WEBVTT 00:00:00.030 --> 00:00:02.400 The following content is provided under a Creative 00:00:02.400 --> 00:00:03.780 Commons license. 00:00:03.780 --> 00:00:06.020 Your support will help MIT OpenCourseWare 00:00:06.020 --> 00:00:10.090 continue to offer high quality educational resources for free. 00:00:10.090 --> 00:00:12.670 To make a donation or to view additional materials 00:00:12.670 --> 00:00:16.580 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.580 --> 00:00:17.740 at osw.mit.edu. 00:00:40.850 --> 00:00:43.100 PROFESSOR: All right, let's just take 10 more seconds. 00:00:58.330 --> 00:01:02.540 All right, so someone want to explain 00:01:02.540 --> 00:01:05.650 why this is the correct answer? 00:01:05.650 --> 00:01:09.260 And we have a syringe highlighter. 00:01:09.260 --> 00:01:12.046 You probably never had something like this before. 00:01:17.090 --> 00:01:21.690 AUDIENCE: OK, so the fourth excited state is n equals 5. 00:01:21.690 --> 00:01:26.460 And then IE is opposite of the negative number shown. 00:01:26.460 --> 00:01:28.532 So it would be a positive reaction. 00:01:28.532 --> 00:01:29.240 PROFESSOR: Right. 00:01:29.240 --> 00:01:32.220 So IE is always going to be positive. 00:01:32.220 --> 00:01:35.600 And you have to pay attention to what n equals when 00:01:35.600 --> 00:01:38.390 you're in the excited state. 00:01:41.460 --> 00:01:44.750 So we've been talking about the hydrogen atom and binding 00:01:44.750 --> 00:01:45.600 energies. 00:01:45.600 --> 00:01:47.950 What comes out of the Schrodinger equation? 00:01:47.950 --> 00:01:50.020 We have the binding energies that come out. 00:01:50.020 --> 00:01:51.914 And we also have wave functions. 00:01:51.914 --> 00:01:53.330 So today we're going to be talking 00:01:53.330 --> 00:01:55.040 about wave functions, which are often 00:01:55.040 --> 00:01:57.500 referred to as orbitals in chemistry, 00:01:57.500 --> 00:02:00.130 for the hydrogen atom. 00:02:00.130 --> 00:02:03.820 So when you solve the Schrodinger equation, 00:02:03.820 --> 00:02:07.880 you get out this information about wave functions. 00:02:07.880 --> 00:02:11.610 And what comes out of it is these quantum numbers. 00:02:11.610 --> 00:02:15.170 And we already saw quantum number n coming out. 00:02:15.170 --> 00:02:16.860 But there are three quantum numbers 00:02:16.860 --> 00:02:19.640 that are going to come out of the Schrodinger equation. 00:02:19.640 --> 00:02:21.620 And those three quantum numbers are 00:02:21.620 --> 00:02:26.280 necessary to describe the wave function or the orbital. 00:02:26.280 --> 00:02:30.650 So we have n, the principle quantum number. 00:02:30.650 --> 00:02:32.950 We've already talked about that. 00:02:32.950 --> 00:02:36.750 And we've already seen that n is an integer. 00:02:36.750 --> 00:02:38.930 So I'll just put that down here. 00:02:38.930 --> 00:02:46.760 So n can equal 1, 2, 3, on to infinity. 00:02:46.760 --> 00:02:51.620 So this describes the energy level or the shell. 00:02:51.620 --> 00:02:55.650 Then we have l, which we haven't talked about yet. 00:02:55.650 --> 00:02:58.870 So that's the angular momentum quantum number. 00:02:58.870 --> 00:03:01.440 So it tells you about the angular momentum. 00:03:01.440 --> 00:03:04.260 It also tells you about the subshell 00:03:04.260 --> 00:03:07.990 or the shape of the orbital. 00:03:07.990 --> 00:03:11.760 And so l is related to n. 00:03:11.760 --> 00:03:22.580 And it can be 0, 1, 2, 3, onward to n minus 1. 00:03:22.580 --> 00:03:27.340 So its biggest number is n minus 1. 00:03:27.340 --> 00:03:33.150 Then we have m, the magnetic quantum number. 00:03:33.150 --> 00:03:37.065 And we often see this also listed as m sub 00:03:37.065 --> 00:03:43.040 l because m is related back to l. 00:03:43.040 --> 00:03:50.150 And this is equal to minus l, dot, dot, dot, to 0, 00:03:50.150 --> 00:03:55.190 dot, dot, dot to plus l. 00:03:55.190 --> 00:03:59.370 And m describes the behavior in a magnetic field. 00:03:59.370 --> 00:04:02.150 It also describes the orientation 00:04:02.150 --> 00:04:06.380 of the orbital with respect to an axes. 00:04:06.380 --> 00:04:10.540 And it tells you about the specific orbital in question. 00:04:10.540 --> 00:04:16.209 So we need all three of these to describe any orbital. 00:04:16.209 --> 00:04:22.410 All right, so let's look at this in a slightly other way. 00:04:22.410 --> 00:04:26.220 So we're going to have lots of different sort of nomenclatures 00:04:26.220 --> 00:04:27.870 for the same thing. 00:04:27.870 --> 00:04:30.761 So to describe an orbital, we need those three quantum 00:04:30.761 --> 00:04:31.260 numbers. 00:04:31.260 --> 00:04:34.490 We need n, l, and m. 00:04:34.490 --> 00:04:43.360 And this can also be expressed as our wave function sub nlm. 00:04:43.360 --> 00:04:45.370 And again, we talked about this last time. 00:04:45.370 --> 00:04:47.400 We're going to talk more about it. 00:04:47.400 --> 00:04:49.970 So our wave function is also described 00:04:49.970 --> 00:04:54.560 by r, the radius, and theta and phi, which are two angles. 00:04:54.560 --> 00:04:57.450 And we're going to talk a lot about those today. 00:04:57.450 --> 00:05:00.900 So the wave function for the ground state 00:05:00.900 --> 00:05:05.950 is abbreviated wave function sub 1, 0, 0. 00:05:05.950 --> 00:05:07.540 Because it's the ground state. 00:05:07.540 --> 00:05:12.330 So n equals 1, and l and m are 0. 00:05:12.330 --> 00:05:15.450 So what you see down here, the 1, 0, 0, 00:05:15.450 --> 00:05:20.940 refers back to what is n, what is l, what is m. 00:05:20.940 --> 00:05:24.320 And this also has another name. 00:05:24.320 --> 00:05:27.150 So in the terminology of chemists, 00:05:27.150 --> 00:05:34.910 we call the wave function 1, 0, 0 1s, or the 1s orbital. 00:05:34.910 --> 00:05:37.800 So let's look again now at the same things 00:05:37.800 --> 00:05:40.200 we just talked about, but going through kind 00:05:40.200 --> 00:05:43.360 of chemistry lingo. 00:05:43.360 --> 00:05:48.380 So again, n describes the shell or the energy level. 00:05:48.380 --> 00:05:53.930 Again, it's integers, 1, 2, 3, et cetera. 00:05:53.930 --> 00:05:57.630 l in chemistry lingo, the subshell or the shape 00:05:57.630 --> 00:05:58.930 of the orbital. 00:05:58.930 --> 00:06:01.960 And instead of listing it this way, 00:06:01.960 --> 00:06:04.890 we have another way to list it if we're a chemist, 00:06:04.890 --> 00:06:10.120 and that is s, p, d, f, et cetera. 00:06:10.120 --> 00:06:13.810 So chemists like numbers, but we also throw in some letters 00:06:13.810 --> 00:06:15.890 every once in a while. 00:06:15.890 --> 00:06:20.310 And then m, again, designates this orbital orientation 00:06:20.310 --> 00:06:22.700 or the specific orbital. 00:06:22.700 --> 00:06:24.920 So for s, there's only s. 00:06:24.920 --> 00:06:29.620 It doesn't have any other designation, 00:06:29.620 --> 00:06:31.090 as we'll talk more about later. 00:06:31.090 --> 00:06:36.200 But for p, we start having suborbitals. 00:06:36.200 --> 00:06:38.030 And there is a difference in terms 00:06:38.030 --> 00:06:40.550 of the orientation of this. 00:06:40.550 --> 00:06:44.870 So we have px, py, pz. 00:06:44.870 --> 00:06:46.740 So that's what m tells us about. 00:06:46.740 --> 00:06:49.020 So if we have all three of these numbers, 00:06:49.020 --> 00:06:50.570 we get down to the specific orbital, 00:06:50.570 --> 00:06:54.220 we can say oh, that's pz, for example. 00:06:54.220 --> 00:06:58.880 So we need all of these three numbers to define the orbital. 00:06:58.880 --> 00:07:02.370 And this is in then the chemistry lingo. 00:07:02.370 --> 00:07:06.010 All right, also a little bit more chemistry lingo. 00:07:06.010 --> 00:07:11.090 So here we have l equals 0. 00:07:11.090 --> 00:07:13.053 So that is the s orbital. 00:07:16.230 --> 00:07:21.810 When l equals 1, that's the p orbital. 00:07:25.290 --> 00:07:30.020 l equals 2 is the d orbital. 00:07:30.020 --> 00:07:33.880 And l equals 3 is the f orbital. 00:07:33.880 --> 00:07:37.460 And frankly we don't really go much beyond that. 00:07:37.460 --> 00:07:39.010 And in this part of the course, we're 00:07:39.010 --> 00:07:43.680 really only going to be talking mostly about s and p orbitals. 00:07:43.680 --> 00:07:47.390 We get to d orbitals around Thanksgiving time. 00:07:47.390 --> 00:07:49.050 So you can look forward to that. 00:07:49.050 --> 00:07:51.770 And pretty much we're not going to really talk about f orbitals 00:07:51.770 --> 00:07:53.279 very much at all. 00:07:53.279 --> 00:07:55.070 You'll need to know some things about them, 00:07:55.070 --> 00:07:58.670 but we're not going to go into them in any kind of detail. 00:07:58.670 --> 00:08:01.640 All right, so if we keep going then, 00:08:01.640 --> 00:08:06.340 we can think about l equals 1 or our p orbitals. 00:08:06.340 --> 00:08:15.260 And then when l equals 1, then m can equal 0 plus 1 or minus 1. 