Calculating the pH of a buffer made by a diprotic acid and its double salt

The concentration of the diprotic acid (tartaric acid) will be constant at $$\pu{ 0.1 M}$$. I want to find the concentration of its double salt (potassium sodium tartrate) needed to create a buffer of $$\pu{ pH }$$=x (for example,x= 4.0). How would I do this? Do I have to use the Henderson-Hasselbalch equation? Make certain assumptions?

I tried to set up two equilibrium expressions with $$K_\mathrm{a1}$$ and $$K_\mathrm{a2}$$ as follows:

$$K_\mathrm{a1} =\frac{[\ce{C4H5O6-}][\ce{H+}]}{[\ce{C4H6O6}]}$$ and

$$K_\mathrm{a2} =\frac{[\ce{C4H4O6^{2-}}][\ce{H+}]}{[\ce{C4H5O6-}]}$$

Can I multiply $$K_\mathrm{a1}$$ and $$K_\mathrm{a1}$$ to eliminate$$[\ce{C4H5O6-}]$$, and then get the concentration of $$\ce{C4H4O6^{2-}}$$ necessary by plugging in $$\pu{0.1M}$$ for$$[\ce{C4H6O6}]$$ and the target $$\pu{pH}$$ in the appropriate form in $$[\ce{H+}$$]? I feel like I'm making some big assumptions here, though.

(Also, please forgive me; I don't know how to use LaTeX or Tex.)

If I try to make the assumption that the second dissociation does not occur as it is too weak (which it isn't) I wouldn't have the concentration of $$\ce{C4H4O6^{2-}}$$ in my equation, which is what I'm trying to find.

Can I multiply Ka1 and Ka1 to eliminate [$$\ce{C4H5O6−}$$], and then get the concentration of C4H4O62− necessary by plugging in 0.1 M for [C4H6O6] and the target pH in the appropriate form in [H+]

No, because $$\ce{C4H5O6−}$$ is a one of the major species. In fact, if you add the tartaric acid and its double salt at equimolar ratios, $$\ce{C4H5O6−}$$ will be the only major species. As a first approximation, you can let this reaction go to completion. If you have an excess of the acid, there will be a mixture of $$\ce{C4H5O6−}$$ and acid as major species. If you have the double salt in excess, there will be a mixture of the double salt and $$\ce{C4H5O6−}$$ as major species.

Do I have to use the Henderson-Hasselbalch equation? Make certain assumptions?

If your ratio of reactants and products is such that you have a conjugate acid: base pair as the major species, you can use Henderson-Hasselbalch as the next step to estimate the pH. (If your major species are $$\ce{C4H6O6}$$ and $$\ce{C4H5O6−}$$, use $$K_{a1}$$, if they are $$\ce{C4H5O6-}$$ and $$\ce{C4H4O6^2−}$$ use $$K_{a2}$$.) In some cases, that's a good approximation, in others, not. You can test that by calculating the concentration of all species (based on the approximate pH) and then calculate Q for the two acid base equilibria to see how well they match K.

I would take different approach.

1. From given $$pH$$ and acidity constants, calculate ratios of the particular forms of the tartaric acid.
2. From given total molar amount, you get the molar amounts for the particular forms.
3. From molar amount of particular forms, you can get molar amount of a hydroxide to be added to the tartaric acid, or of a strong acid to be added to the tartrate.
4. Important thing is to know this is only approximation, as acidity constants are meant for activities and we assume activity coefficients are equal to 1. You may want to search for, how to calculate activity coefficients for ions.
5. If the final goal is practical, better is to take a tabelized compositions of verified buffer solutions, or to adjust the final $$pH$$ with help of calibrated $$pH$$ meter.