1 00:00:00 --> 00:00:01 2 00:00:01 --> 00:00:02 The following content is provided under a Creative 3 00:00:02 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:10 offer high-quality educational resources for free. 6 00:00:10 --> 00:00:13 To make a donation or view additional materials from 7 00:00:13 --> 00:00:15 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:15 --> 00:00:17 at ocw.mit.edu. 9 00:00:17 --> 00:00:22 PROFESSOR: All right. 10 00:00:22 --> 00:00:24 As you're settling in, why don't you take 10 more seconds 11 00:00:24 --> 00:00:26 to answer the clicker question. 12 00:00:26 --> 00:00:29 This is the last question we'll see in class on the 13 00:00:29 --> 00:00:32 photoelectric effect, so hopefully we can have a very 14 00:00:32 --> 00:00:36 high success rate here to show we are all ready to move on 15 00:00:36 --> 00:00:42 with our lives here. 16 00:00:42 --> 00:00:43 OK, good. 17 00:00:43 --> 00:00:46 So, most of you did get the answer correct. 18 00:00:46 --> 00:00:50 For those of you that didn't, you, of course, can ask your 19 00:00:50 --> 00:00:52 TA's about this in recitation, they'll always have a 20 00:00:52 --> 00:00:53 copy of these slides. 21 00:00:53 --> 00:00:56 But just to point out the confusion can be we've 22 00:00:56 --> 00:00:58 actually switched what the question is here. 23 00:00:58 --> 00:01:02 What the information we gave was the work function, which is 24 00:01:02 --> 00:01:05 what we've been giving before, but now we gave you the kinetic 25 00:01:05 --> 00:01:08 energy of the ejected electron, so you just need to rearrange 26 00:01:08 --> 00:01:10 your equation so now you're solving for the incoming 27 00:01:10 --> 00:01:12 energy, which would mean that you need to add those 28 00:01:12 --> 00:01:13 two energies together. 29 00:01:13 --> 00:01:17 So, hopefully everyone that didn't get this right, can look 30 00:01:17 --> 00:01:19 at it again and think about asking it's just asking the 31 00:01:19 --> 00:01:21 question in a different kind of a way. 32 00:01:21 --> 00:01:21 All right. 33 00:01:21 --> 00:01:24 So, we can switch over to the class notes. 34 00:01:24 --> 00:01:26 So today, we're going to start talking about 35 00:01:26 --> 00:01:27 the hydrogen atom. 36 00:01:27 --> 00:01:30 Now that we have our Schrodinger equation for the 37 00:01:30 --> 00:01:33 hydrogen atom, we can talk about it very specifically in 38 00:01:33 --> 00:01:37 terms of binding energies and also in terms of orbitals. 39 00:01:37 --> 00:01:40 And we talked about on Wednesday the conditions that 40 00:01:40 --> 00:01:43 allowed us to use quantum mechanics, which then enable 41 00:01:43 --> 00:01:45 us to have the Schrodinger equation, which we can apply, 42 00:01:45 --> 00:01:48 and part of that is the wave particle duality of 43 00:01:48 --> 00:01:50 light and matter. 44 00:01:50 --> 00:01:56 So there was a good question in Wednesday's class about the de 45 00:01:56 --> 00:01:59 Broglie wavelength and if it can actually go to infinity. 46 00:01:59 --> 00:02:02 So I just wanted to address that quickly before we move on, 47 00:02:02 --> 00:02:05 and actually address another thing about dealing with 48 00:02:05 --> 00:02:08 wavelengths of particles that sometimes comes up. 49 00:02:08 --> 00:02:16 So, the question that we had in last class, was if we have 50 00:02:16 --> 00:02:19 actually a macroscopic particle, and the velocity 51 00:02:19 --> 00:02:22 let's say starts to approach zero, shouldn't we have the 52 00:02:22 --> 00:02:27 wavelength go to infinity, even if we have a magic board, and 53 00:02:27 --> 00:02:29 even if the mass is really large. 54 00:02:29 --> 00:02:33 So, in most cases, you would think that as the velocity gets 55 00:02:33 --> 00:02:38 very tiny, the mass is still going to be large enough to 56 00:02:38 --> 00:02:42 cancel it out and still make it such that the wavelength is 57 00:02:42 --> 00:02:43 going to be pretty small, right. 58 00:02:43 --> 00:02:47 Because if we think about the h, Planck's constant, here 59 00:02:47 --> 00:02:52 that's measured in 10 to the negative 34 joules per second. 60 00:02:52 --> 00:02:54 So, we would actually need a really, really, really tiny 61 00:02:54 --> 00:02:59 velocity here to actually overcome the size of the mass, 62 00:02:59 --> 00:03:02 if we're talking about macroscopic particles, to have 63 00:03:02 --> 00:03:04 a wavelength that's going to be on the order. 64 00:03:04 --> 00:03:07 So, let's say we're talking about the baseball, have a 65 00:03:07 --> 00:03:08 wavelength of the baseball that's on the order 66 00:03:08 --> 00:03:10 of the baseball. 67 00:03:10 --> 00:03:13 So, if we kind of think about the numbers we would need, we 68 00:03:13 --> 00:03:16 would actually need a velocity that approached something 69 00:03:16 --> 00:03:20 that's about 10 to the negative 30 meters per second. 70 00:03:20 --> 00:03:23 So first of all, that's pretty slow here. 71 00:03:23 --> 00:03:27 It's going to be hard to measure anyway, and in fact, if 72 00:03:27 --> 00:03:30 we're talking about something going 10 to the negative 30, 73 00:03:30 --> 00:03:34 and we're going to observe it using our eyes, so you using 74 00:03:34 --> 00:03:37 visible light to observe something going this slow, 75 00:03:37 --> 00:03:39 we're actually not going to be able to do it because we're 76 00:03:39 --> 00:03:42 limited by the wavelength of light to see how precise we 77 00:03:42 --> 00:03:45 can measure where the actual position is. 78 00:03:45 --> 00:03:48 So let's say we have the wavelength of light somewhere 79 00:03:48 --> 00:03:51 on the order of 10 to the negative 5 meters, being the 80 00:03:51 --> 00:03:55 wavelength of light, we're only going to be able to measure the 81 00:03:55 --> 00:03:57 velocity because of the uncertainty principle 82 00:03:57 --> 00:03:58 to a certain degree. 83 00:03:58 --> 00:04:05 And it turns out it's three orders of magnitude that -- the 84 00:04:05 --> 00:04:07 uncertainty is three orders of magnitude bigger than the 85 00:04:07 --> 00:04:10 velocity that we're actually trying to observe to get to a 86 00:04:10 --> 00:04:13 point where we could see the wavelength, for example, for 87 00:04:13 --> 00:04:16 even a baseball that is moving this slow? 88 00:04:16 --> 00:04:19 So that the more complete answer to the question is that 89 00:04:19 --> 00:04:22 no, we're never going to be able to observe that because of 90 00:04:22 --> 00:04:25 the uncertainty principle it's not possible to observe a 91 00:04:25 --> 00:04:27 velocity that's this slow for a macroscopic object. 92 00:04:27 --> 00:04:32 So, hopefully that kind of clears up that question. 93 00:04:32 --> 00:04:35 And, of course, when the velocity actually is zero, this 94 00:04:35 --> 00:04:38 equation that the de Broglie has put forth is valid for 95 00:04:38 --> 00:04:41 anything that has momentum, so if something does not have any 96 00:04:41 --> 00:04:43 velocity at all, it actually does not have momentum, so you 97 00:04:43 --> 00:04:46 can't apply that equation anyway. 98 00:04:46 --> 00:04:51 And another thing that came up, and it came up in remembering 99 00:04:51 --> 00:04:54 as I was writing your problem set for this week, which will 100 00:04:54 --> 00:04:57 be posted sometime this afternoon, your problem set 2, 101 00:04:57 --> 00:05:01 is when we're talking about wavelengths of particles, and 102 00:05:01 --> 00:05:03 for specifically for electrons sometimes, you're asked to 103 00:05:03 --> 00:05:05 calculate what the energy is. 104 00:05:05 --> 00:05:08 And I just want to remind everyone, so this is a separate 105 00:05:08 --> 00:05:14 thought here, that we often use the energy where energy is 106 00:05:14 --> 00:05:18 equal to h c divided by wavelength. 107 00:05:18 --> 00:05:21 So if we're talking about, for example, we know the wavelength 108 00:05:21 --> 00:05:24 of an electron and we're trying to find the energy or vice 109 00:05:24 --> 00:05:28 versa, is this an equation we can use to do that? 110 00:05:28 --> 00:05:31 What do you think? 