WEBVTT 00:00:00.000 --> 00:00:02.415 align:middle line:90% [SQUEAKING] 00:00:02.415 --> 00:00:04.347 align:middle line:90% [RUSTLING] 00:00:04.347 --> 00:00:07.728 align:middle line:90% [CLICKING] 00:00:07.728 --> 00:00:10.693 align:middle line:90% 00:00:10.693 --> 00:00:12.235 align:middle line:84% GIAN PAOLO BERETTA: OK, good morning. 00:00:12.235 --> 00:00:16.490 align:middle line:90% 00:00:16.490 --> 00:00:20.630 align:middle line:84% So let me remind you where we were. 00:00:20.630 --> 00:00:23.900 align:middle line:84% We are dealing with systems with chemical reactions. 00:00:23.900 --> 00:00:28.910 align:middle line:84% We talked about the role of the activated complex 00:00:28.910 --> 00:00:34.430 align:middle line:84% in the mechanism of converting reactants into products 00:00:34.430 --> 00:00:38.850 align:middle line:90% as a result of collisions. 00:00:38.850 --> 00:00:49.070 align:middle line:84% And in order to do that, then we did chemical equilibrium 00:00:49.070 --> 00:00:51.085 align:middle line:84% derived from the principle of maximum entropy. 00:00:51.085 --> 00:00:55.680 align:middle line:90% 00:00:55.680 --> 00:01:00.090 align:middle line:84% So if you remember, one of the key modeling assumptions 00:01:00.090 --> 00:01:04.260 align:middle line:84% that we had to make was to integrate 00:01:04.260 --> 00:01:12.550 align:middle line:84% what we call the simple system model, which others call 00:01:12.550 --> 00:01:18.230 align:middle line:84% the thermodynamic limit or the many particle 00:01:18.230 --> 00:01:24.200 align:middle line:84% model, in which you remember that the idea was 00:01:24.200 --> 00:01:27.920 align:middle line:84% that, essentially, we could add or remove 00:01:27.920 --> 00:01:34.490 align:middle line:84% partitions in our system without that affecting the properties. 00:01:34.490 --> 00:01:41.700 align:middle line:84% Here, we add also the fact that we add and remove or, say, 00:01:41.700 --> 00:01:43.740 align:middle line:90% turn on and off-- 00:01:43.740 --> 00:01:48.700 align:middle line:84% very conceptual idea-- the chemical reactions. 00:01:48.700 --> 00:01:53.560 align:middle line:84% And that doesn't affect the properties-- of course, 00:01:53.560 --> 00:01:59.040 align:middle line:84% it affects the time evolution because it will inhibit or allow 00:01:59.040 --> 00:02:00.880 align:middle line:90% chemical reactions to proceed. 00:02:00.880 --> 00:02:03.360 align:middle line:84% But for the instantaneous picture 00:02:03.360 --> 00:02:06.840 align:middle line:84% that we take when we measure properties of our mixture 00:02:06.840 --> 00:02:14.220 align:middle line:84% with chemical reactions, that the reactions are on or off 00:02:14.220 --> 00:02:19.900 align:middle line:84% doesn't make a lot of a difference for many particles 00:02:19.900 --> 00:02:25.140 align:middle line:84% because the role of the activated complexes 00:02:25.140 --> 00:02:31.880 align:middle line:84% is minute, since they are very few, since they are very 00:02:31.880 --> 00:02:43.630 align:middle line:84% short-lived, and also because the collisions that produce 00:02:43.630 --> 00:02:52.290 align:middle line:84% chemical reactions are much fewer than the collisions that 00:02:52.290 --> 00:03:00.560 align:middle line:84% simply redistribute energy among the molecules. 00:03:00.560 --> 00:03:07.410 align:middle line:84% And so as part of our assumption here, 00:03:07.410 --> 00:03:09.390 align:middle line:84% that we can model the properties as 00:03:09.390 --> 00:03:13.860 align:middle line:84% if it were a stable equilibrium state with the actions turned 00:03:13.860 --> 00:03:23.320 align:middle line:84% off, is also the assumption that the non-chemical collisions are 00:03:23.320 --> 00:03:26.660 align:middle line:84% much faster, are much more abundant, 00:03:26.660 --> 00:03:31.120 align:middle line:84% and therefore equilibrate energy much faster 00:03:31.120 --> 00:03:39.520 align:middle line:84% than the effective chemical collisions. 00:03:39.520 --> 00:03:51.750 align:middle line:84% So that all is in order to allow us to essentially represent 00:03:51.750 --> 00:03:56.640 align:middle line:84% this family of would-be equilibrium states 00:03:56.640 --> 00:04:01.150 align:middle line:84% if you didn't have chemical reactions, the blue curves. 00:04:01.150 --> 00:04:05.900 align:middle line:84% And this family has one parameter 00:04:05.900 --> 00:04:07.300 align:middle line:84% if you have one chemical reaction 00:04:07.300 --> 00:04:13.690 align:middle line:84% or has as many parameters as there are chemical reactions. 00:04:13.690 --> 00:04:17.680 align:middle line:84% So you shift from one to the other by changing the reaction 00:04:17.680 --> 00:04:20.300 align:middle line:90% coordinate epsilon. 00:04:20.300 --> 00:04:26.030 align:middle line:84% And the envelope of these stable equilibrium state curves 00:04:26.030 --> 00:04:29.450 align:middle line:84% is the stable equilibrium state curve 00:04:29.450 --> 00:04:32.160 align:middle line:84% for the system with chemical reactions. 00:04:32.160 --> 00:04:36.020 align:middle line:84% And that's where the stable equilibrium 00:04:36.020 --> 00:04:40.870 align:middle line:84% states are with the reactions on, 00:04:40.870 --> 00:04:47.120 align:middle line:84% and that's where we calculate the maximum. 00:04:47.120 --> 00:04:49.590 align:middle line:84% We apply the maximum entropy principle. 00:04:49.590 --> 00:04:51.650 align:middle line:84% And the maximum entropy principle 00:04:51.650 --> 00:04:54.890 align:middle line:84% implies that the derivatives of how 00:04:54.890 --> 00:05:00.200 align:middle line:84% the entropy varies with respect to those parameters must be 0. 00:05:00.200 --> 00:05:03.946 align:middle line:84% And these derivatives are what we call the affinities. 00:05:03.946 --> 00:05:07.200 align:middle line:90% 00:05:07.200 --> 00:05:10.250 align:middle line:84% These are the entropic affinities. 00:05:10.250 --> 00:05:19.640 align:middle line:84% Some others call affinity this times minus the temperature. 00:05:19.640 --> 00:05:23.790 align:middle line:90% 00:05:23.790 --> 00:05:28.530 align:middle line:84% That would be the energetic affinity. 00:05:28.530 --> 00:05:36.150 align:middle line:84% And then, by applying the definition, 00:05:36.150 --> 00:05:40.380 align:middle line:84% the simple definition of affinity, which is this one, 00:05:40.380 --> 00:05:45.890 align:middle line:84% and applying the expressions for the chemical potentials 00:05:45.890 --> 00:05:52.390 align:middle line:84% that we have worked out in several cases, 00:05:52.390 --> 00:05:57.160 align:middle line:84% we end up with the law of mass action. 00:05:57.160 --> 00:05:59.500 align:middle line:84% This is in general, so it's a nonequilibrium law 00:05:59.500 --> 00:06:06.760 align:middle line:84% of mass action, which becomes the equilibrium one when you 00:06:06.760 --> 00:06:10.090 align:middle line:90% set the affinities equal to 0. 00:06:10.090 --> 00:06:16.480 align:middle line:84% And, as you have seen, we've defined various forms 00:06:16.480 --> 00:06:19.680 align:middle line:84% that we all call equilibrium constants. 00:06:19.680 --> 00:06:22.450 align:middle line:84% For example, this one is a dimensionless equilibrium 00:06:22.450 --> 00:06:26.290 align:middle line:84% constant, which is simply e to the minus 00:06:26.290 --> 00:06:31.610 align:middle line:84% the Gibbs free energy of the reaction divided by RT. 00:06:31.610 --> 00:06:37.390 align:middle line:84% And this other one is the equilibrium constant. 00:06:37.390 --> 00:06:42.620 align:middle line:84% In terms of temperature and also pressure for a gas. 00:06:42.620 --> 00:06:44.690 align:middle line:90% This is for the gas. 00:06:44.690 --> 00:06:50.110 align:middle line:84% And so this is the pressure dependence. 00:06:50.110 --> 00:06:55.600 align:middle line:84% Or you often work with concentrations, 00:06:55.600 --> 00:07:01.240 align:middle line:84% the molarity-- so how many moles per volume you have per liter. 00:07:01.240 --> 00:07:06.580 align:middle line:84% And so you use this symbol for those concentrations. 00:07:06.580 --> 00:07:09.130 align:middle line:90% And then you have this form. 00:07:09.130 --> 00:07:13.815 align:middle line:84% So will we proceed using more often now this form here. 00:07:13.815 --> 00:07:18.420 align:middle line:90% 00:07:18.420 --> 00:07:21.000 align:middle line:84% And, of course, these various forms 00:07:21.000 --> 00:07:25.020 align:middle line:84% of the chemical equilibrium constant 00:07:25.020 --> 00:07:28.110 align:middle line:90% are related to one another. 00:07:28.110 --> 00:07:32.080 align:middle line:84% As an example, before we proceed, 00:07:32.080 --> 00:07:34.770 align:middle line:84% I wanted to just mention this one, which 00:07:34.770 --> 00:07:42.800 align:middle line:84% is that which happens in pure water, the fact 00:07:42.800 --> 00:07:49.240 align:middle line:90% that water ionizes itself. 00:07:49.240 --> 00:07:55.030 align:middle line:84% And so there is an equilibrium between the water molecules 00:07:55.030 --> 00:08:00.990 align:middle line:84% and the formation of the hydronium or the hydroxide ions. 00:08:00.990 --> 00:08:05.500 align:middle line:90% 00:08:05.500 --> 00:08:10.390 align:middle line:84% And as you know, the abundance of these ions 00:08:10.390 --> 00:08:22.750 align:middle line:84% defines the scale of acidity of an aqueous solution. 00:08:22.750 --> 00:08:29.030 align:middle line:84% In fact, what we call pH is minus the log base 00:08:29.030 --> 00:08:37.780 align:middle line:84% 10 of the concentration of the H3O plus ions. 00:08:37.780 --> 00:08:43.360 align:middle line:84% And, sometimes, you call also pOH minus the log 00:08:43.360 --> 00:08:46.360 align:middle line:90% of this concentration. 00:08:46.360 --> 00:08:48.440 align:middle line:84% Now since this is a concentration, 00:08:48.440 --> 00:08:52.100 align:middle line:84% so it is dimensional, it is mole per liter, 00:08:52.100 --> 00:08:55.610 align:middle line:84% you have to divide here by the dimension. 00:08:55.610 --> 00:08:59.300 align:middle line:84% So this is put under the logarithmic number, 00:08:59.300 --> 00:09:00.680 align:middle line:90% which is dimensional. 00:09:00.680 --> 00:09:04.330 align:middle line:84% So you have to take the dimensions away or use 00:09:04.330 --> 00:09:06.040 align:middle line:84% the convention that you always going 00:09:06.040 --> 00:09:11.410 align:middle line:84% to use those dimensions because, usually, logarithms 00:09:11.410 --> 00:09:16.890 align:middle line:84% like to have nondimensional numbers as arguments. 00:09:16.890 --> 00:09:22.740 align:middle line:84% And here is how the law of mass action 00:09:22.740 --> 00:09:27.170 align:middle line:84% looks when you write it for that equation. 00:09:27.170 --> 00:09:30.920 align:middle line:84% See the stoichiometric coefficients are minus 2 00:09:30.920 --> 00:09:32.270 align:middle line:90% for water. 00:09:32.270 --> 00:09:36.813 align:middle line:84% So when you get to this exponent here, it's a minus 2. 00:09:36.813 --> 00:09:38.480 align:middle line:84% So that's why the concentration of water 00:09:38.480 --> 00:09:41.720 align:middle line:90% goes to the denominator. 00:09:41.720 --> 00:09:48.020 align:middle line:84% And then plus 1 and plus 1 for the other two 00:09:48.020 --> 00:09:54.080 align:middle line:84% molecules/ions so that they go to the numerator with the power 00:09:54.080 --> 00:09:55.740 align:middle line:90% 1. 00:09:55.740 --> 00:10:00.000 align:middle line:90% So this is the product of those. 00:10:00.000 --> 00:10:04.230 align:middle line:90% And that has to be equal to-- 00:10:04.230 --> 00:10:08.160 align:middle line:84% at equilibrium has to be equal to the equilibrium 00:10:08.160 --> 00:10:12.590 align:middle line:84% constant based on concentrations. 00:10:12.590 --> 00:10:15.080 align:middle line:84% Now in this business, therefore, you 00:10:15.080 --> 00:10:21.280 align:middle line:84% need to compute the concentration of pure water. 00:10:21.280 --> 00:10:24.520 align:middle line:84% And that's defined by how many molecules you have per liter. 00:10:24.520 --> 00:10:29.350 align:middle line:84% You can write it as the density over the molecular weight. 00:10:29.350 --> 00:10:34.330 align:middle line:84% And, for example, at 25 degrees C, 00:10:34.330 --> 00:10:37.570 align:middle line:84% the density is 997 grams per liter. 00:10:37.570 --> 00:10:41.710 align:middle line:84% Molecular weight is 18 gram per mole. 00:10:41.710 --> 00:10:47.060 align:middle line:84% And so that concentration is 55.3 moles per liter. 00:10:47.060 --> 00:10:48.075 align:middle line:90% You need it squared. 00:10:48.075 --> 00:10:52.330 align:middle line:90% 00:10:52.330 --> 00:10:57.430 align:middle line:84% So in order not to carry these along in the practice 00:10:57.430 --> 00:10:59.890 align:middle line:90% of chemical engineering, often-- 00:10:59.890 --> 00:11:02.670 align:middle line:90% 00:11:02.670 --> 00:11:07.950 align:middle line:84% almost always-- you find this product of these equilibrium 00:11:07.950 --> 00:11:12.420 align:middle line:84% constant times the square of the water concentration-- 00:11:12.420 --> 00:11:19.760 align:middle line:84% you find it called the Kw, the equilibrium constant for water. 00:11:19.760 --> 00:11:21.710 align:middle line:84% And then, therefore, this equation 00:11:21.710 --> 00:11:25.970 align:middle line:84% rewrites this way that the product of the concentrations 00:11:25.970 --> 00:11:29.240 align:middle line:84% of the hydronium and the hydroxide ions 00:11:29.240 --> 00:11:32.610 align:middle line:90% is equal to this constant. 00:11:32.610 --> 00:11:35.040 align:middle line:90% Which constant is not. 00:11:35.040 --> 00:11:38.220 align:middle line:84% Because you remember that the equilibrium constants vary 00:11:38.220 --> 00:11:47.520 align:middle line:84% a lot with temperature, and also this constant has dimensions 00:11:47.520 --> 00:11:53.940 align:middle line:84% because, for this particular reaction, usually also 00:11:53.940 --> 00:11:58.770 align:middle line:84% the equilibrium constant based on concentration is dimensional. 00:11:58.770 --> 00:12:03.240 align:middle line:84% It has dimensions of the overall concentration 00:12:03.240 --> 00:12:08.853 align:middle line:84% to the power equal to the sum of the stoichiometric coefficients. 00:12:08.853 --> 00:12:10.270 align:middle line:84% Now, for this particular reaction, 00:12:10.270 --> 00:12:12.630 align:middle line:84% the sum of the stoichiometric coefficients 00:12:12.630 --> 00:12:17.070 align:middle line:84% is 0 because minus 2 plus 1 plus 1 is 0. 00:12:17.070 --> 00:12:21.080 align:middle line:90% So this is dimensionless. 