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PROFESSOR: So I'll
briefly review

9
00:00:21,850 --> 00:00:24,560
what we were doing last time.

10
00:00:24,560 --> 00:00:28,747
We are gradually
developing the structure

11
00:00:28,747 --> 00:00:30,080
that we need for thermodynamics.

12
00:00:33,870 --> 00:00:40,730
And the first thing that we
did was to state that it wholly

13
00:00:40,730 --> 00:00:46,070
depends on the existence
of equilibrim--

14
00:00:46,070 --> 00:00:50,460
is the first and foremost
thing that we need to know,

15
00:00:50,460 --> 00:00:54,590
is that the objects that we
are studying in thermodynamics

16
00:00:54,590 --> 00:00:56,180
are in equilibrium.

17
00:00:56,180 --> 00:00:58,370
And once they're
in equilibrium, we

18
00:00:58,370 --> 00:01:01,310
can characterize them
by a set of properties.

19
00:01:01,310 --> 00:01:03,330
So there's a list of
coordinates that I

20
00:01:03,330 --> 00:01:05,160
would have the list here.

21
00:01:05,160 --> 00:01:09,460
And we saw that
since we will also

22
00:01:09,460 --> 00:01:13,690
need to think in terms of
mechanical work, a good place

23
00:01:13,690 --> 00:01:16,420
to start-- and since we are
familiar with mechanical

24
00:01:16,420 --> 00:01:21,480
coordinates-- is to enumerate
here the mechanical coordinates

25
00:01:21,480 --> 00:01:23,730
that we would describe
the system with.

26
00:01:23,730 --> 00:01:29,080
And we said that we could,
in principal, characterize

27
00:01:29,080 --> 00:01:31,910
or divide those
coordinates in the form

28
00:01:31,910 --> 00:01:37,750
of some generalized
displacements

29
00:01:37,750 --> 00:01:41,910
and the corresponding
conjugate forces.

30
00:01:41,910 --> 00:01:46,960
And typically, the displacements
are extensive proportional

31
00:01:46,960 --> 00:01:48,320
to the size of your system.

32
00:01:48,320 --> 00:01:50,280
The forces are intensive.

33
00:01:50,280 --> 00:01:53,330
And the work is related to
the product of those two

34
00:01:53,330 --> 00:01:55,600
in some fashion.

35
00:01:55,600 --> 00:01:58,340
One of the things that
I will do throughout

36
00:01:58,340 --> 00:02:03,010
is to emphasize that one of
the main materials that we

37
00:02:03,010 --> 00:02:07,960
are going to look at, and
illustrate the concept with,

38
00:02:07,960 --> 00:02:08,585
is the gas.

39
00:02:08,585 --> 00:02:12,220
And in particular,
the version of the gas

40
00:02:12,220 --> 00:02:14,330
that is the ideal gas.

41
00:02:14,330 --> 00:02:17,120
But forgetting the ideal
part for the time being,

42
00:02:17,120 --> 00:02:21,060
the coordinates that we use
to describe the gas with

43
00:02:21,060 --> 00:02:24,330
is V and pressure.

44
00:02:24,330 --> 00:02:26,360
And actually, we
saw that in order

45
00:02:26,360 --> 00:02:30,830
to ultimately be consistent
with how we will define work,

46
00:02:30,830 --> 00:02:34,670
it is good to think
of minus pressure

47
00:02:34,670 --> 00:02:37,110
as the appropriate force.

48
00:02:37,110 --> 00:02:42,960
And so basically, then if I want
to characterize the equilibrium

49
00:02:42,960 --> 00:02:47,940
state of a gas, I
need, for example,

50
00:02:47,940 --> 00:02:53,670
to specify where I
lie in this P-V plane.

51
00:02:53,670 --> 00:02:57,230
OK so that's the
first thing to know.

52
00:02:57,230 --> 00:03:01,280
But then we also realize
that mechanical coordinates

53
00:03:01,280 --> 00:03:04,880
are insufficient to describe
the properties of the systems

54
00:03:04,880 --> 00:03:06,480
that we encounter.

55
00:03:06,480 --> 00:03:10,110
For example, if we are
thinking in terms of a spring

56
00:03:10,110 --> 00:03:11,980
and we pull on
the spring-- so we

57
00:03:11,980 --> 00:03:15,220
are looking at the relationship
between displacement

58
00:03:15,220 --> 00:03:18,060
and the force-- the result
that we get how easy it

59
00:03:18,060 --> 00:03:21,830
is to pull on this depends
on something else--

60
00:03:21,830 --> 00:03:23,640
depends on temperature.

61
00:03:23,640 --> 00:03:28,460
So somehow, we need to
include these properties that

62
00:03:28,460 --> 00:03:31,690
have to do with heat
and heat transfer

63
00:03:31,690 --> 00:03:34,400
in order to have a complete
description of systems

64
00:03:34,400 --> 00:03:36,160
that are in equilibrium.

65
00:03:36,160 --> 00:03:38,460
And so going along
that direction,

66
00:03:38,460 --> 00:03:42,370
we started with the Zeroth Law.

67
00:03:42,370 --> 00:03:44,050
And the statement
of the Zeroth Law

68
00:03:44,050 --> 00:03:46,440
was the transitivity
of equilibrium--

69
00:03:46,440 --> 00:03:48,170
if two objects
are in equilibrium

70
00:03:48,170 --> 00:03:52,310
with each other they are--
two objects are in equilibrium

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00:03:52,310 --> 00:03:54,690
with a third object,
they're also in equilibrium

72
00:03:54,690 --> 00:03:56,250
with each other.

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00:03:56,250 --> 00:03:59,270
And we saw that
what that means, is

74
00:03:59,270 --> 00:04:03,170
that once I am at some point
in this coordinate space,

75
00:04:03,170 --> 00:04:05,210
I know that there
exists some function.

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00:04:05,210 --> 00:04:07,930
I don't know its
functional form.

77
00:04:07,930 --> 00:04:11,730
This empirical temperature
and being in equilibrium

78
00:04:11,730 --> 00:04:16,260
means that there is
this important function

79
00:04:16,260 --> 00:04:19,810
of the coordinates
of the first system

80
00:04:19,810 --> 00:04:23,510
equal to some other functional
or form potentially,

81
00:04:23,510 --> 00:04:27,130
depending on the coordinates
of the second system.

82
00:04:27,130 --> 00:04:33,650
And this empirical temperature
has to be the same.

83
00:04:33,650 --> 00:04:37,490
We said that this is kind
of like being on a scale,

84
00:04:37,490 --> 00:04:40,740
and having a balance
between different things,

85
00:04:40,740 --> 00:04:43,230
and then they are in
balance with each other.

86
00:04:43,230 --> 00:04:46,800
So we know that there's
something like mass.

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00:04:46,800 --> 00:04:51,860
Now, when we describe this
in the context of the gas,

88
00:04:51,860 --> 00:04:56,890
we noted that for all gases in
the limit that they are dilute,

89
00:04:56,890 --> 00:05:01,960
the isotherms-- the places that
would correspond in this case

90
00:05:01,960 --> 00:05:05,240
to always being in
contact with the object

91
00:05:05,240 --> 00:05:09,620
is at a fixed temperature,
form these hyperboles.

92
00:05:09,620 --> 00:05:15,020
So the isotherms in
the limit of dilute

93
00:05:15,020 --> 00:05:18,925
are of the form PV is
proportional to time span.

94
00:05:21,900 --> 00:05:24,170
We can call that theta.

95
00:05:24,170 --> 00:05:28,110
We can use that, even, to
define the ideal gas temperature

96
00:05:28,110 --> 00:05:32,450
scale, provided that we
choose some proportionality

97
00:05:32,450 --> 00:05:34,880
coefficient which
was selected so

98
00:05:34,880 --> 00:05:37,910
that the temperature
of water, ice,

99
00:05:37,910 --> 00:05:42,330
steam, coexistent at
a particular value.

100
00:05:42,330 --> 00:05:46,370
OK, so that's the specific
property of the ideal gas

101
00:05:46,370 --> 00:05:50,250
as far as its
temperature is concerned.

102
00:05:50,250 --> 00:05:56,830
The next thing that
we did was to look

103
00:05:56,830 --> 00:06:02,250
at how changes are carrying a
system, a particular system.

104
00:06:02,250 --> 00:06:07,710
And we found that if you take
the system from one location

105
00:06:07,710 --> 00:06:12,190
in its coordinate space
to another location,

106
00:06:12,190 --> 00:06:17,730
do it in a manner that does not
involve the exchange of heat.

107
00:06:17,730 --> 00:06:20,170
So basically, you
idealize this thing

108
00:06:20,170 --> 00:06:24,390
and isolate it from
any other source--

109
00:06:24,390 --> 00:06:26,370
that the amount of
work-- and we can then

110
00:06:26,370 --> 00:06:29,180
calculate mechanical work--
the amount of mechanical work

111
00:06:29,180 --> 00:06:32,640
that you do is only a function
of the initial and final

112
00:06:32,640 --> 00:06:35,890
states, and does
not depend on how

113
00:06:35,890 --> 00:06:42,020
you supply the work that
produces this, et cetera.

114
00:06:42,020 --> 00:06:47,390
And so that immediately reminded
us of conservation of energy,

115
00:06:47,390 --> 00:06:54,710
and a suggestion that there
exists this function that

116
00:06:54,710 --> 00:07:00,720
depends on where you are in this
coordinate representation that

117
00:07:00,720 --> 00:07:04,900
gives the total energy
content of the system.

118
00:07:04,900 --> 00:07:08,630
What was important
was that, of course,

119
00:07:08,630 --> 00:07:15,950
we want to relax the condition
of making changes over which

120
00:07:15,950 --> 00:07:19,880
there is no heat
exchange to the system.

121
00:07:19,880 --> 00:07:25,000
And more generally then,
we said that the changes

122
00:07:25,000 --> 00:07:30,080
that we obtain in energy are
the sum of the two components--

123
00:07:30,080 --> 00:07:33,500
the amount of work
that you do on it,

124
00:07:33,500 --> 00:07:38,780
and that we know how to measure
from our various mechanical

125
00:07:38,780 --> 00:07:40,490
prescriptions.

126
00:07:40,490 --> 00:07:44,120
But if there is a shortcoming
from the internal energy

127
00:07:44,120 --> 00:07:47,940
function that we had
constructed previously,

128
00:07:47,940 --> 00:07:51,320
and the work that we compute
for a particular process,

129
00:07:51,320 --> 00:07:54,530
we say that in
that process, there

130
00:07:54,530 --> 00:07:57,930
was some heat that
went into the system.

131
00:07:57,930 --> 00:08:00,640
And it is, again,
important to state

132
00:08:00,640 --> 00:08:06,030
that this dE really only depends
on the initial and final state.

133
00:08:06,030 --> 00:08:09,110
So this depends on state.

134
00:08:09,110 --> 00:08:12,620
While these things-- that's
why we put a bar on them--

135
00:08:12,620 --> 00:08:15,290
depend on the particular
path that we take.

136
00:08:21,790 --> 00:08:24,400
Now again, if you
ask, well, what can

137
00:08:24,400 --> 00:08:28,040
we say in this context
for the ideal gas?

138
00:08:28,040 --> 00:08:38,640
And there is the very
nice experiment by Joule

139
00:08:38,640 --> 00:08:45,570
where he said, let's
isolate our gas, let's say,

140
00:08:45,570 --> 00:08:48,830
into two chambers.

141
00:08:48,830 --> 00:08:53,870
These are kind of rigid
balls, very nicely isolated

142
00:08:53,870 --> 00:08:55,830
from the environment.

143
00:08:55,830 --> 00:09:03,150
Initially, the gas is confined
completely onto one side.

144
00:09:03,150 --> 00:09:06,792
So let's say that the
initial state is over here,

145
00:09:06,792 --> 00:09:09,530
with this volume.

146
00:09:09,530 --> 00:09:12,670
And then we release this.

147
00:09:12,670 --> 00:09:16,010
And so the gas goes and expands.

148
00:09:16,010 --> 00:09:17,625
And we wait sufficiently.

149
00:09:17,625 --> 00:09:19,150
We wait sufficiently.

150
00:09:19,150 --> 00:09:21,650
Ultimately, we're
settled to a place,

151
00:09:21,650 --> 00:09:25,911
let's say over here--
the final state.

152
00:09:25,911 --> 00:09:29,090
Now, it's important
to say that I

153
00:09:29,090 --> 00:09:33,520
don't know the intermediate
state of this transformation.

