All heat engines between twoheat reservoirsare less efficient than a Carnot heat engine operating between thesamereservoirs

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.

Suppose there are two enginesEAEAandEBEBoperating between the given source at temperature T1 and the given sink at temperature T2.

LetEAEAbe any irreversible heat engine andEBEBbe any reversible heat engine. We have to prove that efficiency of heat engineEBEBis more than that of heat engineEAEA.

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

LetWAWAandWBWBbe the work output from the engines and their corresponding heat rejections be(Q–WA)(Q–WA)and(Q–WB)(Q–WB)respectively.

Assume that the efficiency of the irreversible engine be more than the reversible engine i.e.ηA>ηBηA>ηB

Hence,WAQ>WBQWAQ>WBQ

i.eWA>WBWA>WB

Now let us couple both the engines andEBEBis reversed which will act as a heat pump. It receives(Q–WB)(Q–WB)from sink andWAWAfrom irreversible engineEAEAand pumps heat Q to the source at temperature T1.

The net result is that heatWA–WBWA–WBis 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.