00:12:21.080 --> 00:12:24.560 align:middle line:84% It coincides with the equilibrium constant K(T) 00:12:24.560 --> 00:12:27.300 align:middle line:90% without the c. 00:12:27.300 --> 00:12:30.030 align:middle line:84% So this is dimensionless, but this is not. 00:12:30.030 --> 00:12:35.390 align:middle line:84% So the Kw has dimensions of mole squared per liter squared. 00:12:35.390 --> 00:12:39.090 align:middle line:90% 00:12:39.090 --> 00:12:44.070 align:middle line:84% At 25 degrees C, it is almost exactly equal to 10 00:12:44.070 --> 00:12:46.580 align:middle line:90% to the minus 14 in those units. 00:12:46.580 --> 00:12:51.140 align:middle line:90% 00:12:51.140 --> 00:12:58.550 align:middle line:84% And if you have a neutral solution in which there 00:12:58.550 --> 00:13:02.660 align:middle line:84% is no acid or base dissolved that 00:13:02.660 --> 00:13:05.800 align:middle line:84% would capture some of these ions, 00:13:05.800 --> 00:13:12.230 align:middle line:84% they would be equally populated, so their concentrations will 00:13:12.230 --> 00:13:19.300 align:middle line:84% be equal and, therefore, each one takes 10 to the minus 7. 00:13:19.300 --> 00:13:22.810 align:middle line:84% 10 to the minus 7 times 10 to the minus 7 00:13:22.810 --> 00:13:25.270 align:middle line:90% gives 10 to the minus 14. 00:13:25.270 --> 00:13:28.570 align:middle line:84% And 10 to the minus 7 put into the log base 00:13:28.570 --> 00:13:33.610 align:middle line:84% 10 with the minus in front makes pH equals 7. 00:13:33.610 --> 00:13:36.160 align:middle line:84% And that pH equals 7 is, therefore, 00:13:36.160 --> 00:13:42.583 align:middle line:90% the notion of a neutral-- 00:13:42.583 --> 00:13:45.190 align:middle line:90% 00:13:45.190 --> 00:13:47.110 align:middle line:90% like pure water is neutral-- 00:13:47.110 --> 00:13:50.720 align:middle line:90% and it has a pH equal to 7. 00:13:50.720 --> 00:13:53.940 align:middle line:84% At 25 C, though, because if you go at different temperatures-- 00:13:53.940 --> 00:13:56.450 align:middle line:84% for example, you go here, you see 00:13:56.450 --> 00:13:59.690 align:middle line:84% how, for example, at body temperature, which 00:13:59.690 --> 00:14:08.490 align:middle line:84% is 37 degrees C, the Kw is not 10 to the minus 14, 00:14:08.490 --> 00:14:11.820 align:middle line:90% but it's 3.55 times. 00:14:11.820 --> 00:14:15.710 align:middle line:84% And that makes the pH for neutral 6.7-- 00:14:15.710 --> 00:14:18.110 align:middle line:90% at 100 degrees, 6.17. 00:14:18.110 --> 00:14:23.940 align:middle line:90% Or, at 0 degrees, it's 7.47. 00:14:23.940 --> 00:14:29.278 align:middle line:84% So this is one of the applications of equilibrium. 00:14:29.278 --> 00:14:31.195 align:middle line:84% Of course, it's just one of the many examples. 00:14:31.195 --> 00:14:37.390 align:middle line:90% 00:14:37.390 --> 00:14:42.850 align:middle line:84% So now we already talked about the elements of the model 00:14:42.850 --> 00:14:47.360 align:middle line:84% for chemical kinetics, and we want to develop more details. 00:14:47.360 --> 00:14:51.190 align:middle line:84% As you see, we are transitioning from the second part 00:14:51.190 --> 00:14:53.800 align:middle line:84% of the course to the third part because we've already 00:14:53.800 --> 00:14:56.620 align:middle line:84% been talking about nonequilibrium 00:14:56.620 --> 00:14:58.600 align:middle line:84% from the first lecture, I believe. 00:14:58.600 --> 00:15:02.620 align:middle line:90% 00:15:02.620 --> 00:15:06.220 align:middle line:84% But the more we advance in and we 00:15:06.220 --> 00:15:09.370 align:middle line:84% get close to our stable equilibrium, which 00:15:09.370 --> 00:15:12.700 align:middle line:84% will be the end of the course, we'll 00:15:12.700 --> 00:15:18.640 align:middle line:84% talk more and more about nonequilibrium states. 00:15:18.640 --> 00:15:23.260 align:middle line:84% You remember this picture in which we presented 00:15:23.260 --> 00:15:27.110 align:middle line:84% the idea of a potential energy surface, 00:15:27.110 --> 00:15:32.190 align:middle line:84% and that's what we used to define the activated complex. 00:15:32.190 --> 00:15:38.730 align:middle line:84% The potential energy surface represents the energy 00:15:38.730 --> 00:15:43.380 align:middle line:84% at 0 temperature of the various configurations with which you 00:15:43.380 --> 00:15:47.190 align:middle line:84% can find the reactants or the products. 00:15:47.190 --> 00:15:49.680 align:middle line:84% For example, these are the reactants. 00:15:49.680 --> 00:15:51.390 align:middle line:90% These are the products. 00:15:51.390 --> 00:15:58.500 align:middle line:84% And here, in this scenario, and along the path 00:15:58.500 --> 00:16:02.580 align:middle line:84% of steepest ascent here and steepest descent 00:16:02.580 --> 00:16:07.510 align:middle line:84% here, or vice versa, you find the saddle point, 00:16:07.510 --> 00:16:13.070 align:middle line:84% which is the lowest point that allows 00:16:13.070 --> 00:16:18.380 align:middle line:84% you to go from one valley of this landscape 00:16:18.380 --> 00:16:20.100 align:middle line:90% to this other valley. 00:16:20.100 --> 00:16:22.780 align:middle line:90% 00:16:22.780 --> 00:16:31.020 align:middle line:90% Now that's at 0 temperature. 00:16:31.020 --> 00:16:33.930 align:middle line:84% If there is temperature, it means 00:16:33.930 --> 00:16:38.940 align:middle line:84% that part of the energy of the molecules 00:16:38.940 --> 00:16:43.560 align:middle line:90% is in their degrees of freedom-- 00:16:43.560 --> 00:16:49.280 align:middle line:84% translational, rotational, vibrational, electronic. 00:16:49.280 --> 00:16:56.350 align:middle line:84% They all are capable of storing energy in the molecules. 00:16:56.350 --> 00:17:02.650 align:middle line:84% At low temperatures, most of the energy is stored in the-- 00:17:02.650 --> 00:17:06.260 align:middle line:84% let's call them easy degrees of freedom, 00:17:06.260 --> 00:17:11.079 align:middle line:84% which is the translational, the rotational. 00:17:11.079 --> 00:17:14.170 align:middle line:84% As you increase in temperature, also the vibrational 00:17:14.170 --> 00:17:16.060 align:middle line:90% become important. 00:17:16.060 --> 00:17:21.010 align:middle line:84% And if you go even higher, also electronic, 00:17:21.010 --> 00:17:25.060 align:middle line:84% you can excite the electrons and, therefore, they 00:17:25.060 --> 00:17:30.280 align:middle line:84% could jump up and down from the various orbitals that 00:17:30.280 --> 00:17:34.780 align:middle line:84% represent the energy levels of the molecule. 00:17:34.780 --> 00:17:45.310 align:middle line:84% So if, therefore, there is a kinetic energy, first of all, 00:17:45.310 --> 00:17:48.610 align:middle line:84% we've seen that there are collisions. 00:17:48.610 --> 00:17:52.900 align:middle line:84% Molecules, by having kinetic energy, 00:17:52.900 --> 00:17:56.850 align:middle line:90% undergo a lot of collisions. 00:17:56.850 --> 00:17:58.690 align:middle line:84% And, as a result of these collisions, 00:17:58.690 --> 00:18:00.390 align:middle line:84% they put together their kinetic energy, 00:18:00.390 --> 00:18:06.270 align:middle line:84% and they form these activated complexes, sometimes. 00:18:06.270 --> 00:18:08.680 align:middle line:84% Sometimes, they don't form the activated complex. 00:18:08.680 --> 00:18:14.130 align:middle line:84% Sometimes, they just jump on the sides of the valleys, which 00:18:14.130 --> 00:18:16.760 align:middle line:90% have higher energy. 00:18:16.760 --> 00:18:19.830 align:middle line:90% But then they go back. 00:18:19.830 --> 00:18:28.430 align:middle line:84% So they bounce in the valley up in these higher energies. 00:18:28.430 --> 00:18:32.360 align:middle line:84% So the result of collision might be the formation of things 00:18:32.360 --> 00:18:35.010 align:middle line:84% that look like almost activated complexes, 00:18:35.010 --> 00:18:39.170 align:middle line:84% but they're not this particular one 00:18:39.170 --> 00:18:45.200 align:middle line:84% or in this area that would allow going from reactants to products 00:18:45.200 --> 00:18:46.135 align:middle line:90% and vice versa. 00:18:46.135 --> 00:18:51.700 align:middle line:90% 00:18:51.700 --> 00:19:01.430 align:middle line:84% And also, we've seen that, as a result of collision, 00:19:01.430 --> 00:19:04.340 align:middle line:84% the kinetic energy gets momentarily transformed 00:19:04.340 --> 00:19:13.070 align:middle line:84% into elastic energy, like two springs that collide. 00:19:13.070 --> 00:19:15.800 align:middle line:84% So they store the energy for a while, 00:19:15.800 --> 00:19:24.270 align:middle line:84% and then they explode back and shoot the fragments 00:19:24.270 --> 00:19:33.210 align:middle line:84% of the fission of this activated complex in various directions. 00:19:33.210 --> 00:19:37.760 align:middle line:84% These fragments will have a higher energy 00:19:37.760 --> 00:19:39.570 align:middle line:90% than the neighbors. 00:19:39.570 --> 00:19:42.690 align:middle line:84% And, therefore, by the other collisions, 00:19:42.690 --> 00:19:46.610 align:middle line:84% the nonchemical collisions, that energy will get redistributed 00:19:46.610 --> 00:19:52.500 align:middle line:84% and will either heat up if they have higher energies, 00:19:52.500 --> 00:19:55.880 align:middle line:84% but sometimes, also, they also have lower energies 00:19:55.880 --> 00:20:03.350 align:middle line:84% because some of the energy, if the reaction has been effective, 00:20:03.350 --> 00:20:08.980 align:middle line:84% has been retained by the new configuration 00:20:08.980 --> 00:20:16.110 align:middle line:84% of the molecule in the form of potential energy, like here. 00:20:16.110 --> 00:20:24.030 align:middle line:84% So if you go from here in this direction, 00:20:24.030 --> 00:20:31.370 align:middle line:84% so from left to right here, this much potential energy 00:20:31.370 --> 00:20:35.130 align:middle line:84% is stored in the new configuration. 00:20:35.130 --> 00:20:40.000 align:middle line:84% So it came from kinetic energy, went up to form this complex, 00:20:40.000 --> 00:20:42.910 align:middle line:84% and then, as you go down, you release-- 00:20:42.910 --> 00:20:45.930 align:middle line:84% so this difference from here to here 00:20:45.930 --> 00:20:48.750 align:middle line:84% will be the kinetic energy that is released 00:20:48.750 --> 00:20:53.730 align:middle line:84% by the fragments of the explosion, so to speak, 00:20:53.730 --> 00:20:56.790 align:middle line:84% whereas this other difference is not anymore 00:20:56.790 --> 00:20:58.830 align:middle line:84% available as kinetic energy because it 00:20:58.830 --> 00:21:00.485 align:middle line:90% has become potential energy. 00:21:00.485 --> 00:21:04.810 align:middle line:90% 00:21:04.810 --> 00:21:16.585 align:middle line:84% Now the collisions are governed, as you know, 00:21:16.585 --> 00:21:22.216 align:middle line:84% by the Maxwell-Boltzmann distribution of velocities. 00:21:22.216 --> 00:21:26.080 align:middle line:90% 00:21:26.080 --> 00:21:37.640 align:middle line:84% And this distribution-- maybe later on, if I have time, 00:21:37.640 --> 00:21:49.210 align:middle line:84% I'll show you how it can be derived from the maximum entropy 00:21:49.210 --> 00:21:51.580 align:middle line:90% principle. 00:21:51.580 --> 00:21:56.470 align:middle line:84% And so it is the distribution that maximizes the entropy given 00:21:56.470 --> 00:21:57.966 align:middle line:90% certain constraints. 00:21:57.966 --> 00:22:04.530 align:middle line:90% 00:22:04.530 --> 00:22:09.480 align:middle line:84% So that means that these molecules do not all 00:22:09.480 --> 00:22:11.640 align:middle line:90% have the same velocity. 00:22:11.640 --> 00:22:22.480 align:middle line:84% And also, this nice GIF that I took from Wikipedia shows-- 00:22:22.480 --> 00:22:25.900 align:middle line:84% if you look at it carefully, it starts from a nonequilibrium 00:22:25.900 --> 00:22:30.130 align:middle line:84% situation in which there are some particles that have 00:22:30.130 --> 00:22:31.520 align:middle line:90% higher velocities than others. 00:22:31.520 --> 00:22:37.100 align:middle line:84% So it's a slightly skewed distribution. 00:22:37.100 --> 00:22:41.200 align:middle line:84% And it shows how quickly the many collisions 00:22:41.200 --> 00:22:44.770 align:middle line:84% will redistribute and return the distribution 00:22:44.770 --> 00:22:47.810 align:middle line:90% back to Maxwell-Boltzmann. 00:22:47.810 --> 00:22:52.630 align:middle line:90% 00:22:52.630 --> 00:22:58.860 align:middle line:84% This is like a pictorial representation of the idea 00:22:58.860 --> 00:23:05.400 align:middle line:84% that nonchemical collisions is a fast dynamics 00:23:05.400 --> 00:23:12.300 align:middle line:84% because collisions are many and occur many more than chemical 00:23:12.300 --> 00:23:13.080 align:middle line:90% ones. 00:23:13.080 --> 00:23:18.810 align:middle line:84% And, therefore, you can assume that this distribution is almost 00:23:18.810 --> 00:23:21.621 align:middle line:90% immediately-- 00:23:21.621 --> 00:23:25.230 align:middle line:84% it returns almost immediately to Maxwellian 00:23:25.230 --> 00:23:29.250 align:middle line:84% after each chemical reaction has released or absorbed 00:23:29.250 --> 00:23:30.935 align:middle line:90% some local energy. 00:23:30.935 --> 00:23:36.200 align:middle line:90% 00:23:36.200 --> 00:23:46.210 align:middle line:84% So now let's go back to this path from the reactants 00:23:46.210 --> 00:23:50.260 align:middle line:84% from the bottom of the reactants valley to the activated complex, 00:23:50.260 --> 00:23:56.480 align:middle line:84% the saddle point, to the bottom of the products valley. 00:23:56.480 --> 00:24:02.110 align:middle line:84% And here, on the side, I have drawn 00:24:02.110 --> 00:24:05.270 align:middle line:84% not exactly the Maxwell-Boltzmann distribution, 00:24:05.270 --> 00:24:09.670 align:middle line:84% but a distribution that can be obtained 00:24:09.670 --> 00:24:12.730 align:middle line:90% from the Maxwell-Boltzmann. 00:24:12.730 --> 00:24:16.500 align:middle line:84% And it is a distribution of the kinetic energy that is 00:24:16.500 --> 00:24:18.380 align:middle line:90% made available upon collision. 00:24:18.380 --> 00:24:22.490 align:middle line:90% 00:24:22.490 --> 00:24:27.360 align:middle line:84% Because what we need in order to climb this diagram here, 00:24:27.360 --> 00:24:29.640 align:middle line:84% this valley, to go to the saddle point, 00:24:29.640 --> 00:24:34.480 align:middle line:84% we need enough energy for the collision-- 00:24:34.480 --> 00:24:37.200 align:middle line:90% as a result of the collision. 