1 00:00:00,030 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,860 Your support will help MIT OpenCourseWare continue to 4 00:00:06,860 --> 00:00:10,510 offer high quality educational resources for free. 5 00:00:10,510 --> 00:00:13,390 To make a donation, or view additional materials from 6 00:00:13,390 --> 00:00:17,600 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,600 --> 00:00:20,290 ocw.mit.edu. 8 00:00:20,290 --> 00:00:23,010 PROFESSOR: So, the first thing that we did is look at a 9 00:00:23,010 --> 00:00:26,490 simple polymer model not too different from polymer models 10 00:00:26,490 --> 00:00:27,290 that we've seen before. 11 00:00:27,290 --> 00:00:29,770 That is, models for molecular configurations. 12 00:00:29,770 --> 00:00:32,090 The difference in the case that we treated last time, 13 00:00:32,090 --> 00:00:35,470 though, was that instead of having just a handful of 14 00:00:35,470 --> 00:00:39,110 possible configurations with their just, designated 15 00:00:39,110 --> 00:00:40,860 energies, we had essentially an 16 00:00:40,860 --> 00:00:42,760 infinite sequence of levels. 17 00:00:42,760 --> 00:00:45,450 All of evenly spaced energy. 18 00:00:45,450 --> 00:00:49,390 And we went through that and saw what the thermodynamics 19 00:00:49,390 --> 00:00:50,160 worked out to be. 20 00:00:50,160 --> 00:00:53,420 And also with the high and low temperature limits of the 21 00:00:53,420 --> 00:00:55,420 thermodynamics turned out to be. 22 00:00:55,420 --> 00:00:58,830 And the interesting consequence of that is that 23 00:00:58,830 --> 00:01:02,300 that particular choice of configurations and energies 24 00:01:02,300 --> 00:01:08,170 associated with them maps exactly onto the vibrational 25 00:01:08,170 --> 00:01:10,590 levels of an ordinary molecule. 26 00:01:10,590 --> 00:01:14,450 So, of course, when we do the statistical mechanics in the 27 00:01:14,450 --> 00:01:17,110 end, everything depends on the partition function. 28 00:01:17,110 --> 00:01:17,760 Right? 29 00:01:17,760 --> 00:01:22,910 So we're always writing some molecular partition function. 30 00:01:22,910 --> 00:01:30,430 And we've got a sum over our energies for all the states. 31 00:01:30,430 --> 00:01:34,270 And so the only input information we need to know 32 00:01:34,270 --> 00:01:37,650 is, what are the levels and what are their energies. 33 00:01:37,650 --> 00:01:40,160 So, although we started yesterday with a model of 34 00:01:40,160 --> 00:01:44,110 polymer confirmations, once we decided what the energies 35 00:01:44,110 --> 00:01:48,540 were, which was this sequence of energies starting at zero 36 00:01:48,540 --> 00:01:54,250 and going with E, and 2E, and 3E, I think we labeled them 37 00:01:54,250 --> 00:01:58,780 E0, and so on, that's all that mattered as far as the 38 00:01:58,780 --> 00:02:00,880 statistical mechanics was concerned. 39 00:02:00,880 --> 00:02:05,540 And so, since we started, We formulated this by imagining 40 00:02:05,540 --> 00:02:08,460 different confirmations for a polymer, but in fact these are 41 00:02:08,460 --> 00:02:11,830 the vibrational energies of any molecule too. 42 00:02:11,830 --> 00:02:13,250 Of any vibrational mode. 43 00:02:13,250 --> 00:02:16,970 So the results are very generally applicable. 44 00:02:16,970 --> 00:02:19,540 And so we saw what the thermodynamics 45 00:02:19,540 --> 00:02:20,460 worked out to be. 46 00:02:20,460 --> 00:02:22,580 And what the limiting cases were. 47 00:02:22,580 --> 00:02:25,210 And a couple of them, that I'll just review, are, one 48 00:02:25,210 --> 00:02:29,680 that's very important in lots of settings, is that the low 49 00:02:29,680 --> 00:02:32,740 temperature limit when you're down it basically zero degrees 50 00:02:32,740 --> 00:02:44,540 Kelvin, that's where you've got kT way down here. 51 00:02:44,540 --> 00:02:46,870 So there's not nearly enough thermal energy for anything to 52 00:02:46,870 --> 00:02:48,810 get out of even the lowest level. 53 00:02:48,810 --> 00:02:50,260 And so in that case, everything is 54 00:02:50,260 --> 00:02:51,930 in the ground state. 55 00:02:51,930 --> 00:02:55,270 And then the heat capacity, which is this often a very 56 00:02:55,270 --> 00:02:58,030 simple thing to measure, you change the temperature. 57 00:02:58,030 --> 00:03:00,200 And you see that there's no change in the total 58 00:03:00,200 --> 00:03:01,010 vibrational energy. 59 00:03:01,010 --> 00:03:03,040 In other words, the limiting low temperature 60 00:03:03,040 --> 00:03:06,010 heat capacity is zero. 61 00:03:06,010 --> 00:03:15,600 So we saw that for Cv for vibrations. 62 00:03:15,600 --> 00:03:20,130 And the limit that T goes to zero was zero. 63 00:03:20,130 --> 00:03:23,810 And that's because we were in this limit, where everything 64 00:03:23,810 --> 00:03:24,710 is in the ground state. 65 00:03:24,710 --> 00:03:31,100 Which is to say that u vibrational energy in that 66 00:03:31,100 --> 00:03:35,970 limit is zero. 67 00:03:35,970 --> 00:03:38,420 Unlike the case we treated earlier, where we had a 68 00:03:38,420 --> 00:03:39,870 limited number of levels. 69 00:03:39,870 --> 00:03:42,600 Because we treated a simple four-unit polymer with only 70 00:03:42,600 --> 00:03:46,430 four possible confirmations, in this case we have 71 00:03:46,430 --> 00:03:47,810 essentially an infinite number. 72 00:03:47,810 --> 00:03:51,220 So even in the high temperature limit, we don't 73 00:03:51,220 --> 00:03:54,070 have them, there's not a maximum total energy. 74 00:03:54,070 --> 00:03:55,720 If the temperature keeps going up, you'll get 75 00:03:55,720 --> 00:03:56,890 more and more energy. 76 00:03:56,890 --> 00:03:58,210 Because you can keep populating 77 00:03:58,210 --> 00:04:00,270 higher and higher levels. 78 00:04:00,270 --> 00:04:03,690 But the way that happens stops changing. 79 00:04:03,690 --> 00:04:06,030 Because the levels are all evenly spaced. 80 00:04:06,030 --> 00:04:09,840 So if the temperature, instead of being down here, is 81 00:04:09,840 --> 00:04:15,350 somewhere up here, now for sure if we change the 82 00:04:15,350 --> 00:04:18,760 temperature, the energy will continue to change. 83 00:04:18,760 --> 00:04:22,400 But what we've found is that in the high temperature limit, 84 00:04:22,400 --> 00:04:26,010 we had the equipartition of energy result. 85 00:04:26,010 --> 00:04:35,460 And that is, u vibrational in the high temperature limit, 86 00:04:35,460 --> 00:04:40,410 was just, for N molecules, it was NkT. 87 00:04:40,410 --> 00:04:44,240 1/2 kT for each kinetic energy degree of freedom. 88 00:04:44,240 --> 00:04:47,590 1/2 kT for each potential energy degree of freedom. 89 00:04:47,590 --> 00:04:50,680 And since vibrations have each of those, potential and 90 00:04:50,680 --> 00:04:55,220 kinetic energy, it's kT for each molecule, for each degree 91 00:04:55,220 --> 00:04:56,210 of freedom. 92 00:04:56,210 --> 00:04:58,220 That's an incredibly simple, useful thing. 93 00:04:58,220 --> 00:05:01,950 That means, if I've got a molecule in the gas phase and 94 00:05:01,950 --> 00:05:05,570 it's in the high temperature limit for translation, I know 95 00:05:05,570 --> 00:05:09,720 each translational degree of freedom will contribute for 96 00:05:09,720 --> 00:05:12,520 each molecule 1/2 kT. 97 00:05:12,520 --> 00:05:15,170 Or 3/2 kT, in all three dimensions. 98 00:05:15,170 --> 00:05:19,980 So I know the translational energy of a mole of molecules 99 00:05:19,980 --> 00:05:22,520 in the gas phase at room temperature without doing 100 00:05:22,520 --> 00:05:23,040 anything all. 101 00:05:23,040 --> 00:05:24,640 It's 3/2 NkT. 102 00:05:24,640 --> 00:05:28,740 Rotations, if it's a diatomic molecule, will be two 103 00:05:28,740 --> 00:05:30,050 different degrees of freedom. 104 00:05:30,050 --> 00:05:30,740 For rotation. 105 00:05:30,740 --> 00:05:32,920 In two orthogonal planes. 106 00:05:32,920 --> 00:05:35,220 And I'll have 1/2 kT for each. 107 00:05:35,220 --> 00:05:37,750 1/2 NkT for N molecules. 