1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:02 The following content is provided under a Creative 3 00:00:02 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:10 offer high quality educational resources for free. 6 00:00:10 --> 00:00:13 To make a donation, or view additional materials from 7 00:00:13 --> 00:00:16 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16 --> 00:00:20 at ocw.mit.edu. 9 00:00:20 --> 00:00:26 PROFESSOR: So, now we'll start on the last of the main 10 00:00:26 --> 00:00:27 topics in the course. 11 00:00:27 --> 00:00:30 So we've finished most of our macroscopic thermodynamics, and 12 00:00:30 --> 00:00:34 our microscopic approach to it through statistical mechanics. 13 00:00:34 --> 00:00:37 And now, our final topic is kinetics. 14 00:00:37 --> 00:00:40 Kinetics is really a very different kind of topic. 15 00:00:40 --> 00:00:44 Because unlike thermodynamics, thermodynamics tells you all 16 00:00:44 --> 00:00:46 about equilibrium properties. 17 00:00:46 --> 00:00:49 A huge part of the work of this term has been to figure 18 00:00:49 --> 00:00:52 out what equilibrium is. 19 00:00:52 --> 00:00:56 What is the equilibrium state, given some situation. 20 00:00:56 --> 00:00:58 You've got some phases present. 21 00:00:58 --> 00:01:01 Different then, you could go from solid to liquid to gas. 22 00:01:01 --> 00:01:04 Or you've got different chemical constituents 23 00:01:04 --> 00:01:06 together that can react and go back and forth. 24 00:01:06 --> 00:01:09 What are the equilibrium concentrations? 25 00:01:09 --> 00:01:12 But we haven't worried at all about how long it 26 00:01:12 --> 00:01:15 might take to get there. 27 00:01:15 --> 00:01:17 And that's what kinetics does. 28 00:01:17 --> 00:01:21 Kinetics is concerned with rates of reactions, primarily. 29 00:01:21 --> 00:01:23 How long it takes to reach equilibrium. 30 00:01:23 --> 00:01:25 And of course it's super-important. 31 00:01:25 --> 00:01:29 Because if you look at that window glass, it's 32 00:01:29 --> 00:01:31 not in equilibrium. 33 00:01:31 --> 00:01:33 It's silicon dioxide, the equilibrium state 34 00:01:33 --> 00:01:34 would be a crystal. 35 00:01:34 --> 00:01:37 It would be crystal and quartz. 36 00:01:37 --> 00:01:40 Nevertheless, none of us is very worried that on the moment 37 00:01:40 --> 00:01:42 it's likely to suddenly, spontaneously turn into a 38 00:01:42 --> 00:01:45 crystal and be opaque and scatter light and do all 39 00:01:45 --> 00:01:47 that sort of stuff. 40 00:01:47 --> 00:01:50 And in lots of other situations, certainly if you 41 00:01:50 --> 00:01:56 look at any living biological system, including yourselves or 42 00:01:56 --> 00:02:00 your friends or any other one, it's certainly far 43 00:02:00 --> 00:02:01 from equilibrium. 44 00:02:01 --> 00:02:05 And you probably would hope for it to stay that way. 45 00:02:05 --> 00:02:09 So there's an enormous amount of chemistry and processes 46 00:02:09 --> 00:02:10 we're concerned with. 47 00:02:10 --> 00:02:13 Which depend intimately on kinetics in order to 48 00:02:13 --> 00:02:16 work the way they work. 49 00:02:16 --> 00:02:19 They also do depend on thermodynamics and where 50 00:02:19 --> 00:02:20 equilibrium states are. 51 00:02:20 --> 00:02:22 But that doesn't mean they necessarily reach 52 00:02:22 --> 00:02:25 equilibrium states. 53 00:02:25 --> 00:02:28 So we're going to go through kinetics, starting with the 54 00:02:28 --> 00:02:31 simplest examples and working our way up to more 55 00:02:31 --> 00:02:33 complex cases. 56 00:02:33 --> 00:02:37 And just see how we can describe elementary chemical 57 00:02:37 --> 00:02:42 reaction rates and processes. 58 00:02:42 --> 00:03:02 So that's our concern now, is dynamics. 59 00:03:02 --> 00:03:05 How long things take to get to equilibrium. 60 00:03:05 --> 00:03:11 And actually just like macroscopic thermodynamics, 61 00:03:11 --> 00:03:16 kinetics does take an empirical approach to the topic. 62 00:03:16 --> 00:03:19 In other words, it's based on experimental observation. 63 00:03:19 --> 00:03:20 Of macroscopic rates. 64 00:03:20 --> 00:03:23 How long it takes a collection of stuff, a mole of stuff 65 00:03:23 --> 00:03:25 to change chemically. 66 00:03:25 --> 00:03:27 And undergo reaction. 67 00:03:27 --> 00:03:29 And so forth. 68 00:03:29 --> 00:03:35 We often infer molecular mechanisms based on kinetics. 69 00:03:35 --> 00:03:39 And it's hugely important and valuable to do that. 70 00:03:39 --> 00:03:42 But it's also important to recognize that what kinetics 71 00:03:42 --> 00:03:47 can do is show us how we can formulate microscopic 72 00:03:47 --> 00:03:51 mechanisms that might be consistent with our 73 00:03:51 --> 00:03:53 macroscopic kinetics models. 74 00:03:53 --> 00:03:56 But the kinetics models by themselves don't 75 00:03:56 --> 00:03:58 prove the mechanism. 76 00:03:58 --> 00:04:02 And there are all sorts of examples where mechanisms that 77 00:04:02 --> 00:04:07 were proposed and accepted because they were consistent 78 00:04:07 --> 00:04:11 with macroscopic kinetics results turned out to fail. 79 00:04:11 --> 00:04:13 There are other ways to prove mechanisms. 80 00:04:13 --> 00:04:17 You might be able to design direct spectroscopic 81 00:04:17 --> 00:04:20 observations of intermediates and so forth. 