WEBVTT 00:00:00.030 --> 00:00:02.400 The following content is provided under a Creative 00:00:02.400 --> 00:00:03.830 Commons License. 00:00:03.830 --> 00:00:06.840 Your support will help MIT OpenCourseWare continue to 00:00:06.840 --> 00:00:10.510 offer high quality educational resources for free. 00:00:10.510 --> 00:00:13.390 To make a donation or view additional materials from 00:00:13.390 --> 00:00:15.840 hundreds of MIT courses, visit mitopencourseware@ocw.mit.edu. 00:00:21.110 --> 00:00:22.570 PROFESSOR: OK, settle down. 00:00:22.570 --> 00:00:23.980 Let's get started. 00:00:23.980 --> 00:00:29.770 One announcement, yesterday we had our first weekly I would 00:00:29.770 --> 00:00:31.630 say minor celebration. 00:00:31.630 --> 00:00:33.580 And by and large, it went well. 00:00:33.580 --> 00:00:36.840 Please make sure that you go to your assigned recitation. 00:00:36.840 --> 00:00:40.760 If you miss your recitation, you need to get down to see 00:00:40.760 --> 00:00:44.400 Hillary so that we make sure that we have enough copies of 00:00:44.400 --> 00:00:46.780 the weekly quizzes on hand. 00:00:46.780 --> 00:00:49.490 If you've joined the class, you have to check in with her. 00:00:49.490 --> 00:00:52.910 She's down the hall here in room 8-201. 00:00:52.910 --> 00:00:54.310 And you'll be assigned to a section. 00:00:57.640 --> 00:00:59.560 What else do I have by way of announcements? 00:00:59.560 --> 00:01:00.520 Oh yes. 00:01:00.520 --> 00:01:03.460 Just reminding you, this was from 2003. 00:01:03.460 --> 00:01:04.380 See it doesn't change. 00:01:04.380 --> 00:01:09.470 It's the same s-block, p-block, and d-block elements. 00:01:09.470 --> 00:01:12.450 So that's coming up a week from tomorrow, two 00:01:12.450 --> 00:01:14.810 celebrations next. 00:01:14.810 --> 00:01:20.790 And of course the contests, the contests with hot prizes. 00:01:20.790 --> 00:01:22.320 All right. 00:01:22.320 --> 00:01:23.410 Let's get down to business. 00:01:23.410 --> 00:01:27.220 Last day we looked at the Rutherford-Geiger-Marsden 00:01:27.220 --> 00:01:28.400 experiment. 00:01:28.400 --> 00:01:30.820 Oh there's one other one. 00:01:30.820 --> 00:01:33.380 If you look at the readings, there's this one section 00:01:33.380 --> 00:01:34.890 called the archives. 00:01:34.890 --> 00:01:39.530 My predecessor, Professor Wit, wrote a set of lecture notes. 00:01:39.530 --> 00:01:41.880 And they look something like this. 00:01:41.880 --> 00:01:44.580 And some students have said that they find these a little 00:01:44.580 --> 00:01:49.080 more expository in certain sections on certain topics 00:01:49.080 --> 00:01:51.260 than the book to be. 00:01:51.260 --> 00:01:52.660 And I have no preference. 00:01:52.660 --> 00:01:55.520 But if you take a look at this it'll say LN1, 00:01:55.520 --> 00:01:56.810 lecture notes 1. 00:01:56.810 --> 00:01:58.170 If you go to this, read this. 00:01:58.170 --> 00:02:00.060 If you find that's helpful, good. 00:02:00.060 --> 00:02:03.710 If you don't find it helpful, than stick with the book. 00:02:03.710 --> 00:02:05.280 Just letting you know what that is. 00:02:05.280 --> 00:02:06.950 All right, so last day we looked at the 00:02:06.950 --> 00:02:10.250 Rutherford-Geiger-Marsden experiment in which a high 00:02:10.250 --> 00:02:15.290 energy beam of alpha particles bombarded a thin, gold foil. 00:02:15.290 --> 00:02:18.610 And on the basis of the scattering results, namely 00:02:18.610 --> 00:02:22.890 most of the particles went through with minor scattering. 00:02:22.890 --> 00:02:24.990 And a tiny fraction of them were scattered 00:02:24.990 --> 00:02:27.500 through large angles. 00:02:27.500 --> 00:02:31.620 The Thomson plum pudding model was rejected in favor of 00:02:31.620 --> 00:02:34.500 Rutherford's nuclear model of the atom. 00:02:34.500 --> 00:02:38.240 And then subsequently, Bohr came up with the quantitative 00:02:38.240 --> 00:02:41.840 representation off the Rutherford nuclear model. 00:02:41.840 --> 00:02:47.040 And we were partway through the treatment of Bohr last day 00:02:47.040 --> 00:02:47.790 when we adjourned. 00:02:47.790 --> 00:02:51.220 So let's right pick up the thread from where we left off. 00:02:51.220 --> 00:02:56.010 And so just to remind you, the Bohr model is for a 1-electron 00:02:56.010 --> 00:02:58.020 atom gas phase. 00:02:58.020 --> 00:03:01.280 So this is either atomic hydrogen, it could be helium 00:03:01.280 --> 00:03:07.380 plus lithium 2 plus roentgenium 110 plus Its 00:03:07.380 --> 00:03:08.830 doesn't matter how many protons. 00:03:08.830 --> 00:03:11.390 there's always only want electron. 00:03:11.390 --> 00:03:13.500 And it's a planetary model. 00:03:13.500 --> 00:03:16.360 The positive charge concentrated in the nucleus, Z 00:03:16.360 --> 00:03:18.160 is the proton number. 00:03:18.160 --> 00:03:22.910 And at a distance r from the nucleus is a circular orbit in 00:03:22.910 --> 00:03:25.075 which resides 1 electron. 00:03:25.075 --> 00:03:27.780 It has a charge of minus E. 00:03:27.780 --> 00:03:30.180 And I just designated them q1 and q2. 00:03:30.180 --> 00:03:31.360 You could have done it the other way. 00:03:31.360 --> 00:03:33.060 But I had to choose something. 00:03:33.060 --> 00:03:34.590 So we went through and looked at the 00:03:34.590 --> 00:03:35.950 constitutive equations here. 00:03:35.950 --> 00:03:38.670 So first of all, the energy of the system-- 00:03:38.670 --> 00:03:40.960 and this is only going to be the energy the electron. 00:03:40.960 --> 00:03:45.480 Because we assume that the nucleus is far more massive. 00:03:45.480 --> 00:03:48.100 And so we don't have to get into things like reduced mass 00:03:48.100 --> 00:03:48.830 or anything like that. 00:03:48.830 --> 00:03:51.430 So just measure the energy of the electron. 00:03:51.430 --> 00:03:54.210 1/2 mv squared is the newtonian component. 00:03:54.210 --> 00:03:59.910 And then coulombic energy that's stored is z times e 00:03:59.910 --> 00:04:04.140 squared over 4 pi epsilon zero r, where epsilon zero is the 00:04:04.140 --> 00:04:05.430 permittivity of vacuum. 00:04:05.430 --> 00:04:09.160 And it's the factor, the 4 pi epsilon zero, is the factor 00:04:09.160 --> 00:04:12.240 that allows us to take electrostatic energies and put 00:04:12.240 --> 00:04:16.525 them on the same plane as mechanical energies. 00:04:16.525 --> 00:04:19.700 When we run through this, we always end up in joules. 00:04:19.700 --> 00:04:22.480 Then there's a force balanced to make sure that the electron 00:04:22.480 --> 00:04:25.890 neither falls into the nucleus nor flees and 00:04:25.890 --> 00:04:29.490 breaks free of the atom. 00:04:29.490 --> 00:04:32.560 And the force balance is if you put a ball at the end of a 00:04:32.560 --> 00:04:35.830 string, and you whip it around on a tether, you have a 00:04:35.830 --> 00:04:38.040 centrifugal force that's trying to get the ball to 00:04:38.040 --> 00:04:38.760 break away. 00:04:38.760 --> 00:04:41.260 And then the string is pulling in. 00:04:41.260 --> 00:04:45.270 So the pull in, in this case, is the coulombic force. 00:04:45.270 --> 00:04:49.340 And the force that makes the ball want to flee is this mv 00:04:49.340 --> 00:04:50.360 squared over r. 00:04:50.360 --> 00:04:52.510 And that must be net zero. 00:04:52.510 --> 00:04:55.380 Otherwise we're going to have a shift in orbit. 00:04:55.380 --> 00:04:57.530 And then lastly we have the quantum condition. 00:04:57.530 --> 00:05:02.270 And this was the breakthrough of Bohr where he enunciated 00:05:02.270 --> 00:05:06.910 that the quantum condition is going to give us this energy 00:05:06.910 --> 00:05:08.500 level quantization. 00:05:08.500 --> 00:05:12.010 And this was a big departure from what had been in the 00:05:12.010 --> 00:05:16.150 past. The only antecedent idea of this nature was the work by 00:05:16.150 --> 00:05:18.700 Planck who said that light is quantized. 00:05:18.700 --> 00:05:20.900 But as I told you last day, who knows what 00:05:20.900 --> 00:05:22.070 light really is. 00:05:22.070 --> 00:05:26.040 The Newtonian notion of a ball orbiting was 00:05:26.040 --> 00:05:27.780 very compelling here. 00:05:27.780 --> 00:05:32.690 And the notion that the movement of the electron could 00:05:32.690 --> 00:05:37.380 in some way be discontinuous was quite a major departure. 00:05:37.380 --> 00:05:38.790 So I left you last day with three 00:05:38.790 --> 00:05:40.420 equations and three unknowns. 00:05:40.420 --> 00:05:42.370 And what I'm not going to do right now 00:05:42.370 --> 00:05:44.080 is solve the equations. 00:05:44.080 --> 00:05:46.390 Because I've been lecturing long enough to know that's the 00:05:46.390 --> 00:05:49.530 way to kill interest, quench a lecture. 00:05:49.530 --> 00:05:52.570 And so if you really want to go through the algebra, be my 00:05:52.570 --> 00:05:54.250 guest. You're smart enough to do that. 00:05:54.250 --> 00:05:55.650 Instead I'm going to show you the results. 00:05:55.650 --> 00:05:57.790 But those are the three equations that you need. 00:05:57.790 --> 00:05:59.650 So we have an equation in r. 00:05:59.650 --> 00:06:01.960 We have an equation at v, and an equation in e. 00:06:01.960 --> 00:06:03.430 So let's go after them. 00:06:03.430 --> 00:06:07.020 If you first look at the solution for r. 00:06:07.020 --> 00:06:10.640 This is the radius of the orbit of the electron, the 00:06:10.640 --> 00:06:11.805 orbit of the electron. 