Equilibrium

1. Equilibrium in Physical Processes

When a liquid evaporates in a closed container, molecules with relatively higher kinetic energy escape the liquid surface into the vapour phase, while some molecules from the vapour strike the liquid surface and are retained. This gives rise to a constant vapour pressure — an equilibrium in which the number of molecules leaving the liquid equals the number returning from the vapour:

$$\text{H}_2\text{O(l)} \rightleftharpoons \text{H}_2\text{O(vap)}$$

The double half-arrows indicate that both processes occur simultaneously; this is not static equilibrium, but a dynamic one, and the mixture of reactants and products in the equilibrium state is called an equilibrium mixture.

Similar equilibria exist for other physical processes:

• Solid–liquid: ice and water at 273 K and 1 atm remain in dynamic equilibrium; the melting point of any pure substance is the temperature at which its solid and liquid phases coexist at equilibrium.
• Liquid–vapour: the equilibrium vapour pressure of a liquid is constant at a given temperature; a more volatile liquid has a higher vapour pressure and lower boiling point.
• Solid–vapour (sublimation): solid iodine sublimes to give iodine vapour, which condenses back to solid iodine at equilibrium.
• Dissolution of a solid or gas in a liquid: a saturated solution is one in which no more solute can dissolve at a given temperature; dissolved gas in a liquid is governed by Henry's law: the mass of a gas dissolved in a given mass of solvent at any temperature is proportional to the pressure of the gas above the solvent.

General characteristics of physical equilibria: (i) possible only in a closed system at a given temperature; (ii) both opposing processes occur at the same rate (dynamic, but stable); (iii) all measurable properties of the system remain constant; (iv) the magnitude of such quantities at any stage indicates the extent to which the process has proceeded before reaching equilibrium.

2. Dynamic Chemical Equilibrium & the Law of Mass Action

Like physical processes, chemical reactions can also attain a state of equilibrium. Consider a general reversible reaction $A + B \rightleftharpoons C + D$. With passage of time, products $C$ and $D$ accumulate while reactants $A$ and $B$ deplete; the rate of the forward reaction decreases while the rate of the reverse reaction increases, until eventually the two rates become equal and the system reaches a state of chemical equilibrium. Equilibrium can be attained from either direction — starting from pure reactants or from pure products.

The dynamic nature of chemical equilibrium was demonstrated by Haber using isotopic labelling (deuterium): even after the ammonia synthesis reaction appears to have stopped changing in composition, isotope scrambling between H and D atoms continues, proving that the forward and reverse reactions are still occurring, just at equal rates.

The equilibrium law (law of mass action): for a general reversible reaction $aA + bB \rightleftharpoons cC + dD$, at a given temperature,

$$K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b} \tag{7.4}$$

This is called the equilibrium constant expression, and $K_c$ the equilibrium constant (concentrations expressed in mol L⁻¹). The exponents are the stoichiometric coefficients from the balanced equation. The equilibrium constant for the reverse reaction is the reciprocal of that for the forward reaction: $K_c' = 1/K_c$. If a reaction's coefficients are all multiplied by a factor $n$, its equilibrium constant is raised to the power $n$.

3. K_p, K_c and the Relationship Between Them

For reactions involving gases, it is often more convenient to express the equilibrium constant in terms of partial pressures, $K_p$. Using the ideal gas equation ($p = [\text{gas}]RT$), the relationship between $K_p$ and $K_c$ for a general reaction $aA + bB \rightleftharpoons cC + dD$ is:

$$K_p = K_c(RT)^{\Delta n_g} \tag{7.15}$$

where $\Delta n_g$ = (moles of gaseous products) − (moles of gaseous reactants). $K_p = K_c$ only when $\Delta n_g = 0$.

Units of the equilibrium constant: $K_c$ and $K_p$ can have units depending on $\Delta n_g$ (unless the exponents of numerator and denominator are equal, giving no net units), or can be treated as dimensionless quantities if concentrations/pressures are expressed relative to their standard states (1M for solutes, 1 bar for gases).

4. Homogeneous & Heterogeneous Equilibria

In a homogeneous equilibrium, all reactants and products are in the same phase, e.g. $\text{N}_2(g) + 3\text{H}_2(g) \rightleftharpoons 2\text{NH}_3(g)$.

In a heterogeneous equilibrium, reactants and products are in more than one phase, e.g. the thermal dissociation of calcium carbonate:

$$\text{CaCO}_3(s) \rightleftharpoons \text{CaO}(s) + \text{CO}_2(g)$$

Since the molar concentration of a pure solid or pure liquid is constant (independent of the amount present), pure solids and pure liquids are omitted entirely from the equilibrium constant expression — only gases and species in solution appear. For calcium carbonate's decomposition, $K_c' = [\text{CO}_2(g)]$ or equivalently $K_p = p_{\text{CO}_2}$. It must be remembered that for a heterogeneous equilibrium to exist, the pure solids/liquids must still be physically present (however small the amount), even though they don't appear in the K expression.

