GATE 2022 PH – Question 37
Water at 300 K can be brought to 320 K using one of the following processes. Process 1: Water is brought in equilibrium with a reservoir at 320 K directly. Process 2: Water is first brought in equilibrium with a reservoir at 310 K and then with the reservoir at 320 K. Process 3: Water is first brought in equilibrium with a reservoir at 350 K and then with the reservoir at 320 K. The corresponding changes in the entropy of the universe for these processes are ∆Sଵ, ∆Sଶ and ∆Sଷ, respectively. Then
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Correct answer: (B) $\Delta S_3>\Delta S_1>\Delta S_2$
Explanation
The entropy of the universe (system + reservoirs) increases by an amount that depends on how irreversible the heat transfer is. For water of heat capacity $C$ heated from $T_i$ to $T_f$ by a reservoir at $T_r$:
$$\Delta S_{univ}=C\ln\frac{T_f}{T_i}-\frac{C\,(T_f-T_i)}{T_r}.$$
- **Process 1:** a single step with the reservoir at 320 K.
- **Process 2:** two steps (310 K, then 320 K). The temperature differences are smaller, so the heat flows are more nearly reversible and the entropy generated is **smaller** than in process 1.
- **Process 3:** the water goes first to 350 K, which overshoots 320 K, and then has to be cooled back to 320 K by the 320 K reservoir. The temperature differences are larger, so the entropy generated is **larger** than in process 1.
Therefore $\Delta S_3>\Delta S_1>\Delta S_2$ (option B).