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### Course: Physics library > Unit 10

Lesson 3: Laws of thermodynamics- Macrostates and microstates
- Quasistatic and reversible processes
- First law of thermodynamics / internal energy
- More on internal energy
- What is the first law of thermodynamics?
- Work from expansion
- PV-diagrams and expansion work
- What are PV diagrams?
- Proof: U = (3/2)PV or U = (3/2)nRT
- Work done by isothermic process
- Carnot cycle and Carnot engine
- Proof: Volume ratios in a Carnot cycle
- Proof: S (or entropy) is a valid state variable
- Thermodynamic entropy definition clarification
- Reconciling thermodynamic and state definitions of entropy
- Entropy intuition
- Maxwell's demon
- More on entropy
- Efficiency of a Carnot engine
- Carnot efficiency 2: Reversing the cycle
- Carnot efficiency 3: Proving that it is the most efficient

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# Carnot efficiency 3: Proving that it is the most efficient

Proving that a Carnot Engine is the most efficient engine. Created by Sal Khan.

## Want to join the conversation?

- There is a serious issue with this proof. If the Carnot engine was replaced by some other reversible engine of efficiency n (lower than that of a Carnot engine working between the same temperatures), then we could prove that no other engine can exceed this arbitrarily chosen efficiency n. As this must not be correct, this proof is either not correct (although I cannot see any logical error) or some property of engine C, that makes it a Carnot engine rather just any other reversible engine, was assumed without mentioning.

A similar proof I have seen in 'Classical Thermodynamics' by Elwell and Pointon. The issue was not mentioned there, which made me curious and led here. I see this question was asked earlier by Decio Pinto, but he did not get a reply, so I thought I would restate it in my own words. This is a serious flaw and should be explained, otherwise it is not a proof but merely a tricky story, which made many people falsely believe they understood why perpetual motion was not possible.(14 votes)- Sal has just proved that ALL reversible ideal engines have THE SAME efficiency, namely the Carnot efficiency.

He shows in the video that if you assume an engine with higher efficiency you run into a conflict with the 2nd law of thermodynamics.

But just the same: If there was a process with lower efficiency, simply switch the roles of the two engines and you again could pump heat from the cold to the hot reservoir and thus this is also in conflict with the 2nd law.(19 votes)

- What happens if instead of using your friend's "Super Engine" in the final engine you used a Carnot Engine? Would the Second law of thermodynamics still be violated?(9 votes)
- No - in that case, the two engines would cancel each other out and there would be no net work and no net heat transfer. The two engines would do nothing.

It is the equivalent of x being equal to 0 in the video since the efficiency of one Carnot engine = (1+0) times the efficiency of the other.(27 votes)

- Why do the temperatures have to be 500K and 300K yielding a 40% efficency? Couldn't you have a system which had temperatures of 500K and 100K yielding 80% efficency? So then the greater the difference in temperatures the higher the efficency according to the math. What am I missing here?(4 votes)
- I think you are right...but Sal said it - it is the most efficient engine, which operates between two teperatures - this means, that you cant get more efficiency than 80% between your temperatures (500 and 100 K)and you cant get more efficiency than 40% when working between 500 and 300 K ....so he just chose those temperatures randomly(19 votes)

- Hi. I am agreed with the final conclusion, that a system like that would oppose the second law. But I do not understand why we have to call the first engine or the refrigerator by Carnot engine/refrigerator. Let's suppose we have another engine "Y" that performs a different cycle than Carnot one. It takes some heat Q_1 from the hot reservoir, does some work W and leaves Q_2 to the cold reservoir, as well does Carnot engine, but not necessarily the same quantities. We can also make it works as a refrigerator. In the same way as in the video, if we imagine another engine more efficient than Y and couple it with an Y refrigerator, I guess we will reach the same (bad) result. The fact is that I did not need mention Y is a Carnot engine to conclude it. Actually, it looks like I can not couple machines with different efficiencies (such an odd conclusion, I know!). I am sure I am making a mistake, but I can not figure it out. Sorry about some English error. Thanks in advance.(5 votes)
- Actually the title of the video is misleading. Sal did not prove that the Carnot engine is the most efficient one. He proved that it violates the 2nd law of thermodynamics if there were two ideal reversible engines with different efficiencies.

In other words ALL ideal reversible engines have THE SAME efficiency, namely the Carnot efficiency. That is what is proved here!(11 votes)

- Why does the reverse Carnot Engine take in (Q-W)(1+x) ? Why not Q1 - W(1+x)?(6 votes)
- Because it's carnot engine, not the S engine.

At tge end of the last video, he explained about xQ1, xW, xQ2 that x is the same for all three. Q2 is just Q1-W from equation (W=Q1-Q2) so xQ2=x(Q1-W). When x is (1+x) then the term will be (1+x)(Q-W).

