Equilibrium

Reversible Reaction: Reaction that can proceed both forward and backward. Indicated by ⇌.

Chemical Equilibrium: State where forward rate = backward rate; concentrations remain constant (dynamic, not static).

Law of Mass Action (Guldberg-Waage): Rate of reaction $\propto$ product of active masses (concentrations) of reactants.

For: $aA + bB \rightleftharpoons cC + dD$ $$K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}$$

$K_c$ (concentration): Used when concentrations expressed in mol/L. Units depend on $\Delta n = (c+d)-(a+b)$.

$K_p$ (partial pressure): Used for gaseous reactions. Relation: $$K_p = K_c(RT)^{\Delta n_g}$$ $\Delta n_g$ = (gaseous product moles) - (gaseous reactant moles).

• If $\Delta n = 0$: $K_p = K_c$
• If $\Delta n > 0$: $K_p > K_c$
• If $\Delta n < 0$: $K_p < K_c$

$K_x$ (mole fraction): $K_p = K_x(P_{\text{total}})^{\Delta n}$.

Characteristics of Equilibrium Constant:

1. Constant at given temperature
2. Independent of initial concentrations
3. Independent of presence of catalyst
4. Changes with temperature
5. Reverse reaction: $K_{\text{reverse}} = 1/K_{\text{forward}}$
6. Adding reactions: $K_{\text{net}} = K_1 \times K_2$

Magnitude of $K$:

• $K > 10^3$: forward strongly favored (products dominate)
• $K < 10^{-3}$: backward favored (reactants dominate)
• $10^{-3} < K < 10^3$: appreciable concentrations of both

Reaction Quotient ($Q$): Same expression as $K$, but with current (not equilibrium) concentrations.

• $Q < K$: forward shifts (toward products)
• $Q > K$: backward shifts (toward reactants)
• $Q = K$: at equilibrium

Relation with Free Energy: $$\Delta G^\ominus = -RT\ln K = -2.303RT\log K$$ $$\Delta G = \Delta G^\ominus + RT\ln Q$$

Degree of dissociation ($\alpha$): Fraction of molecules dissociating.

For: $PCl_5 \rightleftharpoons PCl_3 + Cl_2$, starting with $n$ moles in volume $V$: $$K_c = \frac{n\alpha^2}{V(1-\alpha)} \quad \text{(if } \alpha \ll 1 \text{: } K_c \approx \alpha^2 \cdot n/V)$$

Le Chatelier's Principle (1884): If a system in equilibrium is disturbed, it shifts in a direction that minimizes the disturbance.

Factors:

Effect of inert gas:

• At constant volume: No effect on equilibrium ($K$ same)
• At constant pressure: Effectively increases volume → shifts toward more gas moles

Effect of temperature:

• Exothermic: $K$ decreases with increasing $T$
• Endothermic: $K$ increases with increasing $T$

Van't Hoff Equation: $$\ln\left(\frac{K_2}{K_1}\right) = \frac{\Delta H^\ominus}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$$

Industrial Applications:

Catalyst:

• Speeds up forward and reverse reactions equally
• Does NOT shift equilibrium
• Only reduces time to reach equilibrium

Heterogeneous Equilibrium: Pure solids and pure liquids don't appear in $K$ expression (their activities = 1). e.g., $CaCO_3(s) \rightleftharpoons CaO(s) + CO_2(g)$: $K_p = P_{CO_2}$.

Theories of Acids and Bases:

Conjugate Pairs: Acid → its conjugate base after losing H⁺. e.g., HCl/Cl⁻, H₂O/OH⁻, NH₃/NH₄⁺.

Strong vs Weak:

• Strong acid (HCl, HBr, HI, HNO₃, H₂SO₄, HClO₄): fully dissociates
• Weak acid (CH₃COOH, HCN, HF): partial dissociation; Ka small
• Strong base (NaOH, KOH, LiOH, group I/II hydroxides): fully dissociates
• Weak base (NH₃, RNH₂): partial; Kb small

Auto-ionization of Water: $H_2O \rightleftharpoons H^+ + OH^-$, $K_w = [H^+][OH^-] = 10^{-14}$ at $25°$C.

pH Scale: $$pH = -\log[H^+], \quad pOH = -\log[OH^-], \quad pH + pOH = 14 \text{ at } 25°C$$

• Acidic: pH < 7
• Neutral: pH = 7
• Basic: pH > 7

Ostwald's Dilution Law (for weak electrolytes): $K_a = c\alpha^2/(1-\alpha)$. If $\alpha \ll 1$: $K_a \approx c\alpha^2$, so $\alpha = \sqrt{K_a/c}$.

For weak acid: $[H^+] = \sqrt{K_aC}$, $pH = -(1/2)(\log K_a + \log C) = (1/2)(pK_a - \log C)$.

For weak base: $pOH = (1/2)(pK_b - \log C)$.

Buffer Solution: Resists change in pH on addition of small amounts of acid/base.

Henderson-Hasselbalch Equation: $$pH = pK_a + \log\frac{[\text{salt}]}{[\text{acid}]} \text{ (acidic buffer)}$$ $$pOH = pK_b + \log\frac{[\text{salt}]}{[\text{base}]} \text{ (basic buffer)}$$

Buffer Capacity: Maximum when $[\text{salt}] = [\text{acid}]$ (or base), giving pH = pKa.

Solubility Product ($K_{sp}$): Ionic product of saturated solution of sparingly soluble salt.

For: $A_xB_y(s) \rightleftharpoons xA^{p+}(aq) + yB^{q-}(aq)$ $$K_{sp} = [A^{p+}]^x[B^{q-}]^y$$

Relation with solubility $s$ (mol/L):

Ionic Product (Q): Calculated like $K_{sp}$ but with current concentrations.

• $Q < K_{sp}$: unsaturated; more solid dissolves
• $Q = K_{sp}$: saturated (equilibrium)
• $Q > K_{sp}$: supersaturated; precipitation occurs

Common Ion Effect: Solubility of a salt is reduced by the presence of a common ion. e.g., AgCl solubility decreases in NaCl solution.

Application: Salt precipitation, qualitative analysis (group separation in salt analysis).

Salt Hydrolysis: Ions of dissolved salt react with water, changing pH.

For salt of weak acid + strong base (e.g., CH₃COONa): $$K_h = K_w/K_a, \quad pH = 7 + (1/2)(pK_a + \log C)$$

For salt of weak base + strong acid (e.g., NH₄Cl): $$K_h = K_w/K_b, \quad pH = 7 - (1/2)(pK_b + \log C)$$

For salt of weak acid + weak base: $pH = 7 + (1/2)(pK_a - pK_b)$ (independent of concentration to first order).

Degree of hydrolysis ($h$): Fraction of salt that hydrolyzes. $h = \sqrt{K_h/C}$.