Re: W+Q deprecated

["John S. Denker" <jsd@MONMOUTH.COM>]

I wrote:

> >    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.

Carl E. Mungan wrote:

> Can you give me an example or two of a reversible process for which
> this division into W and Q fails?

For starters, it fails for anything that isn't
at (or very near) thermal equilibrium.  For
example, a fuel cell converts H2 + O2 to H2O
plus energy _without_ burning the reactants to
form heat. The laws that are fundamental to what
I call thermodynamics (i.e. conservation of
energy and nondecrease of entropy) apply just
fine to fuel cells.  But I have no idea how to
use the W+Q notion at all in this scenario, let
alone why it should be considered primary and
fundamental, the alleged starting point of the
subject.

It also fails in a grand-canonical situation;
you would need additional terms on the RHS of
the "E=W+Q" expression.

As I said in the last note, these problems can
(maybe) be fixed by sufficiently restricting
the domain of discourse, and by adding additional
"laws", but that seems Procrustean at best.
Thermodyamics (if done right) has tremendous
power and generality.  It seems a shame to
amputate it just to make it fit the "W+Q"
approach, when vastly better alternatives are
available.


> Or to put the question another way, is there a problem with the
> standard formula dS = dQ_rev/T which implicitly assumes that Q_rev is
> the same for any reversible path connecting the initial and final
> states of the system (assumed to be equilibrium states)?

The question is remarkably well stated.

I do not define S in terms of "Q" or T.
I consider entropy to be primary and fundamental.
Entropy is well-defined even in cases where the
temperature is zero, unknown, irrelevant, or
undefinable.

Speaking less theoretically:  the implicit assumption
about Q_rev is just plain false.  Q_rev is not independent
of path, unless all paths have the same temperature.
Very often we need to know S under conditions where
the temperature depends on path.

As far as I can tell, any time you are tempted to
write "dQ" alarm bells should go off.

The purpose of a good notation is to make things
that are semantically bogus look syntactically
bogus.  To my eyes, "dQ" is bogus.  In contrast,
it's perfectly fine to write TdS where
  T is a thermodynamic potential, a zero-form
  S is a thermodynamic potential, a zero-form, and
  dS is a thermodynamic potential, a one-form.

But there is no "Q" of any kind such that TdS "=" dQ.
(Except under highly contrived and impractical
restrictions.)

> > You can if you like define
> >        q := TdS

> I'm only happy with this for a reversible process.

Agreed.  I should have said something about
the limited range of validity.

> I agree this is a problem. I take it you're not fond of "inexact
> differential" notation,

Right.
Just to be clear:  We agree it's OK to have an
inexact one-form. TdS is an inexact one-form.
But by definition of "inexact" TdS is not
the exterior derivative of "Q" or anything else.

> ie. my use of dQ_rev above could be written
> "d_bar Q_rev" to emphasize the point that Q is not a function and so
> dQ does not mean a derivative but just a small amount.

I need to see a better definition of Q_rev
before I sign off on that.  I'm skeptical
that any useful Q_rev exists, except in cases
so restricted as to be of negligible practical
value.  And the pedagogical value seems to be
worse than nothing.  Remember my analogy to
cancelling the "6" in 16/64ths.  Even if it
works in one or two cases, it's misleading in
general.