Analysis of the Polarographic Method of Studying Metal Complex Equilibria Leon N. Klatt’ and Russell L. Rouseff Department of Chemistry, University of Georgia, Athens, Ga. 30601 The effect of experimental variables upon the measured coordination numbers and the calculated equilibrium constants have been simulated on a digital computer. Small errors in the measured half-wave potential shifts removed all curvature from the A h z VI. loglo [cx]plot. Noninte ral ligand numbers are calculated from the slope o these plots for chemical systems containing a mixture of complexes. Analysis of experimental data by a least squares fit to DeFord and Hume’s F,(X) function is the preferred procedure. The maximum precision of the calculated equilibrium constants was found to depend only upon the ratio of successive formation constants. By judicious selection of ligand concentrations, errors can be minimized.
9
THEPOLAROGRAPHIC method of studying metal complex equilibria is based upon the fact that the half-wave potentials of metal ions are shifted, usually to more negative values, by complex formation. By measuring this shift as a function of the concentration of the complexing agent, both the formula and formation constant of the metal complex may be evaluated. This method is one of the standard techniques of studying metal complex equilibria (I). The reduction of a complex ion to the metallic state (amalgam) may be represented by
+ + Hg + M (Hg) + jX-b
MXj(n-fb) ne
(1)
Evaluation of experimental data may be carried out by either the Heyrovsky and Ilkovic ( 2 ) or the DeFord and Hume (3) approach. The relationship between the shift in the halfwave potential and the concentration of complexing agent for each method is given by the following equations,
or more metal complex species exist in a particular concentration range of complexing agent. Recently, studies of the composition and formation constants of lead-oxalate complexes were reported (5, 6). Plots of half-wave potentials us. the logarithm of oxalate ,concentration produced a linear relationship in accord with Equation 2 with a ligand number of 1.9 calculated from the slope. This suggests the presence of a single lead-oxalate complex containing two moles of oxalate per mole of Pb (11). Similar results have been reported by other investigators (7,s). However, when these data are analyzed according to the procedure suggested by DeFord and Hume (3),the presence of two leadoxalate complexes, consisting of one and two moles of oxalate per mole of Pb (11), respectively, is indicated. Jain, Kumar, and Gaur (5) suggested that a linear relation between Eliz and log [Cx]is not the only criterion for the presence of a single complex. However, they state that if a single complex is formed, a plot of DeFord and Hume’s Fo (X) function (3) us. CX should be linear. This is valid only for the formation of the MX complex. An analysis of the Heyrovsky and Ilkovic and the DeFord and Hume method of evaluating metal complex equilibria is required in order to clarify this apparent conflict. Meites ( 4 ) and Irving (9) have discussed these approaches, pointing out the assumptions made in the respective derivations as well as the importance of controlling experimental factors, such as liquid junction potential and the uncertainties in the activity coefficients. However, the question of ascertaining the inherent capabilities of this technique, even if these variables were explicitly defined, has not been answered. The objective of this study was to examine this fundamental question. DATA ANALYSIS
The definition of symbols is essentially the same as that used by DeFord and Hume (3) and Meites (4). The difference between Equations 2 and 3 is that the former assumes that for a given concentration range of complexing agent only one metal complex species exists, while the latter considers the possibility of several metal complex species existing simultaneously. Thus, plots of shifts in half-wave potentials us. the logarithm of ligand concentration, at constant ionic strength, will be linear or curved depending upon whether one 1
To whom correspondence should be addressed.
