1
00:00:00 --> 00:00:01
 
 

2
00:00:01 --> 00:00:02
The following content is
provided under a Creative

3
00:00:02 --> 00:00:03
Commons license.
 

4
00:00:03 --> 00:00:06
Your support will help MIT
OpenCourseWare continue to

5
00:00:06 --> 00:00:10
offer high quality educational
resources for free.

6
00:00:10 --> 00:00:13
To make a donation or view
additional materials from

7
00:00:13 --> 00:00:16
hundreds of MIT courses,
visit MIT OpenCourseWare

8
00:00:16 --> 00:00:20
at ocw.mit.edu.
 

9
00:00:20 --> 00:00:24
PROFESSOR: And last time, you
got to see how you can derive

10
00:00:24 --> 00:00:28
macroscopic thermodynamic
results from the microscopic

11
00:00:28 --> 00:00:30
point of view of
statistical mechanics.

12
00:00:30 --> 00:00:34
So in contrast to the way we'd
gone through the course from

13
00:00:34 --> 00:00:38
the beginning, starting from
empirical macroscopic

14
00:00:38 --> 00:00:41
observation and ideas through
macroscopic laws and then

15
00:00:41 --> 00:00:44
working out the consequences of
them, in statistical mechanics

16
00:00:44 --> 00:00:48
you start from a microscopic
model, build up the energy

17
00:00:48 --> 00:00:51
levels that are going to give
you the terms in the partition

18
00:00:51 --> 00:00:54
function, determine the
partition function, and what

19
00:00:54 --> 00:00:57
you saw last time is that
given that, you can

20
00:00:57 --> 00:00:58
calculate everything.
 

21
00:00:58 --> 00:01:00
You can calculate all the
macroscopic properties that

22
00:01:00 --> 00:01:04
ordinarily come from the
thermodynamic laws that

23
00:01:04 --> 00:01:07
were based on empirical
macroscopic observation.

24
00:01:07 --> 00:01:09
So today, I just want
to continue doing some

25
00:01:09 --> 00:01:11
statistical thermodynamics.
 

26
00:01:11 --> 00:01:14
Basically going through a few
examples, just to see how it

27
00:01:14 --> 00:01:18
plays out when we calculate
thermodynamic quantities, based

28
00:01:18 --> 00:01:19
on our microscopic picture.
 

29
00:01:19 --> 00:01:32
So let's just look
at a few cases.

30
00:01:32 --> 00:01:34
And I'll start with a couple
of examples that are

31
00:01:34 --> 00:01:36
entirely entropy driven.
 

32
00:01:36 --> 00:01:39
And we've seen them before from
the macroscopic point of view.

33
00:01:39 --> 00:01:41
What we should hope, of course,
is that we can derive those

34
00:01:41 --> 00:01:43
results from the partition
functions that are

35
00:01:43 --> 00:01:45
appropriate here.
 

36
00:01:45 --> 00:01:57
So, we'll look at entropically
driven processes.

37
00:01:57 --> 00:01:59
And the first one, the simplest
one, maybe, to cover, is just

38
00:01:59 --> 00:02:01
free expansion of a gas.
 

39
00:02:01 --> 00:02:05
So what happens if we've got
particles in a gas that are

40
00:02:05 --> 00:02:07
enclosed in a certain volume.
 

41
00:02:07 --> 00:02:09
And now we let the volume
expand into vacuum

42
00:02:09 --> 00:02:10
on the other side.
 

43
00:02:10 --> 00:02:11
Of course, you know
what'll happen.

44
00:02:11 --> 00:02:14
The volume will be filled
and we've derived the

45
00:02:14 --> 00:02:15
thermodynamics for it before.
 

46
00:02:15 --> 00:02:18
So let's see what happens
in a statistical

47
00:02:18 --> 00:02:20
mechanical treatment.
 

48
00:02:20 --> 00:02:27
So let's say, here's
V1, there's our gas.

49
00:02:27 --> 00:02:29
There's vacuum.
 

50
00:02:29 --> 00:02:33
And now we'll open it up and
just have the bigger volume,

51
00:02:33 --> 00:02:37
V2, and let the gas fill it.
 

52
00:02:37 --> 00:02:41
So I think last time you got
introduced to basically

53
00:02:41 --> 00:02:44
a lattice model for
translational motion.

54
00:02:44 --> 00:02:46
And I started in on this
just a little bit the

55
00:02:46 --> 00:02:48
time before also.
 

56
00:02:48 --> 00:02:53
So this is a simple way to
describe the statistical

57
00:02:53 --> 00:02:55
mechanics of filling
an open volume.

58
00:02:55 --> 00:02:57
Filling a space with molecules.
 

59
00:02:57 --> 00:03:03
And all it does is say that
we're going to divide up our

60
00:03:03 --> 00:03:06
available volume into little
bits that are basically the

61
00:03:06 --> 00:03:09
size of an atom or a molecule,
or whatever the particle

62
00:03:09 --> 00:03:11
is in the gas phase.
 

63
00:03:11 --> 00:03:14
And say, OK, there are
maybe 10 of the 30th of

64
00:03:14 --> 00:03:15
those little volumes.
 

65
00:03:15 --> 00:03:17
The little volume elements are
going to be on the order of an

66
00:03:17 --> 00:03:20
angstrom cubed if it's an atom,
a little bit bigger

67
00:03:20 --> 00:03:22
if it's a molecule.
 

68
00:03:22 --> 00:03:26
And the available volume is on
the order of meters cubed.

69
00:03:26 --> 00:03:30
So that works out, an angstrom
is 10 to the minus 10 meters.

70
00:03:30 --> 00:03:33
And then the cube of that
is 10 to the minus 30th.

71
00:03:33 --> 00:03:36
So if you look at the total
number of little volume

72
00:03:36 --> 00:03:40
elements of this sort, it's on
the order of 10 to the 30th.

73
00:03:40 --> 00:03:55
So, the total volume,
capital V, let's make

74
00:03:55 --> 00:04:00
that distinction clear.
 

75
00:04:00 --> 00:04:03
And in this case, all the
states of the system

76
00:04:03 --> 00:04:06
have equal energies.
 

77
00:04:06 --> 00:04:08
So in other words, it doesn't
matter we're dividing our

78
00:04:08 --> 00:04:17
volume up into imagined
little volume elements.

79
00:04:17 --> 00:04:21
And for both the molecular
and the system states,

80
00:04:21 --> 00:04:23
the energy is the same.
 

81
00:04:23 --> 00:04:25
It doesn't matter whether
the molecules are here,

82
00:04:25 --> 00:04:27
here, here and so forth.
 

83
00:04:27 --> 00:04:29
They're not interacting
in this picture at all.

84
00:04:29 --> 00:04:32
So it doesn't matter
how close or far they

85
00:04:32 --> 00:04:33
are from each other.
 

86
00:04:33 --> 00:04:35
They can't be in
the same volume.

87
00:04:35 --> 00:04:37
So all the energies
are the same.

88
00:04:37 --> 00:04:40
And what that means is, in the
partition function, which is a

89
00:04:40 --> 00:04:43
sum over all these terms with
these Boltzmann factors.

90
00:04:43 --> 00:04:45
That have e to the
minus energy over kT.

91
00:04:45 --> 00:04:48
But energy's all the same
in every one of the terms.

92
00:04:48 --> 00:04:51
So in order to determine the
partition function, to

93
00:04:51 --> 00:04:54
determine Q, all we have to do
is count up the total number of

94
00:04:54 --> 00:04:58
states that are available.
 

95
00:04:58 --> 00:05:03
So, our energy, our molecular
translational energy,

96
00:05:03 --> 00:05:05
we'll just set it to zero.
 

97
00:05:05 --> 00:05:10
Same with our system
translational energy.

98
00:05:10 --> 00:05:11
We'll set that to zero.
 

99
00:05:11 --> 00:05:16
All the microscopic available
states, that is, if I take an

100
00:05:16 --> 00:05:18
individual particle and I say
where it can be, all those

101
00:05:18 --> 00:05:20
states have the same energy.
 

102
00:05:20 --> 00:05:23
If I take a microstate of the
system, you know, the whole

103
00:05:23 --> 00:05:27
collection of the particles,
all of those states also

104
00:05:27 --> 00:05:30
have the same energy.
 