82 00:04:20 --> 00:04:23 And in that case, it often becomes possible to distinguish 83 00:04:23 --> 00:04:25 between different mechanisms. 84 00:04:25 --> 00:04:31 Which might all satisfy the macroscopic kinetics equation. 85 00:04:31 --> 00:04:34 So we use kinetics to infer a mechanism but not 86 00:04:34 --> 00:04:36 necessarily to prove it. 87 00:04:36 --> 00:05:02 Not generally to prove it. 88 00:05:02 --> 00:05:05 We also use kinetics to describe an enormous 89 00:05:05 --> 00:05:07 range of time scales. 90 00:05:07 --> 00:05:13 So, the fastest things that we're sometimes concerned with 91 00:05:13 --> 00:05:16 that might take place on femtosecond time scales, 10 to 92 00:05:16 --> 00:05:19 the minus 13 seconds or so, or 10 to the minus 15 93 00:05:19 --> 00:05:21 seconds, even. 94 00:05:21 --> 00:05:26 So, if we look at the range of time scales we might be 95 00:05:26 --> 00:05:29 concerned with, might go anywhere from about 10 to the 96 00:05:29 --> 00:05:37 minus 15 seconds at the fastest, and might go to 97 00:05:37 --> 00:05:39 enormous scales on the other end. 98 00:05:39 --> 00:05:42 All the way out to maybe 10 to the 10 seconds, which is 99 00:05:42 --> 00:05:45 on the thousands of years. 100 00:05:45 --> 00:05:46 Could be longer than that, sometimes even 101 00:05:46 --> 00:05:51 millions of years. 102 00:05:51 --> 00:05:55 And the formalism that we'll set up applies equally to the 103 00:05:55 --> 00:05:58 full range of time scales. 104 00:05:58 --> 00:06:02 So it can describe an enormous amount of chemical activity. 105 00:06:02 --> 00:06:05 More commonly, for stuff that we're going to compare to, you 106 00:06:05 --> 00:06:11 might measure in the lab. 107 00:06:11 --> 00:06:16 Common time scales range from roughly 10 to the minus 6 108 00:06:16 --> 00:06:20 seconds out to about 10 to the 5th seconds. 109 00:06:20 --> 00:06:30 That is, microseconds to about a day. 110 00:06:30 --> 00:06:32 But there are plenty of examples of going either faster 111 00:06:32 --> 00:06:35 or slower, depending on the need and the experimental 112 00:06:35 --> 00:06:39 equipment available. 113 00:06:39 --> 00:06:41 Let's define a few terms. 114 00:06:41 --> 00:06:44 Let's talk about how we're going to just formulate 115 00:06:44 --> 00:06:46 chemical reaction rates. 116 00:06:46 --> 00:07:07 So, let's just imagine any simple reaction of this sort. 117 00:07:07 --> 00:07:09 So A plus B are going to go to C. 118 00:07:09 --> 00:07:12 There's some rate at which it happens. 119 00:07:12 --> 00:07:18 Note, by the way, it's an arrow in one direction. 120 00:07:18 --> 00:07:21 It's not an equals sign like we've seen before, 121 00:07:21 --> 00:07:24 or a double arrow. 122 00:07:24 --> 00:07:27 So the point I'm emphasizing here is when we talk about 123 00:07:27 --> 00:07:31 reaction rates, unlike equilibria, we're talking 124 00:07:31 --> 00:07:33 about a particular direction. 125 00:07:33 --> 00:07:37 Later on, we will talk about reversible reactions. 126 00:07:37 --> 00:07:41 But there, too, the arrow going this way refers only to one of 127 00:07:41 --> 00:07:43 the chemical reactions that can take place. 128 00:07:43 --> 00:07:48 Namely, in this case, the changing of what was the 129 00:07:48 --> 00:07:51 product now, would be the reactant, C, back into A plus 130 00:07:51 --> 00:07:54 B and it has nothing to do with the rate this way. 131 00:07:54 --> 00:07:59 Measured independently and so on. 132 00:07:59 --> 00:08:15 So the rate, we can look at the rate of disappearance of A. 133 00:08:15 --> 00:08:21 So it's just negative d[A]/dt, where the brackets 134 00:08:21 --> 00:08:23 indicate concentration. 135 00:08:23 --> 00:08:31 Usually in moles per liter. 136 00:08:31 --> 00:08:37 If it's in a gas, then it would be pA, of pressure. 137 00:08:37 --> 00:08:41 So negative d[A]/dt is our rate. 138 00:08:41 --> 00:08:45 And note that that's generally a positive number. 139 00:08:45 --> 00:08:48 We're looking at reactions going, if it's a reaction 140 00:08:48 --> 00:08:50 going in this direction. 141 00:08:50 --> 00:08:53 A is gradually disappearing. 142 00:08:53 --> 00:08:56 So we're going to define our rate this way. 143 00:08:56 --> 00:09:00 To be a positive number. 144 00:09:00 --> 00:09:06 The rate for C is going to be plus d[C]/dt. 145 00:09:06 --> 00:09:08 146 00:09:08 --> 00:09:11 It's also going to be defined in a way that 147 00:09:11 --> 00:09:12 makes it positive. 148 00:09:12 --> 00:09:14 Because in this case, since the reaction's going this way, 149 00:09:14 --> 00:09:19 we're looking at the appearance of C. 150 00:09:19 --> 00:09:24 Now, by stoichiometry , in this case, of course whenever a 151 00:09:24 --> 00:09:26 molecule or a mole of A disappears, a molecule 152 00:09:26 --> 00:09:29 or a mole of C appears. 153 00:09:29 --> 00:09:37 So because of that, in this case, the stoichiometry tells 154 00:09:37 --> 00:09:47 us that that d[C]/dt is equal to negative d[A]/dt. 155 00:09:47 --> 00:09:49 And also equal to negative d[B]/dt. 156 00:09:49 --> 00:09:55 157 00:09:55 --> 00:10:13 And any of those can be used to define the rate of reaction. 158 00:10:13 --> 00:10:16 Now, this is a particularly simple case because I've chosen 159 00:10:16 --> 00:10:18 the case where all of the stoichiometric coefficients 160 00:10:18 --> 00:10:20 are equal to one. 161 00:10:20 --> 00:10:25 So now let's just look at any case that's different. 