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The Adiabatic Two-piston System

Consider the diagram on the left.  An ideal gas is inserted between two movable pistons of mass M and cross-section area A.  Initially, the pistons are clamped at L0 position.  It is then allowed to move  so that the system can evolve into steady state.  Describe the pistons’ motion and how the temperature and the volume of the gas evolve.

Sounds like an undergrad physics’ thermodynamic problem? Well, it’s not or at least not for now.

In a well-written paper that will be published in the American Journal of Physics (September issue), R. Amor and J.P. Esguerra solved the problem and suggested an experiment that could verify their solution [1].

The step by step solution will not be discussed here but the recipe is quite simple.  First is to determine the temperature as a function of the time and obtain from it the velocities and the distance between the pistons also as functions of time.  The authors started with the conservation rules of energy and momentum, and the equation that relates the temperature and the length in an adiabatic process.   With a little tweaking,  the temperature can now be determined and the rest of the quantities follow.

Ok, not so little tweaking. The authors calculated the quantities for 3 cases – purely monatomic, purely diatomic, and mixed monatomic and diatomic ideal gases.  Although for the pure cases the solution may be straight forward,  a special function is needed for the mixed ideal gases’ case.  From their solutions they observed that;

1.  the temperature decreases as an inverse power of the time for large times, with the exponent as a function of the monatomic to diatomic gas ratio,

2.  the pistons’ speeds increase from zero to a maximum value determined by the heat capacity of the gas and the masses of the pistons, and

3.  there is a point of inflection in the temperature and pistons’ speed versus the logarithm of the time plots, which can be interpreted as the start of the steady state behavior.  This point depends on the ratio of the monatomic and diatomic ideal gases.  The point of inflection’s lower limit corresponds to the purely monatomic ideal gas case while its upper limit with the purely diatomic gas case.

The authors are cautious however because their results are expected to be valid only in the quasistatic regime.  That’s is why they suggested an indicator – the ratio between the speed of the pistons  to the root mean square speed of the ideal gas.  This indicator is not new by the way, as it has been used by a lot of people before.  The ratio should be much less than 1, a good rule of thumb is that the ratio should be below 0.10.   This limit is important for the experimental verification which the authors also proposed.  With their calculations using 1L nitrogen N2 gas at T0=300 K, they saw that the quasistatic regime begins to break down when the mass of the piston is about 10g.

The authors say that their derivations can still be refined with the kinetic theory.  They also add, “It would be interesting to determine the behavior for sufficiently small piston masses such that at the time of release, the quasistatic assumption is valid and then breaks down as the system evolves.”

Beyond extensions however, the paper is important because 1) the problem it tackled is rich in concepts,  and 2) a new experiment in upper-level thermodynamics that is proposed here is much welcomed.   How many experiments in thermodynamics can boast of a wealth of concepts that is both simple and direct?

Rumelo Amor is from the University of Sto. Tomas, while Jose Perico Esguerra is from the National Institute of Physics, UP Diliman

[1] R. Amor and J.P. Esguerra, Evolution of ideal gas mixtures confined in an insulated container by two identical pistons, Am. J. Phys. 78 916 (2010).  DOI: 10.1119/1.3428856