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PROFESSOR: So today's topic
is liquid gas condensation

9
00:00:36,920 --> 00:00:40,690
and throughout this
course I have already

10
00:00:40,690 --> 00:00:43,710
mentioned two
different perspectives

11
00:00:43,710 --> 00:00:46,220
on the same
phenomenon, transition

12
00:00:46,220 --> 00:00:48,790
between liquid and gas.

13
00:00:48,790 --> 00:00:54,290
One perspective was when we
looked at the phase diagram

14
00:00:54,290 --> 00:00:56,360
even in the first
lecture I mentioned

15
00:00:56,360 --> 00:01:04,680
that if you look at
something like water

16
00:01:04,680 --> 00:01:06,800
as a function of
pressure and temperature

17
00:01:06,800 --> 00:01:10,710
it can exist in three
different phases.

18
00:01:10,710 --> 00:01:14,800
There is the gas
phase, which you

19
00:01:14,800 --> 00:01:18,010
have at high temperatures
and low pressures.

20
00:01:18,010 --> 00:01:21,360
At low temperatures
and high pressures

21
00:01:21,360 --> 00:01:24,240
you have the liquid phase.

22
00:01:24,240 --> 00:01:31,960
And of course at
very low temperatures

23
00:01:31,960 --> 00:01:37,110
you also have the possibility
of the solid phase.

24
00:01:37,110 --> 00:01:39,920
And our concern
at some point was

25
00:01:39,920 --> 00:01:44,840
to think about the location of
the coexistence of the three

26
00:01:44,840 --> 00:01:52,230
phases of that was set as
the basis of the temperature,

27
00:01:52,230 --> 00:01:56,020
273.16.

28
00:01:56,020 --> 00:01:58,800
Actually what you're
going to focus now

29
00:01:58,800 --> 00:02:02,320
is not this part of
the phase diagram

30
00:02:02,320 --> 00:02:07,480
but the portion that consists
of the transition between liquid

31
00:02:07,480 --> 00:02:10,449
and gas phases.

32
00:02:10,449 --> 00:02:14,150
And this coexistence
line actually

33
00:02:14,150 --> 00:02:20,190
terminates at what is called
the critical point that we will

34
00:02:20,190 --> 00:02:25,080
talk about more today at the
particular value of Tc and Pc.

35
00:02:28,930 --> 00:02:32,230
An equivalent perspective
that we have looked at that

36
00:02:32,230 --> 00:02:35,420
and I just want to make sure
that you have both of these

37
00:02:35,420 --> 00:02:38,940
in mind and know the
relationship between them

38
00:02:38,940 --> 00:02:41,840
is to look at isotherms.

39
00:02:41,840 --> 00:02:45,560
Basically we also
looked at cases

40
00:02:45,560 --> 00:02:51,200
where we looked at isotherms
of pressure versus volume.

41
00:02:51,200 --> 00:02:56,540
And this statement was that
if we look at the system

42
00:02:56,540 --> 00:03:04,160
at high enough temperatures and
low enough pressures so let's

43
00:03:04,160 --> 00:03:10,220
say pick this temperature and
scan along a line such as this,

44
00:03:10,220 --> 00:03:13,370
it would be equivalent
over here to have

45
00:03:13,370 --> 00:03:19,300
some kind of an isotherm that is
kind of potentially a distorted

46
00:03:19,300 --> 00:03:24,090
version of an ideal
gas hyperbola.

47
00:03:24,090 --> 00:03:30,640
Now if we were to look
at another isotherm

48
00:03:30,640 --> 00:03:35,080
that corresponds to
this scan over here then

49
00:03:35,080 --> 00:03:39,190
what would happen is
that it will cross

50
00:03:39,190 --> 00:03:42,360
this line of the
coexistence, part of it

51
00:03:42,360 --> 00:03:44,300
would fall in the
liquid phase, part of it

52
00:03:44,300 --> 00:03:46,400
would fall in the gas phase.

53
00:03:46,400 --> 00:03:49,720
And we draw the
corresponding isotherm,

54
00:03:49,720 --> 00:03:51,750
it would look
something like this.

55
00:03:51,750 --> 00:03:54,110
There would be a
portion that would

56
00:03:54,110 --> 00:03:57,650
correspond to being
in the liquid.

57
00:03:57,650 --> 00:03:59,190
There would be a
portion that would

58
00:03:59,190 --> 00:04:00,875
correspond to being in the gas.

59
00:04:04,520 --> 00:04:09,460
And this line, this point
where you hit this line

60
00:04:09,460 --> 00:04:14,450
would correspond
to the coexistence.

61
00:04:14,450 --> 00:04:20,610
Essentially you could change
the volume of a container

62
00:04:20,610 --> 00:04:27,450
and at high volumes, you
would start with entirety gas.

63
00:04:27,450 --> 00:04:30,070
At the same pressure
and temperature

64
00:04:30,070 --> 00:04:33,440
you could squeeze it
and some of the gas

65
00:04:33,440 --> 00:04:36,590
would get converted
to liquid and you

66
00:04:36,590 --> 00:04:39,580
would have a coexistence
of gas and liquid

67
00:04:39,580 --> 00:04:42,880
until you squeezed it
sufficiently, still maintaining

68
00:04:42,880 --> 00:04:45,310
the same pressure
and temperature

69
00:04:45,310 --> 00:04:47,970
until your container
was fully liquid.

70
00:04:47,970 --> 00:04:51,650
So that's a different
type of isotherm.

71
00:04:51,650 --> 00:04:55,110
And what we discussed
was that basically there

72
00:04:55,110 --> 00:05:03,410
is a coexistence
boundary that separates

73
00:05:03,410 --> 00:05:07,470
the first and the second
types of isotherms.

74
00:05:07,470 --> 00:05:14,380
And presumably in between
there is some trajectory

75
00:05:14,380 --> 00:05:17,570
that would correspond to
basically being exactly

76
00:05:17,570 --> 00:05:22,138
at the boundary and it would
look something like this.

77
00:05:22,138 --> 00:05:27,440
This would be the trajectory
that you would have at Tc.

78
00:05:27,440 --> 00:05:29,930
The location Pc is
the same, of course,

79
00:05:29,930 --> 00:05:32,660
so basically I
can carry this out

80
00:05:32,660 --> 00:05:36,970
so there is the same Pc
that would occur here.

81
00:05:36,970 --> 00:05:41,480
And there would be
some particular Vc

82
00:05:41,480 --> 00:05:46,240
but that depends on the amount
of material that I have.

83
00:05:46,240 --> 00:05:48,240
OK?

84
00:05:48,240 --> 00:05:55,260
So we are going to try to
understand what is happening

85
00:05:55,260 --> 00:06:00,750
here and in particular
we note that suddenly we

86
00:06:00,750 --> 00:06:04,900
have to deal with cases
where there are singularities

87
00:06:04,900 --> 00:06:09,110
in our thermodynamic parameters.

88
00:06:09,110 --> 00:06:11,310
There's some
thermodynamic parameter

89
00:06:11,310 --> 00:06:16,000
that I'm scanning as I
go across the system.

90
00:06:16,000 --> 00:06:18,300
It will not be
varying continuously.

91
00:06:18,300 --> 00:06:21,440
It has these kinds of
discontinuities in it.

92
00:06:21,440 --> 00:06:25,830
And how did discontinuities
appear and how can

93
00:06:25,830 --> 00:06:28,660
we account for these
phase transitions

94
00:06:28,660 --> 00:06:31,610
given the formalisms
that we have developed

95
00:06:31,610 --> 00:06:35,080
for studying
thermodynamic functions.

96
00:06:35,080 --> 00:06:38,460
And in particular
let's, for example,

97
00:06:38,460 --> 00:06:45,000
start with the canonical
prescription where I state,

98
00:06:45,000 --> 00:06:52,210
let's say that I have volume
and temperature of a fixed

99
00:06:52,210 --> 00:06:55,980
number of particles and
I want to figure out

100
00:06:55,980 --> 00:06:59,410
thermodynamic properties
in this perspective.

101
00:06:59,410 --> 00:07:03,020
What I need to calculate
is a partition function

102
00:07:03,020 --> 00:07:06,460
and we've seen that the
partition function is obtained

103
00:07:06,460 --> 00:07:12,140
by integrating over all degrees
of freedom which are basically

104
00:07:12,140 --> 00:07:15,720
the coordinates and momentum
of particles that make up

105
00:07:15,720 --> 00:07:19,030
this gas so I have
to do d cubed p i,

106
00:07:19,030 --> 00:07:26,530
d cubed q i, divide by h cubed,
divide by n factorial because

107
00:07:26,530 --> 00:07:30,710
of the way that we've been
looking at things, of energy

108
00:07:30,710 --> 00:07:36,030
so I have e to the minus
beta h that has a part that

109
00:07:36,030 --> 00:07:43,670
is from the kinetic energy which
I can integrate immediately.

110
00:07:43,670 --> 00:07:47,920
And then it has a part that
is from the potential energy

111
00:07:47,920 --> 00:07:50,490
of the interactions among
all of these particles.

112
00:07:55,581 --> 00:07:57,980
OK?

113
00:07:57,980 --> 00:08:03,610
So somehow if I could
do these integrations--

114
00:08:03,610 --> 00:08:06,280
we already did them for
an ideal gas and nothing

115
00:08:06,280 --> 00:08:08,050
special happened
but presumably if I

116
00:08:08,050 --> 00:08:12,950
can do that for the case of
an interacting gas, buried

117
00:08:12,950 --> 00:08:18,660
within it would be the
properties of this phase

118
00:08:18,660 --> 00:08:21,950
transition and the
singularities, et cetera.

119
00:08:21,950 --> 00:08:24,900
Question is how
does that happen?

120
00:08:24,900 --> 00:08:27,710
So last time we tried
to do shortcuts.

121
00:08:27,710 --> 00:08:31,520
We started with calculating a
first derivative calculation

122
00:08:31,520 --> 00:08:35,500
in this potential and
then try to guess things

123
00:08:35,500 --> 00:08:38,580
and maybe that was
not so satisfactory.

124
00:08:38,580 --> 00:08:41,309
So today we'll take
another approach

125
00:08:41,309 --> 00:08:44,290
to calculating this
partition function where

126
00:08:44,290 --> 00:08:46,980
the approximations and
assumptions are more

127
00:08:46,980 --> 00:08:54,290
clearly stated and we
can see what happens.

128
00:08:54,290 --> 00:08:57,480
So I want to calculate
this partition function.

129
00:08:57,480 --> 00:08:58,440
OK?

130
00:08:58,440 --> 00:09:00,540
So part of that
partition function

131
00:09:00,540 --> 00:09:06,030
that depends on the momentum I
can very easily take care of.

132
00:09:06,030 --> 00:09:09,890
That gives me these
factors of 1 over lambda.

133
00:09:09,890 --> 00:09:18,660
Again as usual my lambda
is h over 2 pi m k T

134
00:09:18,660 --> 00:09:22,740
and then I have to do
the integrations over all

135
00:09:22,740 --> 00:09:29,410
of the coordinates, all of
these coordinate integrations

136
00:09:29,410 --> 00:09:32,160
and the potential that I'm
going to be thinking about,

137
00:09:32,160 --> 00:09:35,780
so again in my U
will be something

138
00:09:35,780 --> 00:09:41,640
like sum over pairs
V of q i minus q j,

139
00:09:41,640 --> 00:09:46,450
where the typical form of this V
as a function of the separation

140
00:09:46,450 --> 00:09:50,830
that I'm going to
look at has presumably

141
00:09:50,830 --> 00:09:55,360
a part that is hard-core
and a part that is attract.

142
00:09:55,360 --> 00:09:58,200
Something like this.

143
00:09:58,200 --> 00:09:59,560
OK?

144
00:09:59,560 --> 00:10:04,900
So given that potential what
are the kinds of configurations

145
00:10:04,900 --> 00:10:06,835
I expect to happen in my system?

146
00:10:06,835 --> 00:10:08,840
Well, let's see.

147
00:10:08,840 --> 00:10:11,640
Let's try to make a diagram.

148
00:10:11,640 --> 00:10:17,150
I have a huge number of
particles that don't come very

149
00:10:17,150 --> 00:10:20,020
close to each other, there's
a hard-core repulsion,

150
00:10:20,020 --> 00:10:24,200
so maybe I can think of them
as having some kind of a size--

151
00:10:24,200 --> 00:10:28,870
marbles, et cetera, and they are
distributed so that they cannot

152
00:10:28,870 --> 00:10:29,950
come close to each other.

153
00:10:33,390 --> 00:10:37,030
More than that are
zero and then I

154
00:10:37,030 --> 00:10:40,480
have to sum over all
configurations that

155
00:10:40,480 --> 00:10:44,280
are compatible where they
are not crossing each other

156
00:10:44,280 --> 00:10:46,550
and calculate e to
the minus beta u.

157
00:10:50,790 --> 00:10:55,860
So let's try to do some kind
of an approximation to this u.

158
00:10:55,860 --> 00:11:01,030
I claim that I can write this
U as follows- I can write it

159
00:11:01,030 --> 00:11:04,840
as one-half, basically
this i less than j,

160
00:11:04,840 --> 00:11:08,370
I can write as one-half
all i not equal to j,

161
00:11:08,370 --> 00:11:11,310
so that's where the
one-half comes from.

162
00:11:11,310 --> 00:11:14,090
I will write it as one-half
but not in this form,

163
00:11:14,090 --> 00:11:20,170
as an integral d cubed
r d cubed r prime n of r

164
00:11:20,170 --> 00:11:28,520
n of r prime v of r minus
r prime where n of r

165
00:11:28,520 --> 00:11:39,125
is sum over all particles asking
whether they are at location r.

