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Question

Two Carnot engines A and B are operated in series. Engine A receives heat from a reservoir at 600 K and rejects heat to a reservoir at temperature T. Engine B receives heat rejected by engine A and in turn rejects it to a reservoir at 100 K. If the efficiencies of the two engines A and B are represented by ηA and ηB, respectively, then what is the value of ηAηB?

A
127
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B
712
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C
125
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D
512
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Solution

The correct option is B 712
Let the output of both engines be W
Let the first engine take Q1 heat as input at temperature T1=600 K and gives out heat Q2 at temperature T the second engine receives Q2 as input ang give out Q3 at temperature T3=100 K to the sink.
Work done by first engine W=Q1Q2
Work done by secondt engine W=Q2Q3
Thus,
Q1Q2=Q2Q3
Dividing both sides by Q1
1Q2Q1=Q2Q1Q3Q1
1TT1=Q2Q1(1Q3Q2)
1TT1=Q2Q1(1T3T)
1TT1=TT1(1T3T)
T1T1=1T3T
T1T+T3T=2
1T(T1+T3)=2
T=(T1+T3)2
T=600+1002=350 K
Now using Carnot engine efficiency formula
ηA=600T600
ηB=T100T
ηAηB=600350600350100350=250600250350=350600=712

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