The Graph automorphism group of the dissociation microequilibrium of polyprotic acids

Abstract

The dissociation micro-states (DMS) of an $N$-protic acid are described using set theory notation. This facilitates the mathematical description of the dissociation micro-equilibrium (DME). In particular, the DME constants are easily obtained in terms of the dissociation equilibrium constants and the molar fractions of the DMSs. Representing of the DMEs in terms of graph theory allows to identify permutations between DMSs that preserve the vertex-edge connectivity of the graph. These permutations, along with their composition, allow us to identify the direct product of the cyclic group $C_2$, and the symmetric group $S_N$, $C_2\times S_N$, as the graph automorphism group of the micro-dissociation of $N$-protic acids.

Figures (9)

• Figure 1: Dissociation graphs of some polyprotic acids. (a) Monoprotic acid; (b) Diprotic acid; (c) Triprotic acid. Red, green and blue edges represent the deprotonation/protonation of protons (1), (2), and (3), respectively, gray edges represent tautomerizations. Dashed edges are used to facilitate the visualization of $G_3$.
• Figure 2: Effect of the permutations $\sigma_{10}$ on the graph $G_1$.
• Figure 3: Effect of the permutations $\sigma_{12,0}$ and $\sigma_{21}$ on the graph $G_2$.
• Figure 4: Effect of the permutations $\sigma_{13,23}\sigma_{1,2}$, $\sigma_{12,13}\sigma_{2,3}$, and $\sigma_{S_3,0}\sigma_{12,1}\sigma_{13,2}\sigma_{23,3}$, on the graph $G_3$.
• Figure 5: Cayley graph for the 3-protic acid dissociation. Red, green and blue edges represent the action of the generators $\sigma_{\mathrm{d}}$, $\sigma_{\mathrm{d}}'$, and $c_2$, respectively.