1
00:00:00,000 --> 00:00:00,610


2
00:00:00,610 --> 00:00:03,280
Let's say we have
two reservoirs.

3
00:00:03,280 --> 00:00:08,250
Let's say up here, I have
my hot reservoir.

4
00:00:08,250 --> 00:00:10,720
It's at th for hot.

5
00:00:10,720 --> 00:00:13,210
And I have my engine.

6
00:00:13,210 --> 00:00:16,940
It's going to be a Carnot
engine, because we'll learn

7
00:00:16,940 --> 00:00:19,140
that no engine is better,
at least from an

8
00:00:19,140 --> 00:00:20,310
efficiency point of view.

9
00:00:20,310 --> 00:00:22,130
We have to be careful when
we say "better."

10
00:00:22,130 --> 00:00:25,870
We have our Carnot engine,
and it operates on heat

11
00:00:25,870 --> 00:00:26,850
differentials.

12
00:00:26,850 --> 00:00:30,830
So it takes some heat from
our hot heat source.

13
00:00:30,830 --> 00:00:33,045
It takes some heat there.

14
00:00:33,045 --> 00:00:34,825
Let's call that q1.

15
00:00:34,825 --> 00:00:38,080
And it does some work.

16
00:00:38,080 --> 00:00:41,220
Work is a good thing, so I'll
make that in green.

17
00:00:41,220 --> 00:00:48,855
It does some work, and then
the surplus energy, the

18
00:00:48,855 --> 00:00:53,400
surplus heat, q2, then goes
to our cold reservoir.

19
00:00:53,400 --> 00:00:56,970
I'll do that right there.

20
00:00:56,970 --> 00:00:59,020
t cold.

21
00:00:59,020 --> 00:01:02,350
Now, I made multiple
insinuations in the previous

22
00:01:02,350 --> 00:01:07,800
video that this is the most
efficient engine that can be

23
00:01:07,800 --> 00:01:14,230
created between two reservoirs,
th and tc.

24
00:01:14,230 --> 00:01:16,540
Now, you come along and say,
no, no, no, no, no.

25
00:01:16,540 --> 00:01:21,360
I know of a friend, he has
invented a new engine that is

26
00:01:21,360 --> 00:01:24,710
more efficient than this engine
between these same two

27
00:01:24,710 --> 00:01:25,400
reservoirs.

28
00:01:25,400 --> 00:01:29,350
And you go, and you proceed to
draw this same type of diagram

29
00:01:29,350 --> 00:01:32,140
for your friend's engine.

30
00:01:32,140 --> 00:01:34,730
You say, look-- let me make it
clear-- this is the same

31
00:01:34,730 --> 00:01:38,850
reservoir, these same reservoirs
we're dealing with.

32
00:01:38,850 --> 00:01:40,750
Actually, I should probably draw
this line all the way,

33
00:01:40,750 --> 00:01:43,250
because I'm going to have to
do multiple engines here.

34
00:01:43,250 --> 00:01:45,930
So the same reservoir we're
dealing with, right?

35
00:01:45,930 --> 00:01:49,910
This is all the hot reservoir,
th th, and this is all the

36
00:01:49,910 --> 00:01:50,830
cold reservoir.

37
00:01:50,830 --> 00:01:52,600
I need space for multiple
engines that we're

38
00:01:52,600 --> 00:01:53,530
going to deal with.

39
00:01:53,530 --> 00:01:56,210
So your friend has an engine.

40
00:01:56,210 --> 00:01:57,460
We'll call it the
super engine.

41
00:01:57,460 --> 00:02:01,240


42
00:02:01,240 --> 00:02:06,430
And your friend's claim-- and
I'll show you why your

43
00:02:06,430 --> 00:02:09,054
friend's claim cannot be true,
if you believe the second law

44
00:02:09,054 --> 00:02:13,140
of thermodynamics-- so your
friend's claim, they have the

45
00:02:13,140 --> 00:02:14,210
super engine.

46
00:02:14,210 --> 00:02:17,760
And they claim that look, I
can actually take in q1.

47
00:02:17,760 --> 00:02:22,270
I can take in that same heat
from this heat source up here.

48
00:02:22,270 --> 00:02:27,300
But I can produce more work
than your Carnot engine.

49
00:02:27,300 --> 00:02:31,460
I could produce 1 plus x.

50
00:02:31,460 --> 00:02:32,520
I don't want to get
too algebraic.

51
00:02:32,520 --> 00:02:35,520
But let's say, you produce w.

