WEBVTT 00:00:00.000 --> 00:00:01.940 align:middle line:90% [SQUEAKING] 00:00:01.940 --> 00:00:03.395 align:middle line:90% [RUSTLING] 00:00:03.395 --> 00:00:04.850 align:middle line:90% [CLICKING] 00:00:04.850 --> 00:00:13.660 align:middle line:90% 00:00:13.660 --> 00:00:16.460 align:middle line:84% GIAN PAOLO BERETTA: So good afternoon. 00:00:16.460 --> 00:00:19.130 align:middle line:90% 00:00:19.130 --> 00:00:28.700 align:middle line:84% And we are-- hopefully, we'll finish the topic 00:00:28.700 --> 00:00:31.300 align:middle line:84% on liquid-vapor equilibria today. 00:00:31.300 --> 00:00:35.220 align:middle line:90% 00:00:35.220 --> 00:00:45.680 align:middle line:84% Now, we have seen that everything is contained 00:00:45.680 --> 00:00:48.740 align:middle line:90% in the chemical potentials. 00:00:48.740 --> 00:00:54.360 align:middle line:84% And so this summarizes, for example, 00:00:54.360 --> 00:00:56.990 align:middle line:84% for a pure substance, when you have the two 00:00:56.990 --> 00:01:02.410 align:middle line:84% phases in mutual equilibrium, like liquid and vapor, 00:01:02.410 --> 00:01:08.920 align:middle line:84% the equality of chemical potentials for the pure liquid 00:01:08.920 --> 00:01:12.620 align:middle line:84% and the pure vapor gives a relation 00:01:12.620 --> 00:01:14.370 align:middle line:84% between pressure and temperature, 00:01:14.370 --> 00:01:20.160 align:middle line:84% which we call p sat, or the inverse function is T sat. 00:01:20.160 --> 00:01:22.350 align:middle line:84% That's what you find in, for example, 00:01:22.350 --> 00:01:27.110 align:middle line:84% the steam tables for water or every other substance, 00:01:27.110 --> 00:01:34.810 align:middle line:84% as this establishes the boiling point at a given pressure. 00:01:34.810 --> 00:01:39.390 align:middle line:84% So the equivalent for a binary mixture is that now we have-- 00:01:39.390 --> 00:01:45.080 align:middle line:84% if both substances are volatile and condensable, 00:01:45.080 --> 00:01:50.430 align:middle line:84% then you have two chemical equilibrium-- 00:01:50.430 --> 00:01:53.040 align:middle line:84% chemical potential equalities that are 00:01:53.040 --> 00:01:56.910 align:middle line:84% required for a mutual stable equilibrium between the two 00:01:56.910 --> 00:01:59.150 align:middle line:90% phases. 00:01:59.150 --> 00:02:01.900 align:middle line:84% Of course, the chemical potential 00:02:01.900 --> 00:02:04.730 align:middle line:84% is now not only a function of temperature and pressure, 00:02:04.730 --> 00:02:06.730 align:middle line:90% but also of composition. 00:02:06.730 --> 00:02:10.630 align:middle line:84% But because the mole fractions add up to 1, 00:02:10.630 --> 00:02:15.160 align:middle line:84% you just need one independent mole fraction 00:02:15.160 --> 00:02:19.040 align:middle line:90% to represent the composition. 00:02:19.040 --> 00:02:22.670 align:middle line:84% But you need one mole fraction for the liquid phase 00:02:22.670 --> 00:02:26.150 align:middle line:84% and another mole fraction for the vapor phase 00:02:26.150 --> 00:02:29.180 align:middle line:84% because they are not necessarily equal. 00:02:29.180 --> 00:02:36.780 align:middle line:84% So these two equalities in the four variables, 00:02:36.780 --> 00:02:46.020 align:middle line:84% T, p, y1f, and y1g, can be solved to give these relations. 00:02:46.020 --> 00:02:49.310 align:middle line:84% So you only have a variance of two, 00:02:49.310 --> 00:02:53.070 align:middle line:84% it gives a phase-rule variance of two, meaning that, 00:02:53.070 --> 00:02:56.870 align:middle line:84% for example, if you fix temperature and pressure, that 00:02:56.870 --> 00:03:00.590 align:middle line:84% automatically fixes a unique composition 00:03:00.590 --> 00:03:02.720 align:middle line:84% for the liquid and a unique composition 00:03:02.720 --> 00:03:06.450 align:middle line:84% for the vapor that are not equal. 00:03:06.450 --> 00:03:11.420 align:middle line:84% And the Raoult's law that we have developed 00:03:11.420 --> 00:03:15.380 align:middle line:84% establishes the relation between the composition of the vapor 00:03:15.380 --> 00:03:17.305 align:middle line:84% and the composition of the liquid. 00:03:17.305 --> 00:03:21.290 align:middle line:90% 00:03:21.290 --> 00:03:24.200 align:middle line:90% You can also rewrite-- 00:03:24.200 --> 00:03:27.270 align:middle line:84% solve these equations for other independent variables. 00:03:27.270 --> 00:03:32.000 align:middle line:84% For example, you keep T and the mole fraction of the liquid, 00:03:32.000 --> 00:03:35.280 align:middle line:84% and you get from these relations a function, 00:03:35.280 --> 00:03:38.900 align:middle line:90% which we call the bubble line. 00:03:38.900 --> 00:03:41.720 align:middle line:84% Or if you instead like to keep the temperature 00:03:41.720 --> 00:03:45.420 align:middle line:84% and the composition of the vapor, 00:03:45.420 --> 00:03:52.137 align:middle line:84% you get the dew line, which can be expressed as pressure 00:03:52.137 --> 00:03:54.220 align:middle line:84% as a function of temperature, but also temperature 00:03:54.220 --> 00:03:57.280 align:middle line:90% as a function of pressure. 00:03:57.280 --> 00:04:01.580 align:middle line:84% So these are all equivalent forms. 00:04:01.580 --> 00:04:06.620 align:middle line:84% And here is sort of a three-dimensional representation 00:04:06.620 --> 00:04:10.190 align:middle line:90% of how these things look. 00:04:10.190 --> 00:04:12.660 align:middle line:84% Of course, in the limit of the pure substances, 00:04:12.660 --> 00:04:17.100 align:middle line:84% you get the usual p-T diagram for that substance. 00:04:17.100 --> 00:04:18.070 align:middle line:90% This is for one. 00:04:18.070 --> 00:04:18.755 align:middle line:90% This is for two. 00:04:18.755 --> 00:04:21.269 align:middle line:90% 00:04:21.269 --> 00:04:24.630 align:middle line:84% But halfway, when the mole fractions 00:04:24.630 --> 00:04:27.510 align:middle line:84% go from between 0 and 1, then you 00:04:27.510 --> 00:04:36.810 align:middle line:84% get these relations, as we have seen, gives rise to gaps. 00:04:36.810 --> 00:04:40.680 align:middle line:84% So to regions that are forbidden, 00:04:40.680 --> 00:04:42.720 align:middle line:90% they cannot be obtained-- 00:04:42.720 --> 00:04:46.025 align:middle line:90% 00:04:46.025 --> 00:04:48.890 align:middle line:84% in which the substance doesn't behave in-- 00:04:48.890 --> 00:04:51.200 align:middle line:84% doesn't exist in stable equilibrium. 00:04:51.200 --> 00:04:54.320 align:middle line:90% It likes to separate. 00:04:54.320 --> 00:05:00.250 align:middle line:84% So today, to finish up, we need to see a little more about this. 00:05:00.250 --> 00:05:04.070 align:middle line:90% 00:05:04.070 --> 00:05:08.450 align:middle line:84% So the first result that we can talk about 00:05:08.450 --> 00:05:14.590 align:middle line:84% is called Henry's law, which is a byproduct of Raoult's law. 00:05:14.590 --> 00:05:21.360 align:middle line:84% So remember, Raoult's law was a relation here, this one. 00:05:21.360 --> 00:05:28.410 align:middle line:84% That gives us, as we just said, the composition of the vapor 00:05:28.410 --> 00:05:30.160 align:middle line:90% phase in equilibrium. 00:05:30.160 --> 00:05:33.480 align:middle line:84% It is a function of the composition 00:05:33.480 --> 00:05:37.350 align:middle line:84% of the liquid phase, with which it is in mutual equilibrium. 00:05:37.350 --> 00:05:40.140 align:middle line:84% And the saturation pressure of the pure substance 00:05:40.140 --> 00:05:44.280 align:middle line:90% enters in Raoult's equation. 00:05:44.280 --> 00:05:49.420 align:middle line:84% We derive this by equating the chemical potentials of substance 00:05:49.420 --> 00:05:51.540 align:middle line:90% one. 00:05:51.540 --> 00:05:57.380 align:middle line:84% And, of course, if you have-- if your mixture is 00:05:57.380 --> 00:06:03.150 align:middle line:84% ideal at all compositions and both substances 00:06:03.150 --> 00:06:09.540 align:middle line:84% are volatile and condensible, you have also Raoult's law 00:06:09.540 --> 00:06:11.380 align:middle line:90% for the second substance. 00:06:11.380 --> 00:06:14.220 align:middle line:84% And so you have the same expression, like here, 00:06:14.220 --> 00:06:18.840 align:middle line:84% only with the subscript 2 instead of 1. 00:06:18.840 --> 00:06:27.390 align:middle line:84% However, very few mixtures in the liquid phase 00:06:27.390 --> 00:06:34.090 align:middle line:84% are really ideal at all compositions. 00:06:34.090 --> 00:06:41.660 align:middle line:84% But we know that in the dilute limit, so 00:06:41.660 --> 00:06:45.170 align:middle line:84% therefore a dilute solution, we argued 00:06:45.170 --> 00:06:52.370 align:middle line:84% that every liquid solution is always in that limit 00:06:52.370 --> 00:06:55.550 align:middle line:90% behaving as ideal. 00:06:55.550 --> 00:06:57.660 align:middle line:90% So that means-- 00:06:57.660 --> 00:07:00.150 align:middle line:84% Let's consider component 1 as the solvent 00:07:00.150 --> 00:07:11.510 align:middle line:84% for which this equation is valid in the limit of y1f-- 00:07:11.510 --> 00:07:15.970 align:middle line:84% That means the liquid composition-- is very 00:07:15.970 --> 00:07:17.390 align:middle line:90% close to the pure substance. 00:07:17.390 --> 00:07:20.445 align:middle line:84% So that's a dilute solution limit. 00:07:20.445 --> 00:07:23.260 align:middle line:90% 00:07:23.260 --> 00:07:28.570 align:middle line:84% Substance 2 does not behave as Raoult's law. 00:07:28.570 --> 00:07:32.470 align:middle line:84% But we will see that now, the next lines 00:07:32.470 --> 00:07:37.330 align:middle line:84% are going-- we're going to need to prove this relation here, 00:07:37.330 --> 00:07:44.000 align:middle line:84% which is Henry's law, which looks similar to Raoult's law, 00:07:44.000 --> 00:07:46.340 align:middle line:84% except that instead of the saturation 00:07:46.340 --> 00:07:49.680 align:middle line:84% pressure of the substance, of the second substance, 00:07:49.680 --> 00:07:54.630 align:middle line:84% you have another function of not just 00:07:54.630 --> 00:07:58.020 align:middle line:84% temperature, but temperature and pressure, which is called 00:07:58.020 --> 00:08:01.030 align:middle line:90% Henry's volatility constant. 00:08:01.030 --> 00:08:03.650 align:middle line:84% And it's going to be given by this relation here. 00:08:03.650 --> 00:08:06.220 align:middle line:90% So how do we obtain this? 00:08:06.220 --> 00:08:14.850 align:middle line:84% And this is valid for any mixture in the dilute solution 00:08:14.850 --> 00:08:15.350 align:middle line:90% limit. 00:08:15.350 --> 00:08:18.230 align:middle line:90% There is a constant like that. 00:08:18.230 --> 00:08:22.320 align:middle line:84% It's called Henry's constant, but it's really something 00:08:22.320 --> 00:08:24.370 align:middle line:84% that it is independent of composition, 00:08:24.370 --> 00:08:27.318 align:middle line:84% and it's a function-- but it is a function of temperature 00:08:27.318 --> 00:08:27.860 align:middle line:90% and pressure. 00:08:27.860 --> 00:08:32.870 align:middle line:84% So it's constant once you fix temperature and pressure. 00:08:32.870 --> 00:08:36.169 align:middle line:90% So where does it come about? 00:08:36.169 --> 00:08:40.929 align:middle line:84% It comes about from the Gibbs-Duhem relations 00:08:40.929 --> 00:08:43.607 align:middle line:84% because they can-- they must always hold. 00:08:43.607 --> 00:08:45.190 align:middle line:84% Do you remember, Gibbs-Duhem relations 00:08:45.190 --> 00:08:48.160 align:middle line:84% are the ones that we use also to derive the Gibbs phase 00:08:48.160 --> 00:08:53.110 align:middle line:84% rule and the Clausius-Clapeyron relations 00:08:53.110 --> 00:08:57.070 align:middle line:90% for the pure substances. 00:08:57.070 --> 00:09:02.560 align:middle line:84% For a binary mixture, the usual summation 00:09:02.560 --> 00:09:05.360 align:middle line:84% becomes just two contributions here. 00:09:05.360 --> 00:09:10.360 align:middle line:84% So this is minus s dT plus v dp plus the summation-- 00:09:10.360 --> 00:09:13.360 align:middle line:84% minus the summation of the mole fractions times the d 00:09:13.360 --> 00:09:18.000 align:middle line:90% of the chemical potentials. 00:09:18.000 --> 00:09:21.270 align:middle line:84% Now, suppose that we fixed temperature and pressure. 00:09:21.270 --> 00:09:24.830 align:middle line:90% So we set dT and dp equal 0. 00:09:24.830 --> 00:09:26.790 align:middle line:84% Then you see, this gives a relation 00:09:26.790 --> 00:09:30.510 align:middle line:84% between how the chemical potential 00:09:30.510 --> 00:09:34.980 align:middle line:84% of the second substance, so of the solute 00:09:34.980 --> 00:09:39.600 align:middle line:84% in the dilute solution, changes as you 00:09:39.600 --> 00:09:44.260 align:middle line:84% have changes in the chemical potential of the solvent. 00:09:44.260 --> 00:09:46.156 align:middle line:90% So they are not independent. 00:09:46.156 --> 00:09:49.110 align:middle line:90% 00:09:49.110 --> 00:09:57.800 align:middle line:84% So since the solvent behaves like this, it's ideal. 00:09:57.800 --> 00:10:03.290 align:middle line:84% So it obeys this ideal relation for chemical potentials. 00:10:03.290 --> 00:10:10.290 align:middle line:84% If you do the differential of that at constant T and p, 00:10:10.290 --> 00:10:14.310 align:middle line:84% you are left with just the differential of log 00:10:14.310 --> 00:10:17.430 align:middle line:90% of y, which is dy over y. 00:10:17.430 --> 00:10:21.430 align:middle line:84% So that's dmu1 in the liquid phase. 00:10:21.430 --> 00:10:23.840 align:middle line:84% So you can substitute in this relation here. 00:10:23.840 --> 00:10:27.020 align:middle line:90% 00:10:27.020 --> 00:10:31.380 align:middle line:84% So you see, this relation can be rewritten this way. 00:10:31.380 --> 00:10:34.320 align:middle line:84% And then here, for this one, you substitute 00:10:34.320 --> 00:10:37.900 align:middle line:84% what we obtained from the ideal solution. 00:10:37.900 --> 00:10:39.720 align:middle line:90% And here is what you get. 00:10:39.720 --> 00:10:43.150 align:middle line:90% You get a similar relation. 00:10:43.150 --> 00:10:46.570 align:middle line:84% So you get that, also, the chemical potential 00:10:46.570 --> 00:10:50.680 align:middle line:84% of the second substance in the liquid solution 00:10:50.680 --> 00:10:54.590 align:middle line:84% behaves logarithmically with the composition. 00:10:54.590 --> 00:10:57.860 align:middle line:90% 00:10:57.860 --> 00:11:02.690 align:middle line:84% So that means that if it is logarithmic-- 00:11:02.690 --> 00:11:05.330 align:middle line:84% see, for the ideal solution, this 00:11:05.330 --> 00:11:07.550 align:middle line:90% is the logarithmic dependence. 00:11:07.550 --> 00:11:10.040 align:middle line:84% And here you have the pure substance. 