results of 150 ft reported in Table I, reference 4. Guiochon (6) in his treatment did not realize the error in Table I of reference 4. When taken into account, his predicted maximizing length (Eq. 14, reference 6) should be about 260 ft which of course is much too long. Table I1 of reference 4 shows that the maximizing length in the case of octane isomers is at 200 ft and at koptof 0.58. This seems to contradict the theory offered here; i.e., kODt2 2. However, calculations using the various experimental values given in that table, show that the relative retention increases in going from L = 300 ft to L = 200 ft, but upon further decreases of the length to 150 feet, a decreases. The reason for this decrease in a with the reduction of temperature, a most unusual behavior in light of the initial increase in a, is not given in Karger's work. It is, however, this decrease in the relative retention term that causes the reduction in the resolution. Neglecting the last entry in Table I1 of reference 4, a and a can be determined and the left hand side of Equation 6 can be shown to be equal to -3.89. From Figure 1, the corresponding kept value is about 3.85. This would correspond to a column length of about 85 ft. We feel that the maximizing length should be at around 85 ft, and not 200 ft as reported in the original work (4). Guiochon's treatment yields 3800 cm as the maximizing length, a value which is also much lower than the experimental one. The unexplained and peculiar experimental decrease in a is the reason for the apparent disagreement between our treatment and Karger's work. The maximum in the resolution in their work is due, not to the normalization mechanisms, but to other unrelated effects. In a similar manner, it can be shown that in the case of the methylnaphthalenes reported previously by this author ( I I ) , the left hand side of Equation 6 is -4.17. Correspondingly, the optimizing k is about 3.7 and the maximizing length calculated from Equation 7 is about 190 ft. This result
agrees well with the experimental trend showing an approach to maximum resolution at 150 ft. In the case of the decane isomers reported by us ( I I ) , the theoretical maximizing k and L are about 3.3 and 55 ft, respectively. The experimental results seem to indicate that the maximum resolution occurs at a column length below 50 ft. In this case, the agreement between theory and experiment is not as good as in the previously discussed work. The explanation might lie in the approximations made in deriving Equation 6. It is thus seen that the relative retention term must be taken into account when one is attempting to increase the resolution via LTTNC. Guiochon's (6) recommendation, to optimize k to 3 in the case of packed columns and to 1.5 for capillary columns, is only a first-order approximation. A more exact prediction of the maximizing length requires an understanding of the interaction-between k , a,and the column length. Equation 6 takes these interactions into account, thus enabling a more direct approach to the maximum resolution. It should be realized that in deriving Equation 6 three approximations were made : the relation between the relative retention term and the temperature as given in Equation 3; the resolution expression as given in Equation 2; and the independence of H of the length and temperature. In view of these assumptions, the agreement between theory and experiment is indeed gratifying. It should be stressed that all the parameters required to solve Equation 6 can be determined experimentally with one column operated at two different temperatures. Once calculated, the researcher can then proceed to obtain maximum resolution without sacrificing analysis time.
RECEIVED for review October 19, 1970. Accepted February 5,1971.
Effect of Direct Current Polarographic Maxima on Alternating Current Waves F. M. Hawkridge and H. H. Bauer Department of Chemistry, Uniuersity of Kentucky, Lexington, K y . 40506
THECHARACTERISTICS of ac polarographic waves in systems where the dc polarograms show maxima have not been systematically studied; however, the behavior of several such systems has been reported. Breyer and Bauer ( I ) noted a splitting of the ac wave of nitrobenzene under conditions where a maximum was formed. Tamamushi and Tanaka ( 2 ) found that ac polarography could be successfully applied in the presence of small amounts of maximum suppressors. Tanaka et al. ( 3 , 4 ) reported cases where the presence of a maximum did not affect the ac wave. On the other hand, Fujiwara et al. (5) (1) B. Breyer and H. H. Bauer, Aust. J. Chem., 9,425 (1956). (2) R. Tamamushi and N. Tanaka, 2.Phys. Chem., 21,89 (1959). (3) N. Tanaka, T. Koizumi, T. Murayama, M. Kodama, and T.