00:24:37.200 --> 00:24:47.470 align:middle line:84% And so this curve, it represents the probability density function 00:24:47.470 --> 00:24:50.650 align:middle line:84% not for velocity, but for kinetic energy 00:24:50.650 --> 00:24:57.220 align:middle line:84% of particles from the Maxwell-Boltzmann distribution 00:24:57.220 --> 00:24:58.010 align:middle line:90% of velocity. 00:24:58.010 --> 00:25:01.740 align:middle line:90% 00:25:01.740 --> 00:25:04.320 align:middle line:84% So as a function of kinetic energy, which is plotted 00:25:04.320 --> 00:25:09.430 align:middle line:90% this way, that's the PDF. 00:25:09.430 --> 00:25:18.270 align:middle line:84% In order for a chemical collision to be effective, 00:25:18.270 --> 00:25:22.920 align:middle line:84% we need that it has at least this much kinetic energy so as 00:25:22.920 --> 00:25:24.830 align:middle line:90% to climb up here. 00:25:24.830 --> 00:25:33.420 align:middle line:84% So all the collisions that have energies above that are good. 00:25:33.420 --> 00:25:38.490 align:middle line:84% So if you do this, the area under this probability 00:25:38.490 --> 00:25:42.600 align:middle line:84% distribution function, which is the integral from this energy 00:25:42.600 --> 00:25:52.930 align:middle line:84% to infinity, that's the overall probability 00:25:52.930 --> 00:25:56.260 align:middle line:84% that your collision has a kinetic energy higher 00:25:56.260 --> 00:26:03.580 align:middle line:84% than this value delta H. So that's 00:26:03.580 --> 00:26:04.820 align:middle line:90% the result of the integral. 00:26:04.820 --> 00:26:09.550 align:middle line:84% So on this side, I need that to be-- 00:26:09.550 --> 00:26:13.240 align:middle line:84% therefore, the kinetic energy that I need 00:26:13.240 --> 00:26:23.150 align:middle line:84% is the enthalpy of the forward reaction or forward half 00:26:23.150 --> 00:26:26.360 align:middle line:84% reaction of formation of the activated 00:26:26.360 --> 00:26:30.670 align:middle line:90% complex from the reactants. 00:26:30.670 --> 00:26:35.050 align:middle line:84% On the other side, I start lower. 00:26:35.050 --> 00:26:37.490 align:middle line:84% I'm in the same box, so-to-speak. 00:26:37.490 --> 00:26:39.970 align:middle line:84% So the molecules are in the same mixture, 00:26:39.970 --> 00:26:43.540 align:middle line:84% same Maxwellian distribution, same temperature. 00:26:43.540 --> 00:26:46.550 align:middle line:84% So this curve is the same as that one, 00:26:46.550 --> 00:26:51.230 align:middle line:90% only that it starts below here. 00:26:51.230 --> 00:26:57.720 align:middle line:84% Therefore, if I want to form the activated complex 00:26:57.720 --> 00:27:01.720 align:middle line:84% from this side, I need this much energy, 00:27:01.720 --> 00:27:08.590 align:middle line:84% the enthalpy of reaction of the backward half reaction. 00:27:08.590 --> 00:27:15.160 align:middle line:84% And so the probability to have the necessary energy 00:27:15.160 --> 00:27:20.590 align:middle line:84% as a result of the collision is this one, this area, which 00:27:20.590 --> 00:27:24.760 align:middle line:84% is equal to this exponential formula. 00:27:24.760 --> 00:27:28.600 align:middle line:84% And this exponential formula is what 00:27:28.600 --> 00:27:29.940 align:middle line:90% is called the Arrhenius law. 00:27:29.940 --> 00:27:33.060 align:middle line:90% 00:27:33.060 --> 00:27:36.930 align:middle line:84% And, of course, from this side, it's smaller. 00:27:36.930 --> 00:27:41.820 align:middle line:84% So there is more probability to get up here from the reactants 00:27:41.820 --> 00:27:55.180 align:middle line:84% than from the products in this case because the delta h plus is 00:27:55.180 --> 00:27:58.120 align:middle line:90% smaller than the delta h minus. 00:27:58.120 --> 00:28:02.970 align:middle line:84% So this is one of the ingredients 00:28:02.970 --> 00:28:10.870 align:middle line:84% in order to formulate a theory or a model 00:28:10.870 --> 00:28:14.210 align:middle line:84% for the rate of chemical reactions, 00:28:14.210 --> 00:28:20.240 align:middle line:84% for the rate at which chemical reactions proceed. 00:28:20.240 --> 00:28:22.580 align:middle line:84% And in our language, therefore, it's 00:28:22.580 --> 00:28:26.690 align:middle line:84% for the rate at which the reaction coordinate changes 00:28:26.690 --> 00:28:29.650 align:middle line:90% with time, epsilon dot. 00:28:29.650 --> 00:28:32.740 align:middle line:90% So the bottom line is here. 00:28:32.740 --> 00:28:41.840 align:middle line:84% Epsilon dot will turn out to be the net result of how many-- 00:28:41.840 --> 00:28:45.080 align:middle line:84% of the epsilon dot of the forward half 00:28:45.080 --> 00:28:49.810 align:middle line:84% reaction and the epsilon dot of the backward half reaction. 00:28:49.810 --> 00:28:52.480 align:middle line:90% 00:28:52.480 --> 00:28:56.670 align:middle line:90% And these ones are done in here. 00:28:56.670 --> 00:28:59.610 align:middle line:84% So this column does the half reaction. 00:28:59.610 --> 00:29:06.100 align:middle line:84% This column does the forward and this does the backward. 00:29:06.100 --> 00:29:10.390 align:middle line:84% So we talked about we can view this thing in terms of this 00:29:10.390 --> 00:29:13.060 align:middle line:90% in terms of probabilities. 00:29:13.060 --> 00:29:16.620 align:middle line:84% So the Arrhenius formula gives us, 00:29:16.620 --> 00:29:19.190 align:middle line:84% as we've discussed, the probability 00:29:19.190 --> 00:29:22.730 align:middle line:84% to have enough collisions, enough kinetic energy 00:29:22.730 --> 00:29:27.100 align:middle line:84% from the collision so that we form the activated complex. 00:29:27.100 --> 00:29:30.480 align:middle line:90% 00:29:30.480 --> 00:29:34.610 align:middle line:84% Then we need another probability. 00:29:34.610 --> 00:29:37.670 align:middle line:84% Once I have the activated complex, which 00:29:37.670 --> 00:29:40.820 align:middle line:84% is that springs that are compressed 00:29:40.820 --> 00:29:45.380 align:middle line:84% and short-lived and ready to explode, 00:29:45.380 --> 00:29:51.890 align:middle line:84% the question is, will it explode towards the reactant valley 00:29:51.890 --> 00:29:53.460 align:middle line:90% or towards the product valley? 00:29:53.460 --> 00:29:57.830 align:middle line:90% 00:29:57.830 --> 00:30:01.350 align:middle line:84% That depends also on how the molecules have collided. 00:30:01.350 --> 00:30:03.450 align:middle line:84% It's not just the kinetic energy. 00:30:03.450 --> 00:30:10.400 align:middle line:84% It is also the angles with which they collide that is important 00:30:10.400 --> 00:30:21.050 align:middle line:84% because that allows to have the right vibrational modes to be 00:30:21.050 --> 00:30:27.290 align:middle line:84% triggered and, therefore, form the new configurations. 00:30:27.290 --> 00:30:32.130 align:middle line:84% So it is complex reasoning about that. 00:30:32.130 --> 00:30:37.610 align:middle line:84% It's not simple to make a theory about this part 00:30:37.610 --> 00:30:44.060 align:middle line:84% of the probability, and, therefore, 00:30:44.060 --> 00:30:48.450 align:middle line:84% you just say, all right, let's call it a probability. 00:30:48.450 --> 00:30:54.080 align:middle line:84% And let's call it A. And there is one for the forward and one 00:30:54.080 --> 00:30:59.280 align:middle line:90% for the backward. 00:30:59.280 --> 00:31:09.210 align:middle line:84% But then you also need another situation to occur, 00:31:09.210 --> 00:31:12.330 align:middle line:84% that it's not enough that the particles have 00:31:12.330 --> 00:31:15.030 align:middle line:84% the proper kinetic energy, but you need 00:31:15.030 --> 00:31:19.090 align:middle line:90% the right particles to collide. 00:31:19.090 --> 00:31:23.620 align:middle line:84% I need, for example, two hydrogens and maybe something 00:31:23.620 --> 00:31:24.290 align:middle line:90% else. 00:31:24.290 --> 00:31:26.440 align:middle line:84% So it'll be a three-body reaction. 00:31:26.440 --> 00:31:29.140 align:middle line:84% So because if I have a collision with the right energy 00:31:29.140 --> 00:31:32.350 align:middle line:84% but the wrong molecules, that's useless. 00:31:32.350 --> 00:31:37.700 align:middle line:84% So you remember that the probability 00:31:37.700 --> 00:31:44.780 align:middle line:84% to have a molecule in a mixture is the mole fraction, 00:31:44.780 --> 00:31:47.060 align:middle line:84% a nice number between 0 and 1, which 00:31:47.060 --> 00:31:48.740 align:middle line:90% represents how many you have. 00:31:48.740 --> 00:31:52.130 align:middle line:84% And so the chance that you have those molecules 00:31:52.130 --> 00:31:54.200 align:middle line:90% is the probability. 00:31:54.200 --> 00:32:00.780 align:middle line:84% And the chance that you have several of them-- 00:32:00.780 --> 00:32:04.420 align:middle line:84% it's the multiplication of the probabilities. 00:32:04.420 --> 00:32:09.920 align:middle line:84% And if I need two molecules of the same kind, 00:32:09.920 --> 00:32:13.300 align:middle line:84% then that probability counts twice. 00:32:13.300 --> 00:32:18.760 align:middle line:84% So it's the mole fraction to the power 2. 00:32:18.760 --> 00:32:21.580 align:middle line:84% And those amounts are precisely what 00:32:21.580 --> 00:32:24.080 align:middle line:84% the stoichiometric coefficients are. 00:32:24.080 --> 00:32:28.070 align:middle line:84% So that is why this product is a product of probabilities 00:32:28.070 --> 00:32:31.070 align:middle line:84% and the exponents of the stoichiometric coefficients 00:32:31.070 --> 00:32:33.710 align:middle line:84% which count properly how many times you 00:32:33.710 --> 00:32:37.460 align:middle line:84% need that kind of molecule to be present in a collision. 00:32:37.460 --> 00:32:42.710 align:middle line:84% Most of the times, effectively, the most probable collisions 00:32:42.710 --> 00:32:46.960 align:middle line:84% are two-body, just two molecules that collide. 00:32:46.960 --> 00:32:48.550 align:middle line:84% Already, the three-body reactions 00:32:48.550 --> 00:32:52.630 align:middle line:84% are much fewer and going four and so on, it's 00:32:52.630 --> 00:32:56.660 align:middle line:90% almost irrelevant. 00:32:56.660 --> 00:32:59.786 align:middle line:90% So this is a bit hypothetical. 00:32:59.786 --> 00:33:02.490 align:middle line:90% 00:33:02.490 --> 00:33:08.490 align:middle line:84% Again, that colleague of mine would call it Disneyland. 00:33:08.490 --> 00:33:14.000 align:middle line:90% But for a theory, that's OK. 00:33:14.000 --> 00:33:18.900 align:middle line:90% 00:33:18.900 --> 00:33:23.220 align:middle line:84% For the practice, it means that you are never 00:33:23.220 --> 00:33:29.010 align:middle line:84% going to work with an overall chemical reaction scheme. 00:33:29.010 --> 00:33:33.790 align:middle line:84% But you will split the chemical reaction in many steps. 00:33:33.790 --> 00:33:37.380 align:middle line:84% So that is why we need so many reactions, like we will see 00:33:37.380 --> 00:33:41.280 align:middle line:84% again in the hydrogen will be 24 reactions just 00:33:41.280 --> 00:33:46.060 align:middle line:84% for the oxidation of hydrogen. It's not just one reaction. 00:33:46.060 --> 00:33:50.650 align:middle line:84% And those detailed kinetic schemes 00:33:50.650 --> 00:33:58.090 align:middle line:84% are mostly done with two-body or three-body reactions. 00:33:58.090 --> 00:34:03.610 align:middle line:84% Well, anyway, these are probabilities. 00:34:03.610 --> 00:34:06.160 align:middle line:84% We assume they are independent events so we 00:34:06.160 --> 00:34:10.179 align:middle line:90% can multiply the probabilities. 00:34:10.179 --> 00:34:15.940 align:middle line:84% And yeah, we can also rewrite the mole fractions because here, 00:34:15.940 --> 00:34:19.429 align:middle line:84% we want to write it in terms of the concentration. 00:34:19.429 --> 00:34:21.489 align:middle line:84% So the mole fractions can be written 00:34:21.489 --> 00:34:25.780 align:middle line:90% in terms of concentrations. 00:34:25.780 --> 00:34:31.020 align:middle line:84% So we multiply this, that, and that, and we say, all right. 00:34:31.020 --> 00:34:37.500 align:middle line:84% We assume that the rate of change of the forward half 00:34:37.500 --> 00:34:42.270 align:middle line:84% reaction is given by the product of these probabilities. 00:34:42.270 --> 00:34:43.480 align:middle line:90% And this is the formula. 00:34:43.480 --> 00:34:47.489 align:middle line:90% This is the multiplication. 00:34:47.489 --> 00:34:52.650 align:middle line:84% And then the traditional naming is that, yes, 00:34:52.650 --> 00:34:54.940 align:middle line:84% you have this product of concentrations 00:34:54.940 --> 00:35:00.650 align:middle line:84% to the stoichiometric coefficients that's singled out, 00:35:00.650 --> 00:35:10.240 align:middle line:84% the prefactor of that is called the rate constant, 00:35:10.240 --> 00:35:11.870 align:middle line:90% which is not really a constant. 00:35:11.870 --> 00:35:14.000 align:middle line:84% Again, it's a function of temperature. 00:35:14.000 --> 00:35:17.370 align:middle line:90% 00:35:17.370 --> 00:35:23.010 align:middle line:84% And it is an Arrhenius factor times a prefactor here. 00:35:23.010 --> 00:35:25.290 align:middle line:84% Clearly, what I've done for the forward 00:35:25.290 --> 00:35:26.980 align:middle line:90% I can do also for the backward. 00:35:26.980 --> 00:35:29.820 align:middle line:84% So you just change the pluses into minuses, 00:35:29.820 --> 00:35:34.450 align:middle line:90% and you get the backward rate. 00:35:34.450 --> 00:35:38.830 align:middle line:84% Of course, here is the Arrhenius term 00:35:38.830 --> 00:35:44.010 align:middle line:84% that is important for the kinetic energy. 00:35:44.010 --> 00:35:51.290 align:middle line:84% And here are the concentrations based on the backward 00:35:51.290 --> 00:35:53.160 align:middle line:90% stoichiometric coefficients. 00:35:53.160 --> 00:35:56.670 align:middle line:84% So, in the end, the net reaction rate 00:35:56.670 --> 00:36:01.170 align:middle line:84% will be the difference between the forward and the backwards. 00:36:01.170 --> 00:36:05.620 align:middle line:84% And when we are at equilibrium, they will be equal to 0-- 00:36:05.620 --> 00:36:07.603 align:middle line:90% I'm sorry, not equal to 0. 00:36:07.603 --> 00:36:09.020 align:middle line:84% They will be equal to one another. 00:36:09.020 --> 00:36:16.990 align:middle line:90% 00:36:16.990 --> 00:36:24.010 align:middle line:84% Notice also, here-- just we need to remind it later on-- 00:36:24.010 --> 00:36:30.520 align:middle line:84% that the way we have defined the half reactions, 00:36:30.520 --> 00:36:34.090 align:middle line:84% these stoichiometric coefficients with the plus 00:36:34.090 --> 00:36:38.163 align:middle line:84% and with the minus are positive numbers. 