Re: irreversible adiabatic compression of an ideal gas

[John Denker <jsd@MONMOUTH.COM>]

At 11:48 AM 10/30/99 -0500, Carl E. Mungan wrote:
> here's a little problem someone threw at me the other day:
>
> An equilibrated monatomic ideal gas is contained in an adiabatic cylinder
> fitted with a piston. A large weight is placed on the piston. What is the
> maximum amount by which the gas could have been compressed when equilbrium
> is again attained?
...
> why is it that if I compress the gas *reversibly*
> (i.e., following the adiabat P*V^1.67) by dribbling the weight onto the
> piston bit by bit, I can squeeze the gas down to nothing (P->infinity =>
> V->0), but cannot if I do it *irreversibly* by suddenly dropping the whole
> weight onto the piston?


This problem is two legs of a "heat engine" cycle in disguise, just in time
for Halloween.  You can umask it as follows:

*) Start out with the gas at pressure P in a cylinder of area A and height H.

*) [The adiabatic leg] Gently place a humongous weight W on it using a
crane, and *adiabatically* lower it using the crane.  The gas will follow
the usual adiabatic law
  P * [AH]^gamma = constant.
During the process, make a note of P as a function of H.  Continue the
process until PA = W;  at that point gently disconnect the crane and get
rid of it.

*) Go back and figure out how much work the crane did:
   crane_work = integral [ W - A*P(H) ] dH

*) [The constant-pressure leg]  To model the fact that the original
specification called for dropping the weight rather than lowering with the
crane, take the crane_work and non-adiabatically add it to the gas (using
an electrical heater or whatever).  This adds heat at constant pressure P =
W/A.  The gas follows the usual heating law and expands to a new volume.

End of story.

The crane_work is the reason why the original problem differs from the
corresponding adiabatic process.


______________________________________________________________
copyright (C) 1999 John S. Denker             jsd@monmouth.com