154
00:09:33,520 --> 00:09:36,070
So I can't say that
somewhere between t

155
00:09:36,070 --> 00:09:38,380
equals to zero when
I open the valve,

156
00:09:38,380 --> 00:09:41,830
and t goes to infinity where
the whole thing is settled.

157
00:09:41,830 --> 00:09:44,255
Zero and infinity points I know.

158
00:09:44,255 --> 00:09:48,495
In between, I can't
really have an idea

159
00:09:48,495 --> 00:09:50,810
of where to put my system.

160
00:09:50,810 --> 00:09:54,780
It is at non-equilibrium
by construction.

161
00:09:54,780 --> 00:09:57,830
But this is a process
that by definition I

162
00:09:57,830 --> 00:10:00,640
follow the path
along which there

163
00:10:00,640 --> 00:10:05,790
was no input either in
the form of work or heat

164
00:10:05,790 --> 00:10:06,510
into the system.

165
00:10:06,510 --> 00:10:08,300
It was isolated.

166
00:10:08,300 --> 00:10:11,470
So all I know for sure is
that however amount of energy

167
00:10:11,470 --> 00:10:15,200
I had initially is
what I have finally.

168
00:10:15,200 --> 00:10:18,790
The observation of
Joule was that if we

169
00:10:18,790 --> 00:10:22,160
do this for the case
of a dilute gas,

170
00:10:22,160 --> 00:10:25,760
I start with some
initial temperature Ti.

171
00:10:25,760 --> 00:10:28,080
I go to some final temperature.

172
00:10:28,080 --> 00:10:30,600
But that final
temperature is the same

173
00:10:30,600 --> 00:10:32,600
as the initial temperature.

174
00:10:32,600 --> 00:10:35,660
And that is actually
why I put both

175
00:10:35,660 --> 00:10:39,390
of these points on
the same isotherm

176
00:10:39,390 --> 00:10:41,240
that I had drawn before.

177
00:10:41,240 --> 00:10:43,897
So although I don't know
anything about the in

178
00:10:43,897 --> 00:10:48,890
between points, I know that this
is the property of the system.

179
00:10:48,890 --> 00:10:52,660
And essentially,
we know, therefore,

180
00:10:52,660 --> 00:10:56,090
that although in
principle I can write E

181
00:10:56,090 --> 00:10:58,960
as a function of
P and V, it must

182
00:10:58,960 --> 00:11:02,650
be only a function
of the product PV,

183
00:11:02,650 --> 00:11:05,790
because it only-- P
changes in the process.

184
00:11:05,790 --> 00:11:07,270
V changes in the process.

185
00:11:07,270 --> 00:11:09,960
But PV remains constant.

186
00:11:09,960 --> 00:11:12,600
Or if you like, E
is really a function

187
00:11:12,600 --> 00:11:16,180
of T, which is related
to the product.

188
00:11:16,180 --> 00:11:21,900
And sorry about being kind
of inconsistent with using T

189
00:11:21,900 --> 00:11:23,410
and theta for temperature.

190
00:11:26,200 --> 00:11:28,430
OK, any questions?

191
00:11:28,430 --> 00:11:33,160
So this is kind of recap
of what we were doing.

192
00:11:33,160 --> 00:11:36,820
All right, now, it
would be good if we

193
00:11:36,820 --> 00:11:39,370
can construct this
function somehow--

194
00:11:39,370 --> 00:11:41,330
at least theoretically, even.

195
00:11:41,330 --> 00:11:45,410
And you would say, well,
if I had this spring,

196
00:11:45,410 --> 00:11:48,090
and I didn't know whether it
was a Hookean or a nonlinear

197
00:11:48,090 --> 00:11:52,500
spring, what I could do
is I could pull on it,

198
00:11:52,500 --> 00:11:55,770
calculate what the force
is, and the displacement,

199
00:11:55,770 --> 00:12:01,450
and integrate f dx and
use the formula that quite

200
00:12:01,450 --> 00:12:07,670
generally, the mechanical
work is sum over i,

201
00:12:07,670 --> 00:12:09,000
let's say Ji dxi.

202
00:12:11,990 --> 00:12:13,690
Now, of course you
know that I can only

203
00:12:13,690 --> 00:12:16,210
do this if I pull
it on this spring

204
00:12:16,210 --> 00:12:20,150
sufficiently slowly, so
that the work that I do

205
00:12:20,150 --> 00:12:24,010
goes into changing the
internal energy of the spring.

206
00:12:24,010 --> 00:12:27,150
And then that would be
a contribution to dE.

207
00:12:27,150 --> 00:12:29,990
If I do that rapidly, then
it will send this spring

208
00:12:29,990 --> 00:12:31,040
into oscillation.

209
00:12:31,040 --> 00:12:36,730
I have no idea, again,
where the system

210
00:12:36,730 --> 00:12:40,020
is in the intermediate stages.

211
00:12:40,020 --> 00:12:43,780
So I can use this
kind of formula

212
00:12:43,780 --> 00:12:47,760
if, in this type of
diagram that indicates

213
00:12:47,760 --> 00:12:52,220
the state of the system, I
proceed sufficiently slowly

214
00:12:52,220 --> 00:12:54,780
so that I can put points
that corresponding

215
00:12:54,780 --> 00:12:57,120
to all intermediate states.

216
00:12:57,120 --> 00:13:04,340
And so that's for things that
are sufficiently slow and close

217
00:13:04,340 --> 00:13:08,500
to equilibrium that
we call quasistatic.

218
00:13:12,230 --> 00:13:16,430
And so in principal, I guess
rather than opening this

219
00:13:16,430 --> 00:13:21,600
immediately, I could
have put a slow piston

220
00:13:21,600 --> 00:13:25,110
here, change its position
slowly, so that at each stage

221
00:13:25,110 --> 00:13:28,120
I can calculate where
I am on the PV diagram.

222
00:13:28,120 --> 00:13:32,660
And I could have calculated
what the work is, and used

223
00:13:32,660 --> 00:13:37,290
this formula to calculate
the contribution of the work

224
00:13:37,290 --> 00:13:40,800
to the change in
internal energy.

225
00:13:40,800 --> 00:13:46,930
OK, now I said that
it would be ideal--

226
00:13:46,930 --> 00:13:50,070
and what we would like
to do, ultimately--

227
00:13:50,070 --> 00:13:56,524
is to have a similar
formula for dQ.

228
00:13:56,524 --> 00:14:05,030
And if we look by analogy,
we see that for W,

229
00:14:05,030 --> 00:14:07,440
you have J's times the X's.

230
00:14:07,440 --> 00:14:09,940
The forces are the
things that tell us

231
00:14:09,940 --> 00:14:13,170
whether systems are in
equilibrium with each other.

232
00:14:13,170 --> 00:14:16,590
So two-- if I have,
let's say, a piston

233
00:14:16,590 --> 00:14:18,960
separating two parts
of the gas, the piston

234
00:14:18,960 --> 00:14:20,950
will not move one
direction or the other

235
00:14:20,950 --> 00:14:22,540
if the pressure
from one direction

236
00:14:22,540 --> 00:14:25,550
is the same as the pressure
from the other direction.

237
00:14:25,550 --> 00:14:29,470
So mechanical equilibrium
tells us that J's are the same.

238
00:14:29,470 --> 00:14:33,000
And it suggests that
if I want to write

239
00:14:33,000 --> 00:14:37,850
a similar type of formula for
heat, that the thing that I

240
00:14:37,850 --> 00:14:41,980
should put here is temperature
or empirical temperature, which

241
00:14:41,980 --> 00:14:45,490
is, again, a measure
of thermal equilibrium.

242
00:14:45,490 --> 00:14:48,700
And then the question
that would immediately

243
00:14:48,700 --> 00:14:52,180
jump onto you is, what
is the conjugate that I

244
00:14:52,180 --> 00:14:54,880
have put for the displacement?

245
00:14:54,880 --> 00:14:57,130
And you know
ultimately that it is

246
00:14:57,130 --> 00:14:59,360
going to end up to be entropy.

247
00:14:59,360 --> 00:15:03,940
And so the next part of
the story is to build that.

248
00:15:03,940 --> 00:15:08,349
And we're going to do that
through the Second Law

249
00:15:08,349 --> 00:15:09,098
of Thermodynamics.

250
00:15:15,930 --> 00:15:18,130
Any questions?

251
00:15:18,130 --> 00:15:19,119
Yes.

252
00:15:19,119 --> 00:15:21,410
AUDIENCE: I mean, this is
kind of a semantics question,

253
00:15:21,410 --> 00:15:24,130
but what always confuses
me in thermodynamics

254
00:15:24,130 --> 00:15:27,780
when we talk about quasistatic
is, I mean, if it isn't

255
00:15:27,780 --> 00:15:32,450
quasistatic, introductory
formation of [INAUDIBLE]

256
00:15:32,450 --> 00:15:34,770
doesn't really give a
means to address it.

257
00:15:34,770 --> 00:15:38,250
So we have to rely on the
fact that it's quasistatic.

258
00:15:38,250 --> 00:15:40,370
And the only way
we could check it

259
00:15:40,370 --> 00:15:42,530
is to assume that what
we're already saying is true

260
00:15:42,530 --> 00:15:43,970
will be true.

261
00:15:43,970 --> 00:15:47,960
So I guess my question is,
in the context of a spring,

262
00:15:47,960 --> 00:15:50,130
like, there's
obviously-- there exists

263
00:15:50,130 --> 00:15:54,374
a physical threshold at which--
that's not infinitely slow,

264
00:15:54,374 --> 00:15:56,290
and which you can get a
reasonable measurement

265
00:15:56,290 --> 00:15:58,540
if you're doing an experiment.

266
00:15:58,540 --> 00:16:01,180
PROFESSOR: So in
all of these things,

267
00:16:01,180 --> 00:16:06,360
I would sort of resort to some
kind of a limiting procedure.

268
00:16:06,360 --> 00:16:10,290
So you could, for example, pull
the spring at some velocity,

269
00:16:10,290 --> 00:16:12,910
and figure out what
the amount of work is,

270
00:16:12,910 --> 00:16:15,400
and gradually reduce
that velocity,

271
00:16:15,400 --> 00:16:18,990
and hope that the formula--
the values-- that you get

272
00:16:18,990 --> 00:16:22,610
are converging to something
as you go to low velocity.

273
00:16:22,610 --> 00:16:25,060
I don't think there's
actually a threshold velocity,

274
00:16:25,060 --> 00:16:28,140
kind of implied that if you
are slower than something, then

275
00:16:28,140 --> 00:16:29,290
it would work.

276
00:16:29,290 --> 00:16:32,910
I think it really only works
in the limit of zero velocity.

277
00:16:32,910 --> 00:16:38,110
And so if you like,
that's kind of procedure

278
00:16:38,110 --> 00:16:40,480
you would use to define
derivatives, right?

279
00:16:40,480 --> 00:16:44,800
So you can't say what's the
meaning of velocity in physics.

280
00:16:49,780 --> 00:16:51,580
So, but it is an idealization.

281
00:16:51,580 --> 00:16:54,180
I said at the beginning
that the very-- actually, I

282
00:16:54,180 --> 00:16:55,930
would say that
more fundamentally,

283
00:16:55,930 --> 00:16:58,440
the thing that is
really an idealization

284
00:16:58,440 --> 00:17:00,690
is this adiabatic walls.

285
00:17:00,690 --> 00:17:03,080
Even the concept
of equilibrium--

286
00:17:03,080 --> 00:17:05,960
what do we see around us
that we are absolutely

287
00:17:05,960 --> 00:17:07,780
100% sure in equilibrium?

288
00:17:07,780 --> 00:17:08,970
Nothing is.

289
00:17:08,970 --> 00:17:12,339
The fate of the universe is that
all of our atoms and molecules

290
00:17:12,339 --> 00:17:15,599
are going to separate out
and go to infinity, if it

291
00:17:15,599 --> 00:17:16,495
expands forever.

292
00:17:16,495 --> 00:17:16,994
Right?

293
00:17:19,720 --> 00:17:28,610
OK, Second Law--
all right, so again,

294
00:17:28,610 --> 00:17:31,620
we want to sort of
set our minds back

295
00:17:31,620 --> 00:17:33,720
to where these laws
were developed,

296
00:17:33,720 --> 00:17:35,720
which is in the 18th century.

297
00:17:35,720 --> 00:17:42,510
And the thing that was of
importance at that time

298
00:17:42,510 --> 00:17:46,440
was sort of Industrial
Revolution-- get things moving.