111 00:05:31 --> 00:05:31 No. 112 00:05:31 --> 00:05:33 Hopefully you're going to say no. 113 00:05:33 --> 00:05:37 And the reason is, and this will come up on the problems 114 00:05:37 --> 00:05:39 and a lot of students end up using this equation, which is 115 00:05:39 --> 00:05:42 why I want to head it off and mention it ahead of time, we 116 00:05:42 --> 00:05:45 can't use an equation because this equation is very 117 00:05:45 --> 00:05:46 specific for light. 118 00:05:46 --> 00:05:50 We know it's very specific for light because in this equation 119 00:05:50 --> 00:05:52 is c, the speed of light. 120 00:05:52 --> 00:05:54 So any time you go to use this equation, if you're trying to 121 00:05:54 --> 00:05:57 use it for an electron, just ask yourself first, does 122 00:05:57 --> 00:06:00 an electron travel at the speed of light? 123 00:06:00 --> 00:06:04 And if your answer is no, your answer will be no, then you 124 00:06:04 --> 00:06:06 just know you can't use this equation here. 125 00:06:06 --> 00:06:09 So instead you'd have to maybe if you start with wavelength, 126 00:06:09 --> 00:06:12 go over there, and then figure out velocity and do something 127 00:06:12 --> 00:06:16 more like kinetic energy equals 1/2 n b squared to get there. 128 00:06:16 --> 00:06:18 So this is just a heads up for as you start 129 00:06:18 --> 00:06:21 your next problem set. 130 00:06:21 --> 00:06:21 All right. 131 00:06:21 --> 00:06:25 So jumping in to having established that, yes, 132 00:06:25 --> 00:06:28 particles have wave-like behavior, even though no, 133 00:06:28 --> 00:06:31 they're not actually photons, we can't use that equation. 134 00:06:31 --> 00:06:36 But we can use equations that describe waves to describe 135 00:06:36 --> 00:06:38 matter, and that's what we're going to be doing today. 136 00:06:38 --> 00:06:40 We're going to be looking at the solutions to the 137 00:06:40 --> 00:06:43 Schrodinger equation for a hydrogen atom, and specifically 138 00:06:43 --> 00:06:46 we'll be looking at the binding energy of the electron 139 00:06:46 --> 00:06:47 to the nucleus. 140 00:06:47 --> 00:06:49 So we'll be looking at the solution to this part of 141 00:06:49 --> 00:06:52 the Schrodinger equation where we're finding e. 142 00:06:52 --> 00:06:54 Then we'll go on, after we've made all of our predictions for 143 00:06:54 --> 00:06:58 what the energy should be, we can actually confirm whether or 144 00:06:58 --> 00:07:00 not we're correct, and we'll do this by looking at photon 145 00:07:00 --> 00:07:04 emission and photon absorption for hydrogen atoms, and we'll 146 00:07:04 --> 00:07:07 actually do a demo with that, too, so we can confirm it 147 00:07:07 --> 00:07:11 ourselves, as well as matching it with the observation 148 00:07:11 --> 00:07:12 of others. 149 00:07:12 --> 00:07:15 And if we have time, we'll move on also to talking about the 150 00:07:15 --> 00:07:18 other part of the solution to the Schrodinger equation, which 151 00:07:18 --> 00:07:21 is psi or this wave function here. 152 00:07:21 --> 00:07:23 And remember I said that wave function is just a 153 00:07:23 --> 00:07:27 representation of the particle, particularly when we're talking 154 00:07:27 --> 00:07:29 about electrons -- we're familiar with the term 155 00:07:29 --> 00:07:34 orbitals. psi is just a description of the orbital. 156 00:07:34 --> 00:07:36 So, we'll start with an introduction to that if 157 00:07:36 --> 00:07:39 we got to it at the end. 158 00:07:39 --> 00:07:42 So, to remind you, when we look at the Schrodinger equation 159 00:07:42 --> 00:07:46 here, we have two parts to it, so when we solve the 160 00:07:46 --> 00:07:49 Schrodinger equation, we're either finding psi, which as I 161 00:07:49 --> 00:07:52 said, is a wave function or an orbital. 162 00:07:52 --> 00:07:57 And in addition to finding psi, we can also solve to find e or 163 00:07:57 --> 00:08:01 to find the energy for any given psi, and these are the 164 00:08:01 --> 00:08:04 binding energies of the electron to the nucleus. 165 00:08:04 --> 00:08:06 And the most important thing about using the Schrodinger 166 00:08:06 --> 00:08:09 equation and getting out our solutions for potential 167 00:08:09 --> 00:08:12 orbitals and potential energies for an electron with the 168 00:08:12 --> 00:08:15 nucleus, is that what we find is that quantum mechanics, and 169 00:08:15 --> 00:08:18 quantum mechanics allowing us to get to the Schrodinger 170 00:08:18 --> 00:08:21 equation, allows us to correctly predict and confirm 171 00:08:21 --> 00:08:25 our observations for what we can actually measure are 172 00:08:25 --> 00:08:27 indeed the energy levels. 173 00:08:27 --> 00:08:29 Here we're talking about a hydrogen atom and that's 174 00:08:29 --> 00:08:30 what we'll focus on today. 175 00:08:30 --> 00:08:33 And it's incredibly precise and we're able to make the 176 00:08:33 --> 00:08:35 predictions and match them with experiment. 177 00:08:35 --> 00:08:38 Also, when we're looking at the Schrodinger equation, it allows 178 00:08:38 --> 00:08:42 us to explain a stable hydrogen atom, which is something that 179 00:08:42 --> 00:08:46 classical mechanics did not allow us to do. 180 00:08:46 --> 00:08:51 So here's the solution for a hydrogen atom, where we have 181 00:08:51 --> 00:08:55 the e term here is equal to everything written in green. 182 00:08:55 --> 00:08:58 We've got a lot of constants in this solution to the hydrogen 183 00:08:58 --> 00:09:01 atom, and we know what most of these mean. 184 00:09:01 --> 00:09:04 But remember that this whole term in green here is what is 185 00:09:04 --> 00:09:07 going to be equal to that binding energy between the 186 00:09:07 --> 00:09:11 nucleus of a hydrogen atom and the electron. 187 00:09:11 --> 00:09:14 So, let's go ahead and define our variables here, they 188 00:09:14 --> 00:09:17 should be familiar to us. 189 00:09:17 --> 00:09:23 We have the mass, first of all, m is equal to m e, so 190 00:09:23 --> 00:09:29 that's the electron mass. 191 00:09:29 --> 00:09:35 We also have e, which is going to be the charge 192 00:09:35 --> 00:09:40 on the electron. 193 00:09:40 --> 00:09:43 In addition to that, we have that epsilon nought value, 194 00:09:43 --> 00:09:45 remember that's the permittivity constant in a 195 00:09:45 --> 00:09:50 vacuum, and basically that is what we use as a conversion 196 00:09:50 --> 00:09:51 factor to get from units. 197 00:09:51 --> 00:09:54 We don't want namely coulombs to units, we want that will 198 00:09:54 --> 00:09:56 allow us to cancel out in this equation. 199 00:09:56 --> 00:09:59 And finally we have Planck's constant here, which 200 00:09:59 --> 00:10:03 we're all familiar with. 201 00:10:03 --> 00:10:07 So, what actually happens when people work with the solution 202 00:10:07 --> 00:10:10 to the Schrodinger equation for a hydrogen atom is that they 203 00:10:10 --> 00:10:13 don't always want to deal with all these constants here, so we 204 00:10:13 --> 00:10:16 can actually group them together and use them as a 205 00:10:16 --> 00:10:20 single new constant, and this new constant is the 206 00:10:20 --> 00:10:21 Rydberg constant. 207 00:10:21 --> 00:10:25 And the Rydberg constant is actually equal to 2 . 208 00:10:25 --> 00:10:30 1 8 times 10 to the negative 18 joules. 209 00:10:30 --> 00:10:33 So when we pull out all of those constants and instead use 210 00:10:33 --> 00:10:36 the Rydberg constant, what it allows us to do is really 211 00:10:36 --> 00:10:38 simplify our energy equation. 212 00:10:38 --> 00:10:41 So now we have that energy is equal to the negative 213 00:10:41 --> 00:10:46 of the Rydberg constant divided by n squared. 214 00:10:46 --> 00:10:49 So, what we have left in our equation is only one part that 215 00:10:49 --> 00:10:52 we haven't explained yet, and that is that n value. 216 00:10:52 --> 00:10:55 And it turns out that when you solve the Schrodinger equation, 217 00:10:55 --> 00:10:58 you find that there are only certain allowed values 218 00:10:58 --> 00:10:59 of this integer n. 219 00:10:59 --> 00:11:07 And those allowed values range anywhere from n equals 1, you 220 00:11:07 --> 00:11:11 can have n equal that 2, 3, and it goes all the way 221 00:11:11 --> 00:11:12 up to infinity. 