chapter is chapter 4. 00:16:41.070 --> 00:16:42.900 align:middle line:84% If you read only one chapter of this book, 00:16:42.900 --> 00:16:47.810 align:middle line:84% pick chapter 4 because it contains all 00:16:47.810 --> 00:16:49.730 align:middle line:84% that we need to actually solve problems. 00:16:49.730 --> 00:16:51.920 align:middle line:84% To actually solve problems, you just 00:16:51.920 --> 00:16:53.630 align:middle line:90% have to understand chapter 4. 00:16:53.630 --> 00:16:54.440 align:middle line:90% And I say just. 00:16:54.440 --> 00:16:56.480 align:middle line:84% It's actually one of the hairier chapters. 00:16:56.480 --> 00:16:57.320 align:middle line:90% But there it is. 00:16:57.320 --> 00:16:59.300 align:middle line:84% And it's all about these coefficient relations 00:16:59.300 --> 00:17:02.600 align:middle line:84% and identifying independent variables 00:17:02.600 --> 00:17:03.920 align:middle line:90% and calculating coefficients. 00:17:03.920 --> 00:17:05.210 align:middle line:90% So let's see how to do that. 00:17:05.210 --> 00:17:07.780 align:middle line:90% 00:17:07.780 --> 00:17:10.480 align:middle line:84% I wouldn't have called this a strategy. 00:17:10.480 --> 00:17:11.829 align:middle line:90% I think it's a little funny. 00:17:11.829 --> 00:17:14.140 align:middle line:84% Besides, it's a tactic, not a strategy. 00:17:14.140 --> 00:17:16.119 align:middle line:84% But that's what it's called in DeHoff. 00:17:16.119 --> 00:17:17.109 align:middle line:90% So I'll stick with it. 00:17:17.109 --> 00:17:21.550 align:middle line:84% The general strategy for deriving 00:17:21.550 --> 00:17:28.010 align:middle line:90% thermodynamic relations. 00:17:28.010 --> 00:17:32.020 align:middle line:90% 00:17:32.020 --> 00:17:34.195 align:middle line:84% This is really best illustrated by an example. 00:17:34.195 --> 00:17:35.570 align:middle line:84% So that's what we're going to do. 00:17:35.570 --> 00:17:43.190 align:middle line:84% Here is the general problem, the general problem statement. 00:17:43.190 --> 00:17:50.230 align:middle line:90% 00:17:50.230 --> 00:18:06.220 align:middle line:84% Given a dependent state variable Z, 00:18:06.220 --> 00:18:23.080 align:middle line:84% independent state variables x and y, calculate-- 00:18:23.080 --> 00:18:26.020 align:middle line:90% so this is a word problem-- 00:18:26.020 --> 00:18:30.124 align:middle line:90% delta V for a given process. 00:18:30.124 --> 00:18:35.820 align:middle line:90% 00:18:35.820 --> 00:18:37.010 align:middle line:90% So this is like-- 00:18:37.010 --> 00:18:42.620 align:middle line:84% probably half of all thermo problems boil down to something 00:18:42.620 --> 00:18:44.198 align:middle line:90% like this. 00:18:44.198 --> 00:18:46.490 align:middle line:84% There's some kind of state variable, which you identify 00:18:46.490 --> 00:18:47.880 align:middle line:90% through the problem statement. 00:18:47.880 --> 00:18:49.422 align:middle line:84% There are some independent variables, 00:18:49.422 --> 00:18:51.048 align:middle line:90% which usually you're not told. 00:18:51.048 --> 00:18:53.090 align:middle line:84% You figure them out through the problem statement 00:18:53.090 --> 00:18:54.830 align:middle line:90% or the situation in hand. 00:18:54.830 --> 00:18:57.450 align:middle line:84% And you want to calculate the change for a given process. 00:18:57.450 --> 00:18:59.520 align:middle line:84% So what example are we going to do? 00:18:59.520 --> 00:19:01.182 align:middle line:90% Example. 00:19:01.182 --> 00:19:23.370 align:middle line:84% Calculate delta V for a process involving a change of pressure 00:19:23.370 --> 00:19:34.510 align:middle line:84% at fixed entropy, which is also is 00:19:34.510 --> 00:19:40.320 align:middle line:90% a reversible adiabatic process. 00:19:40.320 --> 00:19:46.730 align:middle line:90% 00:19:46.730 --> 00:19:50.220 align:middle line:84% This is helpful, remembering that 00:19:50.220 --> 00:19:56.580 align:middle line:84% fixed entropy and reversible adiabatic 00:19:56.580 --> 00:19:58.138 align:middle line:90% processes are related. 00:19:58.138 --> 00:19:59.055 align:middle line:90% That's really helpful. 00:19:59.055 --> 00:20:03.290 align:middle line:90% 00:20:03.290 --> 00:20:05.450 align:middle line:84% So now I'm going to step through this example. 00:20:05.450 --> 00:20:10.730 align:middle line:90% 00:20:10.730 --> 00:20:16.827 align:middle line:84% We want to write down the differential for the change 00:20:16.827 --> 00:20:17.910 align:middle line:90% in the dependent variable. 00:20:17.910 --> 00:20:20.700 align:middle line:90% 00:20:20.700 --> 00:20:28.090 align:middle line:84% And this is going to depend on some coefficient times dP 00:20:28.090 --> 00:20:33.020 align:middle line:90% plus some coefficient times dS. 00:20:33.020 --> 00:20:36.540 align:middle line:90% 00:20:36.540 --> 00:20:39.060 align:middle line:90% And our job is to find-- 00:20:39.060 --> 00:20:50.850 align:middle line:90% 00:20:50.850 --> 00:20:55.110 align:middle line:84% we want to find those coefficients, 00:20:55.110 --> 00:20:56.670 align:middle line:90% and then integrate-- 00:20:56.670 --> 00:21:07.940 align:middle line:90% 00:21:07.940 --> 00:21:10.230 align:middle line:90% this is the approach. 