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Influence of supporting electrolyte activity on formal potentials measured for dissolved internal standards in acetonitrile
www.elsevier.nl/locate/jelechem Journal of Electroanalytical Chemistry 498 (2001) 192– 200
Influence of supporting electrolyte activity on formal potentials measured for dissolved internal standards in acetonitrile Jody Redepenning *,, Efrain Castro-Narro, Guhanand Venkataraman, Eric Mechalke1 Department of Chemistry, Uni6ersity of Nebraska, Lincoln NE 68588 -0304, USA Received 30 March 2000; received in revised form 3 July 2000; accepted 7 August 2000 Dedicated to Professor Fred C. Anson for his distinguished contributions to electrochemistry
Abstract Osmotic coefficients are determined for tetra-n-butylammonium perchorate (TBAClO4) in acetonitrile. Pitzer equations are fit to the data, and the resulting fitting parameters are used to calculate activity coefficients for TBAClO4. The activity coefficients are used to control electrolyte activities so that the junction potential across a permselective anion exchanger is near zero. An electrochemical cell containing this junction is used to examine the influence of electrolyte activity on the formal potentials measured for ferrocene/ferrocenium and decamethylferrocene/decamethylferrocenium. It is clear that the oxidized forms of both redox couples are ion-paired with perchlorate anions in the supporting electrolyte and that this ion-pairing has a significant influence on the formal potentials. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Acetonitrile; Reference; Ferrocene; Ferrocenium; Silver; Activity
1. Introduction
Nernstian in water, i.e., we have been able to get readily interpretable potentials for the following cell
We recently demonstrated the use of a well-described polymer-modified electrode surface to gain a better understanding of electrochemical cell potentials measured in acetonitrile [1]. Our approach in acetonitrile, which we consider to be a test case for work in other solvents, involved the use of simple electrochemical cells to which others can make simple comparisons. In many respects our initial measurements in acetonitrile are analogous to those in water involving the historically important junctionless Pt H2 H+, Cl− AgCl Ag cell. The cell that has proved to be most useful for our purposes is
Ag X M Ag+(H2O), X M ClO4− (H2O)
Ag X M Ag+(CH3CN), X M ClO4− (CH3CN) poly-[Ru(vbpy)3(ClO4)n ] Pt
(1)
where vbpy is 4-methyl-4%-vinyl-2,2%-bipyridine, X is the concentration, and n is 2 or 3. Not only is this system well-behaved in acetonitrile, it is also recognizably * Corresponding author. Fax: +1-402-4729402. E-mail address: [email protected] (J. Redepenning). 1 Present address: Casper College, Casper, WY, USA.
poly-[Ru(vbpy)3(ClO4)n ] Pt
(2)
and to make a direct comparison of the two cells over a broad range of concentrations. Once activity effects were taken into account, the cells represented by Eqs. (1) and (2) were shown to exhibit Nernstian responses consistent with a simple reaction stoichiometry. In both solvents plots of cell potentials versus the log of the silver perchlorate activity gave straight lines with slopes that were indistinguishable from the ideal of − 118.3 mV. Furthermore, these two lines were offset from each other by an amount that was close to the value expected from the Gibbs energy of transfer of silver perchlorate from water to acetonitrile. Although our previous study demonstrated a means by which readily interpretable measurements of cell potentials can be made in acetonitrile, the applicability of this work was lessened to some extent by the bothersome lower potential limit introduced by the necessity of using a silver salt as the supporting electrolyte. The focus of the work described below is to provide a
0022-0728/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 7 2 8 ( 0 0 ) 0 0 3 9 3 - 4
J. Redepenning et al. / Journal of Electroanalytical Chemistry 498 (2001) 192–200
practical experimental alternative to the junctionless electrochemical cells used in our previous work. The reference electrode we describe makes it possible to extend the electrochemical window to that normally expected for acetonitrile. This is accomplished without introducing measurable discrepancies with junctionless electrochemical cells we believe should serve as the benchmark.
2. Background The interpretation of cell potentials measured in nonaqueous solvents has been reviewed by a number of authors [2–7]. Several studies are particularly relevant to the measurement of cell potentials in acetonitrile [1,8 –14]. Our discussion in this work will ultimately lead to the consideration of dissolved redox couples used as internal reference half-cells in non-aqueous electrochemical cells. A brief overview background information is warranted. The idea of using dissolved redox couples as internal reference half-cells is based on Pleskov’s efforts in the late 1940s to find a redox couple that did not interact with the solvent [11,15]. If such a couple was available, its half-cell potential would be independent of the solvent. Research associated with this ideal has appeared sporadically since the time of Pleskov’s initial efforts [16 – 21]. Often cited work by Gritzner and Kuta recommends the use of the ferrocene/ferrocenium couple as an internal reference couple [22]. Adherence to this recommendation has made it much easier to reproduce the results of other research groups and to compare formal potentials measured for other half-cells under the same experimental conditions; however, the use of internal standards to compare formal potentials measured in different media remains problematic. Such comparisons are problematic because the medium generally has non-negligible influences on the half-cell potentials. Popovich addressed the empirical assumption of negligible medium effects in his classic 1970 review of medium effects [23]. He concluded that …the contention that any solute could experience negligible changes in sol6ation energy upon transfer from water to a 6ariety of sol6ents seems implausible as a generalization. There is an entropic problem that is difficult to circumvent. Parker, in the concluding paragraph to his paper titled ‘‘Solvation of ions-enthalpies, entropies and Gibbs energies of transfer’’ succinctly summarizes two important issues [24]: Perhaps the most interesting conclusion from our work on entropies of transfer is confirmation that the
193
unique position of water as a good sol6ent for electrolytes is due to water’s highly de6eloped structure, i.e., to the 6ery fa6ourable entropy changes (relati6e to other sol6ents) accompanying sol6ation of the first kind, when ions are introduced into water. Con6ersely, the reluctance of water to dissol6e significant amounts of uncharged non-polar solutes is due to sol6ation of the second kind i.e. to a 6ery unfa6orable loss of entropy, accompanying the formation of a highly structured water surface about hydrophobic solutes.
If we accept the reasonable assumption that it will be difficult to find dissolved redox couples for which the entropies of transfer (from water to other solvents) are near zero, then it is still possible to find a couple for which the Gibbs energies of transfer are near zero if the enthalpic and entropic contributions fortuitously compensated each other. Of course if such a couple could be found, this fortuitous compensation could occur at only a single temperature. Furthermore, since one should correct the Gibbs energies of transfer for changes in activity, and since the activity coefficients and the extent of ion-pairing are rarely known, appropriate correction factors are not often applied. One noteworthy exception is recent work by Lehmann and Evans, which takes an important step toward InfiniteDilution Voltammetry [25]. This work is an important complement to our efforts at higher concentrations. Data over a broad range of concentrations will be required to interpret adequately the thermodynamic behavior of any couple that might be used as an internal standard. Despite present difficulties associated with the rigorous interpretation of half-cell potentials for dissolved redox standards, and despite the virtual certainty that no dissolved reference couples that are negligibly influenced by medium effects will ever be found, the ease with which dissolved redox couples can be employed as reference half-cells virtually guarantees their continued presence in the electrochemical literature. Fortunately, the hurdles to obtaining a more detailed understanding of cell potentials that incorporate dissolved reference standards are not insurmountable.
3. Experimental Most of the materials, apparatus and procedures used in this work have been described previously [1]. TBAClO4 (tetra-n-butylammonium perchlorate, Fluka), ferrocene (Aldrich, 98%) and decamethylferrocene (Aldrich, 97%) were used as purchased. Radiation-
194
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4. Results and discussion For the cells represented by the shorthand notation in equations Eq. (1) and Eq. (2), we previously found that the cell potentials are described by the following equation [1]: Ecell = {E 0poly-Ru − E 0Ag + /0}+ −
Fig. 1. Reference electrode assembly: (A) silver wire to which external contact is made, (B) threaded main body, (C) internal reference solution in contact with silver wire, (D) open-ended cap that seals membrane on bottom end of electrode, (E) permselective anion exchange membrane (Raipore R-4030).
grafted PTFE anion exchange membranes (Raipore R-4030) were donated by Pall Corporation (East Hills, NY). A schematic diagram of the reference electrode used in these studies is shown in Fig. 1. This design provides a solvent-tight seal between the internal filling solution and the external electrolyte solution. Both the body of the reference electrode and the cap were constructed from Teflon. Discs (6.4 mm diameter) of permselective Raipore membrane were cut from sheets of the material using a common paper punch.
