Theory of Alternating Current Polarography SIR: Reinmuth and Smith (8) present a series of self-nullifying argumentse.g., in the eighth paragraph; between the seventh and final paragraphs-but it appears that they are concerned with two questions: 1. Do treatments of the electrochemical process on the basis of rate theory, Fick’s law diffusion, and sinusoidal boundary conditions lead to two expressions for the phase angle, or is only one of these a mathematicallv valid outcome? 2. In any case, are the results of this simde treatment atmlicable to the sysiems which have-been studied, or must the model be elaborated?
On the first point, the assertion of -4kn (1) is repeated-viz., that only one expression can be obtained. We have discussed the question in some detail in an article to which Reinmuth and Smith referred; in brief, we felt that the use of sinusoidal boundary conditions involves the assumption that the polarization effects ensuing a t higher frequencies can be expressed as a phase shift of a purely sinusoidal quantity, whereas it seems unlikely that such a model could be correct in view of the rectifying properties of the faradaic admittance. Reinmuth and Smith seek to rebut this view in three ways :
“Die Erorterungen sollen sich in dieser Arbeit auf den Fall beschranken, bei dem die Amplitude der uberlagerten Wechselspannung so klein ist, dass die Nichtlinearitat der Elektrodencharakteristik fur Wechselspannung vernachlassigbar ist.” 2. By stating that treatments of the redoxokinetic effect show the rectifying effects to be negligible a t small amplitudes. However, these treatments are based on the same model of the process as are the treatments of the faradaic admittance itself, and it is therefore not surprising that experimental results are at variance with these treatments also, as indicated by Reinmuth and Smith. This state of affairs, of course, supports the view which they criticize: Since the equations are manifestly inapplicable, their prediction of negligible rectifying effects is a t best irrelevant, 3. By misquotation from our papers. Compare the statement of Reinmuth and Smith--“. . .have postulated that one expression may be validly applicable to systems in which the oxidation reaction is more hindered and the other to systems in which reduction is more hindered. . .”-and our original (i?), “The experimental results. . .can be qualitatively interprekd on the basis. . . that neither equation. . .can be correct. . . ”
1. By claiming that rectification has been taken into account by Matsuda. However, this is true only for Matsuda’s treatment of the low-frequency case (6), where the question of two solutions does not arise. In the relevant paper (6), Matsuda states, in contradiction to the assertion of Reinmuth and Smith,
On the second point, Reinmuth and Smith agree with our suggestion that the model on which these treatments are based is too simplified. We suggested the neglect of rectifying properties to be such an oversimplification; other suggestions have also been made which do not exclude our suggestion-eg., adsorption of the electroactive species (3), more sophisticated models of the diffusion process (Y),and effects due to the structure of the double layer (4).
SIR: We comment on the points raised by Bauer (1). Our arguments parenthetically mentioned in his first paragraph have been elaborated in a recently submitted paper (6). In regard to the first question he formulates, the most rigorous treatment of the problem, that of Matsuda (4), does not employ sinusoidal boundary conditions (except for the controlled variable, potential). Indeed, the derived expression for current is not a strictly sinusoidal function, but rather a recurrent series of sinusoidal terms with time-dependent coefficients. However, less rigorous treatments in which sinusoidal boundary conditions are used, in so far as they lead to the valid expression for phase angle, agree closely with the predictions of Natsuda’s treatment. Since the treatment does not employ the sinusoidal boundary
conditions to which Bauer refers, it does not depend OD the implicit assumption to which he objects in his second paragraph. The a priori applicability of this assumption, however, is obvious for any experimental system in which such phase shifts can be measured and Matsuda’s treatment demonstrates, rather than assumes, its validity for the particular theoretical model he considers. In regard to point 1 of the second paragraph, the quotation from Matsuda’s work is misleading. Matsuda’s discussion is limited to cases in which nonlinearity is negligible (which the treatment shows to be true under usual experimental conditions) ; however, his derivation is not so limited and can be manipulated to yield expressions for currents of any amplitude and any harmonic.
