observed experimentally is compared with the calculated value of 1.9 PA, as shown in Table I. The discrepancy most probably arises from our inability to correct for residual current contributions. Using the Cu(I1)-Cu(1) solution described above, the ring electrode is now poteiitiostated at +0.4 V, where the Cu(1) will be oxidized to Cu(I1). At disc potentials more positive than +0.35 V, the ring electrode is shielded, because most of the flux of Cu(I1) to the ring electrode is due to the oxidation at the disc electrode of Cu(1) from solution. The ring electrode current, iR (--0.4, +0.4), is given by i ~ ( + 0 . 4 ,+0.4) = i ~ ( + 0 . 4 ,open) X S
(14)
The experimental value of -26.5 p A shown in Table I agrees satisfactorily with the calculated value of -25.5 pA. As the limiting convective-diffusion current for the reduction of Cu(I1) to Cu(1) is reached at the disc electrode, a large anodic current is observed at the ring electrode. Cu(1) from
the solution is no longer oxidized at the disc electrode and the ring electrode current is now given by i ~ ( + 0 . 4 , -0.15) = i ~ ( + 0 . 4 , open) - N i ~ ( - 0 . 1 5 ) (15) The experimental value shown in Table I of -70.1 pA agrees closely with the calculated value of -68.9 pA. As copper is plated on the disc electrode, the ring electrode becomes shielded. We conclude that the independent potentiostatic control of two indicating electrodes us. a single reference electrode is satisfactorily accomplished by the instrument described above, and that the behavior of the Cu(II)/Cu(I) and Cu(I)/Cu system in 0.5M KCI is in agreement with steady-state ringdisc electrode theory.
RECEIVED for review November 25,1966. Accepted January 26, 1967. The work was supported by the National Science Foundation and the Space Science Center of the University of Minnesota,
Critique of Chronopotentiometry as a Tool for Study of Adsorption Peter James Lingane:L California Institute of Zechnology, Pasadena, Calif. Least squares procedures are described for the analysis of chronopotentiometric data according to four of the several model!; for the relative rates of reaction of the adsorbed and solution species. Extensive synthetic data are analyzed by these procedures and the minimum conditions for the applicability of chronopotentiometry are delineated. It is concluded that chronopotentiometry is of very limited value for the study of adsorption.
ACADEMEHAS ENTHUSIASTICALLY EMBRACED chronopotentiometry because it is, both conceptually and instrumentally, a delightfully simple technique. We have witnessed a tremendous technique-oriented developmental program although the fruits, in terms of boniifide electrochemical discoveries and analytical achievements, have been relatively few. One of the major and most controversial applications of chronopotentiometry has been 'to the study of the manner and extent to which electrochemicdly active species are adsorbed onto the surfaces of electrodes. The central purpose of this paper is to delineate the disadvantages inherent in using chronopotentiometry for this purpose. In addition, considerable effort is devoted toward developing a data analysis procedure capable of distinguishing realistic models for the co-reaction of the adsorbed and diffusing species. The major difficulties with chronopotentiometry arise in the determination of the :hronopotentiometric transition time and in the interpretation of the transition time if several charge consuming processes operate in parallel. A wide variety of methods have been suggested for the determination of chronopotentiometric transition times (I) but there is not at the
moment any consensus as to which is best under any given set of conditions. It is true that the several techniques yield results agreeing within perhaps 10% but this precision is inferior to the precision obtainable with other transient and electroanalytical techniques. The difficulty with parallel charge consuming reactions arises because it is the current which is controlled in a chronopotentiometric experiment. Thus the sum of the rates of the several parallel reactions is controlled while their relative rates are not known. The data must be analyzed in terms of a mathematical model which at least approximates the manner in which the current is distributed among the several parallel reactions. One generally can only guess at this model; in fact, the elucidation of an appropriate model usually dwarfs the original problem both in complexity and in importance. Because of the capacity of the electrode/electrolyte interface, there always exists a double layer charging current in parallel with the other electrode processes. It is unfortunate that this parallel process produces major effects at high current densities where chronopotentiometry holds the most promise for the elucidation of electrochemical and solution kinetics. When engaged in studies of the manner and the extent to which electrochemically active species are adsorbed onto the surfaces of electrodes, we are dealing with parallel reaction paths. The current is distributed among at least three processes:
* Present address, Deptwtment of Chemistry, University of Minnesota, Minneapolis, hlinn.
a) faradaic reaction of adsorbed material b) faradaic reaction of nonadsorbed, diffusing material C) nonfaradaic charging of the electrode/electrolyte interface.
