Thermodynamics - does homogeneity follow from additivity?

[andrewkirk]

In Herbert Callen's text 'Thermodynamics and an introduction to thermostatistics' 2nd edition, he introduces four postulates of thermodynamics in the first chapter. The third postulate incorporates an 'additivity property' which is stated as 'The entropy of a composite system is additive over the constituent subsystems'. He makes clear in the immediately following paragraph that by this he simply means that the entropy of any composite system is equal to the sum of the entropies of its constituent subsystems.

A few paragraphs later on he claims 'The additive property applied to spatially separate subsystems requires the following property: The entropy of a simple system is a homogeneous first order function of the extensive parameters.' He provides no argument to support this claim and I can see no way that it could be derived without assuming additional postulates.

We do have the following postulates that he has made, but they do not seem sufficient to prove Callen's claim:

Postulate 1. There exist equilibrium states that are characterised completely by U, V and $N_k$ for all particle types $k$..

Postulate 2. There exists a function S, called entropy, of the extensive parameters of a composite system, defined for all possible equilibrium states of the system, with the property that adding a constraint cannot increase the entropy.

Postulate 3.
3.a Entropy of a composite system is additive over constituent sub-systems.
3.b Entropy is a differentiable function of the extensive parameters.
3.c Entropy is a monotone increasing function of energy.

Postulate 4.
4. $\frac{\partial U}{\partial S}=0\Rightarrow S=0$

It seems to me that, to justify his claim, Callen would need an additional postulate like

Postulate 5: The fundamental equation of a composite system made up of a number of spatially separated identical systems has the same functional form as the (identical) fundamental equations of the constituent subsystems.

Am I missing something obvious here? Can the claim be proven without any additional postulates? If so how?

 

[andrewkirk]

Surely there's somebody on PF that can answer this, isn't there?

Callen seemed to think the homogeneity was an important point. Is that not the case in real thermodynamics? Am I wasting time on something unimportant?

I also got the impression that Callen was a much-used text. I did the usual web-searching to decide what book to buy before I bought it and I think I saw quite a few recommendations. I bought it a couple of years ago and I can't remember what recommendations I saw. Maybe it's out of date now. It was last revised in 1985 so maybe it has fallen into disuse.