52
00:02:35,520 --> 00:02:40,080
I produce w times 1
plus x of work.

53
00:02:40,080 --> 00:02:41,530
Where x is a positive number.

54
00:02:41,530 --> 00:02:43,490
So he's saying, look,
x is greater than 0.

55
00:02:43,490 --> 00:02:46,190
Whatever that number he
might feed you is.

56
00:02:46,190 --> 00:02:48,000
And then the rest of
the energy that's

57
00:02:48,000 --> 00:02:52,660
left over is what?

58
00:02:52,660 --> 00:02:55,150
It's q1 minus this.

59
00:02:55,150 --> 00:03:01,480
So it's q1 minus w
times 1 plus x.

60
00:03:01,480 --> 00:03:04,930
And just to be clear, q2 right
here, I could rewrite that as

61
00:03:04,930 --> 00:03:07,430
q1 minus w.

62
00:03:07,430 --> 00:03:08,000
Fair enough.

63
00:03:08,000 --> 00:03:09,030
So you look at that.

64
00:03:09,030 --> 00:03:11,320
You come to me with this, and
I say, no, no, no, no.

65
00:03:11,320 --> 00:03:12,750
This cannot be true.

66
00:03:12,750 --> 00:03:16,300
Because if this were, then we
would solve literally all of

67
00:03:16,300 --> 00:03:18,950
the world's energy problems. And
I'm about to show you why

68
00:03:18,950 --> 00:03:21,490
we could solve all of the
world's energy problems, and

69
00:03:21,490 --> 00:03:24,240
we would have a perpetual motion
machine, and be able to

70
00:03:24,240 --> 00:03:26,610
defy all sorts of things
if we had this.

71
00:03:26,610 --> 00:03:28,560
Now, this is my Carnot engine.

72
00:03:28,560 --> 00:03:31,600
But I could devise a reverse
Carnot engine, right?

73
00:03:31,600 --> 00:03:34,800
Let me make a reverse
Carnot engine.

74
00:03:34,800 --> 00:03:38,230
So my reverse Carnot engine
would look like this.

75
00:03:38,230 --> 00:03:43,180


76
00:03:43,180 --> 00:03:46,930
And it's going to do the same
thing, but in reverse.

77
00:03:46,930 --> 00:03:55,070
So instead of producing q1 minus
w here and putting it

78
00:03:55,070 --> 00:04:10,420
into tc, it could take in q1
minus w from our cold source--

79
00:04:10,420 --> 00:04:13,520
so could take that in-- or even
better, let's scale it up

80
00:04:13,520 --> 00:04:14,270
a little bit.

81
00:04:14,270 --> 00:04:20,430
Let's say it takes in q1
minus w times 1 plus x.

82
00:04:20,430 --> 00:04:23,660
So I've just made a slightly
larger reversed Carnot engine.

83
00:04:23,660 --> 00:04:28,580
Now, if I take in that much--
in order to do this in

84
00:04:28,580 --> 00:04:30,180
reverse, I'm going to have to
take in, I'm going to have to

85
00:04:30,180 --> 00:04:32,780
scale up this Carnot engine
and reverse everything.

86
00:04:32,780 --> 00:04:35,860
So instead of producing work,
I'm now going to need work to

87
00:04:35,860 --> 00:04:39,010
go on the other direction, and
I've scaled it up by 1 plus x.

88
00:04:39,010 --> 00:04:43,450
So I'm going to need the
amount of work here

89
00:04:43,450 --> 00:04:44,820
times 1 plus x.

90
00:04:44,820 --> 00:04:47,810


91
00:04:47,810 --> 00:04:53,360
And then I'm going to push q1,
but I've scaled it up.

92
00:04:53,360 --> 00:04:57,180
I'm going to push in
q1 times 1 plus x

93
00:04:57,180 --> 00:04:58,790
into my hot heat source.

94
00:04:58,790 --> 00:05:00,490
And once again, this isn't
defying the laws of

95
00:05:00,490 --> 00:05:01,110
thermodynamics.

96
00:05:01,110 --> 00:05:03,410
I'm taking up some work.

97
00:05:03,410 --> 00:05:06,500
There's work that needs to be
done in order to do this.

98
00:05:06,500 --> 00:05:10,800
But all of a sudden, you come
to me and say, look, this is

99
00:05:10,800 --> 00:05:11,380
an awesome deal.

100
00:05:11,380 --> 00:05:13,080
You have this nice engine
that works this way.

101
00:05:13,080 --> 00:05:14,980
My friend has a super engine.

