Re: Temperature and Pressure at atomic level?

In article <1154511549.700037.197820@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, "tadchem" <tadchem@xxxxxxxxxxx> writes:

mmeron@xxxxxxxxxxxxxxxxxx wrote:
In article <1154466473.218041.267040@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, "tadchem" <tadchem@xxxxxxxxxxx> writes:

mmeron@xxxxxxxxxxxxxxxxxx wrote:

<snip>

So, staying away from exotic
stuff I would say "under common conditions temperature is proportional
to average energy per particle". Note, "energy", not "kinetic
energy". There is nothing separating kinetic energy from any other
kind, here.

Yes there is. Motion.

It so happens that the first (and, unfortunately, often the last)
statistical physics example students encounter is this of an ideal
gas.

If they take physical chemistry (as *every* chemist must, and most
physicists *should*) they also get into rotational, vibrational, and
electronic partition functions, each of which freely exchanges energy
with the other forms, rapidly reaching thermal equilibrium - unless of
course the particles (atoms,. molecules, ions) are *maintained* in a
non-equilibrium state.

Yes, of course they reach equilibrium. None of this contradicts
anything I said.

In this specific case the *only* degrees of freedom are those of
translational motion and, therefore, the *only* energy present is
kinetic. So there, indeed, temperature is just the average kinetic
energy per particle. This, however, is just a very specail, very
simple case. Not the general one.

At thermal equilibrium the temperature is the same for translation,
rotation, vibration, and electronic states. The energy levels will
differ, of course, but the distribution of particles among those energy
levels will be governed by Boltzmann statistics. Only in the cases in
which the spins of the particles becomes relevant to particle energy
will Bose-Einstein or Fermi-Dirac statistics come into play.

Again, non of this contradicts anything I said.

If you'll say "the average kinetic energy of a particle equals (up to
proportionality coefficients etc.) the temperature", this is correct.
If you'll say "temperature is the the average kinetic energy of a
particle", this is misleading, since temperature can be many other
things as well.

Mati Meron | "When you argue with a fool,
meron@xxxxxxxxxxxxxxxxx | chances are he is doing just the same"

So, please enlighten us all.

Assuming the presence of thermal equilibrium, in what more general case
is the average kinetic energy per particle *not* proportional to the
temperature? Or, alternatively, when is temperature *not* proportional
to the average kinetic energy per particle?

I did not say that it is not proportional. Mmind you, cases like this
can be easily found, when you're tlaking about the electrons in solid,
then even under normal conditions the distribution is not even close
to MB, in fact it is Fermi-Dirac to a good approximation. But I do
not intend to engage in nitpicking here. All I mean here is that
there is a difference between "the tmeperature is proportional to the
mean kinetic energy ..." and "the temperature *is* the mean kinetic
energy". The first is fine for most standard situations, the second
conveys the meaning that temperature represents kinetic energy alone,
which is false. Not a big problem for those who know what they're
doing but the second has the potential to completely mislead laymen,
and laymen constitute a big part of the crowd here.

Mati Meron | "When you argue with a fool,
meron@xxxxxxxxxxxxxxxxx | chances are he is doing just the same"
.

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