The work W in the first law is the work done on the system by external forces from the environment. However, it makes more sense in “practical thermodynamics” to use the work done by the system. For example, you want to know how much useful work you can obtain from an expanding gas. The work done by the system is called $W_{\text{s}}$.

Work done by the environment and work done by the system are not mutually exclusive. In fact, they are very simply related by $W_{\text{s}}=-W$. In FIGURE 21.1, force $\vec{F}{}_{\text{gas}}$ due to the gas pressure does work when the piston moves. This is $W_{\text{s}}$, the work done by the system. At the same time, some object in the environment, such as a piston rod, must be pushing inward with force $\vec{F}{}_{\text{ext}}=-\vec{F}{}_{\text{gas}}$ to keep the gas pressure from blowing the piston out. This force does the work W on the system, work that you’ve learned is the negative of the area under the pV curve of the process.

FIGURE 21.1 Forces $\vec{F}{}_{\text{gas}}$ and $\vec{F}{}_{\text{ext}}$ both do work as the piston moves.

Full Alternative Text

Because the forces are equal but opposite, we see that

When a gas expands and pushes the piston out, transferring energy out of the system, we say “the system does work on the environment.” While this may seem to imply that the environment is doing no work on the system, all the phrase means is that $W_{\text{s}}$ is positive and W is negative. Similarly, “the environment does work on the system” means that $W>0$ (energy is transferred into the system) and thus $W_{\text{s}}<0$. Whether we use W or $W_{\text{s}}$ is a matter of convenience.

The first law of thermodynamics $\text{Δ}{E}_{\text{th}}=W+Q$ can be written in terms of $W_{\text{s}}$ as

Any energy transferred into a system as heat is either used to do work or stored within the system as an increased thermal energy.