J. Phys. Chem. 1983, 87, 105-112
the vectors into another octant of the sphere. In all, seven more sets of basis vectors may be generated by finite rotations of the basis set shown in Figure 1A; each set may be cyclically permuted in three ways. The eight sets in the eight octants are represented schematically in Table 11. We assume that the position of the SO2 molecule does not change as we generate these eight sets of Cartesian basis vectors. We also assume that the normal coordinates remain spatially unchanged but the normal-coordinate coefficients will change numerically because of the change in the basis vectors. These numerical changes may be written out by making the appropriate changes in Table IA.The L"matrix corresponding to the second set of basis vectors in Table 11, illustrated in Figure lB, is shown in Table IB. It is a simple matter to calculate c31 and (32 from eq 4 for each set of normal-coordinate coefficients; the results are summarized in Table 11. We know that dp/dQ3 and dp/dQ1 have direction independent of the choice of Cartesian basis vectors and arbitrarily take both to be positive when the basis vectors are defined as in Figure 1. Changes in sign of d p / d Q , and dp/dQ1 because of a change in sign in the corresponding basis vector are also shown in Table 11. It is obvious and not very surprising that the sign of the product in expression 1 depends on the orientation of the basis vectors. It is well-known that all quantum-mechanical integrals representing physical observables must be invariant under rotation of the basis set, but this invariance requires that all operators and wave functions, including their adopted phase conventions, be properly rotated. The present difficulty arises because the problem is divided into two parts: the first part is the analysis of the experimental spectrum utilizing the definition of basis vectors and the
105
phase convention on rotational wave functions adopted by Mills,' and the second part is the introduction of the phase convention of the vibrational wave functions which is adopted by later users of the result and which may well introduce a different definition of Cartesian basis vectors. Any set of Cartesian basis vectors which may be generated by cyclic permutation of the set taken in the analysis of the Coriolis intensity problem will faithfully maintain rotational invariance. If the set of basis vectors is generated by acyclic permutation, the result will be a sign change in (,,, and a completely erroneous interpretation of the result obtained from the analysis of the Coriolis perturbation. Ideally one would use the same phase convention for rotational and vibrational wave functions and the same choice of basis vectors for the Cartesian representation for the entire problem. However, it is a matter of practical convenience to separate the analysis of a Coriolis perturbation in which the sign of expression 1 is experimentally determined from the applications of this result. If any set of basis vectors is used which allows the product ~~,~c(apa/aQ,)(e~b/aQ,,)
to be written in such a way that (a,b,c) are taken cyclically from (x,y,z) the result should be exactly the same as if the same definition of the basis vectors had been retained throughout the problem. If the relation between (a,b,c) and (x,y,z) is acyclic, a sign change must be introduced. This means that one must pay careful attention to the orientation of the Cartesian basis vedors in any application of a quoted sign of a Coriolis perturbation; caueat emptor!
Acknowledgment. This research was supported by a grant from the National Science Foundation (CHE81 15695).
Mediated Electron Transfer Reactions between Redox Centers in Prussian Blue and Reactants in the Solutlon Klngo Itaya;
lsamu Uchlda,t and Shlnobu Toshimat
Research Instltute of Elechical Communication and Department of Applied Chemistry, Facuw of Engineering, Tohoku Universily, Sendai 980, Japan (Received: January 12, 1982; I n Final Form: September 10, 1982)
By means of rotating disk electrodes, kinetic investigations for mediation reactions at Prussian blue (PB), Fe43+[Fe"(CN)6]3, modified electrodes have been carried out to reveal the electron transfer mechanisms in the crystal of Prussian blue. The mediation reactions examined were reduction of Fe(CN)$-, Fe3+,Ru"'(edta), and IrCb2-, and oxidation of Fe(CN)64-.It has been disclosed by the observation of the reduction of IrC162that the PB film has two kinds of electron transfer channels: One of the channels is due to the redox center consisting of low-spin iron ions (Fe"ln) in the PB lattice, working at highly anodic potentials (-1.0 V vs. SCE), and another is due to the redox center of both iron constituents (Fern/" and Fe3+J2+), working at relatively cathodic potentials (-0.2 V vs. SCE). The rate comiants (k, cm/s) for the mediated electron transfer between the reactants and the redox centers in the film have been evaluated by the Koutecky-Levich equation. It has been found that the mediated reactions are first order with respect to the bulk concentration of the reactants in the solution.
Introduction Recently methods for synthesis of the thin films of Prussian blue (PB) have been discovered by Neff et al.' and our TOU UP.^^^ We disclosed an electrochemical method
* Research Institute of Electrical Communication.
