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PROFESSOR: So, last time we
started in on a discussion of

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00:00:23,020 --> 00:00:25,650
a new topic, with was
statistical mechanics.

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00:00:25,650 --> 00:00:30,730
So what we're trying to do now
is revisit the thermodynamics

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00:00:30,730 --> 00:00:33,850
that we've spent most of the
term deriving and trying to

12
00:00:33,850 --> 00:00:36,860
use in a microscopic approach.

13
00:00:36,860 --> 00:00:40,200
So our hope is to be able to
start with a microscopic model

14
00:00:40,200 --> 00:00:42,930
of matter, starting with atoms
and molecules that we

15
00:00:42,930 --> 00:00:44,360
know are out there.

16
00:00:44,360 --> 00:00:47,850
And formulate thermodynamics
starting from that microscopic

17
00:00:47,850 --> 00:00:48,650
point of view.

18
00:00:48,650 --> 00:00:53,940
In contrast to the way it was
first formulated historically,

19
00:00:53,940 --> 00:00:57,020
and the way we've presented it
also, which is as an entirely

20
00:00:57,020 --> 00:01:01,930
empirical subject based on
macroscopic observation and

21
00:01:01,930 --> 00:01:04,090
deduction from that.

22
00:01:04,090 --> 00:01:09,830
So, we looked at the
probabilities that states with

23
00:01:09,830 --> 00:01:13,310
different energies would be
occupied and we inferred that

24
00:01:13,310 --> 00:01:17,120
there would be a simple way to
describe the distribution of

25
00:01:17,120 --> 00:01:20,830
atoms or molecules in states
of different levels.

26
00:01:20,830 --> 00:01:23,130
First of all, although I didn't
state it explicitly,

27
00:01:23,130 --> 00:01:26,420
essentially assumed there is
that if I've got a bunch of

28
00:01:26,420 --> 00:01:30,740
molecules, and I've got states
that they could be in of equal

29
00:01:30,740 --> 00:01:34,330
energy, then the probability
that they would be in one or

30
00:01:34,330 --> 00:01:38,390
another of those states
is the same.

31
00:01:38,390 --> 00:01:41,770
If the energies of the states
are equal, the probabilities

32
00:01:41,770 --> 00:01:43,630
that those states will
be occupied.

33
00:01:43,630 --> 00:01:47,170
That they'll be populated by
molecules, will be equal.

34
00:01:47,170 --> 00:01:50,070
And then we deduced what's
called the Boltzmann

35
00:01:50,070 --> 00:02:07,850
probability distribution.

36
00:02:07,850 --> 00:02:12,540
Which says that the probability
that a molecular

37
00:02:12,540 --> 00:02:28,770
state, i, will be occupied is
proportional to e to the minus

38
00:02:28,770 --> 00:02:33,770
Ei over kT, where little
Ei is the energy of

39
00:02:33,770 --> 00:02:40,560
that molecular state.

40
00:02:40,560 --> 00:02:44,310
And then realizing that the
probability that some state

41
00:02:44,310 --> 00:02:47,430
out there has to be occupied,
we saw that of course if we

42
00:02:47,430 --> 00:02:50,220
sum over all these probabilities
that sum has to

43
00:02:50,220 --> 00:02:52,350
equal one, right?

44
00:02:52,350 --> 00:02:57,560
In other words, the sum
over all of i of P of

45
00:02:57,560 --> 00:03:00,340
i is equal to one.

46
00:03:00,340 --> 00:03:04,250
So that means that we could
write not just that this is

47
00:03:04,250 --> 00:03:08,730
proportional to this, but we
could write that Pi is equal

48
00:03:08,730 --> 00:03:15,200
to e to the minus Ei over
kT divided by the

49
00:03:15,200 --> 00:03:22,890
sum of all such terms.

50
00:03:22,890 --> 00:03:26,140
So now we know how to figure
out how likely it is that a

51
00:03:26,140 --> 00:03:27,430
certain state is occupied.

52
00:03:27,430 --> 00:03:31,690
And what it means is, let's say
we have a whole bunch of

53
00:03:31,690 --> 00:03:37,560
states whose energy is pretty
low compared to kT. kT, the

54
00:03:37,560 --> 00:03:43,030
Boltzmann constant times
temperature is energy units.

55
00:03:43,030 --> 00:03:45,800
So if it's pretty warm, maybe
it's room temperature, maybe

56
00:03:45,800 --> 00:03:46,910
it's warmer.

57
00:03:46,910 --> 00:03:50,350
And there are a whole lot of
states that are accessible

58
00:03:50,350 --> 00:03:54,840
because the energies
are less than kT.

59
00:03:54,840 --> 00:03:55,960
What does that mean?

60
00:03:55,960 --> 00:03:59,660
That means that you could put a
bunch of different i's here

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00:03:59,660 --> 00:04:02,810
whose energies are all quite
low compared to kT.

62
00:04:02,810 --> 00:04:06,090
And they'll all have significant
probabilities.

63
00:04:06,090 --> 00:04:08,680
Now let's decrease the energy.

64
00:04:08,680 --> 00:04:10,170
Go to cold temperature.

65
00:04:10,170 --> 00:04:13,970
So kT becomes really small.

66
00:04:13,970 --> 00:04:19,070
So that e to the minus energy
over kT starts to, if this

67
00:04:19,070 --> 00:04:22,220
gets to be bigger than kT by a
lot, then this whole thing is

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00:04:22,220 --> 00:04:24,700
a very small number.

69
00:04:24,700 --> 00:04:30,600
Suddenly, you get very few
states accessible.

70
00:04:30,600 --> 00:04:35,200
So if we just plot this
distribution, of course it's

71
00:04:35,200 --> 00:04:38,780
easy to do, it's just a
decaying exponential.

72
00:04:38,780 --> 00:04:43,710
So Pi, which is a
function of Ei.

73
00:04:43,710 --> 00:04:56,780
It just looks like this.

74
00:04:56,780 --> 00:05:00,520
And here are a bunch
of states.

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00:05:00,520 --> 00:05:03,590
And if we have a classical
mechanics picture of matter,

76
00:05:03,590 --> 00:05:06,290
then there would just be
continuous states.

77
00:05:06,290 --> 00:05:08,440
And if we have a quantum
mechanical picture of matter,

78
00:05:08,440 --> 00:05:12,000
there might be individual states
with gaps in between

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00:05:12,000 --> 00:05:15,780
the energies.

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00:05:15,780 --> 00:05:20,260
Either way, what happens, what
this is saying is, if we go

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00:05:20,260 --> 00:05:27,940
out to higher and higher
energies, then you have a

82
00:05:27,940 --> 00:05:31,630
smaller and smaller probability
to be in

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00:05:31,630 --> 00:05:34,640
a state like that.

84
00:05:34,640 --> 00:05:40,650
And if you go to low energies,
the probability gets bigger.

85
00:05:40,650 --> 00:05:42,850
And it depends on temperature.

86
00:05:42,850 --> 00:05:47,310
Because it's the ratio between
the energy and kT that

87
00:05:47,310 --> 00:05:50,530
dictates the size
of that term.

88
00:05:50,530 --> 00:05:57,860
So let's say this is moderate
temperature.

89
00:05:57,860 --> 00:06:07,810
If we go to low temperature, it
might look more like this.

90
00:06:07,810 --> 00:06:11,200
Hardly any states can be
occupied because even states

91
00:06:11,200 --> 00:06:16,490
of rather moderate energies,
suddenly now those energies

92
00:06:16,490 --> 00:06:18,090
are much bigger than kT.

93
00:06:18,090 --> 00:06:20,420
In other words, kT is measuring
thermal energy.

94
00:06:20,420 --> 00:06:23,500
It's saying there's not enough
thermal energy to populate, to

95
00:06:23,500 --> 00:06:26,390
knock things into states
whose energy is

96
00:06:26,390 --> 00:06:29,940
much higher than that.

97
00:06:29,940 --> 00:06:33,400
So you have a very precipitous
decay.

98
00:06:33,400 --> 00:06:35,490
If you go to the other
extreme, of very high

99
00:06:35,490 --> 00:06:47,200
temperature, then this will
tend to flatten out.

100
00:06:47,200 --> 00:06:53,640
Eventually it'll decay, but
it may take a long time.

101
00:06:53,640 --> 00:06:57,410
Because now kT is enormous.

102
00:06:57,410 --> 00:07:00,740
So the energy has to
get to be very big.

103
00:07:00,740 --> 00:07:02,940
Before it's bigger than kT.

104
00:07:02,940 --> 00:07:04,590
And as long as it
isn't, then this

105
00:07:04,590 --> 00:07:06,990
exponential term is not small.

106
00:07:06,990 --> 00:07:09,950
So there be a whole bunch of
states that may be occupied.

107
00:07:09,950 --> 00:07:12,850
In other words, a whole bunch
of states that are thermally

108
00:07:12,850 --> 00:07:15,820
accessible at equilibrium.

109
00:07:15,820 --> 00:07:19,300
Systems at thermal equilibrium,
molecules are

110
00:07:19,300 --> 00:07:20,990
getting knocked around
with whatever

111
00:07:20,990 --> 00:07:22,400
thermal energy is available.

112
00:07:22,400 --> 00:07:26,990
Crashing into each other or into
the walls of a vessel.

113
00:07:26,990 --> 00:07:30,790
And there'll be a distribution
of molecular energies.

114
00:07:30,790 --> 00:07:33,340
And that distribution is skewed
either low or high

115
00:07:33,340 --> 00:07:37,710
depending on the temperature.

116
00:07:37,710 --> 00:07:40,330
So that's what that
distribution, the Boltzmann

117
00:07:40,330 --> 00:07:43,970
distribution, is telling us.

118
00:07:43,970 --> 00:07:46,870
This is often called a Boltzmann
factor, because it's

119
00:07:46,870 --> 00:07:53,460
telling what the population of
some particular state is.

120
00:07:53,460 --> 00:07:57,580
OK.

121
00:07:57,580 --> 00:08:01,980
Now, this is just dealing with
individual molecule states.

122
00:08:01,980 --> 00:08:05,660
Then we said, OK, what about
the whole system?

123
00:08:05,660 --> 00:08:08,880
Well, the same kind of relation
holds there, too.

124
00:08:08,880 --> 00:08:12,260
In other words, if these are
individual molecule energies,

125
00:08:12,260 --> 00:08:29,240
now if I look at the entire
system, right well, still,

126
00:08:29,240 --> 00:08:31,850
there's no reason that same
distribution doesn't hold.

127
00:08:31,850 --> 00:08:33,140
And it does.

128
00:08:33,140 --> 00:08:44,800
So in other words, Pi of Ei,
that's the whole system

129
00:08:44,800 --> 00:08:55,870
energy, is e to the minus Ei
over kT, over the sum over i,

130
00:08:55,870 --> 00:08:57,920
Ei over kT.

131
00:08:57,920 --> 00:09:02,920
Now, this i doesn't refer to
a single molecule state.

132
00:09:02,920 --> 00:09:03,920
We're talking about
a whole system.

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00:09:03,920 --> 00:09:06,820
It might be a mole of atoms or
molecules in the gas phase, or

134
00:09:06,820 --> 00:09:09,020
what have you.

135
00:09:09,020 --> 00:09:13,170
It refers to a system state
where the energy, the state of

136
00:09:13,170 --> 00:09:19,370
every one of those atoms or
molecules is specified.

137
00:09:19,370 --> 00:09:30,650
So this is a system
microstate.

138
00:09:30,650 --> 00:09:40,990
Every molecular state
is specified.

139
00:09:40,990 --> 00:09:43,920
So for a mole of stuff, that
means there might be 10 to the

140
00:09:43,920 --> 00:09:51,680
24 or so molecular states that
this single subscript is

141
00:09:51,680 --> 00:09:53,770
indicating.

142
00:09:53,770 --> 00:09:55,640
But the point is, those
states exist.

143
00:09:55,640 --> 00:09:57,050
They have a total energy.

144
00:09:57,050 --> 00:09:59,530
What's the probability that the
whole system will be in

145
00:09:59,530 --> 00:10:00,400
such a state?

146
00:10:00,400 --> 00:10:12,750
Still going to be proportional
to the total system energy.

147
00:10:12,750 --> 00:10:30,170
It turns out that these
summations end up taking on an

148
00:10:30,170 --> 00:10:33,240
enormous importance in
statistical mechanics.

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00:10:33,240 --> 00:10:36,990
And the reason is, as we'll see
shortly, it turns out that

150
00:10:36,990 --> 00:10:41,660
every single macroscopic
thermodynamic function can be

151
00:10:41,660 --> 00:10:45,260
derived by knowing just that.

152
00:10:45,260 --> 00:10:51,330
Just these sums, what are called
partition functions.

153
00:10:51,330 --> 00:10:55,050
So of course they take on
enormous importance.

154
00:10:55,050 --> 00:11:05,540
So, we call them partition
functions because what they're

155
00:11:05,540 --> 00:11:08,620
doing is, they're indicating
how the molecules are

156
00:11:08,620 --> 00:11:11,550
partitioned among the different
available levels.

157
00:11:11,550 --> 00:11:15,280
The molecules or
a whole system.

158
00:11:15,280 --> 00:11:37,850
So the molecular partition
function is labeled little q.

159
00:11:37,850 --> 00:11:47,230
And the system partition
function is labeled big Q.

160
00:11:47,230 --> 00:11:59,740
It's called the canonical
partition function.

161
00:11:59,740 --> 00:12:03,910
And because they are going
to take on such special

162
00:12:03,910 --> 00:12:11,450
importance, let's just look at
some of their properties for

163
00:12:11,450 --> 00:12:18,210
different kinds of systems
and in general.

164
00:12:18,210 --> 00:12:24,000
OK, first of all, let's talk
about units and values.

165
00:12:24,000 --> 00:12:26,370
They are unitless.

166
00:12:26,370 --> 00:12:28,240
This is an exponential
function.

167
00:12:28,240 --> 00:12:30,680
Here are units of energy
and energy.

168
00:12:30,680 --> 00:12:32,110
But this is a unitless
was number.

169
00:12:32,110 --> 00:12:33,970
It's just some number, right?

170
00:12:33,970 --> 00:12:34,860
Could be 1.

171
00:12:34,860 --> 00:12:35,700
Could be 10.

172
00:12:35,700 --> 00:12:36,210
Could be 50.

173
00:12:36,210 --> 00:12:37,700
Whatever, right?

174
00:12:37,700 --> 00:12:41,420
Could be 10 to the 24.

175
00:12:41,420 --> 00:12:49,150
Its magnitude tells you about
more or less how many states

176
00:12:49,150 --> 00:12:51,310
are thermally accessible.

177
00:12:51,310 --> 00:12:52,500
Because, look at it.

178
00:12:52,500 --> 00:12:55,710
And then go back to the example
that I showed you if

179
00:12:55,710 --> 00:13:01,300
lots of these terms are big, are
significant because this

180
00:13:01,300 --> 00:13:02,960
is really a big number, right?

181
00:13:02,960 --> 00:13:04,650
It's really hot.

182
00:13:04,650 --> 00:13:08,300
So lots of states have energies
lower than kT, which

183
00:13:08,300 --> 00:13:12,420
means this is not too small, for
lots and lots of values of

184
00:13:12,420 --> 00:13:14,910
i, then this just
keeps adding up.

185
00:13:14,910 --> 00:13:17,230
And of course, same here.

186
00:13:17,230 --> 00:13:20,020
So the number might
be very large.

187
00:13:20,020 --> 00:13:22,420
But if we're in the low
temperature limit, maybe we're

188
00:13:22,420 --> 00:13:27,960
in such low temperature that
only the lowest possible state

189
00:13:27,960 --> 00:13:28,460
is occupied.

190
00:13:28,460 --> 00:13:30,780
And everything else,
it's just too cold.

191
00:13:30,780 --> 00:13:32,990
There's not enough thermal
energy to occupy anything but

192
00:13:32,990 --> 00:13:35,200
the very lowest state
available.

193
00:13:35,200 --> 00:13:39,520
Well, in that case the lowest
state, this would be one.

194
00:13:39,520 --> 00:13:41,340
Essentially we could
label this zero.

195
00:13:41,340 --> 00:13:44,460
We could put the energy, the
zero of energy there.

196
00:13:44,460 --> 00:13:47,370
Everything else is really big
compared to kT, which means

197
00:13:47,370 --> 00:13:50,120
this exponential gets to be a
really small number for every

198
00:13:50,120 --> 00:13:51,510
state except one.

199
00:13:51,510 --> 00:13:54,350
And this is just equal to one.

200
00:13:54,350 --> 00:13:59,510
In other words, the magnitude
of these numbers tells us

201
00:13:59,510 --> 00:14:02,820
about how many states are
accessible, thermally

202
00:14:02,820 --> 00:14:09,410
accessible, to molecules
or to a whole system.

203
00:14:09,410 --> 00:14:12,970
So let's just go through a
couple of specific examples to

204
00:14:12,970 --> 00:14:19,490
try to make that a little
more concrete.

205
00:14:19,490 --> 00:14:21,900
One is, let's start in
the simplest case.

206
00:14:21,900 --> 00:14:24,470
Which it'll turn out
I just alluded to.

207
00:14:24,470 --> 00:14:29,130
Let's consider a perfect atomic
crystal at essentially

208
00:14:29,130 --> 00:14:40,310
zero degrees Kelvin.

209
00:14:40,310 --> 00:14:41,940
Zero Kelvin.

210
00:14:41,940 --> 00:14:43,990
So every atom.

211
00:14:43,990 --> 00:14:46,130
Or even if it's a molecular
crystal, every molecule,

212
00:14:46,130 --> 00:14:47,230
they're all in the
ground state.

213
00:14:47,230 --> 00:14:50,190
There's no excess
thermal energy.

214
00:14:50,190 --> 00:14:52,380
Every molecule is in
its proper place.

215
00:14:52,380 --> 00:14:56,570
Every atom, if it's
an atomic lattice.

216
00:14:56,570 --> 00:14:59,990
In other words, it's in
the ground state.

217
00:14:59,990 --> 00:15:01,910
That's it.

218
00:15:01,910 --> 00:15:06,490
So again, we can place the
zero where we like.

219
00:15:06,490 --> 00:15:08,030
We'll place it there.

220
00:15:08,030 --> 00:15:12,490
That one, for that one state,
this will be equal to one.

221
00:15:12,490 --> 00:15:25,110
And for everything else
it'll be zero.

222
00:15:25,110 --> 00:15:33,800
So Q is just the sum of e
to the minus Ei over kT.

223
00:15:33,800 --> 00:15:40,420
It's equal to e to the minus
zero over kT plus e to the

224
00:15:40,420 --> 00:15:47,790
minus E1 over kT plus e to
the minus E2 over kT.

225
00:15:47,790 --> 00:15:49,770
But remember, T is
really tiny.

226
00:15:49,770 --> 00:15:52,870
It's almost zero
degrees Kelvin.

227
00:15:52,870 --> 00:15:57,180
So all these things are
much bigger than that.

228
00:15:57,180 --> 00:15:59,160
So this is vanishingly small.

229
00:15:59,160 --> 00:16:00,360
This is vanishingly small.

230
00:16:00,360 --> 00:16:04,900
The whole thing is approximately
equal to one.

231
00:16:04,900 --> 00:16:08,290
That's it.

232
00:16:08,290 --> 00:16:13,890
Also, if we say OK, now what's
the probability of the system

233
00:16:13,890 --> 00:16:15,280
being in a particular state.

234
00:16:15,280 --> 00:16:21,220
Well, we have an expression
for that.

235
00:16:21,220 --> 00:16:25,810
So let's look at P0.

236
00:16:25,810 --> 00:16:33,260
It's e to the minus E0
over kT over the sum.

237
00:16:33,260 --> 00:16:42,180
Which we've just seen.

238
00:16:42,180 --> 00:16:46,840
So it's e to the minus E0 over
to kT divided by e to the

239
00:16:46,840 --> 00:16:57,110
minus E0 over kT plus e to the
minus E1 over kT, and so on.

240
00:16:57,110 --> 00:17:00,270
This is the only term
that's significant.

241
00:17:00,270 --> 00:17:03,470
So it's approximately
equal to one.

242
00:17:03,470 --> 00:17:06,670
Now, while I've got this written
here, let's just make

243
00:17:06,670 --> 00:17:09,370
sure we've got something
clear.

244
00:17:09,370 --> 00:17:12,560
What if we hadn't arbitrarily
set the zero of

245
00:17:12,560 --> 00:17:14,360
energy equal to zero?

246
00:17:14,360 --> 00:17:15,790
I mean, it's arbitrary, right?

247
00:17:15,790 --> 00:17:17,600
We can put the zero of
energy anywhere.

248
00:17:17,600 --> 00:17:19,240
And you might think, well, gee
that's going to have a big

249
00:17:19,240 --> 00:17:21,070
effect on everything.

250
00:17:21,070 --> 00:17:25,560
Well, it would have an effect
on the actual number that we

251
00:17:25,560 --> 00:17:29,700
get for Q. But what you'll see
is that any measurable

252
00:17:29,700 --> 00:17:34,000
quantity that we calculate
won't be affected.

253
00:17:34,000 --> 00:17:36,480
It's only the zero of the energy
scale that's going to

254
00:17:36,480 --> 00:17:37,480
be affected.

255
00:17:37,480 --> 00:17:41,260
So for example, let's look
at this probability.

256
00:17:41,260 --> 00:17:46,030
It doesn't matter where we
put the zero of energy.

257
00:17:46,030 --> 00:17:48,300
This term is still going
to be enormously bigger

258
00:17:48,300 --> 00:17:49,560
then the next one.

259
00:17:49,560 --> 00:17:50,280
And the next one.

260
00:17:50,280 --> 00:17:52,560
And the next one.

261
00:17:52,560 --> 00:17:56,310
So in this sum only this term
is going to matter.

262
00:17:56,310 --> 00:17:58,190
It's going to cancel
with this term.

263
00:17:58,190 --> 00:18:00,630
So whether or not these
individual terms are equal to

264
00:18:00,630 --> 00:18:04,650
one, which happens if we set
this to zero, or whether

265
00:18:04,650 --> 00:18:06,480
they're equal to some
other number.

266
00:18:06,480 --> 00:18:10,330
Still, the probability that the
system is in the lowest

267
00:18:10,330 --> 00:18:13,150
state is one, right?

268
00:18:13,150 --> 00:18:16,770
That doesn't depend on where we
arbitrarily put the zero of

269
00:18:16,770 --> 00:18:18,010
the energy scale.

270
00:18:18,010 --> 00:18:20,240
The state is still going to
be in the ground state, at

271
00:18:20,240 --> 00:18:23,440
essentially zero
degrees Kelvin.

272
00:18:23,440 --> 00:18:25,750
And it'll turn out to be that
way with any property that we

273
00:18:25,750 --> 00:18:26,580
can measure.

274
00:18:26,580 --> 00:18:30,450
Only the zero of this scale
moves if we arbitrarily move

275
00:18:30,450 --> 00:18:32,800
it somewhere.

276
00:18:32,800 --> 00:18:36,945
But the observable, measurable
quantities that we calculate,

277
00:18:36,945 --> 00:18:38,010
they won't change.

278
00:18:38,010 --> 00:18:41,610
Other than that scale.

279
00:18:41,610 --> 00:18:47,520
So that's one simple example.

280
00:18:47,520 --> 00:18:50,260
Now let's look at
another example.

281
00:18:50,260 --> 00:18:53,220
Let's consider a mole of atoms,
roaming around in the

282
00:18:53,220 --> 00:19:06,860
gas phase at room temperature.

283
00:19:06,860 --> 00:19:11,240
OK, so now what I want to do
is just have a simple model

284
00:19:11,240 --> 00:19:14,150
for their translational
motion.

285
00:19:14,150 --> 00:19:16,380
And of course, we could
do that either quantum

286
00:19:16,380 --> 00:19:19,390
mechanically or classically
solve for that.