222 00:11:12 --> 00:11:14 But the important part is that there are only certain allowed 223 00:11:14 --> 00:11:17 values, so for example, you can't have 1 . 224 00:11:17 --> 00:11:18 5 or 2 . 225 00:11:18 --> 00:11:22 3, there are only these interger numbers. 226 00:11:22 --> 00:11:34 And this n here is what we call the principle quantum number. 227 00:11:34 --> 00:11:38 And what we find is when we apply n and plug it in to our 228 00:11:38 --> 00:11:42 energy equation, is that what we see is now we don't just 229 00:11:42 --> 00:11:45 have one distinct answer, we don't just have one possible 230 00:11:45 --> 00:11:48 binding energy of the electron to the nucleus. 231 00:11:48 --> 00:11:52 We're going to find that we actually have a whole bunch of 232 00:11:52 --> 00:11:55 possible, in fact, an infinite number of possible energy 233 00:11:55 --> 00:11:58 levels, and that's easier to see on this energy 234 00:11:58 --> 00:12:00 diagram here. 235 00:12:00 --> 00:12:03 So, let's start with n equals 1, since that's, of 236 00:12:03 --> 00:12:05 course, the simplest case. 237 00:12:05 --> 00:12:08 So, if we have n equals 1, we can plug it into our energy 238 00:12:08 --> 00:12:12 equation here, and find that the binding energy, the e sub 239 00:12:12 --> 00:12:15 n, for n equals 1, it's just going to be equal to the 240 00:12:15 --> 00:12:18 negative of the Rydberg constant, so we can actually 241 00:12:18 --> 00:12:22 graph that on an energy diagram here, and it's going to be down 242 00:12:22 --> 00:12:25 low at the bottom because that's going to be, in fact, 243 00:12:25 --> 00:12:30 the lowest or most negative energy when n equals 1. 244 00:12:30 --> 00:12:33 But we saw from our equation that there's more than just one 245 00:12:33 --> 00:12:36 possible value for n, so we could, for example, have n 246 00:12:36 --> 00:12:42 equals 2, n equals 3, all the way up to n equaling infinity. 247 00:12:42 --> 00:12:46 So what this tells us here is that this is not necessarily 248 00:12:46 --> 00:12:49 the binding energy of the electron in a hydrogen atom, 249 00:12:49 --> 00:12:51 it's also possible that it could, for example, have this 250 00:12:51 --> 00:12:54 energy, it could have this energy up here, it could have 251 00:12:54 --> 00:12:56 some energy way up here. 252 00:12:56 --> 00:12:59 So we have this infinite number of possible binding energies. 253 00:12:59 --> 00:13:02 But the really important point here is that they're quantized. 254 00:13:02 --> 00:13:05 So it's not a continuum of energy that we can have, it's 255 00:13:05 --> 00:13:08 only these punctuated points of energy that are possible. 256 00:13:08 --> 00:13:12 So as I tried to say on the board, we can have n equals 1, 257 00:13:12 --> 00:13:15 but since we can't have n equals 1/2, we actually can't 258 00:13:15 --> 00:13:19 have a binding energy that's anywhere in between these 259 00:13:19 --> 00:13:20 levels that are indicated here. 260 00:13:20 --> 00:13:23 And that's a really important point for something that comes 261 00:13:23 --> 00:13:26 out of solving the Schrodinger equation is this quantization 262 00:13:26 --> 00:13:29 of energy levels. 263 00:13:29 --> 00:13:32 And thanks to our equation simplified here, it's very easy 264 00:13:32 --> 00:13:35 for us to figure out what actually the allowed 265 00:13:35 --> 00:13:36 energy levels are. 266 00:13:36 --> 00:13:40 So for n equals 2, what would the binding energy be? 267 00:13:40 --> 00:13:41 Someone shout it out. 268 00:13:41 --> 00:13:44 Yup. 269 00:13:44 --> 00:13:48 So, I think the compilation of the voices that I heard was 270 00:13:48 --> 00:13:51 negative r h over 2 squared. 271 00:13:51 --> 00:13:55 We can do the same thing for 3, negative r h over 3 squared is 272 00:13:55 --> 00:13:57 going to be our binding energy. 273 00:13:57 --> 00:14:01 For 4, we can go all the way up to infinity, and actually when 274 00:14:01 --> 00:14:05 we get to the point where it's infinity, what we find is 275 00:14:05 --> 00:14:09 the binding energy at that point is going to be zero. 276 00:14:09 --> 00:14:13 And when we get to infinity, what that means is that we now 277 00:14:13 --> 00:14:17 have a free electron, so now the electron has totally 278 00:14:17 --> 00:14:19 separated from the atom. 279 00:14:19 --> 00:14:22 And that makes sense because we're at the point where 280 00:14:22 --> 00:14:25 there's no binding energy keeping it stable. 281 00:14:25 --> 00:14:28 You'll also know that all of these binding energies here 282 00:14:28 --> 00:14:32 are negative, so the negative sign indicates that it's low. 283 00:14:32 --> 00:14:35 It's a more negative energy, it's a lower energy state. 284 00:14:35 --> 00:14:37 So whenever we're thinking about energy states, it's 285 00:14:37 --> 00:14:41 always more stable to be more low in an energy well, so 286 00:14:41 --> 00:14:45 that's why it makes sense that it's favorable, in fact, to 287 00:14:45 --> 00:14:48 have an electron interacting with the nucleus that 288 00:14:48 --> 00:14:52 stabilizes and lowers the energy of that 289 00:14:52 --> 00:14:55 electron by doing so. 290 00:14:55 --> 00:14:58 So, we actually term this n equals 1 state gets a special 291 00:14:58 --> 00:15:02 name, which we call the ground state, and it's called the 292 00:15:02 --> 00:15:05 ground state because it is, in fact, the lowest to the 293 00:15:05 --> 00:15:05 ground that we can get. 294 00:15:05 --> 00:15:09 It's the most negative and most stable energy 295 00:15:09 --> 00:15:11 level that we have. 296 00:15:11 --> 00:15:16 And when we think about kind of in a more out practical sense 297 00:15:16 --> 00:15:19 what we mean by all of these binding energies, another way 298 00:15:19 --> 00:15:22 that we can put it is to give it some physical significance, 299 00:15:22 --> 00:15:26 and the physical significance of binding energies is that 300 00:15:26 --> 00:15:30 they're equal to the negative of the ionization energies. 301 00:15:30 --> 00:15:34 So, for example, in a hydrogen atom, if you take the binding 302 00:15:34 --> 00:15:38 energy, the negative of that is going to be how much energy you 303 00:15:38 --> 00:15:41 have to put in to ionize the hydrogen atom. 304 00:15:41 --> 00:15:45 So, if, for example, we were looking at a hydrogen atom in 305 00:15:45 --> 00:15:49 the case where we have the n equals 1 state, so the electron 306 00:15:49 --> 00:15:53 is in that ground state, the ionization energy, it makes 307 00:15:53 --> 00:15:55 sense, is going to be the difference between the ground 308 00:15:55 --> 00:15:58 state and the energy it takes to be a free electron. 309 00:15:58 --> 00:16:02 When we graph that on our chart here, it becomes clear that 310 00:16:02 --> 00:16:04 yes, in fact, the ionization energy is just the negative of 311 00:16:04 --> 00:16:08 the binding energy, so we can just look over here and 312 00:16:08 --> 00:16:11 figure out what our ionization energy is. 313 00:16:11 --> 00:16:14 So when we're talking about the ground state of a hydrogen 314 00:16:14 --> 00:16:16 atom, our ionization energy is just the negative of the 315 00:16:16 --> 00:16:19 Rydberg constant, so that easy, it's 2 . 316 00:16:19 --> 00:16:23 1 8 times 10 to the negative 18 joules. 317 00:16:23 --> 00:16:26 So, that should make a lot of sense intuitively, because it 318 00:16:26 --> 00:16:30 makes sense that if we need to ionize an atom, we need to put 319 00:16:30 --> 00:16:35 energy into the atom in order to eject that electron, and 320 00:16:35 --> 00:16:37 that energy we need to put in better be the difference 321 00:16:37 --> 00:16:39 between where we are now and where we have to be to 322 00:16:39 --> 00:16:44 be a free electron. 323 00:16:44 --> 00:16:47 So in most cases when we talk about ionization energy, if we 324 00:16:47 --> 00:16:51 don't say anything specific to the state we're talking about, 325 00:16:51 --> 00:16:53 you should always assume that we are, in fact, talking 326 00:16:53 --> 00:16:55 about the ground state. 327 00:16:55 --> 00:16:58 So, oftentimes you'll just be asked about ionization energy. 328 00:16:58 --> 00:17:01 If it doesn't say anything else we do mean n equals 1. 