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PROFESSOR: So you've
been learning about
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statistical mechanics.
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The microscopic underpinnings
of thermodynamics.
12
00:00:29 --> 00:00:40
And last time we ended up
working with the canonical
13
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partition function and showing
that once you have the
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canonical position function,
you have basically every
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thermodynamic quantity that
you've learned how to calculate
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this far in the course.
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But before I start, on the
lecture notes 25, there were a
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couple typos in there that
actually didn't make any
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00:01:00 --> 00:01:03
difference to the final result
because they cancel
20
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each other out.
21
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But it's been corrected
on the web version.
22
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And near the bottom of the page
-- because we're going to be
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mostly using these notes today.
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Near the bottom of the page,
where it says A is equal to
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u minus TS is equal to u.
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That should be a plus
here. dA/dT, volume.
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00:01:27 --> 00:01:31
28
00:01:31 --> 00:01:34
And then further down the next
line, blah blah blah blah blah.
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00:01:34 --> 00:01:36
Minus u over T squared.
30
00:01:36 --> 00:01:37
And this should
be a minus here.
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00:01:37 --> 00:01:40
One over T. dA/dT, V.
32
00:01:40 --> 00:01:44
33
00:01:44 --> 00:01:46
And et cetera.
34
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So luckily these two typos
cancel each other out
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00:01:48 --> 00:01:49
the result is correct.
36
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But there it is.
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OK.
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00:01:57 --> 00:02:03
So last time, then, you saw how
from the canonical partition
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function, you could get
something like the energy.
40
00:02:06 --> 00:02:07
You wrote down an equation.
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The energy is equal to k,
T squared, d log Q / dT.
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00:02:16 --> 00:02:19
Under constant volume.
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And number of particles.
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00:02:25 --> 00:02:32
And then you notice that the
important variables are the
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volume, the number of
particles, and the temperature.
46
00:02:37 --> 00:02:41
And we know that every
thermodynamic quantity has a
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set of natural variables.
48
00:02:46 --> 00:02:49
For the Gibbs free energy
it was the pressure and
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the temperature, the
number of particles.
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00:02:52 --> 00:02:55
And for the volume of the
number of particles and the
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temperature, we know that
that's the Helmholtz energy.
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So the natural variable that we
would associate -- the natural
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thermodynamic variable we would
associate with that set
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00:03:06 --> 00:03:08
of constraints is the
Helmholtz free energy.
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So it becomes interesting,
then, to figure out, how can we
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write the Helmholtz free energy
in terms of the canonical
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partition function?
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00:03:16 --> 00:03:21
They seem to have the same
set of natural variables.
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And that's what you started
doing and what we'll do today.
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00:03:24 --> 00:03:29
And that's where
the typo comes in.
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So let's write what we know
about the Helmholtz free energy
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in terms of the energy u.
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I already wrote it up
there. u minus TS.
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That's the definition of
the Helmholtz free energy.
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00:03:45 --> 00:03:52
And from the fact that dA
is equal to minus p dV,
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minus S dT, plus mu dN.
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We can read out what S,
here, is in terms of the
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Helmholtz free energy.
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S is just dA/dT,
constant V and N.
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So we can plug that in here. u
minus T dA, with a minus sign
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there, so that becomes a plus.
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And hence the typo. dT,
constant V, and N.
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OK.
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00:04:29 --> 00:04:32
So now we have an equation
that relates u and the
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partition function.
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We want an equation
that relates A and the
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partition function.
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If we rearrange this slightly,
we can get that u, then,
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is equal to A minus T,
dA/dT, constant V and N.
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00:04:53 --> 00:04:58
So the question that we could
ask ourselves is -- is there a
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function of A that kind
of looks like that?
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And I know the answer, so I'm
going to give it to you.
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00:05:05 --> 00:05:08
A function of A that looks like
the sum of A minus something
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times the derivative, with
respect to that something.
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00:05:11 --> 00:05:16
We should try to look at
something like the derivative
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d(A/T)/dT, constant V and N.
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00:05:21 --> 00:05:27
So if you take that derivative,
you end up with one over T,
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dA/dT, minus A over T squared.
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00:05:41 --> 00:05:45
But if you look at this result,
here, and you multiply by T
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squared, there's the A, and
there's the T times dA/dT.
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There's the A that -- we
just have the sign wrong.
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So we have -- multiply
this by minus T squared.
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Minus T squared, times
minus T squared.
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00:06:02 --> 00:06:06
And you have the
same thing as here.
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00:06:06 --> 00:06:14
So that tells us then, that u
can be written as minus T
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00:06:14 --> 00:06:22
squared, d(A/T)/dT,
constant volume.
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It's a nice way to relate
those two energies.
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00:06:27 --> 00:06:29
And we have an expression
for u in terms of the
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canonical function.
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00:06:32 --> 00:06:41
And we can then
replace it in here.
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00:06:41 --> 00:06:52
And that gets us, then, that
minus T squared, d(A/T)/dT,
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00:06:52 --> 00:06:55
constant number and
volume is equal to u.
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00:06:55 --> 00:07:02
And u, we saw, was equal to
k T squared d log Q / dT.
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00:07:02 --> 00:07:08
105
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V and N fixed.
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00:07:11 --> 00:07:18
And then we can start -- The
t squareds disappear here.
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00:07:18 --> 00:07:21
And then we have d/dT
on this side and d/dT.
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00:07:21 --> 00:07:22
Let's just take the integral.
109
00:07:22 --> 00:07:27
You take the integral of both
sides, and that gets us that
110
00:07:27 --> 00:07:40
then A over T is equal to k,
log Q, plus the constant
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00:07:40 --> 00:07:41
of integration.
112
00:07:41 --> 00:07:43
And we can take that constant
of integration to be
113
00:07:43 --> 00:07:45
whatever we want.
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00:07:45 --> 00:07:47
Energy is all relative to
some reference point.
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We can take it to be zero.
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00:07:52 --> 00:07:59
A is equal to k T log Q.
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00:07:59 --> 00:08:01
That's a pretty neat result.
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00:08:01 --> 00:08:05
There's the microscopic
underpinning of things.
119
00:08:05 --> 00:08:08
Where we know about atoms, and
energies, and states, and even
120
00:08:08 --> 00:08:11
quantum mechanics, and all
sorts of things goes into here.
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00:08:11 --> 00:08:14
All the microscopic
information goes in here.
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00:08:14 --> 00:08:17
And there's a thermodynamic
variable that only cares
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about the macroscopic
state of matter.
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00:08:18 --> 00:08:22
It doesn't care that
there are atoms there.
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00:08:22 --> 00:08:23
It just cares that you know the
pressure, the volume, the
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00:08:23 --> 00:08:25
temperature, or any
couple variables.
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00:08:25 --> 00:08:30
And you can get a direct
equality without any
128
00:08:30 --> 00:08:31
derivatives or anything.
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00:08:31 --> 00:08:34
Between the macroscopic
and the microscopic.
130
00:08:34 --> 00:08:41
So this is really
pretty remarkable.
131
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And once you have A,
you have everything.
132
00:08:43 --> 00:08:46
Just like before, we said once
you have G, that Gibbs free
133
00:08:46 --> 00:08:48
energy, when we were talking
about things that depend on
134
00:08:48 --> 00:08:49
pressure and temperature.
135
00:08:49 --> 00:08:50
You have everything.
136
00:08:50 --> 00:08:51
It's the same thing here.
