WEBVTT 00:00.930 --> 00:04.090 Prof: Okay, at the end last time we were 00:04.093 --> 00:07.463 looking at how you could possibly know the heat of 00:07.464 --> 00:11.424 atomization of graphite; how much energy it takes to put 00:11.418 --> 00:14.558 a carbon atom into the gas phase from graphite. 00:14.560 --> 00:16.060 Why would you want to know that? 00:16.059 --> 00:20.819 Why is that a key value? 00:20.820 --> 00:21.690 For what purpose? 00:21.690 --> 00:27.430 If you want to be able to know whether you can add together 00:27.432 --> 00:31.592 bonds and get the energy of a molecule, 00:31.590 --> 00:35.380 that means you start with separated atoms and you then see 00:35.378 --> 00:39.368 how much energy is given off when those come together to give 00:39.365 --> 00:43.695 a particular molecule, to make a whole bunch of bonds. 00:43.697 --> 00:44.267 Right? 00:44.270 --> 00:50.200 Now you can measure the energy of a molecule with respect to 00:50.199 --> 00:52.209 CO_2 quite easily. 00:52.210 --> 00:53.370 How do you do that? 00:53.370 --> 00:55.060 Burn it. 00:55.060 --> 00:58.580 And you can measure the CO_2 relative to graphite. 00:58.580 --> 01:00.070 How do you do that? 01:00.070 --> 01:04.390 How do you know the energy of a carbon in -- or the energy of 01:04.391 --> 01:08.281 CO_2 relative to the energy of graphite plus oxygen? 01:08.280 --> 01:12.450 You burn graphite. Okay? 01:12.450 --> 01:15.890 But the last thing you need is the energy of graphite relative 01:15.890 --> 01:16.680 to the atom. 01:16.680 --> 01:21.440 And if you then knew that, then you could complete this 01:21.442 --> 01:24.372 scheme and know, just by burning things, 01:24.373 --> 01:27.593 whether you can add together bond energies to get the energy 01:27.592 --> 01:28.522 of a molecule. 01:28.519 --> 01:32.909 Okay, so how do you get the energy from graphite to an atom? 01:32.910 --> 01:35.650 One way is spectroscopy, and we talked about that last 01:35.646 --> 01:38.336 time, that if you see what the 01:38.337 --> 01:44.317 minimum amount of energy you can put into CO_2 and have it break 01:44.316 --> 01:47.646 into two atoms, that gives you the energy of 01:47.650 --> 01:49.510 the two atoms relative to CO_2 -- 01:49.510 --> 01:52.560 pardon me, relative to CO I should be saying. 01:52.563 --> 01:53.123 Right? 01:53.120 --> 01:56.100 And then you burn CO and get it relative to CO_2, 01:56.096 --> 01:58.386 and then you have everything you need. 01:58.391 --> 01:58.951 Okay? 01:58.950 --> 02:04.110 Okay, except that people who were very smart, 02:04.108 --> 02:08.428 Nobel Prize winners and so on, differed on interpreting this, 02:08.430 --> 02:12.640 because some of them thought that when you break into atoms, 02:12.639 --> 02:14.839 with the light, the atom you get is not the 02:14.837 --> 02:18.317 lowest energy state of the atom, but a higher energy state of 02:18.319 --> 02:18.909 the atom. 02:18.908 --> 02:23.228 So some of the energy you're putting in is going into making 02:23.230 --> 02:24.550 an excited atom. 02:24.550 --> 02:29.060 And the true energy of forming the minimum-energy carbon atom 02:29.055 --> 02:33.555 from graphite is lower than what you do spectroscopically. 02:33.560 --> 02:36.380 So the spectroscopic value is very precise. 02:36.378 --> 02:38.878 You can measure the position, the color of the light, 02:38.883 --> 02:39.753 very accurately. 02:39.750 --> 02:42.630 But they didn't know what it corresponded to, 02:42.626 --> 02:46.546 so they needed a different way to know what the energy of the 02:46.549 --> 02:49.229 carbon atom was relative to graphite. 02:49.229 --> 02:54.159 Now Professor Sharpless said -- you know, he talked about 02:54.163 --> 02:58.073 increasing dimension of carbon; start with an atom, 02:58.073 --> 03:00.193 go to a line, polyacetylene and then to a 03:00.186 --> 03:03.426 bunch of double bonds, and finally to a saturated 03:03.430 --> 03:06.250 hydrocarbon -- he said that carbon atoms are 03:06.254 --> 03:07.584 very hard to come by. 03:07.580 --> 03:09.850 You need a really, really low vacuum to get it. 03:09.854 --> 03:10.254 Right? 03:10.250 --> 03:14.410 That is, the equilibrium constant for carbon atoms coming 03:14.408 --> 03:17.828 together to form bonds is very, very favorable. 03:17.827 --> 03:18.567 Right? 03:18.569 --> 03:19.759 So it's hard to get atoms. 03:19.758 --> 03:22.698 So how are you ever going to get atoms? 03:22.699 --> 03:25.589 Well let's think about that problem here. 03:25.590 --> 03:30.210 So we know that the equilibrium constant at room temperature is 03:30.208 --> 03:32.218 10^-(3/4)ΔE. 03:32.220 --> 03:33.760 So you can measure k. 03:33.759 --> 03:35.469 Then you know ΔE. 03:35.470 --> 03:39.320 So, but if the equilibrium you're trying to measure is 03:39.315 --> 03:41.995 between graphite and carbon atoms, 03:42.000 --> 03:46.760 and the energy to take a carbon atom out of graphite is 170 03:46.758 --> 03:51.648 kilocalories/mol, then the equilibrium constant 03:51.650 --> 03:52.880 is 10^-127. 03:52.883 --> 03:53.783 Right? 03:53.780 --> 03:56.990 And that means, since there are only something 03:56.988 --> 04:00.338 of the order of 10^80 atoms in the universe, 04:00.340 --> 04:03.060 there would not be, at room temperature, 04:03.060 --> 04:06.870 a single carbon atom at equilibrium with graphite if 04:06.873 --> 04:09.793 everything in the universe was graphite. 04:09.792 --> 04:10.542 Right? 04:10.538 --> 04:13.618 So it's not a very favorable equilibrium constant. 04:13.620 --> 04:16.920 What could you do about it, if you wanted to measure the 04:16.920 --> 04:19.980 equilibrium constant, and in that way get the energy 04:19.980 --> 04:20.880 difference? 04:20.879 --> 04:22.899 Any knobs you can twist? 04:22.899 --> 04:24.919 Lucas? 04:24.920 --> 04:25.630 Student: Temperature. 04:25.629 --> 04:27.919 Prof: You could increase the temperature. 04:27.920 --> 04:28.310 Right? 04:28.310 --> 04:31.320 Because it's ΔE/kT. 04:31.319 --> 04:33.439 And remember, we're talking about room 04:33.437 --> 04:34.237 temperature. 04:34.240 --> 04:36.740 So if we're much higher than room temperature, 04:36.738 --> 04:38.848 then the exponent gets much smaller. 04:38.850 --> 04:43.100 So suppose we went to ten times room temperature, 04:43.103 --> 04:44.433 to 3000 Kelvin. 04:44.432 --> 04:45.232 Right? 04:45.230 --> 04:47.340 Then it would be, instead of 3/4ths, 04:47.339 --> 04:50.289 it would be 3/40ths, because the denominator would 04:50.293 --> 04:51.743 be ten times bigger. 04:51.740 --> 04:56.360 And now it would be 1 in 10^13th would be an atom. 04:56.360 --> 04:59.120 And now that's a substantial number, if you're talking about 04:59.115 --> 05:00.045 Avogadro's number. 05:00.050 --> 05:02.870 So if you had something really, really sensitive, 05:02.874 --> 05:05.524 to measure atoms, you might be able to measure 05:05.523 --> 05:07.233 the atoms at equilibrium. 05:07.230 --> 05:10.790 But you have to establish equilibrium at something of the 05:13.201 --> 05:13.901 really hot. 05:13.901 --> 05:14.601 Right? 05:14.600 --> 05:17.440 So you could write the equilibrium constant, 05:17.444 --> 05:21.484 the concentration of atoms over the concentration of graphite, 05:21.480 --> 05:22.540 in this way. 05:22.540 --> 05:25.750 That has to do -- that is the heat of formation, 05:25.745 --> 05:27.585 atoms relative to graphite. 05:27.586 --> 05:28.196 Right? 05:28.199 --> 05:30.979 Now, of course, exactly what one means by the 05:30.975 --> 05:34.945 concentration of graphite needs to be -- you scratch your head a 05:34.947 --> 05:36.337 little bit about that. 05:36.336 --> 05:37.026 Right? 05:37.029 --> 05:38.859 But you don't actually need to know it, 05:38.860 --> 05:42.770 because if you could measure the pressure of the C atoms at 05:42.766 --> 05:46.356 equilibrium with graphite, at very high temperature -- 05:46.355 --> 05:48.835 call that the pressure of carbon, right? 