1
00:00:00,000 --> 00:00:00,016
The following content is
provided under a Creative

2
00:00:00,016 --> 00:00:00,022
Commons license.

3
00:00:00,022 --> 00:00:00,038
Your support will help MIT
OpenCourseWare continue to

4
00:00:00,038 --> 00:00:00,054
offer high quality educational
resources for free.

5
00:00:00,054 --> 00:00:00,072
To make a donation or view
additional materials from

6
00:00:00,072 --> 00:00:00,088
hundreds of MIT courses, visit
MIT OpenCourseWare at

7
00:00:00,088 --> 00:00:00,110
ocw.mit.edu.

8
00:00:00,110 --> 00:00:23,990
PROFESSOR: OK, I want to take
10 more seconds now the

9
00:00:23,990 --> 00:00:27,670
clicker slide.

10
00:00:27,670 --> 00:00:31,680
This is giving us one more try
on the vsper geometries,

11
00:00:31,680 --> 00:00:37,100
because it didn't go so
well on Wednesday.

12
00:00:37,100 --> 00:00:38,150
All right, excellent.

13
00:00:38,150 --> 00:00:40,320
So that is a very good job.

14
00:00:40,320 --> 00:00:43,880
Let's quickly go over why.

15
00:00:43,880 --> 00:00:45,620
We have p h 3.

16
00:00:45,620 --> 00:00:50,040
We told you that phosphorous has
5 valence electrons plus 3

17
00:00:50,040 --> 00:00:52,390
from each of the hydrogens,
so we have a total

18
00:00:52,390 --> 00:00:54,860
of 8 valence electrons.

19
00:00:54,860 --> 00:00:57,300
How many do we need to get
full valence shells

20
00:00:57,300 --> 00:00:58,100
everywhere?

21
00:00:58,100 --> 00:01:00,340
STUDENT: [INAUDIBLE]

22
00:01:00,340 --> 00:01:00,620
PROFESSOR: 14.

23
00:01:00,620 --> 00:01:03,330
So, we need 14 minus 8.

24
00:01:03,330 --> 00:01:09,380
That leaves us with 6
bonding electrons.

25
00:01:09,380 --> 00:01:15,330
And if we put that in our bond
here, we have 1, 2, 3 bonds,

26
00:01:15,330 --> 00:01:18,630
plus we have one lone
pair left over.

27
00:01:18,630 --> 00:01:21,000
So this is our Lewis
structure here.

28
00:01:21,000 --> 00:01:24,940
If these bonds were all
completely of equal distance

29
00:01:24,940 --> 00:01:28,300
apart, whether is was a lone
pair or bonding electrons, the

30
00:01:28,300 --> 00:01:29,990
angles would be 109 .

31
00:01:29,990 --> 00:01:31,130
5 degrees.

32
00:01:31,130 --> 00:01:34,640
But because there's this lone
pair here, it's pushing down

33
00:01:34,640 --> 00:01:38,270
on the other bonds, so we
end up with an angle of

34
00:01:38,270 --> 00:01:40,080
less than 109 .

35
00:01:40,080 --> 00:01:40,970
5 degrees.

36
00:01:40,970 --> 00:01:44,630
All right, so let's switch
over to notes for today.

37
00:01:44,630 --> 00:01:46,710
So we're going to finish
talking about molecular

38
00:01:46,710 --> 00:01:50,410
orbital theory, and then we'll
switch over to discussing

39
00:01:50,410 --> 00:01:54,180
bonding in larger molecules,
even larger than diatomic, so

40
00:01:54,180 --> 00:01:59,690
we'll move on to talking about
valence bond theory and

41
00:01:59,690 --> 00:02:00,010
hybridization.

42
00:02:00,010 --> 00:02:01,510
So, clearly you don't have your
notes in front of you

43
00:02:01,510 --> 00:02:04,230
yet, so you can just listen,
take it all in.

44
00:02:04,230 --> 00:02:07,650
What I'll do is I'll post the
notes filled in to the point

45
00:02:07,650 --> 00:02:10,420
where you actually get your
class notes today.

46
00:02:10,420 --> 00:02:13,930
So, this will be a little bit
more like a seminar to start

47
00:02:13,930 --> 00:02:17,460
with, and a little bit less
like a lecture in class.

48
00:02:17,460 --> 00:02:20,550
But let's go ahead and start
our discussion in terms of

49
00:02:20,550 --> 00:02:23,910
molecular orbital theory.

50
00:02:23,910 --> 00:02:27,650
So where we had left off with
was we'd fully discussed up to

51
00:02:27,650 --> 00:02:31,000
the point of considering
homonuclear diatomic

52
00:02:31,000 --> 00:02:34,370
molecules, so molecules that
both have the same nucleus.

53
00:02:34,370 --> 00:02:37,990
And where we had left off was
we were going to start one

54
00:02:37,990 --> 00:02:41,400
example of thinking about now
where we have a heteronuclear

55
00:02:41,400 --> 00:02:47,080
diatomic molecules, so two
different atoms in terms of

56
00:02:47,080 --> 00:02:48,590
forming the molecule.

57
00:02:48,590 --> 00:02:50,760
But first, I just want to remind
you when we're talking

58
00:02:50,760 --> 00:02:53,440
about molecular orbital theory,
this is treating

59
00:02:53,440 --> 00:02:56,350
electrons as waves, so what
we're actually able to do is

60
00:02:56,350 --> 00:03:00,370
either constructively or
destructively combine atomic

61
00:03:00,370 --> 00:03:01,970
orbitals to form molecular
orbitals.

62
00:03:01,970 --> 00:03:06,080
So you should remember that
any time we combine 2 s

63
00:03:06,080 --> 00:03:09,290
orbitals, what we're going to
find is if we constructively

64
00:03:09,290 --> 00:03:11,420
interfere those two orbitals,
we're going to

65
00:03:11,420 --> 00:03:13,220
form a bonding orbital.

66
00:03:13,220 --> 00:03:16,250
And that's going to be lower
in energy than the two

67
00:03:16,250 --> 00:03:18,350
individual atomic orbitals.

68
00:03:18,350 --> 00:03:21,850
And we call that, for this case,
our sigma 2 s orbital.

69
00:03:21,850 --> 00:03:26,740
In contrast, if we have
destructive interference, what

70
00:03:26,740 --> 00:03:30,300
we're going to form is a sigma
2 s star, and what does the

71
00:03:30,300 --> 00:03:31,630
star designate?

72
00:03:31,630 --> 00:03:32,830
STUDENT: [INAUDIBLE]

73
00:03:32,830 --> 00:03:33,630
PROFESSOR: Anti-bonding, yup.

74
00:03:33,630 --> 00:03:35,440
So it's an Anti-bonding
orbital.

75
00:03:35,440 --> 00:03:38,320
It's going to be higher in
energy than the individual

76
00:03:38,320 --> 00:03:38,950
atomic orbitals.

77
00:03:38,950 --> 00:03:41,300
All right, great.

78
00:03:41,300 --> 00:03:43,860
So, I think we have these
molecular orbital energies

79
00:03:43,860 --> 00:03:46,140
down, so let's move on
to talking about

80
00:03:46,140 --> 00:03:48,390
more complex molecules.

81
00:03:48,390 --> 00:03:51,520
And to do this we're going to
introduce valence bond theory,

82
00:03:51,520 --> 00:03:56,270
and the idea of hybridization
of orbitals.

83
00:03:56,270 --> 00:03:59,420
So the idea behind valence bond
theory is very easy to

84
00:03:59,420 --> 00:04:00,090
understand.

85
00:04:00,090 --> 00:04:03,330
Essentially what you have is
bonds resulting from the

86
00:04:03,330 --> 00:04:05,880
pairing of unpaired electrons.

87
00:04:05,880 --> 00:04:09,910
So the simplest case we can
think of is with h 2 where we

88
00:04:09,910 --> 00:04:14,550
have two unpaired electrons,
each in a 1 s orbital of a

89
00:04:14,550 --> 00:04:16,570
separate h atom.

90
00:04:16,570 --> 00:04:20,120
And if we picture those two
coming together, we form the h

91
00:04:20,120 --> 00:04:21,620
2 molecule.

92
00:04:21,620 --> 00:04:25,110
And again, we have the pairing
of the unpaired electrons, and

93
00:04:25,110 --> 00:04:28,170
we have two orbitals
coming together.

94
00:04:28,170 --> 00:04:31,350
So in molecular orbital theory,
what we did was we

95
00:04:31,350 --> 00:04:33,920
named orbitals based
on their symmetry.

96
00:04:33,920 --> 00:04:37,310
In valence bond theory, the
focus is on discussing the

97
00:04:37,310 --> 00:04:40,210
bonds, but it should look very
familiar to you, because

98
00:04:40,210 --> 00:04:43,040
there's two types of bonds that
we want to discuss here.

99
00:04:43,040 --> 00:04:46,570
We want to discuss sigma
bonds and pi bonds.

100
00:04:46,570 --> 00:04:49,340
So this is very similar to what
we saw in terms of sigma

101
00:04:49,340 --> 00:04:51,160
orbitals and pi orbitals.

102
00:04:51,160 --> 00:04:53,470
So in this first case
here, what we're

103
00:04:53,470 --> 00:04:55,770
seeing is a sigma bond.

104
00:04:55,770 --> 00:04:59,010
And a sigma bond forms any time
you have two orbitals

105
00:04:59,010 --> 00:05:02,250
coming together and interacting
on that

106
00:05:02,250 --> 00:05:05,150
internuclear axis.