00:08:15.260 --> 00:08:20.710 And when m equals 0, that's by definition the pz orbital. 00:08:20.710 --> 00:08:25.910 So when you see m equals 0, that's going to be pz. 00:08:25.910 --> 00:08:29.370 And when m is plus 1 or minus 1, those 00:08:29.370 --> 00:08:32.530 are the px or the py orbitals. 00:08:32.530 --> 00:08:35.090 And this is just something that you need to remember, 00:08:35.090 --> 00:08:37.330 that z is the one that's special. 00:08:37.330 --> 00:08:41.940 It's the one that has m equals 0. 00:08:41.940 --> 00:08:45.630 All right, so we can take all of the nomenclatures 00:08:45.630 --> 00:08:50.660 now and use it to fill in this awesome table. 00:08:50.660 --> 00:08:52.340 So this will help you kind of keep 00:08:52.340 --> 00:08:54.360 track of all the different ways you 00:08:54.360 --> 00:08:56.810 can designate the same things. 00:08:56.810 --> 00:08:58.460 And we'll fill this in. 00:08:58.460 --> 00:09:01.470 So first, state label. 00:09:01.470 --> 00:09:03.234 What do I mean by this? 00:09:03.234 --> 00:09:10.610 By this I mean this one 1, 0, 0 to generate this wave function 00:09:10.610 --> 00:09:15.680 where we have this 1, 0, 0 listed below the wave function 00:09:15.680 --> 00:09:16.380 here. 00:09:16.380 --> 00:09:19.490 And so now this is just a little color coding. 00:09:19.490 --> 00:09:21.460 But it's blank in your handout. 00:09:21.460 --> 00:09:24.380 So n equals 1, so n is first. 00:09:24.380 --> 00:09:26.520 l is the second number. 00:09:26.520 --> 00:09:29.030 And m is the third here. 00:09:29.030 --> 00:09:33.640 So 1, 0, 0, and what kind of orbital is this? 00:09:33.640 --> 00:09:35.316 You can just yell it out. 00:09:35.316 --> 00:09:36.150 AUDIENCE: 1s 00:09:36.150 --> 00:09:39.250 PROFESSOR: Yep, so that's the 1s orbital. 00:09:39.250 --> 00:09:44.070 And so the 1, n equals 1, that's 1s. 00:09:44.070 --> 00:09:48.600 And now we have our binding energies again. 00:09:48.600 --> 00:09:51.550 And so we can write those in two different ways. 00:09:51.550 --> 00:09:54.260 So we saw for the hydrogen atom before what 00:09:54.260 --> 00:09:56.430 comes out of the Schrodinger equation, 00:09:56.430 --> 00:09:59.440 that the binding energy of the electron for the nucleus 00:09:59.440 --> 00:10:04.340 is minus the Rydberg constant RH, divided by n squared. 00:10:04.340 --> 00:10:07.590 And here n is 1, so divided by 1 squared. 00:10:07.590 --> 00:10:11.700 So this is just the value for the Rydberg constant, 00:10:11.700 --> 00:10:12.910 the negative value. 00:10:12.910 --> 00:10:16.140 And binding energies, again, are always negative. 00:10:16.140 --> 00:10:19.000 So we have our first one down. 00:10:19.000 --> 00:10:22.920 So now for the second, what number 00:10:22.920 --> 00:10:25.120 am I going to write here for the state label? 00:10:25.120 --> 00:10:27.858 You can just yell it out. 00:10:27.858 --> 00:10:30.720 Yep, 200 or 2, 0, 0. 00:10:30.720 --> 00:10:32.880 And then you would put it this way 00:10:32.880 --> 00:10:36.210 where the state label is by the wave function. 00:10:36.210 --> 00:10:39.740 What orbital is this-- 2s. 00:10:39.740 --> 00:10:43.120 And then we also know the binding energies for this. 00:10:43.120 --> 00:10:46.670 So here we have minus RH over n squared 00:10:46.670 --> 00:10:48.900 where n is 2, 2 squared. 00:10:48.900 --> 00:10:51.840 And we saw this number last time. 00:10:51.840 --> 00:10:53.220 So we can keep going. 00:10:53.220 --> 00:10:56.390 Now we have 2, 1, 1. 00:10:56.390 --> 00:10:58.330 So we can write that down. 00:10:58.330 --> 00:11:00.240 We can write it both ways. 00:11:00.240 --> 00:11:02.876 What orbital is this? 00:11:02.876 --> 00:11:06.070 AUDIENCE: [INAUDIBLE]. 00:11:06.070 --> 00:11:08.440 PROFESSOR: So it's a 2p. 00:11:08.440 --> 00:11:14.522 And because n is plus 1 and not 0, it's either x or y. 00:11:17.430 --> 00:11:22.940 Do we have a different or the same binding energy here? 00:11:22.940 --> 00:11:26.830 We have the same, right, because it's just over n squared. 00:11:26.830 --> 00:11:30.790 We're still talking about n equals 2, so 2 squared. 00:11:30.790 --> 00:11:33.020 So it's the same value here. 00:11:33.020 --> 00:11:36.090 Now we have m equals 0. 00:11:36.090 --> 00:11:38.760 So we write 2, 1, 0. 00:11:38.760 --> 00:11:41.080 And now what is that orbital? 00:11:41.080 --> 00:11:43.310 AUDIENCE: [INAUDIBLE]. 00:11:43.310 --> 00:11:45.870 PROFESSOR: 2pz, right, because that's m 00:11:45.870 --> 00:11:48.630 equals 0, by the definition I gave you. 00:11:48.630 --> 00:11:51.380 So we know that one for sure. 00:11:51.380 --> 00:11:55.210 And again, the energies are going to be the same. 00:11:55.210 --> 00:12:01.400 And then the last one, so now we write 2, 1, minus 1. 00:12:01.400 --> 00:12:04.930 And now it's again a 2p orbital. 00:12:04.930 --> 00:12:08.180 And it's either y or x. 00:12:08.180 --> 00:12:11.780 And the energies are going to be the same. 00:12:11.780 --> 00:12:13.550 So these are just a table that kind 00:12:13.550 --> 00:12:15.600 of interconverts different ways that you 00:12:15.600 --> 00:12:17.210 will see things written. 00:12:17.210 --> 00:12:20.320 And you'll know if you see it one way, what 00:12:20.320 --> 00:12:21.860 orbital to put down. 00:12:21.860 --> 00:12:24.680 And we can also think about the binding energies 00:12:24.680 --> 00:12:27.410 for those particular orbitals, or for electrons 00:12:27.410 --> 00:12:30.600 in those particular orbitals. 00:12:30.600 --> 00:12:34.383 All right, so why don't you try a clicker question on this? 00:13:10.030 --> 00:13:10.600 10 seconds. 00:13:24.442 --> 00:13:26.520 Ah, excellent. 00:13:26.520 --> 00:13:27.380 Right. 00:13:27.380 --> 00:13:29.230 So you're getting the hang of this. 00:13:29.230 --> 00:13:29.840 It's great. 00:13:29.840 --> 00:13:31.140 Some things, it's always nice when 00:13:31.140 --> 00:13:33.306 there's some things that are pretty straightforward. 00:13:33.306 --> 00:13:35.350 So n equals 5. 00:13:35.350 --> 00:13:40.160 l equals 1, which means p orbital and m equals 0, 00:13:40.160 --> 00:13:40.914 means pz. 00:13:44.390 --> 00:13:47.950 So let's think now about these orbitals again. 00:13:47.950 --> 00:13:51.470 And we looked at that table and saw 00:13:51.470 --> 00:13:54.730 that if we were talking about n equals 2, 00:13:54.730 --> 00:13:57.050 they all seem to have the same energy. 00:13:57.050 --> 00:14:01.240 So for a hydrogen atom-- and it will get more complicated 00:14:01.240 --> 00:14:02.780 when we start talking about things 00:14:02.780 --> 00:14:04.350 with more than one electron. 00:14:04.350 --> 00:14:08.930 But for a hydrogen atom, orbitals that have the same n 00:14:08.930 --> 00:14:11.730 value have the same energy. 00:14:11.730 --> 00:14:15.180 So here we have n equals 1, l equals 0. 00:14:15.180 --> 00:14:17.600 This is our 1s. 00:14:17.600 --> 00:14:22.550 We have n equals 2, our 2s, and our 2p orbitals. 00:14:22.550 --> 00:14:28.660 n equals 3, we have our 3s, 3p, and 3d. 00:14:28.660 --> 00:14:32.840 And in this case, all these orbitals 00:14:32.840 --> 00:14:37.220 are what's known as degenerate with respect to each other. 00:14:37.220 --> 00:14:39.680 They have the same energy. 00:14:39.680 --> 00:14:46.290 And so for any n with a hydrogen atom, or any one electron 00:14:46.290 --> 00:14:51.650 system, for n shells, there n square degenerate-- 00:14:51.650 --> 00:14:54.370 or for any n there are n squared generate orbitals. 00:14:54.370 --> 00:14:56.220 So they're all going to be the same energy. 00:14:56.220 --> 00:14:59.420 And that changes when we go to more complicated systems. 00:14:59.420 --> 00:15:02.770 But for hydrogen, this holds. 00:15:02.770 --> 00:15:05.770 So now I'm going to tell you why you 00:15:05.770 --> 00:15:09.630 should care a little about these energy levels again. 00:15:09.630 --> 00:15:15.110 And today you're going to hear in their own words 00:15:15.110 --> 00:15:20.790 from a graduate student in the physical chemistry division. 00:15:20.790 --> 00:15:21.456 [VIDEO PLAYBACK] 00:15:21.456 --> 00:15:23.940 - My name is Benjamin Ofori-Okai. 00:15:23.940 --> 00:15:25.870 I'm entering my third year of graduate school 00:15:25.870 --> 00:15:28.380 in the chemistry department here at MIT. 00:15:28.380 --> 00:15:29.880 And the work that I've been focusing 00:15:29.880 --> 00:15:31.546 on for the last couple of years involves 00:15:31.546 --> 00:15:35.680 nanoscale magnetic resonance imaging or nano MRI. 00:15:35.680 --> 00:15:37.332 When you think of typical MRI, what 00:15:37.332 --> 00:15:39.540 comes to mind for most people is the image of a brain 00:15:39.540 --> 00:15:41.930 scan or a heart scan or some sort of organ 00:15:41.930 --> 00:15:44.930 scan inside the human body. 00:15:44.930 --> 00:15:47.570 The way that MRI works now, the way that you 00:15:47.570 --> 00:15:50.515 take a picture of anything in your body is you use water. 