166
00:11:42,100 --> 00:11:47,450
So I can pick some
particle opposition here--

167
00:11:47,450 --> 00:11:54,030
let's call it r-- ask whether or
not there is a particle there,

168
00:11:54,030 --> 00:11:58,270
construct the density by
summing over everywhere.

169
00:11:58,270 --> 00:12:00,450
You can convince
yourself that if I

170
00:12:00,450 --> 00:12:03,960
were to substitute
this back over here,

171
00:12:03,960 --> 00:12:07,490
the integrals over r
and r prime can be done

172
00:12:07,490 --> 00:12:11,880
and they sit r and r prime
respectively to sum q i and q j

173
00:12:11,880 --> 00:12:15,596
and I will get back the
sum that I had before.

174
00:12:15,596 --> 00:12:16,095
OK?

175
00:12:19,150 --> 00:12:20,810
So what is that thing doing?

176
00:12:20,810 --> 00:12:25,620
Essentially it says
pick some point, r,

177
00:12:25,620 --> 00:12:30,530
and then look at some
other point, r prime,

178
00:12:30,530 --> 00:12:33,310
ask whether there are
particles in that point

179
00:12:33,310 --> 00:12:37,960
and then sum over all pairs
of points r and r prime.

180
00:12:37,960 --> 00:12:39,790
So I changed my perspective.

181
00:12:39,790 --> 00:12:44,400
Rather than calculating the
energy by looking at particles,

182
00:12:44,400 --> 00:12:47,920
I essentially look
at parts of space,

183
00:12:47,920 --> 00:12:51,940
ask how many
particles there are.

184
00:12:51,940 --> 00:12:54,600
Well, it kind of makes
sense if I coarse-grained

185
00:12:54,600 --> 00:12:57,690
this a little bit that
the density should

186
00:12:57,690 --> 00:13:01,460
be more or less the same
in every single box.

187
00:13:01,460 --> 00:13:10,050
So I make the assumption
of uniform density--

188
00:13:10,050 --> 00:13:12,210
I shouldn't say
assumption, let's

189
00:13:12,210 --> 00:13:21,680
call it an approximation of
uniform density in which I

190
00:13:21,680 --> 00:13:24,630
replace this n of r
by it's average value

191
00:13:24,630 --> 00:13:29,752
which is the number of
particles per [INAUDIBLE].

192
00:13:29,752 --> 00:13:31,730
OK?

193
00:13:31,730 --> 00:13:38,230
Then my U-- I will
take the n's outside.

194
00:13:38,230 --> 00:13:42,830
I will have one-half n
squared integral d cubed

195
00:13:42,830 --> 00:13:46,870
r d cubed r prime v
of r minus r prime.

196
00:13:51,000 --> 00:13:54,650
Again it really is a function
of the relative distance

197
00:13:54,650 --> 00:13:57,950
so I can integrate
over the center of mass

198
00:13:57,950 --> 00:14:00,150
if you like to get
one factor of volumes

199
00:14:00,150 --> 00:14:04,380
so I have one n
squared V and then

200
00:14:04,380 --> 00:14:06,770
I have an integral
over the relative

201
00:14:06,770 --> 00:14:10,760
coordinate d cubed r-- let's
write it as 4 pi r squared d

202
00:14:10,760 --> 00:14:16,150
r, the potential as a
function of separation.

203
00:14:16,150 --> 00:14:20,340
Now the only configurations
that are possible

204
00:14:20,340 --> 00:14:27,370
are ones that don't really
come closer than wherever

205
00:14:27,370 --> 00:14:30,790
the particles are on
top of each other,

206
00:14:30,790 --> 00:14:37,430
so really there is some kind
of a minimum value over here.

207
00:14:37,430 --> 00:14:40,440
The maximum value you could
say is the size of the box

208
00:14:40,440 --> 00:14:43,650
but the typical range of
potentials that we are thinking

209
00:14:43,650 --> 00:14:47,000
is much, much less than
the size of the box.

210
00:14:47,000 --> 00:14:48,940
So for all intents
and purposes I

211
00:14:48,940 --> 00:14:54,790
can set the value of the other
part of the thing to infinity.

212
00:14:54,790 --> 00:15:02,700
So what I'm doing is essentially
I'm integrating this portion

213
00:15:02,700 --> 00:15:09,395
and I can call the result
of doing that to be minus u.

214
00:15:09,395 --> 00:15:10,710
Why minus?

215
00:15:10,710 --> 00:15:13,620
Because it's clearly
the attractive portion

216
00:15:13,620 --> 00:15:16,480
in this picture I'm integrating.

217
00:15:16,480 --> 00:15:20,420
And you can convince yourself
that if I use the potential

218
00:15:20,420 --> 00:15:23,280
that I had before-- yes,
the last time around it

219
00:15:23,280 --> 00:15:27,715
was minus u 0 r 0 over
r to the sixth power,

220
00:15:27,715 --> 00:15:35,400
but this u is actually the
omega that I was using times u 0

221
00:15:35,400 --> 00:15:38,990
but I will keep it as
u to sort of indicate

222
00:15:38,990 --> 00:15:41,560
that it could be a
more general potential.

223
00:15:41,560 --> 00:15:45,210
But the specific potential that
we were working with last time

224
00:15:45,210 --> 00:15:47,660
to calculate the second
visual coefficient

225
00:15:47,660 --> 00:15:48,910
would correspond to that.

226
00:15:48,910 --> 00:15:53,550
So essentially I claim
that the configurations

227
00:15:53,550 --> 00:15:56,720
that I'm interested
will give a contribution

228
00:15:56,720 --> 00:16:00,120
to the energy which is
minus whatever this u 0--

229
00:16:00,120 --> 00:16:03,770
u is n squared V divided by 2.

230
00:16:07,800 --> 00:16:11,350
Now clearly not
all configuration

231
00:16:11,350 --> 00:16:12,610
has the same energy.

232
00:16:12,610 --> 00:16:14,650
I mean that's the
whole thing, I have

233
00:16:14,650 --> 00:16:17,880
to really integrate
overall configurations.

234
00:16:17,880 --> 00:16:21,050
I've sort of looked at
an average contribution.

235
00:16:21,050 --> 00:16:23,930
There will be configurations
where the particles are

236
00:16:23,930 --> 00:16:26,570
more bunched or separate,
differently arranged,

237
00:16:26,570 --> 00:16:29,480
et cetera, and
energy would vary.

238
00:16:29,480 --> 00:16:31,780
But I expect that
most of the time--

239
00:16:31,780 --> 00:16:34,470
I see something like
a gas in this room,

240
00:16:34,470 --> 00:16:36,550
there is a uniform
density typically.

241
00:16:36,550 --> 00:16:38,870
The fluctuations in
density I can ignore

242
00:16:38,870 --> 00:16:41,560
and I will have a
contribution such as this.

243
00:16:41,560 --> 00:16:48,750
So this factor I will replace
by what I have over here.

244
00:16:48,750 --> 00:16:52,070
What I have over here will
give me e to the minus

245
00:16:52,070 --> 00:16:58,090
beta u density squared actually
I can write as N squared

246
00:16:58,090 --> 00:17:05,179
over V squared, one of V's
cancel here, I have 2 V. OK?

247
00:17:08,680 --> 00:17:12,520
So that's the typical value of
this quantity but now I have

248
00:17:12,520 --> 00:17:16,930
to do the integrations
over all of the q's.

249
00:17:16,930 --> 00:17:17,430
Yes?

250
00:17:17,430 --> 00:17:19,550
AUDIENCE: In terms of the
approximation you made--

251
00:17:19,550 --> 00:17:20,175
PROFESSOR: Yes.

252
00:17:20,175 --> 00:17:23,020
AUDIENCE: --the implications are
that if you have a more complex

253
00:17:23,020 --> 00:17:23,535
potential--

254
00:17:23,535 --> 00:17:24,160
PROFESSOR: Yes.

255
00:17:24,160 --> 00:17:26,618
AUDIENCE: --you'll get into
trouble with for a given sample

256
00:17:26,618 --> 00:17:30,130
size, it will be less accurate.

257
00:17:30,130 --> 00:17:31,702
would you agree with that?

258
00:17:31,702 --> 00:17:34,035
PROFESSOR: What do you mean
by a more complex potential?

259
00:17:34,035 --> 00:17:35,250
Until you tell me, I can't--

260
00:17:35,250 --> 00:17:38,370
AUDIENCE: I mean I, guess,
and usually-- I mean,

261
00:17:38,370 --> 00:17:40,362
I guess if you had like
multi-body or maybe

262
00:17:40,362 --> 00:17:41,830
the cell group.

263
00:17:41,830 --> 00:17:42,550
PROFESSOR: OK.

264
00:17:42,550 --> 00:17:46,850
So certainly what I have assumed
here is a two-body potential.

265
00:17:46,850 --> 00:17:49,220
If I had a three-body
potential then I

266
00:17:49,220 --> 00:17:51,930
would have a term that
would be density cubed, yes.

267
00:17:51,930 --> 00:17:54,530
AUDIENCE: But then you would--
my point is that-- and maybe

268
00:17:54,530 --> 00:17:58,870
it's not a place to question--
for a given sample size,

269
00:17:58,870 --> 00:18:01,420
I would imagine there's some
relation between how small

270
00:18:01,420 --> 00:18:04,440
it can be and the complexity
of the potential when you make

271
00:18:04,440 --> 00:18:07,547
this uniform density assumption.

272
00:18:07,547 --> 00:18:08,130
PROFESSOR: OK.

273
00:18:08,130 --> 00:18:12,630
We are always evaluating things
in the thermodynamic limit,

274
00:18:12,630 --> 00:18:16,410
so ultimately I'm always
interested in m and v going

275
00:18:16,410 --> 00:18:21,310
to infinity, while the
ratio of n over v is fixed.

276
00:18:21,310 --> 00:18:22,310
OK?

277
00:18:22,310 --> 00:18:27,060
So things-- there are certainly
problems associated with let's

278
00:18:27,060 --> 00:18:31,220
say extending the range of
this integration to infinity.

279
00:18:31,220 --> 00:18:34,390
We essentially are
not worrying about

280
00:18:34,390 --> 00:18:36,840
the walls of the
container, et cetera.

281
00:18:36,840 --> 00:18:39,970
All of those effects
are proportional to area

282
00:18:39,970 --> 00:18:43,540
and in the thermodynamic limit,
the ratio of area to volume

283
00:18:43,540 --> 00:18:44,530
goes to zero.

284
00:18:44,530 --> 00:18:46,150
I can ignore that.

285
00:18:46,150 --> 00:18:49,070
If I bring things to
become smaller and smaller,

286
00:18:49,070 --> 00:18:51,180
then I need to worry
about a lot of things,

287
00:18:51,180 --> 00:18:54,730
like do particles absorb
on surfaces, et cetera.

288
00:18:54,730 --> 00:18:56,490
I don't want to do that.

289
00:18:56,490 --> 00:18:58,780
But if you are worrying
about complexity

290
00:18:58,780 --> 00:19:03,430
of the potential such as I
assume things to be radially

291
00:19:03,430 --> 00:19:05,610
symmetric, you
say, well actually

292
00:19:05,610 --> 00:19:08,310
if I think about
oxygen molecules,

293
00:19:08,310 --> 00:19:09,500
they are not vertical.

294
00:19:09,500 --> 00:19:12,380
They have a dipole-dipole
interactions potentially,

295
00:19:12,380 --> 00:19:13,190
et cetera.

296
00:19:13,190 --> 00:19:17,610
All of those you can take care
of by doing this integral more

297
00:19:17,610 --> 00:19:18,670
carefully.

298
00:19:18,670 --> 00:19:22,900
Ultimately the value of your
parameter u will be different.

299
00:19:22,900 --> 00:19:25,930
AUDIENCE: But I guess what I'm
getting at is you deviate from

300
00:19:25,930 --> 00:19:29,210
the-- if you're doing it-- if
we're back when computers are

301
00:19:29,210 --> 00:19:31,126
not as fast and you can
essentially approach

302
00:19:31,126 --> 00:19:33,659
the thermodynamic limit and
you're doing a simulation

303
00:19:33,659 --> 00:19:35,200
and you want to use
these as a tool--

304
00:19:35,200 --> 00:19:36,282
PROFESSOR: Mm-hmm?

305
00:19:36,282 --> 00:19:38,490
AUDIENCE: --then you get
into those questions, right?

306
00:19:38,490 --> 00:19:39,115
PROFESSOR: Yes.

307
00:19:39,115 --> 00:19:42,420
AUDIENCE: But this
framework should still work?

308
00:19:42,420 --> 00:19:43,100
PROFESSOR: No.

309
00:19:43,100 --> 00:19:46,170
This framework is
an approximation

310
00:19:46,170 --> 00:19:49,500
intended to answer the
following question--

311
00:19:49,500 --> 00:19:53,630
how is it possible that
singularities can emerge?

312
00:19:53,630 --> 00:19:56,180
And we will see
shortly that the origin

313
00:19:56,180 --> 00:19:58,430
of the emergence
of singularities

314
00:19:58,430 --> 00:20:01,130
is precisely this.

315
00:20:01,130 --> 00:20:05,610
And if I don't have that, then
I don't have singularities.

316
00:20:05,610 --> 00:20:08,320
So truly when you do
a computer simulation

317
00:20:08,320 --> 00:20:13,280
with 10 million particles, you
will not see the singularity.

318
00:20:13,280 --> 00:20:16,530
It emerges only in the
thermodynamic limit

319
00:20:16,530 --> 00:20:21,340
and my point here is to sort
of start from that limit.

320
00:20:21,340 --> 00:20:23,690
Once we understand
that limit, then maybe

321
00:20:23,690 --> 00:20:25,650
we can better
answer your question

322
00:20:25,650 --> 00:20:27,400
about limitations of
computer simulations.

