Entropy As A State Function

As with many other state functions, entropy itself is also a state function. What exactly does it entail for a system to have something as its state function though? Easy - it simply means that the physical property known as a state function depends solely on the state of the system and doesn't depend on the system's past history or how the system came into existence.

I'm not going to go into a lecture here, but I'm just going to give you one question to consider regarding entropy: if entropy is truly a state function, then no matter how the change takes place, regardless of how the system was prepared, we can ignore what the system went through and simply consider the initial and final states right?

Now the above is an absolute truth that applies to all state functions, but now let us consider this scenario (a gedanken, or thought experiment, as I'd like to call it):

I've a piece of alloy, and it's a special alloy, known as β-brass, made up of 50% Zinc and 50% Copper. My question is, if I have β-brass at room temperature and I cool it down slowly and reversibly to 10 K, and alternatively, I cool it down rapidly to 10 K, will the change in entropy be the same?

If not, why? If yes, under what conditions? Answering this question will allow you to understand that entropy is in fact, a special function of a special parameter of life.

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So far, everyone (everyone being a miserable 2 people) have decided that the entropy change must of necessity be the same regardless of how the process is carried out, because entropy is a state function, and must therefore be independent of the path taken, for that is the very essence of a state function (mathematical rigour aside!).

However, I was waiting for an alternative view to this problem - that someone would say that everyone is stubbornly following the Laws of Thermodynamics, why can't everyone see an obvious flaw?

I was hoping for someone to give this alternative point of view:

Dude, look at the situation this way man, under normal conditions at room temperature, the Zinc and Copper atoms are all in a very disordered array, with both types of atoms randomly dispersed amongst atoms of each type. If you cool it down rapidly, then what happens is that you freeze the molecular infrastructure in that disordered form! If you cool it down slowly, you allow rearrangements to take place such that hey, you actually obtain an ordered form, where each Copper atom is surrounded by eight Zinc atoms (and vice versa!). Clearly both methods of cooling will result in different end states, so how can the entropy ever be the same?

By the way, if you're thinking about other possible ways to reject this idea, forget it, I must be right - if you're thinking about volume, both methods result in the same volume, so this can't be the reason for the discrepancy. I've proven that the 2nd Law isn't always right, and must therefore be subject to certain conditions.

I so wanted someone to give this explanation, so I could burst his bubble. But oh well, turns out I'm the one offering this explanation for people to burst my bubble instead. :p

So the question is, what is wrong with his view? Figure it out, and you'll realise that entropy is a very important function of a very important parameter of our lives.

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The key idea here is that entropy is also a function of time! When the person above said that rapid cooling makes a difference as opposed to slow cooling, he is right! When you cool the solid slowly, you allow time for the atoms to move into their fixed ordely positions so that they can assume a regular array by the time the solid hits 10 K. But if you cool it rapidly, the atoms may not have enough time to hit that fixed orderly state, and thus be frozen into a seemingly disordered state.

I said seemingly because that disordered state isn't the actual final state of equilibrium, and it is more accurately known as a metastable state. A metastable state is one where the relaxation time is long as compared to the time where the process governing the evolution of the state is being observed. That is to say, that the disordered array of atoms is actually still undergoing a process, where the atoms are still moving throughout the solid (even though you can't observe it!). As such, you'll have to wait for the atoms to move into their final positions before you make the final measurements!

Hey, that's not fair! You might be thinking that I was giving you a trick question, but think about this: How long should you wait in both instances? If you want a reversible process, theoretically, you need to wait infinitely long. If you want the atoms to go to their positions at 10 K after being cooled rapidly, notice that because the temperature is so low, you'll also have to wait infinitely long. As such, in both instances, you have to wait infinitely long!

This shows that entropy is indeed a function of time - this is best shown using mathematical proofs, but just suffice it to say that entropy is the arrow of time.

And this is embodied in the Second Law of Thermodynamics, where it is said that entropy tends towards a maximum - pardon me, but how can anything tend towards a maximum if it is developing within a process that is governed by time? It shows that entropy is a quantity that evolves with time. We have failed to see this because the original idea of entropy came from macroscopic observations, and not via atomic properties.

Oh well. I think this first post is pretty much a fiasco. :p