WEBVTT 00:00:00.000 --> 00:00:01.491 align:middle line:90% [SQUEAKING] 00:00:01.491 --> 00:00:04.473 align:middle line:90% [RUSTLING] 00:00:04.473 --> 00:00:07.952 align:middle line:90% [CLICKING] 00:00:07.952 --> 00:00:11.727 align:middle line:90% 00:00:11.727 --> 00:00:13.310 align:middle line:84% PROFESSOR: All right, let's get going. 00:00:13.310 --> 00:00:33.790 align:middle line:90% 00:00:33.790 --> 00:00:35.290 align:middle line:84% I mentioned that a lot of what we're 00:00:35.290 --> 00:00:37.990 align:middle line:84% going to do when we model solutions is bookkeeping. 00:00:37.990 --> 00:00:40.173 align:middle line:84% So this is a really important quantity 00:00:40.173 --> 00:00:41.590 align:middle line:84% that we're going to track, and I'm 00:00:41.590 --> 00:00:43.632 align:middle line:84% going to introduce it-- partial molar properties. 00:00:43.632 --> 00:00:47.650 align:middle line:84% And we're going to use the notation as in the text. 00:00:47.650 --> 00:00:51.310 align:middle line:84% So the partial molar property for some generic state 00:00:51.310 --> 00:00:54.430 align:middle line:84% function-- we're using B here because it can stand in for any 00:00:54.430 --> 00:00:56.620 align:middle line:90% state function-- 00:00:56.620 --> 00:00:59.920 align:middle line:90% is the partial of the total B-- 00:00:59.920 --> 00:01:02.020 align:middle line:84% this is an extensive state function-- 00:01:02.020 --> 00:01:09.050 align:middle line:84% with moles of component k at fixed temperature, fixed 00:01:09.050 --> 00:01:15.630 align:middle line:84% pressure, and fixed moles of everything that's not k. 00:01:15.630 --> 00:01:17.700 align:middle line:90% j not equal to k. 00:01:17.700 --> 00:01:20.655 align:middle line:90% So B is an extensive property. 00:01:20.655 --> 00:01:27.120 align:middle line:90% 00:01:27.120 --> 00:01:28.860 align:middle line:90% Extensive property. 00:01:28.860 --> 00:01:31.935 align:middle line:90% n of k is moles of k. 00:01:31.935 --> 00:01:35.480 align:middle line:90% 00:01:35.480 --> 00:01:41.960 align:middle line:84% And B bar of k equals a partial molar property. 00:01:41.960 --> 00:01:44.335 align:middle line:84% Sometimes, I'll just call this PMPs. 00:01:44.335 --> 00:01:53.870 align:middle line:90% 00:01:53.870 --> 00:01:56.810 align:middle line:84% So this is kind of formal looking, 00:01:56.810 --> 00:01:58.640 align:middle line:90% but it's not a hard concept. 00:01:58.640 --> 00:02:03.056 align:middle line:90% 00:02:03.056 --> 00:02:05.272 align:middle line:84% You could think of this in many different ways. 00:02:05.272 --> 00:02:06.980 align:middle line:84% You could think of the extensive property 00:02:06.980 --> 00:02:12.160 align:middle line:84% as the total amount of money in all of our pockets right now. 00:02:12.160 --> 00:02:13.630 align:middle line:84% But we each have different amounts 00:02:13.630 --> 00:02:14.650 align:middle line:90% of money in our pockets. 00:02:14.650 --> 00:02:17.380 align:middle line:84% So our own contribution to the total amount 00:02:17.380 --> 00:02:19.660 align:middle line:84% of money in all of our pockets right now 00:02:19.660 --> 00:02:23.470 align:middle line:84% would be our partial personal property, I guess. 00:02:23.470 --> 00:02:28.730 align:middle line:90% 00:02:28.730 --> 00:02:30.860 align:middle line:84% You can think of this as the total area 00:02:30.860 --> 00:02:33.470 align:middle line:84% in my whole neighborhood, land area. 00:02:33.470 --> 00:02:37.310 align:middle line:84% But each of our houses has a different lot size. 00:02:37.310 --> 00:02:39.350 align:middle line:84% So we each have a different contribution 00:02:39.350 --> 00:02:41.990 align:middle line:84% to the total land area in my neighborhood 00:02:41.990 --> 00:02:44.135 align:middle line:84% over here in West Cambridge and so forth and so on. 00:02:44.135 --> 00:02:46.160 align:middle line:84% So this bookkeeping is kind of formal, 00:02:46.160 --> 00:02:48.350 align:middle line:90% but the concept is familiar. 00:02:48.350 --> 00:02:57.340 align:middle line:90% 00:02:57.340 --> 00:03:02.350 align:middle line:84% We're going to make sure we know what the total differential 00:03:02.350 --> 00:03:05.000 align:middle line:90% of property B is. 00:03:05.000 --> 00:03:06.830 align:middle line:90% We have these coefficients. 00:03:06.830 --> 00:03:17.440 align:middle line:84% dB dT affects P and mole number dT plus dB dP. 00:03:17.440 --> 00:03:22.100 align:middle line:90% That affects T and mole number. 00:03:22.100 --> 00:03:26.150 align:middle line:84% And here's where we get to use the partial molar properties. 00:03:26.150 --> 00:03:31.340 align:middle line:90% Plus B i dni. 00:03:31.340 --> 00:03:35.180 align:middle line:84% So as defined here, it's a coefficient. 00:03:35.180 --> 00:03:37.336 align:middle line:84% Partial molar property is this coefficient. 00:03:37.336 --> 00:03:40.670 align:middle line:90% 00:03:40.670 --> 00:03:42.170 align:middle line:90% And one clarification. 00:03:42.170 --> 00:03:46.590 align:middle line:84% For pure phase, for a pure phase, 00:03:46.590 --> 00:03:48.210 align:middle line:90% you only have one component. 00:03:48.210 --> 00:03:50.130 align:middle line:90% Its molar fraction is 1. 00:03:50.130 --> 00:03:56.130 align:middle line:84% And the partial molar property is simply the molar property. 00:03:56.130 --> 00:03:59.030 align:middle line:90% There's no partial involved. 00:03:59.030 --> 00:04:02.120 align:middle line:84% One thing about PMPs is PMPs are-- 00:04:02.120 --> 00:04:06.870 align:middle line:90% 00:04:06.870 --> 00:04:10.410 align:middle line:90% they're intensive. 00:04:10.410 --> 00:04:12.195 align:middle line:84% They're independent of the system size. 00:04:12.195 --> 00:04:21.089 align:middle line:90% 00:04:21.089 --> 00:04:25.410 align:middle line:84% And I'm going to use some illustrations to tell you 00:04:25.410 --> 00:04:26.160 align:middle line:90% what I mean there. 