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How do we derive the formula of isoelectric point?

pI=(pKa1 + pKa2)/2

I read that isoelectric point is defined as the pH at which degree of protonation in amino group is same as degree of deprotonation at carboxylic group, but I still could not derive the relation.

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The isoelectric point is the pH at which the amino acid is in its zwitterionic form (neutral, typically with a deprotonated carboxylic group and a protonated amino group). The titration of an amino acid is similar as the titration of a diprotic weak acid, with 2 buffer regions at pH=3 and pH=9 (which correspond to the pKa of the alpha carboxylic and amino groups)

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The amino acid is neutral when the carboxylic group is fully deprotonated, and the amino group is still fully protonated. This occurs at the first equivalence. The pH at the first equivalence is given by pH = pI = (pKa1 + pKa2)/2

I'm pretty sure there are more complicated ways and more accurate formulas to find the pH at the first equivalence by considering the 2 equilibria of protonation and deprotonation of the zwitterion.

I hope this helps

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