Chapter 17: Additional Aqueous Equilibria – Study Notes

Additional Aqueous Equilibria

Overview

This chapter explores advanced topics in aqueous equilibria, including the common-ion effect, buffer solutions, acid-base titrations, solubility equilibria, selective precipitation, and qualitative analysis of metal ions. Mastery of these concepts is essential for understanding chemical behavior in solution and for laboratory applications.

The Common-Ion Effect

Definition and Principle

• Common-ion effect: The shift in equilibrium that occurs when a compound containing an ion already present in the solution is added, reducing the solubility or ionization of a solute.

• Generic equilibrium constant: $K_{eq} = \frac{[\text{Products}]}{[\text{Reactants}]}$ (solids are not included in the expression).

• Adding a source of a common ion (e.g., sodium acetate to acetic acid) increases the concentration of that ion, shifting equilibrium and decreasing the concentration of other ions (Le Châtelier's Principle).

Example: Adding sodium acetate to acetic acid:

• $\mathrm{CH_3COOH(aq) \rightleftharpoons H^+(aq) + CH_3COO^-(aq)}$   $K_a = 1.8 \times 10^{-5}$ at 25°C

• $\mathrm{CH_3COONa(aq) \rightarrow Na^+(aq) + CH_3COO^-(aq)}$ (strong electrolyte)

• Increasing $[\mathrm{CH_3COO^-}]$ causes $[\mathrm{H^+}]$ to decrease to maintain $K_a$.

Buffers

Definition and Properties

• Buffer solution: Contains high concentrations of a weak acid and its conjugate base (or weak base and its conjugate acid).

• Resists changes in pH upon addition of small amounts of strong acid or base.

• The pH of a buffer is determined by the ratio and concentrations of the conjugate acid-base pair.

Buffer Synthesis and Action

• Buffers are made by mixing a weak acid (HA) with its conjugate base (A-), or a weak base with its conjugate acid.

• Small additions of H+ or OH- are neutralized by the buffer components, minimally affecting the pH.

Key Equilibrium:

• $\mathrm{HA(aq) \rightleftharpoons H^+(aq) + A^-(aq)}$

• $K_a = \frac{[H^+][A^-]}{[HA]}$

Buffer pH and the Henderson-Hasselbalch Equation

• Rearranging the acid dissociation constant gives:

$K_a = \frac{[H^+][A^-]}{[HA]}$

Taking the negative logarithm:

$-\log[H^+] = -\log K_a - \log \frac{[A^-]}{[HA]}$

$\mathrm{pH} = \mathrm{p}K_a + \log \frac{[A^-]}{[HA]}$

• This is the Henderson-Hasselbalch equation, used to estimate the pH of buffer solutions.

• Valid when $[HA]$ and $[A^-]$ are much greater than $K_a$ (typically $>1000 \times K_a$).

• If not, use an ICE table and the quadratic formula for accuracy.

Buffer Capacity and Range

• Buffer capacity: The amount of acid or base a buffer can neutralize before pH changes significantly.

• Increases with higher concentrations of buffer components.

• Buffer pH range: $\mathrm{p}K_a \pm 1$ (where $[A^-]/[HA]$ is between 0.1 and 10).

Acid-Base Titrations

Principles and Types

• Titration: The process of adding a solution of known concentration (titrant) to a solution of unknown concentration to determine its amount.

• Equivalence point: The point at which moles of acid equal moles of base.

• Types:

  • Acid as titrant (added to base)

  • Base as titrant (added to acid)

Finding the Endpoint

• Endpoint is detected using indicators (color change) or a pH meter.

• Indicators change color over a specific pH range; pH meters provide more precise measurements.

Titration Curves

• Strong acid with strong base: Rapid pH change near equivalence point; initial pH is low.

• Strong base with strong acid: Rapid pH drop near equivalence point; initial pH is high.

• Weak acid with strong base: Higher initial pH, buffer region before equivalence, equivalence point pH > 7.

• Weak acid strength effects: Lower $K_a$ (weaker acid) leads to higher pH at equivalence and smaller pH change near equivalence.

Titrations Using Indicators

• Choose an indicator whose color-change interval overlaps the rapid pH change near the equivalence point.

• Weak acids produce basic solutions at equivalence; weak bases produce acidic solutions at equivalence.

Titrating Polyprotic Acids

• Polyprotic acids lose protons in steps, each with its own equivalence point.

• pH halfway between equivalence points equals $\mathrm{p}K_{a,n}$ for each step.

• At halfway point, $[HA] = [A^-]$, so $\log(1) = 0$.

Solubility-Product Constant ($K_{sp}$)

Solubility Rules and $K_{sp}$

• Solubility rules classify ionic compounds as "soluble" or "insoluble" in water.

• "Soluble": Dissolves in water, no precipitate forms.

• "Insoluble": Does not dissolve, precipitate forms.

• Any ionic compound can form a saturated solution; $K_{sp}$ quantifies the equilibrium between solid and dissolved ions.

Expression and Examples

• For $A_xB_y(s) \rightleftharpoons xA^{y+}(aq) + yB^{x-}(aq)$:

$K_{sp} = [A^{y+}]^x [B^{x-}]^y$

• Examples:

  • $\mathrm{BaSO_4(s) \rightleftharpoons Ba^{2+}(aq) + SO_4^{2-}(aq)}$   $K_{sp} = [Ba^{2+}][SO_4^{2-}]$

  • Low $K_{sp}$ values indicate low solubility (e.g., $K_{sp}$ for $\mathrm{HgS}$ is $2 \times 10^{-53}$).

Solubility Calculations

• Solubility: The concentration of a dissolved solute in a saturated solution (mol/L or g/L).