108 00:05:37,750 --> 00:05:40,560 For vibrations, if I'm in a high temperature limit, then 109 00:05:40,560 --> 00:05:42,960 it'll be kT for each vibrational mode. 110 00:05:42,960 --> 00:05:45,700 For molecules, the vibrational frequencies are usually high, 111 00:05:45,700 --> 00:05:47,580 so that you're not in the high temperature limit. 112 00:05:47,580 --> 00:05:52,180 But for materials where you have vibrations, acoustic 113 00:05:52,180 --> 00:05:54,130 vibrations, those are low frequency. 114 00:05:54,130 --> 00:05:55,690 You can be in the high temperature limit. 115 00:05:55,690 --> 00:05:57,840 And easily are. 116 00:05:57,840 --> 00:06:01,210 So the result for energy for vibrational energy in the high 117 00:06:01,210 --> 00:06:02,920 temperature limit, was NkT. 118 00:06:02,920 --> 00:06:08,250 And so that means that the heat capacity in that same 119 00:06:08,250 --> 00:06:13,530 limit at high temperature was just Nk, derivative with 120 00:06:13,530 --> 00:06:14,960 respect to temperature. 121 00:06:14,960 --> 00:06:17,650 Very simple thing to measure, again. 122 00:06:17,650 --> 00:06:20,360 So that's what we saw last time in the case of both the 123 00:06:20,360 --> 00:06:24,360 conformational model that we treated and the vibrational 124 00:06:24,360 --> 00:06:29,580 energies of molecules onto which that same model maps. 125 00:06:29,580 --> 00:06:32,670 Now what I want to do, is look a little further. 126 00:06:32,670 --> 00:06:35,070 And let me actually write the partition functions too. 127 00:06:35,070 --> 00:06:38,840 Also an important result was that the partition function 128 00:06:38,840 --> 00:06:46,920 itself, q vibrational, in the high temperature limit, was 129 00:06:46,920 --> 00:06:50,800 just kT over E0. 130 00:06:50,800 --> 00:06:57,190 Also very simple result. 131 00:06:57,190 --> 00:07:10,530 E0 is h nu 0 for vibrational mode with frequency nu 0. 132 00:07:10,530 --> 00:07:12,410 And of course, given the partition function you can 133 00:07:12,410 --> 00:07:13,790 calculate everything. 134 00:07:13,790 --> 00:07:15,190 So in the high temperature limit we can 135 00:07:15,190 --> 00:07:16,390 easily calculate things. 136 00:07:16,390 --> 00:07:18,940 In the low temperature limit, it's just one, right? 137 00:07:18,940 --> 00:07:22,080 There's only one allowed possible state in the low 138 00:07:22,080 --> 00:07:22,540 temperature limit. 139 00:07:22,540 --> 00:07:26,310 Everything is in the ground state when it's cold enough. 140 00:07:26,310 --> 00:07:30,270 So the sum of our states just gives us exactly one term 141 00:07:30,270 --> 00:07:32,020 that's of any reasonable value. 142 00:07:32,020 --> 00:07:35,010 That is, the term with this being the lowest energy. 143 00:07:35,010 --> 00:07:38,850 Everything else, the energy is much bigger than kT. 144 00:07:38,850 --> 00:07:45,980 This is a vanishingly small number then. 145 00:07:45,980 --> 00:07:51,660 So just write q, vibrational, in the low temperature limit, 146 00:07:51,660 --> 00:07:53,210 is just one. 147 00:07:53,210 --> 00:07:57,900 OK, that was the case for vibration. 148 00:07:57,900 --> 00:08:02,010 And remember, these low temperature limiting cases, 149 00:08:02,010 --> 00:08:04,900 that's a common case for nearly any degree of freedom. 150 00:08:04,900 --> 00:08:06,520 As long as it's quantized. 151 00:08:06,520 --> 00:08:08,670 At some point, you'll get this situation. 152 00:08:08,670 --> 00:08:11,760 Where the temperature is low compared to even the lowest 153 00:08:11,760 --> 00:08:12,850 excited level. 154 00:08:12,850 --> 00:08:15,920 Of course, what'll happen in the high temperature limit 155 00:08:15,920 --> 00:08:18,050 might vary, depending on the structure 156 00:08:18,050 --> 00:08:19,600 of the energy levels. 157 00:08:19,600 --> 00:08:22,020 In the case of vibrations, it's like this, it'll turn out 158 00:08:22,020 --> 00:08:23,500 rotations do the same thing. 159 00:08:23,500 --> 00:08:28,800 But it's not always as simple as this. 160 00:08:28,800 --> 00:08:34,430 OK, now what I want to do is just go through the next step 161 00:08:34,430 --> 00:08:37,030 of what we can treat, given the statistical 162 00:08:37,030 --> 00:08:38,310 mechanics that we know. 163 00:08:38,310 --> 00:08:40,800 Which is chemical equilibrium. 164 00:08:40,800 --> 00:08:44,180 Why can't we just calculate equilibrium constants based on 165 00:08:44,180 --> 00:08:45,480 what we've seen so far. 166 00:08:45,480 --> 00:08:48,810 If we can calculate the partition functions for 167 00:08:48,810 --> 00:08:51,550 molecules, and they undergo reactions and they're in 168 00:08:51,550 --> 00:08:54,015 equilibrium, we should be able to calculate 169 00:08:54,015 --> 00:08:55,380 the equilibrium constants. 170 00:08:55,380 --> 00:08:57,880 From first principles, just based on the statistical math 171 00:08:57,880 --> 00:08:59,630 we've seen so far. 172 00:08:59,630 --> 00:09:15,340 So let's try to do that. 173 00:09:15,340 --> 00:09:18,710 So it just means that, again, just as always, if we know all 174 00:09:18,710 --> 00:09:21,700 the energy levels and what all the possible states are, 175 00:09:21,700 --> 00:09:24,700 there's no reason we shouldn't be able to set our sights on a 176 00:09:24,700 --> 00:09:28,690 calculation of that sort. 177 00:09:28,690 --> 00:09:31,930 So let's just sketch out what the levels are going to look 178 00:09:31,930 --> 00:09:34,830 like for simple chemical events. 179 00:09:34,830 --> 00:09:43,170 So I just want to draw an energy diagram, here's energy. 180 00:09:43,170 --> 00:09:46,595 And, like we often do with chemical equilibria, I'm going 181 00:09:46,595 --> 00:09:51,090 to set the zero of energy at the separated atoms, and I'm 182 00:09:51,090 --> 00:10:14,000 imagining I've got reactants and products. 183 00:10:14,000 --> 00:10:20,980 So let's make this our products. 184 00:10:20,980 --> 00:10:23,790 Then over here we'll have our reactants and make the 185 00:10:23,790 --> 00:10:32,410 energies different. 186 00:10:32,410 --> 00:10:33,510 Alright. 187 00:10:33,510 --> 00:10:36,150 So there's some amount of binding energy, right? 188 00:10:36,150 --> 00:10:40,470 There's a bond dissociation energy going from the lowest 189 00:10:40,470 --> 00:10:44,800 available level in each molecule to the dissociation 190 00:10:44,800 --> 00:10:47,020 limit where we pulled the atoms apart. 191 00:10:47,020 --> 00:10:50,900 So now I'm going to draw vibrational energy levels 192 00:10:50,900 --> 00:10:51,790 inside the molecule. 193 00:10:51,790 --> 00:10:54,230 Let's imagine, it wouldn't need to be this, but let's 194 00:10:54,230 --> 00:10:56,830 imagine it's just diatomic molecules. 195 00:10:56,830 --> 00:10:59,380 So there's one vibrational mode in each. 196 00:10:59,380 --> 00:11:01,270 Just the stretching mode. 197 00:11:01,270 --> 00:11:06,860 And we've already seen the levels are evenly spaced. 198 00:11:06,860 --> 00:11:09,730 So there's going to be a bunch of evenly spaced levels. 199 00:11:09,730 --> 00:11:12,520 Actually, it stopped being quite evenly spaced once this 200 00:11:12,520 --> 00:11:16,120 stops being a simple harmonic oscillator, but for all the 201 00:11:16,120 --> 00:11:18,970 low lying levels it's pretty close to that. 202 00:11:18,970 --> 00:11:22,430 So those are the available levels. 203 00:11:22,430 --> 00:11:25,560 And there's a dissociation energy. 204 00:11:25,560 --> 00:11:36,150 So minus E, and I'll call this for the products, Ep. 205 00:11:40,780 --> 00:11:45,300 And it's just minus D0 for the products. 206 00:11:45,300 --> 00:11:47,080 Right, in language, terminology 207 00:11:47,080 --> 00:11:48,990 that we've seen before. 208 00:11:48,990 --> 00:11:56,870 And it's the same thing for the reactants. 209 00:11:56,870 --> 00:12:00,260 And really, if this were more than a diatomic molecule, 210 00:12:00,260 --> 00:12:02,200 maybe there would be a bunch of vibrational modes. 211 00:12:02,200 --> 00:12:03,080 But it wouldn't matter. 