1 00:00:00,000 --> 00:00:00,272 The following content is provided under a Creative 2 00:00:00,272 --> 00:00:00,374 Commons license. 3 00:00:00,374 --> 00:00:00,646 Your support will help MIT OpenCourseWare continue to 4 00:00:00,646 --> 00:00:00,918 offer high quality educational resources for free. 5 00:00:00,918 --> 00:00:01,224 To make a donation or view additional materials from 6 00:00:01,224 --> 00:00:01,496 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:01,496 --> 00:00:01,550 ocw.mit.edu. 8 00:00:01,550 --> 00:00:21,930 PROFESSOR: All right. 9 00:00:21,930 --> 00:00:27,070 As everyone finishes getting settled in, why don't you take 10 00:00:27,070 --> 00:00:34,710 10 more seconds on the clicker question here, and let's see 11 00:00:34,710 --> 00:00:37,370 how you did on that this, this is very similar to the clicker 12 00:00:37,370 --> 00:00:41,380 question that we had on Friday. 13 00:00:41,380 --> 00:00:44,250 OK, so let's get started here. 14 00:00:44,250 --> 00:00:47,040 It looks like we are doing a lot better. 15 00:00:47,040 --> 00:00:51,200 We now have 77% getting the correct answer, we only had 16 00:00:51,200 --> 00:00:53,560 about 30-something percent on Friday for a 17 00:00:53,560 --> 00:00:55,630 very similar question. 18 00:00:55,630 --> 00:00:59,740 So if you're not in this 77%, let's quickly go over why, in 19 00:00:59,740 --> 00:01:02,710 fact, this is the correct answer, 0 . 20 00:01:02,710 --> 00:01:05,390 9 times 10 to the negative 18 joules. 21 00:01:05,390 --> 00:01:09,150 So I'm using the same kind of tricky language that we'd used 22 00:01:09,150 --> 00:01:11,560 before, not to trick you, but so that you're not tricked in 23 00:01:11,560 --> 00:01:12,460 the future. 24 00:01:12,460 --> 00:01:15,590 So if we're talking about the fourth excited state, and we 25 00:01:15,590 --> 00:01:18,740 talk instead about principle quantum numbers, what 26 00:01:18,740 --> 00:01:21,340 principle quantum number corresponds to the fourth 27 00:01:21,340 --> 00:01:22,900 excited state of a hydrogen atom. 28 00:01:22,900 --> 00:01:24,130 STUDENT: Five. 29 00:01:24,130 --> 00:01:24,920 PROFESSOR: Five. 30 00:01:24,920 --> 00:01:25,060 OK. 31 00:01:25,060 --> 00:01:27,290 So, hopefully that cleared up for some of you why you got 32 00:01:27,290 --> 00:01:28,740 the wrong answer. 33 00:01:28,740 --> 00:01:32,700 So we know that we're in the n equals 5 state, so we can find 34 00:01:32,700 --> 00:01:34,710 what the binding energy is here. 35 00:01:34,710 --> 00:01:37,990 The ionization energy, of course, is just the negative 36 00:01:37,990 --> 00:01:39,460 of the binding energy. 37 00:01:39,460 --> 00:01:42,040 We know that binding energy is always negative, we know that 38 00:01:42,040 --> 00:01:44,450 ionization energy is always positive. 39 00:01:44,450 --> 00:01:47,370 So hopefully, putting all those things together, if you 40 00:01:47,370 --> 00:01:50,750 looked at this question again we'd get 100% on it, that our 41 00:01:50,750 --> 00:01:53,390 only option here is 0 . 42 00:01:53,390 --> 00:01:55,720 9, and that it's not the negative, it's the positive 43 00:01:55,720 --> 00:01:58,030 version, because we're talking about how much energy we have 44 00:01:58,030 --> 00:02:03,420 to put into the system in order to eject an electron. 45 00:02:03,420 --> 00:02:03,900 All right. 46 00:02:03,900 --> 00:02:06,900 And today we're going to mostly be talking about wave 47 00:02:06,900 --> 00:02:11,170 functions of electrons, but before we get to that, I 48 00:02:11,170 --> 00:02:13,960 wanted to review one last thing that's back on to 49 00:02:13,960 --> 00:02:16,870 Friday's topic, which was when we were solving the 50 00:02:16,870 --> 00:02:19,900 Schrodinger equation, or in fact, using the solution to 51 00:02:19,900 --> 00:02:23,640 the Schrodinger equation for the energy, the binding energy 52 00:02:23,640 --> 00:02:26,000 between an electron and a nucleus. 53 00:02:26,000 --> 00:02:28,950 And when we talked about that, what we found was that we 54 00:02:28,950 --> 00:02:33,360 could actually validate our predicted binding energies by 55 00:02:33,360 --> 00:02:36,140 looking at the emission spectra of the hydrogen atom, 56 00:02:36,140 --> 00:02:39,620 which is what we did as the demo, or we could think about 57 00:02:39,620 --> 00:02:41,610 the absorption spectra as well. 58 00:02:41,610 --> 00:02:44,360 And what we predict as an energy difference between two 59 00:02:44,360 --> 00:02:47,450 levels, we know should correspond to the energy of 60 00:02:47,450 --> 00:02:50,510 light that's either emitted, if we're giving off a photon, 61 00:02:50,510 --> 00:02:53,440 or that's absorbed if we're going to take on a photon and 62 00:02:53,440 --> 00:02:56,040 jump from a lower to a higher energy level. 63 00:02:56,040 --> 00:02:58,760 So we came up with two formulas, which are similar to 64 00:02:58,760 --> 00:03:00,700 the two that I'm showing here. 65 00:03:00,700 --> 00:03:03,940 The formula tells us the frequency of the light that's 66 00:03:03,940 --> 00:03:07,900 emitted or absorbed based on the energy difference between 67 00:03:07,900 --> 00:03:09,810 the two levels that we're going between, that the 68 00:03:09,810 --> 00:03:12,090 electron is transitioning between. 69 00:03:12,090 --> 00:03:14,230 You'll notice that there's a little bit of a difference in 70 00:03:14,230 --> 00:03:16,840 these equations here from the ones from the other day, which 71 00:03:16,840 --> 00:03:19,550 is that you have this z squared value in there. 72 00:03:19,550 --> 00:03:22,580 So these are both called Rydberg formulas for figuring 73 00:03:22,580 --> 00:03:25,330 out the frequency of light emitted or absorbed, and 74 00:03:25,330 --> 00:03:28,180 before we were looking at the Rydberg formula specifically 75 00:03:28,180 --> 00:03:31,740 for the hydrogen atom, and now that we have this z squared 76 00:03:31,740 --> 00:03:35,120 term in the formula here, we're now talking about 77 00:03:35,120 --> 00:03:37,350 absolutely any one electron atom. 78 00:03:37,350 --> 00:03:40,010 And it should make sense where we got this from, because we 79 00:03:40,010 --> 00:03:44,380 know that the binding energy, if we're talking about a 80 00:03:44,380 --> 00:03:52,620 hydrogen atom, what is the binding energy equal to? 81 00:03:52,620 --> 00:03:57,750 Negative Rydberg over what? 82 00:03:57,750 --> 00:03:58,040 Yes. 83 00:03:58,040 --> 00:04:00,890 So, it's negative Rydberg constant over n squared. 84 00:04:00,890 --> 00:04:03,720 But if we're talking more generally about any one 85 00:04:03,720 --> 00:04:10,630 electron atom, now we have a more general equation for the 86 00:04:10,630 --> 00:04:15,300 binding energy, which has this z squared term out in front of 87 00:04:15,300 --> 00:04:18,680 it, right, so it's negative z squared times the Rydberg 88 00:04:18,680 --> 00:04:22,080 constant all over n squared. 89 00:04:22,080 --> 00:04:24,910 So, essentially when we're talking about these equations 90 00:04:24,910 --> 00:04:27,600 up here, all we're doing is talking about the regular 91 00:04:27,600 --> 00:04:30,580 Rydberg formulas, but instead we could go back and re-derive 92 00:04:30,580 --> 00:04:33,370 the equation for any one electron atom, which would 93 00:04:33,370 --> 00:04:36,190 just mean that we put that z squared term in the front. 94 00:04:36,190 --> 00:04:39,110 So when you solve certain types of problems, such as 95 00:04:39,110 --> 00:04:42,580 problems later on in the second half of your p-set, if 96 00:04:42,580 --> 00:04:45,640 you need to talk about the frequency of light emitted or 97 00:04:45,640 --> 00:04:49,200 absorbed for a one electron atom, such as lithium plus 2, 98 00:04:49,200 --> 00:04:51,930 for example, then you would need to plug in z, and 99 00:04:51,930 --> 00:04:55,690 remember the z value for lithium would just be 3. 100 00:04:55,690 --> 00:04:59,190 The z value for hydrogen, of course, is 1, and that's why 101 00:04:59,190 --> 00:05:01,680 this term falls out of that equation when we're talking 102 00:05:01,680 --> 00:05:04,670 specifically about the hydrogen atom. 103 00:05:04,670 --> 00:05:07,160 So, just to finish our review of what we talked about on 104 00:05:07,160 --> 00:05:10,720 Friday, when we're thinking about transitions between two 105 00:05:10,720 --> 00:05:13,700 different states, and we're talking about a situation 106 00:05:13,700 --> 00:05:17,510 where the final state, the n final, is greater than n 107 00:05:17,510 --> 00:05:20,710 initial, in this case, are we talking about absorption or 108 00:05:20,710 --> 00:05:24,690 are we talking about emission? 109 00:05:24,690 --> 00:05:27,200 Hearing a little bit of a mix here. 110 00:05:27,200 --> 00:05:31,190 In fact, we're talking about absorption when n final is 111 00:05:31,190 --> 00:05:32,290 greater than n initial. 112 00:05:32,290 --> 00:05:35,870 We start at this lower energy state and go up -- that means 113 00:05:35,870 --> 00:05:38,870 we need to absorb a photon, we have to take in energy. 114 00:05:38,870 --> 00:05:42,910 Specifically, we have to take in this exact amount of energy 115 00:05:42,910 --> 00:05:46,260 in order to bump the electron up to the higher energy level. 116 00:05:46,260 --> 00:05:49,380 So that means that when instead we start high and go 117 00:05:49,380 --> 00:05:53,610 low, we're dealing with emission where we have excess 118 00:05:53,610 --> 00:05:56,570 energy that the electron's giving off, and that energy is 119 00:05:56,570 --> 00:06:00,050 going to be equal the energy of the photon that is released 120 00:06:00,050 --> 00:06:03,560 and, of course, through our equations we know how to get 121 00:06:03,560 --> 00:06:07,450 from energy to frequency or to wavelength of the photon that 122 00:06:07,450 --> 00:06:09,970 we're talking about. 