2RT ln k 9 [AgClO4] F
RT aRu(III) ln aRu(II) F (3)
Eq. (3) indicates that if one could measure cell potentials at a fixed apoly-Ru(III)/apoly-Ru(II) in solutions containing various electrolyte concentrations, then plots of (E ocell)app, versus log aAgClO4 should give straight lines with a slope of − 118.3 mV. If one uses the cell potential at the midpoint between the peaks of poly[Ru(vbpy)3(ClO4)n ] surface waves as an estimate of the case when aRu(III) = aRu(II), then plots of this cell potential versus log aAgClO4 (See Fig. 2) give lines with slopes that are experimentally indistinguishable from the expected value of − 118.3 mV in water and in acetonitrile. The difference in the y-intercepts can be attributed primarily to the Gibbs energy of transfer of silver perchlorate from water to acetonitrile (often represented by symbolism such as: DGt{AgClO4, w- \s}, where w=water and s= solvent, which in this case is acetonitrile). The value for DGt{AgClO4, w- \s} is a knowable quantity that is estimated to be −21.9 kJ mol − 1 [1]. This accounts for − 0.227 V of the −0.244 V shift; thus, it appears that changing the solvent from water to acetonitrile has little influence on the half-cell potential for the redox polymer, i.e., DGt for the polymer from water to acetonitrile appears to be near zero.
Fig. 2. (A) Squares: Cell potentials in acetonitrile previously reported for the cell represented by Eq. (1), y = −(119.4 91.9)x+603.3 92.5 (B) Circles: Cell potentials in water previously reported for the cell represented by Eq. (2), y = −(117.9 9 0.9)x+359.4 9 1.1 (C) Triangles: Cell potentials in acetonitrile for the cell represented by Eq. (4) when a(TBAClO4) =a(AgClO4), y = − (121.6 9 1.7)x+603.6 92.1. All activities are on the molar scale.
J. Redepenning et al. / Journal of Electroanalytical Chemistry 498 (2001) 192–200
Ecell = {E 0poly-Ru − E 0Ag + /0}+
Table 1 Equilibrium molalities of TBAClO4 and TBAI m(TBAClO4) /mol kg−1 0.0703 0.0928 0.1000 0.1714 0.2718 0.3355 0.3980 0.5255 0.5320 0.6280 0.7790 0.9825 1.0735 1.1930 1.2552 1.3312 1.5214 1.6799 1.7286 1.8634 1.8932 2.2332
m(TBAI)/mol kg−1
% uncertainty (in m(TBAClO4)/m(TBAI))
0.0723 0.0960 0.1038 0.1804 0.2892 0.3550 0.4287 0.5712 0.5806 0.6878 0.8637 1.0944 1.2088 1.3263 1.3997 1.4905 1.7063 1.8853 1.9152 2.1012 2.1146 2.5026
0.81 0.21 0.27 0.45 0.29 0.35 0.61 0.39 0.85 0.46 0.71 0.07 0.24 0.27 0.11 0.20 0.15 0.19 0.12 0.23 0.24 0.43
−
Ej =
− RT F
i
h
ti d ln ai zi
(6)
In the limit in which the transport number for the common anion approaches one, the transport numbers for cations approach zero, and the junction potential between the a and b phases simplifies to the following expression. Ej =
RT RT ln a(ClO− ln a(ClO− )i − 4 4 )h F F
(7)
Substituting this result into Eq. (5) gives, Ecell = {E 0poly-Ru − E 0Ag + /0}+ −
Ag X M AgClO4 Y M TBAClO4 poly-Ru(vbpy)3 i
(5)
where Ej is given by [26]
It seems unlikely that the poly-[Ru(vbpy)3(ClO4)n ] redox polymer will be used as described above to construct reliable reference half-cells that could be used by non-specialists under a variety of different solvent and electrolyte conditions. Even if DGt{polymer, w-\ s} to a number of solvents were near zero, the labor associated with making the monomeric metal complex and then exacting a reproducible polymerization limits the practical utility of this system as a reference halfcell. Still, the recognizably Nernstian behavior of poly[Ru(vbpy)3(ClO4)n ] serves as an important tool in constructing a reliable and readily interpretable reference scheme. A more versatile cell that incorporates the poly-[Ru(vbpy)3(ClO4)n ] half-cell while exhibiting a wider electrochemical window is desirable. To this end we have examined nonaqueous electrochemical cells such as the one shown below in which a liquid junction potential forms across a permselective anion exchange membrane denoted by . h
RT aRu(III) ln F aRu(II)
RT RT ln a(Ag+) − ln a(ClO4−) + Ej h i F F
&
195
-(ClO4)n Pt
(4)
In words, the reference electrode assembly shown in Fig. 1 was placed into an electrolyte solution of TBAClO4 in acetonitrile, and a platinum electrode modified with poly-[Ru(vbpy)3(ClO4)n ] was used as the working electrode. The potential for the cell in Eq. (4) is given by Eq. (5)
RT aRu(III) ln F aRu(II)
2RT ln k 9 [AgClO4] F
(8)
which is the same as Eq. (3). This convenient result occurs because the contribution of perchlorate activity (in the b-phase) to Ej is cancelled by the contribution of the perchlorate activity to the Donnan potential at the interface between the b-phase and the redox polymer. The goal of this research is not to map-out a wide variety of conditions for which the scheme described in Eqs. (4) –(7) and Eq. (8) can be made to work. Instead, we choose to examine conditions that further simplify the interpretation of the cell potentials. A particularly useful set of conditions are present when the perchlorate activities in the a and b phases depicted in Eq. (4) are equal. This is Lingane’s Type 2 junction [27], which occurs when different electrolytes containing a common ion are present at equal activities on either side of a junction. For the unique Type 2 junction in which the transport number for the common anion is one, note (from Eq. (6)) that Ej is zero. Of course the transport number for perchlorate is not exactly one, but by keeping the activity of perchlorate equal on either side of the membrane, the impact of any deviation from ideal permselectivity should be minimized, and the response predicted by Eq. (8) should be realized in the absence of cathodically obstructive silver ions. We must rely on mean ionic activity coefficients (k 9 ) to prepare the electrolyte solutions. Mean ionic activity coefficients for silver perchlorate in acetonitrile were available from our previous study [1], but k 9 for TBAClO4 in acetonitrile were not available, so we measured them using the isopiestic technique [28 –30]. Equilibrium molalities for TBAClO4 and TBAI are shown in Table 1. Mean ionic activity coefficients for TBAClO4 were calculated by treating the data in Table
J. Redepenning et al. / Journal of Electroanalytical Chemistry 498 (2001) 192–200
196
1 as described below. If the osmotic coefficient () for a reference salt (tetra-n-butylammonium iodide, TBAI, in our case) is known at each equilibrium molality, one can determine the unknown osmotic coefficient for the salt with which it is in equilibrium (TBAClO4 in our case) using the equation:
where
mTBAClO4TBAClO4 = mTBAITBAI
(9)
The resulting osmotic coefficients for TBAClO4, which are shown in Fig. 3, can be fit using the semi-empirical Pitzer equation [31,32], which is given by: −1 = f +mB + m 2C
(10)
f = − A I 1/2/(1+ bI 1/2)
(10a)
B = i (0) + i (1)exp[− h1I 1/2]+ i (2)exp[− h2I 1/2]
(10b)
A = (1/3)(2yNAz0)1/2(e 2/4ym0mkT)3/2
(10c)
I= 0.5%mi z 2i
(10d)
and where b is the size parameter, NA is Avogadro’s constant, z0 is the density of the pure solvent, m is the relative permittivity, m0 is the permittivity of free space, k is Boltzmann’s constant, and T is the temperature. The coefficients h1, h2, i (0), i (1), i (2) and C f are adjustable Pitzer parameters used to do the fitting. Activity coefficients can be calculated using the integrated form of Eq. (10) found in Eq. (11) [31,32]: ln k 9 = f k + mB k + m 2C k
(11)
where f k = f − A (2/b)ln(1+ bI 1/2)
(11a)
B k = 2i (0) + x (1) + x (2)
(11b)
x (i ) = (2i ()/h 2i I)[1− exp(− hi I 1/2)(1+ hi I 1/2 − 0.5h 2i I)] i= 1,2 C k = 1.5C
Fig. 3. Osmotic coefficients for TBAClO4. Error bars are one standard deviation in the ratio of the equilibrium molalities. Solid line: least squares fit through the osmotic coefficients using Eq. (10), fitting parameters are found in Table 2. Dotted line: Activity coefficients calculated for TBAClO4 using Eq. (11) with the fitting parameters found in Table 2. Table 2 Pitzer parameters in acetonitrile a Electrolyte
i (0) kg mol−1
i (1) kg mol−1
i (2) kg mol−1
C f kg2 mol−2
AgClO4 TBAClO4 TBAI
0.04745 −0.02262 −0.04090
0.05756 −0.02712 −0.40277
−10.828 −12.387 −14.635
0.02849 0.00480 0.00762
a Use of all seven adjustable parameters to fit the data was unwarranted for each of these data sets. Hence, b, h1, and h2 were set to common values reasonable in each of the solutions. This practice is particularly important if one is to consider solutions containing mixed electrolytes. In each of these cases: b=3.2 kg1/2 mol−1/2, h1 =2.0 kg1/2 mol−1/2, and h2 = 12.0 kg1/2 mol−1/2.