05:48.839 --> 05:52.169 -- that is some constant, and that constant will include 05:52.170 --> 05:55.320 whatever this concentration of graphite should be. 05:55.319 --> 05:58.889 So B. So multiply both sides by that concentration of 05:58.887 --> 05:59.497 graphite. 05:59.504 --> 06:00.264 Right? 06:00.259 --> 06:03.859 So we get some constant on the right, and then that heat of 06:03.862 --> 06:07.032 formation, in the exponent, that we want to find. 06:07.028 --> 06:10.418 So that means we could write -- if we took the log of both 06:10.423 --> 06:10.903 sides. 06:10.899 --> 06:13.949 You have the log of the pressure of carbon atoms, 06:13.949 --> 06:17.699 is the log of whatever B is -- that's some constant -- minus 06:17.699 --> 06:18.969 this other term. 06:18.970 --> 06:22.210 And what that means is that if you -- 06:22.209 --> 06:25.699 that that minus the heat of formation of the carbon atom, 06:25.699 --> 06:30.429 divided by R, is the slope of a plot of the 06:30.434 --> 06:34.904 pressure of carbon atoms, versus 1/T. 06:34.899 --> 06:36.739 Does everyone see that? 06:36.740 --> 06:40.260 In that equation, that equation says that there's 06:40.257 --> 06:43.437 an intercept, which is the log of b, 06:43.439 --> 06:46.079 and a slope, which is -ΔH 06:46.079 --> 06:48.499 formation to carbon, divided by R, 06:48.500 --> 06:50.020 if you plot against 1/T. 06:50.019 --> 06:51.929 Everybody with me on that? 06:51.930 --> 06:54.110 Nod if you see it. 06:54.110 --> 06:56.030 Okay. 06:56.029 --> 07:00.479 So all you need to do is plot the log of the pressure of 07:00.480 --> 07:03.390 carbon atoms, which will be very low, 07:03.394 --> 07:07.044 versus 1/T, at very high temperature -- 07:07.036 --> 07:09.646 right?; of the order of 3000 Kelvin. 07:09.649 --> 07:13.879 So that's not easy to do. 07:13.879 --> 07:19.249 But it was done in 1955 by Chupka and Inghram. 07:19.250 --> 07:24.640 And this is the sketch of their instrument, and I'll show you 07:24.641 --> 07:26.261 how they did it. 07:26.259 --> 07:29.769 First there's a graphite cylinder. 07:29.774 --> 07:30.524 Okay? 07:30.519 --> 07:35.259 And around it is a can made out of tantalum. 07:35.259 --> 07:36.809 Now why tantalum? 07:36.810 --> 07:40.420 Because that's about the highest melting thing you can 07:40.420 --> 07:42.950 get; it's the highest melting metal. 07:42.949 --> 07:47.069 So 3293 Kelvin is its melting point. 07:47.069 --> 07:50.959 So you could really heat the heck out of this thing. 07:50.964 --> 07:51.504 Okay? 07:51.500 --> 07:55.550 And now you surround that with wires, and those wires are made 07:55.553 --> 07:59.013 of tungsten, because that's very high melting too. 07:59.009 --> 08:03.559 And so you connect wire -- you connect electricity to the 08:03.555 --> 08:06.555 tungsten, and also to the tantalum. 08:06.560 --> 08:10.630 So electrons boil off the tungsten and bombard the 08:10.634 --> 08:15.544 tantalum, and when the electrons hit it, they heat it up. 08:15.540 --> 08:19.390 So you can heat the heck out of this tantalum can by bombarding 08:19.387 --> 08:20.627 it with electrons. 08:20.629 --> 08:24.139 And that, of course, heats the graphite that's 08:24.137 --> 08:24.837 inside. 08:24.839 --> 08:29.519 Now, so inside there's going to be a gas of carbon, 08:29.521 --> 08:32.801 in equilibrium with the graphite. 08:32.799 --> 08:34.739 Now it won't just be C atoms. 08:34.740 --> 08:38.260 It'll also be C_2, C_3, C_60, C_70, 08:38.264 --> 08:41.594 and so on; but lots of different forms of 08:41.591 --> 08:42.841 carbon, as a gas. 08:42.840 --> 08:44.590 So you can't just measure the pressure in there; 08:44.590 --> 08:48.160 and indeed it would be tough to measure the pressure inside, 08:49.070 --> 08:53.400 But you at least have these things there at equilibrium. 08:53.399 --> 08:56.319 Now what they did, Chupka and Inghram, 08:56.322 --> 09:00.512 was to drill a tiny hole, in the top of this thing. 09:00.509 --> 09:04.359 And that will allow a little bit of the gas to escape; 09:04.360 --> 09:08.150 not so much that you destroy the equilibrium inside, 09:08.148 --> 09:09.928 just a really slow leak. 09:09.931 --> 09:10.601 Right? 09:10.600 --> 09:14.710 So now, if you could measure the amount of these different 09:14.711 --> 09:18.321 carbon species leaking out, you could know what the 09:18.320 --> 09:20.850 pressure of them was inside the can. 09:20.846 --> 09:21.636 Okay? 09:21.639 --> 09:26.009 Now, so there's a beam of carbon species coming out, 09:26.008 --> 09:28.148 gaseous carbon species. 09:28.149 --> 09:30.419 This is all at very, very high vacuum. 09:30.418 --> 09:30.908 Right? 09:30.908 --> 09:34.028 And then up there, an electron beam comes across 09:34.032 --> 09:37.752 and hits these carbon things, and knocks electrons out of 09:37.751 --> 09:38.351 them. 09:38.350 --> 09:47.730 And that converts them into C+; C_1, C_2, C_3, C_60+. Okay? 09:47.730 --> 09:51.750 So you have a beam coming out of these charged carbon species. 09:51.750 --> 09:54.000 Now why do you want them to be charged? 09:54.000 --> 09:55.240 For two reasons. 09:55.240 --> 09:59.640 One, so you can detect them; because you can detect it when 09:59.644 --> 10:00.984 charge hits a plate. 10:00.980 --> 10:05.670 And the other one is so you can deflect the beams with a 10:05.672 --> 10:06.952 magnetic field. 10:06.951 --> 10:07.721 Right? 10:07.720 --> 10:11.340 So these various carbon species come out, and they go into a 10:11.337 --> 10:14.037 magnetic field, which puts a lateral force on 10:14.035 --> 10:16.055 them and causes them to bend. 10:16.058 --> 10:18.588 And, of course, the lighter they are, 10:18.594 --> 10:23.034 the easier they are to bend if they all have the same charge. 10:23.028 --> 10:26.218 So you're going to get curved trajectories. 10:26.220 --> 10:30.310 C_3+, C_2+, C_1+ will be the most bent. 10:30.308 --> 10:34.548 And now if you put a detector at these different positions, 10:34.553 --> 10:37.343 you can see how much C_1, how much C_2, 10:37.335 --> 10:39.015 how much C_3 there was. 10:39.018 --> 10:39.748 Okay? 10:39.750 --> 10:41.510 So that's how you're going to do it. 10:41.509 --> 10:43.579 But this machine has to be pretty special, 10:43.578 --> 10:45.248 because you're heating it so high. 10:45.245 --> 10:45.745 Lucas? 10:45.750 --> 10:47.130 Student: How do you know the electron beam only 10:47.125 --> 10:47.785 knocks off one electron? 10:47.788 --> 10:50.318 Prof: It can knock off two, but then you get it at a 10:50.317 --> 10:51.447 very different position. 10:51.450 --> 10:52.510 You can tell these things. 10:52.509 --> 10:56.429 This thing is called mass spectroscopy. 10:56.428 --> 10:59.718 But it knocks off one much more often than it knocks off two or 10:59.716 --> 11:00.136 three. 11:00.139 --> 11:03.879 Student: And isn't this kind of the same problem as we 11:03.880 --> 11:07.010 had with the other thing, where the C is ionized? 11:07.009 --> 11:09.099 Prof: Which other thing? 11:09.100 --> 11:10.620 I can't remember what problem -- 11:10.620 --> 11:13.870 Student: You said that we couldn't measure the other 11:13.869 --> 11:17.229 one because it was a high energy state, spectrographically. 11:17.230 --> 11:18.110 Prof: Oh. 11:18.105 --> 11:19.745 Might it be an excited state? 11:19.750 --> 11:19.950 Student: Yeah. 11:19.950 --> 11:21.550 Prof: No because it's in equilibrium. 11:21.549 --> 11:22.669 Student: Okay. 11:22.668 --> 11:24.818 Prof: The excited state would be much, 11:24.817 --> 11:27.497 much less at equilibrium, because it's higher in energy. 11:27.500 --> 11:27.990 Right? 11:27.990 --> 11:30.270 That's a good point though; good that you thought about 11:30.273 --> 11:30.573 that. 11:30.570 --> 11:33.270 Now, so you have to put shielding around this stuff, 11:33.267 --> 11:34.957 so it doesn't melt everything. 11:34.960 --> 11:38.640 So you use tantalum, which is high melting, 11:38.643 --> 11:41.453 and a series of can-inside-a-can, 11:41.448 --> 11:43.118 like Russian dolls. 11:43.115 --> 11:44.075 Right? 