105
00:05:30 --> 00:05:35
So what that means is that
little q is just equal to

106
00:05:35 --> 00:05:39
capital V over little v, right?
 

107
00:05:39 --> 00:05:40
How many states are there?
 

108
00:05:40 --> 00:05:44
Well, I've got my little
volume, and then however many

109
00:05:44 --> 00:05:47
individual cells there are,
that's the number of available

110
00:05:47 --> 00:05:49
states in this model.
 

111
00:05:49 --> 00:05:57
So it's big V over little v, on
the order of 10 to the 30th.

112
00:05:57 --> 00:06:01
And then, big Q, the canonical
partition function for the

113
00:06:01 --> 00:06:06
whole system, it's something
that we've been through before.

114
00:06:06 --> 00:06:10
It's this process of saying,
well, you take the first atom

115
00:06:10 --> 00:06:14
or molecule, and it has any
one of these possible states.

116
00:06:14 --> 00:06:17
So it's on the order of 10
to the 30th possibilities.

117
00:06:17 --> 00:06:18
Then where does the
second one go?

118
00:06:18 --> 00:06:21
Well, there are basically 10 to
the 30th more possibilities.

119
00:06:21 --> 00:06:23
And the third one
has 10 to the 30th.

120
00:06:23 --> 00:06:27
Since there are so many fewer
atoms or molecules then there

121
00:06:27 --> 00:06:31
are volume elements, when we're
dealing with the gas phase, we

122
00:06:31 --> 00:06:34
don't have to worry about the
fact that, well the first

123
00:06:34 --> 00:06:37
million of the atoms
filled some of the sites.

124
00:06:37 --> 00:06:40
So the next ones have fewer
sites available to them.

125
00:06:40 --> 00:06:42
It's true, but it's such a
small fraction that are

126
00:06:42 --> 00:06:47
ever filled that we don't
need to worry about it.

127
00:06:47 --> 00:06:52
So Q is just little q to
the capital N power, the

128
00:06:52 --> 00:06:54
number of particles.
 

129
00:06:54 --> 00:06:59
And then we've seen you have to
divide by N factorial to avoid

130
00:06:59 --> 00:07:03
the overcounting of
configurations that are in fact

131
00:07:03 --> 00:07:05
not distinguishable.
 

132
00:07:05 --> 00:07:09
The whole idea that maybe
there's an atom here

133
00:07:09 --> 00:07:10
and another one here.
 

134
00:07:10 --> 00:07:12
That configuration is
indistinguishable from the

135
00:07:12 --> 00:07:16
configuration with those two
atoms reversed if we're dealing

136
00:07:16 --> 00:07:21
with a mole of identical atoms.
 

137
00:07:21 --> 00:07:22
So, OK.
 

138
00:07:22 --> 00:07:25
There's our capital Q.
 

139
00:07:25 --> 00:07:28
And that's also our
system degeneracy.

140
00:07:28 --> 00:07:33
So the degeneracy, the little g
for the molecular degeneracy,

141
00:07:33 --> 00:07:37
how many molecular states are
there of a certain energy.

142
00:07:37 --> 00:07:39
Well, it's the same thing as q.
 

143
00:07:39 --> 00:07:42
And it's the same thing here
for the system states.

144
00:07:42 --> 00:07:49
The capital Omega is
just equal to that.

145
00:07:49 --> 00:07:55
Well, now let's look at what
happens when we do this.

146
00:07:55 --> 00:07:59
When we expand from
volume V1 to volume V2.

147
00:07:59 --> 00:08:03
So that capital V is
going to change.

148
00:08:03 --> 00:08:10
So capital V1 goes
to capital V2.

149
00:08:10 --> 00:08:11
And we know it's
entropically driven.

150
00:08:11 --> 00:08:15
Let's calculate the
change in entropy.

151
00:08:15 --> 00:08:23
So delta S is just k log
capital Omega 2 minus

152
00:08:23 --> 00:08:27
k log capital Omega 1.
 

153
00:08:27 --> 00:08:30
Now, in the process of just
seeing how the development

154
00:08:30 --> 00:08:34
goes, how you can derive all
these thermodynamic quantities

155
00:08:34 --> 00:08:38
from the partition function,
one of the really most central

156
00:08:38 --> 00:08:40
results concerns the entropy.
 

157
00:08:40 --> 00:08:45
How you can describe the
entropy in terms of the

158
00:08:45 --> 00:08:48
probabilities that the
different states are occupied.

159
00:08:48 --> 00:08:51
And how, in the case like this,
where the states all have the

160
00:08:51 --> 00:08:54
same probability of being
occupied, then you have this

161
00:08:54 --> 00:08:57
very simplified form
for the entropy.

162
00:08:57 --> 00:09:00
Just, if S is k log capital
Omega, where capital

163
00:09:00 --> 00:09:02
Omega is the degeneracy.
 

164
00:09:02 --> 00:09:06
The number of system
states of that energy.

165
00:09:06 --> 00:09:08
An amazing result.
 

166
00:09:08 --> 00:09:13
You told them about Boltzmann's
tombstone and so forth.

167
00:09:13 --> 00:09:15
So it's on there.
 

168
00:09:15 --> 00:09:19
In fact, the ill acceptance
of that result kind of led

169
00:09:19 --> 00:09:23
to the tombstone being
erected when it did.

170
00:09:23 --> 00:09:28
Boltzmann, depressed over the
lack of acceptance of his

171
00:09:28 --> 00:09:30
theories, put himself
to an early death.

172
00:09:30 --> 00:09:33
And this was off a big
part of the reason.

173
00:09:33 --> 00:09:38
But we, generations later, have
come to accept the results

174
00:09:38 --> 00:09:41
that concerned him so deeply.
 

175
00:09:41 --> 00:09:43
So this is our change in
entropy, when we just have this

176
00:09:43 --> 00:09:46
expansion of gas from V1 to V2.
 

177
00:09:46 --> 00:09:52
So it's just k log
omega 2, omega 1.

178
00:09:52 --> 00:09:55
So now let's just put in our
results for the volumes.

179
00:09:55 --> 00:10:04
It's k log V2 over v, I'm
exaggerating the size there.

180
00:10:04 --> 00:10:11
To the N, over N factorial,
over V1 over v to the N

181
00:10:11 --> 00:10:13
divided by N factorial.
 

182
00:10:13 --> 00:10:16
So this is going to turn out to
be a relatively simple case,

183
00:10:16 --> 00:10:18
because the factorials
are going to cancel.

184
00:10:18 --> 00:10:28
So, then we just have
N k, log V2 over V1.

185
00:10:28 --> 00:10:30
Terrific, right?
 

186
00:10:30 --> 00:10:36
Now, remember, k, the Boltzmann
constant, is just the ideal gas

187
00:10:36 --> 00:10:39
constant per molecule
rather than per mole.

188
00:10:39 --> 00:10:43
So this is the same thing as
little N, the number of moles,

189
00:10:43 --> 00:10:49
times R, times log V2 over V1.
 

190
00:10:49 --> 00:10:51
And that should be
a familiar result.

191
00:10:51 --> 00:10:53
That's the change in entropy
in expansion, free expansion

192
00:10:53 --> 00:10:55
of a gas, from V1 to V2.
 

193
00:10:55 --> 00:10:58
So now we've been able to
derive that just based on

194
00:10:58 --> 00:11:01
this really simple
molecular picture.

195
00:11:01 --> 00:11:06
Based on that plus the result
that Boltzmann fought and paid

196
00:11:06 --> 00:11:11
so dearly for, so that we would
have it and understand it.

197
00:11:11 --> 00:11:14
As before, of course note
it's greater than zero.

198
00:11:14 --> 00:11:16
If the volume, if we're
expanding into a bigger

199
00:11:16 --> 00:11:18
volume than before.
 

200
00:11:18 --> 00:11:21
The entropy goes up.
 

201
00:11:21 --> 00:11:25
Also notice one of the other
results that you've seen is

202
00:11:25 --> 00:11:30
that you could relate the
partition function to A, right?

203
00:11:30 --> 00:11:40
The Helmholtz free energy
is minus k T log Q.

204
00:11:40 --> 00:11:43
And S is negative dA/dT.
 

205
00:11:43 --> 00:11:50
 


206
00:11:50 --> 00:11:52
And so this would immediately
give us the same result,

207
00:11:52 --> 00:11:56
because omega's going
to go in here.

