Polarographic evaluation of formation constants for a partly-reversible

Potential step chronocoulometric procedure for the conversion of quasireversible waves into reversible ones. Maria Luisa Foresti , Rolando Guidelli , ...
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Polarographic Evaluation of Formation Constants for a. Partly-Reversible System and a New Systematic Procedure for All- as Well as Partly-Reversible Systems Kozo Momoki and Hisao Ogawal Laboratory f o r Industrial Analytical Chemistry, Faculty of Engineering, Yokohama National University, 2-31-1 Ooka, Minami-ku, Yokohama-shi, Japan

Since being established by DeFord and Hume, the polarographic F-functions method for evaluating complex formation constants has been limited to all-reversible systems only, all the polarograms of which were measured as reversible. Another F-function method is now developed theoretically for partly-reversible systems in which reversible polarograms are measured only for the ligand-containing states with an irreversible one for the ligand-free solution. The new procedure in which only the data measured at the reversible region are necessary is applied to some typical systems. Applicabilities and limitations are discussed while a systematic interpretation of these F-function methods including those for an all-reversible system is presented. With such understanding, a new systematic evaluation procedure for all- and partly-reversible systems is also made possible.

STATISTICAL PROCEDURES for evaluating overall successive formation constants p,(j = 1 , 2 , . , . , N) and their confidence intervals from polarographic data for metal-complex formation systems have been reported in our previous papers ( I , 2). The method was based fundamentally o n the F-functions theory by DeFord and Hume (3),where the following observation equations are solved for P, under the method of least squares: 1

+ CiPl + Ci% + . . + C P P N = .

(Ids/Idi)‘exp[(nF/RT)(E1I2s - E1/2i)]= Foi (1) In the equation, C, represents a concentration of ligand given at M different levels (i = 1, 2, . . . , M> under a constant concentration C,MEof metal ion. Subscript i for the other terms indicates the values measured at a given C, while s the values measured at Ci = 0 under the same constant C Y E . E1j2and Id represent half-wave potential and limiting diffusion current, respectively. Now, Equation 1 can be applied t o systems only when the polarograms measured at ligand-free (Ci = 0) and ligandcontaining (C, > 0) states are both reversible. Such systems may be called here “all-reversible(AR)”. However, some systems consisting of a metal ion such as indium(III), bismuth (111), or zinc(I1) have been known to show irreversible polarograms unless more than a certain amount of appropriate ligand is added to the solutions (4). Since these “partly-reverPresent address, Totsuka Factory, Japan Oils and Fats Co., Ltd., 296 Shimokurata-machi, Totsuka-ku, Yokohama-shi, Japan

(1) K. Momoki, H. Sato, and H. Ogawa, ANAL.CHEM., 39, 1072 (1967). (2) K . Momoki, H. Ogawa, and H. Sato. ibid., 41, 1826 (1969). (3) D. D. DeFord and D. N. H u m , J. Amer. Chem. Soc., 73, 5321 (1951). (4) B. Breyer and H. H. Bauer, “Alternating Current Polarography and Tensametry,” Interscience Publishers, New York, N. Y., 1963.

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sible(PR)” systems lack experimental E1,Zs and Ids values to be applied to Equation 1 , P3 can no longer be evaluated for the systems with our computer technique nor by the conventional graphical method. The present paper deals with the evaluation of P j for PR systems in extending our statistical treatments hitherto developed only for AR systems ( 1 , 2 ) . The problem may, howand I d s ever, be considered as solved if unmeasurable can be estimated as “would-be” Ellis' and Id,’ from experimental data. In fact, some theoretical or experimental approaches to the estimation seemed feasible. However, theoretical treatments proposed (5-8) have been limited t o quasireversible cases only and thus lack general applicabilities t o PR systems. Experimentally, a graphical extrapolation against Ci of El/*(or Id( measured at the reversible region to Ci= 0 has been attempted for such studies (9, IO). Since the changes of Id in the C, region close to 0 are usually very small or unmeasurable, the extrapolation or even a substitution with a measured Id, seems to give reasonable Ids’. For the ), latter case, I d m measured at the minimum C, (denoted as C giving a reversible polarogram for the system can usually be used as the substitute, as will be seen. The changes of Ell2 are rather large as the fact itself is the main basis for such evaluations of P I , and the extrapolation for is thus apt t o cause problems. But, it should be noted anyway that no independent relationship exists theoretically between El/?and C or Id and C, so that the simple extrapolations become more or less arbitrary in spite of their frequent actual uses. Lane et al. (11)tried another extrapolation of Fot to C ,= 0 where, if Equation 1 could still hold for the P R region, thusextrapolated FOOmust become exact 1. Using measured reversible data and a simply assumed Ids’ as above, they estimated Elizs’as one which gave 1 for the Fco in the extrapolations made with variously assumed El 2s values. However, as will be discussed later, the extrapolated FOO must always include experimental as well as drawing errors and cannot be thus taken as exact 1. Graphical extrapolations generally suffer from many uncertainties because of such errors ( I ) . We have treated the problem statistically as in our previous papers (1,2) and obtained a new F-function method in which experimental data measured at the reversible region are only necessary for P R systems. The method is somewhat similar to that of Lane et al. (11) but differs with the statistical con(5) H. Matsuda, Z. Elektrochem., 61,487 (1957); 62, 977 (1958). (6) H. Matsuda and Y. Ayabe, ibid., 63, 1164 (1959). (7) P. J. Gellings, ibid., 66, 477, 481, 799 (1962); 67, 167 (1963). (8) J. Koryta, Chem. Zuesti, 8, 644 (1954). (9) D. Cozzi and S. Vivarelli, Z. Elektrochern., 57, 408 (1953). (10) T. P. Radhakrishnan and A. K. Sundararn, J. Electround. Chem., 5, 124 (1963). (11) T. J. Lane, J. W. Thompson, and J. A. Ryan, J . Amer. Chem. Soc., 81, 3569 (1959).