123 00:06:09,970 --> 00:06:10,470 All right. 124 00:06:10,470 --> 00:06:14,040 So that's all I'm going to say today in terms of solving the 125 00:06:14,040 --> 00:06:17,250 energy part of the Schrodinger equation, so what we're really 126 00:06:17,250 --> 00:06:19,690 going to focus on is the other part of the Schrodinger 127 00:06:19,690 --> 00:06:23,220 equation today, which is solving for psi. 128 00:06:23,220 --> 00:06:26,260 So we're going to for psi, and before that, we're going to 129 00:06:26,260 --> 00:06:28,800 figure out that instead of just that one quantum number 130 00:06:28,800 --> 00:06:30,920 n, we're going to have a few other quantum numbers that 131 00:06:30,920 --> 00:06:33,560 fall out of solving the Schrodinger equation 132 00:06:33,560 --> 00:06:35,150 for what psi is. 133 00:06:35,150 --> 00:06:38,480 We're also going to talk more about what psi actually means. 134 00:06:38,480 --> 00:06:41,010 When we first introduced the Schrodinger equation, what I 135 00:06:41,010 --> 00:06:44,590 told you was think of psi as being some representation of 136 00:06:44,590 --> 00:06:46,030 what an electron is. 137 00:06:46,030 --> 00:06:49,170 We'll get more specific here, more specific even than just 138 00:06:49,170 --> 00:06:50,630 saying you can think of it as an orbital. 139 00:06:50,630 --> 00:06:52,930 We'll really think about what psi means. 140 00:06:52,930 --> 00:06:55,950 And in doing that, we'll also talk about the shapes of h 141 00:06:55,950 --> 00:06:58,590 atom wave functions, specifically the shapes of 142 00:06:58,590 --> 00:07:01,600 orbitals, and then something called radial probability 143 00:07:01,600 --> 00:07:05,060 distribution, which will make sense when we get to it. 144 00:07:05,060 --> 00:07:08,670 But, as I said before that, we have some more quantum numbers 145 00:07:08,670 --> 00:07:12,360 to take care of, because it turns out that when you solve 146 00:07:12,360 --> 00:07:15,790 the Schrodinger equation for psi, these other quantum 147 00:07:15,790 --> 00:07:17,390 numbers have to be the defined. 148 00:07:17,390 --> 00:07:20,190 When we talked about binding energy, we just had one 149 00:07:20,190 --> 00:07:22,840 quantum number that came out of it. 150 00:07:22,840 --> 00:07:25,590 And that quantum number was n, which was our principle 151 00:07:25,590 --> 00:07:29,820 quantum number, and we know that n could be equal to any 152 00:07:29,820 --> 00:07:35,780 integer value, so, 1, 2, 3, all the way up to infinity. 153 00:07:35,780 --> 00:07:39,830 And this quantization that comes out of having n is what 154 00:07:39,830 --> 00:07:42,230 gives us the quantization of different energy levels. 155 00:07:42,230 --> 00:07:45,000 That's why we can't have a continuum of energy, we 156 00:07:45,000 --> 00:07:50,190 actually have those quantized points. 157 00:07:50,190 --> 00:07:53,250 So, it turns out that n is not the only quantum number needed 158 00:07:53,250 --> 00:07:55,460 to describe a wave function, however. 159 00:07:55,460 --> 00:07:58,490 There's two more that you can see come out of it. 160 00:07:58,490 --> 00:08:02,780 And the first is l, and l is our angular momentum quantum 161 00:08:02,780 --> 00:08:06,660 number, and it's called that because it actually dictates 162 00:08:06,660 --> 00:08:10,460 the angular momentum that our electron has in our atom. 163 00:08:10,460 --> 00:08:13,220 And when we talk about l it is a quantum number, so because 164 00:08:13,220 --> 00:08:15,900 it's a quantum number, we know that it can only have discreet 165 00:08:15,900 --> 00:08:19,320 values, it can't just be any value we want, it's very 166 00:08:19,320 --> 00:08:20,620 specific values. 167 00:08:20,620 --> 00:08:24,440 And unlike n, l can start all the way down at 0, and it 168 00:08:24,440 --> 00:08:29,400 increases by integer value, so we go 1, 2, 3, 169 00:08:29,400 --> 00:08:30,980 and all the way up. 170 00:08:30,980 --> 00:08:34,530 But also unlike n, l cannot have just any value, we can't 171 00:08:34,530 --> 00:08:36,420 go into infinity. 172 00:08:36,420 --> 00:08:39,680 L is limited such that the highest value of 173 00:08:39,680 --> 00:08:41,870 l is n minus 1. 174 00:08:41,870 --> 00:08:44,260 We can't get any higher than that. 175 00:08:44,260 --> 00:08:46,665 So, it would be a good question to ask why are we 176 00:08:46,665 --> 00:08:49,010 limited -- clearly there's this relationship between l 177 00:08:49,010 --> 00:08:51,950 and n, and we can't get any higher than n equals one. 178 00:08:51,950 --> 00:08:54,940 We can actually think about why that is, and the reason is 179 00:08:54,940 --> 00:08:57,400 because l is our angular momentum. 180 00:08:57,400 --> 00:09:00,140 It describes the angular momentum of the electron. 181 00:09:00,140 --> 00:09:02,580 So another way to think about that is just the rotational 182 00:09:02,580 --> 00:09:05,400 kinetic energy of our electron. 183 00:09:05,400 --> 00:09:08,960 And we know that n describes the total energy, that total 184 00:09:08,960 --> 00:09:11,750 binding energy of the electron, so the total energy 185 00:09:11,750 --> 00:09:14,170 is going to be equal to potential energy 186 00:09:14,170 --> 00:09:15,820 plus kinetic energy. 187 00:09:15,820 --> 00:09:18,920 So if we say that l is just talking about our kinetic 188 00:09:18,920 --> 00:09:21,990 energy part, our rotational kinetic energy, and we know 189 00:09:21,990 --> 00:09:24,940 that electrons have potential energy, then it makes sense 190 00:09:24,940 --> 00:09:27,820 that l, in fact, can never go higher than n. 191 00:09:27,820 --> 00:09:30,500 And, in fact, it can't even reach n, because then we would 192 00:09:30,500 --> 00:09:33,660 have no potential energy at all in our electron, which is 193 00:09:33,660 --> 00:09:35,130 not correct. 194 00:09:35,130 --> 00:09:37,860 So, that's the second quantum number. 195 00:09:37,860 --> 00:09:44,950 And the third one is called m, it's also m sub l. 196 00:09:44,950 --> 00:09:48,200 This is what we call the magnetic quantum number, and 197 00:09:48,200 --> 00:09:52,900 we won't deal with the fact of its being the magnetic quantum 198 00:09:52,900 --> 00:09:55,980 number here -- that kind of tells us the shape of the 199 00:09:55,980 --> 00:09:59,210 orbital or the way that the electron will behave in a 200 00:09:59,210 --> 00:10:02,360 magnetic field, but what's more relevant to thinking 201 00:10:02,360 --> 00:10:05,500 about the limits of this number is that it's also the z 202 00:10:05,500 --> 00:10:07,860 component of the angular momentum. 203 00:10:07,860 --> 00:10:11,090 So since it's a component of the angular momentum, that 204 00:10:11,090 --> 00:10:14,330 means that it's never going to be able to go higher than l 205 00:10:14,330 --> 00:10:17,180 is, so it makes sense that, for example, it could start at 206 00:10:17,180 --> 00:10:20,530 0 and then go all the way up to l. 207 00:10:20,530 --> 00:10:23,330 But since it is a component it can have a direction, too, so 208 00:10:23,330 --> 00:10:26,280 can go up between negative l and positive l. 209 00:10:26,280 --> 00:10:30,150 So the allowed values for m sub l are going to be negative 210 00:10:30,150 --> 00:10:35,640 l, all the way up to 0, and then up to positive l. 211 00:10:35,640 --> 00:10:40,590 So, if we think of just an example, we could say that 4 l 212 00:10:40,590 --> 00:10:45,740 equals 2, what would be our lowest value of m sub l? 213 00:10:45,740 --> 00:10:46,310 Yup. 214 00:10:46,310 --> 00:10:49,870 So m sub l could equal negative 2, 215 00:10:49,870 --> 00:10:54,410 negative 1, 0, 1 or 2. 216 00:10:54,410 --> 00:11:00,310 So we could have five different values of m sub l. 217 00:11:00,310 --> 00:11:02,410 So, those are our three quantum numbers. 218 00:11:02,410 --> 00:11:06,460 So if, in fact, we want to describe a wave function, we 219 00:11:06,460 --> 00:11:09,110 know that we need to describe it in terms of all three 220 00:11:09,110 --> 00:11:13,430 quantum numbers, and also as a function of our three 221 00:11:13,430 --> 00:11:18,660 positional factors, which are r, the radius, plus the two 222 00:11:18,660 --> 00:11:20,600 angles, theta and phi. 223 00:11:20,600 --> 00:11:23,040 So, we have now a complete description of a wave function 224 00:11:23,040 --> 00:11:24,500 that we can talk about. 225 00:11:24,500 --> 00:11:26,940 So, we can think about what is it that we would call the 226 00:11:26,940 --> 00:11:28,710 ground state wave function. 227 00:11:28,710 --> 00:11:32,120 We knew from Friday, when we talked about energy, that 228 00:11:32,120 --> 00:11:35,730 ground state was that n equals 1 value, that was the lowest 229 00:11:35,730 --> 00:11:37,720 energy, that was the most stable place for 230 00:11:37,720 --> 00:11:39,020 the electron to be. 231 00:11:39,020 --> 00:11:42,510 But now we need to talk about l and m as well. 232 00:11:42,510 --> 00:11:45,450 So now when we talk about a ground state in terms of wave 233 00:11:45,450 --> 00:11:49,390 function, we need to talk about the wave function of 1, 234 00:11:49,390 --> 00:11:54,560 0, 0, and again, as a function of r, theta and phi. 235 00:11:54,560 --> 00:11:57,960 So this is our complete description of the ground 236 00:11:57,960 --> 00:12:00,670 state wave function. 