WEBVTT 00:00:00.000 --> 00:00:00.272 The following content is provided under a Creative 00:00:00.272 --> 00:00:00.374 Commons license. 00:00:00.374 --> 00:00:00.646 Your support will help MIT OpenCourseWare continue to 00:00:00.646 --> 00:00:00.918 offer high quality educational resources for free. 00:00:00.918 --> 00:00:01.224 To make a donation or view additional materials from 00:00:01.224 --> 00:00:01.496 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:01.496 --> 00:00:01.550 ocw.mit.edu. 00:00:01.550 --> 00:00:21.930 PROFESSOR: All right. 00:00:21.930 --> 00:00:27.070 As everyone finishes getting settled in, why don't you take 00:00:27.070 --> 00:00:34.710 10 more seconds on the clicker question here, and let's see 00:00:34.710 --> 00:00:37.370 how you did on that this, this is very similar to the clicker 00:00:37.370 --> 00:00:41.380 question that we had on Friday. 00:00:41.380 --> 00:00:44.250 OK, so let's get started here. 00:00:44.250 --> 00:00:47.040 It looks like we are doing a lot better. 00:00:47.040 --> 00:00:51.200 We now have 77% getting the correct answer, we only had 00:00:51.200 --> 00:00:53.560 about 30-something percent on Friday for a 00:00:53.560 --> 00:00:55.630 very similar question. 00:00:55.630 --> 00:00:59.740 So if you're not in this 77%, let's quickly go over why, in 00:00:59.740 --> 00:01:02.710 fact, this is the correct answer, 0 . 00:01:02.710 --> 00:01:05.390 9 times 10 to the negative 18 joules. 00:01:05.390 --> 00:01:09.150 So I'm using the same kind of tricky language that we'd used 00:01:09.150 --> 00:01:11.560 before, not to trick you, but so that you're not tricked in 00:01:11.560 --> 00:01:12.460 the future. 00:01:12.460 --> 00:01:15.590 So if we're talking about the fourth excited state, and we 00:01:15.590 --> 00:01:18.740 talk instead about principle quantum numbers, what 00:01:18.740 --> 00:01:21.340 principle quantum number corresponds to the fourth 00:01:21.340 --> 00:01:22.900 excited state of a hydrogen atom. 00:01:22.900 --> 00:01:24.130 STUDENT: Five. 00:01:24.130 --> 00:01:24.920 PROFESSOR: Five. 00:01:24.920 --> 00:01:25.060 OK. 00:01:25.060 --> 00:01:27.290 So, hopefully that cleared up for some of you why you got 00:01:27.290 --> 00:01:28.740 the wrong answer. 00:01:28.740 --> 00:01:32.700 So we know that we're in the n equals 5 state, so we can find 00:01:32.700 --> 00:01:34.710 what the binding energy is here. 00:01:34.710 --> 00:01:37.990 The ionization energy, of course, is just the negative 00:01:37.990 --> 00:01:39.460 of the binding energy. 00:01:39.460 --> 00:01:42.040 We know that binding energy is always negative, we know that 00:01:42.040 --> 00:01:44.450 ionization energy is always positive. 00:01:44.450 --> 00:01:47.370 So hopefully, putting all those things together, if you 00:01:47.370 --> 00:01:50.750 looked at this question again we'd get 100% on it, that our 00:01:50.750 --> 00:01:53.390 only option here is 0 . 00:01:53.390 --> 00:01:55.720 9, and that it's not the negative, it's the positive 00:01:55.720 --> 00:01:58.030 version, because we're talking about how much energy we have 00:01:58.030 --> 00:02:03.420 to put into the system in order to eject an electron. 00:02:03.420 --> 00:02:03.900 All right. 00:02:03.900 --> 00:02:06.900 And today we're going to mostly be talking about wave 00:02:06.900 --> 00:02:11.170 functions of electrons, but before we get to that, I 00:02:11.170 --> 00:02:13.960 wanted to review one last thing that's back on to 00:02:13.960 --> 00:02:16.870 Friday's topic, which was when we were solving the 00:02:16.870 --> 00:02:19.900 Schrodinger equation, or in fact, using the solution to 00:02:19.900 --> 00:02:23.640 the Schrodinger equation for the energy, the binding energy 00:02:23.640 --> 00:02:26.000 between an electron and a nucleus. 00:02:26.000 --> 00:02:28.950 And when we talked about that, what we found was that we 00:02:28.950 --> 00:02:33.360 could actually validate our predicted binding energies by 00:02:33.360 --> 00:02:36.140 looking at the emission spectra of the hydrogen atom, 00:02:36.140 --> 00:02:39.620 which is what we did as the demo, or we could think about 00:02:39.620 --> 00:02:41.610 the absorption spectra as well. 00:02:41.610 --> 00:02:44.360 And what we predict as an energy difference between two 00:02:44.360 --> 00:02:47.450 levels, we know should correspond to the energy of 00:02:47.450 --> 00:02:50.510 light that's either emitted, if we're giving off a photon, 00:02:50.510 --> 00:02:53.440 or that's absorbed if we're going to take on a photon and 00:02:53.440 --> 00:02:56.040 jump from a lower to a higher energy level. 00:02:56.040 --> 00:02:58.760 So we came up with two formulas, which are similar to 00:02:58.760 --> 00:03:00.700 the two that I'm showing here. 00:03:00.700 --> 00:03:03.940 The formula tells us the frequency of the light that's 00:03:03.940 --> 00:03:07.900 emitted or absorbed based on the energy difference between 00:03:07.900 --> 00:03:09.810 the two levels that we're going between, that the 00:03:09.810 --> 00:03:12.090 electron is transitioning between. 00:03:12.090 --> 00:03:14.230 You'll notice that there's a little bit of a difference in 00:03:14.230 --> 00:03:16.840 these equations here from the ones from the other day, which 00:03:16.840 --> 00:03:19.550 is that you have this z squared value in there. 00:03:19.550 --> 00:03:22.580 So these are both called Rydberg formulas for figuring 00:03:22.580 --> 00:03:25.330 out the frequency of light emitted or absorbed, and 00:03:25.330 --> 00:03:28.180 before we were looking at the Rydberg formula specifically 00:03:28.180 --> 00:03:31.740 for the hydrogen atom, and now that we have this z squared 00:03:31.740 --> 00:03:35.120 term in the formula here, we're now talking about 00:03:35.120 --> 00:03:37.350 absolutely any one electron atom. 00:03:37.350 --> 00:03:40.010 And it should make sense where we got this from, because we 00:03:40.010 --> 00:03:44.380 know that the binding energy, if we're talking about a 00:03:44.380 --> 00:03:52.620 hydrogen atom, what is the binding energy equal to? 00:03:52.620 --> 00:03:57.750 Negative Rydberg over what? 00:03:57.750 --> 00:03:58.040 Yes. 00:03:58.040 --> 00:04:00.890 So, it's negative Rydberg constant over n squared. 00:04:00.890 --> 00:04:03.720 But if we're talking more generally about any one 00:04:03.720 --> 00:04:10.630 electron atom, now we have a more general equation for the 00:04:10.630 --> 00:04:15.300 binding energy, which has this z squared term out in front of 00:04:15.300 --> 00:04:18.680 it, right, so it's negative z squared times the Rydberg 00:04:18.680 --> 00:04:22.080 constant all over n squared. 00:04:22.080 --> 00:04:24.910 So, essentially when we're talking about these equations 00:04:24.910 --> 00:04:27.600 up here, all we're doing is talking about the regular 00:04:27.600 --> 00:04:30.580 Rydberg formulas, but instead we could go back and re-derive 00:04:30.580 --> 00:04:33.370 the equation for any one electron atom, which would 00:04:33.370 --> 00:04:36.190 just mean that we put that z squared term in the front. 00:04:36.190 --> 00:04:39.110 So when you solve certain types of problems, such as 00:04:39.110 --> 00:04:42.580 problems later on in the second half of your p-set, if 00:04:42.580 --> 00:04:45.640 you need to talk about the frequency of light emitted or 00:04:45.640 --> 00:04:49.200 absorbed for a one electron atom, such as lithium plus 2, 00:04:49.200 --> 00:04:51.930 for example, then you would need to plug in z, and 00:04:51.930 --> 00:04:55.690 remember the z value for lithium would just be 3. 00:04:55.690 --> 00:04:59.190 The z value for hydrogen, of course, is 1, and that's why 00:04:59.190 --> 00:05:01.680 this term falls out of that equation when we're talking 00:05:01.680 --> 00:05:04.670 specifically about the hydrogen atom. 00:05:04.670 --> 00:05:07.160 So, just to finish our review of what we talked about on 00:05:07.160 --> 00:05:10.720 Friday, when we're thinking about transitions between two 00:05:10.720 --> 00:05:13.700 different states, and we're talking about a situation 00:05:13.700 --> 00:05:17.510 where the final state, the n final, is greater than n 00:05:17.510 --> 00:05:20.710 initial, in this case, are we talking about absorption or 00:05:20.710 --> 00:05:24.690 are we talking about emission? 00:05:24.690 --> 00:05:27.200 Hearing a little bit of a mix here. 00:05:27.200 --> 00:05:31.190 In fact, we're talking about absorption when n final is 00:05:31.190 --> 00:05:32.290 greater than n initial. 00:05:32.290 --> 00:05:35.870 We start at this lower energy state and go up -- that means 00:05:35.870 --> 00:05:38.870 we need to absorb a photon, we have to take in energy. 00:05:38.870 --> 00:05:42.910 Specifically, we have to take in this exact amount of energy 00:05:42.910 --> 00:05:46.260 in order to bump the electron up to the higher energy level. 