1
00:00:00 --> 00:00:00
2
00:00:00 --> 00:00:00
The following content is
provided under a Creative
3
00:00:00 --> 00:00:00
Commons license.
4
00:00:00 --> 00:00:00
Your support will help MIT
OpenCourseWare continue to
5
00:00:00 --> 00:00:00
offer high quality educational
resources for free.
6
00:00:00 --> 00:00:01
To make a donation or view
additional materials from
7
00:00:01 --> 00:00:01
hundreds of MIT courses,
visit MIT OpenCourseWare
8
00:00:01 --> 00:00:01
at ocw.mit.edu.
9
00:00:01 --> 00:00:21
PROFESSOR: All right.
10
00:00:21 --> 00:00:27
As everyone finishes getting
settled in, why don't you take
11
00:00:27 --> 00:00:34
10 more seconds on the clicker
question here, and let's see
12
00:00:34 --> 00:00:37
how you did on that this, this
is very similar to the clicker
13
00:00:37 --> 00:00:41
question that we had on Friday.
14
00:00:41 --> 00:00:44
OK, so let's get started here.
15
00:00:44 --> 00:00:47
It looks like we are
doing a lot better.
16
00:00:47 --> 00:00:51
We now have 77% getting the
correct answer, we only had
17
00:00:51 --> 00:00:53
about 30-something percent
on Friday for a very
18
00:00:53 --> 00:00:55
similar question.
19
00:00:55 --> 00:00:59
So if you're not in this 77%,
let's quickly go over why,
20
00:00:59 --> 00:01:02
in fact, this is the
correct answer, 0 .
21
00:01:02 --> 00:01:05
9 times 10 to the
negative 18 joules.
22
00:01:05 --> 00:01:09
So I'm using the same kind of
tricky language that we'd used
23
00:01:09 --> 00:01:11
before, not to trick you,
but so that you're not
24
00:01:11 --> 00:01:12
tricked in the future.
25
00:01:12 --> 00:01:15
So if we're talking about the
fourth excited state, and we
26
00:01:15 --> 00:01:18
talk instead about principle
quantum numbers, what principle
27
00:01:18 --> 00:01:22
quantum number corresponds to
the fourth excited state
28
00:01:22 --> 00:01:22
of a hydrogen atom.
29
00:01:22 --> 00:01:24
STUDENT: Five.
30
00:01:24 --> 00:01:24
PROFESSOR: Five.
31
00:01:24 --> 00:01:25
OK.
32
00:01:25 --> 00:01:26
So, hopefully that cleared
up for some of you why
33
00:01:26 --> 00:01:28
you got the wrong answer.
34
00:01:28 --> 00:01:32
So we know that we're in the n
equals 5 state, so we can find
35
00:01:32 --> 00:01:34
what the binding
energy is here.
36
00:01:34 --> 00:01:37
The ionization energy, of
course, is just the negative
37
00:01:37 --> 00:01:39
of the binding energy.
38
00:01:39 --> 00:01:42
We know that binding energy is
always negative, we know that
39
00:01:42 --> 00:01:44
ionization energy is
always positive.
40
00:01:44 --> 00:01:47
So hopefully, putting all those
things together, if you looked
41
00:01:47 --> 00:01:50
at this question again we'd
get 100% on it, that our
42
00:01:50 --> 00:01:53
only option here is 0 .
43
00:01:53 --> 00:01:55
9, and that it's not the
negative, it's the positive
44
00:01:55 --> 00:01:58
version, because we're talking
about how much energy we have
45
00:01:58 --> 00:02:03
to put into the system in
order to eject an electron.
46
00:02:03 --> 00:02:03
All right.
47
00:02:03 --> 00:02:08
And today we're going to mostly
be talking about wave functions
48
00:02:08 --> 00:02:12
of electrons, but before we get
to that, I wanted to review one
49
00:02:12 --> 00:02:15
last thing that's back on to
Friday's topic, which was when
50
00:02:15 --> 00:02:19
we were solving the Schrodinger
equation, or in fact, using the
51
00:02:19 --> 00:02:22
solution to the Schrodinger
equation for the energy, the
52
00:02:22 --> 00:02:26
binding energy between an
electron and a nucleus.
53
00:02:26 --> 00:02:29
And when we talked about that,
what we found was that we could
54
00:02:29 --> 00:02:33
actually validate our predicted
binding energies by looking at
55
00:02:33 --> 00:02:36
the emission spectra of the
hydrogen atom, which is what we
56
00:02:36 --> 00:02:40
did as the demo, or we could
think about the absorption
57
00:02:40 --> 00:02:41
spectra as well.
58
00:02:41 --> 00:02:44
And what we predict as an
energy difference between two
59
00:02:44 --> 00:02:47
levels, we know should
correspond to the energy of
60
00:02:47 --> 00:02:50
light that's either emitted, if
we're giving off a photon, or
61
00:02:50 --> 00:02:53
that's absorbed if we're going
to take on a photon and
62
00:02:53 --> 00:02:56
jump from a lower to a
higher energy level.
63
00:02:56 --> 00:02:58
So we came up with two
formulas, which are similar to
64
00:02:58 --> 00:03:00
the two that I'm showing here.
65
00:03:00 --> 00:03:03
The formula tells us the
frequency of the light that's
66
00:03:03 --> 00:03:07
emitted or absorbed based on
the energy difference between
67
00:03:07 --> 00:03:10
the two levels that we're going
between, that the electron
68
00:03:10 --> 00:03:12
is transitioning between.
69
00:03:12 --> 00:03:14
You'll notice that there's a
little bit of a difference in
70
00:03:14 --> 00:03:16
these equations here from the
ones from the other day, which
71
00:03:16 --> 00:03:19
is that you have this z
squared value in there.
72
00:03:19 --> 00:03:22
So these are both called
Rydberg formulas for figuring
73
00:03:22 --> 00:03:25
out the frequency of light
emitted or absorbed, and before
74
00:03:25 --> 00:03:28
we were looking at the Rydberg
formula specifically for the
75
00:03:28 --> 00:03:32
hydrogen atom, and now that we
have this z squared term in the
76
00:03:32 --> 00:03:35
formula here, we're now talking
about absolutely any
77
00:03:35 --> 00:03:37
one electron atom.
78
00:03:37 --> 00:03:40
And it should make sense where
we got this from, because we
79
00:03:40 --> 00:03:44
know that the binding energy,
if we're talking about a
80
00:03:44 --> 00:03:52
hydrogen atom, what is the
binding energy equal to?
81
00:03:52 --> 00:03:57
Negative Rydberg over what?
82
00:03:57 --> 00:03:58
Yes.
83
00:03:58 --> 00:04:00
So, it's negative Rydberg
constant over n squared.
84
00:04:00 --> 00:04:03
But if we're talking more
generally about any one
85
00:04:03 --> 00:04:10
electron atom, now we have a
more general equation for the
86
00:04:10 --> 00:04:15
binding energy, which has this
z squared term out in front of
87
00:04:15 --> 00:04:18
it, right, so it's negative z
squared times the Rydberg
88
00:04:18 --> 00:04:22
constant all over n squared.
89
00:04:22 --> 00:04:24
So, essentially when we're
talking about these equations
90
00:04:24 --> 00:04:27
up here, all we're doing is
talking about the regular
91
00:04:27 --> 00:04:30
Rydberg formulas, but instead
we could go back and re-derive
92
00:04:30 --> 00:04:33
the equation for any one
electron atom, which would
93
00:04:33 --> 00:04:36
just mean that we put that z
squared term in the front.
94
00:04:36 --> 00:04:39
So when you solve certain types
of problems, such as problems
95
00:04:39 --> 00:04:43
later on in the second half of
your p-set, if you need to talk
96
00:04:43 --> 00:04:46
about the frequency of light
emitted or absorbed for a one
97
00:04:46 --> 00:04:50
electron atom, such as lithium
plus 2, for example, then you
98
00:04:50 --> 00:04:53
would need to plug in z, and
remember the z value for
99
00:04:53 --> 00:04:55
lithium would just be 3.
100
00:04:55 --> 00:04:59
The z value for hydrogen, of
course, is 1, and that's why
101
00:04:59 --> 00:05:01
this term falls out of that
equation when we're talking
102
00:05:01 --> 00:05:04
specifically about
the hydrogen atom.
103
00:05:04 --> 00:05:07
So, just to finish our review
of what we talked about on
104
00:05:07 --> 00:05:10
Friday, when we're thinking
about transitions between two
105
00:05:10 --> 00:05:14
different states, and we're
talking about a situation where
106
00:05:14 --> 00:05:18
the final state, the n final,
is greater than n initial, in
107
00:05:18 --> 00:05:20
this case, are we talking
about absorption or are we
108
00:05:20 --> 00:05:24
talking about emission?
109
00:05:24 --> 00:05:27
Hearing a little
bit of a mix here.
110
00:05:27 --> 00:05:31
In fact, we're talking about
absorption when n final is
111
00:05:31 --> 00:05:32
greater than n initial.
112
00:05:32 --> 00:05:35
We start at this lower energy
state and go up -- that means
113
00:05:35 --> 00:05:38
we need to absorb a photon,
we have to take in energy.
114
00:05:38 --> 00:05:42
Specifically, we have to take
in this exact amount of energy
115
00:05:42 --> 00:05:46
in order to bump the electron
up to the higher energy level.
116
00:05:46 --> 00:05:50
So that means that when instead
we start high and go low, we're
117
00:05:50 --> 00:05:54
dealing with emission where we
have excess energy that the
118
00:05:54 --> 00:05:57
electron's giving off, and that
energy is going to be equal the
119
00:05:57 --> 00:06:01
energy of the photon that is
released and, of course,
120
00:06:01 --> 00:06:04
through our equations we know
how to get from energy to
121
00:06:04 --> 00:06:07
frequency or to wavelength of
the photon that we're
122
00:06:07 --> 00:06:09
talking about.
123
00:06:09 --> 00:06:10
All right.
124
00:06:10 --> 00:06:14
So that's all I'm going to say
today in terms of solving the
125
00:06:14 --> 00:06:17
energy part of the Schrodinger
equation, so what we're really
126
00:06:17 --> 00:06:19
going to focus on is the other
part of the Schrodinger
127
00:06:19 --> 00:06:23
equation today, which
is solving for psi.
128
00:06:23 --> 00:06:26
So we're going to for psi, and
before that, we're going to
129
00:06:26 --> 00:06:29
figure out that instead of just
that one quantum number n,
130
00:06:29 --> 00:06:31
we're going to have a few other
quantum numbers that fall out
131
00:06:31 --> 00:06:35
of solving the Schrodinger
equation for what psi is.
132
00:06:35 --> 00:06:38
We're also going to talk more
about what psi actually means.
133
00:06:38 --> 00:06:41
When we first introduced the
Schrodinger equation, what I
134
00:06:41 --> 00:06:44
told you was think of psi as
being some representation
135
00:06:44 --> 00:06:46
of what an electron is.
136
00:06:46 --> 00:06:49
We'll get more specific here,
more specific even than just
137
00:06:49 --> 00:06:50
saying you can think
of it as an orbital.
138
00:06:50 --> 00:06:52
We'll really think
about what psi means.
139
00:06:52 --> 00:06:56
And in doing that, we'll also
talk about the shapes of h atom
140
00:06:56 --> 00:06:59
wave functions, specifically
the shapes of orbitals, and
141
00:06:59 --> 00:07:02
then something called radial
probability distribution,
142
00:07:02 --> 00:07:05
which will make sense
when we get to it.
143
00:07:05 --> 00:07:08
But, as I said before that, we
have some more quantum numbers
144
00:07:08 --> 00:07:12
to take care of, because it
turns out that when you solve
145
00:07:12 --> 00:07:15
the Schrodinger equation for
psi, these other quantum
146
00:07:15 --> 00:07:17
numbers have to be the defined.
147
00:07:17 --> 00:07:20
When we talked about binding
energy, we just had one quantum
148
00:07:20 --> 00:07:22
number that came out of it.
149
00:07:22 --> 00:07:25
And that quantum number was n,
which was our principle quantum
150
00:07:25 --> 00:07:30
number, and we know that n
could be equal to any integer
151
00:07:30 --> 00:07:35
value, so, 1, 2, 3, all
the way up to infinity.
152
00:07:35 --> 00:07:39
And this quantization that
comes out of having n is what
153
00:07:39 --> 00:07:42
gives us the quantization
of different energy levels.
154
00:07:42 --> 00:07:45
That's why we can't have a
continuum of energy, we
155
00:07:45 --> 00:07:50
actually have those
quantized points.
156
00:07:50 --> 00:07:53
So, it turns out that n is not
the only quantum number needed
157
00:07:53 --> 00:07:55
to describe a wave
function, however.
158
00:07:55 --> 00:07:58
There's two more that you
can see come out of it.
159
00:07:58 --> 00:08:02
And the first is l, and l is
our angular momentum quantum
160
00:08:02 --> 00:08:06
number, and it's called that
because it actually dictates
161
00:08:06 --> 00:08:10
the angular momentum that our
electron has in our atom.
162
00:08:10 --> 00:08:13
And when we talk about l it is
a quantum number, so because
163
00:08:13 --> 00:08:15
it's a quantum number, we know
that it can only have discreet
164
00:08:15 --> 00:08:19
values, it can't just be
any value we want, it's
165
00:08:19 --> 00:08:20
very specific values.
166
00:08:20 --> 00:08:24
And unlike n, l can start all
the way down at 0, and it
167
00:08:24 --> 00:08:29
increases by integer value, so
we go 1, 2, 3, and
168
00:08:29 --> 00:08:30
all the way up.
169
00:08:30 --> 00:08:34
But also unlike n, l cannot
have just any value, we
170
00:08:34 --> 00:08:36
can't go into infinity.
171
00:08:36 --> 00:08:39
L is limited such that
the highest value
172
00:08:39 --> 00:08:41
of l is n minus 1.
173
00:08:41 --> 00:08:44
We can't get any
higher than that.
174
00:08:44 --> 00:08:46
So, it would be a good question
to ask why are we limited --
175
00:08:46 --> 00:08:49
clearly there's this
relationship between l and n,
176
00:08:49 --> 00:08:51
and we can't get any
higher than n equals one.
177
00:08:51 --> 00:08:54
We can actually think about
why that is, and the
178
00:08:54 --> 00:08:57
reason is because l is
our angular momentum.
179
00:08:57 --> 00:09:00
It describes the angular
momentum of the electron.
180
00:09:00 --> 00:09:02
So another way to think about
that is just the rotational
181
00:09:02 --> 00:09:05
kinetic energy of our electron.
182
00:09:05 --> 00:09:08
And we know that n describes
the total energy, that total
183
00:09:08 --> 00:09:12
binding energy of the electron,
so the total energy is going to
184
00:09:12 --> 00:09:15
be equal to potential energy
plus kinetic energy.
185
00:09:15 --> 00:09:18
So if we say that l is just
talking about our kinetic
186
00:09:18 --> 00:09:21
energy part, our rotational
kinetic energy, and we know
187
00:09:21 --> 00:09:24
that electrons have potential
energy, then it makes
188
00:09:24 --> 00:09:27
sense that l, in fact, can
never go higher than n.
189
00:09:27 --> 00:09:30
And, in fact, it can't even
reach n, because then we would
190
00:09:30 --> 00:09:33
have no potential energy
at all in our electron,
191
00:09:33 --> 00:09:35
which is not correct.
192
00:09:35 --> 00:09:37
So, that's the second
quantum number.
193
00:09:37 --> 00:09:44
And the third one is called
m, it's also m sub l.
194
00:09:44 --> 00:09:50
This is what we call the
magnetic quantum number, and we
195
00:09:50 --> 00:09:52
won't deal with the fact of its
being the magnetic quantum
196
00:09:52 --> 00:09:55
number here -- that kind of
tells us the shape of the
197
00:09:55 --> 00:09:59
orbital or the way that the
electron will behave in a
198
00:09:59 --> 00:10:02
magnetic field, but what's more
relevant to thinking about the
199
00:10:02 --> 00:10:06
limits of this number is that
it's also the z component
200
00:10:06 --> 00:10:07
of the angular momentum.
201
00:10:07 --> 00:10:11
So since it's a component of
the angular momentum, that
202
00:10:11 --> 00:10:14
means that it's never going to
be able to go higher than l is,
203
00:10:14 --> 00:10:17
so it makes sense that, for
example, it could start at
204
00:10:17 --> 00:10:20
0 and then go all
the way up to l.
205
00:10:20 --> 00:10:23
But since it is a component it
can have a direction, too, so
206
00:10:23 --> 00:10:26
can go up between negative
l and positive l.
207
00:10:26 --> 00:10:31
So the allowed values for m sub
l are going to be negative l,
208
00:10:31 --> 00:10:35
all the way up to 0, and
then up to positive l.
209
00:10:35 --> 00:10:40
So, if we think of just an
example, we could say that 4 l
210
00:10:40 --> 00:10:45
equals 2, what would be our
lowest value of m sub l?
211
00:10:45 --> 00:10:46
Yup.
212
00:10:46 --> 00:10:54
So m sub l could equal negative
2, negative 1, 0, 1 or 2.
213
00:10:54 --> 00:11:00
So we could have five
different values of m sub l.
214
00:11:00 --> 00:11:02
So, those are our three
quantum numbers.
215
00:11:02 --> 00:11:06
So if, in fact, we want to
describe a wave function, we
216
00:11:06 --> 00:11:09
know that we need to describe
it in terms of all three
217
00:11:09 --> 00:11:13
quantum numbers, and also as a
function of our three
218
00:11:13 --> 00:11:18
positional factors, which are
r, the radius, plus the two
219
00:11:18 --> 00:11:20
angles, theta and phi.
220
00:11:20 --> 00:11:23
So, we have now a complete
description of a wave function
221
00:11:23 --> 00:11:24
that we can talk about.
222
00:11:24 --> 00:11:26
So, we can think about what is
it that we would call the
223
00:11:26 --> 00:11:28
ground state wave function.
224
00:11:28 --> 00:11:32
We knew from Friday, when we
talked about energy, that
225
00:11:32 --> 00:11:35
ground state was that n equals
1 value, that was the lowest
226
00:11:35 --> 00:11:37
energy, that was the most
stable place for the
227
00:11:37 --> 00:11:39
electron to be.
228
00:11:39 --> 00:11:42
But now we need to talk
about l and m as well.
229
00:11:42 --> 00:11:45
So now when we talk about a
ground state in terms of wave
230
00:11:45 --> 00:11:50
function, we need to talk about
the wave function of 1, 0, 0,
231
00:11:50 --> 00:11:54
and again, as a function
of r, theta and phi.
232
00:11:54 --> 00:11:57
So this is our complete
description of the ground
233
00:11:57 --> 00:12:00
state wave function.
234
00:12:00 --> 00:12:04
So, a lot of you talked about
different types of orbitals in
235
00:12:04 --> 00:12:08
high school, I'm sure, or in
previous courses, and it might
236
00:12:08 --> 00:12:10
be less common that you
actually talked about a
237
00:12:10 --> 00:12:12
wave function that was
labeled like this.
238
00:12:12 --> 00:12:16
We're used to labelling
orbitals as an s, or a p, or a
239
00:12:16 --> 00:12:19
d, for example, but it turns
out that these correlate to
240
00:12:19 --> 00:12:22
those letters that we're
more used to seeing.
241
00:12:22 --> 00:12:28
Does anyone know what the 1,
0, 0 orbital is also called?
242
00:12:28 --> 00:12:28
Yeah.
243
00:12:28 --> 00:12:31
And specfically it's the
1 s, so not just the s,
244
00:12:31 --> 00:12:33
but the 1 s orbital.
245
00:12:33 --> 00:12:35
So, using the terminology of
chemists, which is a good thing
246
00:12:35 --> 00:12:38
to do, because in this course
we are all chemists, we want to
247
00:12:38 --> 00:12:42
make sure that we're not using
just the physical description
248
00:12:42 --> 00:12:44
of the numbers, but that we can
correlate it to what we
249
00:12:44 --> 00:12:48
understand as orbitals, and
instead of 1, 0, 0, we call
250
00:12:48 --> 00:12:49
this the 1 s orbital.
251
00:12:49 --> 00:12:52
The reason that we do this is
because this is another way
252
00:12:52 --> 00:12:53
to completely describe it.
253
00:12:53 --> 00:12:57
The n designates the shell, so
that's what this number is
254
00:12:57 --> 00:13:00
here, we're in the first shell.
255
00:13:00 --> 00:13:03
The l is what we
call the sub shell.
256
00:13:03 --> 00:13:06
And instead of having
a 0 there, what we
257
00:13:06 --> 00:13:08
have here is an s.
258
00:13:08 --> 00:13:10
So, if we look at what the
other sub shells are
259
00:13:10 --> 00:13:12
called, essentially we're
just converting the
260
00:13:12 --> 00:13:14
number to a letter.
261
00:13:14 --> 00:13:19
L equals 0 is s,
what is l equals 1?
262
00:13:19 --> 00:13:20
Um-hmm, it's the p.
263
00:13:20 --> 00:13:25
What about 2? d, and 3?
264
00:13:25 --> 00:13:27
Yup, so 3 is f.
265
00:13:27 --> 00:13:30
So these names, they don't
really make any sense if we're
266
00:13:30 --> 00:13:32
looking at them why they're
called past s p and f, and it
267
00:13:32 --> 00:13:36
turns out that it comes from
spectroscopy terms that are
268
00:13:36 --> 00:13:39
pre-quantum mechanics where,
for example, this is called the
269
00:13:39 --> 00:13:42
sharp line, I think the
principle, the diffuse,
270
00:13:42 --> 00:13:43
and the fundamental.
271
00:13:43 --> 00:13:47
It doesn't even make sense
now, they're not used in
272
00:13:47 --> 00:13:49
spectroscopy anymore, but this
is where the names originally
273
00:13:49 --> 00:13:51
came from and they did stick.
274
00:13:51 --> 00:13:56
So, we being chemists, we'll
call that 1 s instead of 1, 0.
275
00:13:56 --> 00:14:00
In addition to having another
name to denote l, we also have
276
00:14:00 --> 00:14:04
another name for the
m designation here.
277
00:14:04 --> 00:14:13
So, for example, when l is
equal to 0, we're going to find
278
00:14:13 --> 00:14:22
that we have to call -- we have
to specify what m is as well.
279
00:14:22 --> 00:14:23
All right.
280
00:14:23 --> 00:14:27
So, when we have, for example,
l equal to 1, what kind
281
00:14:27 --> 00:14:30
of orbital is this?
282
00:14:30 --> 00:14:31
The p orbital.
283
00:14:31 --> 00:14:34
And for example, we could
also in this case,
284
00:14:34 --> 00:14:36
have m is equal to 0.
285
00:14:36 --> 00:14:39
If m is equal to 0, in this
case we would call it the
286
00:14:39 --> 00:14:44
p z orbital, so we would
have the subscript z here.
287
00:14:44 --> 00:14:50
Similarly, if m is equal to
either plus 1 or minus 1, we
288
00:14:50 --> 00:14:57
would in turn call it the p y
orbital, or the p x orbital.
289
00:14:57 --> 00:15:01
So you should know that any
time m is equal to zero when
290
00:15:01 --> 00:15:04
we are talking about p
orbitals, that it's the p z.
291
00:15:04 --> 00:15:06
The p y and the p x are
actually a bit more
292
00:15:06 --> 00:15:10
complicated, they're linear
combinations of the m plus 1,
293
00:15:10 --> 00:15:13
and the m minus 1 orbital,
where 1 is the positive linear
294
00:15:13 --> 00:15:16
combination, and 1 is the
negative linear combination.
295
00:15:16 --> 00:15:18
You're not responsible for
that, you're not responsible
296
00:15:18 --> 00:15:21
for correlating plus
1 to y, minus 1 to x.
297
00:15:21 --> 00:15:25
Just know that you have plus or
minus 1, for our class, you can
298
00:15:25 --> 00:15:29
call it either x or y, either
is fine, because it's a little
299
00:15:29 --> 00:15:33
bit more complicated than just
the 1:1 translation between,
300
00:15:33 --> 00:15:38
for example, m equals 0
and having a p z orbital.
301
00:15:38 --> 00:15:41
All right.
302
00:15:41 --> 00:15:44
So let's look at some of these
wave functions and make sure
303
00:15:44 --> 00:15:47
that we know how to name all of
them in terms of orbitals and
304
00:15:47 --> 00:15:48
not just in terms
of their numbers.
305
00:15:48 --> 00:15:52
Once we can do that we can go
on and say okay, what actually
306
00:15:52 --> 00:15:54
is a wave function, but first
we need to know how to describe
307
00:15:54 --> 00:15:56
which ones were talking about.
308
00:15:56 --> 00:15:59
So we saw that our lowest,
our ground state wave
309
00:15:59 --> 00:16:01
function is 1, 0, 0.
310
00:16:01 --> 00:16:04
We can call that psi 1,
0, 0 is how we write
311
00:16:04 --> 00:16:07
it as a wave function.
312
00:16:07 --> 00:16:10
We said that's the 1 s orbital.
313
00:16:10 --> 00:16:14
We also know how to figure out
the energy of this orbital, and
314
00:16:14 --> 00:16:17
we know how to figure out the
energy using this formula here,
315
00:16:17 --> 00:16:21
which was the binding energy,
which is negative r h, and
316
00:16:21 --> 00:16:25
instead of n, we can plug it in
because n equals 1, so over 1
317
00:16:25 --> 00:16:28
squared, and the actual
energy is here.
318
00:16:28 --> 00:16:33
So, our next level up that we
can go is going to be the n
319
00:16:33 --> 00:16:38
equals 2 energy level, but we
also have an l and an m
320
00:16:38 --> 00:16:42
value, so our lowest l is
going to be a 0 there.
321
00:16:42 --> 00:16:46
So we'll call that psi
2, 0, 0 wave function.
322
00:16:46 --> 00:16:50
What will we call that
in terms of orbitals?
323
00:16:50 --> 00:16:52
Yup, so that's the 2 s orbital.
324
00:16:52 --> 00:16:55
So something I actually wanted
to point out that I forgot to
325
00:16:55 --> 00:16:58
here is you'll notice that
there's no subscript to the s.
326
00:16:58 --> 00:17:02
We said we have a subscript
to the p, for example, that
327
00:17:02 --> 00:17:04
describes what m is equal to.
328
00:17:04 --> 00:17:07
The reason that we have no
subscript to the s, is because
329
00:17:07 --> 00:17:10
the only possibility for m
when you have an s orbital is
330
00:17:10 --> 00:17:12
that m has to be equal to 0.
331
00:17:12 --> 00:17:15
So we just assume it, you don't
actually have to write it
332
00:17:15 --> 00:17:18
because there is, in fact,
only one possibility.
333
00:17:18 --> 00:17:21
We can also figure out the
energy of this orbital here,
334
00:17:21 --> 00:17:23
and the energy is equal
to the Rydberg constant.
335
00:17:23 --> 00:17:26
The negative of the Rydberg
constant now divided
336
00:17:26 --> 00:17:28
by 2 squared.
337
00:17:28 --> 00:17:30
So we can go on and do this
for any orbital or any state
338
00:17:30 --> 00:17:32
function that we would like to.
339
00:17:32 --> 00:17:36
So, for example, if we talk
about the 2, 1, 1 state label,
340
00:17:36 --> 00:17:39
that's just psi 2, 1, 1.
341
00:17:39 --> 00:17:40
What, in this case,
would be our orbital?
342
00:17:40 --> 00:17:44
2 p what?
343
00:17:44 --> 00:17:49
OK, good, I heard mixed
answers, which is correct.
344
00:17:49 --> 00:17:53
So you can either write 2
p x or 2 p y, whichever
345
00:17:53 --> 00:17:55
one you want is fine.
346
00:17:55 --> 00:17:58
And again, you'll notice that
our energy is absolutely the
347
00:17:58 --> 00:18:02
same for an electron in that 2
p x orbital and in
348
00:18:02 --> 00:18:04
the 2 s orbital.