237 00:12:00,670 --> 00:12:04,270 So, a lot of you talked about different types of orbitals in 238 00:12:04,270 --> 00:12:08,050 high school, I'm sure, or in previous courses, and it might 239 00:12:08,050 --> 00:12:10,580 be less common that you actually talked about a wave 240 00:12:10,580 --> 00:12:12,720 function that was labeled like this. 241 00:12:12,720 --> 00:12:16,480 We're used to labelling orbitals as an s, or a p, or a 242 00:12:16,480 --> 00:12:19,830 d, for example, but it turns out that these correlate to 243 00:12:19,830 --> 00:12:22,640 those letters that we're more used to seeing. 244 00:12:22,640 --> 00:12:28,000 Does anyone know what the 1, 0, 0 orbital is also called? 245 00:12:28,000 --> 00:12:28,450 Yeah. 246 00:12:28,450 --> 00:12:30,700 And specfically it's the 1 s, so not just the 247 00:12:30,700 --> 00:12:33,320 s, but the 1 s orbital. 248 00:12:33,320 --> 00:12:35,770 So, using the terminology of chemists, which is a good 249 00:12:35,770 --> 00:12:38,670 thing to do, because in this course we are all chemists, we 250 00:12:38,670 --> 00:12:42,140 want to make sure that we're not using just the physical 251 00:12:42,140 --> 00:12:44,415 description of the numbers, but that we can correlate it 252 00:12:44,415 --> 00:12:47,570 to what we understand as orbitals, and instead of 1, 0, 253 00:12:47,570 --> 00:12:49,840 0, we call this the 1 s orbital. 254 00:12:49,840 --> 00:12:52,280 The reason that we do this is because this is another way to 255 00:12:52,280 --> 00:12:53,870 completely describe it. 256 00:12:53,870 --> 00:12:57,970 The n designates the shell, so that's what this number is 257 00:12:57,970 --> 00:13:00,070 here, we're in the first shell. 258 00:13:00,070 --> 00:13:03,190 The l is what we call the sub shell. 259 00:13:03,190 --> 00:13:06,300 And instead of having a 0 there, what we 260 00:13:06,300 --> 00:13:08,180 have here is an s. 261 00:13:08,180 --> 00:13:10,920 So, if we look at what the other sub shells are called, 262 00:13:10,920 --> 00:13:12,190 essentially we're just converting 263 00:13:12,190 --> 00:13:14,680 the number to a letter. 264 00:13:14,680 --> 00:13:19,300 L equals 0 is s, what is l equals 1? 265 00:13:19,300 --> 00:13:20,730 Um-hmm, it's the p. 266 00:13:20,730 --> 00:13:25,710 What about 2? d, and 3? 267 00:13:25,710 --> 00:13:27,470 Yup, so 3 is f. 268 00:13:27,470 --> 00:13:30,090 So these names, they don't really make any sense if we're 269 00:13:30,090 --> 00:13:32,990 looking at them why they're called past s p and f, and it 270 00:13:32,990 --> 00:13:36,110 turns out that it comes from spectroscopy terms that are 271 00:13:36,110 --> 00:13:39,420 pre-quantum mechanics where, for example, this is called 272 00:13:39,420 --> 00:13:42,540 the sharp line, I think the principle, the diffuse, and 273 00:13:42,540 --> 00:13:43,610 the fundamental. 274 00:13:43,610 --> 00:13:47,220 It doesn't even make sense now, they're not used in 275 00:13:47,220 --> 00:13:49,920 spectroscopy anymore, but this is where the names originally 276 00:13:49,920 --> 00:13:51,600 came from and they did stick. 277 00:13:51,600 --> 00:13:56,810 So, we being chemists, we'll call that 1 s instead of 1, 0. 278 00:13:56,810 --> 00:14:00,200 In addition to having another name to denote l, we also have 279 00:14:00,200 --> 00:14:04,820 another name for the m designation here. 280 00:14:04,820 --> 00:14:13,300 So, for example, when l is equal to 0, we're going to 281 00:14:13,300 --> 00:14:16,860 find that we have to call -- we have to specify 282 00:14:16,860 --> 00:14:22,780 what m is as well. 283 00:14:22,780 --> 00:14:23,170 All right. 284 00:14:23,170 --> 00:14:28,070 So, when we have, for example, l equal to 1, what kind of 285 00:14:28,070 --> 00:14:30,590 orbital is this? 286 00:14:30,590 --> 00:14:31,250 The p orbital. 287 00:14:31,250 --> 00:14:34,380 And for example, we could also in this case, have 288 00:14:34,380 --> 00:14:36,270 m is equal to 0. 289 00:14:36,270 --> 00:14:40,700 If m is equal to 0, in this case we would call it the p z 290 00:14:40,700 --> 00:14:44,750 orbital, so we would have the subscript z here. 291 00:14:44,750 --> 00:14:50,680 Similarly, if m is equal to either plus 1 or minus 1, we 292 00:14:50,680 --> 00:14:57,850 would in turn call it the p y orbital, or the p x orbital. 293 00:14:57,850 --> 00:15:01,270 So you should know that any time m is equal to zero when 294 00:15:01,270 --> 00:15:04,180 we are talking about p orbitals, that it's the p z. 295 00:15:04,180 --> 00:15:06,360 The p y and the p x are actually a bit more 296 00:15:06,360 --> 00:15:10,280 complicated, they're linear combinations of the m plus 1, 297 00:15:10,280 --> 00:15:13,340 and the m minus 1 orbital, where 1 is the positive linear 298 00:15:13,340 --> 00:15:16,040 combination, and 1 is the negative linear combination. 299 00:15:16,040 --> 00:15:18,540 You're not responsible for that, you're not responsible 300 00:15:18,540 --> 00:15:21,810 for correlating plus 1 to y, minus 1 to x. 301 00:15:21,810 --> 00:15:24,950 Just know that you have plus or minus 1, for our class, you 302 00:15:24,950 --> 00:15:29,060 can call it either x or y, either is fine, because it's a 303 00:15:29,060 --> 00:15:33,520 little bit more complicated than just the 1:1 translation 304 00:15:33,520 --> 00:15:38,170 between, for example, m equals 0 and having a p z orbital. 305 00:15:38,170 --> 00:15:41,850 All right. 306 00:15:41,850 --> 00:15:44,560 So let's look at some of these wave functions and make sure 307 00:15:44,560 --> 00:15:47,120 that we know how to name all of them in terms of orbitals 308 00:15:47,120 --> 00:15:48,950 and not just in terms of their numbers. 309 00:15:48,950 --> 00:15:52,100 Once we can do that we can go on and say okay, what actually 310 00:15:52,100 --> 00:15:54,060 is a wave function, but first we need to know how to 311 00:15:54,060 --> 00:15:56,470 describe which ones were talking about. 312 00:15:56,470 --> 00:15:59,650 So we saw that our lowest, our ground state wave 313 00:15:59,650 --> 00:16:01,750 function is 1, 0, 0. 314 00:16:01,750 --> 00:16:04,750 We can call that psi 1, 0, 0 is how we write 315 00:16:04,750 --> 00:16:07,420 it as a wave function. 316 00:16:07,420 --> 00:16:10,460 We said that's the 1 s orbital. 317 00:16:10,460 --> 00:16:14,380 We also know how to figure out the energy of this orbital, 318 00:16:14,380 --> 00:16:17,560 and we know how to figure out the energy using this formula 319 00:16:17,560 --> 00:16:21,140 here, which was the binding energy, which is negative r h, 320 00:16:21,140 --> 00:16:24,390 and instead of n, we can plug it in because n equals 1, so 321 00:16:24,390 --> 00:16:28,730 over 1 squared, and the actual energy is here. 322 00:16:28,730 --> 00:16:33,060 So, our next level up that we can go is going to be the n 323 00:16:33,060 --> 00:16:39,160 equals 2 energy level, but we also have an l and an m value, 324 00:16:39,160 --> 00:16:42,050 so our lowest l is going to be a 0 there. 325 00:16:42,050 --> 00:16:46,220 So we'll call that psi 2, 0, 0 wave function. 326 00:16:46,220 --> 00:16:50,070 What will we call that in terms of orbitals? 327 00:16:50,070 --> 00:16:52,690 Yup, so that's the 2 s orbital. 328 00:16:52,690 --> 00:16:55,370 So something I actually wanted to point out that I forgot to 329 00:16:55,370 --> 00:16:58,850 here is you'll notice that there's no subscript to the s. 330 00:16:58,850 --> 00:17:02,120 We said we have a subscript to the p, for example, that 331 00:17:02,120 --> 00:17:04,150 describes what m is equal to. 332 00:17:04,150 --> 00:17:07,000 The reason that we have no subscript to the s, is because 333 00:17:07,000 --> 00:17:10,630 the only possibility for m when you have an s orbital is 334 00:17:10,630 --> 00:17:12,770 that m has to be equal to 0. 335 00:17:12,770 --> 00:17:15,280 So we just assume it, you don't actually have to write 336 00:17:15,280 --> 00:17:18,540 it because there is, in fact, only one possibility. 337 00:17:18,540 --> 00:17:21,270 We can also figure out the energy of this orbital here, 338 00:17:21,270 --> 00:17:23,990 and the energy is equal to the Rydberg constant. 339 00:17:23,990 --> 00:17:25,750 The negative of the Rydberg constant now 340 00:17:25,750 --> 00:17:28,110 divided by 2 squared. 341 00:17:28,110 --> 00:17:30,770 So we can go on and do this for any orbital or any state 342 00:17:30,770 --> 00:17:32,430 function that we would like to. 343 00:17:32,430 --> 00:17:36,400 So, for example, if we talk about the 2, 1, 1 state label, 344 00:17:36,400 --> 00:17:39,100 that's just psi 2, 1, 1. 345 00:17:39,100 --> 00:17:40,910 What, in this case, would be our orbital? 346 00:17:40,910 --> 00:17:44,660 2 p what? 347 00:17:44,660 --> 00:17:49,750 OK, good, I heard mixed answers, which is correct. 348 00:17:49,750 --> 00:17:53,390 So you can either write 2 p x or 2 p y, whichever one you 349 00:17:53,390 --> 00:17:55,430 want is fine. 350 00:17:55,430 --> 00:17:58,270 And again, you'll notice that our energy is absolutely the 351 00:17:58,270 --> 00:18:01,870 same for an electron in that 2 p x orbital 352 00:18:01,870 --> 00:18:04,420 and in the 2 s orbital. 353 00:18:04,420 --> 00:18:06,640 So that's true for a hydrogen atom, it doesn't matter if 354 00:18:06,640 --> 00:18:10,570 you're in a p or an s orbital, their energies are the same. 355 00:18:10,570 --> 00:18:13,970 Then we can also talk about the 2, 1, 0 state function, 356 00:18:13,970 --> 00:18:15,880 which would be psi 2, 1, 0. 357 00:18:15,880 --> 00:18:16,670 What is this orbital? 