212 00:12:03,080 --> 00:12:06,110 This would just represent all the vibrational energies. 213 00:12:06,110 --> 00:12:14,090 There's some lowest state available. 214 00:12:14,090 --> 00:12:17,720 So now we've got the dissociation 215 00:12:17,720 --> 00:12:23,530 energy for the reactants. 216 00:12:23,530 --> 00:12:25,580 So those are the energies it takes to 217 00:12:25,580 --> 00:12:26,820 separate the molecule. 218 00:12:26,820 --> 00:12:29,840 But those aren't the only states available to molecules. 219 00:12:29,840 --> 00:12:32,490 Of course, they could have extra vibrational energy. 220 00:12:32,490 --> 00:12:35,400 They're not always in their ground states. 221 00:12:35,400 --> 00:12:37,470 And we're going to calculate partition functions. 222 00:12:37,470 --> 00:12:41,880 In general we would sum over all the available states. 223 00:12:41,880 --> 00:12:43,890 And then just calculate the probability of 224 00:12:43,890 --> 00:12:46,290 being in the state. 225 00:12:46,290 --> 00:12:49,880 One way to look at this, since if there's chemical 226 00:12:49,880 --> 00:12:51,860 equilibrium between the species, it means the 227 00:12:51,860 --> 00:12:54,440 molecules can interconvert. 228 00:12:54,440 --> 00:12:57,190 What that really means is that any of the molecule has access 229 00:12:57,190 --> 00:13:00,670 to any of these states. 230 00:13:00,670 --> 00:13:03,560 Of course, we like to group all these states over here. 231 00:13:03,560 --> 00:13:06,500 Because they correspond to a particular chemical structure. 232 00:13:06,500 --> 00:13:08,480 And we like to group these states over here, because they 233 00:13:08,480 --> 00:13:12,690 correspond to this different chemical structure. 234 00:13:12,690 --> 00:13:15,850 From a statistical mechanics point of view, it's just 235 00:13:15,850 --> 00:13:18,320 states and levels. 236 00:13:18,320 --> 00:13:22,170 And we could just calculate the probability of any 237 00:13:22,170 --> 00:13:25,440 molecule being in any one of the states. 238 00:13:25,440 --> 00:13:28,690 But it's useful to keep them separated like this. 239 00:13:28,690 --> 00:13:31,260 So let's do that. 240 00:13:31,260 --> 00:13:36,750 And let's look at the difference here, the 241 00:13:36,750 --> 00:13:39,870 difference in dissociation energy. 242 00:13:39,870 --> 00:13:41,970 Delta D0. 243 00:13:41,970 --> 00:13:44,680 Let's not put a double arrow. 244 00:13:44,680 --> 00:13:49,100 Let's define the sign of it. 245 00:13:49,100 --> 00:13:54,990 This way, going from products to reactants. 246 00:13:54,990 --> 00:13:58,870 OK, now let's just look at how equilibrium should work. 247 00:13:58,870 --> 00:14:01,670 So now, let's just take a generic reaction. 248 00:14:01,670 --> 00:14:06,760 Little a, our number, our stoichiometric number, A plus 249 00:14:06,760 --> 00:14:17,170 b, and B goes to c C, d D. And we know that delta G0 is minus 250 00:14:17,170 --> 00:14:20,630 RT log of Kp. 251 00:14:24,300 --> 00:14:28,570 And that delta G0 is just the free energy of the products 252 00:14:28,570 --> 00:14:30,470 minus the free energy of the reactants. 253 00:14:30,470 --> 00:14:43,030 So it's c times G C0 plus d G D0 minus a G A0 254 00:14:43,030 --> 00:14:47,760 minus b G equals B0. 255 00:14:47,760 --> 00:14:50,610 So we need to know the free energy of 256 00:14:50,610 --> 00:14:53,820 each one of the species. 257 00:14:53,820 --> 00:14:56,200 And now we know how to calculate that from first 258 00:14:56,200 --> 00:14:58,930 principles, through statistical mechanics. 259 00:14:58,930 --> 00:15:05,850 So we know that G is A plus pV. 260 00:15:05,850 --> 00:15:16,300 A is minus kT log of capital Q. And I'm going to assume 261 00:15:16,300 --> 00:15:18,310 that we're in the gas phase. 262 00:15:18,310 --> 00:15:21,860 And it's an ideal gas, so I'm going to replace pV by NkT. 263 00:15:31,190 --> 00:15:33,260 And what's Q? 264 00:15:33,260 --> 00:15:37,240 Well, we know what Q is. 265 00:15:37,240 --> 00:15:43,510 It's q, little q, to the N power 266 00:15:43,510 --> 00:15:47,280 translational over N factorial. 267 00:15:47,280 --> 00:15:50,430 For atoms, this is all it would be. 268 00:15:50,430 --> 00:15:51,570 But we have molecules. 269 00:15:51,570 --> 00:15:54,920 So we have individual other degrees of freedom besides 270 00:15:54,920 --> 00:15:57,320 translation. 271 00:15:57,320 --> 00:16:04,040 So let me just label those internal, q internal. 272 00:16:04,040 --> 00:16:06,300 And that's also to the N power. 273 00:16:06,300 --> 00:16:09,860 And q internal, if you say, what are those degrees of 274 00:16:09,860 --> 00:16:12,580 freedom, well it's the electronic energy. it's the 275 00:16:12,580 --> 00:16:13,860 vibrational energy. 276 00:16:13,860 --> 00:16:14,700 It's the rotational energy. 277 00:16:14,700 --> 00:16:17,940 And we'll deal with those shortly, but let's just 278 00:16:17,940 --> 00:16:19,720 separate it this way for now. 279 00:16:19,720 --> 00:16:26,890 Just we can keep track of where the N factorial belongs. 280 00:16:26,890 --> 00:16:28,990 And now we're going to use Stirling's approximation for 281 00:16:28,990 --> 00:16:32,090 the N factorial. 282 00:16:32,090 --> 00:16:46,460 So our log of Q, which we need up here, is just N log of q 283 00:16:46,460 --> 00:17:05,210 trans q internal minus log of N factorial. 284 00:17:05,210 --> 00:17:15,790 And that's just equal to N log q trans q internal, minus N 285 00:17:15,790 --> 00:17:24,170 log N. Plus N. And of course, this I'm going to put down 286 00:17:24,170 --> 00:17:26,570 here momentarily. 287 00:17:26,570 --> 00:17:31,810 And now our expression for G is up there. 288 00:17:31,810 --> 00:17:36,510 So it's minus kT log of Q plus NkT. 289 00:17:41,830 --> 00:17:47,190 So these factors of N are going to cancel. 290 00:17:47,190 --> 00:17:50,800 So when I take minus kT log of Q, that's going to be minus 291 00:17:50,800 --> 00:17:52,290 NkT over here. 292 00:17:52,290 --> 00:17:54,780 That's going to cancel the plus NkT here. 293 00:17:54,780 --> 00:17:57,660 So all I'm going to have left is this term and this term. 294 00:17:57,660 --> 00:18:01,950 Which I can combine. 295 00:18:01,950 --> 00:18:14,520 So it's just minus NkT log of q translational q internal 296 00:18:14,520 --> 00:18:18,910 over N. Now these, really, are just the same as q, it's just 297 00:18:18,910 --> 00:18:26,140 minus NkT log of little q over N. I didn't need to separate 298 00:18:26,140 --> 00:18:27,130 that into this product. 299 00:18:27,130 --> 00:18:29,490 I only wanted to do it to be clear where we were getting 300 00:18:29,490 --> 00:18:33,770 this factor of N factorial from. 301 00:18:33,770 --> 00:18:39,030 So now we have an expression for G. And if we know G for 302 00:18:39,030 --> 00:18:41,590 all the species involved in a chemical reaction, we should 303 00:18:41,590 --> 00:18:44,730 be able to calculate the chemical equilibrium. 304 00:18:44,730 --> 00:18:46,770 So let's do it. 305 00:18:46,770 --> 00:18:50,320 So now to do it, let's look a little more closely at what 306 00:18:50,320 --> 00:18:52,330 these internal degrees of freedom are. 307 00:18:52,330 --> 00:18:55,880 Because that's where these important details are 308 00:18:55,880 --> 00:18:56,790 going to come in. 309 00:18:56,790 --> 00:18:58,750 Obviously the equilibrium is going to depend on the 310 00:18:58,750 --> 00:18:59,610 energetics. 311 00:18:59,610 --> 00:19:02,610 How much different are the bonding energies or the 312 00:19:02,610 --> 00:19:05,450 dissociation energies and the molecules involved. 313 00:19:05,450 --> 00:19:08,200 And also, how much different or the other 314 00:19:08,200 --> 00:19:09,300 molecular energy levels. 315 00:19:09,300 --> 00:19:12,710 The vibrations, rotations, and so forth. 316 00:19:12,710 --> 00:19:27,460 So, q internal is the product of rotational, vibrational, 317 00:19:27,460 --> 00:19:30,300 and electronic partition functions. 