162 00:10:25 --> 00:10:32 Let's look at 2A plus B goes to something else, 3C plus D. 163 00:10:32 --> 00:10:36 And just look at the reaction rates that we might see there. 164 00:10:36 --> 00:10:42 So here now, the appearance of C is going to be three times as 165 00:10:42 --> 00:10:45 fast as the appearance of D, for example. 166 00:10:45 --> 00:10:51 And also three times as fast as the disappearance of B. 167 00:10:51 --> 00:10:59 So if we write negative d[B]/dt, we expect that's going 168 00:10:59 --> 00:11:03 to be negative 1/2 d[A]/dt. 169 00:11:03 --> 00:11:05 170 00:11:05 --> 00:11:09 In other words, A is going to disappear twice as fast as B. 171 00:11:09 --> 00:11:15 Every time a molecule of B reacts, two molecules of A do. 172 00:11:15 --> 00:11:21 And that's going to be plus 1/3 d[C]/dt, and every time that 173 00:11:21 --> 00:11:26 happens three molecules of C get formed. 174 00:11:26 --> 00:11:32 And it's going to be plus d[D]/dt. 175 00:11:32 --> 00:11:34 176 00:11:34 --> 00:11:39 One molecule of D gets formed. 177 00:11:39 --> 00:11:42 And so, the reaction rate could be defined in 178 00:11:42 --> 00:11:43 terms of any of these. 179 00:11:43 --> 00:11:46 But the important thing is to keep track of stoichiometry so 180 00:11:46 --> 00:11:50 that the rate as it pertains to each constituent is 181 00:11:50 --> 00:11:56 accounted for correctly. 182 00:11:56 --> 00:12:07 So to generalize, if I have little a of A, and little b of 183 00:12:07 --> 00:12:22 B, going to little c of C, and little d of D, then the rate of 184 00:12:22 --> 00:12:32 reaction can be written as minus one over a d[A]/dt, or 185 00:12:32 --> 00:12:44 minus one over b d[B]/dt, or one over c d[C]/dt, or 186 00:12:44 --> 00:12:47 one over d d[D]/dt. 187 00:12:47 --> 00:13:17 188 00:13:17 --> 00:13:21 So, experimentally, lots of measurements of reaction 189 00:13:21 --> 00:13:22 rates have been made. 190 00:13:22 --> 00:13:26 And now to start on what's seen empirically, basically the 191 00:13:26 --> 00:13:36 following result is extremely common. 192 00:13:36 --> 00:13:41 The rate is equal to some constant times the 193 00:13:41 --> 00:13:46 concentration of A to some power alpha, times the 194 00:13:46 --> 00:13:58 concentration of B to some power beta, and so on. 195 00:13:58 --> 00:14:00 For all reactants. 196 00:14:00 --> 00:14:02 Multiplied together, each concentration taken 197 00:14:02 --> 00:14:04 to some power. 198 00:14:04 --> 00:14:06 Notice no products. 199 00:14:06 --> 00:14:11 Again, we're only looking at a reaction going one way. 200 00:14:11 --> 00:14:16 And if we look at the rate, this is typically what's found. 201 00:14:16 --> 00:14:32 Alpha is called the order of reaction, with respect to A. 202 00:14:32 --> 00:14:39 Beta order with respect to B. 203 00:14:39 --> 00:14:43 And so on. 204 00:14:43 --> 00:14:53 And little k is our rate constant which, just to make 205 00:14:53 --> 00:14:56 clear, since we've been using it extensively in the past few 206 00:14:56 --> 00:15:03 lectures, is completely not equal to the 207 00:15:03 --> 00:15:05 Boltzmann constant. 208 00:15:05 --> 00:15:12 Absolutely no connection between them. 209 00:15:12 --> 00:15:17 Now, alpha and beta, what they are, what they turn out to 210 00:15:17 --> 00:15:22 be, is typically what's determined as a result 211 00:15:22 --> 00:15:24 of kinetic measurement. 212 00:15:24 --> 00:15:42 Typically they're small integers. 213 00:15:42 --> 00:16:00 So, just sort of a typical example, if you look at the 214 00:16:00 --> 00:16:14 reaction of NO and O2, to make NO2, simple reaction, and you 215 00:16:14 --> 00:16:23 look at minus d[O2]/dt, use that as a measure of 216 00:16:23 --> 00:16:25 the reaction rate. 217 00:16:25 --> 00:16:32 What's found is that it's a constant times NO concentration 218 00:16:32 --> 00:16:37 squared, times O2. 219 00:16:37 --> 00:16:37 Simple, right? 220 00:16:37 --> 00:16:42 One of the exponents is two, the other's one. 221 00:16:42 --> 00:16:49 Doesn't seem so surprising mechanistically. 222 00:16:49 --> 00:16:51 But it's not always the case. 223 00:16:51 --> 00:16:54 Even when you have small integers, it's not always the 224 00:16:54 --> 00:16:58 case that the most obvious mechanism you would infer 225 00:16:58 --> 00:17:00 is the real mechanism. 226 00:17:00 --> 00:17:03 And sometimes those exponents don't turn out even 227 00:17:03 --> 00:17:05 to be integers. 228 00:17:05 --> 00:17:13 But here's another example. 229 00:17:13 --> 00:17:23 If you just look it CH3CHO going to methane and carbon 230 00:17:23 --> 00:17:28 monoxide, seems like it would be a pretty straightforward 231 00:17:28 --> 00:17:30 thing, too. 232 00:17:30 --> 00:17:34 So if you measure, for example, the rate of appearance of 233 00:17:34 --> 00:17:47 methane, what you discover is that it's equal to a constant 234 00:17:47 --> 00:17:54 times the concentration of the starting material 235 00:17:54 --> 00:17:59 to the 3/2 power. 236 00:17:59 --> 00:18:02 Not obvious mechanistically why that should be the case. 237 00:18:02 --> 00:18:05 Usually it's telling us something. 238 00:18:05 --> 00:18:08 It's telling us that the reaction mechanism 239 00:18:08 --> 00:18:10 is complicated. 240 00:18:10 --> 00:18:11 It's a multi-step process. 241 00:18:11 --> 00:18:16 Sometimes there could be chain reactions and so forth. 