137
00:08:51 --> 00:08:52
We've got A, we've got u.
138
00:08:52 --> 00:08:54
We have everything.
139
00:08:54 --> 00:08:56
We can calculate every single
thermodynamic variable
140
00:08:56 --> 00:08:58
from then on.
141
00:08:58 --> 00:09:03
For instance, if we want to
have the entropy, S is equal to
142
00:09:03 --> 00:09:07
minus A over T, minus u over T.
143
00:09:07 --> 00:09:08
Where did I get that?
144
00:09:08 --> 00:09:09
I got that from way up here.
145
00:09:09 --> 00:09:10
A is equal to u minus TS.
146
00:09:10 --> 00:09:13
I solved for S in
terms of A and u.
147
00:09:13 --> 00:09:15
I've got expressions in
terms of the canonical
148
00:09:15 --> 00:09:16
function for A and u.
149
00:09:16 --> 00:09:18
Plug that in there.
150
00:09:18 --> 00:09:31
Get k log Q, plus k T, d log
Q / dT, constant number
151
00:09:31 --> 00:09:35
and volume. et cetera.
152
00:09:35 --> 00:09:36
You can get the pressure.
153
00:09:36 --> 00:09:41
You can get the pressure from
the fact that the pressure up
154
00:09:41 --> 00:09:45
here is a derivative of A
with respect to volume.
155
00:09:45 --> 00:09:49
So you take the derivative of A
with respect to volume here.
156
00:09:49 --> 00:09:50
You get the pressure.
157
00:09:50 --> 00:09:52
If you want the
chemical potential.
158
00:09:52 --> 00:09:55
The chemical potential from the
fundamental equation up here is
159
00:09:55 --> 00:09:58
the derivative of A with
respect to the number
160
00:09:58 --> 00:09:59
of particles.
161
00:09:59 --> 00:10:01
You take the derivative of A
with respect to the number of
162
00:10:01 --> 00:10:04
particles, you get the chemical
potential in terms of the
163
00:10:04 --> 00:10:09
canonical partition function.
164
00:10:09 --> 00:10:10
So you've got everything.
165
00:10:10 --> 00:10:11
You've got u.
166
00:10:11 --> 00:10:11
You've got A.
167
00:10:11 --> 00:10:12
You've got p.
168
00:10:12 --> 00:10:13
You've got S.
169
00:10:13 --> 00:10:13
You've got H.
170
00:10:13 --> 00:10:15
You've got G.
171
00:10:15 --> 00:10:17
Name your valuable,
you've got it.
172
00:10:17 --> 00:10:18
You want the heat capacity?
173
00:10:18 --> 00:10:21
It can get you the heat
capacity in terms of the
174
00:10:21 --> 00:10:25
partition function.
175
00:10:25 --> 00:10:29
Any questions?
176
00:10:29 --> 00:10:34
Alright so let's go on.
177
00:10:34 --> 00:10:39
Let's look a little bit closer
at the entropy, in terms of
178
00:10:39 --> 00:10:46
the microscopic theory here.
179
00:10:46 --> 00:10:53
So let's start with S
is A minus u over T.
180
00:10:53 --> 00:10:56
And let's see how
far we can go.
181
00:10:56 --> 00:10:59
Alright.
182
00:10:59 --> 00:11:02
Let's write it in terms of --
let's write A and u in terms
183
00:11:02 --> 00:11:05
of things that we know.
184
00:11:05 --> 00:11:09
And let me just go to the end
to tell you where I'm going,
185
00:11:09 --> 00:11:11
and why I'm going to
make certain changes
186
00:11:11 --> 00:11:13
in my math here.
187
00:11:13 --> 00:11:15
So what we're going to get at
the end is that S is this
188
00:11:15 --> 00:11:23
very nice quantity, which
is minus k, pi log pi.
189
00:11:23 --> 00:11:27
Where the p's are the
microstate probabilities.
190
00:11:27 --> 00:11:30
The probability that your state
is in a particular -- yes?
191
00:11:30 --> 00:11:33
STUDENT: [UNINTELLIGIBLE]
192
00:11:33 --> 00:11:38
PROFESSOR: S is u minus A.
193
00:11:38 --> 00:11:41
Yes. u minus A.
194
00:11:41 --> 00:11:41
Thank you.
195
00:11:41 --> 00:11:48
I should read my notes.
196
00:11:48 --> 00:11:48
OK.
197
00:11:48 --> 00:11:52
So we're going to equate S here
in terms of the probabilities
198
00:11:52 --> 00:11:54
of microstates.
199
00:11:54 --> 00:11:57
And that's going to be --
remember how we talked about
200
00:11:57 --> 00:12:03
S is related to concepts
of randomness, or
201
00:12:03 --> 00:12:05
order, or disorder.
202
00:12:05 --> 00:12:09
So the number of possible
microstates is related to the
203
00:12:09 --> 00:12:12
number of amount of disorder
that you might have.
204
00:12:12 --> 00:12:16
If you have a pure crystal, and
every atom is in its place,
205
00:12:16 --> 00:12:20
then the number of microstates
at zero degree Kelvin is one.
206
00:12:20 --> 00:12:23
So the probability of being
in that microstate is one.
207
00:12:23 --> 00:12:27
And the probability of being in
every other microstate is zero.
208
00:12:27 --> 00:12:29
Alright, so there's a
relationship that we're going
209
00:12:29 --> 00:12:31
to have here, which is
going to be interesting.
210
00:12:31 --> 00:12:34
Let's derive it.
211
00:12:34 --> 00:12:37
That means that we're going
to want to have this somehow
212
00:12:37 --> 00:12:40
pop out of the equation.
213
00:12:40 --> 00:12:42
We're going to want to have
this pop out of the equation.
214
00:12:42 --> 00:12:59
If you remember, pi is the --
pi is e to the minus the Ei
215
00:12:59 --> 00:13:08
over kT, divided by the
partition function Q.
216
00:13:08 --> 00:13:11
So somehow we are going to
have to get this to come out.
217
00:13:11 --> 00:13:12
We're going to have to
have these e to the minus
218
00:13:12 --> 00:13:15
Ei over kT's come out.
219
00:13:15 --> 00:13:15
OK.
220
00:13:15 --> 00:13:18
So let's try to get them
to come out right away.
221
00:13:18 --> 00:13:19
We know u.
222
00:13:19 --> 00:13:22
A way to write u is
the average energy.
223
00:13:22 --> 00:13:29
Which means let's take
one over kT out here.
224
00:13:29 --> 00:13:33
And there's a -- let's
divided by k here.
225
00:13:33 --> 00:13:35
Let's get the k here. kT here.
226
00:13:35 --> 00:13:37
We're going to want to
have a k come out here.
227
00:13:37 --> 00:13:38
There's the k here.
228
00:13:38 --> 00:13:40
So that's one way of
getting it to come out.
229
00:13:40 --> 00:13:48
And then the u is going to
be the average energy. e
230
00:13:48 --> 00:13:52
to the minus Ei, minus kT.
231
00:13:52 --> 00:13:56
That's just writing
the average energy.
232
00:13:56 --> 00:13:58
So the energy times the
probability of having
233
00:13:58 --> 00:14:03
that energy divided
by the normalization.
234
00:14:03 --> 00:14:06
Plus, well, A is just log Q.
235
00:14:06 --> 00:14:09
A over kT is just log Q.
