do not stipulate the inlet pressure at which their flowmeters were calibrated. The special nature of the inhibition titration curve (Figure 1) elucidates another important source of confusion concerning the extent of inhibition by anions. A given absorption value may correspond to as many as three different silicate concentrations. This precludes the determinations of silicate cia its direct inhibiting effect on standard magnesium solutions. To date, most inhibition methods are based on this technique. The reported inhibition titration curve, with its reversals in slope, is not unique to the magnesium-silicate system. Similarly sloped curves were obtained for the titration of silicate with calcium and phosphate with magnesium or calcium. Huber and Crawford (22) observed a similar titration curve for phosphate inhibition of magnesium and attributed these effects to refractory compound stoichiometry.
The inhibition curve is also not unique to the absorption process. Similarly shaped curves were obtained using flame emission to monitor titrations of silicate with magnesium and calcium. Hence, most of the effects and data herein reported apply equally well to flame emission, and the method developed should also be applicable to flame emission. The proposed technique, AAIT, provides a new, unique, and convenient means of studying not only inhibition effects, but other high temperature chemistry as well. The ease of varying stoichiometry and the rapidity with which data may be obtained should result in the revelation of many new aspects of flame chemistry. A particularly attractive aspect of the proposed method is the ease with which it could be automated for the routine determination of large numbers of samples. Work is continuing on the determination of phosphate, sulfate, and other anions by the same general procedure.
(22) C. 0. Huber and W. C. Crawford, Abstracts 160th National Meeting of the American Chemical Society, Chicago, Ill., September 1970, No. A71.
RECEIVEDfor review September 10, 1970. Accepted December 15, 1970. This work supported by Grant No. 16020-DHD from the Federal Water Pollution Control Administration, Department of Interior.
Titration Errors and Curve Shapes in Potentiometric Titrations Employing Ion-Selective Indicator Electrodes Franklin A. Schultz Department of Chemistry, Florida Atlantic University, Boca Raton, Fla. 33432 Calculated titration curves and errors are presented, illustrating the effect of interfering ions on potentiometric titrations which employ ion-selective indicator electrodes. The presence of interfering ions in the sample solution or titrant distorts the titration curve and causes the inflection point to precede the equivalence point. Using an isovalent precipitation titration as a model, the titration error increases as the sample ion concentration decreases and as the interfering ion concentration, solubility product constant, and dilution factor increase. The calculated titration error can be as large as several per cent.
ION-SELECTIVE ELECTRODES have been developed rapidly in recent years and have widespread application in direct potentiometric measurements and as indicator electrodes in potentiometric titrations. In direct potentiometry the need to correct for the presence of interfering ions has long been recognized and is a serious factor limiting the accuracy of the method. For example, with full Nernstian response at 25 "C of 59.2 mV per decade change in concentration, a 4z error in concentration results from an error of only 1.0 mV in the potentiometric measurement. Potentiometric titrations, however, are generally regarded as being more accurate than direct potentiometric measurements and little consideration has been given to the effect of interfering ions on such titrations. An intuitive appraisal of the subject suggests that the presence of interfering ions will distort a titration curve and cause the inflection point to occur at a point other than the equivalence point. If these points are assumed to coincide, an error will result in the titration. Several authors have encountered distorted titration curves 502
ANALYTICAL CHEMISTRY, VOL. 43, NO. 4, APRIL 1971
when performing potentiometric titrations with ion-selective indicator electrodes in the presence of interfering ions (1-5). However, no quantitative treatment of the error involved has been presented to date. This paper describes the effect of interfering ions on potentiometric titration curves obtained with ion-selective indicator electrodes. In earlier work Meites and coworkers (6-8) derived important fundamental relationships regarding the location of the inflection point in acid-base, precipitation, and chelometric titrations. When the effects of dilution were rigorously included, it was a general conclusion that the inflection point and equivalence point of the titration curve do not coincide. These derivations assumed an ideal indicator electrode, however, and the resulting titration errors were small and not likely to be experimentally detectable. Whitfield et al. (9, 10) have calculated titration curves specifically for chelometric titrations of calcium and magnesium with calcium
(1) R. J. Baczuk and R. J. DuBois, ANAL.CHEM., 40, 685 (1968). (2) A. K. Mukherji, Anal. Chim. Acfa,40, 354 (1968). (3) G. A. Rechnitz and T. M. Hseu, ANAL.CHEM., 41, 111 (1969). (4) M. Whitfield and J. V. Leyendekkers, Anal. Chim. Acta, 45, 383 (1969). ( 5 ) J. S. DiGregorio and M. D. Morris, ANAL. CHEM.,42, 94 (1970). (6) L. Meites and J. A. Goldman, Anal. Chim. Acta, 29,472 (1963). (7) Ibid., 30, 18 (1964). (8) L. Meites and T. Meites, ibid., 37,1 (1967). (9) . , M. Whitfield. J. V. Levendekkers, and J. D. Kerr, ibid., 45, 399 (1969). (10) M. Whitfield and J. V. Leyendekkers, ibid.,46,63 (1969).
