Gas Laws in the Light of Kinetic Theory

Kinetic theory's real power is that it does not merely describe pressure and temperature separately -- it shows that the THREE gas laws found empirically, decades earlier, by Boyle, Charles and Avogadro working independently from laboratory observation, all follow as direct mathematical consequences of the SAME single relation, $PV = \tfrac{1}{3}Nm\overline{v^2}$, combined with the kinetic interpretation of temperature, $\overline{KE} \propto T$ (Section 9.5).

Boyle's law ($PV = \text{constant}$ at fixed temperature and fixed amount of gas). At constant $T$, the average kinetic energy per molecule, $\tfrac{1}{2}m\overline{v^2}$, stays fixed (Section 9.5), so $\overline{v^2}$ itself is a constant. With $N$ (the number of molecules) also fixed, the pressure relation $PV = \tfrac{1}{3}Nm\overline{v^2}$ then has every quantity on the right-hand side constant except $P$ and $V$ themselves -- so their product $PV$ must stay constant as either one is varied, which is exactly Boyle's law.

Charles's law ($V \propto T$ at constant pressure and fixed amount of gas). Starting again from $PV = \tfrac{1}{3}Nm\overline{v^2}$, and using $\overline{v^2} = 3k_BT/m$ from Section 9.4-9.5, this becomes $PV = Nk_BT$. At constant $P$ and fixed $N$, this rearranges directly to $V = (Nk_B/P)T$ -- since everything in the bracket is now held constant, $V$ is directly proportional to the absolute temperature $T$, which is Charles's law. …