00:04:26.160 --> 00:04:30.520 align:middle line:90% But let's see. 00:04:30.520 --> 00:04:33.220 align:middle line:84% How do we know they're intensive? 00:04:33.220 --> 00:04:35.950 align:middle line:84% A PMP is an extensive quantity divided 00:04:35.950 --> 00:04:38.340 align:middle line:90% by an extensive quantity. 00:04:38.340 --> 00:04:42.590 align:middle line:84% So if I imagine this is a partial molar property, 00:04:42.590 --> 00:04:47.990 align:middle line:84% and the system is a given size, and I double the system size, 00:04:47.990 --> 00:04:52.340 align:middle line:84% then both the numerator and the denominator here will double. 00:04:52.340 --> 00:04:56.150 align:middle line:84% And the intensive partial molar property will remain unchanged. 00:04:56.150 --> 00:05:00.020 align:middle line:90% 00:05:00.020 --> 00:05:02.480 align:middle line:84% But there are useful consequences of this. 00:05:02.480 --> 00:05:09.500 align:middle line:84% So let's imagine a system made out of triangles and circles. 00:05:09.500 --> 00:05:15.810 align:middle line:84% So for example, let's imagine a system with X circle equals X 00:05:15.810 --> 00:05:17.040 align:middle line:90% triangle. 00:05:17.040 --> 00:05:22.060 align:middle line:84% And it's 50/50, just to be specific. 00:05:22.060 --> 00:05:26.670 align:middle line:84% So let's imagine three different systems here. 00:05:26.670 --> 00:05:36.350 align:middle line:84% Here we have one, two, three, four, five circles. 00:05:36.350 --> 00:05:38.530 align:middle line:90% And we have five triangles. 00:05:38.530 --> 00:05:45.710 align:middle line:90% 00:05:45.710 --> 00:05:54.010 align:middle line:84% And if we have B circle equals 7 and B triangle-- 00:05:54.010 --> 00:05:55.630 align:middle line:90% I'm just making a numbers here. 00:05:55.630 --> 00:05:58.420 align:middle line:84% It doesn't matter what numbers I use. 00:05:58.420 --> 00:06:05.530 align:middle line:84% And for T and P constant, in this case, 00:06:05.530 --> 00:06:14.500 align:middle line:84% the total amount of B equals B circle n circle 00:06:14.500 --> 00:06:19.280 align:middle line:90% plus B triangle n triangle. 00:06:19.280 --> 00:06:29.270 align:middle line:84% And this is 7 times 5 plus 4 times 5. 00:06:29.270 --> 00:06:31.760 align:middle line:84% Let's consider a slightly different case. 00:06:31.760 --> 00:06:35.250 align:middle line:84% Let's make the system a little smaller. 00:06:35.250 --> 00:06:39.580 align:middle line:90% Let's have three moles of each. 00:06:39.580 --> 00:06:44.600 align:middle line:90% 00:06:44.600 --> 00:06:54.500 align:middle line:84% In this case, the total amount of B again is sum B of i n of i 00:06:54.500 --> 00:07:00.230 align:middle line:90% equals 7 times 3 plus 4 times 3. 00:07:00.230 --> 00:07:03.900 align:middle line:90% 00:07:03.900 --> 00:07:05.900 align:middle line:84% And then, of course, you can keep on going here. 00:07:05.900 --> 00:07:11.750 align:middle line:84% Let's imagine one circle and one triangle. 00:07:11.750 --> 00:07:19.300 align:middle line:84% The total amount of B equals 7 plus 4. 00:07:19.300 --> 00:07:22.980 align:middle line:90% 00:07:22.980 --> 00:07:25.090 align:middle line:90% So why am I telling you this? 00:07:25.090 --> 00:07:27.510 align:middle line:84% Why did we just go through this exercise 00:07:27.510 --> 00:07:30.360 align:middle line:90% with circles and triangles? 00:07:30.360 --> 00:07:33.090 align:middle line:90% 00:07:33.090 --> 00:07:34.830 align:middle line:84% It's because we're going to add up 00:07:34.830 --> 00:07:36.692 align:middle line:90% partial molar properties a lot. 00:07:36.692 --> 00:07:38.400 align:middle line:84% We're going to be doing this quite a lot. 00:07:38.400 --> 00:07:40.750 align:middle line:90% And it comes back to this. 00:07:40.750 --> 00:07:46.650 align:middle line:84% The whole is the sum of the contributions of the parts, 00:07:46.650 --> 00:07:49.420 align:middle line:84% according to this bookkeeping scheme. 00:07:49.420 --> 00:07:51.170 align:middle line:84% It's according to this bookkeeping scheme. 00:07:51.170 --> 00:08:02.162 align:middle line:90% 00:08:02.162 --> 00:08:02.870 align:middle line:90% Let's keep going. 00:08:02.870 --> 00:08:06.560 align:middle line:90% 00:08:06.560 --> 00:08:20.420 align:middle line:84% We need some mathematical relations 00:08:20.420 --> 00:08:29.180 align:middle line:84% between the whole that is a total amount of B and the PMPs. 00:08:29.180 --> 00:08:40.105 align:middle line:90% 00:08:40.105 --> 00:08:47.730 align:middle line:84% We're going to consider scaling the system size 00:08:47.730 --> 00:08:48.855 align:middle line:90% by a scaling factor. 00:08:48.855 --> 00:08:53.222 align:middle line:90% 00:08:53.222 --> 00:08:54.680 align:middle line:84% And this is the unit of this thing. 00:08:54.680 --> 00:08:56.190 align:middle line:90% I'll just use lambda. 00:08:56.190 --> 00:09:08.540 align:middle line:84% So imagine B is a function of temperature, pressure, 00:09:08.540 --> 00:09:09.955 align:middle line:90% and the scaled system size. 00:09:09.955 --> 00:09:13.080 align:middle line:90% 00:09:13.080 --> 00:09:16.230 align:middle line:84% Well, if we imagine, let's say, doubling 00:09:16.230 --> 00:09:18.450 align:middle line:84% the amount of moles of everybody, 00:09:18.450 --> 00:09:22.990 align:middle line:84% we simply double all extensive quantities. 00:09:22.990 --> 00:09:24.422 align:middle line:90% And that holds for any-- 00:09:24.422 --> 00:09:25.630 align:middle line:90% it doesn't have to be double. 00:09:25.630 --> 00:09:26.290 align:middle line:90% It could be triple. 00:09:26.290 --> 00:09:26.998 align:middle line:90% It could be half. 00:09:26.998 --> 00:09:28.340 align:middle line:90% It could be any factor. 