1 00:00:00,000 --> 00:00:00,016 The following content is provided under a Creative 2 00:00:00,016 --> 00:00:00,022 Commons license. 3 00:00:00,022 --> 00:00:00,038 Your support will help MIT OpenCourseWare continue to 4 00:00:00,038 --> 00:00:00,054 offer high quality educational resources for free. 5 00:00:00,054 --> 00:00:00,072 To make a donation or view additional materials from 6 00:00:00,072 --> 00:00:00,088 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:00,088 --> 00:00:00,110 ocw.mit.edu. 8 00:00:00,110 --> 00:00:23,020 PROFESSOR: All right. 9 00:00:23,020 --> 00:00:26,120 So, let's get started. 10 00:00:26,120 --> 00:00:28,920 Why doesn't everyone take 10 more seconds to answer this 11 00:00:28,920 --> 00:00:42,310 clicker question here. 12 00:00:42,310 --> 00:00:42,910 All right. 13 00:00:42,910 --> 00:00:43,570 So, good. 14 00:00:43,570 --> 00:00:46,620 It looks like just about everyone is able to go from 15 00:00:46,620 --> 00:00:50,000 the name of an orbital to the state function. 16 00:00:50,000 --> 00:00:51,300 That's important. 17 00:00:51,300 --> 00:00:53,470 And we're actually going to get a little bit deeper in our 18 00:00:53,470 --> 00:00:56,660 clicker questions here, since when you do your problem-set 19 00:00:56,660 --> 00:00:58,530 it won't be quite this straight forward that you'll 20 00:00:58,530 --> 00:01:00,320 be answering this kind of question, but actually you'll 21 00:01:00,320 --> 00:01:03,120 be thinking about how many different orbitals can have 22 00:01:03,120 --> 00:01:06,700 certain state functions or certain orbital names. 23 00:01:06,700 --> 00:01:08,660 So let's go to a second clicker question 24 00:01:08,660 --> 00:01:11,430 here and try one more. 25 00:01:11,430 --> 00:01:14,630 So why don't you tell me how many possible orbitals you can 26 00:01:14,630 --> 00:01:16,640 have in a single atom that have the 27 00:01:16,640 --> 00:01:17,960 following two quantum numbers? 28 00:01:17,960 --> 00:01:21,790 So let's say we have n equals 4, and n sub l equalling 29 00:01:21,790 --> 00:01:23,300 negative 2. 30 00:01:23,300 --> 00:01:26,230 How many different orbitals can you have that have those 31 00:01:26,230 --> 00:01:27,760 two quantum numbers in them? 32 00:01:27,760 --> 00:01:30,100 And this should look kind of familiar to some of the 33 00:01:30,100 --> 00:01:33,260 problems you may have seen on the problem-set if you started 34 00:01:33,260 --> 00:01:38,520 it this weekend. 35 00:01:38,520 --> 00:01:40,580 So this should be something you can do pretty quickly, so 36 00:01:40,580 --> 00:01:53,810 let's take 10 more seconds on that. 37 00:01:53,810 --> 00:01:55,010 All right. 38 00:01:55,010 --> 00:01:57,970 OK, looks like we got the majority, which is a good 39 00:01:57,970 --> 00:01:59,980 start, but we having discrepancy on 40 00:01:59,980 --> 00:02:01,510 what people are thinking. 41 00:02:01,510 --> 00:02:03,060 So, let's go through this one. 42 00:02:03,060 --> 00:02:10,070 So, what we're saying is that we have n equals to 4, and m 43 00:02:10,070 --> 00:02:12,960 sub I being equal to negative 2. 44 00:02:12,960 --> 00:02:16,000 If we have n equals 4, what is the highest value of 45 00:02:16,000 --> 00:02:17,780 l that we can have? 46 00:02:17,780 --> 00:02:18,350 STUDENT: 3. 47 00:02:18,350 --> 00:02:18,620 PROFESSOR: OK. 48 00:02:18,620 --> 00:02:22,260 We can have n 4, l 3, and then, sure, we can have m sub 49 00:02:22,260 --> 00:02:26,430 l equal negative 2 if l equals 3 What's the second value of l 50 00:02:26,430 --> 00:02:27,260 that we can have? 51 00:02:27,260 --> 00:02:28,670 2. 52 00:02:28,670 --> 00:02:29,140 OK. 53 00:02:29,140 --> 00:02:32,920 So we can have this orbital here. 54 00:02:32,920 --> 00:02:36,930 What about l equals 1, can we have this? 55 00:02:36,930 --> 00:02:37,870 No, we can't. 56 00:02:37,870 --> 00:02:42,110 Because if l equals 1, we can not have m sub l equal 57 00:02:42,110 --> 00:02:45,160 negative 2, right, because the magnetic quantum number only 58 00:02:45,160 --> 00:02:49,010 goes from negative l to positive l here. 59 00:02:49,010 --> 00:02:52,000 So that means it's not possible, if we've made these 60 00:02:52,000 --> 00:02:54,870 stipulations in the first place, to have a case 61 00:02:54,870 --> 00:02:56,450 where l equals 1. 62 00:02:56,450 --> 00:02:59,220 So this means we can only have 2 different values of l. 63 00:02:59,220 --> 00:03:02,270 We already know our value of m. 64 00:03:02,270 --> 00:03:05,730 So now we're just counting up our orbitals, an orbital is 65 00:03:05,730 --> 00:03:08,650 completely described by the 3 quantum numbers. 66 00:03:08,650 --> 00:03:12,670 So we end up having 2 orbitals here. 67 00:03:12,670 --> 00:03:18,410 All right, so hopefully if you see any other combination of 68 00:03:18,410 --> 00:03:21,340 quantum numbers, for example, if it doesn't quickly come to 69 00:03:21,340 --> 00:03:23,710 you how many orbitals you have, you can actually try to 70 00:03:23,710 --> 00:03:26,760 write out all the possible orbitals and that should get 71 00:03:26,760 --> 00:03:28,670 you started. 72 00:03:28,670 --> 00:03:29,130 All right. 73 00:03:29,130 --> 00:03:32,630 So today we're going to finish up our discussion of the 74 00:03:32,630 --> 00:03:34,120 hydrogen atom. 75 00:03:34,120 --> 00:03:37,450 We'd started on Monday talking about radial probability 76 00:03:37,450 --> 00:03:39,490 distributions for the s orbitals. 77 00:03:39,490 --> 00:03:42,910 We'll finish that up, and then we're going to move on to 78 00:03:42,910 --> 00:03:44,650 talking about the p orbitals. 79 00:03:44,650 --> 00:03:47,610 We'll start with talking about the shape, just like we did 80 00:03:47,610 --> 00:03:50,440 with the s orbitals, and then move on to those radial 81 00:03:50,440 --> 00:03:53,600 probability distributions and compare the radial probability 82 00:03:53,600 --> 00:03:58,320 at different radius for p orbital versus an s orbital. 83 00:03:58,320 --> 00:04:01,480 And once we do that, we're actually going to move on to 84 00:04:01,480 --> 00:04:03,660 multi-electron atoms. 85 00:04:03,660 --> 00:04:07,400 So, you might have noticed that we will have spent about 86 00:04:07,400 --> 00:04:09,750 6 and 1/2 lectures just getting to the point where we 87 00:04:09,750 --> 00:04:13,590 have only one electron, so we're only up hydrogen so far. 88 00:04:13,590 --> 00:04:15,820 And you might have kind of been projecting ahead and 89 00:04:15,820 --> 00:04:18,830 thinking if we keep up at this pace, pretty much we would 90 00:04:18,830 --> 00:04:22,080 only get to carbon by the end of the semester. 91 00:04:22,080 --> 00:04:24,600 So I will assure you that we will not be spending 6 92 00:04:24,600 --> 00:04:28,690 lectures per atom as we move forward, and in fact, what 93 00:04:28,690 --> 00:04:31,840 we're going to find is that by taking all the principles 94 00:04:31,840 --> 00:04:34,600 we've learned about the hydrogen atom and applying 95 00:04:34,600 --> 00:04:38,820 that to multi-electron atoms, but making a few of changes 96 00:04:38,820 --> 00:04:41,420 and making a few modifications to take into the fact that 97 00:04:41,420 --> 00:04:43,410 we're going to have electron-electron repulsions 98 00:04:43,410 --> 00:04:46,140 going on, we'll be able to think about any multi-electron 99 00:04:46,140 --> 00:04:50,280 atom using these same general ideas, and the Schrodinger 100 00:04:50,280 --> 00:04:52,910 equation ideas that we came up with and have looked at for 101 00:04:52,910 --> 00:04:53,440 the hydrogen atom. 102 00:04:53,440 --> 00:04:56,600 And this is really good news because it's good to get 103 00:04:56,600 --> 00:04:57,350 passed carbon. 104 00:04:57,350 --> 00:05:00,010 I'm an organic chemist, so I love carbon, it's one of my 105 00:05:00,010 --> 00:05:02,390 favorite atoms to talk about, but it would be nice to get to 106 00:05:02,390 --> 00:05:05,400 the point of bonding and even reactions to talk about all 107 00:05:05,400 --> 00:05:07,570 the exciting things we can think about once 108 00:05:07,570 --> 00:05:08,280 we're at that point. 109 00:05:08,280 --> 00:05:11,050 So, one we finish our discussion of how we think 110 00:05:11,050 --> 00:05:16,210 about multi-electron atoms, we can go right on and start 111 00:05:16,210 --> 00:05:18,780 talking about these other things. 112 00:05:18,780 --> 00:05:24,400 All right. 113 00:05:24,400 --> 00:05:27,400 So, let's pick up where we left off, first of all we're 114 00:05:27,400 --> 00:05:30,470 still on the hydrogen atom from Monday. 115 00:05:30,470 --> 00:05:33,260 And on Monday what we were discussing was the solution to 116 00:05:33,260 --> 00:05:36,880 the Schrodinger equation for the wave function. 117 00:05:36,880 --> 00:05:39,930 And we also, when we solved or we looked at the solution to 118 00:05:39,930 --> 00:05:43,290 that Schrodinger equation, what we saw was that we 119 00:05:43,290 --> 00:05:46,270 actually needed three different quantum numbers to 120 00:05:46,270 --> 00:05:48,990 fully describe the wave function of a hydrogen atom or 121 00:05:48,990 --> 00:05:51,550 to fully describe an orbital. 