ANALYTICAL CHEMISTRY, VOL. 43, NO. 12, OCTOBER 1971

sideration in using the curve-fitting extrapolation (I, 2). Thus, the proposed method also provides a means by which E112~’ could be estimated reasonably, although the value itself of is not necessary for evaluating P3. Some actual P R systems are tested, and applicabilities and limitations of the method are discussed. However, one of characteristic features in the present study is that a fundamental understanding of the curve-fitting procedures in these F-function methods for P R and even AR systems has been obtained systematically. Thus, the previous method for AR systems ( I ) is now revised t o attain the statistical rigorousness while a systematic calculation procedure applicable to the both systems is also presented.

MATHEMATICS AND A NEW STATISTICAL PROCEDURE FOR PR SYSTEMS A reversible half-wabe potential El,?,? if measured is written for a system, as

El 2i

=

E,” - (RTjnF)~In(ISojIuo)

(2)

where E,” is the standard potential of the amalgam electrode while I,O and Ionare the diffusion current constants of metal (-aquo) ion at the ligand-free state and metal in the amalgam, respectively. On thr other hand, a reversible half-wave potential ElrPi at a ligand-containing state of a given Ci can also be written, as

Eli 2 r.

=

E,“ - (RT,nF).In(I,o/I,o) -

(RT/nF).In(l

4

C,P,

+ C12P?+

+ C,”P.v)

(3)

where I,[)is the diffusion current constant of metal-complex at the C , . For AR systems, Equation 3 can be combined with Equation 2 , where Equation l is easily deduced under the following usual relationships: (4)

1,‘ = Id,/k’c,%f8= Idl/k

(5)

In these equations, k ‘ is the capillary constant (= r n 2 1 a f 1 iand 6) k is another constant representing k ’ C y 8 where CY8 is given as a constant. Now for PR systems, we cannot obtain Equation 2 but Equation 3. However, the latter alone may be used for the evaluation of p., if the equation is rearranged, as

When Equations 7 , 8 , and 9 are inserted into Equation 6,we obtain

Po’

P j

1_ _ - . (6) I d i ’ exp[(nF/RT)Eiizil _

where Equation 5 is inserted for I I n . In Equation 6, the denominator o n the left side includes Ia0 and E,. Although both of these must have definite values, respectively, for a system under the experimental condition, their values cannot usually be measured. Thus, the whole denominator is better included in unknown terms as variables for the system, by writing as 1 / k I, ’ exp[( nF/RT)E, O] = /Po‘

(7)

and P,/kI,o.exp[(nFIRT)E,o] = Pj.P0’ = P,‘ (8) O n the other hand, the denominator on the right side of Equation 8 can be measured actually and denoted with Fri‘ a s in ~lId~.exp[(nF/RT)Ei~z1I = Fo,’

1.9)

,

.,

+ Ci.~PN’=

FOi’

(10)

=

@,’/PO’

(11)

However, in the actual calculations of Equation 10, the values of the right side observable usually become too large as compared with those for the coefficient terms in the left side, often making accurate solutions difficult. This is because the variables must correspond to the displacements of and I d whereas the above Foil does not include the displacements but and Idias in Equation 9. F o r the situation, we have only divided both sides of Equation 10 by FM‘obtained at C , = C,, as in (Po’/Fom’)

+ Cifj3I’/Fom’)+ . . + Ct.vQ3,~~’/F~m’) =

Foi’/Fh’

=

(rd,/rd,).exp[(nF/RT)(Eiis, - E11z1)1 (12)

where El,2mrepresents a half-wave potential measured at Now, we can solve Equation 12 for P0’/Fh‘and p3‘/Fom‘ as new variables and then divide the latter by the former to obtain P, directly, just as in the relation shown between the variables in Equation 11. Thus, we can still take Equation 10 as the present observation equations with only a change of the definition for Fr,’from Equation 9 to

Furthermore, we can conveniently use the previous program of our 1/Fo2-weight method ( I ) for the above calculation with only slight modifications: the new variable Po’ is introduced while the previous and I d s are replaced with EliZm and Idm, respectively. The relative weight wi for each observation equation is thus written, as in wi =

1/Foi”

= ( I d i / ~ ~ m ) 2 ~ e X p [ 2 ( n F / ~ ~)(E Eiirnt)] iizi

(14)

The variances uPJ2of obtained P, are now calculated (IZ),as in =

~

+ CtPI’ + Ct%‘ +

This is another F-function equation which would now represent the observation equations for PR systems with variables PO’ and P j ’ as well as observable FO’ measured a t given C,. The variables can easily be solved there under the method of least squares as before (I) and will then be inserted into Equation 8 to give final P j , as in

+ AJ,j/Po” - 2A1,>PJ’/Po’3)’(S/f) (15)

(A1,1PJ‘2/PO’4

where A i , jstands for (iJ) element of the inverse matrix for the coefficient matrix of the normal equation derived from Equation 10,frepresents the degree of freedom for the case, and S is the sum of squares of weighted residuals as in S =

Cwi(Po’ i

+ CtPi‘ + . . + CiNP.v’ - Fat')' ,

(16)

The precision in evaluating P j for the P R system will thus be more o r less reduced by the new participant PO’ from A R case if the both systems are measured under similar precisions.

