1 00:00:00,000 --> 00:00:02,410 The following content is provided under a Creative 2 00:00:02,410 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,850 Your support will help MIT OpenCourseWare continue to 4 00:00:06,850 --> 00:00:10,520 offer high quality educational resources for free. 5 00:00:10,520 --> 00:00:13,390 To make a donation or view additional materials from 6 00:00:13,390 --> 00:00:17,430 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,430 --> 00:00:20,360 ocw.mit.edu. 8 00:00:20,360 --> 00:00:24,140 PROFESSOR: So we're going to start directly today and look 9 00:00:24,140 --> 00:00:24,910 at chain reactions. 10 00:00:24,910 --> 00:00:29,220 And explosions. 11 00:00:29,220 --> 00:00:31,070 So chain reactions, how many people have 12 00:00:31,070 --> 00:00:34,740 heard of chain reactions? 13 00:00:34,740 --> 00:00:36,440 Lot of people have heard of chain reactions. 14 00:00:36,440 --> 00:00:40,110 Have seen chain reactions done, in kinetics. 15 00:00:40,110 --> 00:00:41,840 No. 16 00:00:41,840 --> 00:00:46,430 Alright, well, then I'll do chain reactions. 17 00:00:46,430 --> 00:00:48,290 Chain reactions. 18 00:00:48,290 --> 00:00:55,920 Chain reactions are reactions that feed on themselves. 19 00:00:55,920 --> 00:00:57,540 And there are three parts to a chain reaction. 20 00:00:57,540 --> 00:01:05,120 There's an initiation step, which is where your reactant 21 00:01:05,120 --> 00:01:06,580 makes an intermediate. 22 00:01:06,580 --> 00:01:09,040 And that's usually a slow step. 23 00:01:09,040 --> 00:01:17,640 There's a propagation step, where the intermediate reacts 24 00:01:17,640 --> 00:01:20,520 with your reactant, becomes a feedstock. 25 00:01:20,520 --> 00:01:24,260 And then there's some intermediate plus maybe a 26 00:01:24,260 --> 00:01:26,490 first product. 27 00:01:26,490 --> 00:01:32,960 Then there's that second intermediate, can further 28 00:01:32,960 --> 00:01:38,330 react to re-form the initial intermediate. 29 00:01:38,330 --> 00:01:39,260 And that's the loop. 30 00:01:39,260 --> 00:01:41,150 That's the loop of the chain reaction. 31 00:01:41,150 --> 00:01:43,970 Plus maybe a second product. 32 00:01:43,970 --> 00:01:48,760 And then there's a termination step. 33 00:01:48,760 --> 00:01:52,900 Which is where when your intermediate, I1, becomes a 34 00:01:52,900 --> 00:01:54,050 stable product. 35 00:01:54,050 --> 00:01:55,630 Or I2 becomes a stable product. 36 00:01:55,630 --> 00:01:57,320 Basically, where the intermediates 37 00:01:57,320 --> 00:01:58,960 come out of the loop. 38 00:01:58,960 --> 00:02:08,770 So if you look at this loop here, you have your reactant 39 00:02:08,770 --> 00:02:11,940 that comes into the loop. 40 00:02:11,940 --> 00:02:14,020 And then go around the loop. 41 00:02:14,020 --> 00:02:20,420 You create I1, and maybe a product comes out. 42 00:02:20,420 --> 00:02:23,510 Maybe I1 gets terminated, but most likely it keeps going 43 00:02:23,510 --> 00:02:25,120 around the loop. 44 00:02:25,120 --> 00:02:29,310 Creating I2, which reacts with, the reactant at some 45 00:02:29,310 --> 00:02:31,540 point gets included in the circle. 46 00:02:31,540 --> 00:02:33,940 And I1 comes again, and I2, and I1 comes, and I2. 47 00:02:33,940 --> 00:02:36,372 And every time you form I1 or I2, you may 48 00:02:36,372 --> 00:02:38,330 form some of the product. 49 00:02:38,330 --> 00:02:40,010 So this is the initial feed. 50 00:02:40,010 --> 00:02:42,090 Then you go in circles. 51 00:02:42,090 --> 00:02:45,340 Eating up R, spitting out products. 52 00:02:45,340 --> 00:02:46,160 Keep going in circles. 53 00:02:46,160 --> 00:02:47,980 And at some point the termination's going to win, 54 00:02:47,980 --> 00:02:52,870 and the circle stops, and the chain has stopped. 55 00:02:52,870 --> 00:02:57,160 And there are two kinds of chain reactions. 56 00:02:57,160 --> 00:03:00,840 There's a stable chain and an unstable chain, where you get 57 00:03:00,840 --> 00:03:08,310 your explosion. 58 00:03:08,310 --> 00:03:15,650 Stable chain, or also called stationary, chain reaction, is 59 00:03:15,650 --> 00:03:21,040 where the amount of these intermediates, I1 and I2, are 60 00:03:21,040 --> 00:03:23,490 constant in time. 61 00:03:23,490 --> 00:03:27,420 So you don't build up any intermediate. 62 00:03:27,420 --> 00:03:37,110 So I1 constant in time, which is why it's called stationary. 63 00:03:37,110 --> 00:03:48,410 Then you have the unstable, or non-stationary chain. 64 00:03:48,410 --> 00:03:54,090 Where in that case your intermediate 65 00:03:54,090 --> 00:03:59,540 increases in time. 66 00:03:59,540 --> 00:04:06,000 So in the example I gave here, you destroy I1 in the first 67 00:04:06,000 --> 00:04:08,000 step, you create it in the second step. 68 00:04:08,000 --> 00:04:10,045 Comes back, you destroy it, you create it, you destroy it 69 00:04:10,045 --> 00:04:11,680 you create it. 70 00:04:11,680 --> 00:04:14,660 For every I1 that you start out with, you 71 00:04:14,660 --> 00:04:16,310 end up with an I1. 72 00:04:16,310 --> 00:04:18,580 There's no change in the concentration of I1. 73 00:04:18,580 --> 00:04:27,210 But if instead I put a two here, than my reactant creates 74 00:04:27,210 --> 00:04:28,740 an intermediate I1. 75 00:04:28,740 --> 00:04:30,160 I destroy it. 76 00:04:30,160 --> 00:04:34,220 Create a new I2, the I2 decomposes, creates two I1's. 77 00:04:34,220 --> 00:04:41,640 So after the first cycle, let's look at the 78 00:04:41,640 --> 00:04:45,260 concentration of P2 here, after the first cycle here I 79 00:04:45,260 --> 00:04:48,810 have one I1, and make one P2. 80 00:04:48,810 --> 00:04:49,930 But I make two I2's. 81 00:04:49,930 --> 00:04:52,800 So now I have two I1's coming in here. 82 00:04:52,800 --> 00:04:55,960 So I have twice this reaction. 83 00:04:55,960 --> 00:04:57,050 I make two I2's. 84 00:04:57,050 --> 00:05:01,710 These two I2's form two products. 85 00:05:01,710 --> 00:05:06,540 And now four I1's, two times two. 86 00:05:06,540 --> 00:05:08,780 These four I1's go through the process. 87 00:05:08,780 --> 00:05:09,980 I get four P2's. 88 00:05:09,980 --> 00:05:12,610 So third cycle, I get four P2's. 89 00:05:12,610 --> 00:05:15,390 And I get two times four, eight I1's. 90 00:05:15,390 --> 00:05:17,790 These eight I1's go through, and I get eight here. 91 00:05:17,790 --> 00:05:19,980 For cycle, et cetera. 92 00:05:19,980 --> 00:05:24,870 So I end up with something that goes like, so this is two 93 00:05:24,870 --> 00:05:25,840 to the zero power. 94 00:05:25,840 --> 00:05:28,080 This is two to the first, two to the second, two to the 95 00:05:28,080 --> 00:05:29,540 third, et cetera. 96 00:05:29,540 --> 00:05:32,090 Eventually you get two to the 25 or whatever 97 00:05:32,090 --> 00:05:33,650 after so many cycles. 98 00:05:33,650 --> 00:05:36,240 And if these are gas phase products, so if there's an 99 00:05:36,240 --> 00:05:41,370 exothermic process where heat gets created at any of these 100 00:05:41,370 --> 00:05:46,940 steps here, then you end up with a problem. 