Pressure work ($\delta W=pdV$)

Pressure work is mechanical work transferred by pressure forces acting through a moving boundary. For a piston of area $A$ moving through $dx$, $dV=Adx$ and the opposing applied force is $p_{\mathrm{ext}}A$. The work done by the fluid is thereforeThe familiar $\delta W=pdV$ uses the system pressure when it equals the applied boundary pressure, as in a reversible mechanical step. In a free expansion into vacuum, $p_{\mathrm{ext}}=0$ and the boundary work vanishes even if the gas initially has nonzero internal pressure.

In steady incompressible flow, the Bernoulli equation contains $p/\rho$ as flow work per unit mass. This accounts for pressure forces at a control volume's entrances and exits; it is not an independently conserved internal energy of an isolated parcel. The first law of thermodynamics keeps the complete work and energy balance.