What a buffer does and what the equation tells you
A buffer is a solution that resists pH change. It contains meaningful amounts of both a weak acid and its conjugate base, so it has a reservoir of each: added strong acid is absorbed by the base form, added strong base is absorbed by the acid form, and the pH barely moves. Every enzyme assay, every cell culture medium, every chromatography mobile phase and your own blood plasma depends on one.
The Henderson-Hasselbalch equation gives the pH of that solution from two numbers: the pKa of the acid, which is fixed by chemistry, and the ratio of the two forms, which you control. Notice what is absent. The absolute concentrations do not appear — only their ratio does. A 1 M acetate buffer at a 1:1 ratio and a 0.01 M acetate buffer at a 1:1 ratio have the same pH.
What the absolute concentration does control is capacity: how much acid or base the buffer can swallow before the pH shifts. The 1 M buffer holds a hundred times more of both forms, so it takes a hundred times more insult to move it. Choosing a buffer therefore means two separate decisions — pick the pKa to set the pH, and pick the total concentration to set the capacity.
Because the ratio enters through a logarithm, the pH is remarkably insensitive to it. A tenfold change in ratio moves the pH by exactly one unit; a 10% error in your weighing moves it by about 0.04 units. That is the property that makes buffers practical: small mistakes in preparation produce small errors in pH.
Where the equation comes from, and what it assumes
Start with the dissociation equilibrium HA ⇌ H⁺ + A⁻ and its constant Ka = [H⁺][A⁻]/[HA]. Rearrange for [H⁺]: [H⁺] = Ka × [HA]/[A⁻]. Take the negative base-10 logarithm of both sides. Since pH = −log[H⁺] and pKa = −log Ka, and the reciprocal inside flips the sign of the log, you get pH = pKa + log([A⁻]/[HA]). No approximation has been made yet — the equation is an exact restatement of the equilibrium constant.
The approximation enters when you substitute the amounts you weighed out for the equilibrium concentrations. Strictly, [HA] is the acid remaining after some has dissociated, and [A⁻] is the salt you added plus what the acid released. Provided both amounts are large compared with the H⁺ and OH⁻ concentrations — which holds comfortably for buffers above about 1 mM in the pH 3–11 window — the correction is negligible. Outside that window, or at very low concentration, use an exact treatment such as the weak acid pH calculator.
The second approximation is that concentrations replace activities. Real ions in a salty solution are shielded by their neighbours and behave as though they were less concentrated. The consequence is that the apparent pKa shifts with ionic strength: for a monoprotic acid the shift is modest, but for phosphate — where the charge changes from −1 to −2 across the second dissociation — the apparent pKa₂ drops from about 7.20 at zero ionic strength to about 6.8 at 0.1 M ionic strength. If your protocol specifies a precise pH, prepare from the calculation and then adjust with a meter.
Third, pKa depends on temperature, and some buffers care a great deal. Tris changes by roughly −0.028 pH units per Celsius degree, so a Tris buffer adjusted to pH 8.0 on the bench at 25 °C reads near pH 8.6 in a 4 °C cold room. Phosphate and acetate are far less temperature-sensitive. Always adjust the pH at the temperature the buffer will be used at, and record which.
The buffer capacity reported above comes from differentiating the equation: β = 2.303 · C · f · (1 − f), where C is the total concentration and f is the fraction in the base form. It is maximised at f = 0.5 — that is, at pH = pKa — where it equals 0.576 × C. That single result is the reason for the rule that follows.
Worked example: 1.00 L of 0.100 M acetate buffer at pH 5.00
You need a litre of 0.100 M acetate buffer at pH 5.00, made from glacial acetic acid (M = 60.052 g/mol) and anhydrous sodium acetate (M = 82.034 g/mol). Acetic acid has pKa = 4.756 at 25 °C.
- Find the required ratio. pH − pKa = 5.00 − 4.756 = 0.244. So [A⁻]/[HA] = 10^0.244 = 1.7539.
- Convert the ratio to a fraction. The fraction in the base form is 1.7539 ÷ (1 + 1.7539) = 0.6369. The remaining 0.3631 is the acid form.
- Split the total concentration. [A⁻] = 0.100 × 0.6369 = 0.06369 M; [HA] = 0.100 × 0.3631 = 0.03631 M. They sum to 0.100 M, as required.
- Convert to moles for 1.00 L. 0.06369 mol of sodium acetate and 0.03631 mol of acetic acid.
- Convert to masses. Sodium acetate: 0.06369 × 82.034 = 5.225 g. Acetic acid: 0.03631 × 60.052 = 2.181 g, which is 2.08 mL of the glacial liquid at 1.049 g/mL.
- Check the capacity. f = 0.6369, so β = 2.303 × 0.100 × 0.6369 × 0.3631 = 0.0533 mol/(L·pH). Adding 5 mmol of strong acid to a litre would move the pH by roughly 5 ÷ 53.3 ≈ 0.09 units.
Dissolve both in about 900 mL of water, check the pH with a calibrated meter, adjust with a little concentrated acid or base if it is off, then make up to exactly 1.00 L. The final volume adjustment must come last, because adding acid or base changes the volume.
Working the other direction, suppose you already made a buffer with 0.050 M of each form. The ratio is 1.000, its logarithm is zero, and the pH is exactly the pKa, 4.756. That is the half-equivalence point of an acetic acid titration, which is precisely how pKa values are measured in the first place.
Choosing a buffer: the pKa ± 1 rule and what it costs to break it
Select a buffer whose pKa is within one pH unit of your working pH, and as close to it as you can manage. The reason is the capacity curve plotted above, which is a symmetric hump centred on the pKa.