101 00:05:46,940 --> 00:05:50,530 The thing goes out of control. 102 00:05:50,530 --> 00:05:51,640 Sometimes that's a good thing. 103 00:05:51,640 --> 00:05:52,570 If that's what you want to do. 104 00:05:52,570 --> 00:05:54,650 And sometimes it's a bad thing. 105 00:05:54,650 --> 00:05:56,370 Depends on what you want to do. 106 00:05:56,370 --> 00:05:58,220 OK, so we're going to show two examples 107 00:05:58,220 --> 00:05:59,550 of these chain reactions. 108 00:05:59,550 --> 00:06:02,510 One is stable chain and the other one is unstable chain. 109 00:06:02,510 --> 00:06:24,050 And go through the process of how you solve for them. 110 00:06:24,050 --> 00:06:29,720 So the first example is taking, so 111 00:06:29,720 --> 00:06:31,430 let's do a stable chain. 112 00:06:31,430 --> 00:06:38,060 Take acetaldehyde, CH3CHO, that decomposers to form 113 00:06:38,060 --> 00:06:42,270 methane and carbon monoxide. 114 00:06:42,270 --> 00:06:44,740 That's the reaction. 115 00:06:44,740 --> 00:06:46,320 And here are the observations. 116 00:06:46,320 --> 00:06:47,580 There are two observations. 117 00:06:47,580 --> 00:06:52,240 The first one is that methane and carbon dioxide are not the 118 00:06:52,240 --> 00:06:53,400 only products that are formed. 119 00:06:53,400 --> 00:07:05,260 In fact, there's a trace amount of ethane and hydrogen, 120 00:07:05,260 --> 00:07:13,180 dihydrogen are also formed. 121 00:07:13,180 --> 00:07:16,090 That's a clue to something funny is going on here. 122 00:07:16,090 --> 00:07:18,140 That there's a mechanism that's more complicated than 123 00:07:18,140 --> 00:07:21,240 just this decomposing in a first order process. 124 00:07:21,240 --> 00:07:24,460 And the other thing is that the rate, if you do an 125 00:07:24,460 --> 00:07:28,240 experiment and you measure, somehow you do a half time 126 00:07:28,240 --> 00:07:32,270 experiment, or an initial rate experiment, or one of the ways 127 00:07:32,270 --> 00:07:35,320 that we talked about early on, you measure the rate of the 128 00:07:35,320 --> 00:07:41,440 reaction and you find that it's proportional to something 129 00:07:41,440 --> 00:07:43,590 to the 3/2 power. 130 00:07:43,590 --> 00:07:45,310 That's pretty weird. 131 00:07:45,310 --> 00:07:47,560 It's not a simple mechanism. 132 00:07:47,560 --> 00:07:49,970 It's pretty weird. 133 00:07:49,970 --> 00:07:53,100 And so those two observations are clues, usually, that 134 00:07:53,100 --> 00:07:55,740 you've got a chain reaction. 135 00:07:55,740 --> 00:08:00,870 The fact that you have a trace amount of extra things, that's 136 00:08:00,870 --> 00:08:02,880 usually due to a termination reaction. 137 00:08:02,880 --> 00:08:06,460 It's not a major product, it's a minor product that's the 138 00:08:06,460 --> 00:08:07,670 termination of the chain. 139 00:08:07,670 --> 00:08:11,650 And then this funny power here, in the kinetics. 140 00:08:11,650 --> 00:08:13,210 Of the rate of reaction, that's also 141 00:08:13,210 --> 00:08:17,040 usually a chain reaction. 142 00:08:17,040 --> 00:08:20,910 So now we have to figure out if there's a mechanism that 143 00:08:20,910 --> 00:08:23,760 supports those observations. 144 00:08:23,760 --> 00:08:26,730 And I'm going to tell you possible mechanism that 145 00:08:26,730 --> 00:08:27,990 supports those observations. 146 00:08:27,990 --> 00:08:30,550 And you can do experiments to show that you can fish out 147 00:08:30,550 --> 00:08:31,820 certain intermediates. 148 00:08:31,820 --> 00:08:34,790 It's not proved that that's the ultimate mechanism, but 149 00:08:34,790 --> 00:08:43,940 it's at least consistent with these observations. 150 00:08:43,940 --> 00:08:51,490 More hidden chemistry. 151 00:08:51,490 --> 00:08:53,620 OK, so this is the mechanism that's proposed. 152 00:08:53,620 --> 00:08:55,440 And then we'll figure out whether this proposed 153 00:08:55,440 --> 00:08:58,370 mechanism fits the data. 154 00:08:58,370 --> 00:09:01,710 So, the proposed mechanism is that the initiation step 155 00:09:01,710 --> 00:09:07,390 consists of the acetaldehyde, CH3CHO, 156 00:09:07,390 --> 00:09:09,670 decomposing into a radical. 157 00:09:09,670 --> 00:09:13,710 Chain reactions often have radical processes also. 158 00:09:13,710 --> 00:09:18,530 Because the radicals love to make chains. 159 00:09:18,530 --> 00:09:23,660 To the methyl radical, and then the CHO radical. 160 00:09:23,660 --> 00:09:24,720 So this is the first step. 161 00:09:24,720 --> 00:09:27,220 It's the decomposition step, it's very slow. 162 00:09:27,220 --> 00:09:31,310 That molecule doesn't really want to fall apart. 163 00:09:31,310 --> 00:09:34,280 Then there's a propagation step. 164 00:09:34,280 --> 00:09:39,370 And that's the methyl radical plus the acetaldehyde, your 165 00:09:39,370 --> 00:09:46,910 reactant, forming with the system rate k2. 166 00:09:46,910 --> 00:09:52,440 Forming CH3CO radical, plus methane. 167 00:09:52,440 --> 00:09:54,910 So that is their product right here, coming out of the first 168 00:09:54,910 --> 00:09:57,020 step of the propagation. 169 00:09:57,020 --> 00:10:00,840 And then this new intermediate here, so there's this one here 170 00:10:00,840 --> 00:10:01,910 would be I1. 171 00:10:01,910 --> 00:10:03,160 The methyl radical. 172 00:10:03,160 --> 00:10:05,950 And the CH3CO radical here, this is I2. 173 00:10:05,950 --> 00:10:08,280 And you have to use that up in the second step. 174 00:10:08,280 --> 00:10:12,030 CH3CO. 175 00:10:12,030 --> 00:10:14,080 That decomposes on its own. 176 00:10:14,080 --> 00:10:19,190 Rate k3, to form the first intermediate back, the methyl 177 00:10:19,190 --> 00:10:22,580 radical, plus the second product, CO. 178 00:10:22,580 --> 00:10:23,990 And then this intermediate then feeds 179 00:10:23,990 --> 00:10:27,810 the first step back. 180 00:10:27,810 --> 00:10:30,760 And you've got your cycle. 181 00:10:30,760 --> 00:10:35,460 And every point you spit out your two products. 182 00:10:35,460 --> 00:10:37,690 And this could go on forever. 183 00:10:37,690 --> 00:10:40,760 But it doesn't, because at some point those two methyl 184 00:10:40,760 --> 00:10:42,530 radicals, two of those methyl radicals can 185 00:10:42,530 --> 00:10:45,310 get along, can collide. 186 00:10:45,310 --> 00:10:48,040 And in a very easy way. 187 00:10:48,040 --> 00:10:50,860 So we've got a few termination steps. 188 00:10:50,860 --> 00:11:00,560 You can form two times CH3 radical ethane, C2H6. 189 00:11:00,560 --> 00:11:03,810 Rate k4, let's call this k4. 190 00:11:03,810 --> 00:11:09,990 And then there are other side reactions that can happen. 191 00:11:09,990 --> 00:11:13,890 For instance, this radical here, which we 192 00:11:13,890 --> 00:11:15,950 haven't used up. 193 00:11:15,950 --> 00:11:17,310 That can also react. 194 00:11:17,310 --> 00:11:20,620 It's just sitting around, slowly building up as this 195 00:11:20,620 --> 00:11:23,006 thing, very slowly building up, because the first step 196 00:11:23,006 --> 00:11:24,440 here is very slow. 197 00:11:24,440 --> 00:11:26,670 You don't need very much of this methyl radical 198 00:11:26,670 --> 00:11:27,880 to start the chain. 