The most straightforward way to determine Ka from a titration curve is to find the half-equivalence point, where exactly half the acid has been neutralized. At that point, the pH of the solution equals the pKa of the acid, and converting pKa to Ka is a single calculation: Ka = 10⁻ᵖᴷᵃ. The trick is locating that point accurately on your curve.
Why pH Equals pKa at the Half-Equivalence Point
When a weak acid (HA) dissolves in water, it partially dissociates into H⁺ ions and its conjugate base (A⁻). The acid dissociation constant describes this equilibrium:
Ka = [H⁺][A⁻] / [HA]
Rearranging and taking the negative logarithm of both sides gives you the Henderson-Hasselbalch equation:
pH = pKa + log([A⁻] / [HA])
As you add strong base to the acid during titration, the base converts HA into A⁻. At the half-equivalence point, exactly half of the original acid has been converted, so the concentrations of HA and A⁻ are equal. When [A⁻] = [HA], the ratio inside the logarithm is 1, and log(1) = 0. The equation collapses to pH = pKa. No additional math needed. You simply read the pH off your curve at the halfway mark.
Step 1: Find the Equivalence Point
Before you can find the halfway mark, you need to know where the full equivalence point is. On a standard titration curve (pH on the y-axis, volume of base added on the x-axis), the equivalence point sits at the steepest part of the S-shaped curve, where the pH rises most sharply.
For a rough reading, you can eyeball it: look for the inflection point in the steep region. For more precision, two derivative methods are commonly used:
- First derivative method. Plot the change in pH per change in volume (ΔpH/ΔV) against volume. The equivalence point appears as a sharp spike, the volume where the rate of pH change is greatest.
- Second derivative method. Plot the second derivative (Δ²pH/ΔV²) against volume. The equivalence point is where this curve crosses zero. Automated titrators often use this zero-crossing approach, and it typically gives precision to within a fraction of a milliliter.
Record this volume carefully. If your equivalence point is at 30.4 mL of base, your half-equivalence point is at 15.2 mL.
Step 2: Read pH at the Half-Equivalence Point
Go to your titration curve and find the pH value corresponding to half the equivalence point volume. This region of the curve is relatively flat because it falls within the buffer zone, where the solution resists pH changes. That flatness actually works in your favor: small errors in identifying the exact half-volume won’t throw off your pH reading much.
Suppose your equivalence point is at 30 mL and the pH at 15 mL reads 4.74. That means pKa = 4.74, which corresponds to acetic acid.
Step 3: Convert pKa to Ka
The conversion is:
Ka = 10⁻ᵖᴷᵃ
For the acetic acid example: Ka = 10⁻⁴·⁷⁴ = 1.82 × 10⁻⁵. That’s it. If your titration data is clean and your equivalence point is accurate, this value will be close to the literature value.
Polyprotic Acids Have Multiple Ka Values
Acids with more than one ionizable proton produce titration curves with multiple steep rises, each corresponding to a separate equivalence point. A diprotic acid like maleic acid shows two equivalence points; a triprotic acid like citric acid shows three.
Each “titration” on the curve has its own half-equivalence point. The pH at the midpoint of the first steep rise gives you pKa1 (for loss of the first proton). The pH at the midpoint of the second steep rise gives you pKa2. For citric acid, you can extract all three Ka values this way, one from each buffer region.
Identifying where one equivalence region ends and the next begins can be tricky when Ka values are close together, because the steep regions start to merge. In those cases, the first and second derivative methods become essential for separating the individual equivalence points.
What About Weak Bases?
If you’re titrating a weak base with a strong acid instead, the logic is similar but requires one extra step. The half-equivalence point pH still equals the pKa of the conjugate acid that forms during the titration. To get the Kb of the original base, use the relationship:
pKb = 14 – pKa
Then convert: Kb = 10⁻ᵖᴷᵇ. The value 14 comes from the ion product of water at 25 °C, so this conversion only holds at that temperature.
Gran Plots for Better Accuracy
The half-equivalence method is quick but assumes ideal behavior. For more precise work, a Gran plot linearizes the titration data. You plot (Vb × 10⁻ᵖᴴ) on the y-axis against volume of base added (Vb) on the x-axis, using data points collected before the equivalence point. The result is a straight line whose x-intercept gives the equivalence point volume and whose slope is directly related to Ka (specifically, the slope equals -Ka, adjusted for activity coefficients at the solution’s ionic strength).
Gran plots are particularly useful when the titration curve’s inflection point is hard to read, such as when working with very weak acids (pKa above 10 or so) where the steep region is subtle.
Factors That Affect Your Result
The dissociation constant is not truly constant. It shifts with temperature, ionic strength, and solvent composition. Temperature matters most in a teaching lab setting. The pKa of many acids changes measurably between 20 °C and 30 °C, so running the titration at a stable, known temperature is important. Most published Ka values are reported at 25 °C, and comparing your result to literature values only makes sense if your conditions match.
Ionic strength also plays a role. As salt concentration in the solution increases throughout the titration, activity coefficients shift. For a careful determination, you would keep ionic strength roughly constant by adding a background electrolyte before starting. For a standard general chemistry lab, this correction is usually small enough to ignore.
When reporting or comparing Ka values, always note the temperature and conditions. A Ka measured at 37 °C in physiological saline will differ noticeably from one measured at 25 °C in pure water, and neither is “wrong.” They simply describe the acid’s behavior under different conditions.