Titration Curve Point Calculator

Background.

This calculator gives the pH at any point of a monoprotic acid–base titration: before you start, anywhere in the buffer region, exactly at the equivalence point, or well past it. It covers strong acid against strong base, weak acid against strong base, and weak base against strong acid, and it computes all three from a single exact charge-balance equation rather than from the four regional formulas textbooks teach.

That choice is the most important thing about the page, so it belongs here rather than in an FAQ. The usual approach splits the curve into segments — initial equilibrium, Henderson–Hasselbalch through the buffer region, salt hydrolysis at equivalence, excess titrant afterwards — and each segment gets its own formula. The result has singularities at exactly the two points a user is most likely to ask about: Henderson–Hasselbalch contains log([A⁻]/[HA]), which is minus infinity at zero titrant added and plus infinity at the equivalence point. The segments also disagree with each other at the seams, so the curve steps rather than rises, and the step is biggest where the curve is steepest. Robert de Levie made the case against segmentation in the Journal of Chemical Education in 1993, in a paper whose subtitle is 'without using approximations or segmentation'. This page follows it. There are still region labels, but they only describe the answer — they never produce it.

The default is the classic experiment: 25.00 mL of 0.1000 mol/L acetic acid titrated with 0.1000 mol/L sodium hydroxide, with 12.50 mL delivered. The equivalence volume is 25.00 mL, so 12.50 mL is exactly half-equivalence, and the calculator returns pH 4.75646. The pKa used is 4.756, the NIST critically evaluated value — and that near-identity is the reason half-equivalence is how pKa values are measured from a titration curve. The exact answer is 0.00046 above the pKa, not equal to it, because a little of the acid has also ionised on its own account, which the Henderson–Hasselbalch derivation ignores.

The rest of that curve: pH 2.88088 with no titrant added, 4.15621 at 5 mL, 5.35827 at 20 mL, 7.15204 at 24.9 mL, 8.72754 at the 25.00 mL equivalence point, 10.30047 at 25.1 mL, and 12.52288 by 50 mL. The near-vertical stretch between 24.9 and 25.1 mL — over three pH units for a fifth of a millilitre — is why indicator choice matters, and it is exactly where segmented calculations misbehave.

Note that the equivalence point of a weak-acid titration is basic, not neutral: 8.72754 here, because the acetate left in the flask hydrolyses. Only a strong acid against a strong base gives a neutral equivalence point, and even then 'neutral' means pKw/2 rather than 7.00 — it is 7.00 at 25 °C, 6.63 at 50 °C and 6.13 at 100 °C, because pKw is temperature-dependent. Titrating a weak base with a strong acid mirrors the acid case: the default ammonia titration reaches equivalence at pH 5.27296.

Scope. Monoprotic and 1:1 only: one mole of titrant neutralises one mole of analyte. Diprotic acids such as sulfuric or carbonic, and 2:1 titrants such as calcium hydroxide, will give the wrong equivalence volume here. Volumes are taken as additive. Activity coefficients are set to one, there is no correction for ionic strength, and the system is assumed closed, so a solution left open to air and absorbing carbon dioxide will drift from the calculated curve, particularly on the alkaline side.

What is titration curve point calculator?

A titration curve is a plot of pH against the volume of titrant added. Its shape carries most of the information a titration can give: the equivalence volume tells you the analyte's concentration, the steepness at that point tells you whether the titration is feasible and which indicator will work, and the pH at half-equivalence tells you the analyte's pKa.

The curve has a characteristic S shape with four recognisable stretches. At the start the pH is set by the analyte alone. As titrant is added, a weak analyte is progressively converted into its conjugate, and the mixture buffers — this is the flat portion, and it is where Henderson–Hasselbalch applies. Approaching equivalence the buffering collapses, because one of the two components is nearly exhausted, and the pH lurches. Past equivalence the excess titrant dominates and the curve flattens again on a logarithmic approach to the titrant's own pH.

What happens at the equivalence point depends entirely on what is left in the flask. For a strong acid and a strong base, only spectator ions remain and the pH is exactly pKw/2 — 7.00 at 25 °C. For a weak acid titrated with a strong base, the flask contains the conjugate base, which hydrolyses and pushes the pH above pKw/2. For a weak base titrated with a strong acid, the conjugate acid does the reverse. This is why an indicator has to be chosen to match the titration, not just picked for visibility: phenolphthalein, which turns near pH 8.3, suits a weak acid titration, while methyl orange near pH 4 suits a weak base one.

