WEBVTT 00:00:01.000 --> 00:00:04.000 The following content is provided by MIT OpenCourseWare 00:00:04.000 --> 00:00:06.000 under a Creative Commons license. 00:00:06.000 --> 00:00:10.000 Additional information about our license and MIT 00:00:10.000 --> 00:00:15.000 OpenCourseWare in general is available at ocw.mit.edu. 00:00:15.000 --> 00:00:21.000 Last time, we went through the same arguments that Maxwell and 00:00:21.000 --> 00:00:27.000 Boltzmann did to understand the microscopic origin of the ideal 00:00:27.000 --> 00:00:32.000 gas law, PV equal nRT. 00:00:32.000 --> 00:00:38.000 And we saw how Maxwell had hypothesized that the pressure 00:00:38.000 --> 00:00:45.000 of a gas on the walls of some container, if the molecules were 00:00:45.000 --> 00:00:51.000 moving and they collided onto the walls, that pressure must be 00:00:51.000 --> 00:00:59.000 due to the individual impacts of the molecules on the wall. 00:00:59.000 --> 00:01:04.000 That must be due to the change in the momentum of the wall when 00:01:04.000 --> 00:01:07.000 the molecules slammed right into it. 00:01:07.000 --> 00:01:12.000 And that change in momentum over some change in time is this 00:01:12.000 --> 00:01:16.000 force. That force divided by the total 00:01:16.000 --> 00:01:20.000 area is the pressure. And it was using that argument 00:01:20.000 --> 00:01:25.000 that we came up with an expression, just like Maxwell 00:01:25.000 --> 00:01:30.000 did for the pressure times the volume. 00:01:30.000 --> 00:01:34.000 And we see that we were able to write it in terms of the 00:01:34.000 --> 00:01:39.000 velocity of the molecules that were moving in this gas. 00:01:39.000 --> 00:01:44.000 Well, that is very nice because if we have this theoretical 00:01:44.000 --> 00:01:48.000 expression P times V, and we know experimentally that 00:01:48.000 --> 00:01:52.000 P times V equal nRT, if this theory is 00:01:52.000 --> 00:01:57.000 correct, then this quantity n M, average of the velocity 00:01:57.000 --> 00:02:01.000 squared, over three, that better be equal to nRT. 00:02:01.000 --> 00:02:07.000 Or, solving for what we call 00:02:07.000 --> 00:02:11.000 the root mean square velocity, that better be equal to the 00:02:11.000 --> 00:02:15.000 square root of 3RT over M. 00:02:15.000 --> 00:02:19.000 That kinetic theory made a prediction for what the velocity 00:02:19.000 --> 00:02:22.000 ought to be. It took, of course, 00:02:22.000 --> 00:02:26.000 another hundred years before somebody could measure that. 00:02:26.000 --> 00:02:28.000 And, of course, it is correct. 00:02:28.000 --> 00:02:32.000 But what is interesting here is that this model gave us an 00:02:32.000 --> 00:02:37.000 understanding for what temperature is. 00:02:37.000 --> 00:02:41.000 Temperature is a measure of the speed of the molecules in the 00:02:41.000 --> 00:02:44.000 gas. It is also, as we saw last 00:02:44.000 --> 00:02:47.000 time, a measure of their kinetic energy. 00:02:47.000 --> 00:02:52.000 We saw that the kinetic energy is one-half M average of the 00:02:52.000 --> 00:02:56.000 velocity squared. We put 00:02:56.000 --> 00:03:00.000 in our result from kinetic theory. 00:03:00.000 --> 00:03:05.000 That shows us that the average energy is three-halves RT. 00:03:05.000 --> 00:03:08.000 Temperature is the measure of 00:03:08.000 --> 00:03:12.000 the kinetic energy of these molecules in the gas. 00:03:12.000 --> 00:03:15.000 For the first time, there was a microscopic 00:03:15.000 --> 00:03:18.000 understanding of what temperature was. 00:03:18.000 --> 00:03:22.000 Now, what we looked at also, last time, was the 00:03:22.000 --> 00:03:27.000 Maxwell-Boltzmann distribution of velocities or speeds and 00:03:27.000 --> 00:03:32.000 talked about this particular functional form. 00:03:32.000 --> 00:03:35.000 How there was this quadratic dependence right here, 00:03:35.000 --> 00:03:39.000 a low v because of the v squared here and then an 00:03:39.000 --> 00:03:44.000 exponentially decaying tail. There were two parameters, 00:03:44.000 --> 00:03:48.000 the mass and the temperature. We talked about that last time. 00:03:48.000 --> 00:03:52.000 But, of course, if there is a distribution here 00:03:52.000 --> 00:03:56.000 of speeds where this distribution is a probability of 00:03:56.000 --> 00:04:01.000 finding a molecule with a particular speed between v and v 00:04:01.000 --> 00:04:05.000 plus dv, -- -- then there also has to be a 00:04:05.000 --> 00:04:09.000 distribution of energies because the velocity is related to the 00:04:09.000 --> 00:04:11.000 energy. What do we have to do? 00:04:11.000 --> 00:04:14.000 We have to take that Maxwell-Boltzmann speed 00:04:14.000 --> 00:04:18.000 distribution and change the variable to energy. 00:04:18.000 --> 00:04:22.000 We know how to do that. We know how to equate those two 00:04:22.000 --> 00:04:26.000 distribution functions. And, using the Jacobean of the 00:04:26.000 --> 00:04:29.000 transformation that we talked about last time, 00:04:29.000 --> 00:04:34.000 we can calculate what the energy distribution is. 00:04:34.000 --> 00:04:37.000 f of E is our Maxwell-Boltzmann energy 00:04:37.000 --> 00:04:40.000 distribution. It is the probability of 00:04:40.000 --> 00:04:45.000 finding a molecule with a kinetic energy E to E plus dE. 00:04:45.000 --> 00:04:48.000 It also has a decaying exponential term, 00:04:48.000 --> 00:04:53.000 here, with the energy in the argument, and it is multiplied 00:04:53.000 --> 00:04:57.000 by E of the one-half. Let's take a look at what that 00:04:57.000 --> 00:05:01.000 distribution function looks like. 00:05:01.000 --> 00:05:04.000 Here it is. Notice, as we learned last 00:05:04.000 --> 00:05:09.000 time, that the energy has nothing to do with the mass of 00:05:09.000 --> 00:05:12.000 the particle. The only parameter that is 00:05:12.000 --> 00:05:17.000 important is the temperature. At a given temperature, 00:05:17.000 --> 00:05:21.000 all particles, it doesn't matter what their 00:05:21.000 --> 00:05:26.000 mass is, have the same energy. So, all particles at 00:05:26.000 --> 00:05:30.000 degrees Kelvin have a Maxwell-Boltzmann energy 00:05:30.000 --> 00:05:35.000 distribution that looks like this. 00:05:35.000 --> 00:05:39.000 There is a very rapid rise in that distribution function, 00:05:39.000 --> 00:05:42.000 unlike the velocity distribution, 00:05:42.000 --> 00:05:45.000 which increased as v squared. 00:05:45.000 --> 00:05:48.000 It peaks rapidly, and then there is an 00:05:48.000 --> 00:05:52.000 exponentially decaying term in the energy. 