Reinmuth and Smith insist that, of these, posdbilities, adsorption is the only one which should logically be considered and which can account for all observations; an objective basis for this conviction is lacking. In questioning the significance of our experimental results, Reinmuth and Smith gave figures for the ratio of the faradaic and double-layer admittances (0.1 a t 1000 C.P.S. and 0.04 at 5000 c.P.s.); they did not state how these figures were obtained, other than to mention millimolar cadmium. Our results for this system-eg., in 1M sodium sulfate-gave ratios of 1.5 a t 1000 C.P.S. and 0.5 a t 5000 C.P.S. The magnitudes given by Reinmuth and Smith are clearly wrong, as must already have been obvious to anyone conversant with the field. H. H. BAUER Faculty of Agriculture Tne University of Sydney Sydney, Australia LITERATURE CITED
(1) Aten, A. C., doctoral thesis, Free University of Amsterdam, 1959.
(2) Bauer, H. H., Elving, P. J., J. Electroanal. Chem. 2, 53 (1961). (3) Laitinen, H. A., Randles, J. E. B., Trans. Faraday SOC.51,54 (1955). (4) Matsuda, H., J . Phys. Chem. 64, 339 (1960).
(5) . . Matsuda,. H.,, Z . Elektrochem. 61, 489 (1957). (6) Ibid., 62, 977 (1958).
S. G., 2nd Seminar on Electrochemistry, Karaikudi, 1960, Abstract 1.10. (8) Reinmuth, W. H., Smith, D. E., ANAL.CHEM.33,964 (1961). (7) Rangarajan, S. K., Doss, K.
Matsuda’s treatment indicates that the fundamental component of current does not differ significantly from its linear value until the potential reaches such amplitude that the faradaic rectification component of current becomes significant in magnitude. More generally, however, it is self-evident that, whatever model of the electrode process be chosen, the fundamental component of current cannot stray from its linear value to an extent appreciably greater than the sum of the magnitudes of the nonlinear (rectification and higher harmonic) components. With regard to point 2 of Bauer’s second paragraph we reiterate our contentions that theoretically, inapplicability of present treatments to existent anomalous data cannot arise through their neglect of the nonlinearity of the electrode reaction, because their VOL. 33, NO. 12, NOVEMBER 1961
1803
predictions are not substantially altered by explicit consideration of nonlinearity; and experimentally, no published data on these anomalous systems indicate the magnitudes of nonlinear components of the current to be sufficiently large that nonlinearity of the electrode reaction need be taken into account for the adequate theoretical description of their small amplitude a.c. polarographic characteristics in terms of whatever model may be applicable. In regard to point 3 of the second paragraph, our original objection (6) was raised against the following statement of Bauer and Elving ( 8 ) : “Thus, the mathematical treatment can lead equally to Equation 1 or 2. Whether cot I$ for a particular system is greater or
.
less than 1.. depends on whether the oxidation or the reduction process is preferentially hindered or favoured.’’ In regard to the third paragraph, we concur that the conviction cited by Bauer would be without objective basis. Discussion in a recent paper by Tamamushi and Tanaka (7) is pertinent to our observations that adsorption would be a logical explanation of some of the anomalous data of Bauer, Smith, and Elving (9). In regard to the fourth paragraph, our estimations were seriously m i s calculated, although we find them accurate for some systems similar in constitution to those employed by Bauer, Smith, and Elving.
LITERATURE CITED
(1) Bauer, H. H., ANAL.CHEM.33, 1803 (1961). (2) Bauer, H. H., Elving, P. J., J. Ebctroanal. Chem. 2, 83 (1961). (3) Bauer, H. H., Smith, D. L., Elvin , P. J., J . A m . Chem. SOC.82,2094 (19607 (4) Matsuda, H., Z. Ebktrochem. 62, 97f (1958). (5) Reinmuth, W. H., Smith, D. E., ANAL.CHEM.33,964 (1961). (6) Zbid., submitted for publication. (7) Tamamushi, R., Tanaka, N., 2. physik. Chem. (N. F.) 28,158 (1961). W. H. REINMUTH D. E. SMITH’ De artment of Chemistry Cokmbia University New York 27, N. Y. 1 Present address, Department of Chemistry, Northwestern University, Evanston, Ill.