00:21:10.230 --> 00:21:12.390 align:middle line:84% Once you find these coefficients, 00:21:12.390 --> 00:21:17.125 align:middle line:84% then you can integrate this term, which for fixed entropy 00:21:17.125 --> 00:21:18.750 align:middle line:84% gives you the thing you're looking for, 00:21:18.750 --> 00:21:20.237 align:middle line:90% which is the change in volume. 00:21:20.237 --> 00:21:21.320 align:middle line:90% That's what we need to do. 00:21:21.320 --> 00:21:25.000 align:middle line:90% 00:21:25.000 --> 00:21:28.750 align:middle line:84% Questions before I go on to actually do that? 00:21:28.750 --> 00:21:29.250 align:middle line:90% OK. 00:21:29.250 --> 00:21:32.370 align:middle line:90% 00:21:32.370 --> 00:21:34.440 align:middle line:90% Then we're going to start. 00:21:34.440 --> 00:21:41.792 align:middle line:84% 1-- we're going to write the exact differential. 00:21:41.792 --> 00:21:45.950 align:middle line:90% 00:21:45.950 --> 00:21:47.870 align:middle line:90% Rewriting what we did. 00:21:47.870 --> 00:21:53.820 align:middle line:90% dV equals x dP plus Y dS. 00:21:53.820 --> 00:21:58.110 align:middle line:90% 00:21:58.110 --> 00:22:03.650 align:middle line:90% x equals dV dP at s. 00:22:03.650 --> 00:22:09.820 align:middle line:90% Y equals dV dS at p. 00:22:09.820 --> 00:22:11.580 align:middle line:90% OK, good. 00:22:11.580 --> 00:22:18.780 align:middle line:84% Then, we're going to do a change of variables. 00:22:18.780 --> 00:22:21.820 align:middle line:90% 00:22:21.820 --> 00:22:26.790 align:middle line:84% There's a really useful reference, DeHoff table 4.5. 00:22:26.790 --> 00:22:42.415 align:middle line:84% To express dP and dS in terms of dP and dT. 00:22:42.415 --> 00:22:48.020 align:middle line:90% 00:22:48.020 --> 00:22:51.170 align:middle line:84% DeHoff chooses P and T because it's a materials science text. 00:22:51.170 --> 00:22:53.300 align:middle line:84% Another text might have chosen a different two. 00:22:53.300 --> 00:22:55.880 align:middle line:90% You can choose any two. 00:22:55.880 --> 00:22:59.830 align:middle line:84% So we're going to look up in table 4.5. 00:22:59.830 --> 00:23:08.660 align:middle line:84% And it has the coefficients for every state variable 00:23:08.660 --> 00:23:11.780 align:middle line:90% as a function of dP and dT. 00:23:11.780 --> 00:23:14.210 align:middle line:90% So here, this is trivial. 00:23:14.210 --> 00:23:19.280 align:middle line:84% There's going to be no-- this is 0 because dP equals dP. 00:23:19.280 --> 00:23:20.900 align:middle line:90% That's trivial. 00:23:20.900 --> 00:23:22.582 align:middle line:90% The entropy is nontrivial. 00:23:22.582 --> 00:23:25.040 align:middle line:84% We could derive this, but we're not going to take the time. 00:23:25.040 --> 00:23:27.800 align:middle line:84% We're just going to copy it from the table. 00:23:27.800 --> 00:23:32.330 align:middle line:84% It happens to be equal to Cp over temperature times dT 00:23:32.330 --> 00:23:35.780 align:middle line:90% minus V alpha dP. 00:23:35.780 --> 00:23:39.440 align:middle line:90% 00:23:39.440 --> 00:23:40.950 align:middle line:90% That's not obvious to me. 00:23:40.950 --> 00:23:42.530 align:middle line:90% I have to think about it. 00:23:42.530 --> 00:23:45.170 align:middle line:84% This I could figure out if I thought about it long enough. 00:23:45.170 --> 00:23:47.128 align:middle line:84% This, it doesn't make sense to me. 00:23:47.128 --> 00:23:48.170 align:middle line:90% I just have to derive it. 00:23:48.170 --> 00:23:50.380 align:middle line:84% But you don't have to derive it or think about it. 00:23:50.380 --> 00:23:51.630 align:middle line:90% It's right there in the table. 00:23:51.630 --> 00:23:55.820 align:middle line:84% So here is pressure and entropy written now 00:23:55.820 --> 00:23:59.613 align:middle line:84% as dependent variables, dependent on temperature 00:23:59.613 --> 00:24:00.155 align:middle line:90% and pressure. 00:24:00.155 --> 00:24:03.600 align:middle line:90% 00:24:03.600 --> 00:24:06.870 align:middle line:84% So we're going to combine now, and we get the following. 00:24:06.870 --> 00:24:11.435 align:middle line:90% dV equals x-- then we combine-- 00:24:11.435 --> 00:24:16.400 align:middle line:90% 00:24:16.400 --> 00:24:20.320 align:middle line:90% --equals dP-- that's easy-- 00:24:20.320 --> 00:24:22.210 align:middle line:90% plus Y. What's Cs? 00:24:22.210 --> 00:24:29.020 align:middle line:84% Cs equals Cp over T dT minus V alpha dP. 00:24:29.020 --> 00:24:32.950 align:middle line:90% 00:24:32.950 --> 00:24:36.820 align:middle line:84% I just plugged in, substituted, substituted for dS here. 00:24:36.820 --> 00:24:40.877 align:middle line:90% 00:24:40.877 --> 00:24:41.960 align:middle line:90% So that's the second step. 00:24:41.960 --> 00:24:44.640 align:middle line:90% 00:24:44.640 --> 00:24:46.190 align:middle line:90% Then I'm going to collect terms. 00:24:46.190 --> 00:24:48.050 align:middle line:90% The third step-- collect terms. 00:24:48.050 --> 00:24:52.260 align:middle line:90% 00:24:52.260 --> 00:24:52.860 align:middle line:90% Collect terms. 00:24:52.860 --> 00:24:53.860 align:middle line:90% So I have the following. 