358 00:18:16,670 --> 00:18:18,810 Yup. 359 00:18:18,810 --> 00:18:20,360 And there's only one correct answer here, 360 00:18:20,360 --> 00:18:22,590 which is to 2 p z. 361 00:18:22,590 --> 00:18:26,700 Is the energy going to be the same or different as up here? 362 00:18:26,700 --> 00:18:28,150 It's going to be the same energy. 363 00:18:28,150 --> 00:18:30,840 Again, the reason for that is because the energy only 364 00:18:30,840 --> 00:18:33,030 depends on the n value here, it doesn't 365 00:18:33,030 --> 00:18:35,850 depend on l or on m. 366 00:18:35,850 --> 00:18:38,660 So finally, if we talk about our last example of when n is 367 00:18:38,660 --> 00:18:42,240 going to be equal 2, we can have 2, 1 for l and 368 00:18:42,240 --> 00:18:43,840 then minus 1 for m. 369 00:18:43,840 --> 00:18:47,570 We can re-write this as psi 2 1 negative 1. 370 00:18:47,570 --> 00:18:50,570 And then our orbital is going to be just the opposite of 371 00:18:50,570 --> 00:18:52,370 whatever we said it was up here. 372 00:18:52,370 --> 00:18:56,720 So if you said 2 p x the first time, say 2 p y this time. 373 00:18:56,720 --> 00:19:00,240 And again, our energy is going to be the same where we again 374 00:19:00,240 --> 00:19:03,230 only depend on the n value. 375 00:19:03,230 --> 00:19:03,550 All right. 376 00:19:03,550 --> 00:19:06,230 So hopefully we're pretty comfortable naming any type of 377 00:19:06,230 --> 00:19:09,280 wave function using the chemist terminology. 378 00:19:09,280 --> 00:19:11,920 Let's switch to a clicker question and just confirm that 379 00:19:11,920 --> 00:19:13,690 that is, in fact, true. 380 00:19:13,690 --> 00:19:18,300 So what's the corresponding orbital if we talk about this 381 00:19:18,300 --> 00:19:27,530 state, 5, 1, 0? 382 00:19:27,530 --> 00:19:42,180 And you can go ahead and give 10 seconds on that. 383 00:19:42,180 --> 00:19:44,340 OK. 384 00:19:44,340 --> 00:19:45,980 All right, 77%. 385 00:19:45,980 --> 00:19:49,400 So, that's OK, you don't have to memorize things as I speak, 386 00:19:49,400 --> 00:19:52,310 you just need to go back and look at this and make sure you 387 00:19:52,310 --> 00:19:54,890 understand how to name it and that you'll be able to, for 388 00:19:54,890 --> 00:19:58,090 example, by next class, get a similar clicker question 389 00:19:58,090 --> 00:20:01,530 correct, and good job to the 77% that did get it. 390 00:20:01,530 --> 00:20:04,140 So I think we're safe to move on here. 391 00:20:04,140 --> 00:20:07,290 And I just want to point out that now we have these three 392 00:20:07,290 --> 00:20:08,010 quantum numbers. 393 00:20:08,010 --> 00:20:09,940 The reason there are three quantum numbers is we're 394 00:20:09,940 --> 00:20:13,000 describing an orbital in three dimensions, so it makes sense 395 00:20:13,000 --> 00:20:15,360 that we would need to describe in terms of three different 396 00:20:15,360 --> 00:20:16,620 quantum numbers. 397 00:20:16,620 --> 00:20:20,760 And the complete description, as I said, is from n l and m. 398 00:20:20,760 --> 00:20:24,060 And when you talk about n for an orbital, it's talking about 399 00:20:24,060 --> 00:20:27,150 the shell -- that shell is kind of what you picture when 400 00:20:27,150 --> 00:20:29,900 you think of a classical picture of an atom where you 401 00:20:29,900 --> 00:20:32,870 have 1 energy level, the next one is further out, the next 402 00:20:32,870 --> 00:20:34,780 one's further away. 403 00:20:34,780 --> 00:20:37,440 That's kind of your shell that we're discussing. 404 00:20:37,440 --> 00:20:43,700 L is the sub shell here, and then we have m, which is 405 00:20:43,700 --> 00:20:47,420 finally the complete description of the orbital. 406 00:20:47,420 --> 00:20:52,070 And what you can see is that for any n that has an l equals 407 00:20:52,070 --> 00:20:54,720 0, you can see here how there's only one possibility 408 00:20:54,720 --> 00:20:57,250 for and orbital description, and that's why we don't need 409 00:20:57,250 --> 00:21:01,910 to include the m when we're talking about and s orbital. 410 00:21:01,910 --> 00:21:04,350 The other thing that we know, which is what we were just 411 00:21:04,350 --> 00:21:06,870 discussing when we were going through the table is how this 412 00:21:06,870 --> 00:21:08,730 all relates to energy. 413 00:21:08,730 --> 00:21:11,720 And I want to really highlight here we're talking about for a 414 00:21:11,720 --> 00:21:15,280 hydrogen atom -- orbitals with the same n value 415 00:21:15,280 --> 00:21:16,980 have the same energy. 416 00:21:16,980 --> 00:21:19,510 Some of you might be saying in your heads, wait a second, I 417 00:21:19,510 --> 00:21:22,780 happen to know, I happen to remember from high school, 418 00:21:22,780 --> 00:21:25,960 that p orbitals have different energies then, 419 00:21:25,960 --> 00:21:27,620 for example, s orbitals. 420 00:21:27,620 --> 00:21:30,580 And that is not true for one electron atoms. We're going to 421 00:21:30,580 --> 00:21:34,175 get to more complicated atoms eventually where we're going 422 00:21:34,175 --> 00:21:37,450 to have more than one electron in it, but when we're talking 423 00:21:37,450 --> 00:21:40,370 about a single electron atom, we know that the binding 424 00:21:40,370 --> 00:21:43,100 energy is equal to the negative of the Rydberg 425 00:21:43,100 --> 00:21:46,640 constant over n squared, so it's only depends on n. 426 00:21:46,640 --> 00:21:49,260 So, for example, if we're talking about the n equals 2 427 00:21:49,260 --> 00:21:53,240 state, all of these four orbital descriptions are going 428 00:21:53,240 --> 00:21:54,690 to have the same energy. 429 00:21:54,690 --> 00:21:57,560 And we can generalize to figure out, based on any 430 00:21:57,560 --> 00:22:01,710 principle quantum number n, how many orbitals we have of 431 00:22:01,710 --> 00:22:07,460 the same energy, and what we can say is that for any shell 432 00:22:07,460 --> 00:22:09,810 n, there are n squared degenerate orbitals. 433 00:22:09,810 --> 00:22:15,730 And the word degenerate simply means same energy, so you have 434 00:22:15,730 --> 00:22:18,620 n squared orbitals that are of equal energy when they're 435 00:22:18,620 --> 00:22:21,430 degenerate. 436 00:22:21,430 --> 00:22:24,400 So, let's look at where this comes from with an energy 437 00:22:24,400 --> 00:22:27,090 level diagram here. 438 00:22:27,090 --> 00:22:35,210 So what you can see is again, we've got this ground state. 439 00:22:35,210 --> 00:22:38,030 So if we go to the ground state, what you see is we're 440 00:22:38,030 --> 00:22:41,260 at that lowest energy level, and we only have one 441 00:22:41,260 --> 00:22:44,320 possibility for an orbital, because when n equals 1, 442 00:22:44,320 --> 00:22:45,690 that's all we can do. 443 00:22:45,690 --> 00:22:49,950 So that's the 1 s orbital -- we have n squared or 1 444 00:22:49,950 --> 00:22:51,760 degenerate orbitals. 445 00:22:51,760 --> 00:22:57,100 When we talk about the n equals 2 state, we now have 2 446 00:22:57,100 --> 00:23:01,460 squared or 4 degenerate same energy orbitals, and those are 447 00:23:01,460 --> 00:23:03,900 the 2 s orbital. 448 00:23:03,900 --> 00:23:08,590 And then we also have the l being equal to 1 orbital, so 449 00:23:08,590 --> 00:23:12,030 those are going to be the 2 p x, the 2 p z, 450 00:23:12,030 --> 00:23:13,290 and the 2 p y orbital. 451 00:23:13,290 --> 00:23:16,980 All four of these orbitals have the same energy, they're 452 00:23:16,980 --> 00:23:18,060 degenerate. 453 00:23:18,060 --> 00:23:21,990 And as we go up the next energy level, which is based 454 00:23:21,990 --> 00:23:25,250 on n equals 3 principle quantum number, well now we 455 00:23:25,250 --> 00:23:29,045 have again the s, so we have the 3 s orbital, we're going 456 00:23:29,045 --> 00:23:34,350 to have three 3 p orbitals, right, so we'll have 3 p x, 3 457 00:23:34,350 --> 00:23:38,280 p z, and 3 p y, and now we're actually also going to have 458 00:23:38,280 --> 00:23:41,470 five different possible l equals 2 orbitals. 459 00:23:41,470 --> 00:23:44,970 Does anyone remember the l equals 2? 460 00:23:44,970 --> 00:23:46,110 Yes, everyone remembers. 461 00:23:46,110 --> 00:23:46,670 Good. 462 00:23:46,670 --> 00:23:49,640 So we have five possible d orbitals. 463 00:23:49,640 --> 00:23:55,540 We'll call these here the 3 d x y, as the subscript, the 3 d 464 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 465 00:24:03,430 --> 00:24:05,740 minus y squared. 466 00:24:05,740 --> 00:24:10,360 So, what do you need to know here? 467 00:24:10,360 --> 00:24:13,350 What you need to know is that when m equals 468 00:24:13,350 --> 00:24:16,550 0, it's 3 d z squared. 469 00:24:16,550 --> 00:24:17,540 That's it. 470 00:24:17,540 --> 00:24:22,740 Again, these other p -- or the d x y, d y z, those are going 471 00:24:22,740 --> 00:24:25,340 to be those more complicated linear combinations, you don't 472 00:24:25,340 --> 00:24:26,840 need to worry about them. 473 00:24:26,840 --> 00:24:29,250 Eventually you will, at least, need to know the labels and 474 00:24:29,250 --> 00:24:30,630 know a little bit more about them. 475 00:24:30,630 --> 00:24:33,190 And in the second half of this course, Professor Drennen's 476 00:24:33,190 --> 00:24:36,320 going to talk to us about transition metals in depth, 477 00:24:36,320 --> 00:24:38,910 and that's when we'll really delve into d orbitals. 