1 00:00:00,030 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,780 Commons license. 3 00:00:03,780 --> 00:00:06,020 Your support will help MIT OpenCourseWare 4 00:00:06,020 --> 00:00:10,090 continue to offer high quality educational resources for free. 5 00:00:10,090 --> 00:00:12,670 To make a donation or to view additional materials 6 00:00:12,670 --> 00:00:16,580 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,580 --> 00:00:17,740 at osw.mit.edu. 8 00:00:40,850 --> 00:00:43,100 PROFESSOR: All right, let's just take 10 more seconds. 9 00:00:58,330 --> 00:01:02,540 All right, so someone want to explain 10 00:01:02,540 --> 00:01:05,650 why this is the correct answer? 11 00:01:05,650 --> 00:01:09,260 And we have a syringe highlighter. 12 00:01:09,260 --> 00:01:12,046 You probably never had something like this before. 13 00:01:17,090 --> 00:01:21,690 AUDIENCE: OK, so the fourth excited state is n equals 5. 14 00:01:21,690 --> 00:01:26,460 And then IE is opposite of the negative number shown. 15 00:01:26,460 --> 00:01:28,532 So it would be a positive reaction. 16 00:01:28,532 --> 00:01:29,240 PROFESSOR: Right. 17 00:01:29,240 --> 00:01:32,220 So IE is always going to be positive. 18 00:01:32,220 --> 00:01:35,600 And you have to pay attention to what n equals when 19 00:01:35,600 --> 00:01:38,390 you're in the excited state. 20 00:01:41,460 --> 00:01:44,750 So we've been talking about the hydrogen atom and binding 21 00:01:44,750 --> 00:01:45,600 energies. 22 00:01:45,600 --> 00:01:47,950 What comes out of the Schrodinger equation? 23 00:01:47,950 --> 00:01:50,020 We have the binding energies that come out. 24 00:01:50,020 --> 00:01:51,914 And we also have wave functions. 25 00:01:51,914 --> 00:01:53,330 So today we're going to be talking 26 00:01:53,330 --> 00:01:55,040 about wave functions, which are often 27 00:01:55,040 --> 00:01:57,500 referred to as orbitals in chemistry, 28 00:01:57,500 --> 00:02:00,130 for the hydrogen atom. 29 00:02:00,130 --> 00:02:03,820 So when you solve the Schrodinger equation, 30 00:02:03,820 --> 00:02:07,880 you get out this information about wave functions. 31 00:02:07,880 --> 00:02:11,610 And what comes out of it is these quantum numbers. 32 00:02:11,610 --> 00:02:15,170 And we already saw quantum number n coming out. 33 00:02:15,170 --> 00:02:16,860 But there are three quantum numbers 34 00:02:16,860 --> 00:02:19,640 that are going to come out of the Schrodinger equation. 35 00:02:19,640 --> 00:02:21,620 And those three quantum numbers are 36 00:02:21,620 --> 00:02:26,280 necessary to describe the wave function or the orbital. 37 00:02:26,280 --> 00:02:30,650 So we have n, the principle quantum number. 38 00:02:30,650 --> 00:02:32,950 We've already talked about that. 39 00:02:32,950 --> 00:02:36,750 And we've already seen that n is an integer. 40 00:02:36,750 --> 00:02:38,930 So I'll just put that down here. 41 00:02:38,930 --> 00:02:46,760 So n can equal 1, 2, 3, on to infinity. 42 00:02:46,760 --> 00:02:51,620 So this describes the energy level or the shell. 43 00:02:51,620 --> 00:02:55,650 Then we have l, which we haven't talked about yet. 44 00:02:55,650 --> 00:02:58,870 So that's the angular momentum quantum number. 45 00:02:58,870 --> 00:03:01,440 So it tells you about the angular momentum. 46 00:03:01,440 --> 00:03:04,260 It also tells you about the subshell 47 00:03:04,260 --> 00:03:07,990 or the shape of the orbital. 48 00:03:07,990 --> 00:03:11,760 And so l is related to n. 49 00:03:11,760 --> 00:03:22,580 And it can be 0, 1, 2, 3, onward to n minus 1. 50 00:03:22,580 --> 00:03:27,340 So its biggest number is n minus 1. 51 00:03:27,340 --> 00:03:33,150 Then we have m, the magnetic quantum number. 52 00:03:33,150 --> 00:03:37,065 And we often see this also listed as m sub 53 00:03:37,065 --> 00:03:43,040 l because m is related back to l. 54 00:03:43,040 --> 00:03:50,150 And this is equal to minus l, dot, dot, dot, to 0, 55 00:03:50,150 --> 00:03:55,190 dot, dot, dot to plus l. 56 00:03:55,190 --> 00:03:59,370 And m describes the behavior in a magnetic field. 57 00:03:59,370 --> 00:04:02,150 It also describes the orientation 58 00:04:02,150 --> 00:04:06,380 of the orbital with respect to an axes. 59 00:04:06,380 --> 00:04:10,540 And it tells you about the specific orbital in question. 60 00:04:10,540 --> 00:04:16,209 So we need all three of these to describe any orbital. 61 00:04:16,209 --> 00:04:22,410 All right, so let's look at this in a slightly other way. 62 00:04:22,410 --> 00:04:26,220 So we're going to have lots of different sort of nomenclatures 63 00:04:26,220 --> 00:04:27,870 for the same thing. 64 00:04:27,870 --> 00:04:30,761 So to describe an orbital, we need those three quantum 65 00:04:30,761 --> 00:04:31,260 numbers. 66 00:04:31,260 --> 00:04:34,490 We need n, l, and m. 67 00:04:34,490 --> 00:04:43,360 And this can also be expressed as our wave function sub nlm. 68 00:04:43,360 --> 00:04:45,370 And again, we talked about this last time. 69 00:04:45,370 --> 00:04:47,400 We're going to talk more about it. 70 00:04:47,400 --> 00:04:49,970 So our wave function is also described 71 00:04:49,970 --> 00:04:54,560 by r, the radius, and theta and phi, which are two angles. 72 00:04:54,560 --> 00:04:57,450 And we're going to talk a lot about those today. 73 00:04:57,450 --> 00:05:00,900 So the wave function for the ground state 74 00:05:00,900 --> 00:05:05,950 is abbreviated wave function sub 1, 0, 0. 75 00:05:05,950 --> 00:05:07,540 Because it's the ground state. 76 00:05:07,540 --> 00:05:12,330 So n equals 1, and l and m are 0. 77 00:05:12,330 --> 00:05:15,450 So what you see down here, the 1, 0, 0, 78 00:05:15,450 --> 00:05:20,940 refers back to what is n, what is l, what is m. 79 00:05:20,940 --> 00:05:24,320 And this also has another name. 80 00:05:24,320 --> 00:05:27,150 So in the terminology of chemists, 81 00:05:27,150 --> 00:05:34,910 we call the wave function 1, 0, 0 1s, or the 1s orbital. 82 00:05:34,910 --> 00:05:37,800 So let's look again now at the same things 83 00:05:37,800 --> 00:05:40,200 we just talked about, but going through kind 84 00:05:40,200 --> 00:05:43,360 of chemistry lingo. 85 00:05:43,360 --> 00:05:48,380 So again, n describes the shell or the energy level. 86 00:05:48,380 --> 00:05:53,930 Again, it's integers, 1, 2, 3, et cetera. 87 00:05:53,930 --> 00:05:57,630 l in chemistry lingo, the subshell or the shape 88 00:05:57,630 --> 00:05:58,930 of the orbital. 89 00:05:58,930 --> 00:06:01,960 And instead of listing it this way, 90 00:06:01,960 --> 00:06:04,890 we have another way to list it if we're a chemist, 91 00:06:04,890 --> 00:06:10,120 and that is s, p, d, f, et cetera. 92 00:06:10,120 --> 00:06:13,810 So chemists like numbers, but we also throw in some letters 93 00:06:13,810 --> 00:06:15,890 every once in a while. 94 00:06:15,890 --> 00:06:20,310 And then m, again, designates this orbital orientation 95 00:06:20,310 --> 00:06:22,700 or the specific orbital. 96 00:06:22,700 --> 00:06:24,920 So for s, there's only s. 97 00:06:24,920 --> 00:06:29,620 It doesn't have any other designation, 98 00:06:29,620 --> 00:06:31,090 as we'll talk more about later. 99 00:06:31,090 --> 00:06:36,200 But for p, we start having suborbitals. 100 00:06:36,200 --> 00:06:38,030 And there is a difference in terms 101 00:06:38,030 --> 00:06:40,550 of the orientation of this. 102 00:06:40,550 --> 00:06:44,870 So we have px, py, pz. 103 00:06:44,870 --> 00:06:46,740 So that's what m tells us about. 104 00:06:46,740 --> 00:06:49,020 So if we have all three of these numbers, 105 00:06:49,020 --> 00:06:50,570 we get down to the specific orbital, 106 00:06:50,570 --> 00:06:54,220 we can say oh, that's pz, for example. 107 00:06:54,220 --> 00:06:58,880 So we need all of these three numbers to define the orbital. 108 00:06:58,880 --> 00:07:02,370 And this is in then the chemistry lingo. 109 00:07:02,370 --> 00:07:06,010 All right, also a little bit more chemistry lingo. 110 00:07:06,010 --> 00:07:11,090 So here we have l equals 0. 111 00:07:11,090 --> 00:07:13,053 So that is the s orbital. 112 00:07:16,230 --> 00:07:21,810 When l equals 1, that's the p orbital. 113 00:07:25,290 --> 00:07:30,020 l equals 2 is the d orbital. 114 00:07:30,020 --> 00:07:33,880 And l equals 3 is the f orbital. 115 00:07:33,880 --> 00:07:37,460 And frankly we don't really go much beyond that. 116 00:07:37,460 --> 00:07:39,010 And in this part of the course, we're 117 00:07:39,010 --> 00:07:43,680 really only going to be talking mostly about s and p orbitals. 118 00:07:43,680 --> 00:07:47,390 We get to d orbitals around Thanksgiving time. 119 00:07:47,390 --> 00:07:49,050 So you can look forward to that. 120 00:07:49,050 --> 00:07:51,770 And pretty much we're not going to really talk about f orbitals 121 00:07:51,770 --> 00:07:53,279 very much at all. 122 00:07:53,279 --> 00:07:55,070 You'll need to know some things about them, 123 00:07:55,070 --> 00:07:58,670 but we're not going to go into them in any kind of detail. 124 00:07:58,670 --> 00:08:01,640 All right, so if we keep going then, 125 00:08:01,640 --> 00:08:06,340 we can think about l equals 1 or our p orbitals. 126 00:08:06,340 --> 00:08:15,260 And then when l equals 1, then m can equal 0 plus 1 or minus 1. 127 00:08:15,260 --> 00:08:20,710 And when m equals 0, that's by definition the pz orbital. 