(1) C. D. Russell and J. IM. Peterson, J. Electroanul. Chem., 5, 467 (1963).
In order to determine an adsorption isotherm with any precision, it is necessary to first develop an appropriate VOL. 39, NO. 4, APRIL 1967
485
mechanism for the manner in which the current is distributed among these processes. Unfortunately a definitive a priori decision among the many possible models appears to be possible in only a minority of the cases. By working at high values of the current density (and assuming that the effects of double layer charging can be otherwise accounted for), the contribution due to the reaction of the diffusing material to the total charge will be minimized. Thus for all models lim ir = nFrad, i + c3
(1)
This fact has been evoked by Rao et ai. (2), in estimating the amount of “adsorbed oxygen” on a series of electrode surfaces. In general, it has not proven possible to obtain useful information in this fashion. If the data are sufficiently precise, it should be possible to choose the model or models which best describe the data by testing for which of the possible models best “fit” the data. Two chronopotentiometric studies of specific systems (3, 4). have attempted to do just this by visually intercomparing various graphical treatments of the data; the authors of these studies have expressed reservations about the feasibility of their approach. It is difficult to draw any firm conclusions from these studies, however, since realistic models for the reaction of these adsorbed species are not known and were not determined and since neither the magnitude of the experimental error nor the manner in which the qualitative conclusions would be altered if more precise data were available was known. In addition, linearized approximations were employed for certain of the models under conditions where they are not completely valid and it was not demonstrated that the methods used for the determination of the chronopotentiometric transition time were accurate for the study of the systems under investigation. Most of these faults would be common to any experimental investigation of this nature. It is clear that it is necessary to assess the limitations and potentialities for distinguishing the most realistic of the many possible chronopotentiometric models from a less experimental point of view. To this end, extensive synthetic data were generated and analyzed and the ~2 test was employed as a best fit criterion. The responses of the calculated physical and statistical parameters to changes in the solution composition and in the number and precision of the data were examined in detail. As a part of this study, a suitable least squares analysis was developed for the sequential treatment of the data according to the principal chronopotentiometric models. This analysis overcomes some of the deficiencies and irregularities of previously suggested methods of data treatment ( 4 ) . The conclusions of this study are ultimately dependent on the validity of this data analysis and therefore the method is discussed in some detail. DATA ANALYSIS
Of the several possible chronopotentiometric models for the relative rates of reaction of adsorbed and solution species, we shall consider only four in detail. The first of these, the “reacts first” model, considers that the reaction of the ad(2) M. L. B. Rao, J. Damjanovic, and J. O’M. Bockris, J. Phys. Chem., 67, 2508 (1963). (3) H. A. Laitinen and L. M. Chambers, ANAL.CHEM.,36, 5 (1964). (4) S. V. Tatwawadi and A. J. Bard, Ibid., 36, 2 (1964). 486
ANALYTICAL CHEMISTRY
sorbed material has proceeded to completion before the reaction of the solution material begins. This model predicts an adsorption prewave prior to the main chronopotentiometric wave. The sum of both transition times is denoted by T and is related to the current density i and the charge nFr consumed by the reaction of the adsorbed material by the following expression (5-7) n F r = ir
- b2/i
(2)
where b =nFZ/zC/2
(3)
Alternatively, the nondiffusing material might react at a “constant current efficiency” during the experiment (6, 8). This model has been widely used in studies of the cooxidation or reduction of oxides on electrode surfaces (8-1 7) nFr
