Question
The final volume of a system is equal to the initial volume in a certain process. Is the work done by the system necessarily zero? Is it necessarily non-zero.

Answer

Work done by the system is neither necessarily zero nor necessarily non-zero. If in a certain process, the pressure P stays constant, then,$\Delta\text{W}=\text{P}\Delta\text{V}$
$\Rightarrow\text{W}=\text{P}(\text{V}_2-\text{V}_1)$
As $\text{V}_2=\text{V}_1$$\Rightarrow\text{W}=0$
(Initial volume, $V_1$ = Final volume, $V_2$) Hence, it is an isobaric process. Even if P = P(V), net work done will be zero if $V_2= V_1$. In this case, work done is zero. If the system goes through a cyclic process, then initial volume gets equal to the final volume after one cycle. But work done by the gas is non-zero.

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