242 00:18:16 --> 00:18:19 So again, seeing things like this certainly helps us to 243 00:18:19 --> 00:18:23 infer molecular mechanisms. 244 00:18:23 --> 00:18:25 Again, they don't prove molecular mechanisms. 245 00:18:25 --> 00:18:30 But they certainly can be very helpful in suggesting them. 246 00:18:30 --> 00:18:34 And then other means can be used to try to prove them. 247 00:18:34 --> 00:18:36 Including, above all, direct observation of the 248 00:18:36 --> 00:18:40 intermediates that you would expect on the basis of one 249 00:18:40 --> 00:18:47 mechanism or another. 250 00:18:47 --> 00:18:51 Now let's just go through some elementary examples 251 00:18:51 --> 00:18:53 of kinetics one at a time. 252 00:18:53 --> 00:19:11 So let's start with the simplest possible case. 253 00:19:11 --> 00:19:14 Actually, a very rare case, but one that'll help us just set 254 00:19:14 --> 00:19:18 up the formalism of, and the mechanism for us to proceed. 255 00:19:18 --> 00:19:32 So let's talk about zero order reactions. 256 00:19:32 --> 00:19:35 Actually very rare. 257 00:19:35 --> 00:19:42 So, what this means is something like A goes 258 00:19:42 --> 00:19:45 over to products. 259 00:19:45 --> 00:19:57 Make a measurement of d[A]/dt, and discover that it's k time A 260 00:19:57 --> 00:20:00 to the power of zero, that is, it's just k. 261 00:20:00 --> 00:20:04 There's no dependence on the concentration of A. 262 00:20:04 --> 00:20:07 Or anything else. 263 00:20:07 --> 00:20:11 And you have some rate of its disappearance. 264 00:20:11 --> 00:20:15 So here's an example of it. 265 00:20:15 --> 00:20:31 You could have oxalic acid, and it just turns into hydrogen 266 00:20:31 --> 00:20:35 carbon dioxide and carbon dioxide. 267 00:20:35 --> 00:20:41 Things break apart, forms these products. 268 00:20:41 --> 00:20:44 Make a measurement of the disappearance. 269 00:20:44 --> 00:20:47 Seems to have nothing to do with the concentration of it. 270 00:20:47 --> 00:20:49 At least under certain conditions. 271 00:20:49 --> 00:20:57 Now, turns out that there's another element that 272 00:20:57 --> 00:21:02 helps understand this. 273 00:21:02 --> 00:21:06 Turns out that light is needed, it's a photochemical reaction. 274 00:21:06 --> 00:21:08 And then it's easy to see how this can happen. 275 00:21:08 --> 00:21:11 Let's say we have an abundance of the starting material, 276 00:21:11 --> 00:21:14 and not very much light. 277 00:21:14 --> 00:21:17 So every now and then photons bleed in, and every now and 278 00:21:17 --> 00:21:21 then when they're absorbed, it leads to dissociation. 279 00:21:21 --> 00:21:24 In that situation, you'll be limited by the photons, but 280 00:21:24 --> 00:21:27 not by the concentration of the molecules. 281 00:21:27 --> 00:21:32 And in fact, strictly speaking although this is zero order in 282 00:21:32 --> 00:21:36 terms of the chemical constituents, it's not zero 283 00:21:36 --> 00:21:39 order in the photons. 284 00:21:39 --> 00:21:42 So in some sense, in this sort of situation, the photons 285 00:21:42 --> 00:21:45 should be considered one of the reactants. 286 00:21:45 --> 00:21:49 You could write this as this plus h nu plus one photon, 287 00:21:49 --> 00:21:52 goes over to these products. 288 00:21:52 --> 00:21:56 And then if you measure the rate and things are under 289 00:21:56 --> 00:22:00 circumstances like I described, you would discover that in fact 290 00:22:00 --> 00:22:01 yes you'd be photon limited. 291 00:22:01 --> 00:22:03 The rate would depend on how many both photons 292 00:22:03 --> 00:22:06 are coming in. 293 00:22:06 --> 00:22:10 Still, ordinarily, chemical rate equations aren't 294 00:22:10 --> 00:22:12 formulated in those terms. 295 00:22:12 --> 00:22:15 So in the usual formulation, this would still have the 296 00:22:15 --> 00:22:24 appearance of a zero order reaction. 297 00:22:24 --> 00:22:30 Now, how do we write and formulate a solution? 298 00:22:30 --> 00:22:32 So it's simple. 299 00:22:32 --> 00:22:35 We've we've written a differential equation here. 300 00:22:35 --> 00:22:37 It's a pretty straightforward one. 301 00:22:37 --> 00:22:40 Minus d[A]/dt is just a constant. 302 00:22:40 --> 00:22:42 So we can solve it. 303 00:22:42 --> 00:22:45 And typically we'll solve it by rewriting in integral form and 304 00:22:45 --> 00:22:47 then doing the integration as long as we can do 305 00:22:47 --> 00:22:49 the integration. 306 00:22:49 --> 00:22:56 So from here we can write the integral from starting 307 00:22:56 --> 00:23:05 concentration [A]0 to some other concentration, [A], 308 00:23:05 --> 00:23:19 d[A] is equal to minus k integral from zero to t dt. 309 00:23:19 --> 00:23:24 So all we've done is integrate on both sides. 310 00:23:24 --> 00:23:34 And we've assumed a starting concentration. 311 00:23:34 --> 00:23:36 We've assumed an initial condition. 312 00:23:36 --> 00:23:39 And we've also assumed an initial time, which usually 313 00:23:39 --> 00:23:57 we can just call zero. 314 00:23:57 --> 00:23:59 And this is, of course, something we can solve 315 00:23:59 --> 00:24:00 for straight away. 316 00:24:00 --> 00:24:04 So we just have that [A] 317 00:24:04 --> 00:24:15 minus [A]0 is negative kt minus zero, which is minus kt. 318 00:24:15 --> 00:24:21 319 00:24:21 --> 00:24:22 So [A] 320 00:24:22 --> 00:24:27 is minus kt. 321 00:24:27 --> 00:24:29 Plus [A]0. 