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Re: Entropy and states of matter



Regarding John Denker's comments about his thawing/freezing samples A
& B:

...
For the next level of detail, draw the curves
representing the temperature (for each sample)
as a function of added energy. Colored chalk
may help here.
/
/
/~~~~~~~~~~/
/ /
/----------/
/
/
/

The heat capacity of the two solids is about
the same. The heat capacity of the two liquids
is about the same. The latent heat of melting
is about the same. The big difference is that
the melting occurs at a different temperature.
The plateau corresponds to the physical process
of sucking the entropy out during freezing, and
dumping entropy in during melting.

Actually, I think it may be more helpful to consider that the plateau
corresponds to the physical process of removing thermal potential
energy during freezing and acquiring thermal potential energy during
melting (assuming the transition is done under constant temperature
conditions). The latent heat of transition in a first order phase
transition corresponds to a net change in the mutual potential energy
of the constituent particles that accompanies their relocation w.r.t.
each other because their large scale reconfiguration that happens as
a result of the phase change repositions them enough w.r.t. each
other so that their individual interparticle potential energies tend
to change by significant amounts (almost) all in the same direction.
These individual changes then accumulate and change the total
potential energy in the whole system by a macroscopic extensive
amount.

The entropy change that is associated with this change in potential
energy is a consequence of this energy change happening at some
particular temperature. The physical process of changing the entropy
is also associated with the large scale reconfiguration and
redistribution of the particles in space. The high energy phase is
also the high entropy phase because usually in general it requires
more bits of information to precisely determine the microscopic state
of the system when its particles have been pulled (against their
mutual attractive forces) farther away from each other simply because
of all that extra per capita phase space that is now made available
to the particles. This extra (physical and phase) space gives the
particles many more choices in where to be found.

The kinetic energy of the particles in the transition is unaffected
because the transition happens at a fixed temperature. So as the
melting transition proceeds and the system acquires its latent heat/
thermal energy, all of it goes into increasing the mutual potential
energy between the particles. If the transition to the high energy
phase had (at least partially), instead, happened adiabatically then
the system's temperature would *decrease* (as in the case of
evaporative cooling, or salt-induced melting) since in that case the
increased potential energy of the particles comes from a
corresponding decrease in the particles' overall kinetic energy.

If both transitions really had the same latent heat but the two
transitions occurred at two different temperatures, then they would
have different entropy jumps. This is because the entropy change in
the transition is the latent heat of the transition divided by the
(absolute) temperature of that transition (so different transition
temperatures result in different entropy jumps when keeping the
latent heat constant).

...
The argument is incomplete at this point, because
we've been measuring the energy, not directly
measuring the entropy. So we must show the
connection.

The connection is via the *temperature* at which the
transition takes place. The temperature is, by definition,
the proportionality factor between quasistatic infinitesimal
energy increments and quasistatic infinitesimal entropy
increments (under conditions for which no macro-work is done
on the system at hand). The relationship between energy
changes (under no-work conditions) and entropy changes is
quite analogous to that between energy changes (under no heat
conditions) and volume changes. In the first case the
appropriate factor relating the changes is the thermodynamic
temperature, and in the later case the appropriate factor
relating them is the (negative of) the pressure.

We can understand the what the value of the transition temperature is
in physical terms by observing that the transition temperature is
simply the quotient of the latent heat of the transition divided by
its entropy change. So *if* we can quantify how much extra inter-
particle potential energy the particles acquire because of their
redistribution in space when the transition occurs, and *if* we can
also quantify how much extra information is required to specify the
system's microstate (because the particles redistributed themselves
more randomly with now more available phase space per particle) then
the quotient of these two quantities *determines* the temperature of
the transition.

David Bowman
David_Bowman@georgetowncollege.edu