Sakuma, A m / . Chim.Acta, lS,97 (1958). (4) N. Tanaka, R. Tamamushi, and M. Kodama, ibid., 20, 573 (1959). (5) S . Fujiwara, H. Kojima, T. Umezawa, and T. Kugo, J. Elecfroatid.Chem.,26, 53 (1970). 768
ANALYTICAL CHEMISTRY, VOL. 43, NO. 6, MAY 1971
noted that the presence of a dc maximum did affect the ac waves of several irreversibly reduced species, but not those of several reversibly reduced depolarizers. In the present note, we report the characteristics of the ac behavior of Cu(I1) in noncomplexing electrolyte, where the dc wave frequently shows a maximum. Important factors that have not previously been mentioned concern the use of a three-electrode circuit, and the potential of the ac peak in relation to the potential at which the dc maximum begins. EXPERIMENTAL
Apparatus. All polarograms were recorded using a Princeton Applied Research Model 170 Electrochemistry System. Half-cell potentials were measured with a Ballantine Model 355 AC-DC Digital Voltmeter. Procedure. All solutions were de-aerated with prepurified nitrogen for 15 minutes prior to measurement and were thermostated at 25 + 0.1 "C. Salt bridges were made with supporting electrolyte and filter-paper pulp (Whatman Filter
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E vs. S . C . E . ( V O L T S ) Figure 1. Polarograms of 2 X 10-3M Cum), 0.9M L i N 0 3 , pH 1.0,298
Curve 1. AC polarogram, 3-electrode circuitry, V,, Curve 2. DC polarogram, 3-electrode circuitry
=
OK, rn2iat1/6 =
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Paper Pulp Tablets, No. 5272). A platinum gauze electrode was used as the auxiliary electrode. The half-cell potentials were measured between the working electrode and a second reference electrode. Reagents. All chemicals used were reagent grade. The mercury used in the D M E was Bethlehem Instrument Mercury which is triply distilled. RESULTS AND DISCUSSION
It has been pointed out recently (6) that the use of a threeelectrode circuit with potentiostatic control of the half-cell potential of the working electrode permits the process of maximum formation in dc polarography to be investigated in considerably more detail than is possible with traditional two-electrode polarographic apparatus. In the reduction of Cu(II), the polarogram obtained with three-electrode circuitry shows a segment of a normal diffusion-controlled polarographic step followed by a streaming maxinium; the onset of the maximum is marked, at higher concentrations, by a welldefined inflection point on the curve ( A , Figure 1). At potentials anodic to the inflection point, the dc current appears to be diffusion controlled when examined by the criteria of polarographic log-plots and current-us.-time curves at single drops (6). It can also be seen from Figure 1 that the corresponding ac polarographic wave appears to be of normal shape over potentials at which the dc polarographic wave is diffusion controlled (anodic to ca. 0.025 V us. SCE). Consequently, it is not surprising that the concentration- and frequency-dependence of the height of the ac peak are apparently unaffected by the presence of the maximum (cf:Figures 2, 3) since the peak of the ac wave occurs at a potential where no maximum appears on the dc wave. Furthermore, at lower concentrations of depolarizer, there is no apparent distortion of the ac polarographic wave at all even though there is still a maximum on the dc polarographic wave (though smaller than ( 6 ) F. M. Hawkridge. T. W. Holt, and H. H. Bauer, to be published.
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Figure 2. Plot of charge transfer impedance times millimolar Cu(I1) concentration us. Cu(I1) concentration expressed as mM, for the same system shown in Figure 1, with rn21at1/6= 0.952 mg*i3 se&, V,, = 10.0 mV (RMS),f = 100 Hz
that which is present at higher concentrations of depolarizer). While the line shown in Figure 2 (determined by a leastsquares fit) gives about a 5 % change in the normalized chargetransfer impedance in going from a 1.00mM to a 2.00mM concentration of Cu(II), the corresponding change in the dc maximum is from 0.9 pA to 33.2 PA. In other words, alANALYTICAL CHEMISTRY, VOL. 43, NO. 6, MAY 1971
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Resistive faradaic impedance Capacitive faradaic component
though the magnitude of the dc maximum increases markedly with increasing concentration of depolarizer, there is little dependence of the measured charge-transfer impedance on frequency and concentration. The behavior described by Figures 2 and 3 is a necessary (but not sufficient) condition for assuming that the ac wave is not being influenced by the presence of a maximum. Of importance in discussing the effect of maxima on ac polarographic waves is a distinction between the use of traditional two-electrode and of three-electrode circuitry with potentiostatic control. In a two-electrode circuit, the potential applied across the working electrode and the reference electrode [E(applied)] is commonly not the same as the halfcell potential [E(measured)]. Because of the series resistance of the cell (solution resistance, the resistance of the salt bridges, DME capillary resistance, etc.), a portion of the E(applied) appears as ohmic potential drop. Hence, E(measured) is less than E(app1ied) by an amount that is proportional to the series resistance of the cell. Figure 4 shows the differences in the waves obtained with such circuits. It can be seen that ac peak heights and potentials are measurably different. Curve C shows the ac current us. E(measured) for the twoelectrode circuitry and demonstrates the care required in interpreting results obtained with two-electrode circuitry. The effect of streaming associated with a maximum is greater on the dc current than on the ac current. This can be ascribed to the fact that the ac current depends on the instantaneous concentrations of the oxidized and reduced species at the surface of the electrode, whereas the magnitude of the direct current depends on the magnitude and thickness of the concentration gradient around the electrode. While the latter is markedly affected by the streaming that accompanies maximum formation, the instantaneous concentrations
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Figure 4. Polarograms of 1.25 X 10-3M Cu(II), 0.9M L i N 0 3 , p H 1.0, 29S°K, mzIa t1I6 = 1.105 rng2i3sec1i6 Y,, = 10.0 mV (RMS),f = 100 Hz. Curve A . DC polarogram, 3-electrode circuitry.2.0 fiA/division Curve B. AC polarogram, 2-electrode circuitry. AC current cs. E(applied), 0.5 fiA division Curve C. AC polarogram, 2-electrode circuitry. AC current us. E(measured), 0.5 fiA/division Curve D. AC polarogram, 3-electrode circuitry. AC current c x E(applied), 2.0 fiA/division
at the surface are affected less in the case of systems such as Cu(I1) whose dc response is mass transfer controlled (nernstian). This idea agrees with the theoretical work of Sluyters and coworkers on the ac response with a streaming mercury electrode (7,8) which is relevant to the present problem. CONCLUSIONS
The presence of a maximum does not necessarily preclude the possibility of analytical and mechanistic studies of ac polarography when three-electrode potentiostatic control is used. Kinetic and mechanistic studies of the reduction of Cu(1I) by this technique are in progress, and details will be published in due course. RECEIVED for review November 23, 1970. Accepted January 26, 1971. This work was supported by the U. s. Army through the Themis Contract DAAB07-69-C-0366. The work had been initiated with the help of a grant from the National Science Foundation.
(7) A. B. Ijzermans, M. Sluyters-Rehbach, and J. H. Sluyters, Rec. Trau. Chim., 84,729 (1965). (8) A. B. Ijzermans and J. H. Sluyters, ibid.,pp. 740,752.
