WEBVTT 00:00:00.030 --> 00:00:02.400 The following content is provided under a Creative 00:00:02.400 --> 00:00:03.820 Commons license. 00:00:03.820 --> 00:00:06.860 Your support will help MIT OpenCourseWare continue to 00:00:06.860 --> 00:00:10.510 offer high quality educational resources for free. 00:00:10.510 --> 00:00:13.390 To make a donation, or view additional materials from 00:00:13.390 --> 00:00:17.420 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:17.420 --> 00:00:20.760 ocw.mit.edu. 00:00:20.760 --> 00:00:26.920 PROFESSOR: So, now we'll start on the last of the main topics 00:00:26.920 --> 00:00:27.490 in the course. 00:00:27.490 --> 00:00:30.300 So we've finished most of our macroscopic thermodynamics, 00:00:30.300 --> 00:00:32.850 and our microscopic approach to it 00:00:32.850 --> 00:00:34.290 through statistical mechanics. 00:00:34.290 --> 00:00:37.280 And now, our final topic is kinetics. 00:00:37.280 --> 00:00:40.920 Kinetics is really a very different kind of topic. 00:00:40.920 --> 00:00:44.840 Because unlike thermodynamics, thermodynamics tells you all 00:00:44.840 --> 00:00:46.900 about equilibrium properties. 00:00:46.900 --> 00:00:49.980 A huge part of the work of this term has been to figure 00:00:49.980 --> 00:00:52.570 out what equilibrium is. 00:00:52.570 --> 00:00:56.620 What is the equilibrium state, given some situation. 00:00:56.620 --> 00:00:58.290 You've got some phases present. 00:00:58.290 --> 00:01:01.200 Different then, you could go from solid to liquid to gas. 00:01:01.200 --> 00:01:04.570 Or you've got different chemical constituents together 00:01:04.570 --> 00:01:06.360 that can react and go back and forth. 00:01:06.360 --> 00:01:09.670 What are the equilibrium concentrations? 00:01:09.670 --> 00:01:13.230 But we haven't worried at all about how long it might take 00:01:13.230 --> 00:01:15.550 to get there. 00:01:15.550 --> 00:01:17.110 And that's what kinetics does. 00:01:17.110 --> 00:01:21.980 Kinetics is concerned with rates of reactions, primarily. 00:01:21.980 --> 00:01:23.920 How long it takes to reach equilibrium. 00:01:23.920 --> 00:01:25.260 And of course it's super-important. 00:01:25.260 --> 00:01:29.440 Because if you look at that window glass, it's not in 00:01:29.440 --> 00:01:31.360 equilibrium. 00:01:31.360 --> 00:01:33.200 It's silicon dioxide, the equilibrium 00:01:33.200 --> 00:01:34.550 state would be a crystal. 00:01:34.550 --> 00:01:37.210 It would be crystal and quartz. 00:01:37.210 --> 00:01:39.980 Nevertheless, none of us is very worried that on the 00:01:39.980 --> 00:01:42.570 moment it's likely to suddenly, spontaneously turn 00:01:42.570 --> 00:01:45.310 into a crystal and be opaque and scatter light and do all 00:01:45.310 --> 00:01:47.610 that sort of stuff. 00:01:47.610 --> 00:01:50.670 And in lots of other situations, certainly if you 00:01:50.670 --> 00:01:56.600 look at any living biological system, including yourselves 00:01:56.600 --> 00:02:00.210 or your friends or any other one, it's certainly far from 00:02:00.210 --> 00:02:01.140 equilibrium. 00:02:01.140 --> 00:02:05.070 And you probably would hope for it to stay that way. 00:02:05.070 --> 00:02:09.830 So there's an enormous amount of chemistry and processes 00:02:09.830 --> 00:02:10.770 we're concerned with. 00:02:10.770 --> 00:02:14.310 Which depend intimately on kinetics in order to work the 00:02:14.310 --> 00:02:16.210 way they work. 00:02:16.210 --> 00:02:19.160 They also do depend on thermodynamics and where 00:02:19.160 --> 00:02:20.660 equilibrium states are. 00:02:20.660 --> 00:02:22.450 But that doesn't mean they necessarily 00:02:22.450 --> 00:02:25.790 reach equilibrium states. 00:02:25.790 --> 00:02:28.610 So we're going to go through kinetics, starting with the 00:02:28.610 --> 00:02:31.070 simplest examples and working our way up to 00:02:31.070 --> 00:02:33.260 more complex cases. 00:02:33.260 --> 00:02:37.330 And just see how we can describe elementary chemical 00:02:37.330 --> 00:02:42.130 reaction rates and processes. 00:02:42.130 --> 00:03:02.370 So that's our concern now, is dynamics. 00:03:02.370 --> 00:03:05.950 How long things take to get to equilibrium. 00:03:05.950 --> 00:03:11.340 And actually just like macroscopic thermodynamics, 00:03:11.340 --> 00:03:13.790 kinetics does take an empirical 00:03:13.790 --> 00:03:16.390 approach to the topic. 00:03:16.390 --> 00:03:19.480 In other words, it's based on experimental observation. 00:03:19.480 --> 00:03:20.850 Of macroscopic rates. 00:03:20.850 --> 00:03:23.860 How long it takes a collection of stuff, a mole of stuff to 00:03:23.860 --> 00:03:25.130 change chemically. 00:03:25.130 --> 00:03:27.330 And undergo reaction. 00:03:27.330 --> 00:03:29.560 And so forth. 00:03:29.560 --> 00:03:35.730 We often infer molecular mechanisms based on kinetics. 00:03:35.730 --> 00:03:39.400 And it's hugely important and valuable to do that. 00:03:39.400 --> 00:03:42.560 But it's also important to recognize that what kinetics 00:03:42.560 --> 00:03:47.890 can do is show us how we can formulate microscopic 00:03:47.890 --> 00:03:51.310 mechanisms that might be consistent with our 00:03:51.310 --> 00:03:53.810 macroscopic kinetics models. 00:03:53.810 --> 00:03:55.690 But the kinetics models by themselves 00:03:55.690 --> 00:03:58.830 don't prove the mechanism. 00:03:58.830 --> 00:04:02.650 And there are all sorts of examples where mechanisms that 00:04:02.650 --> 00:04:07.540 were proposed and accepted because they were consistent 00:04:07.540 --> 00:04:11.030 with macroscopic kinetics results turned out to fail. 00:04:11.030 --> 00:04:13.790 There are other ways to prove mechanisms. 