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In the last video, I showed
you that the definition of

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efficiency, eta, is the work
that we do given the amount of

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heat we are given
to work with.

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And we showed that for an
engine, that could also be

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rewritten as 1 minus Q2 over
Q1, or essentially, 1 minus

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the heat we output from our
engine divided by the amount

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of heat we input from
our engine.

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Now we applied this formula to
a Carnot cycle, and we said,

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hey, for a Carnot engine,
we could get an

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efficiency of this.

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So let me write this here.

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So the efficiency for Carnot,
eta for Carnot, is 1

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minus T2 over T1.

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To get this result, we had to
use the fact that we were

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dealing with the Carnot cycle
to use, you know, we were

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moving along these isotherms,
and so I was able to take the

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natural log of them and do all
of that, and I was able to get

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this for the efficiency
of a Carnot engine.

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And let me be very clear.

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This is the efficiency that
can only be attained by a

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Carnot engine.

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The other definitions of
efficiency-- so when I just

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defined efficiency as equal to
the work performed divided by

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the heat, let me call it the
heat input-- or when I defined

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it as the net heat in.

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So Q1 minus Q2 over Q1.

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This applies to all
heat engines.

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This is true for all
heat engines,

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including the Carnot engine.

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A heat engine is an engine
that operates on heat.

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I probably should have said
that a while ago.

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And this engine that I made,
this Carnot engine, is

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definitely a engine that is
operating on heat, because

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it's taking heat here,
and later it releases

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the heat down here.

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The cycle just shows what's
happening to that engine.

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And I just want to make
that distinction, too.

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The engine is the actual
physical thing.

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The cycle just describes
what's happening to it.

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So with that said, I said that
this is only true for a Carnot

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heat engine.

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Now, what I'm about to embark
on-- and I don't know if I'm

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going to finish it
in this video.

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It might take into the next
video to do it properly-- is

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to show you that if we're
operating a heat engine

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between two temperature
sources-- so I have my hot

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temperature source, I'll call
that TH for T hot, and it's

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transferring some heat, Q1, and
some other heat is coming

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out at Q2, and I'm performing
some work, and then my other

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cold temperature reservoir,
I'll call that T

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cold, is down here.

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And that's where I'm releasing
the heat, too.

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I'm going to show over the next
few videos that the most

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efficient engine is this
theoretical Carnot engine.

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That no engine can get more
efficient than this.

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So if this is a Carnot
engine, this is the

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most efficient engine.

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Or this is the ideal, where
nothing is lost. Well, I'll go

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into that in more detail.

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No engine can get more efficient
than this Carnot

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heat engine.

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So to get there, to prove it to
you, I'm just going to play

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with the Carnot engine a little
bit, just to show you

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some of the tools that it
has at its disposal.

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So one of the things-- let me
just draw a PV diagram.

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In the Carnot cycle we've done
so far, we've kind of always

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moved in one direction.

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We had our isothermal
expansion.

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It went something like that.

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That was isothermal.

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Then we had our adiabatic
expansion-- and the whole time

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we were going in that
direction-- and

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it went like that.

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Then we had our isothermal
contraction.

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It went something like this.

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And then we had our adiabatic
contraction, to get to where

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we were to begin with.

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So then we went back
like that.

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And the whole time, we
went in this kind

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of clockwise direction.

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We went in the clockwise
direction, and we took in heat

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up here-- because we were doing
work-- we took in heat

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to keep our temperature
constant, and then we released

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heat here to keep our
temperature from

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going up from Q2.

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And so if I were to draw this
another way-- well, I just did

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one like that, but let
me draw it like this.

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I could also depict it like
this, where that's my engine,

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this is my hot reservoir--
let me put this as

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T1-- T1 is up here.

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It transferred Q1 to
my Carnot engine.

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My Carnot engine did some work,
and then left over, it

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transferred into my cold
reservoir, T2,

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it transferred Q2.

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This is another way of depicting
what went on in this

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Carnot cycle.

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And here I've actually
drawn the engine.

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Now, one of the tools I want to
show you is that this is a

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reversible reaction.

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Or that we can take this and
go the other way around.

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And it's dependent upon an
assumption that I threw out a

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long time ago.

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So when I first drew these, I
kind of introduced you to the

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idea of a quasistatic process.

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And quasistatic just means,
look, you do it really slowly,

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so that you can always say that
you're close enough to

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equilibrium that your
macro state

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variables are always defined.

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And that was the whole
justification for dealing with

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pebbles like this.

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Instead of just doing it
wholesale, instead of just

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moving all the pebbles, and just
getting to this state,

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from A to B kind of jumping, I
wanted to do it gradually, so

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that I would be defined at
every point in between.

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That's what quasistatic
did for us.

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And when I actually made the
video quasistatic processes, I

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said, you know, quasistatic
processes, for the most part,

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are reversible.

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And sometimes I used the
words interchangeably.

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Now, by definition, our
theoretical Carnot cycle is

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said to be, not only is it
quasistatic, but it is also

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reversible.

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Which means at any point in
time-- let's say we've moved a

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couple of pebbles, and we've
gotten right here.

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If we want to, if we're in the
mood, we can add some pebbles

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back, and just follow this right
back to where we were.

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That's what reversible means.

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It means you can reverse
something.

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Now, what has to be ideal about
the system in order for

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that to be true?

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Well, it means that the actual
movement of our piston, of

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this movable ceiling, that it
shouldn't have any friction.

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Because if some of the heat is
lost to friction, then when we

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go back, we would have lost
some of our heat.

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Some heat would have been
destroyed, just going from one

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state forward and back.