=
ir
- br*/z
(4)
Thirdly, the adsorbed material might not react until after the concentration of the solution material has been reduced to zero at the electrode surface. Under these conditions, two inflections should be observed in the chronopotentiometric E-t curve, the second corresponding to an adsorption postwave. The sum 7 of the two transition times is given implicitly for this “reacts last” model by (4, 7, 18)
mFr
= i7 arc cos
[”i2 7
11
- 2 [ b y i % - b2)]*‘2/i
(5)
These three mechanisms are also known by the mnemonics AR, SR (adsorbed reactant, solution reactant), SAR (solution and adsorbed reactant), and SR, A R (solution reactant, adsorbed reactant) (19,20). The fourth model considers that the extent of adsorption of material onto the surface of the electrode is proportional to the concentration in the solution phase at the electrode surface and that the rates of the adsorption and desorption processes are so large that the solution and adsorbed species are in equilibrium at all times. Under this “equilibrium-linear isotherm” condition, the current densities and transition times are implicitly related by the following expression (6,7,21)
a =
cdD7ir
(5) H. A. Laitinen, ANAL.CHEM., 33, 1458 (1961). (6) W. Lorenz, 2.Elektrochem., 59,730 (1955). 33, 322 (1961). (7) W. H. Reinrnuth, ANAL.CHEM., (8) J. J. Lingane, J. Electroanal. Chem., 1, 379 (1960); 2, 296 (1961). (9) A. J. Bard, ANAL.CHEM., 35, 340 (1963). (10) T. R. Blackburn and J, J. Lingane,J. Electroanal. Chem., 5,216 (1963). 33,1839(1961). (11) D. G. Davis, ANAL.CHEM., (12) D. H. Evans and J, J. Lingane, J. Electroanal. Chem., 8, 173 (1964). (13) J. J. Lingane and P. J. Lingane, Ibid., 5,411 (1963). (14) M. D. Morris, Ibid., 8, 350 (1964). (15) M. D. Morris and J. J. Lingane, Ibid., 8, 85 (1964). (16) D. G. Peters and J. J. Lingane, Ibid., 2, 1 (1961). (17) D. G. Peters and W. D. Shults, Ibid., 8,200 (1964). (18) F. C. Anson, ANAL.CHEM., 33, 1123 (1961). (19) R. W. Murray, J. Electroanal. Chem., 7 , 242 (1964). (20) R. W. Murray and D. J. Gross, ANAL.CHEM.,38, 392, 405 (1966). (21) R. A. Munson, J . Phys. Chem., 66,727 (1962); J. Electroanal. Chem., 5, 292 (1963).
In order to emphasize the general applicability of least squares analyses to nonlinear as well as linear problems, we will illustrate the general method of analysis employed for each of these models hy considering in detail the analysis according to the “reacts liist” model. Rewriting Equation 5 , we obtain the model or condition equation F =A
- i r arc cos [ E ,i2r
- 11 + 2[B(i2r - B ) ] l / * / i= 0 (7)
which depends in a nonlinear fashion on the parameter B
(=r9)’D).
A = a n F r . The principle of least squares
consists of choosing the parameters A and B so as to minimize the sum
where is the model equation evaluated with a particular datum (i5,r j )and preliminary values (A,,B,) of the parameters. Experimental errors in the datum and the approximate nature of A , and Bo causefi to (differfrom zero.
fi
where E is the magnitude of the relative precision. Equation 12 was employed in the generation of the synthetic data whose analysis is discussed below. One consequence of the weighting scheme implied by Equations 11 and 12 is that the values of the parameters calculated finally by Equation 16 are independent of the magnitude of E . Thus it is possible to estimate the values of the parameters with no a priori knowledge of E. However, the standard deviations associated with the final calculated values of the parameters, Equation 19, and the value of ~ 2 Equation , 20, are inversely proportional to E2. The weighting function appropriate to the “reacts last” model is obtained by combining Equations 11 and 12.
= F(ij,r~,Ao,Bo #) 0 ; j = 1, 2,.
..N
(13) Equation 8 could be minimized directly but a system of nonlinear algebraic equations in the parameters A and B would result. Since the solution to this set of equations would require an iterative process, it is no less general to expand the model equations in a Taylor’s series and to retain only those terms first order in the residuals A A and AB. In this way, the problem is converted to a linear one which can be improved by iteration.