Address correspondence to this author at his present address: Department of Chemical Engineering, Faculty of Engineering, Tohoku University. t Department of Applied Chemistry.
of synthesis for PB on electrodes such as platinum (Pt), glassy carbon (GC),and tin oxide ( S ~ I O ~ )These . ~ ' ~ electrodes were cathodically polarized by a galvanostatic condition (constant current) or a potentiostatic condition (1) Ellis, D.; Eckhoff, M.; Neff, V. D. J.Phys. Chem. 1981,85, 1225. (2) Itaya, K.; Shibayama, K.; Akahoshi, H.; Toshima, S. J . Appl. Phys. 1982, 53, 804. ( 3 ) Itaya, K.; Ataka, T.; Toshima, S. J . Am. Chem. SOC.1982, 104, 4767.
0 0 2 2 - 3 6 5 4 / 8 3 l 2 0 8 7 - 0 1 0 5 ~ 0 ~ . 5 0 l 00 1983 American Chemical Society
106
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Itaya et al.
No. 1, 1983
(constant potential) in solutions of ferric ferri~yanide.~" The water-insoluble PB, Fe43+[Fe"(CN)6]3,was quantitatively formed by the electrochemical reduction meth0d.~8 The electrochemical redox reactions in the PB film showed a very symmetric wave and an asymmetric wave at 0.2 and 0.9 V vs. SCE, respectively, which were dependent on the current density used for the film preparati~n.~ An in situ Mossbauer-effect measurement showed conclusively that the wave at 0.2 V vs. SCE was due to redox reactions of the high-spin iron ions, Fe3+/2+.4 The crystal structure of P B is a face-centered cubic lattice with a cell constant of 10.2 A, indicating a very roomy crystal s t r ~ c t u r e . " ~PB is zeolitic and acts as a molecular sieve with channel diameters of about 3.2 A.7 The cations of NH4+, K+, Rb+, and Cs+ can transport through the PB crystal a t the reductive and oxidative waves at 0.2 V vs. SCE.',, The extraordinarily high stability of the P B film encouraged us to explore potential applications as an electrochromic material2 and as catalysts. Many authors used rotating disk electrode techniques for kinetic studies of the mediation reactions a t polymer-modified electrodes, revealing the electron transfer mechanisms.&12 In the present study, we describe the mediated electron transfer reactions of reactants in solutions at PB-modified electrodes using the same electrochemical technique. The results will explain the redox capabilities of P B as well as the oxidized and reduced forms of PB, i.e., Berlin green (BG) and Prussian white (PW). Experimental Section Pt disk electrodes were mounted into soft glass tubes and GC into Pyrex tubes.20 The sealing was leak tight, producing cavity-free electrodes. All electrodes used as rotating disks had a projected area of 0.031 cm2. The outer diameter of the glass tubes was about 0.7 cm. The surfaces of the disk electrodes were first polished with an alumina slurry (particle size 0.05 pm), and then the electrodes were immersed in concentrated sulfuric acid for more than 1 h for cleaning. An electrode rotator (Nikko-RRD-1) was used. Voltammograms were obtained with a PAR (Princeton Applied Research) Model 173 instrument equipped with a Model 179 digital coulometer. A saturated calomel electrode (SCE) was used as a reference electrode. The cell was purged with pure nitrogen gas. The PB films on the electrodes were prepared in an (4)Itaya, K.;Ataka, T.; Toshima, S.; Shinohara, T. J . Phys. Chem. 1982,86,2415. (5)Herren, F.; Fisher, P.; Ludi, A.; Hag, W . Inorg. Chem. 1980,19, 956. (6)Ludi, A.;Gudel, H. U. Struct. Bonding (Berlin) 1973,14,1 and references cited therein. (7)Robin, M. B.;Day, P. Ado. Inorg. Chem. Radiochem. 1967,10,247 and references cited therein. (8)Oyama, N.; Anson, F. C. Anal. Chem. 1980,52,1192. (9)Shigehara, K.;Oyama, N.; Anson, F. C. Inorg. Chem. 1981,20,518. (10)Albery, W.J.; Foulds, A. W.; Hall, K. J.; Hillman, A. R. J . Electrochem. SOC.1980,127,654. (11)Delamar, M.; Pham, M. C.; Lacaze, P. C.; Dubois, J. E. J . Electroanal. Chem. 1980,108,1. (12)Gough, D.A,; Leypoldt, J. K. Anal. Chem. 1979,51,439. (13)Mukaida, H.; Okuno, H.; Ishimori, T. Nippon Kagaku Zasshi 1965,86,56. (14)Albery, W.J. 'Electrode Kinetica"; Claredon Press: Oxford, 1975. (15)Levich, V. G. 'Physicochemical Hydrodynamics"; Prentice-Hall: Englewood Cliffs, NJ, 1962. (16)Peerce, P. J.;Bard, A. J. J. Electroaml. Chem. 1980,114,89and references cited therein. (17)Fielding, D. F.; Mellor, D. P. J. Chem. Phys. 1954,22, 1155. (18)Inoue, H.; Yanagisawa, S. J. Inorg. Nucl. Chem. 1974,36,1409. (19)Itaya, K.;Uchida, I.; Toshima, S. J. Phys. Chem., submitted. (20) Uchida, I.; Urushibata, H.; Toshima, S. J. Appl. Electrochem. 1982,12, 115
1
I
I
I
1
1.0
I
I
0 V v s SCE
Flgure 1. Cyclic voltammogram of a PBmodified Pt electrode with 5 mC/cm2 as a total charge consumed at the reductive wave at 0.2 V. The film of PB was prepared at a current density of 10 pA/cm2 for 400 s. The electrode potential was scanned at 20 mV/s. The solution used was 1.0 M KCI (pH 4.0) and was purged with pure nitrogen gas.
