A ~ r a ~ h i c a ~ - ~ n d yIterative t i c d Procedure
N e a l G. Sellers and Joseph A. Caruso University of Cincinnoti Cincinnati, Ohio 45221
for Equivalence Point Determinations
Potentiometry is used extensively in monitoring the change in' activity of a species in various titrations ( I ) . Accurate endpoint detection usually requires an abrupt change in the activity of the species of interest at the equivalence point. Several techniques such as derivative versus titrant volume have been devised for the determination of potentiometric endpoints for systems which do not obey this criterion (2). In these cases, accurate endpoint detection requires an inflection point in near proximity to the equivalence point. Graphical differentiation is subject to error and the equivalence point does not coincide with the inflection point if the titration curve is unsymmetrical (3, 4). An iterative procedure has been developed which combines graphical measurements with analytical calculations to give an exact potentiometric endpoint irrespective of the stoichiometry of the reaction and magnitude of the associated equilihrium constants. This procedure retains many of the advantages of Gran's method (5) and requires no extrapolation of experimental data. Refinements such as activity corrections and electrode response error can he incorporated in the calculations if desired. The principles of this procedure are based on the smaller slopes of the titration curve in the pre-equivalence and post-equivalence regions relative to the slope in the vicinity of the equivalence point and the small sensitivity of pH or potential response a t the equivalence point to changes in sample concentration. The use of this procedure requires the assumption that the Nernst equation is applicable to the system and that the junction potentials remain essentially constant throughout the titration. For a typical case an initial estimate of the equivalence point is taken from a point in the equivalence point region. This value then is utilized to calculate ER (which contains reference electrode and junction potentials) and the value of the appropriate equilihrium constant from points in the pre-equivalence point regions and post-equivalence point regions. These values then are used together with the original estimate of the endpoint volume for calculating a new equivalence point potential. The endpoint volume corresponding to this potential is read from the titration curve. This new endpoint volume is used to calculate a new equilibrium constant, K, and equivalence point potential. This cycle is repeated until convergence occurs. In most applications after three cycles the difference between the equivalence point and the endpoint is less than 5 ppt.
Titration systems of a weak acid, precipitate formation, complex formation, and redox couple were selected as examples to illustrate the simplicity and accuracy of this iterative technique. These titration curves were theoretically calculated by using an IBM 360165 computer system with suitable FORTRAN programs. Activity and volume corrections were incorporated within the calculations. Each titration curve was specified by the reaction, equilibrium constant, millimoles of sample, initial volume, normality of titrant, and if necessary, an ER. These specifications are summarized in Table 1. By applying theoretical titration curves, the equivalence point may be predicted exactly and thus the comparison with the calculated endpoint readily can be made. A theoretical titration curve for boric acid titrated with sodium hydroxide is given in Figure 1. A very shallow break is observed in the vicinity of the equivalence point. An initial endpoint volume is estimated to be 18.0 ml. The half-neutralization volume, 9.0 ml, is used to determine a pK, of 9.1 from the titration curve. Using this value for the pK,, an estimated 1.8 meq of acid, and a total volume of 418 ml, the pH a t the equivalence point, 10.37, is given by eqn. (1). pH = 112 [log (MeqHA)
- IogK,
- log
K, - log V,]
where Vt = total volume, and K , = 1.0 x
0
5
35
10
20
25
(1)
lo-".
30
The
35
ML TITRANT Figure 1. Weak acid-base titration.
Table 1..Selected Parameters for PotentiometricTitration Curves
Titration System
Reaction
Acid-base
HAeH++A-
Precipitation
2Ag+ + CrO?
Complex Redox
M+ 2R,
e AglCrOd
Equilibrium Constant
Sample (Meq)
Initial Volume (ml)
Titrant (Normality)
Ea (V)
KO 6.4 X 10-lo Ksn 1.8 x 10-l2
2.0
400
0.10
0.500
2.0
400
0.10
0.500
2.0
400
O.lOD
0.500
2.0
500
0.10
0.500
+ 2L r!ML2+
Kf
+ Ox2s 20x1 + R2
Koxld
1.0 X lo8
1.0 x lo4
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547
endpoint volume corresponding to this pH is 19.9 ml. A second cycle using this endpoint volume gives a final endpoint volume of 20.0 ml. This value coincides with the theoretical equivalence point. Activity corrections are unnecessary in this example because the values of the activity coefficient remain nearly constant from the half-neutralization point to the equivalence point. The values of the endpoint volume,, K., and equivalence point pH for each iterative step are given in Table 1. The titration of Ag+ with Cr042- tO precipitate AgzCrO4 is an example of a 2:l unsymmetrical precipitation titration. The test data are given in Table 1. The solubility product for silver chromate is given by eqn. (2).
K,,
=
IAg+12[Cr0,2-lf12f2
(2)
where fl = the activity coefficient for a monovalent ion, and f2 = the activity coefficient for a divalent ion. Figure 2 is the precipitation titration curve for the titration of
Figure 3. Cornplexomebic titration
equivalence point potential of 280 mv is 19.70 ml. As seen in Table 1, an additional cycle would produce no change in the calculated values. In this example the activity coefficients in the next iterative cycle results in an accurate endpoint. The value of the solubility product differs slightly from the theoretical value because the solubility of the precipitate is not negligible with respect to the excess chromate a t the 34.0 ml point. An example of a complexometric titration curve is shown in Figure 3. The appropriate reaction and parameters are listed in Table 1. A univalent metal, M + , is assumed to complex with two neutral ligands, L. The formation constant is defined by eqn. (6).