208
00:11:56 --> 00:12:00
So we could get this result
from another pathway, too, but

209
00:12:00 --> 00:12:02
a more simple and direct way is
just to start directly from the

210
00:12:02 --> 00:12:07
Boltzmann result for entropy.
 

211
00:12:07 --> 00:12:10
Any questions?
 

212
00:12:10 --> 00:12:15
Alright, let's go one
step more complex.

213
00:12:15 --> 00:12:17
Let's now look at the
entropy of mixing.

214
00:12:17 --> 00:12:19
So now we've got two species.
 

215
00:12:19 --> 00:12:21
Rather than one expanding into
a free volume, and we're going

216
00:12:21 --> 00:12:23
to open a barrier between
them and see them mix.

217
00:12:23 --> 00:12:25
And again, it's still
entropically driven.

218
00:12:25 --> 00:12:27
And we should be able to
calculate the entropy change

219
00:12:27 --> 00:12:31
that we saw before from a
macroscopic perspective.

220
00:12:31 --> 00:12:39
So, let's take a look.
 

221
00:12:39 --> 00:12:48
So now, we have NA molecules
of A in some volume VA.

222
00:12:48 --> 00:12:53
And NB molecules of B,
in some volume VB.

223
00:12:53 --> 00:12:58
And then we're going
to open it up.

224
00:12:58 --> 00:13:02
Then we've got capital
N, NA plus NB.

225
00:13:02 --> 00:13:04
And capital V.
 

226
00:13:04 --> 00:13:11
VA plus VB, right?
 

227
00:13:11 --> 00:13:19
And the whole thing is
happening at constant pressure.

228
00:13:19 --> 00:13:24
Well, then let's look at the
initial and final expressions

229
00:13:24 --> 00:13:25
for the entropy.
 

230
00:13:25 --> 00:13:28
So, S1, at the beginning.
 

231
00:13:28 --> 00:13:37
It's just k log omega
A plus k log omega B.

232
00:13:37 --> 00:13:49
And that's k log VA over
little v to the NA power.

233
00:13:49 --> 00:13:56
Divided by NA factorial
times the same thing for B.

234
00:13:56 --> 00:14:05
VB over little v to the NB
power over NB factorial.

235
00:14:05 --> 00:14:07
So that's our initial entropy.
 

236
00:14:07 --> 00:14:11
The sum of the two entropy
contributions, from

237
00:14:11 --> 00:14:14
the two sides.
 

238
00:14:14 --> 00:14:20
And S2, the final one,
is just k log omega

239
00:14:20 --> 00:14:23
for the whole shebang.
 

240
00:14:23 --> 00:14:27
And that's k log.
 

241
00:14:27 --> 00:14:33
And now we have capital V over
little v to the N power.

242
00:14:33 --> 00:14:37
And now we have a combinatorics
result that I think probably

243
00:14:37 --> 00:14:39
you're all familiar with from
one context or another.

244
00:14:39 --> 00:14:45
That is, now we have to
divide by NA factorial

245
00:14:45 --> 00:14:52
times NB factorial.
 

246
00:14:52 --> 00:14:56
In other words, the amount that
we need to divide by to avoid

247
00:14:56 --> 00:15:02
overcounting is the product
of those two factorials.

248
00:15:02 --> 00:15:05
Just in case it's been a while
since you've seen stuff like

249
00:15:05 --> 00:15:08
that, I've got a couple of
pages further on the notes

250
00:15:08 --> 00:15:10
at the bottom of the page.
 

251
00:15:10 --> 00:15:13
I've just broken that out for a
really simple example where

252
00:15:13 --> 00:15:16
there are just two molecules of
A, and two molecules of B

253
00:15:16 --> 00:15:22
represented by different little
balls in lattice sites.

254
00:15:22 --> 00:15:24
And I just worked it out for
the total where there are only

255
00:15:24 --> 00:15:29
ten total in the mixture.
 

256
00:15:29 --> 00:15:33
The point is, now what happens
is, when you start filling a

257
00:15:33 --> 00:15:37
lattice, but it's not
all the same, right?

258
00:15:37 --> 00:15:39
So now you have A here.
 

259
00:15:39 --> 00:15:40
And A here.
 

260
00:15:40 --> 00:15:41
And B here.
 

261
00:15:41 --> 00:15:42
And B here.
 

262
00:15:42 --> 00:15:45
So, of course if you
interchange A and B here, you

263
00:15:45 --> 00:15:49
don't have an identical state.
 

264
00:15:49 --> 00:15:51
An indistinguishable state.
 

265
00:15:51 --> 00:15:53
That's distinguishable
and needs to be counted.

266
00:15:53 --> 00:15:57
It's only interchanging B with
B that you need to correct for,

267
00:15:57 --> 00:16:00
and interchanging A with A.
 

268
00:16:00 --> 00:16:02
So of course you need to
account for that in a way

269
00:16:02 --> 00:16:05
that's different from how it
turns out if every one of

270
00:16:05 --> 00:16:10
the molecules is identical.
 

271
00:16:10 --> 00:16:13
So it's the product
of these factorials.

272
00:16:13 --> 00:16:17
In the numerator.
 

273
00:16:17 --> 00:16:29
So now delta S is then
just this minus this.

274
00:16:29 --> 00:16:48
So it's k log V over little v
to the NA plus NB over VA

275
00:16:48 --> 00:16:54
divided by little v to the NA
power VB divided by little

276
00:16:54 --> 00:16:58
v to the NB power.
 

277
00:16:58 --> 00:17:02
And, let's see, maybe
I'd better back up.

278
00:17:02 --> 00:17:04
Let me know, that's all.
 

279
00:17:04 --> 00:17:07
And what's happening is, again,
luckily, these factorials

280
00:17:07 --> 00:17:10
are canceling.
 

281
00:17:10 --> 00:17:23
So we just have k log V to
the NA, V to the NB over

282
00:17:23 --> 00:17:33
VA to the NA VB to the NB.
 

283
00:17:33 --> 00:17:36
And now we're going to go back
and use the fact that the

284
00:17:36 --> 00:17:40
pressures are the same,
all the way through.

285
00:17:40 --> 00:17:44
And what that means is that the
volumes must be in the same

286
00:17:44 --> 00:17:50
ratio as the number
of molecules.

287
00:17:50 --> 00:17:56
And what that means is that,
for example, VA over V is

288
00:17:56 --> 00:17:59
the same as NA over N.
 

289
00:17:59 --> 00:18:02
And that's just equal to XA,
the mole fraction of A.

290
00:18:02 --> 00:18:07
And the same for B.
 

291
00:18:07 --> 00:18:11
So we can substitute that.
 

292
00:18:11 --> 00:18:21
And know our delta S just
becomes k log, let's see,

293
00:18:21 --> 00:18:27
let me break this out
from the beginning.

294
00:18:27 --> 00:18:38
And just take minus k log on
XA to the NA power minus

295
00:18:38 --> 00:18:43
k log XB to the NB power.
 

296
00:18:43 --> 00:18:46
All I've done make these
substitutions here.

297
00:18:46 --> 00:18:49
For the ratios of the volume.
 

298
00:18:49 --> 00:18:58
So it's just minus N k XA
log XA plus XB log XB.

299
00:18:58 --> 00:19:02
 


300
00:19:02 --> 00:19:04
Look familiar?
 

301
00:19:04 --> 00:19:07
Same thing we saw
macroscopically before.

302
00:19:07 --> 00:19:09
So, again, we can still use
our microscopic model.

303
00:19:09 --> 00:19:12
And continue to derive the
macroscopic entropy changes.

304
00:19:12 --> 00:19:15
And, of course, for many
of these we can still get

305
00:19:15 --> 00:19:17
our delta G and so forth.
 

306
00:19:17 --> 00:19:21
Of course, it's only entropy
that's going to contribute.

307
00:19:21 --> 00:19:26
OK, let's look at just one
more entropy driven problem.

308
00:19:26 --> 00:19:29
And that is, it's a little
bit different from these.

309
00:19:29 --> 00:19:33
Let's look at the entropy
mixing in a liquid.

310
00:19:33 --> 00:19:37
And the difference between the
gas and the liquid is that in

311
00:19:37 --> 00:19:42
the case of the liquid, every
single cell is filled.

312
00:19:42 --> 00:19:43
It's not like with the gas.
 