(11c) (11d)
The solid line in Fig. 3 is the least squares fit through the experimentally determined osmotic coefficients. The Pitzer parameters that result from this fit are displayed in Table 2 along with the parameters for AgClO4 (determined from data found in our previous work[1]) and Barthel’s values for the TBAI reference salt [32]. The dotted line in Fig. 3 shows the dependence of k 9 for TBAClO4 on molality. This line was calculated using Eq. (11) and the parameters found in Table 2. Mean ionic activity coefficients on the molal scale were converted to mean ionic activity coefficients on the concentration scale as outlined by Rieger [33]. Values of the latter at convenient concentrations of TBAClO4 in acetonitrile are provided in Table 3. Having activity coefficients for AgClO4 and TBAClO4 in acetonitrile, we can now investigate the plausibility of the measurement scheme outlined in Eqs. (4) –(8). The goal is to determine whether significant differences exist between cell potentials measured for the cell in Eq. (1) and those measured for the cell in Eq. (4) when the AgClO4 and TBAClO4 activities are equal on either side of the membrane separator. In line C of Fig. 2 measurements on the cell depicted in Eq. (4) are superimposed on previous measurements made on the cell depicted in Eq. (1). As before, the midpoint between the surface waves for poly-Ru(vbpy)3(ClO4)n was used as an estimate of the condition in which aRu(III) = aRu(II). For this set of experiments, silver perchlorate
J. Redepenning et al. / Journal of Electroanalytical Chemistry 498 (2001) 192–200 Table 3 Molar activity coefficients for Tba ClO4 in acetonitrile Concentration/mol l−1
Molar k 9
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.100
1.000 0.660 0.578 0.532 0.501 0.478 0.460 0.445 0.433 0.422 0.413 0.356 0.326 0.307 0.294 0.284 0.277 0.273 0.269 0.268 0.268
197
We proceed to describe the cell potential of the above cell using a development similar to that outlined by Laitinen and Harris for cases in which metal –ligand interactions influence the electrode potential [34]. In the development presented below standard potentials and activities are used whenever it is possible to do so. Formal potentials and concentrations are used when there is no reasonable alternative. Not knowing activity coefficients for ferrocenium, we write the formal potential for the ferrocene/ferrocenium couple as shown in the equation: E= E°%1 +
RT [Fc+] ln F [Fc0]
(13)
If ferrocenium can ion-pair according to the simple stoichiometry shown in Eq. (14), Fc+ + ClO− 4 X (FcClO4)ip
(14)
then the conditional equilibrium constant for ion pairing is given by K%ip =
aFcClO4 [Fc+]aClO-
(15)
4
Solving for the concentration of ferrocenium in Eq. (15) and substituting the result into Eq. (13) gives:
concentrations ranged nominally from 0.020 to 1.0 M. Note that the slopes of lines A and C are experimentally indistinguishable, suggesting that the stoichiometries of the two cells are the same. Furthermore, the y-intercepts are indistinguishable, suggesting that the presence of the permselective membrane in Eq. (4) does not influence the overall cell potentials and that we have effectively eliminated Ej over a broad range of concentrations. In short, cell potentials measured for the cells depicted in Eq. (1) and Eq. (4) are equal when the electrolyte activities are equal.
where
4.1. Application to reference redox couples
E°%ip = E°%1 +
E= E°%1 +
RT [FcClO4] RT 1 RT 1 ln + ln + ln F F aClO− [Fc0] F K%ip 4
(16) or E= E°%ip +
RT [FcClO4] ln F [Fc0]
(16a)
RT 1 RT 1 ln + ln F K%ip F aClO-
(16b)
4
While poly-[Ru(vbpy)3(ClO4)n ] (and many of its relatives) have proved to be rich ground for the electrochemical research community, the merits of devoting a large research effort to a detailed thermodynamic study of this system for its own sake are dubious. In our view, the preceding results are most important for what they tell us about the special Type 2 junction shown in Eq. (4). The results described above suggest that a cell such as the one shown in Eq. (12) could be used over a broad range of supporting electrolyte concentrations in acetonitrile to deconvolute voltammetric measurements of formal potentials for redox couples such as ferrocene/ferrocenium (usually starting with all ferrocene). Ag a(AgClO4) =X X =a(TBAClO4), [ferrocene] h
i
=[ferrocenium]= Y mM Pt
where Y51 typically.
(12)
and where the activity coefficient for ion-paired FcClO4 is taken to be one. Of course Eq. (13) and Eq. (16) describe the same electrode in different terms. Eq. (13) is more practical at low concentrations of supporting electrolyte when essentially none of the ferrocenium is ion-paired, and Eq. (16) is useful at high electrolyte concentrations when essentially all of the ferrocenium exists as FcClO4. For ferrocene/ferrocenium (and for decamethylferrocene/decamethylferrocenium) in acetonitrile with perchlorate concentrations ranging from about 0.1 to 1.0 M, it appears that the oxidized form of the redox couple is present both as ion-pairs and as free ions. In order to describe the electrode potential with a single equation over a broad range of electrolyte concentrations, it is useful to introduce the definition shown in the equation: [Fc+]+ [FcClO4]= cox
(17)
J. Redepenning et al. / Journal of Electroanalytical Chemistry 498 (2001) 192–200
198
which can be used to rewrite Eq. (16) as E =E°%1 +
RT RT cox − ln (1 +K%ip ·aClO−) ln 4 [Fc0] F F
(18)
The half-cell potential for the Ag Ag+ reference electrode, E= E °ref +
RT ln aAg+ F
(19)
can be combined with Eq. (18) and Ej to give Eq. (20), which is a convenient expression of the cell potential for the cell shown in Eq. (12) Ecell =E°%1 − E°ref + −
RT c ln ox0 F [Fc ]
RT ln (aAg+ +K%ip ·a 2AgClO4) + Ej F
We have measured half-wave potentials for ferrocene/ferrocenium and decamethylferrocene/decamethylferrocenium (in the cell depicted in Eq. (12)) over electrolyte concentrations ranging nominally from 0.01 to 1.0 M in acetonitrile. As was the case for the experiments with poly-[Ru(vbpy)3(ClO4)n ] described above, the activities of AgClO4 and TBAClO4 were always equal on either side of the membrane separator. In no case was there any evidence of ferrocene (or decamethylferrocene) in the reference electrode compartment, nor was there any evidence of silver ion in the working electrode compartment. Fig. 4 shows our experimental results and fits to the data using Eq. (21), which takes the same general form as Eq. (20), Ecell = k1 − 0.0592 log (a+ k2·a 2)
(20)
It is desirable to use Eq. (20) to describe Ecell over a broad range of electrolyte concentrations; however, as the electrolyte concentrations change, so too do the formalisms, i.e., the values of E o% and K%ip change. Consequently, in lieu of activity coefficients for ferrocenium salts in acetonitrile, we are stuck with semi-empirical relationships between cell potentials and electrolyte activities. While such relationships do not provide much-needed thermodynamic rigor concerning half-cell potentials for couples such as ferrocene/ferrocenium, these relationships are quite useful in fitting experimental results and in providing qualitative phenomenological descriptions of those processes that influence the formal potentials.