00:06:11.805 --> 00:06:15.960 If you go through and solve, you'll end up with this, 00:06:15.960 --> 00:06:19.980 Epsilon zero times the square of the Planck constant divided 00:06:19.980 --> 00:06:22.720 by pi times m. 00:06:22.720 --> 00:06:23.920 It's always the electron. 00:06:23.920 --> 00:06:26.530 So this is the radius of the electron orbit. 00:06:26.530 --> 00:06:29.870 This is the mass of the electron times the square of 00:06:29.870 --> 00:06:31.840 the elementary charge-- 00:06:31.840 --> 00:06:33.730 that whole thing I'm going to group-- 00:06:33.730 --> 00:06:37.540 times the square of the quantum number divided by z, 00:06:37.540 --> 00:06:39.720 the proton number. 00:06:39.720 --> 00:06:41.050 So what we see here? 00:06:41.050 --> 00:06:45.640 Well, everything inside the parentheses is constant. 00:06:45.640 --> 00:06:46.440 These are all constant. 00:06:46.440 --> 00:06:48.750 Pi obviously is geometric constant. 00:06:48.750 --> 00:06:50.970 And the rest of these are constants you could look up in 00:06:50.970 --> 00:06:52.790 your table of constants. 00:06:52.790 --> 00:06:56.710 And we noticed that there is a set of solutions to this. 00:06:56.710 --> 00:07:01.260 The radius of the electron can occupy various discrete values 00:07:01.260 --> 00:07:03.100 defined by n. 00:07:03.100 --> 00:07:06.680 So we say that the radius takes on a plurality of values 00:07:06.680 --> 00:07:08.550 a function of n. 00:07:08.550 --> 00:07:12.170 And furthermore, the functionality goes as the 00:07:12.170 --> 00:07:12.910 square of n. 00:07:12.910 --> 00:07:16.330 It's n squared times a constant, where that constant 00:07:16.330 --> 00:07:18.770 is inside those parentheses. 00:07:18.770 --> 00:07:21.310 And we notice that because the r goes as n 00:07:21.310 --> 00:07:23.670 squared it's nonlinear. 00:07:23.670 --> 00:07:24.920 It's nonlinear. 00:07:28.560 --> 00:07:30.350 This is so important I'm going to write it 00:07:30.350 --> 00:07:31.280 down one more time. 00:07:31.280 --> 00:07:41.725 So the radius of the electron orbit takes multiple values. 00:07:41.725 --> 00:07:43.310 It takes multiple values. 00:07:46.490 --> 00:07:49.280 And they're discreet. 00:07:49.280 --> 00:07:51.540 The physicists like to use a different term. 00:07:51.540 --> 00:07:53.070 When something is discretized, the 00:07:53.070 --> 00:07:56.540 physicists say it is quantized. 00:07:56.540 --> 00:07:58.380 So these values are quantized. 00:07:58.380 --> 00:08:07.610 You cannot continuously vary the radius and nonlinear 00:08:07.610 --> 00:08:12.280 values, multiple values. 00:08:12.280 --> 00:08:15.900 So let's plug in. 00:08:15.900 --> 00:08:17.990 Because I want to get a sense of scale. 00:08:17.990 --> 00:08:21.280 So let's look at the simplest one. 00:08:21.280 --> 00:08:24.115 The most primitive 1-electron atom would be hydrogen. 00:08:24.115 --> 00:08:26.280 In which case, Z equals 1. 00:08:26.280 --> 00:08:29.870 So I've just got a proton orbited by an electron. 00:08:29.870 --> 00:08:35.170 So look at atomic hydrogen. 00:08:35.170 --> 00:08:38.950 So in that case, Z equals 1. 00:08:38.950 --> 00:08:42.790 And I'm going to look at n equals 1, which is the lowest 00:08:42.790 --> 00:08:43.710 number here, right? 00:08:43.710 --> 00:08:45.120 R scales as n squared. 00:08:45.120 --> 00:08:49.130 So the lowest value or r is obtained when n equals 1. 00:08:49.130 --> 00:08:52.230 And this is termed the ground state. 00:08:52.230 --> 00:08:55.560 The ground state. 00:08:55.560 --> 00:09:01.210 So I want to ask what is the radius of the electron orbit 00:09:01.210 --> 00:09:03.280 ground state in atomic hydrogen? 00:09:03.280 --> 00:09:07.265 And if I plug in these values, I'll call this r sub 1. 00:09:07.265 --> 00:09:16.590 It turns out to be 5.29 times 10 to the minus 11 meters, or 00:09:16.590 --> 00:09:22.390 0.529 angstroms. I love the angstrom. 00:09:22.390 --> 00:09:23.640 It's a great unit. 00:09:23.640 --> 00:09:24.580 It's a great unit. 00:09:24.580 --> 00:09:26.460 It's not an SI unit. 00:09:26.460 --> 00:09:28.450 But I like the angstrom. 00:09:28.450 --> 00:09:29.510 I'll show you why. 00:09:29.510 --> 00:09:34.410 If you try to express this in SI units, well there's 10 to 00:09:34.410 --> 00:09:36.990 the minus 11 meters. 00:09:36.990 --> 00:09:43.340 The Si units go in units of clusters of 1,000. 00:09:43.340 --> 00:09:45.200 So for example, you've got the meter. 00:09:45.200 --> 00:09:46.660 You've got the kilometer. 00:09:46.660 --> 00:09:48.230 You've got the micrometer. 00:09:48.230 --> 00:09:51.160 You've got 10 to the minus 9 meters, 00:09:51.160 --> 00:09:53.650 which is the nanometer. 00:09:56.250 --> 00:09:57.520 This is a Goldilocks problem. 00:09:57.520 --> 00:09:58.320 This one's too big. 00:09:58.320 --> 00:10:01.330 And then the next one down here is 10 to the minus 12 00:10:01.330 --> 00:10:03.350 meters, which is the picometer. 00:10:03.350 --> 00:10:07.960 So this is either 52.9 picometers, or a 0.0529 00:10:07.960 --> 00:10:09.970 nanometers. 00:10:09.970 --> 00:10:11.000 And that's no good. 00:10:11.000 --> 00:10:14.930 I want numbers like 3, 7, simple to remember. 00:10:14.930 --> 00:10:20.740 So 0.529, this is about 1/2 angstrom. 00:10:20.740 --> 00:10:21.910 It's good to know. 00:10:21.910 --> 00:10:24.280 But you try to publish, you know what happens in a 00:10:24.280 --> 00:10:25.910 scientific literature today? 00:10:25.910 --> 00:10:28.710 The Literary Lions that control the journals, they'll 00:10:28.710 --> 00:10:31.510 circle that and say you have to convert to SI units. 00:10:31.510 --> 00:10:32.930 And so they have to right some goofy 00:10:32.930 --> 00:10:34.680 nanometer thing or something. 00:10:34.680 --> 00:10:37.430 I know you think I'm crazy, but I love the angstrom. 00:10:37.430 --> 00:10:38.100 So there. 00:10:38.100 --> 00:10:39.890 Anyway, so here it is. 00:10:39.890 --> 00:10:43.770 Once you know that this is 0.529, this is on your table 00:10:43.770 --> 00:10:44.660 of constants. 00:10:44.660 --> 00:10:46.550 It's right on your table of constants. 00:10:46.550 --> 00:10:48.880 So you don't have to go and calculate all this stuff. 00:10:48.880 --> 00:10:52.110 Which means if you do your homework with your table of 00:10:52.110 --> 00:10:56.080 constants, you will know where those numbers lie, as opposed 00:10:56.080 --> 00:10:59.450 to opening this thing up for the first time on the first 00:10:59.450 --> 00:11:02.800 celebration of learning on October the 7th, and with 47 00:11:02.800 --> 00:11:04.910 entries and they're tiny, tiny font. 00:11:04.910 --> 00:11:08.060 And you're wondering where is that thing. 00:11:08.060 --> 00:11:09.140 Just a word to wise. 00:11:09.140 --> 00:11:10.860 So now we know what this is. 00:11:10.860 --> 00:11:14.650 We know this is 0.529 angstroms. So now I can write 00:11:14.650 --> 00:11:17.770 an equation for the radius of a 1-electron 00:11:17.770 --> 00:11:20.330 atom anywhere, anytime. 00:11:20.330 --> 00:11:26.600 r of n is going to be equal to a naught which 00:11:26.600 --> 00:11:29.620 is this value here. 00:11:29.620 --> 00:11:31.460 And it is termed the Bohr radius. 00:11:34.790 --> 00:11:37.050 So you can write it as a function of the Bohr radius, 00:11:37.050 --> 00:11:41.750 times the square of n divided by Z. 00:11:41.750 --> 00:11:45.230 So that's for all 1-electron atoms, gas phase. 00:11:45.230 --> 00:11:50.930 And you can see that as Z goes up, the r goes down, which 00:11:50.930 --> 00:11:51.520 makes sense. 00:11:51.520 --> 00:11:56.100 So suppose instead of hydrogen, we talk helium plus 00:11:56.100 --> 00:11:57.020 What's the only difference? 00:11:57.020 --> 00:11:59.580 Helium plus has 2 protons in the nucleus. 00:11:59.580 --> 00:12:03.090 Which means that the coulombic force of attraction between 00:12:03.090 --> 00:12:07.010 the same 1-electron and now a doubly charged nucleus is 00:12:07.010 --> 00:12:09.340 going to be stronger. 00:12:09.340 --> 00:12:11.320 So the first orbit is going to get pulled in. 00:12:11.320 --> 00:12:14.210 And all the other orbits are going to get pulled in. 00:12:14.210 --> 00:12:16.200 By how much are they going to get pulled in? 00:12:16.200 --> 00:12:17.100 By that much. 00:12:17.100 --> 00:12:19.900 So this is the functional representation 00:12:19.900 --> 00:12:21.730 of all of that physics. 00:12:21.730 --> 00:12:23.540 All right, there's three equations, three unknowns. 00:12:23.540 --> 00:12:24.930 Let's look at energy. 00:12:24.930 --> 00:12:27.960 So if you go through and solve for energy, you get this one. 00:12:27.960 --> 00:12:31.840 Minus this big monstrosity, mass of the electron to the 00:12:31.840 --> 00:12:37.390 fourth power of the elementary charge times 8 times the 00:12:37.390 --> 00:12:41.470 square of the permittivity of vacuum times the square of the 00:12:41.470 --> 00:12:47.740 Planck constant, all times the square of the Proton number 00:12:47.740 --> 00:12:52.630 divided by the square of the quantum number. 00:12:52.630 --> 00:12:56.080 And I just to make sure everybody is with me here. 00:12:56.080 --> 00:13:01.430 I always want to write n here. n equals 1, 2, 3, takes on 00:13:01.430 --> 00:13:02.710 integer values. 00:13:02.710 --> 00:13:04.090 And we'll say it again here. 00:13:04.090 --> 00:13:08.530 N equals 1, 2, 3, et cetera, et cetera. 