Aqueous Equilibria: Common-Ion Effect, Buffers, Titrations, and Solubility

The Common-Ion Effect

The common-ion effect is a fundamental principle in aqueous equilibria stating that whenever a weak electrolyte and a strong electrolyte containing a common ion are placed together in a solution, the weak electrolyte ionizes to a lesser extent than it would if it were present alone. This phenomenon is a direct application of Le Châtelier’s Principle. For example, in a solution containing the weak acid acetic acid ($CH_3COOH$), the equilibrium is defined by its partial dissociation. If sodium acetate ($NaCH_3COO$), which is a strong electrolyte, is added to this solution, it dissociates completely to provide a high concentration of acetate ions ($CH_3COO^-$). This increase in the concentration of the product ion (the common ion) causes the equilibrium of the acetic acid to shift toward the reactant side, thereby decreasing the ionization of the acid. This effect significantly influences both acid-base equilibria and solubility equilibria. In terms of electrolytes, weak electrolytes such as weak acids and weak bases dissociate only partially in water, whereas strong electrolytes, including all soluble ionic compounds, strong acids, and strong bases, dissociate completely.

Buffers and Their Mechanism of Action

Buffers are specialized solutions comprising weak conjugate acid-base pairs that possess the unique ability to resist drastic changes in $pH$ when small amounts of acid or base are added. To function effectively, buffers must contain relatively high concentrations of both the acidic and basic components, typically in concentrations of $10^{-3}\,M$ or greater. Ideally, the concentrations of the acid and base should be approximately equal. A buffer works because the added acid ($H^+$) or base ($OH^-$) is neutralized by one of the components of the buffer system. For instance, in a hydrofluoric acid ($HF$) and fluoride ($F^-$) buffer, the addition of a strong base results in the reaction $HF(aq) + OH^-(aq) \rightarrow H_2O(l) + F^-(aq)$, where the presence of the weak acid counteracts the base to minimize the $pH$ increase. Conversely, the addition of a strong acid results in the reaction $F^-(aq) + H^+(aq) \rightarrow HF(aq)$, where the conjugate base counteracts the acid to minimize the $pH$ decrease. Consequently, the $pH$ of the solution remains relatively stable.

The Henderson-Hasselbalch Equation and Buffer Calculations

The calculation of the $pH$ of a buffer solution is facilitated by the Henderson-Hasselbalch equation, which is derived from the initial-change-equilibrium (ICE) table approximations for weak acids. The equation is expressed as $pH = pK_a + \log\left(\frac{[\text{base}]}{[\text{acid}]}\right)$. This formula allows for the direct determination of $pH$ based on the $pK_a$ of the weak acid and the ratio of the concentrations of the conjugate base and acid. To prepare a buffer at a desired $pH$, one must select a conjugate acid-base pair where the $pK_a$ of the acid is as close as possible to the target $pH$. Buffer capacity refers to the specific amount of acid or base a buffer can neutralize before the $pH$ begins to change significantly. Higher concentrations of the conjugate pair result in a higher buffering capacity. The effective $pH$ range for a buffer system is generally considered to be within $\pm 1$ unit of the acid's $pK_a$. Buffering action is considered poor if the concentration of one component is more than 10 times that of the other.

Acid-Base Titrations and Curves

Titration is an analytical technique used to determine the unknown concentration of an acid or base solution by slowly adding a titrant of known concentration. The process continues until the equivalence point is reached, which is the point where the stoichiometrically equivalent amounts of acid and base have reacted. For a titration involving a strong acid and a strong base, the $pH$ at the equivalence point is exactly $7.0$, representing a neutral salt solution like $NaCl(aq)$. Before the equivalence point, the $pH$ is dominated by the excess strong acid or base; at the equivalence point, all $H^+$ is neutralized by $OH^-$; and after the equivalence point, the $pH$ rises (or falls) rapidly due to excess titrant. In a weak acid-strong base titration, the $pH$ at the equivalence point is greater than $7.0$ because the resulting conjugate base (e.g., $CH_3COO^-$) reacts with water to produce $OH^-$ ions. Furthermore, the $pH$ change near the equivalence point is less sharp for weak acid titrations compared to strong acid titrations, and the initial $pH$ is higher. In such titrations, the addition of base to the weak acid initially forms a buffer solution. If a weak base is titrated with a strong acid, the $pH$ at the equivalence point will be less than $7.0$.

Indicators and Selection Criteria

Indicators are weak acids that exhibit different colors than their conjugate base forms, allowing for visual determination of the titration's progress. Examples include Methyl red (color-change interval: 4.2 < pH < 6.0), Bromthymol blue, and Phenolphthalein (color-change interval: 8.3 < pH < 10.0). The choice of an indicator is critical; it must be selected such that its color-change interval overlaps with the steep portion of the titration curve's $pH$ change near the equivalence point. For a weak acid-strong base titration, phenolphthalein is an appropriate choice because the equivalence point typically falls within its range. Methyl red would be a poor choice for this titration as it would change color long before the equivalence point is reached. Conversely, for a weak base-strong acid titration, methyl red is a suitable indicator, while phenolphthalein is unsatisfactory.