I
I
0.6
1
I
1
I
o v
Figure 2. Cyclic voltammogram (A) and current-potential curve (B) for the reduction of 10.2 mM K,Fe(CN), at PB-modified Pt electrodes. The supporting electrolyte was 1.0 M KCI (pH 4.0). 5 mClcm' was obtained for the amount of PB in both experiments. The potentials were scanned at 20 mV/s for the cyclic voltammetry and at less than 5 mV/s for the rotating-disk experiments. The electrode was rotated at 2500 rpm.
aqueous ferric ferricyanide solution of an equal-volume mixture of 20 mM FeCl, (pH 2.0) and 20 mM K,Fe(CN), in 0.01 M HCl as previously The electrodes were cathodically polarized in the above ferric ferricyanide solution by means of a galvanostatic condition where the current density was set to 10-20 pA/cm2. The electrodes, after the deposition of a certain amount of PB, were rinsed in 0.01 M HCl for 1 min. All the PB-modified electrodes were first examined in 1.0 M KCl (pH 4.0) by repeating the potential scan between 0.6 and -0.2 V. After yielding steady voltammograms, they were subjected to further experiments. K21rC16 was obtained from Soekawa Chemical Co. AquoethylenediaminetetraacetatorutheniumUII)(Ru"'(edta)) was prepared and purified from RuC1, as previously de~cribed.~,'~ K,Fe(CN),, K4Fe(CN),, and Fe(NO3),-7Hz0 were all reagent grade. Results and Discussion Electrochemistry of PB-Modified Electrode. Figure 1 shows a typical cyclic voltammogram of a PB-modified Pt
The Journal of Physical Chemistry, Vol. 87, No. 1, 1983
Mediated Electron Transfer Reactions
107
TABLE I: Kinetic Parameters for Electron Exchange between Reactants in Solution and the Redox Centers in Prussian Blue reactants Ef,aV vs. SCE concn, mM E 1 , 2 , bV vs. SCE ikhtc P A k,d cm/s K3Fe(CN)6
0.23 0.23
K4Fe(CN)6
Fe( NO,), RulI1(e d t a ) IrC1,'-
0.45 -0.2 0.78
2.0 5.5 10.2 9.8 1.8 9.2 4.9 0.45 1.5 2.0
1 5 4 i 10 400i 1 5 0.23 0.23 0.2 -0.2
714t 208t 125i 600 i 14x 38t 5oi
0.024, 0.023; 0.023, 0.0070, 0.022, 0.021, 0.00096, 0.028 0.01 1, 0.016,
20 10 10 40 1
5 7 l O O I 10
0.73 ( 0 . 3 1)e
Half-wave potential for the catalyzed reaction a Half-wave potential for each redox couple measured a t a bare electrode. Kinetic currents. Rate constants. e Half-wave potential for the second at a rotating disk electrode (1000 r p m ) . catalyzed reaction.