199 00:11:27,880 --> 00:11:30,940 Then you start chewing up the reactant. 200 00:11:30,940 --> 00:11:35,400 But you're still building up a little bit of this guy here. 201 00:11:35,400 --> 00:11:36,980 This is a side reaction. 202 00:11:36,980 --> 00:11:39,700 It's not a termination reaction, because that 203 00:11:39,700 --> 00:11:43,350 radical's not involved in the chain propagation step. 204 00:11:43,350 --> 00:11:48,360 But that radical can react with some other species, M, 205 00:11:48,360 --> 00:11:51,330 and it could be any of the other species. 206 00:11:51,330 --> 00:11:52,640 Because it's just a chaperone. 207 00:11:52,640 --> 00:11:58,400 It's just a, to help this thing to decompose. 208 00:11:58,400 --> 00:12:04,680 CO plus H radical plus the chaperone back. 209 00:12:04,680 --> 00:12:13,290 And then that hydrogen radical here can itself react with 210 00:12:13,290 --> 00:12:15,610 your reactant. 211 00:12:15,610 --> 00:12:23,520 CH3 acetaldehyde, CH3CHO, with a rate k6. 212 00:12:23,520 --> 00:12:30,630 To form hydrogen gas plus your intermediate, CH3CO. 213 00:12:30,630 --> 00:12:41,060 And this intermediate can feed the step right here. 214 00:12:41,060 --> 00:12:46,490 OK, so in terms of the observations, then, we got the 215 00:12:46,490 --> 00:12:48,540 right products. 216 00:12:48,540 --> 00:12:50,880 Methane, and carbon monoxide. 217 00:12:50,880 --> 00:12:56,360 We've got the right amount of trace stuff, the ethane coming 218 00:12:56,360 --> 00:12:58,150 out from the termination step. 219 00:12:58,150 --> 00:13:04,410 And hydrogen gas coming out from the side reaction. 220 00:13:04,410 --> 00:13:07,470 And I should probably have written CO also forming from 221 00:13:07,470 --> 00:13:10,570 the side reaction - well, CO is a real product, so it's 222 00:13:10,570 --> 00:13:12,895 happening both from the side reactions as well as from the 223 00:13:12,895 --> 00:13:14,270 main reaction. 224 00:13:14,270 --> 00:13:17,230 So we've got all the species taken into account. 225 00:13:17,230 --> 00:13:21,530 And now we've got to show that this mechanism here, in fact, 226 00:13:21,530 --> 00:13:27,560 gives rise to this 3/2 power. 227 00:13:27,560 --> 00:13:31,860 And I know I won't be able to do it on one board. 228 00:13:31,860 --> 00:13:38,190 I want to keep all up, so let me use up. 229 00:13:38,190 --> 00:13:40,480 So the first thing, as we've been doing in kinetics this 230 00:13:40,480 --> 00:13:43,590 whole time is, you've got to write down all your rates. 231 00:13:43,590 --> 00:13:46,250 Once you've got that down, and you haven't made a mistake, 232 00:13:46,250 --> 00:13:49,580 then you basically have the problem solved. 233 00:13:49,580 --> 00:13:53,400 So you write down all your rate laws. 234 00:13:53,400 --> 00:13:55,110 Eventually, what we're interested in showing is the 235 00:13:55,110 --> 00:13:56,790 rate of the reaction, is that power. 236 00:13:56,790 --> 00:13:58,720 So that's write down what the rate of the reaction is. 237 00:13:58,720 --> 00:14:03,330 The rate of reaction is the rate of formation of one of 238 00:14:03,330 --> 00:14:04,370 the products. 239 00:14:04,370 --> 00:14:09,440 Let's say the methane. 240 00:14:09,440 --> 00:14:10,440 So we write down that rate. 241 00:14:10,440 --> 00:14:11,380 How was it formed? 242 00:14:11,380 --> 00:14:15,300 It's formed through the k2 step. 243 00:14:15,300 --> 00:14:17,840 It's formed through this step right here. 244 00:14:17,840 --> 00:14:28,580 Which is a second order in CH3 radical times CH3CHO. 245 00:14:28,580 --> 00:14:33,040 acetaldehyde. 246 00:14:33,040 --> 00:14:34,580 We're going to keep that in the back of our mind. 247 00:14:34,580 --> 00:14:36,190 Because we're going to need this again. 248 00:14:36,190 --> 00:14:39,470 Because we want to show, at the end, that in fact this is, 249 00:14:39,470 --> 00:14:44,150 this rate is proportional to this funny power. 250 00:14:44,150 --> 00:14:47,280 There's the intermediate sitting here. 251 00:14:47,280 --> 00:14:51,440 So what's the rate for this intermediate here is CH3, dt. 252 00:14:51,440 --> 00:14:53,700 There are many places that it's coming from. 253 00:14:53,700 --> 00:14:56,080 It's being created in the first step, and so the hard 254 00:14:56,080 --> 00:14:58,230 part here is bean-counting. 255 00:14:58,230 --> 00:15:00,710 Making sure that you've found all the places where the 256 00:15:00,710 --> 00:15:08,890 species is being created. k1 times acetaldehyde, CH3CHO, 257 00:15:08,890 --> 00:15:11,210 and then it's destroyed through the second step, minus 258 00:15:11,210 --> 00:15:16,230 k2 CH3, better get your signs right also. 259 00:15:16,230 --> 00:15:20,370 So there's the destruction, there's the formation. 260 00:15:20,370 --> 00:15:23,620 Times acetaldehyde, CH3CHO. 261 00:15:23,620 --> 00:15:26,030 Then it gets created again, through the second step of the 262 00:15:26,030 --> 00:15:31,190 propagation, rate k3. 263 00:15:31,190 --> 00:15:34,640 Radical CH3CO. 264 00:15:34,640 --> 00:15:36,370 And then it finally gets destroyed in 265 00:15:36,370 --> 00:15:38,700 the termination step. 266 00:15:38,700 --> 00:15:41,270 We've got to put that in, even though it may be a very small 267 00:15:41,270 --> 00:15:44,740 amount compared to these other amounts there. 268 00:15:44,740 --> 00:15:47,470 So the termination step is k4. 269 00:15:47,470 --> 00:15:53,460 And it's a second order process, CH3 squared. 270 00:15:53,460 --> 00:15:54,850 We've got another intermediate. 271 00:15:54,850 --> 00:15:57,190 We want to write down all our intermediates here. 272 00:15:57,190 --> 00:15:58,940 Because they're all related, it's the 273 00:15:58,940 --> 00:16:00,000 intermediates sitting here. 274 00:16:00,000 --> 00:16:00,770 So there are going to be a couple of 275 00:16:00,770 --> 00:16:02,150 differential equations. 276 00:16:02,150 --> 00:16:04,380 So we've got to write down the equation 277 00:16:04,380 --> 00:16:12,990 for that CH3CO radical. 278 00:16:12,990 --> 00:16:15,940 And again, you look at your mechanism. 279 00:16:15,940 --> 00:16:17,550 And you find out where it's coming from. 280 00:16:17,550 --> 00:16:19,350 It's being created in this step. 281 00:16:19,350 --> 00:16:21,110 Destroyed in this step. 282 00:16:21,110 --> 00:16:24,320 It's also being created in this step. 283 00:16:24,320 --> 00:16:28,730 But we're going to ignore that side reaction for now. 284 00:16:28,730 --> 00:16:34,630 Just to make our lives a little bit easier. k2 CH3. 285 00:16:34,630 --> 00:16:36,200 You've got to put the termination step in, but the 286 00:16:36,200 --> 00:16:38,380 side reactions are not so important. 287 00:16:38,380 --> 00:16:40,370 CH3. 288 00:16:40,370 --> 00:16:42,620 Because if you don't put the termination step in, then your 289 00:16:42,620 --> 00:16:45,060 reaction's just going to go on forever. 