What is the formula of the titration curve of a triprotic acid versus a strong base?

Consider a solution containing water, a triprotic acid of concentration *C _{a }*and a strong base of concentration

*C*with the following equilibria:

_{b }With reference to charge balance of the above equilibria, the sum of the number of moles of cations *H ^{+}* and

*B*must equal to that of anions

^{+}*H*,

_{2}A^{–}*HA*,

^{2-}*A*and

^{3-}*OH*. As the volume is of the solution is common to all ions,

^{–}Read this article to understand how to arrive at eq23.

With reference to eq20, eq21 and eq22, at any point of the titration, the sum of the number of moles of *H _{3}A*,

*H*,

_{2}A^{–}*HA*and

^{2- }*A*

^{3–}must equal to the number of moles of

*H*if it were undissociated. As the change in volume of the solution is common to all ions,

_{3}Awhere *V _{a}* and

*V*are the volume of triprotic acid in the solution and the volume of strong monoprotic base in the solution respectively. The equilibrium constant equations of eq20, eq21 and eq22 are:

_{b}Substitute eq24c in eq24b

Substitute eq25 in eq24a

Substitute eq26, eq25 and eq24c in eq24 and rearranging,

Substitute eq23 in eq24a and eq24b

Substitute eq7, eq27, eq28, eq29, *K _{w }*= [

*H*][

^{+}*OH*] and [

^{–}*H*] =

^{+}*10*in eq23,

^{-pH}where

Eq30 is the **complete pH titration curve** for a triprotic acid (either strong or weak) versus monoprotic strong base system. We can input it in a mathematical software to generate a curve of pH against *V _{b}*. For example, if we titrate 10 cm

^{3}of 0.200 M of

*H*(

_{3}PO_{4 }*K*= 7.11 x 10

_{a1 }^{-3},

*K*= 6.00 x 10

_{a2 }^{-8 },

*K*= 4.80 x 10

_{a3 }^{-13 }) with 0.100 M of

*NaOH*, we have the following:

The 1^{st} and 2^{nd} stoichiometric points are titrated using methyl orange and thymolphthalein as indicators respectively, while the 3^{rd} stoichiometric point is non-discernible. Another way to titrate phosphoric acid is to first react it with excess cation to completely precipitate the phosphate salt, thereby releasing all *H ^{+}*, which is then titrated with a strong base: