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 --> 00:00:23
PROFESSOR: All right.
 

10
00:00:23 --> 00:00:26
So, let's get started.
 

11
00:00:26 --> 00:00:28
Why doesn't everyone take 10
more seconds to answer this

12
00:00:28 --> 00:00:42
clicker question here.
 

13
00:00:42 --> 00:00:42
All right.
 

14
00:00:42 --> 00:00:43
So, good.
 

15
00:00:43 --> 00:00:46
It looks like just about
everyone is able to go from

16
00:00:46 --> 00:00:50
the name of an orbital
to the state function.

17
00:00:50 --> 00:00:51
That's important.
 

18
00:00:51 --> 00:00:53
And we're actually going to get
a little bit deeper in our

19
00:00:53 --> 00:00:56
clicker questions here, since
when you do your problem-set it

20
00:00:56 --> 00:00:58
won't be quite this straight
forward that you'll be

21
00:00:58 --> 00:01:00
answering this kind of
question, but actually you'll

22
00:01:00 --> 00:01:03
be thinking about how many
different orbitals can have

23
00:01:03 --> 00:01:06
certain state functions or
certain orbital names.

24
00:01:06 --> 00:01:11
So let's go to a second clicker
question here and try one more.

25
00:01:11 --> 00:01:14
So why don't you tell me how
many possible orbitals you can

26
00:01:14 --> 00:01:17
have in a single atom that have
the following two

27
00:01:17 --> 00:01:17
quantum numbers?
 

28
00:01:17 --> 00:01:21
So let's say we have n
equals 4, and n sub l

29
00:01:21 --> 00:01:23
equalling negative 2.
 

30
00:01:23 --> 00:01:26
How many different orbitals can
you have that have those two

31
00:01:26 --> 00:01:27
quantum numbers in them?
 

32
00:01:27 --> 00:01:30
And this should look kind of
familiar to some of the

33
00:01:30 --> 00:01:32
problems you may have seen
on the problem-set if you

34
00:01:32 --> 00:01:38
started it this weekend.
 

35
00:01:38 --> 00:01:40
So this should be something you
can do pretty quickly, so let's

36
00:01:40 --> 00:01:53
take 10 more seconds on that.
 

37
00:01:53 --> 00:01:55
All right.
 

38
00:01:55 --> 00:01:57
OK, looks like we got the
majority, which is a good

39
00:01:57 --> 00:02:00
start, but we having
discrepancy on what

40
00:02:00 --> 00:02:01
people are thinking.
 

41
00:02:01 --> 00:02:03
So, let's go through this one.
 

42
00:02:03 --> 00:02:10
So, what we're saying is that
we have n equals to 4, and m

43
00:02:10 --> 00:02:12
sub I being equal
to negative 2.

44
00:02:12 --> 00:02:15
If we have n equals 4,
what is the highest value

45
00:02:15 --> 00:02:17
of l that we can have?
 

46
00:02:17 --> 00:02:18
STUDENT: 3.
 

47
00:02:18 --> 00:02:18
PROFESSOR: OK.
 

48
00:02:18 --> 00:02:22
We can have n 4, l 3, and then,
sure, we can have m sub l equal

49
00:02:22 --> 00:02:26
negative 2 if l equals 3 What's
the second value of

50
00:02:26 --> 00:02:27
l that we can have?
 

51
00:02:27 --> 00:02:28
2.
 

52
00:02:28 --> 00:02:29
OK.
 

53
00:02:29 --> 00:02:32
So we can have this
orbital here.

54
00:02:32 --> 00:02:36
What about l equals
1, can we have this?

55
00:02:36 --> 00:02:37
No, we can't.
 

56
00:02:37 --> 00:02:42
Because if l equals 1, we can
not have m sub l equal negative

57
00:02:42 --> 00:02:45
2, right, because the magnetic
quantum number only goes from

58
00:02:45 --> 00:02:49
negative l to positive l here.
 

59
00:02:49 --> 00:02:52
So that means it's not
possible, if we've made these

60
00:02:52 --> 00:02:54
stipulations in the first
place, to have a case

61
00:02:54 --> 00:02:56
where l equals 1.
 

62
00:02:56 --> 00:02:59
So this means we can only have
2 different values of l.

63
00:02:59 --> 00:03:02
We already know our value of m.
 

64
00:03:02 --> 00:03:05
So now we're just counting up
our orbitals, an orbital is

65
00:03:05 --> 00:03:08
completely described by
the 3 quantum numbers.

66
00:03:08 --> 00:03:12
So we end up having
2 orbitals here.

67
00:03:12 --> 00:03:18
All right, so hopefully if you
see any other combination of

68
00:03:18 --> 00:03:21
quantum numbers, for example,
if it doesn't quickly come to

69
00:03:21 --> 00:03:24
you how many orbitals you have,
you can actually try to write

70
00:03:24 --> 00:03:26
out all the possible
orbitals and that

71
00:03:26 --> 00:03:28
should get you started.
 

72
00:03:28 --> 00:03:29
All right.
 

73
00:03:29 --> 00:03:32
So today we're going to
finish up our discussion

74
00:03:32 --> 00:03:34
of the hydrogen atom.
 

75
00:03:34 --> 00:03:37
We'd started on Monday talking
about radial probability

76
00:03:37 --> 00:03:39
distributions for
the s orbitals.

77
00:03:39 --> 00:03:42
We'll finish that up, and then
we're going to move on to

78
00:03:42 --> 00:03:44
talking about the p orbitals.
 

79
00:03:44 --> 00:03:47
We'll start with talking about
the shape, just like we did

80
00:03:47 --> 00:03:50
with the s orbitals, and then
move on to those radial

81
00:03:50 --> 00:03:53
probability distributions and
compare the radial probability

82
00:03:53 --> 00:03:58
at different radius for p
orbital versus an s orbital.

83
00:03:58 --> 00:04:01
And once we do that, we're
actually going to move on

84
00:04:01 --> 00:04:03
to multi-electron atoms.
 

85
00:04:03 --> 00:04:07
So, you might have noticed that
we will have spent about 6 and

86
00:04:07 --> 00:04:10
1/2 lectures just getting to
the point where we have only

87
00:04:10 --> 00:04:13
one electron, so we're
only up hydrogen so far.

88
00:04:13 --> 00:04:16
And you might have kind of been
projecting ahead and thinking

89
00:04:16 --> 00:04:19
if we keep up at this pace,
pretty much we would only get

90
00:04:19 --> 00:04:22
to carbon by the end
of the semester.

91
00:04:22 --> 00:04:25
So I will assure you that we
will not be spending 6 lectures

92
00:04:25 --> 00:04:28
per atom as we move forward,
and in fact, what we're going

93
00:04:28 --> 00:04:32
to find is that by taking all
the principles we've learned

94
00:04:32 --> 00:04:36
about the hydrogen atom and
applying that to multi-electron

95
00:04:36 --> 00:04:39
atoms, but making a few of
changes and making a few

96
00:04:39 --> 00:04:41
modifications to take into the
fact that we're going to have

97
00:04:41 --> 00:04:44
electron-electron repulsions
going on, we'll be able to

98
00:04:44 --> 00:04:48
think about any multi-electron
atom using these same general

99
00:04:48 --> 00:04:51
ideas, and the Schrodinger
equation ideas that we came up

100
00:04:51 --> 00:04:53
with and have looked at
for the hydrogen atom.

101
00:04:53 --> 00:04:56
And this is really good
news because it's good

102
00:04:56 --> 00:04:57
to get passed carbon.
 

103
00:04:57 --> 00:05:00
I'm an organic chemist, so I
love carbon, it's one of my

104
00:05:00 --> 00:05:02
favorite atoms to talk about,
but it would be nice to get to

105
00:05:02 --> 00:05:05
the point of bonding and even
reactions to talk about all the

106
00:05:05 --> 00:05:08
exciting things we can think
about once we're at that point.

107
00:05:08 --> 00:05:11
So, one we finish our
discussion of how we think

108
00:05:11 --> 00:05:16
about multi-electron atoms, we
can go right on and start

109
00:05:16 --> 00:05:18
talking about these
other things.

110
00:05:18 --> 00:05:24
All right.
 

111
00:05:24 --> 00:05:27
So, let's pick up where we
left off, first of all

112
00:05:27 --> 00:05:30
we're still on the
hydrogen atom from Monday.

113
00:05:30 --> 00:05:33
And on Monday what we were
discussing was the solution

114
00:05:33 --> 00:05:36
to the Schrodinger equation
for the wave function.

115
00:05:36 --> 00:05:39
And we also, when we solved or
we looked at the solution to

116
00:05:39 --> 00:05:43
that Schrodinger equation, what
we saw was that we actually

117
00:05:43 --> 00:05:47
needed three different quantum
numbers to fully describe

118
00:05:47 --> 00:05:49
the wave function of a
hydrogen atom or to fully

119
00:05:49 --> 00:05:51
describe an orbital.
 

120
00:05:51 --> 00:05:54
We didn't just need that n, not
just the principle quantum

121
00:05:54 --> 00:05:56
number that we needed to
discuss the energy, but we also

122
00:05:56 --> 00:05:59
need to talk about l and m, as
we did in our clicker

123
00:05:59 --> 00:06:01
question up here.
 

124
00:06:01 --> 00:06:03
We also talked about well,
what is that when we say

125
00:06:03 --> 00:06:06
wave function, what does
that actually mean?

126
00:06:06 --> 00:06:09
And first we discussed the fact
that well, in terms of a

127
00:06:09 --> 00:06:11
classical analogy, we don't
really have one for wave

128
00:06:11 --> 00:06:13
function, we can't really think
of a way to picture wave

129
00:06:13 --> 00:06:16
function thinking in
classical terms.

130
00:06:16 --> 00:06:18
But we do have an
interpretation for wave

131
00:06:18 --> 00:06:19
function squared.
 

132
00:06:19 --> 00:06:22
And when we take the wave
function and square it, that's

133
00:06:22 --> 00:06:24
going to be equal to the
probability density of finding

134
00:06:24 --> 00:06:29
an electron at some
point in your atom.

135
00:06:29 --> 00:06:33
And that's useful in terms of
seeing a general shape, but if

136
00:06:33 --> 00:06:36
we're actually interested in
thinking about how far away

137
00:06:36 --> 00:06:38
that electron is from the
nucleus, you can see that

138
00:06:38 --> 00:06:41
instead of talking about
probability density, which is

139
00:06:41 --> 00:06:44
the probability per volume,
instead it would be much more

140
00:06:44 --> 00:06:47
useful to talk about something
called radial probability

141
00:06:47 --> 00:06:51
distribution, or in other words
talking about the probability

142
00:06:51 --> 00:06:55
of finding the electron at some
distance, which we define as r,

143
00:06:55 --> 00:06:59
from the nucleus in a
spherical shell that is just

144
00:06:59 --> 00:07:02
infinitesimally small,
infinitesimally thin at

145
00:07:02 --> 00:07:06
distance or at a thickness that
we'll call a d r here.

146
00:07:06 --> 00:07:10
So, basically what we're saying
is if we take any shell that's

147
00:07:10 --> 00:07:14
at some distance away from the
nucleus, we can think about

148
00:07:14 --> 00:07:16
what the probability is of
finding an electron at that

149
00:07:16 --> 00:07:19
radius, and that's the
definition we gave to the

150
00:07:19 --> 00:07:22
radial probability
distribution.

151
00:07:22 --> 00:07:25
And we can look at the
formula that got us here.

152
00:07:25 --> 00:07:29
This is the radial probability
distribution formula for an s

153
00:07:29 --> 00:07:32
orbital, which is, of course,
dealing with something that's

154
00:07:32 --> 00:07:33
spherically symmetrical.
 

155
00:07:33 --> 00:07:35
It's somewhat different when
we're talking about the p or

156
00:07:35 --> 00:07:38
the d orbitals, and we won't go
into the equation there, but

157
00:07:38 --> 00:07:40
this will give you an idea of
what we're really talking

158
00:07:40 --> 00:07:43
about with this radial
probability distribution.

159
00:07:43 --> 00:07:45
So, what we can do to actually
get a probability instead of a

160
00:07:45 --> 00:07:49
probability density that we're
talking about is to take the

161
00:07:49 --> 00:07:53
wave function squared, which we
know is probability density,

162
00:07:53 --> 00:07:57
and multiply it by the volume
of that very, very thin

163
00:07:57 --> 00:08:01
spherical shell that we're
talking about at distance r.

164
00:08:01 --> 00:08:03
So if we want to talk about the
volume of that, we just talk

165
00:08:03 --> 00:08:07
about the surface area, which
is 4 pi r squared, and

166
00:08:07 --> 00:08:10
we multiply that by
the thickness d r.

167
00:08:10 --> 00:08:14
So if we take this term, which
is a volume term, and multiply

168
00:08:14 --> 00:08:17
it by probability over volume,
what we're going to end up with

169
00:08:17 --> 00:08:21
is an actual probability of
finding our electron at that

170
00:08:21 --> 00:08:24
distance, r, from the nucleus.
 

171
00:08:24 --> 00:08:28
So, the example that we took on
Monday and that we ended with

172
00:08:28 --> 00:08:31
when we ended class, was
looking at the 1 s orbital for

173
00:08:31 --> 00:08:35
hydrogen atom, and what we
could do is we could graph the

174
00:08:35 --> 00:08:39
radial probability as a
function of radius here.

175
00:08:39 --> 00:08:42
And when we do that we can see
this curve, this probability

176
00:08:42 --> 00:08:46
curve, where we have a maximum
probability of finding the

177
00:08:46 --> 00:08:50
electron this far away
from the nucleus.

178
00:08:50 --> 00:08:53
And we call that most probable
radius r sub m p, or

179
00:08:53 --> 00:08:55
most probable radius.
 

180
00:08:55 --> 00:08:57
And what is discussed is that
for a 1 s hydrogen atom, that

181
00:08:57 --> 00:09:01
falls at an a nought distance
away from the nucleus.

182
00:09:01 --> 00:09:05
And remember, a nought, that's
just the Bohr radius, it's

183
00:09:05 --> 00:09:07
a constant -- that's all
we need to worry about.

184
00:09:07 --> 00:09:10
We talked about the Bohr
model and how that told

185
00:09:10 --> 00:09:12
us an exact distance.
 

186
00:09:12 --> 00:09:15
It was a classical model,
right, so we could say the

187
00:09:15 --> 00:09:18
electron is exactly this
far away from the nucleus.

188
00:09:18 --> 00:09:20
We can not do that with quantum
mechanics, the more true

189
00:09:20 --> 00:09:23
picture is the best we can get
to is talk about what the

190
00:09:23 --> 00:09:27
probability is of finding the
electron at any given nucleus.

191
00:09:27 --> 00:09:31
And the most probable one
here is that a nought.

192
00:09:31 --> 00:09:34
The other thing that we looked
at, which I want to stress

193
00:09:34 --> 00:09:37
again and I'll stress it as
many times as I can fit it into

194
00:09:37 --> 00:09:39
lecture, because this is
something that confuses

195
00:09:39 --> 00:09:42
students when they're trying to
identify, for example,

196
00:09:42 --> 00:09:44
different nodes or areas
of no probability.

197
00:09:44 --> 00:09:49
In an orbital is remember that
this area right here at r

198
00:09:49 --> 00:09:51
equals zerio, that
is not a node.

199
00:09:51 --> 00:09:54
We will always have r equals
zero in these radial

200
00:09:54 --> 00:09:57
probability distribution
graphs, and we can think

201
00:09:57 --> 00:09:58
about why that is.
 

202
00:09:58 --> 00:10:00
At first it might be
counter-intuitive because we

203
00:10:00 --> 00:10:04
know the probability density at
the nucleus is the greatest.

204
00:10:04 --> 00:10:07
So the probability of having
an electron at the nucleus

205
00:10:07 --> 00:10:10
in terms of probability per
volume is very, very high.

206
00:10:10 --> 00:10:13
But remember that we need to
multiply it by the volume

207
00:10:13 --> 00:10:16
here, the volume of some
sphere we've defined.

208
00:10:16 --> 00:10:19
And when we define that as r
being equal to zero,

209
00:10:19 --> 00:10:21
essentially we're
multiplying the probability

210
00:10:21 --> 00:10:22
density by zero.
 

211
00:10:22 --> 00:10:26
So that's why we have this zero
point here, and just to point

212
00:10:26 --> 00:10:29
out again and again and again,
it's not a radial node, it's

213
00:10:29 --> 00:10:32
just a point where we're
starting our graph, because

214
00:10:32 --> 00:10:36
we're multiplying it
by r equals zero.

215
00:10:36 --> 00:10:40
So, we can look at other radial
probability distributions

216
00:10:40 --> 00:10:42
of other wave functions
that we talked about.

217
00:10:42 --> 00:10:45
We talked about the wave
function for a 2 s orbital,

218
00:10:45 --> 00:10:48
and also for a 3 s orbital.
 

219
00:10:48 --> 00:10:55
So, let's go ahead and think
about drawing what that would

220
00:10:55 --> 00:10:57
look like in terms of the
radial probability

221
00:10:57 --> 00:10:59
distribution.
 

222
00:10:59 --> 00:11:03
So what we're graphing here
is the radius as a function

223
00:11:03 --> 00:11:05
of radial probability.
 

224
00:11:05 --> 00:11:11
And for a 2 s orbital, you
get a graph that's going to

225
00:11:11 --> 00:11:14
look something like this.
 

226
00:11:14 --> 00:11:16
So, again, we're
starting at zero.

227
00:11:16 --> 00:11:19
We have one node here, and
we can again define that

228
00:11:19 --> 00:11:22
most probable radius.
 

229
00:11:22 --> 00:11:25
And it turns out that for a
2 s orbital, that's equal

230
00:11:25 --> 00:11:29
to 6 times a nought.
 

231
00:11:29 --> 00:11:32
So when we think about what it
is that this radial probability

232
00:11:32 --> 00:11:35
distribution is telling us,
it's telling us that it is most

233
00:11:35 --> 00:11:41
likely that an electron in a 2
s orbital of hydrogen is six

234
00:11:41 --> 00:11:44
times further away from the
nucleus than it is

235
00:11:44 --> 00:11:46
in a 1 s orbital.
 

236
00:11:46 --> 00:11:50
So another way to say that is,
in a sense, if we're thinking

237
00:11:50 --> 00:11:52
about the excited state of a
hydrogen atom, the first

238
00:11:52 --> 00:11:57
excited state, or the n equals
2 state, what we're saying is

239
00:11:57 --> 00:12:01
that it's actually bigger than
the ground state, or the 1 s

240
00:12:01 --> 00:12:03
state of a hydrogen atom.
 

