both free and bound chlorine and syringaldazine with only the free, as indicated by the amperometric procedure. In some waters, where a large percentage of the chlorine exists in the bound form, it is quite possible more free chlorine may be indicated by the o-tolidine-arsenite test than is actually present. The effect of bound chlorine upon the free chlorine values can be minimized by chilling the water before adding the o-tolidine and arsenite reagents ( 2 ) . F o r reasons of convenience, this restriction is rarely used in practice. Either the test is run at room temperature or no attempt is made at estimating the free chlorine content. With the syringaldazine procedure, free chlorine can be measured easily without interference from chloramines
(Tables I1 and 111). The necessity of using a blocking agent, such as arsenite or controlled temperatures, is eliminated. The reagent itself should be prepared fresh each day from a stock alcohol solution of 0.1 syringaldazine. Buffered solutions left standing at room temperature for 24 hours or more showed a gradual decrease in color production. This does not represent a major problem, since the working reagent can be prepared quickly from stock solutions of syringaldazine and phosphate buffer.
x
RECEIVED for review September 14, 1970. Accepted December 21, 1970.
Intrinsic End-Point Errors in Precipitation Titrations with Ion Selective Electrodes Peter W. Carr Department of Chemistry, University of Georgia, Athens, Ga. 30601
Investigations on the effect of ionic interferences in potentiometric precipitation titrations with ion selective electrodes are reported. When the inflection point of the titration curve is used to locate the end point, serious errors (>l%) can result which are often outside the range of the ultimate analytical precision (0.1%) achievable in such analyses. Results are presented in terms of p, a dimensionless variable related to the solubility product and the initial analyte concentration, and b, a parameter related to the concentration and selectivity ratio of interfering ions. I n general, errors result from finite selectivity ratios and from the intrinsic absence of symmetry in the titration curve. The errors are negligible only in the limit of very sparingly soluble materials or in the absence of any interfering ions.
THE PAST SEVERAL YEARS have witnessed a renaissance of potentiometry as an analytical tool (1-3). Most of this is due to the development of both anion and cation selective electrodes which measure the activity rather than the concentration of the species of interest. For thermodynamic and kinetic studies, activity is of principal importance (4). Analytical measurements, however, are mainly concerned with determining the concentration of analyte. Concentration can be measured from a calibration curve, a plot of cell potential us. the decadic logarithm of concentration, from the measured activity and knowledge of activity coefficients, by precision null point potentiometry (5, 6), or by titration. Titration to the inflection point of the potential-volume curve will generally be more precise and accurate than any of the above direct measurement techniques. This method minimizes errors due to variations in activity coefficients, (1) G. A. Rechnitz, Accounts Chem. Res., 3,69 (1970). (2) “Ion Selective Electrodes,” R. A. Durst, Ed., National Bureau of Standards, Special Publication 314 (1969). (3) “Glass Electrodes for Hydrogen and Other Cations: Principles and Practices,” G. Eisenman, Ed., Marcel Dekker, New York, N. Y . ,1967. (4) E. A. Guggenheim, J . Phys. Chem.. 34.1758 (1930). ( 5 ) H. V. Malmstadt and J. D. Winefordner, Anal. Chim. Acta, 20, 283 (1959). (6) R. A. Durst, ANAL.CHEM., 40,931 (1968).
