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Carnot efficiency 2: Reversing the cycle | Thermodynamics | Physics | Khan Academy

December 2, 2019

In the last video, I showed
you that the definition of efficiency, eta, is the work
that we do given the amount of heat we are given
to work with. And we showed that for an
engine, that could also be rewritten as 1 minus Q2 over
Q1, or essentially, 1 minus the heat we output from our
engine divided by the amount of heat we input from
our engine. Now we applied this formula to
a Carnot cycle, and we said, hey, for a Carnot engine,
we could get an efficiency of this. So let me write this here. So the efficiency for Carnot,
eta for Carnot, is 1 minus T2 over T1. To get this result, we had to
use the fact that we were dealing with the Carnot cycle
to use, you know, we were moving along these isotherms,
and so I was able to take the natural log of them and do all
of that, and I was able to get this for the efficiency
of a Carnot engine. And let me be very clear. This is the efficiency that
can only be attained by a Carnot engine. The other definitions of
efficiency– so when I just defined efficiency as equal to
the work performed divided by the heat, let me call it the
heat input– or when I defined it as the net heat in. So Q1 minus Q2 over Q1. This applies to all
heat engines. This is true for all
heat engines, including the Carnot engine. A heat engine is an engine
that operates on heat. I probably should have said
that a while ago. And this engine that I made,
this Carnot engine, is definitely a engine that is
operating on heat, because it’s taking heat here,
and later it releases the heat down here. The cycle just shows what’s
happening to that engine. And I just want to make
that distinction, too. The engine is the actual
physical thing. The cycle just describes
what’s happening to it. So with that said, I said that
this is only true for a Carnot heat engine. Now, what I’m about to embark
on– and I don’t know if I’m going to finish it
in this video. It might take into the next
video to do it properly– is to show you that if we’re
operating a heat engine between two temperature
sources– so I have my hot temperature source, I’ll call
that TH for T hot, and it’s transferring some heat, Q1, and
some other heat is coming out at Q2, and I’m performing
some work, and then my other cold temperature reservoir,
I’ll call that T cold, is down here. And that’s where I’m releasing
the heat, too. I’m going to show over the next
few videos that the most efficient engine is this
theoretical Carnot engine. That no engine can get more
efficient than this. So if this is a Carnot
engine, this is the most efficient engine. Or this is the ideal, where
nothing is lost. Well, I’ll go into that in more detail. No engine can get more efficient
than this Carnot heat engine. So to get there, to prove it to
you, I’m just going to play with the Carnot engine a little
bit, just to show you some of the tools that it
has at its disposal. So one of the things– let me
just draw a PV diagram. In the Carnot cycle we’ve done
so far, we’ve kind of always moved in one direction. We had our isothermal
expansion. It went something like that. That was isothermal. Then we had our adiabatic
expansion– and the whole time we were going in that
direction– and it went like that. Then we had our isothermal
contraction. It went something like this. And then we had our adiabatic
contraction, to get to where we were to begin with. So then we went back
like that. And the whole time, we
went in this kind of clockwise direction. We went in the clockwise
direction, and we took in heat up here– because we were doing
work– we took in heat to keep our temperature
constant, and then we released heat here to keep our
temperature from going up from Q2. And so if I were to draw this
another way– well, I just did one like that, but let
me draw it like this. I could also depict it like
this, where that’s my engine, this is my hot reservoir–
let me put this as T1– T1 is up here. It transferred Q1 to
my Carnot engine. My Carnot engine did some work,
and then left over, it transferred into my cold
reservoir, T2, it transferred Q2. This is another way of depicting
what went on in this Carnot cycle. And here I’ve actually
drawn the engine. Now, one of the tools I want to
show you is that this is a reversible reaction. Or that we can take this and
go the other way around. And it’s dependent upon an
assumption that I threw out a long time ago. So when I first drew these, I
kind of introduced you to the idea of a quasistatic process. And quasistatic just means,
look, you do it really slowly, so that you can always say that
you’re close enough to equilibrium that your
macro state variables are always defined. And that was the whole
justification for dealing with pebbles like this. Instead of just doing it
wholesale, instead of just moving all the pebbles, and just
getting to this state, from A to B kind of jumping, I
wanted to do it gradually, so that I would be defined at
every point in between. That’s what quasistatic
did for us. And when I actually made the
video quasistatic processes, I said, you know, quasistatic
processes, for the most part, are reversible. And sometimes I used the
words interchangeably. Now, by definition, our
theoretical Carnot cycle is said to be, not only is it
quasistatic, but it is also reversible. Which means at any point in
time– let’s say we’ve moved a couple of pebbles, and we’ve
gotten right here. If we want to, if we’re in the
mood, we can add some pebbles back, and just follow this right
back to where we were. That’s what reversible means. It means you can reverse
something. Now, what has to be ideal about
the system in order for that to be true? Well, it means that the actual
movement of our piston, of this movable ceiling, that it
shouldn’t have any friction. Because if some of the heat is