236
00:14:09 --> 00:14:16
So we've managed to extract
this guy out here.
237
00:14:16 --> 00:14:17
OK.
238
00:14:17 --> 00:14:21
Now, this sum, here,
this is sum over i.
239
00:14:21 --> 00:14:25
I'd really like to have this
sum come all the way out.
240
00:14:25 --> 00:14:28
So I've got to find
a way to do that.
241
00:14:28 --> 00:14:36
And it would be nice if I
could find a sum here.
242
00:14:36 --> 00:14:37
Maybe if I multiply by one.
243
00:14:37 --> 00:14:41
If I write one in a funny way.
244
00:14:41 --> 00:14:42
Get a log here.
245
00:14:42 --> 00:14:44
You know, if I've got a couple
logs here, maybe I can
246
00:14:44 --> 00:14:46
combine them to get a ratio.
247
00:14:46 --> 00:14:51
So let's rewrite this Ei
in a funny way, here.
248
00:14:51 --> 00:14:57
Ei is -- I'm just rewriting
it, but in a strange way.
249
00:14:57 --> 00:15:03
Log e to the minus Ei over kT.
250
00:15:03 --> 00:15:06
So if I take the log of e to
the minus the Ei over kT.
251
00:15:06 --> 00:15:08
I get minus Ei over kT.
252
00:15:08 --> 00:15:09
The kT's disappear.
253
00:15:09 --> 00:15:11
So I just get Ei
is equal to Ei.
254
00:15:11 --> 00:15:16
I'm just writing something
that's pretty obvious here.
255
00:15:16 --> 00:15:18
And then we're going to
take that expression,
256
00:15:18 --> 00:15:22
and put it in here.
257
00:15:22 --> 00:15:29
So now I'm going to be able to
write S over k is minus -- So
258
00:15:29 --> 00:15:34
the kT here cancels
out this kT here.
259
00:15:34 --> 00:15:42
Sum over i. e to the
minus Ei over kT.
260
00:15:42 --> 00:15:44
Over Q.
261
00:15:44 --> 00:15:45
That's that term right here.
262
00:15:45 --> 00:15:50
And then I have the Ei,
which is log e to the
263
00:15:50 --> 00:15:53
minus Ei over kT.
264
00:15:53 --> 00:15:55
This whole thing is in
this parenthesis here.
265
00:15:55 --> 00:15:59
And then I have the plus log Q.
266
00:15:59 --> 00:16:14
Plus log Q.
267
00:16:14 --> 00:16:17
OK.
268
00:16:17 --> 00:16:20
So I have this nice thing
here. e to the minus Ei
269
00:16:20 --> 00:16:22
over kT divided by Q.
270
00:16:22 --> 00:16:26
Well that's looking an awful
lot like this pi here.
271
00:16:26 --> 00:16:28
Which is what I'm
trying to get out.
272
00:16:28 --> 00:16:30
I'm trying to get these
pi's coming out.
273
00:16:30 --> 00:16:32
So that's a nice
thing to have here.
274
00:16:32 --> 00:16:34
Now if I could only have a pi
coming out here, somehow,
275
00:16:34 --> 00:16:36
that would be great too.
276
00:16:36 --> 00:16:39
And if I also got a sum
here, that's a sum over i
277
00:16:39 --> 00:16:41
I could sort of combine
everything together.
278
00:16:41 --> 00:16:45
So I'm going to write
one in a funny way.
279
00:16:45 --> 00:16:50
One is equal to the sum
of all probabilities.
280
00:16:50 --> 00:16:53
That's obvious.
281
00:16:53 --> 00:16:55
And I'm going to write
this pi here in a form
282
00:16:55 --> 00:16:58
that looks like this.
283
00:16:58 --> 00:17:05
Sum over all i, e to the minus
Ei over kT, divided by Q.
284
00:17:05 --> 00:17:06
Just writing one.
285
00:17:06 --> 00:17:09
The sum of all probability
is equal to one.
286
00:17:09 --> 00:17:14
And I'm going to take this
one, here, and I'm going
287
00:17:14 --> 00:17:16
to put it right in here.
288
00:17:16 --> 00:17:19
Now log Q doesn't care on
i, so it's just a number.
289
00:17:19 --> 00:17:26
So that allows me to rewrite --
S over k is minus the sum over
290
00:17:26 --> 00:17:33
i. e to the minus Ei
over kT, divided by the
291
00:17:33 --> 00:17:35
partition function.
292
00:17:35 --> 00:17:41
Times the log of e to
the minus Ei over kT.
293
00:17:41 --> 00:17:53
Plus sum over i, e to the
minus Ei over kT over Q.
294
00:17:53 --> 00:17:59
Times log Q.
295
00:17:59 --> 00:17:59
OK good.
296
00:17:59 --> 00:18:01
I'm going to take
these summations.
297
00:18:01 --> 00:18:03
Now everything is
over the sum of Ei.
298
00:18:03 --> 00:18:03
This is great.
299
00:18:03 --> 00:18:06
And there's this factor here.
300
00:18:06 --> 00:18:07
E to the minus Ei over
kT divided by the
301
00:18:07 --> 00:18:08
partition function.
302
00:18:08 --> 00:18:09
That appears in both.
303
00:18:09 --> 00:18:12
I can factor that out. and.
304
00:18:12 --> 00:18:14
Then I have these two logs.
305
00:18:14 --> 00:18:18
Log of this and log of that.
306
00:18:18 --> 00:18:20
That I can also
combine together.
307
00:18:20 --> 00:18:22
This is the log of this.
308
00:18:22 --> 00:18:24
And then there's a minus sign.
309
00:18:24 --> 00:18:27
So if I take the -- it's
going to be the log of this
310
00:18:27 --> 00:18:28
minus the log of that,.
311
00:18:28 --> 00:18:35
It's going to end
up with a ratio.
312
00:18:35 --> 00:18:41
S over k is equal to minus Ei.
313
00:18:41 --> 00:18:48
Taking the summation out. e to
the minus Ei over kT, over Q.
314
00:18:48 --> 00:18:50
That's this term.
315
00:18:50 --> 00:18:52
That term.
316
00:18:52 --> 00:18:55
And then I have the logs.
317
00:18:55 --> 00:18:58
Log of e to the
minus Ei over kT.
318
00:18:58 --> 00:19:00
This is a minus sign.
319
00:19:00 --> 00:19:01
This is a plus sign.
320
00:19:01 --> 00:19:07
That means I divide here by Q.
321
00:19:07 --> 00:19:08
This is great.
322
00:19:08 --> 00:19:08
Look.
323
00:19:08 --> 00:19:11
This e to the minus Ei
over kT over Q. e to the
324
00:19:11 --> 00:19:13
minus Ei over kT over Q.
325
00:19:13 --> 00:19:13
What is that?
326
00:19:13 --> 00:19:19
That's just pi.
327
00:19:19 --> 00:19:20
This is pi.
328
00:19:20 --> 00:19:25
That's the probability
of microstate i.
329
00:19:25 --> 00:19:40
And this is equal to minus
sum over i, pi log pi.
330
00:19:40 --> 00:19:41
There's the k, here.
331
00:19:41 --> 00:19:55
S is equal to minus k, log
sum over i, pi, log pi.
332
00:19:55 --> 00:19:59
Another great result.
333
00:19:59 --> 00:20:02
Now if you system is isolated.
334
00:20:02 --> 00:20:04
If you have an isolated system,
that means that the energy
335
00:20:04 --> 00:20:07
-- You've got your boundary.