287
00:19:19,390 --> 00:19:22,240
We're going to use an
even simpler model.

288
00:19:22,240 --> 00:19:24,780
And this model is going to be
very, very useful for a lot of

289
00:19:24,780 --> 00:19:26,580
the things that we'll treat.

290
00:19:26,580 --> 00:19:29,830
It's called a lattice model.

291
00:19:29,830 --> 00:19:32,300
All it means is, we're
going to divide up

292
00:19:32,300 --> 00:19:33,620
the available volume.

293
00:19:33,620 --> 00:19:36,870
This room, for example,
into zillions

294
00:19:36,870 --> 00:19:38,250
of tiny little elements.

295
00:19:38,250 --> 00:19:39,280
Little volume elements.

296
00:19:39,280 --> 00:19:43,770
Each one about the
size of an atom.

297
00:19:43,770 --> 00:19:46,610
The idea being, we're going to
specify the state of the atom

298
00:19:46,610 --> 00:19:48,560
by saying where is it.

299
00:19:48,560 --> 00:19:50,690
Is it in this lattice sight,
in this one, in this

300
00:19:50,690 --> 00:19:53,520
one, in this one.

301
00:19:53,520 --> 00:20:11,670
So it's a lattice model.

302
00:20:11,670 --> 00:20:13,910
Might be an atom there.

303
00:20:13,910 --> 00:20:19,190
Might be one there.

304
00:20:19,190 --> 00:20:22,650
So we're going to divide
the volume up.

305
00:20:22,650 --> 00:20:37,880
So let's call our atomic volume
little v. Our total

306
00:20:37,880 --> 00:20:43,720
volume big V. And an atomic
volume, it's going to be on

307
00:20:43,720 --> 00:20:48,110
the order of 1 angstroms
cubed.

308
00:20:48,110 --> 00:20:52,540
Or 10 to the minus
30 meters cubed.

309
00:20:52,540 --> 00:20:55,850
And our room, our volume
macroscopic one, might be on

310
00:20:55,850 --> 00:20:59,380
the order of one meter cubed.

311
00:20:59,380 --> 00:21:03,420
Ordinary sort of size.

312
00:21:03,420 --> 00:21:09,730
So now let's figure out our
molecular partition function.

313
00:21:09,730 --> 00:21:13,260
Now, implicit in this, we're
basically saying that the

314
00:21:13,260 --> 00:21:17,280
energy, the translational
energy, is basically zero.

315
00:21:17,280 --> 00:21:20,070
In other words, all these states
have the same energy.

316
00:21:20,070 --> 00:21:23,380
They're just located in
different positions.

317
00:21:23,380 --> 00:21:27,520
At any given instant of time.

318
00:21:27,520 --> 00:21:30,140
And what that means is all
these terms, all these

319
00:21:30,140 --> 00:21:32,350
Boltzmann factors, we just
set them equal to one.

320
00:21:32,350 --> 00:21:34,940
We'll set the zero of energy
there, be done with it.

321
00:21:34,940 --> 00:21:36,260
It's a simple model.

322
00:21:36,260 --> 00:21:38,460
But it's going to give us the
right order of magnitude that

323
00:21:38,460 --> 00:21:39,860
we're after.

324
00:21:39,860 --> 00:21:45,490
So, how many states are there
that are accessible?

325
00:21:45,490 --> 00:21:50,950
Well, on the order of 10
to the 30th, right?

326
00:21:50,950 --> 00:21:57,490
So q, little q, we'll call it
little q translational, it's

327
00:21:57,490 --> 00:22:02,520
just discussing where
things are.

328
00:22:02,520 --> 00:22:05,230
Is on the order of
10 to the 30th.

329
00:22:05,230 --> 00:22:09,000
And if we do a more careful
treatment, if we treat the

330
00:22:09,000 --> 00:22:12,010
translational energy of atoms,
either classically or quantum

331
00:22:12,010 --> 00:22:13,870
mechanically and solve it.

332
00:22:13,870 --> 00:22:15,770
We'll still get, we still
do get, about the

333
00:22:15,770 --> 00:22:20,850
same order of magnitude.

334
00:22:20,850 --> 00:22:23,610
Now let's treat the
whole system.

335
00:22:23,610 --> 00:22:25,890
So, what happens?

336
00:22:25,890 --> 00:22:30,930
What are our total
possible states?

337
00:22:30,930 --> 00:22:33,940
Because we have to add this up
for every possible state.

338
00:22:33,940 --> 00:22:36,720
Well, let's start with
the first atom.

339
00:22:36,720 --> 00:22:37,990
It has to be somewhere.

340
00:22:37,990 --> 00:22:39,990
It has 10 to the 30th
possibilities for

341
00:22:39,990 --> 00:22:41,810
where it can be.

342
00:22:41,810 --> 00:22:43,940
Let's start with the
second atom.

343
00:22:43,940 --> 00:22:45,110
And put it somewhere.

344
00:22:45,110 --> 00:22:47,840
Well, it has 10 to the 30th
minus one, which is still

345
00:22:47,840 --> 00:22:50,190
pretty close to 10
to the 30th.

346
00:22:50,190 --> 00:22:52,170
Let's let's go to
the third atom.

347
00:22:52,170 --> 00:22:53,870
And the fourth.

348
00:22:53,870 --> 00:22:56,220
If we have about a mole of
atoms, let's say 10 to the

349
00:22:56,220 --> 00:23:00,480
24th atoms, that's still going
to occupy a very small

350
00:23:00,480 --> 00:23:01,790
fraction of the site.

351
00:23:01,790 --> 00:23:03,430
Only one in a million.

352
00:23:03,430 --> 00:23:05,530
So we don't have to keep
careful track.

353
00:23:05,530 --> 00:23:08,030
Every one of them, we can just
say, look, there are 10 to the

354
00:23:08,030 --> 00:23:11,370
30th available sites.

355
00:23:11,370 --> 00:23:13,120
Because we're not going to worry
about a change in one in

356
00:23:13,120 --> 00:23:15,340
a millionth.

357
00:23:15,340 --> 00:23:24,860
So what that means is, capital
Q, trans for the system, is 10

358
00:23:24,860 --> 00:23:29,620
to the 30th times
10 to the 30th.

359
00:23:29,620 --> 00:23:31,900
In other words, let's now
take the whole state.

360
00:23:31,900 --> 00:23:34,260
Well, I could put
atom one here.

361
00:23:34,260 --> 00:23:36,120
I have 10 to the 30th
possibilities.

362
00:23:36,120 --> 00:23:37,620
Atom two, I can put
anywhere else.

363
00:23:37,620 --> 00:23:40,640
So the joint probability of
that particular state, for

364
00:23:40,640 --> 00:23:44,270
just the two atoms, is 10th to
the 30th times 10 to the 30th.

365
00:23:44,270 --> 00:23:46,750
It's 10 to the 30th squared.

366
00:23:46,750 --> 00:23:56,470
So this is going
to keep going.

367
00:23:56,470 --> 00:24:00,780
It's going to be 10 to the
30th to the Nth power.

368
00:24:00,780 --> 00:24:02,660
Where N is the number
of atoms, which

369
00:24:02,660 --> 00:24:06,060
is 10 of the 24th.

370
00:24:06,060 --> 00:24:07,860
Huge number, right?

371
00:24:07,860 --> 00:24:09,840
It is a huge number.

372
00:24:09,840 --> 00:24:12,840
And that, too, if we treat the
whole thing classically or

373
00:24:12,840 --> 00:24:15,090
quantum mechanically and
work it all out.

374
00:24:15,090 --> 00:24:16,200
We'll get that number.

375
00:24:16,200 --> 00:24:17,790
Or something on that order.

376
00:24:17,790 --> 00:24:22,000
Because you know there really
is a simply astronomical

377
00:24:22,000 --> 00:24:26,890
number of states accessible to
the whole bunch of atoms or

378
00:24:26,890 --> 00:24:29,670
molecules in this room.

379
00:24:29,670 --> 00:24:32,210
It really is that big.

380
00:24:32,210 --> 00:24:34,330
So in other words, capital
Q is just an

381
00:24:34,330 --> 00:24:37,600
astronomical number.

