The origin of multiply sigmoid curves of pH-dependence. The partitioning of groups among titration pK values.

An acid, HnA, with n ionizing groups is known to have the same titration curve as an equimolar mixture of n hypothetical monobasic acids, whose dissociation constants are known as the 'titration constants' of the real acid. We show that the pH-dependence of any property of HnA is also represented by the sum of one-site titration curves, characterized by these same titration constants. Since one such property is the degree of dissociation of one of the dissociating groups, a fraction of each group shows each of the various titration pK values, so that the group partitions among them. The n groups therefore share the same n titration pK values but differ in the fractions belonging to each. The one H+ ion per molecule that titrates with each pK is thus made up of the fractions, one from each group, that share this pK value. A group may possess a single pK value, in that it contributes virtually all of this pK and almost nothing to the others, only if either (1) in titrates in a different pH range from the other groups or (2) its affinity for H+ is unaffected by their ionization state.