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

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding where Ed Schweber wrote:
> ...
>   BTW: two years ago I had a particularly bright AP class which was
> disappointed that I didn't really prove that average KE per molecule was
> proportional to Kelvin temperature and we had an extended and (I think)
> productive discussion about how the microscopic and macroscopic
> interpretations have an entirely different vocabulary that cannot be equated
> through equations but only by interpretation. This is what physics is (at
> least in my opinion).

It seems to me you were wise to not attempt to prove for your students
that the (thermal equilibrium) avg. translational KE of the center of
mass degrees of freedom of the particles (treated classically) was
proportional to the absolute temperature--*not* because such a proof is
not available, but because the proof goes way beyond the level
appropriate for a HS AP class.  I don't think foregoing the proof is as
much "what physics is", as it is an exercise in good pedagogical
judgment.  As far as the differences in vocabulary and concepts between
macroscopic and microscopic physical descriptions goes, I believe the
field of statistical mechanics does an admirable job of negotiating
between these levels of description, bridging the gaps between them, and
forging a conceptual unification of them--including both their respective
equations and their interpretations.

BTW, The fact that the absolute temperature is proportional (for a
classical system of particles in thermal equilibrium) to the avg.
translational kinetic energy associated with the center of mass degrees
of freedom of those particles is a consequence of the fact that the
kinetic energy of the center of mass degrees of freedom of each of the
particles is a rising *power law function* of the momentum magnitude for
those particles.  If some other different functional relationship held
then the KE would not be proportional to the thermodynamic (absolute)
temperature.  In particular, the equation: E_avg_kinetic = (3/2)*k*T for
a classical particle thermally equilibrated with its environment has the
coefficient 3/2 because of the 3 dimensions of space in which the
particle is allowed to translate in, and because of the fact that the
translational kinetic energy of the particle is proportional to the
*square* of the particle's momentum magnitude.  If, more generally, the
particle was allowed to move in D dimensions and its translational
kinetic energy was proportional to the n-th power of its momentum
magnitude, then the equation would become: E_avg_kinetic =(D/n)*k*T,
which is *still* a proportionality.  The proportionality only breaks down
for: A) nonclassical behavior of the particles due to quantum mechanical
or quantum statistical effects, and/or B) relativisitic effects where the
simple quadratic power law relationship: E_kinetic = p^2/(2*m) is
replaced by: E_kinetic = m*c^2*(sqrt(1 + (p/(m*c))^2) - 1) which is not
a power law except at low enough momentum so that the particle is
effectively Newtonian, or at high enough momentum so the particle is
effectively an ultrarelativistic particle of negligible mass.

David Bowman
David_Bowman@georgetowncollege.edu