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 --> 00:00:17,420 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:17,420 --> 00:00:20,760 ocw.mit.edu. 8 00:00:20,760 --> 00:00:26,920 PROFESSOR: So, now we'll start on the last of the main topics 9 00:00:26,920 --> 00:00:27,490 in the course. 10 00:00:27,490 --> 00:00:30,300 So we've finished most of our macroscopic thermodynamics, 11 00:00:30,300 --> 00:00:32,850 and our microscopic approach to it 12 00:00:32,850 --> 00:00:34,290 through statistical mechanics. 13 00:00:34,290 --> 00:00:37,280 And now, our final topic is kinetics. 14 00:00:37,280 --> 00:00:40,920 Kinetics is really a very different kind of topic. 15 00:00:40,920 --> 00:00:44,840 Because unlike thermodynamics, thermodynamics tells you all 16 00:00:44,840 --> 00:00:46,900 about equilibrium properties. 17 00:00:46,900 --> 00:00:49,980 A huge part of the work of this term has been to figure 18 00:00:49,980 --> 00:00:52,570 out what equilibrium is. 19 00:00:52,570 --> 00:00:56,620 What is the equilibrium state, given some situation. 20 00:00:56,620 --> 00:00:58,290 You've got some phases present. 21 00:00:58,290 --> 00:01:01,200 Different then, you could go from solid to liquid to gas. 22 00:01:01,200 --> 00:01:04,570 Or you've got different chemical constituents together 23 00:01:04,570 --> 00:01:06,360 that can react and go back and forth. 24 00:01:06,360 --> 00:01:09,670 What are the equilibrium concentrations? 25 00:01:09,670 --> 00:01:13,230 But we haven't worried at all about how long it might take 26 00:01:13,230 --> 00:01:15,550 to get there. 27 00:01:15,550 --> 00:01:17,110 And that's what kinetics does. 28 00:01:17,110 --> 00:01:21,980 Kinetics is concerned with rates of reactions, primarily. 29 00:01:21,980 --> 00:01:23,920 How long it takes to reach equilibrium. 30 00:01:23,920 --> 00:01:25,260 And of course it's super-important. 31 00:01:25,260 --> 00:01:29,440 Because if you look at that window glass, it's not in 32 00:01:29,440 --> 00:01:31,360 equilibrium. 33 00:01:31,360 --> 00:01:33,200 It's silicon dioxide, the equilibrium 34 00:01:33,200 --> 00:01:34,550 state would be a crystal. 35 00:01:34,550 --> 00:01:37,210 It would be crystal and quartz. 36 00:01:37,210 --> 00:01:39,980 Nevertheless, none of us is very worried that on the 37 00:01:39,980 --> 00:01:42,570 moment it's likely to suddenly, spontaneously turn 38 00:01:42,570 --> 00:01:45,310 into a crystal and be opaque and scatter light and do all 39 00:01:45,310 --> 00:01:47,610 that sort of stuff. 40 00:01:47,610 --> 00:01:50,670 And in lots of other situations, certainly if you 41 00:01:50,670 --> 00:01:56,600 look at any living biological system, including yourselves 42 00:01:56,600 --> 00:02:00,210 or your friends or any other one, it's certainly far from 43 00:02:00,210 --> 00:02:01,140 equilibrium. 44 00:02:01,140 --> 00:02:05,070 And you probably would hope for it to stay that way. 45 00:02:05,070 --> 00:02:09,830 So there's an enormous amount of chemistry and processes 46 00:02:09,830 --> 00:02:10,770 we're concerned with. 47 00:02:10,770 --> 00:02:14,310 Which depend intimately on kinetics in order to work the 48 00:02:14,310 --> 00:02:16,210 way they work. 49 00:02:16,210 --> 00:02:19,160 They also do depend on thermodynamics and where 50 00:02:19,160 --> 00:02:20,660 equilibrium states are. 51 00:02:20,660 --> 00:02:22,450 But that doesn't mean they necessarily 52 00:02:22,450 --> 00:02:25,790 reach equilibrium states. 53 00:02:25,790 --> 00:02:28,610 So we're going to go through kinetics, starting with the 54 00:02:28,610 --> 00:02:31,070 simplest examples and working our way up to 55 00:02:31,070 --> 00:02:33,260 more complex cases. 56 00:02:33,260 --> 00:02:37,330 And just see how we can describe elementary chemical 57 00:02:37,330 --> 00:02:42,130 reaction rates and processes. 58 00:02:42,130 --> 00:03:02,370 So that's our concern now, is dynamics. 59 00:03:02,370 --> 00:03:05,950 How long things take to get to equilibrium. 60 00:03:05,950 --> 00:03:11,340 And actually just like macroscopic thermodynamics, 61 00:03:11,340 --> 00:03:13,790 kinetics does take an empirical 62 00:03:13,790 --> 00:03:16,390 approach to the topic. 63 00:03:16,390 --> 00:03:19,480 In other words, it's based on experimental observation. 64 00:03:19,480 --> 00:03:20,850 Of macroscopic rates. 65 00:03:20,850 --> 00:03:23,860 How long it takes a collection of stuff, a mole of stuff to 66 00:03:23,860 --> 00:03:25,130 change chemically. 67 00:03:25,130 --> 00:03:27,330 And undergo reaction. 68 00:03:27,330 --> 00:03:29,560 And so forth. 69 00:03:29,560 --> 00:03:35,730 We often infer molecular mechanisms based on kinetics. 70 00:03:35,730 --> 00:03:39,400 And it's hugely important and valuable to do that. 71 00:03:39,400 --> 00:03:42,560 But it's also important to recognize that what kinetics 72 00:03:42,560 --> 00:03:47,890 can do is show us how we can formulate microscopic 73 00:03:47,890 --> 00:03:51,310 mechanisms that might be consistent with our 74 00:03:51,310 --> 00:03:53,810 macroscopic kinetics models. 75 00:03:53,810 --> 00:03:55,690 But the kinetics models by themselves 76 00:03:55,690 --> 00:03:58,830 don't prove the mechanism. 77 00:03:58,830 --> 00:04:02,650 And there are all sorts of examples where mechanisms that 78 00:04:02,650 --> 00:04:07,540 were proposed and accepted because they were consistent 79 00:04:07,540 --> 00:04:11,030 with macroscopic kinetics results turned out to fail. 80 00:04:11,030 --> 00:04:13,790 There are other ways to prove mechanisms. 81 00:04:13,790 --> 00:04:17,060 You might be able to design direct spectroscopic 82 00:04:17,060 --> 00:04:20,640 observations of intermediates and so forth. 83 00:04:20,640 --> 00:04:22,930 And in that case, it often becomes possible to 84 00:04:22,930 --> 00:04:25,480 distinguish between different mechanisms. 85 00:04:25,480 --> 00:04:31,740 Which might all satisfy the macroscopic kinetics equation. 