(21)
where a is the activity of silver perchlorate and k1 and k2 are the adjustable parameters used to obtain the fit. All of those terms that account for the changing E o% and K%ip are included by the influence they have on the values ultimately selected for k1 and k2. This simple two-parameter fit to the data works well for both redox couples, and the model simplifies to the expected limiting cases at high electrolyte concentrations and at low electrolyte concentrations. Our value of k2 is somewhat larger than the association constant (189 6) reported by Yang, et al. who used a five-parameter fit to exchange rate data (between ferrocene and ferrocenium as a function of sodium perchlorate concentration) to estimate the association of ferrocenium and perchlorate in acetonitrile [35]. It is important to note, however, that our estimate of the association constant (k2 in Eq. (21)) should not be compared directly to that of Yang,
Fig. 4. (A – C): Formal potentials for ferrocene/ferrocenium in acetonitrile as a function of AgClO4 activity. (A) High concentration limit, y= −16.5 −(59.2) log(k2 × a 2) where k2 = 46.6, or y= − 115.3− (59.2) log(a 2); (B) Low concentration limit, y = − 16.5 −(59.2) log(a); (C) y= −16.5− (59.2) log(a+ (46.6× a 2)). (D–F): Formal potentials for decamethylferrocene /decamethylferrocenium in acetonitrile as a function of AgClO4 activity. (D) High concentration limit, y= − 537.2− (59.2) log(k2 ×a 2) where k2 =10.8, or y = − 598.4 −(59.2) log(a 2); (E) Low concentration limit, y =( −537.2− (59.2) log(a); (F) y= − 537.2 − (59.2) log(a+ (10.8 × a 2)). All activities are on the molar scale.
J. Redepenning et al. / Journal of Electroanalytical Chemistry 498 (2001) 192–200
et al., whose work did not benefit from knowledge of mean ionic activity coefficients at the high electrolyte concentrations used. Finally, it should also be noted in Fig. 4 that decamethylferrocenium does not ion-pair as extensively with perchlorate as ferrocenium does. This is manifested as a significantly smaller value of k2 for the decamethyl analog. These results are consistent with observations recently reported by Noviandri, et al. [21]. A minor complication arises because we are using voltammetric half-wave potentials (E1/2) to estimate the formal potentials. E1/2 =E o% only when the diffusion coefficient for the oxidized species (Dox) equals that for the reduced species (Dred), a condition that is probably never true exactly. Martin and Unwin have determined diffusion coefficients for ferrocene and ferrocenium in acetonitrile [36]. They measured DFc to be 2.15× 10 − 5 and DFc+ to be 1.72×10 − 5 cm2 s − 1, which makes E1/2 −E o%= − 2.9 mV for cyclic voltammetry at macroscopic planar electrodes [37]; however, their measurement of DFc+ was performed in 0.1 M TBAClO4 and we now know that under these conditions in acetonitrile a significant fraction of ferrocenium exists nominally as ion-pairs. In our hands cyclic voltammograms of ferrocene and decamethylferrocene at 1.0 mm and 10 mm electrodes (at 50 and 5 mV s − 1, respectively) gave half-wave potentials that were experimentally indistinguishable at high and at low supporting electrolyte concentrations. Given the influence of (Dred/Dox)n on E1/2 for voltammetry performed at macroelectrodes and at ultramicroelectrodes [25,37], the approximation that E1/2 =E o% appears to be reasonable for cyclic voltammetric characterizations of these couples.
5. Conclusions The semi-empirical equations obtained above provide a relatively simple means of describing the influence of electrolyte activity on half-wave potentials measured for ferrocene and decamethylferrocene in acetonitrile when perchlorate salts are used as the supporting electrolyte. The extensive ion – pairing associated with the oxidized forms of both couples suggests that these couples should not be used indiscriminately as reference half-cells. Of course the presence of significant ion-pairing comes as no great surprise given the pioneering conductance work by Fuoss and Onsanger beginning in the 1930s (see Ref. [38]). Ion-pairing between ferrocenium and supporting electrolyte anions does not present an insurmountable hurdle to the use of ferrocene/ferrocenium or one of its analogs as a reference half-cell, however. Silver perchlorate also ion-pairs strongly at the concentrations we are using in acetonitrile but this ion-pairing was not a major impediment to our previous studies in junctionless cells. As long as the activity coefficients for the
199
electrolyte are known, it is not necessary to describe ion-pairing explicitly. The ion-pairing is implicit in the activity coefficients [39] and is described by the Pitzer relations even when there is no a priori knowledge of the radial charge density. In fact, Pitzer has noted that as long as the thermodynamic properties are known, no assumptions concerning the existence of discrete chemical species are needed (even if such species do exist) [31b] Such considerations are particularly important if one is to avoid discussion that makes some kind of distinction between ion-pairs and solvent-separated ionpairs. Our future course of action is clear. Osmotic coefficients are needed for pure ferrocenium salts and for mixtures of those salts with other univalent salts for which the osmotic coefficients are known. It should then be possible to use known thermodynamic treatments [40] and simple half-reactions to describe the dependence of the half-cell potential for the ferrocene/ ferrocenium couple on electrolyte activity. The prospects of making such measurements on salts of weakly coordinating anions such as tetrakis(pentafluorophenyl)borate are particularly attractive [41,42]. All of this involves a significant amount of experimental work for the electrochemist, but by completing such work it may finally be possible to resolve important ambiguities that still surround the use of ferrocene/ferrocenium as a reference half-cell.
References [1] J. Redepenning, E. Mechalke, Anal. Chem. 69 (1997) 5094. [2] J.N. Butler, in: P. Delahay (Ed.), Reference Electrodes in Aprotic Organic Solvents in Advances in Electrochemistry and Electrochemical Engineering, vol. 7, Academic Press, New York, 1970, pp. 77 – 175. [3] A.K. Covington, in: R.A. Durst (Ed.), Reference Electrodes in Ion-Selective Electrodes, vol. 314 (Special Publication), National Bureau of Standards, Washingtion D.C, 1969, pp. 107–141 (Chapter 4). [4] G.J. Hills, in: D.J.G. Ives, G.J. Janz (Eds.), Reference Electrodes in Nonaqueous Solutions in Reference Electrodes: Theory and Practice, Academic Press, New York, 1961, pp. 433– 463 (Chapter 10). [5] H. Lund, in: M.M. Baizer, H. Lund (Eds.), Practical Problems in Electrochemistry. V. Reference Electrodes in Organic Electrochemistry: An introduction and a guide, second ed., M. Deker, New York, 1983, pp. 187– 211 (Chapter 5). [6] M. Salomon, in: A.K. Covington, T. Dickinson (Eds.), Thermodynamic Measurements. Part 2. Electrochemical Measurements in Physical Chemistry of Organic Solvent Systems, Plenum Press, New York, 1973, pp. 137– 219. [7] C.K. Mann, Electrochemical Reactions in Nonaqueous Systems, Marcel Dekker, New York, 1960. [8] G.J. Janz, H. Taniguchi, Chem. Revs. 53 (1953) 397. [9] J.F. Coetzee, in: I.M. Kolthoff, P.J. Elving (Eds.), Precipitation Equilibria and Titrations in Aqueous and Nonaqueous Media in Treatise on Analytical Chemistry Part 1, vol. 3, second ed., Wiley, New York, 1983, p. 349 (Chapter 28).