00:13:08.530 --> 00:13:15.020 So again we say we see that e is a function of n. 00:13:15.020 --> 00:13:15.720 It's discretized. 00:13:15.720 --> 00:13:17.890 It's quantized. 00:13:17.890 --> 00:13:18.580 Why? 00:13:18.580 --> 00:13:22.330 Because once you impose the quantum condition here on 00:13:22.330 --> 00:13:24.470 angular momentum, it propagates 00:13:24.470 --> 00:13:25.880 through the entire model. 00:13:25.880 --> 00:13:28.470 So radius this quantized, energy is quantized, you're 00:13:28.470 --> 00:13:29.970 going to see velocity is quantized. 00:13:29.970 --> 00:13:33.740 Because the quantum condition is pervasive. 00:13:33.740 --> 00:13:38.110 So e to the n, and I'm going to take this whole quantity in 00:13:38.110 --> 00:13:42.810 parentheses and just call it giant K. 00:13:42.810 --> 00:13:44.600 These are all positive quantities. 00:13:44.600 --> 00:13:45.720 Mass is positive. 00:13:45.720 --> 00:13:49.010 And squares and fourth powers of numbers must be positive. 00:13:49.010 --> 00:13:56.650 So this is K times Z squared over n squared That's good. 00:13:56.650 --> 00:13:59.210 And we can go and evaluate K. 00:13:59.210 --> 00:14:05.990 And when we evaluate L in SI units, we get 2.18 times 10 to 00:14:05.990 --> 00:14:10.500 the minus 18 joules. 00:14:10.500 --> 00:14:13.450 This is joules per atom. 00:14:13.450 --> 00:14:16.750 Or you can multiply this by Avogadro's number. 00:14:16.750 --> 00:14:19.230 If you multiply it by Avogadro's number, then that 00:14:19.230 --> 00:14:28.270 will give you 1.312 megajoules per mol. 00:14:28.270 --> 00:14:32.700 So that's the energy of the electron in the ground state 00:14:32.700 --> 00:14:34.730 of atomic hydrogen. 00:14:34.730 --> 00:14:39.000 And then we can mediate that with Z and n, and go to 00:14:39.000 --> 00:14:41.870 electrons that are outside the ground state, above the ground 00:14:41.870 --> 00:14:45.160 state, or ground state electrons in atoms that have 00:14:45.160 --> 00:14:47.800 more than 1 proton, or both. 00:14:47.800 --> 00:14:50.630 And so let's take a look at the graphical 00:14:50.630 --> 00:14:51.790 representation of that. 00:14:51.790 --> 00:14:55.030 So instead of Cartesian coordinates, because this is 00:14:55.030 --> 00:14:57.450 spatial distribution, I'm going to go to energy 00:14:57.450 --> 00:15:00.080 coordinates and give you an energy level diagram. 00:15:00.080 --> 00:15:05.700 So again, not to scale. 00:15:05.700 --> 00:15:08.940 Because this thing goes is 1 over the square. 00:15:08.940 --> 00:15:10.300 So that's going to be messy. 00:15:10.300 --> 00:15:12.360 So let's start here. 00:15:12.360 --> 00:15:15.535 And on the left side I'm going to designate the energy. 00:15:15.535 --> 00:15:17.590 And on the right side I'm going to 00:15:17.590 --> 00:15:18.950 designate the quantum number. 00:15:18.950 --> 00:15:20.280 I'm going to start down here. 00:15:20.280 --> 00:15:21.470 That's the lowest energy. 00:15:21.470 --> 00:15:23.810 You see these are all negative values, first of all. 00:15:23.810 --> 00:15:25.670 They're all negative values. 00:15:25.670 --> 00:15:27.910 Because Z is a square, n is a square, and K 00:15:27.910 --> 00:15:29.750 is a positive quantity. 00:15:29.750 --> 00:15:31.600 So n equals 1 is the lowest state. 00:15:31.600 --> 00:15:32.770 It's the ground state. 00:15:32.770 --> 00:15:37.680 It has a value of minus K. 00:15:37.680 --> 00:15:40.660 So I'm going to do this one just for atomic hydrogen. 00:15:40.660 --> 00:15:44.440 So I'm going to write atomic H. 00:15:44.440 --> 00:15:47.900 So now Z equals 1. 00:15:47.900 --> 00:15:51.520 You can do it later for Z equals 2, 3, whatever. 00:15:51.520 --> 00:15:52.850 So this is atomic hydrogen. 00:15:52.850 --> 00:15:56.580 So ground state energy is minus K. 00:15:56.580 --> 00:15:59.700 What happens if we go to n equals 2? n equals 2 it 00:15:59.700 --> 00:16:02.460 becomes K divided by 2 squared 4. 00:16:02.460 --> 00:16:04.000 So it should be 3/4 of the way up. 00:16:04.000 --> 00:16:06.180 I'm not going to go quite 3/4 of the way. 00:16:06.180 --> 00:16:08.710 Because I want to leave room for some fine structure. 00:16:08.710 --> 00:16:10.380 That's why it's not to scale. 00:16:10.380 --> 00:16:13.200 All right so this is minus L over 4. 00:16:13.200 --> 00:16:15.060 What if we go to n equals 3? 00:16:15.060 --> 00:16:16.430 Well it's not symmetric here. 00:16:16.430 --> 00:16:17.150 It's nonlinear. 00:16:17.150 --> 00:16:18.940 But this should be really what? 00:16:18.940 --> 00:16:21.880 Minus K over 3 squared is 9. 00:16:21.880 --> 00:16:22.720 You get the picture. 00:16:22.720 --> 00:16:27.170 3, you can go 4, 5, and so on until n equals infinity. 00:16:27.170 --> 00:16:29.450 What happens when n equals infinity? 00:16:29.450 --> 00:16:33.210 I've got minus K over infinity, which is vanishingly 00:16:33.210 --> 00:16:34.840 small, zero. 00:16:34.840 --> 00:16:38.590 Where is the electron when n equals infinity? 00:16:38.590 --> 00:16:42.740 r is n squared times the Bohr radius. 00:16:42.740 --> 00:16:46.830 That's a great, great distance away. 00:16:46.830 --> 00:16:47.870 What does it mean? 00:16:47.870 --> 00:16:50.810 Physically it means that the electron is so far away that 00:16:50.810 --> 00:16:52.190 it is no longer bound. 00:16:52.190 --> 00:16:55.710 It's no longer part of the atom. 00:16:55.710 --> 00:16:58.910 And when it's no longer part of the atom, and the potential 00:16:58.910 --> 00:17:01.920 energy that's stored is a result of the charges coming 00:17:01.920 --> 00:17:03.630 together from infinity. 00:17:03.630 --> 00:17:05.800 I'm starting to talk like someone out of 802. 00:17:05.800 --> 00:17:09.480 What's the energy if I take 2 charged particles at infinite 00:17:09.480 --> 00:17:10.210 separation? 00:17:10.210 --> 00:17:12.510 I bring them into some finite separation. 00:17:12.510 --> 00:17:13.170 Voila. 00:17:13.170 --> 00:17:13.970 There it is. 00:17:13.970 --> 00:17:15.590 So when they're in infinite separation 00:17:15.590 --> 00:17:17.030 there's no energy stored. 00:17:17.030 --> 00:17:18.850 Hence, you are at that point. 00:17:18.850 --> 00:17:25.000 So n equals infinity means r equals infinity, which means E 00:17:25.000 --> 00:17:25.780 equal zero. 00:17:25.780 --> 00:17:27.590 There's no stored energy. 00:17:27.590 --> 00:17:30.570 So this means the electron is no longer bound. 00:17:33.940 --> 00:17:37.080 And therefore, if it's no longer bound, we 00:17:37.080 --> 00:17:37.930 have a term for it. 00:17:37.930 --> 00:17:39.440 It's called free. 00:17:39.440 --> 00:17:42.290 It's a free electron. 00:17:42.290 --> 00:17:44.480 And if the electron is free, then the 00:17:44.480 --> 00:17:48.565 atom is electron deficient. 00:17:48.565 --> 00:17:50.780 So if the electron is free, that means the 00:17:50.780 --> 00:17:54.810 atom is now an ion. 00:17:54.810 --> 00:17:56.225 Because it's lost an electron. 00:17:56.225 --> 00:17:57.990 It's no longer net neutral. 00:17:57.990 --> 00:18:03.800 Or we say an atom hasn't turned into an ion. 00:18:03.800 --> 00:18:06.200 Or the electron has been ionized. 00:18:12.900 --> 00:18:14.130 What's the energy for that? 00:18:14.130 --> 00:18:15.740 We can calculate what that energy is. 00:18:15.740 --> 00:18:17.760 It would be called the ionization energy. 00:18:17.760 --> 00:18:20.620 So if I started with an electron down in here, and I 00:18:20.620 --> 00:18:23.090 sent it all the way to infinity. 00:18:23.090 --> 00:18:24.490 See that's an energy space. 00:18:24.490 --> 00:18:27.410 Which is the equivalent in Cartesian space to go from 00:18:27.410 --> 00:18:30.510 here to infinity, same idea. 00:18:30.510 --> 00:18:32.040 Do you see the models? 00:18:32.040 --> 00:18:34.460 This is Cartesian. 00:18:34.460 --> 00:18:34.900 It's like a. 00:18:34.900 --> 00:18:37.450 Map This is energy coordinates. 00:18:37.450 --> 00:18:38.740 It's different. 00:18:38.740 --> 00:18:40.210 And you're going to be able to think from 00:18:40.210 --> 00:18:41.600 one model to another. 00:18:41.600 --> 00:18:43.050 What's the energy consequences? 00:18:43.050 --> 00:18:44.980 What are the Cartesian consequences? 00:18:44.980 --> 00:18:46.980 Until we get to the point where there is no Cartesian 00:18:46.980 --> 00:18:47.790 representation. 00:18:47.790 --> 00:18:50.560 Because the abstraction level is too high, we'll have to 00:18:50.560 --> 00:18:51.720 content ourselves with this. 00:18:51.720 --> 00:18:54.000 So get comfortable moving from there to there. 00:18:54.000 --> 00:18:56.550 And then some day we're going to say it's too complicated. 00:18:56.550 --> 00:18:58.240 There's no Cartesian thing. 00:18:58.240 --> 00:18:59.800 We'll be comfortable by then with this. 00:18:59.800 --> 00:19:02.580 OK so now we're taking an electron from here up to here. 00:19:02.580 --> 00:19:06.600 While we ask, what is the ionization energy? 00:19:06.600 --> 00:19:10.370 The ionization energy must equal the delta E of the 00:19:10.370 --> 00:19:11.620 transition. 00:19:14.440 --> 00:19:15.230 So what's that? 00:19:15.230 --> 00:19:18.560 The delta E of the transition is going to equal always E 00:19:18.560 --> 00:19:22.360 final minus E initial. 00:19:22.360 --> 00:19:25.290 E final minus E initial. 00:19:25.290 --> 00:19:29.530 Which is equal to E at infinity minus 00:19:29.530 --> 00:19:32.000 E1, the ground state. 