Acids, Bases & pH

Titration curve calculator

Choose what is in the flask; the exact curve is drawn at once. Drag along it to read the pH.

Filled dots: equivalence points. Hollow dots: half-equivalence points, labelled “pH ≈ pKa” only where the two agree within 0.02. Square: where the chosen indicator changes color. Every point is the equilibrium pH of the mixture for an ideal solution at 25 °C.

How the curve is calculated

For each volume of titrant, the calculator writes down everything that must be true in the flask at equilibrium: each acid’s dissociation constants, the ion product of water, conservation of mass (everything diluted to the analyte volume plus the titrant volume), and the balance of positive and negative charge. That leaves one equation in one unknown, the hydrogen-ion concentration, which it solves numerically. The same expression, rearranged by Robert de Levie, gives the volume needed to reach any pH, and the calculator uses it to find where the indicator changes color[1][2]. There is no stitching of a “buffer region” to an “equivalence region”: one equation covers the whole curve, for polyprotic acids too.

The equivalence point is pure stoichiometry: the titrant volume that supplies exactly as many moles as the analyte has protons to give (or take), Veq = CaVa/Ct for a monoprotic acid. The half-equivalence point is half that volume. There, half of the acid has been converted to its conjugate base, so by the Henderson–Hasselbalch equation the pH is close to the pKa. The calculator labels both the exact pH and the pKa, and writes “≈” between them only when they agree within 0.02. They differ when H₃O⁺ or OH⁻ is no longer scarce next to the acid and base forms: for acids with pKa below about 3, whose own dissociation adds base form; for bases strong enough to take protons from water (pKa above about 11); and for solutions below about 1 mM, where dilution leaves water’s own ions comparable to the buffer pair.

Assumptions and valid range

  • Ideal solution. Activities are taken as concentrations, and the error is never zero. By the Davies equation, at 0.01 M the ideal curve is off by about 0.03 for acetic acid, 0.045 for HCl and 0.12 for phosphoric acid in its middle buffer region; even at 0.001 M it is off by about 0.01 for acetic acid and 0.04 for phosphoric acid. Near 0.1 M a real curve sits roughly 0.1 (acetate) to 0.3 (phosphate) pH units below the ideal one in the buffer region[3][4]. The equivalence volumes are pure stoichiometry and are not affected. Concentrations above 0.1 M are accepted but flagged.
  • 25 °C, pKw 13.994, and the pKa values of Goldberg, Kishore and Lennen[5].
  • Additive volumes, no CO₂ from the air. An open flask titrated slowly with NaOH picks up carbon dioxide, which blurs the end point of weak-acid titrations.
  • Concentrations from 0.0001 to 1 mol/L, analyte volumes from 1 to 100 mL; the curve is drawn to 1.5 times the last equivalence volume, at most 250 mL.

Worked examples

25.0 mL of 0.100 M acetic acid with 0.100 M NaOH (the default). Equivalence needs 2.50 mmol of NaOH, 25.00 mL. At the half-equivalence point, 12.50 mL, the pH is 4.756, equal to the pKa to three decimals. At equivalence the flask holds 0.0500 M sodium acetate, a weak base, and the pH is 8.72, not 7. Phenolphthalein turns pink at 24.99 mL, 0.01 mL before equivalence: well inside one drop.

25.0 mL of 0.100 M phosphoric acid with 0.100 M NaOH. Two clear equivalence points, at 25.00 and 50.00 mL (pH 4.70 and 9.66); the third, at 75.00 mL, gives no visible jump, because the third proton (pKa 12.35) is too weak to be titrated in water. The half-equivalence points sit at 12.50, 37.50 and 62.50 mL. Only the middle one lands on its pKa (pH 7.20). The first is at pH 2.28, not 2.15, because phosphoric acid is strong enough that its own dissociation shifts it; the third is at 11.86, not 12.35, because PO₄³⁻ is a strong enough base to take protons back from water, so a large share of the added hydroxide stays free in solution instead of removing the third proton. The plot writes those two as “pH 2.28, pKa 2.15” and “pH 11.86, pKa 12.35”, not with “≈”.

To watch a titration happen, with a burette, a stopcock and an indicator changing color in the flask, open the titration bench. The link under the calculator carries your analyte, concentrations and indicator there.

Sources

  1. R. de Levie (1993). 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, 209. doi:10.1021/ed070p209. primary
  2. D. Harvey (2016). Analytical Chemistry 2.1, chapter 9.2: Acid–base titrations. DePauw University (open textbook, LibreTexts edition). textbook
  3. R. P. Buck, S. Rondinini, A. K. Covington, F. G. K. Baucke, et al. (2002). Measurement of pH. Definition, standards, and procedures (IUPAC Recommendations 2002). Pure and Applied Chemistry 74, 2169–2200. doi:10.1351/pac200274112169. standard
  4. C. W. Davies (1962). Ion Association. Butterworths, London (the Davies equation for activity coefficients, ch. 3). ISBN 9780408119002. book
  5. R. N. Goldberg, N. Kishore, R. M. Lennen (2002). Thermodynamic quantities for the ionization reactions of buffers. Journal of Physical and Chemical Reference Data 31, 231–370 (NIST). doi:10.1063/1.1416902. dataset Section 8: selected pK values at 298.15 K and 0.1 MPa, used for every pKa on this site.