liquid junction potentials, and spurious interferences which change the activity of the analyte. The cost of this increased accuracy and precision is a decrease in the speed and convenience of measurement. It is widely recognized that ion selective electrodes are not perfectly selective (7,8)and that in actual practice the absolute accuracy of direct potentiometric measurements with such electrodes depends critically upon the presence of potential determining foreign ions. It is generally considered that the presence of these interfering ions in a potentiometric titration ultimately limits the magnitude of the “end point break” and, concomitantly, the precision of measurement. The fact that such interferences can seriously influence the position of the inflection point of a titration curve and thus its analytical accuracy has not yet been widely appreciated (9). As will be shown, such errors often amount to several per cent, well outside the range of the precision (hO.1-0.2 routinely obtainable by potentiometric titration. Although there has been some discussion (10, 11) as to evaluation of the extent of electrode selectivity, there seems to be fairly general agreement that if the liquid junction and reference electrode potentials are constant, then the potential of a cell containing an ion selective electrode is well represented as follows:
x)
E
=
const
+ RT In ( y x C x + B ) ZXF
(1)
In this equation (12), species X is the ion of interest (gen(7) J. W. Ross, Jr., “Solid State and Liquid Membrane Ion Selective
Electrodes,” in Ion Selective Electrodes, R. A. Durst, Ed., National Bureau of Standards, Special Publication 314 (1969). (8) G. A. Rechnitz, Chem. Eng. News, 43 (25), 146 (1967). (9) M. Whitfield and J. V. Leyendekkers, Anal. Chim. Acta, 45,383 (1969). 40,457 (1968). (10) M. S. Frant, ANAL.CHEM., (11) A. Shatkay, ibid., p 458. (12) G. A. Rechnitz, “Analytical Studies on Ion Selective Mem-
brane Electrodes,” in Ion Selective Electrodes, R. A. Durst, Ed., National Bureau of Standards, Special Publication 314 (1969). ANALYTICAL CHEMISTRY, VOL. 43, NO. 3, MARCH 1971
425
erally that ion which the electrode has been designed to measure); zx, yx, and Cx are this ion's charge, activity coefficient, and concentration, respectively. The term B represents the cumulative interference due to all other ions present in the test solution and may be written as: B
I I
e,
5
=
KI(~ZCZ)'"~X
The variable, Kr, is generally termed a selectivity ratio which would be zero for all species if the electrode were perfectly selective for X. At present there are no electrodes, including glass electrodes for pH measurement, which are perfectly selective. In order to compute the effect of a finite selectivity ratio on the inflection point of titration curves, we have adapted a mathematical procedure developed by Meites and Goldman (13, 14) for evaluating the effect of dilution on acid-base and precipitation titration curves. The basic method is to represent, in terms of the relevant derivatives of concentration, the inflection point condition, i.e., the point where the second derivative of potential with respect to the added volume o f titrant or a related variable such as the fraction titrated becomes zero. In the present context:
+ B) = o
d2E - = - RT d 2 In ( Y X C X df zxF df
* 0
a
e,
(4)
0, c m.'
5
When this condition is imposed on the relevant chemical equilibrium, the resultant equations allow computation of the fraction titrated at the end point (fep) from which the titration error (4) defined implicitly below can be determined.
% error K 1004 = lOO(1 - f e p )
I
we,
'
v)
(5)
The presence of the term B in Equation 4 results in major differences between the inflection point and the equivalence point. Consider a generalized precipitation reaction between two species, A , the analyte, and T, the titrant.
v
1
(3)
The fraction titrated, f , is generally defined as the ratio v/v*, L; being the volume of titrant added, and v* the volume of titrant required to reach the equivalence point. Equation 3 is equivalent to:
.e
-
(2)
nA
+ rnT=
ACTm(4 )
(6)
The chemical reaction can be represented by an equation for the concentration of either A or T i n terms of the fraction titrated, which results from considering the formal solubility product, related mass balances, and dilution factors.
.z
C a Y f ' - ~~COACA' - VKgplim= 0
(7)
+ YCT)Y - V K ~ P =" ~0
(8)
and Vcp(ffco~
?+?q
where CoAis the initial concentration of analyte, Coris the concentration of titrant, Y is a stoichiometry coefficient defined as the ratio, nlm, and a and Kspare given below:
0 0 0
I'&,
I1
/I
m
s
u
426
-
a
ANALYTICAL CHEMISTRY, VOL. 43, NO. 3, MARCH 1971
=
Can C T ~
(9)
(13) L. Meites and J. A. Goldman, Anal. Chim. Acfa,29,472(1963). (14) Zbid., 30, 18 (1964).
2.80
2,401
200-
Figure 1. Computed titration curves-plot of -log,, (CA b) us. the fraction titrated for the reaction
+
A
+T=
AT(
160-
4)
Curve a. p = lo-*, b = 0 Curve b. p = 0, b = 10-2 Curve c. p = lo+, b = 10-2
;;
+ lo'
120-
-
n
080-
040-
I
020
(17) df2 The derivatives present above are easily computed from the definition of a. When Equations 14 and 17 are solved simultaneously, one obtains for a given value of v and r algebraic equations for a: in terms of the parameters b and /3. Since in general the equation for the end-point error, 4, is of third and higher order, the detailed results of this and calculation for the most common cases, i.e., v = 1, 2 are presented in Table I. In order to illustrate the effect of the term b on the titration curve and on the position of the inflection point, the data of Figures 1 and 2 were computed. Three cases are disand b = 0; second, p = and cussed-first, p = 6 = 0.01; and last, = 0 with b = 0.01. As indicated by
df2
ex is the normalized variable Cx/CoAand:
Equations 7, 8, and 9 can also be put in dimensionless form by dividing by (CoA)Yflresulting in: CA'(C'4 - a:) - p = 0 (14) VCT(VCT a:)2 - p = 0 (15)
+
+
+T= Curve a.