00:05:46.260 --> 00:05:49.380 So that means that when instead we start high and go 00:05:49.380 --> 00:05:53.610 low, we're dealing with emission where we have excess 00:05:53.610 --> 00:05:56.570 energy that the electron's giving off, and that energy is 00:05:56.570 --> 00:06:00.050 going to be equal the energy of the photon that is released 00:06:00.050 --> 00:06:03.560 and, of course, through our equations we know how to get 00:06:03.560 --> 00:06:07.450 from energy to frequency or to wavelength of the photon that 00:06:07.450 --> 00:06:09.970 we're talking about. 00:06:09.970 --> 00:06:10.470 All right. 00:06:10.470 --> 00:06:14.040 So that's all I'm going to say today in terms of solving the 00:06:14.040 --> 00:06:17.250 energy part of the Schrodinger equation, so what we're really 00:06:17.250 --> 00:06:19.690 going to focus on is the other part of the Schrodinger 00:06:19.690 --> 00:06:23.220 equation today, which is solving for psi. 00:06:23.220 --> 00:06:26.260 So we're going to for psi, and before that, we're going to 00:06:26.260 --> 00:06:28.800 figure out that instead of just that one quantum number 00:06:28.800 --> 00:06:30.920 n, we're going to have a few other quantum numbers that 00:06:30.920 --> 00:06:33.560 fall out of solving the Schrodinger equation 00:06:33.560 --> 00:06:35.150 for what psi is. 00:06:35.150 --> 00:06:38.480 We're also going to talk more about what psi actually means. 00:06:38.480 --> 00:06:41.010 When we first introduced the Schrodinger equation, what I 00:06:41.010 --> 00:06:44.590 told you was think of psi as being some representation of 00:06:44.590 --> 00:06:46.030 what an electron is. 00:06:46.030 --> 00:06:49.170 We'll get more specific here, more specific even than just 00:06:49.170 --> 00:06:50.630 saying you can think of it as an orbital. 00:06:50.630 --> 00:06:52.930 We'll really think about what psi means. 00:06:52.930 --> 00:06:55.950 And in doing that, we'll also talk about the shapes of h 00:06:55.950 --> 00:06:58.590 atom wave functions, specifically the shapes of 00:06:58.590 --> 00:07:01.600 orbitals, and then something called radial probability 00:07:01.600 --> 00:07:05.060 distribution, which will make sense when we get to it. 00:07:05.060 --> 00:07:08.670 But, as I said before that, we have some more quantum numbers 00:07:08.670 --> 00:07:12.360 to take care of, because it turns out that when you solve 00:07:12.360 --> 00:07:15.790 the Schrodinger equation for psi, these other quantum 00:07:15.790 --> 00:07:17.390 numbers have to be the defined. 00:07:17.390 --> 00:07:20.190 When we talked about binding energy, we just had one 00:07:20.190 --> 00:07:22.840 quantum number that came out of it. 00:07:22.840 --> 00:07:25.590 And that quantum number was n, which was our principle 00:07:25.590 --> 00:07:29.820 quantum number, and we know that n could be equal to any 00:07:29.820 --> 00:07:35.780 integer value, so, 1, 2, 3, all the way up to infinity. 00:07:35.780 --> 00:07:39.830 And this quantization that comes out of having n is what 00:07:39.830 --> 00:07:42.230 gives us the quantization of different energy levels. 00:07:42.230 --> 00:07:45.000 That's why we can't have a continuum of energy, we 00:07:45.000 --> 00:07:50.190 actually have those quantized points. 00:07:50.190 --> 00:07:53.250 So, it turns out that n is not the only quantum number needed 00:07:53.250 --> 00:07:55.460 to describe a wave function, however. 00:07:55.460 --> 00:07:58.490 There's two more that you can see come out of it. 00:07:58.490 --> 00:08:02.780 And the first is l, and l is our angular momentum quantum 00:08:02.780 --> 00:08:06.660 number, and it's called that because it actually dictates 00:08:06.660 --> 00:08:10.460 the angular momentum that our electron has in our atom. 00:08:10.460 --> 00:08:13.220 And when we talk about l it is a quantum number, so because 00:08:13.220 --> 00:08:15.900 it's a quantum number, we know that it can only have discreet 00:08:15.900 --> 00:08:19.320 values, it can't just be any value we want, it's very 00:08:19.320 --> 00:08:20.620 specific values. 00:08:20.620 --> 00:08:24.440 And unlike n, l can start all the way down at 0, and it 00:08:24.440 --> 00:08:29.400 increases by integer value, so we go 1, 2, 3, 00:08:29.400 --> 00:08:30.980 and all the way up. 00:08:30.980 --> 00:08:34.530 But also unlike n, l cannot have just any value, we can't 00:08:34.530 --> 00:08:36.420 go into infinity. 00:08:36.420 --> 00:08:39.680 L is limited such that the highest value of 00:08:39.680 --> 00:08:41.870 l is n minus 1. 00:08:41.870 --> 00:08:44.260 We can't get any higher than that. 00:08:44.260 --> 00:08:46.665 So, it would be a good question to ask why are we 00:08:46.665 --> 00:08:49.010 limited -- clearly there's this relationship between l 00:08:49.010 --> 00:08:51.950 and n, and we can't get any higher than n equals one. 00:08:51.950 --> 00:08:54.940 We can actually think about why that is, and the reason is 00:08:54.940 --> 00:08:57.400 because l is our angular momentum. 00:08:57.400 --> 00:09:00.140 It describes the angular momentum of the electron. 00:09:00.140 --> 00:09:02.580 So another way to think about that is just the rotational 00:09:02.580 --> 00:09:05.400 kinetic energy of our electron. 00:09:05.400 --> 00:09:08.960 And we know that n describes the total energy, that total 00:09:08.960 --> 00:09:11.750 binding energy of the electron, so the total energy 00:09:11.750 --> 00:09:14.170 is going to be equal to potential energy 00:09:14.170 --> 00:09:15.820 plus kinetic energy. 00:09:15.820 --> 00:09:18.920 So if we say that l is just talking about our kinetic 00:09:18.920 --> 00:09:21.990 energy part, our rotational kinetic energy, and we know 00:09:21.990 --> 00:09:24.940 that electrons have potential energy, then it makes sense 00:09:24.940 --> 00:09:27.820 that l, in fact, can never go higher than n. 00:09:27.820 --> 00:09:30.500 And, in fact, it can't even reach n, because then we would 00:09:30.500 --> 00:09:33.660 have no potential energy at all in our electron, which is 00:09:33.660 --> 00:09:35.130 not correct. 00:09:35.130 --> 00:09:37.860 So, that's the second quantum number. 00:09:37.860 --> 00:09:44.950 And the third one is called m, it's also m sub l. 00:09:44.950 --> 00:09:48.200 This is what we call the magnetic quantum number, and 00:09:48.200 --> 00:09:52.900 we won't deal with the fact of its being the magnetic quantum 00:09:52.900 --> 00:09:55.980 number here -- that kind of tells us the shape of the 00:09:55.980 --> 00:09:59.210 orbital or the way that the electron will behave in a 00:09:59.210 --> 00:10:02.360 magnetic field, but what's more relevant to thinking 00:10:02.360 --> 00:10:05.500 about the limits of this number is that it's also the z 00:10:05.500 --> 00:10:07.860 component of the angular momentum. 00:10:07.860 --> 00:10:11.090 So since it's a component of the angular momentum, that 00:10:11.090 --> 00:10:14.330 means that it's never going to be able to go higher than l 00:10:14.330 --> 00:10:17.180 is, so it makes sense that, for example, it could start at 00:10:17.180 --> 00:10:20.530 0 and then go all the way up to l. 00:10:20.530 --> 00:10:23.330 But since it is a component it can have a direction, too, so 00:10:23.330 --> 00:10:26.280 can go up between negative l and positive l. 00:10:26.280 --> 00:10:30.150 So the allowed values for m sub l are going to be negative 00:10:30.150 --> 00:10:35.640 l, all the way up to 0, and then up to positive l. 00:10:35.640 --> 00:10:40.590 So, if we think of just an example, we could say that 4 l 00:10:40.590 --> 00:10:45.740 equals 2, what would be our lowest value of m sub l? 00:10:45.740 --> 00:10:46.310 Yup. 00:10:46.310 --> 00:10:49.870 So m sub l could equal negative 2, 00:10:49.870 --> 00:10:54.410 negative 1, 0, 1 or 2. 00:10:54.410 --> 00:11:00.310 So we could have five different values of m sub l. 00:11:00.310 --> 00:11:02.410 So, those are our three quantum numbers. 00:11:02.410 --> 00:11:06.460 So if, in fact, we want to describe a wave function, we 00:11:06.460 --> 00:11:09.110 know that we need to describe it in terms of all three 00:11:09.110 --> 00:11:13.430 quantum numbers, and also as a function of our three 00:11:13.430 --> 00:11:18.660 positional factors, which are r, the radius, plus the two 00:11:18.660 --> 00:11:20.600 angles, theta and phi. 00:11:20.600 --> 00:11:23.040 So, we have now a complete description of a wave function 00:11:23.040 --> 00:11:24.500 that we can talk about. 00:11:24.500 --> 00:11:26.940 So, we can think about what is it that we would call the 00:11:26.940 --> 00:11:28.710 ground state wave function. 00:11:28.710 --> 00:11:32.120 We knew from Friday, when we talked about energy, that 00:11:32.120 --> 00:11:35.730 ground state was that n equals 1 value, that was the lowest 00:11:35.730 --> 00:11:37.720 energy, that was the most stable place for 00:11:37.720 --> 00:11:39.020 the electron to be. 00:11:39.020 --> 00:11:42.510 But now we need to talk about l and m as well. 00:11:42.510 --> 00:11:45.450 So now when we talk about a ground state in terms of wave 00:11:45.450 --> 00:11:49.390 function, we need to talk about the wave function of 1, 00:11:49.390 --> 00:11:54.560 0, 0, and again, as a function of r, theta and phi. 00:11:54.560 --> 00:11:57.960 So this is our complete description of the ground 00:11:57.960 --> 00:12:00.670 state wave function. 