102
00:05:14,980 --> 00:05:16,780
Let's just couple
them together.

103
00:05:16,780 --> 00:05:21,450
Let's take the work that he
produces right here-- he

104
00:05:21,450 --> 00:05:24,580
produces w times 1 plus x, and
that just happens to be the

105
00:05:24,580 --> 00:05:27,410
amount of work that you need
to operate your engine.

106
00:05:27,410 --> 00:05:29,310
So you just feed that
into there.

107
00:05:29,310 --> 00:05:31,630
So what's the net effect
of these two engines?

108
00:05:31,630 --> 00:05:34,420


109
00:05:34,420 --> 00:05:39,750
So let me do another, scroll
a little bit more.

110
00:05:39,750 --> 00:05:42,190
Actually, that might be
the best way to do it.

111
00:05:42,190 --> 00:05:45,580
So let me make sure that we
understand that these are the

112
00:05:45,580 --> 00:05:49,510
same heat sinks or heat sources
that we're using the

113
00:05:49,510 --> 00:05:49,960
whole time.

114
00:05:49,960 --> 00:05:55,680
So that's my hot source, my
cold source is down here.

115
00:05:55,680 --> 00:05:59,400
So if I add our two engines
together-- so if I have a, you

116
00:05:59,400 --> 00:06:02,210
know, let me call it a-- I'm
going to pick a new color.

117
00:06:02,210 --> 00:06:05,320
These colors are getting
monotonous.

118
00:06:05,320 --> 00:06:06,160
Nope.

119
00:06:06,160 --> 00:06:08,760
I wanted to do the
rectangle tool.

120
00:06:08,760 --> 00:06:09,950
There you go.

121
00:06:09,950 --> 00:06:10,780
All right.

122
00:06:10,780 --> 00:06:12,880
So I combine these two
engines together.

123
00:06:12,880 --> 00:06:14,320
Essentially I just put a
big box around them.

124
00:06:14,320 --> 00:06:18,060
They're both operating between
these two heat sources.

125
00:06:18,060 --> 00:06:19,160
These two reservoirs.

126
00:06:19,160 --> 00:06:22,730
So I call this the, you know,
your super engine plus my

127
00:06:22,730 --> 00:06:25,300
reverse Carnot engine.

128
00:06:25,300 --> 00:06:26,990
So what's happening now?

129
00:06:26,990 --> 00:06:32,685
What's the net heat that's
being taken in

130
00:06:32,685 --> 00:06:33,570
or put out of here?

131
00:06:33,570 --> 00:06:38,710
So we have q1-- we have--
let me see.

132
00:06:38,710 --> 00:06:43,670
We have q1 minus w 1 plus x, but
in this direction, we have

133
00:06:43,670 --> 00:06:47,340
q1-- so in this direction,
we could rewrite this.

134
00:06:47,340 --> 00:06:48,620
I want to make sure you're
clear on the algebra.

135
00:06:48,620 --> 00:06:50,490
This could be rewritten
as what?

136
00:06:50,490 --> 00:06:58,360
As q1 times 1 plus x times, or
minus, w times 1 place x.

137
00:06:58,360 --> 00:06:59,090
Right?

138
00:06:59,090 --> 00:07:03,530
Now, if you compare these
terms, this is the

139
00:07:03,530 --> 00:07:05,340
same as this term.

140
00:07:05,340 --> 00:07:07,770
This term is bigger
than this term.

141
00:07:07,770 --> 00:07:08,120
Right?

142
00:07:08,120 --> 00:07:10,060
This term is clearly bigger,
because we're multiplying it

143
00:07:10,060 --> 00:07:11,180
by something larger than 1.

144
00:07:11,180 --> 00:07:12,630
It's bigger than this storm.

145
00:07:12,630 --> 00:07:17,640
So if we combine these two, the
upward movement, or the

146
00:07:17,640 --> 00:07:20,120
amount of heat I'm taking up
from my reverse Carnot, is

147
00:07:20,120 --> 00:07:23,530
going to be greater than the
amount of heat being put in by

148
00:07:23,530 --> 00:07:25,670
your friend's super engine.

149
00:07:25,670 --> 00:07:26,900
And we can actually calculate
the amount.

150
00:07:26,900 --> 00:07:29,440
We can just take this amount
minus that amount, and that's

151
00:07:29,440 --> 00:07:31,640
the net upward movement.

152
00:07:31,640 --> 00:07:36,040
So the net upward movement
from our cold

153
00:07:36,040 --> 00:07:38,470
reservoir is what?