00:05:52.000 --> 00:05:56.000 At 1500 degrees Kelvin, the energy distribution looks 00:05:56.000 --> 00:05:59.000 like this. The energy of the molecules has 00:05:59.000 --> 00:06:04.000 increased. Since this is a probability, 00:06:04.000 --> 00:06:07.000 and the area under these curves has to equal to one, 00:06:07.000 --> 00:06:11.000 if the energy goes up, that is we have more molecules 00:06:11.000 --> 00:06:16.000 out here with higher energies, well, then this maximum value 00:06:16.000 --> 00:06:21.000 for the probabilities has got to go down because we have to keep 00:06:21.000 --> 00:06:25.000 the area under that curve equal to one, since this is a 00:06:25.000 --> 00:06:28.000 probability. What we see here, 00:06:28.000 --> 00:06:33.000 as we raise the energy, is that we have more and more 00:06:33.000 --> 00:06:38.000 molecules with high energies. There are not a lot of them, 00:06:38.000 --> 00:06:42.000 but there are some. And they are going to be really 00:06:42.000 --> 00:06:47.000 important in the example I am going to show you in just a 00:06:47.000 --> 00:06:51.000 moment. Average energy at 600 Kelvin, 00:06:51.000 --> 00:06:55.000 just three-halves RT, is 7.5 kilojoules per mole. 00:06:55.000 --> 00:07:02.000 Average energy at 1500 Kelvin is 18.7 kilojoules per mole. 00:07:02.000 --> 00:07:07.000 But now, I want to show you an example of the importance of 00:07:07.000 --> 00:07:12.000 this Maxwell-Boltzmann distribution function for 00:07:12.000 --> 00:07:18.000 energies to chemical reactions, the importance of it in making 00:07:18.000 --> 00:07:23.000 some kinds of chemical reactions actually work. 00:07:23.000 --> 00:07:29.000 And so the example is going to be this reaction. 00:07:29.000 --> 00:07:32.000 This reaction is called steam reforming of natural gas. 00:07:32.000 --> 00:07:35.000 It is the reaction of methane plus water. 00:07:35.000 --> 00:07:39.000 Natural gas is mostly methane. But this reaction of methane 00:07:39.000 --> 00:07:43.000 and water makes CO and hydrogen. It 00:07:43.000 --> 00:07:47.000 turns out that this reaction does not work in the gas phase. 00:07:47.000 --> 00:07:51.000 That is, if you have a methane molecule and water molecule 00:07:51.000 --> 00:07:56.000 collide, they are just going to collide, bounce apart and go in 00:07:56.000 --> 00:07:59.000 their opposite directions. They are not going to make CO 00:07:59.000 --> 00:08:04.000 and hydrogen. And so what you have to have in 00:08:04.000 --> 00:08:08.000 this reaction to make it go is a catalyst. 00:08:08.000 --> 00:08:11.000 That catalyst is going to be a nickel surface. 00:08:11.000 --> 00:08:17.000 A catalyst is something that will lower the activation energy 00:08:17.000 --> 00:08:21.000 barrier by changing the mechanism of a reaction so that 00:08:21.000 --> 00:08:24.000 the reaction can proceed. In this case, 00:08:24.000 --> 00:08:29.000 what happens is that the methane and the water impinge on 00:08:29.000 --> 00:08:35.000 this nickel metal catalyst. And the methane and the water 00:08:35.000 --> 00:08:38.000 decompose. They fall apart to their 00:08:38.000 --> 00:08:43.000 elements on that nickel surface. And then, once you have carbon, 00:08:43.000 --> 00:08:46.000 hydrogen and oxygen on that nickel surface, 00:08:46.000 --> 00:08:50.000 the atoms rearrange and come off as CO and molecular 00:08:50.000 --> 00:08:53.000 hydrogen. So, that nickel surface is a 00:08:53.000 --> 00:08:56.000 catalyst. We call this the catalytic 00:08:56.000 --> 00:08:59.000 reaction. In particular, 00:08:59.000 --> 00:09:02.000 we call it a heterogeneous catalytic reaction. 00:09:02.000 --> 00:09:08.000 It is heterogeneous because the catalyst is in a different phase 00:09:08.000 --> 00:09:12.000 than the reactants. The catalyst here is a solid. 00:09:12.000 --> 00:09:16.000 The reactants are gases. A homogeneous catalyst is one 00:09:16.000 --> 00:09:21.000 in which the phase of the catalyst and the reactants is 00:09:21.000 --> 00:09:24.000 the same. This is a heterogeneous 00:09:24.000 --> 00:09:28.000 catalytic reaction. It turns out that this reaction 00:09:28.000 --> 00:09:33.000 here is really an important reaction from the standpoint of 00:09:33.000 --> 00:09:39.000 the production of hydrogen. All of the hydrogen that you 00:09:39.000 --> 00:09:42.000 use, you know, if you are doing a laboratory 00:09:42.000 --> 00:09:47.000 experiment and you go get a tank of hydrogen, that hydrogen is 00:09:47.000 --> 00:09:51.000 made by this reaction, by reacting methane and water 00:09:51.000 --> 00:09:55.000 to form that hydrogen. All of our commercial sources 00:09:55.000 --> 00:10:00.000 of hydrogen come from carrying out this reaction. 00:10:00.000 --> 00:10:03.000 For example, if you are in the business of 00:10:03.000 --> 00:10:08.000 ammonia synthesis, which is taking hydrogen and 00:10:08.000 --> 00:10:13.000 nitrogen to make ammonia, which is also carried out on an 00:10:13.000 --> 00:10:17.000 iron surface, another heterogeneous catalytic 00:10:17.000 --> 00:10:19.000 reaction. And ammonia, 00:10:19.000 --> 00:10:24.000 of course, is the starting material for lots of chemicals, 00:10:24.000 --> 00:10:30.000 in particular fertilizers. If you have an ammonia plant, 00:10:30.000 --> 00:10:35.000 right next to the ammonia plant you have a steamer-forming plant 00:10:35.000 --> 00:10:39.000 to make the hydrogen to feed into this reaction. 00:10:39.000 --> 00:10:44.000 And another place where this is useful is in methanol synthesis. 00:10:44.000 --> 00:10:49.000 That is, you can take hydrogen and CO on a copper zinc oxide 00:10:49.000 --> 00:10:54.000 heterogeneous catalyst and make methanol, the starting point for 00:10:54.000 --> 00:11:00.000 gasoline, which now is certainly economically feasible. 00:11:00.000 --> 00:11:05.000 But it turns out that this reaction here is really a hard 00:11:05.000 --> 00:11:09.000 one to carry out. Despite the fact that nickel is 00:11:09.000 --> 00:11:15.000 a catalyst for the reaction and makes it go, the reaction still 00:11:15.000 --> 00:11:20.000 has to be carried out at very high temperatures. 00:11:20.000 --> 00:11:24.000 1500 degrees kelvin is a very high temperature. 00:11:24.000 --> 00:11:29.000 It also needs very high pressure. 