122 00:05:51,550 --> 00:05:54,480 We didn't just need that n, not just the principle quantum 123 00:05:54,480 --> 00:05:56,460 number that we needed to discuss the energy, but we 124 00:05:56,460 --> 00:05:59,750 also need to talk about l and m, as we did in our clicker 125 00:05:59,750 --> 00:06:01,470 question up here. 126 00:06:01,470 --> 00:06:04,200 We also talked about well, what is that when we say wave 127 00:06:04,200 --> 00:06:06,040 function, what does that actually mean? 128 00:06:06,040 --> 00:06:09,070 And first we discussed the fact that well, in terms of a 129 00:06:09,070 --> 00:06:11,470 classical analogy, we don't really have one for wave 130 00:06:11,470 --> 00:06:13,880 function, we can't really think of a way to picture wave 131 00:06:13,880 --> 00:06:17,240 function thinking in classical terms. But we do have an 132 00:06:17,240 --> 00:06:19,610 interpretation for wave function squared. 133 00:06:19,610 --> 00:06:22,280 And when we take the wave function and square it, that's 134 00:06:22,280 --> 00:06:24,940 going to be equal to the probability density of finding 135 00:06:24,940 --> 00:06:29,000 an electron at some point in your atom. 136 00:06:29,000 --> 00:06:33,000 And that's useful in terms of seeing a general shape, but if 137 00:06:33,000 --> 00:06:36,060 we're actually interested in thinking about how far away 138 00:06:36,060 --> 00:06:38,900 that electron is from the nucleus, you can see that 139 00:06:38,900 --> 00:06:41,480 instead of talking about probability density, which is 140 00:06:41,480 --> 00:06:44,590 the probability per volume, instead it would be much more 141 00:06:44,590 --> 00:06:47,400 useful to talk about something called radial probability 142 00:06:47,400 --> 00:06:50,460 distribution, or in other words talking about the 143 00:06:50,460 --> 00:06:54,030 probability of finding the electron at some distance, 144 00:06:54,030 --> 00:06:58,600 which we define as r, from the nucleus in a spherical shell 145 00:06:58,600 --> 00:07:02,670 that is just infinitesimally small, infinitesimally thin at 146 00:07:02,670 --> 00:07:06,760 distance or at a thickness that we'll call a d r here. 147 00:07:06,760 --> 00:07:10,340 So, basically what we're saying is if we take any shell 148 00:07:10,340 --> 00:07:13,770 that's at some distance away from the nucleus, we can think 149 00:07:13,770 --> 00:07:16,920 about what the probability is of finding an electron at that 150 00:07:16,920 --> 00:07:19,270 radius, and that's the definition we gave to the 151 00:07:19,270 --> 00:07:22,290 radial probability distribution. 152 00:07:22,290 --> 00:07:25,370 And we can look at the formula that got us here. 153 00:07:25,370 --> 00:07:29,210 This is the radial probability distribution formula for an s 154 00:07:29,210 --> 00:07:32,150 orbital, which is, of course, dealing with something that's 155 00:07:32,150 --> 00:07:33,650 spherically symmetrical. 156 00:07:33,650 --> 00:07:35,975 It's somewhat different when we're talking about the p or 157 00:07:35,975 --> 00:07:38,170 the d orbitals, and we won't go into the equation there, 158 00:07:38,170 --> 00:07:40,340 but this will give you an idea of what we're really talking 159 00:07:40,340 --> 00:07:43,070 about with this radial probability distribution. 160 00:07:43,070 --> 00:07:45,970 So, what we can do to actually get a probability instead of a 161 00:07:45,970 --> 00:07:49,220 probability density that we're talking about is to take the 162 00:07:49,220 --> 00:07:52,520 wave function squared, which we know is probability 163 00:07:52,520 --> 00:07:57,380 density, and multiply it by the volume of that very, very 164 00:07:57,380 --> 00:07:59,840 thin spherical shell that we're talking about at 165 00:07:59,840 --> 00:08:01,050 distance r. 166 00:08:01,050 --> 00:08:03,480 So if we want to talk about the volume of that, we just 167 00:08:03,480 --> 00:08:07,850 talk about the surface area, which is 4 pi r squared, and 168 00:08:07,850 --> 00:08:10,540 we multiply that by the thickness d r. 169 00:08:10,540 --> 00:08:14,020 So if we take this term, which is a volume term, and multiply 170 00:08:14,020 --> 00:08:17,310 it by probability over volume, what we're going to end up 171 00:08:17,310 --> 00:08:21,060 with is an actual probability of finding our electron at 172 00:08:21,060 --> 00:08:24,800 that distance, r, from the nucleus. 173 00:08:24,800 --> 00:08:27,890 So, the example that we took on Monday and that we ended 174 00:08:27,890 --> 00:08:31,760 with when we ended class, was looking at the 1 s orbital for 175 00:08:31,760 --> 00:08:35,020 hydrogen atom, and what we could do is we could graph the 176 00:08:35,020 --> 00:08:39,170 radial probability as a function of radius here. 177 00:08:39,170 --> 00:08:42,890 And when we do that we can see this curve, this probability 178 00:08:42,890 --> 00:08:46,940 curve, where we have a maximum probability of finding the 179 00:08:46,940 --> 00:08:50,000 electron this far away from the nucleus. 180 00:08:50,000 --> 00:08:54,120 And we call that most probable radius r sub m p, or most 181 00:08:54,120 --> 00:08:55,420 probable radius. 182 00:08:55,420 --> 00:08:57,920 And what is discussed is that for a 1 s hydrogen atom, that 183 00:08:57,920 --> 00:09:01,740 falls at an a nought distance away from the nucleus. 184 00:09:01,740 --> 00:09:05,820 And remember, a nought, that's just the Bohr radius, it's a 185 00:09:05,820 --> 00:09:07,650 constant -- that's all we need to worry about. 186 00:09:07,650 --> 00:09:10,300 We talked about the Bohr model and how that 187 00:09:10,300 --> 00:09:12,600 told us an exact distance. 188 00:09:12,600 --> 00:09:15,740 It was a classical model, right, so we could say the 189 00:09:15,740 --> 00:09:18,290 electron is exactly this far away from the nucleus. 190 00:09:18,290 --> 00:09:20,380 We can not do that with quantum mechanics, the more 191 00:09:20,380 --> 00:09:23,590 true picture is the best we can get to is talk about what 192 00:09:23,590 --> 00:09:26,270 the probability is of finding the electron 193 00:09:26,270 --> 00:09:27,420 at any given nucleus. 194 00:09:27,420 --> 00:09:31,620 And the most probable one here is that a nought. 195 00:09:31,620 --> 00:09:34,130 The other thing that we looked at, which I want to stress 196 00:09:34,130 --> 00:09:36,780 again and I'll stress it as many times as I can fit it 197 00:09:36,780 --> 00:09:39,960 into lecture, because this is something that confuses 198 00:09:39,960 --> 00:09:42,170 students when they're trying to identify, for example, 199 00:09:42,170 --> 00:09:44,910 different nodes or areas of no probability. 200 00:09:44,910 --> 00:09:49,380 In an orbital is remember that this area right here at r 201 00:09:49,380 --> 00:09:51,710 equals zerio, that is not a node. 202 00:09:51,710 --> 00:09:54,750 We will always have r equals zero in these radial 203 00:09:54,750 --> 00:09:57,950 probability distribution graphs, and we can think about 204 00:09:57,950 --> 00:09:58,570 why that is. 205 00:09:58,570 --> 00:10:00,730 At first it might be counter-intuitive because we 206 00:10:00,730 --> 00:10:03,850 know the probability density at the nucleus is the 207 00:10:03,850 --> 00:10:07,010 greatest. So the probability of having an electron at the 208 00:10:07,010 --> 00:10:09,490 nucleus in terms of probability per volume is 209 00:10:09,490 --> 00:10:10,820 very, very high. 210 00:10:10,820 --> 00:10:13,480 But remember that we need to multiply it by the volume 211 00:10:13,480 --> 00:10:16,350 here, the volume of some sphere we've defined. 212 00:10:16,350 --> 00:10:19,220 And when we define that as r being equal to zero, 213 00:10:19,220 --> 00:10:20,560 essentially we're multiplying the 214 00:10:20,560 --> 00:10:22,700 probability density by zero. 215 00:10:22,700 --> 00:10:26,260 So that's why we have this zero point here, and just to 216 00:10:26,260 --> 00:10:29,270 point out again and again and again, it's not a radial node, 217 00:10:29,270 --> 00:10:32,280 it's just a point where we're starting our graph, because 218 00:10:32,280 --> 00:10:36,980 we're multiplying it by r equals zero. 219 00:10:36,980 --> 00:10:39,740 So, we can look at other radial probability 220 00:10:39,740 --> 00:10:41,890 distributions of other wave functions 221 00:10:41,890 --> 00:10:42,750 that we talked about. 222 00:10:42,750 --> 00:10:45,820 We talked about the wave function for a 2 s orbital, 223 00:10:45,820 --> 00:10:48,470 and also for a 3 s orbital. 224 00:10:48,470 --> 00:10:55,270 So, let's go ahead and think about drawing what that would 225 00:10:55,270 --> 00:10:57,610 look like in terms of the radial probability 226 00:10:57,610 --> 00:10:59,390 distribution. 227 00:10:59,390 --> 00:11:03,650 So what we're graphing here is the radius as a function of 228 00:11:03,650 --> 00:11:05,940 radial probability. 229 00:11:05,940 --> 00:11:11,280 And for a 2 s orbital, you get a graph that's going to look 230 00:11:11,280 --> 00:11:14,090 something like this. 231 00:11:14,090 --> 00:11:16,200 So, again, we're starting at zero. 232 00:11:16,200 --> 00:11:20,040 We have one node here, and we can again define that most 233 00:11:20,040 --> 00:11:22,160 probable radius. 234 00:11:22,160 --> 00:11:26,050 And it turns out that for a 2 s orbital, that's equal to 6 235 00:11:26,050 --> 00:11:29,350 times a nought. 236 00:11:29,350 --> 00:11:31,600 So when we think about what it is that this radial 237 00:11:31,600 --> 00:11:34,900 probability distribution is telling us, it's telling us 238 00:11:34,900 --> 00:11:39,400 that it is most likely that an electron in a 2 s orbital of 239 00:11:39,400 --> 00:11:43,960 hydrogen is six times further away from the nucleus than it 240 00:11:43,960 --> 00:11:46,710 is in a 1 s orbital. 