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The kinetic theory derivation of pressure considers $N$ molecules in a cubic container of side $L$. Each molecule with x-component of velocity $v_x$ transfers momentum $2mv_x$ per wall collision, with collision frequency $v_x/(2L)$ on one wall. Summing over all molecules and averaging over three directions (isotropy: $\overline{v_x^2} = \overline{v_y^2} = \overline{v_z^2} = \overline{v^2}/3$) gives $P = \frac{1}{3}\frac{Nm\overline{v^2}}{V} = \frac{1}{3}\rho\overline{v^2}$.

Rewriting: $P = \frac{2}{3}\frac{N}{V}\overline{KE}$ where $\overline{KE} = \frac{1}{2}m\overline{v^2}$ is the average translational KE per molecule. This shows pressure is proportional to the kinetic energy density (kinetic energy per unit volume). Substituting $\overline{KE} = \frac{3}{2}k_BT$ recovers $PV = Nk_BT = nRT$.

Dalton's law follows naturally: in a mixture, each species contributes independently to pressure. $P_{\text{total}} = \sum P_i = \sum N_i k_BT/V$. Crucially, at the same $T$ and $V$, equal numbers of molecules of different gases exert equal pressure — pressure does not depend on molecular mass (Avogadro's insight).