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[10] G.J. Janz, H. Taniguchi, unpublished results sited in reference [8]. [11] V.A. Pleskov, Zhur. Fiz. Khim. 22 (1948) 351. [12] A.I. Popov, D.H. Geske, J. Am. Chem. Soc. 79 (1957) 2074. [13] I.M. Kolthoff, F.G. Thomas, J. Phys. Chem. 69 (1965) 3049. [14] B. Kratochvil, E. Lorah, C. Garber, Anal. Chem. 41 (1969) 1793. [15] V.A. Pleskov, Usp. Khim. 16 (1947) 254. [16] H. Strehblow, Z. Elektrochem. 56 (1952) 827. [17] H.M. Koepp, H. Wendt, H. Strehblow, Z. Elektrochem. 64 (1960) 483. [18] W. Ward, Ph.D. Thesis, Electrode Potential Measurements in Anhydrous Acetonitrile, University of Iowa, 1958. [19] R.R. Gagne, C.A. Koval, G.C. Lisensky, Inorg. Chem. 19 (1980) 2854. [20] R.S. Stojanovic, A.M. Bond, Anal. Chem. 65 (1993) 56. [21] I. Noviandri, R.D. Bolskar, P.A. Lay, C.A. Reed, J. Phys. Chem. B 101 (1997) 6350. [22] G. Gritzner, J. Kuta, Pure Appl. Chem. 54 (1982) 1527. [23] O. Popovich, in: I.M. Kolthoff, P.J. Elving (Eds.), Estimation of Medium Effects for Single Ions in Non-aqueous Solvents. Treatise on Analytical Chemistry Part 1, vol. 1, Wiley, New York, 1970, p. 73. [24] A.J. Parker, Electrochim. Acta. 21 (1976) 671. [25] M.W. Lehmann, D.H. Evans, J. Phys. Chem. B101 (1998) 9928. [26] A.J. Bard, L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980, p. 71. [27] J.J. Lingane, Electroanalytical Chemistry, second ed., Prentice Hall, New York, 1958.
.
[28] D.A. Sinclair, J. Phys. Chem. 37 (1933) 495. [29] R.A. Robinson, D.A. Sinclair, J. Am. Chem. Soc. 56 (1934) 1830. [30] A.R. Gordon, J. Am. Chem. Soc. 65 (1943) 221. [31] (a) K.S. Pitzer, G. Mayorga, J. Phys. Chem. 77 (1973) 2300. (b) K.S. Pitzer, G. Mayorga, J. Solution Chem. 3 (1974) 539. (c) K.S. Pitzer, Acc. Chem. Res. 10 (1977) 371. [32] J. Barthel, W.J. Kunz, J. Solution Chem. 17 (1988) 399. [33] P.H. Rieger, Electrochemistry, Prentice-Hall, Englewood Cliffs, NJ, 1987 p117– 118. [34] H.A. Laitinen, W.E. Harris, Chemical Analysis, second ed., McGraw-Hill, New York, 1975, p. 227. [35] E.S. Yang, M.S. Chan, A.C. Wahl, J. Phys. Chem. 84 (1980) 3094. [36] R.D. Martin, P.R. Unwin, Anal. Chem. 70 (1998) 276. [37] A.J. Bard, L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980, p. 218. [38] R.M. Fuoss, F. Accascina, Electrolytic Conductance, Interscience, New York, 1959. [39] K.S. Johnson, R.M. Pytkowicz, Ion association and activity coefficients in multicomponent solutions and activity coefficients in electrolyte solutions, vol. II, CRC Press, Boca Raton, Florida, 1979, p. 3ff. [40] Y.C. Wu, R.M. Rush, G. Scatchard, J. Phys. Chem. 72 (1968) 4048. [41] M.G. Hill, W.M. Lamanna, K.R. Mann, Inorg. Chem. 30 (1991) 4687. [42] R.J. LeSuer, W.E. Geiger, Angew. Chem. Int. Ed. 39 (2000) 248.
Influence of supporting electrolyte activity on formal potentials measured for dissolved internal standards in acetonitrile Jody Redepenning *,, Efrain Castro-Narro, Guhanand Venkataraman, Eric Mechalke1 Department of Chemistry, Uni6ersity of Nebraska, Lincoln NE 68588 -0304, USA Received 30 March 2000; received in revised form 3 July 2000; accepted 7 August 2000 Dedicated to Professor Fred C. Anson for his distinguished contributions to electrochemistry
Abstract Osmotic coefficients are determined for tetra-n-butylammonium perchorate (TBAClO4) in acetonitrile. Pitzer equations are fit to the data, and the resulting fitting parameters are used to calculate activity coefficients for TBAClO4. The activity coefficients are used to control electrolyte activities so that the junction potential across a permselective anion exchanger is near zero. An electrochemical cell containing this junction is used to examine the influence of electrolyte activity on the formal potentials measured for ferrocene/ferrocenium and decamethylferrocene/decamethylferrocenium. It is clear that the oxidized forms of both redox couples are ion-paired with perchlorate anions in the supporting electrolyte and that this ion-pairing has a significant influence on the formal potentials. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Acetonitrile; Reference; Ferrocene; Ferrocenium; Silver; Activity
1. Introduction
Nernstian in water, i.e., we have been able to get readily interpretable potentials for the following cell
We recently demonstrated the use of a well-described polymer-modified electrode surface to gain a better understanding of electrochemical cell potentials measured in acetonitrile [1]. Our approach in acetonitrile, which we consider to be a test case for work in other solvents, involved the use of simple electrochemical cells to which others can make simple comparisons. In many respects our initial measurements in acetonitrile are analogous to those in water involving the historically important junctionless Pt H2 H+, Cl− AgCl Ag cell. The cell that has proved to be most useful for our purposes is
Ag X M Ag+(H2O), X M ClO4− (H2O)
Ag X M Ag+(CH3CN), X M ClO4− (CH3CN) poly-[Ru(vbpy)3(ClO4)n ] Pt
(1)
where vbpy is 4-methyl-4%-vinyl-2,2%-bipyridine, X is the concentration, and n is 2 or 3. Not only is this system well-behaved in acetonitrile, it is also recognizably * Corresponding author. Fax: +1-402-4729402. E-mail address: [email protected] (J. Redepenning). 1 Present address: Casper College, Casper, WY, USA.
poly-[Ru(vbpy)3(ClO4)n ] Pt
(2)
and to make a direct comparison of the two cells over a broad range of concentrations. Once activity effects were taken into account, the cells represented by Eqs. (1) and (2) were shown to exhibit Nernstian responses consistent with a simple reaction stoichiometry. In both solvents plots of cell potentials versus the log of the silver perchlorate activity gave straight lines with slopes that were indistinguishable from the ideal of − 118.3 mV. Furthermore, these two lines were offset from each other by an amount that was close to the value expected from the Gibbs energy of transfer of silver perchlorate from water to acetonitrile. Although our previous study demonstrated a means by which readily interpretable measurements of cell potentials can be made in acetonitrile, the applicability of this work was lessened to some extent by the bothersome lower potential limit introduced by the necessity of using a silver salt as the supporting electrolyte. The focus of the work described below is to provide a
0022-0728/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 7 2 8 ( 0 0 ) 0 0 3 9 3 - 4
J. Redepenning et al. / Journal of Electroanalytical Chemistry 498 (2001) 192–200
practical experimental alternative to the junctionless electrochemical cells used in our previous work. The reference electrode we describe makes it possible to extend the electrochemical window to that normally expected for acetonitrile. This is accomplished without introducing measurable discrepancies with junctionless electrochemical cells we believe should serve as the benchmark.
2. Background The interpretation of cell potentials measured in nonaqueous solvents has been reviewed by a number of authors [2–7]. Several studies are particularly relevant to the measurement of cell potentials in acetonitrile [1,8 –14]. Our discussion in this work will ultimately lead to the consideration of dissolved redox couples used as internal reference half-cells in non-aqueous electrochemical cells. A brief overview background information is warranted. The idea of using dissolved redox couples as internal reference half-cells is based on Pleskov’s efforts in the late 1940s to find a redox couple that did not interact with the solvent [11,15]. If such a couple was available, its half-cell potential would be independent of the solvent. Research associated with this ideal has appeared sporadically since the time of Pleskov’s initial efforts [16 – 21]. Often cited work by Gritzner and Kuta recommends the use of the ferrocene/ferrocenium couple as an internal reference couple [22]. Adherence to this recommendation has made it much easier to reproduce the results of other research groups and to compare formal potentials measured for other half-cells under the same experimental conditions; however, the use of internal standards to compare formal potentials measured in different media remains problematic. Such comparisons are problematic because the medium generally has non-negligible influences on the half-cell potentials. Popovich addressed the empirical assumption of negligible medium effects in his classic 1970 review of medium effects [23]. He concluded that …the contention that any solute could experience negligible changes in sol6ation energy upon transfer from water to a 6ariety of sol6ents seems implausible as a generalization. There is an entropic problem that is difficult to circumvent. Parker, in the concluding paragraph to his paper titled ‘‘Solvation of ions-enthalpies, entropies and Gibbs energies of transfer’’ succinctly summarizes two important issues [24]: Perhaps the most interesting conclusion from our work on entropies of transfer is confirmation that the
193
unique position of water as a good sol6ent for electrolytes is due to water’s highly de6eloped structure, i.e., to the 6ery fa6ourable entropy changes (relati6e to other sol6ents) accompanying sol6ation of the first kind, when ions are introduced into water. Con6ersely, the reluctance of water to dissol6e significant amounts of uncharged non-polar solutes is due to sol6ation of the second kind i.e. to a 6ery unfa6orable loss of entropy, accompanying the formation of a highly structured water surface about hydrophobic solutes.
