Refinements of formation constants of the cadmium-thiocyanate

Douglas W. Franco , Eduardo A. Neves , Paschoal Senise. Journal of Electroanalytical Chemistry and Interfacial Electrochemistry 1975 60 (3), 341-347 ...
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Refinements of Formation Constants of the CadmiumThiocyanate System from Polarographic Measurements with More Statistical Considerations Kozo Momoki, Hisao Ogawa, and Hisakuni Sat0 Laboratory for Industrial Analytical Chemistry, Faculty of Engineering, Yokohama National University, Ooka-machi, Minami-ku, Yokohama-shi, Japan By statistically analyzing the effects of different supporting media on polarograms measured with a statistically discussed circuit, we can refine the formation constants of the cadmium-thiocyanate system and find a reasonable explanation for the “apparent” nonexistence of Cd(SCN)3- reported previously with potassium nitrate medium. Refined formation constants are presented where possible existences of not only Cd-N03 but also Cd-SCN-N03 and even Cd(SCN)2-N03, besides Cd-(SCN),-a, are indicated. By these studies, the importance of statistical considerations for this evaluation method i s again emphasized.

INOUR previous papers ( I , 2 ) on the calculation of successive formation constants from polarographic data (3) using a digital computer, the most probable values of the constants of the cadmium-thiocyanate system were derived statistically as most reasonable. With these results, the importance of statistical considerations in the evaluation procedure for complex formation constants was demonstrated. The papers ( I , 2), however, gave a critical contradiction to the usual understanding of the p3value for Cd(SCN)3-. Although our computerized calculation with polarographic data by Hume et al. (4) showed p3 = 0 statistically as with our data, the nonexistence of Cd(SCN)3- in the system studied is not easy to accept as a chemical possibility. Thus, whether p3 of the cadmium-thiocyanate system is really zero was left in the previous papers ( I , 2) as a problem for further study. The present work was started with the main purpose of studying this p3 problem by applying further statistical treatments since even if the p3 is not zero, the constant may be so small that whether it is zero or not would have to be tested statistically. Any statistical treatments, however, should be based on the most precise (and accurate as possible) experimental data. Also, it was suggested (2) that even slight changes of an experimental condition can give different values for not only the p3 but also the other 0’s. Because the last studies ( I , 2) concerned primarily the refinement of the calculation procedures in the evaluation method, the other parts to obtain necessary polarographic data and treat them for reasonable p values are now reexamined statistically in this paper. By such treatment, a reasonable explanation for the p3 problem has been achieved, while the most probable constant values for the system have been obtained as refined from these reported previously ( I , 2). Thus, we employ here a simple circuit using undamped recorders for precise measurements (5) and remeasure pre(1) K. Momoki, H. Sato, and H. Ogawa, Bull. Faculty Eng., Yokohama Nat’l Unic., 16, 127 (1967). (2) K. Momoki, H. Sato, and H. Ogawa, ANAL.CHEM., 39, 1072 (1 967). (3) D. D. DeFord and D. N. Hume, J. Amer. Chem. SOC.,73,5321

(1951). (4) D. N. Hume, D. D. DeFord, and G. C. B. Cave, ibid., 73, 5323 (1951). (5) L. Meites, “Polarographic Techniques,” 2nd ed., Interscience Publishers, New York, N. Y., 1965. 1826

RECORDER

f-

DME

\(*hi POTENTIOMETER

+ Figure 1. Circuit I for manual polarograph using a damping capacitor C

cisely the effects of different supporting media on the polarograms of the system studied (6). Our treatments for these problems are not new in polarography, but the treatments are made with sufficient statistical considerations in order to evaluate more reasonable formation constants, whereas most previous work has lacked these necessary statistical treatments. With the polarographic data thus obtained, we then reevaluate the formation constants of the cadmium-thiocyanate system to search an explanation for the p3 problem. Although Senise and Neves (6) took into account the presence of cadmium mononitrate besides cadmium-thiocyanate complexes for the similar situation measured in potassium nitrate medium, we take into account also some mixed complexes. To estimate the possible species of such mixed complexes for the system in nitrate media, we use in part the method of Schaap and McMasters (7). However, our treatments are controlled statistically and have given reasonable results which would not be obtained by simple applications of their method nor those of computer calculations. Mixed complexes including even cadmium-dithiocyanate-mononitrate species have thus been introduced reasonably to explain the fi3 problem, while the other p’s have also been refined statistically. Thus, we have refined the formation constants for the cadmium-thiocyanate system by introducing new statistical treatments into the evaluation procedures which have not heretofore been so statistical. The present paper deals with such experimental, as well as analytical, aspects to refine the previous p values. We emphasize, by these studies, that complex formation constants should be treated more carefully both experimentally and statistically. Chemistry of even weak mixed complexes as treated in the paper could be developed further by such precise experiments as well as by statistical analyses. (6) P. Senise and E. F. de A. Neves, J. Amer. Chem. SOC.,83, 4146

(1961). (7) W. B. Schaap and D. L. McMasters, ibid.,p 4699.

ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969

-

I

P

RECORDER

RECORDER

+

DME STUDENT

POTENTIOMETER

Figure 2. Circuit I1 for manual polarograph using undamped recorders PIand P3

0

I

I

I

t

2t

3t

TIME

Figure 3. Recordings on the undamped recorders PI and P3 shown in Figure 2

STATISTICAL CONSIDERATIONS OF POLAROGRAPHIC MEASURING CIRCUIT A Simple Circuit Using Undamped Recorders. To measure precise dc polarographic data manually for evaluating highprecision formation constants, two circuits usually used are compared with each other statistically. Figure 1 shows a simple manual polarographic circuit (Circuit I) using a damping capacitor C to minimize the oscillation of the electrolytic current as employed in the previous work ( I , 2). In this, the suppressed oscillation of the current is recorded on a recorder, PI(Toa-Denpa's EPR-2T as a microammeter), and the middle point of the recorded small oscillation at a given applied voltage was visually taken as the electrolytic current. The cathode potential is measured with a student potentiometer, P2, and a galvanometer, G, having an appropriate sensitivity as indicated. The oscillation of the potential was thus transferred to cause only the small oscillation of the pointer of G around the galva zero by adjusting P,. The adjusted P2 was then read as the cathode potential when the middle point of the oscillating pointer of G came also visually to be at the galva zero. Although the potential drop by a resistor R3 was further corrected, the cathode potential measured with Circuit I would still include some uncertain potential drops, including junction potentials, caused by cell resistances. Such uncertainties of the potential measurement should naturally be avoided as much as possible to obtain high-precision polarographic data from which high-precision p values are to be evaluated (2). For high-precision measurements, the constant potential devices using operational amplifiers have been constructed in which any potential losses caused by cell resistances can be eliminated (8, 9). However, a simple manual polarograph can be used for precise measurements at minimum uncertainties if the measurement is made at the end of the mercury drop life with an undamped recorder (5). Sawyer et af. (IO) used an X-Y recorder for this, whereas a most simple circuit shown in Figure 2 (Circuit 11) has been successfully used in the present work. In Figure 2, the electrolytic current is measured on an undamped recorder, P8, of a usual millivoltmeter type (Yokogawa's LER-12A) which has a minimum full scale of 1 mV and a span speed of less than 1 sec for the 25-cm full span. The recorder locus of the current is shown in Figure 3 where the current maximum A at the end of the mercury drop life ~

(8) M.T.Kelley, H. C . Jones, and D. J. Fisher, ANAL.CHEM., 31, 1475 (1959). (9) P. Arthur and R. H. Vanderkam, ibid., 33,765 (1961). (IO) 0.T. Sawyer, R. L. Pecsok, and K. K. Jensen, ibid., 30, 481 (1 958).

was read on the chart as the electrolytic current at the given applied voltage. The corresponding cathode potential is also recorded on an undamped recorder, P1,(the same PI as in Figure 1 was used) which has a minimum full scale of 10 mV and a span speed of less than 0.3 sec for the 15-cm full span. The minimum potential B in Figure 3 at the corresponding fall of the mercury drop to the case of A was then marked on the chart. A potentiometer, P2 (the same P2 as in Figure 1 was also used), was adjusted to give an input potential of zero volts for PI at the minimum potential B, and the adjusted PZ was read as the appropriate cathode potential at the given applied voltage. The further correction of the potential drop by a resistor R 4was neglected as small, because only a small R4is needed in Circuit 11. Experimental. Circuit I1 was compared statistically with Circuit I in evaluating p values of the same cadmium-thiocyanate system from the polarographic data obtained. The solutions with the same supporting electrolyte of potassium nitrate as in (2) were measured at the same temperature of 25.0 =k 0.1 "C, where the constant concentration of cadmium ions was 0.956 X 10-aM against the previous 1.20 X 10-3M. The dropping mercury electrode, electrolytic cell, liquid junctions, and two SCE's (2) were essentially the same as in the previous work (2). The residual currents were also corrected with the polarograms measured in the solutions of the supporting medium alone as usual, as in the previous work with Circuit I. The evaluation of half-wave potentials from the polarographic data obtained were carried out with a computer, while the calculations of p values were also made with the computerized l/Foz-weight method ( I , 2). The computer used was the same NEAC-2230 in the previous work as in the following calculations. Results and Discussion. Table I shows the calculated /3 values from polarographic data obtained with Circuit I1 compared with the previous values which are taken from Table I1 in Ref. (2). These results in Table I are obviously very interesting. With Circuit 11, the most probable p3 value is obtained again as statistical zero ( I , 2), as with Circuit I, while generally smaller 9 5 % confidence intervals are also obtained for all p by assuming p3 = 0 than by p3 # 0. The most probable values themselves for each PI, p2, and p4will be noticed as not so much changed by assuming p3 = 0 from those by p3 # 0, with a circuit of either I1 or I. However, each most probable p value with Circuit I1 is obtained as rather considerably different from corresponding /? value with Circuit I. And further, Circuit I1 is found to reduce the confidence intervals

ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969

0

1827

~

~~

~

~~~~~

~~~

~~

Table I. Comparison of Calculated Successive Formation Constants for Cadmium-Thiocyanate System from Polarograms Measured with Circuits I1 and I P1

