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I have a question related to title. Please see the following discussion:

In an irreversible process:

  1. $ΔU_\mathrm{irr} = q_\mathrm{irr} + w_\mathrm{irr}$
  2. $ΔU_\mathrm{rev} = q_\mathrm{rev} + w_\mathrm{rev}$
  3. $ΔU_\mathrm{irr} = ΔU_\mathrm{rev}$ ($ ∵ U$ is a state function)
  4. $|w_\mathrm{rev}| \ge |w_\mathrm{irr}|$; $w_\mathrm{rev} \le w_\mathrm{irr}$
  5. $q_\mathrm{irr} = q_\mathrm{rev} = 0$

And there is a contradiction: $ΔU_\mathrm{irr} = ΔU_\mathrm{rev}, w_\mathrm{rev} \le w_\mathrm{irr}$ How is it possible?

Or how is the gap $|w_\mathrm{rev} - w_\mathrm{irr}|$ generated? Is this related to entropy?

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    $\begingroup$ Delta U_rev <> Delta U_irr, as W_rev <> W_irr. // From the same initial state, you cannot reach the same final state by rev. and irr. adiabatic compression. $\endgroup$
    – Poutnik
    Commented Jun 10 at 13:15
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    $\begingroup$ There is no adiabatic reversible path between the same two end states as far an adiabatic irreversible process. $\endgroup$ Commented Jun 10 at 13:19

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For an ideal gas undergoing an adiabatic free expansion, a simple example of an irreversible adiabatic process, no work is done, so $$q=w=0$$ implying $$\Delta U = w+q = 0$$ and it follows (since the gas is ideal) that T is constant.

For any reversible adiabatic process, on the other hand, work is done, so $$w \ne 0, ~q=0$$ which means that $$\Delta U \ne 0$$ and (since the gas is ideal) T is not constant.

It follows that the final states for reversible and irreversible adiabatic expansions starting from the same state differ, and therefore that $$\Delta U_\mathrm{irr} \ne \Delta U_\mathrm{rev}$$ It follows that point (3) in the question is false (despite $U$ being a state function).

It is possible to describe a reversible path between the states connected by the irreversible adiabatic expansion, but this path cannot involve only an adiabatic expansion. Further comparison of reversible and irreversible processes can be found in other posts on the site, for instance here on how to compute the entropy change for an irreversible adiabatic expansion by identifying a non-adiabatic reversible path between the initial and final states.

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