318 00:19:30,300 --> 00:19:35,370 Remember how we saw that if you can write the energy as a 319 00:19:35,370 --> 00:19:38,030 sum of energies, then the partition functions are 320 00:19:38,030 --> 00:19:38,690 multiplied. 321 00:19:38,690 --> 00:19:44,900 Because, of course, if this is the sum of a rotational plus 322 00:19:44,900 --> 00:19:47,740 vibrational plus electronic energy, then of course I can 323 00:19:47,740 --> 00:19:49,510 just separate out these things. 324 00:19:49,510 --> 00:19:50,480 These are in the exponent. 325 00:19:50,480 --> 00:19:55,200 I can write it as a product. 326 00:19:55,200 --> 00:19:56,670 Same with translations. 327 00:19:56,670 --> 00:20:01,440 I already have separated that. 328 00:20:01,440 --> 00:20:05,370 So now let's look at how these things behave. 329 00:20:05,370 --> 00:20:13,190 Well, for the electronic case, there's really only one 330 00:20:13,190 --> 00:20:15,630 electronic state of interest in general. 331 00:20:15,630 --> 00:20:18,250 And that's the lowest state. 332 00:20:18,250 --> 00:20:20,810 In cases you're extremely familiar with, if it were the 333 00:20:20,810 --> 00:20:23,640 hydrogen atom, it would be down in the 1s orbital. 334 00:20:23,640 --> 00:20:26,550 And it would take a huge amount of energy to get up 335 00:20:26,550 --> 00:20:28,930 into the 2s or 2p orbital. 336 00:20:28,930 --> 00:20:31,810 At ordinary temperatures you never have that. 337 00:20:31,810 --> 00:20:34,620 All the atoms would be in the ground state. 338 00:20:34,620 --> 00:20:38,150 Now, for most molecules, it don't take as much energy as 339 00:20:38,150 --> 00:20:39,790 for the hydrogen atom. 340 00:20:39,790 --> 00:20:43,300 If you have benzene, for example, the ground electronic 341 00:20:43,300 --> 00:20:47,700 state, the lowest electronic state, is quite far below the 342 00:20:47,700 --> 00:20:48,990 first excited state. 343 00:20:48,990 --> 00:20:52,130 Not as much as the hydrogen atom going from 1s to 2s to 344 00:20:52,130 --> 00:20:55,890 2p, but still by much more than ordinary thermal energies 345 00:20:55,890 --> 00:20:57,400 at room temperature. 346 00:20:57,400 --> 00:21:00,900 And what that means is, again, only the ground state term is 347 00:21:00,900 --> 00:21:04,570 going to matter. 348 00:21:04,570 --> 00:21:07,350 Now, sometimes we would define that is this 349 00:21:07,350 --> 00:21:08,090 as the zero of energy. 350 00:21:08,090 --> 00:21:10,690 But since we're doing chemical equilibria, the zero of energy 351 00:21:10,690 --> 00:21:12,670 is up here. 352 00:21:12,670 --> 00:21:18,460 So that epsilon, that energy term, is this amount. 353 00:21:18,460 --> 00:21:20,280 The dissociation energy. 354 00:21:20,280 --> 00:21:22,440 This is minus dissociation energy. 355 00:21:22,440 --> 00:21:24,940 So we're going to have a positive number there. 356 00:21:24,940 --> 00:21:37,590 So q electronic is just e to the D0 over kT. 357 00:21:37,590 --> 00:22:13,880 So that's easy enough. 358 00:22:13,880 --> 00:22:18,280 We also know what the vibrational 359 00:22:18,280 --> 00:22:19,300 partition function is. 360 00:22:19,300 --> 00:22:21,130 We saw it last time. 361 00:22:21,130 --> 00:22:24,200 It's this thing we could write as a simple form. 362 00:22:24,200 --> 00:22:28,080 One over one minus e to the minus E, I'll call it 363 00:22:28,080 --> 00:22:31,750 E vibe, over kT. 364 00:22:31,750 --> 00:22:41,590 It's h nu 0, where nu is the vibrational frequency. 365 00:22:41,590 --> 00:22:45,040 Again, in the case of many molecules, the vibrational 366 00:22:45,040 --> 00:22:47,090 energy is pretty high compared to kT. 367 00:22:47,090 --> 00:22:50,830 So even this often simplifies to be just one. 368 00:22:50,830 --> 00:22:54,270 In other words, all the molecules, in many cases, are 369 00:22:54,270 --> 00:22:58,160 basically all of them are in the ground vibrational state 370 00:22:58,160 --> 00:23:00,830 at room temperature. 371 00:23:00,830 --> 00:23:03,280 Certainly if you look at molecules of high frequencies, 372 00:23:03,280 --> 00:23:07,860 if you look at the hydrogen molecule, H2, the vibrational 373 00:23:07,860 --> 00:23:10,750 frequency is about 4,000 wave numbers. 374 00:23:10,750 --> 00:23:13,590 Remember I mentioned last time, kT at room temperature, 375 00:23:13,590 --> 00:23:15,620 T is 300 Kelvin. 376 00:23:15,620 --> 00:23:18,270 Then kT corresponds to 200 wave numbers. 377 00:23:18,270 --> 00:23:20,010 It's a factor of 2/3. 378 00:23:20,010 --> 00:23:23,170 Much less than 4,000 wave numbers. 379 00:23:23,170 --> 00:23:26,130 In other words, the vibrational energy is much 380 00:23:26,130 --> 00:23:28,280 lower than kT. 381 00:23:28,280 --> 00:23:31,820 So, again, everything would be in the lowest level. 382 00:23:31,820 --> 00:23:34,180 And this just simplifies to one. 383 00:23:34,180 --> 00:23:40,630 In many cases, it's reasonably close to one. 384 00:23:40,630 --> 00:23:54,130 For high vibrational frequencies, nu 0. 385 00:23:54,130 --> 00:24:03,440 Now, we haven't talked about rotation. 386 00:24:03,440 --> 00:24:08,430 But you probably have just an intuitive feeling that at 387 00:24:08,430 --> 00:24:12,490 ordinary temperatures, if I do this, if I wave my hand in the 388 00:24:12,490 --> 00:24:15,670 air, molecules that I happen to intersect are going to 389 00:24:15,670 --> 00:24:17,740 start spinning faster. 390 00:24:17,740 --> 00:24:20,750 In other words, ordinary thermal energies do populate 391 00:24:20,750 --> 00:24:26,300 some number of rotational levels of molecules. 392 00:24:26,300 --> 00:24:28,490 Molecules probably aren't going to start getting 393 00:24:28,490 --> 00:24:31,830 squished together and vibrate harder when I do 394 00:24:31,830 --> 00:24:33,390 something like this. 395 00:24:33,390 --> 00:24:36,400 So molecules might generally still be in the ground 396 00:24:36,400 --> 00:24:39,610 vibrational levels, thermal energy isn't enough to raise 397 00:24:39,610 --> 00:24:41,070 them in many cases. 398 00:24:41,070 --> 00:24:42,920 But rotation, for sure. 399 00:24:42,920 --> 00:24:45,680 They're not all in the lowest level. 400 00:24:45,680 --> 00:24:51,180 Turns out that, in fact, the energy separation between 401 00:24:51,180 --> 00:24:54,780 rotational levels is very small compared to kT at room 402 00:24:54,780 --> 00:24:55,660 temperature. 403 00:24:55,660 --> 00:25:00,090 So just like we saw for the high temperature limit for 404 00:25:00,090 --> 00:25:10,500 vibrations, it turns out that for q rotation in the high 405 00:25:10,500 --> 00:25:16,760 temperature limit, we have the same situation where we have 406 00:25:16,760 --> 00:25:23,430 kT over, now it's a rotational energy. 407 00:25:23,430 --> 00:25:26,970 And it's typically on the order of about one to ten wave 408 00:25:26,970 --> 00:25:32,400 numbers for small to medium sized molecules. 409 00:25:32,400 --> 00:25:35,820 Remember, too, last time when we talked about in the 410 00:25:35,820 --> 00:25:38,940 vibrational levels and said, well, if you're at kT and it's 411 00:25:38,940 --> 00:25:42,440 this big, it's basically telling you roughly how many 412 00:25:42,440 --> 00:25:46,120 levels do you have thermal access to. 413 00:25:46,120 --> 00:25:49,540 Because we saw the same results for vibrations. 414 00:25:49,540 --> 00:25:52,250 It's not so different for rotation. 415 00:25:52,250 --> 00:25:54,560 And it turns out that at ordinary temperatures, you 416 00:25:54,560 --> 00:25:57,930 might have access to a few tens to a few hundreds of 417 00:25:57,930 --> 00:26:02,035 rotational levels, depending on whether they're closer to 418 00:26:02,035 --> 00:26:03,950 one or ten wave numbers or a little higher. 419 00:26:03,950 --> 00:26:08,830 Again, room temperature is 200 wave numbers. 420 00:26:08,830 --> 00:26:13,400 So, that means that, remember q is a unitless number. 421 00:26:13,400 --> 00:26:15,920 You're counting the states, weighted by the energy. 422 00:26:15,920 --> 00:26:18,290 Weighted by the Boltzmann factor. 