1 00:00:01,873 --> 00:00:03,266 - [Voiceover] In the Bohr model of the hydrogen atom, 2 00:00:03,266 --> 00:00:06,313 the one electron of hydrogen is in orbit 3 00:00:06,313 --> 00:00:08,874 around the nucleus at a certain distance, r. 4 00:00:08,874 --> 00:00:10,433 So in the Bohr model, 5 00:00:10,433 --> 00:00:14,196 the electron is in orbit. 6 00:00:14,196 --> 00:00:16,858 In the quantum mechanics version of the hydrogen atom, 7 00:00:16,858 --> 00:00:19,906 we don't know exactly where the electron is, 8 00:00:19,906 --> 00:00:22,120 but we can say with high probability 9 00:00:22,120 --> 00:00:26,021 that the electron is in an orbital. 10 00:00:26,021 --> 00:00:28,520 An orbital is the region of space 11 00:00:28,520 --> 00:00:31,588 where the electron is most likely to be found. 12 00:00:31,588 --> 00:00:34,189 For hydrogen, imagine a sphere, 13 00:00:34,189 --> 00:00:36,849 a three-dimensional volume, a sphere, 14 00:00:36,849 --> 00:00:38,476 around the nucleus. 15 00:00:38,476 --> 00:00:40,385 Somewhere in that region of space, 16 00:00:40,385 --> 00:00:42,742 somewhere in that sphere, we're most likely 17 00:00:42,742 --> 00:00:45,973 to find the one electron of hydrogen. 18 00:00:45,973 --> 00:00:48,959 So we have these two competing visions. 19 00:00:48,959 --> 00:00:52,799 The Bohr model is classical mechanics. 20 00:00:52,799 --> 00:00:54,851 The electron orbits the nucleus 21 00:00:54,851 --> 00:00:56,497 like the planets around the sun, 22 00:00:56,497 --> 00:00:58,996 but quantum mechanics says we don't know 23 00:00:58,996 --> 00:01:01,474 exactly where that electron is. 24 00:01:01,474 --> 00:01:04,766 The Bohr model turns out to be incorrect, 25 00:01:04,766 --> 00:01:07,346 and quantum mechanics has proven to be the best way 26 00:01:07,346 --> 00:01:11,622 to explain electrons in orbitals. 27 00:01:12,786 --> 00:01:14,430 We can describe those electrons in orbitals 28 00:01:14,430 --> 00:01:17,071 using the four quantum numbers. 29 00:01:17,071 --> 00:01:19,733 Let's look at the first quantum number here. 30 00:01:19,733 --> 00:01:23,512 This is called the principal quantum number. 31 00:01:23,512 --> 00:01:26,905 The principal quantum number is symbolized by n. 32 00:01:26,905 --> 00:01:30,440 n is a positive integer, so n could be equal 33 00:01:30,440 --> 00:01:35,296 to one, two, three, and so on. 34 00:01:35,296 --> 00:01:37,571 It indicates the main energy level 35 00:01:37,571 --> 00:01:39,806 occupied by the electron. 36 00:01:40,975 --> 00:01:45,374 This tells us the main energy level. 37 00:01:45,374 --> 00:01:48,810 You might hear this referred to as a shell sometimes, 38 00:01:48,810 --> 00:01:52,825 so we could say what kind of shell the electron is in. 39 00:01:52,825 --> 00:01:55,537 As n increases, the average distance 40 00:01:55,537 --> 00:01:57,894 of the electron from the nucleus increases, 41 00:01:57,894 --> 00:02:00,190 and therefore so does the energy. 42 00:02:00,190 --> 00:02:03,481 For example, if this was our nucleus right here, 43 00:02:03,481 --> 00:02:06,387 and let's talk about n is equal to one. 44 00:02:06,387 --> 00:02:08,539 For n is equal to one, let's say the average 45 00:02:08,539 --> 00:02:11,811 distance from the nucleus is right about here. 46 00:02:11,811 --> 00:02:14,676 Let's compare that with n is equal to two. 47 00:02:14,676 --> 00:02:17,641 n is equal to two means a higher energy level, 48 00:02:17,641 --> 00:02:19,580 so on average, the electron 49 00:02:19,580 --> 00:02:22,119 is further away from the nucleus, 50 00:02:22,119 --> 00:02:25,403 and has a higher energy associated with it. 51 00:02:25,403 --> 00:02:28,966 That's the idea of the principal quantum number. 