290 00:16:45,060 --> 00:16:48,370 That's just not physical. 291 00:16:48,370 --> 00:16:54,460 Minus k3 CH3CO. 292 00:16:54,460 --> 00:16:56,860 Creation and destruction. 293 00:16:56,860 --> 00:16:58,490 OK, so now what do we do? 294 00:16:58,490 --> 00:17:02,820 Well, we know it's a chain reaction. 295 00:17:02,820 --> 00:17:04,190 A stable chain reaction. 296 00:17:04,190 --> 00:17:08,120 Because we don't create more intermediates than 297 00:17:08,120 --> 00:17:09,490 we started out with. 298 00:17:09,490 --> 00:17:14,460 The first step is always very slow. 299 00:17:14,460 --> 00:17:17,450 Initiation at this step is slow. 300 00:17:17,450 --> 00:17:21,340 So we're not making very much of this methyl radical. 301 00:17:21,340 --> 00:17:24,470 We're not making very much of it. 302 00:17:24,470 --> 00:17:27,170 And we're not making any more of it here. 303 00:17:27,170 --> 00:17:29,330 So during this whole time that the chain reaction is 304 00:17:29,330 --> 00:17:33,350 happening, this methyl radical here is in small quantities. 305 00:17:33,350 --> 00:17:35,060 Small quantities and it's not going to change 306 00:17:35,060 --> 00:17:36,440 very much at all. 307 00:17:36,440 --> 00:17:38,530 It's pretty much stationary. 308 00:17:38,530 --> 00:17:39,610 Magic word is stationary. 309 00:17:39,610 --> 00:17:42,480 Stationary means it's not changing. 310 00:17:42,480 --> 00:17:44,040 Not changing, so which approximation 311 00:17:44,040 --> 00:17:45,480 does that belong to? 312 00:17:45,480 --> 00:17:47,270 That's a steady state approximation. 313 00:17:47,270 --> 00:17:48,300 We've got a steady state here. 314 00:17:48,300 --> 00:17:51,100 Chain, which is going blah, blah, blah, blah, stationary. 315 00:17:51,100 --> 00:17:51,900 Steady state. 316 00:17:51,900 --> 00:17:53,370 All sounds the same. 317 00:17:53,370 --> 00:17:55,430 So that's the clue here. 318 00:17:55,430 --> 00:17:57,540 This is going to be a steady state approximation problem. 319 00:17:57,540 --> 00:18:00,420 So we're going to put a steady state here. 320 00:18:00,420 --> 00:18:02,240 We're going to put a steady state here. 321 00:18:02,240 --> 00:18:04,020 We're going to set this equal to zero. 322 00:18:04,020 --> 00:18:07,070 We're going to put a steady state right here. 323 00:18:07,070 --> 00:18:09,860 We're going to put a steady state here. 324 00:18:09,860 --> 00:18:10,860 We're going to set this equal to zero. 325 00:18:10,860 --> 00:18:13,360 And suddenly the problem becomes a 326 00:18:13,360 --> 00:18:16,170 tractable algebraic problem. 327 00:18:16,170 --> 00:18:22,020 Now we have, to solve this, let's do it in one more place. 328 00:18:22,020 --> 00:18:24,490 Let's put it right here. 329 00:18:24,490 --> 00:18:28,760 So now we have two equations, two algebraic equations. 330 00:18:28,760 --> 00:18:30,640 Two unknowns. 331 00:18:30,640 --> 00:18:31,740 These two intermediates. 332 00:18:31,740 --> 00:18:33,280 The concentration of these two 333 00:18:33,280 --> 00:18:34,920 intermediates are the two unknowns. 334 00:18:34,920 --> 00:18:40,830 Two equations, two unknowns, we can solve it. 335 00:18:40,830 --> 00:18:43,460 It can be messy. 336 00:18:43,460 --> 00:18:45,410 But not necessarily tricky. 337 00:18:45,410 --> 00:18:47,360 Just turn the crank. 338 00:18:47,360 --> 00:18:50,080 And computers could do this. 339 00:18:50,080 --> 00:18:55,820 So, let me just write the answer. 340 00:18:55,820 --> 00:19:01,530 Because I don't want to go through the algebra. 341 00:19:01,530 --> 00:19:04,510 After you turn the crank, what you want is to get this out. 342 00:19:04,510 --> 00:19:06,060 You've got your two equations, two unknowns. 343 00:19:06,060 --> 00:19:08,810 You turn the crank. 344 00:19:08,810 --> 00:19:17,190 And you get that form for the concentration of the methyl 345 00:19:17,190 --> 00:19:20,590 radical two k4. 346 00:19:20,590 --> 00:19:27,820 CH3CHO turns out to be to the 1/2 power. 347 00:19:27,820 --> 00:19:29,720 When you do that. 348 00:19:29,720 --> 00:19:32,740 There's the funny power coming up. 349 00:19:32,740 --> 00:19:35,120 That's going to be here. 350 00:19:35,120 --> 00:19:44,070 And then this is what you want to put into here. 351 00:19:44,070 --> 00:19:46,360 So, and there's the acetaldehyde, 352 00:19:46,360 --> 00:19:47,800 sitting right here. 353 00:19:47,800 --> 00:19:52,110 There's one acetaldehyde to the 1/2 power, times 354 00:19:52,110 --> 00:19:54,150 acetaldehyde to the one power, that's the whole 355 00:19:54,150 --> 00:19:58,060 thing to a 3/2 power. 356 00:19:58,060 --> 00:20:00,320 So in fact, the rate is proportion to the 3/2, 357 00:20:00,320 --> 00:20:03,410 according to our mechanism. 358 00:20:03,410 --> 00:20:07,070 And all these rate constants go into it as well. 359 00:20:07,070 --> 00:20:18,330 And what we find is that the rate of reaction is equal to 360 00:20:18,330 --> 00:20:28,170 k2 times square root of k1 over two k4. 361 00:20:28,170 --> 00:20:32,550 CH2CHO to the 3/2 power. 362 00:20:32,550 --> 00:20:35,750 Basically, k prime to an effective rate times that. 363 00:20:35,750 --> 00:20:37,420 It's a typical problem. 364 00:20:37,420 --> 00:20:44,570 Any questions on this one right here? 365 00:20:44,570 --> 00:20:45,000 Yes. 366 00:20:45,000 --> 00:20:59,850 STUDENT:: [INAUDIBLE] 367 00:20:59,850 --> 00:21:02,700 PROFESSOR: Here? 368 00:21:02,700 --> 00:21:04,620 Right here. 369 00:21:04,620 --> 00:21:05,660 STUDENT:: [INAUDIBLE] 370 00:21:05,660 --> 00:21:08,100 PROFESSOR: You know, according to my notes I don't have it, 371 00:21:08,100 --> 00:21:09,180 yes, you're absolutely right. 372 00:21:09,180 --> 00:21:11,440 There's a two here. 373 00:21:11,440 --> 00:21:13,690 Thank you very much. 374 00:21:13,690 --> 00:21:16,200 It's arbitrary. 375 00:21:16,200 --> 00:21:21,690 Where the two comes from, I didn't want to get into this. 376 00:21:21,690 --> 00:21:31,500 Sometimes it's a matter of convention. 377 00:21:31,500 --> 00:21:46,780 When you have a second order reaction, A plus B going to C, 378 00:21:46,780 --> 00:21:51,840 you write your rate of the reaction, as let's write it 379 00:21:51,840 --> 00:22:00,716 here, k times A times B. Sometimes we have 2A goes to 380 00:22:00,716 --> 00:22:07,190 C. You have a rate k2, and the rate of the reaction is, you 381 00:22:07,190 --> 00:22:17,810 can write it as dC/dt, which is minus k2 times A squared. 382 00:22:17,810 --> 00:22:24,080 But if you write dA/dt, because of the stoichiometry. 383 00:22:24,080 --> 00:22:33,890 dA/dt concentration for, so you have that C is equal to A0 384 00:22:33,890 --> 00:22:40,820 minus A, twice. 385 00:22:40,820 --> 00:22:48,950 So dC/dt is equal to minus two dA/dt. 386 00:22:48,950 --> 00:22:52,800 So minus 1/2 dC/dt is equal to dA/dt. 387 00:22:52,800 --> 00:22:59,980 So dA/dt is equal to minus 1/2 k2 A squared. 388 00:22:59,980 --> 00:23:03,110 And instead of having the 1/2 here, what we do is we 389 00:23:03,110 --> 00:23:05,650 redefine our rate constant. 390 00:23:05,650 --> 00:23:09,180 So that's k2 prime is equal to twice k2. 