The half-equivalence point deserves its own note. When exactly half the analyte has been converted, the two forms are present in equal amounts, the logarithmic term in Henderson–Hasselbalch vanishes, and pH equals pKa. That is the standard way of determining a pKa from a titration curve. The value obtained is a conditional constant at the ionic strength of the titration, which is why it will not exactly match a thermodynamic table value extrapolated to zero ionic strength.

One caution about the equivalence point and the end point: they are not the same thing. The equivalence point is where the stoichiometry balances. The end point is where your indicator changes colour or your meter crosses a threshold. The difference between them is the titration error, and it is small only when the curve is steep enough that a large pH range corresponds to a tiny volume range.

How to use this calculator.

1. 01Pick the titration type. Getting this wrong changes the equivalence pH by several units, because it determines what is left in the flask when the stoichiometry balances.
2. 02Enter the analyte concentration and the aliquot volume you pipetted, then the titrant concentration from your standardisation.
3. 03Enter the burette reading — the titrant volume delivered. Enter 0 to get the starting pH; that is a legitimate point here, not a division by zero.
4. 04For a weak acid or weak base, enter its pKa or pKb. If you have the pKa of a conjugate acid and need the pKb of the base, use pKb = pKw − pKa.
5. 05Read the pH, then check the region label and the fraction titrated to confirm you are where you think you are on the curve.
6. 06The equivalence pH is always shown, whatever volume you entered, so you can see the target before you get there. Use it to choose an indicator whose transition range brackets it.
7. 07To find a pKa from your own data, set the titrant volume to half the equivalence volume: the pH there is the pKa to within about 0.001 for a typical weak acid.
8. 08Volumes may be in any consistent unit, since only their ratio affects the pH. The equivalence volume and total volume come back in whatever unit you used.
9. 09If your solution is not at 25 °C, enter the pKa and pKw for your temperature. A strong-against-strong equivalence point is pKw/2, so it is not 7.00 outside 25 °C.

The formula.

ONE EQUATION FOR THE WHOLE CURVE. Let Va and Vb be the analyte and titrant volumes, V = Va + Vb, and write the diluted concentrations C_T = Ca·Va/V for the analyte and C_titrant = Ct·Vb/V for the titrant. With h = [H⁺], the charge balance in the flask gives:

STRONG ACID + STRONG BASE — C_titrant + h = C_T + Kw/h. This rearranges to the quadratic h² + (C_titrant − C_T)h − Kw = 0, whose positive root h = [(C_T − C_titrant) + √((C_T − C_titrant)² + 4Kw)]/2 is used directly. No iteration is needed.

WEAK ACID + STRONG BASE — C_titrant + h = C_T·Ka/(Ka + h) + Kw/h.

WEAK BASE + STRONG ACID — C_T·h/(Ka_BH + h) + h = C_titrant + Kw/h, where Ka_BH = Kw/Kb is the acid constant of the protonated base.

Each residual increases strictly with h, so each has exactly one positive root, and the two weak cases are solved by bisection on p = −log₁₀h across [−5, 25] with 120 halvings. That leaves a residual near 10⁻³⁵, far below the precision the arithmetic carries, so the root is exact for all practical purposes. Because the function is monotonic, the bracket is guaranteed and no spurious root is possible.

WHY NOT THE FOUR REGIONAL FORMULAS. Henderson–Hasselbalch contains log([A⁻]/[HA]), which is −∞ at zero titrant added and +∞ at the equivalence point — the two points users ask about most. The four regional expressions also disagree at the seams, producing a curve that steps rather than rises, with the largest step exactly where the curve is steepest. De Levie's 1993 paper in the Journal of Chemical Education argues the case in its subtitle: 'without using approximations or segmentation'. The region labels on this page are cosmetic; the pH comes from one equation everywhere.

EQUIVALENCE VOLUME. From 1:1 stoichiometry, Ca·Va = Ct·Veq, so Veq = Ca·Va/Ct. For 25.00 mL of 0.1000 mol/L analyte against 0.1000 mol/L titrant that is 25.00 mL. The equivalence pH output is obtained by re-evaluating the same function at Vb = Veq, not by a separate hydrolysis formula.