313
00:19:43 --> 00:19:46
And that makes a difference
in how the states

314
00:19:46 --> 00:19:47
need to be counted.
 

315
00:19:47 --> 00:19:50
Because here, remember
how I said when we were

316
00:19:50 --> 00:19:51
filling this lattice.
 

317
00:19:51 --> 00:19:53
Well, the first molecules
goes somewhere.

318
00:19:53 --> 00:19:54
And there are 10 to the
30th possibilities.

319
00:19:54 --> 00:19:56
And the second molecule goes
somewhere and there are 10

320
00:19:56 --> 00:19:58
to the 30th possibilities.
 

321
00:19:58 --> 00:20:01
And there are always 10 to the
30th possibilities because

322
00:20:01 --> 00:20:04
there's so few molecules that
you never have to worry about

323
00:20:04 --> 00:20:06
the fact that it's getting
filled up, so there would

324
00:20:06 --> 00:20:12
become fewer possibilities
for that last molecule.

325
00:20:12 --> 00:20:15
And the reason again is because
there are so few molecules that

326
00:20:15 --> 00:20:20
essentially, all the cells are
open and available even

327
00:20:20 --> 00:20:22
to the last molecule.
 

328
00:20:22 --> 00:20:23
Because maybe there's
a mole of molecules.

329
00:20:23 --> 00:20:26
Maybe there are 10 to
the 24 molecules.

330
00:20:26 --> 00:20:30
And there are 10 to the
30th lattice sites.

331
00:20:30 --> 00:20:32
So there might be one in a
million lattice sites occupied,

332
00:20:32 --> 00:20:35
even at the very end
of the procedure.

333
00:20:35 --> 00:20:39
But of course for the liquid,
that's definitely not the case.

334
00:20:39 --> 00:20:42
All the available
volume is filled.

335
00:20:42 --> 00:20:44
So by the time you get to that
last molecule, it has one and

336
00:20:44 --> 00:20:47
only one space it can go into.
 

337
00:20:47 --> 00:20:51
So you have to count for the
diminishment of the available

338
00:20:51 --> 00:20:56
sites as molecules
are placed in them.

339
00:20:56 --> 00:20:57
OK.
 

340
00:20:57 --> 00:21:01
So it's a different
kind of combinatorics

341
00:21:01 --> 00:21:04
problem that results.
 

342
00:21:04 --> 00:21:07
So, OK.
 

343
00:21:07 --> 00:21:12
Let's look at liquid mixture.
 

344
00:21:12 --> 00:21:22
So now, it's going to
look like, and they're

345
00:21:22 --> 00:21:29
all filled up.
 

346
00:21:29 --> 00:21:39
I sure used fewer sites maybe,
but that'll be easier here.

347
00:21:39 --> 00:21:43
We've got open circles.
 

348
00:21:43 --> 00:21:46
So these two are going to mix.
 

349
00:21:46 --> 00:21:52
And now we're going to
have filled mixture.

350
00:21:52 --> 00:21:55
So we're going to have a bigger
volume, but it's still going to

351
00:21:55 --> 00:21:58
be filled and all the
positions are random.

352
00:21:58 --> 00:22:01
So we have to consider all
the possible configurations

353
00:22:01 --> 00:22:02
that we have.
 

354
00:22:02 --> 00:22:03
OK.
 

355
00:22:03 --> 00:22:08
So we still should go through
the same part of the procedure.

356
00:22:08 --> 00:22:10
So let's call this A.
 

357
00:22:10 --> 00:22:14
And this, B.
 

358
00:22:14 --> 00:22:18
So at first, already the
situation is different.

359
00:22:18 --> 00:22:22
If we say what's the
entropy of system A.

360
00:22:22 --> 00:22:27
Well, in this simple
model the entropy of the

361
00:22:27 --> 00:22:30
pure liquid is zero.
 

362
00:22:30 --> 00:22:32
Now, of course, that's
not realistic.

363
00:22:32 --> 00:22:34
But it's going to turn
out to be suitable.

364
00:22:34 --> 00:22:37
Because the change in entropy
that we go from here

365
00:22:37 --> 00:22:38
to the mixture.
 

366
00:22:38 --> 00:22:41
The other contributions
to entropy are going to

367
00:22:41 --> 00:22:44
more or less be the same
from beginning to end.

368
00:22:44 --> 00:22:47
What's going to be
significantly different is

369
00:22:47 --> 00:22:51
simply the fact that in the
mixture you have the random

370
00:22:51 --> 00:22:53
positions that can be occupied.
 

371
00:22:53 --> 00:22:54
So that's going to be the
term that matters, the

372
00:22:54 --> 00:22:56
contribution that matters.
 

373
00:22:56 --> 00:23:02
So entropy is just
k log Omega A.

374
00:23:02 --> 00:23:03
There's only one system state.
 

375
00:23:03 --> 00:23:05
All of the lattice sites
are filled, and they're

376
00:23:05 --> 00:23:07
all filled with A.
 

377
00:23:07 --> 00:23:08
So there's only
one way to do it.

378
00:23:08 --> 00:23:10
And of course, it doesn't
matter if you interchange them.

379
00:23:10 --> 00:23:13
They're indistinguishable.
 

380
00:23:13 --> 00:23:16
So this is zero.
 

381
00:23:16 --> 00:23:18
This is one.
 

382
00:23:18 --> 00:23:19
There's only one state.
 

383
00:23:19 --> 00:23:27
Same, of course, for B.
 

384
00:23:27 --> 00:23:33
So that's our initial entropy
in this simple model.

385
00:23:33 --> 00:23:38
Well then, delta S of
the mixture is just

386
00:23:38 --> 00:23:39
S of the mixture.
 

387
00:23:39 --> 00:23:45
We don't need to worry
about the original parts.

388
00:23:45 --> 00:23:51
Now, this is going to be k
log of N factorial over NA

389
00:23:51 --> 00:23:55
factorial NB factorial,
like we've seen before.

390
00:23:55 --> 00:23:58
That's not different
from before.

391
00:23:58 --> 00:24:02
Except that unlike before, we
don't have the convenient and

392
00:24:02 --> 00:24:06
simple cancellation
of the factorials.

393
00:24:06 --> 00:24:09
That happened before,
because they were there.

394
00:24:09 --> 00:24:12
In other words, they were there
in the entropy contribution in

395
00:24:12 --> 00:24:15
the gas phase as well,
in the pure gas.

396
00:24:15 --> 00:24:19
They're only there
here in the liquid.

397
00:24:19 --> 00:24:21
And that means we have
to deal with them.

398
00:24:21 --> 00:24:24
But it turns out it's not
too bad to deal with them.

399
00:24:24 --> 00:24:31
And we're going to use this
Stirling's approximation

400
00:24:31 --> 00:24:32
that I introduced before.
 

401
00:24:32 --> 00:24:38
So the log of N factorial
is approximately equal

402
00:24:38 --> 00:24:45
to N log N minus N.
 

403
00:24:45 --> 00:24:49
So let's just break that out
then, and use the Stirling's

404
00:24:49 --> 00:24:53
approximation for each
of the factorial terms.

405
00:24:53 --> 00:25:03
So then we have S for
the mixture is N k log

406
00:25:03 --> 00:25:09
N minus N k minus.
 

407
00:25:09 --> 00:25:11
And now we'll do the
bottom, the numerator.

408
00:25:11 --> 00:25:20
NA k log NA minus NA times
k, that's this part.

409
00:25:20 --> 00:25:22
And now we'll do this one.
 

410
00:25:22 --> 00:25:31
So it's plus NB k log
NB minus NB times k.

411
00:25:31 --> 00:25:35
But NA and NB, of
course, are just N.

412
00:25:35 --> 00:25:43
So these will cancel.
 

413
00:25:43 --> 00:25:55
So then we're left with NA plus
NB times k log N minus NA --

414
00:25:55 --> 00:25:58
And I just want to write
this this way so that I can

415
00:25:58 --> 00:26:00
then separate the terms.
 

416
00:26:00 --> 00:26:11
Minus NA k log NA
minus NB k log NB.

417
00:26:11 --> 00:26:16
And now I just want to combine
the easily combined terms.

418
00:26:16 --> 00:26:30
So it's NA k log N over NA
plus NB k log N over NB.

419
00:26:30 --> 00:26:32
Looks like we're home.
 