241 00:11:46,710 --> 00:11:50,110 So another way to say that is, in a sense, if we're thinking 242 00:11:50,110 --> 00:11:52,620 about the excited state of a hydrogen atom, the first 243 00:11:52,620 --> 00:11:57,510 excited state, or the n equals 2 state, what we're saying is 244 00:11:57,510 --> 00:12:01,430 that it's actually bigger than the ground state, or the 1 s 245 00:12:01,430 --> 00:12:03,350 state of a hydrogen atom. 246 00:12:03,350 --> 00:12:05,770 And when we say bigger, remember this is not a 247 00:12:05,770 --> 00:12:08,240 classical description we're talking about. 248 00:12:08,240 --> 00:12:10,130 We are talking about probability, but what we're 249 00:12:10,130 --> 00:12:13,710 saying is that most probable radius is further away from 250 00:12:13,710 --> 00:12:16,280 the nucleus. 251 00:12:16,280 --> 00:12:20,020 So we can also look at this in terms of the 3 s orbital. 252 00:12:20,020 --> 00:12:26,940 And in this case, we have a graph that looks 253 00:12:26,940 --> 00:12:28,060 something like this. 254 00:12:28,060 --> 00:12:32,010 So you can draw that into your notes. 255 00:12:32,010 --> 00:12:36,420 And again, we can define what that most probable radius is, 256 00:12:36,420 --> 00:12:37,910 that distance at which we're most 257 00:12:37,910 --> 00:12:40,840 likely to find an electron. 258 00:12:40,840 --> 00:12:43,280 And in the case of the 3 s orbital, that's going to be 259 00:12:43,280 --> 00:12:44,940 equal to 11 . 260 00:12:44,940 --> 00:12:48,540 5 times a nought. 261 00:12:48,540 --> 00:12:51,800 So again, what we're saying here is that it is most likely 262 00:12:51,800 --> 00:12:55,580 in the 3 s orbital that we would find the electron 11 and 263 00:12:55,580 --> 00:12:59,330 1/2 times further away from the nucleus than we would in a 264 00:12:59,330 --> 00:13:03,110 around state hydrogen atom. 265 00:13:03,110 --> 00:13:05,470 And I just want to point out here in terms of things that 266 00:13:05,470 --> 00:13:08,950 you're responsible for, you should know that the most 267 00:13:08,950 --> 00:13:11,530 probable radius for a 1 s hydrogen 268 00:13:11,530 --> 00:13:13,720 atom is equal a nought. 269 00:13:13,720 --> 00:13:18,040 And you should know that a 2 s is larger than that, and a 3 s 270 00:13:18,040 --> 00:13:21,200 is even larger, and of course, hopefully as we go to 4 and 5, 271 00:13:21,200 --> 00:13:23,380 you would be able to guess that those are going to get 272 00:13:23,380 --> 00:13:24,250 even larger. 273 00:13:24,250 --> 00:13:26,410 But you're not responsible for knowing specifically 274 00:13:26,410 --> 00:13:27,460 that it's 11 . 275 00:13:27,460 --> 00:13:28,510 5 times greater. 276 00:13:28,510 --> 00:13:33,270 You just need to know the trend there. 277 00:13:33,270 --> 00:13:36,990 Another thing to point out in these two graphs is that we do 278 00:13:36,990 --> 00:13:40,540 have nodes, and we figured out last time, we calculated how 279 00:13:40,540 --> 00:13:42,700 many nodes we should have in a 2 s orbital. 280 00:13:42,700 --> 00:13:46,410 And in terms of radial nodes, we expect to see one node. 281 00:13:46,410 --> 00:13:49,330 And how many nodes do you see in the 3 s orbital? two, good. 282 00:13:49,330 --> 00:13:53,690 I'm glad to hear that no one counted this r 283 00:13:53,690 --> 00:13:54,930 equal zero as a node. 284 00:13:54,930 --> 00:13:58,980 So we expect to see two nodes right here in the 3 s orbital. 285 00:13:58,980 --> 00:14:02,250 And we can calculate that with the formula that we used, 286 00:14:02,250 --> 00:14:09,920 which was just n minus l minus 1 equals the number of nodes. 287 00:14:09,920 --> 00:14:12,100 Or we could just look at the radial probability 288 00:14:12,100 --> 00:14:15,910 distribution itself and see how many nodes there are. 289 00:14:15,910 --> 00:14:19,430 So if we're looking at these two situations here and we're 290 00:14:19,430 --> 00:14:22,560 actually thinking of them from a more classical standpoint, 291 00:14:22,560 --> 00:14:25,220 which is natural for us to do because we live our lives in 292 00:14:25,220 --> 00:14:27,740 the every day world, not thinking about things on the 293 00:14:27,740 --> 00:14:30,320 atomic size scale all the time, most of us. 294 00:14:30,320 --> 00:14:35,070 So, for example, if we were to look at this 3 s orbital here, 295 00:14:35,070 --> 00:14:38,710 you might have the question of how this can be, because we're 296 00:14:38,710 --> 00:14:42,490 saying that, for example, we have probability of having an 297 00:14:42,490 --> 00:14:45,450 electron here, an electron can also be way out 298 00:14:45,450 --> 00:14:46,950 at this radius here. 299 00:14:46,950 --> 00:14:49,260 But what we're saying is there's a node here, so that 300 00:14:49,260 --> 00:14:52,440 there's no probability of finding an electron between 301 00:14:52,440 --> 00:14:53,820 those two points. 302 00:14:53,820 --> 00:14:56,440 So you can think of it, if we were to just think of it as a 303 00:14:56,440 --> 00:14:59,230 straight line that we were going across, essentially what 304 00:14:59,230 --> 00:15:04,030 we're saying is that we're getting from point a to point 305 00:15:04,030 --> 00:15:07,260 c without ever getting through point b. 306 00:15:07,260 --> 00:15:10,540 So, that can be a little bit confusing for us to think 307 00:15:10,540 --> 00:15:13,830 about, and when it's a very good question you might, in 308 00:15:13,830 --> 00:15:17,610 fact, say well, maybe there's not zero probability here, 309 00:15:17,610 --> 00:15:20,600 maybe it's just this teeny, teeny, tiny number, and in 310 00:15:20,600 --> 00:15:23,580 fact, sometimes an electron can get through, it's just 311 00:15:23,580 --> 00:15:27,020 very low probability so that's why we never really see it. 312 00:15:27,020 --> 00:15:28,680 And in fact, that's not the answer. 313 00:15:28,680 --> 00:15:31,670 The answer is, in fact, there is zero, absolutely zero 314 00:15:31,670 --> 00:15:33,880 probability of finding a electron here. 315 00:15:33,880 --> 00:15:37,100 So basically we're saying yes, we can go from point a to 316 00:15:37,100 --> 00:15:39,990 point c without ever going through point b. 317 00:15:39,990 --> 00:15:42,180 That might seem confusing if you're thinking about 318 00:15:42,180 --> 00:15:45,430 particles, but remember we're talking about the wave-like 319 00:15:45,430 --> 00:15:47,000 nature of electrons. 320 00:15:47,000 --> 00:15:51,310 So, the quantum mechanical interpretation is that we can, 321 00:15:51,310 --> 00:15:55,820 in fact, have probability density here and probability 322 00:15:55,820 --> 00:16:00,730 density there, without having any probability of having the 323 00:16:00,730 --> 00:16:02,700 electron in the space between. 324 00:16:02,700 --> 00:16:05,230 And you can think about that if you think about a standing 325 00:16:05,230 --> 00:16:08,080 wave, for example, where you can have amplitude at many 326 00:16:08,080 --> 00:16:11,370 different values of x, so an amplitude at many different 327 00:16:11,370 --> 00:16:14,000 distances, but you also have areas where 328 00:16:14,000 --> 00:16:15,680 there is a 0 amplitude. 329 00:16:15,680 --> 00:16:18,880 So, remember this makes sense if you just think of it as a 330 00:16:18,880 --> 00:16:22,490 wave and forget the particle part of it for right now, 331 00:16:22,490 --> 00:16:25,780 because that would be very upsetting to think about and 332 00:16:25,780 --> 00:16:27,760 that's not, in fact, what's going on, we're talking about 333 00:16:27,760 --> 00:16:28,940 quantum mechanics here. 334 00:16:28,940 --> 00:16:29,130 Yes. 335 00:16:29,130 --> 00:16:36,300 STUDENT: [INAUDIBLE] 336 00:16:36,300 --> 00:16:37,230 PROFESSOR: Oh, I'm sorry. 337 00:16:37,230 --> 00:16:40,030 So it's n minus l minus 1. 338 00:16:40,030 --> 00:16:46,000 So here we have 3 minus l equals 0, because it's an s 339 00:16:46,000 --> 00:16:57,160 orbital, minus 1, so we have two radial nodes here. 340 00:16:57,160 --> 00:16:57,560 OK. 341 00:16:57,560 --> 00:17:02,990 So let's actually go to a clicker question now on radial 342 00:17:02,990 --> 00:17:04,960 probability distributions. 343 00:17:04,960 --> 00:17:08,860 So I mentioned you should be able to identify both how many 344 00:17:08,860 --> 00:17:11,550 nodes you have and what a graph might look like of 345 00:17:11,550 --> 00:17:13,940 different radial probability distributions. 346 00:17:13,940 --> 00:17:17,380 So here, what I'd like you to do is identify the correct 347 00:17:17,380 --> 00:17:21,100 radial probability distribution plot for a 5 s 348 00:17:21,100 --> 00:17:23,890 orbital, and also make sure that it matches up with the 349 00:17:23,890 --> 00:17:33,600 right number of radial nodes that you would expect. 350 00:17:33,600 --> 00:17:33,860 All right. 351 00:17:33,860 --> 00:17:36,820 Let's take 10 more seconds on that, this should be a quick 352 00:17:36,820 --> 00:17:47,680 identification for us to do. 353 00:17:47,680 --> 00:17:48,070 All right. 354 00:17:48,070 --> 00:17:51,910 So it looks like 82% got the correct answer here. 355 00:17:51,910 --> 00:17:55,790 So, you should know that there's four radial nodes, 356 00:17:55,790 --> 00:18:00,430 right, we have 5 minus 1 minus l -- is there a question? 