423 00:26:18,290 --> 00:26:22,070 And for rotations, it's a number that might be on the 424 00:26:22,070 --> 00:26:34,320 order of 100 or so. 425 00:26:34,320 --> 00:26:38,520 Order of magnitude estimate, that's all. 426 00:26:38,520 --> 00:26:44,130 And of course we've seen the translational partition 427 00:26:44,130 --> 00:26:54,530 function is on the order of 10 to the 30th, right? 428 00:26:54,530 --> 00:26:55,890 Enormous number. 429 00:26:55,890 --> 00:26:59,450 In other words, in the simple lattice model that we've use 430 00:26:59,450 --> 00:27:02,390 to describe translation, just breaking up the available 431 00:27:02,390 --> 00:27:04,350 volume into little pieces. 432 00:27:04,350 --> 00:27:04,930 Counting. 433 00:27:04,930 --> 00:27:08,170 Well, OK, you have something on the order of 10 to the 30th 434 00:27:08,170 --> 00:27:09,920 possible locations. 435 00:27:09,920 --> 00:27:12,370 And if you treat this properly, quantum 436 00:27:12,370 --> 00:27:14,310 mechanically, for the translations, there's actually 437 00:27:14,310 --> 00:27:19,850 a magnitude of the number is similar. 438 00:27:19,850 --> 00:27:23,640 Now I just want to go through a very simple example. 439 00:27:23,640 --> 00:27:28,760 Just treating a particular generic reaction. 440 00:27:28,760 --> 00:27:30,760 And look at what the equilibrium constant is. 441 00:27:30,760 --> 00:27:33,620 Working it through, given what we've seen so far. 442 00:27:33,620 --> 00:27:37,880 So I just want to simplify it by having all the 443 00:27:37,880 --> 00:27:41,840 stoichiometric coefficients equal to one. 444 00:27:41,840 --> 00:27:45,730 It's not a great complication, if we don't do that. 445 00:27:45,730 --> 00:27:50,050 But it's a little bit simpler. 446 00:27:50,050 --> 00:27:56,860 So let's take a molecule that's A-B plus C-D. And now 447 00:27:56,860 --> 00:27:59,740 let's break bonds and reform them. 448 00:27:59,740 --> 00:28:05,950 So we get A-C plus B-D, right? 449 00:28:05,950 --> 00:28:08,640 So we're going to do something simple. 450 00:28:08,640 --> 00:28:11,560 And since I've got all the stoichiometric coefficients 451 00:28:11,560 --> 00:28:13,860 equal to one, we've got the same number of molecules as 452 00:28:13,860 --> 00:28:16,810 reactants and products, that'll make things a little 453 00:28:16,810 --> 00:28:18,060 bit easier. 454 00:28:18,060 --> 00:28:21,770 Just in the sense that if we look at the contribution of 455 00:28:21,770 --> 00:28:24,380 the translational energies at room temperature, they're 456 00:28:24,380 --> 00:28:25,120 going to be the same. 457 00:28:25,120 --> 00:28:27,420 For the reactants and the products. 458 00:28:27,420 --> 00:28:30,600 Any of the molecules is going to have a translational 459 00:28:30,600 --> 00:28:33,320 partition function on the order of 10 to the 30th, at 460 00:28:33,320 --> 00:28:35,710 room temperature and an ordinary volume. 461 00:28:35,710 --> 00:28:40,550 And nothing's going to change from reactants to products. 462 00:28:40,550 --> 00:28:45,490 And for vibrations, let's assume, as is often the case, 463 00:28:45,490 --> 00:28:48,710 that the vibrational frequencies are fairly high. 464 00:28:48,710 --> 00:28:51,140 So all the partition functions for the vibrations 465 00:28:51,140 --> 00:28:53,100 are equal to one. 466 00:28:53,100 --> 00:28:54,270 Like we've got written up there. 467 00:28:54,270 --> 00:28:56,330 For all four of the molecules. 468 00:28:56,330 --> 00:28:59,150 So then nothing is going to change in the vibrations. 469 00:28:59,150 --> 00:29:03,400 Going from reactants to products. 470 00:29:03,400 --> 00:29:05,720 Finally, the rotations. 471 00:29:05,720 --> 00:29:08,130 So, for the rotations our partition function is 472 00:29:08,130 --> 00:29:13,320 something on the order of, I think I meant to write 100, 473 00:29:13,320 --> 00:29:15,910 not ten here. 474 00:29:15,910 --> 00:29:17,040 As our order of magnitude. 475 00:29:17,040 --> 00:29:17,950 That's what it's going to be. 476 00:29:17,950 --> 00:29:21,640 It's a number on that order. 477 00:29:21,640 --> 00:29:25,130 And if we assume that the masses of the atoms involved 478 00:29:25,130 --> 00:29:28,550 are comparable, then we can cheat a little bit and say 479 00:29:28,550 --> 00:29:32,290 that's also, those numbers are also, going to be comparable 480 00:29:32,290 --> 00:29:35,250 for the reactants and the products. 481 00:29:35,250 --> 00:29:36,760 We don't have to do that. 482 00:29:36,760 --> 00:29:39,950 We could put it in, and rather easily calculate it. 483 00:29:39,950 --> 00:29:42,940 But I'm going to make life simple in this way and just 484 00:29:42,940 --> 00:29:46,620 work through how to carry through the calculation. 485 00:29:46,620 --> 00:29:51,480 And I think it'll be clear how to put in different partition 486 00:29:51,480 --> 00:29:53,700 functions for those quantities if they are different from 487 00:29:53,700 --> 00:29:55,000 each other. 488 00:29:55,000 --> 00:30:03,920 OK, so let's try it. 489 00:30:03,920 --> 00:30:16,520 Then, our delta G0, let's go back over here. 490 00:30:16,520 --> 00:30:23,160 Here's delta G, let me just put the relationship up. 491 00:30:23,160 --> 00:30:28,070 So we've got an expression for G. Which is, let's start from 492 00:30:28,070 --> 00:30:28,690 back there. 493 00:30:28,690 --> 00:30:35,690 It's minus. 494 00:30:35,690 --> 00:30:40,570 And now for each one of the substances, it's Ni 495 00:30:40,570 --> 00:30:47,290 kT log qi over Ni. 496 00:30:47,290 --> 00:30:49,950 I'm just going to put this in molar terms. 497 00:30:49,950 --> 00:30:55,110 So it's minus little ni RT. 498 00:30:55,110 --> 00:31:00,240 Log of qi over capital Ni. 499 00:31:00,240 --> 00:31:02,300 And I only want to do that because I want to express this 500 00:31:02,300 --> 00:31:04,150 in free energy. 501 00:31:04,150 --> 00:31:05,840 This is for substance i. 502 00:31:05,840 --> 00:31:08,420 Free energy per mole. 503 00:31:08,420 --> 00:31:15,190 So it's just minus RT log of qi over Ni. 504 00:31:17,850 --> 00:31:19,560 That's G per mole. 505 00:31:19,560 --> 00:31:25,240 That's Gi naught per mole. 506 00:31:25,240 --> 00:31:30,880 Plus RT log pi over p0. 507 00:31:30,880 --> 00:31:33,010 Where p0 is an atmosphere, basically. 508 00:31:33,010 --> 00:31:39,150 One bar. 509 00:31:39,150 --> 00:31:45,820 So that means Gi0 bar is minus RT. 510 00:31:45,820 --> 00:31:49,960 And I'm going to take this minus this and combine them. 511 00:31:49,960 --> 00:31:57,740 So it's log of qi over Ni, that's this part. 512 00:31:57,740 --> 00:32:00,080 And now I've got log of pi over p, and I'm just going to 513 00:32:00,080 --> 00:32:01,730 use the ideal gas law. 514 00:32:01,730 --> 00:32:03,440 So I'm going to use pV is nRT. 515 00:32:05,980 --> 00:32:15,920 So then I've got Ni kT over my factor of 516 00:32:15,920 --> 00:32:23,350 p0, times the volume. 517 00:32:23,350 --> 00:32:30,780 So that's Gi0 for each one of the substances. 518 00:32:30,780 --> 00:32:33,110 The Ni's cancel. 519 00:32:33,110 --> 00:32:47,740 And I've got minus RT log of qi kT over p0 times V. Now, qi 520 00:32:47,740 --> 00:32:51,460 is just the total molecular partition function. 521 00:32:51,460 --> 00:32:55,310 It's this product of q trans times q rotational times q 522 00:32:55,310 --> 00:33:00,020 vibrational times q electronic. 523 00:33:00,020 --> 00:33:16,370 And now I need delta G. So delta G0, is just minus RT. 524 00:33:16,370 --> 00:33:18,660 And now it's just, I'm going to take this for each one of 525 00:33:18,660 --> 00:33:22,450 the substances, for the products, minus the reactants. 526 00:33:22,450 --> 00:33:24,010 And I'm going to combine the log terms. 527 00:33:24,010 --> 00:33:25,160 This is the same, and I'm going to have 528 00:33:25,160 --> 00:33:26,430 this in every term. 529 00:33:26,430 --> 00:33:27,410 So what's going to happen? 530 00:33:27,410 --> 00:33:32,440 Well, all this stuff is going to cancel. 531 00:33:32,440 --> 00:33:34,320 Now, that doesn't necessarily happen. 