Solubility Equilibria and the Solubility Product Constant ($K_{sp}$)

Solubility equilibria describe the balance between an undissolved ionic solid and its dissolved ions in a saturated solution. The equilibrium expression for this process is governed by the solubility product constant ($K_{sp}$). For example, for the dissolution of barium sulfate, $BaSO_4(s) \rightleftharpoons Ba^{2+}(aq) + SO_4^{2-}(aq)$, the constant is $K_{sp} = [Ba^{2+}][SO_4^{2-}] = 1.1 \times 10^{-10}$ at $25^{\circ}C$. Solubility is a quantitative measure of the amount of solute that dissolves to form a saturated solution, typically expressed in $g/L$ or as molar solubility in $mol/L$. While molar solubility can change depending on the presence of other ions or changes in $pH$, the $K_{sp}$ value is a constant for a specific solute at a given temperature. To predict qualitative solubility, one can use solubility guidelines: compounds containing $NO_3^-$, $CH_3COO^-$, and most $Cl^-$, $Br^-$, $I^-$, and $SO_4^{2-}$ are soluble (with specific exceptions like $Ag^+$, $Hg_2^{2+}$, and $Pb^{2+}$); whereas most $S^{2-}$, $CO_3^{2-}$, $PO_4^{3-}$, and $OH^-$ are insoluble, except when paired with alkali metals or $NH_4^+$.

Factors Affecting Solubility

Three primary factors influence the solubility of ionic compounds: the presence of common ions, the $pH$ of the solution, and the presence of complexing agents. According to the common-ion effect, the addition of an ion already present in the equilibrium will shift the equilibrium toward the solid precipitate, thereby decreasing solubility. For instance, adding $F^-$ to a saturated solution of $CaF_2$ will reduce the solubility of the salt. Solubility is also highly dependent on $pH$; in general, the solubility of compounds containing basic anions (the conjugate bases of weak acids) increases as the acidity of the solution increases. This occurs because the added protons ($H^+$) react with the basic anion, removing it from the solubility equilibrium and driving the dissolution of more solid. For example, $Ni(OH)_2$ is more soluble in acidic solutions than in basic solutions because $OH^-$ is a basic anion. Similarly, lead(II) fluoride ($PbF_2$) solubility increases with acidity because $F^-$ is the conjugate base of a weak acid ($HF$), whereas lead(II) chloride ($PbCl_2$) solubility is largely unaffected by $pH$ because $Cl^-$ is the conjugate base of a strong acid ($HCl$).

Questions & Discussion

Practice Question: What is the $pH$ of a solution made by adding $0.30\,mol$ of acetic acid and $0.30\,mol$ of sodium acetate to enough water to make $1.0\,L$ of solution?

Practice Question: Compare the $pH$ found in the previous question to the $pH$ of a $0.30\,M$ acetic acid solution.

Practice Question: What is the $pH$ of a buffer that is $0.12\,M$ in lactic acid, $CH_3CH(OH)COOH$, and $0.10\,M$ in sodium lactate? (Note: $K_a$ of lactic acid = $1.4 \times 10^{-4}$).

Practice Question: How many grams of $Na_2CO_3$ must be added to a $1.5\,L$ solution of $0.20\,M\,NaHCO_3$ to create a buffer at $pH\,10.00$? ($K_a$ for $HCO_3^- = 4.7 \times 10^{-11}$).

Practice Question: Calculate the $pH$ of a solution where $10.00\,mL$ of $0.100\,M\,NaOH$ is added to $25\,mL$ of $0.100\,M\,CH_3COOH$.

Practice Question: A buffer is made by adding $0.300\,mol\,CH_3COOH$ and $0.300\,mol$ of $NaCH_3COO$ to enough water to make a $1.00\,L$ solution ($K_a = 1.8 \times 10^{-5}$). Calculate the $pH$ after $5.0\,mL$ of $4.0\,M\,NaOH$ is added. Compare this to the $pH$ change if the same amount of base was added to $1\,L$ of pure water.

Practice Question: Determine the $pH$ during the titration of $30.0\,mL$ of $0.10\,M\,KOH$ with $0.10\,M\,HBr$ at the following stages: (a) $0.00\,mL$ titrant added; (b) $29.00\,mL$ titrant added; (c) the volume required to reach the equivalence point; (d) $31.00\,mL$ titrant added.

Practice Question: Consider the titration of $20.0\,mL$ of $0.175\,M\,HClO_3$ with $0.150\,M\,Ca(OH)_2$. What is the $pH$ at the equivalence point and after $30.00\,mL$ of $Ca(OH)_2$ have been added?

Practice Question: $25.0\,mL$ of $0.100\,M\,CH_3COOH$ is titrated with $0.100\,M\,NaOH$. Calculate the $pH$ at: (a) $0.00\,mL$; (b) $10.0\,mL$; (c) $12.5\,mL$; (d) $25.0\,mL$; (e) $30.0\,mL$ of $NaOH$ added.

Practice Question: Write $K_{sp}$ expressions for various reactions and calculate the solubility of $CaF_2$ in $g/L$ ($K_{sp} = 3.9 \times 10^{-11}$ at $25^{\circ}C$). Determine the molar solubility of $CaF_2$ in (a) $0.010\,M\,Ca(NO_3)_2$ and (b) $0.010\,M\,NaF$.