Buffer Calculator (Henderson–Hasselbalch)
Phosphate, Tris, HEPES, MES, citrate and more: recipes and pH at your temperature.
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Result
Recipe
Forms of the buffer at this pH
The same buffer at other temperatures and diluted
Same amounts in the same volume; the pK values follow the temperature and the ionic strength follows the dilution.
How it was worked out
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About the Buffer Calculator (Henderson–Hasselbalch)
Work out a buffer the way a lab makes it. Choose the buffer — phosphate, Tris, HEPES, MES, MOPS, TES, acetate, citrate, glycine, borate, carbonate and others — the pH, the concentration, the volume and the temperature, and get the grams of each form to weigh out, or the volume of NaOH or HCl that brings one form to the pH. Or enter what is in a mixture and get its pH.
The pK values are the evaluated ones of the NIST review of buffer thermodynamics, corrected to your temperature with their enthalpies and heat capacities (Tris falls by about 0.03 pH unit per °C) and to the ionic strength of your buffer with the Debye–Hückel equation, which is why 100 mM phosphate at pH 7.4 needs far more of the basic form than the textbook pKa of 7.2 suggests. Every pH is solved exactly from the charge balance, with Henderson–Hasselbalch shown beside it, and you also get the buffer capacity, the share of each form and the pH of the same buffer at 4, 20, 25 and 37 °C and diluted tenfold.
With a Pro pass, copy the recipe or download it as a CSV, or unlock this one result. Without a pass the free preview shows the buffer pH or the base-to-acid ratio in full, with the first rows of each table and the first step of the working; every other figure is hidden.
How to use it
- Choose Mix two forms, Adjust with acid or base or pH of a mixture, then the buffer.
- Enter the target pH, the buffer concentration (both forms together), the final volume and the temperature at which you will use the buffer.
- For two forms, pick the acid form and the base form you have: each hydrate has its own molar mass. To adjust one form, pick it and the strength of your NaOH or HCl.
- Read the recipe, the pK′ at your ionic strength, the buffer capacity and how the pH moves with temperature and dilution; check the final pH with a calibrated meter.
- Copy the recipe or download it as a CSV — with a Pro pass, or after unlocking this result.
Examples
Mix two forms: NaH₂PO₄·H₂O and Na₂HPO₄·7H₂O
2.603 g NaH₂PO₄·H₂O and 21.75 g Na₂HPO₄·7H₂O (81 % of the phosphate as HPO₄²⁻); pK′ 6.77 at an ionic strength of 0.26 M
Without the ionic-strength correction (pKa 7.20) the split would be 61 % : 39 % and the buffer would read well below 7.4.
Adjust with acid: Tris base and 1 M HCl
6.057 g Tris base and 29.0 mL of 1 M HCl; the same buffer reads pH 8.64 at 4 °C and 7.68 at 37 °C
Adjust with base: HEPES (acid form) and 1 M NaOH
5.958 g HEPES and 10.8 mL of 1 M NaOH (0.43 equivalents)
Mix two forms: acetic acid and sodium acetate trihydrate
1.890 g acetic acid and 9.326 g CH₃COONa·3H₂O
pH of a mixture: 25 mM KH₂PO₄ and 25 mM Na₂HPO₄ at 25 °C
pH 6.88; NIST certifies 6.864 for this buffer at 0.025 mol/kg each
Common uses
- Preparing phosphate, Tris, HEPES or MES buffers for biochemistry, cell culture and molecular biology
- Making a buffer that holds its pH at 4 °C or 37 °C, not only on the bench
- Checking what pH a mixture of salts and acid really has before you measure it
- Chemistry coursework on Henderson–Hasselbalch, buffer capacity and activity, with the working shown
How the pH is worked out
- Henderson–Hasselbalch: pH = pK′ + log₁₀([base] ÷ [acid]); shown for every result, from the amounts you mix
- Exact: the charge balance — every form of the buffer, the sodium, potassium and chloride ions, H⁺ and OH⁻ — solved for the pH, so dilute buffers, buffers far from their pK and polyprotic ones such as phosphate and citrate are right too
- Temperature: pK(T) = pK(25 °C) − ΔH°/(R ln 10)·(1/298.15 K − 1/T) − ΔC_p°/(R ln 10)·(298.15 K/T − 1 + ln(T/298.15 K)), the equation used by the NIST review, with its enthalpy ΔH° and heat capacity ΔC_p° for each step; water’s ion product follows the IAPWS formulation
- Ionic strength: I = ½Σcz², and log γ = −(A/ln 10)·z²·√I ÷ (1 + 1.6√I), the extended Debye–Hückel equation of the NIST review, with the constant A worked out for the temperature; pK′ = pK + log γ(base) − log γ(acid), so phosphate (HPO₄²⁻/H₂PO₄⁻) falls by about 0.3 at I = 0.1 while Tris rises slightly
- Buffer capacity: β = dC/dpH, the strong base or acid (mmol/L) that moves the pH by one unit, at most about 0.58 × the buffer concentration at pH = pK′
Buffers and their pK at 25 °C (I = 0)
- Phosphate 2.15, 7.20, 12.35 · citrate 3.13, 4.76, 6.40 · acetate 4.76 · phthalate 2.95, 5.41
- MES 6.27 · Bis-tris 6.48 · imidazole 6.99 · MOPS 7.18 · TES 7.55 · HEPES 7.56 · Tris 8.07 · Tricine 8.14 · Bicine 8.33
- Ammonia 9.25 · borate 9.24 · CHES 9.39 · glycine 2.35, 9.78 · carbonate 6.35, 10.33 · CAPS 10.50
These are thermodynamic values, for zero ionic strength. Catalogue values are often given at about 0.1 M, so they read lower for phosphate, MES and HEPES and higher for Tris: the calculator applies the correction for your own buffer.