00:12:00.670 --> 00:12:04.270 So, a lot of you talked about different types of orbitals in 00:12:04.270 --> 00:12:08.050 high school, I'm sure, or in previous courses, and it might 00:12:08.050 --> 00:12:10.580 be less common that you actually talked about a wave 00:12:10.580 --> 00:12:12.720 function that was labeled like this. 00:12:12.720 --> 00:12:16.480 We're used to labelling orbitals as an s, or a p, or a 00:12:16.480 --> 00:12:19.830 d, for example, but it turns out that these correlate to 00:12:19.830 --> 00:12:22.640 those letters that we're more used to seeing. 00:12:22.640 --> 00:12:28.000 Does anyone know what the 1, 0, 0 orbital is also called? 00:12:28.000 --> 00:12:28.450 Yeah. 00:12:28.450 --> 00:12:30.700 And specfically it's the 1 s, so not just the 00:12:30.700 --> 00:12:33.320 s, but the 1 s orbital. 00:12:33.320 --> 00:12:35.770 So, using the terminology of chemists, which is a good 00:12:35.770 --> 00:12:38.670 thing to do, because in this course we are all chemists, we 00:12:38.670 --> 00:12:42.140 want to make sure that we're not using just the physical 00:12:42.140 --> 00:12:44.415 description of the numbers, but that we can correlate it 00:12:44.415 --> 00:12:47.570 to what we understand as orbitals, and instead of 1, 0, 00:12:47.570 --> 00:12:49.840 0, we call this the 1 s orbital. 00:12:49.840 --> 00:12:52.280 The reason that we do this is because this is another way to 00:12:52.280 --> 00:12:53.870 completely describe it. 00:12:53.870 --> 00:12:57.970 The n designates the shell, so that's what this number is 00:12:57.970 --> 00:13:00.070 here, we're in the first shell. 00:13:00.070 --> 00:13:03.190 The l is what we call the sub shell. 00:13:03.190 --> 00:13:06.300 And instead of having a 0 there, what we 00:13:06.300 --> 00:13:08.180 have here is an s. 00:13:08.180 --> 00:13:10.920 So, if we look at what the other sub shells are called, 00:13:10.920 --> 00:13:12.190 essentially we're just converting 00:13:12.190 --> 00:13:14.680 the number to a letter. 00:13:14.680 --> 00:13:19.300 L equals 0 is s, what is l equals 1? 00:13:19.300 --> 00:13:20.730 Um-hmm, it's the p. 00:13:20.730 --> 00:13:25.710 What about 2? d, and 3? 00:13:25.710 --> 00:13:27.470 Yup, so 3 is f. 00:13:27.470 --> 00:13:30.090 So these names, they don't really make any sense if we're 00:13:30.090 --> 00:13:32.990 looking at them why they're called past s p and f, and it 00:13:32.990 --> 00:13:36.110 turns out that it comes from spectroscopy terms that are 00:13:36.110 --> 00:13:39.420 pre-quantum mechanics where, for example, this is called 00:13:39.420 --> 00:13:42.540 the sharp line, I think the principle, the diffuse, and 00:13:42.540 --> 00:13:43.610 the fundamental. 00:13:43.610 --> 00:13:47.220 It doesn't even make sense now, they're not used in 00:13:47.220 --> 00:13:49.920 spectroscopy anymore, but this is where the names originally 00:13:49.920 --> 00:13:51.600 came from and they did stick. 00:13:51.600 --> 00:13:56.810 So, we being chemists, we'll call that 1 s instead of 1, 0. 00:13:56.810 --> 00:14:00.200 In addition to having another name to denote l, we also have 00:14:00.200 --> 00:14:04.820 another name for the m designation here. 00:14:04.820 --> 00:14:13.300 So, for example, when l is equal to 0, we're going to 00:14:13.300 --> 00:14:16.860 find that we have to call -- we have to specify 00:14:16.860 --> 00:14:22.780 what m is as well. 00:14:22.780 --> 00:14:23.170 All right. 00:14:23.170 --> 00:14:28.070 So, when we have, for example, l equal to 1, what kind of 00:14:28.070 --> 00:14:30.590 orbital is this? 00:14:30.590 --> 00:14:31.250 The p orbital. 00:14:31.250 --> 00:14:34.380 And for example, we could also in this case, have 00:14:34.380 --> 00:14:36.270 m is equal to 0. 00:14:36.270 --> 00:14:40.700 If m is equal to 0, in this case we would call it the p z 00:14:40.700 --> 00:14:44.750 orbital, so we would have the subscript z here. 00:14:44.750 --> 00:14:50.680 Similarly, if m is equal to either plus 1 or minus 1, we 00:14:50.680 --> 00:14:57.850 would in turn call it the p y orbital, or the p x orbital. 00:14:57.850 --> 00:15:01.270 So you should know that any time m is equal to zero when 00:15:01.270 --> 00:15:04.180 we are talking about p orbitals, that it's the p z. 00:15:04.180 --> 00:15:06.360 The p y and the p x are actually a bit more 00:15:06.360 --> 00:15:10.280 complicated, they're linear combinations of the m plus 1, 00:15:10.280 --> 00:15:13.340 and the m minus 1 orbital, where 1 is the positive linear 00:15:13.340 --> 00:15:16.040 combination, and 1 is the negative linear combination. 00:15:16.040 --> 00:15:18.540 You're not responsible for that, you're not responsible 00:15:18.540 --> 00:15:21.810 for correlating plus 1 to y, minus 1 to x. 00:15:21.810 --> 00:15:24.950 Just know that you have plus or minus 1, for our class, you 00:15:24.950 --> 00:15:29.060 can call it either x or y, either is fine, because it's a 00:15:29.060 --> 00:15:33.520 little bit more complicated than just the 1:1 translation 00:15:33.520 --> 00:15:38.170 between, for example, m equals 0 and having a p z orbital. 00:15:38.170 --> 00:15:41.850 All right. 00:15:41.850 --> 00:15:44.560 So let's look at some of these wave functions and make sure 00:15:44.560 --> 00:15:47.120 that we know how to name all of them in terms of orbitals 00:15:47.120 --> 00:15:48.950 and not just in terms of their numbers. 00:15:48.950 --> 00:15:52.100 Once we can do that we can go on and say okay, what actually 00:15:52.100 --> 00:15:54.060 is a wave function, but first we need to know how to 00:15:54.060 --> 00:15:56.470 describe which ones were talking about. 00:15:56.470 --> 00:15:59.650 So we saw that our lowest, our ground state wave 00:15:59.650 --> 00:16:01.750 function is 1, 0, 0. 00:16:01.750 --> 00:16:04.750 We can call that psi 1, 0, 0 is how we write 00:16:04.750 --> 00:16:07.420 it as a wave function. 00:16:07.420 --> 00:16:10.460 We said that's the 1 s orbital. 00:16:10.460 --> 00:16:14.380 We also know how to figure out the energy of this orbital, 00:16:14.380 --> 00:16:17.560 and we know how to figure out the energy using this formula 00:16:17.560 --> 00:16:21.140 here, which was the binding energy, which is negative r h, 00:16:21.140 --> 00:16:24.390 and instead of n, we can plug it in because n equals 1, so 00:16:24.390 --> 00:16:28.730 over 1 squared, and the actual energy is here. 00:16:28.730 --> 00:16:33.060 So, our next level up that we can go is going to be the n 00:16:33.060 --> 00:16:39.160 equals 2 energy level, but we also have an l and an m value, 00:16:39.160 --> 00:16:42.050 so our lowest l is going to be a 0 there. 00:16:42.050 --> 00:16:46.220 So we'll call that psi 2, 0, 0 wave function. 00:16:46.220 --> 00:16:50.070 What will we call that in terms of orbitals? 00:16:50.070 --> 00:16:52.690 Yup, so that's the 2 s orbital. 00:16:52.690 --> 00:16:55.370 So something I actually wanted to point out that I forgot to 00:16:55.370 --> 00:16:58.850 here is you'll notice that there's no subscript to the s. 00:16:58.850 --> 00:17:02.120 We said we have a subscript to the p, for example, that 00:17:02.120 --> 00:17:04.150 describes what m is equal to. 00:17:04.150 --> 00:17:07.000 The reason that we have no subscript to the s, is because 00:17:07.000 --> 00:17:10.630 the only possibility for m when you have an s orbital is 00:17:10.630 --> 00:17:12.770 that m has to be equal to 0. 00:17:12.770 --> 00:17:15.280 So we just assume it, you don't actually have to write 00:17:15.280 --> 00:17:18.540 it because there is, in fact, only one possibility. 00:17:18.540 --> 00:17:21.270 We can also figure out the energy of this orbital here, 00:17:21.270 --> 00:17:23.990 and the energy is equal to the Rydberg constant. 00:17:23.990 --> 00:17:25.750 The negative of the Rydberg constant now 00:17:25.750 --> 00:17:28.110 divided by 2 squared. 00:17:28.110 --> 00:17:30.770 So we can go on and do this for any orbital or any state 00:17:30.770 --> 00:17:32.430 function that we would like to. 00:17:32.430 --> 00:17:36.400 So, for example, if we talk about the 2, 1, 1 state label, 00:17:36.400 --> 00:17:39.100 that's just psi 2, 1, 1. 00:17:39.100 --> 00:17:40.910 What, in this case, would be our orbital? 00:17:40.910 --> 00:17:44.660 2 p what? 00:17:44.660 --> 00:17:49.750 OK, good, I heard mixed answers, which is correct. 00:17:49.750 --> 00:17:53.390 So you can either write 2 p x or 2 p y, whichever one you 00:17:53.390 --> 00:17:55.430 want is fine. 00:17:55.430 --> 00:17:58.270 And again, you'll notice that our energy is absolutely the 00:17:58.270 --> 00:18:01.870 same for an electron in that 2 p x orbital 00:18:01.870 --> 00:18:04.420 and in the 2 s orbital. 00:18:04.420 --> 00:18:06.640 So that's true for a hydrogen atom, it doesn't matter if 00:18:06.640 --> 00:18:10.570 you're in a p or an s orbital, their energies are the same. 00:18:10.570 --> 00:18:13.970 Then we can also talk about the 2, 1, 0 state function, 00:18:13.970 --> 00:18:15.880 which would be psi 2, 1, 0. 00:18:15.880 --> 00:18:16.670 What is this orbital? 