The efficiency of a Carnot engine is $50\%$ and the temperature of the sink is $500\,K$. If the temperature of the source is kept constant and its efficiency is to be raised to $60\%$,then the required temperature of the sink will be ........... $K$.

$A$ heat engine operating between temperatures $T_1$ and $T_2$ has an efficiency of $\frac{1}{6}$. When $T_2$ is lowered by $62 \ K$,its efficiency increases to $\frac{1}{3}$. Then $T_1$ and $T_2$ respectively are:

Let $\eta_{1}$ be the efficiency of a Carnot engine at $T_{H}=447^{\circ}C$ and $T_{L}=147^{\circ}C$,while $\eta_{2}$ is the efficiency at $T_{H}=947^{\circ}C$ and $T_{L}=47^{\circ}C$. The ratio $\frac{\eta_{1}}{\eta_{2}}$ will be:

$A$ Carnot engine operates with a source at $227^{\circ}C$ and a sink at $27^{\circ}C$. If the source supplies $50 \ kJ$ of heat energy,the work done by the engine is (in $kJ$)

The $p-V$ diagram of a Carnot's engine is shown in the graph below. The engine uses $1$ mole of an ideal gas as the working substance. From the graph,the area enclosed by the $p-V$ diagram is equal to the net work done by the engine. Given that the heat supplied to the gas is $8000 \ J$,calculate the net work done by the engine. (Note: The efficiency of a Carnot engine is $\eta = 1 - \frac{T_2}{T_1} = \frac{W}{Q_1}$) (in $J$)

Two Carnot engines $A$ and $B$ are operated in series. The first one,$A,$ receives heat at $T_1 = 600 \, K$ and rejects heat to a reservoir at temperature $T_2.$ The second engine $B$ receives the heat rejected by the first engine and,in turn,rejects heat to a reservoir at $T_3 = 400 \, K.$ Calculate the temperature $T_2$ if the work outputs of the two engines are equal. (in $K$)