86 00:04:31,740 --> 00:04:35,600 So we use kinetics to infer a mechanism but not necessarily 87 00:04:35,600 --> 00:04:36,100 to prove it. 88 00:04:36,100 --> 00:05:02,320 Not generally to prove it. 89 00:05:02,320 --> 00:05:06,430 We also use kinetics to describe an enormous range of 90 00:05:06,430 --> 00:05:07,590 time scales. 91 00:05:07,590 --> 00:05:13,530 So, the fastest things that we're sometimes concerned with 92 00:05:13,530 --> 00:05:16,250 that might take place on femtosecond time scales, 10 to 93 00:05:16,250 --> 00:05:18,840 the minus 13 seconds or so, or 10 to the 94 00:05:18,840 --> 00:05:21,770 minus 15 seconds, even. 95 00:05:21,770 --> 00:05:26,600 So, if we look at the range of time scales we might be 96 00:05:26,600 --> 00:05:29,290 concerned with, might go anywhere from about 10 to the 97 00:05:29,290 --> 00:05:37,580 minus 15 seconds at the fastest, and might go to 98 00:05:37,580 --> 00:05:39,700 enormous scales on the other end. 99 00:05:39,700 --> 00:05:43,080 All the way out to maybe 10 to the 10 seconds, which is on 100 00:05:43,080 --> 00:05:45,090 the thousands of years. 101 00:05:45,090 --> 00:05:46,660 Could be longer than that, sometimes 102 00:05:46,660 --> 00:05:51,570 even millions of years. 103 00:05:51,570 --> 00:05:55,520 And the formalism that we'll set up applies equally to the 104 00:05:55,520 --> 00:05:58,160 full range of time scales. 105 00:05:58,160 --> 00:06:02,350 So it can describe an enormous amount of chemical activity. 106 00:06:02,350 --> 00:06:05,740 More commonly, for stuff that we're going to compare to, you 107 00:06:05,740 --> 00:06:11,100 might measure in the lab. 108 00:06:11,100 --> 00:06:16,120 Common time scales range from roughly 10 to the minus 6 109 00:06:16,120 --> 00:06:20,250 seconds out to about 10 to the 5th seconds. 110 00:06:20,250 --> 00:06:30,160 That is, microseconds to about a day. 111 00:06:30,160 --> 00:06:32,240 But there are plenty of examples of going either 112 00:06:32,240 --> 00:06:35,190 faster or slower, depending on the need and the experimental 113 00:06:35,190 --> 00:06:39,320 equipment available. 114 00:06:39,320 --> 00:06:41,080 Let's define a few terms. 115 00:06:41,080 --> 00:06:44,150 Let's talk about how we're going to just formulate 116 00:06:44,150 --> 00:06:46,130 chemical reaction rates. 117 00:06:46,130 --> 00:07:07,150 So, let's just imagine any simple reaction of this sort. 118 00:07:07,150 --> 00:07:11,030 So A plus B are going to go to C. There's some rate at which 119 00:07:11,030 --> 00:07:12,360 it happens. 120 00:07:12,360 --> 00:07:18,960 Note, by the way, it's an arrow in one direction. 121 00:07:18,960 --> 00:07:20,970 It's not an equals sign like we've seen 122 00:07:20,970 --> 00:07:24,250 before, or a double arrow. 123 00:07:24,250 --> 00:07:27,990 So the point I'm emphasizing here is when we talk about 124 00:07:27,990 --> 00:07:31,600 reaction rates, unlike equilibria, we're talking 125 00:07:31,600 --> 00:07:33,820 about a particular direction. 126 00:07:33,820 --> 00:07:37,520 Later on, we will talk about reversible reactions. 127 00:07:37,520 --> 00:07:41,140 But there, too, the arrow going this way refers only to 128 00:07:41,140 --> 00:07:43,920 one of the chemical reactions that can take place. 129 00:07:43,920 --> 00:07:48,240 Namely, in this case, the changing of what was the 130 00:07:48,240 --> 00:07:51,025 product now, would be the reactant, C, back into A plus 131 00:07:51,025 --> 00:07:54,490 B and it has nothing to do with the rate this way. 132 00:07:54,490 --> 00:07:59,840 Measured independently and so on. 133 00:07:59,840 --> 00:08:09,790 So the rate, we can look at the rate of disappearance of 134 00:08:09,790 --> 00:08:21,600 A. So it's just negative d[A]/dt, where the brackets 135 00:08:21,600 --> 00:08:23,390 indicate concentration. 136 00:08:23,390 --> 00:08:31,960 Usually in moles per liter. 137 00:08:31,960 --> 00:08:37,540 If it's in a gas, then it would be pA, of pressure. 138 00:08:37,540 --> 00:08:41,260 So negative d[A]/dt is our rate. 139 00:08:41,260 --> 00:08:45,960 And note that that's generally a positive number. 140 00:08:45,960 --> 00:08:48,590 We're looking at reactions going, if it's a reaction 141 00:08:48,590 --> 00:08:50,430 going in this direction. 142 00:08:50,430 --> 00:08:53,790 A is gradually disappearing. 143 00:08:53,790 --> 00:08:56,690 So we're going to define our rate this way. 144 00:08:56,690 --> 00:09:00,480 To be a positive number. 145 00:09:00,480 --> 00:09:06,360 The rate for C is going to be plus d[C]/dt. 146 00:09:08,900 --> 00:09:11,640 It's also going to be defined in a way 147 00:09:11,640 --> 00:09:12,810 that makes it positive. 148 00:09:12,810 --> 00:09:14,920 Because in this case, since the reaction's going this way, 149 00:09:14,920 --> 00:09:19,510 we're looking at the appearance of C. 150 00:09:19,510 --> 00:09:24,460 Now, by stoichiometry , in this case, of course whenever 151 00:09:24,460 --> 00:09:27,150 a molecule or a mole of A disappears, a molecule or a 152 00:09:27,150 --> 00:09:29,500 mole of C appears. 153 00:09:29,500 --> 00:09:37,940 So because of that, in this case, the stoichiometry tells 154 00:09:37,940 --> 00:09:47,480 us that that d[C]/dt is equal to negative d[A]/dt. 155 00:09:47,480 --> 00:09:49,800 And also equal to negative d[B]/dt. 156 00:09:55,060 --> 00:10:01,180 And any of those can be used to 157 00:10:01,180 --> 00:10:13,520 define the rate of reaction. 158 00:10:13,520 --> 00:10:15,670 Now, this is a particularly simple case because I've 159 00:10:15,670 --> 00:10:17,510 chosen the case where all of the stoichiometric 160 00:10:17,510 --> 00:10:20,010 coefficients are equal to one. 161 00:10:20,010 --> 00:10:25,140 So now let's just look at any case that's different. 162 00:10:25,140 --> 00:10:32,760 Let's look at 2A plus B goes to something else, 3C plus D. 163 00:10:32,760 --> 00:10:36,820 And just look at the reaction rates that we might see there. 