# Calculating the pH of a mixture of Na2HPO4 and Na3PO4?

I know there are more questions about this on the forum, but I was just wondering: when we mix both solutions, would we have to consider the equilibria corresponding to $\mathrm{p}K_\mathrm{a2}$ and $\mathrm{p}K_\mathrm{a3}$, or just the latter?

• Actually you need $\ce{{pKa}_3}$. – MaxW Jan 16 '17 at 17:32
• Sorry, I wrote the wrong values! I just updated the question. – Bee Jan 16 '17 at 18:33

The whole truth is that any time that you add any phosphate ion into an aqueous solution, then you will have all four phosphate species ($\ce{H3PO4}$, $\ce{H2PO4^-}$, $\ce{HPO4^{2-}}$, and $\ce{PO4^{3-}}$) in solution. The relative amounts depend on the 3 equilibrium equations, and the total concentration of all of the phosphate species.

For phosphoric acid the three pKa's are different enough so that only two phosphate species will have a "significant" concentration at whatever pH the solution is at. Here is where the answer gets fuzzy. What is "significant"? To account for 99% of the species (2 significant figures) is typically good enough and at most two species would need to be considered. However if you want to account for 99.99999999999% of the species (12 significant figures), then you're going to have to consider all four phosphate species. So the gist is how many significant figures do you need to consider in the calculations?

What is typically done is to simply the four equilibrium equations to the two "significant" ones (maybe only 1 species at high or low pH's), and then calculate the concentrations of the last two species using the found concentrations of the first two.

In reality there is another consideration. If you look at the pKa values for phosphoric acid, Wikipedia lists pKa1 = 2.148, pKa2 = 7.198, and pKa3 = 12.319. There are only three significant figures in each of these equilibrium constants. (Only the mantissa counts, not the characteristic.) So you can only have three significant figures for any given phosphate species.

So, to three significant figures, for any sort of mixture of $\ce{Na2HPO4}$ and $\ce{Na3PO4}$ salts you'll need to consider both $pKa_2$ and $pKa_3$ and you'll end up with a quadratic equation to solve. The simplifying assumption is that $$\ce{[H3PO4] + [H2PO4^-] << [HPO4^{2-}] + [PO4^{3-}]}$$

An "exact" iterative solution, considering all four species, can easily be solved via a computer program, but it is really messy to do such a calculation by hand.

As a final check you should calculate the concentrations for $\ce{H3PO4}$ and $\ce{H2PO4^-}$ and verify that the assumption holds. (For phosphoric acid if the assumption doesn't hold to three significant figures, then normalizing the calculated values would yield a "good enough" result to 3 significant figures.)

This mixture can be considered as phosphate buffer as pKa3 of phosphoric acid is very high ( a weak acid). In Henderson equation for acidic buffer we will use pKa3 because Na2HPO4 acting as weak acid and Na3PO4 as its salt with strong base.