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 nonzero?

Answer 1

Work done by the system is neither necessarily zero nor necessarily non-zero.

If in a certain process, the pressure P stays constant, then
$$\begin{gathered} ∆W=P∆V \\ \Rightarrow W=P(V_2-V_1) \\ \mathrm{As}{V}_2=V_1 \\ \Rightarrow W=0 \end{gathered}$$
(Initial volume, V₁ = Final volume, V₂)
Hence, it is an isobaric process.

Even if P = P(V), net work done will be zero if V₂ = V₁. 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.