How to Find the Equivalence Point on a Titration Curve

To find the equivalence point on a titration curve, plot pH against titrant volume and read the volume at the steepest part of the rise, the inflection point. For most acid–base titrations that point very nearly coincides with the equivalence point [1]. When the curve is shallow or the reading must be precise, confirm it with a first derivative (a spike at the equivalence point) or a second derivative (a crossing of the volume axis). This guide walks through each step with a worked numeric example from Harvey’s Analytical Chemistry 2.1 and states what each method can and cannot do.

The equivalence point is the stoichiometric one: it is reached when the analyte and titrant have reacted in exactly equal stoichiometric amounts. It is a theoretical value, not a measurement [1]. What you actually locate on a curve or with an indicator is an estimate of it, the end point.

What you need

• Paired readings of titrant volume (mL) and pH, recorded after each addition. A calibrated pH meter is the usual source; see what a pH meter is and how it works.
• Many readings close together in the region where pH rises rapidly. Derivative methods work well only if sufficient data are recorded during the rapid increase in pH near the equivalence point [1].
• Graph paper or a spreadsheet to plot pH on the vertical axis and volume on the horizontal axis, and to compute differences between successive readings.

Steps

1. Plot pH against titrant volume. Join the points as a smooth curve. Include the flat region before the rise, the rise itself, and the flat region after it.
2. Find the steep section. The section where pH changes most rapidly for each added millilitre is the region that contains the equivalence point.
3. Mark the inflection point. The inflection point is the point of greatest slope. For most acid–base titrations it very nearly coincides with the equivalence point [1]. Read the titrant volume at that point on the horizontal axis. Treat this as a first estimate: Harvey calls it the simplest method for finding the end point and also the least accurate, particularly when the curve has a shallow slope at the equivalence point [1].
4. Confirm with the first derivative. Approximate the slope as ΔpH/ΔV between successive readings, and assign each result to the average of the two volumes. Plot those values against volume. Because the slope is at its maximum at the inflection point, the first derivative shows a spike at the equivalence point [1].
5. Use the second derivative for a sharper read. Repeat the same difference calculation on the first-derivative values. The second derivative may be more useful than the first because the equivalence point is where the curve intersects the volume axis [1].

Worked example: computing the derivatives

Harvey’s example uses three consecutive readings from the titration of 50.0 mL of 0.050 M CH₃COOH with 0.10 M NaOH, taken close to the equivalence point [1].

Titrant volume (mL)	pH
23.65	6.00
23.91	6.10
24.13	6.20

1. First pair: ΔpH/ΔV = (6.10 − 6.00) / (23.91 − 23.65) = 0.10 / 0.26 = 0.385. Assign it to the average volume, 23.78 mL [1].
2. Second pair: ΔpH/ΔV = (6.20 − 6.10) / (24.13 − 23.91) = 0.10 / 0.22 = 0.455, assigned to 24.02 mL [1].
3. Second derivative: the change between the two first-derivative values, divided by the change in their volumes, is (0.455 − 0.385) / (24.02 − 23.78), which Harvey reports as 0.292 [1].

Notice what is lost at each stage: three readings give two first-derivative values and one second-derivative value [1]. That is why a dense run of data around the rise matters. With a derivative plot, you read the volume at the spike (first derivative) or at the volume-axis crossing (second derivative) rather than judging the steepest point by eye.

Which method to use

Method	What it does	Stated strength or limit [1]
Inflection point on the pH curve	Read the volume at the point of greatest slope	Simplest, but the least accurate; the inflection point must be present and easy to identify
First derivative	Plot ΔpH/ΔV against volume	Shows a spike at the equivalence point; needs sufficient data during the rapid pH increase
Second derivative	Plot the change in the first derivative against volume	Can be more useful than the first derivative because the equivalence point intersects the volume axis
Gran plot	Rearranges data from part of the curve into a straight line	Converts a portion of a titration curve into a straight line
Visual indicator	A colour change marks the end point	The indicator’s entire pH range must fall within the rapid change in pH at the equivalence point

What can go wrong

• Treating the end point as the equivalence point. The end point is an experimental result that represents the best estimate of the equivalence point. Any difference between the two is a source of determinate error, and it is even possible that an equivalence point has no useful end point [1]. For the error sources in a real titration, see sources of errors in titration.
• No clear inflection point. The principal limitation of the inflection-point method is that the inflection point must be present; for some titrations it may be missing or difficult to find [1]. A derivative plot can help when a normal curve is poorly defined [1].
• Too few readings near the rise. Derivative methods work well only with sufficient data during the rapid pH increase, which is usually not a problem with an automatic titrator [1].
• Small bias from the inflection point. An inflection point precedes its equivalence point by a small amount, with the error approaching 0.1% for weak acids or bases with dissociation constants smaller than 10⁻⁹, or for very dilute solutions [1].
• Wrong indicator. The colour change marks the end point only if it occurs at or near the equivalence point, so the indicator’s whole pH range must sit inside the steep part of the curve [1].

For a weak acid titrated with a strong base the shape of the curve and the choice of indicator differ from a strong acid–strong base titration; see titration of a weak acid with a strong base.

When to get professional help

Ask the instructor, the method owner or the laboratory’s quality lead when the curve has no recognisable steep section, when replicate titrations give end points that disagree by more than your method allows, or when the result will be reported against a specification. Those decisions belong to whoever owns the method, not to the curve-reading steps above.

FAQ

What is the equivalence point of titration?

It is the point at which the analyte and titrant have reacted in exactly equal stoichiometric amounts. It is a theoretical value, not an experimental one [1].

What is a titration end point?

The end point is the experimental result: the volume at which you observe the change (an indicator colour change or the inflection on a curve). It represents your best estimate of the equivalence point [1].

How to find pKa on a titration curve?

Locate the equivalence-point volume using the steps above, halve it, and read the pH of the curve at that half volume. For a weak acid, that pH is the pKa. The full method, with its assumptions, is in reading pKa from the half-titration point. The same method answers how to determine, calculate or find pKa from a titration curve.

Is the inflection point the same as the equivalence point?

Not exactly. For most acid–base titrations it very nearly coincides with it, and it precedes it by a small amount [1].

Sources

1. Harvey, D. Analytical Chemistry 2.1, section 9.2, Acid–Base Titrations (“Selecting and Evaluating the End point”), LibreTexts: https://chem.libretexts.org/Courses/Northeastern_University/CHEM_1000:_General_Chemistry/09:_Titrimetric_Methods/9.2:_AcidBase_Titrations. Checked 29 October 2026.

Related reading on CASRAI: gravimetric titration.