164 00:10:36,820 --> 00:10:42,120 So here now, the appearance of C is going to be three times 165 00:10:42,120 --> 00:10:45,170 as fast as the appearance of D, for example. 166 00:10:45,170 --> 00:10:51,650 And also three times as fast as the disappearance of B. So 167 00:10:51,650 --> 00:11:00,360 if we write negative d[B]/dt, we expect that's going to be 168 00:11:00,360 --> 00:11:03,000 negative 1/2 d[A]/dt. 169 00:11:05,720 --> 00:11:09,020 In other words, A is going to disappear twice as fast as B. 170 00:11:09,020 --> 00:11:15,110 Every time a molecule of B reacts, two molecules of A do. 171 00:11:15,110 --> 00:11:21,640 And that's going to be plus 1/3 d[C]/dt, and every time 172 00:11:21,640 --> 00:11:26,260 that happens three molecules of C get formed. 173 00:11:26,260 --> 00:11:32,180 And it's going to be plus d[D]/dt. 174 00:11:34,900 --> 00:11:39,100 One molecule of D gets formed. 175 00:11:39,100 --> 00:11:42,970 And so, the reaction rate could be defined in terms of 176 00:11:42,970 --> 00:11:43,600 any of these. 177 00:11:43,600 --> 00:11:46,470 But the important thing is to keep track of stoichiometry so 178 00:11:46,470 --> 00:11:50,080 that the rate as it pertains to each constituent is 179 00:11:50,080 --> 00:11:56,690 accounted for correctly. 180 00:11:56,690 --> 00:12:07,550 So to generalize, if I have little a of A, and little b of 181 00:12:07,550 --> 00:12:22,130 B, going to little c of C, and little d of D, then the rate 182 00:12:22,130 --> 00:12:32,870 of reaction can be written as minus one over a d[A]/dt, or 183 00:12:32,870 --> 00:12:45,230 minus one over b d[B]/dt, or one over c d[C]/dt, or one 184 00:12:45,230 --> 00:12:47,040 over d d[D]/dt. 185 00:13:17,860 --> 00:13:21,560 So, experimentally, lots of measurements of reaction rates 186 00:13:21,560 --> 00:13:22,420 have been made. 187 00:13:22,420 --> 00:13:26,190 And now to start on what's seen empirically, basically 188 00:13:26,190 --> 00:13:36,660 the following result is extremely common. 189 00:13:36,660 --> 00:13:41,740 The rate is equal to some constant times the 190 00:13:41,740 --> 00:13:46,570 concentration of A to some power alpha, times the 191 00:13:46,570 --> 00:13:58,170 concentration of B to some power beta, and so on. 192 00:13:58,170 --> 00:14:00,000 For all reactants. 193 00:14:00,000 --> 00:14:02,430 Multiplied together, each concentration 194 00:14:02,430 --> 00:14:04,720 taken to some power. 195 00:14:04,720 --> 00:14:06,700 Notice no products. 196 00:14:06,700 --> 00:14:11,490 Again, we're only looking at a reaction going one way. 197 00:14:11,490 --> 00:14:13,760 And if we look at the rate, this is 198 00:14:13,760 --> 00:14:16,680 typically what's found. 199 00:14:16,680 --> 00:14:32,830 Alpha is called the order of reaction, with respect to A. 200 00:14:32,830 --> 00:14:43,320 Beta order with respect to B. And so on. 201 00:14:43,320 --> 00:14:53,470 And little k is our rate constant which, just to make 202 00:14:53,470 --> 00:14:56,940 clear, since we've been using it extensively in the past few 203 00:14:56,940 --> 00:15:03,320 lectures, is completely not equal to 204 00:15:03,320 --> 00:15:05,340 the Boltzmann constant. 205 00:15:05,340 --> 00:15:12,420 Absolutely no connection between them. 206 00:15:12,420 --> 00:15:18,150 Now, alpha and beta, what they are, what they turn out to be, 207 00:15:18,150 --> 00:15:23,150 is typically what's determined as a result of kinetic 208 00:15:23,150 --> 00:15:24,950 measurement. 209 00:15:24,950 --> 00:15:42,730 Typically they're small integers. 210 00:15:42,730 --> 00:16:00,690 So, just sort of a typical example, if you look at the 211 00:16:00,690 --> 00:16:14,870 reaction of NO and O2, to make NO2, simple reaction, and you 212 00:16:14,870 --> 00:16:23,360 look at minus d[O2]/dt, use that as a measure of the 213 00:16:23,360 --> 00:16:25,530 reaction rate. 214 00:16:25,530 --> 00:16:31,390 What's found is that it's a constant times NO 215 00:16:31,390 --> 00:16:37,110 concentration squared, times O2. 216 00:16:37,110 --> 00:16:37,940 Simple, right? 217 00:16:37,940 --> 00:16:42,430 One of the exponents is two, the other's one. 218 00:16:42,430 --> 00:16:49,260 Doesn't seem so surprising mechanistically. 219 00:16:49,260 --> 00:16:51,720 But it's not always the case. 220 00:16:51,720 --> 00:16:54,750 Even when you have small integers, it's not always the 221 00:16:54,750 --> 00:16:58,560 case that the most obvious mechanism you would infer is 222 00:16:58,560 --> 00:17:00,330 the real mechanism. 223 00:17:00,330 --> 00:17:03,280 And sometimes those exponents don't turn 224 00:17:03,280 --> 00:17:05,810 out even to be integers. 225 00:17:05,810 --> 00:17:13,220 But here's another example. 226 00:17:13,220 --> 00:17:23,630 If you just look it CH3CHO going to methane and carbon 227 00:17:23,630 --> 00:17:28,480 monoxide, seems like it would be a pretty straightforward 228 00:17:28,480 --> 00:17:30,000 thing, too. 229 00:17:30,000 --> 00:17:34,120 So if you measure, for example, the rate of 230 00:17:34,120 --> 00:17:45,140 appearance of methane, what you discover is that it's 231 00:17:45,140 --> 00:17:50,500 equal to a constant times the concentration of the starting 232 00:17:50,500 --> 00:17:59,240 material to the 3/2 power. 233 00:17:59,240 --> 00:18:02,490 Not obvious mechanistically why that should be the case. 234 00:18:02,490 --> 00:18:05,640 Usually it's telling us something. 235 00:18:05,640 --> 00:18:09,140 It's telling us that the reaction mechanism is 236 00:18:09,140 --> 00:18:10,200 complicated. 237 00:18:10,200 --> 00:18:11,970 It's a multi-step process. 238 00:18:11,970 --> 00:18:16,170 Sometimes there could be chain reactions and so forth. 239 00:18:16,170 --> 00:18:19,400 So again, seeing things like this certainly helps us to 240 00:18:19,400 --> 00:18:23,000 infer molecular mechanisms. 241 00:18:23,000 --> 00:18:25,820 Again, they don't prove molecular mechanisms. 