Kinetic Theory — Advanced Multi-Step Practice (Hard)

Quick Recap — Advanced Kinetic Theory (Multi-step)

• rms speed numbers: $v_{rms}=\sqrt{\dfrac{3RT}{M}}=\sqrt{\dfrac{3k_BT}{m}}$. Ratio between two gases at the same $T$: $\dfrac{v_1}{v_2}=\sqrt{\dfrac{M_2}{M_1}}$.
• Mixture specific heats: $C_{V,mix}=\dfrac{n_1C_{V1}+n_2C_{V2}}{n_1+n_2}$; then $\gamma_{mix}=\dfrac{C_{P,mix}}{C_{V,mix}}$.
• Energy scale: average KE per molecule $=\dfrac32 k_BT$ (e.g. $\approx6.2\times10^{-21}$ J at 300 K).
• Graham's effusion: rate $\propto\dfrac{1}{\sqrt M}$.

Worked mini-example. A mixture of 1 mol monatomic ($C_V=\tfrac32 R$) and 1 mol diatomic ($C_V=\tfrac52 R$): $C_{V,mix}=\dfrac{\tfrac32R+\tfrac52R}{2}=2R$, $C_{P,mix}=3R$, so $\gamma_{mix}=\dfrac{3R}{2R}=1.5$.