00:15:50.515 --> 00:15:51.890 And the reason that you use water 00:15:51.890 --> 00:15:54.660 is because it's made up of hydrogen atoms and oxygen 00:15:54.660 --> 00:15:55.390 atoms. 00:15:55.390 --> 00:15:58.370 And hydrogen atoms actually generate a magnetic signal. 00:15:58.370 --> 00:16:01.030 And so you can take a picture of that. 00:16:01.030 --> 00:16:06.290 The idea behind nano MRI is that you want to take a picture. 00:16:06.290 --> 00:16:08.060 You want to do the same kind of imaging, 00:16:08.060 --> 00:16:11.480 but on a considerably smaller scale. 00:16:11.480 --> 00:16:13.350 We have this probe which is sensitive 00:16:13.350 --> 00:16:16.432 to local magnetic fields. 00:16:16.432 --> 00:16:17.890 And the way that the probe works is 00:16:17.890 --> 00:16:19.610 that you have these electrons. 00:16:19.610 --> 00:16:21.540 There's a ground state for these electrons 00:16:21.540 --> 00:16:24.230 and two excited states for these electrons, which are actually 00:16:24.230 --> 00:16:25.490 degenerate with each other. 00:16:25.490 --> 00:16:28.250 And degenerate means that they just have the exact same energy 00:16:28.250 --> 00:16:29.460 level. 00:16:29.460 --> 00:16:31.430 As you move the probe around, anything 00:16:31.430 --> 00:16:34.060 that's in the environment that generates a magnetic field 00:16:34.060 --> 00:16:35.900 will change what the energy levels 00:16:35.900 --> 00:16:38.350 of these two excited states is. 00:16:38.350 --> 00:16:40.180 So when you're far away, there's no change 00:16:40.180 --> 00:16:41.456 and they're exactly the same. 00:16:41.456 --> 00:16:42.830 And as you get closer and closer, 00:16:42.830 --> 00:16:45.120 these levels start to split. 00:16:45.120 --> 00:16:46.832 And what we actually care about is 00:16:46.832 --> 00:16:48.790 what is the splitting between these two levels, 00:16:48.790 --> 00:16:53.650 because that's what tells us what the magnetic field is. 00:16:53.650 --> 00:16:55.550 In traditional MRI, the probe that we 00:16:55.550 --> 00:16:57.530 use, the thing that measures the fields, 00:16:57.530 --> 00:16:58.850 itself is very, very big. 00:16:58.850 --> 00:17:00.790 It's person sized. 00:17:00.790 --> 00:17:03.430 The probe that we're using in this nano MRI 00:17:03.430 --> 00:17:04.995 is nanometer sized. 00:17:04.995 --> 00:17:07.119 So this gives us the ability to look at things that 00:17:07.119 --> 00:17:08.750 are on the nanometer scale. 00:17:08.750 --> 00:17:11.660 And to give you a sense of size, that's like 1/10,000 the width 00:17:11.660 --> 00:17:13.410 of a human hair. 00:17:13.410 --> 00:17:17.990 So that includes viruses, cells, parts of proteins, 00:17:17.990 --> 00:17:19.480 not just the entire protein. 00:17:19.480 --> 00:17:22.481 And on top of that, we'll be able to look within objects. 00:17:22.481 --> 00:17:24.730 So you're not just sensitive to what's on the surface. 00:17:24.730 --> 00:17:26.869 You can actually see how are things-- 00:17:26.869 --> 00:17:27.869 what's the constitution? 00:17:27.869 --> 00:17:30.220 What's the makeup of things within the object that you 00:17:30.220 --> 00:17:31.330 want to image? 00:17:31.330 --> 00:17:33.550 So the long term goal, the one thing 00:17:33.550 --> 00:17:36.820 that I'd really love to see this technology be able to do 00:17:36.820 --> 00:17:38.910 is say, OK, we've got this virus. 00:17:38.910 --> 00:17:40.450 Let's just see how it works. 00:17:40.450 --> 00:17:42.280 Let's watch it in real time. 00:17:42.280 --> 00:17:44.990 Let's see if we can see how it attaches to cells 00:17:44.990 --> 00:17:48.270 and invades them and ultimately kills them. 00:17:48.270 --> 00:17:51.580 [END PLAYBACK] 00:17:51.580 --> 00:17:54.470 PROFESSOR: OK, so I always think this is a great time of year 00:17:54.470 --> 00:17:58.210 to show this video because pretty much viruses, I think, 00:17:58.210 --> 00:17:59.970 start to be on people's minds. 00:17:59.970 --> 00:18:03.320 Everyone has sinuses and colds and other things going on. 00:18:03.320 --> 00:18:07.310 And so understanding, we're still very far away 00:18:07.310 --> 00:18:10.550 from having a real cure for the common cold. 00:18:10.550 --> 00:18:19.430 So I think it's very timely to be talking about, 00:18:19.430 --> 00:18:21.620 talking about this research. 00:18:21.620 --> 00:18:25.000 I'll also use this to remind myself to tell you 00:18:25.000 --> 00:18:29.430 that if you qualify for extra time on the exam, 00:18:29.430 --> 00:18:31.410 you should get me your form for the exam. 00:18:31.410 --> 00:18:34.420 And it reminded me to say that because Ben, 00:18:34.420 --> 00:18:36.010 who is a former TA for this class, 00:18:36.010 --> 00:18:38.100 always proctors the extra time folks. 00:18:38.100 --> 00:18:40.340 So you'll get to meet him in real life 00:18:40.340 --> 00:18:44.830 if you qualify for extra time on exams. 00:18:44.830 --> 00:18:48.560 So hydrogen is in fact important. 00:18:48.560 --> 00:18:51.590 I'm excited to get on to elements that 00:18:51.590 --> 00:18:53.550 have more than one electron. 00:18:53.550 --> 00:18:56.530 But hydrogen actually does turn out to be extremely important. 00:18:56.530 --> 00:18:59.677 A lot of imaging, as you heard from Ben, is based on hydrogen. 00:18:59.677 --> 00:19:01.510 So we're spending a lot of time on hydrogen, 00:19:01.510 --> 00:19:05.500 but hydrogen really, really is an important element. 00:19:05.500 --> 00:19:11.320 So continuing on now, what is the significance 00:19:11.320 --> 00:19:12.583 of this wave function? 00:19:15.100 --> 00:19:17.780 Why do we care about this? 00:19:17.780 --> 00:19:20.120 And so really, we're interested in trying 00:19:20.120 --> 00:19:23.010 to understand not just how tightly the electron is 00:19:23.010 --> 00:19:27.420 bound to the nucleus, but kind of how the electrons exist 00:19:27.420 --> 00:19:29.700 around the nucleus. 00:19:29.700 --> 00:19:32.250 And so the wave function really gets at this. 00:19:32.250 --> 00:19:36.470 It gets at the probability density, the likelihood 00:19:36.470 --> 00:19:41.150 that you'll find an electron at a certain location, 00:19:41.150 --> 00:19:43.970 the probability per unit volume. 00:19:43.970 --> 00:19:46.820 And again, this is a three dimensional problem. 00:19:46.820 --> 00:19:50.840 So our wave function depends on a radius r. 00:19:50.840 --> 00:19:54.510 But it also depends on two angles, the theta and phi. 00:19:54.510 --> 00:19:57.510 And so you can kind of think of those as latitude and longitude 00:19:57.510 --> 00:19:58.350 if you will. 00:19:58.350 --> 00:20:02.640 And so we want to know what the probability is 00:20:02.640 --> 00:20:07.760 that an electron will be at a certain r, theta, and phi 00:20:07.760 --> 00:20:13.350 position in a particular small unit volume in that area. 00:20:13.350 --> 00:20:17.840 How well can we understand where the electron is? 00:20:17.840 --> 00:20:21.900 And this gives rise to a lot of the properties of the elements. 00:20:21.900 --> 00:20:27.380 So probability density, density per unit volume. 00:20:27.380 --> 00:20:30.590 So really, when we're talking about where electrons are, 00:20:30.590 --> 00:20:33.170 we're thinking about a shape of an orbital, 00:20:33.170 --> 00:20:37.670 a shape of a probability density of where that electron might 00:20:37.670 --> 00:20:38.620 be. 00:20:38.620 --> 00:20:42.500 So now we're going to think about shapes. 00:20:42.500 --> 00:20:46.120 So we can define a wave function in terms 00:20:46.120 --> 00:20:50.250 of two properties, a radial wave function and an angular wave 00:20:50.250 --> 00:20:51.030 function. 00:20:51.030 --> 00:20:54.270 So again, the wave function has these three things. 00:20:54.270 --> 00:20:57.220 We are considered with a radius and these two angles. 00:20:57.220 --> 00:20:59.990 So we can rewrite this, breaking up 00:20:59.990 --> 00:21:03.240 these two different components-- the radial component that 00:21:03.240 --> 00:21:05.510 depends on the radius-- so that's easy to remember, 00:21:05.510 --> 00:21:08.750 radial, radius-- and the angular component 00:21:08.750 --> 00:21:10.930 that depends on the angles. 00:21:10.930 --> 00:21:13.860 So the nomenclature here is pretty good. 00:21:13.860 --> 00:21:16.870 All right, so we have these two components. 00:21:16.870 --> 00:21:20.140 So now I'm going to show you a table that 00:21:20.140 --> 00:21:22.540 is largely from your book. 00:21:22.540 --> 00:21:24.040 Don't let it scare you. 00:21:24.040 --> 00:21:27.210 You do not need to memorize any of these things. 00:21:27.210 --> 00:21:28.930 And I'm showing this to you because I 00:21:28.930 --> 00:21:33.800 want you to believe me about certain properties of these two 00:21:33.800 --> 00:21:34.490 functions. 00:21:34.490 --> 00:21:35.610 So here they are solved. 00:21:35.610 --> 00:21:37.020 You can look them up. 00:21:37.020 --> 00:21:39.840 Actually I think we just typed a new copy of this 00:21:39.840 --> 00:21:41.070 so it was easier to see. 00:21:41.070 --> 00:21:44.002 If you find any typos, please let me know. 00:21:44.002 --> 00:21:45.710 But there's a couple of important points. 