11:44.080 --> 11:47.310 So the inside most one is very, very hot; 11:47.308 --> 11:49.528 then a little less hot, a little less hot. 11:49.529 --> 11:49.909 Okay? 11:49.909 --> 11:50.919 So you shield it. 11:50.918 --> 11:54.118 And now you need to know the temperature inside. 11:54.120 --> 11:57.700 And you do it by drilling a tiny hole through those shields 11:57.698 --> 12:00.968 so you can have what's called an optical pyrometer. 12:00.970 --> 12:03.490 That's just something that looks at the color of something 12:03.485 --> 12:04.365 that's really hot. 12:04.370 --> 12:07.420 And something that's hot gives -- even something that's cold -- 12:07.417 --> 12:09.037 gives off black-body radiation. 12:09.038 --> 12:13.168 And the color has to do with the temperature; 12:13.168 --> 12:16.068 you know, there's red heat, white heat, blue heat, 12:16.068 --> 12:16.778 and so on. 12:16.778 --> 12:19.548 So by measuring the color, you can see what the 12:19.552 --> 12:21.062 temperature was inside. 12:21.058 --> 12:23.728 And that window, that the light comes through, 12:23.734 --> 12:26.354 has to be made of quartz, not Pyrex glass. 12:26.350 --> 12:27.230 You know why? 12:27.230 --> 12:28.870 Student: Pyrex would melt. 12:28.870 --> 12:31.400 Prof: Because Pyrex glass would melt, 12:31.397 --> 12:32.687 even at that distance. 12:32.690 --> 12:33.220 Right? 12:33.220 --> 12:36.060 So you use quartz glass. 12:36.059 --> 12:38.019 Okay, so that's what you do. 12:38.019 --> 12:41.629 And here's a graph of the pressure of these various 12:41.625 --> 12:43.415 species, measured with that mass 12:43.418 --> 12:45.678 spectrometer, at different temperatures, 12:45.682 --> 12:48.102 measured as one over temperature here. 12:48.100 --> 12:52.450 So it goes from 2150 Kelvin to 2450 Kelvin, and the pressure 12:52.446 --> 12:53.326 increases. 12:53.330 --> 12:57.250 And it's plotted as the log of the intensity of the signal; 12:57.250 --> 12:59.780 that is, the pressure times temperature, because the 12:59.783 --> 13:02.123 pressure has to be corrected for temperature. 13:02.120 --> 13:04.810 Because if you have the same number of things giving the 13:04.812 --> 13:07.652 pressure, but they're hotter, they'll be pressing harder on 13:07.650 --> 13:08.190 things. 13:08.190 --> 13:12.000 It's the intensity of the signal times the temperature 13:11.998 --> 13:13.578 that you plot there. 13:13.580 --> 13:17.080 And from the slope, you can see up there at the 13:17.083 --> 13:20.973 top, that for C_1 you get -- the slope says it's 171 13:20.970 --> 13:24.940 kilocalories/mol; Q.E.D.***Right? 13:24.940 --> 13:26.970 Now you know which was the right one, measured by 13:26.966 --> 13:27.596 spectroscopy. 13:27.600 --> 13:29.920 It was the one that said 171. 13:29.918 --> 13:32.628 So this experiment, actually measuring the 13:32.625 --> 13:35.855 equilibrium between carbon atoms and graphite, 13:35.860 --> 13:39.440 by a really gargantuan kind of experiment, 13:39.440 --> 13:42.180 is what settled the question finally. 13:42.178 --> 13:45.258 So that when you look at the appendix of this book, 13:45.259 --> 13:46.659 Streitwieser, Heathcock and Kosower, 13:46.658 --> 13:49.608 you find that there are heats of formation for atoms and 13:49.606 --> 13:51.956 radicals, measured by spectroscopy and 13:51.960 --> 13:53.060 things like this. 13:53.059 --> 13:56.389 And there you see carbon, 170.9. 13:56.389 --> 13:58.759 And that was done by Professor Chupka, who was in this 13:58.758 --> 14:01.528 department and who used to come and tell people how he did this 14:01.528 --> 14:02.198 experiment. 14:02.200 --> 14:05.000 But as you can see, he passed away in 2007. 14:05.000 --> 14:06.820 So thanks to Professor Chupka. 14:06.820 --> 14:10.090 And the nice thing is, once that's done, 14:10.091 --> 14:11.101 it's done. 14:11.100 --> 14:14.090 Now you know it and you just plug it in when you burn your 14:14.085 --> 14:16.385 stuff and want to know what its energy is. 14:16.389 --> 14:19.529 You can get it relative to carbon atoms in the gas phase. 14:19.528 --> 14:23.048 Okay, now how good are these spectroscopic experiments? 14:23.049 --> 14:24.409 Well this is a neat thing. 14:24.408 --> 14:28.298 The heat of atomization of methane, measured in the ways 14:28.298 --> 14:32.468 we've just been talking about, is 397.5 kilocalories/mol. 14:32.470 --> 14:37.430 Now that comes from mating a carbon atom with four hydrogen 14:37.426 --> 14:41.106 atoms; that is 397.5. Right? 14:41.110 --> 14:43.710 So we know what the average bond energy. 14:43.710 --> 14:44.980 There are four such bonds. 14:44.980 --> 14:48.330 So each one's worth 99.4 kilocalories/mol. 14:48.330 --> 14:51.570 So about 100 kilocalories/mol for a C-H bond; 14:51.570 --> 14:53.630 that's convenient to remember. 14:53.628 --> 14:54.038 Okay? 14:54.038 --> 14:58.638 But that's not how much it costs to take a single hydrogen 14:58.640 --> 15:00.660 atom away from methane. 15:00.658 --> 15:04.388 Taking a single hydrogen atom away from methane, 15:04.389 --> 15:07.239 the so-called "bond dissociation energy", 15:07.240 --> 15:10.180 which is the actual experimental energy it takes to 15:10.183 --> 15:14.043 do some particular process -- average bond energy is just an 15:14.042 --> 15:16.382 average, but the individual ones are not 15:16.384 --> 15:19.444 the same -- taking a hydrogen away from 15:19.442 --> 15:22.912 methane is 104.99, plus or minus 0.03 15:22.908 --> 15:24.588 kilocalories/mol. 15:24.590 --> 15:26.270 Close, but not the same thing. 15:26.269 --> 15:28.569 And then you have CH_3. 15:28.570 --> 15:31.390 If you take a second H off that, it's 110.4. 15:31.389 --> 15:34.839 The next one is 101.3, and the final one, 15:34.841 --> 15:37.781 taking H away from C, is only 80.9. 15:37.775 --> 15:38.635 Right? 15:38.639 --> 15:41.439 Now these are done by spectroscopy. 15:41.440 --> 15:44.820 And Barney Ellison may come and talk to us in the spring; 15:44.820 --> 15:47.710 he's often traveling through and talks about how he measured 15:47.714 --> 15:48.454 these things. 15:48.450 --> 15:50.240 But those are done by spectroscopy. 15:50.240 --> 15:54.130 But the neat thing about it is if you add all those four 15:54.134 --> 15:57.824 numbers together -- so, pardon me, I was going to say 15:57.818 --> 16:00.438 no individual bond equals the average. 16:00.440 --> 16:01.290 Right? 16:01.288 --> 16:04.698 But if you add them together you get 397.5, 16:04.700 --> 16:07.380 which is precisely the average. 16:07.379 --> 16:09.289 So these are very good experiments. 16:09.288 --> 16:14.518 So we know, through heroic spectroscopy and this work of 16:14.524 --> 16:18.314 Chupka and Inghram, we know what these energies 16:18.307 --> 16:23.557 are, bond energies, and bond dissociation energies. 16:23.558 --> 16:26.468 So here are average bond energies in a table that you 16:26.474 --> 16:29.224 have at the end of this organic chemistry text. 16:29.220 --> 16:31.920 And it says a carbon-hydrogen bond is 99; 16:31.918 --> 16:33.448 and now you see where we get that. 16:33.450 --> 16:38.390 And you see that a carbon-carbon bond is 83. 16:38.389 --> 16:42.719 But the second carbon-carbon bond, in a double bond, 16:42.721 --> 16:43.651 is only 63. 16:43.654 --> 16:44.424 Right? 16:44.419 --> 16:45.239 Why is it weaker? 16:45.240 --> 16:52.270 Why is the second bond of a carbon-carbon double bond weaker 16:52.273 --> 16:54.303 than the first? 16:54.299 --> 16:56.519 Pardon me? 16:56.519 --> 17:00.079 Devin, what do you say? 17:00.080 --> 17:01.270 Student: The overlap. 17:01.269 --> 17:03.589 Prof: Yeah, you have bad overlap between 17:03.594 --> 17:05.014 the π electrons. 17:05.009 --> 17:08.329 In fact, the first, the single bond of a double 17:08.334 --> 17:12.314 bond, is probably stronger than a normal single bond. 17:12.308 --> 17:14.378 Can anybody see why that would be? 17:14.380 --> 17:17.280 Student: More s character. 