WEBVTT 00:00:00.000 --> 00:00:01.452 align:middle line:90% [SQUEAKING] 00:00:01.452 --> 00:00:01.952 align:middle line:90% 00:00:01.952 --> 00:00:03.904 align:middle line:90% [RUSTLING] 00:00:03.904 --> 00:00:05.856 align:middle line:90% [CLICKING] 00:00:05.856 --> 00:00:12.125 align:middle line:90% 00:00:12.125 --> 00:00:14.000 align:middle line:84% RAFAEL JARAMILLO: And all right, so let's go, 00:00:14.000 --> 00:00:16.340 align:middle line:90% ideal gas processes. 00:00:16.340 --> 00:00:23.950 align:middle line:90% 00:00:23.950 --> 00:00:26.350 align:middle line:90% How to motivate this? 00:00:26.350 --> 00:00:28.450 align:middle line:84% It's not just that we like ideal gases 00:00:28.450 --> 00:00:30.992 align:middle line:84% because they have a simple equation of state. 00:00:30.992 --> 00:00:32.950 align:middle line:84% What we're going to see in a couple of lectures 00:00:32.950 --> 00:00:37.390 align:middle line:84% is that the process of mixing ideal gases 00:00:37.390 --> 00:00:43.742 align:middle line:84% is a model for the process of mixing real materials. 00:00:43.742 --> 00:00:45.700 align:middle line:84% So in case you're wondering why are we spending 00:00:45.700 --> 00:00:47.230 align:middle line:90% all this time on gases-- 00:00:47.230 --> 00:00:50.590 align:middle line:84% and ideal gases, for that matter-- 00:00:50.590 --> 00:00:53.083 align:middle line:84% there is a material science motivation for it. 00:00:53.083 --> 00:00:53.810 align:middle line:90% OK. 00:00:53.810 --> 00:00:56.980 align:middle line:84% Let's start with reversible adiabatic expansion. 00:00:56.980 --> 00:00:59.480 align:middle line:84% You looked at this a little bit at the last lecture. 00:00:59.480 --> 00:01:02.350 align:middle line:84% I want to look at it some more because it is really important. 00:01:02.350 --> 00:01:07.910 align:middle line:90% 00:01:07.910 --> 00:01:09.720 align:middle line:90% Reversible adiabatic expansion. 00:01:09.720 --> 00:01:23.230 align:middle line:84% So what to choose for independent variables? 00:01:23.230 --> 00:01:26.930 align:middle line:90% 00:01:26.930 --> 00:01:29.270 align:middle line:84% So this thought exercise again, you have a problem, 00:01:29.270 --> 00:01:32.270 align:middle line:84% you want to calculate something about that process. 00:01:32.270 --> 00:01:35.940 align:middle line:84% What are you going to choose for independent variables? 00:01:35.940 --> 00:01:38.440 align:middle line:90% Well, adiabatic means no heat. 00:01:38.440 --> 00:01:41.110 align:middle line:90% 00:01:41.110 --> 00:01:45.070 align:middle line:90% And we said it's reversible. 00:01:45.070 --> 00:01:46.030 align:middle line:90% So what does that mean? 00:01:46.030 --> 00:01:48.600 align:middle line:90% Is anything fixed? 00:01:48.600 --> 00:01:50.313 align:middle line:84% And what does that imply we should use 00:01:50.313 --> 00:01:51.480 align:middle line:90% for an independent variable? 00:01:51.480 --> 00:01:55.050 align:middle line:90% 00:01:55.050 --> 00:01:58.182 align:middle line:84% AUDIENCE: Pressure and temperature. 00:01:58.182 --> 00:02:00.015 align:middle line:84% RAFAEL JARAMILLO: So here's a rule of thumb. 00:02:00.015 --> 00:02:03.000 align:middle line:90% 00:02:03.000 --> 00:02:15.620 align:middle line:84% If something is fixed, meaning constant, 00:02:15.620 --> 00:02:19.990 align:middle line:90% then it's independent. 00:02:19.990 --> 00:02:26.703 align:middle line:90% 00:02:26.703 --> 00:02:27.745 align:middle line:90% Let's call this stubborn. 00:02:27.745 --> 00:02:31.393 align:middle line:90% 00:02:31.393 --> 00:02:32.810 align:middle line:84% If something is fixed or constant, 00:02:32.810 --> 00:02:34.520 align:middle line:90% you can also say it's stubborn. 00:02:34.520 --> 00:02:37.170 align:middle line:84% Something that's stubborn is independent. 00:02:37.170 --> 00:02:41.900 align:middle line:84% So what is a state variable, which is fixed, 00:02:41.900 --> 00:02:46.130 align:middle line:84% for a adiabatic reversible process? 00:02:46.130 --> 00:02:47.630 align:middle line:90% AUDIENCE: Would it be entropy? 00:02:47.630 --> 00:02:49.490 align:middle line:84% RAFAEL JARAMILLO: Entropy, right. 00:02:49.490 --> 00:02:52.280 align:middle line:84% This is a process for dS equals 0, 00:02:52.280 --> 00:02:57.620 align:middle line:84% which means we need to use x as an independent variable. 00:02:57.620 --> 00:03:01.510 align:middle line:84% If something stubborn, it's independent. 00:03:01.510 --> 00:03:02.530 align:middle line:90% All right. 00:03:02.530 --> 00:03:03.880 align:middle line:90% And I said expansion. 00:03:03.880 --> 00:03:07.030 align:middle line:90% 00:03:07.030 --> 00:03:13.050 align:middle line:90% So you could say P or V change. 00:03:13.050 --> 00:03:19.330 align:middle line:84% Let's use P. So these are going to be our independent variables 00:03:19.330 --> 00:03:23.030 align:middle line:90% here, S and P. 00:03:23.030 --> 00:03:31.690 align:middle line:84% So we want to find the equation of state that describes, 00:03:31.690 --> 00:03:35.070 align:middle line:84% let's say, temperature as a function of entropy 00:03:35.070 --> 00:03:36.085 align:middle line:90% and pressure. 00:03:36.085 --> 00:03:37.710 align:middle line:84% Just for example, let's say the problem 00:03:37.710 --> 00:03:40.440 align:middle line:84% asked, calculate the temperature change 00:03:40.440 --> 00:03:47.250 align:middle line:84% for a reversible adiabatic process moving from pressure 1 00:03:47.250 --> 00:03:49.020 align:middle line:84% to pressure 2, something like that. 00:03:49.020 --> 00:03:52.210 align:middle line:84% So that would be a useful equation of state to find. 00:03:52.210 --> 00:03:52.710 align:middle line:90% All right. 00:03:52.710 --> 00:03:54.440 align:middle line:84% So now this is going to be more practice 00:03:54.440 --> 00:03:55.440 align:middle line:90% on the general strategy. 00:03:55.440 --> 00:03:59.210 align:middle line:84% That's really one of the reasons I'm doing this here. 00:03:59.210 --> 00:04:00.910 align:middle line:90% So you see this. 00:04:00.910 --> 00:04:10.530 align:middle line:84% Use general strategy to find dT equals-- 00:04:10.530 --> 00:04:19.500 align:middle line:84% let's say M dS plus N dP, where M and N 00:04:19.500 --> 00:04:24.300 align:middle line:90% are the unknown coefficients. 00:04:24.300 --> 00:04:32.370 align:middle line:90% 00:04:32.370 --> 00:04:34.760 align:middle line:90% So then we substitute in. 00:04:34.760 --> 00:04:36.300 align:middle line:90% We do a change of variables. 00:04:36.300 --> 00:04:42.660 align:middle line:84% We change from S and P to T and P, as we did the last time. 00:04:42.660 --> 00:04:47.430 align:middle line:90% 00:04:47.430 --> 00:04:48.670 align:middle line:90% Pardon me. 00:04:48.670 --> 00:04:53.130 align:middle line:90% Minus V alpha dP plus NdP. 00:04:53.130 --> 00:04:55.770 align:middle line:90% 00:04:55.770 --> 00:05:04.310 align:middle line:84% Change-- that's a change of variables. 00:05:04.310 --> 00:05:05.675 align:middle line:90% And then we gather terms. 00:05:05.675 --> 00:05:20.190 align:middle line:90% 00:05:20.190 --> 00:05:21.120 align:middle line:90% Just gathering terms. 00:05:21.120 --> 00:05:29.660 align:middle line:90% 00:05:29.660 --> 00:05:45.540 align:middle line:84% Then we can say by inspection, M Cp over T equals 1. 00:05:45.540 --> 00:05:46.600 align:middle line:90% Why is that? 00:05:46.600 --> 00:05:48.011 align:middle line:90% dT has to equal dT. 00:05:48.011 --> 00:05:51.560 align:middle line:90% 00:05:51.560 --> 00:05:52.760 align:middle line:90% So this got to be equal 1. 00:05:52.760 --> 00:05:56.030 align:middle line:90% This has to be equal to 1. 00:05:56.030 --> 00:05:58.940 align:middle line:90% And so it's equal-- 00:05:58.940 --> 00:06:04.470 align:middle line:90% M equals T over Cp. 00:06:04.470 --> 00:06:04.970 align:middle line:90% All right. 