299
00:17:46,440 --> 00:17:50,390
You need energy, and you
would get energy from coal,

300
00:17:50,390 --> 00:17:54,730
and so you need to
convert heat to work.

301
00:17:54,730 --> 00:18:00,160
And this conversion of heat
to work is important to you.

302
00:18:00,160 --> 00:18:07,440
So typical thing that you
have for your coal engine

303
00:18:07,440 --> 00:18:11,380
is you have something
that is a source of heat.

304
00:18:11,380 --> 00:18:16,090
So there's, for example, a
fire that you're burning.

305
00:18:16,090 --> 00:18:23,490
You have some kind
of a machine that

306
00:18:23,490 --> 00:18:27,280
has components, such as a
piston, that is being heated.

307
00:18:27,280 --> 00:18:30,610
And during this process,
you extract a certain amount

308
00:18:30,610 --> 00:18:34,950
of heat from your fire.

309
00:18:34,950 --> 00:18:37,100
And then typically,
you realize that you

310
00:18:37,100 --> 00:18:44,590
will be releasing a
certain amount of energy

311
00:18:44,590 --> 00:18:47,920
back to the atmosphere,
causing pollution.

312
00:18:47,920 --> 00:18:52,110
So there is a coal sink.

313
00:18:52,110 --> 00:18:54,500
But in the process,
you will be extracting

314
00:18:54,500 --> 00:18:57,970
a certain amount of work.

315
00:18:57,970 --> 00:19:02,770
And what you want
to do is to make

316
00:19:02,770 --> 00:19:04,840
the best use of this set up.

317
00:19:04,840 --> 00:19:10,930
So you're concerned
with efficiency,

318
00:19:10,930 --> 00:19:15,160
which is defined to
be the amount of work

319
00:19:15,160 --> 00:19:20,190
that you're going to extract
out of a certain amount of heat

320
00:19:20,190 --> 00:19:27,180
that you expend in
your coal, or whatever.

321
00:19:27,180 --> 00:19:31,800
And again, based on
conservation of energy,

322
00:19:31,800 --> 00:19:40,380
we expect that this W to be
less than QH by an amount

323
00:19:40,380 --> 00:19:43,390
that you are setting
to the exhaust.

324
00:19:43,390 --> 00:19:47,980
And because of that, this
eta has to be less than

325
00:19:47,980 --> 00:19:50,415
or equal to 1, in principal.

326
00:19:55,440 --> 00:20:05,050
Now, another device that is
using the same rough principles

327
00:20:05,050 --> 00:20:07,475
is the same thing
going in reverse,

328
00:20:07,475 --> 00:20:08,475
which is a refrigerator.

329
00:20:16,190 --> 00:20:21,740
So in order to
cool this room, you

330
00:20:21,740 --> 00:20:26,120
use electricity or
some other means

331
00:20:26,120 --> 00:20:34,120
of doing work on a machine whose
job it is to extract heat out

332
00:20:34,120 --> 00:20:35,010
of the room.

333
00:20:37,690 --> 00:20:40,362
So let's say this is our room.

334
00:20:40,362 --> 00:20:44,550
We want to extract the heat.

335
00:20:44,550 --> 00:20:47,400
But of course, that heat
has to go somewhere.

336
00:20:47,400 --> 00:20:51,860
So this has to have
some kind of exhaust

337
00:20:51,860 --> 00:20:58,020
putting this again to the
atmosphere or somewhere.

338
00:21:00,610 --> 00:21:04,380
So how good is this
thing performing?

339
00:21:04,380 --> 00:21:10,040
So there is a measure of the
performance-- figure of merit,

340
00:21:10,040 --> 00:21:13,590
if you like, for
this-- which we will

341
00:21:13,590 --> 00:21:20,560
label by omega, which is
how much heat you were

342
00:21:20,560 --> 00:21:26,520
able to remove from the room
given some amount of work

343
00:21:26,520 --> 00:21:28,590
or energy that was put in.

344
00:21:28,590 --> 00:21:37,180
And again, because of
conservation, this W

345
00:21:37,180 --> 00:21:39,140
is going to be QH minus QC.

346
00:21:44,660 --> 00:21:47,240
Now, this particular
number, which

347
00:21:47,240 --> 00:21:50,850
is a useful measure of how
well your refrigerator works,

348
00:21:50,850 --> 00:21:53,160
has no particular constraint.

349
00:21:53,160 --> 00:21:54,390
It can be less than 1.

350
00:21:54,390 --> 00:21:56,290
It can be larger than 1.

351
00:21:56,290 --> 00:22:00,350
So again, you want it to
be as large as possible,

352
00:22:00,350 --> 00:22:03,960
and in principle, it
can be 200-- whatever.

353
00:22:09,930 --> 00:22:15,760
So clearly, already
these are very much

354
00:22:15,760 --> 00:22:18,410
idealizations of some
complicated process

355
00:22:18,410 --> 00:22:20,520
that is going on.

356
00:22:20,520 --> 00:22:22,900
And the thing that
is interesting

357
00:22:22,900 --> 00:22:27,740
is that you can take
a look at how well you

358
00:22:27,740 --> 00:22:31,500
are able to do these
processes, and then

359
00:22:31,500 --> 00:22:34,820
make this equation
that we wrote up here

360
00:22:34,820 --> 00:22:37,880
to be an exact
differential form.

361
00:22:37,880 --> 00:22:42,160
So how you do that is an
interesting thing that I think

362
00:22:42,160 --> 00:22:45,310
is worth repeating
and exploring.

363
00:22:45,310 --> 00:22:50,750
And essentially, you do
that by formulating things

364
00:22:50,750 --> 00:22:52,760
that are possible
or not possible

365
00:22:52,760 --> 00:22:55,060
through the Second
Law, and then doing

366
00:22:55,060 --> 00:22:57,760
some mathematical manipulations.

367
00:22:57,760 --> 00:23:02,480
And we will use two
formulations of the Second Law

368
00:23:02,480 --> 00:23:05,070
due to Kevin and Clausius.

369
00:23:05,070 --> 00:23:11,710
So I will indicate them
either by K for Kelvin--

370
00:23:11,710 --> 00:23:15,260
so Kelvin's statement
of the Second Law

371
00:23:15,260 --> 00:23:32,230
is the following-- no process
is possible whose sole result

372
00:23:32,230 --> 00:23:42,770
is complete conversion
of heat to work.

373
00:23:49,560 --> 00:23:55,810
So what that is, is a statement
about how good an engine

374
00:23:55,810 --> 00:23:57,150
you can make.

375
00:23:57,150 --> 00:24:01,030
He says that you cannot make
an engine that takes a certain

376
00:24:01,030 --> 00:24:04,910
amount of heat and completely
makes it to work without having

377
00:24:04,910 --> 00:24:07,770
any waste QC.

378
00:24:07,770 --> 00:24:18,070
So basically, it says that
there is no ideal engine,

379
00:24:18,070 --> 00:24:21,549
and that your eta has
to be less than 1.

380
00:24:27,910 --> 00:24:31,020
There is a second
variant of this,

381
00:24:31,020 --> 00:24:37,640
which is due to Clausius-- says
that no process is possible

382
00:24:37,640 --> 00:24:53,520
whose sole result is transfer
of heat from cold to hot.

383
00:25:01,250 --> 00:25:07,800
So if you want to make
an ideal refrigerator,

384
00:25:07,800 --> 00:25:10,680
what you want to do is
to extract more and more

385
00:25:10,680 --> 00:25:16,520
QC for less and less
W. So the ideal version

386
00:25:16,520 --> 00:25:20,240
would be where essentially,
the limit of zero

387
00:25:20,240 --> 00:25:23,940
work, you would still
be able to transfer heat

388
00:25:23,940 --> 00:25:28,335
from the room to the outside
and get an ideal refrigerator.

389
00:25:32,850 --> 00:25:36,126
And Clausius says that
it's not possible.

390
00:25:36,126 --> 00:25:44,860
So these are two statements
of the Second Law.

391
00:25:44,860 --> 00:25:49,670
There are other versions and
statements of the Second Law,

392
00:25:49,670 --> 00:25:56,870
but these are kind of most close
to this historical perspective.

393
00:25:56,870 --> 00:26:00,710
And then the question is,
well, which one of them

394
00:26:00,710 --> 00:26:02,010
do you want to use?

395
00:26:02,010 --> 00:26:06,060
And since I will be
kind of switching

396
00:26:06,060 --> 00:26:08,660
between using one
or the other, it

397
00:26:08,660 --> 00:26:11,270
would be good if I can
show that they really

398
00:26:11,270 --> 00:26:13,660
are the same statement,
and I'm really

399
00:26:13,660 --> 00:26:16,920
not making use of two
different statements when

400
00:26:16,920 --> 00:26:20,400
I use one or the
other alternative.

401
00:26:20,400 --> 00:26:27,610
So what I really want to do is
to show that these are really,

402
00:26:27,610 --> 00:26:30,180
logically, the same statement.

403
00:26:30,180 --> 00:26:33,670
And essentially,
what that means is

404
00:26:33,670 --> 00:26:39,010
that the two statements
are equivalent.

405
00:26:39,010 --> 00:26:43,584
And two statements are
equivalent to each other

406
00:26:43,584 --> 00:26:49,950
if one of them being
incorrect-- which I've indicated

407
00:26:49,950 --> 00:26:56,540
by K with a bar on it-- implies
that the other one is correct,

408
00:26:56,540 --> 00:27:01,930
and simultaneously,
the other way around.

409
00:27:06,070 --> 00:27:10,360
And since this does not take
more than a few minutes,

410
00:27:10,360 --> 00:27:15,790
and is a nice exercise, I
think it's worth showing this.

411
00:27:15,790 --> 00:27:20,790
So let's say that
somebody came to you

412
00:27:20,790 --> 00:27:24,770
and said that I have
constructed a machine that

413
00:27:24,770 --> 00:27:27,040
violates Clausius.

414
00:27:27,040 --> 00:27:30,430
We'll call that K bar.

415
00:27:30,430 --> 00:27:32,490
So what does that machine do?

416
00:27:32,490 --> 00:27:43,060
What that machine does
is it converts heat, Q,

417
00:27:43,060 --> 00:27:49,870
completely to work, without
needing to exhaust anything.

418
00:27:49,870 --> 00:27:54,155
So this W is this Q, and
there's no exhaust machine.

419
00:27:54,155 --> 00:27:56,890
He says, OK I have this machine.

420
00:27:56,890 --> 00:28:03,160
I say, OK, what I will do
is I will use that work

421
00:28:03,160 --> 00:28:05,490
to run a refrigerator.

422
00:28:05,490 --> 00:28:08,320
So I connect that
to a refrigerator.

423
00:28:13,270 --> 00:28:27,040
And that work will be used to
take heat, QC, out of some room

424
00:28:27,040 --> 00:28:31,490
and exhaust it to
the atmosphere.

425
00:28:31,490 --> 00:28:36,220
And I will choose
the exhaust location

426
00:28:36,220 --> 00:28:38,900
to be the same as
the source of heat

427
00:28:38,900 --> 00:28:45,596
that I had for my
anti-Kelvin machine.

428
00:28:45,596 --> 00:28:46,095
OK?

429
00:28:49,810 --> 00:28:55,386
So then if you look at how
the sum of these two machines

430
00:28:55,386 --> 00:29:03,590
is operating together, from
the perspective of looking

431
00:29:03,590 --> 00:29:05,930
at the sum of the two
machines, the work

432
00:29:05,930 --> 00:29:08,030
is just an internal operation.

433
00:29:08,030 --> 00:29:09,640
You don't really care.

434
00:29:09,640 --> 00:29:15,230
So as far as the net thing is
concerned, what you're seeing

435
00:29:15,230 --> 00:29:20,330
is that there is no work
involved externally.

436
00:29:20,330 --> 00:29:21,770
It's all internal.

437
00:29:21,770 --> 00:29:23,590
But what is happening
is that you're

438
00:29:23,590 --> 00:29:31,390
pulling QC out of the room,
and exhausting, presumably,

439
00:29:31,390 --> 00:29:38,310
what you have here-- QH
minus Q to the other side.

440
00:29:38,310 --> 00:29:42,295
This, again, by conservation of
energy must be the same as QC,

441
00:29:42,295 --> 00:29:44,090
right?