128 00:08:20,710 --> 00:08:25,910 So when you see m equals 0, that's going to be pz. 129 00:08:25,910 --> 00:08:29,370 And when m is plus 1 or minus 1, those 130 00:08:29,370 --> 00:08:32,530 are the px or the py orbitals. 131 00:08:32,530 --> 00:08:35,090 And this is just something that you need to remember, 132 00:08:35,090 --> 00:08:37,330 that z is the one that's special. 133 00:08:37,330 --> 00:08:41,940 It's the one that has m equals 0. 134 00:08:41,940 --> 00:08:45,630 All right, so we can take all of the nomenclatures 135 00:08:45,630 --> 00:08:50,660 now and use it to fill in this awesome table. 136 00:08:50,660 --> 00:08:52,340 So this will help you kind of keep 137 00:08:52,340 --> 00:08:54,360 track of all the different ways you 138 00:08:54,360 --> 00:08:56,810 can designate the same things. 139 00:08:56,810 --> 00:08:58,460 And we'll fill this in. 140 00:08:58,460 --> 00:09:01,470 So first, state label. 141 00:09:01,470 --> 00:09:03,234 What do I mean by this? 142 00:09:03,234 --> 00:09:10,610 By this I mean this one 1, 0, 0 to generate this wave function 143 00:09:10,610 --> 00:09:15,680 where we have this 1, 0, 0 listed below the wave function 144 00:09:15,680 --> 00:09:16,380 here. 145 00:09:16,380 --> 00:09:19,490 And so now this is just a little color coding. 146 00:09:19,490 --> 00:09:21,460 But it's blank in your handout. 147 00:09:21,460 --> 00:09:24,380 So n equals 1, so n is first. 148 00:09:24,380 --> 00:09:26,520 l is the second number. 149 00:09:26,520 --> 00:09:29,030 And m is the third here. 150 00:09:29,030 --> 00:09:33,640 So 1, 0, 0, and what kind of orbital is this? 151 00:09:33,640 --> 00:09:35,316 You can just yell it out. 152 00:09:35,316 --> 00:09:36,150 AUDIENCE: 1s 153 00:09:36,150 --> 00:09:39,250 PROFESSOR: Yep, so that's the 1s orbital. 154 00:09:39,250 --> 00:09:44,070 And so the 1, n equals 1, that's 1s. 155 00:09:44,070 --> 00:09:48,600 And now we have our binding energies again. 156 00:09:48,600 --> 00:09:51,550 And so we can write those in two different ways. 157 00:09:51,550 --> 00:09:54,260 So we saw for the hydrogen atom before what 158 00:09:54,260 --> 00:09:56,430 comes out of the Schrodinger equation, 159 00:09:56,430 --> 00:09:59,440 that the binding energy of the electron for the nucleus 160 00:09:59,440 --> 00:10:04,340 is minus the Rydberg constant RH, divided by n squared. 161 00:10:04,340 --> 00:10:07,590 And here n is 1, so divided by 1 squared. 162 00:10:07,590 --> 00:10:11,700 So this is just the value for the Rydberg constant, 163 00:10:11,700 --> 00:10:12,910 the negative value. 164 00:10:12,910 --> 00:10:16,140 And binding energies, again, are always negative. 165 00:10:16,140 --> 00:10:19,000 So we have our first one down. 166 00:10:19,000 --> 00:10:22,920 So now for the second, what number 167 00:10:22,920 --> 00:10:25,120 am I going to write here for the state label? 168 00:10:25,120 --> 00:10:27,858 You can just yell it out. 169 00:10:27,858 --> 00:10:30,720 Yep, 200 or 2, 0, 0. 170 00:10:30,720 --> 00:10:32,880 And then you would put it this way 171 00:10:32,880 --> 00:10:36,210 where the state label is by the wave function. 172 00:10:36,210 --> 00:10:39,740 What orbital is this-- 2s. 173 00:10:39,740 --> 00:10:43,120 And then we also know the binding energies for this. 174 00:10:43,120 --> 00:10:46,670 So here we have minus RH over n squared 175 00:10:46,670 --> 00:10:48,900 where n is 2, 2 squared. 176 00:10:48,900 --> 00:10:51,840 And we saw this number last time. 177 00:10:51,840 --> 00:10:53,220 So we can keep going. 178 00:10:53,220 --> 00:10:56,390 Now we have 2, 1, 1. 179 00:10:56,390 --> 00:10:58,330 So we can write that down. 180 00:10:58,330 --> 00:11:00,240 We can write it both ways. 181 00:11:00,240 --> 00:11:02,876 What orbital is this? 182 00:11:02,876 --> 00:11:06,070 AUDIENCE: [INAUDIBLE]. 183 00:11:06,070 --> 00:11:08,440 PROFESSOR: So it's a 2p. 184 00:11:08,440 --> 00:11:14,522 And because n is plus 1 and not 0, it's either x or y. 185 00:11:17,430 --> 00:11:22,940 Do we have a different or the same binding energy here? 186 00:11:22,940 --> 00:11:26,830 We have the same, right, because it's just over n squared. 187 00:11:26,830 --> 00:11:30,790 We're still talking about n equals 2, so 2 squared. 188 00:11:30,790 --> 00:11:33,020 So it's the same value here. 189 00:11:33,020 --> 00:11:36,090 Now we have m equals 0. 190 00:11:36,090 --> 00:11:38,760 So we write 2, 1, 0. 191 00:11:38,760 --> 00:11:41,080 And now what is that orbital? 192 00:11:41,080 --> 00:11:43,310 AUDIENCE: [INAUDIBLE]. 193 00:11:43,310 --> 00:11:45,870 PROFESSOR: 2pz, right, because that's m 194 00:11:45,870 --> 00:11:48,630 equals 0, by the definition I gave you. 195 00:11:48,630 --> 00:11:51,380 So we know that one for sure. 196 00:11:51,380 --> 00:11:55,210 And again, the energies are going to be the same. 197 00:11:55,210 --> 00:12:01,400 And then the last one, so now we write 2, 1, minus 1. 198 00:12:01,400 --> 00:12:04,930 And now it's again a 2p orbital. 199 00:12:04,930 --> 00:12:08,180 And it's either y or x. 200 00:12:08,180 --> 00:12:11,780 And the energies are going to be the same. 201 00:12:11,780 --> 00:12:13,550 So these are just a table that kind 202 00:12:13,550 --> 00:12:15,600 of interconverts different ways that you 203 00:12:15,600 --> 00:12:17,210 will see things written. 204 00:12:17,210 --> 00:12:20,320 And you'll know if you see it one way, what 205 00:12:20,320 --> 00:12:21,860 orbital to put down. 206 00:12:21,860 --> 00:12:24,680 And we can also think about the binding energies 207 00:12:24,680 --> 00:12:27,410 for those particular orbitals, or for electrons 208 00:12:27,410 --> 00:12:30,600 in those particular orbitals. 209 00:12:30,600 --> 00:12:34,383 All right, so why don't you try a clicker question on this? 210 00:13:10,030 --> 00:13:10,600 10 seconds. 211 00:13:24,442 --> 00:13:26,520 Ah, excellent. 212 00:13:26,520 --> 00:13:27,380 Right. 213 00:13:27,380 --> 00:13:29,230 So you're getting the hang of this. 214 00:13:29,230 --> 00:13:29,840 It's great. 215 00:13:29,840 --> 00:13:31,140 Some things, it's always nice when 216 00:13:31,140 --> 00:13:33,306 there's some things that are pretty straightforward. 217 00:13:33,306 --> 00:13:35,350 So n equals 5. 218 00:13:35,350 --> 00:13:40,160 l equals 1, which means p orbital and m equals 0, 219 00:13:40,160 --> 00:13:40,914 means pz. 220 00:13:44,390 --> 00:13:47,950 So let's think now about these orbitals again. 221 00:13:47,950 --> 00:13:51,470 And we looked at that table and saw 222 00:13:51,470 --> 00:13:54,730 that if we were talking about n equals 2, 223 00:13:54,730 --> 00:13:57,050 they all seem to have the same energy. 224 00:13:57,050 --> 00:14:01,240 So for a hydrogen atom-- and it will get more complicated 225 00:14:01,240 --> 00:14:02,780 when we start talking about things 226 00:14:02,780 --> 00:14:04,350 with more than one electron. 227 00:14:04,350 --> 00:14:08,930 But for a hydrogen atom, orbitals that have the same n 228 00:14:08,930 --> 00:14:11,730 value have the same energy. 229 00:14:11,730 --> 00:14:15,180 So here we have n equals 1, l equals 0. 230 00:14:15,180 --> 00:14:17,600 This is our 1s. 231 00:14:17,600 --> 00:14:22,550 We have n equals 2, our 2s, and our 2p orbitals. 232 00:14:22,550 --> 00:14:28,660 n equals 3, we have our 3s, 3p, and 3d. 233 00:14:28,660 --> 00:14:32,840 And in this case, all these orbitals 234 00:14:32,840 --> 00:14:37,220 are what's known as degenerate with respect to each other. 235 00:14:37,220 --> 00:14:39,680 They have the same energy. 236 00:14:39,680 --> 00:14:46,290 And so for any n with a hydrogen atom, or any one electron 237 00:14:46,290 --> 00:14:51,650 system, for n shells, there n square degenerate-- 238 00:14:51,650 --> 00:14:54,370 or for any n there are n squared generate orbitals. 239 00:14:54,370 --> 00:14:56,220 So they're all going to be the same energy. 240 00:14:56,220 --> 00:14:59,420 And that changes when we go to more complicated systems. 241 00:14:59,420 --> 00:15:02,770 But for hydrogen, this holds. 242 00:15:02,770 --> 00:15:05,770 So now I'm going to tell you why you 243 00:15:05,770 --> 00:15:09,630 should care a little about these energy levels again. 244 00:15:09,630 --> 00:15:15,110 And today you're going to hear in their own words 245 00:15:15,110 --> 00:15:20,790 from a graduate student in the physical chemistry division. 246 00:15:20,790 --> 00:15:21,456 [VIDEO PLAYBACK] 247 00:15:21,456 --> 00:15:23,940 - My name is Benjamin Ofori-Okai. 248 00:15:23,940 --> 00:15:25,870 I'm entering my third year of graduate school 249 00:15:25,870 --> 00:15:28,380 in the chemistry department here at MIT. 250 00:15:28,380 --> 00:15:29,880 And the work that I've been focusing 251 00:15:29,880 --> 00:15:31,546 on for the last couple of years involves 252 00:15:31,546 --> 00:15:35,680 nanoscale magnetic resonance imaging or nano MRI. 253 00:15:35,680 --> 00:15:37,332 When you think of typical MRI, what 254 00:15:37,332 --> 00:15:39,540 comes to mind for most people is the image of a brain 255 00:15:39,540 --> 00:15:41,930 scan or a heart scan or some sort of organ 256 00:15:41,930 --> 00:15:44,930 scan inside the human body. 257 00:15:44,930 --> 00:15:47,570 The way that MRI works now, the way that you 258 00:15:47,570 --> 00:15:50,515 take a picture of anything in your body is you use water. 259 00:15:50,515 --> 00:15:51,890 And the reason that you use water 260 00:15:51,890 --> 00:15:54,660 is because it's made up of hydrogen atoms and oxygen 261 00:15:54,660 --> 00:15:55,390 atoms. 