391 00:23:09,180 --> 00:23:12,620 You put a two here, a prime here. 392 00:23:12,620 --> 00:23:14,210 Get rid of this here. 393 00:23:14,210 --> 00:23:16,850 And there's the two sitting right here. 394 00:23:16,850 --> 00:23:18,580 Purely convention. 395 00:23:18,580 --> 00:23:20,070 Not something to really worry about. 396 00:23:20,070 --> 00:23:21,620 You've just got to have it right at the beginning, and 397 00:23:21,620 --> 00:23:28,610 keep that two in, if you're going to put it in there. 398 00:23:28,610 --> 00:23:31,540 Does that make sense? 399 00:23:31,540 --> 00:23:34,680 It's the stoichiometry, it's because of this two sitting 400 00:23:34,680 --> 00:23:35,920 right here. 401 00:23:35,920 --> 00:23:37,230 Then you have a choice. 402 00:23:37,230 --> 00:23:40,020 You can either say the rate of the reaction is the rate of 403 00:23:40,020 --> 00:23:42,110 formation of this thing. 404 00:23:42,110 --> 00:23:44,270 Or you can say the rate of reaction is the rate of 405 00:23:44,270 --> 00:23:50,030 destruction of A. And then if you say that this is the rate, 406 00:23:50,030 --> 00:23:53,540 and you don't want to have any factors in here, then the two 407 00:23:53,540 --> 00:23:54,040 shows up here. 408 00:23:54,040 --> 00:23:55,870 If you want to say this is the rate, you don't want any 409 00:23:55,870 --> 00:23:57,290 factors in here, then you're going to have a 410 00:23:57,290 --> 00:23:59,420 different rate constant. 411 00:23:59,420 --> 00:24:03,090 Purely arbitrary choice. 412 00:24:03,090 --> 00:24:04,900 Good question, though. 413 00:24:04,900 --> 00:24:11,520 Any other questions? 414 00:24:11,520 --> 00:24:12,310 OK. 415 00:24:12,310 --> 00:24:19,895 So, there's another thing about the chain reactions, a 416 00:24:19,895 --> 00:24:23,730 bit of nomenclature that's interesting to talk about. 417 00:24:23,730 --> 00:24:30,630 It's the chain length. 418 00:24:30,630 --> 00:24:33,780 And that nomenclature is due to the fact that many of these 419 00:24:33,780 --> 00:24:37,190 chain reactions are used for polymer reactions. 420 00:24:37,190 --> 00:24:39,890 Where you use radical polymerization and then you 421 00:24:39,890 --> 00:24:45,920 have, basically, in a polymer reaction you'll have an 422 00:24:45,920 --> 00:24:49,860 initiator and then you'll add a monomer 423 00:24:49,860 --> 00:24:52,330 through a radical process. 424 00:24:52,330 --> 00:24:54,600 And another monomer will add through a radical process. 425 00:24:54,600 --> 00:24:57,310 Another monomer will add, another monomer will add, 426 00:24:57,310 --> 00:24:58,670 another monomer will add, and then 427 00:24:58,670 --> 00:25:02,990 you're going to terminate. 428 00:25:02,990 --> 00:25:04,310 And then you end up with a chain. 429 00:25:04,310 --> 00:25:06,520 A polymer chain. 430 00:25:06,520 --> 00:25:10,300 So your product, instead of being molecular products, the 431 00:25:10,300 --> 00:25:12,900 product is adding a monomer to the chain 432 00:25:12,900 --> 00:25:14,820 of the growing polymer. 433 00:25:14,820 --> 00:25:19,460 And then it's meaningful to talk about chain length. 434 00:25:19,460 --> 00:25:24,380 So that's where the nomenclature comes from. 435 00:25:24,380 --> 00:25:27,020 It's a number of cycles of the reaction before you get a 436 00:25:27,020 --> 00:25:29,070 termination step. 437 00:25:29,070 --> 00:25:40,950 So we can define the chain length as the number of 438 00:25:40,950 --> 00:25:51,230 propagation steps per initiation step. 439 00:25:51,230 --> 00:25:54,590 That gives you the chain. 440 00:25:54,590 --> 00:25:56,770 Each one of these is a propagation step. 441 00:25:56,770 --> 00:26:01,970 And there's the initiation step here. 442 00:26:01,970 --> 00:26:04,010 And this could also be termination step. 443 00:26:04,010 --> 00:26:16,010 In in a stable chain, or termination step. 444 00:26:16,010 --> 00:26:17,650 There's a termination step here. 445 00:26:17,650 --> 00:26:20,310 For every initiation step in a stable chain, you've got a 446 00:26:20,310 --> 00:26:23,590 termination step. 447 00:26:23,590 --> 00:26:24,760 Because you're not building up intermediates. 448 00:26:24,760 --> 00:26:26,600 So you create it. 449 00:26:26,600 --> 00:26:28,950 At the end of the chain you terminate it. 450 00:26:28,950 --> 00:26:31,160 You have as many initiation as termination steps. 451 00:26:31,160 --> 00:26:33,640 So you have a choice here, in the algebra. 452 00:26:33,640 --> 00:26:37,390 You can use either one. 453 00:26:37,390 --> 00:26:40,740 So another way, instead of doing numbers here, 454 00:26:40,740 --> 00:26:41,820 you can also rate. 455 00:26:41,820 --> 00:26:44,920 This is also the rate of propagation divided by the 456 00:26:44,920 --> 00:26:46,000 rate of initiation. 457 00:26:46,000 --> 00:26:47,752 You just take the derivative of this with respect to time 458 00:26:47,752 --> 00:26:49,110 of the top and bottom. 459 00:26:49,110 --> 00:26:50,800 Same thing. 460 00:26:50,800 --> 00:26:58,940 So we can also define it as rate of propagation divided by 461 00:26:58,940 --> 00:27:02,700 rate of, let's say, termination. 462 00:27:02,700 --> 00:27:07,100 Or initiation, whichever is your favorite one to use to 463 00:27:07,100 --> 00:27:08,370 make it simple. 464 00:27:08,370 --> 00:27:12,760 So, for instance, in our example here, the rate of 465 00:27:12,760 --> 00:27:21,770 propagation is the rate, also the rate of product formation. 466 00:27:21,770 --> 00:27:26,490 Since we have a product formation. 467 00:27:26,490 --> 00:27:29,390 Rate of propagation is the rate of forming 468 00:27:29,390 --> 00:27:31,760 this methane here. 469 00:27:31,760 --> 00:27:35,720 So in our example here, the rate of forming the methane is 470 00:27:35,720 --> 00:27:36,680 the rate reaction here. 471 00:27:36,680 --> 00:27:38,080 This is this right here. 472 00:27:38,080 --> 00:27:44,240 So you can plug this here on top. 473 00:27:44,240 --> 00:27:46,980 I may not write it down because I don't want to spend 474 00:27:46,980 --> 00:27:48,320 time writing too much down. 475 00:27:48,320 --> 00:27:50,640 So this guy goes right in there. 476 00:27:50,640 --> 00:27:57,460 And then the rate of termination is going to be, 477 00:27:57,460 --> 00:28:02,760 the rate of termination is, well, let's not use 478 00:28:02,760 --> 00:28:04,570 termination because that has intermediates in it it. 479 00:28:04,570 --> 00:28:07,070 Let's use initiation. 480 00:28:07,070 --> 00:28:09,250 Since we have a choice. 481 00:28:09,250 --> 00:28:11,310 Rate of initiation. 482 00:28:11,310 --> 00:28:12,720 Either one works fine. 483 00:28:12,720 --> 00:28:21,900 Initiation is k1 times the acetaldehyde. k1 times CH3CHO. 484 00:28:21,900 --> 00:28:28,740 So there's the 3/2 power, which, where is it? 485 00:28:28,740 --> 00:28:36,730 There's 1/2 plus, so 1/2 plus one is the 3/2 power here. 486 00:28:36,730 --> 00:28:38,760 So there's CH3CHO to the 3/2 power. 487 00:28:38,760 --> 00:28:46,540 CH3CHO to the 3/2 power, then some effective rate k prime. 