00:19:32.000 --> 00:19:34.540 Well E infinity, we just said, is zero. 00:19:34.540 --> 00:19:38.630 And the ground state energy is equal to minus K. 00:19:38.630 --> 00:19:40.860 So minus minus K is K. 00:19:40.860 --> 00:19:44.120 So you also get the energy. 00:19:44.120 --> 00:19:47.090 From infinity down to ground state is 00:19:47.090 --> 00:19:48.960 the ionization energy. 00:19:48.960 --> 00:19:58.050 So we can define the ionization energy in terms of 00:19:58.050 --> 00:19:59.610 this transition. 00:19:59.610 --> 00:20:06.280 Define ionization energy as the minimum energy to remove 00:20:06.280 --> 00:20:23.430 an electron from the ground state of an 00:20:23.430 --> 00:20:24.733 atom in a gas phase. 00:20:31.020 --> 00:20:32.640 So that means there's no solids, no liquids. 00:20:32.640 --> 00:20:33.980 There's no work function here. 00:20:33.980 --> 00:20:36.410 There's no lattice energy, and so on. 00:20:36.410 --> 00:20:40.340 So there's a definition of the ionization energy. 00:20:40.340 --> 00:20:42.880 And we can be a little bit more elaborate. 00:20:42.880 --> 00:20:44.300 Even though right now I'm just going to do a 00:20:44.300 --> 00:20:45.380 little break here. 00:20:45.380 --> 00:20:46.620 I don't want to mislead people. 00:20:46.620 --> 00:20:49.620 But just an aside. 00:20:49.620 --> 00:20:51.110 I'm nonlinear. 00:20:51.110 --> 00:20:53.070 I can have multiple conversations at once. 00:20:53.070 --> 00:20:54.910 And you are capable of stacking. 00:20:54.910 --> 00:20:56.130 So we're going to break now. 00:20:56.130 --> 00:20:58.160 We're not going to talk about 1-electron atom. 00:20:58.160 --> 00:21:01.900 We're going to follow the thread of ionization energy. 00:21:01.900 --> 00:21:03.310 I'm going to take lithium. 00:21:03.310 --> 00:21:08.250 Lithium in its normal state has 3 protons, 3 electrons. 00:21:08.250 --> 00:21:10.140 So I'm going to take lithium gas. 00:21:10.140 --> 00:21:13.800 And I'm going to ionize it and make lithium plus. 00:21:13.800 --> 00:21:16.230 So this is a lithium plus ion in the gas phase. 00:21:16.230 --> 00:21:17.760 It's still got 2 electrons. 00:21:17.760 --> 00:21:19.330 So this isn't Bohr model stuff. 00:21:19.330 --> 00:21:21.880 But anyway, here we are. 00:21:21.880 --> 00:21:26.470 So the energy for this action would be called 00:21:26.470 --> 00:21:27.690 the ionization energy. 00:21:27.690 --> 00:21:31.437 Because I took a neutral atom, and I pulled an electron out 00:21:31.437 --> 00:21:32.910 of the ground state, and so on. 00:21:32.910 --> 00:21:34.140 So this is an ionization energy. 00:21:34.140 --> 00:21:36.790 But now I can continue this process. 00:21:36.790 --> 00:21:39.860 And I can take Lithium plus. 00:21:39.860 --> 00:21:42.900 And I can lose an electron from that, which will than 00:21:42.900 --> 00:21:45.970 give me lithium 2 plus. 00:21:45.970 --> 00:21:48.150 And this is called also an ionization energy. 00:21:48.150 --> 00:21:50.710 This is called the second ionization energy. 00:21:50.710 --> 00:21:52.320 So this is the first ionization energy. 00:21:52.320 --> 00:21:55.650 But just as when you write an equation, when the coefficient 00:21:55.650 --> 00:21:59.450 is 1, you don't write the 1. 00:21:59.450 --> 00:22:01.110 I don't write 1 lithium here. 00:22:01.110 --> 00:22:02.360 I know it's 1. 00:22:02.360 --> 00:22:04.460 This is the second ionization energy. 00:22:04.460 --> 00:22:06.700 And then I can keep stripping away electrons. 00:22:06.700 --> 00:22:11.010 And I can take lithium 2 plus in the gas phase, and take 00:22:11.010 --> 00:22:16.120 away that electron leaving me with just the lithium nucleus, 00:22:16.120 --> 00:22:19.060 lithium 3 plus, plus electron. 00:22:19.060 --> 00:22:23.260 And that's called the third ionization energy. 00:22:23.260 --> 00:22:25.340 OK, now what can we say? 00:22:25.340 --> 00:22:29.530 What's the relationship here between any of this except the 00:22:29.530 --> 00:22:31.860 definition and the Bohr model? 00:22:31.860 --> 00:22:32.250 well? 00:22:32.250 --> 00:22:35.810 The Bohr model applies only to 1-electron atoms. Are there 00:22:35.810 --> 00:22:40.270 any 1-electron atoms on this board? 00:22:40.270 --> 00:22:42.460 So we can calculate the energy, the 00:22:42.460 --> 00:22:43.820 third ionization energy. 00:22:43.820 --> 00:22:45.730 We can get that from the Bohr model. 00:22:48.330 --> 00:22:49.820 You can do it in your head. 00:22:49.820 --> 00:22:51.360 You can do it in your head right? 00:22:51.360 --> 00:22:55.520 It's just K times Z squared right here. 00:22:55.520 --> 00:22:56.630 It's going from 1. 00:22:56.630 --> 00:23:02.480 So if this is 2.18, it's going to be 9 times that trivially. 00:23:02.480 --> 00:23:08.040 OK, so this is just 3 squared times K. 00:23:08.040 --> 00:23:11.860 And this when you have to get from the literature. 00:23:11.860 --> 00:23:14.000 So you have to go to primary sources. 00:23:14.000 --> 00:23:15.060 Which is why you're going to learn how to 00:23:15.060 --> 00:23:17.530 use the proper database. 00:23:17.530 --> 00:23:20.070 And this one here also you get from the literature. 00:23:20.070 --> 00:23:23.020 But this is on your periodic table. 00:23:23.020 --> 00:23:26.040 Your periodic table, one of the data points it gives is 00:23:26.040 --> 00:23:28.960 the first ionization energy of all of the elements. 00:23:28.960 --> 00:23:31.790 And so even though lithium normally is a solid at room 00:23:31.790 --> 00:23:35.140 temperature, the ionization energy for lithium as given on 00:23:35.140 --> 00:23:36.945 your periodic table is for this reaction. 00:23:36.945 --> 00:23:40.100 It's for the gas. 00:23:40.100 --> 00:23:40.400 Anyway. 00:23:40.400 --> 00:23:42.660 So that's little aside. 00:23:42.660 --> 00:23:46.410 All right, the last quantity that we could get from the 00:23:46.410 --> 00:23:49.690 Bohr model is v, the velocity. 00:23:49.690 --> 00:23:50.730 And I solve for the velocity. 00:23:50.730 --> 00:23:52.240 We don't talk about this very much. 00:23:52.240 --> 00:23:56.950 But we're going to do so once today. 00:23:56.950 --> 00:23:58.210 And here it is. 00:23:58.210 --> 00:23:59.820 So I went through the algebra. 00:23:59.820 --> 00:24:05.780 And you get nh over 2 pi mr, where r is the radius. 00:24:05.780 --> 00:24:07.020 There's the quantum number. 00:24:07.020 --> 00:24:08.600 This is quantized as well. 00:24:08.600 --> 00:24:09.830 So I regrouped this. 00:24:09.830 --> 00:24:12.800 And I already have a nice, cool expression for r in terms 00:24:12.800 --> 00:24:14.070 of the Bohr radius. 00:24:14.070 --> 00:24:16.910 So I use that because that's on the table. 00:24:16.910 --> 00:24:20.690 So this is 2 pi times the mass of the electron times the Bohr 00:24:20.690 --> 00:24:28.240 radius, 1/2 angstrom, times Z proton number divided by n, 00:24:28.240 --> 00:24:32.810 where n equals 1, 2, 3, and so on. 00:24:32.810 --> 00:24:34.980 So let's again get a sense of scale. 00:24:34.980 --> 00:24:38.160 So let's try for sense of scale. 00:24:41.540 --> 00:24:45.950 Let's do velocity of the ground state electron in 00:24:45.950 --> 00:24:47.350 atomic hydrogen. 00:24:47.350 --> 00:24:51.080 So that means Z equals 1, n equals 1. 00:24:51.080 --> 00:24:52.760 So I plug in the numbers. 00:24:52.760 --> 00:24:59.800 And I get v1 for hydrogen, atomic hydrogen, gives me 2.18 00:24:59.800 --> 00:25:02.870 times 10 to the 6 meters for second. 00:25:06.870 --> 00:25:07.180 I don't know. 00:25:07.180 --> 00:25:07.950 Is that fast? 00:25:07.950 --> 00:25:08.880 Is that slow? 00:25:08.880 --> 00:25:09.260 I don't know. 00:25:09.260 --> 00:25:10.490 But I do know this much. 00:25:10.490 --> 00:25:14.890 I know that the speed of light is equal to 3 times 10 to the 00:25:14.890 --> 00:25:16.943 8 meters per second. 00:25:20.500 --> 00:25:24.080 So 10 to the 8 divided by 10 to the 6. 00:25:24.080 --> 00:25:27.280 2 and 3, that's roughly 1, speaking as an engineer. 00:25:27.280 --> 00:25:28.620 Who cares? 00:25:28.620 --> 00:25:32.540 So this is about 1% of the speed of light. 00:25:35.840 --> 00:25:36.790 That's pretty good. 00:25:36.790 --> 00:25:38.960 That gives me something I can hang on to. 00:25:38.960 --> 00:25:42.900 I would say that if this thing is zipping around at 1% of the 00:25:42.900 --> 00:25:50.090 speed of light, I would say that's relatively fast. One 00:25:50.090 --> 00:25:52.060 more time, 1% of the speed of light. 00:25:52.060 --> 00:25:58.030 That's relatively fast. 00:25:58.030 --> 00:26:01.560 Remember last day I told you that Bohr simply dismissed the 00:26:01.560 --> 00:26:06.850 concept of the use of classical electrodynamics down 00:26:06.850 --> 00:26:08.270 to atomic dimensions. 00:26:08.270 --> 00:26:11.840 Well here's another example of why a lot of these assumptions 00:26:11.840 --> 00:26:13.250 aren't going to work so well. 00:26:13.250 --> 00:26:17.330 We're talking about the ground state electron velocity in 00:26:17.330 --> 00:26:21.140 this putative planetary model of a 1-electron atom. 00:26:21.140 --> 00:26:23.010 You're already getting into relativistic effects. 00:26:23.010 --> 00:26:25.120 So, just another example. 00:26:25.120 --> 00:26:29.592 By the way, why do we use the letter c for speed of light? 00:26:29.592 --> 00:26:31.420 It comes from the latin word 00:26:31.420 --> 00:26:35.530 celeritas, which means swiftness. 00:26:35.530 --> 00:26:38.200 And we get the modern word acceleration, 00:26:38.200 --> 00:26:40.150 deceleration from that. 00:26:40.150 --> 00:26:41.150 OK. 00:26:41.150 --> 00:26:41.430 All right. 