323
00:20:31,580 --> 00:20:32,080
OK?

324
00:20:35,950 --> 00:20:39,260
So I've said that there's
basically configurations

325
00:20:39,260 --> 00:20:42,350
that typically give
the [INAUDIBLE].

326
00:20:42,350 --> 00:20:43,840
But how many configurations?

327
00:20:43,840 --> 00:20:47,590
I can certainly move
these particles around.

328
00:20:47,590 --> 00:20:52,650
And therefore I have to see what
the value of these integrations

329
00:20:52,650 --> 00:20:54,500
are.

330
00:20:54,500 --> 00:20:55,860
OK?

331
00:20:55,860 --> 00:21:00,980
So I will do so, as follows--
I will say that the first

332
00:21:00,980 --> 00:21:03,940
particle, if there
was nobody else--

333
00:21:03,940 --> 00:21:08,040
so I have to do this
integration over q 1, q 2, q 3,

334
00:21:08,040 --> 00:21:11,900
and I have some constraints in
the space that the q's cannot

335
00:21:11,900 --> 00:21:15,440
come closer to each
other than this distance.

336
00:21:15,440 --> 00:21:18,730
So what I'm going to
try to calculate now

337
00:21:18,730 --> 00:21:22,680
is how this factor of v to the
n that you would have normally

338
00:21:22,680 --> 00:21:27,510
put here for ideal
gases gets modified.

339
00:21:27,510 --> 00:21:30,410
Well I say that the first
particle that I put in the box

340
00:21:30,410 --> 00:21:33,100
can explore the entire space.

341
00:21:33,100 --> 00:21:37,030
If I had two particles,
the second particle

342
00:21:37,030 --> 00:21:39,640
could explore everything
except the region that

343
00:21:39,640 --> 00:21:43,670
is excluded by the
first particle.

344
00:21:43,670 --> 00:21:46,080
The second particle
that I put in

345
00:21:46,080 --> 00:21:49,860
can explore the space
minus the region that

346
00:21:49,860 --> 00:21:53,220
is excluded by the
first two particles.

347
00:21:53,220 --> 00:21:59,890
And the last, the end
particle, the space

348
00:21:59,890 --> 00:22:02,970
except the region
that is excluded

349
00:22:02,970 --> 00:22:05,340
by the first n minus 1.

350
00:22:05,340 --> 00:22:07,980
This is an approximation, right?

351
00:22:07,980 --> 00:22:12,650
So if I have three
particles, the space

352
00:22:12,650 --> 00:22:15,430
that is excluded for
the third particle

353
00:22:15,430 --> 00:22:18,840
can be more than this
if the two particles

354
00:22:18,840 --> 00:22:20,430
are kind of close to each other.

355
00:22:20,430 --> 00:22:22,030
If you have two
billiard balls that

356
00:22:22,030 --> 00:22:25,490
are kind of close to each
other the region between them

357
00:22:25,490 --> 00:22:28,700
is also excluded for
the first particle.

358
00:22:28,700 --> 00:22:33,680
We are throwing that out and
one can show that that is really

359
00:22:33,680 --> 00:22:35,610
throwing out things
that are higher

360
00:22:35,610 --> 00:22:40,010
order in the ratio of omega
over V So if you're right,

361
00:22:40,010 --> 00:22:44,930
this is really an
expansion in omega over V

362
00:22:44,930 --> 00:22:47,660
and I have calculated
things correctly

363
00:22:47,660 --> 00:22:50,040
to order of omega over V.

364
00:22:50,040 --> 00:22:53,910
And if I am consistent
with that and I multiply

365
00:22:53,910 --> 00:22:58,840
all of these things together,
the first term is V to the N

366
00:22:58,840 --> 00:23:05,600
and then I have 1 minus omega
plus 2 omega plus all the way

367
00:23:05,600 --> 00:23:12,940
to N minus 1 omega which then
basically sums out to 1 plus 2

368
00:23:12,940 --> 00:23:18,590
plus 3 all the way to N minus
1, which is N minus 1 over 2

369
00:23:18,590 --> 00:23:20,215
in the large N limit.

370
00:23:20,215 --> 00:23:23,786
It's the same thing
as N squared over 2

371
00:23:23,786 --> 00:23:29,750
and this whole thing is
raised to the power-- OK

372
00:23:29,750 --> 00:23:35,290
this whole thing I can
approximate by V minus N

373
00:23:35,290 --> 00:23:36,865
omega over 2 squared.

374
00:23:39,660 --> 00:23:41,300
There are various
ways of seeing this.

375
00:23:41,300 --> 00:23:46,320
I mean one way of seeing this
is to pair things one from one

376
00:23:46,320 --> 00:23:50,430
end and one from the
other end and then

377
00:23:50,430 --> 00:23:53,040
multiply them together.

378
00:23:53,040 --> 00:23:54,590
When you multiply
them you will have

379
00:23:54,590 --> 00:23:58,450
a term that would go be
V squared, a term that

380
00:23:58,450 --> 00:24:00,780
would be proportionate
to V omega

381
00:24:00,780 --> 00:24:04,520
and would be the sum of
the coefficients of omega

382
00:24:04,520 --> 00:24:06,530
from the two.

383
00:24:06,530 --> 00:24:10,220
And you can see that
if I pick, say, alpha

384
00:24:10,220 --> 00:24:12,640
term from this
side N minus alpha

385
00:24:12,640 --> 00:24:14,730
minus one term from
the other side,

386
00:24:14,730 --> 00:24:20,560
add them up, the alphas
cancel out, N and N minus 1

387
00:24:20,560 --> 00:24:24,150
are roughly the same, so
basically the square of two

388
00:24:24,150 --> 00:24:31,190
of them is the same as
the square of V minus N

389
00:24:31,190 --> 00:24:33,850
omega over 2.

390
00:24:33,850 --> 00:24:38,764
And then I can repeat that
for all pairs and I get this.

391
00:24:38,764 --> 00:24:39,263
Sorry.

392
00:24:42,970 --> 00:24:47,650
So the statement is that
the effect of the excluded

393
00:24:47,650 --> 00:24:57,000
volumes since it is joint
effect of mutually excluding

394
00:24:57,000 --> 00:25:01,810
each other is that V to the
N that I have for ideal gas,

395
00:25:01,810 --> 00:25:04,950
each one of them can go
over the entire place,

396
00:25:04,950 --> 00:25:10,450
gets replaced by V minus
something to the power of N

397
00:25:10,450 --> 00:25:14,240
and that something
is N omega over 2.

398
00:25:17,810 --> 00:25:20,120
Plus higher orders
in powers of omega.

399
00:25:23,228 --> 00:25:23,728
OK?

400
00:25:28,520 --> 00:25:33,630
So I will call this a
mean field estimate.

401
00:25:39,900 --> 00:25:44,300
Really, it's an average
density estimate

402
00:25:44,300 --> 00:25:46,840
but this kind of
approximation is typically

403
00:25:46,840 --> 00:25:50,890
done for magnetic systems
and in that context

404
00:25:50,890 --> 00:25:54,840
the name of mean
field has stuck.

405
00:25:54,840 --> 00:26:00,850
And the ultimate result is that
my estimation for the partition

406
00:26:00,850 --> 00:26:09,210
function is 1 over
N factorial lambda

407
00:26:09,210 --> 00:26:14,660
to the power of 3 N V
minus N omega over 2

408
00:26:14,660 --> 00:26:20,370
raised to the power of N
because of the excluded volume

409
00:26:20,370 --> 00:26:22,960
and because of the Boltzmann
weight of the attraction

410
00:26:22,960 --> 00:26:25,070
between particles
I have a term which

411
00:26:25,070 --> 00:26:30,620
is e to the minus beta
U N squared over 2V.

412
00:26:37,480 --> 00:26:40,420
OK?

413
00:26:40,420 --> 00:26:44,740
So the approximations
are clear on what

414
00:26:44,740 --> 00:26:50,765
set of assumptions I get this
estimate for the partition

415
00:26:50,765 --> 00:26:53,450
function.

416
00:26:53,450 --> 00:26:55,730
Once I have the
partition function,

417
00:26:55,730 --> 00:27:00,060
I can calculate the
pressure because log z is

418
00:27:00,060 --> 00:27:01,890
going to be related
to free energy,

419
00:27:01,890 --> 00:27:05,500
the derivative of the free
energy respect to volume

420
00:27:05,500 --> 00:27:09,450
will give me the pressure,
and you can ultimately

421
00:27:09,450 --> 00:27:14,410
check that the formula
is that beta P is the log

422
00:27:14,410 --> 00:27:20,720
Z by d V. That's
the correct formula.

423
00:27:20,720 --> 00:27:24,490
And because we are doing this
in the canonical formalism

424
00:27:24,490 --> 00:27:27,490
that I emphasized
over here, let's

425
00:27:27,490 --> 00:27:31,080
call this P that we get through
this process beta P canonical.

426
00:27:33,625 --> 00:27:34,970
OK?

427
00:27:34,970 --> 00:27:40,060
So when I take log z,
there's a bunch of terms

428
00:27:40,060 --> 00:27:41,990
and I don't really
care about because they

429
00:27:41,990 --> 00:27:44,810
don't have V dependence
from N factorial lambda

430
00:27:44,810 --> 00:27:48,830
to the power of 3N,
but I have N times log

431
00:27:48,830 --> 00:27:52,260
of V minus N omega over 2.

432
00:27:52,260 --> 00:27:54,830
So that has V dependence.

433
00:27:54,830 --> 00:27:59,120
Take the derivative of log
of V minus N omega over 2,

434
00:27:59,120 --> 00:28:00,580
what do I get?

435
00:28:00,580 --> 00:28:03,760
I get V minus N omega over 2.

436
00:28:07,220 --> 00:28:10,000
The other theorem
when I take the log

437
00:28:10,000 --> 00:28:15,020
I have a minus beta
U N squared over 2 V

438
00:28:15,020 --> 00:28:17,160
so then I take a
derivative of this 1

439
00:28:17,160 --> 00:28:20,480
over V. I will
get minus 1 over V

440
00:28:20,480 --> 00:28:23,500
squared so I have
a term here which

441
00:28:23,500 --> 00:28:33,090
is beta U over 2 N
squared over V squared.

442
00:28:37,930 --> 00:28:39,380
OK?

443
00:28:39,380 --> 00:28:41,930
Let's multiply by
k T so that we have

444
00:28:41,930 --> 00:28:47,220
the formula for P canonical.

445
00:28:47,220 --> 00:28:56,940
P canonical is then N k
T V minus N omega over 2

446
00:28:56,940 --> 00:29:02,750
and then I have-- I think I made
a sign over here-- yes the sign

447
00:29:02,750 --> 00:29:06,880
error that I made
is as follows-- note

448
00:29:06,880 --> 00:29:11,350
that the potential is attractive
so I have a minus U here.

449
00:29:11,350 --> 00:29:15,820
So when I exponentiate
that, it becomes a plus here

450
00:29:15,820 --> 00:29:20,260
and will be a plus
here and therefore I

451
00:29:20,260 --> 00:29:27,250
will have a minus here
and I will have a minus U

452
00:29:27,250 --> 00:29:32,740
over 2 times the
density squared.

453
00:29:32,740 --> 00:29:34,720
OK?

454
00:29:34,720 --> 00:29:36,600
So what did we arrive at?

455
00:29:36,600 --> 00:29:39,042
We arrived at the van
der Waals equation.

456
00:29:44,494 --> 00:29:44,994
OK?

457
00:29:47,970 --> 00:29:53,330
So we discussed
already last time

458
00:29:53,330 --> 00:30:00,030
around that if I now look
at the isotherms pressure

459
00:30:00,030 --> 00:30:02,970
volume at different
temperatures,

460
00:30:02,970 --> 00:30:05,980
at high temperatures
I have no problem.

461
00:30:05,980 --> 00:30:10,500
I will get things that look
reasonable except if I have

462
00:30:10,500 --> 00:30:17,220
to terminate at something that
is related to this excluded

463
00:30:17,220 --> 00:30:22,500
volume, whereas if I go to
low temperatures, what happens

464
00:30:22,500 --> 00:30:25,910
is that the kind of
isotherms that I get

465
00:30:25,910 --> 00:30:31,630
have a structure which
is kind of like this

466
00:30:31,630 --> 00:30:38,350
and incorporates a region
which violates everything

467
00:30:38,350 --> 00:30:40,630
that we know about
thermodynamics,

468
00:30:40,630 --> 00:30:43,880
is not stable, et cetera.

469
00:30:43,880 --> 00:30:46,420
OK?

470
00:30:46,420 --> 00:30:48,310
So what happened here?

471
00:30:52,230 --> 00:30:53,920
I did a calculation.

472
00:30:53,920 --> 00:30:58,000
You can see every step of
the calculation over there.

473
00:30:58,000 --> 00:30:59,370
Why does it give me nonsense?

474
00:31:07,350 --> 00:31:09,500
Everything that you need
to know is on the board.

475
00:31:18,221 --> 00:31:18,720
Suggestions?

476
00:31:22,110 --> 00:31:22,866
Yep?

477
00:31:22,866 --> 00:31:24,657
AUDIENCE: The assumption
of uniform density

478
00:31:24,657 --> 00:31:26,360
in a phase transition?

479
00:31:26,360 --> 00:31:27,580
PROFESSOR: Right.

480
00:31:27,580 --> 00:31:31,030
That's the picture
that I put over here.

481
00:31:31,030 --> 00:31:34,510
If I draw the box that
corresponds to what

482
00:31:34,510 --> 00:31:37,750
is happening over there,
what's going on in the box?

