How Can You Prove the Inflection Point on a Titration Graph with Two pKa Values?

[apchemstudent]

I know this is a chem question in general, but the steps needed to solve this are more mathematical.

This is not a homework question, but my prof came up with the concept of determining the inflection point on a titration graph with 2 pKas. He simply told us that it was the average of the 2 pKa values. However, he also just mentioned that it can also be proven through calculations, but he never showed us how.

I'm just curious as to how this can be proven, and remember, this is not a homework question. Can some one show this to me? Thanks.

The example we were talking about in class was glycine :P, again. so the COOH group's pKa value was 2.4, and the NH3(+) group's pKa value was 9.6.

I just don't know how to prove it...

here's the henderson hasselbach equation that we use

pH = pKa + log(conjugate base/conjugate acid) to determine the pH at different conjugate base to acid ratios.

the dissociation of COOH(conjugate acid) <-> COO(-) (conjugate base) + H(+)
dissociation of NH3(+)(conjugate acid) <-> NH2(conjugate base) + H(+)

I hope this helps as a background. Please help, thanks.

 

[shaner-baner]

you are talking about inflection points, so I assume you know calculus. If you don't I'm not sure you'll understand the answer. First, you need a function that describes the titration curve, then you find possible points of inflections by setting the second derivative equal to 0. Finally you check to see if what you found were really p.i.'s. I probably don't have enough background to come up with the specific equation.