00:21:45.710 --> 00:21:49.380 So on this side, we have the radial wave function, 00:21:49.380 --> 00:21:52.620 and over here we have the angular wave function, 00:21:52.620 --> 00:21:55.070 for various values of n and l. 00:21:55.070 --> 00:21:58.010 So again, not an exhaustive list here. 00:21:58.010 --> 00:22:01.890 And a lot of these are written in terms of a0, 00:22:01.890 --> 00:22:09.960 which is the Bohr radius, which is a constant, 52.9 picometers. 00:22:09.960 --> 00:22:14.750 All right, so now let's just consider the ground state. 00:22:14.750 --> 00:22:16.990 So we'll start with that lowest energy 00:22:16.990 --> 00:22:20.860 state or most stable state, the 1s orbital for the hydrogen 00:22:20.860 --> 00:22:21.640 atom. 00:22:21.640 --> 00:22:25.360 So we have our wave function 1, 0, 0 here. 00:22:25.360 --> 00:22:27.700 And this is 1s up here. 00:22:27.700 --> 00:22:29.990 Again, n equals 1. l equals 0. 00:22:29.990 --> 00:22:31.960 So that's 1s. 00:22:31.960 --> 00:22:34.520 And z for hydrogen atom is 1. 00:22:34.520 --> 00:22:38.280 So I've gotten rid of all z's to make it a little simpler. 00:22:38.280 --> 00:22:40.470 So here we have the radial wave function 00:22:40.470 --> 00:22:44.730 times the angular wave function, which is listed up here. 00:22:44.730 --> 00:22:47.590 And the thing that I really want you to notice 00:22:47.590 --> 00:22:52.440 is that for all of the s orbitals, this is a constant. 00:22:52.440 --> 00:22:57.590 So this is always the angular component for all s orbitals. 00:22:57.590 --> 00:23:03.630 And in fact, there are no angular components in there. 00:23:03.630 --> 00:23:09.390 So all 1s, 2s, 3s, all have this same constant. 00:23:09.390 --> 00:23:11.890 And that leads to a very important property 00:23:11.890 --> 00:23:14.270 of s orbitals, which is that they're 00:23:14.270 --> 00:23:16.800 spherically symmetrical. 00:23:16.800 --> 00:23:20.580 In other words, they're independent of those angles, 00:23:20.580 --> 00:23:22.950 of theta and phi. 00:23:22.950 --> 00:23:26.010 And so that means that the probability of finding 00:23:26.010 --> 00:23:28.470 the electron away from the nucleus 00:23:28.470 --> 00:23:31.170 is just going to depend on r. 00:23:31.170 --> 00:23:33.680 There's only r in this equation. 00:23:33.680 --> 00:23:36.070 The angles are not part of the equation. 00:23:36.070 --> 00:23:40.250 So s is spherically symmetrical. 00:23:40.250 --> 00:23:42.250 The probability of finding the electron 00:23:42.250 --> 00:23:45.480 just depends on the radius. 00:23:45.480 --> 00:23:49.650 So we can draw a picture, or multiple pictures, 00:23:49.650 --> 00:23:52.060 of what that could look like. 00:23:52.060 --> 00:23:55.060 And these are three common plots. 00:23:55.060 --> 00:23:57.020 So I'll tell you that on your handout, 00:23:57.020 --> 00:23:59.030 the plots are listed on one page, 00:23:59.030 --> 00:24:01.670 and then the plots are shown on the next page. 00:24:01.670 --> 00:24:05.010 And I'm going to kind of go back and forth between things. 00:24:05.010 --> 00:24:08.360 So the plots-- don't have to write this down. 00:24:08.360 --> 00:24:09.540 They're on the other page. 00:24:09.540 --> 00:24:13.230 But if you want to pay attention to which kind of plot 00:24:13.230 --> 00:24:16.280 goes with which plot. 00:24:16.280 --> 00:24:19.310 So these are three different ways to, quote, visualize. 00:24:19.310 --> 00:24:24.100 And some people say, can you give me another visualization? 00:24:24.100 --> 00:24:27.130 We're really just trying to think about probabilities 00:24:27.130 --> 00:24:28.880 of finding electrons here. 00:24:28.880 --> 00:24:32.224 And so you can't sort of take a picture of an orbital. 00:24:32.224 --> 00:24:33.890 So these are just different ways to help 00:24:33.890 --> 00:24:38.370 people think about that possible distribution of electrons 00:24:38.370 --> 00:24:39.850 around the nucleus. 00:24:39.850 --> 00:24:42.880 All right, so one thing that everyone's feeling pretty good 00:24:42.880 --> 00:24:47.410 about is that it should be spherically symmetric 00:24:47.410 --> 00:24:49.710 hole for an s orbital. 00:24:49.710 --> 00:24:51.700 And so we have a circle. 00:24:51.700 --> 00:24:54.270 And so the probability density, which 00:24:54.270 --> 00:24:57.670 is shown in this plot-- and the probability density parts 00:24:57.670 --> 00:25:00.480 are basically just dots where the more concentrated 00:25:00.480 --> 00:25:02.550 the dots are, the higher the probability 00:25:02.550 --> 00:25:05.930 density for that particular-- the probability 00:25:05.930 --> 00:25:08.470 for that particular volume exists. 00:25:08.470 --> 00:25:12.190 So in here there are sort of more dots and then less dots 00:25:12.190 --> 00:25:13.660 as you come out. 00:25:13.660 --> 00:25:17.450 And so that is a circle, which is what? 00:25:17.450 --> 00:25:18.950 It's symmetrical. 00:25:18.950 --> 00:25:21.710 So you can always recognize a 1s. 00:25:21.710 --> 00:25:23.650 You have this symmetrical thing. 00:25:23.650 --> 00:25:29.240 So this is the wave function squared, is this probability 00:25:29.240 --> 00:25:31.000 density plot. 00:25:31.000 --> 00:25:34.180 Another kind of plot that you can see 00:25:34.180 --> 00:25:36.770 looks at the radial wave function 00:25:36.770 --> 00:25:40.750 plotted against the distance r here, 00:25:40.750 --> 00:25:43.120 distance from the nucleus. 00:25:43.120 --> 00:25:46.800 And then a third kind of plot is another probability plot, 00:25:46.800 --> 00:25:48.290 like this one up here. 00:25:48.290 --> 00:25:51.850 But instead of the dots indicating the higher 00:25:51.850 --> 00:25:55.300 probability density, you have a radial probability 00:25:55.300 --> 00:25:56.750 distribution. 00:25:56.750 --> 00:26:01.490 And so at the nucleus, at 0, well then 00:26:01.490 --> 00:26:02.960 the probability goes up. 00:26:02.960 --> 00:26:05.410 The electron is not going to crash into the nucleus, 00:26:05.410 --> 00:26:07.860 so it won't be right on top of the nucleus. 00:26:07.860 --> 00:26:10.350 But as you get out a little bit farther away, 00:26:10.350 --> 00:26:12.560 there's a high probability that it's there. 00:26:12.560 --> 00:26:14.630 And then that decreases again. 00:26:14.630 --> 00:26:16.380 So the top one and the bottom one 00:26:16.380 --> 00:26:19.900 both talk about the probability of finding an electron 00:26:19.900 --> 00:26:21.920 in a particular unit. 00:26:21.920 --> 00:26:24.880 And I'll give you just a little more definition of this. 00:26:24.880 --> 00:26:28.830 And this is on the same page above those different plots. 00:26:28.830 --> 00:26:32.470 So the radial probability distribution 00:26:32.470 --> 00:26:34.470 reports on the probability of finding 00:26:34.470 --> 00:26:37.750 an electron in the spherical shell 00:26:37.750 --> 00:26:41.410 at some little distance dr from the origin. 00:26:41.410 --> 00:26:43.100 And one thing that comes out of this, 00:26:43.100 --> 00:26:47.670 which is pretty important, is the most probable value 00:26:47.670 --> 00:26:50.980 for that distance r, which is denoted 00:26:50.980 --> 00:26:55.890 rmp, so most probable distance. 00:26:55.890 --> 00:27:00.930 And for a hydrogen atom, this is a0, the Bohr radius. 00:27:00.930 --> 00:27:06.190 And you can see it expressed in different units over here. 00:27:06.190 --> 00:27:11.080 And from the plot, that will be the top part of the plot, 00:27:11.080 --> 00:27:13.080 the most probable distance. 00:27:13.080 --> 00:27:17.190 In this case, that's the Bohr radius for the hydrogen atom. 00:27:17.190 --> 00:27:19.760 So we have now these three different kinds 00:27:19.760 --> 00:27:21.670 of plots that you'll see. 00:27:21.670 --> 00:27:23.990 And I want to point out that they're different plots. 00:27:23.990 --> 00:27:27.270 Sometimes people are thinking that there is sort of one plot 00:27:27.270 --> 00:27:30.260 and they're trying to read one of them as probability density, 00:27:30.260 --> 00:27:32.220 and that's not what it is. 00:27:32.220 --> 00:27:35.100 So we'll look at these again. 00:27:35.100 --> 00:27:37.260 All right, so going back and we'll just 00:27:37.260 --> 00:27:40.070 look at them again now that we sort of talked about what 00:27:40.070 --> 00:27:43.580 all of them are, again, we have our sort 00:27:43.580 --> 00:27:48.800 of dot density, probability density plot, our wave function 00:27:48.800 --> 00:27:54.240 plot, and our radial probability distribution plot. 00:27:54.240 --> 00:28:00.990 And for 1s, we have the dots closer to the nucleus here. 00:28:00.990 --> 00:28:03.250 Probability goes up and goes down. 00:28:03.250 --> 00:28:05.910 And here, you're thinking about this 00:28:05.910 --> 00:28:09.870 as the amplitude of finding an electron as you move away 00:28:09.870 --> 00:28:12.710 from the nucleus. 