Unit 3: Equilibria: Chemical and Ionic Principles

Chemical Equilibrium

Free Energy Change in a Chemical Reaction

Chemical equilibrium is a dynamic state where the rates of forward and reverse reactions are equal, and the net change in concentration of reactants and products is zero. The spontaneity and direction of a chemical reaction are governed by the change in Gibbs free energy (ΔG).

ΔG = ΔG⁰ + RTlnQ

• ΔG: Gibbs free energy change under non-standard conditions. It determines the spontaneity of a reaction at any given moment.
• ΔG⁰: Standard Gibbs free energy change, where all reactants and products are in their standard states (1 atm for gases, 1 M for solutions, pure solids/liquids).
• R: Ideal gas constant (8.314 J mol^-1 K^-1).
• T: Absolute temperature in Kelvin.
• Q: Reaction quotient, which has the same form as the equilibrium constant but uses non-equilibrium concentrations or partial pressures.

At equilibrium, ΔG = 0 and Q = K (equilibrium constant). Thus, the equation becomes:

0 = ΔG⁰ + RTlnK
ΔG⁰ = -RTlnK

This equation establishes a direct relationship between the standard free energy change and the equilibrium constant.

Thermodynamic Derivation of the Law of Chemical Equilibrium

The law of chemical equilibrium states that for a reversible reaction at constant temperature, the ratio of the product of molar concentrations (or partial pressures) of products to that of reactants, each raised to the power of their stoichiometric coefficients, is constant. This constant is known as the equilibrium constant (K).

Consider a general reversible reaction:

aA + bB ⇌ cC + dD

The Gibbs free energy change for this reaction under non-standard conditions is given by:

ΔG = ΔG⁰ + RTlnQ

where Q is the reaction quotient, Q = ([C]^c[D]^d) / ([A]^a[B]^b) (for concentrations) or Q = (P_C^c P_D^d) / (P_A^a P_B^b) (for partial pressures).

At equilibrium, the system is at its lowest free energy, meaning ΔG = 0, and the reaction quotient Q becomes the equilibrium constant K.

0 = ΔG⁰ + RTlnK
ΔG⁰ = -RTlnK

This equation confirms that K is a constant for a given reaction at a specific temperature, as ΔG⁰ is constant at a constant temperature. This is the thermodynamic derivation of the law of chemical equilibrium.

Distinction between ΔG and ΔG⁰

Le Chatelier’s Principle

Le Chatelier’s principle states that if a change of condition is applied to a system in equilibrium, the system will shift in a direction that relieves the stress.

Common stresses include changes in concentration, pressure (for gaseous reactions), and temperature.

• Effect of Concentration: If the concentration of a reactant is increased, the equilibrium shifts to the right (towards products) to consume the added reactant. If the concentration of a product is increased, the equilibrium shifts to the left (towards reactants).
• Effect of Pressure/Volume (for gaseous reactions):
  • Increase in pressure (decrease in volume): The equilibrium shifts towards the side with fewer moles of gas to reduce the pressure.
  • Decrease in pressure (increase in volume): The equilibrium shifts towards the side with more moles of gas to increase the pressure.
  • If the number of moles of gas on both sides is equal, pressure change has no effect.
• Effect of Temperature:
  • Increase in temperature: The equilibrium shifts in the endothermic direction (to absorb the added heat).
  • Decrease in temperature: The equilibrium shifts in the exothermic direction (to release heat).
• Effect of Catalyst: A catalyst increases the rates of both forward and reverse reactions equally, leading to faster attainment of equilibrium but does not change the position of equilibrium or the value of K.
• Effect of Inert Gas Addition:
  • Constant Volume: No effect on partial pressures or concentrations, so no shift in equilibrium.
  • Constant Pressure: Volume increases, causing the equilibrium to shift towards the side with more moles of gas (similar to decreasing pressure).

Relationships between K_p, K_c and K_x for Reactions Involving Ideal Gases

For a general gaseous equilibrium: aA_(g) + bB_(g) ⇌ cC_(g) + dD_(g)

1. Relationship between K_p and K_c:

K_c is the equilibrium constant expressed in terms of molar concentrations (mol/L).
K_p is the equilibrium constant expressed in terms of partial pressures (atm or bar).