00:11:29.000 --> 00:11:33.000 And one of the reasons why we don't have a hydrogen economy is 00:11:33.000 --> 00:11:36.000 because it is so difficult to make. 00:11:36.000 --> 00:11:39.000 Hydrogen can be done, and it is done in all of the 00:11:39.000 --> 00:11:43.000 commercial processes that I mentioned to you, 00:11:43.000 --> 00:11:46.000 but it is still hard. 1500 degrees kelvin, 00:11:46.000 --> 00:11:48.000 high pressures. The question is, 00:11:48.000 --> 00:11:52.000 why is it so hard? Why is there this barrier here? 00:11:52.000 --> 00:11:56.000 Why do we need to raise the temperature of the gas in order 00:11:56.000 --> 00:12:02.000 to get this reaction to go? Well, let's take a look at 00:12:02.000 --> 00:12:05.000 that. What there exists here in this 00:12:05.000 --> 00:12:09.000 problem is what is called an activation energy barrier to 00:12:09.000 --> 00:12:13.000 making the reaction go. It turns out that this 00:12:13.000 --> 00:12:18.000 activation energy barrier is really in the very first step of 00:12:18.000 --> 00:12:22.000 the reaction, which is pulling the methane 00:12:22.000 --> 00:12:27.000 apart, pulling that first hydrogen atom off of the methane 00:12:27.000 --> 00:12:31.000 molecule. What I represent right here is 00:12:31.000 --> 00:12:36.000 the energy of the reaction as a function of the reaction, 00:12:36.000 --> 00:12:40.000 and the reaction here is just the first step. 00:12:40.000 --> 00:12:45.000 It is taking methane gas, pulling the hydrogen off so 00:12:45.000 --> 00:12:50.000 that you have a methyl radical stuck to the surface and you 00:12:50.000 --> 00:12:54.000 have a hydrogen atom stuck to the surface. 00:12:54.000 --> 00:12:59.000 It is just breaking that first C-H bond. 00:12:59.000 --> 00:13:04.000 What you can see here is that you have to put 50 kilojoules of 00:13:04.000 --> 00:13:08.000 energy into that reaction in order to make it go. 00:13:08.000 --> 00:13:12.000 You have to put that 50 kilocalories of energy in first, 00:13:12.000 --> 00:13:17.000 before you get any energy back. You can see that this is 00:13:17.000 --> 00:13:20.000 exothermic. But you have to put this energy 00:13:20.000 --> 00:13:23.000 in, in order to get that reaction to go. 00:13:23.000 --> 00:13:29.000 Well, how do we know that? Let me back up a minute. 00:13:29.000 --> 00:13:31.000 Here is where the Maxwell-Boltzmann energy 00:13:31.000 --> 00:13:35.000 distribution is important. What I did was I took those 00:13:35.000 --> 00:13:39.000 Boltzmann energy distributions that I plotted for you at 00:13:39.000 --> 00:13:43.000 Kelvin and 1500 Kelvin and turned them on their side. 00:13:43.000 --> 00:13:47.000 This is essentially equal to the probability of finding a 00:13:47.000 --> 00:13:51.000 molecule at a particular energy versus the energy. 00:13:51.000 --> 00:13:55.000 Here is the Maxwell-Boltzmann distribution at 600 degrees 00:13:55.000 --> 00:13:57.000 Kelvin. It is peaked to very low 00:13:57.000 --> 00:14:01.000 energies. And, if you look here in this 00:14:01.000 --> 00:14:05.000 Maxwell-Boltzmann tail, there are not very many 00:14:05.000 --> 00:14:10.000 molecules that have enough energy to get over that barrier. 00:14:10.000 --> 00:14:14.000 However, if we raise the temperature of the gas to 00:14:14.000 --> 00:14:18.000 degrees Kelvin, then now you can see that there 00:14:18.000 --> 00:14:23.000 are more molecules here in the high energy part of this tail. 00:14:23.000 --> 00:14:27.000 And it is these molecules at these high energies that 00:14:27.000 --> 00:14:32.000 actually can get over this barrier to dissociation of the 00:14:32.000 --> 00:14:36.000 methane molecule. They are not many of those 00:14:36.000 --> 00:14:39.000 molecules with high energy, but there are some. 00:14:39.000 --> 00:14:42.000 And that is important, because what happens is when 00:14:42.000 --> 00:14:45.000 they have enough energy, they react and they leave. 00:14:45.000 --> 00:14:49.000 And then this Boltzmann distribution re-equilibrates. 00:14:49.000 --> 00:14:52.000 If the high energy molecules leave, then there are lots of 00:14:52.000 --> 00:14:56.000 collisions here that kick some more molecules up to the high 00:14:56.000 --> 00:15:00.000 energy part of the tail, and they react. 00:15:00.000 --> 00:15:04.000 And so it works. That is why it is important. 00:15:04.000 --> 00:15:09.000 How do we really know that there is this barrier here to 00:15:09.000 --> 00:15:15.000 the dissociation of methane, and why is there this barrier? 00:15:15.000 --> 00:15:21.000 What is the physical origin of this barrier to pulling this C-H 00:15:21.000 --> 00:15:25.000 bond apart? Well, one way we could know 00:15:25.000 --> 00:15:30.000 that there was this barrier there for sure is to do the 00:15:30.000 --> 00:15:35.000 following. If we had a way to take methane 00:15:35.000 --> 00:15:38.000 gas and make it have just a single energy, 00:15:38.000 --> 00:15:41.000 not the Maxwell-Boltzmann distribution of energies that 00:15:41.000 --> 00:15:46.000 have lots of different energies, which have lots of different 00:15:46.000 --> 00:15:49.000 energies in them, and you saw how broad those 00:15:49.000 --> 00:15:51.000 curves were. But if we had a way to make 00:15:51.000 --> 00:15:56.000 methane gas with just say energy E sub 1, whatever that is, 00:15:56.000 --> 00:15:59.000 we could then take those molecules with just those 00:15:59.000 --> 00:16:02.000 energies, aim them at this nickel surface, 00:16:02.000 --> 00:16:06.000 and see if the methane fell apart. 00:16:06.000 --> 00:16:09.000 And, if it didn't, then we would prepare molecules 00:16:09.000 --> 00:16:13.000 at some higher energy and see if they fell apart. 00:16:13.000 --> 00:16:16.000 If they didn't then we would go to higher energy. 00:16:16.000 --> 00:16:20.000 We would keep going until we got to the top of the barrier. 00:16:20.000 --> 00:16:24.000 Well, how do you do that? How do you make molecules with 00:16:24.000 --> 00:16:28.000 a particular energy? Because I just showed you a 00:16:28.000 --> 00:16:32.000 Maxwell-Boltzmann energy distribution at 300 Kelvin, 00:16:32.000 --> 00:16:37.000 600 Kelvin. There are a lot of energies 00:16:37.000 --> 00:16:41.000 present. How do we make molecules with 00:16:41.000 --> 00:16:46.000 just one energy? We have a way to do that by 00:16:46.000 --> 00:16:52.000 using some beam techniques and high pressure adiabatic 00:16:52.000 --> 00:16:57.000 expansions. Basically, the way it works is 00:16:57.000 --> 00:17:01.000 this. What we are going to do is take 00:17:01.000 --> 00:17:07.000 a tube here, which is going to have a high pressure of methane 00:17:07.000 --> 00:17:10.000 in it. Then we are going to punch a 00:17:10.000 --> 00:17:14.000 little hole in that tube. This tube is actually sitting 00:17:14.000 --> 00:17:19.000 in a vacuum, so we are going to expand the methane from that 00:17:19.000 --> 00:17:24.000 tube into the vacuum. We are going to squirt it out. 00:17:24.000 --> 00:17:29.000 It is going to become a beam of molecules. 