00:28:12.710 --> 00:28:14.380 So 1s is pretty simple. 00:28:14.380 --> 00:28:17.120 And I think these plots are a lot more meaningful when we go 00:28:17.120 --> 00:28:21.890 on to look at other orbitals. 00:28:21.890 --> 00:28:24.130 So let's think about those other orbitals. 00:28:24.130 --> 00:28:26.390 And we'll finish the other plots. 00:28:26.390 --> 00:28:30.256 So this is just-- you can actually stay, in this case. 00:28:30.256 --> 00:28:32.130 So we're going a lot of back and forth today. 00:28:32.130 --> 00:28:36.250 So here is your table that we had before. 00:28:36.250 --> 00:28:38.770 And here's 1s. 00:28:38.770 --> 00:28:40.030 Here's 2s. 00:28:40.030 --> 00:28:41.340 Here's 3s. 00:28:41.340 --> 00:28:44.770 These terms are in fact different, as you can see. 00:28:44.770 --> 00:28:48.370 But the angular term, as we mentioned before, 00:28:48.370 --> 00:28:49.580 is still the same. 00:28:49.580 --> 00:28:53.370 So that means 2s and 3s are still symmetrical. 00:28:53.370 --> 00:28:55.430 So we're still thinking about the probability 00:28:55.430 --> 00:28:58.440 of finding an electron in some volume 00:28:58.440 --> 00:29:02.300 as just going out as a distance of r. 00:29:02.300 --> 00:29:06.860 So let's look now at the three plots, and compare those plots. 00:29:06.860 --> 00:29:10.260 And this is the one on your handouts we looked at. 00:29:10.260 --> 00:29:11.200 I showed you this. 00:29:11.200 --> 00:29:15.750 And now we have all of these three plots together here. 00:29:15.750 --> 00:29:17.970 And in the comparison of these three, 00:29:17.970 --> 00:29:20.220 I think it helps differentiate what 00:29:20.220 --> 00:29:22.940 you're seeing in these plots. 00:29:22.940 --> 00:29:27.000 So important point, they're spherical. 00:29:27.000 --> 00:29:31.280 1s, 2s, 3s, they're all spherical. 00:29:31.280 --> 00:29:34.760 And here we see the dot density increase. 00:29:34.760 --> 00:29:37.930 And then the dot density goes to 0. 00:29:37.930 --> 00:29:41.250 And that's known as a node. 00:29:41.250 --> 00:29:46.210 So a node is a value of r or theta or phi 00:29:46.210 --> 00:29:48.810 for which the wave function and wave function 00:29:48.810 --> 00:29:52.730 squared, or the probability density, is 0. 00:29:52.730 --> 00:29:56.760 And in this particular case, the type of node that we're seeing 00:29:56.760 --> 00:29:59.050 is a radial node. 00:29:59.050 --> 00:30:02.730 And so that's a value of r for which the wave function, 00:30:02.730 --> 00:30:08.450 wave function squared probability density is 0. 00:30:08.450 --> 00:30:10.250 So it goes to 0. 00:30:10.250 --> 00:30:13.050 We have a node, a radial node. 00:30:13.050 --> 00:30:15.490 Then there's more probability. 00:30:15.490 --> 00:30:19.310 And then it increases, and then starts decreasing again. 00:30:19.310 --> 00:30:24.510 And so if you plot this with the radial wave function versus r, 00:30:24.510 --> 00:30:26.530 you see it go down. 00:30:26.530 --> 00:30:29.590 And it crosses the zero line here. 00:30:29.590 --> 00:30:31.120 And that's the node. 00:30:31.120 --> 00:30:34.200 And that's at 2a0. 00:30:34.200 --> 00:30:35.780 And then it goes back up. 00:30:35.780 --> 00:30:37.650 And this plot often bothers people. 00:30:37.650 --> 00:30:40.770 They're saying, what, there's now negative probability? 00:30:40.770 --> 00:30:43.610 No, these are not the probability diagrams. 00:30:43.610 --> 00:30:46.260 This is thinking about the amplitude 00:30:46.260 --> 00:30:47.880 of finding an electron. 00:30:47.880 --> 00:30:49.940 So we don't have to worry. 00:30:49.940 --> 00:30:53.550 It can have a positive or a negative phase to it. 00:30:53.550 --> 00:30:58.220 And if you look at this plot, the radial probability 00:30:58.220 --> 00:31:00.650 distribution plot, then you'll see 00:31:00.650 --> 00:31:04.610 that actually the radius, the most probable radius 00:31:04.610 --> 00:31:07.060 is in this region over here. 00:31:07.060 --> 00:31:11.560 And you see that this is concentrated dots up here. 00:31:11.560 --> 00:31:15.130 So if we think about these two, which are really probability 00:31:15.130 --> 00:31:18.080 distribution diagrams, we're thinking about the probability 00:31:18.080 --> 00:31:19.690 of finding an electron. 00:31:19.690 --> 00:31:23.840 You have a probability in here close to the nucleus. 00:31:23.840 --> 00:31:25.420 Then you get a node. 00:31:25.420 --> 00:31:28.520 And then you have another probability, high probability 00:31:28.520 --> 00:31:30.140 of finding the electron. 00:31:30.140 --> 00:31:33.930 In fact that's the most probable radius here for 2s. 00:31:33.930 --> 00:31:35.730 And then it decreases. 00:31:35.730 --> 00:31:39.330 So this line shows you what a radial node 00:31:39.330 --> 00:31:42.160 looks like in all three plots. 00:31:42.160 --> 00:31:47.760 In this probability diagram, wave function 00:31:47.760 --> 00:31:51.650 squared plot, it looks like there's just an empty space, 00:31:51.650 --> 00:31:52.930 no dots at all. 00:31:52.930 --> 00:31:56.050 Down here, it's where it crosses the line. 00:31:56.050 --> 00:32:00.430 And in the bottom plot, it is where you go up and down 00:32:00.430 --> 00:32:04.080 and again touches the line before going back up. 00:32:04.080 --> 00:32:06.410 So you should be able to look at these plots 00:32:06.410 --> 00:32:10.610 and think about what they mean. 00:32:10.610 --> 00:32:14.140 For 3s, we see the same thing. 00:32:14.140 --> 00:32:20.210 But now we have an intense spot in the middle near the nucleus. 00:32:20.210 --> 00:32:22.540 That is indicated down here. 00:32:22.540 --> 00:32:26.130 There is probability of finding the electron near the nucleus. 00:32:26.130 --> 00:32:27.510 Then there's a node. 00:32:27.510 --> 00:32:30.220 And that's in this plot where it crosses the line 00:32:30.220 --> 00:32:33.710 and in this plot where you have the empty space. 00:32:33.710 --> 00:32:37.360 Then you have more probability of finding the electron. 00:32:37.360 --> 00:32:39.850 You have another bump here. 00:32:39.850 --> 00:32:44.210 And then we have another node, indicated by touching the zero 00:32:44.210 --> 00:32:45.920 line here, touching here. 00:32:45.920 --> 00:32:48.090 That's at 7.1a0. 00:32:48.090 --> 00:32:50.730 And then we have more probability 00:32:50.730 --> 00:32:53.500 of finding the electron. 00:32:53.500 --> 00:33:00.560 And this is where the most probable radius is at 11.5. 00:33:00.560 --> 00:33:03.630 So again, you need to be able to look at these diagrams 00:33:03.630 --> 00:33:07.050 and recognize what constitutes a radial node. 00:33:07.050 --> 00:33:09.660 And a node is a place where there 00:33:09.660 --> 00:33:12.490 is no probability that you're going to find an electron. 00:33:16.710 --> 00:33:20.820 So now let's think about how many nodes, or radial 00:33:20.820 --> 00:33:24.020 nodes you should have when you have 00:33:24.020 --> 00:33:26.890 different types of orbitals. 00:33:26.890 --> 00:33:30.710 And this is just a similar diagram to what I just showed. 00:33:30.710 --> 00:33:35.730 This is the wave function squared, probability diagram. 00:33:35.730 --> 00:33:39.170 And now instead of blue you have orange dots, 00:33:39.170 --> 00:33:43.900 but otherwise should be the same-- so for 1s, for 2s, 00:33:43.900 --> 00:33:45.980 and 3s. 00:33:45.980 --> 00:33:49.680 So for the 1s orbital, we can calculate 00:33:49.680 --> 00:33:51.700 how many radial nodes that we should 00:33:51.700 --> 00:33:57.540 have by using this handy formula, n minus 1 minus l. 00:33:57.540 --> 00:34:01.430 So for 1s we have 1 minus 1. 00:34:01.430 --> 00:34:03.230 And l is 0. 00:34:03.230 --> 00:34:05.990 So we have zero radial nodes. 00:34:05.990 --> 00:34:08.290 And we can see that from that diagram 00:34:08.290 --> 00:34:11.040 there are zero radial nodes. 00:34:11.040 --> 00:34:14.790 2s now-- 2, n is 2. 00:34:14.790 --> 00:34:19.139 Minus 1, minus 0-- so that's one radial node. 00:34:19.139 --> 00:34:22.040 And the radial node, again, in this kind of diagram 00:34:22.040 --> 00:34:23.810 is the empty space. 00:34:23.810 --> 00:34:26.330 And that radial node is at 2a0. 00:34:29.679 --> 00:34:37.010 For 3s, we have n equals 3 minus 1 minus l, which is still 0. 00:34:37.010 --> 00:34:39.070 So we have two radial nodes. 00:34:39.070 --> 00:34:44.630 And so again, the empty space here at 1.9a0 and then 00:34:44.630 --> 00:34:48.070 at 7.1a0. 00:34:48.070 --> 00:34:50.469 So why don't you give this a try now 00:34:50.469 --> 00:34:55.210 and tell me what kind of radial nodes you would expect for 4p. 00:35:20.090 --> 00:35:21.377 OK, 10 seconds. 00:35:21.377 --> 00:35:22.293 These are pretty fast. 00:35:38.020 --> 00:35:39.470 Yep. 00:35:39.470 --> 00:35:44.240 So again, we have to do n, which is 4, minus 1. 00:35:44.240 --> 00:35:48.530 And then what is l in this case-- 1. 00:35:48.530 --> 00:35:51.250 So that gives you 2. 