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I am now reading about how to calculate entropy changes of a system undergoing a process, and the textbook says that to be able to calculate such a change, you need to imagine that the system is undergoing an internally reversible process between the initial and final states. Does this mean that the process can be externally irreversible? And if that is true, can you give me an example of an internally reversible but externally irreversible process?

My understanding of the situation tells me that for a process to be internally reversible, it also should be externally reversible, because if for instance we had a system exchanging heat at a finite temperature difference, then the process will not be quasi-static and therefore not internally reversible.

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An example of an internally reversible but externally irreversible process is if you are heating a solid using an ideal reservoir at a constant (higher) temperature, with heat flowing through an insulating layer of very low, but finite, thermal conductivity. If the solid is considered the system, and the insulating layer plus reservoir is considered the surroundings, then the system experiences an internally (essential quasi static) reversible heating while the surroundings experience an irreversible change.

Another example is isothermal expansion of a gas against against at piston with kinetic friction.

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