# Maximum velocity of piston pushed by a gas on adiabatic irreversible expansion [closed]

I am currently studying for my country's IPhO selection process and thermodynamics has to be one of my favorites right now. However I came across a very odd problem that is making me revise a lot of the theory.

The point is that a monoatomic gas is compressed in a cylinder of cross-section area $S$ and the gas goes up to the height $l$ where a piston of mass $m$ is being held. The pressure of the gas is $P_0$.

Once the piston is released, the gas adiabatically expands, pushing the piston up. After moving a distance $h$, the piston reaches its maximum velocity. There is no is no friction or atmospheric pressure and the gravitational acceleration is g. The problem asked me for the maximum velocity of the piston and I have no idea what to do. Also, it has given me this piece of information that I have no idea how to use either:$$P_0 ~>~ mg \frac{5h+3l}{3Sl} \,.$$

My attempts and trials:

When in adiabatic processes, I am used to the work done in reversible processes. In this case, it would be very easy to calculate the maximum velocity using the kinetic energy theorem since the work would be given by the first law of thermodynamics as internal energy variation would be $\frac{nR \Delta Τ}{\gamma - 1}$ and this could be calculated using problem data. However, I cannot use $PV = nRT$ for the maximum-velocity-moment since the gas is not in equilibrium, I suppose.

I asked a chemistry undergrad about it and he told me that the work would be ${P}_{\text{ext}} \, \Delta V$ where ${P}_{\text{ext}}$ is external pressure. This doesn't seem to be useful since the pressure exerted by the piston changes in a way I have not been able to analytically describe. Also, I haven't understood why this equation could be applied to an adiabatic irreversible case.

I have also tried the First Law of Thermodynamics, $Q = \Delta U + τ,$ so $τ = - \Delta U .$ But I am unable to use $PV = nRT$ for the reason I mentioned before, so finding $\Delta T$ is something I'm pretty sure I can't do. The lack of equilibrium in the gas is making this very hard for me. Also, I do not know how to use the inequality given above.

Another trial was: the maximum velocity is achieved when the forces balance. So $PS = mg$ where $P$ is the pressure of the gas in the maximum-velocity-moment. This doesn't seem applicable since the pressure shouldn't be uniform in the gas in a non-equilibrium state.

I am stuck with this problem and I believe wherever I am getting it wrong is probably some theoretical issue. Please, send ideas and correct my theoretical mistake. Any help is appreciated.

## 1 Answer

Your concerns are well-founded. If the process is irreversible, you are not going to be able to analyze this in full rigor. However, suppose you are willing to assume that the piston is sufficiently massive that, at least during the first oscillation of the piston, the behavior of the gas can be approximated as being adiabatic and reversible. Then, at the maximum velocity point, the gas pressure times the area will be equal to the weight of the piston. You then determine the work done by the gas up to that point and the change in potential energy of the piston in order to determine the maximum kinetic energy. This is very similar to a problem in simple harmonic motion involving a spring and mass.

• I see. So is there no way to solve this if I don't assume that the process is reversible? If I assume reversibility then the problem indeed becomes much easier. – João Vítor G. Lima May 29 '18 at 2:41
• It can be solved if the process in irreversible by solving the differential thermal energy balance equation in conjunction with the Navier Stokes equations within the deforming gas, but this would require the use of computational fluid dynamics. This is obviously not what they had in mind when they posed this problem. So, in lieu of that, you're stuck with approximating the behavior of the gas as adiabatic and reversible. But, you were very perceptive in recognizing the conceptual issues with this problem statement. – Chet Miller May 29 '18 at 2:50
• By the way, I might also mention that the piston can't keep oscillating forever, because the deformation really is irreversible, and viscous stresses in the gas will cause the oscillation to gradually be damped. But, on the first oscillation, assuming reversibility may not be too bad an approximation. – Chet Miller May 29 '18 at 3:07
• Indeed, the problem wanted me to assume the process to be reversible. After I did, it became a lot easier and I found the correct answer. – João Vítor G. Lima May 29 '18 at 18:00