00:06:04.970 --> 00:06:10.730 align:middle line:84% And likewise, if we have T as an independent variable-- 00:06:10.730 --> 00:06:14.030 align:middle line:84% dependent, sorry-- and the two independent variables here are 00:06:14.030 --> 00:06:15.380 align:middle line:90% T and P-- 00:06:15.380 --> 00:06:16.670 align:middle line:90% it's kind of silly-- 00:06:16.670 --> 00:06:18.680 align:middle line:90% this thing has to be 0. 00:06:18.680 --> 00:06:22.420 align:middle line:84% dT has to be equal to dT, end of story. 00:06:22.420 --> 00:06:27.495 align:middle line:90% So N minus MV alpha equals 0. 00:06:27.495 --> 00:06:30.780 align:middle line:90% 00:06:30.780 --> 00:06:37.933 align:middle line:84% And so N equals TV alpha over Cp. 00:06:37.933 --> 00:06:39.600 align:middle line:84% All right, so these are the coefficients 00:06:39.600 --> 00:06:42.760 align:middle line:90% that we were looking for. 00:06:42.760 --> 00:06:45.993 align:middle line:84% I did this more quickly than I did it on Monday. 00:06:45.993 --> 00:06:47.910 align:middle line:84% Again, there's lots more examples in the book, 00:06:47.910 --> 00:06:49.722 align:middle line:84% but this is the general strategy. 00:06:49.722 --> 00:06:51.180 align:middle line:84% So we found our coefficients there. 00:06:51.180 --> 00:06:52.513 align:middle line:90% That's what we were looking for. 00:06:52.513 --> 00:06:56.810 align:middle line:90% 00:06:56.810 --> 00:06:58.450 align:middle line:90% Now we can plug in. 00:06:58.450 --> 00:07:00.085 align:middle line:84% We can use properties of the ideal gas. 00:07:00.085 --> 00:07:11.250 align:middle line:90% 00:07:11.250 --> 00:07:12.960 align:middle line:84% What are some properties of ideal gases? 00:07:12.960 --> 00:07:25.180 align:middle line:84% V equals RT over P. Let's just keep the number of moles 00:07:25.180 --> 00:07:28.870 align:middle line:84% equals 1 first because it's easier that way and also 00:07:28.870 --> 00:07:32.050 align:middle line:84% because I did this thing where I used 00:07:32.050 --> 00:07:34.970 align:middle line:84% N and M, because the textbook does it that way. 00:07:34.970 --> 00:07:36.970 align:middle line:84% I don't like to do it that way because N is also 00:07:36.970 --> 00:07:37.820 align:middle line:90% the number of moles. 00:07:37.820 --> 00:07:40.570 align:middle line:84% So we're going to just pretend I didn't do that. 00:07:40.570 --> 00:07:41.320 align:middle line:90% All right. 00:07:41.320 --> 00:07:43.960 align:middle line:84% And alpha equals 1/T. We calculated 00:07:43.960 --> 00:07:45.700 align:middle line:90% on that first or second day. 00:07:45.700 --> 00:07:47.650 align:middle line:84% And another result, which you're probably 00:07:47.650 --> 00:07:49.150 align:middle line:90% familiar with from the book-- 00:07:49.150 --> 00:07:51.350 align:middle line:84% we haven't explicitly mentioned in class-- 00:07:51.350 --> 00:07:58.080 align:middle line:84% which is that Cp equals 5/2 R for a monatomic gas, monatomic. 00:07:58.080 --> 00:08:03.560 align:middle line:90% 00:08:03.560 --> 00:08:06.070 align:middle line:90% All right. 00:08:06.070 --> 00:08:09.490 align:middle line:84% So now we can calculate the thing which we are looking for. 00:08:09.490 --> 00:08:17.980 align:middle line:84% dT at fixed entropy equals dT dP at fixed entropy dP. 00:08:17.980 --> 00:08:19.900 align:middle line:84% And this we have from the previous board. 00:08:19.900 --> 00:08:22.180 align:middle line:84% That's what our general strategy gave us. 00:08:22.180 --> 00:08:27.720 align:middle line:90% TV alpha over Cp dP. 00:08:27.720 --> 00:08:30.630 align:middle line:90% 00:08:30.630 --> 00:08:37.959 align:middle line:84% And we can plug in stuff from the ideal gas-- equals TRT Cp 00:08:37.959 --> 00:08:38.459 align:middle line:90% TP. 00:08:38.459 --> 00:08:44.690 align:middle line:90% 00:08:44.690 --> 00:08:45.190 align:middle line:90% All right. 00:08:45.190 --> 00:08:48.742 align:middle line:84% We're using ideal gas properties there. 00:08:48.742 --> 00:09:00.040 align:middle line:84% And we're going to simplify R over Cp T/P dP equals-- 00:09:00.040 --> 00:09:05.380 align:middle line:90% you can also say 2/5 T/P dP. 00:09:05.380 --> 00:09:06.830 align:middle line:90% I'm writing off the edge there. 00:09:06.830 --> 00:09:07.540 align:middle line:90% Sorry about that. 00:09:07.540 --> 00:09:12.750 align:middle line:90% 00:09:12.750 --> 00:09:13.990 align:middle line:90% I can separate variables. 00:09:13.990 --> 00:09:15.995 align:middle line:84% This is a separable differential equation. 00:09:15.995 --> 00:09:17.745 align:middle line:84% This is a separable differential equation. 00:09:17.745 --> 00:09:18.585 align:middle line:90% There's dT. 00:09:18.585 --> 00:09:19.920 align:middle line:90% There's dP. 00:09:19.920 --> 00:09:22.800 align:middle line:84% There's T. There's P. It's separable. 00:09:22.800 --> 00:09:26.490 align:middle line:90% Separate and integrate. 00:09:26.490 --> 00:09:31.200 align:middle line:90% 00:09:31.200 --> 00:09:42.320 align:middle line:84% DT over T equals R over Cp dP over p. 00:09:42.320 --> 00:09:48.650 align:middle line:84% Integrate that, and I get T final over T initial 00:09:48.650 --> 00:10:02.128 align:middle line:84% equals P final over P initial R over Cp. 00:10:02.128 --> 00:10:07.290 align:middle line:90% 00:10:07.290 --> 00:10:12.900 align:middle line:84% So I gave you this in the P set and in a lecture 00:10:12.900 --> 00:10:14.140 align:middle line:90% a couple of lectures ago. 00:10:14.140 --> 00:10:17.820 align:middle line:84% And then on Monday, we did something like this. 00:10:17.820 --> 00:10:20.520 align:middle line:84% We used the general strategy, and we reduced it 00:10:20.520 --> 00:10:21.840 align:middle line:90% to the case of ideal gases. 00:10:21.840 --> 00:10:25.110 align:middle line:84% And then today, we just went a little farther 00:10:25.110 --> 00:10:26.940 align:middle line:90% because we went a little faster. 00:10:26.940 --> 00:10:28.980 align:middle line:84% And we actually derived an expression 00:10:28.980 --> 00:10:33.433 align:middle line:84% for the adiabatic reversible expansion of an ideal gas. 00:10:33.433 --> 00:10:35.100 align:middle line:84% So we've kind of seen the same thing now 00:10:35.100 --> 00:10:37.810 align:middle line:90% at three different levels. 00:10:37.810 --> 00:10:41.460 align:middle line:84% And this is just one expression which you'll find. 00:10:41.460 --> 00:10:44.910 align:middle line:84% If you look up on Wikipedia ideal gas adiabatic process, 00:10:44.910 --> 00:10:47.970 align:middle line:90% you'll find several expressions. 00:10:47.970 --> 00:10:49.140 align:middle line:90% They're all equivalent. 00:10:49.140 --> 00:10:53.300 align:middle line:90% 00:10:53.300 --> 00:10:54.695 align:middle line:90% There are alternative forms. 00:10:54.695 --> 00:11:04.070 align:middle line:90% 00:11:04.070 --> 00:11:24.680 align:middle line:84% There are alternative forms for adiabatic of the original IG. 00:11:24.680 --> 00:11:27.800 align:middle line:90% PV to gamma equals constant. 00:11:27.800 --> 00:11:30.890 align:middle line:90% That's an alternative form. 00:11:30.890 --> 00:11:36.297 align:middle line:84% TV to the gamma minus 1 equals constant. 00:11:36.297 --> 00:11:38.005 align:middle line:84% These are simply related by [INAUDIBLE].. 00:11:38.005 --> 00:11:40.030 align:middle line:84% It does use the ideal gas equation of state. 00:11:40.030 --> 00:11:42.220 align:middle line:84% You can get from one to the other. 00:11:42.220 --> 00:11:46.660 align:middle line:84% Here again, gamma equals the ratio of Cp over Cv, which 00:11:46.660 --> 00:11:49.390 align:middle line:84% we know is greater than 1 for physical reasons 00:11:49.390 --> 00:11:51.460 align:middle line:90% we've discussed before. 00:11:51.460 --> 00:11:56.350 align:middle line:84% Or if you want, P final over P initial 00:11:56.350 --> 00:12:01.540 align:middle line:90% equals V initial over V final. 