ESTIMATION OF E112a’ As might be noticed in Equation 10, (30’ would be FOO’extrapolated to Ci = 0 from reversible For‘and thus represent the “would-be” reversible ligand-free state for P R system. And further, Equation 11 shows that P j t o be finally deter(12) W. E. Deming, “Statistical Adjustment of Data,” John Wiley and Sons, New York, N. Y., 1943.

ANALYTICAL CHEMISTRY, VOL. 43, NO. 12, OCTOBER 1971

1665

r-7

suction

to SCE(NoCi1

to cell

-0.971

Figure 1. Electrolytic salt bridge of a three-way tap type

0

- (RT/nF). h ( p 0 ‘ I d s ’1

(17)

where PO’ is the original one defined in Equation 7. If PO’ is taken as the working variable in the actual observation equations as mentioned, Equation 17 will be reduced to

EI’z~‘ = Elllm- (RT/nF).ln/h’

(18)

where I d s ‘ = I d m is also assumed for actual uses. This equation was used throughout this paper for estimating E1/ps’. The variance of estimated El,ls’will be (RT/nF)*(l/Po’ ) 2 , (ig0/* (1 9) where the term by the variance of measuring Idm is neglected as small compared with the above by that of evaluating PO’. UE,,~”!

=

EXPERIMENTAL Zinc(I1)-thiocyanate, indium(II1)-bromide, and indium (111)-thiocyanate systems as three PR systems showing small, medium, and large pj values have been studied under the proposed method. A manual polarographic circuit with the three-electrode cell was used as described previously (2). The SCE’s saturated with NaCl were again employed (2) for all the solutions containing Na+ as the supporting cation. Thus, the potential values in this paper are mostly those us. this type of SCE while only those for Zn (NO&KSCNK N O B system are cs. SCE saturated ordinarily with KCI. Correspondingly, most liquid junctions were made with Na+ while K + was used only for the above zinc system. T o improve the reproducibility of the electrolytic salt bridges, one of three-way tap types (13) was used as in Figure 1. It was (13) G. Mattock, “pH Measurement and Titration,” Heywood and Co. Ltd., London, 1961, p 177. 1666

1

2 .o

1

[SCN-)

mined cannot be evaluated without Po’. Therefore, the proposed method is based primarily o n the estimation of the “would-be” reversible ligand-free state through Po ’. For this, the curve-fitting extrapolation is necessarily made with experimcntal data measured in the reversible region only. It should be noted, however, that the above estimation does not directly mean that of E I I p s ’ oI rd s ’ . We d o not need these values for evaluating p,, but especially the value of could be used for some references if estimated properly, as will be seen. As for I d s ’ , the simple extrapolation or even Ids’ = I d m seems t o be used for the estimation as mentioned. A simple estimation of is also possible if “would-be” quantities involved can be given some reasonable assumptions. We assume that Elrls’can be written with “would-be” Iso in the same form as in Equation 2 while the Is0’ can also be written with Ids’ just as in Equation 4. Combining these assumptions with Equation 7, we obtain EIIlS ’=

I

1.0

Figure 2. El .,-C, cyanate system

plots for zinc(ll)-thio-

0 Our data measured in NaClO, medium 0 X

Our data measured in KNO3 medium Data by R . E. Frank and D. N. Hume (14) measured in K N 0 3 medium

especially useful when different sorts of ligand for a metal ion in the same medium were measured and compared, as will be seen. The capillary constant k ‘ of the D M E used was 1.82 in a deaerated solution of 2.OM NaCIO, at an applied potential of -0.5 V. All the polarographic data were measured at a temperature of 25.0 T 0.1”C under an ionic strength of 2.0 maintained mostly by NaCI04. All chemicals of reagent grade were used without further purifications, except some mentioned below. Sodium perchlorate and sodium thiocyanate were recrystallized once from their reagent chemicals and dissolved into conductivity waters. The solutions of indium(II1) and zinc(I1) perchlorates were prepared from their precipitated hydroxides which were then dissolved into perchloric acid solutions. All the indium(II1) solutions used were added as 0.1M with perchloric acid to prevent hydrolysis. The concentrations of the indium and zinc solutions measured polarographically were determined as 5.83 X lO-‘M and 7.76 X lO-4M, respectively, by EDTA titrations. The concentrations of sodium bromide and sodium thiocyanate solutions were checked argentometrically for their stock solutions. The computer calculations were carried out with NEAC2230, IBM-1130, or HITAC-5020 whichever was available for our need.

EXPERIMENTAL RESULTS The polarographic data of the zinc(I1)-thiocyanate system measured in KNO,{and NaC104 media are shown in Figure 2 with reference data reported in the same K N O Bmedium (14). The latter medium was studied previously by us also ( 1 5 ) . Our present data show that the C,, is at 0.1M for the K N O J medium while at 0.05M for the NaC10, medium. The results of the computer calculations are shown in Table I in which some of the P, values reported by other authors are also given for comparison. The E, ?,’ was estimated as -0.9799 i 0.0008 V for the NaCIO, medium while -0.9994 + 0.0017 V for the K N 0 3medium. Indium(II1)-bromide system was measured in the NaC104 medium containing 0.1M HCIO, (9), and an irreversible wave (14) R . E. Frank and D. N. Hurne, J . Amer. Chen7. Soc.. 75, 1736 ( 1953). (15) K. Momoki, H. Sato, and H. Ogawa. Bid/. Faculty Eug.. Yokohama Nai. Uuic., 16, 127 (1967).

ANALYTICAL CHEMISTRY, VOL. 43, NO. 12, OCTOBER 1971

Table I. Comparison of Evaluated Formation Constants for Zinc(I1)-Thiocyanate System with Some Previous Data Ref. (14 (16)

(17) This study This study

Method Temp.