532 00:33:34,320 --> 00:33:36,550 That happens because of the stoichiometric 533 00:33:36,550 --> 00:33:37,640 numbers being the same. 534 00:33:37,640 --> 00:33:40,120 Otherwise I'd have to take them to that the power of the 535 00:33:40,120 --> 00:33:42,880 stoichiometric number. 536 00:33:42,880 --> 00:33:45,920 But in this case they're just going to simply cancel. 537 00:33:45,920 --> 00:33:53,360 So I'm going to have left is the ratio of my partition 538 00:33:53,360 --> 00:33:56,870 function for molecule AC. 539 00:33:56,870 --> 00:34:00,480 Partition function for molecule BD. 540 00:34:00,480 --> 00:34:03,830 Partition function for molecule AB, and 541 00:34:03,830 --> 00:34:07,760 for molecule CD. 542 00:34:07,760 --> 00:34:11,680 Products, reactants. 543 00:34:11,680 --> 00:34:13,040 Pretty simple, because I know how to 544 00:34:13,040 --> 00:34:15,460 calculate all this stuff. 545 00:34:15,460 --> 00:34:19,400 And again I'd suggested some simplifying assumptions for 546 00:34:19,400 --> 00:34:20,990 what those partition functions are. 547 00:34:20,990 --> 00:34:23,850 But it wouldn't be hard to calculate each one of them if 548 00:34:23,850 --> 00:34:27,150 I just had all the relevant energy levels. 549 00:34:27,150 --> 00:34:29,070 To make this simple, we're going to assume that the 550 00:34:29,070 --> 00:34:33,420 rotational energies are the same for all the molecules. 551 00:34:33,420 --> 00:34:35,640 It wouldn't need to be, it wouldn't be hard, to plug in 552 00:34:35,640 --> 00:34:36,880 different values. 553 00:34:36,880 --> 00:34:39,440 We're going to assume we're in the low temperature limit for 554 00:34:39,440 --> 00:34:40,310 vibrations. 555 00:34:40,310 --> 00:34:42,570 So q vibrational for each one of these 556 00:34:42,570 --> 00:34:46,420 things is equal to one. 557 00:34:46,420 --> 00:34:50,240 The translational contributions, are equal for 558 00:34:50,240 --> 00:34:51,540 all of them. 559 00:34:51,540 --> 00:34:55,420 So the only thing that's different is the electronic 560 00:34:55,420 --> 00:34:56,040 contribution. 561 00:34:56,040 --> 00:34:57,980 And that's different. 562 00:34:57,980 --> 00:35:03,230 Because, of course, you have different electronic energies. 563 00:35:03,230 --> 00:35:06,410 The binding energies for the products and the reactants 564 00:35:06,410 --> 00:35:08,550 aren't in general going to be equal. 565 00:35:08,550 --> 00:35:13,750 And in most cases, it is the energetics that dominate the 566 00:35:13,750 --> 00:35:15,150 equilibrium constants. 567 00:35:15,150 --> 00:35:18,480 Of course, that is, it's the electronic energies that 568 00:35:18,480 --> 00:35:19,530 usually dominate. 569 00:35:19,530 --> 00:35:22,190 Of course, the other things do matter. 570 00:35:22,190 --> 00:35:23,420 But it's not unusual for the 571 00:35:23,420 --> 00:35:26,650 electronic energies to dominate. 572 00:35:26,650 --> 00:35:27,830 So here's delta G0. 573 00:35:27,830 --> 00:35:33,090 Of course, this is minus RT log of Kp. 574 00:35:33,090 --> 00:35:38,250 So there's our expression for our equilibrium constants. 575 00:35:38,250 --> 00:35:41,090 So it's just this product. 576 00:35:41,090 --> 00:35:44,100 So let's just put in the partition functions for the 577 00:35:44,100 --> 00:35:45,540 electronic part. 578 00:35:45,540 --> 00:35:53,380 It's e to the D0 over kT for molecule AC. 579 00:35:53,380 --> 00:36:11,140 Times e to the D0 for molecule BD over kT. 580 00:36:11,140 --> 00:36:21,530 Divided by e to the D0 for AB over kT, times e to the D0 for 581 00:36:21,530 --> 00:36:25,030 molecule CD over kT. 582 00:36:25,030 --> 00:36:36,240 So the whole thing is just e to the D0 AC, plus D0 BD minus 583 00:36:36,240 --> 00:36:49,740 D0 AB minus D0 CD over kT. 584 00:36:49,740 --> 00:36:57,090 Or in other words, it's e to the delta D0 over kT. 585 00:36:57,090 --> 00:37:00,930 That's the whole story. 586 00:37:00,930 --> 00:37:06,620 So it's a super-simple calculation to execute. 587 00:37:06,620 --> 00:37:08,580 And for lots of molecules, we know the 588 00:37:08,580 --> 00:37:10,800 dissociation energies. 589 00:37:10,800 --> 00:37:12,720 These are measured, determined, either 590 00:37:12,720 --> 00:37:14,990 thermochemically or spectroscopically for a great 591 00:37:14,990 --> 00:37:16,600 many molecules. 592 00:37:16,600 --> 00:37:20,070 So in fact, we have the information we need to do that 593 00:37:20,070 --> 00:37:21,000 calculation. 594 00:37:21,000 --> 00:37:24,040 And if we want to worry about the vibrational and rotational 595 00:37:24,040 --> 00:37:26,310 levels, typically we have that information, too, 596 00:37:26,310 --> 00:37:28,360 spectroscopically. 597 00:37:28,360 --> 00:37:32,910 So we know what the relevant energy levels are to do the 598 00:37:32,910 --> 00:37:33,950 whole calculation. 599 00:37:33,950 --> 00:37:37,340 So what it means is, we can calculate equilibrium 600 00:37:37,340 --> 00:37:41,980 constants for ordinary chemical reactions just from 601 00:37:41,980 --> 00:37:42,810 first principles. 602 00:37:42,810 --> 00:37:45,230 Knowing the energy levels of the available 603 00:37:45,230 --> 00:37:49,560 states of the molecules. 604 00:37:49,560 --> 00:37:58,490 Any questions? 605 00:37:58,490 --> 00:38:03,730 In your notes I've included another example. 606 00:38:03,730 --> 00:38:06,730 It's a little bit of an extra example. 607 00:38:06,730 --> 00:38:08,110 You can go through it if you'd like. 608 00:38:08,110 --> 00:38:11,560 It turns out to have a kind of 609 00:38:11,560 --> 00:38:13,280 attractive closed form solution. 610 00:38:13,280 --> 00:38:16,910 Which is the case of a chemical reaction like an 611 00:38:16,910 --> 00:38:21,200 isomerization inside a crystalline solid. 612 00:38:21,200 --> 00:38:23,840 So you can imagine you've got a crystal of some molecule, 613 00:38:23,840 --> 00:38:27,050 many crystals, thermally over time, they might decompose. 614 00:38:27,050 --> 00:38:29,360 They might have reaction processes that can take place. 615 00:38:29,360 --> 00:38:31,240 So I've imagined you've got uni-molecular 616 00:38:31,240 --> 00:38:32,930 reactions that can occur. 617 00:38:32,930 --> 00:38:34,450 And in that case you start with a pure 618 00:38:34,450 --> 00:38:36,040 crystal of one species. 619 00:38:36,040 --> 00:38:38,350 And you end up with a mixture of some number of the original 620 00:38:38,350 --> 00:38:40,300 species, and some number of new species. 621 00:38:40,300 --> 00:38:42,580 And there's an equilibrium constant that'll say where 622 00:38:42,580 --> 00:38:44,480 that should be. 623 00:38:44,480 --> 00:38:48,020 So again, it'll depend on the energies. 624 00:38:48,020 --> 00:38:52,760 On the dissociation energies of the different species. 625 00:38:52,760 --> 00:38:55,390 And there's also a kind of mixing term, because 626 00:38:55,390 --> 00:38:59,310 effectively now you've got the different species located at 627 00:38:59,310 --> 00:39:00,690 different places in the crystal. 628 00:39:00,690 --> 00:39:04,060 Those are distinguishable states, in the solid. 629 00:39:04,060 --> 00:39:07,790 And you need to count those. 630 00:39:07,790 --> 00:39:13,430 But the one other example I want to work through is a 631 00:39:13,430 --> 00:39:15,240 little bit different from chemical equilibria. 632 00:39:15,240 --> 00:39:17,670 It's phase equilibria. 633 00:39:17,670 --> 00:39:22,120 Why shouldn't we be able to calculate phase diagrams? 634 00:39:22,120 --> 00:39:25,020 So in the earlier part of the course, you went through and 635 00:39:25,020 --> 00:39:28,360 saw the macroscopic thermodynamic treatment. 636 00:39:28,360 --> 00:39:31,470 The equilibrium constants and chemical equilibria. 637 00:39:31,470 --> 00:39:35,980 So of course you saw the whole delta G0 as minus RT log Kp 638 00:39:35,980 --> 00:39:37,710 and so forth. 639 00:39:37,710 --> 00:39:39,830 And you've seen now we can actually calculate all that 640 00:39:39,830 --> 00:39:42,030 from first principles. 641 00:39:42,030 --> 00:39:43,420 What about phase equilibria? 