(9)
wj is the weighting funciion and serves to discount those values off, associated with large experimental uncertainties. An unweighted or, more properly, an equally weighted procedure in which all the w, are taken to be equal can only be validly applied to the analysis of data which are known to be each associated with the same absolute uncertainty. If certain of the data pairs, are known less precisely, the corresponding values of jj will tend to be large and will predominate in the sum s. In an equally weighted procedure, the least squares adjusted values for the parameters A and B would be principally determined by that portion of the data which is least precise! The measured variables in chronopotentiometry are the time independent current density i and the transition time r. The weighting function appropriate to chronopotentiometry is
uTj2;and
are the variances associated with the jth values of r and i. The currenl. density is generally known to a much higher precision than the corresponding value of the transition time; therefore, it may be possible to assume that all of the experimental error is localized in the measured values of the transition times. Under this condition the weighting function would become
- AA B = Bo - AB
A = A,
This approach possesses the additional advantage that it renders identical the formalisms for the solution of linear and nonlinear problems (with the exception that the final adjusted values of the parameters are obtained on the first iteration for a nonlinear problem). The residuals AA and AB are obtained from the matrix equation (22,23)
where
and
DET (The current must protlably be regulated to better than 0.1 per cent if
($)’
uri2
is to be much less than
5
(E),’
UTf2
the whole range of transition times.) The relative precision for individual transition times measurements is generally remarkably constant-frequently varying by less than a factor of two over a thousand-fold range of transition times. Under these conditions, it is possible to write uT2= ET)^
(12)
(15)
= ~ W ~ F A ~ F A ~ Z W~F ( ~B W ~ F~ BF~A ~ F B(17) ~)~
Improved values of the parameters A and B are calculated from Equation 15 and the process is repeated until the calculated values of the parameters become constant. Note that it will be necessary to iterate linear problems if the weighting functions are functions of the parameters. Thus it is necessary to “cycle on the weights” in the analysis of data (22) W. E. Deming, “The Statistical Adjustment of Data,” Wiley, New York, 1943, (Republished by Dover, New York, 1964) Chapter X. (23) W. C. Hamilton, “Statistics in Physical Science,” Ronald
Press, New York, 1964, Chapters 2 and 4.
VOL. 39, NO. 4, APRIL 1967
487
Table 1. Models for Least Squares Analysis 1. “reacts first” model A = nFr
B =
(nF)s nD
FA = 1 FB = l/i Fr = -i 2. “reacts last” model
A = nnFr
B =
FA = 1 FB = 2[(i2r
(?)* n D
generally sufficient to achieve convergence. The iterative process was also started by using as preliminary values of the parameters the values estimated according to the “constant current efficiency” model but this procedure was less satisfactory. In the case of the nonlinear “equilibrium-linear isotherm” model, the initial values of the parameters r and D were taken to be equal to the values estimated according to the “constant current efficiency” model. The standard deviations associated with the calculated values of the parameters are estimated according to the following expression (22).
- B)/Bll/*/i
FT = -iarccos
SA’ = Z W J F B ~ F B ~ I D ~
(” - I) i2s
se2
5. “equilibrium-linearisotherm” model A
=r
FA = 1
B = D
2a
- -+ (2a2- 1) exp a’ erfc a
6
-C 2 D FT = -exp a2 erfc a r
a
=cQm
according to the linear “constant current efficiency” model; this process converges very rapidly, on the second or at most the third cycle. The only remaining problem is the way in which A, and Bo, the initial values of the parameters, are chosen. If the analysis is to be routine, it is important that the method of choosing A , and Bo be systematic and capable of producing sufficiently good values that the iterative process diverges only infrequently. For linear problems, it is generally convenient to choose A, and Bo equal to zero since this results in a tremendous simplification of the algebraic manipulations and a corresponding decrease in computation time. In the case of the nonlinear “reacts last” model, it was decided to calculate the initial values of the parameters m F r and
tF)2
n D according to an approximate “reacts last”
model due to Lorenz (14).
