INTERACTIVE COMPANION · EQUILIBRIUM CHEMISTRY · SPRING 2025

The Buffer Bench

A live model accompanying “What a Buffer Is Actually Resisting”

Two benches sit below. The first builds any single weak-acid buffer you like and computes its capacity exactly, with no Henderson-Hasselbalch step anywhere. The second sets the carbon dioxide system up twice at blood pH, once in a stoppered bottle and once with a lung attached, and reports the ratio.

Neither calculator approximates anything. Both solve the full charge balance by bisection, the same way analysis/buffer-capacity.py does, and at their default settings they reproduce the article's headline numbers to the digits printed there.

Model 1. Any buffer you like

Capacity is the slope of a titration curve. Add Cb moles of strong base per litre to a solution holding C moles of weak acid, write down charge balance, and you get the amount of base that lands the solution at a given hydrogen ion concentration h:

Cb(h) = Kw/h − h + C·Ka/(Ka + h)

Differentiating with respect to pH, and using dh/dpH = −ln10·h, gives the capacity:

beta(h) = ln10 · [ Kw/h + h + C·Ka·h/(Ka + h)² ]

The plot draws that. The peak is marked, the half-capacity window is shaded, and the readout reports what a real dose of strong base does to the pH at whatever point you have picked. That last number is not computed from the slope. The page adds the base to the model solution and re-solves the equilibrium for the new pH, which is why the linear estimate and the exact answer drift apart as you raise the dose.

Acid dissociation constant, pKa 7.000 Start at 7.000, where the peak sits exactly at the pKa. Drag to 5.000 and watch the peak land at pH 4.99965 instead of 5, which is the water term and not a rounding error. Push to 2.000 and the peak disappears entirely: the strong-acid wall is steeper than the buffer, so there is no maximum to find. Total buffer concentration 0.1000 mol/L Logarithmic, from 0.1 mM to 1 M. The peak height is exactly proportional to concentration, so the whole hill should scale without changing shape. Drop to 0.0001 and the water floor starts to show: about 0.8% of the capacity is then the solvent. Working pH 7.000 Move this away from the peak and watch the retained fraction fall. At one pH unit out you have 33.06% of the peak left, not half. The half-capacity edges are 0.7656 units either side, which is log of 3 plus 2 root 2. Dose of strong base 1.0 mmol/L Negative is strong acid. At 1.0 mmol/L on the defaults the exact answer and the linear estimate agree to five decimal places. At 10 mmol/L they part company, because the capacity changes over the interval you are crossing.

peak at pH: – beta at peak: – beta here: – retained: – half window: – pH after dose: –

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default peak: –
ln(10)·C/4: –
analytic half-width: –

At the defaults this bench prints a peak of 0.0575651 mol/L per pH unit against an analytic ln10·C/4 of 0.0575646, the difference being the solvent, and a half-capacity window running from pH 6.234424 to 7.765576. Both figures appear in the article's results table, and both come from the same arithmetic.

Model 2. The bottle against the lung

Same chemistry, two boundary conditions. In a stoppered bottle the total dissolved carbonate is conserved, so acid converts bicarbonate into dissolved carbon dioxide and the buffer consumes itself. With respiration switched on, dissolved carbon dioxide is pinned by the partial pressure in the lung, and the capacity becomes

beta_open = ln10 · [ Kw/h + h + K1·S/h + 4·K1·K2·S/h² ]

where S = alpha · pCO₂ is held constant. The third term is just ln10 · [HCO₃−], which is why the open capacity rises with pH instead of peaking: at fixed partial pressure, every pH unit multiplies the bicarbonate by ten.

Blood pH 7.400 7.40 is arterial. Drag down to 7.25, a real acidosis, and the ratio falls to about 14.6 because you have moved closer to the pK1' where the closed system is strongest. The open system gets worse too, but more slowly. Arterial pCO₂ 40.0 mmHg This scales both capacities together, so watch the ratio barely move while both absolute numbers swing. Between 30 and 55 mmHg the ratio changes by less than 0.01. Almost nothing else in this model is that indifferent. Apparent pK₁' 6.100 The clinical value is 6.10. This is the single most sensitive input in the whole study: drop it to 6.03 and the ratio climbs past 22. Try 6.5 to see what blood would be like if carbonic acid were a weaker acid than it is. Apparent pK₂' 9.800 Genuinely uncertain for plasma. The article carries a uniform prior from 9.80 to 10.40 through a Monte Carlo for exactly this reason. Sliding across that range moves the ratio by about 5%, which is larger than most people expect from an ion present at 0.1 mmol/L. Respiration The whole article in one control. Switching it off does not change a single equilibrium constant or a single concentration. It changes only what is held fixed while acid arrives, and the capacity falls by a factor of nineteen. Acid load 10.0 mmol/L Added to one litre and then solved exactly, with no linearisation. On the defaults, 10 mmol/L takes the open system from 7.400 to 7.176 and the closed system to 6.220. One of those is survivable.

bicarbonate: – beta open: – beta closed: – ratio: – pH after load: –

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simple prediction: –
model ratio: –
total carbonate: –

The simple prediction shown above is 1 + 10^(pH − pK₁'), which is what you get by dropping the carbonate ion and the water terms from both capacities. At the defaults it reads 20.953 against the full model's 19.498. The gap is the carbonate ion, and it is the reason the article's Monte Carlo puts a wide prior on pK₂'.

Three things worth doing

Set model 1 to pKa 7.000 and slide the working pH to 8.000. The retained fraction reads 0.3306, which is 40/121 exactly, and the article's arithmetic section derives that with a pencil.

Set the dose in model 1 to 20 mmol/L and compare the exact pH against the linear estimate. The slope is an honest guide near the peak and a liar once you have left it.

In model 2, switch respiration off and then slide pCO₂ from 40 to 80. Nothing much happens to the ratio, because both systems scale together. Switch respiration back on and slide pK₁' instead. That is where the physiology actually lives.