00:04:13.790 --> 00:04:17.060 You might be able to design direct spectroscopic 00:04:17.060 --> 00:04:20.640 observations of intermediates and so forth. 00:04:20.640 --> 00:04:22.930 And in that case, it often becomes possible to 00:04:22.930 --> 00:04:25.480 distinguish between different mechanisms. 00:04:25.480 --> 00:04:31.740 Which might all satisfy the macroscopic kinetics equation. 00:04:31.740 --> 00:04:35.600 So we use kinetics to infer a mechanism but not necessarily 00:04:35.600 --> 00:04:36.100 to prove it. 00:04:36.100 --> 00:05:02.320 Not generally to prove it. 00:05:02.320 --> 00:05:06.430 We also use kinetics to describe an enormous range of 00:05:06.430 --> 00:05:07.590 time scales. 00:05:07.590 --> 00:05:13.530 So, the fastest things that we're sometimes concerned with 00:05:13.530 --> 00:05:16.250 that might take place on femtosecond time scales, 10 to 00:05:16.250 --> 00:05:18.840 the minus 13 seconds or so, or 10 to the 00:05:18.840 --> 00:05:21.770 minus 15 seconds, even. 00:05:21.770 --> 00:05:26.600 So, if we look at the range of time scales we might be 00:05:26.600 --> 00:05:29.290 concerned with, might go anywhere from about 10 to the 00:05:29.290 --> 00:05:37.580 minus 15 seconds at the fastest, and might go to 00:05:37.580 --> 00:05:39.700 enormous scales on the other end. 00:05:39.700 --> 00:05:43.080 All the way out to maybe 10 to the 10 seconds, which is on 00:05:43.080 --> 00:05:45.090 the thousands of years. 00:05:45.090 --> 00:05:46.660 Could be longer than that, sometimes 00:05:46.660 --> 00:05:51.570 even millions of years. 00:05:51.570 --> 00:05:55.520 And the formalism that we'll set up applies equally to the 00:05:55.520 --> 00:05:58.160 full range of time scales. 00:05:58.160 --> 00:06:02.350 So it can describe an enormous amount of chemical activity. 00:06:02.350 --> 00:06:05.740 More commonly, for stuff that we're going to compare to, you 00:06:05.740 --> 00:06:11.100 might measure in the lab. 00:06:11.100 --> 00:06:16.120 Common time scales range from roughly 10 to the minus 6 00:06:16.120 --> 00:06:20.250 seconds out to about 10 to the 5th seconds. 00:06:20.250 --> 00:06:30.160 That is, microseconds to about a day. 00:06:30.160 --> 00:06:32.240 But there are plenty of examples of going either 00:06:32.240 --> 00:06:35.190 faster or slower, depending on the need and the experimental 00:06:35.190 --> 00:06:39.320 equipment available. 00:06:39.320 --> 00:06:41.080 Let's define a few terms. 00:06:41.080 --> 00:06:44.150 Let's talk about how we're going to just formulate 00:06:44.150 --> 00:06:46.130 chemical reaction rates. 00:06:46.130 --> 00:07:07.150 So, let's just imagine any simple reaction of this sort. 00:07:07.150 --> 00:07:11.030 So A plus B are going to go to C. There's some rate at which 00:07:11.030 --> 00:07:12.360 it happens. 00:07:12.360 --> 00:07:18.960 Note, by the way, it's an arrow in one direction. 00:07:18.960 --> 00:07:20.970 It's not an equals sign like we've seen 00:07:20.970 --> 00:07:24.250 before, or a double arrow. 00:07:24.250 --> 00:07:27.990 So the point I'm emphasizing here is when we talk about 00:07:27.990 --> 00:07:31.600 reaction rates, unlike equilibria, we're talking 00:07:31.600 --> 00:07:33.820 about a particular direction. 00:07:33.820 --> 00:07:37.520 Later on, we will talk about reversible reactions. 00:07:37.520 --> 00:07:41.140 But there, too, the arrow going this way refers only to 00:07:41.140 --> 00:07:43.920 one of the chemical reactions that can take place. 00:07:43.920 --> 00:07:48.240 Namely, in this case, the changing of what was the 00:07:48.240 --> 00:07:51.025 product now, would be the reactant, C, back into A plus 00:07:51.025 --> 00:07:54.490 B and it has nothing to do with the rate this way. 00:07:54.490 --> 00:07:59.840 Measured independently and so on. 00:07:59.840 --> 00:08:09.790 So the rate, we can look at the rate of disappearance of 00:08:09.790 --> 00:08:21.600 A. So it's just negative d[A]/dt, where the brackets 00:08:21.600 --> 00:08:23.390 indicate concentration. 00:08:23.390 --> 00:08:31.960 Usually in moles per liter. 00:08:31.960 --> 00:08:37.540 If it's in a gas, then it would be pA, of pressure. 00:08:37.540 --> 00:08:41.260 So negative d[A]/dt is our rate. 00:08:41.260 --> 00:08:45.960 And note that that's generally a positive number. 00:08:45.960 --> 00:08:48.590 We're looking at reactions going, if it's a reaction 00:08:48.590 --> 00:08:50.430 going in this direction. 00:08:50.430 --> 00:08:53.790 A is gradually disappearing. 00:08:53.790 --> 00:08:56.690 So we're going to define our rate this way. 00:08:56.690 --> 00:09:00.480 To be a positive number. 00:09:00.480 --> 00:09:06.360 The rate for C is going to be plus d[C]/dt. 00:09:08.900 --> 00:09:11.640 It's also going to be defined in a way 00:09:11.640 --> 00:09:12.810 that makes it positive. 00:09:12.810 --> 00:09:14.920 Because in this case, since the reaction's going this way, 00:09:14.920 --> 00:09:19.510 we're looking at the appearance of C. 00:09:19.510 --> 00:09:24.460 Now, by stoichiometry , in this case, of course whenever 00:09:24.460 --> 00:09:27.150 a molecule or a mole of A disappears, a molecule or a 00:09:27.150 --> 00:09:29.500 mole of C appears. 00:09:29.500 --> 00:09:37.940 So because of that, in this case, the stoichiometry tells 00:09:37.940 --> 00:09:47.480 us that that d[C]/dt is equal to negative d[A]/dt. 00:09:47.480 --> 00:09:49.800 And also equal to negative d[B]/dt. 00:09:55.060 --> 00:10:01.180 And any of those can be used to 00:10:01.180 --> 00:10:13.520 define the rate of reaction. 