107
00:05:05,150 --> 00:05:08,830
So we talk about a sigma bond
as being cylindrically

108
00:05:08,830 --> 00:05:11,980
symmetric about the bond axis,
and it's important to point

109
00:05:11,980 --> 00:05:15,950
out that it has no nodal plane
across this bond axis.

110
00:05:15,950 --> 00:05:17,840
This is in direct contrast
to when we're

111
00:05:17,840 --> 00:05:20,000
thinking about pi bonds.

112
00:05:20,000 --> 00:05:25,520
So pi bonds have electron
density both above and below

113
00:05:25,520 --> 00:05:29,560
the bond axis, but they actually
have a nodal plane at

114
00:05:29,560 --> 00:05:31,940
this z, this bond axis here.

115
00:05:31,940 --> 00:05:34,990
And remember for this class,
we always define z as the

116
00:05:34,990 --> 00:05:37,570
internuclear or the bond axis.

117
00:05:37,570 --> 00:05:40,700
So it might look like here, if
you don't understand about p

118
00:05:40,700 --> 00:05:42,950
orbitals, which I know all you
do, but if someone else was

119
00:05:42,950 --> 00:05:45,100
just looking and seeing, it kind
of looks like there's two

120
00:05:45,100 --> 00:05:46,090
bonds here.

121
00:05:46,090 --> 00:05:49,320
There's not two bonds, that's
one pi bond, and the reason is

122
00:05:49,320 --> 00:05:52,440
because it's 2 p orbitals coming
together, and remember

123
00:05:52,440 --> 00:05:55,805
p orbitals have electron density
above and below the

124
00:05:55,805 --> 00:05:58,290
axis, so when they come
together, it kind of looks

125
00:05:58,290 --> 00:06:00,670
like one bonds, but essentially
what we have here

126
00:06:00,670 --> 00:06:03,270
is one pi bond.

127
00:06:03,270 --> 00:06:07,330
So let's think about how we can
classify single and double

128
00:06:07,330 --> 00:06:09,360
and triple bonds, which is
what we're really used to

129
00:06:09,360 --> 00:06:11,920
dealing with in terms
of these sigma bonds

130
00:06:11,920 --> 00:06:13,750
and these pi bonds.

131
00:06:13,750 --> 00:06:16,340
So, if we take a look at what
a single bond is, and let me

132
00:06:16,340 --> 00:06:19,720
grab some molecules here.

133
00:06:19,720 --> 00:06:22,570
If we're talking about a single
bond, we're talking

134
00:06:22,570 --> 00:06:27,070
about 2 orbitals overlapping
in the internuclear axis.

135
00:06:27,070 --> 00:06:29,860
So if we have a single bond
here, would you consider that

136
00:06:29,860 --> 00:06:33,300
a sigma bond or a pi bond?

137
00:06:33,300 --> 00:06:33,830
STUDENT: [INAUDIBLE]

138
00:06:33,830 --> 00:06:35,510
PROFESSOR: Right, it's
a sigma bond.

139
00:06:35,510 --> 00:06:38,020
Essentially what we're
seeing is overlapping

140
00:06:38,020 --> 00:06:40,450
in this z axis here.

141
00:06:40,450 --> 00:06:44,130
In contrast, if we talk about a
double bond, what we're now

142
00:06:44,130 --> 00:06:47,450
talking about is having
both a sigma bond

143
00:06:47,450 --> 00:06:49,750
and also one pi bond.

144
00:06:49,750 --> 00:06:53,210
And I apologize, I intended to
set this up right before

145
00:06:53,210 --> 00:06:55,840
class, but that didn't
happen today.

146
00:06:55,840 --> 00:06:59,760
All right, so what we see here
is we have our sigma bond

147
00:06:59,760 --> 00:07:03,470
that's along the internuclear
axis here, but we also have a

148
00:07:03,470 --> 00:07:07,210
pi bond, because each of these
atoms now has electrons in

149
00:07:07,210 --> 00:07:09,510
it's in a p orbital, so we're
going to overlap of electron

150
00:07:09,510 --> 00:07:13,540
density above and
below the bond.

151
00:07:13,540 --> 00:07:16,900
So that's exactly what our
definition of a pi bond is, so

152
00:07:16,900 --> 00:07:20,160
we have one sigma bond,
and one pi bond.

153
00:07:20,160 --> 00:07:22,310
So now let's think about
a triple bond.

154
00:07:22,310 --> 00:07:26,400
A triple bond, again is going to
have one sigma bond on the

155
00:07:26,400 --> 00:07:28,130
internuclear axis.

156
00:07:28,130 --> 00:07:29,930
How many pi bonds would
you expect?

157
00:07:29,930 --> 00:07:31,410
STUDENT: [INAUDIBLE]

158
00:07:31,410 --> 00:07:32,670
PROFESSOR: Two, great.

159
00:07:32,670 --> 00:07:34,460
So, we're going to
see two pi bonds.

160
00:07:34,460 --> 00:07:38,120
The first one will be above
and below the bond axis is

161
00:07:38,120 --> 00:07:40,650
where we'll see the electron
density, and the second will

162
00:07:40,650 --> 00:07:44,050
be perpendicular to that, so it
will be a density in front

163
00:07:44,050 --> 00:07:45,860
of and behind the bond axis.

164
00:07:45,860 --> 00:07:48,680
So we can kind of flip it this
way -- this will be one pi

165
00:07:48,680 --> 00:07:50,690
bond, this will be another
interacting

166
00:07:50,690 --> 00:07:51,740
between these p orbitals.

167
00:07:51,740 --> 00:07:56,040
All right, so that's really all
there is to thinking about

168
00:07:56,040 --> 00:07:58,400
valence bond theory in
terms of the most

169
00:07:58,400 --> 00:08:00,160
simple explanation here.

170
00:08:00,160 --> 00:08:02,500
But what we're going is we're
going to start trying to apply

171
00:08:02,500 --> 00:08:05,200
it to a molecule, and I actually
picked a molecule

172
00:08:05,200 --> 00:08:07,630
that it's not going to work for,
even though it would work

173
00:08:07,630 --> 00:08:10,410
even just at this level for
many, many molecules.

174
00:08:10,410 --> 00:08:13,510
And I picked looking at methane
so we could see if

175
00:08:13,510 --> 00:08:16,470
there are other factors that
we're not considering, that we

176
00:08:16,470 --> 00:08:19,920
need to maybe tweak our model
a little bit, and I think

177
00:08:19,920 --> 00:08:24,150
we'll find that we do if we take
a look at a polyatomic

178
00:08:24,150 --> 00:08:27,660
molecule, methane, so c h 4.

179
00:08:27,660 --> 00:08:32,560
So let's think about methane
using valence bond theory.

180
00:08:32,560 --> 00:08:36,010
So, using our simple valence
bond theory, what we would

181
00:08:36,010 --> 00:08:40,790
expect is that we want to pair
up any unpaired electrons in

182
00:08:40,790 --> 00:08:44,580
methane with unpaired electrons
from hydrogen and

183
00:08:44,580 --> 00:08:46,280
form bonds.

184
00:08:46,280 --> 00:08:49,410
But what we see we have is that
we only have two unpaired

185
00:08:49,410 --> 00:08:51,000
electrons here.

186
00:08:51,000 --> 00:08:55,180
Because we have paired set in
a 2 s orbital, so all we're

187
00:08:55,180 --> 00:08:58,360
left essentially is two
electrons that are available

188
00:08:58,360 --> 00:08:59,200
for bonding.

189
00:08:59,200 --> 00:09:01,780
So this should immediately look
like a problem because we

190
00:09:01,780 --> 00:09:04,850
know, in fact, that methane is
tetravalent, and this is

191
00:09:04,850 --> 00:09:07,130
telling us it's only divalent.

192
00:09:07,130 --> 00:09:10,380
Essentially it would only allow
for us to bond to two

193
00:09:10,380 --> 00:09:14,650
hydrogen atoms. So if it did
this, it now looks like, from

194
00:09:14,650 --> 00:09:17,610
looking at the paired electrons
that we have a

195
00:09:17,610 --> 00:09:21,930
stable structure here, and our
structure is not c h 4, it's a

196
00:09:21,930 --> 00:09:25,740
stable structure of c h 2, and
it will actually predict,

197
00:09:25,740 --> 00:09:29,740
also, what this h c h
bond angle it is.

198
00:09:29,740 --> 00:09:32,510
So according to this model
what is that bond angle?

199
00:09:32,510 --> 00:09:35,660
STUDENT: [INAUDIBLE]

200
00:09:35,660 --> 00:09:37,760
PROFESSOR: One more time.

201
00:09:37,760 --> 00:09:39,250
OK, I hear a mix.

202
00:09:39,250 --> 00:09:42,630
So, according to this model,
what we're seeing is a bond

203
00:09:42,630 --> 00:09:44,150
angle of 90 degrees.

204
00:09:44,150 --> 00:09:48,060
What do you know the bond
angle should be?

205
00:09:48,060 --> 00:09:49,270
It's 109 .

206
00:09:49,270 --> 00:09:51,370
5 is what we would expect
for methane because it's

207
00:09:51,370 --> 00:09:54,370
tetravalent, but here we're just
seeing something that's

208
00:09:54,370 --> 00:09:56,740
divalent, and they're both
in p orbitals that are

209
00:09:56,740 --> 00:09:58,290
perpendicular to each other.

210
00:09:58,290 --> 00:10:01,620
So what we're predicting is a
bond angle of 90 degrees.

211
00:10:01,620 --> 00:10:05,790
This is totally wrong, this is
the wrong picture altogether.