242 00:18:25,820 --> 00:18:30,250 But they certainly can be very helpful in suggesting them. 243 00:18:30,250 --> 00:18:34,390 And then other means can be used to try to prove them. 244 00:18:34,390 --> 00:18:36,890 Including, above all, direct observation of the 245 00:18:36,890 --> 00:18:40,400 intermediates that you would expect on the basis of one 246 00:18:40,400 --> 00:18:47,840 mechanism or another. 247 00:18:47,840 --> 00:18:51,190 Now let's just go through some elementary examples of 248 00:18:51,190 --> 00:18:53,510 kinetics one at a time. 249 00:18:53,510 --> 00:19:11,900 So let's start with the simplest possible case. 250 00:19:11,900 --> 00:19:14,390 Actually, a very rare case, but one that'll help us just 251 00:19:14,390 --> 00:19:16,410 set up the formalism of, and the 252 00:19:16,410 --> 00:19:18,500 mechanism for us to proceed. 253 00:19:18,500 --> 00:19:32,680 So let's talk about zero order reactions. 254 00:19:32,680 --> 00:19:35,890 Actually very rare. 255 00:19:35,890 --> 00:19:42,690 So, what this means is something like A 256 00:19:42,690 --> 00:19:45,070 goes over to products. 257 00:19:45,070 --> 00:19:57,715 Make a measurement of d[A]/dt, and discover that it's k time 258 00:19:57,715 --> 00:20:00,710 A to the power of zero, that is, it's just k. 259 00:20:00,710 --> 00:20:04,630 There's no dependence on the concentration of A. Or 260 00:20:04,630 --> 00:20:07,050 anything else. 261 00:20:07,050 --> 00:20:11,300 And you have some rate of its disappearance. 262 00:20:11,300 --> 00:20:15,880 So here's an example of it. 263 00:20:15,880 --> 00:20:29,850 You could have oxalic acid, and it just turns into 264 00:20:29,850 --> 00:20:35,350 hydrogen carbon dioxide and carbon dioxide. 265 00:20:35,350 --> 00:20:41,760 Things break apart, forms these products. 266 00:20:41,760 --> 00:20:44,150 Make a measurement of the disappearance. 267 00:20:44,150 --> 00:20:47,960 Seems to have nothing to do with the concentration of it. 268 00:20:47,960 --> 00:20:49,730 At least under certain conditions. 269 00:20:49,730 --> 00:20:57,580 Now, turns out that there's another element that helps 270 00:20:57,580 --> 00:21:02,610 understand this. 271 00:21:02,610 --> 00:21:04,575 Turns out that light is needed, it's a 272 00:21:04,575 --> 00:21:06,770 photochemical reaction. 273 00:21:06,770 --> 00:21:08,920 And then it's easy to see how this can happen. 274 00:21:08,920 --> 00:21:11,940 Let's say we have an abundance of the starting material, and 275 00:21:11,940 --> 00:21:14,540 not very much light. 276 00:21:14,540 --> 00:21:17,940 So every now and then photons bleed in, and every now and 277 00:21:17,940 --> 00:21:21,200 then when they're absorbed, it leads to dissociation. 278 00:21:21,200 --> 00:21:24,660 In that situation, you'll be limited by the photons, but 279 00:21:24,660 --> 00:21:27,240 not by the concentration of the molecules. 280 00:21:27,240 --> 00:21:32,040 And in fact, strictly speaking although this is zero order in 281 00:21:32,040 --> 00:21:36,300 terms of the chemical constituents, it's not zero 282 00:21:36,300 --> 00:21:39,390 order in the photons. 283 00:21:39,390 --> 00:21:42,260 So in some sense, in this sort of situation, the photons 284 00:21:42,260 --> 00:21:45,160 should be considered one of the reactants. 285 00:21:45,160 --> 00:21:49,860 You could write this as this plus h nu plus one photon, 286 00:21:49,860 --> 00:21:52,150 goes over to these products. 287 00:21:52,150 --> 00:21:56,510 And then if you measure the rate and things are under 288 00:21:56,510 --> 00:21:59,770 circumstances like I described, you would discover 289 00:21:59,770 --> 00:22:01,680 that in fact yes you'd be photon limited. 290 00:22:01,680 --> 00:22:03,520 The rate would depend on how many both 291 00:22:03,520 --> 00:22:06,220 photons are coming in. 292 00:22:06,220 --> 00:22:10,390 Still, ordinarily, chemical rate equations aren't 293 00:22:10,390 --> 00:22:12,420 formulated in those terms. 294 00:22:12,420 --> 00:22:15,190 So in the usual formulation, this would still have the 295 00:22:15,190 --> 00:22:24,660 appearance of a zero order reaction. 296 00:22:24,660 --> 00:22:30,880 Now, how do we write and formulate a solution? 297 00:22:30,880 --> 00:22:32,950 So it's simple. 298 00:22:32,950 --> 00:22:35,980 We've we've written a differential equation here. 299 00:22:35,980 --> 00:22:37,690 It's a pretty straightforward one. 300 00:22:37,690 --> 00:22:40,750 Minus d[A]/dt is just a constant. 301 00:22:40,750 --> 00:22:42,000 So we can solve it. 302 00:22:42,000 --> 00:22:44,920 And typically we'll solve it by rewriting in integral form 303 00:22:44,920 --> 00:22:47,570 and then doing the integration as long as we can do the 304 00:22:47,570 --> 00:22:49,550 integration. 305 00:22:49,550 --> 00:22:56,090 So from here we can write the integral from starting 306 00:22:56,090 --> 00:23:07,440 concentration [A]0 to some other concentration, [A], d[A] 307 00:23:07,440 --> 00:23:19,310 is equal to minus k integral from zero to t dt. 308 00:23:19,310 --> 00:23:24,230 So all we've done is integrate on both sides. 309 00:23:24,230 --> 00:23:34,790 And we've assumed a starting concentration. 310 00:23:34,790 --> 00:23:36,770 We've assumed an initial condition. 311 00:23:36,770 --> 00:23:39,130 And we've also assumed an initial time, which usually we 312 00:23:39,130 --> 00:23:57,450 can just call zero. 313 00:23:57,450 --> 00:23:59,390 And this is, of course, something we can solve for 314 00:23:59,390 --> 00:24:00,560 straight away. 315 00:24:00,560 --> 00:24:12,840 So we just have that [A] minus [A]0 is negative kt minus 316 00:24:12,840 --> 00:24:15,030 zero, which is minus kt. 317 00:24:21,540 --> 00:24:27,560 So [A] is minus kt. 318 00:24:27,560 --> 00:24:29,430 Plus [A]0. 319 00:24:29,430 --> 00:24:31,870 In other words, the concentration of [A] at any 320 00:24:31,870 --> 00:24:35,150 time is given by the initial concentration, minus the rate 321 00:24:35,150 --> 00:24:36,100 constant times time. 322 00:24:36,100 --> 00:24:39,900 It decays linearly in time. 323 00:24:39,900 --> 00:24:51,810 So we can sketch that. 324 00:24:51,810 --> 00:24:54,020 This is our initial concentration. 