Theoretical Analysis of One-One ~om~lexometric Titration Curves Wang ~ianlin; Liu Fengling, and Feng ~ i r n i n ~ ~ Department of Chemistry, Shandong Normal University, Jinan 250014, P.R.China
Complexometric titration is one of the most important omis of titrimetric analvsi~.Titration curves indicate the Lesults of chemical reactions involved in the titrations. Students often encounter problems with this subject, such as criteria for existence of the maximum breakpoint (the point having the greatest slope on a titration curve) and location of the maximum breakpoint relative to the equivalence point. These topics are of great importance in the theory of complexometric titration. Although many excellent monographs and textbooks on complexometric titrations have been published (149, no direct answer can be found to the above-mentioned problems. As far as the authors are aware, this kind ofwork has not yet been done. This article takes one-one complexometric titration as an example of how to approach these problems. Topics such as the sharpness index of complexometric titration curves and the calculation of complexometric titration errors are also discussed briefly. Relationship between pMand Q A solution of ligand L is used as titrant, and a solution of metal ion M is the titrand. The chemical reaction involved in the one-one complexometric titration is
Eliminating the volume quantities VL and V, from eq 2 by combining eqs 2-5, we get
This can be rearranged as a quadratic in [MI.
Equation 7 can be solved to obtain [MI. The positive real root for [MI is given by
With the operatorp, eq 8 can be converted to pM = log 2 +log K + log (CL+ QCM)
where M is the metal ion; L is the ligand species; and ML is the complex species formed. (Ionic charges are omitted for generality and simplicity.) At any point of the titration curve, the titrated equivalent fraction Q can be defined by
where (Q 5 0); VL is the volume of the titrant added; V, is the initial volume of the titrand; and CL and CM are the initial molar concentrations of the titrant and the titrand. (Before titration, Q = 0, and at the equivalence point, Q =
.,
which represents one-one complexometric titration curves. Expression of pM as a function of VL can be obtained by substitution of eq 2 for Q i n eq 9. However, eq 9 is more convenient for use in the later discussion. Maximum Breakpoint
The maximum breakpoint
1.1
All side reactions and ionic strength effects are neglected, and it is assumed that the volume of the mixture of titrant and titrand is VL+ VM.When the the equilibrium of eq 1is established at any point of the titration curve, the following relations are valid.
is the point with the greatest slope on a titration curve. It is important to know its properties when one investigates the feasibility of a given complexometric titration system. A o n w n e complexometric titration system in which the maximum breakpoint exists and coincides with the equivalence point is thmretically ideal. It is not a good choice to fmd the maximum breakpoint by directly deriving
where [L], [MI, and [MLI refer to equilibrium concentrations of ligand L, metal ion M, and complex ML in the mixture; and K is the formation constant of the complex ML or the concentration equilibrium constant of the eq 1. 'Author to whom correspondence should be addressed. 2~resent address: Shandong School of Mechanical Industry, Jinan 250014, P.R.China.
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Journal of Chemical Education
and
by differentiating eq 9 with respect to Q because the mathematics are not straightfonvard. An alternative is to examine the point
1.0
@L.
s
which leads to mathematical simplicity Substituting eq 10 into eq 6, we get
[MI=
0.6
(10)
with corresponding pMq Figure 1. Comparison of pM, and Ombp (b) pMw;(c) Qmbp; (d)G; and Ow as afunctionof log ~ ? a pMmbp; )
Differentiating eq 11 with respect topM, we get eqs 12-14.
+ (I" 1 0 ) ~ c
C,= C, = O . l O O O M.
Thus. eo 18 is a snecified necessarv wndition for existence of ( d ~ l d p ~ ) , ~ ~(dpMIdQ),,. -or I n a similar way, eq + KC; - KCLCM- C L ) ~ O - *+~~ ( C +LC M ) ~ O 16~ can ~ be ~ specified as eq 19. + 1+ ~ 1 0 - ~ ~ ) ~ 1 0 - ~ ~ CM(KCL CM(l+KCd3+ 4KCM(l+ (13) + *(I+ KCL)(KCLCM +KC: + CL+ 6CM)-ZpM
Because the right-hand side of eq 12 is always positive, so is (dpM)l(dQ).Thus, pM increases with Q for Q > 0, which is a consequence of both complexation and dilution due to addition of the titrant. The necessary and sufficient condition for existence of (dQldpM),, or (dpMldQ),, is eqs 15 and 16.
