[MaxShlochz]

I've been watching the KhanAcademy chemistry playlist for quite some while and Sal Khan gave many proofs in order to finally get to the 2nd law of thermodynamics.

I understand every single step from the road taken, BUT I'm missing only one important thing which I have much trouble understanding.

I know that in a Carnot Cycle or more generally, a reversible process, S (Entropy) is a valid state variable.
Simply because we can prove that the ΔS=0 after each and every cycle of the Carnot Cycle, which means it doesn't change no matter what or how many paths we take. (Obviously we're assuming that there's no friction or any other energy loss)

In addition, if we look at the statistical mechanics definition, we'll be able to find that S is the number of different states the system can take on. (N*k*ln(Ω))

So what do we do after that to get to the Entropy definition that we're familiar with?
We try to prove that the ΔS of the Carnot Cycle perspective is an equivalent definition to the ΔS of the statistical mechanics point of view.

So after all of this background, (because it just really bothers me and I want to be understood 100%)
My questions:

If we prove that the ΔS of a reversible process has the equivalent definition of the statistical mechanics point of view, Does that mean that we only proved that the ΔS is the change the system has for the number of states it can have only in REVERSIBLE processes or in a quasistatic processes or all of the processes?

And if so, how does it even help to us understand our world better?
I was told that there are no true reversible processes so how does it even help our understanding about Entropy in general?


I hope I didn't make any logical mistakes writing this, I hope I'm understood correctly, I'll appreciate any kind of help, this is killing me.

[curiouscat]

And if so, how does it even help to us understand our world better?
I was told that there are no true reversible processes so how does it even help our understanding about Entropy in general?

It does put a hard upper bound on how well a process can do. You cannot do better than a reversible process so it sets an envelope as to how well you might operate and also lets you compare a real world process to an ideal one.

[MaxShlochz]

It does put a hard upper bound on how well a process can do. You cannot do better than a reversible process so it sets an envelope as to how well you might operate and also lets you compare a real world process to an ideal one.

ok, that does make sense that since we surely can't do better than a reversible process and we can't practically build a fully reversible process anyway. (Like a Carnot Engine)

This I understand.

But what still some what boggles my mind is that we proved that Entropy is a valid state variable only in a reversible state, and with the proof of the statistical mechanics, (which involves combinatorics) we managed to derive the 2nd law of themodynamics: ΔS universe≥ 0 or ΔS universe = ΔS1 + ΔS2;
ΔS1 + ΔS2≥0 .

Which is understandable but it seems to me that it only applies if we're talking about a reversible system so why does the 2nd law of thermodynamics applies to all systems? (or at least that's what I figured)

[curiouscat]

Which is understandable but it seems to me that it only applies if we're talking about a reversible system so why does the 2nd law of thermodynamics applies to all systems? (or at least that's what I figured)

Because it is an inequality.

[MaxShlochz]

Because it is an inequality.

I think I'm getting there now, can you please give a good example of this inequality of Entropy in a system?