Equilibrium

1. Dynamic Nature of Equilibrium & Law of Chemical Equilibrium

• Dynamic Equilibrium: A reversible state where the rate of the forward reaction equals the rate of the reverse reaction ($r_f = r_b$). Concentrations of reactants and products remain constant over time, though molecular transformations continue across both directions.
• Law of Mass Action (Guldberg & Waage): At a given temperature, for a reversible reaction $a\text{A} + b\text{B} \rightleftharpoons c\text{C} + d\text{D}$, the equilibrium constant is expressed as:

  $$K_c = \frac{[\text{C}]^c [\text{D}]^d}{[\text{A}]^a [\text{B}]^b}$$

• Mathematical Properties of Equilibrium Constants:
  • Reversing an equation inverts $K$: $K_{\text{rev}} = \frac{1}{K}$.
  • Multiplying coefficients by $n$ raises $K$ to the $n$-th power: $K' = K^n$.
  • Adding two chemical equations multiplies their equilibrium constants: $K_{\text{net}} = K_1 \times K_2$.

2. Relationships between $K_p$, $K_c$, and Reaction Quotient ($Q$)

• $K_p$ and $K_c$ Relationship:

  $$K_p = K_c (RT)^{\Delta n_g}$$

  Where $\Delta n_g = (\text{Moles of gaseous products}) - (\text{Moles of gaseous reactants})$.
  • If $\Delta n_g = 0 \implies K_p = K_c$ (e.g., $\text{H}_2(g) + \text{I}_2(g) \rightleftharpoons 2\text{HI}(g)$).
  • If $\Delta n_g > 0 \implies K_p > K_c$ (e.g., $\text{PCl}_5(g) \rightleftharpoons \text{PCl}_3(g) + \text{Cl}_2(g)$).
  • If $\Delta n_g < 0 \implies K_p < K_c$ (e.g., $\text{N}_2(g) + 3\text{H}_2(g) \rightleftharpoons 2\text{NH}_3(g)$).
• Direction Prediction via Reaction Quotient ($Q$):
  • $Q_c < K_c$: Forward reaction proceeds ($r_f > r_b$, net reactants $\rightarrow$ products).
  • $Q_c = K_c$: System is at dynamic equilibrium.
  • $Q_c > K_c$: Reverse reaction proceeds ($r_b > r_f$, net products $\rightarrow$ reactants).
• Thermodynamic Link:

  $$\Delta G = \Delta G^\circ + RT \ln Q \quad \implies \quad \Delta G^\circ = -RT \ln K = -2.303 RT \log K$$

3. Le Chatelier's Principle & Industrial Optimisation

If a constraint (concentration, pressure, volume, or temperature) is applied to a system at equilibrium, the equilibrium shifts in the direction that counteracts the effect of the constraint.

4. Acid-Base Theories & Auto-Ionisation of Water

• Three Classical Definitions:
  1. Arrhenius: Acids dissociate to produce $\text{H}^+(aq)$; bases dissociate to produce $\text{OH}^-(aq)$ in water.
  2. Brønsted-Lowry: Acids are proton ($\text{H}^+$) donors; bases are proton ($\text{H}^+$) acceptors. A conjugate acid-base pair differs by exactly one proton:

     $$\text{Acid} \rightleftharpoons \text{Conjugate Base} + \text{H}^+$$

  3. Lewis: Acids are electron-pair acceptors (electron deficient: $\text{BF}_3, \text{AlCl}_3, \text{H}^+$); bases are electron-pair donors ($\text{NH}_3, \text{H}_2\text{O}, \text{OH}^-$).
• Ionic Product of Water ($K_w$):

  $$2\text{H}_2\text{O}(l) \rightleftharpoons \text{H}_3\text{O}^+(aq) + \text{OH}^-(aq) \quad \implies \quad K_w = [\text{H}^[\text{OH}^-] = 1.0 \times 10^{-14}\text{ at 298 K}$$

  $$\text{p}K_w = \text{pH} + \text{pOH} = 14$$

  Because auto-protolysis of water is endothermic, $K_w$ increases with temperature, decreasing the neutral pH point below 7 at elevated temperatures.

5. Ionisation of Weak Electrolytes & Ostwald's Dilution Law

For a weak acid $\text{HA}$ with initial concentration $C$ and degree of ionisation $\alpha$:

$$\text{HA} \rightleftharpoons \text{H}^+ + \text{A}^- \quad \implies \quad K_a = \frac{C\alpha^2}{1 - \alpha}$$

For weak electrolytes where $\alpha \ll 1$ (typically $\alpha < 0.05$):

$$\alpha = \sqrt{\frac{K_a}{C}} \quad \text{and} \quad [\text{H}^ = C\alpha = \sqrt{K_a \cdot C} \quad \implies \quad \text{pH} = \frac{1}{2}(\text{p}K_a - \log C)$$

• Common Ion Effect: The suppression of the degree of dissociation of a weak electrolyte by the addition of a strong electrolyte providing a common ion (e.g., adding $\text{CH}_3\text{COONa}$ to $\text{CH}_3\text{COOH}$ lowers $[\text{H}^$ and raises pH).