electrode prepared with a low current density, 10 pA/cm2, in a 1.0 M KC1 solution. The total amount of the charge consumed at the reductive wave at 0.2 V was about 5 mC/cm2, which was measured coulometrically under a potential scan from 0.6 to -0.2 V vs. SCE. The electron transfer mechanisms for the waves at 0.2 and 0.9 V have been formulated as follows, assuming the formula of Fe43+[Fe11(CN)6]3 for PB: 2-4 wave at 0.2 V Fe43f[Fe11(CN)6]3 + 4e-
+ 4K+
K4+Fe,2+[Fe11(CN)6]3 (1)
Lo A
wave at 0.9 V Fe2+[Fe11(CN)6]3 - 3e-
+ 3A- e Fe43+[Fe11'(CN)6A-]3
(2) where A- represents the anions of the supporting electrolyte used. It should be stressed that the waves of 0.2 and 0.9 V are due to the redox reactions for the high-spin iron ions, Fe3+/2+,and the low-spin iron ions, Fe111/11(CN)6*/4-, in the PB film as previously di~cussed.~ Figure 2A shows a cyclic voltammogram at a PB-modified electrode obtained in the supporting electrolyte (1 M KC1, pH 4) containing 10.2 mM K,Fe(CN)6. The reduction of Fem(CN)6*was superimposed on the reduction wave for PB. In order to separate the reduction process, we employed steady-state measurements at PB-modified electrodes. As seen in the upper trace (Figure 2B), the steady-state voltammogram for the reduction of Fe"'(CN)6* commenced at the onset potential of PB reduction. Figure 3 shows the oxidation reaction of Fe"(CN):at the PB-modified electrode. Obviously, the oxidation is superimposed on the oxidation wave of Prussian white, the reduced form of PB. The half-wave redox potential (Ef)of Fe111/11(CN)~-/4measured at a bare Pt electrode was 0.23 V vs. SCE in this medium (1M KC1, pH 4). The identical values of the E l j 2 were obtained for the reduction and oxidation reactions taking place at PB-modified electrodes as shown in Table I. In this case the E , value is very close to the peak potential (0.2 V vs. SCd) of PB. These findings strongly suggest that the reduction and oxidation processes of Fer11/1r(CN)63-/4couple are mediated at the waves of PB (reduction) and of PW (oxidation), respectively. The concentrations of Fe111(CN)63-and Fe"(CN),*- in Figures 2 and 3 were almost the same. However, it can be seen that both limiting currents shown in the figures are quite different in spite of the same experimental condition. According to the Koutecky-Levich equation, the limiting currents me not entirely transport controlled when kinetic currents, i.e., mediation reactions in this case, are
i
0.6
1
v 1
1
o
I
v
Figure 3. Cyclic voltammogram (A) and current-potential curve (B) for the oxidation of 9.8 mM K,Fe(CN),. Experimental conditions are the same as those of Figure 2.
involved. The apparent difference in the limiting current will be explained later on the basis of differences in the rates of the mediation reactions. Figure 4 shows a cyclic voltammogram in a 9.2 mM Fe3+ solution obtained at a PB-modified Pt electrode. The supporting electrolyte was 1.0 M KC1 and the pH was adjusted to 2.0 by hydrochloric acid in order to prevent any hydrolysis of Fe3+ and Fe2+. The upper traces in Figure 4 show the current-potential curves for the reductions of Fe3+at bare Pt (Figure 4B) and PB-modified Pt (Figure 4C) rotating disk electrodes, respectively. The half-wave redox potential of the Fe3+12+couple was measured as 0.47 V vs. SCE at Pt, while the E l j 2at PB-modified Pt was 0.2 V vs. SCE. Some important results are found in this experiment. The reduction of Fe3+was superimposed on the reductive wave of PB just as seen in the reduction of Fem(CN):- shown in Figure 2. However, the Elj2of the reduction of Fe3+at a PB-modified electrode is largely shifted cathodically by about 0.2 V, approaching the peak potential of PB. No reduction current was observed at the Ell2of the Fe3+I2+couple on the PB-modified electrode. The reoxidation wave for Fe2+produced is not clearly observed until 0.75 V vs. SCE which is very near
100
Itaya et al.
The Journal of Physical Chemistry, Vol. 87,No. 1, 1983 r
4OPA I
I
\
L A
1
I
I
1
I
1
0.6
1
I
1
o
v
Figure 4. Cyclic voltammogram (A) and current-potential curves for the reduction of 9.2 mM Fe3+at bare (B) and PB-modified (A, C) Pt electrodes. The supporting electrolyte was 1.O M KCI (pH 2.0). Other conditions are as in Figure 2.
