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I have a gas transitioning adiabatically between A ($P_1$, $V_1$) and B ($P_2$, $V_2$) where $P_1>P_2$ and $V_2>V_1$. The question is to determine the net work done on the gas if the gas is first expanded reversibly from A to B ($w = dE = C_v(T_2-T_1)$), and then compressed irreversibly from B to A ($w = -P_{ext}(V_1-V_2)$) at a constant external pressure defined by A. In this scenario, simply looking at the areas under the graphs the net work should be positive.

I am trying to reconcile this with $dE$ for the gas. For the roundtrip transition (A to B to A), $dE = 0$. And if we take each step as adiabatic, then $dE = w$ for each step, but as I have described above you would end up with two different values for $dE$ for each step, thus $dE$ not equal to zero. My logic is flawed somewhere. If I compress irreversibly would the transition still be adiabatic? Alternatively, is the original scenario flawed: can I have an irreversible adiabatic transition?

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In irreversible adiabatic expansion, the volume change is more as compared to that in reversible adiabatic expansion (for the same final pressure value and same initial conditions). Thus, the assumption of getting back to A from B by irreversible adiabatic process is wrong.

This change in volume can be understood by this simple example: Suppose you have gas in an adiabatic container with a piston above where a large number of blocks are kept. Then, for a reversible adiabatic process, you remove a block and wait for the system to attain equilibrium so that the gas nearby gets enough time to compensate the energy loss from previous collision with piston. Whereas, for an irreversible adiabatic expansion, you remove many blocks instantaneously to reduce the pressure, so the gas near the piston doesn't get enough time to regain energy and thus doesn't collide efficiently and hence does lesser work as compared to that in reversible adiabatic expansion. That implies greater change in volume for the same pressure change in irreversible adiabatic expansion as compared to in reversible adiabatic expansion.

You may even try understanding it by writing equation for work done.

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