# Total entropy generated for a sudden expansion of an ideal gas

Topic: Adiabatic, irreversible expansion of an ideal gas.

Suppose you have an ideal gas in an insulated piston-cylinder arrangement originally at $$P_0,T_0,V_0$$. The piston is massless and frictionless. The pressure on the piston is originally $$P_0$$ and is suddenly dropped to $$P_F$$. The gas expands irreversibly. As shown elsewhere, the final conditions and entropy change of the gas can be calculated with

\begin{align*} T_{F}&=\left[1-\frac{\gamma-1}{\gamma}\frac{P_0-P_F}{P_0}\right]T_0\\ V_{F}&=\left[1-\frac{\gamma-1}{\gamma}\frac{P_0-P_F}{P_0}\right]\frac{P_0}{P_F}V_0\\ \Delta S_{\text{sys}}&=Mc_v\ln(T_{F}/T_0)+MR\ln(V_{F}/V_0) \end{align*}

My question is, what is the total entropy generated for the universe in this process? We have \begin{align*} \Delta S_{\text{uni}}&=\Delta S_\text{sys}+\Delta S_\text{sur}\\ &=\left(\int\frac{\delta Q}{T}+S_{\text{gen}}\right)_{\text{sys}}+\left(\int\frac{\delta Q}{T}+S_{\text{gen}}\right)_{\text{sur}}\\ &=\left(0+S_{\text{gen}}\right)_{\text{sys}}+\left(0+S_{\text{gen}}\right)_{\text{sur}} \end{align*}

Evidently, $$S_{\text{gen,sys}}=\Delta S_{\text{sys}}=Mc_v\ln(T_{F}/T_0)+MR\ln(V_{F}/V_0)$$

But what is $$S_{\text{gen,sur}}$$? Is it possible to determine this?

• This is a very interesting and thought-provoking question. The answer depends on the details of how the surroundings pressure $P_F$ is imposed on the external face of the piston. What are your thoughts on this? Commented Oct 15, 2022 at 11:21
• After reading your discussion below and some of the referenced papers, I'm content to let the pressure be decreased by removing a weight and assuming that outside the piston-cylinder arrangement is a vacuum! It is a complicated problem indeed if you assume a partition rather than a piston. It seems that most of my inability to approach the problem stemmed from not fully defining the surroundings. Commented Oct 15, 2022 at 15:01

The way you defined the "surroundings" of your system is that it is a purely mechanical constraint, therefore has no entropy, or rather its entropy whatever was before does not change during or after the process it participates in. To get the entropy change in the surroundings you need to define your process to which your system is subjected so that there is heat exchange, or rather entropy exchange between the system and its surroundings, so the energy exchange cannot be purely adiabatic.

• Suppose that it is ideally adiabatic. What then? Commented Oct 15, 2022 at 11:23
• @ChetMiller The way I account for a "system" and for its "surroundings" is to define the "system" by boundary conditions. The question assumed an adiabatic process with a pure mechanical interaction, say, with a weight in a gravitating field. The gravitating field is the surroundings then and the piston/cylinder is assumed to be an heat insulator. Such procedure cannot return the system to its original state for there is no adiabatic irreversible cycle while a reversible adiabatic process (cycle or not) is just a mechanical interaction without any thermal significance. Commented Oct 15, 2022 at 11:38
• All that means to me is that you are assuming that the entropy generated in the surroundings (if it is done mechanically) is negligible. This seems reasonable to me. But what if the surroundings consists of just the air in a sealed, adiabatic, room, initially at 1 bar pressure? This air is pressing on the outside face of the piston. What then? Commented Oct 15, 2022 at 11:57
• @ChetMiller Now that question is an old conundrum: how to account for an adiabatic partition within a system that is in adiabatic envelope. And that problem has an enormous literature under the heading "adiabatic piston", here are a few publications on the subject without even mentioning the various contradicting arguments in the usual textbooks. My view is that the problem of "adiabatic partition/piston" is a problem of mathematical idealization, similar to the equilibration of voltages between capacitors. Commented Oct 15, 2022 at 12:57
• Here is a forum discussion I participated in regarding the adiabatic partition problem. physicsforums.com/threads/…. The outcome of this discussion was that, in order to solve the problem fully, one has to include consideration of the transport processes occurring within each compartment (i.e., viscous deformation and convective /conductive heat transfer). However, we arrived at what we believed to be a good approximation. Commented Oct 15, 2022 at 13:48