00:11:10.040 --> 00:11:15.260 align:middle line:84% So the constant here is the pure substance. 00:11:15.260 --> 00:11:16.070 align:middle line:90% Oops. 00:11:16.070 --> 00:11:22.220 align:middle line:84% Whereas, for this other case, if you integrate this relation, 00:11:22.220 --> 00:11:24.530 align:middle line:84% you get something similar to the one 00:11:24.530 --> 00:11:28.550 align:middle line:84% up there, but with a different integration constant 00:11:28.550 --> 00:11:33.860 align:middle line:84% that is not necessarily equal to the pure component chemical 00:11:33.860 --> 00:11:35.470 align:middle line:90% potential. 00:11:35.470 --> 00:11:39.030 align:middle line:84% It is equal to the pure component chemical 00:11:39.030 --> 00:11:45.460 align:middle line:84% potential in the limit of ideal solution at all compositions. 00:11:45.460 --> 00:11:47.670 align:middle line:84% But in general, this is some other function 00:11:47.670 --> 00:11:49.800 align:middle line:84% of temperature and pressure, which 00:11:49.800 --> 00:11:53.350 align:middle line:90% nevertheless can be measured. 00:11:53.350 --> 00:11:58.170 align:middle line:84% And therefore, if you use this relation to obtain-- 00:11:58.170 --> 00:12:00.880 align:middle line:84% like we did for obtaining Raoult's law, 00:12:00.880 --> 00:12:04.640 align:middle line:90% you obtain Henry's law. 00:12:04.640 --> 00:12:06.950 align:middle line:84% And here is the explicit expression 00:12:06.950 --> 00:12:10.280 align:middle line:84% that derives for Henry's constant in terms 00:12:10.280 --> 00:12:12.960 align:middle line:84% of the saturation pressure of the second substance. 00:12:12.960 --> 00:12:14.580 align:middle line:90% There is this correction factor. 00:12:14.580 --> 00:12:19.380 align:middle line:84% And this correction factor can be very large. 00:12:19.380 --> 00:12:31.540 align:middle line:84% For example, one important example in some practices is 00:12:31.540 --> 00:12:38.500 align:middle line:84% the one for nitrogen. So Henry's constant has a value which is 00:12:38.500 --> 00:12:41.560 align:middle line:90% 85,000 atmospheres. 00:12:41.560 --> 00:12:43.350 align:middle line:90% It's a very high pressure. 00:12:43.350 --> 00:12:45.960 align:middle line:90% 00:12:45.960 --> 00:12:48.330 align:middle line:90% From the point of view of-- 00:12:48.330 --> 00:12:51.990 align:middle line:84% graphic point of view, suppose this one is 00:12:51.990 --> 00:12:54.320 align:middle line:90% mole fraction of the solvent. 00:12:54.320 --> 00:12:59.570 align:middle line:84% Say that we are talking about water. 00:12:59.570 --> 00:13:04.920 align:middle line:84% So this is-- the dilute limit is in this area. 00:13:04.920 --> 00:13:09.510 align:middle line:84% So mole fraction of water is very close to 1. 00:13:09.510 --> 00:13:13.290 align:middle line:84% And so what is missing to get to 1 00:13:13.290 --> 00:13:16.570 align:middle line:84% is the mole fraction of the solute. 00:13:16.570 --> 00:13:25.570 align:middle line:84% And suppose solute is nitrogen. The Henry's slope-- 00:13:25.570 --> 00:13:31.150 align:middle line:84% if you had ideal behavior, this line 00:13:31.150 --> 00:13:33.290 align:middle line:90% represents the partial pressure. 00:13:33.290 --> 00:13:37.940 align:middle line:84% So it's the product of mole fraction and pressure. 00:13:37.940 --> 00:13:41.870 align:middle line:84% And it's a straight line because that's a linear relation. 00:13:41.870 --> 00:13:46.520 align:middle line:84% That takes you to the saturation pressure 00:13:46.520 --> 00:13:54.390 align:middle line:84% of nitrogen. Since this constant is very high, 00:13:54.390 --> 00:13:56.220 align:middle line:84% the Henry's constant is very high, 00:13:56.220 --> 00:13:59.620 align:middle line:84% the actual slope here is much steeper. 00:13:59.620 --> 00:14:03.115 align:middle line:90% 00:14:03.115 --> 00:14:12.850 align:middle line:84% And that is sort of the tangent of this non-ideal behavior 00:14:12.850 --> 00:14:17.520 align:middle line:90% bubble line. 00:14:17.520 --> 00:14:21.250 align:middle line:90% Now, why nitrogen is important? 00:14:21.250 --> 00:14:25.945 align:middle line:90% For example, for divers-- 00:14:25.945 --> 00:14:31.150 align:middle line:84% you know that there's this problem of decompression. 00:14:31.150 --> 00:14:39.240 align:middle line:84% And if you dive and you stay for a long time at high pressures, 00:14:39.240 --> 00:14:39.940 align:middle line:90% nitrogen-- 00:14:39.940 --> 00:14:42.750 align:middle line:90% 00:14:42.750 --> 00:14:51.990 align:middle line:84% you see, here, if I increase the pressure, the mole fraction 00:14:51.990 --> 00:14:55.390 align:middle line:84% of nitrogen that can stay in the liquid, 00:14:55.390 --> 00:14:59.160 align:middle line:84% so let's say the-- let's call it the solubility of nitrogen 00:14:59.160 --> 00:15:04.460 align:middle line:90% in the blood gets higher. 00:15:04.460 --> 00:15:11.370 align:middle line:84% So the longer you stay, the more nitrogen goes into your blood. 00:15:11.370 --> 00:15:16.530 align:middle line:90% And this happens in the lungs. 00:15:16.530 --> 00:15:22.410 align:middle line:84% Now, if you come up to sea surface too quickly 00:15:22.410 --> 00:15:29.890 align:middle line:84% and you don't let the necessary time to adjust back 00:15:29.890 --> 00:15:35.950 align:middle line:84% through the lungs, this nitrogen will 00:15:35.950 --> 00:15:41.150 align:middle line:84% evaporate from your bloodstream and form bubbles all over. 00:15:41.150 --> 00:15:47.110 align:middle line:84% And these bubbles typically collect in joints 00:15:47.110 --> 00:15:51.340 align:middle line:84% and gives you a lot of pain, especially when 00:15:51.340 --> 00:15:52.370 align:middle line:90% you move the joints. 00:15:52.370 --> 00:15:55.220 align:middle line:84% And that's why the disease is called, also, the bends. 00:15:55.220 --> 00:15:57.270 align:middle line:90% When you bend, you suffer. 00:15:57.270 --> 00:16:01.320 align:middle line:90% 00:16:01.320 --> 00:16:02.910 align:middle line:84% But of course, the Henry's constant 00:16:02.910 --> 00:16:05.340 align:middle line:84% has also some other applications that 00:16:05.340 --> 00:16:11.050 align:middle line:84% also have to do with atmospheric and ocean science. 00:16:11.050 --> 00:16:14.730 align:middle line:90% And so this is-- 00:16:14.730 --> 00:16:25.750 align:middle line:84% for example, it determines the composition 00:16:25.750 --> 00:16:29.610 align:middle line:90% of clouds, cloud droplets. 00:16:29.610 --> 00:16:33.210 align:middle line:84% And also, it is important in how, 00:16:33.210 --> 00:16:36.450 align:middle line:84% for example, some pollutants are absorbed 00:16:36.450 --> 00:16:40.700 align:middle line:84% in wastewater management plants, treatment plants. 00:16:40.700 --> 00:16:44.325 align:middle line:90% 00:16:44.325 --> 00:16:46.810 align:middle line:84% So this was an example of application 00:16:46.810 --> 00:16:51.520 align:middle line:84% of the Gibbs-Duhem relation, which has 00:16:51.520 --> 00:16:53.810 align:middle line:90% some practical applications. 00:16:53.810 --> 00:16:59.120 align:middle line:90% 00:16:59.120 --> 00:17:04.349 align:middle line:84% Before we go on, I have to stress again something 00:17:04.349 --> 00:17:06.790 align:middle line:90% that we have already discussed. 00:17:06.790 --> 00:17:09.390 align:middle line:84% So I'm not going to go into too many details 00:17:09.390 --> 00:17:12.300 align:middle line:84% here, but I leave it, this slide, 00:17:12.300 --> 00:17:18.210 align:middle line:90% for reference because when we-- 00:17:18.210 --> 00:17:25.810 align:middle line:84% like we're going to do next, we deal with models of mixtures 00:17:25.810 --> 00:17:33.060 align:middle line:84% to describe this non-ideal behavior, 00:17:33.060 --> 00:17:36.846 align:middle line:84% we are going to have to be concerned 00:17:36.846 --> 00:17:41.560 align:middle line:84% with the stability or the solubility limits, 00:17:41.560 --> 00:17:44.020 align:middle line:84% the stability conditions, for these 00:17:44.020 --> 00:17:48.220 align:middle line:84% mixtures because sometimes, for example, liquids, you know, 00:17:48.220 --> 00:17:56.850 align:middle line:84% oil and water, don't mix, at least not at all compositions. 00:17:56.850 --> 00:18:02.880 align:middle line:84% So that is an effect of the fact that at certain compositions, 00:18:02.880 --> 00:18:07.560 align:middle line:84% a homogeneous mixture of oil and water 00:18:07.560 --> 00:18:11.496 align:middle line:84% is unstable, at most compositions. 00:18:11.496 --> 00:18:15.960 align:middle line:90% 00:18:15.960 --> 00:18:20.940 align:middle line:84% All right, well, so our models will give expressions 00:18:20.940 --> 00:18:23.580 align:middle line:90% for the Gibbs free energy. 00:18:23.580 --> 00:18:26.040 align:middle line:84% And you can express the Gibbs free energy 00:18:26.040 --> 00:18:32.700 align:middle line:84% as the total value or the specific value. 00:18:32.700 --> 00:18:37.860 align:middle line:84% And of course, if we are talking about large systems 00:18:37.860 --> 00:18:41.010 align:middle line:84% with large numbers of particles, so we 00:18:41.010 --> 00:18:42.910 align:middle line:90% can apply the Euler relation. 00:18:42.910 --> 00:18:48.240 align:middle line:84% And this is the fact that the specific Gibbs free energy 00:18:48.240 --> 00:18:50.130 align:middle line:84% is essentially-- you can write it 00:18:50.130 --> 00:18:55.080 align:middle line:84% as the weighted sum of the chemical potentials weighted 00:18:55.080 --> 00:18:59.730 align:middle line:84% by the mole fractions of the components. 00:18:59.730 --> 00:19:01.695 align:middle line:84% But we have to remember that-- you see, 00:19:01.695 --> 00:19:06.420 align:middle line:84% you might be tempted to say that, therefore, the chemical 00:19:06.420 --> 00:19:12.420 align:middle line:84% potential is the derivative of little g with respect to y, 00:19:12.420 --> 00:19:13.950 align:middle line:90% but it is not. 00:19:13.950 --> 00:19:16.250 align:middle line:84% That's why you have to be careful. 00:19:16.250 --> 00:19:19.410 align:middle line:84% The chemical potential, when you want to write it in terms 00:19:19.410 --> 00:19:22.290 align:middle line:84% of the Gibbs free energy, is the partial of the Gibbs free 00:19:22.290 --> 00:19:28.720 align:middle line:84% energy, capital G, with respect to n at constant T, p, 00:19:28.720 --> 00:19:30.570 align:middle line:90% and all the other n's. 00:19:30.570 --> 00:19:38.690 align:middle line:84% And if you do all the correct steps, here it is. 00:19:38.690 --> 00:19:43.120 align:middle line:84% Here is the expression in terms of the specific Gibbs 00:19:43.120 --> 00:19:45.560 align:middle line:90% free energy. 00:19:45.560 --> 00:19:47.750 align:middle line:90% There are lots of contributions. 00:19:47.750 --> 00:19:53.450 align:middle line:84% It's not just the partial of g with respect to y. 00:19:53.450 --> 00:19:56.320 align:middle line:90% There are other stuff. 00:19:56.320 --> 00:20:01.930 align:middle line:90% For example, if you assume-- 00:20:01.930 --> 00:20:05.290 align:middle line:84% if you apply this to the Gibbs free energy 00:20:05.290 --> 00:20:12.130 align:middle line:84% of the ideal solution, which we discussed last time also, 00:20:12.130 --> 00:20:19.600 align:middle line:84% which is given only by this term plus the mixing entropy type 00:20:19.600 --> 00:20:24.720 align:middle line:84% term, and you do this derivative here, 00:20:24.720 --> 00:20:31.230 align:middle line:84% you see that this derivative of the lowercase g with respect 00:20:31.230 --> 00:20:35.190 align:middle line:84% to the mole fraction is equal to the pure component 00:20:35.190 --> 00:20:36.760 align:middle line:90% plus something like that. 00:20:36.760 --> 00:20:41.430 align:middle line:84% So there is an RT term, too many, 00:20:41.430 --> 00:20:47.680 align:middle line:84% which makes it different from the ideal chemical potential. 00:20:47.680 --> 00:20:50.230 align:middle line:90% 00:20:50.230 --> 00:20:53.210 align:middle line:84% But if you put this expression into here, 00:20:53.210 --> 00:20:56.110 align:middle line:84% then you get the proper expression 00:20:56.110 --> 00:20:58.360 align:middle line:90% for the chemical potential. 00:20:58.360 --> 00:21:01.840 align:middle line:84% For a binary mixture, which is something-- so 00:21:01.840 --> 00:21:06.760 align:middle line:84% these are relations that we will need in a few moments. 00:21:06.760 --> 00:21:12.220 align:middle line:84% For a binary mixture, the specific Gibbs free energy 00:21:12.220 --> 00:21:14.270 align:middle line:84% is function of only one mole fraction, 00:21:14.270 --> 00:21:20.620 align:middle line:84% of course, because the other is 1 minus y1. 00:21:20.620 --> 00:21:25.510 align:middle line:84% The chemical potential, if you use this expression, 00:21:25.510 --> 00:21:27.015 align:middle line:90% can be written this way. 00:21:27.015 --> 00:21:30.230 align:middle line:90% 00:21:30.230 --> 00:21:32.450 align:middle line:84% And for the second component, it can 00:21:32.450 --> 00:21:38.400 align:middle line:84% be written this way, in terms of the derivative of little g 00:21:38.400 --> 00:21:40.410 align:middle line:90% with respect to y. 00:21:40.410 --> 00:21:47.520 align:middle line:84% And so where do we need these two expressions? 00:21:47.520 --> 00:21:49.880 align:middle line:84% We need them when we want to apply the conditions 00:21:49.880 --> 00:21:53.980 align:middle line:84% for mutual equilibrium between two phases 00:21:53.980 --> 00:21:57.130 align:middle line:84% because, if we have the two phases in equilibrium, 00:21:57.130 --> 00:22:00.220 align:middle line:84% component 1 must have the same chemical 00:22:00.220 --> 00:22:04.160 align:middle line:84% potential in both phases, and component 2 also. 00:22:04.160 --> 00:22:07.620 align:middle line:90% So this gives two relations. 00:22:07.620 --> 00:22:11.270 align:middle line:84% And these two relations can be written-- 00:22:11.270 --> 00:22:14.270 align:middle line:84% from these two relations, you can obtain these-- 00:22:14.270 --> 00:22:17.570 align:middle line:84% by substituting the expression above-- these two expressions 00:22:17.570 --> 00:22:19.830 align:middle line:90% here. 00:22:19.830 --> 00:22:25.720 align:middle line:84% So it tells you that the partial of g with respect to y 00:22:25.720 --> 00:22:28.090 align:middle line:84% must be equal for the two phases. 00:22:28.090 --> 00:22:31.540 align:middle line:84% Even if this derivative doesn't represent 00:22:31.540 --> 00:22:34.810 align:middle line:84% the chemical potential, but only a part of it, 00:22:34.810 --> 00:22:39.020 align:middle line:84% still, it has to be equal for the two phases. 