Titration Errors in Chelometric Titrations Ion-Selective Indicator Electrodes Franklin A. Schultz Department of Chemistry, Florida Atlantic University, Boca Raton, Fla. 33432
T w o RECENT REPORTS (I, 2) have described the error in potentiometric titrations which employ ion-selective indicator electrodes in the presence of interfering ions. These ions, when either present in the sample or introduced with the titrant, are sensed by the electrode and distort the titration curve, causing the inflection point to fail to coincide with the equivalence point. Previous treatments considered the case of precipitation titrations. This work reports the titration error in chelometric titrations and considers the effect of interfering ions bearing a charge different than that of the sample ion. This is a common circumstance in chelometric titrations of divalent metal ions where univalent cations may be employed in a buffer or as the counter ion in the titrant. THEORY The chemical equilibrium is represented as the dissociation of the complex MY MYF!M+Y
(5)
where A , is the value of k,{C,)"/'sat the beginning of the titration ( V y = 0) for each ion, and p , is the metal ion to interfering ion charge ratio ( n / z i ) . For interfering ions present in the titrant
where Li is the value of ki(Ci)"/" for each ion in the titrant solution. A complete expression for electrode potential during the chelometric titration of M with a metal ionselective electrode is therefore
(1)
which has the equilibrium constant
In the titration V M Oml of a CM"Fsolution of M are titrated with V y ml of a C y F solution of complexing agent. Solving the appropriate mass-balance equations for M and Y yields the following equation for metal ion concentration during the titration
where (4)
In these equations C is the initial sample concentration (C,"), r is the dilution factor (CM"/CY), a n d f i s the fraction titrated ( C y V y / C MVoM"). An expression equivalent to Equations 3 and 4 was presented originally by Meites and Meites (3). Potentiometric interferences occur as terms of the form k,(Cl)n"., where k , , C,, and z , are the selectivity coefficient, concentration, and charge, respectively, of the interfering ion, and n is the charge of the metal ion. C , is an instantaneous value which varies throughout the titration as a result of dilution. For interfering ions present initially in the sample, the potentiometric interference is given by (1) F. A . Schultz. ANAL.CHEM.. ~~, , 43., 502 (1971). , , ~
~
(2) P. W. Carr, ibid.,p 425. (3) L. Meites and T. Meites, A n d . Chin7. Acta, 37, 1 (1967).
where S is the Nernst factor, 2.303RTIF. An expression similar to Equation 7 may be used to calculate the error of precipitation titrations in the presence of ions of various charge types. For the precipitation titration of X with an X-ion selective electrode, the expression for M in Equation 7 is replaced by the expression for [Xn-] in Equations 10 and 11 in Reference I . In all treatments, activities are equated with concentrations, and variations in ionic strength and liquid junction potential are neglected. Titration errors are calculated as described previously ( I ) using an IBM 360/40 computer. An explicit equation for the second derivative of Equation 7 (d*E/dfl)is examined at increments o f f = 0.001 for a change of sign. Linear extrapolation is used to estimate the titration error to the nearest 0.01 %. RESULTS AND DISCUSSION Table I compares titration errors of chelometric and precipitation titrations. Under equivalent experimental conditions the error is significantly smaller in a chelometric titration. This is consistent with the greater precision inherent in a chelometric titration relative to an ion-combination titration (3). Consideration of the ion charge ratio, p , becomes important only when dilution strongly influences the titration error. For interfering ions in the sample solution the difference between the errors at p = 0.5, 1.0, and 2.0 is slight. When interfering ions are introduced with the titrant, the dilution factor and ion charge ratio become important variabies since the interference term for this case is of the form rpL. Table I shows that the error is larger when a divalent ion
