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. $\text{T}=\frac{2{\text{T}}_1{\text{T}}_2}{{\text{T}}_1+{\text{T}}_2}$
• B. $\text{T}=\frac{{\text{T}}_1+{\text{T}}_2}{2}$
• C. $\text{T}=\sqrt{{\text{T}}_1{\text{T}}_2}$
• D. $T=0$

Answer 1

Answer: (B)

$Q_1:$ Heat input to $I^{st}$ engine
$Q:$ Heat rejected from $I^{st}$ engine
$Q_2:$ Heat rejected from $I{I}^{nd}$ engine
Work done by $I^{st}$ engine $=$ work done by $I{I}^{nd}$ engine.
$Q_1-Q=Q-Q_2$
$2Q=Q_1+Q_2$
$2=\frac{T_1}{T}+\frac{T_2}{T}$
$\text{T}=\frac{{\text{T}}_1+{\text{T}}_2}{2}$