The important parameter of Equations 2 and 3 that determines the observed half-wave potential shift is PjCx3’. Since for a given chemical system the p’s are fixed, although unknown, the magnitude of the observed potential shift is determined by the Cx range employed in a particular study. Thus, the number of p’s and the precision thereof that may be evaluated from the experimental data (AE1i2as a function of CX)is controlled principally by the C, range investigated and errors associated with measuring the shift in half-wave potentials. The procedure used in the evaluation of the HeyrovskyIlkovic relationship involved the generation of synthetic ligand concentration us. half-wave potential data via Equation 3 , ( 5 ) D. S. Jain, A. Kumar, and J. N. Gaur, J . Electroanal. Chem.,
(1) F. J. C. Rossotti and H. Rossotti, “The Determination of Stability Constants,” McGraw-Hill, New York, N. y., 1961, pp 171-188. (2) J. Heyrovsky and D. Ilkovic, Collect. Czech. Chem. Commun., 7, 198 (1935). ( 3 ) D. D. DeFord and D. N. Hume, J. Amer. Chem. Soc., 73, 5321 (1951). (4) L. Meites, “Polarographic Techniques,” 2nd ed., Interscience, New York, N. Y . , 1965,pp 267-284. 1234
17,201 (1968). (6) R. L. Rouseff, M. S. Thesis, Southern Illinois University, Carbondale, Ill., 1968. (7) I. M. Kolthoff, R. W. Perlich, and D. Weiblen, J. Phys. Chem., 46, 561 (1942). (8) A. I. Vogel, “Quantitative Inorganic Analysis,” John Wiley and Sons, New York, N. Y . , 1961, pp 1031-1032. (9) H. Irving, “Advances in Polarography,” Vol. 1 , Pergamon Press, New York, N. Y . , 1960, pp 42-67.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
for assumed values of the overall formation constant, followed by the least-squares analysis of these data according to Equation 2. This approach may be justified by the following considerations: For the metal ion reductions expressed by Reaction 1, Equation 3 is the general relationship for the shift in half-wave potential as a function of complexing agent concentration, both for the number of complexes formed and the magnitude of the formation constant; and the work of Jain et al. (5) and Rouseff (6) suggests that Equation 2 may be insensitive to stepwise complex formation. The least-squares analysis of the synthetic data was carried out under two conditions. For the first condition, no experimental error was introduced. This provided the minimum properties that a chemical system must possess before adherence to Equation 2 is obtained, The second condition involved the introduction of experimental errors into the calculated half-wave potential data by selecting fifty sequential random normal deviates (RND) of zero mean and unit standard deviation (10) and defining
(4) is the standard deviation of the shift in half-wave potential determined by the precision of (E1& and (El& but independent of the magnitude of their difference, AEwLe., U A E , , ? = Using this procedure, the generated data appear to be drawn from normal populations with uniform standard deviation. Analysis of the half-wave potential ligand concentration data according to the DeFord and Hume approach involves plotting the exponential form of Equation 3 us. Cx. This curve is an Nth degree polynomial whosejth coefficient is the formation constant of MX,. A weighted least-squares curve fitting algorithm employing ideas presented by Momoki, Sato, and Ogawa (11) and Varga (12) was developed to carry out this analysis (13). All activity coefficients were assumed constant. For analysis of Equation 2, pj and Cx values were selected such that PICX’>> 1 . The lower limit of Cx was 0.01, which would require experimental work using total metal ion concentration of 0.2 m M or less. The range of ligand concentrations was varied by changing the increment in Cx. The maximum value for CX was arbitrarily set at 1.5 M . All calculations were carried out using 50 data points. A random distribution in the signs of the least-squares residuals and the magnitude of the computed variance was the criteria used to determine when the data could be described by a particular mathematical function, When several functions appeared to describe a given set of data, the F test (14) was applied to the variances calculated from the sum of the squares of the least-squares residuals. If the F test failed to reject any of these functions at the 95 % confidence limit, the one with the least number of adjustable parameters (pj’s) was selected to describe the data. UAE,,,
~TQE,,,.
RESULTS AND DISCUSSION
For the purposes of this study, metal complex equilibria were divided into various cases according to the relative (10) H. Hyrenius and R. Gustafsson, “Tables of Normal and LogNormal Deviates,” Skriftserie Publications, Goteborg, Sweden, 1962. (11) K. Momoki, H. Sato, and H. Ogawa, ANAL.CHEM., 39, 1072 (1967). (12) Louis Varga, ibid., 41, 323 (1969). (13) L. N. Klatt, unpublished research. (14) W. C. Hamilton, “Statistics in Physical Science,” Ronald Press, New York, N. Y . , 1964, pp 84-89.
