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My textbook (Engel Physical Chemistry 3rd Edition) says:

For any irreversible process in an isolated system, there is a unique direction of spontaneous change: $\Delta S > 0$ for the spontaneous process, $\Delta S < 0$ for the opposite or nonspontaneous direction of change, and $\Delta S = 0$ only for a reversible process. In a quasi-static reversible process, there is no direction of spontaneous change because the system is proceeding along a path, each step of which corresponds to an equilibrium state.

It is my understanding that when a system is broken up into a system and the surroundings, $\Delta S + \Delta S_\text{surroundings} = \Delta S_\text{total} = 0$ for a reversible process.

I was doing the following question when I got stuck:

An ideal gas sample containing $\pu{2.50 mol}$ for which $C_{V,m} =3/2R$ undergoes the following reversible cyclical process from an initial state characterized by $T = \pu{450 K}$ and $P = \pu{1.00 bar}$:

a. It is expanded reversibly and adiabatically until the volume doubles.

b. It is reversibly heated at constant volume until $T$ increases to $\pu{450 K}$.

c. The pressure is increased in an isothermal reversible compression until $P = \pu{1.00 bar}$.

Calculate $q, w, \Delta U, \Delta H, \Delta S, \Delta S_\text{surroundings}$, and $\Delta S_\text{total}$ for each step in the cycle, and for the total cycle. The temperature of the surroundings is $\pu{300 K}$.

I was attempting the process denoted in question b, and my answers matched the books answers:

Answer toquestion by the book

However, when I thought about the values, I was confused as to why $\Delta S_\text{total}$ is negative. Isn't it true that in the reversible process, $\Delta S_\text{total} = 0$?

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    $\begingroup$ How can you heat to 450 K if the temperature of the surrounding is 300 K? $\endgroup$ – Karsten Theis May 17 at 21:58
  • $\begingroup$ Perhaps there is an electrical heating coil inside the system. Are you saying the question is incorrect? $\endgroup$ – Cyclopropane May 17 at 22:19
  • $\begingroup$ Even if there is an electrical heater coil inside the system, topologically it is part of the surroundings (i.w., outaide the system) and you need to consider the heat transfer at the coil temperature. $\endgroup$ – Chet Miller May 17 at 23:45
  • $\begingroup$ Furthermore, there is nothing reversible about how an electrical heater operates. $\endgroup$ – Chet Miller May 18 at 0:14
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    $\begingroup$ @DrPepper In your solution to b) you wrote $w = 0$. That would exclude any non-expansion work such as the electrical work done on the system if you want the heating coil inside the system. "Are you saying the question is incorrect?". No, I am saying I do not understand the question. A p-V diagram would help, with each step on the diagram labeled to clarify the interaction with the surrounding. $\endgroup$ – Karsten Theis May 18 at 12:30

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