~NTERPRETATIONOF
2209
ELECTRODE ADMITTANCE
On the Interpretation of Electrode Admittance in the Case of
Specific Reactant Adsorption. Application to the Bis (ethylenediamine)cobalt (111-11) System by M. Sluyters-Rehbach and J. H. Sluyters" Laboratory of Analflkal Chemistry, State University, Utrecht, The Netherlands
(Received December 4 , 1970)
Publkatwn costs borne completely by The Journal of Physical Chemistry
The results of a recent study by Sherwood and Laitinen of the bis(ethylenediamine)cobalt(III-11) electrode reaction are reconsidered. I t turns out that the impedance data can be well explained on the assumption of an infinitely fast charge transfer with reactant adsorption. The literature on the mathematical formulation of this problem is briefly commented upon.
Introduction Recently, Sherwood and Laitinen' have presented in this journal a study of the admittance of the bis(ethy1enediamine) cobalt (111)- bis (ethylenediamine) cobalt (11) system, which evidently belongs to the class of systems that exhibit specific reactant adsorption. Their interpretation of experimental results, which is inspired by our work in this field, gives us the idea that there is still confusion in literature about the problem how to handle these systems. Therefore it was thought worthwhile to make some comments, mainly meant as a short review of the equations describing the electrode admittance in the case of reactant adsorption and the possibility of using simplified expressions. As a consequence of these considerations it appeared possible to reinterpret Sherwood and Laitinen's data, the results of which will be presented in this paper.
Further, one deals with the surface excesses r0and r R of the redox components, which are functions of co, cR, and E
where i stands for either 0 or R. In the early stage, after Delahay had pointed out the above mentioned inseparability,2 he derived expressions for the interfacial admittance,a y' iy", however in fact using the expression
+
d_r _, _ _d r_, d_E dt d E dt
Theory I n general the equivalent admittance of an electrodesolution interface has to account for the electrode process which is occurring and for the charging of the electrical double layer. If there is no reactant adsorption, both processes can be treated separately and the double layer contribution has only to do with the charge density y i,
=
3
dt
(4)
instead of eq 3, and eq 1 instead of eq 2 . We showed4 that his results could be written in the simple form w=/2
y' = w1/2
Y"
= u
P + l
T p2 + 2 p + 2 + 1 p2+2p+2
P
p2
+ 2p + 2
(54
+
E = dy -d d E dt
where i, is the charging current. If there is reactant adsorption, the charging process and the faradaic process can no longer be separated,2 because in the first place q will also be a function of the concentrations of the redox components
(1) P. J. Sherwood and H. A . Laitinen, J . Phys. Chem., 74, 1757 (1970). (2) P. Delahay, ihid., 70, 2373 (1966). (3) P. Delahay and G. G. Susbielles, ihid., 70, 3150 (1966). (4) M. Sluyters-Rehbach, B. Timmer, and J. H. Sluyters, J . Electroanal. Chem., 15, 151 (1967).