00:36:38.163 --> 00:36:43.100 align:middle line:90% 00:36:43.100 --> 00:36:54.340 align:middle line:84% And they are related to the overall reaction 00:36:54.340 --> 00:36:59.122 align:middle line:84% stoichiometric coefficients in this way, 00:36:59.122 --> 00:37:03.530 align:middle line:84% the one of the backward minus the one of the forward, at least 00:37:03.530 --> 00:37:06.650 align:middle line:84% according to our convention that puts everything on the left hand 00:37:06.650 --> 00:37:11.210 align:middle line:84% side and with a positive stoichiometric coefficients 00:37:11.210 --> 00:37:14.680 align:middle line:90% for what we call the products. 00:37:14.680 --> 00:37:19.070 align:middle line:84% Of course, you can always switch point of view and call what-- 00:37:19.070 --> 00:37:22.030 align:middle line:84% the products, call them reactants, and then you'll 00:37:22.030 --> 00:37:23.130 align:middle line:90% change all the signs. 00:37:23.130 --> 00:37:28.280 align:middle line:90% 00:37:28.280 --> 00:37:42.230 align:middle line:84% So, for example, here is the rate constants 00:37:42.230 --> 00:37:46.940 align:middle line:84% for the set of 24 reactions that we already 00:37:46.940 --> 00:38:00.580 align:middle line:84% mentioned that is the most trusted, detailed kinetic scheme 00:38:00.580 --> 00:38:10.300 align:middle line:84% for the oxidation of hydrogen in pure oxygen. As you see here, 00:38:10.300 --> 00:38:18.790 align:middle line:84% they are listed from number 11 to 24. 00:38:18.790 --> 00:38:20.320 align:middle line:90% They are two-body reactions. 00:38:20.320 --> 00:38:25.070 align:middle line:84% You see, on both sides, you have two molecules. 00:38:25.070 --> 00:38:30.380 align:middle line:84% So, in order for this reaction to form its activated complex, 00:38:30.380 --> 00:38:36.830 align:middle line:84% you have to have a collision of an oxygen atom and a hydrogen 00:38:36.830 --> 00:38:38.690 align:middle line:90% molecule. 00:38:38.690 --> 00:38:40.820 align:middle line:84% And if it is effective, that will 00:38:40.820 --> 00:38:45.415 align:middle line:90% form a hydrogen atom and an OH. 00:38:45.415 --> 00:38:50.430 align:middle line:90% And here is the prefactor. 00:38:50.430 --> 00:38:53.580 align:middle line:84% Actually, the prefactor-- what we call the A of T now, 00:38:53.580 --> 00:39:00.240 align:middle line:84% here it is written as a constant times T to the power b-- 00:39:00.240 --> 00:39:03.220 align:middle line:84% this is the power of the temperature. 00:39:03.220 --> 00:39:07.260 align:middle line:90% These are experimental values. 00:39:07.260 --> 00:39:18.630 align:middle line:84% And the activation energy, so the Arrhenius threshold 00:39:18.630 --> 00:39:20.150 align:middle line:90% is listed here. 00:39:20.150 --> 00:39:28.960 align:middle line:90% 00:39:28.960 --> 00:39:34.540 align:middle line:84% It is also noteworthy that these three-body reactions, instead, 00:39:34.540 --> 00:39:43.500 align:middle line:84% you see on the first 10 reactions require three bodies, 00:39:43.500 --> 00:39:49.430 align:middle line:84% three-body collisions on either one side or the other or both-- 00:39:49.430 --> 00:39:51.390 align:middle line:90% just one side or the other. 00:39:51.390 --> 00:39:55.780 align:middle line:90% 00:39:55.780 --> 00:39:59.175 align:middle line:84% And they're represented with zero activation energy. 00:39:59.175 --> 00:40:05.600 align:middle line:90% 00:40:05.600 --> 00:40:08.840 align:middle line:84% We'll actually have, later on hopefully 00:40:08.840 --> 00:40:12.710 align:middle line:84% today, another example in which this list of reactions 00:40:12.710 --> 00:40:14.950 align:middle line:90% will be analyzed somehow. 00:40:14.950 --> 00:40:19.210 align:middle line:90% 00:40:19.210 --> 00:40:23.890 align:middle line:84% Now let's see what are the consequences of this model 00:40:23.890 --> 00:40:25.670 align:middle line:90% that we have developed. 00:40:25.670 --> 00:40:30.780 align:middle line:84% So we just said we want to represent 00:40:30.780 --> 00:40:37.080 align:middle line:84% reaction rates this way, forward and backward. 00:40:37.080 --> 00:40:44.710 align:middle line:84% Notice that all the ingredients are positive because, after all, 00:40:44.710 --> 00:40:47.220 align:middle line:84% it's a product of probabilities and you cannot have negative 00:40:47.220 --> 00:40:47.920 align:middle line:90% probabilities. 00:40:47.920 --> 00:40:52.050 align:middle line:84% So these are both positive epsilon plus and epsilon 00:40:52.050 --> 00:40:54.220 align:middle line:90% minus are positive. 00:40:54.220 --> 00:40:58.940 align:middle line:90% 00:40:58.940 --> 00:41:01.700 align:middle line:84% Remember also that we defined and we just 00:41:01.700 --> 00:41:05.350 align:middle line:84% recalled the nonequilibrium law of mass action. 00:41:05.350 --> 00:41:10.320 align:middle line:90% 00:41:10.320 --> 00:41:21.360 align:middle line:84% So let's now compute the ratio between the backward 00:41:21.360 --> 00:41:25.760 align:middle line:90% and the forward reaction rates. 00:41:25.760 --> 00:41:29.890 align:middle line:90% So we substitute from up there. 00:41:29.890 --> 00:41:34.420 align:middle line:84% We notice that this product, on the numerator 00:41:34.420 --> 00:41:36.700 align:middle line:84% and the denominator, you can also 00:41:36.700 --> 00:41:42.890 align:middle line:84% write-- since the argument of the exponent are the same, 00:41:42.890 --> 00:41:50.180 align:middle line:84% you can also write it as Ni to the nu minus minus nu plus. 00:41:50.180 --> 00:41:54.890 align:middle line:84% That's N to the stoichiometric coefficient 00:41:54.890 --> 00:41:56.860 align:middle line:90% of the overall reaction. 00:41:56.860 --> 00:42:00.160 align:middle line:90% So you can make this step. 00:42:00.160 --> 00:42:03.160 align:middle line:84% And then we can substitute, from the non-equilibrium law 00:42:03.160 --> 00:42:07.560 align:middle line:84% of mass action, because this is valid always, 00:42:07.560 --> 00:42:10.980 align:middle line:84% and you get this expression here. 00:42:10.980 --> 00:42:15.060 align:middle line:84% Now this expression here is general, 00:42:15.060 --> 00:42:18.690 align:middle line:84% at least given this kind of modeling 00:42:18.690 --> 00:42:21.940 align:middle line:90% for the kinetics is general. 00:42:21.940 --> 00:42:24.310 align:middle line:84% Therefore, it applies also at equilibrium. 00:42:24.310 --> 00:42:30.220 align:middle line:84% So also, when the affinity is equal to 0. 00:42:30.220 --> 00:42:36.190 align:middle line:84% So when the affinity is equal to 0, this e to the 0 is 1. 00:42:36.190 --> 00:42:41.890 align:middle line:84% And also, at equilibrium, the rate of the backward 00:42:41.890 --> 00:42:44.590 align:middle line:84% is equal to the rate of the forward. 00:42:44.590 --> 00:42:48.300 align:middle line:84% They are not 0, but they are equal to one another. 00:42:48.300 --> 00:42:50.920 align:middle line:90% So it's a dynamic equilibrium. 00:42:50.920 --> 00:42:54.980 align:middle line:90% 00:42:54.980 --> 00:42:58.460 align:middle line:84% So this is one, and also, the left-hand side 00:42:58.460 --> 00:43:02.550 align:middle line:90% is one at equilibrium. 00:43:02.550 --> 00:43:09.810 align:middle line:84% And so that gives us a relation between the backward 00:43:09.810 --> 00:43:16.175 align:middle line:84% forward reaction rate constants and the equilibrium constant. 00:43:16.175 --> 00:43:21.280 align:middle line:90% 00:43:21.280 --> 00:43:23.985 align:middle line:84% This is called the principle of detailed balance. 00:43:23.985 --> 00:43:27.530 align:middle line:90% 00:43:27.530 --> 00:43:32.880 align:middle line:84% It's useful because you don't, as you have seen, for example, 00:43:32.880 --> 00:43:35.330 align:middle line:90% we mentioned that-- 00:43:35.330 --> 00:43:41.660 align:middle line:84% in the list that I gave for the hydrogen rates, 00:43:41.660 --> 00:43:48.280 align:middle line:84% we gave only the k plus, the forward. 00:43:48.280 --> 00:43:53.730 align:middle line:84% That's because from the equilibrium constant, which 00:43:53.730 --> 00:43:56.520 align:middle line:84% you can compute from the thermodynamic properties, 00:43:56.520 --> 00:44:00.600 align:middle line:84% like the Gibbs free energy of the reaction, 00:44:00.600 --> 00:44:04.560 align:middle line:84% given the forward rate, you can compute, 00:44:04.560 --> 00:44:09.850 align:middle line:84% using this principle, the backward rate. 00:44:09.850 --> 00:44:11.510 align:middle line:90% So they're not independent. 00:44:11.510 --> 00:44:15.980 align:middle line:90% 00:44:15.980 --> 00:44:20.080 align:middle line:90% That means I can, say-- 00:44:20.080 --> 00:44:24.420 align:middle line:84% well, that means that this combination here, this prefactor 00:44:24.420 --> 00:44:26.030 align:middle line:90% of the exponent, is equal to 1. 00:44:26.030 --> 00:44:28.730 align:middle line:90% 00:44:28.730 --> 00:44:33.290 align:middle line:84% And so, in general, I obtain that the ratio 00:44:33.290 --> 00:44:43.340 align:middle line:84% of the backward to the forward rates is equal to e to the minus 00:44:43.340 --> 00:44:44.890 align:middle line:84% the affinity over the gas constant. 00:44:44.890 --> 00:44:49.470 align:middle line:90% 00:44:49.470 --> 00:44:51.605 align:middle line:84% If you take the logarithm of that-- 00:44:51.605 --> 00:44:54.340 align:middle line:90% 00:44:54.340 --> 00:44:58.000 align:middle line:84% or, actually, if you take the logarithm 00:44:58.000 --> 00:45:01.450 align:middle line:84% of the inverse of that, so the logarithm of the forward 00:45:01.450 --> 00:45:08.820 align:middle line:84% to the negative rate, which by this formula is equal, 00:45:08.820 --> 00:45:12.360 align:middle line:84% therefore, to the affinity divided by R, 00:45:12.360 --> 00:45:15.990 align:middle line:84% we typically call that the degree of disequilibrium 00:45:15.990 --> 00:45:18.730 align:middle line:90% of the reaction. 00:45:18.730 --> 00:45:21.840 align:middle line:84% So, in other words, it's another way of-- we 00:45:21.840 --> 00:45:24.010 align:middle line:84% can call it affinity, but you can also call it 00:45:24.010 --> 00:45:27.670 align:middle line:90% a degree of disequilibrium. 00:45:27.670 --> 00:45:32.190 align:middle line:90% And it's related to this ratio. 00:45:32.190 --> 00:45:35.260 align:middle line:84% Clearly, when the ratio is equal to 1 at equilibrium, 00:45:35.260 --> 00:45:38.880 align:middle line:84% the logarithm is 0, and the degree of disequilibrium is 0. 00:45:38.880 --> 00:45:44.180 align:middle line:84% And the degree of disequilibrium can be positive and negative. 00:45:44.180 --> 00:45:54.480 align:middle line:90% 00:45:54.480 --> 00:46:02.480 align:middle line:84% So let's turn, for example, to other results that follow 00:46:02.480 --> 00:46:06.140 align:middle line:90% from what we've just seen. 00:46:06.140 --> 00:46:09.545 align:middle line:84% Think of the energy balance for a system-- 00:46:09.545 --> 00:46:12.350 align:middle line:90% 00:46:12.350 --> 00:46:14.480 align:middle line:84% for an isolated box in which there 00:46:14.480 --> 00:46:16.970 align:middle line:84% are chemical reactions going on, but there 00:46:16.970 --> 00:46:29.340 align:middle line:84% are no exchanges of energy or particles with the outside 00:46:29.340 --> 00:46:30.780 align:middle line:90% nor entropy also. 00:46:30.780 --> 00:46:36.300 align:middle line:84% So the rate of entropy change is equal only 00:46:36.300 --> 00:46:39.640 align:middle line:84% to the rate of entropy produced by irreversibility. 00:46:39.640 --> 00:46:42.510 align:middle line:90% 00:46:42.510 --> 00:46:44.045 align:middle line:90% Now the rate of entropy change-- 00:46:44.045 --> 00:46:47.310 align:middle line:90% 00:46:47.310 --> 00:46:50.810 align:middle line:84% it's the partial of the entropy with respect to time. 00:46:50.810 --> 00:46:54.330 align:middle line:84% Now in that scenario, for the model that we have given-- 00:46:54.330 --> 00:46:58.080 align:middle line:84% we just reviewed in early slides today 00:46:58.080 --> 00:47:02.460 align:middle line:90% the expression for the entropy. 00:47:02.460 --> 00:47:05.710 align:middle line:84% In our model, in our simple system model, 00:47:05.710 --> 00:47:10.690 align:middle line:84% it's a function of many things, including the epsilons, 00:47:10.690 --> 00:47:12.420 align:middle line:90% the reaction coordinates. 00:47:12.420 --> 00:47:14.730 align:middle line:84% If the epsilons are the only things that 00:47:14.730 --> 00:47:18.540 align:middle line:84% change with time because the energy, the volume, 00:47:18.540 --> 00:47:22.830 align:middle line:84% and the everything else is fixed, 00:47:22.830 --> 00:47:25.290 align:middle line:84% then the rate of change of the entropy 00:47:25.290 --> 00:47:32.010 align:middle line:84% is the partial of S with respect to epsilon times epsilon dot. 00:47:32.010 --> 00:47:34.140 align:middle line:84% But the partial of S with respect to epsilon 00:47:34.140 --> 00:47:35.840 align:middle line:90% is the affinity. 00:47:35.840 --> 00:47:39.600 align:middle line:84% So we have these expressions here. 00:47:39.600 --> 00:47:43.030 align:middle line:84% And since the system is isolated, 00:47:43.030 --> 00:47:49.130 align:middle line:84% that also represents the entropy produced by irreversibility. 00:47:49.130 --> 00:47:52.526 align:middle line:90% And it better be nonnegative. 00:47:52.526 --> 00:47:55.690 align:middle line:90% 00:47:55.690 --> 00:48:01.090 align:middle line:84% In fact, if you write epsilon dot like we have written 00:48:01.090 --> 00:48:07.882 align:middle line:84% as a dot plus minus dot minus multiplied by the affinity, 00:48:07.882 --> 00:48:09.340 align:middle line:84% if you remember that the affinity-- 00:48:09.340 --> 00:48:13.410 align:middle line:90% we just defined it as equal to-- 00:48:13.410 --> 00:48:15.780 align:middle line:84% I mean, not defined-- we just discovered 00:48:15.780 --> 00:48:23.290 align:middle line:84% that it is equal to R times the degree of this equilibrium. 00:48:23.290 --> 00:48:26.620 align:middle line:90% 00:48:26.620 --> 00:48:29.860 align:middle line:84% We get this expression, which gives us 00:48:29.860 --> 00:48:31.600 align:middle line:84% the rate of entropy change, which 00:48:31.600 --> 00:48:34.240 align:middle line:84% has to be equal to the rate of entropy produced 00:48:34.240 --> 00:48:36.260 align:middle line:84% by irreversibility for the closed system, 00:48:36.260 --> 00:48:37.760 align:middle line:90% for the isolated system. 00:48:37.760 --> 00:48:43.360 align:middle line:84% And this one, in fact, is a combination 00:48:43.360 --> 00:48:48.500 align:middle line:84% of numbers that is always positive. 