00:10:13.520 --> 00:10:15.670 Now, this is a particularly simple case because I've 00:10:15.670 --> 00:10:17.510 chosen the case where all of the stoichiometric 00:10:17.510 --> 00:10:20.010 coefficients are equal to one. 00:10:20.010 --> 00:10:25.140 So now let's just look at any case that's different. 00:10:25.140 --> 00:10:32.760 Let's look at 2A plus B goes to something else, 3C plus D. 00:10:32.760 --> 00:10:36.820 And just look at the reaction rates that we might see there. 00:10:36.820 --> 00:10:42.120 So here now, the appearance of C is going to be three times 00:10:42.120 --> 00:10:45.170 as fast as the appearance of D, for example. 00:10:45.170 --> 00:10:51.650 And also three times as fast as the disappearance of B. So 00:10:51.650 --> 00:11:00.360 if we write negative d[B]/dt, we expect that's going to be 00:11:00.360 --> 00:11:03.000 negative 1/2 d[A]/dt. 00:11:05.720 --> 00:11:09.020 In other words, A is going to disappear twice as fast as B. 00:11:09.020 --> 00:11:15.110 Every time a molecule of B reacts, two molecules of A do. 00:11:15.110 --> 00:11:21.640 And that's going to be plus 1/3 d[C]/dt, and every time 00:11:21.640 --> 00:11:26.260 that happens three molecules of C get formed. 00:11:26.260 --> 00:11:32.180 And it's going to be plus d[D]/dt. 00:11:34.900 --> 00:11:39.100 One molecule of D gets formed. 00:11:39.100 --> 00:11:42.970 And so, the reaction rate could be defined in terms of 00:11:42.970 --> 00:11:43.600 any of these. 00:11:43.600 --> 00:11:46.470 But the important thing is to keep track of stoichiometry so 00:11:46.470 --> 00:11:50.080 that the rate as it pertains to each constituent is 00:11:50.080 --> 00:11:56.690 accounted for correctly. 00:11:56.690 --> 00:12:07.550 So to generalize, if I have little a of A, and little b of 00:12:07.550 --> 00:12:22.130 B, going to little c of C, and little d of D, then the rate 00:12:22.130 --> 00:12:32.870 of reaction can be written as minus one over a d[A]/dt, or 00:12:32.870 --> 00:12:45.230 minus one over b d[B]/dt, or one over c d[C]/dt, or one 00:12:45.230 --> 00:12:47.040 over d d[D]/dt. 00:13:17.860 --> 00:13:21.560 So, experimentally, lots of measurements of reaction rates 00:13:21.560 --> 00:13:22.420 have been made. 00:13:22.420 --> 00:13:26.190 And now to start on what's seen empirically, basically 00:13:26.190 --> 00:13:36.660 the following result is extremely common. 00:13:36.660 --> 00:13:41.740 The rate is equal to some constant times the 00:13:41.740 --> 00:13:46.570 concentration of A to some power alpha, times the 00:13:46.570 --> 00:13:58.170 concentration of B to some power beta, and so on. 00:13:58.170 --> 00:14:00.000 For all reactants. 00:14:00.000 --> 00:14:02.430 Multiplied together, each concentration 00:14:02.430 --> 00:14:04.720 taken to some power. 00:14:04.720 --> 00:14:06.700 Notice no products. 00:14:06.700 --> 00:14:11.490 Again, we're only looking at a reaction going one way. 00:14:11.490 --> 00:14:13.760 And if we look at the rate, this is 00:14:13.760 --> 00:14:16.680 typically what's found. 00:14:16.680 --> 00:14:32.830 Alpha is called the order of reaction, with respect to A. 00:14:32.830 --> 00:14:43.320 Beta order with respect to B. And so on. 00:14:43.320 --> 00:14:53.470 And little k is our rate constant which, just to make 00:14:53.470 --> 00:14:56.940 clear, since we've been using it extensively in the past few 00:14:56.940 --> 00:15:03.320 lectures, is completely not equal to 00:15:03.320 --> 00:15:05.340 the Boltzmann constant. 00:15:05.340 --> 00:15:12.420 Absolutely no connection between them. 00:15:12.420 --> 00:15:18.150 Now, alpha and beta, what they are, what they turn out to be, 00:15:18.150 --> 00:15:23.150 is typically what's determined as a result of kinetic 00:15:23.150 --> 00:15:24.950 measurement. 00:15:24.950 --> 00:15:42.730 Typically they're small integers. 00:15:42.730 --> 00:16:00.690 So, just sort of a typical example, if you look at the 00:16:00.690 --> 00:16:14.870 reaction of NO and O2, to make NO2, simple reaction, and you 00:16:14.870 --> 00:16:23.360 look at minus d[O2]/dt, use that as a measure of the 00:16:23.360 --> 00:16:25.530 reaction rate. 00:16:25.530 --> 00:16:31.390 What's found is that it's a constant times NO 00:16:31.390 --> 00:16:37.110 concentration squared, times O2. 00:16:37.110 --> 00:16:37.940 Simple, right? 00:16:37.940 --> 00:16:42.430 One of the exponents is two, the other's one. 00:16:42.430 --> 00:16:49.260 Doesn't seem so surprising mechanistically. 00:16:49.260 --> 00:16:51.720 But it's not always the case. 00:16:51.720 --> 00:16:54.750 Even when you have small integers, it's not always the 00:16:54.750 --> 00:16:58.560 case that the most obvious mechanism you would infer is 00:16:58.560 --> 00:17:00.330 the real mechanism. 00:17:00.330 --> 00:17:03.280 And sometimes those exponents don't turn 00:17:03.280 --> 00:17:05.810 out even to be integers. 00:17:05.810 --> 00:17:13.220 But here's another example. 00:17:13.220 --> 00:17:23.630 If you just look it CH3CHO going to methane and carbon 00:17:23.630 --> 00:17:28.480 monoxide, seems like it would be a pretty straightforward 00:17:28.480 --> 00:17:30.000 thing, too. 00:17:30.000 --> 00:17:34.120 So if you measure, for example, the rate of 00:17:34.120 --> 00:17:45.140 appearance of methane, what you discover is that it's 00:17:45.140 --> 00:17:50.500 equal to a constant times the concentration of the starting 00:17:50.500 --> 00:17:59.240 material to the 3/2 power. 00:17:59.240 --> 00:18:02.490 Not obvious mechanistically why that should be the case. 00:18:02.490 --> 00:18:05.640 Usually it's telling us something. 00:18:05.640 --> 00:18:09.140 It's telling us that the reaction mechanism is 00:18:09.140 --> 00:18:10.200 complicated. 00:18:10.200 --> 00:18:11.970 It's a multi-step process. 