and divalent metal ion-selective indicator electrodes. Distortion of the titration curves was evident in the presence of interfering ions, but n o calculation of titration errors was attempted. A general treatment is presented here which describes the effects of electrode selectivity, interfering ion concentration, sample ion concentration, titration-reaction equilibrium constant, and dilution on potentiometric titrations employing ion-selective indicator electrodes. Equations for titration curves including these variables are derived and examined using computer techniques to indicate the nature of curve shapes and the magnitudes of errors that can be expected in experimental situations. Calculations are carried out using an isovalent precipitation titration as a model, but the results are indicative of behavior that can be expected for other types of titration reactions.
and
(7) Two additional terms useful in representing the progress of the titration are f ; the fraction titrated
and r , the dilution factor r = -C X O
Substitution of Equations 2, 6, 7, 8, and 9 into Equation 5 results in a quadratic equation in [Xn-] which has the solution
THEORY
[Xn-]
The model chosen for the following calculations is the isovalent precipitation titration of X"- with Mn+ in which an anion-selective electrode is used as the indicator electrode. The chemical equilibrium is Mn+
+ Xn-
kj
MX(s)
K
=
K s p = [MnT][X"-]
(2)
The electrode potential is given by the equation
where ax"- is the activity of X"- and ui,zi, and ki are the activities, charges, and selectivity coefficients of the interfering ions. In this treatment, the ionic strength and liquid junction potential are assumed to remain constant so that a simplification can be made by equating individual ionic activities with concentrations. If the Nernst factor, 2.303RTl F, is represented by S, Equation 3 can be rearranged to
where [X"-] and Ci are the concentrations of X"- and interfering ions, respectively, during the titration. In identifying the source of anions other than Xn- which may be sensed by the electrode, either the interfering ions are assumed to be present only in the sample solution 0 1 only in the titrant. The general case, corresponding to most practical situations, considers both the sample and titrant as sources of interferences. The solutions of all three cases follow. Case I. Interfering Ions Present Only in Sample. In order to analyze clearly the nature of the titration curves, a continuous function relating the electrode potential to [X"-] is required. This is derived by assuming the sample to be V X " ml of a CX'F solution of AX and the titrant to be a C N F solution of MY. Electroneutrality with regard t o the species involved in the titration is stated as [M"+]
+ [Anf] = [x"-] + [Y"-]
=
(5)
Assuming that VI[ ml of titrant are required t o reach the equivalence point, the concentrations of An+ and Y n - during the titration are given by
+
cy
.\/cy2
+ 4K
2
(10)
where cy=
(1)
which has the solubility concentration product
(9)
cM
(1 -f)cxO 1 rf
+
Equations 10 and 11 describe the concentration of Xnthroughout the titration in terms of the variables f, r, CX", and K, and are the same as the solution given by Meites and Goldman (7) in their treatment of precipitation titration curves. The interfering ions are assumed to be chemically unreactive in this system, and their concentrations therefore are diminished only by dilution. If all ions are assumed to be isovalent (zi = n), the term
represents the initial level of potentiometric interferences in the sample solution. During the titration, dilution decreases A by the factor Vxo/(VXo VA,). Combining this expression with Equations 8 and 9 yields the following equation for the instantaneous level of interferences in the sample solution.