442
00:29:44,090 --> 00:29:48,990
So if you have a machine
that violates Kelvin,

443
00:29:48,990 --> 00:29:53,090
you connect it to a
refrigerator, and what you have

444
00:29:53,090 --> 00:29:56,520
is a machine that transfers
heat from a cold-air

445
00:29:56,520 --> 00:30:01,870
to a hot-air body,
and violates Clausius.

446
00:30:01,870 --> 00:30:07,440
So that's the first part
we have established.

447
00:30:07,440 --> 00:30:09,610
And to establish
the second part,

448
00:30:09,610 --> 00:30:19,230
let's say that we have a
machine that violates Clausius.

449
00:30:23,960 --> 00:30:29,880
And this machine takes a
certain amount of heat, Q,

450
00:30:29,880 --> 00:30:36,670
from the room, and deposits
it to the hotter outside.

451
00:30:36,670 --> 00:30:40,990
What we will do
immediately is to take

452
00:30:40,990 --> 00:30:48,040
some of this heat that has
been generated by this machine

453
00:30:48,040 --> 00:30:57,240
and use it in a regular engine
to create some amount of work.

454
00:30:57,240 --> 00:31:02,200
And then this has to take
an amount of heat QH,

455
00:31:02,200 --> 00:31:08,060
deposit an amount of heat, QC,
which we will select to be,

456
00:31:08,060 --> 00:31:12,760
essentially, the temperature
or the room from which

457
00:31:12,760 --> 00:31:16,880
the anti Clausius
machine is operating.

458
00:31:16,880 --> 00:31:20,020
And then we will run this
engine several times,

459
00:31:20,020 --> 00:31:24,010
or fractions of times, et
cetera, and the other machine

460
00:31:24,010 --> 00:31:32,160
several times, ensuring that
QC is the same as Q. That

461
00:31:32,160 --> 00:31:38,230
is, the engine puts
out as much dumped heat

462
00:31:38,230 --> 00:31:43,330
as this anti-Clausius machine
is extracting from the room.

463
00:31:43,330 --> 00:31:50,550
Then if you look at the
combined system, what

464
00:31:50,550 --> 00:31:58,280
we see is that the combined
system is doing the following--

465
00:31:58,280 --> 00:32:00,540
this was equivalent to this.

466
00:32:00,540 --> 00:32:08,110
This is equivalent
to a combined system

467
00:32:08,110 --> 00:32:10,640
that does a certain
amount of work.

468
00:32:10,640 --> 00:32:14,010
So there is a W
that is coming out.

469
00:32:14,010 --> 00:32:17,620
There is no heat
exchange with the room,

470
00:32:17,620 --> 00:32:20,690
because we ensured that these
two heats are completely

471
00:32:20,690 --> 00:32:21,940
balanced.

472
00:32:21,940 --> 00:32:24,080
So you can sort of think
of them as something

473
00:32:24,080 --> 00:32:27,210
that is internal to
this bigger machine.

474
00:32:27,210 --> 00:32:29,920
And so what is happening
is that there's

475
00:32:29,920 --> 00:32:35,550
a certain amount of heat, QH
minus Q that is taken here

476
00:32:35,550 --> 00:32:39,890
that has to be equal to W,
by conservation of energy.

477
00:32:39,890 --> 00:32:42,400
And we have converted
heat entirely

478
00:32:42,400 --> 00:32:47,100
to work, violating
Kelvin's statement.

479
00:32:47,100 --> 00:32:49,984
So basically, we
have proved this.

480
00:33:27,552 --> 00:33:28,135
Any questions?

481
00:33:32,190 --> 00:33:34,920
OK, one more step-- there
is actually a couple

482
00:33:34,920 --> 00:33:39,805
more steps left before we get to
construct our entropy function.

483
00:33:43,190 --> 00:33:45,090
And the next step is
the Carnot Engine.

484
00:33:53,950 --> 00:33:58,570
So this is yet another
one of these idealizations

485
00:33:58,570 --> 00:34:02,220
that-- again, theoretical
construct, which you

486
00:34:02,220 --> 00:34:05,570
can try to approach as in
some limiting procedure.

487
00:34:08,219 --> 00:34:18,139
A Carnot engine is any engine
with the following properties--

488
00:34:18,139 --> 00:34:43,380
that is, one, reversible,
two, operates in a cycle,

489
00:34:43,380 --> 00:34:56,219
and three, all inputs--
all heat inputs--

490
00:34:56,219 --> 00:35:00,990
outputs at two temperatures.

491
00:35:10,150 --> 00:35:16,680
OK, so let's go
through the conditions.

492
00:35:16,680 --> 00:35:24,780
Reversible implies
that at any stage,

493
00:35:24,780 --> 00:35:30,810
if I change the directions
of inputs and outputs,

494
00:35:30,810 --> 00:35:33,230
it will just go backward.

495
00:35:33,230 --> 00:35:50,573
So can go forward, backward,
by reversing inputs, outputs.

496
00:35:56,850 --> 00:36:04,700
And so again, think of
the case of our spring

497
00:36:04,700 --> 00:36:09,430
that we were pushing
back and forth.

498
00:36:09,430 --> 00:36:15,690
In order for it to have this
property of reversibility,

499
00:36:15,690 --> 00:36:22,300
I should not only do it pulling
down and up sufficiently slowly

500
00:36:22,300 --> 00:36:26,230
so that at each stage, I
can define its tension,

501
00:36:26,230 --> 00:36:28,910
but also as far
as its connection

502
00:36:28,910 --> 00:36:32,430
to the outside world, I have
to do it frictionlessly,

503
00:36:32,430 --> 00:36:36,290
so that if I were to reverse
the direction of the force

504
00:36:36,290 --> 00:36:40,290
that I have, it will either go
forward or backward, reversing

505
00:36:40,290 --> 00:36:41,990
the path that it has.

506
00:36:41,990 --> 00:36:45,270
If there is any friction
that is involved,

507
00:36:45,270 --> 00:36:48,270
then I cannot do that.

508
00:36:48,270 --> 00:36:51,530
So essentially,
this reversibility

509
00:36:51,530 --> 00:36:58,760
is some kind of a generalization
of the frictionless condition

510
00:36:58,760 --> 00:37:00,105
that we would use in mechanics.

511
00:37:05,870 --> 00:37:07,330
The cycle part is easy.

512
00:37:10,910 --> 00:37:14,460
Essentially, it says
that the start and end

513
00:37:14,460 --> 00:37:15,400
points are the same.

514
00:37:24,430 --> 00:37:30,920
And it kind of harks
back to this statement

515
00:37:30,920 --> 00:37:35,240
that is part of either
one of these formulations

516
00:37:35,240 --> 00:37:38,630
of the Second Law--
that is the net result

517
00:37:38,630 --> 00:37:41,010
or sole result is something.

518
00:37:41,010 --> 00:37:43,710
So we want the
engine to essentially

519
00:37:43,710 --> 00:37:47,020
be back to where it
was, so that when

520
00:37:47,020 --> 00:37:49,820
we are looking at changes
that are involved,

521
00:37:49,820 --> 00:37:52,450
we say, OK, the engine is
not part of the equation.

522
00:37:52,450 --> 00:37:55,560
It is where it was
at the beginning

523
00:37:55,560 --> 00:37:59,490
and at the end of the cycle.

524
00:37:59,490 --> 00:38:05,110
Now there's something
that I didn't emphasize,

525
00:38:05,110 --> 00:38:09,900
but I was-- and actually,
I was not really using,

526
00:38:09,900 --> 00:38:13,170
but you may have
thought I was using--

527
00:38:13,170 --> 00:38:18,070
is that in all of these
pictures that I drew,

528
00:38:18,070 --> 00:38:21,990
I said there is a hot place
and there's a cold place.

529
00:38:21,990 --> 00:38:26,000
Now, I didn't specify
exactly what these are,

530
00:38:26,000 --> 00:38:29,820
and whether this corresponds
to a particular temperature.

531
00:38:29,820 --> 00:38:33,700
Now, for the case of the Carnot
engine, I have to be precise.

532
00:38:33,700 --> 00:38:37,820
I have to say that this is
always at one temperature.

533
00:38:37,820 --> 00:38:40,480
This is always at the
other temperature.

534
00:38:40,480 --> 00:38:46,450
So these Carnot engines
are defined implicitly

535
00:38:46,450 --> 00:38:49,280
with two temperature
labels corresponding

536
00:38:49,280 --> 00:38:53,640
to the hot part
and the cold part.

537
00:38:53,640 --> 00:38:56,320
In principle, everything
else that I talked about

538
00:38:56,320 --> 00:38:59,066
could have had a
range of temperatures.

539
00:38:59,066 --> 00:39:01,710
But the Carnot engine says,
OK, just two temperatures.

540
00:39:07,370 --> 00:39:11,820
OK, so for the Carnot
engine, if I can now

541
00:39:11,820 --> 00:39:18,910
draw a diagram that would
be kind of straight line

542
00:39:18,910 --> 00:39:21,170
version of what I had before.

543
00:39:21,170 --> 00:39:27,700
And the Carnot engine will be
extracting a certain amount

544
00:39:27,700 --> 00:39:33,100
of heat, QH, from
here, QC from here,

545
00:39:33,100 --> 00:39:36,150
doing a certain amount of work.

546
00:39:36,150 --> 00:39:40,910
And since it is reversible, I
could actually run it backward.

547
00:39:40,910 --> 00:39:47,890
I could have W. I
could have QH, QC,

548
00:39:47,890 --> 00:39:50,290
and it would work
as a refrigerator.

549
00:40:05,080 --> 00:40:10,780
Now, it is good to have
at least one example--

550
00:40:10,780 --> 00:40:15,590
that such a theoretical
construct as a Carnot engine

551
00:40:15,590 --> 00:40:20,120
can be made based on
things that we have so far.

552
00:40:20,120 --> 00:40:24,060
And we can be
explicit, and show that

553
00:40:24,060 --> 00:40:26,386
for the case of the ideal gas.

554
00:40:37,010 --> 00:40:44,950
So we said that a gas I
can represent in equivalent

555
00:40:44,950 --> 00:40:48,230
by a point in the
pressure-volume diagram.

556
00:40:51,480 --> 00:40:56,580
So let's say that
what is the working

557
00:40:56,580 --> 00:40:59,870
substance of this
Carnot engine is a gas.

558
00:40:59,870 --> 00:41:04,420
And part of this
whole story is that it

559
00:41:04,420 --> 00:41:09,010
should be working between two
isotherms-- the TH and TC.

560
00:41:09,010 --> 00:41:11,490
But we've already
established that we

561
00:41:11,490 --> 00:41:16,460
can have isotherms for the
ideal gas that correspond

562
00:41:16,460 --> 00:41:22,300
to these curves that
correspond to PV as constant.

563
00:41:22,300 --> 00:41:25,490
So I could really choose
one of the isotherms

564
00:41:25,490 --> 00:41:28,110
that we discussed before
corresponds to TH.

565
00:41:28,110 --> 00:41:30,530
One corresponds to TC.

566
00:41:30,530 --> 00:41:40,740
And I can imagine taking a gas
whose initial state is point A,

567
00:41:40,740 --> 00:41:47,810
and expanding it while
maintaining it at this isotherm

568
00:41:47,810 --> 00:41:54,360
TH, and ending up
at some point B.

569
00:41:54,360 --> 00:41:58,920
So as I'm expanding this,
let's say we have put this gas.

570
00:41:58,920 --> 00:42:02,510
I have this piston
that contains this gas.

571
00:42:02,510 --> 00:42:06,500
My TH and TC
correspond to two baths

572
00:42:06,500 --> 00:42:09,630
that correspond to, if you
like, lakes-- a temperature

573
00:42:09,630 --> 00:42:11,480
TH, lakes at temperature TC.

574
00:42:11,480 --> 00:42:14,890
I took my piston, put
it into the hot bath,

575
00:42:14,890 --> 00:42:18,550
make it expand up to some point.

576
00:42:18,550 --> 00:42:21,160
And in the process,
there is certainly

577
00:42:21,160 --> 00:42:25,610
a certain amount of heat that
will go into the system, QH.

578
00:42:28,530 --> 00:42:36,260
So I have taken QH
out of the hot source.

579
00:42:36,260 --> 00:42:39,290
Now, next thing
that I want to do

580
00:42:39,290 --> 00:42:44,092
is to take this, and put
it into the colder lake.

581
00:42:44,092 --> 00:42:49,340
But what the process is,
is I have to do it in a way

582
00:42:49,340 --> 00:42:52,270
that there is no heat exchange.

