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Re: W+Q deprecated (was: Kinetic Energy elephant)



I wrote:
>> None of the above is sufficient to salvage the
>> ghastly "thermodynamic" formula
>> delta E = W + Q
>>
>> That formula is a Bad Idea and no amount of
>> tinkering with the definition of W will fix it.

Carl responded:
> I credit you with getting me to slightly loosen my attachment to this
> standard version of the FLT.
>
> However, isn't it just fine for reversible processes?

That's a good question. Two or three answers:
1) To answer the letter (but not the spirit)
of the question, no, just saying "reversible"
isn't enought of a restriction to make the
W+Q law work as a starting point for thermodynamics.
2) More to the spirit of the question, you
could add a few additional "un-numbered" laws and
restrict the domain of discourse to include
Carnot-cycle engines and not much else, in which
case the W+Q problems probably wouldn't bite you
too badly.

Now to answer a slightly different question, even
if it were "just fine" for reversible processes,
I would still find it objectionable.

To explain where I'm coming from, here's a
greatly-exaggerated example. Suppose we
want to simplify the fraction
16
----
64

Cancel the "6" upstairs and downstairs to get
1
---
4

It's the right answer -- but that doesn't make
it "just fine" in my book. The procedure is
completely bogus and doesn't generalize at
all well.

> Or is your dislike of this formula based on the idea that real-world
> processes are irreversible? But don't we often make idealizations in
> intro physics?

Not just intro -- all of physics involves lots
of approximations.

To zeroth order, I suppose any approximation is
better than knowing nothing at all.

But what separates science from wild-ass guessing
is to make a _controlled_ approximation. At the
very least one needs to know when the approximation
is likely to hold and when it isn't.

It is particularly galling to hear the W+Q
expression called the "first law of thermodynamics".
Thermodynamics is a respectable subject and
doesn't deserve to be slandered that way. Maybe
the W+Q law should be called the first pseudo-law
of reversible thermo-babydynamics.

==========

One of the meta-laws of science is that I
shouldn't complain about the imperfections of
some approximation unless I have something
better to offer.

In this case there is something so much better
that I feel empowered to complain loudly about
the W+Q notion. Specifically: focusing on the
conservation of energy and the nondecrease of
entropy seems like a hands-down winner. It just
works. Reversible or irreversible. Easier to
teach, easier to learn, easier to use, fewer
limitations and exceptions to worry about.....

Every time you write down W+Q in class it
tempts the typical student to think there is
some quantity W and some quantity Q that are
separately conserved. They're almost force
to think in those terms, because they don't
have any better way to think about it. And
it's not just students; I've seen big-time
physics professors blithely assume that W
and Q were thermodynamic potentials (functions
of state) which is totally bogus. (Maybe
not quite as bogus as cancelling the 6 in
16/64, but certainly not something to be
proud of.)

It's hard just to explain how deep the problem
is. We don't have any very good terminology
for discussing the real issues.

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

The following is a teaser. It's something I'm
working on. It isn't finished, and there's no
guarantee that I can ever make it work 100%.
But it feels like I'm on the right track.

The recent Clifford Algebra adventures got me
thinking about differential forms. I suspect
the machinery of differential forms can be
used to shed light on elementary thermodynamics.

We borrow from Mr. Clifford and friends the
idea of a graded algebra. The basic thermodynamic
potentials like E, T, S, P, V ... are grade=0
objects, i.e. zero-forms. Note that W and Q
are definitely _excluded_ from this list because
they are not zero-forms; see below.

The exterior derivative dE is then a one-form.
Similarly dT, dS, dP, dV ... are one-forms.
They are, by construction, exact one-forms,
meaning that each is the exterior derivative
of some zero-form.

We can multiply these critters. PdV is a one-form.
TdS is a one-form. These are actually wedge products
e.g. T/\dS but there's no harm in eliding the wedge
when one of the operands is a grade=0 object.

We can assert
dE = TdS - PdV
which is an equation among one-forms.

You can if you like define
q := TdS
and
w := PdV
where I am very intentionally writing w and q
in lower case. Both of these critters are
one-forms. They are !!not!! exact one-forms.
There is no zero-form "Q" or "W" such that
q "=" dQ
or w "=" dW

Note the heavy use of scare quotes.
See reference for a picture of a non-exact
one-form.

I intend to duck the issue of whether w or
"W" should be called work, or whether q or
"Q" should be called heat. I prefer to
discuss physics ideas, not terminological
holy war.

Taking stock, we see that we can write
dE = w + q
as an equation among one-forms, but we must
!!not!! write any of the following improper
expressions:
E "=" w + q
E "=" W + Q
E "=" dW + dQ
dE "=" dW + dQ

The creepy thing is that practically everywhere
I look I see improper expressions like that. It
tells me that a lot of people aren't thinking
very clearly about the subject.

Reference:
For a half-baked crash introduction to one-forms,
including some possibly-helpful illustrations, see
http://www.monmouth.com/~jsd/physics/thermo-forms.htm

Reminder: this is not finished. Suggestions
on how to make it advance in a good direction
are being accepted.