[kankerfist]

This question is on my study guide:
In a Na₃PO₄ solution with an initial concentration of 0.03M, what is the final concentration of [PO₄^3-] when the pH of this solution is 12?  Only considering the equilibrium:
PO₄^3- + H₂O --> HPO₄^2- + OH^- : pKb = 1.35

Solution attempt:
First I assumed that 0.03M of Na₃PO₄ completely dissolves resulting in 0.03M PO₄^3-
I had the impression that Kb = [HPO₄^2-][OH^-]/[PO₄^3-]
given pKb = 1.35, then Kb = .0447.  Also,  I had the impression that [HPO₄^2-]=[OH^-] 
So plugging in the given values results in:
.0447 = [X][X]/[.03]
then X = 0.0366.  Because [OH] = 0.0366, pH can be calculated by 14 - pOH, which finally results in a pH of 10.7.  So how can the initial 0.03M Na₃PO₄ wind up with a pH of 12?  If any of my above assumptions is incorrect, please let me know.  Any tips would be appreciated!

[kankerfist]

Maybe I should have posted this in the inorganic chem section    The study guide is for a college class, but I remember that high school chem is when I was most thoroughly taught weak acid/base topics.

[Yggdrasil]

[HPO₄^2-]=[OH^-]

This is the incorrect assumption because some OH^- exists in solution before the reaction occurs.  However, since you know that pH = 12 at equilibrium, you can calculate [OH^-].  Now you can use your Kb equation along with one other simple equation to find the equilibrium concentration of phosphate.

[Borek]

kankerfist - as Yggdrasil already wrote your assumption in general is wrong, you were not told that this solutions contains only Na₃PO₄. Could be the pH was adjusted by NaOH or H₃PO₄ added. However, I understand where you confusion comes from, question wording is far from perfect IMHO.

But - pH of the pure Na₃PO₄ solution is not 10.7. Check your math.

[kankerfist]

Thanks for the reply.  My professor speaks only a little english so the test questions are sometimes hard to understand. 

I've considered the above and found an error:

[OH-] concentration from the Na₃PO₄ solution is 0.0366 so I did the following:

pOH = -ln[OH-] / ln(10) = log₁₀(0.0366) = 1.436
pH = 14 - pOH = 14 - 1.436 = 12.56

If 0.03 M Na₃PO₄ fully dissolves, it results in a solution of 0.09 M Na⁺ and 0.03 M PO₄^3- and water.  So wouldn't the PO₄^3- ions that react each result in 1 HPO₄^2- and 1 OH^- ion?  And those that didn't would leave H₂O intact, so I don't see where HPO₄^2- could become more/less concentrated than OH^-.  If I can figure this one last thing out then I'll be all done studying for my final tomorrow