Ranking ΔS of the surroundings

Problem

Four heat exchanges $Q$ between the system and a reservoir at temperature $T$: (A) $Q=500\ \text{J}$ given to the reservoir at $T=300\ \text{K}$; (B) $Q=1000\ \text{J}$ given to the reservoir at $T=300\ \text{K}$; (C) $Q=500\ \text{J}$ given to the reservoir at $T=500\ \text{K}$; (D) $Q=500\ \text{J}$ withdrawn from the reservoir at $T=300\ \text{K}$. Rank $\Delta S$ of the reservoir from smallest to largest.

Solution

The reservoir exchanges heat at constant temperature, so $\Delta S_\text{res}=\dfrac{Q}{T}$ with the sign of the heat: positive if the reservoir absorbs it (heat given TO the reservoir), negative if it gives it up (heat withdrawn FROM the reservoir). $\Delta S_A=\frac{+500}{300}\approx +1{,}67\;\text{J/K}$ $\Delta S_B=\frac{+1000}{300}\approx +3{,}33\;\text{J/K}$ $\Delta S_C=\frac{+500}{500}=+1{,}00\;\text{J/K}$ $\Delta S_D=\frac{-500}{300}\approx -1{,}67\;\text{J/K}$ Ranking from smallest (negative) to largest: $\ev{D < C < A < B}$

Links

Topics: Entropy and the second law Concepts: Entropy · Second law of thermodynamics Skills: Ranking reasoning