Summary

Ideal gas equation (from the kinetic theory): $PV = \frac{1}{3} m N \overline{v^2}$, where $m$ is mass of one molecule, $N$ is number of molecules, and $\overline{v^2}$ is mean square speed. This leads to $PV = nRT$ (the macroscopic ideal gas law).

Pressure is due to molecular collisions with the walls: $P = \frac{1}{3} \rho \overline{v^2}$, where $\rho$ is density. The average kinetic energy per molecule is $\frac{1}{2} m \overline{v^2} = \frac{3}{2} k_B T$, with $k_B = 1.38 \times 10^{-23} \, \text{J/K}$.

Root-mean-square speed: $v_{\text{rms}} = \sqrt{\overline{v^2}} = \sqrt{\frac{3RT}{M}}$, where $M$ is molar mass. For a given gas, $v_{\text{rms}} \propto \sqrt{T}$.

Degrees of freedom ($f$): monatomic ($f=3$), diatomic at moderate temperatures ($f=5$: 3 translational + 2 rotational), polyatomic ($f=6$ for non-linear). Each degree contributes $\frac{1}{2} k_B T$ per molecule to internal energy.

Internal energy of $n$ moles: $U = n \cdot \frac{f}{2} RT$. For a monatomic gas, $U = \frac{3}{2} nRT$.

Molar specific heats: $C_V = \frac{f}{2}R$, $C_P = C_V + R = \frac{f+2}{2}R$. Ratio $\gamma = \frac{C_P}{C_V} = 1 + \frac{2}{f}$.

Mean free path $\lambda = \frac{1}{\sqrt{2} \pi d^2 n_v}$, where $d$ is molecular diameter and $n_v$ is number density. $\lambda$ is inversely proportional to pressure (at constant $T$). …