A Carnot engine takes $$3 \times {10^6}{{\,cal}}{\text{.}}$$   of heat from a reservoir at $${27^ \circ }C,$$  and gives it to a sink at $${27^ \circ }C.$$  The work done by the engine is

Solution :
$$\eqalign{ & \eta = 1 - \frac{{{T_2}}}{{{T_1}}} \cr & = 1 - \frac{{\left( {273 + 27} \right)}}{{\left( {273 + 627} \right)}} \cr & = 1 - \frac{{300}}{{900}} \cr & = 1 - \frac{1}{3} \cr & = \frac{2}{3} \cr & {\text{But }}\eta = \frac{W}{Q} \cr & \therefore \,\,\frac{W}{Q} = \frac{2}{3} \cr & \Rightarrow \,\,W = \frac{2}{3} \times Q \cr & = \frac{2}{3} \times 3 \times {10^6} \cr & = 2 \times {10^6}\,{{cal}}{\text{.}} \cr & {\text{ = }}2 \times {10^6} \times 4.2\,J \cr & = 8.4 \times {10^6}\,J \cr} $$