00:12:01.540 --> 00:12:03.120 align:middle line:90% I think we've seen that already. 00:12:03.120 --> 00:12:04.330 align:middle line:90% So these are all equivalent. 00:12:04.330 --> 00:12:07.177 align:middle line:90% These are all the same thing. 00:12:07.177 --> 00:12:08.260 align:middle line:90% And we know that we have-- 00:12:08.260 --> 00:12:12.320 align:middle line:90% 00:12:12.320 --> 00:12:26.210 align:middle line:84% on a VP plot, if we have isotherms, 00:12:26.210 --> 00:12:30.180 align:middle line:84% the adiabat is going to be something steeper. 00:12:30.180 --> 00:12:41.630 align:middle line:84% So if these are isotherms, this is the adiabat. 00:12:41.630 --> 00:12:47.400 align:middle line:90% 00:12:47.400 --> 00:12:48.140 align:middle line:90% Right. 00:12:48.140 --> 00:12:50.100 align:middle line:90% OK. 00:12:50.100 --> 00:12:51.525 align:middle line:90% So we've seen this before. 00:12:51.525 --> 00:12:55.780 align:middle line:90% 00:12:55.780 --> 00:12:57.790 align:middle line:84% There's an interesting alternative derivation 00:12:57.790 --> 00:13:02.228 align:middle line:84% of this that does not use the general strategy. 00:13:02.228 --> 00:13:03.270 align:middle line:90% I think it's interesting. 00:13:03.270 --> 00:13:05.840 align:middle line:90% You should check it out. 00:13:05.840 --> 00:13:07.688 align:middle line:90% DeHoff section is-- 00:13:07.688 --> 00:13:08.480 align:middle line:90% I wrote this down-- 00:13:08.480 --> 00:13:11.510 align:middle line:90% 4.2.4. 00:13:11.510 --> 00:13:17.770 align:middle line:84% He uses the fact that for an ideal gas, 00:13:17.770 --> 00:13:19.840 align:middle line:84% the internal energy depends only on temperature. 00:13:19.840 --> 00:13:25.820 align:middle line:84% And you can also start from this expression and derive these. 00:13:25.820 --> 00:13:30.087 align:middle line:84% So if you're interested, let's do that. 00:13:30.087 --> 00:13:30.670 align:middle line:90% Let's do that. 00:13:30.670 --> 00:13:35.140 align:middle line:90% 00:13:35.140 --> 00:13:38.860 align:middle line:90% dQ equals 0 alternative. 00:13:38.860 --> 00:13:44.900 align:middle line:90% 00:13:44.900 --> 00:13:47.840 align:middle line:90% Alternatively, dQ equals 0. 00:13:47.840 --> 00:13:52.790 align:middle line:90% That means that dU equals work-- 00:13:52.790 --> 00:13:54.470 align:middle line:90% because there's no Q-- 00:13:54.470 --> 00:13:55.865 align:middle line:90% equals minus PdV. 00:13:55.865 --> 00:13:59.500 align:middle line:90% 00:13:59.500 --> 00:14:00.000 align:middle line:90% All right. 00:14:00.000 --> 00:14:05.410 align:middle line:90% 00:14:05.410 --> 00:14:30.330 align:middle line:84% Previous-- we're going to use that. 00:14:30.330 --> 00:14:38.130 align:middle line:84% And we will use both of these combined, substituting Cv dT 00:14:38.130 --> 00:14:49.730 align:middle line:84% equals minus p dV equals minus RT V dV. 00:14:49.730 --> 00:14:51.320 align:middle line:90% Again, separate and integrate. 00:14:51.320 --> 00:14:59.860 align:middle line:90% 00:14:59.860 --> 00:15:03.760 align:middle line:84% We got-- well, this is separable, right? 00:15:03.760 --> 00:15:09.175 align:middle line:90% Cv dT over T equals-- 00:15:09.175 --> 00:15:12.280 align:middle line:90% 00:15:12.280 --> 00:15:17.350 align:middle line:84% minus R dV over V. And we can integrate that, 00:15:17.350 --> 00:15:21.430 align:middle line:84% and we get T final over T initial 00:15:21.430 --> 00:15:28.330 align:middle line:84% equals V initial over V final R over Cv-- 00:15:28.330 --> 00:15:30.410 align:middle line:90% which, again, is the same thing. 00:15:30.410 --> 00:15:33.400 align:middle line:84% So we're dancing around the same thing here. 00:15:33.400 --> 00:15:34.070 align:middle line:90% Good. 00:15:34.070 --> 00:15:34.570 align:middle line:90% OK. 00:15:34.570 --> 00:15:37.300 align:middle line:90% 00:15:37.300 --> 00:15:39.650 align:middle line:84% So by now, you've probably had more than enough 00:15:39.650 --> 00:15:45.050 align:middle line:84% of reversible adiabatic expansion of an ideal gas. 00:15:45.050 --> 00:15:46.670 align:middle line:90% Let's do isothermal expansion. 00:15:46.670 --> 00:16:03.100 align:middle line:90% 00:16:03.100 --> 00:16:03.600 align:middle line:90% All right. 00:16:03.600 --> 00:16:07.710 align:middle line:84% Isothermal expansion-- this is our second one. 00:16:07.710 --> 00:16:14.320 align:middle line:84% Here, same question I asked in the beginning, what to use 00:16:14.320 --> 00:16:16.060 align:middle line:90% four independent variables? 00:16:16.060 --> 00:16:24.235 align:middle line:90% 00:16:24.235 --> 00:16:26.110 align:middle line:84% What should we use for independent variables? 00:16:26.110 --> 00:16:33.200 align:middle line:90% 00:16:33.200 --> 00:16:36.140 align:middle line:84% AUDIENCE: Temperature and then either pressure or volume 00:16:36.140 --> 00:16:36.950 align:middle line:90% again. 00:16:36.950 --> 00:16:39.990 align:middle line:84% RAFAEL JARAMILLO: Good, thank you. 00:16:39.990 --> 00:16:45.330 align:middle line:90% Isothermal dT equals 0. 00:16:45.330 --> 00:16:49.660 align:middle line:84% That means you use T. Temperature is stubborn. 00:16:49.660 --> 00:16:51.610 align:middle line:90% It's independent. 00:16:51.610 --> 00:16:54.720 align:middle line:84% And expansion, you're going to control something. 00:16:54.720 --> 00:16:56.250 align:middle line:84% Something that's controlled sounds 00:16:56.250 --> 00:17:01.390 align:middle line:84% like it should be an independent variable expansion. 00:17:01.390 --> 00:17:05.594 align:middle line:84% So we're going to-- in this case, we use pressure. 00:17:05.594 --> 00:17:11.510 align:middle line:84% We could use volume, but we'll use pressure. 00:17:11.510 --> 00:17:12.690 align:middle line:90% All right. 00:17:12.690 --> 00:17:22.770 align:middle line:84% And what I want to do now is find the equation of state 00:17:22.770 --> 00:17:28.713 align:middle line:84% Gibbs free energy T and P. So this 00:17:28.713 --> 00:17:30.630 align:middle line:84% is an example where-- let's say someone said-- 00:17:30.630 --> 00:17:33.000 align:middle line:84% or your problem statement boil down to, 00:17:33.000 --> 00:17:34.768 align:middle line:84% we have an isothermal expansion and we 00:17:34.768 --> 00:17:36.810 align:middle line:84% want to calculate the change of Gibbs free energy 00:17:36.810 --> 00:17:40.250 align:middle line:84% for some process going from pressure 1 to pressure 2. 00:17:40.250 --> 00:17:42.770 align:middle line:84% That's a very common scenario that you'll find. 00:17:42.770 --> 00:17:45.900 align:middle line:84% To do this calculation is very, very common. 00:17:45.900 --> 00:17:49.120 align:middle line:90% So let's see how to do it. 00:17:49.120 --> 00:17:52.180 align:middle line:84% We want to find a change, so we want to use calculus. 00:17:52.180 --> 00:17:57.420 align:middle line:84% So let's use the proper differential form. 00:17:57.420 --> 00:18:04.230 align:middle line:90% 00:18:04.230 --> 00:18:12.420 align:middle line:84% dG equals-- well dG equals minus SdT plus P dV. 00:18:12.420 --> 00:18:14.580 align:middle line:90% That, we know. 00:18:14.580 --> 00:18:17.310 align:middle line:84% But we also said this was isothermal. 00:18:17.310 --> 00:18:20.670 align:middle line:90% 00:18:20.670 --> 00:18:24.898 align:middle line:90% So this is really pretty simple. 00:18:24.898 --> 00:18:26.565 align:middle line:84% It's a single-variable calculus problem. 00:18:26.565 --> 00:18:29.058 align:middle line:90% 00:18:29.058 --> 00:18:30.600 align:middle line:84% I point this out over and over again, 00:18:30.600 --> 00:18:32.780 align:middle line:84% so forgive me for pointing out again. 00:18:32.780 --> 00:18:35.750 align:middle line:84% This is why if something is stubborn or something is fixed, 00:18:35.750 --> 00:18:37.460 align:middle line:90% it's independent. 00:18:37.460 --> 00:18:40.820 align:middle line:84% Not only does that make sense, it also makes your life easier. 