Acids, Bases & pH

Buffer calculator

Henderson–Hasselbalch and the exact answer side by side, for acetate, phosphate and Tris.

Switch to “a recipe for a pH” for the amounts to weigh out. Drag along the colored scale to change the ratio or the target. The activity estimate uses the Davies equation; it is an estimate, not a measurement.

How to calculate the pH of a buffer

A buffer is a weak acid and its conjugate base in comparable amounts. The Henderson–Hasselbalch equation gives its pH from the ratio of the two as you mixed them:

pH = pKa + log10([A⁻]/[HA])(B.1)

For 0.05 M acetic acid with 0.05 M sodium acetate the ratio is 1, the logarithm is 0, and the pH is the pKa, 4.756. Double the acetate and the pH rises by log 2 = 0.30. The equation assumes that the amounts at equilibrium are the amounts you weighed out. The calculator also solves the full charge balance, which drops that assumption[1]; the two agree to 0.01 for ordinary buffers and part company when the buffer is very dilute, very lopsided, or made from an acid with a pKa below about 3 (the error map in Chapter 4 shows where).

Why a pH meter reads lower: activity

Both of those numbers are exact for an ideal solution, in which ions do not feel each other. Real ions do, and a pH meter responds to the hydrogen ion’s activity, not its concentration. The effect grows with the ionic strength I and with the square of each ion’s charge, so it is largest for phosphate, whose base form HPO₄²⁻ carries a charge of −2. The size is not small. The IUPAC primary standard buffer, 0.025 mol/kg each of KH₂PO₄ and Na₂HPO₄, measures pH 6.865 at 25 °C; Henderson–Hasselbalch and the ideal charge balance both give 7.198, 0.33 too high[2].

The calculator estimates the correction with the Davies equation[3],

log10 γ = −A z² (√I / (1 + √I) − 0.3 I),   A ≈ 0.51 at 25 °C(B.2)

applied to every ion, with the ionic strength solved self-consistently. Tested against the IUPAC standard buffers it lands within 0.015 of every one this site can describe:

IUPAC standard (25 °C)certified pHideal modelDavies estimate
0.025 KH₂PO₄ + 0.025 Na₂HPO₄6.8657.1986.876
0.008695 KH₂PO₄ + 0.03043 Na₂HPO₄7.4137.7427.420
0.01 borax9.1809.2379.176
0.025 NaHCO₃ + 0.025 Na₂CO₃10.01210.32910.003

It is an estimate. The Davies equation ignores ion pairing and the differences between ions of the same charge; it is good to a few hundredths of a pH unit up to an ionic strength of about 0.1 M and is used here up to about 0.5 M, beyond which the calculator gives no activity estimate at all[3][4]. The IUPAC values are certified on the molal scale (mol/kg); at these strengths molal and molar differ by well under 1 %.

Temperature

The pKa of a buffer moves with temperature according to its enthalpy of ionization, ΔrH°, and that enthalpy itself changes with temperature according to the heat capacity of ionization, ΔrCp°. The calculator uses both, in the integrated van ’t Hoff equation with ΔrCp° held constant, and the values tabulated by Goldberg, Kishore and Lennen[5]:

ln K(T) = ln K(T0) − ΔrH°R(1T − 1T0) + ΔrCp°R(T0T − 1 + ln TT0)(B.3)
PairpKa at 25 °CΔrH°, kJ/molΔrCp°, J/(K mol)change per °C near 25 °C
acetic acid / acetate4.756−0.41−142+0.0002
H₂PO₄⁻ / HPO₄²⁻7.1983.6−230−0.0021
Tris·H⁺ / Tris8.07247.45−59−0.028

Tris is the one to watch: a Tris buffer made up to pH 8.07 at 25 °C is near 8.71 at 4 °C and near 7.75 at 37 °C. Acetate and phosphate barely move near room temperature, but not because their pKa is fixed: both pass through a minimum (acetic acid near 25 °C, H₂PO₄⁻ near 40 °C) and rise again, by about 0.02 (acetate) and 0.1 (phosphate) at 0 °C. That curvature is the heat-capacity term; leave it out, as the simpler constant-ΔH form does, and the pKa is off by 0.047 for phosphate and 0.029 for acetate at 0 °C (0.038 and 0.023 at 50 °C). Water’s Kw follows the IAPWS formulation[6]. The calculator accepts 0–50 °C.

How good is the result? Against the IUPAC phosphate standards, which are certified from 0 to 50 °C[2], the calculator’s Davies estimate is within 0.017 of the 1:1 standard and within 0.027 of the 1:3.5 standard at every temperature in that range (the largest gaps are at 50 °C; at 25 °C they are 0.011 and 0.007). Treat its temperature-corrected values as good to two or three hundredths, then check with a meter calibrated at the working temperature.

Worked examples

Acetate, pH 4.76, 100 mL at 0.1 M (recipe mode). Henderson–Hasselbalch says equal amounts: 0.299 g acetic acid and 0.412 g sodium acetate. Mixed that way, the solution would read about 4.67 on a calibrated meter, because a meter reads activity. The calculator’s recommended recipe moves about 5 % of the acetic acid over to acetate (0.454 g sodium acetate), so that it reads 4.76.

Phosphate, pH 7.40, 100 mL at 0.1 M. Henderson–Hasselbalch puts 61 % of the phosphate in the HPO₄²⁻ form; allowing for activity, about 80 %. The difference, 1.13 g against 0.87 g of Na₂HPO₄ in 100 mL, is why phosphate recipes made from the equation alone need so much adjusting. (At this strength the ionic strength is 0.26 M, so treat the activity figure as good to a few hundredths.)

Every recipe ends the same way: dissolve, check with a calibrated pH meter, adjust with a little strong acid or base, and make up to volume. The calculator gets you close; the meter gets you there.

Sources

  1. D. Harvey (2016). Analytical Chemistry 2.1, chapter 6: Equilibrium chemistry (6.7 Solving equilibrium problems; 6.8 Buffer solutions; 6.9 Activity effects). DePauw University (open textbook, LibreTexts edition). textbook
  2. 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
  3. C. W. Davies (1962). Ion Association. Butterworths, London (the Davies equation for activity coefficients, ch. 3). ISBN 9780408119002. book
  4. R. A. Robinson, R. H. Stokes (1959). Electrolyte Solutions, 2nd ed. (revised). Butterworths, London; reprinted Dover 2002 (Debye–Hückel constant, ch. 7; activity coefficients of HCl, Appendix 8.10). ISBN 9780486422251. 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.
  6. H. Arcis and A. H. Harvey (2025). Ionization constant of water (tables for the CRC Handbook of Chemistry and Physics, 106th edition). National Institute of Standards and Technology / CRC Press. dataset Values computed from the IAPWS R11-24 formulation; we use the saturated-liquid table, 0–100 °C.