382
00:24:37,600 --> 00:24:42,080
And it is the case.

383
00:24:42,080 --> 00:24:42,950
OK.

384
00:24:42,950 --> 00:24:47,590
There's an important
sort of nuance

385
00:24:47,590 --> 00:24:51,430
that we need to introduce.

386
00:24:51,430 --> 00:24:53,150
And it's the following.

387
00:24:53,150 --> 00:24:54,880
Turns out, it is an astronomical
number.

388
00:24:54,880 --> 00:24:58,530
But a tiny bit less astronomical
than what I've

389
00:24:58,530 --> 00:25:00,300
treated so far would indicate.

390
00:25:00,300 --> 00:25:02,300
So let's look a little
more carefully.

391
00:25:02,300 --> 00:25:08,910
What I've said is that Q
translational is little q

392
00:25:08,910 --> 00:25:12,130
translational, that is, that
10 to the 30th number.

393
00:25:12,130 --> 00:25:15,150
To the Nth power.

394
00:25:15,150 --> 00:25:17,220
But there's one failing here.

395
00:25:17,220 --> 00:25:23,030
Which is, when I got that, when
I decided that if I have

396
00:25:23,030 --> 00:25:26,000
just the first two atoms I've
got 10th to the 30th times 10

397
00:25:26,000 --> 00:25:30,090
to the 30th possible states, the
trouble with that is then

398
00:25:30,090 --> 00:25:32,830
if I keep counting all the
possible atoms starting with

399
00:25:32,830 --> 00:25:37,980
each one, I'll double-count
it, right?

400
00:25:37,980 --> 00:25:41,360
In other words, what if I
interchange those two atoms?

401
00:25:41,360 --> 00:25:43,680
Well, those states are
identical, right?

402
00:25:43,680 --> 00:25:45,730
Indistinguishable.

403
00:25:45,730 --> 00:25:51,700
And it's only distinguishable
states that count.

404
00:25:51,700 --> 00:25:56,550
When you specify these
things, these are

405
00:25:56,550 --> 00:26:00,140
indicating distinct states.

406
00:26:00,140 --> 00:26:01,560
In some way, at least in

407
00:26:01,560 --> 00:26:04,020
principle, measurably different.

408
00:26:04,020 --> 00:26:07,150
If the atom's are identical, in
other words, if it's a mole

409
00:26:07,150 --> 00:26:10,810
of the same stuff, I don't have
a way of distinguishing

410
00:26:10,810 --> 00:26:11,620
between those two.

411
00:26:11,620 --> 00:26:14,080
I have to correct for that.

412
00:26:14,080 --> 00:26:17,050
And when I come to the third
atom, I have to correct for

413
00:26:17,050 --> 00:26:18,480
all the possible interchanges.

414
00:26:18,480 --> 00:26:20,730
Of course, it's three
factorial.

415
00:26:20,730 --> 00:26:23,230
And in general, it's
N factorial.

416
00:26:23,230 --> 00:26:26,430
So this result needs
to be modified.

417
00:26:26,430 --> 00:26:37,330
This is true for distinguishable
particles.

418
00:26:37,330 --> 00:26:40,170
In other words, if I had all
different atoms, so I could

419
00:26:40,170 --> 00:26:42,980
label them all, then I don't
have that correction.

420
00:26:42,980 --> 00:26:45,570
Because then there's really a
difference between that state

421
00:26:45,570 --> 00:26:48,400
and the state with the two
atoms interchanged.

422
00:26:48,400 --> 00:26:50,780
But if it's a mole of identical
atoms, that's no

423
00:26:50,780 --> 00:26:51,580
longer the case.

424
00:26:51,580 --> 00:26:58,250
So then, Q translational is
little q translational to the

425
00:26:58,250 --> 00:27:04,190
N power divided by
N factorial.

426
00:27:04,190 --> 00:27:07,370
It's still going to be an
absolutely enormous number.

427
00:27:07,370 --> 00:27:09,560
But it's going to be a little
less absolutely enormous than

428
00:27:09,560 --> 00:27:13,170
it was a minute ago.

429
00:27:13,170 --> 00:27:15,930
Now finally, I just want
to introduce a handy

430
00:27:15,930 --> 00:27:19,650
approximation to N factorial
that's going to turn out to be

431
00:27:19,650 --> 00:27:22,450
very useful again and again.

432
00:27:22,450 --> 00:27:31,850
And that's called Stirling's
approximation.

433
00:27:31,850 --> 00:27:40,750
For the log of a big number,
ln N factorial is N log N

434
00:27:40,750 --> 00:27:48,670
minus N. If we take e to those,
both sides, then we

435
00:27:48,670 --> 00:27:53,910
find that N factorial is equal
to e to the minus N

436
00:27:53,910 --> 00:28:02,030
N to the nth power.

437
00:28:02,030 --> 00:28:07,160
So this, then, is approximately
equal to q

438
00:28:07,160 --> 00:28:14,980
translational to
the Nth power.

439
00:28:14,980 --> 00:28:21,030
Over N to the N times e to the
minus N. Now let's put the

440
00:28:21,030 --> 00:28:24,910
numbers back in.

441
00:28:24,910 --> 00:28:32,280
There's our 10 to the 30th to
the 10 to the 24th power.

442
00:28:32,280 --> 00:28:35,880
And now we're going to diminish
that just a bit.

443
00:28:35,880 --> 00:28:37,090
It's N to the Nth power.

444
00:28:37,090 --> 00:28:47,600
So it's 10 to the 24th to the
10 to the 24th power.

445
00:28:47,600 --> 00:28:55,350
Times e to the minus 10
to the 24th power.

446
00:28:55,350 --> 00:28:58,040
So, we can cancel
something here.

447
00:28:58,040 --> 00:29:05,620
So we have 10 to the 6th to the
10 to the 24th power times

448
00:29:05,620 --> 00:29:14,740
e to the 10 to the 24th
power. e is about 10

449
00:29:14,740 --> 00:29:17,480
to the 0.4th power.

450
00:29:17,480 --> 00:29:23,630
So this whole thing is about 10
to the 6.4th power to the

451
00:29:23,630 --> 00:29:28,590
10 to the 24th power
It's still a

452
00:29:28,590 --> 00:29:31,850
pretty respectable number.

453
00:29:31,850 --> 00:29:33,920
You could still say
astronomical.

454
00:29:33,920 --> 00:29:35,820
But maybe astronomical,
but not quite to

455
00:29:35,820 --> 00:29:36,790
the edge of the universe.

456
00:29:36,790 --> 00:29:39,490
Whereas the one before maybe hit
the edge of the universe

457
00:29:39,490 --> 00:29:44,880
and then beyond.

458
00:29:44,880 --> 00:29:46,430
So that's our second example.

459
00:29:46,430 --> 00:29:48,970
And again, part of what I'm
trying to do here is just

460
00:29:48,970 --> 00:29:50,320
introduce some ideas.

461
00:29:50,320 --> 00:29:51,840
Things like this
way of modeling

462
00:29:51,840 --> 00:29:53,170
positions and so forth.

463
00:29:53,170 --> 00:29:55,340
But also, again, orders
of magnitude.

464
00:29:55,340 --> 00:29:58,470
How big are these numbers.

465
00:29:58,470 --> 00:30:00,280
For different sorts
of systems.

466
00:30:00,280 --> 00:30:02,660
So let's do one more example.

467
00:30:02,660 --> 00:30:07,170
Now, let's consider a
polymer in a liquid.

468
00:30:07,170 --> 00:30:08,760
And it has different
configurations.

469
00:30:08,760 --> 00:30:10,620
And they might be a little
bit different in energy.

470
00:30:10,620 --> 00:30:13,630
For example, some configurations
might bring

471
00:30:13,630 --> 00:30:16,310
neighboring regions of the
polymer into proximity where

472
00:30:16,310 --> 00:30:18,280
they could hydrogen bond.

473
00:30:18,280 --> 00:30:22,710
And the point is that then the
molecular energies involved

474
00:30:22,710 --> 00:30:23,940
will change a little bit.

