# How do I calculate the amount of polyprotic acid and base needed to achieve a certain buffer pH?

e.g. Citric Acid and Trisodium Citrate

$\ce{H3C6H5O7 <=>[ka_1] H2C6H5O7- + H+ <=>[ka_2] HC6H5O7^2- + H+ <=>[ka_3] C6H5O7^3- + H+}$

Using the definition of Ka and substituting for intermediates, I can get the relation:

$$\ce{[C6H5O7^3-] = (ka1\times ka2\times ka3\times [H3C6H5O7])/[H+]^3}$$

Taking the negative log of both sides gives:

$$\mathrm{-\log[}\ce{C6H5O7^3-}\mathrm{] = pka1+pka2+pka3-\log[\ce{H3C6H5O7}]-3\times pH}$$

$$\mathrm{\log(\frac{[\ce{H3C6H5O7}]}{[\ce{C6H5O7^3-}]}) = pka1+pka2+pka3-3\times pH}$$

$$\mathrm{\frac{[\ce{H3C6H5O7}]}{[\ce{C6H5O7^3-}]} = 10^{(pka1 + pka2 + pka3 - 3\times pH)}}$$

$$\mathrm{[\ce{H3C6H5O7}] = 10^{(pka1+pka2+pka3-3\times pH)} \times [\ce{C6H5O7^3-}]}$$

For Citric Acid's pkas of $3.13,~4.76,~6.4$, this equation implies that you need Trisodium Citrate at 5000x the concentration of Citric Acid to get a pH of 6, which makes no sense as that's essentially pure Sodium Citrate, which has a pH in the $8-9$ range. What am I doing wrong here?

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• HINT - Look at the stoichiometry. At pH 3.13 citric acid equals sodium citrate. At pH 4.76 sodium citrate equals disodium citrate. At pH 6.4 disodium citrate equals trisodium citrate. – MaxW Aug 23 '18 at 3:16