329 00:17:01 --> 00:17:04 But, in fact, we can also talk about the ionization energy 330 00:17:04 --> 00:17:07 of different states of the hydrogen atom or of any atom. 331 00:17:07 --> 00:17:11 So, for example, we could talk about the n equals 2 state, so 332 00:17:11 --> 00:17:14 that's this state here, and it's also what we could call 333 00:17:14 --> 00:17:15 the first excited state. 334 00:17:15 --> 00:17:18 So we have the ground state, and if we excite an electron 335 00:17:18 --> 00:17:21 into the next closest state, we're at the first excited 336 00:17:21 --> 00:17:24 state, or the n equals 2 state. 337 00:17:24 --> 00:17:27 So, we can now calculate the ionization energy here. it's an 338 00:17:27 --> 00:17:31 easy calculation -- we're just taking the negative of the 339 00:17:31 --> 00:17:33 binding energy, again that makes sense, because it's this 340 00:17:33 --> 00:17:35 difference in energy here. 341 00:17:35 --> 00:17:39 So what we get is that the binding energy, when it's 342 00:17:39 --> 00:17:42 negative, the ionization energy is 5 . 343 00:17:42 --> 00:17:46 4 5 times 10 to the negative 19 joules. 344 00:17:46 --> 00:17:49 So we should be able to think about these binding energies 345 00:17:49 --> 00:17:51 and figure out the ionization energy for any state 346 00:17:51 --> 00:17:52 that were asked about. 347 00:17:52 --> 00:17:55 So if we can switch over to a clicker question here 348 00:17:55 --> 00:17:58 and we'll let you do that. 349 00:17:58 --> 00:18:02 And what we're asking you to do is now tell us what the 350 00:18:02 --> 00:18:05 ionization energy is of a hydrogen atom that is in 351 00:18:05 --> 00:18:21 its third excited state. 352 00:18:21 --> 00:18:21 All right. 353 00:18:21 --> 00:18:36 Let's take 10 more seconds on that. 354 00:18:36 --> 00:18:36 OK. 355 00:18:36 --> 00:18:37 Interesting. 356 00:18:37 --> 00:18:40 Usually the majority is correct, but actually what you 357 00:18:40 --> 00:18:43 did was illustrate a point that I really wanted to stress and 358 00:18:43 --> 00:18:46 there's no better way to stress it then to get it incorrect, 359 00:18:46 --> 00:18:49 especially when it doesn't count, it doesn't hurt so bad. 360 00:18:49 --> 00:18:51 So, if you want to switch back over to the notes, we'll 361 00:18:51 --> 00:18:56 explain why, in fact, the correct answer is 4. 362 00:18:56 --> 00:19:00 So the key word here is that we asked you to identify 363 00:19:00 --> 00:19:02 the third excited state. 364 00:19:02 --> 00:19:06 So, what white is n equal to for the third excited state? 365 00:19:06 --> 00:19:07 4 OK. 366 00:19:07 --> 00:19:11 So that explains probably most of the confusion here and you 367 00:19:11 --> 00:19:13 just want to be careful when you're reading the problems 368 00:19:13 --> 00:19:15 that that's what you read correctly. 369 00:19:15 --> 00:19:18 I think everyone would now get the clicker question correct. 370 00:19:18 --> 00:19:23 So, the third excited state, is n equal to 4, because n equals 371 00:19:23 --> 00:19:26 2 is first excited, 3 is second excited, 4 is third 372 00:19:26 --> 00:19:28 excited state. 373 00:19:28 --> 00:19:31 So now we can just take the negative of that binding 374 00:19:31 --> 00:19:34 energy here, and I've just rounded up here or 1 . 375 00:19:34 --> 00:19:37 4 times 10 to the negative 19 joules. 376 00:19:37 --> 00:19:41 So, I noticed that a few, a very, very small proportion of 377 00:19:41 --> 00:19:44 you, did type in selections that were negative 378 00:19:44 --> 00:19:45 ionization energies. 379 00:19:45 --> 00:19:48 And I'll just say it right now you can absolutely never have a 380 00:19:48 --> 00:19:50 negative ionization energy, so that's good to 381 00:19:50 --> 00:19:52 remember as well. 382 00:19:52 --> 00:19:54 And intuitively, it should make sense, right, because 383 00:19:54 --> 00:19:57 ionization energy is the amount of energy you need to put in to 384 00:19:57 --> 00:20:00 eject an electron from an atom. 385 00:20:00 --> 00:20:02 So you don't want to put in a negative energy, that's not 386 00:20:02 --> 00:20:05 going to help you out, you need to put in positive energy to 387 00:20:05 --> 00:20:07 get an electron out of the system. 388 00:20:07 --> 00:20:09 So that's why you'll find binding energies are always 389 00:20:09 --> 00:20:12 negative, and ionization energies are always going to be 390 00:20:12 --> 00:20:15 positive, or you could look at the equation and see it 391 00:20:15 --> 00:20:17 from there as well. 392 00:20:17 --> 00:20:18 All right. 393 00:20:18 --> 00:20:21 So, using the equation we'd initially discussed, the 394 00:20:21 --> 00:20:26 negative r sub h over n squared, we could figure out 395 00:20:26 --> 00:20:29 all of the different ionization energies and binding energies 396 00:20:29 --> 00:20:32 for a hydrogen atom, and it turns out if we change the 397 00:20:32 --> 00:20:36 equation only slightly to add a negative z squared in there, 398 00:20:36 --> 00:20:40 so, negative z squared times the Rydberg constant over n 399 00:20:40 --> 00:20:44 squared, now let's us calculate energy levels for absolutely 400 00:20:44 --> 00:20:47 any atom as long as this one important stipulation, it 401 00:20:47 --> 00:20:50 only has 1 electron in it. 402 00:20:50 --> 00:20:52 So basically we're dealing with hydrogen atoms and then we're 403 00:20:52 --> 00:20:55 going to be dealing with ions. 404 00:20:55 --> 00:20:58 So, for example, a helium plus 1 ion has 1 405 00:20:58 --> 00:21:01 electron at z equal 2. 406 00:21:01 --> 00:21:06 A lithium 2 plus ion has 1 electron, it has z equals 3, so 407 00:21:06 --> 00:21:08 if we were to plug in, we would just do z squared up 408 00:21:08 --> 00:21:11 here, or 3 squared. 409 00:21:11 --> 00:21:14 Terbium 64 plus, another 1 electron atom. 410 00:21:14 --> 00:21:17 What is z for that? 411 00:21:17 --> 00:21:17 Yup. 412 00:21:17 --> 00:21:19 65. 413 00:21:19 --> 00:21:23 So again, terbium 64 plus, not an ion we probably will run 414 00:21:23 --> 00:21:27 into, but if we did, we could, in fact, calculate all of the 415 00:21:27 --> 00:21:31 energy levels for it using this equation here. 416 00:21:31 --> 00:21:33 And the difference between the equation, the reason that that 417 00:21:33 --> 00:21:37 z squared comes in there is because if you go back to your 418 00:21:37 --> 00:21:40 notes from Wednesday, and you look at the long written out 419 00:21:40 --> 00:21:43 form of the Schrodinger equation for a hydrogen atom, 420 00:21:43 --> 00:21:47 or any 1 electron atom, you see the last term there is a 421 00:21:47 --> 00:21:50 coulomb potential energy between the electron 422 00:21:50 --> 00:21:51 and the nucleus. 423 00:21:51 --> 00:21:55 So, of course, when we have a charge on the nucleus equal to 424 00:21:55 --> 00:21:58 1, as we do in a hydrogen atom, the z is equal to 1, so it 425 00:21:58 --> 00:22:01 drops out there, but normally we would have to include the 426 00:22:01 --> 00:22:04 full charge on the nucleus, which is equal to z or the 427 00:22:04 --> 00:22:06 atomic number times the electron. 428 00:22:06 --> 00:22:09 So even if we strip an atom of all of its electrons, we still 429 00:22:09 --> 00:22:13 have that same amount of positive charge in the nucleus. 430 00:22:13 --> 00:22:17 So, this allows us to look at a bunch of different atoms, of 431 00:22:17 --> 00:22:20 course, limited to the fact that it has to be a 432 00:22:20 --> 00:22:23 1 electron atom. 433 00:22:23 --> 00:22:27 So, now that we can calculate the binding energies, we can 434 00:22:27 --> 00:22:31 think about is this, in fact, what matches up with what's 435 00:22:31 --> 00:22:34 been observed, or, in fact, could we predict what we will 436 00:22:34 --> 00:22:37 observe in different kinds of situations now that we know how 437 00:22:37 --> 00:22:42 to use the binding energy, and hopefully we can and we will. 438 00:22:42 --> 00:22:46 So one thing we could do is we could look at the different 439 00:22:46 --> 00:22:50 wavelengths of light that are emitted by hydrogen atoms that 440 00:22:50 --> 00:22:51 are excited to a higher state. 