154
00:07:38,470 --> 00:07:40,500
It's this value minus
this value.

155
00:07:40,500 --> 00:07:47,450
So minus q1 minus w 1 plus x.

156
00:07:47,450 --> 00:07:48,973
If we take a minus, we're
going to subtract it.

157
00:07:48,973 --> 00:07:50,800
So it's a minus and a plus.

158
00:07:50,800 --> 00:07:52,450
These cancel out.

159
00:07:52,450 --> 00:07:55,610
This minus cancels out with--
so this first term could be

160
00:07:55,610 --> 00:08:00,010
rewritten as q1 plus q1x.

161
00:08:00,010 --> 00:08:00,830
Right?

162
00:08:00,830 --> 00:08:01,910
We could rewrite it that way.

163
00:08:01,910 --> 00:08:03,400
So this cancels out with that.

164
00:08:03,400 --> 00:08:06,050
And so the net upward movement
when we combine the two

165
00:08:06,050 --> 00:08:11,270
engines is q1 times x.

166
00:08:11,270 --> 00:08:12,440
Fair enough.

167
00:08:12,440 --> 00:08:13,640
Now what about the
work transfer?

168
00:08:13,640 --> 00:08:16,380
Well, whatever work this guy
produces is exactly the amount

169
00:08:16,380 --> 00:08:17,090
of work that I need.

170
00:08:17,090 --> 00:08:20,540
So no outside work has to
be done on the system.

171
00:08:20,540 --> 00:08:21,650
It just works.

172
00:08:21,650 --> 00:08:23,960
This guy produces work, this
guy uses the work.

173
00:08:23,960 --> 00:08:25,730
Now what's in that
heat transferred

174
00:08:25,730 --> 00:08:28,175
up to our hot reservoir?

175
00:08:28,175 --> 00:08:30,690


176
00:08:30,690 --> 00:08:33,140
What's the amount of heat?

177
00:08:33,140 --> 00:08:35,090
Well, it's the difference
between these two.

178
00:08:35,090 --> 00:08:38,419
And this is clearly a larger
number than this one, so the

179
00:08:38,419 --> 00:08:40,230
upward movement dominates.

180
00:08:40,230 --> 00:08:42,299
So what's this minus that?

181
00:08:42,299 --> 00:08:48,750
So this can be rewritten
as q1 plus q1x, right?

182
00:08:48,750 --> 00:08:50,380
I just distributed the q1.

183
00:08:50,380 --> 00:08:51,580
We're going to subtract
that out.

184
00:08:51,580 --> 00:08:53,400
Minus q1.

185
00:08:53,400 --> 00:08:54,330
You're left with q1x.

186
00:08:54,330 --> 00:08:56,090
So the net movement,
when we combine the

187
00:08:56,090 --> 00:08:59,700
two engines, is q1x.

188
00:08:59,700 --> 00:09:01,580
So what's happening here?

189
00:09:01,580 --> 00:09:05,770
I have no external energy or
work has to be expended into

190
00:09:05,770 --> 00:09:06,795
this system.

191
00:09:06,795 --> 00:09:11,890
And it's just taking heat from
a cold body, and it's moving

192
00:09:11,890 --> 00:09:13,970
it to a warm body.

193
00:09:13,970 --> 00:09:15,140
And it does this indefinitely.

194
00:09:15,140 --> 00:09:16,260
It'll do this as much
as I want to.

195
00:09:16,260 --> 00:09:17,290
I can just build a bigger one.

196
00:09:17,290 --> 00:09:19,840
It'll do it on even a larger
and larger scale.

197
00:09:19,840 --> 00:09:24,520
So if you think about it, I
could heat my house with ice

198
00:09:24,520 --> 00:09:26,090
by just making the ice colder.

199
00:09:26,090 --> 00:09:32,270
I could create steam from things
that arbitrarily cold.

200
00:09:32,270 --> 00:09:36,570
This goes against the second
law of thermodynamics.

201
00:09:36,570 --> 00:09:39,770
The net entropy in this
world is going down.

202
00:09:39,770 --> 00:09:41,620
Because what's happening here?

203
00:09:41,620 --> 00:09:45,700
This is just a straight up
transfer of q1x from a cold

204
00:09:45,700 --> 00:09:47,230
body to a hot body.

205
00:09:47,230 --> 00:09:50,340
So what's the net change
in entropy here?

206
00:09:50,340 --> 00:09:53,060
The change in entropy
of the universe.

