A room air conditioner is a Carnot cycle based heat engine run is reverse. An amount of heat `Q_(2)` is absorbed from the room at a temperature `T_(2)` into coils having a working gas (these gases are not good for environment!). The gas is compressed adiabatically to the outside temperature `T_(1)`. Then the gas is compressed isothermally in the unit outside the room, giving off an amount of heat `Q_(1)`. The gas expands adiabatically back to the temperature`T_(2)` and the cycle is repeated. The electric motor electric consumes power P.
(i) Find the maximum rate at which heat can be removed from the room.
(i) Heat flows into the room at a constant rate of `k DeltaT` where k is a constant and `Delta T` is temperature difference between the outside and inside of the room. Find the smallest possible room temperature in terms of `T_(1), k` and P.

Answer

`(i) P((T_(2))/(T_(1)-T_(2)))`
`(ii) T_(1)-P/k [sqrt(1+(4kT_(1))-1]`