00:26:41.430 --> 00:26:45.430 So the Bohr model, we've now rolled it all out. 00:26:45.430 --> 00:26:46.890 We have the energy portrait. 00:26:46.890 --> 00:26:49.230 We have the radii, discrete. 00:26:49.230 --> 00:26:50.750 We have quantization. 00:26:50.750 --> 00:26:52.660 And we have velocities if we ever want to 00:26:52.660 --> 00:26:53.550 look at those again. 00:26:53.550 --> 00:26:56.120 Now what's the next thing we do in science? 00:26:56.120 --> 00:27:00.110 We compare the predictions of the Bohr model with data. 00:27:00.110 --> 00:27:01.640 Are there any data to support this? 00:27:01.640 --> 00:27:04.370 Because remember, all Rutherford said was plum 00:27:04.370 --> 00:27:05.690 pudding doesn't make sense. 00:27:05.690 --> 00:27:08.070 Instead I'm going to concentrate the positive mass 00:27:08.070 --> 00:27:08.950 in the center. 00:27:08.950 --> 00:27:12.030 And then Bohr came along and said, not only is it going to 00:27:12.030 --> 00:27:15.350 be a planetary model, I'm going to have circular orbits. 00:27:15.350 --> 00:27:17.600 So now we've gone a long way from Geiger-Marsden. 00:27:17.600 --> 00:27:19.700 So is there any data? 00:27:19.700 --> 00:27:23.620 Well there were data in 1853. 00:27:23.620 --> 00:27:26.500 Remember, Bohr published this in 1913. 00:27:26.500 --> 00:27:30.820 In 1853 there was a spectroscopist by the name-- 00:27:30.820 --> 00:27:32.570 I'm going to tell you his name-- 00:27:32.570 --> 00:27:36.360 in Uppsala, Sweden. 00:27:36.360 --> 00:27:38.570 And his name was Angstrom. 00:27:42.720 --> 00:27:45.710 Angstrom was doing experiments on hydrogen in 00:27:45.710 --> 00:27:48.140 gas discharge tubes. 00:27:48.140 --> 00:27:58.680 So he measured emissions from gas discharge tube. 00:28:01.200 --> 00:28:04.810 And it was filled with various gases 00:28:04.810 --> 00:28:08.890 including atomic hydrogen. 00:28:08.890 --> 00:28:12.550 And in order to take his data, what he used 00:28:12.550 --> 00:28:15.050 was this device here-- 00:28:15.050 --> 00:28:17.330 there's the Bohr radius, just an example. 00:28:17.330 --> 00:28:19.740 He used the prism spectrograph. 00:28:19.740 --> 00:28:21.230 So here's a gas discharge tube. 00:28:21.230 --> 00:28:23.300 And I'm going to show you the physics of that in a second. 00:28:23.300 --> 00:28:24.890 Basically you've got a pair of electrodes. 00:28:24.890 --> 00:28:26.135 You've got gas in the tube. 00:28:26.135 --> 00:28:30.080 And as this cartoon shows, you apply a potential across the 00:28:30.080 --> 00:28:30.750 electrodes. 00:28:30.750 --> 00:28:33.170 And beyond a certain threshold potential, the 00:28:33.170 --> 00:28:34.860 tube begins to glow. 00:28:34.860 --> 00:28:36.530 And the glow goes in all directions. 00:28:36.530 --> 00:28:38.110 And a blinds you when you're in the lab. 00:28:38.110 --> 00:28:40.410 So what you do, is you cover this up a bit. 00:28:40.410 --> 00:28:41.760 You have a narrow slit. 00:28:41.760 --> 00:28:44.260 And then you force the light to come through in a thin 00:28:44.260 --> 00:28:46.880 ribbon, and then expose it to a prism. 00:28:46.880 --> 00:28:50.850 What the prism does, is it takes the light and breaks it 00:28:50.850 --> 00:28:54.490 into its components, sort of rainbow-like, and magnifies 00:28:54.490 --> 00:28:56.630 the difference. 00:28:56.630 --> 00:28:59.560 As refraction goes, different wavelengths will refract 00:28:59.560 --> 00:29:00.550 different amounts. 00:29:00.550 --> 00:29:02.420 And then you shoot this across the room. 00:29:05.280 --> 00:29:06.430 There's two counters. 00:29:06.430 --> 00:29:09.340 One is a scintillation screen and an army of graduate 00:29:09.340 --> 00:29:10.980 students who sit there in the dark. 00:29:10.980 --> 00:29:13.690 But they're no good because you can't stick them into the 00:29:13.690 --> 00:29:14.570 publication. 00:29:14.570 --> 00:29:17.120 You need to have data that people will rely upon. 00:29:17.120 --> 00:29:19.460 So instead you use a photographic plate. 00:29:19.460 --> 00:29:23.740 And if you put even a tiny, tiny angle of separation 00:29:23.740 --> 00:29:26.380 across a great enough distance, you start to get 00:29:26.380 --> 00:29:28.090 enough line splitting that you can see. 00:29:28.090 --> 00:29:29.030 And then you go backwards. 00:29:29.030 --> 00:29:30.230 And you know the geometry here. 00:29:30.230 --> 00:29:32.870 And you can figure out what the wavelength must have been 00:29:32.870 --> 00:29:34.710 to go this distance, et cetera, et cetera. 00:29:34.710 --> 00:29:38.070 And these are all color coded, not because they had color 00:29:38.070 --> 00:29:41.400 film in those days, but just to let you know that 656 00:29:41.400 --> 00:29:45.150 nanometers, if you were the graduate student sitting 00:29:45.150 --> 00:29:47.470 there, you'd see a red line, a green line, a blue line, and a 00:29:47.470 --> 00:29:49.330 violent line. 00:29:49.330 --> 00:29:50.580 So that's how he made the measurements. 00:29:54.070 --> 00:29:56.140 And he published those measurements. 00:29:56.140 --> 00:30:00.510 And so they lay. 00:30:00.510 --> 00:30:03.620 And then the story gets a little thicker. 00:30:03.620 --> 00:30:09.310 In 1885, there's a Swiss high school math teacher. 00:30:09.310 --> 00:30:10.720 I mean, I can't make this stuff. 00:30:10.720 --> 00:30:12.030 This is true story. 00:30:12.030 --> 00:30:15.040 There's a Swiss high school math teacher by the name of J. 00:30:15.040 --> 00:30:15.310 J. 00:30:15.310 --> 00:30:16.290 Balmer. 00:30:16.290 --> 00:30:16.720 We had J. 00:30:16.720 --> 00:30:16.940 J. 00:30:16.940 --> 00:30:17.440 Thomson. 00:30:17.440 --> 00:30:18.070 Now we've got J. 00:30:18.070 --> 00:30:18.280 J. 00:30:18.280 --> 00:30:19.300 Balmer. 00:30:19.300 --> 00:30:19.630 And J. 00:30:19.630 --> 00:30:19.770 J. 00:30:19.770 --> 00:30:22.620 Balmer, he loved to play with numbers. 00:30:22.620 --> 00:30:25.640 And he was studying this set of lines. 00:30:25.640 --> 00:30:27.730 And he was trying to come up with a pattern. 00:30:27.730 --> 00:30:28.840 Can you see a pattern there? 00:30:28.840 --> 00:30:33.160 410 434 486, 656, do they go squares? 00:30:33.160 --> 00:30:35.180 Are they primes? 00:30:35.180 --> 00:30:36.380 What's the pattern there? 00:30:36.380 --> 00:30:39.440 So Balmer puzzled over this for awhile. 00:30:39.440 --> 00:30:42.480 And he finally came up with the equation to represent 00:30:42.480 --> 00:30:43.650 those lines. 00:30:43.650 --> 00:30:49.640 So he studied Angstrom's data found the pattern. 00:30:53.150 --> 00:30:54.870 And here's the pattern that he found. 00:30:54.870 --> 00:30:58.450 He said that those are wavelengths. 00:30:58.450 --> 00:31:04.720 If I take instead wave number, nu bar is called wave number, 00:31:04.720 --> 00:31:09.660 which is the reciprocal of the wavelength. 00:31:09.660 --> 00:31:13.430 So if I take the reciprocal of the wavelength, I end up with 00:31:13.430 --> 00:31:17.980 those 4 lines conforming to a series that goes like this. 00:31:17.980 --> 00:31:22.780 1 over 2 squared minus 1 over n squared, where n 00:31:22.780 --> 00:31:27.260 equals 3, 4, 5, 6. 00:31:27.260 --> 00:31:30.510 And there's a constant here which we're going 00:31:30.510 --> 00:31:32.330 to designate r. 00:31:32.330 --> 00:31:34.790 And the value of R-- 00:31:34.790 --> 00:31:37.480 I'm going to put that on the next board-- 00:31:37.480 --> 00:31:43.040 the value of R as expressed in SI units today, would be 1.1 00:31:43.040 --> 00:31:46.740 times 10 to the 7 reciprocal meters. 00:31:46.740 --> 00:31:48.390 Wavelength is a meter. 00:31:48.390 --> 00:31:50.370 Reciprocal wavelength or wave number must 00:31:50.370 --> 00:31:52.120 be reciprocal meters. 00:31:52.120 --> 00:31:56.160 So how this all of this support the Bohr model? 00:31:56.160 --> 00:31:58.450 Well in order to explain it, I've got a first tell you what 00:31:58.450 --> 00:32:01.200 the physics of the gas discharge tube are. 00:32:01.200 --> 00:32:06.520 So let's go inside the gas discharge tube and understand 00:32:06.520 --> 00:32:07.200 those physics. 00:32:07.200 --> 00:32:09.410 So here's the gas discharge tube. 00:32:09.410 --> 00:32:12.830 It's made are borosilicate glass. 00:32:12.830 --> 00:32:15.390 And we fill it with gas. 00:32:15.390 --> 00:32:19.530 And in this case, the gas is going to contain among other 00:32:19.530 --> 00:32:21.660 things hydrogen. 00:32:21.660 --> 00:32:25.120 So this is a hydrogen gas phase atom. 00:32:25.120 --> 00:32:27.340 And this is probably at low pressure. 00:32:27.340 --> 00:32:29.330 And then I said I need electrode. 00:32:29.330 --> 00:32:31.200 I'll get the electrodes inside. 00:32:31.200 --> 00:32:33.320 So I've got to have a really good glass blower who can make 00:32:33.320 --> 00:32:35.330 a glass to metal seal, and have a 00:32:35.330 --> 00:32:37.670 feedthrough to an electrode. 00:32:37.670 --> 00:32:39.800 So this is still a vacuum seal. 00:32:39.800 --> 00:32:41.420 How do they get the gas in the first place? 00:32:41.420 --> 00:32:42.550 We don't show you this in the books. 00:32:42.550 --> 00:32:44.430 I'll tell you because I did this in my Ph.D. 00:32:44.430 --> 00:32:47.430 What you do is you have a little side tube here. 00:32:47.430 --> 00:32:49.380 You evacuate. 00:32:49.380 --> 00:32:52.300 This goes to a vacuum pump. 00:32:52.300 --> 00:32:54.190 And then over here you have a gas source. 