Equilibrium Constants

$K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b} \quad \text{(equilibrium concentrations only; omit pure solids and liquids)}$

$K_p = K_c(RT)^{\Delta n} \quad \text{where } \Delta n = \text{moles gaseous products} - \text{moles gaseous reactants}$

$\Delta G^\circ = -RT\ln K = -2.303\,RT\log K$

$\text{When } K > 1: \Delta G^\circ < 0 \text{ (product-favored)}; \quad K < 1: \Delta G^\circ > 0 \text{ (reactant-favored)}$

pH and pOH

$pH = -\log[H^+] \qquad pOH = -\log[OH^-] \qquad pH + pOH = 14 \text{ (at 25°C only)}$

$K_w = [H^+][OH^-] = 10^{-14} \text{ at 25°C}$

Weak Acid / Base

$K_a = C\alpha^2 \approx C\alpha^2 \text{ (when } \alpha \ll 1\text{)} \Rightarrow [H^+] = \sqrt{K_a \cdot C}$

$K_a \times K_b = K_w = 10^{-14} \text{ (conjugate pair at 25°C)}$

Henderson-Hasselbalch

$pH = pK_a + \log\frac{[\text{A}^-]}{[\text{HA}]} = pK_a + \log\frac{[\text{salt}]}{[\text{acid}]}$

$pOH = pK_b + \log\frac{[\text{BH}^+]}{[\text{B}]} = pK_b + \log\frac{[\text{salt}]}{[\text{base}]}$

Salt Hydrolysis

$\text{Weak acid + Strong base: } pH = 7 + \tfrac{1}{2}pK_a + \tfrac{1}{2}\log C$

$\text{Strong acid + Weak base: } pH = 7 - \tfrac{1}{2}pK_b - \tfrac{1}{2}\log C$

$\text{Weak acid + Weak base: } pH = 7 + \tfrac{1}{2}pK_a - \tfrac{1}{2}pK_b$

Solubility Product

$$K_{sp} = [$A^{n+}$]^m[$B^{m-}$]^n \quad \text{for } A_mB_n \rightleftharpoons m$A^{n+}$ + n$B^{m-}$$$

$\text{AgCl: } s = \sqrt{K_{sp}} \qquad \text{Ag}_2\text{CrO}_4: K_{sp} = 4s^3 \Rightarrow s = \left(\frac{K_{sp}}{4}\right)^{1/3}$

Key Reactions with Conditions

$N_{2}$(g) + $3H_{2}$(g) --[Fe catalyst, 400-500°C, 200 atm]--> $2NH_{3}$(g) (Haber process — exothermic; high T lowers yield)

AgCl(s) --[water]--> $Ag^{+}$(aq) + $Cl^{-}$(aq) ; Ksp = [$Ag^{+}$][$Cl^{-}$] = $1.8 \times 10^{-10}$