Re: Entropy

[palmer@sfu.ca (Leigh Palmer)]

> 1) Is it easy to show that the following two expressioins are equivalent?
>
>   S = k ln(W)    and      S = Int dSrev = Int dQ/T

The first expression is elegant and useful if one has a system for which
one can count the distinct quantum mechanical eigenstates. It may then be
taken as the definition of the entropy, though of course entropy had been
discovered long before this interpretation (due to Boltzmann and Gibbs)
came along. Thus it is appropriate only for the simplest systems.

The second expression needs a lot more qualification to go along with it.
The first thing that should be said is that it does not define the entropy
of a system; it prescribes a method of comparing the entropies of two
states of the same system and the expression you have written (suitably
qualified) is an expression for the entropy *difference* between those
states, not the entropy of either.

> 2) Is this an important exercise?  Does anyone care?

The exercise of demonstrating the equivalence is meaningless before one
understands thoroughly what one or the other expression means. I often
teach thermodynamics to sophomores and try hard to introduce the concept
to them. I usually fail with the majority, and I'm convinced that I do
so because the word "entropy" gets introduced too early. For some reason
my students think entropy is related somehow to the "state of disorder"
of a system. They have no idea what that means, but they mistakenly think
they do, which complicates my task enormously.

As to whether the exercise is important or not, I would say that no, it
isn't nearly so important as introducing the students to entropy *per se*.
If the students already understand quantum mechanics and the meaning of
adiabatic processes in quantum systems then you can start by teaching them
S = k ln W. Since W is an adiabatic invariant for an ensemble of similarly
(but not identically) prepared systems it is trivial to note that S is a
conserved quantity in adiabatic processes.

Usually the entropy is introduced before students learn quantum mechanics
and so it is much better to teach them the classical form first if there
is to be any hope of teaching them something of value. Of course any
physicist who does not understand both forms has a serious lacuna in his
preparation. The expression S = k ln W is, indeed, carved in stone, but
the student who merely memorizes it will find it of even less use now
than the man who lies buried beneath that stone. Eventually one must
expose the equivalence, and I recommend Fred Reif's book for that purpose.

Leigh