322 00:24:29 --> 00:24:31 In other words, the concentration of [A] 323 00:24:31 --> 00:24:34 at any time is given by the initial concentration, minus 324 00:24:34 --> 00:24:36 the rate constant times time. 325 00:24:36 --> 00:24:39 It decays linearly in time. 326 00:24:39 --> 00:24:51 So we can sketch that. 327 00:24:51 --> 00:24:54 This is our initial concentration. 328 00:24:54 --> 00:25:06 And then it's just going to decline with time after that. 329 00:25:06 --> 00:25:09 And there's our solution. 330 00:25:09 --> 00:25:13 In lots of cases, it's useful to define what's 331 00:25:13 --> 00:25:15 called a half-life. 332 00:25:15 --> 00:25:16 It's just useful. 333 00:25:16 --> 00:25:19 Because it provides some timeframe, a single number 334 00:25:19 --> 00:25:25 that's a timeframe on which the reaction occurs. 335 00:25:25 --> 00:25:28 So, the half-life is just the time that it takes for half of 336 00:25:28 --> 00:25:50 the reactants to disappear. t 1/2 time to react 337 00:25:50 --> 00:25:57 half the reactants. 338 00:25:57 --> 00:25:59 OK, so in this case it's straightforward to see 339 00:25:59 --> 00:26:00 when that happens. 340 00:26:00 --> 00:26:03 In other words, that's the time at which this concentration of 341 00:26:03 --> 00:26:14 A is just equal to [A]0 over two. 342 00:26:14 --> 00:26:26 So we have [A]0 over two is minus kt 1/2 plus [A]0 343 00:26:26 --> 00:26:33 or t 1/2 is equal to [A] 344 00:26:33 --> 00:26:35 over 2k. 345 00:26:35 --> 00:26:38 346 00:26:38 --> 00:26:44 [A]0 over 2k. 347 00:26:44 --> 00:26:48 So there's our half-life. 348 00:26:48 --> 00:27:02 So if we go over here, we can put that in. 349 00:27:02 --> 00:27:13 [A]0 over two, and this time is our half-life. 350 00:27:13 --> 00:27:14 So that's zero order reactions. 351 00:27:14 --> 00:27:17 And, again, zero order reactions are rare. 352 00:27:17 --> 00:27:21 But the procedure we're going to used to solve for kinetics 353 00:27:21 --> 00:27:23 is outlined in this way. 354 00:27:23 --> 00:27:34 And we'll use that again and again. 355 00:27:34 --> 00:27:38 Let's look at first-order kinetics. 356 00:27:38 --> 00:28:03 Let's go over here. 357 00:28:03 --> 00:28:06 Now, first order reactions are quite common. 358 00:28:06 --> 00:28:08 Much, much more common than zero order. 359 00:28:08 --> 00:28:17 So here, you have A goes to products. 360 00:28:17 --> 00:28:20 That's the simplest case. 361 00:28:20 --> 00:28:26 But this time if we measure d[A]/dt, we discover that it's 362 00:28:26 --> 00:28:34 equal to a constant times the concentration of A. 363 00:28:34 --> 00:28:37 There's an important point to note here. 364 00:28:37 --> 00:28:43 What are the units of k, in this case. 365 00:28:43 --> 00:28:48 What do they have to be? 366 00:28:48 --> 00:28:50 Yeah, reciprocal seconds right? 367 00:28:50 --> 00:28:52 The equation has to work. 368 00:28:52 --> 00:28:55 This is concentration per second. 369 00:28:55 --> 00:28:56 This is concentration. 370 00:28:56 --> 00:29:05 This better be per second. 371 00:29:05 --> 00:29:13 Let's just, before we move on completely, look at this. 372 00:29:13 --> 00:29:16 What are its units? 373 00:29:16 --> 00:29:19 Here's the equation. 374 00:29:19 --> 00:29:28 What are the units of k? 375 00:29:28 --> 00:29:31 No, not unitless, because, look at this. 376 00:29:31 --> 00:29:34 This is some concentration unit, typically 377 00:29:34 --> 00:29:35 moles per liter. 378 00:29:35 --> 00:29:37 So this is moles per liter per second, right? 379 00:29:37 --> 00:29:41 It's disappearance of some concentration for time. 380 00:29:41 --> 00:29:42 That's got to be here. 381 00:29:42 --> 00:29:45 All that is here. 382 00:29:45 --> 00:29:51 Moles per liter per second. 383 00:29:51 --> 00:29:55 So in every case, the units of k, the rate constant, have to 384 00:29:55 --> 00:29:59 be figured out on the basis of the specific rate equation. 385 00:29:59 --> 00:30:01 Doesn't have the same units. 386 00:30:01 --> 00:30:10 When the kinetics are of different order. 387 00:30:10 --> 00:30:12 Now, let's solve this using the same approach as before. 388 00:30:12 --> 00:30:15 Namely, this is still a pretty straightforward 389 00:30:15 --> 00:30:16 differential equation. 390 00:30:16 --> 00:30:19 So let's just integrate both sides. 391 00:30:19 --> 00:30:30 So that is going to tell us the integral from [A]0 to [A]. 392 00:30:30 --> 00:30:32 But now we have [A] 393 00:30:32 --> 00:30:35 on this side. 394 00:30:35 --> 00:30:40 So here we just had a constant. 395 00:30:40 --> 00:30:42 And effectively, I didn't write it out. 396 00:30:42 --> 00:30:49 But we effectively wrote this, rewrote this, as d[A] equals 397 00:30:49 --> 00:30:55 minus k dt, and then went from here to the integral. 398 00:30:55 --> 00:30:58 We're going to do the same thing here, except 399 00:30:58 --> 00:30:59 now there's this. 400 00:30:59 --> 00:31:04 So really we're going to have d[A] over [A] 401 00:31:04 --> 00:31:10 is minus k dt. 402 00:31:10 --> 00:31:11 That's what we're going to integrate on both side. 403 00:31:11 --> 00:31:18 We need to have the variables distinct on each side. 404 00:31:18 --> 00:31:24 So we have d[A] over [A]. 405 00:31:24 --> 00:31:31 Equal to minus k integral from zero to t dt. 406 00:31:31 --> 00:31:33 And so, of course, you know how to do this integral. 407 00:31:33 --> 00:31:35 It's going to look like log of [A]. 