Question

Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures, $T_1$ and $T_2.$ The temperature of the hot reservoir of the first engine is $T_1$ and the temperature of the cold reservoir of the second engine is $T_2.T$ is temperature of the sink of first engine which is also the source for the second engine. How is $T$ related to $T_1$ and $T_2,$ if both the engines perform equal amount of work?

• A. $T=0$
• B. $T=\frac{2T_1{T}_2}{T_1+T_2}$
• C. $T=\frac{T_1+T_2}{2}$
• D. $T=\sqrt{T_1{T}_2}$

Answer 1

The correct option is C $T=\frac{T_1+T_2}{2}$

Let $Q_1:$ Heat input to first engine
$Q_C:$ Heat rejected by first engine
$Q_2:$ Heat rejected by second engine
$T_C:$ Lower temperature of first engine
Given, both the engines perform same amount of work,
$W=Q_1-Q_C=Q_C-Q_2$

$\Rightarrow 2Q_C=Q_1+Q_2$

$\Rightarrow 2T_C=T_1+T_2$

$\Rightarrow T_C=\frac{T_1+T_2}{2}$
Final answer: (𝑐)