K,
=
[ML,+l -WILL]"
An estimate of the equivalence point is taken as 22.0 ml or 1.10 mmoles of metal ion. A value of Ek is determined from the initial potential reading and eqn. (7). Figure 2. Precipitation titration
silver ion with chromate ion. The concentration of the silver ion is monitored during the titration. From a point in the equivalence point region the endpoint volume is estimated to be 18.00 ml. The value of Ek is calculated using eqn. (3) and the initial point of the titration curve.
E,
=
E
- 0.0591 log
[M'lf,
(7)
The activity coefficient is assumed to be unity. Using the calculated value of Ek, 494 mv, the estimate of the quantity of M+, 1.1 mmoles, and a point in the post-equivalence point region, 35.0 ml; pKf is calculated from eqn. (8). E - E, - log K, = - zlop [Ll - log [MLJ (8) 0.0591
+
Using the estimated concentration of Ag+, (18.0 X 0.11 400), and an activity coefficient of unity, Ek is found to be 0.501 V. Equation (4), relating the solubility product, chromate ion concentration and potential, is derived by solving eqn. (2) for [Ag+]fi and then substituting this expression into eqn. (3). With a known Ek, a point in the post-equivalence region is selected in order to estimate pK,, from eqn. (4). 2.0 (El - E ) - log f , - log [Cr042-l (4) - log K,, = 0.0591
IF 34.0 ml is selected and the solubility of the precipitate and activity correction are neglected, the concentration of the chromate ion is (34.0 - 18.0) x 0.1/(2 X 434). The value of pKb, is 11.50. At theequivalence point, the silver ion concentration is equal to twice the chromate ion concentration. From this relationship, eqn. (2) and eqn. (3), eqn. (5) is derived which relates the equivalence point potential to the solubility product. From a calculated Ek and a calculated pK,,, the equivalence point potential is determined by eqn. (5). 0.0591 E=Ea+(5) 3 (log K., + log 2.0 + log f d f J The endpoint volume corresponding to the calculated 548
/ Journal of Chemical Education
The dissociation of the complex is ignored when calculating the concentration of the ligand, (35.00 - 22.00) x (0.1/435), and the concentration of the complex, 22.00 x 0.1/(2.0 X 435). After determining pKf and Ek, an equivalence point potential is calculated from eqn. (9).
The cycle is repeated until no further changes occur in the calculated values. The inclusion of activity coefficients in the calculations gives an accurate endpoint as seen in Table 2. The small deviations in the other values in Table 2 from the theoretical values are due to rounding errors and to the assumption that the complex formation is complete. Theoretical values can be obtained by including the incompleteness of the reaction in the calculations. The sharpness exhibited by the break of a redox titration curve is dependent upon the equilibrium constant of the redox couple and the concentrations of the reagents. A redox couple which has a moderate equilibrium constant does not give a sharp break in the vicinity of the equivalence point. This type of system is shown in Figure 4. Sample Rll+ is oxidized to OxI2+ while the titrant OX^+ is reduced to R2+. The reaction, equilibrium constant,
Table 2. Calculated and Theoretical Values for Potentiometric Titrations
Acid-Base
1 2 3
Reeipitation
Complex
Activity c o n 1 Theoretical 1 2 Activity Con l Activity c o n 2 Theoretical 1 2
M . 4 8 7 M Y at 90
t
10
4 Activity c o n 1 Activity Con 2 Theomtied Redm
1
2 Activity c o n 1 Activity c a n 2 Activity c o n 3 Theoretical
'Volts.
Eel, E R ~and , concentrations are given in Table 1. The equivalence point is initially estimated to be 18.0 ml. The , determined from points in the values, Ek; and E R ~are pre-equivalence and post-equivalence point regions and from eqns. (10) and (11). [OxJ.f2 EI, = E - 0.0591 logTR,1 f ,
where f;, f ~ and , f3 represent activity coefficients for monovalent, divalent, and trivalent ions, respectively. As a first approximation, fi, fi, and f3 are assumed equal to unity and the reaction is assumed to be complete. Then Ee, and Eh, are equal to the potential at the half-neutralization volume and twice the neutralization volume, respectively. The value of the equivalence point potential is given by eqn. (12).
20
15
25
ML TITRANT
3
Figure 4. Oxidation-reduction titration.
tial as calculated from the volumes of titrant, 9.0 ml, 36.0 ml and 18.0 ml, respectively, are given in Table 2. Although it is not necessary in order to determine the equivalence point, pK,, can be determined from eqn. (13).
- log K
=
(Ek, - 2Eb)
(13)
X 2.010.0591
The new equivalence point volume is read from the titration curve a t a potential of 562 mv. The iterative cycle then is continued-until no further change occurs. As i n t h e examples above, activity corrections provide an endpoint which coincides exactly with the equivalence point. Rapid and accurate convergence occurred in each of the examples of potentiometric titrations. The initial estimates of the equivalence point purposely were chosen to be in error by 10%. The iterative cycles first were made without activity corrections in order to show the effect on the final endpoint. In practice, the activity coefficients would be calculated from the Dehye-Hiickle Law or the Extended Dehye-Hiickle Equation and would be included in each cycle. Accurate endpoints then may be found within three iterations. Literature Cited (11 Kolfhoff, 1. M., and Funnsn, N. H., "Potentiomeuie Titrationr: John Wiley & Sons,Inc.. New Ymk. 1931. (21 Willard. H.H..MenitC L. L., and Dean, J. A.. " I n a w m e n ~ Methods l of AaslysC,"Van NastrsndReinhold Co., (4thed.l. NewYork, 1966.p. 553. (31 Dslshay. Paul, "lnrtrumantal Analpis." Tho Meemilisn Co., New York. 19.57, p. w
141 Laintinen. H. A.. "Chemical Analysis: MeGraw-Hill Book Ca., New Ymk,
lm, p.
The values of ER,, Ek2. and the equivalence point poten-
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