325 00:24:54,020 --> 00:25:06,890 And then it's just going to decline with time after that. 326 00:25:06,890 --> 00:25:09,540 And there's our solution. 327 00:25:09,540 --> 00:25:13,430 In lots of cases, it's useful to define 328 00:25:13,430 --> 00:25:15,510 what's called a half-life. 329 00:25:15,510 --> 00:25:16,560 It's just useful. 330 00:25:16,560 --> 00:25:19,810 Because it provides some timeframe, a single number 331 00:25:19,810 --> 00:25:25,250 that's a timeframe on which the reaction occurs. 332 00:25:25,250 --> 00:25:28,480 So, the half-life is just the time that it takes for half of 333 00:25:28,480 --> 00:25:47,300 the reactants to disappear. t 1/2 time to 334 00:25:47,300 --> 00:25:57,390 react half the reactants. 335 00:25:57,390 --> 00:25:59,700 OK, so in this case it's straightforward to see when 336 00:25:59,700 --> 00:26:00,230 that happens. 337 00:26:00,230 --> 00:26:02,180 In other words, that's the time at which this 338 00:26:02,180 --> 00:26:14,580 concentration of A is just equal to [A]0 over two. 339 00:26:14,580 --> 00:26:28,750 So we have [A]0 over two is minus kt 1/2 plus [A]0 or t 340 00:26:28,750 --> 00:26:35,460 1/2 is equal to [A] over 2k. 341 00:26:38,100 --> 00:26:44,020 [A]0 over 2k. 342 00:26:44,020 --> 00:26:48,650 So there's our half-life. 343 00:26:48,650 --> 00:27:02,050 So if we go over here, we can put that in. 344 00:27:02,050 --> 00:27:13,280 [A]0 over two, and this time is our half-life. 345 00:27:13,280 --> 00:27:14,820 So that's zero order reactions. 346 00:27:14,820 --> 00:27:17,310 And, again, zero order reactions are rare. 347 00:27:17,310 --> 00:27:21,890 But the procedure we're going to used to solve for kinetics 348 00:27:21,890 --> 00:27:23,150 is outlined in this way. 349 00:27:23,150 --> 00:27:34,420 And we'll use that again and again. 350 00:27:34,420 --> 00:27:38,080 Let's look at first-order kinetics. 351 00:27:38,080 --> 00:28:03,550 Let's go over here. 352 00:28:03,550 --> 00:28:06,360 Now, first order reactions are quite common. 353 00:28:06,360 --> 00:28:08,830 Much, much more common than zero order. 354 00:28:08,830 --> 00:28:17,550 So here, you have A goes to products. 355 00:28:17,550 --> 00:28:20,070 That's the simplest case. 356 00:28:20,070 --> 00:28:26,900 But this time if we measure d[A]/dt, we discover that it's 357 00:28:26,900 --> 00:28:34,350 equal to a constant times the concentration of A. There's an 358 00:28:34,350 --> 00:28:37,170 important point to note here. 359 00:28:37,170 --> 00:28:43,170 What are the units of k, in this case. 360 00:28:43,170 --> 00:28:48,780 What do they have to be? 361 00:28:48,780 --> 00:28:50,230 Yeah, reciprocal seconds right? 362 00:28:50,230 --> 00:28:52,660 The equation has to work. 363 00:28:52,660 --> 00:28:55,090 This is concentration per second. 364 00:28:55,090 --> 00:28:56,530 This is concentration. 365 00:28:56,530 --> 00:29:05,980 This better be per second. 366 00:29:05,980 --> 00:29:13,250 Let's just, before we move on completely, look at this. 367 00:29:13,250 --> 00:29:16,460 What are its units? 368 00:29:16,460 --> 00:29:19,470 Here's the equation. 369 00:29:19,470 --> 00:29:28,490 What are the units of k? 370 00:29:28,490 --> 00:29:31,680 No, not unitless, because, look at this. 371 00:29:31,680 --> 00:29:33,730 This is some concentration unit, 372 00:29:33,730 --> 00:29:35,500 typically moles per liter. 373 00:29:35,500 --> 00:29:37,260 So this is moles per liter per second, right? 374 00:29:37,260 --> 00:29:41,270 It's disappearance of some concentration for time. 375 00:29:41,270 --> 00:29:42,250 That's got to be here. 376 00:29:42,250 --> 00:29:45,490 All that is here. 377 00:29:45,490 --> 00:29:51,410 Moles per liter per second. 378 00:29:51,410 --> 00:29:55,480 So in every case, the units of k, the rate constant, have to 379 00:29:55,480 --> 00:29:59,020 be figured out on the basis of the specific rate equation. 380 00:29:59,020 --> 00:30:01,410 Doesn't have the same units. 381 00:30:01,410 --> 00:30:10,010 When the kinetics are of different order. 382 00:30:10,010 --> 00:30:12,610 Now, let's solve this using the same approach as before. 383 00:30:12,610 --> 00:30:15,000 Namely, this is still a pretty straightforward 384 00:30:15,000 --> 00:30:16,230 differential equation. 385 00:30:16,230 --> 00:30:19,140 So let's just integrate both sides. 386 00:30:19,140 --> 00:30:30,250 So that is going to tell us the integral from [A]0 to [A]. 387 00:30:30,250 --> 00:30:35,340 But now we have [A] on this side. 388 00:30:35,340 --> 00:30:40,130 So here we just had a constant. 389 00:30:40,130 --> 00:30:42,300 And effectively, I didn't write it out. 390 00:30:42,300 --> 00:30:49,110 But we effectively wrote this, rewrote this, as d[A] 391 00:30:49,110 --> 00:30:52,480 equals minus k dt, and then went 392 00:30:52,480 --> 00:30:55,860 from here to the integral. 393 00:30:55,860 --> 00:30:58,460 We're going to do the same thing here, 394 00:30:58,460 --> 00:30:59,420 except now there's this. 395 00:30:59,420 --> 00:31:03,490 So really we're going to have d[A] 396 00:31:03,490 --> 00:31:10,250 over [A] is minus k dt. 397 00:31:10,250 --> 00:31:11,990 That's what we're going to integrate on both side. 398 00:31:11,990 --> 00:31:18,880 We need to have the variables distinct on each side. 399 00:31:18,880 --> 00:31:20,170 So we have d[A] 400 00:31:22,850 --> 00:31:24,660 over [A]. 401 00:31:24,660 --> 00:31:31,510 Equal to minus k integral from zero to t dt. 402 00:31:31,510 --> 00:31:33,590 And so, of course, you know how to do this integral. 403 00:31:33,590 --> 00:31:35,560 It's going to look like log of [A]. 404 00:31:35,560 --> 00:31:39,460 And again, it's taken at [A] or, at [A]0. 405 00:31:39,460 --> 00:31:47,360 So we have log of [A] over [A]0. 