A simple chemical example of hierarchical thermodynamic interactions: the protonation equilibria of inorganic polyprotic acids

A general method for formulating complex thermodynamic systems in terms of hierarchical interactions has been developed, and has been applied in a pre vious analysis to hemoglobin oxygen binding data. Polyprotic acids can be c onsidered a simple chemical model of thermodynamic interaction between liga nd binding events. To further illustrate the hierarchical interaction appro ach it is applied to the analysis of the thermodynamic interactions between proton binding events in inorganic polyprotic acids. pK values for arsenat e, carbonate, chromate, phosphate, phosphite, selenite, sulfide and sulfite were recast into hierarchical interaction terms. The intrinsic K-d,(h) for protonation ranged from 8.8 x 10(-13) (M) for phosphate to 1.3 X 10(-6) (M ) for chromate. Pairwise interactions (Kd,hh) between protonation events ra nged from 1.3 x 10(4) for phosphite to 9.4 X 10(5) for carbonate. Third ord er interactions (K-d,K-hhh) were 0.91 and 0.51 for arsenate and phosphate, respectively, values relatively close to the no interaction value of 1. A p rinciple feature of systems described by hierarchical interactions is that higher order interactions, representing more complex interactions, are less likely to be significant than lower order interactions, and this is furthe r illustrated by these observations from polyprotic acids. The set of signi ficant hierarchical interaction values can be used to predict values for as yet unobserved events, and projected pK values are made for all the polypr otic acids included in this study. Finally, application of this method to t he protonation equilibria of water demonstrates a profound painwise interac tion between protonation events (K-d,K-hh = 1.3 x 10(17)), which is attribu ted to oxygen's small size and lack of polarizability. (C) 2000 Elsevier Sc ience B.V. All rights reserved.