At pH = pKa, the two forms are equal and β = 0.576 × C — its largest possible value. One unit away, the ratio is 10:1, the minor form makes up 9.1% of the total, and β falls to 2.303 × C × 0.0909 × 0.9091 = 0.190 × C, about a third of the peak. Two units away, the minor form is 0.99% of the total and β is 0.0226 × C, a twenty-fifth of the peak. Push further and the solution stops buffering in any useful sense: it has almost nothing left of one form to absorb an insult.
So the practical readings of the outputs above are these. If the reported ratio is between 0.1 and 10, you have a sound buffer. If it is outside that, either pick a different buffer system or raise the total concentration to compensate — though raising concentration has its own costs in ionic strength, enzyme inhibition and cost of reagent.
The buffer capacity figure is directly usable. It tells you the millimoles of strong acid or strong base per litre that shift the pH by one unit, so dividing your expected acid load by β gives the expected pH drift. A reaction that liberates 2 mmol of protons per litre into a buffer with β = 0.053 will move the pH by about 0.038 units — negligible. The same reaction in a buffer with β = 0.002 moves it a full pH unit.
One more consideration that the equation cannot show you: chemical compatibility. Phosphate precipitates calcium, magnesium and many transition metals, and it inhibits several enzymes. Tris has a free amine that reacts with aldehydes and interferes with protein assays. Carbonate buffers lose CO₂ to the air and drift upward. Choose on pKa first, then screen for interference.
pKa values of common laboratory buffers at 25 °C
| Buffer system | pKa | Useful pH range | Notes |
|---|---|---|---|
| Phosphoric acid / H₂PO₄⁻ | 2.15 | 1.1–3.1 | pKa₁ |
| Citric acid (first) | 3.13 | 2.1–4.1 | Triprotic; chelates metals |
| Formic acid / formate | 3.75 | 2.8–4.8 | Volatile, MS-compatible |
| Acetic acid / acetate | 4.76 | 3.8–5.8 | Volatile, cheap |
| MES | 6.15 | 5.2–7.2 | Zwitterionic, low metal binding |
| Carbonic acid / bicarbonate | 6.35 | 5.4–7.4 | Loses CO₂ to air |
| PIPES | 6.80 | 5.8–7.8 | Zwitterionic |
| H₂PO₄⁻ / HPO₄²⁻ | 7.20 | 6.2–8.2 | pKa₂; precipitates Ca²⁺ and Mg²⁺ |
| HEPES | 7.5 | 6.5–8.5 | Standard for cell culture |
| Tris | 8.06 | 7.1–9.1 | Strongly temperature-dependent |
| Boric acid / borate | 9.24 | 8.2–10.2 | Complexes cis-diols |
| Ammonium / ammonia | 9.25 | 8.3–10.3 | Volatile |
| Glycine (amine) | 9.78 | 8.8–10.8 | Second pKa; first is 2.35 |
| HCO₃⁻ / CO₃²⁻ | 10.33 | 9.3–11.3 | pKa₂ of carbonic acid |
HEPES and other Good's buffers are usually quoted between 7.4 and 7.6 depending on temperature and source; use the value supplied with your reagent for precise work.
Mistakes that produce a buffer at the wrong pH
- Using a pKa from a different temperature. Tris shifts by about −0.028 pH units per Celsius degree, so a buffer adjusted at 25 °C and used at 4 °C is roughly 0.6 units higher than intended.
- Ignoring waters of hydration in the molar mass. Sodium acetate trihydrate is 136.08 g/mol against 82.03 for the anhydrous salt. Weighing the trihydrate against the anhydrous number gives 40% less buffer than you planned.
- Picking the wrong pKa of a polyprotic acid. Phosphate has three; the one that governs a pH 7 buffer is pKa₂ = 7.20, not pKa₁ = 2.15. Match the pKa to the dissociation step that is active at your pH.
- Adjusting the volume before adjusting the pH. Titrating to the final pH adds liquid. Make up to a little under the target volume, adjust the pH, then top up.
- Assuming concentration sets the pH. Diluting a buffer twofold leaves the pH nearly unchanged, because both concentrations fall together and only the ratio matters. What halves is the capacity.
- Trusting the equation at high ionic strength. Above roughly 0.1 M supporting electrolyte the activity corrections are real, and they are largest for the multiply-charged phosphate and citrate systems. Verify with a calibrated meter.
- Buffering outside the pKa ± 1 window. At two units away, 99% of the buffer sits in one form and the capacity is about 4% of its peak — an expensive way to make a solution that does not buffer.
Related calculations and when to use something else
The Henderson-Hasselbalch equation applies only when both members of a conjugate pair are present in substantial amounts. For a solution of a weak acid alone, use the weak acid pH calculator, which solves the full quadratic; for a weak base alone, the weak base pH calculator. For a strong acid or base, the concentration itself gives the pH directly through the pH calculator.
The buffer region is also the middle of a titration curve, and the pKa is read off exactly at the half-equivalence volume — where half the acid has been converted to its conjugate base, so the ratio is 1. The acid-base titration calculator handles the equivalence-point stoichiometry and shows where that half-way point falls.
Two related equilibrium tools sit alongside. The equilibrium constant calculator handles the general K expression that Ka is a special case of, and the solubility product calculator covers the case where a buffer component precipitates — which is exactly the failure mode of a phosphate buffer in the presence of calcium.
Finally, remember the biological reason this equation is taught in every physiology course. Blood plasma is buffered by the bicarbonate system at pKa 6.1, well outside the pKa ± 1 rule for plasma pH 7.4. It works anyway because it is an open system: the lungs continuously remove CO₂, so the acid form is regenerated or discarded on demand and the effective capacity is far higher than a closed-system calculation predicts. That is a good reminder that this equation describes a sealed beaker, and a real system with mass transfer can behave better than it says.