K_c = ([C]^c[D]^d) / ([A]^a[B]^b)

K_p = (P_C^c P_D^d) / (P_A^a P_B^b)

Using the ideal gas law, PV = nRT, we have P = (n/V)RT = CRT (where C is molar concentration).

• P_A = [A]RT
• P_B = [B]RT
• P_C = [C]RT
• P_D = [D]RT

Substituting these into the K_p expression:

K_p = (([C]RT)^c([D]RT)^d) / (([A]RT)^a([B]RT)^b)
K_p = (([C]^c[D]^d) / ([A]^a[B]^b)) × (RT)^(c+d)-(a+b)
K_p = K_c(RT)^Δn_g

Where Δn_g = (moles of gaseous products) - (moles of gaseous reactants).

2. Relationship between K_p and K_x (K_n):

K_x (sometimes denoted K_n) is the equilibrium constant expressed in terms of mole fractions.

P_i = x_iP_total

Where P_i is the partial pressure of component i, x_i is its mole fraction, and P_total is the total pressure.

K_p = ((x_CP_total)^c(x_DP_total)^d) / ((x_AP_total)^a(x_BP_total)^b)
K_p = ((x_C^cx_D^d) / (x_A^ax_B^b)) × (P_total)^(c+d)-(a+b)
K_p = K_x(P_total)^Δn_g

Thus, K_x = K_p / (P_total)^Δn_g.

Ionic Equilibria

Strong, Moderate and Weak Electrolytes

An electrolyte is a substance that produces ions when dissolved in a solvent, making the solution electrically conductive. Electrolytes are classified based on their degree of ionization in solution.

Degree of Ionization and Factors Affecting It

The degree of ionization (α) is the fraction of the total number of molecules of an electrolyte that ionize into ions in solution. It ranges from 0 to 1 (or 0% to 100%).

For a weak electrolyte AB ⇌ A⁺ + B^-, if C is the initial concentration, then at equilibrium:

[A⁺] = [B^-] = Cα
[AB] = C(1 - α)

Factors affecting the degree of ionization:

• Nature of Electrolyte: Strong electrolytes have α ≈ 1, while weak electrolytes have small α.
• Nature of Solvent: Solvents with high dielectric constants (like water) promote ionization by reducing the electrostatic attraction between ions.
• Temperature: Ionization is often an endothermic process. Increasing temperature generally increases the degree of ionization.
• Concentration: As the concentration of a weak electrolyte decreases (dilution increases), its degree of ionization increases. This is explained by Ostwald's dilution law.
• Common Ion Effect: The presence of a common ion suppresses the ionization of a weak electrolyte, thereby decreasing its degree of ionization.

Ionization Constant and Ionic Product of Water

Ionization Constants (K_a and K_b):

For a weak acid (HA): HA_(aq) + H₂O_(l) ⇌ H₃O⁺_(aq) + A^-_(aq)

K_a = ([H₃O⁺][A^-]) / [HA]

For a weak base (B): B_(aq) + H₂O_(l) ⇌ BH⁺_(aq) + OH^-_(aq)

K_b = ([BH⁺][OH^-]) / [B]

K_a and K_b are measures of the strength of a weak acid and a weak base, respectively. Larger values indicate stronger acids/bases.

Ionic Product of Water (K_w):

Water undergoes autoionization to a small extent:

H₂O_(l) + H₂O_(l) ⇌ H₃O⁺_(aq) + OH^-_(aq)

K_w = [H₃O⁺][OH^-] or [H⁺][OH^-]

At 25 °C, the value of K_w is approximately 1.0 × 10^-14.

Important Relationship: For a conjugate acid-base pair, K_a × K_b = K_w.

Ionization of Weak Acids and Bases

Weak acids and bases ionize only partially in water, establishing an equilibrium. Their ionization can be calculated using their respective ionization constants (K_a or K_b).

For a weak acid HA:
HA ⇌ H⁺ + A^-

Initial: C 0 0
Change: -Cα +Cα +Cα
Equil: C(1-α) Cα Cα

K_a = (Cα × Cα) / C(1-α) = (Cα²) / (1-α)

If α is very small (typically < 5%), then (1-α) ≈ 1. In this case, K_a ≈ Cα², so α = √ (K_a/C).

Once α is known, [H⁺] = Cα, and pH can be calculated.

