Figure below represents a Carnot engine that works between temperatures $T_{1}$ 400 K and $T_{2}$ 150 K and drives aCarnot refrigerator that works between temperatures $T_{3}$ 325 K and $T_{4}$ 225 K.What is the ratio $\frac{Q 3}{Q 1}$ ?

$\frac{104}{85}$

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$\frac{104}{95}$

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$\frac{85}{104}$

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$\frac{95}{104}$

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Solution

The correct option is C
$\frac{85}{104}$

Let us consider the efficiency of the heat engine first.

$η_{H . E} = 1 - \frac{T_{2}}{T_{1}}$

$= 1 - \frac{150}{400} = \frac{5}{8}$

$∴ W = η_{H . E} \times Q_{1}$

$= \frac{5}{8} Q_{1}$ .......................(1)

Now this work is utilized by the refrigerator to add heat to the hot body from the cold body. The efficiency of this refrigerator is calculated using.

$η_{r} = 1 - \frac{T_{3}}{T_{4}}$

$= 1 - \frac{225}{325}$

$= \frac{4}{13}$ ...........(2)

We know that $η_{r} = \frac{Q_{2}}{w}$

$\Rightarrow Q_{2} = w \times η_{r}$ ............(3)

From $1^{s t}$ law of thermodynamics we know

$Q_{3} = Q_{2} + w$

$= w + w \times η_{r}$ - - - - - - (4)

Substituting for w and $e t a_{r}$ in (4) we get

$Q_{3} = \frac{5}{8} Q_{1} + \frac{5}{8} Q_{1} \times \frac{4}{13}$

$= Q_{1} (\frac{65 + 20}{8 \times 13})$

$= Q_{1} \times \frac{85}{104}$

$\frac{Q_{3}}{Q_{1}} = \frac{85}{104}$

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Figure below represents a Carnot engine that works between temperatures $T_{1}$ 400 K and $T_{2}$ 150 K and drives aCarnot refrigerator that works between temperatures $T_{3}$ 325 K and $T_{4}$ 225 K.What is the ratio $\frac{Q 3}{Q 1}$ ?

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Figure below represents a Carnot engine that works between temperatures T1 400 K and T2 150 K and drives aCarnot refrigerator that works between temperatures T3 325 K and T4 225 K.What is the ratio Q3Q1?

Figure below represents a Carnot engine that works between temperatures $T_{1}$ 400 K and $T_{2}$ 150 K and drives aCarnot refrigerator that works between temperatures $T_{3}$ 325 K and $T_{4}$ 225 K.What is the ratio $\frac{Q 3}{Q 1}$ ?