ZWjFnjF~jlDm
(19)
x2 “goodness of fit” parameter (23-25) is given by (22) x2 = Z w j f 2 - AAZwjhFAj - ABZwlfFBj (20)
The 3. “constant current efficiency”
4. “reacts last” (Lorenz) model
=:
The values of the parameters, derivatives, and weighting functions employed in the least squares analyses according to these various models are summarized in Table I. This general least squares method of estimating the values of the parameters does not require any particular mathematical form for the model to which the data are being fitted. It is certainly true that the calculations become more tedious as the complexity of the problem increases and especially if the parameters appear in a nonlinear fashion but this is of little consequence if the calculations are performed on high-speed computing equipment. The frequent ploy of introducing a suitable approximation so as to linearize the model in order to “simplify” the data treatment is an unnecessary and archaic procedure and one which has undoubtedly introduced a significant bias into many experimental results. A word of caution. No data analysis technique will ever relieve the research worker of his obligation to sift through the results and to consider their reasonableness. An improperly weighted analysis can often be detected simply by plotting the derived function and some of the experimental data it is supposed to describe. Furthermore, a data analysis procedure will always give some numerical answer and one must constantly be on the lookout for impossible or unphysical results. Sometimes too, a variety of circumstances will conspire to cause even the best designed methods of analysis to fail to converge. There is very little one can do under these circumstances short of trying a new method of analysis, or a different set of data. GENERATION OF THE SYNTHETIC DATA
It was decided to apply this method of data analysis to the study of synthetic data in an attempt to gauge the extent to which the estimated values of the parameters depend upon the model according to which they were calculated. The use of synthetic data permits a greater flexibility with respect to the choice of values for the parameters and also eliminates any possible ambiguity associated with the measurement of chronopotentiometric transition times. Synthetic transition time data were generated according to the “reacts first,” “constant current efficiency,” and “reacts last” models. In the absence of adsorpiion, a transition time of one second would correspond to a current density of ~
This procedure proved to be satisfactory for most of the synthetic data presented below as well as for a variety of experimental data. The calculation of five cycles was 488
0
ANALYTICAL CHEMISTRY
(24) W. P. Schaefer, Inorg. Chem., 4, 642 (1965). (25) E. B. Wilson, Jr., “An Introduction to Scientific Research,” McGraw-Hill, New York, 1952, Chapter 8 .
-oo-oo-oo-ow----..oo-oo--o---ow--oo---oowoo--ooo-oo-o--oo--o-ooo-ooo---oo-o--o--o--o-oo-oo-o-oo--o “reacts last” -oo-oo-o--ow--~-..oo-oo--o---ooo--oo---oooow--ooo-oo-o--w--o-ooo-ooo---oo-o--o--o--o-oo-oo-o-oo--o
-
“equilibrium linear isotherm”
------------------~---------~-oo--------ooww-ooooooooooooowoooowwoooowoooooo-o 2. E-lOqb %we’’
-oo-oo-oo-ooo--”1-~oo-w--o-1-ow-1oo---oow~--~o-w-o--oo--o-ooo-ooo---oo-o--o--o--o-oo-oo-o-oo--o “reacts last” ooo-oo-oooooo-o---oo-oo-~o---ow-wo--wooow--~oww-o--oo--o-ooo-oooo--w-o--o--o--o“reacts last” (Lorentz) ooo-oo-oooooo-----oo-oo--o---ooo--oo---ooww--oowoo-o--w--o-ooo-wo---oo-o--o--o--o-oo-oo-o-oo--o “reacts first“ --o-oo-----o------oo-oo--o---ooo--oo---oooow--oooow-o--~--o-ooooooowooo-o--oo-oo-owoooo~o-oo--constant current e:fficiency” --o-oo-o---oo-----oo-oo--o---ooo--oo--~oowoo--oooow-~--oo--o-oowoooo-ow-o--oo-oo-oooowo-o-oo--o “equilibrium - linear isotherm” --0-00-0- --oo-----oo-oo--o---ooo--oo---oo~w--oowoo-o--oo--o-oo~oooo-ooo-o--oo-oo-owo-oo-o001-0 ”
Figure 1. The relative signs of the residuals calculated for the individual values of The data were calculated according to the “reacts last” model C
n FdA
=
T
cm2/sec;nFr
0.001M; D = 6.25 X
= 25
pC/cm2
RESULTS AND DISCUSSION
amp cm-::.
Current densities of i(K) amp cm-2
2 would correspond to transition times of 10-4 -1 second.
The calculation of thr: transition time in the presence of adsorption according to the first two of these models was performed analytically for each value of i ( K ) by rearranging Equation 2 and Equation 3. The values of the transition time corresponding to the “reacts last” model were calculated by the Newton-Raphson iterative procedure (26) and the iteration was continued until the last correction to the calculated transition time was less than 0.1 percent. The data so generated covered a range of about l-10d3 seconds. “Experimental errors” were introduced into the calculated transition time data by selecting 100 sequential normal deviates ( N . D . ) of zero mean and unit standard deviation (27) and multiplying these deviates by the appropriate percentage errorE. Thus ~ e x p=
(1
+ E X N.D.)Ttheor
Table I11 presents some of the values of the parameters estimated from the synthetic data. Calculations were also performed on additional data with relative standard deviations of one and five percent and on data sets of 25 rather than 100 values. These smaller data sets were constructed from the first, fifth, ninth, etc., values of the larger set. Since these five models depend only on the parameters nFr and nCD112,the results in Table I11 actually correspond to more general situations than are suggested by the key in
Table II. Key to Table 111 A. Experimental Conditions:
J 1 2 3 4 5 6
(22)
In this fashion, the generated data appear to be drawn from normal populations wil h uniform relative standard deviations. Because the distribution is normal, the x 2 test, Equation 20, is applicable. Because u t 2 = 0 and u,2 = E2r2,the weighting function given by Equation 12 is applicable.