52 00:02:28,966 --> 00:02:31,689 You're thinking about energy levels or shells, 53 00:02:31,689 --> 00:02:32,431 and you're also thinking about 54 00:02:32,431 --> 00:02:34,642 average distance from the nucleus. 55 00:02:35,559 --> 00:02:37,958 All right, our second quantum number 56 00:02:37,958 --> 00:02:41,959 is called the angular momentum quantum number. 57 00:02:41,959 --> 00:02:44,479 The angular momentum quantum number 58 00:02:44,479 --> 00:02:46,633 is symbolized by l. 59 00:02:46,633 --> 00:02:49,843 l indicates the shape of the orbital. 60 00:02:50,919 --> 00:02:55,229 This will tell us the shape of the orbital. 61 00:02:55,229 --> 00:02:58,764 Values for l are dependent on n, 62 00:02:58,764 --> 00:03:02,868 so the values for l go from zero 63 00:03:02,868 --> 00:03:05,530 all the way up to n minus one, 64 00:03:05,530 --> 00:03:07,797 so it could be zero, one, two, 65 00:03:09,124 --> 00:03:13,571 or however values there are up to n minus one. 66 00:03:13,571 --> 00:03:15,523 For example, let's talk about the first 67 00:03:15,523 --> 00:03:18,042 main energy level, or the first shell. 68 00:03:18,042 --> 00:03:20,460 n is equal to one. 69 00:03:20,460 --> 00:03:23,366 There's only one possible value you could get 70 00:03:23,366 --> 00:03:26,921 for the angular momentum quantum number, l. 71 00:03:26,921 --> 00:03:30,700 n minus one is equal to zero, 72 00:03:31,576 --> 00:03:33,545 so that's the only possible value, 73 00:03:33,545 --> 00:03:35,801 the only allowed value of l. 74 00:03:35,801 --> 00:03:39,458 When l is equal to zero, we call this an s orbital. 75 00:03:39,458 --> 00:03:42,770 This is referring to an s orbital. 76 00:03:42,770 --> 00:03:45,511 The shape of an s orbital is a sphere. 77 00:03:45,511 --> 00:03:48,725 We've already talked about that with the hydrogen atom. 78 00:03:48,725 --> 00:03:50,930 Just imagine this as being a sphere, 79 00:03:50,930 --> 00:03:53,836 so a three-dimensional volume here. 80 00:03:53,836 --> 00:03:56,335 The angular momentum quantum number, l, 81 00:03:56,335 --> 00:03:59,545 since l is equal to zero, that corresponds to an s orbital, 82 00:03:59,545 --> 00:04:02,084 so we know that we're talking about an s orbital here 83 00:04:02,084 --> 00:04:03,335 which is shaped like a sphere. 84 00:04:03,335 --> 00:04:06,708 So the electron is most likely 85 00:04:06,708 --> 00:04:09,409 to be found somewhere in that sphere. 86 00:04:09,409 --> 00:04:11,706 Let's do the next shell. 87 00:04:11,706 --> 00:04:13,250 n is equal to two. 88 00:04:13,250 --> 00:04:17,923 If n is equal to two, what are the allowed values for l? 89 00:04:17,923 --> 00:04:20,279 l goes zero, one, and so on all the way 90 00:04:20,279 --> 00:04:21,599 up to n minus one. 91 00:04:21,600 --> 00:04:24,282 l is equal to zero. 92 00:04:24,282 --> 00:04:26,903 Then n minus one would be equal to one. 93 00:04:26,903 --> 00:04:29,950 So we have two possible values for l. 94 00:04:29,950 --> 00:04:31,413 l could be equal to zero, 95 00:04:31,413 --> 00:04:33,140 and l could be equal to one. 96 00:04:33,877 --> 00:04:36,330 Notice that the number of allowed values for l 97 00:04:36,330 --> 00:04:37,999 is equal to n. 98 00:04:37,999 --> 00:04:39,828 So for example, if n is equal to one, 99 00:04:39,828 --> 00:04:41,819 we have one allowed value. 100 00:04:41,819 --> 00:04:45,221 If n is equal to two, we have two allowed values. 101 00:04:46,132 --> 00:04:49,348 We've already talked about what l is equal to zero, 102 00:04:49,348 --> 00:04:50,444 what that means. 