488 00:28:46,540 --> 00:28:54,250 And you end up with chain length going to the 1/2 power. 489 00:28:54,250 --> 00:28:58,450 So there's some k double prime, so effective rate times 490 00:28:58,450 --> 00:29:02,860 CH3CHO to the 1/2 power. 491 00:29:02,860 --> 00:29:03,710 That's the length of chain. 492 00:29:03,710 --> 00:29:08,240 It depends on the concentration of the reactant. 493 00:29:08,240 --> 00:29:13,190 And the typical chain's on the order of a couple hundred or 494 00:29:13,190 --> 00:29:13,900 something like this, usually. 495 00:29:13,900 --> 00:29:19,030 You always have a termination step at at some point. 496 00:29:19,030 --> 00:29:24,800 Maybe a few thousand if you're doing polymer chemistry and 497 00:29:24,800 --> 00:29:26,580 sometimes millions, if you're very good at 498 00:29:26,580 --> 00:29:29,950 doing polymer chemistry. 499 00:29:29,950 --> 00:29:35,100 OK, any questions on the stable chain? 500 00:29:35,100 --> 00:29:41,820 OK, let's do the explosion now. 501 00:29:41,820 --> 00:29:55,020 And so this is the typical example. 502 00:29:55,020 --> 00:29:56,140 Of an unstable chain. 503 00:29:56,140 --> 00:30:00,960 It's the formation of water from hydrogen and oxygen. 504 00:30:00,960 --> 00:30:05,250 2 H2O. 505 00:30:05,250 --> 00:30:07,750 That's the reaction. 506 00:30:07,750 --> 00:30:09,710 Gas phase reaction. 507 00:30:09,710 --> 00:30:12,360 And the observation is that that, well, you know the 508 00:30:12,360 --> 00:30:12,880 observation. 509 00:30:12,880 --> 00:30:16,590 You put a spark in a balloon that contains hydrogen and 510 00:30:16,590 --> 00:30:20,920 oxygen and get an explosion. 511 00:30:20,920 --> 00:30:24,670 So we're going to have to account for that. 512 00:30:24,670 --> 00:30:27,050 Sometimes you don't get an explosion. 513 00:30:27,050 --> 00:30:28,320 Sometimes it kind of peters out. 514 00:30:28,320 --> 00:30:32,160 You have a drop of water coming out. 515 00:30:32,160 --> 00:30:35,380 Not bad. 516 00:30:35,380 --> 00:30:37,610 And there's a temperature dependence to it also. 517 00:30:37,610 --> 00:30:40,970 So, whatever mechanism we put in there needs to account for 518 00:30:40,970 --> 00:30:44,320 all the pressure dependence, the stoichiometry dependence, 519 00:30:44,320 --> 00:30:46,810 and the temperature dependence. 520 00:30:46,810 --> 00:30:50,190 So this is the mechanism that's proposed. 521 00:30:50,190 --> 00:30:51,550 Initiation. 522 00:30:51,550 --> 00:30:56,830 You have your hydrogen molecule. 523 00:30:56,830 --> 00:31:01,725 Spark, or something slow that decomposes it to 524 00:31:01,725 --> 00:31:03,790 two hydrogen radicals. 525 00:31:03,790 --> 00:31:08,800 This could be a spark plug, and every time you make a 526 00:31:08,800 --> 00:31:11,200 little bit of the radical. 527 00:31:11,200 --> 00:31:15,040 Then you have a branching step. 528 00:31:15,040 --> 00:31:22,100 Where your radical reacts with the oxygen molecule to form a 529 00:31:22,100 --> 00:31:25,210 hydroxyl radical. 530 00:31:25,210 --> 00:31:29,270 Plus an oxygen. 531 00:31:29,270 --> 00:31:36,335 Oxygen atom plus your hydrogen gas, your reactant, reacts to 532 00:31:36,335 --> 00:31:42,540 form more hydroxyl radical plus the initial intermediate. 533 00:31:42,540 --> 00:31:45,580 And then the hydroxyl radical, I don't know why I have 534 00:31:45,580 --> 00:31:51,960 parentheses here, hydroxyl radical reacts with the 535 00:31:51,960 --> 00:32:01,190 reactant again, the hydrogen gas, k3, to form more of the 536 00:32:01,190 --> 00:32:05,500 initial intermediate plus water, your product. 537 00:32:05,500 --> 00:32:09,180 So there are three steps to the branching here. 538 00:32:09,180 --> 00:32:12,040 You get one intermediate, one of the initial intermediates 539 00:32:12,040 --> 00:32:13,420 here coming in. 540 00:32:13,420 --> 00:32:15,240 Forms a second intermediate. 541 00:32:15,240 --> 00:32:17,220 OH radical. 542 00:32:17,220 --> 00:32:19,020 Two OH radicals. 543 00:32:19,020 --> 00:32:22,870 And the OH radicals form the hydrogen. 544 00:32:22,870 --> 00:32:27,140 There's also radical hydrogen that's formed right here. 545 00:32:27,140 --> 00:32:31,270 So you have this formation. 546 00:32:31,270 --> 00:32:35,460 So because you're forming two hydroxyl radicals here, in 547 00:32:35,460 --> 00:32:38,000 this step here you're actually forming the equivalent of two 548 00:32:38,000 --> 00:32:40,230 hydrogen radicals. 549 00:32:40,230 --> 00:32:43,940 So, one hydrogen atom here. 550 00:32:43,940 --> 00:32:47,450 After going through the branching, you end up with 551 00:32:47,450 --> 00:32:50,470 three hydrogen radicals. 552 00:32:50,470 --> 00:32:57,340 So you go from one to three in one cycle. 553 00:32:57,340 --> 00:32:59,880 And the second time around, you go from three to nine. 554 00:32:59,880 --> 00:33:02,740 The next time, you go from nine to 27. 555 00:33:02,740 --> 00:33:04,700 Pretty soon you've got a couple of 556 00:33:04,700 --> 00:33:07,450 billion hydrogen radicals. 557 00:33:07,450 --> 00:33:09,830 And that's the problem right there. 558 00:33:09,830 --> 00:33:12,260 Or the opportunity. 559 00:33:12,260 --> 00:33:13,570 The opportunity to get something 560 00:33:13,570 --> 00:33:18,040 that goes out of control. 561 00:33:18,040 --> 00:33:21,210 And there are termination steps. 562 00:33:21,210 --> 00:33:23,290 That we need to take into account. 563 00:33:23,290 --> 00:33:33,430 The hydrogen radical can hit the wall. 564 00:33:33,430 --> 00:33:36,260 Before it finds something to react with. 565 00:33:36,260 --> 00:33:38,130 And gets stuck. 566 00:33:38,130 --> 00:33:40,640 So if the pressure is low enough, you know 567 00:33:40,640 --> 00:33:41,530 that's going to happen. 568 00:33:41,530 --> 00:33:43,370 You're going to make a hydrogen radical. 569 00:33:43,370 --> 00:33:44,840 Hardly any gas molecules around. 570 00:33:44,840 --> 00:33:46,470 It's going to find a wall, it's going to get stuck. 571 00:33:46,470 --> 00:33:47,690 It's very reactive. 572 00:33:47,690 --> 00:33:51,740 So that's going to terminate the process. 573 00:33:51,740 --> 00:33:56,480 Hydrogen radical could re-find an oxygen and another 574 00:33:56,480 --> 00:33:58,480 chaperone, that's a termolecular process. 575 00:33:58,480 --> 00:33:59,930 That's an unlikely process. 576 00:33:59,930 --> 00:34:02,650 But nevertheless, it could happen. 577 00:34:02,650 --> 00:34:08,520 With another termination step, to form the HO2 radical plus 578 00:34:08,520 --> 00:34:12,300 M. And then that HO2 too radical can react in a side 579 00:34:12,300 --> 00:34:14,200 reaction to form some products, but not 580 00:34:14,200 --> 00:34:15,540 be part of the chain. 581 00:34:15,540 --> 00:34:15,920 Yes. 582 00:34:15,920 --> 00:34:18,510 STUDENT:: [INAUDIBLE] 583 00:34:18,510 --> 00:34:20,030 PROFESSOR: How do we get three hydrogens? 