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For the angularly resolved molecular arrival flux, let $f(v)$ be normalized by ${\int }_0^{\infty }f(v)dv=1$. Isotropic velocity directions have probability $d\Omega /(4\pi )$. Integrating over azimuth gives $(1/2)\sin \theta d\theta$, while the speed normal to a stationary piston is $v\cos \theta$. For incident directions $0\le \theta \le \pi /2$, the arrival flux is thereforeAn elastic collision transfers normal momentum $2mv\cos \theta$ to the piston. Integrating the momentum flux givesNo specific Maxwellian speed distribution was needed for this isotropic kinetic-pressure identity.

Use the figure's $+x$ direction toward the piston and let its inward speed $u>0$ mean piston velocity $-u\hat{\mathbf{x}}$. For reflection from a translating planar piston, in the piston frame the incoming normal velocity is $v_x+u$. Elastic reflection reverses it, and transforming back givesThe exact kinetic-energy change in this ideal massive-piston model is $\Delta K=2mu{v}_x+2m{u}^2$. Thus, to first order in the small piston speed,The first-order collision-flux approximation for a slow piston applies: the incident collision flux changes at order $u$, but its product with the order-$u$ energy increment changes the energy rate only at order $u^2$. Hence one may use the stationary flux at leading order. For piston area $S$,Here $dV/dt=-Su$. This compression work from molecular impacts, with no heat flow, is the first law for work done on the gas by compression. The derivation assumes $u$ small compared with thermal speeds and an approximately isotropic distribution at the piston; rapid compression can produce nonequilibrium pressure and higher-order corrections.