583
00:42:52,270 --> 00:42:57,850
So I have to find the path
that goes from the hot bath

584
00:42:57,850 --> 00:43:02,150
to the cold path-- to
point C, some point C--

585
00:43:02,150 --> 00:43:03,430
without any heat involved.

586
00:43:06,140 --> 00:43:08,280
We'll have to ask
how that's possible,

587
00:43:08,280 --> 00:43:11,040
and what's the
structure of that path.

588
00:43:11,040 --> 00:43:14,480
Once I'm in the
colder bath, then I

589
00:43:14,480 --> 00:43:21,280
can expand my gas
up to some point D.

590
00:43:21,280 --> 00:43:25,620
And clearly, I have to choose
this point D sufficiently

591
00:43:25,620 --> 00:43:30,160
precisely so that
then, when I take

592
00:43:30,160 --> 00:43:35,520
my gas from the cool to
the original hot position,

593
00:43:35,520 --> 00:43:40,620
I end up at precisely the
location that I started,

594
00:43:40,620 --> 00:43:42,480
so that I have
completed a cycle.

595
00:43:45,060 --> 00:43:47,040
So it's necessary
that I should be

596
00:43:47,040 --> 00:43:52,580
able to construct these
two paths, BC and DA, that

597
00:43:52,580 --> 00:43:55,340
correspond to no heat exchange.

598
00:43:55,340 --> 00:43:58,940
And somehow I'm sure that
I can complete this cycle.

599
00:44:02,480 --> 00:44:13,210
So the paths BC
and DA are called

600
00:44:13,210 --> 00:44:20,820
adiabats, or adiabatic
paths, in the sense

601
00:44:20,820 --> 00:44:23,980
that this is what
you would get if you

602
00:44:23,980 --> 00:44:30,840
put your system in a container
with these adiabatic walls that

603
00:44:30,840 --> 00:44:33,580
allows no exchange of heat.

604
00:44:33,580 --> 00:44:37,500
Now clearly, I also want to
do this sufficiently slowly

605
00:44:37,500 --> 00:44:39,910
so that infinitesimally,
I can draw

606
00:44:39,910 --> 00:44:42,510
this path as a series of points.

607
00:44:42,510 --> 00:44:46,180
And so I don't want to do what
was in the Joule expansion

608
00:44:46,180 --> 00:44:50,160
experiment, and suddenly
expand the whole thing.

609
00:44:50,160 --> 00:44:52,150
I have to do it
sufficiently slowly

610
00:44:52,150 --> 00:44:57,890
so that the conditions of this
Carnot engine are satisfied.

611
00:44:57,890 --> 00:45:00,570
So along these paths,
what do I know?

612
00:45:00,570 --> 00:45:04,550
By definition dQ is 0.

613
00:45:04,550 --> 00:45:12,990
And dQ is dE minus
dW along those paths.

614
00:45:12,990 --> 00:45:16,790
And since I'm performing these
paths sufficiently slowly,

615
00:45:16,790 --> 00:45:23,440
I can write the W
as PdV minus PdV.

616
00:45:32,090 --> 00:45:36,170
Now, what you will show
in the problem set is

617
00:45:36,170 --> 00:45:40,680
that I don't need to make any
assumption about the functional

618
00:45:40,680 --> 00:45:47,855
form of energy on P and V. Just
knowing that I'm describing

619
00:45:47,855 --> 00:45:52,310
a system that is characterized
by two coordinates, P and V,

620
00:45:52,310 --> 00:45:56,320
is sufficient for you to
construct these curves.

621
00:45:56,320 --> 00:45:58,360
Turns out that if you
have more than two

622
00:45:58,360 --> 00:46:01,080
coordinates-- three or four
coordinates-- at this time

623
00:46:01,080 --> 00:46:03,640
you don't know that you
will be able to do so.

624
00:46:03,640 --> 00:46:06,750
But we're sticking with this.

625
00:46:06,750 --> 00:46:10,610
For simplicity, I'm going
to choose this-- the form

626
00:46:10,610 --> 00:46:14,675
that I have for the ideal gas,
which I know that it's really

627
00:46:14,675 --> 00:46:18,540
a function only
of the product PV.

628
00:46:18,540 --> 00:46:21,640
This is what we had used before.

629
00:46:21,640 --> 00:46:25,650
And again, so as to
simplify the algebra,

630
00:46:25,650 --> 00:46:31,175
I will use the form that is
applicable to a monoatomic gas.

631
00:46:35,290 --> 00:46:38,050
And this-- I really have
no reason to do this,

632
00:46:38,050 --> 00:46:40,270
except that I want to be
able to get through this

633
00:46:40,270 --> 00:46:43,085
in two minutes with the algebra.

634
00:46:43,085 --> 00:46:47,220
So I will write the
energy to be 3/2 PV.

635
00:46:47,220 --> 00:46:51,590
So then for that particular
choice, what I have here

636
00:46:51,590 --> 00:46:57,210
is d of 3/2 PV plus PdV.

637
00:46:57,210 --> 00:47:09,420
And that's 3/2 PdV plus
V, 3/2 VdP plus PdV,

638
00:47:09,420 --> 00:47:19,600
which is 5/2 PdV
plus 3/2 VdP is 0.

639
00:47:19,600 --> 00:47:22,280
And you can
rearrange that easily

640
00:47:22,280 --> 00:47:35,030
to dP over P plus
5/3 dV over V is 0.

641
00:47:35,030 --> 00:47:40,470
And this is simply the
derivative of log of PV

642
00:47:40,470 --> 00:47:43,700
to the power of 5/3.

643
00:47:43,700 --> 00:47:49,010
And since this
derivative is 0, you

644
00:47:49,010 --> 00:47:53,920
know that along the curves that
correspond to no heat exchange,

645
00:47:53,920 --> 00:47:58,600
you have something like
PV to the power of gamma--

646
00:47:58,600 --> 00:48:08,620
in this case, gamma being
5/3-- that is some constant.

647
00:48:11,730 --> 00:48:16,695
So after you have done your
expansion, all you need is to,

648
00:48:16,695 --> 00:48:19,910
in this diagram, go
continuously along the path

649
00:48:19,910 --> 00:48:24,720
that corresponds to this
formula that we have indicated.

650
00:48:24,720 --> 00:48:27,330
And this formula
describes a path

651
00:48:27,330 --> 00:48:30,230
that is distinct
from the isotherm.

652
00:48:30,230 --> 00:48:32,350
So you start from somewhere.

653
00:48:32,350 --> 00:48:34,400
You start along some path.

654
00:48:34,400 --> 00:48:38,750
You are guaranteed to hit the
other isotherm at some point.

655
00:48:38,750 --> 00:48:40,410
You start from the
starting point.

656
00:48:40,410 --> 00:48:41,840
You go backward.

657
00:48:41,840 --> 00:48:44,950
You're guaranteed to hit
the isotherm at some point.

658
00:48:44,950 --> 00:48:46,490
And then you join
those two points,

659
00:48:46,490 --> 00:48:48,420
and you've completed your cycle.

660
00:48:48,420 --> 00:48:54,340
And so you know that, at least
for the case of the ideal gas,

661
00:48:54,340 --> 00:48:57,100
or actually for any
two-coordinate system,

662
00:48:57,100 --> 00:49:00,870
you can construct
a cardinal cycle.

663
00:49:00,870 --> 00:49:06,070
That is, put your material,
go through a cycle that

664
00:49:06,070 --> 00:49:10,240
is reversible with all heat
exchanges at two temperatures

665
00:49:10,240 --> 00:49:10,740
only.

666
00:49:32,490 --> 00:49:37,464
OK, now why is this
idealization even useful?

667
00:49:37,464 --> 00:49:40,140
Well, we said that
this Carnot engine

668
00:49:40,140 --> 00:49:48,160
is stamped by really two
temperatures-- TH and TC.

669
00:49:48,160 --> 00:49:53,180
And the following
theorem is important,

670
00:49:53,180 --> 00:50:06,410
which says that of all
engines operating only

671
00:50:06,410 --> 00:50:14,900
between TH and TC, the Carnot
engine is most efficient.

672
00:50:27,700 --> 00:50:31,440
So we are going to move
away from this idealization

673
00:50:31,440 --> 00:50:34,955
that we've made, but so
far, only in one small step.

674
00:50:34,955 --> 00:50:38,390
That is, we are still going
to assume that all heat

675
00:50:38,390 --> 00:50:43,470
exchanges are done
between TH and TC.

676
00:50:43,470 --> 00:50:47,270
OK, and so let's
say that we have

677
00:50:47,270 --> 00:50:56,320
some kind of an engine which
does not have to be reversible.

678
00:50:56,320 --> 00:51:00,080
So because it does not
have to be reversible,

679
00:51:00,080 --> 00:51:05,270
it's not a Carnot engine, but
otherwise, operates completely

680
00:51:05,270 --> 00:51:07,100
between these two points.

681
00:51:07,100 --> 00:51:14,000
So it takes, let's say,
heat QH prime, QC prime,

682
00:51:14,000 --> 00:51:21,210
and does a certain
amount of work W.

683
00:51:21,210 --> 00:51:26,250
And so somebody
comes and says, I've

684
00:51:26,250 --> 00:51:30,860
constructed this engine
that is actually very good.

685
00:51:30,860 --> 00:51:33,590
It's better than
your Carnot engine.

686
00:51:33,590 --> 00:51:35,650
I say, no, that's not possible.

687
00:51:35,650 --> 00:51:37,380
And the way that I
will prove it to you

688
00:51:37,380 --> 00:51:42,280
is as follows-- I will take
the output of your engine,

689
00:51:42,280 --> 00:51:46,950
connect it to a Carnot engine,
and since the Carnot engine

690
00:51:46,950 --> 00:51:53,460
can be run backward, it
will act as a refrigerator

691
00:51:53,460 --> 00:51:57,390
and extract heat and
deposit heat precisely

692
00:51:57,390 --> 00:52:02,530
at the two temperatures that
your engine is operating.

693
00:52:02,530 --> 00:52:08,050
So if somebody looks at the
combination of these two,

694
00:52:08,050 --> 00:52:10,830
what do they see?

695
00:52:10,830 --> 00:52:14,100
The thing that they
see for the combination

696
00:52:14,100 --> 00:52:16,980
is that there is
some entity that

697
00:52:16,980 --> 00:52:20,870
is operating between TH and TC.

698
00:52:20,870 --> 00:52:23,400
And there is some
internal amount of work

699
00:52:23,400 --> 00:52:26,440
that is going on, and you
don't care about that.

700
00:52:26,440 --> 00:52:31,970
But there is a certain amount
of heat, QH prime minus QH that

701
00:52:31,970 --> 00:52:35,050
becomes QC prime minus QC.

702
00:52:40,300 --> 00:52:43,260
Now here we invoke Clausius.

703
00:52:46,640 --> 00:52:51,740
Clausius says that if you
see heat going between two

704
00:52:51,740 --> 00:52:56,720
temperatures, it could have only
gone from the direction hotter

705
00:52:56,720 --> 00:53:01,300
to colder, which means
that this amount of heat

706
00:53:01,300 --> 00:53:03,420
has to be positive.

707
00:53:03,420 --> 00:53:08,550
QH prime should be
greater than QH.

708
00:53:08,550 --> 00:53:14,550
So this is why Clausius-- we
know that this has to be case.

709
00:53:19,350 --> 00:53:29,050
So then what I can do, is I can
divide both expressions by W.

710
00:53:29,050 --> 00:53:31,330
And then invert this thing.

711
00:53:31,330 --> 00:53:34,840
And if I invert it, the
inequality gets inverted.

712
00:53:34,840 --> 00:53:42,560
So I get that W over QH
prime is less than W over QH.

713
00:53:42,560 --> 00:53:48,020
And the left-hand
side is the efficiency

714
00:53:48,020 --> 00:53:50,630
of my non-Carnot engine.

715
00:53:50,630 --> 00:53:55,190
And the right-hand side is the
efficiency of my Carnot engine.

716
00:53:55,190 --> 00:53:59,200
Carnot And what I've shown
is that the efficiency

717
00:53:59,200 --> 00:54:01,900
of the non-Carnot engine
has to be less than

718
00:54:01,900 --> 00:54:04,140
or equal to the efficiency
of the Carnot engine.

719
00:54:07,840 --> 00:54:16,360
OK, now, the next step
is the following--

720
00:54:16,360 --> 00:54:36,690
that all Carnot engines
operating between TH and TC

721
00:54:36,690 --> 00:54:37,740
have the same efficiency.