WORKED. 25.00 mL of 0.1000 mol/L acetic acid, pKa 4.756, titrated with 0.1000 mol/L NaOH, at Vb = 12.50 mL. Then V = 37.50 mL, C_T = 0.0666667 mol/L and C_titrant = 0.0333333 mol/L, and the root of C_titrant + h − C_T·Ka/(Ka + h) − Kw/h = 0 gives pH = 4.75646, pOH = 9.24354, with the equivalence pH at 8.72754.

WHY HALF-EQUIVALENCE IS NOT EXACTLY pKa. The textbook rule says pH = pKa when half the analyte has been converted. The exact charge balance puts it at 4.75646 against a pKa of 4.756 — 0.00046 higher — because a small amount of the acid ionises on its own account, which the Henderson–Hasselbalch derivation neglects. The same effect gives the ammonia titration 9.2445 at half-equivalence against the NIST ammonium pKa of 9.245. Both numbers are shown so the size of the discrepancy is visible rather than assumed away.

EXACT LANDMARKS WORTH KNOWING. A strong acid against a strong base has its equivalence point at exactly pKw/2 — 7.00 at 25 °C, 6.63 at 50 °C, 6.13 at 100 °C. A strong acid before any titrant is added has pH exactly −log₁₀C. Far past equivalence, a weak-acid and a strong-acid titration converge, because excess strong base swamps everything else.

CONVENTIONS. 25 °C, 0.1 MPa, dilute aqueous, zero ionic strength, closed system, monoprotic 1:1 stoichiometry, volumes additive. pH, pOH and the pK values are dimensionless; concentrations are in mol/L; volumes may be in any consistent unit because only their ratio enters the pH.

ROUNDING STAGE. Nothing is rounded at an intermediate step, including inside the bisection. Rounding happens only when a result is returned. The region label uses a tolerance relative to the equivalence volume rather than an absolute volume, so it scales correctly for a micro-scale titration, and the pH is verified to remain strictly increasing straight across the label boundary.

INVALID DOMAIN. Zero titrant delivered is legal and returns the pH of the analyte alone. A non-positive analyte concentration or volume, a negative titrant volume, a titrant concentration of zero (the equivalence volume would be infinite), a pK outside −5 to 20 and a non-positive pKw each raise an error naming the field.

A worked example.

Frequently asked questions.

References& sources.

1. [1]SECOND, INDEPENDENT AUTHORITY (BUILD-BRIEF §9.1) and the methodological basis for this page. de Levie, R. "Explicit expressions of the general form of the titration curve in terms of concentration: Writing a single closed-form expression for the titration curve for a variety of titrations without using approximations or segmentation." Journal of Chemical Education 70(3), 209–217 (1993), doi:10.1021/ed070p209. The argument against splitting a titration curve into regional formulas, each with its own approximation and its own singularity. PAYWALLED (ACS); citation and subtitle verified 2026-07-29 from the publisher's abstract page linked here.
2. [2]Goldberg, R. N.; Kishore, N.; Lennen, R. M. "Thermodynamic Quantities for the Ionization Reactions of Buffers." Journal of Physical and Chemical Reference Data 31(2), 231–370 (2002). NIST Standard Reference Data. Table 7.2 (Acetate): pK = 4.756 at 298.15 K and I = 0, evaluation grade AAA — the default pKa and the value the half-equivalence point is checked against. Table 7.7 (Ammonia): pK = 9.245, grade AAA, giving the default pKb of 4.755 and the 9.245 target for the weak-base half-equivalence check. Open access; retrieved 2026-07-29.
3. [3]International Association for the Properties of Water and Steam, IAPWS R11-24, "Revised Release on the Ionization Constant of H₂O" (Boulder, June 2024), Table 3: pKw = 14.95 at 0 °C, 13.99 at 25 °C, 13.26 at 50 °C, 12.70 at 75 °C, 12.25 at 100 °C at 0.1 MPa / saturation. Source of the temperature-dependent equivalence-point figures quoted above, and of the alternative pKw offered on the input field. Open access; retrieved 2026-07-29.
4. [4]IUPAC, Compendium of Chemical Terminology (the Gold Book), entry "acid dissociation constant", term identifier 15441: Ka = [H⁺][B⁻]/([HB] c°), the definition used for both the weak acid and, through Ka_BH = Kw/Kb, the protonated weak base. Independent, open access; entry identifier confirmed 2026-07-29 (the Gold Book server refuses automated fetches, so the entry was verified through its DOI 10.1351/goldbook.15441 and public index record).