Key Equations

Ideal gas law in terms of molecules	$pV=N{k}_{\text{B}}T$
Ideal gas law ratios if the amount of gas is constant	$\frac{p_1{V}_1}{T_1}=\frac{p_2{V}_2}{T_2}$
Ideal gas law in terms of moles	$pV=nRT$
Van der Waals equation	$\left[p+a{\left(\frac{n}{V}\right)}^2\right]\left(V\text{−}nb\right)=nRT$
Pressure, volume, and molecular speed	$pV=\text{}\frac{1}{3}Nm\stackrel{\text{–}}{v^2}$
Root-mean-square speed	$v_{\text{rms}}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3k_{\text{B}}T}{m}}$
Mean free path	$\lambda =\frac{V}{4\sqrt{2}\pi r^{2}N}=\frac{k_{\text{B}}T}{4\sqrt{2}\pi r^{2}p}$
Mean free time	$\tau =\frac{k_{\text{B}}T}{4\sqrt{2}\pi r^{2}p{v}_{\text{rms}}}$
The following two equations apply only to a monatomic ideal gas:
Average kinetic energy of a molecule	$\stackrel{\text{–}}{K}=\frac{3}{2}{k}_{\text{B}}T$
Internal energy	$E_{\text{int}}=\frac{3}{2}N{k}_{\text{B}}T.$
Heat in terms of molar heat capacity at constant volume	$Q=n{C}_V\text{Δ}T$
Molar heat capacity at constant volume for an ideal gas with d degrees of freedom	$C_V=\frac{d}{2}R$
Maxwell–Boltzmann speed distribution	$f(v)=\frac{4}{\sqrt{\pi }}{\left(\frac{m}{2k_{\text{B}}T}\right)}^{3\text{/}2}{v}^2{e}^{\text{−}m{v}^2\text{/}2k_{\text{B}}T}$
Average velocity of a molecule	$\overline{v}=\sqrt{\frac{8}{\pi }\frac{k_{\text{B}}T}{m}}=\sqrt{\frac{8}{\pi }\frac{RT}{M}}$
Peak velocity of a molecule	$v_p=\sqrt{\frac{2k_{\text{B}}T}{m}}=\sqrt{\frac{2RT}{M}}$