212
00:10:05,790 --> 00:10:07,950
If you had your notes, you could
do some fun scribbling

213
00:10:07,950 --> 00:10:10,110
right now, so you can
do that at home.

214
00:10:10,110 --> 00:10:14,320
We're going to need to tweak our
explanation here, and take

215
00:10:14,320 --> 00:10:17,970
into account another factor,
and that factor is the fact

216
00:10:17,970 --> 00:10:21,460
that we know that we must have
four unpaired electrons in

217
00:10:21,460 --> 00:10:24,400
carbon if we're going
to form four bonds.

218
00:10:24,400 --> 00:10:26,580
So the way that we can explain
this is through something

219
00:10:26,580 --> 00:10:28,580
called electron promotion and

220
00:10:28,580 --> 00:10:30,460
hybridization of atomic orbitals.

221
00:10:30,460 --> 00:10:34,440
So let's take a look at
what we mean by this.

222
00:10:34,440 --> 00:10:39,160
So if we take our carbon atom
here, which has two electrons

223
00:10:39,160 --> 00:10:44,130
in the 2 s orbital, and we
promote one of these electrons

224
00:10:44,130 --> 00:10:49,090
into a 2 p orbital, what we see
now is that yes, we do, we

225
00:10:49,090 --> 00:10:50,460
have four unpaired electrons.

226
00:10:50,460 --> 00:10:53,450
So, looking at this,
this might not

227
00:10:53,450 --> 00:10:54,440
look so good for you.

228
00:10:54,440 --> 00:10:58,610
What we're proposing here is
that you take a nice low

229
00:10:58,610 --> 00:11:01,090
energy s electron and
move it into a

230
00:11:01,090 --> 00:11:02,790
higher energy p orbital.

231
00:11:02,790 --> 00:11:05,750
And the truth is that yes, this
costs energy, we're going

232
00:11:05,750 --> 00:11:07,740
up to a higher energy state.

233
00:11:07,740 --> 00:11:10,300
But it doesn't actually cost
as much energy as you might

234
00:11:10,300 --> 00:11:13,350
think, because in this s orbital
here we have a paired

235
00:11:13,350 --> 00:11:16,530
electron situation where we're
moving up to a p orbital where

236
00:11:16,530 --> 00:11:19,040
the electron is no longer
paired, so it won't feel quite

237
00:11:19,040 --> 00:11:22,050
as much electron repulsion, but
nonetheless, this is going

238
00:11:22,050 --> 00:11:23,140
to cost us energy.

239
00:11:23,140 --> 00:11:25,210
So we'll have to think about
where that energy is going to

240
00:11:25,210 --> 00:11:27,530
come from and we'll see
that in just a minute.

241
00:11:27,530 --> 00:11:30,440
But let's assume that this is,
in fact, going to happen.

242
00:11:30,440 --> 00:11:33,760
So now what we have is four
unpaired electrons.

243
00:11:33,760 --> 00:11:36,510
That's great, but it's still not
quite the picture we need,

244
00:11:36,510 --> 00:11:39,430
because actually, all the
electrons are not in equal

245
00:11:39,430 --> 00:11:43,040
orbitals -- one's in an s
orbital, and 3 are in p.

246
00:11:43,040 --> 00:11:46,270
But what we need to remember is
the fact that we're talking

247
00:11:46,270 --> 00:11:47,990
about electrons which
are waves.

248
00:11:47,990 --> 00:11:50,490
When we're talking about
orbitals, we're talking about

249
00:11:50,490 --> 00:11:51,350
wave functions.

250
00:11:51,350 --> 00:11:53,750
So we can actually
constructively and

251
00:11:53,750 --> 00:11:57,320
destructively combine these
waves, these atomic orbitals

252
00:11:57,320 --> 00:11:59,620
to make a hybrid.

253
00:11:59,620 --> 00:12:02,410
So if we go ahead and hybridize
our p orbitals and

254
00:12:02,410 --> 00:12:06,130
our s orbitals, we'll switch
from having these original

255
00:12:06,130 --> 00:12:10,170
orbitals to having something
called hybrid orbitals.

256
00:12:10,170 --> 00:12:13,810
And hybrid orbitals are all
going to be completely equal,

257
00:12:13,810 --> 00:12:16,690
and you'll notice that they're
higher in energy than the s

258
00:12:16,690 --> 00:12:20,400
orbital, and lower in energy
than the p orbital.

259
00:12:20,400 --> 00:12:23,540
That should make sense because
they come from combining s

260
00:12:23,540 --> 00:12:25,360
orbitals and p orbitals.

261
00:12:25,360 --> 00:12:28,080
And specifically, when we give
them a name it's very clear

262
00:12:28,080 --> 00:12:30,690
exactly which orbitals they
come from combining, we're

263
00:12:30,690 --> 00:12:34,540
calling these s p 3 orbitals
-- that's because they come

264
00:12:34,540 --> 00:12:38,230
from combining 1 s orbital
and 3 p orbitals.

265
00:12:38,230 --> 00:12:41,070
You should never get the names
of hybrid orbitals wrong

266
00:12:41,070 --> 00:12:42,720
because it's very
straightforward.

267
00:12:42,720 --> 00:12:46,770
If it has 1 s and 3 p's,
we call it s p 3.

268
00:12:46,770 --> 00:12:49,200
Naming doesn't always make sense
in chemistry, so I like

269
00:12:49,200 --> 00:12:51,320
to point out this is a place
where naming does

270
00:12:51,320 --> 00:12:52,490
make a lot of sense.

271
00:12:52,490 --> 00:12:56,200
All right, so let's consider our
methane situation now that

272
00:12:56,200 --> 00:12:58,850
we have our hybrid orbitals.

273
00:12:58,850 --> 00:13:01,650
So I want to mention also, these
are exactly equivalent,

274
00:13:01,650 --> 00:13:04,540
they're equivalent in energy,
they're equivalent in shape.

275
00:13:04,540 --> 00:13:06,420
The only thing that is different
about these orbitals

276
00:13:06,420 --> 00:13:08,540
is their orientation in space.

277
00:13:08,540 --> 00:13:10,500
So actually, first let's
take a look at how

278
00:13:10,500 --> 00:13:11,890
we got these orbitals.

279
00:13:11,890 --> 00:13:14,710
We got them from combining
again, 1 s

280
00:13:14,710 --> 00:13:17,310
orbital and the 3 p orbitals.

281
00:13:17,310 --> 00:13:21,040
If we hybridize these, what we
end up seeing are these four

282
00:13:21,040 --> 00:13:22,080
hybrid orbitals.

283
00:13:22,080 --> 00:13:24,780
You'll notice the shape is the
same, all that's different is

284
00:13:24,780 --> 00:13:26,360
their orientation.

285
00:13:26,360 --> 00:13:29,510
So essentially, each of these
orbitals come from linear

286
00:13:29,510 --> 00:13:33,330
combinations of all of the
original orbitals, and it's

287
00:13:33,330 --> 00:13:36,260
hard to picture exactly how that
happens, but one that you

288
00:13:36,260 --> 00:13:38,350
can at least start to get an
idea is if you think about

289
00:13:38,350 --> 00:13:42,000
combining the 2 s and the 2 p
z here, which is not quite

290
00:13:42,000 --> 00:13:44,290
accurate because of course,
we're combining all of them.

291
00:13:44,290 --> 00:13:45,530
But it would just give
you a little bit

292
00:13:45,530 --> 00:13:46,580
of an idea of shape.

293
00:13:46,580 --> 00:13:50,050
You can see if we combine the s
with the top lobe of the p,

294
00:13:50,050 --> 00:13:51,810
they're going to constructively
interfere

295
00:13:51,810 --> 00:13:53,720
because they have
the same sign.

296
00:13:53,720 --> 00:13:57,090
So you see in the hybrid orbital
we actually have a

297
00:13:57,090 --> 00:14:01,340
larger lobe on top where they
constructively interfered.

298
00:14:01,340 --> 00:14:04,250
If you compare the s orbital
with the bottom lobe, these

299
00:14:04,250 --> 00:14:05,830
have a different sign
so they're going to

300
00:14:05,830 --> 00:14:07,580
destructively interfere.

301
00:14:07,580 --> 00:14:10,900
So what you see is actually a
diminished lobe on the back

302
00:14:10,900 --> 00:14:12,960
part of this s p 3 orbital.

303
00:14:12,960 --> 00:14:16,580
So s p 3 orbitals always have
one huge lobe and one really

304
00:14:16,580 --> 00:14:17,410
little lobe.

305
00:14:17,410 --> 00:14:20,140
A lot of times when people draw
them, they even only draw

306
00:14:20,140 --> 00:14:22,710
the big lobe just to keep their
paper looking nicer, but

307
00:14:22,710 --> 00:14:27,290
there is that little tiny
lobe on the other side.

308
00:14:27,290 --> 00:14:30,470
All right, so in terms of s
p 3 hybrid orbitals, let's

309
00:14:30,470 --> 00:14:34,690
combine all four together on
one axis, because this is

310
00:14:34,690 --> 00:14:37,680
what's going to happen in
an s p 3 carbon atom.

311
00:14:37,680 --> 00:14:39,480
So in this case, what
would you say that

312
00:14:39,480 --> 00:14:40,650
the angle is here?

313
00:14:40,650 --> 00:14:43,320
STUDENT: [INAUDIBLE]

314
00:14:43,320 --> 00:14:43,770
PROFESSOR: Right, great.