349
00:18:04 --> 00:18:06
So that's true for a hydrogen
atom, it doesn't matter if
350
00:18:06 --> 00:18:10
you're in a p or an s orbital,
their energies are the same.
351
00:18:10 --> 00:18:13
Then we can also talk about
the 2, 1, 0 state function,
352
00:18:13 --> 00:18:15
which would be psi 2, 1, 0.
353
00:18:15 --> 00:18:16
What is this orbital?
354
00:18:16 --> 00:18:18
Yup.
355
00:18:18 --> 00:18:22
And there's only one correct
answer here, which is to 2 p z.
356
00:18:22 --> 00:18:26
Is the energy going to be the
same or different as up here?
357
00:18:26 --> 00:18:28
It's going to be
the same energy.
358
00:18:28 --> 00:18:31
Again, the reason for that is
because the energy only depends
359
00:18:31 --> 00:18:35
on the n value here, it
doesn't depend on l or on m.
360
00:18:35 --> 00:18:38
So finally, if we talk about
our last example of when n is
361
00:18:38 --> 00:18:42
going to be equal 2, we can
have 2, 1 for l and
362
00:18:42 --> 00:18:43
then minus 1 for m.
363
00:18:43 --> 00:18:47
We can re-write this as
psi 2 1 negative 1.
364
00:18:47 --> 00:18:50
And then our orbital is going
to be just the opposite of
365
00:18:50 --> 00:18:52
whatever we said
it was up here.
366
00:18:52 --> 00:18:56
So if you said 2 p x the first
time, say 2 p y this time.
367
00:18:56 --> 00:19:00
And again, our energy is going
to be the same where we again
368
00:19:00 --> 00:19:03
only depend on the n value.
369
00:19:03 --> 00:19:03
All right.
370
00:19:03 --> 00:19:06
So hopefully we're pretty
comfortable naming any type
371
00:19:06 --> 00:19:09
of wave function using
the chemist terminology.
372
00:19:09 --> 00:19:11
Let's switch to a clicker
question and just confirm
373
00:19:11 --> 00:19:13
that that is, in fact, true.
374
00:19:13 --> 00:19:18
So what's the corresponding
orbital if we talk about
375
00:19:18 --> 00:19:27
this state, 5, 1, 0?
376
00:19:27 --> 00:19:42
And you can go ahead and
give 10 seconds on that.
377
00:19:42 --> 00:19:44
OK.
378
00:19:44 --> 00:19:45
All right, 77%.
379
00:19:45 --> 00:19:49
So, that's OK, you don't have
to memorize things as I speak,
380
00:19:49 --> 00:19:52
you just need to go back and
look at this and make sure you
381
00:19:52 --> 00:19:54
understand how to name it and
that you'll be able to, for
382
00:19:54 --> 00:19:58
example, by next class, get a
similar clicker question
383
00:19:58 --> 00:20:01
correct, and good job to
the 77% that did get it.
384
00:20:01 --> 00:20:04
So I think we're safe
to move on here.
385
00:20:04 --> 00:20:06
And I just want to point
out that now we have these
386
00:20:06 --> 00:20:08
three quantum numbers.
387
00:20:08 --> 00:20:09
The reason there are three
quantum numbers is we're
388
00:20:09 --> 00:20:13
describing an orbital in three
dimensions, so it makes sense
389
00:20:13 --> 00:20:15
that we would need to describe
in terms of three different
390
00:20:15 --> 00:20:16
quantum numbers.
391
00:20:16 --> 00:20:20
And the complete description,
as I said, is from n l and m.
392
00:20:20 --> 00:20:24
And when you talk about n for
an orbital, it's talking about
393
00:20:24 --> 00:20:27
the shell -- that shell is kind
of what you picture when you
394
00:20:27 --> 00:20:30
think of a classical picture of
an atom where you have 1 energy
395
00:20:30 --> 00:20:33
level, the next one is further
out, the next one's
396
00:20:33 --> 00:20:34
further away.
397
00:20:34 --> 00:20:37
That's kind of your shell
that we're discussing.
398
00:20:37 --> 00:20:43
L is the sub shell here, and
then we have m, which is
399
00:20:43 --> 00:20:47
finally the complete
description of the orbital.
400
00:20:47 --> 00:20:52
And what you can see is that
for any n that has an l equals
401
00:20:52 --> 00:20:54
0, you can see here how there's
only one possibility for and
402
00:20:54 --> 00:20:58
orbital description, and that's
why we don't need to include
403
00:20:58 --> 00:21:01
the m when we're talking
about and s orbital.
404
00:21:01 --> 00:21:04
The other thing that we know,
which is what we were just
405
00:21:04 --> 00:21:06
discussing when we were going
through the table is how
406
00:21:06 --> 00:21:08
this all relates to energy.
407
00:21:08 --> 00:21:11
And I want to really highlight
here we're talking about for a
408
00:21:11 --> 00:21:15
hydrogen atom -- orbitals with
the same n value have
409
00:21:15 --> 00:21:16
the same energy.
410
00:21:16 --> 00:21:19
Some of you might be saying in
your heads, wait a second, I
411
00:21:19 --> 00:21:23
happen to know, I happen to
remember from high school, that
412
00:21:23 --> 00:21:26
p orbitals have different
energies then, for
413
00:21:26 --> 00:21:27
example, s orbitals.
414
00:21:27 --> 00:21:30
And that is not true for
one electron atoms.
415
00:21:30 --> 00:21:33
We're going to get to more
complicated atoms eventually
416
00:21:33 --> 00:21:36
where we're going to have more
than one electron in it, but
417
00:21:36 --> 00:21:39
when we're talking about a
single electron atom, we know
418
00:21:39 --> 00:21:42
that the binding energy is
equal to the negative of the
419
00:21:42 --> 00:21:44
Rydberg constant over
n squared, so it's
420
00:21:44 --> 00:21:46
only depends on n.
421
00:21:46 --> 00:21:49
So, for example, if we're
talking about the n equals 2
422
00:21:49 --> 00:21:53
state, all of these four
orbital descriptions are going
423
00:21:53 --> 00:21:54
to have the same energy.
424
00:21:54 --> 00:21:58
And we can generalize to figure
out, based on any principle
425
00:21:58 --> 00:22:02
quantum number n, how many
orbitals we have of the same
426
00:22:02 --> 00:22:08
energy, and what we can say is
that for any shell n, there are
427
00:22:08 --> 00:22:09
n squared degenerate orbitals.
428
00:22:09 --> 00:22:15
And the word degenerate simply
means same energy, so you have
429
00:22:15 --> 00:22:18
n squared orbitals that are
of equal energy when
430
00:22:18 --> 00:22:21
they're degenerate.
431
00:22:21 --> 00:22:24
So, let's look at where this
comes from with an energy
432
00:22:24 --> 00:22:27
level diagram here.
433
00:22:27 --> 00:22:35
So what you can see is again,
we've got this ground state.
434
00:22:35 --> 00:22:38
So if we go to the ground
state, what you see is we're at
435
00:22:38 --> 00:22:41
that lowest energy level, and
we only have one possibility
436
00:22:41 --> 00:22:45
for an orbital, because when n
equals 1, that's all we can do.
437
00:22:45 --> 00:22:49
So that's the 1 s orbital
-- we have n squared or
438
00:22:49 --> 00:22:51
1 degenerate orbitals.
439
00:22:51 --> 00:22:57
When we talk about the n equals
2 state, we now have 2 squared
440
00:22:57 --> 00:23:01
or 4 degenerate same energy
orbitals, and those
441
00:23:01 --> 00:23:03
are the 2 s orbital.
442
00:23:03 --> 00:23:08
And then we also have the l
being equal to 1 orbital, so
443
00:23:08 --> 00:23:12
those are going to be the 2
p x, the 2 p z, and
444
00:23:12 --> 00:23:13
the 2 p y orbital.
445
00:23:13 --> 00:23:16
All four of these orbitals
have the same energy,
446
00:23:16 --> 00:23:18
they're degenerate.
447
00:23:18 --> 00:23:22
And as we go up the next energy
level, which is based on n
448
00:23:22 --> 00:23:25
equals 3 principle quantum
number, well now we have again
449
00:23:25 --> 00:23:29
the s, so we have the 3 s
orbital, we're going to have
450
00:23:29 --> 00:23:35
three 3 p orbitals, right, so
we'll have 3 p x, 3 p z, and 3
451
00:23:35 --> 00:23:38
p y, and now we're actually
also going to have five
452
00:23:38 --> 00:23:41
different possible l
equals 2 orbitals.
453
00:23:41 --> 00:23:44
Does anyone remember
the l equals 2?
454
00:23:44 --> 00:23:46
Yes, everyone remembers.
455
00:23:46 --> 00:23:46
Good.
456
00:23:46 --> 00:23:49
So we have five
possible d orbitals.
457
00:23:49 --> 00:23:56
We'll call these here the 3 d x
y, as the subscript, the 3 d y
458
00:23:56 --> 00:24:03
z, the 3 d z squared, the 3 d x
z, and the 3 d x squared
459
00:24:03 --> 00:24:05
minus y squared.
460
00:24:05 --> 00:24:10
So, what do you
need to know here?
461
00:24:10 --> 00:24:13
What you need to know is
that when m equals 0,
462
00:24:13 --> 00:24:16
it's 3 d z squared.
463
00:24:16 --> 00:24:17
That's it.
