Question

Prove that all the reversible heat engines working between the same temperature heat source and sink have the same efficiency

Answer 1

Suppose there are two reversible engines EAEA and EBEB operating between the given source at temperature T1 and the given sink at temperature T2.

We have to prove that efficiency of heat engine EBEB and heat engine EAEA is same.

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

Let WAWA and WBWB be 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 engine EA be more than the engine EBEB
i.e. ηA>ηBηA>ηB
Hence, WAQ=WBQWAQ=WBQ
i.e WA>WBWA>WB
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Now let us couple both the engines and EBEB is reversed which will act as a heat pump. It receives (Q–WB)(Q–WB) from sink and WAWA from engine EAEA and pumps heat Q to the source at temperature T1.

The net result is that heat WA–WBWA–WB is taken from sink and equal amount of work is produce. This violates second law of thermodynamics.

Hence the assumption we made that reversible engine EAEA having higher efficiency than the reversible engine EBEB is wrong.

Similarly we can assume that efficiency of engine EBEB is greater than efficiencyof engine EAEA, this will also result in the violation of the second law of thermodynamics.

Hence it is concluded that reversible engines working between same temperature limits have the same efficiency.


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