00:17:29.000 --> 00:17:34.000 But when you expand a gas from high pressure to low pressure, 00:17:34.000 --> 00:17:38.000 this expansion is called an adiabatic expansion, 00:17:38.000 --> 00:17:42.000 which means the gas cools. I will explain that in a 00:17:42.000 --> 00:17:47.000 moment, but the adiabatic expansion you are going to talk 00:17:47.000 --> 00:17:50.000 a lot about in 5.60. And, if you are a chemical 00:17:50.000 --> 00:17:55.000 engineer, you will talk even more about it past 5.60. 00:17:55.000 --> 00:17:59.000 Here is how it works. You take these molecules and 00:17:59.000 --> 00:18:04.000 squirt them out. Because the pressure right here 00:18:04.000 --> 00:18:08.000 is so high, what happens is there are lots and lots and lots 00:18:08.000 --> 00:18:11.000 of collisions. And so, you can imagine, 00:18:11.000 --> 00:18:15.000 if you have some slow molecule just kind of lumbering along and 00:18:15.000 --> 00:18:20.000 a fast one comes and hits you, what is going to happen is you 00:18:20.000 --> 00:18:23.000 are going to speed up, and the fast one is going to 00:18:23.000 --> 00:18:26.000 slow down. And, if you keep doing that 00:18:26.000 --> 00:18:30.000 again and again and again, all the molecules are going to 00:18:30.000 --> 00:18:35.000 end up with the same energy or the same velocity. 00:18:35.000 --> 00:18:39.000 If you have so many collisions, after a while they are going to 00:18:39.000 --> 00:18:44.000 all have the same velocity or the same energy because one gets 00:18:44.000 --> 00:18:48.000 sped up, one gets slowed down. They are all going to end up 00:18:48.000 --> 00:18:51.000 with the same energy. That is what happens. 00:18:51.000 --> 00:18:56.000 That is how we make a beam of molecules, a source of molecules 00:18:56.000 --> 00:19:00.000 with the same kinetic energy. And basically, 00:19:00.000 --> 00:19:03.000 what is happening then is that before the expansion, 00:19:03.000 --> 00:19:07.000 just look at this curve, here is an actual Boltzmann 00:19:07.000 --> 00:19:10.000 distribution of velocities. It is broad. 00:19:10.000 --> 00:19:14.000 But after this expansion, here is the distribution. 00:19:14.000 --> 00:19:18.000 It is really pretty narrow. If you put a temperature to 00:19:18.000 --> 00:19:21.000 that distribution, you might find that at one 00:19:21.000 --> 00:19:24.000 degree Kelvin, you can really make molecules 00:19:24.000 --> 00:19:30.000 with a single or very narrow distribution in energies. 00:19:30.000 --> 00:19:33.000 That is what we do. We make those molecules with 00:19:33.000 --> 00:19:38.000 this energy E sub 1. Then we have some more tricks 00:19:38.000 --> 00:19:41.000 for changing those energies in a controlled way. 00:19:41.000 --> 00:19:46.000 And we just keep cranking those energies up and watch to see 00:19:46.000 --> 00:19:51.000 when the methane falls apart. And so, you get data that kind 00:19:51.000 --> 00:19:54.000 of looks like this. This is a dissociation 00:19:54.000 --> 00:19:58.000 probability of a methane as a function of its energy. 00:19:58.000 --> 00:20:02.000 This is in kilocalories per mole. 00:20:02.000 --> 00:20:05.000 At some point, say right about here, 00:20:05.000 --> 00:20:10.000 15 kilocalories per mole, the dissociation probability 00:20:10.000 --> 00:20:13.000 skyrockets. All of a sudden, 00:20:13.000 --> 00:20:18.000 you are at a high enough energy to get that methane to fall 00:20:18.000 --> 00:20:22.000 apart. This tells you exactly where 00:20:22.000 --> 00:20:26.000 that barrier is. But now, the question is, 00:20:26.000 --> 00:20:32.000 why is there this barrier? Physically, why is there a 00:20:32.000 --> 00:20:38.000 barrier to pulling the hydrogen off of the carbon when that 00:20:38.000 --> 00:20:43.000 methane molecule comes close to the surface? 00:20:43.000 --> 00:20:48.000 Well, the reason is this. In order to break the 00:20:48.000 --> 00:20:54.000 carbon-hydrogen bond in methane, in order to have enough energy 00:20:54.000 --> 00:21:00.000 to break that bond, what you need to do is you need 00:21:00.000 --> 00:21:05.000 to simultaneously form a nickel-carbon bond and a 00:21:05.000 --> 00:21:10.000 nickel-hydrogen bond. In other words, 00:21:10.000 --> 00:21:17.000 when this methane molecule here comes into some nickel surface 00:21:17.000 --> 00:21:23.000 very slowly, because that methane is tetrahedral and the 00:21:23.000 --> 00:21:30.000 carbon is kind of hidden in the center, the hydrogens interact 00:21:30.000 --> 00:21:35.000 with the nickel. But the carbon does not get in 00:21:35.000 --> 00:21:39.000 close enough to the nickel to start to form a nickel-carbon 00:21:39.000 --> 00:21:42.000 bond. Now, if you speed that methane 00:21:42.000 --> 00:21:46.000 molecule up and you really ram it into the surface, 00:21:46.000 --> 00:21:49.000 upon collision of the molecule with the surface, 00:21:49.000 --> 00:21:54.000 the hydrogens are pushed back, the carbon gets in close enough 00:21:54.000 --> 00:22:00.000 to the nickel to start to form, here, this nickel-carbon bond. 00:22:00.000 --> 00:22:05.000 You form a nickel-hydrogen bond, and now you can break that 00:22:05.000 --> 00:22:09.000 C-H bond. We call this black chemistry. 00:22:09.000 --> 00:22:12.000 The barrier, there, to that reaction 00:22:12.000 --> 00:22:18.000 physically is the amount of energy that you have to put into 00:22:18.000 --> 00:22:22.000 the molecule to push those hydrogens back, 00:22:22.000 --> 00:22:30.000 to bend those hydrogens back to distort this methane molecule. 00:22:30.000 --> 00:22:33.000 Once you do that, the reaction goes. 00:22:33.000 --> 00:22:35.000 You are on a roll there. It goes. 00:22:35.000 --> 00:22:39.000 So, that is the physical origin of this barrier, 00:22:39.000 --> 00:22:43.000 here, to the dissociation of methane. 00:22:43.000 --> 00:22:47.000 You need to put enough kinetic energy into those methane 00:22:47.000 --> 00:22:51.000 molecules in order to get over that barrier. 