Why is maximum work done for reversible paths in isothermal expansion and not minimum work done? [closed]

Question

Work done is given by $W = -P_{ext}dV $ according to convention.

Which should mean that in cases of expansion, work done on the system would be negative. If work done is negative, isn't it wrong to say that work done in isothermal reversible process is maximum? Because in irreversible expansions, area covered is smaller but then, if the value is negative, then wouldn't it be that minimum work is done in case of reversible isothermal expansion and maximum for irreversible case?

Image for clarification. This image is for expansion. The dark gray area is for irreversible process and the light gray + dark gray is the work done by reversible process. Clearly reversible process has higher area and hence higher negative value.

Answer 1

When we talk about the work done in the above equation,

$W=-P_{ext}dV$ ,

the negative sign only tells us about whether the work is done by the system or work is done on the system. So during expansion the work comes out to be negative indicating the work is done by the system. It is just a physical notion and has nothing to do with the actual magnitude, as we have for acceleration or velocity where negative indicates the opposite direction, just like that.