AT( 4) p = 10-3, b = 0 Curve b. p = b = 10-8
A
-
O L O'e8
9
KUN
2
160
140
+ 1 ) j 2 = [(v + ~ ) C A- v(u]~(C'A+ b) d2a: -
+ b) d2Cx -=
Figure 2. Computed second derivative titration b ) / A f 2 us. the curves-plot of -Az log,, (CA fraction titrated for the reaction:
1'20
~ V ( V
+ z x F (ex+
where
100
080
060
Two categories of measurement can be discerned-those in which species A is sensed by the electrode, Le., X = A and those in which the titrant is sensed, X = T. The first case will be described in detail. The second constitutes a straightforward extension. Calculation of the Analytical Error. If the inflection point conditions as stated by Equation 4 with X = A are imposed on the second derivative of Equation 14, the following expression results :
Where r is the ratio ( C 0 ~ / C o p ) When . dilution is negligible, as in titrations with extremely concentrated reagents or coulometric titrations, r is zero. From this stage, it is important to note that Equation 4 results only when certain restrictions are imposed. We will assume that activity coefficients, junction potentials, and the cumulative interference effect, B, are invariant as the titration proceeds even though the volume of the system may change. This is a n unrealistic assumption since B must vary as titrant is added. As will be shown later, this is not a serious problem. The above restrictions allow the reformulation of Equations 1 and 4 as: RT E = const' - In b) (11) (ex
040
'
Ob0
092
Ob4
096
0s;
'
102
104
106
IO8
110
FRACTION TITRATED
-10-
ANALYTICAL CHEMISTRY, VOL. 43, NO. 3, M A R C H 1971
427
'
b
P 10-1 10-2 10-3 10-4 10-6 10-6 10-7 10-8 10-9 10-10 10-11 10-12
Table 11. Analytical Error in Precipitation Titrations with Ion Selective Electrodes Sensitive to the Titrant 0.1 3.16 x 10-2 10-2 3.16 x 10-3 10-3 0 u=‘/2 v = 2 u = y 2 u=2 u = y 2 v=2 u = y 2 u = 2 u=’/z v=2 .=I/$ v-2 -28.85 17.41 -26.77 35.09 -26.02 46.23 -25.77 51.85 -25.69 54.10 -25.65 55.26 -7.78 0.79 -6.48 11.17 -5.87 18.34 -5.64 22.57 -5.56 24.54 -5.53 25.65 -2.36 -3.83 -1.81 2.15 -1.47 6.45 -1.30 9.31 -1.23 10.87 -1.19 11.90 -0.74 -4.21 -0.56 -0.75 -0.43 1.74 -0.34 3.51 -0.29 4.62 -0.26 5.53 -0.23 -3.39 -0.17 -1.36 -0.13 0.08 -0.10 1.12 -0.08 1.83 -0.06 2.57 -0.07 -2.44 -0.06 -1.22 -0.04 -0.38 -0.02 0.65 -0.03 0.22 -0.01 1.19 -1.66 -0.92 -0.42 -0.08 0.17 0.55 -0.64 - 1.09 -0.34 0.01 -0.14 0.25 -0.43 -0.71 -0.24 -0.12 -0.04 0.13 -0.28 -0.46 -0.17 -0.09 -0.04 0.10 -0.18 -0.29 -0.11 -0.06 -0.04 0.10 -0.16 -0.18 -0.07 -0.04 -0.03 0.09 ,
curve b of Figure 1 , the potential changes very rapidly when /3 = 0 as long as f < 1.00; at the equivalence point, all of species A is consumed by the titrant and the potential of the electrode is determined solely by the term B (see Equation 1). The condition of total insolubility and invariance of B forces a discontinuity on the titration curve, causing a point of inflection to occur at the equivalence point and obviating any determinate errors. When b = 0 but /3 is finite (see curve a, Figure l), the potential continues to rise after the equivalence point since there are no interfering ions determining the electrode potential. In this case, of course, the titration curve in the absence of dilution is symmetrical about the equivalence point (see Figure 2, curve b) and there is no end-point error. When both /3 and b are finite, a n end-point error occurs. This is illustrated most clearly in Figure 2, curve b. The data presented here were obtained by computing the titration curve in fraction-titrated increments of 0.005 units. The second derivative was obtained using a central difference approximation. In this case interpolation indicates an end-point error of 1.65 %. If the computation is repeated with A,/ = 0.001 unit, a n error of 1 . 3 8 z is obtained, which is in excellent agreement with the a priori approach discussed later. If the term b is set equal to zero in Equation 17, the equations of Meites and Goldman result. More importantly, if dilution is neglected by setting r = 0 in all equations, then the second derivative of Q with respect to fvanishes and Equation 17 simplifies to:
O
401
? 60
8.0
-LOG,,
Curve a. Precision = 2.30x Curve b. Precision = 1.15% Curve c. Precision = 0.23z In order to assess whether these errors are outside the range of normal precisions, one can calculate the end-point precision from the “sharpness” index, 7 , which has been described extensively by Butler (15).