Circuit I1

Pa

P3

P2

1 1 , O f 1.9 11.1 f1.1 12.6 f 4 . 8 13.2 f 1 . 9

f 0 =0

70.7 f 12.9 - 1 . 7 f 21.9 69.7A 3.5 0 P3 Circuit I(2) Pa # 0 50.4 11.5 - 5 . 8 f 17.4 4 6 . 6 f 5.6 0 Pr = 0 All the formation constants are evaluated as (the most probable value) f ( 9 5 x confidence interval). See Ref. (2). P3

~~~~~~~

~~

*

~

~

~

more than Circuit I did for all /3 by introducing the assumption of p3 = 0. Circuit 11, with the assumption of p3 = 0, is thus shown to give the smallest confidence intervals and the most reasonable probable values for all 4. However, the fact that the most probable values obtained for each p are considerably different between the two circuits should be of much practical importance. The differences of these p values can naturally be traced back to the differences of the polarographic data obtained with the two circuits. Figure 4 shows the displacements of the half-wave potentials by the complex formation (AEl,%= E1/zs - E1izc) as well as the slopes of the log-plot for each El,*$or El/zC, where the plots are made against the concentration of thiocyanate ions. El,zs indicates the half-wave potential measured without the ligand while El,zcthose with the ligand. Each E1/zSor Ellzcmeasured with Circuit I1 was determined from 16 points of the measurements made on each wave, while with Circuit I from 9 points. Because the same number of such measurements is better employed in Figure 4 for comparison, 9 points are arbitrarily chosen there also for data with Circuit 11. In Figure 4, slight differences of measured AEI~z between the two circuits are noticeable in the middle range of [SCN-] than at smaller and larger [SCN-1. This fact may be responsible for the larger difference of the p~ value rather than that of p1 or p4between the two circuits as shown in Table I. Yet only about 2 mV should be the maximum difference in the observed AEll2to yield such different p values. On the other hand, the slopes of the log-plot show a clear difference between the two circuits. Except at [SCN-] = 0, all the slopes measured with Circuit I have definitely smaller values than the ideal one of 29.58 mV for n = 2 and would not be favored from the thermodynamic standpoint. Because these measured values will be reduced further by applying the correction for cell resistances, if the latter can be evaluated, the slopes measured with Circuit I must become more unfavored. On the other hand, the dopes measured with Circuit I1 occupy thermodynamically favorable positions in Figure 4 with less deviations among these plotted points than those among the slopes with Circuit I. The slope at [SCN-] = 0 measured by Circuit I agrees almost with that by Circuit 11. Thus, the former circuit might be given somewhat deformed oscillations for the pointer of the galvanometer G when the measurements were made with the additions of thiocyanate ions. Somewhat biased potentials toward minus direction with some large deviations must have thus resulted with Circuit I. From the above observations, it will readily be expected that the polarographic measurements at least of the present system can be made more precisely with Circuit I1 than with Circuit I. This point was ensured further by statistical comparisons of their precisions in two ways. First, the estimated variances ug1lZ2of the half-wave potentials derived from 1828

76.6 f 9 . 9 75.8 f 2 . 2 87.8 f 7 . 1 85.5 f 3 . 1

[SCN-) Figure 4. Displacements (AEljJ of half-wave potentials by complex formation and slopes of the log-plot

-. 0.

- -.

with Circuit 11; AEI/* with Circuit I slope with Circuit 11; 0.slope with Circuit I

AEljZ

each log-plot line (2) are plotted against [SCN-] as in Figure 5 . The same 9 points of the measurements as in Figure 4 are 1 2 ~ .5 employed for evaluating each of these ~ ~ ~ Figure clearly indicates that Circuit I1 can give more precise halfwave potentials for the log-plots with less deviations than Circuit I. Second, the deviations of measured log-terms around each log-plot line are subjected to the F-test for significant difference between data by the two circuits. Sixteen points of the measurements were used for Circuit I1 while 9 points were used for Circuit I. Experimental F for nl (Circuit 11) = 133 and n? (Circuit I) = 154 was obtained as 3.57 which obviously exceeds theoretical F at 1 significance level of 1.53 for nl = n2 = 120. Again, these F values show that Circuit I is far less precise than Circuit I1 and the latter circuit should be preferred to the former for the purpose of obtaining high-precision /3 values. Thus, more reasonable values of the most probable /3 with smaller confidence intervals have been found to be obtained with the simple, but precise, Circuit I1 than with previous Circuit I. The /3 values with Circuit I1 in Table I are thus

ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969

2.0

- I X

lo

loot

>

&

E

0

I;I"SO

T

0

(sc ri] Figure 5. Variances of half-wave potentials derived from each log-plot line

a. with Circuit II;

0

1.0

2.0

Figure 6. Differences of measured polarograms by different supporting media for cadmiumthiocyanate system

a. NaCIO4;

0. with Circuit I

A.

thought to be refined from the previous values with Circuit I in Ref. (2). Although the serious p 3 problem for the system still remains as unsolved with Circuit 11, it should be noted that only about 2 mV of differences in measured AE1,2 can cause such considerable differences in the @ values evaluated for even the same complex formation system. With this, we would like to emphasize again the importance of careful experimental work which is necessary for high-precision complex formation studies.

SODIUM PERCHLORATE AND SODIUM NITRATE AS THE SUPPORTING ELECTROLYTES

