Understanding a proof of Carnot's theorem.

In summary, the proof establishes that if there exists a hypothetical engine that is more efficient than a Carnot engine, then the Carnot engine cannot be made to work using the same hot and cold reservoirs.
  • #1
Fallen Seraph
33
0
[SOLVED] Understanding a proof of Carnot's theorem.

I'm having trouble understand this proof of Carnot's theorem, and I would appreciate it if someone could point out where my reasoning goes wrong.

The proof reads thusly:


Suppose there exists a a hypothetical engine with a greater efficiency than a Carnot engine.

Consider this engine working from the same hot and cold reservoirs as a Carnot engine.

Adjust the cycle of the Carnot engine such that its work output == that of the hypothetical engine == W

Since the Carnot engine is reversable, we can turn it into a refrigerator that takes in work W from the hypothetical engine and energy Q2 from the cold reservoir and then outputs energy Q1 into the hot reservoir.

Let the energy taken from the hot reservoir by the hypothetical engine == P1.

We have that the efficiency of the hypothetical engine is greater than that of the Carnot one, so

W/P1>W/Q1

=>

Q1>P1.

This means that our construction is taking heat from the cold reservoir, and depositing it in the hot one. Which violates the 2nd law, and thus proves the theorem.


My problem with it is that is seems to imply that either the second law is wrong, or the Carnot engine is not the most efficient because:

Why not just get an engine that is less efficient than the Carnot one and construct the same device as was made in the proof except with the Carnot engine in the place that the hypothetical one occupied in the proof, and the less efficient engine in the place of the Carnot?

This, as far as I can see, would also move heat from the cold reservoir to the hot one.

What am I missing?

Thanks in advance.
 
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  • #2
Fallen Seraph said:
What am I missing?

Any process with efficiency less than Carnot efficiency is not reversible, for example.
 
  • #3
Ah! So it is!

Thank you.
 
  • #4


This Carnot engine works as refrigerator. So, shouldn't it eficiency be Q2/W and not W/Q1?
 
  • #5
that's not efficiency,that's co-efficient of performance..
we are working with efficiency here so if we take coefficient of performance we must have to convert it into efficiency through the relation between them finally..:)
 

1. What is Carnot's theorem and why is it important?

Carnot's theorem is a fundamental principle in thermodynamics that states that the efficiency of any heat engine operating between two temperatures is determined by the ratio of the temperatures. It is important because it provides a theoretical maximum limit for the efficiency of heat engines, and helps us understand the principles behind how they work.

2. How does Carnot's theorem relate to the laws of thermodynamics?

Carnot's theorem is closely related to the first and second laws of thermodynamics. The first law states that energy cannot be created or destroyed, and the second law states that heat cannot spontaneously flow from a colder body to a hotter one. Carnot's theorem helps us understand the implications of these laws in terms of the efficiency of heat engines.

3. What is the mathematical proof for Carnot's theorem?

The mathematical proof for Carnot's theorem involves using the laws of thermodynamics, specifically the concept of entropy, to show that the efficiency of a heat engine cannot exceed the Carnot efficiency, which is given by the ratio of the absolute temperatures of the hot and cold reservoirs.

4. How is Carnot's theorem applied in real-world scenarios?

Carnot's theorem is applied in real-world scenarios in the design and optimization of heat engines, such as steam engines and gas turbines. It helps engineers determine the most efficient operating conditions and improve the efficiency of these engines.

5. Are there any limitations to Carnot's theorem?

While Carnot's theorem is a fundamental principle in thermodynamics, it has some limitations. It assumes idealized conditions, such as a perfectly reversible engine, and does not take into account factors such as friction and heat loss. Additionally, it only applies to heat engines operating between two temperatures, and does not consider more complex systems.

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