441 00:22:51 --> 00:22:55 So what we'll do in a few minutes here is try 442 00:22:55 --> 00:22:56 this with hydrogen. 443 00:22:56 --> 00:23:03 So we'll take h 2 and we'll run -- or actually we'll have h 2 444 00:23:03 --> 00:23:05 filled in an evacuated glass tube. 445 00:23:05 --> 00:23:09 When we increase the potential between the 2 electrodes that 446 00:23:09 --> 00:23:12 we have in the tube -- we actually split the h 2 into the 447 00:23:12 --> 00:23:15 individual hydrogen atoms, and not only do that, but 448 00:23:15 --> 00:23:17 also excite the atoms. 449 00:23:17 --> 00:23:20 So when you just run across an atom in the street, you can 450 00:23:20 --> 00:23:22 assume it's going to be in its most stable ground state, 451 00:23:22 --> 00:23:25 that's where the electron would be, but when we add energy to 452 00:23:25 --> 00:23:28 the system, we can actually excite it up into all different 453 00:23:28 --> 00:23:32 sorts of higher states -- n equals 6, n equals 10, any 454 00:23:32 --> 00:23:33 of those higher states. 455 00:23:33 --> 00:23:36 But that only happens momentarily, because, of 456 00:23:36 --> 00:23:39 course, if you have an energy in a higher energy level, it's 457 00:23:39 --> 00:23:41 going to want to drop back down to that lower or 458 00:23:41 --> 00:23:42 more stable level. 459 00:23:42 --> 00:23:46 And when it does that it's going to give off some energy 460 00:23:46 --> 00:23:48 equal to the difference between those two levels. 461 00:23:48 --> 00:23:52 And that will be associated with a wavelength if it 462 00:23:52 --> 00:23:54 releases the energy in terms of a photon. 463 00:23:54 --> 00:23:56 So that's what we'll look at in a few minutes. 464 00:23:56 --> 00:23:58 There's some important things to point out about what 465 00:23:58 --> 00:24:00 is happening here. 466 00:24:00 --> 00:24:03 Just to visualize exactly what we're saying, what we're saying 467 00:24:03 --> 00:24:07 is when we have an energy in a higher energy level, so let's 468 00:24:07 --> 00:24:11 say energy level, this initial level high up here, and it 469 00:24:11 --> 00:24:15 drops down to a lower final level, what we find is that the 470 00:24:15 --> 00:24:19 photon that is going to be emitted is going to be emitted 471 00:24:19 --> 00:24:22 with the exact energy, and the important term here 472 00:24:22 --> 00:24:24 is the exact. 473 00:24:24 --> 00:24:29 That is the difference between these two energy states. 474 00:24:29 --> 00:24:32 That makes sense because we're losing energy, we're going to a 475 00:24:32 --> 00:24:34 level lower level, so we can give off that extra in 476 00:24:34 --> 00:24:36 the form of light. 477 00:24:36 --> 00:24:38 And we can actually write the equation for what we would 478 00:24:38 --> 00:24:40 expect the energy for the light to be. 479 00:24:40 --> 00:24:44 So this delta energy here is very simply the energy of 480 00:24:44 --> 00:24:49 the initial state minus the energy of the final state. 481 00:24:49 --> 00:24:52 This is a little bit generic, we're not actually specifying 482 00:24:52 --> 00:24:55 the states here, but we could, we know we can calculate the 483 00:24:55 --> 00:24:57 energy from any of the states. 484 00:24:57 --> 00:25:00 So, for example, let's say we excited the hydrogen atom such 485 00:25:00 --> 00:25:04 that the electron was starting in the n equals 6 state, 486 00:25:04 --> 00:25:06 so that's our n initial. 487 00:25:06 --> 00:25:09 And we drop down to the n equals 2 state, or the 488 00:25:09 --> 00:25:11 first excited state. 489 00:25:11 --> 00:25:14 Then we would be able to change our equation to make it a 490 00:25:14 --> 00:25:17 little bit more specific and say that delta energy here is 491 00:25:17 --> 00:25:21 equal to energy of n equals 6, minus the energy of 492 00:25:21 --> 00:25:24 the n equals 2 state. 493 00:25:24 --> 00:25:28 When we talk about that frequency of light that's going 494 00:25:28 --> 00:25:31 to be emitted, it's not too commonly that we'll actually 495 00:25:31 --> 00:25:32 talk about it in terms of energy. 496 00:25:32 --> 00:25:35 A lot of times we talk about light in terms of its frequency 497 00:25:35 --> 00:25:39 or it's wavelength, but that's OK, because we know how to 498 00:25:39 --> 00:25:42 convert from energy to frequency, so we can do that 499 00:25:42 --> 00:25:46 here as well, where our frequency is just our energy 500 00:25:46 --> 00:25:49 divided by Planck's constant, and since here we're talking 501 00:25:49 --> 00:25:52 about a delta energy, we're going to talk about the 502 00:25:52 --> 00:25:56 frequency as being equal to delta energy over h. 503 00:25:56 --> 00:26:00 Or we can say the frequency is going to be equal to the energy 504 00:26:00 --> 00:26:05 initial minus the energy final all over Planck's constant. 505 00:26:05 --> 00:26:07 So this means that we can go directly from the energy 506 00:26:07 --> 00:26:10 between two levels to the frequency of the photon 507 00:26:10 --> 00:26:14 that's emitted when you go between those levels. 508 00:26:14 --> 00:26:17 What we can also do is say something about the wavelength 509 00:26:17 --> 00:26:20 as well because we know the relationship between energy 510 00:26:20 --> 00:26:22 and frequency and wavelength. 511 00:26:22 --> 00:26:25 So, in the first case here, let's say we go from a high 512 00:26:25 --> 00:26:29 level to a low level, let's say we go from five to one. 513 00:26:29 --> 00:26:32 If we have a large energy difference here, are we 514 00:26:32 --> 00:26:36 going to have a high or low frequency? 515 00:26:36 --> 00:26:36 Good. 516 00:26:36 --> 00:26:38 A high frequency. 517 00:26:38 --> 00:26:40 If we have a high frequency, what about the wavelength, 518 00:26:40 --> 00:26:43 long or short? 519 00:26:43 --> 00:26:43 All right. 520 00:26:43 --> 00:26:44 Good. 521 00:26:44 --> 00:26:45 So we should always be able to keep these 522 00:26:45 --> 00:26:47 relationships in mind. 523 00:26:47 --> 00:26:50 So, similarly in a case where instead we have a small energy 524 00:26:50 --> 00:26:54 difference, we're going to have a low frequency, which means 525 00:26:54 --> 00:27:01 that we're going to have a long wavelength here. 526 00:27:01 --> 00:27:04 So now we can go ahead and try observing some 527 00:27:04 --> 00:27:06 of this ourselves. 528 00:27:06 --> 00:27:09 So what we're actually going to do is this experiment that I 529 00:27:09 --> 00:27:13 explained here where we're going to excite hydrogen atoms 530 00:27:13 --> 00:27:17 such that they're electrons go into these higher energy 531 00:27:17 --> 00:27:20 levels, and then we're going to see if we can actually see 532 00:27:20 --> 00:27:23 individual wavelengths that come out of that that 533 00:27:23 --> 00:27:25 correspond with the energy difference. 534 00:27:25 --> 00:27:30 So, our detection devices are a little bit limited here today, 535 00:27:30 --> 00:27:34 we're actually only going to be using our eyes, so that means 536 00:27:34 --> 00:27:36 that we need to stick with the visible range of the 537 00:27:36 --> 00:27:38 electromagnetic spectrum. 538 00:27:38 --> 00:27:40 Actually that simplifies things, because that really 539 00:27:40 --> 00:27:42 cuts down on the number of wavelengths that we're going 540 00:27:42 --> 00:27:44 to be trying to observe here. 541 00:27:44 --> 00:27:47 So it turns out that in the visible range, when you figure 542 00:27:47 --> 00:27:50 out the differences between energy levels, in hydrogen 543 00:27:50 --> 00:27:53 atoms, there's only 4 wavelengths that fall in the 544 00:27:53 --> 00:27:55 visible range of the spectrum. 545 00:27:55 --> 00:27:58 So hopefully, when we turn out the lights, we're going to turn 546 00:27:58 --> 00:28:03 on this lamp here, which has hydrogen in it and we're going 547 00:28:03 --> 00:28:05 to excite that hydrogen. 548 00:28:05 --> 00:28:08 You'll see light coming out, but it's, of course, going to 549 00:28:08 --> 00:28:11 be bulk light -- you're not going to be able to tell the 550 00:28:11 --> 00:28:12 individual wavelengths. 