• Use $K_{sp}$ to calculate molar solubility and vice versa.

Common-Ion Effect on Solubility

• Adding a common ion decreases the solubility of a salt (Le Châtelier's Principle).

• For $\mathrm{CaF_2(s) \rightleftharpoons Ca^{2+}(aq) + 2F^-(aq)}$, adding $Ca^{2+}$ or $F^-$ reduces solubility.

pH Effect on Solubility

• If a salt contains the conjugate of a weak acid or base, changing pH affects solubility.

• Example: $\mathrm{Mg(OH)_2(s) \rightleftharpoons Mg^{2+}(aq) + 2OH^-(aq)}$; lowering pH (adding acid) increases solubility by removing $OH^-$.

Coordination Complexes and Solubility

Coordination Complexes

• Formed when metal ions bond with molecules or ions called ligands.

• Ligands can be monodentate (one donor atom) or multidentate (multiple donor atoms, called chelators).

• Complexes may be neutral or charged; if charged, counterions are present.

Complexation and Solubility

• Formation of complexes can dramatically increase the solubility of otherwise insoluble salts.

• Example: $\mathrm{AgCl}$ is insoluble, but adding $NH_3$ forms $\mathrm{Ag(NH_3)_2^+}$, increasing solubility.

• Formation constant ($K_f$): $K_f = \frac{[\mathrm{Ag(NH_3)_2^+}]}{[\mathrm{Ag^+}][\mathrm{NH_3}]^2}$

Formation Constants Table

The following table summarizes some common formation constants ($K_f$) for metal-ligand complexes:

Complex Ion	$K_f$	Chemical Equation
Ag(NH3)2+	$1.7 \times 10^7$	Ag+ + 2 NH3 → Ag(NH3)2+
Cu(NH3)42+	$5.0 \times 10^{12}$	Cu2+ + 4 NH3 → Cu(NH3)42+
Zn(NH3)42+	$2.8 \times 10^9$	Zn2+ + 4 NH3 → Zn(NH3)42+
Fe(CN)64-	$1.0 \times 10^{35}$	Fe2+ + 6 CN- → Fe(CN)64-
Al(OH)4-	$1.9 \times 10^{15}$	Al3+ + 4 OH- → Al(OH)4-
Ni(NH3)62+	$2.0 \times 10^8$	Ni2+ + 6 NH3 → Ni(NH3)62+
Co(NH3)62+	$5.0 \times 10^4$	Co2+ + 6 NH3 → Co(NH3)62+
Fe(OH)4-	$4.0 \times 10^{14}$	Fe3+ + 4 OH- → Fe(OH)4-
Zn(OH)42-	$3.0 \times 10^{15}$	Zn2+ + 4 OH- → Zn(OH)42-

Additional info: Table values inferred from standard formation constant tables.

Amphoterism

Definition and Examples

• Amphoteric substances: Metal oxides or hydroxides that dissolve in both acidic and basic solutions.

• Examples: $\mathrm{Al(OH)_3}$, $\mathrm{Cr(OH)_3}$, $\mathrm{Zn(OH)_2}$, $\mathrm{Sn(OH)_2}$

• Non-amphoteric: $\mathrm{Fe(OH)_2}$, $\mathrm{Fe(OH)_3}$ (rust) – dissolve only in acid, not base.

Precipitation and Selective Precipitation

Precipitation by Solution Mixing

• Mixing solutions may produce a precipitate if the product of ion concentrations ($Q$) exceeds $K_{sp}$.

• Common-ion effect can influence which compounds precipitate.

Predicting Precipitation

• Calculate $Q$ for possible precipitates and compare to $K_{sp}$:

  • If $Q > K_{sp}$: Precipitate forms.

  • If $Q < K_{sp}$: No precipitate.

  • If $Q = K_{sp}$: Solution is saturated (precipitation unlikely).

Selective Precipitation

• Used to separate ions in solution by adding a counterion that forms a precipitate with only one ion (based on $K_{sp}$ values).

• Example: Adding $Cl^-$ to a solution containing $Ag^+$ and $Pb^{2+}$; $AgCl$ ($K_{sp} = 1.8 \times 10^{-10}$) precipitates before $PbCl_2$ ($K_{sp} = 1.7 \times 10^{-5}$).

Qualitative Analysis of Metal Ions

Group Separation by Precipitation

• Metal ions are separated in groups by selective precipitation:

Step	Group	Examples
1	Ag+, Pb2+, Hg22+	Precipitate as chlorides
2	Cu2+, Cd2+, Bi3+, Pb2+, Hg2+, As3+, Sb3+, Sn2+	Precipitate as sulfides
3	Al3+, Fe3+, Cr3+, Zn2+, Ni2+, Mn2+, Co2+	Precipitate as hydroxides
4	Mg2+, Ca2+, Sr2+, Ba2+	Precipitate as carbonates or sulfates

Each step involves adding a reagent to selectively precipitate a group of ions, followed by filtration and further analysis.

Summary Table: Key Equations

Concept	Equation
Acid dissociation constant	$K_a = \frac{[H^+][A^-]}{[HA]}$
Henderson-Hasselbalch	$\mathrm{pH} = \mathrm{p}K_a + \log \frac{[A^-]}{[HA]}$
Solubility product	$K_{sp} = [A^{y+}]^x [B^{x-}]^y$
Formation constant	$K_f = \frac{[\text{complex ion}]}{[\text{metal ion}][\text{ligand}]^n}$
Quadratic formula (for ICE tables)	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Additional info: These notes synthesize and expand upon the provided slides and text, adding definitions, context, and examples for clarity and completeness.