357 00:18:00,430 --> 00:18:03,360 STUDENT: [INAUDIBLE] 358 00:18:03,360 --> 00:18:06,420 PROFESSOR: It is very difficult for me to draw 359 00:18:06,420 --> 00:18:07,750 graphs on the computer. 360 00:18:07,750 --> 00:18:09,340 That's a good point, I'm sorry. 361 00:18:09,340 --> 00:18:12,110 This was my best attempt at hitting zero and not having 362 00:18:12,110 --> 00:18:13,580 the graph go down there. 363 00:18:13,580 --> 00:18:16,940 I'm not the most gifted at drawing on the computer. 364 00:18:16,940 --> 00:18:20,590 So yes, it should be zero at zero, but I made 365 00:18:20,590 --> 00:18:22,330 the line too thick. 366 00:18:22,330 --> 00:18:24,680 So, assuming -- if anyone got it wrong because of that, 367 00:18:24,680 --> 00:18:26,240 that's my apologies, that's my fault. 368 00:18:26,240 --> 00:18:29,410 But you should see that there are four radial nodes here 369 00:18:29,410 --> 00:18:33,290 since we have a 5 s orbital. 370 00:18:33,290 --> 00:18:38,160 And also that we know that the zero does not count as a node, 371 00:18:38,160 --> 00:18:41,170 if per se I actually had managed to hit zero in drawing 372 00:18:41,170 --> 00:18:46,030 that, so the correct answer would be the bottom one there. 373 00:18:46,030 --> 00:18:49,470 So, you should be able to generally identify and draw 374 00:18:49,470 --> 00:18:51,670 the general form of these radial probability 375 00:18:51,670 --> 00:18:52,730 distributions. 376 00:18:52,730 --> 00:18:56,010 Obviously we don't expect you to know exactly what the 377 00:18:56,010 --> 00:18:58,560 distances are, but you should be able to compare them 378 00:18:58,560 --> 00:19:00,010 relatively. 379 00:19:00,010 --> 00:19:00,350 Yes? 380 00:19:00,350 --> 00:19:06,660 STUDENT: [INAUDIBLE] 381 00:19:06,660 --> 00:19:10,260 PROFESSOR: No, they actually don't, and when you 382 00:19:10,260 --> 00:19:10,940 graph it all out. 383 00:19:10,940 --> 00:19:12,220 You can see this if you look at some examples 384 00:19:12,220 --> 00:19:13,410 in your book, actually. 385 00:19:13,410 --> 00:19:16,390 So this doesn't fall, for example, at 6 a nought, but 386 00:19:16,390 --> 00:19:18,010 that's a really good question. 387 00:19:18,010 --> 00:19:21,480 And the trend always is that the probability gets smaller 388 00:19:21,480 --> 00:19:26,490 with each of the peaks as you're drawing them. 389 00:19:26,490 --> 00:19:26,760 All right. 390 00:19:26,760 --> 00:19:29,380 So we can switch back to our notes. 391 00:19:29,380 --> 00:19:33,590 So we got our clicker question set there. 392 00:19:33,590 --> 00:19:36,390 And so now we can move on to thinking about p orbitals, we 393 00:19:36,390 --> 00:19:38,380 now have two ways to talk about p orbitals. 394 00:19:38,380 --> 00:19:40,520 We can talk about the wave function squared, the 395 00:19:40,520 --> 00:19:43,850 probability density, or we can talk about the radial 396 00:19:43,850 --> 00:19:46,990 probability distribution. 397 00:19:46,990 --> 00:19:50,070 So when we talk about p orbitals, it's similar to 398 00:19:50,070 --> 00:19:53,720 talking about s orbitals, and the difference lies, and now 399 00:19:53,720 --> 00:19:57,470 we have a different value for l, so l equals 1 for a p 400 00:19:57,470 --> 00:20:00,490 orbital, and we know if we have l equal 1, we can have 401 00:20:00,490 --> 00:20:05,240 three different total orbitals that have sub-shell of l 402 00:20:05,240 --> 00:20:06,650 equalling 1. 403 00:20:06,650 --> 00:20:11,300 So we can have, if we have the final quantum number m equal 404 00:20:11,300 --> 00:20:13,710 plus 1 or minus 1, we're dealing with a 405 00:20:13,710 --> 00:20:16,420 p x or a p y orbital. 406 00:20:16,420 --> 00:20:18,650 Remember, we don't do a one-to-one correlation, 407 00:20:18,650 --> 00:20:22,930 because p x and p y are some linear combination of the m 408 00:20:22,930 --> 00:20:25,550 plus 1 and m minus 1 orbital. 409 00:20:25,550 --> 00:20:28,040 And if we talk about m equals 0, we're 410 00:20:28,040 --> 00:20:29,650 looking at the p z orbital. 411 00:20:29,650 --> 00:20:36,810 And the significant difference between s orbitals and p 412 00:20:36,810 --> 00:20:40,190 orbitals that comes from the fact that we do have angular 413 00:20:40,190 --> 00:20:43,350 momentum here in these p orbitals, is that p orbital 414 00:20:43,350 --> 00:20:47,130 wave functions do, in fact, have theta and phi dependence. 415 00:20:47,130 --> 00:20:49,320 So they do have an angular dependence that 416 00:20:49,320 --> 00:20:52,280 we're talking about. 417 00:20:52,280 --> 00:20:54,430 And what I'm showing here is not on your notes, if you're 418 00:20:54,430 --> 00:20:56,420 interested you can look it up in your book. 419 00:20:56,420 --> 00:20:58,870 This is a table that's directly from your book, and 420 00:20:58,870 --> 00:21:01,580 what it's just showing is the wave function for a bunch of 421 00:21:01,580 --> 00:21:02,030 different orbitals. 422 00:21:02,030 --> 00:21:04,480 I mentioned last time that there was this 423 00:21:04,480 --> 00:21:05,840 list in your book. 424 00:21:05,840 --> 00:21:09,170 And what I want to point out here is this angular 425 00:21:09,170 --> 00:21:13,020 dependence for the p orbitals for the l equals 1 orbital. 426 00:21:13,020 --> 00:21:17,880 So, first, if I point out when l equals 0, when we have an s 427 00:21:17,880 --> 00:21:21,410 orbital, what you see is that angular part of the wave 428 00:21:21,410 --> 00:21:23,600 function is equal to a constant. 429 00:21:23,600 --> 00:21:26,250 So, remember we can break up the total wave function into 430 00:21:26,250 --> 00:21:30,470 the radial part and the angular part. 431 00:21:30,470 --> 00:21:31,860 When we look at this angular part, we see that it's always 432 00:21:31,860 --> 00:21:35,280 the square root of 1 over 4 pi, it doesn't matter what the 433 00:21:35,280 --> 00:21:39,550 angle is, it's not dependent on the angle. 434 00:21:39,550 --> 00:21:43,410 In contrast when we're looking at a p orbital, so any time l 435 00:21:43,410 --> 00:21:46,960 is equal to 1, and you look at angular part of the wave 436 00:21:46,960 --> 00:21:49,750 function here, what you see is the wave function either 437 00:21:49,750 --> 00:21:52,450 depends on theta or is dependent on 438 00:21:52,450 --> 00:21:54,370 both theta and phi. 439 00:21:54,370 --> 00:21:57,210 So we do, in fact, have a dependence on what the angle 440 00:21:57,210 --> 00:22:01,890 is of the electron as we define it in the orbital. 441 00:22:01,890 --> 00:22:06,480 So what this means is that unlike s orbitals, p orbitals 442 00:22:06,480 --> 00:22:09,690 are not spherically symmetrical -- they don't have 443 00:22:09,690 --> 00:22:13,770 the exact same shape at any radius from the nucleus. 444 00:22:13,770 --> 00:22:18,330 And these shapes of p orbitals probably do look familiar to 445 00:22:18,330 --> 00:22:20,680 you, most of you, I'm sure, have seen some sort of picture 446 00:22:20,680 --> 00:22:22,670 of p orbitals before. 447 00:22:22,670 --> 00:22:25,270 So what I want to point out about them is that they're 448 00:22:25,270 --> 00:22:28,790 made up of two nodes, and what you can see is that nodes are 449 00:22:28,790 --> 00:22:30,650 shown in different colors here and those 450 00:22:30,650 --> 00:22:32,220 are different phases. 451 00:22:32,220 --> 00:22:35,750 Sometimes you see this written when you see p orbitals, one 452 00:22:35,750 --> 00:22:38,210 is written as plus, one is written in minus. 453 00:22:38,210 --> 00:22:40,430 That's not a positive and negative charge, that's 454 00:22:40,430 --> 00:22:43,760 actually a phase, and that arises from the wave equation. 455 00:22:43,760 --> 00:22:46,190 Remember when we have waves we can have positive or a 456 00:22:46,190 --> 00:22:46,910 negative amplitude. 457 00:22:46,910 --> 00:22:50,690 When we talk about p orbitals the phase of the orbital 458 00:22:50,690 --> 00:22:54,170 becomes important once we talk about bonding, which hopefully 459 00:22:54,170 --> 00:22:55,920 you were happy to hear at the beginning of class 460 00:22:55,920 --> 00:22:57,200 we will get to soon. 461 00:22:57,200 --> 00:23:00,410 And it turns out that when you constructively have two p 462 00:23:00,410 --> 00:23:03,700 orbitals interfere, and when I say constructively, I mean 463 00:23:03,700 --> 00:23:06,140 they're both either positive or they're both the negative 464 00:23:06,140 --> 00:23:08,250 lobes, that's when you got bonding. 465 00:23:08,250 --> 00:23:12,170 Whereas if the phases where mismatched, you 466 00:23:12,170 --> 00:23:13,760 would not get bonding. 467 00:23:13,760 --> 00:23:16,380 So, that's going to be important later when we get to 468 00:23:16,380 --> 00:23:18,980 bonding, but just take note of it now, we have two nodes, 469 00:23:18,980 --> 00:23:24,490 each with a separate phase -- or we have two lobes, excuse 470 00:23:24,490 --> 00:23:26,020 me, each with a separate phase. 471 00:23:26,020 --> 00:23:28,760 And when we look at this, it's actually split by what's 472 00:23:28,760 --> 00:23:31,980 called a nodal plane, which is pointed out in light orange 473 00:23:31,980 --> 00:23:34,910 here on this picture, but what we just mean is that there is 474 00:23:34,910 --> 00:23:37,970 this whole plane that separates the two lobes where 475 00:23:37,970 --> 00:23:41,250 there is absolutely no electron density. 