Work done in an irreversible process I was given a problem to identify whether the given statement is true or false.

And the statement was:

"Consider an ideal gas which is expanded from $(P_{1},V_{1},T_{1})$ to $(P_{2},V_{2},T_{2}).$ Then, the work done on the gas is maximum when it is compressed back irreversibly from $ (P_{2},V_{2}) $ to $ (P_{1},V_{1})  $ against a constant pressure $P_1$".

I concluded that the statement is false, as:


It is nowhere mentioned what is greater $P_1$ or $P_2$ as there is no information on how the expansion process was done.
Even if I assume that $ P_1 &gt; P_2$, still during irreversible compression I can vary the pressure such that the pressure goes beyond $P_1$ in the middle of the process but finally returns to the required final state $ (P_{1},V_{1})  $ , which would mean that I am doing more work than what I would be doing if I keep the pressure constant.


However the answer says that the statement is true.

Where am I wrong?
It is tricky wording. Usually when you compress something you want to do so as efficiently as possible. The minimum amount of work would be required if you compressed this reversibly.

If for some reason, as in the question, you want to waste your energy and overkill the work required (i.e., do things the hard way), then compress the gas IRREVERSIBLY. that will make it take the maximum amount of energy to get the desired compression.

The engineer in my cringes at this :)