00:48:48.500 --> 00:49:01.080 align:middle line:84% You can find that for any x and y, you have this inequality. 00:49:01.080 --> 00:49:08.000 align:middle line:90% 00:49:08.000 --> 00:49:12.220 align:middle line:84% So at least we are happy that we are not 00:49:12.220 --> 00:49:16.310 align:middle line:84% violating the principle of entropy non-decrease. 00:49:16.310 --> 00:49:20.060 align:middle line:84% And therefore, we accept this model as a reasonable one, 00:49:20.060 --> 00:49:22.010 align:middle line:84% otherwise people would not be using it. 00:49:22.010 --> 00:49:25.730 align:middle line:90% 00:49:25.730 --> 00:49:31.050 align:middle line:84% We can also look at this expression here. 00:49:31.050 --> 00:49:35.310 align:middle line:90% And notice that we could call-- 00:49:35.310 --> 00:49:38.730 align:middle line:84% so the contribution of to the entropy change 00:49:38.730 --> 00:49:41.250 align:middle line:84% or the entropy production for the closed 00:49:41.250 --> 00:49:46.110 align:middle line:84% system or the isolated system has 00:49:46.110 --> 00:49:48.820 align:middle line:84% a positive term and a negative term, 00:49:48.820 --> 00:49:54.320 align:middle line:84% or at least a term with a plus sign and term with a minus sign. 00:49:54.320 --> 00:49:59.090 align:middle line:84% And, actually, since we said that these epsilon pluses 00:49:59.090 --> 00:50:04.550 align:middle line:84% and minuses are always positive, whereas the affinity can 00:50:04.550 --> 00:50:06.150 align:middle line:90% be positive or negative. 00:50:06.150 --> 00:50:10.430 align:middle line:90% 00:50:10.430 --> 00:50:13.990 align:middle line:84% So what we do is, we call these two contributions this way. 00:50:13.990 --> 00:50:19.520 align:middle line:84% We call it S plus, S dot plus, and S dot minus. 00:50:19.520 --> 00:50:22.850 align:middle line:84% So we rewrite that expression for the entropy change 00:50:22.850 --> 00:50:26.130 align:middle line:90% as S dot plus minus S dot minus. 00:50:26.130 --> 00:50:34.400 align:middle line:90% 00:50:34.400 --> 00:50:40.970 align:middle line:84% Of course, we notice here that these results 00:50:40.970 --> 00:50:44.360 align:middle line:90% in always being positive. 00:50:44.360 --> 00:50:51.790 align:middle line:84% But sometimes, it's positive because S plus 00:50:51.790 --> 00:50:54.480 align:middle line:90% is greater than S minus. 00:50:54.480 --> 00:50:59.980 align:middle line:84% But sometimes, it's positive because S minus 00:50:59.980 --> 00:51:02.550 align:middle line:90% is more negative than S plus. 00:51:02.550 --> 00:51:08.360 align:middle line:90% 00:51:08.360 --> 00:51:10.310 align:middle line:90% In fact, we can-- 00:51:10.310 --> 00:51:15.200 align:middle line:84% by this definition, we see that S minus over S plus 00:51:15.200 --> 00:51:18.150 align:middle line:84% is also equal to the ratio of (the backward to) the forward. 00:51:18.150 --> 00:51:21.620 align:middle line:84% So it's essentially related to the degree of disequilibrium 00:51:21.620 --> 00:51:23.190 align:middle line:90% of the overall reaction. 00:51:23.190 --> 00:51:25.820 align:middle line:84% Or you could also take it as the ratio 00:51:25.820 --> 00:51:30.530 align:middle line:84% of the exponentials of the degrees of the disequilibrium 00:51:30.530 --> 00:51:35.056 align:middle line:90% of the half reactions. 00:51:35.056 --> 00:51:39.200 align:middle line:90% 00:51:39.200 --> 00:51:47.510 align:middle line:84% Expressions of this sort, have in the recent literature-- 00:51:47.510 --> 00:51:50.420 align:middle line:90% let's say, 15 years-- 00:51:50.420 --> 00:51:54.350 align:middle line:84% being called fluctuation relations, 00:51:54.350 --> 00:51:57.050 align:middle line:90% there are many of this sort. 00:51:57.050 --> 00:52:00.640 align:middle line:90% 00:52:00.640 --> 00:52:06.770 align:middle line:84% And here, essentially, you see that the degree 00:52:06.770 --> 00:52:18.090 align:middle line:84% of disequilibrium governs the probability that these numbers 00:52:18.090 --> 00:52:22.475 align:middle line:90% be greater or smaller than 1. 00:52:22.475 --> 00:52:28.090 align:middle line:90% 00:52:28.090 --> 00:52:33.190 align:middle line:84% Sometimes people push a bit too much the interpretation, 00:52:33.190 --> 00:52:38.830 align:middle line:84% they say, this is a process that goes against the second law. 00:52:38.830 --> 00:52:42.850 align:middle line:84% This one goes towards the second law. 00:52:42.850 --> 00:52:45.680 align:middle line:84% But the net balance never violates. 00:52:45.680 --> 00:52:51.590 align:middle line:84% So it's like saying, well, one is more unusual than the other. 00:52:51.590 --> 00:52:54.280 align:middle line:84% But for the second law, what counts 00:52:54.280 --> 00:52:57.390 align:middle line:90% is the overall net effect. 00:52:57.390 --> 00:53:03.460 align:middle line:90% 00:53:03.460 --> 00:53:05.650 align:middle line:84% And here is the list of possibilities 00:53:05.650 --> 00:53:11.920 align:middle line:84% if you want that to be greater than 0, which is always. 00:53:11.920 --> 00:53:19.330 align:middle line:84% If you have a positive affinity, so a positive degree 00:53:19.330 --> 00:53:27.820 align:middle line:84% of disequilibrium, then this is the case 00:53:27.820 --> 00:53:35.170 align:middle line:84% in which the S plus is more positive than the S minus. 00:53:35.170 --> 00:53:37.390 align:middle line:84% Or, if you have the affinity negative 00:53:37.390 --> 00:53:39.680 align:middle line:84% or the degree of disequilibrium negative, 00:53:39.680 --> 00:53:42.310 align:middle line:84% this is the case in which the S minus is 00:53:42.310 --> 00:53:45.070 align:middle line:90% more negative than the S plus. 00:53:45.070 --> 00:53:48.090 align:middle line:84% And, of course, at equilibrium, they are both equal to 0. 00:53:48.090 --> 00:53:59.930 align:middle line:90% 00:53:59.930 --> 00:54:05.930 align:middle line:84% There is another way of getting to the same result, 00:54:05.930 --> 00:54:12.480 align:middle line:90% to the same kinetic theory-- 00:54:12.480 --> 00:54:14.570 align:middle line:90% theory of chemical kinetics. 00:54:14.570 --> 00:54:17.540 align:middle line:90% 00:54:17.540 --> 00:54:24.140 align:middle line:84% And this is a model based on equilibrium considerations. 00:54:24.140 --> 00:54:31.770 align:middle line:84% So let's start from the overall reaction that we are studying, 00:54:31.770 --> 00:54:39.800 align:middle line:84% like this one, for which we recall 00:54:39.800 --> 00:54:42.890 align:middle line:84% what the definition of the affinity is-- 00:54:42.890 --> 00:54:44.840 align:middle line:84% it's the weighted sum of minus nu 00:54:44.840 --> 00:54:52.280 align:middle line:84% over T of the various chemical potentials-- 00:54:52.280 --> 00:54:58.510 align:middle line:84% and here is the law of mass action, the nonequilibrium 00:54:58.510 --> 00:55:01.240 align:middle line:90% law of mass action. 00:55:01.240 --> 00:55:07.520 align:middle line:84% Now let's consider also the half reactions of formation, 00:55:07.520 --> 00:55:12.180 align:middle line:84% now here, for compactness, I call them plus and minus. 00:55:12.180 --> 00:55:17.200 align:middle line:84% So this represents two reactions, the half forward 00:55:17.200 --> 00:55:21.630 align:middle line:84% and the half backward of formation-- 00:55:21.630 --> 00:55:25.880 align:middle line:84% so they are also written in that same format with everything 00:55:25.880 --> 00:55:29.270 align:middle line:90% to the left of the equal sign. 00:55:29.270 --> 00:55:30.985 align:middle line:90% So that's the activated complex. 00:55:30.985 --> 00:55:34.110 align:middle line:90% 00:55:34.110 --> 00:55:37.680 align:middle line:84% This being a reaction as any other, 00:55:37.680 --> 00:55:39.510 align:middle line:90% I can define its affinities. 00:55:39.510 --> 00:55:44.850 align:middle line:84% So I can define the affinity plus or the affinity 00:55:44.850 --> 00:55:46.350 align:middle line:90% of the minus reaction. 00:55:46.350 --> 00:55:50.100 align:middle line:84% I have to be careful that the stoichiometric coefficients are 00:55:50.100 --> 00:55:52.240 align:middle line:90% not only these ones. 00:55:52.240 --> 00:55:55.000 align:middle line:84% But also, there is a stoichiometric coefficient 00:55:55.000 --> 00:55:57.830 align:middle line:84% equal to 1 in front of the activated complex. 00:55:57.830 --> 00:56:00.370 align:middle line:90% So that's this contribution. 00:56:00.370 --> 00:56:05.140 align:middle line:84% So I need to know, or at least to theoretically 00:56:05.140 --> 00:56:11.170 align:middle line:84% assume that we are able to know the chemical potential 00:56:11.170 --> 00:56:15.120 align:middle line:90% of the activated complex. 00:56:15.120 --> 00:56:17.520 align:middle line:84% Whereas these are the usual ones. 00:56:17.520 --> 00:56:24.130 align:middle line:84% I mean, those of the reactants and the products. 00:56:24.130 --> 00:56:27.410 align:middle line:84% And, clearly, also for this reaction, 00:56:27.410 --> 00:56:31.990 align:middle line:84% I can write a nonequilibrium law of mass action 00:56:31.990 --> 00:56:35.200 align:middle line:84% exactly like that one, only using properly 00:56:35.200 --> 00:56:37.370 align:middle line:90% the stoichiometric coefficients. 00:56:37.370 --> 00:56:40.110 align:middle line:90% 00:56:40.110 --> 00:56:43.090 align:middle line:84% The concentration of the activated complex 00:56:43.090 --> 00:56:47.500 align:middle line:84% goes on top because it has a plus stoichiometric coefficient. 00:56:47.500 --> 00:56:52.330 align:middle line:84% In this summation, I put it at the denominator 00:56:52.330 --> 00:56:54.880 align:middle line:84% because, otherwise, I would have to put it 00:56:54.880 --> 00:56:59.620 align:middle line:84% at the numerator with a minus nu plus minus. 00:56:59.620 --> 00:57:05.500 align:middle line:84% And, clearly, I also need to compute 00:57:05.500 --> 00:57:09.460 align:middle line:84% an equilibrium constant for the forward and for the backward 00:57:09.460 --> 00:57:11.270 align:middle line:90% reaction. 00:57:11.270 --> 00:57:18.645 align:middle line:84% So that's pretty general, no assumptions so far. 00:57:18.645 --> 00:57:23.440 align:middle line:90% 00:57:23.440 --> 00:57:28.470 align:middle line:84% We can also compute, therefore, the equilibrium. 00:57:28.470 --> 00:57:31.760 align:middle line:90% 00:57:31.760 --> 00:57:32.910 align:middle line:90% Now here's the trick. 00:57:32.910 --> 00:57:41.180 align:middle line:84% We are going to assume partial equilibrium, only 00:57:41.180 --> 00:57:44.780 align:middle line:84% either for the forward or for the backward reaction. 00:57:44.780 --> 00:57:53.160 align:middle line:84% So we say, suppose the backward reaction is at equilibrium. 00:57:53.160 --> 00:57:56.570 align:middle line:90% We set its affinity equal to 0. 00:57:56.570 --> 00:58:00.440 align:middle line:84% And from this law of mass action, this becomes 1. 00:58:00.440 --> 00:58:06.760 align:middle line:84% And we compute what would be the concentration of the activated 00:58:06.760 --> 00:58:13.608 align:middle line:84% complex if it were in equilibrium with the reactants-- 00:58:13.608 --> 00:58:15.400 align:middle line:84% I'm sorry, with the products because that's 00:58:15.400 --> 00:58:17.570 align:middle line:90% the backward reaction. 00:58:17.570 --> 00:58:21.100 align:middle line:84% So only on that side of the valley, there is equilibrium. 00:58:21.100 --> 00:58:26.770 align:middle line:90% And that gives me N star minus. 00:58:26.770 --> 00:58:29.240 align:middle line:84% I do the same also on the other side, 00:58:29.240 --> 00:58:32.900 align:middle line:84% and that would give me another value for the concentration. 00:58:32.900 --> 00:58:40.130 align:middle line:90% 00:58:40.130 --> 00:58:43.250 align:middle line:84% The actual concentration of the activated complex 00:58:43.250 --> 00:58:48.540 align:middle line:84% is related to those ones through the degrees of disequilibrium 00:58:48.540 --> 00:58:52.450 align:middle line:84% of the forward and backward reactions 00:58:52.450 --> 00:59:00.430 align:middle line:84% because I can simply recognize that this k and this summation 00:59:00.430 --> 00:59:01.640 align:middle line:90% appear up here. 00:59:01.640 --> 00:59:04.360 align:middle line:84% So if I substitute these two in here, 00:59:04.360 --> 00:59:07.500 align:middle line:90% it brings me this relation. 00:59:07.500 --> 00:59:13.380 align:middle line:90% 00:59:13.380 --> 00:59:16.230 align:middle line:84% I also notice that the ratio between what 00:59:16.230 --> 00:59:19.620 align:middle line:84% would be the equilibrium concentration of the activated 00:59:19.620 --> 00:59:24.660 align:middle line:84% complex for the minus or for the plus, the ratio of the two 00:59:24.660 --> 00:59:30.685 align:middle line:84% is equal to, again, the degree of disequilibrium, 00:59:30.685 --> 00:59:32.960 align:middle line:84% or the exponential of the degree of disequilibrium. 00:59:32.960 --> 00:59:34.790 align:middle line:84% So the degree of disequilibrium equilibrium 00:59:34.790 --> 00:59:38.510 align:middle line:84% can be written also, therefore, as the logarithm 00:59:38.510 --> 00:59:44.160 align:middle line:84% of those would-be concentrations, 00:59:44.160 --> 00:59:49.330 align:middle line:84% meaning that at equilibrium, those concentrations are equal. 00:59:49.330 --> 00:59:51.745 align:middle line:90% If they are both-- 00:59:51.745 --> 00:59:54.550 align:middle line:84% at overall at equilibrium, so equilibrium 00:59:54.550 --> 01:00:01.150 align:middle line:84% for the overall reaction, then also, the forward 01:00:01.150 --> 01:00:03.291 align:middle line:84% and the backward are at equilibrium. 01:00:03.291 --> 01:00:06.900 align:middle line:90% 01:00:06.900 --> 01:00:08.590 align:middle line:90% So the idea is-- 01:00:08.590 --> 01:00:13.440 align:middle line:84% so that means that I have an indicator-- which 01:00:13.440 --> 01:00:23.712 align:middle line:90% is this ratio of the unbalance. 01:00:23.712 --> 01:00:26.740 align:middle line:84% We can think of this equilibrium concentrations 01:00:26.740 --> 01:00:34.150 align:middle line:84% as the rate at which the reactants form the activated 01:00:34.150 --> 01:00:34.910 align:middle line:90% complex. 