17:17.278 --> 17:18.838 Prof: It's got more s character, 17:18.839 --> 17:21.309 better overlap; sp^2-sp^2. 17:21.308 --> 17:25.288 So the second one is probably more than 20 kilocalories weaker 17:25.292 --> 17:26.732 than the single one. 17:26.730 --> 17:30.630 But at any rate it's 146, that you add up to get a double 17:30.631 --> 17:35.311 bond, and 200 for a triple bond; which means the third bond is 17:35.310 --> 17:37.780 worth only 54 kilocalories/mol. 17:37.781 --> 17:38.341 Okay? 17:38.338 --> 17:44.948 And in C=O, it's about the same as C-C. 17:44.950 --> 17:47.270 So C-O is eighty-six. 17:47.269 --> 17:50.189 But the double bond, notice, is different in this 17:50.192 --> 17:50.622 case. 17:50.618 --> 17:53.598 Now the second bond is stronger than the first. 17:53.598 --> 18:00.988 So the carbonyl group is an especially stable group. 18:00.986 --> 18:01.996 Okay? 18:02.000 --> 18:05.260 So you have the question, can you sum up these average 18:05.259 --> 18:08.459 bond energies and get useful heats of atomization? 18:08.460 --> 18:11.600 So can you look at a structure and say how stable it's going to 18:11.596 --> 18:11.846 be? 18:11.849 --> 18:14.439 Okay, so let's try it. 18:14.440 --> 18:18.130 So here's heats of atomization by additivity of average bond 18:18.131 --> 18:18.821 energies. 18:18.818 --> 18:21.858 So we have these average bond energies from the table. 18:21.859 --> 18:23.549 A C-C single-bond is 83. 18:23.549 --> 18:25.189 A C-H is 99. 18:25.190 --> 18:28.980 C double bond C is 146,86, 111 and so on. 18:28.980 --> 18:31.260 And we're going to sum them all up to get the heat of 18:31.260 --> 18:34.070 atomization -- compare it with the actual heat of atomization. 18:34.068 --> 18:38.408 Okay, so for ethene, there are four C-H bonds, 18:38.413 --> 18:41.313 there's one C-C double bond. 18:41.309 --> 18:43.509 Add them up, and you get 542. 18:43.509 --> 18:47.549 The actual heat of atomization is 537.7. 18:47.548 --> 18:50.278 So there's an error of 4.3 kilocalories/mol, 18:50.284 --> 18:51.814 which is less than 1%. 18:51.809 --> 18:53.669 That's pretty good. 18:53.670 --> 18:57.610 But on the other hand, ethene probably entered in to 18:57.606 --> 19:00.616 determining these average bond energies. 19:00.616 --> 19:01.386 Right? 19:01.390 --> 19:04.210 So it's not 100% fair. Okay? 19:04.210 --> 19:05.410 How about cyclohexane? 19:05.410 --> 19:09.260 Now we have 6 carbon-carbon single bonds and 12 19:09.259 --> 19:13.529 carbon-hydrogen single bonds: 1686 versus 1680.1. 19:13.528 --> 19:16.778 An error of only 5.9, less than half a percent error. 19:16.778 --> 19:19.268 So pretty good, by adding up bonds. 19:19.269 --> 19:20.439 Cyclohexanol. 19:20.440 --> 19:23.100 Remember we had quite a bit of trouble with these partly 19:23.097 --> 19:25.657 oxidized things before, when we were trying to just do 19:25.660 --> 19:27.400 it on the basis of the elements. 19:27.400 --> 19:31.370 But if you add bonds together, you get within 0.3% of the 19:31.366 --> 19:32.356 right value. 19:32.358 --> 19:37.038 Or if you do glucose, which has lots of oxygens in 19:37.035 --> 19:41.325 it, then you get within again less then 1%; 19:41.329 --> 19:43.209 0.7% of the right value. 19:43.210 --> 19:46.710 So this is pretty darn good; very impressive, 19:46.708 --> 19:48.878 very small errors, to predict these. 19:48.880 --> 19:50.890 But the question is, is it useful? 19:50.890 --> 19:53.810 How accurate does it have to be, to be useful? 19:53.808 --> 19:56.678 So why do you need to know the values? 19:56.680 --> 19:58.720 Because you want to know equilibrium constants. 19:58.720 --> 20:00.850 You want to know which direction a reaction will go, 20:00.854 --> 20:01.444 for example. 20:01.440 --> 20:05.170 Okay, so we know that the -- if we want to calculate an 20:05.166 --> 20:07.996 equilibrium constant, we can do it at room 20:07.997 --> 20:10.687 temperature with this 3/4ths trick. 20:10.690 --> 20:13.950 So the calculated equilibrium constant is whatever we're 20:13.948 --> 20:16.968 calculating here, for energy, between two things. 20:16.970 --> 20:18.780 We have two things: calculate the energy of these, 20:18.778 --> 20:21.428 the energy of these, compare the energies, 20:21.430 --> 20:23.080 and that'll give us the equilibrium constant, 20:23.079 --> 20:24.169 according to this formula. 20:24.170 --> 20:27.430 But notice I'm doing it on the basis of calculation. 20:27.430 --> 20:31.650 Now so that's -- whatever -- the calculated energy is 20:31.648 --> 20:34.568 whatever the true energy would be. 20:34.568 --> 20:38.228 But there's also some error in there. 20:38.228 --> 20:38.938 Okay? 20:38.940 --> 20:42.910 But if you add two exponents, that's the same as multiplying 20:42.909 --> 20:44.389 two things together. 20:44.390 --> 20:47.990 So the calculated equilibrium constant is the true equilibrium 20:47.991 --> 20:49.781 constant -- that's the first part, 20:49.782 --> 20:52.112 the part that it has with ΔH true -- 20:52.108 --> 20:55.218 times the part that has this exponent. 20:55.218 --> 20:55.888 Right? 20:55.890 --> 20:58.490 3/4ths of the ΔH error. 20:58.490 --> 21:03.140 So if you want the error to be small, that factor to be small, 21:03.138 --> 21:06.568 then ΔH error has to be small; 21:06.568 --> 21:12.418 small, not on a percentage basis, but absolutely it has to 21:12.420 --> 21:13.550 be small. 21:13.548 --> 21:19.038 That error, not the percentage error, determines this error 21:19.044 --> 21:19.714 factor. 21:19.707 --> 21:20.557 Right? 21:20.558 --> 21:23.648 So to keep the error less than a factor of ten, 21:23.650 --> 21:27.100 in the equilibrium constant, you need to know the 21:27.099 --> 21:30.909 equilibrium constants within 1.3 kilocalories/mol, 21:30.910 --> 21:34.350 so that 3/4ths of it will be one, and that would mean you'd 21:34.348 --> 21:36.008 be within a factor of ten. 21:36.009 --> 21:37.309 Everybody with me on this? 21:37.308 --> 21:40.758 So you need to do even better than this. 21:40.759 --> 21:44.009 You can't use the average bond energies and get something 21:44.011 --> 21:46.791 that's very useful, because if you're off by 21:46.788 --> 21:49.388 sixteen down here, in the case of sugar, 21:49.390 --> 21:53.060 that means you're off by a factor of 10^12th in predicting 21:53.055 --> 21:57.395 the equilibrium constant, which wouldn't be acceptable 21:57.401 --> 21:58.201 probably. 21:58.199 --> 21:58.819 Okay? 21:58.818 --> 22:04.838 So let's try it with the equilibrium between a ketone and 22:04.836 --> 22:08.746 the so-called enol, which is an isomer of a ketone 22:08.750 --> 22:11.750 in which a hydrogen has been taken from the methyl group on 22:11.749 --> 22:15.839 the right and put on the oxygen, and the double bond moved. Okay? 22:15.838 --> 22:19.528 So that's a very important equilibrium that we'll encounter 22:19.526 --> 22:22.446 when we talk about the chemistry of ketones. 22:22.450 --> 22:26.500 Let's see what the equilibrium constant -- should there be more 22:26.501 --> 22:27.941 ketone or more enol? 22:27.940 --> 22:32.140 Do you have a guess right at the outset of which one would be 22:32.140 --> 22:33.120 more stable? 22:33.118 --> 22:38.908 I would guess the ketone, because of what I just told 22:38.913 --> 22:44.483 you, that the C-O double bond is remarkably stable. 22:44.484 --> 22:45.604 Right? 22:45.598 --> 22:47.698 In the other case you have a C-C double bond. 22:47.700 --> 22:48.800 Okay, let's see. 22:48.798 --> 22:52.318 Now we could add together all the bonds. 22:52.318 --> 22:55.368 But most of them, most bonds are the same between 22:55.365 --> 22:57.645 the starting material and product. 22:57.650 --> 23:00.820 We only need to compare the ones that change. 23:00.816 --> 23:01.316 Okay? 23:01.318 --> 23:04.788 So we've highlighted in red the bonds that change between the 23:04.788 --> 23:06.488 two forms, the two isomers, 23:06.491 --> 23:10.011 because we're interested in the difference in energy between 23:10.006 --> 23:12.996 these two, to get the equilibrium constant. 