00:24:53.860 --> 00:25:09.470 align:middle line:84% I have dV equals Y tP over temperature dT plus X minus YV 00:25:09.470 --> 00:25:11.350 align:middle line:90% alpha dP. 00:25:11.350 --> 00:25:18.178 align:middle line:90% 00:25:18.178 --> 00:25:20.900 align:middle line:84% I still don't have X and Y. I still don't know what they are. 00:25:20.900 --> 00:25:24.210 align:middle line:90% 00:25:24.210 --> 00:25:35.490 align:middle line:84% But I can say by inspection, dV dT at fixed pressure 00:25:35.490 --> 00:25:40.050 align:middle line:84% equals Y Cp over T. That's this coefficient. 00:25:40.050 --> 00:25:43.700 align:middle line:90% 00:25:43.700 --> 00:25:48.830 align:middle line:84% dV by temperature at fixed pressure is that coefficient. 00:25:48.830 --> 00:25:57.450 align:middle line:84% And dV dP at fixed temperature equals this coefficient, X 00:25:57.450 --> 00:26:00.675 align:middle line:90% minus YV alpha. 00:26:00.675 --> 00:26:15.020 align:middle line:90% 00:26:15.020 --> 00:26:16.760 align:middle line:84% We get the coefficients by inspection. 00:26:16.760 --> 00:26:19.930 align:middle line:90% 00:26:19.930 --> 00:26:30.710 align:middle line:84% The fourth step is you compare to the known coefficients, 00:26:30.710 --> 00:26:34.580 align:middle line:90% again referring to table 4.5. 00:26:34.580 --> 00:26:38.300 align:middle line:84% So in table 4.5, you will find the differential form 00:26:38.300 --> 00:26:41.670 align:middle line:84% of volume written as a function of temperature and pressure. 00:26:41.670 --> 00:26:43.070 align:middle line:90% And this is what you'll find. 00:26:43.070 --> 00:26:48.470 align:middle line:90% V alpha dT minus V beta dP. 00:26:48.470 --> 00:26:53.400 align:middle line:90% 00:26:53.400 --> 00:26:56.920 align:middle line:90% So here's the thing. 00:26:56.920 --> 00:27:03.800 align:middle line:84% This coefficient and this thing have to be equal. 00:27:03.800 --> 00:27:07.660 align:middle line:84% And this thing and this thing have 00:27:07.660 --> 00:27:12.070 align:middle line:84% to be equal because they're both the same mathematical objects. 00:27:12.070 --> 00:27:14.027 align:middle line:90% They are these partials. 00:27:14.027 --> 00:27:16.360 align:middle line:84% I always say if it walks like a duck, talks like a duck, 00:27:16.360 --> 00:27:18.250 align:middle line:90% it's the same thing. 00:27:18.250 --> 00:27:20.082 align:middle line:84% Compare the known coefficients and equate. 00:27:20.082 --> 00:27:22.988 align:middle line:90% 00:27:22.988 --> 00:27:24.030 align:middle line:90% So we're going to equate. 00:27:24.030 --> 00:27:25.155 align:middle line:90% We're going to equate them. 00:27:25.155 --> 00:27:30.880 align:middle line:84% And we're to get the following. dV dT at fixed pressure-- 00:27:30.880 --> 00:27:32.850 align:middle line:90% well, that was this-- 00:27:32.850 --> 00:27:36.060 align:middle line:90% equals Y Cp over T-- 00:27:36.060 --> 00:27:37.920 align:middle line:90% but it's also this-- 00:27:37.920 --> 00:27:40.050 align:middle line:90% equals V alpha. 00:27:40.050 --> 00:27:48.380 align:middle line:84% And dV dP at fixed temperature equals this. 00:27:48.380 --> 00:27:51.590 align:middle line:90% X minus Y V alpha. 00:27:51.590 --> 00:27:53.480 align:middle line:90% But it's also this. 00:27:53.480 --> 00:27:56.910 align:middle line:90% It goes minus V beta. 00:27:56.910 --> 00:27:58.940 align:middle line:90% So now, we have our answer. 00:27:58.940 --> 00:27:59.900 align:middle line:90% How? 00:27:59.900 --> 00:28:02.360 align:middle line:84% We have a system of two equations 00:28:02.360 --> 00:28:07.430 align:middle line:84% and two unknowns, X and Y. Two equations and two unknowns. 00:28:07.430 --> 00:28:08.315 align:middle line:90% So now you can solve. 00:28:08.315 --> 00:28:11.310 align:middle line:90% 00:28:11.310 --> 00:28:13.775 align:middle line:90% 5-- solve for unknowns. 00:28:13.775 --> 00:28:21.660 align:middle line:90% 00:28:21.660 --> 00:28:25.350 align:middle line:84% I'll spare you the algebra there because it's pretty easy. 00:28:25.350 --> 00:28:31.960 align:middle line:90% Solving for Y V alpha T over Cp. 00:28:31.960 --> 00:28:36.510 align:middle line:84% And solving for X equals V squared alpha 00:28:36.510 --> 00:28:45.160 align:middle line:90% squared T over Cp minus V beta. 00:28:45.160 --> 00:28:48.040 align:middle line:84% And these are complicated and nonobvious. 00:28:48.040 --> 00:28:50.290 align:middle line:84% That's why you need a tactic, set of tactics, 00:28:50.290 --> 00:28:51.207 align:middle line:90% to get to the answers. 00:28:51.207 --> 00:28:52.707 align:middle line:84% Because if you follow this approach, 00:28:52.707 --> 00:28:54.280 align:middle line:84% you will always get the right answer 00:28:54.280 --> 00:28:57.410 align:middle line:84% even if it doesn't make sense to you. 00:28:57.410 --> 00:28:59.290 align:middle line:90% So here's the final answer. 00:28:59.290 --> 00:29:02.770 align:middle line:84% dV equals V squared alpha squared 00:29:02.770 --> 00:29:14.390 align:middle line:84% T over a Cp minus V beta times dP 00:29:14.390 --> 00:29:21.560 align:middle line:90% plus V alpha T over Cp times dS. 00:29:21.560 --> 00:29:23.180 align:middle line:84% This is the thing we were looking for. 00:29:23.180 --> 00:29:27.730 align:middle line:90% 00:29:27.730 --> 00:29:30.660 align:middle line:90% That's it. 00:29:30.660 --> 00:29:55.610 align:middle line:84% We integrate this to find change delta V for given delta P 00:29:55.610 --> 00:29:58.680 align:middle line:90% at fixed entropy. 