Question #44000

In the spontaneous formation of a dipeptide via Nucleophilic Acyl Substitution (NAS), the reactive intermediate requires the simultaneous presence of an unprotonated amine ($-\ce{NH2}$) as the nucleophile and a protonated carboxylic acid ($-\ce{COOH}$) as the electrophile. Consider an $\alpha$-amino acid in aqueous solution at 298 K with macroscopic acid dissociation constants $pK_{a1} = 2.0$ and $pK_{a2} = 9.0$. Let $f$ be the fraction of the amino acid molecules that exist in the highly reactive, completely neutral (non-zwitterionic) state at its isoelectric point. Given that the intrinsic $pK_a$ for the ionization of the $-\ce{COOH}$ group from this neutral species is 4.0, calculate the value of $-\log_{10}(f)$. (Assume that at the isoelectric point, the total concentration of the amino acid is overwhelmingly dominated by the zwitterionic form).