What is the precise definition of maximum work in the context of electrochemistry and thermodynamics?

Question

I'd like clarification of the meaning of $W_\text{max}$ in the following equations (and thus in the combined result in the third equation): \begin{align} W_\text{max} &= \Delta G\\ W_\text{max} &= -nF\varepsilon\\ \Delta G &= -nF\varepsilon\\ \end{align}

In the first equation, we somehow come to the conclusion that the maximum useful work available from a process is that process' change in Gibbs Free Energy. I ask now what work that means - does it mean work done on the surroundings by release of energy from the system? Does it mean the maximum work done on the system via a chemical process (this makes more sense because it doesn't make sense that an increase in Gibbs Free Energy of a system would do work on anything but the system)?

Secondly, in the electrochemical definition of $W_\text{max}$, we have the point of view of the system; that is, positive work is done on the system and negative by the system. Therefore, we add the negative sign because when cell potential is positive the work done on the system is negative (it does work on the surroundings).

The only logical conclusion I can come to is that the two $W_\text{max}$ definitions must be both with respect to the system's point of view, which means that $W_\text{max} = \Delta G$ refers to the maximum amount of work done on a system during a chemical process (the intuitive interpretation that I posed). However, it still makes me uncomfortable to define "useful work" as work done on the system. How can we harness work from putting energy into something?

Answer 1

"Maximum Additional Work that can be extracted at constant pressure and temperature" to be precise, which may include electrical work, etc. It means the work besides the expansion work.

The negative sign signifies the sign of the charge of the electron, since electrical work is proportional to the charge.

Answer 2

Therefore, we add the negative sign because when cell potential is positive the work done on the system is negative (it does work on the surroundings).

That's correct. This is merely a convention, though. We could define positive and negative work the other way around as long as we also flipped the sign in all the equations that use work: $\Delta G = w$ becomes $\Delta G = -w$; $\Delta U = q + w$ becomes $\Delta U = q - w$; etc.

How can we harness work from putting energy into something?

You can't, you harness the work when you take the energy out; you "store" the work when you put energy in. If, for example, you connect a battery to a DC motor and then have the motor raise a weight, you're doing ("harnassing") work by the cell on the surroundings, and a convention among chemists is to consider this negative work.