Relations between Protonation Constants and Titration Curves in Polyprotic Acids:  A Critical View

Three thermodynamically meaningful pKa values can be defined for polyprotic acids: macroscopic, microscopic, and quasisite pKa values. In this paper, the relation between these pKa values and their relation to titration curves is discussed. Often inflection points of total and individual titration curves or the pH value where the proton binding site is half protonated, so-called pK1/2 values, are used to identify the pKa values of polyprotic acids or of a proton binding site within the polyprotic acids. However, both are generally not identical with the pKa values of a polyprotic acid. The different thermodynamic definitions of pKa values are compared to commonly used ways of obtaining pKa values from titration curves. The inflection points and pK1/2 values are a first good guess for further fitting. However, only fitting titration curves to proper thermodynamic expressions lead to the respective pKa values that are associated with the reaction free energy. A polyprotic acid with N titratable groups has 2N microstates and thus 2N − 1 independent microscopic constants. However, only N2 − N + 1 parameters can be extracted from the titration curves of all individual sites. Because 2N − 1 is greater than N2 − N + 1 for N > 3, it follows that it is impossible to obtain all microscopic constants from the titration curves of all individual sites for polyprotic acids with more than three nonidentical proton binding sites. For N ≤ 3, it is explained how to obtain the microscopic constants from the titration curves of all individual sites using the decoupled sites representation. The method is applied to determine the microscopic constants of DTPA, which has highly irregular titration curves. From the microscopic constants, the state populations are calculated and the reason for the unusually shaped titration curve is explained.