Gas Laws

$PV = \text{constant} \quad \text{(Boyle's Law, const. } T, n\text{)}$

$\frac{V}{T} = \text{constant} \quad \text{(Charles's Law, const. } P, n\text{)}$

$\frac{P}{T} = \text{constant} \quad \text{(Gay-Lussac's Law, const. } V, n\text{)}$

$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \quad \text{(Combined Gas Law, const. } n\text{)}$

$PV = nRT \quad \text{where } R = 0.0821 \text{ L·atm·mol}^{-1}\text{K}^{-1} = 8.314 \text{ J·mol}^{-1}\text{K}^{-1}$

Dalton's Law

$P_{\text{total}} = p_1 + p_2 + p_3 + \cdots$

$p_i = x_i \times P_{\text{total}} \quad \text{where } x_i = \frac{n_i}{n_{\text{total}}}$

Graham's Law

$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} = \sqrt{\frac{\rho_2}{\rho_1}} \quad \text{(rates of diffusion/effusion)}$

$\frac{t_1}{t_2} = \sqrt{\frac{M_1}{M_2}} \quad \text{(effusion times)}$

Molecular Speeds

$v_{rms} = \sqrt{\frac{3RT}{M}} \quad \text{(root mean square speed)}$

$v_{avg} = \sqrt{\frac{8RT}{\pi M}} \quad \text{(average speed)}$

$v_{mp} = \sqrt{\frac{2RT}{M}} \quad \text{(most probable speed)}$

$v_{rms} : v_{avg} : v_{mp} = \sqrt{3} : \sqrt{\frac{8}{\pi}} : \sqrt{2} \approx 1.224 : 1.128 : 1.000$

Kinetic Theory

$\overline{KE} = \frac{3}{2}RT \quad \text{(per mole)}$

$\overline{KE} = \frac{3}{2}k_BT \quad \text{(per molecule, } k_B = 1.38 \times 10^{-23} \text{ J/K)}$

$PV = \frac{1}{3}Nmc^2 = \frac{1}{3}nMv_{rms}^2$

Real Gases — Van der Waals Equation

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

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

$Z = \frac{PV}{nRT} = \frac{PV_m}{RT} \quad \text{(compressibility factor)}$

$T_B = \frac{a}{Rb} \quad \text{(Boyle temperature)}$

Critical Constants

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

$\frac{P_cV_c}{T_c} = \frac{3R}{8} \approx 0.375R$

Gas Density

$d = \frac{PM}{RT} \quad \text{or} \quad M = \frac{dRT}{P}$

Temperature Conversion

$T(\text{K}) = T(°\text{C}) + 273.15$