If we accept the reasonable assumption that it will be difficult to find dissolved redox couples for which the entropies of transfer (from water to other solvents) are near zero, then it is still possible to find a couple for which the Gibbs energies of transfer are near zero if the enthalpic and entropic contributions fortuitously compensated each other. Of course if such a couple could be found, this fortuitous compensation could occur at only a single temperature. Furthermore, since one should correct the Gibbs energies of transfer for changes in activity, and since the activity coefficients and the extent of ion-pairing are rarely known, appropriate correction factors are not often applied. One noteworthy exception is recent work by Lehmann and Evans, which takes an important step toward InfiniteDilution Voltammetry [25]. This work is an important complement to our efforts at higher concentrations. Data over a broad range of concentrations will be required to interpret adequately the thermodynamic behavior of any couple that might be used as an internal standard. Despite present difficulties associated with the rigorous interpretation of half-cell potentials for dissolved redox standards, and despite the virtual certainty that no dissolved reference couples that are negligibly influenced by medium effects will ever be found, the ease with which dissolved redox couples can be employed as reference half-cells virtually guarantees their continued presence in the electrochemical literature. Fortunately, the hurdles to obtaining a more detailed understanding of cell potentials that incorporate dissolved reference standards are not insurmountable.
3. Experimental Most of the materials, apparatus and procedures used in this work have been described previously [1]. TBAClO4 (tetra-n-butylammonium perchlorate, Fluka), ferrocene (Aldrich, 98%) and decamethylferrocene (Aldrich, 97%) were used as purchased. Radiation-
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J. Redepenning et al. / Journal of Electroanalytical Chemistry 498 (2001) 192–200
4. Results and discussion For the cells represented by the shorthand notation in equations Eq. (1) and Eq. (2), we previously found that the cell potentials are described by the following equation [1]: Ecell = {E 0poly-Ru − E 0Ag + /0}+ −
Fig. 1. Reference electrode assembly: (A) silver wire to which external contact is made, (B) threaded main body, (C) internal reference solution in contact with silver wire, (D) open-ended cap that seals membrane on bottom end of electrode, (E) permselective anion exchange membrane (Raipore R-4030).
grafted PTFE anion exchange membranes (Raipore R-4030) were donated by Pall Corporation (East Hills, NY). A schematic diagram of the reference electrode used in these studies is shown in Fig. 1. This design provides a solvent-tight seal between the internal filling solution and the external electrolyte solution. Both the body of the reference electrode and the cap were constructed from Teflon. Discs (6.4 mm diameter) of permselective Raipore membrane were cut from sheets of the material using a common paper punch.
2RT ln k 9 [AgClO4] F
RT aRu(III) ln aRu(II) F (3)
Eq. (3) indicates that if one could measure cell potentials at a fixed apoly-Ru(III)/apoly-Ru(II) in solutions containing various electrolyte concentrations, then plots of (E ocell)app, versus log aAgClO4 should give straight lines with a slope of − 118.3 mV. If one uses the cell potential at the midpoint between the peaks of poly[Ru(vbpy)3(ClO4)n ] surface waves as an estimate of the case when aRu(III) = aRu(II), then plots of this cell potential versus log aAgClO4 (See Fig. 2) give lines with slopes that are experimentally indistinguishable from the expected value of − 118.3 mV in water and in acetonitrile. The difference in the y-intercepts can be attributed primarily to the Gibbs energy of transfer of silver perchlorate from water to acetonitrile (often represented by symbolism such as: DGt{AgClO4, w- \s}, where w=water and s= solvent, which in this case is acetonitrile). The value for DGt{AgClO4, w- \s} is a knowable quantity that is estimated to be −21.9 kJ mol − 1 [1]. This accounts for − 0.227 V of the −0.244 V shift; thus, it appears that changing the solvent from water to acetonitrile has little influence on the half-cell potential for the redox polymer, i.e., DGt for the polymer from water to acetonitrile appears to be near zero.
Fig. 2. (A) Squares: Cell potentials in acetonitrile previously reported for the cell represented by Eq. (1), y = −(119.4 91.9)x+603.3 92.5 (B) Circles: Cell potentials in water previously reported for the cell represented by Eq. (2), y = −(117.9 9 0.9)x+359.4 9 1.1 (C) Triangles: Cell potentials in acetonitrile for the cell represented by Eq. (4) when a(TBAClO4) =a(AgClO4), y = − (121.6 9 1.7)x+603.6 92.1. All activities are on the molar scale.
J. Redepenning et al. / Journal of Electroanalytical Chemistry 498 (2001) 192–200
Ecell = {E 0poly-Ru − E 0Ag + /0}+
Table 1 Equilibrium molalities of TBAClO4 and TBAI m(TBAClO4) /mol kg−1 0.0703 0.0928 0.1000 0.1714 0.2718 0.3355 0.3980 0.5255 0.5320 0.6280 0.7790 0.9825 1.0735 1.1930 1.2552 1.3312 1.5214 1.6799 1.7286 1.8634 1.8932 2.2332
m(TBAI)/mol kg−1
% uncertainty (in m(TBAClO4)/m(TBAI))
0.0723 0.0960 0.1038 0.1804 0.2892 0.3550 0.4287 0.5712 0.5806 0.6878 0.8637 1.0944 1.2088 1.3263 1.3997 1.4905 1.7063 1.8853 1.9152 2.1012 2.1146 2.5026
0.81 0.21 0.27 0.45 0.29 0.35 0.61 0.39 0.85 0.46 0.71 0.07 0.24 0.27 0.11 0.20 0.15 0.19 0.12 0.23 0.24 0.43
−
Ej =
− RT F
i
h
ti d ln ai zi
(6)
In the limit in which the transport number for the common anion approaches one, the transport numbers for cations approach zero, and the junction potential between the a and b phases simplifies to the following expression. Ej =
RT RT ln a(ClO− ln a(ClO− )i − 4 4 )h F F
(7)
Substituting this result into Eq. (5) gives, Ecell = {E 0poly-Ru − E 0Ag + /0}+ −
Ag X M AgClO4 Y M TBAClO4 poly-Ru(vbpy)3 i
(5)
where Ej is given by [26]
It seems unlikely that the poly-[Ru(vbpy)3(ClO4)n ] redox polymer will be used as described above to construct reliable reference half-cells that could be used by non-specialists under a variety of different solvent and electrolyte conditions. Even if DGt{polymer, w-\ s} to a number of solvents were near zero, the labor associated with making the monomeric metal complex and then exacting a reproducible polymerization limits the practical utility of this system as a reference halfcell. Still, the recognizably Nernstian behavior of poly[Ru(vbpy)3(ClO4)n ] serves as an important tool in constructing a reliable and readily interpretable reference scheme. A more versatile cell that incorporates the poly-[Ru(vbpy)3(ClO4)n ] half-cell while exhibiting a wider electrochemical window is desirable. To this end we have examined nonaqueous electrochemical cells such as the one shown below in which a liquid junction potential forms across a permselective anion exchange membrane denoted by . h
RT aRu(III) ln F aRu(II)
RT RT ln a(Ag+) − ln a(ClO4−) + Ej h i F F
&
195
-(ClO4)n Pt
(4)
In words, the reference electrode assembly shown in Fig. 1 was placed into an electrolyte solution of TBAClO4 in acetonitrile, and a platinum electrode modified with poly-[Ru(vbpy)3(ClO4)n ] was used as the working electrode. The potential for the cell in Eq. (4) is given by Eq. (5)
RT aRu(III) ln F aRu(II)
2RT ln k 9 [AgClO4] F
(8)
