Question

Two Carnot engines are operated in succession. The first engine receives heat from a source at `T=800K` and rejects to sink at `T_(2)K`. The second engine receives heat rejected by the first engine and rejects to another sink at `T_(3)= 300K`. If work outputs of the two engines are equal, then find the value of `T_(2)`.

Answer 1

For the first engine,
`Q_(1)-Q_(2)= W and (Q_(2))/(Q_(1))= (T_(2))/(T_(1))`….(i)
For the second engine,
`Q_(2)-Q_(3)= W and (Q_(3))/(Q_(2))=(T_(3))/(T_(2))`....(ii)
`(Q_(3))/(Q_(2))xx(Q_(2))/(Q_(1))=(T_(2))/(T_(1))xx(T_(3))/(T_(2))`
`(Q_(3))/(Q_(1))=(T_(3))/(T_(1))=(300)/(800)=3/8`
Also, from (i) and (ii)
`Q_(1)-Q_(2)= Q_(2)-Q_(3)`
`Q_(1)+Q_(3)=2 Q_(2)`
`1+(Q_(3))/(Q_(1))= 2(Q_(2))/(Q_(1))=(2T_(2))/(T_(1))`
`1+3/8= (2T_(2))/(T_(1))= 11/8`
`T_(2)= (11)/(16)xxT_(1)=(11)/(16)xx800= 550K`