464
00:24:17 --> 00:24:22
Again, these other p -- or the
d x y, d y z, those are going
465
00:24:22 --> 00:24:25
to be those more complicated
linear combinations, you don't
466
00:24:25 --> 00:24:26
need to worry about them.
467
00:24:26 --> 00:24:29
Eventually you will, at least,
need to know the labels and
468
00:24:29 --> 00:24:30
know a little bit
more about them.
469
00:24:30 --> 00:24:33
And in the second half of this
course, Professor Drennen's
470
00:24:33 --> 00:24:36
going to talk to us about
transition metals in depth, and
471
00:24:36 --> 00:24:38
that's when we'll really
delve into d orbitals.
472
00:24:38 --> 00:24:41
For right now, you can kind
of put the d orbitals in
473
00:24:41 --> 00:24:41
the back of your head.
474
00:24:41 --> 00:24:44
You need to know how to think
about them in the same way we
475
00:24:44 --> 00:24:47
think about s and p orbitals,
but for example, you don't yet
476
00:24:47 --> 00:24:50
need to know what all of the
names are except for this
477
00:24:50 --> 00:24:53
3 d z squared here.
478
00:24:53 --> 00:24:56
So we'll wait on that until we
start talking more specifically
479
00:24:56 --> 00:25:00
about atoms where the d orbital
becomes very significant.
480
00:25:00 --> 00:25:03
So, what we can see
is this degeneracy.
481
00:25:03 --> 00:25:07
So what we know now is we can
start thinking about the next
482
00:25:07 --> 00:25:10
step because we can fully
describe the energy of
483
00:25:10 --> 00:25:13
orbitals, and we can fully
describe a complete orbital in
484
00:25:13 --> 00:25:16
terms of its three quantum
numbers, and its three
485
00:25:16 --> 00:25:19
positional variables,
r, theta, and phi.
486
00:25:19 --> 00:25:23
So next we can think about
okay, what is actually a wave
487
00:25:23 --> 00:25:26
function, and for example, what
might the shape of different
488
00:25:26 --> 00:25:29
wave functions be.
489
00:25:29 --> 00:25:31
So essentially, what we're
asking for here is the physical
490
00:25:31 --> 00:25:36
interpretation of psi, of the
value of psi for an electron.
491
00:25:36 --> 00:25:39
And it turns out that the
answer to can we have this
492
00:25:39 --> 00:25:41
physical interpretation of
thinking about what psi
493
00:25:41 --> 00:25:45
means, the answer is
really no, that we can't.
494
00:25:45 --> 00:25:47
There's no classical way
to think about what
495
00:25:47 --> 00:25:49
a wave function is.
496
00:25:49 --> 00:25:52
There's no classical analogy
that explains oh, this is what
497
00:25:52 --> 00:25:56
you can kind of picture when
you picture a wave function.
498
00:25:56 --> 00:25:57
And that's somewhat
inconvenient because we're
499
00:25:57 --> 00:26:00
working with wave functions,
but it's a reality that comes
500
00:26:00 --> 00:26:04
out of quantum mechanics often,
which is that we're describing
501
00:26:04 --> 00:26:07
a world that is so much
different from the world that
502
00:26:07 --> 00:26:09
we observe on a day-to-day
basis, that we're not always
503
00:26:09 --> 00:26:12
going to be able to make those
one-to-one analogies.
504
00:26:12 --> 00:26:14
But luckily we don't have to
worry about how we're going to
505
00:26:14 --> 00:26:17
picture all this, now that I
said that, because even though
506
00:26:17 --> 00:26:19
there's no physical
interpretation for what a wave
507
00:26:19 --> 00:26:22
function is, there is a
physical interpretation for
508
00:26:22 --> 00:26:25
what a wave function
squared means.
509
00:26:25 --> 00:26:29
So when we talk about a wave
function squared, we're taking
510
00:26:29 --> 00:26:32
the square of the wave
function, any one that we
511
00:26:32 --> 00:26:36
specify between n, l and m, at
any position that we specify
512
00:26:36 --> 00:26:38
based on r, theta, and phi.
513
00:26:38 --> 00:26:42
And if we go ahead and square
that, then what we get is a
514
00:26:42 --> 00:26:44
probability density, and
specifically it's the
515
00:26:44 --> 00:26:49
probability of finding an
electron in a certain small
516
00:26:49 --> 00:26:52
defined volume away
from the nucleus.
517
00:26:52 --> 00:26:54
So it's a probability density.
518
00:26:54 --> 00:26:57
The important point here is
it's not just a probability,
519
00:26:57 --> 00:26:59
it's a density, so we know
that it's a probability
520
00:26:59 --> 00:27:01
divided by volume.
521
00:27:01 --> 00:27:04
And the person we have to thank
for actually giving us this
522
00:27:04 --> 00:27:07
more concrete way to think
about what a wave function
523
00:27:07 --> 00:27:10
squared is is Max Born here.
524
00:27:10 --> 00:27:14
And actually after the
Schrodinger equation first was
525
00:27:14 --> 00:27:18
put forth, people had a lot of
discussions about how is it
526
00:27:18 --> 00:27:21
that we can actually interpret
what this wave function means,
527
00:27:21 --> 00:27:24
and a lot of ideas were put
forth, and none of them worked
528
00:27:24 --> 00:27:27
out to match up with
observations until Max Born
529
00:27:27 --> 00:27:29
here came up with the idea that
we just square the wave
530
00:27:29 --> 00:27:32
function, and that's the
probability density of finding
531
00:27:32 --> 00:27:36
an electron in a certain
defined volume.
532
00:27:36 --> 00:27:38
And it's very helpful
because it gives us a
533
00:27:38 --> 00:27:39
way to think about it.
534
00:27:39 --> 00:27:42
We can't actually go ahead and
derive this equation of the
535
00:27:42 --> 00:27:45
wave function squared, because
no one ever derived it, it's
536
00:27:45 --> 00:27:47
just an interpretation, but
it's an interpretation that
537
00:27:47 --> 00:27:50
works essentially perfectly.
538
00:27:50 --> 00:27:52
Ever since this was first
proposed, there has never been
539
00:27:52 --> 00:27:56
any observations that do not
coincide with the idea, that
540
00:27:56 --> 00:28:00
did not match the fact that the
probability density is equal to
541
00:28:00 --> 00:28:02
the wave function squared.
542
00:28:02 --> 00:28:06
So, also about Max Born, just
to give you a little bit of a
543
00:28:06 --> 00:28:09
trivial pursuit type knowledge,
he not only gave us this
544
00:28:09 --> 00:28:12
relationship between wave
function squared, he also
545
00:28:12 --> 00:28:15
gave us Olivia Newton-John.
546
00:28:15 --> 00:28:17
This is her grandfather, I
don't know if you can see
547
00:28:17 --> 00:28:19
from the eyes, I feel like
there's a little bit of
548
00:28:19 --> 00:28:22
a resemblance there.
549
00:28:22 --> 00:28:25
So, I don't know what she grew
up hearing about when she went
550
00:28:25 --> 00:28:27
to her grandparents' house,
but it might have been
551
00:28:27 --> 00:28:29
wave function squared.
552
00:28:29 --> 00:28:33
So, a little tidbit of
knowledge for you that's
553
00:28:33 --> 00:28:36
somewhat trivial.
554
00:28:36 --> 00:28:40
Then back to the non-trivial
knowledge that is not trivial
555
00:28:40 --> 00:28:43
at all, in fact, is OK, how do
we think about this probability
556
00:28:43 --> 00:28:46
density now that we have a
little bit more of an idea.
557
00:28:46 --> 00:28:50
We know that it's a density,
it's not an actual probability.
558
00:28:50 --> 00:28:54
So, one way we could look at it
is by looking at this density
559
00:28:54 --> 00:28:58
dot diagram, where the density
of the dots correlates to
560
00:28:58 --> 00:29:00
the probability density.
561
00:29:00 --> 00:29:05
So, what you see is near the
nucleus, the density is
562
00:29:05 --> 00:29:08
the strongest, the dots
are closest together.
563
00:29:08 --> 00:29:11
As you get far away from the
nucleus, the dots get farther
564
00:29:11 --> 00:29:15
and farther apart, meaning the
probability density at those
565
00:29:15 --> 00:29:19
volumes far away from the
nucleus is going to be quite
566
00:29:19 --> 00:29:22
low, eventually going to almost
zero, although it turns out
567
00:29:22 --> 00:29:25
that it never goes to exactly
zero, so if we're talking about
568
00:29:25 --> 00:29:29
any orbital or any atom, it
never actually ends, it
569
00:29:29 --> 00:29:30
never goes to zerio.
570
00:29:30 --> 00:29:33
But it turns out the
probability is only significant
571
00:29:33 --> 00:29:34
within one angstrom.
572
00:29:34 --> 00:29:37
So you can either say that
electrons are very, very tiny
573
00:29:37 --> 00:29:40
or that they're never ending,
and both are pretty accurate
574
00:29:40 --> 00:29:43
ways to think about
what an atom is.
575
00:29:43 --> 00:29:47
So, that's probability density,
but in terms of thinking about
576
00:29:47 --> 00:29:50
it in terms of actual solutions
to the wave function, let's
577
00:29:50 --> 00:29:52
take a little bit of
a step back here.
578
00:29:52 --> 00:29:55
I have yet to show you the
solution to a wave function for
579
00:29:55 --> 00:29:59
the hydrogen atom, so let me do
that here, and then we'll build
580
00:29:59 --> 00:30:02
back up to probability
densities, and it turns out
581
00:30:02 --> 00:30:05
that if we're talking about any
wave function, we can actually
582
00:30:05 --> 00:30:08
break it up into two
components, which are called
583
00:30:08 --> 00:30:12
the radial wave function and
angular wave function.