+
(1 3) Equations 4, 10, and 13 combine to give the final result for the indicator electrode potential during the precipitation titration of a sample containing interfering ions
Case 11. Interfering Ions Present Only in Titrant. Interfering ions in the titrant may consist of only counter ions Yn-,but a general treatment includes the presence of additional ions. Assuming that all ions have the charge zi = n, the potentiometric level of interferences present in the titrant is defined as L =
C
kt(Ct)titrant
(15 )
z
where (Ci)tltrant refers to concentrations in the titrant solution. Upon addition to the sample, ions from the titrant are diluted by the factor VIr/(Vxo + VN). By combining Equations 8, 9, and 15, the instantaneous level of interferences in the sample solution arising from ions introduced with the titrant is ANALYTICAL CHEMISTRY, VOL. 43, NO. 4, APRIL 1971
503
/,,/ I
,-.
/ '
I 0
I
I
02
04
I
4
I
I
I
06 08 IO 12 f Figure 1. Effect of interfering ion concentration on potentiometric titration curves with interfering ions present only in sample (Case I); C = 1.00 X lo-*, K = 1.00 X 10-8, r = 0.100
Values of A : (l),-, 10-2
0; (Z), -#-.-, 10-6; (31, . . . ', 10-4; (4), - .
.-. ., 10-3; (5),
- - - - -,
Equations 14 and 17 into a single formula. The complete expression for the general case for an entirely isovalent system is
n(E - E') S
I
I
I
Figure 2. Expanded view of potentiometric titration curves in the vicinity of the equivalence point (Case I); C = 1.00 X K = 1.00 X 10-8, r = 0.100 Values of A : (l), --,
( 2 ) , - - - - -, 3
. . . . ., 10-3
x
10-4; (31,
The expression for the indicator electrode potential during a titration in which interfering ions are introduced with the titrant is
n(E - E') S
=
{
+rfL } 1+rf
--log' [X"-]
(17)
Note that the concentration of Xn-during the titration is given by Equations 10 and 11 as long as an isovalent titration reaction is maintained. Case 111. Interfering Ions Present in Sample and Titrant. Inclusion of effects due to interfering ions present in both the sample and the titrant combine the terms derived for 504
ANALYTICAL CHEMISTRY, VOL. 43, NO. 4, APRIL 1971
=
-log
[X"-]+
{
A
_-
l+rf
+ ____ rfz } l+rf
(18)
Calculations. An IBM 360/40 computer employing Fortran language was used to carry out calculations on Equations 14, 17, and 18. Programs and additional data listings are available upon request. Titration curve shapes were evaluated from plots of the function n(E - E')/S at increments o f f = 0.005. The inflection point of each curve was found by examining the explicit equation for the second derivative of each function for a change of sign. A rough search was conducted at increments of f = 0.005 over the interval f = 0.0 to f' = 1.2, and repeated at increments of 0.001 over a narrower range which included the indicated point of inflection. Linear extrapolation between the two points at the change of sign was used to estimate the point of inflection to one more significant digit. Results equivalent to within .f = *0.0001 were obtained in most cases by using the first difference of the first derivative to locate the infl ection point. The largest deviation between the two methods of calculation was 0.0003. Estimates of the inflection point based on calculation of the second difference of the original function proved to be less reliable. RESULTS AND DISCUSSION Case I. Figure 1 illustrates the effects of the parameter A on potentiometric titration curves with interfering ions present in the sample solution. Increasing levels of interference severely flatten the curves and cause the inflection points to become indistinct. The flattening of each curve beyond the equivalence point results from the reduction of the concentration of X"- to below the level of the interfering ion term, A/(1 r f ) , which is approximately constant throughout the titration. The distortion evident with increasing values of A is shown in Figure 2 by an expanded view of several curves with C = K = and r = 0.1. It is clear in all
+
1
0
/---
/------
-------I 0___---
I
I
I
I
I
I
02
04
06
08
IO
12
f
Figure 3. Potentiometric titration curves at varying values of sample ion concentration with interfering ions present only in sample (Case I); A = 1.00 X 10-4, K = 1.00 X r = 0.100 Values of C: (l), (51, * -, i o - 3
-
-. .-. ., 10-l;
(2),
-,
3 X
lo-*;
(3),
- - - -,lo-*; (4), .