-
OAE,,2
1.0 m v
~ ~ - ~ ~ - ~ - - - ~ ~ ~ - - ~ ~ ~ ~ - - - ~ o - - ~ - ~ ~ - ~ - O O Q - - O - - O - - - - O - O o
Figure 1. Signs of residuals for individual values of AE1/ Assumed p’s p1 = 1.00 x 103 p 2 = 1.00 x 105 p3 = 2.00 x 106 pa = 1.00 x 104 0 indicates a positive residual indicates a negative residual
-
magnitudes of the pj’s. A final case involves calculations where pjCxj N 1. Each case was selected so as to approximate known chemical systems. The first case considered involved systems where one species was substantially more stable than all others (pi 2 100 P I , i # j , but j may equal N). Zn(I1)-glycinate and Zn(I1)-ethylenediamine are examples of chemical systems that fall into this category (15). With zero experimental error, an almost relationship insignificant curvature in the AEIIZus. log [CX] was obtained. Residuals were typically 0.2 mV or less. A QAE,,, of 0.1 to 0.3 mV, values smaller than attainable experimentally, removed all curvature. Ligand numbers obtained from the slope of these relationships were essentially equal to j. Analysis of the generated data by a least-squares curve fitting to the DeFord and Hume function indicated the presence of a single complex. Changes in the concentration range did not affect the results. The second case considered involved chemical systems where at least two formation constants differed by approximately one order of magnitude, but were substantially larger than all other formation constants. Hg!II)-Cl- and Cd(I1)NH3 are examples of chemical systems that fall into this category (16). For the condition of zero experimental error, slight curvature was noted in the AEllZL‘S. log [Cx] function. Figure 1 shows the signs of the least-squares residuals for a typical case. The ligand number calculated from the slope of the least-squares line for each case in Figure 1 was 2.8. Analysis of this data via the DeFord and Hume expression indicated the presence of the more stable complexes. Typical results are given in Table I. The complexes of lower stability were not included in the calculated DeFord and Hume function because they did not significantly decrease the variance of the computed curve. Attempts to include these less stable complexes often led to negative coefficients in the DeFord and Hume function; and consequently, these models were discarded because they did not represent a physically meaningful result. (15) L. Meites, “Handbook of Analytical Chemistry,” McGrawHill, New York, N. Y.,1963. (16) J. N. Butler, “Ionic Equilibrium-A Mathematical Approach,” Addison-Wesley, Reading, Mass., 1964.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
1235
~
~
~~
Table I. Typical Evaluation of 0's via Curve Fit to the Exponential Form of Equation 3 Assumed
Assumed pj's p1 = 1.00 x 103 pz = 1.00 x 106
Calculated pj's
UAEIIZ,mV
pa
=
1 , o o i 0.01
x 108
0.5
p1 = 1.00 x 103 pz = 1.00 x 106
pz
=
106
p3
=
1.06 f 0.05 X lo5 2.00 i 0.02 x 106
0.5
pz
= =
p3
=
= p4 = p3
pl pz
=
=
03 =
84 =
1.00 x 108
2.00 x 1.00 x 2.00 x 6.31 X 6.31 X 6.31 X 2.50 x
p1 = 8 2 = 0.0
p3 = 3.16 x pa = 3.98 x p1 = 1.00 x pz = 1.00 x p3 = 2.00 x pi = 50 p2 = 75
104 105 10'' 1013 lor4
=
pi pz
=
50
=
75
P3
=
40
=
6.31 i 0.04 X 10" 5.83 i 0.42 X 1013 6.41 =t0.09 X 1014
= =
3.10 i 0.04 X 10'6 4.10 i 0.09 X 10"
0.5
104
83
1015 1015 103 105 105
84
40
p3
p3
0.5
1.04 i 0.02 x 105 1.97 + 0.04 X 106 01 = 48.5 zk 1.2 pz = 79.5 i 6.2 P3 = 37.4 i 6.3
0.5
pi
1.5
pz
=
p3
=
pz
= =
0.4
39.3 i 2.7 119 f 6
When at least two overall formation constants are approximately equal, but significantly larger than all others, somewhat increased curvature over the previous case was noted in the semilogarithmic plot of Equation 2. Zn(II)-OH- is an example of this type of chemical system (15). The signs of the least-squares residuals showed the same behavior as in Figure 1. A typical case with pl = 2.5 X lo4, pz = 0.0, p3 = 3.16 X 1015,and p4 = 3.98 X 1015resulted in a calculated ligand number of 3.2. A UAE,,,of 1.0 mV removed all curvature from this function, but did not significantly change the calculated ligand number. Analysis of these generated data uiu the DeFord and Hume function produced results very similar to the previous case. Typical results are given in Table 1. The smaller formation constants were generally determinable only when UAE,,,< 0.3 mV. For chemical systems that fall into the latter two cases, the ratio of successive formation constants as well as the concentration range considered greatly influences the p's and their precision that may be evaluated. In Figure 2 are shown typical results. The relative AEljz is the largest AE1,2 calculated for a particular concentration range divided by AEl,z obtained when Cx = 1.5M for a given set of 6's. pj-1 and pj+3were assigned values loa smaller than pj and pj+*. Each point is the result of a calculation using a selected concentration range spanned by 50 data points. Four general results are apparent from Figure 2. First, as the ratio of successive p's increases, the precision of the