which is the same as Eq. (3). This convenient result occurs because the contribution of perchlorate activity (in the b-phase) to Ej is cancelled by the contribution of the perchlorate activity to the Donnan potential at the interface between the b-phase and the redox polymer. The goal of this research is not to map-out a wide variety of conditions for which the scheme described in Eqs. (4) –(7) and Eq. (8) can be made to work. Instead, we choose to examine conditions that further simplify the interpretation of the cell potentials. A particularly useful set of conditions are present when the perchlorate activities in the a and b phases depicted in Eq. (4) are equal. This is Lingane’s Type 2 junction [27], which occurs when different electrolytes containing a common ion are present at equal activities on either side of a junction. For the unique Type 2 junction in which the transport number for the common anion is one, note (from Eq. (6)) that Ej is zero. Of course the transport number for perchlorate is not exactly one, but by keeping the activity of perchlorate equal on either side of the membrane, the impact of any deviation from ideal permselectivity should be minimized, and the response predicted by Eq. (8) should be realized in the absence of cathodically obstructive silver ions. We must rely on mean ionic activity coefficients (k 9 ) to prepare the electrolyte solutions. Mean ionic activity coefficients for silver perchlorate in acetonitrile were available from our previous study [1], but k 9 for TBAClO4 in acetonitrile were not available, so we measured them using the isopiestic technique [28 –30]. Equilibrium molalities for TBAClO4 and TBAI are shown in Table 1. Mean ionic activity coefficients for TBAClO4 were calculated by treating the data in Table
J. Redepenning et al. / Journal of Electroanalytical Chemistry 498 (2001) 192–200
196
1 as described below. If the osmotic coefficient () for a reference salt (tetra-n-butylammonium iodide, TBAI, in our case) is known at each equilibrium molality, one can determine the unknown osmotic coefficient for the salt with which it is in equilibrium (TBAClO4 in our case) using the equation:
where
mTBAClO4TBAClO4 = mTBAITBAI
(9)
The resulting osmotic coefficients for TBAClO4, which are shown in Fig. 3, can be fit using the semi-empirical Pitzer equation [31,32], which is given by: −1 = f +mB + m 2C
(10)
f = − A I 1/2/(1+ bI 1/2)
(10a)
B = i (0) + i (1)exp[− h1I 1/2]+ i (2)exp[− h2I 1/2]
(10b)
A = (1/3)(2yNAz0)1/2(e 2/4ym0mkT)3/2
(10c)
I= 0.5%mi z 2i
(10d)
and where b is the size parameter, NA is Avogadro’s constant, z0 is the density of the pure solvent, m is the relative permittivity, m0 is the permittivity of free space, k is Boltzmann’s constant, and T is the temperature. The coefficients h1, h2, i (0), i (1), i (2) and C f are adjustable Pitzer parameters used to do the fitting. Activity coefficients can be calculated using the integrated form of Eq. (10) found in Eq. (11) [31,32]: ln k 9 = f k + mB k + m 2C k
(11)
where f k = f − A (2/b)ln(1+ bI 1/2)
(11a)
B k = 2i (0) + x (1) + x (2)
(11b)
x (i ) = (2i ()/h 2i I)[1− exp(− hi I 1/2)(1+ hi I 1/2 − 0.5h 2i I)] i= 1,2 C k = 1.5C
Fig. 3. Osmotic coefficients for TBAClO4. Error bars are one standard deviation in the ratio of the equilibrium molalities. Solid line: least squares fit through the osmotic coefficients using Eq. (10), fitting parameters are found in Table 2. Dotted line: Activity coefficients calculated for TBAClO4 using Eq. (11) with the fitting parameters found in Table 2. Table 2 Pitzer parameters in acetonitrile a Electrolyte
i (0) kg mol−1
i (1) kg mol−1
i (2) kg mol−1
C f kg2 mol−2
AgClO4 TBAClO4 TBAI
0.04745 −0.02262 −0.04090
0.05756 −0.02712 −0.40277
−10.828 −12.387 −14.635
0.02849 0.00480 0.00762
a Use of all seven adjustable parameters to fit the data was unwarranted for each of these data sets. Hence, b, h1, and h2 were set to common values reasonable in each of the solutions. This practice is particularly important if one is to consider solutions containing mixed electrolytes. In each of these cases: b=3.2 kg1/2 mol−1/2, h1 =2.0 kg1/2 mol−1/2, and h2 = 12.0 kg1/2 mol−1/2.
(11c) (11d)
The solid line in Fig. 3 is the least squares fit through the experimentally determined osmotic coefficients. The Pitzer parameters that result from this fit are displayed in Table 2 along with the parameters for AgClO4 (determined from data found in our previous work[1]) and Barthel’s values for the TBAI reference salt [32]. The dotted line in Fig. 3 shows the dependence of k 9 for TBAClO4 on molality. This line was calculated using Eq. (11) and the parameters found in Table 2. Mean ionic activity coefficients on the molal scale were converted to mean ionic activity coefficients on the concentration scale as outlined by Rieger [33]. Values of the latter at convenient concentrations of TBAClO4 in acetonitrile are provided in Table 3. Having activity coefficients for AgClO4 and TBAClO4 in acetonitrile, we can now investigate the plausibility of the measurement scheme outlined in Eqs. (4) –(8). The goal is to determine whether significant differences exist between cell potentials measured for the cell in Eq. (1) and those measured for the cell in Eq. (4) when the AgClO4 and TBAClO4 activities are equal on either side of the membrane separator. In line C of Fig. 2 measurements on the cell depicted in Eq. (4) are superimposed on previous measurements made on the cell depicted in Eq. (1). As before, the midpoint between the surface waves for poly-Ru(vbpy)3(ClO4)n was used as an estimate of the condition in which aRu(III) = aRu(II). For this set of experiments, silver perchlorate
J. Redepenning et al. / Journal of Electroanalytical Chemistry 498 (2001) 192–200 Table 3 Molar activity coefficients for Tba ClO4 in acetonitrile Concentration/mol l−1
Molar k 9
0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.100
1.000 0.660 0.578 0.532 0.501 0.478 0.460 0.445 0.433 0.422 0.413 0.356 0.326 0.307 0.294 0.284 0.277 0.273 0.269 0.268 0.268
197
We proceed to describe the cell potential of the above cell using a development similar to that outlined by Laitinen and Harris for cases in which metal –ligand interactions influence the electrode potential [34]. In the development presented below standard potentials and activities are used whenever it is possible to do so. Formal potentials and concentrations are used when there is no reasonable alternative. Not knowing activity coefficients for ferrocenium, we write the formal potential for the ferrocene/ferrocenium couple as shown in the equation: E= E°%1 +
RT [Fc+] ln F [Fc0]
(13)
If ferrocenium can ion-pair according to the simple stoichiometry shown in Eq. (14), Fc+ + ClO− 4 X (FcClO4)ip
(14)
then the conditional equilibrium constant for ion pairing is given by K%ip =
aFcClO4 [Fc+]aClO-
(15)
4
Solving for the concentration of ferrocenium in Eq. (15) and substituting the result into Eq. (13) gives:
concentrations ranged nominally from 0.020 to 1.0 M. Note that the slopes of lines A and C are experimentally indistinguishable, suggesting that the stoichiometries of the two cells are the same. Furthermore, the y-intercepts are indistinguishable, suggesting that the presence of the permselective membrane in Eq. (4) does not influence the overall cell potentials and that we have effectively eliminated Ej over a broad range of concentrations. In short, cell potentials measured for the cells depicted in Eq. (1) and Eq. (4) are equal when the electrolyte activities are equal.
where
4.1. Application to reference redox couples
E°%ip = E°%1 +
E= E°%1 +
RT [FcClO4] RT 1 RT 1 ln + ln + ln F F aClO− [Fc0] F K%ip 4
(16) or E= E°%ip +
RT [FcClO4] ln F [Fc0]
(16a)
RT 1 RT 1 ln + ln F K%ip F aClO-
(16b)
4
While poly-[Ru(vbpy)3(ClO4)n ] (and many of its relatives) have proved to be rich ground for the electrochemical research community, the merits of devoting a large research effort to a detailed thermodynamic study of this system for its own sake are dubious. In our view, the preceding results are most important for what they tell us about the special Type 2 junction shown in Eq. (4). The results described above suggest that a cell such as the one shown in Eq. (12) could be used over a broad range of supporting electrolyte concentrations in acetonitrile to deconvolute voltammetric measurements of formal potentials for redox couples such as ferrocene/ferrocenium (usually starting with all ferrocene). Ag a(AgClO4) =X X =a(TBAClO4), [ferrocene] h
i
=[ferrocenium]= Y mM Pt
where Y51 typically.