315
00:14:43,770 --> 00:14:45,750
So, we've achieved the angle
that we observed, which is

316
00:14:45,750 --> 00:14:47,260
good, which is a 109 .

317
00:14:47,260 --> 00:14:48,600
5.

318
00:14:48,600 --> 00:14:52,170
So we can think about now doing
bonding, and now we have

319
00:14:52,170 --> 00:14:56,130
four equal orbitals with
one electronic each.

320
00:14:56,130 --> 00:14:58,940
So we can bring in four hydrogen
atoms, which will

321
00:14:58,940 --> 00:15:01,500
each contribute another
unpaired electron.

322
00:15:01,500 --> 00:15:05,360
So now what we have
is four bonds.

323
00:15:05,360 --> 00:15:08,000
And we can think about where
we did get that energy for

324
00:15:08,000 --> 00:15:10,370
electron promotion that I
mentioned before where we

325
00:15:10,370 --> 00:15:13,460
moved the electron from
the 2 s to the 2 p.

326
00:15:13,460 --> 00:15:14,490
We get that from bonding.

327
00:15:14,490 --> 00:15:16,510
We're going to release a lot
of energy for bonding, it's

328
00:15:16,510 --> 00:15:18,760
going to more than make up for
the fact that we actually had

329
00:15:18,760 --> 00:15:22,540
to spend some energy to
promote that electron.

330
00:15:22,540 --> 00:15:26,400
So, we can think about now how
do we describe this bond in

331
00:15:26,400 --> 00:15:28,050
valence bond theory.

332
00:15:28,050 --> 00:15:30,890
So the way that you describe
a bond is you describe the

333
00:15:30,890 --> 00:15:33,350
orbitals that the bond comes
from, and also the

334
00:15:33,350 --> 00:15:34,740
symmetry of the bond.

335
00:15:34,740 --> 00:15:37,080
So would you expect this
to be a pi bond or

336
00:15:37,080 --> 00:15:38,340
a sigma bond here?

337
00:15:38,340 --> 00:15:41,210
STUDENT: [INAUDIBLE]

338
00:15:41,210 --> 00:15:43,630
PROFESSOR: OK, so I'm hearing
some mixed answers.

339
00:15:43,630 --> 00:15:45,820
It turns out that it's
a sigma bond.

340
00:15:45,820 --> 00:15:49,350
The reason that it's a sigma
bond is because the s p 3

341
00:15:49,350 --> 00:15:52,090
hybrid orbital is directly
interacting with the 1 s

342
00:15:52,090 --> 00:15:55,810
orbital of the hydrogen atom,
and that's going to happen on

343
00:15:55,810 --> 00:15:58,070
the internuclear axis, they're
just coming together.

344
00:15:58,070 --> 00:16:01,550
Any time two orbitals come
straight on together in that

345
00:16:01,550 --> 00:16:05,690
internuclear axis, you're going
to have a sigma bond.

346
00:16:05,690 --> 00:16:09,120
So if we go ahead and name this
bond, what we're going to

347
00:16:09,120 --> 00:16:15,750
name it is sigma, because that's
the -- basically the

348
00:16:15,750 --> 00:16:17,620
shape of the bond or
that's how our

349
00:16:17,620 --> 00:16:19,100
bond is coming together.

350
00:16:19,100 --> 00:16:21,330
And then we're going to name the
atomic orbitals that make

351
00:16:21,330 --> 00:16:26,050
it up, and it's being made up
of a carbon 2 s p 3 orbital,

352
00:16:26,050 --> 00:16:30,090
and a hydrogen 1 s orbital.

353
00:16:30,090 --> 00:16:32,570
All right, so let's think of
a case now that's getting a

354
00:16:32,570 --> 00:16:33,720
little bit more complicated.

355
00:16:33,720 --> 00:16:36,090
We were talking about
methane, which has

356
00:16:36,090 --> 00:16:37,780
only one central atom.

357
00:16:37,780 --> 00:16:40,750
We can also talk about atoms
that have two or more central

358
00:16:40,750 --> 00:16:44,540
atoms. So let's talk about
ethane now, which is c h 2.

359
00:16:44,540 --> 00:16:48,580
So let's take our carbon s p 3
hybridized carbon and just

360
00:16:48,580 --> 00:16:53,490
move it around here so we can
make the z inter- bonding axis

361
00:16:53,490 --> 00:16:56,080
between the two carbons
right here.

362
00:16:56,080 --> 00:16:58,840
So if we still have an
angle of a 109 .

363
00:16:58,840 --> 00:17:03,200
5 degrees, and again, we still
have four unpaired electrons

364
00:17:03,200 --> 00:17:06,850
available for bonding, we can
make one of those bonds with

365
00:17:06,850 --> 00:17:10,550
another s p 3 hybridized carbon,
so we're going to make

366
00:17:10,550 --> 00:17:13,830
up one pair here.

367
00:17:13,830 --> 00:17:16,320
If we think about that, that's
a sigma bond, right, they're

368
00:17:16,320 --> 00:17:22,730
coming together along
the nuclear axis.

369
00:17:22,730 --> 00:17:25,810
We also have six spots available
to form hydrogen

370
00:17:25,810 --> 00:17:28,260
bonds, so we can go ahead
and fill in those

371
00:17:28,260 --> 00:17:30,750
electrons as well.

372
00:17:30,750 --> 00:17:33,950
So in terms of thinking about
ethane, we actually have two

373
00:17:33,950 --> 00:17:38,680
bond types that we're going to
be describing just in terms of

374
00:17:38,680 --> 00:17:42,730
the carbon-carbon bond and
then the carbon h bonds.

375
00:17:42,730 --> 00:17:45,660
So let's talk about ethane and
how we would actually write

376
00:17:45,660 --> 00:17:47,180
these bonds.

377
00:17:47,180 --> 00:17:52,250
If we have the molecule ethane,
then what we're going

378
00:17:52,250 --> 00:17:55,810
to have first is our sigma bond
that we described between

379
00:17:55,810 --> 00:17:57,670
the two carbons.

380
00:17:57,670 --> 00:18:00,000
So it's going to be carbon,
and then what's the

381
00:18:00,000 --> 00:18:01,160
hybridization here?

382
00:18:01,160 --> 00:18:05,500
STUDENT: [INAUDIBLE]

383
00:18:05,500 --> 00:18:06,410
PROFESSOR: All right, start
again, what's the

384
00:18:06,410 --> 00:18:07,530
hybridization of the
carbon atom?

385
00:18:07,530 --> 00:18:10,190
STUDENT: [INAUDIBLE]

386
00:18:10,190 --> 00:18:14,820
PROFESSOR: OK, so it's 2 s p
3, and our second carbon is

387
00:18:14,820 --> 00:18:17,250
also 2 s p 3.

388
00:18:17,250 --> 00:18:21,560
All right, so this is our
first type of bond here.

389
00:18:21,560 --> 00:18:24,420
Our second bond is going to be
between the carbon and the

390
00:18:24,420 --> 00:18:28,380
hydrogen atoms. Is that
a sigma or a pi bond?

391
00:18:28,380 --> 00:18:28,830
STUDENT: [INAUDIBLE]

392
00:18:28,830 --> 00:18:30,050
PROFESSOR: Sigma, good.

393
00:18:30,050 --> 00:18:35,000
So again, our carbon is
going to be 2 s p 3.

394
00:18:35,000 --> 00:18:37,300
And what will our hydrogen be?

395
00:18:37,300 --> 00:18:40,050
1 s -- we don't have to
hybridize it, it already has

396
00:18:40,050 --> 00:18:43,290
only one unpaired electron
in a 1 s orbital.

397
00:18:43,290 --> 00:18:47,060
All right, so that's how
we describe ethane.

398
00:18:47,060 --> 00:18:49,540
We don't have to just stick
with carbon, we can think

399
00:18:49,540 --> 00:18:52,960
about describing other types
of atoms as well using this

400
00:18:52,960 --> 00:18:54,170
hybridization.

401
00:18:54,170 --> 00:18:57,260
For example, we can talk about
nitrogen, and nitrogen has

402
00:18:57,260 --> 00:19:00,100
five valence electrons
shown here.

403
00:19:00,100 --> 00:19:03,410
Would you expect to see electron
promotion in nitrogen

404
00:19:03,410 --> 00:19:05,300
where we pull one of these
2 s electrons into

405
00:19:05,300 --> 00:19:06,950
one of the 2 p orbitals?

406
00:19:06,950 --> 00:19:08,640
STUDENT: [INAUDIBLE]

407
00:19:08,640 --> 00:19:09,330
PROFESSOR: No, good.

408
00:19:09,330 --> 00:19:12,770
So, electron promotion does
not happen in terms of

409
00:19:12,770 --> 00:19:15,250
nitrogen, because it would not
increased our number of

410
00:19:15,250 --> 00:19:16,670
unpaired electrons.

411
00:19:16,670 --> 00:19:19,380
No matter what we do in terms
of promotion, we're always

412
00:19:19,380 --> 00:19:22,460
going to have three unpaired
electrons.

413
00:19:22,460 --> 00:19:25,700
We can still hybridize all these
orbitals, however, so we

414
00:19:25,700 --> 00:19:31,290
can still form four hybrid
orbitals, which are again, 2 s

415
00:19:31,290 --> 00:19:35,060
p 3 hybrid orbitals.

416
00:19:35,060 --> 00:19:38,150
So if we take a look at nitrogen
here, what you'll

417
00:19:38,150 --> 00:19:40,400
notice is we have thre available
for bonding, and we

418
00:19:40,400 --> 00:19:42,890
already have our lone pair
-- one of our orbitals is

419
00:19:42,890 --> 00:19:45,210
already filled up.