408 00:31:35 --> 00:31:37 And again, it's taken at [A] 409 00:31:37 --> 00:31:39 or, at [A]0. 410 00:31:39 --> 00:31:44 So we have log of [A] 411 00:31:44 --> 00:31:47 over [A]0. 412 00:31:47 --> 00:31:48 We're going to have log of [A] 413 00:31:48 --> 00:31:51 minus log of [A]0 coming out. 414 00:31:51 --> 00:31:57 And that's equal to minus kt. 415 00:31:57 --> 00:32:01 So now the kinetics are quite different. 416 00:32:01 --> 00:32:04 We have [A] 417 00:32:04 --> 00:32:15 is equal to [A]0, e to the minus kt. 418 00:32:15 --> 00:32:17 Very important, very common sort of result. 419 00:32:17 --> 00:32:20 It's saying we start with a certain amount of 420 00:32:20 --> 00:32:23 material, and there's an exponential decay of it. 421 00:32:23 --> 00:32:29 So very different from kinetics here, which are just linear. 422 00:32:29 --> 00:32:31 And again the kinetics here, if you imagine that situation 423 00:32:31 --> 00:32:35 where you've got starting material and bleeding in 424 00:32:35 --> 00:32:39 gradually are photons, presumably at the same rate, 425 00:32:39 --> 00:32:41 then sure that material you're going to see the disappearance 426 00:32:41 --> 00:32:44 of it linearly with time. 427 00:32:44 --> 00:32:46 Just depending on the rate at which the photons 428 00:32:46 --> 00:32:47 are coming in. 429 00:32:47 --> 00:32:50 Here it's very different because, presumably, A is 430 00:32:50 --> 00:32:53 required in order to do this reaction. 431 00:32:53 --> 00:32:54 It'll depend on how much. 432 00:32:54 --> 00:32:56 Because of course if there's more of it, you'll just have 433 00:32:56 --> 00:33:02 more at any given time decaying over to products. 434 00:33:02 --> 00:33:29 So you have an exponential decay. 435 00:33:29 --> 00:33:31 So let's plot that. 436 00:33:31 --> 00:33:44 Here's [A], let's make that [A]0. 437 00:33:44 --> 00:33:54 There it is. 438 00:33:54 --> 00:33:56 Now, let's look at what happens to the product. 439 00:33:56 --> 00:34:05 So let's imagine that it's A going to B, so of course minus 440 00:34:05 --> 00:34:14 d[A]/dt is just equal to d[B]/dt, and the rate of 441 00:34:14 --> 00:34:18 appearance of B has to match the rate of disappearance of A. 442 00:34:18 --> 00:34:25 Let's assume that we don't have any of B present at first. 443 00:34:25 --> 00:34:34 So let's make [B]0 equal to zero. 444 00:34:34 --> 00:34:37 Well then, [B] 445 00:34:37 --> 00:34:43 just has to equal [A]0 minus [A], right? 446 00:34:43 --> 00:34:47 All the stuff that's left, all of the A that has 447 00:34:47 --> 00:34:50 disappeared, that's given by this difference. 448 00:34:50 --> 00:34:54 Is just equal to B. 449 00:34:54 --> 00:35:00 So that's just [A]0 minus concentration of A, but 450 00:35:00 --> 00:35:03 that's just given by that. 451 00:35:03 --> 00:35:07 Which is [A]0 e to the minus kt. 452 00:35:07 --> 00:35:14 Or in other words, [B] 453 00:35:14 --> 00:35:24 is just equal to [A]0 times one minus e to the minus kt. 454 00:35:24 --> 00:35:27 So at t equals zero, this is zero and this is one. 455 00:35:27 --> 00:35:32 In other words, this is going to be zero at first. 456 00:35:32 --> 00:35:35 And then it's going to grow in with the same exponential 457 00:35:35 --> 00:35:36 form that this decayed. 458 00:35:36 --> 00:35:37 So, [B] 459 00:35:37 --> 00:35:41 is going to do the exact opposite. 460 00:35:41 --> 00:35:44 It's going to be like this. 461 00:35:44 --> 00:35:48 So this is [B] 462 00:35:48 --> 00:35:53 of t and this is [A] 463 00:35:53 --> 00:36:03 of t equals [A]0 e to the minus kt. 464 00:36:03 --> 00:36:07 Now, it's always useful to, whenever possible, to plot 465 00:36:07 --> 00:36:09 these things linearly. 466 00:36:09 --> 00:36:11 Find a way to plot these as straight lines. 467 00:36:11 --> 00:36:13 And of course in this case it's straightforward to 468 00:36:13 --> 00:36:21 do that as a log plot. 469 00:36:21 --> 00:36:27 So if we take the log of both sides, we know of course 470 00:36:27 --> 00:36:28 that the log of [A] 471 00:36:28 --> 00:36:42 is just minus kt, plus the log of [A]0, so now 472 00:36:42 --> 00:36:45 let's look at that. 473 00:36:45 --> 00:36:51 Make this the log of [A]0 and this is the log of [A] 474 00:36:51 --> 00:36:52 on the axis. 475 00:36:52 --> 00:36:56 That'll be time. 476 00:36:56 --> 00:37:07 So there's just some linear decay now. 477 00:37:07 --> 00:37:10 And the slope is minus k. 478 00:37:10 --> 00:37:12 So experimentally, of course, this'll be done typically as a 479 00:37:12 --> 00:37:18 simple way of determining it. 480 00:37:18 --> 00:37:22 Now, we also can usefully look at the half life, in this case, 481 00:37:22 --> 00:37:37 in the case of first order kinetics. 482 00:37:37 --> 00:37:40 So we have an expression in general for [A]. 483 00:37:40 --> 00:37:44 So if we let [A] 484 00:37:44 --> 00:37:56 equal [A]0 over two, at t equals t 1/2, that tells 485 00:37:56 --> 00:38:08 us that log of [A]0 over two divided by [A]0 is 486 00:38:08 --> 00:38:12 minus kt to the 1/2. 487 00:38:12 --> 00:38:27 But this is just the log of two, or log of two is 488 00:38:27 --> 00:38:35 over k is t to the 1/2. 489 00:38:35 --> 00:38:49 And so t to the 1/2 is just 0.693 over k. 490 00:38:49 --> 00:38:51 So we can write that just generally, completely 491 00:38:51 --> 00:38:53 independent of whatever [A] 492 00:38:53 --> 00:39:00 is, also independent of what [A]0 is, and that makes sense. 