The Journal of Physical Chemistry, Vol. 76, N o . 14, 1971
M. SLUYTERS-REHBACH AND J. H. SLUYTERS
22 10 where
K
=
QO
-nFa
dro dE
a =
+ -nFu + UR
drR dE
(7)
(8) e being the transfer resistance, a. and UR the Warburg coefficients, and c d the double-layer capacity. We analyzed the system Pb2+/Pb(Hg)in 1 M KC1 successfully with eq 5 . The incorrectness of using eq 1 and 4 was soon recognized both by Timmer6 and Delahay6 and a new derivation has been given by Delahay and coworkers,’ now using the proper eq 2 and 3. However, the resulting expressions for the interfacial admittance are very complicated and contain too many parameters to allow an unambiguous analysis of experimental data. It is therefore certainly worthwhile to examine whether in some special case eq 5, or possibly another simplification, may be valid as a good approximation. I n their paper, Sherwood and Laitinen follow a proposal by Timmer8 to assume that the rate constant is so large that the overvoltage is governed by the Nernst equation. This makes one of the three variables cot CR, and E dependent of the others, so that one derivative can be left out in eq 2 and 3, e.g. a0
0%
Further, they assume that either 0 or R is only weakly adsorbed. However, we cannot agree with the conclusions they draw from this assumption. If, for example, R is not or weakly adsorbed, eq 9 for the reduction component becomes
but for the oxidation component one has st’ill dE dt
trations co and CR than on the potential E. This occurs if the adsorption has reached saturation coverage, a case also mentioned by Sherwood and Laitinen. It is, however, not likely that saturation coverage will be reached at the rather low concentrations, required for admittance measurements (note that, when saturation coverage occurs, K should be found independent of concentration). It is rather more likely that the adsorption isotherm can be represented by
rt = kfct (12) Then the above-mentioned condition requires that
in other words, kfmust be strongly potential dependent. Even if this is the case, it is still not allowed to simplify eq 2 . This can be inferred from the Gibbs equation, which we, for the sake of simplicity, m i t e down for 0 being the only adsorbed specie^.^ - d r = qdE From eq 14 it follows that
(g)co (2) bpo
w’/2
The Journal a j Physical Chemistry, Vol. ‘76,No. 14, 19’72
E =
(14)
RT dco E 2 (3)
(15)
Therefore, if (dr0/bE),, = c O ( b k t / b E ) , has , a substantial ) ~ probably not negligible. value, ( b q / b ~is~most The arguments given above lead to the conclusion that on theoretical grounds eq 5 is not suitable to analyze admittance data. Consequently the analysis performed by Sherwood and Laitinen must be rejected, as well as the conclusions they draw from the evaluated parameters. If one wishes to account for both reactant adsorption and charge-transfer control, one should use the rigorous equations given by Delahay.’ If a simpler expression is required, in order to reduce the number of parameters, the only possibility is to assume the validity of the Nernst equation, as was proposed by Timrwier.8 One should be well aware that it is not sufficient to assume that the system is reversible in the dc sense, but that the Nernst equation should hold for the ac perturbations, so that it would be inconsequent to incorporate any contribution of charge transfer in the admittance equations. The equations for Nernstian behavior areS y’ =
and not eq 4, as was stated by Sherwood and Laitinen.l Note that (drO/bCR)E is certainly not zero, because its physical meaning is the change in ro,when CR changes at constant E , ie., with a simultaneous change in co. There is also no reason why eq 2, reduced by the Nernst equation to a similar form as eq 9, should reduce further when r R = 0 or Po = 0. I n our opinion not only eq 9, but even eq 3 could reduce to eq 4 only if ri depends much less on the concen-
+ Fodpo
-
2a
+ ~ ( C L -F CHF) + 2 u + 2 U
u2
(154
(5) B. Timmer, M . Sluyters-Rehbach, and J. H. Sluyters, J . Electronal. Chem., 15, 343 (1967). (6) P. Delahay, K. Holub, G. Susbielles, and G . Tessari, J . Phys. Chem., 71, 779 (1967). (7) K. Holub, C. Tessari, and P. Delahay, ibid., 71, 2612 (1967). (8) €3. Timmer, M. Sluyters-Rehbaoh, and J. H. Sluyters, J . Electroanal. Chem., 18, 93 (1968). (9) M . Sluyters-Rehbach and J. H. Sluyters in “Electroanalytical Chemistry,” Vol. 4, A . J. Bard, Ed., Marcel Dekker, New York, N. Y., 1970.