262 00:15:55,390 --> 00:15:58,370 And hydrogen atoms actually generate a magnetic signal. 263 00:15:58,370 --> 00:16:01,030 And so you can take a picture of that. 264 00:16:01,030 --> 00:16:06,290 The idea behind nano MRI is that you want to take a picture. 265 00:16:06,290 --> 00:16:08,060 You want to do the same kind of imaging, 266 00:16:08,060 --> 00:16:11,480 but on a considerably smaller scale. 267 00:16:11,480 --> 00:16:13,350 We have this probe which is sensitive 268 00:16:13,350 --> 00:16:16,432 to local magnetic fields. 269 00:16:16,432 --> 00:16:17,890 And the way that the probe works is 270 00:16:17,890 --> 00:16:19,610 that you have these electrons. 271 00:16:19,610 --> 00:16:21,540 There's a ground state for these electrons 272 00:16:21,540 --> 00:16:24,230 and two excited states for these electrons, which are actually 273 00:16:24,230 --> 00:16:25,490 degenerate with each other. 274 00:16:25,490 --> 00:16:28,250 And degenerate means that they just have the exact same energy 275 00:16:28,250 --> 00:16:29,460 level. 276 00:16:29,460 --> 00:16:31,430 As you move the probe around, anything 277 00:16:31,430 --> 00:16:34,060 that's in the environment that generates a magnetic field 278 00:16:34,060 --> 00:16:35,900 will change what the energy levels 279 00:16:35,900 --> 00:16:38,350 of these two excited states is. 280 00:16:38,350 --> 00:16:40,180 So when you're far away, there's no change 281 00:16:40,180 --> 00:16:41,456 and they're exactly the same. 282 00:16:41,456 --> 00:16:42,830 And as you get closer and closer, 283 00:16:42,830 --> 00:16:45,120 these levels start to split. 284 00:16:45,120 --> 00:16:46,832 And what we actually care about is 285 00:16:46,832 --> 00:16:48,790 what is the splitting between these two levels, 286 00:16:48,790 --> 00:16:53,650 because that's what tells us what the magnetic field is. 287 00:16:53,650 --> 00:16:55,550 In traditional MRI, the probe that we 288 00:16:55,550 --> 00:16:57,530 use, the thing that measures the fields, 289 00:16:57,530 --> 00:16:58,850 itself is very, very big. 290 00:16:58,850 --> 00:17:00,790 It's person sized. 291 00:17:00,790 --> 00:17:03,430 The probe that we're using in this nano MRI 292 00:17:03,430 --> 00:17:04,995 is nanometer sized. 293 00:17:04,995 --> 00:17:07,119 So this gives us the ability to look at things that 294 00:17:07,119 --> 00:17:08,750 are on the nanometer scale. 295 00:17:08,750 --> 00:17:11,660 And to give you a sense of size, that's like 1/10,000 the width 296 00:17:11,660 --> 00:17:13,410 of a human hair. 297 00:17:13,410 --> 00:17:17,990 So that includes viruses, cells, parts of proteins, 298 00:17:17,990 --> 00:17:19,480 not just the entire protein. 299 00:17:19,480 --> 00:17:22,481 And on top of that, we'll be able to look within objects. 300 00:17:22,481 --> 00:17:24,730 So you're not just sensitive to what's on the surface. 301 00:17:24,730 --> 00:17:26,869 You can actually see how are things-- 302 00:17:26,869 --> 00:17:27,869 what's the constitution? 303 00:17:27,869 --> 00:17:30,220 What's the makeup of things within the object that you 304 00:17:30,220 --> 00:17:31,330 want to image? 305 00:17:31,330 --> 00:17:33,550 So the long term goal, the one thing 306 00:17:33,550 --> 00:17:36,820 that I'd really love to see this technology be able to do 307 00:17:36,820 --> 00:17:38,910 is say, OK, we've got this virus. 308 00:17:38,910 --> 00:17:40,450 Let's just see how it works. 309 00:17:40,450 --> 00:17:42,280 Let's watch it in real time. 310 00:17:42,280 --> 00:17:44,990 Let's see if we can see how it attaches to cells 311 00:17:44,990 --> 00:17:48,270 and invades them and ultimately kills them. 312 00:17:48,270 --> 00:17:51,580 [END PLAYBACK] 313 00:17:51,580 --> 00:17:54,470 PROFESSOR: OK, so I always think this is a great time of year 314 00:17:54,470 --> 00:17:58,210 to show this video because pretty much viruses, I think, 315 00:17:58,210 --> 00:17:59,970 start to be on people's minds. 316 00:17:59,970 --> 00:18:03,320 Everyone has sinuses and colds and other things going on. 317 00:18:03,320 --> 00:18:07,310 And so understanding, we're still very far away 318 00:18:07,310 --> 00:18:10,550 from having a real cure for the common cold. 319 00:18:10,550 --> 00:18:19,430 So I think it's very timely to be talking about, 320 00:18:19,430 --> 00:18:21,620 talking about this research. 321 00:18:21,620 --> 00:18:25,000 I'll also use this to remind myself to tell you 322 00:18:25,000 --> 00:18:29,430 that if you qualify for extra time on the exam, 323 00:18:29,430 --> 00:18:31,410 you should get me your form for the exam. 324 00:18:31,410 --> 00:18:34,420 And it reminded me to say that because Ben, 325 00:18:34,420 --> 00:18:36,010 who is a former TA for this class, 326 00:18:36,010 --> 00:18:38,100 always proctors the extra time folks. 327 00:18:38,100 --> 00:18:40,340 So you'll get to meet him in real life 328 00:18:40,340 --> 00:18:44,830 if you qualify for extra time on exams. 329 00:18:44,830 --> 00:18:48,560 So hydrogen is in fact important. 330 00:18:48,560 --> 00:18:51,590 I'm excited to get on to elements that 331 00:18:51,590 --> 00:18:53,550 have more than one electron. 332 00:18:53,550 --> 00:18:56,530 But hydrogen actually does turn out to be extremely important. 333 00:18:56,530 --> 00:18:59,677 A lot of imaging, as you heard from Ben, is based on hydrogen. 334 00:18:59,677 --> 00:19:01,510 So we're spending a lot of time on hydrogen, 335 00:19:01,510 --> 00:19:05,500 but hydrogen really, really is an important element. 336 00:19:05,500 --> 00:19:11,320 So continuing on now, what is the significance 337 00:19:11,320 --> 00:19:12,583 of this wave function? 338 00:19:15,100 --> 00:19:17,780 Why do we care about this? 339 00:19:17,780 --> 00:19:20,120 And so really, we're interested in trying 340 00:19:20,120 --> 00:19:23,010 to understand not just how tightly the electron is 341 00:19:23,010 --> 00:19:27,420 bound to the nucleus, but kind of how the electrons exist 342 00:19:27,420 --> 00:19:29,700 around the nucleus. 343 00:19:29,700 --> 00:19:32,250 And so the wave function really gets at this. 344 00:19:32,250 --> 00:19:36,470 It gets at the probability density, the likelihood 345 00:19:36,470 --> 00:19:41,150 that you'll find an electron at a certain location, 346 00:19:41,150 --> 00:19:43,970 the probability per unit volume. 347 00:19:43,970 --> 00:19:46,820 And again, this is a three dimensional problem. 348 00:19:46,820 --> 00:19:50,840 So our wave function depends on a radius r. 349 00:19:50,840 --> 00:19:54,510 But it also depends on two angles, the theta and phi. 350 00:19:54,510 --> 00:19:57,510 And so you can kind of think of those as latitude and longitude 351 00:19:57,510 --> 00:19:58,350 if you will. 352 00:19:58,350 --> 00:20:02,640 And so we want to know what the probability is 353 00:20:02,640 --> 00:20:07,760 that an electron will be at a certain r, theta, and phi 354 00:20:07,760 --> 00:20:13,350 position in a particular small unit volume in that area. 355 00:20:13,350 --> 00:20:17,840 How well can we understand where the electron is? 356 00:20:17,840 --> 00:20:21,900 And this gives rise to a lot of the properties of the elements. 357 00:20:21,900 --> 00:20:27,380 So probability density, density per unit volume. 358 00:20:27,380 --> 00:20:30,590 So really, when we're talking about where electrons are, 359 00:20:30,590 --> 00:20:33,170 we're thinking about a shape of an orbital, 360 00:20:33,170 --> 00:20:37,670 a shape of a probability density of where that electron might 361 00:20:37,670 --> 00:20:38,620 be. 362 00:20:38,620 --> 00:20:42,500 So now we're going to think about shapes. 363 00:20:42,500 --> 00:20:46,120 So we can define a wave function in terms 364 00:20:46,120 --> 00:20:50,250 of two properties, a radial wave function and an angular wave 365 00:20:50,250 --> 00:20:51,030 function. 366 00:20:51,030 --> 00:20:54,270 So again, the wave function has these three things. 367 00:20:54,270 --> 00:20:57,220 We are considered with a radius and these two angles. 368 00:20:57,220 --> 00:20:59,990 So we can rewrite this, breaking up 369 00:20:59,990 --> 00:21:03,240 these two different components-- the radial component that 370 00:21:03,240 --> 00:21:05,510 depends on the radius-- so that's easy to remember, 371 00:21:05,510 --> 00:21:08,750 radial, radius-- and the angular component 372 00:21:08,750 --> 00:21:10,930 that depends on the angles. 373 00:21:10,930 --> 00:21:13,860 So the nomenclature here is pretty good. 374 00:21:13,860 --> 00:21:16,870 All right, so we have these two components. 375 00:21:16,870 --> 00:21:20,140 So now I'm going to show you a table that 376 00:21:20,140 --> 00:21:22,540 is largely from your book. 377 00:21:22,540 --> 00:21:24,040 Don't let it scare you. 378 00:21:24,040 --> 00:21:27,210 You do not need to memorize any of these things. 379 00:21:27,210 --> 00:21:28,930 And I'm showing this to you because I 380 00:21:28,930 --> 00:21:33,800 want you to believe me about certain properties of these two 381 00:21:33,800 --> 00:21:34,490 functions. 382 00:21:34,490 --> 00:21:35,610 So here they are solved. 383 00:21:35,610 --> 00:21:37,020 You can look them up. 384 00:21:37,020 --> 00:21:39,840 Actually I think we just typed a new copy of this 385 00:21:39,840 --> 00:21:41,070 so it was easier to see. 386 00:21:41,070 --> 00:21:44,002 If you find any typos, please let me know. 387 00:21:44,002 --> 00:21:45,710 But there's a couple of important points. 