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State and prove Carnot theorem.
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• Carnot theorem states that no heat engine working in a cycle between two constant temperature reservoirs can be more efficient than a reversible engine working between the same reservoirs.

• In other words it means that all the engines operating between a given constant temperature source and a given constant temperature sink, none, has a higher efficiency than a reversible engine.

Proof:

• Suppose there are two engines $E_A$ and $E_B$ operating between the given source at temperature T1 and the given sink at temperature T2.

• Let $E_A$ be any irreversible heat engine and $E_B$ be any reversible heat engine. We have to prove that efficiency of heat engine $E_B$is more than that of heat engine $E_A$.

• Suppose both the heat engines receive same quantity of heat Q from the source at temperature T1.

• Let $W_A$ and $W_B$ be the work output from the engines and their corresponding heat rejections be $(Q – W_A)$ and $(Q – W_B)$ respectively.

• Assume that the efficiency of the irreversible engine be more than the reversible engine i.e. $η_A \gt η_B$

Hence, $\frac{W_A}{Q} \gt \frac{W_B}{Q}$

i.e $W_A \gt W_B$

• Now let us couple both the engines and $E_B$ is reversed which will act as a heat pump. It receives $(Q – W_B)$ from sink and $W_A$ from irreversible engine $E_A$ and pumps heat Q to the source at temperature T1.

• The net result is that heat $W_A – W_B$ is taken from sink and equal amount of work is produce. This violates second law of thermodynamics.

• Hence the assumption we made that irreversible engine having higher efficiency than the reversible engine is wrong.

• Hence it is concluded that reversible engine working between same temperature limits is more efficient than irreversible engine thereby proving Carnot’s theorem.