493 00:39:00 --> 00:39:02 So the point is, if you have something that just 494 00:39:02 --> 00:39:07 spontaneously decays into products, maybe it's 495 00:39:07 --> 00:39:10 just gradual chemical decomposition of something. 496 00:39:10 --> 00:39:12 Spontaneously, without the participation of 497 00:39:12 --> 00:39:15 other constituents. 498 00:39:15 --> 00:39:17 There's a general half-life that can be 499 00:39:17 --> 00:39:19 associated with that. 500 00:39:19 --> 00:39:21 That'll just be related directly to the 501 00:39:21 --> 00:39:23 rate of the decay. 502 00:39:23 --> 00:39:27 So it's knowable and measurable in a simple form. 503 00:39:27 --> 00:39:29 And again, the half-life is always useful. 504 00:39:29 --> 00:39:34 Because it just gives a simple, one-number measure of the rough 505 00:39:34 --> 00:39:38 timescale for things to change. 506 00:39:38 --> 00:39:48 So if we go back to this plot, and then once again here's 507 00:39:48 --> 00:40:04 our half concentration. 508 00:40:04 --> 00:40:13 And here's our t 1/2, just a useful way of summarizing in 509 00:40:13 --> 00:40:14 a simple way, what happens. 510 00:40:14 --> 00:40:15 Yeah? 511 00:40:15 --> 00:40:21 STUDENT: [INAUDIBLE] 512 00:40:21 --> 00:40:24 PROFESSOR: Ooh. 513 00:40:24 --> 00:40:27 Well, I didn't think about it. 514 00:40:27 --> 00:40:31 But it isn't necessary that that'll happen. 515 00:40:31 --> 00:40:32 Sure seems like it must be. 516 00:40:32 --> 00:40:35 When one is half decayed, the other must be half formed as 517 00:40:35 --> 00:40:37 long as there wasn't any B present at first. 518 00:40:37 --> 00:40:40 So yeah, thank you. 519 00:40:40 --> 00:40:45 Well, given that, something needs to move. 520 00:40:45 --> 00:40:49 But let's pretend like I got it right at this intersection. 521 00:40:49 --> 00:40:51 Even though it doesn't look that good. 522 00:40:51 --> 00:40:54 So really it should be there. 523 00:40:54 --> 00:40:58 Thank you. 524 00:40:58 --> 00:41:04 So the single most common example of first order 525 00:41:04 --> 00:41:07 kinetics of this form is radioactive decay. 526 00:41:07 --> 00:41:11 You've got some radioactive isotopic that can spontaneously 527 00:41:11 --> 00:41:14 decay into some other nucleus. 528 00:41:14 --> 00:41:16 And of course, this is measured and half-lives have been 529 00:41:16 --> 00:41:19 tabulated for lots of cases of this sort. 530 00:41:19 --> 00:41:48 So a simple example. 531 00:41:48 --> 00:41:52 Let's look at carbon-14. 532 00:41:52 --> 00:41:56 It's got a nuclear charge of six. 533 00:41:56 --> 00:42:05 It can decay into nitrogen-14 through the loss 534 00:42:05 --> 00:42:11 of an electron. 535 00:42:11 --> 00:42:16 Happens. 536 00:42:16 --> 00:42:17 First order kinetics. 537 00:42:17 --> 00:42:21 Now, in the atmosphere, what happens is this will 538 00:42:21 --> 00:42:24 end up, the 14C ends up getting replenished.. 539 00:42:24 --> 00:42:31 Because from cosmic rays, what can happen is in the 540 00:42:31 --> 00:42:39 atmosphere, you can have your nitrogen plus a neutron will 541 00:42:39 --> 00:42:48 come and form 14C plus a hydrogen atom. 542 00:42:48 --> 00:42:51 So in fact, the overall concentration in the 543 00:42:51 --> 00:42:55 atmosphere of 14C tends to be constant over time. 544 00:42:55 --> 00:43:04 But stuff that's formed down here on Earth, with carbon, its 545 00:43:04 --> 00:43:08 content of 14C decays over time, and it doesn't 546 00:43:08 --> 00:43:12 get replenished. 547 00:43:12 --> 00:43:20 So, let's think back some long time ago. 548 00:43:20 --> 00:43:27 Here's a tree. we can make it pretty. 549 00:43:27 --> 00:43:35 So it's got some concentration of 14C that came from carbon 550 00:43:35 --> 00:43:38 dioxide in the atmosphere. 551 00:43:38 --> 00:43:40 That's what it started with. 552 00:43:40 --> 00:43:43 That's our starting point. 553 00:43:43 --> 00:43:49 At some point later, through natural occurrence or human 554 00:43:49 --> 00:43:55 intervention, that tree became horizontal. 555 00:43:55 --> 00:43:59 Then, and although we're looking back here, let's call 556 00:43:59 --> 00:44:02 this our t equals zero. 557 00:44:02 --> 00:44:16 And our 14C concentration at the time is our 558 00:44:16 --> 00:44:21 concentration at zero. 559 00:44:21 --> 00:44:23 Now let's say, shortly after that, either right after 560 00:44:23 --> 00:44:26 because of deliberate action, or shortly after because it was 561 00:44:26 --> 00:44:32 just discovered, somebody decides to build something 562 00:44:32 --> 00:44:37 using that tree. 563 00:44:37 --> 00:44:48 So, early human craft. 564 00:44:48 --> 00:44:51 And then let's say lots later, depending on your definition of 565 00:44:51 --> 00:45:06 lots, modern human, which can be denoted by this style of 566 00:45:06 --> 00:45:15 hat, makes exciting discovery. 567 00:45:15 --> 00:45:18 Terrific. 568 00:45:18 --> 00:45:27 And would like to know how old is it. 569 00:45:27 --> 00:45:31 How long ago did all that stuff happen. 570 00:45:31 --> 00:45:39 Let's assume this is also approximately t equals zero. 571 00:45:39 --> 00:45:55 Well, so a log of [14C] 572 00:45:55 --> 00:45:59 over [14C]0. 573 00:45:59 --> 00:46:06 So this carbon dating, all it's really doing is measuring that. 574 00:46:06 --> 00:46:08 It gives a number for it. 575 00:46:08 --> 00:46:12 And we know this because we're assuming it hasn't changed 576 00:46:12 --> 00:46:13 any in all those years. 