The data presented in Table I were computed by solving Equations 14 and 18, simultaneously. The resulting equation was solved numerically using Horner’s scheme and the Newton-Raphson technique for polynomials. Table I1 presents the results for the analogous situation in which the titrant species is sensed by the electrode. The general inflection point condition when X = T and r = 0, can be written as:
It is important to note that Equations 18 and 19 hold only at the inflection point. This condition superimposed on Equation 15 was used to generate the data of Table 11. The results for v = 1 are not presented since the errors in this case are identical in magnitude but of opposite sign to the case where X = A and v = 1. 0
20
Figure 3. Plot of the maximum value of b us. log,, p which will permit a titration of the indicated precision in accord with Equation 22
For the case where v by
428
40
P
ANALYTICAL CHEMISTRY, VOL. 43, NO, 3, MARCH 1971
=
1 and dilution is negligible, 7 is given
1 0.4343 v=?b+dF The per cent uncertainty in locating the equivalence point is given approximately by Equations 20 and 21 as follows:
This equation realistically predicts that as /3 and b become larger, the precision decreases. A reasonable estimate of the uncertaintyinpXis 10.01. At 25 “C, this corresponds to a n error of 1 0 . 6 mV(zx = 1) in the end-point potential. Equation 21 was used t o compute the maximum value of b such that the titration can be performed with a precision of 2.30, 1.30, or 0.23 % at a given value of /3. Figure 3 illustrates (15) J. N. Butler, “Ionic Equilibrium, A Mathematical Approach,”
Addison-Wesley, Reading, 1964.
10-2 10-3
10-7 10-8 10-9
28.29b 8.92 4.11 1.97 0.94 0.44 0.21 0.10
10-10
0.04
10-4 10-6 10-6
20.58 5.12 2.39 1.22 0.60 0.29 0.14 0.06 0.03
1
16.61 2.12 0.64 0.38 0.22 0.12 0.06 0.03 0.01
17.64 3.06 1.29 0.72 0.38 0.19 0.10 0.04 0.02
16.28 1.77 0.34 0.17 0.12 0.07
16.17 1.66 0.22 0.07 0.05 0.04 0.02
0.04
16.12 1.60 0.16 0.02
0.02
All data was computed assuming that the electrode is sensitive to the analyte, that
v = 1 and that r = 1. by summing the error due to dilution and the error due to imperfect se-
* Values below or to the right of the double line may be obtained lectivity.