00:09:28.340 --> 00:09:30.175 align:middle line:90% So we have this. 00:09:30.175 --> 00:09:37.500 align:middle line:90% 00:09:37.500 --> 00:09:39.660 align:middle line:84% So if I multiply it by some constant scale 00:09:39.660 --> 00:09:41.790 align:middle line:84% factor, the size of the system, it's 00:09:41.790 --> 00:09:45.150 align:middle line:84% the same thing as multiplying the overall extensive quantity 00:09:45.150 --> 00:09:46.365 align:middle line:90% by that same scale factor. 00:09:46.365 --> 00:09:49.400 align:middle line:90% 00:09:49.400 --> 00:09:50.460 align:middle line:90% I'm going to use this. 00:09:50.460 --> 00:09:52.020 align:middle line:90% So this is not just for fun. 00:09:52.020 --> 00:09:53.360 align:middle line:90% This becomes useful. 00:09:53.360 --> 00:09:57.770 align:middle line:84% But I will call out, for those of you who are mathematically 00:09:57.770 --> 00:10:12.340 align:middle line:84% inclined, that that means that B prime is a homogeneous function 00:10:12.340 --> 00:10:19.540 align:middle line:90% of the Mk's of order one. 00:10:19.540 --> 00:10:21.760 align:middle line:84% So for those of you who do real analysis and stuff, 00:10:21.760 --> 00:10:24.070 align:middle line:90% that comes into it. 00:10:24.070 --> 00:10:24.670 align:middle line:90% Never mind. 00:10:24.670 --> 00:10:26.230 align:middle line:84% So what we're going to do with that 00:10:26.230 --> 00:10:34.830 align:middle line:84% is we're going to take the derivative, 00:10:34.830 --> 00:10:41.885 align:middle line:84% the total derivative, d d lambda of both sides. 00:10:41.885 --> 00:10:45.310 align:middle line:90% 00:10:45.310 --> 00:10:45.870 align:middle line:90% So let's see. 00:10:45.870 --> 00:10:50.640 align:middle line:84% The left-hand side is going to give us 00:10:50.640 --> 00:11:14.440 align:middle line:84% dB d lambda equals dB d lambda n of k. 00:11:14.440 --> 00:11:15.790 align:middle line:90% Just using the chain rule here. 00:11:15.790 --> 00:11:35.210 align:middle line:90% 00:11:35.210 --> 00:11:36.920 align:middle line:90% This is chain rule. 00:11:36.920 --> 00:11:43.880 align:middle line:84% And this is our definition of partial molar property. 00:11:43.880 --> 00:11:45.515 align:middle line:84% And this derivative is really easy. 00:11:45.515 --> 00:11:50.030 align:middle line:90% 00:11:50.030 --> 00:12:05.860 align:middle line:84% Right-hand side, d d lambda times lambda B T P n of k. 00:12:05.860 --> 00:12:08.230 align:middle line:84% And that is also an easy derivative. 00:12:08.230 --> 00:12:13.330 align:middle line:90% That's B T P n of k. 00:12:13.330 --> 00:12:17.110 align:middle line:84% So I took the total derivative of the left-hand side 00:12:17.110 --> 00:12:19.540 align:middle line:90% on the right-hand side. 00:12:19.540 --> 00:12:21.450 align:middle line:84% And I'm going to then compare them. 00:12:21.450 --> 00:12:46.460 align:middle line:90% 00:12:46.460 --> 00:12:48.380 align:middle line:84% Again, this is exactly what we were doing 00:12:48.380 --> 00:12:50.210 align:middle line:90% with circles and triangles. 00:12:50.210 --> 00:12:56.250 align:middle line:84% The total extensive quantity for a system 00:12:56.250 --> 00:12:58.740 align:middle line:84% is the sum of the partial molar properties 00:12:58.740 --> 00:13:00.400 align:middle line:90% times the mole numbers. 00:13:00.400 --> 00:13:05.640 align:middle line:84% So again, circles and triangles, scaling with system size. 00:13:05.640 --> 00:13:12.930 align:middle line:84% And this is called an Euler equation. 00:13:12.930 --> 00:13:17.272 align:middle line:90% 00:13:17.272 --> 00:13:19.730 align:middle line:84% I think it's the only one we're going to use in this class. 00:13:19.730 --> 00:13:22.660 align:middle line:84% So you don't have to worry so much about that there's 00:13:22.660 --> 00:13:24.490 align:middle line:84% this category of equations called 00:13:24.490 --> 00:13:25.720 align:middle line:90% Euler equations in thermo. 00:13:25.720 --> 00:13:28.990 align:middle line:84% But you might encounter them, if not now, then in a later class. 00:13:28.990 --> 00:13:31.120 align:middle line:90% And that's what this is. 00:13:31.120 --> 00:13:35.840 align:middle line:84% It is the consequence of that first-order homogeneous 00:13:35.840 --> 00:13:36.340 align:middle line:90% property. 00:13:36.340 --> 00:13:39.200 align:middle line:90% 00:13:39.200 --> 00:13:41.840 align:middle line:84% Where is it stated a little more colloquially? 00:13:41.840 --> 00:13:59.190 align:middle line:84% Extensive properties of a solution phase 00:13:59.190 --> 00:14:18.230 align:middle line:84% are made up of the mole weighted PMPs of the components. 00:14:18.230 --> 00:14:24.020 align:middle line:90% 00:14:24.020 --> 00:14:25.440 align:middle line:90% That's what it means. 00:14:25.440 --> 00:14:29.810 align:middle line:84% So for example-- this is the one we'll be using almost 00:14:29.810 --> 00:14:33.330 align:middle line:84% to the exclusion of any other in this class-- 00:14:33.330 --> 00:14:36.570 align:middle line:84% the Gibbs free entry of a system is 00:14:36.570 --> 00:14:48.960 align:middle line:84% going to be the sum of these PMPs times the mole numbers. 00:14:48.960 --> 00:14:51.540 align:middle line:84% And because Gibbs free energy is so special, 00:14:51.540 --> 00:14:57.240 align:middle line:84% we have a special expression for the molar Gibbs free energy 00:14:57.240 --> 00:15:00.620 align:middle line:90% and its chemical potential. 00:15:00.620 --> 00:15:12.497 align:middle line:84% So the Gibbs free energy, PMP, is the chemical potential. 00:15:12.497 --> 00:15:14.080 align:middle line:84% So you've seen this expression before. 00:15:14.080 --> 00:15:24.250 align:middle line:90% 00:15:24.250 --> 00:15:25.680 align:middle line:84% So if you ever see someone add up 00:15:25.680 --> 00:15:27.270 align:middle line:84% the Gibbs free energy of a system, 00:15:27.270 --> 00:15:29.700 align:middle line:84% and they just jump right to this, 00:15:29.700 --> 00:15:32.380 align:middle line:90% you know where it comes from. 00:15:32.380 --> 00:15:34.140 align:middle line:84% So sum of the chemical potentials 00:15:34.140 --> 00:15:35.385 align:middle line:90% weighted by the mole numbers. 