336
00:20:07 --> 00:20:10
The boundary doesn't let
energy go in and out.
337
00:20:10 --> 00:20:13
Doesn't let the number of
particles go in and out.
338
00:20:13 --> 00:20:17
Every single microstate is
going to have the same energy.
339
00:20:17 --> 00:20:19
If this system is isolated.
340
00:20:19 --> 00:20:22
The only thing you're going
to change is the positions
341
00:20:22 --> 00:20:23
of the particles.
342
00:20:23 --> 00:20:25
Or their vibrational energy.
343
00:20:25 --> 00:20:25
Or something.
344
00:20:25 --> 00:20:28
But let's just stick
with translation.
345
00:20:28 --> 00:20:30
You're just going to
change the positions.
346
00:20:30 --> 00:20:34
You're not going to
change the energy.
347
00:20:34 --> 00:20:42
So, if the system is isolated,
then the degeneracy of your
348
00:20:42 --> 00:20:45
energy is just a number of ways
that you can flip the
349
00:20:45 --> 00:20:46
positions around.
350
00:20:46 --> 00:20:49
Indistinguishable ways.
351
00:20:49 --> 00:20:54
So the probability is just one
over the number of possible
352
00:20:54 --> 00:21:00
ways of switching positions
around for your particles.
353
00:21:00 --> 00:21:30
So for an isolated system,
all microstates have
354
00:21:30 --> 00:21:35
the same energy.
355
00:21:35 --> 00:21:39
We can set that equal to zero
as our reference point.
356
00:21:39 --> 00:21:44
And the probability of being in
any one microstate is just one
357
00:21:44 --> 00:21:48
over the number of possible
ways of rearranging things.
358
00:21:48 --> 00:21:53
So the probability of been in
any one microstate is one over
359
00:21:53 --> 00:21:56
the number of my microstates.
360
00:21:56 --> 00:22:01
They all have the same energy.
361
00:22:01 --> 00:22:07
Where this is the degeneracy.
362
00:22:07 --> 00:22:14
So now when you plug that
in here, S is minus k.
363
00:22:14 --> 00:22:22
Sum over all microstates
from one to blah, one
364
00:22:22 --> 00:22:26
over blah, log of blah.
365
00:22:26 --> 00:22:27
This is a number.
366
00:22:27 --> 00:22:29
It can come out.
367
00:22:29 --> 00:22:33
The sum of one to omega.
368
00:22:33 --> 00:22:34
Of one over omega.
369
00:22:34 --> 00:22:35
Is omega times omega.
370
00:22:35 --> 00:22:37
It's one.
371
00:22:37 --> 00:22:40
The one over omega -- the
log of one over omega is
372
00:22:40 --> 00:22:41
minus the log of omega.
373
00:22:41 --> 00:22:53
The negative signs cancel
out. k log capital omega.
374
00:22:53 --> 00:23:04
For an isolated system.
375
00:23:04 --> 00:23:08
You've seen this
before, probably.
376
00:23:08 --> 00:23:10
This is called the
Boltzmann equation.
377
00:23:10 --> 00:23:13
And that is what is
on his tombstone.
378
00:23:13 --> 00:23:16
If you go to - I think it's
in Germany somewhere.
379
00:23:16 --> 00:23:17
Is it in Germany?
380
00:23:17 --> 00:23:18
Do you guys know?
381
00:23:18 --> 00:23:19
STUDENT: Austria.
382
00:23:19 --> 00:23:20
PROFESSOR: Austria, thank you.
383
00:23:20 --> 00:23:21
I knew it was that
part of the world.
384
00:23:21 --> 00:23:25
If you go to Austria, to some
famous cemetery, and go look
385
00:23:25 --> 00:23:27
for the tombstone that says
S is equal to log omega.
386
00:23:27 --> 00:23:33
That's where
Boltzmann is buried.
387
00:23:33 --> 00:23:34
OK.
388
00:23:34 --> 00:23:37
So this picture of
-- yes question?
389
00:23:37 --> 00:23:44
STUDENT: Can you repeat the
argument for making that
390
00:23:44 --> 00:23:44
one over omega go away?
391
00:23:44 --> 00:23:46
PROFESSOR: Making the one
over omega go away here?
392
00:23:46 --> 00:23:50
Because you're taking the
sum of all states from one,
393
00:23:50 --> 00:23:52
i equals one to omega.
394
00:23:52 --> 00:23:54
So it's one over our omega plus
one over omega plus one over
395
00:23:54 --> 00:23:57
omega plus -- omega times.
396
00:23:57 --> 00:23:58
And this is just a number.
397
00:23:58 --> 00:23:59
So it comes out.
398
00:23:59 --> 00:24:01
It's not in the sum.
399
00:24:01 --> 00:24:13
So the sum, here,
is all by itself.
400
00:24:13 --> 00:24:21
Any other questions?
401
00:24:21 --> 00:24:21
OK.
402
00:24:21 --> 00:24:30
That's why we talk about
entropy as being this
403
00:24:30 --> 00:24:34
fundamental property that
tells you about the number
404
00:24:34 --> 00:24:35
of available states.
405
00:24:35 --> 00:24:36
That's what it is.
406
00:24:36 --> 00:24:40
You've got this connection now
between this variable, which is
407
00:24:40 --> 00:24:43
sort of hard to really
intuitively understand, when
408
00:24:43 --> 00:24:45
you're talking about
thermodynamics.
409
00:24:45 --> 00:24:47
And this is much easier to
understand here, in terms of
410
00:24:47 --> 00:24:52
the available ways of
distributing your energy, or
411
00:24:52 --> 00:24:54
your particles, in
this case here.
412
00:24:54 --> 00:24:59
In different bins.
413
00:24:59 --> 00:25:06
OK any questions?
414
00:25:06 --> 00:25:13
So the next topic is we're
going to work a little bit
415
00:25:13 --> 00:25:15
with the partition functions.
416
00:25:15 --> 00:25:21
And see how when you have
systems that have multiple
417
00:25:21 --> 00:25:25
degrees of freedom, where each
degree of freedom has a
418
00:25:25 --> 00:25:26
different kind of energy.
419
00:25:26 --> 00:25:30
Let's say translation,
rotation, vibration.
420
00:25:30 --> 00:25:32
Then you can have a partition
function for each of
421
00:25:32 --> 00:25:34
these degrees of freedom.
422
00:25:34 --> 00:25:38
And whereas the energies of the
degrees of freedom add up, the
423
00:25:38 --> 00:25:41
partition functions
get multiplied.
424
00:25:41 --> 00:25:45
So it's the separation of the
partition functions into
425
00:25:45 --> 00:25:50
subsystem partition functions.
426
00:25:50 --> 00:25:59
So so far, we've written for
translation partition function.
427
00:25:59 --> 00:26:05
That the system partition
function is the molecular
428
00:26:05 --> 00:26:09
partition function
to the Nth power.
429
00:26:09 --> 00:26:13
If you have distinguishable
particles.
430
00:26:13 --> 00:26:18
And you have to divide
by N factorial.
431
00:26:18 --> 00:26:22
If you can swap particles
without knowing the difference.
432
00:26:22 --> 00:26:25
This is the number of ways of
swapping N particles with
433
00:26:25 --> 00:26:40
each other, so it's
indistinguishable particles.
434
00:26:40 --> 00:26:40
OK.
