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Systematic approach for JEE kinetic theory problems:

Speed problems: Identify which speed is needed (RMS, average, or most probable). Use the appropriate formula with $T$ in kelvin and $M$ in kg/mol. For ratios, most quantities cancel: $v \propto \sqrt{T/M}$.

Energy problems: Translational KE per molecule = $\frac{3}{2}k_BT$ (always, regardless of gas type). Total internal energy depends on DOF: $U = \frac{f}{2}nRT$. Don't confuse translational KE with total internal energy for polyatomic gases.

Pressure problems: Use $P = \frac{1}{3}\rho\overline{v^2}$ when density and speed are given. Use $PV = nRT$ when moles and temperature are given. For mixtures, add partial pressures.

Mean free path: $\lambda = k_BT/(\sqrt{2}\pi d^2 P)$. State clearly whether $T$ or $P$ is held constant before claiming proportionality.

Common traps: (1) Using $M$ in g/mol instead of kg/mol in $\sqrt{3RT/M}$. (2) Confusing translational KE ($\frac{3}{2}k_BT$) with total energy ($\frac{f}{2}k_BT$). (3) Averaging $\gamma$ values instead of computing $C_v$ and $C_p$ separately for mixtures. (4) Forgetting that temperature is in kelvin (not Celsius) in all kinetic theory formulas.