Titration curves Knowledge-Based Study

Study notes

1. What is a Titration Curve?

   A titration curve is a graphical representation of the pH of an analyte solution plotted as a function of the volume of titrant added. These curves provide essential insights into acid–base equilibria, enabling the determination of the equivalence point, the acid dissociation constant ($K_a$) or base dissociation constant ($K_b$), and the optimal selection of acid–base indicators.

   Key terminology to distinguish before analyzing specific curves:

   • Equivalence Point: The stoichiometric point in a titration where moles of hydronium ions ($\text{H}_3\text{O}^+$) exactly equal moles of hydroxide ions ($\text{OH}^-$).
   • End Point: The experimental point at which an indicator permanently changes color, ideally coinciding closely with the equivalence point.
   • Buffer Region: The domain where significant concentrations of both a weak acid and its conjugate base (or a weak base and its conjugate acid) coexist, resisting significant changes in pH upon further titrant addition.

   Profile of a Strong Acid–Strong Base Curve

   When a strong acid (such as $\text{HCl}$) is titrated with a strong base (such as $\text{NaOH}$), both species undergo complete dissociation in aqueous solution.

   Key Regions of the Curve

   • Initial pH: Determined solely by the concentration of the strong acid: $$\text{pH} = -\log[\text{H}_3\text{O}^+]$$
   • Before the Equivalence Point: Excess $\text{H}_3\text{O}^+$ remains. The pH increases very gradually over a wide range of added titrant volume because the solution has high hydronium ion concentration and volume dilution effects dominate.
   • Equivalence Point: The only species present in significant amounts are spectator ions (such as $\text{Na}^+$ and $\text{Cl}^-$) and water. Because neither ion undergoes hydrolysis, the equivalence point occurs at exactly $\text{pH} = 7.00$ at $25\text{ }^\circ\text{C}$. Near this point, the pH rises steeply over a very small volume increment.
   • After the Equivalence Point: Excess $\text{OH}^-$ dictates the solution composition. The pH is governed by the residual hydroxide concentration: $$\text{pOH} = -\log[\text{OH}^-] \quad \text{and} \quad \text{pH} = 14.00 - \text{pOH}$$

   Characteristics of Weak Acid Titrations

   Titrating a weak acid (such as $\text{CH}_3\text{COOH}$) with a strong base (such as $\text{NaOH}$) produces a curve distinctly different from strong acid titrations due to incomplete ionization and conjugate base hydrolysis.

   Essential Regions and Principles

   • Initial pH: Higher than that of a strong acid of equal concentration. Calculated using the acid's $K_a$ value via an equilibrium expression: $$K_a = \frac{[\text{H}_3\text{O}^+][\text{A}^-]}{[\text{HA}]}$$
   • Buffer Region: Addition of strong base converts $\text{HA}$ into its conjugate base $\text{A}^-$. The mixture of unreacted $\text{HA}$ and produced $\text{A}^-$ forms a buffer solution described by the Henderson–Hasselbalch equation: $$\text{pH} = \text{p}K_a + \log\left(\frac{[\text{A}^-]}{[\text{HA}]}\right)$$
   • Half-Equivalence Point: Exactly halfway to the equivalence point, half of the initial $\text{HA}$ has been neutralized, meaning $[\text{HA}] = [\text{A}^-]$. Consequently: $$\text{pH} = \text{p}K_a$$
   • Equivalence Point: All $\text{HA}$ is converted into the conjugate base $\text{A}^-$. Because $\text{A}^-$ hydrolyzes water ($\text{A}^- + \text{H}_2\text{O} ightleftharpoons \text{HA} + \text{OH}^-$), the solution is basic at equivalence: $\text{pH} > 7.00$.
   • Post-Equivalence: Dominated by excess strong base, converging closely to the strong acid–strong base trajectory.

   Weak Base Titrations

   When titrating a weak base (such as $\text{NH}_3$) with a strong acid (such as $\text{HCl}$):

   • The curve begins at a moderately basic pH and slopes downward.
   • A buffer region exists where $\text{B}$ and $\text{BH}^+$ coexist, with $\text{pH} = \text{p}K_a(\text{BH}^+)$ at the half-equivalence point.
   • At the equivalence point, the solution contains the conjugate acid $\text{BH}^+$, which undergoes hydrolysis to produce $\text{H}_3\text{O}^+$. Therefore, $\text{pH} < 7.00$ at the equivalence point.

   Polyprotic Acid Curves

   Acids capable of donating more than one proton (such as $\text{H}_2\text{SO}_3$ or $\text{H}_3\text{PO}_4$) undergo stepwise neutralization:

   • A distinct equivalence point and steep vertical rise occur for each dissociable proton, provided the successive ionization constants differ by at least $10^4$ ($K_{a1} / K_{a2} \ge 10^4$).
   • The first half-equivalence point yields $\text{pH} = \text{p}K_{a1}$, and the second half-equivalence point yields $\text{pH} = \text{p}K_{a2}$.

   Matching Indicators to the Equivalence Point

   Acid–base indicators are weak organic acids whose protonated and deprotonated forms display different colors. An indicator changes color over a characteristic transition range approximately equal to $\text{p}K_{\text{In}} \pm 1$.

   Selection Criteria

   • An indicator must undergo its color transition within the steep, vertical section of the titration curve containing the equivalence point.
   • Strong Acid–Strong Base: The vertical inflection spans roughly pH 4 to 10. Thus, either bromothymol blue ($\text{p}K_{\text{In}} \approx 7.1$) or phenolphthalein ($\text{p}K_{\text{In}} \approx 9.4$) serves as a suitable indicator.
   • Weak Acid–Strong Base: The equivalence point is basic ($\text{pH} > 7$). Phenolphthalein is suitable because its transition range (pH 8.2–10.0) coincides with the vertical jump. Methyl red or methyl orange would change color prematurely in the buffer region.
   • Weak Base–Strong Acid: The equivalence point is acidic ($\text{pH} < 7$). Methyl red ($\text{p}K_{\text{In}} \approx 5.1$) or bromocresol green is appropriate.