What a buffer is
A buffer is a solution of a weak acid, HA, together with its conjugate base, A⁻: acetic acid with sodium acetate, or dihydrogen phosphate with hydrogen phosphate. Add a strong acid and the base takes up the protons (A⁻ + H⁺ → HA); add a strong base and the acid gives them up (HA + OH⁻ → A⁻ + H₂O). Either way the added acid or base is swapped for a small shift in the ratio of A⁻ to HA rather than a large change in free H⁺, so the pH barely moves. Blood is held at pH 7.4 this way, mainly by the bicarbonate buffer, and nearly every biochemistry experiment is run in one.
The Henderson-Hasselbalch equation
pH = pKa + log10([A⁻] ÷ [HA])
- pKa = −log10 Ka, the pH at which the acid is half dissociated.
- Equal amounts of acid and base: pH = pKa. Ten times more base: pKa + 1. Ten times more acid: pKa − 1.
- Because only the ratio appears, diluting a buffer barely changes its pH; it does weaken it.
A worked example
100 mL of acetate buffer has 5 mmol acetic acid and 5 mmol sodium acetate (100 mM total, pKa 4.76), so pH = 4.76. Add 2 mL of 1 M HCl, 2 mmol of H⁺:
- The H⁺ turns 2 mmol of acetate into acetic acid: acetate 5 − 2 = 3 mmol, acetic acid 5 + 2 = 7 mmol.
- pH = 4.76 + log10(3 ÷ 7) = 4.76 − 0.37 = 4.39.
The same 2 mmol in 100 mL of pure water gives 2 mmol in 102 mL, [H⁺] = 0.0196 M, pH = 1.71: a drop of over five units against the buffer's 0.37.
Buffer capacity
Buffer capacity, β, is how much strong acid or base, in moles per liter, it takes to change the pH by one unit:
β = 2.303 (Kw ÷ [H⁺] + [H⁺] + CT Ka [H⁺] ÷ (Ka + [H⁺])2)
The last term, from the buffer itself, peaks at pH = pKa, where it is 2.303 × CT ÷ 4 = 0.576 CT. So a buffer works best at its pKa, works reasonably within about 1 unit of it, and is proportional to its total concentration CT: 100 mM resists ten times as much as 10 mM. Once one form is used up the buffer has nothing left to give, and the pH runs away as it does in water.
Choosing a buffer
- Acetate, pKa 4.76: pH 3.8 to 5.8.
- Bicarbonate, pKa 6.35 (for CO₂ + H₂O ⇌ H⁺ + HCO₃⁻): blood and cell culture media, where dissolved CO₂ is kept in balance with the air.
- Phosphate, pKa 7.20 (second dissociation): pH 6.2 to 8.2; phosphate-buffered saline.
- Tris, pKa 8.07 at 25 °C: molecular biology, pH 7 to 9; its pKa falls about 0.03 per °C, so a Tris buffer made at room temperature is about 0.6 units more basic in the cold room.
- Ammonium/ammonia, pKa 9.25: pH 8.3 to 10.3.
Pick one whose pKa is within 1 of the pH you need, at a concentration that gives enough capacity for the acid or base the experiment will produce, and that does not interfere (phosphate precipitates calcium; Tris reacts with aldehydes).
Using the simulation
The Buffer Simulator puts a buffer and the same volume of water side by side. Each press of a button adds the same amount of 1 M HCl or NaOH to both. The liquids are colored like universal indicator, the buffer's molecules turn from blue (acid form) to yellow (base form), and red or purple dots show the free H₃O⁺ or OH⁻, one per 0.1 mM. Things to try:
- Add acid one milliliter at a time and watch the buffer's pH creep while the water's crashes; then keep going until the buffer's base form runs out and its pH drops too.
- Lower the concentration to 10 mM: the buffer gives out after a tenth of the acid.
- Start with a ratio of 100 (log ratio 2): the buffer resists acid well but base hardly at all, because it is already almost all base.
- Compare the Henderson-Hasselbalch estimate with the exact pH as the buffer nears exhaustion.
What the model assumes
- Concentrations, not activities. In real solutions ions interact, so their effective concentrations (activities) are lower, by 10 to 25% at 0.1 M ionic strength. Measured pKa values shift with ionic strength; phosphate's falls to about 6.8 in typical buffers.
- One acid-base pair per buffer. Phosphate is treated as H₂PO₄⁻/HPO₄²⁻ alone and bicarbonate as CO₂/HCO₃⁻ alone; their other dissociations (pKa 2.15 and 12.35 for phosphate, 10.3 for bicarbonate) are left out.
- A closed beaker at 25 °C, with Kw = 10−14. For bicarbonate no CO₂ escapes to the air.
- Exact proton balance. The pH is solved from CT[H⁺] ÷ ([H⁺] + Ka) + [H⁺] − Kw ÷ [H⁺] = (protons added) ÷ V, which holds at any pH, including when the buffer is exhausted.
- Additions mix instantly and add their volume; the starting volume is 100 mL and the acid and base are 1 M.
Edge cases
- Exhausted buffer: when one form is gone, Henderson-Hasselbalch gives the log of zero or a negative number and is undefined; the exact solution carries on and shows the pH following the excess strong acid or base.
- Very dilute buffers (a few mM) near pH 7: water's own H⁺ and OH⁻ are no longer negligible, and the simple equation drifts from the exact value.
- pH far from the pKa: at a ratio of 100 or 1/100 the buffer has little capacity in one direction and Henderson-Hasselbalch is less accurate.
- Pure water changes pH by about 5 units for a millimolar addition, which is why unbuffered solutions are unusable for biochemistry.
Where the model stops being right
- High ionic strength. Above about 0.1 M, activity corrections (Debye–Hückel, Davies) are needed, and concentrated buffers can be off by several tenths of a pH unit from this model.
- Temperature. pKa values change with temperature, Tris by about −0.03 per °C and phosphate much less; the model uses 25 °C values.
- Open systems. Bicarbonate in contact with air exchanges CO₂: blood keeps its pH by breathing CO₂ out, a far stronger effect than the closed beaker shows.
- Polyprotic and mixed systems. Citrate, phosphate across its full range and biological fluids with many buffers need every pKa; proteins themselves buffer through their side chains.
- Real pH meters read activity, not concentration, and need calibration; the number on the meter may differ from a calculated pH by 0.05 to 0.1 even when everything is right.
Calculators
Make up a buffer at a chosen pH with the Buffer Calculator, work out the pH of acid and base solutions with the pH Calculator, and see a full titration, where the buffer region is the flat part of the curve, in the Titration Curve Simulator. How to Calculate pH covers strong and weak acids by hand.