Question

Two Carnot engines A and B are operated in succession. The first one, A receives heat from a source at ${T_1} = 800\,{\text{K}}$ and rejects to a sink at ${T_2}\,{\text{K}}$.The second engine B receives heat rejected by the first engine and rejects to another sink at ${T_3} = 300\,{\text{K}}$. If the work outputs of two engines are equal, then the value of ${T_2}\,$ is
A. $489.4\,{\text{K}}$
B. ${\text{469}}{\text{.4}}\,{\text{K}}$
C. $449.4\,{\text{K}}$
D. $429.4\,{\text{K}}$

Answer 1

We are asked to find the value of ${T_2}\,$ when the Carnot engines A and B have equal efficiency. To solve this problem, you will need to recall the formula to find efficiency of a Carnot engine. Then find the efficiencies of Carnot engines A and B separately and then equate them to find the value of ${T_2}\,$.

Complete step by step answer:
Given,
For Carnot engine A,
Temperature of the source is ${T_1} = 800\,{\text{K}}$
Temperature of the sink is ${T_2}\,{\text{K}}$
For Carnot engine B,
Temperature of the source is ${T_2}\,{\text{K}}$
Temperature of the sink is ${T_3} = 300\,{\text{K}}$
The efficiencies of Carnot engines A and B are equal.
The efficiency of a Carnot engine is given by the formula,
$\eta = 1 - \dfrac{{{T_{{\text{sink}}}}}}{{{T_{{\text{source}}}}}}$ …………....(i)
where ${T_{{\text{source}}}}$ is the temperature of the source and ${T_{{\text{sink}}}}$ is the temperature of the sink.

Now, we will find the efficiencies of both the engines A and B and equate them to find the value of ${T_2}\,$.
Efficiency of Carnot engine A using equation (i) is,
${\eta _A} = 1 - \dfrac{{{T_2}}}{{{T_1}}}$
Putting the values of ${T_1}$ we get,
${\eta _A} = 1 - \dfrac{{{T_2}}}{{800}}$ ……………....(ii)
Efficiency of Carnot engine B using equation (ii) is,
${\eta _B} = 1 - \dfrac{{{T_3}}}{{{T_2}}}$
Putting the value of ${T_3}$ we get,
${\eta _B} = 1 - \dfrac{{300}}{{{T_2}}}$ ………………...(iii)
Since the efficiencies of both the engines are equal so, we equate the efficiencies of Carnot engine A and B ,
${\eta _A} = {\eta _B}$
Putting the values of ${\eta _A}$ and ${\eta _B}$ from equation (ii) and (iii) respectively we get,
$1 - \dfrac{{{T_2}}}{{800}} = 1 - \dfrac{{300}}{{{T_2}}}$
$\Rightarrow \dfrac{{{T_2}}}{{800}} = \dfrac{{300}}{{{T_2}}}$
$\Rightarrow {T_2}^2 = 800 \times 300$
$\Rightarrow {T_2}^2 = 240000$
$\Rightarrow {T_2} = \sqrt {240000}$
$\therefore {T_2} = 489.4\,{\text{K}}$
Therefore, the value of ${T_2}$ is$489.4\,{\text{K}}$.

Hence, the correct answer is option A.

Note: The efficiency of a Carnot engine depends only on the temperatures of the source and the sink and is independent of the working substance. Carnot’s theorem states that no heat engine working between two temperatures can have more efficiency than a Carnot engine working between the same two temperatures.