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According to Wikipedia, a reversible process is "a process whose direction can be "reversed" by inducing infinitesimal changes to some property of the system via its surroundings, with no increase in entropy."

However, for isothermal processes, any reversible heat added to the system at constant temperature increases entropy. So, a reversible process of heating is causing a change in entropy. This seems to contradict the definition above.

Where have I gone wrong?

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The entropy does not change if the process is reversible and the system is closed. In your example you are taking into account only the subsystem which indeed has its entropy increased. If you consider also the neighbourhood then you see that its entropy change is negative (loosing heat) and that precisely cancels the entropy change of the subsystem. The entropy of the whole closed system does not change.

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  • $\begingroup$ What do you mean by neighborhood? $\endgroup$
    – sangstar
    Dec 2, 2017 at 21:29
  • $\begingroup$ @sangstar everything that is exchanging heat with your system. $\endgroup$
    – Diracology
    Dec 2, 2017 at 21:39
  • $\begingroup$ Could you give a physical example? I'm having trouble understanding it from that definition. $\endgroup$
    – sangstar
    Dec 2, 2017 at 21:42
  • $\begingroup$ @sangstar a heat engine under Carnot cycle. The gas is the system whereas the hot and cold thermal reservoirs (sources) form the neighbourhood. Another example: the system you mentioned. It gain heat isothermically. Well something has to be losing heat. That is the neighbourhood. $\endgroup$
    – Diracology
    Dec 2, 2017 at 21:46
  • $\begingroup$ The hot and cold sources form the neighborhood? But how can their entropy change be negative? I thought they were thermal reservoirs, so their temperature can't change. $\endgroup$
    – sangstar
    Dec 2, 2017 at 22:24
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When one says that entropy does not increase for a reversible process, the entropy they are talking about is the total entropy of an isolated system. If we consider the universe as our isolated system total entropy is equal to the entropy of system + the entropy of the surroundings.

So, for a reversible process where heat is added into the system, the entropy of the system does increase however the entropy of the surroundings in any reversible process = -(entropy of the system)and ultimately, the total entropy is zero.

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