4.3 Titration curves and equivalence points

Titration Curves

Titration curves plot pH (y-axis) against the volume of titrant added (x-axis), giving you a visual map of what happens during an acid-base neutralization. They're how you identify equivalence points, buffer regions, and the pKa of weak acids. The shape of the curve tells you a lot about the strength of the acid and base involved.

Construction of Titration Curves

Strong acid-strong base titrations (like HCl + NaOH) produce the most straightforward curves:

• The curve starts at a low pH because the strong acid fully dissociates, releasing a high concentration of $H_3O^+$ ions.
• As you add strong base, the pH rises slowly at first, then shoots up steeply near the equivalence point. This steep jump happens because you're going from a tiny excess of acid to a tiny excess of base in just a few drops.
• The equivalence point sits at pH 7 because the products are just water and a neutral salt (like NaCl). Neither ion hydrolyzes, so the solution is neutral.
• Past the equivalence point, pH continues to climb as excess $OH^-$ accumulates.

Weak acid-strong base titrations (like $CH_3COOH$ + NaOH) look noticeably different:

• The initial pH is higher than in a strong acid titration because the weak acid only partially dissociates.
• Before the equivalence point, you get a buffer region where the weak acid and its conjugate base coexist (e.g., acetic acid and sodium acetate). In this region, pH changes gradually because the buffer resists large pH shifts.
• The equivalence point occurs at pH > 7. Why? At the equivalence point, all the weak acid has been converted to its conjugate base ($CH_3COO^-$), which hydrolyzes in water to produce $OH^-$ ions, making the solution basic.
• After the equivalence point, pH rises sharply as excess NaOH dominates.

Equivalence Point Significance

The equivalence point is where the moles of acid exactly equal the moles of base, meaning complete neutralization has occurred. This is distinct from the endpoint, which is where an indicator changes color (ideally these are close together, but they're not the same thing).

Where the equivalence point falls on the pH scale depends on the titration type:

• Strong acid + strong base: pH = 7 (neutral salt formed)
• Weak acid + strong base: pH > 7 (conjugate base hydrolyzes)
• Strong acid + weak base: pH < 7 (conjugate acid hydrolyzes)

The equivalence point is what makes titrations useful for standardization, which means determining an unknown concentration. If you know the exact volume and concentration of your titrant at the equivalence point, you can calculate the moles of the unknown analyte using stoichiometry.

Weak Acid pKa Determination

The pKa is the negative log of the acid dissociation constant: $pK_a = -\log(K_a)$. It tells you how readily a weak acid donates a proton. Lower pKa = stronger weak acid.

To find pKa from a titration curve:

1. Identify the equivalence point (the midpoint of the steepest rise).
2. Find the half-equivalence point, which is at exactly half the volume of titrant needed to reach the equivalence point.
3. Read the pH at that half-equivalence point. That pH equals the pKa.

This works because at the half-equivalence point, exactly half the weak acid has been neutralized, so $[HA] = [A^-]$. Plugging into the Henderson-Hasselbalch equation:

$pH = pK_a + \log\frac{[A^-]}{[HA]}$

When $[A^-] = [HA]$, the log term becomes $\log(1) = 0$, so $pH = pK_a$.

For example, acetic acid has a $pK_a$ of 4.76, so at the half-equivalence point of an acetic acid titration, the pH will be 4.76. Knowing the pKa also lets you predict the pH of buffer solutions containing that acid and its conjugate base.

Monoprotic vs. Polyprotic Acid Titrations

Monoprotic acids (like HCl or $CH_3COOH$) have one ionizable proton, so their titration curves show a single equivalence point and one steep rise.

Polyprotic acids (like $H_2CO_3$ with 2 protons, or $H_3PO_4$ with 3) lose their protons in separate steps, and each step has its own $K_a$. This produces a titration curve with distinct features:

• Multiple equivalence points, one for each ionizable proton. $H_3PO_4$ has three.
• Buffer regions between each equivalence point, where the partially deprotonated species and its conjugate coexist.
• A half-equivalence point before each equivalence point, where $pH = pK_a$ for that specific ionization step.

The equivalence points are only clearly separated on the curve if the $K_a$ values for successive ionizations differ by a factor of at least ~$10^3$ to $10^4$. When $K_a$ values are too close together, the steps blend into each other and individual equivalence points become hard to distinguish. Each equivalence point also sits at a different pH, reflecting the different $K_a$ values of each ionization step.