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Re: KE & temperature (was: Newton's 3rd law? ...)



Regarding Rick's disagreement:
I have to strongly disagree with David here. The derivation of absolute
temperature (as a measure of the average translational kinetic energy per
molecule due to random motion) IS, IMO, very accessible to AP level students
and IS a very good example of how a concept such as temperature that seems
far afield from their basic kinematic and dynamic studies can be shown to
arise from just such studies.

The concept of temperature does not arise from either "kinematic" nor
"dynamic studies". The concept of thermodynamic temperature is defined
as the rate of change of a system's internal thermal energy w.r.t. the
change in its entropy where the change is taken both as a quasistatic
perturbation which preserves equilibrium, *and* under conditions such
that no work is done by (or on) the system during the change. This
concept has little, if anything, to do with both dynamics and kinematics.
It is a (quasi)static equilibrium statistical concept. The particular
dynamics that drives both the approach to equilibrium and thermal
fluctuations in equilibrium is not a part of the meaning of the
temperature. What *is* relevant to the concept of temperature is how
fast the relative number of allowed accessible microscopic states
increases as the system's internal energy increases. I contend that the
proof of how this concept relates (i.e. proportionally) to the mean per
particle translational kinetic energy is beyond the ken of typical HS AP
students.

To be sure, the Algebra text derivations do
not consider any of David's subtleties and usually starts with an assumption
of the Universal Gas Law as part of the derivation.

The assumption of the Ideal Gas law cannot demonstrate the result you wish
to believe is derived in the "Algebra text derivations", if, for no other
reason, their assumption of the Ideal Gas law automatically restricts any
conceivable validity to just the case ideal gases rather than the general
result. The actual result of the general proportionality of
thermodynamic temperature to the mean translational kinetic energy
applies to any general classical substance with particles whose centers
of mass are allowed to move (at least some tiny bit, as in a crystal
lattice) obeying *any* equation of state, and has *nothing* necessarily
to do with ideal gases as such per se.

What is necessary for the Ideal gas law to hold is that the particles
that make up the system must not exert any net forces on each other no
matter how close those particles get to each other, and that the
distribution of positions of each particle must be independent of the
locations of each all the other particles. This has *nothing* to do
with the Equipartition-ish result of the absolute temperature being
proportional to the average translational kinetic energy. One concept
(i.e. KE/temp proportionality) has to do with the power law dependence of
the translational kinetic energy on the particle momentum magnitude, and
the other (Ideal Gas law) has to do with the absence of interparticle
potential energy terms between the locations of the centers of mass of
the system's unbound particles in the system's Hamiltonian. These are
completely different concepts and relate to *different* microscopic
degrees of freedom (i.e. one relates to translational momentum
degrees of freedom and kinetic energy, and the other relates
to positional degrees of freedom and potential energy, or more
precisely, the absense of such a potential energy).

Also since those "Algebra text derivations" start with the Ideal gas law,
(which explicitly involves the temperature) they presume the concept of
temperature is already somehow related to the product of the pressure
and the volume per particle. This means they presume a meaning for the
absolute temperature which is a priori proportional to to the product
of the pressure and the per-particle volume. What the "derivation"
merely does by its kinetic arguments is simply to relate the pressure to
2/3 of the translational kinetic energy density for a substance obeying
the positional independence conditions of an Ideal Gas. A substitution
of the Ideal gas law into this relationship produces the desired result.

This "derivation" was already *admitted* by Ed to not really prove the
desired result, and his (2 yr) previous AP class was even sharp enough
to call him on it. In discussing this swindle with his class Ed
appealed to some sort of principle that real physics involved making
plausibly interpreted jumps of logic between a microscopic and
maroscopic description. Even though I disagree with Ed about this view
of the essence the physics in this situation, I certainly agree with
him that such a 'derivation' is not a real proof of the claim that the
mean per particle translational kinetic energy is proportional to the
thermodynamic temperature. But it is probably all that one can
expect to do for a HS AP class, since the real proof is too hard for
them to follow as it involves calculating the canonical partition
function for the general system which involves the use of certain kinds
of math that the students are not expected to have yet seen.

But that being said,
looking at the wall collisions as a source of the pressure, the change in
momentum in such collisions, the time of flight between ends of the box,
etc., etc. is a very important exercise for this level of student to work
through. It really shows how to work out new physics from old, basic
principles.

The proportionality of the mean translational KE to the absolute
temperature holds just as well for systems (such as for internally bound
crystals) that exert *no* pressure on their environment--walls or not.

David Bowman
David_Bowman@georgetowncollege.edu