00:15:35.385 --> 00:15:38.520 align:middle line:90% 00:15:38.520 --> 00:15:41.010 align:middle line:84% Unfortunately, we have to continue with more equations. 00:15:41.010 --> 00:16:00.555 align:middle line:90% 00:16:00.555 --> 00:16:01.930 align:middle line:84% So what we're doing here is we're 00:16:01.930 --> 00:16:04.300 align:middle line:84% building up to these useful expressions, 00:16:04.300 --> 00:16:05.890 align:middle line:90% which you can use-- 00:16:05.890 --> 00:16:07.898 align:middle line:84% you will use on the homework, for instance-- 00:16:07.898 --> 00:16:09.940 align:middle line:84% without necessarily knowing where they come from. 00:16:09.940 --> 00:16:14.890 align:middle line:90% 00:16:14.890 --> 00:16:17.340 align:middle line:84% So we have an Euler equation, which 00:16:17.340 --> 00:16:22.260 align:middle line:84% is like the one we just derived, which 00:16:22.260 --> 00:16:29.290 align:middle line:84% is the total amount of B equals the sum of the partial molar 00:16:29.290 --> 00:16:32.718 align:middle line:90% B times the mole numbers. 00:16:32.718 --> 00:16:34.510 align:middle line:84% And then we're going to use the chain rule. 00:16:34.510 --> 00:16:38.210 align:middle line:90% 00:16:38.210 --> 00:16:42.140 align:middle line:84% And we're going to calculate the total derivative dB. 00:16:42.140 --> 00:16:44.765 align:middle line:84% And that's going to be equal to chain rule stuff. 00:16:44.765 --> 00:16:47.330 align:middle line:90% 00:16:47.330 --> 00:16:56.510 align:middle line:90% Bk dn of k plus n of k d d of k. 00:16:56.510 --> 00:16:59.240 align:middle line:84% And that's relatively straightforward. 00:16:59.240 --> 00:17:01.370 align:middle line:84% And we're going to use a coefficient relation. 00:17:01.370 --> 00:17:07.480 align:middle line:90% 00:17:07.480 --> 00:17:12.760 align:middle line:84% The coefficient relation is that dB at constant temperature 00:17:12.760 --> 00:17:14.940 align:middle line:90% and pressure equals-- 00:17:14.940 --> 00:17:23.660 align:middle line:90% 00:17:23.660 --> 00:17:25.579 align:middle line:84% that was from the beginning, when we defined 00:17:25.579 --> 00:17:27.119 align:middle line:90% the partial model property. 00:17:27.119 --> 00:17:28.702 align:middle line:84% And so what we're going to do is we're 00:17:28.702 --> 00:17:37.210 align:middle line:84% going to simply compare these expressions, this and this. 00:17:37.210 --> 00:17:40.345 align:middle line:84% And we find that one of these two sums is 0. 00:17:40.345 --> 00:18:09.710 align:middle line:90% 00:18:09.710 --> 00:18:12.680 align:middle line:84% This is called a Gibbs-Duhem equation. 00:18:12.680 --> 00:18:15.080 align:middle line:84% And by the way, I really would not 00:18:15.080 --> 00:18:17.360 align:middle line:84% be spending the time on this if it didn't end up 00:18:17.360 --> 00:18:19.080 align:middle line:90% in a really useful place. 00:18:19.080 --> 00:18:20.630 align:middle line:84% So if you're wondering why on Earth 00:18:20.630 --> 00:18:24.380 align:middle line:84% you sign up for a class on analysis, all I can say 00:18:24.380 --> 00:18:26.480 align:middle line:90% is this is a class on thermo. 00:18:26.480 --> 00:18:27.475 align:middle line:90% And this is part of it. 00:18:27.475 --> 00:18:28.850 align:middle line:84% And I'm including it because it's 00:18:28.850 --> 00:18:31.970 align:middle line:84% going to take us somewhere really useful. 00:18:31.970 --> 00:18:33.140 align:middle line:90% Enough excuse making. 00:18:33.140 --> 00:18:34.280 align:middle line:90% This is thermo. 00:18:34.280 --> 00:18:37.290 align:middle line:84% So n dbk, what on Earth does this mean? 00:18:37.290 --> 00:18:40.430 align:middle line:84% It means that you can't change all the partial molar 00:18:40.430 --> 00:18:43.810 align:middle line:84% properties independently of each other. 00:18:43.810 --> 00:18:49.390 align:middle line:84% If you have two components, let's say, the sum runs over 2, 00:18:49.390 --> 00:18:52.450 align:middle line:84% there's a linkage between how the partial molar 00:18:52.450 --> 00:18:55.960 align:middle line:84% properties of one change when the partial molar properties 00:18:55.960 --> 00:18:58.920 align:middle line:90% of the other change. 00:18:58.920 --> 00:18:59.993 align:middle line:90% There's this equation. 00:18:59.993 --> 00:19:01.035 align:middle line:90% They're related to those. 00:19:01.035 --> 00:19:07.130 align:middle line:90% 00:19:07.130 --> 00:19:08.840 align:middle line:90% Let me say that. 00:19:08.840 --> 00:19:12.520 align:middle line:90% 00:19:12.520 --> 00:19:32.590 align:middle line:84% The Gibbs-Duhem equations are constraints on the variation 00:19:32.590 --> 00:19:34.420 align:middle line:90% of intensive properties. 00:19:34.420 --> 00:19:43.360 align:middle line:90% 00:19:43.360 --> 00:19:51.200 align:middle line:84% Or in other words, they express how the partial molar 00:19:51.200 --> 00:19:54.320 align:middle line:90% properties covary. 00:19:54.320 --> 00:19:57.200 align:middle line:90% That's a term from optimization. 00:19:57.200 --> 00:19:59.420 align:middle line:84% They covary with system composition. 00:19:59.420 --> 00:20:11.490 align:middle line:90% 00:20:11.490 --> 00:20:13.950 align:middle line:84% In other words, it's a differential equation. 00:20:13.950 --> 00:20:20.630 align:middle line:90% 00:20:20.630 --> 00:20:23.165 align:middle line:84% An equation that describes how things covary with each other 00:20:23.165 --> 00:20:24.290 align:middle line:90% is a differential equation. 00:20:24.290 --> 00:20:25.040 align:middle line:90% That's what it is. 00:20:25.040 --> 00:20:28.890 align:middle line:90% 00:20:28.890 --> 00:20:34.290 align:middle line:84% So now we're going to move towards partial molar 00:20:34.290 --> 00:20:35.800 align:middle line:84% properties of mixing and so forth. 