722
00:54:49,470 --> 00:54:55,620
And the statement is, suppose
I have two different Carnot

723
00:54:55,620 --> 00:54:58,690
engines-- Carnot engine
one and Carnot engine two.

724
00:54:58,690 --> 00:55:01,150
I can use one to run
the other one backward,

725
00:55:01,150 --> 00:55:04,840
and I would get that the
efficiency of Carnot engine one

726
00:55:04,840 --> 00:55:07,610
is less than or equal to Carnot
engine two, or the other way

727
00:55:07,610 --> 00:55:10,750
around-- that two is less
than or equal to one.

728
00:55:10,750 --> 00:55:12,520
And hence, the
only possibility is

729
00:55:12,520 --> 00:55:17,990
that they are the same, which
is actually interesting,

730
00:55:17,990 --> 00:55:20,740
because I told you that
Carnot engines are stamped

731
00:55:20,740 --> 00:55:25,330
by two temperatures, TH and TC.

732
00:55:25,330 --> 00:55:28,390
And what it says
is that they all

733
00:55:28,390 --> 00:55:31,510
have an efficiency that
is a function, therefore,

734
00:55:31,510 --> 00:55:33,710
of these two temperatures only.

735
00:55:33,710 --> 00:55:37,260
It cannot depend on whether I
constructed my Carnot engine

736
00:55:37,260 --> 00:55:40,920
out of an ideal gas.

737
00:55:40,920 --> 00:55:43,260
I constructed it
out of a rubber band

738
00:55:43,260 --> 00:55:48,040
that I was pulling or pushing,
or any other substance

739
00:55:48,040 --> 00:55:51,360
in the world that
you can think about,

740
00:55:51,360 --> 00:55:55,720
as long as that substance I can
construct some kind of a Carnot

741
00:55:55,720 --> 00:56:01,200
cycle, irrespective of what the
coordinate systems are, what

742
00:56:01,200 --> 00:56:03,710
the isotherms look
like, et cetera.

743
00:56:03,710 --> 00:56:07,130
Just working between two
temperatures and two isotherms

744
00:56:07,130 --> 00:56:11,140
tells me that this
reversible cycle

745
00:56:11,140 --> 00:56:15,740
will have this efficiency
relating the amount of work

746
00:56:15,740 --> 00:56:18,550
and the amount of heat
that these exchange as you

747
00:56:18,550 --> 00:56:19,950
go around.

748
00:56:19,950 --> 00:56:22,190
OK?

749
00:56:22,190 --> 00:56:26,340
Now, that's important
because previously we

750
00:56:26,340 --> 00:56:28,770
wanted to define
our temperature.

751
00:56:28,770 --> 00:56:33,260
And we had to rely on
some property of the gas.

752
00:56:33,260 --> 00:56:37,290
We said OK, empirically we
observe that for the gas,

753
00:56:37,290 --> 00:56:39,940
you have this relationship
that when it is dilute,

754
00:56:39,940 --> 00:56:45,170
PV is proportional to T and we
use that property of the gas

755
00:56:45,170 --> 00:56:49,105
to construct our ideal
gas temperature scale.

756
00:56:49,105 --> 00:56:54,260
Well, that's OK, but it
relies on some observations

757
00:56:54,260 --> 00:56:57,430
additionally to everything
else that we had.

758
00:56:57,430 --> 00:57:01,470
Whereas now we have the
means to define temperatures

759
00:57:01,470 --> 00:57:03,760
irrespective of any material.

760
00:57:03,760 --> 00:57:06,555
This is much more of
a universal quantity.

761
00:57:06,555 --> 00:57:08,670
It' doesn't depend
on any substance,

762
00:57:08,670 --> 00:57:10,640
any particular structure.

763
00:57:10,640 --> 00:57:14,130
So it's a much
better way to rely

764
00:57:14,130 --> 00:57:17,190
for constructing a
temperature scale.

765
00:57:17,190 --> 00:57:22,060
So in some sense, at
least theoretically,

766
00:57:22,060 --> 00:57:24,690
we are going to abandon
our previous temperature

767
00:57:24,690 --> 00:57:27,565
scale based on properties
of the ideal gas,

768
00:57:27,565 --> 00:57:31,717
and reconstruct the temperature
scale based on this.

769
00:57:31,717 --> 00:57:32,300
Any questions?

770
00:57:36,050 --> 00:57:37,610
OK.

771
00:57:37,610 --> 00:57:42,530
So this is what will be called
the thermodynamic temperature

772
00:57:42,530 --> 00:57:43,030
scale.

773
00:57:50,020 --> 00:57:56,540
And what it amounts
to is that somehow I

774
00:57:56,540 --> 00:58:01,100
use the efficiency
between two temperatures

775
00:58:01,100 --> 00:58:04,960
or vice versa-- use that
there are two temperatures.

776
00:58:04,960 --> 00:58:07,500
Let's say one of them is
a reference temperature.

777
00:58:07,500 --> 00:58:11,570
I derive the temperature
of the other one

778
00:58:11,570 --> 00:58:14,470
by the maximum efficiency
of all engines that

779
00:58:14,470 --> 00:58:16,660
can operate between
the reference

780
00:58:16,660 --> 00:58:20,150
temperature and the temperature
that I wish to measure.

781
00:58:20,150 --> 00:58:21,790
It's T.

782
00:58:21,790 --> 00:58:24,330
But that means I--
before doing that, I

783
00:58:24,330 --> 00:58:27,340
need to know at least something,
or some kind of a postulate

784
00:58:27,340 --> 00:58:31,230
about the structure of this
function of two variables.

785
00:58:31,230 --> 00:58:35,580
And I cannot write any
arbitrary function.

786
00:58:35,580 --> 00:58:39,870
It has to satisfy certain
properties and symmetries.

787
00:58:39,870 --> 00:58:44,430
And I can see what
those properties are

788
00:58:44,430 --> 00:58:49,320
by putting a couple of
these engines in series.

789
00:58:49,320 --> 00:58:53,490
So let's imagine that we
have a structure where

790
00:58:53,490 --> 00:58:57,120
I have the highest
temperature T1,

791
00:58:57,120 --> 00:59:04,270
and intermediate temperature T2,
and the lowest temperature T3.

792
00:59:04,270 --> 00:59:09,790
And I put one Carnot engine
to operate between these two

793
00:59:09,790 --> 00:59:14,510
temperatures, and one to operate
between these temperatures.

794
00:59:14,510 --> 00:59:18,350
And the first one is
going to take heat

795
00:59:18,350 --> 00:59:20,374
that I will called Q1.

796
00:59:25,370 --> 00:59:32,650
And release a heat that I will
call Q2, in the process doing

797
00:59:32,650 --> 00:59:37,960
a certain amount of work that
I will call W between 1 and 2.

798
00:59:37,960 --> 00:59:44,820
Now what I will do is I will use
the entirety of that heat, Q2,

799
00:59:44,820 --> 00:59:50,240
to run the second Carnot
engine to temperature T3,

800
00:59:50,240 --> 00:59:54,670
reducing here the
amount of heat, Q3,

801
00:59:54,670 --> 00:59:56,895
and doing a certain
amount of work-- W23.

802
01:00:00,210 --> 01:00:03,580
Clearly what I want
to do is to say

803
01:00:03,580 --> 01:00:09,610
that when I look at the
combination of these things,

804
01:00:09,610 --> 01:00:15,110
what happens at temperature
2 is an intermediate.

805
01:00:15,110 --> 01:00:19,780
And the whole thing is
equivalent to a Carnot engine

806
01:00:19,780 --> 01:00:22,570
operating between T1 and T3.

807
01:00:22,570 --> 01:00:26,830
Again, the whole
thing is reversible,

808
01:00:26,830 --> 01:00:29,140
operates between
two temperatures,

809
01:00:29,140 --> 01:00:30,720
and since each
one of them can be

810
01:00:30,720 --> 01:00:33,660
made to go back to where
they started, is a cycle.

811
01:00:33,660 --> 01:00:37,764
So the whole thing is really
equivalent to another Carnot

812
01:00:37,764 --> 01:00:48,770
engine that takes
Q1, deposits Q3, does

813
01:00:48,770 --> 01:00:51,810
a certain amount
of work, W13, which

814
01:00:51,810 --> 01:00:54,752
is the sum of W12 plus W23.

815
01:00:58,600 --> 01:01:02,670
So clearly, these two
different perspectives

816
01:01:02,670 --> 01:01:05,870
on the same operation
will constrain

817
01:01:05,870 --> 01:01:09,350
some-- give some
constraint between the form

818
01:01:09,350 --> 01:01:11,420
of the functions that
will be describing

819
01:01:11,420 --> 01:01:13,940
the efficiencies
of Carnot engines.

820
01:01:13,940 --> 01:01:17,270
So let's follow
that mathematically.

821
01:01:17,270 --> 01:01:22,910
OK, so the first
one-- what do we have?

822
01:01:22,910 --> 01:01:24,940
By conservation
of energy, I have

823
01:01:24,940 --> 01:01:31,510
that Q2 is-- so this
is-- let me be precise.

824
01:01:31,510 --> 01:01:44,100
This is Carnot engine 1-- tells
me that Q2 is Q1 minus W12.

825
01:01:44,100 --> 01:01:46,240
All right, so how
much heat I have here

826
01:01:46,240 --> 01:01:48,680
is the difference between
this and the amount of work

827
01:01:48,680 --> 01:01:51,830
that I did--
conservation of energy.

828
01:01:51,830 --> 01:01:58,110
W12 is the efficiency times Q.
That was how it was defined.

829
01:01:58,110 --> 01:02:04,140
So it is 1 minus the efficiency
operating between T1 and T2

830
01:02:04,140 --> 01:02:05,870
times Q1.

831
01:02:09,000 --> 01:02:11,530
What do we know about
Carnot engine number 2?

832
01:02:14,260 --> 01:02:17,200
Carnot engine
number 2 says that I

833
01:02:17,200 --> 01:02:22,920
can look at Q3 as being
the difference between Q2

834
01:02:22,920 --> 01:02:26,340
minus the work that
is done over here.

835
01:02:26,340 --> 01:02:30,910
The work is related to the
heat through multiplying

836
01:02:30,910 --> 01:02:37,070
by the efficiency that is
operating between T2 and T3.

837
01:02:37,070 --> 01:02:41,690
And if I substitute for Q2
from the first equation, what

838
01:02:41,690 --> 01:02:42,250
do I get?

839
01:02:42,250 --> 01:02:47,120
I will get Q2 being Q1,
1 minus efficiency T1,

840
01:02:47,120 --> 01:02:50,970
T2, multiplying by the
second bracket that

841
01:02:50,970 --> 01:02:53,590
is 1 minus efficiency T2 T3.

842
01:03:02,300 --> 01:03:08,180
But I can also look at the
composite Carnot engine-- that

843
01:03:08,180 --> 01:03:12,330
is the sum total
of them represented

844
01:03:12,330 --> 01:03:15,030
by the diagram on the right.

845
01:03:15,030 --> 01:03:18,190
And what that states-- just
writing the whole thing

846
01:03:18,190 --> 01:03:26,270
same way-- is that this
Q3 is the same as Q1

847
01:03:26,270 --> 01:03:33,690
minus this amount of W13,
which is the same thing as Q1,

848
01:03:33,690 --> 01:03:37,528
1 minus the efficiency
between T1 and T3.

849
01:03:41,930 --> 01:03:46,170
So suddenly you see that
I have two equations that

850
01:03:46,170 --> 01:03:55,920
relate Q3 and Q1-- essentially
two formulations of this ratio.

851
01:03:55,920 --> 01:04:08,330
And what that says is that the
efficiency functions calculated

852
01:04:08,330 --> 01:04:11,930
between pairs of
temperatures, when

853
01:04:11,930 --> 01:04:16,480
we look at triplet of them,
have to be related by 1 minus

854
01:04:16,480 --> 01:04:23,200
efficiency between T1 and T3 is
1 minus efficiency between T1

855
01:04:23,200 --> 01:04:28,582
and T2, 1 minus efficiency
between T2 and T3.

856
01:04:32,200 --> 01:04:37,080
So when we are constructing
our temperature scale

857
01:04:37,080 --> 01:04:41,350
based on properties of
this efficiency function,

858
01:04:41,350 --> 01:04:44,210
we better choose and
efficiency function

859
01:04:44,210 --> 01:04:46,890
that satisfies this equality.

860
01:04:49,880 --> 01:04:54,580
And very roughly again, where
did this type of equality

861
01:04:54,580 --> 01:04:57,550
come from?

