Concentrations of conjugate bases at specific pH levels

[Shafty]

The pKa for the dissociation of H3PO4 is 2.15. What is the concentration of H2PO4-1 (in M) at pH 3.21 if the original concentration of the phosphate was 2.37 M?

My Attempt:

Key Information:

pH final: 3.21
Initial Molarity of phosphate: 2.37 M
pKa: 2.15

I started with the first dissociation of the weak acid into its proton and conjugate base.
The stoicheometry is as follows:

H3PO4 <-----> H3O+ + H2PO4-

I am only looking at the first dissociation constant K1.

pKa = -log Ka

2.15 = -log Ka = 10^-2.15 = 7.08e-3

-----This is where I am stuck------

When the question refers to the "original concentration of the phosphate", is it referring to the conjugate base or the original acid? When I try to set up an "ICE table" plugging the given molarity into the "Initial" conjugate base column, I end up with too many unknowns and can not solve quadratically. Yet when I plug the Molarity into the [HA] column I get values that do not make sense. I understand that:

Ka= [H+][A-]/[HA]

If I gloss over this middle step I can continue to find the concentration of H2PO4- ions in the final solution. Since the first dissociation is 1:1, the concentration of H+ ions will be the same as the concentration of H2PO4- ions. Therefore:

pH = 3.21 = 10^-3.21 = 6.17e-4 M

I feel like I am missing a crucial part of this problem (obviously, since my attempts have yielded wrong answers). I have solved for the original pH of the solution at 2.37 M (0.89), I just don't know how to pull all the pieces together to solve the problem. Any help would be greatly appreciated. Thank you in advance.

 

[PhaseShifter]

The problem isn't very clear, unless they mean they initially started with $$PO_{4}^{3-}$$ in solution. If that's the case you need a few more constants. (and it seems a little early in the semester to be working with competing equilibria, so I suspect it's not that.)

The concentration of $$H^{+}$$ may not be the same as the $$H_{2}PO_{4}^{-}$$ concentration if the pH was adjusted to 3.21.

I would try "initial phosphate concentration" $$=[H_{3}PO_{4}]+[H_{2}PO_{4}^{-}]$$ and see what the quadratic equation gives.

 

[Borek]

As PhaseShifter suggests, treat 1M as a total concentration of all forms of pohosphoric acid.

No need for quadratic though. Take a look here:

acid base titration indicators

Don't pay attention to the fact it describes indicator, formula derived works for ANY acid.

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