00:29:58.680 --> 00:30:04.330 align:middle line:90% 00:30:04.330 --> 00:30:05.830 align:middle line:90% Again, this is mechanics. 00:30:05.830 --> 00:30:07.480 align:middle line:84% This will work for you even if you 00:30:07.480 --> 00:30:10.370 align:middle line:84% don't have a physical intuition for every step. 00:30:10.370 --> 00:30:13.660 align:middle line:90% This strategy works. 00:30:13.660 --> 00:30:15.010 align:middle line:90% Question. 00:30:15.010 --> 00:30:18.190 align:middle line:84% For a real material, when you are-- 00:30:18.190 --> 00:30:20.420 align:middle line:90% this is very realistic. 00:30:20.420 --> 00:30:22.690 align:middle line:84% I mean, now we go from multivariable calculus 00:30:22.690 --> 00:30:24.400 align:middle line:90% to money on the line. 00:30:24.400 --> 00:30:29.650 align:middle line:84% You are calculating the effect of a pressure change 00:30:29.650 --> 00:30:31.780 align:middle line:90% on the volume of components. 00:30:31.780 --> 00:30:34.330 align:middle line:84% Let's say you're going to send this into outer space. 00:30:34.330 --> 00:30:37.285 align:middle line:84% And you have a system made out of components. 00:30:37.285 --> 00:30:39.160 align:middle line:84% And the components are made out of materials. 00:30:39.160 --> 00:30:41.243 align:middle line:84% And you want to make sure that the pressure change 00:30:41.243 --> 00:30:44.980 align:middle line:84% that your device experience isn't going to cause the thing 00:30:44.980 --> 00:30:46.900 align:middle line:90% to fall apart. 00:30:46.900 --> 00:30:48.120 align:middle line:90% So now this matters a lot. 00:30:48.120 --> 00:30:50.790 align:middle line:84% You need to know how to do that calculation. 00:30:50.790 --> 00:30:51.720 align:middle line:90% Here's how. 00:30:51.720 --> 00:30:52.440 align:middle line:90% Question. 00:30:52.440 --> 00:30:55.371 align:middle line:90% Where are you going to get this? 00:30:55.371 --> 00:30:57.150 align:middle line:90% This is the integrant. 00:30:57.150 --> 00:30:59.220 align:middle line:84% How are you going to evaluate the integrant? 00:30:59.220 --> 00:31:02.685 align:middle line:90% 00:31:02.685 --> 00:31:04.310 align:middle line:84% How are you're probably going to end up 00:31:04.310 --> 00:31:07.383 align:middle line:84% evaluating the integrant when you're working at SpaceX, 00:31:07.383 --> 00:31:09.050 align:middle line:84% and you need the answer in five minutes? 00:31:09.050 --> 00:31:16.760 align:middle line:90% 00:31:16.760 --> 00:31:18.050 align:middle line:90% Let me set up a straw man. 00:31:18.050 --> 00:31:20.600 align:middle line:84% Are you going to do density functional theory calculations 00:31:20.600 --> 00:31:23.960 align:middle line:84% to calculate the thermal expansion coefficient, 00:31:23.960 --> 00:31:27.200 align:middle line:84% and specific heat, and volume, and compressibility 00:31:27.200 --> 00:31:28.940 align:middle line:84% for your material from first principles? 00:31:28.940 --> 00:31:32.080 align:middle line:90% 00:31:32.080 --> 00:31:34.360 align:middle line:84% AUDIENCE: You're going to probably look at databases. 00:31:34.360 --> 00:31:35.110 align:middle line:90% PROFESSOR: Databases. 00:31:35.110 --> 00:31:35.610 align:middle line:90% Thanks. 00:31:35.610 --> 00:31:36.813 align:middle line:90% That's my favorite word. 00:31:36.813 --> 00:31:38.230 align:middle line:84% You're going to look at databases. 00:31:38.230 --> 00:31:39.640 align:middle line:90% This stuff is in databases. 00:31:39.640 --> 00:31:44.422 align:middle line:84% You're going to parameterize what you need using databases. 00:31:44.422 --> 00:31:45.630 align:middle line:90% You're going to plug that in. 00:31:45.630 --> 00:31:47.550 align:middle line:84% And you're going to do this integral numerically. 00:31:47.550 --> 00:31:49.175 align:middle line:84% The computer is going to do it for you. 00:31:49.175 --> 00:31:51.600 align:middle line:84% You're going to make sure you have good sources of data. 00:31:51.600 --> 00:31:54.142 align:middle line:84% You're going to make sure the data is inputted correctly. 00:31:54.142 --> 00:31:56.100 align:middle line:84% You're going to make sure your units are right. 00:31:56.100 --> 00:31:57.960 align:middle line:84% And then the computer is going to do this for you 00:31:57.960 --> 00:31:59.190 align:middle line:90% as a numerical integration. 00:31:59.190 --> 00:32:02.520 align:middle line:84% That's how it works in the real world. 00:32:02.520 --> 00:32:06.825 align:middle line:84% So this hairy-looking challenge is why we hammer on this stuff 00:32:06.825 --> 00:32:07.940 align:middle line:90% in 020. 00:32:07.940 --> 00:32:09.802 align:middle line:90% And they make you talk to the-- 00:32:09.802 --> 00:32:11.730 align:middle line:84% you get to meet the librarians and such 00:32:11.730 --> 00:32:15.630 align:middle line:84% because being able to go from those data resources to answers 00:32:15.630 --> 00:32:17.410 align:middle line:90% is really an important skill. 00:32:17.410 --> 00:32:20.220 align:middle line:90% It's really an important skill. 