Question

A Carnot engine $C_{1}$ operates between temperature $T_{1}$ and $T_{2}\left(T_{1} >T_{2}\right) .$ A second Carnot engine $C_{2}$ uses all the heat rejected by the engine $C_{1}$ and operates between temperature $T_{2}$ and $T_{3}$ (where $T_{2} >T_{3}$ ). The efficiency of this combined $\left(C_{1}\right.$ and $C_{2}$ together $)$ engine is

• A. $1-\frac{T_{3}}{T_{1}}$
• B. $1-\frac{T_{3}}{T_{2}}$
• C. $1-\frac{\left(T_{2}+T_{3}\right)}{T_{1}}$
• D. $1-\left(1-\frac{T_{2}}{T_{2}}\right)\left(1-\frac{T_{3}}{T_{2}}\right)$

Answer 1

Answer: (C)

Efficiency of Carnot's engine $C_{1}$ is given as
$\eta_{l}=1-\frac{T_{2}}{T_{l}} \ldots$ (i)
where, $T_{2}=$ temperature of $\sin k$
and $T_{1}=$ temperature of source.
Similarly, efficiency of second Carnot's engine,
$\eta_{2}=1-\frac{T_{3}}{T_{2}} \ldots$ (ii)
The efficiencyof combined engine $\left(C_{1}\right.$ and $\left.C_{2}\right)$ is given as
$\eta=\eta_{l}+\eta_{2}=1-\frac{T_{2}}{T_{1}}+1-\frac{T_{3}}{T_{2}}$
$=2-\left(\frac{T_{2}}{T_{1}}+\frac{T_{3}}{T_{2}}\right)$