420
00:26:32 --> 00:26:45
Now we just have N k XA log
XA plus XB log XB right?

421
00:26:45 --> 00:26:47
This NA over N is just XA.
 

422
00:26:47 --> 00:26:52
NB over is XB.
 

423
00:26:52 --> 00:26:55
And then I've divided the same
thing to make XA's over here.

424
00:26:55 --> 00:26:58
And take the total
number of moles out.

425
00:26:58 --> 00:26:59
Alright.
 

426
00:26:59 --> 00:27:02
Look familiar?
 

427
00:27:02 --> 00:27:04
Again, the same thing derived
now in a way that's a bit

428
00:27:04 --> 00:27:06
different from what we
did in the gas phase.

429
00:27:06 --> 00:27:09
So now I've got the entropy of
mixing even in a condensed

430
00:27:09 --> 00:27:10
phase the liquid.
 

431
00:27:10 --> 00:27:14
And again, the reason this
simple model works is because

432
00:27:14 --> 00:27:18
although a real liquid,
certainly a pure liquid has a

433
00:27:18 --> 00:27:21
finite entropy, a substantial
amount of disorder, that's

434
00:27:21 --> 00:27:24
present, that kind of disorder.
 

435
00:27:24 --> 00:27:26
In other words, the fact that
you don't really have the

436
00:27:26 --> 00:27:30
molecules sitting in sites on a
lattice but there's disorder.

437
00:27:30 --> 00:27:35
There's also rotational
disorder if it's a molecule.

438
00:27:35 --> 00:27:37
There are various
contributions to entropy.

439
00:27:37 --> 00:27:43
But those are all comparable in
the starting and final states.

440
00:27:43 --> 00:27:46
The thing that's changed is
simply the interchanging

441
00:27:46 --> 00:27:47
of A and B molecules.
 

442
00:27:47 --> 00:27:50
That's introduced a new kind
of disorder that didn't

443
00:27:50 --> 00:27:52
used to be present.
 

444
00:27:52 --> 00:27:55
A very big difference in the
number of states at the

445
00:27:55 --> 00:27:57
end from the beginning.
 

446
00:27:57 --> 00:27:59
And that's what we've
calculated and captured here.

447
00:27:59 --> 00:28:04
And that's why such a simple
model still works OK.

448
00:28:04 --> 00:28:09
Any questions about these
entropy driven cases?

449
00:28:09 --> 00:28:11
OK.
 

450
00:28:11 --> 00:28:15
Now let's move on and talk
about the cases which are more

451
00:28:15 --> 00:28:17
commonly encountered, where the
states aren't all

452
00:28:17 --> 00:28:19
the same energy.
 

453
00:28:19 --> 00:28:21
Of course, here in the liquid,
just like in the gas, we

454
00:28:21 --> 00:28:25
haven't treated interactions
between the molecules.

455
00:28:25 --> 00:28:27
We've assumed that they're
equal between A and B, as

456
00:28:27 --> 00:28:30
they are between A
and A and B and B.

457
00:28:30 --> 00:28:32
So in this case then all the
energy to the states are the

458
00:28:32 --> 00:28:34
same and it's just a
counting problem.

459
00:28:34 --> 00:28:37
How many states are
there available.

460
00:28:37 --> 00:28:40
As soon as the energies are
different, then of course

461
00:28:40 --> 00:28:43
we need to account for all
those Boltzmann factors.

462
00:28:43 --> 00:28:47
Those e to the minus energy
over kT factors become part of

463
00:28:47 --> 00:28:49
the, they weight the counting.
 

464
00:28:49 --> 00:29:15
And we have to do that.
 

465
00:29:15 --> 00:29:20
So, what I want to do is go
back to this simple polymer

466
00:29:20 --> 00:29:25
configurational model that
we introduced before.

467
00:29:25 --> 00:29:29
And this sort of model, in one
form or another, we're going

468
00:29:29 --> 00:29:34
to see a few times in the
rest of this treatment.

469
00:29:34 --> 00:29:38
The reason, really, is because
it's a simple and also

470
00:29:38 --> 00:29:45
realistic way of portraying a
system that has a finite number

471
00:29:45 --> 00:29:49
of well-defined energies.
 

472
00:29:49 --> 00:29:51
If you don't have that,
of course you could

473
00:29:51 --> 00:29:53
do the treatment.
 

474
00:29:53 --> 00:29:57
In a, sort of, classical
mechanic sense, a continuum

475
00:29:57 --> 00:30:00
of states, all with
different energies.

476
00:30:00 --> 00:30:01
It's hard.
 

477
00:30:01 --> 00:30:05
Because then those sum in the
partition function become

478
00:30:05 --> 00:30:07
integrals over all the
different configurations

479
00:30:07 --> 00:30:08
and possibilities.
 

480
00:30:08 --> 00:30:10
It's much more
complicated to do.

481
00:30:10 --> 00:30:14
It's much more straightforward
if you can identify discrete

482
00:30:14 --> 00:30:17
states and add over them.
 

483
00:30:17 --> 00:30:21
Now, someday you'll take
quantum mechanics.

484
00:30:21 --> 00:30:25
And then you'll see that the
states are discrete, even if

485
00:30:25 --> 00:30:29
we're talking about translation
or rotation and so forth.

486
00:30:29 --> 00:30:31
So you'll have discrete
quantum mechanical energy

487
00:30:31 --> 00:30:34
levels and so forth.
 

488
00:30:34 --> 00:30:35
You haven't had that yet.
 

489
00:30:35 --> 00:30:38
And so that's not the starting
point that we're going to use.

490
00:30:38 --> 00:30:40
Instead, we're going to use a
starting point that goes

491
00:30:40 --> 00:30:44
through the exact same
formalism, which is just the

492
00:30:44 --> 00:30:47
discrete states available
in molecules that have

493
00:30:47 --> 00:30:50
multiple configurations.
 

494
00:30:50 --> 00:31:11
So we're going to have
unequal energy states.

495
00:31:11 --> 00:31:25
And what we're envisioning
is molecular or polymer

496
00:31:25 --> 00:31:28
configurations.
 

497
00:31:28 --> 00:31:42
So here we have our states.
 

498
00:31:42 --> 00:31:46
And just like before, the idea
is that if you have two

499
00:31:46 --> 00:31:49
sub-units that are in
proximity, there's some kind of

500
00:31:49 --> 00:31:50
favorable interaction.
 

501
00:31:50 --> 00:31:52
It could be hydrogen bonding.
 

502
00:31:52 --> 00:31:53
Could be different.
 

503
00:31:53 --> 00:31:56
But the point is, there's some
sort of interaction there

504
00:31:56 --> 00:32:12
that reduces the energy.
 

505
00:32:12 --> 00:32:14
And then there are other
configurations that

506
00:32:14 --> 00:32:16
just don't have that.
 

507
00:32:16 --> 00:32:32
Because the sub-units aren't
in proximity to each other.

508
00:32:32 --> 00:32:34
And it's not hard to work out
that in this very simple

509
00:32:34 --> 00:32:37
model, these are the only
configurations that

510
00:32:37 --> 00:32:39
are available.
 

511
00:32:39 --> 00:32:42
The only distinct
configurations.

512
00:32:42 --> 00:32:50
So then, our molecular energy,
E, we can define as zero here.

513
00:32:50 --> 00:32:53
Before we defined it as minus E
int, but I'm going to, it's a

514
00:32:53 --> 00:32:56
little more convenient to make
this the zero of energy, and

515
00:32:56 --> 00:33:03
then this is plus the
interaction for each

516
00:33:03 --> 00:33:04
of these states.
 

517
00:33:04 --> 00:33:09
Of course, we can put the zero
of energy wherever we prefer.

518
00:33:09 --> 00:33:15
And then we have the
degeneracy is one.

519
00:33:15 --> 00:33:21
And in this case it's three.
 

520
00:33:21 --> 00:33:28
So now we can just write out
the configurational partition

521
00:33:28 --> 00:33:33
function for the molecules and
also the canonical partition

522
00:33:33 --> 00:33:34
function for the system.
 

523
00:33:34 --> 00:33:39
So q configurational.
 

524
00:33:39 --> 00:33:40
And we're just going to
sum over the states.

525
00:33:40 --> 00:33:50
So it's e to the zero over
kT, for the lowest state.