475
00:30:23,940 --> 00:30:26,790
Because of some sorts of
interactions that are possible

476
00:30:26,790 --> 00:30:28,180
in some configurations.

477
00:30:28,180 --> 00:30:33,410
And not in other
configurations.

478
00:30:33,410 --> 00:30:36,360
So what I'm trying to do is
introduce a very simple

479
00:30:36,360 --> 00:30:38,750
framework through which we
might be able to look at

480
00:30:38,750 --> 00:30:42,880
things like protein folding,
or DNA hydrogen bonding.

481
00:30:42,880 --> 00:30:43,610
Things like this.

482
00:30:43,610 --> 00:30:46,290
And just in a simple way model
how those work and what the

483
00:30:46,290 --> 00:31:15,950
forces are that drive them.

484
00:31:15,950 --> 00:31:22,960
So we'll think about polymer
configurations.

485
00:31:22,960 --> 00:31:28,310
So let's look at a few
configurations.

486
00:31:28,310 --> 00:31:33,000
Here is going to be one.

487
00:31:33,000 --> 00:31:37,780
I'm going to label
this one here.

488
00:31:37,780 --> 00:32:02,750
And then here are
a few others.

489
00:32:02,750 --> 00:32:05,740
So I've labeled them this way
so that you can see how they

490
00:32:05,740 --> 00:32:07,740
are distinct from each other.

491
00:32:07,740 --> 00:32:12,850
This one would have a possible
interaction.

492
00:32:12,850 --> 00:32:14,690
So we'll label its energy.

493
00:32:14,690 --> 00:32:25,290
So this is molecular state
i, here's going to

494
00:32:25,290 --> 00:32:28,890
be our energy, Ei.

495
00:32:28,890 --> 00:32:33,120
And let's call this one
negative e int, for an

496
00:32:33,120 --> 00:32:36,600
interaction energy
that's favorable.

497
00:32:36,600 --> 00:32:44,490
And these will be zero.

498
00:32:44,490 --> 00:32:46,250
Let's just put that there.

499
00:32:46,250 --> 00:32:56,030
Zero, zero, zero.

500
00:32:56,030 --> 00:33:01,630
And let's also indicate
the degeneracy.

501
00:33:01,630 --> 00:33:03,890
How many states, different
states are there

502
00:33:03,890 --> 00:33:07,880
with the same energy.

503
00:33:07,880 --> 00:33:09,830
And that's called gi.

504
00:33:09,830 --> 00:33:11,650
And here it's one.

505
00:33:11,650 --> 00:33:16,560
And here are these three.

506
00:33:16,560 --> 00:33:20,240
So that's the framework
of our model.

507
00:33:20,240 --> 00:33:25,700
Well, so what's our molecular
partition function for this

508
00:33:25,700 --> 00:33:29,760
configurational degree
of freedom?

509
00:33:29,760 --> 00:33:37,720
Well, we can label it little
q configurational.

510
00:33:37,720 --> 00:33:43,920
So we're going to sum over these
states. e to the minus

511
00:33:43,920 --> 00:33:50,950
Ei for the different
configurations over kT.

512
00:33:50,950 --> 00:33:57,400
So it's e to the
e int over kT.

513
00:33:57,400 --> 00:34:02,380
Plus three times e to the zero
over kT, we'll get all those

514
00:34:02,380 --> 00:34:04,770
other three terms.

515
00:34:04,770 --> 00:34:05,850
So that's it.

516
00:34:05,850 --> 00:34:12,730
It's e to the e int over
kT plus three.

517
00:34:12,730 --> 00:34:16,040
Remember, the interaction energy
is negative e int.

518
00:34:16,040 --> 00:34:22,640
It's a favorable interaction.
e int is a positive number.

519
00:34:22,640 --> 00:34:25,120
So that's our result.

520
00:34:25,120 --> 00:34:32,760
Now we've described the
probabilities in terms of the

521
00:34:32,760 --> 00:34:34,380
states that can be occupied.

522
00:34:34,380 --> 00:34:35,480
That is, we've added it up.

523
00:34:35,480 --> 00:34:38,000
But of course, even the way
I wrote it just out of

524
00:34:38,000 --> 00:34:40,250
convenience, I didn't
actually write out

525
00:34:40,250 --> 00:34:41,350
each term in the sum.

526
00:34:41,350 --> 00:34:42,860
In the notes I actually
did that.

527
00:34:42,860 --> 00:34:45,620
But of course it's not necessary
to write e to the

528
00:34:45,620 --> 00:34:47,980
zero over kT plus e to the zero
over kT and write that

529
00:34:47,980 --> 00:34:50,360
three times for each
of these three.

530
00:34:50,360 --> 00:34:54,330
Rather, it's convenient to
group them together.

531
00:34:54,330 --> 00:34:57,190
And the point I'm illustrating
here is that in our

532
00:34:57,190 --> 00:35:02,500
expression, for the partition
function, we've written that

533
00:35:02,500 --> 00:35:06,060
in terms of the individual
states.

534
00:35:06,060 --> 00:35:09,440
But we doing a sum over the
individual molecular states.

535
00:35:09,440 --> 00:35:14,090
But we could also sum
over energy levels.

536
00:35:14,090 --> 00:35:17,210
Including the degeneracy.

537
00:35:17,210 --> 00:35:21,950
So we could say, let's not do
the sum over every state.

538
00:35:21,950 --> 00:35:24,370
After all, what if there are a
hundred states that had the

539
00:35:24,370 --> 00:35:24,950
same energy.

540
00:35:24,950 --> 00:35:26,830
Rather than just three.

541
00:35:26,830 --> 00:35:28,740
Gets to be kind of
painful, right?

542
00:35:28,740 --> 00:35:31,230
Instead let's just sum
over energy levels.

543
00:35:31,230 --> 00:35:34,500
They're all going to be the
same factor anyway.

544
00:35:34,500 --> 00:35:37,160
And then we'll have, we'll
just explicitly write the

545
00:35:37,160 --> 00:35:40,140
degeneracy in there.

546
00:35:40,140 --> 00:35:51,380
So, we can write the partition
function as a

547
00:35:51,380 --> 00:35:57,740
sum over energy level.

548
00:35:57,740 --> 00:36:06,170
So, in other words q is the
sum over i, e to the

549
00:36:06,170 --> 00:36:11,500
minus Ei over kT.

550
00:36:11,500 --> 00:36:22,390
That's individual molecular
states i.

551
00:36:22,390 --> 00:36:27,880
But we also could write it as
the sum over i, where this now

552
00:36:27,880 --> 00:36:36,070
is molecular energy levels i.

553
00:36:36,070 --> 00:36:40,590
And then we need to incorporate
the degeneracy gi

554
00:36:40,590 --> 00:36:44,380
e to the minus Ei over kT.

555
00:36:44,380 --> 00:36:47,220
Same thing, of course.

556
00:36:47,220 --> 00:36:49,130
But again, sometimes much
more convenient to

557
00:36:49,130 --> 00:36:53,150
write things this way.

558
00:36:53,150 --> 00:36:56,760
And of course, we can write
the probabilities of the

559
00:36:56,760 --> 00:36:59,920
occupancies in the
same way, too.

560
00:36:59,920 --> 00:37:07,190
That is, we could talk about the
Pi in terms of individual

561
00:37:07,190 --> 00:37:17,190
states. e to the minus Ei over
kT over q, the whole sum.

562
00:37:17,190 --> 00:37:35,650
Or Pi summing over energy
levels, gi e to the minus Ei

563
00:37:35,650 --> 00:37:40,450
over kT, divided by q.

564
00:37:40,450 --> 00:37:42,280
Here, of course, this
is bigger if

565
00:37:42,280 --> 00:37:43,230
it's degenerate, right?

566
00:37:43,230 --> 00:37:46,010
It's saying that the probability
of being in this

567
00:37:46,010 --> 00:37:49,450
energy level is three times the
probability of it being in

568
00:37:49,450 --> 00:37:53,540
any individual one
of the states.

569
00:37:53,540 --> 00:37:55,410
But sometimes it's useful
to keep track of

570
00:37:55,410 --> 00:37:57,120
things in this way.