INTERPRETATION OF ELECTRODE ADMITTANCE
2211
Table I Co(III), mM
0.5 0.5 0.5 0.5 0.5
{::E 1.283 3.701 3.765
CO(II),
mM
0 0 0 0 0 0 0
2,467 0 0
u‘
-E, mV
425 430 436 440 445 446 446 448 435 440
x
sed2
CLF, rF/ornz
CHF, rF/oml
5 1 1 5=k1 6.5 f1.5 6 1 1 8&1 24 =k 2 20 f 3 -28 f 5 ~ 2 =k05 14 f 1
36 1 0 . 3 36.0 f0 . 3 36.0 f 0 . 3 35.5 f 0.1 35.3 i 0 . 2 66 f 1 59 1 1 97 f 4 99 =k 5 92.5 i 1.5
23 12 22 Z t 1 2 3 . 5 =k 1 . 5 73.0 f 1 35.0 f 1 27f 1 26.5 & 1 36 f 2 38 f 5 1Of2
lo’,
with
+ = coDo1/2+ cRDR1’’
(19)
Also with these equations the Pb2+-Pb(Hg) system in KC1 could be described very ~ l l .This is not surprising in view of the similarity in frequency dependence of eq 5 and 15, although the meaning of the parameters is quite different. Reinterpretation of the Bis(ethy1enediamine)cobalt Data Sherwood and Laitinen presented in their Table I1 a set of data (0.5 mM Co(II1) at -440 mV), which we tried to fit to eq 15. When this appeared to be successful, Dr. Sherwood and Professor Laitinen kindly supplied also the data pertaining to the measurements indicated in their Table I. Using the B values, listed in that table, we could establish a good fit to eq 15, except for the cases 1.283 mM Co(II1) 2.467 mM Co(I1) at -448 mM and 3.701 mM Co(II1) at -435 mV. It should be noted that the data for these cases showed a considerable scatter. Based on these results, which are tabulated in Table I, the following remarks can be made. 1. The experimental data of Shermood and Laitinen indicate that the bis(diethylenediamine)Co(III-11) system belongs to the class of electrode reactions which can be described by the most simple model of infinitely
+
a, ohm
oms
ge0-1’~
666.9 656.4 659.0 676 0 704.2 361.7 335.0 96.68 89.02 89.81 I
rapid charge transfer combined with reactant adsorption. This means that it is pointless to invoke Delahay’s more rigorous equations, since extension of the number of parameters can never lead to unambiguous solutions. 2. It is remarkable that CHF is found to be lower than the double layer capacitance c d in the pure supporting electrolyte: cf. CHF = 23-26 pF cm-2 for 0.5 and 1.061 mM Co(III), CHF = 10 pF for 3.766 mM Co(II1) (a very good fit was obtained here, opposed to the other high concentration cases) and Cd = 32-30 pF cm+. This phenomenon has not been observed before, as in the metal ion-amalgam systems, investigated by us, CHF = c d was always found. The question arises which of the two terms in eq 17 is responsible for this. It seems reasonable that the second term, which mainly represents the conversion of Red into Ox at constant I’, will have a positive sign, or is equal to zero if r R = 0. Therefore one is apt to conclude that the first term, representing the change in q with E at constant amount of reactant adsorption, is smaller than the original double-layer capacity, which represents the change in q with E in the absence of adsorption of the Co complex. 3. Sherwood and Laitinen observed a depression in the double-layer capacitance, which is comparable to our observation of a difference between Cd and CHF. They conclude further that this depression indicates a stronger adsorption of the Co(II1) complex. The exact meaning of CHFin our interpretation makes such a conclusion rather impossible. It seems to us that difference in adsorption of the ox- and red-component can be better investigated by considering the doublelayer capacitances outside, or in any case at the extreme ends, of the faradaic region. The present data do not allow this. Besides, one would intuitively assume that both species are adsorbed in comparative amounts, since the complexing agent is responsible for the adsorption. 4. Equation 10 in Sherwood and Laitinen’s paper, derived by Timmer,6is still based on the erroneous concept connected to eq 5 to 8 of our present communicaThe JOUTnal of Physical Chemistry, Vol. 76, N o . 14, 1971
NOTES
2212
tion. The model, pertaining to eq 16-19, leads to a better version, v ~ z . ~ (20)
The value ( d 2 y / b E 2 )= -13.87 pF/cm2 reported by Sherwood and Laitinen for 1.061 mM Co(II1) a t - 446 mV leads, combined with CLF = 60 MuF/cm2 (see Table I), to a new value for ro rR, nakeiy 0.54 x 10-10 mol/cm2,
This equation implies that the interfacial tension y is measured at varying E with constant C o d % C R d G , i.k., under normal polarographic conditions.
Acknowledgment. We wish to express our gratitude to Professor H. A. Laitinen and Dr. P. Sherwood for supplying the impedance data used for the calculations of the data in Table I.