00:22:39.020 --> 00:22:45.260 align:middle line:84% And plus, it represents a slope of a geometrical construction 00:22:45.260 --> 00:22:48.360 align:middle line:90% that looks like this. 00:22:48.360 --> 00:22:49.670 align:middle line:90% So if I have-- 00:22:49.670 --> 00:23:01.160 align:middle line:84% I plot g with respect to mole fraction, y, sometimes, 00:23:01.160 --> 00:23:06.890 align:middle line:84% we will see-- you get expressions-- 00:23:06.890 --> 00:23:07.930 align:middle line:90% plots like this. 00:23:07.930 --> 00:23:12.440 align:middle line:90% 00:23:12.440 --> 00:23:16.290 align:middle line:84% We will see that the condition for mutual equilibrium, 00:23:16.290 --> 00:23:23.740 align:middle line:84% the first one, this one, says that there are going to be-- 00:23:23.740 --> 00:23:28.200 align:middle line:84% if you find a line, which is tangent both here and there-- 00:23:28.200 --> 00:23:32.390 align:middle line:90% 00:23:32.390 --> 00:23:42.800 align:middle line:84% almost-- this slope is dg dy, also there. 00:23:42.800 --> 00:23:49.100 align:middle line:84% But here it's evaluated at y prime first-- 00:23:49.100 --> 00:23:50.280 align:middle line:90% phase one. 00:23:50.280 --> 00:23:54.085 align:middle line:84% And here it is evaluated at y double prime. 00:23:54.085 --> 00:23:58.400 align:middle line:90% 00:23:58.400 --> 00:24:02.780 align:middle line:84% So finding this tangent line identifies 00:24:02.780 --> 00:24:08.850 align:middle line:84% those two mole fractions that are in mutual equilibrium. 00:24:08.850 --> 00:24:11.480 align:middle line:90% 00:24:11.480 --> 00:24:15.680 align:middle line:90% And also, see, this is g prime. 00:24:15.680 --> 00:24:18.910 align:middle line:90% And that's g double prime. 00:24:18.910 --> 00:24:21.750 align:middle line:90% 00:24:21.750 --> 00:24:26.760 align:middle line:84% And as you see, the equation of this line 00:24:26.760 --> 00:24:30.765 align:middle line:90% is precisely this equation here. 00:24:30.765 --> 00:24:33.900 align:middle line:90% 00:24:33.900 --> 00:24:38.820 align:middle line:84% So that construction helps us identify 00:24:38.820 --> 00:24:42.180 align:middle line:84% for a given temperature-- because here we 00:24:42.180 --> 00:24:45.890 align:middle line:84% have fixed a given temperature and pressure. 00:24:45.890 --> 00:24:48.570 align:middle line:90% 00:24:48.570 --> 00:24:51.665 align:middle line:84% It helps identify these two points. 00:24:51.665 --> 00:24:58.850 align:middle line:90% 00:24:58.850 --> 00:25:07.160 align:middle line:84% So, again, that means, suppose I fix now the pressure. 00:25:07.160 --> 00:25:11.700 align:middle line:90% And I take that temperature. 00:25:11.700 --> 00:25:15.450 align:middle line:84% So at this temperature, T, I'm going 00:25:15.450 --> 00:25:19.780 align:middle line:84% to plot these two points that I have found, 00:25:19.780 --> 00:25:21.660 align:middle line:90% y prime and y double prime. 00:25:21.660 --> 00:25:25.430 align:middle line:90% 00:25:25.430 --> 00:25:35.910 align:middle line:84% This represents the two phases in mutual equilibrium. 00:25:35.910 --> 00:25:38.730 align:middle line:90% So this would be, say, y1-- 00:25:38.730 --> 00:25:45.090 align:middle line:84% for example, for the phase, this would be y1f 00:25:45.090 --> 00:25:47.910 align:middle line:84% and this is y1g if you're talking 00:25:47.910 --> 00:25:51.940 align:middle line:90% about liquid-vapor equilibrium. 00:25:51.940 --> 00:25:54.070 align:middle line:84% Now, if I change the temperature and make 00:25:54.070 --> 00:25:56.800 align:middle line:84% another plot of the same kind, I'm 00:25:56.800 --> 00:26:02.200 align:middle line:84% going to identify by the same way 00:26:02.200 --> 00:26:07.590 align:middle line:84% the two other compositions that correspond 00:26:07.590 --> 00:26:11.070 align:middle line:84% to the mutual equilibrium and so on, 00:26:11.070 --> 00:26:16.260 align:middle line:84% until I get to a temperature in which these points are going 00:26:16.260 --> 00:26:17.080 align:middle line:90% to coincide. 00:26:17.080 --> 00:26:20.900 align:middle line:84% So I'm going to obtain a curve like that. 00:26:20.900 --> 00:26:24.660 align:middle line:90% 00:26:24.660 --> 00:26:28.740 align:middle line:84% Maybe it's a bit smoother in practice. 00:26:28.740 --> 00:26:31.425 align:middle line:90% This is called a critical point. 00:26:31.425 --> 00:26:37.770 align:middle line:90% 00:26:37.770 --> 00:26:39.390 align:middle line:90% It's critical in what sense? 00:26:39.390 --> 00:26:44.810 align:middle line:84% It's critical in the sense that above that temperature, 00:26:44.810 --> 00:26:48.890 align:middle line:84% you can see that all compositions are 00:26:48.890 --> 00:26:52.370 align:middle line:90% possible for equilibrium. 00:26:52.370 --> 00:26:58.520 align:middle line:84% Whereas, below here, there is a gap. 00:26:58.520 --> 00:27:04.520 align:middle line:90% So this area here is forbidden. 00:27:04.520 --> 00:27:06.490 align:middle line:84% And it's called the miscibility gap. 00:27:06.490 --> 00:27:09.440 align:middle line:90% 00:27:09.440 --> 00:27:13.285 align:middle line:84% There are no stable situations inside here. 00:27:13.285 --> 00:27:17.670 align:middle line:90% 00:27:17.670 --> 00:27:19.930 align:middle line:84% But as we have seen for the pure substance, 00:27:19.930 --> 00:27:23.130 align:middle line:84% we may have metastable situations, which 00:27:23.130 --> 00:27:28.206 align:middle line:90% is now what we want to analyze. 00:27:28.206 --> 00:27:30.960 align:middle line:90% 00:27:30.960 --> 00:27:35.750 align:middle line:84% So let's try to do that construction with one 00:27:35.750 --> 00:27:39.410 align:middle line:84% of the models that we have assumed, 00:27:39.410 --> 00:27:43.580 align:middle line:84% the simplest, the one-parameter Margules model 00:27:43.580 --> 00:27:47.090 align:middle line:84% for the binary mixture, so just parameter A. 00:27:47.090 --> 00:27:53.960 align:middle line:84% So the Gibbs free energy is the sum of an ideal part 00:27:53.960 --> 00:27:56.010 align:middle line:90% and an excess part. 00:27:56.010 --> 00:28:01.340 align:middle line:90% The ideal part is given here. 00:28:01.340 --> 00:28:04.760 align:middle line:84% Something-- it can be computed from the pure substance 00:28:04.760 --> 00:28:05.790 align:middle line:90% properties-- 00:28:05.790 --> 00:28:11.000 align:middle line:84% something is related to the entropy of mixing. 00:28:11.000 --> 00:28:16.700 align:middle line:84% And then here, the excess part is related to the composition 00:28:16.700 --> 00:28:24.710 align:middle line:84% and to this constant A, which you remember is essentially 00:28:24.710 --> 00:28:31.250 align:middle line:84% the mixing enthalpy of the two components, which in this model 00:28:31.250 --> 00:28:36.600 align:middle line:84% are equal, for the two components. 00:28:36.600 --> 00:28:38.810 align:middle line:90% So that's the model. 00:28:38.810 --> 00:28:47.650 align:middle line:84% Now we apply the stability conditions 00:28:47.650 --> 00:28:50.180 align:middle line:84% because you remember that for mutual-- 00:28:50.180 --> 00:28:55.870 align:middle line:84% for stable equilibrium, we derived a while ago 00:28:55.870 --> 00:29:00.160 align:middle line:84% general conditions for stability that are not just-- 00:29:00.160 --> 00:29:03.260 align:middle line:84% yes, they are theoretical things, 00:29:03.260 --> 00:29:09.400 align:middle line:84% but which have practical consequences and that, 00:29:09.400 --> 00:29:14.040 align:middle line:84% therefore, allow to describe some physical reality. 00:29:14.040 --> 00:29:16.930 align:middle line:90% 00:29:16.930 --> 00:29:23.410 align:middle line:84% So the second differential of g, if I move out 00:29:23.410 --> 00:29:26.170 align:middle line:84% from a state in which the system is 00:29:26.170 --> 00:29:29.560 align:middle line:84% in mutual equilibrium with a reservoir at temperature T 00:29:29.560 --> 00:29:36.090 align:middle line:84% and p, and I move out by changing the composition 00:29:36.090 --> 00:29:43.070 align:middle line:84% while keeping T and p constant, this differential 00:29:43.070 --> 00:29:44.280 align:middle line:90% must be greater than 0. 00:29:44.280 --> 00:29:50.140 align:middle line:84% So that implies that if I keep, also, 00:29:50.140 --> 00:29:54.410 align:middle line:84% n among the various possibilities, I can-- 00:29:54.410 --> 00:29:57.410 align:middle line:84% of changing the composition, I'm taking the simplest. 00:29:57.410 --> 00:30:01.390 align:middle line:84% So I take--I keep constant the total amount and just change 00:30:01.390 --> 00:30:03.740 align:middle line:90% the relative composition. 00:30:03.740 --> 00:30:08.060 align:middle line:84% So I just changed the mole fraction of component 1. 00:30:08.060 --> 00:30:14.230 align:middle line:84% And this second differential gives 00:30:14.230 --> 00:30:17.420 align:middle line:84% rise-- well, maybe here, more than an equal sign, 00:30:17.420 --> 00:30:21.550 align:middle line:84% there should be an arrow because this is not an equality. 00:30:21.550 --> 00:30:25.960 align:middle line:84% It's wrong mathematically, but because here we 00:30:25.960 --> 00:30:35.060 align:middle line:84% should have also dy1 squared, which being squared 00:30:35.060 --> 00:30:38.100 align:middle line:90% has a positive sign. 00:30:38.100 --> 00:30:42.150 align:middle line:84% So it doesn't-- it is always positive. 00:30:42.150 --> 00:30:43.270 align:middle line:90% So I have to correct that. 00:30:43.270 --> 00:30:49.660 align:middle line:84% And so this is the second derivative. 00:30:49.660 --> 00:30:52.260 align:middle line:84% And on this graph, the second derivative 00:30:52.260 --> 00:30:57.490 align:middle line:90% is the convexity of the curve. 00:30:57.490 --> 00:30:59.500 align:middle line:90% It has to be positive. 00:30:59.500 --> 00:31:03.400 align:middle line:84% So we see here that it's positive here, it's positive 00:31:03.400 --> 00:31:06.220 align:middle line:84% here, it's positive up to a certain point here, 00:31:06.220 --> 00:31:09.250 align:middle line:84% but then it becomes negative, the second derivative. 00:31:09.250 --> 00:31:13.810 align:middle line:84% So from where it becomes negative to where 00:31:13.810 --> 00:31:20.240 align:middle line:84% it comes back positive, this is the region of instability. 00:31:20.240 --> 00:31:27.080 align:middle line:84% And these two points here are going to define-- 00:31:27.080 --> 00:31:32.930 align:middle line:84% see, they're not-- they are at different mole 00:31:32.930 --> 00:31:38.970 align:middle line:84% fractions than the two phases in mutual equilibrium. 00:31:38.970 --> 00:31:43.160 align:middle line:84% So see, this is a different mole fraction and a different mole 00:31:43.160 --> 00:31:43.770 align:middle line:90% fraction. 00:31:43.770 --> 00:31:50.460 align:middle line:84% So this will identify another curve inside here. 00:31:50.460 --> 00:31:55.120 align:middle line:84% So the locus of all these other points, these other curve, 00:31:55.120 --> 00:32:00.330 align:middle line:84% is what we will call the spinodal curve. 00:32:00.330 --> 00:32:03.030 align:middle line:90% So usually, one calls-- 00:32:03.030 --> 00:32:04.603 align:middle line:90% this one is called binodal. 00:32:04.603 --> 00:32:07.200 align:middle line:90% 00:32:07.200 --> 00:32:09.780 align:middle line:84% And the one inside, that's the spinodal. 00:32:09.780 --> 00:32:20.577 align:middle line:90% 00:32:20.577 --> 00:32:23.960 align:middle line:84% So we have the expression for the Gibbs free energy. 00:32:23.960 --> 00:32:25.630 align:middle line:90% We can take derivatives. 00:32:25.630 --> 00:32:30.440 align:middle line:84% Let me skip the steps and go to the result here. 00:32:30.440 --> 00:32:32.590 align:middle line:90% This is the second derivative. 00:32:32.590 --> 00:32:35.200 align:middle line:84% It's in terms of the mole fractions, RT, 00:32:35.200 --> 00:32:40.910 align:middle line:84% and A, the constant A. For this to be greater than 0-- 00:32:40.910 --> 00:32:44.290 align:middle line:90% 00:32:44.290 --> 00:32:45.800 align:middle line:90% and see, this subtracts. 00:32:45.800 --> 00:32:51.990 align:middle line:84% So you need that A should not be too large. 00:32:51.990 --> 00:32:54.840 align:middle line:84% The maximum value of this fraction is 1-- 00:32:54.840 --> 00:32:57.670 align:middle line:90% is 2. 00:32:57.670 --> 00:33:01.390 align:middle line:84% No, it's 4 because half and half. 00:33:01.390 --> 00:33:09.140 align:middle line:84% And therefore, A equals 2 is the maximum allowed value. 00:33:09.140 --> 00:33:15.680 align:middle line:84% So the stability condition is satisfied if A is less than 2. 00:33:15.680 --> 00:33:19.645 align:middle line:84% The critical situation is when A is equal to 2. 00:33:19.645 --> 00:33:22.180 align:middle line:90% 00:33:22.180 --> 00:33:35.420 align:middle line:84% And if A is greater than 2, you can still 00:33:35.420 --> 00:33:44.050 align:middle line:84% satisfy this condition, but only in a certain range. 00:33:44.050 --> 00:33:45.910 align:middle line:90% See, this is the expression. 00:33:45.910 --> 00:33:48.950 align:middle line:84% It's like a second order equation. 00:33:48.950 --> 00:33:53.270 align:middle line:84% So you can solve it, and you find this range. 00:33:53.270 --> 00:33:59.680 align:middle line:84% So this is this mole fraction and that 00:33:59.680 --> 00:34:04.952 align:middle line:84% is this other mole fraction as a function of parameter A. 00:34:04.952 --> 00:34:09.110 align:middle line:84% So this is one part of the condition for-- 00:34:09.110 --> 00:34:13.620 align:middle line:84% it's a necessary condition for stability. 00:34:13.620 --> 00:34:19.120 align:middle line:84% But the other condition is that the two phases are 00:34:19.120 --> 00:34:20.770 align:middle line:90% in mutual stable equilibrium. 00:34:20.770 --> 00:34:23.489 align:middle line:84% And therefore, you also have the conditions that the chemical 00:34:23.489 --> 00:34:26.500 align:middle line:90% potentials must be equal. 00:34:26.500 --> 00:34:30.550 align:middle line:84% And these two conditions essentially 00:34:30.550 --> 00:34:34.960 align:middle line:84% correspond to this construction of the tangent line 00:34:34.960 --> 00:34:37.070 align:middle line:90% that we just described. 00:34:37.070 --> 00:34:39.980 align:middle line:90% 00:34:39.980 --> 00:34:46.400 align:middle line:84% So these equations here, if you fix an A, 00:34:46.400 --> 00:34:49.820 align:middle line:84% you find the mole fractions of the phases 00:34:49.820 --> 00:34:51.840 align:middle line:90% in mutual equilibrium. 00:34:51.840 --> 00:34:54.590 align:middle line:90% 00:34:54.590 --> 00:34:57.480 align:middle line:84% This is easy enough that you can implement it, 00:34:57.480 --> 00:35:01.080 align:middle line:90% for example, in an Excel file. 00:35:01.080 --> 00:35:08.850 align:middle line:84% So you see the Gibbs free energy, little g molar, 00:35:08.850 --> 00:35:11.440 align:middle line:90% is given by this relation. 