526
00:33:50 --> 00:33:52
And then there are three states
that are going to have this

527
00:33:52 --> 00:33:58
term, e to the minus E
interaction over kT, where E

528
00:33:58 --> 00:34:00
int is a positive number.
 

529
00:34:00 --> 00:34:03
In other words, the probability
of any one of these states is a

530
00:34:03 --> 00:34:06
little bit lower
than this state.

531
00:34:06 --> 00:34:09
Because this state
has lower energy.

532
00:34:09 --> 00:34:12
Remember what I mentioned
earlier, which is although the

533
00:34:12 --> 00:34:16
probability of any one of these
states is lower than the

534
00:34:16 --> 00:34:21
probability of this state, the
probability of this energy is

535
00:34:21 --> 00:34:24
likely to be higher than the
probability of this energy.

536
00:34:24 --> 00:34:26
Because there's only one
of these states and the

537
00:34:26 --> 00:34:28
degeneracy here is three.
 

538
00:34:28 --> 00:34:31
So there are three
possibilities in which the

539
00:34:31 --> 00:34:33
molecule could have this
energy, and only one this.

540
00:34:33 --> 00:34:39
So if, basically, kT is bigger
than E int, so in other words,

541
00:34:39 --> 00:34:44
if this term isn't very much
smaller than one, then of

542
00:34:44 --> 00:34:48
course this will be
bigger than this.

543
00:34:48 --> 00:34:51
So of course this is one
plus three e to the

544
00:34:51 --> 00:34:59
minus E int over kT.
 

545
00:34:59 --> 00:35:05
And now we have our capital Q,
our canonical configurational

546
00:35:05 --> 00:35:07
partition function.
 

547
00:35:07 --> 00:35:11
And that's just little
q configurational

548
00:35:11 --> 00:35:12
to the Nth power.
 

549
00:35:12 --> 00:35:16
And like you've seen so far, in
various cases where the

550
00:35:16 --> 00:35:19
molecules are separate
non-interacting molecules, this

551
00:35:19 --> 00:35:26
is just the molecular partition
function taken capital N times.

552
00:35:26 --> 00:35:30
If all the molecules are
behaving independently,

553
00:35:30 --> 00:35:31
then you sum over
all of those states.

554
00:35:31 --> 00:35:33
And you see that each one
of these is just taken

555
00:35:33 --> 00:35:34
again and again.
 

556
00:35:34 --> 00:35:36
As we've seen implicitly
in all these treatments

557
00:35:36 --> 00:35:38
so far as well.
 

558
00:35:38 --> 00:35:40
Now, in the translational
case where you

559
00:35:40 --> 00:35:41
interchange particles.
 

560
00:35:41 --> 00:35:45
Like particles, you have to
divide by N factorial. .

561
00:35:45 --> 00:35:49
But the different
configurations don't do that.

562
00:35:49 --> 00:35:53
If I've got a system where a
molecule over here is in this

563
00:35:53 --> 00:35:55
configuration and a molecule
somewhere else is in this

564
00:35:55 --> 00:35:58
one, and now they change
configurations, that's a

565
00:35:58 --> 00:36:01
distinguishable state.
 

566
00:36:01 --> 00:36:04
So there's no N factorial
involved here.

567
00:36:04 --> 00:36:07
In the configurational
partition function.

568
00:36:07 --> 00:36:13
So, fine, then it's just one
plus three e to the minus E

569
00:36:13 --> 00:36:17
int over kT to the Nth power.
 

570
00:36:17 --> 00:36:19
That's Q.
 

571
00:36:19 --> 00:36:22
And once we know Q, as you've
seen, we know everything.

572
00:36:22 --> 00:36:26
And so we can immediately
start in deriving all of

573
00:36:26 --> 00:36:27
the thermodynamics, right?
 

574
00:36:27 --> 00:36:33
And the place to start is A,
Helmholtz free energy, or

575
00:36:33 --> 00:36:35
configurational free energy.
 

576
00:36:35 --> 00:36:40
Because that's the simplest
relation, just as minus kT log

577
00:36:40 --> 00:36:46
of capital Q configurational.
 

578
00:36:46 --> 00:36:54
So it's minus N kT one
plus three e to the

579
00:36:54 --> 00:37:21
minus E int over kT.
 

580
00:37:21 --> 00:37:25
I'm going to rewrite A
configurational in terms of

581
00:37:25 --> 00:37:28
beta rather than kT, just
because there are going to

582
00:37:28 --> 00:37:29
be a lot of these factors.
 

583
00:37:29 --> 00:37:35
So it's minus N kT one
plus three e to the minus

584
00:37:35 --> 00:37:39
beta E interaction.
 

585
00:37:39 --> 00:37:41
And that's our A.
 

586
00:37:41 --> 00:37:43
Everything's going to
follow from that.

587
00:37:43 --> 00:37:45
Before I write some of the
other results, one thing to

588
00:37:45 --> 00:37:50
notice is, it has N in it.
 

589
00:37:50 --> 00:37:54
In other words, there's a
free energy per molecule.

590
00:37:54 --> 00:38:01
The total free energy is
just something times N.

591
00:38:01 --> 00:38:03
And that's because all the
molecules in this model are

592
00:38:03 --> 00:38:06
independently behaving.
 

593
00:38:06 --> 00:38:11
So whatever their average free
energy is, that's going to add

594
00:38:11 --> 00:38:13
up and give us the total.
 

595
00:38:13 --> 00:38:15
They're all acting
independently.

596
00:38:15 --> 00:38:18
And in fact, we could have
gotten this directly from

597
00:38:18 --> 00:38:24
minus kT log little
q configurational.

598
00:38:24 --> 00:38:28
And if we look at energy,
regular energy, u,

599
00:38:28 --> 00:38:35
configurational, it's minus
d log capital Q with

600
00:38:35 --> 00:38:39
respect to beta.
 

601
00:38:39 --> 00:38:42
V and N.
 

602
00:38:42 --> 00:38:44
And I'll just write the result.
 

603
00:38:44 --> 00:38:54
It's N times three E int e to
the minus beta E int over one

604
00:38:54 --> 00:39:00
plus three e to the
minus beta E int.

605
00:39:00 --> 00:39:06
And once again, the feature I
want to point out is that

606
00:39:06 --> 00:39:09
there's a factor of N here.
 

607
00:39:09 --> 00:39:13
Again, there's an
energy per molecule.

608
00:39:13 --> 00:39:20
And so if we want, we can also
just write the average E,

609
00:39:20 --> 00:39:25
little E, configurational.
 

610
00:39:25 --> 00:39:29
It's just u
configurational over N.

611
00:39:29 --> 00:39:32
It's the same as this
without the factor of N.

612
00:39:32 --> 00:39:40
Not only that, again, we could
get this directly from the

613
00:39:40 --> 00:39:42
molecular partition
function up there.

614
00:39:42 --> 00:39:46
From little q configurational.
 

615
00:39:46 --> 00:39:48
We'd have the same result.
 

616
00:39:48 --> 00:39:50
Exactly as it should be.
 

617
00:39:50 --> 00:39:55
So if we wrote E as we've seen
in the past, it's just the sum

618
00:39:55 --> 00:40:03
over that energy times the
probability for each state.

619
00:40:03 --> 00:40:08
And that's just the sum
of Ei, e to the minus

620
00:40:08 --> 00:40:13
beta Ei over little q.
 

621
00:40:13 --> 00:40:19
And if you put in the terms,
you immediately get this.

622
00:40:19 --> 00:40:22
Here's our little q, and this
has the interaction energy

623
00:40:22 --> 00:40:29
brought out, multiplied here.
 

624
00:40:29 --> 00:40:33
So the point is, in a case like
this, where you have a bunch of

625
00:40:33 --> 00:40:37
independently behaving
particles, the totals for

626
00:40:37 --> 00:40:41
these, for quantities like
energy, are simply for the

627
00:40:41 --> 00:40:44
system energy, it's just the
average particle energy times

628
00:40:44 --> 00:40:46
the number of particles.
 

629
00:40:46 --> 00:40:48
The number of molecules.
 

630
00:40:48 --> 00:40:52
And remember, before, I spoke a
little bit about the fact that,

631
00:40:52 --> 00:40:55
well, if you look at one
individual molecule and another

632
00:40:55 --> 00:40:57
and another, the energy
will fluctuate.

633
00:40:57 --> 00:40:59
It'll vary considerably.
 