862
01:04:57,550 --> 01:05:05,940
It originated in the fact that
this 1 minus eta was the ratio

863
01:05:05,940 --> 01:05:08,810
of two Q's.

864
01:05:08,810 --> 01:05:12,250
This was the ratio
of Q2 over Q1,

865
01:05:12,250 --> 01:05:14,860
and then the other
one was Q1 over Q3,

866
01:05:14,860 --> 01:05:16,360
and you would
multiply them, and you

867
01:05:16,360 --> 01:05:19,520
would get Q2 over Q3, et cetera.

868
01:05:19,520 --> 01:05:22,790
And that gives you
a hint as to how

869
01:05:22,790 --> 01:05:25,290
you should construct
this function

870
01:05:25,290 --> 01:05:28,420
to satisfy the
property that you want.

871
01:05:28,420 --> 01:05:37,210
If I write the function as the
ratio of some function of T2

872
01:05:37,210 --> 01:05:40,580
divided by some
function of T1, it

873
01:05:40,580 --> 01:05:43,380
would cancel in
precisely the same way

874
01:05:43,380 --> 01:05:46,080
that when you were
multiplying Q2's and Q1's,

875
01:05:46,080 --> 01:05:48,330
the cancellations occur.

876
01:05:48,330 --> 01:05:49,580
OK?

877
01:05:49,580 --> 01:05:54,960
So what I need to
do is to postulate

878
01:05:54,960 --> 01:05:58,870
that this is of this form.

879
01:05:58,870 --> 01:06:02,970
And then actually, I'm
free to choose any function

880
01:06:02,970 --> 01:06:06,620
or form for F that I want.

881
01:06:06,620 --> 01:06:09,830
So here we need to
make a convention.

882
01:06:09,830 --> 01:06:12,790
And the convention
that we make--

883
01:06:12,790 --> 01:06:19,740
so this is-- this
was not required--

884
01:06:19,740 --> 01:06:22,200
is that this is really the
functional form is linear.

885
01:06:25,500 --> 01:06:29,090
So this defines for you the
thermodynamic temperature

886
01:06:29,090 --> 01:06:29,590
scale.

887
01:06:37,920 --> 01:06:39,390
Yes.

888
01:06:39,390 --> 01:06:41,890
AUDIENCE: Yes, the
part, the step right

889
01:06:41,890 --> 01:06:46,650
before the convention, where
you wrote F of T1 over T1,

890
01:06:46,650 --> 01:06:51,446
is this also an assumption,
or is it a necessity?

891
01:06:54,870 --> 01:07:00,080
PROFESSOR: I'm,
let's say, 99.9% sure

892
01:07:00,080 --> 01:07:02,790
that it's a
mathematical necessity.

893
01:07:02,790 --> 01:07:11,270
I can't-- at this stage, think
of precisely how to make that--

894
01:07:11,270 --> 01:07:13,810
actually, yes, I
can make that rigor.

895
01:07:13,810 --> 01:07:20,700
So essentially think of writing
this in the following way-- 1

896
01:07:20,700 --> 01:07:31,220
minus eta of T1 and T2 equals
to 1 minus eta of T1 and T3,

897
01:07:31,220 --> 01:07:35,940
1 minus eta of T2 and T3.

898
01:07:35,940 --> 01:07:36,932
OK?

899
01:07:36,932 --> 01:07:40,410
The left-hand side
depends on T1 and T2.

900
01:07:40,410 --> 01:07:44,810
The right-hand side is the
ratio of two functional forms

901
01:07:44,810 --> 01:07:49,080
involving T1 and T2 precisely
as you have over here,

902
01:07:49,080 --> 01:07:53,510
as long as you regard
T3 to be some arbitrary

903
01:07:53,510 --> 01:07:56,460
parameter or constant.

904
01:07:56,460 --> 01:07:57,300
OK?

905
01:07:57,300 --> 01:07:59,422
So that's the proof.

906
01:08:03,278 --> 01:08:07,640
All right, unfortunately,
it's like doing magic.

907
01:08:07,640 --> 01:08:12,269
Once you reveal the trick, it
becomes not so interesting.

908
01:08:12,269 --> 01:08:14,077
Yes.

909
01:08:14,077 --> 01:08:15,702
AUDIENCE: Does that
mean more precisely

910
01:08:15,702 --> 01:08:20,927
if we assumed that
T2 and T1 using F

911
01:08:20,927 --> 01:08:23,790
are empirical
temperatures, and we define

912
01:08:23,790 --> 01:08:28,616
the different temperature to
be F [INAUDIBLE] 2, something

913
01:08:28,616 --> 01:08:29,588
like that?

914
01:08:32,100 --> 01:08:37,540
PROFESSOR: OK, what I wanted to
say next maybe answers to that.

915
01:08:37,540 --> 01:08:39,859
So why don't I give that
answer first, and then

916
01:08:39,859 --> 01:08:41,210
come back to you?

917
01:08:41,210 --> 01:08:45,540
Is that so far, we have
defined two temperatures.

918
01:08:45,540 --> 01:08:48,420
There is the thermodynamic
temperature scale

919
01:08:48,420 --> 01:08:55,000
based on these efficiencies and
there is one other statement,

920
01:08:55,000 --> 01:08:59,000
which is that this only
defines the ratios.

921
01:08:59,000 --> 01:09:02,180
So I need to also pick
a reference temperature.

922
01:09:07,260 --> 01:09:10,649
And for reference temperature,
I pick the temperature

923
01:09:10,649 --> 01:09:16,229
of the coexistence
of ice, steam, water,

924
01:09:16,229 --> 01:09:20,710
to be 273.16 degrees
K, which is what

925
01:09:20,710 --> 01:09:23,300
I have for the ideal gas scale.

926
01:09:23,300 --> 01:09:25,340
Because if I don't
do that, I only

927
01:09:25,340 --> 01:09:29,279
know things up to some
proportionality constant.

928
01:09:29,279 --> 01:09:35,590
If I say that I pick that
reference temperature, then

929
01:09:35,590 --> 01:09:41,330
in principle what I could do
is, if somebody gives me a bath,

930
01:09:41,330 --> 01:09:45,620
I can run a Carnot
engine between that bath

931
01:09:45,620 --> 01:09:49,210
of unknown temperature
and this reference point,

932
01:09:49,210 --> 01:09:53,330
calculate the efficiency,
1 minus the efficiency,

933
01:09:53,330 --> 01:09:56,510
use this ratio--
273.16-- and I have

934
01:09:56,510 --> 01:09:59,590
the temperature
of my new object.

935
01:09:59,590 --> 01:10:02,470
So we've completed that
thermodynamic temperature

936
01:10:02,470 --> 01:10:03,460
scale.

937
01:10:03,460 --> 01:10:07,160
If I understood correctly
the question that was posed,

938
01:10:07,160 --> 01:10:12,260
is, well, what about the
empirical temperature?

939
01:10:12,260 --> 01:10:20,120
And I was going to answer
something different, which

940
01:10:20,120 --> 01:10:22,060
is not really what
you were asking.

941
01:10:22,060 --> 01:10:25,922
So maybe you can repeat
your question one more time.

942
01:10:25,922 --> 01:10:30,610
AUDIENCE: If the, all
of our computations

943
01:10:30,610 --> 01:10:35,295
where we assume that instead
of T's, there are some thetas.

944
01:10:35,295 --> 01:10:36,416
PROFESSOR: Right.

945
01:10:36,416 --> 01:10:41,960
AUDIENCE: And which means
that we use the [INAUDIBLE]

946
01:10:41,960 --> 01:10:45,710
at last we define the
thermodynamic temperature,

947
01:10:45,710 --> 01:10:47,502
it seems to have more precise.

948
01:10:47,502 --> 01:10:48,460
PROFESSOR: That's fine.

949
01:10:48,460 --> 01:10:53,400
So you're saying that what I
did with the Zeroth Law was

950
01:10:53,400 --> 01:10:58,350
to state that for somebody that
exists in thermal equilibrium

951
01:10:58,350 --> 01:11:02,600
with a bath, I can
characterize it with the theta.

952
01:11:02,600 --> 01:11:04,840
And I can go through
the whole argument

953
01:11:04,840 --> 01:11:07,450
that I had over here
by constructing,

954
01:11:07,450 --> 01:11:12,040
let's say, Carnot
engines that operate

955
01:11:12,040 --> 01:11:16,480
between empirical temperatures
defined as theta H and theta C.

956
01:11:16,480 --> 01:11:19,610
And nowhere in the
process have I given you

957
01:11:19,610 --> 01:11:23,850
a number for what theta H and
theta C is up to this point.

958
01:11:23,850 --> 01:11:27,680
And at this point, I
defined efficiency,

959
01:11:27,680 --> 01:11:30,040
which I have established
is only a function

960
01:11:30,040 --> 01:11:33,110
of the empirical temperature
operating between two

961
01:11:33,110 --> 01:11:37,380
isotherms, as some way
through this process

962
01:11:37,380 --> 01:11:40,660
and this definition, giving
an actual numerical value.

963
01:11:40,660 --> 01:11:42,408
So that's fine.

964
01:11:42,408 --> 01:11:43,356
Yes.

965
01:11:43,356 --> 01:11:47,622
AUDIENCE: So should you also
define temperature [INAUDIBLE]

966
01:11:47,622 --> 01:11:50,735
with ideal gas and
infinite expansion?

967
01:11:50,735 --> 01:11:51,360
PROFESSOR: Yes.

968
01:11:51,360 --> 01:11:56,340
AUDIENCE: And doesn't it define
what convention for effective

969
01:11:56,340 --> 01:11:58,252
we need to pick here?

970
01:12:01,190 --> 01:12:05,050
So for all the definitions to
be consistent with each other?

971
01:12:05,050 --> 01:12:07,600
PROFESSOR: OK, so now that
becomes a different question.

972
01:12:07,600 --> 01:12:09,880
So I think the
first question was,

973
01:12:09,880 --> 01:12:13,430
let's not define any temperature
scale up to this point.

974
01:12:13,430 --> 01:12:16,480
And this is the first time that
we define a temperature scale.

975
01:12:16,480 --> 01:12:17,980
Now, your question
is the one that I

976
01:12:17,980 --> 01:12:20,230
wanted to originally
answer, so thank you

977
01:12:20,230 --> 01:12:22,420
for the question, which
is that I actually

978
01:12:22,420 --> 01:12:26,210
did define for you a temperature
scale through some property

979
01:12:26,210 --> 01:12:27,670
of the ideal gas.

980
01:12:27,670 --> 01:12:31,080
And indeed, that
property of the ideal gas

981
01:12:31,080 --> 01:12:34,710
I implicitly used here in
the shape of the isotherms

982
01:12:34,710 --> 01:12:38,540
that I had and
constructing this energy

983
01:12:38,540 --> 01:12:41,430
functional in this fashion.

984
01:12:41,430 --> 01:12:44,880
Now you say well,
OK, you defined

985
01:12:44,880 --> 01:12:46,530
temperature two different ways.

986
01:12:46,530 --> 01:12:48,980
Are they consistent
with each other?

987
01:12:48,980 --> 01:12:51,520
And you will have
a problem set that

988
01:12:51,520 --> 01:12:54,160
will ask the following
question-- let's

989
01:12:54,160 --> 01:13:01,120
run an ideal gas Carnot cycle
between ideal gas temperatures

990
01:13:01,120 --> 01:13:05,540
theta H and theta C. OK?

991
01:13:05,540 --> 01:13:08,860
Now you can-- and I
have drawn for you

992
01:13:08,860 --> 01:13:11,810
the form of this Carnot cycle.

993
01:13:11,810 --> 01:13:16,420
Using PdV, you can calculate
what amount of work

994
01:13:16,420 --> 01:13:19,050
goes into this at
different stages.

995
01:13:19,050 --> 01:13:22,640
And the net work would be
the area of this cycle.

996
01:13:22,640 --> 01:13:26,180
So W would give you
the area of the cycle.

997
01:13:26,180 --> 01:13:29,810
You can calculate what QH is.

998
01:13:29,810 --> 01:13:31,940
And you can calculate
the efficiency

999
01:13:31,940 --> 01:13:38,810
of this as a function
theta H and theta C.

1000
01:13:38,810 --> 01:13:41,930
And what you will find, if
you do things correctly,

1001
01:13:41,930 --> 01:13:48,770
is that using all of these
definitions for the shapes

1002
01:13:48,770 --> 01:13:51,510
of these isotherms,
you can calculate

1003
01:13:51,510 --> 01:13:53,150
what the efficiency is.

