Clausius' Theorem and Entropy

[laser1]

Homework Statement

textbook

Relevant Equations

Clausius' Theorem

Okay, I agree with this logic. However, if we consider a reversible section first, then an irreversible section, I get the following:
$$\frac{dQ_{rev}}{T} \leq \frac{dQ}{T} $$ which is the opposite to equation (14.8). Why is this? Is it "somehow" not viable to think of a reversible section than an irreversible one? Thanks!

 

[Chestermiller]

First of all, the T in the equation is supposed to be $T_S$, the temperature of the surroundings (assumed to consist of one or more ideal isothermal reservoirs). Secondly, the final equation is supposed to involve two closely neighboring thermodynamic equilibrium states, and the two paths between these states, and the two paths between these states do not have to match one another. I would never have written down those equations in terms of differentials. Can you live with only the integral from of the equation, without accepting the differential form. I have a rule I follow that has never failed me: Never express the changes during an irreversible process in terms of just differentials when comparing reversible and irreversible paths.

 

[laser1]

First of all, the T in the equation is supposed to be $T_S$, the temperature of the surroundings (assumed to consist of one or more ideal isothermal reservoirs). Secondly, the final equation is supposed to involve two closely neighboring thermodynamic equilibrium states, and the two paths between these states, and the two paths between these states do not have to match one another. I would never have written down those equations in terms of differentials. Can you live with only the integral from of the equation, without accepting the differential form. I have a rule I follow that has never failed me: Never express the changes during an irreversible process in terms of just differentials when comparing reversible and irreversible paths.

So why do I get the opposite answer if we consider a reversible section first, and then an irreversible one? I don't get what you are saying here.

 

[Chestermiller]

You are aware that the signs of the dQ’s have flipped and that the irreversible path cannot be run in reverse, right?

 

[laser1]

You are aware that the signs of the dQ’s have flipped

Oh right my bad, that was what I was missing, thanks!

"the irreversible path cannot be run in reverse" - This I don't understand. Isn't that what the textbook is doing? It goes irreversibly from A to B, then reversibly back from B to A.

If it is going reversibly from A to B, and irreversibly back from B to A, how is that any different?

 

[Philip Koeck]

Oh right my bad, that was what I was missing, thanks!

"the irreversible path cannot be run in reverse" - This I don't understand. Isn't that what the textbook is doing? It goes irreversibly from A to B, then reversibly back from B to A.

If it is going reversibly from A to B, and irreversibly back from B to A, how is that any different?

In an irreversible process from A to B the total entropy (system plus surroundings) will increase.
If there was an irreversible process from B to A the total entropy would have to decrease so this process can't exist.

 

[laser1]

In an irreversible process from A to B the total entropy (system plus surroundings) will increase.
If there was an irreversible process from B to A the total entropy would have to decrease so this process can't exist.

I completely agree, but entropy hasn't been defined yet. The image in the original post was in the process of "deriving" entropy. I am trying to see the book's argument if you get me

 

[Philip Koeck]

I completely agree, but entropy hasn't been defined yet. The image in the original post was in the process of "deriving" entropy. I am trying to see the book's argument if you get me

Not sure if that helps, but think of a free expansion. You can't reverse it and get a free compression.
On the other hand you can reverse the reversible isothermal expansion that runs between the same states as the free expansion.
It's just an example, but it's the best I can do at the moment.

What you're looking for might be (logically) circular. The whole concept of entropy comes from studying processes that can and can't be reversed. That certain processes can't be reversed is just something we know from experience and entropy makes this quantitative.

 

[Philip Koeck]

"the irreversible path cannot be run in reverse" - This I don't understand. Isn't that what the textbook is doing? It goes irreversibly from A to B, then reversibly back from B to A.

If it is going reversibly from A to B, and irreversibly back from B to A, how is that any different?

Think of the free expansion and the reversible isothermal expansion from state A to B.
You can reverse the isothermal, but not the free expansion.
Reversing the free expansion would mean that you'd have to achieve a compression without doing work on the gas. The molecules would have to all move to one part of the cylinder without being forced to.
From experience we know that doesn't happen just like many other things don't happen, heat going from cold to hot (without work) for example.

Of course the reason for this is that it's statistically very unlikely. So you have to look to statistical physics for an explanation. However, Clausius and colleagues didn't have that. They just stated that certain things don't happen whereas others do and tried to connect this to a new concept, which we now know as entropy.