00:22:51.000 --> 00:22:56.000 And we do that on a practical scale by raising the temperature 00:22:56.000 --> 00:23:01.000 of the gas. And on a microscopic scale, 00:23:01.000 --> 00:23:06.000 this is what is happening. You can imagine that my 00:23:06.000 --> 00:23:11.000 students and I did this experiment 15 years ago. 00:23:11.000 --> 00:23:16.000 But then, we said if this barrier here is the energy 00:23:16.000 --> 00:23:22.000 required to deform that methane molecule, then we should be able 00:23:22.000 --> 00:23:26.000 to do this experiment. This experiment is the 00:23:26.000 --> 00:23:31.000 following. We are going to take our nickel 00:23:31.000 --> 00:23:34.000 surface, here, and lower the temperature of 00:23:34.000 --> 00:23:38.000 the surface to 47 kelvin. At that temperature, 00:23:38.000 --> 00:23:42.000 what we can do is we can freeze a layer of methane on the 00:23:42.000 --> 00:23:44.000 surface. We call it fizzy-sorb, 00:23:44.000 --> 00:23:47.000 a layer of methane on the surface. 00:23:47.000 --> 00:23:51.000 It will just stick there. And then, if this barrier is 00:23:51.000 --> 00:23:55.000 the energy required to deform or to start the molecule, 00:23:55.000 --> 00:23:59.000 then in principle I could take a hammer and pound that molecule 00:23:59.000 --> 00:24:04.000 into the correct shape for the transition state that leads to 00:24:04.000 --> 00:24:09.000 dissociation. It sounds like a simple idea, 00:24:09.000 --> 00:24:14.000 and it is, except that we cannot really take that hammer. 00:24:14.000 --> 00:24:19.000 But we can take an argon atom. We can freeze the methane onto 00:24:19.000 --> 00:24:23.000 the surface, and now we come in with an argon atom. 00:24:23.000 --> 00:24:27.000 That is just a big ball. Or, a xenon atom or a krypton 00:24:27.000 --> 00:24:30.000 atom. And we know how to accelerate 00:24:30.000 --> 00:24:35.000 xenon or krypton. We don't accelerate it too 00:24:35.000 --> 00:24:38.000 much, 50 kilocalories, 70 kilocalories per mole, 00:24:38.000 --> 00:24:42.000 something like that. And then we can bring it in. 00:24:42.000 --> 00:24:47.000 What happens is that the impact of the collision on that methane 00:24:47.000 --> 00:24:51.000 causes that methane molecule to compress, distort, 00:24:51.000 --> 00:24:56.000 gets it into the configuration of the transition state that 00:24:56.000 --> 00:25:00.000 leads to the methane falling apart. 00:25:00.000 --> 00:25:02.000 And so, you get the same result. 00:25:02.000 --> 00:25:05.000 The question, there, is just getting the 00:25:05.000 --> 00:25:09.000 energy into the molecule to actually distort it, 00:25:09.000 --> 00:25:14.000 to deform it so that you can make the nickel-carbon bond and 00:25:14.000 --> 00:25:18.000 the nickel-hydrogen bond. That is the key. 00:25:18.000 --> 00:25:22.000 My students and I, after we had spent many years 00:25:22.000 --> 00:25:26.000 doing this experiment and spent a lot of money on this 00:25:26.000 --> 00:25:31.000 experiment, said we have proven for sure that when a bug flies 00:25:31.000 --> 00:25:36.000 into the windshield of your car, you do get the same result as 00:25:36.000 --> 00:25:41.000 if you hit the bug on the windshield of your car with a 00:25:41.000 --> 00:25:46.000 bug swatter. That is what we get. 00:25:46.000 --> 00:25:51.000 This is an example of one chemical reaction for which we 00:25:51.000 --> 00:25:55.000 know the physical origin of the barrier. 00:25:55.000 --> 00:26:01.000 There are very few chemical reactions for which we know 00:26:01.000 --> 00:26:06.000 anything about the physical origin of a barrier to a 00:26:06.000 --> 00:26:11.000 reaction. Now, what I want to talk about 00:26:11.000 --> 00:26:18.000 is go back to kinetic theory, because the kinetic theory is 00:26:18.000 --> 00:26:25.000 going to allow us to calculate a couple of more quantities that 00:26:25.000 --> 00:26:31.000 are of interest to us. And one of those quantities is 00:26:31.000 --> 00:26:37.000 the collision frequency. And let me just write that on 00:26:37.000 --> 00:26:41.000 the board here. What I am going to want to 00:26:41.000 --> 00:26:47.000 calculate is something called Z1, or I am going to call it Z1. 00:26:47.000 --> 00:26:53.000 It is the number of collisions that a molecule makes in a gas 00:26:53.000 --> 00:26:58.000 per unit time. And our unit time is going to 00:26:58.000 --> 00:27:06.000 be seconds. This is the collision frequency 00:27:06.000 --> 00:27:13.000 of a single molecule in a gas. 00:27:28.000 --> 00:27:29.000 How am I going to calculate the quantity using the kinetic 00:27:29.000 --> 00:27:30.000 theory approach that we have been talking about? 00:27:30.000 --> 00:27:33.000 What I am going to do is this. I am going to take some gas, 00:27:33.000 --> 00:27:38.000 and the molecules in that gas are represented by these blue 00:27:38.000 --> 00:27:42.000 circles right here. They have a diameter D. 00:27:42.000 --> 00:27:47.000 And then, within that gas, I am going to imagine this 00:27:47.000 --> 00:27:52.000 lighter blue cylinder. That cylinder is an imaginary 00:27:52.000 --> 00:27:55.000 construct. We are going to use it to 00:27:55.000 --> 00:28:00.000 calculate this collision frequency. 00:28:00.000 --> 00:28:04.000 What it is going to be is the collision volume. 00:28:04.000 --> 00:28:09.000 I have set the diameter of that cylinder to be equal to two 00:28:09.000 --> 00:28:12.000 times the diameter of the molecule. 00:28:12.000 --> 00:28:17.000 I have set this upright. And the bottom line is that all 00:28:17.000 --> 00:28:22.000 of the molecules that instantaneously happen to be in 00:28:22.000 --> 00:28:26.000 this cylinder when a molecule comes through, 00:28:26.000 --> 00:28:32.000 all of those molecules are actually going to be hit. 00:28:32.000 --> 00:28:37.000 They are going to suffer a collision with another molecule. 00:28:37.000 --> 00:28:42.000 And I am going to use that, the number of molecules in this 00:28:42.000 --> 00:28:46.000 volume, to calculate this collision frequency. 00:28:46.000 --> 00:28:51.000 That is what I am going to do. Snapshot in time, 00:28:51.000 --> 00:28:56.000 those molecules are frozen there, except there is one 00:28:56.000 --> 00:29:02.000 smart-alecky molecule that comes cruising on through. 00:29:02.000 --> 00:29:04.000 Here he comes. Bam, bam, bam. 00:29:04.000 --> 00:29:10.000 Hits these three molecules, here, because they are more 00:29:10.000 --> 00:29:14.000 than halfway into that collision cylinder. 