103 00:04:50,444 --> 00:04:53,046 l is equal to zero means an s orbital, 104 00:04:53,046 --> 00:04:54,935 shaped like a sphere. 105 00:04:54,935 --> 00:04:57,677 Now, in the second main energy level, 106 00:04:57,677 --> 00:05:00,786 or the second shell, we have another value for l. 107 00:05:00,786 --> 00:05:02,737 l is equal to one. 108 00:05:02,737 --> 00:05:06,455 When l is equal to one, we're talking about a p orbital. 109 00:05:06,455 --> 00:05:09,848 l is equal to one means a p orbital. 110 00:05:09,848 --> 00:05:12,855 The shape of a p orbital is a little bit strange, 111 00:05:12,855 --> 00:05:16,104 so I'll attempt to sketch it in here. 112 00:05:16,104 --> 00:05:18,625 You might hear several different terms for this. 113 00:05:18,625 --> 00:05:20,311 Imagine this is a volume. 114 00:05:20,311 --> 00:05:23,887 This is a three-dimensional region in here. 115 00:05:23,887 --> 00:05:27,524 You could call these dumbbell shaped or bow-tie, 116 00:05:27,524 --> 00:05:29,554 whatever makes the most sense to you. 117 00:05:30,338 --> 00:05:32,603 This is the orbital, this is the region of space 118 00:05:32,603 --> 00:05:34,614 where the electron is most likely 119 00:05:34,614 --> 00:05:38,036 to be found if it's found in a p orbital here. 120 00:05:38,543 --> 00:05:42,555 Sometimes you'll hear these called sub-shells. 121 00:05:43,600 --> 00:05:48,054 If n is equal to two, if we call this a shell, 122 00:05:49,232 --> 00:05:52,198 then we would call these sub-shells. 123 00:05:52,198 --> 00:05:54,270 These are sub-shells here. 124 00:05:54,270 --> 00:05:55,956 Again, we're talking about orbitals. 125 00:05:55,956 --> 00:05:57,663 l is equal to zero is an s orbital. 126 00:05:57,663 --> 00:06:00,670 l is equal to one is a p orbital. 127 00:06:00,670 --> 00:06:03,332 Let's look at the next quantum number. 128 00:06:03,332 --> 00:06:05,180 Let's get some more space down here. 129 00:06:05,180 --> 00:06:08,187 This is the magnetic quantum number, 130 00:06:08,187 --> 00:06:11,214 symbolized my m sub l here. 131 00:06:11,214 --> 00:06:15,136 m sub l indicates the orientation 132 00:06:15,136 --> 00:06:18,407 of an orbital around the nucleus. 133 00:06:19,437 --> 00:06:23,101 This tells us the orientation of that orbital. 134 00:06:23,671 --> 00:06:27,409 The values for ml depend on l. 135 00:06:29,055 --> 00:06:33,121 ml is equal to any integral value 136 00:06:33,121 --> 00:06:38,121 that goes from negative l to positive l. 137 00:06:39,338 --> 00:06:41,206 That sounds a little bit confusing. 138 00:06:43,052 --> 00:06:46,906 Let's go ahead and do the example of l is equal to zero. 139 00:06:46,906 --> 00:06:49,223 l is equal to zero up here. 140 00:06:49,223 --> 00:06:51,215 Let's go ahead and write that down here. 141 00:06:51,215 --> 00:06:52,576 If l is equal to zero, 142 00:06:52,576 --> 00:06:56,729 what are the allowed values for ml? 143 00:06:56,741 --> 00:06:58,021 There's only one, right? 144 00:06:58,021 --> 00:06:58,935 There's only one. 145 00:06:58,935 --> 00:07:02,511 The only possible value we could have here is zero. 146 00:07:02,511 --> 00:07:04,583 When l is equal to zero ... 147 00:07:05,736 --> 00:07:07,145 Let me use a different color here. 148 00:07:07,145 --> 00:07:08,590 If l is equal to zero, we know we're 149 00:07:08,590 --> 00:07:10,947 talking about an s orbital. 150 00:07:10,947 --> 00:07:12,115 When l is equal to zero, we're talking about 151 00:07:12,115 --> 00:07:15,509 an s orbital, which is shaped like a sphere. 