For a weak base B:
B + H₂O ⇌ BH⁺ + OH^-

Initial: C 0 0
Change: -Cα +Cα +Cα
Equil: C(1-α) Cα Cα

K_b = (Cα × Cα) / C(1-α) = (Cα²) / (1-α)

Similarly, if α is very small, K_b ≈ Cα², so α = √ (K_b/C).

Once α is known, [OH^-] = Cα, and pOH can be calculated, then pH.

pH Scale

The pH scale is a logarithmic scale used to specify the acidity or basicity of an aqueous solution. It is defined as the negative logarithm (base 10) of the molar concentration of hydrogen ions (H⁺).

pH = -log₁₀[H⁺] (or -log₁₀[H₃O⁺])

Similarly, pOH is defined as:

pOH = -log₁₀[OH^-]

The relationship between pH and pOH is derived from the ionic product of water:

[H⁺][OH^-] = K_w = 1.0 × 10^-14 (at 25 °C)

-log([H⁺][OH^-]) = -log(1.0 × 10^-14)
-log[H⁺] - log[OH^-] = 14
pH + pOH = 14

pH Scale Interpretation (at 25 °C):

• pH < 7: Acidic solution ([H⁺] > [OH^-])
• pH = 7: Neutral solution ([H⁺] = [OH^-] = 1.0 × 10^-7 M)
• pH > 7: Basic (Alkaline) solution ([H⁺] < [OH^-])

Common Ion Effect

The common ion effect is the suppression of the ionization of a weak electrolyte caused by the addition of a strong electrolyte containing an ion common to the weak electrolyte.

Example: Acetic acid (CH₃COOH) and Sodium acetate (CH₃COONa)

Weak acid equilibrium: CH₃COOH_(aq) ⇌ H⁺_(aq) + CH₃COO^-_(aq)

Sodium acetate is a strong electrolyte that dissociates completely:

CH₃COONa_(aq) → Na⁺_(aq) + CH₃COO^-_(aq)

When sodium acetate is added to a solution of acetic acid, the concentration of the common ion, acetate (CH₃COO^-), increases. According to Le Chatelier's principle, the equilibrium of the weak acid will shift to the left, reducing the ionization of CH₃COOH and thus decreasing the [H⁺] concentration and increasing the pH.

Salt Hydrolysis

Salt hydrolysis is the reaction of an ion (cation or anion) of a salt with water to produce an acidic or basic solution. This reaction changes the pH of the solution.

The type of hydrolysis depends on the nature of the parent acid and base from which the salt is formed.

1. Salt of Strong Acid and Strong Base (e.g., NaCl, KNO₃)

• Neither cation nor anion hydrolyzes.
• The solution remains neutral (pH ≈ 7).

2. Salt of Strong Acid and Weak Base (e.g., NH₄Cl, CuSO₄)

• The cation (conjugate acid of weak base) hydrolyzes.
• NH₄⁺_(aq) + H₂O_(l) ⇌ NH_3(aq) + H₃O⁺_(aq)
• The solution is acidic (pH < 7).
• Hydrolysis Constant (K_h): K_h = K_w / K_b
• Degree of Hydrolysis (h): h = √ (K_h / C) = √ (K_w / (K_bC))
• [H⁺]: [H⁺] = √ (K_hC) = √ ((K_w/K_b)C)
• pH: pH = (1/2)(pK_w - pK_b - logC)

3. Salt of Weak Acid and Strong Base (e.g., CH₃COONa, KCN)

• The anion (conjugate base of weak acid) hydrolyzes.
• CH₃COO^-_(aq) + H₂O_(l) ⇌ CH₃COOH_(aq) + OH^-_(aq)
• The solution is basic (pH > 7).
• Hydrolysis Constant (K_h): K_h = K_w / K_a
• Degree of Hydrolysis (h): h = √ (K_h / C) = √ (K_w / (K_aC))
• [OH^-]: [OH^-] = √ (K_hC) = √ ((K_w/K_a)C)
• pH: pH = (1/2)(pK_w + pK_a + logC)

4. Salt of Weak Acid and Weak Base (e.g., CH₃COONH₄)

• Both cation and anion hydrolyze. The pH depends on the relative strengths of the acid and base (K_a and K_b).
• Hydrolysis Constant (K_h): K_h = K_w / (K_aK_b)
• Degree of Hydrolysis (h): h = √ (K_h) = √ (K_w / (K_aK_b)) (independent of concentration, C, at low concentrations)
• [H⁺]: [H⁺] = √ (K_w × K_a / K_b)
• pH: pH = (1/2)(pK_w + pK_a - pK_b)

Buffer Solutions

A buffer solution is an aqueous solution consisting of a mixture of a weak acid and its conjugate base, or a weak base and its conjugate acid. It resists changes in pH upon the addition of small amounts of acid or base.