00:18:16.670 --> 00:18:18.810 Yup. 00:18:18.810 --> 00:18:20.360 And there's only one correct answer here, 00:18:20.360 --> 00:18:22.590 which is to 2 p z. 00:18:22.590 --> 00:18:26.700 Is the energy going to be the same or different as up here? 00:18:26.700 --> 00:18:28.150 It's going to be the same energy. 00:18:28.150 --> 00:18:30.840 Again, the reason for that is because the energy only 00:18:30.840 --> 00:18:33.030 depends on the n value here, it doesn't 00:18:33.030 --> 00:18:35.850 depend on l or on m. 00:18:35.850 --> 00:18:38.660 So finally, if we talk about our last example of when n is 00:18:38.660 --> 00:18:42.240 going to be equal 2, we can have 2, 1 for l and 00:18:42.240 --> 00:18:43.840 then minus 1 for m. 00:18:43.840 --> 00:18:47.570 We can re-write this as psi 2 1 negative 1. 00:18:47.570 --> 00:18:50.570 And then our orbital is going to be just the opposite of 00:18:50.570 --> 00:18:52.370 whatever we said it was up here. 00:18:52.370 --> 00:18:56.720 So if you said 2 p x the first time, say 2 p y this time. 00:18:56.720 --> 00:19:00.240 And again, our energy is going to be the same where we again 00:19:00.240 --> 00:19:03.230 only depend on the n value. 00:19:03.230 --> 00:19:03.550 All right. 00:19:03.550 --> 00:19:06.230 So hopefully we're pretty comfortable naming any type of 00:19:06.230 --> 00:19:09.280 wave function using the chemist terminology. 00:19:09.280 --> 00:19:11.920 Let's switch to a clicker question and just confirm that 00:19:11.920 --> 00:19:13.690 that is, in fact, true. 00:19:13.690 --> 00:19:18.300 So what's the corresponding orbital if we talk about this 00:19:18.300 --> 00:19:27.530 state, 5, 1, 0? 00:19:27.530 --> 00:19:42.180 And you can go ahead and give 10 seconds on that. 00:19:42.180 --> 00:19:44.340 OK. 00:19:44.340 --> 00:19:45.980 All right, 77%. 00:19:45.980 --> 00:19:49.400 So, that's OK, you don't have to memorize things as I speak, 00:19:49.400 --> 00:19:52.310 you just need to go back and look at this and make sure you 00:19:52.310 --> 00:19:54.890 understand how to name it and that you'll be able to, for 00:19:54.890 --> 00:19:58.090 example, by next class, get a similar clicker question 00:19:58.090 --> 00:20:01.530 correct, and good job to the 77% that did get it. 00:20:01.530 --> 00:20:04.140 So I think we're safe to move on here. 00:20:04.140 --> 00:20:07.290 And I just want to point out that now we have these three 00:20:07.290 --> 00:20:08.010 quantum numbers. 00:20:08.010 --> 00:20:09.940 The reason there are three quantum numbers is we're 00:20:09.940 --> 00:20:13.000 describing an orbital in three dimensions, so it makes sense 00:20:13.000 --> 00:20:15.360 that we would need to describe in terms of three different 00:20:15.360 --> 00:20:16.620 quantum numbers. 00:20:16.620 --> 00:20:20.760 And the complete description, as I said, is from n l and m. 00:20:20.760 --> 00:20:24.060 And when you talk about n for an orbital, it's talking about 00:20:24.060 --> 00:20:27.150 the shell -- that shell is kind of what you picture when 00:20:27.150 --> 00:20:29.900 you think of a classical picture of an atom where you 00:20:29.900 --> 00:20:32.870 have 1 energy level, the next one is further out, the next 00:20:32.870 --> 00:20:34.780 one's further away. 00:20:34.780 --> 00:20:37.440 That's kind of your shell that we're discussing. 00:20:37.440 --> 00:20:43.700 L is the sub shell here, and then we have m, which is 00:20:43.700 --> 00:20:47.420 finally the complete description of the orbital. 00:20:47.420 --> 00:20:52.070 And what you can see is that for any n that has an l equals 00:20:52.070 --> 00:20:54.720 0, you can see here how there's only one possibility 00:20:54.720 --> 00:20:57.250 for and orbital description, and that's why we don't need 00:20:57.250 --> 00:21:01.910 to include the m when we're talking about and s orbital. 00:21:01.910 --> 00:21:04.350 The other thing that we know, which is what we were just 00:21:04.350 --> 00:21:06.870 discussing when we were going through the table is how this 00:21:06.870 --> 00:21:08.730 all relates to energy. 00:21:08.730 --> 00:21:11.720 And I want to really highlight here we're talking about for a 00:21:11.720 --> 00:21:15.280 hydrogen atom -- orbitals with the same n value 00:21:15.280 --> 00:21:16.980 have the same energy. 00:21:16.980 --> 00:21:19.510 Some of you might be saying in your heads, wait a second, I 00:21:19.510 --> 00:21:22.780 happen to know, I happen to remember from high school, 00:21:22.780 --> 00:21:25.960 that p orbitals have different energies then, 00:21:25.960 --> 00:21:27.620 for example, s orbitals. 00:21:27.620 --> 00:21:30.580 And that is not true for one electron atoms. We're going to 00:21:30.580 --> 00:21:34.175 get to more complicated atoms eventually where we're going 00:21:34.175 --> 00:21:37.450 to have more than one electron in it, but when we're talking 00:21:37.450 --> 00:21:40.370 about a single electron atom, we know that the binding 00:21:40.370 --> 00:21:43.100 energy is equal to the negative of the Rydberg 00:21:43.100 --> 00:21:46.640 constant over n squared, so it's only depends on n. 00:21:46.640 --> 00:21:49.260 So, for example, if we're talking about the n equals 2 00:21:49.260 --> 00:21:53.240 state, all of these four orbital descriptions are going 00:21:53.240 --> 00:21:54.690 to have the same energy. 00:21:54.690 --> 00:21:57.560 And we can generalize to figure out, based on any 00:21:57.560 --> 00:22:01.710 principle quantum number n, how many orbitals we have of 00:22:01.710 --> 00:22:07.460 the same energy, and what we can say is that for any shell 00:22:07.460 --> 00:22:09.810 n, there are n squared degenerate orbitals. 00:22:09.810 --> 00:22:15.730 And the word degenerate simply means same energy, so you have 00:22:15.730 --> 00:22:18.620 n squared orbitals that are of equal energy when they're 00:22:18.620 --> 00:22:21.430 degenerate. 00:22:21.430 --> 00:22:24.400 So, let's look at where this comes from with an energy 00:22:24.400 --> 00:22:27.090 level diagram here. 00:22:27.090 --> 00:22:35.210 So what you can see is again, we've got this ground state. 00:22:35.210 --> 00:22:38.030 So if we go to the ground state, what you see is we're 00:22:38.030 --> 00:22:41.260 at that lowest energy level, and we only have one 00:22:41.260 --> 00:22:44.320 possibility for an orbital, because when n equals 1, 00:22:44.320 --> 00:22:45.690 that's all we can do. 00:22:45.690 --> 00:22:49.950 So that's the 1 s orbital -- we have n squared or 1 00:22:49.950 --> 00:22:51.760 degenerate orbitals. 00:22:51.760 --> 00:22:57.100 When we talk about the n equals 2 state, we now have 2 00:22:57.100 --> 00:23:01.460 squared or 4 degenerate same energy orbitals, and those are 00:23:01.460 --> 00:23:03.900 the 2 s orbital. 00:23:03.900 --> 00:23:08.590 And then we also have the l being equal to 1 orbital, so 00:23:08.590 --> 00:23:12.030 those are going to be the 2 p x, the 2 p z, 00:23:12.030 --> 00:23:13.290 and the 2 p y orbital. 00:23:13.290 --> 00:23:16.980 All four of these orbitals have the same energy, they're 00:23:16.980 --> 00:23:18.060 degenerate. 00:23:18.060 --> 00:23:21.990 And as we go up the next energy level, which is based 00:23:21.990 --> 00:23:25.250 on n equals 3 principle quantum number, well now we 00:23:25.250 --> 00:23:29.045 have again the s, so we have the 3 s orbital, we're going 00:23:29.045 --> 00:23:34.350 to have three 3 p orbitals, right, so we'll have 3 p x, 3 00:23:34.350 --> 00:23:38.280 p z, and 3 p y, and now we're actually also going to have 00:23:38.280 --> 00:23:41.470 five different possible l equals 2 orbitals. 00:23:41.470 --> 00:23:44.970 Does anyone remember the l equals 2? 00:23:44.970 --> 00:23:46.110 Yes, everyone remembers. 00:23:46.110 --> 00:23:46.670 Good. 00:23:46.670 --> 00:23:49.640 So we have five possible d orbitals. 00:23:49.640 --> 00:23:55.540 We'll call these here the 3 d x y, as the subscript, the 3 d 00:23:55.540 --> 00:24:03.430 y z, the 3 d z squared, the 3 d x z, and the 3 d x squared 00:24:03.430 --> 00:24:05.740 minus y squared. 00:24:05.740 --> 00:24:10.360 So, what do you need to know here? 00:24:10.360 --> 00:24:13.350 What you need to know is that when m equals 00:24:13.350 --> 00:24:16.550 0, it's 3 d z squared. 00:24:16.550 --> 00:24:17.540 That's it. 00:24:17.540 --> 00:24:22.740 Again, these other p -- or the d x y, d y z, those are going 00:24:22.740 --> 00:24:25.340 to be those more complicated linear combinations, you don't 00:24:25.340 --> 00:24:26.840 need to worry about them. 00:24:26.840 --> 00:24:29.250 Eventually you will, at least, need to know the labels and 00:24:29.250 --> 00:24:30.630 know a little bit more about them. 00:24:30.630 --> 00:24:33.190 And in the second half of this course, Professor Drennen's 00:24:33.190 --> 00:24:36.320 going to talk to us about transition metals in depth, 00:24:36.320 --> 00:24:38.910 and that's when we'll really delve into d orbitals. 