551 00:28:12 --> 00:28:15 So what our TAs, actually if they can come down now, are 552 00:28:15 --> 00:28:19 going to pass out to you is these either defraction 553 00:28:19 --> 00:28:23 goggles, or just a little plate, and you're going to be 554 00:28:23 --> 00:28:26 splitting that light into its individual wavelengths. 555 00:28:26 --> 00:28:30 And the glasses, there aren't enough to go around for all of 556 00:28:30 --> 00:28:32 you, so that's why there's plates. 557 00:28:32 --> 00:28:35 And though glasses do look way cooler, the plates work a 558 00:28:35 --> 00:28:37 little better, so either you or your neighbor should try to 559 00:28:37 --> 00:28:39 have one of the plates in case one of you can't 560 00:28:39 --> 00:28:40 see all the lines. 561 00:28:40 --> 00:28:49 So our TAs will pass these around for us. 562 00:28:49 --> 00:28:53 And I also want to point out, it's guaranteed pretty much 563 00:28:53 --> 00:28:56 you'll be able to see these three here in the visible range 564 00:28:56 --> 00:28:59 -- you may or may not be able to see, sometimes it's hard to 565 00:28:59 --> 00:29:03 see that one that's getting near the UV end of our 566 00:29:03 --> 00:29:04 visible spectrum. 567 00:29:04 --> 00:29:14 So we won't worry if we can't see that. 568 00:29:14 --> 00:29:24 I'll take one actually. 569 00:29:24 --> 00:29:53 Can you raise your hand if you if you still need one? 570 00:29:53 --> 00:29:56 All right, so TA's walk carefully now, I'm going to 571 00:29:56 --> 00:29:58 shut the lights down here. 572 00:29:58 --> 00:30:10 All right, so we do still have some little bits of ambient 573 00:30:10 --> 00:30:12 light, so you might see a slight amount of the 574 00:30:12 --> 00:30:14 full continuum. 575 00:30:14 --> 00:30:17 But if you look through your plate, and actually especially 576 00:30:17 --> 00:30:20 if you kind of look off to the side, hopefully you'll be 577 00:30:20 --> 00:30:25 able to see the individual lines of the spectrum. 578 00:30:25 --> 00:30:27 Is everyone seeing that? 579 00:30:27 --> 00:30:28 Yeah, pretty much. 580 00:30:28 --> 00:30:29 OK. 581 00:30:29 --> 00:30:30 Can anyone not see it? 582 00:30:30 --> 00:30:32 Does anyone need -- actually I can't even tell if 583 00:30:32 --> 00:30:33 you raise your hand. 584 00:30:33 --> 00:30:36 So ask your neighbor if you can't see it and get one of the 585 00:30:36 --> 00:30:39 plates if you're having trouble seeing with the glasses. 586 00:30:39 --> 00:30:42 So this should match up with the spectrum that we saw. 587 00:30:42 --> 00:30:46 And actually keep those glasses with you. 588 00:30:46 --> 00:30:49 We'll turn the light back on for a second here. 589 00:30:49 --> 00:30:53 And hydrogen atom is what we're learning about, so that's 590 00:30:53 --> 00:30:54 the most relevant here. 591 00:30:54 --> 00:30:58 But just to show you that each atom does have its own set of 592 00:30:58 --> 00:31:02 spectral lines, just for fun we'll look at neon also so you 593 00:31:02 --> 00:31:17 can have a comparison point. 594 00:31:17 --> 00:31:17 All right. 595 00:31:17 --> 00:31:20 So, this is probably a familiar color having seen many neon 596 00:31:20 --> 00:31:28 lights around everywhere. 597 00:31:28 --> 00:31:31 So you see with the neon is there's just a lot more lines 598 00:31:31 --> 00:31:35 in that orange part of the spectrum then compared to the 599 00:31:35 --> 00:31:39 hydrogen, and that's really what gives you that neon 600 00:31:39 --> 00:31:41 color in the neon signs. 601 00:31:41 --> 00:31:45 That's the true color of a neon being excited, sometimes neon 602 00:31:45 --> 00:31:49 signs are painted with other compounds. 603 00:31:49 --> 00:31:50 All right, does everyone have their fill of 604 00:31:50 --> 00:31:51 seeing the neon lines? 605 00:31:51 --> 00:31:53 STUDENT: No. 606 00:31:53 --> 00:31:56 PROFESSOR: All right, let's take two more seconds 607 00:31:56 --> 00:32:01 to look at neon then. 608 00:32:01 --> 00:32:02 All right. 609 00:32:02 --> 00:32:08 So our special effects portion of the class is over. 610 00:32:08 --> 00:32:10 And what you see when we see it with our eye, which is all the 611 00:32:10 --> 00:32:13 wavelengths, of course, mixed together, is whichever those 612 00:32:13 --> 00:32:15 wavelengths is most intense. 613 00:32:15 --> 00:32:18 So, when we looked at the individual neon lines, it was 614 00:32:18 --> 00:32:21 the orange colors that was most intense, which is why we were 615 00:32:21 --> 00:32:25 seeing kind of a general orange glow with the neon, which is 616 00:32:25 --> 00:32:29 different from what we see with the hydrogen. 617 00:32:29 --> 00:32:35 All right, so we can, in fact, observe individual lines. 618 00:32:35 --> 00:32:37 There's nothing more exciting to see with your glasses 619 00:32:37 --> 00:32:38 on, while you look nice. 620 00:32:38 --> 00:32:42 You can take those off if you wish to, or you can try to just 621 00:32:42 --> 00:32:46 be splitting the light in the room until the TAs grab your 622 00:32:46 --> 00:32:48 glasses, either is fine. 623 00:32:48 --> 00:32:51 It turns out that we are far from the first people, although 624 00:32:51 --> 00:32:53 it felt exciting, we did not discover this for the 625 00:32:53 --> 00:32:56 first time here today. 626 00:32:56 --> 00:32:57 In fact, J.J. 627 00:32:57 --> 00:33:02 Balmer, who was a school teacher in the 1800s, was the 628 00:33:02 --> 00:33:05 first to describe these lines that could be seen 629 00:33:05 --> 00:33:07 from hydrogen. 630 00:33:07 --> 00:33:11 And he saw the same lines that we saw here today, and although 631 00:33:11 --> 00:33:16 he could not explain, even start to explain why you saw 632 00:33:16 --> 00:33:19 only these specific lines and not a whole continuum 633 00:33:19 --> 00:33:20 of the light. 634 00:33:20 --> 00:33:22 Right, we already have an idea because we just talked about 635 00:33:22 --> 00:33:24 energy levels, we know there's only certain allowed energy 636 00:33:24 --> 00:33:27 levels, but at the time there's no reason J.J. 637 00:33:27 --> 00:33:30 Balmer should have known this, and in fact he didn't, but he 638 00:33:30 --> 00:33:33 still came up with a quantitative way to describe 639 00:33:33 --> 00:33:34 what was going on. 640 00:33:34 --> 00:33:38 He came up with this equation here where what he found was 641 00:33:38 --> 00:33:42 that he could explain the wavelengths of these different 642 00:33:42 --> 00:33:47 lines by multiplying 1 over 4 minus 1 over some integer 643 00:33:47 --> 00:33:51 n, and multiplying it by this number, 3 . 644 00:33:51 --> 00:33:55 29 times 10 to the 15 Hertz, and he found that this was 645 00:33:55 --> 00:34:00 true where n was some integer value -- 3, 4, 5, or 6. 646 00:34:00 --> 00:34:11 So he could explain it quantitatively in terms of 647 00:34:11 --> 00:34:14 putting an equation with it, but he couldn't explain what 648 00:34:14 --> 00:34:15 was actually going on. 649 00:34:15 --> 00:34:19 But we, having learned about energy levels, having had the 650 00:34:19 --> 00:34:22 Schrodinger equation solved for us to understand what's going 651 00:34:22 --> 00:34:26 on, can, in fact, explain what happened when we saw 652 00:34:26 --> 00:34:28 these different colors. 653 00:34:28 --> 00:34:31 So, what we know is happening is that were having transitions 654 00:34:31 --> 00:34:37 from some excited states to a more relaxed lower, more stable 655 00:34:37 --> 00:34:39 state in the hydrogen atom. 656 00:34:39 --> 00:34:43 And it turns out what we can detect visibly with our eyes is 657 00:34:43 --> 00:34:46 in the visible range, and that means that the final 658 00:34:46 --> 00:34:49 state is n equals 2. 659 00:34:49 --> 00:34:52 Because you see how n equals 1 is so much further away, and 660 00:34:52 --> 00:34:55 actually that's not to scale, it's actually much, much 661 00:34:55 --> 00:34:58 further down in the energy well, such that the energy of 662 00:34:58 --> 00:35:02 the light is so great that it's going to be in ultraviolet very 663 00:35:02 --> 00:35:04 high energy, high frequency range. 664 00:35:04 --> 00:35:07 So we can't actually see any of that, it's too high 665 00:35:07 --> 00:35:08 energy for us to see. 666 00:35:08 --> 00:35:11 So everything we see is going to be where we have the final 667 00:35:11 --> 00:35:15 energy state being n equals 2. 