152 00:07:15,509 --> 00:07:17,244 If you think about that, we have only one 153 00:07:17,244 --> 00:07:20,092 allowed value for the magnetic quantum number. 154 00:07:20,092 --> 00:07:21,880 That tells us the orientation, 155 00:07:21,880 --> 00:07:23,749 so there's only one orientation 156 00:07:23,749 --> 00:07:26,390 for that orbital around the nucleus. 157 00:07:26,390 --> 00:07:28,544 And that makes sense, because a sphere 158 00:07:28,544 --> 00:07:31,185 has only one possible orientation. 159 00:07:31,185 --> 00:07:35,907 If you think about this as being an xyz axis, 160 00:07:35,907 --> 00:07:37,788 (clears throat) excuse me, 161 00:07:37,788 --> 00:07:39,148 and if this is a sphere, 162 00:07:39,148 --> 00:07:42,766 there's only one way to orient that sphere in space. 163 00:07:42,766 --> 00:07:46,731 So that's the idea of the magnetic quantum number. 164 00:07:48,004 --> 00:07:52,945 Let's do the same thing for l is equal to one. 165 00:07:52,945 --> 00:07:54,814 Let's look at that now. 166 00:07:54,814 --> 00:07:57,313 If we're considering l is equal to one ... 167 00:07:57,313 --> 00:07:59,080 Let me use a different color here. 168 00:07:59,080 --> 00:08:01,315 l is equal to one. 169 00:08:01,315 --> 00:08:02,595 Let's write that down here. 170 00:08:02,595 --> 00:08:04,830 If l is equal to one, what are the allowed 171 00:08:04,830 --> 00:08:07,634 values for the magnetic quantum number? 172 00:08:07,634 --> 00:08:09,625 ml is equal to -- 173 00:08:09,625 --> 00:08:12,937 This goes from negative l to positive l, 174 00:08:12,937 --> 00:08:16,451 so any integral value from negative l to positive l. 175 00:08:16,451 --> 00:08:18,869 Negative l would be negative one, 176 00:08:18,869 --> 00:08:20,169 so let's go ahead and write this in here. 177 00:08:20,169 --> 00:08:24,801 We have negative one, zero, and positive one. 178 00:08:24,801 --> 00:08:27,647 So we have three possible values. 179 00:08:27,647 --> 00:08:31,061 When l is equal to one, we have three possible values 180 00:08:31,061 --> 00:08:33,052 for the magnetic quantum number, 181 00:08:33,052 --> 00:08:36,303 one, two, and three. 182 00:08:36,303 --> 00:08:39,919 The magnetic quantum number tells us the orientations, 183 00:08:39,919 --> 00:08:43,373 the possible orientations of the orbital 184 00:08:43,373 --> 00:08:45,445 or orbitals around the nucleus here. 185 00:08:45,445 --> 00:08:48,330 So we have three values for the magnetic quantum number. 186 00:08:48,330 --> 00:08:52,069 That means we get three different orientations. 187 00:08:52,069 --> 00:08:55,074 We already said that when l is equal to one, 188 00:08:55,074 --> 00:08:57,391 we're talking about a p orbital. 189 00:08:57,391 --> 00:08:59,850 A p orbital is shaped like a dumbbell here, 190 00:08:59,850 --> 00:09:02,268 so we have three possible orientations 191 00:09:02,268 --> 00:09:04,889 for a dumbbell shape. 192 00:09:04,889 --> 00:09:07,835 If we went ahead and mark these axes here, 193 00:09:07,835 --> 00:09:10,151 let's just say this is x axis, 194 00:09:10,151 --> 00:09:14,804 y axis, and the z axis here. 195 00:09:16,474 --> 00:09:19,943 We could put a dumbbell on the x axis like that. 196 00:09:19,943 --> 00:09:22,448 Again, imagine this as being a volume. 197 00:09:22,448 --> 00:09:23,993 This would be a p orbital. 198 00:09:23,993 --> 00:09:26,187 We call this a px orbital. 199 00:09:26,187 --> 00:09:29,742 It's a p orbital and it's on the x axis here. 200 00:09:29,742 --> 00:09:32,201 We have two more orientations. 