The p-V Diagram

Recall the work-done integral from Section 4.5, $W = \int_{V_i}^{V_f} p\,dV$ (Eq. 4.2). This integral can only be evaluated once the specific relationship between $p$ and $V$ — that is, the path — connecting the initial and final volumes is known. Graphically, this integral is exactly the area under the p-V curve between the initial and final volumes — so the area under any segment of a p-V curve directly represents the work done by the gas during that segment of the process.

A gas confined to a cylinder with a movable, frictionless, massless piston can change its volume in several distinct ways, each producing a visually distinct p-V curve:

• Expansion with varying pressure — the piston moves outward while the pressure changes continuously along the way; the curve slopes downward as volume rises, and the area beneath it is positive (the gas does positive work).
• Compression with varying pressure — the piston moves inward while the pressure changes continuously; the curve traces from a larger to a smaller volume, and the corresponding work is negative (work is done ON the gas).
• Expansion at constant pressure — the p-V curve is simply a horizontal straight line (parallel to the volume axis) running from an initial volume $V_1$ to a larger final volume $V_2$; the rectangular area beneath this flat line gives $W = p(V_2 - V_1)$ directly, since here — and only here — the pressure can be pulled outside the integral in Eq. (4.2). …