-(”’> dE2
=
n2F2(TOUR
CLF
- - __ RT u2
(r0
+
rR)
+
+
NOTES
A n Investigation of Aqueous Mixtures of Nonionic Surfactants by Membrane Osmometry
by D. Attwood,* P. H. Elworthy, and S. B. Kayne School of Pharmaceutical Sciences, University of Strathclyde, Glasgow, C.1. United Kingdom (Received July IS, 1970) Publication costs borne completely by The Journal of Physical Chemistry
Recently, Co11112 and Attwood, Elworthy, and K a ~ n ehave ~ , ~reported the use of membrane osmometry in the examination of aqueous micellar systems of single micelle-forming components. This present investigation reports on the application of this technique to the study of micellization in a mixture of two nonionic surfactants. Most of the published work on mixed micellar systems has been concerned with mixtures consisting of an ionic and a nonionic component and has been reviewed previously.6 Evidence for the existence of a mixed micelle in such systems was first provided by the electrophoretic studies of Nakagawa and Inoue,6 and this has been substantiated by subsequent investigations of these systems. Similarly, electrophoretic measurements’ on mixtures of ionic surfactants have indicated mixed micelle formation, and the composition of the mixed micenes formed in mixtures of sodium decyl and dodecyl sulfates and u-phenylpentyl- and w-phenyloctyltrimethylammonium bromides have been deduced from conductivitys and nmrs studies, respectively. Most of the reported work on mixtures of nonionic surfactants has been restricted to the determination of the critical micelle concentration (cmc) lo and assumes rather than establishes that the nonionic components interact completely to form mixed micelles. The small-system thermodynamics of Hill1’ has been The Journal of Physical Chemistry, Vol, 76, No. 14, 1971
applied to mixed micellar systems of nonionic surfactants12*18 and an equation has been derived enabling the prediction of the aggregation number of a perfect mixed micellar system. I n this paper the number-average micellar weights, M,, of mixed solutions of n-hexadecyl nonaoxyethylene monoether (C1,ng) and n-dodecyl hexaoxyethylene monoether (Clzne)are determined as a function of the composition of the system. The M , values are compared with values calculated assuming the two components to exist independently in solution and with values predicted by small-system thermodynamics.
Experimental Section Materials. The samples of n-hexadecyl nonaoxyethylene monoether and n-dodecyl hexaoxyethylene monoether prepared previously4 were used in this investigation. Membrane Osmometry. Measurements were made on a Hewlett-Packard 503 high-speed osmometer using (1) H.Coll, J . Amer. Oil Chem. Soc., 46, 593 (1969). (2) H . Coll, J . Phys. Chem., 74, 520 (1970). (3) D. Attwood, P. H . Elworthy, and S. B. Kayne, J . Pharm. Pharmacol., 21, 619 (1969). (4) D. Attwood, P. H. Elworthy, and S. B. Kayne, J . Phys. Chem., 74, 3529 (1970). (5) P.Becher in “Nonionic Surfactants,” M . J. Schick, Ed., Marcel Dekker, New York, N. Y., 1967, p 508. (6) T. Nakagawa and H. Inoue, J . Chem. Soc. Jap., 78, 636 (1957). (7) H.W.Hoyer and A. Marmo, J . Phys. Chem., 65, 1807 (1961). (8) K. J. Mysels and R. J. Otter, J . Colloid Sei., 16, 462 (1961). (9) H.Inoue and T. Nakagawa, J . Phys. Chem., 70, 1108 (1966). (10) K . Shinoda, T. Nakagawa, B. Tamamushi, and T. Isemura, “Colloidal Surfactants: Some Physicochemical Properties,” Ac& demic Press, Xew York, N. Y.,1963, p 68. (11) T.L.Hill, “Thermodynamics of Small Systems,” Vol. 1 and 2, W. A. Benjamin, New York, N. Y., 1964. (12) D . G.Hall and B. A. Pethica in “Nonionic Surfactants,” M. J. Schick, Ed., Marcel Dekker, New York, N. Y., 1967,p 516. (13) D. G.Hall, Trans. Faraday Soc., 66, 1351, 1359 (1970).