00:35:51.250 --> 00:35:58.674 All right, so 4 minus 1 minus 1 or 2 radial nodes. 00:35:58.674 --> 00:36:00.340 All right, don't put your clickers away. 00:36:00.340 --> 00:36:02.300 Let's try something else. 00:36:05.380 --> 00:36:10.120 So now tell me which of these is correct both in terms 00:36:10.120 --> 00:36:13.080 of the indicated number of radial nodes 00:36:13.080 --> 00:36:16.940 and in terms of the plot for a 5s orbital. 00:37:04.650 --> 00:37:06.436 All right, let's just do 10 more seconds. 00:37:22.980 --> 00:37:30.310 We're varying it up in terms of the plots. 00:37:30.310 --> 00:37:40.970 So maybe someone want to say what the right answer is here? 00:37:40.970 --> 00:37:41.470 Yeah? 00:37:46.440 --> 00:37:48.960 AUDIENCE: So by the formula we just did, 00:37:48.960 --> 00:37:51.270 that has four radial nodes. 00:37:51.270 --> 00:37:54.060 And if you look at the graph of one, there's three, 00:37:54.060 --> 00:37:56.260 and then there's another one at the origin. 00:37:56.260 --> 00:37:59.110 So that's four radial nodes. 00:37:59.110 --> 00:38:01.040 Right? 00:38:01.040 --> 00:38:02.400 Right? 00:38:02.400 --> 00:38:04.730 PROFESSOR: Actually, I just realized 00:38:04.730 --> 00:38:06.650 that-- let me count here. 00:38:06.650 --> 00:38:12.300 So this answer here, we should have four radial nodes. 00:38:12.300 --> 00:38:16.960 That is correct because we have n minus 1 minus l. 00:38:16.960 --> 00:38:20.000 Actually, I think this is going to this-- 00:38:20.000 --> 00:38:21.500 this should be going to this answer, 00:38:21.500 --> 00:38:27.250 because if we count 1, 2, 3, 4. 00:38:27.250 --> 00:38:29.700 Sorry, the new plot is highly confusing. 00:38:29.700 --> 00:38:30.930 I have to count. 00:38:30.930 --> 00:38:33.855 So the one at the origin should actually not count. 00:38:33.855 --> 00:38:34.980 AUDIENCE: It doesn't count? 00:38:34.980 --> 00:38:36.526 PROFESSOR: This is not a node. 00:38:41.270 --> 00:38:45.835 So we have 1, 2, 3, 4, should be our four radial nodes. 00:38:45.835 --> 00:38:49.640 Because that's a nucleus, and there isn't one there. 00:38:49.640 --> 00:38:51.450 But that doesn't count as a node. 00:38:51.450 --> 00:38:54.780 So this should be here. 00:38:54.780 --> 00:38:56.360 I guess that's-- right. 00:38:56.360 --> 00:39:03.220 But thank you very much, and [INAUDIBLE], here. 00:39:06.990 --> 00:39:10.049 You were brave enough to answer. 00:39:10.049 --> 00:39:11.090 Yeah, there's a question? 00:39:15.880 --> 00:39:18.490 AUDIENCE: Should there also be a certain number of peaks 00:39:18.490 --> 00:39:22.044 in the graph as well as nodes? 00:39:22.044 --> 00:39:22.710 PROFESSOR: Yeah. 00:39:22.710 --> 00:39:28.840 So if you look at the peaks, these are really hard to draw. 00:39:28.840 --> 00:39:31.420 And I think that's partly what the problem is. 00:39:31.420 --> 00:39:36.450 But when we look later in the handout 00:39:36.450 --> 00:39:41.070 where they're drawn a little bit more carefully, 00:39:41.070 --> 00:39:42.650 it does increase. 00:39:42.650 --> 00:39:44.590 So there are different numbers. 00:39:44.590 --> 00:39:47.940 So we'll have nodes going down here. 00:39:47.940 --> 00:39:51.080 But then we'll have more distributions. 00:39:51.080 --> 00:39:55.250 But often the ones as you go along, 00:39:55.250 --> 00:40:00.370 it does indicate where the most probable radius is 00:40:00.370 --> 00:40:05.660 as the taller ones, and that it's usually drawn at the end. 00:40:05.660 --> 00:40:07.960 So we have some plots and I'll point this out later. 00:40:12.272 --> 00:40:14.230 We're going to look at more plots, don't worry. 00:40:21.320 --> 00:40:22.900 So if anyone's good at drawing those, 00:40:22.900 --> 00:40:25.571 let me know, because they're really hard to draw. 00:40:25.571 --> 00:40:27.320 So a lot of them are copied from the book, 00:40:27.320 --> 00:40:29.760 but then they don't copy very well. 00:40:29.760 --> 00:40:32.740 So let's consider other kinds of nodes. 00:40:32.740 --> 00:40:35.180 And we're going to come back to radial nodes. 00:40:35.180 --> 00:40:38.030 All right, so what about p orbitals? 00:40:38.030 --> 00:40:39.730 So here we have our table again. 00:40:39.730 --> 00:40:42.270 These are our p orbitals over here. 00:40:45.050 --> 00:40:51.880 And we have our n equals 2 cases here and our l equals 1. 00:40:51.880 --> 00:40:58.460 So these are x, y, and z-- so our 3p orbitals over here. 00:40:58.460 --> 00:41:01.120 And the important point is not to memorize 00:41:01.120 --> 00:41:02.580 what these values are. 00:41:02.580 --> 00:41:06.220 But now all of a sudden we have dependence on angles. 00:41:06.220 --> 00:41:10.400 So we're going to have an angular component to these. 00:41:10.400 --> 00:41:13.480 And that means the probability density 00:41:13.480 --> 00:41:17.150 as you go out from the nucleus doesn't just 00:41:17.150 --> 00:41:18.830 depend on r anymore. 00:41:18.830 --> 00:41:22.260 It depends on theta and phi, which 00:41:22.260 --> 00:41:29.530 are sort of the equivalent to latitude and longitude, 00:41:29.530 --> 00:41:32.280 if you're thinking about geography. 00:41:32.280 --> 00:41:34.500 All right, so let's see what that looks like. 00:41:34.500 --> 00:41:36.420 So that means then the p orbitals 00:41:36.420 --> 00:41:42.870 are not spherically symmetric, because it depends on angle. 00:41:42.870 --> 00:41:47.080 So you just don't go out and have the probability depend 00:41:47.080 --> 00:41:48.960 on the radius and it's symmetrical 00:41:48.960 --> 00:41:51.642 in all the different directions. 00:41:51.642 --> 00:41:53.350 And here are what some of them look like. 00:41:53.350 --> 00:41:55.930 These figures are in your handouts. 00:41:55.930 --> 00:41:57.122 Here are some other figures. 00:41:59.880 --> 00:42:04.060 So the orbitals consists of two lobes. 00:42:04.060 --> 00:42:07.740 So you could view this as a lobe up here and a lobe down here. 00:42:07.740 --> 00:42:11.900 Or you have these lobes as these two different colors over here. 00:42:11.900 --> 00:42:15.710 And the lobes are separated by a nodal plane. 00:42:15.710 --> 00:42:19.630 And the nodal plane is a plane on which the probability 00:42:19.630 --> 00:42:23.550 of finding the electrons is 0. 00:42:23.550 --> 00:42:27.200 So in the top drawing, the nodal plane is drawn as a plane. 00:42:27.200 --> 00:42:30.670 And in the bottom drawings, you don't see a plane. 00:42:30.670 --> 00:42:33.130 You just see an empty space between the lobes. 00:42:33.130 --> 00:42:36.550 So empty space here, empty space here, empty space there. 00:42:36.550 --> 00:42:38.680 And so if it helps you to kind of think 00:42:38.680 --> 00:42:42.570 about an actual plane in between, that's good. 00:42:42.570 --> 00:42:43.990 Or you can just think that there's 00:42:43.990 --> 00:42:47.370 a break between these nodes. 00:42:47.370 --> 00:42:49.960 And again, the nodal plane, there's 00:42:49.960 --> 00:42:55.300 no probability of finding an electron in the nodal planes. 00:42:55.300 --> 00:42:57.990 And the nodal planes are at the nucleus. 00:42:57.990 --> 00:43:00.360 Therefore, there is zero probability 00:43:00.360 --> 00:43:03.440 of finding a p electron at the nucleus. 00:43:03.440 --> 00:43:05.560 s can get pretty close to the nucleus. 00:43:05.560 --> 00:43:09.980 But with a p orbital, there's a nodal plane there. 00:43:09.980 --> 00:43:14.530 No electrons are going to be at the nucleus. 00:43:14.530 --> 00:43:18.350 So now if you're going out from the nucleus, the probability 00:43:18.350 --> 00:43:21.280 of an electron, finding it, if you're 00:43:21.280 --> 00:43:23.660 going out in this direction, you're 00:43:23.660 --> 00:43:25.010 not going to do very well. 00:43:25.010 --> 00:43:26.700 If you're going in this direction, 00:43:26.700 --> 00:43:28.110 you should do a lot better. 00:43:28.110 --> 00:43:30.560 So here the angular components really matter. 00:43:30.560 --> 00:43:32.960 That defines the shape of the orbital. 00:43:32.960 --> 00:43:35.110 And where you're going, what direction you're 00:43:35.110 --> 00:43:36.810 going in, what angles you're going 00:43:36.810 --> 00:43:39.250 in matters in terms of whether you're going 00:43:39.250 --> 00:43:43.020 to find that electron or not. 00:43:43.020 --> 00:43:45.680 So another way to think about this 00:43:45.680 --> 00:43:50.140 in sort of these nodal planes-- so here we'll 00:43:50.140 --> 00:43:52.860 just define what plane it is. 00:43:52.860 --> 00:43:55.120 So we have our pz orbital. 00:43:55.120 --> 00:43:58.790 That's a nodal plane then in x and y. 00:43:58.790 --> 00:44:01.680 And so x and y are over here. 00:44:01.680 --> 00:44:04.620 Our px orbital is going to be in-- 00:44:04.620 --> 00:44:10.070 or the nodal plane is going to be in yz plane, so over here. 00:44:10.070 --> 00:44:15.410 And py will be in xz plane. 00:44:15.410 --> 00:44:17.300 So again, these nodal planes, there's 00:44:17.300 --> 00:44:20.300 no electron density there. 