00:32:54.190 --> 00:32:55.440 You evacuate. 00:32:55.440 --> 00:32:57.520 Put in the gas to whatever pressure you want. 00:32:57.520 --> 00:33:00.230 And then the glass blower disconnects. 00:33:00.230 --> 00:33:02.170 This pulls this down to a reduced 00:33:02.170 --> 00:33:05.200 pressure, and seals it. 00:33:05.200 --> 00:33:06.880 But the books don't show you that. 00:33:06.880 --> 00:33:09.650 That's a secret. 00:33:09.650 --> 00:33:11.310 So we've got a gas at low pressure. 00:33:11.310 --> 00:33:13.320 And I've got an electrode over here and an 00:33:13.320 --> 00:33:14.270 electrode over here. 00:33:14.270 --> 00:33:18.055 And they're connected to a variable voltage power supply. 00:33:18.055 --> 00:33:20.200 So I'm going to put an arrow with the V meaning it's 00:33:20.200 --> 00:33:21.070 variable voltage. 00:33:21.070 --> 00:33:22.650 I can change the voltage. 00:33:22.650 --> 00:33:24.730 And this convention, this is the negative side. 00:33:24.730 --> 00:33:26.680 So this means the electrons leave the power 00:33:26.680 --> 00:33:27.950 supply and go like this. 00:33:27.950 --> 00:33:30.300 Which means this electrode will be negative. 00:33:30.300 --> 00:33:32.530 And this electrode will be positive. 00:33:32.530 --> 00:33:35.390 And thanks to Michael Faraday, we will call 00:33:35.390 --> 00:33:37.270 this one the cathode. 00:33:37.270 --> 00:33:39.100 And this one we will call the anode. 00:33:41.880 --> 00:33:43.080 Now what happens? 00:33:43.080 --> 00:33:45.660 We start turning up the pressure, turning up the 00:33:45.660 --> 00:33:46.650 voltage rather. 00:33:46.650 --> 00:33:47.820 Low pressure here, but 00:33:47.820 --> 00:33:49.890 electrical pressure is voltage. 00:33:49.890 --> 00:33:53.760 The voltage gets high enough, eventually the electrons will 00:33:53.760 --> 00:33:56.170 boil off the cathode. 00:33:56.170 --> 00:33:58.030 And this is a gas at low pressure. 00:33:58.030 --> 00:34:00.690 And they will accelerate from rest and go all the way across 00:34:00.690 --> 00:34:03.120 the tube and crash into the anode. 00:34:03.120 --> 00:34:05.130 And we complete the circuit. 00:34:05.130 --> 00:34:07.770 But if there's a gas in here, some of these electrons are 00:34:07.770 --> 00:34:10.000 going to hit the gas molecules. 00:34:10.000 --> 00:34:12.330 And when they hit the gas molecules, if they have enough 00:34:12.330 --> 00:34:17.760 energy to do so, they will cause electrons inside the gas 00:34:17.760 --> 00:34:19.730 molecules to be excited. 00:34:23.170 --> 00:34:26.740 And if they're excited enough, the electrons will jump up to 00:34:26.740 --> 00:34:28.100 a higher energy level. 00:34:28.100 --> 00:34:30.180 But they can't be sustained. 00:34:30.180 --> 00:34:32.410 Because this is a ballistic collision. 00:34:32.410 --> 00:34:33.220 It's a one of. 00:34:33.220 --> 00:34:34.520 It's like a bowling alley. 00:34:34.520 --> 00:34:37.910 One ball, one pin, one impact. 00:34:37.910 --> 00:34:39.030 Now the pin is in the air. 00:34:39.030 --> 00:34:40.490 What happens to the pin? 00:34:40.490 --> 00:34:41.540 It falls back down. 00:34:41.540 --> 00:34:41.870 Why? 00:34:41.870 --> 00:34:43.690 Because gravity pulls it down. 00:34:43.690 --> 00:34:47.030 In this case, you've got the energetics pulling the 00:34:47.030 --> 00:34:48.170 electron back down. 00:34:48.170 --> 00:34:50.760 Now the electron goes from high energy to low energy. 00:34:50.760 --> 00:34:54.380 And when that happens, the energy difference is given off 00:34:54.380 --> 00:34:57.200 in the form of a photon. 00:34:57.200 --> 00:35:01.690 So I get photon emission when this falls back down. 00:35:01.690 --> 00:35:05.730 And this photon has a wavelength. 00:35:05.730 --> 00:35:09.820 What I'm going to show you is that the set of lines that you 00:35:09.820 --> 00:35:14.860 get from exactly this configuration using this 00:35:14.860 --> 00:35:19.100 equation give you the Balmer series. 00:35:19.100 --> 00:35:23.710 So now you've got a model that Bohr postulated for atomic 00:35:23.710 --> 00:35:28.460 hydrogen on 1-electron atom that exactly predicts that set 00:35:28.460 --> 00:35:29.680 of 4 lines. 00:35:29.680 --> 00:35:33.670 Which were measured 50 years before. 00:35:33.670 --> 00:35:35.160 So let's go. 00:35:35.160 --> 00:35:38.520 So first of all let's get the energy here. 00:35:38.520 --> 00:35:40.220 And I'm going to get the energy of this electron. 00:35:40.220 --> 00:35:44.110 This electron I'm going to call a ballistic electron. 00:35:44.110 --> 00:35:46.080 Why do I call it a ballistic electron? 00:35:46.080 --> 00:35:47.580 Because it's not bound. 00:35:47.580 --> 00:35:48.240 It's free. 00:35:48.240 --> 00:35:51.790 It boils off the cathode, flies through free space, and 00:35:51.790 --> 00:35:52.850 crashes into an anode. 00:35:52.850 --> 00:35:55.050 Clearly it's not part of an atom. 00:35:55.050 --> 00:35:57.270 But there's a second electron in this story. 00:35:57.270 --> 00:35:59.950 And it's the ground state electron in hydrogen. 00:35:59.950 --> 00:36:01.200 And it lives here. 00:36:04.080 --> 00:36:06.070 So what's the energy of the ballistic electron? 00:36:10.490 --> 00:36:14.750 Well that's just 1/2 mv squared And where did it get 00:36:14.750 --> 00:36:15.640 its energy from? 00:36:15.640 --> 00:36:18.070 It got its energy from the power supply. 00:36:18.070 --> 00:36:20.330 And what's the electrostatic energy? 00:36:20.330 --> 00:36:24.200 It's the product of the charge on the species times the 00:36:24.200 --> 00:36:27.270 voltage through which it was accelerated. 00:36:27.270 --> 00:36:29.710 So away we go. 00:36:29.710 --> 00:36:32.190 I know the charge on the electron is minus E. 00:36:32.190 --> 00:36:34.300 Whatever the voltage is there, 1 volt, 10 00:36:34.300 --> 00:36:35.960 volts, 100 volts, whatever. 00:36:35.960 --> 00:36:37.560 Away we go. 00:36:37.560 --> 00:36:40.630 By the way, I'm going to show you just one other thing in 00:36:40.630 --> 00:36:42.580 terms of order of magnitude. 00:36:42.580 --> 00:36:46.290 The kinds of voltages you see along here are 1 volt, 10 00:36:46.290 --> 00:36:47.650 volts, that sort of thing. 00:36:47.650 --> 00:36:57.160 So suppose, to get an order of magnitude, suppose we had a 00:36:57.160 --> 00:37:02.070 species of charge E. 00:37:02.070 --> 00:37:05.255 So in other words, it's only 1 times the elementary charge. 00:37:05.255 --> 00:37:14.370 A species of charge E influenced by 00:37:14.370 --> 00:37:18.350 voltage of 1 volt. 00:37:18.350 --> 00:37:23.040 So this is 1 in the voltage units, and 1 in the elementary 00:37:23.040 --> 00:37:24.000 charge units. 00:37:24.000 --> 00:37:25.310 How much energy would that be? 00:37:25.310 --> 00:37:28.260 That's equivalent to making 1 volt and accelerate an 00:37:28.260 --> 00:37:33.110 electron from rest across this gap. 00:37:33.110 --> 00:37:37.400 And the result would be, the energy then would simply equal 00:37:37.400 --> 00:37:46.260 1.6 times 10 to the minus 19 coulombs times 1 volt. 00:37:46.260 --> 00:37:47.720 And what's the energy going to be? 00:37:47.720 --> 00:37:49.330 Well I've got coulombs times volts. 00:37:49.330 --> 00:37:51.550 And I don't know how I convert one to the other. 00:37:51.550 --> 00:37:52.200 I don't have to. 00:37:52.200 --> 00:37:53.170 Why not? 00:37:53.170 --> 00:37:54.570 Because that's an SI unit. 00:37:54.570 --> 00:37:56.040 And that's an SI unit. 00:37:56.040 --> 00:37:57.110 This is an energy. 00:37:57.110 --> 00:37:59.576 So with impunity, I write 1.6 times 10 to 00:37:59.576 --> 00:38:01.400 the minus 19 joules. 00:38:01.400 --> 00:38:04.820 That's the beauty of SI units. 00:38:04.820 --> 00:38:05.820 So that's a good news. 00:38:05.820 --> 00:38:06.560 I know it's joules. 00:38:06.560 --> 00:38:08.640 The bad news is I hate this number. 00:38:08.640 --> 00:38:10.850 It's a stupid number, 1.6 times 10 to the minus 19. 00:38:10.850 --> 00:38:11.450 it's crazy. 00:38:11.450 --> 00:38:14.920 Why don't I come up with a number like 3, 7? 00:38:14.920 --> 00:38:17.440 So what I could do, is I could define. 00:38:17.440 --> 00:38:21.070 I could define a unit such that when the elementary 00:38:21.070 --> 00:38:26.300 charge is accelerated across the unit voltage, I would call 00:38:26.300 --> 00:38:30.470 that unit 1 electron volt. 00:38:30.470 --> 00:38:32.650 And so somebody thought of this before me. 00:38:32.650 --> 00:38:35.190 And hence, this is the unit of the electron volt. 00:38:35.190 --> 00:38:38.790 It takes these crazy things that we've been spewing here 00:38:38.790 --> 00:38:42.510 up until now, and rationalizes them into numbers that people 00:38:42.510 --> 00:38:44.060 can carry around in their heads. 00:38:44.060 --> 00:38:45.380 So what's K now? 00:38:45.380 --> 00:38:48.990 K is 2.18 times 10 to the minus 18 joules. 00:38:48.990 --> 00:38:49.960 Yuck! 00:38:49.960 --> 00:38:51.700 Let's convert that to electron volts. 00:38:51.700 --> 00:38:55.100 So I divide by 1.6 times 10 to the minus 19. 00:38:55.100 --> 00:38:58.600 And I got 13.6 electron volts. 00:38:58.600 --> 00:39:02.090 You'll remember that on your death bed, ionization energy 00:39:02.090 --> 00:39:04.440 of atomic hydrogen. 00:39:04.440 --> 00:39:06.980 Maybe we don't have to give out tables of constants. 00:39:06.980 --> 00:39:08.300 You just know this stuff. 00:39:08.300 --> 00:39:09.510 It's OK. 00:39:09.510 --> 00:39:10.430 All right. 