the onset of the oxidation wave of PB itself. The above results indicate clearly that the electrochemical reduction of Fe3+is actually mediated by the electroactive centers in the PB film. The PB film seems to have no porous structure to allow Fe3+as well as Fez+to move through the film to reach the Pt surface. A scanning electron micrograph (SEM) of a film prepared at low current density revealed a very uniform pinhole free film structure with the expected thickness. The result shown in Figure 4 can be regarded as a direct proof that the PB film has pinhole free structure. Therefore, the electron transfer reactions can be explained as mediated electron transfer reactions between the redox centers in the film and the reactants in the solution. This implies that Fez+is oxidized to Fe3+by the oxidized form of PB, Berlin green, at 0.75 V vs. SCE as shown in Figure 4. Figure 5A shows a cyclic voltammogram for Ru'"(edta) at a bare GC electrode in a 1.0 M KC1 solution (pH 3.7). The half-wave redox potential of the RuIn/I1(edta)couple, which is -0.2 V vs. SCE, is located at about 0.4 V more negative than the peak potential of the reduction wave of PB. This Ellz is, therefore, positioned at the tailing part of the reduction wave as shown in Figure 5B. The upper trace (Figure 5C) obtained at PB-modified GC shows that the steady-state reduction current of Ru"'(edta) commenced at about 0 V and then increased with an irreversible manner at -0.1 V. A limiting current is clearly observed at -0.4 V, but its value was quite below the diffusion-limited current observed at a bare GC electrode at the same rotation speed. This result suggests that the tailing part of PB still has reducing power, but the available redox centers are largely limited (details will be discussed later). Figure 6A shows a cyclic voltammogram for the reduction of IrC16'- at a PB-modified Pt in a solution of 0.5 M K2S04. The Ef of the IrC162-/3-couple determined from Figure 6B was 0.78 V vs. SCE, which was closely located at the foot of the oxidation wave of PB as shown in Figure 6A. In the cyclic voltammogram, it can be seen that the redox wave due to the IrC&'-/* couple is superimposed on
I
1
0
0.6
1
-0.4 V
Figure 5. Cyclic voltammograms (A, B) and a current-potential curve for the reduction of 4.9 mM Ru"'(edta) at bare (A) and PB-modified (B, C) GC electrodes. The supporting electrolyte was 1.0 M KCI (pH 3.7). Other Conditions are as in Figure 2.
I
I
I
0.6 Figure 6.
I
I
L
1
o
v
Cyclic voltammogram (A, B) and a current-potential curve
for the reduction of 2 mM K,IrCI, at bare (B) and PB-modified (A, C) Pt electrodes.
The supporting electrolyte was 0.5 M K2S04(pH 4.0). Other conditions are as in Figure 2.
the oxidation wave of PB at 0.9 V and another shoulder is also observed at the onset potential (-0.4 V) of the reduction of PB. The upper trace (Figure 6C) shows that the reduction of IrCb2- commences at about 0.8 V vs. SCE and is limited at about 0.6 V. A distinct plateau is observed in the potential range from 0.6 to 0.45 V. Then the current for the reduction of IrCls2- again increases at 0.4 V which is coincident with the onset of the reduction of PB itself and shows the second plateau at a potential more negative than 0.3 V. The magnitude of the current at the second plateau was exactly the same as that obtained at a bare Pt rotating electrode. This indicates that the reduction of IrCb2- at a potential more negative than 0.3 V is a diffusion-limited process. An interesting feature of the IrC1,2- case is that the reduction of IrCls2- is first limited by the rate of the electron exchange between IrC16*-and the redox centers in the film, Fe43+[Feu1/11(CN)6]3, and then limited by the diffusion rate of IrCb2- in the solution. The redox center of Fel"/I1 in PB is definitely responsible for the reduction of IrC16'- at potentials more positive than 0.4 V, because
The Journal of Physical Chemistry. Vol. 87,No. 1, 1983 109
Mediated Electron Transfer Reactions
I
1
:F
- FBI I-
100
---P
Crystal
So'ution
Flgure 7. Illustrative depiction of the reduction of IrC1,'- at different electrode potentials: (A) at 0.6 V vs. SCE; (B) at -0.2 V vs. SCE.
another redox center (Fe3+i2+)cannot be involved in electron transfer reactions at this potential range. The electrochemical reduction of the high-spin iron ions, Fe3+, in the PB film occurs at potentials more negative than 0.4 V as shown in Figure 1. Therefore, the low-spin iron ions, Fen, must be the mediators for the reduction of IrC12- at a potential more positive than 0.4 V. The reduced form of PB, Prussian white, is a strong reductant for the reactant of IrCb2- with respect to their redox potentials. This explains why the reduction of IrCb2- at a potential more negative than 0.4 V is limited by the diffusion of IrC12to the electrode surface. Figure 7 illustrates the mediated electron transfer reactions for the reductions of at 0.6 and -0.2 v, respectively. In this illustration, we would particularly like to stress that the electron transfer channels in the PB film are due to the redox reactions of the low-spin iron ions, FeIIIlII, at 0.6 V and to both iron ions, i.e., FeI1I/I1and Fe3+I2+,at -0.2 V, respectively. Kinetics of the Reactions at the PB-Modified Electrodes. The kinetics of the electrode processes have been investigated by rotating disk electrodes.'*J5 For the simple reaction Ox
+ ne + Red
(3)
the limiting current (ilev)is expressed by the Levich equation15 ilev= 1.554nFAD2~3v-1~6Cow1/2
(4)
where D, u , C,, and w are the diffusion coefficient, the kinematic viscosity, the bulk concentration, and the rotation speed in hertz, respectively. When the electron transfer reaction is limited by the rate of the electron exchange reaction between the reactants in the solution and the redox centers in the PB film, the limiting current (iL), which is not entirely transport controlled, can be expressed by the Koutecky-Levich equation8-12915 l/iL
=
l/ile"
+ l/ikin
(5)
20
60
40
80
r D "'2
B
0.05
c
/
0.01
0.03
0.05
rpmk
Flgure 8. Limiting current vs. (rotating speed)"' (A) and KouteckyLevich plots (B) for the reduction of Fe1"(CN),3- at a PB-modified electrode. Closed points (0)were obtained at a bare Pt electrode. Concentrations of FeI1'(CN)2-: (a) 2, (b) 5.5, and (c) 10.2 mM.
