Re: W-->E-->Q-->U, at my level of teaching

[John Denker <jsd@MONMOUTH.COM>]

At 11:58 AM 11/1/99 -0500, Ludwik Kowalski wrote:

[a long summary]

I agree with 99% of this.

In particular:

> Thus the question of how much of the lost KE -->"heat"
> and how much --> "work" can not be answered in general.
> The answer depends on the collision mechanisms involved
> and on the acceptable time scale. A collision mechanism of
> some kind is always present and the W/Q question can not
> be answered without examining its details. Even in cotton
> some threads behave as "compressed springs".

Yes!

==============

But at the very end things seem to go off the track:

> What is heat? It is thermal energy which is transferred
> spontaneously from one body to another.

That definition of heat seems to be catching, like the flu.

Objection #1:  It begs the question of what is "thermal energy".

Objection #2:  Although there is a set of problems for which that
definition works OK, if it is taken too literally (and alas students often
take definitions literally) then it leads to problems.  To wit:  If heat
consists only of energy that is *transferred* from one body to another,
then the transfer "from" must equal the transfer "to" and we conclude that
the total amount of heat in the world is conserved!

This syllogism (equal transfer implies conservation) may not be what you
intended, but it is an inference the students can reasonably draw and most
likely will draw.

Now, according to the Encyclopedia Britannica, the _caloric_ theory fell
into disuse precisely because it held that caloric was conserved.  We know
that heat is not conserved.  There simply *must* be interactions that
involve some heat-like quantity that is not merely transferred from one
body to another.

NOTE:  It may be possible to partially "paper over" this difficulty by
renaming things so that it is called "heat" when it is being transferred
and calling it "thermal energy" at other times.  But I would argue against
this on several grounds.
  a) Why use different words for something that is fundamentally the same
physical quantity?  In particular, since heat is considered a four-letter
word in some circles, why not get rid of it entirely and continue the
discussion using only the term "thermal energy"?

  The definition cited above can't be used *at all* until we define thermal
energy.  Then, once we know what thermal energy is, we can greatly shorten
the definition of heat by keeping the part that says "heat is thermal
energy" and crossing out the rest.  (If you want to restrict heat to
connote a _change_ in thermal energy that's OK with me.  It makes no
difference.)

  So this is the specific constructive suggestion:  If one defines "heat"
to be synonymous with "thermal energy" (or a change therein) whether or not
it is being transferred from another body, then there really is no problem
with the nomenclature or the physics.

  b) From a pedagogical point of view, it is unreasonable to expect
students to learn a distinction between "heat" and "thermal energy".  It is
unreasonable (and quite unnecessary) to teach them to say the microwaves in
the oven don't heat the butter.  You know that as soon as they leave the
classroom they will revert to saying the microwaves heat the butter.

I've been saying for years that the microwaves heat the butter, and the
gods haven't struck me down yet.  And it hasn't in the least impeded my
ability to do thermodynamics calculations.


> What is work?
> It is F*x, or the integral of F(x)*dx, to be more general.
> To keep things simple, I am referring to a force acting on
> a rigid body and passing through its center of mass along
> the x axis.

That definition works fine within its stated limits, because of the careful
restriction to a rigid body.

It leaves open the question of how to define work for not-quite rigid
bodies.  We cannot discuss heat production (and maybe not even heat
transfer) without at least a nod to the nonrigid internal degrees of freedom.

  Specific constructive suggestion:  when considering not-quite rigid
bodies, work can be computed as the macroscopic force dotted into the
macroscopic displacement.  This explicitly excludes random small forces and
random small displacements.  We can say that the object is macroscopically
rigid but microscopically full of random wiggles.

This posting still hasn't produced a workable distinction between heat and
work.  So how is thermal energy different from non-thermal energy?

The answer, of course, is entropy.

In particular, consider an apparatus consisting of a pair of coaxial
counter-rotating disks.  There is zero total linear momentum, zero total
angular momentum, and initially lots of rotational kinetic energy......
Then I throw a switch and the two disks brake against each other.  The
macroscopic kinetic energy goes away, and thermal energy appears.

In the initial low-entropy situation, the coordinates of the atoms in the
disk are given by 10^25 equations in one unknown.  That is, if I tell you
the rotation rate you know what all the atoms are doing.  In contrast, in
the later high-entropy case, there are 10^25 equations in roughly 10^25
unknowns.  The energy has not changed.  Only the entropy has changed.

(The previous paragraph is a statistical-mechanics statement, not a
classical thermodynamics statement.  There's a classical version, but I
don't feel like going into it.)

IMHO if students don't grok the relationship between the spinning disks and
the warm disks, then they don't know anything about thermodynamics.

The notion that we can distinguish one type of energy from another based on
how it got there is guaranteed to cause problems.  If I show you a warm
disk, it really doesn't matter whether the warmness was "spontaneously
transferred from another body".  The only thing that matters is that there
is kinetic energy in there, and it is in a high-entropy form.

On this topic, here's an analogy:  Energy is a state variable.  If I show
you a disk on the second floor, to evaluate its potential energy content
all you need to know is its height and weight;  you don't need to know
whether I carried it up the stairs or took the elevator.  Similarly entropy
is (plus or minus a couple of quibbles) a state variable.  If I show you a
liter of helium, to evaluate its entropy all you need to know is its
pressure and temperature.  You shouldn't care whether the entropy flowed in
from some other body or was created _in situ_.

Note that work is an energy change, not necessarily an energy transfer.  In
a dissipative situation, work on body A need not be balanced by work on
some other body B.  Work is not conserved.  Heat is not conserved.  Energy
is conserved.

To those who say this point of view is heretical, I refer to Feynman vol I
equation 44.1 which refers to the U, Q, and W of a single system.  I also
refer to equation 44.10 which says
        Q = S T
where we have an absolute measure for T and (via the third law) an absolute
measure for S.  Therefore we have an absolute measure of Q which can very
reasonably be interpreted as the absolute thermal energy content of the
system.  Maybe I'm a heretic, but I'm happy with the company I keep.

=====

To summarize:

a) What is thermal energy?  Qualitatively and intuitively, the warm disks
have thermal energy whereas the spinning disks had nonthermal energy.  More
formally, thermal energy is microscopically random kinetic energy.
(Quantitative definitions are available, but that's beyond the scope of
this email.)

b) What is heat?  Either thermal energy (Q) or a change in thermal energy (dQ).

c) What is work?  A change in an object's non-thermal energy.

I recommend these definitions as simple, non-circular, workable, consistent
with each other, reasonably consistent with vernacular usage, and
consistent with the usage of thoughtful experts.

______________________________________________________________
copyright (C) 1999 John S. Denker             jsd@monmouth.com