Kinetic Theory — Ideal Gas & Molecular Speeds (Medium)

Quick Recap — Ideal Gas & Kinetic Interpretation

• Ideal gas law: $PV=nRT=Nk_BT$ (with $k_B=\dfrac{R}{N_A}$).
• Pressure (kinetic): $P=\dfrac13\rho\,v_{rms}^2=\dfrac13\dfrac{Nm}{V}v_{rms}^2$.
• Energy: average translational KE per molecule $=\dfrac32 k_BT$; per mole $=\dfrac32 RT$ — it depends only on temperature, not on the gas.
• Speeds: $v_{rms}=\sqrt{\dfrac{3RT}{M}}\propto\sqrt{\dfrac{T}{M}}$; also $v_{rms}:v_{avg}:v_{mp}=\sqrt3:\sqrt{\tfrac{8}{\pi}}:\sqrt2\approx1.73:1.60:1.41$.

Worked mini-example. For oxygen ($M=0.032$ kg/mol) at 300 K: $v_{rms}=\sqrt{\dfrac{3RT}{M}}=\sqrt{\dfrac{3(8.31)(300)}{0.032}}\approx483$ m/s.