00:18:11.970 --> 00:18:16.170 Sometimes there could be chain reactions and so forth. 00:18:16.170 --> 00:18:19.400 So again, seeing things like this certainly helps us to 00:18:19.400 --> 00:18:23.000 infer molecular mechanisms. 00:18:23.000 --> 00:18:25.820 Again, they don't prove molecular mechanisms. 00:18:25.820 --> 00:18:30.250 But they certainly can be very helpful in suggesting them. 00:18:30.250 --> 00:18:34.390 And then other means can be used to try to prove them. 00:18:34.390 --> 00:18:36.890 Including, above all, direct observation of the 00:18:36.890 --> 00:18:40.400 intermediates that you would expect on the basis of one 00:18:40.400 --> 00:18:47.840 mechanism or another. 00:18:47.840 --> 00:18:51.190 Now let's just go through some elementary examples of 00:18:51.190 --> 00:18:53.510 kinetics one at a time. 00:18:53.510 --> 00:19:11.900 So let's start with the simplest possible case. 00:19:11.900 --> 00:19:14.390 Actually, a very rare case, but one that'll help us just 00:19:14.390 --> 00:19:16.410 set up the formalism of, and the 00:19:16.410 --> 00:19:18.500 mechanism for us to proceed. 00:19:18.500 --> 00:19:32.680 So let's talk about zero order reactions. 00:19:32.680 --> 00:19:35.890 Actually very rare. 00:19:35.890 --> 00:19:42.690 So, what this means is something like A 00:19:42.690 --> 00:19:45.070 goes over to products. 00:19:45.070 --> 00:19:57.715 Make a measurement of d[A]/dt, and discover that it's k time 00:19:57.715 --> 00:20:00.710 A to the power of zero, that is, it's just k. 00:20:00.710 --> 00:20:04.630 There's no dependence on the concentration of A. Or 00:20:04.630 --> 00:20:07.050 anything else. 00:20:07.050 --> 00:20:11.300 And you have some rate of its disappearance. 00:20:11.300 --> 00:20:15.880 So here's an example of it. 00:20:15.880 --> 00:20:29.850 You could have oxalic acid, and it just turns into 00:20:29.850 --> 00:20:35.350 hydrogen carbon dioxide and carbon dioxide. 00:20:35.350 --> 00:20:41.760 Things break apart, forms these products. 00:20:41.760 --> 00:20:44.150 Make a measurement of the disappearance. 00:20:44.150 --> 00:20:47.960 Seems to have nothing to do with the concentration of it. 00:20:47.960 --> 00:20:49.730 At least under certain conditions. 00:20:49.730 --> 00:20:57.580 Now, turns out that there's another element that helps 00:20:57.580 --> 00:21:02.610 understand this. 00:21:02.610 --> 00:21:04.575 Turns out that light is needed, it's a 00:21:04.575 --> 00:21:06.770 photochemical reaction. 00:21:06.770 --> 00:21:08.920 And then it's easy to see how this can happen. 00:21:08.920 --> 00:21:11.940 Let's say we have an abundance of the starting material, and 00:21:11.940 --> 00:21:14.540 not very much light. 00:21:14.540 --> 00:21:17.940 So every now and then photons bleed in, and every now and 00:21:17.940 --> 00:21:21.200 then when they're absorbed, it leads to dissociation. 00:21:21.200 --> 00:21:24.660 In that situation, you'll be limited by the photons, but 00:21:24.660 --> 00:21:27.240 not by the concentration of the molecules. 00:21:27.240 --> 00:21:32.040 And in fact, strictly speaking although this is zero order in 00:21:32.040 --> 00:21:36.300 terms of the chemical constituents, it's not zero 00:21:36.300 --> 00:21:39.390 order in the photons. 00:21:39.390 --> 00:21:42.260 So in some sense, in this sort of situation, the photons 00:21:42.260 --> 00:21:45.160 should be considered one of the reactants. 00:21:45.160 --> 00:21:49.860 You could write this as this plus h nu plus one photon, 00:21:49.860 --> 00:21:52.150 goes over to these products. 00:21:52.150 --> 00:21:56.510 And then if you measure the rate and things are under 00:21:56.510 --> 00:21:59.770 circumstances like I described, you would discover 00:21:59.770 --> 00:22:01.680 that in fact yes you'd be photon limited. 00:22:01.680 --> 00:22:03.520 The rate would depend on how many both 00:22:03.520 --> 00:22:06.220 photons are coming in. 00:22:06.220 --> 00:22:10.390 Still, ordinarily, chemical rate equations aren't 00:22:10.390 --> 00:22:12.420 formulated in those terms. 00:22:12.420 --> 00:22:15.190 So in the usual formulation, this would still have the 00:22:15.190 --> 00:22:24.660 appearance of a zero order reaction. 00:22:24.660 --> 00:22:30.880 Now, how do we write and formulate a solution? 00:22:30.880 --> 00:22:32.950 So it's simple. 00:22:32.950 --> 00:22:35.980 We've we've written a differential equation here. 00:22:35.980 --> 00:22:37.690 It's a pretty straightforward one. 00:22:37.690 --> 00:22:40.750 Minus d[A]/dt is just a constant. 00:22:40.750 --> 00:22:42.000 So we can solve it. 00:22:42.000 --> 00:22:44.920 And typically we'll solve it by rewriting in integral form 00:22:44.920 --> 00:22:47.570 and then doing the integration as long as we can do the 00:22:47.570 --> 00:22:49.550 integration. 00:22:49.550 --> 00:22:56.090 So from here we can write the integral from starting 00:22:56.090 --> 00:23:07.440 concentration [A]0 to some other concentration, [A], d[A] 00:23:07.440 --> 00:23:19.310 is equal to minus k integral from zero to t dt. 00:23:19.310 --> 00:23:24.230 So all we've done is integrate on both sides. 00:23:24.230 --> 00:23:34.790 And we've assumed a starting concentration. 00:23:34.790 --> 00:23:36.770 We've assumed an initial condition. 00:23:36.770 --> 00:23:39.130 And we've also assumed an initial time, which usually we 00:23:39.130 --> 00:23:57.450 can just call zero. 00:23:57.450 --> 00:23:59.390 And this is, of course, something we can solve for 00:23:59.390 --> 00:24:00.560 straight away. 00:24:00.560 --> 00:24:12.840 So we just have that [A] minus [A]0 is negative kt minus 00:24:12.840 --> 00:24:15.030 zero, which is minus kt. 00:24:21.540 --> 00:24:27.560 So [A] is minus kt. 00:24:27.560 --> 00:24:29.430 Plus [A]0. 