Carnot's theorem
Proof
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Carnot's theorem uses the concept of a Carnot engine. Carnot's theorem is another form of the second law of thermodynamics and is a consequence of the second law.
Theorem:
1. All heat engines operating between two temperatures are less efficient than the Carnot heat engine operating between the same two temperatures.
2. All Carnot heat engines operating between the same two temperatures are equally efficient, regardless of the processes and the substances (solids, liquids, gases) used.
Efficiency:
Carnot engine is a reversible engine that transfers heat between a hot tank at a constant temperature T1 and a cold sink at a constant temperature T2. The engine absorbs heat Q1 at the hot tank , does useful work W and transfers heat Q2.
Efficiency = η = W / Q1 = (Q1 - Q2)/Q1 = 1 - Q2/Q1 = 1 - T2/T1
Proof:
Proof is done by contradiction. See the diagram. Let us have two engines. One is more efficient than the other. Both are connected between the same two infinite bodies of heat at temperatures T1 and T2.
The more efficient heat engine η1 drives the other η2 to act as a heat pump. There is no energy received from external environment. Both machines operate on the energy during the heat transfer from the hot body to the cold body. If the η2 > η1 then heat will flow backwards into the hot reservoir.
As η2 is a heat pump it is a reversible engine. We know Q and W are positive. Let us apply the energy conservation principle. Energy is input in engine E1 and output via engine E2.
E1_in = Q = (1 - η1) Q + η1 Q
E2_out = η1 Q + η1 Q * (1/η2 - 1) = η2 Q /η1
We know that η1 = W1 / Q and
η2 = W2 / Q = η1 Q / [η1 Q/ η2 ]
See the denominator of the above. We find that the heat delivered by the engine 2, is more than what it is given. That means that extract heat is being transferred from the cold engine to the hot engine from nowhere (ie., no external energy given to it). Thus its figure of merit seems to be incorrect. It seems to be violating the second law of Thermodynamics. Thus we arrive at the contradiction.
So η1 must be same as η2.
Also irreversible engines are less efficient compared to Carnot reversible engines. Thus all engines are less efficient than Carnot heat engines.
Theorem:
1. All heat engines operating between two temperatures are less efficient than the Carnot heat engine operating between the same two temperatures.
2. All Carnot heat engines operating between the same two temperatures are equally efficient, regardless of the processes and the substances (solids, liquids, gases) used.
Efficiency:
Carnot engine is a reversible engine that transfers heat between a hot tank at a constant temperature T1 and a cold sink at a constant temperature T2. The engine absorbs heat Q1 at the hot tank , does useful work W and transfers heat Q2.
Efficiency = η = W / Q1 = (Q1 - Q2)/Q1 = 1 - Q2/Q1 = 1 - T2/T1
Proof:
Proof is done by contradiction. See the diagram. Let us have two engines. One is more efficient than the other. Both are connected between the same two infinite bodies of heat at temperatures T1 and T2.
The more efficient heat engine η1 drives the other η2 to act as a heat pump. There is no energy received from external environment. Both machines operate on the energy during the heat transfer from the hot body to the cold body. If the η2 > η1 then heat will flow backwards into the hot reservoir.
As η2 is a heat pump it is a reversible engine. We know Q and W are positive. Let us apply the energy conservation principle. Energy is input in engine E1 and output via engine E2.
E1_in = Q = (1 - η1) Q + η1 Q
E2_out = η1 Q + η1 Q * (1/η2 - 1) = η2 Q /η1
We know that η1 = W1 / Q and
η2 = W2 / Q = η1 Q / [η1 Q/ η2 ]
See the denominator of the above. We find that the heat delivered by the engine 2, is more than what it is given. That means that extract heat is being transferred from the cold engine to the hot engine from nowhere (ie., no external energy given to it). Thus its figure of merit seems to be incorrect. It seems to be violating the second law of Thermodynamics. Thus we arrive at the contradiction.
So η1 must be same as η2.
Also irreversible engines are less efficient compared to Carnot reversible engines. Thus all engines are less efficient than Carnot heat engines.
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