406 00:31:47,360 --> 00:31:51,620 We're going to have log of [A] minus log of [A]0 coming out. 407 00:31:51,620 --> 00:31:57,250 And that's equal to minus kt. 408 00:31:57,250 --> 00:32:01,470 So now the kinetics are quite different. 409 00:32:01,470 --> 00:32:15,200 We have [A] is equal to [A]0, e to the minus kt. 410 00:32:15,200 --> 00:32:17,620 Very important, very common sort of result. 411 00:32:17,620 --> 00:32:20,600 It's saying we start with a certain amount of material, 412 00:32:20,600 --> 00:32:23,640 and there's an exponential decay of it. 413 00:32:23,640 --> 00:32:26,110 So very different from kinetics here, 414 00:32:26,110 --> 00:32:29,210 which are just linear. 415 00:32:29,210 --> 00:32:31,730 And again the kinetics here, if you imagine that situation 416 00:32:31,730 --> 00:32:35,420 where you've got starting material and bleeding in 417 00:32:35,420 --> 00:32:39,010 gradually are photons, presumably at the same rate, 418 00:32:39,010 --> 00:32:41,720 then sure that material you're going to see the disappearance 419 00:32:41,720 --> 00:32:44,590 of it linearly with time. 420 00:32:44,590 --> 00:32:46,790 Just depending on the rate at which the 421 00:32:46,790 --> 00:32:47,980 photons are coming in. 422 00:32:47,980 --> 00:32:50,860 Here it's very different because, presumably, A is 423 00:32:50,860 --> 00:32:53,250 required in order to do this reaction. 424 00:32:53,250 --> 00:32:54,110 It'll depend on how much. 425 00:32:54,110 --> 00:32:56,780 Because of course if there's more of it, you'll just have 426 00:32:56,780 --> 00:33:02,640 more at any given time decaying over to products. 427 00:33:02,640 --> 00:33:29,250 So you have an exponential decay. 428 00:33:29,250 --> 00:33:31,330 So let's plot that. 429 00:33:31,330 --> 00:33:44,570 Here's [A], let's make that [A]0. 430 00:33:44,570 --> 00:33:54,030 There it is. 431 00:33:54,030 --> 00:33:56,250 Now, let's look at what happens to the product. 432 00:33:56,250 --> 00:34:05,270 So let's imagine that it's A going to B, so of course minus 433 00:34:05,270 --> 00:34:14,150 d[A]/dt is just equal to d[B]/dt, and the rate of 434 00:34:14,150 --> 00:34:17,625 appearance of B has to match the rate of disappearance of 435 00:34:17,625 --> 00:34:20,780 A. Let's assume that we don't have any of 436 00:34:20,780 --> 00:34:25,190 B present at first. 437 00:34:25,190 --> 00:34:34,700 So let's make [B]0 equal to zero. 438 00:34:34,700 --> 00:34:43,950 Well then, [B] just has to equal [A]0 minus [A], right? 439 00:34:43,950 --> 00:34:48,260 All the stuff that's left, all of the A that has disappeared, 440 00:34:48,260 --> 00:34:50,560 that's given by this difference. 441 00:34:50,560 --> 00:35:00,400 Is just equal to B. So that's just [A]0 minus concentration 442 00:35:00,400 --> 00:35:03,480 of A, but that's just given by that. 443 00:35:03,480 --> 00:35:07,690 Which is [A]0 e to the minus kt. 444 00:35:07,690 --> 00:35:20,530 Or in other words, [B] is just equal to [A]0 times one minus 445 00:35:20,530 --> 00:35:24,460 e to the minus kt. 446 00:35:24,460 --> 00:35:27,840 So at t equals zero, this is zero and this is one. 447 00:35:27,840 --> 00:35:32,750 In other words, this is going to be zero at first. 448 00:35:32,750 --> 00:35:35,550 And then it's going to grow in with the same exponential form 449 00:35:35,550 --> 00:35:36,760 that this decayed. 450 00:35:36,760 --> 00:35:41,160 So, [B] is going to do the exact opposite. 451 00:35:41,160 --> 00:35:44,890 It's going to be like this. 452 00:35:44,890 --> 00:35:57,550 So this is [B] of t and this is [A] of t equals [A]0 e to 453 00:35:57,550 --> 00:36:03,270 the minus kt. 454 00:36:03,270 --> 00:36:07,760 Now, it's always useful to, whenever possible, to plot 455 00:36:07,760 --> 00:36:09,510 these things linearly. 456 00:36:09,510 --> 00:36:11,060 Find a way to plot these as straight lines. 457 00:36:11,060 --> 00:36:13,690 And of course in this case it's straightforward to do 458 00:36:13,690 --> 00:36:21,470 that as a log plot. 459 00:36:21,470 --> 00:36:27,760 So if we take the log of both sides, we know of course that 460 00:36:27,760 --> 00:36:42,740 the log of [A] is just minus kt, plus the log of [A]0, so 461 00:36:42,740 --> 00:36:45,100 now let's look at that. 462 00:36:45,100 --> 00:36:51,540 Make this the log of [A]0 and this is the log of [A] 463 00:36:51,540 --> 00:36:52,260 on the axis. 464 00:36:52,260 --> 00:36:56,020 That'll be time. 465 00:36:56,020 --> 00:37:07,210 So there's just some linear decay now. 466 00:37:07,210 --> 00:37:10,010 And the slope is minus k. 467 00:37:10,010 --> 00:37:12,590 So experimentally, of course, this'll be done typically as a 468 00:37:12,590 --> 00:37:18,450 simple way of determining it. 469 00:37:18,450 --> 00:37:21,850 Now, we also can usefully look at the half life, in this 470 00:37:21,850 --> 00:37:37,740 case, in the case of first order kinetics. 471 00:37:37,740 --> 00:37:40,130 So we have an expression in general for [A]. 472 00:37:40,130 --> 00:37:55,580 So if we let [A] equal [A]0 over two, at t equals t 1/2, 473 00:37:55,580 --> 00:38:08,830 that tells us that log of [A]0 over two divided by [A]0 is 474 00:38:08,830 --> 00:38:12,220 minus kt to the 1/2. 475 00:38:12,220 --> 00:38:29,020 But this is just the log of two, or log of two is over k 476 00:38:29,020 --> 00:38:35,960 is t to the 1/2. 477 00:38:35,960 --> 00:38:49,030 And so t to the 1/2 is just 0.693 over k. 478 00:38:49,030 --> 00:38:51,640 So we can write that just generally, completely 479 00:38:51,640 --> 00:38:56,020 independent of whatever [A] is, also independent of what 480 00:38:56,020 --> 00:39:00,200 [A]0 is, and that makes sense. 481 00:39:00,200 --> 00:39:02,490 So the point is, if you have something that just 482 00:39:02,490 --> 00:39:07,410 spontaneously decays into products, maybe it's just 483 00:39:07,410 --> 00:39:10,010 gradual chemical decomposition of something. 