435
00:26:40 --> 00:26:52
So now let's say that -- Let's
just make sure that -- Let's
436
00:26:52 --> 00:26:55
say that your energy of
your system -- yes?
437
00:26:55 --> 00:27:09
STUDENT: It says that when
system's not isolated,
438
00:27:09 --> 00:27:09
[UNINTELLIGIBLE].
439
00:27:09 --> 00:27:10
PROFESSOR: Where
does it say that?
440
00:27:10 --> 00:27:11
In the notes?
441
00:27:11 --> 00:27:19
STUDENT: [UNINTELLIGIBLE]
442
00:27:19 --> 00:27:21
PROFESSOR: Let me
see the notes here.
443
00:27:21 --> 00:27:45
What does it say?
444
00:27:45 --> 00:27:49
So that sentence has to
do with this guy here.
445
00:27:49 --> 00:27:52
Basically it says that if
you've got a huge number
446
00:27:52 --> 00:27:56
particles, the average
energy is a given number.
447
00:27:56 --> 00:28:00
And the fluctuations around
that average are very small.
448
00:28:00 --> 00:28:07
And so, the system behaves
as if it's isolated.
449
00:28:07 --> 00:28:11
So when you have a system which
is not isolated, then energy
450
00:28:11 --> 00:28:12
can come in and out
of the system.
451
00:28:12 --> 00:28:15
So in principle, over
time, you could have huge
452
00:28:15 --> 00:28:16
energy fluctuations.
453
00:28:16 --> 00:28:18
As energy comes out,
or energy comes in.
454
00:28:18 --> 00:28:23
And if you have a countable
number of molecules in your
455
00:28:23 --> 00:28:27
system, then if one molecule
suddenly captures a lot of
456
00:28:27 --> 00:28:31
energy, then the whole system
energy will go up a lot.
457
00:28:31 --> 00:28:35
But if you have ten to the 24th
molecules, if one molecule
458
00:28:35 --> 00:28:39
suddenly gains a lot of energy,
the system energy doesn't care.
459
00:28:39 --> 00:28:42
So small fluctuations -- or big
fluctuations in the small
460
00:28:42 --> 00:28:43
number of molecules doesn't
make any difference
461
00:28:43 --> 00:28:45
to the total energy.
462
00:28:45 --> 00:28:48
And so you can still
use this then.
463
00:28:48 --> 00:28:49
It's good enough.
464
00:28:49 --> 00:28:57
STUDENT: So how long is that
good enough [UNINTELLIGIBLE]
465
00:28:57 --> 00:29:00
PROFESSOR: Well if it's
accountable, a handful of
466
00:29:00 --> 00:29:02
things, and it's not valid.
467
00:29:02 --> 00:29:04
If it's ten to the 24th and
it's valid, and somewhere
468
00:29:04 --> 00:29:05
in between it breaks down.
469
00:29:05 --> 00:29:06
Then I don't know
what the answer is.
470
00:29:06 --> 00:29:09
But usually if you have
a thermodynamic system,
471
00:29:09 --> 00:29:11
then it's big enough.
472
00:29:11 --> 00:29:12
That's what
thermodynamics is about.
473
00:29:12 --> 00:29:14
Where you don't really care
that you have atoms there.
474
00:29:14 --> 00:29:16
You don't even know
you have atoms there.
475
00:29:16 --> 00:29:23
So it's big enough.
476
00:29:23 --> 00:29:25
Good question.
477
00:29:25 --> 00:29:31
Alright, so now let's take
our microstate energy here.
478
00:29:31 --> 00:29:35
And our microstate energy
is the sum of all the
479
00:29:35 --> 00:29:39
molecular energies Ei.
480
00:29:39 --> 00:29:59
So it's the sum of all energies
E sub, over all the atoms.
481
00:29:59 --> 00:30:03
And each one of these energies,
if it's a molecular energies,
482
00:30:03 --> 00:30:08
can be indexed by a quantum
number of some sort.
483
00:30:08 --> 00:30:13
So it would be the sum
over all energies.
484
00:30:13 --> 00:30:18
So quantum number for particle
one, n1 is some sort of quantum
485
00:30:18 --> 00:30:21
number. n2 is some sort of
quantum number. n3 is some
486
00:30:21 --> 00:30:23
sort of quantum number.
487
00:30:23 --> 00:30:26
And then you have
all the molecules.
488
00:30:26 --> 00:30:31
En1 plus En2 plus
En3, et cetera.
489
00:30:31 --> 00:30:34
So this is the energy from
molecule one, energy from
490
00:30:34 --> 00:30:36
molecule two, energy from
molecule three, energy
491
00:30:36 --> 00:30:37
from molecule four.
492
00:30:37 --> 00:30:40
And that little n tells
you which energy state
493
00:30:40 --> 00:30:42
that molecule is.
494
00:30:42 --> 00:30:47
And the sum of all
these energies is your
495
00:30:47 --> 00:30:49
microstate energy.
496
00:30:49 --> 00:30:51
As long as you can write this
this way, then you're allowed
497
00:30:51 --> 00:30:57
to write this this way.
498
00:30:57 --> 00:30:59
So that basically means that
they're not interacting
499
00:30:59 --> 00:31:00
with each other.
500
00:31:00 --> 00:31:02
They're independent
from each other.
501
00:31:02 --> 00:31:03
In this case here.
502
00:31:03 --> 00:31:11
So now if I write Q in terms of
the sum over all microstates
503
00:31:11 --> 00:31:16
Ei. e to the minus Ei over kT.
504
00:31:16 --> 00:31:19
I'm going to replace this
Ei here with the sum over
505
00:31:19 --> 00:31:22
all these energies here.
506
00:31:22 --> 00:31:24
And so the sum over all
microstates, then, becomes
507
00:31:24 --> 00:31:28
the sum over all possible
combinations of quantum
508
00:31:28 --> 00:31:37
numbers. n1, n2,
n3, n4, et cetera.
509
00:31:37 --> 00:31:40
All the possible ways
of getting molecule
510
00:31:40 --> 00:31:41
one in some state.
511
00:31:41 --> 00:31:43
All the possible ways of
getting molecule two in
512
00:31:43 --> 00:31:45
some quantum number state.
513
00:31:45 --> 00:31:50
And then e to the minus --
and instead of capital Ei,
514
00:31:50 --> 00:31:54
I'm going to write the
molecular energies.
515
00:31:54 --> 00:32:00
En1, plus En2, plus epsilon
n3 plus et cetera.
516
00:32:00 --> 00:32:06
And then divide by kT.
517
00:32:06 --> 00:32:10
Basically I'm going to prove
that this is a fine statement
518
00:32:10 --> 00:32:15
to make, as long as you can
write the energy as a sum
519
00:32:15 --> 00:32:26
of component energies.
520
00:32:26 --> 00:32:29
OK so now this term here,
e to the minus En1, only
521
00:32:29 --> 00:32:31
cares about this sum here.
522
00:32:31 --> 00:32:33
En2, that's molecule
number two, only cares
523
00:32:33 --> 00:32:35
about this sum here.
524
00:32:35 --> 00:32:37
Molecule number three only
cares about the sum over all
525
00:32:37 --> 00:32:39
possible quantum numbers
connected to molecule
526
00:32:39 --> 00:32:40
number three.
527
00:32:40 --> 00:32:47
So I can factor out all these
sums into a factor of sums.