388 00:21:45,710 --> 00:21:49,380 So on this side, we have the radial wave function, 389 00:21:49,380 --> 00:21:52,620 and over here we have the angular wave function, 390 00:21:52,620 --> 00:21:55,070 for various values of n and l. 391 00:21:55,070 --> 00:21:58,010 So again, not an exhaustive list here. 392 00:21:58,010 --> 00:22:01,890 And a lot of these are written in terms of a0, 393 00:22:01,890 --> 00:22:09,960 which is the Bohr radius, which is a constant, 52.9 picometers. 394 00:22:09,960 --> 00:22:14,750 All right, so now let's just consider the ground state. 395 00:22:14,750 --> 00:22:16,990 So we'll start with that lowest energy 396 00:22:16,990 --> 00:22:20,860 state or most stable state, the 1s orbital for the hydrogen 397 00:22:20,860 --> 00:22:21,640 atom. 398 00:22:21,640 --> 00:22:25,360 So we have our wave function 1, 0, 0 here. 399 00:22:25,360 --> 00:22:27,700 And this is 1s up here. 400 00:22:27,700 --> 00:22:29,990 Again, n equals 1. l equals 0. 401 00:22:29,990 --> 00:22:31,960 So that's 1s. 402 00:22:31,960 --> 00:22:34,520 And z for hydrogen atom is 1. 403 00:22:34,520 --> 00:22:38,280 So I've gotten rid of all z's to make it a little simpler. 404 00:22:38,280 --> 00:22:40,470 So here we have the radial wave function 405 00:22:40,470 --> 00:22:44,730 times the angular wave function, which is listed up here. 406 00:22:44,730 --> 00:22:47,590 And the thing that I really want you to notice 407 00:22:47,590 --> 00:22:52,440 is that for all of the s orbitals, this is a constant. 408 00:22:52,440 --> 00:22:57,590 So this is always the angular component for all s orbitals. 409 00:22:57,590 --> 00:23:03,630 And in fact, there are no angular components in there. 410 00:23:03,630 --> 00:23:09,390 So all 1s, 2s, 3s, all have this same constant. 411 00:23:09,390 --> 00:23:11,890 And that leads to a very important property 412 00:23:11,890 --> 00:23:14,270 of s orbitals, which is that they're 413 00:23:14,270 --> 00:23:16,800 spherically symmetrical. 414 00:23:16,800 --> 00:23:20,580 In other words, they're independent of those angles, 415 00:23:20,580 --> 00:23:22,950 of theta and phi. 416 00:23:22,950 --> 00:23:26,010 And so that means that the probability of finding 417 00:23:26,010 --> 00:23:28,470 the electron away from the nucleus 418 00:23:28,470 --> 00:23:31,170 is just going to depend on r. 419 00:23:31,170 --> 00:23:33,680 There's only r in this equation. 420 00:23:33,680 --> 00:23:36,070 The angles are not part of the equation. 421 00:23:36,070 --> 00:23:40,250 So s is spherically symmetrical. 422 00:23:40,250 --> 00:23:42,250 The probability of finding the electron 423 00:23:42,250 --> 00:23:45,480 just depends on the radius. 424 00:23:45,480 --> 00:23:49,650 So we can draw a picture, or multiple pictures, 425 00:23:49,650 --> 00:23:52,060 of what that could look like. 426 00:23:52,060 --> 00:23:55,060 And these are three common plots. 427 00:23:55,060 --> 00:23:57,020 So I'll tell you that on your handout, 428 00:23:57,020 --> 00:23:59,030 the plots are listed on one page, 429 00:23:59,030 --> 00:24:01,670 and then the plots are shown on the next page. 430 00:24:01,670 --> 00:24:05,010 And I'm going to kind of go back and forth between things. 431 00:24:05,010 --> 00:24:08,360 So the plots-- don't have to write this down. 432 00:24:08,360 --> 00:24:09,540 They're on the other page. 433 00:24:09,540 --> 00:24:13,230 But if you want to pay attention to which kind of plot 434 00:24:13,230 --> 00:24:16,280 goes with which plot. 435 00:24:16,280 --> 00:24:19,310 So these are three different ways to, quote, visualize. 436 00:24:19,310 --> 00:24:24,100 And some people say, can you give me another visualization? 437 00:24:24,100 --> 00:24:27,130 We're really just trying to think about probabilities 438 00:24:27,130 --> 00:24:28,880 of finding electrons here. 439 00:24:28,880 --> 00:24:32,224 And so you can't sort of take a picture of an orbital. 440 00:24:32,224 --> 00:24:33,890 So these are just different ways to help 441 00:24:33,890 --> 00:24:38,370 people think about that possible distribution of electrons 442 00:24:38,370 --> 00:24:39,850 around the nucleus. 443 00:24:39,850 --> 00:24:42,880 All right, so one thing that everyone's feeling pretty good 444 00:24:42,880 --> 00:24:47,410 about is that it should be spherically symmetric 445 00:24:47,410 --> 00:24:49,710 hole for an s orbital. 446 00:24:49,710 --> 00:24:51,700 And so we have a circle. 447 00:24:51,700 --> 00:24:54,270 And so the probability density, which 448 00:24:54,270 --> 00:24:57,670 is shown in this plot-- and the probability density parts 449 00:24:57,670 --> 00:25:00,480 are basically just dots where the more concentrated 450 00:25:00,480 --> 00:25:02,550 the dots are, the higher the probability 451 00:25:02,550 --> 00:25:05,930 density for that particular-- the probability 452 00:25:05,930 --> 00:25:08,470 for that particular volume exists. 453 00:25:08,470 --> 00:25:12,190 So in here there are sort of more dots and then less dots 454 00:25:12,190 --> 00:25:13,660 as you come out. 455 00:25:13,660 --> 00:25:17,450 And so that is a circle, which is what? 456 00:25:17,450 --> 00:25:18,950 It's symmetrical. 457 00:25:18,950 --> 00:25:21,710 So you can always recognize a 1s. 458 00:25:21,710 --> 00:25:23,650 You have this symmetrical thing. 459 00:25:23,650 --> 00:25:29,240 So this is the wave function squared, is this probability 460 00:25:29,240 --> 00:25:31,000 density plot. 461 00:25:31,000 --> 00:25:34,180 Another kind of plot that you can see 462 00:25:34,180 --> 00:25:36,770 looks at the radial wave function 463 00:25:36,770 --> 00:25:40,750 plotted against the distance r here, 464 00:25:40,750 --> 00:25:43,120 distance from the nucleus. 465 00:25:43,120 --> 00:25:46,800 And then a third kind of plot is another probability plot, 466 00:25:46,800 --> 00:25:48,290 like this one up here. 467 00:25:48,290 --> 00:25:51,850 But instead of the dots indicating the higher 468 00:25:51,850 --> 00:25:55,300 probability density, you have a radial probability 469 00:25:55,300 --> 00:25:56,750 distribution. 470 00:25:56,750 --> 00:26:01,490 And so at the nucleus, at 0, well then 471 00:26:01,490 --> 00:26:02,960 the probability goes up. 472 00:26:02,960 --> 00:26:05,410 The electron is not going to crash into the nucleus, 473 00:26:05,410 --> 00:26:07,860 so it won't be right on top of the nucleus. 474 00:26:07,860 --> 00:26:10,350 But as you get out a little bit farther away, 475 00:26:10,350 --> 00:26:12,560 there's a high probability that it's there. 476 00:26:12,560 --> 00:26:14,630 And then that decreases again. 477 00:26:14,630 --> 00:26:16,380 So the top one and the bottom one 478 00:26:16,380 --> 00:26:19,900 both talk about the probability of finding an electron 479 00:26:19,900 --> 00:26:21,920 in a particular unit. 480 00:26:21,920 --> 00:26:24,880 And I'll give you just a little more definition of this. 481 00:26:24,880 --> 00:26:28,830 And this is on the same page above those different plots. 482 00:26:28,830 --> 00:26:32,470 So the radial probability distribution 483 00:26:32,470 --> 00:26:34,470 reports on the probability of finding 484 00:26:34,470 --> 00:26:37,750 an electron in the spherical shell 485 00:26:37,750 --> 00:26:41,410 at some little distance dr from the origin. 486 00:26:41,410 --> 00:26:43,100 And one thing that comes out of this, 487 00:26:43,100 --> 00:26:47,670 which is pretty important, is the most probable value 488 00:26:47,670 --> 00:26:50,980 for that distance r, which is denoted 489 00:26:50,980 --> 00:26:55,890 rmp, so most probable distance. 490 00:26:55,890 --> 00:27:00,930 And for a hydrogen atom, this is a0, the Bohr radius. 491 00:27:00,930 --> 00:27:06,190 And you can see it expressed in different units over here. 492 00:27:06,190 --> 00:27:11,080 And from the plot, that will be the top part of the plot, 493 00:27:11,080 --> 00:27:13,080 the most probable distance. 494 00:27:13,080 --> 00:27:17,190 In this case, that's the Bohr radius for the hydrogen atom. 495 00:27:17,190 --> 00:27:19,760 So we have now these three different kinds 496 00:27:19,760 --> 00:27:21,670 of plots that you'll see. 497 00:27:21,670 --> 00:27:23,990 And I want to point out that they're different plots. 498 00:27:23,990 --> 00:27:27,270 Sometimes people are thinking that there is sort of one plot 499 00:27:27,270 --> 00:27:30,260 and they're trying to read one of them as probability density, 500 00:27:30,260 --> 00:27:32,220 and that's not what it is. 501 00:27:32,220 --> 00:27:35,100 So we'll look at these again. 502 00:27:35,100 --> 00:27:37,260 All right, so going back and we'll just 503 00:27:37,260 --> 00:27:40,070 look at them again now that we sort of talked about what 504 00:27:40,070 --> 00:27:43,580 all of them are, again, we have our sort 505 00:27:43,580 --> 00:27:48,800 of dot density, probability density plot, our wave function 506 00:27:48,800 --> 00:27:54,240 plot, and our radial probability distribution plot. 507 00:27:54,240 --> 00:28:00,990 And for 1s, we have the dots closer to the nucleus here. 508 00:28:00,990 --> 00:28:03,250 Probability goes up and goes down. 509 00:28:03,250 --> 00:28:05,910 And here, you're thinking about this 510 00:28:05,910 --> 00:28:09,870 as the amplitude of finding an electron as you move away 511 00:28:09,870 --> 00:28:12,710 from the nucleus. 512 00:28:12,710 --> 00:28:14,380 So 1s is pretty simple. 