Types of Buffer Solutions:

• Acidic Buffer: Consists of a weak acid and a salt of that weak acid with a strong base (e.g., CH₃COOH + CH₃COONa). It maintains an acidic pH.
• Basic Buffer: Consists of a weak base and a salt of that weak base with a strong acid (e.g., NH₄OH + NH₄Cl). It maintains a basic pH.

Mechanism of Buffer Action (for an acidic buffer CH₃COOH/CH₃COO^-):

• When a small amount of strong acid (H⁺) is added, the H⁺ ions react with the conjugate base: CH₃COO^- + H⁺ → CH₃COOH. The H⁺ ions are consumed, and pH change is minimized.
• When a small amount of strong base (OH^-) is added, the OH^- ions react with the weak acid: CH₃COOH + OH^- → CH₃COO^- + H₂O. The OH^- ions are consumed, and pH change is minimized.

Henderson-Hasselbalch Equation:

For an acidic buffer (Weak Acid / Salt of Weak Acid):

pH = pK_a + log₁₀([Salt] / [Acid])

For a basic buffer (Weak Base / Salt of Weak Base):

pOH = pK_b + log₁₀([Salt] / [Base])
pH = 14 - pOH

Solubility and Solubility Product of Sparingly Soluble Salts – Applications of Solubility Product Principle

Solubility (s):

Solubility (s) refers to the maximum amount of solute that can dissolve in a given amount of solvent at a specific temperature to form a saturated solution. For sparingly soluble salts, it is often expressed in mol/L.

Solubility Product (K_sp):

The solubility product constant (K_sp) is the equilibrium constant for the dissolution of a sparingly soluble ionic compound in water. It represents the product of the molar concentrations of its constituent ions, each raised to the power of their stoichiometric coefficients, in a saturated solution.

For a general sparingly soluble salt, A_xB_y:

A_xB_y(s) ⇌ xA^y+_(aq) + yB^x-_(aq)

K_sp = [A^y+]^x[B^x-]^y

If 's' is the molar solubility of A_xB_y, then [A^y+] = xs and [B^x-] = ys.

K_sp = (xs)^x(ys)^y = x^xy^ys^(x+y)

Examples of K_sp relations:

• Type AB (e.g., AgCl): AgCl_(s) ⇌ Ag⁺_(aq) + Cl^-_(aq)
  K_sp = [Ag⁺][Cl^-] = s × s = s²
• Type AB₂ (e.g., PbI₂): PbI_2(s) ⇌ Pb²⁺_(aq) + 2I^-_(aq)
  K_sp = [Pb²⁺][I^-]² = s × (2s)² = 4s³
• Type A₂B₃ (e.g., Fe₂S₃): Fe₂S_3(s) ⇌ 2Fe³⁺_(aq) + 3S^2-_(aq)
  K_sp = [Fe³⁺]²[S^2-]³ = (2s)² × (3s)³ = 4s² × 27s³ = 108s⁵

Applications of Solubility Product Principle:

• Predicting Precipitation: The ion product (Q_sp) is calculated using initial (non-equilibrium) concentrations.
  • If Q_sp < K_sp: Solution is unsaturated; no precipitation will occur.
  • If Q_sp = K_sp: Solution is saturated; equilibrium exists.
  • If Q_sp > K_sp: Solution is supersaturated; precipitation will occur until Q_sp = K_sp.
• Common Ion Effect on Solubility: The solubility of a sparingly soluble salt decreases significantly in the presence of a common ion from another soluble salt. According to Le Chatelier's principle, adding a product ion shifts the dissolution equilibrium to the left, decreasing the solubility of the sparingly soluble salt.
• Selective Precipitation: This technique is used to separate ions from a solution by adding a reagent that selectively precipitates one ion while others remain in solution. This is achieved by controlling the concentration of the precipitating agent such that the K_sp of one compound is exceeded while others are not.