241
00:12:03 --> 00:12:05
And when we say bigger,
remember this is not a

242
00:12:05 --> 00:12:08
classical description
we're talking about.

243
00:12:08 --> 00:12:10
We are talking about
probability, but what we're

244
00:12:10 --> 00:12:13
saying is that most probable
radius is further away

245
00:12:13 --> 00:12:16
from the nucleus.
 

246
00:12:16 --> 00:12:20
So we can also look at this
in terms of the 3 s orbital.

247
00:12:20 --> 00:12:26
And in this case, we
have a graph that looks

248
00:12:26 --> 00:12:28
something like this.
 

249
00:12:28 --> 00:12:32
So you can draw that
into your notes.

250
00:12:32 --> 00:12:36
And again, we can define what
that most probable radius is,

251
00:12:36 --> 00:12:38
that distance at which
we're most likely to

252
00:12:38 --> 00:12:40
find an electron.
 

253
00:12:40 --> 00:12:43
And in the case of the 3
s orbital, that's going

254
00:12:43 --> 00:12:44
to be equal to 11 .
 

255
00:12:44 --> 00:12:48
5 times a nought.
 

256
00:12:48 --> 00:12:51
So again, what we're saying
here is that it is most likely

257
00:12:51 --> 00:12:55
in the 3 s orbital that we
would find the electron 11 and

258
00:12:55 --> 00:12:59
1/2 times further away from the
nucleus than we would in a

259
00:12:59 --> 00:13:03
around state hydrogen atom.
 

260
00:13:03 --> 00:13:05
And I just want to point out
here in terms of things that

261
00:13:05 --> 00:13:08
you're responsible for, you
should know that the most

262
00:13:08 --> 00:13:12
probable radius for a 1 s
hydrogen atom is

263
00:13:12 --> 00:13:13
equal a nought.
 

264
00:13:13 --> 00:13:18
And you should know that a 2 s
is larger than that, and a 3 s

265
00:13:18 --> 00:13:21
is even larger, and of course,
hopefully as we go to 4 and 5,

266
00:13:21 --> 00:13:23
you would be able to guess that
those are going to

267
00:13:23 --> 00:13:24
get even larger.
 

268
00:13:24 --> 00:13:26
But you're not responsible
for knowing specifically

269
00:13:26 --> 00:13:27
that it's 11 .
 

270
00:13:27 --> 00:13:28
5 times greater.
 

271
00:13:28 --> 00:13:33
You just need to know
the trend there.

272
00:13:33 --> 00:13:36
Another thing to point out in
these two graphs is that we do

273
00:13:36 --> 00:13:40
have nodes, and we figured out
last time, we calculated

274
00:13:40 --> 00:13:42
how many nodes we should
have in a 2 s orbital.

275
00:13:42 --> 00:13:46
And in terms of radial nodes,
we expect to see one node.

276
00:13:46 --> 00:13:49
And how many nodes do you see
in the 3 s orbital? two, good.

277
00:13:49 --> 00:13:53
I'm glad to hear that
no one counted this r

278
00:13:53 --> 00:13:54
equal zero as a node.
 

279
00:13:54 --> 00:13:58
So we expect to see two nodes
right here in the 3 s orbital.

280
00:13:58 --> 00:14:02
And we can calculate that with
the formula that we used, which

281
00:14:02 --> 00:14:09
was just n minus l minus 1
equals the number of nodes.

282
00:14:09 --> 00:14:12
Or we could just look at the
radial probability distribution

283
00:14:12 --> 00:14:15
itself and see how
many nodes there are.

284
00:14:15 --> 00:14:19
So if we're looking at these
two situations here and we're

285
00:14:19 --> 00:14:22
actually thinking of them from
a more classical standpoint,

286
00:14:22 --> 00:14:25
which is natural for us to do
because we live our lives in

287
00:14:25 --> 00:14:27
the every day world, not
thinking about things on the

288
00:14:27 --> 00:14:30
atomic size scale all
the time, most of us.

289
00:14:30 --> 00:14:35
So, for example, if we were to
look at this 3 s orbital here,

290
00:14:35 --> 00:14:38
you might have the question of
how this can be, because we're

291
00:14:38 --> 00:14:42
saying that, for example, we
have probability of having an

292
00:14:42 --> 00:14:45
electron here, an electron can
also be way out at

293
00:14:45 --> 00:14:46
this radius here.
 

294
00:14:46 --> 00:14:49
But what we're saying is
there's a node here, so that

295
00:14:49 --> 00:14:51
there's no probability
of finding an electron

296
00:14:51 --> 00:14:53
between those two points.
 

297
00:14:53 --> 00:14:56
So you can think of it, if we
were to just think of it as a

298
00:14:56 --> 00:14:59
straight line that we were
going across, essentially what

299
00:14:59 --> 00:15:04
we're saying is that we're
getting from point a to point c

300
00:15:04 --> 00:15:07
without ever getting
through point b.

301
00:15:07 --> 00:15:10
So, that can be a little bit
confusing for us to think

302
00:15:10 --> 00:15:13
about, and when it's a very
good question you might, in

303
00:15:13 --> 00:15:17
fact, say well, maybe there's
not zero probability here,

304
00:15:17 --> 00:15:20
maybe it's just this teeny,
teeny, tiny number, and in

305
00:15:20 --> 00:15:24
fact, sometimes an electron can
get through, it's just very low

306
00:15:24 --> 00:15:27
probability so that's why
we never really see it.

307
00:15:27 --> 00:15:28
And in fact, that's
not the answer.

308
00:15:28 --> 00:15:31
The answer is, in fact, there
is zero, absolutely zero

309
00:15:31 --> 00:15:33
probability of finding
a electron here.

310
00:15:33 --> 00:15:37
So basically we're saying yes,
we can go from point a to point

311
00:15:37 --> 00:15:39
c without ever going
through point b.

312
00:15:39 --> 00:15:42
That might seem confusing if
you're thinking about

313
00:15:42 --> 00:15:45
particles, but remember we're
talking about the wave-like

314
00:15:45 --> 00:15:47
nature of electrons.
 

315
00:15:47 --> 00:15:51
So, the quantum mechanical
interpretation is that we can,

316
00:15:51 --> 00:15:55
in fact, have probability
density here and probability

317
00:15:55 --> 00:16:00
density there, without having
any probability of having the

318
00:16:00 --> 00:16:02
electron in the space between.
 

319
00:16:02 --> 00:16:05
And you can think about that if
you think about a standing

320
00:16:05 --> 00:16:08
wave, for example, where you
can have amplitude at many

321
00:16:08 --> 00:16:11
different values of x, so an
amplitude at many different

322
00:16:11 --> 00:16:14
distances, but you also have
areas where there

323
00:16:14 --> 00:16:15
is a 0 amplitude.
 

324
00:16:15 --> 00:16:18
So, remember this makes sense
if you just think of it as a

325
00:16:18 --> 00:16:22
wave and forget the particle
part of it for right now,

326
00:16:22 --> 00:16:25
because that would be very
upsetting to think about and

327
00:16:25 --> 00:16:27
that's not, in fact, what's
going on, we're talking about

328
00:16:27 --> 00:16:28
quantum mechanics here.
 

329
00:16:28 --> 00:16:29
Yes.
 

330
00:16:29 --> 00:16:36
STUDENT: [INAUDIBLE]
 

331
00:16:36 --> 00:16:37
PROFESSOR: Oh, I'm sorry.
 

332
00:16:37 --> 00:16:40
So it's n minus l minus 1.
 

333
00:16:40 --> 00:16:46
So here we have 3 minus l
equals 0, because it's an s

334
00:16:46 --> 00:16:57
orbital, minus 1, so we have
two radial nodes here.

335
00:16:57 --> 00:16:57
OK.
 

336
00:16:57 --> 00:17:02
So let's actually go to a
clicker question now on radial

337
00:17:02 --> 00:17:04
probability distributions.
 

338
00:17:04 --> 00:17:08
So I mentioned you should be
able to identify both how many

339
00:17:08 --> 00:17:11
nodes you have and what a graph
might look like of different

340
00:17:11 --> 00:17:13
radial probability
distributions.

341
00:17:13 --> 00:17:17
So here, what I'd like you to
do is identify the correct

342
00:17:17 --> 00:17:22
radial probability distribution
plot for a 5 s orbital, and

343
00:17:22 --> 00:17:24
also make sure that it matches
up with the right number of

344
00:17:24 --> 00:17:33
radial nodes that
you would expect.

345
00:17:33 --> 00:17:33
All right.
 

346
00:17:33 --> 00:17:36
Let's take 10 more seconds on
that, this should be a quick

347
00:17:36 --> 00:17:47
identification for us to do.
 

348
00:17:47 --> 00:17:48
All right.
 

349
00:17:48 --> 00:17:51
So it looks like 82% got
the correct answer here.

350
00:17:51 --> 00:17:55
So, you should know that
there's four radial nodes,

351
00:17:55 --> 00:18:00
right, we have 5 minus 1 minus
l -- is there a question?

352
00:18:00 --> 00:18:03
STUDENT: [INAUDIBLE]
 

353
00:18:03 --> 00:18:06
PROFESSOR: It is very difficult
for me to draw graphs

354
00:18:06 --> 00:18:07
on the computer.
 

