A Carnot's engine has an efficiency of $25 \%$ when its sink is at $27^{\circ} \mathrm{C}$. If it has to be…

A Carnot's engine has an efficiency of $25 \%$ when its sink is at $27^{\circ} \mathrm{C}$. If it has to be increased to $40 \%$, what should be the temperature of the sink keeping the temperature of the source constant?
  1. $320 \mathrm{~K}$
  2. $375 \mathrm{~K}$
  3. $240 \mathrm{~K}$
  4. $300 \mathrm{~K}$

Solution

Efficiency of Carnot's engine, $ \eta=25 \%=0.25 $ Temperature of sink, $ T_2=27^{\circ} \mathrm{C}=273+27=300 \mathrm{~K} $ Temperature of source $=T_1$ $ \begin{array}{ll} \therefore & \eta=1-\frac{T_2}{T_1} \\ \Rightarrow & 0.25=1-\frac{T_2}{T_1} \\ \Rightarrow & \frac{T_2}{T_1}=1-0.25=0.75 \\ \Rightarrow & T_1=\frac{T_2}{0.75}=\frac{300}{0.75}=400 \mathrm{~K} \\ \Rightarrow & T_1=400 \mathrm{~K} \end{array} $ When efficiency, $\eta^{\prime}=40 \%=0.4$ $ \begin{array}{ll} \text { then, } & 1-\frac{T_2^{\prime}}{T_1}=0.4 \Rightarrow \frac{T_2^{\prime}}{T_1}=1-0.4=0.6 \\ \Rightarrow & T_2^{\prime}=0.6 T_1=0.6 \times 400=240 \mathrm{~K} \end{array} $

Asked in: AP EAMCET 2020 (22 Sep Shift 2)

Practice more Thermodynamics questions on Aicharya