Summary

This chapter built up WBCHSE Unit 9's kinetic theory of gases in the order the syllabus lists it. Assumptions (Section 9.2): molecules are point-like, in ceaseless random motion, exert no force on each other except during brief, perfectly elastic collisions, and obey Newton's laws throughout. Pressure (Section 9.3): deriving the momentum delivered per collision and summing over all molecules and all directions gives $P = \tfrac{1}{3}\rho\overline{v^2} = \tfrac{1}{3}\dfrac{Nm}{V}\overline{v^2}$, the pressure exerted purely by molecular collisions with the container wall. RMS speed (Section 9.4): combining this pressure relation with the ideal gas equation gives $v_{rms} = \sqrt{\overline{v^2}} = \sqrt{3RT/M} = \sqrt{3k_BT/m}$, increasing with $\sqrt{T}$ and decreasing with $\sqrt{M}$; the Maxwell-Boltzmann distribution shows the full spread of molecular speeds, ordered $v_p < \bar{v} < v_{rms}$. Kinetic interpretation of temperature (Section 9.5): average translational kinetic energy per molecule is $\tfrac{1}{2}m\overline{v^2} = \tfrac{3}{2}k_BT$, giving absolute temperature a direct mechanical meaning, and absolute zero the meaning of zero molecular translational motion. Gas laws from kinetic theory (Section 9.6): Boyle's law, Charles's law, and Avogadro's law all follow directly from $PV = Nk_BT$ examined under different conditions held fixed in turn. Degrees of freedom (Section 9.7): monatomic $f=3$ (translation only); rigid diatomic $f=5$ (3 translation + 2 rotation); non-linear triatomic $f=6$ (3 translation + 3 rotation), vibration ignored at moderate temperature. Law of equipartition of energy (Section 9.8, statement only): each degree of freedom carries average energy $\tfrac{1}{2}k_BT$ per molecule, giving $U=\tfrac{f}{2}nRT$, $C_v=\tfrac{f}{2}R$, $C_p=C_v+R$ (Mayer's relation), and $\gamma=C_p/C_v=(f+2)/f$ -- $5/3$ for monatomic, $7/5$ for diatomic, $4/3$ for non-linear triatomic gases. Mean free path (Section 9.9): $\lambda = \dfrac{1}{\sqrt{2}\,\pi d^2 n}$, the average distance travelle …