01:00:34.910 --> 01:00:37.580 align:middle line:84% And if those rates are unbalanced, 01:00:37.580 --> 01:00:40.900 align:middle line:84% there is more formation of activated complex from one side 01:00:40.900 --> 01:00:42.970 align:middle line:84% and from the other, and, therefore, 01:00:42.970 --> 01:00:47.590 align:middle line:84% there is more chance that the reaction proceeds in order 01:00:47.590 --> 01:00:50.240 align:middle line:90% to balance this. 01:00:50.240 --> 01:00:56.300 align:middle line:84% That is why we can assume that the reaction 01:00:56.300 --> 01:01:05.470 align:middle line:84% rate, of the forward, is equal to the forward equilibrium 01:01:05.470 --> 01:01:10.330 align:middle line:84% of the activated complex times a coefficient, which represents 01:01:10.330 --> 01:01:18.190 align:middle line:84% the frequency with which the activated complex decays 01:01:18.190 --> 01:01:24.700 align:middle line:84% in the forward direction and, conversely, for the minus. 01:01:24.700 --> 01:01:28.980 align:middle line:90% So that is the assumption. 01:01:28.980 --> 01:01:33.150 align:middle line:84% And, with that assumption, we get-- 01:01:33.150 --> 01:01:37.330 align:middle line:84% if you substitute this equilibrium expression, 01:01:37.330 --> 01:01:46.410 align:middle line:84% you get this formula, which is very similar to-- 01:01:46.410 --> 01:01:50.010 align:middle line:84% it has something that depends on temperature, which is still 01:01:50.010 --> 01:01:52.630 align:middle line:90% what we call the rate constant. 01:01:52.630 --> 01:01:59.570 align:middle line:84% So it's another way of writing the rate constant, essentially, 01:01:59.570 --> 01:02:01.806 align:middle line:90% from a different reasoning-- 01:02:01.806 --> 01:02:05.780 align:middle line:90% 01:02:05.780 --> 01:02:08.850 align:middle line:84% not in terms of probabilities, but in terms of equilibrium. 01:02:08.850 --> 01:02:14.320 align:middle line:90% 01:02:14.320 --> 01:02:18.170 align:middle line:90% And, of course, we can apply-- 01:02:18.170 --> 01:02:21.670 align:middle line:84% we can do the same as same ratio that we 01:02:21.670 --> 01:02:25.035 align:middle line:84% did in order to compute the principle of detailed balance. 01:02:25.035 --> 01:02:29.030 align:middle line:90% 01:02:29.030 --> 01:02:30.840 align:middle line:90% So this is in general. 01:02:30.840 --> 01:02:33.090 align:middle line:84% If you evaluate this at equilibrium, 01:02:33.090 --> 01:02:36.000 align:middle line:84% when epsilon minus and plus are equal 01:02:36.000 --> 01:02:41.210 align:middle line:84% and they are equal to epsilon 0, so this is equal to 1. 01:02:41.210 --> 01:02:46.410 align:middle line:84% The affinity is equal to 0, so e is equal to 1. 01:02:46.410 --> 01:02:53.470 align:middle line:84% That implies that k minus must be equal to k plus, 01:02:53.470 --> 01:02:56.070 align:middle line:84% so that this frequency of forward and backward 01:02:56.070 --> 01:02:59.345 align:middle line:90% decay is equal on both sides-- 01:02:59.345 --> 01:03:02.050 align:middle line:90% 01:03:02.050 --> 01:03:13.020 align:middle line:84% symmetrical-- which gives us some nice understanding. 01:03:13.020 --> 01:03:16.480 align:middle line:84% So we could have used here a single k. 01:03:16.480 --> 01:03:19.220 align:middle line:90% 01:03:19.220 --> 01:03:23.960 align:middle line:84% And, therefore, you see that this forward and backward 01:03:23.960 --> 01:03:28.910 align:middle line:84% reactions are proportional by the same constant 01:03:28.910 --> 01:03:34.530 align:middle line:84% to these would-be concentrations of the activated complexes. 01:03:34.530 --> 01:03:57.500 align:middle line:90% 01:03:57.500 --> 01:04:01.200 align:middle line:84% Let's return to the rate of entropy production. 01:04:01.200 --> 01:04:06.650 align:middle line:90% 01:04:06.650 --> 01:04:12.280 align:middle line:84% And this is for the case in which-- 01:04:12.280 --> 01:04:17.860 align:middle line:84% we did it before for the case in which we have only one chemical 01:04:17.860 --> 01:04:20.680 align:middle line:90% reaction. 01:04:20.680 --> 01:04:25.660 align:middle line:84% Most of the times, we have many chemical reactions, 01:04:25.660 --> 01:04:33.040 align:middle line:84% like the 24 for hydrogen or the 350 for methane and so on. 01:04:33.040 --> 01:04:36.730 align:middle line:90% So we don't have only 1 epsilon. 01:04:36.730 --> 01:04:43.530 align:middle line:84% We have many epsilons, and z is the number of reactions. 01:04:43.530 --> 01:04:45.970 align:middle line:84% So the rate of change for the entropy, 01:04:45.970 --> 01:04:47.910 align:middle line:84% which for the isolated system is also 01:04:47.910 --> 01:04:50.230 align:middle line:84% the entropy produced by irreversibility, 01:04:50.230 --> 01:04:54.780 align:middle line:84% is a summation of affinities, each 01:04:54.780 --> 01:04:58.150 align:middle line:84% multiplied by each rate, which we can write also 01:04:58.150 --> 01:04:59.175 align:middle line:90% in this vector form. 01:04:59.175 --> 01:05:03.840 align:middle line:90% 01:05:03.840 --> 01:05:20.360 align:middle line:84% Now here, sometimes, we realize that not all reactions 01:05:20.360 --> 01:05:25.720 align:middle line:84% are important in the sense that we 01:05:25.720 --> 01:05:30.730 align:middle line:84% need to describe their kinetics because some reactions are 01:05:30.730 --> 01:05:39.470 align:middle line:84% so fast that regardless of the kinetics, 01:05:39.470 --> 01:05:45.050 align:middle line:90% they equilibrate very quickly. 01:05:45.050 --> 01:05:55.310 align:middle line:84% So we can save ourself some time or complication if we assume 01:05:55.310 --> 01:05:57.740 align:middle line:90% that they are at equilibrium. 01:05:57.740 --> 01:06:03.320 align:middle line:84% So essentially, what we do is to split the set of z reactions 01:06:03.320 --> 01:06:08.070 align:middle line:84% into a set of slow and a set of fast reactions. 01:06:08.070 --> 01:06:10.646 align:middle line:84% I split the summation into two summations. 01:06:10.646 --> 01:06:13.770 align:middle line:90% 01:06:13.770 --> 01:06:19.260 align:middle line:84% Of course, the proportionality relation is this one. 01:06:19.260 --> 01:06:20.680 align:middle line:90% It's also a summation. 01:06:20.680 --> 01:06:24.380 align:middle line:84% So we have split the summation also in here. 01:06:24.380 --> 01:06:28.210 align:middle line:90% 01:06:28.210 --> 01:06:34.240 align:middle line:84% And slow means that we can assume 01:06:34.240 --> 01:06:36.880 align:middle line:84% that this part of the composition 01:06:36.880 --> 01:06:41.890 align:middle line:84% is slowly varying as compared to this contribution here 01:06:41.890 --> 01:06:42.765 align:middle line:90% of the fast reaction. 01:06:42.765 --> 01:06:47.280 align:middle line:90% 01:06:47.280 --> 01:06:58.640 align:middle line:84% And then we assume that the epsilons of the fast reactions, 01:06:58.640 --> 01:07:04.230 align:middle line:84% are at any instant of time the equilibrium epsilons. 01:07:04.230 --> 01:07:10.320 align:middle line:84% So we use the law of mass action to compute the equilibrium 01:07:10.320 --> 01:07:17.900 align:middle line:84% composition only of the subset of fast reactions, 01:07:17.900 --> 01:07:21.260 align:middle line:84% assuming that the slow ones are frozen. 01:07:21.260 --> 01:07:27.020 align:middle line:84% We temporarily freeze the slow reactions. 01:07:27.020 --> 01:07:31.890 align:middle line:84% So that fixes this part of the composition. 01:07:31.890 --> 01:07:36.190 align:middle line:84% And we say, all right, for this-- 01:07:36.190 --> 01:07:39.425 align:middle line:84% so this become the initial amounts, like the na's. 01:07:39.425 --> 01:07:42.610 align:middle line:90% 01:07:42.610 --> 01:07:45.280 align:middle line:84% So for these initial amounts, what 01:07:45.280 --> 01:07:50.490 align:middle line:84% would be the equilibrium composition 01:07:50.490 --> 01:07:52.110 align:middle line:84% for the stoichiometric coefficients 01:07:52.110 --> 01:07:54.280 align:middle line:90% of the fast reactions? 01:07:54.280 --> 01:07:55.940 align:middle line:84% And that's what we've been doing. 01:07:55.940 --> 01:08:00.010 align:middle line:84% We apply the law of mass action in order to compute 01:08:00.010 --> 01:08:07.943 align:middle line:84% compositions, which can be cast as values of the reaction rates 01:08:07.943 --> 01:08:09.610 align:middle line:84% (I meant, coordinates), the epsilon 0's. 01:08:09.610 --> 01:08:16.328 align:middle line:90% 01:08:16.328 --> 01:08:19.600 align:middle line:84% Of course, this has to be done at every instant of time 01:08:19.600 --> 01:08:24.649 align:middle line:84% because slowly, the slow reactions do advance. 01:08:24.649 --> 01:08:26.600 align:middle line:90% They also advance. 01:08:26.600 --> 01:08:29.910 align:middle line:84% Actually, they are the bottleneck of the mechanism. 01:08:29.910 --> 01:08:32.510 align:middle line:84% They are the rate-controlling reactions. 01:08:32.510 --> 01:08:36.870 align:middle line:84% The slow ones are the ones that control the rate of the others. 01:08:36.870 --> 01:08:43.250 align:middle line:84% When you have a car jam, the slow cars 01:08:43.250 --> 01:08:48.229 align:middle line:84% ahead impede the ones behind to go faster, 01:08:48.229 --> 01:08:49.229 align:middle line:90% even if they are faster. 01:08:49.229 --> 01:08:53.670 align:middle line:84% But as soon as they get there, they get stuck. 01:08:53.670 --> 01:08:57.930 align:middle line:84% So, essentially, this is what the fast reaction do. 01:08:57.930 --> 01:09:01.609 align:middle line:90% They get stuck in queue of-- 01:09:01.609 --> 01:09:04.550 align:middle line:84% they wait for the slow reactions to change something, and then 01:09:04.550 --> 01:09:05.394 align:middle line:90% immediately adapt. 01:09:05.394 --> 01:09:11.040 align:middle line:90% 01:09:11.040 --> 01:09:16.689 align:middle line:84% So you may say, that's interesting. 01:09:16.689 --> 01:09:22.370 align:middle line:84% But if the reactions the fast reactions are at equilibrium, 01:09:22.370 --> 01:09:26.140 align:middle line:84% it means that their affinities are equal to 0. 01:09:26.140 --> 01:09:31.060 align:middle line:84% So in the expression for the entropy change, 01:09:31.060 --> 01:09:35.050 align:middle line:84% the affinities of the fast reactions are, by this model, 01:09:35.050 --> 01:09:41.450 align:middle line:84% assumed to be all equal to 0, which is true. 01:09:41.450 --> 01:09:44.220 align:middle line:90% 01:09:44.220 --> 01:09:48.720 align:middle line:84% But one might be led to conclude, oh, 01:09:48.720 --> 01:09:51.990 align:middle line:84% so the fast reactions don't contribute 01:09:51.990 --> 01:09:54.090 align:middle line:90% to the entropy production? 01:09:54.090 --> 01:09:56.940 align:middle line:90% Because this sum is equal to 0? 01:09:56.940 --> 01:09:59.760 align:middle line:84% No, that would be a wrong conclusion 01:09:59.760 --> 01:10:08.230 align:middle line:84% because these assumptions makes the epsilons 01:10:08.230 --> 01:10:15.060 align:middle line:84% of the fast reactions appear in the affinities of the slow ones. 01:10:15.060 --> 01:10:21.810 align:middle line:90% 01:10:21.810 --> 01:10:25.530 align:middle line:84% So here, this says that the affinity of the fast reactions 01:10:25.530 --> 01:10:30.390 align:middle line:84% is equal to 0, which is the set of equations 01:10:30.390 --> 01:10:35.230 align:middle line:84% that you have to solve in order to compute this epsilon 0 fast. 01:10:35.230 --> 01:10:41.830 align:middle line:84% Once you find it, you put it in the expression, the analogous 01:10:41.830 --> 01:10:44.290 align:middle line:84% expression, for the slow reactions, 01:10:44.290 --> 01:10:48.320 align:middle line:84% which depend on energy volume and the amounts. 01:10:48.320 --> 01:10:54.280 align:middle line:84% Now the amounts are given by the proportionality relation here. 01:10:54.280 --> 01:10:57.540 align:middle line:84% With the epsilon, instead of being 01:10:57.540 --> 01:10:59.940 align:middle line:84% now a function of time this way, it 01:10:59.940 --> 01:11:02.340 align:middle line:90% is a function of time this way. 01:11:02.340 --> 01:11:07.320 align:middle line:84% So I substitute this epsilon with the result 01:11:07.320 --> 01:11:09.160 align:middle line:90% of the equilibrium computation. 01:11:09.160 --> 01:11:12.180 align:middle line:90% 01:11:12.180 --> 01:11:18.940 align:middle line:84% And so now you see that the fast reactions advance. 01:11:18.940 --> 01:11:23.230 align:middle line:84% So these epsilon 0's advance following the slow ones, 01:11:23.230 --> 01:11:28.020 align:middle line:90% and their advance changes-- 01:11:28.020 --> 01:11:33.120 align:middle line:84% affects the affinities of the slow reactions. 01:11:33.120 --> 01:11:37.870 align:middle line:84% So, essentially, we have moved the effects of this summation, 01:11:37.870 --> 01:11:40.120 align:middle line:90% which is actually fast. 01:11:40.120 --> 01:11:46.260 align:middle line:84% So it's producing, immediately, the entropy. 01:11:46.260 --> 01:11:54.035 align:middle line:84% And we built it in the affinities of the slow reaction. 01:11:54.035 --> 01:11:58.520 align:middle line:90% 01:11:58.520 --> 01:12:03.880 align:middle line:84% So with this model, at least, you 01:12:03.880 --> 01:12:12.280 align:middle line:84% find that it's useless to know the reaction rate constants-- 01:12:12.280 --> 01:12:16.780 align:middle line:84% so the Arrhenius factors and all of that for the reactions 01:12:16.780 --> 01:12:24.200 align:middle line:84% that you feel are faster and, therefore, equilibrated. 01:12:24.200 --> 01:12:29.190 align:middle line:90% 01:12:29.190 --> 01:12:39.280 align:middle line:84% So here is an example of a nozzle expansion. 01:12:39.280 --> 01:12:44.630 align:middle line:84% So here, we have, at the entrance 01:12:44.630 --> 01:12:48.350 align:middle line:84% before this throat of the nozzle-- 01:12:48.350 --> 01:12:50.690 align:middle line:90% this is a supersonic nozzle-- 01:12:50.690 --> 01:12:56.570 align:middle line:84% this throat has, on the left-hand side, not drawn here, 01:12:56.570 --> 01:13:02.210 align:middle line:84% a combustion chamber in which you have produced the-- 01:13:02.210 --> 01:13:06.008 align:middle line:84% you burnt hydrogen in pure oxygen. 01:13:06.008 --> 01:13:08.300 align:middle line:84% And so you have the products of combustion, essentially 01:13:08.300 --> 01:13:13.430 align:middle line:90% water and a few other things-- 01:13:13.430 --> 01:13:16.490 align:middle line:84% the other eight species that are in that scheme 01:13:16.490 --> 01:13:22.430 align:middle line:84% that will appear again in the next slide. 01:13:22.430 --> 01:13:26.480 align:middle line:84% So these products of combustion are the adiabatic flame 01:13:26.480 --> 01:13:27.270 align:middle line:90% temperature. 01:13:27.270 --> 01:13:34.380 align:middle line:84% They are at a certain pressure, about 25 atmospheres. 