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

A general method for formulating complex thermodynamic systems in terms of hierarchical interactions has been developed, and has been applied in a previous analysis to hemoglobin oxygen binding data. Polyprotic acids can be considered a simple chemical model of thermodynamic interaction between ligand binding events. To further illustrate the hierarchical interaction approach it is applied to the analysis of the thermodynamic interactions between proton binding events in inorganic polyprotic acids. pK values for arsenate, 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 (K(d,hh)) between protonation events ranged from 1.3 x 10(4) for phosphite to 9.4 x 10(5) for carbonate. Third order interactions (K(d,hhh)) were 0.91 and 0.51 for arsenate and phosphate, respectively, values relatively close to the no interaction value of 1. A principle 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 further illustrated by these observations from polyprotic acids. The set of significant hierarchical interaction values can be used to predict values for as yet unobserved events, and projected pK values are made for all the polyprotic acids included in this study. Finally, application of this method to the protonation equilibria of water demonstrates a profound pairwise interaction between protonation events (K(d,hh) = 1.3 x 10(17)), which is attributed to oxygen's small size and lack of polarizability.