00:20:35.800 --> 00:20:39.218 align:middle line:84% And we will finally get to graphical interpretation, which 00:20:39.218 --> 00:20:40.510 align:middle line:90% is why we're doing all of this. 00:20:40.510 --> 00:20:51.470 align:middle line:84% So for example, Gibbs-Duhem for Gibbs free energy. 00:20:51.470 --> 00:20:57.720 align:middle line:90% 00:20:57.720 --> 00:21:00.150 align:middle line:84% If anyone's wondering, what's with Gibbs? 00:21:00.150 --> 00:21:02.576 align:middle line:84% How come he gets his name on everything? 00:21:02.576 --> 00:21:04.530 align:middle line:84% He's one of the most important and prolific 00:21:04.530 --> 00:21:06.690 align:middle line:90% American scientists ever. 00:21:06.690 --> 00:21:09.690 align:middle line:84% And he was a thermodynamics researcher, obviously, 00:21:09.690 --> 00:21:11.650 align:middle line:90% in the late 19th century. 00:21:11.650 --> 00:21:15.090 align:middle line:90% And he spent his career at Yale. 00:21:15.090 --> 00:21:16.350 align:middle line:90% And yeah. 00:21:16.350 --> 00:21:18.128 align:middle line:84% His name is all over classical thermo. 00:21:18.128 --> 00:21:20.880 align:middle line:90% 00:21:20.880 --> 00:21:24.670 align:middle line:84% Euler, Euler gives us the following. 00:21:24.670 --> 00:21:27.550 align:middle line:84% The total of Gibbs free energy is 00:21:27.550 --> 00:21:35.040 align:middle line:84% the sum of mu of k n of k, the chain rule. 00:21:35.040 --> 00:21:39.010 align:middle line:90% 00:21:39.010 --> 00:21:54.195 align:middle line:84% It gives us dG equals mu of k dn of k plus n of k d mu of k. 00:21:54.195 --> 00:21:57.240 align:middle line:90% 00:21:57.240 --> 00:22:02.660 align:middle line:84% And then we combine the combined statement. 00:22:02.660 --> 00:22:05.590 align:middle line:90% 00:22:05.590 --> 00:22:09.340 align:middle line:84% The combined statement is a coefficient relation. 00:22:09.340 --> 00:22:26.070 align:middle line:84% dG equals minus s dT plus VdP plus mu of k dn of k. 00:22:26.070 --> 00:22:32.270 align:middle line:84% And I'm going to use these, and I get the following. 00:22:32.270 --> 00:22:35.770 align:middle line:90% 00:22:35.770 --> 00:22:47.734 align:middle line:84% n of k du of k equals minus S dT plus V dP. 00:22:47.734 --> 00:22:50.730 align:middle line:90% 00:22:50.730 --> 00:22:58.860 align:middle line:84% Or if I want to divide through by the total number of moles, 00:22:58.860 --> 00:23:01.490 align:middle line:90% I get an equivalent statement. 00:23:01.490 --> 00:23:08.090 align:middle line:84% Mole fraction d mu of k equals molar entropy. 00:23:08.090 --> 00:23:16.260 align:middle line:90% 00:23:16.260 --> 00:23:20.790 align:middle line:90% These are also gives qm. 00:23:20.790 --> 00:23:25.900 align:middle line:90% 00:23:25.900 --> 00:23:27.400 align:middle line:84% So again, you see what these are is 00:23:27.400 --> 00:23:30.790 align:middle line:84% they're constraints on how the intensive properties are 00:23:30.790 --> 00:23:32.560 align:middle line:90% varying. 00:23:32.560 --> 00:23:34.970 align:middle line:84% Because intensive properties have pressure, temperature, 00:23:34.970 --> 00:23:37.660 align:middle line:84% and chemical potential, we can't all vary independently 00:23:37.660 --> 00:23:38.300 align:middle line:90% of each other. 00:23:38.300 --> 00:23:39.786 align:middle line:90% We have these constraints. 00:23:39.786 --> 00:23:48.617 align:middle line:90% 00:23:48.617 --> 00:23:49.200 align:middle line:90% Let's move on. 00:23:49.200 --> 00:24:12.370 align:middle line:90% 00:24:12.370 --> 00:24:14.640 align:middle line:84% And this is really like actuarial science. 00:24:14.640 --> 00:24:18.000 align:middle line:84% We just keep on coming up with these bookkeeping schemes. 00:24:18.000 --> 00:24:20.590 align:middle line:90% And we have to stick with it. 00:24:20.590 --> 00:24:26.510 align:middle line:84% So we're going to have something called the mixing quantities. 00:24:26.510 --> 00:24:31.050 align:middle line:84% So the change due to mixing is going 00:24:31.050 --> 00:24:39.730 align:middle line:84% to be the total quantity in the solution minus 00:24:39.730 --> 00:24:41.620 align:middle line:84% what you started with before you made 00:24:41.620 --> 00:24:45.670 align:middle line:84% the solution, the pure components. 00:24:45.670 --> 00:24:48.710 align:middle line:90% 00:24:48.710 --> 00:24:53.120 align:middle line:84% We talked about this last time, that, in general, the whole 00:24:53.120 --> 00:24:57.013 align:middle line:84% is not simply the sum of the parts when you make a solution. 00:24:57.013 --> 00:24:58.430 align:middle line:84% And there were some examples given 00:24:58.430 --> 00:25:01.600 align:middle line:84% of materials which, for example, get 00:25:01.600 --> 00:25:04.218 align:middle line:84% bigger than you would expect or smaller than you would expect 00:25:04.218 --> 00:25:04.760 align:middle line:90% and so forth. 00:25:04.760 --> 00:25:09.290 align:middle line:84% So we're just going to capture that change here. 00:25:09.290 --> 00:25:17.250 align:middle line:84% And we're going to write the total B using partial molar 00:25:17.250 --> 00:25:23.210 align:middle line:84% properties, as we just learned how to do, Bk n of k. 00:25:23.210 --> 00:25:26.030 align:middle line:84% And then we're going to introduce even more notation 00:25:26.030 --> 00:25:29.270 align:middle line:84% and say, for the pure components, 00:25:29.270 --> 00:25:35.160 align:middle line:90% you had pure molar properties. 00:25:35.160 --> 00:25:39.087 align:middle line:90% 00:25:39.087 --> 00:25:40.170 align:middle line:90% And we've got group terms. 00:25:40.170 --> 00:25:57.420 align:middle line:90% 00:25:57.420 --> 00:26:05.620 align:middle line:84% And we're going to define that as delta B partial molar 00:26:05.620 --> 00:26:08.850 align:middle line:90% property due to mixing. 