I
To locate (dpMldQ),,, first check whether the point exists using eqs 17 and 19. If it really exists, calculate the value of pM corresponding to the maximum breakpoint pMmbpThen substitute the value ofpMmbpforpM in eq 11 to evaluate Q a t the maximum breakpoint, Q,b, The coincidence of the maximum breakpoint with the equivalence point happens only when Qmbp= Qes. As an example, a simple complexometric titration system with CL = CM= 0.1000 M and various values of K is considered here. For K = 1x lo1, the condition of eq 18 is not satisfied. Therefore the maximup breakpoint does not exist. For K = 1x lo2, the wndition of eq 18 can be satisfied, but no positive root of eq 17 exists. The maximum breakpoint does not exist either. When K is too small, the dilution effect bewmes dominant in the titration. So, the maximum breakpoint does not exist for those cases. and Qmbpare lotted in Figure 1for valResults ofpMmbP ues of K from 1x lo3 to 1x 10' m t h correspondingpMeg and Qeqfor comparison. In the given case, we find that the maximum breakpoint with smaller K precedes the equivalence point, and the two points become closer a s K increases.
J'.
Equations 15 and 16 can be specified in physical significance knowing eqs 13 and 14. Setting (d2Q)l(dpM?= 0 is equivalent to equating the numerator of the righthand side of eq 13 to zero. This gives eq 17, which requires eq 18 because all the other terms on the left-hand side of eq 17, determined by their physical significance in the wmplexometric titration, are positive.
Sharpness Index The steepness of a titration curve at a certain value of the equivalent fraction Q can be characterized by a wrresponding value of the sharpness index 11 (61, which is defined as the magnitude of the slope of the titration curve pM = KQ), which can also be wnsidered as the reciprocal of the differential capacity of a p M buffer. Volume 70 Number 10 October 1993
797
The sharpness index a t the equivalence point qeqcan be used to give a qualitative or rough quantitative estlmate of the titration error. When the differences are small, eq 20 can be substituted for differentials and gives
Near the equivalence point, eq 23 can be changed to eq 24 to estimate titration error.
Also when CL = CM= C, the following can be deduced from eq 9. ApM =pMep-pM,, = log (1+ Q,)
+ log ((1+ KC!)^'^ - 1)
Figure 2. Sharpness index q as a functionof titrated equivalent frac tion O; (a) log K = 3; (b) log K = 4; (c)log K= 5; CL = CM= 0.1000M.
This gives the relationship among The sharpness index represented by eq 20 can be evaluated by substituting eqs 12 and 9. As a n example, a few curves of sharpness index q vs. Q for the simple systems of CL= CM = 0.1000 M and several values of K are shown in Figure 2. As shown, the maximum sharpness index, q approaches the equivalence point, and its value increases with the value of K. Titration Error
Calculation of the titration error (TE)is one of the most important applications of equilibrium principles to titrimetric analysis. Titration error results when the endpoint of the titration differs from the equivalence point. Based on the discussion above, titration error can be expressed by TE=Q,,-Q
eq
= Qep - 1
. .
(21)
where the subscri~ten denotes a auantitv a t the end of the titration. S u b ~ t i t u t i n g p Mfor ~ pM in eq 11yields the expression for QeD.Combining it with eq 21, we get the followingequation for the titration error.
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Journal of Chemical Education
ApM, a quantity determinedhy the given endpoint detection method of a complexometric titration O n - . a quantity related to titration error TE .KC,' chamcte&ic properties of a given complexametric titration system Equation 25 is very useful. For example, from it we can easily make a computer program to construct a 4pM-TElog KC diagram. Knowing two of the values 4pM, TE,and log KC, one can find the third. Acknowledgment This paper was supported in part by a research grant of Shandong Normal University Science Foundation. The authors thank Dr. W. Th. Kok of University ofAmsterdam for his help and advice. Literature Cited 1. Ringborn,A Complpurtion inAnaiytir.l Chmisby; Wiley: NewYork, 1963. 2. Schwanenbach, G.;Flssehka, H.CamplPmmfric Tlfmtion, 5th 4 . ;Methuen: Inn-
don, 1969. 3. Inuedy, d,AndytiedAppli~fiiiofComplaEqullib~o; Ellis Howmd: Chicheater, 1976. 4 . Laitinen, H. A. ChemrmlAmly8iq MaGraw-HWNew Yark 1960:pp 220-246. 5. Kolth~ff,I. M ; Sandell, E. B.:Meeha3I.E. J.; Bruekenstein,S. QUontiLofiveche",ieo1 AnolyeG, 4th ed.; Maemillan: New York,1969; pp 126751. 6. Sueha, L.;Kotrly, St. Solution Equilibeo in Anolytid CheMatty; Van Noatrand Reinhold: Landon, 1912;p 271.
