Equilibrium

In this chapter: equilibrium in physical and chemical processes, the law of chemical equilibrium, K_c and K_p, homogeneous and heterogeneous equilibria, reaction quotient, K and ΔG, Le Chatelier's principle, acids and bases (Arrhenius, Brönsted-Lowry, Lewis), ionic product of water and pH, K_a and K_b, common ion effect, hydrolysis of salts, buffer solutions and solubility product.

Equilibrium is dynamic

When a reaction is carried out in a closed vessel, the forward rate falls as reactants are used up and the backward rate rises as products build up. When the two rates become equal, concentrations stop changing and the system is at equilibrium. Both reactions are still going on; this is why it is called dynamic equilibrium. The same equilibrium is reached from either the reactant side or the product side.

Physical processes show equilibrium too: ice and water at 273 K and 1 atm (rate of melting = rate of freezing), water and its vapour in a closed vessel (constant vapour pressure), and a saturated solution with undissolved solid. For a gas dissolving in a liquid, Henry's law says the mass of gas dissolved in a given mass of solvent is proportional to the pressure of the gas above the solvent; this is why a soda bottle fizzes when it is opened.

Equilibrium constant

For a general reaction aA + bB ⇌ cC + dD, the law of chemical equilibrium gives

• K has a fixed value for a reaction at a given temperature; it changes only when temperature changes.
• For the reverse reaction, K′ = 1/K. If the equation is multiplied by n, the new constant is Kⁿ. If equations are added, their constants are multiplied.
• Heterogeneous equilibria: the concentrations of pure solids and pure liquids are constant and are left out. For CaCO₃(s) ⇌ CaO(s) + CO₂(g), K_p = p_CO2.
• Extent of reaction: K_c > 10³, products predominate; K_c < 10⁻³, reactants predominate; in between, both are present in appreciable amounts.

Worked example: 0.50 mol each of H₂ and I₂ are heated in a 1.0 L vessel. At equilibrium, [HI] = 0.78 mol L⁻¹. Find K_c for H₂(g) + I₂(g) ⇌ 2HI(g).
Solution: 0.78 mol of HI needs 0.39 mol each of H₂ and I₂, so [H₂] = [I₂] = 0.50 − 0.39 = 0.11 mol L⁻¹. K_c = (0.78)² / (0.11 × 0.11) = 0.608/0.0121 ≈ 50. Here Δn = 0, so K_p = K_c.

Reaction quotient, Q

Q has the same expression as K but uses concentrations at any moment, not only at equilibrium. If Q < K, the reaction goes forward; if Q > K, it goes backward; if Q = K, the system is at equilibrium. K is linked to Gibbs energy by

Le Chatelier's principle

If a system at equilibrium is disturbed by a change in concentration, pressure or temperature, the equilibrium shifts in the direction that tends to undo the change.

In the Haber process, N₂(g) + 3H₂(g) ⇌ 2NH₃(g), ΔH = −92.38 kJ mol⁻¹. High pressure favours ammonia (4 moles of gas give 2), and low temperature favours it too, but the reaction is then too slow. In practice about 500 °C and 200 atm are used with an iron catalyst.

Acids, bases and salts

• Arrhenius: an acid gives H⁺ (as H₃O⁺) in water; a base gives OH⁻.
• Brönsted-Lowry: an acid is a proton donor and a base is a proton acceptor. An acid and the species formed after it loses a proton make a conjugate acid-base pair (HCl/Cl⁻, NH₄⁺/NH₃). A strong acid has a weak conjugate base. Water can act as either, so it is amphoteric.
• Lewis: an acid accepts an electron pair and a base donates one. BF₃, AlCl₃, H⁺, Co³⁺ and Mg²⁺ are Lewis acids; NH₃, H₂O and OH⁻ are Lewis bases.

Ionic product of water and pH

Pure water at 298 K has [H₃O⁺] = 1.0 × 10⁻⁷ M and pH 7. K_w increases with temperature, so neutral water is below pH 7 when hot. A change of one pH unit means a tenfold change in [H⁺].

Weak acids and bases

For a weak acid HX of concentration c and degree of ionisation α, K_a = cα²/(1 − α) ≈ cα² when α is small, so [H⁺] = cα = √K_ac. The same form holds for a weak base with K_b and [OH⁻]. For a conjugate pair, K_a × K_b = K_w and pK_a + pK_b = 14. The smaller the pK_a, the stronger the acid. Polybasic acids ionise in steps, and K_a1 > K_a2 > K_a3, because it is harder to remove a proton from a negative ion.

Worked example: Find the pH of 0.10 M acetic acid (K_a = 1.8 × 10⁻⁵).
Solution: [H⁺] = √(1.8 × 10⁻⁵ × 0.10) = √(1.8 × 10⁻⁶) = 1.34 × 10⁻³ M. pH = −log(1.34 × 10⁻³) = 3 − 0.127 = 2.87. Check: α = 1.34 × 10⁻³/0.10 = 0.013, small enough for the approximation.

Acid strength: down a group, the H−A bond gets weaker, so acid strength rises: HF < HCl < HBr < HI. Across a period, it rises with the electronegativity of A: CH₄ < NH₃ < H₂O < HF.

Common ion effect and hydrolysis of salts

Adding an ion that is already present in the equilibrium shifts it back. Sodium acetate added to acetic acid supplies acetate ions and suppresses the ionisation of the acid, so [H⁺] falls.

Buffer solutions

A buffer resists change in pH on dilution or on adding small amounts of acid or base. A mixture of acetic acid and sodium acetate buffers around pH 4.75; ammonium hydroxide and ammonium chloride buffer around pH 9.25. For an acidic buffer,

Solubility product

For a sparingly soluble salt in contact with its saturated solution, such as BaSO₄(s) ⇌ Ba²⁺(aq) + SO₄²⁻(aq), K_sp = [Ba²⁺][SO₄²⁻]. If the molar solubility is S, then K_sp = S² for a 1:1 salt, and for a salt A_xB_y, K_sp = x^xy^yS^x+y (for example, 4S³ for CaF₂). If the ionic product exceeds K_sp, the salt precipitates. A common ion lowers the solubility: AgCl is less soluble in NaCl solution than in pure water.

Common mistakes: (1) Including solids and pure liquids in the K expression. (2) Forgetting that only temperature changes the value of K; concentration and pressure changes only shift the position. (3) Using pH = −log c for a weak acid; that works only for strong acids. (4) Taking Δn in K_p = K_c(RT)^Δn over all species instead of gases only. (5) Writing K_sp = S² for salts like CaF₂ or Ag₂CrO₄, which need 4S³.

JEE and NEET focus

• Writing K_c and K_p, converting between them, and K for reversed, multiplied or added equations.
• Equilibrium concentration problems from initial amounts and degree of dissociation.
• Le Chatelier's principle for concentration, pressure, inert gas, temperature and catalyst, applied to the Haber process.
• pH of strong acids and bases, weak acids and bases, and mixtures; K_a, K_b and pK values.
• Nature of salt solutions, buffer pH by the Henderson equation, and conjugate acid-base pairs.
• K_sp and solubility for different salt types, precipitation and the common ion effect.

Practice questions