GATE 2023 ME – Question 22
A heat engine extracts heat ($Q_H$) from a thermal reservoir at a temperature of 1000 K and rejects heat ($Q_L$) to a thermal reservoir at a temperature of 100 K, while producing work ($W$). Which one of the combinations of [$Q_H$, $Q_L$ and $W$] given is allowed?
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Correct answer: (B) $Q_H = 2000$ J, $Q_L = 750$ J, $W = 1250$ J
Explanation
A valid combination must satisfy the first law, $W = Q_H - Q_L$, and the second law, $\frac{Q_L}{T_L} - \frac{Q_H}{T_H} \ge 0$. Option A fails the first law ($2000 - 500 = 1500 \ne 1000$) and D too ($6000 - 600 = 5400 \ne 5500$). Option B satisfies it ($2000 - 750 = 1250$), and the entropy generation is $\frac{750}{100} - \frac{2000}{1000} = 5.5 > 0$, so it is allowed. Option C satisfies the first law, but $\frac{500}{100} - \frac{6000}{1000} = -1 < 0$ violates the second law (its efficiency of 91.7% exceeds the Carnot limit of 90%).