634
00:40:59 --> 00:41:01
Of course, this is a
simple case with only

635
00:41:01 --> 00:41:02
two possible energies.
 

636
00:41:02 --> 00:41:05
But in a case where there may
be many more possible energies,

637
00:41:05 --> 00:41:09
the molecular energies
may vary quite widely.

638
00:41:09 --> 00:41:11
Still, there will be a
well-defined average.

639
00:41:11 --> 00:41:13
And then the system energy
would simply be the total

640
00:41:13 --> 00:41:16
number of molecules times that.
 

641
00:41:16 --> 00:41:18
Where they're all independent.
 

642
00:41:18 --> 00:41:21
Where all the energies are
independent of each other.

643
00:41:21 --> 00:41:25
And, of course, the system
energy fluctuates a great deal

644
00:41:25 --> 00:41:30
less than the individual
molecule energies.

645
00:41:30 --> 00:41:36
OK, so that's our energy term.
 

646
00:41:36 --> 00:41:39
And I think I won't
write out the results.

647
00:41:39 --> 00:41:41
They're in your
notes for entropy.

648
00:41:41 --> 00:41:45
For chemical potential.
 

649
00:41:45 --> 00:41:46
They're all there.
 

650
00:41:46 --> 00:41:48
And again, the same
point would hold.

651
00:41:48 --> 00:41:52
They all scale with N.
 

652
00:41:52 --> 00:41:55
But I want to talk a little
bit about the heat capacity.

653
00:41:55 --> 00:41:59
The expression for
it isn't so simple.

654
00:41:59 --> 00:42:04
So let me just right
Cv configurational.

655
00:42:04 --> 00:42:12
It's du configurational /
dT, at constant V and N.

656
00:42:12 --> 00:42:15
And again, the details are
worked out in the notes.

657
00:42:15 --> 00:42:20
So I won't write out the
whole derivation of it.

658
00:42:20 --> 00:42:24
And even writing out the
results, I'm almost

659
00:42:24 --> 00:42:26
reluctant to do it.
 

660
00:42:26 --> 00:42:27
But I'll put it up.
 

661
00:42:27 --> 00:42:32
Three E int over k T squared,
because there are some things

662
00:42:32 --> 00:42:33
I want to point out about it.
 

663
00:42:33 --> 00:42:45
N, and then one plus three e to
the minus beta E int minus E

664
00:42:45 --> 00:42:52
int e to the minus beta E int.
 

665
00:42:52 --> 00:43:02
Minus e to the minus beta,
I have to get rid of this.

666
00:43:02 --> 00:43:14
Minus e to the minus beta E int
times negative three E int

667
00:43:14 --> 00:43:19
e to the minus beta E int.
 

668
00:43:19 --> 00:43:30
All over one plus three e
to the minus beta E int,

669
00:43:30 --> 00:43:33
that whole thing squared.
 

670
00:43:33 --> 00:43:36
And this is just, of course,
the kind of messy results that

671
00:43:36 --> 00:43:39
comes from taking this
derivative with respect to

672
00:43:39 --> 00:43:43
temperature and doing the chain
rule to get it with respect

673
00:43:43 --> 00:43:44
to beta and so forth.
 

674
00:43:44 --> 00:43:48
So it looks like a little bit
of an intractable, or at

675
00:43:48 --> 00:43:50
least a little bit of a
complicated, function.

676
00:43:50 --> 00:43:52
And the detailed functional
form is complicated.

677
00:43:52 --> 00:43:56
But what I want to emphasize
is that it has simple limits

678
00:43:56 --> 00:43:58
that are very easy to
understand physically.

679
00:43:58 --> 00:44:01
And that are important to
understand for lots of systems.

680
00:44:01 --> 00:44:09
So I just want to look at the
limits of the heat capacity

681
00:44:09 --> 00:44:11
at low and high temperature.
 

682
00:44:11 --> 00:44:15
And this is something that
recurs in statistical

683
00:44:15 --> 00:44:18
mechanics, in an enormous
number of systems where you

684
00:44:18 --> 00:44:20
have simplified limits.
 

685
00:44:20 --> 00:44:21
And they're really important.
 

686
00:44:21 --> 00:44:26
Because what's going
to matter is this.

687
00:44:26 --> 00:44:30
Maybe I shouldn't have covered
up those configurations.

688
00:44:30 --> 00:44:35
So there's a big difference in
what happens when kT is much

689
00:44:35 --> 00:44:37
bigger than this energy.
 

690
00:44:37 --> 00:44:40
Where you know that under those
conditions - let's say it's

691
00:44:40 --> 00:44:43
hot, and this is a small
energy difference.

692
00:44:43 --> 00:44:46
Then the probabilities
are essentially equal.

693
00:44:46 --> 00:44:48
That the molecules are
in this state or in

694
00:44:48 --> 00:44:49
any of these states.
 

695
00:44:49 --> 00:44:51
Because the energy is so tiny
compared to the thermal energy.

696
00:44:51 --> 00:44:54
The molecules are continually
being kicked around between

697
00:44:54 --> 00:44:56
all the states, among
all those states.

698
00:44:56 --> 00:45:00
So the high temperature limit,
physically, is one that's

699
00:45:00 --> 00:45:01
simple to understand.
 

700
00:45:01 --> 00:45:04
Where now you're really just
going to revert to the cases

701
00:45:04 --> 00:45:06
that we've treated before,
where all the energies are

702
00:45:06 --> 00:45:08
effectively the same.
 

703
00:45:08 --> 00:45:11
Because compared
to kT, they are.

704
00:45:11 --> 00:45:14
And in the low temperature
limit, now go to

705
00:45:14 --> 00:45:15
the opposite limit.
 

706
00:45:15 --> 00:45:17
Let's say kT is much,
much, smaller than the

707
00:45:17 --> 00:45:19
interaction energy.
 

708
00:45:19 --> 00:45:22
So that now this term
is really small.

709
00:45:22 --> 00:45:24
Because this is much
bigger than this.

710
00:45:24 --> 00:45:27
So, what it means physically
is all the molecules are

711
00:45:27 --> 00:45:28
in the ground state.
 

712
00:45:28 --> 00:45:30
The probability of this
is basically one.

713
00:45:30 --> 00:45:33
The probability of being in
any of these states is zero.

714
00:45:33 --> 00:45:35
And that's also a
simple result.

715
00:45:35 --> 00:45:38
And there are lots and lots
of cases where one of those

716
00:45:38 --> 00:45:41
limits really obtains.
 

717
00:45:41 --> 00:45:44
If you look at molecules
moving around in the gas

718
00:45:44 --> 00:45:46
phase or in a liquid.
 

719
00:45:46 --> 00:45:47
And you say, well, OK,
let's think about their

720
00:45:47 --> 00:45:49
rotational motion.
 

721
00:45:49 --> 00:45:51
Rotational energies are small.
 

722
00:45:51 --> 00:45:54
At room temperature, kT's
much bigger than them.

723
00:45:54 --> 00:45:57
You immediately go to the
high temperature limit.

724
00:45:57 --> 00:45:59
Now let's go to a small
molecule and look at the

725
00:45:59 --> 00:46:00
vibrational energy.
 

726
00:46:00 --> 00:46:03
In most cases, the vibrational
frequency is pretty,

727
00:46:03 --> 00:46:05
molecules are pretty stiff.
 

728
00:46:05 --> 00:46:07
And in many cases you can
just say, look, forget it.

729
00:46:07 --> 00:46:09
All the molecules are
in the ground state.

730
00:46:09 --> 00:46:13
And again, in the opposite,
but also simple limit

731
00:46:13 --> 00:46:14
ends up holding.
 

732
00:46:14 --> 00:46:17
Or molecular electronic
states, right?

733
00:46:17 --> 00:46:20
If you've got benzene or
hydrogen atoms in, say, at room

734
00:46:20 --> 00:46:23
temperature, how many hydrogen
atoms thermally are going to be

735
00:46:23 --> 00:46:28
up in the n equals two state,
the 2p state or the 2s state?

736
00:46:28 --> 00:46:29
Forget it.
 

737
00:46:29 --> 00:46:31
There's not nearly enough
thermal energy to do that.

738
00:46:31 --> 00:46:36
So these simpler limiting cases
play a huge role in simplifying

739
00:46:36 --> 00:46:38
statistical mechanics and the
calculations from

740
00:46:38 --> 00:46:39
them generally.
 