. . .,3 x
10-3;
1-
I
I
1
I
I
I
I
Figure 4. Potentiometric titration curves at varying values of solubility product constant with interfering ions present only in sample (Case I); A = 1.00 X 10-4, C = 1.00 x r = 0.100 Values of K: (l), -, -.-.- ,10-6
10-1*; (2),
-. .-. ., 10-lo;
cases that the inflection point does not coincide with the equivalence point (f = 1.00). Figures 3 and 4, respectively, depict the results of varying the sample ion concentration, C (expressed earlier as CX"), and the solubility product constant, K . Instances of relatively large values of C and small values of K lead to sharply plateaued titration curves. Lower values o f ' C and larger values of K lead to more rounded titration curves which are also distorted. Variation of the dilution factor from r = 0.01 to r = 1.00 has relatively little effect on the shape of the titration curves. Table I lists the per cent titration error for various combinations of values of A , C, K , and r for Case I potentiometric titrations. The per cent error is equal to (f - 1 ) X 100 evaluated at the inflection point of each titration curve. The titration error is always negative (inflection point precedes the equivalence point) and can be as large as several per cent. The magnitude of the observed titration error is
(3),
- - - - -, io-*;
(4),
. . . . ., 10-7; (s),
much larger than the error shown to arise from the effects of dilution in the absence of interfering ions. In Table I the error when A = 0 is always less than 0.01% except K = 10-8, and r = 0.1 in which an for the case C = is found. When A = 0 the error becomes error of -0.01 as large as -0.88% for C = loA3,K = and r = 0.1 or for C = K = and r = 0.1. Errors in the absence of interfering ions due to dilution have been discussed by Meites and Goldman (6, 7), and the values calculated here are in agreement with their work. The results in Table I indicate that an error of significant magnitude can occur under not extreme experimental conditions where titration curves of relatively normal sigmoid appearance are observed. Therefore, unless suitable corrections are made, considerable uncertainty may accompany potentiometric titrations which employ ion-selective indicator electrodes in the presence of interfering ions. Decreasing values of C and increasing values of A , K ,
ANALYTICAL CHEMISTRY, VOL. 43, NO. 4, APRIL 1971
505
~
~~
Table I. Per Cent Titration Error in Potentiometric Titrations with Interfering Ions Present Only in Sample (Case I)
c
= K = r =
A
0 1.00 x 10-5 3.00x 10-6 1.00x 10-4 3.00x 10-4 1.00x 10-3 3.00x 10-3 1.00x 10-2
-0.01%
-0.18 -0.42 -0.91 -1.59 -2.64 -3.99 -6.13' A K r
C
1.00x 3.00x 1.00x 3.00x 1.00 x
a. Interfering Ion Concentration Varied 1.00x 10-2 c = 1.00x 10-1 1.00X 10-8 K = 1.00X 10-8 0.100 r = 0.100
10-1 10-2 10-2 10-3 10-3
c K r
0.00%
b. Sample Ion Concentration Varied 1.00 x 10-4 A = 1.00x 10-5 1.00X K = 1.00x 10-5 = 0.100 I' = 0.100 -0.09% -0.02% -0.06 -0.30 -0.91 -0.18 -3.06 -0.66 -9.58 -2.48
A = LOO x 10-3 K = 1.00X r = 0.100 -0.10% -0.10 -0.17 -0.46 - 1.34a
= =
c. Equilibrium Constant Varied A = 1.00x 10-3 c= c = 1.00x 10-2 r = 0.100 r = 0.100 -0.10% -0.17% -0.27 -0.61 -0.51 -1.29 -0.91 -2.64 -1.42 -5.17 -2.48 -9.58
x 10-4 1.00x 10-2
r 0.010 0.100 0.300 1.000 a Potential difference betweenf = 0.00 andf * No inflection point.