00:24:29.430 --> 00:24:31.870 In other words, the concentration of [A] at any 00:24:31.870 --> 00:24:35.150 time is given by the initial concentration, minus the rate 00:24:35.150 --> 00:24:36.100 constant times time. 00:24:36.100 --> 00:24:39.900 It decays linearly in time. 00:24:39.900 --> 00:24:51.810 So we can sketch that. 00:24:51.810 --> 00:24:54.020 This is our initial concentration. 00:24:54.020 --> 00:25:06.890 And then it's just going to decline with time after that. 00:25:06.890 --> 00:25:09.540 And there's our solution. 00:25:09.540 --> 00:25:13.430 In lots of cases, it's useful to define 00:25:13.430 --> 00:25:15.510 what's called a half-life. 00:25:15.510 --> 00:25:16.560 It's just useful. 00:25:16.560 --> 00:25:19.810 Because it provides some timeframe, a single number 00:25:19.810 --> 00:25:25.250 that's a timeframe on which the reaction occurs. 00:25:25.250 --> 00:25:28.480 So, the half-life is just the time that it takes for half of 00:25:28.480 --> 00:25:47.300 the reactants to disappear. t 1/2 time to 00:25:47.300 --> 00:25:57.390 react half the reactants. 00:25:57.390 --> 00:25:59.700 OK, so in this case it's straightforward to see when 00:25:59.700 --> 00:26:00.230 that happens. 00:26:00.230 --> 00:26:02.180 In other words, that's the time at which this 00:26:02.180 --> 00:26:14.580 concentration of A is just equal to [A]0 over two. 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 00:26:28.750 --> 00:26:35.460 1/2 is equal to [A] over 2k. 00:26:38.100 --> 00:26:44.020 [A]0 over 2k. 00:26:44.020 --> 00:26:48.650 So there's our half-life. 00:26:48.650 --> 00:27:02.050 So if we go over here, we can put that in. 00:27:02.050 --> 00:27:13.280 [A]0 over two, and this time is our half-life. 00:27:13.280 --> 00:27:14.820 So that's zero order reactions. 00:27:14.820 --> 00:27:17.310 And, again, zero order reactions are rare. 00:27:17.310 --> 00:27:21.890 But the procedure we're going to used to solve for kinetics 00:27:21.890 --> 00:27:23.150 is outlined in this way. 00:27:23.150 --> 00:27:34.420 And we'll use that again and again. 00:27:34.420 --> 00:27:38.080 Let's look at first-order kinetics. 00:27:38.080 --> 00:28:03.550 Let's go over here. 00:28:03.550 --> 00:28:06.360 Now, first order reactions are quite common. 00:28:06.360 --> 00:28:08.830 Much, much more common than zero order. 00:28:08.830 --> 00:28:17.550 So here, you have A goes to products. 00:28:17.550 --> 00:28:20.070 That's the simplest case. 00:28:20.070 --> 00:28:26.900 But this time if we measure d[A]/dt, we discover that it's 00:28:26.900 --> 00:28:34.350 equal to a constant times the concentration of A. There's an 00:28:34.350 --> 00:28:37.170 important point to note here. 00:28:37.170 --> 00:28:43.170 What are the units of k, in this case. 00:28:43.170 --> 00:28:48.780 What do they have to be? 00:28:48.780 --> 00:28:50.230 Yeah, reciprocal seconds right? 00:28:50.230 --> 00:28:52.660 The equation has to work. 00:28:52.660 --> 00:28:55.090 This is concentration per second. 00:28:55.090 --> 00:28:56.530 This is concentration. 00:28:56.530 --> 00:29:05.980 This better be per second. 00:29:05.980 --> 00:29:13.250 Let's just, before we move on completely, look at this. 00:29:13.250 --> 00:29:16.460 What are its units? 00:29:16.460 --> 00:29:19.470 Here's the equation. 00:29:19.470 --> 00:29:28.490 What are the units of k? 00:29:28.490 --> 00:29:31.680 No, not unitless, because, look at this. 00:29:31.680 --> 00:29:33.730 This is some concentration unit, 00:29:33.730 --> 00:29:35.500 typically moles per liter. 00:29:35.500 --> 00:29:37.260 So this is moles per liter per second, right? 00:29:37.260 --> 00:29:41.270 It's disappearance of some concentration for time. 00:29:41.270 --> 00:29:42.250 That's got to be here. 00:29:42.250 --> 00:29:45.490 All that is here. 00:29:45.490 --> 00:29:51.410 Moles per liter per second. 00:29:51.410 --> 00:29:55.480 So in every case, the units of k, the rate constant, have to 00:29:55.480 --> 00:29:59.020 be figured out on the basis of the specific rate equation. 00:29:59.020 --> 00:30:01.410 Doesn't have the same units. 00:30:01.410 --> 00:30:10.010 When the kinetics are of different order. 00:30:10.010 --> 00:30:12.610 Now, let's solve this using the same approach as before. 00:30:12.610 --> 00:30:15.000 Namely, this is still a pretty straightforward 00:30:15.000 --> 00:30:16.230 differential equation. 00:30:16.230 --> 00:30:19.140 So let's just integrate both sides. 00:30:19.140 --> 00:30:30.250 So that is going to tell us the integral from [A]0 to [A]. 00:30:30.250 --> 00:30:35.340 But now we have [A] on this side. 00:30:35.340 --> 00:30:40.130 So here we just had a constant. 00:30:40.130 --> 00:30:42.300 And effectively, I didn't write it out. 00:30:42.300 --> 00:30:49.110 But we effectively wrote this, rewrote this, as d[A] 00:30:49.110 --> 00:30:52.480 equals minus k dt, and then went 00:30:52.480 --> 00:30:55.860 from here to the integral. 00:30:55.860 --> 00:30:58.460 We're going to do the same thing here, 00:30:58.460 --> 00:30:59.420 except now there's this. 00:30:59.420 --> 00:31:03.490 So really we're going to have d[A] 00:31:03.490 --> 00:31:10.250 over [A] is minus k dt. 00:31:10.250 --> 00:31:11.990 That's what we're going to integrate on both side. 00:31:11.990 --> 00:31:18.880 We need to have the variables distinct on each side. 00:31:18.880 --> 00:31:20.170 So we have d[A] 00:31:22.850 --> 00:31:24.660 over [A]. 00:31:24.660 --> 00:31:31.510 Equal to minus k integral from zero to t dt. 00:31:31.510 --> 00:31:33.590 And so, of course, you know how to do this integral. 