It is now well known in polarography that even the same complex formation system often gives different polarograms when using different supporting electrolytes (5). DeFord and Andersen (11) emphasized the necessity of considering the effects of ionic strength and ionic environment in the polarographic investigation of complex formation. Accordingly, different groups of @ values can naturally be obtained for one system if the polarograms are obtained in different supporting media. The cadmium-thiocyanate system was measured in sodium perchlorate medium by Senise and Neves (6) who then derived PI = 25, f12 = 75, f13 = 85, and p4 = 240 as considerably different from those in the previous potassium nitrate medium shown in Table I. Although they ascribed (6) this discrepancy to the formation of CdN03+ (12,13) according to Vanderzee and Dawson (14), it is especially to be noted that they gave a definite plus value for p 3 against our result with K N 0 3 medium. This point was suspected to be related to the p3 problem. Thus in this section, sodium perchlorate is first re-examined as the supporting electrolyte for measuring the cadmium(11) D. D. DeFord and D. L. Andersen, J. Amer. Chem. SOC.,72, 3918 (1950). (12) E. C . Righellato and C. W. Davis, Trans. Faraday SOC.,26, 592 (1930). (13) I. Leden, 2.Physik. Chem., 188A, 160 (1940). (14) C . E. Vanderzee and H. J. Dawson, Jr., J. Amer. Chem. Soc., 75, 5659 (1953).

0. NaN03; X. KNO3(2) NaClO, by Senise and Neve ( 6 )

thiocyanate system with the high-precision Circuit 11. At the same time, N a N 0 3 is also used as another supporting electrolyte because anion effects of these electrolytes on the complex formation can better be compared under a common cation. Unfortunately, potassium cations cannot be used as the common cation because of a small solubility of potassium perchlorate. For the same reason, the two SCE's used in the K N 0 3 medium (2) were replaced in the new media with saturated NaC1-calomel electrodes and sodium salts were also used in the agar-agar junction bridges in each supporting medium. In the experiments, cadmium perchlorate was used in the NaCIOa medium and prepared from reagent-grade cadmium nitrate through the hydroxide, which was then dissolved into perchloric acid solution. Reagent-grade cadmium nitrate was directly used in the NaN0, medium. The concentrations of cadmium ions in the solutions were determined with EDTA titration and found to be 0.976 X 10-3M in the NaC104 medium and 0.952 X 10-3M in the NaN03 medium. All of the polarograms were measured with Circuit I1 at the ionic strength of 2.0 and a temperature of 25.0 =t0.1 "C, using the same electrolytic cell and dropping mercury electrode as in the previous section. Figure 6 shows differences of the measured polarograms by the different supporting media. As expected, large differences of AEl/*are seen between the plots for the NaC104 and NaN0, media, while the differences between the NaN03 and KNOB media are only very slight. Because these electrolytes are all simple uni-univalent salts and, on the abscissa, they are continuously replaced with changing NaSCN which is also a uni-univalent salt, the activity coefficients of cadmium ions in such solutions may not be changed so much, except by some complex formation. The junction potentials involved in measuring or E1/zCmay be changed to some extent from a plot to the others, but most of these terms in one supporting medium would be canceled out in deriving each AEli2by subtraction. It will be noticed that the rather large differences of A E ~ , z between the NaC104and NaNO, media come mainly from the difference in for cadmium aquo complexes between the two media, as shown also in Figure 6. The latter difference

ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969

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Table 11. Comparison of Calculated Successive Formation Constants for Cadmium-Thiocyanate System from Polarograms Measured in Different Supporting Media

Medium NaCIOl

P2

PI

83 Pa

NaC10p

83 f P3 =

graphical (6) NaN03

83 Pa

18.3 f 1 . 4 14.6 i 1 . 4 23.3 f 4 . 2 20.6 f 3.5 25 11.9 C 1 . 3 10.6 f 1 . 1

0 = 0 f

0 0

# 0 = 0

Pa

P3

137 f 13 178 f 6 7 7 . 5 i 45.8 115 f 19 75 6 3 . 3 f 10.0 75.2 f 3.9

87.6 f 26.4 0 93 f 105 0

85 2 0 . 6 i 16.6 0

178 221 238 2S5

f 14 C 6

f 56 C 20 240

49.1 f 7 . 0 5 7 . 6 f 1.9

Polarographic data given in Ref. (6) were subjected to the computer calculation.

F

Figure 8. F-function plots for cadmium-nitrate system

I

I

I

0

2,b

I

I. 0 [SCN-)

Figure 7. Weighted residuals between measured and calculated FOvalues in the NaC104 medium

calculated Fo with assuming p3 # 0 for calculated Fo with assuming p3 = 0

0. for

X.

of about 15 mV seems too large to be simply considered as coming from the differences of junction potentials and activity coefficients. And further, this El,z. in the NaC104 medium is actually located more positive, by some 15 mV, than that in the NaN03 medium. This offers a basis for further analyses of the constant values in these media, although the KNO, medium cannot be compared directly with the others for the experimental reason, as will be seen. With these different polarographic data, the p values calculated for the systems also differed in the different supporting media, as seen in Table 11. Some interesting features will soon be noticed. The improvement of the 95% confidence intervals by assuming ,f33 = 0 seems to hold for the results in the new media. It should be noted, however, that p3 in the NaC104 medium is obviously given a definitely plus value which is very close to p3 = 85 given by Senise and Neves (6). On the other hand, their polarographic data which Table 111. Sum of Squares of Weighted Residuals (Fitness Value) between Measured and Assumed Fo Values

Assumption Medium NaC104 NaN03 KNOI

1830

*

A.

@3