00:24:38.910 --> 00:24:41.360 For right now, you can kind of put the d orbitals in the back 00:24:41.360 --> 00:24:41.840 of your head. 00:24:41.840 --> 00:24:44.990 You need to know how to think about them in the same way we 00:24:44.990 --> 00:24:47.950 think about s and p orbitals, but for example, you don't yet 00:24:47.950 --> 00:24:51.520 need to know what all of the names are except for this 3 d 00:24:51.520 --> 00:24:53.340 z squared here. 00:24:53.340 --> 00:24:56.070 So we'll wait on that until we start talking more 00:24:56.070 --> 00:24:58.910 specifically about atoms where the d orbital becomes very 00:24:58.910 --> 00:25:00.890 significant. 00:25:00.890 --> 00:25:03.660 So, what we can see is this degeneracy. 00:25:03.660 --> 00:25:07.130 So what we know now is we can start thinking about the next 00:25:07.130 --> 00:25:10.150 step because we can fully describe the energy of 00:25:10.150 --> 00:25:13.890 orbitals, and we can fully describe a complete orbital in 00:25:13.890 --> 00:25:16.270 terms of its three quantum numbers, and its three 00:25:16.270 --> 00:25:19.830 positional variables, r, theta, and phi. 00:25:19.830 --> 00:25:23.050 So next we can think about okay, what is actually a wave 00:25:23.050 --> 00:25:25.990 function, and for example, what might the shape of 00:25:25.990 --> 00:25:29.130 different wave functions be. 00:25:29.130 --> 00:25:31.370 So essentially, what we're asking for here is the 00:25:31.370 --> 00:25:34.876 physical interpretation of psi, of the value of psi for 00:25:34.876 --> 00:25:36.210 an electron. 00:25:36.210 --> 00:25:39.560 And it turns out that the answer to can we have this 00:25:39.560 --> 00:25:42.640 physical interpretation of thinking about what psi means, 00:25:42.640 --> 00:25:45.190 the answer is really no, that we can't. 00:25:45.190 --> 00:25:48.310 There's no classical way to think about what a wave 00:25:48.310 --> 00:25:49.460 function is. 00:25:49.460 --> 00:25:52.580 There's no classical analogy that explains oh, this is what 00:25:52.580 --> 00:25:56.080 you can kind of picture when you picture a wave function. 00:25:56.080 --> 00:25:57.960 And that's somewhat inconvenient because we're 00:25:57.960 --> 00:26:00.940 working with wave functions, but it's a reality that comes 00:26:00.940 --> 00:26:04.050 out of quantum mechanics often, which is that we're 00:26:04.050 --> 00:26:07.120 describing a world that is so much different from the world 00:26:07.120 --> 00:26:09.170 that we observe on a day-to-day basis, that we're 00:26:09.170 --> 00:26:10.450 not always going to be able to make 00:26:10.450 --> 00:26:12.240 those one-to-one analogies. 00:26:12.240 --> 00:26:14.670 But luckily we don't have to worry about how we're going to 00:26:14.670 --> 00:26:17.440 picture all this, now that I said that, because even though 00:26:17.440 --> 00:26:19.720 there's no physical interpretation for what a wave 00:26:19.720 --> 00:26:22.830 function is, there is a physical interpretation for 00:26:22.830 --> 00:26:25.620 what a wave function squared means. 00:26:25.620 --> 00:26:29.300 So when we talk about a wave function squared, we're taking 00:26:29.300 --> 00:26:32.140 the square of the wave function, any one that we 00:26:32.140 --> 00:26:36.810 specify between n, l and m, at any position that we specify 00:26:36.810 --> 00:26:38.920 based on r, theta, and phi. 00:26:38.920 --> 00:26:42.040 And if we go ahead and square that, then what we get is a 00:26:42.040 --> 00:26:44.710 probability density, and specifically it's the 00:26:44.710 --> 00:26:49.280 probability of finding an electron in a certain small 00:26:49.280 --> 00:26:52.300 defined volume away from the nucleus. 00:26:52.300 --> 00:26:54.680 So it's a probability density. 00:26:54.680 --> 00:26:57.260 The important point here is it's not just a probability, 00:26:57.260 --> 00:26:59.530 it's a density, so we know that it's a probability 00:26:59.530 --> 00:27:01.690 divided by volume. 00:27:01.690 --> 00:27:04.420 And the person we have to thank for actually giving us 00:27:04.420 --> 00:27:07.010 this more concrete way to think about what a wave 00:27:07.010 --> 00:27:10.570 function squared is is Max Born here. 00:27:10.570 --> 00:27:14.820 And actually after the Schrodinger equation first was 00:27:14.820 --> 00:27:18.150 put forth, people had a lot of discussions about how is it 00:27:18.150 --> 00:27:21.250 that we can actually interpret what this wave function means, 00:27:21.250 --> 00:27:24.260 and a lot of ideas were put forth, and none of them worked 00:27:24.260 --> 00:27:27.440 out to match up with observations until Max Born 00:27:27.440 --> 00:27:29.700 here came up with the idea that we just square the wave 00:27:29.700 --> 00:27:32.350 function, and that's the probability density of finding 00:27:32.350 --> 00:27:36.130 an electron in a certain defined volume. 00:27:36.130 --> 00:27:38.660 And it's very helpful because it gives us a way 00:27:38.660 --> 00:27:39.730 to think about it. 00:27:39.730 --> 00:27:42.410 We can't actually go ahead and derive this equation of the 00:27:42.410 --> 00:27:45.655 wave function squared, because no one ever derived it, it's 00:27:45.655 --> 00:27:47.960 just an interpretation, but it's an interpretation that 00:27:47.960 --> 00:27:50.120 works essentially perfectly. 00:27:50.120 --> 00:27:52.380 Ever since this was first proposed, there has never been 00:27:52.380 --> 00:27:56.360 any observations that do not coincide with the idea, that 00:27:56.360 --> 00:27:59.680 did not match the fact that the probability density is 00:27:59.680 --> 00:28:02.480 equal to the wave function squared. 00:28:02.480 --> 00:28:06.020 So, also about Max Born, just to give you a little bit of a 00:28:06.020 --> 00:28:08.910 trivial pursuit type knowledge, he not only gave us 00:28:08.910 --> 00:28:13.010 this relationship between wave function squared, he also gave 00:28:13.010 --> 00:28:15.420 us Olivia Newton-John. 00:28:15.420 --> 00:28:17.950 This is her grandfather, I don't know if you can see from 00:28:17.950 --> 00:28:19.980 the eyes, I feel like there's a little bit of 00:28:19.980 --> 00:28:22.640 a resemblance there. 00:28:22.640 --> 00:28:25.086 So, I don't know what she grew up hearing about when she went 00:28:25.086 --> 00:28:27.210 to her grandparents' house, but it might have been wave 00:28:27.210 --> 00:28:29.080 function squared. 00:28:29.080 --> 00:28:33.290 So, a little tidbit of knowledge for you that's 00:28:33.290 --> 00:28:36.140 somewhat trivial. 00:28:36.140 --> 00:28:40.320 Then back to the non-trivial knowledge that is not trivial 00:28:40.320 --> 00:28:43.190 at all, in fact, is OK, how do we think about this 00:28:43.190 --> 00:28:45.140 probability density now that we have a little 00:28:45.140 --> 00:28:46.510 bit more of an idea. 00:28:46.510 --> 00:28:50.120 We know that it's a density, it's not an actual 00:28:50.120 --> 00:28:50.860 probability. 00:28:50.860 --> 00:28:53.850 So, one way we could look at it is by looking at this 00:28:53.850 --> 00:28:58.610 density dot diagram, where the density of the dots correlates 00:28:58.610 --> 00:29:00.720 to the probability density. 00:29:00.720 --> 00:29:05.960 So, what you see is near the nucleus, the density is the 00:29:05.960 --> 00:29:08.200 strongest, the dots are closest together. 00:29:08.200 --> 00:29:11.250 As you get far away from the nucleus, the dots get farther 00:29:11.250 --> 00:29:15.810 and farther apart, meaning the probability density at those 00:29:15.810 --> 00:29:19.490 volumes far away from the nucleus is going to be quite 00:29:19.490 --> 00:29:22.610 low, eventually going to almost zero, although it turns 00:29:22.610 --> 00:29:25.060 out that it never goes to exactly zero, so if we're 00:29:25.060 --> 00:29:28.640 talking about any orbital or any atom, it never actually 00:29:28.640 --> 00:29:30.770 ends, it never goes to zerio. 00:29:30.770 --> 00:29:32.590 But it turns out the probability is only 00:29:32.590 --> 00:29:34.810 significant within one angstrom. 00:29:34.810 --> 00:29:37.470 So you can either say that electrons are very, very tiny 00:29:37.470 --> 00:29:40.070 or that they're never ending, and both are pretty accurate 00:29:40.070 --> 00:29:43.730 ways to think about what an atom is. 00:29:43.730 --> 00:29:47.360 So, that's probability density, but in terms of 00:29:47.360 --> 00:29:50.150 thinking about it in terms of actual solutions to the wave 00:29:50.150 --> 00:29:52.390 function, let's take a little bit of a step back here. 00:29:52.390 --> 00:29:55.870 I have yet to show you the solution to a wave function 00:29:55.870 --> 00:29:58.680 for the hydrogen atom, so let me do that here, and then 00:29:58.680 --> 00:30:01.760 we'll build back up to probability densities, and it 00:30:01.760 --> 00:30:03.820 turns out that if we're talking about any wave 00:30:03.820 --> 00:30:06.500 function, we can actually break it up into two 00:30:06.500 --> 00:30:09.950 components, which are called the radial wave function and 00:30:09.950 --> 00:30:12.230 angular wave function. 00:30:12.230 --> 00:30:15.040 So, essentially we're just breaking it up into two parts 00:30:15.040 --> 00:30:18.270 that can be separated, and the part that is only dealing with 00:30:18.270 --> 00:30:22.030 the radius, so it's only a function of the radius of the 00:30:22.030 --> 00:30:24.520 electron from the nucleus. 