Two ideal Carnot engines operate in cascade (all heat given up by… - Vidyadip

MCQ

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, $\mathrm{T}_{1}$ and $\mathrm{T}_{2} .$ The temperature of the hot reservoir of the first engine is $\mathrm{T}_{1}$ and the temperature of the cold reservoir of the second engine is $\mathrm{T}_{2} . T$ is temperature of the sink of first engine which is also the source for the second engine. How is $T$ related to $\mathrm{T}_{1}$ and $\mathrm{T}_{2}$, if both the engines perform equal amount of work?

A
$\mathrm{T}=\frac{2 \mathrm{T}_{1} \mathrm{T}_{2}}{\mathrm{T}_{1}+\mathrm{T}_{2}}$
B
$\mathrm{T}=\sqrt{\mathrm{T}_{1} \mathrm{T}_{2}}$
✓
$\mathrm{T}=\frac{\mathrm{T}_{1}+\mathrm{T}_{2}}{2}$
D
$T=0$

✓

Answer

Correct option: C.

$\mathrm{T}=\frac{\mathrm{T}_{1}+\mathrm{T}_{2}}{2}$

c
$\frac{\mathrm{Q}_{\mathrm{H}}}{\mathrm{Q}_{\mathrm{L}}}=\frac{\mathrm{T}_{1}}{\mathrm{T}}$ and $\mathrm{W}=\mathrm{Q}_{\mathrm{H}}-\mathrm{Q}_{\mathrm{L}}$

$\frac{\mathrm{Q}_{\mathrm{L}}}{\mathrm{Q}_{\mathrm{L}}^{\prime}}=\frac{\mathrm{T}}{\mathrm{T}_{2}}$ and $\mathrm{W}=\mathrm{Q}_{\mathrm{L}}-\mathrm{Q}_{\mathrm{L}}$

From $(1)$ and $(2)$

we get $\mathrm{T}=\frac{\mathrm{T}_{1}+\mathrm{T}_{2}}{2}$

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