355
00:18:07 --> 00:18:09
That's a good point, I'm sorry.
 

356
00:18:09 --> 00:18:12
This was my best attempt at
hitting zero and not having

357
00:18:12 --> 00:18:13
the graph go down there.
 

358
00:18:13 --> 00:18:16
I'm not the most gifted at
drawing on the computer.

359
00:18:16 --> 00:18:20
So yes, it should be zero
at zero, but I made

360
00:18:20 --> 00:18:22
the line too thick.
 

361
00:18:22 --> 00:18:24
So, assuming -- if anyone got
it wrong because of that,

362
00:18:24 --> 00:18:26
that's my apologies,
that's my fault.

363
00:18:26 --> 00:18:29
But you should see that there
are four radial nodes here

364
00:18:29 --> 00:18:33
since we have a 5 s orbital.
 

365
00:18:33 --> 00:18:38
And also that we know that the
zero does not count as a node,

366
00:18:38 --> 00:18:41
if per se I actually had
managed to hit zero in drawing

367
00:18:41 --> 00:18:46
that, so the correct answer
would be the bottom one there.

368
00:18:46 --> 00:18:49
So, you should be able to
generally identify and draw the

369
00:18:49 --> 00:18:52
general form of these radial
probability distributions.

370
00:18:52 --> 00:18:56
Obviously we don't expect you
to know exactly what the

371
00:18:56 --> 00:18:58
distances are, but you should
be able to compare

372
00:18:58 --> 00:19:00
them relatively.
 

373
00:19:00 --> 00:19:00
Yes?
 

374
00:19:00 --> 00:19:06
STUDENT: [INAUDIBLE]
 

375
00:19:06 --> 00:19:10
PROFESSOR: No, they actually
don't, and when you

376
00:19:10 --> 00:19:10
graph it all out.
 

377
00:19:10 --> 00:19:12
You can see this if you
look at some examples

378
00:19:12 --> 00:19:13
in your book, actually.
 

379
00:19:13 --> 00:19:16
So this doesn't fall, for
example, at 6 a nought, but

380
00:19:16 --> 00:19:18
that's a really good question.
 

381
00:19:18 --> 00:19:21
And the trend always is that
the probability gets smaller

382
00:19:21 --> 00:19:26
with each of the peaks
as you're drawing them.

383
00:19:26 --> 00:19:26
All right.
 

384
00:19:26 --> 00:19:29
So we can switch
back to our notes.

385
00:19:29 --> 00:19:33
So we got our clicker
question set there.

386
00:19:33 --> 00:19:36
And so now we can move on to
thinking about p orbitals,

387
00:19:36 --> 00:19:38
we now have two ways to
talk about p orbitals.

388
00:19:38 --> 00:19:40
We can talk about the wave
function squared, the

389
00:19:40 --> 00:19:43
probability density, or we
can talk about the radial

390
00:19:43 --> 00:19:46
probability distribution.
 

391
00:19:46 --> 00:19:50
So when we talk about p
orbitals, it's similar to

392
00:19:50 --> 00:19:53
talking about s orbitals, and
the difference lies, and now we

393
00:19:53 --> 00:19:57
have a different value for l,
so l equals 1 for a p orbital,

394
00:19:57 --> 00:20:01
and we know if we have l equal
1, we can have three different

395
00:20:01 --> 00:20:06
total orbitals that have
sub-shell of l equalling 1.

396
00:20:06 --> 00:20:11
So we can have, if we have the
final quantum number m equal

397
00:20:11 --> 00:20:14
plus 1 or minus 1, we're
dealing with a p x

398
00:20:14 --> 00:20:16
or a p y orbital.
 

399
00:20:16 --> 00:20:19
Remember, we don't do a
one-to-one correlation, because

400
00:20:19 --> 00:20:23
p x and p y are some linear
combination of the m plus

401
00:20:23 --> 00:20:25
1 and m minus 1 orbital.
 

402
00:20:25 --> 00:20:28
And if we talk about m
equals 0, we're looking

403
00:20:28 --> 00:20:29
at the p z orbital.
 

404
00:20:29 --> 00:20:36
And the significant difference
between s orbitals and p

405
00:20:36 --> 00:20:40
orbitals that comes from the
fact that we do have angular

406
00:20:40 --> 00:20:43
momentum here in these p
orbitals, is that p orbital

407
00:20:43 --> 00:20:47
wave functions do, in fact,
have theta and phi dependence.

408
00:20:47 --> 00:20:49
So they do have an
angular dependence that

409
00:20:49 --> 00:20:52
we're talking about.
 

410
00:20:52 --> 00:20:54
And what I'm showing here is
not on your notes, if you're

411
00:20:54 --> 00:20:56
interested you can look
it up in your book.

412
00:20:56 --> 00:20:59
This is a table that's directly
from your book, and what it's

413
00:20:59 --> 00:21:01
just showing is the wave
function for a bunch of

414
00:21:01 --> 00:21:02
different orbitals.
 

415
00:21:02 --> 00:21:04
I mentioned last time
that there was this

416
00:21:04 --> 00:21:05
list in your book.
 

417
00:21:05 --> 00:21:10
And what I want to point out
here is this angular dependence

418
00:21:10 --> 00:21:13
for the p orbitals for
the l equals 1 orbital.

419
00:21:13 --> 00:21:17
So, first, if I point out when
l equals 0, when we have an s

420
00:21:17 --> 00:21:21
orbital, what you see is that
angular part of the wave

421
00:21:21 --> 00:21:23
function is equal
to a constant.

422
00:21:23 --> 00:21:26
So, remember we can break up
the total wave function into

423
00:21:26 --> 00:21:30
the radial part and
the angular part.

424
00:21:30 --> 00:21:31
When we look at this angular
part, we see that it's always

425
00:21:31 --> 00:21:35
the square root of 1 over 4
pi, it doesn't matter what

426
00:21:35 --> 00:21:39
the angle is, it's not
dependent on the angle.

427
00:21:39 --> 00:21:43
In contrast when we're looking
at a p orbital, so any time l

428
00:21:43 --> 00:21:46
is equal to 1, and you look at
angular part of the wave

429
00:21:46 --> 00:21:49
function here, what you see is
the wave function either

430
00:21:49 --> 00:21:52
depends on theta or is
dependent on both

431
00:21:52 --> 00:21:54
theta and phi.
 

432
00:21:54 --> 00:21:57
So we do, in fact, have a
dependence on what the angle

433
00:21:57 --> 00:22:01
is of the electron as we
define it in the orbital.

434
00:22:01 --> 00:22:06
So what this means is that
unlike s orbitals, p orbitals

435
00:22:06 --> 00:22:10
are not spherically symmetrical
-- they don't have the exact

436
00:22:10 --> 00:22:13
same shape at any radius
from the nucleus.

437
00:22:13 --> 00:22:18
And these shapes of p orbitals
probably do look familiar to

438
00:22:18 --> 00:22:20
you, most of you, I'm sure,
have seen some sort of picture

439
00:22:20 --> 00:22:22
of p orbitals before.
 

440
00:22:22 --> 00:22:25
So what I want to point out
about them is that they're made

441
00:22:25 --> 00:22:29
up of two nodes, and what you
can see is that nodes are shown

442
00:22:29 --> 00:22:32
in different colors here and
those are different phases.

443
00:22:32 --> 00:22:35
Sometimes you see this written
when you see p orbitals, one is

444
00:22:35 --> 00:22:38
written as plus, one
is written in minus.

445
00:22:38 --> 00:22:40
That's not a positive and
negative charge, that's

446
00:22:40 --> 00:22:43
actually a phase, and that
arises from the wave equation.

447
00:22:43 --> 00:22:46
Remember when we have waves
we can have positive or

448
00:22:46 --> 00:22:46
a negative amplitude.
 

449
00:22:46 --> 00:22:50
When we talk about p orbitals
the phase of the orbital

450
00:22:50 --> 00:22:54
becomes important once we talk
about bonding, which hopefully

451
00:22:54 --> 00:22:56
you were happy to hear at the
beginning of class we

452
00:22:56 --> 00:22:57
will get to soon.
 

453
00:22:57 --> 00:23:00
And it turns out that when you
constructively have two p

454
00:23:00 --> 00:23:03
orbitals interfere, and when I
say constructively, I mean

455
00:23:03 --> 00:23:06
they're both either positive or
they're both the negative

456
00:23:06 --> 00:23:08
lobes, that's when
you got bonding.

457
00:23:08 --> 00:23:12
Whereas if the phases where
mismatched, you would

458
00:23:12 --> 00:23:13
not get bonding.
 

459
00:23:13 --> 00:23:16
So, that's going to be
important later when we get to

460
00:23:16 --> 00:23:19
bonding, but just take note of
it now, we have two nodes, each

461
00:23:19 --> 00:23:24
with a separate phase -- or we
have two lobes, excuse me, each

462
00:23:24 --> 00:23:26
with a separate phase.
 

463
00:23:26 --> 00:23:29
And when we look at this, it's
actually split by what's called

464
00:23:29 --> 00:23:32
a nodal plane, which is pointed
out in light orange here on

465
00:23:32 --> 00:23:35
this picture, but what we just
mean is that there is this

466
00:23:35 --> 00:23:38
whole plane that separates the
two lobes where there is

467
00:23:38 --> 00:23:41
absolutely no electron density.
 