P2

Pa

P4

PO

b6

7 f 3 8 150

I f 1 16 220

20 iz 2 20 103

17

4.3

2.2 f 0.8

23.5 f 1 . 1

0

52.8 f 3.7

28

3 . 4 f 1.4

9 . 6 f 1.0

0

21.4 f 3 . 3

33

Medium

30

2(KN03)

MHg

25 20

3(NaCI04) 3.3?(KN03)

PO1

25

2(NaC104)

PO1

25

2(KN03)

PO1 POI

PI

3 f 0.5 0.73 29

M 11

Table 11. Polarographic Data for Partly-Reversible In(C10&NaBr-NaCI04 System Slope of Slope of u ~ X ~ the, log-~ ~ ~ U E ~ , ~ ,X * the log[Br-] M -El,2,, V loo, V 2 plot, mV Id,, r A [Br-] M -Em,, V IOo, Vz plot, mV 0.943 20.88 5.85 0.600 0.5483 0.931 0.0200 0.5105 20.46 0.878 20.70 5.85 0.700 0.5510 0.340 0,0300 0.5134 20.78 1.74 20.95 5.85 0.5158 0.800 0.5535 0.805 20.96 0.04oO 20.88 5.85 0.900 0.5175 0.729 0.5557 1.40 21.32 0.0500 20,97 1 .00 0.5574 0.826 5.89 0.5194 1.85 20.87 0,0600 20.88 5.92 1.10 1.66 0.5221 0.965 0.5590 20.98 0,0800 21 .00 5.94 1.95 1.20 0.5606 0.5246 0.654 21.03 0.100 1.26 20,63 5.92 1.30 0.5627 20.52 0.5291 1 .29 0.150 20.80 1.98 6.00 1.40 20.55 0.5323 0.204 0.5640 0.200 5.97 1.50 20.76 0.5653 0.824 20.71 0.5353 0.699 0.250 20.87 6.05 1.60 1.31 20.51 0.5668 0.5378 1.03 0.300 6.07 1.70 0.5681 0.949 0.909 21.27 21.16 0,5403 0.350 0,582 20.71 6.01 1.80 1.78 21.27 0.5690 0.400 0.5418 0,844 20.98 6.10 1.90 0.5703 0.927 20.87 0.500 0.5452 All the solutions are made as 0.1M with HC104. The final ionic strength is kept at 2.0. The temperature is 25.0 i 0.1 "C.

Idi,

6.00 6.13 6.10 6.13 6.12 6.08 6.11 6.02 6.03 6.10 6.04 6.09 6.18 6.09

Table 111. Evaluation of Formation Constants for Partly-Reversible In(C104)3-NaBr-NaC104 System No. 1

2 3 4

P1

*

138 83 135 f 63 151 f 63 161 f 56

P2

508 f 149 512 f 118 503 f 123 509 f 127

P3

180 f 1080 126 3z 418 307 f 212 359 f 125

was observed for the ligand-free solution (18) as reversible ones for the solutions with C, = 0.02M. These data are shown in Table I1 and Figure 3. The results of the calculations are shown in Table I11 in which combination No. 4 of PI, /32, and p4 was found best as before (2). The Eliza'was estimated as -0.4977 f 0.0026 V. A considerably strong indium(II1)-thiocyanate system (10) was measured in NaClO, medium and the polarographic data were obtained with C, = 0.02M as in Figure 4 and Table IV. However, the straight applications of the proposed method t o the data could not give any reasonable combination of pj for the system, as will be discussed.

04 85 f 1570 170 f 337 15 f 54 0

-0.58

sm x 103)

06 -14 f 247

P6

11 f 1060 -47 f 103 0 0

0.4833 0.4614 0.4570 0.4442 ( M = 28)

0 0 0

I

X

-0.54

Ey .V -0.50 I

1

DISCUSSIONS

Applicabilities and Limitations of the Proposed P R Method. The zinc(I1)-thiocyanate system treated once as AR (14) gives us PR polarograms as in our previous study (15). However, the AR treatment made even by the conventional graphical method (14) is shown in Table I to give quite similar /3, values to those in the same K N 0 3medium with our PR computer technique. This may come from the fact that the system is considerably weak and needs only a mild extrapolation for the evaluation in the C,region close to 0 as in the (16) H. G. Tsiang and K. H. Hsu,Krxue Torigbao, 1959, 331. (17) A. M. Golub and G. D. Ivanochenko, Zh. Neorg. Fhim., 3, 333 (1958). (18) E. D. Moorhead and W. M. MacNevin, ANAL. CHEM.,34, 269 (1962).

0

I

I

I

I .o

I 2.0

(BFJ

Figure 3. E112z-C,plots for indium(II1)-bromide measured in NaCIOl medium

system

0 Our data x Data by D. Coni and S. Vivarelli (9)

plots in Figure 2. Only if reasonably continuous data are measured for some appropriate C,, the evaluation of such a weak system may not be affected much by the treatment as AR nor PR. Our estimated are also plotted in Figure 2 to show that such smooth extrapolations have been carried out for reasonable evaluations. In Table I, the same p3 = 0 are seen for both our media in which different p2 and p4, respectively, are given. Since the

ANALYTICAL CHEMISTRY, VOL. 43, NO. 12, OCTOBER 1971

1667

Table IV. Polarographic Data for Partly-Reversible In(C104)3-NaSCN-NaC104 System

-0.61

u -0.57

.v

f

’*

-0.4s

I

0

E i s f o r indlum(lnl-bromide ryrtem I

I

1.0

I

I 2.0

[SCFC) or [ B S I

Figure 4. E,,,,-C, plots for indium(III)-thiocyanate system measured in NaCIO4medium 0 Our data 0 Data by T. P.Radhakrishnanand A. K. Sundaram (IO) x Our data for indium(II1)-bromide system (same as in Figure 3) for comparison