these results. A cursory examination of the data of Tables I and I1 and Figure 3 indicates that the errors caused by finite values of are outside the precision of measurement. Equation 22 demonstrates that the interference ratio has a very deleterious effect o n the precision of analysis, e . g . , when p = and b = 0, a precision of 0.005% would result, i.e., if volumetric errors are negligible; however, and b = 0.1, the precision can not be better when p = than about 0.5 %. Effect of Dilution on the Titration Curve. I n any practical titration, with the exception of coulometric titrations, the volume of the system will change perceptively because of the addition of titrant. This increase can be minimized by using a titrant concentration, C O T , which is much larger than the analyte concentration, C o A . This must be recommended in titrations with ion selective electrodes, particularly under circumstances where the interference ratio is finite. This is mandated by the fact that dilution effects always act to decrease the magnitude of the potential break, Le., decrease 7, and second often increase the discrepancy between the inflection point and equivalence point, at least when v = 1 . The data presented in Table I11 were computed from Equation 17, using v = r = 1 . It should be emphasized that this treatment is not exact, since we assumed earlier in the derivation of Equation 17 that the term b was independent of the fraction titrated. This is not precisely true, since as the volume changes, the concentration of species contributing to b will vary; however, the approach does allow an approximate computation of the effect of dilution on the analytical accuracy. RESULTS AND DISCUSSION
A comparison of the data of Tables I and I1 and Figure 3 indicates that the errors caused by finite selectivity ratios are well outside the range of the computed end-point uncertainty. The magnitude of these errors is not surprising. Even though the selectivity term may be small with respect to C o A ,in the vicinity of the equivalence point b may be larger than P A and, therefore, be the predominant term in Equation 11 which defines the inflection point. I n order to discuss the data of Tables I and 11, we will refer to a symmetry error, Le., the end-point error, when b = 0 and a selectivity error, i.e., the difference between the symmetry error and the computed error, when b is finite. The two principal variables which must be considered are v and whether the electrode is sensitive to species A or T. When v = 1 the system is quite simple. There is n o symmetry error and the selectivity errors are equal in magnitude but opposite in sign; the inflection point precedes the equivalence point when X = A and follows the equivalence point when X = T. At a given value of 6, the error decreases
monotonically as p decreases, the error being zero when the solubility is vanishingly small. Similarly at a given value of p, the error decreases monotonically to zero as b approaches zero. Whenever v # 1, the error in the absence of any interfering materials is not zero but depends strongly o n the parameter 0. One can show from the work of Meites and Goldman (14) that the error in the absence of dilution will be given by
Equation 23 indicates that the symmetry error will be positive when v > 1 and negative when v < 1. It should be obvious that when v is other than unity, the magnitude of the symmetry error generally requires that the precipitate be rather insoluble (small K S p o) r that the titration be carried out a t high concentration (large CoA),since either circumstance tends to decrease /3 and, therefore, the symmetry error. The selectivity error is positive when species A is measured and negative when T i s measured. As witnessed by the data of Tables I and 11, only under those circumstances where the two errors, i.e., selectivity and symmetry, reinforce one another will the overall error be a monotonically decreasing function of p. This occurs when X = A and v > 1 or when X = T and v < 1. Under these circumstances the sign of the error is invariant and the error decreases uniformly toward zero as p approaches zero. Under the alternate set of circumstances, Le., X = T and v > 1 or X = A and v < 1, the error does not decrease monotonically as p decreases and the sign of the error is not invariant, although in the limit as /3 approaches zero, the error goes to zero. This results because the two errors are of opposite sign and either one or the other will predominate for a given value of p and b. It is quite interesting to note that in those cases where the sign of q5 oscillates, the error becomes negligibly small at larger values of /3 than in those cases where the sign is invariant. This, somewhat surprisingly, suggests that the order of reagent addition may improve the accuracy of the titration. F o r example, consider the following titration process: 2A
+ T%
Under these circumstances v below.
=
A2T
2, m = 1 , and
(24)
is defined
Now interchange the titrant and reagent species, keep all else constant, i.e., use the same electrode, electrolyte, initial concentration, etc. The reaction becomes : A
+ 2 T e AT,
ANALYTICAL CHEMISTRY, VOL. 43, NO. 3, MARCH 1971
(26) 429
with v = 1/2, m = 2, and /3 is redefined as:
l/zKspl'z (C0'4)3/2
P 2 =
This may be summarized as:
Pz
=
( CO a ) 3' 2 *P1 4KsP1I2
~