23:13.000 --> 23:16.170 Okay, so these are the numbers I took from the table, 23:16.165 --> 23:19.695 that you see on the top left there: 179 for C-O double bond 23:19.696 --> 23:20.546 and so on. 23:20.548 --> 23:25.728 And I sum them up and that's 361, for those bonds. 23:25.730 --> 23:29.460 And the new set of bonds, in the enol, 23:29.459 --> 23:30.769 sum to 343. 23:30.769 --> 23:34.179 So the ketone indeed is more stable, it appears, 23:34.183 --> 23:36.003 by 18 kilocalories/mol. 23:36.000 --> 23:40.550 So 18 kilocalories/mol means that you have a factor of 23:40.550 --> 23:43.940 10^13th; the equilibrium constant is 23:43.936 --> 23:44.776 10^13.5. 23:44.779 --> 23:47.539 So it should lie, for practical purposes, 23:47.536 --> 23:50.426 entirely in the direction of the ketone. 23:50.430 --> 23:54.600 However, if you do it experimentally, 23:54.597 --> 24:00.737 you find that the equilibrium constant is only 10^7th, 24:00.736 --> 24:02.126 not 10^14th. 24:02.125 --> 24:03.395 Right? 24:03.400 --> 24:08.170 So the true energy difference is 9.3 kilocalories; 24:08.170 --> 24:12.930 not the eighteen that we got by adding bonds together. 24:12.931 --> 24:13.651 Right? 24:13.650 --> 24:19.940 So that means we're going to have to deal with addressing why 24:19.939 --> 24:22.559 the enol is too stable. 24:22.558 --> 24:26.348 It's 9 kilocalories/mol too stable, compared to our 24:26.346 --> 24:30.356 predictions, on the basis of adding bonds together. 24:30.359 --> 24:31.559 Now why? 24:31.558 --> 24:36.868 Well one thing is that we -- that those bonds that we 24:36.872 --> 24:40.212 cancelled, the C-H bonds that didn't 24:40.214 --> 24:44.654 change, in fact did change between the starting material 24:44.647 --> 24:46.177 and the product. 24:46.180 --> 24:48.420 Why could I say that they did change? 24:48.420 --> 24:49.870 In both cases, on both sides, 24:49.865 --> 24:52.755 there are single carbon-carbon, carbon-hydrogen bonds. 24:52.759 --> 24:54.639 How can I say they change? 24:54.641 --> 24:55.221 Angela? 24:55.220 --> 24:58.080 Student: Well with the ketone, they're sp^3 24:58.077 --> 24:58.727 hybridized. 24:58.730 --> 24:59.140 Prof: Ah ha. 24:59.140 --> 25:01.410 Student: In the enol they're sp^2 hybridized. 25:01.410 --> 25:02.240 Prof: Right. 25:02.240 --> 25:03.640 They're changing hybridization. 25:03.640 --> 25:05.850 Actually, yeah, they go from sp^3 to 25:05.847 --> 25:08.317 sp^2, on the carbon, as you go across. 25:08.318 --> 25:11.828 And the sp^2's on the right should be more stable. 25:11.830 --> 25:12.270 Okay? 25:12.269 --> 25:14.719 So the sp^2-H should be stronger. 25:14.720 --> 25:18.450 So these things that I was saying cancelled do not actually 25:18.445 --> 25:21.525 cancel, if we take hybridization into account. 25:21.529 --> 25:23.149 So that's one factor. 25:23.150 --> 25:28.280 And there's another as well, which is you have that unshared 25:28.282 --> 25:31.942 pair, on the top right here, on the oxygen, 25:31.938 --> 25:34.808 is adjacent to a double bond. 25:34.808 --> 25:38.778 That means that this high HOMO can be stabilized by the 25:38.775 --> 25:41.635 π* low LUMO; it'll overlap. 25:41.640 --> 25:46.110 That isn't a possibility here, where the unshared pairs on the 25:46.108 --> 25:50.138 oxygen did not overlap with the π* orbital. 25:50.140 --> 25:54.370 So you get intramolecular HOMO/LUMO mixing in the enol 25:54.373 --> 25:57.173 that you don't get in the ketone; 25:57.170 --> 26:00.710 which will help stabilize the enol, with that -- we could draw 26:00.711 --> 26:02.281 that resonance structure. 26:02.278 --> 26:06.648 So those two things together make up that 9 kilocalorie 26:06.652 --> 26:08.982 error; or at least we can think -- 26:08.979 --> 26:10.969 they contribute to it at least. 26:10.970 --> 26:15.170 So constitutional energy, what we would get by adding 26:15.166 --> 26:19.656 bonds together, has to be corrected for various 26:19.661 --> 26:23.081 "effects", we'll call them, 26:23.077 --> 26:26.167 such as resonance, that's what we just looked at, 26:26.170 --> 26:31.090 like this HOMO/LUMO thing, such as hybridization changes, 26:31.086 --> 26:34.946 or such as strain, as in the case of axial 26:34.949 --> 26:37.169 methylcyclohexane, that we looked at. 26:37.170 --> 26:41.310 So there are lots and lots of these corrections that you have 26:41.314 --> 26:44.884 to apply to this model, where you add together bond 26:44.883 --> 26:48.563 energies in order to predict the energies of a particular 26:48.560 --> 26:49.350 molecule. 26:49.348 --> 26:52.748 But for many cases now, you can do a pretty good job of 26:52.750 --> 26:57.170 predicting these things, and actually not do so bad at 26:57.173 --> 27:01.123 predicting relative energies of isomers, 27:01.118 --> 27:03.108 and therefore equilibrium constants. 27:03.108 --> 27:07.548 And these effects, of course, are a polite name 27:07.547 --> 27:08.507 for error. 27:08.510 --> 27:09.380 Right? 27:09.380 --> 27:13.020 They're correcting -- various ways of correcting errors that 27:13.018 --> 27:16.718 you think there should be in this scheme of just adding bonds 27:16.718 --> 27:17.518 together. 27:17.519 --> 27:21.539 Now, energy determines what can happen. 27:21.538 --> 27:24.578 Things always move toward equilibrium. 27:24.582 --> 27:25.242 Right? 27:25.240 --> 27:29.210 So if the ratio of two things is something, 27:29.210 --> 27:31.100 but the equilibrium ratio is different, 27:31.098 --> 27:33.938 the ratio will always move toward that, 27:33.940 --> 27:39.160 toward the equilibrium, if it's in isolation. 27:39.160 --> 27:43.630 But there's another equally important thing is how fast will 27:43.628 --> 27:44.688 it go there? 27:44.690 --> 27:48.820 And that, as we've seen before, can be approximated as 27:48.817 --> 27:52.397 10^13th/second, times this same kind of factor, 27:52.398 --> 27:54.578 relating to the barrier. 27:54.578 --> 28:01.958 Now both of these things suggest that being low in energy 28:01.959 --> 28:03.009 is good. 28:03.012 --> 28:04.202 Right? 28:04.200 --> 28:06.180 You favor things that are low in energy. 28:06.180 --> 28:09.060 But you might ask why? 28:09.058 --> 28:11.988 That's not what people say about money. 28:11.990 --> 28:15.880 They don't say the less money you have the better. 28:15.875 --> 28:16.505 Right? 28:16.509 --> 28:18.389 Why the less energy the better? 28:18.390 --> 28:22.390 This is a really interesting case, and it has to do with 28:22.393 --> 28:23.343 statistics. 28:23.338 --> 28:26.288 And especially at Yale we should talk about this, 28:26.288 --> 28:30.198 because in 1902, when Yale celebrated it's 28:30.202 --> 28:33.292 bicentennial, they published a number of 28:33.288 --> 28:35.868 books showing off the scholarship of Yale; 28:35.869 --> 28:37.999 as you can see here. 28:38.000 --> 28:40.540 And the most important of those books, 28:40.538 --> 28:42.938 by about 500 miles, was this one: 28:42.942 --> 28:47.152 Elementary Principles in Statistical Mechanics and the 28:47.147 --> 28:50.227 Rational Foundation of Thermodynamics, 28:50.227 --> 28:52.327 by J. Willard Gibbs. 28:52.329 --> 28:55.479 So it's statistical mechanics. 28:55.480 --> 28:59.570 It's trying to understand the behavior of chemical substances, 28:59.567 --> 29:01.507 on the basis of statistics. 29:01.509 --> 29:06.889 Now when you do this, you get exponents. 29:06.890 --> 29:11.960 And the organization of our presentation here is going to 29:11.961 --> 29:17.581 have to do with three different ways in which statistics enters 29:17.577 --> 29:21.137 into exponents, for purposes of doing 29:21.136 --> 29:22.216 equilibrium. 29:22.220 --> 29:26.140 So there's the Boltzmann Factor; that's what we've been talking 29:26.140 --> 29:28.130 about, the 10^(3/4 ΔH), 29:28.125 --> 29:30.275 that's called the Boltzmann Factor. 29:30.278 --> 29:32.048 It includes the Boltzmann Constant. 