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

[John Denker <jsd@av8n.com>]

On 06/15/2007 09:22 AM, Stefan Jeglinski wrote in part:
> ....  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.

Hmmm.  I'm not sure that's wrong ... but let's just say I
wouldn't have done it that way.

I prefer a more pictorial, spatial approach to thermodynamics.
I imagine there is a point (representing the system) moving in
an abstract space.  At every point in this space, there are
definite values of all the variables E, T, S, V, P, N, H, G, F,
et cetera.  This includes dependent as well as independent
variables, treating all variables on an equal footing.

As the point moves from point A to point B, the energy changes
from E(A) to E(B). I don't find it necessary or helpful to
partition the energy change into an entropy-related piece and
a temperature-related piece. I might say things like
 -- the energy changed because the energy changed.
 -- the energy changed because we pushed a bunch of
  energy across the boundary of the system, using the
  heater resistor.

Now it is *also* true that when we heat liquid water, while
the energy is changing from E(A) to E(B), we can additionally
observe that:
 -- the entropy is changing from S(A) to S(B),
 -- the temperature is changing from T(A) to T(B),
 -- the pressure is changing from P(A) to P(B),
 -- the enthalpy is changing from H(A) to H(B),
 -- the compressibility is changing from K(a) to K(B),
 -- and a jillion other things.

but in my picture, those additional observations are not
necessary to the analysis.  In particular, in heat capacity
experiments, it is traditional and convenient (albeit not
obligatory) to choose T, V, and X as the independent variables,
where V is the volume, T is the temperature, and X is the
fraction of ice in the water, i.e. X=0 for the liquid.

Having chosen variables {T, V, X}, it is natural (and normally
goes without saying) that we use {dT, dV, dX} as basis vectors.

Given these chosen quantities, you are allowed to expand dE
in terms of dT, dV, and dX ... but remember this expansion
depends on a choice, and is therefore not fundamental physics.
We are talking about decomposing a vector into components
according to a chosen basis.  The vector is fundamental
physics;  the components are not fundamental.  You are free
to choose whatever basis you like, but remember that others
may choose differently.

*** So you can see my point:  Talking about an entropy-related
    piece of the energy is not necessarily wrong, but it makes
    about as much sense as talking about a compressibility-related
    piece.  It is unnecessary, unconventional, and non-fundamental.

Also BTW, if you choose an /overcomplete/ basis, additional weird
things are going to happen, including (for starters) non-unique
expansions, which are even more obviously non-fundamental.