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

  $1-\frac{\left(T_2+T_3\right)}{T_1}$
• Correct answer
  D

  $2 - \left(\frac{T_2}{T_1} + \frac{T_3}{T_2}\right)$

Solution

Efficiency of Carnot's engine $C_1$ is given as
where, $T_2=$ temperature of sink and $\quad T_1=$ temperature of source. Similarly, efficiency of second Carnot's engine,
The efficiency of combined engine $\left(C_1\right.$ and $\left.C_2\right)$ is given as $\eta=\eta_1+\eta_2=1-\frac{T_2}{T_1}+1-\frac{T_3}{T_2}=2-\left(\frac{T_2}{T_1}+\frac{T_3}{T_2}\right)$

Hence, the correct answer is option: 'D'.