6. Salt Hydrolysis & Buffer Systems

Salt Hydrolysis Formulations at 298 K

1. Salt of Weak Acid + Strong Base ($\text{CH}_3\text{COONa}$): Anionic hydrolysis produces alkaline solutions:

   $$\text{pH} = 7 + \frac{1}{2}(\text{p}K_a + \log C)$$

2. Salt of Strong Acid + Weak Base ($\text{NH}_4\text{Cl}$): Cationic hydrolysis produces acidic solutions:

   $$\text{pH} = 7 - \frac{1}{2}(\text{p}K_b + \log C)$$

3. Salt of Weak Acid + Weak Base ($\text{CH}_3\text{COONH}_4$): Both ions hydrolyse; pH is concentration-independent:

   $$\text{pH} = 7 + \frac{1}{2}(\text{p}K_a - \text{p}K_b)$$

Buffer Solutions & Henderson-Hasselbalch Equations

Solutions that resist changes in pH upon addition of small amounts of strong acid or alkali:

• Acidic Buffer (Weak Acid + Conjugate Salt, e.g., $\text{CH}_3\text{COOH} + \text{CH}_3\text{COONa}$):

  $$\text{pH} = \text{p}K_a + \log\left(\frac{[\text{Salt}]}{[\text{Acid}]}\right)$$

• Basic Buffer (Weak Base + Conjugate Salt, e.g., $\text{NH}_4\text{OH} + \text{NH}_4\text{Cl}$):

  $$\text{pOH} = \text{p}K_b + \log\left(\frac{[\text{Salt}]}{[\text{Base}]}\right) \quad \implies \quad \text{pH} = 14 - \text{pOH}$$

7. Solubility Product ($K_{sp}$) & Precipitation Criteria

For a general sparingly soluble salt $\text{M}_x\text{X}_y(s) \rightleftharpoons x\text{M}^{p+}(aq) + y\text{X}^{q-}(aq)$ with molar solubility $s$:

$$K_{sp} = [\text{M}^{p+}]^x [\text{X}^{q-}]^y = (xs)^x (ys)^y = x^x y^y s^{x+y}$$

High-Yield Stoichiometric Mappings

• Binary Salt ($1:1$, $\text{AgCl}, \text{BaSO}_4$): $K_{sp} = s^2 \implies s = \sqrt{K_{sp}}$
• Ternary Salt ($1:2$ or $2:1$, $\text{Mg(OH)}_2, \text{Ag}_2\text{CrO}_4, \text{CaF}_2$): $K_{sp} = 4s^3 \implies s = \left(\frac{K_{sp}}{4}\right)^{1/3}$
• Quaternary Salt ($1:3$, $\text{Fe(OH)}_3, \text{Al(OH)}_3$): $K_{sp} = 27s^4 \implies s = \left(\frac{K_{sp}}{27}\right)^{1/4}$
• Pentameric Salt ($2:3$, $\text{As}_2\text{S}_3$): $K_{sp} = 108s^5 \implies s = \left(\frac{K_{sp}}{108}\right)^{1/5}$
• Heptameric Salt ($3:4$, $\text{Zr}_3(\text{PO}_4)_4$): $K_{sp} = 6912s^7 \implies s = \left(\frac{K_{sp}}{6912}\right)^{1/7}$

Applications in Qualitative Cation Analysis

• Group II Cation Separation: $\text{H}_2\text{S}$ is passed in the presence of dilute $\text{HCl}$. The common ion $\text{H}^+$ suppresses the dissociation of $\text{H}_2\text{S}$, keeping $[\text{S}^{2-}]$ low so that only the low-$K_{sp}$ Group II sulphides ($\text{CuS}, \text{PbS}, \text{Bi}_2\text{S}_3$) precipitate, leaving Group IV sulphides ($\text{ZnS}, \text{MnS}$) in solution.
• Group III Cation Separation: $\text{NH}_4\text{OH}$ is added in the presence of $\text{NH}_4\text{Cl}$. Common ion $\text{NH}_4^+$ limits $[\text{OH}^-]$ to selectively precipitate Group III hydroxides ($\text{Fe(OH)}_3, \text{Al(OH)}_3$) without exceeding the $K_{sp}$ of Group IV/V cations.