476 00:23:41,250 --> 00:23:43,790 So, the wave function at all of these points in this plane 477 00:23:43,790 --> 00:23:45,980 is equal to zero, so therefore, also the wave 478 00:23:45,980 --> 00:23:50,930 function squared is going to be equal to zero. 479 00:23:50,930 --> 00:23:56,020 So, if we say that in this entire plane we have zero 480 00:23:56,020 --> 00:23:58,810 probability of finding a p electron anywhere in the 481 00:23:58,810 --> 00:24:02,420 plane, the plane goes directly through the nucleus in every 482 00:24:02,420 --> 00:24:05,660 case but a p orbital, so what we can also say is that there 483 00:24:05,660 --> 00:24:13,850 is zero probability of finding a p electron at the nucleus. 484 00:24:13,850 --> 00:24:17,320 So, again we can use these probability density plots, 485 00:24:17,320 --> 00:24:23,160 which are just a plot of psi squared, where the density of 486 00:24:23,160 --> 00:24:26,630 the dots is proportional to the density, the probability 487 00:24:26,630 --> 00:24:28,730 density, at that point. 488 00:24:28,730 --> 00:24:31,850 So what we can say is look at each of these separately, so 489 00:24:31,850 --> 00:24:36,430 if we start with looking at the 2 p z orbital, the highest 490 00:24:36,430 --> 00:24:40,000 probability of finding an electron in the 2 p z orbital, 491 00:24:40,000 --> 00:24:42,180 is going to be along this z-axis. 492 00:24:42,180 --> 00:24:44,770 We can see that right here. 493 00:24:44,770 --> 00:24:48,630 And in terms of thinking about the phase of this p orbital, 494 00:24:48,630 --> 00:24:50,520 the phase is going to be positive 495 00:24:50,520 --> 00:24:52,250 anywhere where z is positive. 496 00:24:52,250 --> 00:24:55,780 So we would say we have a positive phase here and a 497 00:24:55,780 --> 00:24:57,140 negative phase there. 498 00:24:57,140 --> 00:24:59,200 Remember, that's going to become important when we talk 499 00:24:59,200 --> 00:25:01,030 about bonding, we don't need to worry about it 500 00:25:01,030 --> 00:25:03,280 too much right now. 501 00:25:03,280 --> 00:25:07,420 We can also think about where the nodal plane is in this p z 502 00:25:07,420 --> 00:25:11,060 orbital, so how would we define the nodal plane here? 503 00:25:11,060 --> 00:25:16,580 What would the nodal plane be? 504 00:25:16,580 --> 00:25:18,950 So, it's the x-y plane, you can see there's no electron 505 00:25:18,950 --> 00:25:20,940 density anywhere there. 506 00:25:20,940 --> 00:25:23,760 And similarly, actually, if we're looking at our polar 507 00:25:23,760 --> 00:25:27,540 coordinates here, what we see is it's any place where theta 508 00:25:27,540 --> 00:25:31,080 is equal to 0 is what's going to put up on the x-y plane. 509 00:25:31,080 --> 00:25:33,900 So another way to define the nodal plane is where theta is 510 00:25:33,900 --> 00:25:36,820 equal to 90 degrees. 511 00:25:36,820 --> 00:25:40,450 So let's look now at the 2 p x orbital. 512 00:25:40,450 --> 00:25:43,140 This is the probability density map, so we're talking 513 00:25:43,140 --> 00:25:44,680 about the square here. 514 00:25:44,680 --> 00:25:48,320 The highest probability now is going to be along the x-axis, 515 00:25:48,320 --> 00:25:52,060 so that means we're going to have a positive wave function 516 00:25:52,060 --> 00:25:54,420 every place where x is positive. 517 00:25:54,420 --> 00:25:56,290 What is the nodal plane in this case? 518 00:25:56,290 --> 00:25:57,990 Um-hmm. 519 00:25:57,990 --> 00:26:02,030 So, it's going to be the y z nodal plane, or in other 520 00:26:02,030 --> 00:26:04,460 words, we can say it's any place where phi 521 00:26:04,460 --> 00:26:06,360 is equal to 90 degrees. 522 00:26:06,360 --> 00:26:09,190 So you can see if you take phi, and you move it over 90 523 00:26:09,190 --> 00:26:11,580 degrees, we're right here in the y z plane. 524 00:26:11,580 --> 00:26:13,820 Anywhere where that's the case we're going to have no 525 00:26:13,820 --> 00:26:17,530 probability density of finding an electron. 526 00:26:17,530 --> 00:26:20,910 And finally, we can look at the 2 p y, so the highest 527 00:26:20,910 --> 00:26:23,590 probability is going to be along the y-axis. 528 00:26:23,590 --> 00:26:26,360 It's going to be positive in terms of its wave function or 529 00:26:26,360 --> 00:26:29,590 in terms of its phase anywhere where y is positive. 530 00:26:29,590 --> 00:26:33,520 And the nodal plane's going to be in the x z plane, or again, 531 00:26:33,520 --> 00:26:37,430 anywhere where phi is going to be equal to 0, that takes us 532 00:26:37,430 --> 00:26:41,440 to the x z plane. 533 00:26:41,440 --> 00:26:44,550 So, let me get a little bit more specific about what we 534 00:26:44,550 --> 00:26:47,590 mean by nodal plane and where the idea of nodal plane comes 535 00:26:47,590 --> 00:26:51,000 from, and nodal planes arise from any place you have 536 00:26:51,000 --> 00:26:52,830 angular nodes. 537 00:26:52,830 --> 00:26:55,430 So we talked about radial nodes when we're doing these 538 00:26:55,430 --> 00:26:58,350 radial probability density diagrams here. 539 00:26:58,350 --> 00:27:00,930 You can also have angular notes, and when we talk about 540 00:27:00,930 --> 00:27:04,460 an anglar node, what we're talking about is values of 541 00:27:04,460 --> 00:27:07,570 theta or values of phi at which the wave function, and 542 00:27:07,570 --> 00:27:10,270 therefore, the wave function squared, or the probability 543 00:27:10,270 --> 00:27:13,400 density are going to be equal to zero. 544 00:27:13,400 --> 00:27:17,570 So, you remember from last time radial nodes are values 545 00:27:17,570 --> 00:27:21,040 of r at which the wave function and wave function 546 00:27:21,040 --> 00:27:23,680 squared are zero, so the difference is now we're just 547 00:27:23,680 --> 00:27:26,910 talking about the angular part of the wave function. 548 00:27:26,910 --> 00:27:29,440 And, in fact, these are the only two types of nodes that 549 00:27:29,440 --> 00:27:32,600 we're going to be describing, so we can actually calculate 550 00:27:32,600 --> 00:27:35,490 both the total number of notes and the number of each type of 551 00:27:35,490 --> 00:27:39,170 node we should expect to see in any type of orbital. 552 00:27:39,170 --> 00:27:41,980 And our equation for total nodes is just the principle 553 00:27:41,980 --> 00:27:43,810 quantum number minus 1. 554 00:27:43,810 --> 00:27:46,690 And when we talk about angular nodes, the number of angular 555 00:27:46,690 --> 00:27:50,820 nodes we have in an orbital is going to be equal to l. 556 00:27:50,820 --> 00:27:54,490 So that's why we saw, for example, in the p orbitals we 557 00:27:54,490 --> 00:27:57,470 had one angular node in each p orbital, because l 558 00:27:57,470 --> 00:27:58,880 is equal to 1 there. 559 00:27:58,880 --> 00:28:01,650 And we talked about the equation you can use for 560 00:28:01,650 --> 00:28:07,520 radial nodes last time, and that's just n minus 1 minus l. 561 00:28:07,520 --> 00:28:10,460 You can go ahead and use that equation, or you could figure 562 00:28:10,460 --> 00:28:12,690 it out every time, because if you know the total number of 563 00:28:12,690 --> 00:28:15,450 nodes, and you know the angular node number, then you 564 00:28:15,450 --> 00:28:17,160 know how many nodes you're going to have left. 565 00:28:17,160 --> 00:28:18,710 So you don't really have to memorize that. 566 00:28:18,710 --> 00:28:22,760 So, let's go ahead and just do a few of these. 567 00:28:22,760 --> 00:28:24,990 They're pretty straight forward to do and it gives us 568 00:28:24,990 --> 00:28:27,430 an idea what kind of nodal structure we can 569 00:28:27,430 --> 00:28:28,990 expect it an orbital. 570 00:28:28,990 --> 00:28:34,050 So for a 2 s orbital, how many total nodes will we have? 571 00:28:34,050 --> 00:28:38,700 Yup, I heard one, so 2 minus 1, one total node. 572 00:28:38,700 --> 00:28:41,190 Angular nodes, we're not going to have any of those, we'll 573 00:28:41,190 --> 00:28:45,820 have zero, l equals 0, so we have zero angular nodes. 574 00:28:45,820 --> 00:28:51,240 And in terms of radial nodes, we have 2 minus 1 minus 0, so 575 00:28:51,240 --> 00:28:54,130 what we have is one radial node. 576 00:28:54,130 --> 00:28:56,720 So, what you find with the s orbital, and this is general 577 00:28:56,720 --> 00:29:00,610 for all s orbitals is that all of your nodes end up being 578 00:29:00,610 --> 00:29:01,370 radial nodes. 579 00:29:01,370 --> 00:29:02,990 That has to be the case because l 580 00:29:02,990 --> 00:29:06,490 equals 0 for s orbitals. 581 00:29:06,490 --> 00:29:08,990 Let's look now at a p orbital, so how many total 582 00:29:08,990 --> 00:29:13,720 nodes do we have here? 583 00:29:13,720 --> 00:29:18,630 Yup, so one total node, 2 minus 1 is 1, and that means 584 00:29:18,630 --> 00:29:23,160 since l is equal to 1, we have one angular nodes, and that 585 00:29:23,160 --> 00:29:26,710 leaves us with how many radial nodes? 586 00:29:26,710 --> 00:29:28,620 Yup, zero radial nodes. 587 00:29:28,620 --> 00:29:32,440 So, for a 2 p orbital, all the nodes actually turn out to be 588 00:29:32,440 --> 00:29:35,070 angular nodes. 589 00:29:35,070 --> 00:29:37,900 So, let's have you try one more, if we can switch over 590 00:29:37,900 --> 00:29:41,270 and talk about a 3 d orbital. 