468
00:23:41 --> 00:23:43
So, the wave function at all of
these points in this plane is

469
00:23:43 --> 00:23:46
equal to zero, so therefore,
also the wave function squared

470
00:23:46 --> 00:23:50
is going to be equal to zero.
 

471
00:23:50 --> 00:23:56
So, if we say that in this
entire plane we have zero

472
00:23:56 --> 00:23:59
probability of finding a p
electron anywhere in the plane,

473
00:23:59 --> 00:24:03
the plane goes directly through
the nucleus in every case but a

474
00:24:03 --> 00:24:06
p orbital, so what we can also
say is that there is zero

475
00:24:06 --> 00:24:13
probability of finding a p
electron at the nucleus.

476
00:24:13 --> 00:24:17
So, again we can use these
probability density plots,

477
00:24:17 --> 00:24:23
which are just a plot of psi
squared, where the density of

478
00:24:23 --> 00:24:26
the dots is proportional to
the density, the probability

479
00:24:26 --> 00:24:28
density, at that point.
 

480
00:24:28 --> 00:24:31
So what we can say is look at
each of these separately, so if

481
00:24:31 --> 00:24:36
we start with looking at the 2
p z orbital, the highest

482
00:24:36 --> 00:24:39
probability of finding an
electron in the 2 p z

483
00:24:39 --> 00:24:42
orbital, is going to
be along this z-axis.

484
00:24:42 --> 00:24:44
We can see that right here.
 

485
00:24:44 --> 00:24:48
And in terms of thinking about
the phase of this p orbital,

486
00:24:48 --> 00:24:51
the phase is going to
be positive anywhere

487
00:24:51 --> 00:24:52
where z is positive.
 

488
00:24:52 --> 00:24:55
So we would say we have a
positive phase here and

489
00:24:55 --> 00:24:57
a negative phase there.
 

490
00:24:57 --> 00:24:59
Remember, that's going to
become important when we talk

491
00:24:59 --> 00:25:01
about bonding, we don't
need to worry about it

492
00:25:01 --> 00:25:03
too much right now.
 

493
00:25:03 --> 00:25:07
We can also think about where
the nodal plane is in this p

494
00:25:07 --> 00:25:11
z orbital, so how would we
define the nodal plane here?

495
00:25:11 --> 00:25:16
What would the nodal plane be?
 

496
00:25:16 --> 00:25:18
So, it's the x-y plane, you
can see there's no electron

497
00:25:18 --> 00:25:20
density anywhere there.
 

498
00:25:20 --> 00:25:23
And similarly, actually, if
we're looking at our polar

499
00:25:23 --> 00:25:27
coordinates here, what we see
is it's any place where theta

500
00:25:27 --> 00:25:31
is equal to 0 is what's going
to put up on the x-y plane.

501
00:25:31 --> 00:25:33
So another way to define the
nodal plane is where theta

502
00:25:33 --> 00:25:36
is equal to 90 degrees.
 

503
00:25:36 --> 00:25:40
So let's look now at
the 2 p x orbital.

504
00:25:40 --> 00:25:43
This is the probability
density map, so we're talking

505
00:25:43 --> 00:25:44
about the square here.
 

506
00:25:44 --> 00:25:48
The highest probability now is
going to be along the x-axis,

507
00:25:48 --> 00:25:52
so that means we're going to
have a positive wave function

508
00:25:52 --> 00:25:54
every place where
x is positive.

509
00:25:54 --> 00:25:56
What is the nodal
plane in this case?

510
00:25:56 --> 00:25:57
Um-hmm.
 

511
00:25:57 --> 00:26:02
So, it's going to be the y z
nodal plane, or in other words,

512
00:26:02 --> 00:26:06
we can say it's any place where
phi is equal to 90 degrees.

513
00:26:06 --> 00:26:08
So you can see if you take
phi, and you move it over

514
00:26:08 --> 00:26:11
90 degrees, we're right
here in the y z plane.

515
00:26:11 --> 00:26:13
Anywhere where that's the
case we're going to have

516
00:26:13 --> 00:26:17
no probability density
of finding an electron.

517
00:26:17 --> 00:26:20
And finally, we can look at
the 2 p y, so the highest

518
00:26:20 --> 00:26:23
probability is going to
be along the y-axis.

519
00:26:23 --> 00:26:26
It's going to be positive in
terms of its wave function or

520
00:26:26 --> 00:26:29
in terms of its phase anywhere
where y is positive.

521
00:26:29 --> 00:26:33
And the nodal plane's going to
be in the x z plane, or again,

522
00:26:33 --> 00:26:37
anywhere where phi is going to
be equal to 0, that takes

523
00:26:37 --> 00:26:41
us to the x z plane.
 

524
00:26:41 --> 00:26:44
So, let me get a little bit
more specific about what we

525
00:26:44 --> 00:26:47
mean by nodal plane and where
the idea of nodal plane comes

526
00:26:47 --> 00:26:50
from, and nodal planes
arise from any place

527
00:26:50 --> 00:26:52
you have angular nodes.
 

528
00:26:52 --> 00:26:55
So we talked about radial nodes
when we're doing these radial

529
00:26:55 --> 00:26:58
probability density
diagrams here.

530
00:26:58 --> 00:27:00
You can also have angular
notes, and when we talk about

531
00:27:00 --> 00:27:04
an anglar node, what we're
talking about is values of

532
00:27:04 --> 00:27:07
theta or values of phi at
which the wave function, and

533
00:27:07 --> 00:27:10
therefore, the wave function
squared, or the probability

534
00:27:10 --> 00:27:13
density are going to
be equal to zero.

535
00:27:13 --> 00:27:19
So, you remember from last time
radial nodes are values of r at

536
00:27:19 --> 00:27:22
which the wave function and
wave function squared are zero,

537
00:27:22 --> 00:27:24
so the difference is now we're
just talking about the angular

538
00:27:24 --> 00:27:26
part of the wave function.
 

539
00:27:26 --> 00:27:29
And, in fact, these are the
only two types of nodes that

540
00:27:29 --> 00:27:32
we're going to be describing,
so we can actually calculate

541
00:27:32 --> 00:27:35
both the total number of notes
and the number of each type of

542
00:27:35 --> 00:27:39
node we should expect to see
in any type of orbital.

543
00:27:39 --> 00:27:41
And our equation for total
nodes is just the principle

544
00:27:41 --> 00:27:43
quantum number minus 1.
 

545
00:27:43 --> 00:27:46
And when we talk about angular
nodes, the number of angular

546
00:27:46 --> 00:27:50
nodes we have in an orbital
is going to be equal to l.

547
00:27:50 --> 00:27:54
So that's why we saw, for
example, in the p orbitals we

548
00:27:54 --> 00:27:57
had one angular node in each
p orbital, because l

549
00:27:57 --> 00:27:58
is equal to 1 there.
 

550
00:27:58 --> 00:28:02
And we talked about the
equation you can use for radial

551
00:28:02 --> 00:28:07
nodes last time, and that's
just n minus 1 minus l.

552
00:28:07 --> 00:28:10
You can go ahead and use that
equation, or you could figure

553
00:28:10 --> 00:28:12
it out every time, because if
you know the total number of

554
00:28:12 --> 00:28:15
nodes, and you know the angular
node number, then you know

555
00:28:15 --> 00:28:17
how many nodes you're
going to have left.

556
00:28:17 --> 00:28:18
So you don't really
have to memorize that.

557
00:28:18 --> 00:28:22
So, let's go ahead and
just do a few of these.

558
00:28:22 --> 00:28:25
They're pretty straight forward
to do and it gives us an idea

559
00:28:25 --> 00:28:28
what kind of nodal structure
we can expect it an orbital.

560
00:28:28 --> 00:28:34
So for a 2 s orbital, how many
total nodes will we have?

561
00:28:34 --> 00:28:38
Yup, I heard one, so 2
minus 1, one total node.

562
00:28:38 --> 00:28:41
Angular nodes, we're not going
to have any of those, we'll

563
00:28:41 --> 00:28:45
have zero, l equals 0, so we
have zero angular nodes.

564
00:28:45 --> 00:28:51
And in terms of radial nodes,
we have 2 minus 1 minus 0,

565
00:28:51 --> 00:28:54
so what we have is
one radial node.

566
00:28:54 --> 00:28:56
So, what you find with the s
orbital, and this is general

567
00:28:56 --> 00:28:59
for all s orbitals is that
all of your nodes end

568
00:28:59 --> 00:29:01
up being radial nodes.
 

569
00:29:01 --> 00:29:06
That has to be the case because
l equals 0 for s orbitals.

570
00:29:06 --> 00:29:08
Let's look now at a p
orbital, so how many total

571
00:29:08 --> 00:29:13
nodes do we have here?
 

572
00:29:13 --> 00:29:18
Yup, so one total node, 2 minus
1 is 1, and that means since

573
00:29:18 --> 00:29:23
l is equal to 1, we have one
angular nodes, and that leaves

574
00:29:23 --> 00:29:26
us with how many radial nodes?
 

575
00:29:26 --> 00:29:28
Yup, zero radial nodes.
 

576
00:29:28 --> 00:29:32
So, for a 2 p orbital, all
the nodes actually turn

577
00:29:32 --> 00:29:35
out to be angular nodes.
 

578
00:29:35 --> 00:29:37
So, let's have you try one
more, if we can switch over

579
00:29:37 --> 00:29:41
and talk about a 3 d orbital.
 