00:29:14.000 --> 00:29:20.000 What I know is this smart-aleck was moving at some average 00:29:20.000 --> 00:29:24.000 velocity, which I am going to call vbar. 00:29:24.000 --> 00:29:27.000 In a time T, this molecule, 00:29:27.000 --> 00:29:32.000 since it is moving with an average velocity vbar, 00:29:32.000 --> 00:29:36.000 in a time t, it is traveling a length vbar 00:29:36.000 --> 00:29:41.000 times t. 00:29:41.000 --> 00:29:44.000 Velocity times time, that is going to give you a 00:29:44.000 --> 00:29:47.000 length. And so, I am going to set, 00:29:47.000 --> 00:29:50.000 for convenience, this time equal to one second 00:29:50.000 --> 00:29:53.000 because that is going to be my unit of time. 00:29:53.000 --> 00:29:58.000 I want to do this per second. I am going to set this equal to 00:29:58.000 --> 00:30:02.000 one second. Therefore, this molecule is 00:30:02.000 --> 00:30:07.000 going to travel a length vbar in one second. 00:30:07.000 --> 00:30:13.000 And that is the length that I am going to make the collision 00:30:13.000 --> 00:30:16.000 cylinder. I am going to make that 00:30:16.000 --> 00:30:22.000 collision cylinder be vbar long. That is the distance traveled 00:30:22.000 --> 00:30:28.000 in one second by this smart-alecky molecule. 00:30:28.000 --> 00:30:34.000 Now, all I have to do is I have got to take the volume of the 00:30:34.000 --> 00:30:40.000 collision cylinder and multiply it by all the molecules that 00:30:40.000 --> 00:30:45.000 happen to be at the volume at that particular time. 00:30:45.000 --> 00:30:51.000 Because that volume represents the volume swept out in one 00:30:51.000 --> 00:30:55.000 second by this smart-alecky molecule. 00:30:55.000 --> 00:31:00.000 What is the volume of the cylinder? 00:31:00.000 --> 00:31:03.000 I set the diameter to be equal to 2d. 00:31:03.000 --> 00:31:08.000 The volume of the cylinder is the cross-sectional area times 00:31:08.000 --> 00:31:11.000 the length. That cross-sectional area, 00:31:11.000 --> 00:31:16.000 then, is pi r squared, but r here is equal to d, 00:31:16.000 --> 00:31:21.000 just to make it confusing. The area is pi d squared. 00:31:21.000 --> 00:31:24.000 The length is vbar. 00:31:24.000 --> 00:31:28.000 That is the volume of the cylinder, pi d squared vbar. 00:31:28.000 --> 00:31:34.000 So, we have the volume of the 00:31:34.000 --> 00:31:37.000 cylinder. Now what we are going to need 00:31:37.000 --> 00:31:42.000 to know is the density of the molecules in this cylinder, 00:31:42.000 --> 00:31:46.000 which is the density of the molecules in the gas. 00:31:46.000 --> 00:31:50.000 The cylinder is not special. The cylinder is an imaginary 00:31:50.000 --> 00:31:55.000 construct that I put in there to help me calculate the collision 00:31:55.000 --> 00:31:58.000 frequency. I need the density of the 00:31:58.000 --> 00:32:03.000 molecules in the gas or in the cylinder. 00:32:03.000 --> 00:32:07.000 The density of the molecules I am going to set as equal to N, 00:32:07.000 --> 00:32:10.000 the total number of molecules in this volume, 00:32:10.000 --> 00:32:13.000 divided by the volume of the gas. 00:32:13.000 --> 00:32:17.000 I am going to call that rho. N over V is going to be equal 00:32:17.000 --> 00:32:21.000 to rho. That is the number of molecules 00:32:21.000 --> 00:32:25.000 per cubic meter. That is the density of the gas. 00:32:25.000 --> 00:32:29.000 This is the symbol that I am going to use from now on for 00:32:29.000 --> 00:32:34.000 density defined as molecules per cubic meter. 00:32:34.000 --> 00:32:39.000 Then the collision frequency is simply going to be the density 00:32:39.000 --> 00:32:43.000 of the molecules times the volume of the cylinder. 00:32:43.000 --> 00:32:47.000 The collision frequency Z1, here, is this volume of the 00:32:47.000 --> 00:32:52.000 cylinder, pi d squared vbar times the density. 00:32:52.000 --> 00:32:53.000 Why? 00:32:53.000 --> 00:32:58.000 Because I set it up so that every molecule in that cylinder 00:32:58.000 --> 00:33:04.000 would suffer a collision. And that cylinder is the length 00:33:04.000 --> 00:33:08.000 that a molecule travels in one second. 00:33:08.000 --> 00:33:11.000 There is the density times the volume. 00:33:11.000 --> 00:33:16.000 The meters cubed disappears. What I have left is rho, 00:33:16.000 --> 00:33:20.000 the density, pi d squared vbar. 00:33:20.000 --> 00:33:24.000 This is collisions per second. 00:33:24.000 --> 00:33:27.000 So, I have my collision frequency. 00:33:27.000 --> 00:33:33.000 This is the single-molecule collision frequency. 00:33:33.000 --> 00:33:37.000 This is the frequency of collisions that one molecule 00:33:37.000 --> 00:33:40.000 makes with the other gas molecules. 00:33:40.000 --> 00:33:44.000 That is important. We are going to do a different 00:33:44.000 --> 00:33:47.000 kind of collision frequency in just a moment. 00:33:47.000 --> 00:33:52.000 But now, it turns out that there is another factor here 00:33:52.000 --> 00:33:57.000 that I had left out because I haven't done as sophisticated of 00:33:57.000 --> 00:34:02.000 an analysis as I could. And that is that I made the 00:34:02.000 --> 00:34:07.000 assumption in my picture before that all the other molecules 00:34:07.000 --> 00:34:11.000 were frozen in time and that there was only one smart-aleck 00:34:11.000 --> 00:34:15.000 that was moving around, cruising through. 00:34:15.000 --> 00:34:17.000 The reality is they are all moving. 00:34:17.000 --> 00:34:23.000 And what I really have to do is I have to take into account the 00:34:23.000 --> 00:34:27.000 relative velocities of the molecules, the relative speeds. 00:34:27.000 --> 00:34:31.000 I can do that. There is a little more 00:34:31.000 --> 00:34:36.000 sophisticated analysis in doing that, but I could do that. 00:34:36.000 --> 00:34:40.000 And, if we do that, this makes a difference of the 00:34:40.000 --> 00:34:44.000 square root of two. I am just going to put that 00:34:44.000 --> 00:34:49.000 square root of two in there right now, and later on you will 00:34:49.000 --> 00:34:54.000 be able to see where that comes from, in a later course. 