Equilibrium — Notes

Physical equilibria (closed system, fixed T): solid ⇌ liquid at the melting point; liquid ⇌ vapour (vapour pressure); solid ⇌ vapour; solute ⇌ solution in a saturated solution (solubility); gas ⇌ dissolved gas (Henry’s law: solubility ∝ pressure).

Chemical equilibrium is dynamic: the forward and reverse rates are equal and the macroscopic properties (concentration, colour, pressure) stay constant. It is reached from either side, needs a closed system, and is not altered in position by a catalyst, which only speeds its attainment.

For $a\text{A} + b\text{B} \rightleftharpoons c\text{C} + d\text{D}$: $K_c = \dfrac{[\text{C}]^c[\text{D}]^d}{[\text{A}]^a[\text{B}]^b}$. Pure solids and pure liquids are omitted (e.g. for $\ce{CaCO3(s) <=> CaO(s) + CO2(g)}$, $K_c = [\text{CO}_2]$).

For gases $K_p$ uses partial pressures, and $K_p = K_c(RT)^{\Delta n}$ with $\Delta n = n_\text{gaseous products} - n_\text{gaseous reactants}$ ($R = 0.08314\ \text{L bar K}^{-1}\text{mol}^{-1}$ for $p$ in bar).

Manipulation: reverse → $1/K$; multiply coefficients by $n$ → $K^n$; add equations → $K_1K_2$. A large K (≫ 1) means products are favoured; small K means reactants are favoured.

Reaction quotient $Q$: same form as $K$ with the present concentrations. $Q < K$: net forward; $Q = K$: equilibrium; $Q > K$: net reverse.

ICE table: Initial, Change (−ax, +cx), Equilibrium; substitute into $K$ and solve for $x$.

A system at equilibrium responds to a disturbance by partly opposing it.

• Concentration: adding a reactant (or removing a product) shifts the equilibrium forward; the opposite shifts it back. $K$ is unchanged.
• Pressure: increasing pressure (decreasing volume) shifts the equilibrium towards the side with fewer moles of gas; no effect if $\Delta n = 0$. $K$ is unchanged.
• Temperature: raising T favours the endothermic direction; lowering T favours the exothermic direction. This is the only change that alters $K$.
• Catalyst: speeds both directions equally; no shift, no change in $K$.
• Inert gas: at constant volume, no shift; at constant pressure the volume grows, favouring the side with more gas moles.

Haber process $\ce{N2 + 3H2 <=> 2NH3}$, $\Delta H^\circ = -92\ \text{kJ}$: high pressure and low temperature favour ammonia; about 700 K, high pressure and an iron catalyst are a compromise for rate.

• Arrhenius: acid gives $\ce{H+}$, base gives $\ce{OH-}$ in water.
• Brønsted–Lowry: acid = proton donor, base = proton acceptor; each has a conjugate partner ($\ce{HCl}/\ce{Cl-}$, $\ce{NH4+}/\ce{NH3}$); water and $\ce{HCO3-}$ are amphoteric. A strong acid has a very weak conjugate base.
• Lewis: acid = electron-pair acceptor ($\ce{BF3}$, $\ce{H+}$), base = electron-pair donor ($\ce{NH3}$, $\ce{OH-}$).

$K_w = [\text{H}^+][\text{OH}^-] = 1.0 \times 10^{-14}$ at 298 K (increases with temperature). $\text{pH} = -\log[\text{H}^+]$, $\text{pH} + \text{pOH} = 14$ at 298 K. A tenfold change in $[\text{H}^+]$ is one pH unit. For a conjugate pair $K_aK_b = K_w$, i.e. $pK_a + pK_b = pK_w$.

Weak acid HA: $K_a = c\alpha^2/(1-\alpha) \approx c\alpha^2$; $\alpha = \sqrt{K_a/c}$ (rises on dilution); $[\text{H}^+] = \sqrt{K_ac}$. Weak base: $[\text{OH}^-] = \sqrt{K_bc}$. Use the approximation only when $\alpha \ll 1$ (below about 5%).

Common ion effect: adding a common ion suppresses the ionisation of a weak electrolyte (and the solubility of a sparingly soluble salt).

Hydrolysis: salt of strong acid + strong base: neutral; weak acid + strong base: basic ($K_h = K_w/K_a$, $[\text{OH}^-] = \sqrt{K_hc}$); strong acid + weak base: acidic; weak acid + weak base: depends on $K_a$ and $K_b$.

Buffers resist a change in pH on adding small amounts of acid or base. Acidic buffer: weak acid + its salt; basic buffer: weak base + its salt. Henderson: $\text{pH} = pK_a + \log([\text{salt}]/[\text{acid}])$; at equal concentrations pH = $pK_a$; useful range about $pK_a \pm 1$.

For $\ce{A_xB_y(s) <=> xA^{y+} + yB^{x-}}$ in a saturated solution, $K_{sp} = [\text{A}^{y+}]^x[\text{B}^{x-}]^y$. With molar solubility $s$: AB, $K_{sp} = s^2$; $\text{AB}_2$ or $\text{A}_2\text{B}$, $K_{sp} = 4s^3$; $\text{A}_3\text{B}$, $K_{sp} = 27s^4$.

A common ion lowers solubility: for AgCl in $c$ mol L⁻¹ NaCl, $s = K_{sp}/c$.

Precipitation: compare the ionic product $Q_{sp}$ (using concentrations after mixing) with $K_{sp}$: $Q_{sp} > K_{sp}$, precipitate forms; $Q_{sp} = K_{sp}$, saturated; $Q_{sp} < K_{sp}$, no precipitate. Values used: $K_{sp}(\text{AgCl}) = 1.8 \times 10^{-10}$, $K_{sp}(\text{BaSO}_4) = 1.1 \times 10^{-10}$, $K_{sp}(\text{Ag}_2\text{CrO}_4) = 1.1 \times 10^{-12}$ at 298 K.