00:39:10.430 --> 00:39:14.380 Now one last thing about this. 00:39:14.380 --> 00:39:16.380 So this has got gas in it. 00:39:16.380 --> 00:39:17.720 This is the cathode. 00:39:17.720 --> 00:39:23.270 And this beam of electrons, back in the 1800s, there was a 00:39:23.270 --> 00:39:25.750 popular term, it was called the ray. 00:39:25.750 --> 00:39:28.410 So instead of a beam of light, people refer 00:39:28.410 --> 00:39:30.460 to the ray of light. 00:39:30.460 --> 00:39:33.700 So then when they got to particle beams, they talk to 00:39:33.700 --> 00:39:35.210 them as rays. 00:39:35.210 --> 00:39:39.890 So this is now not an electron beam, it's an electron ray. 00:39:39.890 --> 00:39:41.550 And it comes off the cathode. 00:39:41.550 --> 00:39:43.500 And it's in a vacuum tube. 00:39:43.500 --> 00:39:49.715 So this could be called a cathode ray tube, a CRT. 00:39:49.715 --> 00:39:51.650 Now see, I could flatten this. 00:39:51.650 --> 00:39:53.450 And I could spray it with phosphors. 00:39:53.450 --> 00:39:56.880 And then I could put some charge plates here. 00:39:56.880 --> 00:39:58.670 The electrons have a negative charge. 00:39:58.670 --> 00:40:01.440 So if I charge these plates, and I was clever about how I 00:40:01.440 --> 00:40:04.490 charge them and varied the charge, I could raster the 00:40:04.490 --> 00:40:07.920 electron beam like this about 30 times a second all up and 00:40:07.920 --> 00:40:08.850 down the screen. 00:40:08.850 --> 00:40:11.370 And then I could put some program signal in there. 00:40:11.370 --> 00:40:12.430 And I could sit here. 00:40:12.430 --> 00:40:13.680 And I could watch TV. 00:40:17.630 --> 00:40:21.830 It all started with the gas discharge tube. 00:40:21.830 --> 00:40:25.930 It says nothing about the content unfortunately. 00:40:25.930 --> 00:40:28.660 Very nice physics, but no content. 00:40:28.660 --> 00:40:29.030 All right. 00:40:29.030 --> 00:40:32.940 So now we've got the electron. 00:40:32.940 --> 00:40:35.060 The electron is moving, the ballistic electron. 00:40:35.060 --> 00:40:36.690 Now I want to look at what happens when the ballistic 00:40:36.690 --> 00:40:39.640 electron smashes into one of those hydrogen atoms. So let's 00:40:39.640 --> 00:40:41.340 go back over here. 00:40:41.340 --> 00:40:44.330 So here's the incident particle. 00:40:44.330 --> 00:40:47.250 And it's going to be an electron in this case. 00:40:47.250 --> 00:40:48.900 So I'm going to designate this electron. 00:40:48.900 --> 00:40:52.860 This is my ballistic electron. 00:40:52.860 --> 00:40:57.810 So here's the incident electron. 00:40:57.810 --> 00:41:00.500 And this is ballistic, just to be clear. 00:41:00.500 --> 00:41:02.800 It's the ballistic incident electron. 00:41:02.800 --> 00:41:04.750 Now this is a mixed metaphor here. 00:41:04.750 --> 00:41:08.390 Because I'm representing this in Cartesian space. 00:41:08.390 --> 00:41:10.790 But I've moved into energy space. 00:41:10.790 --> 00:41:12.360 So some people are going to get really upset. 00:41:12.360 --> 00:41:13.430 Because they're going to say, well this is Cartesian, but 00:41:13.430 --> 00:41:14.480 this isn't. 00:41:14.480 --> 00:41:15.615 It doesn't matter. 00:41:15.615 --> 00:41:17.060 It's my lecture. 00:41:17.060 --> 00:41:17.800 It's my model. 00:41:17.800 --> 00:41:19.050 It works. 00:41:21.180 --> 00:41:22.860 I'm the professor. 00:41:22.860 --> 00:41:24.700 So we've got this mixed metaphor here. 00:41:24.700 --> 00:41:26.300 But anyways, it helps a lot. 00:41:26.300 --> 00:41:28.565 So what happens when this thing comes in? 00:41:28.565 --> 00:41:32.610 It depends on how much energy it has. 00:41:32.610 --> 00:41:38.680 Now if the incident energy, if E incident is 00:41:38.680 --> 00:41:40.400 tiny, nothing happens. 00:41:40.400 --> 00:41:43.470 This thing just zooms right on through. 00:41:43.470 --> 00:41:47.860 But if the incident energy, if E of the incident ballistic 00:41:47.860 --> 00:41:52.830 electron is greater than delta E for any 00:41:52.830 --> 00:41:54.900 transition that's feasible-- 00:41:54.900 --> 00:41:57.470 and in this case I'm going to assume I don't have thermal 00:41:57.470 --> 00:41:58.700 distribution of electrons. 00:41:58.700 --> 00:42:00.890 If I gave you Avogadro's number of hydrogen atoms 00:42:00.890 --> 00:42:02.810 because of the thermal distribution of energies-- and 00:42:02.810 --> 00:42:04.310 we'll come back to this later-- 00:42:04.310 --> 00:42:07.740 there might actually be, at any moment, some electrons 00:42:07.740 --> 00:42:10.110 that are thermally excited above the ground state. 00:42:10.110 --> 00:42:11.710 We're going to forget about that for now. 00:42:11.710 --> 00:42:15.590 We're going to spiral up the learning curve here. 00:42:15.590 --> 00:42:17.920 So first time we're going to assume all the electrons are 00:42:17.920 --> 00:42:19.270 in the ground state. 00:42:19.270 --> 00:42:24.520 If I don't enough energy to go from n equals 1 to n equals 2, 00:42:24.520 --> 00:42:25.330 nothing happens. 00:42:25.330 --> 00:42:27.760 If I have more than enough energy to go for n equals 1 to 00:42:27.760 --> 00:42:30.200 n equals 2, I will take that energy. 00:42:30.200 --> 00:42:33.530 And the electron will jump, steal that amount of energy, 00:42:33.530 --> 00:42:36.010 and then this thing moves on-- 00:42:36.010 --> 00:42:38.700 and I'm purposely making this vector shorter than the 00:42:38.700 --> 00:42:39.800 incident vector-- 00:42:39.800 --> 00:42:41.510 with that amount of energy raw. 00:42:41.510 --> 00:42:43.820 And this is called the scattered electron. 00:42:43.820 --> 00:42:45.590 Go back to the bowling ball analogy. 00:42:45.590 --> 00:42:48.440 The bowling ball comes in with a certain energy, hits the 00:42:48.440 --> 00:42:50.190 pin, continues to roll. 00:42:50.190 --> 00:42:53.420 But you know that there's a loss of kinetic energy in the 00:42:53.420 --> 00:42:54.100 bowling ball. 00:42:54.100 --> 00:42:55.040 That's what we're seeing here. 00:42:55.040 --> 00:42:57.920 It's purely ballistic. 00:42:57.920 --> 00:43:00.680 Now let's say we do have more than enough energy. 00:43:00.680 --> 00:43:04.620 Suppose I have enough energy to go from n equals 1 halfway 00:43:04.620 --> 00:43:07.560 between n equals 2 and n equals 3. 00:43:07.560 --> 00:43:10.340 There's only n equals 1, n equals 2. 00:43:10.340 --> 00:43:14.850 I can't take the electron up to n equals 2.3. 00:43:14.850 --> 00:43:15.630 It's unallowed. 00:43:15.630 --> 00:43:17.460 These are the only allowed states. 00:43:17.460 --> 00:43:21.390 So that differential amount of energy then resides with the 00:43:21.390 --> 00:43:23.380 electron that's ballistic. 00:43:23.380 --> 00:43:24.610 And it moves on here. 00:43:24.610 --> 00:43:26.690 So we've got conservation of energy here. 00:43:26.690 --> 00:43:32.370 We can say that E incident will then equal the energy 00:43:32.370 --> 00:43:36.570 that's lost in the transition plus the energy that's still 00:43:36.570 --> 00:43:40.070 left with the scattered electron. 00:43:40.070 --> 00:43:43.880 And we can calculate what that transitional energy is. 00:43:43.880 --> 00:43:46.530 That transitional energy is going to equal 00:43:46.530 --> 00:43:51.840 1/2 mv squared incident. 00:43:51.840 --> 00:43:55.160 This is the velocity, the incident electron. 00:43:55.160 --> 00:43:58.670 What's the energy to go from n equals 1 to n equals n? 00:43:58.670 --> 00:44:03.990 Whatever it is, its minus K times Z squared If it's 00:44:03.990 --> 00:44:05.470 hydrogen, Z is 1. 00:44:05.470 --> 00:44:08.280 It's going to be 1 over nf. 00:44:08.280 --> 00:44:11.780 The final quantum number squared minus 1 over the 00:44:11.780 --> 00:44:14.690 square of the initial quantum number. 00:44:14.690 --> 00:44:17.860 And then, what's left over after this has been robbed 00:44:17.860 --> 00:44:22.260 from the incident ballistic energy is 1/2 mv squared of 00:44:22.260 --> 00:44:25.890 the scattered ballistic electron. 00:44:25.890 --> 00:44:39.490 And the quantization dictates that only if E incident is 00:44:39.490 --> 00:44:44.720 greater than delta E going 1 up to n-- here I'm assuming 00:44:44.720 --> 00:44:45.950 everything is in its ground state. 00:44:45.950 --> 00:44:47.630 Later on we're going to be more sophisticated. 00:44:47.630 --> 00:44:49.980 But for first time through, all are 00:44:49.980 --> 00:44:50.830 ground state electrons. 00:44:50.830 --> 00:44:54.040 I have to have enough energy to go at least to n equals 2. 00:44:54.040 --> 00:44:55.760 I can go to n equals 3, m equals 4. 00:44:55.760 --> 00:44:59.490 In principle, if this thing had more than 13.6 electron 00:44:59.490 --> 00:45:01.320 volts, what would happen? 00:45:01.320 --> 00:45:03.060 It would kick this electron out. 00:45:03.060 --> 00:45:04.350 Gone! 00:45:04.350 --> 00:45:08.110 And you'd have 2 free electrons. 00:45:08.110 --> 00:45:13.620 So if it's greater than this, then the consequence is 00:45:13.620 --> 00:45:14.870 electron promotion. 00:45:19.850 --> 00:45:22.490 So we're moving along. 00:45:22.490 --> 00:45:24.890 But this excited state is unstable. 00:45:24.890 --> 00:45:28.600 This excited state is unstable because it's like the bowling 00:45:28.600 --> 00:45:29.450 pin that got thrown up. 00:45:29.450 --> 00:45:30.070 So what happens? 00:45:30.070 --> 00:45:32.460 The electron standing up there on n equals 2, for example, 00:45:32.460 --> 00:45:34.310 looking down to n equals 1. 