528
00:32:47 --> 00:32:51
Is equal to the sum over
quantum number n1. e to
529
00:32:51 --> 00:32:58
the minus epsilon of
n1, divided by kT.
530
00:32:58 --> 00:33:10
Times n2 e to the minus epsilon
n2 over kT, times n3 e to the
531
00:33:10 --> 00:33:13
minus epsilon n3 divided
by kT, et cetera.
532
00:33:13 --> 00:33:15
And now each one of these
is basically the molecular
533
00:33:15 --> 00:33:16
partition function.
534
00:33:16 --> 00:33:21
These are all the possible
energies of that molecule.
535
00:33:21 --> 00:33:24
And the sum over all possible
energies times e to the minus
536
00:33:24 --> 00:33:26
E over kT is the partition
function for the molecule.
537
00:33:26 --> 00:33:31
So we have q for molecule one
times q for molecule two.
538
00:33:31 --> 00:33:32
And they're all the same.
539
00:33:32 --> 00:33:34
There are N of them.
540
00:33:34 --> 00:33:36
Plus q to the N.
541
00:33:36 --> 00:33:42
So just, in a way, clarified
that the reason why we're able
542
00:33:42 --> 00:33:47
to write this system partition
function, in terms of the
543
00:33:47 --> 00:33:51
molecular partition functions,
with N of them, to the Nth
544
00:33:51 --> 00:33:54
power, is because we were
able to separate out
545
00:33:54 --> 00:33:55
the energy here.
546
00:33:55 --> 00:33:58
In terms of the independent
molecular energies.
547
00:33:58 --> 00:34:01
Where this is saying the
molecules don't interact
548
00:34:01 --> 00:34:02
with each other.
549
00:34:02 --> 00:34:04
And are independent
from each other.
550
00:34:04 --> 00:34:07
And then the one over N
factorial comes in, so that
551
00:34:07 --> 00:34:12
you don't over count, for
translation, the positions.
552
00:34:12 --> 00:34:21
They're indistinguishable.
553
00:34:21 --> 00:34:22
OK.
554
00:34:22 --> 00:34:26
So now we can have -- Actually
we're not -- This basic concept
555
00:34:26 --> 00:34:32
of the partition function
multiplying each other, if the
556
00:34:32 --> 00:34:37
energies add, is not limited to
going from the molecular
557
00:34:37 --> 00:34:40
partition functions to the
system partition function.
558
00:34:40 --> 00:34:43
You can also look at the
molecular partition
559
00:34:43 --> 00:34:45
function itself.
560
00:34:45 --> 00:34:49
And if the energy, the
molecular energy, can be
561
00:34:49 --> 00:34:53
written in terms of a sum of
energies of different degrees
562
00:34:53 --> 00:34:56
of freedom, for instance, the
energy of a molecule could be
563
00:34:56 --> 00:34:58
the energy of the vibration,
plus the energy of the
564
00:34:58 --> 00:35:01
translation, plus the energy of
the rotation, plus the energy
565
00:35:01 --> 00:35:03
of the magnetic field, plus the
energy of the electric
566
00:35:03 --> 00:35:04
field. et cetera.
567
00:35:04 --> 00:35:07
You have many energies that can
add up with each other to
568
00:35:07 --> 00:35:10
create the molecular energy.
569
00:35:10 --> 00:35:14
And what we're going to be able
to write, then, is that this
570
00:35:14 --> 00:35:17
molecular partition function
itself can be written in terms
571
00:35:17 --> 00:35:23
of a product of partition
functions for the sub parts
572
00:35:23 --> 00:35:25
of the molecular energy.
573
00:35:25 --> 00:35:40
So let me clarify
that statement.
574
00:35:40 --> 00:35:46
So if I can write my molecular
energy, epsilon, is equal to a
575
00:35:46 --> 00:35:51
translational energy, plus a
vibrational energy, plus a
576
00:35:51 --> 00:35:55
rotational energy, plus every
other little energies that
577
00:35:55 --> 00:35:58
you can think of that are
independent of each other.
578
00:35:58 --> 00:36:03
Then using the same argument we
used to show that Q is the
579
00:36:03 --> 00:36:07
multiplication of these
molecular partition function,
580
00:36:07 --> 00:36:13
we can write that the molecular
partition function, little q,
581
00:36:13 --> 00:36:18
is just the multiplication of
the degree of freedom partition
582
00:36:18 --> 00:36:20
functions -- molecular
partition functions.
583
00:36:20 --> 00:36:23
The translational partition
function times the vibrational
584
00:36:23 --> 00:36:28
partition function, times the
rotational partition
585
00:36:28 --> 00:36:29
function, et cetera.
586
00:36:29 --> 00:36:32
If the energies add, then
the partition functions
587
00:36:32 --> 00:36:36
multiply each other.
588
00:36:36 --> 00:36:38
And that's going to be powerful
because when we look at
589
00:36:38 --> 00:36:43
something like a polymer or DNA
or protein or something,
590
00:36:43 --> 00:36:44
in solution.
591
00:36:44 --> 00:36:46
And we're going to be looking
at the configurations possible
592
00:36:46 --> 00:36:52
for that polymer or that
biopolymer, then we'll know
593
00:36:52 --> 00:36:57
that the energy of that polymer
in solution is going to be --
594
00:36:57 --> 00:37:01
we'll be able to approximate
it as the energy of the
595
00:37:01 --> 00:37:03
configuration for that polymer.
596
00:37:03 --> 00:37:08
The different ways that you can
fold the protein, for instance.
597
00:37:08 --> 00:37:10
Plus everything else.
598
00:37:10 --> 00:37:14
The energy of everything else.
599
00:37:14 --> 00:37:15
OK.
600
00:37:15 --> 00:37:18
So if the configurational
energy can be separated from
601
00:37:18 --> 00:37:20
the sum of all vibration
energies of all the
602
00:37:20 --> 00:37:22
bonds in that polymer.
603
00:37:22 --> 00:37:27
The way that polymer interacts
with -- The way that
604
00:37:27 --> 00:37:30
the solution itself
interacts with itself.
605
00:37:30 --> 00:37:34
Then if we can do this and we
can do this approximation most
606
00:37:34 --> 00:37:38
of the time, then we'll be able
to take the partition function
607
00:37:38 --> 00:37:41
for the polymer, and write it
as the configurational
608
00:37:41 --> 00:37:43
partition function times
the partition function
609
00:37:43 --> 00:37:45
for everything else.
610
00:37:45 --> 00:37:55
And we'll find that this part
here will tend to factor out.
611
00:37:55 --> 00:37:58
We won't have to
worry about it.
612
00:37:58 --> 00:38:01
And that this will carry all
the important information that
613
00:38:01 --> 00:38:05
we'll need to know to see
about changes in the system.
614
00:38:05 --> 00:38:07
Changes in Gibbs free
energy, changes in the
615
00:38:07 --> 00:38:08
chemical potential.
616
00:38:08 --> 00:38:12
Everything will be related
to this partition function.
617
00:38:12 --> 00:38:14
This subsystem.
618
00:38:14 --> 00:38:19
And because of the fact that
you can factor them out, then
619
00:38:19 --> 00:38:21
this thing will end
up dropping out.
620
00:38:21 --> 00:38:30
And this will become
the important factor.
621
00:38:30 --> 00:38:30
OK.
622
00:38:30 --> 00:38:55
Let's do a quick example.
623
00:38:55 --> 00:38:56
OK.