# 0

0.853 X 0.783 X 0.431 X

B. @ 3 = 0 2.07 X 1.73 X 0.432 X

Ratio @/A)

2.4 2.2 1 .o

'

differed only slightly from ours (Figure 6) were subjected to the computer calculation and found to give a p3 value almost statistically equal to zero, even assuming p3 # 0, as shown in Table 11. Also in this table, even the NaNO, medium is shown to give not statistical zero but a slightly plus value for p3. Although these observations again suggest bases for further analyses, only the results of statistical tests to check the existence of the p3 species in the new media will here be described. First for the NaC104 medium, weighted residuals between measured and correspondingly calculated Fo are plotted against [SCN-] as in Figure 7. The calculated FO values are derived from the evaluated p values under the assumption of either p3 = 0 or p3 + 0, and the plottings are made to show the distribution of the measured FO values around each calculated FO curve. The plotted points for p3 = 0 in Figure 7 are seen to be distributed with less of the desired randomness around the abscissa than those for p3 # 0. This suggests that the latter assumption would have given a calculated Fo curve around which the measured Fo values are distributed more at random, and thus the more reasonable ,f3 values would be obtained. Because the sum of squares of such weighted residuals for each assumption gives each fitness value of the measured FO values, the sums of squares of the ordinate values for p3 = 0 or p3 # 0 as in Figure 7 were secondly calculated for the three media and compared with each other in Table 111. The table indicates that these fitness values are improved by employing the assumption of p3 z 0 rather than p3 = 0 only for the new media. This again suggests that the assumption of p3 = 0 can be applied only in the K N 0 3 medium, while that of p3 # 0 must be favored more than = 0 in the new media though the NaN03 medium did not give a clear result in the above distribution test. Thus, considerably different /3 values of the same cadmium-

ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969

thiocyanate system obtained with the three different supporting media are statistically ensured now as reasonable. For such statistical estimations, not only most probable fl values and their confidence intervals as in the previous papers ( I , 2) but distributions and fitness values of measured FO values around assumed Focurves have been newly considered. These improve the sureness of the statistical considerations as mentioned above. Although further treatments are still required to solve the P3 problem more quantitatively, such treatments themselves should also be based on these high-precision polarographic data as well as thus-refined P values with sufficient statistical considerations. REFINEMENT OF THE FORMATION CONSTANTS FOR CADMIUM-THIOCYANATE SYSTEM WITH CONSIDERING MIXED COMPLEXES IN THE NITRATE MEDIA AND THE NATURE OF THE 63 PROBLEM

The observations in the above section, especially in Figure 6, suggest that nitrate ions in the nitrate media might take part also in the cadmium-thiocyanate complex formations. The additional complexes, if formed, may be related to our result in which the assumption of P3 # 0 is more favored in the new NaC104 and NaN03 media than P3 = 0, while the latter assumption seems merely "apparent" in the original K N 0 3 medium. To explain the difference between K N 0 3 and NaCIOl media for the complex formation, Senise and Neves (6) considered simply cadmium mononitrate (12, 13) with the same kind of calculation as for cadmium-chloro complexes made by Vanderzee and Dawson (14). Now with our refined measuring and computer techniques, possible complex species containing nitrate ions, even as mixed complexes in these media, could be sought more thoroughly. In the following treatments, perchlorate ions are assumed to be incapable of forming complexes at all (15), in accordance with usual treatments in complex formation studies. In addition, the complex formations of nitrate ions will be analyzed mainly in N a N 0 3 and N a N 0 3 NaC104 media for the experimental reason mentioned previously. In the K N 0 3 medium, nitrate complexes can not be measured directly, but will be discussed in the final stages of this section. To begin with, the complex formation between cadmium and nitrate ions was checked polarographically. The measured solutions were kept at the ionic strength 2.0 with NaC104 where N a N 0 3 was added increasingly up to 2.OM in replacing [C104-] with [NO3-]. The constant concentration of cadmium ions was the same as in the previous NaC104 medium while the polarograms were measured with Circuit I1 as in the following experiments. Because of very weak complex formation between cadmium and nitrate ions as expected, only 14.5 mV was again observed as the maximum AE1/z at the maximum [NO3-] of 2.OM. The F-function plots gave Fl a linear line which is almost parallel to the [N03-]-axis as in Figure 8, which indicates that mono- and di-nitrate complexes are formed in the solutions. These polarographic data were then subjected to the computer calculation, where KO = 0.78 =t 0.05 for Cd-NO3 and KO' = 0.14 f 0.04 for Cd-(NO& were obtained as the formation constants. The corresponding value given with conductometry by Righellato and Davis (12) was KO= 0.374 as the true constant, while that with potentiometry by Leden (13) was KO = 1.3 as the conditional one at the ionic strength 3. Vanderzee and Dawson (14) also gave KO = 0.7 f 0.3

+

( 1 5 ) F. A. Cotton and D. L. Weaver, ibid., 87,4189 (1965).

