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Thermodynamics 2nd Law Confusion

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    Ok, I'm looking over my notes and my lecturer has given an example of proving a process is irreversible. This is how it goes:

    Assume a sudden compression (called process 1-2). Then assume the same compression, but slow and reversible (called process 1-2R). One of the tests is to check that W_12 < W_12R (which is to be proved wrong), and another is to show that W_12 is not equal to W_12R. This would then imply that W_12 had to be greater than W_12R, implying irreversibility.

    I am confused with the first test, showing that W_12 < W_12R.

    He says, from the 1st law, Q-W=\Delta U. Therefore -W_{12}=\Delta U_{12} as the process is adiabatic. Then, similarly -W_{12R}=\Delta U_{12R}. The next step then says that this implies T_{12} &lt; T_{12R}. I do not understand this. Surely as U = C_vT, C_vT_{12}=-U_{12} (and similarly for the 12R process). Therefore, -C_vT_{12}&lt;-C_vT_{12}. Then when you remove the minus signs, the inequality sign flips, and you get the opposite to what my lecture proposed.

    Can someone tell me what I am doing wrong here?

    EDIT: Also, can someone explain why, if W_12 > W_12R this implies irreversibility?

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