# Why can't adiabatic changes decrease entropy for a closed system?

I read that adiabatic changes in a closed system can never decrease the entropy.

To try to understand why, I thought of two scenerios: reversible processes, and irreversible processes.

For a reversible process where $$ds = dq_{rev}/T$$

any process done adiabatically means that the heat transfer is 0, and therefore this equality becomes 0. So far, what I read holds up.

However, for an irreversible process, I'm not sure why it can't decrease entropy. Can anyone prove to me why an irreversible adiabatic process on a closed system will always increase the entropy of that system?

Please note, this isn't a homework problem that I'm trying to get an answer for; I'm trying to understand the ideas inside-out and came across this statement online, which has made me curious.

EDIT:

I just thought of a possibility of why. According to the Clausius inequality:

$$ds ≥ dq_{irrev}/T$$

So if there is an adiabatic change, dq (and q) equals 0. So, if ds ≥ 0, the entropy change is positive or zero.

However, assuming this is correct, this is just the mathematical explanation/proof. Can anyone provide a conceptual and intuitive explanation of why?