0%= 0 for the NaC104 medium is against the observation in the cadmium system (2), the existence of additional nitrate complexes which may be correspondingly expected even in the present KN03 medium can no longer be analyzed with the previous procedure (2). The present p3 = 0 may be ascribed mainly to the fact that less precision must be expected as mentioned for the proposed method in applying to the less strong system than in the previous AR case. The evaluation errors (and measurement errors) might then be too large to point out such small contributions if any of these p3 species to the measured data. Anyway, the system is very weak and may be at a limit of the polarographic evaluation, especially with the PR method. Also, we d o not think that ps and pBcan be so easily evaluated (16) as in Table I for such a weak system. In the indium(II1Fbromide system, steeper curves in the C , region close to 0 than in the above system are observed for the plots shown in Figure 3. This shows that the system is much stronger, and presents the problem of the more difficult extrapolations for the evaluation. The difficulty is reflected in a somewhat large confidence interval given for our El,zs’ but might be also in the previous study (9) which gave too large p1 = 6.3 X l o 3and p2 = 6.3 x 10‘. Thus, we carefully inspected their polarographic data, shown also in Figure 3, and found that the data were measured mostly as being reasonably continuous except in the C,region very close to 0. The latter region was thought as important for the difficult extrapolation, where they used a plot measured at C, as low as 4 X 10-4M but only erroneously because the C M Ewas then given at 10-3~(1,3). The proposed method will be favored particularly for such steep cases, because the plots at such small C,are not necessary, but still a reasonable ligand-free state is obtainable for a PR system almost unequivocally due to the least squares curve-fitting method. The p, values of No. 4 in Table 111 could be compared with those reported (19) for the complex studies (19-25). Seemingly large confidence intervals given (19) E. A. Burns and D. N. Hurne, J. Amer. Cltem. SOC.,79, 2704 (1957). (20) H. Irving and F. J. C. Rossotti, J . Cliern. SOC.,1955, 1927, 1938, 1946. (21) B. G. F. Carleson and H. Irving, ibid., 1954, 4390. (22) L. A. Woodward and P. T. Bill, ibid., 1955, 1699. (23) L. A. Woodward and M. J. Taylor, ibid., 1960, 4473. (24) T. H. Cannon and R . E. Richards, Trans. Faraday SOC.,1967, 1378. (25) M. J. Taylor, J . Chem. SOC.,( A ) , 1968, 1780.

1668

x~

SloDe of

the-log, ~ ~

[SCN-1, M

- E I , ~ ;V,

loo V2

plot, rnV

Idi,FA

0.0200 0.0100

0.5178 0.5227 0,5267 0,5296 0.5325 0.5369 0.5411 0.5445 0.5474 0.5517 0.5555 0,5609 0,5655 0.5689 0.5720 0.5752 0.5782 0.5809 0.5876 0.5916 0.5950 0.5981 0.6010 0,6037 0.6064 0.6091 0.6114 0.6135 0.6158 0.6177 0.6196

1.05 1.33 1.29 1.73 1.08 1.19 1.01 0.223 0.838 1.16 1.84 1.56 0.533 1.42 0.459 1.11 0.879 1.44 1.63 2.70 1.26 1.24 1.67 1.49 2.86 1 .80 1.52 0.668 0,543 0.499 0.528

21.70 21.30 21.40 21.37 21.28 21.57 21.98 21.57 21.44 21.20 21.34 21.54 21.54 21.20 21,54 21.70 21.74 21.35 21,17 21.12 20.99 21.09 21.02 21.08 21.44 21.55 21.41 21.15 21.39 21.13 21. IO

5.85 5.85 5.84 5.85 5.85 5.85 5.86 5.85 5.85 5.83 5.84 5.86 5.85 5.84 5.86 5.86 5.86 5.85 5.84 5.84 5.84 5.83 5.83 5.82 5.85 5.85 5.82 5.81 5.81 5.82 5.83

0 . W

-0.53

~

0,0500 0.0600 0.0800 0. 100 0.120 0.140 0.170 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.700 0.800 0.900 1 .OO 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90

All the solutions are made as 0 . 1 M with HCIO,. The final ionic strength is kept at 2.0. The temperature is 25.0 3~ 0.1 “C.

for our p, may be unavoidable for the system having such p, values amount to several hundreds as indicated. However, far steeper extrapolation which was necessary for the considerably strong indium(II1tthiocyanate system as shown in Figure 4 could not be handled adequately even with the proposed method. In fact, no reasonable combinations of p, have been obtained in the PR calculations made for the polarographic data in Table IV, where the unreasonable combinations have been caused by obtaining negative Po’. We also checked the recent data for the same study (10) using the PR method as well as the 1/Fo2-weight method (1). We used only their reversible data in the former while in the latter and I d s f estimated with the simple extrapolations their E1,2af were used in addition to the reversible data. And, although their data themselves were questionable in using ammonium thiocyanate as the thiocyanate ion source for the complex formation in the NaCIOl medium, we obtained again only minus (30’. The cause of obtaining these negative PO’ has been discussed to be related with the far steep plots as in Figure 4 giving the difficult extrapolations, as will be seen in the next section where a n interpretation of the present Ffunctions PR procedure will also be discussed. However, a practical method to evaluate p, more reasonably for this system has been thought as probably possible because the present ligand-free state would be quite the same as that of the last indium(II1)-bromide system which was measured in the same NaCIO, medium. Actually, the comparison of and I d , in the the data in Tables IV and I1 shows that the ElrL, C, regions close to 0 seem to converge to the same values, respectively, at C, = 0. We checked and assured this point experimentally again where the electrolytic salt bridge shown in Figure 1 was found as quite useful. Thus, assuming that the above unreasonable p, were based only on wrong estima-