642 00:39:43,420 --> 00:39:47,830 What you saw before, we just drew phase diagrams. 643 00:39:47,830 --> 00:39:51,580 And they had boiling points or lines of boiling. 644 00:39:51,580 --> 00:39:54,360 Or in other words, liquid-solid and liquid-gas 645 00:39:54,360 --> 00:39:56,970 and solid-gas equilibria. 646 00:39:56,970 --> 00:40:02,410 And what we did is, given where those lines fell, you 647 00:40:02,410 --> 00:40:05,260 can calculate things. 648 00:40:05,260 --> 00:40:07,510 But where do the lines fall? 649 00:40:07,510 --> 00:40:10,440 What's the boiling point or the sublimation temperature of 650 00:40:10,440 --> 00:40:13,370 some material at a particular pressure? 651 00:40:13,370 --> 00:40:15,450 Well, we didn't offer any prescription 652 00:40:15,450 --> 00:40:16,780 for calculating that. 653 00:40:16,780 --> 00:40:19,730 You had to take that from measurement and then given 654 00:40:19,730 --> 00:40:22,730 that, you could use the Clausius-Clapeyron equation 655 00:40:22,730 --> 00:40:25,550 and so forth and look at the way things behaved if you 656 00:40:25,550 --> 00:40:28,730 moved along a line in a phase diagram. 657 00:40:28,730 --> 00:40:31,170 But there was no prescription offered to calculate where 658 00:40:31,170 --> 00:40:35,350 exactly would that line be on the phase diagram. 659 00:40:35,350 --> 00:40:38,650 But again, if you know all the energies of the possible 660 00:40:38,650 --> 00:40:42,820 states, in the solid, in the liquid and the gas, 661 00:40:42,820 --> 00:40:45,890 statistical mechanics shows us that we can calculate the 662 00:40:45,890 --> 00:40:46,960 equilibrium between those. 663 00:40:46,960 --> 00:40:48,990 Which is to say we know we can calculate 664 00:40:48,990 --> 00:40:51,090 where those lines belong. 665 00:40:51,090 --> 00:40:56,330 So I want to just go through maybe one of a couple of 666 00:40:56,330 --> 00:40:58,400 relatively simple cases. 667 00:40:58,400 --> 00:41:02,450 I want to go through the case of a solid-solid equilibrium. 668 00:41:02,450 --> 00:41:05,150 Let's imagine we have two solid phases and an 669 00:41:05,150 --> 00:41:06,770 equilibrium between them. 670 00:41:06,770 --> 00:41:08,320 And this problem was on the problem set. 671 00:41:08,320 --> 00:41:13,000 How many of you did it? 672 00:41:13,000 --> 00:41:15,540 Everybody did it. 673 00:41:15,540 --> 00:41:19,330 Did you all get to the end of it? 674 00:41:19,330 --> 00:41:23,940 Who got to the end of it? 675 00:41:23,940 --> 00:41:25,510 Some of you got to the end of it. 676 00:41:25,510 --> 00:41:29,310 If you did, congratulations, it's not so trivial to work on 677 00:41:29,310 --> 00:41:31,160 for the first time. 678 00:41:31,160 --> 00:41:36,090 So I'll just briefly go through that problem. 679 00:41:36,090 --> 00:41:37,360 And show how it works. 680 00:41:37,360 --> 00:41:42,400 So, it's similar, of course, to the case 681 00:41:42,400 --> 00:41:43,810 that we're doing now. 682 00:41:43,810 --> 00:41:46,200 To the case we just did, of chemical equilibria. 683 00:41:46,200 --> 00:41:49,270 The difference, though, is that it's cooperative. 684 00:41:49,270 --> 00:41:56,380 The solid, at least in the way I formulated that problem, you 685 00:41:56,380 --> 00:41:59,190 either have the crystal in phase one or phase alpha, or 686 00:41:59,190 --> 00:41:59,960 in phase beta. 687 00:41:59,960 --> 00:42:02,140 And there's nothing in between. 688 00:42:02,140 --> 00:42:03,830 There are lots of systems like that. 689 00:42:03,830 --> 00:42:06,880 Not just crystals, you could go from 690 00:42:06,880 --> 00:42:09,330 different forms of DNA. 691 00:42:09,330 --> 00:42:11,230 Act like that. 692 00:42:11,230 --> 00:42:15,710 Where you have super coiled DNA and regular DNA. 693 00:42:15,710 --> 00:42:19,760 And you actually can have some of the DNA in each. 694 00:42:19,760 --> 00:42:22,510 But it's actually very cooperative. 695 00:42:22,510 --> 00:42:26,170 So there's a huge tendency to either be all in one, or be 696 00:42:26,170 --> 00:42:26,860 all in the other. 697 00:42:26,860 --> 00:42:30,850 Or at least very very, large pieces of it like that. 698 00:42:30,850 --> 00:42:32,890 Lots of other systems act like that. 699 00:42:32,890 --> 00:42:35,850 In other words, unlike the case where we're thinking 700 00:42:35,850 --> 00:42:39,030 about chemical equilibrium among molecules in the gas 701 00:42:39,030 --> 00:42:42,000 phase, these two molecules over here crash into each 702 00:42:42,000 --> 00:42:44,060 other and they react. 703 00:42:44,060 --> 00:42:45,370 Nothing else cares. 704 00:42:45,370 --> 00:42:47,560 The other whole mole of molecules does whatever they 705 00:42:47,560 --> 00:42:48,630 were doing. 706 00:42:48,630 --> 00:42:51,070 And a little later some other pair of molecules happen to 707 00:42:51,070 --> 00:42:53,640 crash into each other and maybe react or maybe don't. 708 00:42:53,640 --> 00:42:56,330 In other words, there's no cooperativity at all. 709 00:42:56,330 --> 00:43:00,500 What happens to some pair of reactants and products, 710 00:43:00,500 --> 00:43:03,040 everything else is completely independent of it. 711 00:43:03,040 --> 00:43:05,860 That's not the case when you have, in many cases, a phase 712 00:43:05,860 --> 00:43:06,460 transition. 713 00:43:06,460 --> 00:43:08,740 You can see it by eye, often. 714 00:43:08,740 --> 00:43:13,060 If you do something like super cool water. 715 00:43:13,060 --> 00:43:14,380 This is a kind of fun experiment. 716 00:43:14,380 --> 00:43:17,160 If you take dry ice, and you've got a 717 00:43:17,160 --> 00:43:18,020 little bit of water. 718 00:43:18,020 --> 00:43:21,520 Right or a little pot of water. 719 00:43:21,520 --> 00:43:23,250 And you put some dry ice into it. 720 00:43:23,250 --> 00:43:26,840 So you can actually get the water to be below the freezing 721 00:43:26,840 --> 00:43:28,940 temperature and it won't freeze yet. 722 00:43:28,940 --> 00:43:31,380 Then you drop a crystal in there and 723 00:43:31,380 --> 00:43:33,880 boom, it all freezes. 724 00:43:33,880 --> 00:43:35,790 Very cooperatively. 725 00:43:35,790 --> 00:43:38,500 Lots of phase transitions behave like that. 726 00:43:38,500 --> 00:43:52,350 So let's just see how it works. 727 00:43:52,350 --> 00:44:04,130 In this case, it's really similar to what we just did. 728 00:44:04,130 --> 00:44:09,670 We can treat it, here's phase alpha. 729 00:44:09,670 --> 00:44:14,220 Here's phase beta. 730 00:44:14,220 --> 00:44:18,900 And the crystal, the interactions between molecules 731 00:44:18,900 --> 00:44:20,130 or the atoms in the crystal are 732 00:44:20,130 --> 00:44:21,830 different in the two phases. 733 00:44:21,830 --> 00:44:23,900 So effectively, the binding energy is different. 734 00:44:23,900 --> 00:44:26,440 In other words, if you say, now let's think of the energy 735 00:44:26,440 --> 00:44:29,980 it would take to evaporate all the atoms or molecules and let 736 00:44:29,980 --> 00:44:31,040 them loose in the gas phase. 737 00:44:31,040 --> 00:44:34,780 That's the analog of dissociation for the molecule. 738 00:44:34,780 --> 00:44:36,840 You're pulling everything apart from the crystal and 739 00:44:36,840 --> 00:44:38,940 separating them all. 740 00:44:38,940 --> 00:44:41,030 So that's what we'll call these energies. 741 00:44:41,030 --> 00:44:48,370 This'll be minus E alpha. 742 00:44:48,370 --> 00:44:55,950 And this will be minus E beta. 743 00:44:55,950 --> 00:44:58,440 And then there are vibrational energies. 744 00:44:58,440 --> 00:45:02,740 So the lattice has a bunch of vibrational energies. 745 00:45:02,740 --> 00:45:05,440 We can assume they're evenly spaced. 746 00:45:05,440 --> 00:45:07,040 And they're going to be different. 747 00:45:07,040 --> 00:45:08,460 They're not the same in the two 748 00:45:08,460 --> 00:45:10,630 different crystalline forms. 749 00:45:10,630 --> 00:45:11,810 That's typically the case. 750 00:45:11,810 --> 00:45:14,090 You have a phase transition from one crystal to another. 