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

I'm in the odd position of agreeing with everything David says below, except
the statement that he disagrees with me (except as noted below). I didn't
intend my post to indicate any disagreement; but to be interpreted as a
parenthetical comment. The original post was clear as to its intent. David
has pointed out that I was being a bit rash in saying "only to an object
without structure"; and called me on that point. The equation in question
certainly applies if we are talking about the average translational kinetic
energy degrees of freedom of the center-of-mass of the molecules in
question, *even if they have structure*.

I only wanted to indicate that other degrees of freedom may be considered
and related to temperature as well; and in fact must be when one considering
properties like specific heat; (which of course wasn't the point of David's
post, which is why I viewed my comment as being parenthetical).

Joel

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