00:18:40.820 --> 00:18:43.743 align:middle line:84% When you have multivariable calculus 00:18:43.743 --> 00:18:45.410 align:middle line:84% and you have an opportunity to reduce it 00:18:45.410 --> 00:18:48.240 align:middle line:84% to single-variable calculus, you should take that opportunity. 00:18:48.240 --> 00:18:49.380 align:middle line:90% So here's an example. 00:18:49.380 --> 00:18:52.640 align:middle line:84% If you have two terms on the right-hand side 00:18:52.640 --> 00:18:56.690 align:middle line:84% and you can set one of them to 0, the math gets easier. 00:18:56.690 --> 00:18:59.320 align:middle line:84% That's why a correct choice of independent variables 00:18:59.320 --> 00:19:00.890 align:middle line:90% is so important. 00:19:00.890 --> 00:19:02.440 align:middle line:90% This boils is way down here. 00:19:02.440 --> 00:19:08.945 align:middle line:84% PdV for an ideal gas is RT P dP-- 00:19:08.945 --> 00:19:10.900 align:middle line:90% wait, I'm sorry. 00:19:10.900 --> 00:19:11.800 align:middle line:90% No one called me out. 00:19:11.800 --> 00:19:15.940 align:middle line:90% 00:19:15.940 --> 00:19:16.440 align:middle line:90% Right. 00:19:16.440 --> 00:19:19.310 align:middle line:90% 00:19:19.310 --> 00:19:22.160 align:middle line:84% Gibbs free energy, independent variables are T and P. 00:19:22.160 --> 00:19:25.170 align:middle line:84% And I get to ignore half of the right-hand side. 00:19:25.170 --> 00:19:26.930 align:middle line:90% So things simplify. 00:19:26.930 --> 00:19:31.970 align:middle line:84% And this is pretty easy integral. 00:19:31.970 --> 00:19:40.110 align:middle line:84% So dG equals G final minus G initial-- 00:19:40.110 --> 00:19:43.380 align:middle line:90% we'll call that delta G-- 00:19:43.380 --> 00:19:50.880 align:middle line:84% equals RT plug P final over P initial. 00:19:50.880 --> 00:19:56.800 align:middle line:90% 00:19:56.800 --> 00:20:01.870 align:middle line:84% The math is easy when we set the problem up right. 00:20:01.870 --> 00:20:07.420 align:middle line:84% I'm going to put two boxes around this. 00:20:07.420 --> 00:20:09.970 align:middle line:84% The math-- oh it's like a-- it's a Christmas equation. 00:20:09.970 --> 00:20:12.970 align:middle line:84% The math you just saw is very easy, 00:20:12.970 --> 00:20:16.510 align:middle line:84% but this is something which will stick in your mind. 00:20:16.510 --> 00:20:20.350 align:middle line:84% Because you will see expressions like this in other classes 00:20:20.350 --> 00:20:22.370 align:middle line:90% over and over and over again. 00:20:22.370 --> 00:20:25.750 align:middle line:84% So, for example, if you do electrochemistry, 00:20:25.750 --> 00:20:31.100 align:middle line:84% this equation sets the voltage of your device, 00:20:31.100 --> 00:20:33.280 align:middle line:90% be it a reactor or a battery. 00:20:33.280 --> 00:20:36.990 align:middle line:84% This is called the Nernst equation in another context. 00:20:36.990 --> 00:20:40.440 align:middle line:84% If you do chemical engineering and you calculate change 00:20:40.440 --> 00:20:42.510 align:middle line:90% in chemical potential with 00:20:42.510 --> 00:20:44.948 align:middle line:84% fugacity-- we're not going to use that word in this class 00:20:44.948 --> 00:20:45.990 align:middle line:90% because this is course 3. 00:20:45.990 --> 00:20:47.820 align:middle line:84% But if you're in course 10, it's like fugacity this, 00:20:47.820 --> 00:20:50.340 align:middle line:84% fugacity that-- you're going to have log of fugacities, 00:20:50.340 --> 00:20:52.782 align:middle line:84% and it's going to be the same equation-- 00:20:52.782 --> 00:20:53.740 align:middle line:90% and so forth and so on. 00:20:53.740 --> 00:20:55.440 align:middle line:84% This is really foundational, so that's 00:20:55.440 --> 00:20:57.315 align:middle line:90% why I put it in two colors. 00:20:57.315 --> 00:20:59.190 align:middle line:84% AUDIENCE: Sorry, right have a quick question. 00:20:59.190 --> 00:21:03.840 align:middle line:84% Are we assuming that we're using molar volume here? 00:21:03.840 --> 00:21:08.580 align:middle line:84% RAFAEL JARAMILLO: I've kept N equals 1 moles. 00:21:08.580 --> 00:21:11.900 align:middle line:90% N equals 1 for simplicity here. 00:21:11.900 --> 00:21:12.900 align:middle line:90% AUDIENCE: OK, thank you. 00:21:12.900 --> 00:21:15.180 align:middle line:90% RAFAEL JARAMILLO: Yeah. 00:21:15.180 --> 00:21:17.820 align:middle line:90% So thanks for clarifying. 00:21:17.820 --> 00:21:18.330 align:middle line:90% Yeah, sure. 00:21:18.330 --> 00:21:20.030 align:middle line:84% It's molar volume because it's 1 moles. 00:21:20.030 --> 00:21:21.780 align:middle line:84% I just didn't want to carry an extra term. 00:21:21.780 --> 00:21:23.710 align:middle line:84% If you wanted to, you could put an N there. 00:21:23.710 --> 00:21:26.590 align:middle line:84% And then you have an N there, and that would be fine. 00:21:26.590 --> 00:21:28.890 align:middle line:90% So let me we put a go to N here. 00:21:28.890 --> 00:21:31.290 align:middle line:84% Here's an N, and you can put an N there. 00:21:31.290 --> 00:21:33.620 align:middle line:84% And everything else is the same, same thing. 00:21:33.620 --> 00:21:39.110 align:middle line:90% 00:21:39.110 --> 00:21:39.610 align:middle line:90% Good. 00:21:39.610 --> 00:21:43.850 align:middle line:90% 00:21:43.850 --> 00:21:44.640 align:middle line:90% All right. 00:21:44.640 --> 00:21:46.700 align:middle line:84% Now, again, I don't expect you to have 00:21:46.700 --> 00:21:49.095 align:middle line:84% the skies part and the angels singing now, 00:21:49.095 --> 00:21:51.720 align:middle line:84% because we haven't gotten to the usefulness of this expression. 00:21:51.720 --> 00:21:53.178 align:middle line:84% But I want it to lodge in your mind 00:21:53.178 --> 00:21:56.240 align:middle line:84% because we're going to come back to this over and over. 00:21:56.240 --> 00:22:04.020 align:middle line:84% All right, let's talk about another ideal gas process, 00:22:04.020 --> 00:22:10.900 align:middle line:90% adiabatic free expansion. 00:22:10.900 --> 00:22:14.470 align:middle line:90% 00:22:14.470 --> 00:22:18.202 align:middle line:84% So this is a little bit of a kooky example when you first 00:22:18.202 --> 00:22:19.785 align:middle line:84% see it, if you haven't seen it before. 00:22:19.785 --> 00:22:22.820 align:middle line:90% 00:22:22.820 --> 00:22:24.012 align:middle line:90% So just bear with me. 00:22:24.012 --> 00:22:24.970 align:middle line:90% So here's what we have. 00:22:24.970 --> 00:22:26.290 align:middle line:90% Here's the situation. 00:22:26.290 --> 00:22:32.390 align:middle line:90% 00:22:32.390 --> 00:22:34.772 align:middle line:90% We're going to have a piston. 00:22:34.772 --> 00:22:36.105 align:middle line:90% So we're going to have a piston. 00:22:36.105 --> 00:22:45.840 align:middle line:90% 00:22:45.840 --> 00:22:53.190 align:middle line:84% And this is a cylinder, and it's thermally insulated. 00:22:53.190 --> 00:22:54.570 align:middle line:90% So I'm drawing insulation. 00:22:54.570 --> 00:23:02.930 align:middle line:90% 00:23:02.930 --> 00:23:06.060 align:middle line:90% Insulated. 00:23:06.060 --> 00:23:08.240 align:middle line:84% Pardon me, that's not very clear-- insulated. 00:23:08.240 --> 00:23:11.960 align:middle line:90% And let's make the gas orange. 00:23:11.960 --> 00:23:18.800 align:middle line:90% 00:23:18.800 --> 00:23:20.390 align:middle line:90% Simple picture, right? 00:23:20.390 --> 00:23:23.930 align:middle line:84% So what we're going to do is we're 00:23:23.930 --> 00:23:39.070 align:middle line:84% going to withdraw the piston instantaneously, 00:23:39.070 --> 00:23:43.870 align:middle line:84% instantaneously, so much faster than the gas molecules 00:23:43.870 --> 00:23:51.240 align:middle line:84% can move, definitely much faster than their sound velocity. 