584
00:30:12 --> 00:30:15
So, essentially we're just
breaking it up into two parts
585
00:30:15 --> 00:30:18
that can be separated, and the
part that is only dealing with
586
00:30:18 --> 00:30:22
the radius, so it's only a
function of the radius of the
587
00:30:22 --> 00:30:24
electron from the nucleus.
588
00:30:24 --> 00:30:28
And we abbreviate that by
calling it r, which is
589
00:30:28 --> 00:30:31
specified by two quantum
numbers, and an l as a
590
00:30:31 --> 00:30:34
function of little r, radius.
591
00:30:34 --> 00:30:37
And we have the angular wave
function, which is specified by
592
00:30:37 --> 00:30:41
l and m, and it's a function of
the two angles when we're
593
00:30:41 --> 00:30:45
describing the position of the
electron, so theta and phi.
594
00:30:45 --> 00:30:49
So, let's look at what this
actually is for what we're
595
00:30:49 --> 00:30:51
showing here is the
1 s hydrogen atom.
596
00:30:51 --> 00:30:53
If you look in your book
there's a whole table of
597
00:30:53 --> 00:30:56
different solutions to the
Schrodinger equation for
598
00:30:56 --> 00:30:58
several different
wave functions.
599
00:30:58 --> 00:31:01
So this is the 1 s, you can
look it up if you're interested
600
00:31:01 --> 00:31:05
for the 2 s, or 3 s, or 5 s, or
whatever you're curious about.
601
00:31:05 --> 00:31:08
But what I'm going to show you
here is the 1 s solution.
602
00:31:08 --> 00:31:11
So you can see there's this
radial part here, and you have
603
00:31:11 --> 00:31:14
the angular part, you can
combine the two parts to get
604
00:31:14 --> 00:31:16
the total wave function.
605
00:31:16 --> 00:31:20
And what you can see is we
have this new constant that
606
00:31:20 --> 00:31:22
we haven't seen before.
607
00:31:22 --> 00:31:26
So what do you see in
there that is new?
608
00:31:26 --> 00:31:26
Yeah.
609
00:31:26 --> 00:31:27
This a sub nought.
610
00:31:27 --> 00:31:30
That's a new constant
for us in this course.
611
00:31:30 --> 00:31:35
This is what's called the Bohr
radius, and we'll explain --
612
00:31:35 --> 00:31:39
hopefully we'll get to it today
where this Bohr radius name
613
00:31:39 --> 00:31:42
comes from, but for now what
you need to know is just that
614
00:31:42 --> 00:31:45
it's a constant, just treat it
like a constant, and it turns
615
00:31:45 --> 00:31:47
out to be equal to 52 .
616
00:31:47 --> 00:31:51
9 pekameters or about
1/2 an angstrom.
617
00:31:51 --> 00:31:54
The more important thing that I
want you to notice when you're
618
00:31:54 --> 00:32:02
looking at this wave equation
for a 1 s h atom, is the fact
619
00:32:02 --> 00:32:05
that if you look at the angular
component of the wave function,
620
00:32:05 --> 00:32:08
you'll notice that
it's a constant.
621
00:32:08 --> 00:32:12
It doesn't depend on theta,
it doesn't depend on phi.
622
00:32:12 --> 00:32:15
No matter where you specify
your electron is in terms of
623
00:32:15 --> 00:32:18
those two angles, it doesn't
matter the angular part of
624
00:32:18 --> 00:32:21
your wave function is
going to be the same.
625
00:32:21 --> 00:32:23
So, what does that mean for us?
626
00:32:23 --> 00:32:26
Well, essentially what that
tells is that these s orbitals
627
00:32:26 --> 00:32:28
are spherically symmetrical.
628
00:32:28 --> 00:32:29
That should make sense,
right, because they're
629
00:32:29 --> 00:32:31
only dependent on r.
630
00:32:31 --> 00:32:34
How far you are away from the
nucleus in terms of a radius,
631
00:32:34 --> 00:32:37
they don't depend at all on
those two angles, they're
632
00:32:37 --> 00:32:42
independent of theta and
they're independent of phi.
633
00:32:42 --> 00:32:45
So, what I'm showing in
this picture here is
634
00:32:45 --> 00:32:48
just an electron cloud
that you can see.
635
00:32:48 --> 00:32:51
Think of it as a
probability density plot.
636
00:32:51 --> 00:32:55
And what here is just a graph
of the 1 s wave function going
637
00:32:55 --> 00:32:59
across some radius defined this
way, and you can see that the
638
00:32:59 --> 00:33:03
probability -- well, this is
the wave function, so we would
639
00:33:03 --> 00:33:05
have to square it and think
about the probability.
640
00:33:05 --> 00:33:11
So this squared at the origin
is going to be a very high
641
00:33:11 --> 00:33:14
probability, and it decays off
as you get farther and farther
642
00:33:14 --> 00:33:18
away from the nucleus or from
the center, and that's
643
00:33:18 --> 00:33:21
independent of the angle.
644
00:33:21 --> 00:33:24
So, let's look at these
probability plots of different
645
00:33:24 --> 00:33:34
s orbitals here, and up top
here, we have the probability
646
00:33:34 --> 00:33:38
density plot and what you can
see is what I just said, a very
647
00:33:38 --> 00:33:41
high probability density in the
nucleus, decays as you go out.
648
00:33:41 --> 00:33:45
And what is plotted below is
the actual wave function, so
649
00:33:45 --> 00:33:48
you can see it starts very high
and then the decays down.
650
00:33:48 --> 00:33:51
More interesting is to look
at the 2 s wave function.
651
00:33:51 --> 00:33:55
So, if we look at the bottom
here and the actual plot of the
652
00:33:55 --> 00:33:58
wave function, we see it starts
high, very positive, and it
653
00:33:58 --> 00:34:01
goes down and it eventually
hits zero, and goes through
654
00:34:01 --> 00:34:05
zero and then becomes negative
and then never quite hits zero
655
00:34:05 --> 00:34:07
again, although it
approaches zero.
656
00:34:07 --> 00:34:10
So, at this place where it hits
zero, that means that the
657
00:34:10 --> 00:34:12
square of the wave function is
also going to be zero, right.
658
00:34:12 --> 00:34:17
So we can see if we look at the
probability density plot, we
659
00:34:17 --> 00:34:20
can see there's a place where
the probability density of
660
00:34:20 --> 00:34:22
finding an electron anywhere
there is actually
661
00:34:22 --> 00:34:25
going to be zero.
662
00:34:25 --> 00:34:29
So we can think of a third case
where we have the 3 s orbital,
663
00:34:29 --> 00:34:32
and in the 3 s orbital we see
something similar, we start
664
00:34:32 --> 00:34:35
high, we go through zero, where
there will now be zero
665
00:34:35 --> 00:34:38
probability density, as we
can see in the in the
666
00:34:38 --> 00:34:40
density plot graph.
667
00:34:40 --> 00:34:43
Then we go negative and we go
through zero again, which
668
00:34:43 --> 00:34:47
correlates to the second area
of zero, that shows up also in
669
00:34:47 --> 00:34:50
our probability density plot,
and then we're positive again
670
00:34:50 --> 00:34:55
and approach zero as we
go to infinity for r.
671
00:34:55 --> 00:34:59
So, what this means is that
when we're looking at an actual
672
00:34:59 --> 00:35:02
wave function, we're treating
it as a wave, right, so waves
673
00:35:02 --> 00:35:08
can have both magnitude, but
they can also have a direction,
674
00:35:08 --> 00:35:10
so they can be either
positive or negative.
675
00:35:10 --> 00:35:13
So, for example, if we were
looking at the actual wave
676
00:35:13 --> 00:35:15
function, we would say that
these parts here have a
677
00:35:15 --> 00:35:20
positive amplitude, and in here
we have a negative amplitude.
678
00:35:20 --> 00:35:23
And when we're looking at the
probability density graphs, it
679
00:35:23 --> 00:35:27
doesn't make a difference, it's
okay, It has no meaning for our
680
00:35:27 --> 00:35:29
actual plot there, because
we're squaring it, so it
681
00:35:29 --> 00:35:32
doesn't matter whether it's
negative or positive, all that
682
00:35:32 --> 00:35:34
matters is the magnitude.
683
00:35:34 --> 00:35:37
But when we're thinking about
actual wave behavior of
684
00:35:37 --> 00:35:39
electrons, it's just important
to keep in the back of our head
685
00:35:39 --> 00:35:42
that some areas have positive
amplitude and some
686
00:35:42 --> 00:35:43
have negative.
687
00:35:43 --> 00:35:45
So we'll talk about this
more we get into p orbitals
688
00:35:45 --> 00:35:48
and bonding is where it's
going to become an issue.
689
00:35:48 --> 00:35:50
So I just want to kind of
introduce that idea here.
690
00:35:50 --> 00:35:54
Because if we think about wave
behavior of electrons and we're
691
00:35:54 --> 00:35:57
forming bonds, then what we
have to do is have constructive
692
00:35:57 --> 00:36:00
interference of 2 different
electrons, right, to form a
693
00:36:00 --> 00:36:04
bond, we want to and together
those probabilities.
694
00:36:04 --> 00:36:07
So we want to have constructive
interference to form a bond,
695
00:36:07 --> 00:36:10
whereas if we had destructive
interference, we would
696
00:36:10 --> 00:36:12
not be forming a bond.
