Re: The 2nd Law of Thermodynamics - Irrelevant to Origins

David Ewan Kahana <dek@xxxxxxx> wrote:

Thermodynamics has to be just about the most subtle subject
in all of physics. There really ought to be a law against even
trying to discuss it without understanding that from the outset.

Nah. It is no more subtle than the concept of energy...


It seems to have taken such a long time for the essential
content to become even reasonably clear to people that we are
left with an enormous historical residue of understandings and
half-understandings to deal with. Even lots of physicists remain
deeply confused about the subject and it's all too easy to make
serious errors.

It isn't that the initial pioneers were so wrong-headed or anything
of that sort. It's that the roots of the subject were in describing
the behaviour of heat engines, and were so intuitive and practical,
while the applications turned out to be so enormously general.

Grumph. Warning: historical tidbit follows.

The major problem was that the Caloric theory was in vogue
during the development of thermodynamics. Nevertheless,
Sadie Carnot gave the Second Law quite correctly in 1824,
long before the Law of Conservation of energy was formulated.

Developments went in several directions. Maxwell, for
example, seems to have thought in geometric terms and
his "Theory of Heat" (written for non-specialists) relies
heavily on it. Both Clausius and Clapeyron used a more
or less "modern" approach, using calculus.

Even Caratheodory went only so far in reducing the subject to
its essentials, and he introduced all sorts of structures and
requirements that aren't really central.

Caratheodory reduced thermodynamics to the study of
certain differential forms. That's yet another technique.
We are up to what, four now? Carnot and logic, Maxwell
and geometry, C&C rather simple calculus, and Caratheodory,
differential forms.

How the subject generalizes
to chemical systems was the input of Gibbs,

That's six,


and his roots were
really on the side of statistical mechanics. And thermodynamics
is just not complete without discussing its applicability to chemical
transformations. It is very general, perhaps THE most generallly
applicable branch of the physical sciences, and therein lies the
problem.

It seems that everybody and his mother is willing to
opine on the subject, even when they are no doubt well
aware that they have no idea even what the word
entropy is really supposed to mean. And the fact that
creationists have taken hold of it and draw all sorts of insane
conclusions from it just makes things worse.

One of the very best short treatments of the subject that
I have seen was in an article that I think was called
`A Guide to Entropy and the Second Law,' by Elliot Lieb
and a co-author, whose name I don't remember. It's a
very mathematical approach, probably not much
use for a beginner, but it really reduces the subject to
its essentials.

If I had my way, we would do as they do, and replace
the second law and most of the other laws by an entropy
principle, and the axioms of thermodynamics by certain
statements about the relationships between macroscopic
thermodynamic states of systems and the scaling
properties thereof. If the subject is stated in this fashion the
need to talk about vague, though intuitively clear
notions like `heat,' `hot,' and `cold' is eliminated.

I'd not do that. The beauty of thermodynamics is that
it is based on two postulates (three, if you wish) and
a couple of axioms. One only need start with the concepts
of energy and mechanical work. Heat is readily defined
in a very measurable way from these.

This leads very directly to considerations of chemical
reactions and makes the relationship between them and,
for example, the energy minimum principle rather obvious.

We chemists routinely do it in 10 or 11 weeks, even to
classes composed of the mathematically challenged.


Their statement of the entropy principle involved
the notion of one state being adiabatically accessible
from another, which is defined as a special kind of
ordering relation on the states of systems.

That is s silly physicist way of doing things. Too
often clever and not enough attention to clarity.

Seriously. One has to watch that.

Their entropy principle went something like this, and
please pardon any inaccuracies that I may introduce:

If you think that's going to be easy for the average
junior to grasp. you are very wrong.