00:31:33.590 --> 00:31:35.560 It's going to look like log of [A]. 00:31:35.560 --> 00:31:39.460 And again, it's taken at [A] or, at [A]0. 00:31:39.460 --> 00:31:47.360 So we have log of [A] over [A]0. 00:31:47.360 --> 00:31:51.620 We're going to have log of [A] minus log of [A]0 coming out. 00:31:51.620 --> 00:31:57.250 And that's equal to minus kt. 00:31:57.250 --> 00:32:01.470 So now the kinetics are quite different. 00:32:01.470 --> 00:32:15.200 We have [A] is equal to [A]0, e to the minus kt. 00:32:15.200 --> 00:32:17.620 Very important, very common sort of result. 00:32:17.620 --> 00:32:20.600 It's saying we start with a certain amount of material, 00:32:20.600 --> 00:32:23.640 and there's an exponential decay of it. 00:32:23.640 --> 00:32:26.110 So very different from kinetics here, 00:32:26.110 --> 00:32:29.210 which are just linear. 00:32:29.210 --> 00:32:31.730 And again the kinetics here, if you imagine that situation 00:32:31.730 --> 00:32:35.420 where you've got starting material and bleeding in 00:32:35.420 --> 00:32:39.010 gradually are photons, presumably at the same rate, 00:32:39.010 --> 00:32:41.720 then sure that material you're going to see the disappearance 00:32:41.720 --> 00:32:44.590 of it linearly with time. 00:32:44.590 --> 00:32:46.790 Just depending on the rate at which the 00:32:46.790 --> 00:32:47.980 photons are coming in. 00:32:47.980 --> 00:32:50.860 Here it's very different because, presumably, A is 00:32:50.860 --> 00:32:53.250 required in order to do this reaction. 00:32:53.250 --> 00:32:54.110 It'll depend on how much. 00:32:54.110 --> 00:32:56.780 Because of course if there's more of it, you'll just have 00:32:56.780 --> 00:33:02.640 more at any given time decaying over to products. 00:33:02.640 --> 00:33:29.250 So you have an exponential decay. 00:33:29.250 --> 00:33:31.330 So let's plot that. 00:33:31.330 --> 00:33:44.570 Here's [A], let's make that [A]0. 00:33:44.570 --> 00:33:54.030 There it is. 00:33:54.030 --> 00:33:56.250 Now, let's look at what happens to the product. 00:33:56.250 --> 00:34:05.270 So let's imagine that it's A going to B, so of course minus 00:34:05.270 --> 00:34:14.150 d[A]/dt is just equal to d[B]/dt, and the rate of 00:34:14.150 --> 00:34:17.625 appearance of B has to match the rate of disappearance of 00:34:17.625 --> 00:34:20.780 A. Let's assume that we don't have any of 00:34:20.780 --> 00:34:25.190 B present at first. 00:34:25.190 --> 00:34:34.700 So let's make [B]0 equal to zero. 00:34:34.700 --> 00:34:43.950 Well then, [B] just has to equal [A]0 minus [A], right? 00:34:43.950 --> 00:34:48.260 All the stuff that's left, all of the A that has disappeared, 00:34:48.260 --> 00:34:50.560 that's given by this difference. 00:34:50.560 --> 00:35:00.400 Is just equal to B. So that's just [A]0 minus concentration 00:35:00.400 --> 00:35:03.480 of A, but that's just given by that. 00:35:03.480 --> 00:35:07.690 Which is [A]0 e to the minus kt. 00:35:07.690 --> 00:35:20.530 Or in other words, [B] is just equal to [A]0 times one minus 00:35:20.530 --> 00:35:24.460 e to the minus kt. 00:35:24.460 --> 00:35:27.840 So at t equals zero, this is zero and this is one. 00:35:27.840 --> 00:35:32.750 In other words, this is going to be zero at first. 00:35:32.750 --> 00:35:35.550 And then it's going to grow in with the same exponential form 00:35:35.550 --> 00:35:36.760 that this decayed. 00:35:36.760 --> 00:35:41.160 So, [B] is going to do the exact opposite. 00:35:41.160 --> 00:35:44.890 It's going to be like this. 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 00:35:57.550 --> 00:36:03.270 the minus kt. 00:36:03.270 --> 00:36:07.760 Now, it's always useful to, whenever possible, to plot 00:36:07.760 --> 00:36:09.510 these things linearly. 00:36:09.510 --> 00:36:11.060 Find a way to plot these as straight lines. 00:36:11.060 --> 00:36:13.690 And of course in this case it's straightforward to do 00:36:13.690 --> 00:36:21.470 that as a log plot. 00:36:21.470 --> 00:36:27.760 So if we take the log of both sides, we know of course that 00:36:27.760 --> 00:36:42.740 the log of [A] is just minus kt, plus the log of [A]0, so 00:36:42.740 --> 00:36:45.100 now let's look at that. 00:36:45.100 --> 00:36:51.540 Make this the log of [A]0 and this is the log of [A] 00:36:51.540 --> 00:36:52.260 on the axis. 00:36:52.260 --> 00:36:56.020 That'll be time. 00:36:56.020 --> 00:37:07.210 So there's just some linear decay now. 00:37:07.210 --> 00:37:10.010 And the slope is minus k. 00:37:10.010 --> 00:37:12.590 So experimentally, of course, this'll be done typically as a 00:37:12.590 --> 00:37:18.450 simple way of determining it. 00:37:18.450 --> 00:37:21.850 Now, we also can usefully look at the half life, in this 00:37:21.850 --> 00:37:37.740 case, in the case of first order kinetics. 00:37:37.740 --> 00:37:40.130 So we have an expression in general for [A]. 00:37:40.130 --> 00:37:55.580 So if we let [A] equal [A]0 over two, at t equals t 1/2, 00:37:55.580 --> 00:38:08.830 that tells us that log of [A]0 over two divided by [A]0 is 00:38:08.830 --> 00:38:12.220 minus kt to the 1/2. 00:38:12.220 --> 00:38:29.020 But this is just the log of two, or log of two is over k 00:38:29.020 --> 00:38:35.960 is t to the 1/2. 00:38:35.960 --> 00:38:49.030 And so t to the 1/2 is just 0.693 over k. 00:38:49.030 --> 00:38:51.640 So we can write that just generally, completely 00:38:51.640 --> 00:38:56.020 independent of whatever [A] is, also independent of what 00:38:56.020 --> 00:39:00.200 [A]0 is, and that makes sense. 00:39:00.200 --> 00:39:02.490 So the point is, if you have something that just 00:39:02.490 --> 00:39:07.410 spontaneously decays into products, maybe it's just 00:39:07.410 --> 00:39:10.010 gradual chemical decomposition of something. 