(12)
and where the activity coefficient for ion-paired FcClO4 is taken to be one. Of course Eq. (13) and Eq. (16) describe the same electrode in different terms. Eq. (13) is more practical at low concentrations of supporting electrolyte when essentially none of the ferrocenium is ion-paired, and Eq. (16) is useful at high electrolyte concentrations when essentially all of the ferrocenium exists as FcClO4. For ferrocene/ferrocenium (and for decamethylferrocene/decamethylferrocenium) in acetonitrile with perchlorate concentrations ranging from about 0.1 to 1.0 M, it appears that the oxidized form of the redox couple is present both as ion-pairs and as free ions. In order to describe the electrode potential with a single equation over a broad range of electrolyte concentrations, it is useful to introduce the definition shown in the equation: [Fc+]+ [FcClO4]= cox
(17)
J. Redepenning et al. / Journal of Electroanalytical Chemistry 498 (2001) 192–200
198
which can be used to rewrite Eq. (16) as E =E°%1 +
RT RT cox − ln (1 +K%ip ·aClO−) ln 4 [Fc0] F F
(18)
The half-cell potential for the Ag Ag+ reference electrode, E= E °ref +
RT ln aAg+ F
(19)
can be combined with Eq. (18) and Ej to give Eq. (20), which is a convenient expression of the cell potential for the cell shown in Eq. (12) Ecell =E°%1 − E°ref + −
RT c ln ox0 F [Fc ]
RT ln (aAg+ +K%ip ·a 2AgClO4) + Ej F
We have measured half-wave potentials for ferrocene/ferrocenium and decamethylferrocene/decamethylferrocenium (in the cell depicted in Eq. (12)) over electrolyte concentrations ranging nominally from 0.01 to 1.0 M in acetonitrile. As was the case for the experiments with poly-[Ru(vbpy)3(ClO4)n ] described above, the activities of AgClO4 and TBAClO4 were always equal on either side of the membrane separator. In no case was there any evidence of ferrocene (or decamethylferrocene) in the reference electrode compartment, nor was there any evidence of silver ion in the working electrode compartment. Fig. 4 shows our experimental results and fits to the data using Eq. (21), which takes the same general form as Eq. (20), Ecell = k1 − 0.0592 log (a+ k2·a 2)
(20)
It is desirable to use Eq. (20) to describe Ecell over a broad range of electrolyte concentrations; however, as the electrolyte concentrations change, so too do the formalisms, i.e., the values of E o% and K%ip change. Consequently, in lieu of activity coefficients for ferrocenium salts in acetonitrile, we are stuck with semi-empirical relationships between cell potentials and electrolyte activities. While such relationships do not provide much-needed thermodynamic rigor concerning half-cell potentials for couples such as ferrocene/ferrocenium, these relationships are quite useful in fitting experimental results and in providing qualitative phenomenological descriptions of those processes that influence the formal potentials.
(21)
where a is the activity of silver perchlorate and k1 and k2 are the adjustable parameters used to obtain the fit. All of those terms that account for the changing E o% and K%ip are included by the influence they have on the values ultimately selected for k1 and k2. This simple two-parameter fit to the data works well for both redox couples, and the model simplifies to the expected limiting cases at high electrolyte concentrations and at low electrolyte concentrations. Our value of k2 is somewhat larger than the association constant (189 6) reported by Yang, et al. who used a five-parameter fit to exchange rate data (between ferrocene and ferrocenium as a function of sodium perchlorate concentration) to estimate the association of ferrocenium and perchlorate in acetonitrile [35]. It is important to note, however, that our estimate of the association constant (k2 in Eq. (21)) should not be compared directly to that of Yang,
Fig. 4. (A – C): Formal potentials for ferrocene/ferrocenium in acetonitrile as a function of AgClO4 activity. (A) High concentration limit, y= −16.5 −(59.2) log(k2 × a 2) where k2 = 46.6, or y= − 115.3− (59.2) log(a 2); (B) Low concentration limit, y = − 16.5 −(59.2) log(a); (C) y= −16.5− (59.2) log(a+ (46.6× a 2)). (D–F): Formal potentials for decamethylferrocene /decamethylferrocenium in acetonitrile as a function of AgClO4 activity. (D) High concentration limit, y= − 537.2− (59.2) log(k2 ×a 2) where k2 =10.8, or y = − 598.4 −(59.2) log(a 2); (E) Low concentration limit, y =( −537.2− (59.2) log(a); (F) y= − 537.2 − (59.2) log(a+ (10.8 × a 2)). All activities are on the molar scale.
J. Redepenning et al. / Journal of Electroanalytical Chemistry 498 (2001) 192–200
et al., whose work did not benefit from knowledge of mean ionic activity coefficients at the high electrolyte concentrations used. Finally, it should also be noted in Fig. 4 that decamethylferrocenium does not ion-pair as extensively with perchlorate as ferrocenium does. This is manifested as a significantly smaller value of k2 for the decamethyl analog. These results are consistent with observations recently reported by Noviandri, et al. [21]. A minor complication arises because we are using voltammetric half-wave potentials (E1/2) to estimate the formal potentials. E1/2 =E o% only when the diffusion coefficient for the oxidized species (Dox) equals that for the reduced species (Dred), a condition that is probably never true exactly. Martin and Unwin have determined diffusion coefficients for ferrocene and ferrocenium in acetonitrile [36]. They measured DFc to be 2.15× 10 − 5 and DFc+ to be 1.72×10 − 5 cm2 s − 1, which makes E1/2 −E o%= − 2.9 mV for cyclic voltammetry at macroscopic planar electrodes [37]; however, their measurement of DFc+ was performed in 0.1 M TBAClO4 and we now know that under these conditions in acetonitrile a significant fraction of ferrocenium exists nominally as ion-pairs. In our hands cyclic voltammograms of ferrocene and decamethylferrocene at 1.0 mm and 10 mm electrodes (at 50 and 5 mV s − 1, respectively) gave half-wave potentials that were experimentally indistinguishable at high and at low supporting electrolyte concentrations. Given the influence of (Dred/Dox)n on E1/2 for voltammetry performed at macroelectrodes and at ultramicroelectrodes [25,37], the approximation that E1/2 =E o% appears to be reasonable for cyclic voltammetric characterizations of these couples.
5. Conclusions The semi-empirical equations obtained above provide a relatively simple means of describing the influence of electrolyte activity on half-wave potentials measured for ferrocene and decamethylferrocene in acetonitrile when perchlorate salts are used as the supporting electrolyte. The extensive ion – pairing associated with the oxidized forms of both couples suggests that these couples should not be used indiscriminately as reference half-cells. Of course the presence of significant ion-pairing comes as no great surprise given the pioneering conductance work by Fuoss and Onsanger beginning in the 1930s (see Ref. [38]). Ion-pairing between ferrocenium and supporting electrolyte anions does not present an insurmountable hurdle to the use of ferrocene/ferrocenium or one of its analogs as a reference half-cell, however. Silver perchlorate also ion-pairs strongly at the concentrations we are using in acetonitrile but this ion-pairing was not a major impediment to our previous studies in junctionless cells. As long as the activity coefficients for the
199
electrolyte are known, it is not necessary to describe ion-pairing explicitly. The ion-pairing is implicit in the activity coefficients [39] and is described by the Pitzer relations even when there is no a priori knowledge of the radial charge density. In fact, Pitzer has noted that as long as the thermodynamic properties are known, no assumptions concerning the existence of discrete chemical species are needed (even if such species do exist) [31b] Such considerations are particularly important if one is to avoid discussion that makes some kind of distinction between ion-pairs and solvent-separated ionpairs. Our future course of action is clear. Osmotic coefficients are needed for pure ferrocenium salts and for mixtures of those salts with other univalent salts for which the osmotic coefficients are known. It should then be possible to use known thermodynamic treatments [40] and simple half-reactions to describe the dependence of the half-cell potential for the ferrocene/ ferrocenium couple on electrolyte activity. The prospects of making such measurements on salts of weakly coordinating anions such as tetrakis(pentafluorophenyl)borate are particularly attractive [41,42]. All of this involves a significant amount of experimental work for the electrochemist, but by completing such work it may finally be possible to resolve important ambiguities that still surround the use of ferrocene/ferrocenium as a reference half-cell.
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