571
00:37:57,120 --> 00:38:03,780
What it shows you too is that,
remember when we looked at,

572
00:38:03,780 --> 00:38:05,760
did I erase it?

573
00:38:05,760 --> 00:38:07,060
I guess it's gone.

574
00:38:07,060 --> 00:38:13,300
When we looked at this
probability distribution for

575
00:38:13,300 --> 00:38:18,550
the occupancies of the levels,
of course, what it says is at

576
00:38:18,550 --> 00:38:22,990
any temperature, the lowest
level is the most probable.

577
00:38:22,990 --> 00:38:24,220
For intermediate temperatures.

578
00:38:24,220 --> 00:38:29,730
For low temperatures, for
high temperatures.

579
00:38:29,730 --> 00:38:32,590
At high temperature, it might be
only a little more probable

580
00:38:32,590 --> 00:38:34,950
than the next one over, and the
next one, and so forth.

581
00:38:34,950 --> 00:38:37,690
But the very lowest state
always has the highest

582
00:38:37,690 --> 00:38:41,040
probability.

583
00:38:41,040 --> 00:38:44,400
But the lowest energy doesn't
always have the highest

584
00:38:44,400 --> 00:38:46,330
probability because
of degeneracies.

585
00:38:46,330 --> 00:38:49,310
There might be many states
with an energy up here.

586
00:38:49,310 --> 00:38:53,020
And the probability of any one
of them is only a little lower

587
00:38:53,020 --> 00:38:54,700
than the probability of
the lowest energy.

588
00:38:54,700 --> 00:38:55,840
That could be the case here.

589
00:38:55,840 --> 00:38:57,210
Let's say, the interaction
energy

590
00:38:57,210 --> 00:39:00,610
isn't enormously strong.

591
00:39:00,610 --> 00:39:07,090
So there is some energetic
favoring of this state.

592
00:39:07,090 --> 00:39:10,300
Maybe there are 10% at
room temperature.

593
00:39:10,300 --> 00:39:13,420
Maybe it turns out there are 10%
more molecules like this

594
00:39:13,420 --> 00:39:15,830
than in any one of
these states.

595
00:39:15,830 --> 00:39:18,110
But of course, these three
altogether mean that there are

596
00:39:18,110 --> 00:39:21,310
many more molecules at
this energy than

597
00:39:21,310 --> 00:39:32,620
at the lowest energy.

598
00:39:32,620 --> 00:39:37,030
Now of course we could do the
same thing for the canonical

599
00:39:37,030 --> 00:39:37,880
partition function.

600
00:39:37,880 --> 00:39:41,160
Not just the molecular one.

601
00:39:41,160 --> 00:39:58,370
So in other words, capital Q sum
over i system microstates.

602
00:39:58,370 --> 00:40:03,300
e to the minus capital
Ei over kT.

603
00:40:03,300 --> 00:40:08,350
But we could also write it
as a sum over energies.

604
00:40:08,350 --> 00:40:17,300
Sum over system energies.

605
00:40:17,300 --> 00:40:19,390
Ei.

606
00:40:19,390 --> 00:40:22,950
And then we have to include
the degeneracy.

607
00:40:22,950 --> 00:40:35,500
Capital Omega i e to the
minus Ei over kT.

608
00:40:35,500 --> 00:40:46,680
Degeneracy of the system
energy Ei.

609
00:40:46,680 --> 00:40:54,040
Little g here is the
degeneracy of

610
00:40:54,040 --> 00:41:03,140
molecular energy Ei.

611
00:41:03,140 --> 00:41:05,970
Now, same form.

612
00:41:05,970 --> 00:41:08,210
Just an important difference,
though.

613
00:41:08,210 --> 00:41:12,120
This number, this little gi, is
typically a small number.

614
00:41:12,120 --> 00:41:16,500
It could be one, it
could be a few.

615
00:41:16,500 --> 00:41:18,590
This number is usually
astronomical.

616
00:41:18,590 --> 00:41:22,420
That's basically, that is,
what we calculated here.

617
00:41:22,420 --> 00:41:26,220
In other words, how many total
system states are there with a

618
00:41:26,220 --> 00:41:27,760
particular energy.

619
00:41:27,760 --> 00:41:31,230
Well, in many, many, many
cases the answer is some

620
00:41:31,230 --> 00:41:32,930
astronomical number.

621
00:41:32,930 --> 00:41:36,130
So this number might often
be between one and ten.

622
00:41:36,130 --> 00:41:38,530
This number might be
10 to the 24th.

623
00:41:38,530 --> 00:41:43,230
It might be 10 to the 24th
to a large power.

624
00:41:43,230 --> 00:41:46,940
And, of course, that has a big
effect on the way things end

625
00:41:46,940 --> 00:41:47,890
up working.

626
00:41:47,890 --> 00:41:51,730
In statistical mechanics
and in thermodynamics.

627
00:41:51,730 --> 00:41:55,650
A lot of thermodynamics results
are the way they are

628
00:41:55,650 --> 00:42:00,900
because you have so many
possible states with a

629
00:42:00,900 --> 00:42:02,890
particular energy.

630
00:42:02,890 --> 00:42:06,950
That that energy can be strongly
favored just by

631
00:42:06,950 --> 00:42:09,160
virtue of the number of
states that there are.

632
00:42:09,160 --> 00:42:11,720
Just the way, in a very small
way, this energy might be

633
00:42:11,720 --> 00:42:14,350
favored just because there are
more states with it then there

634
00:42:14,350 --> 00:42:15,290
are states here.

635
00:42:15,290 --> 00:42:17,610
But again, with the
system, it's not a

636
00:42:17,610 --> 00:42:18,740
factor of three to one.

637
00:42:18,740 --> 00:42:21,340
It might be a factor of
10 to the 24th to

638
00:42:21,340 --> 00:42:23,740
the power of something.

639
00:42:23,740 --> 00:42:27,760
It might be just enormously
larger than other

640
00:42:27,760 --> 00:42:30,100
possibilities that'll tend
to put the energy

641
00:42:30,100 --> 00:42:40,500
at a certain place.

642
00:42:40,500 --> 00:42:43,370
And just to continue, of course,
the same thing goes

643
00:42:43,370 --> 00:42:46,930
for the system probabilities.

644
00:42:46,930 --> 00:42:52,490
Pi, right, which is e to the
minus Ei over kT divided by

645
00:42:52,490 --> 00:43:04,180
capital Q. If this is a sum over
states, or omega i e to

646
00:43:04,180 --> 00:43:17,640
the minus Ei over kT over Q,
let's write this over, if now

647
00:43:17,640 --> 00:43:19,250
we're calculating
the probability

648
00:43:19,250 --> 00:43:25,230
of an energy level.

649
00:43:25,230 --> 00:43:27,360
And it's important to make
the distinction.

650
00:43:27,360 --> 00:43:31,560
Because in many cases, that's
what we care about.

651
00:43:31,560 --> 00:43:34,620
What's the energy
of the system?

652
00:43:34,620 --> 00:43:38,440
And we often don't care about
exactly where's that molecule

653
00:43:38,440 --> 00:43:40,340
and that one, and that one,
and that one, right?

654
00:43:40,340 --> 00:43:44,060
The individual states that might
be involved that would

655
00:43:44,060 --> 00:43:51,600
comprise that energy.

656
00:43:51,600 --> 00:43:57,240
Now, what I want to do is start
deriving thermodynamics.

657
00:43:57,240 --> 00:44:01,630
Like I promised, we're going
to be able to derive every

658
00:44:01,630 --> 00:44:03,590
thermodynamic quantity
if we just know

659
00:44:03,590 --> 00:44:06,270
the partition function.

660
00:44:06,270 --> 00:44:09,780
So now I just want
to show that that

661
00:44:09,780 --> 00:44:32,020
might really be true.

662
00:44:32,020 --> 00:44:43,840
So, the point is that from Q
we're going to get all of our

663
00:44:43,840 --> 00:44:44,870
thermodynamics.

664
00:44:44,870 --> 00:44:47,670
And let's start with
the energy.

665
00:44:47,670 --> 00:44:50,000
Remember u, right?