741
00:46:39 --> 00:46:42
OK, so let's just
see what happens.

742
00:46:42 --> 00:46:54
So, this we want, so
this we can move down.

743
00:46:54 --> 00:47:06
So our limiting cases, when T
goes to zero, not going to

744
00:47:06 --> 00:47:09
work out what happens to all
the betas, beta gets

745
00:47:09 --> 00:47:11
big in that case.
 

746
00:47:11 --> 00:47:18
But the result is that Cv
configurational goes to zero.

747
00:47:18 --> 00:47:20
That's the limit where, like
I've described, here is

748
00:47:20 --> 00:47:24
E interaction, or E int.
 

749
00:47:24 --> 00:47:29
Here is zero.
 

750
00:47:29 --> 00:47:33
And your kT is
barely above zero.

751
00:47:33 --> 00:47:35
So when all the molecules are
here, none of them has enough

752
00:47:35 --> 00:47:38
thermal energy to be up here.
 

753
00:47:38 --> 00:47:42
So why should the heat
capacity be zero?

754
00:47:42 --> 00:47:45
The heat capacity is du/dT.
 

755
00:47:45 --> 00:47:48
It's zero because, here's kT.
 

756
00:47:48 --> 00:47:51
Let's say I increment
kT up a little bit.

757
00:47:51 --> 00:47:55
I just heat the system
a tiny, tiny bit.

758
00:47:55 --> 00:47:58
An infinitesimal amount, to
look at the derivative.

759
00:47:58 --> 00:48:01
Once I do this, how many
molecules are in this

760
00:48:01 --> 00:48:04
higher level now?
 

761
00:48:04 --> 00:48:05
Still zero, right?
 

762
00:48:05 --> 00:48:07
There's still not nearly
enough thermal energy to

763
00:48:07 --> 00:48:09
have any molecules up here.
 

764
00:48:09 --> 00:48:11
So what, was the derivative
of the energy with respect

765
00:48:11 --> 00:48:12
to the temperature?
 

766
00:48:12 --> 00:48:14
I changed the temperature.
 

767
00:48:14 --> 00:48:16
The energy didn't change a bit.
 

768
00:48:16 --> 00:48:22
And that means the heat
capacity is zero.

769
00:48:22 --> 00:48:27
Now let's look at
the other limit.

770
00:48:27 --> 00:48:29
High temperature limit.
 

771
00:48:29 --> 00:48:30
Really, it doesn't
need to be infinity.

772
00:48:30 --> 00:48:34
It's really just T greater
than, much bigger than, kT is

773
00:48:34 --> 00:48:36
much bigger than E interaction.
 

774
00:48:36 --> 00:48:39
And in this case, kT
is much less than the

775
00:48:39 --> 00:48:41
interaction energy.
 

776
00:48:41 --> 00:48:44
So it doesn't need to be
nearly that extreme.

777
00:48:44 --> 00:48:51
Well, what you find out is
the heat capacity is zero.

778
00:48:51 --> 00:48:52
Now, it's zero.
 

779
00:48:52 --> 00:49:02
Because we're in the
following limit.

780
00:49:02 --> 00:49:09
Now kT is way up here.
 

781
00:49:09 --> 00:49:15
Compared to kT, E interaction
is way down here.

782
00:49:15 --> 00:49:16
So, what happens?
 

783
00:49:16 --> 00:49:21
It means that this is so hot,
that term, The e to the

784
00:49:21 --> 00:49:26
minus E int over kT,
forget it, it's one.

785
00:49:26 --> 00:49:29
And so the probability, in
other words, that the molecules

786
00:49:29 --> 00:49:31
are in this state or this
state are essentially equal.

787
00:49:31 --> 00:49:33
So now let's say I raise
the temperature a

788
00:49:33 --> 00:49:34
little bit higher.
 

789
00:49:34 --> 00:49:36
What happens to those
probabilities?

790
00:49:36 --> 00:49:38
What changes?
 

791
00:49:38 --> 00:49:38
Right.
 

792
00:49:38 --> 00:49:39
Nothing changes.
 

793
00:49:39 --> 00:49:43
So what happened to the energy
when I raised the temperature?

794
00:49:43 --> 00:49:44
Of course, nothing
happened to it.

795
00:49:44 --> 00:49:46
Which means the heat
capacity is zero.

796
00:49:46 --> 00:49:53
Now, the first limit that
I described, this one

797
00:49:53 --> 00:49:56
is almost universal.
 

798
00:49:56 --> 00:50:00
For any system where you have
quantized level, you can always

799
00:50:00 --> 00:50:04
eventually get to a low enough
temperature that you're

800
00:50:04 --> 00:50:05
in the first limit.
 

801
00:50:05 --> 00:50:07
Where the kT is lower
than, by far, than the

802
00:50:07 --> 00:50:09
first excited level.
 

803
00:50:09 --> 00:50:11
Everything's in
the ground state.

804
00:50:11 --> 00:50:14
And so you reach this limit.
 

805
00:50:14 --> 00:50:18
This one, it's only cases
where you have a finite

806
00:50:18 --> 00:50:20
number of levels.
 

807
00:50:20 --> 00:50:22
Then you have the same high
temperature limit to it.

808
00:50:22 --> 00:50:27
In other words, the reason this
result happens is because there

809
00:50:27 --> 00:50:30
aren't a bunch of other levels
up here that eventually get

810
00:50:30 --> 00:50:33
to be comparable to kT.
 

811
00:50:33 --> 00:50:36
And not all kinds of systems
or degrees of freedom

812
00:50:36 --> 00:50:37
are like that.
 

813
00:50:37 --> 00:50:41
This one is, because there's a
finite number, four in this

814
00:50:41 --> 00:50:44
case, of configurations.
 

815
00:50:44 --> 00:50:48
So there's just
nothing above there.

816
00:50:48 --> 00:50:50
There's another really
important kind of degree

817
00:50:50 --> 00:50:51
of freedom like that.
 

818
00:50:51 --> 00:50:54
And that's spin.
 

819
00:50:54 --> 00:50:55
Think of proton spins.
 

820
00:50:55 --> 00:50:58
It's plus 1/2 or it's
minus 1/2, and that's it.

821
00:50:58 --> 00:51:02
You can't put any more
spin energy into it.

822
00:51:02 --> 00:51:05
Just like you can't put any
more configurational energy

823
00:51:05 --> 00:51:07
into the system than
to be in this state.

824
00:51:07 --> 00:51:08
That's it.
 

825
00:51:08 --> 00:51:13
So in other words, the maximum
possible energy is finite.

826
00:51:13 --> 00:51:16
Of course, lots of other
degrees of freedom are

827
00:51:16 --> 00:51:17
different from that.
 

828
00:51:17 --> 00:51:18
If you think a molecule's
rotating, they could

829
00:51:18 --> 00:51:21
always spin faster.
 

830
00:51:21 --> 00:51:23
Vibrating, they can
always vibrate harder,

831
00:51:23 --> 00:51:25
translate faster.
 

832
00:51:25 --> 00:51:31
Those degrees of freedom
won't have this limit.

833
00:51:31 --> 00:51:34
They also will have a simple
high temperature limit, but not

834
00:51:34 --> 00:51:36
zero, because, of course, if
there are always more levels,

835
00:51:36 --> 00:51:40
and I keep increasing kT, then
I'll have thermal energy to

836
00:51:40 --> 00:51:42
go into those higher
and higher levels.

837
00:51:42 --> 00:51:44
It'll still go up.
 

838
00:51:44 --> 00:51:48
But for systems with a finite
number of possible levels, and

839
00:51:48 --> 00:51:51
a finite amount of total
energy, for degrees of freedom

840
00:51:51 --> 00:51:55
like that, once I get to a
temperature higher than any of

841
00:51:55 --> 00:51:57
that stuff, then forget it.
 

842
00:51:57 --> 00:51:59
You can't change the
energy anymore thermally.

843
00:51:59 --> 00:52:01
So your heat capacity is zero.
 

844
00:52:01 --> 00:52:04
You can change the temperature
and nothing further happens.

845
00:52:04 --> 00:52:06
OK, next time we'll see what
happens when you do have

846
00:52:06 --> 00:52:10
continuing basically
unbounded possible energies.

847
00:52:10 --> 00:52:10