624
00:38:56 --> 00:39:00
So this is the example
of having a very
625
00:39:00 --> 00:39:02
very, short polymer.
626
00:39:02 --> 00:39:06
Containing three monomers.
627
00:39:06 --> 00:39:13
Which can be in two
configurations.
628
00:39:13 --> 00:39:17
And the energies are the same
for these two configurations.
629
00:39:17 --> 00:39:21
So the configurational
partition function, which you
630
00:39:21 --> 00:39:26
would generally write as the
sum of e to the minus Ei for
631
00:39:26 --> 00:39:29
that configuration, divided by
kT, plus e to the -- So we
632
00:39:29 --> 00:39:35
would usually write it as e to
the minus epsilon one over kT,
633
00:39:35 --> 00:39:41
plus e to the minus epsilon two
over kT, plus et cetera.
634
00:39:41 --> 00:39:43
They're all the same energy.
635
00:39:43 --> 00:39:45
And there are two
configurations.
636
00:39:45 --> 00:39:47
The degeneracy is two.
637
00:39:47 --> 00:39:50
So you can write this as
the degeneracy of the
638
00:39:50 --> 00:39:55
configuration, times the
energy of the configuration.
639
00:39:55 --> 00:40:00
And you can set your energy
reference to be zero.
640
00:40:00 --> 00:40:01
You can choose whatever
you want it to be.
641
00:40:01 --> 00:40:03
And zero is a good number.
642
00:40:03 --> 00:40:05
So that e to the zero
is equal to one.
643
00:40:05 --> 00:40:07
So that configuration partition
function is just the
644
00:40:07 --> 00:40:23
degeneracy, which is equal to
two, in this case here.
645
00:40:23 --> 00:40:32
So now let's calculate the
molecular and canonical
646
00:40:32 --> 00:40:39
partition functions for an
ideal gas of these
647
00:40:39 --> 00:40:41
molecules here.
648
00:40:41 --> 00:40:47
And it's usually interesting to
use a lattice model as a guide.
649
00:40:47 --> 00:40:50
And so in this lattice model
here, you would divide space
650
00:40:50 --> 00:40:55
up into little cells.
651
00:40:55 --> 00:40:57
Pieces in two dimensions,
but in reality it would
652
00:40:57 --> 00:40:58
be three dimensions.
653
00:40:58 --> 00:41:03
And then you place your
molecules in lattice sites.
654
00:41:03 --> 00:41:06
Something like this.
655
00:41:06 --> 00:41:10
And then you end up counting
the number of ways of arranging
656
00:41:10 --> 00:41:12
the molecules on the lattice.
657
00:41:12 --> 00:41:16
And let's say that we have
N molecules that are
658
00:41:16 --> 00:41:18
in the gas phase.
659
00:41:18 --> 00:41:24
And the molecular volume,
i.e. the size of the
660
00:41:24 --> 00:41:28
lattice site is v.
661
00:41:28 --> 00:41:31
That's the molecular volume.
662
00:41:31 --> 00:41:36
And N times v is
the total volume.
663
00:41:36 --> 00:41:41
That's the total volume
occupied by the particles.
664
00:41:41 --> 00:41:43
The total volume is the
number of lattice sites.
665
00:41:43 --> 00:41:56
V is the total volume.
666
00:41:56 --> 00:42:01
And we're going to assume
that all particles, all
667
00:42:01 --> 00:42:03
molecules have the same
translation energy.
668
00:42:03 --> 00:42:05
It's an adequate approximation.
669
00:42:05 --> 00:42:08
We're going to set
that equal to zero.
670
00:42:08 --> 00:42:11
So all molecules at any
position here has the
671
00:42:11 --> 00:42:15
same E translation.
672
00:42:15 --> 00:42:20
And we're going to set
that equal to zero.
673
00:42:20 --> 00:42:23
So the translational
partition function.
674
00:42:23 --> 00:42:33
The molecular transitional
partition function is -- Well
675
00:42:33 --> 00:42:35
there's only one energy.
676
00:42:35 --> 00:42:36
It's zero.
677
00:42:36 --> 00:42:40
So we only care about the
degeneracy of that molecule.
678
00:42:40 --> 00:42:45
That molecule could be in here,
or it could be here, it could
679
00:42:45 --> 00:42:48
be here, it could be here, it
could be anywhere, right?
680
00:42:48 --> 00:42:52
The number of ways of putting
that molecule on the lattice
681
00:42:52 --> 00:42:55
is the number of lattice
sites available.
682
00:42:55 --> 00:42:58
Which is basically the
molecular volume.
683
00:42:58 --> 00:42:59
Which is the total
volume divided by the
684
00:42:59 --> 00:43:02
molecular volume.
685
00:43:02 --> 00:43:15
The total volume is -- right.
686
00:43:15 --> 00:43:22
So the total volume is the
number of lattice sites
687
00:43:22 --> 00:43:26
times the volume of
each lattice site.
688
00:43:26 --> 00:43:30
So the total volume divided by
the small volume is the total
689
00:43:30 --> 00:43:32
number of lattice sites.
690
00:43:32 --> 00:43:34
And the number of choices of
putting that one molecule is
691
00:43:34 --> 00:43:39
anywhere on the lattice.
692
00:43:39 --> 00:43:44
That's your degeneracy.
693
00:43:44 --> 00:43:50
So now if I look at the total
molecular partition function,
694
00:43:50 --> 00:43:52
it's going to be the
multiplication of the
695
00:43:52 --> 00:43:55
configurational partition
function and the translational
696
00:43:55 --> 00:43:58
partition function.
697
00:43:58 --> 00:44:18
At each site, the molecule
could have two configurations.
698
00:44:18 --> 00:44:22
So q, for the molecule,
that's q translational.
699
00:44:22 --> 00:44:23
I'm going to ignore
all vibrations,
700
00:44:23 --> 00:44:24
rotation, et cetera.
701
00:44:24 --> 00:44:26
I'm going to assume that there
are two degrees of freedom.
702
00:44:26 --> 00:44:29
Translational one, which is
basically the positional one.
703
00:44:29 --> 00:44:31
And then the configurational
one, which is internal
704
00:44:31 --> 00:44:36
to the molecule.
705
00:44:36 --> 00:44:38
So this one is V over v.
706
00:44:38 --> 00:44:39
That's the degeneracy.
707
00:44:39 --> 00:44:41
Capital V over little v.
708
00:44:41 --> 00:44:45
The degeneracy of placing the
molecule on the lattice.
709
00:44:45 --> 00:44:49
And the configuration is
the degeneracy of how
710
00:44:49 --> 00:44:52
the molecule folds.
711
00:44:52 --> 00:44:52
Yes.
712
00:44:52 --> 00:45:01
STUDENT: [UNINTELLIGIBLE]
713
00:45:01 --> 00:45:05
PROFESSOR: The notes are wrong.
714
00:45:05 --> 00:45:10
So usually capital V is large
and little v is small.
715
00:45:10 --> 00:45:18
So in the notes, if we have it
in reverse, we should fix that.
716
00:45:18 --> 00:45:21
This is lecture 25, right?
717
00:45:21 --> 00:45:22
So we have q.
718
00:45:22 --> 00:45:23
No the notes seem to be right.
719
00:45:23 --> 00:45:31
Total volume is capital V,
molecular volume is little v.
720
00:45:31 --> 00:45:33
Where is it that
wrong in the notes?
