• id: JTHERM-02-N02
• title: Derivation of Gas Pressure from Kinetic Theory
• tags: pressure, momentum-transfer, derivation

Pressure arises from momentum transfer when molecules collide with container walls. For $N$ molecules in volume $V$, each of mass $m$ with mean square speed $\overline{v^2}$: $P = \frac{1}{3}\frac{Nm\overline{v^2}}{V} = \frac{1}{3}\rho\overline{v^2}$, where $\rho = Nm/V$ is the gas density. The factor $\frac{1}{3}$ arises because the three spatial components of velocity contribute equally on average: $\overline{v_x^2} = \overline{v_y^2} = \overline{v_z^2} = \frac{1}{3}\overline{v^2}$. This derivation connects the macroscopic observable (pressure) to microscopic molecular motion.