00:26:08.850 --> 00:26:09.350 align:middle line:90% Oof. 00:26:09.350 --> 00:26:20.200 align:middle line:90% 00:26:20.200 --> 00:26:29.720 align:middle line:90% Change in PMP due to mixing. 00:26:29.720 --> 00:26:32.600 align:middle line:90% My goodness. 00:26:32.600 --> 00:26:35.300 align:middle line:90% What is this? 00:26:35.300 --> 00:26:36.290 align:middle line:90% Let me give an analogy. 00:26:36.290 --> 00:26:37.623 align:middle line:90% I don't know whether this helps. 00:26:37.623 --> 00:26:42.940 align:middle line:90% 00:26:42.940 --> 00:26:46.870 align:middle line:84% Let's imagine this is personal space. 00:26:46.870 --> 00:26:52.900 align:middle line:84% We're calculating here the area occupied by a crowd of people. 00:26:52.900 --> 00:26:54.820 align:middle line:84% And you could define a personal space 00:26:54.820 --> 00:26:58.390 align:middle line:84% as the exclusive area around everybody such 00:26:58.390 --> 00:27:01.570 align:middle line:84% that the total area is everyone's personal space added 00:27:01.570 --> 00:27:02.360 align:middle line:90% up. 00:27:02.360 --> 00:27:03.820 align:middle line:90% Does that makes sense? 00:27:03.820 --> 00:27:06.340 align:middle line:90% Imagine a crowd of people. 00:27:06.340 --> 00:27:10.510 align:middle line:84% What this is is an expression, acknowledgment, 00:27:10.510 --> 00:27:17.500 align:middle line:84% that your personal space in a given crowd 00:27:17.500 --> 00:27:21.600 align:middle line:90% varies with the crowd. 00:27:21.600 --> 00:27:25.430 align:middle line:90% So if you are-- 00:27:25.430 --> 00:27:28.170 align:middle line:90% let's see. 00:27:28.170 --> 00:27:32.430 align:middle line:84% If your crowd is a dispersal of people 00:27:32.430 --> 00:27:37.700 align:middle line:84% in Killian Court on a sunny day, you're 00:27:37.700 --> 00:27:39.890 align:middle line:84% going to have a lot of personal space per person. 00:27:39.890 --> 00:27:42.450 align:middle line:90% 00:27:42.450 --> 00:27:44.460 align:middle line:84% It would be very, very strange to go up 00:27:44.460 --> 00:27:47.730 align:middle line:84% to somebody who is sitting there, having a picnic, 00:27:47.730 --> 00:27:49.900 align:middle line:84% and to sit so that your shoulders are rubbing, 00:27:49.900 --> 00:27:51.150 align:middle line:90% and you don't know the person. 00:27:51.150 --> 00:27:51.990 align:middle line:90% Total stranger. 00:27:51.990 --> 00:27:53.550 align:middle line:84% Killian Court's wide open and sunny. 00:27:53.550 --> 00:27:55.342 align:middle line:84% And you go sit with your shoulders rubbing. 00:27:55.342 --> 00:27:57.420 align:middle line:90% And that's the space you picked. 00:27:57.420 --> 00:28:00.810 align:middle line:90% That would be very unusual. 00:28:00.810 --> 00:28:05.340 align:middle line:84% But same people in a crowded subway car or on a dance floor, 00:28:05.340 --> 00:28:10.380 align:middle line:84% you might have a very different personal space. 00:28:10.380 --> 00:28:12.020 align:middle line:84% So your personal space is not fixed 00:28:12.020 --> 00:28:16.500 align:middle line:90% by who you are as a person. 00:28:16.500 --> 00:28:22.410 align:middle line:84% It's determined by the solution you find yourself in. 00:28:22.410 --> 00:28:24.480 align:middle line:84% And different solutions will result 00:28:24.480 --> 00:28:27.720 align:middle line:84% in different personal space, space per-- 00:28:27.720 --> 00:28:28.710 align:middle line:90% area per person. 00:28:28.710 --> 00:28:31.830 align:middle line:84% And we have this term, delta B of k, 00:28:31.830 --> 00:28:35.610 align:middle line:84% personal property of mixing, to capture how that changes. 00:28:35.610 --> 00:28:37.410 align:middle line:84% So how does your personal space change 00:28:37.410 --> 00:28:42.293 align:middle line:84% from your ideal situation, where you're in a field by yourself? 00:28:42.293 --> 00:28:43.710 align:middle line:84% If you're in a crowded subway car, 00:28:43.710 --> 00:28:48.090 align:middle line:84% you're surrounded by a crowd of people who you like. 00:28:48.090 --> 00:28:49.650 align:middle line:84% Maybe your personal space shrinks. 00:28:49.650 --> 00:28:51.750 align:middle line:84% Or you're surrounded by a crowd of people who you don't really 00:28:51.750 --> 00:28:52.250 align:middle line:90% like. 00:28:52.250 --> 00:28:54.955 align:middle line:84% Maybe your personal space expands. 00:28:54.955 --> 00:28:56.330 align:middle line:84% And there's going to be analogies 00:28:56.330 --> 00:28:57.455 align:middle line:90% for all of these in thermo. 00:28:57.455 --> 00:29:00.770 align:middle line:84% When a molecule is surrounded by other molecules that it likes 00:29:00.770 --> 00:29:04.400 align:middle line:84% or that it wants to bond with, its purse's volume 00:29:04.400 --> 00:29:06.028 align:middle line:90% is going to shrink a little bit. 00:29:06.028 --> 00:29:08.070 align:middle line:84% Its volume per molecule will shrink a little bit. 00:29:08.070 --> 00:29:10.790 align:middle line:84% And if a molecule is surrounded by molecules 00:29:10.790 --> 00:29:14.820 align:middle line:84% that it doesn't want to bond with, 00:29:14.820 --> 00:29:18.356 align:middle line:84% its partial molar volume will expand a little bit. 00:29:18.356 --> 00:29:22.128 align:middle line:84% So these concepts depend on the surroundings. 00:29:22.128 --> 00:29:30.140 align:middle line:90% 00:29:30.140 --> 00:29:34.700 align:middle line:84% So let's make this specific again, Gibbs free energy. 00:29:34.700 --> 00:29:41.680 align:middle line:90% 00:29:41.680 --> 00:29:43.810 align:middle line:84% The total Gibbs free energy of a system 00:29:43.810 --> 00:29:49.200 align:middle line:84% is equal to the sum of partial molar Gibbs 00:29:49.200 --> 00:29:53.960 align:middle line:84% free energy times mole number is-- 00:29:53.960 --> 00:29:56.660 align:middle line:84% because Gibbs is special, it has its own term, chemical 00:29:56.660 --> 00:30:00.100 align:middle line:90% potential, times mole number. 