591 00:29:41,270 --> 00:29:44,000 So, I'm asking very specifically about radial 592 00:29:44,000 --> 00:29:47,090 nodes here, how many radial nodes does a hydrogen atom 3 d 593 00:29:47,090 --> 00:29:48,200 orbital have? 594 00:29:48,200 --> 00:29:56,470 So, you can go ahead and take 10 seconds on that. 595 00:29:56,470 --> 00:30:09,530 All right. 596 00:30:09,530 --> 00:30:12,550 So most of you got that, though there is this little 597 00:30:12,550 --> 00:30:16,090 sub-set we have thinking that we have one, so let's actually 598 00:30:16,090 --> 00:30:17,950 write this out here. 599 00:30:17,950 --> 00:30:24,450 So if we have a 3 d orbital, we're talking about n minus l 600 00:30:24,450 --> 00:30:28,180 minus 1, what is n equal to? 601 00:30:28,180 --> 00:30:30,180 What is l equal to? 602 00:30:30,180 --> 00:30:33,560 OK. and 1 is equal to 1. 603 00:30:33,560 --> 00:30:37,010 So, it turns out that we have zero nodes that we're dealing 604 00:30:37,010 --> 00:30:39,070 with when we're talking about a 3 d orbital. 605 00:30:39,070 --> 00:30:43,860 OK. 606 00:30:43,860 --> 00:30:46,400 So we should be able to figure this out for any orbital that 607 00:30:46,400 --> 00:30:50,490 we're discussing, and when we can figure out especially 608 00:30:50,490 --> 00:30:53,580 radial nodes, we have a good head start on going ahead and 609 00:30:53,580 --> 00:30:57,070 thinking about drawing radial probability distributions. 610 00:30:57,070 --> 00:30:59,700 We did it for the s orbitals, we can also do it for the p, 611 00:30:59,700 --> 00:31:00,850 we can do it for the d. 612 00:31:00,850 --> 00:31:03,200 All we have to figure out is how many nodes we're dealing 613 00:31:03,200 --> 00:31:05,170 with and then we can get the general shape 614 00:31:05,170 --> 00:31:07,170 of the graph here. 615 00:31:07,170 --> 00:31:10,620 So, let's actually compare the radial probability 616 00:31:10,620 --> 00:31:13,910 distribution of p orbitals to what we've already looked at, 617 00:31:13,910 --> 00:31:16,580 which are s orbitals, and we'll find that we can get 618 00:31:16,580 --> 00:31:20,410 some information out of comparing these graphs. 619 00:31:20,410 --> 00:31:23,860 So if we draw the 2 p orbital, what we just figured out was 620 00:31:23,860 --> 00:31:25,870 there should be zero radial nodes, so 621 00:31:25,870 --> 00:31:27,950 that's what we see here. 622 00:31:27,950 --> 00:31:30,200 The other thing that I want you to notice, is if you look 623 00:31:30,200 --> 00:31:34,010 at the most probable radius, for the 2 s orbital it's 624 00:31:34,010 --> 00:31:37,560 actually out further away from the nucleus than it is for the 625 00:31:37,560 --> 00:31:39,320 2 p orbital. 626 00:31:39,320 --> 00:31:43,420 So what we can say here is that the 2 p is less than or 627 00:31:43,420 --> 00:31:46,930 smaller than the 2 s orbital. 628 00:31:46,930 --> 00:31:49,590 So think about what that means, we're, of course, not 629 00:31:49,590 --> 00:31:51,580 talking about this in classical terms, so what it 630 00:31:51,580 --> 00:31:55,390 means if we have an electron in the 2 p orbital, it's more 631 00:31:55,390 --> 00:31:58,190 likely, the probability is that will be closer to the 632 00:31:58,190 --> 00:32:03,080 nucleus than it would be if it were in the 2 s orbital. 633 00:32:03,080 --> 00:32:06,720 We can also take a look the 3 s, which we have looked at 634 00:32:06,720 --> 00:32:09,440 before, and we figured out that that should have two 635 00:32:09,440 --> 00:32:10,880 radial nodes. 636 00:32:10,880 --> 00:32:14,850 We can look at the 2 p, which should have one radial node, 637 00:32:14,850 --> 00:32:18,980 and we just figured it out for the, excuse me, for the 3 p 638 00:32:18,980 --> 00:32:22,840 has one radial node, and for the 3 d here, we should have 639 00:32:22,840 --> 00:32:25,370 zero radial nodes, we just calculated that. 640 00:32:25,370 --> 00:32:28,570 So again, what we see is the same pattern where the most 641 00:32:28,570 --> 00:32:32,070 probable radius, if we talk about it in terms of the d, 642 00:32:32,070 --> 00:32:35,270 that's going to be smaller then for the p, and the 3 p 643 00:32:35,270 --> 00:32:38,330 most probable radius is going to be closer to the nucleus 644 00:32:38,330 --> 00:32:43,200 than it is for the 3 s most probable radius that we're 645 00:32:43,200 --> 00:32:45,160 looking at. 646 00:32:45,160 --> 00:32:47,960 So, there are 2 different things that we can compare 647 00:32:47,960 --> 00:32:50,950 when we're comparing graphs of radial probability 648 00:32:50,950 --> 00:32:54,660 distribution, and the first thing we can do is think about 649 00:32:54,660 --> 00:32:57,500 well, how does the radius change, or the most probable 650 00:32:57,500 --> 00:33:00,540 radius change when we're increasing n, when we're 651 00:33:00,540 --> 00:33:03,860 increasing the principle quantum number here? 652 00:33:03,860 --> 00:33:07,290 So, from going from the shell of n equals 2, let's say, to 653 00:33:07,290 --> 00:33:09,090 the shell of n equals 3. 654 00:33:09,090 --> 00:33:13,500 And what we find is we're going from about or exactly a 655 00:33:13,500 --> 00:33:17,410 6 a nought here, to almost three times that when we're 656 00:33:17,410 --> 00:33:19,650 going from 2 s to 3 s. 657 00:33:19,650 --> 00:33:22,940 So we say if n increases, the orbital size is 658 00:33:22,940 --> 00:33:23,990 also going to increase. 659 00:33:23,990 --> 00:33:26,140 And when we talk about size, I'm again just going to say 660 00:33:26,140 --> 00:33:29,260 the stipulation that we're talking about, probability -- 661 00:33:29,260 --> 00:33:32,420 we're not talking about an absolute classical concept 662 00:33:32,420 --> 00:33:36,210 here, but in general we're going to picture it being much 663 00:33:36,210 --> 00:33:40,920 further away from the nucleus as we move up in terms of n. 664 00:33:40,920 --> 00:33:44,210 The other thing that we took note as is what happens as l 665 00:33:44,210 --> 00:33:47,890 increases, and specifically as l increases for any given the 666 00:33:47,890 --> 00:33:49,280 principle quantum number. 667 00:33:49,280 --> 00:33:52,890 So if we're keeping n the same, we look and what we saw 668 00:33:52,890 --> 00:33:56,340 was that size actually decreases as we increase the 669 00:33:56,340 --> 00:33:58,610 value of l. 670 00:33:58,610 --> 00:34:02,230 So, I want to contrast that with another concept that 671 00:34:02,230 --> 00:34:06,330 seemed to be opposing ideas, and that is thinking about not 672 00:34:06,330 --> 00:34:10,000 how far away the most probable radius is, but thinking about 673 00:34:10,000 --> 00:34:13,546 how close an electron can get to the nucleus if it's 674 00:34:13,546 --> 00:34:14,380 actually in that orbital. 675 00:34:14,380 --> 00:34:18,370 And what we'll find is that we actually see the opposite. 676 00:34:18,370 --> 00:34:24,150 So if we compare l increasing here, so a 3 s to a 3 p to a 3 677 00:34:24,150 --> 00:34:28,550 d, what we find is that it's only in the s orbital that we 678 00:34:28,550 --> 00:34:31,920 have a significant probability of actually getting very close 679 00:34:31,920 --> 00:34:33,140 to the nucleus. 680 00:34:33,140 --> 00:34:37,130 So, if I kind of circle where the probability gets somewhat 681 00:34:37,130 --> 00:34:39,910 substantial here, you can see we're much closer to the 682 00:34:39,910 --> 00:34:44,390 nucleus at the s orbital than we are for the p, then when we 683 00:34:44,390 --> 00:34:46,350 are for the d. 684 00:34:46,350 --> 00:34:51,150 So, the size still for an s orbital is larger than for a d 685 00:34:51,150 --> 00:34:54,330 orbital, but what we say is that an s electron can 686 00:34:54,330 --> 00:34:56,950 actually penetrate closer to the nucleus. 687 00:34:56,950 --> 00:34:59,940 There's some probability that it can get very, very close 688 00:34:59,940 --> 00:35:01,850 the nucleus, and that probability is actually 689 00:35:01,850 --> 00:35:05,470 substantial. 690 00:35:05,470 --> 00:35:11,140 A kind of consequence of this is if we're thinking about a 691 00:35:11,140 --> 00:35:13,880 multi-electron atom, which we'll get to in a minute where 692 00:35:13,880 --> 00:35:16,810 electrons can shield each other from the pull of the 693 00:35:16,810 --> 00:35:19,780 nucleus, we're going to say that the electrons in the s 694 00:35:19,780 --> 00:35:22,110 orbitals are actually the least shielded. 695 00:35:22,110 --> 00:35:24,520 And the reason that they're the least sheilded is because 696 00:35:24,520 --> 00:35:26,650 they can get closest to the nucleus, so we can think of 697 00:35:26,650 --> 00:35:29,430 them as not getting blocked by a bunch of other electron, 698 00:35:29,430 --> 00:35:31,010 because there's some probability that they can 699 00:35:31,010 --> 00:35:33,970 actually work their way all the way in to the nucleus. 700 00:35:33,970 --> 00:35:35,450 So, this is a concept that's going to 701 00:35:35,450 --> 00:35:36,990 become really important. 702 00:35:36,990 --> 00:35:39,520 Soon when we're talking about multi-electron atoms, and I 703 00:35:39,520 --> 00:35:41,870 just want to introduce it here, that it is sort of 704 00:35:41,870 --> 00:35:45,090 opposing ideas that even though the s is the biggest 705 00:35:45,090 --> 00:35:47,440 and it's most likely that the electron's going to be 706 00:35:47,440 --> 00:35:50,280 furthest away from the nucleus, that's also the 707 00:35:50,280 --> 00:35:54,500 orbital in which the electron can, in fact, penetrate 708 00:35:54,500 --> 00:35:56,370 closest. 