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Re: CONSERVATION OF ENERGY



[Note, the long post below was composed before I noticed that Leigh came out
of retirement on the CONSERVATION OF ENERGY thread.]

I'm (re-)entering this discussion hoping to make a few points and then make
a clean escape.

Brian Whatcott quotes Ludiwk and Leigh and commented:
At 22:05 7/14/97 -0700, Leigh Palmer wrote:
[Ludwik]
Why is it misleading to say that "energy is the ability to do work"?

[Leigh]
It's wrong. It doesn't mean anything. Now I retire from the fray.

Leigh

Leigh will not be satisfied it seems by anything less than this:
"Energy: the property of a system that is a measure of its capacity for
doing work"

You might be excused for mistaking Ludwik's construction for the dictionary
definition that I take to be Leigh's view.

Since Leigh has, wisely, "retired from the fray", I thought that I would
make another attempt at speaking for him (I had thought that I was pretty
good at this, but Doug Craigen has taken the art form to a whole new
level).

The reason that saying "energy is the ability to do work" is wrong is that
this common dictionary definition phrase comes closer to describing *free
energy* (or availibility, as Tom Wayburn likes to call it) than it does to
describing actual (internal) energy. The reason that "it doesn't mean
anything" is that the concept of (thermodynamic) work requires the prior
concept of energy for its definition. The elementary definition for
work of *force times distance increment* is inadequate to describe all the
generalized ways that thermodynamic work can be accomplished, and even the
generalized versions of this require the concept of energy to properly
formulate the generalized concept of a generalized thermodynamic force
anyway. So I doubt that Leigh would even be satisfied with Brian's
definition.

In a different post Tom Wayburn flattered me by quoting a definition of
energy that I submitted here over 2 years ago. I suppose the fact that
this quote is so old is testimony to how much the topics of heat, work, and
energy are perennial favorites around here. They even compete well with
tidal bulges for staying power on the oldies charts.

However, Tom's subsequent discussion shows that he misunderstood my
definition. (At least Ludwik recognized that he didn't understand Tom's
understanding of understanding.) When I said that energy was "that
quantity in a dynamical system that generates an infinitesimal displacement
of that system's microstate in time" I was *not* referring to *any* work
concept at all. Rather, I was describing the system's Hamiltonian function
(or operator in QM) whose value is the system's energy. The phrase
"infinitesimal displacement ... in time" has nothing to do with work, but
rather refers to an infinitesimal canonical transformation generated by the
Hamiltonian which infinitesimally evolves the dynamical (i.e. microscopic)
state of the system forward (or backward, to be precise) in time. When
viewed as an infinitesimal passive transformation, this "displacement in
time" is *sort of* the opposite of the concept of a "virtual displacement"
which is also often introduced in Dynamics. A virtual displacement of a
dynamical system is a hypothetical small change in one or more dynamical
parameter(s) (i.e. degrees of freedom) occurring at a *fixed time*. (The
"clock" or time parameter of the system is stopped, and then someone tweeks
the system in some way, and then the "clock" is restarted and the system's
dynamical evolution resumes. Virtual work is the work done on the sytem by
a virtual displacement.) The kind of "displacement in time" that I was
describing was a sort of the opposite, yet still hypothetical, passive
transformation. Here, at a given time the system's dynamical state is
"frozen" and the system's "clock" is incremented and then it is restarted
with its normal dynamical evolution to follow. (It's sort of like
resetting your watch to Standard Time in the Fall when previously on
Daylight Savings Time.) The dynamical quantity that generates such an
infinitesimal temporal canonical transformation is the Hamiltonian whose
value is the energy. This tendency of the Hamiltonian to generate the
temporal evolution of the system's dynamical state is seen in the Liouville
equation in classical mechanics and in the Schroedinger equation in Quantum
Mechanics. For further information consult any book on dynamics that
treats canonical transformations and Hamilton-Jacobi theory. For the
quantum version consult Dirac's book on QM or any equivalent treatment.

Ludwik Kowalski wrote a list of questions for Leigh, some of which I'll
take a shot at during his retirement.