ANALYTICAL CHEMISTRY, VOL. 43, NO. 12, OCTOBER 1971

Table V. Evaluation of Formation Constants for Partly-Reversible Indium(II1)-Thiocyanate System No. la

Slf

Method

BI

A

-213 +71 62 f76 (332 f22) (351 f19) (275 f27) (392 f 17)

B4

PI

62

- 29000

- 789

- 101

Br

- 16200 f7820 0

B6

951

( X 103) 0 . I626

f2820 0 2.650 10700 3690 f8240 f1930 (11600 (0.2864) ( - 7630 (51900 3a Bb (6790 f8060) f24100) f7640) f858) (0.3162) (28600 0 (62200 (5890 4” Bh 14410) f3850) f616) (33800 0 0 (2.859) (9100 5” Bh f753) f3770) (4140 (0,6271) 6* Bh (76700 (41600 0 i1030) f433) f1660) 7a B* (336 (6580 (54200 (55000 0 (9240 (0.2672) rt17) f525) f3320) 14550) i1170) 1060 - 16500 29400 - 64900 - 447 0.2331 8c A -126 f2490 *Ill00 rt I5 f250 f16700 +I1400 (90900 (-4170 (80 (3610 (- 34800 (107oM3 ( 1 ,974) 9c Bd 3: 19) i949) i12700) i55800) f90100) i46200) A The proposed method for partly-reversiblesystems shown in this paper. B The 1/Fo2-weightmethod ( I ) . Values obtained with this method are expressed in parentheses as shown because they are tentative due to the weight problem involved. Our experimental data in Table IV are used. E,12n’ = -0.4976> V and I d a ‘ = 5.85 pA from the indium(II1)-bromide system in the paper are used. c The experimental data reported in Ref. I O are used. d El . 2 a ’ = -0.4833 V and Ida’ = 3.55 p A graphically estimated in Ref. 10 are used. The f values in the table are the standard deviations, not the 9573 confidence intervals which are used in the other tables throughout the paper.

2a

A

f1020

f10800

fl1100 32200 i19800 (62400 f23600) (28500 f8230) (88100 f3220) 0

0

‘8

tions of the “would-be” reversible ligand-free state, we bor= O.497b5V and Ids’ = le,,,= 5.85 FA from rerowed measured and reestimated data for the easier indium(II1)bromide system. With the other E I l 2and Zo,! taken from Table IV, we calculated possible 0, values under the 1/Fo2-weight method. As a result, the evaluation seemed improved and some reasonable combinations of P , were obtained as in Table V in which some unsuccessful data are also shown for comparison. In the table, the combinations No. 4, 6, and 7 seem to have similar possibilities to be most reasonable while No. 5 is not unreasonable but shows much larger S/fto be less favorable than the former three. The actual combination may thus be almost in these three with a small chance for No. 5 , although we have not been able to choose any one as the best. Since the relative weights cannot easily be treated in the above procedure in which the data for the two independent systems are combined, the values presented in Table V are still tentative. The study of the indium(II1)-thiocyanate system probably shows a limitation of the polarographic evaluation method for such a strong PR system. Thus, the above result is presented as possible only within the experimental and calculation limitations. A possibility of using experimental data of a weaker system for evaluating another stronger system is also suggested as a n application of the PR method. An Interpretation of the Proposed PR Procedure in Relating to Obtaining Minus Po’. As mentioned earlier, the present curve-fitting treatment is made actually on the plots between FO,‘and C,, where Fotfis defined in Equation 13. The difficult extrapolations which have been repeatedly mentioned in the paper are in fact on this sort of plot. A continuous curve must then be obtained as in Figure 5 in which Fo,‘is assigned as 1 at C , = C, and will become increasingly larger as C, increases. At smaller C , than C , where n o plots are given, Fo,’ decreases also continuously with a value less than 1 o n the extrapolated curve. The curve will finally reach a n intercept with the ordinate at C , = 0, where the intercept would be given on the ordinate as

PO’ =

( Z d m / l d s ’ ) . exp[(nF/RT)(El/2,

-

El/zS1 ’1

(20)

Since this right-side has a plus value anyway, the intercept should come somewhere between 0 and 1 o n the ordinate. We can interpret the present curve-fitting treatment as mentioned above, where the intercept must be especially marked. Namely, by the least squares calculation, the Po’ should be solved as taking a value between 0 and 1 to give reasonable P,. However, the allowed range between 0 and 1 for the intercept is rather narrow compared with usual distributions of the plots given at larger C , , as is also seen in Figure 5. Therefore, the intercept is apt to come to the minus side of the ordinate giving minus Po’, particularly when the plots become considerably steep by the system of considerable strength. The trend will of course be assisted by the poor precisions which must be expected in measuring and calculating such strong PR systems. Why the above indium(II1)-thiocyanate system has been evaluated as giving minus PO’ may thus be explained. Figure 5 also shows that weaker systems are comparatively easier for extrapolations to give reasonable b,. It should be noted that, although the problem of the difficult extrapolation appears to concern many steep plots only in the C, region close to 0, the extrapolated intercept should also decide the whole curve-fitting even at larger C,. Giving a estimated in another system as in the last section corresponds to indicating the intercept for other plotted points prior to the curve-fitting process. And this would make the curve-fitting for such a difficult system more reasonable if the ElizF’ can be given properly. The case may, however, be noticed as similar to one for AR system, as will be seen.