01:13:34.380 --> 01:13:41.040 align:middle line:84% And the throat is so designed so as 01:13:41.040 --> 01:13:45.540 align:middle line:84% to achieve the maximum speed that when you have Mach number-- 01:13:45.540 --> 01:13:51.640 align:middle line:84% so when the flow is at Mach number equal to 1, 01:13:51.640 --> 01:13:57.660 align:middle line:84% that's the maximum speed allowed by the combination 01:13:57.660 --> 01:14:01.240 align:middle line:84% of the entropy, momentum, and energy balance, 01:14:01.240 --> 01:14:05.160 align:middle line:84% plus the equation of state for the gas. 01:14:05.160 --> 01:14:14.140 align:middle line:84% That's fluid mechanics, combined with thermodynamics. 01:14:14.140 --> 01:14:17.400 align:middle line:84% But, in any case, so you have Mach 1, and then as 01:14:17.400 --> 01:14:23.310 align:middle line:84% a result of the expansion here, if you 01:14:23.310 --> 01:14:27.330 align:middle line:84% have a well-designed nozzle, the fluid 01:14:27.330 --> 01:14:31.020 align:middle line:84% will accelerate and become supersonic. 01:14:31.020 --> 01:14:35.380 align:middle line:84% For example, here it would exit at Mach 4. 01:14:35.380 --> 01:14:46.510 align:middle line:84% Of course, that means that the thermal energy gets transformed 01:14:46.510 --> 01:14:48.920 align:middle line:90% into kinetic energy. 01:14:48.920 --> 01:14:53.370 align:middle line:84% So the gas cools down, and also the pressure goes down. 01:14:53.370 --> 01:14:56.770 align:middle line:90% 01:14:56.770 --> 01:15:03.400 align:middle line:84% This occurs pretty quickly because of the high velocities-- 01:15:03.400 --> 01:15:06.070 align:middle line:90% this expansion. 01:15:06.070 --> 01:15:11.350 align:middle line:84% And the rapidity is important here 01:15:11.350 --> 01:15:24.880 align:middle line:84% because the time that the collisions require 01:15:24.880 --> 01:15:34.960 align:middle line:84% to re-equilibrate energy is not enough. 01:15:34.960 --> 01:15:42.820 align:middle line:84% Given the short residence time, those collisions cannot keep up 01:15:42.820 --> 01:15:45.160 align:middle line:90% with equilibrium. 01:15:45.160 --> 01:15:50.030 align:middle line:90% 01:15:50.030 --> 01:15:53.340 align:middle line:84% Here, at the entrance of the nozzle, 01:15:53.340 --> 01:15:55.660 align:middle line:90% we have chemical equilibrium. 01:15:55.660 --> 01:15:59.560 align:middle line:84% So the composition here is that of chemical equilibrium. 01:15:59.560 --> 01:16:05.230 align:middle line:90% And this is the temperature. 01:16:05.230 --> 01:16:10.370 align:middle line:84% As it expands, you would think that 01:16:10.370 --> 01:16:14.600 align:middle line:84% the chemically effective collisions are fast enough 01:16:14.600 --> 01:16:17.480 align:middle line:84% to keep up and maintain chemical equilibrium. 01:16:17.480 --> 01:16:26.910 align:middle line:84% If they did, you would follow this blue dotted line-- 01:16:26.910 --> 01:16:27.670 align:middle line:90% dashed line. 01:16:27.670 --> 01:16:31.090 align:middle line:90% 01:16:31.090 --> 01:16:37.720 align:middle line:84% In reality, they follow the red line. 01:16:37.720 --> 01:16:43.150 align:middle line:84% That's at least according to the 24 reactions, 8 species 01:16:43.150 --> 01:16:44.430 align:middle line:90% detailed chemical scheme. 01:16:44.430 --> 01:16:50.450 align:middle line:90% 01:16:50.450 --> 01:16:52.160 align:middle line:84% They are not completely frozen, though. 01:16:52.160 --> 01:16:56.310 align:middle line:84% Some reactions do occur because if you try to compute, 01:16:56.310 --> 01:17:04.910 align:middle line:84% assume that the composition here remains the same all through, 01:17:04.910 --> 01:17:09.890 align:middle line:84% you would obtain this result, when you assume 01:17:09.890 --> 01:17:12.020 align:middle line:90% all the reactions frozen. 01:17:12.020 --> 01:17:14.118 align:middle line:90% So they are not frozen. 01:17:14.118 --> 01:17:14.785 align:middle line:90% They are active. 01:17:14.785 --> 01:17:19.910 align:middle line:90% 01:17:19.910 --> 01:17:27.400 align:middle line:84% So here are the 24 reactions that we have seen. 01:17:27.400 --> 01:17:31.720 align:middle line:84% And now we look at what each one does 01:17:31.720 --> 01:17:35.245 align:middle line:84% by looking at the individual degree of this equilibrium 01:17:35.245 --> 01:17:36.120 align:middle line:90% that we have defined. 01:17:36.120 --> 01:17:39.830 align:middle line:90% 01:17:39.830 --> 01:17:46.790 align:middle line:84% And we find something, which is a bit surprising if you don't 01:17:46.790 --> 01:17:49.660 align:middle line:90% think a lot about this, that-- 01:17:49.660 --> 01:17:52.670 align:middle line:90% 01:17:52.670 --> 01:17:54.260 align:middle line:84% what's not surprising is that you 01:17:54.260 --> 01:17:58.520 align:middle line:84% have some reactions for which the degree of disequilibrium 01:17:58.520 --> 01:18:02.390 align:middle line:90% remains practically 0. 01:18:02.390 --> 01:18:04.140 align:middle line:90% So they remain at equilibrium. 01:18:04.140 --> 01:18:09.270 align:middle line:84% These are the ones that are pretty fast. 01:18:09.270 --> 01:18:12.160 align:middle line:90% 01:18:12.160 --> 01:18:19.090 align:middle line:84% And if you group them here, like 11, 13, 14, 18, 20, 21, 23, 01:18:19.090 --> 01:18:21.310 align:middle line:90% it's these ones. 01:18:21.310 --> 01:18:23.090 align:middle line:90% They are two-body reactions. 01:18:23.090 --> 01:18:32.080 align:middle line:90% 01:18:32.080 --> 01:18:37.330 align:middle line:84% They have certain features, these FO minus FV. 01:18:37.330 --> 01:18:43.210 align:middle line:84% FO represents the free oxygen and FV the free valence 01:18:43.210 --> 01:18:45.210 align:middle line:90% of the molecules. 01:18:45.210 --> 01:18:48.840 align:middle line:90% And this difference, turns out-- 01:18:48.840 --> 01:18:52.230 align:middle line:84% I frankly don't know why that particular difference. 01:18:52.230 --> 01:18:56.910 align:middle line:84% I'm not enough a chemical engineer or a chemist 01:18:56.910 --> 01:18:58.935 align:middle line:90% to understand that. 01:18:58.935 --> 01:19:02.560 align:middle line:84% I wish Professor Keck was here to tell us. 01:19:02.560 --> 01:19:14.790 align:middle line:90% 01:19:14.790 --> 01:19:18.420 align:middle line:84% What's surprising here is that there 01:19:18.420 --> 01:19:19.900 align:middle line:90% are other groups of reactions. 01:19:19.900 --> 01:19:22.955 align:middle line:84% So the reactions behave like groups. 01:19:22.955 --> 01:19:26.120 align:middle line:90% 01:19:26.120 --> 01:19:32.980 align:middle line:84% For example, 7, 8, 9, and 10 build up 01:19:32.980 --> 01:19:34.920 align:middle line:90% a degree of disequilibrium. 01:19:34.920 --> 01:19:40.970 align:middle line:84% So that means that the kinetics cannot keep up and, therefore, 01:19:40.970 --> 01:19:42.590 align:middle line:90% you build a-- 01:19:42.590 --> 01:19:46.820 align:middle line:90% keep up to reach equilibrium. 01:19:46.820 --> 01:19:53.290 align:middle line:84% And, therefore, you build disequilibrium. 01:19:53.290 --> 01:19:55.270 align:middle line:84% That's the degree of disequilibrium 01:19:55.270 --> 01:19:58.980 align:middle line:90% as you go downstream. 01:19:58.980 --> 01:20:01.140 align:middle line:84% But these four reactions, which are 01:20:01.140 --> 01:20:04.470 align:middle line:84% very different from one another, yet they 01:20:04.470 --> 01:20:13.980 align:middle line:84% follow the same departure from equilibrium. 01:20:13.980 --> 01:20:16.500 align:middle line:90% 01:20:16.500 --> 01:20:20.530 align:middle line:84% And notice that this one are three-body reactions. 01:20:20.530 --> 01:20:24.640 align:middle line:84% These are reactions that require the collision of three 01:20:24.640 --> 01:20:25.610 align:middle line:90% molecules. 01:20:25.610 --> 01:20:27.610 align:middle line:90% They are much less frequent. 01:20:27.610 --> 01:20:32.470 align:middle line:84% And that is why this doesn't occur fast enough 01:20:32.470 --> 01:20:34.220 align:middle line:90% to keep up with equilibrium. 01:20:34.220 --> 01:20:38.790 align:middle line:90% 01:20:38.790 --> 01:20:44.932 align:middle line:84% But they do not change the difference between free oxygen 01:20:44.932 --> 01:20:45.640 align:middle line:90% and free valence. 01:20:45.640 --> 01:20:48.470 align:middle line:90% 01:20:48.470 --> 01:20:52.310 align:middle line:84% So, essentially, these reactions are 01:20:52.310 --> 01:20:57.530 align:middle line:84% subject to what we would call the same bottleneck, which 01:20:57.530 --> 01:21:00.200 align:middle line:90% is the three-body reactions. 01:21:00.200 --> 01:21:05.120 align:middle line:84% You need a three-body reaction for them 01:21:05.120 --> 01:21:06.720 align:middle line:90% to-- a three-body collision. 01:21:06.720 --> 01:21:08.600 align:middle line:90% And those are infrequent. 01:21:08.600 --> 01:21:11.000 align:middle line:84% And that is characterized by a change 01:21:11.000 --> 01:21:18.530 align:middle line:84% in the overall molecular number because you see go from 3 to 2. 01:21:18.530 --> 01:21:22.220 align:middle line:84% So you have collision of three particles, 01:21:22.220 --> 01:21:27.650 align:middle line:84% and after the activated complex breaks down, 01:21:27.650 --> 01:21:32.360 align:middle line:90% it produces two fragments. 01:21:32.360 --> 01:21:35.715 align:middle line:84% Then we have another set, another group. 01:21:35.715 --> 01:21:37.530 align:middle line:90% Which is these ones-- 01:21:37.530 --> 01:21:38.440 align:middle line:90% these ones. 01:21:38.440 --> 01:21:41.300 align:middle line:90% These are two-body reactions. 01:21:41.300 --> 01:21:44.930 align:middle line:84% But their bottleneck is that these reactions 01:21:44.930 --> 01:21:50.340 align:middle line:84% change, this magic difference between free oxygen 01:21:50.340 --> 01:21:51.750 align:middle line:90% and free valence. 01:21:51.750 --> 01:21:56.880 align:middle line:84% They change it by the same amount. 01:21:56.880 --> 01:21:59.420 align:middle line:90% 01:21:59.420 --> 01:22:03.390 align:middle line:84% This is what they have in common. 01:22:03.390 --> 01:22:06.890 align:middle line:84% So they have to face the same bottleneck, changing that. 01:22:06.890 --> 01:22:16.280 align:middle line:84% And apparently that requires probably 01:22:16.280 --> 01:22:23.390 align:middle line:84% something of the mechanism that is common, even 01:22:23.390 --> 01:22:26.780 align:middle line:90% if the molecules are different. 01:22:26.780 --> 01:22:32.370 align:middle line:84% And that difference plays a role in the kinetics and it's slow. 01:22:32.370 --> 01:22:34.330 align:middle line:90% It slows down. 01:22:34.330 --> 01:22:37.480 align:middle line:90% So those cannot keep up. 01:22:37.480 --> 01:22:42.430 align:middle line:84% And then you also-- if you look long enough, 01:22:42.430 --> 01:22:46.900 align:middle line:84% you may recognize that this other group 01:22:46.900 --> 01:22:51.390 align:middle line:84% follows a curve which is the sum of the curves below. 01:22:51.390 --> 01:22:57.310 align:middle line:84% So if you take the sum of this curve and of that curve, 01:22:57.310 --> 01:22:59.690 align:middle line:90% you get that. 01:22:59.690 --> 01:23:02.090 align:middle line:84% So the degree of disequilibrium of these ones 01:23:02.090 --> 01:23:04.220 align:middle line:84% at any instant of time is equal to the sum 01:23:04.220 --> 01:23:08.090 align:middle line:84% of the degrees of disequilibrium that has built up 01:23:08.090 --> 01:23:09.620 align:middle line:90% for the other two mechanisms. 01:23:09.620 --> 01:23:12.860 align:middle line:84% And, in fact, this is the group of reactions here 01:23:12.860 --> 01:23:15.720 align:middle line:84% that is subject to both bottlenecks. 01:23:15.720 --> 01:23:18.230 align:middle line:90% They require three collisions-- 01:23:18.230 --> 01:23:21.210 align:middle line:90% I mean, a three-body collision-- 01:23:21.210 --> 01:23:26.210 align:middle line:84% plus they require to change the difference between free oxygen 01:23:26.210 --> 01:23:27.610 align:middle line:90% and free valence. 01:23:27.610 --> 01:23:31.580 align:middle line:84% And that's why they play double-duty, so-to-speak, 01:23:31.580 --> 01:23:35.470 align:middle line:84% and they remain behind by the sum of these degrees 01:23:35.470 --> 01:23:36.842 align:middle line:90% of disequilibrium. 01:23:36.842 --> 01:23:40.540 align:middle line:90% 01:23:40.540 --> 01:23:46.300 align:middle line:84% So the idea, if you go back one moment, 01:23:46.300 --> 01:23:51.340 align:middle line:84% the idea of the rate controlled constrained equilibrium model 01:23:51.340 --> 01:23:52.990 align:middle line:90% is to assign-- 01:23:52.990 --> 01:23:58.660 align:middle line:84% instead of using all the 24 reaction rates-- 01:23:58.660 --> 01:24:03.760 align:middle line:84% and, actually, here, this shows that you get very good results 01:24:03.760 --> 01:24:10.750 align:middle line:84% with only rates for four constraints. 01:24:10.750 --> 01:24:15.190 align:middle line:84% Two are just the element conservation. 01:24:15.190 --> 01:24:17.180 align:middle line:90% And the others are-- 01:24:17.180 --> 01:24:21.730 align:middle line:84% one is the number of molecules conservation, which 01:24:21.730 --> 01:24:25.780 align:middle line:84% is conserved by the two-body collisions 01:24:25.780 --> 01:24:28.530 align:middle line:84% and not by the three-body collisions. 01:24:28.530 --> 01:24:32.850 align:middle line:84% And the other is that difference between the free 01:24:32.850 --> 01:24:36.690 align:middle line:90% valence and free oxygen. 01:24:36.690 --> 01:24:41.460 align:middle line:84% And that means that you can obtain the exact match 01:24:41.460 --> 01:24:44.310 align:middle line:90% of the experimental results. 01:24:44.310 --> 01:24:48.140 align:middle line:84% Instead of solving 24 rate equations, 01:24:48.140 --> 01:24:49.940 align:middle line:90% you just solve 2 plus 2. 01:24:49.940 --> 01:24:58.520 align:middle line:84% So that makes 4 with a big gain, especially if you're doing also 01:24:58.520 --> 01:25:00.630 align:middle line:90% computational fluid mechanics. 01:25:00.630 --> 01:25:03.600 align:middle line:84% And, therefore, in addition to the chemical kinetics, 01:25:03.600 --> 01:25:11.780 align:middle line:84% you have to solve the fluid mechanics of the problem, where 01:25:11.780 --> 01:25:16.520 align:middle line:84% if you have to deal with turbulence and so on, 01:25:16.520 --> 01:25:19.910 align:middle line:84% it makes the problem already complicated enough 01:25:19.910 --> 01:25:25.190 align:middle line:84% that savings in the number of differential equations you 01:25:25.190 --> 01:25:29.982 align:middle line:84% have to solve simultaneously makes a lot of a difference. 