Two carnot engines A and B are operated in series. The first one A receivesheat at 900 K and reject to a reservoir at temperature T K. The secondengine B receives the heat rejected by the first engine and in turn rejectsto a heat reservoir at 400 K calculate the temperature T when(i) The efficiencies of the two engines are equal(ii) The work output of the two engines are equal

Let the output of both engines be W

Let the 1st engine take Q1 heat as input at temperature T1 = 900K and gives out heat Q2 at temperature T the second engine receive Q2 as input and give out Q3 at temperature T3 = 400k to the sink.

Work done by first engine W =  Q1 – Q2

Work done by 2nd engine W = Q2 – Q3

Thus,

 Q1 – Q2 = Q2 – Q3

Dividing both sides by Q1

1 – Q2/Q1  = Q2/Q1 – Q3/Q1

=> 1-T/T1 = Q2/ Q1(1- Q3/ Q2)

=> 1-T/T1 = Q2/ Q1(1- T3/ T)

=>  1-T/T1 = T/T1 (1- T3/ T)

=> T1/T – 1 = 1 - T3/ T

=> T1/T + T3/ T = 2

=> 1/T (T1 + T3) = 2

=>T = (T1 + T3)/2

=> T = (900+400)/2 = 650 K

Regards

Arun (askIITians forum expert)