Testing statistical thermo (was: Re: TdS is not dQ or d(anything))

[Pentcho Valev <pvalev@BAS.BG>]

Jim Green wrote:

>   10:48 10 05 2003 , the following was  received:
> > On 05/10/2003 12:14 PM, Bob Sciamanda wrote:
> > > ... the term TdS represents the heat transfer dQrev.
> >
> > I know practically every reference writes
> >         TdS "=" dQ
> > but it makes my skin crawl every time I see it.
>
> Folks, John and I don't always agree, but we do agree on this point.  [I
> emphasize that in my view there is no such thing as "heat
> transfer".  "Heat" is not a substance.  It is a form of "work".]
>
> The problem arises when France did poorly in a war with England -- partly
> because of England's railway transportation system.  Carnot, a diplomat's
> son, could see the advantage of improving the efficiency of steam engines
> if France were to do better the next time.  [Alas they never have.  And
> England has babysat them to this very day.]
>
> So like a good amateur physicist Carnot set up a simple model --  the
> ubiquitous adiabatic cylinder with a piston and an ideal gas -- and based
> his thermodynamics on Aristotle -- both terribly unfortunate because due to
> the awkward momentum of physicists, we still have this unhelpful model in
> every science textbook today.
>
> Now purely just by bald coincidence, the work done by the ideal piston in
> this ideal system doesn't change the system entropy!  [It would be very
> nice if someone could give an example where the entropy of a system could
> be calculated due to any other mechanical macroscopic work where the
> entropy change is zero.]
>
> And because Carnot -- and his followers -- based on Aristotle -- thought of
> "heat" as a substance that flowed into the system, they came to the
> conclusion that dS=deltaQ/T [Note there is no such thing as dQ].

Jim, you certainly have a point but you are not right to generalize and deny
everything. dQ is a meaningful quantity - e.g. it is the energy increase in a
system in contact with hotter surroundings. Also, Carnot was a genius - perhaps
the greatest logician (together with Mendel) among natural scientists. The fact
that one of his premises proved wrong in the end does not matter. Let me give
an example showing where the problem really lies - this will simultaneously be
an easy test of the validity of statistical thermodynamics.
     Statistical thermodynamics is based on the following argument. Let

E = SUM n_i e_i                               /1/

be the total energy of a system of particles, n_i is the number of particles in
the ith state (or energy level) and e_i is the energy of that state. By
differentiating /1/ textbooks obtain

dE = SUM e_i dn_i  + SUM n_i de_i                 /2/

Further, textbooks juxtapose /2/ and the expression for the first law

dE = dQ + dW                                     /3/

and immediately obtain

dQ = SUM e_i dn_i                             /4/

dW = SUM n_i de_i                          /5/

Here the idea that heat is ESSENTIALLY different from work is clearly
expressed.
     Let us test eq. /2/ by considering an analogous situation. We have a large
group of people each of which possesses a certain amount of money. Let

M = SUM n_i m_i                            /6/

be the total amount of money in the group and n_i is the number of people whose
money is m_i. Through exchanging money with the outside world M slightly
changes and the change can be expressed in two ways:

dM = SUM m_i dn_i  +  SUM n_i dm_i                   /7/

dM = SUM m_i dn_i                               /8/

In my view /8/ is correct whereas /7/ is meaningless. If so and if the analogy
is correct, statistical thermodynamics rests on a wrong foundation. The reason
is that heat is regarded as essentially different from work.

Pentcho