00:35:11.440 --> 00:35:13.540 align:middle line:84% It's the sum of the two contributions, 00:35:13.540 --> 00:35:24.080 align:middle line:84% the ideal, which is this, and the excess, which is that. 00:35:24.080 --> 00:35:30.130 align:middle line:84% In this case, for A less than 2 at all compositions, 00:35:30.130 --> 00:35:34.560 align:middle line:84% the second derivative of g is positive and we have stability. 00:35:34.560 --> 00:35:40.360 align:middle line:84% So in general, this A depends on temperature. 00:35:40.360 --> 00:35:46.000 align:middle line:84% So at these temperatures, the two liquids 00:35:46.000 --> 00:35:47.040 align:middle line:90% are completely miscible. 00:35:47.040 --> 00:35:49.700 align:middle line:90% 00:35:49.700 --> 00:35:54.830 align:middle line:84% As you decrease the temperature, the value of A changes, 00:35:54.830 --> 00:36:00.950 align:middle line:84% and eventually you reach the critical value, A equals 2. 00:36:00.950 --> 00:36:06.070 align:middle line:84% At that moment, you still are OK in terms of stability. 00:36:06.070 --> 00:36:13.650 align:middle line:84% But at the minimum here, you have a second order derivative 00:36:13.650 --> 00:36:14.435 align:middle line:90% equal to 0. 00:36:14.435 --> 00:36:18.110 align:middle line:90% 00:36:18.110 --> 00:36:19.640 align:middle line:90% Not a minimum at this-- 00:36:19.640 --> 00:36:23.510 align:middle line:84% at some point, you get a second order, second derivative, 00:36:23.510 --> 00:36:24.420 align:middle line:90% equal to 0. 00:36:24.420 --> 00:36:27.800 align:middle line:90% 00:36:27.800 --> 00:36:30.290 align:middle line:84% Of course, for these plots, we had 00:36:30.290 --> 00:36:32.780 align:middle line:84% to assume something about the properties 00:36:32.780 --> 00:36:34.380 align:middle line:90% of the pure substances. 00:36:34.380 --> 00:36:39.650 align:middle line:84% So I assumed these values, 0.2 for one substance 00:36:39.650 --> 00:36:45.120 align:middle line:90% and 0.4 for the other. 00:36:45.120 --> 00:36:54.290 align:middle line:84% If you set a lower temperature, so you go to A greater than 2, 00:36:54.290 --> 00:36:59.690 align:middle line:84% then you get partial miscibility. 00:36:59.690 --> 00:37:06.770 align:middle line:90% You get this miscibility gap. 00:37:06.770 --> 00:37:11.890 align:middle line:84% So from the spinodal composition here to the spinodal composition 00:37:11.890 --> 00:37:14.110 align:middle line:84% there, for the liquid and the vapor, 00:37:14.110 --> 00:37:18.160 align:middle line:84% there is this entire range of compositions 00:37:18.160 --> 00:37:24.420 align:middle line:84% in which the mixture doesn't want to stay. 00:37:24.420 --> 00:37:25.350 align:middle line:90% So if you find-- 00:37:25.350 --> 00:37:30.510 align:middle line:84% if you manage to produce a mixture with this composition 00:37:30.510 --> 00:37:33.160 align:middle line:84% at these conditions of temperature and pressure, 00:37:33.160 --> 00:37:38.220 align:middle line:84% it will separate spontaneously in search for stability. 00:37:38.220 --> 00:37:44.450 align:middle line:90% And it will form two phases. 00:37:44.450 --> 00:37:52.030 align:middle line:84% It will form a phase which is rich in component 1, 00:37:52.030 --> 00:37:57.700 align:middle line:84% this one, and a phase which is lean in component 1, which 00:37:57.700 --> 00:37:59.080 align:middle line:90% is this. 00:37:59.080 --> 00:38:04.750 align:middle line:84% And eventually, it will reach mutual equilibrium-- 00:38:04.750 --> 00:38:09.520 align:middle line:84% it will phase separate to two phases that 00:38:09.520 --> 00:38:12.260 align:middle line:84% are in mutual equilibrium at these compositions. 00:38:12.260 --> 00:38:16.260 align:middle line:90% 00:38:16.260 --> 00:38:19.020 align:middle line:84% These equations are, again, that constructions 00:38:19.020 --> 00:38:20.170 align:middle line:90% that we have already done. 00:38:20.170 --> 00:38:23.160 align:middle line:90% 00:38:23.160 --> 00:38:30.560 align:middle line:84% So if you-- by some means, for example, one possibility 00:38:30.560 --> 00:38:36.830 align:middle line:84% is to start from a mixture at a temperature above critical, 00:38:36.830 --> 00:38:39.080 align:middle line:90% say here. 00:38:39.080 --> 00:38:43.790 align:middle line:84% And then you find a way to quench it, 00:38:43.790 --> 00:38:48.770 align:middle line:84% much like what we discussed also for the van der Waals 00:38:48.770 --> 00:38:50.910 align:middle line:90% spinodal decomposition. 00:38:50.910 --> 00:38:54.310 align:middle line:84% So you quench to this point here. 00:38:54.310 --> 00:38:59.010 align:middle line:90% 00:38:59.010 --> 00:39:00.930 align:middle line:90% So here is the temperature. 00:39:00.930 --> 00:39:09.890 align:middle line:84% At this point, the instability triggers what you see here. 00:39:09.890 --> 00:39:16.450 align:middle line:84% It triggers the fact that from homogeneous, the phases 00:39:16.450 --> 00:39:21.580 align:middle line:84% begin to separate because substance 1 either 00:39:21.580 --> 00:39:27.600 align:middle line:84% wants to be close to a lot of substance 1 00:39:27.600 --> 00:39:31.090 align:middle line:90% or a little of substance 1. 00:39:31.090 --> 00:39:37.670 align:middle line:90% So you get diffusion process. 00:39:37.670 --> 00:39:45.800 align:middle line:84% And this is fast because, as we have seen for the van der Waals 00:39:45.800 --> 00:39:49.670 align:middle line:84% case, we have a lot of free energy 00:39:49.670 --> 00:39:58.040 align:middle line:84% available to create the surface that separates the two phases. 00:39:58.040 --> 00:40:04.503 align:middle line:84% And then once that happens, coalescence makes-- 00:40:04.503 --> 00:40:07.800 align:middle line:90% 00:40:07.800 --> 00:40:12.540 align:middle line:84% uses up that surface energy and tries 00:40:12.540 --> 00:40:18.720 align:middle line:84% to find configurations of lesser free energy 00:40:18.720 --> 00:40:28.260 align:middle line:84% by making drops or bubbles, drops, in this case, 00:40:28.260 --> 00:40:31.640 align:middle line:90% bigger and bigger. 00:40:31.640 --> 00:40:37.160 align:middle line:84% And also here, if you do the separation, 00:40:37.160 --> 00:40:41.300 align:middle line:84% if you do this quenching at exactly 00:40:41.300 --> 00:40:49.920 align:middle line:84% the critical composition, then the mixture, the separation, 00:40:49.920 --> 00:40:55.000 align:middle line:90% as you see, doesn't either-- 00:40:55.000 --> 00:40:59.290 align:middle line:90% neither do drops nor-- 00:40:59.290 --> 00:41:03.130 align:middle line:84% doesn't do-- yeah, in the van der Waals, 00:41:03.130 --> 00:41:06.370 align:middle line:84% it's neither drops nor bubbles here. 00:41:06.370 --> 00:41:11.900 align:middle line:90% It's drops in all cases. 00:41:11.900 --> 00:41:13.630 align:middle line:84% But you see, instead of drops, it's 00:41:13.630 --> 00:41:25.710 align:middle line:84% continuous regions that became gross and grosser. 00:41:25.710 --> 00:41:31.830 align:middle line:84% If you instead are off critical on either side, 00:41:31.830 --> 00:41:39.695 align:middle line:84% you get drops of one into a continuous region of the other. 00:41:39.695 --> 00:41:45.096 align:middle line:90% 00:41:45.096 --> 00:41:49.980 align:middle line:84% Now, there is also this region here 00:41:49.980 --> 00:41:55.710 align:middle line:84% of metastable states between the spinodal and the binodal curve. 00:41:55.710 --> 00:41:59.050 align:middle line:90% There is metastability. 00:41:59.050 --> 00:42:06.010 align:middle line:84% And these metastable states are interesting because sometimes 00:42:06.010 --> 00:42:10.310 align:middle line:84% they can be used for applications. 00:42:10.310 --> 00:42:13.605 align:middle line:90% For example, this is-- 00:42:13.605 --> 00:42:16.400 align:middle line:90% 00:42:16.400 --> 00:42:23.980 align:middle line:84% I know that one of you is Greek, and so certainly knows 00:42:23.980 --> 00:42:26.140 align:middle line:90% this ouzo effect. 00:42:26.140 --> 00:42:33.180 align:middle line:84% But in the Mediterranean, there is these liqueurs 00:42:33.180 --> 00:42:36.520 align:middle line:90% that have very different names. 00:42:36.520 --> 00:42:37.860 align:middle line:90% So let's see. 00:42:37.860 --> 00:42:40.320 align:middle line:84% I wrote them down because I forget. 00:42:40.320 --> 00:42:43.740 align:middle line:84% In France, you get Anisette or Pastis. 00:42:43.740 --> 00:42:46.130 align:middle line:90% And in Italy, Sambuca. 00:42:46.130 --> 00:42:48.390 align:middle line:90% That one I know. 00:42:48.390 --> 00:42:51.150 align:middle line:90% Turkey is Raki. 00:42:51.150 --> 00:42:55.440 align:middle line:84% In Colombia, there is some kinds of aguardiente, 00:42:55.440 --> 00:43:02.490 align:middle line:84% which is flavored with anisette, with anise. 00:43:02.490 --> 00:43:04.150 align:middle line:90% Mexico, also, has its own. 00:43:04.150 --> 00:43:14.820 align:middle line:84% And also, some of the Egypt and Israel have Arak. 00:43:14.820 --> 00:43:16.685 align:middle line:90% So essentially, this is-- 00:43:16.685 --> 00:43:21.030 align:middle line:90% 00:43:21.030 --> 00:43:28.046 align:middle line:84% the funny thing of these liquors is that they are transparent. 00:43:28.046 --> 00:43:31.790 align:middle line:90% 00:43:31.790 --> 00:43:33.980 align:middle line:84% And when you mix it with water, which 00:43:33.980 --> 00:43:38.560 align:middle line:84% is another transparent liquid, they become-- 00:43:38.560 --> 00:43:45.660 align:middle line:84% they give you a milky kind of opaque thing. 00:43:45.660 --> 00:43:51.720 align:middle line:84% So that's always sort of a mysterious effect 00:43:51.720 --> 00:43:53.345 align:middle line:90% if you see the first time. 00:43:53.345 --> 00:43:56.500 align:middle line:90% 00:43:56.500 --> 00:43:59.170 align:middle line:84% It turns out that this effect, it 00:43:59.170 --> 00:44:02.780 align:middle line:84% has been named the ouzo effect only relatively recently, 00:44:02.780 --> 00:44:06.870 align:middle line:84% maybe 20 years ago, when somebody-- 00:44:06.870 --> 00:44:15.750 align:middle line:84% some real scientist visited the Mediterranean, but-- 00:44:15.750 --> 00:44:19.470 align:middle line:84% and it is not spinodal decomposition. 00:44:19.470 --> 00:44:25.980 align:middle line:84% It is just the fact that you start, you generate a mixture-- 00:44:25.980 --> 00:44:27.990 align:middle line:90% this has to do-- 00:44:27.990 --> 00:44:30.940 align:middle line:84% this is more complicated because it's not a binary mixture. 00:44:30.940 --> 00:44:33.280 align:middle line:90% It is a three-component mixture. 00:44:33.280 --> 00:44:40.490 align:middle line:84% It has water, it has alcohol, which is ethanol, 00:44:40.490 --> 00:44:48.800 align:middle line:84% and it has a little amount of this oily thing that you extract 00:44:48.800 --> 00:44:58.070 align:middle line:84% from the seeds of a plant, of a flower that is called anisette 00:44:58.070 --> 00:45:00.840 align:middle line:90% or something like that. 00:45:00.840 --> 00:45:02.260 align:middle line:84% No, I don't have the name of that. 00:45:02.260 --> 00:45:05.940 align:middle line:90% 00:45:05.940 --> 00:45:13.230 align:middle line:84% I think the chemical composition of this oily thing that 00:45:13.230 --> 00:45:17.530 align:middle line:84% gives the flavor is trans-anethole, 00:45:17.530 --> 00:45:19.070 align:middle line:90% and it's C10H12O. 00:45:19.070 --> 00:45:21.660 align:middle line:90% 00:45:21.660 --> 00:45:26.370 align:middle line:84% You find all these in this nice article that 00:45:26.370 --> 00:45:30.120 align:middle line:84% give the scientific explanation, and from which I 00:45:30.120 --> 00:45:32.000 align:middle line:90% have extracted these pictures. 00:45:32.000 --> 00:45:34.600 align:middle line:90% 00:45:34.600 --> 00:45:42.460 align:middle line:84% So the idea here is that you have three substances. 00:45:42.460 --> 00:45:45.190 align:middle line:90% So let me just see. 00:45:45.190 --> 00:46:00.430 align:middle line:84% They call it DVB in that graph, that plays the role of the anise 00:46:00.430 --> 00:46:03.100 align:middle line:90% oil, but I don't think-- 00:46:03.100 --> 00:46:06.910 align:middle line:84% they prefer to do it with a polymer that has-- 00:46:06.910 --> 00:46:12.680 align:middle line:84% polymeric oil that-- so they use divinylbenzene. 00:46:12.680 --> 00:46:22.910 align:middle line:84% So this is divinylbenzene, at least in their experiment. 00:46:22.910 --> 00:46:30.670 align:middle line:84% So let's say this is substance C. Now, it 00:46:30.670 --> 00:46:38.480 align:middle line:84% turns out that this substance is soluble in alcohol. 00:46:38.480 --> 00:46:41.370 align:middle line:90% 00:46:41.370 --> 00:46:44.335 align:middle line:90% It's not soluble in water. 00:46:44.335 --> 00:46:53.400 align:middle line:90% 00:46:53.400 --> 00:47:01.396 align:middle line:84% So let's call water B and alcohol A. 00:47:01.396 --> 00:47:08.810 align:middle line:84% So essentially, here we have the mole fraction 00:47:08.810 --> 00:47:11.470 align:middle line:90% of the organic oil. 00:47:11.470 --> 00:47:13.750 align:middle line:90% That's our C. 00:47:13.750 --> 00:47:20.380 align:middle line:84% And this is the mole fraction of the ethanol, which 00:47:20.380 --> 00:47:29.690 align:middle line:84% is our A. This is how the equivalent of this diagram 00:47:29.690 --> 00:47:34.670 align:middle line:84% looks at a fixed temperature as a function 00:47:34.670 --> 00:47:35.960 align:middle line:90% of these compositions. 00:47:35.960 --> 00:47:39.980 align:middle line:90% 00:47:39.980 --> 00:47:41.810 align:middle line:84% Of course, the mole fraction of water 00:47:41.810 --> 00:47:50.020 align:middle line:84% is 1 minus the sum of these two, hence the triangular line here. 00:47:50.020 --> 00:47:53.385 align:middle line:84% So up here, we got just pure water. 00:47:53.385 --> 00:47:57.145 align:middle line:90% 00:47:57.145 --> 00:48:00.440 align:middle line:84% If you plot this, it is very skewed. 00:48:00.440 --> 00:48:02.870 align:middle line:84% If you plot it in a logarithmic scale, 00:48:02.870 --> 00:48:06.640 align:middle line:84% in the composition of the organic oil, 00:48:06.640 --> 00:48:08.680 align:middle line:90% this is how it looks. 00:48:08.680 --> 00:48:09.870 align:middle line:90% It opens up. 00:48:09.870 --> 00:48:13.065 align:middle line:90% 00:48:13.065 --> 00:48:17.910 align:middle line:90% So what you typically have is-- 00:48:17.910 --> 00:48:28.590 align:middle line:84% ouzo is 40% by volume, meaning about 35% or 34% by weight. 00:48:28.590 --> 00:48:30.960 align:middle line:90% And here is by weight. 00:48:30.960 --> 00:48:34.260 align:middle line:90% So this is the alcohol content. 00:48:34.260 --> 00:48:44.700 align:middle line:84% And let's say the content of this oily anisette organic oil 00:48:44.700 --> 00:48:48.330 align:middle line:84% extracted from these flowers, from these seeds, 00:48:48.330 --> 00:48:51.690 align:middle line:90% suppose it's this much. 00:48:51.690 --> 00:48:56.950 align:middle line:84% What happens is when you add water, you dilute the solution. 