=
d. Dilution Factor Varied A = 1.00x 10-4 c = 1.00x 10-2 K = 1.00X 10-8 -0.87% -0.91 -0.99 -1.25 1.00less than 0.50S / / ImV.
and r lead to a larger titration error. Variation of the sample ion concentration has the most pronounced effect. With A = K = lo-*, and r = 0.1 the error varies from -0.09% at C = 10-1 to -9.58x at C = The interfering ion concentration and solubility product constant also exert a large influence on the titration error but not to the degree of the sample ion concentration over a small range of values. The dilution parameter only slightly influences the error over a range of two orders of magnitude. Although the titration error may become large, there exists a wide range of conditions under which the error may be confined to 1 % or less. These conditions require that A 6 C K 6 and r 6 0.3. A general correlation exists between the shape of the titration curve and the magnitude of the titration error; the error increases as the curve takes on a more flattened appearance. The titration error is generally smallest when the titration curve displays an unusual heel-shape at the equivalence point. This feature corresponds to conditions of large sample ion concentration or small solubility product constant. Regardless of other conditions, however, the
>
506
ANALYTICAL CHEMISTRY, VOL. 43, NO. 4, APRIL 1971
1.00x 10-2 1.00X 10-12 0.100 0.00% -0.09 -0.09 -0.10 -0.10 -0.17 -0.20 -0.30a
-0.02 -0.05 -0.09 -0.17 -0.27 -0.40 -0.61
A = 1.00
K 1 .00x 10-12 1 .OO x 10-10 1.00x 10-9 1.00x 10-8 I .oo x 10-7 1 .oo x 10-6
= = =
A
c
K
= = =
A
c Y
1.00X 10-j 1.00x 10-3 = 0.100 -0.27% -0.91 -1.42 -2.48 -10.33
=
=
. . . a .b 1.00 x 10-3 1.00x 10-2 1.00X -2.50% -2.64 -2.93 -3.86
interfering ion concentration may become so large as to cause the inflection point to disappear or become difficult to locate experimentally. Conditions under which no inflection is observed are indicated in Tables 1-111, as are titration curves for which the potential variation from f = 0 to f' = 1 is less than 0.50 Sin mV. Beyond this arbitrary limit it is anticipated that detection of a n inflection point will be experimentally difficult because of the limitations of precision in potentiometric measurements. Case 11. Titration curves calculated for Case I1 are very similar in appearance to those calculated for Case I. These titration curves are comparable in shape when the factor rL in Case I1 is equal in magnitude to A in Case I. In the vicinity of the equivalence point, the titration curves from both cases are virtually superimposable. Beyond the equivalence point Case I1 titration curves exhibit a small, broad maximum. This arises from the fact that addition of titrant increases rather than dilutes the concentration of interfering ions. The size of the maximum is variable, but it is always less than 0.10 S/n mV forL 6 lo-'. Case I1 titration errors are listed in Table 11. Paralleling
Table 11. Per Cent Titration Error in Potentiometric Titrations with Interfering Ions Present Only in Titrant (Case 11) c. Equilibrium Constant Varied a. Interfering Ion Concentration Varied c = 1.00 x 10-2 L = 1.00 x 10-3 K = 1.00 x 10-8 c = 1.00 x 10-2 L r = 0.100 K r = 0.100 0 -0.01% 1.00 x 1 0 - 1 2 -0.10% 1.00 x 10-6 -0.03 1.00 x 10-10 -0.28 3.00 x 10-6 -0.07 1.00 x 10-9 -0.52 1.00 x 10-4 -0.19 1 .OO x 10-8 -0.94 3.00 x 10-4 -0.44 1.m x 10-7 -1.58 1.00 x 10-3 -0.94 1.00 x 10-6 -3.11 3.00 x 10-3 -1.67 d. Dilution Factor Varied -2.94 1.00 x 10-a 1.00 x 10-1 . . .a,* L = 1.00 x 10-4 L = 1.00 x 10-3 c = 1.00 x 10-2 c = 1.00 x 10-2 b. Sample Ion Concentration Varied r K = 1.00 X 10-8 K = 1.00 X 10-8 L = 1.00 x 10-3 0.010 -0.02% -0.18% K = 1.00 X 0. 100 -0.19 -0.94 C r = 0.100 0.300 -0.47 -1.85 1.00 x 10-1 -0.09% 1.Ooo -1.33 -4.37 3.00 x 10-2 -0.31' 1.00 x 10-2 -0.94 -3.44 3.00 x 10-3 1.00 x 10-8 -12.87 a Potential difference between f = 0.00 and f = 1.00 less than 0.50 Sin mV. * No inflection point. ~