721
00:45:33 --> 00:45:46
STUDENT: [UNINTELLIGIBLE]
722
00:45:46 --> 00:45:46
PROFESSOR: Oh look at that.
723
00:45:46 --> 00:45:51
My notes are different
than yours.
724
00:45:51 --> 00:45:53
My notes are right.
725
00:45:53 --> 00:45:57
OK well it's obviously
right on the web.
726
00:45:57 --> 00:45:58
Because this is the
latest version.
727
00:45:58 --> 00:46:03
Alright so flip those big
V's and little v's then.
728
00:46:03 --> 00:46:03
Huh.
729
00:46:03 --> 00:46:08
I thought they were
from the same pile.
730
00:46:08 --> 00:46:08
OK.
731
00:46:08 --> 00:46:15
So this is your molecular
partition function.
732
00:46:15 --> 00:46:20
And then when you look at the
system, the system partition
733
00:46:20 --> 00:46:24
function can also be separated
into a translation and the
734
00:46:24 --> 00:46:27
configuration for the system.
735
00:46:27 --> 00:46:31
We know what you need to do
is take all the molecular
736
00:46:31 --> 00:46:34
partition functions, the
transitional ones, and
737
00:46:34 --> 00:46:37
-- to the N factor.
738
00:46:37 --> 00:46:39
The number of particles.
739
00:46:39 --> 00:46:42
But now you have degeneracy.
740
00:46:42 --> 00:46:45
You've over counted.
741
00:46:45 --> 00:46:50
So you need to divide
this N factorial.
742
00:46:50 --> 00:46:53
You also have the system
partition function for
743
00:46:53 --> 00:46:56
the configurations.
744
00:46:56 --> 00:47:01
And that's q
configuration to the N.
745
00:47:01 --> 00:47:03
Except here we don't need to
divide by one over N
746
00:47:03 --> 00:47:07
factorial, because we're not
over counting here.
747
00:47:07 --> 00:47:13
The over counting only happens
when you're placing identical
748
00:47:13 --> 00:47:18
particles in a lattice, and you
can swap them without
749
00:47:18 --> 00:47:19
making a difference.
750
00:47:19 --> 00:47:21
Here we talking about
configurations.
751
00:47:21 --> 00:47:23
When we're talking about
configurations, we're not
752
00:47:23 --> 00:47:26
talking about placing the
identical particles
753
00:47:26 --> 00:47:27
in different spots.
754
00:47:27 --> 00:47:31
We're just looking at these
two configurations here.
755
00:47:31 --> 00:47:33
And then the next particle
is two configurations.
756
00:47:33 --> 00:47:37
The next particle is
two configurations.
757
00:47:37 --> 00:47:38
So this is really important.
758
00:47:38 --> 00:47:41
That this N over factorial only
comes into play when you're
759
00:47:41 --> 00:47:43
talking about the translational
degree of freedom.
760
00:47:43 --> 00:47:46
Not the other
degrees of freedom.
761
00:47:46 --> 00:47:48
And now the total system
partition function is the
762
00:47:48 --> 00:47:51
multiplication of these two.
763
00:47:51 --> 00:47:57
It ends up being capital V over
v, to the N power, over N
764
00:47:57 --> 00:48:06
factorial, times two to the N.
765
00:48:06 --> 00:48:07
OK.
766
00:48:07 --> 00:48:10
In general you could extend
this analysis to include
767
00:48:10 --> 00:48:15
vibrations, rotations, energy
in a magnetical field,
768
00:48:15 --> 00:48:18
electric field, et cetera.
769
00:48:18 --> 00:48:24
Any questions?
770
00:48:24 --> 00:48:26
Alright the next thing that
you're going to do then is to
771
00:48:26 --> 00:48:31
use this concept, as a sort of
example, as a way to begin to
772
00:48:31 --> 00:48:34
calculate things that you've
already calculated before.
773
00:48:34 --> 00:48:38
For instance, if you look at an
expansion of an ideal gas, can
774
00:48:38 --> 00:48:41
we now calculate the
entropy change.
775
00:48:41 --> 00:48:45
Not based on thermodynamics,
but based on the
776
00:48:45 --> 00:48:46
statistical mechanics.
777
00:48:46 --> 00:48:49
On the microscopic description
that we've just gone through.
778
00:48:49 --> 00:48:53
And it turns out that
that's what happens.
779
00:48:53 --> 00:48:54
That you can do that.
780
00:48:54 --> 00:48:55
You get the same answer.
781
00:48:55 --> 00:48:57
Thank god you get
the same answer.
782
00:48:57 --> 00:49:00
Otherwise we'd be
in big trouble.
783
00:49:00 --> 00:49:02
So I'm just going to set
up the problem, because I
784
00:49:02 --> 00:49:03
won't have time to do it.
785
00:49:03 --> 00:49:06
And then you can do it next
time, when Keith comes back.
786
00:49:06 --> 00:49:11
So the problem is going to be
the usual problem of having
787
00:49:11 --> 00:49:17
a volume V1 of a gas on
one side, and a vaccuum
788
00:49:17 --> 00:49:23
expanding to volume V2, gas.
789
00:49:23 --> 00:49:29
And asking what is
the entropy change.
790
00:49:29 --> 00:49:37
And you know that from thermo,
that delta S in this case here
791
00:49:37 --> 00:49:43
is nR log V2 over V1, when
the temperature is constant.
792
00:49:43 --> 00:49:44
That's going to be our answer.
793
00:49:44 --> 00:49:45
It has to be our answer.
794
00:49:45 --> 00:49:47
But this time, instead of
knowing the answer, we're
795
00:49:47 --> 00:49:51
going to calculate it
from microscopics.
796
00:49:51 --> 00:49:54
So what you do is you start
out with your initial state.
797
00:49:54 --> 00:50:01
You ask what is -- you write
down the molecular volume V.
798
00:50:01 --> 00:50:02
The total volume here is V1.
799
00:50:02 --> 00:50:10
800
00:50:10 --> 00:50:14
You assume that all
molecules have the same
801
00:50:14 --> 00:50:15
translational energy.
802
00:50:15 --> 00:50:17
You set that equal to zero.
803
00:50:17 --> 00:50:20
The system translational
energy is equal to zero.
804
00:50:20 --> 00:50:27
And so the entropy for this gas
here is just the number of ways
805
00:50:27 --> 00:50:30
of placing the molecule
in the lattice.
806
00:50:30 --> 00:50:34
This model that we have
of space being separated
807
00:50:34 --> 00:50:35
into little cells.
808
00:50:35 --> 00:50:43
And so S is k log omega, where
omega is the number of ways of
809
00:50:43 --> 00:50:45
placing the molecules
in the lattice.
810
00:50:45 --> 00:50:49
Which is basically
k log V, over v.
811
00:50:49 --> 00:51:03
Where this is the number
of lattice sites.
812
00:51:03 --> 00:51:05
OK.
813
00:51:05 --> 00:51:07
And what you're going
to do next time, then,
814
00:51:07 --> 00:51:10
is start from here.
815
00:51:10 --> 00:51:11
Calculate what it is before.
816
00:51:11 --> 00:51:14
Calculate what it is after,
and turn the crank, and
817
00:51:14 --> 00:51:16
get to the right answer.
818
00:51:16 --> 00:51:19
Then you're going to do the
same thing for liquids.
819
00:51:19 --> 00:51:25
And that'll be it for this
simple statistical mechanics.
820
00:51:25 --> 00:51:26