The kinetic current (ikin)is expressed by the following equation:
ikin= nFAkCo
(6)
where k is the rate constant (cm/s) for the reaction between the reactants in the solution and the redox centers in the PB film. It is assumed that the mediated electron transfer reactions are first order with respect to the bulk concentration of the reactavts in the solution. Figure 8A shows observed limiting current (iL) vs. rotation speed data for the reduction of Fe111(CN)63at bare and PB-modified electrodes. Perfectly straight lines are observed at a bare electrode. On the other hand, the limiting currents at a PB-modified electrode deviate from the straight line with increasing rotation speed. This discrepancy indicates clearly that the limiting currents
The Journal of Physical Chemistry, Vol. 87, No. 1, 7983
110
0.0 3
001 1
I
I
I
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I
I
20
005
Itaya et al.
rpv''2
I
1
I 40
r pm
I
I
60 '/2
Figure 9. Limiting current vs. (rotating speed)"' (a, b) and a Koutecky-Levich plot (c) for the oxidation of 9.8 mM K,Fe(CN)6 at a Pt-modified Pt electrode. Closed points ( 0 )were obtained at a bare Pt electrode.
measured a t a PB-modified electrode are not entirely transport controlled, but partly controlled by the mediated electron transfer kinetics. The variation of the limiting currents with rotation speed are plotted in Figure 8B according to the Koutecky-Levich equation (eq 5). The expected linearity was obtained, and the slopes of these lines were proportionated to the reciprocal of the bulk concentration of Fe"'(CN),3-. From the intercepts of Figure 8B, which correspond to infinite rotation speed, the rate constants (k,cm/s) for the mediated electron transfer between Fe1r1(CN)63-and the reduced form of PB, K,Fe,2+[Fe"(CN),],, were calculated as 0.024, 0.0235, and 0.0234 cm/s in solutions of 2, 5.5, and 10.2 mM, respectively. This result indicates that the rate constant evaluated from the Koutecky-Levich equation is independent of the bulk concentration of the reactant in the solution, as has been assumed. Figures 9-12 show limiting currents (iL) vs. rotation speed data and the Koutecky-Levich plots for the oxidation of Fen(CN),4-,the reductions of Fe3+,Rum(edta),and IrCl:-, respectively. In all the cases, the limiting currents observed at the PB-modified electrodes were explained by the Koutecky-Levich equation. In the case of the reduction of IrCl?-, the limiting currents observed at a bare Pt electrode a t 0.6 V vs. SCE were essentially equal to those values of the PB-modified electrode at 0 V vs. SCE up to the highest rotation speed of 10 000 rpm employed here. Table I summarizes the kinetic parameters obtained in this study. The rate constant of about 0.007 cm/s for the oxidation of Fer1(CN)2-is obviously less than that for the reduction of Fe"'(CN),3-. This notable difference can explain the difference between the anodic and cathodic limiting currents observed in Figures 2 and 3. It is noteworthy that the Ef of the Fe111/11(CN)63-/4couple is somewhat more positive than the peak potential of the reductive wave of PB. That is, the reduced form of P B (PW) is a stronger reductant than Fe"(CNIG4-and correspondingly the oxidized form of PW, i.e., PB, is a weaker oxidant than Fe111(CN)63-.The energetic consideration qualitatively explains the difference in the rate constant. A much lower rate constant was evaluated for the reduction of Ru"'(edta), being attributable to its more negative Efthan the peak potential of P B itself. Judging from the energetic standpoint, we may assume that Ru"(edta) is such a stronger reductant than the reduced form of P B that the electron transfer from the reduced
.
20
50
rDm
"2
LB 007
c O'OS.
7-d
0.03
/
f
/
0.03
0.01
0.05
r p mJ+
Figure 10. Limiting current vs. (rotating speed)"' (A) and KouteckyLevich plots (6) for the reduction of Fe3+at a PB-modified Pt electrode. Closed points (0)were obtained at a bare Pt electrode. Concentratins of Fe3+: (a) 9.2 and (b) 1.8 mM.