~~
~~~
~
Table 111. Per Cent Titration Error in Potentiometric Titrations with Interfering Ions Present in Both Sample and Titrant (Case III) a. Interfering Ion Concentration in Titrant Varied with Constant Interfering Ion Concentration in Sample A = 1.00 x 10-5 A = 1.00 x 10-4 A = 1.00 x 10-3 c = 1.00 x 10-2 c = 1.00 x 10-2 c = 1.00 x 10-2 K = 1.00 X loT8 K = 1.00 X 10-8 K = 1.00 X 10-8 L r = 0.100 r = 0.100 r = 0.100 -0.18% -0.91% -2.642 0 1.00 x 10-5 -0.19 -0.91 -2.64 3.00 x 10-5 -0.22 -0.92 -2.64 1.00 x 10-4 -0.32 -0.96 -2.65 3.00 x 10-4 -0.53 -1.06 -2.68 1.00 x 10-3 -0.99 -1.34 -2.77 3.00 x 10-3 -1.69 -1.89 -3.01 1.00 x 10-2 -2.95 -3.04 -3.77 b. Interfering Ion Concentration in Sample Varied with Constant Interfering Ion Concentration in Titrant L = 1.00 x 10-4 L = 1.00 x 10-3 L = 1.00 x 10-2
c = 1.00 x 10-2 K = 1.00 x 10-8 A r = 0.100 0 -0.18% 1.00 x 10-5 -0.32 3.00 x 10-5 -0.52 1.00 x 10-4 -0.96 3.00 x 10-4 -1.61 1.00 x 10-3 -2.65 3.00 x 10-3 -4.00 1.00 x 10-2 -6.14" a Potential difference betweenf = 0.00 andf = 1.00 less than 0.50 S/nmV. the behavior of curve shapes, titration errors for Case I and Case 11 are roughly equal under similar conditions if rL = A. Variations in C and K have similar qualitative and quantitative effects in both cases. However, the error in Case I1 is consistently, although only slightly, larger than the corresponding error in Case 1. Case 111. The general case considers interfering ions to be present in both the sample and the titrant. The calculated titration curves and titration errors are not greatly different from the results for the two individual cases. If either of the two terms Al(1 rf) or rfL/(l rf) predominates in
+
+
c = 1.00 x 10-2 K = 1.00 X 10-8 r = 0.100 -0.94% -0.99 -1.08 -1.34 -1.83 -2.77 -4.07 -6.20"
c
= 1.00
x 10-2
K = 1.00 X r = 0.100
-2.91% -2.95 -2.97 -3.04 -3.23 -3.77 -4.82 -6 86a
Equation 18, the results follow the calculations for the particular individual case. If the two terms are approximately equal in magnitude they make approximately equal contributions in determining the titration error and curve shape. Table 111 presents titration errors for Case 111. The error is essentially a function of A alone when A 2 10 rL, and a function of L alone when rL 2 10 A. When the two interfering ion terms are of equivalent magnitude ( A = rL), the error in Case 111 ranges between 68 and 8 6 z of the sum of errors from Cases I and 11. The error in the general case is therefore somewhat less than the directly additive error ANALYTICAL CHEMISTRY, VOL. 43, NO. 4, APRIL 1971
507
from the individual cases. As before, the titration error is smaller at larger sample ion concentration values and smaller solubility product constant values. Experimental Considerations. The most important factor determining the extent to which interfering ions affect a given titration is the selectivity coefficient of the indicator electrode. Most of the recently developed electrodes of the crystal membrane type exhibit very high selectivity for the ion of interest, and it is unlikely that any significant error can be incurred using these electrodes. However, many commercially available glass