I am trying to understand the following commentary I found in Wikipedia about this paradox:
Now a door in the container wall is opened to allow the gas particles to mix between the containers. No macroscopic changes occur, as the system is in equilibrium. The entropy of the gas in the two-container system can be easily calculated, but if the equation is not extensive, the entropy would not be $2S$. In fact, the non-extensive entropy quantity defined and studied by Gibbs would predict additional entropy. Closing the door then reduces the entropy again to $2S$, in supposed violation of the Second Law of Thermodynamics.
I understand the reason why the entropy doesn't change (macroscopic at least). The state the system was before the removal of the wall is no different then that after the removal,simply because we cannot tell whether a particle of the left or right side moved in the other direction.
But the commentary I highlighted is what confuses me, therefore I have some questions about it:
First I tried to calculate the entropy of the system when we remove the thin wall, in the simple case where the 2 chambers have same gas, same nr. of particles, same $T$, $P$, $V$. I got as a results:
$\Delta S =2nR\,\ln2$ and since there is no exchange of heat with the environment or work done by the system or from the system, the entropy of the environment doesn't change. And as a consequence the total entropy (system + environment) > 0, which implies an irreversible process. I hope my understanding is correct up to this point.
Then the text proceeds with he following segment : " ... but if the equation is not extensive, the entropy would not be $2S$." What does it mean? What is the meaning of an extensive equation?
"In fact, the non-extensive entropy quantity defined and studied by Gibbs would predict additional entropy." This I don't understand at all.
" Closing the door then reduces the entropy again to $2S$, in supposed violation of the Second Law of Thermodynamics." Is this a violation of the 2nd law because the entropy which was $S =2nR\ln2 + 2S$ is reduced to $2S$. How would I be able to calculate this reduced entropy?
Furthermore the following is also said in the wikipedia article about this:
"Suppose the two volumes are separated by a barrier in the beginning. Removing or reinserting the wall is reversible, but the entropy increases when the barrier is removed by the amount $\Delta S =2nR\ln2 >0$ which is in contradiction to thermodynamics if you re-insert the barrier. This is the Gibbs paradox."
What is reversible here? I have never considered the removal or addition of a barrier as a process, and I cannot also understand how it is qualified as a reversible, process I guess? The only thing that I do remember is when we study a gas in one side of a volume and we remove the barrier, in this case the expansion of the gas in the whole volume is an irreversible process, but never has anything been said regarding the removal of the barrier.
I hope someone can help me understand what this paradox is about and how entropy changes, which violates the 2nd law?
Edit: And why would (supposedly) entropy change two $2S$ ($S$ for each sub system) when we reinstall the barrier and not to something else, something not symmetrical?