697
00:36:12 --> 00:36:14
So that's where you have to
think about whether it's
698
00:36:14 --> 00:36:15
positive or negative.
699
00:36:15 --> 00:36:17
You don't have to think about
it right now, but you might
700
00:36:17 --> 00:36:19
have heard in high school
talking about p orbitals, the
701
00:36:19 --> 00:36:23
phase, sometimes you mark a p
orbital as being a plus
702
00:36:23 --> 00:36:24
sign or negative sign.
703
00:36:24 --> 00:36:27
Did any of you do that
in high school at all?
704
00:36:27 --> 00:36:28
A little bit, yeah.
705
00:36:28 --> 00:36:31
So, that's having to do with
the actual wave function.
706
00:36:31 --> 00:36:34
So, that'll become more
relevant later, bonding
707
00:36:34 --> 00:36:36
actually, a couple
lectures down the road.
708
00:36:36 --> 00:36:39
But I just want to introduce it
here while we do, in fact, have
709
00:36:39 --> 00:36:42
the wave function
plots up here.
710
00:36:42 --> 00:36:45
But a real key in looking at
these plots is where we, in
711
00:36:45 --> 00:36:47
fact, did go through
zer and have this zero
712
00:36:47 --> 00:36:49
probability density.
713
00:36:49 --> 00:36:54
We call that a node, and a
node, more specifically, is any
714
00:36:54 --> 00:36:59
value of either r, the radius,
or the two angles for which the
715
00:36:59 --> 00:37:02
wave function, and that also
means the wave function squared
716
00:37:02 --> 00:37:06
or the probability density, is
going to be equal to zero.
717
00:37:06 --> 00:37:10
So, we can see in our
1 s orbital, how many
718
00:37:10 --> 00:37:12
nodes do we have?
719
00:37:12 --> 00:37:13
There's no nodes, yeah.
720
00:37:13 --> 00:37:16
It looks like we hit zero, but
we actually don't -- remember
721
00:37:16 --> 00:37:18
that we never go all the way to
zero, so there's these little
722
00:37:18 --> 00:37:22
points if we were to look
really carefully at an accurate
723
00:37:22 --> 00:37:26
probability density plot, it
would never actually hit zero.
724
00:37:26 --> 00:37:29
And then, for example, how
many nodes do we have in
725
00:37:29 --> 00:37:31
the 3 s orbital? two.
726
00:37:31 --> 00:37:32
That's correct.
727
00:37:32 --> 00:37:35
So we have two nodes
in the 3 s orbital.
728
00:37:35 --> 00:37:39
We can actually specify where
those nodes are, which is
729
00:37:39 --> 00:37:40
written on your notes.
730
00:37:40 --> 00:37:45
For the 2 s orbital, at 2 a
nought, so it's just 2 times
731
00:37:45 --> 00:37:49
that constant a nought,
which is the Bohr radius.
732
00:37:49 --> 00:37:51
And for the 3 s, we
have one at 1 .
733
00:37:51 --> 00:37:53
9 a nought, and one at 7 .
734
00:37:53 --> 00:37:55
1 a nought.
735
00:37:55 --> 00:37:58
We can also specify what kind
of node we're talking about.
736
00:37:58 --> 00:38:02
We'll introduce in the next
course angular nodes, but today
737
00:38:02 --> 00:38:05
we're just going to be talking
about radial nodes, and a
738
00:38:05 --> 00:38:10
radial node is a value for r at
which psi, and therefore, also
739
00:38:10 --> 00:38:14
the probability psi squared is
going to be equal to zero.
740
00:38:14 --> 00:38:18
So, when we're talking about an
s orbital, since there is no
741
00:38:18 --> 00:38:21
angular dependence, and it only
depends on r, every single one
742
00:38:21 --> 00:38:23
of our nodes is actually going
to specifically be a radial
743
00:38:23 --> 00:38:27
node, right, because these are,
for example, this 2 a nought is
744
00:38:27 --> 00:38:31
a value of r, a value of the
radius, no matter which way you
745
00:38:31 --> 00:38:34
go around at which there's
going to be a node at which
746
00:38:34 --> 00:38:37
there is zero probability
density of finding
747
00:38:37 --> 00:38:40
an electron there.
748
00:38:40 --> 00:38:42
So, it's very easy to
calculate, however, the number
749
00:38:42 --> 00:38:44
of radial nodes, and this works
not just for s orbitals, but
750
00:38:44 --> 00:38:47
also for p orbitals, or d
orbitals, or whatever kind of
751
00:38:47 --> 00:38:49
work of orbitals you
want to discuss.
752
00:38:49 --> 00:38:52
And that's just to take the
principle quantum number and
753
00:38:52 --> 00:38:56
subtract it by 1, and then
also subtract from that
754
00:38:56 --> 00:38:58
your l quantum number.
755
00:38:58 --> 00:39:03
So what you can do for a 1 s is
just take 1 minus 1 and then
756
00:39:03 --> 00:39:07
l is equal to 0, so you
have zero radial nodes.
757
00:39:07 --> 00:39:09
And that matches up
with what we saw.
758
00:39:09 --> 00:39:14
If we try this for the 2 s,
we have 2 minus 1 minus 0.
759
00:39:14 --> 00:39:17
So what we should expect to see
is one radial node, and that is
760
00:39:17 --> 00:39:21
what we see here in the
probability density plot.
761
00:39:21 --> 00:39:26
And then if we think about 3 s,
we want to start with 3, we
762
00:39:26 --> 00:39:30
subtract 1, again l is equal
to 0, so minus 0 and we
763
00:39:30 --> 00:39:34
have two radial nodes.
764
00:39:34 --> 00:39:36
So, this should be pretty
straight forward, let's see if
765
00:39:36 --> 00:39:39
we can get close to a 100% on
this one, which is how many
766
00:39:39 --> 00:39:49
radial nodes does a
4 p orbital have?
767
00:39:49 --> 00:40:04
And let's give 10 seconds on
that, make you think fast here.
768
00:40:04 --> 00:40:09
OK, so most people were
correct, or well, the majority,
769
00:40:09 --> 00:40:11
at least, were correct.
770
00:40:11 --> 00:40:15
And seeing that it's a 4 p has
two nodes -- let's just write
771
00:40:15 --> 00:40:18
this out since not everyone
did get it correct.
772
00:40:18 --> 00:40:21
So, if we're talking about a 4
p orbital, and our equation is
773
00:40:21 --> 00:40:28
n minus 1 minus l, the
principle quantum number is 4,
774
00:40:28 --> 00:40:31
1 is 1 -- what is l
for a p orbital?
775
00:40:31 --> 00:40:33
STUDENT: 1.
776
00:40:33 --> 00:40:34
PROFESSOR: 1.
777
00:40:34 --> 00:40:36
So, I tricked you a little, I
guess I didn't put an s up
778
00:40:36 --> 00:40:37
there and that's what we had
been talking about, so that
779
00:40:37 --> 00:40:39
was probably the issue.
780
00:40:39 --> 00:40:47
But what we find is that
we have two radial nodes.
781
00:40:47 --> 00:40:47
All right.
782
00:40:47 --> 00:40:50
So we can switch back
to our notes here.
783
00:40:50 --> 00:40:54
So, doing those probability
density dot graphs, we can get
784
00:40:54 --> 00:40:57
an idea of the shape of those
orbitals, we know that they're
785
00:40:57 --> 00:40:59
spherically symmetrical.
786
00:40:59 --> 00:41:02
We're not going to talk about p
orbitals today, we're going to
787
00:41:02 --> 00:41:05
talk about p orbitals
exclusively on Friday, and as I
788
00:41:05 --> 00:41:07
said, d orbitals you'll get to
with Professor Drennen.
789
00:41:07 --> 00:41:12
But we can also think when
we're talking about wave
790
00:41:12 --> 00:41:15
function squared, what we're
really talking about is the
791
00:41:15 --> 00:41:19
probability density, right, the
probability in some volume.
792
00:41:19 --> 00:41:21
But there's also a way to get
rid of the volume part and
793
00:41:21 --> 00:41:24
actually talk about the
probability of finding an
794
00:41:24 --> 00:41:30
electron at some certain area
within the atom, and this
795
00:41:30 --> 00:41:35
is what we do using
radial probability
796
00:41:35 --> 00:41:37
distribution graphs.
797
00:41:37 --> 00:41:40
And what that is the
probability of finding an
798
00:41:40 --> 00:41:44
electron in some shell where we
define the thickness as d r,
799
00:41:44 --> 00:41:47
some distance, r,
from the nucleus.
800
00:41:47 --> 00:41:49
So, think about what
we're saying here.
801
00:41:49 --> 00:41:52
We're saying the probability of
finding an electron at some
802
00:41:52 --> 00:41:56
distance from the nucleus in
some very thin shell that
803
00:41:56 --> 00:41:58
we describe by d r.
804
00:41:58 --> 00:42:00
So if you think of a shell, you
can actually just think of an
805
00:42:00 --> 00:42:03
egg shell, that's probably the
easiest way to think of it,
806
00:42:03 --> 00:42:05
where the yolk, if you really
maybe make it a lot smaller
807
00:42:05 --> 00:42:07
might be the nucleus.
808
00:42:07 --> 00:42:09
And let's also make our
egg perfectly symmetric
809
00:42:09 --> 00:42:11
and perfectly round.
810
00:42:11 --> 00:42:14
But still, when we're talking
about the radial probability
811
00:42:14 --> 00:42:18
distribution, what we actually
want to think about is what's
812
00:42:18 --> 00:42:22
the probability of finding
the electron in that shell?