00:32:20.220 --> 00:32:23.350 align:middle line:84% There is one case where you don't need to do any of that. 00:32:23.350 --> 00:32:26.370 align:middle line:84% And we'll discuss this a little more next lecture. 00:32:26.370 --> 00:32:30.567 align:middle line:84% And you can work this out on your own if you're curious. 00:32:30.567 --> 00:32:32.400 align:middle line:84% It's also worked out plenty in the textbook. 00:32:32.400 --> 00:32:38.100 align:middle line:84% For an ideal gas, as with many things, it simplifies. 00:32:38.100 --> 00:32:40.950 align:middle line:84% So you don't need any databases to evaluate 00:32:40.950 --> 00:32:41.970 align:middle line:90% this for an ideal gas. 00:32:41.970 --> 00:32:43.680 align:middle line:90% It simplifies to the following. 00:32:43.680 --> 00:32:54.790 align:middle line:84% X equals minus V over P times Cp minus R or Cp 00:32:54.790 --> 00:33:02.140 align:middle line:84% is also equal to V over P Cv over Cp. 00:33:02.140 --> 00:33:09.900 align:middle line:84% And this leads to an expression, which before we just gave you. 00:33:09.900 --> 00:33:20.020 align:middle line:84% P final over P initial equals V initial over V final. 00:33:20.020 --> 00:33:21.640 align:middle line:90% This is where that comes from. 00:33:21.640 --> 00:33:24.730 align:middle line:84% And if you use this expression for adiabatic process 00:33:24.730 --> 00:33:26.410 align:middle line:84% for an ideal gas, it comes from this. 00:33:26.410 --> 00:33:30.480 align:middle line:90% 00:33:30.480 --> 00:33:32.673 align:middle line:84% So I'm not going to take time to derive that now. 00:33:32.673 --> 00:33:34.340 align:middle line:84% It's kind of an interesting calculation. 00:33:34.340 --> 00:33:37.490 align:middle line:84% You can just evaluate all this stuff. 00:33:37.490 --> 00:33:42.690 align:middle line:84% And I'll just mention that in p set 2, in p set 2, 00:33:42.690 --> 00:33:45.000 align:middle line:84% you're doing a related problem, which 00:33:45.000 --> 00:33:49.310 align:middle line:84% is you're calculating change in Gibbs free energy 00:33:49.310 --> 00:33:53.870 align:middle line:84% with current pressure and entropy. 00:33:53.870 --> 00:33:58.160 align:middle line:84% So we just did volume with pressure and entropy. 00:33:58.160 --> 00:34:00.050 align:middle line:84% And in p set 2, you're going to do Gibbs 00:34:00.050 --> 00:34:01.550 align:middle line:90% with pressure and entropy. 00:34:01.550 --> 00:34:04.640 align:middle line:84% But the overall what they call the general strategy 00:34:04.640 --> 00:34:05.390 align:middle line:90% is the same. 00:34:05.390 --> 00:34:14.172 align:middle line:90% 00:34:14.172 --> 00:34:15.630 align:middle line:84% So for the remaining minutes, we're 00:34:15.630 --> 00:34:16.770 align:middle line:84% going to get a little bit specific 00:34:16.770 --> 00:34:18.430 align:middle line:84% and talk about Gibbs free strategy. 00:34:18.430 --> 00:34:25.850 align:middle line:90% 00:34:25.850 --> 00:34:27.440 align:middle line:90% Let's talk about equilibrium-- 00:34:27.440 --> 00:34:40.739 align:middle line:90% 00:34:40.739 --> 00:34:43.000 align:middle line:90% fixed pressure and temperature. 00:34:43.000 --> 00:34:45.090 align:middle line:84% So now we're reprocessing materials. 00:34:45.090 --> 00:34:52.590 align:middle line:84% Gibbs free energy equals U plus PV minus TS is minimized. 00:34:52.590 --> 00:34:59.600 align:middle line:90% 00:34:59.600 --> 00:35:02.370 align:middle line:84% Let's play with a differential form, dG. 00:35:02.370 --> 00:35:15.160 align:middle line:84% So I'm going to use the chain rule. dG [AUDIO OUT] dS 00:35:15.160 --> 00:35:19.210 align:middle line:90% minus S dT. 00:35:19.210 --> 00:35:20.985 align:middle line:90% And that was just chain rule. 00:35:20.985 --> 00:35:30.020 align:middle line:90% 00:35:30.020 --> 00:35:33.400 align:middle line:84% And now, I can use the combined statement. 00:35:33.400 --> 00:35:43.565 align:middle line:84% dU equals TdS minus PdV plus mu of i dN of i. 00:35:43.565 --> 00:35:50.320 align:middle line:90% That is all dU. 00:35:50.320 --> 00:35:51.700 align:middle line:90% I carry the rest of the terms. 00:35:51.700 --> 00:36:01.390 align:middle line:84% Plus PdV plus VdP minus TdS minus S dT. 00:36:01.390 --> 00:36:03.376 align:middle line:84% So here I use combined statement. 00:36:03.376 --> 00:36:14.070 align:middle line:90% 00:36:14.070 --> 00:36:16.010 align:middle line:84% And I'm going to see a bunch of things cancel. 00:36:16.010 --> 00:36:17.800 align:middle line:84% I'm going to see a bunch of things cancel. 00:36:17.800 --> 00:36:21.170 align:middle line:90% So we have TdS cancels TdS. 00:36:21.170 --> 00:36:25.780 align:middle line:90% 00:36:25.780 --> 00:36:27.490 align:middle line:90% Minus PdV cancels PdV. 00:36:27.490 --> 00:36:30.050 align:middle line:90% 00:36:30.050 --> 00:36:31.610 align:middle line:84% And so this simplifies a little bit. 00:36:31.610 --> 00:36:32.960 align:middle line:90% And I get the following. 00:36:32.960 --> 00:36:45.600 align:middle line:84% Equals minus SdT plus VdP plus mu i dN of i. 00:36:45.600 --> 00:36:46.290 align:middle line:90% Good. 00:36:46.290 --> 00:36:48.420 align:middle line:84% This is what we wanted because I told you 00:36:48.420 --> 00:36:53.040 align:middle line:84% the natural variables for Gibbs were temperature, pressure, 00:36:53.040 --> 00:36:54.730 align:middle line:90% and particle number. 