What does Entropy mean, other than just "disorder of the system"? Could it be called "Tendency to move towards Thermal Equilibrium"?

Question

What is meant by Entropy? My professor tells me "disorder"(I make no comments on that, except that it really adds no real understanding). The way the concept of entropy kept developing in class, I was forced to conclude that the statement "Entropy of an isolated system always increases" is equivalent to the statement "An isolated system always reaches thermodynamic equilibrium (among its parts, i.e. all parts of the isolated system reach the same temperature)". If that is so, why invent Entropy? Why not just say "Thermal Equilibrium"? Then, I understand that $ΔS=\frac{q_{rev}}{T}$. Meaning, if the reaction is reversible, Entropy change may be interpreted as heat exchanged divided by temperature at which heat flows. Other than this I have no understanding of $S$ or $ΔS$.

So, What is meant by "Entropy"? And why is $ΔS=\frac{q_{rev}}{T}$? And what is up with this thing called "Disorder"?

Answer 1

"Measurement of Disorder" is not a particularly good metaphor for entropy. Your title suggestion is closer to a sounder notion: in these terms I would call entropy a measure of "progress towards thermodynamic equilibrium".

What the entropy of a system's macrostate actually measures is the likelihood of that macrostate, i.e. the logarithm of the number of quantum (microscopic) states that are consistent with an observed macrostate, assuming all possible microscopic states are equally probable.

Another way of saying this is that it is the size of the maximally compressed document or codeword, in bits, that you would need to specify the system's exact state given that you already know it is in the observed macrostate. Actually we multiply the bit length by a constant to make this definition consistent with the classical, Clausius definition that you cite, but a document size in bytes is a good thought picture to hold on to.

I say more about these ideas in my anser here.

The reason I think one could think, in your words, of entropy as a "progress" towards equilibrium is that in any thermodynamic system studied in all undergrad statistical mechanics there are so many possible states that the law of large numbers enforces the following statement: in any such thermodynamic system, there are states that look very much like the maximum likelihood state and almost nothing else, as I discuss in my answer here. Thus, if a system beginning in any state undergoes a random walk in state space, then almost certainly it will wind up near the maximum likelihood, equilibrium state.