7 8 9 10 11 12
B.
25.0 25.0 25.0 25.0 25.0 25.0 100.0 100.0 100.0 100.0 100.0 100.0
D X cm2/sec 6.25 6.25 6.25 25.0 25.0 25.0 6.25 6.25 6.25 25.0 25.0 25.0
nC,meq/liter 10.0 1.o 0.10 10.0 1 .o 0.10 10.0 1.0 0.10 10.0 1.o 0.10
Models:
K (26) J. Mathews and R. iL. Walker, “Mathematical Methods of Physics,” Benjamin, Inc., New York, 1964, p. 339. (27) Rand Corporation, “ A Million Random Digits and 100,ooO Normal Deviates,” Free Press, Glencoe, Ill., 1955. Nos. 04020 to 04119.
nFr, pC/cm2
1 2 3 4 5
Model “reacts first” “reacts last” “constant current efficiency” “reacts last” (Lorenz) “equilibrium-linear isotherm”
VOL. 39, NO. 4, APRIL 1967
489
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49 1
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ANALYTICAL CHEMISTRY
d d d
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Table 11. Thus, for example, the results corresponding to J = 1 are actually characteristic of any situation in which nFr is 25 pC/cm2 and the product nCD1i2equals 25 X equivalent/cm2 set" *. Let us first examine the behavior of x2. It is clear from these results that the x2 test (23-25) picks out the model according to which the synthetic data were generated if the relative standard deviation of r is less than about one or two per cent. This qualitative conclusion is not affected if the number of data is reduced to 25. Figure 1 illustrates the relative signs of the data residuals (22), that is the amount by which an individual “experimental” value of must be adjusted to force that data point to coincide with the least squares line, resulting from the analysis according to each of the models. The string designated as “true” corresponds to the experimental errors which were introduced into the data; this string would be unknown in an actual experimental study. Considering first the one per cent data, the signs of the residuals appear to vary in a random fashion only for the “reacts last” model, the model according to which the data were actually generated. The signs of the residuals calculated for the other models exhibit distinct trends which indicate that the data fit these models less well. With ten per cent datii, it is no longer possible to state that the residuals calculated by one model vary in a more random fashion than those calculated for the others. We have achieved a situation where the experimental errors are large compared to the differences in the behavior indicated for the different models and this “fit” test is no longer applicable. It is for just this same reason that the x 2 test does not distinguish among the different models at this level of experimental error. We are able to comment favorably on the criterion suggested by Murray and Gross (20) that the estimated values of the diffusion coefficient must be independent of the solution concentration of the reactant. Consider a hypothetical adsorption isotherm in which izFr increases from 25 to 100 pC/cm2 when the bulk concentration of the reactant increases from 0.1 to 10 mM. This situation would correspond to entries 3 and 7 or 6 and 10 of Table 111. It is clear that inconstancy of the estimated values of the diffusion coefficients with changes in the bulk concentration of the reactant is sufficient evidence to demonstrate that a particular model is unrealistic. This conclusion appears to be valid with 100 data points if the relative precision of the data is better than about 10 per cent. With 25 data points, the use of this criterion is restricted to data whose relative precision is better than about 5 per cent. Table I11 indicates that the divergence among the values of D estimated according to the various models is largest in dilute solutions while the divergence among the estimated values of nFr is largest in the more concentrated solutions. As the solution concentration decreases, the estimated values of nFr fall into two classes: the class characteristic of the “reacts first” model and the class characteristic of the other three models. Therefore, if it is desired to obtain an approximate value for nFr in a dilute solution, it is of less consequence which model is chosen since the differences among the various models are probably no more than 25-50 per cent. CONCLUSION
The above analysis has demonstrated the following limited conditions under which it should be possible to use internal criteria to distinguish the most realistic chronopotentiometric model.