483
00:31:37,750 --> 00:31:40,100
So there is a box.

484
00:31:40,100 --> 00:31:44,860
I have particles in it
and in the region where

485
00:31:44,860 --> 00:31:47,640
this nonsense is
happening, what is actually

486
00:31:47,640 --> 00:31:48,930
happening in reality?

487
00:31:48,930 --> 00:31:51,390
What is actually
happening in reality

488
00:31:51,390 --> 00:31:54,380
is that I get some
of the particles

489
00:31:54,380 --> 00:31:58,220
to condense into
liquid drops somewhere

490
00:31:58,220 --> 00:31:59,880
and then there's
basically the rest

491
00:31:59,880 --> 00:32:03,950
of them floating
around as a gas.

492
00:32:03,950 --> 00:32:05,360
Right?

493
00:32:05,360 --> 00:32:12,560
So clearly I cannot, for this
configuration that I have over

494
00:32:12,560 --> 00:32:18,730
here, calculate energy
and contribution this way.

495
00:32:18,730 --> 00:32:26,460
So coexistence implies
non-uniform density.

496
00:32:33,885 --> 00:32:34,385
OK?

497
00:32:37,270 --> 00:32:41,120
So what do we do?

498
00:32:41,120 --> 00:32:44,100
I want to carry out
this calculation.

499
00:32:44,100 --> 00:32:46,490
I want to stay as
much as possible

500
00:32:46,490 --> 00:32:50,070
and within this framework
and I can do it,

501
00:32:50,070 --> 00:32:52,679
but I need to do
one other thing.

502
00:32:52,679 --> 00:32:53,345
Any suggestions?

503
00:32:58,860 --> 00:32:59,930
Yes?

504
00:32:59,930 --> 00:33:03,670
AUDIENCE: Try to describe
both densities simultaneously?

505
00:33:03,670 --> 00:33:06,695
PROFESSOR: Try to
describe both densities--

506
00:33:06,695 --> 00:33:08,320
AUDIENCE: You know
the amount of liquid

507
00:33:08,320 --> 00:33:10,310
and the amount of
vapor on average--

508
00:33:13,280 --> 00:33:14,610
PROFESSOR: I guess I could.

509
00:33:14,610 --> 00:33:17,341
I have an easier
way of doing things.

510
00:33:17,341 --> 00:33:17,840
OK.

511
00:33:17,840 --> 00:33:20,540
So the easier way that
I have on doing things

512
00:33:20,540 --> 00:33:23,860
is we have emphasized
that in thermodynamics, we

513
00:33:23,860 --> 00:33:26,570
can look at different
perspectives.

514
00:33:26,570 --> 00:33:30,780
Problem with this
perspective of the canonical

515
00:33:30,780 --> 00:33:34,880
is that I know that I will
encounter this singularity.

516
00:33:34,880 --> 00:33:35,540
I know that.

517
00:33:35,540 --> 00:33:38,450
There's phase diagrams
like this exist.

518
00:33:38,450 --> 00:33:43,070
Whereas if I use this
prescription I have one.

519
00:33:43,070 --> 00:33:45,370
So what's the difference
is this prescription

520
00:33:45,370 --> 00:33:49,240
is better mind in terms of
pressure and temperature.

521
00:33:49,240 --> 00:33:52,500
So what I want to
do is to replace

522
00:33:52,500 --> 00:33:55,380
the canonical
perspective with what

523
00:33:55,380 --> 00:34:00,080
I will call the isobaric-- it's
the Gibbs canonical ensemble.

524
00:34:00,080 --> 00:34:04,220
In this context it's
called isobaric ensemble

525
00:34:04,220 --> 00:34:08,469
where rather than describing
things at fixed volume,

526
00:34:08,469 --> 00:34:11,560
I describe things
at fixed pressure.

527
00:34:11,560 --> 00:34:14,010
So I tell you what
pressure, temperature,

528
00:34:14,010 --> 00:34:16,710
and the number of
particles are and then

529
00:34:16,710 --> 00:34:21,440
I calculate the corresponding
Gibbs partition function.

530
00:34:25,030 --> 00:34:27,449
Why does that help?

531
00:34:27,449 --> 00:34:30,690
Why that helps is because of
what you see on the left side

532
00:34:30,690 --> 00:34:33,900
diagram is if I
wanted to maintain

533
00:34:33,900 --> 00:34:36,120
this system at the
fixed pressure rather

534
00:34:36,120 --> 00:34:38,810
than a fixed volume,
I would replace

535
00:34:38,810 --> 00:34:42,900
one of the walls of this
container with a piston

536
00:34:42,900 --> 00:34:45,520
and then I would apply a
particular value of pressure.

537
00:34:48,060 --> 00:34:50,620
Then you can see that a
situation such as this

538
00:34:50,620 --> 00:34:53,179
does not occur.

539
00:34:53,179 --> 00:34:58,300
The piston will move
so that ultimately I

540
00:34:58,300 --> 00:35:00,650
have a uniform density.

541
00:35:00,650 --> 00:35:05,820
If I'm choosing my
pressure to be up here,

542
00:35:05,820 --> 00:35:07,100
then it would be all liquid.

543
00:35:07,100 --> 00:35:10,700
If I choose my pressure to be
down here, it will be all gas.

544
00:35:10,700 --> 00:35:13,220
So I have this piston on top.

545
00:35:13,220 --> 00:35:16,250
It does not tolerate
coexistence.

546
00:35:16,250 --> 00:35:19,260
It will either keep
everybody in the gas phase

547
00:35:19,260 --> 00:35:21,590
or it will compress,
change the volume

548
00:35:21,590 --> 00:35:25,970
so that everybody becomes
liquid so what I have done

549
00:35:25,970 --> 00:35:28,520
is I have gotten
rid of the volume.

550
00:35:28,520 --> 00:35:32,670
And I do expect that
in this ensemble as I

551
00:35:32,670 --> 00:35:35,270
change pressure and
go through this line,

552
00:35:35,270 --> 00:35:40,340
there is a discontinuity that
I should observe in volume.

553
00:35:40,340 --> 00:35:41,790
OK?

554
00:35:41,790 --> 00:35:43,660
So again just think
of the physics.

555
00:35:43,660 --> 00:35:45,860
Put the piston, what's
going to happen?

556
00:35:45,860 --> 00:35:49,050
It's going to be either
one or the other.

557
00:35:49,050 --> 00:35:52,780
And whether it is one or the
other, I'm at uniform density

558
00:35:52,780 --> 00:35:56,150
and so I should be able to
use that approximation that I

559
00:35:56,150 --> 00:35:58,034
have over there.

560
00:35:58,034 --> 00:36:00,330
OK?

561
00:36:00,330 --> 00:36:04,100
So mathematics,
what does that mean?

562
00:36:04,100 --> 00:36:07,240
This Gibbs version
of the partition

563
00:36:07,240 --> 00:36:10,330
function in this
isobaric ensemble,

564
00:36:10,330 --> 00:36:14,060
you don't fix the volume
you just integrate over

565
00:36:14,060 --> 00:36:19,200
all possible volumes weighted
by this e to the minus beta P V

566
00:36:19,200 --> 00:36:22,130
that does the Laplace
transform from one ensemble

567
00:36:22,130 --> 00:36:26,550
to the other of the
partition function

568
00:36:26,550 --> 00:36:35,730
which is at some fixed
volume, V T N. OK?

569
00:36:35,730 --> 00:36:47,106
And for this we use the
uniform density approximation.

570
00:36:53,060 --> 00:36:58,980
So my Gibbs partition
function, function

571
00:36:58,980 --> 00:37:06,530
of P T and N is the integral 0
to infinity d V e to the minus

572
00:37:06,530 --> 00:37:14,720
beta P V and I guess I have
what would go over here

573
00:37:14,720 --> 00:37:24,850
would be log Z of V T N. OK?

574
00:37:24,850 --> 00:37:28,160
Again this is answer
to previous question,

575
00:37:28,160 --> 00:37:31,310
I am in the thermodynamic limit.

576
00:37:31,310 --> 00:37:37,030
I expect that this which gives
essentially the probabilities

577
00:37:37,030 --> 00:37:39,500
or the weights of
different volumes

578
00:37:39,500 --> 00:37:43,390
is going to be dominated
by a single volume that

579
00:37:43,390 --> 00:37:47,120
makes the largest contribution
to the thermodynamics

580
00:37:47,120 --> 00:37:49,110
or to this integration.

581
00:37:49,110 --> 00:37:51,710
But this integration
should be subtle point

582
00:37:51,710 --> 00:37:56,780
like so what I would do is I
will call whatever is appearing

583
00:37:56,780 --> 00:38:02,560
here to be some function of V
because I'm integrating over V

584
00:38:02,560 --> 00:38:05,790
and I'm going to
look for its extreme.

585
00:38:05,790 --> 00:38:12,450
So basically I expect that this
because of this subtle point

586
00:38:12,450 --> 00:38:19,580
will be e to the psi of V
that maximizes this weight.

587
00:38:23,720 --> 00:38:24,220
OK?

588
00:38:28,310 --> 00:38:34,270
So what I have to do is I
have to take the psi by d V

589
00:38:34,270 --> 00:38:36,990
and set that to 0.

590
00:38:36,990 --> 00:38:45,830
Well part of it is simply minus
beta P. The other part of it

591
00:38:45,830 --> 00:39:00,340
is log Z by d V but the log
Z by d V I have up there.

592
00:39:00,340 --> 00:39:02,290
It's the same Z that
I'm calculating.

593
00:39:02,290 --> 00:39:04,980
It is beta P canonical.

594
00:39:04,980 --> 00:39:09,710
So this is none
other than minus beta

595
00:39:09,710 --> 00:39:14,980
the P that I in this
ensemble have set out,

596
00:39:14,980 --> 00:39:18,529
minus this P canonical that
I calculated before which

597
00:39:18,529 --> 00:39:19,820
is some function of the volume.

598
00:39:24,500 --> 00:39:26,600
Again, generally, it's fine.

599
00:39:26,600 --> 00:39:29,550
We expect the
different ensembles

600
00:39:29,550 --> 00:39:33,080
to correspond to each
other and essentially that

601
00:39:33,080 --> 00:39:38,670
says that most of the time
you will see that you're

602
00:39:38,670 --> 00:39:44,060
going to get the pressure
calculated in this fashion

603
00:39:44,060 --> 00:39:49,510
and calculated canonically
to be the same thing.

604
00:39:49,510 --> 00:39:54,970
Except that sometimes we seem
to have ambiguity because what

605
00:39:54,970 --> 00:39:58,760
I should do is given that I
have some particular value

606
00:39:58,760 --> 00:40:06,750
of pressure that I emphasize
exists in my isobaric ensemble,

607
00:40:06,750 --> 00:40:09,020
I calculate what
the corresponding V

608
00:40:09,020 --> 00:40:11,500
is by solving this equation.

609
00:40:11,500 --> 00:40:13,710
What is this
equation graphically?

610
00:40:13,710 --> 00:40:17,420
It says pick your
P and figure out

611
00:40:17,420 --> 00:40:23,560
what volume in these
canonical curves, intersected.

612
00:40:23,560 --> 00:40:28,250
So the case that I drew
has clearly one answer.

613
00:40:28,250 --> 00:40:32,720
I can go up here
I have one answer.

614
00:40:32,720 --> 00:40:36,990
But what do I do when I have
a situation such as this?

615
00:40:36,990 --> 00:40:38,600
I have three answers.

616
00:40:43,010 --> 00:40:50,500
So in this ensemble I
say what the pressure is

617
00:40:50,500 --> 00:40:52,670
and I ask, well,
what's the volume?

618
00:40:52,670 --> 00:40:56,590
It says you should solve this
equation to find the volume.

619
00:40:56,590 --> 00:40:58,670
I solve the equation
graphically and I

620
00:40:58,670 --> 00:41:01,880
have now three
possible solutions.

621
00:41:01,880 --> 00:41:02,988
What does that mean?

622
00:41:06,270 --> 00:41:12,910
All this equation says is that
these solutions are extrema.

623
00:41:12,910 --> 00:41:15,940
I set the derivative to 0.

624
00:41:15,940 --> 00:41:22,060
The task here is to find if
I have multiple solutions,

625
00:41:22,060 --> 00:41:27,310
the one that gives me the
largest value over here.

626
00:41:27,310 --> 00:41:29,910
So what could be happening?

627
00:41:29,910 --> 00:41:34,140
What is happening is
that when I'm integrating

628
00:41:34,140 --> 00:41:37,610
as a function of
V, this integrand

629
00:41:37,610 --> 00:41:41,910
which is e to psi of V. Right?

630
00:41:41,910 --> 00:41:45,510
An integrand in the
curve let's call

631
00:41:45,510 --> 00:41:49,070
these curves over
here number one.

632
00:41:49,070 --> 00:41:52,240
In the case number one
there's a clear solution

633
00:41:52,240 --> 00:41:59,640
is if I'm scanning in volume,
there is a particular location

634
00:41:59,640 --> 00:42:05,530
that maximizes this side of V
and the corresponding volume

635
00:42:05,530 --> 00:42:07,710
is unambiguous.

636
00:42:07,710 --> 00:42:09,900
If I go and look
at the curve let's

637
00:42:09,900 --> 00:42:16,035
say up here let's call it
that case number three.

638
00:42:16,035 --> 00:42:21,210
For case number three it
also hits the blue curve

639
00:42:21,210 --> 00:42:25,420
unambiguously at one point
which is at much lower volume

640
00:42:25,420 --> 00:42:27,720
and so presumably
I have a situation

641
00:42:27,720 --> 00:42:30,200
such as this for my number two.

