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*From*: John Denker <jsd@av8n.com>*Date*: Fri, 15 Jun 2007 16:56:13 -0400

Replacement version, part 2.

On 06/15/2007 03:09 PM, Stefan Jeglinski wrote in part:

> [1] dE = (dE/dS)(const T) dS + (dE/dT)(const S) dT

> [2] dE = (dE/dV)(const T) dV + (dE/dT)(const V) dT = CvdT (for dV = 0)

> [3] dE = (dE/dS)(const V) dS + (dE/dV)(const S) dV = TdS (for dV = 0)

....

> Now we turn to the phase change for which dT = 0. For this case, Eq 2

> implies that there can be no phase change without a volume change,

> unless (dE/dV)(const T) is exactly zero. In fact, more than implying

> so, it seems to constitute a proof of sorts (?)

No. The problem is that equations [2] and [3] are not powerful

enough to handle this case. They are incomplete. We need a third

term on the RHS of equation [3]. The conventional choice is a term

involving dX, where X is the fraction of ice in the water ... but

choosing S as the "extra" variable is perfectly allowable; see below.

The experiment is set up so that

-- V is an independent variable (or independent non-variable if you like)

-- E is an independent variable

-- T is an easily-observed dependent variable

-- X (i.e. the fraction of ice in the water) can be considered

an observable dependent variable.

-- My recommendation is to treat S as a dependent variable.

Experimentally, there is (roughly speaking) a distinction between

independent and dependent variables. Conceptually, though, it

usually pays to treat all variables on an equal footing.

> Anyway, for the phase change Eq 1 would seem to be my choice here,

> since I don't a priori know how the volume changes.

You can continue to hold constant the volume of the ice+water system.

This is the least of our worries.

> I know that dE

> will be the latent heat of vaporization for the water-steam phase

> change, or that of fusion for the water-ice phase change. But to use

> Eq 1, I shall need to find out what thermodynamic quantity(s)

> represents (dE/dS)(const T). Or use an even better relationship

> (E(S,P) and enthalpy comes to mind obviously).

It is a pain in the neck to observe S operationally. There is

no handy entropyometer. Therefore it is more practical to measure

something else, and calculate S at a later stage ... but

conceptually, S is a perfectly well-behaved variable, as

illustrated at

http://www.av8n.com/physics/img48/s-t-phase-change.png

> My greater concern at this point is the observation about linear

> independence of, say, T and S in Eq 1. How does one avoid this

> pitfall in general?

Assuming dT and dS etc. are not simply multiples of one another,

which is almost always OK (though you have to check), then the

key to having a linearly independent basis set is to not have

*too many* vectors in the set.

Again, it really helps to imagine the "system point" moving in

thermodynamic state space. Here's how I visualize it:

http://www.av8n.com/physics/img48/s-t-phase-change.png

That is, in the calorimeter, in the pure liquid phase, as we

change the energy the point moves along a path of changing E,

changing T, unchanging V, and unchanging X=0. Then when we

hit the phase transition, there is a *corner* in the path.

During the transition, the point moves along a path of

changing E, unchanging T, unchanging V, and changing X.

To repeat, at the corner one variable (X) becomes "alive" and

another (T) becomes "dead". The contours of constant S cross

at an angle to the path, such that S continues to be a smooth

function of E on both sides of the corner, i.e. in both the

one-phase and two-phase regions.

This is tricky, because T and X are "natural" variables at

experiment-time ... but they are badly behaved at analysis-time.

I know I've been talking a lot about linear independence, but

I think in this case the problem is elsewhere.

-- Equation [1] and equation [2] have a problem with the dE/dT

term in the two-phase region. You can't evaluate dE/dT when

there is no possible dT.

++ Equation [3] dodges this bullet, and is the conventional way

to attack this problem.

**References**:**[Phys-l] thermo differential and extensive/intensive variables***From:*Stefan Jeglinski <jeglin@4pi.com>

**Re: [Phys-l] thermo differential and extensive/intensive variables***From:*John Denker <jsd@av8n.com>

**Re: [Phys-l] thermo differential and extensive/intensive variables***From:*Stefan Jeglinski <jeglin@4pi.com>

**Re: [Phys-l] thermo differential and extensive/intensive variables***From:*John Denker <jsd@av8n.com>

**Re: [Phys-l] thermo differential and extensive/intensive variables***From:*Stefan Jeglinski <jeglin@4pi.com>

**Re: [Phys-l] thermo differential and extensive/intensive variables***From:*John Denker <jsd@av8n.com>

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