201 00:09:32,201 --> 00:09:34,293 We could put, again, if this is x, 202 00:09:34,293 --> 00:09:36,366 this is y, and this is z, 203 00:09:38,135 --> 00:09:41,122 we could put a dumbbell here on the y axis. 204 00:09:41,122 --> 00:09:44,231 There's our second possible orientation. 205 00:09:44,231 --> 00:09:47,603 Finally, if this is x, this is y, and this is z, 206 00:09:47,603 --> 00:09:52,154 of course we could put a dumbbell on the z axis, like that. 207 00:09:52,154 --> 00:09:55,265 This would be a pz orbital. 208 00:09:56,080 --> 00:09:58,821 We could write a pz orbital here, 209 00:09:58,821 --> 00:10:02,145 and then this one right here would be a py orbital. 210 00:10:03,514 --> 00:10:07,337 We have three orbitals, we have three p orbitals here, 211 00:10:07,337 --> 00:10:10,097 one for each axis. 212 00:10:12,051 --> 00:10:14,062 Let's go to the last quantum number. 213 00:10:14,062 --> 00:10:17,618 The last quantum number is the spin quantum number. 214 00:10:17,618 --> 00:10:21,559 The spin quantum number is m sub s here. 215 00:10:21,559 --> 00:10:24,810 When it says spin, I'm going to put this in quotations. 216 00:10:24,810 --> 00:10:29,097 This seems to imply that an electron 217 00:10:29,097 --> 00:10:31,555 is spinning on an axis. 218 00:10:31,555 --> 00:10:33,059 That's not really what's happening, 219 00:10:33,059 --> 00:10:35,293 but let me just go ahead and draw that in here. 220 00:10:35,293 --> 00:10:36,716 I could have an electron ... 221 00:10:36,716 --> 00:10:39,723 Let me draw two different versions here. 222 00:10:40,953 --> 00:10:44,183 I could have an electron spin like a top, if you will, 223 00:10:44,183 --> 00:10:46,681 this way, or I could have an electron 224 00:10:46,681 --> 00:10:49,608 spin around that axis going this way. 225 00:10:49,608 --> 00:10:53,021 Again, this is not actually what's happening in reality. 226 00:10:53,021 --> 00:10:56,230 The electrons don't really spin on an axis like a top, 227 00:10:56,230 --> 00:10:58,324 but it does help me to think about 228 00:10:58,324 --> 00:11:00,477 the fact that we have two possible values 229 00:11:00,477 --> 00:11:02,794 for this spin quantum number. 230 00:11:02,794 --> 00:11:04,280 You could spin one way, 231 00:11:04,280 --> 00:11:07,021 so we could say the spin quantum number 232 00:11:07,021 --> 00:11:10,616 is equal to a positive one-half. 233 00:11:10,616 --> 00:11:13,401 Usually you hear that called spin up, 234 00:11:13,401 --> 00:11:15,480 so spin up, and we'll symbolize this 235 00:11:15,480 --> 00:11:19,137 with an arrow going up in later videos here. 236 00:11:19,137 --> 00:11:21,656 Then the other possible value for the spin 237 00:11:21,656 --> 00:11:24,155 quantum number, so the spin quantum number 238 00:11:24,155 --> 00:11:27,223 is equal to a negative one-half. 239 00:11:27,223 --> 00:11:30,535 You usually hear that referred to as spin down, 240 00:11:30,535 --> 00:11:33,075 and you could put an arrow going down. 241 00:11:33,075 --> 00:11:36,407 Again, electrons aren't really spinning 242 00:11:36,407 --> 00:11:38,398 in a physical sense like this, 243 00:11:38,398 --> 00:11:41,811 but, again, if you think about two possible ways 244 00:11:41,811 --> 00:11:43,213 for an electron to spin, 245 00:11:43,213 --> 00:11:46,016 then you get these two different, 246 00:11:46,016 --> 00:11:48,536 these two possible spin quantum numbers, 247 00:11:48,536 --> 00:11:51,258 so positive one-half or negative one-half. 248 00:11:51,258 --> 00:11:54,225 Those are the four quantum numbers, 249 00:11:54,225 --> 00:11:56,642 and we're going to use those to, again, 250 00:11:56,642 --> 00:00:00,000 think about electrons in orbitals.