00:36:54.730 --> 00:36:56.250 align:middle line:90% And that's what we get. 00:36:56.250 --> 00:36:57.960 align:middle line:84% That's what the calculus gives us. 00:36:57.960 --> 00:37:01.425 align:middle line:90% 00:37:01.425 --> 00:37:03.050 align:middle line:84% So there's a little clue there into how 00:37:03.050 --> 00:37:04.880 align:middle line:90% the Legendre transform works. 00:37:04.880 --> 00:37:06.600 align:middle line:90% But again, we won't go there. 00:37:06.600 --> 00:37:23.620 align:middle line:84% So the independent variables are T, P, and N sub i. 00:37:23.620 --> 00:37:26.490 align:middle line:90% And what are the coefficients? 00:37:26.490 --> 00:37:27.150 align:middle line:90% Somebody? 00:37:27.150 --> 00:37:27.650 align:middle line:90% Please? 00:37:27.650 --> 00:37:33.970 align:middle line:90% 00:37:33.970 --> 00:37:36.092 align:middle line:90% AUDIENCE: S, V, and mu of i. 00:37:36.092 --> 00:37:36.800 align:middle line:90% PROFESSOR: Right. 00:37:36.800 --> 00:37:42.770 align:middle line:90% Minus S, V, and mu of i. 00:37:42.770 --> 00:37:45.770 align:middle line:90% I'll make a note here. 00:37:45.770 --> 00:37:54.108 align:middle line:84% A lot of times, in this class, I use implicit summation. 00:37:54.108 --> 00:37:55.400 align:middle line:90% You probably picked up on that. 00:37:55.400 --> 00:38:03.860 align:middle line:84% But this implies i, mu of i, and i. 00:38:03.860 --> 00:38:06.860 align:middle line:84% It's just a little bit more compact. 00:38:06.860 --> 00:38:08.838 align:middle line:84% There's no i index on the left-hand side. 00:38:08.838 --> 00:38:11.130 align:middle line:84% So that means it has to resolve on the right-hand side. 00:38:11.130 --> 00:38:13.490 align:middle line:90% So you implicitly sum. 00:38:13.490 --> 00:38:24.160 align:middle line:84% And by inspection, by inspection, minus S 00:38:24.160 --> 00:38:29.800 align:middle line:84% describes the change of Gibbs with temperature 00:38:29.800 --> 00:38:36.170 align:middle line:84% at fixed pressure and particle number. 00:38:36.170 --> 00:38:40.670 align:middle line:84% Volume describes the change of Gibbs 00:38:40.670 --> 00:38:44.240 align:middle line:84% with pressure at fixed temperature and particle 00:38:44.240 --> 00:38:45.240 align:middle line:90% number. 00:38:45.240 --> 00:38:47.510 align:middle line:84% And finally, our favorite-- chemical potential 00:38:47.510 --> 00:38:52.070 align:middle line:84% describes the change of Gibbs with particle number i 00:38:52.070 --> 00:38:55.340 align:middle line:84% have fixed temperature, pressure, and particles 00:38:55.340 --> 00:38:58.109 align:middle line:90% j not equal to i. 00:38:58.109 --> 00:39:02.890 align:middle line:90% 00:39:02.890 --> 00:39:04.890 align:middle line:84% So already, we can say something about the slope 00:39:04.890 --> 00:39:07.590 align:middle line:90% of these things. 00:39:07.590 --> 00:39:09.780 align:middle line:90% On a slope of-- 00:39:09.780 --> 00:39:10.290 align:middle line:90% let's see. 00:39:10.290 --> 00:39:12.980 align:middle line:90% 00:39:12.980 --> 00:39:15.980 align:middle line:90% Here's pressure. 00:39:15.980 --> 00:39:17.900 align:middle line:90% And here's Gibbs free energy. 00:39:17.900 --> 00:39:21.875 align:middle line:84% And we're fixing temperature and particle number. 00:39:21.875 --> 00:39:25.300 align:middle line:90% 00:39:25.300 --> 00:39:26.972 align:middle line:90% What's the slope? 00:39:26.972 --> 00:39:27.680 align:middle line:90% What's the slope? 00:39:27.680 --> 00:39:28.910 align:middle line:90% Is it positive or negative? 00:39:28.910 --> 00:39:34.265 align:middle line:84% Gibbs versus pressure for fixed temperature and particle 00:39:34.265 --> 00:39:34.940 align:middle line:90% number. 00:39:34.940 --> 00:39:45.020 align:middle line:90% 00:39:45.020 --> 00:39:46.070 align:middle line:90% AUDIENCE: It's positive. 00:39:46.070 --> 00:39:48.320 align:middle line:84% PROFESSOR: Positive because volume is always positive. 00:39:48.320 --> 00:39:50.750 align:middle line:90% Can't have negative volume. 00:39:50.750 --> 00:39:52.747 align:middle line:90% So that's good. 00:39:52.747 --> 00:39:54.830 align:middle line:84% Being able to pick off the slope and the curvature 00:39:54.830 --> 00:39:56.580 align:middle line:84% of these things is really useful. 00:39:56.580 --> 00:39:58.610 align:middle line:84% It's really, really useful, as you'll see. 00:39:58.610 --> 00:39:59.310 align:middle line:90% What about this? 00:39:59.310 --> 00:40:02.980 align:middle line:90% What about temperature? 00:40:02.980 --> 00:40:07.630 align:middle line:84% Change of Gibbs with temperature for fixed pressure and particle 00:40:07.630 --> 00:40:09.300 align:middle line:90% number. 00:40:09.300 --> 00:40:10.760 align:middle line:90% AUDIENCE: Negative. 00:40:10.760 --> 00:40:13.190 align:middle line:84% PROFESSOR: Yeah, it's negative because entropy is strictly 00:40:13.190 --> 00:40:13.890 align:middle line:90% positive. 00:40:13.890 --> 00:40:15.770 align:middle line:84% So negative entropy is strictly negative. 00:40:15.770 --> 00:40:18.620 align:middle line:90% 00:40:18.620 --> 00:40:22.630 align:middle line:84% You got to think about these slopes. 