We consider an ideal gas expansion in which the piston moves at speeds comparable to or greater than the average particle speed. We obtain a limiting expression for the small temperature change that results from this process. This example can help students enrich their understanding of the kinetic theory of gases and the meaning of temperature. Sample questions are included.

1.

The “sudden expansion into vacuum” problems considered in thermal physics texts proceed by means of a stopcock or similar device, so as to avoid the particle collisions with the piston that are the focus here and that lead to ΔU0. Note that we are not concerned here with accounting for the additional cooling that takes place due to deviations of the gas from ideal.

2.
In the context of a one-particle gas,
H. S.
Leff
offered a different, and in our view much less straightforward, analysis of collisions between a particle and a moving piston. See “
Thermodynamic insights from a one-particle gas
,”
Am. J. Phys.
63
(
10
),
895
905
(
1995
).
3.

In reality, atomic or molecular collisions with solid walls do not follow the simple model of an incoming billiard ball rebounding elastically from a featureless hard surface. Rather, an incoming particle will bond briefly to the piston surface and then be re-emitted a moment later. Therefore, the most recent particle to collide with the piston and the next particle to rebound from the piston will likely be different particles. Of course, the energy books remain balanced over time, and we shall simplify reality by following individual particle trajectories through instantaneous collisions with the piston and cylinder walls.

4.
For a related compression problem, see
D. S.
Lemons
, “
Nonadiabatic compression of a cold gas
,”
Am. J. Phys.
50
(
7
),
607
609
(
1982
). Lemons analyzes the rapid compression of an initially cold ideal gas. Taking the gas to be initially cold (that is, motionless) obviates the need to take into account the Maxwellian distribution of velocities considered here.
5.
P. M.
Bellan
, “
A microscopic, mechanical derivation of the adiabatic gas relation
,”
Am. J. Phys.
72
(
5
),
679
682
(
2004
).
6.

Not even all of region A1(1) contributes appreciably to the result. Only those particles with x0>0.2L move sufficiently slow to be well represented in the Maxwell-Boltzmann distribution to make a difference.

7.
H.
Fair
, “
The electromagnetic launch technology revolution
,” Magnetics Magazine, Winter 2003, available at ⟨magneticsmagazine.com⟩.
8.
E. H.
Kennard
,
Kinetic Theory of Gases
(
McGraw-Hill
, New York,
1938
). Sec. 100.
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