584 00:34:20,030 --> 00:34:22,790 Because there are two hydroxyl radicals here. 585 00:34:22,790 --> 00:34:25,000 And so in this step here, you actually have two hydroxyl 586 00:34:25,000 --> 00:34:33,120 radicals to form two, so I could multiply this by two. 587 00:34:33,120 --> 00:34:40,420 But you usually don't do that. 588 00:34:40,420 --> 00:34:45,110 So this already tells you what conditions you're going to 589 00:34:45,110 --> 00:34:47,840 get, with something where the termination is fast, compared 590 00:34:47,840 --> 00:34:49,110 to the propagation. 591 00:34:49,110 --> 00:34:50,640 Because you have low pressure. 592 00:34:50,640 --> 00:34:53,310 If you can't get that hydrogen radical to react with 593 00:34:53,310 --> 00:34:56,420 something before hitting the wall, you're going to be OK. 594 00:34:56,420 --> 00:34:58,240 You're not getting an explosion. 595 00:34:58,240 --> 00:35:01,080 This tells you that a very high pressure, where the 596 00:35:01,080 --> 00:35:04,840 probability of having a three-body collision becomes 597 00:35:04,840 --> 00:35:08,210 high enough, then you're going to terminate 598 00:35:08,210 --> 00:35:09,410 the chain as well. 599 00:35:09,410 --> 00:35:13,390 Those hydrogen radicals are going to find oxygen molecules 600 00:35:13,390 --> 00:35:18,580 and some other molecule and form this radical before you 601 00:35:18,580 --> 00:35:20,610 propagate the chain. 602 00:35:20,610 --> 00:35:24,700 At least some reasonably high pressure. 603 00:35:24,700 --> 00:35:28,730 So that's the clue here. 604 00:35:28,730 --> 00:35:29,540 What's the strategy? 605 00:35:29,540 --> 00:35:33,050 The strategy is to assume that everything's going to be fine. 606 00:35:33,050 --> 00:35:34,860 That is, stable. 607 00:35:34,860 --> 00:35:35,790 That's going to be the strategy. 608 00:35:35,790 --> 00:35:37,030 And then we're going to do a proof by 609 00:35:37,030 --> 00:35:38,340 contradiction, basically. 610 00:35:38,340 --> 00:35:41,820 We can assume that the chain is stable. 611 00:35:41,820 --> 00:35:44,710 That we can use the steady state approximation. 612 00:35:44,710 --> 00:35:47,090 We're going to apply the steady state approximation. 613 00:35:47,090 --> 00:35:49,240 We're going to find where that breaks down. 614 00:35:49,240 --> 00:35:52,410 And where it breaks down means that it wasn't right. 615 00:35:52,410 --> 00:35:54,660 That the intermediate concentration was too high. 616 00:35:54,660 --> 00:35:57,190 And that means that we're going to get an explosion. 617 00:35:57,190 --> 00:36:08,190 That's the logic. 618 00:36:08,190 --> 00:36:11,180 So strategy is a proof by contradiction, basically. 619 00:36:11,180 --> 00:36:20,550 Proof by contradiction. 620 00:36:20,550 --> 00:36:24,410 So we're going to assume that the propagation, or the 621 00:36:24,410 --> 00:36:26,970 intermediates, are in small quantities. 622 00:36:26,970 --> 00:36:28,830 We're going to assume that we can use the steady state 623 00:36:28,830 --> 00:36:36,000 approximation, the d[O]/dt steady state is equal to zero. 624 00:36:36,000 --> 00:36:44,140 That the hydroxyl radical, d[OH]/dt, steady state is 625 00:36:44,140 --> 00:36:45,380 equal to zero. 626 00:36:45,380 --> 00:36:54,006 And that d[H]/dt steady state is equal to zero. 627 00:36:54,006 --> 00:36:54,980 These are going to be our assumptions. 628 00:36:54,980 --> 00:36:58,900 That all the intermediates are in small quantities. 629 00:36:58,900 --> 00:37:05,510 And they don't change very much. 630 00:37:05,510 --> 00:37:08,080 And now we're going to write down, you write down the rates 631 00:37:08,080 --> 00:37:09,650 that correspond to these guys. 632 00:37:09,650 --> 00:37:11,910 And then you solve. 633 00:37:11,910 --> 00:37:14,300 So I'm only going to do this first one here. 634 00:37:14,300 --> 00:37:18,270 So you write dO/dt, you see where it gets formed. 635 00:37:18,270 --> 00:37:20,550 It gets formed in the first step here. 636 00:37:20,550 --> 00:37:26,730 So it's going to be k1 times H radical times O2, and it gets 637 00:37:26,730 --> 00:37:33,980 destroyed in the second step here, minus k2 638 00:37:33,980 --> 00:37:37,400 times O times H2. 639 00:37:37,400 --> 00:37:38,790 You set the steady state approximation, you get a 640 00:37:38,790 --> 00:37:39,380 steady state. 641 00:37:39,380 --> 00:37:43,230 Steady state here, you set that equal to zero. 642 00:37:43,230 --> 00:37:50,280 You solve for O, radical O, steady state. 643 00:37:50,280 --> 00:38:02,930 And you find that it's k1 times H times O2 times k2 H2. 644 00:38:02,930 --> 00:38:05,040 Then you do the same thing for the other radicals. 645 00:38:05,040 --> 00:38:13,080 You solve for OH steady state. 646 00:38:13,080 --> 00:38:15,290 And you get some expression which is in the notes. 647 00:38:15,290 --> 00:38:16,183 I'm not going to write it down, it's 648 00:38:16,183 --> 00:38:17,820 going to get too messy. 649 00:38:17,820 --> 00:38:19,940 But the expression that I will write down is the final 650 00:38:19,940 --> 00:38:23,460 expression, which we're going to be taking into account. 651 00:38:23,460 --> 00:38:31,460 So, this expression here, for the oxygen atom. 652 00:38:31,460 --> 00:38:34,050 It has the radical in there. 653 00:38:34,050 --> 00:38:38,020 So it's not, we'd like something that's just in terms 654 00:38:38,020 --> 00:38:43,830 of the rates and either products or reactants. 655 00:38:43,830 --> 00:38:49,270 And this hydroxyl radical, the concentration turns out, it 656 00:38:49,270 --> 00:38:54,520 will also be a function of the concentration of 657 00:38:54,520 --> 00:38:57,300 the hydrogen radical. 658 00:38:57,300 --> 00:39:02,470 So what we really want, I don't want to cover this up, 659 00:39:02,470 --> 00:39:05,440 let me go on this board here. 660 00:39:05,440 --> 00:39:14,780 So when you solve for this, the hydrogen atom is where you 661 00:39:14,780 --> 00:39:19,320 get something that's free of intermediates. 662 00:39:19,320 --> 00:39:21,534 And has just rates in it. 663 00:39:21,534 --> 00:39:24,740 There's a termination rate here going in the denominator. 664 00:39:24,740 --> 00:39:28,270 Another termination rate in the denominator. 665 00:39:28,270 --> 00:39:33,600 Times O2 times M, where M could be anything, really. 666 00:39:33,600 --> 00:39:36,810 Minus, and there's the creation rate of the first 667 00:39:36,810 --> 00:39:37,900 propagation step. 668 00:39:37,900 --> 00:39:42,960 O2 here. 669 00:39:42,960 --> 00:39:44,630 So now we have to see whether or not our 670 00:39:44,630 --> 00:39:46,420 approximation was valid. 671 00:39:46,420 --> 00:39:47,530 We made an approximation. 672 00:39:47,530 --> 00:39:49,860 We said the steady state approximation was valid. 673 00:39:49,860 --> 00:39:53,770 If it is valid, then this concentration, this pressure 674 00:39:53,770 --> 00:39:56,810 here, needs to be small. 675 00:39:56,810 --> 00:39:59,680 Needs to be small at all times. 676 00:39:59,680 --> 00:40:03,760 If it's not small, then steady state is not valid, we have a 677 00:40:03,760 --> 00:40:07,670 branching non-stable chain and we've got an explosion. 