00:34:54.000 --> 00:34:59.000 That actually is the collision frequency of a single molecule 00:34:59.000 --> 00:35:01.000 in the gas. 00:35:07.000 --> 00:35:12.000 I am also interested in another quantity, which is the total 00:35:12.000 --> 00:35:16.000 collision frequency. That was the single collision 00:35:16.000 --> 00:35:21.000 frequency, but I am also interested in knowing how many 00:35:21.000 --> 00:35:26.000 collisions are occurring in the entire gas per unit time, 00:35:26.000 --> 00:35:32.000 the total collision frequency. You know why I am interested in 00:35:32.000 --> 00:35:36.000 that number? I am interested in that number 00:35:36.000 --> 00:35:41.000 because that is going to be the upper limit to any reaction 00:35:41.000 --> 00:35:44.000 rate. A reaction in the gas phase or 00:35:44.000 --> 00:35:49.000 in solution cannot happen any faster than the molecules 00:35:49.000 --> 00:35:52.000 collide. They have to collide before a 00:35:52.000 --> 00:35:57.000 reaction is going to occur. The total collision frequency, 00:35:57.000 --> 00:36:03.000 the importance of that number is that it is the upper limit to 00:36:03.000 --> 00:36:08.000 a reaction rate. Let's calculate that. 00:36:08.000 --> 00:36:13.000 Z is going to be equal to the collision frequency of one 00:36:13.000 --> 00:36:17.000 molecule, Z1, times the total number of 00:36:17.000 --> 00:36:21.000 molecules in the gas, N. 00:36:21.000 --> 00:36:25.000 That is what N stands for. But since each collision 00:36:25.000 --> 00:36:31.000 involves two molecules, I am going to have to multiply 00:36:31.000 --> 00:36:36.000 this by one-half. Otherwise, I am going to over 00:36:36.000 --> 00:36:41.000 count the number of collisions because each collision involves 00:36:41.000 --> 00:36:44.000 two molecules. The total collision frequency 00:36:44.000 --> 00:36:48.000 here is one-half times N times Z1. 00:36:48.000 --> 00:36:52.000 Therefore, if I go and plug in my expression for Z1 and N and 00:36:52.000 --> 00:36:56.000 simplify things, my total collision frequency 00:36:56.000 --> 00:37:01.000 here is one over the square root of two times N times rho pi d 00:37:01.000 --> 00:37:06.000 squared times the average speed. 00:37:06.000 --> 00:37:11.000 That is my total collisions per 00:37:11.000 --> 00:37:15.000 second. This is not rocket science. 00:37:15.000 --> 00:37:20.000 This is easy. What I want you to realize is 00:37:20.000 --> 00:37:26.000 that you do have to understand what N is, what rho is, 00:37:26.000 --> 00:37:32.000 what d is, what vbar is. And I am always surprised on an 00:37:32.000 --> 00:37:38.000 exam, when we are going to give you this equation, 00:37:38.000 --> 00:37:43.000 that students don't actually know what rho is, 00:37:43.000 --> 00:37:48.000 or N is, or v is, or d is. 00:37:48.000 --> 00:37:51.000 This is easy. You do just have to understand, 00:37:51.000 --> 00:37:54.000 this is the total number of molecules in the gas, 00:37:54.000 --> 00:37:58.000 this is the density of the gas in molecules per cubic meter, 00:37:58.000 --> 00:38:02.000 the diameter of the molecule, the average speed. 00:38:02.000 --> 00:38:07.000 That was just a helpful hint. This is the upper limit to the 00:38:07.000 --> 00:38:10.000 reaction rate. When somebody tells you, 00:38:10.000 --> 00:38:15.000 the rate of this reaction is so and so many molecules per cubic 00:38:15.000 --> 00:38:18.000 meter per second, what you can do is a 00:38:18.000 --> 00:38:24.000 back-of-the-envelope calculation to see whether or not they are 00:38:24.000 --> 00:38:30.000 telling you the reaction rate is greater than this number. 00:38:30.000 --> 00:38:35.000 If it is, you can say ah-ha, got you, it cannot possibly be. 00:38:35.000 --> 00:38:40.000 Really important. If a reaction has this rate, 00:38:40.000 --> 00:38:44.000 it will mean that the probability of the reaction 00:38:44.000 --> 00:38:49.000 occurring is one. If the reaction has that rate, 00:38:49.000 --> 00:38:52.000 we call that the gas kinetic rate. 00:38:52.000 --> 00:38:55.000 That is a term that we also use. 00:38:55.000 --> 00:38:59.000 Then, finally, one other quantity from the gas 00:38:59.000 --> 00:39:03.000 kinetic theory. Yes? 00:39:10.000 --> 00:39:13.000 She asked, what velocity, here, would you use? 00:39:13.000 --> 00:39:17.000 You actually are going to, in that particular case, 00:39:17.000 --> 00:39:21.000 have to take the average of the average velocities. 00:39:21.000 --> 00:39:24.000 In other words, if you had two different 00:39:24.000 --> 00:39:29.000 molecules that were reacting for this average velocity here, 00:39:29.000 --> 00:39:33.000 you are going to have to use the average of the average 00:39:33.000 --> 00:39:38.000 velocity. And then your error bars on the 00:39:38.000 --> 00:39:44.000 experiment will always be large enough to take that into 00:39:44.000 --> 00:39:47.000 consideration. Good question. 00:39:47.000 --> 00:39:52.000 One other quantity, something called the mean free 00:39:52.000 --> 00:39:55.000 path. What I want to know, 00:39:55.000 --> 00:40:00.000 here, is on the average, how far does the molecule 00:40:00.000 --> 00:40:07.000 travel in the gas before it suffers a collision? 00:40:07.000 --> 00:40:09.000 That is what a mean free path is. 00:40:09.000 --> 00:40:13.000 A mean free path in solution. Sometimes you will do a solid 00:40:13.000 --> 00:40:17.000 state physics course and will hear about the electron mean 00:40:17.000 --> 00:40:20.000 free path. A mean free path is always the 00:40:20.000 --> 00:40:22.000 distance traveled between collisions. 00:40:22.000 --> 00:40:26.000 That is what that is. And we are going to call it 00:40:26.000 --> 00:40:30.000 lambda. This is not wavelength anymore. 00:40:30.000 --> 00:40:33.000 We just changed our definition of lambda. 00:40:33.000 --> 00:40:36.000 Lambda, here, is the average distance between 00:40:36.000 --> 00:40:40.000 collisions. How are we going to calculate 00:40:40.000 --> 00:40:43.000 that? What we are going to do is take 00:40:43.000 --> 00:40:47.000 the average distance that a molecule travels per unit time, 00:40:47.000 --> 00:40:52.000 per second, and are going to divide it by the number of 00:40:52.000 --> 00:40:55.000 collisions that occur per second. 00:40:55.000 --> 00:40:59.000 And you are going to see that the per seconds are going to 00:40:59.000 --> 00:41:03.000 cancel here. We are going to have the 00:41:03.000 --> 00:41:08.000 average distance per collision. That is what we are after. 00:41:08.000 --> 00:41:13.000 This is easy to do because the average distance traveled per 00:41:13.000 --> 00:41:16.000 second is simply the average velocity. 00:41:16.000 --> 00:41:20.000 It is meters per second. The average distance traveled 00:41:20.000 --> 00:41:24.000 per second. The number of collisions per 00:41:24.000 --> 00:41:27.000 second that happened, we just calculated that. 00:41:27.000 --> 00:41:33.000 That was Z1. We can substitute in Z1 vbar, 00:41:33.000 --> 00:41:38.000 vbar cancels, and what we are left with is 00:41:38.000 --> 00:41:46.000 one over the square root of 2 rho pi d squared. 