pKa from Titration Curve Calculator

This calculator finds the pKa of a weak acid from the pH at the half-equivalence point of a titration curve. You enter the pH values and volumes from your titration data, and the calculator identifies the equivalence and half-equivalence points to return the pKa value. The result is the pH at the half-equivalence point, which equals the pKa for a monoprotic weak acid.

Formula

For a weak acid titrated with a strong base, the Henderson-Hasselbalch equation relates pH and pKa:

pH = pKa + log([A⁻]/[HA])

At the half-equivalence point, half of the acid has been neutralized, so [A⁻] = [HA]. The log term becomes log(1) = 0, and therefore:

pKa = pH at half-equivalence point

In the calculator above, the pKa is estimated by interpolating the pH at the volume that is exactly halfway between the starting volume and the equivalence volume. If you have already identified the half-equivalence volume, you can simply read the pH at that volume from your titration curve.

Worked Example

Using the default values: pH before equivalence = 4.75, pH after equivalence = 9.25, volume at half-equivalence = 25 mL, volume at equivalence = 50 mL.

Step 1: Determine the fraction of the way to equivalence at the half-equivalence point: 25 mL / 50 mL = 0.5.

Step 2: Interpolate the pH at that fraction: pH = 4.75 + (9.25 – 4.75) × 0.5 = 4.75 + 2.25 = 7.00.

Step 3: The pKa is the pH at the half-equivalence point, so pKa ≈ 7.00.

The calculator returns 7.00, which matches the manual calculation.

How to Read a Titration Curve

A titration curve plots pH (y-axis) against the volume of titrant added (x-axis). For a weak acid–strong base titration, the curve has a characteristic S-shape. The key points are:

• Initial pH: pH of the weak acid solution before any base is added.
• Buffer region: the relatively flat part of the curve before the steep rise; pH changes slowly here.
• Equivalence point: the middle of the steep vertical section, where moles of base added equal moles of acid initially present. For a weak acid, the pH at equivalence is above 7 (basic).
• Half-equivalence point: the point at exactly half the volume of titrant needed to reach equivalence. At this point, pH = pKa.

To find the half-equivalence point, first locate the equivalence point (the steepest part of the curve). Then divide the volume of titrant at equivalence by two, and read the pH at that volume on the curve. That pH is your pKa.

How to Choose the Equivalence Point

Accurately identifying the equivalence point is crucial because the half-equivalence point depends on it. Common methods include:

• Visual inspection: Look for the steepest part of the curve—the point of maximum slope. This is often the centre of the vertical section.
• First derivative: Plot ΔpH/ΔV vs. V. The peak of this derivative corresponds to the equivalence point.
• Second derivative: Plot Δ²pH/ΔV² vs. V. The point where this crosses zero (from positive to negative) is the equivalence point.
• Gran plot: A linearised plot of (V × 10^(-pH)) vs. V for the region before equivalence; the x-intercept gives the equivalence volume.

For a monoprotic weak acid, the half-equivalence volume is simply half of the equivalence volume. For polyprotic acids, there are multiple equivalence points and multiple half-equivalence points, each corresponding to a different pKa.

Common Mistakes and Pitfalls

• Using the wrong volume: Make sure you use the volume of titrant added, not the total volume of the solution.
• Confusing equivalence and half-equivalence: The equivalence point is where moles of base equal moles of acid; the half-equivalence point is half that volume. They are not the same.
• Ignoring dilution: The pH at the half-equivalence point is independent of concentration, but only if the acid is not too dilute or too concentrated. Extremely dilute solutions may show a less distinct buffer region.
• Assuming pKa = pH at half-equivalence for all acids: This holds only for monoprotic weak acids. For diprotic or polyprotic acids, each ionisation step has its own pKa, and the relationship becomes more complex.
• Rounding too early: Keep enough significant figures when reading pH values from the curve, as small errors in pH can shift the pKa estimate.

Limitations of the Method

The half-equivalence method is a quick and useful estimate, but it has limitations:

• Precision: The accuracy depends on how well you can read the pH at the half-equivalence volume. A steep curve makes the equivalence point easier to spot, but the buffer region may be flat, making the pH reading less sensitive.
• Ideal behaviour: The Henderson-Hasselbalch equation assumes ideal solution behaviour and that the acid is not too concentrated. At high concentrations, activity coefficients become important.
• Polyprotic acids: For acids with multiple ionisable protons, the half-equivalence points may overlap if the pKa values are close (within about 3 units). In such cases, the simple relationship fails.
• Weak acid strength: If the acid is extremely weak (pKa > 12) or very strong (pKa < 0), the titration curve may not show a clear buffer region, making the method unreliable.

For more precise pKa determination, you would use a nonlinear least-squares fit of the full titration curve to a theoretical model, which accounts for dilution, activity, and all equilibria.

Frequently Asked Questions

Why does pH equal pKa at the half-equivalence point?

At the half-equivalence point, exactly half of the weak acid has been converted to its conjugate base, so [HA] = [A⁻]. Plugging that into the Henderson-Hasselbalch equation gives pH = pKa + log(1) = pKa. This is a direct consequence of the equilibrium expression for a weak acid.

Can I use this method for a strong acid?

No. Strong acids fully dissociate, so there is no buffer region and the concept of pKa is not applicable in the same way. The half-equivalence method works only for weak acids (or weak bases) that exist in equilibrium with their conjugate species.

What if my titration curve has multiple equivalence points?

If you have a polyprotic acid (like phosphoric acid), each ionisation step has its own equivalence and half-equivalence point. The first half-equivalence point gives pKa1, the second gives pKa2, and so on. However, if the pKa values are close together (within about 3 units), the steps may overlap and the method becomes less accurate.

How do I find the equivalence point if my data is noisy?

Use the first derivative method: calculate ΔpH/ΔV between consecutive data points and plot it against the average volume. The peak of this derivative curve marks the equivalence point. This is more robust than visual estimation. If the derivative is still noisy, you can smooth the data first or use a Gran plot.

Does the concentration of the acid affect the pKa value?

In theory, pKa is a constant for a given acid at a given temperature, independent of concentration. However, at very high concentrations, activity effects can cause the apparent pKa to shift. In typical undergraduate titrations (0.1 M or less), the effect is negligible.

What is the difference between pKa and pH?

pH measures the acidity of a solution (the concentration of H⁺ ions). pKa is a property of a specific acid, describing how readily it donates a proton. A lower pKa means a stronger acid. pKa is constant for a given acid, while pH changes depending on the solution composition.

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in pH1 | pH before equivalence | - | 4.75 | 0
in pH2 | pH after equivalence | - | 9.25 | 0
in V1 | Volume at half-equivalence | mL | 25 | 0
in V2 | Volume at equivalence | mL | 50 | 0
out pka | pKa (pH at half-equivalence) | - | pH1 + (pH2 - pH1) * (V1 / V2) | 2
out eq_ph | pH at equivalence | - | pH2 | 2