b)
C)
600
5 I4
(u
428
z
< W
bZ
343 257
1 W
8
171
\ 86
0 (
I
I
I
I
0.4
0.6
0.8
1.0
0.6
0.8
RE LAT I V E
1.0
A E 1/2
Figure 2. Dependence of relative standard deviation of individual formation constants upon observed half-wave potential shifts A Pi 0 Pj+1 0 Bj+2 A . P j = 6.31 X 10'; p j : p j + 1 : p j + 2 = 1 : 10 : 100 B. P j = 6.31 10"; :p j + 1 :Pj + 2 = 1 :10 100 C. B j 1 X IO8; @ j : p j 4- 1 :pj+2 = 1 :75 :5625
1236
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
smaller constants decreases while that of the largest p slowly increases. Second, the relative precision of the smaller p’s goes through a minimum, while up of the larger constant decreases monotonically. This minimum error was found to be independent of the magnitude of the successiveB’s, but dependent only upon their ratio, and always occurred when the ligand concentration spanned 0.01 to OSM. Furthermore, it was
The evaluation of Equation 2 suggests several important conclusions. First, Equation 2 is generally insensitive to stepwise metal complex equilibria and even the most carefully
conducted experiments will not produce appreciable curvature in the semilogarithmic plots of this expression unless the condition that pjCx’ >> 1 is not met. Considerable discussion in the literature (9, 19) has dealt with segmented curves in plots of Equation 2 when a mixture of complexes may exist. However, during the course of these computations, segmented semilogarithmic plots were never obtained. The apparent occurrence of a segmented curve from experimental data can only be attributed to the failure of the experimentalist to obtain sufficient data to thoroughly define this relationship, or failure to meet the conditions imposed upon Equation 2 (20). When the dominant successiveformation constants differed by less than 50, nonintegral ligand numbers were calculated from the slope of Equation 2. However, the presence of reasonable experimental error removed all curvature from this function but did not significantly change its slope. Meites (4) states that when this function is linear, a single complex predominates over the entire concentration range investigated, and that nonintegral ligand numbers cannot be ascribed to the existence of a mixture of complexes. This is in conflict with the above results and, in fact, a nonintegral ligand number clearly indicates the presence of a mixture of complexes. A linear relationship between A&z and log [CX]may be considered as a necessary condition, but alone is not a sufficient criterion to show that a single complex predominates over the particular ligand concentration range investigated. However, the condition of a linear relationship between AEllr and log [CX] and an integral ligand number, evaluated from its slope, constitute necessary and sufficient criteria to show the existence of a single predominant complex. The results shown in Table I and Figure 2, plus numerous additional computations, of the evaluation of Equation 3 indicated that the DeFord and Hume treatment of metal complex equilibria is the preferred expression upon which data analysis should be based. A least-squares curve fitting procedure should be used to evaluate the overall formation constants directly from F, (X), and not by thegraphical procedure suggested by DeFord and Hume. This permits statistical treatment of errors and intracomparison of results. However, other information concerning the possible species that exist must be considered and often this is the only way of arriving at the correct set of equilibria that may exist. Rossotti and Rossotti ( I ) , Irving (9), and Crow and Westwood (19) have discussed Equation 3 in terms of relating the measured half-wave potential data to the concentration of X at the electrode surface instead of the bulk concentration, This procedure must be questioned seriously, because if X is not present in large excess the shape of the polarographic wave, and consequently the half-wave potential, depends upon the total metal ion concentration as well as the surface ligand concentration with the result that the half-wave potential of the uncomplexed ion and the complexed ion cannot be readily related. This problem has been treated by Macovschi (21) and the original paper should be consulted for details. It should be noted that the polarographic method of studying metal complex equilibria based upon Equation 3 is somewhat limited when compared to the potentiometric method in spite of the fact that the latter method is based upon the same mathematical function. This is true because the potentiometric method does not require a large ligand to metal
(17) D. N. Hume, D. D. DeFord, and G. C. B. Cave, J. Amer. Chem. Soc., 73, 5323 (1951). (18) A. F. Krivis, G. R. Supp, and R. L. Doerr, ANAL.CHEM., 38, 936 (1966).
