Define Carnot Theorem and also give its proof.

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 > η_B. Hence,
W_A /Q>W_B /Q
I.e. W_A > 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_Afrom 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.