513 00:28:14,380 --> 00:28:17,120 And I think these plots are a lot more meaningful when we go 514 00:28:17,120 --> 00:28:21,890 on to look at other orbitals. 515 00:28:21,890 --> 00:28:24,130 So let's think about those other orbitals. 516 00:28:24,130 --> 00:28:26,390 And we'll finish the other plots. 517 00:28:26,390 --> 00:28:30,256 So this is just-- you can actually stay, in this case. 518 00:28:30,256 --> 00:28:32,130 So we're going a lot of back and forth today. 519 00:28:32,130 --> 00:28:36,250 So here is your table that we had before. 520 00:28:36,250 --> 00:28:38,770 And here's 1s. 521 00:28:38,770 --> 00:28:40,030 Here's 2s. 522 00:28:40,030 --> 00:28:41,340 Here's 3s. 523 00:28:41,340 --> 00:28:44,770 These terms are in fact different, as you can see. 524 00:28:44,770 --> 00:28:48,370 But the angular term, as we mentioned before, 525 00:28:48,370 --> 00:28:49,580 is still the same. 526 00:28:49,580 --> 00:28:53,370 So that means 2s and 3s are still symmetrical. 527 00:28:53,370 --> 00:28:55,430 So we're still thinking about the probability 528 00:28:55,430 --> 00:28:58,440 of finding an electron in some volume 529 00:28:58,440 --> 00:29:02,300 as just going out as a distance of r. 530 00:29:02,300 --> 00:29:06,860 So let's look now at the three plots, and compare those plots. 531 00:29:06,860 --> 00:29:10,260 And this is the one on your handouts we looked at. 532 00:29:10,260 --> 00:29:11,200 I showed you this. 533 00:29:11,200 --> 00:29:15,750 And now we have all of these three plots together here. 534 00:29:15,750 --> 00:29:17,970 And in the comparison of these three, 535 00:29:17,970 --> 00:29:20,220 I think it helps differentiate what 536 00:29:20,220 --> 00:29:22,940 you're seeing in these plots. 537 00:29:22,940 --> 00:29:27,000 So important point, they're spherical. 538 00:29:27,000 --> 00:29:31,280 1s, 2s, 3s, they're all spherical. 539 00:29:31,280 --> 00:29:34,760 And here we see the dot density increase. 540 00:29:34,760 --> 00:29:37,930 And then the dot density goes to 0. 541 00:29:37,930 --> 00:29:41,250 And that's known as a node. 542 00:29:41,250 --> 00:29:46,210 So a node is a value of r or theta or phi 543 00:29:46,210 --> 00:29:48,810 for which the wave function and wave function 544 00:29:48,810 --> 00:29:52,730 squared, or the probability density, is 0. 545 00:29:52,730 --> 00:29:56,760 And in this particular case, the type of node that we're seeing 546 00:29:56,760 --> 00:29:59,050 is a radial node. 547 00:29:59,050 --> 00:30:02,730 And so that's a value of r for which the wave function, 548 00:30:02,730 --> 00:30:08,450 wave function squared probability density is 0. 549 00:30:08,450 --> 00:30:10,250 So it goes to 0. 550 00:30:10,250 --> 00:30:13,050 We have a node, a radial node. 551 00:30:13,050 --> 00:30:15,490 Then there's more probability. 552 00:30:15,490 --> 00:30:19,310 And then it increases, and then starts decreasing again. 553 00:30:19,310 --> 00:30:24,510 And so if you plot this with the radial wave function versus r, 554 00:30:24,510 --> 00:30:26,530 you see it go down. 555 00:30:26,530 --> 00:30:29,590 And it crosses the zero line here. 556 00:30:29,590 --> 00:30:31,120 And that's the node. 557 00:30:31,120 --> 00:30:34,200 And that's at 2a0. 558 00:30:34,200 --> 00:30:35,780 And then it goes back up. 559 00:30:35,780 --> 00:30:37,650 And this plot often bothers people. 560 00:30:37,650 --> 00:30:40,770 They're saying, what, there's now negative probability? 561 00:30:40,770 --> 00:30:43,610 No, these are not the probability diagrams. 562 00:30:43,610 --> 00:30:46,260 This is thinking about the amplitude 563 00:30:46,260 --> 00:30:47,880 of finding an electron. 564 00:30:47,880 --> 00:30:49,940 So we don't have to worry. 565 00:30:49,940 --> 00:30:53,550 It can have a positive or a negative phase to it. 566 00:30:53,550 --> 00:30:58,220 And if you look at this plot, the radial probability 567 00:30:58,220 --> 00:31:00,650 distribution plot, then you'll see 568 00:31:00,650 --> 00:31:04,610 that actually the radius, the most probable radius 569 00:31:04,610 --> 00:31:07,060 is in this region over here. 570 00:31:07,060 --> 00:31:11,560 And you see that this is concentrated dots up here. 571 00:31:11,560 --> 00:31:15,130 So if we think about these two, which are really probability 572 00:31:15,130 --> 00:31:18,080 distribution diagrams, we're thinking about the probability 573 00:31:18,080 --> 00:31:19,690 of finding an electron. 574 00:31:19,690 --> 00:31:23,840 You have a probability in here close to the nucleus. 575 00:31:23,840 --> 00:31:25,420 Then you get a node. 576 00:31:25,420 --> 00:31:28,520 And then you have another probability, high probability 577 00:31:28,520 --> 00:31:30,140 of finding the electron. 578 00:31:30,140 --> 00:31:33,930 In fact that's the most probable radius here for 2s. 579 00:31:33,930 --> 00:31:35,730 And then it decreases. 580 00:31:35,730 --> 00:31:39,330 So this line shows you what a radial node 581 00:31:39,330 --> 00:31:42,160 looks like in all three plots. 582 00:31:42,160 --> 00:31:47,760 In this probability diagram, wave function 583 00:31:47,760 --> 00:31:51,650 squared plot, it looks like there's just an empty space, 584 00:31:51,650 --> 00:31:52,930 no dots at all. 585 00:31:52,930 --> 00:31:56,050 Down here, it's where it crosses the line. 586 00:31:56,050 --> 00:32:00,430 And in the bottom plot, it is where you go up and down 587 00:32:00,430 --> 00:32:04,080 and again touches the line before going back up. 588 00:32:04,080 --> 00:32:06,410 So you should be able to look at these plots 589 00:32:06,410 --> 00:32:10,610 and think about what they mean. 590 00:32:10,610 --> 00:32:14,140 For 3s, we see the same thing. 591 00:32:14,140 --> 00:32:20,210 But now we have an intense spot in the middle near the nucleus. 592 00:32:20,210 --> 00:32:22,540 That is indicated down here. 593 00:32:22,540 --> 00:32:26,130 There is probability of finding the electron near the nucleus. 594 00:32:26,130 --> 00:32:27,510 Then there's a node. 595 00:32:27,510 --> 00:32:30,220 And that's in this plot where it crosses the line 596 00:32:30,220 --> 00:32:33,710 and in this plot where you have the empty space. 597 00:32:33,710 --> 00:32:37,360 Then you have more probability of finding the electron. 598 00:32:37,360 --> 00:32:39,850 You have another bump here. 599 00:32:39,850 --> 00:32:44,210 And then we have another node, indicated by touching the zero 600 00:32:44,210 --> 00:32:45,920 line here, touching here. 601 00:32:45,920 --> 00:32:48,090 That's at 7.1a0. 602 00:32:48,090 --> 00:32:50,730 And then we have more probability 603 00:32:50,730 --> 00:32:53,500 of finding the electron. 604 00:32:53,500 --> 00:33:00,560 And this is where the most probable radius is at 11.5. 605 00:33:00,560 --> 00:33:03,630 So again, you need to be able to look at these diagrams 606 00:33:03,630 --> 00:33:07,050 and recognize what constitutes a radial node. 607 00:33:07,050 --> 00:33:09,660 And a node is a place where there 608 00:33:09,660 --> 00:33:12,490 is no probability that you're going to find an electron. 609 00:33:16,710 --> 00:33:20,820 So now let's think about how many nodes, or radial 610 00:33:20,820 --> 00:33:24,020 nodes you should have when you have 611 00:33:24,020 --> 00:33:26,890 different types of orbitals. 612 00:33:26,890 --> 00:33:30,710 And this is just a similar diagram to what I just showed. 613 00:33:30,710 --> 00:33:35,730 This is the wave function squared, probability diagram. 614 00:33:35,730 --> 00:33:39,170 And now instead of blue you have orange dots, 615 00:33:39,170 --> 00:33:43,900 but otherwise should be the same-- so for 1s, for 2s, 616 00:33:43,900 --> 00:33:45,980 and 3s. 617 00:33:45,980 --> 00:33:49,680 So for the 1s orbital, we can calculate 618 00:33:49,680 --> 00:33:51,700 how many radial nodes that we should 619 00:33:51,700 --> 00:33:57,540 have by using this handy formula, n minus 1 minus l. 620 00:33:57,540 --> 00:34:01,430 So for 1s we have 1 minus 1. 621 00:34:01,430 --> 00:34:03,230 And l is 0. 622 00:34:03,230 --> 00:34:05,990 So we have zero radial nodes. 623 00:34:05,990 --> 00:34:08,290 And we can see that from that diagram 624 00:34:08,290 --> 00:34:11,040 there are zero radial nodes. 625 00:34:11,040 --> 00:34:14,790 2s now-- 2, n is 2. 626 00:34:14,790 --> 00:34:19,139 Minus 1, minus 0-- so that's one radial node. 627 00:34:19,139 --> 00:34:22,040 And the radial node, again, in this kind of diagram 628 00:34:22,040 --> 00:34:23,810 is the empty space. 629 00:34:23,810 --> 00:34:26,330 And that radial node is at 2a0. 630 00:34:29,679 --> 00:34:37,010 For 3s, we have n equals 3 minus 1 minus l, which is still 0. 631 00:34:37,010 --> 00:34:39,070 So we have two radial nodes. 632 00:34:39,070 --> 00:34:44,630 And so again, the empty space here at 1.9a0 and then 633 00:34:44,630 --> 00:34:48,070 at 7.1a0. 634 00:34:48,070 --> 00:34:50,469 So why don't you give this a try now 635 00:34:50,469 --> 00:34:55,210 and tell me what kind of radial nodes you would expect for 4p. 636 00:35:20,090 --> 00:35:21,377 OK, 10 seconds. 637 00:35:21,377 --> 00:35:22,293 These are pretty fast. 638 00:35:38,020 --> 00:35:39,470 Yep. 639 00:35:39,470 --> 00:35:44,240 So again, we have to do n, which is 4, minus 1. 640 00:35:44,240 --> 00:35:48,530 And then what is l in this case-- 1. 641 00:35:48,530 --> 00:35:51,250 So that gives you 2. 642 00:35:51,250 --> 00:35:58,674 All right, so 4 minus 1 minus 1 or 2 radial nodes. 