642
00:42:34,470 --> 00:42:37,860
Now given these
two cases it is not

643
00:42:37,860 --> 00:42:44,040
surprising that the middle
one, let's say number two here,

644
00:42:44,040 --> 00:42:49,070
corresponds to a situation
where you have two maximums.

645
00:42:49,070 --> 00:42:52,790
So generically,
presumably, case number two

646
00:42:52,790 --> 00:42:57,840
corresponds to
something such as this.

647
00:42:57,840 --> 00:42:58,640
OK?

648
00:42:58,640 --> 00:43:02,920
There are three solutions to
setting the derivative to 0.

649
00:43:02,920 --> 00:43:04,570
There are three extrema.

650
00:43:04,570 --> 00:43:08,130
Two extrema
correspond to maxima.

651
00:43:08,130 --> 00:43:11,350
One extrema corresponds
to the minimum

652
00:43:11,350 --> 00:43:14,810
and falls between the
two and clearly that

653
00:43:14,810 --> 00:43:19,610
corresponds to this portion
that we say is unstable.

654
00:43:19,610 --> 00:43:23,610
It's clearly unstable because
it's the least likely place

655
00:43:23,610 --> 00:43:25,250
that you're going
to find something.

656
00:43:25,250 --> 00:43:25,930
Right?

657
00:43:25,930 --> 00:43:29,410
Not the least likely but if you
go a little bit to either side,

658
00:43:29,410 --> 00:43:32,790
the probabilities
really increase.

659
00:43:32,790 --> 00:43:33,290
Yes?

660
00:43:33,290 --> 00:43:36,180
AUDIENCE: I think your
indices don't correspond

661
00:43:36,180 --> 00:43:38,500
on the plot you've just
drawn and on the [INAUDIBLE].

662
00:43:38,500 --> 00:43:44,682
Like 1, 2, 3 in parentheses,
in two different cases

663
00:43:44,682 --> 00:43:47,620
they don't correspond to it.

664
00:43:47,620 --> 00:43:50,567
PROFESSOR: Let's
pick number one.

665
00:43:50,567 --> 00:43:52,523
AUDIENCE: Number
one is when we have

666
00:43:52,523 --> 00:43:54,724
a singular maximum
is the biggest.

667
00:43:54,724 --> 00:43:56,390
PROFESSOR: With the
biggest volume, yes?

668
00:43:56,390 --> 00:43:56,940
AUDIENCE: No.

669
00:43:56,940 --> 00:43:58,340
Number two should be--

670
00:43:58,340 --> 00:43:59,260
PROFESSOR: Oh, yes.

671
00:43:59,260 --> 00:44:02,020
That's right.

672
00:44:02,020 --> 00:44:08,860
So this is number three
and this is number two.

673
00:44:08,860 --> 00:44:14,560
And for number two I indicated
that there are maxima that I

674
00:44:14,560 --> 00:44:18,720
labelled one, two, and three
without the parenthesis that

675
00:44:18,720 --> 00:44:20,610
would presumably
correspond to these.

676
00:44:24,820 --> 00:44:25,320
OK?

677
00:44:28,150 --> 00:44:33,590
So then you see there
is now no ambiguity.

678
00:44:36,200 --> 00:44:40,710
I have to pick
among the three that

679
00:44:40,710 --> 00:44:44,110
occur over there, the three
solutions that correspond

680
00:44:44,110 --> 00:44:47,390
that occur from the
van der Waals equation.

681
00:44:47,390 --> 00:44:52,130
The one that would give the
highest value for this function

682
00:44:52,130 --> 00:44:54,780
psi.

683
00:44:54,780 --> 00:44:59,580
Actually what is psi if you
think about this ensemble?

684
00:44:59,580 --> 00:45:07,325
This ensemble is going to
be dominated by minus beta P

685
00:45:07,325 --> 00:45:14,310
V at the location of the
V that is thermodynamic.

686
00:45:14,310 --> 00:45:19,530
This Z is e to the minus beta
E, so there's a minus beta E

687
00:45:19,530 --> 00:45:24,750
and there's omega which
gives me a beta T S.

688
00:45:24,750 --> 00:45:28,460
So it is this combination
of thermodynamic quantities

689
00:45:28,460 --> 00:45:33,520
which if you go and look at your
extensivity characterization

690
00:45:33,520 --> 00:45:37,900
is related to the
chemical potential.

691
00:45:37,900 --> 00:45:42,590
So the value of the
psi at the maximum

692
00:45:42,590 --> 00:45:46,360
is directly related to
the chemical potential

693
00:45:46,360 --> 00:45:51,180
and finding which one of
these has the largest value

694
00:45:51,180 --> 00:45:54,580
corresponds to which solution
has the lowest chemical

695
00:45:54,580 --> 00:45:55,770
potential.

696
00:45:55,770 --> 00:45:59,320
Remember we were drawing this
curve last time around where

697
00:45:59,320 --> 00:46:03,230
we integrated van der
Waals equation to calculate

698
00:46:03,230 --> 00:46:05,050
the chemical
potential and then we

699
00:46:05,050 --> 00:46:07,570
have multiple solutions
for chemical potential.

700
00:46:07,570 --> 00:46:09,843
We picked the lowest one.

701
00:46:09,843 --> 00:46:13,491
Well, here's the justification.

702
00:46:13,491 --> 00:46:13,990
OK?

703
00:46:16,730 --> 00:46:17,230
All right.

704
00:46:20,480 --> 00:46:22,370
So now what's happening?

705
00:46:22,370 --> 00:46:26,170
So the question
that I ask is how

706
00:46:26,170 --> 00:46:32,130
can doing integrals such as
this give you some singularity

707
00:46:32,130 --> 00:46:35,210
as you're scanning in say
pressure or temperature

708
00:46:35,210 --> 00:46:37,660
along the lines such as this?

709
00:46:37,660 --> 00:46:43,200
And now we have the
mechanism because presumably

710
00:46:43,200 --> 00:46:47,530
as I go from here to here
the maximum that corresponds

711
00:46:47,530 --> 00:46:53,580
to the liquid volume will get
replaced with the maximum that

712
00:46:53,580 --> 00:46:57,010
corresponds to the gas
volume or vice versa.

713
00:46:57,010 --> 00:47:04,770
So essentially if you have
a curve such as two then

714
00:47:04,770 --> 00:47:11,960
the value of your Z which is
also e to the minus beta U N

715
00:47:11,960 --> 00:47:20,840
as we said is determined by the
contribution from here or here.

716
00:47:20,840 --> 00:47:22,480
Well, let's write both of them.

717
00:47:22,480 --> 00:47:26,810
I have e to the
psi corresponding

718
00:47:26,810 --> 00:47:30,920
to the V of the gas plus
e to the psi corresponding

719
00:47:30,920 --> 00:47:33,580
to V the liquid.

720
00:47:33,580 --> 00:47:37,040
Now both of these
quantities are quantities

721
00:47:37,040 --> 00:47:42,550
that in the large N limit
will be exponentially large.

722
00:47:42,550 --> 00:47:46,390
So one or the
other will dominate

723
00:47:46,390 --> 00:47:49,550
and the mechanism of
the phase transition

724
00:47:49,550 --> 00:47:53,190
is that as I change
parameters, as I change my P,

725
00:47:53,190 --> 00:47:56,510
I go from a situation
that is like this

726
00:47:56,510 --> 00:47:59,100
to a situation
that is like this.

727
00:47:59,100 --> 00:48:03,270
The two maxima change heights.

728
00:48:03,270 --> 00:48:03,770
OK?

729
00:48:06,350 --> 00:48:10,926
It is because you are
in the large N limit

730
00:48:10,926 --> 00:48:15,640
that this can be expressed
either as being this

731
00:48:15,640 --> 00:48:19,260
or as being the other
and not as a mixture

732
00:48:19,260 --> 00:48:21,410
because the mixture you
can write it that way

733
00:48:21,410 --> 00:48:23,930
but it's a negligible
contribution

734
00:48:23,930 --> 00:48:25,880
from one or the other.

735
00:48:25,880 --> 00:48:30,040
This is the answer I was giving
in connection with computer

736
00:48:30,040 --> 00:48:31,540
simulations.

737
00:48:31,540 --> 00:48:35,330
Computer simulations, let's
say you have 1,000 particles.

738
00:48:35,330 --> 00:48:40,170
You have e to the 1,000 times
something, e to the 1,000

739
00:48:40,170 --> 00:48:42,600
times another thing,
and if you, for example,

740
00:48:42,600 --> 00:48:47,760
look at the curves for
something like the density

741
00:48:47,760 --> 00:48:52,990
you will find that you will
start with the gas density.

742
00:48:52,990 --> 00:48:56,700
You ultimately have to
go to the liquid density

743
00:48:56,700 --> 00:48:58,960
in the true
thermodynamic limit there

744
00:48:58,960 --> 00:49:01,310
will be a discontinuity here.

745
00:49:01,310 --> 00:49:04,580
If you do a computer
simulation with finite end,

746
00:49:04,580 --> 00:49:08,810
you will get a continuous
curve joining one to the other.

747
00:49:08,810 --> 00:49:12,500
As you make your size of
your system simulated bigger,

748
00:49:12,500 --> 00:49:14,920
this becomes
sharper and sharper,

749
00:49:14,920 --> 00:49:18,960
but truly the
singularity will emerge

750
00:49:18,960 --> 00:49:22,290
only in the N goes
to infinity limit.

751
00:49:22,290 --> 00:49:26,820
So phase transitions, et
cetera, mathematically really

752
00:49:26,820 --> 00:49:30,130
exist only for infinite
number of particles.

753
00:49:30,130 --> 00:49:33,310
Well, of course, for
10 to the 23 particles

754
00:49:33,310 --> 00:49:35,340
the resolution you
would need over

755
00:49:35,340 --> 00:49:38,010
here to see that it's
actually going continuously

756
00:49:38,010 --> 00:49:41,760
from one to the other
is immeasurably small.

757
00:49:46,066 --> 00:49:46,565
OK?

758
00:49:49,220 --> 00:49:51,820
Any questions about that?

759
00:49:51,820 --> 00:49:53,250
I guess there was
one other thing

760
00:49:53,250 --> 00:49:56,270
that maybe we should note.

761
00:49:56,270 --> 00:50:01,350
So if I ask what is the
pressure that corresponds

762
00:50:01,350 --> 00:50:04,050
to the location of
this phase transition,

763
00:50:04,050 --> 00:50:07,500
so what is the value of
the pressure that separates

764
00:50:07,500 --> 00:50:11,620
the liquid and gas, the
location of the singularity?

765
00:50:11,620 --> 00:50:19,450
I would have to find Pc from
the following observation,

766
00:50:19,450 --> 00:50:23,810
that that's the pressure
at which the two

767
00:50:23,810 --> 00:50:27,800
psi's, psi of the
gas is the same thing

768
00:50:27,800 --> 00:50:31,220
as the psi of the liquid.

769
00:50:31,220 --> 00:50:33,160
Right?

770
00:50:33,160 --> 00:50:34,010
All right.

771
00:50:34,010 --> 00:50:39,312
Or alternatively, the difference
between them is equal to 0.

772
00:50:39,312 --> 00:50:40,780
OK?

773
00:50:40,780 --> 00:50:42,720
And the difference
between-- I can

774
00:50:42,720 --> 00:50:46,220
write this kind of
trivially as the integral

775
00:50:46,220 --> 00:50:48,800
of the derivative
of the function.

776
00:50:48,800 --> 00:50:57,860
So d V d psi by d V integrated
between the V of the liquid

777
00:50:57,860 --> 00:50:59,060
and then the V of the gas.

778
00:51:03,020 --> 00:51:06,440
That integral should be 0.

779
00:51:06,440 --> 00:51:11,450
d psi by d V we established,
is this quantity.

780
00:51:11,450 --> 00:51:25,420
It is beta integral from
V liquid to V gas d V P

781
00:51:25,420 --> 00:51:31,600
canonical of V minus this
P transition that I want,

782
00:51:31,600 --> 00:51:34,620
has to be zero.

783
00:51:34,620 --> 00:51:35,970
OK?

784
00:51:35,970 --> 00:51:38,470
So what do I have to do?

785
00:51:38,470 --> 00:51:44,420
Since that we identify the
location of your Pc such

786
00:51:44,420 --> 00:51:49,630
that if you integrate the
canonical minus this Pc

787
00:51:49,630 --> 00:51:54,320
all the way from the V of
the gas to V of the liquid,

788
00:51:54,320 --> 00:51:59,110
this integral will give you 0.

789
00:51:59,110 --> 00:52:02,300
So this is the
Maxwell construction

790
00:52:02,300 --> 00:52:04,719
that you were using last time.

791
00:52:09,340 --> 00:52:12,430
Again, it is
equivalent to the fact

792
00:52:12,430 --> 00:52:17,520
that we stated that really this
object is the same or related

793
00:52:17,520 --> 00:52:21,430
to the chemical potential, so
I'm requiring that the chemical

794
00:52:21,430 --> 00:52:27,080
potential should be the same
as I go through the transition.

795
00:52:34,340 --> 00:52:34,840
Yes?

796
00:52:34,840 --> 00:52:38,720
AUDIENCE: Could you repeat
why minus finite size

797
00:52:38,720 --> 00:52:40,680
does not in phase transition?

798
00:52:40,680 --> 00:52:41,405
PROFESSOR: OK.

799
00:52:41,405 --> 00:52:41,904
Yes.

800
00:52:46,990 --> 00:52:55,690
So let's imagine that I
have two possibilities that

801
00:52:55,690 --> 00:53:01,360
appear as exponentials but
the number that is appearing

802
00:53:01,360 --> 00:53:05,800
in the exponential which is the
analog of N makes one of them

803
00:53:05,800 --> 00:53:10,560
to be positive so I have
e to the N u and the other

804
00:53:10,560 --> 00:53:11,660
to be negative.