709 00:35:56,370 --> 00:35:56,690 All right. 710 00:35:56,690 --> 00:35:59,390 So I think we are, in fact, ready to move on to 711 00:35:59,390 --> 00:36:04,190 multi-electron atoms, and what happens is when we solved the 712 00:36:04,190 --> 00:36:07,550 relativistic version of the Schrodinger equation and we're 713 00:36:07,550 --> 00:36:10,790 discussing more than one electron, we actually have a 714 00:36:10,790 --> 00:36:13,880 fourth quantum number that falls out and that we need to 715 00:36:13,880 --> 00:36:16,000 deal with and this is called the electron 716 00:36:16,000 --> 00:36:17,530 spin quantum number. 717 00:36:17,530 --> 00:36:20,330 And I promise, this is the last quantum number that we'll 718 00:36:20,330 --> 00:36:22,750 be introducing. 719 00:36:22,750 --> 00:36:26,490 And this spin magnetic quantum number we abbreviate as m sub 720 00:36:26,490 --> 00:36:32,580 s, so that's to differentiate from m sub l. 721 00:36:32,580 --> 00:36:36,320 And when you solved the relativistic form of the 722 00:36:36,320 --> 00:36:38,730 Schrodinger equation, what you end up with is that you can 723 00:36:38,730 --> 00:36:41,580 have two possible values for the magnetic 724 00:36:41,580 --> 00:36:43,580 spin quantum number. 725 00:36:43,580 --> 00:36:47,940 You can have it equal to plus 1/2, and that's what we call 726 00:36:47,940 --> 00:36:51,730 spin up, or you can have it equal to minus 1/2, which is 727 00:36:51,730 --> 00:36:53,760 what we call spin down. 728 00:36:53,760 --> 00:36:56,970 So, there's two kind of cartoons shown here that give 729 00:36:56,970 --> 00:36:59,420 you a little bit of an idea of what this quantum 730 00:36:59,420 --> 00:37:00,660 number tells us. 731 00:37:00,660 --> 00:37:04,870 And this spin is an intrinsic quality of the electron, it's 732 00:37:04,870 --> 00:37:08,080 a property that is intrinsic in all particles, just like we 733 00:37:08,080 --> 00:37:10,970 would say mass is intrinsic or charge is intrinsic. 734 00:37:10,970 --> 00:37:13,750 Spin is also an intrinsic property. 735 00:37:13,750 --> 00:37:16,140 One way to think about it, if we want to use a classical 736 00:37:16,140 --> 00:37:18,610 analogy, which often helps to give us an idea of what's 737 00:37:18,610 --> 00:37:21,760 going on, is the spin of an electron, we can picture it 738 00:37:21,760 --> 00:37:24,010 rotating on its own axis. 739 00:37:24,010 --> 00:37:27,350 So that's kind of what's shown in these pictures here. 740 00:37:27,350 --> 00:37:29,350 So you can see, if it's spinning on its own axis in 741 00:37:29,350 --> 00:37:31,960 this direction we'd call it spin up, where as this way it 742 00:37:31,960 --> 00:37:34,010 would be what we call spin down. 743 00:37:34,010 --> 00:37:36,910 So, it turns out there's not actually a good classical 744 00:37:36,910 --> 00:37:40,140 analogy for spin, we can't really think of it like that, 745 00:37:40,140 --> 00:37:43,300 but if that helps give you an idea of what's going on here 746 00:37:43,300 --> 00:37:45,750 then it's valuable maybe to consider it spinning on its 747 00:37:45,750 --> 00:37:48,960 own axis, even though that's not technically what's exactly 748 00:37:48,960 --> 00:37:49,980 happening here. 749 00:37:49,980 --> 00:37:52,570 But the reason that I like that analogy is that it points 750 00:37:52,570 --> 00:37:56,130 out a very important part of spin, and that's the idea that 751 00:37:56,130 --> 00:37:58,570 it's a description of the electron. 752 00:37:58,570 --> 00:38:01,490 It is not dependent on the actual orbital. 753 00:38:01,490 --> 00:38:04,830 So we can completely describe an orbital with just using 754 00:38:04,830 --> 00:38:08,100 three quantum numbers, but we have this fourth quantum 755 00:38:08,100 --> 00:38:10,760 number that describes something about the electron 756 00:38:10,760 --> 00:38:13,840 that's required for now a complete description of the 757 00:38:13,840 --> 00:38:16,550 electron, and that's the idea of spin. 758 00:38:16,550 --> 00:38:19,490 So, we need to actually add on this fourth quantum number, 759 00:38:19,490 --> 00:38:25,000 and it's either going to be plus 1/2 or negative 1/2. 760 00:38:25,000 --> 00:38:27,230 So, we can talk a little bit actually, because it's an 761 00:38:27,230 --> 00:38:30,650 interesting story about where the idea of spin came from, 762 00:38:30,650 --> 00:38:33,590 and it was actually first proposed by two very young 763 00:38:33,590 --> 00:38:37,310 scientists at the time, George Uhlenbeck shown here, and 764 00:38:37,310 --> 00:38:40,370 Samuel Goudsmit who's here, and they're with a friend, and 765 00:38:40,370 --> 00:38:43,680 I can't remember who that is, but he did not have anything 766 00:38:43,680 --> 00:38:46,210 to do with discovering spin. 767 00:38:46,210 --> 00:38:48,310 And what you can see in this picture is that these are 768 00:38:48,310 --> 00:38:50,060 actually, they're pretty young guys in this picture. 769 00:38:50,060 --> 00:38:51,910 I think this is taken about two years after they 770 00:38:51,910 --> 00:38:53,930 discovered the fourth quantum number. 771 00:38:53,930 --> 00:38:56,500 So hopefully, you can picture yourself at this age in a 772 00:38:56,500 --> 00:39:00,010 similar situation with an anonymous friend and think 773 00:39:00,010 --> 00:39:02,780 this is something, kind of observations maybe you can 774 00:39:02,780 --> 00:39:04,820 make as well. 775 00:39:04,820 --> 00:39:07,320 And what they were doing when they discovered that there 776 00:39:07,320 --> 00:39:10,100 must be this fourth quantum number is they were looking at 777 00:39:10,100 --> 00:39:14,730 the emission spectrum of sodium, and, so specifically, 778 00:39:14,730 --> 00:39:17,630 they were looking at the frequencies, and if we think 779 00:39:17,630 --> 00:39:20,600 about the frequencies of sodium, it was already known 780 00:39:20,600 --> 00:39:22,930 at this time that you could calculate what those would be 781 00:39:22,930 --> 00:39:25,110 based on the difference in the energy levels -- this happened 782 00:39:25,110 --> 00:39:27,060 in about 1925. 783 00:39:27,060 --> 00:39:29,570 So they actually knew exactly what they were expecting. 784 00:39:29,570 --> 00:39:31,930 So, let's say they were expecting to see one certain 785 00:39:31,930 --> 00:39:36,150 frequency or one line in the spectrum at this point here. 786 00:39:36,150 --> 00:39:40,740 It turns out that what they actually observed, so this is 787 00:39:40,740 --> 00:39:44,830 the actual of what they saw, is if they were expecting 788 00:39:44,830 --> 00:39:47,890 their line at some given frequency, which I'll show by 789 00:39:47,890 --> 00:39:52,720 this dotted line here, what they actually saw was two 790 00:39:52,720 --> 00:39:56,570 lines, and one was just the teeniest tiniest bit of a 791 00:39:56,570 --> 00:39:59,690 higher frequency than what was expected, and one was just, 792 00:39:59,690 --> 00:40:01,660 just below what they expected. 793 00:40:01,660 --> 00:40:04,460 And if we're talking about things in spectroscopy terms 794 00:40:04,460 --> 00:40:08,840 here, this is what we call a doublet. 795 00:40:08,840 --> 00:40:11,190 So it's centered at this frequency that was expected, 796 00:40:11,190 --> 00:40:14,350 but it's actually split into two different frequencies. 797 00:40:14,350 --> 00:40:16,930 And this was an amazing observation that they made. 798 00:40:16,930 --> 00:40:19,620 They were totally surprised and excited, and they were 799 00:40:19,620 --> 00:40:22,100 thinking how could this happen, where did we got this 800 00:40:22,100 --> 00:40:23,650 split doublet from. 801 00:40:23,650 --> 00:40:26,600 And what they could come up with, what they reasoned, is 802 00:40:26,600 --> 00:40:29,130 that there must be some intrinsic property within the 803 00:40:29,130 --> 00:40:32,270 electron, because we know that this describes the complete 804 00:40:32,270 --> 00:40:37,780 energy of the orbital should give us one single frequency. 805 00:40:37,780 --> 00:40:40,810 And that the fact that it split into two was telling 806 00:40:40,810 --> 00:40:43,970 them that there must be some new property to the electron, 807 00:40:43,970 --> 00:40:45,930 and what we call that now is either being 808 00:40:45,930 --> 00:40:47,880 spin up or spin down. 809 00:40:47,880 --> 00:40:51,420 But at the time, they didn't have a well-formed name for 810 00:40:51,420 --> 00:40:54,210 it, they were just saying OK, there's this fourth quantum 811 00:40:54,210 --> 00:40:57,480 number, there's this intrinsic property in the electron. 812 00:40:57,480 --> 00:40:59,910 So, where the story gets kind of unfortunate, but also a 813 00:40:59,910 --> 00:41:03,000 little bit more interesting is the fact that well, they did 814 00:41:03,000 --> 00:41:06,030 publish what they observed, and they did write that up. 815 00:41:06,030 --> 00:41:08,220 They put it in a pretty low impact paper. 