Key Equations

Ideal gas law in terms of molecules	$p V = N k_{B} T$
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	$p V = n R T$
Van der Waals equation	$[p + a {(\frac{n}{V})}^{2}] (V − n b) = n R T$
Pressure, volume, and molecular speed	$p V = \frac{1}{3} N m \overset{–}{v^{2}}$
Root-mean-square speed	$v_{rms} = \sqrt{\frac{3 R T}{M}} = \sqrt{\frac{3 k_{B} T}{m}}$
Mean free path	$λ = \frac{V}{4 \sqrt{2} π r^{2} N} = \frac{k_{B} T}{4 \sqrt{2} π r^{2} p}$
Mean free time	$τ = \frac{k_{B} T}{4 \sqrt{2} π r^{2} p v_{rms}}$
The following two equations apply only to a monatomic ideal gas:
Average kinetic energy of a molecule	$\overset{–}{K} = \frac{3}{2} k_{B} T$
Internal energy	$E_{int} = \frac{3}{2} N k_{B} T .$
Heat in terms of molar heat capacity at constant volume	$Q = n C_{V} Δ T$
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	$f (v) = \frac{4}{\sqrt{π}} {(\frac{m}{2 k_{B} T})}^{3 / 2} v^{2} e^{− m v^{2} / 2 k_{B} T}$
Average velocity of a molecule	$\bar{v} = \sqrt{\frac{8}{π} \frac{k_{B} T}{m}} = \sqrt{\frac{8}{π} \frac{R T}{M}}$
Peak velocity of a molecule	$v_{p} = \sqrt{\frac{2 k_{B} T}{m}} = \sqrt{\frac{2 R T}{M}}$