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The entropy change of a system is given by $$\mathrm dS=\frac{\mathrm dq_\text{rev}}{T}$$ which at constant temperature is $$\Delta S=\frac{q_\text{rev}}{T}$$ I also learned that the entropy change of the surroundings is $$\Delta S_\text{surr}=-\frac{q_\text{actual}}{T}$$ Why do we use $q_\text{rev}$ for the system but $q_\text{actual}$ for the surroundings? How do we derive $$\Delta S_\text{surr}=-\frac{q_\text{actual}}{T}$$

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  • $\begingroup$ The equation $\Delta S_{surr}=-q_{actual}/T$ is based on the assumption that the surroundings consists of an infinite reservoir at constant temperature T (i.e., a reservoir with an infinite capacity to absorb heat without its temperature changing). So any heat transfer to the reservoir will always take place at the temperature T. In this equation, T is temperature of the reservoir, and not the temperature of the system. The only place that the temperature of the reservoir matches the temperature of the system is at the boundary between them. $\endgroup$ – Chet Miller Nov 1 '15 at 23:04

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