Cadmium(I I) /Cadmium (Mercury) Exchange an Double Layer Structure Noel A. Hampson Department of Chemistry, Loughborough University of Technology, Loughborough, Leicestershire, England
David Larkin’ Department of Chemistry, The University of Texas at Austin, Austin, Texas, 78712
THEREHAS
BEEN in the past considerable success in using the Randles-Ershler approach to electrode reactions. The success of this method depends on the possibility of measuring the double layer impedance (and the electrolyte resistance) -CL-RE- in separate experiments and abstracting these from the electrode impedance considered as ;
Table I. Typical Values of io and R, Obtained at Various Cd(Hg) Concentrations Using the Computer Method. 1.0 M NaC104, 3.6 x 106M cm+ Cd(II), 23 “C Cd(1I)M Cd(Hg)M i,A cm-+ R& cm-s X lo5~ m X- 108 ~ G L ~ cm-2 F 0.16 31 .O 61.2 3.6 1.3 0.17 31 .O 61.0 3.6 1.9 0.16 31 .o 60.8 3.6 3.0 60.6 0.17 31.0 3.6 3.8 60,4 0.16 31 .O 3.6 4.9 0.16 31.0 60.3 3.6 5.7
CL are then selected until the condition that the inphase com1 ponent RRminus the out of phase component __ of the faraa CR
so that the faradaic impedance;
1
-W-R,-
yields RDwhen -W- is forced to shrink to zero at infinite frequency (CO-112 = 0). This approach has been satisfactory for electrodes where the true surface area is known. Unfortunately, this has meant that for all practical purposes the method has been restricted to mercury and mercury amalgam electrodes. The Randles analysis has been extended to solid metals (where the true surface area is unknown) by Farr and Hampson ( I ) using a computer technique. Essentially this method entails measuring the electrode impedance as a resistance and capacitance in series;
-le* the measured electrode impedance is then converted by a number of parallel and series transformations to the electrode impedance considered as ;
daic impedance is constant, and that - is zero at infinite UCR frequency. When these two conditions are met;
This method has proved successful in a number of studies using solid metals ( I , 4 , 5). We have previously reported results on the Cd(IT)/Cd(Hg) (6) system obtained by using the Randles approach, RE and C L being obtained in separate experiments. In the present paper we give some further data on this system which have been analyzed using the computer method in addition to the classical method. The results obtained from the two approaches are different and the implication of this appears to have considerable importance to the study of fast electrode reactions. EXPERIMENTAL AND RESULTS The impedance bridge, the electrolyte cell, and purification procedure have all been described elsewhere (1, 6). Figure 1 1
RD
by the steps described by Randles (2, 3). Values of RE and
shows typical RR and __ V * C O - ” ~ plots obtained at different CR Cd(Hg) concentrations. In Table I are given values of io (the exchange current) and R, (the frequency independent component of the electrode impedance) at varying Cd(Hg) concentrations. The value of io = 165 i 10 mA 3mP2was obtained over the concentration range studied and implies that a = 0. DISCUSSION
To whom all correspondence should be sent. ~~
(1) J. P. G. Farr and N. A. Hampson, Trans. Faraday SOC.,62, 3493 (1966). (2) J. E. B. Randles, “Transactions of the Symposium on Electrode Processes,” E. Yeager, Ed., John Wiley and Son, New York, N. Y.,1961, p 209. (3) H. A. Laitinen and J. E. B. Randles, Trans. Faraday SOC.,51, 54 (1955).
The fundamental difference in the two models used by us to analyze these results is that different criteria have been used (4) J. P. G. Farr and N. A. Hampson, ibid., 62, 3502 (1966). (5) N. A. Hampson and D. Larkin, ibid., 65, 1660 (1969). (6) N. A. Hampson and D. Larkin, J. Electroanal. Chem., 18, 401 (1968).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 14, DECEMBER 1970
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stant) which compares with the values of Randles @),Barker (7), and Bauer et al(8). It is clear that such effects would only show up if Rn is small (Le,, fast reactions); if RD was a decade larger, then the effect would be indistinguishable from the experimental uncertainty. A further point is that the effect is most apparent at low frequency; at high frequency, the capacitative path presumably accounts for most of the current Aow and this is unaffected by coupling. These measurements indicate that the value ofRDisobtained only for the case of the most unreactive systems. For (more concentrated) faster systems the electrode model can be represented in the form,
4
RT 1 RT R - - X ; = - X
- zF
0
[zF]*
1 k Coli-” C’“
where R , T , z , F , and a have their usual significance and Co’ and CR’are the concentrations of the oxidized and reduced species, respectively, though it is possible to assign a value to RD,the faradaic path
4
0
-w-
Figure 1. RRand 1,’. CRus. ,-lI2 plots for various Cd(Hg) concentrations; 1.OM NaClOa, 3.6 X ¶06M Cd(II), 23 “C
to match the model of the electrode interphase with the experimental results. Essentially this has resulted in the use of two different models of the electrode interphase. However, it would seem that the different results obtained must by necessity each represent at least a partial solution to the real electrode interphase. The earlier reported values in our opinion represent a limiting case for the model of the electrode interphase (that at infinite dilution) while the values reported here represent the true kinetic behavior of the electrode. Unfortunately, the results reported here are complicated by the fact that it has not been possible to use the exact model of the electrode interphase. The computer results, however we feel, must produce a better fit of the model of the electrode interphase than the earlier treatment. It is significant that the value of RD obtained is similar for both techniques with the less reactive systems. At greater Cd(Hg) concentrations, the value of the electrolyte resistance predicted from the computer method becomes progressively less than that obtained in the supporting electrolyte alone (and used to solve Randles-Ershler model). This results in an ‘enhanced’ value of RD. In other words, it would appear that a coupling of the electrode double layer impedance and the exchange reaction (resistance) occurs. Calculation of a by the computer method indicates a: = 0 and i;’ (the rate con-
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io
0
R,-
R,-
in reality cannot be exactly resolved experimentally into its component parts. The value of RD obtained in the Randles type of experiments will clearly yield a close enough value of the rate constant ‘72 for most reactions; for fast reactions, it yields the limiting value, that at infinite dilution. Physically this occurs when the relaxation time of the charge transfer process is within an order of magnitude of that of the double layer charging process. CONCLUSION
It must be emphasized that whether or not an alternative mathematical method of interpreting the experimental data would yield a different answer i s not the point of this work; rather whether the application of another criterion of electrode behavior would yield a different answer is the point under investigation. The Randles-Ershler method assumes that RB and CL are constant; in the present investigation we are taking the correct behavior of the Warburg impedance as our criterion. RECEIVED for review June 18, 1970. Accepted September 23, 1970. ~~
(7) G. C. Barker, “Transactions of the Symposium on Electrode Processes,” E. Yeager, Ed., John Wiley and Son, New York, N. Y.,1961, p 325. (8) H. H. Bauer, D. L. Smith, and P. J. Elvhg, J . Amer. Clzem. Soc., 82, 2094 (1960).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 14, D E C E M B E R 1970