5. Reaction Quotient Q & Applications of K

The reaction quotient, $Q_c$, has the same mathematical form as $K_c$, but uses concentrations at any arbitrary point in time, not necessarily at equilibrium:

$$Q_c = \frac{[C]^c[D]^d}{[A]^a[B]^b} \tag{7.20 (general form)}$$

Comparing $Q_c$ to $K_c$ predicts the direction in which a reaction will proceed to reach equilibrium:

• If $Q_c > K_c$, the reaction proceeds in the direction of reactants (reverse reaction).
• If $Q_c < K_c$, the reaction proceeds in the direction of products (forward reaction).
• If $Q_c = K_c$, the reaction mixture is already at equilibrium.

Predicting the extent of a reaction from K: if $K_c > 10^3$, products predominate (reaction proceeds nearly to completion); if $K_c < 10^{-3}$, reactants predominate (reaction proceeds only slightly); if $K_c$ is between $10^{-3}$ and $10^3$, appreciable concentrations of both reactants and products are present at equilibrium.

6. Le Chatelier's Principle

Le Chatelier's principle: a change in any of the factors that determine the equilibrium conditions of a system will cause the system to change in such a manner as to reduce or counteract the effect of the change. This applies to all physical and chemical equilibria.

• Effect of concentration: adding a reactant/product shifts equilibrium in the direction that consumes it; removing a reactant/product shifts equilibrium in the direction that replenishes it.
• Effect of pressure (volume) change: relevant only when $\Delta n_g \neq 0$. Increasing pressure (decreasing volume) shifts equilibrium toward the side with fewer moles of gas; decreasing pressure shifts it toward the side with more moles of gas.
• Effect of an inert gas: at constant volume, adding an inert gas does not change any reacting species' partial pressure or concentration, so the equilibrium is undisturbed. (At constant pressure, however, the volume must increase to accommodate the inert gas, which does dilute the reacting species and shift the equilibrium toward the side with more moles of gas.)
• Effect of a catalyst: a catalyst provides a new lower-energy pathway, but speeds up the forward and reverse reactions by exactly the same factor. It reduces the time taken to reach equilibrium, but does not affect the equilibrium composition or the value of K at all.
• Effect of temperature: unlike the other factors, changing temperature does change the value of K itself. For an exothermic reaction, K decreases as temperature increases; for an endothermic reaction, K increases as temperature increases (Le Chatelier's principle treats heat as a "reactant" for endothermic, a "product" for exothermic reactions).

7. Acids, Bases & Ionization Constants

Ionic equilibrium is the equilibrium established between ions and unionized molecules in aqueous solution. Acids, bases and salts are electrolytes; strong electrolytes ionize almost completely, weak electrolytes only partially.

Three definitions of acids and bases, in increasing order of generality:

• Arrhenius: an acid dissociates in water to give H⁺(aq); a base gives OH⁻(aq).
• Brønsted–Lowry: an acid is a proton donor; a base is a proton acceptor. When an acid donates a proton, what remains is its conjugate base; when a base accepts a proton, what forms is its conjugate acid. A conjugate acid-base pair differs by exactly one proton, and a strong acid has a weak conjugate base (and vice versa).
• Lewis: an acid is an electron-pair acceptor; a base is an electron-pair donor. This is the broadest definition — it explains species like BF₃ (no proton at all) acting as an acid by accepting an electron pair from NH₃.

Ionization constant of water: water can act as both acid and base (it is amphoteric). Its self-ionization, $2\text{H}_2\text{O(l)} \rightleftharpoons \text{H}_3\text{O}^+(aq) + \text{OH}^-(aq)$, gives the ionic product of water:

$$K_w = [\text{H}^[\text{OH}^-] = 1.0 \times 10^{-14} \text{ M}^2 \text{ (at 298 K)}$$

Taking negative logarithms: $\text{p}K_w = \text{pH} + \text{pOH} = 14$ (at 298 K). $K_w$ is temperature-dependent, so this value of 14 (and the "neutral pH = 7" rule) strictly holds only at 298 K.

pH scale: $\text{pH} = -\log[\text{H}^$. Acidic solutions have pH < 7, basic solutions pH > 7, neutral solutions pH = 7 (at 298 K).