00:23:51.240 --> 00:23:54.690 align:middle line:84% I made it bigger by accident, didn't mean to do that. 00:23:54.690 --> 00:23:57.600 align:middle line:84% Here's the original position of the piston. 00:23:57.600 --> 00:24:00.325 align:middle line:84% And I'm going to pull it back very quickly. 00:24:00.325 --> 00:24:01.575 align:middle line:90% So it's now going to sit here. 00:24:01.575 --> 00:24:09.817 align:middle line:90% 00:24:09.817 --> 00:24:10.525 align:middle line:90% The same picture. 00:24:10.525 --> 00:24:21.160 align:middle line:90% 00:24:21.160 --> 00:24:27.950 align:middle line:84% And the point is that at T equals 0-- let's call it at-- 00:24:27.950 --> 00:24:30.800 align:middle line:84% instantaneously, none of the gas molecules 00:24:30.800 --> 00:24:32.622 align:middle line:90% have had time to move anywhere. 00:24:32.622 --> 00:24:35.280 align:middle line:90% 00:24:35.280 --> 00:24:38.180 align:middle line:84% So the gas molecules are occupying the space here. 00:24:38.180 --> 00:24:45.330 align:middle line:84% And in the space evacuated by the piston, we have a vacuum. 00:24:45.330 --> 00:24:47.860 align:middle line:90% 00:24:47.860 --> 00:24:49.110 align:middle line:90% So that's the setup. 00:24:49.110 --> 00:24:50.580 align:middle line:84% Like I said, it's a little kooky. 00:24:50.580 --> 00:24:53.700 align:middle line:84% There are actual, useful, real-world scenarios 00:24:53.700 --> 00:24:55.350 align:middle line:90% that approximate this. 00:24:55.350 --> 00:24:58.920 align:middle line:84% But for now, it's just a thought experiment. 00:24:58.920 --> 00:24:59.420 align:middle line:90% All right. 00:24:59.420 --> 00:25:03.125 align:middle line:84% So what's going to happen spontaneously next? 00:25:03.125 --> 00:25:06.330 align:middle line:90% 00:25:06.330 --> 00:25:09.090 align:middle line:84% Gas will spontaneously do something. 00:25:09.090 --> 00:25:15.902 align:middle line:90% 00:25:15.902 --> 00:25:16.610 align:middle line:90% AUDIENCE: Expand? 00:25:16.610 --> 00:25:18.027 align:middle line:84% AUDIENCE: Diffuse into the vacuum? 00:25:18.027 --> 00:25:21.560 align:middle line:90% 00:25:21.560 --> 00:25:28.005 align:middle line:84% RAFAEL JARAMILLO: Freely expand into the free volume. 00:25:28.005 --> 00:25:31.910 align:middle line:90% 00:25:31.910 --> 00:25:35.300 align:middle line:84% It's kind of spontaneous to expand into the free volume. 00:25:35.300 --> 00:25:41.190 align:middle line:84% And this word spontaneously in thermodynamics 00:25:41.190 --> 00:25:43.200 align:middle line:90% it also means irreversibly. 00:25:43.200 --> 00:25:47.920 align:middle line:90% 00:25:47.920 --> 00:25:50.580 align:middle line:84% So when you see one, I want you to think of the other. 00:25:50.580 --> 00:25:53.220 align:middle line:84% If you remember, we did this with the baby book, right? 00:25:53.220 --> 00:25:57.290 align:middle line:84% This was the baby book from as well. 00:25:57.290 --> 00:26:00.410 align:middle line:84% This gets beyond 0.2.0, but somebody said diffuse. 00:26:00.410 --> 00:26:04.040 align:middle line:84% The process here would be initially something 00:26:04.040 --> 00:26:06.030 align:middle line:90% that's super diffusive. 00:26:06.030 --> 00:26:10.140 align:middle line:84% Because initially, the molecules that are moving to the right 00:26:10.140 --> 00:26:13.410 align:middle line:84% encounter no collisions with any molecules coming from the left 00:26:13.410 --> 00:26:15.300 align:middle line:84% or coming from the right, moving to the left. 00:26:15.300 --> 00:26:17.280 align:middle line:84% So you can imagine almost, like, a shock wave 00:26:17.280 --> 00:26:20.820 align:middle line:84% of expansion, followed by a diffusive mixing process-- 00:26:20.820 --> 00:26:23.590 align:middle line:84% so anyway, again, just because somebody mentioned diffusion. 00:26:23.590 --> 00:26:24.465 align:middle line:90% So this is the setup. 00:26:24.465 --> 00:26:28.560 align:middle line:90% 00:26:28.560 --> 00:26:29.060 align:middle line:90% All right. 00:26:29.060 --> 00:26:30.260 align:middle line:90% You got that picture? 00:26:30.260 --> 00:26:32.380 align:middle line:84% So let's calculate something about that process. 00:26:32.380 --> 00:26:41.160 align:middle line:84% Let's calculate the work and heat during free expansion. 00:26:41.160 --> 00:26:45.560 align:middle line:90% 00:26:45.560 --> 00:26:46.460 align:middle line:90% Start with the work. 00:26:46.460 --> 00:26:49.370 align:middle line:90% 00:26:49.370 --> 00:26:56.630 align:middle line:84% Work equals minus dV times P. And here's 00:26:56.630 --> 00:27:05.800 align:middle line:84% the sort of kooky thing which we tell you, but it's true. 00:27:05.800 --> 00:27:07.540 align:middle line:84% The pressure of the expansion is 0 00:27:07.540 --> 00:27:15.726 align:middle line:84% because the gas is expanding into-- 00:27:15.726 --> 00:27:19.980 align:middle line:84% or, here, better say-- against vacuum. 00:27:19.980 --> 00:27:25.170 align:middle line:90% 00:27:25.170 --> 00:27:28.530 align:middle line:84% So the gas is doing no work in order to expand. 00:27:28.530 --> 00:27:30.000 align:middle line:90% There's nothing pushing back. 00:27:30.000 --> 00:27:33.280 align:middle line:90% 00:27:33.280 --> 00:27:35.150 align:middle line:90% So work equals 0. 00:27:35.150 --> 00:27:38.180 align:middle line:90% 00:27:38.180 --> 00:27:41.120 align:middle line:90% What about heat? 00:27:41.120 --> 00:27:43.220 align:middle line:84% AUDIENCE: I have a quick question about homework. 00:27:43.220 --> 00:27:44.178 align:middle line:90% RAFAEL JARAMILLO: Yeah. 00:27:44.178 --> 00:27:48.930 align:middle line:84% AUDIENCE: So since it's like an integral dV and the volume, 00:27:48.930 --> 00:27:50.335 align:middle line:90% isn't the volume changing? 00:27:50.335 --> 00:27:50.760 align:middle line:84% RAFAEL JARAMILLO: Volume's changing. 00:27:50.760 --> 00:27:52.218 align:middle line:84% AUDIENCE: So, like, mathematically, 00:27:52.218 --> 00:27:53.360 align:middle line:90% how would it work out to 0? 00:27:53.360 --> 00:27:54.860 align:middle line:84% RAFAEL JARAMILLO: The pressure is 0. 00:27:54.860 --> 00:27:55.900 align:middle line:90% AUDIENCE: Is it the pressure? 00:27:55.900 --> 00:27:56.240 align:middle line:90% Oh, I see. 00:27:56.240 --> 00:27:56.750 align:middle line:90% Sorry. 00:27:56.750 --> 00:27:58.880 align:middle line:84% RAFAEL JARAMILLO: If you think about-- 00:27:58.880 --> 00:28:03.890 align:middle line:84% work comes back to energy, which is forced by distance. 00:28:03.890 --> 00:28:05.490 align:middle line:90% Go way back to the beginning. 00:28:05.490 --> 00:28:09.477 align:middle line:84% So if you're pushing something, something's 00:28:09.477 --> 00:28:10.310 align:middle line:90% pushing back, right? 00:28:10.310 --> 00:28:12.620 align:middle line:84% That's Newton's third law or something like that. 00:28:12.620 --> 00:28:17.300 align:middle line:84% If nothing's pushing back, there's no work. 00:28:17.300 --> 00:28:19.060 align:middle line:90% So that's one reason it's free. 00:28:19.060 --> 00:28:21.370 align:middle line:84% I mean, this term free expansion is as old as the hill, 00:28:21.370 --> 00:28:24.515 align:middle line:84% so I'm not going to change the term even though sometimes I 00:28:24.515 --> 00:28:25.390 align:middle line:90% think it's confusing. 00:28:25.390 --> 00:28:28.820 align:middle line:84% But that's one way to think of the meaning of free. 00:28:28.820 --> 00:28:30.320 align:middle line:90% The expansion comes for free. 00:28:30.320 --> 00:28:33.370 align:middle line:84% You don't have to do any work to move into that space. 00:28:33.370 --> 00:28:33.870 align:middle line:90% All right. 00:28:33.870 --> 00:28:35.910 align:middle line:84% So dV here is finite, but the pressure 00:28:35.910 --> 00:28:39.130 align:middle line:84% is 0, the pressure during the expansion. 00:28:39.130 --> 00:28:42.880 align:middle line:84% And heat Q equals 0 because I said it was adiabatic. 