CASE OF AR SYSTEM AND A NEW SYSTEMATIC PROCEDURE FOR AR AS WELL AS PR SYSTEMS WITH A STATISTICAL REVISE FOR THE PREVIOUS AR METHOD ( I ) When the C, of a PR system is reduced to 0, the system will naturally become as A R where the curve as in Figure 5 could still be drawn with the intercept a priori given a t 1 o n the ordinate. The A R case is thus thought to be at a limit of the P R system a t C, = 0 and the above interpretation of the proposed PR procedure must be systematically extended also to

ANALYTICAL CHEMISTRY, VOL. 43, NO. 12, OCTOBER 1971

0

1669

However, the mistake has been corrected only by extending the above PR procedure to the limit at C, = 0 as the AR case, as follows: At the limit, Equation 17 can be written now for AR system, as

I:

PO‘ =

.Lo

IC

L

lz 5

exp[(nF/RT)El/p,l

(21) This equation obviously represents the AR ligand-free state while the above discussion corresponds to the fact that the right-side has been considered as given a fixed value in the previous treatment. But, from the statistical situation now made clear, we should better treat the whole Equation 21 as a new observation equation with a variable PO’ representing the AR ligand-free state. The ligand-containing states even for the AR case can naturally be represented by Equation 10 with variables Bo’ and P,’ where F o ~ ’is again defined by Equation 9. Thus, for the above rigorous statistical requirement to be fulfilled, we should use these M 1 equations of Equations 21 and 10, instead of M equations in Equation 1, as the members of the AR observation equations. However, the same sort of transformation as used to obtain Equation 12 from Equation 10 could be applied again to the present Equations 21 and 10, deriving l//ds‘

+

1.0

0 &

0

1

I

0. I [SCN-) or [ET)

0.05

I

O.l!

Figure 5. Plots showing actual curve-fittings for these F-functions methods 0 0 X

---

In(III)-NaSCN-NaC104 by us (with data in Table IV) In(III)-NaBr-NaCI04 by us (with data in Table 11) I~(III)-NHISCN-N~CIO~by T. P Radhakrishnan and A. K Sundararn (IO) Cd(II>-NaSCN-NaClO, by us ( 2 ) (all-reversible system)

the AR system. Therefore, even a systematic evaluation procedure to be applied to the both systems could become available under the systematic interpretation. Since Equation 1 for the AR system is very similar to the PR one in Equations 10 and 13, the extension of the latter to the former may be mathematically conceived. However, the key point of the new procedure is not simply mathematical but statistical. Thus, the new procedure applicable even to AR system has not been attained without through the above PR procedure and considerations. We have thus restudied the previous AR treatment (1) with Figure 5 in mind and realized that one more statistical consideration of importance was necessary for making the systematic procedure possible. Namely, any points plotted as in Figure 5 must not, in speaking strictly, be fixed but statistically variable because experimental errors should always be included in the plots. In fact, the plots at C,= C, in the last section were not fixed at exact 1 and the situation was already taken into account in the above actual PR calculation. However, in the previous AR method ( I ) , the intercept a priori given has been plotted erroneously at the fixed 1 as if including no errors. Namely, the same values of EliZsand Ida,respectively, have almost doubtlessly been given as if without errors for all the members of the AR observation equations as in Equation 1. But, each or I d s has usually been measured once or twice (and averaged) quite similarly to other E l i z , or I d , SO that E I ~ , had to have a similar precision to ElizIas Ida to I d % * With the data thus having similar precisions, the observation equations should have been written in which and I d a are both variable from a member equation to the others as El/?,and I d , . Otherwise, Ellzaand I d a must have been measured with specially high precisions as compared with Eliz,and I d , . Thus, and I d a have we have made a statistical mistake in which not been treated as similar stochastic variables to El,?,and I d , , respectively, against the experiment. 1670

PO”

+ C,P,” +

for i

=

1 , 2,

C12PO”

. ..

+

Po“

(22)

= 1

+

ClNPN”

= Fo,” =

,M, where the relation exists again as in

B,”lPo”

(24) Equations 22 and 23 are the final M 1 observation equations to be used for the AR system. The rigorous statistical revise required for the previous M members as in Equation 1 is here indicated: The 0-order term of C, is changed from the fixed 1 in Equation 1 to a variable PO” in Equation 23 while the variable is particularly designated in Equation 22 to be located only very close to 1. Thus, Figure 5 can also be applied systematically to AR systems if the ordinate is read now as F,,”. Correspondingly, the Fo,extrapolation method to the fixed 1 at C , = 0 for PR case (11, IS) should have been erroneous. Also, a procedure (26, 27) in which only the M members, just as in Equation 23 with the 0-order term as a variable, were used for the AR system must not be correct. Finally, Equation 22 can obviously be included in Equation 23 if the subscript i in the latter is only assumed to take also 0 for C, = 0. Therefore, only Equation 23 can be used for AR system with i = 0, 1, . . , , M while for PR case with i = 1,2, , M . Of course, for the latter, Ellzrin Equation 23 should be replaced with El/*, as IdFwith I d r n , where m is usually taken for i = 1. Thus, we have obtained a new systematic F-function procedure based only on Equation 23 and applicable to both AR and PR systems. We compared this systematic procedure (method I) with the 1/Fo2-weightmethod (method 11) as well as the method based o n the M members only of Equation 24 (method 111) in evaluating the cadmium(I1)-thiocyanate AR systems in K N 0 3medium ( 1 ) (system A), in N a N 0 3 medium ( 2 ) (system B), and in NaCIOl medium (2) (system C). The results are shown in Table VI in which the differences between methods I and I1 are only very slight. Both methods anyway use plotted points at C, = 0, unfixed (method I) and fixed (method 11), where the necessary curve-fittings are not so steep since the systems are rather weak. Therefore, the similarities in the evaluated values are reasonable as long as the experimental data are measured reasonably. The result shows that the previous values for these systems are not necessarily revised. =

PI

+

,

(26) H. Irving in “Advances in Polarography,” Vol. I, I. S. Longmuir, Ed., Pergamon Press, Oxford, 1960, p 52. (27) D. Inman, I. Regan, and B. Girling, J. Chrm. SOC.,1964, 348.