as the conditional constant at the ionic strength 3 in the above indirect evaluation. No one has reported about Cd-(NO& yet. It should be noted, however, that the experimental validity of such small constant values as here obtained often becomes very questionable. Under the strict conditions required for the F-function method from polarographic measurements (2, 3), too weak a system can be given erroneous or rather experimentally undefined constant values. For this reason, the existence of the above very small KO' must not easily be concluded if only with the present experiment. On the other hand, the KO seems to be assumed as more possible at a value around 1 because the similar KOvalues have been given with the different methods as mentioned above. With expecting that at least mononitrate species will thus be probable, possible complex species containing nitrate ions in the nitrate medium were then checked further with the method proposed by Schaap and McMasters (7). Their method is based on considering mixed complexes such as Cd-(SCN),-(N03), in addition to usual Cd(SCN)j and writing the half-wave potentials El,2cJof the solutions in a form with additional terms for the former species to Elizcfor the latter. The displacements of El/zC'by the complex formation from El,zS,the latter for aquo complex, can be written similarly, as AEiiz' = Ei/zs -

(

1

In

EiizcJ =

[

(RT/nF) ln(ld,'/Ids) m

n

+ c PjcZj+ c c I:,

+ (1)

K ~ , C ~ ~ C Y ~ ) ]

n=O n=l

where the last term represents the additional mixed complexes term in which K,, is the overall successive formation constant for Cd-(SCN),-(NO,),. The above KO and KO' thus correspond now to KoL and KOZ,respectively. And also, C, and C y represent the concentrations of thiocyanate and nitrate ions, respectively. Now, it will be noticed that Equation 1 can give a series of the F-functions, if the measurements are made with varying C, at a constant C y under the constant concentration of cadmium ions, as ~

0

= %

1

+

3

3=1

P~c,'+

m

n

m=O n = l

(Ids/Idcr

KmnCzmCyn =

- exp[(nF/RT). AE112,'l

(2)

here the suffix i shows the ones measured at varying C,r with the constant Cy. We assume here that the maximum coordination number of cadmium ions is always 4 as in the previous work (1, 2). This allowsj 5 4 and m n 5 4 in Equation 2, and the latter can be rewritten as

+

Fo, = A

4-B - C , , + C * C z , ,+ D * C , , a+ E*C,,4

(3)

where A J: 1 B = Pi C = Pz

D

=

P 3

+ KoiCr + KozCu* + Ko3Cy3 + Ko4Cu4 + k i C u + KizCr2 + Ki3Cu3 + K ~ C Y+ KzzCu2 + &ICY

(4)

E = Pa Equation 3 obviously indicates that the parameters A , B, C, D , and E can be evaluated graphically o r by computer calculation just as in the ordinary F-function method with C, as the variable. This evaluation is made possible in the

ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969

1831

above experiments in which C, is varied at the constant CY. Since each parameter has a form of F-functions with CY as the variable in Equation 4, the values of K,, could also be obtained with the F-function procedures. Schaap and McMasters (7) thus showed that, if the above experiments of changing C, are repeated at several levels of CY,the formation constants Kmnof additional species can be evaluated also graphically. However, the evaluation of these parameters even at one level of C y in the F-function method with changing C, may sometimes be questioned for their precision (2). The method which further requires such repeated evaluations at several Cy levels will thus be safe to be used rather qualitatively except for some simpler systems. In the present study, this

method was applied only to check the sorts of possible additional species as follows: CY was taken at 5 levels from 0.2M to 1.OM in every 0.2M while C, was changed from OM to 1.OM to take 12 different values at each CY level. The ionic strength of the solutions was kept again at 2.0 with NaC104 while the constant concentration of cadmium ions was given as the same as in the previous NaC104 medium. The polarograms were measured and the values of the parameters were then evaluated by the computer calculation as in Table IV. In Table IV, only A and B are noticed as dependent almost linearly on C y ,whereas the dependence of the other parameters on Cy cannot be discussed because of the poor precision obtained. The latter result seems to come mainly

Table IV. Estimations of Parameter Values in Equation 3 at 5 Levels of CY by Varying C, CY,M A B C D 1 8 0 k 46 -4.4 i 100 0.200 1.3 f 0.3 17f 7 124 f 68 176 f 150 0.400 1.6 f 0.4 22 f 10 117 f 83 176 f 180 0.600 1.5 f 0.5 29 i 13 172 f 100 109 i 220 0.800 1 . 9 f 0.6 22 i 15 112 f 110 212 f 230 1.Ooo 1.7 f 0.7 37 f 16 Cy represents the concentration of nitrate ions while C, that of thiocyanate ions.

E

227 f 67 93 f 97 109 f 120 116 i 140 98 + 150

-

Table V. Polarographic Data for Cadmium-Thiocyanate-Nitrate System Slope of [SCN-1 [Nod-] -E111 u ~ l i 2 tx 109 the Log-Plot Id l.lOVt 30.98 mV 6.80 FA OM OM 0.5594 V 6.70 0.857 30.66 0.0500 0.200 0.5716 6.68 1.18 30.89 0,0500 0.600 0.5745 0.620 30.40 6.65 0.400 0.5805 0.1Ooo 30.53 6.58 1 .Ooo 0.5835 1.17 0.1Ooo 0.740 30.38 6.55 1.80 0.5868 0.1Ooo 6.54 0.885 30.54 0.200 0.600 0.5924 6.56 1.69 30.63 0.200 0.800 0.5931 6.45 1.67 30.28 1.30 0.5953 0.200 6.50 0.958 30.20 0.200 1.70 0.5969 6.45 1.79 30.27 0.200 0.5998 0.300 6.45 2.37 30.41 1.Ooo 0.6022 0.300 6.50 5.75 30.87 1.50 0.6032 0.300 30.80 6.48 0.400 0.6076 1.92 0.400 0.909 30.78 6.47 0.400 0.800 0.6086 6.40 1.OO 30.64 1.40 0.6098 0.400 6.40 1.48 30.62 1.60 0.6105 0.400 6.44 0.999 30.69 0.600 0.6139 0.500 30.61 6.40 1.20 0.6154 1.32 0.500 6.40 0.709 31.07 0.200 0.6186 0.600 6.38 1.01 30.70 1 .Ooo 0.6206 0.600 1.59 30.61 6.36 1.30 0.6212 0.600 6.41 0.551 30.47 0.400 0.6239 0.700 6.42 1.26 31.03 0.800 0.6245 0.700 6.41 0.748 31 .OO 0.600 0.6286 0.800 6.34 0.986 30.46 1.20 0.6298 0.800 6.30 0.961 30.70 0.200 0.6322 0.900 6.33 1.66 31.04 1.Ooo 0.6337 0.900 6.33 1.55 30.74 0.400 0.6362 1.Ooo 6.40 1.28 30.93 0.800 0.6365 1.Ooo 6.28 2.95 30.99 0.300 0.6430 1.20 6.30 0.684 30.36 0.700 0.6435 1.20 6.27 1.80 30 63 0 . 100 0.6457 1.30 6.26 0.606 30.62 0.500 0.6460 1.30 6.26 1.60 30.31 0.700 0.6464 1.30 6.25 1.29 30.67 0.500 0.6495 1.40 6.25 0.553 30.74 0.300 0.6546 1.60 6.24 0.293 30.89 0 0.6579 1.70 6.21 0.987 30.58 0. 100 0.6596 1.80 6.19 0.867 30.82 0 0.6625 1.90 Each wave was measured at 16 points. The ionic strength of the solutions was kept at 2.0 with NaC104. The temperature was 25.0 f 0.1 I