484 00:39:10,010 --> 00:39:12,860 Spontaneously, without the participation of other 485 00:39:12,860 --> 00:39:15,160 constituents. 486 00:39:15,160 --> 00:39:17,380 There's a general half-life that can be 487 00:39:17,380 --> 00:39:19,670 associated with that. 488 00:39:19,670 --> 00:39:21,370 That'll just be related directly to 489 00:39:21,370 --> 00:39:23,690 the rate of the decay. 490 00:39:23,690 --> 00:39:27,170 So it's knowable and measurable in a simple form. 491 00:39:27,170 --> 00:39:29,320 And again, the half-life is always useful. 492 00:39:29,320 --> 00:39:34,070 Because it just gives a simple, one-number measure of 493 00:39:34,070 --> 00:39:38,950 the rough timescale for things to change. 494 00:39:38,950 --> 00:39:50,940 So if we go back to this plot, and then once again here's our 495 00:39:50,940 --> 00:40:04,050 half concentration. 496 00:40:04,050 --> 00:40:13,620 And here's our t 1/2, just a useful way of summarizing in a 497 00:40:13,620 --> 00:40:14,970 simple way, what happens. 498 00:40:14,970 --> 00:40:15,160 Yeah? 499 00:40:15,160 --> 00:40:21,670 STUDENT: [INAUDIBLE] 500 00:40:21,670 --> 00:40:24,180 PROFESSOR: Ooh. 501 00:40:24,180 --> 00:40:27,440 Well, I didn't think about it. 502 00:40:27,440 --> 00:40:31,260 But it isn't necessary that that'll happen. 503 00:40:31,260 --> 00:40:32,420 Sure seems like it must be. 504 00:40:32,420 --> 00:40:35,370 When one is half decayed, the other must be half formed as 505 00:40:35,370 --> 00:40:37,470 long as there wasn't any B present at first. 506 00:40:37,470 --> 00:40:40,010 So yeah, thank you. 507 00:40:40,010 --> 00:40:45,800 Well, given that, something needs to move. 508 00:40:45,800 --> 00:40:49,480 But let's pretend like I got it right at this intersection. 509 00:40:49,480 --> 00:40:51,840 Even though it doesn't look that good. 510 00:40:51,840 --> 00:40:54,490 So really it should be there. 511 00:40:54,490 --> 00:40:58,950 Thank you. 512 00:40:58,950 --> 00:41:04,120 So the single most common example of first order 513 00:41:04,120 --> 00:41:07,950 kinetics of this form is radioactive decay. 514 00:41:07,950 --> 00:41:09,900 You've got some radioactive isotopic that can 515 00:41:09,900 --> 00:41:14,100 spontaneously decay into some other nucleus. 516 00:41:14,100 --> 00:41:16,590 And of course, this is measured and half-lives have 517 00:41:16,590 --> 00:41:19,610 been tabulated for lots of cases of this sort. 518 00:41:19,610 --> 00:41:48,110 So a simple example. 519 00:41:48,110 --> 00:41:52,550 Let's look at carbon-14. 520 00:41:52,550 --> 00:41:56,920 It's got a nuclear charge of six. 521 00:41:56,920 --> 00:42:04,460 It can decay into nitrogen-14 through 522 00:42:04,460 --> 00:42:11,860 the loss of an electron. 523 00:42:11,860 --> 00:42:16,170 Happens. 524 00:42:16,170 --> 00:42:17,720 First order kinetics. 525 00:42:17,720 --> 00:42:21,660 Now, in the atmosphere, what happens is this will end up, 526 00:42:21,660 --> 00:42:24,990 the 14C ends up getting replenished.. 527 00:42:24,990 --> 00:42:31,250 Because from cosmic rays, what can happen is in the 528 00:42:31,250 --> 00:42:39,730 atmosphere, you can have your nitrogen plus a neutron will 529 00:42:39,730 --> 00:42:48,600 come and form 14C plus a hydrogen atom. 530 00:42:48,600 --> 00:42:51,450 So in fact, the overall concentration in the 531 00:42:51,450 --> 00:42:55,000 atmosphere of 14C tends to be constant over time. 532 00:42:55,000 --> 00:43:04,140 But stuff that's formed down here on Earth, with carbon, 533 00:43:04,140 --> 00:43:09,090 its content of 14C decays over time, and it doesn't get 534 00:43:09,090 --> 00:43:12,480 replenished. 535 00:43:12,480 --> 00:43:20,670 So, let's think back some long time ago. 536 00:43:20,670 --> 00:43:27,640 Here's a tree. we can make it pretty. 537 00:43:27,640 --> 00:43:35,060 So it's got some concentration of 14C that came from carbon 538 00:43:35,060 --> 00:43:38,470 dioxide in the atmosphere. 539 00:43:38,470 --> 00:43:40,420 That's what it started with. 540 00:43:40,420 --> 00:43:43,660 That's our starting point. 541 00:43:43,660 --> 00:43:49,680 At some point later, through natural occurrence or human 542 00:43:49,680 --> 00:43:55,220 intervention, that tree became horizontal. 543 00:43:55,220 --> 00:43:59,910 Then, and although we're looking back here, let's call 544 00:43:59,910 --> 00:44:02,520 this our t equals zero. 545 00:44:02,520 --> 00:44:16,880 And our 14C concentration at the time is our 546 00:44:16,880 --> 00:44:21,060 concentration at zero. 547 00:44:21,060 --> 00:44:23,650 Now let's say, shortly after that, either right after 548 00:44:23,650 --> 00:44:26,130 because of deliberate action, or shortly after because it 549 00:44:26,130 --> 00:44:32,650 was just discovered, somebody decides to build something 550 00:44:32,650 --> 00:44:37,630 using that tree. 551 00:44:37,630 --> 00:44:48,200 So, early human craft. 552 00:44:48,200 --> 00:44:51,640 And then let's say lots later, depending on your definition 553 00:44:51,640 --> 00:45:03,000 of lots, modern human, which can be denoted by this style 554 00:45:03,000 --> 00:45:15,830 of hat, makes exciting discovery. 555 00:45:15,830 --> 00:45:18,140 Terrific. 556 00:45:18,140 --> 00:45:27,590 And would like to know how old is it. 557 00:45:27,590 --> 00:45:31,330 How long ago did all that stuff happen. 558 00:45:31,330 --> 00:45:39,070 Let's assume this is also approximately t equals zero. 559 00:45:39,070 --> 00:45:55,680 Well, so a log of [14C] 560 00:45:55,680 --> 00:45:59,080 over [14C]0. 561 00:45:59,080 --> 00:46:03,500 So this carbon dating, all it's really doing 562 00:46:03,500 --> 00:46:06,300 is measuring that. 563 00:46:06,300 --> 00:46:08,330 It gives a number for it. 564 00:46:08,330 --> 00:46:12,420 And we know this because we're assuming it hasn't changed any 565 00:46:12,420 --> 00:46:13,660 in all those years. 566 00:46:13,660 --> 00:46:14,930 In the atmosphere. 