00:40:22.630 --> 00:40:25.000 align:middle line:84% If Gibbs free energy increases when we squeeze, 00:40:25.000 --> 00:40:27.308 align:middle line:90% that's related to the enthalpy. 00:40:27.308 --> 00:40:28.600 align:middle line:90% That's related to the enthalpy. 00:40:28.600 --> 00:40:32.500 align:middle line:84% When we squeeze something, we're increasing the energy content 00:40:32.500 --> 00:40:34.375 align:middle line:84% of all the bonds that make up that thing. 00:40:34.375 --> 00:40:36.980 align:middle line:90% 00:40:36.980 --> 00:40:40.130 align:middle line:84% When we squeeze something, we're increasing the entropy content 00:40:40.130 --> 00:40:43.440 align:middle line:84% of all the bonds that make up that thing. 00:40:43.440 --> 00:40:46.115 align:middle line:84% So this is Gibbs free energy getting higher as we squeeze. 00:40:46.115 --> 00:40:50.110 align:middle line:90% 00:40:50.110 --> 00:40:52.600 align:middle line:84% When we heat something, Gibbs decreases. 00:40:52.600 --> 00:40:57.445 align:middle line:84% When we heat something, we're increasing the entropy. 00:40:57.445 --> 00:41:01.530 align:middle line:84% We're Increasing the amount of mixedupness. 00:41:01.530 --> 00:41:07.050 align:middle line:84% And Gibbs depends negatively with entropy. 00:41:07.050 --> 00:41:09.490 align:middle line:90% Well see this in many examples. 00:41:09.490 --> 00:41:11.697 align:middle line:84% So up with pressure, down with temperature. 00:41:11.697 --> 00:41:17.603 align:middle line:90% 00:41:17.603 --> 00:41:19.520 align:middle line:84% What about conditions that we're interrupting? 00:41:19.520 --> 00:41:26.700 align:middle line:84% At fixed temperature T, at fixed temperature and pressure, 00:41:26.700 --> 00:41:29.010 align:middle line:90% dT equals 0. 00:41:29.010 --> 00:41:30.760 align:middle line:90% That's pretty easy. 00:41:30.760 --> 00:41:32.985 align:middle line:90% And dP equals 0. 00:41:32.985 --> 00:41:36.110 align:middle line:90% 00:41:36.110 --> 00:41:48.950 align:middle line:84% That means the equilibrium, the equilibrium condition, 00:41:48.950 --> 00:42:02.720 align:middle line:84% reduces to dG equals minus SdT plus VdP plus mu of i dN of i. 00:42:02.720 --> 00:42:04.670 align:middle line:90% And we're holding that fixed. 00:42:04.670 --> 00:42:07.360 align:middle line:90% And we're holding that fixed. 00:42:07.360 --> 00:42:10.510 align:middle line:84% That means it basically reduces to this-- 00:42:10.510 --> 00:42:19.777 align:middle line:84% equals 0 because G is min at equilibrium. 00:42:19.777 --> 00:42:21.360 align:middle line:84% So if you think back to calculus, when 00:42:21.360 --> 00:42:24.630 align:middle line:90% you would calculate extrema. 00:42:24.630 --> 00:42:27.330 align:middle line:84% The extrema positions of a function 00:42:27.330 --> 00:42:29.550 align:middle line:90% are where its slope is 0. 00:42:29.550 --> 00:42:32.410 align:middle line:90% 00:42:32.410 --> 00:42:34.930 align:middle line:84% So equilibrium condition is that the change of Gibbs 00:42:34.930 --> 00:42:37.480 align:middle line:90% with particle number i is 0. 00:42:37.480 --> 00:42:41.490 align:middle line:90% 00:42:41.490 --> 00:42:42.990 align:middle line:84% So for most of the cases of interest 00:42:42.990 --> 00:42:44.820 align:middle line:84% to us as practicing materials scientists, 00:42:44.820 --> 00:42:51.755 align:middle line:84% that is multi-component, heterogeneous systems. 00:42:51.755 --> 00:43:00.560 align:middle line:90% 00:43:00.560 --> 00:43:20.710 align:middle line:84% This becomes dG sum over i component k dN i component k. 00:43:20.710 --> 00:43:23.000 align:middle line:84% I'm introducing a new notation here. 00:43:23.000 --> 00:43:29.305 align:middle line:90% Superscript labels phases. 00:43:29.305 --> 00:43:35.930 align:middle line:90% 00:43:35.930 --> 00:43:42.440 align:middle line:84% And the subscript labels components. 00:43:42.440 --> 00:43:51.350 align:middle line:90% 00:43:51.350 --> 00:43:53.860 align:middle line:84% So now we can picture something in our minds. 00:43:53.860 --> 00:43:57.580 align:middle line:84% We have a phase boundary here, two phases. 00:43:57.580 --> 00:44:02.530 align:middle line:84% And each phase has two components. 00:44:02.530 --> 00:44:14.737 align:middle line:90% 00:44:14.737 --> 00:44:16.320 align:middle line:84% This was a purple-rich phase, and this 00:44:16.320 --> 00:44:19.680 align:middle line:90% was an orange-rich phase. 00:44:19.680 --> 00:44:22.200 align:middle line:84% And evaluating the equilibrium condition 00:44:22.200 --> 00:44:26.880 align:middle line:84% is going to come down to knowing how the components can exchange 00:44:26.880 --> 00:44:30.390 align:middle line:84% between phases and what the effect of that exchange 00:44:30.390 --> 00:44:33.420 align:middle line:90% is on the chemical potential. 00:44:33.420 --> 00:44:34.368 align:middle line:90% Let me write that out. 00:44:34.368 --> 00:45:21.760 align:middle line:90% 00:45:21.760 --> 00:45:24.088 align:middle line:84% What determines whether or not components can 00:45:24.088 --> 00:45:25.255 align:middle line:90% exchange between the phases? 00:45:25.255 --> 00:45:28.400 align:middle line:90% 00:45:28.400 --> 00:45:29.810 align:middle line:90% It's the boundary. 