00:45:34.310 --> 00:45:35.080 And it falls. 00:45:35.080 --> 00:45:38.290 When it falls it gives off radiation. 00:45:38.290 --> 00:45:41.170 And that radiation is conservation of energy there. 00:45:41.170 --> 00:45:45.640 And what we know is when it gives off an energy of the 00:45:45.640 --> 00:45:51.240 emitted photon, the energy of the emitted photon must equal 00:45:51.240 --> 00:45:55.895 delta E of the transition falling from 2 to 1. 00:45:55.895 --> 00:45:57.630 And we know how to calculate that. 00:45:57.630 --> 00:45:59.890 That's just that thing flipped around. 00:45:59.890 --> 00:46:02.290 And this thing is equal to what? 00:46:02.290 --> 00:46:04.570 This is equal to h nu. 00:46:04.570 --> 00:46:08.700 According to Planck it's hc over lambda is 00:46:08.700 --> 00:46:11.620 equal to hc nu bar. 00:46:11.620 --> 00:46:12.720 You know what this one is. 00:46:12.720 --> 00:46:18.070 This is minus KZ squared 1 over-- in this case 00:46:18.070 --> 00:46:19.200 it's going to be-- 00:46:19.200 --> 00:46:23.700 1 over nf squared 1 over 1 squared minus 1 over, in this 00:46:23.700 --> 00:46:29.350 case, 1 over 2 squared And you can generalize this 1 over nf. 00:46:29.350 --> 00:46:30.980 So where am I going with this? 00:46:30.980 --> 00:46:33.275 Well I'm going to flip all of this around. 00:46:33.275 --> 00:46:36.780 And when I flip it all around, what I'm going to end up with 00:46:36.780 --> 00:46:38.900 is this equation here. 00:46:38.900 --> 00:46:40.660 And better than that, I'm going to end 00:46:40.660 --> 00:46:41.980 up with this equation. 00:46:41.980 --> 00:46:45.260 And I'm going to end up with this as the constant. 00:46:45.260 --> 00:46:49.080 And when I get that, we're going to say Bohr has done it. 00:46:49.080 --> 00:46:53.070 The data support the theory. 00:46:53.070 --> 00:46:55.920 So that's what we're going to do. 00:46:55.920 --> 00:47:01.020 But I think we're going to stop at this point today. 00:47:01.020 --> 00:47:03.900 So let me just jump here. 00:47:03.900 --> 00:47:06.660 I mentioned to you that if you go here on your periodic 00:47:06.660 --> 00:47:09.120 table, there's the 13.6 electron volts. 00:47:09.120 --> 00:47:15.310 In the case of lithium, this is 5.4 electron volts. 00:47:15.310 --> 00:47:17.330 So you can see all the various values. 00:47:17.330 --> 00:47:19.720 This by the way, people no noise. 00:47:19.720 --> 00:47:20.210 No noise. 00:47:20.210 --> 00:47:22.100 It's 11:52. 00:47:22.100 --> 00:47:24.190 I'm holding court until 11:55. 00:47:24.190 --> 00:47:26.360 I'm simply changing topics. 00:47:26.360 --> 00:47:28.060 It's not, oh this is the part where I 00:47:28.060 --> 00:47:29.290 can talk to my neighbor. 00:47:29.290 --> 00:47:30.820 What are the rules? 00:47:30.820 --> 00:47:31.880 No talking. 00:47:31.880 --> 00:47:32.880 No food. 00:47:32.880 --> 00:47:37.160 No horseplay until 11:55. 00:47:37.160 --> 00:47:39.580 Then, still no horseplay, gentle 00:47:39.580 --> 00:47:42.910 talking, no food, no drink. 00:47:42.910 --> 00:47:46.290 This is a sacred space. 00:47:46.290 --> 00:47:47.390 I'm not kidding you. 00:47:47.390 --> 00:47:48.490 Do you know why? 00:47:48.490 --> 00:47:53.400 In this secular America, this is sacred space because this 00:47:53.400 --> 00:47:55.750 is where people learn. 00:47:55.750 --> 00:47:59.530 The lecture hall is sacred space. 00:47:59.530 --> 00:48:03.020 Now, here's the 13.6 electron volts. 00:48:03.020 --> 00:48:05.430 And there's the 1.6 times 10 to the minus 19 joules. 00:48:05.430 --> 00:48:09.770 And if you multiply those 2, you'll get the 2.18 over here. 00:48:09.770 --> 00:48:10.970 All right. 00:48:10.970 --> 00:48:13.760 Here's a cartoon showing the photon, higher energy orbit, 00:48:13.760 --> 00:48:17.020 lower energy orbit, electron emission transition, right out 00:48:17.020 --> 00:48:17.900 of your book. 00:48:17.900 --> 00:48:20.130 And there's a postulate 6. 00:48:20.130 --> 00:48:23.340 And we're going to finish this up at the beginning of the 00:48:23.340 --> 00:48:24.410 Friday lecture. 00:48:24.410 --> 00:48:32.830 So here's the whole series from n equals 3 to 2, n equals 00:48:32.830 --> 00:48:34.920 4 to 2, and so on. 00:48:34.920 --> 00:48:37.280 See how that works. 00:48:37.280 --> 00:48:38.530 OK. 00:48:41.310 --> 00:48:44.210 One of the things that we've learned here, is that it 00:48:44.210 --> 00:48:47.730 doesn't matter what the incident energy is here. 00:48:47.730 --> 00:48:51.360 The emission is characteristic of the energy 00:48:51.360 --> 00:48:52.920 levels inside the gas. 00:48:55.680 --> 00:48:57.730 Instead of using an incident electron, I could use an 00:48:57.730 --> 00:48:58.710 incident proton. 00:48:58.710 --> 00:49:01.260 I could use an incident alpha particle. 00:49:01.260 --> 00:49:03.130 I could use an incident neutron. 00:49:03.130 --> 00:49:06.020 Anything that has enough energy to kick this up from n 00:49:06.020 --> 00:49:09.250 equals 1 to n equals 2 will result in photon emission of 00:49:09.250 --> 00:49:10.320 this frequency. 00:49:10.320 --> 00:49:15.830 That means the set of those lines is unique to the target. 00:49:15.830 --> 00:49:18.720 And this is the beginning of chemical analysis. 00:49:18.720 --> 00:49:23.050 I can use this to characterize species, to it, stars. 00:49:23.050 --> 00:49:25.960 How do we analyze the composition a stars? 00:49:25.960 --> 00:49:29.550 Well, do we send a NASA spaceship out 25 light-years 00:49:29.550 --> 00:49:32.390 and grab some gas and bring it back to the lab? 00:49:32.390 --> 00:49:32.920 No. 00:49:32.920 --> 00:49:36.480 All we've got is the spectrograph. 00:49:36.480 --> 00:49:37.620 The star is hot. 00:49:37.620 --> 00:49:43.120 That means thermal excitation and cascading 00:49:43.120 --> 00:49:44.760 down with photon emission. 00:49:44.760 --> 00:49:48.760 And the lines we get are related to the energy levels 00:49:48.760 --> 00:49:49.470 within the stars. 00:49:49.470 --> 00:49:52.220 So from a distance of a 100 light-years, I can tell you 00:49:52.220 --> 00:49:53.680 what the composition is. 00:49:53.680 --> 00:49:56.330 And if there's two gases there, what if there's 00:49:56.330 --> 00:49:58.060 hydrogen and helium? 00:49:58.060 --> 00:49:59.650 the helium lines will be there. 00:49:59.650 --> 00:50:02.410 And they'll be superimposed on the hydrogen lines unless they 00:50:02.410 --> 00:50:04.380 lie directly on top of one another. 00:50:04.380 --> 00:50:06.990 I'm going to be able to figure out what's there. 00:50:06.990 --> 00:50:08.920 That's how it works. 00:50:08.920 --> 00:50:14.190 Now here's a story about an astronomer, Cecilia Payne. 00:50:14.190 --> 00:50:15.830 She's the first woman graduate student in 00:50:15.830 --> 00:50:17.116 astronomy at Harvard. 00:50:17.116 --> 00:50:20.900 She went on to chair the Faculty of Arts and Sciences, 00:50:20.900 --> 00:50:24.480 awarded tenure, but denied a professorship for 18 years 00:50:24.480 --> 00:50:26.150 because she was a woman. 00:50:26.150 --> 00:50:28.690 And in her thesis, she was the first person to figure out to 00:50:28.690 --> 00:50:32.830 the sun is dominantly hydrogen, not iron, which is 00:50:32.830 --> 00:50:34.270 what most astronomers thought. 00:50:34.270 --> 00:50:34.520 Why? 00:50:34.520 --> 00:50:37.110 Well, because the earth is made of iron. 00:50:37.110 --> 00:50:39.180 Meteorites are made of iron. 00:50:39.180 --> 00:50:40.590 The whole universe must be made of iron. 00:50:40.590 --> 00:50:42.300 Never mind the fact that the sun is glowing. 00:50:45.310 --> 00:50:46.950 So here's how spectroscopy works. 00:50:46.950 --> 00:50:47.700 Look at this. 00:50:47.700 --> 00:50:48.830 See, what does this mean? 00:50:48.830 --> 00:50:50.580 This is an analogy. 00:50:50.580 --> 00:50:51.490 What could that mean? 00:50:51.490 --> 00:50:52.700 Well, they said iron again. 00:50:52.700 --> 00:50:53.770 I know this is misspelled. 00:50:53.770 --> 00:50:56.610 But maybe it's just a glitch in the instrumentation. 00:50:56.610 --> 00:50:58.600 So most people would look at that and say, 00:50:58.600 --> 00:50:59.900 the message is iron. 00:50:59.900 --> 00:51:01.560 But it's not about the word. 00:51:01.560 --> 00:51:03.180 It's about the pattern. 00:51:03.180 --> 00:51:05.350 It's about the pattern. 00:51:05.350 --> 00:51:06.820 And that's what's spectroscopy is. 00:51:06.820 --> 00:51:11.700 By the way, if you somehow didn't catch this lecture. 00:51:11.700 --> 00:51:14.340 And you walked in, and all you saw was those four lines. 00:51:14.340 --> 00:51:16.040 You know those four lines. 00:51:16.040 --> 00:51:18.390 That set of lines is characteristic of atomic 00:51:18.390 --> 00:51:20.220 hydrogen, and nothing else. 00:51:20.220 --> 00:51:21.700 By the way, those lines are very faint. 00:51:21.700 --> 00:51:23.220 This is not to scale. 00:51:23.220 --> 00:51:25.390 And they're so faint, they're ghostlike. 00:51:25.390 --> 00:51:27.790 And what is the latin word for ghost? 00:51:27.790 --> 00:51:28.820 Specter. 00:51:28.820 --> 00:51:30.260 So what is a spectrum? 00:51:30.260 --> 00:51:32.940 It is a set of ghostlike lines. 00:51:32.940 --> 00:51:35.730 To this day, the term spectroscopy refers to the 00:51:35.730 --> 00:51:40.540 ability to study data that are so faint they're ghostlike. 00:51:40.540 --> 00:51:42.710 All right, we'll see you on Friday.