Chapter 18: Acid-Base Equilibria Study Guide: Comprehensive Equilibria, pH, and Definitions

Chapter Overview and Reference Materials

• Core Topics Covered:     * Acids and Bases in Water.     * Auto-ionization of Water and the pH Scale.     * Proton Transfer and the Brønsted-Lowry Acid-Base Definition.     * Solving Problems Involving Weak-Acid Equilibria.     * Weak Bases and Their Relation to Weak Acids.     * Acid-Base Properties of Salt Solutions.     * Electron-Pair Donation and the Lewis Acid-Base Definition.
• Recommended Readings:     * Silberberg and Amateis: Sections 18.1 \text{ – } 18.5, 18.7 \text{ – } 18.8, $18.10$.     * Silberberg: Sections 18.1 \text{ – } 18.5, 18.7 \text{ – } 18.8.

Fundamental Definitions: Acids and Bases in Water

• General Reactivity:     * Most acid and base applications involve water solutions.     * Molecular Equation: $HX(aq) + MOH(aq) \rightarrow MX(aq) + H_2O(l)$.     * Net Ionic Equation: $H^+(aq) + OH^-(aq) \rightarrow H_2O(l)$.
• The Arrhenius Acid-Base Definition:     * Definition is based on behavior specifically in water.     * Arrhenius Acid: A substance with an $H$ in its formula that reacts in water to form $H_3O^+$.     * Arrhenius Base: A substance with $OH$ in its formula that dissociates in water to yield $OH^-$.     * Neutralization: When they react, they undergo a neutralization reaction: $H^+(aq) + OH^-(aq) \rightarrow H_2O(l)$.     * Standard Enthalpy of Reaction: $\Delta H^\circ_{rxn} = -55.9\,kJ/mol$.     * Limitations: The definition is not always true; for example, $NH_3$ and $K_2CO_3$ behave as bases but do not contain $OH$ in their formulas.

Variation in Acid Strength and the Dissociation Constant ($K_a$)

• Acid Classification: Acids and bases are classified by strength based on the amount of $H_3O^+$ produced, which depends on the dissociation of $H^+$.
• Strong Acids:     * Dissociate completely into ions: $HA(g \text{ or } l) + H_2O(l) \rightarrow H_3O^+(aq) + A^-(aq)$.     * For strong acids, at equilibrium, $[HA]_{eq} \approx 0$ and the equilibrium constant $K_c \gg 1$.
• Weak Acids:     * Only partially dissociate: $HA(aq) + H_2O(l) \rightleftharpoons H_3O^+(aq) + A^-(aq)$.     * For weak acids, the equilibrium constant $K_c$ is small.
• The Acid-Dissociation Constant ($K_a$):     * Standard Equilibrium Constant: $K_c = \frac{[H_3O^+][A^-]}{[HA][H_2O]}$.     * Because the concentration of water is large ($1000\,g / 18.02\,g/mol = 55.5\,M$) compared to other species, it is considered constant.     * $K_a = K_c[H_2O] = \frac{[H_3O^+][A^-]}{[HA]}$.
• Guidelines for Relative Strengths:     * Strong Acids:         1. Halide acids: $HCl$, $HBr$, and $HI$ (Note: $HF$ is weak).         2. Oxoacids: Number of oxygen atoms is 2 or more than the number of ionizable hydrogen atoms (e.g., $HNO_3$, $H_2SO_4$).     * Weak Acids:         1. Halide acid: $HF$.         2. Acids where $H$ is not bonded to halides or oxoanions.         3. Oxoacids: Number of oxygen atoms is equal to or 1 more than the number of ionizable hydrogen atoms (e.g., $HClO$, $H_3PO_4$).         4. Carboxylic acids: General formula $R\text{-}COOH$ (e.g., $CH_3COOH$).     * Strong Bases: Water-soluble compounds containing $O^{2-}$ or $OH^-$. Cations are usually Group 1A metals ($Li$, $Na$, $K$, $Rb$, $Cs$) or Group 2A metals ($Ca$, $Sr$, $Ba$).     * Weak Bases: Compounds with electron-rich nitrogen atoms (lone pairs), such as Ammonia ($NH_3$) and Amines ($R\text{-}NH_2$, $R_2\text{-}NH$, or $R_3N$, e.g., ethylamine, dimethylamine, tripropylamine).