membrane, liquid membrane, and solid membrane ion-exchange electrodes display only moderate selectivities and are likely to be used under conditions where a measurable error exists. In several recent reports liquid membrane anion-selective electrodes are used as indicator electrodes in precipitation titrations (1, 5, 11). A comparison of calculated and experimental errors is of interest in these cases; however, exact calculated values can be obtained only in some cases because experimental conditions in the literature are not always stated fully enough to allow evaluation of all the necessary parameters. Also, the calculations must be made using selectivity coefficients stated by the manufacturers (12, / 3 ) , which are approximate values and may vary for individual electrodes. In several experimental cases-the titration of perchlorate and tetrafluoroborate with no interferences in the sample solution ( / I ) and the titration of perchlorate with fluoride or sulfate in the sample solution (/)-the calculated error is no larger than the uncertainty in the titration. Hence, no meaningful comparison can be made for these examples. However, for the titration of perchlorate in the presence of chloride, bromide, or nitrate ( I ) and the titration of nitrate (11) M. J. Smith and S. E. Manahan, Anal. Chim. Acta, 48, 315 (1969). (12) “A Guide to Specific Ion Electrodes and Instrumentation,” Orion Research, Inc., Cambridge, Mass., 1969. (13) J. W. Ross, Jr., in “Ion-Selective Electrodes,” R. A. Durst, Ed., National Bureau of Standards Special Publication 314, U. S. Government Printing Office, Washington, D. C., 1969, Chapter 2.
508
ANALYTICAL CHEMISTRY, VOL. 43, NO. 4, APRIL 1971
in the presence of large amounts of sulfate (5), positive systematic errors are observed in contrast to negative calculated errors. Distortion of titration curves by potentiometric interferences cannot account for positive titration errors. Supersaturation, coprecipitation, and preferential adsorption are likely to be responsible for the positive bias in the experiments, and any or all of these are superimposed over the distortion and error resulting from potentiometric interferences. The magnitudes of the titration errors calculated in this work and encountered experimentally indicate that the accuracy obtainable from potentiometric titrations with ion-selective indicator electrodes must be appraised cautiously. It is advisable that titrants be standardized with solutions as similar in composition as possible to the samples being analyzed. Also, the equivalence point must be located in a reliable and reproducible fashion. These empirical procedures would seem to be minimum necessary precautions when performing titrations with electrodes of moderate selectivity in the presence of interfering ions. Note that variation of electrode selectivity with solution composition (14, 15) has not been included as a factor in the derivations of this paper. Additional calculations and an experimental study of titration errors under carefully defined conditions are being conducted in this laboratory and will be reported at a future date. ACKNOWLEDGMENT
It is a pleasure to acknowledge the assistance of Paul Viebrock in writing the computer programs. RECEIVED for review September 14, 1970. Accepted January 7, 1971. Presented in part at the ACS Southeast-Southwest Regional Meeting, New Orleans, La., December 1970. (14) K. Srinivasan and G. A. Rechnitz, ANAL.CHEM.,41, 1203 (1969). (15) M. Whitfield and J. V. Leyendekkers, ibid., 42,444(1970).