0.01
r
I
I
0.03 I
I
0.05 I
-',*
rpm
PA
r
1
loo
60
20
20
40
r pm
60
"2
Figure 11. Limiting current vs. (rotating speed)"' (a, b) and a Koutecky-Levich plot (c) for the reduction of 4.9 mM Ru"'(edta) at a PB-modified GC electrode. Closed points ( 0 )were obtained at a bare GC electrode.
form of P B to Ru"'(edta) cannot take place. However, it is said that surface-bound electroactive species have some
The Journal of Physical Chemistry, Vol. 87, No. 1, 1983
Mediated Electron Transfer Reactions
TABLE 11: Dependence of Limiting Currents for t h e Reductions of Fe(CN),3-, 1rCl6*-,and Ru"'(edta) o n t h e Amount of Prussian Blue
/
A P A
40
111
integrated charges: mC/cmZ Fe(CN),32 5 10 20
30
72 85 71
iL, P A
IrCI,*' 23 21 25
Rulll(edta)d 11 8 7
a The charges consumed a t the reductive wave a t 0 . 2 V vs. SCE in 1 . 0 M KC1 ( p H 4.0). The limiting currents for the reduction of 5 mM K,Fe(CN), (1.0 M KCL, pH 4.0). The limiting currents for the reduction of 2 mM K,IrCI, (0.5 M K,SO,, pH 4.0). T h e limiting currents The electrodes for the reduction of 5 mM Ru"'(edta). were rotated a t 2500 rpm.
20
10
50
20
r p m '/2
0.01
0.03
0.05
rpmJ'2
Figure 12. Limiting current vs. (rotatin speed)'" (A) and KouteckyLevlch pbts (B) for the reduction of IC$-at a PBmodified Pt electrode at the electrode potential of 0.6 V vs. SCE. Closed points (0)were obtained at a bare Pt electrode. Concentrationsof ICI:-: (a) 2, (b) 1.5, and (c) 0.45 mM.
repulsive interactions16 and a distribution of the species with different Eo values is c~nceivable.~ Broadening of the surface waves suggests this Eodistribution. Our values of the half-height peak width were about 180 mV instead of 90 mV expected for ideal surface-bound specie^.^ We believe that the small tailing of the reductive wave of PB, involving different EO, caused the mediated electron transfer between the high-spin iron redox center and Rum(edta). The workable redox centers are so limited that the limiting current is very small. Another possible interpretation for the Ru'"(edta) case is that the reactant penetrates the film to react directly at the electrode surface. However, further examination is needed to consider this alternative. Dependence of the Kinetics of the Reactants on the Thickness of the PB Film. Anson et al.8,ghave assumed that the mediated reaction is also first order with respect to the concentration of the electroactive species in the polymer films. In the case of the reactions at polymer-
modified electrodes, the reactants in the solution can be expected to transport through the porous structures of the polymer films. This seems to explain that the mediated electron transfer reactions are first order with respect to the concentration of the electroactive species in the polymer films. However, the PB films prepared by the electrochemical method described here have quite a uniform thickness over the substrate surface according to SEM observation. Besides this observation, the crystal structures of the film cannot be changed under both the reduction and the oxidation of PB, because nearly the same unit cell constants have been observed in the crystals of PB, Prussian white, and Berlin green.6p7 Table I1 shows the limiting currents obtained for the reductions of Fe1'1(CN)63-,IrCl?-, and Ru"'(edta) when changing the amount of PB on the electrode. No appreciable dependence of the limiting currents on the thickness of the PB film was observed. Note that the thickness of the PB film was about 1000 A for a charge density of 5 mC/cm2. The above result seems to be direct evidence that the mediated electron transfer reactions between the reactants and the redox centers take place at the interface between the solution and the PB film. If the reactant penetration into the film is predominant, the penetration rate will be slower with increasing the film thickness, resulting in the smaller limiting current. It has already been shown in our previous paper3 that the cations (K+,Rb+, Cs+, and NH4+)can transport through the crystal of PB, but other cations such as Li+, Ba2+, and tetraethylammonium (TEA+)cannot, because of the zeolitic nature of PB. PB is zeolitic and acts as a molecular sieve with channel diameters of about 3.2 A. The radii calculated from the limiting mobility by Stokes' law are 2.37, 1.83, 1.25,1.18,1.19,1.25,2.81,and 2.88 A for Li+, Na+, K+, Rb+, Cs+, NH4+,TEA', and Ba2+,re~pectively.~This result suggests that the redox couples examined here cannot be expected to transport through the PB crystal. According to the classification of Robin and Day, PB is one of the class I1 mixed valence complexe~.~ There have been several investigations on the electrical properties of a series of compounds of the type Fe,[M(CN),], (M = Fe(II), Ru(II), and Os(I1)). In accordance with its class I1 assignment, PB and its analogues have room-temperature resistivities of about lo7 D cm and negative temperature coefficients of re~istance.'~J~ If PB acts as a semiconductor, reaction kinetics would be controlled by charge carrier densities present in the conduction and/or valence bands and by the position of the Fermi level. However, we did not observe any photocurrent under illumination of a He-Ne laser (632.8 nm) at PB-modified Pt and GC
J. Phys. Chem. 1983,87, 112-118
112
electrodes in 1.0 M KCl solution containing various redox couples examined here. We believe that the electron transfer kinetics characteristic to the semiconductorliquid junction is quite minor in the present study. Further study on photoeffects is
under inve~tigati0n.l~ Registry No. Fe(CN)6s,1340862-3;Fe, 7439-89-6;Rum(edta), 21687-44-5; IrC162-, 16918-91-5; Fe(CN),4-, 13408-63-4; PB, 14038-43-8;Pt, 7440-06-4;BG, 14433-93-3;PW, 81681-39-2.
