Relations between protonation constants and titration curves in polyprotic acids: A critical view

Three thermodynamically meaningful pK(a) values can be defined for polyprotic acids: macroscopic, microscopic, and quasisite pK(a) values. In this paper, the relation between these pK(a) 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 pK(1/2) values, are used to identify the pK(a) values of polyprotic acids or of a proton binding site within the polyprotic acids. However, both are generally not identical with the pK(a) values of a polyprotic acid. The different thermodynamic definitions of pK(a) values are compared to commonly used ways of obtaining pK(a) values from titration curves. The inflection points and pK(1/2) values are a first good guess for further fitting. However, only fitting titration curves to proper thermodynamic expressions lead to the respective pK(a) values that are associated with the reaction free energy. A polyprotic acid with N titratable groups has 2(N) microstates and thus 2(N) - 1 independent microscopic constants. However, only N-2 - N + 1 parameters can be extracted from the titration curves of all individual sites. Because 2(N) - 1 is greater than N-2 - 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 less than or equal to 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.