29:32.048 --> 29:36.298 Then there are things that have to do with entropy, 29:36.297 --> 29:40.457 which often seems to be a very confusing topic. 29:40.460 --> 29:43.000 And finally there's a thing called the Law of Mass Action. 29:43.000 --> 29:46.920 And all of these things have exponents in them. 29:46.920 --> 29:49.940 And if you understand how the exponents behave, 29:49.942 --> 29:52.112 you understand what's going on. 29:52.108 --> 29:55.188 So let's look first at the Boltzmann Factor. 29:55.190 --> 30:00.300 So here's Ludwig Boltzmann, who committed suicide in 1906. 30:00.298 --> 30:03.808 And this is his important paper on "The Relationship 30:03.807 --> 30:07.247 between the Second Law of Thermodynamics and Probability 30:07.252 --> 30:08.822 Calculations" -- 30:08.818 --> 30:11.808 so statistics -- "Regarding the Laws of 30:11.807 --> 30:14.377 Thermal Equilibrium," in 1877. 30:14.380 --> 30:20.210 And his key equation is S = k lnW . So log relates to an 30:20.205 --> 30:24.055 exponential, and we'll see why that is. 30:24.058 --> 30:27.798 And here's his tombstone in the cemetery in Vienna. 30:27.798 --> 30:31.828 And you'll notice, up at the top there, 30:31.825 --> 30:33.725 S = k ln W. 30:33.731 --> 30:34.581 Okay? 30:34.578 --> 30:39.658 So what Boltzmann considered was the implication of random 30:39.661 --> 30:41.891 distribution of energy. 30:41.890 --> 30:44.660 Suppose you have a certain amount of energy, 30:44.662 --> 30:47.432 in a system, but it's distributed at random. 30:47.434 --> 30:48.084 Right? 30:48.079 --> 30:49.829 So purely statistically. 30:49.828 --> 30:51.858 Then how should it be distributed? 30:51.858 --> 30:55.078 How much energy should any particular molecule have, 30:55.082 --> 30:56.222 is the question. 30:56.220 --> 31:00.640 And we can visualize this in a simple case, which is very like 31:00.640 --> 31:04.630 what he did, except he did it analytically and in a much 31:04.625 --> 31:05.925 bigger system. 31:05.930 --> 31:09.350 But just using four containers, which are like molecules, 31:09.348 --> 31:12.828 and each one can have a certain amount of energy in it. 31:12.828 --> 31:16.318 And we'll consider the energy to be -- to come in bits. 31:16.318 --> 31:19.638 He used that idea, that there were bits of energy 31:19.636 --> 31:23.436 to be distributed among molecules, or degrees of freedom 31:23.438 --> 31:24.888 within molecules. 31:24.890 --> 31:27.750 He didn't think that energy came in bits, 31:27.750 --> 31:30.160 but it made it possible to do the statistics, 31:30.160 --> 31:32.960 and then he just took the limit when these bits get very, 31:32.960 --> 31:35.480 very, very, very small, so that it becomes like a 31:35.482 --> 31:39.032 continuum of stuff, like a whole sand of energy 31:39.029 --> 31:39.559 bits. 31:39.558 --> 31:43.418 But anyhow, let's just count up how many different ways there 31:43.421 --> 31:46.641 are of putting three different bits of energy -- 31:46.640 --> 31:48.950 or actually not different, they're all the same -- 31:48.950 --> 31:53.570 but three bits of energy into the red container. 31:53.570 --> 31:54.260 Okay? 31:54.259 --> 31:57.619 So if you -- how many complexions -- 31:57.618 --> 31:59.878 that's what, any particular arrangement he 31:59.884 --> 32:02.854 called a complexion -- how many different complexions 32:02.848 --> 32:05.798 have a certain number of bits in the first container? 32:05.798 --> 32:10.108 Well suppose you put all three of those energy bits into the 32:10.105 --> 32:11.415 first container. 32:11.420 --> 32:14.450 How many different ways are there of doing that? 32:14.450 --> 32:16.840 Just one. Okay? 32:16.838 --> 32:20.208 But suppose you put only two into the first container? 32:20.210 --> 32:23.010 Now how many different ways are there of arranging it so that 32:23.009 --> 32:24.829 there are two in the first container? 32:24.829 --> 32:26.739 How many different ways? 32:26.740 --> 32:30.930 There are three: 1,2, 3. 32:30.930 --> 32:33.980 So there'll be three ways of putting two bits in the first 32:33.978 --> 32:34.618 container. 32:34.618 --> 32:37.288 How many of putting one in the first container? 32:37.289 --> 32:40.739 Well we put one there; there's 1,2, 32:40.740 --> 32:45.680 3,4, 5,6 ways of doing it. 32:45.680 --> 32:47.010 Okay? 32:47.009 --> 32:49.169 So there's six ways of putting one in there. 32:49.170 --> 32:51.690 And how many of putting none at all in there? 32:51.690 --> 33:04.770 1,2, 3,4, 5,6, 7,8, 9,10. Okay? 33:04.769 --> 33:06.579 So there are 10 ways of doing that. 33:06.578 --> 33:10.128 So let's make a graph and see how many ways -- what's the 33:10.125 --> 33:14.045 probability that you'll have a certain number of bits of energy 33:14.049 --> 33:15.759 in the first container? 33:15.759 --> 33:21.919 Okay, it looks like that: 10,6, 3,1. 33:21.920 --> 33:27.710 And what does that curve look like, if you made a plot of 33:27.705 --> 33:28.425 that? 33:28.430 --> 33:32.000 What type of curve does it look like? 33:32.000 --> 33:34.540 Is it a straight line? 33:34.538 --> 33:38.158 Anybody got a name for it, or something that looks a 33:38.164 --> 33:39.804 little bit like that? 33:39.799 --> 33:42.069 Student: Exponential. 33:42.068 --> 33:43.528 Prof: It's exponential decay. 33:43.529 --> 33:46.819 Now it's not truly exponential, in this case, 33:46.824 --> 33:49.674 of three bits among four containers. 33:49.670 --> 33:54.880 But if you do 30 bits among 20 containers, then it looks like 33:54.876 --> 33:57.736 that, and there is an exponential. 33:57.741 --> 33:58.611 Right? 33:58.608 --> 34:02.698 So what Boltzmann was able to do, to show mathematically, 34:02.700 --> 34:05.110 was that the limit, when you have very many, 34:05.108 --> 34:09.898 very small energy bits, is truly an exponential. 34:09.900 --> 34:13.430 So the probability of having a certain amount of energy, 34:13.429 --> 34:15.869 in a degree of freedom, is exponential: 34:15.869 --> 34:19.399 e^-(whatever that energy is, divided by kT). 34:19.400 --> 34:22.610 So Boltzmann showed that that was the limit for lots of 34:22.606 --> 34:24.266 infinitesimal energy bits. 34:24.268 --> 34:27.438 And the idea behind it is quite clear. 34:27.440 --> 34:31.200 If all the complexions for a given total are equally likely 34:31.197 --> 34:34.177 -- and that's what he assumes; it's random, 34:34.179 --> 34:38.729 they can be any place they want to be -- then shifting energy 34:38.726 --> 34:42.286 into any one degree of freedom, of one molecule, 34:42.289 --> 34:43.729 is disfavored. 34:43.730 --> 34:46.320 Because when you put more in one molecule, 34:46.315 --> 34:49.595 there are fewer ways to distribute the rest among the 34:49.596 --> 34:50.286 others. 34:50.289 --> 34:53.359 So there'll be fewer ways, the more you put in this one. 34:53.360 --> 34:54.290 Is that clear? 34:54.289 --> 34:56.109 Because that's really the key concept. 34:56.110 --> 35:00.070 The more bits of energy you put in this one, the fewer different 35:00.070 --> 35:03.590 ways there are of permuting what's left among the others. 35:03.590 --> 35:04.220 Right? 35:04.219 --> 35:06.589 And it's exponential. Right? 35:06.590 --> 35:10.300 So if you have fewer ways among the others, then it's less 35:10.297 --> 35:10.747 likely. 35:10.753 --> 35:11.343 Right? 35:11.340 --> 35:15.680 So it turns out that if you do this, the average energy is 35:19.329 --> 35:24.089 which is to say that k, the Boltzmann Constant, 35:24.088 --> 35:27.678 relates temperature to average energy; 35:27.679 --> 35:31.939 which is to say that temperature is 35:31.940 --> 35:33.450 average energy. 35:33.449 --> 35:39.009 Temperature and average energy at equilibrium are the same 35:39.009 --> 35:40.569 thing -- okay? 35:40.570 --> 35:41.990 -- for each degree of freedom. 35:41.989 --> 35:44.989 You can put -- what did we mean by these little buckets into 35:44.994 --> 35:46.424 which we could put energy? 