# 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? 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. Oct 15, 2022 at 15:01