ANALYTICAL CHEMISTRY, VOL. 43, NO. 11, SEPTEMBER 1971
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~~
Table I. Titration Errors in Chelometric and Precipitation Titrations with Ion-Selective Indicator Electrodeso Interfering ions in sample A 0
10-6 10-4 10-3 10-2
p = 0.5 -0.01 % -0.19 -0.93 -2.70 -6.46
Precipitation titration p = 1.0 -0.01% -0.18 -0.91 -2.64 -6.13
p = 2.0
-0.01% -0.17 -0.86 -2.52 -5.61
p
=
Chelometric titration 1.0 p = 2.0
-0.09
0.00 -0.09
-0.27 -0.59 -1.30
-0.26 -0.58 -1.20
0.00%
Interfering ions in titrant L 0 10-6 lo-‘ 10-3
10-2
p = 0.5 -0.01%
-0.07 -0.46 -1.71 -4.19
Precipitation titration p = 1.0 -0.01%
-0.03 -0.19 -0.94 -2.94
Table 11. Etrect of Sample Ion Concentration and Equilibrium Constant on the Titration Error in Chelometric Titrationsa Sample ion concentration varied K = 1.0 X 10-10 K = 1.0 x 10-8 1.0 x 10-1 -0.10% -0.15% 3.0 x -0.10 -0.29 1 . 0 x 10-2 -0.16 -0.59 3.0 x 10-3 -0.29 -1.30 1.0 x 1 0 - 3 -0.59 -2.68 Equilibrium constant varied K c = 1 . 0 x 1 0 - 2 c = 1 . 0 x 10-3 1.0 x 10-12 -0.10% -0.16% 1.0 x 10-10 -0.16 -0.59 1.0 x 1 0 - 0 -0.29 -1.25 1.0 x 10-8 -0.59 -2.68 1 . 0 x 10-7 -1.23 -5.71 -2.52 - 12.09 1.0 x 10-6 a = 5.0 x 10-4; A~ = 5.0 x 10-4; L, = 0.0; L~ = 1.0 x low4;r = 0.10; p l = 1; p~ = 2.
C
interference is added to a univalent ion sample than when a univalent interference is added to a divalent sample. The latter case is the normal circumstance in chelometric titrations. In usual experimental practice, the error in chelometric titrations contributed by interfering ions introduced with the titrant should be negligible. Variation of the titration error with sample ion concentration and equilibrium constant (the inverse of the conditional formation constant) is shown in Table 11. Values of C, K , A , and L are chosen to correspond to conditions encountered in the analysis of calcium in sea water (4). The approximate sodium and magnesium ion concentrations in (4) M. Whitfield, J. V. Leyendekkers, and J. D. Kerr, Anal. Chim. Acta. 45, 399 (1969).
1524
Chelometric titration p = 2.0
-0.01% -0.01 -0.03 -0.18 -0.92
p = 1.0 0.00%
-0.02
p = 2.0 0.00%
0.00
-0.09
-0.02
-0.27 -0.65
-0.26
-0.09
sea water are 0.5Mand 0.05M, respectively, and the optimum selectivity coefficients of a calcium ion electrode for these ions are kCsNo. i% 1.6 x 10-3 and kcan, E 1 X lo-* (5). These values combined with the above concentrations give A, S Therefore, if one maintains A, 6 i
i
C Z and K 6 lo-*, the titration error is always less than 1 %. If electrode selectivity diminishes (6), however, the titration error increases correspondingly. From Table I, it can be seen that a tenfold increase in A increases the error by about a factor of two. The variables C and K also affect the titration error. A threefold decrease in C and a tenfold increase in K each increases the error by a factor of two. Titration errors arising from potentiometric interferences, as discussed in this and previous work ( I , 2), are most serious when the end point is evaluated from the usual presentation of electrode potential us. titrant volume. These errors can be circumvented by application of the Gran plot technique (7, 8). In this approach the end point is located by extrapolation from points early in the titration where the ratio of sample ion to interfering ion concentration is more favorable than it is at the equivalence point. ACKNOWLEDGMENT Paul Viebrock wrote the computer programs for this work. RECEIVED for review March 25, 1971. Accepted June 2,1971. ( 5 ) J. W. Ross, Jr. in “Ion-Selective Electrodes,” R. A. Durst, Ed., NBS Special Publication 314, U. S. Government Printing
Office, Washington, D. C., 1969, Chapter 2. (6) G . A. Rechnitz and Z. F. Lin, ANAL.CHEM., 40,696 (1968). (7) G . Gran, Analyst, 77, 661 (1952). (8) Specific Ion Electrode Technology Newsletter, Orion Research, Inc., Cambridge, Mass., November-December 1970, pp 49-55.
ANALYTICAL CHEMISTRY, VOL. 43, NO. 11, SEPTEMBER 1971