(19) D. R. Crow and J. V. Westwood, Quart. Rev., 19, 57 (1965). (20) J. H. Smith, A. M. Cruickshank, J. T. Donaghue, and J. F. Pysz, Jr., Znorg. Chem., 1, 148 (1962). (21) M. E. Macovschi, J. Elecrroanal. Chem., 20, 393 (1969).
empirically observed that the minimum % re1
nF
u ~ ~ / u A E,~,
RT was linearly related to the ratio of successive formation constants by the following expressions: min
zre1
u ~ ~ / u A EnF ~ , ,=
4.5 P1+1
RT min
zre1 UB~+I/UAE,,,,
+ 28
(5)
+ 154
(6)
Pr
nF Pl+2 - - 2.7 RT p5+1
These expressions are valid for /3 ratios from 1 to lo2 and predict the errors with an accuracy of approximately 3 for all 0’s greater than IO4, with the exception that the accuracy of Equation 5 decreases to about 10% for a P j of about 102. Third, if for reasons such as solubility, pH, or ionic strength buffers, one is unable to work in a particular concentration region shifts) precision in the calculated p’s may be sacrificed or may be gained. For example, chemical systems with p’s similar to those shown in Figure 2a,b result in greater errors if the solubility of the ligand was low, whereas, for a system having p’s similar to those of Figure 2c, one would be advised to use a maximum ligand concentration of OSM. Finally, chemical systems with small formation constants have a larger dynamic range of measurable half-wave potential shifts. Dynamic range is defined as the difference between the minimum and maximum AEI,z observable for a given chemical system. These lower and upper limits are determined by the ligand concentration range accessible to the experimentalist. The final case involved calculations in which &, and the initial values of CX were selected such that P3CX1 rvl. Polarographic studies of metal complex equilibria in which curvature is noted in the experimentally determined AE1i2us. log [Cx] plots generally fall into this category (17, 18). Extensive curvature was noted when the generated data was analyzed according to Equation 2. This is expected, since the conditions imposed upon Equation 2 were not met. Error perturbation calculations of Equation 2 were not carried out for this case. When the generated data were analyzed according to the DeFord and Hume function, all the significant formation constants were determined whenever reasonable UAB,,, values were assumed. However, as UAE,,, became greater than 1.5 mV, the smaller formation constants could not be included in the calculated model because negative coefficients were obtained or the standard deviation of the smaller formation constants were larger than the calculated value of Typical results are given in Table I. CONCLUSIONS
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
1237
ratio, resulting in a wider accessible dynamic range for the measured variable. Comparing the polarographic with the spectrophotometric method of studying association equilibria, errors of cornparable magnitude are indicated (22). (22) K. Conrow, G . D. Johnson, and R. E. Bowen, J. Amera Chem. Soc., 86, 1025 (1964).
RECEIVEDfor review July 17, 1969. Accepted June 18, 1970. Presented in part at the 157th National Meeting, American Chemical Society, Minneapolis, Minn., April 1969. Work supported in part by the Petroleum Research Fund through Grant No. PRF 1396-G2.3. Funds for commter time Provided by the University of Georgia Computer Center are gratefully acknowledged.
A Complexometric Titration for the Determination of Sodium Ion James D. Carr and D. G . Swartzfager Department of Chemistry, University of Nebraska, Lincoln, Neb. 68508 A method for the direct complexometric titration of sodium ion with CyDTA in the presence of other alkali metal ions, alkaline earth metal ions, transition metals, and rare earth metal ions is described. The method is accurate within 2% for sodium ion concentrations as low as molar and utilizes a sodium ion specific electrode for end-point detection. The stability constants for the 1:l complexes of sodium and potassium with CyDTA are determined to be 2.5 =t0.4 X lo4 and 33 + 2, respectively. The value of the pK4 of CyDTA is determined to be 13.17 0.08.