478 00:24:38,910 --> 00:24:41,360 For right now, you can kind of put the d orbitals in the back 479 00:24:41,360 --> 00:24:41,840 of your head. 480 00:24:41,840 --> 00:24:44,990 You need to know how to think about them in the same way we 481 00:24:44,990 --> 00:24:47,950 think about s and p orbitals, but for example, you don't yet 482 00:24:47,950 --> 00:24:51,520 need to know what all of the names are except for this 3 d 483 00:24:51,520 --> 00:24:53,340 z squared here. 484 00:24:53,340 --> 00:24:56,070 So we'll wait on that until we start talking more 485 00:24:56,070 --> 00:24:58,910 specifically about atoms where the d orbital becomes very 486 00:24:58,910 --> 00:25:00,890 significant. 487 00:25:00,890 --> 00:25:03,660 So, what we can see is this degeneracy. 488 00:25:03,660 --> 00:25:07,130 So what we know now is we can start thinking about the next 489 00:25:07,130 --> 00:25:10,150 step because we can fully describe the energy of 490 00:25:10,150 --> 00:25:13,890 orbitals, and we can fully describe a complete orbital in 491 00:25:13,890 --> 00:25:16,270 terms of its three quantum numbers, and its three 492 00:25:16,270 --> 00:25:19,830 positional variables, r, theta, and phi. 493 00:25:19,830 --> 00:25:23,050 So next we can think about okay, what is actually a wave 494 00:25:23,050 --> 00:25:25,990 function, and for example, what might the shape of 495 00:25:25,990 --> 00:25:29,130 different wave functions be. 496 00:25:29,130 --> 00:25:31,370 So essentially, what we're asking for here is the 497 00:25:31,370 --> 00:25:34,876 physical interpretation of psi, of the value of psi for 498 00:25:34,876 --> 00:25:36,210 an electron. 499 00:25:36,210 --> 00:25:39,560 And it turns out that the answer to can we have this 500 00:25:39,560 --> 00:25:42,640 physical interpretation of thinking about what psi means, 501 00:25:42,640 --> 00:25:45,190 the answer is really no, that we can't. 502 00:25:45,190 --> 00:25:48,310 There's no classical way to think about what a wave 503 00:25:48,310 --> 00:25:49,460 function is. 504 00:25:49,460 --> 00:25:52,580 There's no classical analogy that explains oh, this is what 505 00:25:52,580 --> 00:25:56,080 you can kind of picture when you picture a wave function. 506 00:25:56,080 --> 00:25:57,960 And that's somewhat inconvenient because we're 507 00:25:57,960 --> 00:26:00,940 working with wave functions, but it's a reality that comes 508 00:26:00,940 --> 00:26:04,050 out of quantum mechanics often, which is that we're 509 00:26:04,050 --> 00:26:07,120 describing a world that is so much different from the world 510 00:26:07,120 --> 00:26:09,170 that we observe on a day-to-day basis, that we're 511 00:26:09,170 --> 00:26:10,450 not always going to be able to make 512 00:26:10,450 --> 00:26:12,240 those one-to-one analogies. 513 00:26:12,240 --> 00:26:14,670 But luckily we don't have to worry about how we're going to 514 00:26:14,670 --> 00:26:17,440 picture all this, now that I said that, because even though 515 00:26:17,440 --> 00:26:19,720 there's no physical interpretation for what a wave 516 00:26:19,720 --> 00:26:22,830 function is, there is a physical interpretation for 517 00:26:22,830 --> 00:26:25,620 what a wave function squared means. 518 00:26:25,620 --> 00:26:29,300 So when we talk about a wave function squared, we're taking 519 00:26:29,300 --> 00:26:32,140 the square of the wave function, any one that we 520 00:26:32,140 --> 00:26:36,810 specify between n, l and m, at any position that we specify 521 00:26:36,810 --> 00:26:38,920 based on r, theta, and phi. 522 00:26:38,920 --> 00:26:42,040 And if we go ahead and square that, then what we get is a 523 00:26:42,040 --> 00:26:44,710 probability density, and specifically it's the 524 00:26:44,710 --> 00:26:49,280 probability of finding an electron in a certain small 525 00:26:49,280 --> 00:26:52,300 defined volume away from the nucleus. 526 00:26:52,300 --> 00:26:54,680 So it's a probability density. 527 00:26:54,680 --> 00:26:57,260 The important point here is it's not just a probability, 528 00:26:57,260 --> 00:26:59,530 it's a density, so we know that it's a probability 529 00:26:59,530 --> 00:27:01,690 divided by volume. 530 00:27:01,690 --> 00:27:04,420 And the person we have to thank for actually giving us 531 00:27:04,420 --> 00:27:07,010 this more concrete way to think about what a wave 532 00:27:07,010 --> 00:27:10,570 function squared is is Max Born here. 533 00:27:10,570 --> 00:27:14,820 And actually after the Schrodinger equation first was 534 00:27:14,820 --> 00:27:18,150 put forth, people had a lot of discussions about how is it 535 00:27:18,150 --> 00:27:21,250 that we can actually interpret what this wave function means, 536 00:27:21,250 --> 00:27:24,260 and a lot of ideas were put forth, and none of them worked 537 00:27:24,260 --> 00:27:27,440 out to match up with observations until Max Born 538 00:27:27,440 --> 00:27:29,700 here came up with the idea that we just square the wave 539 00:27:29,700 --> 00:27:32,350 function, and that's the probability density of finding 540 00:27:32,350 --> 00:27:36,130 an electron in a certain defined volume. 541 00:27:36,130 --> 00:27:38,660 And it's very helpful because it gives us a way 542 00:27:38,660 --> 00:27:39,730 to think about it. 543 00:27:39,730 --> 00:27:42,410 We can't actually go ahead and derive this equation of the 544 00:27:42,410 --> 00:27:45,655 wave function squared, because no one ever derived it, it's 545 00:27:45,655 --> 00:27:47,960 just an interpretation, but it's an interpretation that 546 00:27:47,960 --> 00:27:50,120 works essentially perfectly. 547 00:27:50,120 --> 00:27:52,380 Ever since this was first proposed, there has never been 548 00:27:52,380 --> 00:27:56,360 any observations that do not coincide with the idea, that 549 00:27:56,360 --> 00:27:59,680 did not match the fact that the probability density is 550 00:27:59,680 --> 00:28:02,480 equal to the wave function squared. 551 00:28:02,480 --> 00:28:06,020 So, also about Max Born, just to give you a little bit of a 552 00:28:06,020 --> 00:28:08,910 trivial pursuit type knowledge, he not only gave us 553 00:28:08,910 --> 00:28:13,010 this relationship between wave function squared, he also gave 554 00:28:13,010 --> 00:28:15,420 us Olivia Newton-John. 555 00:28:15,420 --> 00:28:17,950 This is her grandfather, I don't know if you can see from 556 00:28:17,950 --> 00:28:19,980 the eyes, I feel like there's a little bit of 557 00:28:19,980 --> 00:28:22,640 a resemblance there. 558 00:28:22,640 --> 00:28:25,086 So, I don't know what she grew up hearing about when she went 559 00:28:25,086 --> 00:28:27,210 to her grandparents' house, but it might have been wave 560 00:28:27,210 --> 00:28:29,080 function squared. 561 00:28:29,080 --> 00:28:33,290 So, a little tidbit of knowledge for you that's 562 00:28:33,290 --> 00:28:36,140 somewhat trivial. 563 00:28:36,140 --> 00:28:40,320 Then back to the non-trivial knowledge that is not trivial 564 00:28:40,320 --> 00:28:43,190 at all, in fact, is OK, how do we think about this 565 00:28:43,190 --> 00:28:45,140 probability density now that we have a little 566 00:28:45,140 --> 00:28:46,510 bit more of an idea. 567 00:28:46,510 --> 00:28:50,120 We know that it's a density, it's not an actual 568 00:28:50,120 --> 00:28:50,860 probability. 569 00:28:50,860 --> 00:28:53,850 So, one way we could look at it is by looking at this 570 00:28:53,850 --> 00:28:58,610 density dot diagram, where the density of the dots correlates 571 00:28:58,610 --> 00:29:00,720 to the probability density. 572 00:29:00,720 --> 00:29:05,960 So, what you see is near the nucleus, the density is the 573 00:29:05,960 --> 00:29:08,200 strongest, the dots are closest together. 574 00:29:08,200 --> 00:29:11,250 As you get far away from the nucleus, the dots get farther 575 00:29:11,250 --> 00:29:15,810 and farther apart, meaning the probability density at those 576 00:29:15,810 --> 00:29:19,490 volumes far away from the nucleus is going to be quite 577 00:29:19,490 --> 00:29:22,610 low, eventually going to almost zero, although it turns 578 00:29:22,610 --> 00:29:25,060 out that it never goes to exactly zero, so if we're 579 00:29:25,060 --> 00:29:28,640 talking about any orbital or any atom, it never actually 580 00:29:28,640 --> 00:29:30,770 ends, it never goes to zerio. 581 00:29:30,770 --> 00:29:32,590 But it turns out the probability is only 582 00:29:32,590 --> 00:29:34,810 significant within one angstrom. 583 00:29:34,810 --> 00:29:37,470 So you can either say that electrons are very, very tiny 584 00:29:37,470 --> 00:29:40,070 or that they're never ending, and both are pretty accurate 585 00:29:40,070 --> 00:29:43,730 ways to think about what an atom is. 586 00:29:43,730 --> 00:29:47,360 So, that's probability density, but in terms of 587 00:29:47,360 --> 00:29:50,150 thinking about it in terms of actual solutions to the wave 588 00:29:50,150 --> 00:29:52,390 function, let's take a little bit of a step back here. 589 00:29:52,390 --> 00:29:55,870 I have yet to show you the solution to a wave function 590 00:29:55,870 --> 00:29:58,680 for the hydrogen atom, so let me do that here, and then 591 00:29:58,680 --> 00:30:01,760 we'll build back up to probability densities, and it 592 00:30:01,760 --> 00:30:03,820 turns out that if we're talking about any wave 593 00:30:03,820 --> 00:30:06,500 function, we can actually break it up into two 594 00:30:06,500 --> 00:30:09,950 components, which are called the radial wave function and 595 00:30:09,950 --> 00:30:12,230 angular wave function. 596 00:30:12,230 --> 00:30:15,040 So, essentially we're just breaking it up into two parts 597 00:30:15,040 --> 00:30:18,270 that can be separated, and the part that is only dealing with 598 00:30:18,270 --> 00:30:22,030 the radius, so it's only a function of the radius of the 599 00:30:22,030 --> 00:30:24,520 electron from the nucleus. 