Partial Molar Volume from the Hard-Sphere Mixture Model B. Leet Department of Chemistry, University of Kansas, Lawrence, Kansas 66045 and Physlcal Sciences Laboratory, Dlvision of Computer Research & Technology, National Institutes of Health, Bethesda, Maryland 20205 (Received: February 16, 1982: In Find Form: July 26, 1982)
Following Klapper (Biochim.Biophys. Acta, 229, 557-66 (1971)),an expression for the partial molar volume, uz, of the solute in infinitely dilute hard-sphere binary mixtures is derived from a very accurate equation of state for the hard-sphere mixtures (Mansoori et al., J . Chem. Phys., 54,1523-5 (1971)). If the result is written as bZ = (4a/3)(Rz+ a ) 3where Rz is the radius of the solute, a is a slowly varying function of Rz and depends primarily on the two properties of the solvent-the size of the solvent molecule and the packing density of the pure solvent. For a system that models the aqueous solution, a ranges from 0.62 to 0.51 A as the size of the solute varies from half to twice that of the solvent. According to this hard-sphere mixture model, the large volume change that occurs when a hydrophobic solute is transferred from a common nonpolar solvent to water is due to the small size of water molecules. According to the same model, wherein the interior of a globular protein molecule is considered as a pure hard-sphere fluid, the volume change upon protein denaturation is governed as much or more by the difference in the packing density between the protein interior and water as it is by the difference in the sue of an average amino acid residue and a water molecule. An Appendix introduces a new packing density function and gives a precise description of the notion of the cavity around a solute of any shape.
Introduction The partial molar volume (PMV) of a solute in a solution is a thermodynamic quantity and, despite the geometrical nature of the concept of volume, it is not possible to calculate PMV directly from simple geometrical consideration. However, for the simplest system, the hard-sphere mixture, a highly accurate equation of state is now available’ from which an expression for the PMV can be obtained that should be very accurate. This expression is important because it does not explicitly contain temperature or pressure and hence correctly describes the purely geometrical aspect of PMV. Thus, by comparing with experimental data, the equation should yield information on the relative importance of geometrical packing effect on volume. It turns out2 that the geometrical packing effect essentially determines the entire volume behavior as long as the solute is spherical and nonpolar. This fortunate circumstance enables one to examine a number of interesting aspects of volume behavior of such molecules. Derivation The equation of state for a hard-sphere mixture has the following form:'^^,^
where A’ =
td
‘Address all correspondence to the National Institutes of Health address.
Here, P, T , and V are the pressure, temperature, and volume of the system, respectively, k is the Boltzmann constant, m is the number of species in the system, and Ri,Ni,and pi are the radius, total number, and the number density, respectively, of the hard-sphere species i. The parameter h has the value of 0, 1,or 3 depending on how the equation is derived. Scaled particle theory3 gives eq 1 with h = 0. The equation of state can also be derived from the exact solution of the approximate Percus-Yevick equation for the radial distribution function of a hardsphere mixture! When the compressibility relation is used with this solution, eq 1 is obtained with h = 0 while the use of the virial theorem gives the same equation with h = 3. The equation with h = 1represents a weighted average of the above two results, but is also derivable empirically from the computer simulation data.’ It is considered to be very accurate. (1) G. A. Manmri, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr., J. Chem. Phys., 54, 152 (1971). (2) M. H. Klapper, Biochim. Biophys. Acta, 229, 557 (1971). (3)J. L.Lebowitz, E. Helfand, and E. Praestgaard. J. Chem. Phys., 43,774 (1965). (4)J. L. Lebowitz, Phys. Reu. A , 133, 895 (1964).
This article not subject to U.S. Copyright. Publlshed 1983 by the American Chemical Society