1. It must be demonstrated that the experimental transition times do not contain a bias, especially at high current densities, due to double layer charging and other effects. 2. A statistically meaningful method of data analysis and a data set of 25-50 points must be employed. 3. Sufficient replicates must be obtained at each data point to determine the standard deviation characteristic of that data point or to demonstrate the functional dependence of the standard deviation on the values of the transition time. It is mandatory that preliminary calculations be performed to determine that the weighting functions chosen are approximately valid under the given experimental conditions before these weighting functions are applied to the analysis of the experimental data.
4. The relative standard deviation of r should be as small as possible and must be less than about five per cent. The one per cent level needed for the application of the x 2 test is probably not accessible except under the most auspicious circumstances. 5 . The entire analysis must be repeated over a wide range of solution concentrations and the estimated values of the diffusion coefficient must be independent of this variation. It should be clear that exceedingly large amounts of data would be required to determine the appropriate model, and accurate values for r in the fashion outlined here. To implement this program would be so tedious that its use would not expedite the study of the extent of adsorption by chronopotentiometry . In view of the lack of any general method for determining the mechanism by which the adsorbed material is reduced and in view of the lack of consensus about the way in which accurate transition time measurements should be made, I submit that chronopotentiometry is an inefficient and time consuming technique of limited value for the study of adsorption. There is now a good deal of evidence to suggest that adsorbed material can sometimes desorb in the double layer charging region of a chronopotentiogram (3, 23, 29). This possibility significantly complicates the application of chronopotentiometry to the study of adsorption for not only must the research worker consider this possibility in interpreting his data but he must also assume some mechanism for the desorption process in order to estimate the extent of the initial adsorption ! These criticisms are of major importance since there is another technique, chronocoulometry, whereby much the same information can be obtained more efficiently (30-32). Since chronocoulometry is a potentiostatic technique, one need only assume that the reactionof the adsorbed and of the diffusing species proceed independently in order to separate out the one from the other. No detailed mechanistic assumptions are needed and therefore the data analysis is less equivocal. Perhaps most importantly of all, chronocoulometry has yielded to ready automation so that large amounts of data can be obtained and analyzed in a very efficient fashion (32-34).
(28) R. A. Osteryoung and J. H. Christie, ANAL.CHEM.,38, 1620 (1966). (29) Zbid.,submitted to ANAL.CHEM.,1966. (30) J. H. Christie, J. Elecrroanal. Chem., in press, 1967. (31) F. C. Anson, ANAL.CHEM.,38,54 (1966). (32) F.C. Anson, J. H. Christie, and R. A. Osteryoung, J. Elecfroanal. Chem., in press, 1967. (33) G. Lauer and R. A. Osteryoung, ANAL.CHEM.,38, 1106 (1966). (34) G. Lauer, Ph.D Thesis, California Institute of Technology, Pasadena, Calif., 1967. VOL. 39,
NO. 4,
APRIL 1967
493
I am realistic enough to recognize that it is essentially impossible to discourage research oriented toward the use of chronopotentiometry for the study of adsorption. I suggest, therefore, that this research could most profitably be directed towards discovering an accurate method for the determination of chronopotentiometric transition times. If a consensus can be reached on this point, and if it proves possible to automate this method, then chronopotentiometry might gain moderate stature as a tool for the study of adsorption.
ACKNOWLEDGMENT The author is grateful for the frequent and cogent criticisms of Fred C. Anson, Joseph H. Christie, and William P. Schaefer. RECEIVED for review August 4, 1966. Accepted February 9, 1967. This work was supported in part by a predoctoral fellowship from the U. S. Public Health Service, Division of General Medical Sciences. Contribution No. 3399 from the Gates and Crellin Laboratories of Chemistry.
Use of Organic Additives to Induce Selective Liquid-Liquid Extraction of Niobium with Thenoyltrifluoroacetone Aart Jurriaansel and Fletcher L. Moore Analytical Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tenn.
