Potentiometric Titration of Phosphoric Acid

Acids that contain more than one acidic (ionizable) hydrogen (proton) are called polyprotic or polybasic acids. The dissociation of polyprotic acids occurs in a stepwise fashion, one proton lost at a time. For example, the generic triprotic acid will dissociate as shown in Reactions (1) through (3). The equilibrium constants for these reactions are symbolized by K_a. The trailing subscript "a" indicates that the equilibrium constant describes an acid dissociation reaction. The trailing subscript "n" is written as a number and indicates which proton is being dissociated.

H₃A + H₂0 <-----> H₃O⁺ + H₂A^-     K_a1 = [H₃O⁺] [H₂A^-] / [H₃A]     (1) H₂A^- + H₂0 <-----> H₃O⁺ + HA^2-     K_a2 = [ H₃O⁺] [HA^2-]/ [H₂A^-]     (2)
HA^2- + H₂0 <-----> H₃O⁺ + A^3-     K_a3 = [ H₃O⁺] [A^3-] / [HA^2-]     (3)

Reactions such as those shown in Reactions (4) and (5) are sometimes used as the basis of chemical calculations, but these reactions are generally not considered to be representative of what is actually occurring in solution. These reactions can be useful at times even though they are not rigorously correct chemically. The equilibrium constant for Reaction (4) is the product of K_a1 and K_a2 while the equilibrium constant for Reaction (5) is the product of K_a1, K_a2, and K_a3. Prove this to yourself.

H₃A + 2H₂0 <-----> 2H₃O⁺ + HA^2-    (4) H₃A + 3H₂0 <-----> 3H₃O⁺ + A^3-     (5)

Remember that the numerical value of an equilibrium constant is an indication of how far to the right a reaction proceeds. The greater the magnitude of the equilibrium constant, the more complete the reaction. Generally the numerical value of K_a1 is larger than that of K_a2, which is larger than K_a3 This is reasonable when one considers that successive protons are removed from species with progressively higher negative charges.
Polyprotic acids are called polybasic because they have more than one conjugate base. A conjugate base is a species that is produced when an acid loses a proton. For example, the species H₂A^-, HA^2-, and A^3- are all conjugate bases of H₃A. The completely deprotonated acid, A^3-, is a weak base. The other two conjugate bases are also weak bases, but in addition they have one or more acidic protons. As a result these species are amphiprotic or amphoteric, which means that they can act as either an acid or a base. Molecules that are partially deprotonated polyprotic acids are called acid salts. A common acid salt is sodium bicarbonate, NaHCO₃, which is found in baking soda.
When a strong base is added to a solution of a polyprotic acid, the protons of the acid are neutralized in a stepwise fashion. That is, Reaction (6) will occur first until all of the H₃A is used up. Then Reaction (7) will begin and will continue until all of the H₂A^- is gone. Then Reaction (8) will occur until all of the HA^2- is gone. This will only be true when the successive dissociation constants are different by a large enough factor and when all of the acidic species are strong enough. For example, phosphoric acid has K_a values that are different by a large enough factor to allow it to react with a strong base in a stepwise fashion, however, the value of K_a3 for phosphoric acid is so small that the last proton of phosphoric acid is extremely difficult to remove, and the reaction of phosphoric acid that is analogous to Reaction (8) essentially will not occur in aqueous solution.

H₃A + OH^- <-----> H₂A^- + H₂0     (6) H₂A^- + OH^- <-----> HA^2- + H₂0     (7)
HA^2- + OH^- <-----> A^3- + H₂0     (8)

The titration curve for the titration of a polyprotic acid such as phosphoric acid will have the form shown in Figure 1. This curve is a plot of pH (-log[H₃0⁺]) versus the volume of base added. The first point on the curve corresponds to a solution of phosphoric acid only. The pH at this point is due solely to the phosphoric acid in the solution. As soon as some base is added, some H₂PO₄^- is produced. The solution now contains both phosphoric acid (a weak acid) and its conjugate base dihydrogen phosphate, and thus it is a buffer. A buffer is a solution that is resistant to change in pH upon dilution and addition of acid and base. This will be the situation from the initial point to the first equivalence point, and therefore this region of the curve is called the first buffer zone. The pH of the point midway between the first and second equivalence points is equal to the negative logarithm of K_a1, pK_a1. At the first equivalence point, all of the H₃PO₄ has been neutralized and only H₂P0₄^- is present in the solution. When more base is added, some of the H₂P0₄^- is neutralized and some HP0₄^2- is produced. This solution contains both dihydrogen phosphate (a weak acid) and its conjugate base hydrogen phosphate, and thus is a buffer. At all points between the first and second equivalence points, the solution will contain these two species, and therefore this portion of the curve is called the second buffer zone. The pH of the solution at the point halfway between the second and third equivalence points is equal to pK_a2. At the second equivalence point the only species in the solution is HPO₄^2-. The region between the second and third equivalence points is called the third buffer zone because the solution contains both the weak acid HPO₄^2- and its conjugate base PO₄^3-. Halfway between the second and third equivalence points the pH of the solution is equal to pK_a3. At the third equivalence point, only the weak base PO₄^3- is in the solution. After the third equivalence point, the solution contains phosphate ion and excess unreacted OH^-.

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