00:30:24.520 --> 00:30:28.460 And we abbreviate that by calling it r, which is 00:30:28.460 --> 00:30:31.290 specified by two quantum numbers, and an l as a 00:30:31.290 --> 00:30:34.130 function of little r, radius. 00:30:34.130 --> 00:30:37.330 And we have the angular wave function, which is specified 00:30:37.330 --> 00:30:41.010 by l and m, and it's a function of the two angles 00:30:41.010 --> 00:30:43.430 when we're describing the position of the electron, so 00:30:43.430 --> 00:30:45.780 theta and phi. 00:30:45.780 --> 00:30:49.060 So, let's look at what this actually is for what we're 00:30:49.060 --> 00:30:51.840 showing here is the 1 s hydrogen atom. 00:30:51.840 --> 00:30:53.780 If you look in your book there's a whole table of 00:30:53.780 --> 00:30:56.330 different solutions to the Schrodinger equation for 00:30:56.330 --> 00:30:58.570 several different wave functions. 00:30:58.570 --> 00:31:00.700 So this is the 1 s, you can look it up if you're 00:31:00.700 --> 00:31:04.810 interested for the 2 s, or 3 s, or 5 s, or whatever you're 00:31:04.810 --> 00:31:05.040 curious about. 00:31:05.040 --> 00:31:08.150 But what I'm going to show you here is the 1 s solution. 00:31:08.150 --> 00:31:11.670 So you can see there's this radial part here, and you have 00:31:11.670 --> 00:31:14.370 the angular part, you can combine the two parts to get 00:31:14.370 --> 00:31:16.170 the total wave function. 00:31:16.170 --> 00:31:20.470 And what you can see is we have this new constant that we 00:31:20.470 --> 00:31:22.000 haven't seen before. 00:31:22.000 --> 00:31:26.030 So what do you see in there that is new? 00:31:26.030 --> 00:31:26.220 Yeah. 00:31:26.220 --> 00:31:27.520 This a sub nought. 00:31:27.520 --> 00:31:30.410 That's a new constant for us in this course. 00:31:30.410 --> 00:31:35.950 This is what's called the Bohr radius, and we'll explain -- 00:31:35.950 --> 00:31:38.940 hopefully we'll get to it today where this Bohr radius 00:31:38.940 --> 00:31:41.930 name comes from, but for now what you need to know is just 00:31:41.930 --> 00:31:44.630 that it's a constant, just treat it like a constant, and 00:31:44.630 --> 00:31:47.050 it turns out to be equal to 52 . 00:31:47.050 --> 00:31:51.620 9 pekameters or about 1/2 an angstrom. 00:31:51.620 --> 00:31:54.510 The more important thing that I want you to notice when 00:31:54.510 --> 00:32:01.540 you're looking at this wave equation for a 1 s h atom, is 00:32:01.540 --> 00:32:04.570 the fact that if you look at the angular component of the 00:32:04.570 --> 00:32:08.160 wave function, you'll notice that it's a constant. 00:32:08.160 --> 00:32:12.200 It doesn't depend on theta, it doesn't depend on phi. 00:32:12.200 --> 00:32:15.120 No matter where you specify your electron is in terms of 00:32:15.120 --> 00:32:18.160 those two angles, it doesn't matter the angular part of 00:32:18.160 --> 00:32:21.090 your wave function is going to be the same. 00:32:21.090 --> 00:32:23.220 So, what does that mean for us? 00:32:23.220 --> 00:32:26.370 Well, essentially what that tells is that these s orbitals 00:32:26.370 --> 00:32:28.190 are spherically symmetrical. 00:32:28.190 --> 00:32:30.050 That should make sense, right, because they're only 00:32:30.050 --> 00:32:31.190 dependent on r. 00:32:31.190 --> 00:32:34.880 How far you are away from the nucleus in terms of a radius, 00:32:34.880 --> 00:32:37.910 they don't depend at all on those two angles, they're 00:32:37.910 --> 00:32:42.960 independent of theta and they're independent of phi. 00:32:42.960 --> 00:32:45.820 So, what I'm showing in this picture here is just an 00:32:45.820 --> 00:32:48.280 electron cloud that you can see. 00:32:48.280 --> 00:32:51.720 Think of it as a probability density plot. 00:32:51.720 --> 00:32:55.880 And what here is just a graph of the 1 s wave function going 00:32:55.880 --> 00:32:59.810 across some radius defined this way, and you can see that 00:32:59.810 --> 00:33:02.955 the probability -- well, this is the wave function, so we 00:33:02.955 --> 00:33:05.820 would have to square it and think about the probability. 00:33:05.820 --> 00:33:11.480 So this squared at the origin is going to be a very high 00:33:11.480 --> 00:33:14.880 probability, and it decays off as you get farther and farther 00:33:14.880 --> 00:33:18.160 away from the nucleus or from the center, and that's 00:33:18.160 --> 00:33:21.370 independent of the angle. 00:33:21.370 --> 00:33:24.400 So, let's look at these probability plots of different 00:33:24.400 --> 00:33:34.660 s orbitals here, and up top here, we have the probability 00:33:34.660 --> 00:33:38.230 density plot and what you can see is what I just said, a 00:33:38.230 --> 00:33:40.060 very high probability density in the nucleus, 00:33:40.060 --> 00:33:41.670 decays as you go out. 00:33:41.670 --> 00:33:45.200 And what is plotted below is the actual wave function, so 00:33:45.200 --> 00:33:48.390 you can see it starts very high and then the decays down. 00:33:48.390 --> 00:33:51.750 More interesting is to look at the 2 s wave function. 00:33:51.750 --> 00:33:54.620 So, if we look at the bottom here and the actual plot of 00:33:54.620 --> 00:33:57.980 the wave function, we see it starts high, very positive, 00:33:57.980 --> 00:34:01.380 and it goes down and it eventually hits zero, and goes 00:34:01.380 --> 00:34:04.480 through zero and then becomes negative and then never quite 00:34:04.480 --> 00:34:07.150 hits zero again, although it approaches zero. 00:34:07.150 --> 00:34:10.050 So, at this place where it hits zero, that means that the 00:34:10.050 --> 00:34:12.650 square of the wave function is also going to be zero, right. 00:34:12.650 --> 00:34:17.110 So we can see if we look at the probability density plot, 00:34:17.110 --> 00:34:20.100 we can see there's a place where the probability density 00:34:20.100 --> 00:34:22.880 of finding an electron anywhere there is actually 00:34:22.880 --> 00:34:25.370 going to be zero. 00:34:25.370 --> 00:34:28.460 So we can think of a third case where we have the 3 s 00:34:28.460 --> 00:34:31.580 orbital, and in the 3 s orbital we see something 00:34:31.580 --> 00:34:35.040 similar, we start high, we go through zero, where there will 00:34:35.040 --> 00:34:38.040 now be zero probability density, as we can see in the 00:34:38.040 --> 00:34:40.250 in the density plot graph. 00:34:40.250 --> 00:34:43.850 Then we go negative and we go through zero again, which 00:34:43.850 --> 00:34:47.170 correlates to the second area of zero, that shows up also in 00:34:47.170 --> 00:34:50.850 our probability density plot, and then we're positive again 00:34:50.850 --> 00:34:55.740 and approach zero as we go to infinity for r. 00:34:55.740 --> 00:34:58.830 So, what this means is that when we're looking at an 00:34:58.830 --> 00:35:01.960 actual wave function, we're treating it as a wave, right, 00:35:01.960 --> 00:35:05.620 so waves can have both magnitude, but they can also 00:35:05.620 --> 00:35:08.380 have a direction, so they can be 00:35:08.380 --> 00:35:10.420 either positive or negative. 00:35:10.420 --> 00:35:13.160 So, for example, if we were looking at the actual wave 00:35:13.160 --> 00:35:15.880 function, we would say that these parts here have a 00:35:15.880 --> 00:35:18.180 positive amplitude, and in here we 00:35:18.180 --> 00:35:20.040 have a negative amplitude. 00:35:20.040 --> 00:35:23.460 And when we're looking at the probability density graphs, it 00:35:23.460 --> 00:35:26.650 doesn't make a difference, it's okay, It has no meaning 00:35:26.650 --> 00:35:29.800 for our actual plot there, because we're squaring it, so 00:35:29.800 --> 00:35:32.290 it doesn't matter whether it's negative or positive, all that 00:35:32.290 --> 00:35:34.040 matters is the magnitude. 00:35:34.040 --> 00:35:37.080 But when we're thinking about actual wave behavior of 00:35:37.080 --> 00:35:39.170 electrons, it's just important to keep in the back of our 00:35:39.170 --> 00:35:42.250 head that some areas have positive amplitude and some 00:35:42.250 --> 00:35:43.350 have negative. 00:35:43.350 --> 00:35:46.210 So we'll talk about this more we get into p orbitals and 00:35:46.210 --> 00:35:48.090 bonding is where it's going to become an issue. 00:35:48.090 --> 00:35:50.990 So I just want to kind of introduce that idea here. 00:35:50.990 --> 00:35:54.340 Because if we think about wave behavior of electrons and 00:35:54.340 --> 00:35:57.160 we're forming bonds, then what we have to do is have 00:35:57.160 --> 00:36:00.620 constructive interference of 2 different electrons, right, to 00:36:00.620 --> 00:36:04.380 form a bond, we want to and together those probabilities. 00:36:04.380 --> 00:36:07.110 So we want to have constructive interference to 00:36:07.110 --> 00:36:10.530 form a bond, whereas if we had destructive interference, we 00:36:10.530 --> 00:36:12.240 would not be forming a bond. 00:36:12.240 --> 00:36:14.590 So that's where you have to think about whether it's 00:36:14.590 --> 00:36:15.580 positive or negative. 