Re: Thermo/Sidebar

[John Denker <jsd@MONMOUTH.COM>]

At 02:58 PM 11/25/99 -0700, Jim Green wrote:

> Are you suggesting an "ideal" (ie reversible ie coefficient of restitution
> of unity) spring (or say a rubber ball) with a platform mounted upon it --
> such that sand might be poured upon the platform quasi-statically?

That's consistent with what I was suggesting.

Note that a real-world metal spring has the nice property that there can be
very little internal dissipation, especially if things are done reasonably
slowly and the strain is not too large.

> A calculation of deltaW seems straight forward but how would one calculate
> deltaS?

Uhhh, how about delta_Q over T?

> Can one calculate a density of states?

A) Yes.

B) Your original challenge (Thu, 25 Nov 1999 11:47:32 -0700) concerned
*classical* thermodynamics.  I thought the whole point of classical
thermodynamics (if there is a point, which is questionable) was to see how
much one could do without saying anything about the microscopics.

Therefore I did not optimize my examples to make statistical mechanics
calculations (e.g. density of states) easy or intuitive.  If that's what
you want, then item (5) below is a better choice.

> The following don't seem macroscopic to me:
>
> >3) What about the thermodynamics of charging a battery?

A battery has a macroscopic voltage and temperature the same way a
gas-cylinder has a macroscopic pressure and temperature.  (To say it the
other way:  If a battery is not macroscopic, then a gas-cylinder isn't
either;  just think of all those nasty microscopic gas molecules.)

My point is that you can analyze the macroscopic voltage-versus-temperature
relationships the same way you would analyze
pressure-versus-temperature.  As the saying goes, "the same equations have
the same solutions".

> I must confess that I don't know much about the following:
>
> >4) What about the thermodynamics of a dilution refrigerator, e.g. the
> >venerable Electrolux cycle, or the 3He-4He dilution refrigerator?
> >
> >5) Perhaps simplest of all, what about the thermodynamics of an adiabatic
> >demagnetization refrigerator?
>
> In what sense are they macroscopic the way the classical cylinder is?

In one part of the cycle, the demag stage absorbs heat at constant
temperature.  In another part of the cycle, it changes temperature at
constant entropy.  It's just the same as the gas-cylinder story, except
that the external variable is magnetic field, not cylinder-volume.  The
same equations have the same solutions.

>   How would I calculate deltaQ, deltaW and deltaS for a cycle?

dQ and dS are the usual things.  One typically chooses H as the independent
variable, so dW = MdH.

Reasonable references include:

 Lounasmaa           Experimental Principles and
                     Methods Below 1K              Academic Press 1974

 White               Experimental Techniques in
                     Low-Temperature Physics       O.U.P. 3rd Ed. 1979

 Betts               Refrigeration and Thermometry
                     Below One Kelvin              Sussex Univ. Press, 1976

Otherwise, be careful...  There are lots of remarkably clueless discussions
of this topic out there.

> TX for the suggestions.

Happy to help.