580
00:29:41 --> 00:29:44
So, I'm asking very
specifically about radial nodes

581
00:29:44 --> 00:29:46
here, how many radial nodes
does a hydrogen atom

582
00:29:46 --> 00:29:48
3 d orbital have?
 

583
00:29:48 --> 00:29:56
So, you can go ahead and
take 10 seconds on that.

584
00:29:56 --> 00:30:09
All right.
 

585
00:30:09 --> 00:30:13
So most of you got that, though
there is this little sub-set we

586
00:30:13 --> 00:30:16
have thinking that we have
one, so let's actually

587
00:30:16 --> 00:30:17
write this out here.
 

588
00:30:17 --> 00:30:24
So if we have a 3 d orbital,
we're talking about n minus l

589
00:30:24 --> 00:30:28
minus 1, what is n equal to?
 

590
00:30:28 --> 00:30:30
What is l equal to?
 

591
00:30:30 --> 00:30:33
OK. and 1 is equal to 1.
 

592
00:30:33 --> 00:30:37
So, it turns out that we have
zero nodes that we're dealing

593
00:30:37 --> 00:30:39
with when we're talking
about a 3 d orbital.

594
00:30:39 --> 00:30:43
OK.
 

595
00:30:43 --> 00:30:46
So we should be able to figure
this out for any orbital that

596
00:30:46 --> 00:30:50
we're discussing, and when we
can figure out especially

597
00:30:50 --> 00:30:53
radial nodes, we have a good
head start on going ahead and

598
00:30:53 --> 00:30:57
thinking about drawing radial
probability distributions.

599
00:30:57 --> 00:30:59
We did it for the s orbitals,
we can also do it for the

600
00:30:59 --> 00:31:00
p, we can do it for the d.
 

601
00:31:00 --> 00:31:03
All we have to figure out is
how many nodes we're dealing

602
00:31:03 --> 00:31:05
with and then we can get the
general shape of

603
00:31:05 --> 00:31:07
the graph here.
 

604
00:31:07 --> 00:31:11
So, let's actually compare the
radial probability distribution

605
00:31:11 --> 00:31:14
of p orbitals to what we've
already looked at, which are s

606
00:31:14 --> 00:31:17
orbitals, and we'll find that
we can get some information out

607
00:31:17 --> 00:31:20
of comparing these graphs.
 

608
00:31:20 --> 00:31:23
So if we draw the 2 p orbital,
what we just figured out was

609
00:31:23 --> 00:31:26
there should be zero
radial nodes, so that's

610
00:31:26 --> 00:31:27
what we see here.
 

611
00:31:27 --> 00:31:30
The other thing that I want you
to notice, is if you look at

612
00:31:30 --> 00:31:34
the most probable radius, for
the 2 s orbital it's actually

613
00:31:34 --> 00:31:37
out further away from
the nucleus than it is

614
00:31:37 --> 00:31:39
for the 2 p orbital.
 

615
00:31:39 --> 00:31:43
So what we can say here is
that the 2 p is less than or

616
00:31:43 --> 00:31:46
smaller than the 2 s orbital.
 

617
00:31:46 --> 00:31:49
So think about what that means,
we're, of course, not talking

618
00:31:49 --> 00:31:52
about this in classical terms,
so what it means if we have an

619
00:31:52 --> 00:31:56
electron in the 2 p orbital,
it's more likely, the

620
00:31:56 --> 00:31:59
probability is that will be
closer to the nucleus than it

621
00:31:59 --> 00:32:03
would be if it were
in the 2 s orbital.

622
00:32:03 --> 00:32:06
We can also take a look the 3
s, which we have looked at

623
00:32:06 --> 00:32:09
before, and we figured out
that that should have

624
00:32:09 --> 00:32:10
two radial nodes.
 

625
00:32:10 --> 00:32:14
We can look at the 2 p, which
should have one radial node,

626
00:32:14 --> 00:32:19
and we just figured it out for
the, excuse me, for the 3 p has

627
00:32:19 --> 00:32:22
one radial node, and for the 3
d here, we should have zero

628
00:32:22 --> 00:32:25
radial nodes, we just
calculated that.

629
00:32:25 --> 00:32:28
So again, what we see is the
same pattern where the most

630
00:32:28 --> 00:32:32
probable radius, if we talk
about it in terms of the d,

631
00:32:32 --> 00:32:35
that's going to be smaller then
for the p, and the 3 p most

632
00:32:35 --> 00:32:38
probable radius is going to be
closer to the nucleus than it

633
00:32:38 --> 00:32:45
is for the 3 s most probable
radius that we're looking at.

634
00:32:45 --> 00:32:48
So, there are 2 different
things that we can compare when

635
00:32:48 --> 00:32:50
we're comparing graphs of
radial probability

636
00:32:50 --> 00:32:54
distribution, and the first
thing we can do is think about

637
00:32:54 --> 00:32:57
well, how does the radius
change, or the most probable

638
00:32:57 --> 00:33:00
radius change when we're
increasing n, when we're

639
00:33:00 --> 00:33:03
increasing the principle
quantum number here?

640
00:33:03 --> 00:33:07
So, from going from the shell
of n equals 2, let's say,

641
00:33:07 --> 00:33:09
to the shell of n equals 3.
 

642
00:33:09 --> 00:33:14
And what we find is we're going
from about or exactly a 6 a

643
00:33:14 --> 00:33:17
nought here, to almost three
times that when we're

644
00:33:17 --> 00:33:19
going from 2 s to 3 s.
 

645
00:33:19 --> 00:33:22
So we say if n increases,
the orbital size is

646
00:33:22 --> 00:33:23
also going to increase.
 

647
00:33:23 --> 00:33:26
And when we talk about size,
I'm again just going to say the

648
00:33:26 --> 00:33:29
stipulation that we're talking
about, probability -- we're not

649
00:33:29 --> 00:33:33
talking about an absolute
classical concept here, but in

650
00:33:33 --> 00:33:37
general we're going to picture
it being much further away from

651
00:33:37 --> 00:33:40
the nucleus as we move
up in terms of n.

652
00:33:40 --> 00:33:44
The other thing that we took
note as is what happens as l

653
00:33:44 --> 00:33:47
increases, and specifically as
l increases for any given the

654
00:33:47 --> 00:33:49
principle quantum number.
 

655
00:33:49 --> 00:33:53
So if we're keeping n the same,
we look and what we saw was

656
00:33:53 --> 00:33:58
that size actually decreases as
we increase the value of l.

657
00:33:58 --> 00:34:02
So, I want to contrast that
with another concept that

658
00:34:02 --> 00:34:06
seemed to be opposing ideas,
and that is thinking about not

659
00:34:06 --> 00:34:10
how far away the most probable
radius is, but thinking about

660
00:34:10 --> 00:34:13
how close an electron can
get to the nucleus if it's

661
00:34:13 --> 00:34:14
actually in that orbital.
 

662
00:34:14 --> 00:34:18
And what we'll find is that we
actually see the opposite.

663
00:34:18 --> 00:34:24
So if we compare l increasing
here, so a 3 s to a 3 p to a 3

664
00:34:24 --> 00:34:28
d, what we find is that it's
only in the s orbital that we

665
00:34:28 --> 00:34:31
have a significant probability
of actually getting very

666
00:34:31 --> 00:34:33
close to the nucleus.
 

667
00:34:33 --> 00:34:37
So, if I kind of circle where
the probability gets somewhat

668
00:34:37 --> 00:34:39
substantial here, you can see
we're much closer to the

669
00:34:39 --> 00:34:44
nucleus at the s orbital than
we are for the p, then

670
00:34:44 --> 00:34:46
when we are for the d.
 

671
00:34:46 --> 00:34:51
So, the size still for an s
orbital is larger than for a d

672
00:34:51 --> 00:34:54
orbital, but what we say is
that an s electron can actually

673
00:34:54 --> 00:34:56
penetrate closer
to the nucleus.

674
00:34:56 --> 00:35:00
There's some probability that
it can get very, very close the

675
00:35:00 --> 00:35:05
nucleus, and that probability
is actually substantial.

676
00:35:05 --> 00:35:11
A kind of consequence of this
is if we're thinking about a

677
00:35:11 --> 00:35:13
multi-electron atom, which
we'll get to in a minute where

678
00:35:13 --> 00:35:17
electrons can shield each other
from the pull of the nucleus,

679
00:35:17 --> 00:35:20
we're going to say that the
electrons in the s orbitals are

680
00:35:20 --> 00:35:22
actually the least shielded.
 

681
00:35:22 --> 00:35:24
And the reason that they're the
least sheilded is because they

682
00:35:24 --> 00:35:27
can get closest to the nucleus,
so we can think of them as not

683
00:35:27 --> 00:35:29
getting blocked by a bunch of
other electron, because there's

684
00:35:29 --> 00:35:32
some probability that they can
actually work their way all

685
00:35:32 --> 00:35:33
the way in to the nucleus.
 

686
00:35:33 --> 00:35:35
So, this is a concept
that's going to become

687
00:35:35 --> 00:35:36
really important.
 

688
00:35:36 --> 00:35:39
Soon when we're talking about
multi-electron atoms, and I

689
00:35:39 --> 00:35:42
just want to introduce it here,
that it is sort of opposing

690
00:35:42 --> 00:35:45
ideas that even though the s is
the biggest and it's most

691
00:35:45 --> 00:35:48
likely that the electron's
going to be furthest away from

692
00:35:48 --> 00:35:51
the nucleus, that's also the
orbital in which the

693
00:35:51 --> 00:35:56
electron can, in fact,
penetrate closest.