00:36:15.580 --> 00:36:17.540 You don't have to think about it right now, but you might 00:36:17.540 --> 00:36:19.580 have heard in high school talking about p orbitals, the 00:36:19.580 --> 00:36:23.420 phase, sometimes you mark a p orbital as being a plus sign 00:36:23.420 --> 00:36:24.200 or negative sign. 00:36:24.200 --> 00:36:27.980 Did any of you do that in high school at all? 00:36:27.980 --> 00:36:28.950 A little bit, yeah. 00:36:28.950 --> 00:36:31.900 So, that's having to do with the actual wave function. 00:36:31.900 --> 00:36:34.130 So, that'll become more relevant later, bonding 00:36:34.130 --> 00:36:36.840 actually, a couple lectures down the road. 00:36:36.840 --> 00:36:39.180 But I just want to introduce it here while we do, in fact, 00:36:39.180 --> 00:36:42.630 have the wave function plots up here. 00:36:42.630 --> 00:36:45.100 But a real key in looking at these plots is where we, in 00:36:45.100 --> 00:36:47.590 fact, did go through zer and have this 00:36:47.590 --> 00:36:49.520 zero probability density. 00:36:49.520 --> 00:36:54.300 We call that a node, and a node, more specifically, is 00:36:54.300 --> 00:36:58.790 any value of either r, the radius, or the two angles for 00:36:58.790 --> 00:37:01.460 which the wave function, and that also means the wave 00:37:01.460 --> 00:37:04.370 function squared or the probability density, is going 00:37:04.370 --> 00:37:06.910 to be equal to zero. 00:37:06.910 --> 00:37:10.310 So, we can see in our 1 s orbital, how 00:37:10.310 --> 00:37:12.910 many nodes do we have? 00:37:12.910 --> 00:37:13.970 There's no nodes, yeah. 00:37:13.970 --> 00:37:16.500 It looks like we hit zero, but we actually don't -- remember 00:37:16.500 --> 00:37:18.720 that we never go all the way to zero, so there's these 00:37:18.720 --> 00:37:21.460 little points if we were to look really carefully at an 00:37:21.460 --> 00:37:25.080 accurate probability density plot, it would never 00:37:25.080 --> 00:37:26.540 actually hit zero. 00:37:26.540 --> 00:37:29.780 And then, for example, how many nodes do we have in the 3 00:37:29.780 --> 00:37:31.810 s orbital? two. 00:37:31.810 --> 00:37:32.890 That's correct. 00:37:32.890 --> 00:37:35.740 So we have two nodes in the 3 s orbital. 00:37:35.740 --> 00:37:39.610 We can actually specify where those nodes are, which is 00:37:39.610 --> 00:37:40.760 written on your notes. 00:37:40.760 --> 00:37:45.420 For the 2 s orbital, at 2 a nought, so it's just 2 times 00:37:45.420 --> 00:37:49.080 that constant a nought, which is the Bohr radius. 00:37:49.080 --> 00:37:51.040 And for the 3 s, we have one at 1 . 00:37:51.040 --> 00:37:53.560 9 a nought, and one at 7 . 00:37:53.560 --> 00:37:55.340 1 a nought. 00:37:55.340 --> 00:37:58.770 We can also specify what kind of node we're talking about. 00:37:58.770 --> 00:38:02.570 We'll introduce in the next course angular nodes, but 00:38:02.570 --> 00:38:05.460 today we're just going to be talking about radial nodes, 00:38:05.460 --> 00:38:09.820 and a radial node is a value for r at which psi, and 00:38:09.820 --> 00:38:12.420 therefore, also the probability psi squared is 00:38:12.420 --> 00:38:14.460 going to be equal to zero. 00:38:14.460 --> 00:38:17.830 So, when we're talking about an s orbital, since there is 00:38:17.830 --> 00:38:20.890 no angular dependence, and it only depends on r, every 00:38:20.890 --> 00:38:23.420 single one of our nodes is actually going to specifically 00:38:23.420 --> 00:38:26.180 be a radial node, right, because these are, for 00:38:26.180 --> 00:38:29.620 example, this 2 a nought is a value of r, a value of the 00:38:29.620 --> 00:38:33.610 radius, no matter which way you go around at which there's 00:38:33.610 --> 00:38:36.170 going to be a node at which there is zero probability 00:38:36.170 --> 00:38:40.030 density of finding an electron there. 00:38:40.030 --> 00:38:42.260 So, it's very easy to calculate, however, the number 00:38:42.260 --> 00:38:44.660 of radial nodes, and this works not just for s orbitals, 00:38:44.660 --> 00:38:47.680 but also for p orbitals, or d orbitals, or whatever kind of 00:38:47.680 --> 00:38:49.910 work of orbitals you want to discuss. 00:38:49.910 --> 00:38:52.940 And that's just to take the principle quantum number and 00:38:52.940 --> 00:38:57.800 subtract it by 1, and then also subtract from that your l 00:38:57.800 --> 00:38:58.700 quantum number. 00:38:58.700 --> 00:39:03.030 So what you can do for a 1 s is just take 1 minus 1 and 00:39:03.030 --> 00:39:07.280 then l is equal to 0, so you have zero radial nodes. 00:39:07.280 --> 00:39:09.400 And that matches up with what we saw. 00:39:09.400 --> 00:39:14.370 If we try this for the 2 s, we have 2 minus 1 minus 0. 00:39:14.370 --> 00:39:17.420 So what we should expect to see is one radial node, and 00:39:17.420 --> 00:39:21.730 that is what we see here in the probability density plot. 00:39:21.730 --> 00:39:26.200 And then if we think about 3 s, we want to start with 3, we 00:39:26.200 --> 00:39:31.030 subtract 1, again l is equal to 0, so minus 0 and we have 00:39:31.030 --> 00:39:34.120 two radial nodes. 00:39:34.120 --> 00:39:36.220 So, this should be pretty straight forward, let's see if 00:39:36.220 --> 00:39:39.900 we can get close to a 100% on this one, which is how many 00:39:39.900 --> 00:39:49.440 radial nodes does a 4 p orbital have? 00:39:49.440 --> 00:39:51.310 And let's give 10 seconds on that, make 00:39:51.310 --> 00:40:04.710 you think fast here. 00:40:04.710 --> 00:40:08.910 OK, so most people were correct, or well, the 00:40:08.910 --> 00:40:11.630 majority, at least, were correct. 00:40:11.630 --> 00:40:15.450 And seeing that it's a 4 p has two nodes -- let's just write 00:40:15.450 --> 00:40:18.310 this out since not everyone did get it correct. 00:40:18.310 --> 00:40:21.810 So, if we're talking about a 4 p orbital, and our equation is 00:40:21.810 --> 00:40:28.030 n minus 1 minus l, the principle quantum number is 4, 00:40:28.030 --> 00:40:31.680 1 is 1 -- what is l for a p orbital? 00:40:31.680 --> 00:40:33.560 STUDENT: 1. 00:40:33.560 --> 00:40:34.190 PROFESSOR: 1. 00:40:34.190 --> 00:40:36.250 So, I tricked you a little, I guess I didn't put an s up 00:40:36.250 --> 00:40:37.910 there and that's what we had been talking about, so that 00:40:37.910 --> 00:40:39.220 was probably the issue. 00:40:39.220 --> 00:40:47.230 But what we find is that we have two radial nodes. 00:40:47.230 --> 00:40:47.500 All right. 00:40:47.500 --> 00:40:50.890 So we can switch back to our notes here. 00:40:50.890 --> 00:40:54.800 So, doing those probability density dot graphs, we can get 00:40:54.800 --> 00:40:57.760 an idea of the shape of those orbitals, we know that they're 00:40:57.760 --> 00:40:59.640 spherically symmetrical. 00:40:59.640 --> 00:41:02.140 We're not going to talk about p orbitals today, we're going 00:41:02.140 --> 00:41:04.970 to talk about p orbitals exclusively on Friday, and as 00:41:04.970 --> 00:41:07.710 I said, d orbitals you'll get to with Professor Drennen. 00:41:07.710 --> 00:41:12.490 But we can also think when we're talking about wave 00:41:12.490 --> 00:41:15.570 function squared, what we're really talking about is the 00:41:15.570 --> 00:41:16.940 probability density, right, the 00:41:16.940 --> 00:41:19.130 probability in some volume. 00:41:19.130 --> 00:41:21.850 But there's also a way to get rid of the volume part and 00:41:21.850 --> 00:41:24.443 actually talk about the probability of finding an 00:41:24.443 --> 00:41:30.710 electron at some certain area within the atom, and this is 00:41:30.710 --> 00:41:35.090 what we do using radial probability 00:41:35.090 --> 00:41:37.550 distribution graphs. 00:41:37.550 --> 00:41:40.360 And what that is the probability of finding an 00:41:40.360 --> 00:41:43.880 electron in some shell where we define the thickness as d 00:41:43.880 --> 00:41:47.540 r, some distance, r, from the nucleus. 00:41:47.540 --> 00:41:49.510 So, think about what we're saying here. 00:41:49.510 --> 00:41:52.740 We're saying the probability of finding an electron at some 00:41:52.740 --> 00:41:56.390 distance from the nucleus in some very thin shell that we 00:41:56.390 --> 00:41:58.300 describe by d r. 00:41:58.300 --> 00:42:00.700 So if you think of a shell, you can actually just think of 00:42:00.700 --> 00:42:02.870 an egg shell, that's probably the easiest way to think of 00:42:02.870 --> 00:42:05.460 it, where the yolk, if you really maybe make it a lot 00:42:05.460 --> 00:42:07.230 smaller might be the nucleus. 00:42:07.230 --> 00:42:09.790 And let's also make our egg perfectly symmetric and 00:42:09.790 --> 00:42:11.200 perfectly round. 00:42:11.200 --> 00:42:14.910 But still, when we're talking about the radial probability 00:42:14.910 --> 00:42:18.140 distribution, what we actually want to think about is what's 00:42:18.140 --> 00:42:22.150 the probability of finding the electron in that shell? 