00:39:10.010 --> 00:39:12.860 Spontaneously, without the participation of other 00:39:12.860 --> 00:39:15.160 constituents. 00:39:15.160 --> 00:39:17.380 There's a general half-life that can be 00:39:17.380 --> 00:39:19.670 associated with that. 00:39:19.670 --> 00:39:21.370 That'll just be related directly to 00:39:21.370 --> 00:39:23.690 the rate of the decay. 00:39:23.690 --> 00:39:27.170 So it's knowable and measurable in a simple form. 00:39:27.170 --> 00:39:29.320 And again, the half-life is always useful. 00:39:29.320 --> 00:39:34.070 Because it just gives a simple, one-number measure of 00:39:34.070 --> 00:39:38.950 the rough timescale for things to change. 00:39:38.950 --> 00:39:50.940 So if we go back to this plot, and then once again here's our 00:39:50.940 --> 00:40:04.050 half concentration. 00:40:04.050 --> 00:40:13.620 And here's our t 1/2, just a useful way of summarizing in a 00:40:13.620 --> 00:40:14.970 simple way, what happens. 00:40:14.970 --> 00:40:15.160 Yeah? 00:40:15.160 --> 00:40:21.670 STUDENT: [INAUDIBLE] 00:40:21.670 --> 00:40:24.180 PROFESSOR: Ooh. 00:40:24.180 --> 00:40:27.440 Well, I didn't think about it. 00:40:27.440 --> 00:40:31.260 But it isn't necessary that that'll happen. 00:40:31.260 --> 00:40:32.420 Sure seems like it must be. 00:40:32.420 --> 00:40:35.370 When one is half decayed, the other must be half formed as 00:40:35.370 --> 00:40:37.470 long as there wasn't any B present at first. 00:40:37.470 --> 00:40:40.010 So yeah, thank you. 00:40:40.010 --> 00:40:45.800 Well, given that, something needs to move. 00:40:45.800 --> 00:40:49.480 But let's pretend like I got it right at this intersection. 00:40:49.480 --> 00:40:51.840 Even though it doesn't look that good. 00:40:51.840 --> 00:40:54.490 So really it should be there. 00:40:54.490 --> 00:40:58.950 Thank you. 00:40:58.950 --> 00:41:04.120 So the single most common example of first order 00:41:04.120 --> 00:41:07.950 kinetics of this form is radioactive decay. 00:41:07.950 --> 00:41:09.900 You've got some radioactive isotopic that can 00:41:09.900 --> 00:41:14.100 spontaneously decay into some other nucleus. 00:41:14.100 --> 00:41:16.590 And of course, this is measured and half-lives have 00:41:16.590 --> 00:41:19.610 been tabulated for lots of cases of this sort. 00:41:19.610 --> 00:41:48.110 So a simple example. 00:41:48.110 --> 00:41:52.550 Let's look at carbon-14. 00:41:52.550 --> 00:41:56.920 It's got a nuclear charge of six. 00:41:56.920 --> 00:42:04.460 It can decay into nitrogen-14 through 00:42:04.460 --> 00:42:11.860 the loss of an electron. 00:42:11.860 --> 00:42:16.170 Happens. 00:42:16.170 --> 00:42:17.720 First order kinetics. 00:42:17.720 --> 00:42:21.660 Now, in the atmosphere, what happens is this will end up, 00:42:21.660 --> 00:42:24.990 the 14C ends up getting replenished.. 00:42:24.990 --> 00:42:31.250 Because from cosmic rays, what can happen is in the 00:42:31.250 --> 00:42:39.730 atmosphere, you can have your nitrogen plus a neutron will 00:42:39.730 --> 00:42:48.600 come and form 14C plus a hydrogen atom. 00:42:48.600 --> 00:42:51.450 So in fact, the overall concentration in the 00:42:51.450 --> 00:42:55.000 atmosphere of 14C tends to be constant over time. 00:42:55.000 --> 00:43:04.140 But stuff that's formed down here on Earth, with carbon, 00:43:04.140 --> 00:43:09.090 its content of 14C decays over time, and it doesn't get 00:43:09.090 --> 00:43:12.480 replenished. 00:43:12.480 --> 00:43:20.670 So, let's think back some long time ago. 00:43:20.670 --> 00:43:27.640 Here's a tree. we can make it pretty. 00:43:27.640 --> 00:43:35.060 So it's got some concentration of 14C that came from carbon 00:43:35.060 --> 00:43:38.470 dioxide in the atmosphere. 00:43:38.470 --> 00:43:40.420 That's what it started with. 00:43:40.420 --> 00:43:43.660 That's our starting point. 00:43:43.660 --> 00:43:49.680 At some point later, through natural occurrence or human 00:43:49.680 --> 00:43:55.220 intervention, that tree became horizontal. 00:43:55.220 --> 00:43:59.910 Then, and although we're looking back here, let's call 00:43:59.910 --> 00:44:02.520 this our t equals zero. 00:44:02.520 --> 00:44:16.880 And our 14C concentration at the time is our 00:44:16.880 --> 00:44:21.060 concentration at zero. 00:44:21.060 --> 00:44:23.650 Now let's say, shortly after that, either right after 00:44:23.650 --> 00:44:26.130 because of deliberate action, or shortly after because it 00:44:26.130 --> 00:44:32.650 was just discovered, somebody decides to build something 00:44:32.650 --> 00:44:37.630 using that tree. 00:44:37.630 --> 00:44:48.200 So, early human craft. 00:44:48.200 --> 00:44:51.640 And then let's say lots later, depending on your definition 00:44:51.640 --> 00:45:03.000 of lots, modern human, which can be denoted by this style 00:45:03.000 --> 00:45:15.830 of hat, makes exciting discovery. 00:45:15.830 --> 00:45:18.140 Terrific. 00:45:18.140 --> 00:45:27.590 And would like to know how old is it. 00:45:27.590 --> 00:45:31.330 How long ago did all that stuff happen. 00:45:31.330 --> 00:45:39.070 Let's assume this is also approximately t equals zero. 00:45:39.070 --> 00:45:55.680 Well, so a log of [14C] 00:45:55.680 --> 00:45:59.080 over [14C]0. 00:45:59.080 --> 00:46:03.500 So this carbon dating, all it's really doing 00:46:03.500 --> 00:46:06.300 is measuring that. 00:46:06.300 --> 00:46:08.330 It gives a number for it. 00:46:08.330 --> 00:46:12.420 And we know this because we're assuming it hasn't changed any 00:46:12.420 --> 00:46:13.660 in all those years. 00:46:13.660 --> 00:46:14.930 In the atmosphere. 