Exploring Entropy Transfer Across System Boundaries

[lifeson22]

I've noticed a terrific number of authors talk about "entropy transfer" across a system boundary.

But entropy is defined as log(multiplicity), and is a measure of available states to a system in a given configuration. We can transfer mass, charge, energy, from one system to another, and thus destroy A available states in one system and produce B available states in the other.

But the available states depend on the composition and configuration of the system, and vary from one to another. When I transfer a particle among states, I know that the particle I started out with in one system is the particle I ended up with in the other. But if I were to talk of transferring "available states" among systems, that would not be the case - not only would the numbers be different, the states themselves would be different. Introducing a Cu ion to an insulating crystal is very different from introducing it to a conducting copper wire - not only will the number of new available states be in general different from each other, the states themselves will very different from each other. In the insulating crystal the valence band is completely filled and the ionic charge will go into the conduction band which sits a band gap above the valence band. In the conducting copper wire, the valance band is no completely filled, and that's where the ionic charge will go (I know, it's not *this* simple, but you get the point - different states, not just different numbers of states).

So how in the world can we possibly be talking about transferring entropy across systems, when there is no correlation between what one system gives up and the other receives? I cam take one particle from system A and give it to an infinite number of different systems, and the new states created will in general be all different from each other - and hence the entropy changes will so differ.

Imagine I have an apple in my hand and I "give it" to you - but the moment you receive it you find you now have a corn stalk in your hand instead. I can't be entitled to say I transferred anything to you because I can't say what I transferred. I didn't transfer an apple because you never received one, and I never transferred a corn stalk because I never had one to transfer. If my apple exists as such only so long as it's in my hand, I can't possibly talk of transferring apples to anybody, because "apple" is a function of belonging to me - the moment I try to give it to somebody else, "apple" is no longer, and something else appears.

So, I find it absolutely and completely and utterly senseless to talk about transferring entropy. What we *can* transfer is mass, charge, energy, etc - we can then analyze the effect of changing these quantities on the multiplicity of the system. But we cannot be able to transfer states.

Opinions?

 

[Gear300]

So what you're saying is that entropy is an established measure that is not transferable but is changeable, right?

 

[Mapes]

A very well-explained and thought-provoking question. Isn't it true, though, that one would only expect equal entropy transfer (i.e., the decrease in one system exactly equals the increase in the other system) in the context of reversible heat transfer driven by an infinitesimal temperature gradient? In other contexts, the entropy change is generally unequal; the copper ions in your example have a different partial molar entropy in different systems (presumably only their nuclear entropy is unchanged). In another example, an atom in a solid has a different partial entropy than that same atom after evaporating an instant later to make up a gas. Here one only has to be comfortable with the fact that the solid now has fewer states, the gas more, but that the number of states gained and lost is not necessarily equal.

But now returning to the reversible heat transfer case, one might wonder why the amount of entropy gained and lost is theorized to be equal. This is something I'd like to think about more.

 

[lifeson22]

So what you're saying is that entropy is an established measure that is not transferable but is changeable, right?

That's right.

Following Kittel, the multiplicity for a system of N particles, each of which with only two states available ("up" or "down" spin) is simply

g=$$\frac{N!}{N_{up}!N_{down}!}$$

If I add one particle (say spin up), we have:

g=$$\frac{(N+1)!}{(N+1)_{up}!N_{down}!}$$

Since entropy is log(g), the actual entropy change depends not simply on the number and type of particles I added to this simple system, but on the number and type of particles the system had originally (N, N_up, N_down).

When I take one particle from system A and add it to B, it makes no sense to say I transferred entropy. What I *did* do was transfer a particle of a certain type, and in the process changed *independently* the multiplicity of system A and that of system B - but by different measures, independently of each other.

I can talk of transfer of mass, charge, energy, etc, but I don't see how I can possibly talk of transferring entropy. What I *can* do is talk of the independent changes I caused to the entropies of the two systems.


Entropy transfer is so ingrained in engineering thermodynamics textbooks that it's impossible to escape it. But *does* it make *any* sense to begin with? When thermodynamics was first established the mere concept of an atom was in dispute, and entropy wasn't well understood - it was treated as just another property of matter, something transferable like energy. But given what we know about entropy today - that it is a very well-defined function of a system's multiplicity of states), we can't talk about entropy transfer anymore than we can talk about transfer of states - and then we run into the problem that the states "transferred" differ both in number and characteristics (different energies, for example), and that these are a strictly a function of the system itself - so that a system of A particles of type A will see a different change in number and type of available states than would a system of B particles of type B, upon adding the same one particle to both.

As Gear300 said, we can change the entropies of the two systems, independently, upon transferring something that *can be transferred (like mass or energy), but we *can't* transfer entropy itself.

 

[lifeson22]

But now returning to the reversible heat transfer case, one might wonder why the amount of entropy gained and lost is theorized to be equal. This is something I'd like to think about more.

A reversible process is one in which system+surroundings (the universe) gains no net entropy. But one can see an entropy increase at the expense of the other while observing this relationship: only if the entropy increase in one is equal to the entropy loss in the other.

A reversible process merely happens to be one in which the total number of available states in system+surroundings does not increase, so that it's easy to say they were all "transferred".

When the process becomes irreversible, many engineering thermodynamics authors talk about two components of entropy change: a transferred component and spontaneously created component to account for the disparity.

My argument is that no states can be transferred: that A states are destroyed from system A upon removal of a particle, and B different states are created in system B upon addition of that same particle.

 

[Mapes]

The states are an imaginary construct, though, while the matter is real. I see your point in the case of a particle; its entropy can be very different in different systems, so the concept of an entropy transfer is dubious. But if one were to carry a chunk of metal into a room (with all its nuclear, electronic, and crystalline arrangements unaffected), it seems awfully pedantic to complain about the statement that entropy was transferred into the system.