In this video, i think he's trying to say that such cannot happen where Q1, (1+x)W, Q2, where Q2 is ,from W=Q1-Q2, Q2=Q1-W, where Q1=xQ1 and W=(1+x)W, so Q2=Q1-(1+x)W.(1 vote)

- Nice video!

What's the motivation for using the clever argument of combining the super engine with the reverse carnot to derive a contradiction? It all seems kind of magical :)(4 votes) - Is it possible for you to relate this to a PV diagram if this has something to do with it?(4 votes)
- Can we have a reversible engine less efficient than a Carnot engine with same source and sink?(3 votes)
- No, all reversible engines have the same efficiency.(1 vote)

- Why did you choose the inputted Heat in the Reverse Carnot engine to also be multiplied by (1+x)? This video might have made sense if you left it as the pure reverse of the Super Engine (Q2 = Q1-W(1+x)), but of course the point would not have been made.(2 votes)
- Because you can scale the heat inputs for the engine up. The only characteristic is the efficiency of the engine and by scaling it in such a manner the efficiency doesn't change.(1 vote)

- since the Carnot engine is at equilibrium it does not add entropy to the universe?(2 votes)

## Video transcript

Let's say we have
two reservoirs. Let's say up here, I have
my hot reservoir. It's at th for hot. And I have my engine. It's going to be a Carnot
engine, because we'll learn that no engine is better,
at least from an efficiency point of view. We have to be careful when
we say "better." We have our Carnot engine,
and it operates on heat differentials. So it takes some heat from
our hot heat source. It takes some heat there. Let's call that q1. And it does some work. Work is a good thing, so I'll
make that in green. It does some work, and then
the surplus energy, the surplus heat, q2, then goes
to our cold reservoir. I'll do that right there. t cold. Now, I made multiple
insinuations in the previous video that this is the most
efficient engine that can be created between two reservoirs,
th and tc. Now, you come along and say,
no, no, no, no, no. I know of a friend, he has
invented a new engine that is more efficient than this engine
between these same two reservoirs. And you go, and you proceed to
draw this same type of diagram for your friend's engine. You say, look-- let me make it
clear-- this is the same reservoir, these same reservoirs
we're dealing with. Actually, I should probably draw
this line all the way, because I'm going to have to
do multiple engines here. So the same reservoir we're
dealing with, right? This is all the hot reservoir,
th th, and this is all the cold reservoir. I need space for multiple
engines that we're going to deal with. So your friend has an engine. We'll call it the
super engine. And your friend's claim-- and
I'll show you why your friend's claim cannot be true,
if you believe the second law of thermodynamics-- so your
friend's claim, they have the super engine. And they claim that look, I
can actually take in q1. I can take in that same heat
from this heat source up here. But I can produce more work
than your Carnot engine. I could produce 1 plus x. I don't want to get
too algebraic. But let's say, you produce w. I produce w times 1
plus x of work. Where x is a positive number. So he's saying, look,
x is greater than 0. Whatever that number he
might feed you is. And then the rest of
the energy that's left over is what? It's q1 minus this. So it's q1 minus w
times 1 plus x. And just to be clear, q2 right
here, I could rewrite that as q1 minus w. Fair enough. So you look at that. You come to me with this, and
I say, no, no, no, no. This cannot be true. Because if this were, then we
would solve literally all of the world's energy problems. And
I'm about to show you why we could solve all of the
world's energy problems, and we would have a perpetual motion
machine, and be able to defy all sorts of things
if we had this. Now, this is my Carnot engine. But I could devise a reverse
Carnot engine, right? Let me make a reverse
Carnot engine. So my reverse Carnot engine
would look like this. And it's going to do the same
thing, but in reverse. So instead of producing q1 minus
w here and putting it into tc, it could take in q1
minus w from our cold source-- so could take that in-- or even
better, let's scale it up a little bit. Let's say it takes in q1
minus w times 1 plus x. So I've just made a slightly
larger reversed Carnot engine. Now, if I take in that much--
in order to do this in reverse, I'm going to have to
take in, I'm going to have to scale up this Carnot engine
and reverse everything. So instead of producing work,
I'm now going to need work to go on the other direction, and
I've scaled it up by 1 plus x. So I'm going to need the
amount of work here times 1 plus x. And then I'm going to push q1,
but I've scaled it up. I'm going to push in
q1 times 1 plus x into my hot heat source. And once again, this isn't
defying the laws of thermodynamics. I'm taking up some work. There's work that needs to be
done in order to do this. But all of a sudden, you come
to me and say, look, this is an awesome deal. You have this nice engine
that works this way. My friend has a super engine. Let's just couple
them together. Let's take the work that he
produces right here-- he produces w times 1 plus x, and
that just happens to be the amount of work that you need
to operate your engine. So you just feed that
into there. So what's the net effect
of these two engines? So let me do another, scroll
a little bit more. Actually, that might be
the best way to do it. So let me make sure that we