Auto-ionization of Water and the pH Scale

• The Ion-Product Constant for Water ($K_w$):     * Water undergoes self-ionization: $2H_2O(l) \rightleftharpoons H_3O^+(aq) + OH^-(aq)$.     * $K_w = [H_3O^+][OH^-] = 1.0 \times 10^{-14}$ at $25^\circ C$.
• Relationship between Species:     * $[H_3O^+] = \frac{K_w}{[OH^-]}$.     * $[OH^-] = \frac{K_w}{[H_3O^+]}$.
• The pH Scale:     * $[H_3O^+]$ in aqueous solutions varies from $\sim 10\,M$ to $\sim 10^{-15}\,M$.     * $pH = -\log_{10}[H_3O^+]$.     * $pOH = -\log_{10}[OH^-]$.     * $pK_w = pH + pOH = 14.00$ (at $25^\circ C$.
• Sample Problem 18.2:     * Measured concentration: $[H_3O^+] = 3.0 \times 10^{-4}\,M$.     * Calculate $[OH^-]$ at $25^\circ C$.     * Determine if the solution is neutral, basic, or acidic.

Brønsted-Lowry Acid-Base Definition

• Arrhenius Limitations:     * Does not account for substances yielding $OH^-$ without containing $OH$.     * Does not account for reactions in the gas phase, non-aqueous solutions, or heterogeneous mixtures.
• Brønsted-Lowry Definition:     * Acid: A proton donor (any species that donates an $H^+$). It must contain an $H$ in its formula. (All Arrhenius acids are BL acids).     * Base: A proton acceptor. It must contain a lone pair of electrons to bind $H^+$. (All Arrhenius bases are BL bases, but not all BL bases are Arrhenius).
• Amphiprotic Water: Water can behave as both an acid and a base.
• Conjugate Acid-Base Pairs:     * The definition focuses on the transfer between reactants and products.     * Example: $H_2S$ and $HS^-$ are a pair; $NH_3$ and $NH_4^+$ are a pair.     * Reaction: $Acid_1 + Base_2 \rightleftharpoons Base_1 + Acid_2$.
• Relative Strength and Reaction Direction:     * A reaction proceeds spontaneously towards the formation of a weaker acid and base.     * If the forward reaction forms weaker species, $K_c > 1$.     * Plasma (weak acids and relatively strong bases) vs. Cola ($pH \approx 2.5$, high $K_a$, low $pK_a$).

Solving Problems Involving Weak-Acid Equilibria

• Problem Types:     * Type 1: Given equilibrium concentrations, find $K_a$.     * Type 2: Given $K_a$ and initial concentrations, find equilibrium concentrations.
• Approach (from Chapter 17):     1. Balance the chemical equation.     2. Denote $x$ as the change in concentration.     3. Construct an ICE (Initial, Change, Equilibrium) table.     4. Note: $[H_3O^+]$ from auto-ionization of water is approximately $0$.     5. Note: $[HA]$ is relatively constant if $K_a$ is small.     6. Substitute values into the $K_a$ expression to solve for $x$.
• Polyprotic Acids:     * Acids with more than one ionizing proton (e.g., $H_3PO_4$).     * Successive dissociation steps:         * 1: $H_3PO_4(aq) + H_2O(l) \rightleftharpoons H_2PO_4^-(aq) + H_3O^+(aq)$ ($K_{a1} = 7.2 \times 10^{-3}$).         * 2: $H_2PO_4^-(aq) + H_2O(l) \rightleftharpoons HPO_4^{2-}(aq) + H_3O^+(aq)$ ($K_{a2} = 6.3 \times 10^{-8}$).         * 3: $HPO_4^{2-}(aq) + H_2O(l) \rightleftharpoons PO_4^{3-}(aq) + H_3O^+(aq)$ ($K_{a3} = 4.2 \times 10^{-13}$).     * Key Rule: $K_{a1} > K_{a2} > K_{a3}$.