00:28:42.880 --> 00:28:50.920 align:middle line:84% So the change of energy during adiabatic free expansion, 00:28:50.920 --> 00:29:03.680 align:middle line:84% this implies dT equals 0 for ideal gas. 00:29:03.680 --> 00:29:04.400 align:middle line:90% Right? 00:29:04.400 --> 00:29:07.670 align:middle line:84% For an ideal gas, remember temperature and energy 00:29:07.670 --> 00:29:10.720 align:middle line:90% are one to one. 00:29:10.720 --> 00:29:13.890 align:middle line:84% So if there's no change of energy, 00:29:13.890 --> 00:29:17.920 align:middle line:84% there's no change of temperature. 00:29:17.920 --> 00:29:18.580 align:middle line:90% OK. 00:29:18.580 --> 00:29:21.760 align:middle line:90% Let's keep talking about this. 00:29:21.760 --> 00:29:39.750 align:middle line:84% Adiabatic free expansion is spontaneous, 00:29:39.750 --> 00:29:46.195 align:middle line:84% so delta entropy has to be positive. 00:29:46.195 --> 00:29:47.570 align:middle line:84% That goes back a couple lectures. 00:29:47.570 --> 00:29:48.830 align:middle line:90% It's spontaneous. 00:29:48.830 --> 00:29:50.752 align:middle line:84% That means that it has to change the entropy, 00:29:50.752 --> 00:29:51.835 align:middle line:90% has to change the entropy. 00:29:51.835 --> 00:29:56.060 align:middle line:84% And it has to change it with a positive sign. 00:29:56.060 --> 00:29:59.130 align:middle line:90% How do we calculate-- 00:29:59.130 --> 00:30:05.340 align:middle line:90% how to calculate delta S? 00:30:05.340 --> 00:30:06.420 align:middle line:90% How do we calculate it? 00:30:06.420 --> 00:30:12.360 align:middle line:90% 00:30:12.360 --> 00:30:22.811 align:middle line:84% We want to find a differential form and integrate. 00:30:22.811 --> 00:30:26.960 align:middle line:90% 00:30:26.960 --> 00:30:35.370 align:middle line:84% dS equals-- delta S equals dS equals something. 00:30:35.370 --> 00:30:35.870 align:middle line:90% All right. 00:30:35.870 --> 00:30:38.960 align:middle line:84% So here's one of the first cases where we're going to, 00:30:38.960 --> 00:30:41.600 align:middle line:84% in this class, see how we can calculate 00:30:41.600 --> 00:30:45.140 align:middle line:84% the results of processes using thermo 00:30:45.140 --> 00:30:48.930 align:middle line:84% even if we can't calculate the specific processes themselves. 00:30:48.930 --> 00:30:50.910 align:middle line:84% So we have this spontaneous process, 00:30:50.910 --> 00:30:55.678 align:middle line:84% which involves this nutty expansion into a vacuum, 00:30:55.678 --> 00:30:56.720 align:middle line:90% takes place very quickly. 00:30:56.720 --> 00:30:59.275 align:middle line:90% 00:30:59.275 --> 00:31:00.650 align:middle line:84% That doesn't sound like something 00:31:00.650 --> 00:31:01.700 align:middle line:90% we know how to calculate. 00:31:01.700 --> 00:31:04.550 align:middle line:84% We can't use PV equals nRT because during that process, 00:31:04.550 --> 00:31:07.650 align:middle line:84% the system is not at equilibrium. 00:31:07.650 --> 00:31:09.200 align:middle line:84% So the challenge is, can we think 00:31:09.200 --> 00:31:14.820 align:middle line:84% of an equivalent reversible process that 00:31:14.820 --> 00:31:16.185 align:middle line:90% brings us to the same endpoint? 00:31:16.185 --> 00:31:27.990 align:middle line:90% 00:31:27.990 --> 00:31:38.860 align:middle line:84% Reversible process with same initial and final states. 00:31:38.860 --> 00:31:41.380 align:middle line:90% 00:31:41.380 --> 00:31:47.290 align:middle line:84% So we're going to have start, end. 00:31:47.290 --> 00:31:49.390 align:middle line:84% And we have this free expansion process, 00:31:49.390 --> 00:31:51.015 align:middle line:84% which I'll draw as being kind of nutty. 00:31:51.015 --> 00:31:57.400 align:middle line:90% 00:31:57.400 --> 00:32:02.930 align:middle line:84% And we want to come up with a reversible process 00:32:02.930 --> 00:32:05.550 align:middle line:84% with the same starting and ending points. 00:32:05.550 --> 00:32:09.030 align:middle line:84% Because if we can do that, we can calculate delta S. 00:32:09.030 --> 00:32:11.280 align:middle line:84% And we'll know it's the same, regardless 00:32:11.280 --> 00:32:14.520 align:middle line:84% of whether I take this path or that path. 00:32:14.520 --> 00:32:16.940 align:middle line:84% So let's imagine a reversible process. 00:32:16.940 --> 00:32:18.540 align:middle line:84% What do we know about this process? 00:32:18.540 --> 00:32:23.240 align:middle line:90% We know that delta T equals 0. 00:32:23.240 --> 00:32:25.780 align:middle line:90% 00:32:25.780 --> 00:32:27.850 align:middle line:90% We know that delta T equals 0. 00:32:27.850 --> 00:32:29.560 align:middle line:90% So let's consider isothermal. 00:32:29.560 --> 00:32:42.190 align:middle line:90% 00:32:42.190 --> 00:32:45.220 align:middle line:84% As before, isothermal process, no change in temperature. 00:32:45.220 --> 00:32:48.170 align:middle line:84% We're going to use T as an independent variable. 00:32:48.170 --> 00:32:49.270 align:middle line:90% We'll do expansion. 00:32:49.270 --> 00:32:55.050 align:middle line:90% 00:32:55.050 --> 00:32:56.460 align:middle line:90% Let's use volume this time. 00:32:56.460 --> 00:33:00.870 align:middle line:90% 00:33:00.870 --> 00:33:02.520 align:middle line:90% Isothermal expansion. 00:33:02.520 --> 00:33:11.480 align:middle line:90% 00:33:11.480 --> 00:33:14.080 align:middle line:84% So this isn't saying that the adiabatic free expansion 00:33:14.080 --> 00:33:15.730 align:middle line:84% is the same as isothermal expansion. 00:33:15.730 --> 00:33:21.960 align:middle line:84% But if I want to calculate a change in state variables, 00:33:21.960 --> 00:33:24.480 align:middle line:84% I can use this path instead of this path. 00:33:24.480 --> 00:33:31.040 align:middle line:90% 00:33:31.040 --> 00:33:32.000 align:middle line:90% OK. 00:33:32.000 --> 00:33:34.170 align:middle line:90% So does anybody want to-- 00:33:34.170 --> 00:33:34.670 align:middle line:90% let's see. 00:33:34.670 --> 00:33:39.290 align:middle line:90% I have some-- let's see. 00:33:39.290 --> 00:33:42.030 align:middle line:84% I want to pause here before we do that calculation. 00:33:42.030 --> 00:33:44.030 align:middle line:84% I'm going to pause here before that calculation. 00:33:44.030 --> 00:33:46.790 align:middle line:90% 00:33:46.790 --> 00:33:49.208 align:middle line:90% Because that was a lot of stuff. 00:33:49.208 --> 00:33:50.750 align:middle line:84% I'll take questions before moving on. 00:33:50.750 --> 00:33:56.170 align:middle line:90% 00:33:56.170 --> 00:33:57.970 align:middle line:84% AUDIENCE: Can you go back and explain again 00:33:57.970 --> 00:34:01.222 align:middle line:90% why heat equals 0? 00:34:01.222 --> 00:34:02.680 align:middle line:84% RAFAEL JARAMILLO: For this process? 00:34:02.680 --> 00:34:03.762 align:middle line:90% AUDIENCE: Yes. 00:34:03.762 --> 00:34:04.720 align:middle line:90% RAFAEL JARAMILLO: Yeah. 00:34:04.720 --> 00:34:09.319 align:middle line:90% And part of this is-- 00:34:09.319 --> 00:34:13.852 align:middle line:90% 00:34:13.852 --> 00:34:14.810 align:middle line:90% shoot, I was about to-- 00:34:14.810 --> 00:34:20.040 align:middle line:90% 00:34:20.040 --> 00:34:22.120 align:middle line:84% just the way the problem was set up. 00:34:22.120 --> 00:34:26.250 align:middle line:84% So I told you the setup is a little bit kooky. 00:34:26.250 --> 00:34:28.214 align:middle line:84% I'm imagining a thermally-insulated piston. 00:34:28.214 --> 00:34:31.150 align:middle line:90% 00:34:31.150 --> 00:34:36.310 align:middle line:84% So a thermally-insulated piston, it's in the title, adiabatic. 00:34:36.310 --> 00:34:39.310 align:middle line:84% So thermal insulation, heat equals 0. 00:34:39.310 --> 00:34:45.139 align:middle line:84% So for that process, for that process, work equals 0, 00:34:45.139 --> 00:34:46.040 align:middle line:90% and heat equals 0. 