"C.

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ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969

No. 1 2 3 4

Table VI. Evaluation of Formation Constants for Cadmium-Thiocyanate-Nitrate System P Z P3 Pa Kat Ki I K21 K31 1 7 . 9 f 4.3 1 3 9 f 39 89 f 76 162 f 34 1.3 f 0 . 6 8 . 0 f 8 . 4 40 f 35 -5.8 f 41 PI

17.7f4.1 1 9 . 6 f 5.0 5 . 8 f 8.0 5 18.3 f 1 . 4 a Sum of squares of

142f33 1 1 6 f 39 2 5 5 f 52 1 3 7 f 13

83.3f64.3 163 f 66 -29 f 105 8 7 . 6 f 26.2

164f30 126% 31 1 9 7 f 54 1 7 8 f 14

1.28f0.60 8 . 6 f 6 . 9 0 . 4 f 0.6 22 f 4 2.7 f 0 . 6

35.6f16.2

SSRa 0.0170 0.0171 0.0277 0.110 0 .oO853

residuals. Table VII. Comparison of Measured and Calculated p* Values in the NaN03 Medium P2*

PI*

Measured" 11.9 f 1 . 3 Calculated 10 0 Taken from Table I1 for comparison.

from the experimental condition at the given ionic strength of 2.0 under which larger C, and C y suited for estimating such weak successive complexes could not be given. The former result appears more likely to be consistent with the previous observation in which only mononitrate species formed with cadmium ions will become probable. Thus, we reach the assumption that only mononitrate species can take part in the cadmium-thiocyanate complex formation in the nitrate media. All the species Cd-(SCN),NO3 can be considered there under the first assumption which is now reduced to m = 3. This drops the terms more than quadratic from Equation 4, but the above method by Schaap and McMasters will not be applied directly because of its poor precision expected also from Table IV. Accordingly, we decided to use our computer calculation to solve directly the following equation as the observation equation for P j as well as Kml with newly measured polarographic data: 1

+ PlC, + PaC,2 + P3CZ3 + + KoiCv + KiiCXy + KziCz2Cy+ K3iCSaCy = Fo P4CZ4

(5)

In the new measurements, the ionic strength of the solutions was kept also at 2.0 with NaC104 and the constant concentration of cadmium ions was again taken at 0.976 X 10-3M as in the previous NaC104 medium. Forty different combinations of C,([SCN-] as NaSCN) and CY([NO3-] as NaN03) were taken to measure the polarograms as in Table V which also shows the obtained data. These data were then subjected to the computer calculations where, as in the previous paper (2), several calculations were repeated with dropping some terms from Equation 5 to see what a combination of such species can be given most reasonable constant values. Many other combinations including even di- and tri-nitrate species were similarly tested but without appreciable results. The values evaluated for the combinations including only mononitrate species seemed most valuable and are tabulated in Table VI. In Table VI, it will be noticed that Numbers 1 and 4 are meaningless because they contain minus values in their most probable constant values. Numbers 1 and 4 are discarded also by inspecting the SSR values shown in the table. In spite of the fact that such values are usually reduced by increasing the number of variables, the value of Number 1 is not improved so much from that of Number 2. Number 1 is thus unfavored while Number 4 is also unfavored because of its large SSR value. To select then either of remaining Numbers 2 or 3 as more reasonable, another standpoint should be considered.

P4*

P3*

63.3 f 10.0 56

20.6 f 16.6 15

49.1 f 7.0 50

Namely, these constant values are derived from polarographic data measured with NaC104 as the main supporting medium where perchlorate ions are assumed fundamentally as not taking part in any complex formations. Thus in Table VI, the P values themselves evaluated for the cadmiumthiocyanate complexes even in the presence of nitrate ions must not differ from those evaluated in the NaC104 medium without nitrate ions. Because the latter @ values are shown as Number 5 in the table for comparison, it is evident that Number 2 would be most favored in giving the best general agreement between each P value with that in Number 5 , along with the reasonably small SSR value given for the former combination. This is quite an unexpected result because we now have to believe the existences of not only Cd-NO3 but Cd-SCN-NO3 and even Cd-(SCN)2-N03. If, however, the result tells the truth, the P, and Kml values of Number 2 in Table VI must explain quantitatively the differences of the most probable 0 values by the supporting media in Table 11. If such a quantitative explanation is made possible, we would not only obtain sure proof for the validity of the above unexpected result, but would supplement our previous papers for the critical P3 problem. Now consider that the half-wave potential El/lSof cadmium ions in the NaC104 medium without thiocyanate nor nitrate ions is assumed as that for the pure cadmium aquo complex. When we measured the polarograms in the NaN03 medium for Table 11, the half-wave potential measured without thiocyanate ions was not equal to El/2$ but actually to which should have already been that of the cadmium mononitrate complex. E1,2s' was thus displaced negatively from E1,Zs as shown in Figure 6 and should have been written as

+

EI/Z~' = E1/z9 - (RT/nF>[ln(Zgd'/Zd ln(1

+ K o ~ C Y " )(6)~