678 00:40:07,670 --> 00:40:11,100 So let's look at the different possibilities here. 679 00:40:11,100 --> 00:40:14,050 What are the possibilities? 680 00:40:14,050 --> 00:40:18,020 Well, the first possibility is that as we saw intuitively, 681 00:40:18,020 --> 00:40:21,330 when we wrote down this equation here, the first 682 00:40:21,330 --> 00:40:25,120 possibility is that we're at low pressure. 683 00:40:25,120 --> 00:40:31,610 Where at low pressure, then, k1 times oxygen is very small. 684 00:40:31,610 --> 00:40:33,820 Because the concentration of oxygen is small, the pressure 685 00:40:33,820 --> 00:40:37,530 is small. k5 times the pressure of oxygen times the 686 00:40:37,530 --> 00:40:39,740 pressure of M is small also. 687 00:40:39,740 --> 00:40:45,660 So k5 T O M, O2 times M is small. 688 00:40:45,660 --> 00:40:50,530 And they're both, then, much smaller than k4 times the 689 00:40:50,530 --> 00:40:53,710 time. k4 termination. 690 00:40:53,710 --> 00:40:56,140 The hydrogen radical hits the wall before 691 00:40:56,140 --> 00:40:58,020 finding anything else. 692 00:40:58,020 --> 00:41:00,190 End of chain. 693 00:41:00,190 --> 00:41:02,110 So in the equation here, these two guys are 694 00:41:02,110 --> 00:41:05,290 small compared to here. 695 00:41:05,290 --> 00:41:08,750 RI, the initiation rate, is very small. 696 00:41:08,750 --> 00:41:11,170 You control that by your spark. 697 00:41:11,170 --> 00:41:12,760 Once a second, once a minute. 698 00:41:12,760 --> 00:41:15,580 Could be really, slow. 699 00:41:15,580 --> 00:41:17,660 Steady state is good. 700 00:41:17,660 --> 00:41:21,720 We don't have an unstable chain. 701 00:41:21,720 --> 00:41:24,920 No explosion here. 702 00:41:24,920 --> 00:41:25,790 Medium pressure. 703 00:41:25,790 --> 00:41:30,240 You've got a medium pressure where two k1 times oxygen 704 00:41:30,240 --> 00:41:32,330 concentration is approximately. 705 00:41:32,330 --> 00:41:33,900 You're building up your concentration. 706 00:41:33,900 --> 00:41:35,540 This is getting bigger here. 707 00:41:35,540 --> 00:41:37,200 This is a constant. 708 00:41:37,200 --> 00:41:38,720 This is getting bigger. 709 00:41:38,720 --> 00:41:43,525 This is also getting bigger, but you've got this rate here 710 00:41:43,525 --> 00:41:44,300 and this rate here. 711 00:41:44,300 --> 00:41:49,380 At some point, at some point, this rate here, two k1 times 712 00:41:49,380 --> 00:41:56,710 oxygen concentration, is going to be on the order of k4 T 713 00:41:56,710 --> 00:42:04,780 plus k5 T times oxygen times M. At some point it's going to 714 00:42:04,780 --> 00:42:07,550 be like that. 715 00:42:07,550 --> 00:42:10,790 And then you're in big trouble. 716 00:42:10,790 --> 00:42:13,020 You're in big trouble, because you've got a denominator here 717 00:42:13,020 --> 00:42:13,900 that's really tiny. 718 00:42:13,900 --> 00:42:17,575 Could be even, if you hit it right on the nail, you get 719 00:42:17,575 --> 00:42:19,500 zero in the denominator and then you're in real trouble 720 00:42:19,500 --> 00:42:20,650 from the math perspective. 721 00:42:20,650 --> 00:42:23,370 But from the physical science perspective, it's 722 00:42:23,370 --> 00:42:24,250 not being in trouble. 723 00:42:24,250 --> 00:42:26,270 Is just that your approximation is wrong. 724 00:42:26,270 --> 00:42:28,770 The approximation that you were in a 725 00:42:28,770 --> 00:42:30,460 stationary state was wrong. 726 00:42:30,460 --> 00:42:33,420 And so that means that in this case here, if you want to have 727 00:42:33,420 --> 00:42:37,170 an explosion, you've got it. 728 00:42:37,170 --> 00:42:40,240 So you adjust everything until you're in a situation. 729 00:42:40,240 --> 00:42:47,050 And then finally, you keep increasing the pressure. 730 00:42:47,050 --> 00:42:51,280 And then if you increase it high enough, then this guy 731 00:42:51,280 --> 00:42:55,320 here, which goes as the square of the pressures, is going to 732 00:42:55,320 --> 00:42:56,120 overwhelm this. 733 00:42:56,120 --> 00:42:58,810 Which goes linearly with pressure. 734 00:42:58,810 --> 00:43:03,700 And k5, the termination step, O2 times M, is going to be 735 00:43:03,700 --> 00:43:10,530 bigger than two k1 times the pressure of O2. 736 00:43:10,530 --> 00:43:13,110 And then you've got something small again. 737 00:43:13,110 --> 00:43:18,610 This is where that's hydrogen radical here is able to 738 00:43:18,610 --> 00:43:26,360 undergo a termolecular process before finding a reactant. 739 00:43:26,360 --> 00:43:34,570 Then steady state is valid. 740 00:43:34,570 --> 00:43:34,655 OK. 741 00:43:34,655 --> 00:43:36,980 At very high pressures, you get in trouble again, because 742 00:43:36,980 --> 00:43:44,850 this hydroxyl radical, this HO2 radical, so at very high 743 00:43:44,850 --> 00:43:51,220 pressure. then what happens is that you have this side 744 00:43:51,220 --> 00:44:01,440 reaction, HO2 radical plus hydrogen goes to H2O plus 745 00:44:01,440 --> 00:44:04,650 hydroxyl radical. 746 00:44:04,650 --> 00:44:08,320 So there's a probability that this radical here finds a 747 00:44:08,320 --> 00:44:09,370 hydrogen molecule. 748 00:44:09,370 --> 00:44:13,230 And then feeds into the propagation through this 749 00:44:13,230 --> 00:44:15,810 hydroxyl radical that's the result. 750 00:44:15,810 --> 00:44:19,100 And then that keeps the chain propagating. 751 00:44:19,100 --> 00:44:21,470 So at very high pressure, you're in trouble again. 752 00:44:21,470 --> 00:44:26,320 And you get an explosion again. 753 00:44:26,320 --> 00:44:37,640 So you can plot this as a function of the pressure. 754 00:44:37,640 --> 00:44:39,080 And we're going to put pressure, actually, 755 00:44:39,080 --> 00:44:40,850 on the y axis here. 756 00:44:40,850 --> 00:44:42,056 Let's put pressure here. 757 00:44:42,056 --> 00:44:42,810 Put temperature here. 758 00:44:42,810 --> 00:44:44,760 And so for a given temperature, let's call it 759 00:44:44,760 --> 00:44:49,080 temperature T prime, at low pressure there's no explosion. 760 00:44:49,080 --> 00:44:53,390 Then there's a range of pressures. 761 00:44:53,390 --> 00:44:54,290 Where there's an explosion. 762 00:44:54,290 --> 00:44:55,100 Medium pressures. 763 00:44:55,100 --> 00:44:58,080 And then a high pressure termination wins again. 764 00:44:58,080 --> 00:45:01,520 And then if it's really high you get explosion again. 765 00:45:01,520 --> 00:45:04,500 You do this for, as a function of temperature. 766 00:45:04,500 --> 00:45:07,190 And what you find that is a diagram, looks like a phase 767 00:45:07,190 --> 00:45:10,930 diagram, that looks something like this. 768 00:45:10,930 --> 00:45:15,140 Where at high temperature, you get an explosion at low 769 00:45:15,140 --> 00:45:24,850 temperatures, and this steady state is valid. 770 00:45:24,850 --> 00:45:35,060 OK, any questions on this? 771 00:45:35,060 --> 00:45:38,220 Don't try this in your kitchen. 772 00:45:38,220 --> 00:45:42,310 Go to the desert if you want to do this. 773 00:45:42,310 --> 00:45:44,470 Or don't do it either. 774 00:45:44,470 --> 00:45:45,720 OK.