2)
I wrote that c is a constant, that no mass is lost, and that T is a state
variable. Thus the product m*c*dT is path independent. Leigh called this
quantity heat. But thermodynamics textbooks say that heat is not a state
function; it is a path-dependent quantity. That is why I think that the
phrase "a change in thermal energy" is more appropriate for c*m*dT. I am
still not convinced that this triple product, called dH (ethalpy) by
chemists, should be called heat. Enthalpy is a state function. Are we to
tell students, who studied chemistry before physics, that "ethalpy is heat"?
Enthalpy and energy are identical in our specific (vacuum) situation. Nobody
responded to that line of reasonning. Is it correct or not?

Ludwik, but you *did* implicitly specify a path for the quantity c*m*dT
when you stated that the external pressure for the process was constant
(i.e. 0). For quasistatic processes which have constant pressure paths
the heat absorbed is the *change* in the enthalpy. (Actually, since your
system is a pair of solids rather than a fluid the pressure is not such a
good parameter to specify, rather the stress field over the system's
surface is more relevant.) When you write c*m*dT the dT in the expression
gives the expression away as a(n infinitesimal) process. It is *not* a
state function. It is *at best* a *change* in a state function (such as
the enthalphy, in this case). For arbitrary paths such processes do not
necessarily represent changes in a given state function. I suggest looking
up the distinction between an inexact differential and an exact (or total)
differential. It is true for a fluid under zero pressure in equilibrium
that the enthalphy becomes just the internal energy, but this case is not
possible to achieve with less than an infinite volume. A liquid must
evaporate into a gas once the pressure is lowered (at constant temperature)
below its triple point pressure, and a gas has a positive pressure when
confined in any finite volume. For a solid, as I said previously, the
pressure is not such a relevant parameter. The stress over the solid's
surface plays the role for the solid that the pressure exerted on a fluid
by a container plays for the fluid. In the solid case the an analog of
the enthalphy is defined by adding to the internal energy U a quantity
(analogous to p*V) which is the volume integral of the trace of the tensor
product of the stress and the strain. For your problem the stress (and
strain) by the interfacial contact area between the block and the plate is
different than elsewhere (assuming a nonzero normal force of contact
necessary to produce the frictional force to begin with). As the sliding
process occurs the stress field is *not* held constant, so the enthalpy-
like quantity's change is not identical to the heat absorbed/evolved
anyway. As was said before (I think) by others commenting on this problem,
to the extent that the composite block-plate ssytem is isolated from the
rest of the universe to that extent the system's internal energy does not
change, the heat absorbed by the system is zero, and the work done by/on
the system is also zero. The process of the block stopping and the
system's temperature raising is entirely one the equilibration of an
isolated nonequilibrium system which results in the internal generation of
net entropy as the anamolously large energy contained in a degree of
freedom (a component of the total momentum of the block's center of mass)
is redistributed more equitably among the other degrees of freedom of the
system.

3)
Another line of reasoning was that "heat is that part of internal
energy which is transfered through a system boundary due to a difference
of temperatures". Leigh says "that definition makes the first law of
thermodynamics redundant. There should be no mention of internal energy
in the definition of heat." Please explain the redundancy pronouncement.

The internal energy of a system *cannot* be split up into a heat part and
a work part. Once heat or work is done to a system and the acquired
*change* in the internal energy is accomplished, the system doesn't
remember how its energy changed and no record is kept of energy changes due
to different previous processes. (I think Leigh's redundancy quip
concerned either your misuse of the term heat or misapprehension of what
internal energy is.)

4)
.... How can you say, that "there is no heat involved in the
problem you stated"? How can I describe the so-called "work-heat equivalency"
experiment of Joule without saying that heat is involved?

For the answer to the first question see the last part of my answer to 2)
above. When discussing the Joule experiment and work-heat equivalency,
try saying that when a system is worked dissipatively by an outside agency
(such as weights connected to a paddle wheel by a pulley) that the net
work done by that agency results in the same change in the system's
macrostate (as monitored by, say, a thermometer) as would be accomplished if
instead the system were heated (rather than worked) by the corresponding
equivalent of heat (say by building a fire under it).

6) Leigh:
....
The difference between saying "equal to the change" and "the change is
equal" is beyond my grasp; please elaborate, if you think it is an
important part of your argument.
....

I thought that Leigh was clear when he said "m*c*dT is equal to the change
in the internal energy" that he meant that their *values* were equal *in
this case*, and when he said "not that it *is* the change" that these two
quantities are not the same thing *by definition*, but are conceptually
different things.

David Bowman
dbowman@gtc.georgetown.ky.us