420
00:19:45,210 --> 00:19:49,010
So we can add three hydrogen
atoms here, and fill in our

421
00:19:49,010 --> 00:19:50,900
other orbitals right here.

422
00:19:50,900 --> 00:19:54,230
So if we do this and we form
the molecule ammonia, let's

423
00:19:54,230 --> 00:19:57,040
switch to a clicker question,
and have you tell me what the

424
00:19:57,040 --> 00:19:59,510
bond angle is going to
be in ammonia --

425
00:19:59,510 --> 00:20:02,490
the h n h bond angle.

426
00:20:02,490 --> 00:20:05,520
Actually, let me draw it on
the board as you look --

427
00:20:05,520 --> 00:20:08,220
actually, can you put the class
notes on, since you

428
00:20:08,220 --> 00:20:10,000
don't actually have your
notes to refer to.

429
00:20:10,000 --> 00:20:11,710
So there's the class
notes there.

430
00:20:11,710 --> 00:20:17,830
All right, this should be a
pretty quick thing for you to

431
00:20:17,830 --> 00:20:32,640
figure out, so let's just
take 10 seconds on this.

432
00:20:32,640 --> 00:20:33,270
OK, great.

433
00:20:33,270 --> 00:20:36,060
Even thinking quickly, most
of you got it correct.

434
00:20:36,060 --> 00:20:40,410
So what we see is on ammonia
here, we know that it's less

435
00:20:40,410 --> 00:20:41,730
than a 109 .

436
00:20:41,730 --> 00:20:46,400
5, it's actually 107, so
it's less than a 109 .

437
00:20:46,400 --> 00:20:48,810
5, because of that lone
pair pushing down

438
00:20:48,810 --> 00:20:50,430
in the bonding electrons.

439
00:20:50,430 --> 00:20:52,460
And what is the shape,
for one more

440
00:20:52,460 --> 00:20:59,450
clicker question on ammonia?

441
00:20:59,450 --> 00:21:14,790
Let's take 10 seconds again,
this should be pretty quick.

442
00:21:14,790 --> 00:21:15,810
All right, pretty good.

443
00:21:15,810 --> 00:21:17,090
So, 70% of you.

444
00:21:17,090 --> 00:21:19,180
We'd like to get
this up higher.

445
00:21:19,180 --> 00:21:21,740
The shape is actually
trigonal pyramidal.

446
00:21:21,740 --> 00:21:23,940
And you need to just remember
your shapes.

447
00:21:23,940 --> 00:21:26,460
If they're not obvious to you
what they're called, you need

448
00:21:26,460 --> 00:21:28,170
to just study them
and learn them.

449
00:21:28,170 --> 00:21:31,780
So it's trigonal because we have
these three atoms that

450
00:21:31,780 --> 00:21:35,330
are bound to the central atom
here, and if you picture it,

451
00:21:35,330 --> 00:21:37,060
it's actually shaped
like a pyramid.

452
00:21:37,060 --> 00:21:38,700
So it's trigonal pyramidal.

453
00:21:38,700 --> 00:21:42,490
That's what we call when we have
three bonding atoms and

454
00:21:42,490 --> 00:21:46,860
one lone pair.

455
00:21:46,860 --> 00:21:47,190
All right.

456
00:21:47,190 --> 00:21:50,630
So we can switch all the way
back to our notes here.

457
00:21:50,630 --> 00:21:53,200
And the last thing we can think
about is how do we name

458
00:21:53,200 --> 00:21:56,030
this n h bond, and again,
we just name

459
00:21:56,030 --> 00:21:57,380
it based on it symmetry.

460
00:21:57,380 --> 00:22:01,380
It's a sigma bond, and it's
going to be -- no.

461
00:22:01,380 --> 00:22:05,860
OK, it's going to be nitrogen
2 s p 3, because it's a

462
00:22:05,860 --> 00:22:09,930
nitrogen atom, and then
hydrogen 1 s.

463
00:22:09,930 --> 00:22:11,520
So, I don't even have to worry
because you're not writing

464
00:22:11,520 --> 00:22:14,040
this down, so I can just fix it
when I post the notes and

465
00:22:14,040 --> 00:22:17,160
no one will ever know, except
that this is not

466
00:22:17,160 --> 00:22:17,410
OpenCourseWare.

467
00:22:17,410 --> 00:22:22,500
So let's switch to thinking
about oxygen

468
00:22:22,500 --> 00:22:23,990
hybridization here.

469
00:22:23,990 --> 00:22:27,570
So in oxygen we have a similar
situation where, in fact, we

470
00:22:27,570 --> 00:22:30,050
are not going to promote any
of the electrons because we

471
00:22:30,050 --> 00:22:33,680
have two lone pair electrons
no matter what we do.

472
00:22:33,680 --> 00:22:37,460
So when we hybridize our
orbitals, we're going to end

473
00:22:37,460 --> 00:22:42,840
up with again, four hybrid
orbitals, 4 s p 3 orbitals,

474
00:22:42,840 --> 00:22:45,890
and what we'll see is that two
of these are already going to

475
00:22:45,890 --> 00:22:49,590
be filled up with a paired
electrons, so we're only going

476
00:22:49,590 --> 00:22:51,750
to have 2 orbitals with an
unpaired electron available

477
00:22:51,750 --> 00:22:53,780
for bonding.

478
00:22:53,780 --> 00:22:56,750
So let's think about water here
as our simplest example

479
00:22:56,750 --> 00:22:57,970
with oxygen.

480
00:22:57,970 --> 00:23:02,260
So we can have our two hydrogen
atoms come in here,

481
00:23:02,260 --> 00:23:05,260
and what we will find is now
that we have all of our

482
00:23:05,260 --> 00:23:08,870
orbitals filled up -- so
thinking about what this angle

483
00:23:08,870 --> 00:23:13,120
is here, would you expect it
to be less than or greater

484
00:23:13,120 --> 00:23:16,770
than what we saw for
ammonia before?

485
00:23:16,770 --> 00:23:17,730
STUDENT: Less than.

486
00:23:17,730 --> 00:23:18,810
PROFESSOR: Good, good, it's
going to be less than, and

487
00:23:18,810 --> 00:23:20,320
it's going to be less
than because now we

488
00:23:20,320 --> 00:23:22,470
have two lone pairs.

489
00:23:22,470 --> 00:23:24,220
So since we have two lone
pairs, we're going to be

490
00:23:24,220 --> 00:23:27,210
pushing down even further on
the bonding electrons, so

491
00:23:27,210 --> 00:23:29,900
we're going to smoosh those
bonds even closer together.

492
00:23:29,900 --> 00:23:31,980
The bond, it turns
out, is 104 .

493
00:23:31,980 --> 00:23:36,040
5 degrees, that h o h bond.

494
00:23:36,040 --> 00:23:40,590
So in terms of naming our o h
bond, good, it's right here.

495
00:23:40,590 --> 00:23:45,610
So it's going to be a sigma
bond, and we have oxygen 2 s p

496
00:23:45,610 --> 00:23:51,770
3 and hydrogen 1 s.

497
00:23:51,770 --> 00:23:54,650
And the geometry, which I didn't
ask you, is going to be

498
00:23:54,650 --> 00:23:56,030
bent for this molecule.

499
00:23:56,030 --> 00:23:59,300
All right, so that's s p 3
hybridization, but those

500
00:23:59,300 --> 00:24:02,300
aren't the only type of hybrid
orbitals that we can form.

501
00:24:02,300 --> 00:24:04,800
Let's take a look at what
happens if instead of

502
00:24:04,800 --> 00:24:07,940
combining all four orbitals, we
just combine three of those

503
00:24:07,940 --> 00:24:10,660
orbitals, and what we'll
end up with is s p 2

504
00:24:10,660 --> 00:24:12,120
hybridization.

505
00:24:12,120 --> 00:24:16,920
So in s p 2 hybridization,
instead of combining all four,

506
00:24:16,920 --> 00:24:18,460
we're just combining
two of the p

507
00:24:18,460 --> 00:24:20,540
orbitals with the s orbital.

508
00:24:20,540 --> 00:24:22,630
So what we're going to
end up with now is

509
00:24:22,630 --> 00:24:24,480
three hybrid orbitals.

510
00:24:24,480 --> 00:24:28,240
And what happens to this last
p orbital is nothing at all,

511
00:24:28,240 --> 00:24:29,540
we just get it back.

512
00:24:29,540 --> 00:24:32,990
So we end up with 1 p orbital
completely untouched, and

513
00:24:32,990 --> 00:24:36,960
three hybrid s p 2 orbitals.

514
00:24:36,960 --> 00:24:39,550
So again, we can think
of an example here.

515
00:24:39,550 --> 00:24:44,170
So let's take boron, for
example, and this has -- it

516
00:24:44,170 --> 00:24:46,690
starts off with three
valence electrons.

517
00:24:46,690 --> 00:24:49,350
Would you expect to see electron
promotion for boron?

518
00:24:49,350 --> 00:24:50,790
STUDENT: Yes.

519
00:24:50,790 --> 00:24:53,890
PROFESSOR: Yeah, absolutely.
if we move up one of our

520
00:24:53,890 --> 00:24:56,150
electrons into an empty p
orbital, what were going to

521
00:24:56,150 --> 00:24:56,440
see is now we have three
unpaired electrons that are

522
00:24:56,440 --> 00:24:57,910
ready for bonding.