00:45:29.810 --> 00:45:31.790 align:middle line:84% The property of the boundary tells us 00:45:31.790 --> 00:45:34.970 align:middle line:84% what components can exchange between phases. 00:45:34.970 --> 00:45:36.710 align:middle line:90% And that can be engineered. 00:45:36.710 --> 00:45:39.890 align:middle line:84% So there's a lot of engineering around choosing boundaries 00:45:39.890 --> 00:45:42.410 align:middle line:84% to get the desired equilibrium condition. 00:45:42.410 --> 00:45:45.340 align:middle line:90% 00:45:45.340 --> 00:45:48.160 align:middle line:84% But much of what comes later in this class 00:45:48.160 --> 00:46:06.317 align:middle line:84% is calculating composition dependence, 00:46:06.317 --> 00:46:08.650 align:middle line:84% calculating the composition dependence of those chemical 00:46:08.650 --> 00:46:09.820 align:middle line:90% potentials. 00:46:09.820 --> 00:46:11.200 align:middle line:84% And we're going to develop models 00:46:11.200 --> 00:46:13.940 align:middle line:84% for that and graphical tools for that, 00:46:13.940 --> 00:46:16.090 align:middle line:84% finally leading to binary phase diagrams. 00:46:16.090 --> 00:46:23.500 align:middle line:90% 00:46:23.500 --> 00:46:24.250 align:middle line:90% One last note. 00:46:24.250 --> 00:46:27.052 align:middle line:90% 00:46:27.052 --> 00:46:38.110 align:middle line:84% Going to do one last note about independent variables, 00:46:38.110 --> 00:46:39.940 align:middle line:90% sometimes called natural. 00:46:39.940 --> 00:46:50.160 align:middle line:90% 00:46:50.160 --> 00:47:02.030 align:middle line:84% If a state variable is regulated, regulated, 00:47:02.030 --> 00:47:04.220 align:middle line:90% what does that mean? 00:47:04.220 --> 00:47:07.400 align:middle line:90% Controlled. 00:47:07.400 --> 00:47:08.660 align:middle line:90% Regulations are controls. 00:47:08.660 --> 00:47:11.350 align:middle line:90% 00:47:11.350 --> 00:47:13.000 align:middle line:84% If you go to a McMaster-Carr, you 00:47:13.000 --> 00:47:16.120 align:middle line:84% can buy a regulator for pressure. 00:47:16.120 --> 00:47:17.710 align:middle line:84% If you go to Home Depot, you can get 00:47:17.710 --> 00:47:18.877 align:middle line:90% a regulator for temperature. 00:47:18.877 --> 00:47:20.900 align:middle line:90% We call it a thermostat. 00:47:20.900 --> 00:47:25.460 align:middle line:84% And those things that you would buy have analogs in-- 00:47:25.460 --> 00:47:29.090 align:middle line:84% not the real world, but in R&D and materials science, as well. 00:47:29.090 --> 00:47:30.507 align:middle line:84% If a state variable is regulated-- 00:47:30.507 --> 00:47:32.840 align:middle line:84% that's why I'm using that term, because it will help you 00:47:32.840 --> 00:47:33.560 align:middle line:90% in the future. 00:47:33.560 --> 00:47:38.720 align:middle line:84% If it's regulated, then it must be independent. 00:47:38.720 --> 00:47:46.000 align:middle line:90% 00:47:46.000 --> 00:47:47.940 align:middle line:84% So choosing your independent variables 00:47:47.940 --> 00:47:52.750 align:middle line:84% is one of the major intellectual challenges of thermo. 00:47:52.750 --> 00:47:55.340 align:middle line:84% It helps even know that that's the challenge you're facing. 00:47:55.340 --> 00:47:57.340 align:middle line:84% Knowing that that's the challenge you're facing, 00:47:57.340 --> 00:47:59.140 align:middle line:90% it gives you a good start. 00:47:59.140 --> 00:48:18.400 align:middle line:84% Identifying independent and dependent variables 00:48:18.400 --> 00:48:30.530 align:middle line:84% from the problem at hand, that's key to success. 00:48:30.530 --> 00:48:33.190 align:middle line:90% 00:48:33.190 --> 00:48:37.720 align:middle line:84% So for example, if the problem statement 00:48:37.720 --> 00:48:45.920 align:middle line:84% says something about fixed pressure, 00:48:45.920 --> 00:48:56.450 align:middle line:84% then you need to choose pressure as an independent variable. 00:48:56.450 --> 00:49:02.500 align:middle line:90% 00:49:02.500 --> 00:49:14.478 align:middle line:84% If the problem says adiabatic, you need to choose what? 00:49:14.478 --> 00:49:16.770 align:middle line:84% What should be your independent variable if the problem 00:49:16.770 --> 00:49:18.630 align:middle line:90% statement said adiabatic? 00:49:18.630 --> 00:49:21.890 align:middle line:84% What should be one of your two independent variables? 00:49:21.890 --> 00:49:22.880 align:middle line:90% AUDIENCE: Entropy. 00:49:22.880 --> 00:49:23.922 align:middle line:90% PROFESSOR: Yeah, entropy. 00:49:23.922 --> 00:49:26.970 align:middle line:90% 00:49:26.970 --> 00:49:39.040 align:middle line:84% And if the problem statement says in a thermal bath, 00:49:39.040 --> 00:49:41.860 align:middle line:84% then you need to choose temperature 00:49:41.860 --> 00:49:47.530 align:middle line:84% as one of your independent variables and so on. 00:49:47.530 --> 00:49:50.313 align:middle line:84% So this is like a key for turning problems 00:49:50.313 --> 00:49:51.730 align:middle line:84% that you encounter on problem sets 00:49:51.730 --> 00:49:56.130 align:middle line:84% or in the real world into things you can solve using thermo. 00:49:56.130 --> 00:49:58.000 align:middle line:90%