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# Re: [Phys-l] thermo differential and extensive/intensive variables

• From: Stefan Jeglinski <jeglin@4pi.com>
• Date: Fri, 15 Jun 2007 09:22:33 -0400

(going backwards here)

Here's my attempt to formulate thermodynamics without the usual
hogwash:
http://www.av8n.com/physics/thermo-laws.htm

I was reading this, but missed a link to your partial derivatives page (maybe it's there) you also cited. So obviously, my example is doubly wrong:

E(S,T) -> dE = (dE/dS)(const T) dS + (dE/dT)(const S) dT = (!T)dS + (!Cv)dT

(! means "not")

Some peculiar things happens if *all* of the variables are intensive,
but other than that, you can choose any combination of intensive and
extensive variables. Even the all-intensive case makes sense if you
are careful, and is actually kinda elegant.

So it might seem that the most general way of approaching thermo problems is to assume that all variables are always included, for example E = E(S,V,N,P,T), and an equation of state (eg PV=NkT). I haven't looked at this yet, but I don't recall thermo texts with this approach. At first glance it would appear that this approach would not lead to elegant relationships, but this may be a premature conclusion.

---------------

The reason I was reviewing this was to reconsider the simple heating/cooling of water (for example) without resorting to the dE = d'Q + d'W formulation, as per previous discussions here. To use J Denker's terminology, uncramped vs cramped (if I followed it correctly). But I'm finding the going tough. The standard treatment is simply "you heat the water, d'Q = CvdT, and if you want the entropy change, solve TdS = CvdT." My perhaps naive approach is to describe the process non-mathematically as

1. Add/remove energy to/from the water
2. Some of the energy goes to changing the entropy of the water
3. Some of the energy goes to changing the temperature of the water

I would then like to give this a more formal mathematical expression. Importantly, I would like it be valid for a phase change, during which all of the energy goes to changing the entropy, and none to changing the temperature.

Stefan Jeglinski