>  dE = (dE/dV)(const T) dV + (dE/dT)(const V) dT = CvdT (for dV = 0)
>  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  and  are not powerful
enough to handle this case. They are incomplete. We need a third
term on the RHS of equation . 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.
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  and equation  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  dodges this bullet, and is the conventional way
to attack this problem.