Titration Curves & Half-Equivalence Points

The half-equivalence point occurs when exactly half of the volume of titrant required to reach the equivalence point has been added. At this specific volume, half of the original weak acid has been converted into its conjugate base.

Chemically, this is the point where the concentration of the weak acid $[HA]$ is exactly equal to the concentration of its conjugate base $[A^-]$. This equality creates a unique mathematical condition in the equilibrium expression.

For a weak base titration, the half-equivalence point is where the concentration of the weak base $[B]$ equals the concentration of its conjugate acid $[HB^+]$, leading to the relationship $pOH = pK_b$.

The relationship at the half-equivalence point is derived from the Henderson-Hasselbalch equation: $pH = pK_a + \log\left(\frac{[A^-]}{[HA]}\right)$ Since $[A^-] = [HA]$ at this point, the ratio is 1, and $\log(1) = 0$.

Consequently, at the half-equivalence point, the measured pH is equal to the $pK_a$ of the weak acid. This makes titration curves a primary experimental method for determining dissociation constants.

This principle also applies to polyprotic acids, which exhibit multiple half-equivalence points. Each midpoint between successive equivalence points corresponds to the $pK_a$ of the specific proton being removed.

Strong Acid/Base Curves: These curves do not have a half-equivalence point with chemical significance like weak species do. Because strong acids dissociate completely, they do not form a conjugate buffer system, and the pH changes rapidly from the start.

pH at Equivalence: For a strong acid-strong base titration, the pH at equivalence is exactly 7.0. For a weak acid-strong base titration, the pH at equivalence is greater than 7.0 due to the hydrolysis of the resulting conjugate base.

Identify the Volume First: Always find the equivalence point volume ($V_{eq}$) by locating the center of the vertical inflection. Then, divide that volume by two to find the half-equivalence point volume ($V_{1/2}$).

Read the pKa Directly: Once you have identified $V_{1/2}$, move vertically to the curve and then horizontally to the y-axis. The pH value at that point is your experimental $pK_a$.

Check the Equivalence pH: If the pH at the equivalence point is not 7, the analyte is a weak species. If $pH > 7$, you are titrating a weak acid; if $pH < 7$, you are titrating a weak base.

Polyprotic Patterns: For polyprotic acids, ensure you are using the correct half-equivalence point for the specific $pK_a$ requested (e.g., the first midpoint for $pK_{a1}$, the second for $pK_{a2}$).