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

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Joel's comment:
> ...
> It should be pointed out that the above proportionality
>
> E_avg_kinetic =(D/n)*k*T
>
> applies only to an object without structure; i.e. one for which there are no
> rotational degrees of freedom or potential energy degrees of freedom that
> can "store" energy.

I disagree.  The above equation is true for the translational degrees of
freedom for the center of mass of a microscopic particle as long as the
two provisos A) classical, & B) power law dependence for the
translational KE on the momentum magnitude hold, *regardless* of what the
potential energy situation those particles may find themselves in.  The
object can be a solid, liquid, gas, mixture, plasma, pure substance, etc.,
and the particle can be a monatomic gas atom, a simple molecule, a
complicated molecule, or even a protein molecule in the very complicated
environment in a living cell.  The presence of rotational degrees of
freedom and configurational degrees of freedom that have potential energy
of various sorts are perfectly compatible with the above equation, *as
long as* we agree to interpret the l.h.s. of the above equation as
including *only* the translational kinetic energy of the center of mass
of the particle as a whole.  It is *not* meant to include any other
energy that may be possessed by other degrees of freedom in a possibly
complicated system. And when I wrote that equation I *did* make that
stipulation.

I thought I carefully made the point in my post that the equation was
only a consideration of the kinetic energy of the *translational*
degrees of freedom for the particles' centers of masses.  Certainly, if
one wants to consider the temperature dependence of the average energy of
some of the other microscopic degrees of freedom for the system, one may
do so.  But such consideration was beyond the scope of my post.  Also if
one *did* include the temperature dependence of the energy of some other
degrees of freedom (say, because one was interested in some other
contributions to the total specific heat, for instance) such as any
rotational or vibrational modes that might be present, then each one
would make its own contribution to the specific heat, but such degrees of
freedom do not necessarily have their energy proportional to the absolute
temperature.

In the case of vibrational modes the Equipartition Theorem only holds for
temperatures much hotter than the Debye temperature for a solid or the
Einstein temperature for a vibration mode in an isolated molecule (so
that such modes can be treated as fully classical degrees of freedom) but
not so hot that nonlinear anharmonic terms in the interparticle force
laws become important so that a non-power law dependence for the energy
becomes significant (and at which point the probability of ionization
might become nonnegligible).

In the case of rotational degrees of freedom, they tend to be classical
and obey the Equipartition result *as long as* the temperature is high
enough so that k*T >> (h^2)/I where I is the moment of inertia for the
rotational mode of interest.  Of course, if the temperaure is high
enough (and excites sufficiently high rotational angular momenta states
that) the moment of inertia becomes angular momentum dependent, and the
rotational kinetic energy is then no longer a quadratic power law
function of the angular momenta, and then the temperature proportionality
for the rotational kinetic energy no longer holds.

> The modifications necessary follow the lines of
> discussion that David outlined.  Any degree of freedom proportional to the
> square of a generalized coordinate or its conjugate momenta contribute
> (1/2)*k*T to the average energy.  So the typical rotational degrees of
> freedom, of the form (1/2)*I_1*omega_1^2, do so; and SHO potential energy
> degrees of freedom, good ole (1/2)*K*x^2 contribute to the average energy.
> Different power laws follow for these as well as per David's comments.

All this is true assuming the degrees of freedom can be treated
classically (as quadratic degrees of freedom).  But I explicitly did not
want to consider the contributions of such energy sources in the equation
above and are not relevant for my comments in my previous post.  If we
wanted to calculate the heat capacity for our system we *would* need to
include all such sources. But such consideration is not needed if we only
wish to claim that the translational kinetic energy is directly
proportional to the absolute temperature.  If other energy modes are
approximately proportional to the absolute temperature as well, then
that's just fine, but they are not relevant for the claim made solely for
the translational degrees of freedom.

David Bowman
David_Bowman@georgetowncollege.edu