Concept of maximum work

Eq. (4.5) shows that the amount of work performed by a system is governed by the opposing force $(P_{ext})$. The larger the opposing force, the more work is done by the system to overcome it.

If the opposing force is zero, no work is involved. With an increase of the opposing force from zero, more work will be needed by the system. When the opposing force reaches its maximum, the system performs maximum work. With an opposing force being greatest, more effort would be needed to overcome it.

Thus, when the opposing force $(P_{ext})$ becomes greater than the driving force $(P)$, the process gets reversed. Since the opposing force cannot be greater than the driving force, it should be the maximum.

If the pressure $P$ of the gas differs from $P_{ext}$ by a quantity $\Delta P$, then $P - P_{ext} = \Delta P$, and $P_{ext} = P - \Delta P$. The Eq. (4.5) then becomes

$W = -(P - \Delta P)\,\Delta V$ …