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So the assumption that we have
to make in order for the

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Carnot cycle to be reversible
is that it's frictionless.

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So the Carnot heat engine, this
theoretical engine, is a

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frictionless engine, which is
theoretically impossible.

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To be completely frictionless.

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To be-- but we'll talk more
about that in the future.

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So if you have a completely
frictionless engine, and it's

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quasistatic, it's
also reversible.

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So if we want to do
it reversible,

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what does that mean?

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It means I could start in this
state, my state A that I've

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labelled before, but instead
of going around that way, I

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could go around the other way.

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So what I could do first, is I
could adiabatically expand

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first-- so maybe let me redraw
it, so I do it the other way.

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So I could reverse
this reaction.

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And it would happen the
exact same way.

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And that's an artifact of that
I'm always in equilibrium, and

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that my system is frictionless,
that I don't

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lose energy just going
back and forth.

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So I could start at state
A here, and then I could

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adiabatically contract.

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Adiabatic contraction would look
something like this, and

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it'll get to that state.

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Then I can isothermically
expand.

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So I'm going like this.

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And as I isothermically expand--
so I'm going like

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this, I'm all along an
isotherm-- I'd doing some

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isothermic expansion-- so in
this case, I'm doing work, but

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I'm doing work isothermically,
right?

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At some cold isotherm.

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Let's call it T2, right?

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Just like this was T2.

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So in this case, if I'm
expanding, and I'm staying at

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T2, and I'm sitting
on top of my T2

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reservoir, heat is coming.

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This area under the curve,
the work I'm

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doing, is the heat added.

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This is Q2, and that is given
to me by my T2 reservoir.

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So everything is going
in reverse.

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That's the whole idea.

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Then I adiabatically contract,
like that, and then I

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isothermically contract,
like that, to get

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back where I started.

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When I isothermically contract,
what's happening?

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Work is being done to me, so now
all of this area over here

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will be negative.

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And in order to keep my
temperature constant, I have

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to release heat.

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So I'm releasing heat, but
I'm doing it at a high

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temperature.

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So I'm releasing it into
my T1 reservoir.

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So it's the exact same thing
as it happened before, but

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since when I go in a reverse
direction, some

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work is being applied.

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So now, when you look at it this
way, when you figure out

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all of the areas,
the area in here

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will actually be negative.

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And the reason why I'm saying
that is because the positive

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work values are going
to be this.

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This is going to be the
positive, what I'm doing in

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blue right here.

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And the negative work
values are going to

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be all of this stuff.

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So if you wanted to figure out
the total work done, it's

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going to be negative.

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So what's happening, if I run
the Carnot cycle in reverse--

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so I'll call it the Carnot
refrigerator.

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No, that's not what
I wanted to do.

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I'll call it Carnot reverse.

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But it's handy that R also
stands for refrigerator.

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This is the Carnot engine.

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It does work by using heat, by
taking advantage of the heat

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difference between this hot--
you could view this as the T

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hot and the T cold.

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Now, a reverse Carnot engine, or
maybe you call it a Carnot

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refrigerator, does
the opposite.

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That's exactly what I
just drew over here.

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What it does is, it starts with
a cold body-- I'll call

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that T cold, or T2-- it takes
some smaller amount of heat

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from the cold body.

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Some work has to be input
into the system

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in order to do this.

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And then it puts more heat-- you
can kind of view it as a

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combination of this work and
this heat taken from the cold

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body-- and it gives it
to the warm body.

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Sorry.

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This is Q2, and it
gives it Q1.

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So everything just happens
completely in reverse.

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And that's just a byproduct
of, this is reversible.

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So I can just go and I can do,
if this is the way we went

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before, when we're an engine,
if we want to be a

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refrigerator, we go the other
direction, and everything just

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gets reversed.

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And I want you to really
understand

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that this is doable.

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That there's nothing
wrong with this.

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You might say, doesn't this
defy the second law of

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00:11:52,080 --> 00:11:53,230
thermodynamics?

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We're taking heat from a cold
body to a warm body?

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And my answer will be the
same thing I said

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on my entropy videos.

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I said, well, no.

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We're applying some work.

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This is a refrigerator.

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So some work has to be done
in order to do this.

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And whatever object that is
doing the work-- it may be

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some, in the case of your
refrigerator, it's a

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compressor.

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That is adding more entropy to
the universe than the entropy

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that's being destroyed
by our refrigerator.

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So this does not defy the second
law of thermodynamics.

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Now, I want to make another
point about the Carnot engine.

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Let me take the reverse
Carnot engine.

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Let me call it the Carnot
refrigerator.

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So if we take that-- and this is
really just more math than

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anything else.

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If we're taking in Q2, adding
in some work, and producing

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Q1, we can scale this
up arbitrarily.

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If we take x times Q2 in, and
we put in x times W in, then

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we're going to put x times Q1
into our top reservoir.

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And that makes sense,
because these are

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just arbitrary numbers.

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For example, if we have two
Carnot engines in parallel,

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you can just kind of view that
whole thing as two Carnot

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engines doing it together, so
all of these would be 2s.

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If we had three Carnot engines
doing it together, all of

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those would be three, you could
just view them as one

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collective engine.

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Now, with that said, I think
I've laid the framework for at

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least the ideas that will let us
show that the Carnot engine

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is the most efficient engine
that's able to be produced.

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And given that the Carnot
engine's efficiency is this--

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and we're going to prove that
it's the most efficient

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engine-- this becomes the upper
bound on efficiency for

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any engine that anybody
can or will ever make.

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And I'll kind of
do the crowning

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touch in the next video.