00:34:46.040 --> 00:34:50.635 align:middle line:90% 00:34:50.635 --> 00:34:52.510 align:middle line:84% Remember, those are process variables, right? 00:34:52.510 --> 00:34:55.179 align:middle line:84% So that means that a different process 00:34:55.179 --> 00:34:57.850 align:middle line:84% might have different process variables that 00:34:57.850 --> 00:35:01.580 align:middle line:84% still go between the same start and end state. 00:35:01.580 --> 00:35:08.654 align:middle line:90% 00:35:08.654 --> 00:35:11.550 align:middle line:84% So I'm going to calculate delta S next, 00:35:11.550 --> 00:35:13.520 align:middle line:84% but I want to gather more questions. 00:35:13.520 --> 00:35:17.160 align:middle line:90% 00:35:17.160 --> 00:35:19.040 align:middle line:84% AUDIENCE: So for isothermal expansion, 00:35:19.040 --> 00:35:22.910 align:middle line:84% previously, you had used P as your independent variable. 00:35:22.910 --> 00:35:25.880 align:middle line:84% Is there a reason why you're changing it to V now? 00:35:25.880 --> 00:35:27.820 align:middle line:84% Or is there a way that we should-- 00:35:27.820 --> 00:35:29.403 align:middle line:84% RAFAEL JARAMILLO: It's about the same. 00:35:29.403 --> 00:35:35.930 align:middle line:84% For ideal gas, of course, PV equals RT. 00:35:35.930 --> 00:35:39.050 align:middle line:84% For ideal gas with N equals 1, PV equals RT. 00:35:39.050 --> 00:35:44.310 align:middle line:84% And for isothermal processes, this is just a constant. 00:35:44.310 --> 00:35:46.520 align:middle line:84% So you can always swap P and V. They just 00:35:46.520 --> 00:35:48.693 align:middle line:84% end up being the inverse of each other. 00:35:48.693 --> 00:35:49.360 align:middle line:90% AUDIENCE: I see. 00:35:49.360 --> 00:35:50.110 align:middle line:90% OK, thank you. 00:35:50.110 --> 00:35:51.230 align:middle line:84% RAFAEL JARAMILLO: That's always easy. 00:35:51.230 --> 00:35:53.397 align:middle line:84% I just did this because before, I did the other way, 00:35:53.397 --> 00:35:55.660 align:middle line:90% and just see it both ways. 00:35:55.660 --> 00:35:59.968 align:middle line:90% 00:35:59.968 --> 00:36:01.510 align:middle line:84% AUDIENCE: I have a question-- sorry-- 00:36:01.510 --> 00:36:05.200 align:middle line:84% going back to the reversible adiabatic expansion. 00:36:05.200 --> 00:36:08.830 align:middle line:84% And you mentioned the step by inspection. 00:36:08.830 --> 00:36:12.940 align:middle line:84% And that was when we identified the variables M and N. Would 00:36:12.940 --> 00:36:15.940 align:middle line:84% you mind clarifying how you got to there? 00:36:15.940 --> 00:36:17.320 align:middle line:90% RAFAEL JARAMILLO: Sure. 00:36:17.320 --> 00:36:20.648 align:middle line:90% So that was back here, right? 00:36:20.648 --> 00:36:21.440 align:middle line:90% That was back here? 00:36:21.440 --> 00:36:21.980 align:middle line:90% AUDIENCE: Yes. 00:36:21.980 --> 00:36:22.938 align:middle line:90% RAFAEL JARAMILLO: Yeah. 00:36:22.938 --> 00:36:35.420 align:middle line:84% So let's talk about coefficients, coefficients. 00:36:35.420 --> 00:36:41.495 align:middle line:84% I'll ask you, tell me, what is dT at fixed pressure? 00:36:41.495 --> 00:36:45.200 align:middle line:90% 00:36:45.200 --> 00:36:47.152 align:middle line:84% Obviously, that's 1, but forget about the fact 00:36:47.152 --> 00:36:48.110 align:middle line:90% that you know that's 1. 00:36:48.110 --> 00:36:49.580 align:middle line:90% Just pick it off of this line. 00:36:49.580 --> 00:36:51.015 align:middle line:90% What is dT dT at fixed pressure? 00:36:51.015 --> 00:37:02.888 align:middle line:90% 00:37:02.888 --> 00:37:03.680 align:middle line:90% I'm glad you asked. 00:37:03.680 --> 00:37:08.087 align:middle line:84% Because this is a very common mental exercise in thermal, 00:37:08.087 --> 00:37:10.670 align:middle line:84% and maybe you're less familiar with it coming into this class. 00:37:10.670 --> 00:37:13.460 align:middle line:84% All right, differential form, dependent variable, 00:37:13.460 --> 00:37:15.830 align:middle line:84% independent variable, coefficients. 00:37:15.830 --> 00:37:18.600 align:middle line:84% The coefficients are the partial differentials. 00:37:18.600 --> 00:37:23.240 align:middle line:84% So dT and fixed pressure is this coefficient. 00:37:23.240 --> 00:37:28.170 align:middle line:90% 00:37:28.170 --> 00:37:37.050 align:middle line:84% Likewise, dT dP at fixed temperature 00:37:37.050 --> 00:37:40.275 align:middle line:90% equals N minus MV alpha. 00:37:40.275 --> 00:37:45.000 align:middle line:90% 00:37:45.000 --> 00:37:48.660 align:middle line:84% Now, this is sort of silly in some sense, a silly example. 00:37:48.660 --> 00:37:49.620 align:middle line:90% I mean, it's useful. 00:37:49.620 --> 00:37:51.510 align:middle line:90% The result is useful. 00:37:51.510 --> 00:37:59.010 align:middle line:84% It's a little bit of a silly example because dT has to be 1. 00:37:59.010 --> 00:38:01.440 align:middle line:84% You don't need to go to table 4.5 in DeHoff for that. 00:38:01.440 --> 00:38:03.720 align:middle line:84% You can just go ahead and say that's 1. 00:38:03.720 --> 00:38:05.548 align:middle line:84% And dT dP at fixed temperature, what's 00:38:05.548 --> 00:38:07.215 align:middle line:84% the dT when temperature is not changing? 00:38:07.215 --> 00:38:11.910 align:middle line:90% 00:38:11.910 --> 00:38:13.380 align:middle line:90% AUDIENCE: Oh, that would be 0. 00:38:13.380 --> 00:38:14.880 align:middle line:84% RAFAEL JARAMILLO: Has to be 0, yeah. 00:38:14.880 --> 00:38:18.030 align:middle line:84% So what we have here is we have a system of equations, right? 00:38:18.030 --> 00:38:22.035 align:middle line:84% We have two equations and two unknowns. 00:38:22.035 --> 00:38:24.123 align:middle line:90% 00:38:24.123 --> 00:38:25.540 align:middle line:84% And that's what I'm doing here, is 00:38:25.540 --> 00:38:27.457 align:middle line:84% I'm solving for those two unknowns given those 00:38:27.457 --> 00:38:28.500 align:middle line:90% two equations. 00:38:28.500 --> 00:38:30.250 align:middle line:84% The example that you're doing in the P set 00:38:30.250 --> 00:38:33.910 align:middle line:90% is a little bit less-- 00:38:33.910 --> 00:38:38.140 align:middle line:84% I don't want to call this silly, but it 00:38:38.140 --> 00:38:41.860 align:middle line:84% does require that you look up the coefficients in table 4.5. 00:38:41.860 --> 00:38:44.500 align:middle line:84% It's not as immediately apparent what 00:38:44.500 --> 00:38:47.890 align:middle line:84% these should be in the problem you're working on the P set. 00:38:47.890 --> 00:38:49.510 align:middle line:84% But that's what I meant by inspection. 00:38:49.510 --> 00:38:51.302 align:middle line:84% It means you look at the differential form, 00:38:51.302 --> 00:38:56.170 align:middle line:84% you pick off the partial differentials, 00:38:56.170 --> 00:38:59.110 align:middle line:84% and you equate them to what you know them to be. 00:38:59.110 --> 00:39:00.700 align:middle line:90% In this case, it's 1 and 0. 00:39:00.700 --> 00:39:05.128 align:middle line:90% 00:39:05.128 --> 00:39:06.420 align:middle line:90% AUDIENCE: OK, that makes sense. 00:39:06.420 --> 00:39:07.290 align:middle line:90% Thank you so much. 00:39:07.290 --> 00:39:08.990 align:middle line:90% RAFAEL JARAMILLO: Yeah, thanks. 00:39:08.990 --> 00:39:16.540 align:middle line:84% So to finish here, coefficients, coefficients, those 00:39:16.540 --> 00:39:17.530 align:middle line:90% are coefficients. 00:39:17.530 --> 00:39:21.200 align:middle line:90% 00:39:21.200 --> 00:39:22.790 align:middle line:84% Let me calculate the change in entropy 00:39:22.790 --> 00:39:26.720 align:middle line:84% here, because I think that it's another example of using 00:39:26.720 --> 00:39:28.610 align:middle line:90% these differential forms. 00:39:28.610 --> 00:39:30.390 align:middle line:84% And so maybe it will help right now. 00:39:30.390 --> 00:39:33.785 align:middle line:84% So let me give you let me give you something. 