643 00:35:58,674 --> 00:36:00,340 All right, don't put your clickers away. 644 00:36:00,340 --> 00:36:02,300 Let's try something else. 645 00:36:05,380 --> 00:36:10,120 So now tell me which of these is correct both in terms 646 00:36:10,120 --> 00:36:13,080 of the indicated number of radial nodes 647 00:36:13,080 --> 00:36:16,940 and in terms of the plot for a 5s orbital. 648 00:37:04,650 --> 00:37:06,436 All right, let's just do 10 more seconds. 649 00:37:22,980 --> 00:37:30,310 We're varying it up in terms of the plots. 650 00:37:30,310 --> 00:37:40,970 So maybe someone want to say what the right answer is here? 651 00:37:40,970 --> 00:37:41,470 Yeah? 652 00:37:46,440 --> 00:37:48,960 AUDIENCE: So by the formula we just did, 653 00:37:48,960 --> 00:37:51,270 that has four radial nodes. 654 00:37:51,270 --> 00:37:54,060 And if you look at the graph of one, there's three, 655 00:37:54,060 --> 00:37:56,260 and then there's another one at the origin. 656 00:37:56,260 --> 00:37:59,110 So that's four radial nodes. 657 00:37:59,110 --> 00:38:01,040 Right? 658 00:38:01,040 --> 00:38:02,400 Right? 659 00:38:02,400 --> 00:38:04,730 PROFESSOR: Actually, I just realized 660 00:38:04,730 --> 00:38:06,650 that-- let me count here. 661 00:38:06,650 --> 00:38:12,300 So this answer here, we should have four radial nodes. 662 00:38:12,300 --> 00:38:16,960 That is correct because we have n minus 1 minus l. 663 00:38:16,960 --> 00:38:20,000 Actually, I think this is going to this-- 664 00:38:20,000 --> 00:38:21,500 this should be going to this answer, 665 00:38:21,500 --> 00:38:27,250 because if we count 1, 2, 3, 4. 666 00:38:27,250 --> 00:38:29,700 Sorry, the new plot is highly confusing. 667 00:38:29,700 --> 00:38:30,930 I have to count. 668 00:38:30,930 --> 00:38:33,855 So the one at the origin should actually not count. 669 00:38:33,855 --> 00:38:34,980 AUDIENCE: It doesn't count? 670 00:38:34,980 --> 00:38:36,526 PROFESSOR: This is not a node. 671 00:38:41,270 --> 00:38:45,835 So we have 1, 2, 3, 4, should be our four radial nodes. 672 00:38:45,835 --> 00:38:49,640 Because that's a nucleus, and there isn't one there. 673 00:38:49,640 --> 00:38:51,450 But that doesn't count as a node. 674 00:38:51,450 --> 00:38:54,780 So this should be here. 675 00:38:54,780 --> 00:38:56,360 I guess that's-- right. 676 00:38:56,360 --> 00:39:03,220 But thank you very much, and [INAUDIBLE], here. 677 00:39:06,990 --> 00:39:10,049 You were brave enough to answer. 678 00:39:10,049 --> 00:39:11,090 Yeah, there's a question? 679 00:39:15,880 --> 00:39:18,490 AUDIENCE: Should there also be a certain number of peaks 680 00:39:18,490 --> 00:39:22,044 in the graph as well as nodes? 681 00:39:22,044 --> 00:39:22,710 PROFESSOR: Yeah. 682 00:39:22,710 --> 00:39:28,840 So if you look at the peaks, these are really hard to draw. 683 00:39:28,840 --> 00:39:31,420 And I think that's partly what the problem is. 684 00:39:31,420 --> 00:39:36,450 But when we look later in the handout 685 00:39:36,450 --> 00:39:41,070 where they're drawn a little bit more carefully, 686 00:39:41,070 --> 00:39:42,650 it does increase. 687 00:39:42,650 --> 00:39:44,590 So there are different numbers. 688 00:39:44,590 --> 00:39:47,940 So we'll have nodes going down here. 689 00:39:47,940 --> 00:39:51,080 But then we'll have more distributions. 690 00:39:51,080 --> 00:39:55,250 But often the ones as you go along, 691 00:39:55,250 --> 00:40:00,370 it does indicate where the most probable radius is 692 00:40:00,370 --> 00:40:05,660 as the taller ones, and that it's usually drawn at the end. 693 00:40:05,660 --> 00:40:07,960 So we have some plots and I'll point this out later. 694 00:40:12,272 --> 00:40:14,230 We're going to look at more plots, don't worry. 695 00:40:21,320 --> 00:40:22,900 So if anyone's good at drawing those, 696 00:40:22,900 --> 00:40:25,571 let me know, because they're really hard to draw. 697 00:40:25,571 --> 00:40:27,320 So a lot of them are copied from the book, 698 00:40:27,320 --> 00:40:29,760 but then they don't copy very well. 699 00:40:29,760 --> 00:40:32,740 So let's consider other kinds of nodes. 700 00:40:32,740 --> 00:40:35,180 And we're going to come back to radial nodes. 701 00:40:35,180 --> 00:40:38,030 All right, so what about p orbitals? 702 00:40:38,030 --> 00:40:39,730 So here we have our table again. 703 00:40:39,730 --> 00:40:42,270 These are our p orbitals over here. 704 00:40:45,050 --> 00:40:51,880 And we have our n equals 2 cases here and our l equals 1. 705 00:40:51,880 --> 00:40:58,460 So these are x, y, and z-- so our 3p orbitals over here. 706 00:40:58,460 --> 00:41:01,120 And the important point is not to memorize 707 00:41:01,120 --> 00:41:02,580 what these values are. 708 00:41:02,580 --> 00:41:06,220 But now all of a sudden we have dependence on angles. 709 00:41:06,220 --> 00:41:10,400 So we're going to have an angular component to these. 710 00:41:10,400 --> 00:41:13,480 And that means the probability density 711 00:41:13,480 --> 00:41:17,150 as you go out from the nucleus doesn't just 712 00:41:17,150 --> 00:41:18,830 depend on r anymore. 713 00:41:18,830 --> 00:41:22,260 It depends on theta and phi, which 714 00:41:22,260 --> 00:41:29,530 are sort of the equivalent to latitude and longitude, 715 00:41:29,530 --> 00:41:32,280 if you're thinking about geography. 716 00:41:32,280 --> 00:41:34,500 All right, so let's see what that looks like. 717 00:41:34,500 --> 00:41:36,420 So that means then the p orbitals 718 00:41:36,420 --> 00:41:42,870 are not spherically symmetric, because it depends on angle. 719 00:41:42,870 --> 00:41:47,080 So you just don't go out and have the probability depend 720 00:41:47,080 --> 00:41:48,960 on the radius and it's symmetrical 721 00:41:48,960 --> 00:41:51,642 in all the different directions. 722 00:41:51,642 --> 00:41:53,350 And here are what some of them look like. 723 00:41:53,350 --> 00:41:55,930 These figures are in your handouts. 724 00:41:55,930 --> 00:41:57,122 Here are some other figures. 725 00:41:59,880 --> 00:42:04,060 So the orbitals consists of two lobes. 726 00:42:04,060 --> 00:42:07,740 So you could view this as a lobe up here and a lobe down here. 727 00:42:07,740 --> 00:42:11,900 Or you have these lobes as these two different colors over here. 728 00:42:11,900 --> 00:42:15,710 And the lobes are separated by a nodal plane. 729 00:42:15,710 --> 00:42:19,630 And the nodal plane is a plane on which the probability 730 00:42:19,630 --> 00:42:23,550 of finding the electrons is 0. 731 00:42:23,550 --> 00:42:27,200 So in the top drawing, the nodal plane is drawn as a plane. 732 00:42:27,200 --> 00:42:30,670 And in the bottom drawings, you don't see a plane. 733 00:42:30,670 --> 00:42:33,130 You just see an empty space between the lobes. 734 00:42:33,130 --> 00:42:36,550 So empty space here, empty space here, empty space there. 735 00:42:36,550 --> 00:42:38,680 And so if it helps you to kind of think 736 00:42:38,680 --> 00:42:42,570 about an actual plane in between, that's good. 737 00:42:42,570 --> 00:42:43,990 Or you can just think that there's 738 00:42:43,990 --> 00:42:47,370 a break between these nodes. 739 00:42:47,370 --> 00:42:49,960 And again, the nodal plane, there's 740 00:42:49,960 --> 00:42:55,300 no probability of finding an electron in the nodal planes. 741 00:42:55,300 --> 00:42:57,990 And the nodal planes are at the nucleus. 742 00:42:57,990 --> 00:43:00,360 Therefore, there is zero probability 743 00:43:00,360 --> 00:43:03,440 of finding a p electron at the nucleus. 744 00:43:03,440 --> 00:43:05,560 s can get pretty close to the nucleus. 745 00:43:05,560 --> 00:43:09,980 But with a p orbital, there's a nodal plane there. 746 00:43:09,980 --> 00:43:14,530 No electrons are going to be at the nucleus. 747 00:43:14,530 --> 00:43:18,350 So now if you're going out from the nucleus, the probability 748 00:43:18,350 --> 00:43:21,280 of an electron, finding it, if you're 749 00:43:21,280 --> 00:43:23,660 going out in this direction, you're 750 00:43:23,660 --> 00:43:25,010 not going to do very well. 751 00:43:25,010 --> 00:43:26,700 If you're going in this direction, 752 00:43:26,700 --> 00:43:28,110 you should do a lot better. 753 00:43:28,110 --> 00:43:30,560 So here the angular components really matter. 754 00:43:30,560 --> 00:43:32,960 That defines the shape of the orbital. 755 00:43:32,960 --> 00:43:35,110 And where you're going, what direction you're 756 00:43:35,110 --> 00:43:36,810 going in, what angles you're going 757 00:43:36,810 --> 00:43:39,250 in matters in terms of whether you're going 758 00:43:39,250 --> 00:43:43,020 to find that electron or not. 759 00:43:43,020 --> 00:43:45,680 So another way to think about this 760 00:43:45,680 --> 00:43:50,140 in sort of these nodal planes-- so here we'll 761 00:43:50,140 --> 00:43:52,860 just define what plane it is. 762 00:43:52,860 --> 00:43:55,120 So we have our pz orbital. 763 00:43:55,120 --> 00:43:58,790 That's a nodal plane then in x and y. 764 00:43:58,790 --> 00:44:01,680 And so x and y are over here. 765 00:44:01,680 --> 00:44:04,620 Our px orbital is going to be in-- 766 00:44:04,620 --> 00:44:10,070 or the nodal plane is going to be in yz plane, so over here. 767 00:44:10,070 --> 00:44:15,410 And py will be in xz plane. 768 00:44:15,410 --> 00:44:17,300 So again, these nodal planes, there's 769 00:44:17,300 --> 00:44:20,300 no electron density there. 