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irreversible adiabatic compression of an ideal gas



It seems like thermodynamic issues generate a lot of heat and work on this
list, so 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?

The purported answer is 60%, obtained as follows. Since Q=0, delta U = W.
But delta U = 1.5 (Pf*Vf - Pi*Vi). All fine so far. Now for the sticky
part. Put W = -P*delta V, where P is the external pressure of mg/A
(neglecting atmospheric pressure if m is big enough). That is, P = Pf
(since the gas eventually equilibrates to that pressure) and delta V = Vf -
Vi. Take the limit as Pf -> infinity to get the desired answer.

I immediately object that W does not appear to be the work done on the gas.
Surely I want P to be the gas pressure, not the external pressure. But the
gas pressure is not well-defined during this irreversible process. In fact,
the gravitational force downward on the weight is greater than Pi*A from
the gas pushing back, so the piston is going to pick up kinetic energy and
oscillate about the final equilibrium position. I need to add in some
friction to damp this out. But the friction warms up the presumably
roughened walls near the piston, so now we'll get heating and it's not
clear the process can still be called adiabatic.

Looks like a mess. However, thinking it through again.... Suppose I assume
all the "frictional heat" (or whatever your preferred terminology is)
eventually goes into the gas. Then, I started out with gravitational PE of
-mg*delta h (negative because the piston goes down) and everything at rest.
This "external energy" all ends up in the delta U change in internal energy
of the gas. But mg*delta h = Pf*delta V. So the purported solution seems to
be correct.

I welcome your analysis of the problem, adding whatever ancillary
assumptions you think necessary. What still nags at me is the following: If
the above is correct, 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?

Dr. Carl E. Mungan, Assistant Professor http://www.uwf.edu/~cmungan/
Dept. of Physics, University of West Florida, Pensacola, FL 32514-5751
office: 850-474-2645 (secretary -2267, FAX -3323) email: cmungan@uwf.edu