Buffer pH

A conjugate pair only buffers usefully while both members are present in comparable amounts. The usual working window is a ratio between 0.1 and 10, which is pKa -/+ 1. Outside that the buffer capacity on one side of the pH has collapsed, and the pH you read will drift hard with a drop of titrant even though the arithmetic still returns a number.

Where recipe mode's quadratic comes from. For Ca mol/L of HA plus Cs mol/L of NaA the total acetate is Ct = Ca + Cs. Charge balance gives [A-] = h + Cs - Kw/h and mass balance then gives [HA] = Ct - [A-] = Ca - h + Kw/h; notice the salt cancels out of [HA], because the sodium ions are what balance the protons the acid gives up. Substituting into Ka = h[A-]/[HA] and multiplying through by h collects into a cubic in h that carries the water term Kw. Setting only the Kw term to zero - the one approximation here - leaves the exact quadratic h^2 + (Ka + Cs) h - Ka Ca = 0. It is solved in the numerically stable rationalised form h = 2 Ka Ca / ((Ka + Cs) + sqrt((Ka +Cs)^2 + 4 Ka Ca)), never as (-b + sqrt(disc))/2, because that subtraction cancels to zero when Ca is small or comparable to Cs. Kw = 1e-14 at 25 C; the water term shifts the answer by less than 1e-4 pH over the whole useful range, which is why dropping it is safe and why it is dropped.