00:46:14.930 --> 00:46:18.330 It's different down here, because the tree or the boat 00:46:18.330 --> 00:46:19.810 wasn't replenished. 00:46:19.810 --> 00:46:23.870 Not nearly as many cosmic rays fell on it. 00:46:23.870 --> 00:46:31.320 So that's minus log of t. 00:46:31.320 --> 00:46:40.420 And it turns out the t 1/2 is 5,760 years. 00:46:40.420 --> 00:46:43.800 Amazing, that this can be known down to ten years. 00:46:43.800 --> 00:46:45.250 But it is. 00:46:45.250 --> 00:46:55.700 So that says, k his 0.693 divided by t to the 1/2, which 00:46:55.700 --> 00:47:04.160 is one over 8,312 years. 00:47:04.160 --> 00:47:12.470 So there's our answer, then. 00:47:12.470 --> 00:47:19.760 The time, how long ago that happened, is just minus 8,312 00:47:19.760 --> 00:47:27.710 years times the log of [14C] 00:47:27.710 --> 00:47:29.670 in the artifact. 00:47:29.670 --> 00:47:34.120 Divided by the log of, by [14C] 00:47:34.120 --> 00:47:36.090 in the air. 00:47:36.090 --> 00:47:38.600 And the assumption again is that this is the same now as 00:47:38.600 --> 00:47:41.230 it was back then. 00:47:41.230 --> 00:47:43.090 And there's a bigger number than this, so it's a positive 00:47:43.090 --> 00:47:43.980 number overall. 00:47:43.980 --> 00:47:47.870 So in a fairly straightforward way, we'll use first order 00:47:47.870 --> 00:47:51.480 kinetics to determine the lifetime of 00:47:51.480 --> 00:47:52.290 something like this. 00:47:52.290 --> 00:47:54.820 Because we know the rate constant. 00:47:54.820 --> 00:48:01.950 And everything else follows. 00:48:01.950 --> 00:48:02.370 Let's see. 00:48:02.370 --> 00:48:04.220 Next there's going to be second order kinetics. 00:48:04.220 --> 00:48:07.290 But let me just stop here and say just a word about the exam 00:48:07.290 --> 00:48:08.410 on Wednesday. 00:48:08.410 --> 00:48:13.630 So I've handed out an information sheet about it. 00:48:13.630 --> 00:48:17.640 But I don't have anything very different to say this time 00:48:17.640 --> 00:48:20.460 than I've had to say about previous exams. 00:48:20.460 --> 00:48:23.350 Solve lots of problems. 00:48:23.350 --> 00:48:25.250 Go over the homework. 00:48:25.250 --> 00:48:28.040 Go over practice problems. 00:48:28.040 --> 00:48:31.660 Try last year's exam as a sample exam when 00:48:31.660 --> 00:48:32.630 you're ready to do it. 00:48:32.630 --> 00:48:35.190 Try it under test conditions. 00:48:35.190 --> 00:48:37.770 There's nothing on the exam that you're going to look at 00:48:37.770 --> 00:48:41.940 and say oh my God, how was I supposed to know that we ought 00:48:41.940 --> 00:48:42.600 to study that. 00:48:42.600 --> 00:48:45.230 There won't be any surprises. 00:48:45.230 --> 00:48:49.910 Most of the exam questions so far, not all have been easy, 00:48:49.910 --> 00:48:52.680 but I think they've been more or less plain vanilla in the 00:48:52.680 --> 00:48:55.280 sense that they're right more or less down the middle of 00:48:55.280 --> 00:48:56.630 what we're trying to teach. 00:48:56.630 --> 00:49:01.940 And not very many are taking off little peripheral elements 00:49:01.940 --> 00:49:02.680 of the class. 00:49:02.680 --> 00:49:04.860 And it's not going to be any different here. 00:49:04.860 --> 00:49:07.650 So if you can just do the problem solving and get 00:49:07.650 --> 00:49:10.030 familiar enough with it that you're just good at it, so 00:49:10.030 --> 00:49:12.970 that you can do it with a reasonable speed, you're going 00:49:12.970 --> 00:49:16.890 to be fine. 00:49:16.890 --> 00:49:20.770 Also try to work on just understanding the 00:49:20.770 --> 00:49:21.520 underpinnings. 00:49:21.520 --> 00:49:24.580 Especially of statistical mechanics. 00:49:24.580 --> 00:49:27.280 That's where when you have these true-false or multiple 00:49:27.280 --> 00:49:30.160 choice questions, these sort of thought exercises. 00:49:30.160 --> 00:49:31.990 Those are really hard. 00:49:31.990 --> 00:49:35.460 Because they go a little bit beyond problem solving. 00:49:35.460 --> 00:49:36.840 If you could do the problem solving, 00:49:36.840 --> 00:49:38.540 you're going to do fine. 00:49:38.540 --> 00:49:41.990 But it's always useful if you can also formulate things. 00:49:41.990 --> 00:49:46.510 And for that, it's never easy to just tell you here's what 00:49:46.510 --> 00:49:49.860 you have to do, to get good at this. 00:49:49.860 --> 00:49:50.980 Because you have to think about it. 00:49:50.980 --> 00:49:53.750 And it's always hard. 00:49:53.750 --> 00:49:56.710 But to be sure, if you can just try when you review the 00:49:56.710 --> 00:50:00.200 statistical mechanics, especially, and you review the 00:50:00.200 --> 00:50:02.830 expressions that you use and how you solve problems with 00:50:02.830 --> 00:50:06.540 them, the next step in studying is to try to think, 00:50:06.540 --> 00:50:08.280 OK, do I really understand where that 00:50:08.280 --> 00:50:09.930 expression came from. 00:50:09.930 --> 00:50:12.480 And why it makes sense physically. 00:50:12.480 --> 00:50:14.610 And if you can do that, then so much the better. 00:50:14.610 --> 00:50:18.530 Then you'll be in an even stronger position. 00:50:18.530 --> 00:50:22.885 The info sheet gives a handful of equations that we do expect 00:50:22.885 --> 00:50:26.450 you to come in with those on your fingertips. 00:50:26.450 --> 00:50:27.690 There are not very many. 00:50:27.690 --> 00:50:30.370 Mostly we'll provide expressions that you'll need, 00:50:30.370 --> 00:50:33.110 and the important thing is that you know, understand, 00:50:33.110 --> 00:50:34.060 where they apply. 00:50:34.060 --> 00:50:36.170 And how to use them. 00:50:36.170 --> 00:50:38.530 So, good luck on Wednesday.