00:48:56.950 --> 00:49:04.320 align:middle line:84% And so the fraction of ethanol goes down from 35, say, to-- 00:49:04.320 --> 00:49:10.670 align:middle line:84% you cross this line, and you get into the metastable region. 00:49:10.670 --> 00:49:20.060 align:middle line:84% In that metastable region, you get an immediate separation 00:49:20.060 --> 00:49:27.630 align:middle line:84% or formation of these minute droplets, which then, 00:49:27.630 --> 00:49:34.290 align:middle line:84% because you-- so you release the excess Gibbs free energy that 00:49:34.290 --> 00:49:37.590 align:middle line:90% forms those minute droplets. 00:49:37.590 --> 00:49:45.710 align:middle line:84% Those droplets are the reason why, then, light 00:49:45.710 --> 00:49:52.500 align:middle line:84% does not get through, gets diffused by these droplets. 00:49:52.500 --> 00:49:59.030 align:middle line:84% So that's why the liquid loses the transparency to light. 00:49:59.030 --> 00:50:02.360 align:middle line:90% The reasons are these droplets. 00:50:02.360 --> 00:50:05.310 align:middle line:84% But differently from spinodal decomposition, 00:50:05.310 --> 00:50:10.220 align:middle line:84% if you had generated this kind of situation 00:50:10.220 --> 00:50:13.220 align:middle line:84% from the unstable state, the separation 00:50:13.220 --> 00:50:18.650 align:middle line:84% would have been much more vigorous and it would generate-- 00:50:18.650 --> 00:50:25.290 align:middle line:84% have more energy, so generate bigger drops in the end. 00:50:25.290 --> 00:50:29.760 align:middle line:84% And eventually, the mixture will separate. 00:50:29.760 --> 00:50:33.320 align:middle line:84% Or maybe, in any case, it will form bigger drops. 00:50:33.320 --> 00:50:37.820 align:middle line:84% Whereas, in the ouzo effect, you generate-- you 00:50:37.820 --> 00:50:40.680 align:middle line:90% produce very small droplets. 00:50:40.680 --> 00:50:44.900 align:middle line:84% They remain in sort of an emulsion that 00:50:44.900 --> 00:50:48.030 align:middle line:90% is relatively stable. 00:50:48.030 --> 00:50:51.325 align:middle line:84% In other words, it survives hours. 00:50:51.325 --> 00:50:54.970 align:middle line:90% 00:50:54.970 --> 00:51:02.090 align:middle line:84% And in some cases, this kind of process 00:51:02.090 --> 00:51:06.080 align:middle line:84% is wanted because there are applications in which you want 00:51:06.080 --> 00:51:11.710 align:middle line:84% to generate emulsions, which is precisely 00:51:11.710 --> 00:51:19.520 align:middle line:84% a mixture of a solvent with a lot of small droplets in it. 00:51:19.520 --> 00:51:22.800 align:middle line:84% Like when you do vinaigrette, that's an emulsion. 00:51:22.800 --> 00:51:30.330 align:middle line:84% And many times, in order to produce emulsions, 00:51:30.330 --> 00:51:32.250 align:middle line:90% you need surfactants. 00:51:32.250 --> 00:51:40.100 align:middle line:84% But this is instead a way to produce emulsions 00:51:40.100 --> 00:51:43.640 align:middle line:90% without surfactants. 00:51:43.640 --> 00:51:48.680 align:middle line:84% And chemical industry is full of processes 00:51:48.680 --> 00:51:55.910 align:middle line:84% that go back and forth with these kind of effects 00:51:55.910 --> 00:52:03.140 align:middle line:84% in order to separate or mix and generate things. 00:52:03.140 --> 00:52:10.320 align:middle line:84% Also, I think that this method has also 00:52:10.320 --> 00:52:15.390 align:middle line:84% been used in order to generate scaffolds 00:52:15.390 --> 00:52:22.215 align:middle line:84% for biological applications, for example, because if you-- 00:52:22.215 --> 00:52:27.715 align:middle line:84% once you have this emulsion, you freeze it in such a way that-- 00:52:27.715 --> 00:52:31.000 align:middle line:90% 00:52:31.000 --> 00:52:34.870 align:middle line:84% if you freeze it in such a way that the bulk phase remains 00:52:34.870 --> 00:52:40.250 align:middle line:84% liquid and the droplets become solid, 00:52:40.250 --> 00:52:43.480 align:middle line:84% then you've generated small spheres. 00:52:43.480 --> 00:52:49.540 align:middle line:84% So in this-- this is a way in which you produce small spheres. 00:52:49.540 --> 00:52:53.200 align:middle line:90% Or conversely, if you-- 00:52:53.200 --> 00:52:56.360 align:middle line:84% perhaps not these ones, because they are not connected. 00:52:56.360 --> 00:53:03.010 align:middle line:84% And so if you freeze this and the liquid droplets 00:53:03.010 --> 00:53:07.720 align:middle line:84% remain liquid, you still have droplets and you are not-- 00:53:07.720 --> 00:53:10.340 align:middle line:84% there is no path for the liquid to go away. 00:53:10.340 --> 00:53:13.720 align:middle line:84% But if you use a situation like this one, 00:53:13.720 --> 00:53:17.170 align:middle line:84% generated by a spinodal decomposition, and you 00:53:17.170 --> 00:53:21.310 align:middle line:90% freeze, then you-- 00:53:21.310 --> 00:53:25.810 align:middle line:84% because there is a path in which, say, the dark phase 00:53:25.810 --> 00:53:31.730 align:middle line:84% becomes solid and the liquid in the white phase can drain away. 00:53:31.730 --> 00:53:37.960 align:middle line:84% And then you produce a porous material, which can be used-- 00:53:37.960 --> 00:53:41.320 align:middle line:84% a high surface material that can be 00:53:41.320 --> 00:53:46.490 align:middle line:84% used for enhancing or studying biological, 00:53:46.490 --> 00:53:50.090 align:middle line:84% but also for other applications in which you 00:53:50.090 --> 00:53:55.806 align:middle line:84% need a lot of surface for things to happen, like catalysts. 00:53:55.806 --> 00:53:59.120 align:middle line:90% 00:53:59.120 --> 00:54:10.640 align:middle line:84% All right, so with this, I think we can terminate the mixture 00:54:10.640 --> 00:54:14.880 align:middle line:90% stuff, at least of that kind. 00:54:14.880 --> 00:54:21.840 align:middle line:84% But we start with mixtures that are subject also 00:54:21.840 --> 00:54:23.680 align:middle line:90% to chemical reactions. 00:54:23.680 --> 00:54:28.040 align:middle line:90% 00:54:28.040 --> 00:54:33.470 align:middle line:84% So this is the two main situations 00:54:33.470 --> 00:54:41.310 align:middle line:84% that we will approach at the beginning here. 00:54:41.310 --> 00:54:45.120 align:middle line:84% It's two kinds of systems in which we 00:54:45.120 --> 00:54:46.980 align:middle line:90% have chemical reactions. 00:54:46.980 --> 00:54:52.835 align:middle line:84% One is a closed system and the other is an open system. 00:54:52.835 --> 00:54:57.420 align:middle line:90% 00:54:57.420 --> 00:55:00.310 align:middle line:84% So this is a model for a batch process. 00:55:00.310 --> 00:55:06.700 align:middle line:84% And this is a model for a steady state flow process, or flow 00:55:06.700 --> 00:55:07.200 align:middle line:90% reactor. 00:55:07.200 --> 00:55:10.270 align:middle line:90% 00:55:10.270 --> 00:55:18.280 align:middle line:84% In the closed system situation, you have a box. 00:55:18.280 --> 00:55:23.620 align:middle line:84% The application that I like to have in mind 00:55:23.620 --> 00:55:25.940 align:middle line:84% is the internal combustion engine. 00:55:25.940 --> 00:55:30.310 align:middle line:84% So you imagine that this wall is the piston 00:55:30.310 --> 00:55:35.780 align:middle line:84% of the cylinder of your internal combustion engine. 00:55:35.780 --> 00:55:38.710 align:middle line:84% And so this wall goes up and down. 00:55:38.710 --> 00:55:44.720 align:middle line:84% And at the top dead center, it will be at a certain volume. 00:55:44.720 --> 00:55:49.784 align:middle line:84% And suppose the piston stops there for a while. 00:55:49.784 --> 00:55:56.380 align:middle line:84% So at that moment, you have a mixture inside your combustion 00:55:56.380 --> 00:56:06.190 align:middle line:84% chamber, which is a mixture of air and your fuel, 00:56:06.190 --> 00:56:10.930 align:middle line:84% and some water vapor sometimes because air is humid. 00:56:10.930 --> 00:56:13.240 align:middle line:84% And so there is a water vapor content 00:56:13.240 --> 00:56:17.130 align:middle line:84% also at the beginning of the combustion process. 00:56:17.130 --> 00:56:20.100 align:middle line:84% It has some temperature, some pressure, 00:56:20.100 --> 00:56:23.520 align:middle line:84% because the piston has compressed that mixture. 00:56:23.520 --> 00:56:26.830 align:middle line:84% Of course, since we are in a stable equilibrium state, 00:56:26.830 --> 00:56:29.790 align:middle line:90% it also has chemical potentials. 00:56:29.790 --> 00:56:32.760 align:middle line:84% And if we want to compute the energy and the entropy, 00:56:32.760 --> 00:56:39.090 align:middle line:84% because they will enter in the energy and entropy balances, 00:56:39.090 --> 00:56:42.660 align:middle line:84% we know how to do it because we've dealt with mixtures 00:56:42.660 --> 00:56:43.700 align:middle line:90% for a long time now. 00:56:43.700 --> 00:56:47.140 align:middle line:90% 00:56:47.140 --> 00:56:50.090 align:middle line:84% So the energy in general is simply given. 00:56:50.090 --> 00:56:55.400 align:middle line:84% Well, simply, is given by the weighted sum weighted 00:56:55.400 --> 00:57:01.800 align:middle line:84% on the amounts of the partial properties. 00:57:01.800 --> 00:57:04.630 align:middle line:84% And the partial properties depend on composition. 00:57:04.630 --> 00:57:07.500 align:middle line:84% And here we can apply all our theories 00:57:07.500 --> 00:57:13.820 align:middle line:84% about ideal mixtures or other models and so on. 00:57:13.820 --> 00:57:16.910 align:middle line:84% And the same holds also for the other properties, 00:57:16.910 --> 00:57:19.148 align:middle line:84% like entropy, which is the other one that is 00:57:19.148 --> 00:57:20.440 align:middle line:90% needed for the entropy balance. 00:57:20.440 --> 00:57:23.991 align:middle line:90% 00:57:23.991 --> 00:57:27.870 align:middle line:84% Now, what's new here is that next, 00:57:27.870 --> 00:57:35.020 align:middle line:84% you sort of turn on chemical reactions. 00:57:35.020 --> 00:57:39.190 align:middle line:84% So the composition is allowed to change 00:57:39.190 --> 00:57:46.500 align:middle line:84% according to one or more chemical reaction mechanisms. 00:57:46.500 --> 00:57:48.420 align:middle line:90% We will describe them. 00:57:48.420 --> 00:57:52.750 align:middle line:84% I'll go into detail of this description in the next slide, 00:57:52.750 --> 00:57:55.655 align:middle line:84% but we will describe it by this general notation. 00:57:55.655 --> 00:57:59.010 align:middle line:90% 00:57:59.010 --> 00:58:04.650 align:middle line:84% And what will happen is that the process, the fact that you turn 00:58:04.650 --> 00:58:06.510 align:middle line:84% on the possibility of a chemical reaction, 00:58:06.510 --> 00:58:14.310 align:middle line:84% makes the state evolve towards equilibrium 00:58:14.310 --> 00:58:15.935 align:middle line:90% and the composition will change. 00:58:15.935 --> 00:58:19.220 align:middle line:90% 00:58:19.220 --> 00:58:21.830 align:middle line:84% So in the end, you will have a composition 00:58:21.830 --> 00:58:23.720 align:middle line:84% that is different from the beginning. 00:58:23.720 --> 00:58:25.970 align:middle line:90% So here you have the reactants. 00:58:25.970 --> 00:58:30.900 align:middle line:84% So R stands for reactants here, not for reservoir. 00:58:30.900 --> 00:58:35.090 align:middle line:90% And P stands for products. 00:58:35.090 --> 00:58:36.870 align:middle line:84% See, these are the products of combustion, 00:58:36.870 --> 00:58:43.460 align:middle line:84% for example, in which case you have air, of which you've 00:58:43.460 --> 00:58:48.800 align:middle line:84% consumed some of the oxygen. And then 00:58:48.800 --> 00:58:57.035 align:middle line:84% you have carbon dioxide, water vapor, and maybe some nitrogen 00:58:57.035 --> 00:58:57.535 align:middle line:90% oxides. 00:58:57.535 --> 00:59:05.140 align:middle line:90% 00:59:05.140 --> 00:59:09.730 align:middle line:84% Because the reaction occurs irreversibly, 00:59:09.730 --> 00:59:11.500 align:middle line:84% spontaneously, there will be some entropy 00:59:11.500 --> 00:59:14.330 align:middle line:90% produced by irreversibility. 00:59:14.330 --> 00:59:17.960 align:middle line:84% You can compute once you have-- or at least in terms 00:59:17.960 --> 00:59:22.390 align:middle line:84% of the final composition and the final temperature and pressure, 00:59:22.390 --> 00:59:25.500 align:middle line:84% you can compute the energy and the entropy the same way. 00:59:25.500 --> 00:59:28.390 align:middle line:90% 00:59:28.390 --> 00:59:31.720 align:middle line:84% Only, of course, relative to the new composition. 00:59:31.720 --> 00:59:37.880 align:middle line:84% Maybe not all the final things are known. 00:59:37.880 --> 00:59:43.610 align:middle line:84% If you can use up only the energy and the entropy balance, 00:59:43.610 --> 00:59:46.020 align:middle line:84% and perhaps also the volume balance, 00:59:46.020 --> 00:59:50.060 align:middle line:84% if you assume that these volume remains constant or relatively 00:59:50.060 --> 00:59:55.010 align:middle line:84% constant because that reaction at the top dead center 00:59:55.010 --> 00:59:59.690 align:middle line:84% occurs very quickly, and so the piston doesn't move a lot, 00:59:59.690 --> 01:00:03.020 align:middle line:84% then you have three equations, energy, entropy, 01:00:03.020 --> 01:00:04.910 align:middle line:84% and volume balance, with which you 01:00:04.910 --> 01:00:06.860 align:middle line:90% can solve for three unknowns. 01:00:06.860 --> 01:00:10.850 align:middle line:84% So the various possible problems that can be solved 01:00:10.850 --> 01:00:15.530 align:middle line:90% are variations of this thing. 01:00:15.530 --> 01:00:21.810 align:middle line:84% And of course, even though that happens quickly, 01:00:21.810 --> 01:00:31.990 align:middle line:84% maybe there is some exchange of heat between your mixture 01:00:31.990 --> 01:00:36.360 align:middle line:84% and the walls of the container, like the cylinder, 01:00:36.360 --> 01:00:39.700 align:middle line:84% because the internal combustion engines are cooled, 01:00:39.700 --> 01:00:41.920 align:middle line:90% water cooled or air cooled. 01:00:41.920 --> 01:00:46.890 align:middle line:84% And so some of the energy can be exchanged with heat 01:00:46.890 --> 01:00:50.410 align:middle line:90% to the wall temperature. 01:00:50.410 --> 01:00:56.200 align:middle line:84% And also, if the piston is moving while this is happening, 01:00:56.200 --> 01:01:04.390 align:middle line:84% there may also be some work due to the pressure times the volume 01:01:04.390 --> 01:01:04.890 align:middle line:90% change. 