The isoelectric point, pI, of an amino acid is the pH at which the predominant species in solution is the zwitterionic form of the amino acid and only small but equal concentrations of positively and negatively charged forms of the amino acid are present. It follows that at the isoelectric point, the average charge on the amino acid is zero. Show that (a) pI = 1/2(pKa1 + pKa2) for amino acids with side chains that are neither acidic nor basic (such as glycine and alanine), (b) pI = 1/2(pKa1 + pKa2) for amino acids with acidic side chains (such as aspartic acid and glutamic acid), and (c) pI = 1/2(pKa2 + pKa3) for amino acids with basic side chains (such as lysine and histidine), where pKa1, pKa2, and pKa3 are given in Table 4.6. Hint: See Case study 4.4 and Section 4.10.