ideal gas law: $pV=nRT$

gas constant $R=8.31\textrm{J}/\textrm{mol}\cdot \textrm{K}$

volume work of expansion at constant pressure: $W=\int \frac{nRT}{V}dV=nRT \, \mathrm{ln}(\frac{V_f}{V_i})$

gas pressure: $p=\frac{nMv_{\textrm{rms}}^2}{3V}$

translational kinetic energy: $\overline{K}=\frac{3}{2}kT$

Boltzman constant $k=R/N_A$

mean free path: $\lambda =\frac{1}{\sqrt{2}\pi dN/V}$\footnote{$d$: diameter, $N$: number of molecules.}

Maxwell's speed distribution: $P(v)=4\pi (\frac{M}{2 \pi RT})^{3/2}v^2e^{-Mv^2/2RT}$

most propable speed: $v_p=\sqrt{\frac{2RT}{M}}$

average speed: $\overline{v}=\sqrt{\frac{8RT}{\pi M}}$

rms speed: $v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}$

internal energy of monoatomic gas: $E_{\mathrm{int}}=(nN_A)\overline{K}=\frac{3}{2}nRT$

monoatom: 3/2 ($f=1$)

diatom: 5/2 ($f=2$)

5-atom: 3 ($f=5$)

molar specific heat of monoatomic gas at constant volume: $C_v=\frac{3}{2}R=12.5\,\textrm{J/molK}$

constant volume, change in internal energy:

$\Delta E_{\textrm{int}}=Q=nC_v(T_f-T_i)$

molar specific heat of monoatomic gas at constant pressure: $C_p-C_v=R$

heat: $Q=nC_p(T_f-T_i)$

work: $W=nR(T_f-T_i)$

law of adiabatic expansion: $pV^\gamma =\textrm{constant}$, or $TV^{\gamma -1}=\textrm{constant}$

$\gamma =C_p/C_v=1+2/f$