577 00:46:13 --> 00:46:14 In the atmosphere. 578 00:46:14 --> 00:46:18 It's different down here, because the tree or the 579 00:46:18 --> 00:46:19 boat wasn't replenished. 580 00:46:19 --> 00:46:23 Not nearly as many cosmic rays fell on it. 581 00:46:23 --> 00:46:31 So that's minus log of t. 582 00:46:31 --> 00:46:40 And it turns out the t 1/2 is 5,760 years. 583 00:46:40 --> 00:46:43 Amazing, that this can be known down to ten years. 584 00:46:43 --> 00:46:45 But it is. 585 00:46:45 --> 00:46:55 So that says, k his 0.693 divided by t to the 1/2, which 586 00:46:55 --> 00:47:04 is one over 8,312 years. 587 00:47:04 --> 00:47:12 So there's our answer, then. 588 00:47:12 --> 00:47:19 The time, how long ago that happened, is just minus 8 589 00:47:19 --> 00:47:27 years times the log of [14C] 590 00:47:27 --> 00:47:29 in the artifact. 591 00:47:29 --> 00:47:34 Divided by the log of, by [14C] 592 00:47:34 --> 00:47:36 in the air. 593 00:47:36 --> 00:47:38 And the assumption again is that this is the same 594 00:47:38 --> 00:47:41 now as it was back then. 595 00:47:41 --> 00:47:42 And there's a bigger number than this, so it's a 596 00:47:42 --> 00:47:43 positive number overall. 597 00:47:43 --> 00:47:47 So in a fairly straightforward way, we'll use first order 598 00:47:47 --> 00:47:51 kinetics to determine the lifetime of something 599 00:47:51 --> 00:47:52 like this. 600 00:47:52 --> 00:47:54 Because we know the rate constant. 601 00:47:54 --> 00:48:01 And everything else follows. 602 00:48:01 --> 00:48:02 Let's see. 603 00:48:02 --> 00:48:04 Next there's going to be second order kinetics. 604 00:48:04 --> 00:48:06 But let me just stop here and say just a word about 605 00:48:06 --> 00:48:08 the exam on Wednesday. 606 00:48:08 --> 00:48:13 So I've handed out an information sheet about it. 607 00:48:13 --> 00:48:17 But I don't have anything very different to say this time than 608 00:48:17 --> 00:48:20 I've had to say about previous exams. 609 00:48:20 --> 00:48:23 Solve lots of problems. 610 00:48:23 --> 00:48:25 Go over the homework. 611 00:48:25 --> 00:48:28 Go over practice problems. 612 00:48:28 --> 00:48:31 Try last year's exam as a sample exam when 613 00:48:31 --> 00:48:32 you're ready to do it. 614 00:48:32 --> 00:48:35 Try it under test conditions. 615 00:48:35 --> 00:48:37 There's nothing on the exam that you're going to look at 616 00:48:37 --> 00:48:41 and say oh my God, how was I supposed to know that we 617 00:48:41 --> 00:48:42 ought to study that. 618 00:48:42 --> 00:48:45 There won't be any surprises. 619 00:48:45 --> 00:48:49 Most of the exam questions so far, not all have been easy, 620 00:48:49 --> 00:48:52 but I think they've been more or less plain vanilla in the 621 00:48:52 --> 00:48:55 sense that they're right more or less down the middle of 622 00:48:55 --> 00:48:56 what we're trying to teach. 623 00:48:56 --> 00:49:01 And not very many are taking off little peripheral 624 00:49:01 --> 00:49:02 elements of the class. 625 00:49:02 --> 00:49:04 And it's not going to be any different here. 626 00:49:04 --> 00:49:07 So if you can just do the problem solving and get 627 00:49:07 --> 00:49:10 familiar enough with it that you're just good at it, so that 628 00:49:10 --> 00:49:16 you can do it with a reasonable speed, you're going to be fine. 629 00:49:16 --> 00:49:20 Also try to work on just understanding 630 00:49:20 --> 00:49:21 the underpinnings. 631 00:49:21 --> 00:49:24 Especially of statistical mechanics. 632 00:49:24 --> 00:49:27 That's where when you have these true-false or multiple 633 00:49:27 --> 00:49:30 choice questions, these sort of thought exercises. 634 00:49:30 --> 00:49:31 Those are really hard. 635 00:49:31 --> 00:49:35 Because they go a little bit beyond problem solving. 636 00:49:35 --> 00:49:36 If you could do the problem solving, you're 637 00:49:36 --> 00:49:38 going to do fine. 638 00:49:38 --> 00:49:41 But it's always useful if you can also formulate things. 639 00:49:41 --> 00:49:46 And for that, it's never easy to just tell you here's what 640 00:49:46 --> 00:49:49 you have to do, to get good at this. 641 00:49:49 --> 00:49:50 Because you have to think about it. 642 00:49:50 --> 00:49:53 And it's always hard. 643 00:49:53 --> 00:49:56 But to be sure, if you can just try when you review the 644 00:49:56 --> 00:50:00 statistical mechanics, especially, and you review the 645 00:50:00 --> 00:50:02 expressions that you use and how you solve problems with 646 00:50:02 --> 00:50:07 them, the next step in studying is to try to think, OK, do I 647 00:50:07 --> 00:50:09 really understand where that expression came from. 648 00:50:09 --> 00:50:12 And why it makes sense physically. 649 00:50:12 --> 00:50:14 And if you can do that, then so much the better. 650 00:50:14 --> 00:50:18 Then you'll be in an even stronger position. 651 00:50:18 --> 00:50:22 The info sheet gives a handful of equations that we do expect 652 00:50:22 --> 00:50:26 you to come in with those on your fingertips. 653 00:50:26 --> 00:50:27 There are not very many. 654 00:50:27 --> 00:50:30 Mostly we'll provide expressions that you'll need, 655 00:50:30 --> 00:50:33 and the important thing is that you know, understand, 656 00:50:33 --> 00:50:34 where they apply. 657 00:50:34 --> 00:50:36 And how to use them. 658 00:50:36 --> 00:50:38 So, good luck on Wednesday.