694
00:35:56 --> 00:35:56
All right.
 

695
00:35:56 --> 00:35:59
So I think we are, in fact,
ready to move on to

696
00:35:59 --> 00:36:04
multi-electron atoms, and what
happens is when we solved the

697
00:36:04 --> 00:36:07
relativistic version of the
Schrodinger equation and we're

698
00:36:07 --> 00:36:10
discussing more than one
electron, we actually have a

699
00:36:10 --> 00:36:13
fourth quantum number that
falls out and that we need to

700
00:36:13 --> 00:36:16
deal with and this is called
the electron spin

701
00:36:16 --> 00:36:17
quantum number.
 

702
00:36:17 --> 00:36:20
And I promise, this is the
last quantum number that

703
00:36:20 --> 00:36:22
we'll be introducing.
 

704
00:36:22 --> 00:36:25
And this spin magnetic quantum
number we abbreviate as

705
00:36:25 --> 00:36:32
m sub s, so that's to
differentiate from m sub l.

706
00:36:32 --> 00:36:36
And when you solved the
relativistic form of the

707
00:36:36 --> 00:36:38
Schrodinger equation, what you
end up with is that you can

708
00:36:38 --> 00:36:42
have two possible values for
the magnetic spin

709
00:36:42 --> 00:36:43
quantum number.
 

710
00:36:43 --> 00:36:47
You can have it equal to plus
1/2, and that's what we call

711
00:36:47 --> 00:36:51
spin up, or you can have it
equal to minus 1/2, which

712
00:36:51 --> 00:36:53
is what we call spin down.
 

713
00:36:53 --> 00:36:56
So, there's two kind of
cartoons shown here that give

714
00:36:56 --> 00:36:59
you a little bit of an idea of
what this quantum

715
00:36:59 --> 00:37:00
number tells us.
 

716
00:37:00 --> 00:37:05
And this spin is an intrinsic
quality of the electron, it's a

717
00:37:05 --> 00:37:08
property that is intrinsic in
all particles, just like we

718
00:37:08 --> 00:37:10
would say mass is intrinsic
or charge is intrinsic.

719
00:37:10 --> 00:37:13
Spin is also an
intrinsic property.

720
00:37:13 --> 00:37:16
One way to think about it, if
we want to use a classical

721
00:37:16 --> 00:37:18
analogy, which often helps to
give us an idea of what's going

722
00:37:18 --> 00:37:22
on, is the spin of an electron,
we can picture it rotating

723
00:37:22 --> 00:37:24
on its own axis.
 

724
00:37:24 --> 00:37:27
So that's kind of what's shown
in these pictures here.

725
00:37:27 --> 00:37:29
So you can see, if it's
spinning on its own axis in

726
00:37:29 --> 00:37:31
this direction we'd call it
spin up, where as this

727
00:37:31 --> 00:37:34
way it would be what
we call spin down.

728
00:37:34 --> 00:37:36
So, it turns out there's not
actually a good classical

729
00:37:36 --> 00:37:40
analogy for spin, we can't
really think of it like that,

730
00:37:40 --> 00:37:43
but if that helps give you an
idea of what's going on here

731
00:37:43 --> 00:37:45
then it's valuable maybe to
consider it spinning on its

732
00:37:45 --> 00:37:48
own axis, even though that's
not technically what's

733
00:37:48 --> 00:37:49
exactly happening here.
 

734
00:37:49 --> 00:37:52
But the reason that I like that
analogy is that it points out a

735
00:37:52 --> 00:37:56
very important part of spin,
and that's the idea that it's a

736
00:37:56 --> 00:37:58
description of the electron.
 

737
00:37:58 --> 00:38:01
It is not dependent on
the actual orbital.

738
00:38:01 --> 00:38:04
So we can completely describe
an orbital with just using

739
00:38:04 --> 00:38:08
three quantum numbers, but we
have this fourth quantum number

740
00:38:08 --> 00:38:11
that describes something about
the electron that's required

741
00:38:11 --> 00:38:14
for now a complete description
of the electron, and

742
00:38:14 --> 00:38:16
that's the idea of spin.
 

743
00:38:16 --> 00:38:19
So, we need to actually add on
this fourth quantum number,

744
00:38:19 --> 00:38:25
and it's either going to be
plus 1/2 or negative 1/2.

745
00:38:25 --> 00:38:27
So, we can talk a little bit
actually, because it's an

746
00:38:27 --> 00:38:30
interesting story about where
the idea of spin came from, and

747
00:38:30 --> 00:38:34
it was actually first proposed
by two very young scientists at

748
00:38:34 --> 00:38:37
the time, George Uhlenbeck
shown here, and Samuel Goudsmit

749
00:38:37 --> 00:38:41
who's here, and they're with a
friend, and I can't remember

750
00:38:41 --> 00:38:43
who that is, but he did
not have anything to do

751
00:38:43 --> 00:38:46
with discovering spin.
 

752
00:38:46 --> 00:38:48
And what you can see in this
picture is that these are

753
00:38:48 --> 00:38:50
actually, they're pretty
young guys in this picture.

754
00:38:50 --> 00:38:52
I think this is taken about two
years after they discovered

755
00:38:52 --> 00:38:53
the fourth quantum number.
 

756
00:38:53 --> 00:38:56
So hopefully, you can picture
yourself at this age in a

757
00:38:56 --> 00:39:00
similar situation with an
anonymous friend and think

758
00:39:00 --> 00:39:02
this is something, kind of
observations maybe you

759
00:39:02 --> 00:39:04
can make as well.
 

760
00:39:04 --> 00:39:07
And what they were doing when
they discovered that there must

761
00:39:07 --> 00:39:10
be this fourth quantum number
is they were looking at the

762
00:39:10 --> 00:39:14
emission spectrum of sodium,
and, so specifically, they were

763
00:39:14 --> 00:39:18
looking at the frequencies, and
if we think about the

764
00:39:18 --> 00:39:21
frequencies of sodium, it was
already known at this time that

765
00:39:21 --> 00:39:23
you could calculate what those
would be based on the

766
00:39:23 --> 00:39:27
difference in the energy levels
-- this happened in about 1925.

767
00:39:27 --> 00:39:29
So they actually knew exactly
what they were expecting.

768
00:39:29 --> 00:39:31
So, let's say they were
expecting to see one certain

769
00:39:31 --> 00:39:36
frequency or one line in the
spectrum at this point here.

770
00:39:36 --> 00:39:40
It turns out that what they
actually observed, so this is

771
00:39:40 --> 00:39:45
the actual of what they saw, is
if they were expecting their

772
00:39:45 --> 00:39:48
line at some given frequency,
which I'll show by this dotted

773
00:39:48 --> 00:39:54
line here, what they actually
saw was two lines, and one was

774
00:39:54 --> 00:39:57
just the teeniest tiniest bit
of a higher frequency than what

775
00:39:57 --> 00:40:01
was expected, and one was just,
just below what they expected.

776
00:40:01 --> 00:40:04
And if we're talking about
things in spectroscopy

777
00:40:04 --> 00:40:08
terms here, this is
what we call a doublet.

778
00:40:08 --> 00:40:11
So it's centered at this
frequency that was expected,

779
00:40:11 --> 00:40:14
but it's actually split into
two different frequencies.

780
00:40:14 --> 00:40:16
And this was an amazing
observation that they made.

781
00:40:16 --> 00:40:20
They were totally surprised and
excited, and they were thinking

782
00:40:20 --> 00:40:22
how could this happen,
where did we got this

783
00:40:22 --> 00:40:23
split doublet from.
 

784
00:40:23 --> 00:40:26
And what they could come up
with, what they reasoned, is

785
00:40:26 --> 00:40:29
that there must be some
intrinsic property within the

786
00:40:29 --> 00:40:32
electron, because we know that
this describes the complete

787
00:40:32 --> 00:40:37
energy of the orbital should
give us one single frequency.

788
00:40:37 --> 00:40:41
And that the fact that it split
into two was telling them that

789
00:40:41 --> 00:40:44
there must be some new property
to the electron, and what

790
00:40:44 --> 00:40:47
we call that now is either
being spin up or spin down.

791
00:40:47 --> 00:40:51
But at the time, they didn't
have a well-formed name for it,

792
00:40:51 --> 00:40:54
they were just saying OK,
there's this fourth quantum

793
00:40:54 --> 00:40:57
number, there's this intrinsic
property in the electron.

794
00:40:57 --> 00:40:59
So, where the story gets kind
of unfortunate, but also a

795
00:40:59 --> 00:41:03
little bit more interesting is
the fact that well, they did

796
00:41:03 --> 00:41:06
publish what they observed,
and they did write that up.

797
00:41:06 --> 00:41:08
They put it in a pretty
low impact paper.

798
00:41:08 --> 00:41:12
I think it was in French, so it
wasn't really hitting all the

799
00:41:12 --> 00:41:15
scientific community of the
time in any major way.

800
00:41:15 --> 00:41:19
And they didn't put their
explanation of what they

801
00:41:19 --> 00:41:21
thought was going on,
it just sort of was

802
00:41:21 --> 00:41:23
observing what they saw.
 