00:41:46.000 --> 00:41:49.000 That is meters. 00:41:49.000 --> 00:41:56.000 That is the average distance the molecule traveled per 00:41:56.000 --> 00:42:01.000 collision. This is another general generic 00:42:01.000 --> 00:42:06.000 quantity useful in many cases, also in other kinds of cases 00:42:06.000 --> 00:42:10.000 that I just mentioned in solid state physics. 00:42:10.000 --> 00:42:16.000 What I want to do now is that we said we were going to start 00:42:16.000 --> 00:42:19.000 talking about the motions of molecules. 00:42:19.000 --> 00:42:24.000 We have taken care of, now, the translational motion. 00:42:24.000 --> 00:42:29.000 Now I want to start talking about the internal degrees of 00:42:29.000 --> 00:42:33.000 freedom. And let's do that. 00:42:33.000 --> 00:42:38.000 Let me just see if I have something, here. 00:42:38.000 --> 00:42:43.000 Let me raise the board, here. 00:42:58.000 --> 00:43:02.000 We have talked about this molecule, which is a set of 00:43:02.000 --> 00:43:07.000 atoms that are bonded together. And, as we always say, 00:43:07.000 --> 00:43:12.000 they are bonded together for the mutual comfort of their 00:43:12.000 --> 00:43:15.000 electrons. And we saw that it is not 00:43:15.000 --> 00:43:17.000 frozen. It moves. 00:43:17.000 --> 00:43:20.000 It has kinetic energy. It has velocity. 00:43:20.000 --> 00:43:25.000 The velocity generates this macroscopic property of 00:43:25.000 --> 00:43:30.000 pressure. But molecules also jiggle. 00:43:30.000 --> 00:43:33.000 For example, we are going to take here 00:43:33.000 --> 00:43:39.000 nitrogen, triple bond. That triple bond actually 00:43:39.000 --> 00:43:44.000 functions like a spring. We can think of that bond as a 00:43:44.000 --> 00:43:49.000 spring between the two atoms. That spring can stretch, 00:43:49.000 --> 00:43:54.000 and that spring can compress. When we tell you the 00:43:54.000 --> 00:44:00.000 equilibrium bond length of some molecule is some number, 00:44:00.000 --> 00:44:05.000 it is the equilibrium bond length. 00:44:05.000 --> 00:44:09.000 It is not the bond length when the molecule is stretched. 00:44:09.000 --> 00:44:14.000 It is not the bond length when the molecule is compressed. 00:44:14.000 --> 00:44:19.000 This motion is the vibrational motion of the molecule. 00:44:24.000 --> 00:44:30.000 So, we have that kind of internal motion in a molecule. 00:44:30.000 --> 00:44:33.000 Molecules also tumble. For example, 00:44:33.000 --> 00:44:38.000 let's take our nitrogen molecule, again. 00:44:38.000 --> 00:44:45.000 It rotates around an axis. Going through the center of the 00:44:45.000 --> 00:44:49.000 mass, it rotates around this axis. 00:44:49.000 --> 00:44:54.000 This nitrogen molecule also rotates around an axis, 00:44:54.000 --> 00:45:00.000 here, perpendicular to the board. 00:45:00.000 --> 00:45:06.000 It rotates in that direction. This is the rotational motion 00:45:06.000 --> 00:45:11.000 of the molecule. The different ways in which we 00:45:11.000 --> 00:45:15.000 can have this motion are called modes. 00:45:15.000 --> 00:45:21.000 This is a rotational mode. This is vibrational mode. 00:45:21.000 --> 00:45:28.000 We did not use the word before in talking about translation, 00:45:28.000 --> 00:45:33.000 but those are translational modes. 00:45:33.000 --> 00:45:40.000 Sometimes, we also call these modes degrees of freedom. 00:45:40.000 --> 00:45:49.000 A molecule has translational modes or degrees of freedom, 00:45:49.000 --> 00:45:57.000 vibrational degrees of freedom, and rotational degrees of 00:45:57.000 --> 00:46:00.000 freedom. Now, in general, 00:46:00.000 --> 00:46:10.000 if you have an N-atom molecule, you have 3N total modes. 00:46:10.000 --> 00:46:17.000 When I say total modes that means the sum of translation, 00:46:17.000 --> 00:46:24.000 rotation and vibration. Of those 3N total modes, 00:46:24.000 --> 00:46:30.000 three of them are translational modes. 00:46:35.000 --> 00:46:38.000 Why do we have three translational modes? 00:46:38.000 --> 00:46:43.000 We have three translational modes because we are operating 00:46:43.000 --> 00:46:48.000 in a three-dimensional space. If we have a molecule here, 00:46:48.000 --> 00:46:53.000 that molecule has motion in the x direction, in the y direction, 00:46:53.000 --> 00:46:57.000 and also in the z direction. Those are the three 00:46:57.000 --> 00:47:02.000 translational modes of the molecule. 00:47:02.000 --> 00:47:08.000 What that means then, if three of the modes are used 00:47:08.000 --> 00:47:13.000 up for translation, then we must have 3N minus 3 00:47:13.000 --> 00:47:20.000 internal modes. 3N minus 3 must be the number 00:47:20.000 --> 00:47:27.000 of modes that is leftover for the internal degrees of freedom 00:47:27.000 --> 00:47:33.000 for rotation and also for vibration. 00:47:33.000 --> 00:47:39.000 And what we are going to see, next time, is that these 00:47:39.000 --> 00:47:46.000 internal degrees of freedom, these internal modes are going 00:47:46.000 --> 00:47:51.000 to be quantized. That is, this molecule is going 00:47:51.000 --> 00:47:59.000 to be able to vibrate with only certain amounts of energy. 00:47:59.000 --> 00:48:02.000 There is going to be a ground vibrational state. 00:48:02.000 --> 00:48:07.000 There is going to be a first excited vibrational state. 00:48:07.000 --> 00:48:12.000 A second excited vibrational state, just like we talked about 00:48:12.000 --> 00:48:17.000 the energy levels of a hydrogen atom, where the electron was 00:48:17.000 --> 00:48:22.000 bound with different amounts of energy dictated by a principle 00:48:22.000 --> 00:48:26.000 quantum number. The vibrational degree of 00:48:26.000 --> 00:48:30.000 freedom is also going to be dictated by a vibrational 00:48:30.000 --> 00:48:33.000 quantum number. In addition, 00:48:33.000 --> 00:48:38.000 the rotations of the molecules, they are also quantized. 00:48:38.000 --> 00:48:43.000 A molecule is going to be able to rotate with only one energy, 00:48:43.000 --> 00:48:46.000 or another energy, or another energy, 00:48:46.000 --> 00:48:49.000 but not any energies in between. 00:48:49.000 --> 00:48:52.000 There are discrete rotational states. 00:48:52.000 --> 00:48:56.000 We are going to be talking about another kind of quantum 00:48:56.000 --> 00:48:58.000 number. In that case, 00:48:58.000 --> 00:49:04.000 a rotational quantum number. That is where we are going to 00:49:04.451 --> 00:49:07.000 pick up on Monday. See you then.