35:46.420 --> 35:48.970 We had a way of putting energy into the molecule, 35:48.969 --> 35:52.439 like stretching this bond, or stretching this bond, 35:52.440 --> 35:55.690 or bending some bonds, or torsion, or something like 35:55.693 --> 35:56.143 that. 35:56.139 --> 36:00.779 Now truly, we deal with quantum states. 36:00.780 --> 36:03.720 So you put energy into different quantum states and you 36:03.715 --> 36:06.975 count up the quantum states, to see how likely things are. 36:06.980 --> 36:11.320 Okay, so that's where the Boltzmann Factor is. 36:11.320 --> 36:14.990 That exponential, that e to the -- exponential 36:14.987 --> 36:19.707 ¾ ΔH comes just because that's what you 36:19.713 --> 36:20.613 expect. 36:20.610 --> 36:23.910 If you randomly distribute things, it'll come out that way. 36:23.909 --> 36:26.769 Now how about the entropy factor? 36:26.769 --> 36:28.129 And this one is fun. 36:28.130 --> 36:32.300 Feynman, in his wonderful Lectures on Physics, 36:32.300 --> 36:38.080 says: "It is the change from an ordered arrangement to a 36:38.083 --> 36:42.053 disordered arrangement, which is the source of 36:42.045 --> 36:43.365 irreversibly." 36:43.369 --> 36:47.319 Have you heard this said, that entropy is disorder, 36:47.320 --> 36:52.220 and that you increase entropy in order to increase disorder? 36:52.219 --> 36:55.149 Okay, so that's what Feynman is saying here. 36:55.150 --> 36:57.810 "The change from an ordered arrangement to a 36:57.809 --> 36:59.419 disordered arrangement." 36:59.416 --> 36:59.856 Okay. 36:59.860 --> 37:03.110 Now here are two arrangements of the same number of dots. 37:03.110 --> 37:06.600 Which one is more ordered, left or right? 37:06.599 --> 37:08.719 Students: Left. 37:08.719 --> 37:09.759 Prof: Okay. 37:09.755 --> 37:11.825 You know I'm setting you up, right? 37:11.829 --> 37:15.219 So you're -- but I know what you would have voted for. 37:15.217 --> 37:15.727 Right? 37:15.730 --> 37:18.260 So I'm not going to ask you to vote. 37:18.255 --> 37:18.755 Okay? 37:18.760 --> 37:23.530 But look at the one on the right, from a different point of 37:23.532 --> 37:24.112 view. 37:24.110 --> 37:31.190 > 37:31.190 --> 37:35.180 So what do you conclude? 37:35.179 --> 37:37.029 Which one is more ordered? 37:37.030 --> 37:42.380 The one on the right is just as ordered as the one on the left, 37:42.375 --> 37:45.125 but we didn't perceive the order. 37:45.134 --> 37:45.914 Okay? 37:45.909 --> 37:50.139 And that's like constellations; you know, the shepherds lay out 37:50.141 --> 37:53.971 and saw bears and dragons and things, in the sky -- right? 37:53.969 --> 37:56.019 -- and thought they were ordered. 37:56.021 --> 37:56.471 Okay? 37:56.469 --> 38:00.559 Now disorder, reversibility and Couette flow. 38:00.559 --> 38:05.399 Now I brought an experiment to do here, and it's -- but I'm not 38:05.402 --> 38:07.592 really sure it would work. 38:07.590 --> 38:10.100 So what I'm going to do -- because I didn't practice it 38:10.097 --> 38:10.837 before I came. 38:10.840 --> 38:12.850 I've done it before, but I didn't practice it today, 38:12.853 --> 38:15.143 and my pipette broke and I had to get a new one made before 38:15.144 --> 38:15.544 class. 38:15.539 --> 38:17.949 So if you want to see that, come after class and we'll try 38:17.945 --> 38:18.955 the actual experiment. 38:18.960 --> 38:21.730 But I'm going to show you a movie of it instead. 38:21.730 --> 38:24.590 Here. 38:24.590 --> 38:26.640 So this is the same thing here. 38:26.639 --> 38:28.839 What it is, is a -- well you'll see in the movie. 38:28.840 --> 38:30.990 I'll just start it up. 38:30.989 --> 38:33.159 > 38:33.159 --> 38:36.179 Prof: Okay, so it's a glass rod that goes 38:36.177 --> 38:38.037 up inside a glass cylinder. 38:38.039 --> 38:41.659 So there's like a doughnut inside, right? 38:41.659 --> 38:44.389 So I'm now going to pour Karo syrup in there; 38:44.389 --> 38:46.259 I brought Karo syrup with me to show you. 38:46.260 --> 38:51.660 So it's in that annulus between the rod and the cylinder. 38:51.664 --> 38:52.344 Okay? 38:52.340 --> 39:00.140 And now I'm going to take some yellow dye and put a strip of it 39:00.135 --> 39:02.545 -- first I'm going to mark it, 39:02.547 --> 39:04.777 so I can tell -- I'm going to rotate that 39:04.775 --> 39:06.725 outside cylinder, so you can see that it's 39:06.726 --> 39:07.196 rotating. 39:07.199 --> 39:11.319 And I'm putting a strip of yellow dye between the rod and 39:11.317 --> 39:12.417 the cylinder. 39:12.420 --> 39:14.150 Everybody see what I'm doing? 39:14.150 --> 39:16.880 And then I'm going to stir it up. 39:16.880 --> 39:20.420 And the way I'm going to stir it is by rotating the outside 39:20.418 --> 39:21.088 cylinder. 39:21.090 --> 39:24.500 Okay, so we'll zoom in and you'll see the watch, 39:24.498 --> 39:27.618 not very well, but to show that I'm not just 39:27.615 --> 39:30.365 running the movie backward or anything. 39:30.371 --> 39:31.171 Okay? 39:31.170 --> 39:34.170 Okay, now I start rotating the outside. 39:34.170 --> 39:37.320 So there's one rotation, two rotations, 39:37.318 --> 39:38.808 three rotations. 39:38.809 --> 39:39.979 So now it's all mixed up. 39:39.980 --> 39:41.480 And now watch. 39:41.480 --> 39:44.360 I'm unrotating. 39:44.360 --> 39:47.750 39:47.750 --> 39:50.160 And it comes back. 39:50.159 --> 39:53.879 So if you want to see that happen, we'll try it after class 39:53.880 --> 39:54.330 here. 39:54.329 --> 39:56.879 Student: Oh wow. 39:56.880 --> 39:58.480 Prof: So you unmix things. 39:58.480 --> 40:01.940 That doesn't sound like entropy is working right. 40:01.936 --> 40:02.436 Okay? 40:02.440 --> 40:08.270 Now here's the way it actually happened. 40:08.268 --> 40:10.558 So there was the syrup, between the rod and the 40:10.561 --> 40:13.011 cylinder -- a look down from the top -- and 40:13.012 --> 40:15.052 we put a strip of ink in between, 40:15.050 --> 40:17.040 and then started rotating. 40:17.039 --> 40:19.739 And as we rotated, the ink spread out, 40:19.735 --> 40:20.455 like this. 40:20.463 --> 40:21.123 Right? 40:21.119 --> 40:24.309 Because the outer part moved, and the inner part didn't move, 40:24.306 --> 40:26.376 where it was in contact with the rod. 40:26.380 --> 40:29.120 So after I'd done three rotations, it looked like that. 40:29.119 --> 40:33.609 It wasn't really evenly mixed up. 40:33.610 --> 40:36.320 It's just that when we looked at it, it looked like it was 40:36.318 --> 40:37.268 mixed up -- right? 40:37.269 --> 40:38.169 -- when we looked through it. 40:38.170 --> 40:41.790 And now when we unrotate it, the whole thing -- nothing 40:41.793 --> 40:44.953 diffused and molecules didn't move at random. 40:44.949 --> 40:47.609 They just got spread out that way, but in a particular way. 40:47.610 --> 40:51.960 So it came back again. Okay? 40:51.960 --> 40:55.490 So the rotated state only seemed to be 40:55.489 --> 40:56.389 disordered. 40:56.389 --> 40:58.749 So that's the basis of the trick. 40:58.750 --> 40:59.340 Right? 40:59.340 --> 41:02.810 But that raises a very fundamental question. 41:02.809 --> 41:07.869 If disorder is in the mind of the beholder -- 41:07.869 --> 41:10.049 in this case, or in the case of that 41:10.050 --> 41:13.260 dinosaur, connect the dots -- if disorder 41:13.255 --> 41:17.215 is in the mind of the beholder, how can it measure a 41:17.222 --> 41:19.612 fundamental property, like entropy, 41:19.612 --> 41:22.112 if it depends on who's looking at it, 41:22.110 --> 41:25.830 to say whether it's disordered or not? 41:25.833 --> 41:26.643 Right? 41:26.639 --> 41:30.369 In fact, a disordered arrangement is an oxymoron, 41:30.369 --> 41:34.409 because arrangement is arrangement, and disordered is 41:34.411 --> 41:35.661 not arrangement. 41:35.655 --> 41:36.505 Right? 41:36.510 --> 41:39.810 So how can you have a disordered arrangement, 41:39.809 --> 41:42.059 if the shepherd sees a dragon? 41:42.059 --> 41:42.659 Okay? 41:42.659 --> 41:46.059 The situation favored at equilibrium, 41:46.059 --> 41:50.979 by entropy, is one where particles have diffused every 41:50.980 --> 41:55.210 which-away, not into a coiled up piece of 41:55.208 --> 42:00.818 paper like the yellow thing, or not into a dinosaur. 