Determination of Small Amounts of Calcium in Magnesium O x i d e SIR: The determination of calcium and magnesium with (ethylenedinitril0)tetracetic acid (EDTA) is qui’ck and reasonably accurate. Lewis and Melnick (2) evaluated the conventional EDTA method and compared results with those obtained by the classical gravimetric procedures. However, their
Table l.
work was limited to analyses of mixtures containing high ratios of calcium to magnesium. The EDTA method has been used with some success for routine determination of calcium in magnesium oxide for our plant process control. However, there was not good agreement be-
Calcium Determination in Known Solutions of Calcium and Magnesium Chloride
Un-
Slow Precipitation-EDTA Solution “,OH-EDTA Analyst I1 Analyst I 1 Caa found in 0.127 0.134 0.162 0.129 0.137 0.132 2 duplicates, % 0.166 0.177 0.193 0.197 0.189 0.202 0 200 0 197 0.184 0.206 0.204 0.220 3 4 01208 0:iio 0:223 01% 0:241 0.232 5 0.235 0.235 0.237 0.257 0.262 0.270 Av. % Ca, i? 0.189 0.202 0.209 0.004 0.014 0,008 Std. dev. of method. So Coefficient of variation of analytical method, 100 2.1 7.0 3.8 Coefficient of variation from true mean, 100 -10.9 -4.7 -1.4 known
($)
(e)
Conventional EDTA 0.046 0.040 0.069 0.075 0.075 0.069 0.079 0.089 0.117 0.100 0.076 0.007 9.2
-64.2
a % calcium added.to each solution: 0:140 (l),0.200 (2), 0.220 (3), 0.239 (4), 0.260 (5). Average per cent calcium ( p ) for all solutions was 0.212. Values reported on MgO basie.
Table II.
Calcium Determination in Magnesium Oxide
“,OH-EDTA Ca found in duplicates, 0.265 0.272 % 0.286 0.265 0.322 0.329 0.300 0.322 0.264 0.286 0.291 Av. % Ca, 2 0,009 Std. dev. of method, So Coefficient of variation of analytical method, 100 3.1
(9)
1804
e
ANALYTICAL CHEMISTRY
Slow Precipitation-EDTA Analyst I Analyst I1 0.272 0.270 0.272 0.236 0.285 0.257 0.300 0.304 0.343 0.350 0.357 0.372 0.386 0.357 0.343 0.380 0.265 0.307 0.272 0.293 0.313 0.309 0.013 0.013 4.2
4.2
Conventional EDTA 0,096 0.134 0.160 0.189 0.176 0.188 0.169 0.143 0.122 0.165 0.154 0.024 15.6
tween replicate calcium determinations when the percentage of calcium in magnesium oxide was less than 0.5%. For example, a comparison of the analysis of 20 duplicate spot samples showed that the relative standard deviation of the conventional EDTA (9) method was greater than 15%. It also revealed that the average value obtained for the calcium was approximately two thirds that obtained by the classical precipitation method (1). On the premise that part of the calcium was being coprecipitated with the magnesium hydroxide in the EDTA method, a study was undertaken to eliminate this possible source of error. EXPERIMENTAL
Two EDTA approaches were evaluated. The first used ammonium hydroxide to remove most of the magnesium as magnesium hydroxide before determination of the calcium. The second attempted to precipitate the magnesium as its hydroxide, as homogeneously as possible, by the slow dropwise addition of a buffer solution before the determination of the calcium. A M M O N I U M HYDROXIDE MODIFICATION
Accurately weigh duplicate samples of magnesium oxide containing 2 to 3 mg. of Ca, transfer to 250-ml. beakers, and add 25 ml. of distilled water to each sample and just enough 6N hydrochloric acid to dissolve the samples. Heat the solutions to boiling and slowly add concentrated ammonium hydroxide until a slight excess is present to precipitate magnesium hydroxide. Continue to boil the samples for a few minutes, then cool and centrifuge. Decant the clear solutions into 250ml. Erlenmeyer flasks. Wash the precipitate in the centrifuge tubes with distilled water, recentrifuge, and com-
