Why should this thermodynamics problem be indeterminate?

Question

Callen asks us to consider the following

A cylinder of length $L$ and cross-sectional area $A$ is divided into two equal-volume chambers by a piston, held at the midpoint of the cylinder by a setscrew. One chamber of the cylinder contains $N$ moles of a monatomic ideal gas at temperature $T_0$ • This same chamber contains a spring connected to the piston and to the end-wall of the cylinder; the unstretched length of the spring is $L/2$, so that it exerts no force on the piston when the piston is at its initial midpoint position. The force constant of the spring is $K_{spr}$; The other chamber of the cylinder is evacuated. The setscrew is suddenly removed. Find the volume and temperature of the gas when equilibrium is achieved. Assume the walls and the piston to be adiabatic and the heat capacities of the spring, piston, and walls to be negligible. Discuss the nature of the processes that lead to the final equilibrium state. If there were gas in each chamber of the cylinder the problem as stated would be indeterminate! Why?

I have completed the problem except for the bolded part. I suppose there are two ways of approaching this question depending on the interpretation. In interpretation (1), we assume that the initial gas pressures are identical. In interpretation (2), we do not have that the initial gas pressures are identical.

(1) Why should such a system be indeterminate? Denote the relevant forces by $F_{spr}, F_{g1}$, and $F_{g2}$. The initial state is surely the final (equilibrium) state once the setscrew is removed, since it is characterized by the equality $$F_{spr} + F_{g1} + F_{g2} = 0 + F_{g1} + F_{g2} = (P_1 - P_2)A = 0.$$ Is Callen saying that in general there could be other equilibria and that small fluctuations (which are outside the scope of classical thermodynamic theory, or so I thought?) could shift the system to other equilibria?

(2) Obviously the initial state is not an equilibrium state, but how can I see in this case that there is no (unique?) final equilibrium state?

Answer 1

Regarding your question (1), I think it is related to the notorious conundrum (paradox?) of the adiabatic piston. Callen has two other problems related to it, and on page 51 he writes:

The case of a moveable adiabatic (rather than diathermal) wall presents a unique problem with subtleties that are best discussed after the formalism is developed more fully; we shall return to that case in Problem 2.7-3 and in Problem 5.1-2.

I suggest that you consult Gruber where this problem is analyzed in all its gory detail; it is a very readable work, I should add. Gruber and his coworkers subsequently published several other pieces at varying degree of complexity on the same subject.

Instead of rehashing what Gruber wrote, while I agree he does offer an effective solution to the problem, I discuss another approach to it because I think the issue really lies somewhere else. To me the real problem is in the aggressive (?) over-idealization of the concept of reversibility, constraints and reservoirs. I alluded this to you yesterday while referring to it as "singular perturbation".

To illustrate it I assume you know the so-called two-capacitor paradox. To remind you it is about connecting directly in parallel a charged capacitor with an uncharged capacitor of equal size and then noticing that from symmetry, in equilibrium, both the voltages and charges must be equal, so that either we have lost half the energy or we have lost half the charge. The usual explanation is that when you turn the switch on to connect the two capacitors, you get sparks, radiation etc. and that is not included in the energy balance. I think this explanation is completely backwards because there is no radiation in Kirchhoff's laws that describe abstract elements defined by constitutive relationships $i=C\frac{dv}{dt}, v=RI, v=L\frac{di}{dt}$ with impressed independent sources $v_s, i_s$, AND, of equal importance, a set of connection rules, namely all initial conditions must be such that both voltages and currents must be continuous and stay continuous throughout. The last requirement immediately disallows switching a charged capacitor on to a discharged one because their voltages are different and the resulting initial current is infinite, not continuous. Before somebody dismisses KVL and KIL as some approximation to the true Maxwell's equations, I would like to remind him/her that Kirchhoff's laws may be history's most tested theory of any theory, as the integrated circuits in digital computers with trillions of resistors, capacitors transistors testify to its absolutely phenomenal accuracy. These ICs are all designed using Kirchhoff's laws but they never connect one charged capacitor on to a discharged one with an ideal switch.

The moment you add to the switch an infinitesimal resistance just to show that it is not ideal, you will have fully restored both charge and energy conservation, and of course all design software would have that miniscule resistor added in it when it is needed. So the problem is in the unrealistic idealization and illicit couplings of the various abstract items. The most amazing result in the solution of the capacitor paradox is that it does not matter how large or how small that resistance of the switch or any of the connecting wires between the capacitors is as long as it is not zero: the dissipated energy in that resistance is always half the initial stored energy. The problem, the paradox, happens only when that connecting resistance is assumed to be exactly zero. This is the singular perturbation I was referring to.

Now what does this have to do with the adiabatic piston problem? There it is assumed that in a cylinder two fluids are separated by a moveable ideal frictionless, massless and adiabatic wall. Some claim that it will oscillate and never relax, some claim that temperatures may not equalize but pressures will, etc. I say that there is no real adiabatic wall, and there must be some amount of diabaticity leading to the equalization of temperatures and per force the equalization of pressures, as well, and problem solved.

In my view, what is missing is a clear set of connection principles/rules that would not allow a massless adiabatic wall to separate two thermodynamic fluids whose state depend on the hydrostatic pressure and temperature, exactly the parameters that are not allowed to be in communication by the idealized wall. There is no difficulty in the 1st part of your question because there the massless adiabatic wall is separating a mechanical system (elastic spring) from a thermodynamic system (gas). Because the spring has no temperature and it is characterized by its length, the mechanical properties by themselves are enough to specify the equilibrium.