523
00:24:57,910 --> 00:25:05,880
So, if we hybridize just these
three orbitals, what we're

524
00:25:05,880 --> 00:25:10,240
going to end up with is our
s p 2 hybrid orbitals.

525
00:25:10,240 --> 00:25:12,860
Again, the name is very
straightforward, it comes from

526
00:25:12,860 --> 00:25:17,130
1 s and 2 p orbital, so
it will be s p 2.

527
00:25:17,130 --> 00:25:20,250
And again, you might be thinking
well, why didn't we

528
00:25:20,250 --> 00:25:23,010
actually hybridize this
2 p y orbital.

529
00:25:23,010 --> 00:25:26,180
It doesn't actually have an
electron in it, so we don't

530
00:25:26,180 --> 00:25:28,720
have to worry about whether it's
very high in energy or

531
00:25:28,720 --> 00:25:30,720
not, we don't care that
it's high in energy.

532
00:25:30,720 --> 00:25:33,930
What we do care about is the
energy of our orbitals that

533
00:25:33,930 --> 00:25:37,680
have electrons in them, and if
we combined all four of the

534
00:25:37,680 --> 00:25:40,480
orbitals, then our hybrid
orbitals would have more p

535
00:25:40,480 --> 00:25:43,930
character to them, so they'd
actually be higher in energy.

536
00:25:43,930 --> 00:25:46,240
So if we don't have to hybridize
one of the p

537
00:25:46,240 --> 00:25:49,150
orbitals, we can actually end
up with a lower energy

538
00:25:49,150 --> 00:25:53,320
situation, because now these
s p 2 orbitals are 1/3 s

539
00:25:53,320 --> 00:25:59,660
character, and only 2/3 p
character, instead of 3/4.

540
00:25:59,660 --> 00:26:04,690
So we end up with 3 s p 2 hybrid
orbitals, so we can

541
00:26:04,690 --> 00:26:09,420
think about what would happen
here in terms of bonding, and

542
00:26:09,420 --> 00:26:14,880
if we think about how to get
our bonds as far away as

543
00:26:14,880 --> 00:26:16,860
possible from each other, what
we're going to have is the

544
00:26:16,860 --> 00:26:18,600
trigonal planer situation.

545
00:26:18,600 --> 00:26:21,950
So if you picture, for example,
b h 3, it's going to

546
00:26:21,950 --> 00:26:23,750
look like this.

547
00:26:23,750 --> 00:26:26,610
All of our electrons are in our
bonds, we want to got them

548
00:26:26,610 --> 00:26:29,960
a 120 degrees away from each
other, that's as far away as

549
00:26:29,960 --> 00:26:31,220
we can get them.

550
00:26:31,220 --> 00:26:34,310
Keep in mind we do have this p
orbital here and it's coming

551
00:26:34,310 --> 00:26:35,620
right out at us.

552
00:26:35,620 --> 00:26:38,350
And this p orbital is here,
but it's empty, it doesn't

553
00:26:38,350 --> 00:26:41,220
have any electrons in it, that's
why we don't have to

554
00:26:41,220 --> 00:26:43,720
worry about it in terms of
getting our electrons as far

555
00:26:43,720 --> 00:26:45,810
away from each other
as possible.

556
00:26:45,810 --> 00:26:49,060
So what we'll have here is a
trigonal planar case, and you

557
00:26:49,060 --> 00:26:52,330
can see that we only have three
electrons that are set

558
00:26:52,330 --> 00:26:55,950
for bonding, so we'll add three
hydrogens, and for b h

559
00:26:55,950 --> 00:26:58,650
3, we'll get a stable
structure here.

560
00:26:58,650 --> 00:27:02,070
So, remember, boron was one of
those exceptions to our Lewis

561
00:27:02,070 --> 00:27:04,780
structure rules where it was
perfectly happy not having a

562
00:27:04,780 --> 00:27:05,760
full octet.

563
00:27:05,760 --> 00:27:09,660
So this can tell you why it's so
happy with only having six

564
00:27:09,660 --> 00:27:10,830
electrons around it.

565
00:27:10,830 --> 00:27:16,830
All right, so if we think about
b h bond here, again,

566
00:27:16,830 --> 00:27:20,000
it's the sigma bond, and we're
going to say it's a boron 2 s

567
00:27:20,000 --> 00:27:26,470
p 2 hybrid orbital interacting
with a hydrogen 1 s orbital.

568
00:27:26,470 --> 00:27:29,400
So let's take a look at another
case where we have s p

569
00:27:29,400 --> 00:27:31,510
2 hybridization, we can
actually also have

570
00:27:31,510 --> 00:27:33,190
it happen in carbon.

571
00:27:33,190 --> 00:27:35,970
So if we think about having
it happen in carbon, we're

572
00:27:35,970 --> 00:27:38,630
starting with the situation
where we've already promoted

573
00:27:38,630 --> 00:27:42,500
our electron into a 2 p orbital
here, and what we're

574
00:27:42,500 --> 00:27:46,340
going to do is just combine the
s and two of the p's, so

575
00:27:46,340 --> 00:27:50,470
we'll end up with electrons
in one of each three

576
00:27:50,470 --> 00:27:52,090
s p 2 hybrid orbitals.

577
00:27:52,090 --> 00:27:55,890
But unlike the case with boron
where we had an empty p

578
00:27:55,890 --> 00:27:59,520
orbital, we're actually going
to have an electron in the p

579
00:27:59,520 --> 00:28:02,920
orbital of carbon as well.

580
00:28:02,920 --> 00:28:05,790
So again, if we think about
that shape of that carbon

581
00:28:05,790 --> 00:28:08,110
atom, it's going to be trigonal
planar, it's going to

582
00:28:08,110 --> 00:28:12,500
have bond angles of 120 degrees,
because we have this

583
00:28:12,500 --> 00:28:16,350
set up of having three
hybrid orbitals.

584
00:28:16,350 --> 00:28:18,790
So let's take a look at what
actually happens if we're

585
00:28:18,790 --> 00:28:22,990
talking about a carbon-carbon
double bond, such as in

586
00:28:22,990 --> 00:28:27,060
ethene, c 2 h 4, we're going
to have a double bond.

587
00:28:27,060 --> 00:28:30,310
If we have a double bond, we
know we need to have only one

588
00:28:30,310 --> 00:28:33,180
sigma bond, and we're also going
to have one pi bond.

589
00:28:33,180 --> 00:28:35,080
So it already should make
sense why we have that p

590
00:28:35,080 --> 00:28:37,350
orbital there, in order to form
a pi bond, we're going to

591
00:28:37,350 --> 00:28:38,770
need a p orbital.

592
00:28:38,770 --> 00:28:42,740
So if you picture this as our
s p 2 carbon atom where we

593
00:28:42,740 --> 00:28:47,510
have three hybrid orbitals,
and then one p y orbital

594
00:28:47,510 --> 00:28:49,090
coming right out at us.

595
00:28:49,090 --> 00:28:51,840
So again, we picture the same
thing as we pictured with the

596
00:28:51,840 --> 00:28:53,770
boron there.

597
00:28:53,770 --> 00:28:57,600
If we have, coming along this
z axis, another carbon atom,

598
00:28:57,600 --> 00:29:00,060
we can actually form one
bond between the two

599
00:29:00,060 --> 00:29:01,670
carbon atoms there.

600
00:29:01,670 --> 00:29:07,850
So if we picture how this
happens, what we have here if

601
00:29:07,850 --> 00:29:19,680
these are our 2 s p 2 carbon
atoms -- so here we have s p 2

602
00:29:19,680 --> 00:29:25,410
hybrid carbon, and here we have
s p 2 hybrid carbon atom.

603
00:29:25,410 --> 00:29:29,180
These 2 are going to come
together like this, and the

604
00:29:29,180 --> 00:29:31,450
first bond that we're going to
form is going to be a sigma

605
00:29:31,450 --> 00:29:32,950
bond, right, so we
see that here.

606
00:29:32,950 --> 00:29:36,020
If we're looking head on, we
see they form a sigma bond.

607
00:29:36,020 --> 00:29:38,440
We can also look at them coming
in from the side, and

608
00:29:38,440 --> 00:29:40,770
that's what I tried to depict
here where you can actually

609
00:29:40,770 --> 00:29:43,250
see in pink is the p orbital.

610
00:29:43,250 --> 00:29:45,850
So we can also show them coming
together this way, so

611
00:29:45,850 --> 00:29:48,460
now you're looking at it where
you can see the p orbital, and

612
00:29:48,460 --> 00:29:52,430
maybe just see well one
of the hydrogen atoms.

613
00:29:52,430 --> 00:29:57,630
So we can have four total
hydrogens bonding here, and we

614
00:29:57,630 --> 00:29:59,340
can think about how
to describe these

615
00:29:59,340 --> 00:30:01,000
carbon-carbon bonds.

616
00:30:01,000 --> 00:30:05,050
So in the first case of this
first bond here that I've put

617
00:30:05,050 --> 00:30:07,460
in a square, what type
of a bond is this, is

618
00:30:07,460 --> 00:30:08,650
the sigma or pi?

619
00:30:08,650 --> 00:30:10,210
STUDENT: Sigma.

620
00:30:10,210 --> 00:30:11,210
PROFESSOR: Yup, it's
a sigma bond.

621
00:30:11,210 --> 00:30:12,870
We're having two
orbitals coming

622
00:30:12,870 --> 00:30:15,530
together on the bond axis.

623
00:30:15,530 --> 00:30:20,210
So we'll call this sigma, and
it's between two s p 2 hybrid

624
00:30:20,210 --> 00:30:26,560
carbon atoms. So it's stigma
carbon s p 2, carbon s p 2.