*
1 :1 complex of measurable stability with potassium (7). Since complexes of CyDTA are usually more stable by at least an order of magnitude than those of d,l-PDTA (8), the possibility of appreciable complexation between potassium and CyDTA was also investigated. The stability constant of sodium-CyDTA is measured potentiometrically and that of potassium-CyDTA is calculated indirectly. Kxacg
+ C Y D T A - ~ NaCyDTA-3 KKC~ K+ + CyDTAKCyDTA-
Na+ THERECENT DEVELOPMENT of the sodium ion selective electrode offers a fast, convenient, and relatively inexpensive method for the determination of sodium. The technique, however, suffers from the variation of the sodium ion activity with solution composition. In this work a method for the direct complexometric titration of sodium ion in the presence of excesses of other alkali metals, alkaline earths, transition metals, and rare earth metal ions is described. The method is sensitive to concentrations at least as low as molar with accuracy and relative precision of less than 2%, utilizing a sodium ion electrode as the means of endpoint detection. A variation on the technique also affords a method for the titration of lithium ion utilizing sodium ion as an indicator. Various other chemical schemes have been proposed for sodium analysis but many are imprecise and require tedious and frequent standardization for reliable results. The method described employs trans-l,2-diaminocyclohexane-N,N,N’,N’-tetraacetic acid (abbreviated here as CyDTA or Cy, abbreviated elsewhere as DCTA) as the complexing ligand. The sodium complexation of the aminocarboxylate multidentate ligands has been illustrated in several contexts in the past, but usually as a handicap to be overcome rather than as a useful analytical reaction (1-6). Recently it has been demonstrated that d,l(l,2-propylenedinitri1o)tetraacetic acid (abbreviated as d,l-PDTA) forms a (1) R. J. Kula, D. T. Sawyer, S. I. Chan, and C. M. Finley, J. Amer. Chem. SOC., 85, 2930 (1963). (2) J. L. Sudmeier and C. N. Reilley, Inorg. Chem., 5, 1047 (1966). (3) R. J. Kula and G. H. Reed, ANAL.CHEM., 38, 697 (1966). (4) J. D. Carr, K. Torrance, C. J. Cruz, and C. N. Reilley, ibid., 39, 1358 (1967). ( 5 ) G . Schwarzenbach and H. Ackermann, Helv. Chim. Acta, 30, 1798 (1947). (6) V. Palaty, Can. J. Chem., 41, 18 (1962). 1238
0
(1) (2)
Other alkali metal ions (except lithium) form much weaker complexes than does sodium and do not interfere with the titration. Alkaline earths, transition metal ions, and rare earths form much stronger complexes than sodium and may be titrated first to a visual or conventional potentiometric end point prior to the sodium end point. Though not directly demonstrated in this work, the tetraanion of CyDTA appears to be the only species which reacts with sodium ion to an appreciable extent. In order to have a large fraction of the total CyDTA species present as the tetraanion, the solution pH must be maintained considerably above the pK4 of CyDTA [calculated from the accepted best value (9) and ionization enthalpy (8) to be 12.29 at 25 “C, p = 0.1 with KNOB]. Potassium hydroxide, potassium orthophosphate, cesium hydroxide, and piperidine, all were used to raise the pH to these values without introduction of substantial sodium impurity. EXPERIMENTAL
All potentiometric measurements involving the sodium ion specific electrode (Corning Model No. 476210) were made with a Corning Model 12 expanded scale pH meter, The titrant was introduced with a 2-ml capacity Gilmont micrometer buret. When simultaneous pH measurements were required, a Sargent Model LS pH meter equipped with a Corning semimicro combination electrode was employed. Solutions and Reagents. All chemicals used were reagent grade whenever possible. The CyDTA was obtained from Apparatus.
~~
~
(7) J. L. Sudmeier and A. J. Senzel, ANAL,CHEM., 40, 1693 (1968). (8) L. G. Sillen and A. E. Martell, “Stability Constants of Metal Ion Complexes,” Special Publication No. 17, The Chemical Society, London, 1964. (9) G . Anderegg, Helv. Chim. Acra, 46, 1833 (1963).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
