Consider a Carnot cycle operating between $T_1 = 500\, K$ and $T_2 = 300\, K$ producing $1\, kJ$ of mechanical work per cycle. Find the heat transferred to the engine by the reservoirs.

(A) The temperature of the source is $T_1 = 500\, K$ and the temperature of the sink is $T_2 = 300\, K$.
The work done per cycle is $W = 1\, kJ = 1000\, J$.
The efficiency of a Carnot engine is given by $\eta = 1 - \frac{T_2}{T_1}$.
Substituting the values,$\eta = 1 - \frac{300}{500} = 1 - 0.6 = 0.4$.
Since efficiency $\eta = \frac{W}{Q_1}$,the heat transferred to the engine from the source is $Q_1 = \frac{W}{\eta}$.
$Q_1 = \frac{1000\, J}{0.4} = 2500\, J$.
The heat rejected to the sink is $Q_2 = Q_1 - W = 2500\, J - 1000\, J = 1500\, J$.