751 00:45:14,090 --> 00:45:16,110 The lattice vibrational frequencies aren't 752 00:45:16,110 --> 00:45:17,130 the same any more. 753 00:45:17,130 --> 00:45:18,740 Speed of sound is different. 754 00:45:18,740 --> 00:45:20,660 Things are different. 755 00:45:20,660 --> 00:45:21,330 And that's all. 756 00:45:21,330 --> 00:45:24,810 That's the only thing that's different. 757 00:45:24,810 --> 00:45:40,100 So these are binding energies per atom. 758 00:45:40,100 --> 00:45:47,600 So now, let's just try to calculate Q. And what we are 759 00:45:47,600 --> 00:45:51,270 supposed to get out of this is, what's the phase 760 00:45:51,270 --> 00:45:52,770 transition temperature? 761 00:45:52,770 --> 00:45:55,620 That's the thing that we didn't have any prescription 762 00:45:55,620 --> 00:45:57,200 for calculating before. 763 00:45:57,200 --> 00:45:58,930 We just said, here it is. 764 00:45:58,930 --> 00:46:00,060 Here's the boiling point. 765 00:46:00,060 --> 00:46:01,510 Here's the melting point, or whatever the 766 00:46:01,510 --> 00:46:02,850 phase transition was. 767 00:46:02,850 --> 00:46:05,620 Or if we drew a phase diagram, here's the line. 768 00:46:05,620 --> 00:46:07,930 But now we're going to try to calculate where that is. 769 00:46:07,930 --> 00:46:09,720 Or what those temperatures are. 770 00:46:09,720 --> 00:46:16,730 So, Q for either phase it's just e to the E over kT. 771 00:46:16,730 --> 00:46:18,000 Just like we saw before for the 772 00:46:18,000 --> 00:46:20,240 dissociation energies of molecules. 773 00:46:20,240 --> 00:46:22,510 To the Nth power. 774 00:46:22,510 --> 00:46:23,620 No N factorial any more. 775 00:46:23,620 --> 00:46:26,580 There's no translation. 776 00:46:26,580 --> 00:46:28,920 Times the vibrational part. 777 00:46:28,920 --> 00:46:32,540 This is the electronic partition function. 778 00:46:32,540 --> 00:46:34,810 And then there's the vibrational part. 779 00:46:34,810 --> 00:46:40,690 And that's one over one minus e to the minus h nu, what's 780 00:46:40,690 --> 00:46:44,560 called E over kT, for the lattice I'm assuming it's just 781 00:46:44,560 --> 00:46:47,410 one frequency. 782 00:46:47,410 --> 00:46:53,590 So this energy is h nu E. And of course, it's different for 783 00:46:53,590 --> 00:46:57,880 the alpha and beta phases. 784 00:46:57,880 --> 00:46:59,950 And I'm going to assume we're in the high temperature limit. 785 00:46:59,950 --> 00:47:02,800 Which is often the case for lattice vibrations. 786 00:47:02,800 --> 00:47:06,880 So it's e to the, little E over kT. 787 00:47:06,880 --> 00:47:09,310 For our electronic part, to the Nth power. 788 00:47:09,310 --> 00:47:11,070 And we've seen what the high temperature limit is. 789 00:47:11,070 --> 00:47:15,100 It's kT over h nu E. And that's also, 790 00:47:15,100 --> 00:47:17,760 that to the 3N power. 791 00:47:17,760 --> 00:47:19,980 I should have written this here. 792 00:47:19,980 --> 00:47:24,630 In other words if you say, how many vibrations are there in 793 00:47:24,630 --> 00:47:28,770 the lattice, well if there are N atoms, each atom in the gas 794 00:47:28,770 --> 00:47:30,420 phase would have three degrees of freedom. 795 00:47:30,420 --> 00:47:32,310 Translational degrees of freedom. 796 00:47:32,310 --> 00:47:35,090 In the crystal, it can't freely translate. 797 00:47:35,090 --> 00:47:36,860 But those degrees of freedom are still there. 798 00:47:36,860 --> 00:47:38,680 The atoms can all move. 799 00:47:38,680 --> 00:47:41,410 So now, those are the lattice vibrations. 800 00:47:41,410 --> 00:47:42,930 When the atoms try to move, they vibrate 801 00:47:42,930 --> 00:47:44,920 against each other. 802 00:47:44,920 --> 00:47:47,580 So how many different modes are there? 803 00:47:47,580 --> 00:47:48,650 They may be degenerate. 804 00:47:48,650 --> 00:47:50,550 They may all have the same energy. 805 00:47:50,550 --> 00:47:52,600 But those modes are all still there. 806 00:47:52,600 --> 00:47:55,690 Actually, they're the acoustic modes of the crystal. 807 00:47:55,690 --> 00:47:58,850 And there are 3N of them. 808 00:47:58,850 --> 00:48:01,160 One for each translational degree or freedom 809 00:48:01,160 --> 00:48:02,990 that each atom had. 810 00:48:02,990 --> 00:48:06,060 OK, so that's it. 811 00:48:06,060 --> 00:48:13,210 A, minus kT log of Q, so it's just minus 812 00:48:13,210 --> 00:48:16,670 NE, and the kT cancels. 813 00:48:16,670 --> 00:48:29,900 Minus 3NkT log of kT over h nu E. Second part. 814 00:48:29,900 --> 00:48:33,810 And then we can calculate the chemical potential. 815 00:48:33,810 --> 00:48:40,670 It's just the dA/dN, d constant T and V. 816 00:48:40,670 --> 00:48:41,850 So this goes away. 817 00:48:41,850 --> 00:48:48,260 It's just minus little E. This goes away, it's minus 3kT log 818 00:48:48,260 --> 00:48:55,440 of kT over h nu E. Well, basically we just finished. 819 00:48:55,440 --> 00:48:57,390 At the phase transition temperature for the two 820 00:48:57,390 --> 00:49:02,350 phases, those things have to be equal. 821 00:49:02,350 --> 00:49:03,550 And that's all there is to it. 822 00:49:03,550 --> 00:49:18,040 So, at, we'll call it Tc, the phase transition temperature, 823 00:49:18,040 --> 00:49:26,400 I guess I called it T1 here, mu alpha is equal to mu beta. 824 00:49:26,400 --> 00:49:30,820 So this stuff for alpha is equal to this for beta. 825 00:49:30,820 --> 00:49:32,540 That's it. 826 00:49:32,540 --> 00:49:33,390 So what does it say? 827 00:49:33,390 --> 00:49:44,320 It says E beta minus E alpha, these terms, equals 3kT1 log 828 00:49:44,320 --> 00:49:50,650 of nu E beta over nu E alpha. 829 00:49:50,650 --> 00:49:52,520 Solve for T1. 830 00:49:52,520 --> 00:49:54,400 That's it. 831 00:49:54,400 --> 00:50:02,760 It's E beta minus E alpha over 3k log nu E 832 00:50:02,760 --> 00:50:06,830 beta over nu E alpha. 833 00:50:06,830 --> 00:50:09,220 There is our phase transition temperature. 834 00:50:09,220 --> 00:50:11,570 That's the whole story. 835 00:50:11,570 --> 00:50:15,700 So if we know the electronic energy that binds the crystal 836 00:50:15,700 --> 00:50:18,850 together, usually something that can be measured easily, 837 00:50:18,850 --> 00:50:21,190 so it's known for many materials. 838 00:50:21,190 --> 00:50:23,850 And if we know the lattice vibrational frequency. 839 00:50:23,850 --> 00:50:26,080 Also something that's pretty routinely 840 00:50:26,080 --> 00:50:29,110 measured, we're done. 841 00:50:29,110 --> 00:50:34,430 So again, from a very simple first principles approach, we 842 00:50:34,430 --> 00:50:38,570 can calculate that phase transition temperature. 843 00:50:38,570 --> 00:50:42,910 We could do it for a solid-gas equilibria too, if we said OK, 844 00:50:42,910 --> 00:50:45,300 let's think of sublimation of the solid. 845 00:50:45,300 --> 00:50:48,000 And now we know how to calculate the chemical 846 00:50:48,000 --> 00:50:50,380 potential in the gas phase. 847 00:50:50,380 --> 00:50:51,780 We can have those be equal. 848 00:50:51,780 --> 00:50:53,540 That's actually worked through in your notes. 849 00:50:53,540 --> 00:50:55,130 Again, it's an extra problem. 850 00:50:55,130 --> 00:50:57,410 If you would like to take a look at it, it's a 851 00:50:57,410 --> 00:50:59,680 straightforward calculation. 852 00:50:59,680 --> 00:51:02,470 The point is, we can calculate those phase equilibria. 853 00:51:02,470 --> 00:51:03,820 Liquids, I will say, are harder. 854 00:51:03,820 --> 00:51:05,980 Just because it's harder to define and know all the 855 00:51:05,980 --> 00:51:07,480 energies available. 856 00:51:07,480 --> 00:51:10,910 Doable, approximately. 857 00:51:10,910 --> 00:51:13,920 But the point is, we can actually now locate all those 858 00:51:13,920 --> 00:51:16,040 lines that we sort of arbitrarily drew 859 00:51:16,040 --> 00:51:17,910 on the phase diagrams. 860 00:51:17,910 --> 00:51:21,120 And make sense of what the temperatures and what the 861 00:51:21,120 --> 00:51:26,220 pressures are, for that matter, where they occur. 862 00:51:26,220 --> 00:51:27,690 Any questions?