207
00:09:53,060 --> 00:10:02,430
Well, the hot body is gaining
some heat, so it's q1x over

208
00:10:02,430 --> 00:10:05,135
the temperature of the hot body,
and then the cold body

209
00:10:05,135 --> 00:10:06,430
is losing the same amount.

210
00:10:06,430 --> 00:10:11,450
So it's minus q1x over
the cold body.

211
00:10:11,450 --> 00:10:17,110
Now, this is a bigger number
than this is, right?

212
00:10:17,110 --> 00:10:19,150
Because the denominator
is smaller.

213
00:10:19,150 --> 00:10:20,130
This is a cold body.

214
00:10:20,130 --> 00:10:22,490
Its temperature in Kelvin will
be a smaller number.

215
00:10:22,490 --> 00:10:26,355
So this is going to be less than
0, which the second law

216
00:10:26,355 --> 00:10:28,240
of thermodynamics tells
us cannot be.

217
00:10:28,240 --> 00:10:30,610
The entropy cannot shrink
in the universe.

218
00:10:30,610 --> 00:10:33,140
This whole thing is an
independent system, and the

219
00:10:33,140 --> 00:10:34,180
entropy is shrinking.

220
00:10:34,180 --> 00:10:36,900
And we can make the entropy
shrink arbitrarily if we just

221
00:10:36,900 --> 00:10:38,480
scale up our x's enough.

222
00:10:38,480 --> 00:10:42,670
So this is why the Carnot engine
is the most efficient

223
00:10:42,670 --> 00:10:43,500
engine possible.

224
00:10:43,500 --> 00:10:46,690
Because if anyone claimed to
have a more efficient engine,

225
00:10:46,690 --> 00:10:49,650
you could couple it with a
reverse Carnot engine, and

226
00:10:49,650 --> 00:10:53,110
then create this perpetual
reverse-- I guess you could

227
00:10:53,110 --> 00:10:55,450
call it a perpetual
refrigeration machine that

228
00:10:55,450 --> 00:10:59,610
just out of the blue creates
anti-entropy from anywhere,

229
00:10:59,610 --> 00:11:01,970
and it would be this perpetual
energy source that creates

230
00:11:01,970 --> 00:11:03,460
energy out of nothing.

231
00:11:03,460 --> 00:11:06,390
And so this is just something
that cannot be done in our

232
00:11:06,390 --> 00:11:08,830
world, especially if you believe
the second law of

233
00:11:08,830 --> 00:11:10,060
thermodynamics.

234
00:11:10,060 --> 00:11:13,620
So the most efficient engine
is the Carnot engine, where

235
00:11:13,620 --> 00:11:17,810
its efficiency is described as 1
minus the temperature of the

236
00:11:17,810 --> 00:11:20,380
cold body divided
by the hot body.

237
00:11:20,380 --> 00:11:22,920
So if I have two temperature
reservoirs, let's say that my

238
00:11:22,920 --> 00:11:30,590
hot one is at 500 Kelvin, and my
cold one is at 300 Kelvin,

239
00:11:30,590 --> 00:11:33,910
and I have some engine that
takes heat from there, and

240
00:11:33,910 --> 00:11:35,690
transfers it there, and
does some work.

241
00:11:35,690 --> 00:11:38,350
The most efficient engine, if
I were to remove all the

242
00:11:38,350 --> 00:11:42,030
friction in the engine, the
highest efficiency I could get

243
00:11:42,030 --> 00:11:47,390
would be 1 minus 300 Kelvin over
500 Kelvin, which is 1

244
00:11:47,390 --> 00:11:53,360
minus 3/5, which is 2/5, which
is equal to 0.4, which is

245
00:11:53,360 --> 00:11:56,650
equal to 40%.

246
00:11:56,650 --> 00:11:59,720
So if someone tells you that
they made an engine that

247
00:11:59,720 --> 00:12:04,570
operates between a reservoir
that's 500 Kelvin and 300

248
00:12:04,570 --> 00:12:07,090
Kelvin, and they say, oh, I've
achieved 41% percent

249
00:12:07,090 --> 00:12:07,520
efficiency.

250
00:12:07,520 --> 00:12:09,460
I've really polished
the thing well.

251
00:12:09,460 --> 00:12:11,250
You know that they are lying.

252
00:12:11,250 --> 00:12:14,150
So anyway, hopefully you found
that reasonably interesting,

253
00:12:14,150 --> 00:00:00,000
and I'll see you in
the next video.