An acid-base titration is a common laboratory technique used to determine the unknown concentration of an acid or a base in a solution. This process involves the careful, measured addition of a titrant (a solution of known concentration) into the analyte solution until the reaction between them is complete. A titration curve plots the solution’s pH against the volume of titrant added, providing a visual record of the reaction progress. The full equivalence point marks the stage where the moles of titrant exactly match the moles of analyte, signifying stoichiometric completion. Before reaching this point, the half-equivalence point offers unique chemical insights and helps characterize the weak acid or weak base being studied.
Defining the Half-Equivalence Point
The half-equivalence point is defined by the volume of titrant added relative to the full equivalence point. It is the stage where exactly half the volume of titrant required to reach complete neutralization has been introduced. For instance, if the equivalence point is reached after adding 20.0 milliliters of base, the half-equivalence point occurs after the addition of 10.0 milliliters.
This volume addition results in a distinct stoichiometric ratio. The titrant reacts with exactly half of the initial weak acid (or weak base) analyte, generating an equal amount of the conjugate species. This means the concentration of the weak acid remaining, [HA], is equal to the concentration of its conjugate base formed, [A\(^-\)]. This \(1:1\) ratio is the chemical signature of the half-equivalence point.
Locating the Half-Equivalence Point on a Titration Curve
Locating this point requires first identifying the full equivalence point on the titration curve, which is characterized by the steepest vertical rise in pH. The volume of titrant corresponding to this inflection point is recorded as the equivalence volume, \(V_{eq}\). The half-equivalence volume, \(V_{1/2}\), is then calculated by dividing the equivalence volume by two (\(V_{1/2} = V_{eq}/2\)).
Once \(V_{1/2}\) is determined, the corresponding pH value on the curve is the pH at the half-equivalence point. The region surrounding this point is noticeably flat, exhibiting the shallowest slope. This flatness indicates that the solution is acting as an effective buffer, resisting large changes in pH upon the addition of small amounts of titrant. The ability to buffer the pH is maximized when the concentrations of the acid and its conjugate base are equal, a condition met at the half-equivalence point.
The presence of this extended flat region, often called the buffer region, is a characteristic feature of titrations involving a weak acid or a weak base. A strong acid-strong base titration does not exhibit this distinct half-equivalence point or buffer region because both reactants fully dissociate, preventing the formation of a stable conjugate acid-base pair. Analyzing the shape of the curve, particularly the location of this shallow slope, is a practical method for confirming the identity of a weak acid or base.
The Significance of pH Equal to pKa
The primary chemical significance of the half-equivalence point is the direct relationship established between the solution’s pH and the acid dissociation constant, or pKa, of the analyte. The pKa value measures an acid’s strength, representing the negative logarithm of the acid dissociation constant (\(K_a\)). This relationship is quantified by the Henderson-Hasselbalch equation, which describes the pH of a buffer solution: \(\text{pH} = \text{pKa} + \log([\text{A}^-]/[\text{HA}])\).
At the half-equivalence point, the remaining concentration of the weak acid, [HA], is equal to the concentration of the conjugate base, [A\(^-\)], that has been formed. Because the concentrations are equal, the ratio of the conjugate base to the weak acid, \([\text{A}^-]/[\text{HA}]\), becomes \(1\).
When this ratio is substituted into the Henderson-Hasselbalch equation, the logarithm term simplifies significantly. The term \(\log(1)\) is equal to zero, effectively eliminating the second part of the equation. This mathematical simplification results in the expression \(\text{pH} = \text{pKa}\). Therefore, measuring the pH of the solution at the half-equivalence point allows a chemist to directly determine the pKa of the weak acid being titrated. This direct relationship provides a simple and accurate method for characterizing the acid.
Calculating Component Concentrations at the Half-Equivalence Point
Determining the concentrations of the species involved at the half-equivalence point is a straightforward calculation based on the \(1:1\) stoichiometric ratio. The initial moles of the weak acid analyte, \(\text{mol}_{\text{HA, initial}}\), must be known. When the half-equivalence volume of titrant is added, exactly half of the initial moles of the acid have been neutralized.
This neutralization consumes half of the weak acid and simultaneously produces an equal number of moles of the conjugate base. If the initial moles of the weak acid were \(N\), then the moles of weak acid remaining, \(\text{mol}_{\text{HA}}\), and the moles of conjugate base formed, \(\text{mol}_{\text{A}^-}\), are both equal to \(N/2\).
The concentration of each species is found by dividing the moles by the total volume of the solution, which includes the initial volume and the added half-equivalence volume of titrant. Since the moles of [HA] and [A\(^-\)] are equal and dissolved in the same total volume, their molar concentrations must also be equal: \([\text{HA}] = [\text{A}^-]\). This \(1:1\) ratio links the experimentally measured pH value at this point directly to the intrinsic pKa value of the weak acid.