Checked against standard buffers
With the ionic-strength correction, the calculator gives pH 9.177 for 0.01 mol/kg borax (NIST certifies 9.195), 6.876 for 0.025 mol/kg KH₂PO₄ + 0.025 mol/kg Na₂HPO₄ (6.864), 7.420 for 0.008 695 + 0.030 43 mol/kg (7.416) and 4.015 for 0.05 mol/kg potassium hydrogen phthalate (4.005), all at 25 °C: within 0.02 pH unit.
Sources
- R. N. Goldberg, N. Kishore and R. M. Lennen, Thermodynamic Quantities for the Ionization Reactions of Buffers, J. Phys. Chem. Ref. Data 31, 231 (NIST; PDF): every pK, ΔH° and ΔC_p°, the temperature equation and the activity model
- N. E. Good et al., Hydrogen Ion Buffers for Biological Research, Biochemistry 5, 467: the Good’s buffers (MES, HEPES, TES, Tricine, Bicine …)
- L. J. Henderson and K. A. Hasselbalch: the equation that bears their names
- IAPWS, Revised Release on the Ionization Constant of H₂O: K_w at the temperature
- C. G. Malmberg and A. A. Maryott, Dielectric Constant of Water from 0 to 100 °C, J. Res. Natl. Bur. Stand. 56, 1: for the Debye–Hückel constant
- NIST Standard Reference Materials 186g, 187f and 185i: the certified pH values used for checking
- Further reading: R. J. Beynon and J. S. Easterby, Buffer Solutions: The Basics
Limitations
- The activity coefficients come from the extended Debye–Hückel equation of the NIST review, which is good to an ionic strength of about 0.1–0.5 mol/L; for concentrated stocks treat the pH as an estimate.
- Concentrations in mol/L are used as molalities; in dilute water solutions at 25 °C the two differ by about 0.3 %, or 0.001 in pK.
- The temperature correction is reliable near room temperature; below 5 °C or above 45 °C it is an extrapolation. The first pK of HEPES and of Bicine have no enthalpy and stay at their 25 °C values (they hardly matter at the usual pH).
- Ions that bind the buffer (calcium or magnesium with phosphate or citrate, metal ions with Tris) and carbon dioxide from the air are not modelled.
- Hygroscopic salts and hydrates of uncertain water content weigh out wrong: check the final pH with a calibrated meter.
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Frequently asked questions
What do I get without a pass?
Without a pass, Buffer Calculator (Henderson–Hasselbalch) shows the buffer pH or the base-to-acid ratio in full, with the first rows of each table and the first step of the working; every other figure is hidden. Until you unlock it, the result can’t be downloaded or copied. A Pro, Premium or Ultimate pass, a one-time payment that never renews, unlocks the full result. The pricing page lists the passes and their prices.
How do I make a phosphate buffer at pH 7.4?
For 1 L of 0.1 M sodium phosphate at 25 °C: dissolve 2.603 g NaH₂PO₄·H₂O and 21.75 g Na₂HPO₄·7H₂O in about 800 mL of water, check the pH with a calibrated meter and make up to 1 L. The ionic strength (0.26 M) lowers the pK′ to 6.77, so 81 % of the phosphate has to be HPO₄²⁻.
Why does my Tris buffer change pH in the cold room?
The pK of Tris falls by about 0.028 per °C (its ionization enthalpy is 47.45 kJ/mol). A 50 mM Tris-HCl buffer made to pH 8.0 at 25 °C reads 8.64 at 4 °C and 7.68 at 37 °C. Set the temperature here to the one you will use the buffer at.
What is the Henderson–Hasselbalch equation?
pH = pKa + log₁₀([A⁻] ÷ [HA]): the pH of a buffer from the ratio of its base and acid forms. It assumes those amounts are not changed by the buffer’s own ionization, which fails for very dilute buffers and far from the pKa, and it needs the pK′ at your ionic strength to match a pH meter. The calculator solves the full equilibrium and shows the equation’s answer beside it.
What is pK′, and why is it not the pKa in my textbook?
pK′ is the pK at your temperature and ionic strength, the one that makes Henderson–Hasselbalch match a pH meter. Ions screen each other, so in 0.1 M salt the pK′ of phosphate is about 6.8 rather than 7.20, while that of Tris goes up slightly; the size of the shift depends on the charges of the two forms.
What is buffer capacity?
The amount of strong acid or base, per litre, that changes the pH by one unit: β = dC/dpH. It is largest at pH = pK′, where it is about 0.58 times the buffer concentration (58 mM per pH unit for a 100 mM buffer), and it falls to a third of that one unit away.
Which buffer should I use?
One whose pK′ is within about one unit of your pH, that does not react with what you study: MES or Bis-tris around pH 6, MOPS, HEPES or phosphate around 7–7.5, Tris or Tricine around 8, borate or CHES around 9, CAPS around 10.5. The hint under the buffer list gives its useful range at your temperature.