666
00:44:50,000 --> 00:44:54,400
That's our system energy.

667
00:44:54,400 --> 00:45:00,290
It's an average energy.

668
00:45:00,290 --> 00:45:04,660
It's an average of the energy
that we would get by looking

669
00:45:04,660 --> 00:45:09,450
at the states that the system
might be occupying.

670
00:45:09,450 --> 00:45:12,660
So we can write it
as a sum over i.

671
00:45:12,660 --> 00:45:17,900
Of Pi times Ei.

672
00:45:17,900 --> 00:45:20,640
In other words, it's going to be
determined by the energy of

673
00:45:20,640 --> 00:45:23,625
any system state times the
probability that the system is

674
00:45:23,625 --> 00:45:25,160
in that state.

675
00:45:25,160 --> 00:45:28,910
Add them all up.

676
00:45:28,910 --> 00:45:31,560
Well, we know what that is.

677
00:45:31,560 --> 00:45:45,630
It's the sum i of Ei, e to the
minus Ei over kT divided by Q.

678
00:45:45,630 --> 00:45:48,220
Now, I'm going to just make
a simple substitution.

679
00:45:48,220 --> 00:45:52,880
I'm going to use the term beta
to mean one over kT.

680
00:45:52,880 --> 00:45:55,000
I'm really just using that so I
don't need to use the chain

681
00:45:55,000 --> 00:45:57,470
rule a billion times and
doing derivatives.

682
00:45:57,470 --> 00:46:04,070
So I'm going to write this is
sum over i Ei e to the minus

683
00:46:04,070 --> 00:46:16,920
beta Ei over Q.

684
00:46:16,920 --> 00:46:20,710
So Q, then, in this term is
just a sum over i e to the

685
00:46:20,710 --> 00:46:23,180
minus beta Ei.

686
00:46:23,180 --> 00:46:24,190
So.

687
00:46:24,190 --> 00:46:25,520
Now I do want to take
some derivatives.

688
00:46:25,520 --> 00:46:40,310
If I take dQ / d beta, keeping
V and N constant, then I get

689
00:46:40,310 --> 00:46:43,910
derivative with respect
to beta.

690
00:46:43,910 --> 00:46:47,760
Sum over i e to the
minus beta Ei.

691
00:46:47,760 --> 00:46:52,260
So that's going to bring
out Ei, right?

692
00:46:52,260 --> 00:46:57,930
So it's going to bring out minus
Ei minus the sum over i,

693
00:46:57,930 --> 00:47:01,100
Ei e to the minus beta Ei.

694
00:47:01,100 --> 00:47:03,050
That obviously looks
like a handy thing.

695
00:47:03,050 --> 00:47:21,880
Because that looks
like that term.

696
00:47:21,880 --> 00:47:29,445
Then, our average energy,
remember, it's one over Q. Sum

697
00:47:29,445 --> 00:47:34,580
of i, Ei e to the
minus beta Ei.

698
00:47:34,580 --> 00:47:45,650
So now, it's just minus one
over q, dQ / d beta.

699
00:47:45,650 --> 00:47:53,180
So that's just the same as
minus d log Q / d beta.

700
00:47:53,180 --> 00:47:55,540
And now I'm going to
use the chain rule.

701
00:47:55,540 --> 00:48:11,260
So it's minus d log Q /
dT times dT / d beta.

702
00:48:11,260 --> 00:48:18,950
All at constant V and N. Now I
can easily get dT / d beta.

703
00:48:18,950 --> 00:48:25,720
Because d beta / dT is just
minus one over kT squared.

704
00:48:25,720 --> 00:48:30,020
It's just the derivative of one
over kT with respect to T.

705
00:48:30,020 --> 00:48:38,400
So finally, my average E, which
is u, is just kT squared

706
00:48:38,400 --> 00:48:51,070
times d log Q / dT constant
V, N. That's terrific.

707
00:48:51,070 --> 00:48:55,950
In other words, if I have an
expression for Q, I know the

708
00:48:55,950 --> 00:48:58,280
partition function, and I
can calculate it at any

709
00:48:58,280 --> 00:48:59,790
temperature.

710
00:48:59,790 --> 00:49:02,600
I just need to take log of it,
take its derivative with

711
00:49:02,600 --> 00:49:04,420
respect to temperature.

712
00:49:04,420 --> 00:49:08,460
Multiply it by k T squared
and I've got my energy.

713
00:49:08,460 --> 00:49:16,110
Not very complicated, right?

714
00:49:16,110 --> 00:49:19,610
So in other words, macroscopic
thermodynamic properties come

715
00:49:19,610 --> 00:49:22,490
straight out of our microscopic
model of

716
00:49:22,490 --> 00:49:24,070
statistical mechanics.

717
00:49:24,070 --> 00:49:29,130
Statistical thermodynamics.

718
00:49:29,130 --> 00:49:31,620
Now I'm just going to state the
next result, just because

719
00:49:31,620 --> 00:49:35,320
I want to get there and it'll be
followed up more next time.

720
00:49:35,320 --> 00:49:36,830
But it's the following.

721
00:49:36,830 --> 00:49:42,550
You can see, of course, our Q
is a function of V and N and

722
00:49:42,550 --> 00:49:44,470
T. It's a function of V because
in principle the

723
00:49:44,470 --> 00:49:46,790
energies that are going
into all this can be

724
00:49:46,790 --> 00:49:50,530
a function of volume.

725
00:49:50,530 --> 00:49:53,970
What thermodynamic function
is naturally a

726
00:49:53,970 --> 00:49:55,660
function of N, V, T?

727
00:49:55,660 --> 00:49:58,350
Who remembers?

728
00:49:58,350 --> 00:49:59,650
Gibbs free energy?

729
00:49:59,650 --> 00:50:05,460
Helmholtz free energy?

730
00:50:05,460 --> 00:50:06,700
Enthalpy?

731
00:50:06,700 --> 00:50:09,640
Which one?

732
00:50:09,640 --> 00:50:13,660
Nobody knows.

733
00:50:13,660 --> 00:50:18,640
What's a function of N, V, and
T. Or, V and T, is what we

734
00:50:18,640 --> 00:50:20,000
really formulate it as.

735
00:50:20,000 --> 00:50:22,060
N was introduced later.

736
00:50:22,060 --> 00:50:23,190
The Helmholtz free energy.

737
00:50:23,190 --> 00:50:26,470
Thank you.

738
00:50:26,470 --> 00:50:29,600
What that suggests is that
actually the simplest and most

739
00:50:29,600 --> 00:50:33,930
natural connection between Q and
macroscopic thermodynamics

740
00:50:33,930 --> 00:50:36,050
is to the Helmholtz
free energy.

741
00:50:36,050 --> 00:50:40,860
And the result that you'll see
derived next time is A is just

742
00:50:40,860 --> 00:50:51,120
minus kT log of Q. What
a simple result.

743
00:50:51,120 --> 00:50:52,770
And you'll see the derivations
in your notes.

744
00:50:52,770 --> 00:50:55,670
You can see it's a couple
of lines, right?

745
00:50:55,670 --> 00:50:58,490
But of course, if you know A,
and you know E, you know

746
00:50:58,490 --> 00:50:59,230
everything, right?

747
00:50:59,230 --> 00:51:00,960
Because S is in there.

748
00:51:00,960 --> 00:51:06,330
And then other combinations
can give S and G and mu.

749
00:51:06,330 --> 00:51:07,110
And p.

750
00:51:07,110 --> 00:51:08,900
And anything else you want.

751
00:51:08,900 --> 00:51:11,330
So that's the point, is that
all of macroscopic

752
00:51:11,330 --> 00:51:12,560
thermodynamics follows.

753
00:51:12,560 --> 00:51:16,690
And you'll see that elaborated
more next time.

754
00:51:16,690 --> 00:51:20,430
And in addition, a very simple
natural expression for the

755
00:51:20,430 --> 00:51:24,470
entropy in terms of the states
available follows, that we've

756
00:51:24,470 --> 00:51:25,410
alluded to before.

757
00:51:25,410 --> 00:51:26,880
And now you'll see
it played out.

758
00:51:26,880 --> 00:51:28,130
Right.