Ionization constant of a weak acid HX, $K_a = \dfrac{[\text{H}^[\text{X}^-]}{[\text{HX}]}$, and of a weak base MOH, $K_b = \dfrac{[\text{M}^[\text{OH}^-]}{[\text{MOH}]}$. Larger $K_a$ (or $K_b$) means a stronger acid (or base). For a conjugate acid-base pair: $K_a \times K_b = K_w$, or equivalently $\text{p}K_a + \text{p}K_b = \text{p}K_w = 14$.

8. Degree of Ionization, Polybasic Acids & Common Ion Effect

Let $\alpha$ be the degree of ionization (the fraction of the initial concentration $c$ that ionizes) of a weak acid HX. Then:

$$K_a = \frac{c\alpha^2}{1-\alpha}$$

When $\alpha \ll 1$, this simplifies to $K_a \approx c\alpha^2$, giving Ostwald's dilution law: $\alpha \approx \sqrt{K_a/c}$ — the degree of ionization of a weak electrolyte increases on dilution.

Di- and polybasic acids (e.g. oxalic acid, sulphuric acid, phosphoric acid) ionize in successive steps, each with its own ionization constant $K_{a_1}, K_{a_2}, K_{a_3}, \ldots$. Successive ionization constants are always much smaller than the preceding one ($K_{a_1} \gg K_{a_2} \gg K_{a_3}$), because it is progressively harder to remove a positively charged proton from an increasingly negatively charged species.

Factors affecting acid strength: within a group, H–A bond strength decreases down the group as atomic size of A increases, so acid strength increases (e.g. HF << HCl < HBr < HI). Within a period, electronegativity of A increases, so acid strength increases (e.g. CH₄ < NH₃ < H₂O < HF).

Common ion effect: a shift in an ionic equilibrium caused by adding a substance that provides an ion already present in that equilibrium — a direct application of Le Chatelier's principle. Adding sodium acetate to acetic acid solution suppresses acetic acid's own ionization (via the common acetate ion), decreasing $[\text{H}^$ and raising the pH.

9. Hydrolysis of Salts & Buffer Solutions

Hydrolysis of salts: the pH of a salt solution depends on the strength of the acid and base it was formed from:

• Strong acid + strong base (e.g. NaCl): no hydrolysis, solution is neutral, pH = 7.
• Weak acid + strong base (e.g. CH₃COONa): the anion hydrolyses, solution is basic, pH > 7; $\text{pH} = 7 + \tfrac12 \text{p}K_a + \tfrac12 \log C$.
• Strong acid + weak base (e.g. NH₄Cl): the cation hydrolyses, solution is acidic, pH < 7.
• Weak acid + weak base (e.g. CH₃COONH₄): both ions hydrolyse; $\text{pH} = 7 + \tfrac12(\text{p}K_a - \text{p}K_b)$, which can be above or below 7 depending on which is larger.

Buffer solutions resist change in pH on dilution, or on addition of small amounts of acid or alkali. An acidic buffer is made from a weak acid and its salt (e.g. acetic acid + sodium acetate); a basic buffer from a weak base and its salt (e.g. ammonium hydroxide + ammonium chloride). The Henderson–Hasselbalch equation for an acidic buffer:

$$\text{pH} = \text{p}K_a + \log\frac{[\text{Salt}]}{[\text{Acid}]} \tag{7.40}$$

When $[\text{Salt}] = [\text{Acid}]$, $\text{pH} = \text{p}K_a$ exactly — this is also the pH at the half-neutralisation point of a weak acid titrated with a strong base, giving a practical way to measure $\text{p}K_a$ experimentally.

10. Solubility Equilibria & K_sp

For a sparingly soluble salt $\text{M}_x\text{X}_y$, the equilibrium between the undissolved solid and its ions in a saturated solution gives the solubility product constant:

$$K_{sp} = [\text{M}^{p+}]^x[\text{X}^{q-}]^y \tag{7.44}$$

If the molar solubility is $S$, then $[\text{M}^{p+}] = xS$ and $[\text{X}^{q-}] = yS$, so $K_{sp} = x^x y^y S^{(x+y)}$.

Ionic product ($Q_{sp}$) vs. solubility product ($K_{sp}$): $Q_{sp}$ is the same expression evaluated for any solution, while $K_{sp}$ is specifically its value at equilibrium (saturation).

• If $Q_{sp} > K_{sp}$: solution is supersaturated → precipitation occurs.
• If $Q_{sp} < K_{sp}$: solution is unsaturated → more solid can dissolve.
• If $Q_{sp} = K_{sp}$: solution is exactly saturated (at equilibrium).

Common ion effect on solubility: adding a common ion (e.g. NaCl to a saturated AgCl solution, providing extra Cl⁻) decreases the solubility of the sparingly soluble salt, since $K_{sp}$ must remain constant — used industrially, e.g. to purify sodium chloride, or to precipitate soap from solution by adding common salt.