600 00:30:24,520 --> 00:30:28,460 And we abbreviate that by calling it r, which is 601 00:30:28,460 --> 00:30:31,290 specified by two quantum numbers, and an l as a 602 00:30:31,290 --> 00:30:34,130 function of little r, radius. 603 00:30:34,130 --> 00:30:37,330 And we have the angular wave function, which is specified 604 00:30:37,330 --> 00:30:41,010 by l and m, and it's a function of the two angles 605 00:30:41,010 --> 00:30:43,430 when we're describing the position of the electron, so 606 00:30:43,430 --> 00:30:45,780 theta and phi. 607 00:30:45,780 --> 00:30:49,060 So, let's look at what this actually is for what we're 608 00:30:49,060 --> 00:30:51,840 showing here is the 1 s hydrogen atom. 609 00:30:51,840 --> 00:30:53,780 If you look in your book there's a whole table of 610 00:30:53,780 --> 00:30:56,330 different solutions to the Schrodinger equation for 611 00:30:56,330 --> 00:30:58,570 several different wave functions. 612 00:30:58,570 --> 00:31:00,700 So this is the 1 s, you can look it up if you're 613 00:31:00,700 --> 00:31:04,810 interested for the 2 s, or 3 s, or 5 s, or whatever you're 614 00:31:04,810 --> 00:31:05,040 curious about. 615 00:31:05,040 --> 00:31:08,150 But what I'm going to show you here is the 1 s solution. 616 00:31:08,150 --> 00:31:11,670 So you can see there's this radial part here, and you have 617 00:31:11,670 --> 00:31:14,370 the angular part, you can combine the two parts to get 618 00:31:14,370 --> 00:31:16,170 the total wave function. 619 00:31:16,170 --> 00:31:20,470 And what you can see is we have this new constant that we 620 00:31:20,470 --> 00:31:22,000 haven't seen before. 621 00:31:22,000 --> 00:31:26,030 So what do you see in there that is new? 622 00:31:26,030 --> 00:31:26,220 Yeah. 623 00:31:26,220 --> 00:31:27,520 This a sub nought. 624 00:31:27,520 --> 00:31:30,410 That's a new constant for us in this course. 625 00:31:30,410 --> 00:31:35,950 This is what's called the Bohr radius, and we'll explain -- 626 00:31:35,950 --> 00:31:38,940 hopefully we'll get to it today where this Bohr radius 627 00:31:38,940 --> 00:31:41,930 name comes from, but for now what you need to know is just 628 00:31:41,930 --> 00:31:44,630 that it's a constant, just treat it like a constant, and 629 00:31:44,630 --> 00:31:47,050 it turns out to be equal to 52 . 630 00:31:47,050 --> 00:31:51,620 9 pekameters or about 1/2 an angstrom. 631 00:31:51,620 --> 00:31:54,510 The more important thing that I want you to notice when 632 00:31:54,510 --> 00:32:01,540 you're looking at this wave equation for a 1 s h atom, is 633 00:32:01,540 --> 00:32:04,570 the fact that if you look at the angular component of the 634 00:32:04,570 --> 00:32:08,160 wave function, you'll notice that it's a constant. 635 00:32:08,160 --> 00:32:12,200 It doesn't depend on theta, it doesn't depend on phi. 636 00:32:12,200 --> 00:32:15,120 No matter where you specify your electron is in terms of 637 00:32:15,120 --> 00:32:18,160 those two angles, it doesn't matter the angular part of 638 00:32:18,160 --> 00:32:21,090 your wave function is going to be the same. 639 00:32:21,090 --> 00:32:23,220 So, what does that mean for us? 640 00:32:23,220 --> 00:32:26,370 Well, essentially what that tells is that these s orbitals 641 00:32:26,370 --> 00:32:28,190 are spherically symmetrical. 642 00:32:28,190 --> 00:32:30,050 That should make sense, right, because they're only 643 00:32:30,050 --> 00:32:31,190 dependent on r. 644 00:32:31,190 --> 00:32:34,880 How far you are away from the nucleus in terms of a radius, 645 00:32:34,880 --> 00:32:37,910 they don't depend at all on those two angles, they're 646 00:32:37,910 --> 00:32:42,960 independent of theta and they're independent of phi. 647 00:32:42,960 --> 00:32:45,820 So, what I'm showing in this picture here is just an 648 00:32:45,820 --> 00:32:48,280 electron cloud that you can see. 649 00:32:48,280 --> 00:32:51,720 Think of it as a probability density plot. 650 00:32:51,720 --> 00:32:55,880 And what here is just a graph of the 1 s wave function going 651 00:32:55,880 --> 00:32:59,810 across some radius defined this way, and you can see that 652 00:32:59,810 --> 00:33:02,955 the probability -- well, this is the wave function, so we 653 00:33:02,955 --> 00:33:05,820 would have to square it and think about the probability. 654 00:33:05,820 --> 00:33:11,480 So this squared at the origin is going to be a very high 655 00:33:11,480 --> 00:33:14,880 probability, and it decays off as you get farther and farther 656 00:33:14,880 --> 00:33:18,160 away from the nucleus or from the center, and that's 657 00:33:18,160 --> 00:33:21,370 independent of the angle. 658 00:33:21,370 --> 00:33:24,400 So, let's look at these probability plots of different 659 00:33:24,400 --> 00:33:34,660 s orbitals here, and up top here, we have the probability 660 00:33:34,660 --> 00:33:38,230 density plot and what you can see is what I just said, a 661 00:33:38,230 --> 00:33:40,060 very high probability density in the nucleus, 662 00:33:40,060 --> 00:33:41,670 decays as you go out. 663 00:33:41,670 --> 00:33:45,200 And what is plotted below is the actual wave function, so 664 00:33:45,200 --> 00:33:48,390 you can see it starts very high and then the decays down. 665 00:33:48,390 --> 00:33:51,750 More interesting is to look at the 2 s wave function. 666 00:33:51,750 --> 00:33:54,620 So, if we look at the bottom here and the actual plot of 667 00:33:54,620 --> 00:33:57,980 the wave function, we see it starts high, very positive, 668 00:33:57,980 --> 00:34:01,380 and it goes down and it eventually hits zero, and goes 669 00:34:01,380 --> 00:34:04,480 through zero and then becomes negative and then never quite 670 00:34:04,480 --> 00:34:07,150 hits zero again, although it approaches zero. 671 00:34:07,150 --> 00:34:10,050 So, at this place where it hits zero, that means that the 672 00:34:10,050 --> 00:34:12,650 square of the wave function is also going to be zero, right. 673 00:34:12,650 --> 00:34:17,110 So we can see if we look at the probability density plot, 674 00:34:17,110 --> 00:34:20,100 we can see there's a place where the probability density 675 00:34:20,100 --> 00:34:22,880 of finding an electron anywhere there is actually 676 00:34:22,880 --> 00:34:25,370 going to be zero. 677 00:34:25,370 --> 00:34:28,460 So we can think of a third case where we have the 3 s 678 00:34:28,460 --> 00:34:31,580 orbital, and in the 3 s orbital we see something 679 00:34:31,580 --> 00:34:35,040 similar, we start high, we go through zero, where there will 680 00:34:35,040 --> 00:34:38,040 now be zero probability density, as we can see in the 681 00:34:38,040 --> 00:34:40,250 in the density plot graph. 682 00:34:40,250 --> 00:34:43,850 Then we go negative and we go through zero again, which 683 00:34:43,850 --> 00:34:47,170 correlates to the second area of zero, that shows up also in 684 00:34:47,170 --> 00:34:50,850 our probability density plot, and then we're positive again 685 00:34:50,850 --> 00:34:55,740 and approach zero as we go to infinity for r. 686 00:34:55,740 --> 00:34:58,830 So, what this means is that when we're looking at an 687 00:34:58,830 --> 00:35:01,960 actual wave function, we're treating it as a wave, right, 688 00:35:01,960 --> 00:35:05,620 so waves can have both magnitude, but they can also 689 00:35:05,620 --> 00:35:08,380 have a direction, so they can be 690 00:35:08,380 --> 00:35:10,420 either positive or negative. 691 00:35:10,420 --> 00:35:13,160 So, for example, if we were looking at the actual wave 692 00:35:13,160 --> 00:35:15,880 function, we would say that these parts here have a 693 00:35:15,880 --> 00:35:18,180 positive amplitude, and in here we 694 00:35:18,180 --> 00:35:20,040 have a negative amplitude. 695 00:35:20,040 --> 00:35:23,460 And when we're looking at the probability density graphs, it 696 00:35:23,460 --> 00:35:26,650 doesn't make a difference, it's okay, It has no meaning 697 00:35:26,650 --> 00:35:29,800 for our actual plot there, because we're squaring it, so 698 00:35:29,800 --> 00:35:32,290 it doesn't matter whether it's negative or positive, all that 699 00:35:32,290 --> 00:35:34,040 matters is the magnitude. 700 00:35:34,040 --> 00:35:37,080 But when we're thinking about actual wave behavior of 701 00:35:37,080 --> 00:35:39,170 electrons, it's just important to keep in the back of our 702 00:35:39,170 --> 00:35:42,250 head that some areas have positive amplitude and some 703 00:35:42,250 --> 00:35:43,350 have negative. 704 00:35:43,350 --> 00:35:46,210 So we'll talk about this more we get into p orbitals and 705 00:35:46,210 --> 00:35:48,090 bonding is where it's going to become an issue. 706 00:35:48,090 --> 00:35:50,990 So I just want to kind of introduce that idea here. 707 00:35:50,990 --> 00:35:54,340 Because if we think about wave behavior of electrons and 708 00:35:54,340 --> 00:35:57,160 we're forming bonds, then what we have to do is have 709 00:35:57,160 --> 00:36:00,620 constructive interference of 2 different electrons, right, to 710 00:36:00,620 --> 00:36:04,380 form a bond, we want to and together those probabilities. 711 00:36:04,380 --> 00:36:07,110 So we want to have constructive interference to 712 00:36:07,110 --> 00:36:10,530 form a bond, whereas if we had destructive interference, we 713 00:36:10,530 --> 00:36:12,240 would not be forming a bond. 714 00:36:12,240 --> 00:36:14,590 So that's where you have to think about whether it's 715 00:36:14,590 --> 00:36:15,580 positive or negative. 