01:01:04.890 --> 01:01:07.760 align:middle line:90% 01:01:07.760 --> 01:01:11.260 align:middle line:84% So that's the setup of the closed system, 01:01:11.260 --> 01:01:13.300 align:middle line:90% so internal combustion engine. 01:01:13.300 --> 01:01:20.830 align:middle line:84% Here instead is a steady state burner, or reactor. 01:01:20.830 --> 01:01:24.730 align:middle line:84% Again, if I have in mind combustion of hydrocarbons, 01:01:24.730 --> 01:01:27.660 align:middle line:84% this is a steady state burner, which 01:01:27.660 --> 01:01:32.910 align:middle line:90% is typically inside a power-- 01:01:32.910 --> 01:01:37.660 align:middle line:84% combustion fossil fuel power plant. 01:01:37.660 --> 01:01:42.720 align:middle line:84% Here, the idea is that you have a flow, bulk flow. 01:01:42.720 --> 01:01:47.380 align:middle line:84% One, if all the substances are pre-mixed. 01:01:47.380 --> 01:01:52.980 align:middle line:84% Or maybe you have several if you have some of the substances that 01:01:52.980 --> 01:01:55.080 align:middle line:90% are injected, for example, air. 01:01:55.080 --> 01:01:58.260 align:middle line:84% You could have an injection of air 01:01:58.260 --> 01:02:01.890 align:middle line:84% through one stream, so one bulk flow interaction, 01:02:01.890 --> 01:02:07.980 align:middle line:84% and then your fuel injected from another pipe that goes in. 01:02:07.980 --> 01:02:11.170 align:middle line:84% So I indicate here a single arrow, 01:02:11.170 --> 01:02:16.110 align:middle line:84% but of course, you may have many. 01:02:16.110 --> 01:02:20.040 align:middle line:84% The properties for bulk flow interactions-- 01:02:20.040 --> 01:02:24.240 align:middle line:84% what enters in the energy balance, as you remember, 01:02:24.240 --> 01:02:27.040 align:middle line:90% is the enthalpy. 01:02:27.040 --> 01:02:32.110 align:middle line:84% And so we need the enthalpy, the rate of enthalpy, 01:02:32.110 --> 01:02:38.830 align:middle line:84% brought in by this inlet bulk flow interaction, which 01:02:38.830 --> 01:02:43.510 align:middle line:84% has to do with the partial enthalpies 01:02:43.510 --> 01:02:45.850 align:middle line:84% of the components in the mixtures 01:02:45.850 --> 01:02:47.450 align:middle line:90% for those that are mixed. 01:02:47.450 --> 01:02:51.220 align:middle line:84% And that would be the specific if they are 01:02:51.220 --> 01:02:53.360 align:middle line:90% injected as pure components. 01:02:53.360 --> 01:02:56.620 align:middle line:90% 01:02:56.620 --> 01:03:00.050 align:middle line:90% And the same for the outlet. 01:03:00.050 --> 01:03:04.500 align:middle line:84% So the outlet here would be the hot product, for example, 01:03:04.500 --> 01:03:06.006 align:middle line:90% the hot products of combustion. 01:03:06.006 --> 01:03:09.640 align:middle line:90% 01:03:09.640 --> 01:03:12.210 align:middle line:84% But in other configurations, it could be-- 01:03:12.210 --> 01:03:15.070 align:middle line:90% 01:03:15.070 --> 01:03:20.020 align:middle line:84% if this reactor contains not only just-- 01:03:20.020 --> 01:03:23.050 align:middle line:84% it's not adiabatic, but it contains also 01:03:23.050 --> 01:03:28.190 align:middle line:84% the machinery, and the thermal cycle that extracts the energy, 01:03:28.190 --> 01:03:28.760 align:middle line:90% and so on. 01:03:28.760 --> 01:03:32.140 align:middle line:84% This would be just the products that 01:03:32.140 --> 01:03:36.540 align:middle line:84% get exhausted at the chimney or the stack of your power plant. 01:03:36.540 --> 01:03:41.140 align:middle line:90% 01:03:41.140 --> 01:03:45.450 align:middle line:84% All right, and so then, in that case, 01:03:45.450 --> 01:03:48.290 align:middle line:84% we may have heat interactions and work interactions 01:03:48.290 --> 01:03:58.410 align:middle line:84% while this complex power plant is evolving. 01:03:58.410 --> 01:04:02.610 align:middle line:84% There is, of course, entropy produced by irreversibility. 01:04:02.610 --> 01:04:08.510 align:middle line:84% And again, the new aspect that we need to deal with 01:04:08.510 --> 01:04:13.935 align:middle line:84% is the fact that inside here, a chemical reaction or chemical 01:04:13.935 --> 01:04:18.020 align:middle line:84% reactions can occur, changing the composition 01:04:18.020 --> 01:04:23.040 align:middle line:90% from the inlet to the outlet. 01:04:23.040 --> 01:04:26.700 align:middle line:84% The energy balance equation for a steady state 01:04:26.700 --> 01:04:29.460 align:middle line:84% says that the rate of change of the energy 01:04:29.460 --> 01:04:32.920 align:middle line:90% is 0 because it's steady state. 01:04:32.920 --> 01:04:36.370 align:middle line:90% And therefore that must be-- 01:04:36.370 --> 01:04:40.990 align:middle line:84% so the rate at which the energies are exchanged 01:04:40.990 --> 01:04:44.230 align:middle line:90% in and out must balance to 0. 01:04:44.230 --> 01:04:48.370 align:middle line:84% So the enthalpy of the inflow-- this 01:04:48.370 --> 01:04:52.860 align:middle line:84% is positive-- the enthalpy of the outflow. 01:04:52.860 --> 01:04:58.310 align:middle line:90% The heat flow, positive if-- 01:04:58.310 --> 01:05:03.720 align:middle line:84% if it has the arrow inside, it means positive, if received, 01:05:03.720 --> 01:05:06.940 align:middle line:84% minus the work, which has the arrow outward. 01:05:06.940 --> 01:05:12.250 align:middle line:84% And so it's positive if it is produced. 01:05:12.250 --> 01:05:15.615 align:middle line:84% And similarly for the entropy balance. 01:05:15.615 --> 01:05:18.170 align:middle line:90% 01:05:18.170 --> 01:05:20.810 align:middle line:84% The rate of change of the entropy of the reactor, 01:05:20.810 --> 01:05:24.490 align:middle line:84% because we are making a balance equation for this reactor 01:05:24.490 --> 01:05:30.450 align:middle line:84% here, if it is steady state, the properties don't change, 01:05:30.450 --> 01:05:35.240 align:middle line:84% so the rate of entropy change is 0. 01:05:35.240 --> 01:05:41.510 align:middle line:84% And so then you have a balance between the net amount 01:05:41.510 --> 01:05:50.510 align:middle line:84% of entropy that comes in and out and the entropy that 01:05:50.510 --> 01:05:52.650 align:middle line:90% is produced by irreversibility. 01:05:52.650 --> 01:05:57.500 align:middle line:90% 01:05:57.500 --> 01:06:04.600 align:middle line:84% Now, if you look at this ones, if you look at these equations 01:06:04.600 --> 01:06:11.080 align:middle line:84% and compare them, you are often tempted to, for example, 01:06:11.080 --> 01:06:18.000 align:middle line:84% rewrite these ones with HP minus HR on the left-hand side, which 01:06:18.000 --> 01:06:23.630 align:middle line:84% is OK, just a legitimate mathematical step. 01:06:23.630 --> 01:06:28.663 align:middle line:84% However, it should be--you shouldn't be confused 01:06:28.663 --> 01:06:30.080 align:middle line:84% because, otherwise, you say, well, 01:06:30.080 --> 01:06:31.930 align:middle line:90% why do I use enthalpy here? 01:06:31.930 --> 01:06:33.300 align:middle line:90% Why do we use energy there? 01:06:33.300 --> 01:06:37.210 align:middle line:90% 01:06:37.210 --> 01:06:40.250 align:middle line:84% You're not using enthalpy here the same way. 01:06:40.250 --> 01:06:42.700 align:middle line:90% It doesn't play the role of-- 01:06:42.700 --> 01:06:46.760 align:middle line:84% the energy, the rate of energy change, is this 0 here. 01:06:46.760 --> 01:06:48.410 align:middle line:90% That 0 is important. 01:06:48.410 --> 01:06:54.830 align:middle line:84% So if you do that mathematical step, you sort of kill a 0. 01:06:54.830 --> 01:06:58.900 align:middle line:84% That is important and gives you a reminder 01:06:58.900 --> 01:07:02.490 align:middle line:84% that you're talking about a steady state process. 01:07:02.490 --> 01:07:07.360 align:middle line:90% 01:07:07.360 --> 01:07:10.680 align:middle line:90% How about notation? 01:07:10.680 --> 01:07:18.430 align:middle line:84% So let's take an example of the reaction in which you 01:07:18.430 --> 01:07:28.140 align:middle line:84% have two hydrogen molecules, this and that, and an oxygen 01:07:28.140 --> 01:07:30.060 align:middle line:90% molecule. 01:07:30.060 --> 01:07:37.860 align:middle line:84% And they get reassembled by this mechanism 01:07:37.860 --> 01:07:40.925 align:middle line:90% into two molecules of water. 01:07:40.925 --> 01:07:47.860 align:middle line:90% 01:07:47.860 --> 01:07:57.000 align:middle line:84% So I use the equal sign here to say that this refers simply 01:07:57.000 --> 01:07:57.970 align:middle line:90% to the mechanism. 01:07:57.970 --> 01:08:02.900 align:middle line:84% It doesn't say which direction it will go. 01:08:02.900 --> 01:08:06.740 align:middle line:90% It simply defines the mechanism. 01:08:06.740 --> 01:08:10.160 align:middle line:84% And therefore, you can go one direction, but also 01:08:10.160 --> 01:08:10.980 align:middle line:90% the opposite. 01:08:10.980 --> 01:08:14.710 align:middle line:90% 01:08:14.710 --> 01:08:19.020 align:middle line:84% And there is, therefore, some arbitrary choice. 01:08:19.020 --> 01:08:22.420 align:middle line:84% You could put on the left-hand side hydrogen and oxygen. 01:08:22.420 --> 01:08:24.450 align:middle line:84% Or you could put on the left-hand side 01:08:24.450 --> 01:08:29.069 align:middle line:90% the water, the symbol for water. 01:08:29.069 --> 01:08:30.020 align:middle line:90% Either one is OK. 01:08:30.020 --> 01:08:33.930 align:middle line:90% 01:08:33.930 --> 01:08:36.779 align:middle line:84% Once you have written that this way, 01:08:36.779 --> 01:08:41.399 align:middle line:84% you can also put them all on the left-hand side, 01:08:41.399 --> 01:08:49.529 align:middle line:84% and with the stoichiometric coefficients positive for what 01:08:49.529 --> 01:08:52.350 align:middle line:84% you will consider the products of the reaction 01:08:52.350 --> 01:09:00.210 align:middle line:84% and negative for those that you will consider the reactants. 01:09:00.210 --> 01:09:04.569 align:middle line:90% 01:09:04.569 --> 01:09:10.705 align:middle line:84% Now, the meaning of the stoichiometric coefficients 01:09:10.705 --> 01:09:17.330 align:middle line:84% are that they determine the possible changes in composition. 01:09:17.330 --> 01:09:20.270 align:middle line:84% So if I start with a composition which 01:09:20.270 --> 01:09:26.490 align:middle line:84% is given by these numbers, anything, 01:09:26.490 --> 01:09:31.439 align:middle line:84% and then I want to see how the composition, that composition, 01:09:31.439 --> 01:09:35.760 align:middle line:84% can change according to this reaction mechanism and only 01:09:35.760 --> 01:09:41.410 align:middle line:90% this one, then you can write-- 01:09:41.410 --> 01:09:46.779 align:middle line:84% these changes are not completely arbitrary 01:09:46.779 --> 01:09:50.649 align:middle line:84% because you have to have the fact 01:09:50.649 --> 01:09:53.740 align:middle line:84% that the ratio of the change of one 01:09:53.740 --> 01:09:58.720 align:middle line:84% component to the respective stoichiometric coefficient 01:09:58.720 --> 01:10:02.390 align:middle line:84% must be equal to the same ratio for all the other components. 01:10:02.390 --> 01:10:05.530 align:middle line:84% So this ratio, we are going to call 01:10:05.530 --> 01:10:09.220 align:middle line:84% the reaction coordinate or change in reaction coordinate. 01:10:09.220 --> 01:10:11.140 align:middle line:84% That defines the reaction coordinates, 01:10:11.140 --> 01:10:14.810 align:middle line:84% the change over the stoichiometric coefficient. 01:10:14.810 --> 01:10:16.900 align:middle line:84% Notice that the stoichiometric coefficient 01:10:16.900 --> 01:10:22.540 align:middle line:84% must be taken with the sign, the proper sign, 01:10:22.540 --> 01:10:27.000 align:middle line:84% negative for the reactants and positive for the products. 01:10:27.000 --> 01:10:31.270 align:middle line:90% 01:10:31.270 --> 01:10:36.840 align:middle line:84% So with that in mind, then in order 01:10:36.840 --> 01:10:40.470 align:middle line:84% to introduce a notation that is typically 01:10:40.470 --> 01:10:47.610 align:middle line:84% used in software that deals with the calculations 01:10:47.610 --> 01:10:54.740 align:middle line:84% of chemical equilibrium or chemical kinetics, 01:10:54.740 --> 01:11:01.920 align:middle line:84% we can define a symbol, A sub i, A sub 1, A sub 2, A sub 3, 01:11:01.920 --> 01:11:05.680 align:middle line:84% to represent the chemical symbols in the list. 01:11:05.680 --> 01:11:16.855 align:middle line:90% So for example, we make a table. 01:11:16.855 --> 01:11:19.830 align:middle line:90% 01:11:19.830 --> 01:11:23.190 align:middle line:90% 1, 2, 3. 01:11:23.190 --> 01:11:30.030 align:middle line:84% We decide to call the first component to make it H2. 01:11:30.030 --> 01:11:36.420 align:middle line:84% The second component is oxygen and the third is water. 01:11:36.420 --> 01:11:38.460 align:middle line:84% So the stoichiometric coefficient, 01:11:38.460 --> 01:11:43.350 align:middle line:84% we use the symbol nu sub i for the stoichiometric coefficients. 01:11:43.350 --> 01:11:53.110 align:middle line:84% It's minus 2 for hydrogen, minus 1 for oxygen, and 2 for water. 01:11:53.110 --> 01:11:57.560 align:middle line:90% 01:11:57.560 --> 01:12:04.110 align:middle line:84% And then you see that this equation here 01:12:04.110 --> 01:12:07.320 align:middle line:84% that represents the chemical reaction mechanism 01:12:07.320 --> 01:12:10.890 align:middle line:84% translates into this summation, which we can write 01:12:10.890 --> 01:12:12.020 align:middle line:90% in the summation this way. 01:12:12.020 --> 01:12:15.970 align:middle line:90% 01:12:15.970 --> 01:12:27.780 align:middle line:84% The advantage of that is that so I have, say, 01:12:27.780 --> 01:12:31.050 align:middle line:90% this for my first reaction. 01:12:31.050 --> 01:12:32.670 align:middle line:90% But suppose I have-- 01:12:32.670 --> 01:12:36.420 align:middle line:84% because in general, life is not simple 01:12:36.420 --> 01:12:39.900 align:middle line:84% and very few things can be described with just 01:12:39.900 --> 01:12:43.520 align:middle line:90% a single chemical reaction. 01:12:43.520 --> 01:12:49.680 align:middle line:84% For example, even for that formation of water, 01:12:49.680 --> 01:12:55.140 align:middle line:84% just for the combustion of hydrogen, 01:12:55.140 --> 01:13:00.860 align:middle line:84% I think that the mechanism-- you have to consider at least, 01:13:00.860 --> 01:13:03.990 align:middle line:84% I forget now, probably eight mechanisms (actually, it's 24, 01:13:03.990 --> 01:13:04.490 align:middle line:90% see later). 01:13:04.490 --> 01:13:08.490 align:middle line:84% And for the combustion of methane, 01:13:08.490 --> 01:13:14.480 align:middle line:84% it's probably 250 or more that play a role 01:13:14.480 --> 01:13:16.950 align:middle line:90% and determine the kinetics. 01:13:16.950 --> 01:13:18.850 align:middle line:84% Different thing is for equilibrium. 