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Question

Two Carnot engines are working between $$400K$$ and $$300K, 300K$$ and $$200K$$ respectively. Find the ratio of the efficiency of first system to the second system.


Solution

System 1:
Temperature of source, $$T_{1} = 400K$$
Temperature of sink, $$T_{2} = 300K$$
$$\therefore$$ efficiency of engine is,
$$\eta_{1} = 1 - \dfrac{T_2}{T_1} = 1 - \dfrac{300}{400} = 0.25$$
System 2:
Temperature of source, $$T_{1} = 300K$$
Temperature of sink, $$T_{2} = 200K$$
$$\therefore$$ efficiency of this engine is,
$$\eta_{2} = 1 - \dfrac{T_2}{T_1} = 1 - \dfrac{200}{300}  = 0.33$$
$$\therefore$$ Ratio of efficiencies is,
$$\dfrac{\eta_1}{\eta_2} = \dfrac{0.25}{0.33} = \dfrac{1 / 4}{1 / 3} = \dfrac{3}{4} = 0.75$$

Physics

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