625
00:30:26,560 --> 00:30:29,750
What about this second bond here
where we're going to have

626
00:30:29,750 --> 00:30:32,470
interaction of 2 p orbitals,
is that sigma or pi?

627
00:30:32,470 --> 00:30:34,140
STUDENT: Pi.

628
00:30:34,140 --> 00:30:34,820
PROFESSOR: Pi, great.

629
00:30:34,820 --> 00:30:38,040
So our second bond is going
to be a pi bond.

630
00:30:38,040 --> 00:30:40,990
And again, this is between the
p orbitals, these are not

631
00:30:40,990 --> 00:30:43,940
hybrid orbitals, so when we name
this bond we're going to

632
00:30:43,940 --> 00:30:47,770
name it as a pi bond here,
because it's between two p

633
00:30:47,770 --> 00:30:51,150
orbitals, and it's going to be
between the carbon 2 p y

634
00:30:51,150 --> 00:30:54,280
orbital, and the other
carbon 2 p y orbital.

635
00:30:54,280 --> 00:30:57,740
Remember, we didn't hybridize
the 2 p y orbital, so that's

636
00:30:57,740 --> 00:31:00,490
what we have left over to
form these pi bonds.

637
00:31:00,490 --> 00:31:03,430
All right.

638
00:31:03,430 --> 00:31:07,160
So in addition to having these
two carbon bonds, we actually

639
00:31:07,160 --> 00:31:11,890
also have four carbon hydrogen
bonds in addition to our

640
00:31:11,890 --> 00:31:13,370
carbon-carbon bonds.

641
00:31:13,370 --> 00:31:17,280
So why don't you tell me what
the valence bond description

642
00:31:17,280 --> 00:31:19,520
would be of these carbon
hydrogen bonds?

643
00:31:19,520 --> 00:31:45,280
So let's take 10 seconds
on that.

644
00:31:45,280 --> 00:31:45,880
OK, great.

645
00:31:45,880 --> 00:31:48,660
So most and you got it, so we
can switch to the notes and

646
00:31:48,660 --> 00:31:50,860
let's talk about this here.

647
00:31:50,860 --> 00:31:55,230
So in terms of the carbon
hydrogen bond, it's a sigma

648
00:31:55,230 --> 00:31:59,520
bond, because we define it --
any time we are bonding to an

649
00:31:59,520 --> 00:32:01,940
atom, we have to keep redefining
our bond axis to

650
00:32:01,940 --> 00:32:04,040
whatever two atoms we're
talking about.

651
00:32:04,040 --> 00:32:07,540
So it's along the bond axis and
it's between a carbon s p

652
00:32:07,540 --> 00:32:10,950
2 hybrid, and then the hydrogen
is just a 1 s orbital

653
00:32:10,950 --> 00:32:12,680
that we're combining here.

654
00:32:12,680 --> 00:32:17,290
So those are our three types
of bonds in ethene.

655
00:32:17,290 --> 00:32:19,340
One thing that I want to
mention that is really

656
00:32:19,340 --> 00:32:22,690
important is once you have
double bonds, what happens

657
00:32:22,690 --> 00:32:25,900
between those two atoms in the
molecule is they can no longer

658
00:32:25,900 --> 00:32:28,380
rotate in relation
to each other.

659
00:32:28,380 --> 00:32:29,950
So you can think about
why that is.

660
00:32:29,950 --> 00:32:33,170
When we have just a single bond
in them molecule, you

661
00:32:33,170 --> 00:32:35,780
have all the free rotation you
want, you can just spin it

662
00:32:35,780 --> 00:32:37,930
around, there's nothing
keeping it in place.

663
00:32:37,930 --> 00:32:41,010
But once you have a double
bond here, we have our pi

664
00:32:41,010 --> 00:32:43,050
bond, as well as
our sigma bond.

665
00:32:43,050 --> 00:32:46,710
So there's electron density
above the bond

666
00:32:46,710 --> 00:32:47,940
and below the bond.

667
00:32:47,940 --> 00:32:51,310
So if I try to rotate my 2
atoms, you see that I have to

668
00:32:51,310 --> 00:32:54,470
break that pi bond, because they
need to be lined up so

669
00:32:54,470 --> 00:32:56,680
that the electron density
can overlap.

670
00:32:56,680 --> 00:33:00,550
So in order to rotate a double
bond, you have to actually

671
00:33:00,550 --> 00:33:02,980
break the pi bond, so
essentially what you're doing

672
00:33:02,980 --> 00:33:04,420
is breaking the double bond.

673
00:33:04,420 --> 00:33:07,880
So really, you can not ever
rotate a double bond, it makes

674
00:33:07,880 --> 00:33:09,820
your molecule very rigid.

675
00:33:09,820 --> 00:33:13,470
This is incredibly important
because if you picture having

676
00:33:13,470 --> 00:33:16,780
a double bond in a very large
molecule, you could have all

677
00:33:16,780 --> 00:33:19,920
sorts of other atoms off this
way and all sorts of other

678
00:33:19,920 --> 00:33:22,710
atoms off this way, and you can
picture the shape would be

679
00:33:22,710 --> 00:33:26,300
very different if you have one
confirmation versus another

680
00:33:26,300 --> 00:33:27,600
confirmation.

681
00:33:27,600 --> 00:33:31,280
So it's very important that the
double bond locks it in a

682
00:33:31,280 --> 00:33:32,610
particular conformation.

683
00:33:32,610 --> 00:33:35,980
This completely could change if
you were to flip from one

684
00:33:35,980 --> 00:33:37,350
to the other conformation
which can

685
00:33:37,350 --> 00:33:39,240
happen in chemical reactions.

686
00:33:39,240 --> 00:33:41,550
If you were to make that change
you would find that the

687
00:33:41,550 --> 00:33:44,460
molecule now has completely
different biological and

688
00:33:44,460 --> 00:33:46,390
chemical properties.

689
00:33:46,390 --> 00:33:48,880
So it's very important to be
keeping in mind that any time

690
00:33:48,880 --> 00:33:51,650
you see a double bond, you
have a pi bond there, so

691
00:33:51,650 --> 00:33:56,160
you're not going to see any
rotation around the bond axis.

692
00:33:56,160 --> 00:33:59,620
All right, so let's think of a
more complicated example of

693
00:33:59,620 --> 00:34:01,590
having a double bond, and
maybe a more interesting

694
00:34:01,590 --> 00:34:03,960
example, and this is talking
about benzene.

695
00:34:03,960 --> 00:34:06,380
I think most and you have talked
a little bit about

696
00:34:06,380 --> 00:34:09,550
benzene over this past
week in recitation.

697
00:34:09,550 --> 00:34:13,520
Benzene is a ring that's made up
of six carbon atoms and six

698
00:34:13,520 --> 00:34:16,760
hydrogen atoms. So let's picture
what this looks like

699
00:34:16,760 --> 00:34:18,660
here, and we'll start
with four and we'll

700
00:34:18,660 --> 00:34:19,430
add in our last two.

701
00:34:19,430 --> 00:34:23,480
So essentially, we have two
ethene or ethylene molecules

702
00:34:23,480 --> 00:34:28,690
here to start with where these
blue are our 2 s p 2 hybrid

703
00:34:28,690 --> 00:34:32,000
orbitals, so you can see that
for each carbon atom, one is

704
00:34:32,000 --> 00:34:35,150
already used up binding to
another carbon atom.

705
00:34:35,150 --> 00:34:39,160
If we think about bringing in
those last two carbons, what

706
00:34:39,160 --> 00:34:43,220
you can see is that for every
carbon, two of its hybrid

707
00:34:43,220 --> 00:34:48,500
orbitals are being used to
bond to other carbons.

708
00:34:48,500 --> 00:34:50,750
So that leaves each carbon
with only one

709
00:34:50,750 --> 00:34:53,370
hybrid orbital left.

710
00:34:53,370 --> 00:34:56,320
And if we think about the six
hydrogens, now each of those

711
00:34:56,320 --> 00:34:59,380
are going to bind by combining
one of the carbon hybrid

712
00:34:59,380 --> 00:35:03,160
orbitals to a 1 s orbital
of hydrogen.

713
00:35:03,160 --> 00:35:07,400
So, if we think about what bonds
are in this molecule, we

714
00:35:07,400 --> 00:35:11,960
actually have six of these
sigma carbon s p 2,

715
00:35:11,960 --> 00:35:14,160
carbon s p 2 bonds.

716
00:35:14,160 --> 00:35:18,660
We also have carbon s p
2 hydrogen 1 s bonds.

717
00:35:18,660 --> 00:35:21,370
How many of those do we have?

718
00:35:21,370 --> 00:35:23,980
Yup, we also have six of these,
because we have six

719
00:35:23,980 --> 00:35:25,770
carbon hydrogen bonds.

720
00:35:25,770 --> 00:35:28,900
So that's two of our types of
bonds in benzene, and we have

721
00:35:28,900 --> 00:35:31,610
one type left, and that's
going to actually be the

722
00:35:31,610 --> 00:35:35,570
double bond or the pi bond that
forms between some of

723
00:35:35,570 --> 00:35:37,140
these p orbitals.

724
00:35:37,140 --> 00:35:41,400
So we can have one bond here
between this carbon's p

725
00:35:41,400 --> 00:35:43,410
orbital and this carbon's
p orbital.

726
00:35:43,410 --> 00:35:46,180
So let's have a clicker question
here on how many

727
00:35:46,180 --> 00:35:50,980
total pi bonds do you expect
to see in benzene?