Dissociation Constants ($K_a$) for Selected Acids (at 298 K)

Weak Bases and Relative Strengths

• Base Ionization Constant ($K_b$):     * $K_b = \frac{[BH^+][OH^-]}{[B]}$.     * Example: $NH_3(aq) + H_2O(l) \rightleftharpoons NH_4^+(aq) + OH^-(aq)$.     * $K_b = 1.76 \times 10^{-5}$ at $25^\circ C$.
• Relation between Conjugate Pairs:     * The sum of the acid and base dissociation reactions results in the auto-ionization of water.     * $K_a \times K_b = K_w$.     * Proof: $\frac{[H_3O^+][A^-]}{[HA]} \times \frac{[HA][OH^-]}{[A^-]} = [H_3O^+][OH^-] = K_w$.
• Anions of Weak Acids: These act as weak bases. For example, in a solution of $NaF$, $Na^+$ is a spectator ion, and the reaction is $F^-(aq) + H_2O(l) \rightleftharpoons HF(aq) + OH^-(aq)$.

Acid-Base Properties of Salt Solutions

• pH Effects of Dissolved Salts:     * Neutral Solution: Salt from a strong acid anion and strong base cation (e.g., $NaCl$).     * Acidic Solution:         1. Salt from a strong acid anion and weak base cation (e.g., $NH_4Cl$).         2. Salt from a strong acid anion and a small, highly charged metal cation (e.g., $AlCl_3$).     * Basic Solution: Salt from a weak acid anion and a strong base cation (e.g., $CH_3COONa$).
• Salts of Weak Acids and Weak Bases: The result depends on the relative values of $K_a$ and $K_b$:     * If $K_a < K_b$, the solution is basic.     * If $K_a > K_b$, the solution is acidic.
• Amphiprotic Anions: Solution $pH$ is determined by comparing the $K_a$ and $K_b$ of the specific ion reacting with water.

Lewis Acid-Base Definition

• The Concept: Focuses on electron pairs rather than protons. It is the most general definition, encompassing Arrhenius and Brønsted-Lowry definitions.
• Lewis Base: Any species that donates an electron pair to form a covalent bond.
• Lewis Acid: Any species that accepts an electron pair to form a covalent bond.
• Adduct: The product of a Lewis acid-base reaction, which contains one extra covalent bond.
• Examples:     1. Electron Deficient Species: $H^+ + :NH_3 \rightarrow NH_4^+$ or $BF_3 + :NH_3 \rightarrow F_3B\text{-}NH_3$.     2. Polar Multiple Bonds: Molecules like $SO_2$ or $CO_2$ reacting with bases. For example, $OH^-$ attacking the oxygen in $SO_2$.     3. Metal Ions: Metal ions acting as Lewis acids to form hydrated ions: $M^{2+} + 6H_2O(l) \rightleftharpoons M(H_2O)_6^{2+}(aq)$.

Practical Examples and Sample Problems

• Sample Problem 18.5: Calculate $[H_3O^+]$, $pH$, $[OH^-]$, and $pOH$ for:     1. $0.30\,M \, HNO_3$ (used for etching copper).     2. $0.0042\,M \, Ca(OH)_2$ (used in leather tanning).
• Sample Problem 18.7: Propanoic acid ($CH_3CH_2COOH$, $HPr$) is used to retard mold in foods. Determine the $pH$ of $0.10\,M \, HPr$. ($K_a = 1.3 \times 10^{-5}$).
• Sample Problem 18.10: Dimethylamine $(CH_3)_2NH$ has a $K_b$ of $5.9 \times 10^{-4}$. Calculate the $pH$ of a $1.5\,M$ solution.
• Sample Problem 18.12: Predict acidity for aqueous solutions of:     1. Potassium perchlorate ($KClO_4$).     2. Sodium benzoate ($C_6H_5COONa$).     3. Chromium (III) nitrate ($Cr(NO_3)_3$).
• Sample Problem 18.13: Determine if zinc formate, $Zn(HCOO)_2$, at $25^\circ C$ is acidic, basic, or neutral. $K_a$ of $Zn(H_2O)_6^{2+} = 1 \times 10^{-9}$.
• Past Exam Question: The $pH$ of a $0.250\,M$ aqueous $HF$ solution is $2.04$. Calculate $K_a$ for this weak acid and use it to find the $pH$ of a $0.0001\,M$ aqueous solution of $HF$.