Q.Using the kinetic theory relation between pressure and mean square speed, show that Boyle's law ($PV = \text{constant}$ at fixed temperature) follows directly from kinetic theory.

From kinetic theory, $PV = \tfrac{1}{3}Nm\overline{v^2}$. At constant temperature, the average translational kinetic energy per molecule, $\tfrac{1}{2}m\overline{v^2}$, stays fixed (Section 9.5), which means $\overline{v^2}$ itself must stay fixed (since $m$ is fixed for a given gas). For a fixed amount of gas, $N$ is also fixed.

With $N$, $m$ and $\overline{v^2}$ all constant, the entire right-hand side of $PV = \tfrac{1}{3}Nm\overline{v^2}$ is a fixed number -- call it $k$. So

$PV = k = \text{constant}$ …

Hold $T$ (hence $\overline{v^2}$) and $N$ fixed in $PV=\tfrac{1}{3}Nm\overline{v^2}$ and observe t …

• Forgetting to justify WHY $\overline{v^2}$ stays constant at fixed $T$ -- it is because average kinetic energy, and so $\overline{v^2}$, depends only on temperature (Section 9.5), not on pressure or volume directly. …