Why is the Work Done indeterministic when we know $∆V=0$?

Question

Given: In a process on a closed system in closed container, the initial pressure and volume are equal to the final pressure and volume. Comment about the Net Work Done by the System, Net Heat given to the system.

Given Solution:

As Heat is path function, so here it is not determined if the heat supplied is zero or non-zero.

Since Work done is a path function, so here it is not determined if the Work Done by the system is zero or non-zero.

I have understood the Heat part but confused about the Work part. Here, I am concerned only about the pressure-volume work.

My Solution:

The net work done by the system is zero because the container is closed so, $∆V = 0$.

Considering the nature of Work done [Path Function] and we do not know about the path of the process, so we may say that Work done is indeterministic but we know that W = P.∆V and ∆V = 0 so how can there be a case in the given problem where Work is non-zero ?

What should be the correct conclusion about the Work done ?

Answer 1

The answer given in your textbook is correct. The correct expression for work is $W=\int PdV$ and not $W=P\Delta V$, which is only valid for an isobaric process. Alternatively, you can simply apply the first law of thermodynamics to find out if work was done. Since $U$, internal energy, is a state function, $\Delta U$ is definitely $0$. Also you say you understand that $\Delta Q$ may be non-zero. Hence, from the first law, is it not obvious that work can be non-zero too?