00:44:20.300 --> 00:44:23.330 And these arise from these angular nodes 00:44:23.330 --> 00:44:25.190 in the wave function. 00:44:25.190 --> 00:44:28.090 So angular nodes then or these angular 00:44:28.090 --> 00:44:32.440 nodal planes are values of theta and phi 00:44:32.440 --> 00:44:38.060 for which the wave function, wave function squared are 0. 00:44:38.060 --> 00:44:41.060 So this is very different from the s case 00:44:41.060 --> 00:44:43.810 where we only had radial nodes. 00:44:43.810 --> 00:44:46.990 But now, when in the p orbitals where 00:44:46.990 --> 00:44:51.870 the angular component matters, they're angular nodes as well. 00:44:51.870 --> 00:44:56.820 So we can think about how to calculate the angular nodes. 00:44:56.820 --> 00:45:02.990 So total nodes is going to be equal to n minus 1. 00:45:02.990 --> 00:45:06.870 The angular nodes is l. 00:45:06.870 --> 00:45:15.840 And as we saw before, the radial nodes are n minus 1 minus l. 00:45:15.840 --> 00:45:19.310 So let's have more practice in calculating these. 00:45:19.310 --> 00:45:22.200 And then we'll look at some more diagrams. 00:45:22.200 --> 00:45:27.410 So for 2s, total nodes-- and you can just yell this out. 00:45:27.410 --> 00:45:28.910 Total nodes will be what? 00:45:28.910 --> 00:45:30.180 AUDIENCE: 1 00:45:30.180 --> 00:45:33.460 PROFESSOR: 1-- 2 minus 1 or 1. 00:45:33.460 --> 00:45:35.200 Angular nodes are? 00:45:35.200 --> 00:45:36.260 AUDIENCE: 0 00:45:36.260 --> 00:45:37.300 PROFESSOR: 0. 00:45:37.300 --> 00:45:38.780 For 1s, there is none. 00:45:38.780 --> 00:45:41.510 And if you forget, l equals 0 there. 00:45:41.510 --> 00:45:43.530 Radial nodes is going to be? 00:45:43.530 --> 00:45:46.530 AUDIENCE: 1 00:45:46.530 --> 00:45:51.800 PROFESSOR: Right, 2 minus 1 minus 0, or 1. 00:45:51.800 --> 00:45:56.830 All right, let's try 3-- or sorry, 2p is next. 00:45:56.830 --> 00:46:00.000 Total nodes? 00:46:00.000 --> 00:46:03.520 1 again, so 2 minus 1 or 1. 00:46:03.520 --> 00:46:05.620 Angular nodes? 00:46:05.620 --> 00:46:07.850 1-- l equals 1 here. 00:46:07.850 --> 00:46:10.840 And radial node? 00:46:10.840 --> 00:46:14.920 Right, 2 minus 1 minus 1, or 0. 00:46:14.920 --> 00:46:17.370 So since there's only one total node, 00:46:17.370 --> 00:46:19.450 if you figured out there was one angular node, 00:46:19.450 --> 00:46:22.650 you could even realize that there had to be zero there. 00:46:22.650 --> 00:46:25.990 It's a way to check maybe your equations. 00:46:25.990 --> 00:46:29.840 All right, so let's try for 3d now. 00:47:03.192 --> 00:47:03.900 How are we doing? 00:47:16.454 --> 00:47:18.162 All right, let's just do 10 more seconds. 00:47:38.270 --> 00:47:43.220 And let's just work that out over here. 00:47:43.220 --> 00:47:51.390 So total nodes for 3d, we have 3 minus 1 or 2. 00:47:51.390 --> 00:47:56.650 Angular nodes, l equals 2 for d. 00:47:56.650 --> 00:48:04.450 So radial nodes, we have 3 minus 1 minus 2, or 0. 00:48:04.450 --> 00:48:07.890 All right, so bring these handouts on Wednesday 00:48:07.890 --> 00:48:11.310 because we need to go back and look at more radial probability 00:48:11.310 --> 00:48:12.480 diagrams. 00:48:12.480 --> 00:48:14.150 And talk more about nodes. 00:48:36.230 --> 00:48:38.320 All right, let's just do 10 more seconds. 00:48:59.200 --> 00:49:04.070 OK, good job everyone. 00:49:04.070 --> 00:49:06.330 Let's look through this a little bit. 00:49:06.330 --> 00:49:08.420 And you can sort of-- everyone can help. 00:49:08.420 --> 00:49:10.330 Yell out some responses. 00:49:10.330 --> 00:49:12.380 So this was 2s. 00:49:12.380 --> 00:49:15.500 And that was the correct answer. 00:49:15.500 --> 00:49:20.540 Which type of orbital is this-- 2p. 00:49:20.540 --> 00:49:23.780 And if you couldn't read this information here, 00:49:23.780 --> 00:49:26.140 you should have been able to read the information 00:49:26.140 --> 00:49:28.000 about the nodes. 00:49:28.000 --> 00:49:32.340 What equation is that for nodes? 00:49:32.340 --> 00:49:36.010 Yeah, n minus 1 minus l, for what kind of nodes? 00:49:36.010 --> 00:49:36.840 AUDIENCE: Radial. 00:49:36.840 --> 00:49:38.210 PROFESSOR: Radial nodes, right. 00:49:38.210 --> 00:49:43.710 So if you know what it means if l equals 0 versus l equals 1, 00:49:43.710 --> 00:49:45.580 and you knew this was l, then you 00:49:45.580 --> 00:49:48.840 could tell if it was an s orbital or a p orbital. 00:49:48.840 --> 00:49:53.970 And then whether it was 2 or 3p is from the n. 00:49:53.970 --> 00:49:55.570 So even if you couldn't read this, 00:49:55.570 --> 00:49:59.350 if you knew that expression, then you were OK. 00:49:59.350 --> 00:50:03.440 What kind of orbital was in plot C? 00:50:03.440 --> 00:50:05.610 This was a 3s. 00:50:05.610 --> 00:50:07.680 l equals 0. 00:50:07.680 --> 00:50:11.205 And then this is a what, 3p and? 00:50:14.390 --> 00:50:17.390 l equals 2. 00:50:17.390 --> 00:50:18.730 Louder. 00:50:18.730 --> 00:50:20.260 D, right? 00:50:20.260 --> 00:50:27.960 So do 3px, 3py, and 3pz have different plots? 00:50:27.960 --> 00:50:31.830 No, they wouldn't have different plots. 00:50:31.830 --> 00:50:35.470 So we'll continue to look at this. 00:50:35.470 --> 00:50:38.210 And we're going to be starting with the handout 00:50:38.210 --> 00:50:39.370 from last time. 00:50:39.370 --> 00:50:41.710 And so let's continue with Monday 00:50:41.710 --> 00:50:48.650 and continue with these radial probability distributions. 00:50:48.650 --> 00:50:51.210 So this is again from Monday, page 6. 00:50:51.210 --> 00:50:53.400 We're talking about orbital size. 00:50:53.400 --> 00:50:55.920 And we've already looked at this a little bit today. 00:50:55.920 --> 00:50:58.700 So we should be able to go through this now 00:50:58.700 --> 00:51:00.040 in a little bit more detail. 00:51:00.040 --> 00:51:01.960 You've already thought about it. 00:51:01.960 --> 00:51:05.110 So here we have the 2s orbital. 00:51:05.110 --> 00:51:07.400 And we're going to have one node using 00:51:07.400 --> 00:51:12.920 our equation that you just told me, n minus 1 minus l. 00:51:12.920 --> 00:51:19.690 And when we go from 2s to 2p, here we have no radial nodes. 00:51:19.690 --> 00:51:23.090 And we can look at r and p, which 00:51:23.090 --> 00:51:26.940 is the radius of the maximal probability of finding 00:51:26.940 --> 00:51:28.250 an electron. 00:51:28.250 --> 00:51:33.480 And you can note that when you go from the 2s to the 2p, 00:51:33.480 --> 00:51:36.300 the radius actually decreases. 00:51:36.300 --> 00:51:42.090 So the most probable radius for 2p is less than that of 2s. 00:51:42.090 --> 00:51:47.020 Now let's consider the 3, n equals 3. 00:51:47.020 --> 00:51:52.200 So we have the 3s situation over here. 00:51:52.200 --> 00:51:53.710 And so l equals 0. 00:51:53.710 --> 00:51:56.290 We have two nodes here. 00:51:56.290 --> 00:52:00.730 And now if you look at the radius, the axis over here, 00:52:00.730 --> 00:52:03.610 you'll see that the most probable for 2s 00:52:03.610 --> 00:52:08.450 is close to 5a0, where a0 is the Bohr radius. 00:52:08.450 --> 00:52:12.470 And over here you're talking between 10 and 15. 00:52:12.470 --> 00:52:16.660 So we see an increase in size going this way. 00:52:16.660 --> 00:52:23.590 And then when we go from 3s to 3p-- so here we have 3 minus 1 00:52:23.590 --> 00:52:25.690 minus l, which is 1. 00:52:25.690 --> 00:52:33.400 So we have one node, down to 3d, 3 minus 1 minus 2, zero nodes. 00:52:33.400 --> 00:52:36.910 And you see that there is a decrease here 00:52:36.910 --> 00:52:40.320 in the most probable radius. 00:52:40.320 --> 00:52:43.960 So, OK, interesting. 00:52:43.960 --> 00:52:51.540 All right, so 3d has the smallest, next 3p, next 3s. 00:52:51.540 --> 00:52:54.700 So there's two different trends we're seeing. 00:52:54.700 --> 00:53:00.330 One, as we increase l within the same n number, 00:53:00.330 --> 00:53:06.440 and one going from a smaller value of n to a larger value, 00:53:06.440 --> 00:53:10.950 and then again within the 3, within the n value 00:53:10.950 --> 00:53:13.040 as we change l. 00:53:13.040 --> 00:53:17.580 So again, to say the same thing in a different way, 00:53:17.580 --> 00:53:24.930 as n increases from 2 to 3, the radius, most probable radius 00:53:24.930 --> 00:53:27.300 or the size increases. 00:53:27.300 --> 00:53:31.040 So from here to here we have an increase in size. 00:53:33.850 --> 00:53:35.550 I just want to make sure people have 00:53:35.550 --> 00:53:41.560 time to kind of get all of this down, but it should be good. 00:53:41.560 --> 00:53:43.490 I have a little picture that just shows 00:53:43.490 --> 00:53:46.200 they're very different in size. 00:53:46.200 --> 00:53:49.260 So we'll go back to this again. 00:53:49.260 --> 00:53:54.220 And then as I also said, as l increases for a given n-- 00:53:54.220 --> 00:53:59.090 so from l equals 0 to l equals 1 here, 00:53:59.090 --> 00:54:02.880 then we have a decrease in the size. 00:54:02.880 --> 00:54:07.060 So you can see the most probable radius moves over. 00:54:07.060 --> 00:54:09.615 And then here is another within n. 00:54:09.615 --> 00:54:11.670 And n equals 3. 