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, . Therefore as the process is adiabatic. Then, similarly . The next step then says that this implies . I do not understand this. Surely as , (and similarly for the 12R process). Therefore, . 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?