[Bob Sacamano]

Why is work a state function?

My instructor used the example of lifting a box straight up 1 m vs moving it up and down and all around the room with the net movement being up 1 m. Why is there any difference in the work between these two situations since the displacement of the box is the same in each case (ie. shouldn't the net work be the same in each case)?

[FreeTheBee]

It is not. Work depends on the path taken. The potential energy difference of the box will depend on the position before and after lifting it. The amount of work done on the box depends on the path from the initial to final position.

[Bob Sacamano]

What do you mean by "the work done on the box"?

Say if you raise the box, you are doing positive work on the box and gravity is doing negative work on the box. What is the work done on the box?

[david.helmly]

when you push the box you do work on it.
when the box pushes you the box does work on you.

both statements are true in reverse if you switch the signs (i.e. negative work)

the net work done on the box by all sources (e.g. you, gravity, the wind) is given by the equation:
w = F*d
work equals force(s) [on box] times distance moved

where
F = m*a
force equals mass [of box] times acceleration [of box]

[Bob Sacamano]

Moving the box horizontally will not give any net work. It is true that you accelerate the box to start it moving horizontally but the at the final position the box is static. Therefore when you stop the box its acceleration will be negative and negative work will be done on the box causing the net work done in the horizontal plane to be zero.

Work will be done in the vertical plane. However that work will be equal to the difference in potential energy of the box which is a state function.

So.... How is work a state function?

Note: I have neglected all frictional forces in my logic.

[Schrödinger]

The definition work done in physics is force times displacement. It doesn't matter how you get that final displacement of 1m. i.e., even if you have moved the box through a distance of 1000 light years up and down and finally up by 1m, the net displacement is just 1m.

Work, by the way, is not a state function always. It is a state function only in a conservative field. i.e, when the force creating the field is conservative in nature.

[Bob Sacamano]

Aha, so why is work considered a state function in thermodynamics?

[Borek]

You were already told at the very beginning - work is not a state function.

In very specific case of gravitational fields, if everything else is neglected, potential energy of the body depends only on its state - that is, distance from the mass center. Hence potential energy is a state function in this case. As changes in the potential energy will be identical to the work done, work will also depend only on the state (position), so it will behave like a state function. But that's only in a very specfic case, in general work is NOT a function of state, as amount of work required to move body from one place to another depends on the path.

[Schrödinger]

Aha, so why is work considered a state function in thermodynamics?

In thermodynamics, work done in a reversible process is a state function. Work done in an irreversible process depends on the way the state is attained.

[Borek]

work done in a reversible process is a state function

Good point.

The only problem being - in the real world there is no such thing as reversible process

[FreeTheBee]

I didn't think of the conservative force field, but had the words 'path integral' ringing in my head

Although the work in a reversible process is a state function, I think it is tricky when first learning this within the context of thermodynamics. At least for me it was. One has to distinguish between the initial and final state of a system and the state it is in when the work is performed. For example, reversibly expanding a gas at ambient temperature, or the sequence of heating the gas at fixed volume, expanding it, cooling it back to ambient temperature at the new volume, will give different work. The initial and final state of the system are the same for both processes, but the work was done between a different initial and final state, at a higher pressure in this case.