42:00.820 --> 42:04.030 Every which-away; the key word is 'every'. 42:04.030 --> 42:07.300 That's what's statistical about it. 42:07.300 --> 42:11.200 A disordered arrangement is a code word for a collection of 42:11.202 --> 42:14.702 random distributions, whose individual structures are 42:14.702 --> 42:15.782 not obvious. 42:15.780 --> 42:18.040 So if a thing looks like, you know, a regular lattice 42:18.043 --> 42:19.313 like that, I say, "Ah ha, 42:19.306 --> 42:20.826 that's a regular lattice." 42:20.829 --> 42:24.309 But if it looks like this, I don't say it's exactly that; 42:24.309 --> 42:26.729 I say it's disordered, by which I mean I can't tell 42:26.728 --> 42:28.858 the difference between that one and this one, 42:28.858 --> 42:30.018 or this one, or this one. 42:30.018 --> 42:30.598 Right? 42:30.599 --> 42:34.149 So there are a whole bunch of those arrangements that I count 42:34.148 --> 42:36.218 together when I say 'disordered'. 42:36.219 --> 42:37.719 It's a collective word. 42:37.719 --> 42:40.239 So if all of them are equally likely, 42:40.239 --> 42:43.679 it's much more likely to have disordered -- 42:43.679 --> 42:46.829 many, many, many arrangements -- than the particular ordered 42:46.826 --> 42:50.336 ones that we're thinking about, even if they're all equally 42:50.344 --> 42:50.854 likely. 42:50.849 --> 42:52.009 So that's the idea. 42:52.010 --> 42:55.820 It is favored at equilibrium because it includes so many 42:55.818 --> 42:57.688 individual distributions. 42:57.690 --> 43:03.120 So entropy is actually counting, in disguise. 43:03.119 --> 43:06.639 You count all these different arrangements, 43:06.639 --> 43:08.279 or all the different quantum levels, 43:08.280 --> 43:12.190 and the more you have, under a certain name, 43:12.190 --> 43:15.890 the higher the entropy associated with that name is. 43:15.889 --> 43:20.199 So, for example, a very common value of the 43:20.202 --> 43:26.672 entropy difference between two things is 1.377 entropy units. 43:26.670 --> 43:29.560 That seems a weird number, right? 43:29.559 --> 43:36.189 Now 1.377 happens to be R times the natural log 43:36.193 --> 43:37.323 of two. 43:37.320 --> 43:44.730 Now, consider the difference in entropy between gauche and 43:44.730 --> 43:46.290 anti-butane. 43:46.289 --> 43:47.329 Okay? 43:47.329 --> 43:53.719 So the equilibrium constant is e^(-ΔG/RT). 43:53.719 --> 43:55.019 Do you remember what G is? 43:55.018 --> 44:00.338 That's the Gibbs Free Energy, which includes both heat, 44:00.340 --> 44:03.280 both the kind of things -- bond energy that we've been talking 44:03.277 --> 44:06.337 about -- and entropy is included in 44:06.338 --> 44:07.428 there too. 44:07.429 --> 44:10.319 So we can split it apart, into the part that has to do 44:10.324 --> 44:11.924 with heat -- or enthalpy, 44:11.916 --> 44:15.016 the ΔH between the two things, 44:15.018 --> 44:18.298 gauche and anti -- and TΔS, 44:18.300 --> 44:20.310 the part that has to do with entropy. 44:20.309 --> 44:24.749 I suspect you've seen this G=H+TS before; 44:24.750 --> 44:26.440 H-TS before. 44:26.440 --> 44:29.070 But let's just split it apart. 44:29.070 --> 44:31.550 Since they're in the exponent we can multiply two things 44:31.552 --> 44:32.052 together. 44:32.050 --> 44:34.880 So we have the first part, the one we've been dealing 44:34.882 --> 44:36.412 with, 3/4thsΔH. 44:36.409 --> 44:36.899 Right? 44:36.900 --> 44:39.550 And then we have the red part, that has to do with entropy. 44:39.550 --> 44:41.320 But you can simplify that. 44:41.320 --> 44:45.150 How can you simplify the part that has to do with entropy, 44:45.146 --> 44:46.486 right off the bat? 44:46.489 --> 44:47.309 Student: Cancel the T's. 44:47.309 --> 44:49.299 Prof: Cancel the T's. 44:49.300 --> 44:52.200 Okay, so it's actually ΔS/R. 44:52.199 --> 44:58.749 Now suppose that the value of ΔS is Rln2; 44:58.750 --> 45:01.710 which I said was a very common entropy difference. 45:01.706 --> 45:02.186 Right? 45:02.190 --> 45:04.360 Now you can simplify it further. 45:04.360 --> 45:08.050 Can you see how to simplify it further, for that particular 45:08.052 --> 45:09.392 entropy difference? 45:09.389 --> 45:11.489 Well obviously the R's cancel. 45:11.489 --> 45:15.509 And what's E raised to the power ln2? 45:15.510 --> 45:16.410 Student: Two. 45:16.409 --> 45:17.929 Prof: Two. 45:17.932 --> 45:22.682 So actually what that is, is our 3/4thsΔH 45:22.684 --> 45:23.854 times two. 45:23.849 --> 45:28.869 So when you see 1.377 entropy units, that's somebody who likes 45:28.873 --> 45:33.573 math telling you that's there a factor of two involved. 45:33.570 --> 45:36.440 That sounds more reasonable. 45:36.440 --> 45:37.050 Right? 45:37.054 --> 45:37.674 Two. 45:37.670 --> 45:42.590 Why should there be a factor of two, that favors gauche- over 45:42.590 --> 45:43.740 anti-butane? 45:43.739 --> 45:47.129 45:47.130 --> 45:48.700 Yes? 45:48.699 --> 45:50.239 Student: There are twice as many gauches. 45:50.239 --> 45:52.649 Prof: There are twice as many gauches as there are 45:52.648 --> 45:54.888 anti's, because it can be right-handed or left-handed 45:54.885 --> 45:55.355 gauche. 45:55.360 --> 45:59.840 So you see what a crock this is, to say that the entropy 45:59.840 --> 46:04.570 difference between gauche- and anti-butane is 1.377 entropy 46:04.565 --> 46:05.375 units? 46:05.380 --> 46:08.900 It's just that there are twice as many of one as the other. 46:08.902 --> 46:09.392 Right? 46:09.389 --> 46:12.859 So that, the fact that ΔS occurs in an 46:12.864 --> 46:17.174 exponent, is just a complicated way of telling you that there's 46:17.172 --> 46:18.912 a statistical factor. 46:18.909 --> 46:21.979 You have to count how many of these things there are. 46:21.976 --> 46:22.386 Okay? 46:22.389 --> 46:24.269 Because you have two gauche butanes. 46:24.269 --> 46:27.619 So the conclusion; it just means a factor of two. 46:27.619 --> 46:32.179 And then that the equilibrium constant depends on temperature, 46:32.175 --> 46:36.725 because of ΔH, not because of ΔS. 46:36.730 --> 46:41.090 Often people think that because the free energy is H and 46:41.092 --> 46:45.102 TS, that therefore the entropy thing is changing as 46:45.101 --> 46:46.581 T changes. 46:46.579 --> 46:49.889 But in fact that's not true, because you divide by T 46:49.894 --> 46:51.724 to get anything out of it again. 46:51.724 --> 46:52.244 Right? 46:52.239 --> 46:55.499 So what really changes with temperature is the contribution 46:55.503 --> 46:56.913 due to ΔH. 46:56.909 --> 47:01.829 So sometimes that's just used to obscure what is fundamentally 47:01.829 --> 47:02.959 very simple. 47:02.960 --> 47:06.240 Okay, we're going to stop here. 47:06.239 --> 47:10.279 And just so everybody's on the same page, we'll have the final 47:10.277 --> 47:11.797 lecture on Wednesday. 47:11.800 --> 47:15.130 But then I'll be here at class time on Friday too, 47:15.126 --> 47:19.196 and we can have a discussion then, to review for the exam. 47:19.199 --> 47:21.059 And I'm willing to have another one. 47:21.063 --> 47:21.653 I forget. 47:21.650 --> 47:23.990 When did I say? On Monday night. 47:23.989 --> 47:27.439 Now do people have -- there are not exams at night, 47:27.443 --> 47:29.243 are there; or are there? 47:29.239 --> 47:33.459 Does anybody have a -- is Monday night okay to have the 47:33.456 --> 47:34.156 review? 47:34.159 --> 47:37.439 It's probably the best time to have it, so you have a full day 47:37.440 --> 47:39.000 after that before the exam. 47:39.000 --> 47:41.160 So I'll get a room for next Monday night, 47:41.157 --> 47:43.527 a week from tonight, for a review session. 47:43.530 --> 47:45.720 But also I'll be here on Friday at lecture time. 47:45.719 --> 47:47.409 So we'll see you. 47:47.409 --> 47:50.469 If anybody wants to see this experiment, we'll do it. 47:50.469 --> 47:55.999