Quasistatic work and path dependence

For a piston with external resisting pressure $p_{\mathrm{ext}}$, the pressure-volume work done on the gas is minus the integral of external pressure over volume change. Expansion has positive volume change and negative work on the gas; compression has the opposite signs. This expression assumes a specified mechanical boundary and separates other work channels such as electrical stirring. When piston kinetic energy or friction in its surroundings matters, identify which body receives each contribution before applying a gas-only energy balance.

For a mechanically reversible quasistatic piston, the pressure difference needed to move it tends to zero and gas pressure can replace external pressure in the limiting integral. A rapid expansion may have no well-defined uniform gas pressure during the motion. Drawing an equilibrium curve between its endpoints does not turn that curve into the actual trajectory. In free expansion into vacuum, external pressure and boundary work vanish even though an ideal gas eventually occupies a larger volume.

In a quasistatic pressure-volume diagram the work by the gas is the signed area under its path. If two routes connect the same initial and final states, their work difference equals the signed enclosed area. A constant-volume segment contributes no pressure-volume work, although pressure and temperature can change substantially along it. An expansion performed at higher pressure transfers more work outward than the same volume change at lower pressure. This is a geometrical explanation of path dependence.

The notation $\delta W$ deliberately differs from the exact differential $dU$ of internal energy. Integrating $dU$ around a closed equilibrium cycle gives zero, but integrating work need not. One cannot define “work contained in the gas” by assigning a W value to each state. The same objection applies to heat: both describe transfers during a specified process. Their sum can nevertheless equal a state-function change because the first law constrains their combined accounting.

Quasistatic means the system passes sufficiently near equilibrium that its state coordinates remain meaningful. Reversible additionally means the complete process can be undone with no residual change in the system and surroundings. Dry friction can dissipate energy even for a slowly moving piston. Heat transfer across a finite temperature difference also produces irreversibility, even when each large body stays internally equilibrated. Merely allowing more time does not remove a persistent force or temperature gap.

A reversible idealization requires eliminating those dissipative mechanisms and driving through infinitesimal imbalances. It provides a useful limiting path and a benchmark for work, not a claim that practical apparatus can complete a finite cycle at finite rate with no losses. When comparing protocols, first specify endpoints, then the mechanical path and contacts, and finally which irreversibilities remain. This order prevents “quasistatic” from being used as permission to apply every reversible heat or entropy formula.