Re: [Phys-L] temperature versus energy

[John Denker <jsd@av8n.com>]

On 02/02/2018 10:59 AM, Anthony Lapinski wrote:

> I'm teaching this (heat) right now. I call temperature a measure of the
> average kinetic energy of the RANDOM motion of the atoms in an object.

I wouldn't have said that.  That's one of the misconceptions
that Hewitt was selling in this month's TPT Figuring Physics.

Like most pernicious misconceptions, it is sorta tangentially
related to the right answer.  However, "sorta half right" is
not the standard we expect of fundamental physics concepts.

Equipartition tells us there should be ½kT of energy for
each quadratic degree of freedom, so the translational
KE will be proportional to temperature.... HOWEVER that
is not all of the energy, nor even all of the kinetic
energy.

The distinction is not important for a monotonic ideal
gas....  HOWEVER there is a lot more to physics than
monatomic ideal gases.

For a harmonic oscillator, on average half of the energy
(thermal or otherwise) is KE and the other half is PE.
Both the position and the momentum are quadratic degrees
of freedom.  So for a particle trapped in a parabolic
potential in D dimensions, the thermal energy is D kT.
That's D/2 kT of KE plus D/2 of PE.  Perhaps even more
interestingly, the heat capacity of an ordinary solid
is well explained in terms of thermal phonons, and each
phonon mode is a harmonic oscillator (to an excellent
approximation), so the heat capacity is 50% PE and 50% KE.

  By way of contrast, for a monatomic ideal gas the
  translational degrees of freedom are quadratic.
  The positional degrees of freedom are not quadratic,
  so the ½kT rule does not apply, and when you look
  into it you find that they contribute essentially
  nothing.

  By way of contrast in another direction, for a
  polyatomic gas (e.g. oxygen or nitrogen), temperature
  is a terrible measure of the microscopic kinetic
  energy, because it doesn't account for the rotational
  KE.

==========

Perhaps most importantly:  Even though temperature is
kinda sorta "related" to KE, this is absolutely not
the defining property.  Please don't tell students
anything that would suggest this is a defining property.

There are plenty of situations where the KE is irrelevant,
undefinable, or non-quadratic.  Hint:  Applied fields
and/or quantum mechanics and/or relativity.  One readily-
observable consequence is the fact that the specific heat
of polyatomic gases is a nontrivial function of temperature.
There is absolutely no way to explain this short of invoking
non-quadratic degrees of freedom.

===========

At the high-school level, you could say that the defining
property of temperature is that at equilibrium, everything
has the same temperature -- in situations where the various
subsystems can exchange energy and entropy.  This is not
ideal, but it's about the best you can do without additional
complexity.

As a corollary, if you have a main system in equilibrium
with an ideal monatomic gas thermometer, temperature is
correlated with the KE of the atoms /in the thermometer/
... but what that says about the main system is much more
complicated.  /All/ the degrees of freedom of the main
system will equilibrate, be they KE, PE, quadratic, or
otherwise.  A tall column of gas in equilibrium in a
gravitational field is a nice easy-to-visualize way of
introducing nontrivial PE.  The heat capacity of such
a gas is not what you would get without the field.  Note
that the earth's atmosphere is nowhere near equilibrium,
as is obvious from the fact that it is not isothermal.

At a more advanced level, temperature is defined as the
slope of the energy-versus-entropy curve.  Moore&Schroeder
have lovely diagrams of this; calculus helps but is not
strictly necessary.  It is then an immediate corollary to
say that /equilibrium is isothermal/ whenever the various
subsystems can exchange energy and entropy.  There's no
other way to conserve both entropy and energy.  (Entropy
is conserved at equilibrium.)

On 02/02/2018 11:38 AM, bernard cleyet wrote:

> Is not expansion non-linear?  Are expansion methods comparable to
> non-expansion methods (speed of sound- E-M radiation, etc.)
> comparable, because the the expansion is small, so a linear “first
> term” is “OK”???

Sometimes the linear approximation is OK.

For thermometers that work over a wide range, the
manufacturer fudges the graduations to make up for
the nonlinearity.