Work Done During a Thermodynamic Process

Consider a system that starts at state $A$, with coordinates $(V_i, p_i)$ on a p-V diagram, and ends at state $B$, with coordinates $(V_f, p_f)$, where $V_f > V_i$ and $p_f < p_i$. There are many different paths — many different sequences of intermediate states — by which the system could be taken from $A$ to $B$; three representative ones illustrate the point.

Path 1 is a single smooth curve directly connecting $A$ to $B$, along which both pressure and volume change together continuously. Because the volume increases throughout, the work done by the system along this path is positive, and its value is given by the area under this curve.

Path 2 takes a two-step, L-shaped route through an intermediate corner state $C$: first the volume increases from $V_i$ to $V_f$ at the constant pressure $p_i$ (a horizontal leg from $A$ to $C$), and only then does the pressure fall from $p_i$ to $p_f$ at the now-constant volume $V_f$ (a vertical leg from $C$ to $B$). Since a constant-volume leg contributes no work at all (Section 4.6.3), the total work done along path 2 is just the rectangular area under the horizontal leg, at the higher pressure $p_i$.

Path 3 takes the mirror-image two-step route: first the pressure falls from $p_i$ to $p_f$ at the constant volume $V_i$ (a vertical leg), and only then does the volume increase from $V_i$ to $V_f$ at the now-constant, lower pressure $p_f$ (a horizontal leg). Again the vertical leg contributes nothing, so the total work is the rectangular area under the horizontal leg — but this time swept out at the lower pressure $p_f$, giving a visibly smaller work value than path 2. …