716 00:36:15,580 --> 00:36:17,540 You don't have to think about it right now, but you might 717 00:36:17,540 --> 00:36:19,580 have heard in high school talking about p orbitals, the 718 00:36:19,580 --> 00:36:23,420 phase, sometimes you mark a p orbital as being a plus sign 719 00:36:23,420 --> 00:36:24,200 or negative sign. 720 00:36:24,200 --> 00:36:27,980 Did any of you do that in high school at all? 721 00:36:27,980 --> 00:36:28,950 A little bit, yeah. 722 00:36:28,950 --> 00:36:31,900 So, that's having to do with the actual wave function. 723 00:36:31,900 --> 00:36:34,130 So, that'll become more relevant later, bonding 724 00:36:34,130 --> 00:36:36,840 actually, a couple lectures down the road. 725 00:36:36,840 --> 00:36:39,180 But I just want to introduce it here while we do, in fact, 726 00:36:39,180 --> 00:36:42,630 have the wave function plots up here. 727 00:36:42,630 --> 00:36:45,100 But a real key in looking at these plots is where we, in 728 00:36:45,100 --> 00:36:47,590 fact, did go through zer and have this 729 00:36:47,590 --> 00:36:49,520 zero probability density. 730 00:36:49,520 --> 00:36:54,300 We call that a node, and a node, more specifically, is 731 00:36:54,300 --> 00:36:58,790 any value of either r, the radius, or the two angles for 732 00:36:58,790 --> 00:37:01,460 which the wave function, and that also means the wave 733 00:37:01,460 --> 00:37:04,370 function squared or the probability density, is going 734 00:37:04,370 --> 00:37:06,910 to be equal to zero. 735 00:37:06,910 --> 00:37:10,310 So, we can see in our 1 s orbital, how 736 00:37:10,310 --> 00:37:12,910 many nodes do we have? 737 00:37:12,910 --> 00:37:13,970 There's no nodes, yeah. 738 00:37:13,970 --> 00:37:16,500 It looks like we hit zero, but we actually don't -- remember 739 00:37:16,500 --> 00:37:18,720 that we never go all the way to zero, so there's these 740 00:37:18,720 --> 00:37:21,460 little points if we were to look really carefully at an 741 00:37:21,460 --> 00:37:25,080 accurate probability density plot, it would never 742 00:37:25,080 --> 00:37:26,540 actually hit zero. 743 00:37:26,540 --> 00:37:29,780 And then, for example, how many nodes do we have in the 3 744 00:37:29,780 --> 00:37:31,810 s orbital? two. 745 00:37:31,810 --> 00:37:32,890 That's correct. 746 00:37:32,890 --> 00:37:35,740 So we have two nodes in the 3 s orbital. 747 00:37:35,740 --> 00:37:39,610 We can actually specify where those nodes are, which is 748 00:37:39,610 --> 00:37:40,760 written on your notes. 749 00:37:40,760 --> 00:37:45,420 For the 2 s orbital, at 2 a nought, so it's just 2 times 750 00:37:45,420 --> 00:37:49,080 that constant a nought, which is the Bohr radius. 751 00:37:49,080 --> 00:37:51,040 And for the 3 s, we have one at 1 . 752 00:37:51,040 --> 00:37:53,560 9 a nought, and one at 7 . 753 00:37:53,560 --> 00:37:55,340 1 a nought. 754 00:37:55,340 --> 00:37:58,770 We can also specify what kind of node we're talking about. 755 00:37:58,770 --> 00:38:02,570 We'll introduce in the next course angular nodes, but 756 00:38:02,570 --> 00:38:05,460 today we're just going to be talking about radial nodes, 757 00:38:05,460 --> 00:38:09,820 and a radial node is a value for r at which psi, and 758 00:38:09,820 --> 00:38:12,420 therefore, also the probability psi squared is 759 00:38:12,420 --> 00:38:14,460 going to be equal to zero. 760 00:38:14,460 --> 00:38:17,830 So, when we're talking about an s orbital, since there is 761 00:38:17,830 --> 00:38:20,890 no angular dependence, and it only depends on r, every 762 00:38:20,890 --> 00:38:23,420 single one of our nodes is actually going to specifically 763 00:38:23,420 --> 00:38:26,180 be a radial node, right, because these are, for 764 00:38:26,180 --> 00:38:29,620 example, this 2 a nought is a value of r, a value of the 765 00:38:29,620 --> 00:38:33,610 radius, no matter which way you go around at which there's 766 00:38:33,610 --> 00:38:36,170 going to be a node at which there is zero probability 767 00:38:36,170 --> 00:38:40,030 density of finding an electron there. 768 00:38:40,030 --> 00:38:42,260 So, it's very easy to calculate, however, the number 769 00:38:42,260 --> 00:38:44,660 of radial nodes, and this works not just for s orbitals, 770 00:38:44,660 --> 00:38:47,680 but also for p orbitals, or d orbitals, or whatever kind of 771 00:38:47,680 --> 00:38:49,910 work of orbitals you want to discuss. 772 00:38:49,910 --> 00:38:52,940 And that's just to take the principle quantum number and 773 00:38:52,940 --> 00:38:57,800 subtract it by 1, and then also subtract from that your l 774 00:38:57,800 --> 00:38:58,700 quantum number. 775 00:38:58,700 --> 00:39:03,030 So what you can do for a 1 s is just take 1 minus 1 and 776 00:39:03,030 --> 00:39:07,280 then l is equal to 0, so you have zero radial nodes. 777 00:39:07,280 --> 00:39:09,400 And that matches up with what we saw. 778 00:39:09,400 --> 00:39:14,370 If we try this for the 2 s, we have 2 minus 1 minus 0. 779 00:39:14,370 --> 00:39:17,420 So what we should expect to see is one radial node, and 780 00:39:17,420 --> 00:39:21,730 that is what we see here in the probability density plot. 781 00:39:21,730 --> 00:39:26,200 And then if we think about 3 s, we want to start with 3, we 782 00:39:26,200 --> 00:39:31,030 subtract 1, again l is equal to 0, so minus 0 and we have 783 00:39:31,030 --> 00:39:34,120 two radial nodes. 784 00:39:34,120 --> 00:39:36,220 So, this should be pretty straight forward, let's see if 785 00:39:36,220 --> 00:39:39,900 we can get close to a 100% on this one, which is how many 786 00:39:39,900 --> 00:39:49,440 radial nodes does a 4 p orbital have? 787 00:39:49,440 --> 00:39:51,310 And let's give 10 seconds on that, make 788 00:39:51,310 --> 00:40:04,710 you think fast here. 789 00:40:04,710 --> 00:40:08,910 OK, so most people were correct, or well, the 790 00:40:08,910 --> 00:40:11,630 majority, at least, were correct. 791 00:40:11,630 --> 00:40:15,450 And seeing that it's a 4 p has two nodes -- let's just write 792 00:40:15,450 --> 00:40:18,310 this out since not everyone did get it correct. 793 00:40:18,310 --> 00:40:21,810 So, if we're talking about a 4 p orbital, and our equation is 794 00:40:21,810 --> 00:40:28,030 n minus 1 minus l, the principle quantum number is 4, 795 00:40:28,030 --> 00:40:31,680 1 is 1 -- what is l for a p orbital? 796 00:40:31,680 --> 00:40:33,560 STUDENT: 1. 797 00:40:33,560 --> 00:40:34,190 PROFESSOR: 1. 798 00:40:34,190 --> 00:40:36,250 So, I tricked you a little, I guess I didn't put an s up 799 00:40:36,250 --> 00:40:37,910 there and that's what we had been talking about, so that 800 00:40:37,910 --> 00:40:39,220 was probably the issue. 801 00:40:39,220 --> 00:40:47,230 But what we find is that we have two radial nodes. 802 00:40:47,230 --> 00:40:47,500 All right. 803 00:40:47,500 --> 00:40:50,890 So we can switch back to our notes here. 804 00:40:50,890 --> 00:40:54,800 So, doing those probability density dot graphs, we can get 805 00:40:54,800 --> 00:40:57,760 an idea of the shape of those orbitals, we know that they're 806 00:40:57,760 --> 00:40:59,640 spherically symmetrical. 807 00:40:59,640 --> 00:41:02,140 We're not going to talk about p orbitals today, we're going 808 00:41:02,140 --> 00:41:04,970 to talk about p orbitals exclusively on Friday, and as 809 00:41:04,970 --> 00:41:07,710 I said, d orbitals you'll get to with Professor Drennen. 810 00:41:07,710 --> 00:41:12,490 But we can also think when we're talking about wave 811 00:41:12,490 --> 00:41:15,570 function squared, what we're really talking about is the 812 00:41:15,570 --> 00:41:16,940 probability density, right, the 813 00:41:16,940 --> 00:41:19,130 probability in some volume. 814 00:41:19,130 --> 00:41:21,850 But there's also a way to get rid of the volume part and 815 00:41:21,850 --> 00:41:24,443 actually talk about the probability of finding an 816 00:41:24,443 --> 00:41:30,710 electron at some certain area within the atom, and this is 817 00:41:30,710 --> 00:41:35,090 what we do using radial probability 818 00:41:35,090 --> 00:41:37,550 distribution graphs. 819 00:41:37,550 --> 00:41:40,360 And what that is the probability of finding an 820 00:41:40,360 --> 00:41:43,880 electron in some shell where we define the thickness as d 821 00:41:43,880 --> 00:41:47,540 r, some distance, r, from the nucleus. 822 00:41:47,540 --> 00:41:49,510 So, think about what we're saying here. 823 00:41:49,510 --> 00:41:52,740 We're saying the probability of finding an electron at some 824 00:41:52,740 --> 00:41:56,390 distance from the nucleus in some very thin shell that we 825 00:41:56,390 --> 00:41:58,300 describe by d r. 826 00:41:58,300 --> 00:42:00,700 So if you think of a shell, you can actually just think of 827 00:42:00,700 --> 00:42:02,870 an egg shell, that's probably the easiest way to think of 828 00:42:02,870 --> 00:42:05,460 it, where the yolk, if you really maybe make it a lot 829 00:42:05,460 --> 00:42:07,230 smaller might be the nucleus. 830 00:42:07,230 --> 00:42:09,790 And let's also make our egg perfectly symmetric and 831 00:42:09,790 --> 00:42:11,200 perfectly round. 832 00:42:11,200 --> 00:42:14,910 But still, when we're talking about the radial probability 833 00:42:14,910 --> 00:42:18,140 distribution, what we actually want to think about is what's 834 00:42:18,140 --> 00:42:22,150 the probability of finding the electron in that shell? 