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JEE Main 2019 (10 Jan, Shift 1) · Heat Engines, Refrigerators & Second Law

Heat Engines, Refrigerators & Second Law question from JEE Main 2019 (10 Jan, Shift 1)

Q60JEE Main 2019 · 10 JAN · SHIFT 1Difficult

Heat Engines, Refrigerators & Second Law

Three Carnot engines operate in series between a heat source at a temperature T1 and a heat sink at temperature T4 (see figure). There are two other reservoirs at temperature T2 and T3, as shown, with T1 > T2 > T3 > T4. The three engines are equally efficient if -

Figure

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Answer: (d)

  1. Equal Carnot efficiencies mean equal ratios: T2T1=T3T2=T4T3=r\frac{T_2}{T_1} = \frac{T_3}{T_2} = \frac{T_4}{T_3} = r.
  2. Multiplying: r3=T4T1⇒r=(T4T1)1/3r^3 = \frac{T_4}{T_1} \Rightarrow r = \left(\frac{T_4}{T_1}\right)^{1/3}
  3. T2=T1r=(T12T4)1/3T_2 = T_1 r = (T_1^2T_4)^{1/3}, T3=T1r2=(T1T42)1/3T_3 = T_1 r^2 = (T_1T_4^2)^{1/3}
  4. So the answer is (d).
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