01:13:18.850 --> 01:13:20.996 align:middle line:84% But for the kinetics, you need many mechanisms. 01:13:20.996 --> 01:13:24.990 align:middle line:90% 01:13:24.990 --> 01:13:32.670 align:middle line:84% So for example, suppose we have, also, another important equation 01:13:32.670 --> 01:13:40.380 align:middle line:84% for combustion, which is the dissociation of CO2 01:13:40.380 --> 01:13:46.780 align:middle line:90% in CO and oxygen. 01:13:46.780 --> 01:13:51.320 align:middle line:84% If, together with my initial reaction, 01:13:51.320 --> 01:13:58.900 align:middle line:84% which is 2H2 plus O2 equals H2O, which we already represented 01:13:58.900 --> 01:14:02.780 align:middle line:84% in our list, if we also have that one, 01:14:02.780 --> 01:14:12.070 align:middle line:84% we can keep going in our list of chemical species that are 01:14:12.070 --> 01:14:15.670 align:middle line:90% relevant to our problem. 01:14:15.670 --> 01:14:18.670 align:middle line:90% We add CO2 and CO. 01:14:18.670 --> 01:14:22.160 align:middle line:84% Of course, they don't play a role in the first reaction, 01:14:22.160 --> 01:14:25.510 align:middle line:84% so we'll have stoichiometric coefficients for reaction 1. 01:14:25.510 --> 01:14:27.820 align:middle line:90% Let's say this is reaction 1. 01:14:27.820 --> 01:14:34.030 align:middle line:84% So I'm putting a superscript 1 for the first reaction. 01:14:34.030 --> 01:14:43.720 align:middle line:84% And then I put here the chemical-- 01:14:43.720 --> 01:14:46.310 align:middle line:84% stoichiometric coefficients for the second reaction. 01:14:46.310 --> 01:14:50.860 align:middle line:84% So for the second reaction, I have plus 1, 01:14:50.860 --> 01:14:53.050 align:middle line:90% since it's a product for CO. 01:14:53.050 --> 01:14:56.380 align:middle line:90% So CO is 1. 01:14:56.380 --> 01:15:03.460 align:middle line:90% Oxygen, 1/2, so 0.5 positive. 01:15:03.460 --> 01:15:08.600 align:middle line:84% Then CO2, it's on the left-hand side, so it's negative. 01:15:08.600 --> 01:15:11.980 align:middle line:90% It's minus 1. 01:15:11.980 --> 01:15:20.190 align:middle line:84% Of course, the others are 0 because, in reaction 1, 01:15:20.190 --> 01:15:24.270 align:middle line:90% there is no play by CO2 and CO. 01:15:24.270 --> 01:15:31.740 align:middle line:84% And reaction 2, there is no play by water and hydrogen. 01:15:31.740 --> 01:15:36.420 align:middle line:84% And so you see, I have just put a j there. 01:15:36.420 --> 01:15:42.455 align:middle line:84% And I have a system of as many chemical reactions as we want. 01:15:42.455 --> 01:15:45.980 align:middle line:90% 01:15:45.980 --> 01:15:49.160 align:middle line:84% Now, the proportionality relations, 01:15:49.160 --> 01:15:52.520 align:middle line:84% namely these relations here that represent 01:15:52.520 --> 01:15:57.020 align:middle line:84% the role of stoichiometric coefficients 01:15:57.020 --> 01:16:01.790 align:middle line:84% in order to represent what changes in composition 01:16:01.790 --> 01:16:05.510 align:middle line:84% are possible, are written this way. 01:16:05.510 --> 01:16:14.280 align:middle line:84% Or also, in generic terms, like delta n over nu sub i equals 01:16:14.280 --> 01:16:25.030 align:middle line:84% delta epsilon, which we can write more explicitly, 01:16:25.030 --> 01:16:30.730 align:middle line:84% for example, if we--like in the previous examples, 01:16:30.730 --> 01:16:36.160 align:middle line:84% like in the batch process, the composition goes from that 01:16:36.160 --> 01:16:40.550 align:middle line:84% of the reactants to that of the products. 01:16:40.550 --> 01:16:48.250 align:middle line:84% So the change delta n is equal to final minus initial. 01:16:48.250 --> 01:16:57.970 align:middle line:84% So this is nP minus nR is equal to nu sub epsilon sorry, nu 01:16:57.970 --> 01:17:01.190 align:middle line:90% sub i times epsilon. 01:17:01.190 --> 01:17:03.940 align:middle line:90% And this epsilon here could be-- 01:17:03.940 --> 01:17:06.960 align:middle line:90% 01:17:06.960 --> 01:17:10.260 align:middle line:90% it's actually a delta. 01:17:10.260 --> 01:17:13.650 align:middle line:84% It's the difference between the value of the reaction coordinate 01:17:13.650 --> 01:17:17.190 align:middle line:84% that describes the products and the value of the reaction 01:17:17.190 --> 01:17:22.670 align:middle line:84% coordinate that describes the reactants composition. 01:17:22.670 --> 01:17:33.960 align:middle line:84% If we have instead a steady state flow continuous reaction, 01:17:33.960 --> 01:17:38.650 align:middle line:84% then we talk about the rate of change of the composition. 01:17:38.650 --> 01:17:40.300 align:middle line:90% So we take-- we define-- 01:17:40.300 --> 01:17:42.040 align:middle line:90% we divide this by dt. 01:17:42.040 --> 01:17:47.185 align:middle line:84% So delta n by delta t is the stoichiometric coefficient times 01:17:47.185 --> 01:17:48.680 align:middle line:90% delta epsilon over delta t. 01:17:48.680 --> 01:17:52.620 align:middle line:90% Let's call it epsilon dot. 01:17:52.620 --> 01:17:55.860 align:middle line:84% And so you find that the rate at which the-- 01:17:55.860 --> 01:17:58.020 align:middle line:84% the difference between the rate at which 01:17:58.020 --> 01:18:02.220 align:middle line:84% the products and the reactants come out and in 01:18:02.220 --> 01:18:05.640 align:middle line:84% is related to this rate of change of the reaction 01:18:05.640 --> 01:18:08.820 align:middle line:84% coordinate, which somehow is related 01:18:08.820 --> 01:18:12.320 align:middle line:90% to the reaction rate or rates. 01:18:12.320 --> 01:18:17.820 align:middle line:90% 01:18:17.820 --> 01:18:22.540 align:middle line:84% And in fact, do I have it in the next slide? 01:18:22.540 --> 01:18:24.580 align:middle line:90% No. 01:18:24.580 --> 01:18:48.100 align:middle line:84% So in fact, so we have n dot iP is equal to n dot iR plus-- 01:18:48.100 --> 01:18:50.590 align:middle line:84% so if you have only one reaction, 01:18:50.590 --> 01:18:57.190 align:middle line:84% it's the summation of epsilon dot times nu i. 01:18:57.190 --> 01:19:03.310 align:middle line:84% But if I have several reactions, then each one 01:19:03.310 --> 01:19:07.220 align:middle line:90% has its own reaction coordinate. 01:19:07.220 --> 01:19:12.610 align:middle line:84% And I have to sum up the contribution 01:19:12.610 --> 01:19:16.275 align:middle line:84% to the change of the various reactions in play. 01:19:16.275 --> 01:19:23.752 align:middle line:90% 01:19:23.752 --> 01:19:25.720 align:middle line:90% So now let's go back-- 01:19:25.720 --> 01:19:28.980 align:middle line:84% sorry, let's go back here to the structure 01:19:28.980 --> 01:19:33.540 align:middle line:90% of the balance equations. 01:19:33.540 --> 01:19:37.230 align:middle line:84% In the balance equations, we have 01:19:37.230 --> 01:19:40.710 align:middle line:84% to deal with differences between the energy, 01:19:40.710 --> 01:19:42.960 align:middle line:84% between products and reactants, the entropy 01:19:42.960 --> 01:19:45.760 align:middle line:84% between products and reactants, also here, 01:19:45.760 --> 01:19:50.790 align:middle line:84% the enthalpy between reactants and products, the entropy. 01:19:50.790 --> 01:20:01.760 align:middle line:84% These differences, yes, it is true that we can write 01:20:01.760 --> 01:20:06.110 align:middle line:84% and we know how to write them, except that we 01:20:06.110 --> 01:20:09.590 align:middle line:84% have to learn a proper way to deal with the fact 01:20:09.590 --> 01:20:13.580 align:middle line:84% that the composition is changing here. 01:20:13.580 --> 01:20:15.500 align:middle line:84% So some molecules are disappearing 01:20:15.500 --> 01:20:19.850 align:middle line:84% and some are appearing that have different properties. 01:20:19.850 --> 01:20:23.300 align:middle line:84% And therefore, we have to be careful in how 01:20:23.300 --> 01:20:30.350 align:middle line:84% we take a reference value for these properties. 01:20:30.350 --> 01:20:33.700 align:middle line:90% And this is done here. 01:20:33.700 --> 01:20:37.240 align:middle line:84% For example, of those differences, 01:20:37.240 --> 01:20:41.840 align:middle line:84% let's take one, which is the energy. 01:20:41.840 --> 01:20:45.610 align:middle line:84% So we write the energy difference between products 01:20:45.610 --> 01:20:50.540 align:middle line:90% and reactants, as we said. 01:20:50.540 --> 01:20:52.960 align:middle line:84% So this is just the implementation 01:20:52.960 --> 01:20:58.420 align:middle line:84% of what we said in terms of partial properties. 01:20:58.420 --> 01:21:06.690 align:middle line:84% Now, let's add and subtract something, which will not 01:21:06.690 --> 01:21:10.800 align:middle line:84% change, therefore, the result, and which we select to add 01:21:10.800 --> 01:21:18.850 align:middle line:84% and subtract for each constituent 01:21:18.850 --> 01:21:22.510 align:middle line:84% the specific property of pure component of that constituent 01:21:22.510 --> 01:21:24.550 align:middle line:90% at some reference. 01:21:24.550 --> 01:21:27.460 align:middle line:84% And here, let's say, at some reference, temperature 01:21:27.460 --> 01:21:30.700 align:middle line:90% and pressure. 01:21:30.700 --> 01:21:34.450 align:middle line:84% I'm going to call the pressure p0 because, typically, it's 01:21:34.450 --> 01:21:35.200 align:middle line:90% taken-- 01:21:35.200 --> 01:21:38.500 align:middle line:84% the standard pressure is 1 atmosphere. 01:21:38.500 --> 01:21:41.080 align:middle line:84% Whereas, here for temperature, let's 01:21:41.080 --> 01:21:43.990 align:middle line:84% use the generic temperature for the moment. 01:21:43.990 --> 01:21:52.040 align:middle line:84% If it will be the standard temperature, 298.15, 01:21:52.040 --> 01:21:55.340 align:middle line:90% we will denote it by T0. 01:21:55.340 --> 01:21:58.850 align:middle line:84% But for the moment, this can be done at any temperature. 01:21:58.850 --> 01:22:01.820 align:middle line:84% And this temperature is not necessarily 01:22:01.820 --> 01:22:08.800 align:middle line:90% equal to either TR or TP. 01:22:08.800 --> 01:22:10.580 align:middle line:90% So we add and subtract. 01:22:10.580 --> 01:22:11.970 align:middle line:90% So here it is. 01:22:11.970 --> 01:22:15.910 align:middle line:84% So we subtract from this summation. 01:22:15.910 --> 01:22:19.800 align:middle line:90% We subtract from that summation. 01:22:19.800 --> 01:22:23.070 align:middle line:84% But then we realize that since we put it in this summation 01:22:23.070 --> 01:22:28.180 align:middle line:90% here, it is multiplied by nP. 01:22:28.180 --> 01:22:30.730 align:middle line:90% And here it is multiplied by nR. 01:22:30.730 --> 01:22:42.320 align:middle line:84% So we have to add up what we subtracted, so this multiplied 01:22:42.320 --> 01:22:47.762 align:middle line:84% by the corresponding amounts here. 01:22:47.762 --> 01:22:50.210 align:middle line:84% So in this summation that compensates 01:22:50.210 --> 01:22:56.900 align:middle line:84% for those subtractions we recognize the difference 01:22:56.900 --> 01:23:00.750 align:middle line:84% between composition, between products and reactants. 01:23:00.750 --> 01:23:04.670 align:middle line:84% And that's related to the reaction 01:23:04.670 --> 01:23:11.160 align:middle line:84% coordinate through the proportionality relations. 01:23:11.160 --> 01:23:13.740 align:middle line:84% So the epsilon doesn't have a subscript. 01:23:13.740 --> 01:23:17.410 align:middle line:90% It goes out of the summation. 01:23:17.410 --> 01:23:19.660 align:middle line:84% And here, we are left with the summation 01:23:19.660 --> 01:23:21.700 align:middle line:84% of the stoichiometric coefficients, which 01:23:21.700 --> 01:23:24.250 align:middle line:90% multiply this reference-- 01:23:24.250 --> 01:23:30.430 align:middle line:84% these pure component properties, specific properties 01:23:30.430 --> 01:23:34.110 align:middle line:84% at those reference of temperature and pressure. 01:23:34.110 --> 01:23:41.190 align:middle line:84% This summation, we call it the energy 01:23:41.190 --> 01:23:44.220 align:middle line:84% of the reaction at standard pressure 01:23:44.220 --> 01:23:45.850 align:middle line:90% and at that temperature. 01:23:45.850 --> 01:23:48.420 align:middle line:84% And we denote it by this symbol here. 01:23:48.420 --> 01:23:53.340 align:middle line:84% So that defines the energy of the reaction. 01:23:53.340 --> 01:23:58.990 align:middle line:84% You can do that for all the other differences here. 01:23:58.990 --> 01:24:01.040 align:middle line:84% For example, this is for the enthalpy, 01:24:01.040 --> 01:24:05.090 align:middle line:90% the rate of enthalpies. 01:24:05.090 --> 01:24:10.945 align:middle line:84% Add and subtract the pure component specific enthalpy. 01:24:10.945 --> 01:24:15.900 align:middle line:90% 01:24:15.900 --> 01:24:19.800 align:middle line:84% In this final term, you get the proportionality relation 01:24:19.800 --> 01:24:21.390 align:middle line:90% for the continuous process. 01:24:21.390 --> 01:24:23.650 align:middle line:84% Epsilon dot goes out of the summation. 01:24:23.650 --> 01:24:30.270 align:middle line:84% This summation, you also call it the enthalpy of the reaction 01:24:30.270 --> 01:24:31.470 align:middle line:90% at-- 01:24:31.470 --> 01:24:33.930 align:middle line:84% it is the enthalpy of the reaction 01:24:33.930 --> 01:24:37.770 align:middle line:84% because it refers to a specific reaction 01:24:37.770 --> 01:24:40.890 align:middle line:84% because it depends on the stoichiometric coefficients 01:24:40.890 --> 01:24:43.380 align:middle line:90% of that reaction. 01:24:43.380 --> 01:24:45.990 align:middle line:84% So I'll have an enthalpy of this reaction 01:24:45.990 --> 01:24:48.380 align:middle line:84% and another enthalpy for that reaction. 01:24:48.380 --> 01:24:51.920 align:middle line:84% So each reaction has its own value 01:24:51.920 --> 01:24:54.780 align:middle line:84% for these properties of reaction. 01:24:54.780 --> 01:24:57.380 align:middle line:84% We call them properties of reaction. 01:24:57.380 --> 01:24:58.500 align:middle line:90% And here is the list. 01:24:58.500 --> 01:25:01.530 align:middle line:84% You can do that for all properties, 01:25:01.530 --> 01:25:05.780 align:middle line:84% including energy, entropy, enthalpy, Gibbs free energy, 01:25:05.780 --> 01:25:09.120 align:middle line:90% and even volume. 01:25:09.120 --> 01:25:19.400 align:middle line:84% And of course, since these are pure component properties, 01:25:19.400 --> 01:25:23.330 align:middle line:84% the relations that hold between properties 01:25:23.330 --> 01:25:28.290 align:middle line:84% also hold between the properties of reaction. 01:25:28.290 --> 01:25:33.620 align:middle line:84% So since the Gibbs free energy, you can write it as h minus Ts, 01:25:33.620 --> 01:25:36.350 align:middle line:84% it turns out that the Gibbs free energy of reaction 01:25:36.350 --> 01:25:40.130 align:middle line:84% is equal to the enthalpy of reaction minus T 01:25:40.130 --> 01:25:44.600 align:middle line:90% times the entropy of reaction. 