lost to friction, then when we go back, we would have lost
some of our heat. Some heat would have been
destroyed, just going from one state forward and back. So the assumption that we have
to make in order for the Carnot cycle to be reversible
is that it’s frictionless. So the Carnot heat engine, this
theoretical engine, is a frictionless engine, which is
theoretically impossible. To be completely frictionless. To be– but we’ll talk more
about that in the future. So if you have a completely
frictionless engine, and it’s quasistatic, it’s
also reversible. So if we want to do
it reversible, what does that mean? It means I could start in this
state, my state A that I’ve labelled before, but instead
of going around that way, I could go around the other way. So what I could do first, is I
could adiabatically expand first– so maybe let me redraw
it, so I do it the other way. So I could reverse
this reaction. And it would happen the
exact same way. And that’s an artifact of that
I’m always in equilibrium, and that my system is frictionless,
that I don’t lose energy just going
back and forth. So I could start at state
A here, and then I could adiabatically contract. Adiabatic contraction would look
something like this, and it’ll get to that state. Then I can isothermically
expand. So I’m going like this. And as I isothermically expand–
so I’m going like this, I’m all along an
isotherm– I’d doing some isothermic expansion– so in
this case, I’m doing work, but I’m doing work isothermically,
right? At some cold isotherm. Let’s call it T2, right? Just like this was T2. So in this case, if I’m
expanding, and I’m staying at T2, and I’m sitting
on top of my T2 reservoir, heat is coming. This area under the curve,
the work I’m doing, is the heat added. This is Q2, and that is given
to me by my T2 reservoir. So everything is going
in reverse. That’s the whole idea. Then I adiabatically contract,
like that, and then I isothermically contract,
like that, to get back where I started. When I isothermically contract,
what’s happening? Work is being done to me, so now
all of this area over here will be negative. And in order to keep my
temperature constant, I have to release heat. So I’m releasing heat, but
I’m doing it at a high temperature. So I’m releasing it into
my T1 reservoir. So it’s the exact same thing
as it happened before, but since when I go in a reverse
direction, some work is being applied. So now, when you look at it this
way, when you figure out all of the areas,
the area in here will actually be negative. And the reason why I’m saying
that is because the positive work values are going
to be this. This is going to be the
positive, what I’m doing in blue right here. And the negative work
values are going to be all of this stuff. So if you wanted to figure out
the total work done, it’s going to be negative. So what’s happening, if I run
the Carnot cycle in reverse– so I’ll call it the Carnot
refrigerator. No, that’s not what
I wanted to do. I’ll call it Carnot reverse. But it’s handy that R also
stands for refrigerator. This is the Carnot engine. It does work by using heat, by
taking advantage of the heat difference between this hot–
you could view this as the T hot and the T cold. Now, a reverse Carnot engine, or
maybe you call it a Carnot refrigerator, does
the opposite. That’s exactly what I
just drew over here. What it does is, it starts with
a cold body– I’ll call that T cold, or T2– it takes
some smaller amount of heat from the cold body. Some work has to be input
into the system in order to do this. And then it puts more heat– you
can kind of view it as a combination of this work and
this heat taken from the cold body– and it gives it
to the warm body. Sorry. This is Q2, and it
gives it Q1. So everything just happens
completely in reverse. And that’s just a byproduct
of, this is reversible. So I can just go and I can do,
if this is the way we went before, when we’re an engine,
if we want to be a refrigerator, we go the other
direction, and everything just gets reversed. And I want you to really
understand that this is doable. That there’s nothing
wrong with this. You might say, doesn’t this
defy the second law of thermodynamics? We’re taking heat from a cold
body to a warm body? And my answer will be the
same thing I said on my entropy videos. I said, well, no. We’re applying some work. This is a refrigerator. So some work has to be done
in order to do this. And whatever object that is
doing the work– it may be some, in the case of your
refrigerator, it’s a compressor. That is adding more entropy to
the universe than the entropy that’s being destroyed
by our refrigerator. So this does not defy the second
law of thermodynamics. Now, I want to make another
point about the Carnot engine. Let me take the reverse
Carnot engine. Let me call it the Carnot
refrigerator. So if we take that– and this is
really just more math than anything else. If we’re taking in Q2, adding
in some work, and producing Q1, we can scale this
up arbitrarily. If we take x times Q2 in, and
we put in x times W in, then we’re going to put x times Q1
into our top reservoir. And that makes sense,
because these are just arbitrary numbers. For example, if we have two
Carnot engines in parallel, you can just kind of view that
whole thing as two Carnot engines doing it together, so
all of these would be 2s. If we had three Carnot engines
doing it together, all of those would be three, you could
just view them as one collective engine. Now, with that said, I think
I’ve laid the framework for at least the ideas that will let us
show that the Carnot engine is the most efficient engine
that’s able to be produced. And given that the Carnot
engine’s efficiency is this– and we’re going to prove that
it’s the most efficient engine– this becomes the upper
bound on efficiency for any engine that anybody
can or will ever make. And I’ll kind of
do the crowning touch in the next video.