With reference to Tables I and 11, it is clear that whenever Pz < pl,the analytical error must be decreased by a n interchange of the analyte and titrant species. This situation will result whenever : If p 2 > p1 the error may o r may not decrease depending upon the specified values of b, 01, and 6 2 . Of course, when Equation 29 is realized, P1 and PZ would be so large that the titration is impractical. Indeed as has been pointed out (14), an inflection point may not occur when Coa is extremely low. It should be noted that the magnitude of the end-point error depends strongly o n the reaction stoichiometry and upon the initial concentration of analyte, since both /3 and b are as indicated by Equations 13 and 16 related to eoA. Generally, the error increases as the analyte becomes more dilute, since this increases both /3 and 6. The solubility product, K,,, has been formulated rather unusually (see Equation 3), and for the titration of a divalent cation with a divalent anion has a value equal to the square root of the more conventionally defined solubility product. All else being equal, errors in such titrations will be much larger than in a similar titration involving univalent species. The magnitude of the errors computed here is such as t o be a serious limitation in the use of ion selective electrodes as inflection point indicators in titrations involving only moderately insoluble salts. Every attempt should be made to assess the presence of ions whose selectivity coefficients are large and to minimize their concentration o r replace them with ions which interfere less significantly. In general, it appears that the addition of an electrolyte for masking interferences and maintaining ionic strength and or junction potentials must, in terms of these errors, be viewed with suspicion. It is instructive to compare these data for precipitation titrations to Whitfield and Leyendekker's (9) results o n the effect of ion selectivity ratios on the complexometric titration of calcium and magnesium. They computed a number of theoretical titration curves, Le., plots of electrode potential us. f; and examined the influence of finite interference ratios on these curves. They found that the inflection point coincided with the equivalence point whenever a large end-point break (2-3 p X units) occurred. It is difficult to contrast our work with theirs since the reactions studied are fundamentally different and the range of equilibrium constants investigated was quite narrow (lO1o-lO1z). Second, a comparison of Figures 1 and 2 of this work indicates that conventional titration curves are not particularly sensitive to this effect, although second derivative titration curves are very sensitive. It may be that the cases studied by these workers represent the limit of large equilibrium constants where any deviations are negligibly small. A number of alternative approaches to the problem of finite interference ratios are possible. A first approach is titration to a predetermined equivalence point potential. I n this case, one must minimize errors due to changes in activity coefficients, junction potential, and spurious materials which can alter the activity of the analyte. A second ap430
ANALYTICAL CHEMISTRY, VOL. 43, NO. 3, MARCH 1971
proach is the method of Gran (16, 17) plots, where the analyte concentration is computed from the measured electrode potential and plotted against the volume of titrant added. A linear titration curve results, whose extrapolated branches intersect to yield an end point free of these errors. As b becomes larger, the precision of such a technique would be seriously impaired. Table 111 presents a compilation of analytical errors for v = 1 titration with r = 1, Le., equal titrant and initial analyte concentrations. These data were computed using Equations 14 and 17 and approximately correcting for the effect of dilution o n the term b in Equation 17 by assuming that the effective value of b around the inflection point was exactly one half its initial value. The last column of Table 11, which is in agreement with the equation of Meites and Goldman (13), indicates that even when the interference ratio is zero, considerable end-point errors will occur. This results because the titration curve becomes quite asymmetrical about the inflection point due to the distortions caused by dilution. Those values of the titration error in Table I1 which fall below or to the right of the double line can be obtained by merely adding the dilution error (Table 111, b = 0) to the selectivity error (Table I, v = 1). In all cases the two values differ by no more than 0.3 in their absolute value. In the light of the data, one can assume that in the limit of either a small b or 6, the two errors are virtually independent and are, therefore, additive. When both /3 and b are large, the coupling of the two effects is stronger and the computed error is larger than the simple sum of the two independent errors. CONCLUSIONS
Titrations with ion selective electrodes are not free of errors related to the selectivity ratio. Finite concentrations of interfering ions can affect both the analytical accuracy and precision when the titration end point is taken as the inflection point of the potential-volume curve. In general, the determinate errors are often well outside the range of the achievable precision whenever the precipitate has a finite solubility. End-point errors are largest when the reaction is not symmetrical (v + 1). Such reactions require extremely small solubility products in order to avoid both symmetry and selectivity errors. Within the range of practical titrations, the error is neglior b < 3 X The magnitude gible as long as P 5 of the errors and the effect of b o n the analytical precision suggest that such titrations should be carried out with minimal supporting electrolyte concentration and at the greatest possible concentrations of both analyte and titrant. Work is in progress in these laboratories to assess the effect of selectivity errors on complex formation titrations and to develop experimental alternatives to the method of end-point location. ACKNOWLEDGMENT
The author thanks Leon N. Klatt of this department for many interesting and fruitful discussions in connection with this work. Funds for computer time provided by the University of Georgia Computer Center are gratefully acknowledged. RECEIVED for review October 26, 1970. Accepted December 14. 1970. (16) G. Gran, Acta Chem. S c a d . , 4, 559 (1950). (17) T.Anfalt, D. Dryssen, and D. Jagner, A i d . Clzim. Acfa, 43,
487 (1968).