01:25:44.600 --> 01:25:53.220 align:middle line:84% Or since the enthalpy is equal to energy plus p v, u plus p v, 01:25:53.220 --> 01:25:56.410 align:middle line:84% there is a relation also between the enthalpy of reaction, 01:25:56.410 --> 01:25:59.915 align:middle line:84% the energy of reaction, and the volume of reaction. 01:25:59.915 --> 01:26:03.300 align:middle line:90% 01:26:03.300 --> 01:26:05.190 align:middle line:84% This is for a generic temperature 01:26:05.190 --> 01:26:09.910 align:middle line:84% T. At standard temperature and pressure, also 01:26:09.910 --> 01:26:11.230 align:middle line:90% the temperature is fixed. 01:26:11.230 --> 01:26:15.750 align:middle line:84% We drop the symbol function of T. 01:26:15.750 --> 01:26:19.620 align:middle line:84% And for simplicity, we just, by convention, 01:26:19.620 --> 01:26:24.250 align:middle line:84% say, well, if there is no dependence of T, 01:26:24.250 --> 01:26:27.110 align:middle line:90% it means that we are at T0. 01:26:27.110 --> 01:26:34.680 align:middle line:84% So the 0 here has two meanings, pressure and also temperature 01:26:34.680 --> 01:26:35.915 align:middle line:90% are at standard values. 01:26:35.915 --> 01:26:39.150 align:middle line:90% 01:26:39.150 --> 01:26:41.670 align:middle line:84% Now, it turns out that if you want 01:26:41.670 --> 01:26:49.730 align:middle line:84% to compute the properties of reaction, you see, 01:26:49.730 --> 01:26:54.850 align:middle line:84% as we've said, every reaction has its own properties. 01:26:54.850 --> 01:26:58.860 align:middle line:90% Should them be all measured? 01:26:58.860 --> 01:27:05.170 align:middle line:84% So if you are interested in one reaction, 01:27:05.170 --> 01:27:08.340 align:middle line:84% you have to find the experimental values 01:27:08.340 --> 01:27:12.730 align:middle line:84% from somebody who measured the properties for your reaction? 01:27:12.730 --> 01:27:17.070 align:middle line:84% Well, the answer is no, fortunately, 01:27:17.070 --> 01:27:18.870 align:middle line:84% because of the Hess relation, which 01:27:18.870 --> 01:27:25.540 align:middle line:84% we will prove in a moment, whereby it turns out 01:27:25.540 --> 01:27:29.770 align:middle line:84% that there is a minimal set of measurements 01:27:29.770 --> 01:27:34.450 align:middle line:84% that you have to make, which is the measurement 01:27:34.450 --> 01:27:37.550 align:middle line:84% of the reactions of formations of the various components. 01:27:37.550 --> 01:27:43.630 align:middle line:84% Once you have those, using these relations, Hess relations, 01:27:43.630 --> 01:27:50.620 align:middle line:84% you can find the properties of reaction for any reaction 01:27:50.620 --> 01:27:56.490 align:middle line:84% by just taking the weighted sum with the stoichiometric 01:27:56.490 --> 01:27:59.670 align:middle line:84% coefficients of your reaction, of the reactions 01:27:59.670 --> 01:28:04.860 align:middle line:84% or formations of the various molecules that 01:28:04.860 --> 01:28:06.485 align:middle line:90% play in your reaction. 01:28:06.485 --> 01:28:09.330 align:middle line:90% 01:28:09.330 --> 01:28:15.225 align:middle line:84% So essentially, here, I'm interested in energy. 01:28:15.225 --> 01:28:21.620 align:middle line:90% 01:28:21.620 --> 01:28:24.440 align:middle line:90% Yeah, the f goes before. 01:28:24.440 --> 01:28:27.020 align:middle line:90% fi0. 01:28:27.020 --> 01:28:30.680 align:middle line:84% And then I'll have to list somewhere here 01:28:30.680 --> 01:28:33.500 align:middle line:84% the values of the energies of formation 01:28:33.500 --> 01:28:34.770 align:middle line:90% for these various components. 01:28:34.770 --> 01:28:37.490 align:middle line:84% And these ones, I find tabulated in the handbooks 01:28:37.490 --> 01:28:40.015 align:middle line:90% of chemical physics. 01:28:40.015 --> 01:28:44.850 align:middle line:90% 01:28:44.850 --> 01:28:47.100 align:middle line:84% Before we prove the Hess relation, 01:28:47.100 --> 01:28:58.018 align:middle line:84% let's see the meaning of this delta u and delta s-- 01:28:58.018 --> 01:29:03.330 align:middle line:84% well, actually, this is for enthalpy and entropy. 01:29:03.330 --> 01:29:05.810 align:middle line:84% And so let's consider one particular reaction, 01:29:05.810 --> 01:29:10.970 align:middle line:84% like that in which a hydrogen molecule splits 01:29:10.970 --> 01:29:13.480 align:middle line:90% into two hydrogen atoms. 01:29:13.480 --> 01:29:21.790 align:middle line:90% 01:29:21.790 --> 01:29:26.830 align:middle line:84% So here I'm drawing the E versus S, or actually, 01:29:26.830 --> 01:29:32.890 align:middle line:84% the enthalpy versus S curve, for the stable equilibrium 01:29:32.890 --> 01:29:39.700 align:middle line:84% states of atomic hydrogen, this is atomic hydrogen, 01:29:39.700 --> 01:29:44.550 align:middle line:84% in the amount of one, fixed amount, 1 mole, 01:29:44.550 --> 01:29:50.250 align:middle line:84% and at standard pressure, 1 atmosphere. 01:29:50.250 --> 01:30:01.060 align:middle line:84% This other curve represents half a mole of molecular hydrogen, 01:30:01.060 --> 01:30:04.715 align:middle line:90% H2, also at standard pressure. 01:30:04.715 --> 01:30:08.810 align:middle line:90% 01:30:08.810 --> 01:30:22.040 align:middle line:84% Then I take a slope equal to 298.15 kelvin and search 01:30:22.040 --> 01:30:26.535 align:middle line:84% for the point on this curve where I have that temperature. 01:30:26.535 --> 01:30:28.770 align:middle line:84% It turns out that the derivative of that, 01:30:28.770 --> 01:30:32.300 align:middle line:84% of H with respect to S at constant p, 01:30:32.300 --> 01:30:34.950 align:middle line:90% is equal to the temperature. 01:30:34.950 --> 01:30:39.030 align:middle line:84% So also on this diagram, the slope is the temperature. 01:30:39.030 --> 01:30:42.450 align:middle line:90% 01:30:42.450 --> 01:30:45.090 align:middle line:90% And here, I find it here. 01:30:45.090 --> 01:30:48.425 align:middle line:84% And for this other substance, I find it there. 01:30:48.425 --> 01:30:52.440 align:middle line:90% 01:30:52.440 --> 01:31:00.910 align:middle line:84% Now, these two curves until yesterday or-- 01:31:00.910 --> 01:31:07.090 align:middle line:84% until we didn't consider chemical reactions, 01:31:07.090 --> 01:31:11.030 align:middle line:90% were sort of independent. 01:31:11.030 --> 01:31:14.940 align:middle line:84% If we needed to describe the properties of this molecule, 01:31:14.940 --> 01:31:22.070 align:middle line:84% we just draw this curve anywhere, taking an arbitrary 0 01:31:22.070 --> 01:31:23.770 align:middle line:90% for the enthalpy. 01:31:23.770 --> 01:31:26.410 align:middle line:90% It didn't matter. 01:31:26.410 --> 01:31:29.770 align:middle line:84% But now, instead, what matters is the fact 01:31:29.770 --> 01:31:33.710 align:middle line:84% that with a chemical reaction mechanism, 01:31:33.710 --> 01:31:38.450 align:middle line:84% you can go from one substance to the other. 01:31:38.450 --> 01:31:43.520 align:middle line:84% So you cannot arbitrarily choose reference values 01:31:43.520 --> 01:31:50.785 align:middle line:84% for the properties for both, because-- 01:31:50.785 --> 01:31:55.670 align:middle line:90% 01:31:55.670 --> 01:31:58.760 align:middle line:90% for example, if you look at-- 01:31:58.760 --> 01:32:01.140 align:middle line:84% suppose we were at 0 temperature. 01:32:01.140 --> 01:32:09.300 align:middle line:90% 01:32:09.300 --> 01:32:19.010 align:middle line:84% In order to split this molecule of molecular hydrogen 01:32:19.010 --> 01:32:25.030 align:middle line:84% into two hydrogen, you have to break a bond, chemical bond. 01:32:25.030 --> 01:32:29.400 align:middle line:84% And the strength of this chemical bond 01:32:29.400 --> 01:32:32.130 align:middle line:84% corresponds to an energy that you 01:32:32.130 --> 01:32:37.980 align:middle line:84% have to provide in order to detach these two atoms that 01:32:37.980 --> 01:32:42.770 align:middle line:90% compose the molecule. 01:32:42.770 --> 01:32:49.050 align:middle line:84% This bond energy is what you have to plot here. 01:32:49.050 --> 01:32:52.760 align:middle line:84% So that is the number that allows 01:32:52.760 --> 01:32:58.170 align:middle line:84% you to place in relative position the two curves because, 01:32:58.170 --> 01:33:02.590 align:middle line:90% this value, you measure. 01:33:02.590 --> 01:33:09.250 align:middle line:84% And therefore, these two curves must be at this distance here. 01:33:09.250 --> 01:33:13.210 align:middle line:84% Then using the properties of the substances, 01:33:13.210 --> 01:33:18.470 align:middle line:84% of the pure substances, here, using specific heats and so on, 01:33:18.470 --> 01:33:21.190 align:middle line:84% you can compute what would be the value of the enthalpy 01:33:21.190 --> 01:33:24.080 align:middle line:90% at this temperature. 01:33:24.080 --> 01:33:26.750 align:middle line:84% And then also, using the specific heats 01:33:26.750 --> 01:33:31.190 align:middle line:84% of this other substance, you can integrate 01:33:31.190 --> 01:33:37.690 align:middle line:84% the equation, the fundamental relation, and get to this point. 01:33:37.690 --> 01:33:42.100 align:middle line:84% These correspond to the standard conditions. 01:33:42.100 --> 01:33:50.770 align:middle line:84% And the enthalpy of formation of atomic hydrogen, so 01:33:50.770 --> 01:33:56.590 align:middle line:84% of this state here, from molecular hydrogen 01:33:56.590 --> 01:34:01.560 align:middle line:84% is given by the difference in the enthalpy between here 01:34:01.560 --> 01:34:02.730 align:middle line:90% and there. 01:34:02.730 --> 01:34:05.940 align:middle line:84% This is the amount of energy that you 01:34:05.940 --> 01:34:11.190 align:middle line:84% have to provide in a steady state reactor, where 01:34:11.190 --> 01:34:16.890 align:middle line:84% as input you have hydrogen at standard conditions, 01:34:16.890 --> 01:34:23.400 align:middle line:84% and as output you have atomic hydrogen also 01:34:23.400 --> 01:34:24.850 align:middle line:90% at standard conditions. 01:34:24.850 --> 01:34:33.720 align:middle line:90% 01:34:33.720 --> 01:34:40.730 align:middle line:84% And this also, there is a difference in entropy. 01:34:40.730 --> 01:34:43.250 align:middle line:84% This difference in entropy corresponds 01:34:43.250 --> 01:34:46.806 align:middle line:90% to the entropy of formation. 01:34:46.806 --> 01:34:57.871 align:middle line:90% 01:34:57.871 --> 01:35:06.520 align:middle line:84% Now, the strength of the bonds, we will return on this graph, 01:35:06.520 --> 01:35:08.830 align:middle line:90% but just-- 01:35:08.830 --> 01:35:18.410 align:middle line:84% the strength of the bonds is also describable 01:35:18.410 --> 01:35:24.040 align:middle line:84% using, for example, a Lennard-Jones potential. 01:35:24.040 --> 01:35:27.730 align:middle line:84% But the strength and the distances in play 01:35:27.730 --> 01:35:30.210 align:middle line:90% are very different. 01:35:30.210 --> 01:35:38.730 align:middle line:84% For example, this would be the Lennard-Jones potential that 01:35:38.730 --> 01:35:50.180 align:middle line:84% describes the covalent bond, which is what happens when this 01:35:50.180 --> 01:35:57.235 align:middle line:84% hydrogen atom gets close to the oxygen and gets to share-- 01:35:57.235 --> 01:35:59.870 align:middle line:90% 01:35:59.870 --> 01:36:02.420 align:middle line:90% they get to share two electrons. 01:36:02.420 --> 01:36:08.240 align:middle line:84% And these two electrons form a negative charge 01:36:08.240 --> 01:36:16.960 align:middle line:84% enough to cover up the repulsion between the positive charges 01:36:16.960 --> 01:36:19.730 align:middle line:90% of the nuclei. 01:36:19.730 --> 01:36:23.380 align:middle line:84% So then this is the repulsion of the nuclei, 01:36:23.380 --> 01:36:31.470 align:middle line:84% this is the attraction, and this is very heavy dip. 01:36:31.470 --> 01:36:37.080 align:middle line:90% It is 459 kilojoules per mole. 01:36:37.080 --> 01:36:41.070 align:middle line:84% You compare it with the van der Waals forces 01:36:41.070 --> 01:36:48.030 align:middle line:84% that you can calculate for water molecules in-- 01:36:48.030 --> 01:36:50.100 align:middle line:84% that describes the liquid and vapor phase 01:36:50.100 --> 01:36:54.240 align:middle line:84% for water molecules using the Lennard-Jones fluid 01:36:54.240 --> 01:36:56.970 align:middle line:90% model that we discussed. 01:36:56.970 --> 01:37:01.860 align:middle line:90% The value of the epsilon is-- 01:37:01.860 --> 01:37:06.930 align:middle line:90% you see it's-- what is it? 01:37:06.930 --> 01:37:09.830 align:middle line:90% 4,000 times smaller. 01:37:09.830 --> 01:37:12.140 align:middle line:90% 5,000 times smaller. 01:37:12.140 --> 01:37:17.180 align:middle line:84% So much that in order to show it in the plot, 01:37:17.180 --> 01:37:21.710 align:middle line:84% I have to multiply it here by a factor of 1,000. 01:37:21.710 --> 01:37:23.610 align:middle line:90% And still it's smaller. 01:37:23.610 --> 01:37:29.190 align:middle line:84% And also, the minimum here for the van der 01:37:29.190 --> 01:37:34.680 align:middle line:84% Waals forces, the attraction between molecules, 01:37:34.680 --> 01:37:36.720 align:middle line:84% which are also of electrostatic nature, 01:37:36.720 --> 01:37:38.520 align:middle line:90% but from a different source-- 01:37:38.520 --> 01:37:45.180 align:middle line:90% 01:37:45.180 --> 01:37:51.000 align:middle line:84% has very large-- much larger characteristic distances. 01:37:51.000 --> 01:38:00.080 align:middle line:84% So it's like three Angstrom here or 300 picometers here. 01:38:00.080 --> 01:38:05.040 align:middle line:84% And instead, for the covalent bond, it's of the order of 74. 01:38:05.040 --> 01:38:09.600 align:middle line:84% So they have to get much closer in order 01:38:09.600 --> 01:38:15.930 align:middle line:84% for these much stronger forces to get in play 01:38:15.930 --> 01:38:17.470 align:middle line:90% and hold the molecule. 01:38:17.470 --> 01:38:26.660 align:middle line:84% So this is-- to say that, yes, you can use Lennard-Jones. 01:38:26.660 --> 01:38:29.090 align:middle line:84% Although, it's not really a great model 01:38:29.090 --> 01:38:39.790 align:middle line:84% for these kind of bonds, but it gives you the first order idea. 01:38:39.790 --> 01:38:44.910 align:middle line:84% But you have to realize, also, of the different order 01:38:44.910 --> 01:38:48.660 align:middle line:84% of magnitude, of the strengths of the bonds with respect 01:38:48.660 --> 01:38:52.120 align:middle line:90% to the van der Waals forces. 01:38:52.120 --> 01:38:58.610 align:middle line:84% So yeah, let's stop here, I guess, yeah. 01:38:58.610 --> 01:39:14.000 align:middle line:90%