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  • Reply KingRobbStark September 19, 2009 at 12:45 am


  • Reply Jo K April 20, 2010 at 3:41 pm

    Two questions mr khan. 1. Why did you call the first isotherm in the carnot refrigerator adiabatic contraction? If the volume increases isn't that expansion rather?
    2. On this isotherm, is there a temperature temperature drop to a value of T<T2?

  • Reply COD5252 April 10, 2011 at 6:48 pm

    contraction means compression

  • Reply rubixcube101 October 26, 2012 at 5:43 am

    thanks a bunch… your videos are awesome! 🙂

  • Reply 27062753 April 26, 2013 at 12:38 am

    8:12 is an adiabatic expansion

  • Reply dbc80 May 30, 2013 at 11:37 am

    at around 11:45 the T2 on the top should be T1

  • Reply Vipul Verma October 15, 2014 at 3:00 pm

    Timeline : 8:17
    Since we are adiabatically contracting, the volume of the air should reduce. But here the volume is increasing as we move from Step 1 to 2. 
    Can you please clear out this confusion?

  • Reply Rituraj Borah October 24, 2015 at 7:47 pm

    well..I feel like Sal was in a kind of a hurry..

  • Reply Siam May 21, 2017 at 8:55 pm

    Thank you

  • Reply Melvin Paul January 15, 2018 at 6:30 am

    you are great bro just keep d'ing

  • Reply Alfonsina Cipollone April 8, 2018 at 4:20 am

    If only he would have said which is the most efficient engine…
    Hahah kidding thanks mate ur help is appreciated

  • Reply Bill Bunn March 16, 2019 at 12:52 am

    Sal. Love these videos. One question about assumptions, in general. In lots of these videos we use the pebble thing, which is an excellent way to understand things. I wondered, though, why the removal of the pebble or the replacement of the pebble is not considered in our calculations. Is that not also a form of work? This came to mind after your Maxwell's demon video, and you mention that the demon is part of the Q consideration. Not suggesting you're a demon, at all. Maybe I should go back and look at your intro to the pebble moment. Anyhow, just a minor question. I love your instruction!

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