803
00:41:23 --> 00:41:26
These were very young
scientists, of course, so what

804
00:41:26 --> 00:41:28
you would expect that they
would do, which makes sense, is

805
00:41:28 --> 00:41:31
go to someone more established
in their field, because they

806
00:41:31 --> 00:41:35
have the completely radical
revolutionary idea, let's just

807
00:41:35 --> 00:41:38
run it by someone before we go
ahead and publish this paper

808
00:41:38 --> 00:41:40
that makes this huge statement
about this fourth

809
00:41:40 --> 00:41:41
quantum number.
 

810
00:41:41 --> 00:41:44
So the person they chose to
talk to, and I think it was

811
00:41:44 --> 00:41:47
just Goudsmit that went to him
and discussed it is Wolfgang

812
00:41:47 --> 00:41:49
Pauli, which is shown here.
 

813
00:41:49 --> 00:41:52
So how many before five
minutes ago had heard

814
00:41:52 --> 00:41:52
the name Goudsmit?
 

815
00:41:52 --> 00:41:55
All right, a couple.
 

816
00:41:55 --> 00:41:56
OK, cool.
 

817
00:41:56 --> 00:41:59
How about Pauli, like the
Pauli exclusion principle?

818
00:41:59 --> 00:42:00
Hmm, OK.
 

819
00:42:00 --> 00:42:05
So, Pauli seems to be getting
a little bit of fame that

820
00:42:05 --> 00:42:06
you'll see in a second here.
 

821
00:42:06 --> 00:42:09
So, it turns out they go and
they discuss their idea with

822
00:42:09 --> 00:42:14
Pauli, and what Pauli tells
them is that this idea is

823
00:42:14 --> 00:42:17
ridiculous, that it's rubbish,
if they go ahead and try to

824
00:42:17 --> 00:42:19
publish this their scientific
careers are ruined.

825
00:42:19 --> 00:42:21
They can pretty much pack it up
and go home, because everyone's

826
00:42:21 --> 00:42:23
going to think they're
ridiculous, no one will believe

827
00:42:23 --> 00:42:26
what they say, and it's a
stupid idea anyway, is

828
00:42:26 --> 00:42:28
basically the gist of
this conversation.

829
00:42:28 --> 00:42:34
And in chemistry, just like in
any discipline, you have all

830
00:42:34 --> 00:42:37
types of scientists, but also
all types of personalities, and

831
00:42:37 --> 00:42:40
unfortunately Pauli had a
personality that was known for,

832
00:42:40 --> 00:42:42
first of all, being very
arrogant, but also the very

833
00:42:42 --> 00:42:45
unfortunate trait of taking
other people's scientific

834
00:42:45 --> 00:42:46
ideas as his own.
 

835
00:42:46 --> 00:42:50
And as the story goes, as
Goudsmit was leaving and the

836
00:42:50 --> 00:42:54
door with slamming, Wolfgang
Pauli was already writing down

837
00:42:54 --> 00:42:58
this idea into a scientific
paper of the idea of a

838
00:42:58 --> 00:42:59
fourth quantum number.
 

839
00:42:59 --> 00:43:03
And, in fact, he did make some
more strides, he was a

840
00:43:03 --> 00:43:06
brilliant thinker, maybe he put
it more articulately than those

841
00:43:06 --> 00:43:08
two younger scientists
could have.

842
00:43:08 --> 00:43:11
But now, it has come to light
that they are the ones that do

843
00:43:11 --> 00:43:14
get credit for first really
coming up with this idea of a

844
00:43:14 --> 00:43:17
spin quantum number, and it's
interesting to think about how

845
00:43:17 --> 00:43:21
the politics work in different
discoveries, as well as the

846
00:43:21 --> 00:43:22
discoveries themselves.
 

847
00:43:22 --> 00:43:24
You see that a lot with Nobel
prizes, there's usually a nice

848
00:43:24 --> 00:43:27
little scandal, a nice little
interesting story behind who

849
00:43:27 --> 00:43:33
else was responsible for the
Nobel Prize-worthy discovery.

850
00:43:33 --> 00:43:37
So, here, Pauli came out on
top, we say, and he's known for

851
00:43:37 --> 00:43:41
the Pauli exclusion principle,
which tells us that no two

852
00:43:41 --> 00:43:43
electrons in the same atom can
have the same four

853
00:43:43 --> 00:43:45
quantum numbers.
 

854
00:43:45 --> 00:43:49
So let's just think exactly
what this means, and that means

855
00:43:49 --> 00:43:53
that if we take away function
and we define it in terms of n,

856
00:43:53 --> 00:43:58
l and m sub l, what we're
defining here is the complete

857
00:43:58 --> 00:44:03
description of an orbital.
 

858
00:44:03 --> 00:44:06
In contrast, if we're taking
the wave function and

859
00:44:06 --> 00:44:12
describing it in terms of n, l,
m sub l, and now also, the

860
00:44:12 --> 00:44:15
spin, what are we
describing here?

861
00:44:15 --> 00:44:16
An electron.
 

862
00:44:16 --> 00:44:20
So now we have the complete
description of an electron

863
00:44:20 --> 00:44:21
within an orbital.
 

864
00:44:21 --> 00:44:26
So, that's an important
distinction to make -- what

865
00:44:26 --> 00:44:28
three quantum numbers tell us,
versus what the fourth quantum

866
00:44:28 --> 00:44:32
number can fill in for us
in terms of information.

867
00:44:32 --> 00:44:35
So what that means is that
we're limited in any atom to

868
00:44:35 --> 00:44:39
having two electrons per
orbital, right, because for any

869
00:44:39 --> 00:44:42
orbital we can either have a
spin up electron, a spin

870
00:44:42 --> 00:44:44
down electron, or both.
 

871
00:44:44 --> 00:44:48
So, if we look at neon just as
an example, neon has ten

872
00:44:48 --> 00:44:53
electron in it, and once we
look at all the orbitals

873
00:44:53 --> 00:44:55
written out here, this is
probably a familiar thing for

874
00:44:55 --> 00:44:59
you to look at, but it's
important to think about why,

875
00:44:59 --> 00:45:02
in fact, we don't just put all
ten electrons, why wouldn't

876
00:45:02 --> 00:45:04
they just want to go in
that ground state, that

877
00:45:04 --> 00:45:05
lowest state, right?
 

878
00:45:05 --> 00:45:08
That should be the most stable,
the lowest energy orbital for

879
00:45:08 --> 00:45:11
it to be in, and the reason
they can't do that is because

880
00:45:11 --> 00:45:14
of the Pauli exclusion
principle -- the idea that all

881
00:45:14 --> 00:45:18
of the electrons have to have a
different set of four quantum

882
00:45:18 --> 00:45:22
numbers, so only two of them
can have the same set of three

883
00:45:22 --> 00:45:25
quantum numbers here, because
for m sub s, we're only

884
00:45:25 --> 00:45:28
left with two options.
 

885
00:45:28 --> 00:45:31
So let's try a clicker question
and thinking about the

886
00:45:31 --> 00:45:33
Pauli exclusion principle.
 

887
00:45:33 --> 00:45:36
It might look a little bit
similar to a question we just

888
00:45:36 --> 00:45:40
saw, but hopefully you'll
find that it is, in fact,

889
00:45:40 --> 00:45:45
not the same question.
 

890
00:45:45 --> 00:45:47
So you can go ahead and
take 10 seconds on that.

891
00:45:47 --> 00:46:02
OK, great.
 

892
00:46:02 --> 00:46:07
So, most of you recognize that
there are four different

893
00:46:07 --> 00:46:09
possibilities of there's four
different electrons that can

894
00:46:09 --> 00:46:11
have those two quantum numbers.
 

895
00:46:11 --> 00:46:13
Actually the easiest way
is probably to bring

896
00:46:13 --> 00:46:15
this down here.
 

897
00:46:15 --> 00:46:17
And the next highest percentage
of you thought that we

898
00:46:17 --> 00:46:19
still only had two.
 

899
00:46:19 --> 00:46:22
So, remember we solved this
problem earlier in the

900
00:46:22 --> 00:46:24
class, but we were
talking about orbitals.

901
00:46:24 --> 00:46:27
So there's two different
orbitals that can have these

902
00:46:27 --> 00:46:30
three quantum numbers, but if
we're talking about electrons,

903
00:46:30 --> 00:46:34
we can also talk about m sub s,
so if we have two orbitals, how

904
00:46:34 --> 00:46:36
many electrons can
we have total?

905
00:46:36 --> 00:46:37
Yeah.
 

906
00:46:37 --> 00:46:42
So we have two orbitals, or
four electrons that can have

907
00:46:42 --> 00:46:44
that set of quantum numbers.
 

908
00:46:44 --> 00:46:46
So you'll notice in your
problem-set, sometimes you're

909
00:46:46 --> 00:46:49
asked for a number of orbitals
with a set of quantum numbers,

910
00:46:49 --> 00:46:52
sometimes you're asked for a
number of electrons for a

911
00:46:52 --> 00:46:53
set of quantum numbers.
 

912
00:46:53 --> 00:46:55
So make sure first that you
read the question carefully,

913
00:46:55 --> 00:46:59
and realize the difference
that is between the two.

914
00:46:59 --> 00:47:01
So, that's where
we'll end today.

915
00:47:01 --> 00:47:03
So on Friday, we'll start
with talking about the

916
00:47:03 --> 00:47:06
wave functions for the
multi-electron atoms.

917
00:47:06 --> 00:47:07