1004
01:13:53,150 --> 01:13:54,910
And the answer for
your efficiency

1005
01:13:54,910 --> 01:13:57,370
will come out to be
theta H minus theta

1006
01:13:57,370 --> 01:14:01,500
C divided by theta H, which is
precisely what you would have

1007
01:14:01,500 --> 01:14:04,780
expected based on
the Carnot cycle.

1008
01:14:04,780 --> 01:14:07,860
So then you can
establish that the ratio

1009
01:14:07,860 --> 01:14:11,910
of theta H to theta C
defined through the ideal gas

1010
01:14:11,910 --> 01:14:16,160
temperature scale is the same
as the ratio of the temperatures

1011
01:14:16,160 --> 01:14:18,640
if you had defined
through the thermodynamic.

1012
01:14:18,640 --> 01:14:23,300
So the two scales, once
you set the same point

1013
01:14:23,300 --> 01:14:26,210
for both of them,
become identical.

1014
01:14:31,580 --> 01:14:34,540
Other questions?

1015
01:14:34,540 --> 01:14:42,690
OK, so we went
through all of this

1016
01:14:42,690 --> 01:14:48,970
in order to ultimately
get back to the story

1017
01:14:48,970 --> 01:14:53,320
that I was constructing at the
beginning of the lecture, which

1018
01:14:53,320 --> 01:15:01,950
was that I'd like to have the
form ultimately for my energy.

1019
01:15:01,950 --> 01:15:04,300
I want to construct the energy.

1020
01:15:04,300 --> 01:15:07,550
And if I was looking only
at mechanical systems,

1021
01:15:07,550 --> 01:15:09,920
and there was no
temperature in the world,

1022
01:15:09,920 --> 01:15:14,300
I would construct it
as sum over i Ji dxi.

1023
01:15:14,300 --> 01:15:16,550
But I know that
that's not enough,

1024
01:15:16,550 --> 01:15:19,120
because there are processes
by which I can change

1025
01:15:19,120 --> 01:15:23,650
the state by not doing any
mechanical work-- go from one

1026
01:15:23,650 --> 01:15:26,460
point in the diagram to
another point in the diagram.

1027
01:15:26,460 --> 01:15:29,860
So there must be something else.

1028
01:15:29,860 --> 01:15:33,600
Now we said that it kind
of makes a lot of sense

1029
01:15:33,600 --> 01:15:38,140
that something else should be
something like a temperature.

1030
01:15:38,140 --> 01:15:40,670
And before that,
really, the only thing

1031
01:15:40,670 --> 01:15:44,440
that we had that was
a form of temperature

1032
01:15:44,440 --> 01:15:48,040
was the ideal gas temperature,
or the empirical temperature

1033
01:15:48,040 --> 01:15:50,330
that we had defined
through the Zeroth Law.

1034
01:15:50,330 --> 01:15:55,710
And neither of them had any
nice connection to heat.

1035
01:15:55,710 --> 01:15:58,330
Through this Carnot
engines, et cetera you

1036
01:15:58,330 --> 01:16:02,400
have established some kind of
a connection between heats,

1037
01:16:02,400 --> 01:16:04,250
temperatures, et cetera.

1038
01:16:04,250 --> 01:16:06,305
And that's why this
thermodynamic temperature

1039
01:16:06,305 --> 01:16:08,210
scale is useful.

1040
01:16:08,210 --> 01:16:10,890
It is independent
of any material.

1041
01:16:10,890 --> 01:16:13,820
And it's really the T
that I should put here.

1042
01:16:13,820 --> 01:16:18,250
And so the next question
is, what should I put here?

1043
01:16:18,250 --> 01:16:22,970
And that will also come
through the Second Law.

1044
01:16:22,970 --> 01:16:27,100
I will develop that mostly
the next time around,

1045
01:16:27,100 --> 01:16:31,620
but I'll give you in the next
two or three minutes a preview.

1046
01:16:31,620 --> 01:16:36,850
So the statement that we
had was that the efficiency

1047
01:16:36,850 --> 01:16:40,190
of any engine is less than
the Carnot engine, right?

1048
01:16:40,190 --> 01:16:44,200
So let's imagine that we
have some kind of a exchange

1049
01:16:44,200 --> 01:16:48,070
process going between TH and TC.

1050
01:16:48,070 --> 01:16:51,330
And there is some
engine-- I don't

1051
01:16:51,330 --> 01:16:58,930
know what that engine is--
that takes heat QH here, QC

1052
01:16:58,930 --> 01:17:01,400
here, and does a
certain amount of work.

1053
01:17:04,590 --> 01:17:14,990
Now I know that W
over QH is less than

1054
01:17:14,990 --> 01:17:19,140
or equal to 1 minus TH over TC.

1055
01:17:22,960 --> 01:17:28,347
And W is really the same
thing as QH minus QC.

1056
01:17:32,010 --> 01:17:34,912
So actually, I could
also have written this

1057
01:17:34,912 --> 01:17:43,380
as 1 minus QC over QH is
less than 1 minus TC over TC.

1058
01:17:43,380 --> 01:17:50,826
I can eliminate the 1,
and rearrange things, yes?

1059
01:17:50,826 --> 01:17:52,820
AUDIENCE: It's TC over TH.

1060
01:17:52,820 --> 01:17:55,550
PROFESSOR: 1 minus TC over
TH, thank you very much.

1061
01:17:58,910 --> 01:18:05,890
Yes, and if you ever
forget-- not that I forgot--

1062
01:18:05,890 --> 01:18:11,440
I miswrote-- but something to
remember is that ultimately,

1063
01:18:11,440 --> 01:18:18,370
I was kind of hinting this--
if you ever forget Q's are

1064
01:18:18,370 --> 01:18:21,960
proportional to T's.

1065
01:18:21,960 --> 01:18:30,260
So what I have is that ideally,
QH will be proportional to TH.

1066
01:18:30,260 --> 01:18:32,940
QC will be proportional to TC.

1067
01:18:32,940 --> 01:18:34,710
So the difference
between them would

1068
01:18:34,710 --> 01:18:37,270
be proportional to
the differences.

1069
01:18:37,270 --> 01:18:41,110
Also, that means that if I were
to rearrange this slightly,

1070
01:18:41,110 --> 01:18:57,450
what I would get is that QH
over TH plus minus QC over TC

1071
01:18:57,450 --> 01:18:58,641
has to be negative.

1072
01:19:01,530 --> 01:19:05,710
So this is just a rearrangement
of the whole thing.

1073
01:19:05,710 --> 01:19:10,350
But this rearrangement--
I wrote this as minus

1074
01:19:10,350 --> 01:19:16,980
QC so as to look at things from
the perspective of the engine.

1075
01:19:16,980 --> 01:19:20,850
So the engine is something
that goes through a cycle.

1076
01:19:20,850 --> 01:19:24,920
As part of that cycle,
it does some work.

1077
01:19:24,920 --> 01:19:27,460
But what this
expression has to do

1078
01:19:27,460 --> 01:19:31,030
is the heat that
goes into the system.

1079
01:19:31,030 --> 01:19:34,770
So I can regard minus
QC as the heat that

1080
01:19:34,770 --> 01:19:40,220
goes into the engine coming from
a reservoir of temperature TC,

1081
01:19:40,220 --> 01:19:42,290
QH going through
the engine, coming

1082
01:19:42,290 --> 01:19:46,530
from a reservoir
of temperature TH.

1083
01:19:46,530 --> 01:19:49,500
And you have a
relation such as this

1084
01:19:49,500 --> 01:19:53,310
because the efficiency
of any type of engine

1085
01:19:53,310 --> 01:19:57,240
has to be less than this
efficiency of the Carnot engine

1086
01:19:57,240 --> 01:20:01,090
that is stamped by the
corresponding temperatures.

1087
01:20:01,090 --> 01:20:03,900
OK, now this is kind of limited.

1088
01:20:03,900 --> 01:20:08,550
And in order to be able to
construct a general formula,

1089
01:20:08,550 --> 01:20:12,140
we need to be able
to make statements

1090
01:20:12,140 --> 01:20:16,640
that's are relevant to
arbitrary complex cycles

1091
01:20:16,640 --> 01:20:20,440
and complex behaviors.

1092
01:20:20,440 --> 01:20:25,960
So what we will
start next time is

1093
01:20:25,960 --> 01:20:29,110
to prove something
that ultimately

1094
01:20:29,110 --> 01:20:31,880
will allow us to
quantify entropy.

1095
01:20:31,880 --> 01:20:38,650
It's called Clausius'
Theorem, which

1096
01:20:38,650 --> 01:20:44,310
states the following-- imagine
some kind of a generalization

1097
01:20:44,310 --> 01:20:49,845
of your engine that takes place
in some multidimensional space.

1098
01:20:49,845 --> 01:20:53,220
So rather than really
thinking about an engine,

1099
01:20:53,220 --> 01:20:56,730
I want to think
about some substance.

1100
01:20:56,730 --> 01:21:00,630
This was an ideal gas run
through some nice cycle.

1101
01:21:00,630 --> 01:21:03,250
But I want to think
about some substance that

1102
01:21:03,250 --> 01:21:07,140
is described by multiple
coordinate system.

1103
01:21:07,140 --> 01:21:11,768
I take the system
from some point A,

1104
01:21:11,768 --> 01:21:15,440
and the only requirement
that I place on it

1105
01:21:15,440 --> 01:21:18,590
is that ultimately,
I do something,

1106
01:21:18,590 --> 01:21:23,070
and I come back to
A. So it is a cycle.

1107
01:21:23,070 --> 01:21:26,340
What I don't even
require is that this

1108
01:21:26,340 --> 01:21:31,590
is a quasistatic process, so
that the intermediate stages

1109
01:21:31,590 --> 01:21:35,820
are well-defined in
this coordinate space--

1110
01:21:35,820 --> 01:21:42,560
OK, so for any arbitrary
cyclic transformation.

1111
01:21:42,560 --> 01:21:47,110
OK, now this
transformation-- we'll do

1112
01:21:47,110 --> 01:21:51,770
is that at various stages--
just like that simple example--

1113
01:21:51,770 --> 01:21:53,500
we'll do work on
the environment,

1114
01:21:53,500 --> 01:21:55,370
take work from the environment.

1115
01:21:55,370 --> 01:21:58,330
But most importantly
from our perspective,

1116
01:21:58,330 --> 01:22:03,300
there will be heat
input into the system

1117
01:22:03,300 --> 01:22:07,290
at various stages of the cycle.

1118
01:22:07,290 --> 01:22:11,410
And I'm going to look at it
from the perspective of what

1119
01:22:11,410 --> 01:22:14,850
goes into the system,
just like in this engine.

1120
01:22:14,850 --> 01:22:18,320
So sometimes maybe this
dQ is negative-- means

1121
01:22:18,320 --> 01:22:21,250
that the system is
really releasing heat,

1122
01:22:21,250 --> 01:22:25,360
just like this engine
was releasing the heat.

1123
01:22:25,360 --> 01:22:29,690
Now, Clausius' Theorem
says that if you integrate

1124
01:22:29,690 --> 01:22:34,120
all around the cycle,
these elements of heat

1125
01:22:34,120 --> 01:22:39,430
that you input to the system
throughout various stages

1126
01:22:39,430 --> 01:22:44,980
of the cyclic transformation,
and divide them by some T.

1127
01:22:44,980 --> 01:22:51,300
And this is kind of something
that needs definition.

1128
01:22:51,300 --> 01:22:54,200
And S is supposed to
parametrize the cycle.

1129
01:22:54,200 --> 01:22:58,580
Let's say S goes from 0 to 1
as you go across the cycle.

1130
01:22:58,580 --> 01:23:00,790
Generalizing that is negative.

1131
01:23:04,330 --> 01:23:08,090
Next time, we will see if this
system is non-equilibrium,

1132
01:23:08,090 --> 01:23:11,850
what exactly I mean
by this T of S.

1133
01:23:11,850 --> 01:23:16,690
And we will see, somehow,
that once I define this,

1134
01:23:16,690 --> 01:23:19,470
by doing part of
this, I can actually

1135
01:23:19,470 --> 01:23:24,610
define an entropy function
integral from A to B dQ over S,

1136
01:23:24,610 --> 01:23:30,020
and complete the definition of
what needs to be put for the Q

1137
01:23:30,020 --> 01:23:36,450
in order to give you an
expression for T. OK.