567 00:46:14,930 --> 00:46:18,330 It's different down here, because the tree or the boat 568 00:46:18,330 --> 00:46:19,810 wasn't replenished. 569 00:46:19,810 --> 00:46:23,870 Not nearly as many cosmic rays fell on it. 570 00:46:23,870 --> 00:46:31,320 So that's minus log of t. 571 00:46:31,320 --> 00:46:40,420 And it turns out the t 1/2 is 5,760 years. 572 00:46:40,420 --> 00:46:43,800 Amazing, that this can be known down to ten years. 573 00:46:43,800 --> 00:46:45,250 But it is. 574 00:46:45,250 --> 00:46:55,700 So that says, k his 0.693 divided by t to the 1/2, which 575 00:46:55,700 --> 00:47:04,160 is one over 8,312 years. 576 00:47:04,160 --> 00:47:12,470 So there's our answer, then. 577 00:47:12,470 --> 00:47:19,760 The time, how long ago that happened, is just minus 8,312 578 00:47:19,760 --> 00:47:27,710 years times the log of [14C] 579 00:47:27,710 --> 00:47:29,670 in the artifact. 580 00:47:29,670 --> 00:47:34,120 Divided by the log of, by [14C] 581 00:47:34,120 --> 00:47:36,090 in the air. 582 00:47:36,090 --> 00:47:38,600 And the assumption again is that this is the same now as 583 00:47:38,600 --> 00:47:41,230 it was back then. 584 00:47:41,230 --> 00:47:43,090 And there's a bigger number than this, so it's a positive 585 00:47:43,090 --> 00:47:43,980 number overall. 586 00:47:43,980 --> 00:47:47,870 So in a fairly straightforward way, we'll use first order 587 00:47:47,870 --> 00:47:51,480 kinetics to determine the lifetime of 588 00:47:51,480 --> 00:47:52,290 something like this. 589 00:47:52,290 --> 00:47:54,820 Because we know the rate constant. 590 00:47:54,820 --> 00:48:01,950 And everything else follows. 591 00:48:01,950 --> 00:48:02,370 Let's see. 592 00:48:02,370 --> 00:48:04,220 Next there's going to be second order kinetics. 593 00:48:04,220 --> 00:48:07,290 But let me just stop here and say just a word about the exam 594 00:48:07,290 --> 00:48:08,410 on Wednesday. 595 00:48:08,410 --> 00:48:13,630 So I've handed out an information sheet about it. 596 00:48:13,630 --> 00:48:17,640 But I don't have anything very different to say this time 597 00:48:17,640 --> 00:48:20,460 than I've had to say about previous exams. 598 00:48:20,460 --> 00:48:23,350 Solve lots of problems. 599 00:48:23,350 --> 00:48:25,250 Go over the homework. 600 00:48:25,250 --> 00:48:28,040 Go over practice problems. 601 00:48:28,040 --> 00:48:31,660 Try last year's exam as a sample exam when 602 00:48:31,660 --> 00:48:32,630 you're ready to do it. 603 00:48:32,630 --> 00:48:35,190 Try it under test conditions. 604 00:48:35,190 --> 00:48:37,770 There's nothing on the exam that you're going to look at 605 00:48:37,770 --> 00:48:41,940 and say oh my God, how was I supposed to know that we ought 606 00:48:41,940 --> 00:48:42,600 to study that. 607 00:48:42,600 --> 00:48:45,230 There won't be any surprises. 608 00:48:45,230 --> 00:48:49,910 Most of the exam questions so far, not all have been easy, 609 00:48:49,910 --> 00:48:52,680 but I think they've been more or less plain vanilla in the 610 00:48:52,680 --> 00:48:55,280 sense that they're right more or less down the middle of 611 00:48:55,280 --> 00:48:56,630 what we're trying to teach. 612 00:48:56,630 --> 00:49:01,940 And not very many are taking off little peripheral elements 613 00:49:01,940 --> 00:49:02,680 of the class. 614 00:49:02,680 --> 00:49:04,860 And it's not going to be any different here. 615 00:49:04,860 --> 00:49:07,650 So if you can just do the problem solving and get 616 00:49:07,650 --> 00:49:10,030 familiar enough with it that you're just good at it, so 617 00:49:10,030 --> 00:49:12,970 that you can do it with a reasonable speed, you're going 618 00:49:12,970 --> 00:49:16,890 to be fine. 619 00:49:16,890 --> 00:49:20,770 Also try to work on just understanding the 620 00:49:20,770 --> 00:49:21,520 underpinnings. 621 00:49:21,520 --> 00:49:24,580 Especially of statistical mechanics. 622 00:49:24,580 --> 00:49:27,280 That's where when you have these true-false or multiple 623 00:49:27,280 --> 00:49:30,160 choice questions, these sort of thought exercises. 624 00:49:30,160 --> 00:49:31,990 Those are really hard. 625 00:49:31,990 --> 00:49:35,460 Because they go a little bit beyond problem solving. 626 00:49:35,460 --> 00:49:36,840 If you could do the problem solving, 627 00:49:36,840 --> 00:49:38,540 you're going to do fine. 628 00:49:38,540 --> 00:49:41,990 But it's always useful if you can also formulate things. 629 00:49:41,990 --> 00:49:46,510 And for that, it's never easy to just tell you here's what 630 00:49:46,510 --> 00:49:49,860 you have to do, to get good at this. 631 00:49:49,860 --> 00:49:50,980 Because you have to think about it. 632 00:49:50,980 --> 00:49:53,750 And it's always hard. 633 00:49:53,750 --> 00:49:56,710 But to be sure, if you can just try when you review the 634 00:49:56,710 --> 00:50:00,200 statistical mechanics, especially, and you review the 635 00:50:00,200 --> 00:50:02,830 expressions that you use and how you solve problems with 636 00:50:02,830 --> 00:50:06,540 them, the next step in studying is to try to think, 637 00:50:06,540 --> 00:50:08,280 OK, do I really understand where that 638 00:50:08,280 --> 00:50:09,930 expression came from. 639 00:50:09,930 --> 00:50:12,480 And why it makes sense physically. 640 00:50:12,480 --> 00:50:14,610 And if you can do that, then so much the better. 641 00:50:14,610 --> 00:50:18,530 Then you'll be in an even stronger position. 642 00:50:18,530 --> 00:50:22,885 The info sheet gives a handful of equations that we do expect 643 00:50:22,885 --> 00:50:26,450 you to come in with those on your fingertips. 644 00:50:26,450 --> 00:50:27,690 There are not very many. 645 00:50:27,690 --> 00:50:30,370 Mostly we'll provide expressions that you'll need, 646 00:50:30,370 --> 00:50:33,110 and the important thing is that you know, understand, 647 00:50:33,110 --> 00:50:34,060 where they apply. 648 00:50:34,060 --> 00:50:36,170 And how to use them. 649 00:50:36,170 --> 00:50:38,530 So, good luck on Wednesday.