00:30:00.100 --> 00:30:05.250 align:middle line:84% And according to our new bookkeeping scheme, 00:30:05.250 --> 00:30:08.700 align:middle line:84% this is the chemical potential of the pure stuff plus 00:30:08.700 --> 00:30:13.630 align:middle line:84% however it changed when it entered the solution phase. 00:30:13.630 --> 00:30:18.430 align:middle line:90% 00:30:18.430 --> 00:30:25.510 align:middle line:84% And then we'll divide through by the total mole count. 00:30:25.510 --> 00:30:28.390 align:middle line:84% And we get Gibbs free energy equals-- 00:30:28.390 --> 00:30:50.470 align:middle line:90% 00:30:50.470 --> 00:31:03.730 align:middle line:84% And again, this is the sum of the pure components. 00:31:03.730 --> 00:31:11.740 align:middle line:90% 00:31:11.740 --> 00:31:21.200 align:middle line:84% And this is change, the change upon mixing. 00:31:21.200 --> 00:31:29.960 align:middle line:90% 00:31:29.960 --> 00:31:31.070 align:middle line:90% We have two more slides. 00:31:31.070 --> 00:31:32.737 align:middle line:84% And we're going to get to the very, very 00:31:32.737 --> 00:31:36.655 align:middle line:84% useful graphical interpretation of all this. 00:31:36.655 --> 00:31:38.030 align:middle line:84% How do we calculate these things? 00:31:38.030 --> 00:31:49.990 align:middle line:90% 00:31:49.990 --> 00:31:53.800 align:middle line:84% So you're to do a little bit of this on the p set. 00:31:53.800 --> 00:32:00.950 align:middle line:84% So we're going to do the case of chemical potential 00:32:00.950 --> 00:32:08.020 align:middle line:84% in a binary system, binary system. 00:32:08.020 --> 00:32:12.190 align:middle line:84% So binary system-- delta G due to mixing 00:32:12.190 --> 00:32:20.700 align:middle line:84% equals delta mu of 1 X1 plus delta mu of 2 X2. 00:32:20.700 --> 00:32:22.470 align:middle line:90% And you have two components. 00:32:22.470 --> 00:32:23.730 align:middle line:90% That's good. 00:32:23.730 --> 00:32:27.480 align:middle line:90% And so this gets-- 00:32:27.480 --> 00:32:30.480 align:middle line:84% you really have to know your d's of thermodynamics here. 00:32:30.480 --> 00:32:33.170 align:middle line:90% 00:32:33.170 --> 00:32:36.590 align:middle line:84% The change of the Gibbs free energy 00:32:36.590 --> 00:32:41.730 align:middle line:84% of mixing for fixed pressure and temperature as a system 00:32:41.730 --> 00:32:42.615 align:middle line:90% composition varies. 00:32:42.615 --> 00:32:54.520 align:middle line:90% 00:32:54.520 --> 00:32:57.190 align:middle line:84% So again, the change of the Gibbs free energy 00:32:57.190 --> 00:33:00.850 align:middle line:84% of mixing at fixed pressure and temperature as a new system 00:33:00.850 --> 00:33:01.690 align:middle line:90% composition varies. 00:33:01.690 --> 00:33:10.060 align:middle line:90% 00:33:10.060 --> 00:33:11.230 align:middle line:90% What about the chain rule? 00:33:11.230 --> 00:33:17.640 align:middle line:90% 00:33:17.640 --> 00:33:26.320 align:middle line:84% Well, that thing equals 0 if I give 2n. 00:33:26.320 --> 00:33:28.450 align:middle line:84% So I'm left with these two terms here. 00:33:28.450 --> 00:33:33.340 align:middle line:84% And then I'm going to use X1 plus X2 00:33:33.340 --> 00:33:36.490 align:middle line:90% equals 1, which means what? 00:33:36.490 --> 00:33:39.700 align:middle line:90% d X1 equals minus d X2. 00:33:39.700 --> 00:33:41.350 align:middle line:90% It's a binary system. 00:33:41.350 --> 00:33:44.780 align:middle line:84% So there's only one composition variable. 00:33:44.780 --> 00:33:48.480 align:middle line:90% And get the following. 00:33:48.480 --> 00:33:55.770 align:middle line:90% D dT G dT delta G mix d X2. 00:33:55.770 --> 00:33:59.310 align:middle line:84% I chose X2 here as my independent variable. 00:33:59.310 --> 00:34:02.400 align:middle line:84% At fixed pressure and temperature 00:34:02.400 --> 00:34:11.070 align:middle line:84% equals delta mu of 2 minus delta mu of 1. 00:34:11.070 --> 00:34:16.120 align:middle line:90% 00:34:16.120 --> 00:34:26.200 align:middle line:84% And then I'm going to eliminate delta mu of 1 using the above. 00:34:26.200 --> 00:34:29.840 align:middle line:90% 00:34:29.840 --> 00:34:32.290 align:middle line:90% What's the above? 00:34:32.290 --> 00:34:33.100 align:middle line:90% That's the above. 00:34:33.100 --> 00:34:35.679 align:middle line:90% 00:34:35.679 --> 00:34:37.420 align:middle line:90% And I get the following. 00:34:37.420 --> 00:34:52.610 align:middle line:84% Delta mu of 2 equals delta G mix plus 1 minus X2 d 00:34:52.610 --> 00:34:57.600 align:middle line:90% delta G mix d X2. 00:34:57.600 --> 00:35:00.170 align:middle line:90% 00:35:00.170 --> 00:35:02.045 align:middle line:84% And believe it or not, this is really useful. 00:35:02.045 --> 00:35:06.730 align:middle line:90% 00:35:06.730 --> 00:35:16.450 align:middle line:90% And likewise for delta mu of 1. 00:35:16.450 --> 00:35:19.450 align:middle line:84% And let me just show you why for each one, 00:35:19.450 --> 00:35:21.070 align:middle line:84% and we can come back for questions. 00:35:21.070 --> 00:35:24.725 align:middle line:84% It's really useful because we have a graphical interpretation 00:35:24.725 --> 00:35:25.225 align:middle line:90% of this. 00:35:25.225 --> 00:35:32.650 align:middle line:90% 00:35:32.650 --> 00:35:43.750 align:middle line:84% Graphical interpretation of PMPs using solution models. 00:35:43.750 --> 00:35:53.130 align:middle line:90% 00:35:53.130 --> 00:36:07.190 align:middle line:84% So we're going to do this for a case of delta G mix and delta 00:36:07.190 --> 00:36:07.970 align:middle line:90% and delta mu k. 00:36:07.970 --> 00:36:10.098 align:middle line:90% So let's draw a plot. 00:36:10.098 --> 00:36:11.390 align:middle line:90% I'll show you what I mean here. 00:36:11.390 --> 00:36:32.100 align:middle line:90% 00:36:32.100 --> 00:36:33.100 align:middle line:90% Here's a solution model. 00:36:33.100 --> 00:36:34.280 align:middle line:90% I'm just going to draw anything. 