835 00:42:22,150 --> 00:42:24,690 Think of it as that egg shell part. 836 00:42:24,690 --> 00:42:28,320 So, we can do that by using this equation, which is for s 837 00:42:28,320 --> 00:42:30,970 orbitals where the radial probability distribution is 838 00:42:30,970 --> 00:42:34,540 going to be equal to 4 pi r squared times the wave 839 00:42:34,540 --> 00:42:36,900 function squared, d r. 840 00:42:36,900 --> 00:42:44,320 That should make sense to us, because when we talk about a 841 00:42:44,320 --> 00:42:52,260 wave function, we're talking about a probability divided by 842 00:42:52,260 --> 00:42:53,800 a volume, because we're talking about 843 00:42:53,800 --> 00:42:55,750 a probability density. 844 00:42:55,750 --> 00:43:00,240 So if we actually go ahead and multiply it by the volume of 845 00:43:00,240 --> 00:43:03,570 our shell, then we end up just with probability, which is 846 00:43:03,570 --> 00:43:06,520 kind of a nicer term to be thinking about here. 847 00:43:06,520 --> 00:43:09,070 So, of course, if we're talking about a perfectly 848 00:43:09,070 --> 00:43:12,600 spherical shell at some distance, thickness, d r, we 849 00:43:12,600 --> 00:43:16,430 talk about it as 4 pi r squared d r, so we just 850 00:43:16,430 --> 00:43:20,780 multiply that by the probability density. 851 00:43:20,780 --> 00:43:25,740 We can graph out what this is where we're graphing the 852 00:43:25,740 --> 00:43:31,410 radial probability density as a function of the radius. 853 00:43:31,410 --> 00:43:35,620 And what you see is that at zero, you start at zero. 854 00:43:35,620 --> 00:43:39,790 And so, the radial probability density at the nucleus is 855 00:43:39,790 --> 00:43:42,840 going to be zero, even though we know the probability 856 00:43:42,840 --> 00:43:46,390 density at the nucleus is very high, that's actually where is 857 00:43:46,390 --> 00:43:49,180 the highest. The reason in our radial probability 858 00:43:49,180 --> 00:43:56,060 distributions we start -- the reason, if you look at the 859 00:43:56,060 --> 00:43:58,880 zero point on the radius that we start at zero is because 860 00:43:58,880 --> 00:44:02,810 we're multiplying the probability density by some 861 00:44:02,810 --> 00:44:05,560 volume, and when we're not anywhere from the nucleus, 862 00:44:05,560 --> 00:44:07,620 that volume is defined as zero. 863 00:44:07,620 --> 00:44:09,450 So, it's a little bit artificial that we're seeing 864 00:44:09,450 --> 00:44:11,020 that zero point there. 865 00:44:11,020 --> 00:44:13,980 So, actually I want you to go ahead in your notes and circle 866 00:44:13,980 --> 00:44:17,820 that zero point and write "not a node." This is not a node 867 00:44:17,820 --> 00:44:19,660 because a node is where we actually have 868 00:44:19,660 --> 00:44:21,410 no probability density. 869 00:44:21,410 --> 00:44:24,370 So this, where we start at zero is not a node, is the 870 00:44:24,370 --> 00:44:27,060 first thing to point out. 871 00:44:27,060 --> 00:44:30,370 And as we get further and further from the radius, the 872 00:44:30,370 --> 00:44:32,930 volume we're multiplying it by actually gets bigger and 873 00:44:32,930 --> 00:44:35,480 bigger, because you can see how the volume of that little 874 00:44:35,480 --> 00:44:38,100 thin shell is going to get larger and larger as you get 875 00:44:38,100 --> 00:44:39,180 further away. 876 00:44:39,180 --> 00:44:43,470 So there's some distance where the probability of actually 877 00:44:43,470 --> 00:44:45,800 finding an electron there is going to be your maximum 878 00:44:45,800 --> 00:44:46,830 probability. 879 00:44:46,830 --> 00:44:50,020 And that's what we label as r sub m p, or your 880 00:44:50,020 --> 00:44:52,970 most probable radius. 881 00:44:52,970 --> 00:44:55,900 This is the point at which your probability is highest 882 00:44:55,900 --> 00:44:57,470 for finding an electron. 883 00:44:57,470 --> 00:45:01,870 This is equal to a sub nought for a hydrogen atom, and we 884 00:45:01,870 --> 00:45:05,260 remember that that's just our Bohr radius, which is 0 . 885 00:45:05,260 --> 00:45:11,450 5 2 9 angstroms. And basically, what that means is 886 00:45:11,450 --> 00:45:15,390 you can actually find an electron anywhere going away 887 00:45:15,390 --> 00:45:18,090 from the nucleus, but you're most likely to find that you 888 00:45:18,090 --> 00:45:21,360 have the highest probability at a distance of a sub nought, 889 00:45:21,360 --> 00:45:23,970 or the Bohr radius. 890 00:45:23,970 --> 00:45:25,920 So, I said I'd tell you a little bit more about where 891 00:45:25,920 --> 00:45:30,010 this Bohr radius came from, and it came from a model of 892 00:45:30,010 --> 00:45:34,320 the atom that pre-dated quantum mechanics, and Neils 893 00:45:34,320 --> 00:45:38,230 Bohr is who came up with the idea of the Bohr radius, and 894 00:45:38,230 --> 00:45:41,870 here is hanging out with Einstein, so he had some 895 00:45:41,870 --> 00:45:44,980 pretty good company that he kept. 896 00:45:44,980 --> 00:45:48,060 And what you need to remember when we're thinking about this 897 00:45:48,060 --> 00:45:51,450 model of the atom is that in 1911 it had already been 898 00:45:51,450 --> 00:45:53,880 discovered that we have an electron, and we have a 899 00:45:53,880 --> 00:45:56,750 nucleus, and there needs to be some way that those two hang 900 00:45:56,750 --> 00:45:59,960 together, but it was not for another 15 years that we 901 00:45:59,960 --> 00:46:02,970 actually had the Schrodinger equation that allowed us to 902 00:46:02,970 --> 00:46:07,130 understand the interaction fully between the electron and 903 00:46:07,130 --> 00:46:07,630 the nucleus. 904 00:46:07,630 --> 00:46:10,900 So all that Bohr, for example, had to go on at this point was 905 00:46:10,900 --> 00:46:14,560 a more classical picture of the atom, as you can see on 906 00:46:14,560 --> 00:46:17,840 the left side of the screen there, which is the idea that 907 00:46:17,840 --> 00:46:20,680 the electrons actually somehow just orbiting the nucleus. 908 00:46:20,680 --> 00:46:23,440 And even though he could figure out that this wasn't 909 00:46:23,440 --> 00:46:26,840 possible, he still used this as a starting point, and what 910 00:46:26,840 --> 00:46:31,170 he did know was that these energy levels that were within 911 00:46:31,170 --> 00:46:34,270 hydrogen atom were quantized. and he knew this the same way 912 00:46:34,270 --> 00:46:37,230 that we saw it in the last class, which is when we viewed 913 00:46:37,230 --> 00:46:40,200 the difference spectra coming out from the hydrogen, and we 914 00:46:40,200 --> 00:46:43,130 also did it for neon, but we saw in the hydrogen atom that 915 00:46:43,130 --> 00:46:46,130 it was very discreet energy levels that we could observe. 916 00:46:46,130 --> 00:46:47,950 He knew the same thing that had been 917 00:46:47,950 --> 00:46:49,440 observed by that point. 918 00:46:49,440 --> 00:46:53,490 So, what he did was kind of impose a quantum mechanical 919 00:46:53,490 --> 00:46:56,270 model, not a full one, just the idea that those energy 920 00:46:56,270 --> 00:47:00,220 levels were quantized on to the classical picture of an 921 00:47:00,220 --> 00:47:02,890 atom that has a discreet orbit. 922 00:47:02,890 --> 00:47:06,130 And what he came out with when he did some calculations is 923 00:47:06,130 --> 00:47:09,710 that there's the radius that he could calculate was equal 924 00:47:09,710 --> 00:47:13,190 to this number a sub nought, which is what we call the Bohr 925 00:47:13,190 --> 00:47:17,550 radius, and it turns out that the Bohr radius happens to be 926 00:47:17,550 --> 00:47:21,220 the radius most probable for a hydrogen atom. 927 00:47:21,220 --> 00:47:23,850 And the reason we won't talk any more about this Bohr model 928 00:47:23,850 --> 00:47:26,120 is because, of course, it's not correct. 929 00:47:26,120 --> 00:47:28,580 So we're not going to spend too much time on it here. 930 00:47:28,580 --> 00:47:31,860 But we can see, for example, one reason or one way in which 931 00:47:31,860 --> 00:47:33,010 is not correct. 932 00:47:33,010 --> 00:47:37,130 Because what it tells is that we can figure out exactly what 933 00:47:37,130 --> 00:47:41,110 the radius of an electron and a nucleus are 934 00:47:41,110 --> 00:47:42,550 in a hydrogen atom. 935 00:47:42,550 --> 00:47:45,100 That's a deterministic way of doing things, that's what you 936 00:47:45,100 --> 00:47:46,890 get from classical mechanics. 937 00:47:46,890 --> 00:47:49,840 But the reality that we know from our quantum mechanical 938 00:47:49,840 --> 00:47:53,050 model, is that we can't know exactly what the radius is, 939 00:47:53,050 --> 00:47:56,640 all we can say is what the probability is of the radius 940 00:47:56,640 --> 00:47:58,980 being at certain different points. so, that's a more 941 00:47:58,980 --> 00:48:01,620 complete quantum mechanical picture of 942 00:48:01,620 --> 00:48:03,360 what is going on here. 943 00:48:03,360 --> 00:48:05,800 So if we superimpose our radial probability 944 00:48:05,800 --> 00:48:09,290 distribution onto the Bohr radius, we see it's much more 945 00:48:09,290 --> 00:48:11,400 complicated than just having a discreet radius. 946 00:48:11,400 --> 00:48:14,760 We can actually have any radius, but some radii just 947 00:48:14,760 --> 00:48:17,710 have much, much smaller probabilities of actually 948 00:48:17,710 --> 00:48:20,890 being significant or not. 949 00:48:20,890 --> 00:48:24,050 So, I think we're a little bit out of time today, but we'll 950 00:48:24,050 --> 00:48:26,670 start next class with thinking about drawing radial 951 00:48:26,670 --> 00:48:28,840 probability distributions of more than 952 00:48:28,840 --> 00:48:31,270 just the 1 s orbital.