Feasibility of Complexometric Back-Titrations Carlo M a d and Mario Fioranl Institute of Analytical Chemistry, Universita di Padova, 35131 Padova, Italy The use of back-titrations in complexometric quantitative analysis, and the interpretation on their results, seem to have long suffered from a ~ o ~ u lmisconcention. ar In fact. it is i r e q u e h y claimed thatihe cation, which'is used to backtitrate the chrlatina acent (e.a., EDTA) added in excess to the sample, must g&ea weaker complex than that given by the analyte cation, thus preventing the titration of EDTA hound by the latter (see, e.g., refs 1-6). However, this claim leaves out the fact that, in principle, any reaction that is followed by a second step or by a second reaction can give precise titrations, provided it goes to completion before the second step or reaction does start. Recent studies on the theoretical shape of potentiometric back-titrations, and on the factors on which the accuracy and the precision of back-titrations with visual (chromatic) endpoint depend (7-91, have clearly shown that, when the back-titrant cation N gives, with the chelating agent Y, a stronger complex than the analyte cation M, the back-titration gives a sharper endpoint than in the opposite case. A thorough examination of the properties of back-titrations requires either an elaborate mathematical treatment (7-9) or appropriate logarithmic diagrams (9). However, approximate equations for the equilibrium concentrations of free back-titrant, [N], can be easily obtained for hack-titrations with 1:l reaction ratios (as i t is normally the case) in the most "unfavorable" (according to usual claims) conditions, that is, when the conditional stability constant of NY, KN,is much higher than that of MY, KM.Of course, this last complex must be initially stable. In obtaining the equations for [N], the dilution effect is neglected for convenience; then the total concentrations of reactants, CM and Cy, can be assumed to he equal to their initial values, and CN to he proportional to the added titrant volume and to the titration ratio. Such equations adequately represent large parts of the titration curve, and correctly describe their main features. Only restricted sections around the equivalence points require more complex calculations (7,9). In the first part of the titration, the proper back-titration (for CN < CY- CM), N+Y-NY
(1)
the titraut cation N reacts only with free Y, and the analyte cation remains completely complexed. Then the mass balances of M and of N can be written in approximate forms
+ [MY] -- [MY]
(2)
C, = [Nl + [NY] = [NY]
(3)
CM= [MI
As a consequence, the mass balance of the chelating agent becomes C, = [Y] + [MY] + [NY] [Y] + CM CN (4)
--
+
Substituting the simplified eqs 3 and 4 for [NY] and [Y] in the expression of KN,one obtains
~ a t half titration of In particular, when CN= (Cy - C M ) /(i.e., the Y excess) the result becomes
After the equivalence point of the back-titration, for Cy CM > KM,N completely displaces M from its complex N+MY-NY+M
(7)
The simplified eq 3 is still valid. Free Y is negligible in the mass balance of the chelating agent Hence the mass balance of M, in which [MI can no more he neglected, becomes CM-- [MI +C,-CN
(9)
From the ratio between the stability constants of MY and NY,
and the simplified mass balance eqs 3,8, and 9 one obtains
(i.e., [MI = [MY], M half displaced), When CN= Cy - 0 . 5 C ~ the result is Finally, after M has been completely displaced from its chelate (for CN> Cy), the concentration of free N is equal to the excess concentration of N over Y The increase of log[N] between the first and the second part of the titration is, on subtraction of eq 6 from eq 12, Alog[N] = logKM+ Log(Cy - 0.5CM)
(14)
This quantity can he assumed as a measure of the "jump" of loelNl around the eauivalence noint of the back-titration (sbk i & u r e ~~. q u a t i n n14 shows ;hat this jump depmds essrntinlly un the value of the stahilitv constant of the analvte metal chelate and is quite independent from the stability constant of the titrant metal complex (when this last is higher). Provided that KMand the analytical concentrations of Y and of M are not too low, the jump is high enoughfor the titration to be feasible, no matter how high KN is. The figure shows a complete titration curve for NY 10,000 times more stable than MY. For comparison, the figure also shows the curves for the titration either by N or by M of Y alone (in an amount equal toits excess in the back-titration). The log[N] jump at the equivalence point of the back-titration is clearly comparable with the log[M] jump a t the equivalence point of the titration of Y by M (which corresponds in height to that of the direct titration of M by Y). The total jump (reactions 1 7) is equal to that of the titration of Y alone by N. Among indicators giving a sufficiently accurate endpoint in the direct titration of Y alone by N (i.e., of N by Y), only those changing color a t the lower (more negative) log[N] values are suitable for back-titrations with N . In the case represented in the figure, an indicator changing color exactly
+
Volume 63 Number 2
February 1966
121
from KM.WhenKNis lower t h a n K ~the . titration is feasible only when KN is not too low; then any indicator suitahle for the direct titration of N can he used for the hack-titration. A comment is also needed on kinetic effects in hack-titrations. When hack-titrations with KN > KMhave been found to he feasible in practice, their feasibility has been ascribed to a verv slow dis~lacementreaction (7). . . According to the foregoing deductions, this condition is not necessary: however, it can o