805
00:53:11,660 --> 00:53:17,358
So my object would be
something like this.

806
00:53:17,358 --> 00:53:19,270
OK?

807
00:53:19,270 --> 00:53:25,270
Then what is this?

808
00:53:25,270 --> 00:53:31,050
This is related to a factor of
2 to hyperbolic cosine of N u.

809
00:53:31,050 --> 00:53:31,550
OK?

810
00:53:34,530 --> 00:53:38,700
So this is like some kind
of a partition function

811
00:53:38,700 --> 00:53:41,000
and what we are
interested is something

812
00:53:41,000 --> 00:53:42,465
like the log of this quantity.

813
00:53:45,050 --> 00:53:47,550
And maybe if I take a
derivative with respect

814
00:53:47,550 --> 00:53:50,480
to u of this I
will get something

815
00:53:50,480 --> 00:53:53,500
like the tangent of N u.

816
00:53:53,500 --> 00:53:54,000
OK?

817
00:53:54,000 --> 00:53:56,450
So some kind-- something
like a tangent and u

818
00:53:56,450 --> 00:53:58,420
would be something
like the expectation

819
00:53:58,420 --> 00:54:00,330
value of this quantity.

820
00:54:00,330 --> 00:54:04,210
It's either plus u
or minus u and it

821
00:54:04,210 --> 00:54:09,980
occurs-- so the average of
it would be e to the N u

822
00:54:09,980 --> 00:54:16,370
minus e to the minus N u e to
the N u plus e to the minus N u

823
00:54:16,370 --> 00:54:20,210
and so u would be some
kind of an expectation

824
00:54:20,210 --> 00:54:22,500
value of the quantity
such as this.

825
00:54:22,500 --> 00:54:26,940
Now what is this function look
like the function tangent.

826
00:54:26,940 --> 00:54:32,710
So for positive values it goes
to plus 1, for negative values

827
00:54:32,710 --> 00:54:37,500
it goes to minus 1 and it's a
perfectly continuous function.

828
00:54:37,500 --> 00:54:40,350
It goes to 0.

829
00:54:40,350 --> 00:54:41,630
All right?

830
00:54:41,630 --> 00:54:44,020
That's the tangent function.

831
00:54:44,020 --> 00:54:48,490
Now suppose I calculate
this for N of 10,

832
00:54:48,490 --> 00:54:50,450
this is the curve that I drew.

833
00:54:50,450 --> 00:54:54,640
Suppose I draw this
same curve for N of 100.

834
00:54:54,640 --> 00:54:56,860
For N of 100 I
claim that the curve

835
00:54:56,860 --> 00:55:03,330
will look something like
this because the slope

836
00:55:03,330 --> 00:55:05,900
that you have at the
origin for the tangent

837
00:55:05,900 --> 00:55:10,110
grows like N so basically
becomes steeper and steeper

838
00:55:10,110 --> 00:55:14,200
but ultimately has to
saturate one or the other.

839
00:55:14,200 --> 00:55:20,030
So what-- for any, for even N of
1,000 then it's a finite slope.

840
00:55:20,030 --> 00:55:24,510
The slope here is related to
1,000 U 1,000, but it's finite.

841
00:55:24,510 --> 00:55:26,650
It is only in the
limit where N goes

842
00:55:26,650 --> 00:55:30,250
to infinity you would say
that the function is either

843
00:55:30,250 --> 00:55:32,950
minus 1 or plus 1
depending on whether U

844
00:55:32,950 --> 00:55:35,350
is positive or negative.

845
00:55:35,350 --> 00:55:38,840
So the same thing
happens as I scan over

846
00:55:38,840 --> 00:55:42,720
here and ask what
is the density?

847
00:55:42,720 --> 00:55:46,940
The density looks precisely
like this tangent function.

848
00:55:46,940 --> 00:55:49,680
For finite values of
the number of particles

849
00:55:49,680 --> 00:55:52,090
the density would do
something like this.

850
00:55:52,090 --> 00:55:55,684
It is only for infinite
number of particles

851
00:55:55,684 --> 00:55:57,100
that the density
is discontinuous.

852
00:56:02,401 --> 00:56:02,900
Yes?

853
00:56:02,900 --> 00:56:06,288
AUDIENCE: Sir, on the plot
which you drew initially

854
00:56:06,288 --> 00:56:10,160
on that board, what are the
axis over which this scans?

855
00:56:13,255 --> 00:56:14,130
PROFESSOR: Over here?

856
00:56:14,130 --> 00:56:15,300
AUDIENCE: Yeah.

857
00:56:15,300 --> 00:56:16,841
Just what values
are physical values?

858
00:56:19,147 --> 00:56:19,730
PROFESSOR: OK.

859
00:56:19,730 --> 00:56:20,590
Pressure, right?

860
00:56:20,590 --> 00:56:26,670
So what I said was that if
I scan across pressure what

861
00:56:26,670 --> 00:56:30,130
I will see is a
discontinuity, right?

862
00:56:30,130 --> 00:56:30,840
So if--

863
00:56:30,840 --> 00:56:32,590
AUDIENCE: You said
that this is the result

864
00:56:32,590 --> 00:56:34,740
of a numerical experiment with
finite number of particles.

865
00:56:34,740 --> 00:56:35,890
So what is the parameter
of the experiment,

866
00:56:35,890 --> 00:56:36,848
and what is the result.

867
00:56:41,190 --> 00:56:44,160
PROFESSOR: We have the
density so you could have,

868
00:56:44,160 --> 00:56:46,680
for example, exactly
the situation that I--

869
00:56:46,680 --> 00:56:47,980
AUDIENCE: [INAUDIBLE]?

870
00:56:47,980 --> 00:56:50,150
PROFESSOR: Pressure.

871
00:56:50,150 --> 00:56:53,540
This box that I'm
showing you over here,

872
00:56:53,540 --> 00:56:55,060
you simulate on the computer.

873
00:56:55,060 --> 00:57:00,050
You have a box of finite volume,
you put a piston on top of it,

874
00:57:00,050 --> 00:57:02,879
you simulate 1,000
particles inside.

875
00:57:02,879 --> 00:57:03,420
AUDIENCE: OK.

876
00:57:03,420 --> 00:57:03,920
Thank you.

877
00:57:03,920 --> 00:57:07,180
PROFESSOR: Then you scan
as a function of pressure,

878
00:57:07,180 --> 00:57:09,330
you plug the density.

879
00:57:09,330 --> 00:57:11,860
Now, of course, if you
want to simulate this,

880
00:57:11,860 --> 00:57:14,420
you have to increase both
the number of particles

881
00:57:14,420 --> 00:57:19,230
and the volume of the box so
that the average density is

882
00:57:19,230 --> 00:57:21,488
fixed and then you would
see something like this.

883
00:57:25,700 --> 00:57:26,260
OK?

884
00:57:26,260 --> 00:57:27,093
Any other questions?

885
00:57:31,630 --> 00:57:35,625
OK, let's stick
with this picture.

886
00:57:45,470 --> 00:57:51,190
I kind of focused on the
occurrence of the singularity,

887
00:57:51,190 --> 00:57:56,460
but now let's see the exact
location of the singularity.

888
00:57:56,460 --> 00:58:03,246
So I sort of started this
lecture by labeling Pc, Tc, Vc.

889
00:58:03,246 --> 00:58:05,480
Well, what are their values?

890
00:58:05,480 --> 00:58:06,410
OK.

891
00:58:06,410 --> 00:58:12,140
So something that you
know from stability

892
00:58:12,140 --> 00:58:17,000
is that the isotherms
are constrained such

893
00:58:17,000 --> 00:58:21,590
that delta P delta V
has to be negative.

894
00:58:21,590 --> 00:58:23,470
OK?

895
00:58:23,470 --> 00:58:32,680
Or writing delta V as a
function of V or delta V,

896
00:58:32,680 --> 00:58:38,740
this is d P by d V along an
isotherm delta V plus d 2

897
00:58:38,740 --> 00:58:42,500
P by d V squared along
the isotherm delta

898
00:58:42,500 --> 00:58:49,710
V squared-- this is one-half
plus one-sixth d cubed P d

899
00:58:49,710 --> 00:58:53,630
V cubed along the
isotherm and so forth.

900
00:58:53,630 --> 00:59:00,498
This whole thing
has to be negative.

901
00:59:00,498 --> 00:59:00,998
OK?

902
00:59:05,440 --> 00:59:13,960
Now generically I pick a point
and a small value of delta V

903
00:59:13,960 --> 00:59:17,280
and the statement is
that along the isotherm

904
00:59:17,280 --> 00:59:21,220
this derivative
has to be negative.

905
00:59:21,220 --> 00:59:28,420
So generically what
I have is that d P

906
00:59:28,420 --> 00:59:33,430
by d V for a physical
isotherm has to be negative

907
00:59:33,430 --> 00:59:37,390
and indeed the reason we
don't like that portion

908
00:59:37,390 --> 00:59:41,890
is because it violates
the stability condition.

909
00:59:41,890 --> 00:59:44,080
Well, clearly that
will be broken

910
00:59:44,080 --> 00:59:48,760
at one point the isotherm that
corresponds to T equals to Tc

911
00:59:48,760 --> 00:59:51,340
is not a generic isotherm.

912
00:59:51,340 --> 00:59:58,120
That's an isotherm that at some
point it comes in tangentially.

913
00:59:58,120 --> 01:00:06,020
So there is over here a
point where d P by d V is 0

914
01:00:06,020 --> 01:00:07,940
and we already discussed this.

915
01:00:07,940 --> 01:00:16,250
If d P by d V is 0 then
the second derivative

916
01:00:16,250 --> 01:00:20,500
in such an expansion
must also be 0.

917
01:00:20,500 --> 01:00:21,000
Why?

918
01:00:21,000 --> 01:00:25,396
Because if it is
nonzero, irrespective

919
01:00:25,396 --> 01:00:29,275
of whether it is positive or
negative, since it multiplies

920
01:00:29,275 --> 01:00:34,510
that V cubed, I will be
able to pick a delta V such

921
01:00:34,510 --> 01:00:37,610
that the sine will be
violated by the delta V

922
01:00:37,610 --> 01:00:40,290
of positive or
delta V of negative.

923
01:00:40,290 --> 01:00:45,990
And clearly what it says is
that over here the simplest

924
01:00:45,990 --> 01:00:50,060
curve that you can have
should not have a second order

925
01:00:50,060 --> 01:00:51,560
therm which would
be like a parabola

926
01:00:51,560 --> 01:00:54,260
but should be like a cubic.

927
01:00:54,260 --> 01:01:02,520
And that the sine of the
cubic should be appropriate

928
01:01:02,520 --> 01:01:06,150
the negative so that delta V to
the fourth is always positive,

929
01:01:06,150 --> 01:01:08,520
then you have the right thing.

930
01:01:08,520 --> 01:01:10,530
OK?

931
01:01:10,530 --> 01:01:18,010
So given that information I
have sufficient parameters,

932
01:01:18,010 --> 01:01:22,770
information to calculate what
the location of Pc, Tc, Vc

933
01:01:22,770 --> 01:01:27,990
are for the van der Waals or
other approximate equation

934
01:01:27,990 --> 01:01:29,620
of state such as
the one that you

935
01:01:29,620 --> 01:01:32,310
will encounter in the problems.

936
01:01:32,310 --> 01:01:34,430
So let's stick with
the van der Waals

937
01:01:34,430 --> 01:01:46,290
so the van der Waals equation
is that my P is-- actually

938
01:01:46,290 --> 01:01:55,400
let's divide by N so it
would be volume per particle

939
01:01:55,400 --> 01:02:00,920
so I have introduced V to be
the volume per particle which

940
01:02:00,920 --> 01:02:03,670
is the inverse of the
density, if you like,

941
01:02:03,670 --> 01:02:05,900
so that is intensive.

942
01:02:05,900 --> 01:02:09,100
V minus-- there's
some excluded volume,

943
01:02:09,100 --> 01:02:11,420
let's call it
through the volume b

944
01:02:11,420 --> 01:02:13,570
and then I have
minus something that

945
01:02:13,570 --> 01:02:17,250
goes like the inverse
of this excluded volume,

946
01:02:17,250 --> 01:02:19,760
let's write it as a.

947
01:02:19,760 --> 01:02:22,440
Just I write those two
parameters as a and b

948
01:02:22,440 --> 01:02:25,060
because I will be
taking many derivatives

949
01:02:25,060 --> 01:02:29,021
and I don't want to write
anything more complicated.

950
01:02:29,021 --> 01:02:29,520
OK.

951
01:02:29,520 --> 01:02:30,880
So that's the equation.

952
01:02:30,880 --> 01:02:35,620
We said that the location
of this critical point

953
01:02:35,620 --> 01:02:39,260
is obtained by the
requirement that d P by d V

954
01:02:39,260 --> 01:02:42,160
should be 0 so what is that?

955
01:02:42,160 --> 01:02:50,430
It is minus k V T divided by
V minus b squared plus 2 a

956
01:02:50,430 --> 01:02:54,920
over V cubed so I
took one derivative.

957
01:02:54,920 --> 01:02:58,510
And the second derivative
is now constrained also,

958
01:02:58,510 --> 01:03:03,090
so that's going to
give me plus 2 k T V

959
01:03:03,090 --> 01:03:10,130
minus b cubed minus
6 a V to the fourth.

960
01:03:10,130 --> 01:03:12,040
OK?

961
01:03:12,040 --> 01:03:17,580
Now if I'm at a critical
point-- so I put c over here,

962
01:03:17,580 --> 01:03:22,950
this will be 0, and
this will also be 0.