understand that these are the same heat sinks or heat sources
that we're using the whole time. So that's my hot source, my
cold source is down here. So if I add our two engines
together-- so if I have a, you know, let me call it a-- I'm
going to pick a new color. These colors are getting
monotonous. Nope. I wanted to do the
rectangle tool. There you go. All right. So I combine these two
engines together. Essentially I just put a
big box around them. They're both operating between
these two heat sources. These two reservoirs. So I call this the, you know,
your super engine plus my reverse Carnot engine. So what's happening now? What's the net heat that's
being taken in or put out of here? So we have q1-- we have--
let me see. We have q1 minus w 1 plus x, but
in this direction, we have q1-- so in this direction,
we could rewrite this. I want to make sure you're
clear on the algebra. This could be rewritten
as what? As q1 times 1 plus x times, or
minus, w times 1 place x. Right? Now, if you compare these
terms, this is the same as this term. This term is bigger
than this term. Right? This term is clearly bigger,
because we're multiplying it by something larger than 1. It's bigger than this storm. So if we combine these two, the
upward movement, or the amount of heat I'm taking up
from my reverse Carnot, is going to be greater than the
amount of heat being put in by your friend's super engine. And we can actually calculate
the amount. We can just take this amount
minus that amount, and that's the net upward movement. So the net upward movement
from our cold reservoir is what? It's this value minus
this value. So minus q1 minus w 1 plus x. If we take a minus, we're
going to subtract it. So it's a minus and a plus. These cancel out. This minus cancels out with--
so this first term could be rewritten as q1 plus q1x. Right? We could rewrite it that way. So this cancels out with that. And so the net upward movement
when we combine the two engines is q1 times x. Fair enough. Now what about the
work transfer? Well, whatever work this guy
produces is exactly the amount of work that I need. So no outside work has to
be done on the system. It just works. This guy produces work, this
guy uses the work. Now what's in that
heat transferred up to our hot reservoir? What's the amount of heat? Well, it's the difference
between these two. And this is clearly a larger
number than this one, so the upward movement dominates. So what's this minus that? So this can be rewritten
as q1 plus q1x, right? I just distributed the q1. We're going to subtract
that out. Minus q1. You're left with q1x. So the net movement,
when we combine the two engines, is q1x. So what's happening here? I have no external energy or
work has to be expended into this system. And it's just taking heat from
a cold body, and it's moving it to a warm body. And it does this indefinitely. It'll do this as much
as I want to. I can just build a bigger one. It'll do it on even a larger
and larger scale. So if you think about it, I
could heat my house with ice by just making the ice colder. I could create steam from things
that arbitrarily cold. This goes against the second
law of thermodynamics. The net entropy in this
world is going down. Because what's happening here? This is just a straight up
transfer of q1x from a cold body to a hot body. So what's the net change
in entropy here? The change in entropy
of the universe. Well, the hot body is gaining
some heat, so it's q1x over the temperature of the hot body,
and then the cold body is losing the same amount. So it's minus q1x over
the cold body. Now, this is a bigger number
than this is, right? Because the denominator
is smaller. This is a cold body. Its temperature in Kelvin will
be a smaller number. So this is going to be less than
0, which the second law of thermodynamics tells
us cannot be. The entropy cannot shrink
in the universe. This whole thing is an
independent system, and the entropy is shrinking. And we can make the entropy
shrink arbitrarily if we just scale up our x's enough. So this is why the Carnot engine
is the most efficient engine possible. Because if anyone claimed to
have a more efficient engine, you could couple it with a
reverse Carnot engine, and then create this perpetual
reverse-- I guess you could call it a perpetual
refrigeration machine that just out of the blue creates
anti-entropy from anywhere, and it would be this perpetual
energy source that creates energy out of nothing. And so this is just something
that cannot be done in our world, especially if you believe
the second law of thermodynamics. So the most efficient engine
is the Carnot engine, where its efficiency is described as 1
minus the temperature of the cold body divided
by the hot body. So if I have two temperature
reservoirs, let's say that my hot one is at 500 Kelvin, and my
cold one is at 300 Kelvin, and I have some engine that
takes heat from there, and transfers it there, and
does some work. The most efficient engine, if
I were to remove all the friction in the engine, the
highest efficiency I could get would be 1 minus 300 Kelvin over
500 Kelvin, which is 1 minus 3/5, which is 2/5, which
is equal to 0.4, which is equal to 40%. So if someone tells you that
they made an engine that operates between a reservoir
that's 500 Kelvin and 300 Kelvin, and they say, oh, I've
achieved 41% percent efficiency. I've really polished
the thing well. You know that they are lying. So anyway, hopefully you found
that reasonably interesting, and I'll see you in
the next video.