`There exists a real valued function on all states of
all systems, including simple and compound systems,
called entropy (S), such that:

(A) S is monotonic. For states X and Y of a system
which are comparable, we have:

S(X) <= S(Y) if and only if X < Y

But first you will have to define X and Y very carefully.
According to your notation S is a function of X, but in
general S is a function of a fair number of variables,
the exact number of which isn't known until you can
state the Gibbs Phase Rule.


(B) S is additive and extensive. If X and Y are states
of two systems, and if the pair (X,Y) denotes a
state of the compound system, then:

S(X,Y) = S(X) + S(Y)

and

For each real r > 0 and state X and its scaled
copy rX contained in G(r), where G is the state
space of the system and G(r) the scaled state
space:

S(rX) = r S(x)

The relation `<' between states, means that one
state is adiabatically accessible from another, and
one has to give that a physical interpretation.

Adiabatic accessibility is a dog. Very hard to see
without an extensive set of arguments.

The meanings and physical interpretations of all of
the words used, such as comparable, have also to
be given and specific explanatory examples have
to be provided.

But in a sense, the really crucial point is just the
existence of an entropy function universally across
all systems, and which has the necessary properties.

The first statement about adiabatic accesibillty can
easily be replaced by one about irreversible and
reversible adiabatic processes.

The very non-trivial physical content is all in the
scaling, extensivity and additivity properties of the
entropy, and there is clearly no need this way for a
restriction to closed systems to aid in the formal
construction of the subject, because the statements
apply just as well to compound systems.

The entropy principle as given can be shown
to imply the existence of adiabats, along which
entropy is constant, and after this the temperature
can be proven to be a consequence of and directly
related to the existence of the entropy.

The whole question then becomes what requirements
on the ordering of adiabatic accessibility, on the state
space, are sufficient to guarantee the existence of the
entropy function. Simple and reasonable axioms can
be given which are sufficient to guarantee it.

In this kind of an approach, it's very clear that the
notion of entropy is what is central, while idealized
notions like closed systems are practically entirely
bypassed, and it's clear the subject lends itself to
a local treatment.

The Clausius inequality and all the other statements
of the law all will of course follow. All of the other
thermodynamic potentials arise from the notions
of energy, entropy, temperature, pressure, volume,
and any other pairs of intensive and extensive state
variables.

I don't really expect that the way in which the subject
is taught should or can change much, though.

But, in any case, people who have never even
heard of the Clausius inequality shouldn't even
begin to quibble about what is the most general
form of the second law.

David

I'm laughing. I gave up attempting to critique your
choice, but I doubt that very many students would find
your setup transparent.

H. B. Callen attempted a similar thing in his book
"Thermodynamics". (Yes, he's a physicist.) Check the
*FIRST* edition. The Second edition is an attempt to
be all things to all peoples and, in my opinion, fails.

The problem is the following. Most of us have to teach
thermo to students who have had only a single year of
calculus -- indeed, we often have to teach a modest amount
about the calculus of many variables.

Chemists have developed methods of doing all this that
leads to your results as conclusions, which makes them,
I think, more palatable.

To chemists the notion of _intensive_ and _extensive_
variables is natural and it is just a step to show that
the former are homogenous functions of degree 0 while the
latter are homogenous of degree 1. Euler's theorem on
homogenous functions is easy to prove. With that,
one has the scale factors that you want above.

But more to the point, by keeping thermodynamics to
its roots, one can understand why the development of
relativity did not change thermo at all.

Some like to tie thermo and stat mech together, but that
seems to me to be a mistake. If our view of quantum
mechanics changes, stat mech will change, but not thermo.

I've taught it stat mech first and thermo first. The
stat mech first approach suffers from the lack of material
to compute -- since things like free energies and entropy
are not yet known. So I prefer to teach it last.

Curiously, crazy creationist statements on the Second Law
sometimes turn up in my exams with a request to explain
why the statement is right if it is right or wrong if it
is wrong. Students do surprisingly well.

All of which tend to show that education is sometimes an
antidote to ignorance.

----- Paul J. Gans

.

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