00:54:11.670 --> 00:54:15.870 We see, again, this decrease. 00:54:15.870 --> 00:54:18.300 So those are the two trends that you 00:54:18.300 --> 00:54:21.740 observe when you look at these radial probability 00:54:21.740 --> 00:54:23.010 distributions. 00:54:23.010 --> 00:54:25.130 So for exam one next week, you should 00:54:25.130 --> 00:54:27.790 be able to draw distributions like this. 00:54:27.790 --> 00:54:30.470 You should be able to tell me how many radial 00:54:30.470 --> 00:54:34.430 nodes you have for different types of orbitals. 00:54:34.430 --> 00:54:37.230 And you should know these trends in size. 00:54:37.230 --> 00:54:40.740 So I think in the exam instructions 00:54:40.740 --> 00:54:43.700 it says up to a 5 case. 00:54:43.700 --> 00:54:46.060 You don't have to go on forever to be able to draw them, 00:54:46.060 --> 00:54:47.685 but you should be able to look at these 00:54:47.685 --> 00:54:51.440 and tell what kind of orbital it is and where the nodes are, 00:54:51.440 --> 00:54:54.760 be able to draw where the nodes are-- one node here, 00:54:54.760 --> 00:54:57.020 one, two, one node here. 00:54:57.020 --> 00:55:00.870 This kind of thing will be on the exam next week. 00:55:00.870 --> 00:55:03.500 So there's something that's a little counterintuitive when 00:55:03.500 --> 00:55:05.870 it comes to this size issue. 00:55:05.870 --> 00:55:09.220 And that has to do with how this correlates 00:55:09.220 --> 00:55:12.320 to the amount of shielding, and as we see later, 00:55:12.320 --> 00:55:14.160 to the energy levels. 00:55:14.160 --> 00:55:18.020 So only electrons in the s state here 00:55:18.020 --> 00:55:21.190 really have any kind of substantial probability 00:55:21.190 --> 00:55:23.330 that they'll be close to the nucleus. 00:55:23.330 --> 00:55:26.550 So we have this little blip over here 00:55:26.550 --> 00:55:29.840 that is close to the nucleus, that at are very small 00:55:29.840 --> 00:55:33.120 radii, very small values of r. 00:55:33.120 --> 00:55:35.990 Even though the most probable is out here, 00:55:35.990 --> 00:55:40.380 if we compare 3s to 3p and look at where the electrons are 00:55:40.380 --> 00:55:43.590 that are closest to the nucleus, they're quite a bit 00:55:43.590 --> 00:55:46.210 farther away than in the 3s. 00:55:46.210 --> 00:55:48.820 Or there's more probability that there's 00:55:48.820 --> 00:55:50.690 going to be some closer here. 00:55:50.690 --> 00:55:54.530 And then the closest probability over here for these electrons 00:55:54.530 --> 00:55:56.340 is quite a bit farther away. 00:55:56.340 --> 00:55:59.070 So we see these circles kind of move out. 00:55:59.070 --> 00:56:01.900 So even though the overall radius, 00:56:01.900 --> 00:56:05.310 the sort of size of the whole thing is decreasing, 00:56:05.310 --> 00:56:06.860 the probability that there are going 00:56:06.860 --> 00:56:10.190 to be electrons really close is actually 00:56:10.190 --> 00:56:12.420 going in the opposite direction. 00:56:12.420 --> 00:56:15.840 And so what this means is that s electrons 00:56:15.840 --> 00:56:18.980 are the least shielded because there's 00:56:18.980 --> 00:56:23.100 higher probability that they'll be some close to the nucleus. 00:56:23.100 --> 00:56:26.500 There's more penetration close to the nucleus. 00:56:26.500 --> 00:56:30.202 So s electrons are the least shielded. 00:56:30.202 --> 00:56:31.910 And we're going to come back to this when 00:56:31.910 --> 00:56:34.120 we move on to today's handout. 00:56:34.120 --> 00:56:36.020 This is really important in terms of thinking 00:56:36.020 --> 00:56:38.400 about the energy levels. 00:56:38.400 --> 00:56:40.280 And I'm going to have these diagrams 00:56:40.280 --> 00:56:42.230 on the handout for today. 00:56:42.230 --> 00:56:44.350 So we'll see them again. 00:56:44.350 --> 00:56:46.960 All right, so before we move to that handout, 00:56:46.960 --> 00:56:49.740 we've got to finish our quantum numbers 00:56:49.740 --> 00:56:53.500 and talk about electron spin. 00:56:53.500 --> 00:56:56.830 So the fourth quantum number describes 00:56:56.830 --> 00:56:59.130 the spin on the electron. 00:56:59.130 --> 00:57:03.460 And we already saw the magnetic quantum number m. 00:57:03.460 --> 00:57:05.310 We saw m sub l. 00:57:05.310 --> 00:57:07.790 And now we have m sub s. 00:57:07.790 --> 00:57:10.470 And the s stands for spin. 00:57:10.470 --> 00:57:15.250 So there's some nomenclature that actually makes sense. 00:57:15.250 --> 00:57:20.550 So there are two possible spin values for an electron. 00:57:20.550 --> 00:57:27.580 And s can equal plus 1/2, spin up, or minus 1/2, spin down. 00:57:27.580 --> 00:57:29.870 And here are some little pictures of that. 00:57:32.730 --> 00:57:38.050 So this ms term, this spin magnetic quantum number, 00:57:38.050 --> 00:57:41.140 completes the description of the electron. 00:57:41.140 --> 00:57:43.980 But it's not dependent on the orbital. 00:57:43.980 --> 00:57:46.280 To describe an orbital completely, 00:57:46.280 --> 00:57:48.330 you only need three quantum numbers. 00:57:48.330 --> 00:57:52.720 But to describe the electron, you need four. 00:57:52.720 --> 00:57:55.990 And that is shown, again, here on this picture, 00:57:55.990 --> 00:57:57.070 or on this slide. 00:57:57.070 --> 00:57:58.670 You need three quantum numbers. 00:57:58.670 --> 00:58:04.410 You need n, l, and m sub l to describe the quantum number, 00:58:04.410 --> 00:58:07.200 describe the orbital completely. 00:58:07.200 --> 00:58:09.710 But you need a fourth one, this m sub 00:58:09.710 --> 00:58:12.310 s to describe the electron. 00:58:12.310 --> 00:58:16.840 So if you see wave function n, l, m sub l, 00:58:16.840 --> 00:58:19.930 you say that's telling me what the orbital is. 00:58:19.930 --> 00:58:23.580 And if we add the m sub s, then you look at that 00:58:23.580 --> 00:58:25.450 and say oh, that's going to tell me 00:58:25.450 --> 00:58:29.350 all the way to the electron what is going on. 00:58:32.040 --> 00:58:37.260 So this final quantum number led to what 00:58:37.260 --> 00:58:42.490 we know as Pauli's exclusion principle, which 00:58:42.490 --> 00:58:47.300 is that no two electrons can have the same four quantum 00:58:47.300 --> 00:58:48.250 numbers. 00:58:48.250 --> 00:58:51.760 They can't have the same-- no two electrons can have 00:58:51.760 --> 00:58:54.910 the same spin, in other words. 00:58:54.910 --> 00:58:58.940 So if we are drawing a configuration for neon 00:58:58.940 --> 00:59:02.200 with 10 electrons, we are going to have 00:59:02.200 --> 00:59:06.130 with one electron being up spin, the next one 00:59:06.130 --> 00:59:07.910 is going to be down. 00:59:07.910 --> 00:59:11.100 Because if we had two of these both going up, 00:59:11.100 --> 00:59:14.700 they would have the same four quantum numbers. 00:59:14.700 --> 00:59:18.780 And that's not allowed by Pauli's exclusion principle. 00:59:18.780 --> 00:59:22.440 So when you have two here, one spin up, 00:59:22.440 --> 00:59:24.460 one spin down in an orbital, then 00:59:24.460 --> 00:59:27.840 we say that those electrons are paired. 00:59:27.840 --> 00:59:31.250 And an important thing that kind of comes out of all of this 00:59:31.250 --> 00:59:35.270 is that one orbital can't hold more than two electrons. 00:59:35.270 --> 00:59:38.550 If it did, there'd be another electron 00:59:38.550 --> 00:59:40.770 that would have the same four quantum numbers. 00:59:40.770 --> 00:59:43.470 Because you need three quantum numbers to describe 00:59:43.470 --> 00:59:45.580 the electron, or the orbital. 00:59:45.580 --> 00:59:50.180 We need three to describe, say, that it's n equals 1, and then 00:59:50.180 --> 00:59:51.990 its s state. 00:59:51.990 --> 00:59:55.190 So we need those other ones to describe the orbital and then 00:59:55.190 --> 00:59:56.820 the fourth one to describe the spin. 00:59:56.820 --> 00:59:58.540 So if we add another electron, you'd 00:59:58.540 --> 01:00:00.100 have two that were spin up, say. 01:00:00.100 --> 01:00:01.500 And that just wouldn't work. 01:00:01.500 --> 01:00:06.050 So you cannot have more than two electrons in the same orbital. 01:00:06.050 --> 01:00:07.760 And this makes a lot of sense when 01:00:07.760 --> 01:00:11.670 you think about why you would be putting electrons in orbitals 01:00:11.670 --> 01:00:12.650 that are higher energy. 01:00:12.650 --> 01:00:16.820 Why not just keep putting him in the low energy orbital? 01:00:16.820 --> 01:00:18.540 And it's because you can't do that. 01:00:18.540 --> 01:00:21.670 You can't put more than two electrons in. 01:00:21.670 --> 01:00:24.610 And so therefore once you've filled a lower energy orbital, 01:00:24.610 --> 01:00:28.990 you've got to move up to the next lowest energy orbital.