A new, highly selective method for the liquid-liquid extraction of niobium(V) is described. The method is based on the ability of n-butanol in the aqueous phase to enhance the formation of an extractable niobium chelate with thenoyltrifluoroacetone. Niobium can be recovered quantitatively from aqueous solutions of concentrated hydrochloric acid or hydrochloric acidsulfuric acid mixtures. Excellent separation of niobium from most metal ions is effected with 0.5M thenoyltrifluoroacetone-xylene solution. FORSEVERAL YEARS chemists have been puzzled by the inextractability of niobium from aqueous acidic solutions with 2-thenoyltrifluoroacetone (TTA). The highly selective extraction of zirconium with TTA (1, 2) from acidic solutions is extensively used for analytical and preparative purposes. Niobium is essentially inextractable under the conditions employed for zirconium. Ordinarily, one would expect the more highly-charged niobium ion with its smaller ionic radius to exhibit even higher extractability than zirconium with TTA. The negligible chelation and extraction of niobium from aqueous solution were postulated to be a reflection of its pronounced hydration characteristics. About 1954 it was observed (3) that 95Nbtracer formed by the decay of 95Zrtracer in 0.5M TTA-xylene solution was extremely difficult to strip from the organic phase with various strong acids. Such behavior suggested that niobium does form a highly stable chelate complex with TTA in the absence of water, Further, several exploratory experiments at that time indicated that the extraction of g5Nbtracer with 0.5M TTA-xylene increased from 3 x in 3.6M hydrochloric acid to 45 %in 3.6M hydrochloric acid-37 (vjv) acetone mixtures. Recently we have studied the extraction behavior of niobium with TTA from mixed aqueous-organic solutions. These studies have led to a new, highly selective liquid-liquid extraction method for niobium. 1 Present address, South African Atomic Energy Board, Private Bag 256, Pretoria, Republic of South Africa.
(1) F. L. Moore, ANAL.CHEM.,28,997 (1956). (2) F. L. Moore, “Metals Analysis with TTA,” Symposium on Solvent Extraction in the Analysis of Metals, ASTM Spec. Publ. No. 238 (1958). (3) F. L. Moore, unpublished data, Oak Ridge National Labora-
tory, Oak Ridge, Tenn, 1954.
494
ANALYTICAL CHEMISTRY
EXPERIMENTAL Apparatus. An internal sample methane proportional counter was used for alpha counting. A NaI(T1) well-type gamma scintillation counter, x2 inches, was used for gamma counting. Reagents. 2-Thenoyltrifluoroacetone (TTA, M.W. = 222) is available from Columbia Organic Chemicals Co., Columbia, S. C. Other chemicals used were analytical reagent grade. Procedure. Pipet 1 ml of g5Nbtracer (1.5 X IO6 gamma c.p.m.) into a 50-ml glass centrifuge tube. Add appropriate amounts of distilled water, hydrochloric acid, and butanol (or other organic solvent under study) to obtain 5 ml of a solution which is 7N hydrochloric acid. Extract the aqueous phase for 5 minutes with 5 ml of 0.5M TTA-xylene solution, using high speed motor stirrers equipped with glass paddles. After the extraction, centrifuge the tubes in a clinical centrifuge for 2 minutes. Count 1-mI aliquots of each phase for S5Nbradioactivity in a well-type gamma scintillation counter.
RESULTS AND DISCUSSION
Preliminary experiments verified that the presence of acetone in aqueous hydrochloric acid solutions enhanced the extraction of 95Nb tracer with 0.5M TTA-xylene solutions. However, the coextraction of acetone produced large volume changes, rendering further studies difficult. Use of 0.5M TTA dissolved in the relatively aqueous insoluble ketone, methyl isobutyl ketone, resulted in about 80 extraction of 95Nb from 7N hydrochloric acid solution. Further experiments with methyl isobutyl ketone containing no TTA gave similar recoveries, suggesting that the niobium extracted as an ion association chloro complex and not as a chelate. At this time we observed that various alcohols were considerably more effective than ketones for enhancing the chelation and extraction of niobium with TTA. Moreover, volume changes of the phases were negligible. The extraction of 96Nbtracer with 0.5M TTA-xylene from aqueous solutions of 7N hydrochloric acid containing varying amounts of methanol, ethanol, propanol, and butanol is shown in Figure 1. The extractability of 95Nb tracer increases with increasing concentration of alcohol in the aqueous phase. With methanol or ethanol the extraction behavior was somewhat erratic; however, with propanol or butanol results were quite reproducible above 10 volume %.
x