813
00:42:22 --> 00:42:24
Think of it as that
egg shell part.
814
00:42:24 --> 00:42:28
So, we can do that by using
this equation, which is for s
815
00:42:28 --> 00:42:30
orbitals where the radial
probability distribution is
816
00:42:30 --> 00:42:34
going to be equal to 4 pi r
squared times the wave
817
00:42:34 --> 00:42:36
function squared, d r.
818
00:42:36 --> 00:42:44
That should make sense to us,
because when we talk about a
819
00:42:44 --> 00:42:52
wave function, we're talking
about a probability divided by
820
00:42:52 --> 00:42:55
a volume, because we're talking
about a probability density.
821
00:42:55 --> 00:43:00
So if we actually go ahead and
multiply it by the volume of
822
00:43:00 --> 00:43:03
our shell, then we end up just
with probability, which is kind
823
00:43:03 --> 00:43:06
of a nicer term to be
thinking about here.
824
00:43:06 --> 00:43:09
So, of course, if we're talking
about a perfectly spherical
825
00:43:09 --> 00:43:13
shell at some distance,
thickness, d r, we talk about
826
00:43:13 --> 00:43:17
it as 4 pi r squared d r, so we
just multiply that by the
827
00:43:17 --> 00:43:20
probability density.
828
00:43:20 --> 00:43:26
We can graph out what this is
where we're graphing the radial
829
00:43:26 --> 00:43:31
probability density as a
function of the radius.
830
00:43:31 --> 00:43:35
And what you see is that at
zero, you start at zero.
831
00:43:35 --> 00:43:39
And so, the radial probability
density at the nucleus is going
832
00:43:39 --> 00:43:43
to be zero, even though we know
the probability density at the
833
00:43:43 --> 00:43:47
nucleus is very high, that's
actually where is the highest.
834
00:43:47 --> 00:43:50
The reason in our radial
probability distributions we
835
00:43:50 --> 00:43:56
start -- the reason, if you
look at the zero point on the
836
00:43:56 --> 00:43:59
radius that we start at zero is
because we're multiplying the
837
00:43:59 --> 00:44:04
probability density by some
volume, and when we're not
838
00:44:04 --> 00:44:07
anywhere from the nucleus, that
volume is defined as zero.
839
00:44:07 --> 00:44:09
So, it's a little bit
artificial that we're seeing
840
00:44:09 --> 00:44:11
that zero point there.
841
00:44:11 --> 00:44:13
So, actually I want you to go
ahead in your notes and circle
842
00:44:13 --> 00:44:17
that zero point and write "not
a node." This is not a node
843
00:44:17 --> 00:44:19
because a node is where
we actually have no
844
00:44:19 --> 00:44:21
probability density.
845
00:44:21 --> 00:44:24
So this, where we start at
zero is not a node, is the
846
00:44:24 --> 00:44:27
first thing to point out.
847
00:44:27 --> 00:44:30
And as we get further and
further from the radius, the
848
00:44:30 --> 00:44:32
volume we're multiplying it by
actually gets bigger and
849
00:44:32 --> 00:44:35
bigger, because you can see how
the volume of that little thin
850
00:44:35 --> 00:44:37
shell is going to get larger
and larger as you
851
00:44:37 --> 00:44:39
get further away.
852
00:44:39 --> 00:44:43
So there's some distance where
the probability of actually
853
00:44:43 --> 00:44:45
finding an electron there
is going to be your
854
00:44:45 --> 00:44:46
maximum probability.
855
00:44:46 --> 00:44:50
And that's what we label
as r sub m p, or your
856
00:44:50 --> 00:44:52
most probable radius.
857
00:44:52 --> 00:44:55
This is the point at which
your probability is highest
858
00:44:55 --> 00:44:57
for finding an electron.
859
00:44:57 --> 00:45:01
This is equal to a sub nought
for a hydrogen atom, and we
860
00:45:01 --> 00:45:05
remember that that's just our
Bohr radius, which is 0 .
861
00:45:05 --> 00:45:08
5 2 9 angstroms.
862
00:45:08 --> 00:45:12
And basically, what that means
is you can actually find an
863
00:45:12 --> 00:45:16
electron anywhere going away
from the nucleus, but you're
864
00:45:16 --> 00:45:19
most likely to find that you
have the highest probability at
865
00:45:19 --> 00:45:23
a distance of a sub nought,
or the Bohr radius.
866
00:45:23 --> 00:45:25
So, I said I'd tell you a
little bit more about where
867
00:45:25 --> 00:45:30
this Bohr radius came from, and
it came from a model of the
868
00:45:30 --> 00:45:35
atom that pre-dated quantum
mechanics, and Neils Bohr is
869
00:45:35 --> 00:45:38
who came up with the idea of
the Bohr radius, and here is
870
00:45:38 --> 00:45:42
hanging out with Einstein, so
he had some pretty good
871
00:45:42 --> 00:45:44
company that he kept.
872
00:45:44 --> 00:45:48
And what you need to remember
when we're thinking about this
873
00:45:48 --> 00:45:51
model of the atom is that in
1911 it had already been
874
00:45:51 --> 00:45:53
discovered that we have an
electron, and we have a
875
00:45:53 --> 00:45:56
nucleus, and there needs to be
some way that those two hang
876
00:45:56 --> 00:45:59
together, but it was not for
another 15 years that we
877
00:45:59 --> 00:46:02
actually had the Schrodinger
equation that allowed us to
878
00:46:02 --> 00:46:06
understand the interaction
fully between the electron
879
00:46:06 --> 00:46:07
and the nucleus.
880
00:46:07 --> 00:46:10
So all that Bohr, for example,
had to go on at this point was
881
00:46:10 --> 00:46:14
a more classical picture of the
atom, as you can see on the
882
00:46:14 --> 00:46:18
left side of the screen there,
which is the idea that the
883
00:46:18 --> 00:46:20
electrons actually somehow
just orbiting the nucleus.
884
00:46:20 --> 00:46:24
And even though he could figure
out that this wasn't possible,
885
00:46:24 --> 00:46:27
he still used this as a
starting point, and what he did
886
00:46:27 --> 00:46:31
know was that these energy
levels that were within
887
00:46:31 --> 00:46:34
hydrogen atom were quantized.
and he knew this the same way
888
00:46:34 --> 00:46:37
that we saw it in the last
class, which is when we viewed
889
00:46:37 --> 00:46:40
the difference spectra coming
out from the hydrogen, and we
890
00:46:40 --> 00:46:43
also did it for neon, but we
saw in the hydrogen atom that
891
00:46:43 --> 00:46:46
it was very discreet energy
levels that we could observe.
892
00:46:46 --> 00:46:49
He knew the same thing that had
been observed by that point.
893
00:46:49 --> 00:46:53
So, what he did was kind of
impose a quantum mechanical
894
00:46:53 --> 00:46:56
model, not a full one, just the
idea that those energy levels
895
00:46:56 --> 00:47:00
were quantized on to the
classical picture of an atom
896
00:47:00 --> 00:47:02
that has a discreet orbit.
897
00:47:02 --> 00:47:06
And what he came out with when
he did some calculations is
898
00:47:06 --> 00:47:09
that there's the radius that he
could calculate was equal to
899
00:47:09 --> 00:47:13
this number a sub nought, which
is what we call the Bohr
900
00:47:13 --> 00:47:17
radius, and it turns out that
the Bohr radius happens to be
901
00:47:17 --> 00:47:21
the radius most probable
for a hydrogen atom.
902
00:47:21 --> 00:47:23
And the reason we won't talk
any more about this Bohr
903
00:47:23 --> 00:47:26
model is because, of
course, it's not correct.
904
00:47:26 --> 00:47:28
So we're not going to spend
too much time on it here.
905
00:47:28 --> 00:47:31
But we can see, for example,
one reason or one way in
906
00:47:31 --> 00:47:33
which is not correct.
907
00:47:33 --> 00:47:37
Because what it tells is that
we can figure out exactly what
908
00:47:37 --> 00:47:42
the radius of an electron and a
nucleus are in a hydrogen atom.
909
00:47:42 --> 00:47:45
That's a deterministic way of
doing things, that's what you
910
00:47:45 --> 00:47:46
get from classical mechanics.
911
00:47:46 --> 00:47:49
But the reality that we know
from our quantum mechanical
912
00:47:49 --> 00:47:53
model, is that we can't know
exactly what the radius is,
913
00:47:53 --> 00:47:56
all we can say is what the
probability is of the radius
914
00:47:56 --> 00:47:58
being at certain different
points. so, that's a more
915
00:47:58 --> 00:48:02
complete quantum mechanical
picture of what is
916
00:48:02 --> 00:48:03
going on here.
917
00:48:03 --> 00:48:07
So if we superimpose our radial
probability distribution onto
918
00:48:07 --> 00:48:10
the Bohr radius, we see it's
much more complicated than just
919
00:48:10 --> 00:48:11
having a discreet radius.
920
00:48:11 --> 00:48:14
We can actually have any
radius, but some radii just
921
00:48:14 --> 00:48:17
have much, much smaller
probabilities of actually
922
00:48:17 --> 00:48:20
being significant or not.
923
00:48:20 --> 00:48:24
So, I think we're a little bit
out of time today, but we'll
924
00:48:24 --> 00:48:26
start next class with thinking
about drawing radial
925
00:48:26 --> 00:48:31
probability distributions of
more than just the 1 s orbital.