770 00:44:20,300 --> 00:44:23,330 And these arise from these angular nodes 771 00:44:23,330 --> 00:44:25,190 in the wave function. 772 00:44:25,190 --> 00:44:28,090 So angular nodes then or these angular 773 00:44:28,090 --> 00:44:32,440 nodal planes are values of theta and phi 774 00:44:32,440 --> 00:44:38,060 for which the wave function, wave function squared are 0. 775 00:44:38,060 --> 00:44:41,060 So this is very different from the s case 776 00:44:41,060 --> 00:44:43,810 where we only had radial nodes. 777 00:44:43,810 --> 00:44:46,990 But now, when in the p orbitals where 778 00:44:46,990 --> 00:44:51,870 the angular component matters, they're angular nodes as well. 779 00:44:51,870 --> 00:44:56,820 So we can think about how to calculate the angular nodes. 780 00:44:56,820 --> 00:45:02,990 So total nodes is going to be equal to n minus 1. 781 00:45:02,990 --> 00:45:06,870 The angular nodes is l. 782 00:45:06,870 --> 00:45:15,840 And as we saw before, the radial nodes are n minus 1 minus l. 783 00:45:15,840 --> 00:45:19,310 So let's have more practice in calculating these. 784 00:45:19,310 --> 00:45:22,200 And then we'll look at some more diagrams. 785 00:45:22,200 --> 00:45:27,410 So for 2s, total nodes-- and you can just yell this out. 786 00:45:27,410 --> 00:45:28,910 Total nodes will be what? 787 00:45:28,910 --> 00:45:30,180 AUDIENCE: 1 788 00:45:30,180 --> 00:45:33,460 PROFESSOR: 1-- 2 minus 1 or 1. 789 00:45:33,460 --> 00:45:35,200 Angular nodes are? 790 00:45:35,200 --> 00:45:36,260 AUDIENCE: 0 791 00:45:36,260 --> 00:45:37,300 PROFESSOR: 0. 792 00:45:37,300 --> 00:45:38,780 For 1s, there is none. 793 00:45:38,780 --> 00:45:41,510 And if you forget, l equals 0 there. 794 00:45:41,510 --> 00:45:43,530 Radial nodes is going to be? 795 00:45:43,530 --> 00:45:46,530 AUDIENCE: 1 796 00:45:46,530 --> 00:45:51,800 PROFESSOR: Right, 2 minus 1 minus 0, or 1. 797 00:45:51,800 --> 00:45:56,830 All right, let's try 3-- or sorry, 2p is next. 798 00:45:56,830 --> 00:46:00,000 Total nodes? 799 00:46:00,000 --> 00:46:03,520 1 again, so 2 minus 1 or 1. 800 00:46:03,520 --> 00:46:05,620 Angular nodes? 801 00:46:05,620 --> 00:46:07,850 1-- l equals 1 here. 802 00:46:07,850 --> 00:46:10,840 And radial node? 803 00:46:10,840 --> 00:46:14,920 Right, 2 minus 1 minus 1, or 0. 804 00:46:14,920 --> 00:46:17,370 So since there's only one total node, 805 00:46:17,370 --> 00:46:19,450 if you figured out there was one angular node, 806 00:46:19,450 --> 00:46:22,650 you could even realize that there had to be zero there. 807 00:46:22,650 --> 00:46:25,990 It's a way to check maybe your equations. 808 00:46:25,990 --> 00:46:29,840 All right, so let's try for 3d now. 809 00:47:03,192 --> 00:47:03,900 How are we doing? 810 00:47:16,454 --> 00:47:18,162 All right, let's just do 10 more seconds. 811 00:47:38,270 --> 00:47:43,220 And let's just work that out over here. 812 00:47:43,220 --> 00:47:51,390 So total nodes for 3d, we have 3 minus 1 or 2. 813 00:47:51,390 --> 00:47:56,650 Angular nodes, l equals 2 for d. 814 00:47:56,650 --> 00:48:04,450 So radial nodes, we have 3 minus 1 minus 2, or 0. 815 00:48:04,450 --> 00:48:07,890 All right, so bring these handouts on Wednesday 816 00:48:07,890 --> 00:48:11,310 because we need to go back and look at more radial probability 817 00:48:11,310 --> 00:48:12,480 diagrams. 818 00:48:12,480 --> 00:48:14,150 And talk more about nodes. 819 00:48:36,230 --> 00:48:38,320 All right, let's just do 10 more seconds. 820 00:48:59,200 --> 00:49:04,070 OK, good job everyone. 821 00:49:04,070 --> 00:49:06,330 Let's look through this a little bit. 822 00:49:06,330 --> 00:49:08,420 And you can sort of-- everyone can help. 823 00:49:08,420 --> 00:49:10,330 Yell out some responses. 824 00:49:10,330 --> 00:49:12,380 So this was 2s. 825 00:49:12,380 --> 00:49:15,500 And that was the correct answer. 826 00:49:15,500 --> 00:49:20,540 Which type of orbital is this-- 2p. 827 00:49:20,540 --> 00:49:23,780 And if you couldn't read this information here, 828 00:49:23,780 --> 00:49:26,140 you should have been able to read the information 829 00:49:26,140 --> 00:49:28,000 about the nodes. 830 00:49:28,000 --> 00:49:32,340 What equation is that for nodes? 831 00:49:32,340 --> 00:49:36,010 Yeah, n minus 1 minus l, for what kind of nodes? 832 00:49:36,010 --> 00:49:36,840 AUDIENCE: Radial. 833 00:49:36,840 --> 00:49:38,210 PROFESSOR: Radial nodes, right. 834 00:49:38,210 --> 00:49:43,710 So if you know what it means if l equals 0 versus l equals 1, 835 00:49:43,710 --> 00:49:45,580 and you knew this was l, then you 836 00:49:45,580 --> 00:49:48,840 could tell if it was an s orbital or a p orbital. 837 00:49:48,840 --> 00:49:53,970 And then whether it was 2 or 3p is from the n. 838 00:49:53,970 --> 00:49:55,570 So even if you couldn't read this, 839 00:49:55,570 --> 00:49:59,350 if you knew that expression, then you were OK. 840 00:49:59,350 --> 00:50:03,440 What kind of orbital was in plot C? 841 00:50:03,440 --> 00:50:05,610 This was a 3s. 842 00:50:05,610 --> 00:50:07,680 l equals 0. 843 00:50:07,680 --> 00:50:11,205 And then this is a what, 3p and? 844 00:50:14,390 --> 00:50:17,390 l equals 2. 845 00:50:17,390 --> 00:50:18,730 Louder. 846 00:50:18,730 --> 00:50:20,260 D, right? 847 00:50:20,260 --> 00:50:27,960 So do 3px, 3py, and 3pz have different plots? 848 00:50:27,960 --> 00:50:31,830 No, they wouldn't have different plots. 849 00:50:31,830 --> 00:50:35,470 So we'll continue to look at this. 850 00:50:35,470 --> 00:50:38,210 And we're going to be starting with the handout 851 00:50:38,210 --> 00:50:39,370 from last time. 852 00:50:39,370 --> 00:50:41,710 And so let's continue with Monday 853 00:50:41,710 --> 00:50:48,650 and continue with these radial probability distributions. 854 00:50:48,650 --> 00:50:51,210 So this is again from Monday, page 6. 855 00:50:51,210 --> 00:50:53,400 We're talking about orbital size. 856 00:50:53,400 --> 00:50:55,920 And we've already looked at this a little bit today. 857 00:50:55,920 --> 00:50:58,700 So we should be able to go through this now 858 00:50:58,700 --> 00:51:00,040 in a little bit more detail. 859 00:51:00,040 --> 00:51:01,960 You've already thought about it. 860 00:51:01,960 --> 00:51:05,110 So here we have the 2s orbital. 861 00:51:05,110 --> 00:51:07,400 And we're going to have one node using 862 00:51:07,400 --> 00:51:12,920 our equation that you just told me, n minus 1 minus l. 863 00:51:12,920 --> 00:51:19,690 And when we go from 2s to 2p, here we have no radial nodes. 864 00:51:19,690 --> 00:51:23,090 And we can look at r and p, which 865 00:51:23,090 --> 00:51:26,940 is the radius of the maximal probability of finding 866 00:51:26,940 --> 00:51:28,250 an electron. 867 00:51:28,250 --> 00:51:33,480 And you can note that when you go from the 2s to the 2p, 868 00:51:33,480 --> 00:51:36,300 the radius actually decreases. 869 00:51:36,300 --> 00:51:42,090 So the most probable radius for 2p is less than that of 2s. 870 00:51:42,090 --> 00:51:47,020 Now let's consider the 3, n equals 3. 871 00:51:47,020 --> 00:51:52,200 So we have the 3s situation over here. 872 00:51:52,200 --> 00:51:53,710 And so l equals 0. 873 00:51:53,710 --> 00:51:56,290 We have two nodes here. 874 00:51:56,290 --> 00:52:00,730 And now if you look at the radius, the axis over here, 875 00:52:00,730 --> 00:52:03,610 you'll see that the most probable for 2s 876 00:52:03,610 --> 00:52:08,450 is close to 5a0, where a0 is the Bohr radius. 877 00:52:08,450 --> 00:52:12,470 And over here you're talking between 10 and 15. 878 00:52:12,470 --> 00:52:16,660 So we see an increase in size going this way. 879 00:52:16,660 --> 00:52:23,590 And then when we go from 3s to 3p-- so here we have 3 minus 1 880 00:52:23,590 --> 00:52:25,690 minus l, which is 1. 881 00:52:25,690 --> 00:52:33,400 So we have one node, down to 3d, 3 minus 1 minus 2, zero nodes. 882 00:52:33,400 --> 00:52:36,910 And you see that there is a decrease here 883 00:52:36,910 --> 00:52:40,320 in the most probable radius. 884 00:52:40,320 --> 00:52:43,960 So, OK, interesting. 885 00:52:43,960 --> 00:52:51,540 All right, so 3d has the smallest, next 3p, next 3s. 886 00:52:51,540 --> 00:52:54,700 So there's two different trends we're seeing. 887 00:52:54,700 --> 00:53:00,330 One, as we increase l within the same n number, 888 00:53:00,330 --> 00:53:06,440 and one going from a smaller value of n to a larger value, 889 00:53:06,440 --> 00:53:10,950 and then again within the 3, within the n value 890 00:53:10,950 --> 00:53:13,040 as we change l. 891 00:53:13,040 --> 00:53:17,580 So again, to say the same thing in a different way, 892 00:53:17,580 --> 00:53:24,930 as n increases from 2 to 3, the radius, most probable radius 893 00:53:24,930 --> 00:53:27,300 or the size increases. 894 00:53:27,300 --> 00:53:31,040 So from here to here we have an increase in size. 895 00:53:33,850 --> 00:53:35,550 I just want to make sure people have 896 00:53:35,550 --> 00:53:41,560 time to kind of get all of this down, but it should be good. 897 00:53:41,560 --> 00:53:43,490 I have a little picture that just shows 898 00:53:43,490 --> 00:53:46,200 they're very different in size. 899 00:53:46,200 --> 00:53:49,260 So we'll go back to this again. 900 00:53:49,260 --> 00:53:54,220 And then as I also said, as l increases for a given n-- 901 00:53:54,220 --> 00:53:59,090 so from l equals 0 to l equals 1 here, 902 00:53:59,090 --> 00:54:02,880 then we have a decrease in the size. 903 00:54:02,880 --> 00:54:07,060 So you can see the most probable radius moves over. 904 00:54:07,060 --> 00:54:09,615 And then here is another within n. 905 00:54:09,615 --> 00:54:11,670 And n equals 3. 