963
01:03:25,860 --> 01:03:29,010
So the conditions for
the critical point

964
01:03:29,010 --> 01:03:39,350
are that first of all k B Tc
divided by Vc minus b cubed

965
01:03:39,350 --> 01:03:44,300
should be 2 a over
Vc cubed and 2

966
01:03:44,300 --> 01:03:51,680
k B Tc Vc minus b to the
fourth power should be 6 a

967
01:03:51,680 --> 01:03:55,958
divided by Vc to
the fourth power.

968
01:03:55,958 --> 01:03:58,940
Hmm?

969
01:03:58,940 --> 01:04:05,460
So this is two equations for two
unknowns, which are Tc and Vc.

970
01:04:05,460 --> 01:04:08,990
We can actually reduce
it to one variable

971
01:04:08,990 --> 01:04:12,580
by dividing these two
equations with each other.

972
01:04:12,580 --> 01:04:14,520
The division of
the left-hand sides

973
01:04:14,520 --> 01:04:20,580
will give me Vc minus b
in the numerator divided

974
01:04:20,580 --> 01:04:24,370
by 2 in the denominator.

975
01:04:24,370 --> 01:04:31,850
The ratio of these two
is simply Vc over 3.

976
01:04:31,850 --> 01:04:34,610
That's a one variable equation.

977
01:04:34,610 --> 01:04:35,575
Yes?

978
01:04:35,575 --> 01:04:38,808
AUDIENCE: Where you have
written the equations in terms

979
01:04:38,808 --> 01:04:42,266
Vc minus b and Vc, it should
come in different powers.

980
01:04:42,266 --> 01:04:45,593
So our first equation should
be Vc minus b squared.

981
01:04:45,593 --> 01:04:46,218
PROFESSOR: Yes.

982
01:04:46,218 --> 01:04:47,134
AUDIENCE: [INAUDIBLE].

983
01:04:53,640 --> 01:04:55,680
PROFESSOR: OK.

984
01:04:55,680 --> 01:05:01,330
Magically the ratio
does not get affected

985
01:05:01,330 --> 01:05:05,450
and therefore neither
does the answer

986
01:05:05,450 --> 01:05:11,560
which is Vc which is 3 b
so that's one parameter.

987
01:05:11,560 --> 01:05:15,840
I can substitute that
Vc into this equation

988
01:05:15,840 --> 01:05:17,710
and get what k B Tc is.

989
01:05:22,150 --> 01:05:26,730
And hopefully you
can see that this

990
01:05:26,730 --> 01:05:30,860
will become 2 b
squared, which is 4b

991
01:05:30,860 --> 01:05:37,250
squared so this gives me a
factor of 8 a, ratio of these

992
01:05:37,250 --> 01:05:43,720
will give me one factor
of b, so that's k B Tc.

993
01:05:43,720 --> 01:05:49,220
And then if I substitute
those in the first equation,

994
01:05:49,220 --> 01:05:54,800
I will get the value
of Pc and I happen

995
01:05:54,800 --> 01:05:58,800
to know that the answer--
oops, this is going to be 27,

996
01:05:58,800 --> 01:06:03,410
I forgot because 3
cubed will give me 27.

997
01:06:03,410 --> 01:06:07,950
And Pc here will
give me 27 also.

998
01:06:07,950 --> 01:06:12,242
I believe b squared a and
there will be a factor of 1.

999
01:06:16,602 --> 01:06:17,102
OK.

1000
01:06:21,948 --> 01:06:22,448
OK?

1001
01:06:26,829 --> 01:06:28,370
Really the only
thing that actually I

1002
01:06:28,370 --> 01:06:30,610
wanted to get out
of this, you can

1003
01:06:30,610 --> 01:06:34,120
do this for any equation
of state, et cetera.

1004
01:06:34,120 --> 01:06:37,680
Point is that we
now have something

1005
01:06:37,680 --> 01:06:40,060
to compare to experiment.

1006
01:06:40,060 --> 01:06:48,180
We have a dimension less
ratio Pc Vc divided by k B Tc.

1007
01:06:48,180 --> 01:06:53,820
So I multiply these two
numbers, I will get one-ninth.

1008
01:06:53,820 --> 01:06:57,550
And then I have
here a factor of 8

1009
01:06:57,550 --> 01:07:02,250
so the whole thing becomes
a factor of 3 over 8

1010
01:07:02,250 --> 01:07:06,420
which is 0.375.

1011
01:07:06,420 --> 01:07:08,570
OK?

1012
01:07:08,570 --> 01:07:14,390
So the point is that we
constructed this van der Waals

1013
01:07:14,390 --> 01:07:20,130
equation through some
reasonable description of a gas.

1014
01:07:20,130 --> 01:07:23,120
You would imagine that
practically any type of gas

1015
01:07:23,120 --> 01:07:27,170
that you have will have some
kind of excluded volume so that

1016
01:07:27,170 --> 01:07:30,510
was one of the parameters
that we used in calculating

1017
01:07:30,510 --> 01:07:34,070
our estimation for the
partition function,

1018
01:07:34,070 --> 01:07:37,040
ultimately getting this omega.

1019
01:07:37,040 --> 01:07:40,470
And there's some kind of
attraction at large distance

1020
01:07:40,470 --> 01:07:44,170
that's been integrated
over the entire range,

1021
01:07:44,170 --> 01:07:46,580
here was this factor
of U. You would

1022
01:07:46,580 --> 01:07:50,340
expect that to be quite generic.

1023
01:07:50,340 --> 01:07:54,160
And once we sort of put
this into this machinery

1024
01:07:54,160 --> 01:07:57,110
that generic thing makes
a particular prediction.

1025
01:07:57,110 --> 01:08:02,620
It says that pick any gas,
find it's critical point--

1026
01:08:02,620 --> 01:08:04,860
critical point occurs
with for some value

1027
01:08:04,860 --> 01:08:09,210
of Pc density or
inverse density, Vc,

1028
01:08:09,210 --> 01:08:12,910
and some characteristic
[INAUDIBLE] pressure k B Tc.

1029
01:08:12,910 --> 01:08:17,600
This is a dimensionless ratio
you can calculate for any gas

1030
01:08:17,600 --> 01:08:20,950
and based on this
semi-reasonable assumption

1031
01:08:20,950 --> 01:08:24,000
that we made, no matter what
you do you should come up

1032
01:08:24,000 --> 01:08:26,979
the number that is around this.

1033
01:08:26,979 --> 01:08:28,220
OK?

1034
01:08:28,220 --> 01:08:29,519
So what you find in experiment?

1035
01:08:33,450 --> 01:08:46,585
You find that the values of this
combination from 0.28 to 0.33,

1036
01:08:46,585 --> 01:08:49,399
so that's kind of a range
that you find for this.

1037
01:08:49,399 --> 01:08:52,180
So first of all, you
don't find that it

1038
01:08:52,180 --> 01:08:54,770
is the same for all gases.

1039
01:08:54,770 --> 01:08:57,140
There is some variation.

1040
01:08:57,140 --> 01:09:00,830
And that it's not
this value of 3 8.

1041
01:09:00,830 --> 01:09:05,130
So there's certainly
now questions

1042
01:09:05,130 --> 01:09:08,310
to be asked about the
approximation, how

1043
01:09:08,310 --> 01:09:12,160
you could make it better.

1044
01:09:12,160 --> 01:09:14,529
You could say, OK,
actually most of the gases,

1045
01:09:14,529 --> 01:09:17,170
you don't have
spherical potentials.

1046
01:09:17,170 --> 01:09:21,010
You have things that
are diatomic that shape

1047
01:09:21,010 --> 01:09:22,399
may be important.

1048
01:09:22,399 --> 01:09:24,470
And so the estimation
that we have

1049
01:09:24,470 --> 01:09:30,750
for the energy for the omega
et cetera is too approximate.

1050
01:09:30,750 --> 01:09:32,649
There is something
to that because you

1051
01:09:32,649 --> 01:09:38,560
find that you get more or less
the same value in this range

1052
01:09:38,560 --> 01:09:42,500
for gases that are similar, like
helium, krypton, neon, argon,

1053
01:09:42,500 --> 01:09:45,020
et cetera, they're kind
of like each other.

1054
01:09:45,020 --> 01:09:48,080
Oxygen, nitrogen, these
diatomic molecules

1055
01:09:48,080 --> 01:09:51,260
are kind of like each other.

1056
01:09:51,260 --> 01:09:51,859
OK?

1057
01:09:51,859 --> 01:09:56,740
So there is some
hope to maybe get

1058
01:09:56,740 --> 01:09:58,659
the more universal description.

1059
01:09:58,659 --> 01:09:59,950
So what is it that we're after?

1060
01:09:59,950 --> 01:10:03,960
I mean the thing that is so
nice about the ideal gas law

1061
01:10:03,960 --> 01:10:07,620
is that it doesn't matter what
material you are looking at.

1062
01:10:07,620 --> 01:10:09,870
You make it
sufficiently dilute, you

1063
01:10:09,870 --> 01:10:12,660
know exactly what the
equation of state is.

1064
01:10:12,660 --> 01:10:16,270
It will be good if
we could extend that.

1065
01:10:16,270 --> 01:10:19,820
If we could say something
about interacting gases

1066
01:10:19,820 --> 01:10:23,150
that also maybe depends
on just a few parameters.

1067
01:10:23,150 --> 01:10:25,960
So you don't have to go
and do a huge calculation

1068
01:10:25,960 --> 01:10:27,980
for the case of
each gas, but you

1069
01:10:27,980 --> 01:10:34,060
have something that has the
sense of universality to it.

1070
01:10:34,060 --> 01:10:36,680
So clearly the van
der Waals equation

1071
01:10:36,680 --> 01:10:41,480
is a step in that direction,
but it is not quite good enough.

1072
01:10:41,480 --> 01:10:45,440
So people said,
well, maybe what we

1073
01:10:45,440 --> 01:10:49,420
should do is increase
the number of parameters,

1074
01:10:49,420 --> 01:10:52,520
because currently we are
constructing everything

1075
01:10:52,520 --> 01:10:54,540
based on two
parameters, the excluded

1076
01:10:54,540 --> 01:10:57,520
volume and some
integrated attraction.

1077
01:10:57,520 --> 01:11:00,720
So those are the two parameters
and with two parameters

1078
01:11:00,720 --> 01:11:04,680
you really can only
fit two things.

1079
01:11:04,680 --> 01:11:08,940
What we see that the ratio of
Pc Vc over k T is not fixed,

1080
01:11:08,940 --> 01:11:11,950
it's as a range, so
maybe what we should do

1081
01:11:11,950 --> 01:11:14,650
is to go to three parameters.

1082
01:11:14,650 --> 01:11:20,470
So this whole is
captured by the search

1083
01:11:20,470 --> 01:11:22,701
for law of corresponding states.

1084
01:11:26,390 --> 01:11:32,710
So the hope is that let's
do a three parameter system.

1085
01:11:32,710 --> 01:11:35,170
Which parameters
should I choose?

1086
01:11:35,170 --> 01:11:38,020
Well, clearly the
different systems

1087
01:11:38,020 --> 01:11:43,130
are characterized by Pc Tc
and Nc T at critical point,

1088
01:11:43,130 --> 01:11:48,470
so maybe what I should do is I
should measure all pressures,

1089
01:11:48,470 --> 01:11:52,420
made them dimensionless
by dividing by Pc

1090
01:11:52,420 --> 01:11:55,570
and hope that there is some
universal function that

1091
01:11:55,570 --> 01:11:59,970
relates that to the ratio of
all temperatures divided by Tc,

1092
01:11:59,970 --> 01:12:04,200
all densities or inverse
densities divided

1093
01:12:04,200 --> 01:12:06,680
by the corresponding Vc.

1094
01:12:06,680 --> 01:12:12,870
So is there such a curve so
that that was a whole and you go

1095
01:12:12,870 --> 01:12:16,500
and play around with this curve
and it is with this suggestion

1096
01:12:16,500 --> 01:12:20,010
and you can convince yourself
very easily that it cannot be

1097
01:12:20,010 --> 01:12:21,080
the case.

1098
01:12:21,080 --> 01:12:22,910
And one easy way
to think about it

1099
01:12:22,910 --> 01:12:25,730
is that if this
was the case, then

1100
01:12:25,730 --> 01:12:28,220
all of our virial
expansions and all

1101
01:12:28,220 --> 01:12:31,660
of the perturbative expansions
that we had should also

1102
01:12:31,660 --> 01:12:35,200
somehow collapse together
with a few parameters.

1103
01:12:35,200 --> 01:12:37,990
Whereas all of these
complicated diagrams that we

1104
01:12:37,990 --> 01:12:42,030
had with the cycles and
shapes and different things

1105
01:12:42,030 --> 01:12:44,870
of the diagrams are really,
you can calculate them.

1106
01:12:44,870 --> 01:12:47,040
They're completely
independent integrals.

1107
01:12:47,040 --> 01:12:49,550
They will give you
results that should not

1108
01:12:49,550 --> 01:12:52,710
be collapsible
into a single form.

1109
01:12:52,710 --> 01:12:58,100
So this was a nice suggestion
what in reality does not work.

1110
01:13:02,790 --> 01:13:05,236
So why am I telling you this?

1111
01:13:05,236 --> 01:13:07,980
The reason I'm
telling you this is

1112
01:13:07,980 --> 01:13:14,280
that surprisingly in the
vicinity of this point

1113
01:13:14,280 --> 01:13:16,470
it does work.

1114
01:13:16,470 --> 01:13:18,800
In the vicinity
of this point you

1115
01:13:18,800 --> 01:13:21,940
can get the huge number
of different system gases,

1116
01:13:21,940 --> 01:13:26,220
krypton, argon, carbon
dioxide, mixtures,

1117
01:13:26,220 --> 01:13:28,610
a whole lot of things.

1118
01:13:28,610 --> 01:13:33,290
And if you appropriately rescale
your pressure, temperature,

1119
01:13:33,290 --> 01:13:36,400
et cetera, for all
of these gases,

1120
01:13:36,400 --> 01:13:39,770
they come together and fall
on exactly the same type

1121
01:13:39,770 --> 01:13:42,080
of curve of over here.

1122
01:13:42,080 --> 01:13:42,920
OK?

1123
01:13:42,920 --> 01:13:46,870
So there is universality,
just not over the entire phase

1124
01:13:46,870 --> 01:13:52,040
diagram but in the vicinity
of this critical point.

1125
01:13:52,040 --> 01:13:55,010
And so what is special about
that, and why does that work?

1126
01:13:57,550 --> 01:13:59,520
OK?

1127
01:13:59,520 --> 01:14:05,680
So I give you first an
argument that says, well,

1128
01:14:05,680 --> 01:14:06,760
it really should work.

1129
01:14:06,760 --> 01:14:07,910
It shouldn't be a surprise.

1130
01:14:11,250 --> 01:14:17,080
And then show that the simple
argument is in fact wrong.

1131
01:14:17,080 --> 01:14:24,290
But let's go through because we
already put the elements of it

1132
01:14:24,290 --> 01:14:26,200
over there.

1133
01:14:26,200 --> 01:14:35,670
So let me try to figure out
what the shape of these pressure

1134
01:14:35,670 --> 01:14:40,140
versus volume curves are
going to be for isotherms that

1135
01:14:40,140 --> 01:14:44,200
correspond to different
temperatures close to Tc.

1136
01:14:44,200 --> 01:14:47,670
So what I want to
do is to calculate P

1137
01:14:47,670 --> 01:14:49,680
as a function of
volume, actually there's

1138
01:14:49,680 --> 01:14:52,780
reduced volume by the
number of particles,

1139
01:14:52,780 --> 01:14:58,520
and P and I'll do the following.

1140
01:14:58,520 --> 01:15:02,170
I will write this
as an expansion

1141
01:15:02,170 --> 01:15:10,507
but in the following
form-- P of Vc T plus d

1142
01:15:10,507 --> 01:15:23,230
P by d V at Vc T times V
minus, Vc plus one-half V 2

1143
01:15:23,230 --> 01:15:32,284
P by d V squared at Vc T minus
Vc squared plus one-sixth d

1144
01:15:32,284 --> 01:15:41,510
cubed P d V cubed at Vc and T
V minus Vc cubed and so forth.

1145
01:15:41,510 --> 01:15:45,140
So it's a function
of two arguments.

1146
01:15:45,140 --> 01:15:49,370
Really it has an expansion
in both deviations

1147
01:15:49,370 --> 01:15:55,450
from this critical point
along the V direction

1148
01:15:55,450 --> 01:16:00,440
so going away from Vc as
well as going T direction,

1149
01:16:00,440 --> 01:16:05,940
going to away from Tc but I
organize it in this fashion

1150
01:16:05,940 --> 01:16:08,860
and realize that actually
these derivatives and all

1151
01:16:08,860 --> 01:16:12,720
of the coefficients will be
a function of T minus Tc.

1152
01:16:12,720 --> 01:16:17,590
So for example P
of Vc and T starts

1153
01:16:17,590 --> 01:16:21,780
with being whatever value I
have at that critical point

1154
01:16:21,780 --> 01:16:24,560
and then if I go to higher
temperatures the pressure will

1155
01:16:24,560 --> 01:16:25,660
go up.

1156
01:16:25,660 --> 01:16:30,230
So there's some coefficient
proportional to T minus Tc

1157
01:16:30,230 --> 01:16:35,170
plus higher order in T minus
Tc and so forth and actually I

1158
01:16:35,170 --> 01:16:37,156
expect this alpha
to be positive.

1159
01:16:40,460 --> 01:16:50,390
The next coefficient d P by
d V Vc T T what do I expect?

1160
01:16:50,390 --> 01:16:57,680
Well, right at this point
d P by d V there's slope.

1161
01:16:57,680 --> 01:17:00,760
This has to be 0 for
the critical isotherm.

1162
01:17:00,760 --> 01:17:08,375
So it starts with 0 and
then if I go and look

1163
01:17:08,375 --> 01:17:12,930
at a curve that corresponds
to high temperatures because

1164
01:17:12,930 --> 01:17:17,830
of the stability, the
slope better be positive.

1165
01:17:17,830 --> 01:17:22,060
So I have a T minus
Tc plus our order

1166
01:17:22,060 --> 01:17:31,000
that I know that a is
positive because of a-- a

1167
01:17:31,000 --> 01:17:33,710
is negative because
of stability.

1168
01:17:33,710 --> 01:17:35,452
So let's write it
in this fashion.

1169
01:17:39,950 --> 01:17:41,890
OK?

1170
01:17:41,890 --> 01:17:43,295
What about the
second derivative?

1171
01:17:48,790 --> 01:17:50,830
Well, we said that if
I look at the point

1172
01:17:50,830 --> 01:17:53,710
where the first derivative is
0, the second derivative better

1173
01:17:53,710 --> 01:17:57,810
be 0, so it starts also with 0.

1174
01:17:57,810 --> 01:18:00,760
And then there's some
coefficient to order

1175
01:18:00,760 --> 01:18:03,680
of T minus Tc and
then higher order

1176
01:18:03,680 --> 01:18:06,560
and actually I don't know
anything about the sign of P.

1177
01:18:06,560 --> 01:18:08,600
It can be both
positive or negative,

1178
01:18:08,600 --> 01:18:11,620
we don't know what it is.

1179
01:18:11,620 --> 01:18:21,160
Now the third derivative
there is no reason

1180
01:18:21,160 --> 01:18:23,250
that it should start at 0.

1181
01:18:23,250 --> 01:18:26,880
It will be some negative
number again in order

1182
01:18:26,880 --> 01:18:31,020
to ensure stability and there
will be higher order terms

1183
01:18:31,020 --> 01:18:34,430
in T minus Tc what we have
is structure such as this.

1184
01:18:51,560 --> 01:18:55,340
So putting everything
together the statement

1185
01:18:55,340 --> 01:19:00,940
is that if I look in the
vicinity of the critical point,

1186
01:19:00,940 --> 01:19:05,430
ask what should
pressure look like?

1187
01:19:05,430 --> 01:19:07,960
You say, OK, it has to
start with a constant

1188
01:19:07,960 --> 01:19:10,270
that we called Pc.

1189
01:19:10,270 --> 01:19:18,790
It has this linear increase
that I put over there.

1190
01:19:18,790 --> 01:19:22,560
The first derivative
is some negative number

1191
01:19:22,560 --> 01:19:27,900
that is proportional to T minus
Tc multiplying by V minus Vc.

1192
01:19:31,370 --> 01:19:38,140
The third coefficient
is negative-- oh

1193
01:19:38,140 --> 01:19:41,400
and there's also a second--
there's the second order

1194
01:19:41,400 --> 01:19:51,920
coefficient and then there
will be high order terms.

1195
01:19:51,920 --> 01:19:58,770
So there should be generically
an expansion such as this.

1196
01:19:58,770 --> 01:20:03,720
Say, well, OK, fine,
what have we learned?

1197
01:20:03,720 --> 01:20:07,130
I say, well, OK, let me tell
you about something that we're

1198
01:20:07,130 --> 01:20:08,620
going to experimentally measure.

1199
01:20:08,620 --> 01:20:12,020
Let's look at the
compressibility, kappa.

1200
01:20:12,020 --> 01:20:19,730
Kappa is minus 1 over V d V
by d P evaluated at whatever

1201
01:20:19,730 --> 01:20:22,600
temperature you are looking at.

1202
01:20:22,600 --> 01:20:28,970
So if I look at this-- actually
I have stated what d P by d V

1203
01:20:28,970 --> 01:20:29,665
is.

1204
01:20:29,665 --> 01:20:33,330
d V by d P will be
the inverse of that,

1205
01:20:33,330 --> 01:20:36,170
so this is going to
give me in the vicinity

1206
01:20:36,170 --> 01:20:39,830
of this critical
point Vc a T minus Tc.

1207
01:20:43,990 --> 01:20:47,910
So the statement is
this is now something

1208
01:20:47,910 --> 01:20:52,080
that I can go and ask
my experimentalist

1209
01:20:52,080 --> 01:20:56,360
friends to measure,
go and calculate

1210
01:20:56,360 --> 01:20:59,400
the compressibility
of this gas, plot it

1211
01:20:59,400 --> 01:21:03,080
as a function of temperature,
and the prediction is

1212
01:21:03,080 --> 01:21:09,640
that the compressibility
will diverge at T goes to Tc.

1213
01:21:09,640 --> 01:21:10,730
OK?

1214
01:21:10,730 --> 01:21:17,020
So the prediction is that
it is 1 over a T minus Tc.

1215
01:21:17,020 --> 01:21:17,550
OK.

1216
01:21:17,550 --> 01:21:19,900
So they go and do
the experiment,

1217
01:21:19,900 --> 01:21:22,470
they do the experiment
for a huge number

1218
01:21:22,470 --> 01:21:25,930
of different systems,
not just one system,

1219
01:21:25,930 --> 01:21:29,180
and they come back
and say it is correct.

1220
01:21:29,180 --> 01:21:35,180
The compressibility does
diverge but what we find

1221
01:21:35,180 --> 01:21:39,240
is that irrespective of
what gas we are looking at,

1222
01:21:39,240 --> 01:21:42,220
and this is a very
universal, it goes

1223
01:21:42,220 --> 01:21:45,279
with an exponent like this.

1224
01:21:45,279 --> 01:21:47,600
OK?

1225
01:21:47,600 --> 01:21:53,670
So question is maybe you made
a mistake or you did something

1226
01:21:53,670 --> 01:21:56,080
or whatever, really
the big puzzle

1227
01:21:56,080 --> 01:22:03,260
is why do such different gases
all over the spectrum and even

1228
01:22:03,260 --> 01:22:06,910
some things that are not gases
but other types of things

1229
01:22:06,910 --> 01:22:08,890
all have the same diverges?

1230
01:22:08,890 --> 01:22:12,823
Where did this number
1.24 come from?

1231
01:22:12,823 --> 01:22:14,737
OK?

1232
01:22:14,737 --> 01:22:18,962
AUDIENCE: So there isn't a
range like there was 0.375.

1233
01:22:18,962 --> 01:22:20,720
It all depends on 1.24.

1234
01:22:20,720 --> 01:22:24,390
PROFESSOR: All hit 1.24.

1235
01:22:24,390 --> 01:22:26,040
Irrespective of
[? order. ?] It's

1236
01:22:26,040 --> 01:22:27,270
a different quantity, though.

1237
01:22:27,270 --> 01:22:30,510
I mean, it's not the
number, it's an exponent.

1238
01:22:30,510 --> 01:22:34,090
It's this, a functional form.

1239
01:22:34,090 --> 01:22:37,400
Why did this functional
form come about?

1240
01:22:37,400 --> 01:22:39,760
Another thing that
you predict is

1241
01:22:39,760 --> 01:22:42,770
the shape of the curve at Tc.

1242
01:22:42,770 --> 01:22:47,660
So what we have said is
that at T close to Tc,

1243
01:22:47,660 --> 01:22:52,690
the shape of this isotherm is
essentially that P minus Pc is

1244
01:22:52,690 --> 01:22:57,150
cubic, it's proportional
to V minus Vc cubed.

1245
01:22:57,150 --> 01:23:00,220
And the way that we said
that is OK at that point

1246
01:23:00,220 --> 01:23:03,250
there's neither a slope
nor a second derivative,

1247
01:23:03,250 --> 01:23:05,740
so it should be cubic.

1248
01:23:05,740 --> 01:23:09,346
They say, OK, what do
you find in experiment?

1249
01:23:09,346 --> 01:23:11,860
You go and do the
experiment and again

1250
01:23:11,860 --> 01:23:15,980
for a whole huge range of
materials all of the data

1251
01:23:15,980 --> 01:23:23,650
collapses and you find the curve
that is something like this,

1252
01:23:23,650 --> 01:23:26,550
is an exponent that
is very close to 5

1253
01:23:26,550 --> 01:23:28,090
but is not exactly 5.

1254
01:23:32,370 --> 01:23:40,540
And again, why not 3 and why
is it always the same number 5?

1255
01:23:40,540 --> 01:23:43,380
How do you understand that?

1256
01:23:43,380 --> 01:23:46,760
Well, I'll tell you why not 3.

1257
01:23:46,760 --> 01:23:47,620
Why not 3?

1258
01:23:47,620 --> 01:23:49,690
All of the things
that I did here

1259
01:23:49,690 --> 01:23:52,040
was based on the
assumption that I

1260
01:23:52,040 --> 01:23:55,060
could make an
analytical expansion.

1261
01:23:55,060 --> 01:23:57,820
The whole idea of
writing a Taylor series

1262
01:23:57,820 --> 01:24:01,950
is based on making
an analytical series.

1263
01:24:01,950 --> 01:24:05,160
Who told you that you can
do an analytic series?

1264
01:24:05,160 --> 01:24:07,200
Experiment tells you
that it is actually

1265
01:24:07,200 --> 01:24:10,130
something non-analytic.

1266
01:24:10,130 --> 01:24:12,680
Why this form of
non-analyticity?

1267
01:24:12,680 --> 01:24:15,070
Why it's universality?

1268
01:24:15,070 --> 01:24:16,770
We won't explain in this course.

1269
01:24:16,770 --> 01:24:20,070
If you want to find
out come to a 334.