00:36:34.280 --> 00:36:35.072 align:middle line:90% I'm going to draw-- 00:36:35.072 --> 00:36:36.610 align:middle line:84% imagine someone gave you this data, 00:36:36.610 --> 00:36:41.475 align:middle line:84% or you figured this out yourself, something like that. 00:36:41.475 --> 00:36:42.475 align:middle line:90% Here's a solution model. 00:36:42.475 --> 00:36:45.730 align:middle line:90% 00:36:45.730 --> 00:36:46.570 align:middle line:90% Something. 00:36:46.570 --> 00:36:50.335 align:middle line:84% It's some trend of thermodynamics with what? 00:36:50.335 --> 00:36:53.780 align:middle line:90% With system composition. 00:36:53.780 --> 00:36:54.460 align:middle line:90% So this is X2. 00:36:54.460 --> 00:36:58.170 align:middle line:90% 00:36:58.170 --> 00:37:00.430 align:middle line:90% So we have some solution model. 00:37:00.430 --> 00:37:04.188 align:middle line:84% And we're going to imagine sitting here at-- 00:37:04.188 --> 00:37:05.126 align:middle line:90% let's see. 00:37:05.126 --> 00:37:08.410 align:middle line:90% 00:37:08.410 --> 00:37:12.000 align:middle line:84% Let's imagine sitting here at this composition. 00:37:12.000 --> 00:37:16.515 align:middle line:90% 00:37:16.515 --> 00:37:18.390 align:middle line:84% Let me just go ahead and draw these tangents, 00:37:18.390 --> 00:37:20.223 align:middle line:84% and then we'll come back to why they matter. 00:37:20.223 --> 00:37:23.310 align:middle line:90% 00:37:23.310 --> 00:37:26.520 align:middle line:84% So there's a point on the curve and my overall system 00:37:26.520 --> 00:37:28.490 align:middle line:90% composition. 00:37:28.490 --> 00:37:32.500 align:middle line:84% This intercept here I'm going to label. 00:37:32.500 --> 00:37:42.580 align:middle line:84% This label, this over here, and also label this over here. 00:37:42.580 --> 00:37:44.020 align:middle line:90% I'm going to label. 00:37:44.020 --> 00:38:00.390 align:middle line:84% I'm going to give these number labels P, S, R, and Q. 00:38:00.390 --> 00:38:03.810 align:middle line:90% So here's my system composition. 00:38:03.810 --> 00:38:09.070 align:middle line:90% 00:38:09.070 --> 00:38:11.060 align:middle line:90% Here's my system composition. 00:38:11.060 --> 00:38:19.330 align:middle line:84% There's a tangent line, the tangent line. 00:38:19.330 --> 00:38:23.425 align:middle line:90% And we have a solution model. 00:38:23.425 --> 00:38:32.540 align:middle line:90% 00:38:32.540 --> 00:38:42.970 align:middle line:84% So delta G mix is just at the height of this curve, 00:38:42.970 --> 00:38:45.300 align:middle line:90% is this QR. 00:38:45.300 --> 00:38:48.270 align:middle line:90% 00:38:48.270 --> 00:38:54.850 align:middle line:84% That's that length, the curve evaluated at P. 00:38:54.850 --> 00:39:05.640 align:middle line:90% Quantity 1 minus X2 equals PR. 00:39:05.640 --> 00:39:08.900 align:middle line:90% 00:39:08.900 --> 00:39:09.920 align:middle line:90% It's just that length. 00:39:09.920 --> 00:39:12.760 align:middle line:90% 00:39:12.760 --> 00:39:21.190 align:middle line:84% And the slope, d delta G mix d X2. 00:39:21.190 --> 00:39:22.360 align:middle line:90% Rise over run. 00:39:22.360 --> 00:39:26.620 align:middle line:90% So the rise is RS. 00:39:26.620 --> 00:39:28.150 align:middle line:90% And the run is PR. 00:39:28.150 --> 00:39:32.380 align:middle line:90% 00:39:32.380 --> 00:39:36.190 align:middle line:84% And so I'm going to plug these geometries 00:39:36.190 --> 00:39:39.505 align:middle line:84% into the expression, which we derived on the previous slide. 00:39:39.505 --> 00:39:50.460 align:middle line:84% And you have QR plus PR times RS over PR. 00:39:50.460 --> 00:39:53.610 align:middle line:84% And this length segments cancel each other. 00:39:53.610 --> 00:39:59.310 align:middle line:84% And you get QR plus RS, so this distance 00:39:59.310 --> 00:40:04.960 align:middle line:84% plus this distance, which, of course, equals QS. 00:40:04.960 --> 00:40:07.500 align:middle line:90% 00:40:07.500 --> 00:40:09.400 align:middle line:90% What does that mean? 00:40:09.400 --> 00:40:31.340 align:middle line:84% It means that this point here, this intercept, is delta mu 2. 00:40:31.340 --> 00:40:36.610 align:middle line:90% 00:40:36.610 --> 00:40:45.780 align:middle line:90% And likewise for delta mu 1. 00:40:45.780 --> 00:40:51.460 align:middle line:84% You have just over here is delta mu 1. 00:40:51.460 --> 00:40:54.220 align:middle line:90% So right. 00:40:54.220 --> 00:40:59.410 align:middle line:84% That means if we have a solution model, some data, some curve, 00:40:59.410 --> 00:41:01.750 align:middle line:84% if we draw the tangent to that, we 00:41:01.750 --> 00:41:03.640 align:middle line:84% can pick out the partial molar properties 00:41:03.640 --> 00:41:06.580 align:middle line:84% of component 1, component 2, by the intercept 00:41:06.580 --> 00:41:10.630 align:middle line:84% of the tangent on the left-hand axis and the right-hand axis. 00:41:10.630 --> 00:41:13.820 align:middle line:90% 00:41:13.820 --> 00:41:19.100 align:middle line:84% And so that's where I'm going to leave it for today. 00:41:19.100 --> 00:41:21.140 align:middle line:84% Either this is kind of mysterious 00:41:21.140 --> 00:41:24.410 align:middle line:84% and wasn't worth the effort, in which case 00:41:24.410 --> 00:41:27.060 align:middle line:90% I would ask you just to hold on. 00:41:27.060 --> 00:41:29.910 align:middle line:84% Or it now it starts to make sense 00:41:29.910 --> 00:41:32.280 align:middle line:84% if you've seen the common tangent 00:41:32.280 --> 00:41:34.510 align:middle line:84% construction before, in which case, 00:41:34.510 --> 00:41:38.040 align:middle line:84% I'll say, all right, well, we'll get there. 00:41:38.040 --> 00:41:39.800 align:middle line:90% Right. 00:41:39.800 --> 00:41:43.420 align:middle line:84% So this was a bunch of math around solution modeling. 00:41:43.420 --> 00:41:46.000 align:middle line:90%