where CY"is the concentration of nitrate ions given when was measured. Thus, the displacements of the half-wave potentials Elizc measured in the NaN03 medium should have been taken not from as in Equation 1 but actually from Foshown in Equation 2 should then have been written as

Fo

=

(Zds'/Zde').exp[(nF/RT)(E~/29' - E~/P~')I =

/

(1

+

4

PIC,' j-1

+

2

\ I

Kmlczmcy)/(l m=O

+ KOlCY")

(7)

Since Cyo= 2.0 and CY = 2.0 - C, were given experimentally for the NaNO, medium as mentioned previously, these

ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969

1833

relations are inserted into Equation 7 where Fo is obviously obtained in a form expanded in the order of C,’s powers. The coefficients in the C,’s power terms contain only thermodynamic constants such as & and Kml and can thus be expressed with new notations Pj* in writing FOas

Fo,= 1

+

P1*Czi

+ P2*Czi2 +

P3*Czi3

+ P4*Czt4

(8)

where the suffix i indicates the ones measured at C, = Czd, and

- Kol + 2.0K11)/(1 + 2.OKoi)

PI*

= @I

P2* P3*

=

P4*

= P4/(1

+

+

(Pz - Kii 2.0K21)/(1 2.OKoi) = @3 - Kzi)/(l 2.OK01)

+

(9)

+ 2.oKoi)

This means that, when we evaluated the constant values

for the NaN03 medium in Table I1 by applying Foi values measured with changing Czi, we calculated actually Equation 8 for these @,* not for & which were thought at that time. Thus, the P, values described in Table I1 are in fact the measured &* values, while the /3,* values can obviously be calculated in Equation 9 by applying the P, values of Number 5 and the Kml values of Number 2 both in Table VI. The comparison between the measured and calculated Pi* values is shown in Table VI1 which gives surprisingly good agreement between each corresponding @,* value. The agreements are quite surprising because we used the /3, and especially Kml values showing not so high precision in Table VI for the calculation. Thus, the formations of not only Cd-N03but Cd-SCN-N03 and even Cd-(SCN)2-N03 species, besides Cd-(SCN)j, in the N a N 0 3 medium have been revealed quantitatively as more reasonable, although unexpected. The most refined P values for the cadmium-thiocyanate system should thus be those shown in Number 5 in Table VI which were measured with the NaC104 supporting medium. The cadmium-thiocyanate complex formation in the NaN0, medium must be stabilized by the formation of the above additional complex species containing nitrate ions. For the K N 0 3 medium, such a quantitative discussion is prevented unfortunately by the experimental difficulty mentioned previously. However, the similar species including mixed complexes may also be probable even in the KNOI medium, since such differences as between Na+ and K+ with the same NO3- in evaluating formation constants have been shown to be only slight (16). The slight differences shown in Table I1 between these two media might come mainly from the differences of activity coefficients and junction potentials. If the same complex species as obtained above in the NaNO, medium can be assumed also in the K N 0 3 medium, it will be noticed that the nature of our P3 problem can be reasoned (16) V. E. Mironov, Zhur. Neorg. Khim.,6 , 659 (1961): C. A., 56, 111836 (1962).

1834

at least qualitatively. Because we evaluated actually the

p*’s not 4’s the last time for the KNO, medium, the P3* value might have been given as almost zero or a slightly minus value in Equation 9 by the true P3 value which happened to be nearly equal to, or slightly smaller than, the KZ1value. This must have made us assume P3 = 0 for Cd(SCN),- in the previous papers (1, 2). But now we believe more firmly that the P3 = 0 is thus only “apparent” in the K N 0 3medium and not essential for cadmium-thiocyanate complex formation itself. Each of the P, and Kml values seems to be still different slightly or considerably, depending on the sort of each complex species between the NaNO, and KN03 media. However, further quantitative analyses of such differences may not be made soon, although the association constants have been considered for these cases (17). CONCLUSION

We have treated experimental and analytical refinements of the formation constants for the cadmium-thiocyanate system from polarographic measurements with considerable statistical considerations. By statistically analyzing the effects of different supporting media on the polarograms measured with a statistically discussed circuit, we can refine the previous formation constants of the system in finding a reasonable explanation for the P3 problem. The following concentration formation constants at the ionic strength 2.0 are presented as refined by assuming that perchlorate ions do not form complexes and sodium ions are given as the supporting cation: PI = 18.3 + 1.4 for Cd-SCN, pz = 137 ==! 13 for Cd-(SCN)2, P3 = 87.6 i 26.2 for Cd(SCN)3, p4 = 178 i 14 for Cd-(SCN)4, KO, = 1.28 f 0.60 for Cd-N03, Kll = 8.6 i 6.9 for Cd-SCN-N03, and Kzl = 35.6 f 16.2 for Cd-(SCN)z-N03,where all the constants are overall and the charges are omitted. The new complexes, especially the mixed ones, are thought as stabilizing the cadmium-thiocyanate complex formations in the nitrate media while having given “apparent” P3 = 0 for the system in the KNO, medium. Thus, the importance of statistical treatment in the polarographic evaluation method of complex formation constants is again emphasized. Still more chemical interpretation and discussion of such treatments, as well as the presented constant values, would naturally be required for true understanding of the complex formation system studied, although many difficulties are expected. The present paper is a contribution to promote such chemical understanding, and we hope it is to be revised further by future studies.

RECEIVED for review January 3, 1969. Accepted May 23, 1969. (17) V. E. Mironov, Zhur. Neorg. Khim., 8, 764 (1963): C. A., 59, 3360d (1963).

ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969