728
00:35:50,980 --> 00:35:56,950
Oh good, so it's left up -- the
notes are left up on this

729
00:35:56,950 --> 00:35:57,690
screen right now.

730
00:35:57,690 --> 00:36:16,980
All right, so let's take
10 seconds on that.

731
00:36:16,980 --> 00:36:17,530
All right, great.

732
00:36:17,530 --> 00:36:20,920
So, most of you saw that what
we would expect to see is a

733
00:36:20,920 --> 00:36:22,540
three bond, some of
you thought six.

734
00:36:22,540 --> 00:36:25,010
So let's take a look at
why three is correct.

735
00:36:25,010 --> 00:36:29,305
So, what we end up having
is three of these pi --

736
00:36:29,305 --> 00:36:32,930
2 p y 2 p y bonds, we
can have one between

737
00:36:32,930 --> 00:36:34,130
these two carbons here.

738
00:36:34,130 --> 00:36:36,260
Can we have one between
these two carbons here

739
00:36:36,260 --> 00:36:37,310
if we have one here?

740
00:36:37,310 --> 00:36:38,280
STUDENT: No.

741
00:36:38,280 --> 00:36:38,670
PROFESSOR: No, we can't.

742
00:36:38,670 --> 00:36:42,230
We're already using it up in
this pi bond here, so that

743
00:36:42,230 --> 00:36:45,130
means we're limited to only
two other spots on the

744
00:36:45,130 --> 00:36:47,250
molecule, so we have three.

745
00:36:47,250 --> 00:36:49,810
But, of course, what you saw in
recitation, and hopefully,

746
00:36:49,810 --> 00:36:53,240
what you can now think very
quickly by looking at this, is

747
00:36:53,240 --> 00:36:56,720
that this is not the only
configuration of pi bonds that

748
00:36:56,720 --> 00:36:58,670
we could have in benzene.

749
00:36:58,670 --> 00:37:00,770
There's absolutely no reason
I couldn't have switched it

750
00:37:00,770 --> 00:37:04,710
around and said that instead the
pi orbitals form between

751
00:37:04,710 --> 00:37:08,120
these atoms instead of those
first atoms I showed.

752
00:37:08,120 --> 00:37:11,390
So, let's look at this in a
more simple structure here

753
00:37:11,390 --> 00:37:15,110
where we have the two possible
forms of benzene, and the

754
00:37:15,110 --> 00:37:17,250
reality is is that it's
going to be some

755
00:37:17,250 --> 00:37:18,410
combination of the two.

756
00:37:18,410 --> 00:37:21,830
This is resonance, this is a
case of a resonance structure.

757
00:37:21,830 --> 00:37:25,260
So what we see is that those six
pi electrons are actually

758
00:37:25,260 --> 00:37:29,620
going to be de-localized around
all six of those atoms.

759
00:37:29,620 --> 00:37:33,330
So if you think about any one of
these carbon-carbon bonds,

760
00:37:33,330 --> 00:37:36,000
what type of a bond would
you expect that to be?

761
00:37:36,000 --> 00:37:38,460
What would the bond order
be for this bond?

762
00:37:38,460 --> 00:37:40,360
STUDENT: [INAUDIBLE]

763
00:37:40,360 --> 00:37:41,320
PROFESSOR: Yup, it's going
to be a 1 and 1/2 bond.

764
00:37:41,320 --> 00:37:45,510
It's a 1 and 1/2 because it's
halfway between a double bond

765
00:37:45,510 --> 00:37:46,590
and a single bond.

766
00:37:46,590 --> 00:37:49,080
So, of course, this is resonance
so we can go ahead

767
00:37:49,080 --> 00:37:51,980
and put our resonance notation
in there to indicate that

768
00:37:51,980 --> 00:37:53,680
benzene is a resonance
structure.

769
00:37:53,680 --> 00:37:55,420
All right.

770
00:37:55,420 --> 00:37:58,530
So let's quickly talk about our
last type of hybridization

771
00:37:58,530 --> 00:38:00,710
that we're going to discuss
today, which is s p

772
00:38:00,710 --> 00:38:01,160
hybridization.

773
00:38:01,160 --> 00:38:06,640
So s p hybridization comes now
from when we're combining an s

774
00:38:06,640 --> 00:38:08,180
orbital now with only
one p orbital.

775
00:38:08,180 --> 00:38:11,700
So let's take a look at
this with carbon.

776
00:38:11,700 --> 00:38:15,450
And if we hybridize these
orbitals in carbon, what we

777
00:38:15,450 --> 00:38:19,060
end up with is having two hybrid
orbitals, and then

778
00:38:19,060 --> 00:38:21,750
we're going to be left with two
of our p orbitals that are

779
00:38:21,750 --> 00:38:25,770
each going to have an electron
associated in them.

780
00:38:25,770 --> 00:38:28,350
So again, looking at the
shapes, now we're just

781
00:38:28,350 --> 00:38:32,720
combining two, we've got these
two equal hybrid orbitals plus

782
00:38:32,720 --> 00:38:34,540
these 2 p orbitals here.

783
00:38:34,540 --> 00:38:38,830
So let's take the case of
acetylene where we have two

784
00:38:38,830 --> 00:38:41,430
carbon atoms that are going to
be triple bonded to each

785
00:38:41,430 --> 00:38:43,420
other, each are bonded
to a carbon

786
00:38:43,420 --> 00:38:45,290
and then to one hydrogen.

787
00:38:45,290 --> 00:38:47,660
So this is a little bit trickier
to look at and see

788
00:38:47,660 --> 00:38:50,290
what it means, but essentially
we have two hybrid orbitals,

789
00:38:50,290 --> 00:38:53,870
which are shown in blue here,
and then we have one p orbital

790
00:38:53,870 --> 00:38:56,590
that's left alone that's going
up and down on the page.

791
00:38:56,590 --> 00:38:58,820
And then that second p orbital's
actually coming

792
00:38:58,820 --> 00:39:01,650
right out at you, it's coming
out of the screen at you.

793
00:39:01,650 --> 00:39:05,070
So, if we think about this z
bonding axis between the two

794
00:39:05,070 --> 00:39:09,170
carbon atoms, we can picture
overlap of those s p hybrid

795
00:39:09,170 --> 00:39:13,400
orbitals, and then we can also
picture bonding to hydrogen.

796
00:39:13,400 --> 00:39:19,380
So, in a settling, what is
the bond angle here?

797
00:39:19,380 --> 00:39:21,700
This is the easiest question
all day, what is the bond

798
00:39:21,700 --> 00:39:24,760
angle between all of these?

799
00:39:24,760 --> 00:39:26,130
Great, so it's 180 degrees.

800
00:39:26,130 --> 00:39:30,800
So, if we think about the bonds
that are forming -- oh I

801
00:39:30,800 --> 00:39:33,870
see our TAs are here, so you
can start handing them out,

802
00:39:33,870 --> 00:39:37,530
because we have two minutes
left to go.

803
00:39:37,530 --> 00:39:44,530
So, as they're very quietly
handing out your class notes,

804
00:39:44,530 --> 00:39:47,670
let's think about what this bond
is here, this boxed bond,

805
00:39:47,670 --> 00:39:50,380
is it a pi bond or
a sigma bond?

806
00:39:50,380 --> 00:39:51,860
It's going to be a sigma bond.

807
00:39:51,860 --> 00:39:56,120
So, we have sigma 2
s p, carbon 2 s p.

808
00:39:56,120 --> 00:39:59,070
So they're two s p
bonds combining.

809
00:39:59,070 --> 00:40:01,740
Now let's think about this first
pi bond, which will be

810
00:40:01,740 --> 00:40:04,500
above and below the
bonding axis.

811
00:40:04,500 --> 00:40:07,330
Is this pi or sigma?

812
00:40:07,330 --> 00:40:08,540
This is pi.

813
00:40:08,540 --> 00:40:13,080
So we're talking about pi carbon
2 p x, because it's the

814
00:40:13,080 --> 00:40:17,720
x axes combining to
carbon 2 p x.

815
00:40:17,720 --> 00:40:22,830
And the last bond that we have
here is a carbon-carbon bond,

816
00:40:22,830 --> 00:40:25,530
and this is our last p orbitals

817
00:40:25,530 --> 00:40:26,470
that are coming together.

818
00:40:26,470 --> 00:40:29,200
These are the ones that are
coming right out at you, so

819
00:40:29,200 --> 00:40:32,470
this is going to be on
a second pi orbital.

820
00:40:32,470 --> 00:40:36,580
So this will be pi carbon
2 p y, carbon 2 p y.

821
00:40:36,580 --> 00:40:38,880
All right.

822
00:40:38,880 --> 00:40:42,000
So, we'll stop here today.

823
00:40:42,000 --> 00:40:44,520
Just stay in your seats for
another 30 seconds as they're

824
00:40:44,520 --> 00:40:46,040
handing out your notes.

825
00:40:46,040 --> 00:40:48,830
I want to mention that you're
going to get problem-set five

826
00:40:48,830 --> 00:40:52,060
is posted today, and I'll
write which ones you can

827
00:40:52,060 --> 00:40:55,430
already do so far, because you
don't have class on Monday.

828
00:40:55,430 --> 00:40:58,760
But remember that you do have
recitation on Tuesday, so that

829
00:40:58,760 --> 00:41:02,660
could be very helpful with the
problem-set, so be sure to go

830
00:41:02,660 --> 00:41:06,090
to recitation on Tuesday, and
have a great long weekend.