# Complexometric titration - equivalence point calculation

At equivalence point we have just a solution of complex, and calculation of concentration of the ion is very similar to the precipitation titration case, we just have to account for the complex concentration. For that we have to know formation constant and complex stoichiometry. We could derive a formula similar to that derived in precipitation titration section, but as it was already signalled its use will be very limited. Most chelating reagents are weak acids and/or bases, so you quite often have to account for pH and speciation of ligand forms.

What is pAg at the equivalence point if the 0.01 M solution of cyanide is titrated with AgNO_{3} solution? Assume pH is so high CN^{-} hydrolysis can be neglected, ignore dilution effects. log K_{f} = 21.1.

First of all - please note, that we are asked to calculate not concentration of titrated substance, but concentration of titrant (Ag^{+}). Second, in the Liebig-Dénigès method of cyanides determination (which the question asks about), titrated solution is alkalized with ammonia, so the pH is high and assumption about lack of the hydrolysis is close to the reality.

Equation of the complexation reaction is

Ag^{+} + 2CN^{-} → Ag(CN)_{2}^{-}

and the formation constant is

At the equivalence point there are stoichiometric amounts of metal and ligans mixed so solution doesn't differ from the one prepared just by dissolving complex - and that means that

so formation constant can be written as

This can be easily solved for [Ag^{+}]. However, to calculate concentration of Ag^{+} we need one additional assumption - that Ag(CN)_{2}^{-} concentration can be treated as constant. Formation constant is relatively high, so this assumption is probably correct - but we will check it once we calculate Ag^{+} concentration.

As is obvious from the stoichiometry of the reaction, concentration of the complex is half that of initial cyanide concentration (in the real experiment it would be even lower, but we were asked to ignore dilution effects).

and finally

It is time to check if our assumption was correct. Seems it was. Concentration of silver is 2×10^{-6} that of the complex concentration, which means we can safely ignore changes of the complex dissociation due to its dissociation.

What is pZn at equivalence point if 0.005 M ZnCl_{2} solution is titrated with 0.01 M solution of EDTA at pH 10? Complex formation constant log K_{f} = 16.8, pK_{a1}=2.07, pK_{a2}=2.75, pK_{a3}=6.24, pK_{a4}=10.34.

At the equivalence point we have just a solution of complex ZnEDTA^{2-}. Complex dissociates according to the reaction

ZnEDTA^{2-} → Zn^{2+} + EDTA^{4-}

and the formation constant is

Problem is, EDTA^{4-} is a weak acid. Once the complex dissociates, ligand reacts with water, its concentration goes down and complex dissociates further. To be able to calculate concentration of Zn^{2+} we have to account for EDTA^{4-} hydrolyzis.

How to do it? Let's start taking a look at given pK_{a} values. We know that when pH=pK_{a} concentrations of acid and conjugate base are identical. When pH changes by one unit, ratio of concentrations changes tenfold. That means that we can safely ignore pK_{a1}, pK_{a2} and pK_{a3} - pH is so high concentrations of these acids will be neglectfully low. But what about pK_{a4}?

This one can be not ignored, we are just 0.34 pH unit below. Let's take a look at acid dissociation constant:

After some rearranging (very similar to that used when deriving Henderson-Hasselbalch equation) this equation takes form

which allows easy calculation of ratio of concentrations of acid and conjugate base. Note, that the same equation can be used for any acid.

From the stoichiometry and mass balance we also know that concentration of Zn^{2+} present in the solution must be identical to sum of concentrations of both EDTA^{4-} and HEDTA^{3-}:

We can easily get rid of [HEDTA^{3-}] - it can be expressed in terms of known pH, pK_{a4} and yet unknown [EDTA^{4-}]:

(notice sign change in exponent) and substituted into mass balance:

or

Combined with complex formation constant:

In general we should also account for the fact, that some amount of ZnEDTA^{2-} complex dissociated, but we won't - formation constant is high enough so that we can treat [ZnEDTA^{2-}] as constant. To be sure we will check correctness of the assumption in the final step of calculations.

Solving for [Zn^{2+}]:

The only thing we need now is the equivalence point concentration of ZnEDTA^{2-} complex. We started with 0.005 M solution of Zn^{2+} and 0.01 M solution of EDTA. As these substances react 1:1 to each V_{Zn} we had to add V_{Zn}/2 of EDTA solution (compare concentrations if you don't know why), final volume is 1.5V_{Zn} and complex concentration is 0.005V_{Zn}/1.5V_{Zn}. Combining this concentration with pH and equation derived above, we get

and

Finally, we should check if our assumption that concentration of the complex can be treated as constant is valid - and it obviously is, as concentration of dissociation products is about 8 orders of magnitude lower.