668 00:35:15 --> 00:35:19 So if we think about, for example, this red line here, 669 00:35:19 --> 00:35:23 which energy state or which principle quantum number 670 00:35:23 --> 00:35:30 do you think that our electron started in? 671 00:35:30 --> 00:35:30 STUDENT: Three. 672 00:35:30 --> 00:35:31 PROFESSOR: Good. 673 00:35:31 --> 00:35:33 So, it's going to be in 3, because that's the shortest 674 00:35:33 --> 00:35:37 energy difference we can have, and the red is the longest 675 00:35:37 --> 00:35:40 wavelength we can see -- those 2 are inversely related, 676 00:35:40 --> 00:35:42 so it must be n equals 3. 677 00:35:42 --> 00:35:47 What about the kind of cyan-ish, blue-green one? 678 00:35:47 --> 00:35:50 Yup. so n equals 4 for that one. 679 00:35:50 --> 00:35:53 Similarly we can go on, match up the others. 680 00:35:53 --> 00:35:56 So n equals 5 for the bluish-purple and the 681 00:35:56 --> 00:35:58 violet is n equals 6. 682 00:35:58 --> 00:36:01 And again, that matches up, because the violet, or getting 683 00:36:01 --> 00:36:06 really close to the UV range here has the longest energy, so 684 00:36:06 --> 00:36:09 the highest frequency, and that's going to be the shortest 685 00:36:09 --> 00:36:11 wavelength and we can see here it is, in fact, the shortest 686 00:36:11 --> 00:36:15 wavelength that we can actually see. 687 00:36:15 --> 00:36:18 So, we can see if we can come up with the same 688 00:36:18 --> 00:36:19 equation that J.J. 689 00:36:19 --> 00:36:24 Balmer came up with by actually starting with what we know and 690 00:36:24 --> 00:36:26 working our way that way instead of coming up from the 691 00:36:26 --> 00:36:29 other direction, which he did, which was just to 692 00:36:29 --> 00:36:31 explain what he saw. 693 00:36:31 --> 00:36:33 So, if we start instead with talking about the energy 694 00:36:33 --> 00:36:36 levels, we can relate these to frequency, because we already 695 00:36:36 --> 00:36:40 said that frequency is related to, or it's equal to the 696 00:36:40 --> 00:36:44 initial energy level here minus the final energy level there 697 00:36:44 --> 00:36:47 over Planck's constant to get us to frequency. 698 00:36:47 --> 00:36:51 And we also have the equation that comes out of Schrodinger 699 00:36:51 --> 00:36:55 equation that tell us exactly what that binding energy is, 700 00:36:55 --> 00:36:57 that binding energy is just equal to the negative of the 701 00:36:57 --> 00:37:01 Rydberg constant over n squared. 702 00:37:01 --> 00:37:05 So that means that our frequency is going to equal, if 703 00:37:05 --> 00:37:10 we plug in e n into the initial and final energy here, 1 over 704 00:37:10 --> 00:37:14 Planck's constant times negative r h over n initial 705 00:37:14 --> 00:37:19 squared, minus negative r h over n final squared. 706 00:37:19 --> 00:37:22 So we have an equation that should relate how we can 707 00:37:22 --> 00:37:27 actually calculate the frequency to what J.J. 708 00:37:27 --> 00:37:29 Balmer observed. 709 00:37:29 --> 00:37:32 We can simplify this equation by pulling up the r h and 710 00:37:32 --> 00:37:35 getting rid of some of these negatives here by saying the 711 00:37:35 --> 00:37:38 frequency is going to be equal to the Rydberg constant divided 712 00:37:38 --> 00:37:44 by Planck's constant all times 1 over n final squared -- this 713 00:37:44 --> 00:37:46 is just to switch the signs around and get rid of some 714 00:37:46 --> 00:37:53 negatives -- minus 1 over n initial squared. 715 00:37:53 --> 00:37:56 We can plug this in further when we're talking about the 716 00:37:56 --> 00:38:00 visible part of the light spectrum, because we know that 717 00:38:00 --> 00:38:04 for n final equals 2, then that would mean we plug in 2 squared 718 00:38:04 --> 00:38:06 here, so what we get is 1 over 4. 719 00:38:06 --> 00:38:10 So this is our final equation, and this is actually called the 720 00:38:10 --> 00:38:13 Balmer series, which was named after Balmer, and this tells us 721 00:38:13 --> 00:38:18 the frequency of any of the lights where we start with an 722 00:38:18 --> 00:38:21 electron in some higher energy level and we drop down to an 723 00:38:21 --> 00:38:23 n final that's equal to 2. 724 00:38:23 --> 00:38:25 So it's a more specific version of the equation where we 725 00:38:25 --> 00:38:28 have the n final equal to 2. 726 00:38:28 --> 00:38:31 And it turns out that actually we find that this matches up 727 00:38:31 --> 00:38:36 perfectly with Balmer's equation, because the value of 728 00:38:36 --> 00:38:39 r h, the Rydberg constant divided by the Planck's 729 00:38:39 --> 00:38:42 constant is actually -- it's also another constant, so we 730 00:38:42 --> 00:38:46 can write it as this kind of strange looking cursive r here. 731 00:38:46 --> 00:38:49 Unfortunately, this is also call the Rydberg constant, so 732 00:38:49 --> 00:38:50 it's a little bit confusing. 733 00:38:50 --> 00:38:55 But really it means the Rydberg constant divided by h, 734 00:38:55 --> 00:38:57 and that's equal to 3 . 735 00:38:57 --> 00:38:59 29 times 10 to the 15 per second. 736 00:38:59 --> 00:39:04 So if you remember what the equation Balmer found was this 737 00:39:04 --> 00:39:06 number multiplied by this here. 738 00:39:06 --> 00:39:10 So, we found the exact same equation, but just now starting 739 00:39:10 --> 00:39:12 from understanding the difference between 740 00:39:12 --> 00:39:14 energy levels. 741 00:39:14 --> 00:39:17 So, sometimes you'll find the Rydberg constant in different 742 00:39:17 --> 00:39:21 forms, but just make sure you pay attention to units because 743 00:39:21 --> 00:39:23 then you won't mess them up, because this is in inverse 744 00:39:23 --> 00:39:26 seconds here, the other Rydberg constant is in joules, so 745 00:39:26 --> 00:39:29 you'll be able to use what you need depending on how you're 746 00:39:29 --> 00:39:33 using that constant. 747 00:39:33 --> 00:39:37 So in talking about the hydrogen atom, they actually 748 00:39:37 --> 00:39:42 have different names for different series, which means 749 00:39:42 --> 00:39:46 in terms of different n values that we end in. 750 00:39:46 --> 00:39:48 So we talked about what we could see with visible light, 751 00:39:48 --> 00:39:51 we said that's actually the Balmer series. 752 00:39:51 --> 00:39:54 So anything that goes from a higher energy level to 2 is 753 00:39:54 --> 00:39:57 going to be falling within the Balmer series, which is in the 754 00:39:57 --> 00:40:00 visible range of the spectrum. 755 00:40:00 --> 00:40:03 We can think about the Lyman series, which 756 00:40:03 --> 00:40:05 is where n equals 1. 757 00:40:05 --> 00:40:08 We know that that's going to be a higher energy difference, so 758 00:40:08 --> 00:40:12 that means that we're going to be in the UV range. 759 00:40:12 --> 00:40:15 We can also go in the opposite direction. 760 00:40:15 --> 00:40:18 So, for example, when n equals 3, that's called the Paschen 761 00:40:18 --> 00:40:21 series, and these are named after basically the people that 762 00:40:21 --> 00:40:24 first discovered these different lines and 763 00:40:24 --> 00:40:27 characterized them, and this is in the near IR range. 764 00:40:27 --> 00:40:31 And again, the n equals 4 is the Brackett series, 765 00:40:31 --> 00:40:33 and that's an IR range. 766 00:40:33 --> 00:40:36 I think there's names for even a few more levels up. 767 00:40:36 --> 00:40:39 You don't need to know those, but just because it's a special 768 00:40:39 --> 00:40:42 case with the hydrogen atom, they do tend to be named -- the 769 00:40:42 --> 00:40:45 most important, of course, tends to be the Balmer series 770 00:40:45 --> 00:40:47 because that's what we can actually see being emitted 771 00:40:47 --> 00:40:51 from the hydrogen atom. 772 00:40:51 --> 00:40:55 So, now we should be able to relate what we know about 773 00:40:55 --> 00:41:00 different frequencies and different wavelengths, so Darcy 774 00:41:00 --> 00:41:06 can you switch us over to a clicker question here? 775 00:41:06 --> 00:41:09 And we can also talk about the difference between what's 776 00:41:09 --> 00:41:13 happening when we have emission, and we're going to 777 00:41:13 --> 00:41:14 switch over to absorption. 778 00:41:14 --> 00:41:17 So, we just talked about emission, so before we head 779 00:41:17 --> 00:41:20 into absorption, if you can answer this clicker question in 780 00:41:20 --> 00:41:23 terms of what do you think absorption means having just 781 00:41:23 --> 00:41:27 discussed hydrogen emission here? 782 00:41:27 --> 00:41:31 So we have four choices in terms of initial and final 783 00:41:31 --> 00:41:33 energy levels, and also what it means in terms of the electron 784 00:41:33 --> 00:41:36 -- whether it's gaining energy or whether it's going 785 00:41:36 --> 00:41:39 to be emitting energy? 786 00:41:39 --> 00:41:42 So, why do you take 10 seconds on that, we'll 787 00:41:42 --> 00:41:54 make it a quick one. 788 00:41:54 --> 00:41:55 All right, great. 789 00:41:55 --> 00:41:58 So already just from knowing the emission part, we can 790 00:41:58 --> 00:42:00 figure out what absorption probably means. 791 00:42:00 --> 00:42:04 Absorption is just the opposite of emission, so instead of 792 00:42:04 --> 00:42:10 starting at a high energy level and dropping down, when we 793 00:42:10 --> 00:42:14 absorb light we start low and we absorb energy to bring 794 00:42:14 --> 00:42:17 ourselves up to an n final that's higher. 795 00:42:17 --> 00:42:21 And instead of having the electron giving off energy as 796 00:42:21 --> 00:42:23 a photon, instead now the electron is going to take in 797 00:42:23 --> 00:42:29 energy from light and move up to that higher level. 798 00:42:29 --> 00:42:31 So now we're going to be talking briefly about 799 00:42:31 --> 00:42:33 photon absorption here. 800 00:42:33 --> 00:42:37 So again, this is just stating the same thing, and it could 801 00:42:37 --> 00:42:41 take in a long wavelength light, which would give it just 802 00:42:41 --> 00:42:44 a little bit of energy, maybe just enough to head up one 803 00:42:44 --> 00:42:48 energy level or two, or we could take in a high energy 804 00:42:48 --> 00:42:51 photon, and that means that the electron is going to get 805 00:42:51 --> 00:42:56 to move up to an even higher energy level. 806 00:42:56 --> 00:43:00 And again, we can talk about the same relationship here, so 807 00:43:00 --> 00:43:02 it's a very similar equation to the Rydberg equation that we 808 00:43:02 --> 00:43:06 saw earlier, except now what you see is the n initial and 809 00:43:06 --> 00:43:08 the n final are swapped places. 810 00:43:08 --> 00:43:12 So instead now we have r h over Planck's constant times 1 over 811 00:43:12 --> 00:43:17 n initial squared minus n final squared. 812 00:43:17 --> 00:43:20 And what you want to keep in mind is that whatever you're 813 00:43:20 --> 00:43:23 dealing with, whether it's absorption or emission, the 814 00:43:23 --> 00:43:25 frequency of the light is always going to be a positive 815 00:43:25 --> 00:43:27 number, so you always want to make sure what's inside these 816 00:43:27 --> 00:43:29 brackets here turns out to be positive. 817 00:43:29 --> 00:43:32 So that's just a little bit of a check for yourself, and it 818 00:43:32 --> 00:43:36 should make sense because what you're doing is you're 819 00:43:36 --> 00:43:39 calculating the difference between energy levels, so you 820 00:43:39 --> 00:43:42 just need to flip around which you put first to end up with a 821 00:43:42 --> 00:43:44 positive number here, and that's a little bit of a check 822 00:43:44 --> 00:43:48 that you can do what yourself. 823 00:43:48 --> 00:43:51 So let's do a sample calculation now using this 824 00:43:51 --> 00:43:55 Rydberg formula, and we'll switch back to emission, and 825 00:43:55 --> 00:43:57 the reason that we'll do that is because it would be nice to 826 00:43:57 --> 00:44:01 actually approve what we just saw here and calculate the 827 00:44:01 --> 00:44:04 frequency of one of our lines in the wavelength of one 828 00:44:04 --> 00:44:05 of the lines we saw. 829 00:44:05 --> 00:44:09 So what we'll do is this problem here, which is let's 830 00:44:09 --> 00:44:11 calculate out what the wavelength of radiation would 831 00:44:11 --> 00:44:17 be emitted from a hydrogen atom if we start at the n equals 3 832 00:44:17 --> 00:44:22 level and we go down to the n equals 2 level. 833 00:44:22 --> 00:44:29 So what we need to do here is use the Rydberg formula, and 834 00:44:29 --> 00:44:32 actually you'll be given the Rydberg formulas in both forms, 835 00:44:32 --> 00:44:38 both or absorption and emission on the exams. if you don't want 836 00:44:38 --> 00:44:41 to use that, you can also derive it as we did every time, 837 00:44:41 --> 00:44:44 it should intuitively make sense how we got there. 838 00:44:44 --> 00:44:47 But the exams are pretty short, so we don't want you doing that 839 00:44:47 --> 00:44:49 every time, so we'll save the 2 minutes and give you the 840 00:44:49 --> 00:44:52 equations directly, but it's still important to 841 00:44:52 --> 00:44:53 know how to use them. 842 00:44:53 --> 00:44:57 So, we can get from these energy differences to frequency 843 00:44:57 --> 00:45:04 by frequency is equal to r sub h over Planck's constant times 844 00:45:04 --> 00:45:12 1 over n final squared minus 1 over n initial squared. 845 00:45:12 --> 00:45:16 So let's actually just simplify this to the other version of 846 00:45:16 --> 00:45:19 the Rydberg constant, since we can use that here. 847 00:45:19 --> 00:45:24 So kind of that strange cursive r, and our n final is 2, so 848 00:45:24 --> 00:45:30 1 over 2 squared minus n initial, so 1 over 3 squared. 849 00:45:30 --> 00:45:33 So, our frequency of light is going to be equal 850 00:45:33 --> 00:45:40 to r times 5 over 36. 851 00:45:40 --> 00:45:43 But when we were actually looking at our different 852 00:45:43 --> 00:45:46 wavelengths, what we associate mostly with color is the wave 853 00:45:46 --> 00:45:49 length of the light and not the frequency of the light, 854 00:45:49 --> 00:45:52 so let's look at wavelength instead. 855 00:45:52 --> 00:45:57 We know that wavelength is equal to c over nu. 856 00:45:57 --> 00:46:03 We can plug in what we have for nu, so we have 36 c divided by 857 00:46:03 --> 00:46:10 5, and that cursivey Rydberg constant, and that gives us 36 858 00:46:10 --> 00:46:12 times the speed of light, 2 . 859 00:46:12 --> 00:46:23 998 times 10 the 8 meters per second, all over 5 times 3 . 860 00:46:23 --> 00:46:31 29 times 10 to the 15 per second. 861 00:46:31 --> 00:46:34 So, what we end up getting when we do this calculation is 862 00:46:34 --> 00:46:39 the wavelength of light, which is equal to 6 . 863 00:46:39 --> 00:46:44 57 times 10 to the negative 7 meters, or if we convert 864 00:46:44 --> 00:46:49 that to nanometers, we have 657 nanometers. 865 00:46:49 --> 00:46:52 So does anyone remember what range of light 657 866 00:46:52 --> 00:46:55 nanometers falls in? 867 00:46:55 --> 00:46:56 What color? 868 00:46:56 --> 00:46:57 STUDENT: Red. 869 00:46:57 --> 00:46:59 PROFESSOR: Yeah, it's in the red range. 870 00:46:59 --> 00:47:00 So that's promising. 871 00:47:00 --> 00:47:05 We did, in fact, see red in our spectrum, and it turns out that 872 00:47:05 --> 00:47:08 that's exactly the wavelength that we see is that we're 873 00:47:08 --> 00:47:10 at 657 nanometers. 874 00:47:10 --> 00:47:15 So it turns out that we can, in fact, use the energy levels to 875 00:47:15 --> 00:47:18 predict, and we could if we wanted to do them for all of 876 00:47:18 --> 00:47:20 the different wavelengths of light that we observed, and 877 00:47:20 --> 00:47:22 also all the different wavelengths of light that can 878 00:47:22 --> 00:47:26 be detected, even if we can't observe them. 879 00:47:26 --> 00:47:29 All right, so that's what we're going to cover in terms of the 880 00:47:29 --> 00:47:32 energy portion of the Schrodinger equation. 881 00:47:32 --> 00:47:36 I mentioned that we can also solve for psi here, which is 882 00:47:36 --> 00:47:40 the wave function, and we're running a little short on time, 883 00:47:40 --> 00:47:44 so we'll start on Monday with solving for the wave function.