816 00:41:08,220 --> 00:41:11,740 I think it was in French, so it wasn't really hitting all 817 00:41:11,740 --> 00:41:15,930 the scientific community of the time in any major way. 818 00:41:15,930 --> 00:41:19,310 And they didn't put their explanation of what they 819 00:41:19,310 --> 00:41:21,500 thought was going on, it just sort of was 820 00:41:21,500 --> 00:41:23,860 observing what they saw. 821 00:41:23,860 --> 00:41:26,210 These were very young scientists, of course, so what 822 00:41:26,210 --> 00:41:28,650 you would expect that they would do, which makes sense, 823 00:41:28,650 --> 00:41:30,850 is go to someone more established in their field, 824 00:41:30,850 --> 00:41:33,100 because they have the completely radical 825 00:41:33,100 --> 00:41:36,760 revolutionary idea, let's just run it by someone before we go 826 00:41:36,760 --> 00:41:39,540 ahead and publish this paper that makes this huge statement 827 00:41:39,540 --> 00:41:41,560 about this fourth quantum number. 828 00:41:41,560 --> 00:41:44,520 So the person they chose to talk to, and I think it was 829 00:41:44,520 --> 00:41:47,290 just Goudsmit that went to him and discussed it is Wolfgang 830 00:41:47,290 --> 00:41:49,480 Pauli, which is shown here. 831 00:41:49,480 --> 00:41:52,030 So how many before five minutes ago had 832 00:41:52,030 --> 00:41:52,840 heard the name Goudsmit? 833 00:41:52,840 --> 00:41:55,620 All right, a couple. 834 00:41:55,620 --> 00:41:56,850 OK, cool. 835 00:41:56,850 --> 00:41:59,580 How about Pauli, like the Pauli exclusion principle? 836 00:41:59,580 --> 00:42:00,750 Hmm, OK. 837 00:42:00,750 --> 00:42:05,480 So, Pauli seems to be getting a little bit of fame that 838 00:42:05,480 --> 00:42:06,710 you'll see in a second here. 839 00:42:06,710 --> 00:42:09,870 So, it turns out they go and they discuss their idea with 840 00:42:09,870 --> 00:42:14,580 Pauli, and what Pauli tells them is that this idea is 841 00:42:14,580 --> 00:42:17,120 ridiculous, that it's rubbish, if they go ahead and try to 842 00:42:17,120 --> 00:42:19,150 publish this their scientific careers are ruined. 843 00:42:19,150 --> 00:42:21,560 They can pretty much pack it up and go home, because 844 00:42:21,560 --> 00:42:23,710 everyone's going to think they're ridiculous, no one 845 00:42:23,710 --> 00:42:25,955 will believe what they say, and it's a stupid idea anyway, 846 00:42:25,955 --> 00:42:28,960 is basically the gist of this conversation. 847 00:42:28,960 --> 00:42:34,050 And in chemistry, just like in any discipline, you have all 848 00:42:34,050 --> 00:42:36,880 types of scientists, but also all types of personalities, 849 00:42:36,880 --> 00:42:39,850 and unfortunately Pauli had a personality that was known 850 00:42:39,850 --> 00:42:42,280 for, first of all, being very arrogant, but also the very 851 00:42:42,280 --> 00:42:45,160 unfortunate trait of taking other people's scientific 852 00:42:45,160 --> 00:42:46,930 ideas as his own. 853 00:42:46,930 --> 00:42:50,330 And as the story goes, as Goudsmit was leaving and the 854 00:42:50,330 --> 00:42:54,850 door with slamming, Wolfgang Pauli was already writing down 855 00:42:54,850 --> 00:42:58,590 this idea into a scientific paper of the idea of a fourth 856 00:42:58,590 --> 00:42:59,840 quantum number. 857 00:42:59,840 --> 00:43:03,150 And, in fact, he did make some more strides, he was a 858 00:43:03,150 --> 00:43:06,440 brilliant thinker, maybe he put it more articulately than 859 00:43:06,440 --> 00:43:09,490 those two younger scientists could have. But now, it has 860 00:43:09,490 --> 00:43:11,660 come to light that they are the ones that do get credit 861 00:43:11,660 --> 00:43:14,860 for first really coming up with this idea of a spin 862 00:43:14,860 --> 00:43:17,600 quantum number, and it's interesting to think about how 863 00:43:17,600 --> 00:43:21,120 the politics work in different discoveries, as well as the 864 00:43:21,120 --> 00:43:22,160 discoveries themselves. 865 00:43:22,160 --> 00:43:24,550 You see that a lot with Nobel prizes, there's usually a nice 866 00:43:24,550 --> 00:43:27,680 little scandal, a nice little interesting story behind who 867 00:43:27,680 --> 00:43:33,070 else was responsible for the Nobel Prize-worthy discovery. 868 00:43:33,070 --> 00:43:37,170 So, here, Pauli came out on top, we say, and he's known 869 00:43:37,170 --> 00:43:40,770 for the Pauli exclusion principle, which tells us that 870 00:43:40,770 --> 00:43:43,810 no two electrons in the same atom can have the same four 871 00:43:43,810 --> 00:43:45,770 quantum numbers. 872 00:43:45,770 --> 00:43:49,330 So let's just think exactly what this means, and that 873 00:43:49,330 --> 00:43:52,360 means that if we take away function and we define it in 874 00:43:52,360 --> 00:43:57,930 terms of n, l and m sub l, what we're defining here is 875 00:43:57,930 --> 00:44:03,180 the complete description of an orbital. 876 00:44:03,180 --> 00:44:06,150 In contrast, if we're taking the wave function and 877 00:44:06,150 --> 00:44:12,580 describing it in terms of n, l, m sub l, and now also, the 878 00:44:12,580 --> 00:44:15,640 spin, what are we describing here? 879 00:44:15,640 --> 00:44:16,450 An electron. 880 00:44:16,450 --> 00:44:20,580 So now we have the complete description of an electron 881 00:44:20,580 --> 00:44:21,780 within an orbital. 882 00:44:21,780 --> 00:44:26,160 So, that's an important distinction to make -- what 883 00:44:26,160 --> 00:44:28,930 three quantum numbers tell us, versus what the fourth quantum 884 00:44:28,930 --> 00:44:32,390 number can fill in for us in terms of information. 885 00:44:32,390 --> 00:44:35,420 So what that means is that we're limited in any atom to 886 00:44:35,420 --> 00:44:38,860 having two electrons per orbital, right, because for 887 00:44:38,860 --> 00:44:42,380 any orbital we can either have a spin up electron, a spin 888 00:44:42,380 --> 00:44:44,670 down electron, or both. 889 00:44:44,670 --> 00:44:48,980 So, if we look at neon just as an example, neon has ten 890 00:44:48,980 --> 00:44:53,100 electron in it, and once we look at all the orbitals 891 00:44:53,100 --> 00:44:55,920 written out here, this is probably a familiar thing for 892 00:44:55,920 --> 00:44:59,310 you to look at, but it's important to think about why, 893 00:44:59,310 --> 00:45:02,600 in fact, we don't just put all ten electrons, why wouldn't 894 00:45:02,600 --> 00:45:04,860 they just want to go in that ground state, that lowest 895 00:45:04,860 --> 00:45:05,940 state, right? 896 00:45:05,940 --> 00:45:08,240 That should be the most stable, the lowest energy 897 00:45:08,240 --> 00:45:11,060 orbital for it to be in, and the reason they can't do that 898 00:45:11,060 --> 00:45:13,470 is because of the Pauli exclusion principle -- the 899 00:45:13,470 --> 00:45:17,410 idea that all of the electrons have to have a different set 900 00:45:17,410 --> 00:45:20,850 of four quantum numbers, so only two of them can have the 901 00:45:20,850 --> 00:45:24,140 same set of three quantum numbers here, because for m 902 00:45:24,140 --> 00:45:28,220 sub s, we're only left with two options. 903 00:45:28,220 --> 00:45:31,510 So let's try a clicker question and thinking about 904 00:45:31,510 --> 00:45:33,490 the Pauli exclusion principle. 905 00:45:33,490 --> 00:45:36,760 It might look a little bit similar to a question we just 906 00:45:36,760 --> 00:45:41,300 saw, but hopefully you'll find that it is, in fact, not the 907 00:45:41,300 --> 00:45:45,490 same question. 908 00:45:45,490 --> 00:45:47,650 So you can go ahead and take 10 seconds on that. 909 00:45:47,650 --> 00:46:02,170 OK, great. 910 00:46:02,170 --> 00:46:07,080 So, most of you recognize that there are four different 911 00:46:07,080 --> 00:46:09,870 possibilities of there's four different electrons that can 912 00:46:09,870 --> 00:46:11,590 have those two quantum numbers. 913 00:46:11,590 --> 00:46:13,200 Actually the easiest way is probably to 914 00:46:13,200 --> 00:46:15,050 bring this down here. 915 00:46:15,050 --> 00:46:17,740 And the next highest percentage of you thought that 916 00:46:17,740 --> 00:46:19,220 we still only had two. 917 00:46:19,220 --> 00:46:22,740 So, remember we solved this problem earlier in the class, 918 00:46:22,740 --> 00:46:24,560 but we were talking about orbitals. 919 00:46:24,560 --> 00:46:27,100 So there's two different orbitals that can have these 920 00:46:27,100 --> 00:46:30,190 three quantum numbers, but if we're talking about electrons, 921 00:46:30,190 --> 00:46:34,090 we can also talk about m sub s, so if we have two orbitals, 922 00:46:34,090 --> 00:46:36,740 how many electrons can we have total? 923 00:46:36,740 --> 00:46:37,000 Yeah. 924 00:46:37,000 --> 00:46:42,400 So we have two orbitals, or four electrons that can have 925 00:46:42,400 --> 00:46:44,430 that set of quantum numbers. 926 00:46:44,430 --> 00:46:46,670 So you'll notice in your problem-set, sometimes you're 927 00:46:46,670 --> 00:46:49,650 asked for a number of orbitals with a set of quantum numbers, 928 00:46:49,650 --> 00:46:52,400 sometimes you're asked for a number of electrons for a set 929 00:46:52,400 --> 00:46:53,110 of quantum numbers. 930 00:46:53,110 --> 00:46:55,180 So make sure first that you read the question carefully, 931 00:46:55,180 --> 00:46:59,830 and realize the difference that is between the two. 932 00:46:59,830 --> 00:47:01,620 So, that's where we'll end today. 933 00:47:01,620 --> 00:47:04,140 So on Friday, we'll start with talking about the wave 934 00:47:04,140 --> 00:47:06,860 functions for the multi-electron atoms.