Gas Laws

$P_1V_1 = P_2V_2 \quad \text{(Boyle, constant } T, n\text{)}$

$\frac{V_1}{T_1} = \frac{V_2}{T_2} \quad \text{(Charles, constant } P, n\text{)}$

$\frac{P_1}{T_1} = \frac{P_2}{T_2} \quad \text{(Gay-Lussac, constant } V, n\text{)}$

$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \quad \text{(Combined gas law)}$

$PV = nRT, \quad R = 0.0821\ \frac{\text{L·atm}}{\text{mol·K}} = 8.314\ \frac{\text{J}}{\text{mol·K}}$

Dalton's Law

$P_\text{total} = p_1 + p_2 + \cdots + p_n, \quad p_i = x_i \cdot P_\text{total}$

Graham's Law

$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}, \qquad \frac{t_1}{t_2} = \sqrt{\frac{M_1}{M_2}}$

Molecular Speeds

$v_{mp} = \sqrt{\frac{2RT}{M}}, \quad v_{avg} = \sqrt{\frac{8RT}{\pi M}}, \quad v_{rms} = \sqrt{\frac{3RT}{M}}$

$v_{mp} : v_{avg} : v_{rms} = \sqrt{2} : \sqrt{\frac{8}{\pi}} : \sqrt{3} \approx 1 : 1.128 : 1.224$

Kinetic Energy

$KE_\text{avg} = \frac{3}{2}RT \quad \text{(per mole)}; \quad KE = \frac{3}{2}k_BT \quad \text{(per molecule)}$

Van der Waals Equation

$\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$

$\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT \quad \text{(per mole, } V_m = V/n\text{)}$

Compressibility Factor

$Z = \frac{PV}{nRT} = \frac{PV_m}{RT}; \quad Z = 1\ (\text{ideal}); \quad Z < 1\ (\text{attraction}); \quad Z > 1\ (\text{repulsion})$

Boyle Temperature and Critical Constants

$T_B = \frac{a}{Rb}; \quad T_c = \frac{8a}{27Rb}; \quad P_c = \frac{a}{27b^2}; \quad V_c = 3b$

$\frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375$

Molar Mass from Density

$M = \frac{dRT}{P}; \quad M = 22.4 \times d \text{ at STP (d in g/L)}$