ANALYTICAL CHEMISTRY, VOL. 43, NO. 12, OCTOBER 1971

Table VI. Evaluations of Formation Constants for All-Reversible Cadmium(I1)-Thiocyanate Systems by Three Different Calculation Methods Calcd System Ma method .f (BO’or BO) 81 BZ P3 P4 ~ ( ~ 1 0 3 ) 11 I 7 ( P o ’ = 0.9994 1 4 . 6 f 1 . 8 46.0Ik 5 . 1 0 88.6f 4.1 0.20694 A(1) f0.0340) (10Y 11 7 1 4 . 6 f 1 . 7 45.9 f 5 . 0 0 88.6Ik 2 . 6 0,20699 (10P 111 6 ( B o = -0.0278 21.4 f 8 . 3 36.7 Ik 11.9 0 90.9 f 3 . 5 ,0.14325 10.1240) B(2) 12 I 7 (~o’=l.O007 11.9f1.5 6 3 . 3 i z l O . O 2 0 . 5 k 1 6 . 7 4 9 . 1 I k 7 . 1 0.11173 f0.0250) (1lY I1 7 11.9iz1.4 63.3f10.1 20.6f16.7 4 9 . 1 & 7 . 0 0,11181 Ill 6 ( P o = 1.2769 8.1Ik6.4 76.4f24.0 5.9f29.2 5 3 . 9 + 1 0 . 4 0,09599 (1lY Ik0.4618) C(2) 37 I 32 ( B o ’ = 1.0058 1 8 . 0 f 1 . 9 137+ 13 86.2iz27.3 1 7 7 f 14 0.26549 f0.0326) Wb I1 32 18.3 f 1 . 4 137 f 13 8 7 , 6 f 26.4 178 k 13 0.26658 (36Y 111 31 (Po 1.1597 14.8 & 3 . 8 155 f 21 63.0 iz 35.8 187 k 16 0.24414 =to.1642) A System Cd(Il)-KSCN-KN03. The experimental data are taken from Ref. 1. B System Cd(I1)-NaSCN-NaNOa. The experimental data are taken from Ref. 2. C System Cd(I1)-NaSCN-NaCIO,. The experimental data are taken from Ref. 2. 1 The systematic method described in this text. 11 The l/Fo*-weightmethod with slight changes in calculation processes from Ref. 1. 111 The method in which O-order term of C,in the equation for Foi is taken as a variable Po as in Refs. 26 and 27. The number of experimental data set ( I d and E,,*) used as the realizations of corresponding stochastic variables for the evaluation, The experimental data for the ligand-free state are treated as without errors. 0

The results with method I11 show the importance of plotting the point at C i = 0 for the A R curve-fitting. Although the method gives the least S/fand thus the best fits around the Fo-curves among the tested procedures, unreasonable intercepts and low precisions are obtained because the plotted points at C i = 0 are not given. The evaluated values for system C with method I1 show that the effect of the no-plot at C i = 0 on the curve-fittings is minimized naturally because the M is given as much ldrger than in A and B, as might be expected also from Figure 5 . The observation will serve as

another support for the above interpretation of these F-function methods systematically with Figure 5. Thus, the present study for PR systems has given considerable understanding of the polarographic F-function methods with introducing another systematic F-function procedure applicable to both AR and PR systems. RECEIVED for review December 15, 1970. Accepted June 16, 1971.

Extension of the Microcoulometric Determination of Total Bound Nitrogen in Hydrocarbons and Water L. A. Fabbro, L. A. Filachek, and R. L. Iannacone Cities Seroice Oil Company, Cranhury, N . J .

R. T . Moore, R. J . Joyce, Y. Takahashi, and M. E. Riddle Dohrmann Dioision, Enoirotech Corp., Mountain View, Gal$ The microcoulometric determination of total bound nitrogen in hydrocarbons add water has been extended in application, range, and precision through modification of system parameters and improvement in sampling techniques. For low ppm nitrogen samples, the precision has been improved to =t0.03 ppm or 3.00J0, whichever is larger, through increased sample size, system parameter optimization and by separating the needle/septum blank from the sample peak. The use of ,a Solid/Liquid Sample Inlet has permitted accurate nitrogen determination for high boiling petrochemical distillates and has facilitated the separation of volatile and nonvolatile nitrogen-bearing compounds in waste water analysis. Through extensive modification of the system parameters, the sampling capacity for solid samples has been extended to accomodate sample sizes up to 10 mg with a total nitrogen content of up to 100 pg in a single five-minute analysis.

MANYDIFFERENT SITUATIONS exist where it is important t o measure the concentration of total bound nitrogen in hydrocarbons and water. In the petrochemical industry, the determination of nitrogen is important because of catalyst poisoning by the nitrogen. In the feed and grain industry, the nitrogen concentration is a measure of the protein content (or food value) of the product. Quality control of nitrogenous pharmaceutical products can often be effected through total nitrogen determinations. Eutrophication of water resources is critically dependent upon total nitrogen content since algae growth is dependent, in part, upon the total nitrogen content. Municipal sewage plants can continuously monitor internal processes by checking total nitrogen as well as ammoniacal nitrogen ( I ) . (1) D. K. Albert, ANAL.CHEM.,39, 1113 (1967).

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