FIGURE 19.9 shows two different processes that take a gas from an initial state i to a final state f. Although the initial and final states are the same, the work done during these two processes is not the same. The work done during an ideal-gas process depends on the path followed through the pV diagram.

FIGURE 19.9 The work done during an ideal-gas process depends on the path.

You may recall that “work is independent of the path,” but that referred to a different situation. In Chapter 10, we found that the work done by a conservative force is independent of the physical path of the object through space. For an ideal-gas process, the “path” is a sequence of thermodynamic states on a pV diagram. It is a figurative path because we can draw a picture of it on a pV diagram, but it is not a literal path.

The path dependence of work has an important implication for multistep processes. For a process $1\to 2\to 3$, the work has to be calculated separately for each step in the process, giving $W_{\text{tot}}=W_{1\text{to}2}+W_{2\text{to}3}$. In most cases, calculating the work for a process that goes directly from 1 to 3 will given an incorrect answer. The initial and final states are the same, but the work is not the same because work depends on the path followed through the pV diagram.