00:39:33.785 --> 00:39:37.040 align:middle line:90% 00:39:37.040 --> 00:39:43.170 align:middle line:84% We have T and V, and I want to calculate delta S. 00:39:43.170 --> 00:39:52.940 align:middle line:90% So I need to integrate dS-- 00:39:52.940 --> 00:39:53.440 align:middle line:90% sorry. 00:39:53.440 --> 00:40:02.610 align:middle line:84% I need to integrate dS, which is going to be dS d-- 00:40:02.610 --> 00:40:05.050 align:middle line:84% no change in temperature-- so dV, dS 00:40:05.050 --> 00:40:10.248 align:middle line:90% dV at fixed temperature dV. 00:40:10.248 --> 00:40:11.790 align:middle line:84% This is the thing I need to integrate 00:40:11.790 --> 00:40:14.640 align:middle line:84% to get my answer, which means I need 00:40:14.640 --> 00:40:16.600 align:middle line:90% to be able to write this down. 00:40:16.600 --> 00:40:18.420 align:middle line:84% I need to know what is this coefficient. 00:40:18.420 --> 00:40:21.270 align:middle line:90% 00:40:21.270 --> 00:40:22.990 align:middle line:90% That's what I need to know. 00:40:22.990 --> 00:40:24.870 align:middle line:84% So the implied differential form here 00:40:24.870 --> 00:40:34.980 align:middle line:84% is dS equals dS dV at fixed temperature dV plus dS as dT 00:40:34.980 --> 00:40:36.960 align:middle line:90% at fixed volume dT. 00:40:36.960 --> 00:40:41.442 align:middle line:90% 00:40:41.442 --> 00:40:42.900 align:middle line:84% I'm going to skip a couple of steps 00:40:42.900 --> 00:40:45.270 align:middle line:84% here because we have 8 minutes remaining. 00:40:45.270 --> 00:40:49.850 align:middle line:84% So I'm going to use the general strategy-- 00:40:49.850 --> 00:40:51.800 align:middle line:84% or I think this particular case is also 00:40:51.800 --> 00:40:54.380 align:middle line:84% just worked in the book, so I could also go look it up. 00:40:54.380 --> 00:40:58.820 align:middle line:84% And I get dS equals-- and we have to copy this-- 00:40:58.820 --> 00:41:07.616 align:middle line:84% Cp over T minus V alpha squared over beta. 00:41:07.616 --> 00:41:10.460 align:middle line:84% A lot of times, again, these coefficients are non-obvious. 00:41:10.460 --> 00:41:12.290 align:middle line:84% You just have to trust that you're 00:41:12.290 --> 00:41:15.780 align:middle line:84% doing the process correctly and use the result. OK. 00:41:15.780 --> 00:41:18.320 align:middle line:84% So this happens to be what you get. 00:41:18.320 --> 00:41:22.640 align:middle line:90% 00:41:22.640 --> 00:41:26.030 align:middle line:90% So for dT equals 0-- 00:41:26.030 --> 00:41:30.537 align:middle line:84% again, good choice of an independent variable because I 00:41:30.537 --> 00:41:32.120 align:middle line:84% get to ignore half the right-hand side 00:41:32.120 --> 00:41:34.010 align:middle line:90% of the equation. 00:41:34.010 --> 00:41:42.490 align:middle line:84% For dT equals 0, dS equals alpha over beta dV. 00:41:42.490 --> 00:41:47.070 align:middle line:90% 00:41:47.070 --> 00:42:00.670 align:middle line:84% For an ideal gas, we have alpha equals 1/T. Beta equals 1/P. 00:42:00.670 --> 00:42:05.680 align:middle line:90% dS equals P over T dV-- 00:42:05.680 --> 00:42:16.440 align:middle line:90% 00:42:16.440 --> 00:42:20.650 align:middle line:84% again, using ideal gas equation of state. 00:42:20.650 --> 00:42:24.340 align:middle line:84% So this is pretty simple in the end. 00:42:24.340 --> 00:42:27.515 align:middle line:90% I can integrate dS. 00:42:27.515 --> 00:42:30.050 align:middle line:90% 00:42:30.050 --> 00:42:37.310 align:middle line:84% It's going to be the integral by dV of R/V 00:42:37.310 --> 00:42:46.550 align:middle line:84% equals simply R log V2 over V1 dS. 00:42:46.550 --> 00:42:48.068 align:middle line:90% This is my answer. 00:42:48.068 --> 00:42:49.610 align:middle line:84% This is the answer I was looking for, 00:42:49.610 --> 00:42:59.340 align:middle line:84% the change of entropy for a reversible process isothermal 00:42:59.340 --> 00:43:00.240 align:middle line:90% expansion. 00:43:00.240 --> 00:43:10.630 align:middle line:90% 00:43:10.630 --> 00:43:13.310 align:middle line:90% And here's an important thing. 00:43:13.310 --> 00:43:15.040 align:middle line:90% It's greater than 0. 00:43:15.040 --> 00:43:16.675 align:middle line:90% So what does this mean? 00:43:16.675 --> 00:43:23.310 align:middle line:90% 00:43:23.310 --> 00:43:42.640 align:middle line:84% If I start with my gas molecules here in vacuum on the right, 00:43:42.640 --> 00:43:54.950 align:middle line:84% spontaneous process is going to bring me to gas molecules 00:43:54.950 --> 00:43:55.901 align:middle line:90% everywhere. 00:43:55.901 --> 00:44:01.130 align:middle line:90% 00:44:01.130 --> 00:44:09.740 align:middle line:84% V2 over V1 greater than 1, delta S greater than 0. 00:44:09.740 --> 00:44:11.450 align:middle line:90% Spontaneous entropy generation. 00:44:11.450 --> 00:44:18.180 align:middle line:90% 00:44:18.180 --> 00:44:20.640 align:middle line:84% What happens if I tried to run this movie backwards? 00:44:20.640 --> 00:44:24.090 align:middle line:90% 00:44:24.090 --> 00:44:29.430 align:middle line:84% If I ran this movie backwards, everything 00:44:29.430 --> 00:44:32.670 align:middle line:84% would be the same in terms of the calculation, 00:44:32.670 --> 00:44:34.980 align:middle line:84% except V2 and V1 would be switched. 00:44:34.980 --> 00:44:38.150 align:middle line:84% The final volume would be smaller 00:44:38.150 --> 00:44:39.350 align:middle line:90% than the initial volume. 00:44:39.350 --> 00:44:45.090 align:middle line:90% 00:44:45.090 --> 00:44:47.210 align:middle line:84% So the change of entropy would be negative. 00:44:47.210 --> 00:44:54.110 align:middle line:90% 00:44:54.110 --> 00:44:58.510 align:middle line:84% So it's our experience that you won't spontaneously 00:44:58.510 --> 00:45:02.260 align:middle line:84% have the gas molecules uniformly distributed in the box 00:45:02.260 --> 00:45:03.850 align:middle line:84% and then going through a situation 00:45:03.850 --> 00:45:06.280 align:middle line:84% where they leave a vacuum on the right-hand side. 00:45:06.280 --> 00:45:07.780 align:middle line:90% That's not our experience. 00:45:07.780 --> 00:45:12.620 align:middle line:90% 00:45:12.620 --> 00:45:22.630 align:middle line:84% Here's uniform gas molecules going to-- 00:45:22.630 --> 00:45:24.790 align:middle line:84% that red pen is causing trouble here-- 00:45:24.790 --> 00:45:25.950 align:middle line:90% going to a situation. 00:45:25.950 --> 00:45:30.540 align:middle line:90% 00:45:30.540 --> 00:45:36.490 align:middle line:84% But they segregated themselves spontaneously on one side. 00:45:36.490 --> 00:45:40.840 align:middle line:90% 00:45:40.840 --> 00:45:45.230 align:middle line:84% We know this would never happen spontaneously. 00:45:45.230 --> 00:45:48.230 align:middle line:84% Again, this sort of goes back to the baby book we discussed. 00:45:48.230 --> 00:45:51.500 align:middle line:84% This happens spontaneously, this doesn't happen spontaneously. 00:45:51.500 --> 00:45:54.890 align:middle line:84% Now we have some math to support that. 00:45:54.890 --> 00:45:56.910 align:middle line:84% We have something called the second law. 00:45:56.910 --> 00:45:58.760 align:middle line:84% We know that spontaneously, entropy 00:45:58.760 --> 00:46:01.530 align:middle line:84% has to always increase or stay the same. 00:46:01.530 --> 00:46:03.590 align:middle line:90% It can never decrease. 00:46:03.590 --> 00:46:06.280 align:middle line:84% And now the math of this backs us up. 00:46:06.280 --> 00:46:10.450 align:middle line:84% In this case, over here, we have V2 over V1 00:46:10.450 --> 00:46:12.580 align:middle line:90% would be less than 1. 00:46:12.580 --> 00:46:17.520 align:middle line:84% And delta S would be less than 0. 00:46:17.520 --> 00:46:19.550 align:middle line:90% So that doesn't happen. 00:46:19.550 --> 00:46:23.000 align:middle line:90%