01:25:29.982 --> 01:25:32.830 align:middle line:90% 01:25:32.830 --> 01:25:41.850 align:middle line:84% Now this same idea also applies here. 01:25:41.850 --> 01:25:50.990 align:middle line:84% Again, a nozzle-- you take products of combustion of some 01:25:50.990 --> 01:25:55.580 align:middle line:84% other fuel so that you obtain, here, 01:25:55.580 --> 01:26:02.120 align:middle line:84% a mixture at equilibrium in which you have carbon dioxide, 01:26:02.120 --> 01:26:09.610 align:middle line:84% nitrogen, and a little bit of water-- 01:26:09.610 --> 01:26:16.310 align:middle line:84% 90% nitrogen, about 10% CO2, and 1% water. 01:26:16.310 --> 01:26:23.080 align:middle line:84% Water, I think, acts as a catalyzer of the reactions 01:26:23.080 --> 01:26:25.620 align:middle line:84% that we're going to describe here. 01:26:25.620 --> 01:26:28.200 align:middle line:90% 01:26:28.200 --> 01:26:34.200 align:middle line:84% Now also in this nozzle, the important effect 01:26:34.200 --> 01:26:38.245 align:middle line:84% is that there is a degree of disequilibrium that builds up, 01:26:38.245 --> 01:26:43.330 align:middle line:84% like we have seen, only that here, the disequilibrium is not 01:26:43.330 --> 01:26:49.450 align:middle line:84% between molecular species or a certain chemical reaction. 01:26:49.450 --> 01:26:53.420 align:middle line:84% It's within the molecules-- in particular, 01:26:53.420 --> 01:26:59.670 align:middle line:84% the molecules of CO2, which are the key molecules that 01:26:59.670 --> 01:27:04.140 align:middle line:84% are interesting in this application. 01:27:04.140 --> 01:27:08.940 align:middle line:84% It's the balance in how the energy is distributed 01:27:08.940 --> 01:27:11.910 align:middle line:84% between the translational rotational 01:27:11.910 --> 01:27:16.180 align:middle line:84% and vibrational levels of the CO2 molecule. 01:27:16.180 --> 01:27:24.970 align:middle line:84% What happens here is that as the rapid expansion occurs, but also 01:27:24.970 --> 01:27:28.370 align:middle line:84% here, we have sonic and Mach 4 at the exit. 01:27:28.370 --> 01:27:32.840 align:middle line:84% So that's similar to the other problem. 01:27:32.840 --> 01:27:37.570 align:middle line:84% What happens is that here, you have 01:27:37.570 --> 01:27:43.310 align:middle line:84% a lot of random kinetic energy of the thermal agitation 01:27:43.310 --> 01:27:47.000 align:middle line:84% of the molecules, and the nozzle takes that kinetic energy 01:27:47.000 --> 01:27:54.020 align:middle line:84% and transforms it into a stream, the kinetic energy of the flow, 01:27:54.020 --> 01:27:57.436 align:middle line:90% like a bulk velocity. 01:27:57.436 --> 01:28:02.470 align:middle line:90% 01:28:02.470 --> 01:28:08.650 align:middle line:84% And if the collision mechanisms were very effective, 01:28:08.650 --> 01:28:14.620 align:middle line:84% as soon as the translational degrees of freedom 01:28:14.620 --> 01:28:18.140 align:middle line:84% cooled down because of the expansion, 01:28:18.140 --> 01:28:21.430 align:middle line:84% you see the temperature goes from high to low, 01:28:21.430 --> 01:28:24.800 align:middle line:84% so that means the thermal kinetic energy goes down. 01:28:24.800 --> 01:28:28.510 align:middle line:84% It transforms into flow kinetic energy. 01:28:28.510 --> 01:28:30.070 align:middle line:90% You can see that here. 01:28:30.070 --> 01:28:32.620 align:middle line:84% This is the flow of kinetic energy. 01:28:32.620 --> 01:28:33.580 align:middle line:90% It goes up. 01:28:33.580 --> 01:28:38.180 align:middle line:84% And this is the thermal kinetic energy that goes down. 01:28:38.180 --> 01:28:41.630 align:middle line:84% And this is the vibrational energy, 01:28:41.630 --> 01:28:45.600 align:middle line:84% the energy in the vibrational modes of the molecules. 01:28:45.600 --> 01:28:52.230 align:middle line:84% Those also, if the effectiveness was very high, 01:28:52.230 --> 01:28:58.950 align:middle line:84% would lose energy quickly and transfer the vibrational energy 01:28:58.950 --> 01:29:03.000 align:middle line:90% into the kinetic and rotation. 01:29:03.000 --> 01:29:05.830 align:middle line:90% 01:29:05.830 --> 01:29:11.150 align:middle line:84% But this mechanism is relatively slow. 01:29:11.150 --> 01:29:19.400 align:middle line:84% So the degree of disequilibrium builds up 01:29:19.400 --> 01:29:21.673 align:middle line:90% in the vibrational levels. 01:29:21.673 --> 01:29:24.640 align:middle line:90% 01:29:24.640 --> 01:29:27.670 align:middle line:84% For example, this is the lower level. 01:29:27.670 --> 01:29:29.970 align:middle line:90% It gets cooled down. 01:29:29.970 --> 01:29:36.310 align:middle line:84% But there is an upper level, the upper vibrational level, 01:29:36.310 --> 01:29:42.250 align:middle line:84% that remains sort of frozen for a long time downstream. 01:29:42.250 --> 01:29:46.960 align:middle line:90% 01:29:46.960 --> 01:29:51.340 align:middle line:84% So why is that important for this application? 01:29:51.340 --> 01:29:53.950 align:middle line:84% For this application, what counts 01:29:53.950 --> 01:29:57.520 align:middle line:84% is the vibrational levels of the CO2 molecule. 01:29:57.520 --> 01:30:04.000 align:middle line:84% And the CO2 molecules, being a molecule aligned this way, 01:30:04.000 --> 01:30:05.940 align:middle line:90% can vibrate in various ways. 01:30:05.940 --> 01:30:09.130 align:middle line:90% 01:30:09.130 --> 01:30:16.310 align:middle line:90% This is a symmetric stretching. 01:30:16.310 --> 01:30:20.100 align:middle line:90% This is a bending mode. 01:30:20.100 --> 01:30:22.210 align:middle line:90% And this is asymmetric. 01:30:22.210 --> 01:30:27.690 align:middle line:84% So it goes distant from one side, 01:30:27.690 --> 01:30:30.660 align:middle line:84% while it goes shorter on the other side, 01:30:30.660 --> 01:30:33.630 align:middle line:90% and keeps going like that. 01:30:33.630 --> 01:30:36.450 align:middle line:84% Now each of these modes have frequencies. 01:30:36.450 --> 01:30:42.140 align:middle line:84% And these frequencies are quantized because we 01:30:42.140 --> 01:30:44.520 align:middle line:90% live in a quantum world. 01:30:44.520 --> 01:30:54.410 align:middle line:84% And these are like harmonic oscillators, springs. 01:30:54.410 --> 01:30:59.020 align:middle line:84% A mass and a spring, according to quantum mechanics, 01:30:59.020 --> 01:31:01.820 align:middle line:84% cannot live in a continuous set of levels, 01:31:01.820 --> 01:31:03.590 align:middle line:90% but the levels are quantized. 01:31:03.590 --> 01:31:08.100 align:middle line:90% 01:31:08.100 --> 01:31:10.560 align:middle line:84% And so the molecule-- if you give more energy 01:31:10.560 --> 01:31:14.090 align:middle line:84% to the molecule, you need to give 01:31:14.090 --> 01:31:19.340 align:middle line:84% it enough to jump to another level. 01:31:19.340 --> 01:31:21.090 align:middle line:90% You cannot give it continuously. 01:31:21.090 --> 01:31:25.220 align:middle line:90% 01:31:25.220 --> 01:31:29.190 align:middle line:84% And the same is, if you have excited some high level, 01:31:29.190 --> 01:31:33.090 align:middle line:84% in order for that to release energy, 01:31:33.090 --> 01:31:35.335 align:middle line:84% it has to jump down to another level. 01:31:35.335 --> 01:31:42.080 align:middle line:90% 01:31:42.080 --> 01:31:45.410 align:middle line:84% The important one is this level here, 01:31:45.410 --> 01:31:51.160 align:middle line:90% which is at this wavelength. 01:31:51.160 --> 01:31:53.050 align:middle line:90% Wavelength is a frequency. 01:31:53.050 --> 01:31:56.780 align:middle line:84% There is a relation between frequency and wavelength. 01:31:56.780 --> 01:32:03.910 align:middle line:84% So the frequency of vibration is related to the wavelength. 01:32:03.910 --> 01:32:10.600 align:middle line:84% And here, it is important to look at the wavelength because 01:32:10.600 --> 01:32:16.390 align:middle line:84% of photons, because of light, we will see in a moment. 01:32:16.390 --> 01:32:20.290 align:middle line:84% So the important thing that happens here 01:32:20.290 --> 01:32:23.470 align:middle line:84% is that this vibrational level, which 01:32:23.470 --> 01:32:28.780 align:middle line:84% is a high level, will be, say, up somewhere here, 01:32:28.780 --> 01:32:33.730 align:middle line:84% is relatively highly populated in the hot gases 01:32:33.730 --> 01:32:36.550 align:middle line:84% before they enter the nozzle because we 01:32:36.550 --> 01:32:37.880 align:middle line:90% are at high temperature. 01:32:37.880 --> 01:32:40.860 align:middle line:90% 01:32:40.860 --> 01:32:49.260 align:middle line:84% Not only-- that level is very close-- 01:32:49.260 --> 01:32:54.140 align:middle line:84% I mean, that level for CO2 is very close to a similar level 01:32:54.140 --> 01:32:56.360 align:middle line:90% for nitrogen molecules. 01:32:56.360 --> 01:32:59.640 align:middle line:84% So by simple collision, the nitrogen molecules 01:32:59.640 --> 01:33:05.220 align:middle line:84% can populate that CO2 particular level because they are tuned, 01:33:05.220 --> 01:33:05.790 align:middle line:90% so-to-speak. 01:33:05.790 --> 01:33:08.010 align:middle line:84% They are exactly the same height. 01:33:08.010 --> 01:33:14.030 align:middle line:84% So it is a good chance that you transfer the energy of one 01:33:14.030 --> 01:33:16.015 align:middle line:90% to the other one. 01:33:16.015 --> 01:33:18.650 align:middle line:90% 01:33:18.650 --> 01:33:24.410 align:middle line:84% And that's why they chose nitrogen and CO2. 01:33:24.410 --> 01:33:31.020 align:middle line:84% And this one, when it decays, it decays to this level. 01:33:31.020 --> 01:33:35.640 align:middle line:84% Now every time a vibrational level decays to a lower level, 01:33:35.640 --> 01:33:38.470 align:middle line:90% it emits a photon. 01:33:38.470 --> 01:33:47.650 align:middle line:84% If that happens in just a random situation, 01:33:47.650 --> 01:33:53.640 align:middle line:84% those photons would go in all directions randomly. 01:33:53.640 --> 01:33:56.260 align:middle line:90% 01:33:56.260 --> 01:34:01.910 align:middle line:84% That is called the spontaneous emission. 01:34:01.910 --> 01:34:10.070 align:middle line:84% But if that decay happens in a situation in which there 01:34:10.070 --> 01:34:13.440 align:middle line:84% are other photons with the same frequency, so other photons that 01:34:13.440 --> 01:34:19.010 align:middle line:84% have exactly the same frequency, these other photons 01:34:19.010 --> 01:34:26.920 align:middle line:84% stimulate the emission of this decay 01:34:26.920 --> 01:34:32.290 align:middle line:84% by producing photons that have exactly the same frequency 01:34:32.290 --> 01:34:33.670 align:middle line:90% and direction. 01:34:33.670 --> 01:34:36.490 align:middle line:90% This is what happens. 01:34:36.490 --> 01:34:38.250 align:middle line:84% This is called stimulated emission. 01:34:38.250 --> 01:34:42.040 align:middle line:90% 01:34:42.040 --> 01:34:50.630 align:middle line:84% And this is the objective of this entire construction. 01:34:50.630 --> 01:34:53.290 align:middle line:90% Because by doing that-- 01:34:53.290 --> 01:34:55.300 align:middle line:90% let's see if have another-- 01:34:55.300 --> 01:34:59.140 align:middle line:84% by doing that, this picture shows you 01:34:59.140 --> 01:35:02.470 align:middle line:90% the details of how big this is. 01:35:02.470 --> 01:35:08.940 align:middle line:84% So the nozzle is about a meter here. 01:35:08.940 --> 01:35:11.460 align:middle line:90% 01:35:11.460 --> 01:35:16.080 align:middle line:84% So what they do, essentially, is to put-- 01:35:16.080 --> 01:35:19.570 align:middle line:90% I'm sorry, this is the diffuser. 01:35:19.570 --> 01:35:22.050 align:middle line:90% The nozzle is here. 01:35:22.050 --> 01:35:24.640 align:middle line:84% These are the nozzles, several of them. 01:35:24.640 --> 01:35:26.170 align:middle line:90% This is the expansion. 01:35:26.170 --> 01:35:29.220 align:middle line:84% And this is the downstream area in which we have 01:35:29.220 --> 01:35:31.540 align:middle line:90% that degree of disequilibrium. 01:35:31.540 --> 01:35:37.020 align:middle line:84% So in which I have still populated the high levels, 01:35:37.020 --> 01:35:40.300 align:middle line:84% this is called inversion of population. 01:35:40.300 --> 01:35:42.210 align:middle line:84% It's the inverse of what it would 01:35:42.210 --> 01:35:46.810 align:middle line:84% be at equilibrium at the local values of temperature. 01:35:46.810 --> 01:35:54.407 align:middle line:84% But with that inversion, it turns out-- 01:35:54.407 --> 01:35:55.990 align:middle line:84% if you have an inversion of population 01:35:55.990 --> 01:35:58.000 align:middle line:84% it means that this one is more populated 01:35:58.000 --> 01:36:03.340 align:middle line:84% and, therefore, the rate at which you produce photons 01:36:03.340 --> 01:36:08.560 align:middle line:84% by decay is higher than the typical rate at which photons 01:36:08.560 --> 01:36:14.450 align:middle line:84% get absorbed in order to repopulate that one. 01:36:14.450 --> 01:36:18.350 align:middle line:84% So you have a net release of photons. 01:36:18.350 --> 01:36:23.700 align:middle line:84% And if you put that between mirrors 01:36:23.700 --> 01:36:25.710 align:middle line:84% so that these photons go back and forth 01:36:25.710 --> 01:36:28.020 align:middle line:84% and stimulate the emission, then you 01:36:28.020 --> 01:36:32.040 align:middle line:84% get the building up of light, coherent light-- photons 01:36:32.040 --> 01:36:33.910 align:middle line:84% with the same frequency, same direction. 01:36:33.910 --> 01:36:36.640 align:middle line:90% So that's a laser. 01:36:36.640 --> 01:36:38.800 align:middle line:84% And this is actually a power laser 01:36:38.800 --> 01:36:41.620 align:middle line:84% because, usually, the lasers that you 01:36:41.620 --> 01:36:45.730 align:middle line:84% have use for-- a pointer laser for PowerPoint 01:36:45.730 --> 01:36:50.950 align:middle line:84% is, I don't know, microwatts or milliwatts. 01:36:50.950 --> 01:36:53.840 align:middle line:90% I don't know-- very small power. 01:36:53.840 --> 01:36:59.470 align:middle line:84% But with this one-- and we're talking about 1970s-- 01:36:59.470 --> 01:37:02.800 align:middle line:84% the output was continuous output of light 01:37:02.800 --> 01:37:09.340 align:middle line:84% of 60 kilowatts, which is a huge amount of power, especially if-- 01:37:09.340 --> 01:37:12.160 align:middle line:84% This is that the kind of light with which you 01:37:12.160 --> 01:37:17.130 align:middle line:90% can do laser cutting of metals. 01:37:17.130 --> 01:37:21.990 align:middle line:84% And this is all due to the fact that you 01:37:21.990 --> 01:37:25.960 align:middle line:84% managed to establish a degree of disequilibrium. 01:37:25.960 --> 01:37:31.260 align:middle line:84% And I think I would stop here for today. 01:37:31.260 --> 01:37:43.000 align:middle line:90%