00:42:22.150 --> 00:42:24.690 Think of it as that egg shell part. 00:42:24.690 --> 00:42:28.320 So, we can do that by using this equation, which is for s 00:42:28.320 --> 00:42:30.970 orbitals where the radial probability distribution is 00:42:30.970 --> 00:42:34.540 going to be equal to 4 pi r squared times the wave 00:42:34.540 --> 00:42:36.900 function squared, d r. 00:42:36.900 --> 00:42:44.320 That should make sense to us, because when we talk about a 00:42:44.320 --> 00:42:52.260 wave function, we're talking about a probability divided by 00:42:52.260 --> 00:42:53.800 a volume, because we're talking about 00:42:53.800 --> 00:42:55.750 a probability density. 00:42:55.750 --> 00:43:00.240 So if we actually go ahead and multiply it by the volume of 00:43:00.240 --> 00:43:03.570 our shell, then we end up just with probability, which is 00:43:03.570 --> 00:43:06.520 kind of a nicer term to be thinking about here. 00:43:06.520 --> 00:43:09.070 So, of course, if we're talking about a perfectly 00:43:09.070 --> 00:43:12.600 spherical shell at some distance, thickness, d r, we 00:43:12.600 --> 00:43:16.430 talk about it as 4 pi r squared d r, so we just 00:43:16.430 --> 00:43:20.780 multiply that by the probability density. 00:43:20.780 --> 00:43:25.740 We can graph out what this is where we're graphing the 00:43:25.740 --> 00:43:31.410 radial probability density as a function of the radius. 00:43:31.410 --> 00:43:35.620 And what you see is that at zero, you start at zero. 00:43:35.620 --> 00:43:39.790 And so, the radial probability density at the nucleus is 00:43:39.790 --> 00:43:42.840 going to be zero, even though we know the probability 00:43:42.840 --> 00:43:46.390 density at the nucleus is very high, that's actually where is 00:43:46.390 --> 00:43:49.180 the highest. The reason in our radial probability 00:43:49.180 --> 00:43:56.060 distributions we start -- the reason, if you look at the 00:43:56.060 --> 00:43:58.880 zero point on the radius that we start at zero is because 00:43:58.880 --> 00:44:02.810 we're multiplying the probability density by some 00:44:02.810 --> 00:44:05.560 volume, and when we're not anywhere from the nucleus, 00:44:05.560 --> 00:44:07.620 that volume is defined as zero. 00:44:07.620 --> 00:44:09.450 So, it's a little bit artificial that we're seeing 00:44:09.450 --> 00:44:11.020 that zero point there. 00:44:11.020 --> 00:44:13.980 So, actually I want you to go ahead in your notes and circle 00:44:13.980 --> 00:44:17.820 that zero point and write "not a node." This is not a node 00:44:17.820 --> 00:44:19.660 because a node is where we actually have 00:44:19.660 --> 00:44:21.410 no probability density. 00:44:21.410 --> 00:44:24.370 So this, where we start at zero is not a node, is the 00:44:24.370 --> 00:44:27.060 first thing to point out. 00:44:27.060 --> 00:44:30.370 And as we get further and further from the radius, the 00:44:30.370 --> 00:44:32.930 volume we're multiplying it by actually gets bigger and 00:44:32.930 --> 00:44:35.480 bigger, because you can see how the volume of that little 00:44:35.480 --> 00:44:38.100 thin shell is going to get larger and larger as you get 00:44:38.100 --> 00:44:39.180 further away. 00:44:39.180 --> 00:44:43.470 So there's some distance where the probability of actually 00:44:43.470 --> 00:44:45.800 finding an electron there is going to be your maximum 00:44:45.800 --> 00:44:46.830 probability. 00:44:46.830 --> 00:44:50.020 And that's what we label as r sub m p, or your 00:44:50.020 --> 00:44:52.970 most probable radius. 00:44:52.970 --> 00:44:55.900 This is the point at which your probability is highest 00:44:55.900 --> 00:44:57.470 for finding an electron. 00:44:57.470 --> 00:45:01.870 This is equal to a sub nought for a hydrogen atom, and we 00:45:01.870 --> 00:45:05.260 remember that that's just our Bohr radius, which is 0 . 00:45:05.260 --> 00:45:11.450 5 2 9 angstroms. And basically, what that means is 00:45:11.450 --> 00:45:15.390 you can actually find an electron anywhere going away 00:45:15.390 --> 00:45:18.090 from the nucleus, but you're most likely to find that you 00:45:18.090 --> 00:45:21.360 have the highest probability at a distance of a sub nought, 00:45:21.360 --> 00:45:23.970 or the Bohr radius. 00:45:23.970 --> 00:45:25.920 So, I said I'd tell you a little bit more about where 00:45:25.920 --> 00:45:30.010 this Bohr radius came from, and it came from a model of 00:45:30.010 --> 00:45:34.320 the atom that pre-dated quantum mechanics, and Neils 00:45:34.320 --> 00:45:38.230 Bohr is who came up with the idea of the Bohr radius, and 00:45:38.230 --> 00:45:41.870 here is hanging out with Einstein, so he had some 00:45:41.870 --> 00:45:44.980 pretty good company that he kept. 00:45:44.980 --> 00:45:48.060 And what you need to remember when we're thinking about this 00:45:48.060 --> 00:45:51.450 model of the atom is that in 1911 it had already been 00:45:51.450 --> 00:45:53.880 discovered that we have an electron, and we have a 00:45:53.880 --> 00:45:56.750 nucleus, and there needs to be some way that those two hang 00:45:56.750 --> 00:45:59.960 together, but it was not for another 15 years that we 00:45:59.960 --> 00:46:02.970 actually had the Schrodinger equation that allowed us to 00:46:02.970 --> 00:46:07.130 understand the interaction fully between the electron and 00:46:07.130 --> 00:46:07.630 the nucleus. 00:46:07.630 --> 00:46:10.900 So all that Bohr, for example, had to go on at this point was 00:46:10.900 --> 00:46:14.560 a more classical picture of the atom, as you can see on 00:46:14.560 --> 00:46:17.840 the left side of the screen there, which is the idea that 00:46:17.840 --> 00:46:20.680 the electrons actually somehow just orbiting the nucleus. 00:46:20.680 --> 00:46:23.440 And even though he could figure out that this wasn't 00:46:23.440 --> 00:46:26.840 possible, he still used this as a starting point, and what 00:46:26.840 --> 00:46:31.170 he did know was that these energy levels that were within 00:46:31.170 --> 00:46:34.270 hydrogen atom were quantized. and he knew this the same way 00:46:34.270 --> 00:46:37.230 that we saw it in the last class, which is when we viewed 00:46:37.230 --> 00:46:40.200 the difference spectra coming out from the hydrogen, and we 00:46:40.200 --> 00:46:43.130 also did it for neon, but we saw in the hydrogen atom that 00:46:43.130 --> 00:46:46.130 it was very discreet energy levels that we could observe. 00:46:46.130 --> 00:46:47.950 He knew the same thing that had been 00:46:47.950 --> 00:46:49.440 observed by that point. 00:46:49.440 --> 00:46:53.490 So, what he did was kind of impose a quantum mechanical 00:46:53.490 --> 00:46:56.270 model, not a full one, just the idea that those energy 00:46:56.270 --> 00:47:00.220 levels were quantized on to the classical picture of an 00:47:00.220 --> 00:47:02.890 atom that has a discreet orbit. 00:47:02.890 --> 00:47:06.130 And what he came out with when he did some calculations is 00:47:06.130 --> 00:47:09.710 that there's the radius that he could calculate was equal 00:47:09.710 --> 00:47:13.190 to this number a sub nought, which is what we call the Bohr 00:47:13.190 --> 00:47:17.550 radius, and it turns out that the Bohr radius happens to be 00:47:17.550 --> 00:47:21.220 the radius most probable for a hydrogen atom. 00:47:21.220 --> 00:47:23.850 And the reason we won't talk any more about this Bohr model 00:47:23.850 --> 00:47:26.120 is because, of course, it's not correct. 00:47:26.120 --> 00:47:28.580 So we're not going to spend too much time on it here. 00:47:28.580 --> 00:47:31.860 But we can see, for example, one reason or one way in which 00:47:31.860 --> 00:47:33.010 is not correct. 00:47:33.010 --> 00:47:37.130 Because what it tells is that we can figure out exactly what 00:47:37.130 --> 00:47:41.110 the radius of an electron and a nucleus are 00:47:41.110 --> 00:47:42.550 in a hydrogen atom. 00:47:42.550 --> 00:47:45.100 That's a deterministic way of doing things, that's what you 00:47:45.100 --> 00:47:46.890 get from classical mechanics. 00:47:46.890 --> 00:47:49.840 But the reality that we know from our quantum mechanical 00:47:49.840 --> 00:47:53.050 model, is that we can't know exactly what the radius is, 00:47:53.050 --> 00:47:56.640 all we can say is what the probability is of the radius 00:47:56.640 --> 00:47:58.980 being at certain different points. so, that's a more 00:47:58.980 --> 00:48:01.620 complete quantum mechanical picture of 00:48:01.620 --> 00:48:03.360 what is going on here. 00:48:03.360 --> 00:48:05.800 So if we superimpose our radial probability 00:48:05.800 --> 00:48:09.290 distribution onto the Bohr radius, we see it's much more 00:48:09.290 --> 00:48:11.400 complicated than just having a discreet radius. 00:48:11.400 --> 00:48:14.760 We can actually have any radius, but some radii just 00:48:14.760 --> 00:48:17.710 have much, much smaller probabilities of actually 00:48:17.710 --> 00:48:20.890 being significant or not. 00:48:20.890 --> 00:48:24.050 So, I think we're a little bit out of time today, but we'll 00:48:24.050 --> 00:48:26.670 start next class with thinking about drawing radial 00:48:26.670 --> 00:48:28.840 probability distributions of more than 00:48:28.840 --> 00:48:31.270 just the 1 s orbital.