906 00:54:11,670 --> 00:54:15,870 We see, again, this decrease. 907 00:54:15,870 --> 00:54:18,300 So those are the two trends that you 908 00:54:18,300 --> 00:54:21,740 observe when you look at these radial probability 909 00:54:21,740 --> 00:54:23,010 distributions. 910 00:54:23,010 --> 00:54:25,130 So for exam one next week, you should 911 00:54:25,130 --> 00:54:27,790 be able to draw distributions like this. 912 00:54:27,790 --> 00:54:30,470 You should be able to tell me how many radial 913 00:54:30,470 --> 00:54:34,430 nodes you have for different types of orbitals. 914 00:54:34,430 --> 00:54:37,230 And you should know these trends in size. 915 00:54:37,230 --> 00:54:40,740 So I think in the exam instructions 916 00:54:40,740 --> 00:54:43,700 it says up to a 5 case. 917 00:54:43,700 --> 00:54:46,060 You don't have to go on forever to be able to draw them, 918 00:54:46,060 --> 00:54:47,685 but you should be able to look at these 919 00:54:47,685 --> 00:54:51,440 and tell what kind of orbital it is and where the nodes are, 920 00:54:51,440 --> 00:54:54,760 be able to draw where the nodes are-- one node here, 921 00:54:54,760 --> 00:54:57,020 one, two, one node here. 922 00:54:57,020 --> 00:55:00,870 This kind of thing will be on the exam next week. 923 00:55:00,870 --> 00:55:03,500 So there's something that's a little counterintuitive when 924 00:55:03,500 --> 00:55:05,870 it comes to this size issue. 925 00:55:05,870 --> 00:55:09,220 And that has to do with how this correlates 926 00:55:09,220 --> 00:55:12,320 to the amount of shielding, and as we see later, 927 00:55:12,320 --> 00:55:14,160 to the energy levels. 928 00:55:14,160 --> 00:55:18,020 So only electrons in the s state here 929 00:55:18,020 --> 00:55:21,190 really have any kind of substantial probability 930 00:55:21,190 --> 00:55:23,330 that they'll be close to the nucleus. 931 00:55:23,330 --> 00:55:26,550 So we have this little blip over here 932 00:55:26,550 --> 00:55:29,840 that is close to the nucleus, that at are very small 933 00:55:29,840 --> 00:55:33,120 radii, very small values of r. 934 00:55:33,120 --> 00:55:35,990 Even though the most probable is out here, 935 00:55:35,990 --> 00:55:40,380 if we compare 3s to 3p and look at where the electrons are 936 00:55:40,380 --> 00:55:43,590 that are closest to the nucleus, they're quite a bit 937 00:55:43,590 --> 00:55:46,210 farther away than in the 3s. 938 00:55:46,210 --> 00:55:48,820 Or there's more probability that there's 939 00:55:48,820 --> 00:55:50,690 going to be some closer here. 940 00:55:50,690 --> 00:55:54,530 And then the closest probability over here for these electrons 941 00:55:54,530 --> 00:55:56,340 is quite a bit farther away. 942 00:55:56,340 --> 00:55:59,070 So we see these circles kind of move out. 943 00:55:59,070 --> 00:56:01,900 So even though the overall radius, 944 00:56:01,900 --> 00:56:05,310 the sort of size of the whole thing is decreasing, 945 00:56:05,310 --> 00:56:06,860 the probability that there are going 946 00:56:06,860 --> 00:56:10,190 to be electrons really close is actually 947 00:56:10,190 --> 00:56:12,420 going in the opposite direction. 948 00:56:12,420 --> 00:56:15,840 And so what this means is that s electrons 949 00:56:15,840 --> 00:56:18,980 are the least shielded because there's 950 00:56:18,980 --> 00:56:23,100 higher probability that they'll be some close to the nucleus. 951 00:56:23,100 --> 00:56:26,500 There's more penetration close to the nucleus. 952 00:56:26,500 --> 00:56:30,202 So s electrons are the least shielded. 953 00:56:30,202 --> 00:56:31,910 And we're going to come back to this when 954 00:56:31,910 --> 00:56:34,120 we move on to today's handout. 955 00:56:34,120 --> 00:56:36,020 This is really important in terms of thinking 956 00:56:36,020 --> 00:56:38,400 about the energy levels. 957 00:56:38,400 --> 00:56:40,280 And I'm going to have these diagrams 958 00:56:40,280 --> 00:56:42,230 on the handout for today. 959 00:56:42,230 --> 00:56:44,350 So we'll see them again. 960 00:56:44,350 --> 00:56:46,960 All right, so before we move to that handout, 961 00:56:46,960 --> 00:56:49,740 we've got to finish our quantum numbers 962 00:56:49,740 --> 00:56:53,500 and talk about electron spin. 963 00:56:53,500 --> 00:56:56,830 So the fourth quantum number describes 964 00:56:56,830 --> 00:56:59,130 the spin on the electron. 965 00:56:59,130 --> 00:57:03,460 And we already saw the magnetic quantum number m. 966 00:57:03,460 --> 00:57:05,310 We saw m sub l. 967 00:57:05,310 --> 00:57:07,790 And now we have m sub s. 968 00:57:07,790 --> 00:57:10,470 And the s stands for spin. 969 00:57:10,470 --> 00:57:15,250 So there's some nomenclature that actually makes sense. 970 00:57:15,250 --> 00:57:20,550 So there are two possible spin values for an electron. 971 00:57:20,550 --> 00:57:27,580 And s can equal plus 1/2, spin up, or minus 1/2, spin down. 972 00:57:27,580 --> 00:57:29,870 And here are some little pictures of that. 973 00:57:32,730 --> 00:57:38,050 So this ms term, this spin magnetic quantum number, 974 00:57:38,050 --> 00:57:41,140 completes the description of the electron. 975 00:57:41,140 --> 00:57:43,980 But it's not dependent on the orbital. 976 00:57:43,980 --> 00:57:46,280 To describe an orbital completely, 977 00:57:46,280 --> 00:57:48,330 you only need three quantum numbers. 978 00:57:48,330 --> 00:57:52,720 But to describe the electron, you need four. 979 00:57:52,720 --> 00:57:55,990 And that is shown, again, here on this picture, 980 00:57:55,990 --> 00:57:57,070 or on this slide. 981 00:57:57,070 --> 00:57:58,670 You need three quantum numbers. 982 00:57:58,670 --> 00:58:04,410 You need n, l, and m sub l to describe the quantum number, 983 00:58:04,410 --> 00:58:07,200 describe the orbital completely. 984 00:58:07,200 --> 00:58:09,710 But you need a fourth one, this m sub 985 00:58:09,710 --> 00:58:12,310 s to describe the electron. 986 00:58:12,310 --> 00:58:16,840 So if you see wave function n, l, m sub l, 987 00:58:16,840 --> 00:58:19,930 you say that's telling me what the orbital is. 988 00:58:19,930 --> 00:58:23,580 And if we add the m sub s, then you look at that 989 00:58:23,580 --> 00:58:25,450 and say oh, that's going to tell me 990 00:58:25,450 --> 00:58:29,350 all the way to the electron what is going on. 991 00:58:32,040 --> 00:58:37,260 So this final quantum number led to what 992 00:58:37,260 --> 00:58:42,490 we know as Pauli's exclusion principle, which 993 00:58:42,490 --> 00:58:47,300 is that no two electrons can have the same four quantum 994 00:58:47,300 --> 00:58:48,250 numbers. 995 00:58:48,250 --> 00:58:51,760 They can't have the same-- no two electrons can have 996 00:58:51,760 --> 00:58:54,910 the same spin, in other words. 997 00:58:54,910 --> 00:58:58,940 So if we are drawing a configuration for neon 998 00:58:58,940 --> 00:59:02,200 with 10 electrons, we are going to have 999 00:59:02,200 --> 00:59:06,130 with one electron being up spin, the next one 1000 00:59:06,130 --> 00:59:07,910 is going to be down. 1001 00:59:07,910 --> 00:59:11,100 Because if we had two of these both going up, 1002 00:59:11,100 --> 00:59:14,700 they would have the same four quantum numbers. 1003 00:59:14,700 --> 00:59:18,780 And that's not allowed by Pauli's exclusion principle. 1004 00:59:18,780 --> 00:59:22,440 So when you have two here, one spin up, 1005 00:59:22,440 --> 00:59:24,460 one spin down in an orbital, then 1006 00:59:24,460 --> 00:59:27,840 we say that those electrons are paired. 1007 00:59:27,840 --> 00:59:31,250 And an important thing that kind of comes out of all of this 1008 00:59:31,250 --> 00:59:35,270 is that one orbital can't hold more than two electrons. 1009 00:59:35,270 --> 00:59:38,550 If it did, there'd be another electron 1010 00:59:38,550 --> 00:59:40,770 that would have the same four quantum numbers. 1011 00:59:40,770 --> 00:59:43,470 Because you need three quantum numbers to describe 1012 00:59:43,470 --> 00:59:45,580 the electron, or the orbital. 1013 00:59:45,580 --> 00:59:50,180 We need three to describe, say, that it's n equals 1, and then 1014 00:59:50,180 --> 00:59:51,990 its s state. 1015 00:59:51,990 --> 00:59:55,190 So we need those other ones to describe the orbital and then 1016 00:59:55,190 --> 00:59:56,820 the fourth one to describe the spin. 1017 00:59:56,820 --> 00:59:58,540 So if we add another electron, you'd 1018 00:59:58,540 --> 01:00:00,100 have two that were spin up, say. 1019 01:00:00,100 --> 01:00:01,500 And that just wouldn't work. 1020 01:00:01,500 --> 01:00:06,050 So you cannot have more than two electrons in the same orbital. 1021 01:00:06,050 --> 01:00:07,760 And this makes a lot of sense when 1022 01:00:07,760 --> 01:00:11,670 you think about why you would be putting electrons in orbitals 1023 01:00:11,670 --> 01:00:12,650 that are higher energy. 1024 01:00:12,650 --> 01:00:16,820 Why not just keep putting him in the low energy orbital? 1025 01:00:16,820 --> 01:00:18,540 And it's because you can't do that. 1026 01:00:18,540 --> 01:00:21,670 You can't put more than two electrons in. 1027 01:00:21,670 --> 01:00:24,610 And so therefore once you've filled a lower energy orbital, 1028 01:00:24,610 --> 01:00:28,990 you've got to move up to the next lowest energy orbital.