Does a closed system with fixed entropy imply an isolated system?

Question

Just first to clarify, my university notation is $$\text{change in entropy} = \text{entropy flow} + \text{internal production of entropy}$$ I am confused with the term fixed entropy. Does that mean change in entropy is zero? Or just $dQ/T = 0$ but there can still be internal entropy production? Because if fixed entropy means that change in entropy is zero then why can't they say an isolated reversible system instead of closed fixed entropy system?

I found this definition on wikipedia where it states: minimum energy principle - for a closed system with fixed entropy the total energy is minimised at equilibrium. But in my book, it says for an isolated system the total energy is minimised at equilibrium. Does this mean isolated = closed system with fixed entropy?

Also, maximum entropy principle: for an isolated system, entropy is maximised at equilibrium - does that mean there can still be an internal entropy production but it has to be bounded basically? Because it doesn't stress whether it has to be reversible or irreversible. Thanks

Answer 1

A closed system with fixed entropy does not imply an isolated system.

Fixed entropy means that $dS = 0$, the entropy of the system does not change. A closed system is one in which no mass can be transferred in or out of the system but heat and/or work can be exchanged in and out of the system. The change in energy of a closed system can then be described as

$$ dU = \delta Q - \delta W \ .$$

$\delta Q$ is the amount of heat added to the system and $\delta W$ is the amount of work performed by the system on its surroundings. If $dS=0$ then $\delta Q = 0$ and the above simplifies to

$$ dU = -\delta W \ .$$

So for a closed system at fixed entropy the change in energy goes as the work done in a reversible process.

In the case of an isolated system the internal energy is constant $dU = 0 $. This means that in an isolated system $\delta Q = \delta W$. If you consider an isolated system being composed of two or more sub-systems then it is clear that $\delta Q$ and $\delta W$ may not end being equal to zero for each sub-system.

Only in specific scenarios will all of these values match up for both an isolated and a closed system, like in the case of $\delta Q = \delta W = 0$. Its important to keep that distinction between closed or isolated because possibly in further questions those differences may play an important role.